that_o the_o line_n of_o be_v make_v equal_a to_o the_o line_n ad_fw-la which_o be_v the_o diameter_n of_o the_o square_n abcd_o of_o which_o square_v the_o line_n ab_fw-la be_v a_o side_n it_o be_v certain_a that_o the_o â—ide_v of_o a_o square_n be_v incommensurable_a in_o length_n to_o the_o diameter_n of_o the_o same_o square_n if_o there_o be_v yet_o find_v any_o one_o superficies_n which_o measure_v the_o two_o square_n abcd_o and_o efgh_a as_o here_o do_v the_o triangle_n abdella_n or_o the_o triangle_n acd_v note_v in_o the_o square_n abcd_o or_o any_o of_o the_o four_o triangle_n note_v in_o the_o square_a efgh_o as_o appear_v somewhat_o more_o manifest_o in_o the_o second_o example_n in_o the_o declaration_n of_o the_o last_o definition_n go_v before_o the_o line_n of_o be_v also_o a_o rational_a line_n note_v that_o these_o line_n which_o here_o be_v call_v rational_a line_n be_v not_o rational_a line_n of_o purpose_n or_o by_o supposition_n as_o be_v the_o first_o rational_a line_n but_o be_v rational_a only_o by_o reason_n of_o relation_n and_o comparison_n which_o they_o have_v unto_o it_o because_o they_o be_v commensurable_a unto_o it_o either_o in_o length_n and_o power_n or_o in_o power_n only_o far_o here_o be_v to_o be_v note_v that_o these_o word_n length_n and_o power_n and_o power_n only_o be_v join_v only_o with_o these_o wordeâ—_n commensurable_a or_o incommensurable_a and_o be_v never_o join_v with_o these_o word_n rational_a or_o irrational_a so_o that_o no_o line_n can_v be_v call_v rational_a in_o length_n or_o in_o power_n nor_o like_o wise_a can_v they_o be_v call_v irrational_a in_o length_n or_o in_o power_n wherein_o undoubted_o campanus_n be_v deceive_v book_n who_o use_v those_o word_n &_o speech_n indifferent_o cause_v &_o bring_v in_o great_a obscurity_n to_o the_o proposition_n and_o demonstration_n of_o this_o book_n which_o he_o shall_v easy_o see_v which_o mark_v with_o diligence_n the_o demonstration_n of_o campanus_n in_o this_o book_n 7_o line_n which_o be_v incommensurable_a to_o the_o rational_a line_n be_v call_v irrational_a definition_n by_o line_n incommensurable_a to_o the_o rational_a line_n suppose_v in_o this_o place_n he_o understand_v such_o as_o be_v incommensurable_a unto_o it_o both_o in_o length_n and_o in_o power_n for_o there_o be_v no_o line_n incommensurable_a in_o power_n only_o for_o it_o can_v be_v that_o any_o line_n shall_v so_o be_v incommensurable_a in_o power_n only_o that_o they_o be_v not_o also_o incommensurable_a in_o length_n what_o so_o ever_o line_n be_v incommensurable_a in_o power_n the_o same_o be_v also_o incommensurable_a in_o length_n neither_o can_v euclid_n here_o in_o this_o place_n mean_a line_n incommensurable_a in_o length_n only_o for_o in_o the_o definition_n before_o he_o call_v they_o rational_a line_n neither_o may_v they_o be_v place_v amongst_o irrational_a line_n wherefore_o it_o remain_v that_o in_o this_o diffintion_n he_o speak_v only_o of_o those_o line_n which_o be_v incommensurable_a to_o the_o rational_a line_n first_o give_v and_o suppose_v both_o in_o length_n and_o in_o power_n which_o by_o all_o mean_n be_v incommensurable_a to_o the_o rational_a line_n &_o therefore_o most_o apt_o be_v they_o call_v irrational_a line_n this_o definition_n be_v easy_a to_o be_v understand_v by_o that_o which_o have_v be_v say_v before_o yet_o for_o the_o more_o plainness_n see_v this_o example_n let_v the_o â—â—rst_a rational_a line_n suppose_v be_v the_o line_n ab_fw-la who_o square_a or_o quadrate_n let_v be_v abcd._n and_o let_v there_o be_v give_v a_o other_o line_n of_o which_o lâ—t_v be_v to_o the_o rational_a line_n incommensurable_a in_o length_n and_o power_n so_o that_o let_v no_o one_o line_n measure_v the_o length_n of_o the_o two_o line_n ab_fw-la and_o of_o and_o let_v the_o square_n of_o the_o line_n of_o be_v efgh_a now_o if_o also_o there_o be_v no_o one_o superficies_n which_o measure_v the_o two_o square_n abcd_o and_o efgh_a as_o be_v suppose_v to_o be_v in_o this_o example_n they_o be_v the_o line_n of_o a_o irrational_a line_n which_o word_n irrational_a as_o before_o do_v this_o word_n rational_a mislike_v many_o learned_a in_o this_o knowledge_n of_o geometry_n flussates_n uncertain_a as_o he_o leave_v the_o word_n rational_a and_o in_o stead_n thereof_o use_v this_o word_n certain_a so_o here_o he_o leave_v the_o word_n irrational_a and_o use_v in_o place_n thereof_o this_o word_n uncertain_a and_o ever_o name_v these_o line_n uncertain_a line_n petrus_n montaureus_n also_o mislike_v the_o word_n irrational_a will_v rather_o have_v they_o to_o be_v call_v surd_a line_n yet_o because_o this_o word_n irrational_a have_v ever_o by_o custom_n and_o long_a use_n so_o general_o be_v receive_n he_o use_v continual_o the_o same_o in_o greek_a such_o line_n be_v call_v 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d alogoi_n which_o signify_v nameless_a unspeakable_a uncertain_a in_o determinate_a line_n and_o with_o out_o proportion_n not_o that_o these_o irrational_a line_n have_v no_o proportion_n at_o all_o either_o to_o the_o first_o rational_a line_n or_o between_o themselves_o but_o be_v so_o name_v for_o that_o their_o proportion_n to_o the_o rational_a line_n can_v be_v express_v in_o number_n that_o be_v undoubted_o very_o untrue_a which_o many_o write_v that_o their_o proportion_n be_v unknown_a both_o to_o we_o and_o to_o nature_n be_v it_o not_o think_v you_o a_o thing_n very_o absurd_a to_o say_v that_o there_o be_v any_o thing_n in_o nature_n and_o produce_v by_o nature_n to_o be_v hide_v from_o nature_n and_o not_o to_o be_v know_v of_o nature_n it_o can_v not_o be_v say_v that_o their_o proportion_n be_v utter_o hide_v and_o unknown_a to_o we_o much_o less_o unto_o nature_n although_o we_o can_v geve_v they_o their_o name_n and_o distinct_o express_v they_o by_o number_n otherwise_o shall_v euclid_n have_v take_v all_o this_o travel_n and_o wonderful_a diligence_n bestow_v in_o this_o bookeâ—_n in_o vain_a and_o to_o no_o useâ—_n in_o which_o he_o do_v nothing_o ellâ—_n but_o teach_v the_o propriety_n and_o passion_n of_o these_o irrational_a linesâ—_n and_o show_v the_o proportion_n which_o they_o have_v the_o one_o to_o the_o other_o here_o be_v also_o to_o be_v note_v which_o thing_n also_o tartalea_n have_v before_o diligent_o notedâ—_n that_o campanus_n and_o many_o other_o writer_n of_o geometryâ—_n over_o much_o â—â—â—ed_a and_o be_v deceive_v in_o that_o they_o write_v and_o teach_v that_o all_o these_o line_n who_o square_n be_v not_o signify_v and_o may_v be_v express_v by_o a_o square_a number_n although_o they_o may_v by_o any_o other_o number_n as_o by_o 11._o 12._o 14._o and_o such_o other_o not_o square_a number_n be_v irrational_a line_n which_o be_v manifest_o repugnant_a to_o the_o ground_n and_o principle_n of_o euclid_n who_o will_v that_o all_o line_n which_o be_v commensurable_a to_o the_o rational_a line_n whether_o it_o be_v in_o length_n and_o power_n or_o in_o power_n only_o shall_v be_v rational_a undoubted_o this_o have_v be_v one_o of_o the_o chief_a and_o great_a cause_n of_o the_o wonderful_a confusion_n and_o darkness_n of_o this_o book_n book_n which_o so_o have_v toss_v and_o turmoil_v the_o wit_n of_o all_o both_o writer_n and_o reader_n master_n and_o scholar_n and_o so_o overwhelm_v they_o that_o they_o can_v not_o with_o out_o infinite_a travel_n and_o sweat_v attain_v to_o the_o truth_n and_o perfect_a understanding_n thereof_o definition_n 8_o the_o square_n which_o be_v describe_v of_o the_o rational_a right_a line_n suppose_v be_v rational_a until_o this_o definition_n have_v euclid_n set_v forth_o the_o nature_n and_o propriety_n of_o the_o first_o kind_n of_o magnitude_n namely_o of_o line_n how_o they_o be_v rational_a or_o irrational_a now_o he_o begin_v to_o â—hew_v how_o the_o second_o kind_n of_o magnitude_n namely_o superficies_n be_v one_o to_o the_o other_o rational_a or_o irrational_a this_o definition_n be_v very_a plain_n suppose_v the_o line_n ab_fw-la to_o be_v the_o rational_a line_n have_v his_o part_n and_o division_n certain_o know_v the_o square_a of_o which_o line_n let_v be_v the_o square_a abcd._n now_o because_o it_o be_v the_o square_a of_o the_o rational_a line_n ab_fw-la it_o be_v also_o call_v rational_a and_o as_o the_o line_n ab_fw-la be_v the_o first_o rational_a line_n unto_o which_o other_o line_n compare_v be_v count_v rational_a or_o irrational_a so_o be_v the_o quadrat_fw-la or_o square_v thereof_o the_o â—irst_n rational_a superficies_n unto_o which_o all_o other_o square_n or_o figure_n compare_v be_v count_v and_o name_v rational_a or_o irrational_a 9_o such_o which_o be_v commensurable_a unto_o it_o be_v rational_a definition_n in_o this_o definition_n where_o it_o be_v say_v such_o as_o be_v commensurable_a to_o the_o square_n of_o the_o rational_a line_n be_v not_o understand_v only_o other_o square_n or_o
a_o assumpt_n forasmuch_o as_o in_o the_o eight_o book_n in_o the_o 26._o proposition_n it_o be_v prove_v that_o like_o plain_a number_n have_v that_o proportion_n the_o one_o to_o the_o other_o that_o a_o square_a number_n have_v to_o a_o square_a number_n and_o likewise_o in_o the_o 24._o of_o the_o same_o book_n it_o be_v prove_v that_o if_o two_o number_n have_v that_o proportion_n the_o one_o to_o the_o other_o eight_o that_o a_o square_a number_n have_v to_o a_o square_a number_n those_o number_n be_v like_o plain_a number_n hereby_o it_o be_v manifest_a that_o unlike_o plain_a number_n that_o be_v who_o side_n be_v not_o proportional_a have_v not_o that_o proportion_n the_o one_o to_o the_o other_o that_o a_o square_a number_n have_v to_o a_o square_a number_n for_o if_o they_o have_v than_o shall_v they_o be_v like_o plain_a number_n which_o be_v contrary_a to_o the_o supposition_n wherefore_o unlike_o plain_a number_n have_v not_o that_o proportion_n the_o one_o to_o the_o other_o that_o a_o square_a number_n have_v to_o a_o square_a number_n and_o therefore_o square_n which_o have_v that_o proportion_n the_o one_o to_o the_o other_o that_o unlike_o plain_a number_n have_v shall_v have_v their_o side_n incommensurable_a in_o length_n by_o the_o last_o part_n of_o the_o former_a proposition_n for_o that_o those_o square_n have_v not_o that_o proportion_n the_o one_o to_o the_o other_o that_o a_o square_a number_n have_v to_o a_o square_a number_n ¶_o the_o 8._o theorem_a the_o 10._o proposition_n if_o four_o magnitude_n be_v proportional_a and_o if_o the_o first_o be_v commensurable_a unto_o the_o second_o the_o three_o also_o shall_v be_v commensurable_a unto_o the_o four_o and_o if_o the_o first_o be_v incommensurable_a unto_o the_o second_o the_o three_o shall_v also_o be_v incommensurable_a unto_o the_o four_o svppose_v that_o these_o four_o magnitude_n a_o b_o c_o d_o be_v proportional_a as_o a_o be_v to_o b_o so_o let_v c_o be_v to_o d_o and_o let_v a_o be_v commensurable_a unto_o b._n then_o i_o say_v that_o c_z be_v also_o commensurable_a unto_o d._n part_n for_o forasmuch_o as_o a_o be_v commensurable_a unto_o b_o it_o have_v by_o the_o five_o of_o the_o ten_o that_o proportion_n that_o number_n have_v to_o number_v but_o as_o a_o be_v to_o b_o so_o be_v c_o to_o d._n wherefore_o c_z also_o have_v unto_o d_z that_o proportion_n that_o number_n have_v to_o number_v wherefore_o c_o be_v commensurable_a unto_o d_z by_o the_o 6._o of_o the_o ten_o but_o now_o suppose_v that_o the_o magnitude_n a_o be_v incommensurable_a unto_o the_o magnitude_n b._n partâ—_n then_o i_o say_v that_o the_o magnitude_n c_o also_o be_v incommensurable_a unto_o the_o magnitude_n d._n for_o forasmuch_o as_o a_o be_v incommensurable_a unto_o b_o therefore_o by_o the_o 7._o of_o this_o book_n a_o have_v not_o unto_o b_o such_o proportion_n as_o number_n have_v to_o number_v but_o as_o a_o be_v to_o b_o so_o be_v c_o to_o d._n wherefore_o c_o have_v not_o unto_o d_z such_o proportion_n as_o number_n have_v to_o number_v wherefore_o by_o the_o 8._o of_o the_o ten_o c_o be_v incommensurable_a unto_o d._n if_o therefore_o there_o be_v four_o magnitude_n proportional_a and_o if_o the_o first_o be_v commensurable_a unto_o the_o second_o the_o three_o also_o shall_v be_v commensurable_a unto_o the_o four_o and_o if_o the_o first_o be_v incommensurable_a unto_o the_o second_o the_o three_o shall_v also_o be_v incommensurable_a unto_o the_o four_o which_o be_v require_v to_o be_v prove_v ¶_o a_o corollary_n add_v by_o montaureus_n if_o there_o be_v four_o line_n proportional_a and_o if_o the_o two_o first_o or_o the_o two_o last_o be_v commensurable_a in_o power_n only_o the_o other_o two_o also_o shall_v be_v commensurable_a in_o power_n only_o corollary_n this_o be_v prove_v by_o the_o 22._o of_o the_o six_o and_o by_o this_o ten_o proposition_n and_o this_o corollary_n euclid_n use_v in_o the_o 27._o and_o 28._o proposition_n of_o this_o book_n and_o in_o other_o proposition_n also_o ¶_o the_o 3._o problem_n the_o 11._o proposition_n unto_o a_o right_a line_n first_o set_v and_o give_v which_o be_v call_v a_o rational_a line_n to_o find_v out_o two_o right_a line_n incommensurable_a the_o one_o in_o length_n only_o and_o the_o other_o in_o length_n and_o also_o in_o power_n svppose_v that_o the_o right_a line_n first_o set_v and_o give_v which_o be_v call_v a_o rational_a line_n of_o purpose_n be_v a._n it_o be_v require_v unto_o the_o say_a line_n a_o to_o find_v out_o two_o right_a line_n incommensurable_a the_o one_o in_o length_n only_o the_o other_o both_o in_o length_n and_o in_o power_n give_v take_v by_o that_o which_o be_v add_v after_o the_o 9_o proposition_n of_o this_o book_n two_o number_n b_o and_o c_o not_o have_v that_o proportion_n the_o one_o to_o the_o other_o that_o a_o square_a number_n have_v to_o a_o square_a number_n that_o be_v let_v they_o not_o be_v like_o plain_a number_n for_o like_o plain_a number_n by_o the_o 26._o of_o the_o eight_o have_v that_o proportion_n the_o one_o to_o the_o other_o that_o a_o square_a number_n have_v to_o a_o square_a number_n and_o as_o the_o number_n b_o be_v to_o the_o number_n c_o so_fw-mi let_v the_o square_n of_o the_o line_n a_o be_v unto_o the_o square_n of_o a_o other_o line_n namely_o of_o d_z how_o to_o do_v this_o be_v teach_v in_o the_o assumpt_n put_v before_o the_o 6._o proposition_n of_o this_o book_n wherefore_o the_o square_a of_o the_o line_n a_o be_v unto_o the_o square_n of_o the_o line_n d_o commensurable_a by_o the_o six_o of_o the_o ten_o and_o forasmuch_o as_o the_o number_n b_o have_v not_o unto_o the_o number_n c_o that_o proportion_n that_o a_o square_a number_n have_v to_o a_o square_a number_n therefore_o the_o square_a of_o the_o line_n a_o have_v not_o unto_o the_o square_n of_o the_o line_n d_z that_o proportion_n that_o a_o square_a number_n have_v to_o a_o number_n wherefore_o by_o the_o 9_o of_o the_o ten_o the_o line_n a_o be_v unto_o the_o line_n d_o incommensurable_a in_o length_n only_o and_o so_o be_v find_v out_o the_o first_o line_n namely_o d_z incommensurable_a in_o length_n only_o to_o the_o line_n give_v a._n give_v again_o take_v by_o the_o 13._o of_o the_o six_o the_o mean_a proportional_a between_o the_o line_n a_o and_o d_o and_o let_v the_o same_o be_v e._n wherefore_o as_o the_o line_n a_o be_v to_o the_o line_n d_o so_o be_v the_o square_a of_o the_o line_n a_o to_o the_o square_n of_o the_o line_n e_o by_o the_o corollary_n of_o the_o 20._o of_o the_o six_o but_o the_o line_n a_o be_v unto_o the_o line_n d_o incommensurable_a in_o length_n wherefore_o also_o the_o square_a of_o the_o line_n a_o be_v unto_o the_o square_n of_o the_o line_n e_o incommensurable_a by_o the_o second_o part_n of_o the_o former_a proposition_n now_o forasmuch_o as_o the_o square_n of_o the_o line_n a_o be_v incommensurable_a to_o the_o square_n of_o the_o line_n e_o it_o follow_v by_o the_o definition_n of_o incommensurable_a line_n that_o the_o line_n a_o be_v incommensurable_a in_o power_n to_o the_o line_n e._n wherefore_o unto_o the_o right_a line_n give_v and_o first_o set_v a_o which_o be_v a_o rational_a line_n and_o which_o be_v suppose_v to_o have_v such_o division_n and_o so_o many_o part_n as_o you_o listen_v to_o conceive_v in_o mind_n as_o in_o this_o example_n 11_z whereunto_o as_o be_v declare_v in_o the_o 5._o definition_n of_o this_o book_n may_v be_v compare_v infinite_a other_o line_n either_o commensurable_a or_o incommensurable_a be_v find_v out_o the_o line_n d_o incommensurable_a in_o length_n only_o wherefore_o the_o line_n d_o be_v rational_a by_o the_o six_o definition_n of_o this_o book_n for_o that_o it_o be_v incommensurable_a in_o length_n only_o to_o the_o line_n a_o which_o be_v the_o first_o line_n set_v and_o be_v by_o supposition_n rational_a there_o be_v also_o find_v out_o the_o line_n e_o which_o be_v unto_o the_o same_o line_n a_o incommensurable_a not_o only_o in_o length_n but_o also_o in_o power_n which_o line_n e_o compare_v to_o the_o rational_a line_n a_o be_v by_o the_o definition_n irrational_a for_o euclid_n always_o call_v those_o line_n irrational_a which_o be_v incommensurable_a both_o in_o length_n and_o in_o power_n to_o the_o line_n first_o set_v and_o by_o supposition_n rational_a ¶_o the_o 9_o theorem_a the_o 12._o proposition_n magnitude_n commensurable_a to_o one_o and_o the_o self_n same_o magnitude_n be_v also_o commensurable_a the_o one_o to_o the_o other_o svppose_v that_o either_o of_o these_o magnitude_n a_o and_o b_o be_v commensurable_a unto_o the_o magnitude_n c_o then_o i_o say_v that_o the_o magnitude_n a_o be_v commensurable_a unto_o the_o magnitude_n b._n construction_n for_o forasmuch_o as_o the_o
a_o square_n which_o be_v require_v to_o be_v do_v ¶_o the_o 15._o theorem_a the_o 18._o proposition_n if_o there_o be_v two_o right_a line_n unequal_a and_o if_o upon_o the_o great_a be_v apply_v a_o parallelogram_n equal_a unto_o the_o four_o part_n of_o the_o square_n of_o less_o and_o want_v in_o figure_n by_o a_o square_a if_o also_o the_o parallelogram_n thus_o apply_v divide_v the_o line_n whereupon_o it_o be_v apply_v into_o part_n incommensurable_a in_o length_n the_o great_a line_n shall_v be_v in_o power_n more_o than_o the_o less_o line_n by_o the_o square_n of_o a_o line_n incommensurable_a in_o length_n unto_o the_o great_a line_n and_o if_o the_o great_a line_n be_v in_o power_n more_o than_o the_o less_o line_n by_o the_o square_n of_o a_o line_n incommensurable_a in_o length_n unto_o the_o great_a and_o if_o also_o upon_o the_o great_a be_v apply_v a_o parallelogram_n equal_a unto_o the_o four_o part_n of_o the_o square_n of_o the_o less_o and_o want_v in_o figure_n by_o a_o square_n then_o shall_v it_o divide_v the_o great_a line_n into_o part_n incommensurable_a in_o length_n but_o now_o suppose_v that_o the_o line_n bc_o be_v in_o power_n more_o than_o the_o line_n a_o by_o the_o square_n of_o a_o line_n incommensurable_a in_o length_n unto_o bc._n former_a and_o upon_o the_o line_n bc_o let_v there_o be_v apply_v a_o parallelogram_n equal_a unto_o the_o four_o part_n of_o the_o square_n of_o the_o line_n a_o and_o want_v in_o figure_n by_o a_o square_a and_o let_v the_o say_a parallelogram_n be_v that_o which_o be_v contain_v under_o the_o line_n bd_o &_o dc_o then_o must_v we_o prove_v that_o the_o line_n bd_o be_v unto_o the_o line_n dc_o incommensurable_a in_o length_n the_o same_o order_n of_o construction_n and_o demonstration_n be_v keep_v we_o may_v in_o like_a sort_n prove_v that_o the_o line_n bc_o be_v in_o power_n more_o than_o the_o line_n a_o by_o the_o square_n of_o the_o line_n fd._n but_o now_o by_o supposition_n the_o line_n bc_o be_v in_o power_n more_o than_o the_o line_n a_o by_o the_o square_n of_o a_o line_n incommensurable_a in_o length_n unto_o bc._n wherefore_o the_o line_n bc_o be_v unto_o the_o line_n fd_v incommensurable_a in_o length_n wherefore_o the_o line_n compose_v of_o bf_o and_o dc_o take_v as_o one_o line_n shall_v be_v incommensurable_a in_o length_n to_o the_o line_n fd_o by_o the_o second_o part_n of_o the_o 16._o of_o the_o ten_o wherefore_o also_o by_o the_o first_o part_n of_o the_o same_o the_o line_n bc_o shall_v be_v incommensurable_a in_o length_n to_o the_o line_n compose_v of_o the_o line_n bf_o and_o dc_o but_o the_o line_n compose_v of_o the_o line_n bf_o and_o dc_o be_v commensurable_a in_o length_n to_o the_o line_n dc_o for_o that_o bf_o as_o before_o have_v be_v prove_v be_v equal_a to_o dc_o wherefore_o the_o line_n bc_o be_v incommensurable_a in_o length_n to_o the_o line_n dc_o by_o the_o 13._o of_o the_o ten_o wherefore_o by_o the_o second_o part_n of_o the_o 16._o of_o the_o ten_o the_o line_n bd_o be_v incommensurable_a in_o length_n unto_o the_o line_n dc_o if_o therefore_o there_o be_v two_o right_a line_n unequal_a and_o if_o upon_o the_o great_a be_v apply_v a_o parallelogram_n equal_a unto_o the_o four_o part_n of_o the_o square_n of_o the_o less_o line_n &_o want_v in_o â—igure_n by_o a_o square_a if_o also_o the_o parallelogram_n thus_o apply_v divide_v the_o line_n whereupon_o it_o be_v apply_v into_o part_n incommensurable_a in_o length_n the_o great_a line_n shall_v be_v in_o power_n more_o than_o the_o less_o line_n by_o the_o square_n of_o a_o line_n incommensurable_a in_o length_n unto_o the_o great_a and_o if_o the_o great_a line_n be_v in_o power_n more_o than_o the_o less_o by_o the_o square_n of_o a_o line_n incommensurable_a in_o length_n unto_o the_o great_a and_o if_o also_o upon_o the_o great_a be_v apply_v a_o parallelogram_n equal_a unto_o the_o four_o part_n of_o the_o square_n of_o the_o less_o line_n and_o want_v in_o figure_n by_o a_o square_a then_o shall_v it_o divide_v the_o great_a line_n into_o part_n incommensurable_a in_o length_n which_o be_v require_v to_o be_v demonstrate_v this_o proposition_n may_v also_o be_v demonstrate_v by_o the_o former_a proposition_n namely_o the_o first_o part_n of_o this_o by_o the_o second_o part_n of_o the_o former_a and_o the_o second_o part_n of_o this_o by_o the_o first_o part_n of_o the_o former_a by_o a_o argument_n lead_v to_o a_o absurdity_n absurdity_n for_o as_o touch_v the_o first_o part_n of_o this_o proposition_n the_o line_n bc_o contain_v in_o power_n more_o than_o the_o line_n a_o by_o the_o square_n of_o the_o line_n fd_o if_o the_o line_n bg_o be_v not_o incommensurable_a unto_o the_o line_n fd_o then_o be_v it_o commensurable_a unto_o it_o wherefore_o by_o the_o second_o part_n of_o the_o 17._o proposition_n the_o line_n bd_o and_o dc_o also_o be_v commensurable_a which_o be_v impossible_a for_o they_o be_v suppose_v to_o be_v incommensurable_a so_o likewise_o as_o touch_v the_o second_o part_n of_o the_o same_o the_o line_n bc_o contain_v in_o power_n more_o than_o the_o line_n a_o by_o the_o square_n of_o the_o line_n fd_o if_o the_o line_n db_o be_v not_o incommensurable_a to_o the_o line_n dc_o then_o be_v it_o commensurable_a unto_o it_o wherefore_o by_o the_o first_o part_n of_o the_o â—â—_o proposition_n the_o line_n bc_o and_o fd_o be_v also_o commensurable_a which_o be_v absurd_a for_o the_o line_n bc_o and_o fd_o be_v suppose_v to_o be_v incommensurable_a which_o be_v require_v to_o be_v prove_v ¶_o a_o assumpt_n forasmuch_o as_o it_o have_v be_v prove_v that_o line_n commensurable_a in_o length_n assumpt_n be_v always_o also_o commensurable_a in_o power_n but_o line_n commensurable_a in_o power_n be_v not_o always_o commensurable_a in_o length_n but_o may_v be_v in_o length_n both_o commensurable_a and_o also_o incommensurable_a it_o be_v manifest_a that_o if_o unto_o the_o line_n propound_v which_o be_v call_v rational_a of_o purpose_n a_o certain_a line_n be_v commensurable_a in_o length_n it_o ought_v to_o be_v call_v rational_a and_o commensurable_a unto_o it_o not_o only_o in_o length_n but_o also_o in_o power_n for_o line_n commensurable_a in_o length_n be_v also_o always_o commensurable_a in_o power_n but_o if_o unto_o the_o line_n propound_v which_o be_v call_v rational_a of_o purpose_n a_o certain_a line_n be_v commensurable_a in_o power_n then_o if_o it_o be_v also_o commensurable_a unto_o it_o in_o length_n it_o be_v call_v rational_a and_o commensurable_a unto_o it_o both_o in_o length_n and_o in_o power_n but_o again_o if_o unto_o the_o say_a line_n give_v which_o be_v call_v rational_a a_o certain_a line_n be_v commensurable_a in_o power_n and_o incommensurable_a in_o length_n that_o also_o be_v call_v rational_a commensurable_a in_o power_n only_o a_o annotation_n of_o proclus_n he_o call_v those_o line_n rational_a which_o be_v unto_o the_o rational_a line_n first_o set_v commensurable_a in_o length_n &_o in_o power_n or_o in_o power_n only_o and_o there_o be_v also_o other_o right_a line_n which_o be_v unto_o the_o rational_a line_n first_o set_v incommensurable_a in_o length_n and_o be_v unto_o it_o commensurable_a in_o power_n only_o and_o therefore_o they_o be_v call_v rational_a &_o commensurable_a the_o one_o to_o the_o otherâ—_n for_o which_o cause_n they_o be_v rational_a but_o even_o these_o line_n may_v be_v commensurable_a the_o one_o to_o the_o other_o either_o in_o length_n and_o therefore_o in_o power_n or_o else_o in_o power_n only_o now_o if_o they_o be_v commensurable_a in_o length_n then_o be_v those_o line_n call_v rational_a commensurable_a in_o length_n but_o yet_o so_o that_o they_o be_v understand_v to_o be_v in_o power_n commensurable_a but_o if_o they_o be_v commensurable_a the_o one_o to_o the_o other_o in_o power_n only_o they_o also_o be_v call_v rational_a commensurable_a in_o power_n only_o ¶_o a_o corollary_n and_o that_o two_o line_n or_o more_o be_v rational_a and_o commensurable_a in_o length_n to_o the_o rational_a line_n first_o set_v be_v also_o commensurable_a the_o one_o to_o the_o other_o in_o length_n hereby_o it_o be_v manifest_a montaureuâ—_n for_o forasmuch_o as_o they_o be_v rational_a and_o commensurable_a in_o length_n to_o the_o rational_a line_n first_o set_v but_o those_o magnitude_n which_o be_v commensurable_a to_o one_o and_o the_o self_n same_o magnitude_n be_v also_o commensurable_a the_o one_o to_o the_o other_o by_o the_o 12._o of_o the_o ten_o wherefore_o the_o rational_a line_n commensurable_a in_o length_n to_o the_o rational_a line_n first_o set_v be_v also_o commensurable_a in_o length_n the_o one_o to_o the_o other_o and_o as_o touch_v those_o which_o be_v rational_a commensurable_a in_o power_n only_o to_o the_o rational_a line_n first_o set_v they_o also_o must_v needs_o be_v at_o the_o least_o commensurable_a in_o power_n the_o
one_o to_o the_o other_o for_o forasmuch_o as_o their_o square_n be_v rational_a they_o shall_v be_v commensurable_a to_o the_o square_n of_o the_o rational_a line_n first_o set_v wherefore_o by_o the_o 12._o of_o this_o book_n they_o be_v also_o commensurable_a the_o one_o to_o the_o other_o wherefore_o their_o line_n be_v at_o the_o least_o commensurable_a in_o power_n the_o one_o to_o the_o other_o and_o it_o be_v possible_a also_o that_o they_o may_v be_v commensurable_a in_o length_n the_o one_o to_o the_o other_o for_o suppose_v that_o a_o be_v a_o rational_a liâ—e_z first_o set_v and_o let_v the_o line_n b_o be_v unto_o the_o same_o rational_a line_n a_o commensurable_a in_o power_n only_o that_o be_v incommensurable_a in_o length_n unto_o it_o let_v there_o be_v also_o a_o other_o line_n c_o commensurable_a in_o length_n to_o the_o line_n b_o which_o be_v possible_a by_o the_o principle_n of_o this_o book_n now_o by_o the_o 13._o of_o the_o ten_o it_o be_v manifest_a that_o the_o line_n c_o be_v incommensurable_a in_o length_n unto_o the_o line_n a._n but_o the_o square_n of_o the_o line_n a_o be_v commensurable_a to_o the_o square_n of_o the_o line_n b_o by_o supposition_n and_o the_o square_a of_o the_o line_n c_o be_v also_o commensurable_a to_o the_o square_n of_o the_o line_n b_o by_o supposition_n wherefore_o by_o the_o 12._o of_o this_o book_n the_o square_a of_o the_o line_n c_o be_v commensurable_a to_o the_o square_n of_o the_o line_n a._n wherefore_o by_o the_o definition_n the_o line_n c_o shall_v be_v rational_a commensurable_a in_o power_n only_o to_o the_o line_n a_o as_o also_o be_v the_o line_n b._n wherefore_o there_o be_v give_v two_o rational_a line_n commensurable_a in_o power_n only_o to_o the_o rational_a line_n first_o set_v and_o commensurable_a in_o length_n the_o one_o to_o the_o other_o here_o be_v to_o be_v note_v which_o thing_n also_o we_o before_o note_v in_o the_o definition_n that_o campane_n and_o other_o which_o follow_v he_o bring_v in_o these_o phrase_n of_o speech_n to_o call_v some_o line_n rational_a in_o power_n only_o cause_n and_o other_o some_o rational_a in_o length_n and_o in_o power_n which_o we_o can_v find_v that_o euclid_n ever_o use_v for_o these_o word_n in_o length_n and_o in_o power_n be_v never_o refer_v to_o rationality_n or_o irrationalitie_n but_o always_o to_o the_o commensurabilitie_n or_o incommensurablitie_n of_o line_n which_o pervert_v of_o word_n as_o be_v there_o declare_v have_v much_o increase_v the_o difficulty_n and_o obscureness_n of_o this_o book_n book_n and_o now_o i_o think_v it_o good_a again_o to_o put_v you_o in_o mind_n that_o in_o these_o proposition_n which_o follow_v we_o must_v ever_o have_v before_o our_o eye_n the_o rational_a line_n first_o set_v note_n unto_o which_o other_o line_n compare_v be_v either_o rational_a or_o irrational_a according_a to_o their_o commensurability_n or_o incommensurabilitie_n ¶_o the_o 16._o theorem_a the_o 19_o proposition_n a_o rectangle_n figure_n comprehend_v under_o right_a line_n commensurable_a in_o length_n be_v rational_a according_a to_o one_o of_o the_o foresay_a way_n be_v rational_a svppose_v that_o this_o rectangle_n figure_n ac_fw-la be_v comprehend_v under_o these_o right_a line_n ab_fw-la and_o bc_o be_v commensurable_a in_o length_n and_o rational_a according_a to_o one_o of_o the_o foresay_a way_n then_o i_o say_v that_o the_o superficies_n ac_fw-la be_v rational_a describe_v by_o the_o 46._o oâ—_n the_o first_o upon_o the_o line_n ab_fw-la a_o square_a ad._n construction_n wherefore_o that_o square_a ad_fw-la be_v rational_a by_o the_o definition_n demonstration_n and_o forasmuch_o as_o the_o line_n ab_fw-la be_v commensurable_a in_o length_n unto_o the_o line_n bc_o and_o the_o line_n ab_fw-la be_v equal_a unto_o the_o line_n bd_o therefore_o the_o line_n bd_o be_v commensurable_a in_o length_n unto_o the_o line_n bc._n and_o as_o the_o line_n bd_o be_v to_o the_o line_n bc_o so_o be_v the_o square_a dam_n to_o the_o superficies_n ac_fw-la by_o the_o first_o of_o the_o six_o but_o it_o be_v prove_v that_o the_o line_n bd_o be_v commensurable_a unto_o the_o line_n bc_o wherefore_o by_o the_o 10._o of_o the_o ten_o the_o square_a dam_n be_v commensurable_a unto_o the_o rectangle_n superficies_n ac_fw-la but_o the_o square_a dam_n be_v rational_a wherefore_o the_o rectangle_n superficies_n ac_fw-la also_o be_v rational_a by_o the_o definition_n a_o rectangle_n figure_n therefore_o comprehend_v under_o right_a line_n commensurable_a in_o length_n be_v rational_a accord_v to_o one_o of_o the_o foresay_a way_n be_v rational_a which_o be_v require_v to_o be_v prove_v proposition_n where_o as_o in_o the_o former_a demonstration_n the_o square_n be_v describe_v upon_o the_o less_o line_n we_o may_v also_o demonstrate_v the_o proposition_n if_o we_o describe_v the_o square_n upon_o the_o great_a line_n and_o that_o after_o this_o manner_n suppose_v that_o the_o rectangle_n superficies_n bc_o be_v contain_v of_o these_o unequal_a line_n ab_fw-la and_o ac_fw-la which_o let_v be_v rational_a commensurable_a the_o one_o to_o the_o other_o in_o length_n and_o let_v the_o line_n ac_fw-la be_v the_o great_a case_n and_o upon_o the_o line_n ac_fw-la describe_v the_o square_a dc_o then_o i_o say_v that_o the_o parallelogram_n bc_o be_v rational_a length_n for_o the_o line_n ac_fw-la be_v commensurable_a in_o length_n unto_o the_o line_n ab_fw-la by_o supposition_n and_o the_o line_n dam_n be_v equal_a to_o the_o line_n ac_fw-la wherefore_o the_o line_n dam_n be_v commensurable_a in_o length_n to_o the_o line_n ab_fw-la but_o what_o proportion_n the_o line_n dam_n have_v to_o the_o line_n ab_fw-la the_o same_o have_v the_o square_a dc_o to_o the_o paraâ—lelogramme_n câ—_n by_o the_o first_o of_o the_o six_o wherefore_o by_o the_o 10._o of_o this_o book_n the_o square_a dc_o be_v commensurable_a to_o the_o parallelogram_n cb._n but_o it_o be_v manifest_a that_o the_o square_a dc_o be_v rational_a for_o that_o it_o be_v the_o square_a of_o a_o rational_a line_n namely_o ac_fw-la wherefore_o by_o the_o definition_n the_o parallelogram_n also_o cb_o be_v rational_a moreover_o forasmuch_o as_o those_o two_o former_a demonstration_n seem_v to_o speak_v of_o that_o parallelogram_n which_o be_v make_v of_o two_o line_n of_o which_o any_o one_o may_v be_v the_o liâ—e_z first_o set_v which_o be_v call_v the_o first_o rational_a line_n from_o which_o we_o say_v ought_v to_o be_v take_v the_o measure_n of_o the_o other_o line_n compare_v unto_o it_o and_o the_o other_o be_v commensurable_a in_o length_n to_o the_o same_o first_o rational_a line_n length_n which_o be_v the_o first_o kind_n of_o rational_a line_n commensurable_a in_o length_n i_o think_v it_o good_a here_o to_o set_v a_o other_o case_n of_o the_o other_o kind_n of_o rational_a line_n of_o line_n i_o say_v rational_a commensurable_a in_o length_n compare_v to_o a_o other_o rational_a line_n first_o set_v to_o declare_v the_o general_a truth_n of_o this_o theorem_a and_o that_o we_o may_v see_v that_o this_o particle_n according_a to_o any_o of_o the_o foresay_a way_n be_v not_o here_o in_o vain_a put_v now_o then_o suppose_v first_o a_o rational_a line_n ab_fw-la let_v there_o be_v also_o a_o parallelogram_n cd_o contain_v under_o the_o line_n ce_fw-fr and_o ed_z which_o line_n let_v be_v rational_a that_o be_v commensurable_a in_o length_n to_o the_o â—irst_n rational_a line_n propound_v ab_fw-la howbeit_o let_v those_o two_o line_n ce_fw-fr and_o ed_z be_v diverse_a and_o unequal_a line_n unto_o the_o first_o rational_a line_n ab_fw-la then_o i_o say_v that_o the_o parallelogram_n cd_o be_v rational_a case_n describe_v the_o square_a of_o the_o line_n de_fw-fr which_o let_v be_v df._n first_o it_o be_v manifest_a by_o the_o 12._o of_o this_o book_n that_o the_o line_n ce_fw-fr &_o ed_z be_v commensurable_a in_o length_n the_o one_o to_o the_o other_o for_o either_o of_o they_o be_v suppose_v to_o be_v commensurable_a in_o length_n unto_o the_o line_n ab_fw-la but_o the_o line_n ed_z be_v equal_a to_o the_o line_n ef._n wherefore_o the_o line_n ce_fw-fr be_v commensurable_a in_o length_n to_o the_o line_n bf_o but_o 〈◊〉_d the_o line_n ce_fw-fr be_v â—o_o the_o line_n â—_o f_o â—o_o be_v the_o parallelogram_n cd_o to_o the_o square_a df_o by_o the_o first_o of_o the_o six_o wherefore_o by_o the_o 10._o of_o this_o book_n the_o parallelogram_n cd_o shall_v be_v commensurable_a to_o the_o square_a df._n but_o the_o square_a df_o be_v commensurable_a to_o the_o square_n of_o the_o line_n ab_fw-la which_o be_v the_o first_o rational_a line_n propound_v wherefore_o by_o the_o 12._o of_o this_o book_n the_o parallelogram_n cd_o be_v commensurable_a to_o the_o square_n of_o the_o line_n ab_fw-la but_o the_o square_n of_o
the_o line_n ab_fw-la be_v rational_a by_o the_o definition_n wherefore_o by_o the_o definition_n also_o of_o rational_a figure_n the_o parallelogram_n cd_o shall_v be_v rational_a now_o rest_v a_o other_o caâ—e_n of_o the_o third_o kind_n of_o rational_a line_n commensurable_a in_o length_n the_o one_o to_o the_o other_o which_o be_v to_o the_o rational_a line_n ab_fw-la first_o set_v commensurable_a in_o power_n only_o and_o yet_o be_v therefore_o rational_a line_n and_o let_v the_o line_n ce_fw-fr and_o ed_z be_v commensurable_a in_o length_n the_o one_o to_o the_o other_o length_n now_o then_o let_v the_o self_n same_o construction_n remain_v that_o be_v in_o the_o former_a so_o that_o let_v the_o line_n ce_fw-fr and_o ed_z be_v rational_a commensurable_a in_o power_n only_o unto_o the_o line_n ab_fw-la but_o let_v they_o be_v commensurable_a in_o length_n the_o one_o to_o the_o other_o then_o i_o say_v that_o in_o this_o case_n also_o the_o parallelogram_n cd_o be_v rational_a first_o it_o may_v be_v prove_v as_o before_o that_o the_o parallelogram_n cd_o be_v commensurable_a to_o the_o square_a df._n wherefore_o by_o the_o 12._o of_o this_o book_n the_o parallelogram_n cd_o shall_v be_v commensurable_a to_o the_o square_n of_o the_o line_n abâ—_n but_o the_o square_n of_o the_o line_n ab_fw-la be_v rational_a case_n wherefore_o by_o the_o definition_n the_o parallelogram_n cd_o shall_v be_v also_o rational_a this_o case_n be_v well_o to_o be_v note_v for_o it_o serve_v to_o the_o demonstration_n and_o understanding_n of_o the_o 25._o proposition_n of_o this_o book_n ¶_o the_o 17._o theorem_a the_o 20._o proposition_n if_o upon_o a_o rational_a line_n be_v apply_v a_o rational_a rectangle_n parallelogram_n the_o other_o side_n that_o make_v the_o breadth_n thereof_o shall_v be_v a_o rational_a line_n and_o commensurable_a in_o length_n unto_o that_o line_n whereupon_o the_o rational_a parallelogram_n be_v apply_v svppose_v that_o this_o rational_a rectangle_n parallelogram_n ac_fw-la be_v apply_v upon_o the_o line_n ab_fw-la which_o let_v be_v rational_a according_a to_o any_o one_o of_o the_o foresay_a way_n whether_o it_o be_v the_o first_o rational_a line_n set_v proposition_n or_o any_o other_o line_n commensurable_a to_o the_o rational_a line_n first_o set_v and_o that_o in_o length_n and_o in_o power_n or_o in_o power_n only_o for_o one_o of_o these_o three_o way_n as_o be_v declare_v in_o the_o assumpt_n put_v before_o the_o 19_o proposition_n of_o this_o book_n be_v a_o line_n call_v rational_a and_o make_v in_o breadth_n the_o line_n bc._n then_o i_o say_v that_o the_o line_n bc_o be_v rational_a and_o commensurable_a in_o length_n unto_o the_o line_n basilius_n construction_n desrcribe_v by_o the_o 46._o of_o the_o first_o upon_o the_o line_n basilius_n a_o square_a ad._n wherefore_o by_o the_o 9_o definition_n of_o the_o ten_o the_o square_a ad_fw-la be_v rational_a but_o the_o parallelogram_n ac_fw-la also_o be_v rational_a by_o supposition_n demonstration_n wherefore_o by_o the_o conversion_n of_o the_o definition_n of_o rational_a figure_n or_o by_o the_o 12._o of_o this_o book_n the_o square_a dam_n be_v commensurable_a unto_o the_o parallelogram_n ac_fw-la but_o as_o the_o square_a dam_n be_v to_o the_o parallelogram_n ac_fw-la so_o be_v the_o line_n db_o to_o the_o line_n bc_o by_o the_o first_o of_o the_o six_o wherefore_o by_o the_o 10._o of_o the_o ten_o the_o line_n db_o be_v commensurable_a unto_o the_o line_n bc._n but_o the_o line_n db_o be_v equal_a unto_o the_o line_n basilius_n wherefore_o the_o line_n ab_fw-la be_v commensurable_a unto_o the_o line_n bc._n but_o the_o line_n ab_fw-la be_v rational_a wherefore_o the_o line_n bc_o also_o be_v rational_a and_o commensurable_a in_o length_n unto_o the_o line_n basilius_n if_o therefore_o upon_o a_o rational_a line_n be_v apply_v a_o rational_a rectangle_n parallelogram_n the_o other_o side_n that_o make_v the_o breadth_n thereof_o shall_v be_v a_o rational_a line_n commensurable_a in_o length_n unto_o that_o line_n whereupon_o the_o rational_a parallelogram_n be_v apply_v which_o be_v require_v to_o be_v demonstrate_v ¶_o a_o assumpt_n a_o line_n contain_v in_o power_n a_o irrational_a superficies_n be_v irrational_a assumpt_n suppose_v that_o the_o line_n ab_fw-la contain_v in_o power_n a_o irrational_a superficies_n that_o be_v let_v the_o square_v describe_v upon_o the_o line_n ab_fw-la be_v equal_a unto_o a_o irrational_a superficies_n then_o i_o say_v that_o the_o line_n ab_fw-la be_v irrational_a for_o if_o the_o line_n ab_fw-la be_v rational_a they_o shall_v the_o square_a of_o the_o line_n ab_fw-la be_v also_o rational_a for_o so_o be_v it_o put_v in_o the_o definition_n but_o by_o supposition_n it_o be_v not_o wherefore_o the_o line_n ab_fw-la be_v irrational_a a_o line_n therefore_o contain_v in_o power_n a_o irrational_a superficies_n be_v irrational_a ¶_o the_o 18._o theorem_a the_o 21._o proposition_n a_o rectangle_n figure_n comprehend_v under_o two_o rational_a right_a line_n commensurable_a in_o power_n only_o be_v irrational_a and_o the_o line_n which_o in_o power_n contain_v that_o rectangle_n figure_n be_v irrational_a &_o be_v call_v a_o medial_a line_n svppose_v that_o this_o rectangle_n figure_n ac_fw-la be_v comprehend_v under_o these_o rational_a right_a line_n ab_fw-la and_o bc_o commensurable_a in_o power_n only_o then_o i_o say_v that_o the_o superficies_n ac_fw-la be_v irrational_a and_o the_o line_n which_o contain_v it_o in_o power_n be_v irrational_a and_o be_v call_v a_o medial_a line_n construction_n describe_v by_o the_o 46._o of_o the_o first_o upon_o the_o line_n ab_fw-la a_o square_a ad._n demonstration_n wherefore_o the_o square_a ad_fw-la be_v rational_a and_o forasmuch_o as_o the_o line_n ab_fw-la be_v unto_o the_o line_n bc_o incommensurable_a in_o length_n for_o they_o be_v suppose_v to_o be_v commensurable_a in_o power_n only_o and_o the_o line_n ab_fw-la be_v equal_a unto_o the_o line_n bd_o therefore_o also_o the_o lineâ—_z bd_o be_v unto_o the_o line_n bc_o incommensurable_a in_o length_n and_o 〈◊〉_d â—hâ—_n linâ—_n 〈…〉_o be_v to_o the_o line_n â—_o c_o so_fw-mi 〈◊〉_d the_o square_a ad_fw-la to_o the_o parallelogram_n ac_fw-la by_o the_o first_o of_o the_o fiuâ—_o wherefore_o by_o the_o 10._o of_o the_o ten_o the_o square_a dam_n be_v unto_o the_o parallelogram_n ac_fw-la incommensurable_a but_o the_o square_a dam_n be_v rational_a wherefore_o the_o parallelogram_n ac_fw-la be_v irrational_a wherefore_o also_o the_o line_n that_o contain_v the_o superficies_n ac_fw-la in_o power_n that_o be_v who_o square_a be_v equal_a unto_o the_o parallelogram_n ac_fw-la be_v by_o the_o assumpt_n go_v before_o irrational_a and_o it_o be_v call_v a_o medial_a line_n for_o that_o the_o square_n which_o be_v make_v of_o it_o be_v equal_a to_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o and_o therefore_o it_o be_v by_o the_o second_o part_n of_o the_o 17._o of_o the_o six_o a_o mean_a proportional_a line_n between_o the_o line_n ab_fw-la and_o bc._n a_o rectangle_n figure_n therefore_o comprehend_v under_o rational_a right_a line_n which_o be_v commensurable_a in_o power_n only_o be_v irrational_a and_o the_o line_n which_o in_o power_n contain_v that_o rectangle_n figure_n be_v irrational_a and_o be_v call_v a_o medial_a line_n at_o this_o proposition_n do_v euclid_n first_o entreat_v of_o the_o generation_n and_o production_n of_o irrational_a line_n and_o here_o he_o search_v out_o the_o first_o kind_n of_o they_o which_o he_o call_v a_o medial_a line_n and_o the_o definition_n thereof_o be_v full_o gather_v and_o take_v out_o of_o this_o 21._o proposition_n which_o be_v this_o a_o medial_a line_n be_v a_o irrational_a line_n who_o square_a be_v equal_a to_o a_o rectangled_a figure_n contain_v of_o two_o rational_a line_n commensurable_a in_o power_n only_o line_n it_o be_v call_v a_o medial_a line_n as_o theon_n right_o say_v for_o two_o cause_n first_o for_o that_o the_o power_n or_o square_v which_o it_o producethâ—_n be_v equal_a to_o a_o medial_a superficies_n or_o parallelogram_n for_o as_o that_o line_n which_o produce_v a_o rational_a square_n be_v call_v a_o rational_a line_n and_o that_o line_n which_o produce_v a_o irrational_a square_n or_o a_o square_n equal_a to_o a_o irrational_a figure_n general_o be_v call_v a_o irrational_a line_n so_o iâ—_z thaâ—_z line_n which_o produce_v a_o medial_a square_n or_o a_o square_n equal_a to_o a_o medial_a superficies_n call_v by_o special_a name_n a_o medial_a line_n second_o it_o be_v call_v a_o medial_a line_n because_o it_o be_v a_o mean_a proportional_a between_o the_o two_o line_n commensurable_a in_o power_n only_o which_o comprehend_v the_o medial_a superficies_n ¶_o a_o corollary_n add_v by_o flussates_n a_o rectangle_n parallelogram_n contain_v under_o a_o rational_a line_n and_o a_o irrational_a line_n be_v irrational_a corollary_n for_o if_o the_o line_n ab_fw-la be_v rational_a and_o
if_o the_o line_n cb_o be_v irrational_a they_o shall_v be_v incommensurable_a but_o as_o the_o line_n bd_o which_o be_v equal_a to_o the_o line_n basilius_n be_v to_o the_o line_n bc_o so_o be_v the_o square_a ad_fw-la to_o the_o parallelogram_n ac_fw-la wherefore_o the_o parallelogram_n ac_fw-la shall_v be_v incommensurable_a to_o the_o square_a ad_fw-la which_o be_v rational_a for_o that_o the_o line_n ab_fw-la whereupon_o it_o be_v describe_v be_v suppose_v to_o be_v rational_a wherefore_o the_o parallelogram_n ac_fw-la which_o be_v contain_v under_o the_o rational_a line_n ab_fw-la and_o the_o irrational_a line_n bc_o be_v irrational_a ¶_o a_o assumpt_n if_o there_o be_v two_o right_a line_n as_o the_o first_o be_v to_o the_o second_o so_o be_v the_o square_n which_o be_v describe_v upon_o the_o first_o to_o the_o parallelogram_n which_o be_v contain_v under_o the_o two_o right_a line_n suppose_v that_o there_o be_v two_o right_a line_n ab_fw-la and_o bc._n book_n then_o i_o say_v that_o as_o the_o line_n ab_fw-la be_v to_o the_o line_n bc_o so_o be_v the_o square_a of_o the_o line_n ab_fw-la â—â—_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc._n describe_v by_o the_o 46._o of_o the_o first_o upon_o the_o line_n ab_fw-la a_o square_a ad._n and_o make_v perfect_a the_o parallelogram_n ac_fw-la now_o for_o that_o as_o the_o line_n ab_fw-la be_v to_o the_o line_n bc_o for_o the_o line_n ab_fw-la be_v equal_a to_o the_o line_n bd_o so_o be_v the_o square_a ad_fw-la to_o the_o parallelogram_n ca_n by_o the_o first_o of_o the_o sixâ—_o and_o ad_fw-la be_v the_o square_n which_o be_v make_v of_o the_o line_n ab_fw-la and_o ac_fw-la be_v that_o which_o be_v contain_v under_o the_o line_n bd_o and_o bc_o that_o be_v under_o the_o line_n ab_fw-la &_o bc_o therefore_o as_o the_o line_n ab_fw-la be_v to_o the_o line_n bc_o so_o be_v the_o square_n describe_v upon_o the_o the_o line_n ab_fw-la to_o the_o rectangle_n figure_n contain_v under_o the_o line_n ab_fw-la &_o bc._n and_o converse_o as_o the_o parallelogram_n which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o be_v to_o the_o square_n of_o the_o line_n ab_fw-la so_o be_v the_o line_n cb_o to_o the_o line_n basilius_n ¶_o the_o 19_o theorem_a the_o 22._o proposition_n if_o upon_o a_o rational_a line_n be_v apply_v the_o square_a of_o a_o medial_a line_n the_o other_o side_n that_o make_v the_o breadth_n thereof_o shall_v be_v rational_a and_o incommensurable_a in_o length_n to_o the_o line_n whereupon_o the_o parallelogram_n be_v apply_v svppose_v that_o a_o be_v a_o medial_a line_n and_o let_v bc_o be_v a_o line_n rational_a and_o upon_o the_o line_n bc_o describe_v a_o rectangle_n parallelogram_n equal_a unto_o the_o square_n of_o the_o line_n a_o and_o let_v the_o same_o be_v bd_o make_v in_o breadth_n the_o line_n cd_o then_o i_o say_v that_o the_o line_n cd_o be_v rational_a and_o incommensurable_a in_o length_n unto_o the_o line_n cb._n 〈◊〉_d for_o forasmuch_o as_o a_o be_v a_o medial_a line_n it_o contain_v in_o power_n by_o the_o 21._o of_o the_o ten_o a_o rectangle_n parallelogram_n comprehend_v under_o rational_a right_a line_n commensurable_a in_o power_n only_o suppose_v that_o be_v contain_v in_o power_n the_o parallelogram_n gf_o and_o by_o supposition_n it_o also_o contain_v in_o power_n the_o parallelogram_n bd._n wherefore_o the_o parallelogram_n bd_o be_v equal_a unto_o the_o parallelogram_n gf_o and_o it_o be_v also_o equiangle_n unto_o it_o for_o that_o they_o be_v each_o rectangle_n but_o in_o parallelogram_n equal_a and_o equiangle_n the_o side_n which_o contain_v the_o equal_a angle_n be_v reciprocal_a by_o the_o 14._o of_o the_o six_o wherefore_o what_o proportion_n the_o line_n bc_o have_v to_o the_o line_n eglantine_n the_o same_o have_v the_o line_n of_o to_o the_o line_n cd_o therefore_o by_o the_o 22._o of_o the_o six_o as_o the_o square_n of_o the_o line_n bc_o be_v to_o the_o square_n of_o the_o line_n eglantine_n so_o be_v the_o square_a of_o the_o line_n of_o to_o the_o square_n of_o the_o line_n cd_o but_o the_o square_n of_o the_o line_n bc_o be_v commensurable_a unto_o the_o square_n of_o the_o line_n eglantine_n by_o supposition_n for_o either_o of_o they_o be_v rational_a wherefore_o by_o the_o the_o 10._o of_o the_o ten_o the_o square_a of_o the_o line_n of_o be_v commensurable_a unto_o the_o square_n of_o the_o linâ—_n cd_o but_o the_o square_n of_o the_o line_n of_o be_v rational_a wherefore_o the_o square_a of_o the_o line_n cd_o be_v likewise_o rational_a wherefore_o the_o line_n cd_o be_v rational_a and_o forasmuch_o as_o the_o line_n of_o be_v incommensurable_a in_o length_n unto_o the_o line_n eglantine_n for_o they_o be_v suppose_v to_o be_v commensurable_a in_o power_n only_o but_o as_o the_o line_n of_o be_v to_o the_o line_n eglantine_n so_o by_o the_o assumpt_n go_v before_o be_v the_o square_a of_o the_o line_n of_o to_o the_o parallelogram_n which_o be_v contain_v under_o the_o line_n of_o and_o eglantine_n wherefore_o by_o the_o 10._o of_o the_o ten_o the_o square_a of_o the_o line_n of_o be_v incommensurable_a unto_o the_o parallelogram_n which_o be_v contain_v under_o the_o line_n fe_o and_o eglantine_n but_o unto_o the_o square_n of_o the_o line_n of_o the_o square_n of_o the_o line_n cd_o be_v commensurable_a for_o it_o be_v prove_v that_o â—ither_o of_o they_o be_v a_o rational_a linâ—_n and_o that_o which_o be_v contain_v under_o the_o line_n dc_o and_o cb_o be_v commensurable_a unto_o that_o which_o be_v contain_v under_o the_o line_n fe_o and_o eglantine_n for_o they_o be_v both_o equal_a to_o the_o square_n of_o the_o line_n a._n wherefore_o by_o the_o 13._o of_o the_o ten_o the_o square_a of_o the_o line_n cd_o be_v incommensurable_a to_o that_o which_o be_v contain_v under_o the_o line_n dc_o and_o cb._n but_o as_o the_o square_n of_o the_o line_n cd_o be_v to_o that_o which_o be_v contain_v under_o the_o line_n dc_o and_o cb_o so_o by_o the_o assumpt_n go_v before_o be_v the_o line_n dc_o to_o the_o line_n cb._n wherefore_o the_o line_n dc_o be_v incommensurable_a in_o length_n unto_o the_o line_n cb._n wherefore_o the_o line_n cd_o be_v rational_a and_o incommensurable_a in_o length_n unto_o the_o line_n cb._n if_o therefore_o upon_o a_o rational_a line_n be_v apply_v the_o square_a of_o a_o medial_a line_n the_o other_o side_n that_o make_v the_o breadth_n thereof_o shall_v be_v rational_a and_o incommensurable_a in_o length_n to_o the_o line_n whereupon_o the_o parallelogram_n be_v apply_v which_o be_v require_v to_o be_v prove_v a_o square_n be_v say_v to_o be_v apply_v upon_o a_o line_n when_o it_o or_o a_o parallelogram_n equal_a unto_o it_o be_v apply_v upon_o the_o say_a line_n line_n if_o upon_o a_o rational_a line_n give_v we_o will_v apply_v a_o rectangle_n parallelogram_n equal_a to_o the_o square_n of_o a_o medial_a line_n give_v and_o so_o of_o any_o line_n give_v we_o must_v by_o the_o 11._o of_o the_o six_o find_v out_o the_o three_o line_n proportional_a with_o the_o rational_a line_n and_o the_o medial_a line_n give_v so_o yet_o that_o the_o rational_a line_n be_v the_o first_o and_o the_o medial_a line_n give_v which_o contain_v in_o power_n the_o square_a to_o be_v apply_v be_v the_o second_o for_o then_o the_o superficies_n contain_v under_o the_o first_o and_o the_o three_o shall_v be_v equal_a to_o the_o square_n of_o the_o middle_a line_n by_o the_o 17._o of_o the_o six_o ¶_o the_o 20._o theorem_a the_o 23._o proposition_n a_o right_a line_n commensurable_a to_o a_o medial_a line_n be_v also_o a_o medial_a line_n svppose_v that_o a_o be_v a_o medial_a line_n and_o unto_o the_o line_n a_o let_v the_o line_n b_o be_v commensurable_a either_o in_o length_n &_o in_o power_n or_o in_o power_n only_o construction_n then_o i_o say_v that_o b_o also_o be_v a_o medial_a line_n let_v there_o be_v put_v a_o rational_a line_n cd_o and_o upon_o the_o line_n cd_o apply_v a_o rectangle_n parallelogram_n ce_fw-fr equal_a unto_o the_o square_n of_o the_o line_n a_o and_o make_v in_o breadth_n the_o line_n ed._n wherefore_o by_o the_o proposition_n go_v before_o the_o line_n ed_z be_v rational_a and_o incommensurable_a in_o length_n unto_o the_o line_n cd_o demonstration_n and_o again_o upon_o the_o line_n cd_o apply_v a_o rectangle_n parallelogram_n cf_o equal_a unto_o the_o square_n of_o the_o line_n b_o and_o make_v in_o breadth_n the_o line_n df._n and_o forasmuch_o as_o the_o line_n a_o be_v commensurable_a unto_o the_o line_n b_o therefore_o the_o square_a of_o the_o line_n a_o be_v commensurable_a to_o the_o square_n of_o the_o line_n b._n but_o the_o parallelogram_n ec_o be_v equal_a to_o the_o square_n of_o the_o linâ—_n a_o and_o the_o parallelogram_n cf_o be_v equal_a to_o
the_o square_n of_o the_o line_n b_o wherefore_o the_o parallelogram_n ec_o be_v commensurable_a unto_o the_o parallelogram_n cf._n but_o as_o the_o parallelogram_n ec_o be_v to_o the_o parallelogram_n cf_o so_o be_v the_o line_n ed_z to_o the_o line_n df_o by_o the_o first_o of_o the_o six_o wherefore_o by_o the_o 10._o of_o the_o ten_o the_o line_n ed_z be_v commensurable_a in_o length_n unto_o the_o line_n df._n but_o the_o line_n ed_z be_v rational_a and_o incommensurable_a in_o length_n unto_o the_o line_n dc_o wherefore_o the_o line_n df_o be_v rational_a and_o incommensurable_a in_o length_n unto_o the_o line_n dc_o by_o the_o 13._o of_o the_o ten_o wherefore_o the_o line_n cd_o and_o df_o be_v rational_a commensurable_a in_o power_n only_o but_o a_o rectangle_n figure_n comprehend_v under_o rational_a right_a line_n commensurable_a in_o power_n only_o be_v by_o the_o â—1_n of_o the_o ten_o irrational_a and_o the_o line_n that_o contain_v it_o in_o power_n be_v irrational_a and_o be_v call_v a_o medial_a line_n wherefore_o the_o line_n that_o contain_v in_o power_n that_o which_o be_v comprehend_v under_o the_o line_n cd_o and_o df_o be_v a_o medial_a line_n but_o the_o line_n b_o contain_v in_o power_n the_o parallelogram_n which_o be_v comprehend_v under_o the_o line_n cd_o and_o df_o wherefore_o the_o line_n b_o be_v a_o medial_a line_n a_o right_a line_n therefore_o commensurable_a to_o a_o medial_a line_n be_v also_o a_o medial_a line_n which_o be_v require_v to_o be_v prove_v ¶_o corollary_n hereby_o it_o be_v manifest_a that_o a_o superficies_n commensurable_a unto_o a_o medial_a superficies_n be_v also_o a_o medial_a superficies_n for_o the_o line_n which_o containâ—_n in_o power_n those_o superficiece_n be_v commensurable_a in_o power_n of_o which_o the_o one_o be_v a_o medial_a line_n by_o the_o definition_n of_o a_o medial_a line_n in_o the_o 21._o of_o this_o ten_o wherefore_o the_o other_o also_o be_v a_o medial_a line_n by_o this_o 23._o proposition_n and_o as_o it_o be_v say_v of_o rational_a line_n so_o also_o be_v it_o to_o be_v say_v oâ—_n medial_a line_n namely_o that_o a_o liâ—e_z commensurable_a to_o a_o medial_a line_n be_v also_o a_o medial_a line_n a_o line_n i_o say_v which_o be_v commensurable_a unto_o a_o medial_a line_n whether_o it_o be_v commensurable_a in_o length_n and_o also_o in_o power_n or_o else_o in_o power_n only_o for_o universal_o it_o be_v true_a that_o line_n commensurable_a in_o length_n be_v also_o commensurable_a in_o power_n now_o if_o unto_o a_o medial_a line_n there_o be_v a_o line_n commensurable_a in_o power_n if_o it_o be_v commensurable_a in_o length_n they_o be_v those_o line_n call_v medial_a line_n commensurable_a in_o length_n &_o in_o power_n but_o if_o they_o be_v commensurable_a in_o power_n only_o thâ—y_o be_v call_v medial_a line_n commensurable_a in_o power_n only_o there_o be_v also_o other_o right_a line_n incommensurable_a in_o length_n to_o the_o medial_a line_n and_o commensurable_a in_o power_n only_o to_o the_o same_o and_o these_o line_n be_v also_o call_v medial_a for_o that_o they_o be_v commensurable_a in_o power_n to_o the_o medial_a line_n and_o in_o aâ—_z muâ—h_o as_o they_o be_v medial_a line_n they_o be_v commensurable_a in_o power_n the_o one_o to_o the_o other_o but_o be_v compare_v the_o one_o to_o the_o other_o they_o may_v be_v commensurable_a either_o in_o length_n and_o therefore_o in_o power_n or_o else_o in_o power_n only_o and_o then_o if_o they_o be_v commensurable_a in_o length_n they_o be_v call_v also_o medial_a line_n commensurable_a in_o length_n and_o so_o consequent_o they_o be_v understand_v to_o be_v commensurable_a in_o power_n but_o iâ—_z they_o be_v commensurable_a in_o power_n only_o yet_o notwithstanding_o they_o also_o be_v call_v medial_a line_n commensurable_a in_o power_n only_o flussates_n after_o this_o proposition_n teach_v how_o to_o come_v to_o the_o understanding_n of_o medial_a superficiece_n and_o line_n by_o surd_a number_n after_o this_o manner_n namely_o to_o express_v the_o medial_a superficiece_n by_o the_o root_n of_o number_n which_o be_v not_o square_a number_n and_o the_o line_n contain_v in_o power_n such_o medial_a superficiece_n by_o the_o root_n of_o root_n of_o number_n not_o square_a medial_a line_n also_o commensurable_a be_v express_v by_o the_o root_n of_o root_n of_o like_a superficial_a number_n but_o yet_o not_o square_a but_o such_o as_o have_v that_o proportion_n that_o the_o square_n of_o square_a number_n have_v for_o the_o root_n of_o those_o number_n and_o the_o root_n of_o root_n be_v in_o proportion_n as_o number_n be_v namely_o if_o the_o square_n be_v proportional_a the_o side_n also_o shall_v be_v proportional_a by_o the_o 22._o of_o the_o six_o but_o medial_a line_n incommensurable_a in_o power_n be_v the_o root_n of_o root_n of_o number_n which_o have_v not_o that_o proportion_n that_o square_a number_n have_v for_o their_o root_n be_v the_o power_n of_o medial_a line_n which_o be_v incommensurable_a by_o the_o 9_o of_o the_o ten_o but_o medial_a line_n commensurable_a in_o power_n only_o be_v the_o root_n of_o root_n of_o number_n which_o have_v that_o proportion_n that_o simple_a square_a number_n have_v and_o not_o which_o the_o square_n of_o square_n have_v for_o the_o root_n which_o be_v the_o power_n of_o the_o medial_a line_n be_v commensurable_a but_o the_o root_n of_o root_n which_o express_v the_o say_v medial_a line_n be_v incommensurable_a wherefore_o there_o may_v be_v find_v out_o infinite_a medial_a line_n incommensurable_a in_o powâ—r_n by_o compare_v infinite_a unlike_o plain_a number_n the_o one_o to_o the_o other_o for_o unlike_o plain_a number_n which_o have_v not_o the_o proportion_n of_o square_a number_n do_v make_v the_o root_n which_o express_v the_o superficiece_n of_o medial_a line_n incommensurable_a by_o the_o 9_o of_o the_o ten_o and_o therefore_o the_o medial_a line_n contain_v in_o power_n those_o superficiece_n be_v incommensurable_a in_o length_n for_o line_n incommensurable_a in_o power_n be_v always_o incommensurable_a in_o length_n by_o the_o corollary_n of_o the_o 9_o of_o the_o ten_o ¶_o the_o 21._o theorem_a the_o 24._o proposition_n a_o rectangle_n parallelogram_n comprehend_v under_o medial_a line_n commensurable_a in_o length_n be_v a_o medial_a rectangle_n parallelogram_n svppose_v that_o the_o rectangle_n parallelogram_n agnostus_n be_v comprehend_v under_o these_o medial_a right_a line_n ab_fw-la and_o bc_o which_o let_v be_v commensurable_a in_o length_n construction_n then_o i_o say_v that_o ac_fw-la be_v a_o medial_a rectangle_n parallelogram_n describe_v by_o the_o 46._o of_o the_o first_o upon_o the_o line_n ab_fw-la a_o square_a ad._n demonstration_n wherefore_o the_o square_a ad_fw-la be_v a_o medial_a superficies_n and_o forasmuch_o as_o the_o line_n ab_fw-la be_v commensurablâ—_n in_o length_n unto_o the_o line_n bc_o and_o the_o line_n ab_fw-la be_v equal_a unto_o the_o line_n bd_o therefore_o the_o line_n bd_o be_v commensurable_a in_o length_n unto_o the_o line_n bc._n but_o 〈◊〉_d the_o line_n db_o be_v to_o the_o line_n bc_o so_o be_v the_o square_a dam_n to_o the_o parallelogram_n ac_fw-la by_o the_o first_o of_o the_o six_o wherefore_o by_o the_o 10._o of_o the_o ten_o the_o square_a dam_n be_v commensurable_a unto_o the_o parallelogram_n ac_fw-la but_o the_o square_a dam_n be_v medial_a for_o that_o it_o be_v describe_v upon_o a_o medial_a line_n wherefore_o ac_fw-la also_o be_v a_o medial_a parallelogram_n by_o the_o former_a corollary_n a_o rectangle_v etc_o etc_o which_o be_v require_v to_o be_v prove_v ¶_o the_o 22â—_n theorem_a the_o 25._o proposition_n a_o rectangle_n parallelogram_n comprehend_v under_o medial_a right_a line_n commensurable_a in_o power_n only_o be_v either_o rational_a or_o medial_a and_o now_o if_o the_o line_n hk_o be_v commensurable_a in_o length_n unto_o the_o line_n he_o that_o be_v unto_o the_o line_n fg_o note_n which_o be_v equal_a to_o the_o line_n he_o than_o by_o the_o 19_o of_o the_o ten_o the_o parallelogram_n nh_o be_v rational_a but_o if_o it_o be_v incommensurable_a in_o length_n unto_o the_o line_n fg_o than_o the_o line_n hk_o and_o he_o be_v rational_a commensurable_a in_o power_n only_o and_o so_o shall_v the_o parallelogram_n hn_o be_v medial_a wherefore_o the_o parallelogram_n hn_o be_v either_o rational_a or_o medial_a but_o the_o parallelogram_n hn_o be_v equal_a to_o the_o parallelogram_n ag._n wherefore_o the_o parallelogram_n ac_fw-la be_v either_o rational_a or_o medial_a a_o rectangle_n parallelogram_n therefore_o comprehend_v under_o medial_a right_a line_n commensurable_a in_o power_n only_o be_v either_o rational_a or_o medial_a which_o be_v require_v to_o be_v demonstrate_v how_o to_o find_v medial_a line_n commensurable_a in_o power_n only_o contain_v a_o rational_a parallelogram_n and_o also_o other_o medial_a line_n commensurable_a in_o power_n contain_v a_o medial_a
parallelogram_n shall_v afterward_o be_v teach_v in_o the_o 27._o and_o 28._o proposition_n of_o this_o book_n ¶_o a_o corollary_n hereby_o it_o be_v manifest_a that_o a_o rectangle_n parallelogram_n contain_v under_o two_o right_a line_n be_v the_o mean_a proportional_a between_o the_o square_n of_o the_o say_v line_n corollary_n as_o it_o be_v manifest_a by_o the_o first_o of_o the_o six_o that_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o be_v the_o mean_a proportional_a between_o the_o square_n ad_fw-la and_o cx_o this_o corollary_n be_v put_v after_o the_o 53._o proposition_n of_o this_o book_n as_o a_o assumpt_n and_o there_o demonstrate_v which_o there_o in_o his_o place_n you_o shall_v find_v but_o because_o it_o follow_v of_o this_o proposition_n so_o evident_o and_o brief_o without_o far_a demonstration_n i_o think_v it_o not_o amiss_o here_o by_o the_o way_n to_o note_v it_o ¶_o the_o 23._o theorem_a the_o 26._o proposition_n a_o medial_a superficies_n exceed_v not_o a_o medial_a superficies_n by_o a_o rational_a superficies_n for_o if_o it_o be_v possible_a let_v ab_fw-la be_v a_o medial_a superficies_n exceed_v ac_fw-la be_v also_o a_o medial_a superficies_n by_o db_o be_v a_o rational_a superficies_n and_o let_v there_o be_v put_v a_o rational_a right_a line_n ef._n construction_n and_o upon_o the_o line_n of_o apply_v a_o rectangle_n parallelogram_n fh_o equal_a unto_o the_o medial_a superficies_n ab_fw-la who_o other_o side_n let_v be_v eh_o and_o from_o the_o parallelogram_n fh_o take_v away_o the_o parallelogram_n fg_o equal_a unto_o the_o medial_a superficies_n ac_fw-la absurdity_n wherefore_o by_o the_o three_o common_a sentence_n the_o residue_n bd_o be_v equal_a to_o the_o residue_n kh_o but_o by_o supposition_n the_o superficies_n db_o be_v rational_a wherefore_o the_o superficies_n kh_o be_v also_o rational_a and_o forasmuch_o as_o either_o of_o these_o superficiece_n ab_fw-la and_o ac_fw-la be_v medial_a and_o ab_fw-la be_v equal_a unto_o fh_o &_o ac_fw-la unto_o fg_o therefore_o either_o of_o these_o superficiece_n fh_a and_o fg_o be_v medial_a and_o they_o be_v apply_v upon_o the_o rational_a line_n ef._n wherefore_o by_o the_o 22._o of_o the_o ten_o either_o of_o these_o line_n he_o and_o eglantine_n be_v rational_a &_o incommensurable_a in_o length_n unto_o the_o line_n ef._n and_o forasmuch_o as_o the_o superficies_n db_o be_v rational_a and_o the_o superficies_n kh_o be_v equal_a unto_o it_o therefore_o kh_o be_v also_o rational_a and_o it_o be_v apply_v upon_o the_o rational_a line_n of_o for_o it_o be_v apply_v upon_o the_o line_n gk_o which_o be_v equal_a to_o the_o line_n of_o wherefore_o by_o the_o 20._o of_o the_o ten_o the_o line_n gh_o be_v rational_a and_o commensurable_a in_o length_n unto_o the_o line_n gk_o but_o the_o line_n gk_o be_v equal_a to_o the_o line_n ef._n wherefore_o the_o line_n gh_o be_v rational_a and_o commensurable_a in_o length_n unto_o the_o line_n ef._n but_o the_o line_n eglantine_n be_v rational_a and_o incommensurable_a in_o length_n to_o the_o line_n ef._n wherefore_o by_o the_o 13._o of_o the_o ten_o the_o line_n eglantine_n be_v incommensurable_a in_o length_n unto_o the_o line_n gh_o and_o as_o the_o line_n eglantine_n be_v to_o the_o line_n gh_o so_o be_v the_o square_a of_o the_o line_n eglantine_n to_o the_o parallelogram_n contain_v under_o the_o line_n eglantine_n and_o gh_o by_o the_o assumpt_n put_v before_o the_o 21._o of_o the_o ten_o wherefore_o by_o the_o 10._o of_o the_o ten_o the_o square_a of_o the_o line_n eglantine_n be_v incommensurable_a unto_o the_o parallelogram_n contain_v under_o the_o line_n eglantine_n and_o gh_o but_o unto_o the_o square_n of_o the_o line_n eglantine_n be_v commensurable_a the_o square_n of_o the_o line_n eglantine_n and_o gh_o for_o either_o of_o they_o be_v rational_a as_o have_v before_o be_v prove_v wherefore_o the_o square_n of_o the_o line_n eglantine_n and_o gh_o be_v incommensurable_a unto_o the_o parallelogram_n contain_v under_o the_o line_n eglantine_n and_o gh_o but_o unto_o the_o parallelogram_n contain_v under_o the_o line_n eglantine_n and_o gh_o be_v commensurable_a that_o which_o be_v contain_v under_o the_o line_n fg_o and_o gh_o twice_o for_o they_o be_v in_o proportion_n the_o one_o to_o the_o other_o as_o number_n be_v to_o number_n namely_o as_o unity_n be_v to_o the_o number_n 2_o or_o as_o 2._o be_v to_o 4_o and_o therefore_o by_o the_o 6._o of_o this_o book_n they_o be_v commensurable_a wherefore_o by_o the_o 13._o of_o the_o ten_o the_o square_n of_o the_o line_n eglantine_n and_o gh_o be_v incommensurable_a unto_o that_o which_o be_v contain_v under_o the_o line_n eglantine_n and_o gh_o twice_o this_o be_v more_o brieâ—ly_o conclude_v by_o the_o corollary_n of_o the_o 13._o of_o the_o ten_o but_o the_o square_n of_o the_o line_n eglantine_n and_o gh_o together_o with_o that_o which_o be_v contain_v under_o the_o line_n eglantine_n and_o gh_o twice_o be_v equal_a to_o the_o square_n of_o the_o line_n eh_o by_o the_o 4._o of_o the_o second_o wherefore_o the_o square_a of_o the_o line_n eh_o be_v incommensurable_a to_o the_o square_n of_o the_o line_n eglantine_n and_o gh_o by_o the_o 16._o of_o the_o ten_o but_o the_o square_n of_o the_o line_n fg_o &_o gh_o be_v rational_a wherefore_o the_o square_a of_o the_o line_n eh_o be_v irrational_a wherefore_o the_o line_n also_o eh_o be_v irrational_a but_o it_o have_v before_o be_v prove_v to_o be_v rational_a which_o be_v impossible_a wherefore_o a_o medial_a superficies_n exceed_v not_o a_o medial_a superficies_n by_o a_o rational_a superficies_n which_o be_v require_v to_o be_v prove_v ¶_o the_o 4._o problem_n the_o 27._o proposition_n to_o find_v out_o medial_a line_n commensurable_a in_o power_n only_o contain_v a_o rational_a parallelogram_n let_v there_o be_v put_v two_o rational_a line_n commensurable_a in_o power_n only_o namely_o a_o and_o b._n construction_n and_o by_o the_o 13._o of_o the_o six_o take_v the_o mean_a proportional_a between_o the_o line_n a_o and_o b_o and_o let_v the_o same_o line_n be_v c._n and_o as_o the_o line_n a_o be_v to_o the_o line_n b_o so_o by_o the_o 12._o of_o the_o six_o let_v the_o line_n c_o be_v to_o the_o line_n d._n demonstration_n and_o forasmuch_o as_o a_o and_o b_o be_v rational_a line_n commensurable_a in_o power_n only_o therefore_o by_o the_o 21._o of_o the_o ten_o that_o which_o be_v contain_v under_o the_o line_n a_o and_o b_o that_o be_v the_o square_a of_o the_o line_n c._n for_o the_o square_n of_o the_o line_n c_o be_v equal_a to_o the_o parallelogram_n contain_v under_o the_o line_n a_o anâ—_z b_o by_o the_o 17._o of_o the_o six_o be_v medial_a â—herfore_o c_z also_o be_v a_o medial_a line_n and_o for_o that_o as_o the_o line_n a_o be_v to_o the_o line_n b_o so_o be_v the_o line_n c_o to_o the_o line_n d_o therefore_o as_o the_o square_n of_o the_o line_n a_o be_v to_o the_o square_n of_o the_o line_n b_o so_o be_v the_o square_a of_o the_o line_n c_o to_o the_o square_n of_o the_o line_n d_o by_o the_o 22._o of_o the_o six_o but_o the_o square_n of_o the_o line_n a_o and_o b_o be_v commensurable_a for_o the_o liâ—â—s_v a_o and_o b_o aâ—e_fw-fr suppose_a to_o be_v rational_a commensurable_a in_o power_n only_o wherefore_o also_o the_o square_n of_o the_o line_n c_o and_o d_o be_v commensurable_a by_o the_o 10._o of_o the_o ten_o wherefore_o the_o line_n c_o and_o d_o be_v commensurable_a in_o power_n only_o and_o c_o be_v a_o medial_a line_n wherefore_o by_o the_o 23._o of_o the_o ten_o d_z also_o be_v a_o medial_a line_n wherefore_o c_z and_o d_o be_v medial_a line_n commensurable_a in_o power_n only_o now_o also_o i_o say_v that_o they_o contain_v a_o rational_a parallelogram_n for_o for_o that_o as_o the_o line_n a_o be_v to_o the_o line_n b_o so_o be_v the_o line_n c_o to_o the_o line_n d_o therefore_o alternate_o also_o by_o the_o 16._o of_o the_o five_o as_o the_o line_n a_o be_v to_o the_o line_n c_o so_o be_v the_o line_n b_o to_o the_o line_n d._n but_o as_o the_o line_n a_o be_v to_o the_o line_n c_o so_o be_v the_o line_n c_o to_o the_o line_n b_o wherefore_o as_o the_o line_n c_o be_v to_o the_o line_n b_o so_o be_v the_o line_n b_o to_o the_o line_n d._n wherefore_o the_o parallelogram_n contain_v under_o the_o line_n c_o and_o d_o be_v equal_a to_o the_o square_n of_o the_o line_n b._n but_o the_o square_n of_o the_o line_n b_o be_v rational_a wherefore_o the_o parallelogram_n which_o be_v contain_v under_o the_o line_n c_o and_o d_o be_v also_o rational_a wherefore_o there_o be_v find_v out_o medial_a line_n commensurablâ—_n in_o powâ—r_n only_o contain_v a_o rational_a parallelogrammeâ—_n which_o 〈◊〉_d require_v to_o be_v do_v the_o 5._o problem_n the_o
28._o proposition_n to_o find_v out_o medial_a right_a line_n commensurable_a in_o power_n only_o contain_v a_o medial_a parallelogram_n let_v there_o be_v put_v three_o rational_a right_a line_n commensurable_a in_o power_n only_o namely_o a_o b_o and_o c_o construction_n and_o by_o the_o 13._o of_o the_o six_o take_v the_o mean_a proportional_a between_o the_o line_n a_o and_o b_o &_o let_v thâ—_z same_o be_v d._n and_o as_o the_o line_n b_o be_v to_o the_o line_n c_o so_o by_o the_o 12._o of_o the_o six_o let_v the_o line_n d_o be_v to_o the_o line_n e._n and_o forasmuch_o as_o the_o line_n a_o and_o b_o be_v rational_a commensurable_a in_o power_n only_o demonstration_n therefore_o by_o the_o 21._o of_o the_o ten_o that_o which_o be_v contain_v under_o the_o line_n a_o and_o b_o that_o be_v the_o square_a of_o the_o line_n d_o be_v medial_a wherefore_o d_z be_v a_o medial_a line_n and_o forasmuch_o as_o the_o line_n b_o and_o c_o be_v commensurable_a in_o power_n only_o and_o as_o the_o line_n b_o be_v to_o the_o line_n c_o so_o be_v the_o line_n d_o to_o the_o line_n e_o wherefore_o the_o line_n d_z and_o e_o be_v commensurable_a in_o power_n only_o by_o the_o corollary_n of_o the_o ten_o of_o this_o book_n but_o d_o be_v a_o medial_a line_n wherefore_o e_o also_o be_v a_o medial_a line_n by_o the_o 23._o of_o this_o book_n wherefore_o d_z &_o e_o be_v medial_a line_n commensurable_a in_o power_n only_o i_o say_v also_o that_o they_o contain_v a_o medial_a parallelogram_n for_o for_o that_o as_o the_o line_n b_o be_v to_o the_o line_n c_o so_o be_v the_o line_n d_o to_o the_o line_n e_o therefore_o alternate_o by_o the_o 16_o of_o the_o five_o as_o the_o line_n b_o be_v to_o the_o line_n d_o so_o be_v the_o line_n c_o to_o the_o line_n e._n but_o as_o the_o line_n b_o be_v to_o the_o line_n d_o so_o be_v the_o line_n d_o to_o the_o line_n aâ—_z by_o converse_n proportion_n which_o be_v prove_v by_o the_o corollary_n of_o the_o four_o of_o the_o five_o wherefore_o as_o the_o line_n d_o be_v to_o the_o line_n a_o so_o be_v the_o line_n c_o to_o the_o line_n e._n wherefore_o that_o which_o be_v contain_v under_o the_o line_n a_o &_o c_o be_v by_o the_o 16._o of_o the_o sixâ—_o equal_a to_o that_o which_o be_v contain_v under_o the_o line_n d_o &_o e._n but_o that_o which_o be_v contain_v under_o the_o line_n a_o and_o c_o be_v medial_a by_o the_o 21._o of_o the_o ten_o wherefore_o that_o which_o be_v contain_v under_o the_o line_n d_o and_o e_o be_v medial_a wherefore_o there_o be_v find_v out_o medial_a line_n commensurable_a in_o power_n only_o contain_v a_o medial_a superficies_n which_o be_v require_v to_o be_v do_v a_o assumpt_n to_o find_v out_o two_o square_a number_n which_o add_v together_o make_v a_o square_a number_n let_v there_o be_v put_v two_o like_o superficial_a number_n ab_fw-la and_o bc_o which_o how_o to_o find_v out_o have_v be_v teach_v after_o the_o 9_o proposition_n of_o this_o book_n and_o let_v they_o both_o be_v either_o even_a number_n or_o odd_a and_o let_v the_o great_a number_n be_v ab_fw-la and_o forasmuch_o as_o if_o from_o any_o even_a number_n be_v take_v away_o a_o even_a number_n or_o from_o a_o odd_a number_n be_v take_v away_o a_o odd_a number_n the_o residue_n shall_v be_v even_o by_o the_o 24._o and_o 26_o of_o the_o nine_o if_o therefore_o from_o ab_fw-la be_v a_o even_a number_n be_v take_v away_o bc_o a_o even_a number_n or_o from_o ab_fw-la be_v a_o odd_a number_n be_v take_v away_o bc_o be_v also_o odd_a the_o residue_n ac_fw-la shall_v be_v even_o divide_v the_o number_n ac_fw-la into_o two_o equal_a part_n in_o d_o wherefore_o the_o number_n which_o be_v produce_v of_o ab_fw-la into_o bc_o together_o with_o the_o square_a number_n of_o cd_o be_v by_o the_o six_o of_o the_o second_o as_o barlaam_n demonstrate_v it_o in_o number_n equal_a to_o the_o square_a number_n of_o bd._n but_o that_o which_o be_v produce_v of_o ab_fw-la into_o bc_o be_v a_o square_a number_n for_o it_o be_v prove_v by_o the_o first_o of_o the_o nine_o that_o if_o two_o like_o plain_a number_n multiply_v the_o one_o the_o other_o produce_v any_o number_n the_o number_n produce_v shall_v be_v a_o square_a number_n wherefore_o there_o be_v find_v out_o two_o square_a number_n the_o one_o be_v the_o square_a number_n which_o be_v produce_v of_o ab_fw-la into_o bc_o and_o the_o other_o the_o square_a number_n produce_v of_o cd_o which_o add_v together_o make_v a_o square_a number_n namely_o the_o square_a number_n produce_v of_o bd_o multiply_v into_o himself_o forasmuch_o as_o they_o be_v demonstrate_v equal_a to_o it_o corollary_n a_o corollary_n â—umber_n and_o hereby_o it_o be_v manifest_a that_o there_o be_v find_v out_o two_o square_a number_n namely_o the_o 〈◊〉_d the_o square_a number_n of_o bd_o and_o the_o other_o the_o square_a number_n of_o cd_o so_o that_o that_o number_n wherein_o the_o one_o exceed_v the_o other_o the_o number_n i_o say_v which_o be_v produce_v of_o ab_fw-la into_o bc_o be_v also_o a_o square_a number_n namely_o when_o aâ—_z &_o bc_o be_v like_o plain_a number_n but_o when_o they_o be_v not_o like_o plain_a number_n then_o be_v there_o find_v out_o two_o square_a number_n the_o square_a number_n of_o bd_o and_o the_o square_a number_n of_o dc_o who_o excess_n that_o be_v the_o number_n whereby_o the_o great_a exceed_v the_o less_o namely_o that_o which_o be_v produce_v of_o ab_fw-la into_o bc_o be_v not_o a_o square_a number_n ¶_o a_o assumpt_n to_o find_v out_o two_o square_a number_n which_o add_v together_o make_v not_o a_o square_a number_n assumpt_n let_v ab_fw-la and_o bc_o be_v like_o plain_a number_n so_o that_o by_o the_o first_o of_o the_o nine_o that_o which_o be_v produce_v of_o ab_fw-la into_o bc_o be_v a_o square_a number_n and_o let_v ac_fw-la be_v a_o even_a number_n and_o divide_v câ—_n into_o two_o equal_a parâ—es_n in_o d._n now_o by_o that_o which_o have_v before_o be_v say_v in_o the_o former_a assumpt_n it_o be_v manifest_a that_o the_o square_a number_n produce_v of_o ab_fw-la into_o bc_o together_o with_o the_o square_a number_n of_o cd_o be_v equal_a to_o the_o square_a number_n of_o bd._n take_v away_o from_o cd_o unity_n de._n wherefore_o that_o which_o be_v produce_v of_o ab_fw-la into_o bc_o together_o with_o the_o square_n of_o ce_fw-fr be_v less_o than_o the_o square_a number_n of_o bd._n now_o then_o i_o say_v that_o the_o square_a numâ—er_n produce_v of_o ab_fw-la into_o bc_o add_v to_o the_o square_a number_n of_o ce_fw-fr make_v not_o a_o square_a number_n for_o if_o they_o do_v make_v a_o square_a number_n than_o that_o square_a number_n which_o they_o make_v be_v either_o great_a they_o the_o square_a number_n of_o be_v or_o equal_a unto_o it_o or_o less_o than_o it_o first_o great_a it_n can_v be_v for_o it_o be_v already_o prove_v that_o the_o square_a number_n produce_v of_o ab_fw-la into_o bc_o together_o with_o the_o square_a number_n of_o ce_fw-fr be_v less_o than_o the_o square_a number_n of_o bd._n but_o between_o the_o square_a number_n of_o bd_o and_o the_o square_a number_n of_o be_v there_o be_v no_o mean_a square_a number_n for_o the_o number_n bd_o exceed_v the_o number_n be_v only_o by_o unity_n which_o unity_n can_v by_o no_o mean_n be_v divide_v into_o number_n or_o if_o the_o number_n produce_v of_o ab_fw-la into_o bc_o together_o with_o the_o square_n of_o the_o number_n ce_fw-fr shall_v be_v great_a than_o the_o square_a of_o the_o number_n be_v then_o shall_v the_o self_n same_o number_n produce_v of_o ab_fw-la into_o bc_o together_o with_o the_o square_n of_o the_o number_n ce_fw-fr be_v equal_a to_o the_o square_n of_o the_o number_n bd_o the_o contrary_a whereof_o be_v already_o prove_v wherefore_o if_o it_o be_v possible_a let_v that_o which_o be_v produce_v of_o ab_fw-la into_o bc_o together_o with_o the_o square_a number_n of_o the_o number_n ce_fw-fr be_v equal_a to_o the_o square_a number_n of_o be._n and_o let_v ga_o be_v double_a to_o unity_n de_fw-fr that_o be_v let_v it_o be_v the_o number_n two_o now_o forasmuch_o as_o the_o whole_a number_n ac_fw-la be_v by_o supposition_n double_a to_o the_o whole_a number_n cd_o of_o which_o the_o number_n agnostus_n be_v double_a to_o unity_n de_fw-fr therefore_o by_o the_o 7._o of_o the_o seven_o the_o residue_n namely_o the_o number_n gc_o be_v double_a to_o the_o residue_n namely_o to_o the_o number_n ec_o wherefore_o the_o number_n gc_o be_v divide_v into_o two_o equal_a part_n in_o e._n wherefore_o that_o which_o be_v produce_v of_o gb_o into_o bc_o together_o with_o the_o square_a number_n of_o ce_fw-fr be_v equal_a to_o the_o square_a number_n
of_o be._n but_o that_o which_o be_v produce_v of_o ab_fw-la into_o bc_o together_o with_o the_o square_a number_n of_o ce_fw-fr be_v suppose_v to_o be_v equal_a to_o the_o square_a number_n of_o be_v wherefore_o that_o which_o be_v produce_v of_o gb_o into_o bc_o together_o with_o the_o square_a number_n of_o ce_fw-fr be_v equal_a to_o that_o which_o be_v produce_v of_o ab_fw-la into_o bc_o together_o with_o the_o square_a number_n of_o ce._n wherefore_o take_v away_o the_o square_a number_n of_o ce_fw-fr which_o be_v common_a to_o they_o both_o the_o number_n ab_fw-la shall_v be_v equal_a to_o the_o number_n gb_o namely_o the_o great_a to_o the_o less_o which_o be_v impossible_a wherefore_o that_o which_o be_v produce_v of_o ab_fw-la into_o bc_o together_o with_o the_o square_a number_n of_o ce_fw-fr be_v not_o equal_a to_o the_o square_a number_n of_o be_v i_o say_v also_o that_o that_o which_o be_v produce_v of_o ab_fw-la into_o bc_o together_o with_o the_o square_a number_n of_o ce_fw-fr be_v not_o less_o than_o the_o square_a number_n of_o be._n for_o if_o it_o be_v possible_a they_o shall_v it_o be_v equal_a to_o some_o square_a number_n less_o than_o the_o square_a number_n of_o be._n wherefore_o let_v the_o number_n produce_v of_o ab_fw-la into_o bc_o together_o with_o the_o square_n of_o the_o number_n ce_fw-fr be_v equal_a to_o the_o square_a number_n of_o bf_o and_o let_v the_o number_n ha_o be_v double_a to_o the_o number_n df._n then_o also_o it_o follow_v that_o the_o number_n hc_n be_v double_a to_o the_o number_n cf_o so_o that_o hc_n also_o be_v divide_v into_o two_o equal_a part_n in_o fletcher_n and_o therefore_o also_o the_o number_n which_o be_v produce_v of_o hd_a into_o bc_o together_o with_o the_o square_a number_n of_o fc_n be_v equal_a to_o the_o square_a number_n of_o the_o number_n bf_o but_o by_o supposition_n the_o number_n which_o be_v produce_v of_o ab_fw-la into_o bc_o together_o with_o the_o square_a number_n of_o ce_fw-fr be_v equal_a to_o the_o square_a number_n of_o bf_o wherefore_o it_o follow_v that_o the_o number_n produce_v of_o ab_fw-la into_o bc_o together_o with_o the_o square_a number_n of_o ce_fw-fr be_v equal_a to_o that_o which_o be_v produce_v of_o hb_o into_o bc_o together_o with_o the_o square_a number_n cf_o which_o be_v impossible_a for_o if_o it_o shall_v be_v equal_a then_o forasmuch_o as_o the_o square_n of_o cf_o be_v less_o than_o the_o square_a of_o ce_fw-fr the_o number_n produce_v of_o hb_o into_o bc_o shall_v be_v great_a than_o thâ—_z number_v produce_v of_o ab_fw-la into_o bc._n and_o so_o also_o shall_v the_o number_n hb_o be_v great_a than_o the_o number_n ab_fw-la when_o yet_o it_o be_v less_o than_o it_o wherefore_o the_o number_n produce_v of_o ab_fw-la into_o bc_o together_o with_o the_o square_a number_n of_o ce_fw-fr be_v not_o less_o than_o the_o square_a number_n of_o â—_o e._n and_o it_o be_v also_o prove_v that_o it_o can_v be_v equal_a to_o the_o square_a number_n of_o be_v neither_o great_a than_o it_o wherefore_o that_o which_o be_v produce_v of_o ab_fw-la into_o bc_o add_v to_o the_o square_a number_n of_o ce_fw-fr make_v not_o a_o square_a number_n and_o although_o it_o be_v possible_a to_o demonstrate_v this_o many_o other_o way_n yet_o this_o seem_v to_o we_o sufficient_a lest_o the_o matter_n be_v over_o long_o shall_v seem_v to_o much_o tedious_a ¶_o the_o 6._o problem_n the_o 29._o proposition_n to_o find_v out_o two_o such_o rational_a right_a line_n commensurable_a in_o power_n only_o that_o the_o great_a shall_v be_v in_o power_n more_o than_o the_o less_o by_o the_o square_n of_o a_o right_a line_n commensurable_a in_o length_n unto_o the_o great_a let_v there_o be_v put_v a_o rational_a line_n ab_fw-la and_o take_v also_o two_o such_o square_a number_n cd_o and_o de_fw-fr construction_n that_o their_o excess_n ce_fw-fr be_v not_o a_o square_a number_n by_o the_o corollary_n of_o the_o first_o assumpt_n of_o the_o 28._o of_o the_o ten_o and_o upon_o the_o line_n ab_fw-la describe_v a_o semicircle_n afb_o and_o by_o the_o corollary_n of_o the_o 6._o of_o the_o ten_o as_o the_o number_n dc_o be_v to_o the_o number_n ce_fw-fr so_o let_v the_o square_n of_o the_o line_n basilius_n be_v to_o the_o square_n of_o the_o line_n af._n demonstration_n and_o draw_v a_o line_n from_o fletcher_n to_o b._n now_o for_o that_o as_o the_o square_n of_o the_o line_n basilius_n be_v to_o the_o square_n of_o the_o line_n of_o so_o be_v the_o number_n cd_o to_o the_o number_n ce_fw-fr therefore_o the_o square_a of_o the_o line_n basilius_n have_v to_o the_o square_n of_o the_o line_n of_o that_o proportion_n that_o the_o number_n cd_o have_v to_o the_o number_n ce._n wherefore_o the_o square_a of_o the_o line_n basilius_n be_v commensurable_a to_o the_o square_n of_o the_o line_n of_o by_o the_o 6._o of_o the_o ten_o but_o the_o square_n of_o the_o line_n ab_fw-la be_v rational_a wherefore_o also_o the_o square_a of_o the_o line_n of_o be_v rational_a wherefore_o also_o the_o line_n of_o be_v rational_a and_o forasmuch_o as_o the_o number_n cd_o have_v not_o unto_o the_o number_n ce_fw-fr that_o proportion_n that_o a_o square_a number_n have_v to_o a_o square_a number_n therefore_o neither_o also_o have_v the_o square_a of_o the_o line_n ab_fw-la to_o the_o square_n of_o the_o line_n of_o that_o proportion_n that_o a_o square_a number_n have_v to_o a_o square_a number_n wherefore_o by_o the_o 9_o of_o the_o ten_o the_o line_n ab_fw-la be_v unto_o the_o line_n of_o incommensurable_a in_o length_n wherefore_o the_o line_n of_o and_o ab_fw-la be_v rational_a commensurable_a in_o power_n only_o and_o for_o that_o as_o the_o number_n dc_o be_v to_o the_o number_n ce_fw-fr so_o be_v the_o square_a of_o the_o line_n ab_fw-la to_o the_o square_n of_o the_o line_n of_o therefore_o by_o conversion_n or_o everse_v proportion_n which_o be_v demonstrate_v by_o the_o corollary_n of_o the_o 19_o of_o the_o five_o as_o the_o number_n cd_o be_v to_o the_o number_n de_fw-fr so_o be_v the_o square_a of_o the_o line_n ab_fw-la to_o the_o square_n of_o the_o line_n bf_o which_o be_v the_o excess_n of_o the_o square_n of_o the_o line_n ab_fw-la above_o the_o square_n of_o the_o line_n of_o by_o the_o assumpt_n put_v before_o the_o 14._o of_o this_o book_n but_o the_o number_n cd_o have_v to_o the_o number_n de_fw-fr that_o proportion_n that_o a_o square_a number_n have_v to_o a_o square_a number_n wherefore_o the_o square_a of_o the_o line_n ab_fw-la have_v to_o the_o square_n of_o the_o line_n bf_o that_o proportion_n that_o a_o square_a numâ—er_n have_v to_o a_o square_a number_n wherefore_o by_o the_o 9_o of_o the_o ten_o the_o line_n ab_fw-la be_v commensurable_a in_o length_n unto_o the_o line_n bf_o and_o by_o the_o 47._o of_o the_o first_o the_o square_a of_o the_o line_n ab_fw-la be_v equal_a to_o the_o square_n of_o the_o line_n of_o and_o fb_o wherefore_o the_o line_n ab_fw-la be_v in_o power_n more_o than_o the_o line_n of_o by_o the_o square_n of_o the_o line_n bf_o which_o be_v commensurable_a in_o length_n unto_o the_o line_n ab_fw-la wherefore_o there_o be_v find_v out_o two_o such_o rational_a line_n commensurable_a in_o power_n only_o namely_o ab_fw-la and_o of_o so_o that_o the_o great_a line_n ab_fw-la be_v in_o power_n more_o than_o the_o less_o line_n of_o by_o the_o square_n of_o the_o line_n fb_o which_o be_v commensurable_a in_o length_n unto_o the_o line_n ab_fw-la which_o be_v require_v to_o be_v do_v ¶_o the_o 7._o theorem_a the_o 30._o proposition_n to_o find_v out_o two_o such_o rational_a line_n commensurable_a in_o power_n only_o cause_n that_o the_o great_a shall_v be_v in_o power_n more_o than_o the_o less_o by_o the_o square_n of_o a_o right_a line_n incommensurable_a in_o length_n to_o the_o great_a let_v there_o be_v put_v a_o rational_a line_n ab_fw-la and_o take_v also_o by_o the_o 2._o assumpt_n of_o the_o 28._o of_o the_o ten_o two_o square_a number_n ce_fw-fr and_o ed_z which_o be_v add_v together_o make_v not_o a_o square_a number_n and_o let_v the_o number_n ce_fw-fr and_o ed_z add_v together_o make_v the_o number_n cd_o construction_n and_o upon_o the_o line_n ab_fw-la describe_v a_o sencircle_v afb_o and_o by_o the_o corollary_n of_o the_o 6._o of_o the_o ten_o as_o the_o number_n dc_o be_v to_o the_o number_n ce_fw-fr so_o let_v the_o square_n of_o the_o line_n ab_fw-la be_v to_o the_o square_n of_o the_o line_n of_o and_o draw_v a_o line_n from_o fletcher_n to_o b._n and_o we_o may_v in_o like_a sort_n demonstration_n as_o we_o do_v in_o the_o former_a proposition_n prove_v that_o the_o line_n basilius_n and_o of_o be_v rational_a commensurable_a in_o power_n only_o and_o for_o that_o as_o the_o number_n dc_o be_v
to_o the_o number_n ce_fw-fr so_o be_v the_o square_a of_o the_o line_n ab_fw-la to_o the_o square_n of_o the_o line_n of_o therefore_o by_o conversion_n by_o the_o corollary_n of_o the_o 19_o of_o the_o five_o as_o the_o number_n cd_o be_v to_o the_o number_n de_fw-fr so_o be_v the_o square_a of_o the_o line_n ab_fw-la to_o the_o square_n to_o the_o line_n fb_o but_o the_o number_n c_o d_o have_v not_o to_o the_o number_n de_fw-fr that_o proportion_n that_o a_o square_a nâ—mbeâ—_n hâ—th_v to_o a_o square_a number_n wherefore_o neither_o also_o the_o square_a of_o the_o line_n ab_fw-la have_v to_o the_o square_n of_o the_o line_n bf_o that_o proportion_n that_o a_o square_a number_n have_v to_o a_o square_a number_n wherefore_o the_o line_n ab_fw-la be_v by_o the_o 9_o of_o the_o ten_o incommensurable_a in_o length_n to_o the_o line_n bf_o and_o the_o line_n ab_fw-la be_v in_o power_n more_o than_o the_o line_n of_o by_o the_o square_n of_o the_o right_a line_n bf_o which_o be_v incommensurable_a in_o length_n unto_o the_o line_n ab_fw-la wherefore_o the_o line_n ab_fw-la and_o of_o be_v rational_a commensurable_a in_o power_n only_o and_o the_o line_n ab_fw-la be_v in_o power_n more_o than_o the_o line_n of_o by_o the_o square_n of_o the_o line_n fb_o which_o be_v commensurable_a in_o length_n unto_o the_o line_n abâ—_n â—_z which_o be_v require_v to_o be_v do_v ¶_o a_o assumpt_n if_o there_o be_v two_o right_a line_n have_v between_o themselves_o any_o proportion_n as_o the_o one_o right_a line_n be_v to_o the_o other_o so_o be_v the_o parallelogram_n contain_v under_o both_o the_o right_a line_n to_o the_o square_n of_o the_o less_o of_o those_o two_o line_n suppose_v that_o these_o two_o right_a ab_fw-la and_o bc_o be_v in_o some_o certain_a proportion_n find_v then_o i_o say_v that_o as_o the_o line_n ab_fw-la be_v to_o the_o line_n bc_o so_o be_v the_o parallelogram_n contain_v under_o ab_fw-la and_o bc_o to_o the_o square_n of_o bc._n describe_v the_o square_a of_o the_o line_n bc_o and_o let_v the_o same_o be_v cd_o and_o make_v perfect_a the_o parallelogram_n ad_fw-la now_o it_o be_v manifest_a that_o as_o the_o line_n ab_fw-la be_v to_o the_o line_n bc_o so_o be_v the_o parallelogram_n ad_fw-la to_o the_o parallelogram_n or_o square_n be_v by_o the_o first_o of_o the_o six_o but_o the_o parallelogram_n ad_fw-la be_v that_o which_o be_v bontain_v under_o the_o line_n ab_fw-la and_o bc_o for_o the_o line_n bc_o be_v equal_a to_o the_o line_n bd_o and_o the_o parallelogram_n be_v be_v the_o square_n of_o the_o line_n bc._n wherefore_o as_o the_o line_n ab_fw-la be_v to_o the_o line_n bc_o so_o be_v the_o parallelogram_n coutain_v under_o the_o line_n ab_fw-la and_o bc_o to_o the_o square_n of_o the_o line_n bc_o which_o be_v require_v to_o be_v prove_v ¶_o the_o 8._o problem_n the_o 31._o proposition_n to_o find_v out_o two_o medial_a line_n commensurable_a in_o power_n only_o comprehend_v a_o rational_a superficies_n so_o that_o the_o great_a shall_v be_v in_o power_n more_o than_o the_o less_o by_o the_o square_n of_o a_o line_n commensurable_a in_o length_n unto_o the_o great_a let_v there_o be_v take_v by_o the_o 29._o of_o the_o ten_o two_o rational_a line_n commensurable_a in_o power_n only_o a_o and_o b_o construction_n so_o that_o let_v the_o line_n a_o be_v the_o great_a be_v in_o power_n more_o than_o the_o line_n b_o be_v the_o less_o by_o the_o square_n of_o a_o line_n commensurable_a in_o length_n unto_o the_o line_n aâ—_n â—_z and_o let_v the_o square_n of_o the_o line_n c_o be_v equal_a to_o the_o parallelogram_n contain_v under_o the_o line_n a_o and_o b_o which_o be_v do_v by_o find_v out_o the_o mean_a proportional_a line_n namely_o the_o line_n c_o between_o the_o line_n a_o and_o b_o by_o the_o 13._o of_o the_o six_o now_o the_o parallelogram_n contain_v under_o the_o line_n a_o and_o b_o be_v medial_a by_o the_o 21._o of_o this_o book_n wherefore_o by_o the_o corollary_n of_o the_o 23._o of_o the_o ten_o the_o square_a also_o of_o the_o line_n c_o be_v medial_a wherefore_o the_o line_n c_o also_o be_v medial_a unto_o the_o square_n of_o the_o line_n b_o let_v the_o parallelogram_n contain_v under_o the_o line_n c_o and_o d_o be_v equal_a by_o find_v out_o a_o three_o line_n proportional_a namely_o the_o line_n d_o to_o the_o two_o line_n c_o and_o b_o by_o the_o 11._o of_o the_o six_o but_o the_o square_n of_o the_o line_n b_o be_v rational_a demonstration_n wherefore_o the_o parallelogram_n contain_v under_o the_o line_n c_o and_o d_o be_v rational_a and_o for_o that_o as_o the_o line_n a_o be_v to_o the_o line_n b_o so_o be_v the_o parallelogram_n contain_v under_o the_o line_n a_o and_o b_o to_o the_o square_n of_o the_o line_n b_o by_o the_o assumpt_n go_v before_o but_o unto_o the_o parallelogram_n contain_v under_o the_o line_n a_o and_o b_o be_v equal_v the_o square_n of_o the_o line_n c_o and_o unto_o the_o square_n of_o the_o line_n b_o be_v equal_a the_o parallelogram_n contain_v under_o the_o line_n c_o and_o d_o as_o it_o have_v now_o be_v prove_v therefore_o as_o the_o line_n a_o be_v to_o the_o line_n b_o so_o be_v the_o square_a of_o the_o line_n c_o to_o the_o parallelogram_n contain_v under_o the_o line_n c_o d._n but_o as_o the_o square_n of_o the_o line_n c_o be_v to_o that_o which_o be_v contain_v under_o the_o line_n c_o and_o d_o so_o be_v the_o line_n c_o to_o the_o line_n d._n wherefore_o as_o the_o line_n a_o be_v to_o the_o line_n b_o so_o be_v the_o line_n c_o to_o the_o line_n d._n but_o by_o supposition_n the_o line_n a_o be_v commensurable_a unto_o the_o line_n b_o in_o power_n only_o wherefore_o by_o the_o 11._o of_o the_o ten_o the_o line_n c_o also_o be_v unto_o the_o line_n d_o commensurable_a in_o power_n only_o but_o the_o line_n c_o be_v medial_a wherefore_o by_o the_o 23â—_n of_o the_o ten_o the_o line_n d_o also_o be_v medial_a and_o for_o that_o as_o the_o line_n a_o be_v to_o the_o line_n b_o so_o be_v the_o line_n c_o to_o the_o line_n d_o but_o the_o line_n a_o be_v in_o power_n more_o than_o the_o line_n b_o by_o the_o square_n of_o a_o line_n commensurable_a in_o length_n unto_o the_o line_n a_o by_o supposition_n wherefore_o the_o line_n c_o also_o be_v in_o power_n more_o than_o the_o line_n d_o by_o the_o square_n of_o a_o line_n commensurable_a in_o length_n unto_o the_o line_n c._n wherefore_o there_o be_v find_v out_o two_o medial_a line_n c_o and_o d_o commensurable_a in_o power_n only_o comprehend_v a_o rational_a superficies_n and_o the_o line_n c_o be_v in_o power_n more_o than_o the_o line_n d_o by_o the_o square_n of_o a_o line_n commensurable_a in_o length_n unto_o the_o line_n c._n and_o in_o like_a sort_n may_v be_v find_v out_o two_o medial_a line_n commensurable_a in_o power_n only_o contain_v a_o rational_a superficies_n so_o that_o the_o great_a shall_v be_v in_o power_n more_o they_o the_o less_o by_o the_o square_n of_o a_o line_n incommensurable_a in_o length_n to_o the_o great_a namely_o when_o the_o line_n a_o be_v in_o power_n more_o they_o the_o line_n b_o by_o the_o square_n of_o a_o line_n incommensurable_n in_o length_n unto_o the_o line_n a_o which_o to_o do_v be_v teach_v by_o the_o 30._o of_o this_o book_n the_o self_n same_o construction_n remain_v that_o part_n of_o this_o proposition_n from_o these_o word_n and_o for_o that_o as_o the_o line_n a_o be_v to_o the_o line_n b_o to_o these_o word_n but_o by_o supposition_n the_o line_n a_o be_v commensurable_a unto_o the_o line_n b_o may_v more_o easy_o be_v demonstrate_v after_o this_o manner_n the_o line_n c_o b_o d_o be_v in_o continual_a proportion_n by_o the_o second_o part_n of_o the_o 17._o of_o the_o six_o but_o the_o line_n a_o c_o d_o be_v also_o in_o continual_a proportion_n by_o the_o same_o wherefore_o by_o the_o 11._o of_o the_o five_o as_o the_o line_n a_o be_v to_o the_o line_n c_o so_o be_v the_o line_n b_o to_o the_o line_n d._n wherefore_o alternate_o as_o the_o line_n a_o be_v to_o the_o line_n b_o so_fw-mi be_v the_o line_n c_o to_o the_o line_n d._n etc_n etc_n which_o be_v require_v to_o be_v do_v ¶_o a_o assumpt_n if_o there_o be_v three_o right_a line_n have_v between_o themselves_o any_o proportion_n as_o the_o first_o be_v to_o the_o three_o so_o be_v the_o parallelogram_n contain_v under_o the_o first_o and_o the_o second_o to_o the_o parallelogram_n contain_v under_o the_o second_o and_o the_o three_o suppose_v that_o these_o three_o line_n ab_fw-la b_o c_o and_o cd_o be_v in_o some_o certain_a proportion_n then_o i_o say_v that_o as_o the_o line_n ab_fw-la be_v to_o
the_o line_n cd_o so_o be_v the_o parallelogram_n contain_v under_o the_o line_n ab_fw-la and_o bc_o to_o the_o parallelogram_n contain_v under_o the_o line_n bc_o and_o cd_o construction_n from_o the_o point_n a_o raise_v up_o unto_o the_o line_n ab_fw-la a_o perpendicular_a line_n ae_n and_o let_v ae_n be_v equal_a to_o the_o line_n bc_o and_o by_o the_o point_n e_o draw_v unto_o the_o line_n ad_fw-la a_o parallel_n line_n eke_o and_o by_o every_o one_o of_o the_o point_n b_o c_o and_o d_o draw_v unto_o the_o line_n ae_n parallel_a line_n bf_o change_z and_o dk_o and_o for_o that_o as_o the_o line_n ab_fw-la be_v to_o the_o line_n bc_o so_o be_v the_o parallelogram_n of_o to_o the_o parallelogram_n bh_o by_o the_o first_o of_o the_o six_o demonstration_n and_o as_o the_o line_n bc_o be_v to_o the_o linâ—_n cd_o so_o be_v the_o parallelogram_n bh_o to_o the_o parallelogram_n ck_o wherefore_o of_o equality_n as_o the_o line_n ab_fw-la be_v to_o the_o line_n cd_o so_o be_v the_o parallelogram_n of_o to_o the_o parallelogram_n ck_o but_o the_o parallelogram_n of_o be_v that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o for_o the_o line_n ae_n be_v put_v equal_a to_o the_o line_n bc._n and_o the_o parallelogram_n ck_o be_v that_o which_o be_v contain_v under_o the_o line_n bc_o and_o cd_o for_o the_o line_n bc_o be_v equal_a to_o the_o line_n change_z for_o that_o the_o line_n change_z be_v equal_a to_o the_o line_n ae_n by_o the_o 34._o of_o the_o first_o if_o therefore_o there_o be_v three_o right_a line_n have_v between_o themselves_o any_o proportion_n as_o the_o first_o be_v to_o the_o three_o so_o be_v the_o parallelogram_n contain_v under_o the_o first_o and_o the_o second_o to_o the_o parallelogram_n contain_v under_o the_o second_o and_o the_o three_o which_o be_v require_v to_o be_v demonstrate_v ¶_o the_o 9_o problem_n the_o 32._o proposition_n to_o find_v out_o two_o medial_a line_n commensurable_a in_o power_n only_o comprehend_v a_o medial_a superficies_n so_o that_o the_o great_a shall_v be_v in_o power_n more_o than_o the_o less_o by_o the_o square_n of_o a_o line_n commensurable_a in_o length_n unto_o the_o great_a let_v there_o be_v take_v three_o rational_a line_n commensurable_a in_o power_n only_o a_o b_o c_o so_fw-mi that_o by_o the_o 29._o of_o the_o ten_o let_v the_o line_n a_o be_v in_o power_n more_o than_o the_o line_n c_o by_o the_o square_n of_o a_o line_n commensurable_a in_o length_n unto_o the_o line_n a._n construction_n and_o unto_o the_o parallelogram_n contain_v under_o the_o line_n a_o &_o b_o let_v the_o square_n of_o the_o line_n d_o be_v equal_a but_o that_o which_o be_v contain_v under_o the_o line_n a_o and_o b_o be_v medial_a wherefore_o by_o the_o corollary_n of_o the_o 23._o of_o the_o ten_o the_o square_a of_o the_o line_n d_o also_o be_v medial_a wherefore_o the_o line_n d_o also_o be_v medial_a and_o unto_o that_o which_o be_v contain_v under_o the_o line_n b_o and_o c_o let_v be_v equal_a that_o which_o be_v contain_v under_o the_o line_n d_o and_o e_o which_o be_v do_v by_o â—inding_v out_o a_o four_o line_n proportional_a unto_o the_o line_n d_o b_o c_o which_o let_v be_v the_o line_n e_o demonstration_n and_o for_o that_o by_o the_o assumpt_n go_v before_o as_o that_o which_o be_v contain_v under_o the_o line_n a_o and_o b_o be_v to_o that_o which_o be_v contain_v under_o the_o line_n b_o and_o c_o so_o be_v the_o line_n a_o to_o the_o line_n c._n but_o unto_o that_o which_o be_v contain_v under_o the_o line_n a_o &_o b_o be_v equal_a the_o square_n of_o the_o line_n d_o and_o unto_o that_o which_o be_v contain_v under_o the_o line_n b_o &_o c_o be_v equal_a that_o which_o be_v contain_v under_o the_o line_n d_o and_o e._n wherefore_o as_o the_o line_n a_o be_v to_o the_o line_n c_o so_fw-mi iâ—_z the_o square_a of_o the_o line_n d_o to_o that_o which_o be_v contain_v under_o the_o line_n d_o and_o e._n but_o as_o the_o square_n of_o the_o line_n d_o be_v to_o that_o which_o be_v contain_v under_o the_o line_n d_o and_o e_o so_o be_v the_o line_n d_o to_o the_o line_n e_o by_o the_o assumpt_n put_v before_o the_o 22._o of_o the_o ten_o wherefore_o as_o the_o line_n a_o be_v to_o the_o line_n c_o so_o be_v the_o line_n d_o to_o the_o line_n e._n but_o the_o line_n a_o be_v unto_o the_o line_n c_o commensurable_a in_o power_n only_o wherefore_o the_o line_n d_o be_v unto_o the_o line_n e_o commensurable_a in_o power_n only_o but_o d_z be_v a_o medial_a line_n wherefore_o by_o the_o 23._o of_o the_o ten_o e_o also_o be_v a_o medial_a line_n and_o for_o that_o as_o the_o line_n a_o be_v to_o the_o line_n c_o so_o be_v the_o line_n d_o to_o the_o line_n e_o and_o the_o line_n a_o be_v in_o power_n more_o than_o the_o line_n c_o by_o the_o square_n of_o a_o line_n commensurable_a in_o length_n unto_o the_o line_n a._n wherefore_o by_o the_o 14._o of_o the_o ten_o d_o be_v in_o power_n more_o than_o e_o by_o the_o square_n of_o a_o line_n commensurable_a in_o length_n unto_o the_o line_n d._n i_o say_v moreover_o that_o that_o which_o be_v contain_v under_o the_o line_n d_o and_o e_o be_v medial_a for_o forasmuch_o as_o that_o which_o be_v contain_v under_o the_o line_n b_o &_o c_o be_v equal_a to_o that_o which_o be_v contain_v under_o the_o line_n d_o and_o e_o but_o that_o which_o be_v contain_v under_o the_o line_n b_o and_o c_o be_v medial_a wherefore_o that_o which_o be_v contain_v under_o the_o line_n d_o and_o e_o be_v also_o medial_a wherefore_o there_o be_v find_v out_o two_o medial_a line_n d_z and_o e_o commensurable_a in_o power_n only_o comprehend_v a_o medial_a superficies_n so_o that_o the_o great_a be_v in_o power_n more_o than_o the_o less_o by_o the_o square_n of_o a_o line_n commensurable_a in_o length_n to_o the_o great_a which_o be_v require_v to_o be_v do_v and_o thus_o iâ—_z iâ—_z euâ—deâ—t_n how_o in_o like_a sort_n may_v be_v find_v out_o two_o medial_a line_n commâ—â—surable_a in_o power_n only_o corollary_n contain_v a_o medial_a superficies_n so_o that_o the_o great_a shall_v be_v in_o power_n more_o theâ—_z the_o less_o by_o the_o square_n of_o a_o line_n incommensurable_a in_o length_n unto_o the_o great_a when_o the_o line_n a_o be_v in_o power_n more_o than_o the_o line_n c_o by_o the_o square_n of_o a_o line_n incommensurable_a in_o length_n unto_o the_o line_n a_o as_o the_o thirty_o teach_v us._n 1._o ¶_o a_o assumpt_n suppose_v that_o there_o be_v a_o rectangle_n triangle_n abc_n have_v the_o angle_n bac_n a_o right_a angle_n and_o â—y_a the_o 12._o of_o the_o first_o from_o the_o point_n a_o to_o the_o right_a line_n bc_o a_o perpendicular_a line_n be_v draw_v ad_fw-la then_o i_o say_v first_o that_o the_o parallelogram_n contain_v under_o the_o line_n câ—_n and_o bd_o be_v equal_a to_o the_o square_n of_o the_o line_n basilius_n second_o i_o say_v that_o the_o parallelogram_n contain_v under_o the_o line_n bc_o and_o cd_o be_v equal_a to_o the_o square_n of_o the_o line_n ca._n three_o i_o say_v that_o the_o parallelogram_n contain_v under_o the_o line_n bd_o and_o dc_o be_v equal_a to_o the_o square_n of_o the_o line_n ad._n and_o fourthly_a i_o say_v that_o the_o parallelogram_n contain_v under_o the_o line_n bc_o &_o ad_fw-la be_v equal_a to_o the_o parallelogram_n contain_v under_o the_o line_n basilius_n &_o ac_fw-la as_o touch_v the_o second_o that_o the_o parellelogramme_n contain_v under_o the_o line_n bâ—_n and_o cd_o be_v equal_a to_o the_o square_n of_o the_o line_n ac_fw-la be_v by_o the_o self_n same_o reason_n prove_v for_o the_o triangle_n abc_n be_v like_a to_o the_o triangle_n adc_o wherefore_o as_o the_o line_n bc_o be_v to_o the_o line_n ac_fw-la dee_n so_o be_v the_o line_n ac_fw-la to_o the_o line_n dc_o corollary_n wherefore_o the_o parallelogram_n contain_v under_o the_o line_n bc_o and_o cd_o be_v equal_a to_o the_o square_n of_o the_o line_n ac_fw-la corollary_n as_o touch_v the_o three_o that_o the_o parallelogram_n contain_v under_o the_o line_n bd_o and_o dc_o be_v equal_a to_o the_o square_n of_o the_o line_n dam_fw-ge be_v thus_o prove_v for_o forasmuch_o as_o if_o in_o a_o rectangle_n triangle_n be_v draw_v from_o the_o right_a angle_n to_o the_o base_a a_o perpendicular_a line_n the_o perpendicular_a so_o draw_v be_v the_o mean_a proportional_a between_o the_o segment_n of_o the_o base_a by_o the_o corollary_n of_o the_o 8._o of_o the_o six_o therefore_o as_o the_o line_n bd_o be_v to_o the_o line_n dam_fw-ge so_o be_v thâ—_z line_n ad_fw-la to_o the_o line_n dc_o wherefore_o by_o the_o 1â—_o of_o the_o six_o the_o parallelogram_n
contain_v under_o the_o line_n bd_o and_o dc_o be_v equal_a to_o the_o square_n of_o the_o line_n da._n as_o touch_v the_o four_o that_o the_o parallelogram_n contain_v under_o the_o line_n bc_o and_o ad_fw-la be_v equal_a to_o the_o parallelogram_n contain_v under_o the_o line_n basilius_n and_o ac_fw-la be_v thus_o prove_v for_o forasmuch_o as_o as_o we_o have_v already_o declare_v the_o triangle_n abc_n be_v like_a and_o therefore_o equiangle_n to_o the_o triangle_n abdella_n therefore_o as_o the_o line_n bc_o be_v to_o the_o line_n ac_fw-la dâ—e_n so_o be_v the_o line_n basilius_n to_o the_o line_n ad_fw-la by_o the_o 4._o of_o the_o six_o corollary_n but_o if_o there_o be_v four_o right_a line_n proportional_a that_o which_o be_v contain_v under_o the_o first_o and_o the_o last_o be_v equal_a to_o that_o which_o be_v contain_v under_o the_o two_o mean_n by_o the_o 16._o of_o the_o six_o wherefore_o that_o which_o be_v contain_v under_o the_o line_n bc_o and_o ad_fw-la be_v equal_a to_o that_o which_o be_v contain_v under_o the_o line_n basilius_n and_o ac_fw-la profitable_a i_o say_v moreover_o that_o if_o there_o be_v make_v a_o parallelogram_n complete_a determination_n contain_v under_o the_o line_n bc_o and_o ad_fw-la which_o let_v be_v ec_o and_o if_o likewise_o be_v make_v complete_a the_o parallelogram_n contain_v under_o the_o line_n basilius_n and_o ac_fw-la which_o let_v be_v of_o it_o may_v by_o a_o other_o way_n be_v prove_v that_o the_o parallelogram_n ec_o be_v equal_a to_o the_o parallelogram_n af._n for_o forasmuch_o as_o either_o of_o they_o be_v double_a to_o the_o triangle_n acb_o by_o the_o 41._o of_o the_o first_o and_o thing_n which_o be_v double_a to_o one_o and_o the_o self_n same_o thing_n be_v equal_a the_o one_o to_o the_o other_o wherefore_o that_o which_o be_v contain_v under_o the_o line_n bc_o and_o ad_fw-la be_v equal_a to_o that_o which_o be_v contain_v under_o the_o line_n basilius_n and_o ac_fw-la 2._o ¶_o a_o assumpt_n if_o a_o right_a line_n be_v divide_v into_o two_o unequal_a part_n as_o the_o great_a part_n be_v to_o the_o less_o assumpt_n so_o be_v the_o parallelogram_n contain_v under_o the_o whole_a line_n and_o the_o great_a part_n to_o the_o parallelogram_n contain_v under_o the_o whole_a line_n and_o the_o less_o part_n this_o assumpt_n differ_v little_a from_o the_o first_o proposition_n of_o the_o six_o book_n 3._o ¶_o a_o assumpt_n if_o there_o be_v two_o unequal_a right_a line_n and_o if_o the_o less_o be_v divide_v into_o two_o equal_a part_n the_o parallelogram_n contain_v under_o the_o two_o unequal_a line_n be_v double_a to_o the_o parallelogram_n contain_v under_o the_o great_a line_n &_o half_a of_o the_o less_o line_n suppose_v that_o there_o be_v two_o unequal_a right_a line_n ab_fw-la and_o bc_o of_o which_o leâ—_z ab_fw-la be_v the_o great_a and_o divide_v the_o line_n bc_o into_o two_o equal_a part_n in_o the_o point_n d._n thâ—n_o i_o say_v that_o the_o parallelogram_n contain_v under_o the_o line_n ab_fw-la &_o bc_o be_v double_a to_o the_o parallelogram_n contain_v under_o the_o line_n ab_fw-la and_o bd._n from_o the_o point_n b_o raise_v up_o upon_o the_o right_a line_n bc_o a_o perpendicular_a line_n be_v and_o let_v be_v be_v equal_a to_o the_o line_n basilius_n and_o draw_v from_o the_o point_n c_o and_o d_o the_o line_n cg_o and_o df_o parallel_v and_o equal_a to_o be_v and_o then_o draw_v the_o right_a line_n gfe_n the_o figure_n be_v complete_a nâ—â—_n for_o that_o aâ—â—he_a line_n db_o be_v to_o the_o line_n dc_o so_o be_v the_o parallelogram_n bf_o to_o the_o parallelogram_n dg_o by_o the_o 1._o of_o the_o six_o therefore_o by_o composition_n of_o proportion_n as_o the_o whole_a line_n bc_o be_v to_o the_o line_n dc_o so_o be_v the_o parallelogram_n bg_o to_o the_o parallelogram_n dg_o by_o the_o 18._o of_o the_o five_o but_o the_o line_n bc_o be_v double_a to_o the_o line_n dc_o wherefore_o the_o parallelogram_n bg_o be_v double_a to_o the_o parallelogram_n dg_o but_o the_o parallâ—logramme_n bg_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o for_o the_o line_n ab_fw-la be_v equal_a to_o the_o line_n be_v and_o the_o parallelogram_n dg_o be_v contain_v under_o the_o line_n ab_fw-la and_o bd_o for_o the_o line_n bd_o be_v equal_a to_o the_o line_n dc_o and_o the_o line_n ab_fw-la to_o the_o line_n df_o which_o be_v require_v to_o be_v demonstrate_v ¶_o the_o 10._o problem_n the_o 33._o proposition_n to_o â—inde_v out_o two_o right_a line_n incommensurable_a in_o power_n who_o square_n add_v together_o make_v a_o rational_a superficies_n and_o the_o parallelogram_n contain_v under_o they_o make_v a_o medial_a superficies_n take_v by_o the_o 30._o of_o the_o ten_o two_o rational_a right_a line_n commensurable_a in_o power_n only_o namely_o ab_fw-la and_o bc_o so_o that_o let_v the_o line_n ab_fw-la be_v the_o great_a be_v in_o power_n more_o than_o the_o line_n bc_o be_v the_o less_o construction_n by_o the_o square_n of_o a_o line_n incommensurable_a in_o length_n unto_o the_o line_n ab_fw-la and_o by_o the_o 10._o of_o the_o first_o divide_v the_o line_n bc_o into_o two_o equal_a part_n in_o the_o point_n d._n and_o upon_o the_o line_n ab_fw-la apply_v a_o parallelogram_n equal_a to_o the_o square_a either_o of_o the_o line_n bd_o or_o of_o the_o line_n dc_o and_o want_v in_o figure_n by_o a_o square_a by_o the_o 28._o of_o the_o six_o and_o let_v that_o parallelogram_n be_v that_o which_o be_v contain_v under_o the_o line_n ae_n and_o ebb_n and_o upon_o the_o line_n ab_fw-la describe_v a_o semicircle_n afb_o and_o by_o the_o 11._o of_o the_o first_o from_o the_o point_n e_o raise_v up_o unto_o the_o line_n ab_fw-la a_o perpendicular_a line_n of_o cut_v the_o circumference_n in_o the_o point_n f._n and_o draw_v line_n from_o a_o to_o fletcher_n demonstration_n and_o from_o fletcher_n to_o b._n and_o forasmuch_o as_o there_o be_v two_o unequal_a right_a line_n ab_fw-la and_o bc_o and_o the_o line_n ab_fw-la be_v in_o power_n more_o than_o the_o line_n bc_o by_o the_o square_n of_o a_o line_n incommensurable_a in_o length_n unto_o ab_fw-la and_o upon_o the_o line_n ab_fw-la be_v apply_v a_o parallelogram_n equal_a to_o the_o four_o part_n oâ—_n the_o square_a of_o the_o line_n bc_o that_o be_v to_o the_o square_n of_o the_o half_a of_o the_o line_n bc_o and_o want_v in_o â—igure_n by_o a_o square_a and_o the_o say_a parallelogram_n be_v that_o which_o be_v contain_v under_o the_o line_n ae_n and_o ebb_n wherefore_o by_o the_o 2._o part_n of_o the_o 18._o of_o the_o ten_o the_o line_n ae_n be_v incommensurable_a in_o length_n unto_o the_o line_n ebb_n but_o as_o the_o line_n ae_n be_v to_o the_o line_n ebb_n so_o be_v the_o parallelogram_n contain_v under_o the_o line_n basilius_n and_o ae_n to_o the_o parallelogram_n contain_v under_o the_o line_n ab_fw-la and_o be_v by_o the_o second_o assumpt_n before_o put_v and_o that_o which_o be_v contain_v under_o the_o line_n basilius_n and_o ae_n be_v equal_a to_o the_o square_n of_o the_o line_n of_o by_o the_o second_o part_n of_o the_o first_o assumpt_v before_o put_v and_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o be_v be_v by_o the_o first_o part_n of_o the_o same_o assumpt_n equal_a to_o the_o square_n of_o the_o line_n bf_o wherefore_o the_o square_a of_o the_o line_n of_o be_v incommensurable_a to_o the_o square_n of_o the_o line_n bf_o wherefore_o the_o line_n of_o and_o bf_o be_v incommensurable_a in_o power_n and_o forasmuch_o as_o ab_fw-la be_v a_o rational_a line_n by_o supposition_n therefore_o by_o the_o 7_o definition_n of_o the_o ten_o the_o square_a of_o the_o line_n ab_fw-la be_v rational_a conclude_v wherefore_o also_o the_o square_n of_o the_o line_n of_o and_o fb_o add_v together_o make_v a_o rational_a superficies_n for_o by_o the_o 47._o of_o the_o first_o they_o be_v equal_a to_o the_o square_n of_o the_o line_n ab_fw-la again_o forasmuch_o as_o by_o the_o three_o part_n of_o the_o first_o assumpt_n go_v before_o that_o which_o be_v contain_v under_o the_o line_n ae_n and_o ebb_n be_v equal_a to_o the_o square_n of_o the_o line_n ef._n but_o by_o supposition_n that_o which_o be_v contain_v under_o the_o line_n ae_n and_o ebb_n be_v equal_a to_o the_o square_n of_o the_o line_n bd._n wherefore_o the_o line_n fe_o be_v equal_a to_o the_o line_n bd._n wherefore_o the_o linâ—_n bg_o be_v double_a to_o the_o line_n â—_o e._n wherefore_o by_o the_o three_o assumpt_v go_v before_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o be_v double_a to_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o ef._n but_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o be_v by_o supposition_n medial_a
wherefore_o by_o the_o corollary_n of_o the_o 23._o of_o the_o ten_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o of_o be_v also_o medial_a but_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o of_o conclude_v be_v by_o the_o last_o part_n of_o the_o first_o assumpt_n go_v before_o equal_a to_o that_o which_o be_v contain_v under_o the_o line_n of_o and_o fb_o wherefore_o that_o which_o be_v contain_v under_o the_o line_n of_o &_o fb_o be_v a_o medial_a superficies_n and_o it_o be_v prove_v that_o that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n of_o and_o fb_o add_v together_o be_v rational_a conclusion_n wherefore_o there_o be_v find_v out_o two_o right_a line_n of_o and_o fb_o incommensurable_a in_o power_n who_o square_n add_v together_o make_v a_o rational_a superficies_n and_o the_o parallelogram_n contain_v under_o they_o be_v a_o medial_a superficiâ—sâ—_n which_o be_v require_v to_o be_v do_v ¶_o the_o 11._o problem_n the_o 34._o proposition_n to_o find_v out_o two_o right_a line_n incommensurable_a in_o power_n who_o square_n add_v together_o make_v a_o medial_a superficies_n and_o the_o parallelogram_n contain_v under_o they_o make_v a_o rational_a superficies_n take_v by_o the_o 31._o of_o the_o ten_o two_o medial_a line_n ab_fw-la and_o bc_o commensurable_a in_o power_n only_o comprehend_v a_o rational_a superficies_n construction_n so_o that_o let_v the_o line_n ab_fw-la be_v in_o power_n more_o than_o the_o line_n bc_o by_o the_o square_n of_o a_o line_n incommensurable_a in_o length_n unto_o the_o line_n ab_fw-la and_o describe_v upon_o the_o line_n ab_fw-la a_o semicircle_n adb_o and_o by_o the_o 10._o of_o the_o first_o divide_v the_o line_n bc_o unto_o two_o equal_a part_n in_o the_o point_n e._n and_o by_o the_o 28._o of_o the_o six_o upon_o the_o line_n ab_fw-la apply_v a_o parallelogram_n equal_a to_o the_o square_n of_o the_o line_n be_v and_o want_v in_o figure_n by_o a_o square_a and_o let_v that_o parallelogram_n be_v that_o which_o be_v contain_v under_o the_o line_n of_o and_o fb_o wherefore_o the_o line_n of_o be_v incommensurable_a in_o length_n unto_o the_o line_n fb_o by_o the_o 2._o part_n of_o the_o 18._o of_o the_o ten_o and_o from_o the_o point_n f_o unto_o the_o right_a line_n ab_fw-la raise_v up_o by_o the_o 11._o of_o the_o first_o a_o perpendicular_a line_n fd_o and_o draw_v line_n from_o a_o to_o d_o and_o from_o d_z to_o b._n demonstration_n and_o forasmuch_o as_o the_o line_n of_o be_v incommensurable_a unto_o the_o line_n fb_o but_o by_o the_o second_o assumpt_v go_v before_o the_o 33._o of_o the_o ten_o as_o the_o line_n of_o be_v to_o the_o line_n fb_o so_o be_v the_o parallelogram_n contain_v under_o the_o line_n basilius_n and_o of_o to_o the_o parallelogram_n contain_v under_o the_o line_n basilius_n and_o bf_o wherefore_o by_o the_o ten_o of_o the_o ten_o that_o which_o be_v contain_v under_o the_o line_n basilius_n and_o of_o be_v incommensurable_a to_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bf_o but_o that_o which_o be_v contain_v under_o the_o line_n basilius_n and_o of_o be_v equal_a to_o the_o square_n of_o the_o line_n ad_fw-la and_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bf_o be_v also_o equal_a to_o the_o square_n of_o the_o line_n db_o by_o the_o second_o part_n of_o the_o first_o assumpt_n go_v before_o the_o 33._o of_o the_o ten_o wherefore_o the_o square_a of_o the_o line_n ad_fw-la be_v incommensurable_a to_o the_o square_n of_o the_o line_n db._n wherefore_o the_o line_n ad_fw-la and_o db_o be_v incommensurable_a in_o power_n and_o forasmuch_o as_o the_o square_n of_o the_o line_n ab_fw-la be_v medial_a therefore_o also_o the_o superficies_n make_v of_o the_o square_n of_o the_o line_n ad_fw-la and_o db_o add_v together_o be_v medial_a conclude_v for_o the_o square_n of_o the_o line_n ad_fw-la and_o db_o be_v by_o the_o 47._o of_o the_o first_o equal_a to_o the_o square_n of_o the_o line_n ab_fw-la and_o forasmuch_o as_o the_o line_n bc_o be_v double_a to_o the_o line_n fd_o as_o it_o be_v prove_v in_o the_o proposition_n go_v before_o therefore_o the_o parallelogram_n contain_v under_o the_o line_n ab_fw-la and_o and_o bc_o be_v double_a to_o the_o parallelogram_n contain_v under_o the_o line_n ab_fw-la and_o fd_v by_o the_o three_o assumpt_v go_v before_o the_o 33._o proposition_n wherefore_o it_o be_v also_o commensurable_a unto_o it_o by_o the_o six_o of_o the_o ten_o but_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o be_v suppose_v to_o be_v rational_a wherefore_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o fd_o be_v also_o rational_a but_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o fd_v be_v equal_a to_o that_o which_o be_v contain_v under_o the_o line_n ad_fw-la and_o db_o by_o the_o last_o part_n of_o the_o first_o assumpt_n go_v before_o the_o 33._o of_o the_o ten_o wherefore_o that_o which_o be_v contain_v under_o the_o line_n ad_fw-la and_o db_o be_v also_o rational_a conclusion_n wherefore_o there_o be_v â—ound_v out_o two_o right_a line_n ad_fw-la and_o db_o incommensurable_a in_o power_n who_o square_n add_v together_o make_v a_o medial_a superficies_n and_o the_o parallelogram_n contain_v under_o they_o make_v a_o rational_a superficies_n which_o be_v require_v to_o be_v do_v ¶_o the_o 12._o problem_n the_o 35._o proposition_n to_o find_v out_o two_o right_a line_n incommensurable_a in_o power_n who_o square_n add_v together_o make_v a_o medial_a superficies_n and_o the_o parallelogram_n contain_v under_o they_o make_v also_o a_o medial_a superficies_n which_o parallelogram_n moreover_o shall_v be_v incommensurable_a to_o the_o superficies_n make_v of_o the_o square_n of_o those_o line_n add_v together_o take_v by_o the_o 32._o of_o the_o ten_o two_o medial_a line_n ab_fw-la and_o bc_o commensurable_a in_o power_n only_o construction_n comprehend_v a_o medial_a superficies_n so_o that_o let_v the_o line_n ab_fw-la be_v in_o power_n more_o than_o the_o line_n bc_o by_o the_o square_n of_o a_o line_n incommensurable_a in_o length_n unto_o the_o line_n ab_fw-la and_o upon_o the_o line_n ab_fw-la describe_v a_o semicircle_n adb_o and_o let_v the_o rest_n of_o the_o construction_n be_v as_o it_o be_v in_o the_o two_o former_a proposition_n and_o forasmuch_o as_o by_o the_o 2_o part_n of_o the_o 18._o of_o the_o ten_o the_o line_n of_o be_v incommensurable_a in_o length_n unto_o the_o line_n fb_o demonstration_n therefore_o the_o line_n ad_fw-la be_v incommensurable_a in_o power_n unto_o the_o line_n db_o by_o that_o which_o be_v demonstrate_v in_o the_o proposition_n go_v before_o and_o forasmuch_o as_o the_o square_n of_o the_o line_n ab_fw-la be_v medial_a therefore_o that_o also_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ad_fw-la and_o db_o which_o square_n be_v equal_a to_o the_o square_n of_o the_o line_n ab_fw-la by_o the_o 47._o of_o the_o first_o be_v medial_a conclude_v and_o forasmuch_o as_o that_o which_o be_v contain_v under_o the_o line_n of_o and_o fb_o be_v equal_a to_o either_o of_o the_o square_n of_o the_o line_n ebb_v and_o fd_o for_o by_o supposition_n the_o parallelogram_n contain_v under_o the_o line_n of_o and_o fb_o be_v equal_a to_o the_o square_n of_o the_o line_n ebb_n and_o the_o same_o parallelogram_n be_v equal_a to_o the_o square_n of_o the_o line_n df_o by_o the_o three_o part_n of_o the_o first_o assumpt_n go_v before_o the_o 33._o of_o the_o ten_o wherefore_o the_o line_n be_v be_v equal_a to_o the_o line_n df._n wherefore_o the_o line_n bc_o be_v double_a to_o the_o line_n fd._n wherefore_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o be_v double_a to_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o fd._n wherefore_o they_o be_v commensurable_a by_o the_o six_o of_o this_o book_n but_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o be_v medial_a by_o supposition_n wherefore_o also_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o fd_o be_v medial_a by_o the_o corollary_n of_o the_o 23_o of_o the_o ten_o but_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o fd_v be_v by_o the_o four_o part_n of_o the_o first_o assumpt_n go_v before_o the_o 33._o of_o the_o ten_o equal_v to_o that_o which_o be_v contain_v under_o the_o line_n ad_fw-la and_o db_o conclude_v wherefore_o that_o which_o be_v contain_v under_o the_o line_n ad_fw-la and_o db_o be_v also_o medial_a and_o forasmuch_o as_o the_o line_n ab_fw-la be_v incommensurable_a in_o length_n unto_o the_o line_n bc._n but_o the_o line_n bc_o be_v commensurable_a in_o length_n unto_o the_o line_n be._n wherefore_o by_o the_o 13â—_n of_o
be_v equal_a to_o that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o twice_o again_o from_o the_o the_o parallelogram_n eke_o take_v away_o the_o parallelogram_n el_n equal_a to_o the_o square_n of_o the_o line_n ad_fw-la and_o db_o which_o be_v less_o than_o the_o square_n of_o the_o line_n ac_fw-la and_o cb._n wherefore_o the_o residue_n namely_o the_o parallelogram_n mk_o be_v equal_a to_o that_o which_o be_v contain_v under_o the_o line_n ad_fw-la and_o db_o twice_o and_o forasmuch_o as_o the_o square_n of_o the_o line_n ac_fw-la and_o cb_o be_v medial_a therefore_o the_o parallelogram_n eglantine_n also_o be_v medial_a impossibility_n and_o it_o be_v apply_v upon_o the_o rational_a line_n efâ—_n â—_z wherefore_o the_o line_n eh_o be_v rational_a and_o incommensurable_a in_o length_n to_o the_o line_n ef._n and_o by_o the_o same_o reason_n the_o parallelogram_n hk_o be_v medial_a for_o that_o which_o be_v equal_a unto_o it_o namely_o that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o twice_o be_v medial_a therefore_o the_o line_n hn_o be_v also_o rational_a and_o incommensurable_a in_o length_n unto_o the_o line_n ef._n and_o forasmuch_o as_o the_o line_n ac_fw-la and_o cb_o be_v medial_a line_n commensurable_a in_o power_n only_o therefore_o the_o line_n ac_fw-la be_v incommensurable_a in_o length_n unto_o the_o line_n cb._n but_o as_o the_o line_n ac_fw-la be_v to_o the_o line_n cb_o so_o be_v the_o square_a of_o the_o line_n ac_fw-la to_o that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o by_o the_o 1._o of_o the_o six_o wherefore_o the_o square_a of_o the_o line_n ac_fw-la be_v incommensurable_a to_o that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb._n butt_n by_o the_o 16._o of_o the_o ten_o unto_o the_o square_n of_o the_o line_n ac_fw-la be_v commensurable_a the_o square_n of_o the_o line_n ac_fw-la and_o cb_o add_v together_o for_o the_o line_n ac_fw-la and_o cb_o be_v commensurable_a in_o power_n only_o and_o unto_o that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o be_v commensurable_a that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o twice_o wherefore_o that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ac_fw-la and_o cb_o be_v incommensurable_a to_o that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o twice_o but_o to_o the_o square_n of_o the_o line_n ac_fw-la and_o cb_o be_v equal_a the_o parallelogram_n eglantine_n and_o to_o that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o twice_o be_v equal_a the_o paralelograme_n hk_o wherefore_o the_o parallelogram_n eglantine_n be_v incommensurable_a to_o the_o parallelogram_n hk_o wherefore_o also_o the_o line_n eh_o be_v incommensurable_a in_o length_n to_o the_o line_n hn._n and_o the_o line_n eh_o and_o hn_o be_v rational_a wherefore_o they_o be_v rational_a commensurable_a in_o power_n only_o but_o if_o two_o rational_a line_n commensurable_a in_o power_n only_o be_v add_v together_o the_o whole_a line_n be_v irrational_a and_o be_v call_v a_o binomial_a line_n by_o the_o 36._o of_o the_o ten_o wherefore_o the_o binomial_a line_n en_fw-fr be_v in_o the_o point_n h_o divide_v into_o his_o name_n and_o by_o the_o same_o reason_n also_o may_v it_o be_v prove_v that_o the_o line_n em_n and_o mn_o be_v rational_a line_n commensurable_a in_o power_n only_o wherefore_o en_fw-fr being_n a_o binomial_a line_n be_v divide_v into_o his_o name_n in_o sundry_a point_n namely_o in_o h_o and_o m_o neither_o be_v the_o line_n eh_o one_o and_o the_o same_o that_o be_v equal_a with_o mn_n for_o the_o square_n of_o the_o line_n ac_fw-la and_o cb_o be_v great_a than_o the_o square_n of_o the_o line_n bd_o and_o ad_fw-la by_o the_o 1._o assumpt_v put_v after_o the_o 41._o of_o the_o ten_o but_o the_o square_n of_o the_o line_n ad_fw-la and_o db_o be_v great_a than_o that_o which_o be_v contain_v under_o the_o line_n ad_fw-la and_o db_o twice_o by_o the_o assumpt_n put_v after_o the_o 39_o of_o the_o ten_o wherefore_o the_o square_n of_o the_o line_n ac_fw-la and_o cb_o that_o be_v the_o parallelogram_n eglantine_n be_v much_o great_a than_o that_o which_o be_v contain_v under_o the_o line_n ad_fw-la and_o db_o twice_o that_o be_v then_o the_o parallelogram_n mk_o wherefore_o by_o the_o first_o of_o the_o six_o the_o line_n eh_o be_v great_a than_o the_o line_n mn_v wherefore_o eh_o be_v not_o one_o and_o the_o same_o with_o mn_n wherefore_o a_o binomial_a line_n be_v in_o two_o sundry_a point_n divide_v into_o his_o name_n which_o be_v impossible_a the_o self_n same_o absurdity_n also_o will_v follow_v if_o the_o line_n ac_fw-la be_v suppose_v to_o be_v less_o than_o the_o line_n db._n a_o second_o binomial_a line_n therefore_o be_v not_o divide_v into_o his_o name_n in_o sundry_a point_n wherefore_o it_o be_v divide_v in_o one_o only_a which_o be_v require_v to_o be_v demonstrate_v ¶_o the_o 33._o theorem_a the_o 45._o proposition_n a_o great_a line_n be_v in_o one_o point_n only_o divide_v into_o his_o name_n let_v ab_fw-la be_v a_o great_a line_n be_v divide_v into_o his_o name_n in_o the_o point_n c_o so_fw-mi that_o let_v the_o line_n ac_fw-la and_o cb_o be_v rational_a incommensurable_a in_o power_n have_v that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ac_fw-la and_o cb_o rational_a and_o that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o medial_a impossibiâ—â—â—e_n then_o i_o say_v that_o the_o line_n ab_fw-la can_v not_o in_o any_o other_o point_n then_o in_o c_o be_v divide_v into_o his_o name_n for_o if_o it_o be_v possible_a let_v it_o be_v divide_v into_o his_o name_n in_o the_o point_n d_o so_fw-mi that_o let_v ad_fw-la and_o db_o be_v line_n incommensurable_a in_o power_n have_v that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ad_fw-la and_o db_o rational_a and_o that_o which_o be_v contain_v under_o the_o line_n ad_fw-la and_o db_o medial_a now_o forasmuch_o as_o how_o much_o the_o square_n of_o the_o line_n ac_fw-la and_o cb_o diâ—â—er_n from_o the_o square_n of_o the_o line_n ad_fw-la and_o db_o so_o much_o differ_v that_o which_o be_v contain_v under_o the_o line_n ad_fw-la and_o db_o twice_o from_o that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o twice_o by_o those_o thing_n which_o have_v beâ—e_v say_v in_o the_o demonstration_n of_o the_o 42._o proposition_n but_o the_o square_n of_o the_o line_n ac_fw-la and_o cb_o exceed_v the_o square_n of_o the_o line_n ad_fw-la and_o db_o by_o a_o rational_a superficies_n for_o they_o be_v either_o of_o they_o rational_a wherefore_o that_o which_o be_v containeâ—â—â—der_v the_o line_n ad_fw-la and_o db_o twice_o exceed_v that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o twice_o by_o a_o rational_a superficies_n when_o as_o either_o of_o they_o be_v a_o medial_a supersâ—â—â—es_n which_o be_v impossible_a by_o the_o 26._o of_o the_o ten_o wherefore_o a_o great_a line_n be_v in_o one_o point_n only_o divide_v into_o his_o name_n which_o be_v require_v to_o be_v prove_v ¶_o the_o 34._o theorem_a the_o 46._o proposition_n a_o line_n contain_v in_o power_n a_o rational_a and_o a_o medial_a be_v in_o one_o point_n only_o divide_v into_o his_o name_n let_v ab_fw-la be_v a_o line_n contain_v in_o power_n a_o rational_a and_o a_o medial_a be_v divide_v into_o his_o name_n in_o the_o point_n c_o so_fw-mi that_o let_v the_o line_n ac_fw-la &_o cb_o be_v incommensurable_a in_o power_n have_v that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ac_fw-la and_o cb_o medial_a and_o that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o rational_a impossibility_n then_o i_o say_v that_o the_o line_n ab_fw-la can_v not_o in_o any_o other_o point_n be_v divide_v into_o his_o name_n but_o only_o in_o the_o point_n c._n for_o if_o it_o be_v possible_a let_v it_o be_v divide_v into_o his_o name_n in_o the_o point_n d_o so_fw-mi that_o let_v the_o line_n ad_fw-la and_o db_o be_v incommensurable_a in_o power_n have_v that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ad_fw-la and_o db_o medial_a and_o that_o which_o be_v contain_v under_o the_o line_n ad_fw-la and_o db_o rational_a now_o forasmuch_o as_o how_o much_o that_o which_o be_v contain_v under_o the_o line_n ad_fw-la &_o db_o twice_o differ_v from_o that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o twice_o so_o much_o differ_v the_o square_n of_o the_o line_n ac_fw-la &_o cb_o add_v together_o from_o the_o square_n of_o the_o line_n ad_fw-la and_o db_o add_v together_o but_o that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o twice_o exceed_v that_o
which_o be_v contain_v under_o the_o line_n ad_fw-la and_o db_o twice_o by_o a_o rational_a superficies_n for_o either_o of_o they_o be_v rational_a wherefore_o also_o the_o square_n of_o the_o line_n ac_fw-la and_o cb_o add_v together_o exceed_v the_o square_n of_o the_o line_n ad_fw-la and_o db_o add_v together_o by_o a_o rational_a superficies_n when_o yet_o each_o of_o they_o be_v a_o medial_a superficies_n which_o be_v impossible_a wherefore_o a_o line_n contain_v in_o power_n a_o rational_a and_o a_o medial_a be_v in_o one_o point_n only_o divide_v into_o his_o name_n which_o be_v require_v to_o be_v demonstrate_v ¶_o the_o 35._o theorem_a the_o 47._o proposition_n a_o line_n contain_v in_o power_n two_o medial_n be_v in_o one_o point_n only_o divide_v into_o his_o name_n svppose_v that_o ab_fw-la be_v a_o line_n contain_v in_o power_n two_o medial_n be_v divide_v into_o his_o name_n in_o the_o point_n c_o so_fw-mi that_o let_v the_o line_n ac_fw-la and_o cb_o be_v incommensurable_a in_o power_n have_v that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ac_fw-la &_o cb_o medial_a and_o that_o also_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o medial_a and_o moreover_o incommensurable_a â—o_o that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ac_fw-la and_o cb._n then_o i_o say_v construction_n that_o the_o line_n ab_fw-la can_v in_o no_o other_o point_n be_v divide_v into_o his_o name_n but_o only_o in_o the_o point_n c._n for_o if_o it_o be_v possible_a let_v it_o be_v divide_v into_o his_o name_n in_o the_o point_n d_o so_fw-mi that_o let_v not_o the_o line_n ac_fw-la be_v one_o and_o the_o same_o that_o be_v equal_a with_o the_o line_n db_o but_o by_o supposition_n let_v the_o line_n ac_fw-la be_v the_o great_a and_o take_v a_o rational_a line_n ef._n and_o by_o the_o 43._o of_o the_o first_o upon_o the_o line_n of_o apply_v a_o rectangle_n parallelogram_n eglantine_n equal_a to_o that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ac_fw-la and_o cb_o and_o likewise_o upon_o the_o line_n hc_n which_o be_v equal_a to_o the_o line_n of_o apply_v the_o parallelogram_n hk_o equal_a to_o that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o twice_o wherefore_o the_o whole_a parallelogram_n eke_o be_v equal_a to_o the_o square_n of_o the_o line_n ab_fw-la absurdity_n again_o upon_o the_o same_o line_n of_o describe_v the_o parallelogram_n el_n equal_a to_o the_o square_n of_o the_o line_n ad_fw-la and_o db._n wherefore_o the_o residue_n namely_o that_o which_o be_v contain_v under_o the_o line_n ad_fw-la and_o db_o twice_o be_v equal_a to_o the_o parallelogram_n remain_v namely_o to_o mk_o and_o forasmuch_o as_o that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ac_fw-la and_o cb_o be_v by_o supposition_n medial_a therefore_o the_o parallelogram_n eglantine_n which_o be_v equal_a unto_o it_o be_v also_o medial_a and_o it_o be_v apply_v upon_o the_o rational_a line_n ef._n wherefore_o by_o the_o 22._o of_o the_o ten_o the_o line_n he_o be_v rational_a and_o incommensurable_a in_o length_n unto_o the_o line_n ef._n and_o by_o the_o same_o reason_n also_o the_o line_n hn_o be_v rational_a and_o incommensurable_a in_o length_n to_o the_o same_o line_n ef._n and_o forasmuch_o as_o that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ac_fw-la and_o cb_o be_v incommensurable_a to_o that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o twice_o for_o it_o be_v suppose_v to_o be_v incommensurable_a to_o that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o once_o therefore_o the_o parallelogram_n eglantine_n be_v incommensurable_a to_o the_o parallelogram_n h_o â—_o wherefore_o the_o line_n eh_o also_o be_v incommensurable_a in_o length_n to_o the_o line_n hn_o and_o they_o be_v rational_a line_n wherefore_o the_o line_n eh_o and_o hn_o be_v rational_a commensurable_a in_o power_n only_o wherefore_o the_o whole_a line_n en_fw-fr be_v a_o binomial_a line_n and_o be_v divide_v into_o his_o name_n in_o the_o point_n h._n and_o in_o like_a sort_n may_v we_o prove_v that_o the_o same_o binomial_a line_n en_fw-fr be_v divide_v into_o his_o name_n in_o the_o point_n m_o and_o that_o the_o line_n eh_o be_v not_o one_o and_o the_o same_o that_o be_v equal_a with_o the_o line_n mn_o as_o it_o be_v prove_v in_o the_o end_n of_o the_o demonstration_n of_o the_o 44._o of_o this_o book_n wherefore_o a_o binomial_a line_n be_v divide_v into_o his_o name_n in_o two_o sundry_a point_n which_o be_v impossible_a by_o the_o 42._o of_o the_o ten_o wherefore_o a_o line_n contain_v in_o power_n two_o medial_n be_v not_o in_o sundry_a point_n divide_v into_o his_o name_n wherefore_o it_o be_v divide_v in_o one_o point_n only_o which_o be_v require_v to_o be_v demonstrate_v ¶_o second_a definition_n it_o be_v show_v before_o that_o of_o binomial_a line_n there_o be_v six_o kind_n line_n the_o definition_n of_o all_o which_o be_v here_o now_o set_v and_o be_v call_v second_o definition_n all_o binomial_a line_n as_o all_o other_o kind_n of_o irrational_a line_n be_v conceive_v consider_v and_o perfect_o understand_v only_o in_o respect_n of_o a_o rational_a line_n who_o part_n as_o before_o be_v teach_v be_v certain_a and_o know_v and_o may_v be_v distinct_o express_v by_o number_n unto_o which_o line_n they_o be_v compare_v this_o rationalâ—_fw-la line_n must_v you_o ever_o have_v before_o your_o eye_n in_o all_o these_o definition_n so_o shall_v they_o all_o be_v â—asie_a enough_o paâ—tâ—s_n a_o binomial_a line_n you_o know_v be_v make_v of_o two_o part_n or_o name_n whereof_o the_o one_o be_v great_a than_o the_o other_o wherefore_o the_o power_n or_o square_v also_o of_o the_o one_o be_v great_a than_o the_o power_n or_o square_a oâ—_n the_o other_o the_o three_o first_o kind_n of_o binomial_a line_n namely_o the_o first_o the_o seconâ—_o &_o the_o three_o be_v produce_v when_o the_o square_a of_o the_o great_a name_n or_o part_n of_o a_o binomâ—all_n eâ—cedeth_v the_o square_a of_o the_o less_o name_n or_o part_n by_o the_o square_n of_o a_o line_n which_o be_v commensurable_a in_o length_n to_o it_o namely_o to_o the_o great_a the_o three_o last_o kind_n namely_o the_o four_o the_o â—iâ—t_n and_o the_o six_o be_v produce_v when_o the_o square_a of_o the_o great_a name_n or_o part_n â—â—â—â—edeth_v the_o square_a of_o the_o less_o name_n or_o part_n by_o the_o square_n of_o a_o line_n incommensurable_a in_o length_n unto_o it_o that_o be_v to_o the_o great_a part_n definition_n a_o first_o binomial_a line_n be_v who_o square_a of_o the_o great_a part_n exceed_v the_o square_a of_o tâ—e_z less_o part_n â—y_a the_o square_n of_o a_o line_n commensurable_a in_o length_n to_o the_o great_a part_n and_o the_o great_a part_n be_v also_o commensurable_a in_o length_n to_o tâ—e_z rational_a line_n first_o set_v as_o lâ—t_v the_o rational_a line_n first_o set_v be_v ab_fw-la who_o part_n be_v distinct_o know_v suppose_v also_o that_o the_o line_n ce_fw-fr be_v a_o binomial_a line_n who_o name_n or_o part_n let_v be_v cd_o and_o de._n and_o let_v the_o square_n of_o the_o line_n cd_o the_o great_a part_n exceed_v the_o square_n of_o the_o line_n de_fw-fr the_o less_o part_n by_o the_o square_n of_o the_o line_n fg_o which_o line_n fg_o let_v bâ—e_o commensurable_a in_o length_n to_o the_o line_n cd_o which_o be_v the_o great_a part_n of_o the_o binomial_a line_n and_o moreover_o let_v the_o line_n cd_o the_o great_a paâ—t_n be_v commensurble_n in_o length_n to_o the_o rational_a line_n first_o set_v namely_o to_o ab_fw-la so_o by_o this_o dâ—â—inition_n the_o binomial_a line_n ce_fw-fr be_v a_o first_o binomial_a line_n definition_n a_o second_o binomial_a line_n be_v when_o the_o square_a of_o the_o great_a part_n exceed_v the_o square_a of_o the_o less_o part_n by_o the_o square_n of_o a_o line_n commensurable_a in_o length_n unto_o it_o and_o the_o less_o part_n be_v commensurable_a in_o length_n to_o the_o rational_a line_n first_o set_v as_o suppose_v ever_o the_o rational_a line_n let_v ce_fw-fr be_v a_o binomial_a line_n divide_v in_o the_o point_n d._n the_o square_n of_o who_o great_a part_n cd_o let_v exceed_v the_o square_n of_o the_o less_o part_n de_fw-fr by_o the_o square_n of_o the_o line_n fg_o which_o line_n â—g_z let_v be_v commensurable_a in_o length_n unto_o the_o line_n cd_o tâ—e_z great_a pâ—â—â—_n oâ—_n the_o binomial_a line_n and_o let_v also_o the_o line_n de_fw-fr the_o less_o part_n of_o the_o binomial_a line_n be_v commensurable_a in_o length_n to_o the_o rational_a line_n first_o set_v ab_fw-la so_o by_o this_o definition_n the_o binomial_a line_n ce_fw-fr be_v a_o second_o binomial_a line_n â—â—â—â—â—â—ition_n a_o three_o binomial_a
line_n be_v when_o the_o square_a of_o the_o great_a part_n exceed_v the_o square_a of_o the_o less_o part_n by_o the_o square_n of_o a_o line_n commensurable_a in_o length_n unto_o it_o and_o neither_o part_n be_v commensurable_a in_o length_n to_o the_o rational_a line_n give_v as_o suppose_v the_o line_n ce_fw-fr to_o be_v a_o binomial_a line_n who_o part_n be_v join_v together_o in_o the_o point_n d_o and_o let_v the_o square_n of_o the_o line_n cd_o the_o great_a part_n exceed_v the_o square_n of_o the_o less_o part_n de_fw-fr by_o the_o square_n of_o the_o line_n fg_o and_o let_v the_o line_n fg_o be_v commensurable_a in_o length_n to_o the_o line_n cd_o the_o great_a part_n of_o the_o binomial_a moreover_o let_v neither_o the_o great_a part_v cd_o nor_o the_o less_o part_n de_fw-fr be_v commensurable_a in_o length_n to_o the_o rational_a line_n ab_fw-la then_o be_v the_o line_n ce_fw-fr by_o this_o definition_n a_o three_o binomial_a line_n a_o four_o binomial_a line_n be_v definition_n when_o the_o square_a of_o the_o great_a part_n exceed_v the_o square_a of_o the_o less_o by_o the_o square_n of_o a_o line_n incommensurable_a in_o length_n unto_o the_o great_a part_n and_o the_o great_a be_v also_o commensurable_a in_o length_n to_o the_o rational_a line_n as_o let_v the_o line_n ce_fw-fr be_v a_o binomial_a line_n who_o part_n let_v be_v cd_o &_o de_fw-fr &_o let_v the_o square_n of_o the_o line_n cd_o the_o great_a part_n exceed_v the_o square_n of_o the_o line_n de_fw-fr the_o less_o by_o the_o square_n of_o the_o line_n fg._n and_o let_v the_o line_n fg_o be_v incommensurable_a in_o length_n to_o the_o line_n cd_o the_o great_a let_v also_o the_o line_n cd_o the_o great_a part_n be_v commensurable_a in_o length_n unto_o the_o rational_a line_n ab_fw-la then_o by_o this_o definition_n the_o line_n ce_fw-fr be_v a_o four_o binomial_a line_n a_o five_o binomial_a line_n be_v definition_n when_o the_o square_a of_o the_o great_a part_n exceed_v the_o square_a of_o the_o less_o part_n by_o the_o square_n of_o a_o line_n incommensurable_a unto_o it_o in_o length_n and_o the_o less_o part_n also_o be_v commensurable_a in_o length_n to_o the_o rational_a line_n give_v as_o suppose_v that_o ce_fw-fr be_v a_o binomial_a line_n who_o great_a part_n let_v be_v cd_o and_o let_v the_o less_o part_n be_v de._n and_o let_v the_o square_n of_o the_o line_n cd_o exceed_v the_o square_n of_o the_o line_n de_fw-fr by_o the_o square_n of_o the_o line_n fg_o which_o let_v be_v incommensurable_a in_o length_n unto_o the_o line_n cd_o the_o great_a part_n of_o the_o binomial_a line_n and_o let_v the_o line_n de_fw-fr the_o second_o part_n of_o the_o binomial_a line_n be_v commensurable_a in_o length_n unto_o the_o rational_a line_n ab_fw-la so_o be_v the_o line_n ce_fw-fr by_o this_o definition_n a_o five_o binomial_a line_n a_o six_o binomial_a line_n be_v definition_n when_o the_o square_a of_o the_o great_a part_n exceed_v the_o square_a of_o the_o less_o by_o the_o square_n of_o a_o line_n incommensurable_a in_o length_n unto_o it_o and_o neither_o part_n be_v commensurable_a in_o length_n to_o the_o rational_a line_n give_v as_o let_v the_o line_n ce_fw-fr be_v a_o binomial_a line_n divide_v into_o his_o name_n in_o the_o point_n d._n the_o square_n of_o who_o great_a part_n cd_o let_v exceed_v the_o square_n of_o the_o less_o part_n de_fw-fr by_o the_o square_n of_o the_o line_n fg_o and_o let_v the_o line_n fg_o be_v incommensurable_a in_o length_n to_o the_o line_n cd_o the_o great_a part_n of_o the_o binomial_a line_n let_v also_o neither_o cd_o the_o great_a part_n nor_o de_fw-fr the_o less_o part_n be_v commensurable_a in_o length_n to_o the_o rational_a line_n ab_fw-la and_o so_o by_o this_o definition_n the_o line_n ce_fw-fr be_v a_o six_o binomial_a line_n so_o you_o see_v that_o by_o these_o definition_n &_o their_o example_n and_o declaration_n all_o the_o kind_n of_o binomial_a line_n be_v make_v very_o plain_a this_o be_v to_o be_v note_v that_o here_o be_v nothing_o speak_v of_o those_o line_n both_o who_o portion_n aâ—e_v commensurable_a in_o length_n unto_o the_o rational_a line_n first_o set_v for_o that_o such_o line_n can_v be_v binomial_a line_n â—or_a binomial_a line_n be_v compose_v of_o two_o rational_a line_n commensurable_a in_o power_n only_o by_o the_o 36._o of_o this_o book_n but_o line_n both_o who_o portion_n be_v commensurable_a in_o length_n to_o the_o rational_a line_n first_o set_v be_v not_o binomial_a line_n for_o that_o the_o part_n of_o such_o line_n shall_v by_o the_o 12._o of_o this_o book_n be_v commensurable_a in_o length_n the_o one_o to_o the_o other_o and_o so_o shall_v they_o not_o be_v such_o line_n as_o be_v require_v to_o the_o composition_n of_o a_o binomial_a line_n moreover_o such_o line_n shall_v not_o be_v irrational_a but_o rational_a for_o that_o they_o be_v commensurable_a tâ—â—ch_o of_o the_o part_n whereof_o they_o be_v compose_v by_o the_o 15._o oâ—_n this_o book_n and_o therefore_o they_o shall_v be_v rational_a for_o that_o the_o line_n which_o composâ—_n they_o be_v rational_a ¶_o the_o 13._o problem_n the_o 48._o proposition_n to_o find_v out_o a_o first_o binomial_a line_n composition_n take_v two_o number_n ac_fw-la and_o cb_o &_o let_v they_o be_v such_o that_o the_o number_n which_o be_v make_v of_o they_o both_o add_v together_o namely_o ab_fw-la have_v unto_o one_o of_o they_o 〈◊〉_d unto_o bc_o that_o proportion_n that_o a_o square_a number_n have_v to_o a_o square_a number_n â—ut_z unto_o the_o other_o namely_o unto_o ca_n let_v it_o not_o have_v that_o proportion_n that_o a_o square_a number_n have_v to_o a_o square_a number_n such_o as_o be_v every_o square_a number_n which_o may_v be_v divide_v into_o a_o square_a number_n and_o into_o a_o number_n not_o square_a construction_n take_v also_o a_o certain_a rational_a line_n and_o let_v the_o same_o be_v d._n and_o unto_o the_o line_n d_o let_v the_o line_n of_o be_v commensurable_a in_o length_n wherefore_o the_o line_n of_o be_v rational_a and_o as_o the_o number_n ab_fw-la be_v to_o the_o number_n ac_fw-la so_o let_v the_o square_n of_o the_o line_n of_o be_v to_o the_o square_n of_o a_o other_o â—iâ—e_n namely_o of_o fg_o by_o the_o corollary_n of_o the_o six_o of_o the_o ten_o wherefore_o the_o square_a of_o the_o line_n of_o have_v to_o the_o square_n of_o the_o line_n fg_o that_o proportion_n that_o number_n have_v to_o number_v wherefore_o the_o square_a of_o the_o line_n of_o be_v commensurable_a to_o the_o square_n of_o the_o line_n fg_o by_o the_o 6._o of_o this_o book_n and_o the_o line_n of_o be_v rational_a demonstration_n wherefore_o the_o line_n fg_o also_o be_v rational_a and_o forasmuch_o as_o the_o number_n ab_fw-la have_v not_o to_o the_o number_n ac_fw-la that_o proportion_n that_o a_o square_a number_n have_v to_o a_o square_a number_n neither_o shall_v the_o square_a of_o the_o line_n of_o have_v to_o the_o square_n of_o the_o line_n fg_o that_o proportion_n that_o a_o square_a number_n have_v to_o a_o square_a number_n wherefore_o the_o line_n of_o be_v incommensurable_a in_o length_n to_o the_o line_n fg_o by_o the_o 9_o of_o this_o book_n wherefore_o the_o line_n of_o and_o fg_o be_v rational_a commensurable_a in_o power_n only_o wherefore_o the_o whole_a line_n eglantine_n be_v a_o binomial_a line_n by_o the_o 36._o of_o the_o ten_o i_o say_v also_o that_o it_o be_v a_o â—irst_n binomial_a line_n for_o for_o that_o as_o the_o 〈◊〉_d basilius_n be_v to_o the_o number_n ac_fw-la so_o be_v the_o square_a of_o the_o line_n of_o to_o the_o square_n of_o the_o line_n â—g_z but_o the_o number_n basilius_n be_v great_a than_o the_o number_n ac_fw-la wherefore_o the_o square_a of_o the_o line_n â—f_n be_v also_o great_a than_o the_o square_a oâ—_n the_o line_n fg._n unto_o the_o square_n of_o the_o line_n of_o let_v the_o square_n of_o the_o line_n fg_o and_o h_o be_v equal_a which_o how_o to_o find_v out_o be_v teach_v in_o the_o assumpt_n put_v â—ftâ—r_v the_o 13._o of_o the_o tâ—nth_o and_o fâ—r_v that_o as_o thâ—_z number_n basilius_n be_v to_o the_o number_n ac_fw-la so_o be_v the_o square_a of_o the_o line_n of_o to_o the_o square_n of_o the_o line_n fg_o therefore_o by_o coâ—uersion_n or_o eâ—ersion_n of_o proportion_n by_o the_o corollary_n of_o the_o 19_o of_o the_o five_o as_o the_o number_n ab_fw-la be_v to_o the_o number_n bc_o so_o be_v the_o square_a of_o the_o line_n of_o to_o the_o square_n of_o the_o line_n h._n but_o the_o number_n ab_fw-la have_v to_o the_o number_n bc_o that_o proportion_n that_o a_o square_a number_n have_v to_o a_o square_a number_n wherefore_o also_o the_o square_a of_o the_o line_n of_o have_v to_o the_o square_n of_o the_o line_n h_o
that_o proportion_n that_o a_o square_a number_n have_v to_o a_o square_a number_n wherefore_o the_o line_n of_o be_v commensurable_a in_o length_n to_o the_o line_n h_o by_o the_o 9_o of_o this_o book_n wherefore_o the_o line_n of_o be_v in_o power_n more_o than_o the_o line_n fg_o by_o the_o square_n of_o a_o line_n commensurable_a in_o length_n to_o the_o line_n ef._n and_o the_o line_n of_o and_o fg_o be_v rational_a commensurable_a in_o power_n only_o and_o the_o line_n of_o be_v commensurable_a in_o length_n to_o the_o rational_a line_n d._n wherefore_o the_o line_n eglantine_n be_v a_o first_o binomial_a line_n which_o be_v require_v to_o be_v do_v the_o 14._o problem_n the_o 49._o proposition_n to_o find_v out_o a_o second_o binomial_a line_n take_v two_o number_n ac_fw-la and_o cb_o and_o let_v they_o be_v such_o that_o the_o number_n make_v of_o they_o both_o add_v together_o namely_o construction_n ab_fw-la have_v unto_o bc_o that_o proportion_n that_o a_o square_a number_n have_v to_o a_o square_a number_n and_o unto_o the_o number_n ca_n let_v it_o not_o have_v that_o proportion_n that_o a_o square_a number_n have_v to_o a_o square_a number_n as_o it_o be_v declare_v in_o the_o former_a proposition_n take_v also_o a_o rational_a line_n and_o let_v the_o same_o be_v d_o and_o unto_o the_o line_n d_o let_v the_o line_n fg_o be_v commensurable_a in_o length_n wherefore_o fg_o be_v a_o rational_a line_n and_o as_o the_o number_n ca_n be_v to_o the_o number_n ab_fw-la so_o let_v the_o square_n of_o the_o line_n gf_o be_v to_o the_o square_n of_o the_o line_n fe_o by_o the_o 6._o of_o the_o ten_o wherefore_o the_o square_a of_o the_o linâ—_n gf_o be_v commensurable_a to_o the_o square_n of_o the_o line_n fe_o wherefore_o also_o fe_o be_v a_o rational_a line_n demonstration_n and_o forasmuch_o as_o the_o number_n ca_n have_v not_o unto_o the_o number_n ab_fw-la that_o proportion_n that_o a_o square_a number_n have_v to_o a_o square_a number_n therefore_o neither_o also_o the_o square_a of_o the_o line_n gf_o have_v to_o the_o square_n of_o the_o line_n fe_o that_o proportion_n that_o a_o square_a number_n have_v to_o a_o square_a number_n wherefore_o the_o line_n gf_o be_v incommensurable_a in_o length_n unto_o the_o line_n fe_o by_o the_o 9_o of_o the_o ten_o wherefore_o the_o line_n fg_o and_o fe_o be_v rational_a commensurable_a in_o power_n only_o wherefore_o the_o whole_a line_n eglantine_n be_v a_o binomial_a line_n i_o say_v moreover_o that_o the_o linâ—_n eglantine_n be_v a_o second_o binomial_a line_n for_o for_o that_o by_o contrary_a proportion_n as_o the_o number_n basilius_n be_v to_o the_o number_n ac_fw-la so_o be_v the_o square_a of_o the_o line_n of_o to_o the_o square_n of_o the_o line_n fg._n but_o the_o number_n basilius_n be_v great_a than_o the_o number_n ac_fw-la wherefore_o also_o the_o square_a of_o the_o line_n of_o be_v great_a than_o the_o square_a of_o the_o line_n fg._n unto_o the_o square_n of_o the_o line_n of_o let_v the_o square_n of_o the_o linâ—s_v gf_o and_o h_o be_v equal_a now_o by_o conversion_n by_o the_o corollary_n of_o the_o 19_o of_o of_o the_o five_o as_o the_o number_n ab_fw-la be_v to_o the_o number_n bc_o so_o be_v the_o square_a of_o the_o line_n of_o to_o the_o square_n of_o the_o line_n h._n but_o the_o number_n ab_fw-la have_v to_o the_o number_n bc_o that_o proportion_n that_o a_o square_a number_n have_v to_o a_o square_a number_n wherefore_o the_o square_a of_o the_o line_n of_o have_v to_o the_o square_n of_o the_o line_n h_o that_o proportion_n that_o a_o square_a number_n have_v to_o a_o square_a number_n wherefore_o by_o the_o 9_o of_o the_o ten_o the_o line_n of_o be_v commensurable_a in_o length_n unto_o the_o line_n h._n wherefore_o the_o line_n of_o be_v in_o power_n more_o than_o the_o line_n fg_o by_o the_o square_n of_o a_o line_n commensurable_a in_o length_n unto_o the_o line_n of_o and_o the_o line_n of_o and_o fg_o be_v rational_a commensurable_a in_o power_n only_o and_o fg_o be_v the_o less_o name_n â—s_v commensurable_a in_o length_n unto_o the_o rational_a line_n give_v namely_o to_o d._n wherefore_o eglantine_n be_v a_o second_o binomial_a line_n which_o be_v require_v to_o be_v do_v ¶_o the_o 15._o problem_n the_o 50._o proposition_n to_o find_v out_o a_o three_o binomial_a line_n construction_n take_v two_o number_n ac_fw-la and_o cb_o and_o let_v they_o be_v such_o that_o the_o number_n make_v of_o they_o both_o add_v together_o namely_o ab_fw-la have_v to_o the_o number_n bc_o that_o proportion_n that_o a_o square_a number_n have_v to_o a_o square_a number_n but_o to_o the_o number_n ac_fw-la let_v it_o not_o have_v that_o proportion_n that_o a_o square_a number_n have_v to_o a_o square_a number_n as_o it_o be_v declare_v in_o the_o two_o former_a and_o take_v also_o some_o other_o number_n that_o be_v not_o a_o square_a number_n and_o let_v the_o same_o be_v d_o and_o let_v not_o the_o number_n d_o have_v either_o to_o the_o number_n basilius_n or_o to_o the_o number_n ac_fw-la that_o proportion_n that_o a_o square_a number_n have_v to_o a_o square_a number_n and_o take_v a_o rational_a line_n and_o let_v the_o the_o same_o be_v e._n and_o as_o the_o number_n d_o be_v to_o the_o number_n ab_fw-la so_o let_v the_o square_n of_o the_o line_n e_o be_v to_o the_o square_n of_o the_o line_n fg._n wherefore_o the_o square_a of_o the_o line_n e_o be_v commensurable_a to_o the_o square_n of_o the_o line_n fg_o but_o the_o line_n e_o be_v rational_a wherefore_o the_o line_n fg_o also_o be_v rational_a and_o for_o that_o the_o number_n d_o have_v not_o to_o the_o number_n ab_fw-la that_o proportion_n that_o a_o square_a number_n have_v to_o a_o square_a number_n neither_o also_o shall_v the_o square_a of_o the_o line_n e_o have_v to_o the_o square_n of_o the_o line_n fg_o that_o proportion_n that_o a_o square_a number_n have_v to_o a_o square_a number_n wherefore_o the_o line_n e_o be_v incommensurable_a in_o length_n to_o the_o line_n fg_o by_o the_o 9_o of_o the_o ten_o now_o again_o as_o the_o number_n ab_fw-la be_v to_o the_o number_n ac_fw-la so_o let_v the_o square_n of_o the_o line_n fg_o be_v to_o the_o square_n of_o the_o line_n gh_o wherefore_o the_o square_a of_o the_o line_n fg_o be_v commensurable_a to_o the_o square_n of_o the_o line_n gh_o demonstration_n and_o the_o line_n fg_o be_v rational_a wherefore_o also_o the_o line_n gh_o be_v rational_a and_o for_o that_o the_o number_n basilius_n have_v not_o to_o the_o number_n ac_fw-la that_o proportion_n that_o a_o square_a number_n have_v to_o a_o square_a number_n therefore_o neither_o also_o have_v the_o square_a of_o the_o line_n fg_o to_o the_o square_n of_o the_o line_n gh_o that_o proportion_n that_o a_o square_a number_n have_v to_o a_o square_a number_n wherefore_o the_o line_n fg_o be_v incommensurable_a in_o length_n to_o the_o line_n gh_o wherefore_o the_o line_n fg_o &_o gh_o be_v rational_a commensurable_a in_o power_n only_o wherefore_o the_o whole_a line_n fh_o be_v a_o binomial_a line_n i_o say_v moreover_o that_o it_o be_v a_o three_o binomial_a line_n for_o for_o that_o as_o the_o number_n d_o be_v to_o the_o number_n ab_fw-la so_o be_v the_o square_a of_o the_o line_n e_o to_o the_o square_n of_o the_o line_n fg_o but_o as_o the_o number_n ab_fw-la be_v to_o the_o number_n ac_fw-la so_o be_v the_o square_a of_o the_o line_n fg_o to_o the_o square_n of_o the_o line_n ghâ—_n therefore_o oâ—_n equality_n by_o the_o 22._o of_o the_o five_o as_o the_o number_n d_z be_v to_o the_o number_n ac_fw-la so_o be_v the_o square_a of_o the_o line_n e_o to_o the_o square_n of_o the_o line_n gh_o but_o the_o number_n d_o have_v not_o to_o the_o number_n a._n c_o that_o proportion_n that_o a_o square_a number_n have_v to_o a_o square_a number_n wherefore_o neither_o also_o have_v the_o square_a of_o the_o line_n e_o to_o the_o square_n of_o the_o line_n gh_o that_o proportion_n thaâ—_z a_o square_a number_n have_v so_o a_o square_a number_n wherefore_o the_o line_n e_o be_v incommensurable_a in_o length_n to_o the_o line_n gh_o and_o for_o that_o as_o the_o number_n ab_fw-la be_v to_o the_o number_n ac_fw-la so_o be_v the_o square_a of_o the_o line_n fg_o to_o the_o square_a oâ—_n the_o line_n gh_o therefore_o the_o square_a of_o the_o line_n fg_o be_v great_a than_o the_o square_a of_o the_o line_n gh_o unto_o the_o square_n of_o the_o line_n fg_o let_v the_o square_n of_o the_o line_n gh_o and_o king_n be_v equal_a wherefore_o by_o averse_a proportion_n by_o the_o corollary_n of_o the_o 19_o of_o the_o â—ift_n as_o the_o number_n ab_fw-la be_v to_o the_o number_n bc_o so_o be_v the_o square_a of_o the_o line_n fg_o to_o
the_o square_n of_o the_o line_n k._n but_o the_o number_n ab_fw-la have_v to_o the_o number_n bc_o that_o proportion_n that_o a_o square_a number_n have_v to_o a_o square_a number_n wherefore_o also_o the_o square_a of_o the_o line_n fg_o have_v to_o the_o square_n of_o the_o line_n king_n that_o proportion_n that_o a_o square_a number_n have_v to_o a_o square_a number_n wherefore_o the_o line_n fg_o be_v commensurable_a in_o length_n to_o the_o line_n k._n wherefore_o the_o line_n fg_o be_v in_o power_n more_o than_o the_o line_n gh_o by_o the_o square_n of_o a_o line_n commensurable_a in_o length_n unto_o it_o and_o the_o line_n fg_o and_o gh_o be_v rational_a commensurable_a in_o power_n only_o and_o neither_o of_o they_o be_v commensurable_a in_o length_n unto_o the_o rational_a line_n e_o wherefore_o the_o line_n fh_o be_v a_o three_o binomial_a line_n which_o be_v require_v to_o be_v do_v ¶_o the_o 16._o problem_n the_o 51._o proposition_n to_o find_v out_o a_o four_o binomial_a line_n take_v two_o number_n ac_fw-la and_o cb_o construction_n &_o let_v they_o be_v such_o that_o the_o number_n make_v of_o they_o both_o add_v together_o namely_o ab_fw-la have_v to_o neither_o of_o the_o number_n ac_fw-la and_o cb_o that_o proportion_n that_o a_o square_a number_n have_v to_o a_o square_a number_n such_o as_o be_v every_o square_a number_n to_o two_o number_n not_o square_a which_o be_v less_o them_z it_o &_o make_v the_o say_v square_a number_n and_o take_v a_o rational_a line_n and_o let_v the_o same_o be_v d._n and_o unto_o the_o line_n d_o let_v the_o line_n of_o be_v commensurable_a in_o length_n demonstration_n wherefore_o of_o be_v a_o rational_a line_n and_o as_o the_o number_n basilius_n be_v to_o the_o number_n ac_fw-la so_o let_v the_o square_n of_o the_o line_n of_o be_v to_o the_o square_n of_o the_o line_n fg._n wherefore_o the_o square_a of_o the_o line_n of_o be_v commensurable_a to_o the_o square_n of_o the_o line_n fg_o and_o the_o line_n bf_o be_v a_o rational_a line_n wherefore_o also_o the_o line_n fg_o be_v a_o rational_a line_n and_o for_o that_o the_o number_n basilius_n have_v not_o to_o the_o number_n ac_fw-la that_o proportion_n that_o a_o square_a number_n have_v to_o a_o square_a number_n neither_o also_o shall_v the_o square_a of_o the_o line_n of_o have_v to_o the_o square_n of_o the_o line_n fg_o that_o proportion_n that_o a_o square_a number_n have_v to_o a_o square_a number_n wherefore_o the_o line_n of_o be_v incommensurable_a in_o length_n to_o the_o line_n fg._n wherefore_o the_o line_n of_o and_o fg_o be_v rational_a commensurable_a in_o power_n only_o wherefore_o the_o whole_a line_n eglantine_n be_v a_o binomial_a line_n i_o say_v moreover_o that_o it_o be_v a_o four_o binomial_a line_n for_o for_o that_o as_o the_o number_n basilius_n be_v to_o the_o number_n ac_fw-la so_o be_v the_o square_a of_o the_o line_n of_o to_o the_o square_n of_o the_o line_n fg._n but_o the_o number_n basilius_n be_v great_a than_o the_o number_n ac_fw-la wherefore_o also_o the_o square_a of_o the_o line_n of_o be_v great_a than_o the_o square_a of_o the_o line_n fg._n unto_o the_o square_n of_o the_o line_n of_o let_v the_o square_n of_o the_o line_n fg_o and_o h_o be_v equal_a wherefore_o by_o conversion_n by_o the_o corollary_n of_o the_o 19_o of_o the_o five_o as_o the_o number_n ab_fw-la be_v to_o the_o number_n bc_o so_o be_v the_o square_a of_o the_o line_n of_o to_o the_o square_n of_o the_o line_n h._n but_o the_o number_n ab_fw-la have_v not_o to_o the_o number_n bc_o that_o proportion_n that_o a_o square_a number_n have_v to_o a_o square_a number_n therefore_o neither_o also_o have_v the_o square_a of_o the_o line_n of_o to_o the_o square_n of_o the_o line_n h_o that_o proportion_n that_o a_o square_a number_n have_v to_o a_o square_a number_n wherefore_o by_o the_o 9_o of_o the_o ten_o the_o line_n of_o be_v incommensurable_a in_o length_n unto_o the_o line_n h._n wherefore_o the_o line_n of_o be_v in_o power_n more_o they_o the_o line_n fg_o by_o the_o square_n of_o a_o line_n incommensurable_a in_o length_n unto_o it_o and_o the_o line_n of_o and_o fg_o be_v rational_a commensurable_a in_o power_n only_o and_o the_o line_n of_o be_v commensurable_a in_o length_n to_o the_o rational_a line_n d._n wherefore_o the_o line_n eglantine_n be_v a_o four_o binomial_a line_n which_o be_v require_v to_o be_v find_v out_o ¶_o the_o 17._o problem_n the_o 52._o proposition_n to_o find_v out_o a_o five_o binomial_a line_n take_v two_o number_n ac_fw-la and_o cb_o and_o let_v they_o be_v such_o that_o the_o number_n ab_fw-la have_v to_o neither_o of_o the_o number_n ac_fw-la or_o cb_o that_o proportion_n that_o a_o square_a number_n have_v to_o a_o square_a number_n construction_n as_o in_o the_o former_a proposition_n and_o take_v a_o rational_a line_n and_o let_v the_o same_o be_v d._n and_o unto_o the_o line_n d_o let_v the_o line_n fg_o be_v commensurable_a in_o length_n wherefore_o the_o line_n fg_o be_v rational_a and_o as_o the_o number_n ca_n be_v to_o the_o number_n ab_fw-la so_o let_v the_o square_n of_o the_o line_n gf_o be_v to_o the_o square_n of_o the_o line_n ef._n demonstration_n wherefore_o the_o square_a of_o the_o line_n gf_o be_v commensurable_a to_o the_o square_n of_o the_o line_n fe_o wherefore_o also_o the_o line_n fe_o be_v rational_a anâ—_z for_o that_o the_o number_n ca_n have_v not_o to_o the_o number_n ab_fw-la that_o proportion_n that_o a_o square_a number_n have_v to_o a_o square_a number_n therefore_o neither_o also_o have_v the_o square_a of_o the_o line_n gf_o to_o the_o square_n of_o the_o line_n fe_o that_o proportion_n that_o a_o square_a number_n have_v to_o a_o square_a number_n wherefore_o by_o the_o 9_o of_o the_o ten_o the_o line_n gf_o be_v incommensurable_a in_o length_n to_o the_o line_n fe_o wherefore_o the_o line_n of_o and_o fg_o be_v rational_a commensurable_a in_o power_n only_o wherefore_o the_o whole_a line_n eglantine_n be_v a_o binomial_a line_n i_o say_v moreover_o that_o it_o be_v a_o five_o binomial_a line_n for_o for_o that_o as_o the_o number_n ca_n be_v to_o the_o number_n ab_fw-la so_o be_v the_o square_a of_o the_o line_n gf_o to_o the_o square_n of_o the_o line_n fe_o therefore_o contrariwise_o as_o the_o number_n basilius_n be_v to_o the_o number_n ac_fw-la so_o be_v the_o square_a of_o the_o line_n of_o to_o the_o square_n of_o the_o line_n fg._n but_o the_o number_n basilius_n be_v great_a than_o the_o number_n ac_fw-la wherefore_o also_o the_o square_a of_o the_o line_n of_o be_v great_a than_o the_o square_a of_o the_o line_n fg._n unto_o the_o square_n of_o the_o line_n of_o let_v the_o square_n of_o the_o line_n fg_o and_o h_o be_v equal_a wherefore_o by_o conversion_n by_o the_o corollary_n of_o the_o 19_o of_o the_o five_o as_o the_o number_n ab_fw-la be_v to_o the_o number_n be_v so_o be_v the_o square_a of_o the_o line_n of_o to_o the_o square_n of_o the_o line_n h._n but_o the_o number_n ab_fw-la have_v not_o to_o the_o number_n bc_o that_o proportion_n that_o a_o square_a number_n have_v to_o a_o square_a number_n wherefore_o neither_o also_o have_v the_o square_a of_o the_o line_n of_o to_o the_o square_n of_o the_o line_n h_o that_o proportion_n that_o a_o square_a number_n have_v to_o a_o square_a number_n wherefore_o by_o the_o 9_o of_o the_o ten_o the_o line_n of_o be_v incommensurable_a in_o length_n to_o the_o line_n h._n wherefore_o the_o line_n of_o be_v in_o power_n more_o than_o the_o line_n fg_o by_o the_o square_n of_o a_o line_n incommensurable_a in_o length_n unto_o it_o and_o the_o line_n of_o and_o fg_o be_v rational_a commensurable_a in_o power_n only_o and_o the_o line_n fg_o be_v the_o less_o name_n be_v commensurable_a in_o length_n to_o the_o rational_a line_n give_v namely_o to_o d._n wherefore_o the_o whole_a line_n eglantine_n be_v a_o five_o binomial_a line_n which_o be_v require_v to_o be_v find_v out_o ¶_o the_o 18._o problem_n the_o 53._o proposition_n to_o find_v out_o a_o six_o binomial_a line_n take_v two_o number_n ac_fw-la &_o cb_o and_o let_v they_o be_v such_o that_o the_o number_n which_o be_v make_v of_o they_o both_o add_v together_o construction_n namely_o ab_fw-la have_v to_o neither_o of_o the_o number_n ac_fw-la or_o cb_o that_o proportion_n that_o a_o square_a number_n have_v to_o a_o square_a number_n take_v also_o any_o other_o number_n which_o be_v not_o a_o square_a number_n and_o let_v the_o same_o be_v d._n and_o let_v not_o the_o number_n d_o have_v to_o any_o one_o of_o these_o number_n ab_fw-la and_o ac_fw-la that_o proportion_n that_o a_o square_a number_n have_v to_o a_o square_a number_n let_v there_o be_v put_v moreover_o a_o rational_a line_n and_o let_v the_o same_o be_v e._n and_o as_o the_o number_n d_o be_v to_o the_o number_n ab_fw-la so_o let_v the_o square_n of_o the_o line_n e_o be_v to_o the_o square_n of_o fg._n wherefore_o by_o the_o 6._o of_o the_o ten_o the_o line_n e_o be_v commensurable_a in_o power_n to_o the_o line_n fg_o &_o the_o line_n e_o be_v rational_a wherefore_o also_o the_o line_n fg_o be_v rational_a and_o for_o that_o the_o number_n d_o have_v not_o â—o_o the_o number_n ab_fw-la that_o proportion_n that_o a_o square_a number_n
have_v to_o a_o square_a number_n therefore_o neither_o also_o shall_v the_o square_a of_o the_o line_n e_o have_v to_o the_o square_n of_o the_o line_n fg_o that_o proportion_n that_o a_o square_a number_n have_v to_o a_o square_a number_n wherefore_o the_o line_n fg_o be_v incommensurable_a in_o length_n to_o the_o line_n e._n again_o as_o the_o number_n basilius_n be_v to_o the_o number_n ac_fw-la so_o let_v the_o square_n of_o the_o line_n fg_o be_v to_o the_o square_n of_o the_o line_n gh_o demonstration_n wherefore_o by_o the_o 6._o of_o the_o ten_o the_o square_a of_o the_o line_n fg_o be_v commensurable_a to_o the_o square_n of_o the_o line_n gh_o and_o the_o square_a of_o the_o line_n fg_o be_v rational_a wherefore_o the_o square_a of_o the_o line_n gh_o be_v also_o rational_a wherefore_o also_o the_o line_n gh_o be_v rational_a and_o for_o that_o the_o number_n ab_fw-la have_v not_o to_o the_o number_n ac_fw-la that_o proportion_n that_o a_o square_a number_n have_v to_o a_o square_a number_n therefore_o neither_o also_o have_v the_o square_a of_o the_o line_n fg_o to_o the_o square_n of_o the_o line_n gh_o that_o proportion_n that_o a_o square_a number_n have_v to_o a_o square_a number_n wherefore_o the_o line_n fg_o be_v incommensurable_a in_o length_n to_o the_o line_n gh_o wherefore_o the_o line_n fg_o and_o gh_o be_v rational_a commensurable_a in_o power_n only_o wherefore_o the_o whole_a line_n fh_o be_v a_o binomial_a line_n i_o say_v moreover_o that_o it_o be_v a_o six_o binomial_a line_n for_o for_o that_o as_o the_o number_n d_o be_v to_o the_o number_n ab_fw-la so_o be_v the_o square_a of_o the_o line_n e_o to_o the_o square_n of_o the_o line_n fg._n and_o as_o the_o number_n basilius_n be_v to_o the_o number_n ac_fw-la so_o be_v the_o square_a of_o the_o line_n fg_o to_o the_o square_n of_o the_o line_n gh_o wherefore_o of_o equality_n by_o the_o 22._o of_o the_o five_o as_o the_o number_n d_z be_v to_o the_o number_n ac_fw-la so_o be_v the_o square_a of_o the_o line_n e_o to_o the_o square_n of_o the_o line_n gh_o but_o the_o number_n d_o have_v not_o to_o the_o number_n ac_fw-la that_o proportion_n that_o a_o square_a number_n have_v to_o a_o square_a number_n wherefore_o neither_o also_o have_v the_o square_a of_o the_o line_n e_o to_o the_o square_n of_o the_o line_n gh_o that_o proportion_n that_o a_o square_a number_n have_v to_o a_o square_a number_n wherefore_o the_o line_n e_o be_v incommensurable_a in_o length_n to_o the_o line_n gh_o and_o it_o be_v already_o prove_v that_o the_o line_n fg_o be_v also_o incommensurable_a in_o length_n to_o the_o line_n e._n wherefore_o either_o of_o these_o line_n fg_o and_o gh_o be_v incommensurable_a in_o length_n to_o the_o line_n e._n and_o for_o that_o as_o the_o number_n â—a_o be_v to_o the_o number_n ac_fw-la so_o be_v the_o square_a of_o the_o line_n fg_o to_o the_o square_n of_o the_o line_n gh_o therefore_o the_o square_n of_o the_o line_n fg_o be_v great_a than_o the_o square_a of_o the_o line_n gh_o unto_o the_o square_n of_o the_o line_n fg_o let_v the_o square_n of_o the_o line_n gh_o and_o king_n be_v equal_a wherefore_o by_o eversion_n of_o proportion_n as_o the_o number_n ab_fw-la be_v to_o the_o number_n bc_o so_o be_v the_o square_a of_o the_o line_n fg_o to_o the_o square_n of_o the_o line_n k._n but_o the_o number_n ab_fw-la have_v not_o to_o the_o number_n bc_o that_o proportion_n that_o a_o square_a number_n have_v to_o a_o square_a number_n wherefore_o neither_o also_o have_v the_o square_a of_o the_o line_n fg_o to_o the_o square_n of_o the_o line_n king_n that_o proportion_n that_o a_o square_a number_n have_v to_o a_o square_a number_n wherefore_o the_o line_n fg_o be_v incommensurable_a in_o length_n unto_o the_o line_n k._n wherefore_o the_o line_n fg_o be_v in_o power_n more_o than_o the_o line_n gh_o by_o the_o square_n of_o a_o line_n incommensurable_a in_o length_n to_o it_o and_o the_o line_n fg_o and_o gh_o be_v rational_a commensurable_a in_o power_n only_o and_o neither_o of_o the_o line_n fg_o &_o gh_o be_v commensurable_a in_o length_n to_o the_o rational_a line_n give_v namely_o to_o e._n wherefore_o the_o line_n fh_o be_v a_o six_o binomial_a line_n which_o be_v require_v to_o be_v find_v out_o ¶_o a_o corollary_n add_v out_o of_o flussates_n by_o the_o 6._o forme_a proposition_n it_o iâ—_z manifest_a hoâ—_n 〈◊〉_d divide_v any_o right_a line_n give_v into_o the_o name_n of_o every_o one_o of_o the_o sixâ—_o foresay_a binomiall_n line_n flussates_n for_o if_o it_o be_v require_v to_o divide_v a_o right_a line_n give_v into_o a_o first_o binomial_a line_n then_o by_o the_o 48â—_n of_o this_o book_n find_v out_o a_o first_o binomial_a line_n and_o this_o right_a line_n be_v so_o find_v out_o divide_v into_o his_o name_n you_o may_v by_o the_o 10._o of_o the_o six_o divide_v the_o right_a line_n give_v in_o like_a sort_n and_o so_o in_o the_o other_o five_o follow_v although_o i_o here_o note_n unto_o you_o this_o corollary_n out_o of_o 〈…〉_o in_o very_a conscience_n and_o of_o grateful_a â—indeâ—_n i_o be_o enforce_v to_o certify_v you_o that_o iâ—_z any_o year_v before_o the_o travail_n of_o flussas_n upon_o euâ—liâ—â—â—_n geometrical_a element_n be_v publish_v the_o order_n how_o to_o divide_v not_o only_o the_o 6._o binomial_a line_n into_o their_o name_n but_o also_o to_o add_v to_o the_o 6._o residâ—â—ls_v their_o due_a part_n â—nd_v fâ—rthermore_o to_o divide_v all_o the_o other_o irrational_a line_n of_o this_o ten_o book_n into_o the_o part_n distinct_a of_o which_o they_o be_v compose_v with_o many_o other_o strange_a conclusion_n mathematical_a to_o the_o better_a understanding_n of_o this_o ten_o book_n and_o other_o mathematical_a book_n most_o necessary_a be_v by_o m._n john_n dee_fw-mi invent_v and_o demonstrate_v mathematicum_fw-la as_o in_o his_o book_n who_o title_n be_v tyrocinium_fw-la mathematicum_fw-la dedicate_v to_o petruâ—_n nonnius_n an._n 1559._o may_v at_o large_a appear_v where_o also_o be_v one_o new_a art_n with_o sundry_a particular_a point_n whereby_o the_o mathematical_a science_n great_o may_v be_v enrich_v which_o his_o book_n i_o hope_v god_n will_v one_o day_n allow_v he_o opportunity_n to_o publish_v with_o diverse_a other_o his_o mathematical_a and_o metaphysical_a labour_n and_o invention_n ¶_o a_o assumpt_n be_v a_o right_a line_n be_v divide_v into_o two_o part_n how_o soever_o the_o rectangle_n parallelogram_n contain_v under_o both_o the_o part_n be_v the_o mean_a proportional_a between_o the_o square_n of_o the_o same_o part_n and_o the_o rectangle_n parallelogram_n contain_v under_o the_o whole_a line_n and_o one_o of_o the_o part_n be_v the_o mean_a proportional_a between_o the_o square_n of_o the_o whole_a line_n and_o the_o square_a of_o the_o say_a part_n book_n suppose_v that_o there_o be_v two_o square_n ab_fw-la and_o bc_o and_o let_v the_o line_n db_o and_o be_v so_o be_v put_v that_o they_o both_o make_v one_o right_a line_n wherefore_o by_o the_o 14._o of_o the_o first_o the_o line_n fb_o and_o bg_o make_v also_o both_o one_o right_a line_n and_o make_v perfect_a the_o parallelogram_n ac_fw-la then_o i_o say_v that_o the_o rectangle_n parallelogram_n dg_o be_v the_o mean_a proportional_a between_o the_o square_n ab_fw-la and_o bc_o and_o moreover_o that_o the_o parallelogram_n dc_o be_v the_o mean_a proportional_a between_o the_o square_n ac_fw-la and_o cb._n first_o the_o parallelogram_n agnostus_n be_v a_o square_a for_o forasmuch_o as_o the_o line_n db_o be_v equal_a to_o the_o line_n bf_o and_o the_o line_n be_v unto_o the_o line_n bg_o therefore_o the_o whole_a line_n de_fw-fr be_v equal_a to_o the_o whole_a line_n fg._n but_o the_o line_n de_fw-fr be_v equal_a to_o either_o of_o these_o line_n ah_o &_o kc_o and_o the_o line_n fg_o be_v equal_a to_o either_o of_o these_o line_n ak_o and_o hc_n by_o the_o 34._o of_o the_o first_o demonstration_n wherefore_o the_o parallelogram_n ac_fw-la be_v equilater_n it_o be_v also_o rectangle_n by_o the_o 29._o of_o the_o first_o wherefore_o by_o the_o 46._o of_o the_o first_o the_o parallelogram_n ac_fw-la be_v a_o square_a now_o for_o that_o as_o the_o line_n fb_o be_v to_o the_o line_n bg_o so_o be_v the_o line_n db_o
wherefore_o the_o line_n mn_a and_o nx_o be_v medial_a line_n commensurable_a in_o power_n only_o now_o i_o say_v moreover_o that_o they_o comprehend_v a_o rational_a superficies_n for_o forasmuch_o as_o by_o supposition_n the_o line_n de_fw-fr be_v commensurable_a in_o length_n to_o either_o of_o these_o line_n ab_fw-la and_o of_o therefore_o the_o line_n fe_o be_v commensurable_a in_o length_n to_o the_o line_n eke_o which_o be_v equal_a to_o the_o line_n ab_fw-la by_o thâ—_z 12._o of_o the_o ten_o and_o either_o of_o these_o line_n of_o and_o eke_o be_v a_o rational_a line_n wherefore_o the_o parallelogram_n el_fw-es that_o be_v the_o parallelogram_n mr._n be_v a_o rational_a superficies_n by_o the_o 19_o of_o the_o ten_o conclude_v but_o the_o parallelogram_n mr._n be_v that_o which_o be_v contain_v under_o the_o line_n mn_a and_o nx_o but_o if_o two_o medial_a line_n commensurable_a in_o power_n only_o and_o comprehend_v a_o rational_a superficies_n be_v add_v together_o the_o whole_a line_n be_v irrational_a and_o be_v call_v a_o first_o bimediall_n by_o the_o 37._o of_o the_o ten_o conclusion_n wherefore_o the_o line_n mx_o be_v a_o first_o bimedial_a line_n which_o be_v require_v to_o be_v demonstrate_v ¶_o the_o 38._o theorem_a the_o 56._o proposition_n if_o a_o superficies_n be_v contain_v under_o a_o rational_a line_n and_o a_o three_o binomial_a line_n the_o line_n that_o contain_v in_o power_n that_o superficies_n be_v irrational_a and_o be_v a_o second_o bimedial_a line_n svppose_v that_o the_o superficies_n abcd_o be_v comprehend_v under_o the_o rational_a line_n ab_fw-la and_o a_o three_o binomial_a line_n ad_fw-la and_o let_v the_o line_n ad_fw-la be_v suppose_v to_o be_v divide_v into_o his_o name_n in_o thâ—_z point_n e_o of_o which_o let_v ae_n be_v the_o great_a name_n then_o i_o say_v that_o the_o line_n that_o contain_v in_o power_n the_o superficies_n ac_fw-la be_v irrational_a and_o be_v a_o second_o bimedial_a line_n let_v the_o same_o construction_n of_o the_o figure_n be_v in_o this_o that_o be_v in_o the_o two_o proposition_n nexâ—_n go_v before_o and_o now_o forasmuch_o as_o the_o line_n ad_fw-la be_v a_o three_o binomial_a line_n therefore_o these_o line_n ae_n and_o ed_z be_v rational_a commensurable_a in_o power_n only_o demonstration_n and_o the_o line_n ae_n be_v in_o power_n more_o they_o the_o line_n ed_z by_o the_o square_n of_o a_o line_n commensurable_a in_o length_n to_o the_o line_n ae_n and_o neither_o of_o the_o line_n ae_n nor_o ed_z be_v commensurable_a in_o length_n to_o the_o line_n ab_fw-la by_o the_o definition_n of_o a_o three_o binomial_a line_n set_v before_o the_o 48._o proposition_n as_o in_o the_o former_a proposition_n it_o be_v demonstrate_v so_o also_o may_v it_o in_o this_o proposition_n be_v prove_v that_o the_o line_n mx_o contain_v in_o power_n the_o superficies_n ac_fw-la and_o that_o the_o line_n mn_a and_o nx_o be_v medial_a line_n commensurable_a in_o power_n only_o wherefore_o the_o line_n mx_o be_v a_o bimedial_a line_n now_o rest_v to_o prove_v that_o it_o be_v a_o second_o bimedial_a line_n forasmuch_o as_o the_o line_n de_fw-fr be_v by_o supposition_n incommensurable_a in_o length_n to_o the_o line_n ab_fw-la that_o be_v to_o the_o line_n eke_o but_o the_o line_n ed_z be_v commensurable_a in_o length_n to_o the_o line_n ef._n wherefore_o by_o the_o 13._o of_o the_o ten_o the_o line_n of_o be_v incommensurable_a in_o length_n to_o the_o line_n eke_o and_o the_o line_n fe_o and_o eke_o be_v rational_a for_o by_o supposition_n the_o line_n ed_z be_v rational_a unto_o which_o the_o line_n fe_o be_v commensurable_a wherefore_o the_o line_n fe_o and_o eke_o be_v rational_a line_n commensurable_a in_o power_n only_o wherefore_o by_o the_o 21._o of_o the_o ten_o the_o parallelogram_n el_fw-es that_o be_v the_o parallelogram_n mr._n which_o be_v contain_v under_o the_o line_n mn_a and_o nx_o be_v a_o medial_a superficies_n wherefore_o that_o which_o be_v contain_v under_o the_o line_n mn_a and_o nx_o be_v a_o medial_a superficies_n wherefore_o the_o line_n mx_o be_v a_o second_o bimedial_a line_n by_o the_o 38._o proposition_n and_o definition_n annex_v thereto_o which_o be_v require_v to_o be_v prove_v ¶_o the_o 39_o theorem_a the_o 57_o proposition_n if_o a_o superficies_n be_v contain_v under_o a_o rational_a line_n and_o a_o four_o binomial_a line_n the_o line_n which_o contain_v in_o power_n that_o superficies_n be_v irrational_a and_o be_v a_o great_a line_n svppose_v that_o the_o superficies_n ac_fw-la be_v comprehend_v under_o a_o rational_a line_n ab_fw-la and_o a_o four_o binomial_a line_n ad_fw-la &_o let_v the_o binomial_a line_n ad_fw-la be_v suppose_v to_o be_v divide_v into_o his_o name_n in_o the_o point_n e_o so_fw-mi that_o let_v the_o line_n ae_n be_v the_o great_a name_n then_o i_o say_v that_o the_o line_n which_o contain_v in_o power_n the_o superficies_n ac_fw-la be_v irrational_a and_o be_v a_o great_a line_n construction_n for_o forasmuch_o as_o the_o line_n ad_fw-la be_v a_o four_o binomial_a line_n therefore_o the_o line_n ae_n and_o ed_z be_v rational_a commensurable_a in_o power_n only_o and_o the_o line_n ae_n be_v in_o power_n more_o than_o the_o line_n ed_z by_o the_o square_n of_o a_o line_n incommensurable_a in_o length_n to_o ae_n and_o the_o line_n ae_n be_v commensurable_a in_o length_n to_o the_o line_n ab_fw-la divide_v by_o the_o 10._o of_o the_o first_o the_o line_n de_fw-fr into_o two_o equal_a part_n in_o the_o point_n f._n and_o upon_o the_o line_n ae_n apply_v a_o parallelogram_n equal_a to_o the_o square_n of_o of_o and_o want_v in_o figure_n by_o a_o square_n and_o let_v the_o same_o parallelogram_n be_v that_o which_o be_v contain_v under_o the_o line_n agnostus_n &_o ge._n wherefore_o by_o the_o second_o part_n of_o the_o 18._o of_o the_o ten_o the_o line_n agnostus_n be_v incommensurable_a in_o length_n to_o the_o line_n eglantine_n draw_v unto_o the_o line_n ab_fw-la by_o the_o point_n g_o e_o f_o parallel_a line_n gh_o eke_o and_o pl_z and_o let_v the_o rest_n of_o the_o construction_n be_v as_o it_o be_v in_o the_o three_o former_a proposition_n demonstration_n now_o it_o be_v manifest_a that_o the_o line_n mx_o contain_v in_o power_n the_o superficies_n ac_fw-la now_o rest_v to_o prove_v that_o the_o line_n mx_o be_v a_o irrational_a line_n and_o a_o great_a line_n forasmuch_o as_o the_o line_n agnostus_n be_v incommensurable_a in_o length_n to_o the_o line_n eglantine_n therefore_o by_o the_o 1._o of_o the_o six_o and_o 11._o of_o the_o ten_o the_o parallelogram_n ah_o be_v incommensurable_a to_o the_o parallelogram_n gk_o that_o be_v the_o square_a sn_a to_o the_o square_a np._n wherefore_o the_o line_n mn_a and_o nk_n be_v incommensurable_a in_o power_n and_o forasmuch_o 〈◊〉_d the_o line_n ae_n be_v commensurable_a in_o length_n to_o the_o rational_a line_n aâ—_z therefore_o the_o parallelogram_n ak_o be_v rational_a and_o it_o be_v equal_a to_o the_o square_n of_o the_o line_n mn_a and_o nx_o wherefore_o that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n mn_a and_o nx_o add_v together_o be_v rational_a and_o forasmuch_o as_o the_o line_n ed_z be_v incommensurable_a in_o length_n to_o the_o line_n ab_fw-la that_o be_v to_o the_o line_n eke_o but_o the_o line_n ed_z be_v commensurable_a in_o length_n to_o the_o line_n of_o therefore_o the_o line_n of_o be_v incommensurable_a in_o length_n to_o the_o line_n eke_o wherefore_o the_o line_n eke_o and_o of_o be_v rational_a commensurable_a in_o power_n only_o wherefore_o by_o the_o 21._o of_o the_o ten_o the_o parallelogram_n le_fw-fr that_o be_v the_o parallelogram_n mr._n be_v medial_a and_o the_o parallelogram_n mr._n be_v that_o which_o be_v contain_v under_o the_o line_n mn_v and_o nx_o wherefore_o that_o which_o be_v contain_v under_o the_o line_n mn_a and_o nx_o be_v medial_a and_o that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n mn_a &_o nx_o be_v prove_v to_o be_v rational_a &_o the_o line_n mn_o be_v demonstrate_v to_o be_v incommensurable_a in_o power_n to_o the_o line_n nx_o but_o if_o two_o line_n incommensurable_a in_o power_n be_v add_v together_o have_v that_o which_o be_v make_v of_o the_o square_n of_o they_o add_v together_o rational_a &_o that_o which_o be_v under_o they_o medial_a the_o whole_a line_n be_v irrational_a and_o be_v call_v a_o great_a line_n by_o the_o 39_o of_o the_o ten_o wherefore_o the_o line_n mx_o be_v irrational_a and_o be_v a_o great_a line_n and_o it_o contain_v in_o power_n the_o superficies_n ac_fw-la which_o be_v require_v to_o be_v demonstrate_v ¶_o the_o 40._o theorem_a the_o 58._o proposition_n if_o a_o superficies_n be_v contain_v under_o a_o rational_a line_n and_o a_o five_o binomial_a line_n the_o line_n which_o contain_v in_o power_n that_o superficies_n be_v
irrational_a and_o be_v a_o line_n contain_v in_o power_n a_o rational_a and_o a_o medial_a superficies_n svppose_v that_o the_o superficies_n agnostus_n be_v contain_v under_o the_o rational_a line_n aâ—_z and_o under_o a_o five_o binomial_a line_n addend_n let_v the_o same_o linâ—_n ad_fw-la be_v suppose_v to_o be_v divide_v into_o his_o name_n in_o the_o point_n e_o so_fw-mi that_o let_v the_o line_n ae_n be_v the_o great_a name_n then_o i_o say_v that_o the_o line_n which_o contain_v in_o power_n the_o superficies_n ac_fw-la be_v irrational_a and_o be_v a_o line_n contain_v in_o power_n a_o rational_a and_o a_o medial_a superficies_n demonstration_n let_v the_o self_n same_o construction_n be_v in_o this_o that_o be_v in_o the_o four_o proposition_n next_o go_v before_o and_o it_o be_v manifest_a that_o the_o line_n mx_o contain_v in_o power_n the_o superficiâ—â—_n ag._n now_o test_v to_o prove_v that_o the_o line_n mx_o be_v a_o line_n contain_v in_o power_n a_o rational_a &_o a_o medial_a superficies_n forasmuch_o as_o the_o line_n agnostus_n be_v incommensurable_a in_o length_n to_o the_o line_n ge_z therefore_o by_o the_o 1._o of_o the_o six_o and_o 10._o of_o the_o ten_o the_o parallelogram_n ah_o be_v incommensurable_a to_o the_o parallelogram_n he_o that_o be_v the_o square_a of_o the_o line_n mn_v to_o the_o square_n of_o the_o line_n nx_o wherefore_o the_o line_n mn_a and_o nx_o be_v incommensurable_a in_o power_n and_o forasmuch_o as_o the_o line_n ad_fw-la iâ—_z a_o fifâ—_n binomial_a line_n and_o his_o less_o name_n or_o part_n be_v the_o line_n ed_z therefore_o the_o line_n ed_z be_v commensurable_a in_o length_n to_o the_o line_n ab_fw-la but_o the_o line_n ae_n be_v incommensurable_a in_o length_n to_o the_o line_n ed._n wherefore_o by_o the_o 13._o of_o the_o ten_o the_o line_n ab_fw-la be_v incommensurable_a in_o length_n to_o the_o line_n ae_n wherefore_o the_o line_n ab_fw-la and_o ae_n be_v rational_a commensurable_a in_o power_n only_o wherefore_o by_o the_o 21._o of_o the_o ten_o the_o parallelogram_n ak_o be_v medial_a that_o be_v that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n mn_a &_o nx_o add_v together_o and_o forasmuch_o as_o the_o line_n de_fw-fr be_v commensurable_a in_o length_n to_o the_o line_n ab_fw-la that_o be_v to_o the_o line_n eke_o but_o the_o line_n de_fw-fr be_v commensurable_a in_o length_n to_o the_o line_n of_o wherefore_o by_o the_o 12._o of_o the_o ten_o the_o line_n of_o be_v also_o commensurable_a in_o length_n to_o the_o line_n eke_o and_o the_o line_n eke_o be_v rational_a wherefore_o by_o the_o 19_o of_o the_o ten_o the_o parallelogram_n el_fw-es that_o be_v the_o parallelogram_n mr._n which_o be_v contain_v under_o the_o line_n mn_a and_o nx_o be_v rational_a wherefore_o the_o line_n mn_a and_o nx_o be_v incommensurable_a in_o power_n have_v that_o which_o be_v compose_v of_o the_o square_n of_o they_o add_v together_o medial_a and_o that_o which_o be_v contain_v under_o they_o rational_a wherefore_o by_o the_o 40._o of_o the_o ten_o the_o whole_a line_n mx_o be_v a_o line_n contain_v in_o power_n a_o rational_a and_o a_o medial_a superficies_n and_o it_o contain_v in_o power_n the_o superficies_n ac_fw-la which_o be_v require_v to_o be_v prove_v ¶_o the_o 41._o theorem_a the_o 59_o proposition_n if_o a_o superficies_n be_v contain_v under_o a_o rational_a line_n and_o a_o six_o binomial_a line_n the_o line_n which_o contain_v in_o power_n that_o superficies_n be_v irrational_a &_o be_v call_v a_o line_n contain_v in_o power_n two_o medial_n svppose_v that_o the_o superficies_n abcd_o be_v contain_v under_o the_o rational_a line_n ab_fw-la and_o under_o a_o six_o binomial_a line_n ad_fw-la and_o let_v the_o line_n ad_fw-la be_v suppose_v to_o be_v divide_v â—â—to_o his_o name_n in_o the_o point_n e_o so_fw-mi that_o let_v the_o line_n ae_n be_v the_o great_a name_n then_o i_o say_v that_o the_o line_n that_o contain_v in_o power_n the_o superficies_n ac_fw-la be_v irrational_a and_o be_v a_o line_n contain_v in_o power_n two_o medial_n demonstration_n let_v the_o self_n same_o construction_n be_v in_o this_o that_o be_v in_o the_o former_a proposition_n now_o it_o be_v manifest_a that_o the_o line_n mx_o contain_v in_o power_n the_o superficies_n ac_fw-la and_o that_o the_o line_n mn_o be_v incommensurable_a in_o power_n to_o the_o line_n nx_o and_o forasmuch_o as_o the_o line_n ae_n be_v incommensurable_a in_o length_n to_o the_o line_n ab_fw-la therefore_o the_o line_n ae_n and_o ab_fw-la be_v rational_a commensurable_a in_o power_n only_o wherefore_o by_o the_o 〈◊〉_d of_o the_o ten_o the_o parallelogrammâ—_n ak_o that_o be_v that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n mn_a and_o nx_o add_v together_o be_v medial_a again_o forasmuch_o as_o the_o line_n ed_z be_v incommensurable_a in_o length_n to_o the_o line_n ab_fw-la therefore_o also_o the_o line_n of_o be_v incommensurable_a in_o length_n to_o the_o line_n eke_o wherefore_o the_o line_n of_o and_o eke_o be_v rational_a commensurable_a in_o power_n only_o wherefore_o the_o parallelogram_n el_fw-es that_o be_v the_o parallelogram_n mr._n which_o be_v contain_v under_o the_o line_n mn_a and_o nx_o be_v medial_a and_o forasmuch_o as_o the_o line_n ae_n be_v incommensurable_a in_o length_n to_o the_o line_n of_o therefore_o the_o parallelogram_n ak_o be_v also_o incommensurable_a to_o the_o parallelogram_n el_fw-es by_o the_o first_o of_o the_o six_o and_o 100l_n of_o the_o ten_o but_o the_o parallelogram_n ak_o be_v equal_a to_o that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n mn_a and_o nx_o add_v together_o and_o the_o parallelogram_n el_n be_v equal_a to_o that_o which_o be_v contain_v under_o the_o line_n mn_a and_o nx_o wherefore_o that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n mn_a and_o nigh_o add_v together_o be_v incommensurable_a to_o that_o which_o be_v contain_v under_o the_o lâ—nes_n mn_n and_o nxâ—_n and_o eâ—â—her_n of_o they_o name_o that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n mn_a and_o nx_o add_v together_o and_o that_o which_o be_v contain_v uâ—der_v the_o line_n mn_a and_o nâ—_z be_v prove_v medial_a and_o the_o line_n mn_a and_o nx_o be_v prove_v incommensurable_a in_o power_n wherefore_o by_o the_o 41._o of_o the_o ten_o the_o whole_a line_n mx_o be_v a_o line_n contain_v in_o power_n two_o medial_n and_o it_o contain_v in_o power_n the_o superfice_n ac_fw-la which_o be_v require_v to_o be_v demonstrate_v a_o assumpt_n if_o a_o right_a line_n be_v divide_v into_o two_o unequal_a part_n the_o square_n which_o be_v make_v of_o the_o unequal_a part_n be_v great_a than_o the_o rectangle_n parallelogram_n contain_v under_o the_o unequal_a part_n twice_o suppose_v that_o ab_fw-la be_v a_o right_a line_n and_o let_v it_o be_v 〈…〉_o point_fw-fr c._n and_o let_v the_o line_n ac_fw-la be_v the_o great_a part_n 〈…〉_o and_o â—b_n be_v great_a they_o that_o which_o be_v contain_v under_o the_o line_n aâ—_z and_o cb_o twice_o dâ—â—idâ—_n by_o the_o 10._o of_o the_o first_o the_o line_n ab_fw-la into_o two_o equal_a part_n in_o the_o point_n d._n now_o forasmuch_o as_o the_o right_a line_n ab_fw-la be_v divide_v into_o two_o equal_a parteâ—_n in_o the_o point_n d_o and_o into_o two_o unequal_a parteâ—_n in_o the_o point_n c_o therefore_o by_o the_o 5._o of_o the_o second_o that_o which_o be_v contain_v under_o the_o line_n 〈…〉_o line_n cd_o be_v equal_a to_o the_o square_n of_o the_o line_n ad._n 〈…〉_o the_o line_n ac_fw-la and_o cb_o omit_v the_o square_a of_o the_o line_n cd_o be_v less_o than_o the_o square_a of_o the_o ad_fw-la by_o the_o 9_o common_a sentence_n and_o the_o seven_o of_o the_o five_o wherefore_o that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o twice_o be_v less_o than_o the_o double_a of_o the_o square_n of_o the_o line_n ad_fw-la that_o be_v them_z twice_o the_o square_a of_o the_o line_n ad_fw-la by_o place_n alternate_a proportion_n and_o the_o 14._o of_o the_o five_o but_o the_o square_n of_o the_o line_n ac_fw-la and_o cb_o be_v double_a to_o the_o square_n of_o the_o line_n ad_fw-la and_o dc_o by_o the_o 9_o of_o the_o second_o therefore_o the_o square_n of_o ac_fw-la and_o cb_o be_v more_o than_o double_a to_o the_o square_n of_o ad_fw-la alone_o leave_v out_o the_o square_n of_o dc_o by_o the_o 8._o of_o the_o five_o but_o the_o parallelogram_n contain_v under_o the_o line_n ac_fw-la and_o â—b_v twice_o be_v prove_v less_o they_o the_o double_a of_o the_o square_n of_o the_o line_n ad._n therefore_o the_o same_o parallelogram_n contain_v under_o the_o line_n ac_fw-la and_o cb_o twice_o be_v much_o less_o than_o the_o square_n of_o the_o line_n ac_fw-la and_o cb._n if_o a_o right_a line_n
the_o definition_n of_o a_o first_o binomial_a line_n seâ—_z before_o the_o 48._o proposition_n of_o this_o book_n the_o line_n dg_o be_v a_o first_o binomial_a line_n which_o be_v require_v to_o be_v prove_v this_o proposition_n and_o the_o five_o follow_v be_v the_o converse_n of_o the_o six_o former_a proposition_n ¶_o the_o 43._o theorem_a the_o 61._o proposition_n the_o square_a of_o a_o first_o bimedial_a line_n apply_v to_o a_o rational_a line_n make_v the_o breadth_n or_o other_o side_n a_o second_o binomial_a line_n svppose_v that_o the_o line_n ab_fw-la be_v a_o first_o bimedial_a line_n and_o let_v it_o be_v suppose_v to_o be_v divide_v into_o his_o part_n in_o the_o point_n c_o of_o which_o let_v ac_fw-la be_v the_o great_a part_n take_v also_o a_o rational_a line_n de_fw-fr and_o by_o the_o 44._o of_o the_o first_o apply_v to_o the_o line_n de_fw-fr the_o parallelogram_n df_o equal_a to_o the_o square_n of_o the_o line_n ab_fw-la &_o make_v in_o breadth_n the_o line_n dg_o construction_n then_o i_o say_v that_o the_o line_n dg_o be_v a_o second_o binomial_a line_n let_v the_o same_o construction_n be_v in_o this_o that_o be_v in_o the_o proposition_n go_v before_o and_o forasmuch_o as_o the_o line_n ab_fw-la be_v a_o first_o bimedial_a line_n demonstration_n and_o be_v divide_v into_o his_o part_n in_o the_o point_n c_o therefore_o by_o the_o 37._o of_o the_o ten_o the_o line_n ac_fw-la and_o cb_o be_v medial_a commensurable_a in_o power_n only_o comprehend_v a_o rational_a superficies_n wherefore_o also_o the_o square_n of_o the_o line_n ac_fw-la and_o cb_o be_v medial_a wherefore_o the_o parallelogram_n dl_o be_v medial_a by_o the_o corollary_n of_o the_o 23._o of_o the_o ten_o and_o it_o be_v apply_v upon_o the_o rational_a line_n de._n wherefore_o by_o the_o 22._o of_o the_o ten_o the_o line_n md_o be_v rational_a and_o incommensurable_a in_o length_n to_o the_o line_n de._n again_o forasmuch_o as_o that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o twice_o be_v rational_a therefore_o also_o the_o parallelogram_n mf_n be_v rational_a and_o it_o be_v apply_v unto_o the_o rational_a line_n ml_o wherefore_o the_o line_n mg_o be_v rational_a and_o commensurable_a in_o length_n to_o the_o line_n ml_o that_o be_v to_o the_o line_n de_fw-fr by_o the_o 20._o of_o the_o ten_o wherefore_o the_o line_n dm_o be_v incommensurable_a in_o length_n to_o the_o line_n mg_o and_o they_o be_v both_o rational_a wherefore_o the_o line_n dm_o and_o mg_o be_v rational_a commensurable_a in_o power_n only_o wherefore_o the_o whole_a line_n dg_o be_v a_o binomial_a line_n line_n now_o rest_v to_o prove_v that_o it_o be_v a_o second_o binomial_a line_n forasmuch_o as_o the_o square_n of_o the_o line_n ac_fw-la and_o cb_o be_v great_a than_o that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o twice_o by_o the_o assumpt_n before_o the_o 60._o of_o this_o book_n therefore_o the_o parallelogram_n dl_o be_v great_a than_o the_o parallelogrrmme_a mf_n wherefore_o also_o by_o the_o first_o of_o the_o six_o the_o line_n dm_o be_v great_a than_o the_o line_n mg_o and_o forasmuch_o as_o the_o square_n of_o the_o line_n ac_fw-la be_v commensurable_a to_o the_o square_n of_o the_o line_n cb_o therefore_o the_o parallelogram_n dh_o be_v commensurable_a to_o the_o parallelogram_n kl_o wherefore_o also_o the_o line_n dk_o be_v commensurable_a in_o length_n to_o the_o line_n km_o and_o that_o which_o be_v contain_v under_o the_o line_n dk_o and_o km_o be_v equal_a to_o the_o square_n of_o the_o line_n mn_o that_o be_v to_o the_o four_o part_n of_o the_o square_n of_o the_o line_n mg_o wherefore_o by_o the_o 17._o of_o the_o ten_o the_o line_n dm_o be_v in_o power_n more_o than_o the_o line_n mg_o by_o the_o square_n of_o a_o line_n commensurable_a in_o length_n unto_o the_o line_n dm_o and_o the_o line_n mg_o be_v commensurable_a in_o length_n to_o the_o rational_a line_n put_v namely_o to_o de._n wherefore_o the_o line_n dg_o be_v a_o second_o binomial_a line_n which_o be_v require_v to_o be_v prove_v ¶_o the_o 44._o theorem_a the_o 62._o proposition_n the_o square_a of_o a_o second_o bimedial_a line_n apply_v unto_o a_o rational_a line_n make_v the_o breadth_n or_o other_o side_n thereof_o a_o three_o binomial_a line_n svppose_v that_o ab_fw-la be_v a_o second_o bimedial_a line_n and_o let_v ab_fw-la be_v suppose_v to_o be_v divide_v into_o his_o part_n in_o the_o point_n c_o so_fw-mi that_o let_v ac_fw-la be_v the_o great_a part_n and_o take_v a_o rational_a line_n de._n and_o by_o the_o 44._o of_o the_o first_o unto_o the_o line_n de_fw-fr apply_v the_o parallelogram_n df_o equal_a to_o the_o square_n of_o the_o line_n ab_fw-la and_o make_v in_o breadth_n the_o line_n dg_o then_o i_o say_v that_o the_o line_n dg_o be_v a_o three_o binomial_a line_n let_v the_o self_n same_o construction_n be_v in_o this_o that_o be_v in_o the_o proposition_n next_o go_v before_o and_o forasmuch_o as_o the_o line_n ab_fw-la be_v a_o second_o bimedial_a line_n construction_n and_o be_v divide_v into_o his_o part_n in_o the_o point_n c_o demonstration_n therefore_o by_o the_o 38._o of_o the_o ten_o the_o line_n ac_fw-la and_o cb_o be_v medial_n commensurable_a in_o power_n only_o comprehend_v a_o medial_a superficies_n wherefore_o †_o that_o which_o be_v make_v of_o the_o square_n of_o the_o line_n ac_fw-la and_o cb_o add_v together_o be_v medial_a and_o it_o be_v equal_a to_o the_o parallelogram_n dl_o by_o construction_n wherefore_o the_o parallelogram_n dl_o be_v medial_a and_o be_v apply_v unto_o the_o rational_a line_n de_fw-fr wherefore_o by_o the_o 22._o of_o the_o ten_o the_o line_n md_o be_v rational_a and_o incommensurable_a in_o length_n to_o the_o line_n de._n and_o by_o the_o like_a reason_n also_o *_o the_o line_n mg_o be_v rational_a and_o incommensurable_a in_o length_n to_o the_o line_n ml_o that_o be_v to_o the_o line_n de._n wherefore_o either_o of_o these_o line_n dm_o and_o mg_o be_v rational_a and_o incommensurable_a in_o length_n to_o the_o line_n de._n and_o forasmuch_o as_o the_o line_n ac_fw-la be_v incommensurable_a in_o length_n to_o the_o line_n cb_o but_o as_o the_o line_n ac_fw-la be_v to_o the_o line_n cb_o so_o by_o the_o assumpt_n go_v before_o the_o 22._o of_o the_o ten_o be_v the_o square_a of_o the_o line_n ac_fw-la to_o that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb._n wherefore_o the_o square_a of_o the_o line_n ac_fw-la be_v incâ—mmmensurable_a to_o that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb._n wherefore_o that_o ‡_o that_o which_o be_v make_v of_o the_o square_n of_o the_o line_n ac_fw-la and_o cb_o add_v together_o be_v incommensurable_a to_o that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o twice_o that_o be_v the_o parallelogram_n dl_o to_o the_o parallelogram_n mf_n wherefore_o by_o the_o first_o of_o the_o six_o and_o 10._o of_o the_o ten_o the_o line_n dm_o be_v incommensurable_a in_o length_n to_o the_o line_n mg_o and_o they_o be_v prove_v both_o rational_a line_n wherefore_o the_o whole_a line_n dg_o be_v a_o binomial_a line_n by_o the_o definition_n in_o the_o 36._o of_o the_o ten_o now_o rest_v to_o prove_v that_o it_o be_v a_o three_o binomial_a line_n as_o in_o the_o former_a proposition_n so_o also_o in_o this_o may_v we_o conclude_v that_o the_o line_n dm_o be_v great_a than_o the_o line_n mg_o and_o that_o the_o line_n dk_o be_v commensurable_a in_o length_n to_o the_o line_n km_o and_o that_o that_o which_o be_v contain_v under_o the_o line_n dk_o and_o km_o be_v equal_a to_o the_o square_n of_o the_o line_n mn_v wherefore_o the_o line_n dm_o be_v in_o power_n more_o than_o the_o line_n mg_o by_o the_o square_n of_o a_o line_n commensurable_a in_o length_n unto_o the_o line_n dm_o and_o neither_o of_o the_o line_n dm_o nor_o mg_o be_v commensurable_a in_o length_n to_o the_o rational_a line_n de._n wherefore_o by_o the_o definition_n of_o a_o three_o binomiâ—ll_v line_v the_o line_n dg_o be_v a_o three_o binomial_a line_n which_o be_v require_v to_o be_v prove_v ¶_o here_o follow_v certain_a annotation_n by_o m._n dee_n make_v upon_o three_o place_n in_o the_o demonstration_n which_o be_v not_o very_o evident_a to_o young_a beginner_n †_o the_o square_n of_o the_o line_n ac_fw-la and_o câ—_n be_v medial_n 〈◊〉_d iâ—_z teach_v after_o the_o 21â—_n of_o this_o ten_o and_o therefore_o forasmuch_o as_o they_o be_v by_o supposition_n commensurable_a the_o one_o to_o the_o other_o by_o the_o 15._o of_o the_o ten_o the_o compound_n of_o they_o both_o be_v commensurable_a to_o each_o part_n but_o the_o part_n be_v medial_n therefore_o by_o the_o corollary_n of_o the_o 23._o of_o the_o ten_o the_o compound_n shall_v be_v
medial_a â—_o for_o that_o mx_o be_v equal_a by_o construction_n to_o that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o which_o be_v prove_v medial_a therefore_o by_o the_o corollary_n of_o the_o 23._o of_o this_o ten_o mx_o be_v medial_a and_o therefore_o by_o the_o same_o corollary_n his_o double_a mf_n be_v medial_a and_o it_o be_v apply_v to_o a_o rational_a line_n ml_o be_v equal_a to_o dâ—_n therefore_o by_o the_o 22._o of_o the_o ten_o the_o line_n mg_o be_v rational_a and_o incommensurable_a in_o length_n to_o ml_o that_o be_v to_o de._n ‡_o because_o the_o compound_n of_o the_o two_o square_n of_o the_o line_n ac_fw-la and_o câ—_n be_v commensurable_a one_o to_o the_o other_o be_v also_o to_o either_o square_n by_o the_o 15._o commensurable_a therefore_o to_o the_o square_n of_o ac_fw-la but_o the_o square_n of_o ac_fw-la be_v prove_v incommensurable_a to_o that_o which_o be_v contain_v under_o ac_fw-la &_o cb_o once_o wherefore_o by_o the_o 13._o of_o the_o ten_o the_o compound_n of_o the_o two_o square_n of_o the_o line_n ac_fw-la and_o cb_o be_v incommensurable_a to_o that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o câ—_n once_o but_o to_o that_o which_o be_v twice_o contain_v under_o the_o same_o line_n ac_fw-la and_o cb_o the_o paralleloganrme_n once_o contain_v be_v commensurable_a for_o it_o be_v as_o 1._o be_v to_o 2._o therefore_o that_o which_o be_v make_v of_o the_o square_n of_o the_o line_n ac_fw-la and_o cb_o be_v incommensurable_a to_o the_o parallelogram_n contain_v under_o ac_fw-la and_o cb_o twice_o by_o the_o say_v 13._o of_o this_o ten_o ¶_o a_o corollary_n dee_n hereby_o it_o be_v evident_a that_o the_o square_n make_v of_o the_o two_o part_n of_o a_o second_o bimedial_a line_n compose_v be_v a_o compound_n medial_a and_o that_o the_o same_o compound_n be_v incommensurable_a to_o the_o parallelogram_n contain_v under_o the_o two_o part_n of_o the_o second_o bimedial_a line_n the_o proof_n hereof_o be_v in_o the_o first_o and_o three_o annotation_n here_o before_o annex_v ¶_o the_o 45._o theorem_a the_o 63._o proposition_n the_o square_a of_o a_o great_a line_n apply_v unto_o a_o rational_a line_n make_v the_o breadth_n or_o other_o side_n a_o four_o binomial_a line_n svppose_v that_o the_o line_n ab_fw-la be_v a_o great_a line_n and_o let_v it_o be_v suppose_v to_o be_v divide_v into_o his_o part_n in_o the_o point_n c_o so_fw-mi that_o let_v ac_fw-la be_v the_o great_a part_n and_o take_v a_o rational_a line_n de._n and_o by_o the_o 44._o of_o the_o first_o unto_o the_o line_n de_fw-fr apply_v the_o parallelogram_n df_o equal_a to_o the_o square_n of_o the_o line_n ab_fw-la and_o make_v in_o breadth_n the_o line_n dg_o then_o i_o say_v that_o the_o line_n dg_o be_v a_o four_o binomial_a line_n let_v the_o self_n same_o construction_n be_v in_o this_o that_o be_v in_o the_o former_a proposition_n construction_n and_o forasmuch_o as_o the_o line_n ab_fw-la be_v a_o great_a line_n &_o be_v divide_v into_o his_o part_n in_o the_o point_n c_o demonstration_n therefore_o the_o line_n ac_fw-la and_o cb_o be_v incommensurable_a in_o power_n have_v that_o which_o be_v make_v of_o the_o square_n of_o they_o add_v together_o rational_a and_o the_o parallelogram_n which_o be_v contain_v under_o they_o medial_a now_o forasmuch_o as_o that_o which_o be_v make_v of_o the_o square_n of_o the_o line_n ac_fw-la and_o cb_o add_v together_o be_v rational_a therefore_o the_o parallelogram_n dl_o be_v rational_a wherefore_o also_o the_o line_n md_o be_v rational_a and_o commensurable_a in_o length_n to_o the_o line_n de_fw-fr by_o the_o 20._o of_o this_o ten_o again_o forasmuch_o as_o that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o twice_o be_v medial_a that_o be_v the_o parallelogram_n mf_n and_o it_o be_v apply_v unto_o the_o rational_a line_n ml_o therefore_o by_o the_o â—2_o of_o the_o ten_o the_o line_n mg_o be_v rational_a and_o incommensurable_a in_o length_n to_o the_o line_n de_fw-fr therefore_o by_o the_o 13._o of_o the_o ten_o the_o line_n dm_o be_v incommensurable_a in_o length_n to_o the_o line_n mg_o wherefore_o the_o line_n dm_o and_o mg_o be_v rational_a commensurable_a in_o power_n only_o wherefore_o the_o whole_a line_n dg_o be_v a_o binomial_a line_n now_o rest_v to_o prove_v that_o it_o be_v also_o a_o four_o binomial_a line_n even_o as_o in_o the_o former_a proposition_n so_o also_o in_o this_o may_v we_o conclude_v that_o the_o line_n dm_o be_v great_a than_o the_o line_n mc_n and_o that_o that_o which_o be_v contain_v under_o the_o line_n dk_o and_o km_o be_v equal_a to_o the_o square_n of_o the_o line_n mn_v now_o forasmuch_o as_o the_o square_n of_o the_o line_n ac_fw-la be_v incommensurable_a to_o the_o square_n of_o the_o line_n cb_o therefore_o the_o parallelogram_n dh_o be_v incommensurable_a to_o the_o parallelogram_n kl_o wherefore_o by_o the_o 1._o of_o the_o six_o and_o 10._o of_o the_o ten_o the_o line_n dk_o be_v incommensurable_a in_o length_n to_o the_o line_n km_o but_o if_o there_o be_v two_o unequal_a right_a line_n and_o if_o upon_o the_o great_a be_v apply_v a_o parallelogram_n equal_a to_o the_o four_o part_n of_o the_o square_n make_v of_o the_o less_o and_o want_v in_o figure_n by_o a_o square_a and_o if_o also_o the_o parallelogram_n thus_o apply_v divide_v the_o line_n whereupon_o it_o be_v apply_v into_o part_n incommensurable_a in_o length_n the_o great_a line_n sâ—all_v be_v in_o power_n more_o than_o the_o less_o by_o the_o square_n of_o a_o line_n incommensurable_a in_o length_n to_o the_o great_a by_o the_o 18._o of_o the_o ten_o wherefore_o the_o line_n dm_o be_v in_o power_n more_o than_o the_o line_n mg_o by_o the_o square_n of_o a_o line_n incommensurable_a in_o length_n to_o dm_o and_o the_o line_n dm_o and_o mg_o be_v prove_v to_o be_v rational_a commensurable_a in_o power_n only_o and_o the_o line_n dm_o be_v commensurable_a in_o length_n to_o the_o rational_a line_n give_v de._n wherefore_o the_o line_n dg_o be_v a_o four_o binomial_a line_n which_o be_v require_v to_o be_v prove_v ¶_o the_o 46._o theorem_a the_o 64._o proposition_n the_o square_a of_o a_o line_n contain_v in_o power_n a_o rational_a and_o a_o medial_a superficies_n apply_v to_o a_o rational_a line_n make_v the_o breadth_n or_o other_o side_n a_o five_o binomial_a line_n svppose_v that_o the_o line_n ab_fw-la be_v a_o line_n contain_v in_o power_n a_o rational_a and_o a_o medial_a superficies_n and_o let_v it_o be_v suppose_v to_o be_v divide_v into_o his_o part_n in_o the_o point_n c_o so_fw-mi that_o let_v ac_fw-la be_v the_o great_a part_n and_o take_v a_o rational_a line_n de._n and_o by_o the_o 44._o of_o the_o first_o unto_o the_o line_n de_fw-fr apply_v the_o parallelogram_n df_o equal_a to_o the_o square_n of_o the_o line_n ab_fw-la and_o make_v in_o breadth_n the_o line_n dg_o then_o i_o say_v that_o the_o line_n dg_o be_v a_o five_o binomial_a line_n construction_n let_v the_o self_n same_o construction_n be_v in_o this_o that_o be_v in_o the_o former_a and_o forasmuch_o as_o ab_fw-la be_v a_o line_n contain_v in_o power_n a_o rational_a and_o a_o medial_a superficies_n demonstration_n and_o be_v divide_v into_o his_o part_n in_o the_o point_n c_o therefore_o the_o line_n ac_fw-la &_o cb_o be_v incommensurable_a in_o power_n have_v that_o which_o be_v make_v of_o the_o square_n of_o they_o add_v together_o medial_a and_o that_o which_o be_v contain_v under_o then_o rational_a now_o forasmuch_o as_o that_o which_o be_v make_v of_o the_o square_n of_o the_o line_n ac_fw-la and_o cb_o add_v together_o be_v medial_a therefore_o also_o the_o parallelogram_n dl_o be_v medial_a wherefore_o by_o the_o 22._o of_o the_o ten_o the_o line_n dm_o be_v rational_a and_o incommensurable_a in_o length_n to_o the_o line_n de._n again_o forasmuch_o as_o that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o twice_o that_o be_v the_o parallelogram_n mf_n be_v rational_a therefore_o by_o the_o 20._o the_o line_n mg_o be_v rational_a &_o commensurable_a in_o length_n to_o the_o line_n de._n wherefore_o by_o the_o 13._o of_o the_o ten_o the_o line_n dm_o be_v incommensurable_a in_o length_n to_o the_o line_n mg_o wherefore_o the_o line_n dm_o and_o mg_o be_v rational_a commensurable_a in_o power_n only_o wherefore_o the_o whole_a line_n dg_o be_v a_o binomial_a line_n i_o say_v moreover_o that_o it_o be_v a_o five_o binomiall_n for_o as_o in_o the_o former_a so_o also_o in_o this_o may_v it_o be_v prove_v that_o that_o which_o be_v contain_v under_o the_o line_n dk_o and_o km_o be_v equal_a to_o the_o square_n of_o mn_a the_o half_a of_o the_o less_o and_o that_o the_o line_n dk_o be_v incommensurable_a in_o length_n
to_o the_o line_n km_o wherefore_o by_o the_o 18._o of_o the_o ten_o the_o line_n dm_o be_v in_o power_n more_o they_o the_o line_n mg_o by_o the_o square_n of_o a_o line_n incommensurable_a in_o length_n to_o the_o line_n dm_o and_o the_o line_n dm_o and_o mg_o be_v rational_a commensurable_a in_o power_n only_o and_o the_o less_o line_n namely_o mg_o be_v commensurable_a in_o length_n to_o the_o rational_a line_n give_v de._n wherefore_o the_o line_n dg_o be_v a_o five_o binomial_a line_n which_o be_v require_v to_o be_v demonstrate_v ¶_o the_o 47._o theorem_a the_o 65._o proposition_n the_o square_a of_o a_o line_n contain_v in_o power_n two_o medial_n apply_v unto_o a_o rational_a line_n make_v the_o breadth_n or_o other_o side_n a_o six_o binomial_a line_n svppose_v that_o the_o line_n ab_fw-la be_v a_o line_n contain_v in_o power_n two_o medial_n and_o let_v it_o be_v suppose_v to_o be_v divide_v into_o his_o part_n in_o the_o point_n c._n and_o take_v a_o rational_a line_n de._n and_o by_o the_o 44._o of_o the_o first_o unto_o the_o rational_a line_n de_fw-fr apply_v the_o parallelogram_n df_o equal_a to_o the_o square_n of_o the_o line_n ab_fw-la and_o make_n in_o breadth_n the_o line_n dg_o then_o i_o say_v that_o the_o line_n dg_o be_v a_o six_o binomial_a line_n construction_n let_v the_o self_n same_o construction_n be_v in_o this_o that_o be_v in_o the_o former_a and_o forasmuch_o as_o the_o line_n ab_fw-la be_v a_o line_n contain_v in_o power_n two_o medial_n demonstration_n and_o be_v divide_v into_o his_o part_n in_o the_o point_n c_o therefore_o the_o line_n ac_fw-la &_o cb_o be_v incommensurable_a in_o power_n have_v that_o which_o be_v make_v of_o the_o square_n of_o they_o add_v together_o medial_a and_o that_o which_o be_v contain_v under_o they_o medial_a and_o moreover_o incommensurable_a to_o that_o which_o be_v make_v of_o the_o square_n of_o they_o add_v together_o wherefore_o by_o those_o thing_n which_o have_v be_v before_o prove_v either_o of_o these_o parallelogram_n dl_o and_o mf_o be_v medial_a and_o either_o of_o they_o be_v apply_v upon_o the_o rational_a line_n de._n wherefore_o by_o the_o 22._o of_o the_o ten_o either_o of_o these_o line_n dm_o and_o mg_o be_v rational_a and_o incommensurable_a in_o length_n to_o the_o line_n de._n and_o forasmuch_o as_o â—hat_n which_o be_v madâ—_n of_o the_o square_n of_o the_o line_n ac_fw-la and_o cb_o add_v together_o be_v incommensurable_a to_o that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o twice_o therefore_o the_o parallelogram_n dl_o be_v incommensurable_a to_o the_o parallelogram_n mf_n wherefore_o by_o the_o 1._o of_o the_o six_o and_o 10._o of_o the_o ten_o the_o line_n dm_o be_v incommensurable_a in_o length_n to_o the_o line_n mg_o wherefore_o the_o line_n dm_o and_o mg_o be_v rational_a commensurable_a in_o power_n only_o wherefore_o the_o whole_a line_n dg_o be_v a_o binomial_a line_n i_o say_v also_o that_o be_v a_o six_o binomial_a line_n for_o even_o as_o in_o the_o other_o proposition_n it_o have_v be_v prove_v so_o also_o in_o this_o may_v it_o be_v prove_v that_o that_o which_o be_v contain_v under_o the_o line_n dk_o and_o km_o be_v equal_a to_o the_o square_n of_o the_o line_n mn_o and_o that_o the_o line_n dk_o be_v incommensurable_a in_o length_n to_o the_o line_n km_o and_o therefore_o by_o the_o 18._o of_o the_o ten_o the_o line_n dm_o be_v in_o power_n more_o than_o the_o line_n mg_o by_o the_o square_n of_o a_o line_n incommensurable_a in_o length_n to_o the_o line_n dm_o and_o neither_o of_o these_o line_n dm_o nor_o mg_o be_v commensurable_a in_o length_n to_o there_o anâ—nall_a line_n give_v de._n wherefore_o the_o line_n dg_o be_v a_o six_o â—â—nomiall_a line_n which_o be_v require_v to_o be_v demonstrate_v ¶_o the_o 48._o theorem_a the_o 66._o proposition_n a_o line_n commensurable_a in_o length_n to_o a_o binomial_a line_n be_v also_o a_o binomial_a line_n of_o the_o self_n same_o order_n svppose_v that_o the_o line_n ab_fw-la be_v a_o binomial_a line_n senary_n and_o unto_o the_o line_n ab_fw-la let_v the_o line_n cd_o be_v commensurable_a in_o length_n then_o i_o say_v that_o the_o line_n cd_o be_v a_o binomial_a line_n and_o of_o the_o self_n same_o order_n that_o the_o line_n ab_fw-la be_v for_o forasmuch_o as_o ab_fw-la be_v â—_o binomial_a line_n let_v it_o be_v divide_v into_o his_o name_n in_o the_o point_n e_o construction_n and_o let_v ae_n be_v the_o great_a name_n wherefore_o the_o line_n ae_n and_o ebb_n be_v rational_a commensurable_a in_o power_n only_o and_o as_o the_o line_n ab_fw-la be_v to_o the_o line_n cd_o so_o by_o the_o 12._o of_o the_o six_o let_v the_o line_n ae_n be_v to_o the_o line_n cf._n demonstration_n wherefore_o by_o the_o 19_o of_o the_o five_o the_o residue_n namely_o the_o line_n ebb_n be_v to_o the_o residue_n namely_o to_o the_o line_n fd_o as_o the_o line_n ab_fw-la be_v to_o the_o line_n cd_o but_o by_o supposition_n the_o line_n ab_fw-la be_v commensurable_a in_o length_n to_o the_o line_n cd_o wherefore_o by_o the_o 10._o of_o the_o ten_o the_o line_n ae_n be_v commensurable_a in_o length_n to_o the_o line_n cf_o and_o the_o line_n ebb_n to_o the_o line_n fd._n and_o the_o line_n ae_n and_o ebb_n be_v rational_a wherefore_o the_o line_n cf_o and_o fd_o be_v also_o rational_a and_o for_o that_o as_o the_o line_n ae_n be_v to_o the_o line_n cf_o so_o be_v the_o line_n ebb_n to_o the_o line_n fd_o therefore_o alternate_o by_o the_o 16._o of_o the_o five_o as_o the_o line_n aâ—e_n be_v to_o the_o line_n ebb_n so_o be_v the_o line_n cf_o to_o the_o line_n fd._n but_o the_o line_n ae_n and_o ebb_n be_v commensurable_a in_o power_n only_o wherefore_o the_o line_n cf_o and_o fd_o be_v also_o commensurable_a in_o power_n only_o and_o they_o be_v rational_a wherefore_o the_o whole_a line_n cd_o be_v a_o binomial_a line_n i_o say_v also_o that_o it_o be_v of_o the_o self_n same_o order_n of_o binomial_a line_n that_o the_o line_n ab_fw-la be_v for_o the_o line_n ae_n be_v in_o power_n more_o than_o the_o line_n ebb_n either_o by_o the_o square_n of_o a_o line_n commensurable_a in_o length_n to_o the_o line_n ae_n or_o by_o the_o square_n of_o a_o line_n incommensurable_a in_o length_n to_o the_o line_n ae_n if_o the_o line_n ae_n be_v in_o power_n more_o than_o the_o line_n ebb_n by_o the_o square_n of_o a_o line_n commensurable_a in_o length_n to_o the_o line_n ae_n the_o line_n also_o cf_o by_o the_o 14._o of_o the_o ten_o shall_v be_v in_o power_n more_o than_o the_o line_n fd_o by_o the_o square_n of_o a_o line_n commensurable_a in_o length_n to_o cf._n and_o if_o the_o lineâ—_z ae_n be_v commensurable_a in_o length_n to_o a_o rational_a line_n give_v the_o line_n cf_o also_o shall_v be_v commensurable_a in_o length_n to_o the_o same_o by_o the_o 12._o of_o the_o ten_o and_o so_o either_o of_o these_o line_n ab_fw-la and_o cd_o be_v a_o first_o binomial_a line_n that_o be_v they_o be_v both_o of_o one_o and_o the_o self_n same_o order_n but_o if_o the_o line_n ebb_n be_v commensurable_a in_o length_n to_o the_o rational_a line_n put_v the_o line_n fd_v also_o shall_v be_v commensurable_a in_o length_n to_o the_o same_o and_o by_o that_o mean_n again_o the_o line_n ab_fw-la and_o cd_o be_v both_o of_o one_o and_o the_o self_n same_o orderâ—_n for_o either_o of_o they_o be_v a_o second_o binomial_a line_n but_o if_o neither_o of_o the_o line_n ae_n nor_o eâ—_n be_v commensurable_a in_o length_n to_o the_o rational_a line_n put_v neither_o also_o of_o these_o line_n cf_o nor_o fd_o shall_v be_v commensurable_a in_o length_n to_o the_o same_o and_o so_o neither_o of_o the_o line_n ab_fw-la and_o cd_o be_v a_o three_o binomial_a line_n but_o if_o the_o line_n ae_n be_v in_o power_n more_o than_o the_o line_n ebb_n by_o the_o square_n of_o a_o line_n incommensurable_a in_o length_n to_o the_o line_n ae_n the_o line_n also_o cf_o shall_v be_v in_o power_n more_o than_o the_o line_n fd_o by_o the_o square_n of_o a_o line_n incommensurable_a in_o length_n to_o the_o line_n cf_o by_o the_o 14._o of_o the_o ten_o and_o then_o if_o the_o line_n ae_n be_v commensurable_a in_o length_n to_o the_o rational_a line_n put_v the_o line_n cf_o also_o shall_v be_v commensurable_a in_o length_n to_o the_o same_o and_o so_o either_o of_o the_o line_n ab_fw-la and_o cd_o shall_v be_v a_o four_o binomial_a line_n and_o if_o the_o line_n ebb_n be_v commensurable_a in_o length_n to_o the_o rational_a line_n give_v the_o line_n fd_v also_o shall_v be_v commensurable_a in_o length_n to_o the_o same_o and_o so_o either_o of_o the_o line_n ab_fw-la and_o cd_o shall_v be_v a_o
either_o of_o these_o superficiece_n ab_fw-la &_o cd_o be_v medial_a therefore_o also_o either_o of_o these_o parallelogram_n eglantine_n and_o high_a be_v medial_a and_o they_o be_v each_o apply_v to_o the_o rational_a line_n of_o make_v in_o breadth_n the_o line_n eh_o and_o hk_o wherefore_o by_o the_o 22._o of_o the_o ten_o either_o of_o these_o line_n eh_o and_o hk_o be_v rational_a and_o incommensurable_a in_o length_n to_o the_o line_n ef._n and_o forasmuch_o as_o the_o superficies_n ab_fw-la be_v incommensurable_a to_o the_o superficies_n cd_o and_o the_o superficies_n ab_fw-la be_v equal_a to_o the_o parallelogram_n eâ—_n and_o the_o superficies_n cd_o to_o the_o parallelogram_n high_a therefore_o the_o parallelogram_n eglantine_n be_v incommensurable_a to_o the_o parallelogram_n high_a but_o by_o the_o 1._o of_o the_o six_o as_o the_o parallelogram_n eglantine_n be_v to_o the_o parallelogram_n high_a so_o be_v the_o line_n ehâ—_n to_o the_o line_n hk_o wherefore_o by_o the_o 10._o of_o the_o ten_o the_o line_n eh_o iâ—â—spansâ—_n hkâ—_n wherefore_o the_o â—â—nes_n eh_o â—nd_v hk_o be_v rational_a commensurable_a in_o power_n only_o wherefore_o the_o whole_a line_n eke_o be_v a_o binomial_a line_n and_o as_o in_o the_o former_a proposition_n so_o alsâ—_n in_o this_o may_v it_o be_v prove_v that_o the_o line_n eh_o be_v great_a than_o the_o line_n hk_o wherefore_o the_o line_n eh_o be_v in_o power_n more_o than_o the_o line_n hk_o either_o by_o the_o square_n of_o a_o line_n commensurable_a in_o length_n to_o the_o line_n eh_o or_o by_o the_o square_n of_o a_o line_n incommensurable_a in_o length_n to_o the_o line_n eh_o first_o let_v it_o be_v great_a by_o the_o square_n of_o a_o line_n commensurable_a in_o length_n unto_o the_o line_n eh_o case_n now_o neither_o of_o these_o line_n eh_o and_o hk_o be_v commensurable_a in_o length_n to_o the_o rational_a line_n give_v ef._n wherefore_o the_o whole_a line_n eke_o be_v a_o three_o binomial_a line_n and_o the_o line_n of_o be_v a_o rational_a line_n but_o if_o a_o superficies_n be_v contain_v under_o a_o rational_a line_n &_o a_o three_o binomial_a line_n the_o line_n that_o contain_v in_o power_n the_o same_o superficies_n be_v by_o the_o 56._o of_o the_o ten_o a_o second_o bimedial_a line_n wherefore_o the_o line_n that_o contain_v in_o power_n the_o superficies_n ei_o that_o be_v the_o superficies_n ad_fw-la be_v a_o second_o bimedial_a line_n case_n but_o now_o suppose_v that_o the_o line_n eh_o be_v in_o power_n more_o than_o the_o line_n hk_o by_o the_o square_n of_o a_o line_n incommensurable_a in_o length_n to_o the_o line_n eh_o and_o forasmuch_o as_o either_o of_o these_o line_n eh_o and_o hk_o be_v incommensurable_a in_o length_n to_o the_o rational_a line_n give_v of_o therefore_o the_o line_n eke_o be_v a_o six_o binomial_a line_n but_o if_o a_o superficies_n be_v contain_v under_o a_o rational_a line_n and_o a_o six_o binomial_a line_n the_o line_n that_o contain_v in_o power_n the_o same_o superficies_n be_v by_o the_o 59_o of_o the_o ten_o a_o line_n contain_v in_o power_n two_o medial_n wherefore_o the_o line_n that_o contain_v in_o power_n the_o superficies_n ad_fw-la be_v a_o line_n contain_v in_o power_n two_o medial_n case_n and_o after_o the_o self_n same_o manner_n if_o the_o superficies_n ab_fw-la be_v less_o than_o the_o superficies_n cd_o may_v we_o prove_v that_o the_o line_n that_o contain_v in_o power_n the_o superficies_n ad_fw-la be_v either_o a_o second_o bimedial_a line_n or_o a_o line_n contain_v in_o power_n two_o medial_n if_o therefore_o two_o medial_a superficiece_n incommensurable_a the_o one_o to_o the_o other_o be_v add_v together_o the_o line_n contain_v in_o power_n the_o whole_a superficies_n be_v one_o of_o the_o two_o irrational_a line_n remain_v namely_o either_o a_o second_o bimedial_a line_n or_o a_o line_n contain_v in_o power_n two_o medial_n which_o be_v require_v to_o be_v prove_v ¶_o a_o corollary_n follow_v of_o the_o former_a proposition_n a_o binomial_a line_n and_o the_o other_o irrational_a line_n follow_v it_o be_v neither_o medial_a line_n nor_o one_o and_o the_o same_o between_o themselves_o for_o the_o square_n of_o a_o medial_a line_n apply_v to_o a_o rational_a line_n corollary_n make_v the_o breadth_n rational_a and_o incommensurale_fw-la in_o length_n to_o the_o rational_a line_n whereunto_o it_o be_v apply_v by_o the_o 22._o of_o the_o ten_o the_o square_a of_o a_o binomial_a line_n apply_v to_o â—_o rational_a line_n make_v the_o breadth_n a_o first_o binomial_a line_n by_o the_o 60._o of_o the_o ten_o the_o square_a of_o a_o first_o bimedial_a line_n apply_v unto_o a_o rational_a line_n make_v the_o breadth_n a_o second_o binomial_a line_n by_o the_o 61._o of_o the_o ten_o the_o square_a of_o a_o second_o bimedial_a line_n apply_v unto_o a_o rational_a line_n make_v the_o breadth_n a_o three_o binomial_a line_n by_o the_o 62._o of_o the_o ten_o the_o square_a of_o a_o great_a line_n apply_v to_o a_o rational_a line_n make_v the_o breadth_n a_o four_o binomial_a line_n by_o the_o 63._o of_o the_o ten_o the_o square_a of_o a_o line_n contain_v in_o power_n a_o rational_a &_o a_o medial_a superficies_n make_v the_o breadth_n a_o five_o binomial_a line_n by_o the_o 64._o of_o the_o ten_o and_o the_o square_a of_o a_o line_n contain_v in_o power_n two_o medial_n apply_v unto_o a_o rational_a line_n make_v the_o breadth_n a_o six_o binomial_a line_n by_o the_o 65._o of_o the_o ten_o see_v therefore_o that_o these_o aforesaid_a breadthes_fw-mi differ_v both_o from_o the_o first_o breadth_n for_o that_o it_o be_v rational_a and_o differ_v also_o the_o one_o from_o the_o other_o for_o that_o they_o be_v binomial_n of_o diverse_a order_n it_o be_v manifest_a that_o those_o irrational_a line_n differ_v also_o the_o one_o from_o the_o other_o here_o begin_v the_o senaries_n by_o substraction_n ¶_o the_o 55._o theorem_a the_o 73._o proposition_n if_o from_o a_o rational_a line_n be_v take_v away_o a_o rational_a line_n commensurable_a in_o power_n only_o to_o the_o whole_a line_n the_o residue_n be_v a_o irrational_a line_n and_o be_v call_v a_o residual_a line_n svppose_v that_o ab_fw-la be_v a_o rational_a line_n and_o from_o ab_fw-la take_v away_o a_o rational_a line_n bc_o commensurable_a in_o power_n only_o to_o the_o whole_a line_n ab_fw-la substraction_n then_o i_o say_v that_o the_o line_n remain_v namely_o ac_fw-la be_v irrational_a and_o be_v call_v a_o residual_a line_n for_o forasmuch_o as_o the_o line_n ab_fw-la be_v incommensurable_a in_o length_n unto_o the_o line_n bc_o and_o by_o the_o assumpt_n go_v before_o the_o 22._o of_o the_o ten_o as_o the_o line_n ab_fw-la be_v to_o the_o line_n bc_o demonstration_n so_o iâ—_z the_o square_a of_o the_o line_n ab_fw-la to_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o wherefore_o by_o the_o 10._o of_o the_o ten_o the_o square_a of_o the_o line_n ab_fw-la be_v incommensurable_a to_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc._n but_o unto_o the_o square_n of_o the_o line_n ab_fw-la be_v commensurable_a the_o square_n of_o the_o line_n ab_fw-la and_o bc_o by_o the_o 15._o of_o the_o ten_o wherefore_o the_o square_n of_o the_o line_n ab_fw-la and_o bc_o be_v incommensurable_a to_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc._n but_o unto_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o be_v commensurable_a that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o twice_o wherefore_o the_o square_n of_o the_o line_n ab_fw-la and_o bc_o be_v incommensurable_a to_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o twice_o but_o the_o square_n of_o the_o line_n ab_fw-la and_o bc_o be_v equal_a to_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o twice_o and_o to_o the_o square_n of_o the_o line_n ac_fw-la by_o the_o 7._o of_o the_o second_o wherefore_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o twice_o together_o with_o the_o square_n of_o the_o line_n ac_fw-la be_v incommensurable_a to_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o twice_o wherefore_o by_o the_o 2_o part_n of_o the_o 16._o of_o the_o ten_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o twice_o be_v incommensurable_a to_o the_o square_n of_o the_o line_n ac_fw-la wherefore_o by_o the_o first_o part_n of_o the_o same_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o twice_o together_o with_o the_o square_n of_o the_o line_n ac_fw-la that_o be_v the_o square_n of_o the_o line_n ab_fw-la and_o bc_o be_v incommensurable_a to_o the_o square_n of_o the_o line_n ac_fw-la but_o the_o square_n of_o the_o line_n ab_fw-la and_o
bc_o be_v rational_a for_o the_o line_n ab_fw-la and_o bc_o be_v put_v to_o be_v rational_a wherefore_o the_o line_n ac_fw-la be_v irrational_a and_o be_v call_v a_o residual_a line_n which_o be_v require_v to_o be_v prove_v an_o other_o demonstration_n after_o campane_n campane_n demonstrate_v this_o proposition_n by_o a_o figure_n more_o brief_o after_o this_o mâ—ner_n campane_n let_v the_o superficies_n eglantine_n be_v equal_a to_o the_o square_n of_o the_o line_n ab_fw-la and_o bc_o add_v together_o which_o shall_v be_v rational_a for_o that_o the_o line_n ab_fw-la and_o bc_o be_v suppose_v to_o be_v rational_a commensurable_a in_o power_n only_o fron_n which_o superficies_n take_v away_o the_o superficies_n df_o equal_a to_o that_o which_o be_v conâ—â—yaâ—d_v under_o the_o line_n ab_fw-la &_o dc_o twice_o which_o shall_v be_v medial_a by_o the_o 21._o of_o this_o book_n now_o by_o the_o 7._o of_o the_o second_o the_o superficies_n fg_o be_v equal_a to_o the_o square_n of_o the_o line_n ac_fw-la and_o forasmuch_o as_o the_o superficies_n eglantine_n be_v incommensurable_a to_o the_o superficies_n df_o for_o that_o the_o one_o be_v rational_a and_o the_o other_o medial_a therefore_o by_o the_o 16._o of_o this_o book_n the_o 〈◊〉_d superficies_n eglantine_n be_v incommensurable_a to_o the_o superficies_n fg._n wherefore_o the_o superficies_n fg_o be_v irrational_a and_o therefore_o the_o line_n ac_fw-la which_o contain_v it_o in_o power_n be_v irrational_a which_o be_v require_v to_o be_v prove_v a_o annotation_n of_o p._n montaâ—reâ—s_n this_o theorem_a teach_v nothing_o else_o but_o that_o that_o portion_n of_o the_o great_a name_n of_o a_o binomial_a line_n which_o remain_v after_o the_o take_n away_o of_o the_o less_o name_n from_o the_o great_a name_n be_v irrational_a which_o be_v call_v a_o residual_a line_n that_o be_v to_o say_v if_o from_o the_o great_a name_n of_o a_o binomial_a line_n which_o great_a name_n be_v a_o rational_a line_n commensurable_a in_o power_n only_o to_o the_o less_o name_n be_v take_v away_o the_o less_o name_n which_o self_n less_o name_n be_v also_o commensurable_a in_o power_n only_o to_o the_o great_a name_n which_o great_a name_n this_o theorem_a call_v the_o whole_a line_n the_o rest_n of_o the_o line_n which_o remain_v be_v irrational_a which_o he_o call_v a_o residual_a line_n wherefore_o all_o the_o line_n which_o be_v entreat_v in_o this_o theorem_a and_o in_o the_o five_o other_o which_o follow_v be_v the_o portion_n remain_v of_o the_o great_a part_n of_o the_o whole_a line_n which_o be_v entreat_v of_o in_o the_o 36.37.38.39.40.41_o proposition_n after_o the_o take_v away_o the_o less_o part_n from_o the_o great_a in_o this_o proposition_n be_v set_v forth_o the_o nature_n of_o the_o eight_o kind_n of_o irrational_a line_n which_o be_v call_v a_o residual_a line_n the_o definition_n whereof_o by_o this_o proposition_n be_v thus_o line_n a_o residual_a line_n be_v a_o irrational_a line_n which_o remain_v when_o from_o a_o rational_a line_n give_v be_v take_v away_o a_o rational_a line_n commensurable_a to_o the_o whole_a line_n in_o power_n only_o ¶_o the_o 56._o theorem_a the_o 74._o proposition_n if_o from_o a_o medial_a line_n be_v take_v away_o a_o medial_a line_n commensurable_a in_o power_n only_o to_o the_o whole_a line_n and_o comprehend_v together_o with_o the_o whole_a line_n a_o rational_a superficies_n the_o residue_n be_v a_o irrational_a line_n and_o be_v call_v a_o first_o medial_a residual_a line_n out_o of_o this_o proposition_n be_v take_v the_o definition_n of_o the_o nine_o kind_n of_o irrational_a line_n which_o be_v call_v a_o first_o residual_a medial_a line_n the_o definition_n whereof_o be_v thus_o line_n a_o first_o residual_a medial_a line_n be_v a_o irrational_a line_n which_o remain_v when_o from_o a_o medial_a line_n be_v take_v away_o a_o medial_a line_n commensurable_a to_o the_o whole_a in_o power_n only_o and_o the_o part_n take_v away_o and_o the_o whole_a line_n contain_v a_o medial_a superficies_n an_o other_o demonstration_n after_o campane_n let_v the_o line_n de_fw-fr be_v rational_a upon_o which_o apply_v the_o superficies_n df_o equal_a to_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o twice_o and_o let_v the_o superficies_n ge_z be_v equal_a to_o that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ab_fw-la and_o bc_o campane_n wherefore_o by_o the_o 7._o of_o the_o second_o the_o superficies_n fg_o be_v equal_a to_o the_o square_n of_o the_o line_n ac_fw-la and_o forasmuch_o as_o by_o supposition_n the_o superficies_n eglantine_n be_v medial_a therefore_o by_o the_o 22._o of_o the_o ten_o the_o line_n dg_o be_v rational_a commensurable_a in_o power_n only_o to_o the_o rational_a line_n de._n and_o forasmuch_o as_o by_o supposition_n the_o superficies_n eh_o be_v rational_a therefore_o by_o the_o 20._o of_o the_o ten_o the_o line_n dh_o be_v rational_a commensurable_a in_o length_n unto_o the_o rational_a line_n de._n wherefore_o the_o line_n dg_o and_o dh_o be_v rational_a commensurable_a in_o power_n only_o by_o the_o assumpt_n put_v before_o the_o 13._o of_o this_o book_n wherefore_o by_o the_o 73_o of_o this_o book_n the_o line_n gh_o be_v a_o residual_a line_n and_o be_v therefore_o irrational_a wherefore_o by_o the_o corollary_n of_o the_o 21._o of_o this_o book_n the_o superficies_n fg_o be_v irrational_a and_o therefore_o the_o line_n ac_fw-la which_o contain_v it_o in_o power_n be_v irrational_a and_o be_v call_v a_o first_o medial_n residual_a line_n ¶_o the_o 57_o theorem_a the_o 75._o proposition_n if_o from_o a_o medial_a line_n be_v take_v away_o a_o medial_a line_n commensurable_a in_o power_n only_o to_o the_o whole_a line_n and_o comprehend_v together_o with_o the_o whole_a line_n a_o medial_a superficies_n the_o residue_n be_v a_o irrational_a line_n and_o be_v call_v a_o second_o medial_a residual_a line_n svppose_v that_o ab_fw-la be_v a_o medial_a line_n and_o from_o ab_fw-la take_v away_o a_o medial_a line_n cb_o commensurable_a in_o power_n only_o to_o the_o whole_a line_n ab_fw-la and_o comprehend_v together_o with_o the_o whole_a line_n ab_fw-la a_o medial_a superficies_n namely_o the_o parallelogram_n contain_v under_o the_o line_n ab_fw-la and_o bc._n construction_n then_o i_o say_v that_o the_o residue_n namely_o the_o line_n ac_fw-la be_v irrational_a and_o be_v call_v a_o second_o medial_a residual_a line_n take_v a_o rational_a line_n diego_n and_o by_o the_o 44._o of_o the_o first_o unto_o the_o line_n diego_n apply_v the_o parallelogram_n de_fw-fr equal_a to_o the_o square_n of_o the_o line_n ab_fw-la &_o bc_o and_o make_v in_o breadth_n the_o line_n dg_o demonstration_n and_o unto_o the_o same_o line_n diego_n apply_v the_o parallelogram_n dh_fw-mi equal_a to_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la &_o bc_o twice_o and_o make_v in_o breadth_n the_o line_n df._n now_o the_o parallelogram_n dh_o be_v less_o than_o the_o parallelogram_n de_fw-fr for_o that_o also_o the_o square_a of_o the_o line_n ab_fw-la and_o bc_o be_v great_a than_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o twice_o by_o the_o square_n of_o the_o line_n ac_fw-la by_o the_o 7._o of_o the_o second_o wherefore_o the_o parallelogram_n remain_v namely_o fe_o be_v equal_a to_o the_o square_n of_o the_o line_n ac_fw-la and_o forasmuch_o as_o the_o square_n of_o the_o line_n ab_fw-la and_o bc_o be_v medial_a therefore_o also_o the_o parallelogram_n de_fw-fr be_v medial_a and_o be_v apply_v to_o the_o rational_a line_n diego_n make_v in_o breadth_n the_o line_n dg_o wherefore_o by_o the_o 22._o of_o the_o ten_o the_o line_n dg_o be_v rational_a and_o incommensurable_a in_o length_n to_o the_o line_n di._n again_o forasmuch_o as_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o be_v medial_a therefore_o also_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o twice_o be_v medial_a but_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o twice_o be_v equal_a to_o the_o parallelogram_n dh_fw-mi wherefore_o the_o parallelogram_n dh_o be_v medial_a and_o be_v apply_v to_o the_o rational_a line_n diego_n make_v in_o breadth_n the_o line_n df._n wherefore_o the_o line_n df_o be_v rational_a and_o incommensurable_a in_o length_n to_o the_o line_n di._n and_o forasmuch_o as_o the_o line_n ab_fw-la and_o bc_o be_v commensurable_a in_o power_n only_o therefore_o the_o line_n ab_fw-la be_v incommensurable_a in_o length_n to_o the_o line_n bc._n wherefore_o by_o the_o assumpt_n go_v before_o the_o 22._o of_o the_o ten_o and_o by_o the_o 10._o of_o the_o ten_o the_o square_a of_o the_o line_n ab_fw-la be_v incommensurable_a to_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc._n but_o unto_o the_o square_n of_o the_o line_n ab_fw-la be_v commensurable_a
the_o square_n of_o ab_fw-la and_o bc_o by_o the_o 15._o of_o the_o ten_o and_o unto_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o be_v commensurable_a to_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o twice_o wherefore_o the_o square_n of_o the_o line_n ab_fw-la and_o bc_o be_v incommensurable_a to_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o twice_o but_o unto_o the_o square_n of_o the_o line_n ab_fw-la and_o bc_o be_v equal_a the_o parallelogram_n de_fw-fr and_o to_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o twice_o be_v equal_a the_o parallelogram_n dh_fw-mi wherefore_o the_o parallelogram_n de_fw-fr be_v incommensurable_a to_o the_o parallelogram_n dh_fw-mi but_o as_o the_o parallelogram_n de_fw-fr be_v to_o the_o parallelogram_n dh_o so_o iâ—_z the_o line_n gd_o to_o the_o line_n df._n wherefore_o the_o line_n gd_o be_v incommensurable_a in_o length_n to_o the_o line_n df._n and_o either_o of_o they_o be_v rational_a wherefore_o the_o line_n gd_o and_o df_o be_v rational_a commensurable_a in_o power_n only_o wherefore_o the_o line_n fg_o be_v a_o residual_a line_n by_o the_o 73._o proposition_n of_o the_o ten_o and_o the_o line_n de_fw-fr be_v a_o rational_a line_n but_o a_o superficies_n comprehend_v under_o a_o rational_a line_n and_o a_o irrational_a line_n be_v irrational_a by_o the_o 21_o of_o the_o teâ—â—â—â—_n and_o the_o line_n which_o contain_v in_o power_n the_o same_o superficies_n be_v irrational_a by_o the_o assumpt_n go_v before_o the_o same_o wherefore_o the_o parallelogram_n fe_o be_v irrational_a but_o the_o line_n ac_fw-la contain_v in_o power_n the_o parallelogram_n fe_o wherefore_o the_o line_n ac_fw-la be_v a_o irrational_a line_n and_o be_v call_v a_o second_o medial_a residual_a line_n and_o this_o second_o medial_a residual_a line_n be_v that_o part_n of_o the_o great_a part_n of_o a_o bimedial_a line_n which_o remain_v after_o the_o take_n away_o of_o the_o less_o part_n from_o the_o great_a which_o be_v require_v to_o be_v prove_v an_o other_o demonstrtion_n more_o brief_a after_o campane_n this_o proposition_n set_v forth_o the_o nature_n of_o the_o ten_o kind_n of_o irrational_a line_n which_o be_v call_v a_o second_o residual_a medial_a line_n which_o be_v thus_o define_v line_n a_o second_o residualâ—_n medial_a line_n be_v a_o irrational_a line_n which_o remain_v when_o from_o a_o medial_a line_n be_v take_v away_o a_o medial_a line_n commensurable_a to_o the_o whole_a in_o power_n only_o and_o the_o part_n take_v away_o &_o the_o whole_a line_n contain_v a_o medial_a superficies_n ¶_o the_o 58._o theorem_a the_o 76._o proposition_n iâ—â—rom_o a_o right_a line_n be_v take_v away_o a_o right_a line_n incommensurable_a in_o power_n to_o the_o whole_a and_o if_o that_o which_o be_v make_v of_o the_o square_n of_o the_o whole_a line_n and_o of_o the_o line_n take_v away_o add_v together_o be_v rational_a and_o the_o parallelogram_n contain_v under_o the_o same_o line_n medial_a the_o line_n remain_v be_v irrational_a and_o be_v call_v a_o less_o line_n in_o this_o proposition_n be_v contain_v the_o definition_n of_o the_o eleven_o kind_n of_o irrational_a line_n which_o be_v call_v a_o less_o line_n who_o definition_n be_v thus_o a_o less_o line_n be_v a_o irrational_a line_n which_o remain_v line_n when_o from_o a_o right_a line_n be_v take_v away_o a_o right_a line_n incommensurable_a in_o power_n to_o the_o whole_a and_o the_o square_a of_o the_o whole_a line_n &_o the_o square_a of_o the_o part_n take_v away_o add_v together_o make_v a_o rational_a superficies_n and_o the_o parallelogram_n contain_v of_o they_o be_v medial_a this_o proposition_n may_v after_o campane_n way_n be_v demonstrate_v if_o you_o remember_v well_o the_o order_n &_o position_n which_o he_o in_o the_o three_o former_a proposition_n use_v ¶_o the_o 19_o theorem_a the_o 77._o proposition_n if_o from_o a_o right_a line_n be_v take_v away_o a_o right_a line_n incommensurable_a in_o power_n to_o the_o whole_a line_n and_o if_o that_o which_o be_v make_v of_o the_o square_n of_o the_o whole_a line_n and_o of_o the_o line_n take_v away_o add_v together_o be_v medial_a and_o the_o parallelogram_n contain_v under_o the_o same_o line_n rational_a the_o line_n remain_v be_v irrational_a and_o be_v call_v a_o line_n make_v with_o a_o rational_a superficies_n the_o whole_a superficies_n medial_a in_o this_o proposition_n be_v declare_v the_o nature_n of_o the_o twelve_o kind_a of_o irrational_a line_n which_o be_v call_v a_o line_n make_v with_o a_o rational_a superficies_n the_o whole_a superficies_n medial_a who_o definition_n be_v thus_o line_n a_o line_n make_v with_o a_o rational_a superficies_n the_o whole_a superficies_n medial_a be_v a_o irrational_a line_n which_o remain_v when_o from_o a_o right_a line_n be_v take_v away_o a_o right_a line_n incommensurable_a in_o power_n to_o the_o whole_a line_n and_o the_o square_a of_o the_o whole_a line_n &_o the_o square_a of_o the_o part_n take_v away_o add_v together_o make_v a_o medial_a superficies_n and_o the_o parallelogram_n contain_v of_o they_o be_v rational_a this_o proposition_n also_o may_v after_o campane_n way_n be_v demonstrate_v observe_v the_o former_a caution_n ¶_o the_o 60._o theorem_a the_o 78._o proposition_n if_o from_o a_o right_a line_n be_v take_v away_o a_o right_a line_n incommensurable_a in_o power_n to_o the_o whole_a line_n and_o if_o that_o which_o be_v make_v of_o the_o square_n of_o the_o whole_a line_n and_o of_o the_o line_n take_v away_o add_v together_o be_v medial_a and_o the_o parallelogram_n contain_v under_o the_o same_o line_n be_v also_o medial_a and_o incommensurable_a to_o that_o which_o be_v make_v of_o the_o square_n of_o the_o say_v line_n add_v together_o the_o line_n remain_v be_v irrational_a and_o be_v call_v a_o line_n make_v with_o a_o medial_a superficies_n the_o whole_a superficies_n medial_a this_o proposition_n may_v thus_o more_o brief_o be_v demonstrate_v forasmuch_o as_o that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ab_fw-la and_o bc_o be_v medial_a and_o that_o also_o which_o be_v contain_v under_o they_o be_v medial_a therefore_o the_o parallelogrammâ—s_n de_fw-fr and_o dh_o which_o be_v equal_a unto_o they_o be_v medial_a but_o a_o medial_a superficies_n exceed_v not_o a_o medial_a superficies_n by_o a_o rational_a superficies_n wherefore_o the_o superficies_n fe_o which_o be_v the_o excess_n of_o the_o medial_a superficies_n de_fw-fr above_o the_o medial_a superficies_n dh_o be_v irrational_a and_o therefore_o the_o line_n ac_fw-la which_o contain_v it_o in_o power_n be_v irrational_a etc_n etc_n in_o this_o proposition_n be_v show_v the_o condition_n and_o nature_n of_o the_o thirteen_o and_o last_o kind_n of_o irrational_a line_n which_o be_v call_v a_o line_n make_v with_o a_o medial_a superficies_n the_o whole_a superficies_n medial_a who_o definition_n be_v thus_o a_o line_n make_v with_o a_o medial_a superficies_n the_o whole_a superficies_n medial_a be_v a_o irrational_a line_n which_o remain_v line_n when_o from_o a_o right_a liâ—e_z be_v take_v away_o a_o right_a line_n incommensurable_a in_o power_n to_o the_o whole_a line_n and_o the_o square_n of_o the_o whole_a line_n and_o of_o the_o line_n take_v away_o add_v together_o make_v a_o medial_a superficies_n and_o the_o parallelogram_n contain_v of_o they_o be_v also_o a_o medial_a superficies_n moreover_o the_o square_n of_o they_o be_v incommensurable_a to_o the_o parallelogram_n contain_v of_o they_o a_o assumpt_n of_o campane_n if_o there_o be_v four_o quantity_n &_o if_o the_o difference_n of_o the_o first_o to_o the_o second_o be_v as_o the_o difference_n of_o the_o three_o to_o the_o four_o then_o alternate_o as_o the_o difference_n of_o the_o first_o be_v to_o the_o three_o so_o be_v the_o difference_n of_o the_o second_o to_o the_o four_o this_o be_v to_o be_v understand_v of_o quantity_n in_o like_a sort_n refer_v the_o one_o to_o the_o other_o that_o be_v if_o the_o first_o be_v great_a than_o the_o second_o campane_n the_o three_o ought_v to_o be_v great_a than_o the_o four_o and_o if_o the_o first_o be_v less_o than_o the_o second_o the_o three_o ought_v to_o be_v less_o than_o the_o four_o and_o be_v also_o to_o be_v understand_v in_o arithmeticiall_a proportionality_n as_o for_o example_n let_v the_o difference_n of_o a_o be_v unto_o b_o as_o the_o difference_n of_o c_o be_v to_o d._n artificial_a then_o i_o say_v that_o as_o the_o difference_n of_o a_o be_v to_o c_o so_o be_v the_o difference_n of_o b_o to_o d._n for_o by_o this_o common_a sentence_n the_o difference_n of_o the_o extreme_n be_v compose_v of_o the_o difference_n of_o the_o extreme_n to_o the_o mean_n the_o difference_n of_o a_o to_o c_z be_v compose_v of_o the_o difference_n of_o a_o
to_o b_o and_o of_o the_o difference_n of_o b_o to_o c._n and_o by_o the_o same_o common_a sentence_n the_o difference_n of_o b_o to_o d_z be_v compose_v of_o the_o difference_n of_o b_o to_o c_z and_o of_o â—he_n difference_n of_o c_o to_o d._n and_o forasmuch_o as_o by_o supposition_n the_o difference_n of_o a_o to_o b_o be_v as_o the_o difference_n of_o c_o to_o d_z and_o the_o difference_n of_o b_o to_o c_z be_v common_a to_o they_o both_o wherefore_o it_o follow_v that_o as_o the_o difference_n of_o a_o be_v to_o c_o so_o be_v the_o difference_n of_o b_o to_o d_z which_o be_v require_v to_o be_v prove_v ¶_o the_o 61._o theorem_a the_o 79._o proposition_n unto_o a_o residual_a line_n can_v be_v join_v one_o only_o right_a line_n rational_a and_o commensurable_a in_o power_n only_o to_o the_o whole_a line_n senary_n let_v ab_fw-la be_v a_o residual_a line_n and_o unto_o it_o let_v the_o line_n bc_o be_v suppose_v to_o be_v join_v so_o that_o let_v the_o line_n ac_fw-la and_o bc_o be_v rational_a commensurable_a in_o power_n only_o then_o i_o say_v that_o unto_o the_o line_n ab_fw-la can_v be_v join_v any_o other_o rational_a line_n commensurable_a in_o power_n only_o to_o the_o whole_a line_n impossibility_n for_o if_o it_o be_v possible_a let_v bd_o be_v such_o a_o line_n add_v unto_o it_o wherefore_o the_o line_n ad_fw-la and_o db_o be_v rational_a commensurable_a in_o power_n only_o and_o forasmuch_o as_o how_o much_o the_o square_n of_o the_o line_n ad_fw-la and_o db_o do_v exceed_v that_o which_o be_v contain_v under_o the_o line_n ad_fw-la and_o db_o twice_o so_o much_o also_o do_v the_o square_n of_o the_o line_n ac_fw-la and_o cb_o exceed_v that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o twice_o for_o the_o excess_n of_o each_o be_v one_o and_o the_o same_o namely_o the_o square_a of_o the_o line_n ab_fw-la by_o the_o 7._o of_o the_o second_o wherefore_o alternate_o by_o the_o â—ormer_a assumpt_n of_o campanus_n how_o much_o the_o square_n of_o the_o line_n ad_fw-la and_o db_o do_v exceed_v the_o square_n of_o the_o line_n ac_fw-la &_o cb_o so_o much_o also_o exceed_v that_o which_o be_v contain_v under_o the_o line_n ad_fw-la and_o db_o twice_o that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o twice_o but_o that_o which_o be_v make_v of_o the_o square_n of_o the_o line_n ad_fw-la and_o db_o add_v together_o exceed_v that_o which_o be_v make_v of_o the_o square_n of_o the_o line_n ac_fw-la and_o cb_o add_v together_o by_o a_o rational_a superficies_n for_o they_o be_v either_o of_o they_o rational_a wherefore_o that_o which_o be_v contain_v under_o the_o line_n ad_fw-la and_o db_o twice_o exceed_v that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o twice_o by_o a_o rational_a superficies_n but_o that_o which_o be_v contain_v under_o the_o line_n ad_fw-la and_o db_o twice_o be_v medial_a for_o it_o be_v commensurable_a to_o that_o which_o be_v contain_v under_o the_o line_n ad_fw-la and_o db_o once_o which_o superficies_n be_v medial_a by_o the_o 21._o of_o the_o ten_o and_o by_o the_o same_o reason_n also_o that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o twice_o be_v medial_a wherefore_o a_o medial_a superficies_n differ_v from_o a_o medial_a superficies_n by_o a_o rational_a superficies_n which_o by_o the_o 26._o of_o the_o ten_o be_v impossible_a wherefore_o unto_o the_o line_n ab_fw-la can_v be_v join_v any_o other_o rational_a line_n beside_o bc_o commensurable_a in_o power_n only_o to_o the_o whole_a line_n wherefore_o unto_o a_o residual_a line_n can_v be_v join_v one_o only_o right_a line_n rational_a and_o commensurable_a in_o power_n only_o to_o the_o whole_a line_n which_o be_v require_v to_o be_v demonstrate_v ¶_o the_o 62._o theorem_a the_o 80._o proposition_n unto_o a_o first_o medial_n residual_a line_n can_v be_v join_v one_o only_o medial_a right_a line_n commensurable_a in_o power_n only_o to_o the_o whole_a line_n and_o comprehend_v with_o the_o whole_a line_n a_o rational_a superficies_n svppose_v that_o ab_fw-la be_v a_o first_o medial_a residual_a line_n &_o unto_o ab_fw-la join_v the_o line_n bc_o so_o that_o let_v the_o line_n ac_fw-la and_o bc_o be_v medial_a commensurable_a in_o power_n only_o &_o let_v that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o bc_o be_v rational_a then_o i_o say_v that_o unto_o the_o line_n ab_fw-la can_v be_v join_v any_o other_o medial_a line_n commensurable_a in_o power_n only_o to_o the_o whole_a line_n and_o comprehend_v together_o with_o the_o whole_a line_n a_o rational_a superficies_n absurdity_n for_o if_o it_o be_v possible_a let_v the_o line_n bd_o be_v such_o a_o line_n wherefore_o the_o line_n ad_fw-la and_o db_o be_v medial_a commensurable_a in_o power_n only_o and_o that_o which_o be_v contain_v under_o the_o line_n ad_fw-la and_o db_o be_v rational_a and_o forasmuch_o as_o how_o much_o the_o square_n of_o the_o line_n ad_fw-la and_o db_o exceed_v that_o which_o be_v contain_v under_o the_o line_n ad_fw-la and_o db_o twice_o so_o much_o also_o exceed_v the_o square_n of_o the_o line_n ac_fw-la &_o bc_o that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o twice_o for_o the_o excess_n of_o each_o be_v one_o and_o the_o same_o namely_o the_o square_a of_o the_o line_n ab_fw-la wherefore_o alternate_o as_o it_o be_v say_v in_o the_o former_a proposition_n how_o much_o the_o square_n of_o the_o line_n ad_fw-la and_o db_o exceed_v the_o square_n of_o the_o line_n ac_fw-la and_o cb_o so_o much_o also_o that_o which_o be_v contain_v under_o the_o line_n ad_fw-la and_o db_o twice_o exceed_v that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o twice_o but_o that_o which_o be_v contain_v under_o the_o line_n ad_fw-la and_o db_o twice_o exceed_v that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o twice_o by_o a_o rational_a superficies_n for_o they_o be_v either_o of_o they_o a_o rational_a supersicy_n wherefore_o that_o which_o be_v make_v of_o the_o square_n of_o the_o line_n ad_fw-la &_o db_o exceed_v that_o which_o be_v make_v of_o the_o square_n of_o the_o line_n ac_fw-la &_o cb_o by_o a_o rational_a superficies_n which_o by_o the_o 26._o of_o the_o ten_o be_v impossible_a for_o they_o be_v either_o of_o they_o medial_a for_o those_o four_o line_n be_v put_v to_o be_v medial_a wherefore_o unto_o a_o first_o medial_a residual_a line_n can_v be_v join_v only_a one_o right_a medial_a line_n commensurable_a in_o power_n only_o to_o the_o whole_a line_n and_o comprehend_v with_o the_o whole_a line_n a_o rational_a superficies_n which_o be_v require_v to_o be_v prove_v ¶_o the_o 63._o theorem_a the_o 81._o proposition_n unto_o a_o second_o medial_a residual_a line_n can_v be_v join_v only_a one_o medial_a right_a line_n commensurable_a in_o power_n only_o to_o the_o whole_a line_n and_o comprehend_v with_o the_o whole_a line_n a_o medial_a superficies_n svppose_v that_o ab_fw-la be_v a_o second_o medial_a residual_a line_n &_o unto_o the_o line_n ab_fw-la join_v the_o line_n bc_o so_o that_o let_v the_o line_n ac_fw-la and_o cb_o be_v medial_a commensurable_a in_o power_n only_o and_o let_v that_o which_o be_v comprehend_v under_o the_o line_n ac_fw-la and_o cb_o be_v medial_a then_o i_o say_v that_o unto_o the_o line_n ab_fw-la can_v not_o be_v join_v any_o other_o medial_a right_a line_n commensurable_a in_o power_n only_o to_o the_o whole_a line_n and_o comprehend_v together_o with_o the_o whole_a line_n a_o medial_a superficies_n for_o if_o it_o be_v possible_a let_v the_o line_n bd_o be_v such_o a_o line_n wherefore_o the_o line_n ad_fw-la &_o db_o be_v medial_a commensurable_a in_o power_n only_o and_o that_o which_o be_v contain_v under_o the_o line_n ad_fw-la and_o db_o be_v also_o medial_a take_v a_o rational_a line_n ef._n construction_n and_o by_o the_o 44._o of_o the_o first_o unto_o the_o line_n of_o apply_v the_o parallelogram_n eglantine_n equal_a to_o the_o square_n of_o the_o line_n ac_fw-la and_o cb_o and_o make_v in_o breadth_n the_o line_n em_n and_o from_o that_o parallelogram_n eglantine_n take_v away_o the_o parallelogram_n hg_o equal_a to_o that_o which_o be_v contain_v under_o ac_fw-la and_o cb_o twice_o and_o make_v in_o breadth_n the_o line_n he_o wherefore_o the_o parallelogram_n remain_v namely_o el_n be_v by_o the_o 7._o of_o the_o second_o equal_a to_o the_o square_n of_o the_o line_n ab_fw-la wherefore_o the_o line_n ab_fw-la contain_v in_o power_n the_o parallelogram_n el._n again_o unto_o the_o line_n of_o apply_v by_o the_o 44._o of_o the_o first_o the_o parallelogram_n ei_o equal_a to_o the_o square_n of_o the_o line_n ad_fw-la and_o db_o and_o make_v in_o breadth_n the_o line_n en_fw
but_o the_o square_n of_o the_o line_n ad_fw-la and_o db_o be_v equal_a to_o that_o which_o be_v contain_v under_o the_o line_n ad_fw-la and_o db_o twice_o and_o to_o the_o square_n of_o the_o line_n ab_fw-la absurdity_n wherefore_o the_o parallelogram_n ei_o be_v equal_a to_o that_o which_o be_v contain_v under_o the_o line_n ad_fw-la &_o db_o twice_o and_o to_o the_o square_n of_o the_o line_n ab_fw-la but_o the_o parallelogram_n el_n be_v equal_a to_o the_o square_n of_o the_o line_n ab_fw-la wherefore_o the_o parallelogram_n remain_v namely_o high_a be_v equal_a to_o that_o which_o be_v contain_v under_o the_o line_n ad_fw-la and_o db_o twice_o and_o forasmuch_o as_o the_o line_n ac_fw-la and_o cb_o be_v â—â—â—iall_a therefore_o the_o square_n also_o of_o the_o line_n ac_fw-la and_o cb_o be_v medial_a and_o they_o be_v equal_a to_o the_o parallelogram_n eglantine_n wherefore_o the_o parallelogram_n eglantine_n be_v by_o that_o which_o be_v speak_v in_o the_o 75._o proposition_n medial_a and_o it_o be_v apply_v unto_o the_o rational_a line_n of_o make_v in_o breadth_n the_o line_n em_n wherefore_o by_o the_o 22._o of_o the_o ten_o the_o line_n em_n be_v rational_a and_o incommensurable_a in_o length_n to_o the_o line_n ef._n again_o forasmuch_o as_o that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o be_v medial_a therefore_o by_o the_o corollary_n of_o the_o 23._o of_o the_o ten_o that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o twice_o be_v also_o medial_a and_o it_o be_v equal_a to_o the_o parallelogram_n hg_o wherefore_o also_o the_o parallelogram_n hg_o be_v medial_a and_o be_v apply_v to_o the_o rational_a line_n of_o make_v in_o breadth_n the_o line_n he_o wherefore_o by_o the_o 22_o of_o the_o ten_o the_o line_n he_o be_v rational_a and_o incommensurable_a in_o length_n to_o the_o line_n ef._n and_o forasmuch_o as_o the_o line_n ac_fw-la and_o cb_o be_v commensurable_a in_o power_n only_o therefore_o the_o line_n ac_fw-la be_v incommensurable_a in_o length_n to_o the_o line_n cb._n but_o as_o the_o line_n ac_fw-la be_v to_o the_o line_n cb_o so_o by_o the_o assumpt_n go_v before_o the_o 22._o of_o the_o ten_o be_v the_o square_a of_o the_o line_n ac_fw-la to_o that_o which_o be_v contain_v under_o the_o line_n ac_fw-la &_o cb._n wherefore_o by_o the_o 10._o of_o the_o ten_o the_o square_a of_o the_o line_n ac_fw-la be_v incommensurable_a to_o that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb._n but_o unto_o the_o square_n of_o the_o line_n ac_fw-la be_v commensurable_a the_o square_n of_o ac_fw-la &_o cb_o and_o unto_o that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o be_v commensurable_a that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o twice_o wherefore_o the_o square_n of_o the_o line_n ac_fw-la &_o cb_o be_v incommensurable_a to_o that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o twice_o but_o unto_o the_o square_n of_o the_o line_n ac_fw-la and_o cb_o be_v equal_a the_o parallelogram_n eglantine_n and_o unto_o that_o which_o be_v contain_v under_o the_o line_n ac_fw-la &_o cb_o twice_o be_v equal_a the_o parallelogram_n gh_o wherefore_o the_o parallelogram_n eglantine_n be_v incommensurable_a to_o the_o parallelogram_n hg_o but_o as_o the_o parallelogram_n eglantine_n be_v to_o the_o parallelogrmme_n hg_o so_o be_v the_o line_n em_n to_o the_o line_n he_o wherefore_o the_o line_n em_n be_v incommensurable_a in_o length_n to_o the_o line_n he_o and_o they_o be_v both_o rational_a line_n wherefore_o the_o line_n em_n and_o mh_o be_v rational_a commensurable_a in_o power_n only_o wherefore_o the_o line_n eh_o be_v a_o residual_a line_n and_o unto_o it_o be_v join_v a_o rational_a line_n he_o commensurable_a in_o power_n only_o to_o the_o whole_a line_n em_n in_o like_a sort_n also_o may_v it_o be_v prove_v that_o unto_o the_o line_n eh_o be_v join_v the_o line_n hn_o be_v also_o rational_a and_o commensurable_a in_o power_n only_o to_o the_o whole_a line_n en_fw-fr wherefore_o unto_o a_o residual_a line_n be_v join_v â—â—re_o than_o one_o only_a line_n commensurable_a in_o power_n only_o to_o the_o whole_a line_n which_o by_o the_o 79._o of_o the_o ten_o be_v impossible_a wherefore_o unto_o a_o second_o medial_a residual_a line_n can_v be_v join_v only_a one_o medial_a right_a line_n commensurable_a in_o power_n only_o to_o the_o whole_a line_n and_o comprehend_v with_o the_o whole_a line_n a_o medial_a superficies_n which_o be_v require_v to_o be_v demonstrate_v ¶_o the_o 64._o theorem_a the_o 82._o proposition_n unto_o a_o less_o line_n can_v be_v join_v only_a one_o right_a line_n incommensurable_a in_o power_n to_o the_o whole_a line_n and_o make_v together_o with_o the_o whole_a line_n that_o which_o be_v make_v of_o their_o square_n add_v together_o rational_a and_o that_o which_o be_v contain_v under_o they_o medial_a svppose_v that_o ab_fw-la be_v a_o less_o line_n and_o to_o ab_fw-la join_v the_o line_n bc_o so_o that_o let_v bc_o be_v such_o a_o line_n as_o be_v require_v in_o the_o theorem_a wherefore_o the_o line_n ac_fw-la and_o cb_o be_v incommensurable_a in_o power_n have_v that_o which_o be_v make_v of_o the_o square_n of_o they_o add_v together_o rational_a and_o that_o which_o be_v contain_v under_o they_o medial_a then_o i_o say_v that_o unto_o ab_fw-la can_v be_v join_v any_o other_o such_o right_a line_n absurdity_n for_o if_o it_o be_v possible_a lâ—t_v the_o line_n bd_o be_v such_o a_o line_n wherefore_o the_o line_n ad_fw-la &_o db_o be_v incommensurable_a in_o power_n have_v that_o which_o be_v make_v of_o the_o square_n of_o they_o add_v together_o rational_a and_o that_o which_o be_v contain_v under_o they_o medial_a and_o for_o that_o how_o much_o the_o square_n of_o the_o line_n ad_fw-la and_o db_o exceed_v the_o square_n of_o the_o line_n ac_fw-la and_o cb_o so_o much_o that_o which_o be_v contain_v under_o the_o line_n ad_fw-la and_o db_o twice_o exceed_v that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o twice_o by_o those_o thing_n which_o be_v speak_v in_o the_o 79._o proposition_n but_o that_o which_o be_v make_v of_o the_o square_n of_o the_o line_n ad_fw-la and_o db_o add_v together_o exceed_v that_o which_o be_v make_v of_o the_o square_n of_o the_o line_n ac_fw-la and_o cb_o add_v together_o by_o a_o rational_a superficies_n for_o they_o be_v either_o of_o they_o rational_a by_o supposition_n wherefore_o that_o which_o be_v contain_v under_o the_o line_n ad_fw-la and_o db_o twice_o exceed_v that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o twice_o by_o a_o rational_a superficies_n which_o by_o the_o 26._o of_o the_o ten_o be_v impossible_a for_o either_o of_o they_o be_v medial_a by_o supposition_n wherefore_o unto_o a_o less_o line_n can_v be_v join_v only_a one_o right_a line_n incommensurable_a in_o power_n to_o the_o whole_a line_n and_o make_v together_o with_o the_o whole_a line_n that_o which_o be_v make_v of_o their_o square_n add_v together_o rational_a and_o that_o which_o be_v contain_v under_o they_o medial_a which_o be_v require_v to_o be_v demonstrate_v ¶_o the_o 65._o theorem_a the_o 83._o proposition_n unto_o a_o line_n make_v with_o a_o rational_a superficies_n the_o whole_a superficies_n medial_a can_v be_v join_v only_a one_o right_a line_n incommensurable_a in_o power_n to_o the_o whole_a line_n and_o make_v together_o with_o the_o whole_a line_n that_o which_o be_v make_v of_o their_o square_n add_v together_o medial_a and_o that_o which_o be_v contain_v under_o they_o rational_a svppose_v that_o ab_fw-la be_v a_o line_n make_v with_o a_o rational_a superficies_n the_o whole_a superficies_n medial_a and_o unto_o it_o let_v the_o line_n bc_o be_v join_v so_o that_o let_v bc_o be_v such_o a_o line_n as_o be_v require_v in_o the_o theorem_a wherefore_o the_o line_n ac_fw-la and_o cb_o be_v incommensurable_a in_o power_n haâ—ing_v that_o which_o be_v make_v of_o the_o square_n of_o the_o line_n ac_fw-la and_o cb_o add_v together_o medial_a and_o that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o rational_a impossibility_n then_o i_o say_v that_o unto_o the_o line_n ab_fw-la can_v be_v join_v any_o other_o such_o line_n for_o if_o it_o be_v possible_a let_v the_o line_n bd_o be_v such_o a_o line_n wherefore_o the_o line_n ad_fw-la and_o db_o be_v incommensurable_a in_o power_n have_v that_o which_o be_v make_v of_o the_o square_n of_o the_o line_n ad_fw-la and_o db_o add_v together_o medial_a and_o that_o which_o be_v contain_v under_o the_o line_n ad_fw-la and_o db_o rational_a now_o for_o that_o how_o much_o the_o square_n of_o the_o line_n ad_fw-la and_o db_o exceed_v the_o square_n of_o the_o line_n ac_fw-la and_o cb_o so_o much_o
three_o be_v produce_v when_o the_o square_a of_o the_o whole_a namely_o of_o that_o which_o be_v make_v of_o the_o residual_a line_n and_o the_o line_n join_v unto_o it_o add_v together_o exceed_v the_o square_a of_o the_o line_n join_v to_o the_o residual_a by_o the_o square_n of_o a_o line_n which_o be_v commensurable_a unto_o it_o in_o length_n and_o as_o the_o three_o last_o kind_n of_o binomial_n namely_o the_o four_o five_o and_o six_o be_v produce_v when_o the_o square_a of_o the_o great_a part_n exceed_v the_o square_a of_o the_o less_o by_o the_o square_n of_o a_o line_n incommensurable_a in_o length_n unto_o it_o even_o so_o the_o three_o last_o kind_n of_o residual_a line_n be_v produce_v when_o the_o square_a of_o the_o whole_a exceed_v the_o square_a of_o the_o line_n adjoin_v by_o the_o square_n of_o a_o line_n incommensurable_a unto_o it_o in_o length_n as_o you_o may_v perceive_v by_o their_o definition_n follow_v definition_n a_o first_o residual_a line_n be_v when_o the_o square_a of_o the_o whole_a exceed_v the_o square_a of_o the_o line_n adjoin_v by_o the_o square_n of_o a_o line_n commensurable_a unto_o it_o in_o length_n and_o also_o the_o whole_a be_v commensurable_a in_o length_n to_o the_o rational_a line_n first_o set_v as_o let_v ab_fw-la be_v a_o rational_a line_n who_o part_n be_v certain_a distinct_a &_o to_o be_v express_v by_o number_n and_o let_v the_o residual_a line_n be_v cd_o and_o let_v the_o line_n join_v unto_o it_o be_v ecâ—_n and_o let_v the_o whole_a be_v compose_v of_o the_o residual_a cd_o and_o the_o line_n adjoin_v ec_o be_v the_o line_n ed_z let_v moreover_o the_o square_a of_o the_o whole_a line_n ed_z exceed_v the_o square_n of_o the_o line_n adjoin_v ec_o by_o the_o square_n of_o the_o line_n f_o which_o line_n fletcher_n let_v be_v commensurable_a in_o length_n to_o the_o whole_a line_n ed_z and_o let_v the_o whole_a line_n ed_z be_v also_o commensurable_a in_o length_n to_o the_o rational_a line_n ab_fw-la then_o be_v the_o residual_a line_n cd_o by_o this_o definition_n a_o first_o residual_a line_n definition_n a_o second_o residual_a line_n be_v when_o the_o square_a of_o the_o whole_a exceed_v the_o square_a of_o the_o line_n adjoin_v by_o the_o square_n of_o a_o line_n commensurable_a unto_o it_o in_o length_n and_o also_o the_o line_n adjoin_v be_v commensurable_a in_o length_n to_o the_o rational_a line_n as_o suppose_v the_o line_n cd_o to_o be_v a_o residual_a and_o let_v the_o line_n adjoin_v unto_o it_o be_v ec_o and_o the_o whole_a make_v of_o they_o both_o let_v be_v the_o line_n ed_z &_o let_v the_o square_n of_o ed_z the_o whole_a line_n exceed_v the_o square_n of_o the_o line_n adjoin_v ec_o by_o the_o square_n of_o the_o line_n fletcher_n and_o let_v the_o line_n f_o be_v commensurable_a in_o length_n to_o the_o whole_a line_n ed_z moreover_o let_v the_o line_n adjoin_v ec_o be_v commensurable_a in_o length_n to_o the_o rational_a line_n ab_fw-la then_o by_o this_o definition_n the_o residual_a line_n cd_o be_v a_o second_o residual_a line_n definition_n a_o three_o residual_a line_n be_v when_o the_o square_a of_o the_o whole_a exceed_v the_o square_a of_o the_o line_n adjoin_v by_o the_o square_n of_o a_o line_n commensurable_a unto_o it_o in_o length_n and_o neither_o the_o whole_a line_n nor_o the_o line_n adjoin_v be_v commensurable_a in_o length_n to_o the_o rational_a line_n as_o the_o former_a supposition_n stand_v suppose_v that_o the_o square_a of_o the_o whole_a line_n ed_z exceed_v the_o square_n of_o the_o line_n adjoin_v ec_o by_o the_o square_n of_o the_o line_n fletcher_n and_o let_v the_o line_n f_o be_v commensurable_a in_o length_n to_o the_o whole_a line_n ec_o and_o let_v neither_o the_o whole_a line_n ed_z nor_o the_o line_n adjoin_v ec_o be_v commensurable_a in_o length_n to_o the_o rational_a line_n ab_fw-la then_o by_o this_o definition_n the_o residual_a line_n cd_o be_v a_o three_o residual_a line_n a_o four_o residual_a line_n be_v definition_n when_o the_o square_a of_o the_o whole_a line_n exceed_v the_o square_a of_o the_o line_n adjoin_v by_o the_o square_n of_o a_o line_n incommensurable_a unto_o it_o in_o length_n and_o the_o whole_a line_n be_v also_o commensurable_a in_o length_n to_o the_o rational_a line_n as_o the_o residual_a line_n be_v as_o before_o cd_o &_o the_o line_n adjoin_v ec_o and_o the_o whole_a ed_z let_v the_o square_n of_o the_o whole_a line_n ed_z exceed_v the_o square_n of_o the_o line_n adjoin_v ec_o by_o the_o square_n of_o the_o line_n fletcher_n and_o let_v the_o line_n f_o be_v incommensurable_a in_o length_n to_o the_o whole_a line_n ed_z and_o let_v ed_z the_o whole_a line_n be_v commensurable_a in_o length_n to_o the_o rational_a line_n ab_fw-la then_o be_v the_o residual_a line_n cd_o by_o this_o declaration_n a_o four_o residual_a line_n a_o five_o residual_a line_n be_v definition_n when_o the_o square_a of_o the_o whole_a line_n exceed_v the_o square_a of_o the_o line_n adjoin_v by_o the_o square_n of_o a_o line_n incommensurable_a unto_o it_o in_o length_n and_o the_o line_n adjoin_v be_v commensurable_a in_o length_n to_o the_o rational_a line_n as_o the_o residual_a line_n be_v cd_o the_o line_n adjoin_v ec_o and_o the_o whole_a line_n ed_z let_v the_o square_n of_o the_o whole_a line_n ed_z exceed_v the_o square_n of_o the_o line_n adjoin_v ec_o by_o the_o square_n of_o the_o line_n fletcher_n and_o let_v the_o line_n f_o be_v incommensurable_a in_o length_n to_o the_o whole_a line_n ed_z and_o let_v also_o ec_o the_o line_n adjoin_v be_v commensurable_a in_o length_n to_o the_o rational_a line_n ae_n then_o shall_v the_o residual_a cd_o be_v by_o this_o definition_n a_o five_o residual_a line_n a_o six_o residual_a line_n be_v when_o the_o square_a of_o the_o whole_a line_n definition_n exceed_v the_o square_a of_o the_o line_n adjoin_v by_o the_o square_n of_o a_o line_n incommensurable_a unto_o it_o in_o length_n and_o neither_o the_o whole_a line_n nor_o the_o line_n adjoin_v be_v commensurable_a in_o length_n to_o the_o rational_a line_n as_o suppose_v the_o residual_a line_n to_o be_v cd_o and_o the_o line_n adjoin_v to_o be_v â—c_z and_o the_o whole_a line_n compose_v of_o they_o let_v be_v ed_z and_o let_v the_o square_n of_o the_o whole_a line_n â—d_a exceed_v the_o square_n of_o the_o line_n adjoin_v by_o the_o square_n of_o the_o line_n fletcher_n which_o line_n fletcher_n let_v be_v incommensurable_a in_o length_n to_o the_o whole_a line_n ed_z moreover_o let_v neither_o the_o whole_a line_n cd_o nor_o the_o line_n adjoin_v ec_o be_v commensurable_a in_o length_n to_o the_o rational_a line_n aâ—_z then_o shall_v the_o residual_a line_n cd_o be_v by_o this_o explication_n a_o six_o residual_a line_n and_o the_o last_o ¶_o the_o 19_o problem_n the_o 85._o proposition_n to_o find_v out_o a_o first_o residual_a line_n take_v a_o rational_a line_n and_o let_v the_o same_o be_v a_o senary_n and_o unto_o it_o let_v the_o line_n bg_o be_v commensurable_a in_o length_n wherefore_o the_o line_n bg_o also_o be_v rational_a and_o take_v two_o square_a number_n de_fw-fr and_o of_o which_o let_v be_v such_o construction_n that_o the_o excess_n of_o the_o great_a namely_o of_o de_fw-fr above_o the_o less_o of_o which_o excess_n let_v be_v the_o number_n df_o be_v no_o square_a number_n by_o the_o corollary_n of_o the_o first_o assumpt_n of_o the_o 28._o of_o the_o ten_o wherefore_o the_o number_n ed_z have_v not_o to_o the_o number_n df_o that_o proportion_n that_o a_o square_a number_n have_v to_o a_o square_a number_n by_o the_o 24._o of_o the_o eight_o and_o as_o the_o number_n ed_z be_v to_o the_o number_n df_o so_o let_v the_o square_n of_o the_o line_n bg_o be_v to_o the_o square_n of_o the_o line_n gc_o by_o the_o corollary_n of_o the_o six_o of_o the_o ten_o demonstration_n wherefore_o the_o square_a of_o the_o line_n bg_o be_v commensurable_a to_o the_o square_n of_o the_o line_n gc_o but_o the_o square_n of_o the_o line_n bg_o be_v rational_a wherefore_o also_o the_o square_a of_o the_o line_n gc_o be_v rational_a wherefore_o the_o line_n gc_o be_v also_o rational_a and_o forasmuch_o as_o the_o number_n ed_z have_v not_o to_o the_o number_n df_o that_o proportion_n that_o a_o square_a number_n have_v to_o a_o square_a number_n therefore_o neither_o also_o have_v the_o square_a oâ—_n the_o line_n bg_o to_o the_o square_n of_o the_o line_n gc_o that_o proportion_n that_o a_o square_a number_n have_v to_o a_o square_a number_n wherefore_o by_o the_o 9_o of_o the_o ten_o the_o line_n bg_o be_v incommensurable_a in_o length_n to_o the_o line_n gc_o and_o they_o be_v both_o rational_a wherefore_o the_o line_n bg_o and_o gc_o be_v rational_a commensurable_a in_o power_n only_o
wherefore_o the_o line_n bc_o be_v a_o residual_a line_n i_o say_v moreover_o that_o it_o be_v a_o first_o residual_a line_n for_o forasmuch_o as_o the_o square_n of_o the_o line_n bg_o be_v great_a than_o the_o square_a of_o the_o line_n gc_o that_o it_o be_v great_a it_o be_v manifest_a for_o by_o supposition_n the_o square_a of_o the_o line_n bg_o be_v to_o the_o square_n of_o the_o line_n gc_o as_o the_o great_a number_n namely_o ed_z be_v to_o the_o number_n df_o unto_o the_o square_n of_o the_o line_n bg_o let_v the_o square_n of_o the_o line_n gc_o and_o h_o be_v equal_a and_o for_o that_o as_o the_o number_n de_fw-fr be_v to_o the_o number_n df_o so_o be_v the_o square_a of_o the_o line_n bg_o to_o the_o square_n of_o the_o line_n gc_o therefore_o by_o conversion_n of_o proportion_n by_o the_o corollary_n of_o the_o 9_o of_o the_o five_o as_o the_o number_n de_fw-fr be_v to_o the_o number_n of_o so_o be_v the_o square_a of_o the_o line_n bg_o to_o the_o square_n of_o the_o line_n h._n but_o the_o number_n de_fw-fr have_v to_o the_o number_n of_o that_o proportion_n that_o a_o square_a number_n have_v to_o a_o square_a number_n for_o either_o of_o they_o be_v a_o square_a number_n wherefore_o also_o the_o square_a of_o the_o line_n bg_o have_v to_o the_o square_n of_o the_o line_n h_o that_o proportion_n that_o a_o square_a numbeâ—_n have_v to_o a_o square_a number_n wherefore_o the_o line_n gb_o be_v commensurable_a in_o length_n to_o the_o line_n h._n wherefore_o the_o line_n gb_o be_v in_o power_n more_o than_o the_o line_n gc_o by_o the_o square_n of_o a_o line_n commensurable_a in_o length_n to_o the_o line_n gb_o and_o the_o whole_a line_n namely_o gb_o be_v commensurable_a in_o length_n to_o the_o rational_a line_n a._n wherefore_o the_o line_n bc_o be_v a_o first_o residual_a line_n wherefore_o there_o be_v find_v out_o a_o first_o residual_a line_n which_o be_v require_v to_o be_v do_v ¶_o the_o 20._o problem_n the_o 86._o proposition_n to_o find_v out_o a_o second_o residual_a line_n construction_n take_v a_o rational_a line_n and_o let_v the_o same_o be_v a_o and_o unto_o it_o let_v the_o line_n gc_o be_v commensurable_a in_o length_n and_o take_v two_o square_a number_n de_fw-fr and_o of_o and_o let_v they_o be_v such_o that_o the_o excess_n of_o the_o great_a namely_o df_o be_v no_o square_a number_n and_o as_o the_o number_n df_o be_v to_o the_o number_n de_fw-fr so_o let_v the_o square_n of_o the_o line_n gc_o be_v to_o the_o square_n of_o the_o line_n gb_o wherefore_o both_o the_o square_n be_v commensurable_a and_o forasmuch_o as_o the_o square_n of_o the_o line_n gc_o be_v rational_a demoâ—stratiâ—â—_n therefore_o the_o square_a of_o the_o line_n bg_o be_v also_o rational_a wherefore_o also_o the_o line_n bg_o be_v rational_a and_o forasmuch_o as_o the_o square_n of_o the_o line_n bg_o &_o gc_o have_v not_o that_o proportion_n the_o one_o to_o the_o other_o that_o a_o square_a number_n have_v to_o a_o square_a number_n therefore_o the_o line_n bg_o and_o gc_o be_v incommensurable_a in_o length_n and_o they_o be_v both_o rational_a wherefore_o the_o line_n bg_o and_o gc_o be_v rational_a commensurable_a in_o power_n only_o wherefore_o the_o line_n bc_o be_v a_o residual_a line_n i_o say_v moreover_o that_o it_o be_v a_o second_o residual_a line_n for_o forasmuch_o as_o the_o square_n of_o the_o line_n bg_o be_v great_a they_o the_o square_n of_o the_o line_n gc_o unto_o the_o square_n of_o the_o line_n bg_o let_v the_o square_n of_o the_o line_n gâ—_n &_o h_o be_v equal_a and_o for_o that_o as_o the_o number_n de_fw-fr be_v to_o the_o number_n df_o so_o be_v the_o square_a of_o the_o line_n gb_o to_o the_o square_n of_o the_o line_n gc_o therefore_o by_o conversion_n of_o proportion_n as_o the_o number_n de_fw-fr be_v to_o the_o number_n of_o so_o be_v the_o square_a of_o the_o line_n bg_o to_o the_o square_n of_o the_o line_n h._n but_o either_o of_o these_o number_n de_fw-fr and_o of_o be_v a_o square_a number_n wherefore_o the_o line_n gb_o be_v commensurable_a in_o length_n to_o the_o line_n h._n wherefore_o the_o line_n bg_o be_v in_o power_n more_o than_o the_o line_n gc_o by_o the_o square_n of_o a_o line_n commensurable_a in_o length_n to_o the_o line_n bg_o and_o the_o line_n gc_o that_o be_v join_v to_o the_o residual_a line_n be_v commensurable_a in_o length_n to_o the_o rational_a line_n a._n wherefore_o the_o line_n bc_o be_v a_o second_o residual_a line_n wherefore_o there_o be_v find_v out_o a_o second_o residual_a line_n which_o be_v require_v to_o be_v do_v ¶_o the_o 21._o problem_n the_o 87._o proposition_n to_o find_v out_o a_o three_o residual_a line_n take_v rational_a line_n &_o let_v the_o same_o be_v a_o and_o take_v three_o number_n e_o b_o c_o construction_n and_o cd_o not_o have_v the_o one_o to_o the_o other_o that_o proportion_n that_o a_o square_a number_n have_v to_o a_o square_a number_n and_o let_v the_o number_n bc_o have_v to_o the_o number_n bd_o that_o proportion_n that_o a_o square_a number_n have_v to_o a_o square_a number_n and_o let_v the_o number_n bc_o be_v great_a than_o the_o number_n cd_o and_o as_o the_o number_n e_o be_v to_o the_o number_n bc_o so_o let_v the_o square_n of_o the_o line_n a_o be_v to_o the_o square_n of_o the_o line_n fg_o and_o as_o the_o number_n bc_o be_v to_o the_o number_n cd_o so_o let_v the_o square_n of_o the_o line_n fg_o be_v to_o the_o square_n of_o the_o line_n hg_o demonstration_n wherefore_o the_o square_a of_o the_o line_n a_o be_v commensurable_a to_o the_o square_n of_o the_o line_n fg._n but_o the_o square_n of_o the_o line_n a_o be_v rational_a wherefore_o also_o the_o square_a of_o the_o line_n fg_o be_v rational_a wherefore_o the_o line_n fg_o be_v also_o rational_a and_o forasmuch_o as_o the_o number_n e_o have_v not_o to_o the_o number_n bc_o that_o proportion_n that_o a_o square_a numberâ—_n have_v to_o a_o square_a number_n therefore_o neither_o also_o have_v the_o square_a of_o the_o line_n a_o to_o the_o square_n of_o the_o line_n fg_o that_o proportion_n that_o a_o square_a number_n have_v to_o a_o square_a number_n wherefore_o the_o line_n a_o be_v incommensurable_a in_o length_n to_o the_o line_n fg._n again_o for_o that_o as_o the_o number_n bc_o be_v to_o the_o number_n cd_o so_o be_v the_o square_a of_o the_o line_n fg_o to_o the_o square_n of_o the_o line_n hg_o therefore_o the_o square_a of_o the_o line_n fg_o be_v commensurable_a to_o the_o square_n of_o the_o line_n hg_o but_o the_o square_n of_o the_o line_n fg_o be_v rational_a wherefore_o also_o the_o square_a of_o the_o line_n hg_o be_v rational_a wherefore_o also_o the_o line_n hg_o be_v rational_a and_o for_o that_o the_o number_n bc_o have_v not_o to_o the_o number_n cd_o that_o proportion_n that_o a_o square_a number_n have_v to_o a_o square_a number_n therefore_o neither_o also_o have_v the_o square_a of_o the_o line_n â—g_z to_o the_o square_n of_o the_o line_n hg_o that_o proportion_n that_o a_o square_a number_n have_v to_o a_o square_a number_n wherefore_o the_o line_n fg_o be_v incommensurable_a in_o length_n to_o the_o line_n hg_o and_o they_o be_v both_o rational_a wherefore_o the_o line_n fg_o &_o hg_o be_v rational_a commensurable_a in_o power_n only_o wherefore_o the_o line_n fh_o be_v a_o residual_a line_n i_o say_v moreover_o that_o it_o be_v a_o three_o residual_a line_n for_o for_o that_o as_o the_o number_n e_o be_v to_o the_o number_n bc_o so_o be_v the_o square_a of_o the_o line_n a_o to_o the_o square_n of_o the_o line_n fg_o and_o as_o the_o number_n â—c_z be_v to_o the_o number_n cd_o so_o be_v the_o square_a of_o the_o line_n fg_o to_o the_o square_n of_o the_o line_n hgâ—_n therefore_o by_o equality_n of_o proportion_n as_o the_o number_n e_o be_v to_o the_o number_n cd_o so_o be_v the_o square_a of_o the_o line_n a_o to_o the_o square_n of_o the_o line_n hgâ—_n but_o the_o number_n e_o have_v â—â—â—_o to_o the_o number_n cd_o that_o proportion_n that_o a_o square_a numâ—â—r_n have_v to_o a_o square_a number_n therefore_o neither_o also_o have_v the_o square_a of_o the_o line_n a_o to_o the_o square_n of_o the_o line_n hg_o that_o proportion_n that_o a_o square_a number_n have_v to_o a_o square_a number_n therefore_o the_o line_n a_o be_v incommensurable_a in_o length_n to_o the_o line_n hg_o wherefore_o neither_o of_o the_o line_n fg_o and_o hg_o be_v commensurable_a in_o length_n to_o the_o rational_a line_n a._n and_o forasmuch_o as_o the_o square_n of_o the_o line_n fg_o be_v great_a than_o the_o square_a of_o the_o line_n hg_o that_o the_o line_n fg_o be_v great_a than_o the_o line_n hg_o it_o be_v
manife_a for_o by_o supposition_n the_o number_n bc_o be_v great_a than_o the_o number_n cd_o unto_o the_o square_n of_o the_o line_n fg_o let_v the_o square_n of_o the_o line_n hg_o &_o kâ—_n be_v equal_a and_o for_o that_o as_o the_o number_n bc_o be_v to_o the_o number_n cd_o so_o be_v the_o square_a of_o the_o line_n fg_o to_o the_o square_n of_o the_o line_n hâ—â—â—erfore_o by_o conversion_n of_o proportion_n as_o the_o number_n bc_o be_v to_o the_o number_n bd_o so_o be_v the_o square_a of_o the_o line_n fg_o to_o the_o square_n of_o the_o line_n k._n but_o the_o number_n bc_o have_v to_o the_o number_n bd_o that_o proportion_n that_o a_o square_a number_n have_v to_o a_o square_a number_n wherefore_o the_o square_a of_o the_o line_n fg_o have_v to_o the_o square_n of_o the_o line_n king_n that_o proportion_n that_o a_o square_a number_n have_v to_o a_o square_a number_n wherefore_o the_o line_n fg_o be_v commensurable_a in_o length_n to_o the_o line_n k._n wherefore_o the_o line_n fg_o be_v in_o power_n more_o than_o the_o line_n hg_o by_o the_o square_n of_o a_o line_n commensurable_a in_o length_n to_o the_o line_n fg_o and_o neither_o of_o the_o line_n fg_o and_o gh_o be_v commensurable_a in_o length_n to_o the_o rational_a line_n a_o when_o yet_o notwithstanding_o either_o of_o the_o line_n fg_o and_o gh_o be_v rational_a wherefore_o the_o line_n fh_o be_v a_o three_o residual_a line_n wherefore_o there_o be_v find_v out_o a_o three_o residual_a line_n which_o be_v require_v to_o be_v do_v ¶_o the_o 22._o problem_n the_o 88_o proposition_n to_o find_v out_o a_o four_o residual_a line_n take_v a_o rational_a line_n and_o let_v the_o same_o be_v a_o and_o unto_o it_o let_v the_o line_n bg_o be_v commensurable_a in_o length_n wherefore_o the_o line_n bg_o be_v rational_a and_o take_v two_o number_n df_o and_o fe_o construction_n and_o let_v they_o be_v such_o that_o the_o whole_a number_n namely_o de_fw-fr have_v to_o neither_o of_o the_o number_n df_o and_o fe_o that_o proportion_n that_o a_o square_a number_n have_v to_o a_o square_a number_n and_o as_o the_o number_n de_fw-fr be_v to_o the_o number_n of_o so_o let_v the_o square_n of_o the_o line_n bg_o be_v to_o the_o square_n of_o the_o line_n gc_o wherefore_o the_o square_a of_o the_o line_n bg_o be_v commensurable_a to_o the_o square_n of_o the_o line_n gc_o wherefore_o also_o the_o square_a of_o the_o line_n gc_o be_v rational_a and_o the_o line_n gc_o be_v also_o rational_a and_o for_o that_o the_o number_n de_fw-fr have_v not_o the_o number_n of_o that_o proportion_n that_o a_o square_a number_n have_v to_o a_o square_a number_n demonstration_n therefore_o the_o line_n bg_o be_v incommensurable_a in_o length_n to_o the_o line_n gc_o and_o they_o be_v both_o rational_a wherefore_o the_o line_n bc_o be_v a_o residual_a line_n i_o say_v moreover_o that_o it_o be_v a_o four_o residual_a line_n for_o forasmuch_o as_o the_o square_n of_o the_o line_n bg_o be_v great_a than_o the_o square_a of_o the_o line_n gc_o unto_o the_o square_n of_o the_o line_n bg_o let_v the_o square_n of_o the_o line_n cg_o and_o h_o be_v equal_a and_o for_o that_o as_o the_o number_n de_fw-fr be_v to_o the_o number_n of_o so_o be_v the_o square_a of_o the_o line_n bg_o to_o the_o square_n of_o the_o line_n gc_o therefore_o by_o conversion_n of_o proportion_n as_o the_o number_n de_fw-fr be_v to_o the_o number_n df_o so_o be_v the_o square_a of_o the_o line_n bg_o to_o the_o square_n of_o the_o line_n h._n but_o the_o number_n de_fw-fr and_o df_o have_v not_o the_o one_o to_o the_o other_o that_o porportion_n that_o a_o square_a number_n have_v to_o a_o square_a number_n wherefore_o the_o line_n bg_o be_v incommensurable_a in_o length_n to_o the_o line_n h._n wherefore_o the_o line_n bg_o be_v in_o power_n more_o than_o the_o line_n gc_o by_o the_o square_n of_o a_o line_n incommensurable_a in_o length_n so_o the_o line_n bg_o and_o the_o whole_a line_n bg_o be_v commensurable_a in_o length_n to_o the_o rational_a line_n a_o wherefore_o the_o line_n bc_o be_v a_o four_o residual_a line_n wherefore_o there_o be_v find_v out_o a_o four_o residual_a line_n which_o be_v require_v to_o be_v do_v ¶_o the_o 23._o problem_n the_o 89._o proposition_n to_o find_v out_o a_o five_o residual_a line_n take_v a_o rational_a line_n and_o let_v the_o same_o be_v a_o construction_n and_o unto_o it_o let_v the_o line_n cg_o be_v commensurable_a in_o length_n wherefore_o the_o line_n cg_o be_v rational_a and_o take_v two_o number_n df_o and_o fe_o which_o let_v be_v such_o that_o the_o number_n de_fw-fr have_v to_o neither_o of_o these_o number_n df_o nor_o fe_o that_o proportion_n that_o a_o square_a number_n have_v to_o a_o square_a number_n and_o as_o the_o number_n fe_o be_v to_o the_o number_n de_fw-fr so_o let_v the_o square_n of_o the_o line_n cg_o be_v to_o the_o square_n of_o the_o line_n bg_o demonstration_n wherefore_o the_o square_a of_o the_o line_n cg_o be_v commensurable_a to_o the_o square_n of_o the_o line_n bgâ—_n wherefore_o the_o square_a of_o the_o line_n bg_o be_v rational_a and_o the_o line_n bg_o be_v also_o rational_a but_o the_o number_n de_fw-fr and_o of_o have_v not_o that_o proportion_n the_o one_o to_o the_o other_o that_o a_o square_a number_n have_v to_o a_o square_a number_n wherefore_o the_o line_n bg_o and_o gc_o be_v rational_a commensurable_a in_o power_n only_o wherâ—fore_o the_o line_n bc_o be_v a_o residual_a line_n i_o say_v moreover_o that_o it_o be_v a_o five_o residual_a line_n for_o forasmuch_o as_o the_o square_n of_o the_o line_n bg_o be_v great_a than_o the_o square_a of_o the_o line_n gc_o unto_o the_o square_n of_o the_o line_n bg_o let_v the_o square_n of_o the_o line_n gc_o and_o h_o be_v equal_a now_o therefore_o for_o that_o as_o the_o number_n de_fw-fr be_v to_o the_o number_n of_o so_o be_v the_o square_a of_o the_o line_n bg_o to_o the_o square_n of_o the_o line_n gc_o therefore_o by_o conversion_n of_o proportion_n at_o the_o number_n de_fw-fr be_v to_o the_o number_n df_o so_o be_v the_o square_a of_o the_o line_n bg_o to_o the_o square_n of_o the_o line_n h._n but_o the_o number_n de_fw-fr &_o df_o have_v not_o that_o proportion_n the_o one_o to_o the_o other_o that_o a_o square_a number_n have_v to_o a_o square_a number_n wherefore_o the_o line_n bg_o be_v incommensurable_a in_o length_n to_o the_o line_n h._n wherefore_o the_o line_n bg_o be_v in_o power_n more_o than_o the_o line_n cg_o by_o the_o square_n of_o a_o line_n incommensurable_a in_o length_n to_o the_o line_n bg_o and_o the_o line_n cg_o which_o be_v join_v to_o the_o residual_a line_n be_v commensurable_a in_o length_n to_o the_o rational_a line_n a._n wherefore_o the_o line_n bc_o be_v a_o five_o residual_a line_n wherefore_o there_o be_v find_v out_o a_o five_o residual_a line_n which_o be_v require_v to_o be_v do_v ¶_o the_o 24._o problem_n the_o 90._o proposition_n to_o find_v out_o a_o six_o residual_a line_n take_v a_o rational_a line_n and_o let_v the_o same_o be_v a_o construction_n and_o take_v three_o number_n e_o bc_o and_o cd_o not_o have_v the_o one_o to_o the_o other_o that_o proportion_n that_o a_o square_a number_n have_v to_o a_o square_a number_n and_o let_v not_o the_o number_n bc_o have_v to_o the_o number_n bd_o that_o proportion_n that_o a_o square_a number_n have_v to_o a_o square_a number_n and_o let_v the_o number_n bc_o be_v great_a than_o the_o number_n cd_o &_o as_o the_o number_n e_o be_v to_o the_o number_n bc_o so_o let_v the_o square_n of_o the_o line_n a_o be_v to_o the_o square_n of_o the_o line_n fg._n and_o as_o the_o number_n bc_o be_v to_o the_o number_n cd_o so_o let_v the_o square_n of_o the_o line_n fg_o be_v to_o the_o square_n of_o the_o line_n gh_o now_o therefore_o for_o that_o as_o the_o number_n e_o be_v to_o the_o number_n bc_o demonstration_n so_o be_v the_o square_a of_o the_o line_n a_o to_o the_o square_n of_o the_o line_n fg_o therefore_o the_o square_a of_o the_o line_n a_o be_v commensurable_a to_o the_o square_n of_o the_o line_n f_o g._n wherefore_o the_o square_a of_o the_o line_n fg_o be_v rational_a and_o the_o line_n fg_o be_v also_o rational_a and_o for_o that_o the_o number_n e_o have_v not_o to_o the_o number_n bc_o that_o proportion_n that_o a_o square_a number_n have_v to_o a_o square_a number_n therefore_o the_o line_n a_o be_v incommensurable_a in_o length_n to_o the_o line_n fg._n again_o for_o that_o as_o the_o number_n bc_o be_v to_o the_o number_n cd_o so_o be_v the_o square_a of_o the_o line_n fg_o to_o the_o square_n of_o the_o line_n gh_o therefore_o the_o square_a of_o the_o line_n
fg_o be_v commensurable_a to_o the_o square_n of_o the_o line_n gh_o but_o the_o square_n of_o the_o line_n fg_o be_v rational_a wherefore_o the_o square_a also_o of_o the_o line_n gh_o be_v rational_a wherefore_o the_o line_n gh_o be_v also_o rational_a and_o for_o that_o the_o number_n bâ—_n have_v not_o to_o the_o number_n cd_o that_o proportion_n that_o a_o square_a number_n have_v to_o a_o square_a number_n therefore_o the_o line_n fg_o be_v incommensurable_a in_o length_n to_o the_o line_n gh_o and_o they_o be_v both_o rational_a wherefore_o the_o line_n fg_o and_o gh_o be_v rational_a commensurable_a in_o power_n only_o wherefore_o the_o line_n fh_o be_v a_o residual_a line_n i_o say_v moreover_o that_o it_o be_v a_o six_o residual_a line_n for_o for_o that_o as_o the_o number_n e_o be_v to_o the_o number_n bc_o so_o be_v the_o square_a of_o the_o line_n a_o to_o the_o square_n of_o the_o line_n fg_o and_o as_o the_o number_n bc_o be_v to_o the_o number_n cd_o so_o be_v the_o square_a of_o the_o line_n fg_o to_o the_o square_n of_o the_o line_n gh_o therefore_o by_o equality_n of_o proportion_n as_o the_o number_n e_o be_v to_o the_o number_n cd_o so_o be_v the_o square_a of_o the_o line_n a_o to_o the_o square_n of_o the_o line_n gh_o but_o the_o number_n e_o have_v not_o to_o the_o number_n cd_o that_o proportion_n that_o a_o square_a number_n have_v to_o a_o square_a number_n wherefore_o the_o line_n a_o be_v incommensurable_a in_o length_n to_o the_o line_n gh_o and_o neither_o of_o these_o line_n fg_o nor_o gâ—_n be_v commensurable_a in_o length_n to_o the_o rational_a line_n a._n and_o forasmuch_o as_o the_o square_n of_o the_o line_n fg_o be_v great_a than_o the_o square_a of_o the_o line_n gh_o unto_o the_o square_n of_o the_o line_n fg_o let_v the_o the_o square_n of_o the_o line_n gh_o and_o king_n be_v equal_a now_o therefore_o for_o that_o as_o the_o number_n bâ—_n be_v to_o the_o number_n cd_o so_o be_v the_o square_a of_o the_o line_n fg_o to_o the_o square_n of_o the_o line_n gh_o therefore_o by_o conversion_n of_o proportion_n as_o the_o number_n bc_o be_v to_o the_o number_n bd_o so_o be_v the_o square_a of_o the_o line_n fg_o to_o the_o square_n of_o the_o line_n k._n but_o the_o number_n bc_o have_v not_o to_o the_o number_n bd_o that_o proportion_n that_o a_o square_a number_n have_v to_o a_o square_a number_n therefore_o the_o line_n fg_o be_v incommensurable_a in_o length_n to_o the_o line_n k._n wherefore_o the_o line_n fg_o be_v in_o power_n more_o than_o the_o line_n gh_o by_o the_o square_n of_o a_o line_n incommensurable_a in_o length_n to_o the_o line_n fg_o and_o neither_o of_o the_o line_n fg_o nor_o gh_o be_v commensurable_a in_o length_n to_o the_o rational_a line_n a._n wherefore_o the_o line_n fh_o be_v a_o six_o residual_a line_n wherefore_o there_o be_v find_v out_o a_o six_o residual_a line_n which_o be_v require_v to_o be_v do_v line_n there_o be_v also_o a_o certain_a other_o ready_a way_n to_o find_v out_o every_o one_o of_o the_o foresaid_a six_o residual_a line_n which_o be_v after_o this_o manner_n suppose_v that_o it_o be_v require_v to_o find_v out_o a_o first_o residual_a line_n take_v a_o first_o binomial_a line_n ac_fw-la &_o let_v the_o great_a name_n thereof_o be_v ab_fw-la and_o unto_o the_o line_n bc_o let_v the_o line_n bd_o be_v equal_a wherefore_o the_o line_n ab_fw-la and_o bc_o that_o be_v the_o line_n ab_fw-la and_o bd_o be_v rational_a commensurable_a in_o power_n only_o and_o the_o line_n ab_fw-la be_v in_o power_n more_o than_o the_o line_n bc_o that_o be_v than_o the_o line_n bd_o by_o the_o square_n of_o a_o line_n commensurable_a in_o length_n to_o the_o line_n ab_fw-la and_o the_o line_n ab_fw-la be_v commensurable_a in_o length_n to_o the_o rational_a line_n give_v for_o the_o line_n ac_fw-la be_v put_v to_o be_v a_o first_o binomial_a line_n wherefore_o the_o line_n ad_fw-la be_v a_o first_o residual_a line_n and_o in_o like_a manner_n may_v you_o find_v out_o a_o second_o a_o three_o a_o four_o a_o five_o and_o a_o six_o residual_a line_n if_o you_o take_v for_o each_o a_o binomial_a line_n of_o the_o same_o order_n ¶_o the_o 67._o theorem_a the_o 91._o proposition_n if_o a_o superficies_n be_v contain_v under_o a_o rational_a line_n &_o a_o first_o residual_a line_n the_o line_n which_o contain_v in_o power_n that_o superficies_n be_v a_o residual_a line_n svppose_v that_o there_o be_v a_o rectangle_n superficies_n ab_fw-la contain_v under_o a_o rational_a line_n ac_fw-la and_o a_o first_o residual_a line_n ad._n senary_n then_o i_o say_v that_o the_o line_n which_o contain_v in_o power_n the_o superficies_n ab_fw-la be_v a_o residual_a line_n for_o forasmuch_o as_o ad_fw-la be_v a_o first_o residual_a line_n construction_n let_v the_o line_n join_v unto_o it_o be_v dg_o by_o the_o line_n join_v unto_o it_o understand_v such_o a_o line_n as_o be_v speak_v of_o in_o the_o end_n of_o the_o 79._o proposition_n wherefore_o the_o line_n agnostus_n and_o gd_o be_v rational_a commensurable_a in_o power_n only_o &_o the_o whole_a line_n agnostus_n be_v commensurable_a in_o length_n to_o the_o rational_a line_n ac_fw-la and_o the_o line_n agnostus_n be_v in_o power_n more_o than_o the_o line_n gd_o by_o the_o square_n of_o a_o line_n commensurable_a in_o length_n unto_o agnostus_n by_o the_o definition_n of_o a_o first_o residual_a line_n divide_v the_o line_n gd_o into_o two_o equal_a part_n in_o the_o point_n e._n and_o upon_o the_o line_n agnostus_n apply_v a_o parallelogram_n equal_a to_o the_o square_n of_o the_o line_n eglantine_n and_o want_v in_o figure_n by_o a_o square_a and_o let_v the_o say_a parallelogram_n be_v that_o which_o be_v contain_v under_o the_o line_n of_o and_o fg._n demonstration_n wherefore_o the_o line_n of_o be_v commensurable_a in_o length_n to_o the_o line_n fg_o by_o the_o 17._o of_o the_o ten_o â—_o and_o by_o the_o point_n e_o f_o and_o g_o draw_v unto_o the_o line_n ac_fw-la these_o parallel_a line_n eh_o fi_n and_o gk_o and_o make_v perfect_a the_o parallelogram_n ak_o and_o forâ—as_v much_o as_o the_o line_n of_o be_v commensurable_a in_o length_n to_o the_o line_n fg_o therefore_o also_o the_o whole_a line_n agnostus_n be_v commensurable_a iâ—_z length_n to_o either_o of_o the_o line_n of_o and_o fg_o by_o the_o 15._o of_o the_o ten_o but_o the_o line_n agnostus_n be_v commensurable_a in_o length_n to_o the_o line_n ac_fw-la wherefore_o either_o of_o the_o line_n of_o and_o fg_o be_v commensurable_a in_o length_n to_o the_o line_n ac_fw-la but_o the_o line_n ac_fw-la be_v rational_a wherefore_o either_o of_o the_o line_n of_o and_o fg_o be_v also_o rational_a wherefore_o by_o the_o 19_o of_o the_o ten_o either_o of_o the_o parallelogram_n ai_fw-fr and_o fk_o be_v also_o rational_a parallelogram_n and_o forasmuch_o as_o the_o line_n de_fw-fr be_v commensurable_a in_o length_n to_o the_o line_n eglantine_n therefore_o also_o by_o the_o 15._o of_o the_o ten_o the_o line_n dg_o be_v commensurable_a in_o length_n to_o either_o of_o the_o line_n de_fw-fr and_o eglantine_n but_o the_o line_n dg_o be_v rational_a wherefore_o either_o of_o the_o line_n de_fw-fr and_o eglantine_n be_v rational_a and_o the_o self_n same_o line_n dg_o be_v incommensurable_a in_o length_n to_o the_o line_n ac_fw-la by_o the_o definition_n of_o a_o first_o residual_a line_n or_o by_o the_o 13._o of_o the_o tenthâ—_n for_o the_o line_n dg_o be_v incommensurable_a in_o length_n to_o the_o line_n agnostus_n which_o line_n agnostus_n be_v commensurable_a in_o length_n to_o the_o line_n ac_fw-la medial_a wherefore_o either_o of_o the_o line_n de_fw-fr and_o eglantine_n be_v rational_a and_o incommensurable_a in_o length_n to_o the_o line_n acâ—_n â—_z wherefore_o by_o the_o 21._o of_o the_o ten_o either_o of_o these_o parallelogram_n dh_o and_o eke_o be_v medial_a unto_o the_o parallelogram_n ai_fw-fr let_v the_o square_v lm_o be_v equal_a and_o unto_o the_o parallelogram_n fk_o let_v the_o square_a nx_n be_v equal_a be_v take_v away_o from_o the_o square_a lmâ—_n â—_z and_o haâ—ing_v the_o angle_n lom_o common_a to_o they_o both_o construction_n and_o to_o do_v this_o there_o must_v be_v find_v out_o the_o mean_a proportional_a between_o the_o line_n fi_n and_o fg._n for_o the_o square_n of_o the_o mean_a proportional_a be_v equal_a to_o the_o parallelogram_n contain_v under_o the_o line_n fi_n and_o fg._n and_o from_o the_o line_n lo_o cut_v of_o a_o line_n equal_a to_o the_o mean_a proportional_a so_o find_v out_o and_o descriâ—e_v the_o square_a thereof_o wherefore_o both_o the_o square_n lm_o and_o nx_o be_v about_o one_o and_o the_o self_n same_o diameter_n by_o the_o 20._o of_o the_o six_o let_v their_o diameter_n be_v or_o and_o describe_v the_o figure_n as_o
it_o be_v hâ—â—e_o sâ—t_a forthâ—_n now_o then_o forasmuch_o as_o the_o parallelogram_n contain_v under_o the_o line_n of_o &_o fg_o be_v equal_a to_o the_o square_n of_o the_o line_n eglantine_n demonstration_n therefore_o by_o the_o 17._o of_o the_o six_o as_o the_o line_n of_o be_v to_o the_o line_n eglantine_n so_o be_v the_o line_n eglantine_n to_o the_o line_n fg._n but_o as_o the_o line_n of_o be_v to_o the_o line_n eglantine_n so_o be_v thâ—_z parallelogram_n a_z to_o the_o parallelogram_n eke_o and_o as_o the_o line_n eglantine_n be_v to_o the_o line_n fg_o so_o be_v the_o parallelogram_n eke_o to_o the_o parallelogram_n fk_o wherefore_o between_o the_o pagramme_n a_z and_o fk_o the_o parallelogram_n eke_o be_v the_o mean_a proportional_a but_o by_o the_o second_o part_n of_o the_o assumpt_n go_v before_o the_o 54._o of_o the_o ten_o between_o the_o square_n lm_o and_o nx_o the_o parallelogram_n mn_o be_v the_o mean_a proportional_a and_o unto_o the_o parallelogram_n ai_fw-fr be_v equal_v the_o square_a lm_o and_o unto_o the_o parallelogram_n fk_o be_v equal_a the_o square_a nx_n by_o construction_n wherefore_o the_o parallelogram_n mn_o be_v equal_a to_o the_o parallelogram_n eke_o by_o the_o 2._o assumpt_n go_v before_o the_o 54._o of_o the_o ten_o but_o the_o parallelogram_n eke_o be_v by_o the_o first_o of_o the_o six_o equal_a to_o the_o parallelogram_n dh_o and_o the_o parallelogram_n mn_o be_v by_o the_o 43._o of_o the_o first_o equal_a to_o the_o parallelogram_n lx._n wherefore_o the_o whole_a parallelogram_n dk_o be_v equal_a to_o the_o gnomon_n vtz_n which_o gnomon_n consist_v of_o those_o parallelogram_n by_o which_o you_o see_v in_o the_o figure_n pass_v a_o portion_n of_o a_o circle_n great_a than_o a_o semicircle_n and_o moreover_o to_o the_o square_a nx_n and_o the_o parallelogram_n ak_o be_v equal_a to_o the_o square_n lm_o and_o nx_o by_o construction_n and_o it_o be_v now_o prove_v that_o the_o parallelogram_n dk_o be_v equal_a to_o the_o gnomon_n vtz_n and_o moreover_o to_o the_o square_a nx_n wherefore_o the_o residue_n namely_o the_o parallelogram_n ab_fw-la be_v equal_a to_o the_o square_a sq_n which_o be_v the_o square_a of_o the_o line_n ln_o consider_v wherefore_o the_o square_a of_o the_o line_n ln_o be_v equal_a to_o the_o parallelogram_n ab_fw-la wherefore_o the_o line_n ln_o contain_v in_o power_n the_o parallelogram_n ab_fw-la i_o say_v moreover_o that_o the_o line_n ln_o be_v a_o residual_a line_n for_o forasmuch_o as_o either_o of_o these_o parallelogram_n a_z and_o fk_o be_v rational_o as_o it_o be_v before_o say_v therefore_o the_o square_n lm_o and_o nx_o which_o be_v equal_a unto_o they_o that_o be_v the_o square_n of_o the_o line_n lo_o and_o on_o be_v rational_a wherefore_o the_o line_n lo_o and_o on_o be_v also_o rational_a again_o forasmuch_o as_o the_o parallelogram_n dh_fw-mi that_o be_v lx_o be_v medial_a therefore_o the_o parallelogram_n lx_o be_v incommensurable_a to_o the_o square_a nx_n wherefore_o by_o the_o 1._o of_o the_o six_o and_o 10._o of_o the_o ten_o the_o line_n lo_o be_v incommensurable_a in_o length_n to_o the_o line_n onâ—_n â—_z and_o they_o be_v both_o rational_a wherefore_o they_o be_v line_n rational_a commensurable_a in_o power_n only_o wherefore_o ln_o be_v a_o residual_a line_n by_o the_o definition_n and_o it_o contain_v in_o power_n the_o paralleloparallelogramme_n ab_fw-la if_o therefore_o a_o superficies_n be_v contain_v under_o a_o rational_a line_n and_o a_o first_o residual_a line_n the_o line_n which_o contain_v in_o power_n that_o superficiesâ—_n be_v a_o residual_a line_n which_o be_v require_v to_o be_v demonstrate_v ¶_o the_o 68_o theorem_a the_o 92._o proposition_n if_o a_o superficies_n be_v contain_v under_o a_o rational_a line_n and_o a_o second_o residual_a line_n the_o line_n which_o contain_v in_o power_n that_o superficies_n be_v a_o first_o medial_a residual_a line_n svppose_v that_o ab_fw-la be_v a_o superficies_n contain_v under_o â—raâ—â—onall_a line_n ac_fw-la and_o a_o second_o residual_a liâ—â—_n ad._n construction_n then_o i_o sayâ—_n that_o the_o line_n thâ—â—_n contain_v in_o power_n the_o superficies_n ab_fw-la be_v a_o first_o medial_a residual_a line_n for_o let_v the_o line_n join_v to_o the_o line_n ad_fw-la be_v dg_o wherefore_o the_o line_n agnostus_n and_o gd_o be_v rational_a commensurable_a in_o power_n only_o and_o the_o line_n that_o be_v join_v to_o the_o residual_a line_n namely_o the_o line_n dg_o be_v commensurable_a in_o length_n to_o the_o rational_a line_n ac_fw-la and_o the_o line_n agnostus_n be_v in_o power_n more_o than_o the_o line_n dg_o by_o the_o square_n of_o a_o line_n commensurable_a in_o length_n to_o the_o line_n ag._n divide_v the_o line_n dg_o into_o two_o equal_a part_n in_o the_o point_n e._n and_o unto_o the_o line_n agnostus_n apply_v a_o parallelogram_n equal_a to_o the_o four_o part_n of_o the_o square_n of_o the_o line_n dg_o that_o be_v equal_a to_o the_o square_n of_o the_o line_n eglantine_n and_o want_v in_o figure_n by_o a_o square_a and_o let_v that_o parallelogram_n be_v that_o which_o be_v contain_v under_o the_o line_n of_o and_o fg._n wherefore_o by_o the_o 1st_a of_o the_o ten_o the_o line_n of_o be_v commensurable_a in_o length_n to_o the_o line_n fg._n and_o by_o the_o point_n e_o f_o and_o g_o draw_v unto_o the_o line_n ac_fw-la these_o parallel_a line_n eh_o fi_n and_o gk_o and_o forasmuch_o as_o the_o line_n of_o be_v commensurable_a in_o length_n to_o the_o line_n fg_o demonstration_n therefore_o the_o whole_a line_n agnostus_n be_v commensurable_a in_o length_n to_o either_o of_o these_o line_n of_o and_o fg._n but_o the_o line_n agnostus_n be_v rational_a and_o incommensurable_a in_o length_n to_o the_o line_n ac_fw-la wherefore_o either_o of_o these_o line_n of_o and_o fg_o be_v rational_a and_o incommensurable_a in_o length_n to_o the_o line_n ac_fw-la wherefore_o either_o of_o thâ—se_a parallelogram_n a_z and_o fk_o be_v by_o the_o 21._o of_o the_o ten_o medial_a medial_a again_o forasmuch_o as_o the_o line_n de_fw-fr be_v commensurable_a in_o length_n to_o the_o line_n eglantine_n therefore_o the_o line_n dg_o be_v commensurable_a in_o length_n to_o either_o of_o these_o line_n de_fw-fr and_o eglantine_n but_o the_o line_n dg_o be_v commensurable_a in_o length_n to_o the_o rational_a line_n ac_fw-la wherefore_o either_o of_o these_o line_n de_fw-fr and_o eglantine_n be_v rational_a and_o commensurable_a in_o length_n to_o the_o line_n ac_fw-la wherefore_o by_o the_o 19_o of_o the_o ten_o either_o of_o these_o parallelogram_n dh_o and_o eke_o be_v rational_a rational_a unto_o the_o parallelogram_n ai_fw-fr describe_v a_o equal_a square_a lm_o construction_n and_o unto_o the_o parallelogram_n fk_o let_v the_o square_a nx_n be_v equal_a as_o in_o the_o proposition_n go_v before_o wherefore_o the_o square_n lm_o and_o nx_o be_v both_o about_o one_o and_o the_o same_o diameter_n demonstration_n let_v the_o diameter_n be_v or_o and_o describe_v the_o figure_n as_o be_v in_o the_o former_a proposition_n express_v now_o therefore_o forasmuch_o as_o the_o parallelogram_n a_z and_o fk_o be_v medial_a be_v thing_n commensurable_a the_o one_o to_o the_o other_o and_o the_o square_n of_o the_o line_n lo_o &_o on_o which_o be_v equal_a to_o those_o parallelogram_n be_v medial_a therefore_o the_o line_n lo_o and_o on_o be_v also_o medial_a commensurable_a in_o power_n and_o it_o be_v manifest_a that_o the_o line_n lo_o and_o on_o be_v commensurable_a in_o power_n for_o their_o square_n be_v commensurable_a and_o those_o square_n namely_o the_o square_n of_o the_o line_n lo_o &_o on_o be_v commensurable_a for_o they_o be_v equal_a to_o the_o parallelogram_n ai_fw-fr and_o fk_o which_o be_v commensurable_a the_o one_o to_o the_o other_o and_o that_o those_o parallelogram_n a_z and_o fk_o be_v commensurable_a the_o one_o to_o the_o other_o hereby_o it_o be_v manifest_a for_o that_o it_o be_v before_o prove_v that_o the_o line_n of_o and_o fg_o be_v commensurable_a in_o length_n wherefore_o by_o the_o 1._o of_o the_o six_o and_o 10._o of_o the_o ten_o the_o parallelogram_n a_z and_o fk_o be_v commensurable_a the_o one_o to_o the_o other_o wherefore_o it_o be_v now_o manifest_a by_o the_o way_n of_o resolution_n that_o the_o line_n lo_o &_o on_o be_v commensurable_a in_o power_n â—_o and_o forasmuch_o as_o the_o parallelogram_n contain_v under_o the_o line_n of_o and_o fg_o be_v equal_a to_o the_o square_n of_o the_o line_n eglantine_n therefore_o as_o the_o line_n of_o be_v to_o the_o line_n eglantine_n so_o be_v the_o line_n eglantine_n to_o the_o line_n fg._n but_o as_o the_o line_n of_o be_v to_o the_o line_n eglantine_n so_o be_v the_o parallelogram_n ai_fw-fr to_o the_o parallelogram_n eke_o and_o as_o the_o line_n eglantine_n be_v to_o the_o line_n fg_o so_o be_v the_o parallelogram_n eke_o to_o the_o parallelogram_n fk_o wherefore_o
the_o parallelogram_n â—k_n be_v the_o meanâ—_z proportional_a between_o the_o parallelogram_n ai_fw-fr and_o fk_o and_o the_o parallelogram_n mn_v iâ—_z also_o the_o mean_a proportional_a between_o the_o square_n lm_o &_o nx_o and_o the_o parallelogram_n ai_fw-fr be_v equal_a to_o the_o square_a lm_o and_o the_o parallelogram_n fk_o be_v equal_a to_o the_o square_a nx_n wherefore_o the_o parallelogram_n mn_o be_v equal_a to_o the_o parallelogram_n eke_o but_o the_o parallelogram_n eke_o be_v equal_a to_o the_o parallelogram_n dh_o and_o the_o parallelogram_n lx_o be_v equal_a to_o the_o parallelogram_n mn_v wherefore_o the_o whole_a parallelogram_n dk_o be_v equal_a to_o the_o gnomon_n vtz_n and_o to_o the_o square_a nx_n wherefore_o the_o residue_n namely_o the_o parallelogram_n ab_fw-la be_v equal_a to_o the_o square_a sq_n that_o be_v to_o the_o square_n of_o the_o line_n ln_o wherefore_o the_o line_n ln_o contain_v in_o power_n the_o superficies_n ab_fw-la discourse_n i_o say_v moreover_o that_o the_o line_n ln_o be_v a_o first_o medial_a residual_a line_n for_o forasmuch_o as_o the_o parallelogram_n eke_o be_v rational_a and_o be_v equal_a to_o the_o parallelogram_n mn_o that_o be_v lx_o therefore_o lx_o that_o be_v the_o parallelogram_n contain_v under_o the_o line_n lo_o and_o on_o be_v rational_a but_o the_o square_a nx_n be_v medial_a for_o it_o be_v already_o prove_v that_o the_o parallelogram_n fk_o which_o be_v equal_a to_o the_o square_a nx_n be_v medial_a wherefore_o the_o parallelogram_n lx_o be_v incommensurable_a to_o the_o square_a nx_n but_o as_o the_o parallelogram_n lx_o be_v to_o the_o square_a nx_n so_o be_v the_o line_n lo_o to_o the_o line_n on_o by_o the_o 1._o of_o the_o six_o wherefore_o by_o the_o 10._o of_o the_o ten_o the_o line_n lo_o and_o on_o be_v incommensurable_a in_o length_n and_o it_o be_v already_o prove_v that_o they_o be_v medial_a commensurable_a in_o power_n wherefore_o the_o line_n lo_o &_o on_o be_v medial_a commensurable_a in_o power_n only_o contain_v a_o rational_a superficies_n wherefore_o the_o line_n ln_o be_v a_o first_o medial_a residual_a line_n and_o contain_v in_o power_n the_o superficies_n ab_fw-la which_o be_v contain_v under_o a_o rational_a line_n and_o a_o second_o residual_a line_n if_o therefore_o a_o superficies_n be_v contain_v under_o a_o rational_a line_n and_o a_o second_o residual_a line_n the_o line_n which_o contain_v in_o power_n that_o superficies_n be_v a_o first_o medial_a residual_a line_n which_o be_v require_v to_o be_v demonstrate_v ¶_o the_o 69._o theorem_a the_o 93._o proposition_n if_o a_o superficies_n be_v contain_v under_o a_o rational_a line_n and_o a_o three_o residual_a line_n the_o line_n that_o contain_v in_o power_n that_o superficies_n be_v a_o second_o medial_a residual_a line_n svppose_v that_o ab_fw-la be_v a_o superficies_n contain_v under_o a_o rational_a line_n ac_fw-la &_o a_o three_o residual_a line_n ad._n then_o i_o say_v that_o the_o line_n which_o contain_v in_o power_n the_o superficies_n ab_fw-la be_v a_o second_o medial_a residual_a line_n let_v the_o line_n join_v unto_o ad_fw-la be_v dg_o wherefore_o the_o line_n agnostus_n and_o gd_o be_v rational_a commensurable_a in_o power_n only_o construction_n and_o neither_o of_o the_o line_n agnostus_n nor_o gd_o be_v commensurable_a in_o length_n to_o the_o rational_a line_n ac_fw-la and_o the_o whole_a line_n ac_fw-la be_v in_o power_n more_o than_o the_o line_n gd_o by_o the_o square_n of_o a_o line_n commensurable_a in_o length_n to_o the_o line_n ag._n let_v the_o rest_n of_o the_o construction_n be_v as_o it_o be_v in_o the_o former_a proposition_n demonstration_n wherefore_o the_o line_n of_o and_o fg_o be_v commensurable_a in_o length_n and_o the_o parallelogram_n ai_fw-fr be_v commensurable_a to_o the_o parallelogram_n fk_o and_o forasmuch_o as_o the_o line_n of_o and_o fg_o be_v commensurable_a in_o length_n therefore_o the_o whole_a line_n agnostus_n be_v commensurable_a in_o length_n to_o either_o of_o these_o line_n of_o and_o fg._n bâ—â—_n the_o linâ—_n agnostus_n be_v rational_a and_o incommensurable_a in_o length_n to_o the_o line_n ag._n wherefore_o either_o of_o these_o line_n of_o and_o fg_o be_v rational_a and_o incommensurable_a in_o length_n to_o the_o line_n ag._n medial_a wherefore_o by_o the_o 21._o of_o the_o ten_o either_o of_o these_o parallelogram_n a_z and_o fk_o be_v medial_a against_o forasmuch_o as_o the_o line_n db_o be_v commensurable_a in_o length_n to_o the_o line_n eglantine_n therefore_o also_o the_o whole_a line_n dg_o be_v commensurable_a in_o length_n to_o either_o of_o these_o line_n de_fw-fr and_o eglantine_n but_o the_o line_n dg_o be_v rational_a commensurable_a in_o power_n only_o to_o the_o line_n ac_fw-la wherefore_o also_o either_o of_o the_o line_n de_fw-fr and_o eglantine_n be_v rational_a and_o commensurable_a in_o power_n only_o to_o the_o line_n ac_fw-la medial_a wherefore_o either_o of_o these_o parallelogram_n dh_o and_o eke_o be_v medial_a again_o forasmuch_o as_o the_o line_n agnostus_n and_o dg_o be_v commensurable_a in_o power_n only_o therefore_o they_o be_v incommensurable_a in_o length_n but_o the_o line_n agnostus_n be_v commensurable_a in_o length_n to_o the_o line_n of_o and_o the_o line_n dg_o be_v commensurable_a in_o length_n to_o the_o line_n ge._n wherefore_o the_o line_n of_o be_v incommensurable_a in_o length_n to_o the_o line_n eglantine_n but_o as_o the_o line_n of_o be_v to_o the_o line_n eglantine_n so_o be_v the_o parallelogram_n ai_fw-fr to_o the_o parallelogram_n eke_o construction_n wherefore_o the_o parallelogram_n ai_fw-fr be_v incommensurable_a to_o the_o parallelogram_n eke_o unto_o the_o parallelogram_n ai_fw-fr describe_v a_o equal_a square_a lm_o and_o unto_o the_o parallelogram_n fk_o describe_v a_o equal_a square_a nx_n and_o describe_v the_o figure_n as_o you_o do_v in_o the_o former_a proposition_n now_o forasmuch_o as_o the_o parallelogram_n contain_v under_o the_o line_n of_o and_o fg_o be_v equal_a to_o the_o square_n of_o the_o line_n eglantine_n therefore_o as_o the_o line_n of_o be_v to_o the_o line_n eglantine_n so_o be_v the_o line_n eglantine_n to_o the_o line_n fg._n but_o as_o the_o line_n of_o be_v to_o the_o line_n eglantine_n so_o be_v the_o parallelogram_n ai_fw-fr to_o the_o parallelogram_n eke_o and_o as_o the_o line_n eglantine_n be_v to_o the_o line_n fg_o so_o be_v the_o parallelogram_n eke_o to_o the_o parallelogram_n fk_o wherefore_o as_o the_o parallelogram_n ai_fw-fr be_v to_o the_o parallelogram_n eke_o so_o be_v the_o parallelogram_n eke_o to_o the_o parallelogram_n fk_o wherefore_o the_o parallelogram_n eke_o be_v the_o mean_a proportional_a between_o the_o parallelogram_n ai_fw-fr and_o fk_o but_o the_o parallelogram_n mn_o be_v the_o mean_a proportional_a between_o the_o square_n lm_o and_o nx_o wherefore_o the_o parallelogram_n eke_o be_v equal_a to_o the_o parallelogram_n mn_v wherefore_o the_o whole_a parallelogram_n dk_o be_v equal_a to_o the_o gnomon_n vtz_n &_o to_o the_o square_a nx_n and_o the_o parallelogram_n ak_o be_v equal_a to_o the_o square_n lm_o and_o nx_o wherefore_o the_o residue_n namely_o the_o parallelogram_n ab_fw-la be_v equal_a to_o the_o square_a q_n that_o be_v find_v to_o the_o square_n of_o the_o line_n ln_o wherefore_o the_o line_n ln_o contain_v in_o power_n the_o superficies_n ab_fw-la i_o say_v moreover_o that_o the_o line_n ln_o be_v a_o second_o medial_a residual_a line_n for_o for_o that_o as_o it_o be_v prove_v the_o parallelogrammeâ—_n ai_fw-fr and_o fk_o be_v medial_a therefore_o the_o square_n that_o be_v equal_a unto_o they_o namely_o the_o square_n of_o the_o line_n lo_o and_o on_o be_v also_o medial_a wherefore_o either_o of_o these_o line_n lo_o and_o on_o be_v medial_a and_o forasmuch_o as_o the_o parallelogram_n ai_fw-fr be_v length_n commensurable_a to_o the_o parallelogram_n fk_o therefore_o the_o square_n that_o be_v equal_a to_o they_o namely_o the_o square_n of_o the_o line_n lo_o and_o on_o be_v also_o commensurable_a again_o forasmuch_o as_o it_o be_v prove_v that_o the_o parallelogram_n ai_fw-fr be_v incommensurable_a to_o the_o parallelogram_n eke_o therefore_o the_o square_a lm_o be_v incommensurable_a to_o the_o parallelogram_n mn_o that_o be_v the_o square_a of_o the_o line_n lo_o to_o the_o parallelogram_n contain_v under_o the_o line_n lo_o &_o on_o wherefore_o also_o the_o ten_o line_n lo_o be_v incommensurable_a in_o length_n to_o the_o line_n on_o wherefore_o the_o line_n lo_o and_o on_o be_v medial_a commensurable_a in_o power_n only_o i_o say_v moreover_o that_o they_o contain_v a_o medial_a superficies_n for_o forasmuch_o as_o it_o be_v prove_v that_o the_o parallelogram_n eke_o be_v medial_a therefore_o the_o parallelogram_n which_o be_v equal_a unto_o it_o namely_o the_o parallelogram_n contain_v under_o the_o line_n lo_o and_o on_o be_v also_o medial_a wherefore_o the_o line_n ln_o be_v a_o second_o medial_a residual_a line_n and_o contain_v in_o power_n the_o
superficies_n ab_fw-la wherefore_o the_o line_n that_o contain_v in_o power_n the_o superficies_n ab_fw-la be_v a_o second_o medial_a residual_a line_n if_o therefore_o a_o superficies_n be_v contain_v under_o a_o rational_a line_n and_o a_o three_o residual_a line_n the_o line_n that_o contain_v iâ—_z power_n that_o superficies_n be_v a_o second_o medial_a residual_a line_n which_o be_v require_v to_o be_v demonstrate_v the_o 70._o theorem_a the_o 94._o proposition_n if_o a_o superficies_n be_v contain_v under_o a_o rational_a line_n and_o a_o four_o residual_a line_n the_o line_n which_o contain_v in_o power_n that_o superficies_n be_v a_o less_o line_n svppose_v that_o there_o be_v a_o superficies_n ab_fw-la contain_v under_o a_o rational_a line_n ac_fw-la and_o a_o forth_o residual_a line_n ad._n then_o i_o say_v that_o the_o line_n which_o contain_v in_o power_n the_o superficies_n ab_fw-la be_v a_o less_o line_n for_o let_v the_o line_n join_v unto_o it_o be_v dg_o wherefore_o the_o line_n agnostus_n and_o dg_o be_v rational_a commensurable_a in_o power_n only_o and_o the_o line_n agnostus_n be_v in_o power_n more_o than_o the_o line_n dg_o by_o the_o square_n of_o a_o line_n incommensurable_a in_o length_n to_o the_o line_n agnostus_n and_o the_o line_n agnostus_n be_v commensurable_a in_o length_n to_o the_o line_n ac_fw-la divide_v the_o line_n dg_o into_o two_o equal_a part_n in_o the_o point_n e._n construction_n and_o unto_o the_o line_n agnostus_n apply_v a_o parallelogram_n equal_a to_o the_o square_n of_o the_o line_n eglantine_n and_o want_v in_o figure_n by_o a_o square_a and_o let_v that_o parallelogram_n be_v that_o which_o be_v contain_v under_o the_o line_n of_o and_o fg._n wherefore_o by_o the_o 18._o of_o the_o ten_o the_o line_n of_o be_v incommensurable_a in_o length_n to_o the_o line_n fg._n draw_v by_o the_o point_n e_o f_o &_o g_o unto_o the_o line_n ac_fw-la and_o db_o these_o parallel_a line_n eh_o fi_n and_o gk_o demonstration_n now_o forasmuch_o as_o the_o line_n agnostus_n be_v rational_a and_o commensurable_a in_o length_n to_o the_o line_n ac_fw-la therefore_o the_o whole_a parallelogram_n ak_o be_v by_o the_o 19_o of_o the_o ten_o rational_a rational_a again_o forasmuch_o as_o the_o line_n dg_o be_v incommensurable_a in_o length_n to_o the_o line_n ac_fw-la for_o if_o the_o line_n dg_o be_v commensurable_a in_o length_n to_o the_o line_n ac_fw-la then_o forasmuch_o as_o the_o line_n agnostus_n be_v commensurable_a in_o length_n to_o the_o same_o line_n ac_fw-la the_o line_n agnostus_n and_o dg_o shall_v be_v commensurable_a in_o length_n the_o one_o to_o the_o other_o when_o yet_o they_o be_v put_v to_o be_v commensurable_a in_o power_n only_o and_o both_o these_o line_n ac_fw-la and_o dg_o be_v rational_a demonstration_n wherefore_o the_o parallelogram_n dk_o be_v medial_a again_o forasmuch_o as_o the_o line_n of_o be_v incommensurable_a in_o length_n to_o the_o line_n fg_o therefore_o the_o parallelogram_n ai_fw-fr be_v incommensurable_a to_o the_o parallelogram_n fk_o unto_o the_o parallelogram_n ai_fw-fr describe_v a_o equal_a square_a lm_o and_o unto_o the_o parallelogram_n fk_o describe_v a_o equal_a square_a nx_n and_o let_v the_o angle_n lom_n be_v common_a to_o both_o those_o square_n wherefore_o the_o square_n lm_o and_o nx_o be_v about_o one_o and_o the_o self_n same_o diameter_n let_v their_o diameter_n be_v or_o and_o describe_v the_o figure_n and_o forasmuch_o as_o the_o parallelogram_n contain_v under_o the_o line_n of_o and_o fg_o be_v equal_a to_o the_o square_n of_o the_o line_n eglantine_n therefore_o proportional_o as_o the_o line_n of_o be_v to_o the_o line_n eglantine_n so_o be_v the_o line_n eglantine_n to_o the_o line_n fg_o but_o as_o the_o line_n of_o be_v to_o the_o line_n eglantine_n so_o be_v the_o parallelogram_n ai_fw-fr to_o the_o parallelogram_n eke_o by_o the_o 1._o of_o the_o six_o and_o as_o the_o line_n eglantine_n be_v to_o the_o line_n fg_o so_o be_v the_o parallelogram_n eke_o to_o the_o parallelogram_n fk_o wherefore_o the_o parallelogram_n eke_o be_v the_o mean_a proportional_a between_o the_o parallelogram_n ai_fw-fr and_o fk_o wherefore_o as_o iâ—_z be_v say_v in_o the_o former_a proposition_n the_o parallelogram_n mn_o be_v equal_a to_o the_o parallelogram_n eke_o but_o the_o parallelogram_n dh_o be_v equal_a to_o the_o parallelogram_n eke_o and_o the_o parallelogram_n mn_v to_o the_o parallelogram_n lx._n wherefore_o the_o whole_a parallelogram_n dk_o be_v equal_a to_o the_o gnomon_n vtz_n and_o to_o the_o square_a nx_n wherefore_o the_o residue_n namely_o the_o parallelogram_n ab_fw-la be_v equal_a to_o the_o residue_n namely_o to_o the_o square_a sq_n find_v that_o be_v to_o the_o square_n of_o the_o line_n ln_o i_o say_v moreover_o that_o ln_o be_v that_o irrational_a line_n which_o be_v call_v a_o less_o line_n for_o forasmuch_o as_o the_o parallelogram_n ak_o be_v rational_a and_o be_v equal_a to_o the_o square_n of_o the_o line_n lo_o and_o on_o therefore_o that_o which_o be_v make_v of_o the_o square_n of_o the_o line_n lo_o and_o on_o add_v together_o be_v rational_a again_o forasmuch_o as_o the_o parallelogram_n dk_o be_v medial_a and_o be_v equal_a to_o that_o which_o be_v contain_v under_o the_o line_n lo_o and_o on_o twice_o therefore_o that_o which_o be_v contain_v under_o the_o line_n lo_o and_o on_o twice_o be_v also_o medial_a and_o forasmuch_o as_o the_o parallelogram_n ai_fw-fr be_v incommensurable_a to_o the_o parallelogram_n fk_o therefore_o the_o square_n which_o be_v equal_a unto_o they_o namely_o the_o square_n of_o the_o line_n lo_o and_o on_o be_v incommensurable_a the_o one_o to_o the_o other_o wherefore_o the_o line_n lo_o and_o on_o be_v incommensurable_a in_o power_n have_v that_o which_o be_v make_v of_o their_o square_n add_v together_o rational_a and_o that_o which_o be_v contain_v under_o they_o twice_o medial_a which_o be_v commensurable_a to_o that_o which_o be_v contain_v under_o they_o once_o wherefore_o that_o which_o be_v contain_v under_o they_o once_o be_v also_o medial_a wherefore_o ln_o be_v that_o irrational_a line_n which_o be_v call_v a_o less_o line_n and_o it_o contain_v in_o power_n the_o superficies_n ab_fw-la if_o therefore_o a_o superficies_n be_v contain_v under_o a_o rational_a line_n and_o a_o four_o residual_a line_n the_o line_n which_o contain_v in_o power_n that_o superficies_n be_v a_o less_o line_n which_o be_v require_v to_o be_v demonstrate_v ¶_o the_o 71._o theorem_a the_o 95._o proposition_n if_o a_o superficies_n be_v contain_v under_o a_o rational_a line_n and_o a_o five_o residual_a line_n the_o line_n that_o contain_v in_o power_n the_o same_o superficies_n be_v a_o line_n make_v with_o a_o rational_a superficies_n the_o whole_a superficies_n medial_a svppose_v that_o there_o be_v a_o superficies_n ab_fw-la contain_v under_o a_o rational_a line_n agnostus_n and_o a_o five_o residual_a line_n ad._n then_o i_o say_v that_o the_o line_n that_o contain_v in_o power_n the_o superficies_n ab_fw-la be_v a_o line_n make_v with_o a_o rational_a superficies_n the_o whole_a superficies_n medial_a for_o unto_o the_o line_n ad_fw-la let_v the_o line_n dg_o be_v join_v which_o shall_v be_v commensurable_a in_o length_n to_o the_o rational_a line_n ac_fw-la and_o let_v the_o rest_n of_o the_o construction_n be_v as_o in_o the_o proposition_n next_o go_v before_o demonstration_n and_o forasmuch_o as_o the_o line_n agnostus_n be_v incommensurable_a in_o length_n to_o the_o line_n ac_fw-la and_o they_o be_v both_o rational_a therefore_o the_o parallelogram_n ak_o be_v medial_a again_o forasmuch_o as_o the_o line_n dg_o be_v rational_a and_o commensurable_a in_o length_n to_o the_o line_n ac_fw-la therefore_o the_o parallelogram_n dk_o be_v rational_a unto_o the_o parallelâ—gramme_n ai_fw-fr describe_v a_o equal_a square_a lm_o and_o unto_o the_o parallelogram_n â—â—_o describe_v a_o equal_a square_a nâ—_z and_o as_o in_o 〈◊〉_d proposition_n next_o go_v before_o so_o also_o in_o this_o may_v we_o prove_v that_o the_o line_n ln_o contain_v in_o power_n the_o superficies_n ab_fw-la ln_o i_o say_v moreover_o that_o that_o line_n ln_o be_v a_o line_n make_v with_o a_o rational_a superficies_n the_o whole_a superficies_n medial_a for_o forasmuch_o as_o the_o parallelogram_n ak_o be_v medial_a therefore_o that_o which_o be_v equal_a unto_o it_o namely_o that_o which_o be_v make_v of_o the_o square_n of_o the_o line_n lo_o and_o on_o add_v together_o be_v also_o medial_a again_o forasmuch_o as_o the_o parallelogram_n dk_o be_v rational_a therefore_o that_o which_o be_v equal_a unto_o it_o namely_o that_o which_o be_v contain_v under_o the_o line_n lo_o and_o on_o twice_o be_v also_o rational_a and_o forasmuch_o as_o the_o line_n of_o be_v incommensurable_a in_o length_n to_o the_o line_n fg_o therefore_o by_o the_o 1._o of_o the_o six_o &_o 10._o of_o the_o ten_o the_o parallelogram_n aâ—_z be_v incommensurable_a to_o the_o
parallelogram_n fk_o wherefore_o also_o the_o square_a of_o the_o line_n lo_o be_v incommensurable_a to_o the_o square_n of_o the_o line_n on_o wherefore_o the_o line_n lo_o and_o on_o be_v incommensurable_a in_o power_n have_v that_o which_o be_v make_v of_o their_o square_n add_v together_o medial_a and_o that_o which_o be_v contain_v under_o they_o twice_o rational_a wherefore_o the_o line_n ln_o be_v that_o irrational_a line_n which_o be_v call_v a_o line_n make_v with_o a_o rational_a superficies_n the_o whole_a superficies_n medial_a and_o it_o contain_v in_o power_n the_o superficies_n ab_fw-la wherefore_o the_o line_n contain_v in_o power_n the_o superficies_n ab_fw-la be_v a_o line_n make_v with_o a_o rational_a superficies_n the_o whole_a superficies_n medial_a if_o therefore_o a_o superficies_n be_v contain_v under_o a_o rational_a line_n &_o a_o five_o residual_a line_n the_o line_n that_o contain_v in_o power_n the_o same_o superficies_n be_v a_o line_n make_v with_o a_o rational_a superficies_n the_o whole_a superficies_n medial_a which_o be_v require_v to_o be_v prove_v ¶_o the_o 72._o theorem_a the_o 96._o proposition_n if_o a_o superficies_n be_v contain_v under_o a_o rational_a line_n and_o a_o six_o residual_a line_n the_o line_n which_o contain_v in_o power_n the_o same_o superficies_n be_v a_o line_n make_v with_o a_o medial_a superficies_n the_o whole_a superficies_n medial_a svppose_v that_o ab_fw-la be_v a_o superficies_n contain_v under_o a_o rational_a line_n ac_fw-la &_o a_o six_o residual_a line_n ad._n then_o i_o say_v shall_v the_o line_n which_o contain_v in_o power_n the_o superficies_n ab_fw-la be_v a_o line_n make_v with_o a_o medial_a superficies_n the_o whole_a superficies_n medial_a for_o unto_o the_o line_n ad_fw-la let_v the_o line_n dg_o be_v join_v demonstration_n and_o let_v the_o rest_n be_v as_o in_o the_o proposition_n go_v before_o and_o forasmuch_o as_o the_o line_n of_o be_v incommensurable_a in_o length_n to_o the_o line_n fg_o therefore_o the_o parallelogram_n ai_fw-fr be_v incommensurable_a to_o the_o parallelogram_n fkâ—_n and_o forasmuch_o as_o the_o line_n agnostus_n and_o ac_fw-la be_v rational_a commensurable_a in_o power_n only_o therefore_o the_o parallelogram_n ak_o be_v medial_a and_o in_o like_a manner_n the_o parallelogram_n dk_o be_v medial_a now_o forasmuch_o as_o the_o line_n agnostus_n and_o gd_o be_v commensurable_a in_o power_n only_o therefore_o they_o be_v incommensurable_a in_o length_n the_o one_o to_o the_o other_o but_o as_o the_o line_n agnostus_n be_v to_o the_o line_n gd_o so_o be_v the_o parallelogram_n ak_o to_o the_o parallelogram_n dk_o therefore_o the_o parallelogram_n ak_o be_v incommensurable_a to_o the_o parallelogram_n dk_o describe_v the_o like_a figure_n that_o be_v describe_v in_o the_o former_a proposition_n and_o we_o may_v in_o like_a sort_n prove_v that_o the_o line_n ln_o contain_v in_o power_n the_o superficies_n ab_fw-la i_o say_v moreover_o that_o it_o be_v a_o line_n make_v wâ—th_v a_o medial_a superficies_n the_o whole_a superficies_n medial_a for_o the_o parallelogram_n ak_o be_v medial_a wherefore_o that_o which_o be_v equal_a unâ—â—_n it_o namely_o that_o which_o be_v make_v of_o the_o sqâ—ares_n of_o the_o line_n lo_o and_o on_o add_v together_o be_v also_o medial_a and_o forasmuch_o as_o the_o parallelogram_n â—k_n be_v medial_a therefore_o that_o which_o be_v equal_a unto_o â—t_a namely_o that_o which_o be_v contain_v under_o the_o line_n lo_o and_o on_o twice_o be_v also_o medial_a and_o forasmuch_o as_o the_o parallogramme_n ak_o be_v incommensurable_a to_o the_o parallelogram_n dk_o therefore_o the_o square_n of_o the_o line_n lo_o and_o on_o be_v incommensurable_a to_o that_o which_o be_v contain_v under_o the_o line_n lo_o and_o on_o twice_o and_o forasmuch_o as_o the_o parallelogram_n ai_fw-fr be_v incommensurable_a to_o the_o parallelogram_n fk_o therefore_o also_o the_o square_a of_o the_o line_n lo_o be_v incommensurable_a to_o the_o square_n of_o the_o line_n on_o wherefore_o the_o line_n lo_o and_o on_o be_v incommensurable_a in_o power_n have_v that_o which_o be_v make_v of_o the_o square_n of_o the_o line_n lo_o and_o on_o medial_a and_o that_o which_o be_v contain_v under_o they_o twice_o medial_a and_o moreover_o that_o which_o be_v make_v of_o the_o square_n of_o they_o be_v incommensurable_a to_o thate_v which_o be_v contain_v under_o they_o twice_o wherefore_o the_o line_n ln_o be_v that_o irrational_a line_n which_o be_v call_v a_o line_n make_v with_o a_o medial_a superficies_n the_o whole_a superficies_n medial_a and_o it_o contain_v in_o power_n the_o superficies_n ab_fw-la wherefore_o the_o line_n which_o contain_v in_o power_n the_o superficies_n ab_fw-la be_v a_o line_n make_v with_o a_o medial_a superficies_n the_o whole_a superficies_n medial_a if_o therefore_o a_o superficies_n be_v contain_v under_o a_o rational_a line_n and_o a_o six_o residual_a line_n the_o line_n which_o contain_v in_o power_n the_o same_o superficies_n be_v a_o line_n make_v which_o a_o medial_a superficies_n the_o whole_a superficies_n mediallâ—_n which_o be_v require_v to_o be_v demonstrate_v ¶_o the_o 73._o theorem_a the_o 97._o proposition_n the_o square_a of_o a_o residual_a line_n apply_v unto_o a_o rational_a line_n senary_a make_v the_o breadth_n or_o other_o side_n a_o first_o reâ—iduall_a line_n svppose_v that_o ab_fw-la be_v a_o residual_a line_n proposition_n and_o let_v cd_o be_v a_o rational_a line_n and_o unto_o the_o line_n cd_o apply_v the_o parallelogram_n ce_fw-fr equal_a to_o the_o square_n of_o the_o line_n ab_fw-la and_o make_v the_o breadth_n the_o line_n cf._n then_o i_o say_v that_o the_o line_n cf_o be_v a_o first_o residual_a lineâ—_z construction_n for_o unto_o the_o line_n ab_fw-la let_v the_o line_n convenient_o join_v be_v suppose_v to_o be_v bâ—_n which_o self_n line_n be_v also_o call_v a_o line_n join_v as_o we_o declare_v in_o the_o end_n of_o the_o 79._o proposition_n wherefore_o the_o â—ines_v agnostus_n and_o gb_o be_v rational_a commensurable_a in_o power_n only_o and_o unto_o the_o line_n cd_o apply_v the_o parallelogram_n ch_n equal_a to_o the_o square_n of_o the_o line_n agnostus_n and_o unto_o the_o line_n kh_o which_o be_v equal_a to_o the_o line_n cd_o apply_v the_o parallelogram_n kl_o equal_a to_o the_o square_n of_o the_o line_n bg_o demonstration_n wherefore_o the_o whole_a parallelogram_n cl_n be_v equal_a to_o the_o square_n of_o the_o linâ—s_n aâ—_z and_o gbâ—_n and_o the_o parallelogrâ—mmâ—_n ce_fw-fr be_v equal_a to_o the_o square_n of_o the_o line_n ab_fw-la wherefore_o the_o parallelogram_n remain_v namely_o the_o parallelogram_n fl_fw-mi be_v equal_a to_o that_o which_o be_v contain_v under_o the_o line_n agnostus_n and_o gb_o twice_o for_o by_o the_o 7._o of_o the_o second_o the_o square_n of_o the_o line_n agnostus_n and_o gb_o be_v equal_a to_o that_o which_o be_v contain_v under_o the_o line_n agnostus_n &_o gb_o twice_o and_o to_o the_o square_n of_o the_o line_n ab_fw-la divide_v the_o line_n fm_o into_o two_o equal_a part_n in_o the_o point_n n._n and_o by_o the_o point_n n_o draw_v unto_o the_o line_n cd_o a_o parallel_a line_n nx_o wherefore_o either_o of_o the_o parallelogram_n fx_n and_o nl_o be_v equal_a to_o that_o which_o be_v contain_v under_o the_o line_n agnostus_n and_o gb_o once_o and_o forasmuch_o as_o the_o square_n of_o the_o line_n agnostus_n and_o gb_o be_v rational_a unto_o which_o square_n the_o parallelogram_n cl_n be_v equal_a therefore_o the_o parallelogram_n cl_n also_o be_v rational_a wherefore_o the_o line_n gm_n be_v rational_a and_o ten_o commensurable_a in_o length_n to_o the_o line_n cd_o again_o forasmuch_o as_o that_o which_o be_v contain_v under_o the_o line_n agnostus_n and_o gb_o twice_o be_v ten_o medial_a therefore_o the_o parallelogram_n equal_a unto_o it_o namely_o the_o paralâ—elogramme_n fl_fw-mi be_v also_o medial_a wherefore_o the_o line_n fm_o be_v rational_a and_o ten_o incommensurable_a in_o length_n to_o the_o line_n cd_o and_o forasmuch_o as_o the_o square_n of_o the_o line_n agnostus_n and_o gb_o be_v rational_a and_o that_o which_o be_v contain_v under_o the_o line_n agnostus_n and_o gb_o twice_o be_v medial_a therefore_o the_o square_n of_o the_o line_n agnostus_n and_o gb_o be_v incommensurable_a to_o that_o which_o be_v contain_v under_o the_o line_n agnostus_n and_o gb_o twice_o but_o unto_o the_o square_n of_o the_o line_n agnostus_n and_o gb_o be_v equal_a the_o parallelogram_n cl_n and_o to_o that_o which_o be_v contain_v under_o the_o line_n agnostus_n and_o gb_o twice_o be_v equal_a the_o parallelogram_n fl_fw-mi wherefore_o the_o parallelogram_n cl_n be_v incommensurable_a to_o the_o parallelogram_n fl._n wherefore_o also_o the_o line_n cm_o be_v incommensurable_a in_o length_n to_o the_o line_n fm_o and_o they_o be_v both_o rational_a wherefore_o the_o line_n cm_o and_o fm_o be_v rational_a commensurable_a in_o power_n only_o and_o therefore_o the_o line_n cf_o be_v
a_o residual_a line_n by_o the_o 73._o of_o the_o ten_o i_o say_v moreover_o that_o it_o be_v a_o first_o residual_a line_n line_n for_o forasmuch_o as_o that_o which_o be_v contain_v under_o the_o line_n agnostus_n and_o gb_o be_v the_o mean_a proportional_a between_o the_o square_n of_o the_o line_n agnostus_n and_o gb_o by_o the_o assumpt_n go_v before_o the_o 54._o of_o the_o ten_o and_o unto_o the_o square_n of_o the_o line_n agnostus_n be_v equal_a the_o parallelogram_n change_z and_o unto_o that_o which_o be_v contain_v under_o the_o line_n agnostus_n and_o gb_o be_v equal_a the_o parallelogram_n nl_o and_o unto_o the_o square_n of_o the_o line_n gb_o be_v equal_a the_o parallelogram_n kl_o wherefore_o the_o parallelogram_n nl_o be_v the_o mean_a proportional_a between_o the_o parallelogram_n ch_n and_o kl_o wherefore_o as_o change_z be_v to_o nl_o so_o iâ—_z nl_o to_o kl_o but_o as_o change_z be_v to_o nl_o so_o be_v the_o line_n ck_o to_o the_o line_n nm_o &_o as_o nl_o be_v to_o kl_o so_o be_v the_o line_n nm_o to_o the_o line_n km_o wherefore_o as_o the_o line_n ck_o be_v to_o the_o line_n nm_o so_o be_v the_o line_n nm_o to_o the_o line_n km_o wherefore_o the_o parallelogram_n contain_v under_o the_o line_n ck_a and_o km_o be_v equal_a to_o the_o square_n of_o the_o line_n nm_o that_o be_v to_o the_o four_o part_n of_o the_o square_n of_o the_o line_n fm_o and_o forasmuch_o as_o the_o square_n of_o the_o line_n agnostus_n be_v commensurable_a to_o the_o square_n of_o the_o line_n gb_o therefore_o the_o parallelogram_n change_z be_v commensurable_a to_o the_o parallelogram_n kl_o but_o as_o change_z be_v to_o kl_o so_o be_v the_o line_n ck_o to_o the_o line_n km_o wherefore_o the_o line_n ck_o be_v commensurable_a in_o length_n to_o the_o line_n km_o wherefore_o by_o the_o 17._o of_o the_o ten_o the_o line_n cm_o be_v in_o power_n more_o than_o the_o line_n fm_o by_o the_o square_n of_o a_o line_n commensurable_a in_o length_n to_o the_o line_n cm_o but_o the_o line_n cm_o be_v commensurable_a in_o length_n to_o the_o rational_a line_n cd_o wherefore_o the_o line_n cf_o be_v a_o fair_a residual_a line_n wherefore_o the_o square_a of_o a_o residual_a line_n apply_v unto_o a_o rational_a line_n make_v the_o breadth_n or_o other_o side_n a_o first_o residual_a line_n which_o be_v require_v to_o be_v demonstrate_v ¶_o the_o â—4_o theorem_a the_o 98._o proposition_n the_o square_a of_o a_o first_o medial_a residual_a line_n apply_v to_o a_o rational_a line_n make_v the_o breadth_n or_o other_o side_n a_o second_o residual_a line_n svppose_v that_o ab_fw-la be_v a_o first_o medial_a residual_a line_n and_o let_v cd_o be_v a_o rational_a line_n and_o unto_o the_o line_n cd_o apply_v the_o parallelogram_n ce_fw-fr equal_a to_o the_o square_n of_o the_o line_n ab_fw-la construction_n and_o make_v in_o breadth_n the_o line_n cf._n then_o i_o say_v that_o the_o line_n cf_o be_v a_o second_o residual_a line_n for_o unto_o the_o line_n ab_fw-la let_v the_o line_n convenient_o join_v be_v suppose_v to_o be_v bg_o wherefore_o the_o line_n agnostus_n and_o bg_o be_v medial_a commensurable_a in_o power_n only_o comprehend_v a_o rational_a superficies_n and_o unto_o the_o line_n cd_o apply_v the_o parallelogram_n ch_n equal_a to_o the_o square_n of_o the_o line_n agnostus_n and_o make_v in_o breadth_n the_o line_n ck_o and_o un-the_a line_n kh_o which_o be_v equal_a to_o the_o line_n cd_o apply_v the_o parallelogram_n kl_o equal_a to_o the_o square_n of_o the_o line_n gb_o and_o make_v in_o breadth_n the_o line_n km_o demonstration_n wherefore_o the_o whole_a parallelogram_n cl_n be_v equal_a to_o both_o the_o square_n of_o the_o line_n agnostus_n and_o gb_o which_o be_v medial_a &_o commensurable_a the_o one_o to_o the_o other_o wherefore_o the_o parallelogram_n ch_n and_o kl_o be_v medial_a and_o commensurable_a the_o one_o to_o the_o other_o wherefore_o by_o the_o 15._o of_o the_o ten_o the_o whole_a parallelogram_n cl_n be_v commensurable_a to_o either_o of_o these_o parallelogram_n ch_n and_o kl_o wherefore_o by_o the_o corollary_n of_o the_o 23._o of_o the_o ten_o the_o whole_a parallelogram_n cl_n be_v also_o medial_a wherefore_o by_o the_o 22._o of_o the_o ten_o the_o line_n cm_o be_v rational_a and_o incommensurable_a in_o length_n to_o the_o line_n cd_o and_o forasmuch_o as_o the_o parallelogram_n cl_n be_v equal_a to_o the_o square_n of_o the_o line_n agnostus_n and_o gbâ—_n and_o the_o square_n of_o the_o line_n agnostus_n and_o gb_o be_v equal_a to_o that_o which_o be_v contain_v under_o the_o line_n agnostus_n and_o gb_o twice_o together_o with_o the_o square_n of_o the_o line_n ab_fw-la by_o the_o 7._o of_o the_o second_o and_o unto_o the_o square_n of_o the_o line_n ab_fw-la be_v equal_a the_o parallelogram_n ce._n wherefore_o the_o residue_n namely_o that_o which_o be_v contain_v under_o the_o line_n agnostus_n and_o gb_o twice_o be_v equal_a to_o the_o residue_n namely_o to_o the_o parallelogram_n fl._n but_o that_o which_o be_v contain_v under_o the_o line_n agnostus_n &_o gb_o twice_o be_v rational_a wherefore_o the_o parallelogram_n fl_fw-mi be_v also_o rational_a wherefore_o the_o line_n fm_o be_v rational_a and_o commensurable_a in_o length_n to_o the_o line_n cd_o by_o the_o 20._o of_o the_o ten_o now_o forasmuch_o as_o the_o parallelogram_n cl_n be_v medial_a and_o the_o parallelogram_n fl_fw-mi be_v rational_a therefore_o they_o be_v incommensurable_a the_o one_o to_o the_o other_o wherefore_o also_o the_o line_n cm_o be_v incommensurable_a in_o length_n to_o the_o line_n fm_o and_o they_o be_v both_o rational_a wherefore_o the_o line_n cf_o be_v a_o residual_a line_n line_n i_o say_v moreover_o that_o it_o be_v a_o second_o residual_a line_n for_o divide_v the_o line_n fm_o into_o two_o equal_a part_n in_o the_o point_n n_o from_o which_o point_n draw_v unto_o the_o line_n cd_o a_o parallel_a line_n nx_o wherefore_o either_o of_o these_o parallelogram_n fx_n and_o nl_o be_v equal_a to_o the_o parallelogram_n contain_v under_o the_o line_n agnostus_n and_o gb_o and_o forasmuch_o as_o the_o parallelogram_n contain_v under_o the_o line_n agnostus_n and_o gb_o be_v the_o mean_a proportional_a between_o the_o square_n of_o the_o line_n agnostus_n and_o bg_o therefore_o the_o parallelogram_n nl_o be_v the_o mean_a proportional_a between_o the_o parallelogram_n ch_n and_o kl_o but_o as_o change_z be_v to_o nl_o so_o be_v the_o line_n ck_o to_o the_o line_n nm_o and_o as_o nl_o be_v to_o kl_o so_o be_v the_o line_n nm_o to_o the_o line_n km_o wherefore_o as_o the_o line_n ck_o be_v to_o the_o line_n nm_o so_o be_v the_o line_n nm_o to_o the_o line_n km_o wherefore_o the_o parallelogram_n contain_v under_o the_o line_n ck_a and_o km_o be_v equal_a to_o the_o square_n of_o the_o line_n nm_o that_o be_v to_o the_o four_o part_n of_o the_o square_n of_o the_o line_n fm_o but_o the_o parallelogram_n change_z be_v commensurable_a to_o the_o parallelogram_n kl_o wherefore_o also_o the_o line_n ck_o be_v commensurable_a in_o length_n to_o the_o line_n km_o wherefore_o by_o the_o 17._o of_o the_o ten_o the_o line_n cm_o be_v in_o power_n more_o than_o the_o line_n fm_o by_o the_o square_n of_o a_o line_n commensurable_a in_o length_n to_o the_o line_n cm_o and_o the_o line_n fm_o which_o be_v the_o line_n convenient_o join_v be_v commensurable_a in_o length_n to_o the_o rational_a line_n cd_o wherefore_o the_o linâ—_n cf_o be_v a_o second_o residual_a line_n wherefore_o the_o square_a of_o a_o first_o medial_a residual_a line_n apply_v to_o a_o rational_a line_n make_v the_o breadth_n or_o other_o side_n a_o second_o residual_a line_n which_o be_v require_v to_o be_v prove_v ¶_o the_o 75._o theorem_a the_o 99_o proposition_n the_o square_a of_o a_o second_o medial_a residual_a line_n apply_v unto_o a_o rational_a line_n make_v the_o breadth_n or_o other_o side_n a_o three_o residual_a line_n svppose_v that_o ab_fw-la be_v a_o second_o medial_a residual_a line_n and_o let_v cd_o be_v a_o rational_a line_n and_o unto_o the_o line_n cd_o apply_v the_o parallelogram_n ce_fw-fr equal_a to_o the_o square_n of_o the_o line_n ab_fw-la and_o make_v in_o breadth_n the_o line_n cf._n construction_n then_o i_o say_v that_o the_o line_n cf_o be_v a_o three_o residual_a line_n for_o unto_o the_o line_n ab_fw-la let_v the_o line_n convenient_o join_v be_v suppose_v to_o be_v bg_o wherefore_o the_o line_n agnostus_n &_o gb_o be_v medial_a commensurable_a in_o power_n only_o contain_v a_o medial_a superficies_n and_o let_v the_o rest_n of_o the_o construction_n be_v as_o in_o the_o proposition_n next_o go_v before_o demonstration_n wherefore_o the_o line_n cm_o be_v rational_a and_o incommensurable_a in_o length_n to_o the_o rational_a line_n cd_o and_o either_o of_o the_o parallelogram_n fx_n
and_o nl_o be_v equal_a to_o that_o which_o be_v contain_v under_o the_o line_n agnostus_n and_o gb_o but_o that_o which_o be_v contain_v under_o the_o line_n agnostus_n &_o gb_o be_v medial_a wherefore_o that_o which_o be_v contain_v under_o the_o line_n agnostus_n and_o gb_o twice_o be_v also_o medial_a wherefore_o the_o whole_a parallelogram_n fl_fw-mi be_v also_o medial_a wherefore_o the_o line_n fm_o be_v rational_a and_o incommensurable_a in_o length_n to_o the_o line_n cd_o and_o forasmuch_o as_o the_o line_n agnostus_n and_o gb_o be_v incommensurable_a in_o length_n therefore_o also_o the_o square_a of_o the_o line_n agnostus_n be_v incommensurable_a to_o the_o parallelogram_n contain_v under_o the_o line_n agnostus_n and_o gb_o but_o unto_o the_o square_n of_o the_o line_n agnostus_n be_v commensurable_a the_o square_n of_o the_o line_n agnostus_n and_o gb_o and_o unto_o the_o parallelogram_n contain_v under_o the_o line_n agnostus_n and_o gb_o be_v commensurable_a that_o which_o be_v contain_v under_o the_o line_n agnostus_n and_o gb_o twice_o wherefore_o the_o square_n of_o the_o line_n agnostus_n and_o gb_o be_v incommensurable_a to_o that_o which_o be_v contain_v under_o the_o line_n agnostus_n and_o gb_o twice_o wherefore_o the_o parallelogram_n which_o be_v equal_a unto_o they_o namely_o the_o parallelogram_n cl_n and_o fl_fw-mi be_v incommensurable_a the_o one_o to_o the_o other_o wherefore_o also_o the_o line_n cm_o be_v incommensurable_a in_o length_n to_o the_o line_n fm_o and_o they_o be_v both_o rational_a wherefore_o the_o line_n cf_o be_v a_o residual_a line_n line_n i_o say_v moreover_o that_o it_o be_v a_o three_o residual_a line_n for_o forasmuch_o as_o the_o square_n of_o the_o line_n agnostus_n that_o be_v the_o parallelogram_n change_z be_v commensurable_a to_o the_o square_n of_o the_o line_n bg_o that_o be_v to_o the_o parallelogram_n kl_o therefore_o the_o line_n ck_o be_v commensurable_a in_o length_n to_o the_o line_n km_o and_o in_o like_a sort_n as_o in_o the_o former_a proposition_n so_o also_o in_o this_o may_v we_o prove_v that_o the_o parallelogram_n contain_v under_o the_o line_n ck_a and_o km_o be_v equal_a to_o the_o square_n of_o the_o line_n nm_o that_o be_v to_o the_o four_o part_n of_o the_o square_n of_o the_o line_n fm_o wherefore_o the_o line_n cm_o be_v in_o power_n more_o than_o the_o line_n fm_o by_o the_o square_n of_o a_o line_n commensurable_a in_o length_n to_o the_o line_n cmâ—_n and_o neither_o of_o the_o line_n cm_o nor_o fm_o be_v commensurable_a in_o length_n to_o the_o rational_a line_n â—d_a wherefore_o the_o line_n cf_o be_v a_o three_o residual_a line_n wherefore_o the_o square_a of_o a_o second_o medial_a residual_a line_n apply_v unto_o a_o rational_a line_n make_v the_o breadth_n or_o other_o side_n a_o three_o residual_a line_n which_o be_v require_v to_o be_v demonstrate_v ¶_o the_o 76._o theorem_a the_o 100_o proposition_n the_o square_a of_o a_o less_o line_n apply_v unto_o a_o rational_a line_n make_v the_o breadth_n or_o other_o side_n a_o four_o residual_a line_n svppose_v that_o ab_fw-la be_v a_o less_o line_n and_o let_v cd_o be_v a_o rational_a line_n and_o unto_o the_o line_n cd_o apply_v the_o parallelogram_n ce_fw-fr equal_a to_o the_o square_n of_o the_o line_n ab_fw-la and_o make_v in_o breadth_n the_o line_n cf._n then_o i_o say_v that_o the_o line_n cf_o be_v a_o four_o residual_a line_n for_o unto_o the_o line_n ab_fw-la let_v the_o line_n convenient_o join_v be_v suppose_v to_o be_v bg_o construction_n wherefore_o the_o line_n agnostus_n &_o gb_o be_v incommensurable_a in_o power_n have_v that_o which_o be_v make_v of_o their_o square_n add_v together_o rational_a and_o that_o which_o be_v contain_v under_o they_o medial_a and_o let_v the_o rest_n of_o the_o construction_n be_v as_o in_o the_o proposition_n go_v before_o wherefore_o the_o whole_a parallelogram_n cl_n be_v rational_a demonstration_n wherefore_o the_o line_n cm_o be_v also_o rational_a and_o commensurable_a in_o length_n to_o the_o line_n cd_o and_o forasmuch_o as_o that_o which_o be_v contain_v under_o the_o line_n agnostus_n and_o gb_o twice_o be_v medial_a therefore_o the_o parallelogram_n which_o be_v equal_a unto_o it_o namely_o the_o parallelogram_n fl_fw-mi be_v also_o medial_a wherefore_o the_o line_n fm_o be_v rational_a and_o incommensurable_a in_o length_n to_o the_o line_n cd_o but_o the_o line_n cm_o be_v commensurable_a in_o length_n to_o the_o line_n cd_o wherefore_o by_o the_o 13._o of_o the_o ten_o the_o line_n cm_o be_v incommensurable_a in_o length_n to_o the_o line_n fm_o and_o they_o be_v both_o rational_a wherefore_o the_o line_n cm_o and_o fm_o be_v rational_a commensurable_a in_o power_n only_o line_n wherefore_o the_o line_n cf_o be_v a_o residual_a line_n i_o say_v moreover_o that_o it_o be_v a_o four_o residual_a line_n for_o forasmuch_o as_o the_o line_n agnostus_n and_o gb_o be_v incommensurable_a in_o power_n therefore_o the_o square_n of_o they_o that_o be_v the_o parallelogram_n which_o be_v equal_a unto_o they_o namely_o the_o parallelogram_n ch_n and_o kl_o be_v incommensurable_a the_o one_o to_o the_o other_o wherefore_o also_o the_o line_n ck_o be_v incommensurable_a in_o length_n to_o the_o line_n km_o and_o in_o like_a sort_n may_v we_o prove_v that_o the_o parallelogram_n contain_v under_o the_o line_n ck_a and_o km_o be_v equal_a to_o the_o square_n of_o the_o line_n nm_o that_o be_v to_o the_o four_o part_n of_o the_o square_n of_o the_o line_n fm_o wherefore_o by_o the_o 18._o of_o the_o ten_o the_o line_n cm_o be_v in_o power_n more_o than_o the_o line_n fm_o by_o the_o square_n of_o a_o line_n incommensurable_a in_o length_n to_o the_o line_n cm_o and_o the_o whole_a line_n cm_o be_v commensurable_a in_o length_n to_o the_o rational_a line_n cd_o wherefore_o the_o line_n cf_o be_v a_o four_o residual_a line_n wherefore_o the_o square_a of_o a_o less_o line_n apply_v unto_o a_o rational_a line_n make_v the_o breadth_n or_o other_o side_n a_o four_o residual_a line_n which_o be_v require_v to_o be_v prove_v ¶_o the_o 77._o theorem_a the_o 101._o proposition_n the_o square_a of_o a_o line_n make_v with_o a_o rational_a superficies_n the_o whole_a superficies_n medial_a apply_v unto_o a_o rational_a line_n make_v the_o breadth_n or_o other_o side_n a_o five_o residual_a line_n svppose_v that_o ab_fw-la be_v a_o line_n make_v with_o a_o rational_a superficies_n the_o whole_a superficies_n medial_a and_o let_v cd_o be_v a_o rational_a line_n and_o unto_o the_o line_n cd_o apply_v the_o parallelogram_n ce_fw-fr equal_a to_o the_o square_n of_o the_o line_n ab_fw-la and_o make_v in_o breadth_n the_o line_n cf._n then_o i_o say_v that_o the_o line_n cf_o be_v a_o five_o residual_a line_n for_o unto_o the_o line_n ab_fw-la let_v the_o line_n convenient_o join_v be_v suppose_v to_o â—e_n bg_o wherefore_o the_o line_n agnostus_n and_o gb_o be_v incommensurable_a in_o power_n have_v that_o which_o be_v make_v of_o their_o square_n add_v together_o medial_a and_o that_o which_o be_v contain_v under_o they_o rational_a let_v the_o rest_n of_o the_o construction_n be_v in_o this_o as_o it_o be_v in_o the_o former_a proposition_n wherefore_o the_o whole_a parallelogram_n cl_n be_v medial_a wherefore_o the_o line_n cm_o be_v rational_a and_o incommensurable_a in_o length_n to_o the_o line_n cd_o and_o either_o of_o the_o parallelogram_n fx_n &_o nl_o be_v rational_o wherefore_o the_o whole_a parallelogram_n fl_fw-mi be_v also_o rational_a wherefore_o also_o the_o line_n fm_o be_v rational_a and_o commensurable_a in_o length_n to_o the_o line_n cd_o and_o forasmuch_o as_o the_o parallelogram_n cl_n be_v medial_a and_o the_o parallelogram_n fl_fw-mi be_v rational_a therefore_o cl_n and_o fl_fw-mi be_v incommensurable_a the_o one_o to_o the_o other_o and_o the_o line_n cm_o be_v incommensurable_a in_o length_n to_o the_o line_n fm_o and_o they_o be_v both_o rational_a wherefore_o the_o line_n cm_o and_o fm_o be_v rational_a commensurable_a in_o power_n only_o wherefore_o the_o line_n gf_o be_v a_o residual_a line_n i_o say_v moreover_o that_o it_o be_v a_o â—iâ—t_o residual_a line_n line_n for_o we_o may_v in_o like_a sort_n prove_v that_o the_o parallelogram_n contain_v under_o the_o line_n ck_a and_o km_o be_v equal_a to_o the_o square_n of_o the_o line_n nm_o that_o be_v to_o the_o four_o part_n of_o the_o square_n of_o the_o line_n fm_o and_o forasmuch_o as_o the_o square_n of_o the_o line_n agnostus_n that_o be_v the_o parallelogram_n change_z be_v incommensurable_a to_o the_o square_n of_o the_o line_n bg_o that_o be_v to_o the_o parallelogram_n kl_o therefore_o the_o line_n ck_o be_v incommensurable_a in_o length_n to_o the_o line_n km_o wherefore_o by_o the_o 18._o of_o the_o ten_o the_o line_n cm_o be_v in_o power_n more_o than_o the_o line_n fm_o by_o the_o square_n of_o a_o
line_n incommensurable_a in_o length_n to_o the_o line_n cm_o and_o the_o line_n convenient_o join_v namely_o the_o line_n fm_o be_v commensurable_a in_o length_n to_o the_o rational_a line_n cd_o wherefore_o the_o line_n cf_o be_v a_o â—iâ—t_o residual_a line_n wherefore_o the_o line_n cf_o be_v a_o five_o residual_a line_n wherefore_o the_o square_a of_o a_o line_n make_v with_o a_o rational_a superficies_n the_o whole_a superficies_n medial_a apply_v unto_o a_o rational_a line_n make_v the_o breadth_n or_o other_o side_n a_o five_o residual_a line_n which_o be_v require_v to_o be_v demonstrate_v ¶_o the_o 78._o theorem_a the_o 102._o proposition_n the_o square_a of_o a_o line_n make_v with_o a_o medial_a superficies_n the_o whole_a superficies_n medial_a apply_v to_o a_o rational_a line_n make_v the_o breadth_n or_o other_o side_n a_o six_o residual_a line_n svppose_v that_o ab_fw-la be_v a_o line_n make_v with_o a_o medial_a superficies_n the_o whole_a superficies_n medial_a construction_n and_o let_v cd_o be_v a_o rational_a line_n and_o unto_o the_o line_n cd_o apply_v the_o parallelogram_n ce_fw-fr equal_a to_o the_o square_n of_o the_o line_n ab_fw-la and_o make_n in_o breadth_n the_o line_n cf._n then_o i_o say_v that_o the_o line_n cf_o be_v a_o six_o residual_a line_n for_o unto_o the_o line_n ab_fw-la let_v the_o line_n convenient_o join_v be_v bg_o wherefore_o the_o line_n agnostus_n and_o bg_o be_v incommensurable_a in_o power_n have_v that_o which_o be_v make_v of_o their_o square_n add_v together_o medial_a &_o that_o which_o be_v contain_v under_o they_o medial_a and_o moreover_o that_o which_o be_v make_v of_o their_o square_n add_v together_o be_v incommensurable_a to_o that_o which_o be_v contain_v under_o they_o let_v the_o rest_n of_o the_o construction_n be_v in_o this_o as_o it_o be_v in_o the_o proposition_n go_v before_o demonstration_n wherefore_o the_o whole_a parallelogram_n cl_n be_v medial_a for_o it_o be_v equal_a to_o that_o which_o be_v make_v of_o the_o square_n of_o the_o line_n agnostus_n &_o gb_o add_v together_o which_o be_v suppose_v to_o be_v medial_a wherefore_o the_o line_n cm_o be_v rational_a and_o incommensurable_a in_o length_n to_o the_o line_n cd_o and_o in_o like_a manner_n the_o parallelogram_n fl_fw-mi be_v medial_a wherefore_o also_o the_o line_n fm_o be_v rational_a and_o incommensurable_a â—n_n length_n to_o the_o line_n cd_o and_o forasmuch_o as_o that_o which_o be_v make_v of_o the_o square_n of_o the_o line_n agnostus_n and_o gb_o add_v together_o be_v incommensurable_a to_o that_o which_o be_v contain_v under_o the_o line_n agnostus_n and_o gb_o twice_o therefore_o the_o parallelogram_n equal_a to_o they_o namely_o the_o parallelogram_n cl_n and_o fl_fw-mi be_v incommensurable_a the_o one_o to_o the_o other_o wherefore_o also_o the_o line_n gm_n and_o fm_o be_v incommensurable_a in_o length_n and_o they_o be_v both_o rational_a wherefore_o they_o be_v rational_a commensurable_a in_o power_n only_o wherefore_o the_o line_n cf_o be_v a_o residual_a line_n i_o say_v moreover_o that_o it_o be_v a_o six_o residual_a line_n residual_a let_v the_o rest_n of_o the_o demonstration_n be_v as_o it_o be_v in_o the_o former_a proposition_n and_o forasmuch_o as_o the_o line_n agnostus_n and_o bg_o be_v incommensurable_a in_o power_n therefore_o their_o square_n that_o be_v the_o parallelogram_n which_o be_v equal_a unto_o they_o namely_o the_o parallelogram_n ch_n and_o kl_o be_v incommensurable_a the_o one_o to_o the_o other_o wherefore_o also_o the_o line_n ck_o be_v incommensurable_a in_o length_n to_o the_o line_n km_o wherefore_o by_o the_o 18._o of_o the_o ten_o the_o line_n cm_o be_v in_o power_n more_o than_o the_o line_n fm_o by_o the_o square_n of_o a_o line_n incommensurable_a in_o length_n to_o the_o line_n cm_o and_o neither_o of_o the_o line_n cm_o nor_o fm_o be_v commensurable_a in_o length_n to_o the_o rational_a line_n cd_o wherefore_o the_o line_n cf_o be_v a_o six_o residual_a line_n wherefore_o the_o square_a of_o a_o line_n make_v with_o a_o medial_a superficies_n the_o whole_a superficies_n medial_a apply_v to_o a_o rational_a line_n make_v the_o breadth_n or_o other_o side_n a_o six_o residual_a line_n which_o be_v require_v to_o be_v demonstrate_v ¶_o the_o 79._o theorem_a the_o 103._o proposition_n a_o line_n commensurable_a in_o length_n to_o a_o residual_a line_n be_v itself_o also_o a_o residual_a line_n of_o the_o self_n same_o order_n svppose_v that_o ab_fw-la be_v a_o residual_a line_n unto_o which_o let_v the_o line_n cd_o be_v commensurable_a in_o length_n senary_n then_o i_o say_v that_o the_o line_n cd_o be_v also_o a_o residual_a line_n and_o of_o the_o self_n same_o order_n of_o residual_a line_n that_o the_o line_n ab_fw-la be_v for_o forasmuch_o as_o the_o line_n ab_fw-la be_v a_o residual_a line_n construction_n let_v the_o line_n convenient_o join_v unto_o it_o be_v suppose_v to_o be_v be._n wherefore_o the_o line_n ae_n and_o be_v be_v rational_a commensurable_a in_o power_n only_o as_o the_o line_n ab_fw-la be_v to_o the_o line_n cd_o so_o by_o the_o 12._o of_o the_o six_o let_v the_o line_n be_v be_v to_o the_o line_n df._n demonstration_n wherefore_o by_o the_o 12._o of_o the_o five_o as_o one_o of_o the_o antecedentes_fw-la be_v to_o one_o of_o the_o consequentes_fw-la so_o be_v all_o the_o antecedentes_fw-la to_o all_o the_o consequentes_fw-la wherefore_o as_o the_o line_n ab_fw-la be_v to_o the_o line_n cd_o so_o be_v the_o whole_a line_n ae_n to_o the_o whole_a line_n cf_o and_o the_o line_n be_v to_o the_o line_n df._n wherefore_o by_o the_o 10._o of_o the_o ten_o the_o line_n ae_n be_v commensurable_a in_o length_n to_o the_o line_n cf_o and_o the_o line_n be_v to_o the_o line_n df._n but_o the_o line_n ae_n be_v rational_a wherefore_o the_o line_n cf_o be_v also_o rational_a and_o in_o like_a sort_n the_o line_n df_o be_v rational_a for_o that_o the_o line_n be_v to_o who_o it_o be_v commensurable_a be_v also_o rational_a and_o for_o that_o as_o the_o line_n be_v be_v to_o the_o line_n ae_n so_o be_v the_o line_n df_o to_o the_o line_n cf._n but_o the_o line_n be_v and_o ae_n be_v commensurable_a in_o power_n only_o wherefore_o the_o line_n cd_o and_o df_o be_v commensurable_a in_o power_n only_o wherefore_o the_o line_n cd_o be_v a_o residual_a line_n line_n i_o say_v moreover_n that_o it_o be_v a_o residual_a line_n of_o the_o self_n same_o order_n that_o the_o line_n ab_fw-la be_v for_o for_o that_o as_o we_o have_v before_o say_v as_o the_o line_n aâ—_z be_v to_o the_o line_n cf_o so_o be_v the_o line_n be_v to_o the_o line_n dfâ—_n therefore_o alternate_o as_o the_o line_n ae_n be_v to_o the_o line_n be_v so_o be_v the_o line_n cf_o to_o the_o line_n df._n but_o the_o line_n ae_n be_v in_o power_n more_o than_o the_o line_n ebb_n either_o by_o the_o square_n of_o a_o line_n commensurable_a in_o length_n to_o the_o line_n ae_n or_o by_o the_o square_n of_o a_o line_n incommensurable_a in_o length_n to_o the_o line_n ae_n if_o ae_n be_v in_o power_n 〈◊〉_d then_o be_v by_o the_o square_n of_o a_o line_n commensurable_a in_o length_n to_o ae_n than_o the_o line_n â—fâ—_n shall_v also_o by_o the_o 14._o of_o the_o ten_o be_v in_o power_n more_o than_o the_o line_n df_o by_o the_o square_n of_o a_o line_n commensurable_a in_o length_n to_o the_o line_n cf_o and_o so_o if_o the_o line_n ae_n be_v commensurable_a in_o length_n to_o the_o rational_a line_n put_v forasmuch_o as_o the_o line_n ae_n be_v commensurable_a in_o length_n to_o the_o line_n cf_o therefore_o by_o the_o 12._o of_o the_o ten_o the_o line_n cf_o shall_v also_o be_v commensurable_a in_o length_n to_o the_o same_o rational_a line_n wherefore_o either_o of_o the_o line_n ab_fw-la and_o cd_o be_v a_o first_o residual_a line_n and_o if_o the_o line_n be_v be_v commensurable_a in_o length_n to_o the_o rational_a line_n put_v forasmuch_o as_o the_o line_n be_v be_v commensurable_a in_o length_n to_o the_o line_n df_o therefore_o the_o line_n df_o shall_v also_o be_v commensurable_a in_o length_n to_o the_o rational_a line_n putâ—_n and_o then_o either_o of_o the_o line_n ab_fw-la and_o cd_o be_v a_o second_o residual_a line_n and_o if_o neither_o of_o the_o line_n ae_n nor_o be_v be_v commensurable_a in_o length_n to_o the_o rational_a line_n put_v then_o neither_o of_o the_o line_n cf_o nor_o df_o shall_v be_v commensurable_a in_o length_n to_o the_o same_o rational_a line_n by_o the_o 13._o of_o the_o ten_o and_o so_o either_o of_o the_o line_n ab_fw-la &_o cd_o be_v a_o three_o residual_a line_n but_o if_o the_o line_n ae_n be_v in_o power_n more_o they_o the_o line_n be_v by_o the_o square_n of_o a_o line_n incommensurable_a in_o length_n to_o the_o line_n ae_n the_o line_n cf_o
medial_n residual_a line_n or_o a_o second_o medial_a residual_a line_n according_a as_o the_o line_n a_o be_v suppose_v to_o be_v which_o be_v require_v to_o be_v prove_v ¶_o the_o 81._o theorem_a the_o 105._o proposition_n a_o line_n commensurable_a to_o a_o less_o line_n be_v itself_o also_o a_o less_o line_n construction_n svppose_v that_o ab_fw-la be_v a_o less_o line_n unto_o who_o let_v the_o line_n cd_o be_v commensurable_a then_o i_o say_v that_o the_o line_n cd_o be_v also_o a_o less_o line_n for_o let_v the_o same_o construction_n be_v in_o this_o that_o be_v in_o the_o former_a proposition_n demonstration_n and_o forasmuch_o as_o the_o line_n ae_n and_o ebb_n be_v incommensurable_a in_o power_n therefore_o by_o the_o 22._o of_o the_o six_o and_o 10._o of_o the_o ten_o the_o line_n cf_o &_o fd_o be_v incommensurable_a in_o power_n again_o by_o the_o 22._o of_o the_o six_o as_o the_o square_n of_o the_o line_n ae_n be_v to_o the_o square_n of_o the_o line_n be_v so_o be_v the_o square_a of_o the_o line_n cf_o to_o the_o square_n of_o the_o line_n df._n wherefore_o by_o composition_n as_o the_o square_n of_o the_o line_n ae_n and_o be_v be_v to_o the_o square_n of_o the_o line_n be_v so_o be_v the_o square_n of_o the_o line_n cf_o and_o df_o to_o the_o square_n of_o the_o line_n df_o and_o alternate_o as_o the_o square_n of_o the_o line_n ae_n and_o be_v be_v to_o the_o square_n of_o the_o line_n cf_o and_o df_o so_o be_v the_o square_a of_o the_o line_n be_v to_o the_o square_n of_o the_o line_n df._n but_o the_o square_n of_o the_o line_n be_v be_v commensurable_a to_o the_o square_n of_o the_o line_n df_o for_o the_o line_n be_v and_o df_o be_v commensurable_a wherefore_o that_o which_o be_v make_v of_o the_o square_n of_o the_o line_n ae_n and_o be_v add_v together_o be_v commensurable_a to_o that_o which_o be_v make_v of_o the_o square_n of_o the_o line_n cf_o and_o df_o add_v together_o but_o that_o which_o be_v make_v of_o the_o square_n of_o the_o line_n ae_n and_o be_v add_v together_o be_v rational_a wherefore_o that_o which_o be_v make_v of_o the_o square_n of_o the_o line_n cf_o and_o df_o add_v together_o be_v also_o rational_a again_o for_o that_o as_o the_o square_n of_o the_o line_n ae_n be_v to_o the_o parallelogram_n contain_v under_o the_o line_n ae_n and_o be_v so_o be_v the_o square_a of_o the_o line_n cf_o to_o the_o parallelogram_n contain_v under_o the_o line_n cf_o and_o df_o as_o we_o declare_v in_o the_o proposition_n next_o go_v before_o therefore_o alternate_o as_o the_o square_n of_o the_o line_n ae_n be_v to_o the_o square_n of_o the_o line_n cf_o so_o be_v the_o parallelogram_n contain_v under_o the_o line_n ae_n and_o be_v to_o the_o parallelogram_n contain_v under_o the_o line_n cf_o and_o df._n but_o the_o square_n of_o the_o line_n ae_n be_v commensurable_a to_o the_o square_n of_o the_o line_n cf_o for_o the_o line_n ae_n &_o cf_o be_v commensurable_a wherefore_o the_o parallelogram_n contain_v under_o the_o line_n ae_n and_o be_v be_v commensurable_a to_o the_o parallelogram_n contain_v under_o the_o line_n cf_o and_o df._n but_o the_o parallelogram_n contain_v under_o the_o line_n ae_n and_o be_v be_v medial_a wherefore_o the_o parallelogram_n contain_v under_o the_o line_n cf_o and_o df_o be_v also_o medial_a wherefore_o the_o line_n cf_o and_o df_o be_v incommensurable_a in_o power_n have_v that_o which_o be_v make_v of_o their_o square_n add_v together_o rational_a and_o the_o parallelogram_n contain_v under_o they_o medial_a wherefore_o the_o line_n cd_o be_v a_o less_o line_n a_o line_n therefore_o commensurable_a to_o a_o less_o line_n be_v itself_o also_o a_o less_o line_n which_o be_v require_v to_o be_v prove_v an_o other_o demonstration_n suppose_v that_o a_o be_v a_o less_o line_n and_o unto_o a_o let_v the_o line_n b_o be_v commensurable_a whether_o in_o length_n and_o power_n or_o in_o power_n only_o then_o i_o say_v that_o b_o be_v a_o less_o line_n take_v a_o rational_a line_n cd_o construction_n and_o unto_o the_o line_n cd_o apply_v by_o the_o 44_o of_o the_o first_o the_o parallelogram_n ce_fw-fr equal_a to_o the_o square_n of_o the_o line_n a_o and_o make_v in_o breadth_n the_o line_n cf._n wherefore_o by_o the_o 100_o proposition_n the_o line_n cf_o be_v a_o four_o residual_a line_n unto_o the_o line_n fe_o apply_v by_o the_o same_o the_o parallelogram_n eh_o equal_a to_o the_o square_n of_o the_o line_n b_o and_o make_v in_o breadth_n the_o line_n fh_o now_o forasmuch_o as_o the_o line_n a_o be_v commensurable_a to_o the_o line_n b_o demonstration_n therefore_o also_o the_o square_a of_o the_o line_n a_o be_v commensurable_a to_o the_o square_n of_o the_o line_n b._n but_o unto_o the_o square_n of_o the_o line_n a_o be_v equal_a the_o parallelogram_n ce_fw-fr &_o unto_o the_o square_n of_o the_o line_n b_o be_v equal_a the_o parallelogram_n eh_o wherefore_o the_o parallelogram_n ce_fw-fr be_v commensurable_a to_o the_o parallelogram_n eh_o but_o as_o the_o parallelogram_n ce_fw-fr be_v to_o the_o parallelogram_n eh_o so_o be_v the_o line_n cf_o to_o the_o line_n fh_o wherefore_o the_o line_n cf_o be_v commensurable_a in_o length_n to_o the_o line_n fh_o but_o the_o line_n cf_o be_v a_o four_o residual_a line_n wherefore_o the_o line_n fh_o be_v also_o a_o four_o residual_a line_n by_o the_o 103._o of_o the_o ten_o and_o the_o line_n fe_o be_v rational_a but_o if_o a_o superficies_n be_v contain_v under_o a_o rational_a line_n and_o a_o four_o residual_a line_n the_o line_n that_o contain_v in_o power_n that_o superficies_n be_v by_o the_o 94._o of_o the_o ten_o a_o less_o line_n but_o the_o line_n b_o contain_v in_o power_n the_o superficies_n eh_o wherefore_o the_o line_n b_o be_v a_o less_o line_n which_o be_v require_v to_o be_v prove_v ¶_o the_o 82._o theorem_a the_o 106._o proposition_n a_o line_n commensurable_a to_o a_o line_n make_v with_o a_o rational_a superficies_n the_o whole_a superficies_n medial_a be_v itself_o also_o a_o line_n make_v with_o a_o rational_a superficies_n the_o whole_a superficies_n medial_a svppose_v that_o ab_fw-la be_v a_o line_n make_v with_o a_o rational_a superficies_n the_o whole_a superficies_n medial_a unto_o who_o let_v the_o line_n cd_o be_v commensurable_a then_o i_o say_v that_o the_o line_n cd_o be_v a_o line_n make_v with_o a_o rational_a superficies_n the_o whole_a superficies_n medial_a construction_n unto_o the_o line_n ab_fw-la let_v the_o line_n convenient_o join_v be_v be._n wherefore_o the_o line_n ae_n and_o ebb_n be_v incommensurable_a in_o power_n have_v that_o which_o be_v make_v of_o their_o square_n add_v together_o medial_a and_o the_o parallelogram_n contain_v under_o they_o rational_a let_v the_o construction_n be_v in_o this_o as_o it_o be_v in_o the_o former_a proposition_n demonstration_n and_o in_o like_a sort_n may_v we_o prove_v that_o as_o the_o line_n ae_n be_v to_o the_o line_n be_v so_o be_v the_o line_n cf_o to_o the_o line_n df_o and_o that_o that_o which_o be_v make_v of_o the_o square_n of_o the_o line_n ae_n and_o be_v add_v together_o be_v commensurable_a to_o that_o which_o be_v make_v of_o the_o square_n of_o the_o line_n cf_o and_o df_o add_v together_o and_o that_o that_o which_o be_v contain_v under_o the_o line_n ae_n and_o ebb_n be_v in_o like_a sort_n commensurable_a to_o that_o which_o be_v contain_v under_o the_o line_n cf_o and_o df._n wherefore_o also_o the_o line_n cf_o and_o df_o be_v commensurable_a in_o power_n have_v that_o which_o be_v make_v of_o their_o square_n add_v together_o medial_a and_o that_o which_o be_v contain_v under_o they_o rational_a wherefore_o the_o line_n cd_o be_v a_o line_n make_v with_o a_o rational_a superficies_n the_o whole_a superficies_n medial_a wherefore_o a_o line_n commensurable_a to_o a_o line_n make_v with_o a_o rational_a superficies_n the_o whole_a superficies_n medial_a be_v itself_o also_o a_o line_n make_v with_o a_o rational_a superficies_n the_o whole_a superficies_n medial_a which_o be_v require_v to_o be_v demonstrate_v an_o other_o demonstration_n suppose_v that_o a_o be_v a_o line_n make_v with_o a_o rational_a superficies_n the_o whole_a superficies_n medial_a and_o unto_o it_o let_v the_o line_n b_o be_v commensurable_a either_o in_o length_n and_o in_o power_n or_o in_o power_n only_o then_o i_o say_v that_o b_o be_v a_o line_n make_v with_o a_o rational_a superficies_n the_o whole_a superficies_n medial_a construction_n take_v a_o rational_a line_n cd_o and_o unto_o the_o line_n cd_o apply_v the_o parallelogram_n ce_fw-fr equal_a to_o the_o square_n of_o the_o line_n a_o and_o make_v in_o breadth_n the_o line_n gf_o wherefore_o by_o the_o 101._o proposition_n the_o line_n cf_o be_v a_o five_o residual_a line_n
of_o the_o ten_o the_o line_n de_fw-fr be_v a_o first_o binomial_a line_n divide_v it_o into_o his_o name_n in_o the_o point_n g._n and_o let_v dg_o be_v the_o great_a name_n wherefore_o the_o line_n dg_o and_o â—e_n be_v rational_a commensurable_a in_o power_n only_o and_o the_o line_n dg_o be_v in_o power_n more_o than_o the_o line_n ge_z by_o the_o square_n of_o a_o line_n commensurable_a in_o length_n to_o the_o line_n dg_o and_o the_o line_n dg_o be_v commensurable_a in_o length_n to_o the_o rational_a line_n put_v dc_o wherefore_o the_o line_n df_o be_v commensurable_a in_o length_n to_o the_o line_n dg_o wherefore_o by_o the_o 13._o of_o the_o ten_o the_o whole_a line_n df_o be_v commensurable_a in_o length_n to_o the_o line_n remain_v namely_o â—o_o the_o line_n gf_o and_o forasmuch_o as_o the_o line_n df_o be_v commensurable_a to_o the_o line_n fg_o but_o the_o line_n fd_o be_v rational_a wherefore_o the_o line_n fg_o be_v also_o rational_a and_o forasmuch_o as_o the_o line_n fd_o be_v commensurable_a in_o length_n to_o the_o line_n fg_o but_o the_o line_n df_o be_v incommensurable_a in_o length_n to_o the_o line_n fe_o wherefore_o the_o line_n fg_o be_v incommensurable_a in_o length_n to_o the_o line_n fe_o by_o the_o 13._o of_o the_o ten_o and_o they_o be_v both_o rational_a line_n wherefore_o the_o line_n gf_o and_o fe_o be_v rational_a commensurable_a in_o power_n only_o wherefore_o by_o the_o 73._o of_o the_o ten_o the_o line_n eglantine_n be_v a_o residual_a line_n but_o it_o be_v also_o rational_a as_o before_o have_v be_v prove_v which_o be_v impossible_a namely_o that_o one_o &_o the_o same_o line_n shall_v be_v both_o rational_a and_o irrational_a wherefore_o a_o residual_a line_n be_v not_o one_o and_o the_o same_o with_o a_o binomial_a line_n that_o be_v be_v not_o a_o binomial_a line_n which_o be_v require_v to_o be_v demonstrate_v corollary_n ¶_o a_o corollary_n a_o residual_a line_n and_o the_o other_o five_o irrational_a line_n follow_v it_o be_v neither_o medial_a line_n nor_o one_o and_o the_o same_o between_o themseluesâ—_n that_o be_v one_o be_v utter_o of_o a_o diverse_a kind_n from_o a_o other_o for_o the_o square_n of_o a_o medial_a line_n apply_v to_o a_o rational_a line_n make_v the_o breadth_n rational_a and_o incommensurable_a in_o length_n to_o the_o rational_a line_n whereunto_o it_o be_v apply_v by_o the_o 22._o of_o the_o ten_o the_o square_a of_o a_o residual_a line_n apply_v to_o a_o rational_a line_n make_v the_o breadth_n a_o first_o residual_a line_n by_o the_o 97._o of_o the_o ten_o the_o square_a of_o a_o first_o medial_a residual_a line_n apply_v to_o a_o rational_a line_n make_v the_o breadth_n a_o second_o residual_a line_n by_o the_o 98._o of_o the_o ten_o the_o square_a of_o a_o second_o medial_a residual_a line_n apply_v unto_o a_o rational_a line_n make_v the_o breadth_n a_o three_o residual_a line_n by_o the_o 99_o of_o the_o ten_o the_o square_a of_o a_o less_o line_n apply_v to_o a_o rational_a line_n make_v the_o breadth_n a_o four_o residual_a line_n by_o the_o 100_o of_o the_o ten_o the_o square_a of_o a_o line_n make_v with_o a_o rational_a superficies_n the_o whole_a superficies_n medial_a apply_v to_o a_o rational_a line_n make_v the_o breadth_n a_o â—ift_n residual_a line_n by_o the_o 101._o of_o the_o ten_o and_o the_o square_a of_o a_o line_n make_v with_o a_o medial_a superficies_n the_o whole_a superficies_n medial_a apply_v to_o a_o rational_a line_n make_v the_o breadth_n a_o six_o residual_a line_n by_o the_o 102._o of_o the_o ten_o now_o forasmuch_o as_o these_o foresay_a side_n which_o be_v the_o breadthes_fw-mi differ_v both_o from_o the_o first_o breadth_n sore_o that_o it_o be_v rational_a and_o differ_v also_o the_o one_o from_o the_o other_o for_o that_o they_o be_v residual_n of_o diverse_a order_n and_o kind_n it_o be_v manifest_a that_o those_o irrational_a line_n differ_v also_o the_o one_o from_o the_o other_o and_o forasmuch_o as_o it_o have_v be_v prove_v in_o the_o 111._o proposition_n that_o 〈◊〉_d residual_a 〈◊〉_d be_v not_o one_o and_o the_o same_o with_o a_o binomial_a line_n and_o it_o have_v also_o be_v prove_v that_o the_o 〈…〉_o of_o a_o residual_a line_n and_o of_o the_o five_o irrational_a line_n that_o follow_v it_o be_v apply_v to_o a_o rational_a line_n do_v make_v their_o breadthes_fw-mi one_o of_o the_o residual_n of_o that_o order_n of_o which_o they_o be_v who_o squareâ—_n be_v apply_v to_o the_o rational_a line_n likewise_o also_o the_o square_n of_o a_o binomial_a line_n and_o of_o the_o five_o irrational_a line_n which_o follow_v it_o be_v apply_v to_o a_o rational_a line_n do_v make_v the_o breadthes_fw-mi one_o of_o the_o binomials_n of_o that_o order_n of_o which_o they_o be_v who_o square_n be_v apply_v to_o the_o rational_a line_n wherefore_o the_o irrational_a line_n which_o follow_v the_o binomial_a line_n and_o the_o irrational_a line_n which_o follow_v the_o residual_a line_n differ_v the_o one_o from_o the_o other_o so_o that_o all_o the_o irrational_a line_n be_v 13._o in_o number_n namely_o these_o 1_o a_o medial_a line_n 2_o a_o binomial_a line_n 3_o a_o first_o bimedial_a line_n 4_o a_o second_o bimedial_a line_n 5_o a_o great_a line_n 6_o a_o line_n contain_v in_o power_n a_o rational_a superficies_n and_o a_o medial_a superficies_n 7_o a_o line_n contain_v in_o power_n two_o medial_a superficiece_n 8_o a_o residual_a line_n 9_o a_o first_o medial_a residual_a line_n 10_o a_o second_o medial_a residual_a line_n 11_o a_o less_o line_n 12_o a_o line_n make_v with_o a_o rational_a superficies_n the_o whole_a superficies_n medial_a 13_o a_o line_n make_v with_o a_o medial_a superficies_n the_o whole_a superficies_n medial_a ¶_o the_o 88_o theorem_a the_o 112._o proposition_n the_o square_a of_o a_o rational_a line_n apply_v unto_o a_o binomial_a line_n make_v the_o breadth_n or_o other_o side_n a_o residual_a line_n who_o name_n be_v commensurable_a to_o the_o name_n of_o the_o binomial_a line_n &_o in_o the_o self_n same_o proportion_n &_o moreover_o that_o residual_a line_n be_v in_o the_o self_n same_o order_n of_o residual_a line_n that_o the_o binomial_a line_n be_v of_o binomial_a line_n svppose_v that_o a_o be_v a_o rational_a line_n prove_v and_o bc_o a_o binomial_a line_n who_o great_a name_n let_v be_v cd_o and_o unto_o the_o square_n of_o the_o line_n a_o let_v the_o parallelogram_n contain_v under_o the_o line_n bc_o and_o of_o so_o that_o of_o be_v the_o breadth_n be_v equal_a then_o i_o say_v that_o of_o be_v a_o residual_a line_n who_o name_n be_v commensurable_a to_o the_o name_n of_o the_o binomial_a line_n bc_o which_o name_n let_v be_v cd_o and_o db_o and_o be_v in_o the_o same_o proportion_n with_o they_o and_o moreover_o the_o line_n of_o be_v in_o the_o self_n same_o order_n of_o residual_a line_n that_o the_o line_n bc_o be_v of_o binomial_a line_n unto_o the_o square_n of_o the_o line_n a_o let_v the_o parallelogram_n contain_v under_o the_o line_n bd_o and_o g_o be_v equal_a construction_n now_o forasmuch_o as_o that_o which_o be_v contain_v under_o the_o line_n bc_o &_o of_o be_v equal_a to_o that_o which_o be_v contain_v under_o the_o line_n bd_o and_o g_o therefore_o reciprocal_o by_o the_o 14._o of_o the_o six_o as_o the_o line_n cb_o be_v to_o the_o bd_o so_o be_v the_o line_n g_o to_o the_o line_n ef._n but_o the_o line_n bc_o be_v great_a than_o the_o line_n bd_o wherefore_o the_o line_n g_o be_v great_a than_o the_o line_n ef._n demonstration_n unto_o the_o line_n g_o let_v the_o line_n eh_o be_v equal_a wherefore_o by_o the_o 11._o of_o the_o five_o as_o the_o line_n cb_o be_v to_o the_o line_n bd_o so_o be_v the_o line_n he_o to_o the_o line_n fe_o wherefore_o by_o division_n by_o the_o 17._o of_o the_o five_o as_o the_o line_n cd_o be_v to_o the_o line_n bd_o so_o be_v the_o line_n hf_o to_o the_o line_n fe_o demonstrate_v as_o the_o line_n hf_o be_v to_o the_o fe_o so_o let_v the_o line_n fk_o be_v to_o the_o line_n ke_o how_o this_o be_v to_o be_v do_v we_o will_v declare_v at_o the_o end_n of_o this_o demonstration_n wherefore_o by_o the_o 12._o of_o the_o five_o the_o whole_a line_n hk_o proportion_n be_v to_o the_o whole_a line_n kf_n as_o the_o line_n fk_o be_v to_o the_o line_n ke_o for_o as_o one_o of_o the_o antecedentes_fw-la be_v to_o one_o of_o the_o consequentes_fw-la so_o be_v all_o the_o antecedentes_fw-la to_o all_o the_o consequentes_fw-la but_o as_o the_o line_n fk_o be_v the_o line_n ke_o so_o be_v the_o line_n cd_o to_o the_o line_n db_o for_o fk_n be_v to_o eke_o as_o hf_o be_v to_o fe_o and_o hf_o be_v to_o fe_o as_o cd_o be_v db_o wherefore_o by_o the_o
construction_n and_o demonstration_n hereof_o be_v word_n for_o word_n to_o be_v take_v as_o it_o stand_v here_o before_o after_o these_o word_n the_o line_n hf_o be_v great_a than_o the_o line_n fe_o ¶_o a_o corollary_n also_o note_v by_o i._n dee_n it_o be_v therefore_o evident_a that_o thus_o be_v three_o right_a line_n in_o our_o handle_n in_o continual_a proportion_n it_o be_v to_o weet_v the_o great_a the_o less_o and_o the_o adjoin_v make_v the_o first_o the_o less_o with_o the_o adjoin_v make_v the_o second_o and_o the_o adjoin_v line_n be_v the_o three_o this_o be_v prove_v in_o the_o begin_n of_o the_o demonstration_n after_o the_o assumpt_n use_v an_o other_o demonstration_n after_o flussas_n take_v a_o rational_a line_n a_o and_o let_v gb_o be_v a_o binomial_a line_n who_o great_a line_n let_v be_v gd_o construction_n and_o upon_o the_o line_n gb_o apply_v by_o the_o 45._o of_o the_o first_o the_o parallelogram_n bz_fw-fr equal_a to_o the_o square_n of_o the_o line_n a_o and_o make_v in_o breadth_n the_o line_n gz_n likewise_o upon_o the_o line_n db_o by_o the_o same_o apply_v the_o parallelogram_n by_o equal_a also_o to_o the_o square_n of_o the_o line_n a_o and_o make_v in_o breadth_n the_o line_n di_fw-it and_o put_v the_o line_n gzt_n equal_a to_o the_o line_n di._n then_o i_o say_v that_o gz_n be_v such_o a_o residual_a line_n as_o be_v require_v in_o the_o proposition_n forasmuch_o as_o the_o parallelogram_n bz_o &_o by_o be_v equal_a therefore_o by_o the_o 14._o of_o the_o six_o reciprocal_o as_o the_o line_n gb_o be_v to_o the_o line_n bd_o so_o be_v the_o line_n diego_n or_o the_o line_n gt_fw-mi which_o be_v equal_a unto_o it_o unto_o the_o line_n gz_n wherefore_o by_o division_n as_o the_o line_n gd_o be_v to_o the_o line_n db_o so_o be_v the_o line_n tz_n to_o the_o line_n zg_v by_o the_o 17._o of_o the_o five_o wherefore_o the_o line_n tz_n be_v great_a than_o the_o line_n zg_v for_o the_o line_n gd_o be_v the_o great_a name_n of_o the_o binomial_a line_n gb_o unto_o the_o line_n zg_v put_v the_o line_n zc_n equal_a and_o as_o the_o line_n tc_n be_v to_o the_o line_n tz_n so_o by_o the_o 11._o of_o the_o six_o let_v the_o line_n zg_n be_v to_o line_n zk_n wherefore_o contrary_a wise_a by_o the_o corollary_n of_o the_o 4._o of_o the_o five_o the_o line_n tz_n be_v to_o the_o line_n tc_n demonstration_n as_o the_o line_n zk_n be_v to_o the_o line_n zg_v wherefore_o by_o conversion_n of_o proportion_n by_o the_o 19_o of_o the_o five_o as_o the_o line_n tz_n be_v to_o the_o line_n zc_n that_o be_v to_o zg_v which_o be_v equal_a unto_o it_o so_o be_v the_o line_n zk_n to_o the_o line_n kg_v but_o the_o line_n tz_n be_v to_o the_o line_n zg_v as_o the_o line_n gd_o be_v to_o the_o line_n db._n wherefore_o by_o the_o 11._o of_o the_o five_o the_o line_n zk_n be_v to_o the_o line_n kg_v as_o the_o line_n gd_o be_v to_o the_o line_n db._n but_o the_o line_n gd_o and_o db_o be_v commensurable_a in_o power_n only_o wherefore_o also_o the_o line_n zk_v and_o kg_v be_v commensurable_a in_o power_n only_o by_o the_o 10._o of_o this_o book_n far_o forasmuch_o as_o the_o line_n tz_n be_v to_o the_o line_n zg_v as_o the_o line_n zk_n be_v to_o the_o line_n kg_v therefore_o by_o the_o 12._o of_o the_o five_o all_o the_o antecedentes_fw-la namely_o the_o whole_a line_n tk_n be_v to_o all_o the_o consequentes_fw-la namely_o to_o the_o line_n kz_o as_o one_o of_o the_o antecedentes_fw-la namely_o the_o line_n zk_n be_v to_o one_o of_o the_o consequentes_fw-la namely_o to_o the_o line_n kg_v wherefore_o the_o line_n zk_n be_v the_o mean_a proportional_a between_o the_o line_n tk_n and_o kg_n and_o therefore_o by_o the_o corollary_n of_o the_o 20._o of_o the_o six_o as_o the_o first_o namely_o the_o line_n tk_n be_v to_o the_o three_o namely_o to_o the_o line_n kg_v so_o be_v the_o square_a of_o the_o line_n tk_n to_o the_o square_n of_o the_o second_o namely_o of_o the_o line_n kz_o and_o forasmuch_o as_o the_o parallelogram_n by_o which_o be_v equal_a to_o the_o square_n of_o the_o rational_a line_n a_o be_v apply_v upon_o the_o rational_a line_n db_o it_o make_v the_o breadth_n diego_n rational_a and_o commensurable_a in_o length_n unto_o the_o line_n db_o by_o the_o 20._o of_o the_o ten_o and_o therefore_o the_o line_n gt_n which_o be_v equal_a unto_o the_o line_n diego_n be_v commensurable_a in_o length_n to_o the_o same_o line_n db._n and_o for_o that_o as_o the_o line_n gd_o be_v to_o the_o line_n db_o so_o be_v the_o line_n kz_o to_o the_o line_n kg_v but_o as_o the_o line_n kz_o be_v to_o the_o line_n kg_v so_o be_v the_o line_n tk_n to_o the_o line_n kz_o therefore_o by_o the_o 11._o of_o the_o five_o as_o the_o line_n gd_o be_v to_o the_o line_n db_o so_o be_v the_o line_n t_o king_n to_o the_o line_n kz_o wherefore_o by_o the_o 22._o of_o the_o six_o as_o the_o square_n of_o the_o line_n gd_o be_v to_o the_o square_n of_o the_o line_n db_o so_o be_v the_o square_a of_o the_o line_n t_o king_n to_o the_o square_n of_o the_o line_n kz_o but_o the_o square_n of_o the_o line_n gd_o be_v commensurable_a to_o the_o square_n of_o the_o line_n db_o for_o the_o name_n gd_o and_o db_o of_o the_o binomial_a line_n gb_o be_v commensurable_a in_o power_n wherefore_o the_o square_a of_o the_o line_n t_o edward_n shall_v be_v commensurable_a to_o the_o square_n of_o the_o line_n kz_o by_o the_o 10._o of_o this_o book_n but_o as_o the_o square_n of_o the_o line_n t_o edward_n be_v to_o the_o square_n of_o the_o line_n kz_o so_o be_v it_o prove_v that_o the_o right_a line_n t_o edward_n be_v to_o the_o right_a line_n kg_v wherefore_o the_o right_a line_n t_o edward_n be_v commensurable_a in_o length_n to_o the_o right_a line_n kg_v wherefore_o it_o be_v also_o commensurable_a in_o length_n to_o the_o line_n tg_n by_o the_o 15._o of_o the_o ten_o which_o line_n tg_n be_v as_o it_o have_v be_v prove_v a_o â—ationall_a line_n and_o equal_a to_o the_o line_n di._n wherefore_o the_o line_n t_o edward_n and_o kg_n be_v rational_a commensurable_a in_o length_n and_o forasmuch_o as_o it_o have_v be_v prove_v that_o the_o line_n z_o edward_n be_v commensurable_a in_o power_n only_o unto_o the_o rational_a line_n kg_v therefore_o the_o line_n z_o edward_n and_o kg_n be_v rational_a commensurable_a in_o power_n only_o demöstration_n wherefore_o the_o line_n gz_n be_v a_o residual_a line_n and_o forasmuch_o as_o the_o rational_a line_n tg_n be_v commensurable_a in_o length_n to_o either_o of_o these_o line_n db_o and_o kg_n wherefore_o the_o line_n db_o &_o kg_n shall_v be_v commensurable_a in_o length_n by_o the_o 12._o of_o the_o ten_o but_o the_o line_n z_o edward_n be_v to_o the_o line_n kg_v as_o the_o line_n gd_o be_v to_o the_o line_n db._n wherefore_o alternate_o by_o the_o 16._o of_o the_o five_o the_o line_n kz_o be_v to_o the_o line_n gd_o as_o the_o line_n kg_n be_v to_o the_o line_n db._n wherefore_o the_o line_n z_o edward_n be_v commensurable_a in_o length_n unto_o the_o line_n gd_o wherefore_o the_o line_n z_o edward_n and_o kg_n the_o name_n of_o the_o residual_a line_n gz_n be_v commensurable_a in_o length_n to_o the_o line_n gd_o and_o db_o which_o be_v the_o name_n of_o the_o binomial_a line_n gb_o and_o the_o line_n z_o edward_n be_v to_o the_o line_n kg_v in_o the_o same_o proportion_n that_o the_o line_n gd_o be_v to_o the_o line_n db._n wherefore_o if_o the_o whole_a line_n z_o edward_n be_v in_o power_n more_o than_o the_o line_n convenient_o join_v kg_n by_o the_o square_n of_o a_o line_n commensurable_a in_o length_n to_o the_o line_n z_o edward_n than_o the_o great_a name_n g_o d_o shall_v be_v in_o power_n more_o than_o the_o less_o name_n db_o by_o the_o square_n of_o a_o line_n commensurable_a in_o length_n to_o the_o line_n gd_o by_o the_o 14._o of_o the_o ten_o and_o if_o the_o line_n z_o edward_n be_v in_o power_n more_o than_o the_o line_n kg_v by_o the_o square_n of_o a_o line_n incommensurable_a in_o length_n to_o the_o line_n z_o edward_n the_o line_n also_o gd_o shall_v be_v in_o power_n more_o than_o the_o line_n db_o by_o the_o square_n of_o a_o line_n incommensurable_a in_o length_n unto_o the_o line_n gd_v by_o the_o same_o proposition_n and_o if_o the_o great_a or_o less_o name_n of_o the_o one_o be_v commensurable_a in_o length_n to_o the_o rational_a line_n put_v the_o great_a of_o lâ—sse_a name_n also_o of_o the_o other_o shall_v be_v commensurable_a in_o length_n to_o the_o same_o rational_a line_n by_o the_o 12._o of_o this_o book_n but_o if_o neither_o name_n
to_o the_o same_o and_o so_o the_o line_n bd_o be_v a_o six_o residual_a line_n and_o the_o line_n kh_o be_v a_o six_o binomial_a line_n wherefore_o kh_o be_v a_o binomial_a line_n who_o name_n kf_a and_o fh_o be_v commensurable_a to_o the_o name_n of_o the_o residual_a line_n bd_o namely_o to_o bc_o and_o cd_o and_o in_o the_o self_n same_o proportion_n and_o the_o binomial_a line_n kh_o be_v in_o the_o self_n same_o order_n of_o binomial_a line_n that_o the_o residual_a bd_o be_v of_o residual_a line_n wherefore_o the_o square_a of_o a_o rational_a line_n apply_v unto_o a_o residual_a line_n make_v the_o breadth_n or_o other_o side_n a_o binomial_a line_n who_o name_n be_v commensurable_a to_o the_o name_n of_o the_o residual_a line_n and_o in_o the_o self_n same_o proportion_n and_o moreover_o the_o binomial_a line_n be_v in_o the_o self_n same_o order_n of_o binomial_a line_n that_o the_o residual_a line_n be_v of_o residual_a line_n which_o be_v require_v to_o be_v demonstrate_v the_o assumpt_v confirm_v now_o let_v we_o declare_v how_o as_o the_o line_n kh_o be_v to_o the_o line_n eh_o so_o to_o make_v the_o line_n hf_o to_o the_o line_n fe_o add_v unto_o the_o line_n kh_o direct_o a_o line_n equal_a to_o he_o and_o let_v the_o whole_a line_n be_v kl_o and_o by_o the_o ten_o of_o the_o six_o let_v the_o line_n he_o be_v divide_v as_o the_o whole_a line_n kl_o be_v divide_v in_o the_o point_n h_o let_v the_o line_n he_o be_v so_o divide_v in_o the_o point_n f._n wherefore_o as_o the_o line_n kh_o be_v to_o the_o line_n hl_o that_o be_v to_o the_o line_n he_o so_o be_v the_o line_n hf_o to_o the_o line_n fe_o an_o other_o demonstration_n after_o flussas_n suppose_v that_o a_o be_v a_o rational_a line_n and_o let_v bd_o be_v a_o residual_a line_n and_o upon_o the_o line_n bd_o apply_v the_o parallelogram_n dt_n equal_a to_o the_o square_n of_o the_o line_n a_o by_o the_o 45._o of_o the_o first_o make_v in_o breadth_n the_o line_n bt_n flussas_n then_o i_o say_v that_o bt_n be_v a_o binominall_a line_n such_o a_o one_o as_o be_v require_v in_o the_o proposition_n forasmuch_o as_o bd_o be_v a_o residual_a line_n let_v the_o line_n convenient_o join_v unto_o it_o be_v gd_o wherefore_o the_o line_n bg_o and_o gd_o be_v rational_a commensurable_a in_o power_n only_o construction_n upon_o the_o rational_a line_n bg_o apply_v the_o parallelogram_n by_o equal_a to_o the_o square_n of_o the_o line_n a_o and_o make_v in_o breadth_n the_o line_n be._n wherefore_o the_o line_n be_v be_v rational_a and_o commensurable_a in_o length_n to_o the_o line_n bg_o by_o the_o 20._o of_o the_o ten_o now_o forasmuch_o as_o the_o parallelogram_n by_o and_o td_v be_v equal_a for_o that_o they_o be_v each_o equal_a to_o the_o square_n of_o the_o line_n a_o demonstration_n therefore_o reciprocal_a by_o the_o 14._o of_o the_o six_o as_o the_o line_n bt_n be_v to_o the_o line_n be_v so_o be_v the_o line_n bg_o to_o the_o line_n bd._n wherefore_o by_o conversion_n of_o proportion_n by_o the_o corollary_n of_o the_o 19_o of_o the_o five_o as_o the_o line_n bt_n be_v to_o the_o line_n te_fw-la so_o be_v the_o line_n bg_o to_o the_o line_n gd_o as_o the_o line_n bg_o be_v to_o the_o line_n gd_o so_o let_v the_o line_n tz_n be_v to_o the_o line_n see_fw-mi by_o the_o corollary_n of_o the_o 10._o of_o the_o six_o wherefore_o by_o the_o 11._o of_o the_o five_o the_o line_n bt_n be_v to_o the_o line_n te_fw-la as_o the_o line_n tz_n be_v to_o the_o line_n ze._n for_o either_o of_o they_o be_v as_o the_o line_n bg_o be_v to_o the_o line_n gd_o wherefore_o the_o residue_n bz_o be_v to_o the_o residue_n zt_n as_o the_o whole_a bt_n be_v to_o the_o whole_a te_fw-la by_o the_o 19_o of_o the_o five_o wherefore_o by_o the_o 11._o of_o the_o five_o the_o line_n bz_o be_v to_o the_o line_n zt_n as_o the_o line_n zt_n be_v to_o the_o line_n ze._n wherefore_o the_o line_n tz_n be_v the_o mean_a proportional_a between_o the_o line_n bz_o and_o ze._n wherefore_o the_o square_a of_o the_o first_o namely_o of_o the_o line_n bz_o be_v to_o the_o square_n of_o the_o second_o namely_o of_o the_o line_n zt_n as_o the_o first_o namely_o the_o line_n bz_o be_v to_o the_o three_o namely_o to_o the_o line_n see_fw-mi by_o the_o corollary_n of_o the_o 20._o of_o the_o six_o and_o for_o that_o as_o the_o line_n bg_o be_v to_o the_o line_n gd_o so_o be_v the_o line_n tz_n to_o the_o line_n see_fw-mi but_o as_o the_o line_n tz_n be_v to_o the_o line_n see_fw-mi so_o be_v the_o line_n bz_o to_o the_o line_n zt_n wherefore_o as_o the_o line_n bg_o be_v to_o the_o line_n gd_o so_o be_v the_o line_n bz_o to_o the_o line_n zt_n by_o the_o 11._o of_o the_o five_o wherefore_o the_o line_n bz_o and_o zt_n be_v commensurable_a in_o power_n only_o as_o also_o be_v the_o line_n bg_o and_o gd_o which_o be_v the_o name_n of_o the_o residual_a line_n bd_o by_o the_o 10._o of_o this_o book_n wherefore_o the_o right_a line_n bz_o and_o see_fw-mi be_v commensurable_a in_o length_n for_o we_o have_v prove_v that_o they_o be_v in_o the_o same_o proportion_n that_o the_o square_n of_o the_o line_n bz_o and_o zt_n be_v and_o therefore_o by_o the_o corollary_n of_o the_o 15._o of_o this_o book_n the_o residue_n be_v which_o be_v a_o rational_a line_n be_v commensurable_a in_o length_n unto_o the_o same_o line_n bz_o wherefore_o also_o the_o line_n bg_o which_o be_v commensurable_a in_o length_n unto_o the_o line_n be_v shall_v also_o be_v commensurable_a in_o length_n unto_o the_o same_o line_n ez_o by_o the_o 12._o of_o the_o ten_o and_o it_o be_v prove_v that_o the_o line_n rz_n be_v to_o the_o line_n zt_n commensurable_a in_o power_n only_o wherefore_o the_o right_a line_n bz_o and_o zt_n be_v rational_a commensurable_a in_o power_n only_o wherefore_o the_o whole_a line_n bt_n be_v a_o binomial_a line_n by_o the_o 36._o of_o this_o book_n and_o for_o that_o as_o the_o line_n bg_o be_v to_o the_o line_n gd_o so_o be_v the_o line_n bz_o to_o the_o line_n zt_n therefore_o alternate_o by_o the_o 16._o of_o the_o five_o the_o line_n bg_o be_v to_o the_o line_n bz_o as_o the_o line_n gd_o be_v to_o the_o line_n zt_n but_o the_o line_n bg_o be_v commensurable_a in_o length_n unto_o the_o line_n bz_o wherefore_o by_o the_o 10._o of_o this_o book_n the_o line_n gd_o be_v commensurable_a in_o length_n unto_o the_o line_n zt_n wherefore_o the_o name_n bg_o and_o gd_o of_o the_o residual_a line_n bd_o be_v commensurable_a in_o length_n unto_o the_o name_n bz_o and_o zt_v of_o the_o binomial_a line_n bt_n and_o the_o line_n bz_o be_v to_o the_o line_n zt_n in_o the_o same_o proportion_n that_o the_o line_n bg_o be_v to_o the_o line_n gd_o as_o before_o it_o be_v more_o manifest_a and_o that_o they_o be_v of_o one_o and_o the_o self_n same_o order_n be_v thus_o prove_v if_o the_o great_a or_o less_o name_n of_o the_o residual_a line_n namely_o the_o right_a line_n bg_o or_o gd_o be_v commensurable_a in_o length_n to_o any_o rational_a line_n put_v the_o great_a name_n also_o or_o less_o namely_o bz_o or_o zt_n shall_v be_v commensurable_a in_o length_n to_o the_o same_o rational_a line_n put_v by_o the_o 12._o of_o this_o book_n and_o if_o neither_o of_o the_o name_n of_o the_o residual_a line_n be_v commensurable_a in_o length_n unto_o the_o rational_a line_n put_v neither_o of_o the_o name_n of_o the_o binomial_a line_n shall_v be_v commensurable_a in_o length_n unto_o the_o same_o rational_a line_n put_v by_o the_o 13._o of_o the_o ten_o and_o if_o the_o great_a name_n bg_o be_v in_o power_n more_o than_o the_o less_o name_n by_o the_o square_n of_o a_o line_n commensurable_a in_o length_n unto_o the_o line_n bg_o the_o great_a name_n also_o bz_o shall_v be_v in_o power_n more_o than_o the_o less_o by_o the_o square_n of_o a_o line_n commensurable_a in_o length_n unto_o the_o line_n bz_o and_o if_o the_o one_o be_v in_o power_n more_o by_o the_o square_n of_o a_o line_n incommensurable_a in_o length_n the_o other_o also_o shall_v be_v in_o power_n more_o by_o the_o square_n of_o a_o line_n incommensurable_a in_o length_n by_o the_o 14._o of_o this_o book_n the_o square_a therefore_o of_o a_o rational_a line_n etc_n etc_n which_o be_v require_v to_o be_v prove_v ¶_o the_o 90._o theorem_a the_o 114._o proposition_n joint_o if_o a_o parallelogram_n be_v contain_v under_o a_o residual_a line_n &_o a_o binomial_a line_n who_o name_n be_v commensurable_a to_o the_o name_n of_o the_o residual_a line_n and_o in_o the_o selâ—e_a same_o proportion_n the_o line_n which_o contain_v in_o power_n
that_o superficies_n be_v rational_a svppose_v that_o a_o parallelogram_n be_v contain_v under_o a_o residual_a line_n ab_fw-la and_o a_o binomial_a line_n cd_o and_o let_v the_o great_a name_n of_o the_o binomial_a line_n be_v ce_fw-fr and_o the_o less_o name_n be_v ed_z and_o let_v the_o name_n of_o the_o binomial_a line_n namely_o ce_fw-fr and_o ed_z be_v commensurable_a to_o the_o name_n of_o the_o residual_a line_n namely_o to_o of_o and_o fâ—_n and_o in_o the_o self_n same_o proportion_n and_o let_v the_o line_n which_o contain_v in_o power_n that_o parallelogram_n be_v g._n then_o i_o say_v that_o the_o line_n g_o be_v rational_a take_v a_o rational_a line_n namely_o h._n and_o unto_o the_o line_n cd_o apply_v a_o parallelogram_n equal_a to_o the_o square_v of_o the_o line_n h_o construction_n and_o make_v in_o breadth_n the_o line_n kl_o wherefore_o by_o the_o 112._o of_o the_o ten_o kl_o be_v a_o residual_a line_n demonstration_n who_o name_n let_v be_v km_o and_o ml_o which_o be_v by_o the_o same_o commensurable_a to_o the_o name_n of_o the_o binomial_a line_n that_o be_v to_o ce_fw-fr and_o ed_z and_o be_v in_o the_o self_n same_o proportion_n but_o by_o position_n the_o line_n ce_fw-fr and_o ed_z be_v commensurable_a to_o the_o line_n of_o and_o fb_o and_o be_v in_o the_o self_n same_o proportion_n wherefore_o by_o the_o 12._o of_o the_o ten_o as_o the_o line_n of_o be_v to_o the_o line_n fbâ—_n so_o be_v the_o line_n km_o to_o the_o line_n ml_o wherefore_o alternate_o by_o the_o 16._o of_o the_o five_o as_o the_o line_n of_o be_v to_o the_o line_n km_o so_o be_v the_o line_n bf_o to_o the_o line_n lm_o wherefore_o the_o residue_n ab_fw-la be_v to_o the_o residue_n kl_o as_o the_o whole_a of_o be_v to_o the_o whole_a km_o but_o the_o line_n of_o be_v commensurable_a to_o the_o line_n km_o for_o either_o of_o the_o line_n of_o and_o km_o be_v commensurable_a to_o the_o line_n ce._n wherefore_o also_o the_o line_n ab_fw-la be_v commensurable_a to_o the_o line_n kl_o and_o as_o the_o line_n ab_fw-la be_v to_o the_o line_n kl_o so_o by_o the_o first_o of_o the_o six_o be_v the_o parallelogram_n contain_v under_o the_o line_n cd_o and_o ab_fw-la to_o the_o parallelogram_n contain_v under_o the_o line_n cd_o and_o kl_o wherefore_o the_o parallelogram_n contain_v under_o the_o line_n cd_o and_o ab_fw-la be_v commensurable_a to_o the_o parallelogram_n contain_v under_o the_o line_n cd_o and_o kl_o but_o the_o parallelogram_n contain_v under_o the_o line_n cd_o and_o kl_o be_v equal_a to_o the_o square_n of_o the_o line_n h._n wherefore_o the_o parallelogram_n contain_v under_o the_o line_n cd_o &_o ab_fw-la be_v commensurable_a to_o the_o square_n of_o the_o line_n h._n but_o the_o parallelogram_n contain_v under_o the_o line_n cd_o and_o ab_fw-la be_v equal_a to_o the_o square_n of_o the_o line_n g._n wherefore_o the_o square_a of_o the_o line_n h_o be_v commensurable_a to_o the_o square_n of_o the_o line_n g._n but_o the_o square_n of_o the_o line_n h_o be_v rational_a wherefore_o the_o square_a of_o the_o line_n g_o be_v also_o rational_a wherefore_o also_o the_o line_n g_o be_v rational_a and_o it_o contain_v in_o power_n the_o parallelogram_n contain_v under_o the_o line_n ab_fw-la and_o cd_o if_o therefore_o a_o parallelogram_n be_v contain_v under_o a_o residual_a line_n and_o a_o binomial_a line_n who_o name_n be_v commensurable_a to_o the_o name_n of_o the_o residual_a line_n and_o in_o the_o self_n same_o proportion_n the_o line_n which_o contain_v in_o power_n that_o superficies_n be_v rational_a which_o be_v require_v to_o be_v prove_v ¶_o corollary_n hereby_o it_o be_v manifest_a that_o a_o rational_a parallelogram_n may_v be_v contain_v under_o irrational_a line_n ¶_o a_o otâ—â—r_n 〈…〉_o flussas_n 〈…〉_o line_n â—d_a whosâ—_n name_v aâ—_z and_o â—d_a let_v be_v commensurable_a in_o length_n unto_o the_o name_n of_o the_o residual_a line_n aâ—_z which_o let_v be_v of_o and_o fb_o and_o let_v the_o liâ—e_z aeâ—_n be_v to_o the_o line_n edâ—_n in_o the_o same_o proportion_n that_o the_o line_n of_o be_v to_o the_o line_n fâ—_n and_o let_v the_o right_a line_n â—_o contain_v in_o power_n the_o superficies_n dâ—_n then_o i_o say_v thaâ—_z the_o liâ—e_z â—_o be_v a_o rational_a linâ—_n 〈…〉_o lâ—ne_n construction_n which_o lâ—â—_n bâ—â—_n and_o upon_o the_o line_n â—â—_o describe_v by_o the_o 4â—_o of_o the_o first_o a_o parallelogram_n equal_a to_o the_o square_n of_o the_o line_n â—â—_o and_o make_n in_o breadth_n the_o line_n dc_o wherefore_o by_o the_o â—12_n of_o this_o book_n cd_o be_v a_o residuâ—ll_a lineâ—_z who_o name_n which_o let_v be_v â—â—_o and_o odd_a shall_v be_v coâ—mensurablâ—_n in_o length_n unto_o the_o name_n aâ—_z and_o â—d_a and_o the_o line_n c_o o_fw-fr shall_v be_v unto_o the_o line_n odd_a demonstration_n in_o the_o same_o proportion_n that_o the_o line_n ae_n be_v to_o the_o line_n edâ—_n but_o as_o the_o line_n aâ—_z be_v to_o the_o line_n â—d_a so_o by_o supposition_n be_v the_o line_n of_o to_o the_o line_n fe_o wherefore_o as_o the_o line_n county_n be_v to_o the_o line_n odd_a so_o be_v the_o line_n of_o to_o the_o line_n fâ—â—_v wherefore_o the_o line_n county_n and_o odd_a be_v commensurable_a with_o the_o line_n aâ—_z and_o â—â—_o by_o the_o â—â—_o of_o this_o book_n wherefore_o the_o residue_n namely_o the_o line_n cd_o be_v to_o the_o residue_n namely_o to_o the_o line_n aâ—_z as_o the_o line_n county_n be_v to_o the_o line_n of_o by_o the_o 19_o of_o the_o five_o but_o it_o be_v prove_v that_o the_o line_n county_n be_v commensurable_a unto_o the_o line_n af._n wherefore_o the_o line_n cd_o be_v commensurable_a unto_o the_o line_n ab_fw-la wherefore_o by_o the_o first_o of_o the_o six_o the_o parallelogram_n ca_n be_v commensurable_a to_o the_o parallelogram_n dâ—_n but_o the_o parallelogram_n â—â—_o iâ—_z by_o construction_n rational_a for_o it_o be_v equal_a to_o the_o square_n of_o the_o rational_a line_n â—_o wherefore_o the_o parallelogram_n â—d_a â—s_n also_o rational_o wherâ—fore_o the_o line_n â—_o which_o by_o supposition_n contain_v in_o power_n the_o superficies_n â—dâ—_n be_v also_o rational_a if_o therefore_o a_o parallelogram_n be_v contain_v etc_o etc_o which_o be_v require_v to_o be_v prove_v ¶_o the_o 91._o theorem_a the_o 115._o proposition_n of_o a_o medial_a line_n be_v produce_v infinite_a irrational_a line_n of_o which_o none_o be_v of_o the_o self_n same_o kind_n with_o any_o of_o those_o that_o be_v before_o svppose_v that_o a_o be_v a_o medial_a line_n then_o i_o say_v that_o of_o the_o line_n a_o may_v be_v produce_v infinite_a irrational_a line_n of_o which_o none_o shall_v be_v of_o the_o self_n same_o kind_n with_o any_o of_o those_o that_o be_v before_o take_v a_o rational_a line_n b._n and_o unto_o that_o which_o be_v contain_v under_o the_o line_n a_o and_o b_o let_v the_o square_n of_o the_o line_n c_o be_v equal_a by_o the_o 14._o of_o the_o second_o â—_o demonstration_n wherefore_o the_o line_n c_o be_v irrational_a for_o a_o superficies_n contain_v under_o a_o rational_a line_n and_o a_o irrational_a line_n be_v by_o the_o assumpt_n follow_v the_o 38._o of_o the_o ten_o irrational_a and_o the_o line_n which_o contain_v in_o power_n a_o irrational_a superficies_n be_v by_o the_o assumpt_n go_v before_o the_o 21._o of_o the_o ten_o irrational_a and_o it_o be_v not_o one_o and_o the_o self_n same_o with_o any_o of_o those_o thirteen_o that_o be_v before_o for_o none_o of_o the_o line_n that_o be_v before_o apply_v to_o a_o rational_a line_n make_v the_o breadth_n medial_a again_o unto_o that_o which_o be_v contain_v under_o the_o line_n b_o and_o c_o let_v the_o square_n of_o d_o be_v equal_a wherefore_o the_o square_a of_o d_o be_v irrational_a wherefore_o also_o the_o line_n d_o be_v irrational_a and_o not_o of_o the_o self_n same_o kind_n with_o any_o of_o those_o that_o be_v before_o for_o the_o square_n of_o none_o of_o the_o line_n which_o be_v before_o apply_v to_o a_o rational_a line_n make_v the_o breadth_n the_o line_n c._n in_o like_a sort_n also_o shall_v it_o so_o follow_v if_o a_o man_n proceed_v infinite_o wherefore_o it_o be_v manifest_a that_o of_o a_o medial_a line_n be_v produce_v infinite_a irrational_a line_n of_o which_o none_o be_v of_o the_o self_n same_o kind_n with_o any_o of_o those_o that_o be_v before_o which_o be_v require_v to_o be_v prove_v an_o other_o demonstration_n demonstration_n suppose_v that_o ac_fw-la be_v a_o medial_a line_n then_o i_o say_v that_o of_o the_o line_n ac_fw-la may_v be_v produce_v infinite_a irrational_a line_n of_o which_o none_o shall_v be_v of_o the_o self_n same_o kind_n with_o any_o of_o those_o irrational_a line_n before_o name_v unto_o the_o line_n ac_fw-la and_o from_o the_o point_n a_o
draw_v by_o the_o 11._o of_o the_o firsâ—_o a_o perpendicular_a line_n ab_fw-la and_o let_v ab_fw-la be_v a_o rational_a line_n and_o make_v perfectâ—_n the_o parallelogram_n bc._n wherefore_o bg_o be_v irrational_a by_o that_o which_o be_v declare_v and_o prove_v in_o manner_n of_o a_o assumpt_n in_o the_o end_n of_o the_o demonstration_n of_o the_o 38_o and_o the_o line_n that_o contain_v it_o iâ—_z power_n be_v also_o irrational_a let_v the_o line_n cd_o contain_v in_o power_n the_o superficies_n bc._n wherefore_o cd_o be_v irrational_a &_o not_o of_o the_o self_n same_o kind_n with_o any_o of_o those_o that_o be_v before_o for_o the_o square_n of_o the_o line_n cd_o apply_v to_o a_o rational_a line_n namely_o ab_fw-la make_v the_o breadth_n a_o medial_a line_n namely_o ac_fw-la but_o the_o square_n of_o none_o of_o the_o foresay_a line_n apply_v to_o a_o rational_a line_n make_v the_o breadth_n a_o medial_a line_n again_o make_v perfect_v the_o parallelogram_n ed._n wherefore_o the_o parallelogram_n ed_z be_v also_o irrational_a by_o the_o say_v assumpt_v in_o the_o end_n of_o the_o 98._o his_o demonstration_n brieâ—ly_o prove_v and_o the_o line_n which_o contain_v it_o in_o power_n be_v irrational_o let_v the_o line_n which_o contain_v it_o in_o power_n be_v df._n wherefore_o df_o be_v irrational_a and_o not_o of_o the_o self_n same_o kind_n with_o any_o of_o the_o foresay_a irrational_a line_n for_o the_o square_n of_o none_o of_o the_o foresay_a irrational_a line_n apply_v unto_o a_o rational_a line_n make_v the_o breadth_n the_o line_n cd_o wherefore_o of_o a_o medial_a line_n be_v produce_v infinite_a irrational_a line_n of_o which_o none_o be_v of_o the_o self_n same_o kind_n with_o any_o of_o those_o that_o be_v before_o which_o be_v require_v to_o be_v demonstrate_v ¶_o the_o 92._o theorem_a the_o 116._o proposition_n now_o let_v we_o prove_v that_o in_o square_a figure_n the_o diameter_n be_v incommensurable_a in_o length_n to_o the_o side_n svppose_v that_o abcd_o be_v a_o square_a and_o let_v the_o diameter_n thereof_o be_v ac_fw-la then_o i_o say_v that_o the_o diameter_n ac_fw-la be_v incommensurable_a in_o length_n to_o the_o side_n ab_fw-la for_o if_o it_o be_v possible_a impossibility_n let_v it_o be_v commensurable_a in_o length_n i_o say_v that_o they_o this_o will_v follow_v that_o one_o and_o the_o self_n same_o number_n shall_v be_v both_o a_o even_a number_n &_o a_o odd_a number_n it_o be_v manifest_a by_o the_o 47._o of_o the_o first_o that_o the_o square_a of_o the_o line_n ac_fw-la be_v double_a to_o the_o square_n of_o the_o line_n ab_fw-la and_o for_o that_o the_o line_n ac_fw-la be_v commensurable_a in_o length_n to_o the_o line_n ab_fw-la by_o supposition_n therefore_o the_o line_n ac_fw-la have_v unto_o the_o line_n ab_fw-la that_o proportion_n that_o a_o number_n have_v to_o a_o number_n by_o the_o 5._o of_o the_o ten_o let_v the_o line_n ac_fw-la have_v unto_o the_o line_n ab_fw-la that_o proportion_n that_o the_o number_n of_o have_v to_o the_o number_n g._n and_o let_v of_o and_o g_z be_v the_o least_o number_n that_o have_v one_o and_o the_o same_o proportion_n with_o they_o wherefore_o of_o be_v not_o unity_n for_o if_o of_o be_v unity_n and_o it_o have_v to_o the_o number_n g_o that_o proportion_n that_o the_o line_n ac_fw-la have_v to_o the_o line_n ab_fw-la and_o the_o line_n ac_fw-la be_v great_a than_o the_o line_n ab_fw-la wherefore_o unity_n of_o be_v great_a than_o the_o number_n g_o which_o be_v impossible_a wherefore_o fe_o be_v not_o unity_n wherefore_o it_o be_v a_o number_n and_o for_o that_o as_o the_o square_n of_o the_o line_n ac_fw-la be_v to_o the_o square_n of_o the_o line_n ab_fw-la so_o be_v the_o square_a number_n of_o the_o number_n of_o to_o the_o square_a number_n of_o the_o number_n g_o for_o in_o each_o be_v the_o proportion_n of_o their_o side_n double_v by_o the_o corollary_n of_o the_o 20_o of_o the_o six_o and_o 11._o of_o the_o eight_o and_o the_o proportion_n of_o the_o line_n ac_fw-la to_o the_o line_n ab_fw-la double_v be_v equal_a to_o the_o proportion_n of_o the_o number_n of_o to_o the_o number_n g_o double_a for_o as_o the_o line_n ac_fw-la be_v to_o the_o line_n ab_fw-la so_o be_v the_o number_n of_o to_o the_o number_n g._n but_o the_o square_n of_o the_o line_n ac_fw-la be_v double_a to_o the_o square_n of_o the_o line_n ab_fw-la wherefore_o the_o square_a number_n produce_v of_o the_o number_n of_o be_v double_a to_o the_o square_a number_n produce_v of_o the_o number_n g._n wherefore_o the_o square_a number_n produce_v of_o of_o be_v a_o even_a number_n wherefore_o of_o be_v also_o a_o even_a number_n for_o if_o of_o be_v a_o odd_a number_n the_o square_a number_n also_o produce_v of_o it_o shall_v by_o the_o 23._o and_o 29._o of_o the_o nine_o be_v a_o odd_a number_n for_o if_o odd_a number_n how_o many_o soever_o be_v add_v together_o and_o if_o the_o multitude_n of_o they_o be_v odd_a the_o whole_a also_o shall_v be_v odd_a wherefore_o of_o be_v a_o even_a number_n divide_v the_o number_n of_o into_o two_o equal_a part_n in_o h._n and_o forasmuch_o as_o the_o number_n of_o and_o g_z be_v the_o least_a number_n in_o that_o proportion_n therefore_o by_o the_o 24._o of_o the_o seven_o they_o be_v prime_a number_n the_o one_o to_o the_o other_o and_o of_o be_v a_o even_a number_n wherefore_o g_z be_v a_o odd_a number_n for_o if_o g_z be_v a_o even_a number_n the_o number_n two_z shall_v measure_v both_o the_o number_n of_o and_o the_o number_n g_z for_o every_o even_a number_n have_v a_o half_a part_n by_o the_o definition_n but_o these_o number_n of_o &_o g_o be_v prime_a the_o one_o to_o the_o other_o wherefore_o it_o be_v impossible_a that_o they_o shall_v be_v measure_v by_o two_o or_o by_o any_o other_o number_n beside_o unity_n wherefore_o g_z be_v a_o odd_a number_n and_o forasmuch_o as_o the_o number_n of_o be_v double_a to_o the_o number_n eh_o therefore_o the_o square_a number_n produce_v of_o of_o be_v quadruple_a to_o the_o square_a number_n produce_v of_o eh_o and_o the_o square_a number_n produce_v of_o of_o be_v double_a to_o the_o square_a number_n produce_v of_o g._n wherefore_o the_o square_a number_n produce_v of_o g_o be_v double_a to_o the_o square_a number_n produce_v of_o eh_o wherefore_o the_o square_a number_n produce_v of_o g_o be_v a_o even_a number_n wherefore_o also_o by_o those_o thing_n which_o have_v be_v before_o speak_v the_o number_n g_z be_v a_o even_a number_n but_o it_o be_v prove_v that_o it_o be_v a_o odd_a number_n which_o be_v impossible_a wherefore_o the_o line_n ac_fw-la be_v not_o commensurable_a in_o length_n to_o the_o line_n ab_fw-la wherefore_o it_o be_v incommensurable_a an_o other_o demonstration_n we_o may_v by_o a_o other_o demonstration_n prove_v that_o the_o diameter_n of_o a_o square_n be_v incommensurable_a to_o the_o side_n thereof_o impossibility_n suppose_v that_o there_o be_v a_o square_a who_o diameter_n let_v be_v a_o and_o let_v the_o side_n thereof_o be_v b._n then_o i_o say_v that_o the_o line_n a_o be_v incommensurable_a in_o length_n to_o the_o line_n b._n for_o if_o it_o be_v possible_a let_v it_o be_v commensurable_a in_o length_n and_o again_o as_o the_o line_n a_o be_v to_o the_o line_n b_o so_o let_v the_o number_n of_o be_v to_o the_o number_n g_o and_o let_v they_o be_v the_o least_o that_o have_v one_o and_o the_o same_o proportion_n with_o they_o wherefore_o the_o number_n of_o and_o g_z be_v prime_a the_o one_o to_o the_o other_o first_o i_o say_v that_o g_z be_v not_o unity_n for_o if_o it_o be_v possible_a let_v it_o be_v unity_n and_o for_o that_o the_o square_a of_o the_o line_n a_o be_v to_o the_o square_n of_o the_o line_n b_o as_o the_o square_a number_n produce_v of_o of_o be_v to_o the_o square_a number_n produce_v of_o g_z as_o it_o be_v prove_v in_o the_o â—ormer_a demonstration_n but_o the_o square_n of_o the_o line_n a_o be_v double_a to_o the_o square_n of_o the_o line_n b._n wherefore_o the_o square_a number_n produce_v of_o of_o be_v double_a to_o the_o square_a number_n produce_v of_o g._n and_o by_o your_o supposition_n g_z be_v unity_n wherefore_o the_o square_a number_n produce_v of_o of_o be_v the_o number_n two_o which_o be_v impossible_a wherefore_o g_z be_v not_o unity_n wherefore_o it_o be_v a_o number_n and_o for_o that_o as_o the_o square_n of_o the_o line_n a_o be_v to_o the_o square_n of_o the_o line_n b_o so_o be_v the_o square_a number_n produce_v of_o of_o to_o the_o square_a number_n produce_v of_o g._n wherefore_o the_o square_a number_n produce_v of_o of_o be_v double_a to_o the_o square_a number_n produce_v of_o g._n wherefore_o the_o square_a number_n produce_v of_o g._n
and_o mean_a proportion_n in_o the_o point_n c_o and_o let_v the_o great_a segment_n thereof_o be_v ac_fw-la then_o i_o say_v that_o either_o of_o the_o line_n ac_fw-la and_o cb_o be_v a_o irrational_a line_n of_o that_o kind_n which_o be_v call_v a_o residual_a extend_v the_o line_n ab_fw-la to_o the_o point_n d_o construction_n and_o let_v the_o line_n ad_fw-la be_v equal_a to_o half_o of_o the_o line_n ab_fw-la now_o forasmuch_o as_o the_o right_a line_n ab_fw-la demonstration_n be_v divide_v by_o a_o extreme_a &_o mean_a proportion_n in_o the_o point_n c_o and_o unto_o the_o great_a segment_n ac_fw-la be_v add_v a_o line_n ad_fw-la equal_a to_o the_o half_a of_o the_o right_a line_n ab_fw-la therefore_o by_o the_o 1._o of_o the_o thirteen_o the_o square_a of_o the_o line_n cd_o be_v quintuple_a to_o the_o square_n of_o the_o line_n ad._n wherefore_o the_o square_a of_o the_o line_n cd_o have_v to_o the_o square_n of_o the_o line_n ad_fw-la that_o proportion_n that_o number_n have_v to_o number_v wherefore_o the_o square_a of_o the_o line_n cd_o be_v commensurable_a to_o the_o square_n of_o the_o line_n ad._n but_o the_o square_n of_o the_o line_n dam_n be_v rational_a for_o the_o line_n dam_n be_v rational_a forasmuch_o as_o it_o be_v the_o half_a of_o the_o rational_a line_n ab_fw-la wherefore_o the_o square_a of_o the_o line_n cd_o be_v rational_a wherefore_o also_o the_o linâ—_n cd_o be_v rational_a and_o forasmuch_o as_o the_o square_n of_o the_o line_n cd_o have_v not_o to_o the_o square_n of_o the_o line_n ad_fw-la that_o proportion_n that_o a_o square_a number_n have_v to_o a_o square_a number_n therefore_o by_o the_o 9_o of_o the_o ten_o the_o line_n cd_o be_v incommensurable_a in_o length_n to_o the_o line_n ad._n wherefore_o the_o line_n cd_o and_o dam_n be_v rational_a commensurable_a in_o power_n only_o wherefore_o the_o line_n ac_fw-la be_v a_o residual_a line_n by_o the_o 73._o of_o the_o ten_o again_o forasmuch_o as_o the_o line_n ab_fw-la be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n and_o the_o great_a segment_n thereof_o be_v ac_fw-la therefore_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la &_o bc_o be_v equal_a to_o the_o square_n of_o the_o line_n ac_fw-la wherefore_o the_o square_a of_o the_o line_n ac_fw-la apply_v to_o the_o rational_a line_n ab_fw-la make_v the_o breadth_n bc._n but_o the_o square_n of_o a_o residual_a line_n apply_v to_o a_o rational_a line_n make_v the_o breadth_n a_o first_o residual_a line_n by_o the_o 97._o of_o the_o ten_o wherefore_o the_o line_n cb_o be_v a_o first_o residual_a line_n and_o it_o be_v prove_v that_o the_o line_n ac_fw-la be_v also_o a_o residual_a line_n if_o therefore_o a_o rational_a right_a line_n be_v divide_v by_o a_o extreme_a and_o meanâ—_z proportion_n either_o of_o the_o segment_n be_v a_o irrational_a line_n of_o that_o kind_n which_o be_v call_v a_o residual_a line_n which_o be_v require_v to_o be_v demonstrate_v ¶_o a_o corollary_n add_v by_o campane_n hereby_o it_o be_v manifest_a that_o if_o the_o great_a segment_n be_v a_o rational_a line_n the_o less_o segment_n shall_v be_v a_o residual_a line_n for_o if_o the_o great_a segment_n ac_fw-la of_o the_o right_a line_n acb_o be_v divide_v into_o two_o equal_a part_n in_o the_o point_n d_o the_o square_a of_o the_o line_n db_o shall_v be_v quintâ—plâ—_n to_o the_o square_n of_o the_o line_n dc_o by_o the_o 3._o of_o this_o book_n and_o forasmuch_o as_o the_o line_n cd_o be_v the_o â—alfe_a of_o the_o rational_a line_n suppose_v ac_fw-la be_v rational_a by_o the_o 6._o definition_n of_o the_o ten_o and_o unto_o the_o square_n of_o the_o line_n dc_o the_o square_a of_o the_o line_n db_o be_v commensurable_a for_o it_o be_v quintuple_a unto_o it_o wherefore_o the_o square_a of_o the_o line_n db_o be_v rational_a wherefore_o also_o the_o line_n db_o be_v rational_a and_o forasmuch_o as_o the_o square_n of_o the_o line_n db_o &_o dc_o be_v not_o in_o proportion_n as_o a_o square_a number_n be_v to_o a_o square_a number_n therefore_o the_o line_n db_o and_o dc_o be_v incommensurable_a in_o length_n by_o the_o 9_o of_o the_o ten_o wherefore_o they_o be_v commensurable_a in_o power_n only_o wherefore_o by_o the_o 73._o of_o the_o ten_o the_o line_n bc_o which_o be_v the_o less_o segment_n be_v a_o residual_a line_n the_o 7._o theorem_a the_o 7._o proposition_n if_o a_o equilater_n pentagon_n have_v three_o of_o his_o angle_n whether_o they_o follow_v in_o order_n or_o not_o in_o order_n equal_v the_o one_o to_o the_o other_o that_o pentagon_n shall_v be_v equiangle_n svppose_v that_o abcde_o be_v a_o equilater_n pentagon_n and_o let_v the_o angle_n of_o the_o say_a pentagon_n proposition_n namely_o first_o three_o angle_n follow_v in_o order_n which_o be_v at_o the_o point_v a_o b_o c_o be_v equal_a the_o one_o to_o the_o other_o then_o i_o say_v that_o the_o pentagon_n abcde_o be_v equiangle_n construction_n draw_v these_o right_a line_n ac_fw-la be_v and_o fd._n now_o forasmuch_o as_o these_o two_o line_n cb_o case_n and_o basilius_n be_v equal_a to_o these_o two_o line_n ba_o and_o ae_n the_o one_o to_o the_o other_o and_o the_o angle_n cba_o be_v equal_a to_o the_o angle_n bae_o demonstration_n therefore_o by_o the_o 4._o of_o the_o first_o the_o base_a ac_fw-la be_v equal_a to_o the_o base_a be_v and_o the_o triangle_n abc_n be_v equal_a to_o the_o triangle_n abe_n and_o the_o rest_n of_o the_o angle_n be_v equal_a to_o the_o rest_n of_o the_o angle_n under_o which_o be_v subtend_v equal_a side_n wherefore_o the_o angle_n bca_o be_v equal_a to_o the_o angle_n bea_fw-mi and_o the_o angle_n abe_n to_o the_o angle_n cab_n wherefore_o also_o the_o side_n of_o be_v equal_a to_o the_o side_n bf_o by_o the_o 6._o of_o the_o first_o and_o it_o be_v prove_v that_o the_o whole_a line_n ac_fw-la be_v equal_a to_o the_o whole_a line_n be._n wherefore_o the_o residue_n cf_o be_v equal_a to_o the_o residue_n fâ—_n and_o the_o line_n cd_o be_v equal_a to_o the_o line_n de._n wherefore_o these_o two_o line_n fc_a and_o cd_o be_v equal_a to_o these_o two_o line_n fe_o &_o ed_z and_o the_o base_a fd_o be_v common_a to_o they_o both_o wherefore_o the_o angle_n fcd_n be_v equal_a to_o the_o angle_n feed_v by_o the_o 8._o of_o the_o first_o and_o it_o be_v prove_v that_o the_o angle_n bca_o be_v equal_a to_o the_o angle_n aeb_fw-mi wherefore_o the_o whole_a angle_n bcd_o be_v equal_a to_o the_o whole_a angle_n aed_n but_o the_o angle_n bcd_o be_v suppose_v to_o be_v equal_a to_o the_o angle_n a_o and_o b._n wherefore_o the_o angle_n aed_n be_v equal_a to_o the_o angle_n a_o and_o b._n in_o like_a sort_n also_o may_v we_o prove_v that_o the_o angle_n cde_o be_v equal_a to_o the_o angle_n a_o and_o b._n wherefore_o the_o pentagon_n abcde_o be_v equiangle_n case_n but_o now_o suppose_v that_o three_o angle_n which_o follow_v not_o in_o order_n be_v equal_a the_o one_o to_o the_o other_o namely_o let_v the_o angle_n a_o c_o d_o be_v equal_a then_o i_o say_v that_o in_o this_o case_n also_o the_o pentagon_n abcde_o be_v equiangle_n draw_v a_o right_a line_n from_o the_o point_n b_o to_o the_o point_n d._n now_o forasmuch_o as_o these_o two_o line_n ba_o and_o aâ—_z be_v equal_a to_o these_o two_o line_n bc_o and_o cd_o and_o they_o comprehend_v equal_a angle_n therefore_o by_o the_o 4._o of_o the_o first_o the_o base_a be_v be_v equal_a to_o the_o base_a bd._n and_o the_o triangle_n abe_n be_v equal_a to_o the_o triangle_n bdc_o and_o the_o rest_n of_o the_o angle_n be_v equal_a to_o the_o rest_n of_o the_o angle_n under_o which_o be_v subtend_v equal_a side_n wherefore_o the_o angle_n aeb_fw-mi be_v equal_a to_o the_o angle_n cdb_n and_o the_o angle_n bed_n be_v equal_a to_o the_o angle_n bde_v by_o the_o 5._o of_o the_o first_o for_o the_o side_n be_v be_v equal_a to_o the_o side_n bd._n wherefore_o the_o whole_a angle_n aed_n be_v equal_a to_o the_o whole_a angle_n cde_o but_o the_o angle_n cde_o be_v suppose_v to_o be_v equal_a to_o the_o angle_n a_o and_o c._n wherefore_o the_o aed_n be_v equal_a to_o the_o angle_n a_o and_o c._n and_o by_o the_o same_o reason_n also_o the_o angle_n abc_n be_v equal_a to_o the_o angle_n a_o c_z and_o d._n wherefore_o the_o pentagon_n abcde_o be_v equiangle_n if_o therefore_o a_o equilater_n pentagon_n have_v three_o of_o his_o angle_n whither_o they_o follow_v in_o order_n or_o not_o in_o order_n equal_v the_o one_o to_o the_o other_o that_o pentagon_n shall_v be_v equiangle_n which_o be_v require_v to_o be_v prove_v the_o 8._o problem_n the_o 8._o proposition_n if_o in_o a_o equilater_n &_o equiangle_n pentagon_n two_o right_a line_n do_v subtend_v two_o of_o
be_v right_a angle_n and_o that_o the_o line_n ac_fw-la be_v double_a to_o the_o line_n cm_o now_o forasmuch_o as_o the_o angle_n alc_n be_v equal_a to_o the_o angle_n amf_n for_o that_o they_o be_v both_o right_a angle_n and_o the_o angle_n lac_n be_v common_a to_o both_o the_o triangle_n alc_n and_o amf_n wherefore_o the_o angle_n remain_v namely_o acl_n be_v equal_a to_o the_o angle_n remain_v afm_n by_o the_o corollary_n of_o the_o 32._o of_o the_o first_o wherefore_o the_o triangle_n acl_n be_v equiangle_n to_o the_o triangle_n amf_n wherefore_o proportional_o by_o the_o 4._o of_o the_o six_o as_o the_o line_n lc_n be_v to_o the_o line_n ca_n so_o be_v the_o line_n mf_n to_o the_o line_n fa._n and_o in_o the_o same_o proportion_n also_o be_v the_o double_n of_o the_o antecedent_n lc_a and_o mf_n by_o the_o 15._o of_o the_o five_o wherefore_o as_o the_o double_a of_o the_o line_n lc_n be_v to_o the_o line_n ca_n so_o be_v the_o double_a of_o the_o line_n mf_n to_o the_o line_n fa._n but_o as_o the_o double_a of_o the_o line_n mf_n be_v to_o the_o line_n favorina_n so_o be_v the_o line_n mf_n to_o the_o half_a of_o the_o line_n favorina_n by_o the_o 15._o of_o the_o five_o wherefore_o as_o the_o double_a of_o the_o line_n lc_n be_v to_o the_o line_n ca_n so_o be_v the_o line_n mf_n to_o the_o half_a of_o the_o line_n favorina_n by_o the_o 11._o of_o the_o five_o and_o in_o the_o same_o proportion_n by_o the_o 15._o of_o the_o five_o be_v the_o half_n of_o the_o consequent_n namely_o of_o ca_n and_o of_o the_o halue_fw-mi of_o the_o line_n af._n wherefore_o as_o the_o double_a of_o the_o line_n lc_n be_v to_o the_o half_a of_o the_o line_n ac_fw-la so_o be_v the_o line_n mf_n to_o the_o four_o part_n of_o the_o line_n fa._n but_o the_o double_a of_o the_o line_n lc_n be_v the_o line_n dc_o and_o the_o half_a of_o the_o line_n ca_n be_v the_o line_n cm_o as_o have_v before_o be_v prove_v and_o the_o four_o part_n of_o the_o line_n favorina_n be_v the_o line_n fk_o for_o the_o line_n fk_o be_v the_o four_o part_n of_o the_o line_n fh_o by_o construction_n wherefore_o as_o the_o line_n dc_o be_v to_o the_o line_n cm_o so_o be_v the_o line_n mf_n to_o the_o line_n fk_o wherefore_o by_o composition_n by_o the_o 18._o of_o the_o five_o as_o both_o the_o line_n dc_o and_o cm_o be_v to_o the_o line_n cm_o so_o be_v the_o whole_a line_n mk_o to_o the_o line_n fk_o wherefore_o also_o by_o the_o 22._o of_o the_o six_o as_o the_o square_n of_o the_o line_n dc_o and_o cm_o be_v to_o the_o square_n of_o the_o line_n cm_o so_o be_v the_o square_a of_o the_o line_n mk_o to_o the_o square_n of_o the_o line_n fk_o and_o forasmuch_o as_o by_o the_o 8._o of_o the_o thirteen_o a_o line_n which_o be_v subtend_v under_o two_o side_n of_o a_o pentagon_n figure_n as_o be_v the_o line_n ac_fw-la be_v divide_v by_o a_o extreme_a &_o mean_a proportion_n the_o great_a segment_n be_v equal_a to_o the_o side_n of_o the_o pentagon_n figure_n that_o be_v unto_o the_o line_n dc_o and_o by_o the_o 1._o of_o the_o thirteen_o the_o great_a segment_n have_v add_v unto_o it_o the_o half_a of_o the_o whole_a be_v in_o power_n quintuple_a to_o the_o square_n make_v of_o the_o half_a of_o the_o whole_a and_o the_o half_a of_o the_o whole_a line_n ac_fw-la be_v the_o line_n cm_o wherefore_o the_o square_a that_o be_v make_v of_o the_o line_n dc_o and_o cm_o that_o be_v of_o the_o great_a segment_n and_o of_o the_o half_a of_o the_o whole_a as_o of_o one_o line_n be_v quintuple_a to_o the_o square_n of_o the_o line_n cm_o that_o be_v of_o the_o half_a of_o the_o whole_a but_o as_o the_o square_n make_v of_o the_o line_n dc_o and_o cm_o as_o of_o one_o line_n be_v to_o the_o square_n of_o the_o line_n cm_o so_o be_v it_o prove_v that_o the_o square_n of_o the_o line_n mk_o be_v to_o the_o square_n of_o the_o line_n fk_o wherefore_o the_o square_a of_o the_o line_n mk_o be_v quintuple_a to_o the_o square_n of_o the_o line_n fk_o but_o the_o square_n of_o the_o line_n kf_o be_v rational_a as_o have_v before_o be_v prove_v wherefore_o also_o the_o square_a of_o the_o line_n mk_o be_v rational_a by_o the_o 9_o definition_n of_o the_o ten_o for_o the_o square_n of_o the_o line_n mk_o have_v to_o the_o square_n of_o the_o line_n kf_o that_o proportion_n that_o number_n have_v to_o number_n namely_o that_o 5._o have_v to_o 1._o and_o therefore_o the_o say_a square_n be_v commensurable_a by_o the_o 6._o of_o the_o ten_o wherefore_o also_o the_o line_n mk_o be_v rational_a and_o forasmuch_o as_o the_o line_n bf_o be_v quadruple_a to_o the_o line_n fk_o for_o the_o semidiameter_n bf_o be_v equal_a to_o the_o semidiameter_n fh_o therefore_o the_o line_n bk_o be_v quintuple_a to_o the_o line_n fk_o wherefore_o the_o square_a of_o the_o line_n bk_o be_v 25._o time_n so_o much_o as_o the_o square_n of_o the_o line_n kf_o by_o the_o corollary_n of_o the_o 20_o of_o the_o six_o but_o the_o square_n of_o the_o line_n mk_o be_v quintuple_a to_o the_o square_n of_o the_o fk_n as_o be_v prove_v wherefore_o the_o square_a of_o the_o line_n bk_o be_v quintuple_a to_o the_o square_n of_o the_o line_n km_o wherefore_o the_o square_v of_o the_o line_n bk_o have_v not_o to_o the_o square_n of_o the_o line_n km_o that_o proportion_n that_o a_o square_a number_n have_v to_o a_o square_a number_n by_o the_o corollary_n of_o the_o 25._o of_o the_o eight_o wherefore_o by_o the_o 9_o of_o the_o ten_o the_o line_n bk_o be_v incommensurable_a in_o length_n to_o the_o line_n km_o and_o either_o of_o the_o line_n be_v rational_a wherefore_o the_o line_n bk_o and_o km_o be_v rational_a commensurable_a in_o power_n only_o but_o if_o from_o a_o rational_a line_n be_v take_v away_o a_o rational_a line_n be_v commensurable_a in_o power_n only_o to_o the_o whole_a that_o which_o remain_v be_v irrational_a and_o be_v by_o the_o 73._o of_o the_o ten_o call_v a_o residual_a line_n wherefore_o the_o line_n mb_n be_v a_o residual_a line_n and_o the_o line_n convenient_o join_v unto_o it_o be_v the_o line_n mk_o now_o i_o say_v that_o the_o line_n bm_n be_v a_o four_o residual_a line_n unto_o the_o excess_n of_o the_o square_n of_o the_o line_n bk_o above_o the_o square_n of_o the_o line_n km_o let_v the_o square_n of_o the_o line_n n_o be_v equal_a which_o excess_n how_o to_o find_v out_o be_v teach_v in_o the_o assumpt_n put_v after_o the_o 13._o proposition_n of_o the_o ten_o wherefore_o the_o line_n bk_o be_v in_o power_n more_o than_o the_o line_n km_o by_o the_o square_n of_o the_o line_n n._n and_o forasmuch_o as_o the_o line_n kf_o be_v commensurable_a in_o length_n to_o the_o line_n fb_o for_o it_o be_v the_o four_o part_n thereof_o therefore_o by_o the_o 16._o of_o the_o ten_o the_o whole_a line_n kb_o be_v commensurable_a in_o length_n to_o the_o line_n fb_o but_o the_o line_n fb_o be_v commensurable_a in_o length_n to_o the_o line_n bh_o namely_o the_o semidiameterâ—_n to_o the_o diameter_n wherefore_o the_o line_n bk_o be_v commensurable_a in_o length_n to_o the_o line_n bh_o by_o the_o 12._o of_o the_o ten_o and_o forasmuch_o as_o the_o square_n of_o the_o line_n bk_o be_v quintuple_a to_o the_o square_n of_o the_o line_n km_o therefore_o the_o square_a of_o the_o line_n bk_o have_v to_o the_o square_n of_o the_o line_n km_o that_o proportion_n that_o five_o have_v to_o one_o wherefore_o by_o conversion_n of_o proportion_n by_o the_o corollary_n of_o the_o 19_o of_o the_o five_o the_o square_a of_o bk_o have_v to_o the_o square_n of_o the_o line_n n_o that_o proportion_n that_o five_o have_v to_o four_o &_o therefore_o it_o have_v not_o that_o proportion_n that_o a_o square_a number_n have_v to_o a_o square_a number_n by_o the_o corollary_n of_o the_o 25._o of_o the_o eight_o wherefore_o the_o line_n bk_o be_v incommensurable_a in_o length_n to_o the_o line_n n_o by_o the_o 9_o of_o the_o ten_o wherefore_o the_o line_n bk_o be_v in_o power_n more_o than_o the_o line_n km_o by_o the_o square_n of_o a_o line_n incommensurable_a in_o length_n to_o the_o line_n bk_o now_o then_o forasmuch_o as_o the_o whole_a line_n bk_o be_v in_o power_n more_o than_o the_o line_n convenient_o join_v namely_o then_o km_o by_o the_o square_n of_o a_o line_n incommensurable_a in_o length_n to_o the_o line_n bk_o and_o the_o whole_a line_n bk_o be_v commensurable_a in_o length_n to_o the_o rational_a line_n give_v bh_o therefore_o the_o line_n mb_n be_v a_o four_o residual_a line_n by_o the_o definition_n of_o a_o four_o residual_a line_n but_o a_o rectangle_n parallelogram_n contain_v under_o a_o rational_a line_n and_o a_o four_o residual_a line_n be_v irrational_a and_o the_o line_n which_o contain_v in_o power_n the_o
give_v wherein_o be_v contain_v the_o former_a solid_n and_o to_o prove_v that_o the_o side_n of_o the_o icosahedron_n be_v a_o irrational_a line_n of_o that_o kind_n which_o be_v call_v a_o less_o line_n demonstration_n now_o forasmuch_o as_o the_o line_n qp_v qv_n qt_n q_n and_o qr_o do_v each_o subtend_v right_a angle_n contain_v under_o the_o side_n of_o a_o equilater_n hexagon_n &_o of_o a_o equilater_n decagon_fw-mi inscribe_v in_o the_o circle_n prstv_n or_o in_o the_o circle_n efghk_n which_o two_a circle_n be_v equal_a therefore_o the_o say_a line_n be_v each_o equal_a to_o the_o side_n of_o the_o pentagon_n inscribe_v in_o the_o foresay_a circle_n by_o the_o 10._o of_o this_o book_n and_o be_v equal_a the_o one_o to_o the_o other_o by_o the_o 4._o of_o the_o first_o for_o all_o the_o angle_n at_o the_o point_n w_n which_o they_o subtend_v be_v right_a angle_n wherefore_o the_o five_o triangle_n qpv_o qpr_fw-la qr_n qst_fw-la and_o qtv_n which_o be_v contain_v under_o the_o say_a line_n qv_n qp_n qr_o q_n qt_n and_o under_o the_o side_n of_o the_o pentagon_n vpr_v be_v equilater_n and_o equal_a to_o the_o ten_o former_a triangle_n and_o by_o the_o same_o reason_n the_o five_o triangle_n opposite_a unto_o they_o namely_o the_o triangle_n yml_n ymn_n ynx_n yxo_n and_o yol_n be_v equilater_n and_o equal_a to_o the_o say_v ten_o triangle_n for_o the_o line_n ill_o ym_fw-fr yn_n yx_n and_o you_o do_v subtend_v right_a angle_n contain_v under_o the_o side_n of_o a_o equilater_n hexagon_n and_o of_o a_o equilater_n decagon_fw-mi inscribe_v in_o the_o circle_n efghk_n which_o be_v equal_a to_o the_o circle_n prstv_n wherefore_o there_o be_v describe_v a_o solid_a contain_v under_o 20._o equilater_n triangle_n wherefore_o by_o the_o last_o definition_n of_o the_o eleven_o there_o be_v describe_v a_o icosahedron_n now_o it_o be_v require_v to_o comprehend_v it_o in_o the_o sphere_n give_v and_o to_o prove_v that_o the_o side_n of_o the_o icosahedron_n be_v a_o irrational_a line_n of_o that_o kind_n which_o be_v call_v a_o less_o line_n forasmuch_o as_o the_o line_n zw_n be_v the_o side_n of_o a_o hexagon_n &_o the_o line_n wq_n be_v the_o side_n of_o a_o decagon_n therefore_o the_o line_n zq_n be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n in_o the_o point_n w_n and_o his_o great_a segment_n be_v zw_n by_o the_o 9_o of_o the_o thirteen_o wherefore_o as_o the_o line_n qz_n be_v to_o the_o line_n zw_n so_o be_v the_o line_n zw_n to_o the_o line_n wq_n but_o the_o zw_n be_v equal_a to_o the_o line_n zl_n by_o construction_n and_o the_o line_n wq_n to_o the_o line_n zy_n by_o construction_n also_o wherefore_o as_o the_o line_n qz_n be_v to_o the_o line_n zl_n so_o be_v the_o line_n zl_n to_o the_o line_n zy_n and_o the_o angle_n qzlâ—_n and_o lzy_n be_v right_a angle_n by_o the_o 2._o definition_n of_o the_o eleven_o if_o therefore_o we_o draw_v a_o right_a line_n from_o the_o point_n l_o to_o the_o point_n qui_fw-fr the_o angle_n ylq_n shall_v be_v a_o right_a angle_n by_o reason_n of_o the_o likeness_n of_o the_o triangle_n ylq_n and_o zlq_n by_o the_o 8._o of_o the_o six_o wherefore_o a_o semicircle_n describe_v upon_o the_o line_n qy_n shall_v pass_v also_o by_o the_o point_n l_o by_o the_o assumpt_n add_v by_o campane_n after_o the_o 13._o of_o this_o book_n and_o by_o the_o same_o reason_n also_o for_o that_o as_o the_o line_n qz_n be_v the_o line_n zw_n so_o be_v the_o line_n zw_n to_o the_o line_n wq_n flussas_n but_o the_o line_n zq_n be_v equal_a to_o the_o line_n yw_n and_o the_o line_n zw_n to_o the_o line_n pw_o wherefore_o as_o the_o line_n yw_n be_v to_o the_o line_n wp_n so_o be_v the_o line_n pw_o to_o the_o line_n wq_n and_o therefore_o again_o if_o we_o draw_v a_o right_a line_n from_o the_o point_n p_o to_o the_o point_n y_fw-fr the_o angle_n ypq_n shall_v be_v a_o right_a angle_n wherefore_o a_o semicircle_n describe_v upon_o the_o line_n qy_n shall_v pass_v also_o by_o the_o point_n p_o by_o the_o former_a assumpt_n &_o if_o the_o diameter_n qy_o abide_v fix_v the_o semicircle_n be_v turn_v round_o about_o until_o it_o come_v to_o the_o self_n same_o place_n from_o whence_o it_o begin_v first_o to_o be_v move_v it_o shall_v pass_v both_o by_o the_o point_n p_o and_o also_o by_o the_o rest_n of_o the_o point_n of_o the_o angle_n of_o the_o icosahedron_n and_o the_o icosahedron_n shall_v be_v comprehend_v in_o a_o sphere_n i_o say_v also_o that_o it_o be_v contain_v in_o the_o sphere_n give_v divide_v by_o the_o 10._o of_o the_o first_o the_o line_n zw_n into_o two_o equal_a part_n in_o the_o point_n a._n and_o forasmuch_o as_o the_o right_a line_n zq_n be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n in_o the_o point_n w_n and_o his_o less_o segment_n be_v qw_o therefore_o the_o segment_n qw_o have_v add_v unto_o it_o the_o half_a of_o the_o great_a segment_n namely_o the_o line_n wa_n be_v by_o the_o 3._o of_o this_o book_n in_o power_n quintuple_a to_o the_o square_n make_v of_o the_o half_a of_o the_o great_a segment_n wherefore_o the_o square_a of_o the_o line_n qa_n be_v quintuple_a to_o the_o square_n of_o the_o line_n â—w_o but_o unto_o the_o square_n of_o the_o qa_n the_o square_a of_o the_o line_n qy_n be_v quadruple_a by_o the_o corollary_n of_o the_o 20._o of_o the_o six_o for_o the_o line_n qy_n be_v double_a to_o the_o line_n qa_n and_o by_o the_o same_o reason_n unto_o the_o square_n of_o the_o wa_n the_o square_a of_o the_o line_n zw_n be_v quadruple_a wherefore_o the_o square_a of_o the_o line_n qy_n be_v quintuple_a to_o the_o square_n of_o the_o line_n zw_n by_o the_o 15._o of_o the_o five_o and_o forasmuch_o as_o the_o line_n ac_fw-la be_v quadruple_a to_o the_o line_n cb_o therefore_o the_o line_n ab_fw-la be_v quintuple_a to_o the_o line_n cb._n but_o as_o the_o line_n ab_fw-la be_v to_o the_o line_n bc_o so_o be_v the_o square_a of_o the_o line_n ab_fw-la to_o the_o square_n of_o the_o line_n bd_o by_o the_o 8_o of_o the_o six_o and_o corollary_n of_o the_o 20._o of_o the_o same_o wherefore_o the_o square_a of_o the_o line_n ab_fw-la be_v quintuple_a to_o the_o square_n of_o the_o line_n bd_o and_o it_o be_v be_v prove_v that_o the_o square_a of_o the_o line_n qy_n be_v quintuple_a to_o the_o square_n of_o the_o line_n zw_n and_o the_o line_n bd_o be_v equal_a to_o the_o line_n zw_n for_o either_o of_o they_o be_v by_o position_n equal_a to_o the_o line_n which_o be_v draw_v from_o the_o centre_n of_o the_o circle_n efghk_v to_o the_o circumference_n wherefore_o the_o line_n ab_fw-la be_v equal_a to_o the_o yq_n but_o the_o line_n ab_fw-la be_v the_o diameter_n of_o the_o sphere_n give_v wherefore_o the_o line_n yq_n which_o be_v prove_v to_o be_v the_o diameter_n of_o the_o sphere_n contain_v the_o icosahedron_n be_v equal_a to_o the_o diameter_n of_o the_o sphere_n give_v wherefore_o the_o icosahedron_n be_v contain_v in_o the_o sphere_n give_v now_o i_o say_v that_o the_o side_n of_o the_o icosahedron_n be_v a_o irrational_a line_n of_o that_o kind_n which_o be_v call_v a_o less_o line_n for_o forasmuch_o as_o the_o diameter_n of_o the_o sphere_n be_v rational_a and_o be_v in_o power_n quintuple_a to_o the_o square_n of_o the_o line_n draw_v from_o the_o centre_n of_o the_o circle_n olmnx_n wherefore_o also_o the_o line_n which_o be_v draw_v from_o the_o centre_n of_o the_o circle_n olmnx_n be_v rational_a wherefore_o the_o diameter_n also_o be_v commensurable_a to_o the_o same_o line_n by_o the_o 6._o of_o the_o ten_o be_v rational_a but_o if_o in_o a_o circle_n have_v a_o rational_a line_n to_o his_o diameter_n be_v describe_v a_o equilater_n pentagon_n the_o side_n of_o the_o pentagon_n be_v by_o the_o 11._o of_o this_o book_n a_o irrational_a line_n of_o that_o kind_n which_o be_v call_v a_o less_o line_n but_o the_o side_n of_o the_o pentagon_n olmnx_n be_v also_o the_o side_n of_o the_o icosahedron_n describe_v as_o have_v before_o be_v prove_v wherefore_o the_o side_n of_o the_o icosahedron_n be_v a_o irrational_a line_n of_o that_o kind_n which_o be_v call_v a_o less_o line_n wherefore_o there_o be_v describe_v a_o icosahedron_n and_o it_o be_v contain_v in_o the_o sphere_n give_v and_o it_o be_v prove_v that_o the_o side_n of_o thou_o icosahedron_n be_v a_o irrational_a line_n of_o that_o kind_n which_o be_v call_v a_o less_o line_n which_o be_v require_v to_o be_v do_v and_o to_o be_v prove_v a_o corollary_n hereby_o it_o be_v manifest_a that_o the_o diameter_n of_o the_o sphere_n be_v in_o power_n quintuple_a to_o the_o line_n which_o be_v draw_v from_o the_o centre_n of_o the_o circle_n to_o
double_a to_o the_o side_n of_o the_o octohedron_n &_o the_o side_n be_v in_o power_n sequitertia_fw-la to_o the_o perpendiclar_a line_n by_o the_o 12._o of_o this_o book_n wherefore_o the_o diameter_n thereof_o be_v in_o power_n duple_fw-fr superbipartiens_fw-la tertias_fw-la to_o the_o perpendicular_a line_n wherefore_o also_o the_o diameter_n and_o the_o perpendicular_a line_n be_v rational_a and_o commensurable_a by_o the_o 6._o of_o the_o ten_o as_o touch_v a_o icosahedron_n it_o be_v prove_v in_o the_o 16._o of_o this_o book_n that_o the_o side_n thereof_o be_v a_o less_o line_n when_o the_o diameter_n of_o the_o sphere_n be_v rational_a and_o forasmuch_o as_o the_o angle_n of_o the_o inclination_n of_o the_o base_n thereof_o be_v contain_v of_o the_o perpendicular_a line_n of_o the_o triangle_n and_o subtend_v of_o the_o right_a line_n which_o subtend_v the_o angle_n of_o the_o pentagon_n which_o contain_v five_o side_n of_o the_o icosahedron_n and_o unto_o the_o perpendicular_a line_n the_o side_n be_v commensurable_a namely_o be_v in_o power_n sesquitertia_fw-la unto_o they_o by_o the_o corollary_n of_o the_o 12._o of_o this_o book_n therefore_o the_o perpendicular_a line_n which_o contain_v the_o angle_n be_v irrational_a line_n namely_o less_o line_n by_o the_o 105._o of_o the_o ten_o book_n and_o forasmuch_o as_o the_o diameter_n contain_v in_o power_n both_o the_o side_n of_o the_o icosahedron_n and_o the_o line_n which_o subtend_v the_o foresay_a angle_n if_o from_o the_o power_n of_o the_o diameter_n which_o be_v rational_a be_v take_v away_o the_o power_n of_o the_o side_n of_o the_o icosahedron_n which_o be_v irrational_a it_o be_v manifest_a that_o the_o residue_n which_o be_v the_o power_n of_o the_o subtend_a line_n shall_v be_v irrational_a for_o if_o it_o shall_v be_v rational_a the_o number_n which_o measure_v the_o whole_a power_n of_o the_o diameter_n and_o the_o part_n take_v away_o of_o the_o subtend_a line_n shall_v also_o by_o the_o 4._o common_a sentence_n of_o the_o seven_o measure_n the_o residue_n namely_o the_o power_n of_o the_o side_n irrational_a which_o be_v irrational_a for_o that_o it_o be_v a_o less_o line_n which_o be_v absurd_a wherefore_o it_o be_v manifest_a that_o the_o right_a line_n which_o compose_v the_o angle_n of_o the_o inclination_n of_o the_o base_n of_o the_o icosahedron_n be_v irrational_a line_n for_o the_o subtend_a line_n have_v to_o the_o line_n contain_v a_o great_a proportion_n than_o the_o whole_a have_v to_o the_o great_a segment_n the_o angle_n of_o the_o inclination_n of_o the_o base_n of_o a_o dodecahedron_n be_v contain_v under_o two_o perpendicular_n of_o the_o base_n of_o the_o dodecahedron_n and_o be_v subtend_v of_o that_o right_a line_n who_o great_a segment_n be_v the_o side_n of_o a_o cube_n inscribe_v in_o the_o dodecahedron_n which_o right_a line_n be_v equal_a to_o the_o line_n which_o couple_v the_o section_n into_o two_o equal_a part_n of_o the_o opposite_a side_n of_o the_o dodecahedron_n and_o this_o couple_a line_n we_o say_v be_v a_o irrational_a line_n for_o that_o the_o diameter_n of_o the_o sphere_n contain_v in_o power_n both_o the_o couple_a line_n and_o the_o side_n of_o the_o dodecahedron_n but_o the_o side_n of_o the_o dodecahedron_n be_v a_o irrational_a line_n namely_o a_o residual_a line_n by_o the_o 17._o of_o this_o book_n wherefore_o the_o residue_n namely_o the_o couple_a line_n be_v a_o irrational_a line_n as_o it_o be_v â—asy_a to_o prove_v by_o the_o 4._o common_a sentence_n of_o the_o seven_o and_o that_o the_o perpendicular_a line_n which_o contain_v the_o angle_n of_o the_o inclination_n be_v irrational_a be_v thus_o prove_v by_o the_o proportion_n of_o the_o subtend_a line_n of_o the_o foresay_a angle_n of_o inclination_n to_o the_o line_n which_o contain_v the_o angle_n be_v find_v out_o the_o obliquity_n of_o the_o angle_n angle_n for_o if_o the_o subtend_a line_n be_v in_o power_n double_a to_o the_o line_n which_o contain_v the_o angle_n then_o be_v the_o angle_n a_o right_a angle_n by_o the_o 48._o of_o the_o first_o but_o if_o it_o be_v in_o power_n less_o than_o the_o double_a it_o be_v a_o acute_a angle_n by_o the_o 23._o of_o the_o second_o but_o if_o it_o be_v in_o power_n more_o than_o the_o double_a or_o have_v a_o great_a proportion_n than_o the_o whole_a have_v to_o the_o great_a segmentâ—_n the_o angle_n shall_v be_v a_o obtuse_a angle_n by_o the_o 12._o of_o the_o second_o and_o 4._o of_o the_o thirteen_o by_o which_o may_v be_v prove_v that_o the_o square_a of_o the_o whole_a be_v great_a than_o the_o double_a of_o the_o square_n of_o the_o great_a segment_n this_o be_v to_o be_v note_v that_o that_o which_o flussas_n have_v here_o teach_v touch_v the_o inclination_n of_o the_o base_n of_o the_o â—ive_a regular_a body_n hypsicles_n teach_v after_o the_o 5_o proposition_n of_o the_o 15._o book_n where_o he_o confess_v that_o he_o receive_v it_o of_o one_o isidorus_n and_o seek_v to_o make_v the_o matter_n more_o clear_a he_o endeavour_v himself_o to_o declare_v that_o the_o angle_n of_o the_o inclination_n of_o the_o solid_n be_v give_v and_o that_o they_o be_v either_o acute_a or_o obtuse_a according_a to_o the_o nature_n of_o the_o solid_a although_o euclid_v in_o all_o his_o 15._o book_n have_v not_o yet_o show_v what_o a_o thing_n give_v be_v wherefore_o flussas_n frame_v his_o demonstration_n upon_o a_o other_o ground_n proceed_v after_o a_o other_o manner_n which_o seem_v more_o plain_a and_o more_o apt_o hereto_o be_v place_v then_o there_o albeit_o the_o reader_n in_o that_o place_n shall_v not_o be_v frustrate_a of_o his_o also_o the_o end_n of_o the_o thirteen_o book_n of_o euclides_n element_n ¶_o the_o fourteen_o book_n of_o euclides_n element_n in_o this_o book_n which_o be_v common_o account_v the_o 14._o book_n of_o euclid_n be_v more_o at_o large_a entreat_v of_o our_o principal_a purpose_n book_n namely_o of_o the_o comparison_n and_o proportion_n of_o the_o five_o regular_a body_n customable_o call_v the_o 5._o figure_n or_o form_n of_o pythagoras_n the_o one_o to_o the_o other_o and_o also_o of_o their_o side_n together_o each_o to_o other_o which_o thing_n be_v of_o most_o secret_a use_n and_o inestimable_a pleasure_n and_o commodity_n to_o such_o as_o diligent_o search_v for_o they_o and_o attain_v unto_o they_o which_o thing_n also_o undoubted_o for_o the_o worthiness_n and_o hardness_n thereof_o for_o thing_n of_o most_o price_n be_v most_o hard_a be_v first_o search_v and_o find_v out_o of_o philosopher_n not_o of_o the_o inferior_a or_o mean_a sort_n but_o of_o the_o deep_a and_o most_o ground_a philosopher_n and_o best_a exercise_v in_o geometry_n and_o albeit_o this_o book_n with_o the_o book_n follow_v namely_o the_o 15._o book_n have_v be_v hitherto_o of_o all_o man_n for_o the_o most_o part_n and_o be_v also_o at_o this_o day_n number_v and_o account_v among_o euclides_n book_n and_o suppose_v to_o be_v two_o of_o he_o namely_o the_o 14._o and_o 15._o in_o order_n as_o all_o exemplar_n not_o only_o new_a and_o late_o set_v abroad_o but_o also_o old_a monument_n write_v by_o hand_n do_v manifest_o witness_v yet_o it_o be_v think_v by_o the_o best_a learned_a in_o these_o day_n that_o these_o two_o book_n be_v none_o of_o euclides_n but_o of_o some_o other_o author_n no_o less_o worthy_a nor_o of_o less_o estimation_n and_o authority_n notwithstanding_o than_o euclid_n apollonius_n a_o man_n of_o deep_a knowledge_n a_o great_a philosopher_n and_o in_o geometry_n marvellous_a who_o wonderful_a book_n write_v of_o the_o section_n of_o cones_fw-la which_o exercise_n &_o occupy_v thewitte_v of_o the_o wise_a and_o best_a learned_a be_v yet_o remain_v be_v think_v and_o that_o not_o without_o just_a cause_n to_o be_v the_o author_n of_o they_o or_o as_o some_o think_v hypsicles_n himself_o for_o what_o can_v be_v more_o plain_o then_o that_o which_o he_o himself_o witness_v in_o the_o preface_n of_o this_o book_n basilides_n of_o tire_n say_v hypsicles_n and_o my_o father_n together_o scan_v and_o peyse_v a_o writing_n or_o book_n of_o apollonius_n which_o be_v of_o the_o comparison_n of_o a_o dodecahedron_n to_o a_o icosahedron_n inscribe_v in_o one_o and_o the_o self_n same_o sphere_n and_o what_o proportion_n these_o figure_n have_v the_o one_o to_o the_o other_o find_v that_o apollonius_n have_v fail_v in_o this_o matter_n but_o afterward_o say_v he_o i_o find_v a_o other_o copy_n or_o book_n of_o apollonius_n wherein_o the_o demonstration_n of_o that_o matter_n be_v full_a and_o perfect_a and_o show_v it_o unto_o they_o whereat_o they_o much_o rejoice_v by_o which_o word_n it_o seem_v to_o be_v manifest_a that_o apollonius_n be_v the_o first_o author_n of_o this_o book_n which_o be_v afterward_o set_v forth_o by_o hypsicles_n for_o so_o his_o own_o word_n after_o in_o
and_o bt_n be_v duple_fw-fr sesquialter_fw-la to_o that_o which_o be_v contain_v unâ—â—r_o the_o line_n az_o &_o kt_n but_o unto_o that_o which_o be_v contain_v under_o the_o line_n az_o and_o kt_n the_o pentagon_n abcig_n be_v prove_v duple_n sesquialter_fw-la wherefore_o the_o pentagon_n abcig_n of_o the_o dodecahedron_n be_v equal_a to_o that_o which_o be_v contain_v under_o the_o perpendicular_a line_n az_o and_o under_o the_o line_n bt_v which_o be_v five_o six_o part_n of_o the_o line_n bg_o ¶_o the_o 7._o proposition_n campane_n a_o right_a line_n divide_v by_o a_o extreme_a and_o mean_a proportion_n what_o proportion_n the_o line_n contain_v in_o power_n the_o whole_a line_n and_o the_o great_a segment_n have_v to_o the_o line_n contain_v in_o power_n the_o whole_a and_o the_o less_o segment_n the_o same_o have_v the_o side_n of_o the_o cube_fw-la to_o the_o side_n of_o the_o icosahedron_n contain_v in_o one_o and_o the_o same_o sphere_n construction_n take_v a_o circle_n abe_n and_o in_o it_o by_o the_o 11._o of_o the_o four_o inscribe_v a_o equilater_n pentagon_n bzech_n and_o by_o the_o second_o of_o the_o same_o a_o equilater_n triangle_n abi_n and_o let_v the_o centre_n thereof_o be_v the_o point_n g._n and_o draw_v a_o line_n from_o g_z to_o b._n and_o divide_v the_o line_n gb_o by_o a_o extreme_a and_o mean_a proportion_n in_o the_o point_n d_o by_o the_o 30._o of_o the_o six_o and_o let_v the_o line_n ml_o contain_v in_o power_n both_o the_o whole_a line_n gb_o and_o his_o less_o segment_n bd_o by_o the_o corollary_n of_o the_o 13._o of_o the_o ten_o and_o draw_v the_o right_a line_n bâ—_n subtend_v the_o angle_n of_o the_o pentagon_n which_o shall_v be_v the_o side_n of_o the_o cube_fw-la by_o the_o corollary_n of_o the_o 17._o of_o the_o thirteen_o â—_o and_o the_o line_n by_o shall_v be_v the_o side_n of_o the_o icosahedron_n and_o the_o line_n â—z_n the_o side_n of_o the_o dodecahedron_n by_o the_o 4._o of_o this_o book_n then_o i_o say_v that_o be_v the_o side_n of_o the_o cube_fw-la be_v to_o by_o the_o side_n of_o the_o icosahedron_n as_o the_o line_n contain_v in_o power_n the_o line_n bg_o &_o gd_o be_v to_o the_o line_n contain_v in_o power_n the_o line_n gb_o and_o bd._n demonstration_n for_o forasmuch_o as_o by_o the_o 12._o of_o the_o thirteen_o the_o line_n by_o be_v in_o power_n triple_a to_o the_o line_n bg_o and_o by_o the_o 4._o of_o the_o same_o the_o square_n of_o the_o line_n gb_o &_o bd_o be_v triple_a to_o the_o square_n of_o the_o line_n gd_o wherefore_o by_o the_o 15._o of_o the_o five_o the_o square_a of_o the_o line_n by_o be_v to_o the_o square_n of_o the_o line_n gb_o &_o bd_o namely_o triple_a to_o treble_v as_o the_o square_n of_o the_o line_n bâ—_n be_v to_o the_o square_n of_o the_o line_n gd_v namely_o as_o one_o be_v to_o one_o but_o as_o the_o square_n of_o the_o line_n bg_o be_v to_o the_o square_n of_o the_o gd_o so_o be_v the_o square_a of_o the_o line_n be_v to_o the_o square_n of_o the_o line_n bz_o for_o the_o line_n bg_o gd_o and_o be_v bz_o be_v in_o one_o and_o the_o same_o proportion_n by_o the_o second_o of_o this_o book_n for_o bz_o be_v the_o great_a segment_n of_o the_o line_n be_v by_o the_o corollary_n of_o the_o 17._o of_o the_o thirteen_o wherefore_o the_o square_a of_o the_o line_n be_v be_v to_o the_o square_n of_o the_o line_n bz_o as_o the_o square_n of_o the_o line_n by_o be_v to_o the_o square_n of_o the_o line_n bg_o and_o bd._n wherefore_o alternate_o the_o square_a of_o the_o line_n be_v be_v to_o the_o square_n of_o the_o line_n by_o as_o the_o square_n of_o the_o line_n bz_o be_v to_o the_o square_n of_o the_o line_n gb_o and_o bd._n but_o the_o square_n of_o the_o line_n bz_o be_v equal_a to_o the_o square_n of_o the_o line_n bg_o and_o gd_o by_o the_o 10._o of_o the_o thirteen_o for_o the_o line_n bg_o be_v equal_a to_o the_o side_n of_o the_o hexagon_n and_o the_o line_n gd_o to_o the_o side_n of_o the_o decagon_n by_o the_o corollary_n of_o the_o 9_o of_o the_o same_o wherefore_o the_o square_n of_o the_o line_n bg_o and_o gd_o be_v to_o the_o square_n of_o the_o line_n gâ—_n and_o bd_o as_o the_o square_n of_o the_o line_n be_v be_v to_o the_o square_n of_o the_o line_n by_o but_o the_o line_n zb_n contain_v in_o power_n the_o line_n bg_o and_o gd_o and_o the_o line_n ml_o contain_v in_o power_n the_o line_n gb_o and_o bd_o by_o construction_n wherefore_o as_o the_o line_n zb_n which_o contain_v in_o power_n the_o whole_a line_n bg_o and_o the_o great_a segment_n gd_o be_v to_o the_o line_n ml_o which_o contain_v in_o in_o power_n the_o whole_a line_n gb_o and_o the_o less_o segment_n bd_o so_o be_v be_v the_o side_n of_o the_o cube_fw-la to_o by_o the_o side_n of_o the_o icosahedron_n by_o the_o 22._o of_o the_o six_o wherefore_o a_o right_a line_n divide_v by_o a_o extreme_a and_o mean_a proportion_n what_o proportion_n the_o line_n contain_v in_o power_n the_o whole_a line_n and_o the_o great_a segment_n have_v to_o the_o line_n contain_v in_o power_n the_o whole_a line_n and_o the_o less_o segment_n the_o same_o have_v the_o side_n of_o the_o cube_fw-la to_o the_o side_n of_o the_o icosahedron_n contain_v in_o one_o and_o the_o same_o sphere_n which_o be_v require_v to_o be_v prove_v ¶_o the_o 8._o proposition_n order_n the_o solid_a of_o a_o dodecahedron_n be_v to_o the_o solid_a of_o a_o icosahedron_n as_o the_o side_n of_o a_o cube_n be_v to_o the_o side_n of_o a_o icosahedron_n all_o those_o solid_n be_v describe_v in_o one_o and_o the_o self_n same_o sphere_n forasmuch_o as_o in_o the_o 4._o of_o this_o book_n it_o have_v be_v prove_v that_o one_o and_o the_o self_n same_o circle_n contain_v both_o the_o triangle_n of_o a_o icosahedron_n and_o the_o pentagon_n of_o a_o dodecahedron_n describe_v in_o one_o and_o the_o self_n same_o sphere_n wherefore_o the_o circle_n which_o contain_v those_o base_n be_v equal_a the_o perpendicular_n also_o which_o be_v draw_v from_o the_o centre_n of_o the_o sphere_n to_o those_o circle_n shall_v be_v equal_a by_o the_o corollary_n of_o the_o assumpt_n of_o the_o 16_o of_o the_o twelve_o and_o therefore_o the_o pyramid_n set_v upon_o the_o base_n of_o those_o solid_n have_v one_o and_o the_o self_n same_o altitude_n for_o the_o altitude_n of_o those_o pyramid_n concurrâ—_n in_o the_o centre_n wherefore_o they_o be_v in_o proportion_n as_o their_o base_n be_v by_o the_o 5._o and_o 6._o of_o the_o twelve_o and_o therefore_o the_o pyramid_n which_o compose_v the_o dodecahedron_n arâ—_n to_o the_o pyramid_n which_o compose_v the_o icosahedron_n as_o the_o base_n be_v which_o base_n be_v the_o superficiece_n of_o those_o solid_n wherefore_o their_o solid_n be_v the_o one_o to_o the_o other_o as_o their_o superficiece_n be_v but_o the_o superficies_n of_o the_o dodecahedron_n be_v to_o the_o superficies_n of_o the_o icosahedron_n as_o the_o side_n of_o the_o cube_fw-la be_v to_o the_o side_n of_o the_o icosahedron_n by_o the_o 6._o of_o this_o book_n wherefore_o by_o the_o 11._o of_o the_o five_o as_o the_o solid_a of_o the_o dodecahedron_n be_v to_o the_o solid_a of_o the_o icosahedron_n so_o be_v the_o side_n of_o the_o cube_fw-la to_o the_o side_n of_o the_o icosahedron_n all_o the_o say_v solid_n be_v inscribe_v in_o one_o and_o the_o self_n same_o sphere_n wherefore_o the_o solid_a of_o a_o dodecahedron_n be_v to_o the_o solid_a of_o a_o icosahedron_n as_o the_o side_n of_o a_o cube_fw-la be_v to_o the_o side_n of_o a_o icosahedron_n all_o those_o solid_n be_v describe_v in_o one_o and_o the_o self_n same_o sphere_n which_o be_v require_v to_o be_v prove_v a_o corollary_n the_o solid_a of_o a_o dodecahedron_n be_v to_o the_o solid_a of_o a_o icosahedron_n campane_n as_o the_o superficiece_n of_o the_o one_o be_v to_o the_o superficiece_n of_o the_o other_o be_v describe_v in_o one_o and_o the_o self_n same_o sphere_n namely_o as_o the_o side_n of_o the_o cube_fw-la be_v to_o the_o side_n of_o the_o icosahedron_n as_o be_v before_o manifest_v for_o they_o be_v resolve_v into_o pyramid_n of_o one_o and_o the_o self_n same_o altitude_n ¶_o the_o 9_o proposition_n if_o the_o side_n of_o a_o equilater_n triangle_n be_v rational_a the_o superficies_n shall_v be_v irrational_a campane_n of_o that_o kind_n which_o be_v call_v medial_a svppose_v that_o abg_o be_v a_o equilater_n triangle_n and_o from_o the_o point_n a_o draw_v unto_o the_o side_n bg_o a_o perpendicular_a line_n ad_fw-la and_o let_v the_o line_n ab_fw-la be_v rational_a then_o i_o say_v that_o the_o superficies_n abg_o be_v medial_a construction_n forasmuch_o as_o the_o line_n ab_fw-la be_v in_o power_n
sesquitertia_fw-la to_o the_o line_n ad_fw-la by_o the_o corollary_n of_o the_o 12._o of_o the_o thirteen_o of_o what_o part_n the_o line_n ab_fw-la contain_v in_o power_n 12_o of_o the_o same_o part_n the_o line_n ad_fw-la contain_v in_o power_n 9_o demonstration_n wherefore_o the_o residue_n bd_o contain_v in_o power_n of_o the_o same_o part_n 3._o â—or_a the_o line_n ab_fw-la contain_v in_o power_n the_o line_n ad_fw-la and_o bd_o by_o the_o 47._o of_o the_o first_o wherefore_o the_o line_n ad_fw-la and_o db_o be_v rational_a and_o commensurable_a to_o the_o rational_a line_n set_v ab_fw-la by_o the_o 6._o of_o the_o ten_o but_o forasmuch_o as_o the_o power_n of_o the_o line_n ad_fw-la be_v to_o the_o power_n of_o the_o line_n db_o in_o that_o proportion_n that_o 9_o a_o square_a number_n be_v to_o 3._o a_o number_n not_o square_a therefore_o they_o be_v not_o in_o the_o proportion_n of_o square_a number_n by_o the_o corollary_n of_o the_o 25._o of_o the_o eight_o and_o therefore_o they_o be_v not_o commensurable_a in_o length_n by_o the_o 9_o of_o the_o ten_o wherefore_o that_o which_o be_v contain_v under_o the_o line_n ad_fw-la and_o db_o which_o be_v rational_a line_n commensurable_a in_o power_n only_o be_v medial_a by_o the_o 22._o of_o the_o ten_o but_o that_o which_o be_v contain_v under_o the_o line_n ad_fw-la and_o db_o be_v double_a to_o the_o triangle_n abdella_n by_o the_o 41._o of_o the_o first_o wherefore_o that_o which_o be_v contain_v under_o the_o line_n ad_fw-la and_o db_o be_v equal_a to_o the_o whole_a triangle_n abg_o which_o be_v double_a to_o the_o triangle_n abdella_n by_o the_o 1._o of_o the_o six_o wherefore_o the_o triangle_n abg_o be_v medial_a if_o therefore_o the_o side_n of_o a_o equilater_n triangle_n be_v rational_a the_o superficies_n shall_v be_v irrational_a of_o that_o kind_n which_o be_v call_v medial_a which_o be_v require_v to_o be_v prove_v a_o corollary_n if_o a_o octohedron_n and_o a_o tetrahedron_n be_v inscribe_v in_o a_o sphere_n who_o diameter_n be_v rational_a campane_n their_o superficiece_n shall_v be_v medial_a for_o those_o superficiece_n consist_v of_o equilater_n triangle_n who_o side_n be_v commensurable_a to_o the_o diameter_n which_o be_v rational_a by_o the_o 13._o and_o 14._o of_o the_o thirteen_o and_o therefore_o they_o be_v rational_a but_o they_o be_v commensurable_a in_o power_n only_o to_o the_o perpendicular_a line_n and_o therefore_o they_o contain_v a_o medial_a triangle_n as_o it_o be_v before_o manifest_v ¶_o the_o 10._o proposition_n campane_n if_o a_o tetrahedron_n and_o a_o octohedron_n be_v inscribe_v in_o one_o and_o the_o self_n same_o sphere_n the_o base_a of_o the_o tetrahedron_n shall_v be_v sesquitertia_fw-la to_o the_o base_a of_o the_o octohedron_n and_o the_o supersiciece_n of_o the_o octohedron_n shall_v be_v sesquialtera_fw-la to_o the_o superficiece_n of_o the_o tetrahedron_n part_n forasmuch_o as_o the_o diameter_n of_o the_o sphere_n be_v in_o power_n sesquialtera_fw-la to_o the_o side_n of_o the_o tetrahedron_n by_o the_o 13._o of_o the_o thirteen_o and_o the_o same_o diameter_n be_v in_o power_n duple_n to_o the_o side_n of_o the_o octohedron_n by_o the_o 14._o of_o the_o same_o book_n therefore_o of_o what_o part_n the_o diameter_n contain_v in_o power_n six_o of_o the_o same_o the_o side_n of_o the_o tetrahedron_n contain_v in_o power_n 4_o and_o of_o the_o same_o the_o side_n of_o the_o octohedron_n contain_v in_o power_n 3._o wherefore_o the_o power_n of_o the_o side_n of_o the_o tetrahedron_n be_v to_o the_o power_n of_o the_o side_n of_o the_o octohedron_n in_o the_o same_o proportion_n that_o 4._o be_v to_o 3_o which_o be_v sesquitertia_fw-la and_o like_o triangle_n which_o be_v the_o base_n of_o the_o solid_n describe_v of_o those_o side_n shall_v have_v the_o one_o to_o the_o other_o the_o same_o proportion_n that_o the_o square_n make_v of_o those_o side_n have_v for_o both_o the_o triangle_n be_v the_o one_o to_o the_o other_o and_o also_o the_o square_n be_v the_o one_o to_o the_o other_o in_o double_a proportion_n of_o that_o in_o which_o the_o side_n be_v by_o the_o 20._o of_o the_o six_o part_n wherefore_o of_o what_o part_n one_o base_a of_o the_o tetrahedron_n be_v 4_o of_o the_o same_o be_v four_o base_n of_o the_o tetrahedron_n 16_o likewise_o of_o what_o part_n of_o the_o same_o one_o base_a of_o the_o octohedron_n be_v 3_o of_o the_o same_o be_v 8._o base_n of_o the_o octohedron_n 24._o wherefore_o the_o base_n of_o the_o octohedron_n be_v to_o the_o base_n of_o the_o tetrahedron_n in_o that_o proportion_n that_o 24._o be_v to_o 16_o which_o be_v sesquialtera_fw-la if_o therefore_o a_o tetrahedron_n and_o a_o octohedron_n be_v inscribe_v in_o one_o and_o the_o self_n same_o sphere_n the_o base_a of_o the_o tetrahedron_n shall_v be_v sesquitertia_fw-la to_o the_o base_a of_o the_o octohedron_n and_o the_o superficiece_n of_o the_o octohedron_n shall_v be_v sesquialtera_fw-la to_o the_o superficiece_n of_o the_o tetrahedron_n which_o be_v require_v to_o be_v prove_v ¶_o the_o 11._o proposition_n a_o tetrahedron_n be_v to_o a_o octohedron_n inscribe_v in_o one_o and_o the_o self_n same_o sphere_n campane_n in_o proportion_n as_o the_o rectangle_n parallelogram_n contain_v under_o the_o line_n which_o contain_v in_o power_n 27._o sixty_o four_o part_n of_o the_o side_n of_o the_o tetrahedron_n &_o under_o the_o line_n which_o be_v subsesquioctaâ—a_n to_o the_o same_o side_n of_o the_o tetrahedron_n be_v to_o the_o square_n of_o the_o diameter_n of_o the_o sphere_n construction_n let_v we_o suppose_v a_o sphere_n who_o diameter_n let_v be_v the_o line_n ab_fw-la and_o let_v the_o centre_n be_v the_o point_n h._n and_o in_o it_o let_v there_o be_v inscribe_v a_o tetrahedron_n adc_o and_o a_o octohedron_n aekbg_n and_o let_v the_o line_n nl_o contain_v in_o power_n â—7_o 64_o of_o ac_fw-la the_o side_n of_o the_o tetrahedron_n and_o let_v the_o line_n ml_o be_v in_o length_n subsesquioctava_fw-la to_o the_o same_o side_n then_o i_o say_v that_o the_o tetrahedron_n acd_o be_v to_o the_o octohedron_n aeb_n as_o the_o rectangle_n parallelogram_n contain_v under_o the_o line_n nl_o and_o lm_o demonstration_n be_v to_o the_o square_n of_o the_o line_n ab_fw-la forasmuch_o as_o the_o line_n draw_v from_o the_o angle_n a_o by_o the_o centre_n h_o perpendicular_o upon_o the_o base_a of_o the_o tetrahedron_n fall_v upon_o the_o centre_n t_o of_o the_o circle_n which_o contain_v that_o base_a and_o make_v the_o right_a line_n ht_v the_o six_o part_n of_o the_o diameter_n ab_fw-la by_o the_o corollary_n of_o the_o 13._o of_o the_o thirteen_o therefore_o the_o line_n ha_o which_o be_v draw_v from_o the_o centre_n to_o the_o circumference_n be_v triple_a to_o the_o line_n ht_o and_o therefore_o the_o whole_a line_n at_o be_v to_o the_o line_n ah_o 〈…〉_o let_v the_o tetrahedron_n adc_o be_v cut_v by_o a_o plain_a ghk_n pass_v by_o the_o centrâ—_n h_o and_o be_v parallel_n unto_o the_o base_a dtc_n by_o the_o corollary_n of_o the_o 11._o of_o the_o eleven_o now_o than_o the_o triangle_n adc_o of_o the_o tetrahedron_n shall_v be_v cut_v by_o the_o right_a line_n kg_v which_o be_v parallel_n to_o the_o line_n dc_o by_o the_o 16._o of_o the_o eleven_o wherefore_o as_o the_o line_n at_o be_v to_o the_o line_n ah_o so_o be_v the_o line_n ac_fw-la to_o the_o line_n agnostus_n by_o the_o 2._o of_o the_o six_o wherefore_o the_o line_n ac_fw-la be_v to_o the_o line_n agnostus_n sesquitertia_fw-la that_o be_v as_o 4._o to_o 3._o and_o forasmuch_o as_o the_o triangle_n adc_o akg_n and_o the_o rest_n which_o be_v cut_v by_o the_o plain_a khg_n be_v like_o the_o one_o to_o the_o other_o by_o the_o 5â—_o of_o the_o six_o the_o pyramid_n adc_o and_o akg_n shall_v be_v like_o the_o one_o to_o the_o other_o by_o the_o 7._o definition_n of_o the_o â—leventh_o wherefore_o they_o be_v in_o triple_a proportion_n of_o that_o in_o which_o the_o side_n ac_fw-la and_o ag_z be_v by_o the_o 8._o of_o the_o twelve_o but_o the_o proportion_n of_o the_o side_n ac_fw-la to_o agnostus_n be_v as_o the_o proportion_n of_o 4._o to_o 3._o now_o then_o if_o by_o the_o 2._o of_o the_o eight_o you_o find_v out_o 4._o of_o the_o lest_o number_n in_o continual_a proportion_n and_o in_o that_o proportion_n that_o 4._o be_v to_o 3_o which_o shall_v be_v 64.48.36_o and_o 27_o it_o be_v manifest_a by_o the_o 15_o definition_n of_o the_o five_o that_o the_o extreme_n â—4_o to_o 27._o be_v in_o triple_a proportion_n of_o that_o in_o which_o the_o proportion_n give_v 4._o to_o 3._o be_v or_o the_o quantity_n of_o the_o proportion_n of_o 4._o to_o 3._o which_o be_v 1._o and_o 1_o â—_o be_v twice_o multiply_v into_o itself_o there_o shall_v be_v produce_v the_o proportion_n of_o 64._o to_o 27._o wherefore_o the_o pyramid_n
the_o whole_a line_n mg_o to_o the_o whole_a line_n ea_fw-la by_o the_o 18._o of_o the_o five_o wherefore_o as_o mg_o the_o side_n of_o the_o cube_fw-la be_v to_o ea_fw-la the_o semidiameter_n so_o be_v the_o line_n fghim_n to_o the_o octohedron_n abkdlc_n inscribe_v in_o one_o &_o the_o self_n same_o sphere_n if_o therefore_o a_o cube_fw-la and_o a_o octohedron_n be_v contain_v in_o one_o and_o the_o self_n same_o sphere_n they_o shall_v be_v in_o proportion_n the_o one_o to_o the_o other_o as_o the_o side_n of_o the_o cube_fw-la be_v to_o the_o semidiameter_n of_o the_o sphere_n which_o be_v require_v to_o be_v demonstrate_v a_o corollary_n distinct_o to_o notify_v the_o power_n of_o the_o side_n of_o the_o five_o solid_n by_o the_o power_n of_o the_o diameter_n of_o the_o sphere_n the_o side_n of_o the_o tetrahedron_n and_o of_o the_o cube_fw-la do_v cut_v the_o power_n of_o the_o diameter_n of_o the_o sphere_n into_o two_o square_n which_o be_v in_o proportion_n double_a the_o one_o to_o the_o other_o the_o octohedron_n cut_v the_o power_n of_o the_o diameter_n into_o two_o equal_a square_n the_o icosahedron_n into_o two_o square_n who_o proportion_n be_v duple_n to_o the_o proportion_n of_o a_o line_n divide_v by_o a_o extreme_a and_o mean_a proportion_n who_o less_o segment_n be_v the_o side_n of_o the_o icosahedron_n and_o the_o dodecahedron_n into_o two_o square_n who_o proportion_n be_v quadruple_a to_o the_o proportion_n of_o a_o line_n divide_v by_o a_o extreme_a and_o mean_a proportion_n who_o less_o segment_n be_v the_o side_n of_o the_o dodecahedron_n for_o ad_fw-la the_o diameter_n of_o the_o sphere_n contain_v in_o power_n ab_fw-la the_o side_n of_o the_o tetrahedron_n and_o bd_o the_o side_n of_o the_o cube_fw-la which_o bd_o be_v in_o power_n half_a of_o the_o side_n ab_fw-la the_o diameter_n also_o of_o the_o sphere_n contain_v in_o power_n ac_fw-la and_o cd_o two_o equal_a side_n of_o the_o octohedron_n but_o the_o diameter_n contain_v in_o power_n the_o whole_a line_n ae_n and_o the_o great_a segment_n thereof_o ed_z which_o be_v the_o side_n of_o the_o icosahedron_n by_o the_o 15._o of_o this_o book_n wherefore_o their_o power_n be_v in_o duple_n proportion_n of_o that_o in_o which_o the_o side_n be_v by_o the_o first_o corollary_n of_o the_o 20._o of_o the_o six_o have_v their_o proportion_n duple_n to_o the_o proportion_n of_o a_o extreme_a &_o mean_a proportion_n far_o the_o diameter_n contain_v in_o power_n the_o whole_a line_n of_o and_o his_o less_o segment_n fd_o which_o be_v the_o side_n of_o the_o dodecahedron_n by_o the_o same_o 15._o of_o this_o book_n wherefore_o the_o whole_a have_v to_o the_o less_o â—_o double_a proportion_n of_o that_o which_o the_o extreme_a have_v to_o the_o mean_a namely_o of_o the_o whole_a to_o the_o great_a segment_n by_o the_o 10._o definition_n of_o the_o five_o it_o follow_v that_o the_o proportion_n of_o the_o power_n be_v double_a to_o the_o double_a proportion_n of_o the_o side_n by_o the_o same_o first_o corollary_n of_o the_o 20._o of_o the_o six_o that_o be_v be_v quadruple_a to_o the_o proportion_n of_o the_o extreme_a and_o of_o the_o mean_a by_o the_o definition_n of_o the_o six_o a_o advertisement_n add_v by_o flussas_n by_o this_o mean_n therefore_o the_o diameter_n of_o a_o sphere_n be_v give_v there_o shall_v be_v give_v the_o side_n of_o every_o one_o of_o the_o body_n inscribe_v and_o forasmuch_o as_o three_o of_o those_o body_n have_v their_o side_n commensurable_a in_o power_n only_o and_o not_o in_o length_n unto_o the_o diameter_n give_v for_o their_o power_n be_v in_o the_o proportion_n of_o a_o square_a number_n to_o a_o number_n not_o square_a wherefore_o they_o have_v not_o the_o proportion_n of_o a_o square_a number_n to_o a_o square_a number_n by_o the_o corollary_n of_o the_o 25._o of_o the_o eight_o wherefore_o also_o their_o side_n be_v incommensurabe_n in_o length_n by_o the_o 9_o of_o the_o ten_o therefore_o it_o be_v sufficient_a to_o compare_v the_o power_n and_o not_o the_o length_n of_o those_o side_n the_o one_o to_o the_o otherâ—_n which_o power_n be_v contain_v in_o the_o power_n of_o the_o diameter_n namely_o from_o the_o power_n of_o the_o diameter_n let_v there_o ble_v take_v away_o the_o power_n of_o the_o cube_fw-la and_o there_o shall_v remain_v the_o power_n of_o the_o tetrahedron_n and_o take_v away_o the_o power_n of_o the_o tetrahedron_n there_o remain_v the_o power_n of_o the_o cube_fw-la and_o take_v away_o from_o the_o power_n of_o the_o diameter_n half_o the_o power_n thereof_o there_o shall_v be_v leave_v the_o power_n of_o the_o side_n of_o the_o octohedron_n but_o forasmuch_o as_o the_o side_n of_o the_o dodecahedron_n and_o of_o the_o icosahedron_n be_v prove_v to_o be_v irrational_a for_o the_o side_n of_o the_o icosahedron_n be_v a_o less_o line_n by_o the_o 16._o of_o the_o thirteen_o and_o the_o side_n of_o the_o dedocahedron_n be_v a_o residual_a line_n by_o the_o 17._o of_o the_o same_o therefore_o those_o side_n be_v unto_o the_o diameter_n which_o be_v a_o rational_a line_n set_v incommensurable_a both_o in_o length_n and_o in_o power_n wherefore_o their_o comparison_n can_v not_o be_v define_v or_o describe_v by_o any_o proportion_n express_v by_o number_n by_o the_o 8._o of_o the_o ten_o neither_o can_v they_o be_v compare_v the_o one_o to_o the_o other_o for_o irrational_a line_n of_o diverse_a kind_n be_v incommensurable_a the_o one_o to_o the_o other_o for_o if_o they_o shall_v be_v commensurable_a they_o shall_v be_v of_o one_o and_o the_o self_n same_o kind_n by_o the_o 103._o and_o 105._o of_o the_o ten_o which_o be_v impossible_a wherefore_o we_o seek_v to_o compare_v they_o to_o the_o power_n of_o the_o diameter_n think_v they_o can_v not_o be_v more_o apt_o express_v then_o by_o such_o proportion_n which_o cut_v that_o rational_a power_n of_o the_o diameter_n according_a to_o their_o side_n namely_o divide_v the_o power_n of_o the_o diameter_n by_o line_n which_o have_v that_o proportion_n that_o the_o great_a segment_n have_v to_o the_o less_o to_o put_v the_o less_o segment_n to_o be_v the_o side_n of_o the_o icosahedron_n &_o devide_v the_o say_a power_n of_o the_o diameter_n by_o line_n have_v the_o proportion_n of_o the_o whole_a to_o the_o less_o segment_n to_o express_v the_o side_n of_o the_o dodecahedron_n by_o the_o less_o segment_n which_o thing_n may_v well_o be_v do_v between_o magnitude_n incommensurable_a the_o end_n of_o the_o fourteen_o book_n of_o euclides_n element_n after_o flussas_n ¶_o the_o fifteen_o book_n of_o euclides_n element_n this_o finetenth_n and_o last_o book_n of_o euclid_n or_o rather_o the_o second_o book_n of_o appollonius_n or_o hypsicles_n book_n teach_v the_o inscription_n and_o circumscription_n of_o the_o five_o regular_a body_n one_o within_o and_o about_o a_o other_o a_o thing_n undouted_o pleasant_a and_o delectable_a in_o mind_n to_o contemplate_v and_o also_o profitable_a and_o necessary_a in_o act_n to_o practice_v for_o without_o practice_n in_o act_n it_o be_v very_o hard_a to_o see_v and_o conceive_v the_o construction_n and_o demonstration_n of_o the_o proposition_n of_o this_o book_n unless_o a_o man_n have_v a_o very_a deep_a sharp_a &_o fine_a imagination_n wherefore_o i_o will_v wish_v the_o diligent_a student_n in_o this_o book_n to_o make_v the_o study_n thereof_o more_o pleasant_a unto_o he_o to_o have_v present_o before_o his_o eye_n the_o body_n form_v &_o frame_v of_o past_v paper_n as_o i_o teach_v after_o the_o definition_n of_o the_o eleven_o book_n and_o then_o to_o draw_v and_o describe_v the_o line_n and_o division_n and_o superficiece_n according_a to_o the_o construction_n of_o the_o proposition_n in_o which_o description_n if_o he_o be_v wary_a and_o diligent_a he_o shall_v find_v all_o thing_n in_o these_o solid_a matter_n as_o clear_v and_o as_o manifest_v unto_o the_o eye_n as_o be_v thing_n before_o teach_v only_o in_o plain_a or_o superficial_a figure_n and_o although_o i_o have_v before_o in_o the_o twelve_o book_n admonish_v the_o reader_n hereof_o yet_o because_o in_o this_o book_n chief_o that_o thing_n be_v require_v i_o think_v it_o shall_v not_o be_v irksome_a unto_o he_o again_o to_o be_v put_v in_o mind_n thereof_o far_o this_o be_v to_o be_v note_v that_o in_o the_o greek_a exemplar_n be_v find_v in_o this_o 15._o book_n only_o 5._o proposition_n which_o 5._o be_v also_o only_o touch_v and_o set_v forth_o by_o hypsicy_n unto_o which_o campane_n add_v 8._o and_o so_o make_v up_o the_o number_n of_o 13._o campane_n undoubted_o although_o he_o be_v very_o well_o learn_v and_o that_o general_o in_o all_o kind_n of_o learning_n yet_o assure_o be_v bring_v up_o in_o a_o time_n of_o rudeness_n when_o all_o good_a letter_n be_v darken_v &_o barberousnes_n have_v
the_o residue_n or_o of_o this_o excess_n but_o a_o pyramid_n be_v to_o the_o same_o cube_fw-la inscribe_v in_o it_o nonecuple_n by_o the_o 30._o of_o this_o book_n wherefore_o the_o dodecahedron_n inscribe_v in_o the_o pyramid_n and_o contain_v the_o same_o cube_fw-la twice_o take_v away_o the_o self_n same_o three_o of_o the_o less_o segment_n and_o moreover_o the_o less_o segment_n of_o the_o less_o segment_n of_o half_a the_o residue_n shall_v contain_v two_o nine_o part_n of_o the_o solid_a of_o the_o pyramid_n of_o which_o nine_o part_v each_o be_v equal_a unto_o the_o cube_fw-la take_v away_o this_o self_n same_o excess_n the_o solid_a therefore_o of_o a_o dodecahedron_n contain_v of_o a_o pyramid_n circumscribe_v about_o it_o two_o nine_o part_n take_v away_o a_o three_o part_n of_o one_o nine_o part_n of_o the_o less_o segment_n of_o a_o line_n divide_v by_o a_o extmere_n and_o mean_a proportion_n and_o moreover_o the_o less_o segment_n of_o the_o less_o segment_n of_o half_a the_o residue_n ¶_o the_o 36._o proposition_n a_o octohedron_n exceed_v a_o icosahedron_n inscribe_v in_o it_o by_o a_o parallelipipedon_n set_v upon_o the_o square_n of_o the_o side_n of_o the_o icosahedron_n and_o have_v to_o his_o altitude_n the_o line_n which_o be_v the_o great_a segment_n of_o half_a the_o semidiameter_n of_o the_o octohedron_n svppose_v that_o there_o be_v a_o octohedron_n abcfpl_n construction_n in_o which_o let_v there_o be_v inscribe_v a_o icosahedron_n hkegmxnudsqtâ—_n by_o the_o â—6_o of_o the_o fifteen_o and_o draw_v the_o diameter_n azrcbroif_n and_o the_o perpendicular_a ko_o parallel_n to_o the_o line_n azr_n then_o i_o say_v that_o the_o octohedron_n abcfpl_n be_v great_a thâ—n_n the_o icosahedron_n inscribe_v in_o it_o by_o a_o parallelipipedon_n set_v upon_o the_o square_n of_o the_o side_n hk_o or_o ge_z and_o have_v to_o his_o altitude_n the_o line_n ko_o or_o rz_n which_o be_v the_o great_a segment_n of_o the_o semidiameter_n ar._n forasmuch_o as_o in_o the_o same_o 16._o it_o have_v be_v prove_v that_o the_o triangle_n kdg_n and_o keq_n be_v describe_v in_o the_o base_n apf_n and_o alf_n of_o the_o octohedron_n demonstration_n therefore_o about_o the_o solid_a angle_n there_o remain_v upon_o the_o base_a feg_fw-mi three_o triangle_n keg_n kfe_n and_o kfg_n which_o contain_v a_o pyramid_n kefg_n unto_o which_o pyramid_n shall_v be_v equal_a and_o like_v the_o opposite_a pyramid_n mefg_n set_v upon_o the_o same_o base_a feg_n by_o the_o 8._o definition_n of_o the_o eleven_o and_o by_o the_o â—ame_n reason_n shall_v there_o at_o every_o solid_a angle_n of_o the_o octohedron_n remain_v two_o pyramid_n equal_a and_o like_a namely_o two_o upon_o the_o base_a ahk_n two_o upon_o the_o base_a bnv_n two_o upon_o the_o base_a dp_n and_o moreover_o two_o upon_o the_o base_a qlt._n now_o they_o there_o shall_v be_v make_v twelve_o pyramid_n set_v upon_o a_o base_a contain_v of_o the_o side_n of_o the_o icosahedron_n and_o under_o two_o leâ—â—e_a segment_n of_o the_o side_n of_o the_o octohedron_n contain_v a_o right_a angle_n as_o for_o example_n the_o base_a gef_n and_o forasmuch_o as_o the_o side_n ge_z subtend_v a_o right_a angle_n be_v by_o the_o 47._o of_o the_o â—irst_n in_o power_n duple_n to_o either_o of_o the_o line_n of_o and_o fg_o and_o so_o the_o â—â—deâ—_n kh_o be_v in_o power_n duple_n to_o either_o of_o the_o side_n ah_o and_o ak_o and_o either_o of_o the_o line_n ah_o ak_o or_o of_o fg_o be_v in_o power_n duple_n to_o either_o of_o the_o line_n az_o or_o zk_v which_o contain_v a_o right_a angle_n make_v in_o the_o triangle_n or_o base_a ahk_n by_o the_o perpendicular_a az_o wherefore_o it_o follow_v that_o the_o side_n ge_z or_o hk_o be_v in_o power_n quadruple_a to_o the_o triangle_n efg_o or_o ahk_n but_o the_o pyramid_n kefg_n have_v his_o base_a efg_o in_o the_o plain_a flbp_n of_o the_o octohedron_n shall_v have_v to_o his_o altitude_n the_o perpendicular_a ko_o by_o the_o 4._o definition_n of_o the_o six_o which_o be_v the_o great_a segment_n of_o the_o semidiameter_n of_o the_o octohedron_n by_o the_o 16._o of_o the_o fifteen_o wherefore_o three_o pyramid_n set_v under_o the_o same_o altitude_n and_o upon_o equal_a base_n shall_v be_v equal_a to_o one_o prism_n set_v upon_o the_o same_o base_a and_o under_o the_o same_o altitude_n by_o the_o 1._o corollary_n of_o the_o 7._o of_o the_o twelve_o wherefore_o 4._o prism_n set_v upon_o the_o base_a gef_n quadruple_v which_o be_v equal_a to_o the_o square_n of_o the_o side_n ge_z and_o under_o the_o altitude_n ko_o or_o rz_n the_o great_a segment_n which_o be_v equal_a to_o ko_o shall_v contain_v a_o solid_a equal_a to_o the_o twelve_o pyramid_n which_o twelve_o pyramid_n make_v the_o excess_n of_o the_o octohedron_n above_o the_o icosahedron_n inscribe_v in_o it_o a_o octohedron_n therefore_o exceed_v a_o icosahedron_n inscribe_v in_o it_o by_o a_o parallelipipedon_n set_v upon_o the_o square_n of_o the_o side_n of_o the_o icosahedron_n and_o have_v to_o his_o altitude_n the_o line_n which_o be_v the_o great_a segment_n of_o half_a the_o semidiameter_n of_o the_o octohedron_n ¶_o a_o corollary_n a_o pyramid_n exceed_v the_o double_a of_o a_o icosahedron_n inscribe_v in_o it_o by_o a_o solid_a set_v upon_o the_o square_n of_o the_o side_n of_o the_o icosahedron_n inscribe_v in_o it_o and_o have_v to_o his_o altitude_n that_o whole_a line_n of_o which_o the_o side_n of_o the_o icosahedron_n be_v the_o great_a segment_n for_o it_o be_v manifest_a by_o the_o 19_o of_o the_o fivetenth_n that_o a_o octohedron_n &_o a_o icosahedron_n inscribe_v in_o it_o be_v inscribe_v in_o one_o &_o the_o self_n same_o pyramid_n it_o have_v moreover_o be_v prove_v in_o the_o 26._o of_o this_o book_n that_o a_o pyramid_n be_v double_a to_o a_o octohedron_n inscribe_v in_o it_o wherefore_o the_o two_o excess_n of_o the_o two_o octohedron_n unto_o which_o the_o pyramid_n be_v equal_a above_o the_o two_o icosahedrons_n inscribe_v in_o the_o say_v two_o octohedron_n be_v bring_v into_o a_o solid_a the_o say_v solid_a shall_v be_v set_v upon_o the_o self_n same_o square_n of_o the_o side_n of_o the_o icosahedron_n and_o shall_v have_v to_o his_o altitude_n the_o perpendicular_a ko_o double_a who_o double_a couple_v the_o opposite_a side_n hk_o and_o xm_o make_v the_o great_a segment_n the_o same_o side_n of_o the_o icosahedron_n by_o the_o first_o and_o second_o corollary_n of_o the_o 14._o of_o the_o fiuâ—â—enâ—h_n the_o 37._o proposition_n if_o in_o a_o triangle_n have_v to_o his_o base_a a_o rational_a line_n set_v the_o side_n be_v commensurable_a in_o power_n to_o the_o base_a and_o from_o the_o top_n be_v draw_v to_o the_o base_a a_o perpendicular_a line_n cut_v the_o base_a the_o section_n of_o the_o base_a shall_v be_v commensurable_a in_o length_n to_o the_o whole_a base_a and_o the_o perpendicular_a shall_v be_v commensurable_a in_o power_n to_o the_o say_v whole_a base_a and_o now_o that_o the_o perpendicular_a ap_n be_v commensurable_a in_o power_n to_o the_o base_a bg_o demonstration_n iâ—_z thus_o prove_v forasmuch_o as_o the_o square_n of_o ab_fw-la be_v by_o supposition_n commensurable_a to_o the_o square_n of_o bg_o and_o unto_o the_o rational_a square_n of_o ab_fw-la be_v commensurable_a the_o rational_a square_n of_o bp_o by_o the_o 12._o of_o the_o eleven_o wherefore_o the_o residue_n namely_o the_o square_a of_o pa_o be_v commensurable_a to_o the_o same_o square_n of_o bp_o by_o the_o 2._o part_n of_o the_o 15._o of_o the_o eleven_o wherefore_o by_o the_o 12._o of_o the_o ten_o the_o square_a of_o pa_o be_v commensurable_a to_o the_o whole_a square_n of_o bg_o wherefore_o the_o perpendicular_a ap_n be_v commensurable_a in_o power_n to_o the_o base_a bg_o by_o the_o 3._o definition_n of_o the_o ten_o which_o be_v require_v to_o be_v prove_v in_o demonstrate_v of_o this_o we_o make_v no_o mention_n at_o all_o of_o the_o length_n of_o the_o side_n ab_fw-la and_o ag_z but_o only_o of_o the_o length_n of_o the_o base_a bg_o for_o that_o the_o line_n bg_o be_v the_o rational_a line_n first_o set_v and_o the_o other_o line_n ab_fw-la and_o ag_z be_v suppose_v to_o be_v commensurable_a in_o power_n only_o to_o the_o line_n bg_o wherefore_o if_o that_o be_v plain_o demonstrate_v when_o the_o side_n be_v commensurable_a in_o power_n only_o to_o the_o base_a much_o more_o easy_o will_v it_o follow_v if_o the_o same_o side_n be_v suppose_v to_o be_v commensurable_a both_o in_o length_n and_o in_o power_n to_o the_o base_a that_o be_v if_o their_o lengthe_n be_v express_v by_o the_o root_n of_o square_a number_n ¶_o a_o corollary_n 1._o by_o the_o former_a thing_n demonstrate_v it_o be_v manifest_a that_o if_o from_o the_o power_n of_o the_o base_a and_o of_o one_o of_o the_o side_n be_v take_v away_o the_o
diameter_n be_v double_a to_o that_o square_n who_o diameter_n it_o be_v corollary_n the_o 34._o theorem_a the_o 48._o proposition_n if_o the_o square_n which_o be_v make_v of_o one_o of_o the_o side_n of_o a_o triangle_n be_v equal_a to_o the_o square_n which_o be_v make_v of_o the_o two_o other_o side_n of_o the_o same_o triangle_n the_o angle_n comprehend_v under_o those_o two_o other_o side_n be_v a_o right_a angle_n svppose_v that_o abc_n be_v a_o triangle_n and_o let_v the_o square_n which_o be_v make_v of_o one_o of_o the_o side_n there_o namely_o of_o the_o side_n bc_o be_v equal_a to_o the_o square_n which_o be_v make_v of_o the_o side_n basilius_n and_o ac_fw-la then_o i_o say_v that_o the_o angle_n bac_n be_v a_o right_a angle_n raise_v up_o by_o the_o 11._o proposition_n from_o the_o point_n a_o unto_o the_o right_a line_n ac_fw-la a_o perpendicular_a line_n ad._n and_o by_o the_o third_o proposition_n unto_o the_o line_n ab_fw-la put_v a_o equal_a line_n ad._n and_o by_o the_o first_o petition_n draw_v a_o right_a line_n from_o the_o point_n d_o to_o the_o poinâ—_n c._n and_o forasmuch_o as_o the_o line_n dam_n be_v equal_a to_o the_o line_n ab_fw-la the_o square_n which_o be_v make_v of_o the_o line_n dam_n be_v equal_a to_o the_o square_n which_o be_v make_v of_o the_o line_n ab_fw-la put_v the_o square_n of_o the_o line_n ac_fw-la common_a to_o they_o both_o wherefore_o the_o square_n of_o the_o line_n dam_n and_o ac_fw-la be_v equal_a to_o the_o square_n of_o the_o line_n basilius_n and_o ac_fw-la but_o by_o the_o proposition_n go_v before_o the_o square_a of_o the_o line_n dc_o be_v equal_a to_o the_o square_n of_o the_o line_n ad_fw-la and_o ac_fw-la for_o the_o angle_n dac_o be_v a_o right_a angle_n and_o the_o square_a of_o bc_o be_v by_o supposition_n equal_a to_o the_o square_n of_o ab_fw-la and_o ac_fw-la wherefore_o the_o square_a of_o dc_o be_v equal_a to_o the_o square_n of_o bc_o wherefore_o the_o side_n dc_o be_v equal_a to_o the_o side_n bc._n and_o forasmuch_o as_o ab_fw-la be_v equal_a to_o ad_fw-la â—nd_n ac_fw-la be_v common_a to_o they_o both_o therefore_o these_o two_o side_n dam_n and_o ac_fw-la be_v equal_a to_o these_o two_o side_n basilius_n and_o ac_fw-la the_o one_o to_o the_o other_o and_o the_o base_a dc_o be_v equal_a to_o the_o base_a be_v wherefore_o by_o the_o 8._o proposition_n the_o angle_n dac_o be_v equal_a to_o the_o angle_n bac_n but_o the_o angle_n dac_o be_v a_o right_a angle_n wherefore_o also_o the_o angle_n bac_n be_v a_o right_a angle_n if_o therefore_o the_o square_n which_o be_v make_v of_o one_o of_o the_o side_n of_o a_o triangle_n be_v equal_a to_o the_o square_n which_o be_v make_v of_o the_o two_o other_o side_n of_o the_o same_o triangle_n the_o angle_n comprehend_v under_o those_o two_o other_o side_n be_v a_o right_a angle_n which_o be_v require_v to_o be_v prove_v former_a this_o proposition_n be_v the_o converse_n of_o the_o former_a and_o be_v of_o pelitarius_n demonstrate_v by_o a_o argument_n lead_v to_o a_o impossibility_n after_o this_o manner_n the_o end_n of_o the_o first_o book_n of_o euclides_n element_n ¶_o the_o second_o book_n of_o euclides_n element_n in_o this_o second_o book_n euclid_n show_v book_n what_o be_v a_o gnomon_n and_o a_o right_a angle_a parallelogram_n also_o in_o this_o book_n be_v set_v forth_o the_o power_n of_o line_n divide_v even_o and_o uneven_o and_o of_o line_n add_v one_o to_o a_o other_o the_o power_n of_o a_o line_n line_n be_v the_o square_a of_o the_o same_o line_n that_o be_v a_o square_a every_o side_n of_o which_o be_v equal_a to_o the_o line_n so_o that_o here_o be_v set_v forth_o the_o quality_n and_o propriety_n of_o the_o square_n and_o right_n line_v figure_n which_o be_v make_v of_o line_n &_o of_o their_o part_n algebra_n the_o arithmetician_n also_o our_o of_o this_o book_n gather_v many_o compendious_a rule_n of_o reckon_n and_o many_o rule_v also_o of_o algebra_n with_o the_o equation_n therein_o use_v the_o ground_n also_o of_o those_o rule_n be_v for_o the_o most_o part_n by_o this_o second_o book_n demonstrate_v book_n this_o book_n moreover_o contain_v two_o wonderful_a proposition_n one_o of_o a_o obtuse_a angle_a triangle_n and_o the_o other_o of_o a_o acute_a which_o with_o the_o aid_n of_o the_o 47._o proposition_n of_o the_o first_o book_n of_o euclid_n which_o be_v of_o a_o rectangle_n triangle_n of_o how_o great_a force_n and_o profit_n they_o be_v in_o matter_n of_o astronomy_n they_o know_v which_o have_v travail_v in_o that_o art_n wherefore_o if_o this_o book_n have_v none_o other_o profit_n be_v side_n only_o for_o these_o 2._o proposition_n sake_n it_o be_v diligent_o to_o be_v embrace_v and_o study_v the_o definition_n 1._o every_o rectangled_a parallelogram_n definition_n be_v say_v to_o be_v contain_v under_o two_o right_a line_n comprehend_v a_o right_a angle_n a_o parallelogram_n be_v a_o figure_n of_o four_o side_n be_v who_o two_o opposite_a or_o contrary_a side_n be_v equal_a the_o one_o to_o the_o other_o there_o be_v of_o parallelogram_n four_o kind_n parallelogram_n a_o square_a a_o figure_n of_o one_o side_n long_o a_o rombus_n or_o diamond_n and_o a_o romboide_n or_o diamond_n like_o figure_n as_o before_o be_v say_v in_o the_o 33._o definition_n of_o the_o first_o book_n of_o these_o four_o sort_n the_o square_a and_o the_o figure_n of_o one_o side_n long_o be_v only_o right_a angle_a parallelogram_n for_o that_o all_o their_o angle_n be_v right_a angle_n and_o either_o of_o they_o be_v contain_v according_a to_o this_o definition_n under_o two_o right_a line_n whiâ—h_o concur_v together_o and_o cause_v the_o right_a angle_n and_o contain_v the_o same_o of_o which_o two_o line_n the_o one_o be_v the_o length_n of_o the_o figure_n &_o the_o other_o the_o breadth_n the_o parallelogram_n be_v imagine_v to_o be_v make_v by_o the_o draught_n or_o motion_n of_o one_o of_o the_o line_n into_o the_o length_n of_o the_o other_o as_o if_o two_o number_n shall_v be_v multiply_v the_o one_o into_o the_o other_o as_o the_o figure_n abcd_o be_v a_o parallelogram_n and_o be_v say_v to_o be_v contain_v under_o the_o two_o right_a line_n ab_fw-la and_o ac_fw-la which_o contain_v the_o right_a angle_n bac_n or_o under_o the_o two_o right_a line_n ac_fw-la and_o cd_o for_o they_o likewise_o contain_v the_o right_a angle_n acd_o of_o which_o 2._o line_n the_o one_o namely_o ab_fw-la be_v the_o length_n and_o the_o other_o namely_o ac_fw-la be_v the_o breadth_n and_o if_o we_o imagine_v the_o line_n ac_fw-la to_o be_v draw_v or_o move_v direct_o according_a to_o the_o length_n of_o the_o line_n ab_fw-la or_o contrary_a wise_a the_o line_n ab_fw-la to_o be_v move_v direct_o according_a to_o the_o length_n of_o the_o line_n ac_fw-la you_o shall_v produce_v the_o whole_a rectangle_n parallelogram_n abcd_o which_o be_v say_v to_o be_v contain_v of_o they_o even_o as_o one_o number_n multiply_v by_o a_o other_o produce_v a_o plain_a and_o right_a angle_a superficial_a number_n as_o you_o see_v in_o the_o figure_n here_o set_v where_o the_o number_n of_o six_o or_o six_o unity_n be_v multiply_v by_o the_o number_n of_o five_o or_o by_o five_o unity_n of_o which_o multiplication_n be_v produce_v 30._o which_o number_n be_v set_v down_o and_o describe_v by_o his_o unity_n represent_v a_o plain_n and_o a_o right_a angle_a number_n wherefore_o even_o as_o equal_a number_n multiple_v by_o equal_a number_n produce_v number_n equal_v the_o one_o to_o the_o other_o so_o rectangle_n parallelogram_n which_o be_v comprehend_v under_o equal_a line_n be_v equal_a the_o one_o to_o the_o other_o definition_n 2._o in_o every_o parallelogram_n one_o of_o those_o parallelogram_n which_o soever_o it_o be_v which_o be_v about_o the_o diameter_n together_o with_o the_o two_o supplement_n be_v call_v a_o gnomon_n those_o particular_a parallelogram_n be_v say_v to_o be_v about_o the_o diameter_n of_o the_o parallelogram_n which_o have_v the_o same_o diameter_n which_o the_o whole_a parallelogram_n have_v and_o supplement_n be_v such_o which_o be_v without_o the_o diameter_n of_o the_o whole_a parallelogram_n as_o of_o the_o parallelogram_n abcd_o the_o partial_a or_o particular_a parallelogram_n gkcf_n and_o ebkh_n be_v parallelogram_n about_o the_o diameter_n for_o that_o each_o of_o they_o have_v for_o his_o diameter_n a_o part_n of_o the_o diameter_n of_o the_o whole_a parallelogram_n as_o ck_v and_o kb_v the_o particular_a diameter_n be_v part_n of_o the_o line_n cb_o which_o be_v the_o diameter_n of_o the_o whole_a parallelogram_n and_o the_o two_o parallelogram_n aegk_n and_o khfd_n be_v supplement_n because_o they_o be_v without_o the_o diameter_n of_o the_o whole_a parallelogram_n namely_o cb._n now_o any_o one_o of_o those_o partial_a parallelogram_n
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thereof_o be_v 〈◊〉_d this_o iâ—_z also_o to_o be_v note_v that_o of_o line_n some_o be_v commensurable_a in_o length_n the_o one_o to_o the_o other_o and_o some_o be_v commensurable_a the_o one_o to_o the_o other_o in_o power_n of_o line_n commensurable_a in_o length_n the_o one_o to_o the_o other_o be_v give_v a_o example_n in_o the_o declaration_n of_o the_o first_o definition_n namely_o the_o line_n a_o and_o b_o which_o be_v commensurable_a in_o length_n one_o and_o the_o self_n measure_n namely_o the_o line_n c_o measure_v the_o length_n of_o either_o of_o they_o of_o the_o other_o kind_n be_v give_v this_o definition_n here_o set_v for_o the_o open_n of_o which_o take_v this_o example_n let_v there_o be_v a_o certain_a line_n namely_o the_o line_n bc_o and_o let_v the_o square_n of_o that_o line_n be_v the_o square_n bcde_v suppose_v also_o a_o other_o line_n namely_o the_o line_n fh_o &_o let_v the_o square_v thereof_o be_v the_o square_a fhik_n and_o let_v a_o certain_a 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nor_o assign_v by_o number_n which_o be_v of_o other_o man_n call_v irrational_a he_o call_v uncertain_a and_o surd_a line_n petrus_n montaureus_n although_o he_o do_v not_o very_o well_o like_a of_o the_o name_n yet_o he_o alter_v it_o not_o but_o use_v it_o in_o all_o his_o book_n likewise_o will_v we_o do_v here_o for_o that_o the_o word_n have_v be_v and_o be_v so_o universal_o receive_v and_o therefore_o will_v we_o use_v the_o same_o name_n and_o call_v it_o a_o rational_a line_n for_o it_o be_v not_o so_o great_a a_o matter_n what_o name_n we_o geve_v to_o thing_n so_o that_o we_o full_o understand_v the_o thing_n which_o the_o name_n signify_v this_o rational_a line_n thus_o here_o define_v be_v the_o ground_n and_o foundation_n of_o all_o the_o proposition_n almost_o of_o this_o whole_a ten_o book_n book_n and_o chief_o from_o the_o ten_o proposition_n forward_o so_o that_o unless_o you_o first_o place_n this_o rational_a line_n and_o have_v a_o special_a and_o continual_a regard_n unto_o it_o before_o you_o begin_v any_o demonstration_n you_o shall_v not_o easy_o understand_v it_o for_o it_o be_v as_o it_o be_v the_o touch_n and_o trial_n of_o all_o other_o line_n by_o which_o it_o be_v know_v whether_o any_o of_o they_o be_v rational_a or_o not_o note_n and_o this_o may_v be_v call_v the_o first_o rational_a line_n purpose_n the_o line_n rational_a of_o purpose_n or_o a_o rational_a line_n set_v in_o the_o first_o place_n and_o so_o make_v distinct_a and_o sever_v from_o other_o rational_a line_n of_o which_o shall_v be_v speak_v afterward_o and_o this_o must_v you_o well_o commit_v to_o memory_n definition_n 6_o line_n which_o be_v commensurable_a to_o this_o line_n whether_o in_o length_n and_o power_n or_o in_o power_n only_o be_v also_o call_v rational_a this_o definition_n need_v no_o declaration_n at_o all_o but_o be_v easy_o perceive_v if_o the_o first_o definition_n be_v remember_v which_o show_v what_o magnitude_n be_v commensurable_a and_o the_o three_o which_o show_v what_o line_n be_v commensurable_a in_o power_n here_o notâ—_n how_o apt_o &_o natural_o euclid_n in_o this_o place_n use_v these_o word_n commensurable_a either_o in_o length_n and_o power_n or_o in_o power_n only_o because_o that_o all_o line_n which_o be_v commensurable_a in_o length_n be_v also_o commensurable_a in_o pour_v when_o he_o speak_v of_o line_n commensurable_a in_o length_n he_o ever_o add_v and_o in_o power_n but_o when_o he_o speak_v of_o line_n commensurable_a in_o power_n he_o add_v this_o word_n only_o and_o add_v not_o this_o word_n in_o length_n as_o he_o in_o the_o other_o add_v this_o word_n in_o power_n for_o not_o all_o line_n which_o be_v commensurable_a in_o power_n be_v straight_o way_n commensurable_a also_o in_o length_n of_o this_o definition_n take_v this_o example_n let_v the_o first_o line_n rational_a of_o purpose_n which_o be_v suppose_v and_o lay_v forth_o who_o part_n be_v certain_a &_o know_v and_o may_v be_v express_v name_v and_o number_v be_v ab_fw-la the_o quadrate_n whereof_o let_v be_v abcd_o then_o suppose_v again_o a_o other_o line_n namely_o the_o line_n of_o which_o let_v be_v commensurable_a both_o in_o length_n and_o in_o power_n to_o the_o first_o rational_a line_n that_o be_v as_o before_o be_v teach_v let_v one_o line_n measure_v the_o length_n of_o each_o line_n and_o also_o lâ—t_v one_o superficies_n measure_v the_o two_o square_n of_o the_o say_v two_o line_n as_o here_o in_o the_o example_n be_v suppose_v and_o also_o appear_v to_o the_o eye_n then_o be_v the_o line_n e_o fletcher_n also_o a_o rational_a line_n moreover_o if_o the_o line_n of_o be_v commensurable_a in_o power_n only_o to_o the_o rational_a line_n ab_fw-la first_o set_v and_o suppose_v so_o that_o no_o one_o line_n do_v measure_v the_o two_o line_n ab_fw-la and_o of_o as_o in_o example_n yâ—_n see_v to_o be_v for_o
quadrates_n but_o all_o other_o kind_n of_o rectiline_a figure_n plain_a plait_n &_o superficiese_n what_o so_o ever_o so_o that_o if_o any_o such_o figure_n be_v commensurable_a unto_o that_o rational_a squareâ—_n it_o be_v also_o rational_a as_o suppose_v that_o the_o square_a of_o the_o rational_a line_n which_o be_v also_o rational_a be_v abcd_o suppose_v 〈◊〉_d so_o some_o other_o square_a as_o the_o square_a efgh_o to_o be_v commensurable_a to_o the_o same_o they_o be_v the_o square_a efgh_o also_o rational_a so_o also_o if_o the_o rectiline_a figure_n klmn_n which_o be_v a_o figure_n on_o the_o one_o side_n long_o be_v commensurable_a unto_o the_o say_a square_a as_o be_v suppose_v in_o this_o exampleâ—_n it_o be_v also_o a_o rational_a superficies_n and_o so_o of_o all_o other_o superficiese_n 10_o such_o which_o be_v incommensurable_a unto_o it_o be_v irrational_a definition_n where_o it_o be_v say_v in_o this_o definition_n such_o which_o be_v incommensurable_a it_o be_v general_o to_o be_v take_v as_o be_v this_o word_n commensurable_a in_o the_o definition_n before_o for_o all_o superficiese_n whether_o they_o be_v square_n or_o figure_n on_o the_o one_o side_n long_o or_o otherwise_o what_o manner_n of_o right_n line_v figure_n so_o ever_o it_o be_v if_o they_o be_v incommensurable_a unto_o the_o rational_a square_n suppose_v they_o be_v they_o irrational_a as_o let_v thâ—_z square_v abcd_o be_v the_o square_n of_o the_o suppose_a rational_a line_n which_o square_n therefore_o be_v also_o rational_a suppose_v also_o also_o a_o other_o square_a namely_o the_o square_a e_o suppose_v also_o any_o other_o figure_n as_o for_o example_n sake_n a_o figure_n of_o one_o side_n long_o which_o let_v be_v f_o now_o if_o the_o square_a e_o and_o the_o figure_n fletcher_n be_v both_o incommensurable_a to_o the_o rational_a square_n abcd_o then_o be_v 〈◊〉_d of_o these_o figure_n e_o &_o fletcher_n irrational_a and_o so_o of_o other_o 11_o and_o these_o line_n who_o powere_n they_o be_v be_v irrational_a if_o they_o be_v square_n then_o be_v their_o side_n irrational_a if_o they_o be_v not_o square_n definition_n but_o some_o other_o rectiline_a figure_n then_o shall_v the_o line_n who_o square_n be_v equal_a to_o these_o rectiline_a figure_n be_v irrational_a suppose_v that_o the_o rational_a square_n be_v abcd._n suppose_v also_o a_o other_o square_a namely_o the_o square_a e_o which_o let_v be_v incommensurable_a to_o the_o rational_a square_n &_o therefore_o be_v it_o irrational_a and_o let_v the_o side_n or_o line_n which_o produce_v this_o square_n be_v the_o line_n fg_o then_o shall_v the_o line_n fg_o by_o this_o definition_n be_v a_o irrational_a line_n because_o it_o be_v the_o side_n of_o a_o irrational_a square_n let_v also_o the_o figure_n h_o be_v a_o figure_n on_o the_o one_o side_n long_o which_o may_v be_v any_o other_o rectiline_a figure_n rectangle_v or_o not_o rectangle_v triangle_n pentagone_fw-mi trapezite_fw-mi or_o what_o so_o ever_o else_o be_v incommensurable_a to_o the_o rational_a square_n abcd_o then_o because_o the_o figure_n h_o be_v not_o a_o square_a it_o have_v no_o side_n or_o root_n to_o produce_v it_o yet_o may_v there_o be_v a_o square_n make_v equal_a unto_o it_o for_o that_o all_o such_o figure_n may_v be_v reduce_v into_o triangle_n and_o so_o into_o square_n by_o the_o 14._o of_o the_o second_o suppose_v that_o the_o square_a queen_n be_v equal_a to_o the_o irrational_a figure_n h._n the_o side_n of_o which_o figure_n q_n let_v be_v the_o line_n kl_o then_o shall_v the_o line_n kl_o be_v also_o a_o irrational_a line_n because_o the_o power_n or_o square_v thereof_o be_v equal_a to_o the_o irrational_a figure_n h_o and_o thus_o conceive_v of_o other_o the_o like_a these_o irrational_a line_n and_o figure_n be_v the_o chief_a matter_n and_o subject_n which_o be_v entreat_v of_o in_o all_o this_o ten_o book_n the_o knowledge_n of_o which_o be_v deep_a and_o secret_a and_o pertain_v to_o the_o high_a and_o most_o worthy_a part_n of_o geometry_n wherein_o stand_v the_o pith_n and_o marry_v of_o the_o hole_n science_n the_o knowledge_n hereof_o bring_v light_n to_o all_o the_o book_n follow_v with_o out_o which_o they_o be_v hard_a and_o can_v be_v at_o all_o understand_v and_o for_o the_o more_o plainness_n you_o shall_v note_v that_o of_o irrational_a line_n there_o be_v diâ—ers_n sort_n and_o kind_n but_o they_o who_o name_n be_v set_v in_o a_o table_n here_o follow_v and_o be_v in_o number_n 13._o be_v the_o chief_a and_o in_o this_o ten_o book_n sufficient_o for_o euclides_n principal_a purpose_n discourse_v on_o a_o medial_a line_n a_o binomial_a line_n a_o first_o bimedial_a line_n a_o second_o bimedial_a line_n a_o great_a line_n a_o line_n contain_v in_o power_n a_o rational_a superficies_n and_o a_o medial_a superficies_n a_o line_n contain_v in_o power_n two_o medial_a superficiece_n a_o residual_a line_n a_o first_o medial_a residual_a line_n a_o second_o medial_a residual_a line_n a_o less_o line_n a_o line_n make_v with_o a_o rational_a superficies_n the_o whole_a superficies_n medial_a a_o line_n make_v with_o a_o medial_a superficies_n the_o whole_a superficies_n medial_a of_o all_o which_o kind_n the_o definition_n together_o with_o there_o declaration_n shall_v be_v set_v here_o after_o in_o their_o due_a place_n ¶_o the_o 1._o theorem_a the_o 1._o proposition_n two_o unequal_a magnitude_n be_v give_v if_o from_o the_o great_a be_v take_v away_o more_o than_o the_o half_a and_o from_o the_o residue_n be_v again_o take_v away_o more_o than_o the_o half_a and_o so_o be_v do_v still_o continual_o there_o shall_v at_o length_n be_v leave_v a_o certain_a magnitude_n lesser_a than_o the_o less_o of_o the_o magnitude_n first_o give_v svppose_v that_o there_o be_v two_o unequal_a magnitude_n ab_fw-la and_o c_o of_o which_o let_v ab_fw-la be_v the_o great_a then_o i_o say_v that_o if_o from_o ab_fw-la be_v take_v away_o more_o than_o the_o half_a construction_n and_o from_o the_o residue_n be_v take_v again_o more_o than_o the_o half_a and_o so_o still_o continual_o there_o shall_v at_o the_o length_n be_v leave_v a_o certain_a magnitude_n lesser_a than_o the_o less_o magnitude_n give_v namely_o then_o c._n for_o forasmuch_o as_o c_z be_v the_o less_o magnitude_n therefore_o c_z may_v be_v so_o multiply_v that_o at_o the_o length_n it_o will_v be_v great_a than_o the_o magnitude_n ab_fw-la by_o the_o 5._o definition_n of_o the_o five_o book_n demonstration_n let_v it_o be_v so_o multiply_v and_o let_v the_o multiplex_n of_o c_z great_a than_o ab_fw-la be_v de._n and_o divide_v de_fw-fr into_o the_o part_n equal_a unto_o c_o which_o let_v be_v df_o fg_o and_o ge._n and_o from_o the_o magnitude_n ab_fw-la take_v away_o more_o than_o the_o half_a which_o let_v be_v bh_o and_o again_o from_o ah_o take_v away_o more_o than_o the_o half_a which_o let_v be_v hk_a and_o so_o do_v continual_o until_o the_o division_n which_o be_v in_o the_o magnitude_n ab_fw-la be_v equal_a in_o multitude_n unto_o the_o division_n which_o be_v in_o the_o magnitude_n de._n so_o that_o let_v the_o division_n ak_o kh_o and_o hb_o be_v equal_a in_o multitude_n unto_o the_o division_n df_o fg_o and_o ge._n and_o forasmuch_o as_o the_o magnitude_n de_fw-fr be_v great_a than_o the_o magnitude_n ab_fw-la and_o from_o de_fw-fr be_v take_v away_o less_o than_o the_o half_a that_o be_v eglantine_n which_o detraction_n or_o take_v away_o be_v understand_v to_o be_v do_v by_o the_o former_a division_n of_o the_o magnitude_n de_fw-fr into_o the_o part_n equal_a unto_o c_o for_o as_o a_o magnitude_n be_v by_o multiplication_n increase_v so_o be_v it_o by_o division_n diminish_v and_o from_o ab_fw-la be_v take_v away_o more_o than_o the_o half_a that_o be_v bh_o therefore_o the_o residue_n gd_o be_v great_a than_o the_o residue_n ha_o which_o thing_n be_v most_o true_a and_o most_o easy_a to_o conceive_v if_o we_o remember_v this_o principle_n that_o the_o residue_n of_o a_o great_a magnitude_n after_o the_o take_v away_o of_o the_o half_a or_o less_o than_o the_o half_a be_v ever_o great_a than_o the_o residue_n of_o a_o less_o magnitude_n after_o the_o take_v away_o of_o more_o than_o the_o half_a and_o forasmuch_o as_o the_o magnitude_n gd_o be_v great_a than_o the_o magnitude_n ha_o and_o from_o gd_o be_v take_v away_o the_o half_a that_o be_v gf_o and_o from_o ah_o be_v take_v away_o more_o than_o the_o half_a that_o be_v hk_o therefore_o the_o residue_n df_o be_v great_a than_o the_o residue_n ak_o by_o the_o foresay_a principle_n but_o the_o magnitude_n df_o be_v equal_a unto_o the_o magnitude_n c_o by_o supposition_n wherefore_o also_o the_o magnitude_n c_o be_v great_a than_o the_o magnitude_n ak_o wherefore_o the_o magnitude_n ak_o be_v less_o than_o the_o magnitude_n c._n wherefore_o of_o the_o magnitude_n
the_o square_a of_o the_o line_n a_o have_v not_o unto_o the_o square_n of_o the_o line_n b_o the_o same_o proportion_n that_o a_o square_a number_n have_v to_o a_o square_a number_n then_o i_o say_v that_o the_o line_n a_o and_o b_o be_v incommensurable_a in_o length_n for_o if_o the_o line_n a_o and_o b_o be_v commensurable_a in_o length_n than_o the_o square_a of_o the_o line_n a_o shall_v have_v unto_o the_o square_n of_o the_o line_n b_o the_o same_o proportion_n that_o a_o square_a number_n have_v to_o a_o square_a number_n by_o the_o first_o part_n of_o this_o proposition_n but_o by_o supposition_n it_o have_v not_o wherefore_o the_o line_n a_o and_o b_o be_v not_o commensurable_a in_o length_n proposition_n wherefore_o they_o be_v incomensurable_a in_o length_n wherefore_o square_n make_v of_o right_a line_n commensura_fw-la in_o length_n have_v that_o proportion_n the_o one_o to_o the_o other_o that_o a_o square_a number_n have_v to_o a_o square_a number_n and_o square_n which_o have_v that_o proportion_n the_o one_o to_o the_o other_o that_o a_o square_a number_n have_v to_o a_o squaâ—e_a number_n shall_v also_o have_v the_o side_n commensurable_a in_o length_n but_o square_n describe_v of_o right_a line_n incommensurable_a in_o length_n have_v not_o that_o proportion_n the_o one_o to_o the_o other_o that_o a_o square_a number_n have_v to_o a_o square_a number_n and_o square_n which_o have_v not_o that_o proportion_n the_o one_o to_o the_o other_o that_o a_o square_a number_n have_v to_o a_o square_a number_n have_v not_o also_o their_o side_n commensurable_a in_o length_n which_o be_v all_o that_o be_v require_v to_o be_v prove_v ¶_o corollary_n hereby_o it_o be_v manifest_a that_o right_a line_n commensurable_a in_o length_n be_v also_o ever_o commensurable_a in_o power_n but_o right_a line_n commensurable_a in_o power_n be_v not_o always_o commensurable_a in_o length_n corollary_n and_o right_a line_n incommensurable_a in_o length_n be_v not_o always_o incommensurable_a in_o power_n but_o right_a line_n incommensurable_a in_o power_n be_v ever_o also_o incommensurable_a in_o length_n for_o forasmuch_o as_o square_n make_v of_o right_a line_n commensurable_a in_o length_n have_v that_o proportion_n the_o one_o to_o the_o other_o corollary_n that_o a_o square_a number_n have_v to_o a_o square_a number_n by_o the_o first_o part_n of_o this_o proposition_n but_o magnitude_n which_o have_v that_o proportion_n the_o one_o to_o the_o other_o that_o number_n simple_o have_v to_o number_n be_v by_o the_o six_o of_o the_o ten_o commensurable_a wherefore_o right_a line_n commensurable_a in_o length_n be_v commensurable_a not_o only_o in_o length_n but_o also_o in_o power_n part_n again_o forasmuch_o as_o there_o be_v certain_a square_n which_o have_v not_o that_o proportion_n the_o one_o to_o the_o other_o that_o a_o square_a number_n have_v to_o a_o square_a number_n but_o yet_o have_v that_o proportion_n the_o one_o to_o the_o other_o which_o number_n simple_o have_v to_o number_v their_o side_n in_o deed_n be_v in_o power_n commensurable_a for_o that_o they_o describe_v square_n which_o have_v that_o proportion_n which_o number_n simple_o have_v to_o number_n which_o square_n be_v therefore_o commensurable_a by_o the_o 6._o of_o this_o book_n but_o the_o say_v side_n be_v incommensurable_a in_o length_n by_o the_o latter_a part_n of_o this_o proposition_n wherâ—fore_o it_o be_v true_a that_o line_n commensurable_a in_o power_n be_v not_o straight_o way_n commensurable_a in_o length_n also_o pâ—rt_n and_o by_o the_o selâ—e_a same_o reason_n be_v prove_v also_o that_o three_o part_n of_o the_o corollary_n that_o line_n incommensurable_a in_o length_n be_v not_o always_o incommensurable_a in_o power_n for_o they_o may_v be_v incommensurable_a in_o length_n but_o yet_o commensurable_a in_o power_n as_o in_o those_o square_n which_o be_v in_o proportion_n the_o one_o to_o the_o other_o as_o number_n be_v to_o number_n but_o not_o as_o a_o square_a number_n be_v to_o a_o square_a number_n part_n but_o right_a line_n incommensurable_a in_o power_n be_v always_o also_o incommensurable_a in_o length_n for_o iâ—_z they_o be_v commensurable_a in_o length_n they_o shall_v also_o be_v commensurable_a in_o power_n by_o the_o first_o part_n of_o this_o corollary_n but_o they_o be_v suppose_v to_o be_v incommensurable_a in_o length_n which_o be_v absurd_a wherefore_o right_a line_n incommensurable_a in_o power_n be_v ever_o incommensurable_a in_o lengthy_a for_o the_o better_a understanding_n of_o this_o proposition_n and_o the_o other_o follow_v i_o have_v here_o add_v certain_a annotation_n take_v out_o of_o montaureus_n montauâ—â—us_fw-la and_o first_o as_o touch_v the_o signification_n oâ—_n word_n and_o term_n herein_o use_v whâ—ch_a arâ—_n such_o that_o unless_o they_o be_v well_o mark_v and_o poise_v the_o matter_n will_v be_v obscure_a and_o hard_a and_o in_o a_o manner_n inexplicable_a first_o this_o you_o must_v note_v that_o line_n to_o be_v commensurable_a in_o length_n and_o line_n to_o be_v in_o proportion_n the_o one_o to_o the_o other_o as_o number_n be_v to_o number_n be_v all_o one_o so_o that_o whatsoever_o line_n be_v commensurable_a in_o length_n be_v also_o in_o proportion_n the_o one_o to_o the_o other_o as_o number_n be_v to_o number_v and_o converse_o what_o so_o ever_o line_n be_v in_o proportion_n the_o one_o to_o the_o other_o as_o number_n be_v to_o number_n be_v also_o commensurable_a in_o length_n as_o it_o be_v manifest_a by_o the_o 5_o and_o 6_o of_o this_o book_n likewise_o line_n to_o be_v incommensurable_a in_o length_n and_o not_o to_o be_v in_o proportion_n the_o one_o to_o the_o other_o as_o number_n be_v to_o number_n be_v all_o one_o as_o it_o be_v manifest_a by_o the_o 7._o and_o 8._o of_o this_o book_n wherefore_o that_o which_o be_v say_v in_o this_o theorem_a aught_o to_o be_v understand_v of_o line_n commensurable_a in_o length_n and_o incommensurable_a in_o length_n this_o moreover_o be_v to_o be_v note_v that_o it_o be_v not_o all_o one_o numbers_z to_o be_v square_a number_n and_o to_o be_v in_o proportion_n the_o one_o to_o the_o other_o as_o a_o square_a number_n be_v to_o a_o square_a number_n for_o although_o square_a number_n be_v in_o proportion_n the_o one_o to_o the_o other_o as_o a_o square_a number_n be_v to_o a_o square_a number_n yet_o be_v not_o all_o those_o number_n which_o be_v in_o proportion_n the_o one_o to_o the_o other_o as_o a_o square_a number_n be_v to_o a_o square_a number_n square_a number_n for_o they_o may_v be_v like_o superficial_a number_n and_o yet_o not_o square_a number_n which_o yet_o be_v in_o proportion_n the_o one_o to_o the_o other_o as_o a_o square_a number_n be_v to_o a_o square_a number_n although_o all_o square_a number_n be_v like_o superficial_a number_n for_o between_o two_o square_a number_n there_o fall_v one_o mean_a proportional_a number_n by_o the_o 11._o of_o the_o eight_o but_o if_o between_o two_o number_n there_o fall_v one_o mean_v proportional_a number_n those_o two_o number_n be_v like_o superficial_a number_n by_o the_o 20._o of_o the_o eight_o so_o also_o if_o two_o number_n be_v in_o proportion_n the_o one_o to_o the_o other_o as_o a_o square_a number_n be_v to_o a_o square_a number_n they_o shall_v be_v like_o superficial_a number_n by_o the_o first_o corollary_n add_v after_o the_o last_o proposition_n of_o the_o eight_o book_n and_o now_o to_o know_v whether_o two_o superficial_a number_n give_v be_v like_o superficial_a number_n or_o no_o no._n it_o be_v thus_o find_v out_o first_o if_o between_o the_o two_o number_n give_v there_o fall_v no_o mean_a proportional_a then_o be_v not_o these_o two_o number_n like_o superficial_a number_n by_o the_o 18._o of_o the_o eight_o but_o if_o there_o do_v fall_v between_o they_o a_o mean_v proportional_a then_o be_v they_o like_v superâ—iciall_a number_n by_o the_o 20._o of_o the_o eight_o moreover_o two_o like_o superficial_a number_n multiply_v the_o one_o into_o the_o other_o do_v produce_v a_o square_a number_n by_o the_o firsâ—_o of_o the_o nine_o wherefore_o if_o they_o do_v not_o produce_v a_o square_a number_n then_o be_v they_o not_o like_o superficial_a number_n and_o if_o the_o one_o be_v multiply_v into_o the_o other_o they_o produce_v a_o square_a number_n then_o be_v they_o like_v superficial_a by_o the_o 2._o of_o the_o nine_o moreover_o if_o the_o say_v two_o superficial_a number_n be_v in_o superperticular_a or_o superbipartient_fw-fr proportion_n then_o be_v they_o not_o like_o superficial_a number_n for_o if_o they_o shall_v be_v like_a then_o shall_v there_o be_v a_o mean_a proportional_a between_o they_o by_o the_o 20._o of_o the_o eight_o but_o that_o be_v contrary_a to_o the_o corollary_n of_o the_o 20._o of_o the_o eight_o and_o the_o easy_o to_o conceive_v the_o demonstration_n follow_v take_v this_o example_n of_o that_o which_o we_o have_v say_v ¶_o
now_o forasmuch_o as_o d_z measure_v ab_fw-la and_o bc_o it_o also_o measure_v the_o whole_a magnitude_n ac_fw-la and_o it_o measure_v ab_fw-la wherefore_o d_z measure_v these_o magnitude_n ca_n and_o ab_fw-la wherefore_o ca_n &_o ab_fw-la be_v commensurable_a and_o they_o be_v suppose_v to_o be_v incommensurableâ—_n which_o be_v impossible_a wherefore_o no_o magnitude_n measure_v ab_fw-la and_o bc._n wherefore_o the_o magnitude_n ab_fw-la and_o bc_o be_v incommensurable_a and_o in_o like_a sort_n may_v they_o be_v prove_v to_o be_v incommensurable_a if_o the_o magnitude_n ac_fw-la be_v suppose_v to_o be_v incommensurable_a unto_o bc._n if_o therefore_o there_o be_v two_o magnitude_n incommensurable_a compose_v the_o whole_a also_o shall_v be_v incommensurable_a unto_o either_o of_o the_o two_o part_n component_a and_o if_o the_o whole_a be_v incommensurable_a to_o one_o of_o the_o part_n component_n those_o first_o magnitude_n shall_v be_v incommensurable_a which_o be_v require_v to_o be_v prove_v ¶_o a_o corollary_n add_v by_o montaureus_n if_o a_o whole_a magnitude_n be_v incommensurable_a to_o one_o of_o the_o two_o magnitude_n which_o make_v the_o whole_a magnitude_n it_o shall_v also_o be_v incommensurable_a to_o the_o other_o of_o the_o two_o magnitude_n for_o if_o the_o whole_a magnitude_n ac_fw-la be_v incommensurable_a unto_o the_o magnitude_n bc_o then_o by_o the_o 2_o part_n of_o this_o 16._o theorem_o the_o magnitude_n ab_fw-la and_o bc_o shall_v be_v incommensurable_a wherefore_o by_o the_o first_o part_n of_o the_o same_o theorem_a the_o magnitude_n ac_fw-la shall_v be_v incommensurable_a to_o either_o of_o these_o magnitude_n ab_fw-la and_o bc._n this_o corollary_n 〈◊〉_d use_v in_o the_o demonstration_n of_o the_o â—3_o theorem_a &_o also_o of_o other_o proposition_n ¶_o a_o assumpt_n if_o upon_o a_o right_a line_n be_v apply_v a_o parallelogram_n want_v in_o figure_n by_o a_o square_n the_o parallelogram_n so_o apply_v be_v equal_a to_o that_o parallelogram_n which_o be_v contain_v under_o the_o segment_n of_o the_o right_a line_n which_o segment_n be_v make_v by_o reason_n of_o that_o application_n this_o assumpt_n i_o before_o add_v as_o a_o corollary_n out_o of_o flussates_n after_o the_o 28._o proposition_n of_o the_o six_o book_n ¶_o the_o 14._o theorem_a the_o 17._o proposition_n if_o there_o be_v two_o right_a line_n unequal_a and_o if_o upon_o the_o great_a be_v apply_v a_o parallelogram_n equal_a unto_o the_o four_o part_n of_o the_o square_n of_o the_o less_o line_n and_o want_v in_o figure_n by_o a_o square_a if_o also_o the_o parallelogram_n thus_o apply_v divide_v the_o line_n where_o upon_o it_o be_v apply_v into_o part_n commensurable_a in_o length_n then_o shall_v the_o great_a line_n be_v in_o power_n more_o than_o the_o less_o by_o the_o square_n of_o a_o line_n commensurable_a in_o length_n unto_o the_o great_a and_o if_o the_o great_a be_v in_o power_n more_o than_o the_o less_o by_o the_o square_n of_o a_o right_a line_n commensurable_a in_o length_n unto_o the_o great_a and_o if_o also_o upon_o the_o great_a be_v apply_v a_o parallelogram_n equal_a unto_o the_o four_o part_n of_o the_o square_n of_o the_o less_o line_n and_o want_v in_o figure_n by_o a_o square_n then_o shall_v it_o divide_v the_o great_a line_n into_o part_n commensurable_a but_o now_o suppose_v that_o the_o line_n bc_o be_v in_o power_n more_o than_o the_o line_n a_o by_o the_o square_n of_o a_o line_n commensurable_a in_o length_n unto_o the_o line_n bc._n first_o and_o upon_o the_o line_n bc_o let_v there_o be_v apply_v a_o rectangle_n parallelogram_n equal_a unto_o the_o four_o part_n of_o the_o square_n of_o the_o line_n a_o and_o want_v in_o figure_n by_o a_o square_a and_o let_v the_o say_a parallelogram_n be_v that_o which_o be_v contain_v under_o the_o line_n bd_o and_o dc_o then_o must_v we_o prove_v that_o the_o line_n bd_o be_v unto_o the_o line_n dc_o commensurable_a in_o length_n the_o same_o construction_n and_o supposition_n that_o be_v before_o remain_v we_o may_v in_o like_a sort_n prove_v that_o the_o line_n bc_o be_v in_o power_n more_o than_o the_o line_n a_o by_o the_o square_n of_o the_o line_n fd._n but_o by_o supposition_n the_o line_n bc_o be_v in_o power_n more_o they_o the_o line_n a_o by_o the_o square_n of_o a_o line_n commensurable_a unto_o it_o in_o length_n wherefore_o the_o line_n bc_o be_v unto_o the_o line_n fd_v commensurable_a in_o length_n wherefore_o the_o line_n compose_v of_o the_o two_o line_n bf_o and_o dc_o be_v commensurable_a in_o length_n unto_o the_o line_n fd_o by_o the_o second_o part_n of_o the_o 15._o of_o the_o ten_o wherefore_o by_o the_o 12._o of_o the_o ten_o or_o by_o the_o first_o part_n of_o the_o 15._o of_o the_o ten_o the_o line_n bc_o be_v commensurable_a in_o length_n to_o the_o line_n compose_v of_o bf_o and_o dc_o but_o the_o whole_a line_n conpose_v bf_o and_o dc_o be_v commensurable_a in_o length_n unto_o dc_o for_o bf_o as_o before_o have_v be_v prove_v be_v equal_a to_o dc_o wherefore_o the_o line_n bc_o be_v commensurable_a in_o length_n unto_o the_o line_n dc_o by_o the_o 12._o of_o the_o ten_o whâ—â—â—fore_o also_o the_o line_n bd_o be_v commensurable_a in_o length_n unto_o the_o line_n dc_o by_o the_o second_o part_n of_o thâ—_z 15._o of_o the_o teâ—th_n if_o therefore_o there_o be_v two_o right_a line_n unequal_a and_o if_o upon_o the_o great_a be_v apply_v a_o parallelogram_n equal_a unto_o the_o four_o part_n of_o the_o square_n of_o the_o less_o and_o want_v in_o figure_n by_o a_o square_a if_o also_o the_o parallelogram_n thus_o apply_v divide_v the_o line_n whereupon_o it_o be_v apply_v into_o part_n commensurable_a in_o length_n then_o shall_v the_o great_a line_n be_v in_o power_n more_o than_o the_o less_o by_o the_o square_n of_o a_o line_n commensurable_a in_o length_n unto_o the_o great_a and_o if_o the_o great_a be_v in_o power_n more_o than_o the_o less_o by_o the_o square_n of_o a_o line_n commensurable_a in_o length_n unto_o the_o great_a and_o if_o also_o upon_o the_o great_a be_v apply_v a_o parallelogram_n equal_a unto_o the_o four_o part_n of_o the_o square_n make_v of_o the_o less_o and_o want_v in_o figure_n by_o a_o square_n then_o shall_v it_o divide_v the_o great_a line_n into_o part_n commensurable_a in_o length_n which_o be_v require_v to_o be_v prove_v campanâ—_n after_o this_o proposition_n reach_v how_o we_o may_v ready_o apply_v upon_o the_o line_n bc_o a_o parallelogram_n equal_a to_o the_o four_o part_n of_o the_o square_n of_o half_n of_o the_o line_n a_o and_o want_v in_o figure_n by_o a_o square_a after_o this_o manner_n divide_v the_o line_n bc_o into_o two_o line_n in_o such_o sort_n that_o half_a of_o the_o line_n a_o shall_v be_v the_o mean_a proportional_a between_o those_o two_o line_n which_o be_v possible_a when_o as_o the_o line_n bc_o be_v suppose_v to_o be_v great_a than_o the_o line_n a_o and_o may_v thus_o be_v do_v proposition_n divide_v the_o line_n bc_o into_o two_o equal_a part_n in_o the_o point_n e_o and_o describe_v upon_o the_o line_n bc_o a_o semicircle_n bhc_n and_o unto_o the_o line_n bc_o and_o from_o the_o point_n c_o erect_v a_o perpâ—dicular_a line_n ck_o and_o put_v the_o line_n ck_o equal_a to_o half_o of_o the_o line_n aâ—_z and_o by_o the_o point_n king_n draw_v unto_o the_o line_n ec_o a_o parallel_n line_n kh_o cut_v the_o semicircle_n in_o the_o point_n h_o which_o it_o must_v needs_o cut_v forasmuch_o as_o the_o line_n bc_o be_v great_a than_o the_o line_n a_o and_o from_o the_o point_n h_o draw_v unto_o the_o line_n bc_o a_o perpendicular_a liâ—e_z hd_v which_o line_n hdâ—_n forasmuch_o as_o by_o the_o 34_o of_o the_o first_o it_o be_v equal_a unto_o the_o line_n kc_o shall_v also_o be_v equal_a to_o half_o of_o the_o line_n a_o draw_v the_o line_n bh_o and_o hc_n now_o then_o by_o the_o â—â—_o of_o the_o three_o the_o angle_n bhc_n be_v a_o right_a aâ—gle_n wherefore_o by_o the_o corollary_n of_o the_o eight_o of_o the_o six_o book_n the_o line_n hd_v be_v the_o mean_a proportional_a between_o the_o line_n bd_o and_o dc_o wherefore_o the_o half_a of_o the_o line_n a_o which_o be_v equal_a unto_o the_o line_n hd_v be_v the_o mean_a proportional_a between_o the_o line_n bd_o and_o dc_o wherefore_o that_o which_o be_v contain_v under_o the_o line_n bd_o and_o dc_o be_v equal_a to_o the_o four_o part_n of_o the_o square_n of_o the_o line_n a._n and_o so_o if_o upon_o the_o line_n bd_o be_v describe_v a_o rectangle_n parallelogram_n have_v his_o other_o side_n equal_a to_o the_o line_n dc_o there_o shall_v be_v apply_v upon_o the_o line_n bc_o a_o rectangle_n parallelogram_n equal_a unto_o the_o square_n of_o half_n of_o the_o line_n a_o and_o want_v in_o figure_n by_o
the_o ten_o the_o line_n ab_fw-la be_v incommensurable_a in_o length_n unto_o the_o line_n be._n wherefore_o the_o square_a of_o the_o line_n ab_fw-la be_v incommensurable_a to_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o be_v by_o the_o first_o of_o the_o six_o and_o 10._o of_o this_o book_n but_o unto_o the_o square_n of_o the_o line_n ab_fw-la be_v equal_a the_o square_n of_o the_o line_n ad_fw-la and_o db_o add_v together_o by_o the_o 47._o of_o the_o first_o and_o unto_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o be_v be_v equal_a that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o fd_v that_o be_v which_o be_v contain_v under_o ad_fw-la and_o db._n for_o the_o parallelogram_n contain_v under_o the_o line_n ab_fw-la and_o fd_o be_v equal_a to_o the_o parallelogram_n contain_v under_o the_o line_n ad_fw-la and_o db_o by_o the_o last_o part_n of_o the_o first_o assumpt_n go_v before_o the_o 33._o of_o this_o ten_o book_n conclude_v wherefore_o that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ad_fw-la and_o db_o be_v incommensurable_a to_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o db._n wherefore_o there_o be_v find_v out_o two_o right_a line_n ad_fw-la and_o db_o incommensurable_a in_o power_n who_o square_n add_v together_o make_v a_o medial_a superficies_n conclusion_n and_o the_o parallelogram_n contain_v under_o they_o make_v also_o a_o medial_a superficies_n which_o parallelogram_n moreover_o be_v incommensurable_a to_o the_o superficies_n compose_v of_o the_o square_n of_o those_o line_n add_v together_o which_o be_v require_v to_o be_v do_v the_o beginning_n of_o the_o senaries_n by_o composition_n ¶_o the_o 2d_o theorem_a the_o 36._o proposition_n if_o two_o rational_a line_n commensurable_a in_o power_n only_o be_v add_v together_o composition_n the_o whole_a line_n be_v irrational_a and_o be_v call_v a_o binomium_n or_o a_o binomial_a line_n 〈…〉_o b_o and_o bc_o be_v incommensurable_a to_o the_o square_n of_o the_o line_n bc._n but_o unto_o the_o parallelogram_n contain_v under_o the_o line_n ab_fw-la and_o bc_o be_v commensurable_a the_o parallelogram_n contain_v under_o ab_fw-la and_o bc_o twice_o by_o the_o 6._o of_o the_o ten_o wherefore_o that_o which_o be_v contain_v under_o ab_fw-la and_o bc_o twice_o be_v incommensurable_a to_o the_o square_n of_o the_o line_n bc_o by_o the_o 13_o of_o the_o ten_o but_o unto_o the_o square_n of_o the_o line_n bc_o be_v commensurable_a that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ab_fw-la and_o bc_o by_o the_o 15._o of_o the_o ten_o for_o by_o supposition_n the_o line_n ab_fw-la and_o bc_o be_v commensurable_a in_o power_n only_o wherefore_o by_o the_o 13._o of_o the_o ten_o that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ab_fw-la and_o bc_o add_v together_o be_v incommensurable_a to_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o twice_o wherefore_o by_o the_o 16._o of_o the_o ten_o that_o which_o be_v contain_v under_o ab_fw-la and_o bc_o twice_o together_o with_o the_o square_n of_o the_o line_n ab_fw-la and_o bc_o which_o by_o the_o 4._o of_o the_o second_o be_v equal_a to_o the_o square_n of_o the_o whole_a line_n ac_fw-la be_v incommensurable_a to_o that_o which_o be_v compose_v of_o the_o square_n of_o ab_fw-la and_o bc_o add_v together_o but_o that_o which_o be_v compose_v of_o the_o square_n of_o ab_fw-la and_o bc_o add_v together_o be_v rational_a for_o it_o be_v commensurable_a to_o either_o of_o the_o square_n of_o the_o line_n ab_fw-la and_o bg_o of_o which_o either_o of_o they_o be_v rational_a by_o supposition_n wherefore_o the_o square_a of_o the_o line_n ac_fw-la be_v by_o the_o 10._o definition_n of_o the_o ten_o irrational_a wherefore_o the_o line_n ac_fw-la also_o be_v irrational_a and_o be_v call_v a_o binomial_a line_n line_n this_o proposition_n show_v the_o generation_n and_o production_n of_o the_o second_o kind_n of_o irrational_a line_n which_o be_v call_v a_o binomium_n or_o a_o binomial_a line_n the_o definition_n whereof_o be_v full_o gather_v out_o of_o this_o proposition_n and_o that_o thus_o a_o binomium_n or_o a_o binomial_a line_n be_v a_o irrational_a line_n compose_v of_o two_o rational_a line_n commensurable_a the_o one_o to_o the_o other_o in_o power_n only_o and_o it_o be_v call_v a_o binomium_n that_o be_v have_v two_o name_n because_o it_o be_v make_v of_o two_o such_o line_n as_o of_o his_o part_n which_o be_v only_o commensurable_a in_o power_n and_o not_o in_o length_n and_o therefore_o each_o part_n or_o line_n or_o at_o the_o least_o the_o one_o of_o they_o as_o touch_v length_n be_v uncertain_a and_o unknown_a wherefore_o be_v join_v together_o their_o quantity_n can_v be_v express_v by_o any_o one_o number_n or_o name_n but_o each_o part_n remain_v to_o be_v several_o name_v in_o such_o sort_n as_o it_o may_v and_o of_o these_o binomial_a line_n there_o be_v six_o several_a kind_n line_n the_o first_o binomiall_n the_o second_o the_o three_o the_o four_o the_o five_o and_o the_o six_o of_o what_o nature_n and_o condition_n each_o of_o these_o be_v shall_v be_v know_v by_o their_o definitious_a which_o be_v afterward_o set_v in_o their_o due_a place_n ¶_o the_o 25._o theorem_a the_o 37._o proposition_n if_o two_o medial_a line_n commensurable_a in_o power_n only_o contain_v a_o rational_a superficies_n be_v add_v together_o the_o whole_a line_n be_v irrational_a and_o be_v call_v a_o first_o bimedial_a line_n let_v these_o two_o medial_a line_n ab_fw-la and_o bc_o be_v commensurable_a in_o power_n only_o and_o contain_v a_o rational_a superficies_n the_o 27._o of_o the_o ten_o teach_v to_o find_v out_o two_o such_o line_n be_v compose_v then_o i_o say_v that_o the_o whole_a line_n ac_fw-la be_v irrational_a for_o as_o 〈◊〉_d say_v in_o the_o proposition_n next_o go_v before_o that_o which_o be_v compose_v of_o the_o square_n of_o the_o 〈◊〉_d ab_fw-la and_o bc_o be_v incommensurable_a to_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o twisâ—_n wherefore_o by_o the_o 16._o of_o the_o ten_o that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ab_fw-la and_o bc_o together_o with_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o twice_o demonstration_n that_o be_v the_o square_a of_o the_o line_n ac_fw-la be_v incommensurable_a to_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o twice_o but_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o twice_o iâ—_z commensurable_a to_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o once_o by_o the_o 6._o of_o the_o ten_o wherefore_o the_o square_a of_o the_o whole_a line_n ac_fw-la be_v by_o the_o 13â—_n of_o the_o ten_o incommensurable_a â—o_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o once_o but_o by_o supposition_n the_o line_n ab_fw-la and_o bc_o comprehend_v a_o rational_a superficies_n wherefore_o the_o square_a of_o the_o whole_a line_n ac_fw-la be_v irrational_a wherefore_o also_o the_o line_n ac_fw-la be_v irrational_a and_o it_o be_v call_v a_o first_o bimedial_a line_n the_o three_o irrational_a line_n which_o be_v call_v a_o first_o bimedial_a line_n be_v show_v by_o this_o proposition_n and_o the_o definition_n thereof_o be_v by_o it_o make_v manifest_a line_n which_o be_v this_o a_o first_o bimedial_a line_n be_v a_o irrational_a line_n which_o be_v compose_v of_o two_o medial_a line_n commensurable_a in_o power_n only_o contain_v a_o rational_a parallelogram_n it_o be_v call_v a_o first_o bimedial_a line_n by_o cause_n the_o two_o medial_a line_n or_o part_n whereof_o it_o be_v compose_v contain_v a_o rational_a superficies_n which_o be_v prefer_v before_o a_o irrational_a ¶_o the_o 26._o theorem_a the_o 38._o proposition_n if_o two_o medial_a line_n commensurable_a in_o power_n only_o contain_v a_o medial_a superficies_n be_v add_v together_o the_o whole_a line_n be_v irrational_a and_o be_v call_v a_o second_o bimedial_a line_n let_v these_o two_o medial_a line_n ab_fw-la and_o bc_o be_v commensurable_a in_o power_n only_o and_o contain_v a_o medial_a superficies_n the_o 28._o of_o the_o ten_o teach_v to_o findâ—_n out_o two_o such_o line_n be_v add_v together_o construction_n then_o i_o say_v that_o the_o whole_a line_n ac_fw-la be_v irrational_a take_v a_o rational_a line_n de._n and_o by_o the_o 44._o of_o the_o first_o upon_o the_o line_n de_fw-fr apply_v the_o parallelogram_n df_o equal_a to_o the_o square_n of_o the_o line_n ac_fw-la who_o be_v other_o side_n let_v be_v the_o line_n dg_o and_o forasmuch_o as_o the_o square_n of_o the_o line_n ac_fw-la be_v by_o the_o 4._o of_o the_o second_o equal_a to_o that_o which_o be_v compose_v of_o
the_o square_n of_o the_o line_n ab_fw-la and_o bc_o demonstration_n together_o with_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o twice_o but_o the_o square_n of_o the_o line_n ac_fw-la be_v equal_a to_o the_o parallelogram_n df._n wherefore_o the_o parallelogram_n df_o be_v equal_a to_o that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ab_fw-la and_o bc_o together_o with_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o twice_o now_o then_o again_o by_o the_o 44_o of_o the_o first_o upon_o the_o line_n de_fw-fr apply_v the_o parallelogram_n eh_o equal_a to_o the_o square_n of_o the_o line_n ab_fw-la and_o bc._n wherefore_o the_o parallelogram_n remain_v namely_o hf_o be_v equal_a to_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o twice_o and_o forasmuch_o as_o either_o of_o these_o line_n ab_fw-la and_o bc_o be_v medial_a therefore_o the_o square_n of_o the_o line_n ab_fw-la and_o bc_o be_v also_o medial_a and_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o twice_o be_v by_o the_o corollary_n of_o the_o â—4_o of_o the_o ten_o medial_a for_o by_o the_o 6._o of_o this_o book_n it_o be_v commensurable_a â—â—_o that_o 〈◊〉_d be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o once_o which_o be_v by_o supposition_n medial_a 〈…〉_o square_n of_o the_o line_n ab_fw-la and_o bc_o be_v equal_a the_o parallelogram_n eh_o and_o unto_o that_o 〈◊〉_d contain_v under_o the_o line_n ab_fw-la and_o bc_o twice_o be_v equal_a the_o parallelogram_n he_o 〈…〉_o either_o of_o these_o parallelogram_n he_o and_o of_o be_v medial_a and_o they_o be_v apply_v upon_o the_o rational_a line_n ed._n wherefore_o by_o the_o 22._o of_o the_o ten_o either_o of_o these_o line_n dh_o and_o hg_o be_v a_o rational_a line_n and_o incommensurable_a in_o length_n unto_o the_o line_n de._n and_o forasmuch_o as_o by_o supposition_n the_o line_n ab_fw-la be_v incommensurable_a in_o length_n unto_o the_o line_n bc._n but_o as_o the_o line_n ab_fw-la be_v to_o the_o line_n be_v â—_z so_o be_v the_o square_a of_o the_o line_n ab_fw-la to_o the_o parallelogram_n which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o by_o the_o first_o of_o the_o six_o wherefore_o by_o the_o 10_o of_o this_o bookâ—_n the_o square_a of_o the_o line_n ab_fw-la be_v incommensurable_a to_o the_o parallelogram_n contain_v under_o the_o line_n ab_fw-la and_o bc._n but_o to_o the_o square_n of_o the_o line_n ab_fw-la be_v commensâ—able_a that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ab_fw-la and_o bc_o by_o the_o 15._o of_o the_o ten_o for_o the_o square_n of_o the_o line_n ab_fw-la and_o bc_o be_v commensurable_a when_o as_o the_o line_n ab_fw-la and_o bc_o be_v put_v to_o be_v commensurable_a in_o power_n only_o and_o to_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o be_v commensurable_a that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o twice_o by_o the_o 6_o of_o the_o ten_o wherefore_o that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ab_fw-la and_o bc_o be_v incommensurable_a to_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o twice_o but_o to_o the_o square_n of_o the_o line_n ab_fw-la and_o bc_o be_v equal_a the_o parallelogram_n eh_o and_o to_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o twice_o be_v equal_a the_o parallelogram_n fh_o wherefore_o the_o parallelogram_n fh_o be_v incommensurable_a to_o the_o parallelogram_n he._n wherefore_o the_o line_n dh_o be_v incommensurable_a in_o length_n to_o the_o line_n hg_o by_o the_o 1_o of_o the_o six_o and_o 10_o of_o this_o book_n and_o it_o be_v prove_v that_o they_o be_v rational_a line_n wherefore_o the_o line_n dh_o &_o hg_o be_v rational_a commensurable_a in_o power_n only_o wherefore_o by_o the_o 36._o of_o the_o ten_o the_o whole_a line_n dg_o be_v irrational_o and_o the_o line_n de_fw-fr be_v rational_a but_o a_o rectangle_n superficies_n comprehend_v under_o a_o rational_a line_n and_o a_o irrational_a line_n be_v by_o the_o corollary_n add_v after_o the_o 21_o of_o the_o ten_o irrational_a wherefore_o the_o superficies_n df_o be_v irrational_a and_o the_o line_n also_o which_o contain_v it_o in_o power_n be_v irrational_a but_o the_o line_n ac_fw-la contain_v in_o power_n the_o superficies_n df._n wherefore_o the_o line_n ac_fw-la be_v irrational_a and_o it_o be_v call_v a_o second_o bimedial_a line_n this_o proposition_n show_v the_o generation_n of_o the_o four_o irrational_a line_n call_v a_o second_o bimedial_a line_n line_n the_o definition_n whereof_o be_v evident_a by_o this_o proposition_n which_o be_v thus_o a_o second_o bimedial_a line_n be_v a_o irrational_a line_n which_o be_v make_v of_o two_o medial_a line_n commensurable_a in_o power_n only_o join_v together_o which_o comprehend_v a_o medial_a superficies_n and_o it_o be_v call_v a_o second_o bimediall_n because_o the_o two_o medial_a line_n of_o which_o it_o be_v compose_v contain_v a_o medial_a superficies_n and_o not_o a_o rational_a now_o a_o medial_a be_v by_o nature_n &_o in_o knowledge_n after_o a_o rational_a ¶_o the_o 27._o theorem_a the_o 39_o proposition_n if_o two_o right_a line_n incommensurable_a in_o power_n be_v add_v together_o have_v that_o which_o be_v compose_v of_o the_o square_n of_o they_o rational_a and_o the_o parallelogram_n contain_v under_o they_o medial_a the_o whole_a right_a line_n be_v irrational_a and_o be_v call_v a_o great_a line_n let_v tâ—ese_n two_o right_a line_n ab_fw-la and_o bc_o be_v incommensurable_a in_o power_n only_o and_o make_v that_o which_o be_v require_v in_o the_o proposition_n the_o 33._o of_o the_o ten_o teach_v to_o find_v out_o two_o such_o line_n be_v add_v together_o then_o i_o say_v that_o the_o whole_a line_n ac_fw-la be_v irrational_a demonstration_n for_o forasmuch_o as_o by_o supposition_n the_o parallelogram_n contain_v under_o the_o line_n ab_fw-la and_o bc_o be_v medial_a therefore_o the_o parallelogram_n contain_v twice_o under_o the_o line_n ab_fw-la and_o bc_o be_v medial_a for_o that_o which_o be_v contain_v under_o ab_fw-la and_o bc_o twice_o be_v commensurable_a to_o that_o which_o be_v contain_v under_o ab_fw-la and_o bc_o once_o by_o the_o 6._o of_o the_o ten_o wherefore_o by_o the_o corollary_n of_o the_o 23._o of_o the_o ten_o that_o which_o be_v contain_v under_o ab_fw-la &_o bc_o twice_o be_v medial_a but_o by_o supposition_n that_o which_o iâ—_z compose_v of_o the_o square_n of_o the_o line_n ab_fw-la and_o bc_o be_v rational_a wherefore_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o twice_o be_v incommensurable_a to_o that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ab_fw-la and_o bc._n wherefore_o by_o the_o 16._o of_o the_o ten_o that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ab_fw-la and_o bc_o together_o with_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la &_o bc_o twice_o which_o be_v by_o the_o 4._o of_o the_o second_o equal_a to_o the_o square_n of_o the_o line_n ac_fw-la be_v incommensurable_a to_o that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ab_fw-la and_o bc._n but_o that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ab_fw-la and_o bc_o be_v rational_a wherefore_o the_o square_a of_o the_o whole_a line_n ac_fw-la be_v irrational_a wherefore_o the_o line_n ac_fw-la also_o be_v irrational_a and_o be_v call_v a_o great_a line_n and_o it_o be_v call_v a_o great_a line_n for_o that_o that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ab_fw-la &_o bc_o which_o be_v rational_a be_v great_a than_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o twice_o which_o be_v medial_a now_o it_o be_v meet_v that_o the_o name_n shall_v be_v give_v according_a to_o the_o property_n of_o the_o rational_a a_o assumpt_n this_o proposition_n teach_v the_o production_n of_o the_o five_o irrational_a line_n which_o be_v call_v a_o great_a line_n which_o be_v by_o the_o sense_n of_o this_o proposition_n thus_o define_v a_o great_a line_n be_v a_o irrational_a line_n which_o be_v compose_v of_o two_o right_a line_n which_o be_v incommensurable_a in_o power_n line_n the_o square_n of_o which_o add_v together_o make_v a_o rational_a superficies_n and_o the_o parallelogram_n which_o they_o contain_v be_v medial_a it_o be_v therefore_o call_v a_o great_a line_n as_o theon_n say_v because_o the_o square_n of_o the_o two_o line_n of_o which_o it_o be_v compose_v add_v together_o be_v rational_a be_v great_a than_o the_o medial_a superficies_n contain_v under_o they_o twice_o and_o it_o be_v convenient_a that_o the_o denomination_n be_v take_v of_o the_o propriety_n of_o the_o
rational_a part_n rather_o than_o of_o the_o medial_a part_n ¶_o the_o 28._o theorem_a the_o 40._o proposition_n if_o two_o right_a line_n incommensurable_a in_o power_n be_v add_v together_o have_v that_o which_o be_v make_v of_o the_o square_n of_o they_o add_v together_o medial_a and_o the_o parallelogram_n contain_v under_o they_o rational_a the_o whole_a right_a line_n be_v irrational_a and_o be_v call_v a_o line_n contain_v in_o power_n a_o rational_a and_o a_o medial_a superficies_n in_o this_o proposition_n be_v teach_v the_o generation_n of_o the_o six_o irrational_a line_n which_o be_v call_v a_o line_n who_o power_n be_v rational_a and_o medial_a the_o definition_n of_o which_o be_v gather_v of_o this_o proposition_n after_o this_o manner_n a_o line_n who_o power_n be_v rational_a and_o medial_a be_v a_o irrational_a line_n which_o be_v make_v of_o two_o right_a line_n incommensurable_a in_o power_n add_v together_o medial_a who_o square_n add_v together_o make_v a_o medial_a superficies_n but_o that_o supersicy_n which_o they_o contain_v be_v rational_a the_o reason_n of_o the_o name_n be_v before_o set_v forth_o in_o the_o proposition_n ¶_o the_o 29._o theorem_a the_o 41._o proposition_n if_o two_o right_a line_n incommensurable_a in_o power_n be_v add_v together_o have_v that_o which_o be_v compose_v of_o the_o square_n of_o they_o add_v together_o medial_a and_o the_o parallelogram_n contain_v under_o they_o medial_a and_o also_o incommensurable_a to_o that_o which_o be_v compose_v of_o the_o square_n of_o they_o add_v togethers_o the_o whole_a right_a line_n be_v irrational_a and_o be_v call_v a_o line_n contain_v in_o power_n two_o medial_n in_o this_o proposition_n be_v teach_v the_o nature_n of_o the_o 7._o kind_n of_o irrational_a line_n which_o be_v call_v a_o line_n who_o power_n be_v two_o medial_n the_o definition_n whereof_o be_v take_v of_o this_o proposition_n after_o this_o manner_n a_o line_n who_o power_n be_v two_o medial_n be_v a_o irrational_a line_n which_o be_v compose_v of_o two_o right_a line_n incommensurable_a in_o power_n medial_n the_o square_n of_o which_o add_v together_o make_v a_o medial_a superficies_n and_o that_o which_o be_v contain_v under_o they_o be_v also_o medial_a and_o moreover_o it_o be_v incommensurable_a to_o that_o which_o be_v compose_v of_o the_o two_o square_n add_v together_o the_o reason_n why_o this_o line_n be_v call_v a_o line_n who_o power_n be_v two_o medial_n be_v before_o in_o the_o end_n of_o the_o demonstration_n declare_v and_o that_o the_o say_v irrational_a line_n be_v divide_v one_o way_n only_o that_o be_v in_o one_o point_n only_o into_o the_o right_a line_n of_o which_o they_o be_v compose_v and_o which_o make_v every_o one_o of_o the_o kind_n of_o those_o irrational_a line_n shall_v straight_o way_n be_v demonstrate_v but_o first_o will_v we_o demonstrate_v two_o assumpte_n here_o follow_v ¶_o a_o assumpt_n take_v a_o right_a line_n and_o let_v the_o same_o be_v ab_fw-la and_o divide_v it_o into_o two_o unequal_a part_n in_o the_o point_n c_o assumpt_n and_o again_o divide_v the_o same_o line_n ab_fw-la into_o two_o other_o unequal_a part_n in_o a_o other_o point_n namely_o in_o d_o and_o let_v the_o line_n ac_fw-la by_o supposition_n be_v great_a than_o the_o line_n db._n then_o i_o say_v that_o the_o square_n of_o the_o line_n ac_fw-la and_o bc_o add_v together_o be_v great_a than_o the_o square_n of_o the_o line_n ad_fw-la and_o db_o add_v together_o divide_v the_o line_n ab_fw-la by_o the_o 10._o of_o the_o first_o into_o two_o equal_a part_n in_o the_o point_n e._n and_o forasmuch_o as_o the_o line_n ac_fw-la be_v great_a than_o the_o line_n db_o take_v away_o the_o line_n dc_o which_o be_v common_a to_o they_o both_o wherefore_o the_o residue_n ad_fw-la be_v great_a than_o the_o residue_n cb_o but_o the_o line_n ae_n be_v equal_a to_o the_o line_n ebb_n wherefore_o the_o line_n de_fw-fr be_v less_o than_o the_o line_n ec_o wherefore_o the_o point_n c_o and_o d_o be_v not_o equal_o distant_a from_o the_o point_n e_o which_o be_v the_o point_n of_o the_o section_n into_o two_o equal_a part_n and_o forasmuch_o as_o by_o the_o 5._o of_o the_o second_o that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o together_o with_o the_o square_n of_o the_o line_n ec_o be_v equal_a to_o the_o square_n of_o the_o line_n ebb_n and_o by_o the_o same_o reason_n that_o which_o be_v contain_v under_o the_o line_n ad_fw-la and_o db_o together_o with_o the_o square_n of_o the_o line_n de_fw-fr be_v also_o equal_a to_o the_o self_n same_o square_n of_o the_o line_n ebb_v wherefore_o that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o together_o with_o the_o square_n of_o the_o line_n ec_o be_v equal_a to_o that_o which_o be_v contain_v under_o the_o line_n ad_fw-la and_o db_o together_o with_o the_o square_n of_o the_o line_n de_fw-fr of_o which_o the_o square_a of_o the_o line_n de_fw-fr be_v less_o than_o the_o square_a of_o the_o line_n ec_o for_o it_o be_v prove_v that_o the_o line_n de_fw-fr be_v less_o than_o the_o line_n ec_o wherefore_o the_o parallelogram_n remain_v contain_v under_o the_o line_n ac_fw-la and_o cb_o be_v less_o they_o the_o parallelogram_n remain_v contain_v under_o the_o line_n ad_fw-la and_o db._n wherefore_o also_o that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o twice_o be_v less_o than_o that_o which_o be_v contain_v under_o the_o line_n ad_fw-la and_o db_o twice_o but_o by_o the_o four_o of_o the_o second_o the_o square_a of_o the_o whole_a line_n ab_fw-la be_v equal_a to_o that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ac_fw-la and_o cb_o together_o with_o that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o twice_o and_o by_o the_o same_o reason_n the_o square_n of_o the_o whole_a line_n ab_fw-la be_v equal_a to_o that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ad_fw-la and_o db_o together_o with_o that_o which_o be_v contain_v under_o the_o line_n ad_fw-la and_o db_o twice_o wherefore_o that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ac_fw-la and_o cb_o together_o with_o that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o twice_o be_v equal_a to_o that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ad_fw-la and_o db_o together_o with_o that_o which_o be_v contain_v under_o the_o line_n ad_fw-la and_o db_o twice_o but_o it_o be_v already_o prove_v that_o that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o twice_o be_v less_o than_o that_o which_o be_v contain_v under_o the_o line_n ad_fw-la &_o db_o twice_o wherefore_o the_o residue_n namely_o that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ac_fw-la and_o cb_o be_v great_a than_o the_o residue_n namely_o then_o that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ad_fw-la and_o db_o which_o be_v require_v to_o be_v demonstrate_v ¶_o a_o assumpt_n a_o rational_a superficies_n exceed_v a_o rational_a superficies_n by_o a_o rational_a superficies_n let_v ad_fw-la be_v a_o rational_a superficies_n and_o let_v it_o exceed_v of_o be_v also_o a_o rational_a superficies_n by_o the_o superficies_n ed._n then_o i_o say_v that_o the_o superficies_n ed_z be_v also_o rational_a for_o the_o parallelogram_n ad_fw-la be_v commensurable_a to_o the_o parallelogram_n of_o for_o that_o either_o of_o they_o be_v rational_a wherefore_o by_o the_o second_o part_n of_o the_o 15._o of_o the_o ten_o the_o parallelogram_n of_o be_v commensurable_a to_o the_o parallelogram_n ed._n but_o the_o the_o parallelogram_n of_o be_v rational_a wherefore_o also_o the_o parallelogram_n ed_z be_v rational_a ¶_o the_o 30._o theorem_a the_o 42._o proposition_n a_o binomial_a line_n be_v in_o one_o point_n only_o divide_v into_o his_o name_n composition_n svppose_v that_o ab_fw-la be_v a_o binomial_a line_n and_o in_o the_o point_n g_o let_v it_o be_v divide_v into_o his_o name_n that_o be_v into_o the_o line_n whereof_o the_o whole_a line_n ab_fw-la be_v compose_v wherefore_o these_o line_n ac_fw-la and_o cb_o be_v rational_a commensurable_a in_o power_n only_o now_o i_o say_v that_o the_o line_n ab_fw-la can_v in_o any_o other_o point_n beside_o c_o be_v divide_v into_o two_o rational_a line_n commensurable_a in_o power_n only_o impossibility_n for_o if_o it_o be_v possible_a let_v it_o be_v divide_v in_o the_o point_n d_o so_fw-mi that_o let_v the_o line_n ad_fw-la and_o de_fw-fr bâ—_z rational_a commensurable_a in_o power_n only_o first_o this_o manifest_a that_o neither_o of_o these_o point_n c_o and_o d_o divide_v the_o right_a line_n ab_fw-la into_o two_o equal_a part_n otherwise_o the_o line_n ac_fw-la and_o cb_o shall_v be_v rational_a commensurable_a in_o
length_n and_o so_o likewise_o shall_v the_o line_n ad_fw-la and_o db_o be_v for_o every_o line_n measure_v itself_o and_o any_o other_o line_n equal_a to_o itself_o moreover_o the_o line_n db_o be_v either_o one_o and_o the_o same_o with_o the_o line_n acâ—_n that_o be_v be_v equal_a to_o the_o line_n ac_fw-la oâ—_n else_o it_o be_v greater-then_a the_o line_n ac_fw-la either_o else_o it_o be_v less_o than_o it_o if_o db_o be_v equal_a to_o the_o line_n ac_fw-la then_o put_v the_o line_n db_o upon_o the_o line_n ac_fw-la each_o end_n of_o the_o one_o shall_v agree_v with_o each_o end_n of_o the_o other_o wherefore_o put_v the_o point_n b_o upon_o the_o point_n a_o the_o point_n d_o also_o shall_v fall_v upon_o the_o point_n c_o and_o the_o line_n ad_fw-la which_o be_v the_o rest_n of_o the_o line_n ac_fw-la shall_v also_o be_v equal_a to_o the_o line_n cb_o which_o be_v the_o rest_n of_o the_o line_n db._n wherefore_o the_o line_n ab_fw-la be_v divide_v into_o his_o name_n in_o the_o point_n c._n and_o so_o also_o shall_v the_o line_n ab_fw-la be_v divide_v in_o the_o point_n d_o be_v divide_v in_o the_o self_n â—ame_n point_n that_o the_o self_n same_o line_n ab_fw-la be_v before_o divide_v in_o the_o point_n c_o which_o be_v contrary_a to_o the_o supposition_n for_o by_o supposition_n it_o be_v divide_v in_o sundry_a point_n namely_o in_o c_o &_o d._n but_o if_o the_o line_n db_o be_v greaterâ—_n the_o the_o line_n ac_fw-la let_v the_o line_n ab_fw-la be_v deâ—ided_v into_o two_o equal_a part_n in_o the_o point_n e._n wherefore_o the_o point_n c_o &_o d_o shall_v not_o equal_o be_v distant_a from_o the_o point_n e_o now_o by_o the_o first_o assupt_v go_v before_o this_o proposition_n that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ad_fw-la &_o db_o be_v great_a they_o that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ac_fw-la &_o cbâ—_n but_o that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ad_fw-la &_o db_o together_o with_o that_o which_o be_v contain_v under_o the_o line_n ad_fw-la &_o db_o twice_o be_v equal_a to_o that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ac_fw-la &_o cb_o together_o with_o that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o twice_o for_o either_o of_o they_o be_v equal_a to_o the_o square_n of_o the_o whole_a line_n ab_fw-la by_o the_o 4._o of_o the_o second_o wherefore_o how_o much_o that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ad_fw-la and_o db_o add_v together_o be_v great_a than_o that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ac_fw-la and_o cb_o add_v together_o so_o much_o be_v that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o twice_o great_a than_o that_o which_o be_v contain_v under_o the_o line_n ad_fw-la and_o db_o twice_o but_o that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ad_fw-la and_o db_o exceed_v that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ac_fw-la and_o cb_o by_o a_o rational_a superficies_n by_o the_o 2._o assumpt_n go_v before_o this_o proposition_n for_o that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ad_fw-la and_o db_o be_v rational_a and_o so_o also_o be_v that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ac_fw-la and_o cb_o for_o the_o line_n ad_fw-la and_o db_o be_v put_v to_o be_v rational_a commensurable_a in_o power_n only_o and_o so_o likewise_o be_v the_o line_n ac_fw-la and_o cb._n wherefore_o also_o that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o twice_o exceed_v that_o which_o be_v contain_v under_o the_o line_n ad_fw-la &_o db_o twice_o by_o a_o rational_a superficies_n when_o yet_o notwithstanding_o they_o be_v both_o medial_a superficiece_n by_o the_o 21._o of_o the_o ten_o which_o by_o the_o 26._o of_o the_o same_o be_v impossible_a and_o if_o the_o line_n db_o be_v less_o than_o the_o line_n ac_fw-la we_o may_v by_o the_o like_a demonstration_n prove_v the_o self_n same_o impossibility_n wherefore_o a_o binomial_a line_n be_v in_o one_o point_n only_o divide_v into_o his_o name_n which_o be_v require_v to_o be_v demonstrate_v 〈…〉_o ollary_n add_v by_o flussates_n two_o ration_n 〈…〉_o surable_a in_o power_n only_o be_v add_v together_o can_v be_v equal_a to_o two_o other_o rational_a line_n 〈…〉_o in_o power_n only_o add_v together_o corollary_n for_o either_o of_o they_o shall_v make_v a_o binomia_fw-la 〈…〉_o so_o shall_v a_o binomial_a line_n be_v divide_v into_o his_o name_n in_o more_o point_n than_o onâ—â—â—ch_v by_o this_o proposition_n be_v prove_v to_o be_v impossible_a the_o like_a shall_v follow_v in_o the_o five_o 〈◊〉_d irrational_a line_n as_o touch_v their_o two_o name_n ¶_o the_o 31._o problem_n the_o 43._o proposition_n a_o first_o bimedial_a line_n be_v in_o one_o point_n only_o divide_v into_o his_o name_n svppose_v that_o ab_fw-la be_v a_o first_o bimedial_a line_n and_o let_v it_o be_v divide_v into_o his_o part_n in_o the_o point_n c_o so_fw-mi that_o let_v the_o line_n ac_fw-la and_o cb_o be_v medial_a commensurable_a in_o power_n only_o and_o contain_v a_o rational_a superficies_n then_o i_o say_v that_o the_o line_n ab_fw-la can_v not_o be_v divide_v into_o his_o name_n in_o any_o other_o point_n then_o in_o c._n for_o if_o it_o be_v possible_a let_v it_o be_v divide_v into_o his_o name_n in_o the_o point_n d_o impossibilâ—â—e_n so_fw-mi that_o let_v ad_fw-la &_o db_o be_v medial_a line_n commensurable_a in_o power_n only_o comprehend_v a_o rational_a superficies_n now_o forasmuch_o as_o how_o much_o that_o which_o be_v contain_v under_o the_o line_n ad_fw-la and_o db_o twice_o diâ—ferreth_v from_o that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o twice_o so_o much_o differ_v that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ad_fw-la and_o db_o from_o that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ac_fw-la and_o cb_o but_o that_o which_o be_v contain_v under_o the_o line_n ad_fw-la and_o db_o twice_o differ_v from_o that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o twice_o by_o a_o rational_a superficies_n by_o the_o second_o assumpt_v go_v before_o the_o 41._o of_o the_o ten_o for_o either_o of_o those_o superficiece_n be_v rational_a wherefore_o that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ac_fw-la and_o cb_o differ_v from_o that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ad_fw-la and_o db_o by_o a_o rational_a superficies_n when_o yet_o they_o be_v both_o medial_a superficiece_n which_o be_v impossible_a wherefore_o a_o first_o bimedial_a line_n be_v in_o one_o point_n only_o divide_v into_o his_o name_n which_o be_v require_v to_o be_v prove_v ¶_o the_o 32._o theorem_a the_o 44._o proposition_n a_o second_o bimedial_a line_n be_v in_o one_o point_n only_o divide_v into_o his_o name_n svppose_v that_o the_o line_n ab_fw-la be_v a_o second_o bimedial_a line_n be_v divide_v into_o his_o name_n in_o the_o point_n c_o so_o that_o let_v the_o line_n ac_fw-la and_o cb_o be_v medial_a line_n commensurable_a in_o power_n only_o comprehend_v a_o medial_a superficies_n it_o be_v manifest_a that_o the_o point_n c_o divide_v not_o the_o whole_a line_n ab_fw-la into_o two_o equal_a part_n impossibility_n for_o the_o line_n ac_fw-la and_o cb_o be_v not_o commensurable_a in_o length_n the_o one_o to_o the_o other_o now_o i_o say_v that_o the_o line_n ab_fw-la can_v be_v divide_v into_o his_o name_n in_o any_o other_o point_n but_o only_o in_o c._n for_o if_o it_o be_v possible_a let_v it_o be_v divide_v into_o his_o name_n in_o the_o point_n d_o so_fw-mi that_o let_v not_o the_o line_n ac_fw-la be_v one_o and_o the_o same_o that_o be_v let_v it_o not_o be_v equal_a with_o the_o line_n db._n but_o let_v it_o be_v great_a than_o it_o now_o it_o be_v manifest_a by_o the_o first_o assumpt_n go_v before_o the_o 42._o proposition_n of_o this_o book_n that_o the_o square_n of_o the_o line_n ac_fw-la and_o cb_o be_v great_a than_o the_o square_n of_o the_o line_n ad_fw-la and_o db._n and_o also_o that_o the_o line_n ad_fw-la and_o db_o be_v medial_a line_n commensurable_a in_o power_n only_o comprehend_v a_o medial_a supersicy_n take_v a_o rational_a line_n ef._n and_o by_o the_o 44._o of_o the_o first_o upon_o the_o line_n of_o apply_v a_o rectangle_n parallelogram_n eke_o equal_a to_o the_o square_n of_o the_o line_n ab_fw-la from_o which_o parallelogram_n take_v away_o the_o parallelogram_n eglantine_n equal_a to_o the_o square_n of_o the_o line_n ac_fw-la and_o cb_o wherefore_o the_o residue_n namely_o the_o parallelogram_n hk_o
to_o the_o line_n be._n but_o as_o the_o line_n fb_o be_v to_o the_o line_n bg_o so_o by_o the_o 1._o of_o the_o six_o be_v the_o parallelogram_n ab_fw-la which_o be_v the_o square_a of_o the_o line_n db_o to_o the_o parallelogram_n dg_o and_o as_o the_o line_n db_o be_v to_o the_o line_n be_v so_o be_v the_o same_o parallelogram_n dg_o to_o the_o parallelogram_n bc_o which_o be_v the_o square_a of_o the_o line_n be._n wherefore_o as_o the_o square_a ab_fw-la be_v to_o the_o parallelogram_n dg_o so_o be_v the_o same_o parallelogram_n dg_o to_o the_o square_a bc._n wherefore_o the_o parallelogram_n dg_o be_v the_o mean_a proportional_a between_o the_o square_n ab_fw-la and_o bc._n i_o say_v moreover_o that_o the_o parallelogram_n dc_o be_v the_o mean_a proportional_a between_o the_o square_n ac_fw-la and_o cb._n for_o for_o that_o as_o the_o line_n ad_fw-la be_v to_o the_o line_n dk_o so_o be_v the_o line_n kg_v to_o the_o line_n gc_o for_o they_o be_v each_o equal_a to_o each_o wherefore_o by_o composition_n by_o the_o 18._o of_o the_o five_o as_o the_o line_n ak_o be_v to_o the_o line_n kd_v so_o be_v the_o line_n kc_o to_o the_o line_n cg_o but_o as_o the_o line_n ak_o be_v to_o the_o line_n kd_v so_o be_v the_o square_a of_o the_o line_n ak_o which_o be_v the_o square_a ac_fw-la to_o the_o parallelogram_n contain_v under_o the_o line_n ak_o and_o kd_v which_o be_v the_o parallelogram_n cd_o and_o as_o the_o line_n kc_o be_v to_o the_o line_n cg_o so_o also_o be_v the_o parallelogram_n dc_o to_o the_o square_n of_o the_o line_n gc_o which_o be_v the_o square_a bc._n wherefore_o as_o the_o square_a ac_fw-la be_v to_o the_o parallelogram_n dc_o so_o be_v the_o parallelogram_n dc_o to_o the_o square_a bc._n wherefore_o the_o parallelogram_n dc_o be_v the_o mean_a proportional_a between_o the_o square_n ac_fw-la and_o bc_o which_o be_v require_v to_o be_v demonstrate_v ¶_o a_o assumpt_n magnitude_n that_o be_v mean_a proportionalls_n between_o the_o self_n same_o or_o equal_a magnitude_n be_v also_o equal_a the_o one_o to_o the_o other_o suppose_v that_o there_o be_v three_o magnitude_n a_o b_o c._n assumpt_n and_o as_o a_o be_v to_o b_o so_o let_v b_o be_v to_o c._n and_o likewise_o as_o the_o same_o magnitude_n a_o be_v to_o d_o so_o let_v d_o be_v to_o the_o same_o magnitude_n c._n then_o i_o say_v that_o b_o and_o d_o be_v equal_a the_o one_o the_o other_o for_o the_o proportion_n of_o a_o unto_o c_o be_v double_a to_o that_o proportion_n which_o a_o have_v to_o b_o by_o the_o 10._o definition_n of_o the_o five_o and_o likewise_o the_o self_n same_o proportion_n of_o a_o to_o c_o be_v by_o the_o same_o definition_n double_a to_o that_o proportion_n which_o a_o have_v to_o d._n but_o magnitude_n who_o equemultiplices_fw-la be_v either_o equal_a or_o the_o self_n same_o be_v also_o equal_a wherefore_o as_o a_o be_v to_o b_o so_o be_v a_o to_o d._n wherefore_o by_o the_o 9_o of_o the_o five_o b_o and_o d_o be_v equal_a the_o one_o to_o the_o other_o so_o shall_v if_o also_o be_v if_o there_o be_v other_o magnitude_n equal_a to_o a_o and_o c_o namely_o e_o and_o f_o between_o which_o let_v the_o magnitude_n d_o be_v the_o mean_a proportional_a ¶_o the_o 36._o theorem_a the_o 54._o proposition_n if_o a_o superficies_n be_v contain_v under_o a_o rational_a line_n &_o a_o first_o binomial_a line_n the_o line_n which_o contain_v in_o power_n that_o superficies_n be_v a_o irrational_a line_n &_o a_o binomial_a line_n composition_n svppose_v that_o the_o superficies_n abcd_o be_v contain_v under_o the_o rational_a line_n ab_fw-la and_o under_o a_o first_o binomial_a line_n ad._n then_o i_o say_v that_o the_o line_n which_o contain_v in_o power_n the_o superficies_n ac_fw-la be_v a_o irrational_a line_n and_o a_o binomial_a line_n construction_n for_o forasmuch_o as_o the_o line_n ad_fw-la be_v a_o first_o binomial_a line_n it_o be_v in_o one_o only_a point_n divide_v into_o his_o name_n by_o the_o 42._o of_o this_o ten_o let_v it_o be_v divide_v into_o his_o name_n in_o the_o point_n e._n and_o let_v ae_n be_v the_o great_a name_n now_o it_o be_v manifest_a that_o the_o line_n ae_n and_o ed_z be_v rational_a commensurable_a in_o power_n only_o and_o that_o the_o line_n ae_n be_v in_o power_n more_o than_o the_o line_n ed_z by_o the_o square_n of_o a_o line_n commensurable_a in_o length_n to_o the_o line_n ae_n and_o moreover_o that_o the_o line_n ae_n be_v commensurable_a in_o length_n to_o the_o rational_a line_n give_v ab_fw-la by_o the_o definition_n of_o a_o first_o binomial_a line_n set_v before_o the_o 48._o proposition_n of_o this_o ten_o divide_v by_o the_o 10._o of_o the_o first_o the_o line_n ed_z into_o two_o equal_a part_n in_o the_o point_n f._n and_o forasmuch_o as_o the_o line_n ae_n be_v in_o power_n more_o than_o the_o line_n ed_z by_o the_o square_n of_o a_o line_n commensurable_a in_o length_n unto_o the_o line_n ae_n therefore_o if_o upon_o the_o great_a line_n namely_o upon_o the_o line_n ae_n be_v apply_v a_o parallelogram_n equal_a to_o the_o four_o part_n of_o the_o square_n of_o the_o less_o line_n that_o be_v to_o the_o square_n of_o the_o line_n of_o &_o want_v in_o form_n by_o a_o square_a it_o shall_v divide_v the_o great_a line_n namely_o ae_n into_o two_o part_n còmmensurable_a in_o length_n the_o one_o to_o the_o other_o by_o the_o second_o part_n of_o the_o 17._o of_o the_o ten_o apply_v therefore_o upon_o the_o line_n ae_n a_o parallelogram_n equal_a to_o the_o square_n of_o the_o line_n of_o and_o want_v in_o form_n by_o a_o square_a by_o the_o 28._o of_o the_o six_o and_o let_v the_o same_o be_v that_o which_o be_v contain_v under_o the_o line_n agnostus_n and_o ge._n wherefore_o the_o line_n agnostus_n be_v commensurable_a in_o length_n to_o the_o line_n ge._n draw_v by_o the_o point_n g_o e_o and_o fletcher_n to_o either_o of_o these_o line_n ab_fw-la and_o dg_v these_o parallel_a line_n gh_o eke_o and_o fl_fw-mi by_o the_o 31._o of_o the_o first_o and_o by_o the_o 14._o of_o the_o second_o unto_o the_o parallelogram_n ah_o describe_v a_o equal_a square_a sn_a and_o unto_o the_o parallelogram_n gk_o describe_v by_o the_o same_o a_o equal_a square_a np._n and_o let_v these_o line_n mn_a &_o nx_o be_v so_o put_v that_o they_o both_o make_v one_o right_a line_n wherefore_o by_o the_o 14._o of_o the_o first_o the_o line_n rn_v and_o no_o make_v also_o both_o one_o right_a line_n demonstration_n make_v perfect_a the_o parallelogram_n sp._n wherefore_o the_o parallelogram_n sparke_n be_v a_o square_a by_o those_o thing_n which_o be_v demonstrate_v after_o the_o determination_n in_o the_o first_o assumpt_n go_v before_o and_o forasmuch_o as_o that_o which_o be_v contain_v under_o the_o line_n agnostus_n and_o ge_z be_v equal_a to_o the_o square_n of_o the_o line_n of_o by_o construction_n therefore_o as_o the_o line_n agnostus_n be_v to_o the_o of_o so_o be_v the_o line_n of_o to_o the_o line_n eglantine_n by_o the_o 14._o or_o 17._o of_o the_o six_o wherefore_o also_o by_o the_o 1._o of_o the_o six_o as_o the_o parallelogram_n ah_o be_v to_o the_o parallelogram_n el_fw-es so_o be_v the_o parallelogram_n el_fw-es to_o the_o parallelogram_n gk_o wherefore_o the_o parallelogram_n el_n be_v the_o mean_a proportional_a between_o the_o parallelogram_n ah_o and_o gk_o but_o the_o parallelogram_n ah_o be_v equal_a to_o the_o square_a sn_a and_o the_o parallelogram_n gk_o be_v equal_a to_o the_o square_a np_n by_o construction_n wherefore_o the_o parallelogram_n el_n be_v the_o mean_a proportional_a between_o the_o square_n sn_a and_o np_a by_o the_o 7._o of_o the_o five_o but_o by_o the_o first_o assumpt_n go_v before_o the_o parallelogram_n mr._n be_v the_o mean_a proportional_a between_o the_o square_n sn_a and_o np._n wherefore_o the_o parallelogram_n mr._n be_v equal_a to_o the_o parallelogram_n el_fw-es by_o the_o last_o assumpt_v go_v before_o but_o the_o parallelogram_n mr._n be_v equal_a to_o the_o parallelogram_n ox_n by_o the_o 43._o of_o the_o first_o and_o the_o parallelogram_n el_n be_v equal_a to_o the_o parallelogram_n fc_o by_o construction_n and_o by_o the_o first_o of_o the_o six_o wherefore_o the_o whole_a parallelogram_n ec_o be_v equal_a to_o the_o two_o parallelogram_n mr._n &_o ox_n and_o the_o parallelogram_n ah_o and_o gk_o be_v equal_a to_o the_o square_n sn_a and_o np_a by_o construction_n wherefore_o the_o whole_a parallelogram_n ac_fw-la be_v equal_a to_o the_o whole_a square_a sparke_n that_o be_v to_o the_o square_n of_o the_o line_n mx_o conclude_v wherefore_o the_o line_n mx_o contain_v in_o power_n the_o parallelogram_n ac_fw-la i_o say_v moreover_o that_o the_o line_n mx_o be_v a_o binomial_a line_n for_o forasmuch_o as_o by_o
the_o 17._o of_o the_o ten_o the_o line_n agnostus_n be_v commensurable_a in_o length_n to_o the_o line_n eglantine_n therefore_o by_o the_o 15._o of_o the_o ten_o the_o whole_a line_n ae_n be_v commensurable_a in_o length_n to_o either_o of_o thâ—se_a line_n agnostus_n and_o ge._n but_o by_o supposition_n the_o line_n ae_n be_v commensurable_a in_o length_n to_o the_o line_n ab_fw-la wherefore_o by_o the_o 12._o of_o the_o ten_o either_o of_o the_o line_n agnostus_n &_o ge_z be_v commensurable_a in_o length_n to_o the_o line_n ab_fw-la but_o the_o line_n ab_fw-la be_v rational_a wherefore_o either_o of_o these_o line_n agnostus_n and_o ge_z be_v rational_a wherefore_o by_o the_o 19_o of_o the_o ten_o either_o of_o these_o parallelogram_n ah_o and_o gk_o be_v rational_a wherefore_o by_o the_o first_o of_o the_o six_o and_o 10._o of_o the_o ten_o the_o parallelogram_n ah_o be_v commensurable_a to_o the_o parallelogram_n gk_o but_o the_o parallelogram_n ah_o be_v equal_a to_o the_o square_a sn_a conclude_v and_o the_o parallelogram_n gk_o be_v equal_a to_o the_o square_a np_n wherefore_o the_o square_n sn_a and_o np_a which_o be_v the_o square_n of_o the_o line_n mn_a and_o nx_o be_v rational_a and_o commensurable_a and_o forasmuch_o as_o by_o supposition_n the_o line_n ae_n be_v incommensurable_a in_o length_n to_o the_o line_n ed._n but_o the_o line_n ae_n be_v commensurable_a in_o length_n to_o the_o line_n ag._n and_o the_o line_n de_fw-fr be_v commensurable_a in_o length_n to_o the_o line_n of_o for_o it_o be_v double_a to_o it_o by_o construction_n wherefore_o by_o the_o 13._o of_o the_o ten_o the_o line_n agnostus_n be_v incommensurable_a in_o length_n to_o the_o line_n ef._n wherefore_o the_o parallelogram_n ah_o be_v incommensurable_a to_o the_o parallelogram_n el._n but_o the_o parallelogram_n ah_o be_v equal_a to_o the_o square_a sn_a and_o the_o parallelogram_n el_n be_v equal_a to_o the_o parallelogram_n mr._n wherefore_o the_o square_a sn_a be_v incommensurable_a to_o the_o parallelogram_n mr._n but_o as_o the_o square_a sn_a be_v to_o the_o parallelogram_n mr._n so_o be_v the_o line_n on_o to_o the_o line_n nr_n by_o the_o 1._o of_o the_o six_o conclude_v wherefore_o the_o line_n on_o be_v incommensurable_a to_o the_o line_n nr_n but_o the_o line_n on_o be_v equal_a to_o the_o line_n mn_o and_o the_o line_n nr_n to_o the_o line_n nx_o wherefore_o the_o line_n mn_o be_v incommensurable_a to_o the_o line_n nx_o and_o it_o be_v already_o prove_v that_o the_o square_n of_o the_o line_n mn_a and_o nx_o be_v rational_a and_o commensurable_a wherefore_o the_o line_n mn_a and_o nx_o be_v rational_a commensurable_a in_o power_n only_o conclusion_n wherefore_o the_o whole_a line_n mx_o be_v a_o binomial_a line_n and_o it_o contain_v in_o power_n the_o parallelogram_n ac_fw-la which_o be_v require_v to_o be_v prove_v ¶_o the_o 37._o theorem_a the_o 55._o proposition_n if_o a_o superficies_n be_v comprehend_v under_o a_o rational_a line_n and_o a_o second_o binomial_a line_n the_o line_n that_o contain_v in_o power_n that_o superficies_n be_v irrational_a and_o be_v a_o first_o bimedial_a line_n svppose_v that_o the_o superficies_n abcd_o be_v contain_v under_o a_o rational_a line_n ab_fw-la and_o under_o a_o second_o binomial_a line_n ad._n then_o i_o say_v that_o the_o line_n that_o contain_v in_o power_n the_o superficies_n ac_fw-la be_v a_o first_o bimedial_a line_n for_o forasmuch_o as_o ad_fw-la be_v a_o second_o binomial_a line_n it_o can_v in_o one_o only_a point_n be_v divide_v into_o his_o name_n by_o the_o 43._o of_o this_o ten_o let_v it_o therefore_o by_o supposition_n be_v divide_v into_o his_o name_n in_o the_o point_n e_o so_fw-mi that_o let_v ae_n be_v the_o great_a name_n wherefore_o the_o line_n ae_n and_o ed_z be_v rational_a commensurable_a in_o power_n only_o and_o the_o line_n ae_n be_v in_o power_n more_o than_o the_o line_n ed_z by_o the_o square_n of_o a_o line_n commensurable_a in_o length_n to_o ae_n and_o the_o less_o name_n namely_o ed_z be_v commensurable_a in_o length_n to_o the_o line_n ab_fw-la by_o the_o definition_n of_o a_o second_o binomial_a line_n set_v before_o the_o 48._o proposition_n of_o this_o ten_o divide_v the_o line_n ed_z by_o the_o ten_o of_o the_o first_o into_o two_o equal_a part_n in_o the_o point_n f._n and_o by_o the_o 28._o of_o the_o six_o upon_o the_o line_n ae_n apply_v a_o parallelogram_n equal_a to_o the_o square_n of_o the_o line_n of_o and_o want_v in_o figure_n by_o a_o square_a and_o let_v the_o same_o parallelogram_n be_v that_o which_o be_v contain_v under_o the_o line_n agnostus_n and_o ge._n wherefore_o by_o the_o second_o part_n of_o the_o 17._o of_o this_o ten_o the_o line_n agnostus_n be_v commensurable_a in_o length_n to_o the_o line_n ge._n and_o by_o the_o 31._o of_o the_o first_o by_o the_o point_n g_o e_o f_o draw_v unto_o the_o line_n ab_fw-la and_o cd_o these_o parallel_a line_n gh_o eke_o &_o fl._n and_o by_o the_o 14._o of_o the_o second_o unto_o the_o parallelogram_n ah_o describe_v a_o equal_a square_a sn_a and_o to_o the_o parallelogram_n gk_o describe_v a_o equal_a square_a np_n and_o let_v the_o line_n mn_a &_o nx_o be_v so_o put_v that_o they_o both_o make_v one_o right_a line_n wherefore_o by_o the_o 14._o of_o the_o first_o the_o line_n also_o rn_v and_o no_o make_v both_o one_o right_a line_n demonstration_n make_v perfect_a the_o parallelogram_n sp._n now_o it_o be_v manifest_a by_o that_o which_o have_v be_v demonstrate_v in_o the_o proposition_n next_o go_v before_o that_o the_o parallelogâ—ame_n mr._n be_v the_o mean_a proportional_a between_o the_o square_n sn_a and_o np_a and_o be_v equal_a to_o the_o parallelogram_n el_fw-es and_o that_o the_o line_n mx_o contain_v in_o power_n the_o superficies_n ac_fw-la now_o rest_v to_o prove_v that_o the_o line_n mx_o be_v a_o first_o bimedial_a line_n forasmuch_o as_o the_o line_n ae_n be_v incommensurable_a in_o length_n to_o the_o line_n ed_z and_o the_o line_n ed_z be_v commensurable_a in_o length_n to_o the_o line_n ab_fw-la therefore_o by_o the_o 13._o of_o the_o ten_o the_o line_n ae_n be_v incommensurable_a in_o length_n to_o the_o line_n ab_fw-la and_o forasmuch_o as_o the_o line_n agnostus_n be_v commensurable_a in_o length_n to_o the_o line_n ge_z therefore_o the_o whole_a line_n ae_n be_v by_o the_o 15._o of_o the_o ten_o commensurable_a in_o length_n to_o either_o of_o these_o line_n agnostus_n and_o ge._n but_o the_o line_n ae_n be_v rational_a wherefore_o either_o of_o these_o line_n agnostus_n and_o ge_z be_v rational_a and_o forasmuch_o as_o the_o line_n ae_n be_v incommensurable_a in_o length_n to_o the_o line_n ab_fw-la but_o the_o line_n ae_n be_v commensurable_a in_o length_n to_o either_o of_o these_o line_n agnostus_n and_o ge_z wherefore_o either_o of_o the_o line_n agnostus_n and_o ge_z be_v incommensurable_a in_o length_n to_o the_o line_n ab_fw-la by_o the_o 13._o of_o the_o ten_o wherefore_o the_o line_n ab_fw-la agnostus_n and_o ge_z be_v rational_a commensurable_a in_o power_n only_o wherefore_o by_o the_o 21._o of_o the_o ten_o either_o of_o these_o parallelogram_n ah_o and_o gk_o be_v a_o medial_a superficies_n wherefore_o also_o either_o of_o these_o square_n sn_a and_o np_n be_v a_o medial_a superficies_n by_o the_o corollary_n of_o the_o 23._o of_o the_o ten_o wherefore_o the_o line_n mn_a &_o nx_o be_v medial_a line_n by_o the_o 21._o of_o this_o ten_o conclude_v and_o forasmuch_o as_o the_o line_n agnostus_n be_v commensurable_a in_o length_n to_o the_o line_n ge_z therefore_o by_o the_o 1._o of_o the_o six_o and_o 11._o of_o the_o ten_o the_o parallelogram_n ah_o be_v commensurable_a to_o the_o parallelogram_n gk_o that_o be_v the_o square_a sn_a to_o the_o square_a np_n that_o be_v the_o square_a of_o the_o line_n mn_v to_o the_o square_n of_o the_o line_n nx_o wherefore_o the_o line_n mn_a and_o nx_o be_v medial_n commensurable_a in_o power_n conclude_v and_o forasmuch_o as_o the_o line_n ae_n be_v incommensurable_a in_o length_n to_o the_o line_n ed_z but_o the_o line_n ae_n be_v commensurable_a in_o length_n to_o the_o line_n agnostus_n and_o the_o line_n ed_z be_v commensurable_a in_o length_n to_o the_o line_n eâ—_n therefore_o by_o the_o 13._o of_o the_o ten_o the_o line_n agnostus_n be_v incommensurable_a in_o length_n to_o the_o line_n eâ—_n wherefore_o by_o the_o â—_o of_o the_o six_o and_o 11._o of_o the_o ten_o the_o parallelogram_n ah_o be_v incommensurable_a to_o the_o parallelogram_n el_fw-es that_o be_v thâ—_z square_v sn_a to_o the_o parallelogram_n mr._n that_o be_v the_o line_n on_o be_v incommensurable_a to_o the_o line_n nr_n that_o be_v the_o line_n mn_v to_o the_o line_n nx_o and_o it_o be_v prove_v that_o the_o line_n mn_a and_o nx_o be_v medial_a line_n commensurablâ—_n in_o power_n concludeâ—_n
therefore_o be_v divide_v into_o two_o unequal_a part_n the_o square_n which_o be_v make_v of_o the_o unequal_a part_n be_v great_a they_o the_o rectangle_n parallelogram_n contain_v under_o the_o unequal_a part_n twice_o which_o be_v require_v to_o be_v demonstrate_v in_o number_n i_o need_v not_o to_o have_v so_o allege_v for_o the_o 17._o of_o the_o seven_o have_v confirm_v the_o double_n to_o be_v one_o to_o the_o other_o as_o their_o single_n be_v but_o in_o our_o magnitude_n it_o likewise_o be_v true_a and_o evident_a by_o alternate_a proportion_n thus_o as_o the_o parallelogram_n of_o the_o line_n ac_fw-la and_o cb_o be_v to_o his_o double_a so_o be_v the_o square_a of_o the_o line_n ad_fw-la to_o his_o double_a each_o be_v half_a wherefore_o alternate_o as_o the_o parallelogram_n be_v to_o the_o square_n so_o be_v the_o parallelogram_n his_o double_a to_o the_o double_a of_o the_o square_n but_o the_o parallelogram_n be_v prove_v less_o than_o thâ—_z square_a wherefore_o his_o double_a be_v less_o than_o the_o square_a his_o double_a by_o the_o 14._o of_o the_o five_o this_o assumpt_n be_v in_o some_o book_n not_o read_v for_o that_o in_o manner_n it_o seem_v to_o be_v all_o one_o with_o that_o which_o be_v put_v after_o the_o 39_o of_o this_o book_n but_o for_o the_o diverse_a manner_n of_o demonstrate_v it_o be_v necessary_a follow_v for_o the_o fear_n of_o invention_n be_v thereby_o further_v and_o though_o zambert_n do_v in_o the_o demonstration_n hereof_o omit_v that_o which_o p._n montaureus_n can_v not_o supplyâ—_n but_o plain_o doubt_v of_o the_o sufficiency_n of_o this_o proof_n yet_o m._n do_v you_o by_o only_a allegation_n of_o the_o due_a place_n of_o credit_n who_o pith_n &_o forceâ—_n theon_n his_o word_n do_v contain_v have_v restore_v to_o the_o demonstration_n sufficient_o both_o light_n and_o authority_n as_o you_o may_v perceive_v and_o chief_o such_o may_v judge_v who_o can_v compare_v this_o demonstration_n here_o thus_o furnish_v with_o the_o greek_a of_o theon_n or_o latin_a translation_n of_o zambert_n ¶_o the_o 42._o theorem_a the_o 60._o proposition_n composition_n the_o square_a of_o a_o binomial_a line_n apply_v unto_o a_o rational_a line_n make_v the_o breadth_n or_o other_o side_n a_o first_o binomial_a line_n svppose_v that_o the_o line_n ab_fw-la be_v a_o binomial_a line_n and_o let_v it_o be_v suppose_v to_o be_v divide_v into_o his_o name_n in_o the_o point_n c_o so_fw-mi that_o let_v ac_fw-la be_v the_o great_a name_n construction_n and_o take_v a_o rational_a line_n deâ—_n â—_z and_o by_o the_o â—4_o of_o the_o first_o unto_o the_o line_n de_fw-fr apply_v a_o rectangle_n parallelogram_n defg_n equal_a to_o the_o square_n of_o theâ—line_n ab_fw-la and_o make_n in_o breadth_n the_o line_n dg_o then_o i_o say_v that_o the_o line_n dg_o be_v a_o first_o binomial_a line_n unto_o the_o line_n de_fw-fr apply_v the_o parallelogram_n dh_fw-mi equal_a to_o the_o square_n of_o the_o line_n ac_fw-la and_o unto_o the_o line_n kh_o which_o be_v equal_a to_o the_o line_n de_fw-fr apply_v the_o parallelogram_n kl_o equal_a to_o the_o square_n of_o the_o line_n bc._n wherefore_o the_o residue_n namely_o that_o which_o be_v contain_v under_o the_o line_n ac_fw-la &_o cb_o twice_o be_v equal_a to_o the_o residue_n namely_o to_o the_o parallelogram_n mf_n by_o 〈◊〉_d 4â—_o of_o the_o second_o divide_v by_o the_o 1â—_o of_o the_o first_o the_o line_n mg_o into_o two_o equal_a part_n in_o the_o point_n n._n demonstration_n and_o by_o the_o 31._o of_o the_o first_o draw_v the_o line_n nx_o parallel_v to_o either_o of_o these_o line_n ml_o and_o gf_o wherefore_o either_o of_o these_o parallelogram_n mx_o and_o nf_n be_v equal_a to_o that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o once_o by_o the_o 15._o of_o the_o five_o and_o forasmuch_o as_o the_o line_n ab_fw-la be_v a_o binomial_a line_n and_o be_v divide_v into_o his_o name_n in_o the_o point_n c_o therefore_o the_o â—ines_n ac_fw-la and_o cb_o be_v rational_a commensurable_a in_o power_n only_o wherefore_o the_o square_n of_o the_o line_n ac_fw-la and_o cb_o be_v rational_a and_o therefore_o commensurable_a the_o one_o to_o the_o other_o wherefore_o by_o the_o 15._o of_o the_o ten_o that_o which_o be_v make_v of_o the_o square_n of_o the_o line_n ac_fw-la and_o cb_o add_v together_o be_v commensurable_a to_o either_o of_o the_o square_n of_o the_o line_n ac_fw-la or_o cb_o wherefore_o that_o which_o be_v make_v of_o the_o square_n of_o the_o lineâ—_z ac_fw-la and_o cb_o add_v together_o be_v rational_a and_o it_o be_v equal_a to_o the_o parallelogram_n dl_o by_o construction_n wherefore_o the_o parallelogram_n dl_o be_v rational_a and_o it_o be_v apply_v unto_o the_o rational_a line_n de_fw-fr wherefore_o by_o the_o 20._o of_o the_o ten_o the_o line_n dm_o be_v rational_a and_o commensurable_a in_o length_n to_o the_o line_n de._n again_o forasmuch_o as_o the_o line_n ac_fw-la and_o cb_o be_v rational_a commensurable_a in_o power_n only_o therefore_o that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o twice_o that_o be_v the_o parallelogram_n mf_n be_v medial_a by_o the_o 21_o of_o the_o ten_o and_o it_o be_v apply_v unto_o the_o rational_a line_n ml_o wherefore_o the_o line_n mg_o be_v rational_a and_o incommensurable_a in_o length_n to_o the_o line_n ml_o by_o the_o 22._o of_o this_o ten_o that_o be_v to_o the_o line_n de._n but_o the_o line_n md_o be_v prove_v rational_a and_o commensurable_a in_o length_n to_o the_o line_n de._n wherefore_o by_o the_o 13._o of_o the_o ten_o the_o line_n dm_o be_v incommensurable_a in_o length_n to_o the_o line_n mg_o wherefore_o the_o line_n dm_o &_o mâ—_n be_v rational_a commensurable_a in_o power_n only_o line_n wherefore_o by_o the_o 36._o of_o the_o ten_o the_o whole_a line_n dg_o be_v a_o binomial_a line_n now_o rest_v to_o prove_v that_o it_o be_v a_o first_o binomial_a line_n forasmuch_o as_o by_o the_o the_o assumpt_n go_v before_o the_o 54._o of_o the_o ten_o that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o be_v the_o mean_a proportional_a between_o the_o square_n of_o the_o line_n ac_fw-la and_o cb_o therefore_o the_o parallelogram_n mx_o be_v the_o mean_a proportional_a between_o the_o parallelogram_n dh_o and_o kl_o wherefore_o as_o the_o parallelogame_n dhâ—_n â—_z be_v to_o the_o parallelogram_n mx_o so_o be_v the_o parallelogram_n mx_o to_o the_o parallelogram_n kl_o that_o be_v as_o the_o line_n dk_o be_v to_o the_o line_n mn_o so_o be_v the_o line_n mn_v to_o the_o line_n mk_o wherefore_o that_o which_o be_v contain_v under_o the_o line_n dk_o and_o km_o be_v equal_a to_o the_o square_n of_o the_o line_n mn_v and_o forasmuch_o as_o the_o square_n of_o the_o line_n ac_fw-la be_v commensurable_a to_o the_o square_n of_o the_o line_n cb_o the_o parallelogram_n dh_o be_v commensurable_a to_o the_o prarallelograme_n kl_o wherefore_o by_o the_o 1._o of_o the_o six_o and_o 10._o of_o the_o ten_o the_o line_n de_fw-fr be_v commensurable_a in_o length_n to_o the_o line_n km_o and_o forasmuch_o as_o the_o square_n of_o the_o line_n ac_fw-la and_o cb_o be_v great_a than_o that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o twice_o by_o the_o assumpt_n go_v before_o this_o proposition_n or_o by_o the_o assumpt_n after_o this_o 39_o of_o the_o ten_o therefore_o the_o parallelogram_n dl_o be_v great_a than_o the_o parallelogram_n mf_n wherefore_o by_o the_o first_o of_o the_o six_o the_o line_n dm_o be_v great_a than_o the_o line_n mg_o and_o that_o which_o be_v contain_v under_o the_o line_n dk_o and_o km_o be_v equal_a to_o the_o square_n of_o the_o line_n mn_o that_o be_v to_o the_o four_o part_n of_o the_o square_n of_o the_o line_n mg_o but_o by_o the_o 17._o of_o the_o ten_o if_o there_o be_v two_o unequal_a right_a line_n and_o if_o upon_o the_o great_a be_v apply_v a_o parallelogram_n equal_a to_o the_o four_o part_n of_o the_o square_n make_v of_o the_o less_o line_n and_o want_v in_o figure_n by_o a_o square_a if_o also_o the_o parallelogram_n thus_o apply_v deâ—ide_v the_o line_n whereupon_o it_o be_v apply_v into_o part_n commensurable_a in_o length_n then_o shall_v the_o great_a line_n be_v in_o power_n more_o than_o the_o less_o by_o the_o square_n of_o a_o line_n commensurable_a in_o length_n to_o the_o great_a wherefore_o the_o line_n dm_o be_v in_o power_n more_o than_o the_o line_n mg_o by_o the_o square_n of_o a_o line_n commensurable_a in_o length_n unto_o the_o line_n dm_o and_o the_o line_n dm_o and_o mg_o be_v prove_v rational_a commensurable_a in_o power_n only_o and_o the_o line_n dm_o be_v prove_v the_o great_a name_n and_o commensurable_a in_o length_n to_o the_o rational_a line_n give_v de._n wherefore_o by_o
five_o binomial_a line_n but_o if_o neither_o of_o the_o line_n ae_n nor_o ebb_n be_v commensurable_a in_o length_n to_o the_o rational_a line_n give_v neither_o also_o of_o the_o line_n cf_o nor_o fd_o shall_v be_v commensurable_a in_o length_n to_o the_o same_o and_o so_o either_o of_o the_o line_n ab_fw-la and_o cd_o shall_v be_v a_o six_o binomial_a line_n a_o line_n therefore_o commensurable_a in_o length_n to_o a_o binomial_a line_n be_v also_o a_o binomial_a line_n of_o the_o self_n same_o order_n which_o be_v require_v to_o be_v prove_v ¶_o the_o 49._o theorem_a the_o 67._o proposition_n a_o line_n commensurable_a in_o length_n to_o a_o bimedial_a line_n be_v also_o a_o bimedial_a line_n and_o of_o the_o self_n same_o order_n svppose_v that_o the_o line_n ab_fw-la be_v a_o bimedial_a line_n and_o unto_o the_o line_n ab_fw-la let_v the_o line_n cd_o be_v commensurable_a in_o length_n then_o i_o say_v that_o the_o line_n cd_o be_v a_o bimedial_a line_n and_o of_o the_o self_n order_n that_o the_o line_n ab_fw-la be_v divide_v the_o line_n ab_fw-la into_o his_o part_n in_o the_o point_n e._n construction_n and_o forasmuch_o as_o the_o line_n ab_fw-la be_v a_o bimedial_a line_n and_o be_v divide_v into_o his_o part_n in_o the_o point_n e_o therefore_o by_o the_o 37._o and_o 38._o of_o the_o ten_o the_o line_n ae_n and_o ebb_n be_v medial_n commensurable_a in_o power_n only_o and_o by_o the_o 12._o of_o the_o six_o as_o the_o line_n ab_fw-la be_v to_o the_o line_n cd_o so_o let_v the_o line_n ae_n be_v to_o the_o line_n cf._n wherefore_o by_o the_o 19_o of_o the_o five_o the_o residue_n demonstration_n namely_o the_o line_n ebb_n be_v to_o the_o residue_n namely_o to_o the_o line_n fd_o as_o the_o line_n ab_fw-la be_v to_o the_o line_n cd_o but_o the_o line_n ab_fw-la be_v commensurable_a in_o length_n to_o the_o line_n cd_o wherefore_o the_o line_n ae_n be_v commensurable_a in_o length_n to_o the_o line_n cf_o and_o the_o line_n ebb_n to_o the_o line_n fd._n now_o the_o line_n ae_n and_o ebb_n be_v medial_a wherefore_o by_o the_o 23._o of_o the_o ten_o the_o line_n cf_o and_o fd_o be_v also_o medial_a and_o for_o that_o as_o the_o line_n ae_n be_v to_o the_o line_n ebb_n so_o be_v the_o line_n cf_o to_o the_o line_n fd._n but_o the_o line_n ae_n and_o ebb_n be_v commensurable_a in_o power_n only_o wherefore_o the_o line_n cf_o and_o fd_o be_v also_o commensurable_a in_o power_n only_o and_o it_o be_v prove_v that_o they_o be_v medial_a wherefore_o the_o line_n cd_o be_v a_o bimedial_a line_n i_o say_v also_o that_o it_o be_v of_o the_o self_n same_o order_n that_o the_o line_n ab_fw-la be_v for_o for_o that_o as_o the_o line_n ae_n be_v to_o the_o line_n ebb_n so_o be_v the_o line_n cf_o to_o the_o line_n fd_o but_o as_o the_o line_n cf_o be_v to_o fd_o so_o be_v the_o square_a of_o the_o line_n cf_o to_o the_o parallelogram_n contain_v under_o the_o line_n cf_o and_o fd_o by_o the_o first_o of_o the_o six_o therefore_o as_o the_o line_n ae_n be_v to_o the_o line_n ebb_n so_o by_o the_o 11._o of_o the_o five_o be_v the_o square_a of_o the_o line_n cf_o to_o the_o parallelogram_n contain_v under_o the_o line_n cf_o and_o fd_o but_o as_o ae_n be_v to_o ebb_v so_o by_o the_o 1._o of_o the_o six_o be_v the_o square_a of_o the_o line_n ae_n to_o the_o parallelogram_n contain_v under_o the_o line_n ae_n and_o ebb_n therefore_o by_o the_o 11._o of_o the_o five_o as_o the_o square_n of_o the_o line_n ae_n be_v to_o that_o which_o be_v contain_v under_o the_o line_n ae_n and_o ebb_n so_o be_v the_o square_a of_o the_o line_n cf_o to_o that_o which_o be_v contain_v under_o the_o line_n cf_o and_o fd._n wherefore_o alternate_o by_o the_o 16._o of_o the_o five_o as_o the_o square_n of_o the_o line_n ae_n be_v to_o the_o square_n of_o the_o line_n cf_o so_o be_v that_o which_o be_v contain_v under_o the_o line_n ae_n and_o ebb_n to_o that_o which_o be_v contain_v under_o the_o line_n cf_o &_o fd._n but_o the_o square_n of_o the_o line_n ae_n be_v commensurable_a to_o the_o square_n of_o the_o line_n cf_o because_o ae_n and_o cf_o be_v commensurable_a in_o length_n wherefore_o that_o which_o be_v contain_v under_o the_o line_n ae_n and_o ebb_n in_o commensurable_a to_o that_o which_o be_v contain_v under_o the_o line_n cf_o and_o fd._n if_o therefore_o that_o which_o be_v contain_v under_o the_o line_n ae_n and_o ebb_n be_v rational_a that_o be_v if_o the_o line_n ab_fw-la be_v a_o first_o bimedial_a line_n that_o also_o which_o be_v contain_v under_o the_o line_n cf_o and_o fd_o be_v rational_a wherefore_o also_o the_o line_n cd_o be_v a_o first_o bimedial_a line_n but_o if_o that_o which_o be_v contain_v under_o the_o line_n ae_n and_o ebb_n be_v medial_a that_o be_v if_o the_o line_n ab_fw-la be_v a_o second_o bimedial_a line_n that_o also_o which_o be_v contain_v under_o the_o line_n cf_o and_o fd_o be_v medial_a wherefore_o also_o the_o line_n cd_o be_v a_o second_o bimedial_a line_n wherefore_o the_o line_n ab_fw-la and_o cd_o be_v both_o of_o one_o and_o the_o self_n same_o order_n which_o be_v require_v to_o be_v prove_v ¶_o a_o corollary_n add_v by_o flussates_n but_o first_o note_v by_o p._n montaâ—reus_n a_o line_n commensurable_a in_o power_n only_o to_o a_o bimedial_a line_n be_v also_o a_o bimedial_a line_n and_o of_o the_o self_n same_o order_n suppose_v that_o ab_fw-la be_v a_o bimedial_a line_n either_o a_o first_o or_o a_o second_o whereunto_o let_v the_o line_n gd_v be_v commensurable_a in_o power_n only_o flussetes_n take_v also_o a_o rational_a line_n ez_o upon_o which_o by_o the_o 45._o of_o the_o first_o apply_v a_o rectangle_n parallelogram_n equal_a to_o the_o square_n of_o the_o line_n ab_fw-la which_o let_v be_v ezfc_a and_o let_v the_o rectangle_n parallelogram_n cfih_n be_v equal_a to_o the_o square_n of_o the_o line_n gd_o and_o forasmuch_o as_o upon_o the_o rational_a line_n ez_o be_v apply_v a_o rectangle_n parallelogram_n of_o equal_a to_o the_o square_n of_o a_o first_o bimedial_a line_n therefore_o the_o other_o side_n thereof_o namely_o ec_o be_v a_o second_o binomial_a line_n by_o the_o 61._o of_o this_o book_n and_o forasmuch_o as_o by_o supposition_n the_o square_n of_o the_o line_n ab_fw-la &_o gd_o be_v commensurable_a therefore_o the_o parallelogram_n of_o and_o ci_o which_o be_v equal_a unto_o they_o be_v also_o commensurable_a and_o therefore_o by_o the_o 1._o of_o the_o six_o the_o line_n ec_o and_o change_z be_v commensurable_a in_o length_n but_o the_o line_n ec_o be_v a_o second_o binomial_a line_n wherefore_o the_o line_n change_z be_v also_o a_o second_o binomial_a line_n by_o the_o 66._o of_o this_o book_n and_o forasmuch_o as_o the_o superficies_n ci_o be_v contain_v under_o a_o rational_a line_n ez_o or_o cf_o and_o a_o second_o binomial_a line_n change_z therefore_o the_o line_n which_o contain_v it_o in_o power_n namely_o the_o line_n gd_o be_v a_o first_o bimedial_a line_n by_o the_o 55._o of_o this_o book_n and_o so_o be_v the_o line_n gd_o in_o the_o self_n same_o order_n of_o bimedial_a line_n that_o the_o line_n ab_fw-la be_v the_o like_a demonstration_n also_o will_v serve_v if_o the_o line_n ab_fw-la be_v suppose_v to_o bâ—_z a_o second_o bimedial_a line_n for_o so_o shall_v it_o make_v the_o breadth_n ec_o a_o three_o binomial_a line_n whereunto_o the_o line_n change_z shall_v be_v commensurable_a in_o length_n and_o therefore_o ch_n also_o shall_v be_v a_o three_o binomial_a line_n by_o mean_n whereof_o the_o line_n which_o contain_v in_o power_n the_o superficies_n ci_o namely_o the_o line_n gd_o shall_v also_o be_v a_o second_o bimedial_a line_n wherefore_o a_o line_n commensurable_a either_o in_o length_n or_o in_o power_n only_o to_o a_o bimedial_a line_n be_v also_o a_o bimedial_a line_n of_o the_o self_n same_o order_n but_o so_o be_v it_o not_o of_o necessity_n in_o binomial_a line_n for_o if_o their_o power_n only_o be_v commensurable_a it_o follow_v not_o of_o necessity_n that_o they_o be_v binomiall_n of_o one_o and_o the_o self_n same_o order_n note_n but_o they_o be_v each_o binomialls_n either_o of_o the_o three_o first_o kind_n or_o of_o the_o three_o last_o as_o for_o example_n suppose_v that_o ab_fw-la be_v a_o first_o binomial_a line_n who_o great_a name_n let_v be_v agnostus_n and_o unto_o ab_fw-la let_v the_o dz_o be_v commensurable_a in_o power_n only_o then_o i_o say_v that_o the_o line_n dz_o be_v not_o of_o the_o self_n same_o order_n that_o the_o line_n ab_fw-la be_v for_o if_o it_o be_v possible_a let_v the_o line_n dz_o be_v of_o the_o self_n same_o order_n that_o the_o line_n ab_fw-la be_v wherefore_o the_o line_n dz_o may_v
in_o like_a sort_n be_v divide_v as_o the_o line_n ab_fw-la be_v by_o that_o which_o have_v be_v demonstrate_v in_o the_o 66._o proposition_n of_o this_o bookeâ—_n let_v it_o be_v so_o divide_v in_o the_o point_n e._n wherefore_o it_o can_v not_o be_v so_o divide_v in_o any_o other_o point_n by_o the_o 42â—_n of_o this_o book_n and_o for_o that_o the_o line_n ab_fw-la â—â—_o to_o the_o line_n dz_o as_o the_o line_n agnostus_n be_v to_o the_o line_n de_fw-fr but_o the_o line_n agnostus_n &_o de_fw-fr namely_o the_o great_a name_n be_v commensurable_a in_o length_n the_o one_o to_o the_o other_o by_o the_o 10._o of_o this_o book_n for_o that_o they_o be_v commensurable_a in_o length_n to_o 〈◊〉_d and_o the_o self_n same_o rational_a line_n by_o the_o first_o definition_n of_o binomial_a line_n wherefore_o the_o line_n ab_fw-la and_o dz_o be_v commensurable_a in_o length_n by_o the_o 13._o of_o this_o book_n but_o by_o supposition_n they_o be_v commensurable_a in_o power_n only_o which_o be_v impossible_a the_o self_n same_o demonstration_n also_o will_v serve_v if_o we_o suppose_v the_o line_n ab_fw-la to_o be_v a_o second_o binomial_a line_n for_o the_o less_o name_n gb_o and_o ez_o be_v commensurable_a in_o length_n to_o one_o and_o the_o self_n same_o rational_a line_n shall_v also_o be_v commensurable_a in_o length_n the_o one_o to_o the_o other_o and_o therefore_o the_o line_n ab_fw-la and_o dz_o which_o be_v in_o the_o self_n same_o proportion_n with_o they_o shall_v also_o be_v commensurable_a in_o length_n the_o one_o to_o the_o other_o which_o be_v contrary_a to_o the_o supposition_n far_o if_o the_o square_n of_o the_o line_n ab_fw-la and_o dz_o be_v apply_v unto_o the_o rational_a line_n cf_o namely_o the_o parallelogram_n et_fw-la and_o hl_o they_o shall_v make_v the_o breadthes_fw-mi change_z and_o hk_o first_o binomial_a line_n of_o what_o order_n soever_o the_o line_n ab_fw-la &_o dz_o who_o square_n be_v apply_v unto_o the_o rational_a line_n be_v by_o the_o 60._o of_o this_o book_n wherefore_o it_o be_v manifest_a that_o under_o a_o rational_a line_n and_o a_o first_o binomial_a line_n be_v confuse_o contain_v all_o the_o power_n of_o binomial_a line_n by_o the_o 54._o of_o this_o book_n wherefore_o the_o only_a commensuration_n of_o the_o power_n do_v not_o of_o necessity_n bring_v forth_o one_o and_o the_o self_n same_o order_n of_o binomial_a line_n the_o self_n same_o thing_n also_o may_v be_v prove_v if_o the_o line_n ab_fw-la and_o dz_o be_v suppose_v to_o be_v a_o four_o or_o five_o binomial_a line_n who_o power_n only_o be_v conmmensurable_a namely_o that_o they_o shall_v as_o the_o first_o bring_v forth_o binomial_a line_n of_o diverse_a order_n now_o forasmuch_o as_o the_o power_n of_o the_o line_n agnostus_n and_o gb_o and_o de_fw-fr and_o ez_o be_v commensurable_a &_o proportional_a it_o be_v manifest_a that_o if_o the_o line_n agnostus_n be_v in_o power_n more_o than_o the_o line_n gb_o by_o the_o square_n of_o a_o line_n commensurable_a in_o length_n unto_o agnostus_n the_o line_n de_fw-fr also_o shall_v be_v in_o power_n more_o than_o the_o line_n ez_o by_o the_o square_n of_o a_o line_n commensurable_a in_o length_n unto_o the_o line_n de_fw-fr by_o the_o 16._o of_o this_o book_n and_o so_o shall_v the_o two_o line_n ab_fw-la and_o dz_o be_v each_o of_o the_o three_o first_o binomial_a line_n but_o if_o the_o line_n agnostus_n be_v in_o power_n more_o than_o the_o line_n gb_o by_o the_o square_n of_o a_o line_n incommensurable_a in_o length_n unto_o the_o line_n agnostus_n the_o line_n de_fw-fr shall_v also_o be_v in_o powâ—r_a 〈◊〉_d than_o the_o line_n ez_o by_o the_o square_n of_o a_o line_n incomensurable_a in_o length_n unto_o the_o line_n de_fw-fr by_o the_o self_n same_o proposition_n and_o so_o shall_v eke_v of_o the_o line_n ab_fw-la and_o dz_o be_v of_o the_o three_o last_o binomial_a line_n but_o why_o it_o be_v not_o so_o in_o the_o three_o and_o six_o binomial_a line_n the_o reason_n be_v for_o that_o in_o they_o neither_o of_o the_o nameâ—_n be_v commensurable_a in_o length_n to_o the_o rational_a line_n put_v fc_n ¶_o the_o 50._o theorem_a the_o 68_o proposition_n a_o line_n commensurable_a to_o a_o great_a line_n be_v also_o a_o great_a line_n svppose_v that_o the_o line_n ab_fw-la be_v a_o great_a line_n and_o unto_o the_o line_n ab_fw-la let_v the_o line_n cd_o be_v commensurable_a construction_n then_o i_o say_v that_o the_o line_n cd_o also_o be_v a_o great_a line_n divide_v the_o line_n ab_fw-la into_o his_o part_n in_o the_o point_n e._n wherefore_o by_o the_o 39_o of_o the_o ten_o the_o line_n ae_n and_o ebb_n be_v incommensurable_a in_o power_n have_v that_o which_o be_v make_v of_o the_o square_n of_o they_o add_v together_o rational_a and_o that_o which_o be_v contain_v under_o they_o medial_a and_o let_v the_o rest_n of_o the_o construction_n be_v in_o this_o as_o it_o be_v in_o the_o former_a and_o for_o that_o as_o the_o line_n ab_fw-la be_v to_o the_o line_n cd_o demonstration_n so_o be_v the_o line_n ae_n to_o the_o line_n cf_o &_o thâ—_z line_n ebb_n to_o the_o line_n fd_o but_o the_o line_n ab_fw-la be_v commensurable_a to_o the_o line_n cd_o by_o supposition_n wherefore_o the_o line_n ae_n be_v commensurable_a to_o the_o line_n cf_o and_o the_o line_n ebb_n to_o the_o line_n fd._n and_o for_o that_o as_o the_o line_n ae_n be_v to_o the_o line_n cf_o so_o be_v the_o line_n ebb_n to_o the_o line_n fd._n therefore_o alternate_o by_o the_o 16._o of_o the_o five_o as_o the_o line_n ae_n be_v to_o the_o line_n ebb_n so_o be_v the_o line_n cf_o to_o the_o line_n fd._n wherefore_o by_o composition_n also_o by_o the_o 18._o of_o the_o five_o as_o the_o line_n ab_fw-la be_v to_o the_o line_n ebb_n so_o be_v the_o line_n cd_o to_o the_o line_n fd._n wherefore_o by_o the_o 22._o of_o the_o six_o as_o the_o square_n of_o the_o line_n ab_fw-la be_v to_o the_o square_n of_o the_o line_n ebb_n so_o be_v the_o square_a of_o the_o line_n cd_o to_o the_o square_n of_o the_o line_n fd._n and_o in_o like_a sort_n may_v we_o prove_v that_o as_o the_o square_n of_o the_o line_n ab_fw-la be_v to_o the_o square_n of_o the_o line_n ae_n so_o be_v the_o square_a of_o the_o line_n cd_o to_o the_o square_n of_o the_o line_n cf._n wherefore_o by_o the_o 11._o of_o the_o five_o as_o the_o square_n of_o the_o line_n ab_fw-la be_v to_o the_o square_n of_o the_o line_n ae_n and_o ebb_n so_o be_v the_o square_a of_o the_o line_n cd_o to_o the_o square_n of_o the_o line_n cf_o and_o fd._n wherefore_o alternate_o by_o the_o 16._o of_o the_o five_o as_o the_o square_n of_o the_o line_n ab_fw-la be_v to_o the_o square_n of_o the_o line_n cd_o so_o be_v the_o square_n of_o the_o line_n ae_n and_o ebb_n to_o the_o square_n of_o the_o line_n cf_o and_o fd._n but_o the_o square_n of_o the_o line_n ab_fw-la be_v commensurable_a to_o the_o square_n of_o the_o line_n cd_o for_o the_o line_n ab_fw-la be_v commensurable_a to_o the_o line_n cd_o by_o supposition_n wherefore_o also_o the_o square_n of_o the_o line_n ae_n and_o ebb_n be_v commensurable_a to_o the_o square_n of_o the_o line_n cf_o and_o fd._n but_o the_o square_n of_o the_o line_n ae_n and_o ebb_n be_v incommensurable_a and_o be_v add_v together_o be_v rational_a wherefore_o the_o square_n of_o the_o line_n cf_o and_o fd_o be_v incommensurable_a &_o be_v add_v together_o be_v also_o rational_a and_o in_o like_a sort_n may_v we_o prove_v that_o that_o which_o be_v contain_v under_o the_o line_n ae_n and_o ebb_n twice_o be_v commensurable_a to_o that_o which_o be_v contain_v under_o the_o line_n cf_o and_o fd_o twice_o but_o that_o which_o be_v contain_v under_o the_o line_n ae_n and_o ebb_n twice_o be_v medial_a wherefore_o also_o that_o which_o be_v contain_v under_o the_o line_n cf_o and_o fd_o twice_o be_v medial_a wherefore_o the_o line_n cf_o and_o fd_o be_v incommensurable_a in_o power_n have_v that_o which_o be_v make_v of_o the_o square_n of_o they_o add_v together_o rational_a and_o that_o which_o be_v contain_v under_o they_o medial_a wherefore_o by_o the_o 39_o of_o the_o ten_o the_o whole_a line_n cd_o be_v irrational_a &_o be_v call_v a_o great_a line_n a_o line_n therefore_o commensurable_a to_o a_o great_a line_n be_v also_o a_o great_a line_n an_o other_o demonstration_n of_o peter_n montaureus_n to_o prove_v the_o same_o suppose_v that_o the_o line_n ab_fw-la be_v a_o great_a line_n and_o unto_o it_o let_v the_o line_n cd_o be_v commensurable_a any_o way_n that_o be_v either_o both_o in_o length_n and_o in_o power_n or_o else_o in_o power_n only_o montaureus_n then_o i_o say_v that_o the_o line_n cd_o also_o be_v a_o great_a
line_n divide_v the_o line_n ab_fw-la into_o his_o part_n in_o the_o point_n e._n and_o let_v the_o rest_n of_o the_o construction_n be_v in_o this_o as_o it_o be_v in_o the_o former_a and_o for_o that_o as_o the_o line_n ab_fw-la be_v to_o the_o line_n cd_o so_o be_v the_o line_n ae_n to_o the_o line_n cf_o and_o the_o line_n ebb_n to_o the_o line_n fd_o therefore_o as_o the_o line_n ae_n be_v to_o the_o line_n cf_o so_o be_v the_o line_n ebb_n to_o the_o line_n fd_o but_o the_o line_n ab_fw-la be_v commensurable_a to_o the_o line_n cd_o wherefore_o also_o the_o line_n ae_n be_v commensurable_a to_o the_o line_n cf_o and_o likewise_o the_o line_n ebb_n to_o the_o line_n fd._n and_o for_o thâ—â—_n as_o the_o line_n ae_n be_v to_o the_o line_n cf_o so_o be_v the_o line_n ebb_n to_o the_o line_n fd_o therefore_o alternate_o as_o the_o line_n ae_n be_v to_o the_o line_n ebb_n so_o be_v the_o line_n cf_o to_o the_o line_n fd._n wherefore_o by_o the_o 22._o of_o the_o six_o as_o the_o square_n of_o the_o line_n ae_n be_v to_o the_o square_n of_o the_o line_n ebb_n so_o be_v the_o square_a of_o the_o line_n cf_o to_o the_o square_n of_o the_o line_n fd._n wherefore_o by_o composition_n by_o the_o 18._o of_o the_o five_o as_o that_o which_o be_v make_v of_o the_o square_n of_o the_o line_n aâ—_z and_o eâ—_n add_v together_o be_v to_o the_o square_n of_o the_o line_n ebb_n so_o be_v that_o which_o be_v make_v of_o the_o squareâ—_n of_o the_o lyneâ—_n câ—_n and_o fd_o add_v together_o to_o the_o square_n of_o the_o line_n fd._n wherefore_o by_o contrary_a proportion_n as_o the_o square_n of_o the_o line_n ebb_n be_v to_o that_o which_o be_v make_v of_o the_o square_n of_o the_o line_n ae_n and_o eâ—_n add_v together_o so_o be_v the_o square_a of_o the_o line_n fd_v to_o that_o which_o be_v make_v of_o the_o square_n of_o the_o line_n cf_o and_o fd_o add_v together_o wherefore_o alternate_o as_o the_o square_n of_o the_o line_n ebb_n be_v to_o the_o square_n of_o the_o line_n fd_o so_o be_v that_o which_o be_v make_v of_o the_o square_n 〈◊〉_d the_o lâ—nes_n ae_n and_o ebb_v add_v together_o to_o that_o which_o be_v make_v of_o the_o square_n of_o the_o line_n cf_o and_o fd_o add_v together_o but_o the_o square_n of_o the_o line_n ebb_n be_v commensurable_a to_o the_o square_n of_o the_o line_n fd_o for_o it_o have_v already_o be_v prove_v that_o the_o line_n ebb_v and_o fd_o be_v commensurable_a wherefore_o that_o which_o be_v make_v of_o the_o square_n of_o the_o line_n ae_n &_o ebb_n add_v together_o be_v commensurable_a to_o that_o which_o be_v make_v of_o the_o square_n of_o câ—_n &_o fd_o add_v together_o but_o that_o which_o be_v make_v of_o the_o square_n of_o the_o line_n ae_n and_o ebb_v add_v together_o be_v rational_a by_o supposition_n wherefore_o that_o which_o be_v make_v of_o the_o square_n of_o the_o line_n cf_o and_o fd_o add_v together_o be_v also_o rational_a and_o as_o the_o line_n ae_n be_v to_o the_o line_n ebb_n so_o be_v the_o line_n cf_o to_o the_o line_n fd_o but_o as_o the_o line_n ae_n be_v to_o the_o line_n ebb_n so_o be_v the_o square_a of_o the_o line_n a_o 〈…〉_o contain_v under_o the_o line_n ae_n and_o ebb_n therefore_o at_o the_o line_n cf_o be_v to_o the_o line_n fd_o so_o be_v the_o square_a of_o the_o line_n ae_n to_o the_o parallelogram_n contain_v under_o the_o line_n ae_n and_o ebb_n &_o as_o the_o line_n cf_o be_v to_o the_o line_n fd_o so_o be_v the_o square_a of_o the_o line_n cf_o to_o the_o parallelogram_n contain_v under_o the_o line_n â—f_a &_o fd._n wherefore_o as_o the_o square_n of_o the_o line_n ae_n be_v to_o the_o parallelogram_n conâ—â—â—â—ed_v under_o the_o line_n ae_n and_o ebb_n so_o be_v the_o square_a of_o the_o line_n cf_o to_o the_o parallelogram_n contain_v under_o the_o line_n cf_o and_o fd._n wherefore_o 〈◊〉_d â—s_v the_o square_a of_o the_o line_n ae_n be_v to_o the_o square_n of_o the_o line_n cf_o so_o be_v the_o parallelogram_n contain_v under_o the_o line_n ae_n and_o ebb_n to_o the_o parallelogram_n 〈◊〉_d undeâ—_n the_o line_n â—â—_o and_o â—â—_o but_o the_o square_n of_o the_o line_n ae_n be_v commensurable_a to_o the_o square_n of_o the_o line_n cf_o for_o it_o be_v already_o prâ—â—â—d_v that_o the_o line_n ae_n and_o cf_o be_v commensurable_a wherefore_o the_o parallelogram_n contain_v under_o the_o line_n ae_n and_o ebb_n be_v commensurable_a to_o the_o parallelogram_n contain_v under_o the_o line_n cf_o and_o fd._n but_o the_o parallelogram_n contain_v under_o the_o line_n ae_n and_o ebb_n be_v medial_a by_o supposition_n wherefore_o the_o parallelogram_n contain_v under_o the_o line_n cf_o and_o â—d_a also_o be_v medial_a and_o as_o it_o have_v already_o be_v prove_v as_o the_o line_n ae_n be_v to_o the_o line_n ebb_n so_o be_v the_o line_n cf_o to_o the_o line_n fd._n but_o the_o line_n ae_n be_v by_o supposition_n incommensurable_a in_o power_n to_o the_o line_n ebb_n wherefore_o by_o the_o 10._o of_o the_o ten_o the_o line_n cf_o be_v incommensurable_a in_o power_n to_o the_o line_n fd._n wherefore_o the_o line_n cf_o and_o fd_o be_v incommensurable_a in_o power_n have_v that_o which_o be_v make_v of_o the_o square_n of_o they_o add_v together_o rational_a and_o that_o which_o be_v contain_v under_o they_o medial_a wherefore_o the_o whole_a line_n cd_o be_v by_o the_o 39_o of_o the_o ten_o a_o great_a line_n wherefore_o a_o line_n commensurable_a to_o a_o great_a line_n be_v also_o a_o great_a line_n which_o be_v require_v to_o be_v demonstrate_v a_o other_o more_o brief_a demonstration_n of_o the_o same_o after_o campane_n suppose_v that_o a_o be_v a_o great_a line_n unto_o which_o let_v the_o line_n b_o be_v commensurable_a either_o in_o length_n and_o power_n or_o in_o power_n only_o campane_n and_o take_v a_o rational_a line_n cd_o and_o upon_o it_o apply_v the_o superficies_n câ—_n equal_a to_o the_o square_n of_o the_o line_n a_o and_o also_o upon_o the_o line_n fe_o which_o be_v equal_a to_o the_o rational_a line_n cd_o apply_v the_o parallelogram_n fg_o equal_a to_o the_o square_n of_o the_o line_n b._n and_o forasmuch_o as_o the_o square_n of_o the_o two_o line_n a_o and_o â—_o be_v commensurable_a by_o supposition_n the_o superficies_n câ—_n shall_v be_v commensurable_a unto_o the_o superficies_n fg_o and_o therefore_o by_o the_o first_o of_o the_o six_o and_o ten_o of_o this_o book_n the_o line_n de_fw-fr be_v commensurable_a in_o length_n to_o the_o line_n gb_o and_o forasmuch_o as_o by_o the_o â—3_o of_o this_o book_n the_o line_n de_fw-fr be_v a_o four_o binomial_a line_n therefore_o by_o the_o â—6_o of_o this_o book_n the_o line_n ge_z be_v also_o a_o four_o binomial_a line_n wherefore_o by_o the_o 57_o of_o this_o book_n the_o line_n b_o which_o contain_v in_o power_n the_o superficies_n fg_o be_v a_o great_a line_n ¶_o the_o 51._o theorem_a the_o 69._o proposition_n a_o line_n commensurable_a to_o a_o line_n contain_v in_o power_n a_o rational_a and_o a_o medial_a be_v also_o a_o line_n contain_v in_o power_n a_o rational_a and_o a_o medial_a svppose_v that_o ab_fw-la be_v a_o line_n contain_v in_o power_n a_o rational_a and_o a_o medial_a and_o unto_o the_o line_n ab_fw-la let_v the_o line_n cd_o be_v commensurable_a whether_o in_o length_n and_o power_n or_o in_o power_n only_o then_o i_o say_v that_o the_o line_n cd_o be_v a_o line_n contain_v in_o power_n a_o rational_a &_o a_o medial_a duide_v the_o line_n ab_fw-la into_o his_o part_n in_o the_o point_n e._n construction_n wherefore_o by_o the_o 40._o of_o the_o ten_o the_o line_n ae_n and_o ebb_n be_v incommensurable_a in_o power_n have_v that_o which_o be_v make_v of_o the_o square_n of_o they_o add_v together_o medial_a and_o that_o which_o be_v contain_v under_o they_o national_a let_v the_o same_o construction_n be_v in_o this_o that_o be_v in_o the_o former_a demonstration_n and_o in_o like_a sort_n we_o may_v prove_v that_o the_o line_n cf_o and_o fd_o be_v incommensurable_a in_o power_n and_o that_o that_o which_o be_v make_v of_o the_o square_n of_o the_o line_n ae_n and_o ebb_n be_v commensurable_a to_o that_o which_o be_v make_v of_o the_o square_n of_o the_o line_n cf_o and_o fd_o and_o that_o that_o also_o which_o be_v contain_v under_o the_o line_n ae_n and_o ebb_n be_v commensurable_a to_o that_o which_o be_v contain_v under_o the_o line_n cf_o and_o fd._n wherefore_o that_o which_o be_v make_v of_o the_o square_n of_o the_o line_n cf_o and_o fd_o be_v medial_a and_o that_o which_o be_v contain_v under_o the_o line_n cf_o and_o fd_o be_v rational_a wherefore_o the_o whole_a line_n cd_o be_v a_o line_n contain_v in_o
power_n a_o rational_a and_o a_o medial_a which_o be_v require_v to_o be_v demonstrate_v an_o other_o demonstration_n of_o the_o same_o after_o campane_n suppose_v that_o ab_fw-la be_v a_o line_n contain_v in_o power_n a_o rational_a and_o a_o medial_a campane_n whereunto_o let_v the_o line_n gd_v be_v commensurable_a either_o in_o length_n and_o power_n or_o in_o power_n only_o then_o i_o say_v that_o the_o line_n gd_o be_v a_o line_n contain_v in_o power_n a_o rational_a and_o a_o medial_a take_v a_o rational_a line_n ez_o upon_o which_o by_o the_o 45._o of_o the_o first_o apply_v a_o rectangle_n parallelogram_n ezfc_n equal_a to_o the_o square_n of_o the_o line_n ab_fw-la and_o upon_o the_o line_n cf_o which_o be_v equal_a to_o the_o line_n ez_o apply_v the_o parallelogram_n fchi_n equal_a to_o the_o square_n of_o the_o line_n gdâ—_n and_o let_v the_o breadth_n of_o the_o say_a parallelogram_n be_v the_o line_n eglantine_n and_o ch._n and_o forasmuch_o as_o the_o line_n ab_fw-la be_v commensurable_a to_o the_o line_n gd_o at_o the_o least_o in_o power_n only_o therefore_o the_o parallelogram_n of_o and_o fh_o which_o be_v equal_a to_o their_o square_n shall_v be_v commensurable_a wherefore_o by_o the_o 1._o of_o the_o six_o the_o right_a line_n ec_o and_o change_z be_v commensurable_a in_o length_n and_o forasmuch_o as_o the_o parallelogram_n of_o which_o be_v equal_a to_o the_o square_n of_o the_o line_n aâ—_z which_o contain_v in_o power_n â—_o rational_a and_o a_o medial_a be_v apply_v upon_o the_o rational_a ez_o make_v in_o breadth_n the_o line_n ec_o therefore_o the_o line_n ec_o be_v a_o five_o binomial_a line_n by_o the_o 64._o of_o this_o book_n unto_o which_o line_n ec_o the_o line_n change_z be_v commensurable_a in_o length_n wherefore_o by_o the_o 66._o of_o this_o book_n the_o line_n change_z be_v also_o a_o five_o binomial_a line_n and_o forasmuch_o as_o the_o superficies_n ci_o be_v contain_v under_o the_o rational_a line_n ez_o that_o be_v cf_o and_o a_o five_o binomall_a line_n change_z therefore_o the_o line_n which_o contain_v in_o power_n the_o superficies_n ci_o which_o by_o supposition_n be_v the_o line_n gd_o be_v a_o line_n contain_v in_o power_n a_o rational_a and_o a_o medial_a by_o the_o 58._o of_o this_o book_n a_o line_n therefore_o commensurable_a to_o a_o line_n contain_v in_o power_n a_o rational_a and_o a_o medial_a etc_n etc_n ¶_o the_o 52._o theorem_a the_o 70._o proposition_n a_o line_n commensurable_a to_o a_o line_n contain_v in_o power_n two_o medial_n be_v also_o a_o line_n contain_v in_o power_n two_o medial_n svppose_v that_o ab_fw-la be_v a_o line_n contain_v in_o power_n two_o medial_n and_o unto_o the_o line_n ab_fw-la let_v the_o line_n cd_o be_v commensurable_a whether_o in_o length_n &_o power_n or_o in_o power_n only_o then_o i_o say_v that_o the_o line_n cd_o be_v a_o line_n contain_v in_o power_n two_o medial_n forasmuch_o as_o the_o line_n ab_fw-la be_v a_o line_n contain_v in_o power_n two_o medial_n construction_n let_v it_o be_v divide_v into_o his_o part_n in_o the_o point_n e._n wherefore_o by_o the_o 41._o of_o the_o ten_o the_o line_n ae_n and_o ebb_n be_v incommensurable_a in_o power_n have_v that_o which_o be_v make_v of_o the_o square_n of_o they_o add_v together_o medial_a and_o that_o also_o which_o be_v contain_v under_o they_o medial_a and_o that_o which_o be_v make_v of_o the_o square_n of_o the_o line_n ae_n &_o ebb_n be_v incommensurable_a to_o that_o which_o be_v contain_v under_o the_o line_n ae_n and_o ebb_n let_v the_o self_n same_o construction_n be_v in_o this_o that_o be_v in_o the_o former_a demonstration_n and_o in_o like_a sort_n may_v we_o prove_v that_o the_o line_n cf_o &_o fd_o be_v incommensurable_a in_o power_n and_o that_o that_o which_o be_v make_v of_o the_o square_n of_o the_o line_n ae_n and_o ebb_v add_v together_o be_v commensurable_a to_o that_o which_o be_v make_v of_o the_o square_n of_o the_o line_n cf_o and_o fd_o add_v together_o and_o that_o that_o also_o which_o be_v contain_v under_o the_o line_n ae_n and_o ebb_n be_v commensurable_a to_o that_o which_o be_v contain_v under_o the_o line_n cf_o and_o fd._n wherefore_o that_o which_o be_v make_v of_o the_o square_n of_o the_o line_n cf_o and_o fd_o be_v medial_a by_o the_o corollary_n of_o the_o 23._o of_o the_o ten_o and_o that_o which_o be_v contain_v under_o the_o line_n cf_o and_o fd_o be_v medial_a by_o the_o same_o corollary_n â—_o and_o moreover_o that_o which_o be_v make_v of_o the_o square_n of_o the_o line_n cf_o &_o fd_o be_v incommensurable_a to_o that_o which_o be_v contain_v under_o the_o line_n cf_o and_o fd._n wherefore_o the_o line_n cd_o be_v a_o line_n contain_v in_o power_n two_o medial_n which_o be_v require_v to_o be_v prove_v ¶_o a_o assumpt_n add_v by_o montaureus_n assumpt_n that_o that_o which_o be_v make_v of_o the_o square_n of_o the_o line_n cf_o and_o fd_o add_v together_o be_v incommensurable_a to_o that_o which_o be_v contain_v under_o the_o line_n cf_o and_o fd_o be_v thus_o prove_v for_o because_o as_o that_o which_o be_v make_v of_o the_o square_n of_o the_o line_n ae_n and_o ebb_v add_v together_o be_v to_o the_o square_n of_o the_o line_n ae_n so_o be_v that_o which_o be_v make_v of_o the_o square_n of_o the_o line_n cf_o and_o fd_o add_v together_o to_o the_o square_n of_o the_o line_n cf_o as_o it_o be_v prove_v in_o the_o proposition_n go_v before_o therefore_o alternate_o as_o that_o which_o be_v make_v of_o the_o square_n of_o ae_n and_o ebb_v add_v together_o be_v to_o that_o which_o be_v make_v of_o the_o square_n of_o cf_o and_o fd_o add_v together_o so_o be_v the_o square_a of_o the_o line_n ae_n to_o the_o square_n of_o the_o line_n cf._n but_o before_o namely_o in_o the_o 68_o proposition_n it_o be_v prove_v that_o as_o the_o square_n of_o the_o line_n ae_n be_v to_o the_o square_n of_o the_o line_n cf_o so_o be_v the_o parallelogram_n contain_v under_o the_o line_n ae_n and_o ebb_n to_o the_o parallelogram_n contain_v under_o the_o line_n cf_o and_o fd._n wherefore_o as_o that_o which_o be_v make_v of_o the_o square_n of_o the_o line_n ae_n and_o ebb_n be_v to_o that_o which_o be_v make_v of_o the_o square_n of_o the_o line_n cf_o and_o fd_o so_o be_v the_o parallelogram_n contain_v under_o the_o line_n ae_n and_o ebb_n to_o the_o parallelogram_n contain_v under_o the_o line_n cf_o and_o fd._n wherefore_o alternate_o as_o that_o which_o be_v make_v of_o the_o square_n of_o the_o line_n ae_n and_o ebb_n be_v to_o the_o parallelogram_n contain_v under_o the_o line_n ae_n and_o ebb_n so_o be_v that_o which_o be_v make_v of_o the_o square_n of_o the_o line_n cf_o and_o fd_o to_o the_o parallelogram_n contain_v under_o the_o line_n cf_o and_o fd._n but_o by_o supposition_n that_o which_o be_v make_v of_o the_o square_n of_o the_o line_n ae_n and_o ebb_n be_v incommensurable_a to_o the_o parallelogram_n contain_v under_o the_o line_n ae_n &_o ebb_n wherefore_o that_o which_o be_v make_v of_o the_o square_n of_o the_o line_n cf_o and_o fd_o add_v together_o be_v incommensurable_a to_o the_o parallelogram_n contain_v under_o the_o line_n cf_o and_o fd_o which_o be_v require_v to_o be_v prove_v an_o other_o demonstration_n after_o campane_n suppose_v that_o ab_fw-la be_v a_o line_n contain_v in_o power_n two_o medial_n whereunto_o let_v the_o line_n gd_v be_v commensurable_a either_o in_o length_n and_o in_o power_n or_o in_o power_n only_o campanâ—_n then_o i_o say_v that_o the_o line_n gd_o be_v a_o line_n contain_v in_o power_n two_o medial_n let_v the_o same_o construction_n be_v in_o this_o that_o be_v in_o the_o former_a and_o forasmuch_o as_o the_o parallelogram_n of_o be_v equal_a to_o the_o square_n of_o the_o line_n ab_fw-la and_o be_v apply_v upon_o a_o rational_a line_n ez_o it_o make_v the_o breadth_n ec_o a_o six_o binomial_a line_n by_o the_o 65._o of_o this_o book_n and_o forasmuch_o as_o the_o parallelogram_n of_o &_o ci_o which_o be_v equal_a unto_o the_o square_n of_o the_o line_n ab_fw-la and_o gd_o which_o be_v suppose_v to_o be_v commensurable_a be_v commensurable_a therefore_o the_o line_n ec_o and_o change_z be_v commensurable_a in_o length_n by_o the_o first_o of_o the_o six_o but_o ec_o be_v a_o six_o binomial_a line_n wherefore_o change_z also_o be_v a_o six_o binomial_a line_n by_o the_o 66._o of_o this_o book_n and_o forasmuch_o as_o the_o superficies_n ci_o be_v contain_v under_o the_o rational_a line_n cf_o and_o a_o six_o binomial_a line_n change_z therefore_o the_o line_n which_o contain_v in_o power_n the_o superficies_n ci_o namely_o the_o line_n gd_o be_v a_o line_n contain_v in_o power_n two_o medial_n by_o the_o 59_o of_o
this_o book_n wherefore_o a_o line_n commensurable_a to_o a_o line_n contain_v in_o power_n two_o medial_n etc_n etc_n a_o annotation_n if_o other_o to_o have_v be_v speak_v of_o six_o senary_n of_o which_o the_o first_o senary_n contain_v the_o production_n of_o irrational_a line_n by_o composition_n the_o second_o the_o division_n of_o they_o namely_o that_o those_o line_n be_v in_o one_o poinâ—_n only_a devideâ—_n the_o three_o the_o find_v out_o of_o binomial_a line_n of_o the_o first_o i_o say_v the_o second_o the_o three_o the_o four_o the_o five_o and_o the_o six_o after_o that_o begin_v the_o forth_o senary_a contain_v the_o difference_n of_o irrational_a line_n between_o themselves_o for_o by_o the_o nature_n of_o every_o one_o of_o the_o binomial_a line_n be_v demonstrate_v the_o difference_n of_o irrational_a line_n the_o five_o entreâ—teth_v of_o the_o application_n of_o the_o square_n of_o every_o irrational_a line_n namely_o what_o irrational_a line_n be_v the_o breadthes_fw-mi of_o every_o superficies_n so_o apply_v in_o the_o six_o senary_a be_v prove_v that_o any_o line_n commensurable_a to_o any_o irrational_a line_n be_v also_o a_o irrational_a line_n of_o the_o same_o nature_n and_o now_o shall_v be_v speak_v of_o the_o seven_o senary_a wherein_o again_o be_v plain_o set_v forth_o the_o rest_n of_o the_o difference_n of_o the_o say_a line_n between_o themselves_o and_o theâ—e_n be_v even_o in_o those_o irrational_a line_n a_o arithmetical_a proportionality_n note_n and_o that_o line_n which_o be_v the_o arithmetical_a mean_a proportional_a between_o the_o part_n of_o any_o irrational_a line_n be_v also_o a_o irrational_a line_n of_o the_o self_n same_o kind_n first_o it_o be_v certain_a that_o there_o be_v a_o arithmetical_a proportion_n between_o those_o part_n for_o suppose_v that_o the_o line_n ab_fw-la be_v any_o of_o the_o foresay_a irrational_a line_n as_o for_o example_n let_v it_o be_v a_o binomial_a line_n &_o let_v it_o be_v divide_v into_o his_o name_n in_o the_o point_n c._n and_o let_v ac_fw-la be_v the_o great_a name_n from_o which_o take_v away_o the_o line_n ad_fw-la equal_a to_o the_o less_o name_n namely_o to_o cb._n and_o divide_v the_o line_n cd_o into_o two_o equal_a part_n in_o the_o point_n e._n it_o be_v manifest_a that_o the_o line_n ae_n be_v equal_a to_o the_o line_n ebb_n let_v the_o line_n fg_o be_v equal_a to_o either_o of_o they_o it_o be_v plain_a that_o how_o much_o the_o line_n ac_fw-la differ_v from_o the_o line_n fg_o so_o much_o the_o same_o line_n fg_o differ_v from_o the_o line_n cb_o for_o in_o each_o be_v the_o difference_n of_o the_o line_n de_fw-fr or_o ec_o which_o be_v the_o property_n of_o arithmetical_a proportionality_n and_o it_o be_v manifest_a that_o the_o line_n fg_o be_v commensurable_a in_o length_n to_o the_o line_n ab_fw-la for_o it_o be_v the_o half_a thereof_o wherefore_o by_o the_o 66._o of_o the_o ten_o the_o line_n fg_o be_v a_o binomial_a line_n and_o after_o the_o self_n same_o manner_n may_v it_o be_v prove_v touch_v the_o rest_n of_o the_o irrational_a line_n ¶_o the_o 53._o theorem_a the_o 71._o proposition_n if_o two_o superficiece_n namely_o a_o rational_a and_o a_o medial_a superficies_n be_v compose_v together_o the_o line_n which_o contain_v in_o power_n the_o whole_a superficies_n be_v one_o of_o these_o four_o irrational_a line_n either_o a_o binomial_a line_n or_o a_o first_o bimedial_a line_n or_o a_o great_a line_n or_o a_o line_n contain_v in_o power_n a_o rational_a and_o a_o medial_a superficies_n but_o now_o let_v the_o line_n eh_o be_v in_o power_n more_o than_o the_o line_n hk_o by_o the_o square_n of_o a_o line_n incommensurable_a in_o length_n to_o the_o line_n eh_o case_n now_o the_o great_a name_n that_o be_v eh_o be_v commensurable_a in_o length_n to_o the_o rational_a line_n give_v ef._n wherefore_o the_o line_n eke_o be_v afourth_o binomial_a line_n and_o the_o line_n of_o be_v rational_a but_o if_o a_o superficies_n be_v contain_v under_o a_o rational_a line_n and_o afourth_o binomial_a line_n the_o line_n that_o contain_v in_o power_n the_o same_o superficies_n be_v by_o the_o 57_o of_o the_o ten_o irrational_a and_o be_v a_o great_a line_n wherefore_o the_o line_n which_o contain_v in_o power_n the_o parallelogram_n ei_o be_v a_o great_a line_n wherefore_o also_o the_o line_n contain_v in_o power_n the_o superficies_n ad_fw-la be_v a_o great_a line_n case_n but_o now_o suppose_v that_o the_o superficies_n ab_fw-la which_o be_v rational_a be_v less_o than_o the_o superficies_n cd_o which_o be_v medial_a wherefore_o also_o the_o parallelogram_n eglantine_n be_v less_o than_o the_o parallelogram_n high_a wherefore_o also_o the_o line_n eh_o be_v less_o than_o the_o line_n hk_o now_o the_o line_n hk_o be_v in_o power_n more_o than_o the_o line_n eh_o either_o by_o the_o square_n of_o a_o line_n commensurable_a in_o length_n to_o the_o line_n hk_o or_o by_o the_o square_n of_o a_o line_n incommensurable_a in_o length_n unto_o the_o line_n hk_o first_o let_v it_o be_v in_o power_n more_o by_o the_o square_n of_o a_o line_n commensurable_a in_o length_n unto_o hk_n case_n now_o the_o less_o name_n that_o be_v eh_o be_v commensurable_a in_o length_n to_o the_o rational_a line_n give_v of_o as_o it_o be_v before_o prove_v wherefore_o the_o whole_a line_n eke_o be_v a_o second_o binomial_a line_n and_o the_o line_n of_o be_v a_o rational_a line_n but_o if_o a_o superficies_n be_v contain_v under_o a_o rational_a line_n and_o a_o second_o binomial_a line_n the_o line_n that_o contain_v in_o power_n the_o same_o superficies_n be_v by_o the_o 55._o of_o the_o ten_o a_o first_o bimedial_a line_n wherefore_o the_o line_n which_o contain_v in_o power_n the_o parallelogram_n ei_o be_v a_o first_o bimedial_a line_n wherefore_o also_o the_o line_n that_o contain_v in_o power_n the_o superficies_n ad_fw-la be_v a_o first_o bimedial_a line_n case_n but_o now_o let_v the_o line_n hk_o be_v in_o power_n more_o than_o the_o line_n eh_o by_o the_o square_n of_o a_o line_n incommensurable_a in_o length_n to_o the_o line_n hk_o now_o the_o less_o name_n that_o be_v eh_o be_v commensurable_a in_o length_n to_o the_o rational_a line_n give_v ef._n wherefore_o the_o whole_a line_n eke_o be_v a_o five_o binomial_a line_n and_o the_o line_n of_o be_v rational_a but_o if_o a_o superficies_n be_v contain_v under_o a_o rational_a line_n and_o a_o five_o binomial_a line_n the_o line_n that_o contain_v in_o power_n the_o same_o superficies_n be_v by_o the_o 58._o of_o the_o ten_o a_o line_n contain_v in_o power_n a_o rational_a and_o a_o medial_a wherefore_o the_o line_n that_o contain_v in_o power_n the_o parallelogram_n ei_o be_v a_o line_n contain_v in_o power_n a_o rational_a and_o a_o medial_a wherefore_o also_o the_o line_n that_o contain_v in_o power_n the_o superficies_n ad_fw-la be_v a_o line_n contain_v in_o power_n a_o rational_a and_o a_o medial_a if_o therefore_o a_o rational_a and_o a_o medial_a superficies_n be_v add_v together_o the_o line_n which_o contain_v in_o power_n the_o whole_a superficies_n be_v one_o of_o these_o four_o irrational_a line_n namely_o either_o a_o binomial_a line_n or_o a_o first_o bimedial_a line_n or_o a_o great_a line_n or_o a_o line_n contain_v in_o power_n a_o rational_a and_o a_o medial_a which_o be_v require_v to_o be_v demonstrate_v ¶_o the_o 54._o theorem_a the_o 72._o proposition_n if_o two_o medial_a superficiece_n incommensurable_a the_o one_o to_o the_o other_o be_v compose_v together_o the_o line_n contain_v in_o power_n the_o whole_a superficies_n be_v one_o of_o the_o two_o irrational_a line_n remain_v namely_o either_o a_o second_o bimedial_a line_n or_o a_o line_n contain_v in_o power_n two_o medial_n let_v these_o two_o medial_a superficiece_n ab_fw-la and_o cd_o be_v incommensurable_a the_o one_o to_o the_o other_o be_v add_v together_o then_o i_o say_v that_o the_o line_n which_o contain_v in_o power_n the_o superficies_n ad_fw-la be_v either_o a_o second_o bimedial_a line_n or_o a_o line_n contain_v in_o power_n two_o medial_n construction_n for_o the_o superficies_n ab_fw-la be_v either_o great_a or_o less_o than_o the_o superficies_n cd_o for_o they_o can_v by_o no_o mean_n be_v equal_a when_o as_o they_o be_v incommensurable_a case_n first_o let_v the_o superficies_n ab_fw-la be_v great_a than_o the_o superficies_n cd_o and_o take_v a_o rational_a line_n ef._n and_o by_o the_o 44._o of_o the_o first_o unto_o the_o line_n of_o apply_v the_o parallelogram_n eglantine_n equal_a to_o the_o superficies_n ab_fw-la and_o make_v in_o breadth_n the_o line_n eh_o and_o unto_o the_o same_o line_n of_o that_o be_v to_o the_o line_n hg_o apply_v the_o parallelogram_n high_a equal_a to_o the_o superficies_n cd_o &_o make_v in_o breadth_n the_o line_n hk_o and_o forasmuch_o as_o
that_o which_o be_v contain_v under_o the_o line_n ad_fw-la and_o db_o twice_o exceed_v that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o twice_o by_o that_o which_o be_v speak_v in_o the_o 79._o proposition_n but_o that_o which_o be_v contain_v under_o the_o line_n ad_fw-la and_o db_o twice_o exceed_v that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o twice_o by_o a_o rational_a superficies_n for_o they_o be_v either_o of_o they_o rational_a by_o supposition_n wherefore_o that_o which_o be_v make_v of_o the_o square_n of_o the_o line_n ad_fw-la and_o db_o add_v together_o exceed_v that_o which_o be_v make_v of_o the_o square_n of_o the_o line_n ac_fw-la and_o cb_o add_v together_o by_o a_o rational_a superficies_n which_o by_o the_o 26._o of_o the_o ten_o be_v impossible_a for_o they_o be_v either_o of_o they_o medial_a by_o supposition_n wherefore_o unto_o the_o line_n ab_fw-la can_v be_v join_v any_o other_o line_n beside_o bc_o make_v that_o which_o be_v require_v in_o the_o proposition_n wherefore_o unto_o a_o line_n make_v with_o a_o rational_a superficies_n the_o whole_a superficies_n medial_a can_v be_v join_v only_a one_o right_a line_n incommensurable_a in_o power_n to_o the_o whole_a line_n and_o make_v together_o with_o the_o whole_a line_n that_o which_o be_v make_v of_o their_o squaâ—es_n add_v together_o medial_a and_o that_o which_o be_v contain_v under_o they_o rational_a which_o be_v require_v to_o be_v prove_v ¶_o the_o 66._o theorem_a the_o 84._o proposition_n unto_o a_o line_n make_v with_o a_o medial_a superficies_n the_o whole_a superficies_n medial_a can_v be_v join_v only_a one_o right_a line_n incommensurable_a in_o power_n to_o the_o whole_a line_n and_o make_v together_o with_o the_o whole_a line_n that_o which_o be_v make_v of_o their_o square_n add_v together_o medial_a and_o that_o which_o be_v contain_v under_o they_o medial_a and_o moreover_o make_v that_o which_o be_v make_v of_o the_o square_n of_o they_o add_v together_o incommensurable_a to_o that_o which_o be_v contain_v under_o they_o svppose_v that_o ab_fw-la be_v a_o line_n make_v with_o a_o medial_a superficies_n the_o whole_a superficies_n medial_a and_o unto_o it_o let_v the_o line_n bc_o be_v join_v so_o that_o let_v bc_o be_v such_o a_o line_n as_o be_v require_v in_o the_o theorem_a wherefore_o the_o line_n ac_fw-la and_o cb_o be_v incommensurable_a in_o power_n have_v that_o which_o be_v make_v of_o the_o square_n of_o the_o line_n ac_fw-la and_o cb_o add_v together_o medial_a &_o that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o medial_a and_o moreover_o that_o which_o be_v make_v of_o the_o square_n of_o the_o line_n ac_fw-la and_o cb_o be_v incommensurable_a so_o that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb._n then_o i_o say_v that_o unto_o the_o line_n ab_fw-la can_v be_v join_v no_o other_o such_o line_n for_o if_o it_o be_v possible_a let_v bd_o be_v such_o a_o line_n wherefore_o the_o line_n ad_fw-la and_o db_o be_v incommensurable_a in_o power_n have_v that_o which_o be_v make_v of_o the_o square_n of_o the_o line_n ad_fw-la and_o db_o add_v together_o medial_a and_o that_o which_o be_v contain_v under_o the_o line_n ad_fw-la and_o db_o medial_a and_o moreover_o that_o which_o be_v make_v of_o the_o square_n of_o the_o line_n ad_fw-la and_o and_o db_o add_v together_o be_v incommensurable_a to_o that_o which_o be_v contain_v under_o the_o line_n ad_fw-la and_o db._n take_v a_o rational_a line_n ef._n construction_n and_o by_o the_o 44._o of_o the_o first_o unto_o the_o line_n of_o apply_v the_o parallelogram_n eglantine_n equal_a to_o the_o square_n of_o the_o line_n ac_fw-la and_o cb_o and_o make_v in_o breadth_n the_o line_n emâ—_n and_o from_o the_o parallelogram_n eglantine_n take_v away_o the_o parallelogram_n hg_o equal_a to_o that_o which_o be_v contain_v under_o the_o line_n ac_fw-la &_o cb_o twice_o and_o make_v in_o breadth_n the_o line_n he_o wherefore_o the_o residue_n namely_o the_o square_a of_o the_o line_n ab_fw-la be_v equal_a to_o the_o parallelogram_n el_fw-es by_o the_o 7._o of_o the_o second_o wherefore_o the_o line_n ab_fw-la contain_v in_o power_n the_o parallelogram_n el._n again_o by_o the_o 44._o of_o the_o first_o unto_o the_o line_n of_o apply_v the_o parallelogram_n eâ—_n equal_a to_o the_o square_n of_o the_o line_n ad_fw-la and_o db_o and_o make_v in_o breadth_n the_o line_n en_fw-fr but_o the_o square_n of_o the_o line_n ab_fw-la be_v equal_a to_o the_o parallelogram_n el._n wherefore_o the_o residue_n namely_o the_o parallelogram_n high_a be_v equal_a to_o that_o which_o be_v contain_v under_o the_o line_n ad_fw-la and_o db_o twice_o abjurdâ—tâ—â—â—_n and_o forasmuch_o as_o that_o which_o be_v make_v of_o the_o square_n of_o the_o line_n ac_fw-la and_o cb_o be_v medial_a and_o be_v equal_a to_o the_o parallelogram_n eglantine_n therefore_o also_o the_o parallelogram_n eglantine_n be_v medial_a and_o it_o be_v apply_v unto_o the_o rational_a line_n of_o make_v in_o breadth_n the_o line_n em_n wherefore_o by_o the_o 22._o of_o the_o ten_o the_o line_n em_n be_v rational_a and_o incommensurable_a in_o length_n to_o the_o line_n ef._n again_o forasmuch_o as_o that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o twice_o be_v medial_a and_o be_v equal_a to_o the_o parallelogram_n hg_o wherefore_o the_o parallelogram_n hâ—g_n be_v medial_a which_o parallelogram_n hg_o be_v apply_v to_o the_o rational_a line_n of_o make_v in_o breadth_n the_o line_n he_o wherefore_o the_o line_n he_o be_v rational_a and_o incommensurable_a in_o length_n to_o the_o line_n ef._n and_o forasmuch_o as_o the_o square_n of_o the_o line_n ac_fw-la and_o cb_o be_v incommensurable_a to_o that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o twice_o therefore_o the_o parallelogram_n eglantine_n be_v incommensurable_a to_o the_o parallelogram_n hg_o wherefore_o the_o line_n em_n be_v incommensurable_a in_o length_n to_o the_o line_n mh_o and_o they_o be_v both_o rational_a wherefore_o the_o line_n em_n and_o mh_o be_v rational_a commensurable_a in_o power_n only_o wherefore_o eh_o be_v a_o residual_a line_n and_o the_o line_n join_v unto_o it_o be_v he_o and_o in_o like_a sort_n may_v we_o prove_v that_o the_o line_n eh_o be_v a_o residual_a line_n and_o that_o the_o line_n hn_o be_v join_v unto_o it_o wherefore_o unto_o a_o residual_a line_n be_v join_v two_o sundry_a line_n be_v each_o commensurable_a in_o power_n only_o so_o the_o whole_a lineâ—_z which_o by_o the_o 79._o of_o the_o ten_o be_v impossible_a wherefore_o unto_o the_o line_n ab_fw-la can_v not_o be_v join_v any_o other_o right_a line_n beside_o the_o line_n bc_o which_o shall_v be_v incommensurable_a in_o power_n to_o the_o whole_a line_n &_o have_v together_o with_o the_o whole_a line_n that_o which_o be_v make_v of_o their_o square_n add_v together_o medial_a and_o that_o which_o be_v contain_v under_o they_o medial_a and_o moreover_o incommensurable_a to_o that_o which_o be_v make_v of_o their_o square_n add_v together_o wherefore_o unto_o a_o line_n make_v with_o a_o medial_a superficies_n the_o whole_a superficies_n medial_a can_v be_v join_v only_a one_o right_a line_n incommensurable_a in_o power_n to_o the_o whole_a line_n and_o make_v together_o with_o the_o whole_a line_n that_o which_o be_v make_v of_o their_o square_n add_v together_o medial_a and_o that_o which_o be_v contain_v under_o they_o medial_a and_o moreover_o make_v that_o which_o be_v make_v of_o the_o square_n of_o they_o add_v together_o incommensurable_a to_o that_o which_o be_v contain_v under_o they_o which_o be_v require_v to_o be_v prove_v ¶_o three_o definition_n as_o of_o binomial_a line_n there_o be_v 6._o diverse_a kind_n so_o also_o of_o residual_a line_n which_o be_v correspondent_a unto_o they_o and_o depend_v of_o they_o for_o a_o residual_a line_n be_v nothing_o else_o as_o be_v before_o say_v but_o that_o which_o remain_v when_o the_o less_o part_n of_o a_o binomial_a line_n be_v take_v from_o the_o great_a part_n or_o name_n thereof_o there_o be_v likewise_o six_o several_a kind_n all_o which_o be_v know_v and_o consider_v in_o comparison_n to_o a_o rational_a line_n set_v forth_o &_o appoint_v and_o these_o residual_a line_n have_v the_o self_n same_o order_n of_o production_n that_o the_o binomial_n have_v line_n for_o as_o the_o three_o first_o kind_n of_o binomial_a line_n namely_o the_o first_o second_o and_o three_o be_v produce_v when_o the_o square_a of_o the_o great_a part_n of_o the_o part_n of_o the_o binomiall_n exceed_v the_o square_a of_o the_o less_o part_n thereof_o by_o the_o square_n of_o a_o line_n commensurable_a unto_o it_o in_o length_n so_o in_o likewise_o the_o first_o three_o kind_n of_o residual_a line_n namely_o the_o first_o second_o and_o
again_o unto_o the_o line_n fe_o apply_v the_o parallelogram_n fg_o equal_a to_o the_o square_n of_o the_o line_n b_o and_o make_v in_o breadth_n the_o line_n fh_o now_o forasmuch_o as_o the_o line_n a_o be_v commensurable_a to_o the_o line_n b_o demonstration_n therefore_o the_o square_a of_o the_o line_n a_o be_v commensurable_a to_o the_o square_n of_o the_o line_n b._n but_o unto_o the_o square_n of_o the_o line_n a_o be_v equal_a the_o parallelogram_n ce_fw-fr and_o unto_o the_o square_n of_o the_o line_n b_o be_v equal_a the_o parallelogram_n fg._n wherefore_o the_o parallelogram_n ce_fw-fr be_v commensurable_a to_o the_o parallelogram_n fg._n wherefore_o the_o line_n cf_o be_v also_o commensurable_a in_o length_n to_o the_o line_n fh_o but_o the_o line_n cf_o be_v a_o five_o residual_a line_n wherefore_o also_o the_o line_n fh_o be_v a_o five_o residual_a line_n and_o the_o line_n fe_o be_v rational_a but_o if_o a_o supersicy_n be_v contain_v under_o a_o rational_a line_n and_o a_o â—ift_n residual_a line_n the_o line_n that_o contain_v in_o power_n that_o superficies_n be_v by_o the_o 95._o of_o the_o ten_o a_o line_n make_v with_o a_o rational_a superficies_n the_o whole_a superficies_n medial_a but_o the_o line_n b_o contain_v in_o power_n the_o parallelogram_n fg._n wherefore_o the_o line_n b_o be_v a_o line_n make_v with_o a_o rational_a superficies_n the_o whole_a superficies_n mediallâ—_n which_o be_v require_v to_o be_v demonstrate_v ¶_o the_o 83._o theorem_a the_o 107._o proposition_n a_o line_n commensurable_a to_o a_o line_n make_v with_o a_o medial_a superficies_n the_o whole_a superficies_n medial_a be_v itself_o also_o a_o line_n make_v with_o a_o medial_a superficies_n the_o whole_a superficies_n medial_a svppose_v that_o ab_fw-la be_v a_o line_n make_v with_o a_o medial_a superficies_n the_o whole_a superficies_n medial_a unto_o who_o let_v the_o line_n cd_o be_v commensurable_a then_o i_o say_v that_o the_o line_n cd_o be_v also_o a_o line_n make_v with_o a_o medial_a superficies_n the_o whole_a superficies_n medial_a for_o unto_o the_o line_n ab_fw-la let_v the_o line_n covenient_o join_v be_v be._n and_o let_v the_o rest_n of_o the_o construction_n be_v in_o this_o as_o it_o be_v in_o the_o former_a proposition_n construction_n wherefore_o the_o line_n ae_n and_o be_v be_v incommensurable_a in_o power_n have_v that_o which_o be_v make_v of_o their_o square_n add_v together_o medial_a demonstration_n and_o that_o which_o be_v contain_v under_o they_o also_o medial_a and_o moreover_o that_o which_o be_v make_v of_o their_o square_n add_v together_o be_v incommensurable_a to_o that_o which_o be_v contain_v under_o they_o but_o the_o line_n ae_n and_o be_v as_o we_o have_v before_o prove_v be_v commensurable_a to_o the_o line_n cf_o &_o df_o and_o that_o which_o be_v make_v of_o the_o square_n of_o the_o line_n ae_n and_o be_v add_v together_o be_v commensurable_a to_o that_o which_o be_v make_v of_o the_o square_n of_o the_o line_n cf_o and_o fd_o add_v together_o and_o the_o parallelogram_n contain_v under_o the_o line_n ae_n and_o be_v be_v commensurable_a to_o the_o parallelogram_n contain_v under_o the_o line_n cf_o and_o df._n wherefore_o the_o line_n cf_o and_o df_o be_v incommensurable_a in_o power_n have_v that_o which_o be_v make_v of_o their_o square_n add_v together_o medial_a and_o that_o which_o be_v contain_v under_o they_o also_o medial_a and_o moreover_o that_o which_o be_v make_v of_o their_o square_n add_v together_o be_v incommensurable_a to_o that_o which_o be_v contain_v under_o they_o wherefore_o the_o line_n cd_o be_v a_o line_n make_v with_o a_o medial_a superficies_n the_o whole_a superficies_n medial_a a_o line_n therefore_o commensurable_a to_o a_o line_n make_v with_o a_o medial_a superficies_n the_o whole_a superficies_n medial_a be_v itself_o also_o a_o line_n make_v with_o a_o medial_a superficies_n the_o whole_a superficies_n medial_a which_o be_v require_v to_o be_v prove_v this_o proposition_n may_v also_o be_v a_o other_o way_n demonstrate_v as_o the_o three_o former_a proposition_n be_v if_o upon_o a_o rational_a line_n you_o apply_v parallelogram_n equal_a to_o the_o square_n of_o the_o line_n ab_fw-la and_o cd_o the_o breadthes_fw-mi of_o which_o parallelogram_n shall_v be_v each_o a_o six_o residual_a line_n and_o therefore_o the_o line_n which_o contain_v they_o in_o power_n namely_o the_o line_n ab_fw-la and_o cd_o shall_v be_v both_o such_o line_n as_o be_v require_v in_o the_o proposition_n which_o be_v easy_a to_o conclude_v mark_v the_o order_n of_o the_o demonstration_n in_o the_o three_o former_a proposition_n ¶_o the_o 84._o theorem_a the_o 108._o proposition_n if_o from_o a_o rational_a superficies_n be_v take_v away_o a_o medialt_n superficies_n the_o line_n which_o contain_v in_o power_n the_o superficies_n remain_v be_v one_o of_o these_o two_o irrational_a line_n namely_o either_o a_o residual_a line_n or_o a_o less_o line_n svppose_v that_o bc_o be_v a_o rational_a superficies_n senary_n and_o from_o it_o take_v away_o a_o medial_a superficies_n namely_o bd._n then_o i_o say_v that_o the_o line_n which_o contain_v in_o power_n the_o superficies_n remain_v namely_o the_o superficies_n ec_o be_v one_o of_o â—hese_fw-mi two_o irrational_a line_n namely_o either_o a_o residual_a line_n or_o a_o less_o line_n take_v a_o rational_a line_n fg._n and_o upon_o fg_o describe_v by_o the_o 44._o of_o the_o first_o a_o rectangle_n parallelogram_n gh_o equal_a to_o the_o superficies_n bc._n constraction_n and_o from_o the_o parallelogram_n gh_o take_v away_o the_o parallelogram_n gk_o equal_a to_o the_o superficies_n bd._n demonstration_n wherefore_o by_o the_o three_o common_a sentence_n the_o superficies_n remain_v namely_o ec_o be_v equal_a to_o the_o parallelogram_n remain_v namely_o to_o lh_o and_o forasmuch_o as_o bc_o be_v rational_a and_o bd_o be_v medial_a and_o bc_o be_v equal_a to_o the_o parallelogram_n gh_o and_o bd_o to_o the_o parallelogram_n gk_o therefore_o gh_o be_v rational_a and_o gk_o be_v medial_a and_o the_o parallelogram_n gh_o be_v apply_v unto_o the_o rational_a line_n fg._n wherefore_o by_o the_o _o of_o the_o ten_o the_o line_n fh_o be_v rational_a and_o commensurable_a in_o length_n to_o the_o line_n fg._n and_o the_o parallelogram_n gk_o be_v also_o apply_v unto_o the_o rational_a line_n fg._n wherefore_o by_o the_o 22._o of_o the_o ten_o the_o line_n fk_o be_v rational_a and_o incommensurable_a in_o length_n to_o the_o line_n fg._n wherefore_o by_o the_o assumpt_n of_o the_o 12._o of_o the_o ten_o the_o line_n fh_o be_v incommensurable_a in_o length_n to_o the_o line_n fk_o and_o they_o be_v both_o rational_a wherefore_o the_o line_n fh_o and_o fk_o be_v rational_a commensurable_a in_o power_n only_o wherefore_o the_o line_n kh_o be_v a_o residual_a line_n and_o the_o line_n convenient_o soyn_v unto_o it_o be_v kf_a now_o the_o line_n fh_o be_v in_o power_n more_o thâ—n_o the_o line_n kf_o either_o by_o the_o square_n of_o a_o line_n commensurable_a in_o length_n to_o the_o line_n fh_o or_o by_o the_o square_n of_o a_o line_n incommensurable_a in_o length_n to_o the_o line_n fh_o first_o let_v it_o be_v in_o power_n more_o than_o the_o line_n fk_o by_o the_o square_n of_o a_o line_n commensurable_a in_o length_n to_o the_o line_n fh_o and_o the_o whole_a line_n fh_o be_v commensurable_a in_o length_n to_o the_o rational_a line_n put_v namely_o to_o fg._n wherefore_o the_o line_n kh_o be_v a_o first_o residual_a line_n but_o if_o a_o superficies_n be_v contain_v under_o a_o rational_a line_n and_o a_o first_o residual_a line_n the_o line_n that_o contain_v in_o power_n that_o superficies_n be_v by_o the_o 91._o of_o the_o ten_o a_o residual_a line_n wherefore_o the_o line_n which_o contain_v in_o power_n lh_o that_o be_v the_o superficies_n ec_o be_v a_o residual_a line_n but_o if_o the_o line_n hf_o be_v in_o power_n more_o than_o the_o line_n fk_o by_o the_o square_n of_o a_o line_n incommensurable_a in_o length_n to_o the_o line_n fh_o and_o the_o whole_a line_n fh_o be_v commensurable_a in_o length_n to_o the_o rational_a line_n give_v fg._n wherefore_o the_o line_n kh_o be_v a_o four_o residual_a line_n but_o a_o line_n contain_v in_o power_n a_o superficies_n contain_v under_o a_o rational_a line_n and_o a_o four_o residual_a line_n as_o a_o less_o line_n by_o the_o 94._o of_o the_o ten_o wherefore_o the_o line_n that_o contain_v in_o power_n the_o superficies_n lh_o that_o be_v the_o superficies_n ec_o be_v a_o less_o line_n if_o therefore_o from_o a_o rational_a superficies_n be_v take_v away_o a_o medial_a superficies_n the_o line_n which_o contain_v in_o power_n the_o superficies_n remain_v be_v one_o of_o these_o two_o irrational_a line_n namely_o either_o a_o residual_a line_n or_o a_o less_o line_n which_o be_v require_v to_o be_v prove_v ¶_o the_o
85._o theorem_a the_o 109._o proposition_n if_o from_o a_o medial_a superficies_n be_v take_v away_o a_o rational_a superficies_n the_o line_n which_o contain_v in_o power_n the_o superficies_n remain_v be_v one_o of_o these_o two_o irrational_a line_n namely_o either_o a_o first_o medial_a residual_a line_n or_o a_o line_n make_v with_o a_o rational_a superficies_n the_o whole_a superficies_n medial_a svppose_v that_o bc_o be_v a_o medial_a superficies_n and_o from_o it_o take_v away_o a_o rational_a superficies_n namely_o bd._n then_o i_o say_v that_o the_o line_n which_o contain_v in_o power_n the_o superficies_n remain_v namely_o the_o superficies_n ec_o be_v one_o of_o these_o two_o irrational_a line_n either_o a_o first_o medial_a residual_a line_n or_o a_o line_n make_v with_o a_o rational_a superficies_n the_o whole_a superficies_n medial_a take_v a_o rational_a line_n fg_o and_o let_v the_o rest_n of_o the_o construction_n be_v in_o this_o as_o it_o be_v in_o the_o former_a proposition_n construction_n wherefore_o it_o follow_v that_o the_o line_n fâ—_n be_v rational_a and_o incommensurable_a in_o length_n to_o the_o line_n fg_o by_o the_o 22._o of_o the_o ten_o demonstration_n and_o that_o the_o line_n kf_o be_v by_o the_o 20._o of_o the_o ten_o rational_a and_o commensurable_a in_o length_n to_o the_o line_n fg._n wherefore_o the_o line_n fh_a and_o fk_o be_v rational_a commensurable_a in_o power_n only_o wherefore_o kh_o be_v a_o residual_a line_n and_o the_o line_n convenient_o join_v unto_o it_o be_v fk_v now_o the_o line_n fh_o be_v in_o power_n more_o than_o the_o line_n fk_o either_o by_o the_o square_n of_o a_o line_n commensurable_a in_o length_n to_o the_o line_n fh_o or_o by_o the_o square_n of_o a_o line_n incommensurable_a in_o length_n unto_o it_o if_o the_o line_n fh_o be_v in_o power_n more_o than_o the_o line_n fk_o by_o the_o square_n of_o a_o line_n commensurable_a in_o length_n to_o the_o line_n fh_o and_o the_o line_n convenient_o join_v unto_o it_o namely_o fk_o be_v commensurable_a in_o length_n to_o the_o rational_a line_n fg._n wherefore_o the_o line_n kh_o be_v a_o second_o residual_a line_n and_o the_o line_n fg_o be_v a_o rational_a line_n but_o a_o line_n contain_v in_o power_n a_o superficies_n comprehend_v under_o a_o rational_a line_n and_o a_o second_o residual_a line_n be_v by_o the_o 92._o of_o the_o ten_o a_o first_o medial_a residual_a line_n wherefore_o the_o line_n thus_o contain_v in_o power_n the_o superficies_n lh_o that_o be_v the_o superficies_n ce_fw-fr be_v a_o first_o medial_a residual_a lne_v but_o if_o the_o line_n hf_o be_v in_o power_n more_o than_o the_o line_n fk_o by_o the_o square_n of_o a_o line_n incommensurable_a in_o length_n to_o the_o line_n fh_o and_o the_o line_n convenient_o join_v namely_o the_o line_n fk_o be_v commensurable_a in_o length_n to_o the_o rational_a line_n put_v namely_o to_o fg_o wherefore_o the_o line_n kh_o be_v a_o â—ift_n residual_a line_n wherefore_o by_o the_o 95._o of_o the_o ten_o the_o line_n that_o contain_v in_o power_n the_o superficies_n lh_o that_o be_v the_o superficies_n ec_o be_v a_o line_n make_v with_o a_o rational_a superficies_n the_o whole_a superficies_n medial_a which_o be_v require_v to_o be_v prove_v ¶_o the_o 86._o theorem_a the_o 110._o proposition_n if_o from_o a_o medial_a superficies_n be_v take_v away_o a_o medial_a superficies_n incommensurable_a to_o the_o whole_a superficies_n the_o line_n which_o contain_v in_o power_n the_o superficies_n which_o remain_v be_v one_o of_o these_o two_o irrational_a line_n namely_o either_o a_o second_o medial_a residual_a line_n or_o a_o line_n make_v with_o a_o medial_a superficies_n the_o whole_a superficies_n medial_a as_o in_o the_o former_a description_n sâ—_n 〈◊〉_d also_o â—ake_v away_o from_o the_o medial_a superficies_n be_v medial_a superficies_n bdâ—_n and_o let_v â—d_a be_v incommensurable_a to_o the_o whole_a superficies_n bc._n then_o i_o say_v that_o the_o line_n which_o contain_v in_o power_n the_o superficies_n ec_o be_v one_o of_o thâ—se_a two_o irrational_a line_n namely_o either_o a_o second_o medial_a residual_a line_n or_o a_o liâ—e_z make_v with_o a_o medial_a superficies_n the_o whole_a superficies_n medial_a demonstration_n for_o forasmuch_o as_o either_o of_o these_o superficieâ—â—s_n bc_o and_o bd_o be_v medial_a and_o bc_o be_v incommensurable_a to_o bd_o it_o follow_v by_o the_o 22._o of_o the_o ten_o thaâ—_z either_o of_o these_o line_n fh_a and_o fk_o be_v rational_a and_o incommensurable_a in_o length_n to_o the_o line_n fg._n and_o forasmuch_o as_o the_o superficies_n bc_o be_v incommensurable_a to_o the_o superficies_n bd_o that_o be_v the_o superficies_n gh_o to_o the_o superficies_n gk_o therefore_o by_o the_o first_o of_o the_o six_o &_o 100l_n of_o the_o ten_o the_o line_n fh_o be_v incommensurable_a in_o length_n to_o the_o line_n fk_o wherefore_o the_o line_n hf_o and_o fk_o be_v rational_a commensurable_a in_o power_n only_o wherefore_o by_o the_o 73._o of_o the_o ten_o the_o line_n kh_o be_v a_o residual_a line_n and_o the_o line_n convenient_o join_v unto_o it_o be_v fk_v now_o the_o line_n hf_o be_v in_o power_n more_o than_o the_o line_n fk_o either_o by_o the_o square_n of_o a_o line_n commensurable_a in_o length_n to_o the_o line_n hf_o or_o by_o the_o square_n of_o a_o line_n incommensurable_a in_o length_n unto_o it_o if_o the_o line_n hf_o be_v in_o power_n more_o than_o the_o line_n fk_o by_o the_o square_n of_o a_o line_n commensurable_a in_o length_n to_o the_o line_n fh_o and_o neither_o of_o the_o line_n hf_o nor_o fk_o iâ—_z commensurable_a to_o the_o rational_a line_n put_v fg._n wherefore_o the_o line_n kh_o be_v a_o three_o residual_a but_o the_o line_n cf_o that_o be_v the_o line_n kl_o be_v rational_a and_o a_o rectangle_n superficies_n contain_v under_o a_o rational_a line_n and_o a_o three_o residual_a line_n be_v irrational_a and_o the_o line_n which_o contain_v in_o power_n that_o superficies_n be_v by_o the_o 93._o of_o the_o ten_o a_o second_o medial_a residual_a line_n wherefore_o the_o line_n that_o contain_v in_o power_n the_o superficies_n lh_o that_o be_v the_o superficies_n ec_o be_v a_o second_o medial_a residual_a line_n but_o if_o the_o line_n hf_o be_v in_o power_n more_o than_o the_o line_n fk_o by_o the_o square_n of_o a_o line_n incommensurable_a in_o length_n to_o the_o line_n fh_o and_o neither_o of_o the_o line_n hf_o nor_o fk_o be_v commensurable_a in_o length_n to_o the_o line_n fg._n wherefore_o the_o line_n hk_o be_v a_o six_o residual_a line_n but_o a_o line_n contain_v in_o power_n a_o superficies_n contain_v under_o a_o rational_a line_n and_o a_o six_o residual_a line_n be_v by_o the_o 96._o of_o the_o ten_o a_o line_n make_v with_o a_o medial_a superficies_n the_o whole_a superficies_n medial_a wherefore_o the_o line_n that_o contain_v in_o power_n the_o superficies_n lh_o that_o be_v the_o superficies_n ec_o be_v a_o line_n make_v with_o a_o medial_a superficies_n the_o whole_a superficies_n medial_a if_o therefore_o from_o a_o medial_a superficies_n be_v take_v away_o a_o medial_a superficies_n incommensurable_a to_o the_o whole_a superficies_n the_o line_n that_o contain_v in_o power_n the_o superficies_n which_o remain_v be_v one_o of_o the_o two_o irrational_a line_n remain_v namely_o either_o a_o second_o medial_a residual_a line_n or_o a_o line_n make_v with_o a_o medial_a superficies_n the_o whole_a superficies_n medial_a which_o be_v require_v to_o be_v prove_v ¶_o the_o 87._o theorem_a the_o 111._o proposition_n a_o residual_a line_n be_v â—ot_n one_o and_o the_o same_o with_o a_o binomial_a line_n svppose_v that_o ab_fw-la be_v a_o residual_a line_n then_o i_o say_v that_o ab_fw-la be_v not_o one_o and_o the_o same_o with_o a_o binomial_a line_n for_o if_o it_o be_v possible_a let_v it_o be_v a_o binomial_a line_n and_o take_v a_o rational_a line_n dc_o construction_n and_o by_o the_o 44._o of_o the_o first_o unto_o the_o line_n cd_o apply_v a_o rectangle_n parallelogram_n ce_fw-fr equal_a to_o the_o square_n of_o the_o line_n ab_fw-la and_o make_v in_o breadth_n the_o line_n de._n impossibility_n and_o forasmuch_o as_o ab_fw-la be_v a_o residual_a line_n therefore_o by_o the_o 97._o of_o the_o ten_o the_o line_n de_fw-fr be_v a_o first_o residual_a line_n let_v the_o line_n convenient_o join_v unto_o it_o be_v eâ—_n wherefore_o the_o line_n d_o fletcher_n and_o fe_o be_v rational_a commensurable_a in_o power_n only_o and_o the_o line_n df_o be_v in_o power_n more_o than_o the_o line_n fe_o by_o the_o square_n of_o a_o line_n commensurable_a in_o length_n to_o the_o line_n de_fw-fr &_o the_o line_n df_o be_v commensurable_a in_o length_n to_o the_o rational_a line_n put_v dc_o again_o forasmuch_o as_o ab_fw-la be_v by_o position_n a_o binomial_a line_n therefore_o by_o the_o 60._o
11._o of_o the_o five_o as_o the_o line_n hk_o be_v to_o the_o line_n kf_o so_o be_v the_o line_n cd_o to_o the_o line_n db._n but_o the_o square_n of_o the_o line_n cd_o be_v commensurable_a to_o the_o square_n of_o the_o line_n db_o wherefore_o by_o the_o 10._o of_o the_o ten_o the_o square_a of_o the_o line_n hk_o be_v commensurable_a to_o the_o square_n of_o the_o line_n fk_o but_o these_o three_o line_n hk_o fk_o and_o eke_o be_v proportional_a in_o continual_a proportion_n as_o it_o have_v already_o be_v prove_v wherefore_o by_o the_o second_o corollary_n of_o the_o 20._o of_o the_o six_o the_o square_a of_o the_o line_n hk_o be_v to_o the_o square_n of_o the_o line_n fk_o as_o the_o line_n hk_o be_v to_o the_o line_n eke_o wherefore_o the_o line_n hk_o be_v commensurable_a in_o length_n to_o the_o line_n eke_o wherefore_o by_o the_o 15._o of_o the_o ten_o the_o line_n he_o be_v commensurable_a in_o length_n to_o the_o line_n fk_o and_o forasmuch_o as_o the_o square_n of_o the_o line_n a_o be_v equal_a to_o that_o which_o be_v contain_v under_o the_o line_n eh_o and_o bd_o but_o the_o square_n of_o the_o line_n a_o be_v rational_a wherefore_o that_o which_o be_v contain_v under_o the_o line_n eh_o and_o bd_o be_v rational_a and_o it_o be_v apply_v unto_o the_o rational_a line_n bd._n wherefore_o by_o the_o 20._o of_o the_o ten_o the_o line_n eh_o be_v rational_a and_o commensurable_a in_o length_n to_o the_o line_n bd._n wherefore_o also_o the_o line_n eke_o which_o be_v commensurable_a in_o length_n to_o the_o line_n he_o be_v rational_a and_o commensurable_a in_o length_n to_o the_o line_n bd._n now_o for_o that_o as_o the_o line_n cd_o be_v to_o the_o line_n db_o so_o be_v tâ—e_z line_n fk_o to_o the_o line_n eke_o for_o it_o be_v before_o prove_v that_o as_o cd_o be_v to_o db_o so_o be_v hf_o to_o fe_o and_o as_o hf_o be_v to_o fe_o so_o be_v fk_v to_o eke_o but_o the_o line_n cd_o and_o db_o be_v commensurable_a in_o power_n only_o wherefore_o by_o the_o 10._o of_o the_o ten_o the_o line_n fk_o and_o ke_o be_v also_o commensurable_a in_o power_n only_o and_o for_o that_o as_o the_o line_n cd_o be_v to_o the_o line_n db_o so_o be_v the_o line_n fk_o to_o the_o line_n eke_o therefore_o by_o contrary_a proportion_n as_o db_o be_v to_o cd_o so_o be_v eke_o to_o fk_v and_o alternate_o as_o db_o be_v to_o eke_o so_o be_v cd_o to_o fk_v but_o the_o line_n bd_o and_o eke_o be_v commensurable_a in_o length_n as_o it_o have_v already_o be_v prove_v wherefore_o also_o the_o line_n cd_o and_o fk_o be_v commensurable_a in_o length_n but_o the_o line_n cd_o be_v rational_a wherefore_o also_o the_o line_n fk_o be_v rational_a wherefore_o the_o line_n fk_o and_o eke_o be_v rational_a commensurable_a in_o power_n only_o wherefore_o the_o line_n fe_o be_v a_o residual_a line_n who_o name_n fk_o and_o ke_o be_v commensurable_a to_o the_o name_n cd_o and_o bd_o of_o the_o binomial_a line_n bc_o and_o in_o the_o same_o proportion_n as_o be_v prove_v determination_n i_o say_v moreover_o that_o it_o be_v a_o residual_a line_n of_o the_o self_n same_o order_n that_o the_o binomial_a line_n be_v for_o the_o line_n cd_o be_v in_o power_n more_o than_o the_o line_n bd_o either_o by_o the_o square_n of_o a_o line_n commensurable_a in_o length_n to_o the_o line_n cd_o or_o by_o the_o square_n of_o a_o line_n incommensurable_a in_o length_n now_o if_o the_o line_n cd_o be_v in_o power_n more_o than_o the_o line_n bd_o by_o the_o square_n of_o a_o line_n commensurable_a in_o length_n unto_o the_o line_n cd_o than_o by_o the_o 13._o of_o the_o ten_o the_o line_n fk_o be_v in_o power_n more_o than_o the_o line_n eke_o by_o the_o square_n of_o a_o line_n commensurable_a in_o length_n to_o the_o line_n fk_o and_o so_o if_o the_o line_n cd_o be_v commensurable_a in_o length_n to_o the_o rational_a line_n put_v the_o line_n fk_o also_o shall_v be_v commensurable_a in_o length_n to_o the_o same_o rational_a line_n wherefore_o then_o the_o line_n bc_o be_v a_o first_o binomial_a line_n &_o the_o line_n fe_o be_v likewise_o a_o first_o residual_a line_n and_o if_o the_o line_n bd_o be_v commensurable_a in_o length_n to_o the_o rational_a line_n the_o line_n â—k_n be_v also_o commensurable_a in_o length_n to_o the_o same_o and_o then_o the_o line_n bc_o be_v a_o second_o binomial_a line_n and_o the_o line_n fe_o a_o second_o residual_a line_n and_o if_o neither_o of_o the_o line_n cd_o nor_o db_o be_v commensurable_a in_o length_n unto_o the_o rational_a line_n neither_o of_o the_o line_n fk_v nor_o eke_o be_v commensurable_a in_o length_n unto_o the_o same_o and_o then_o the_o line_n bc_o be_v a_o three_o binomial_a line_n &_o the_o line_n fe_o be_v a_o three_o residual_a line_n and_o if_o the_o line_n cd_o be_v in_o power_n more_o than_o the_o line_n bd_o by_o the_o square_n of_o a_o line_n incommensurable_a in_o length_n to_o the_o line_n cd_o the_o line_n fk_o be_v also_o by_o the_o 14._o of_o the_o ten_o in_o power_n more_o than_o the_o line_n eke_o by_o the_o square_n of_o a_o line_n incommensurable_a in_o length_n to_o the_o line_n fk_o and_o so_o if_o the_o line_n cd_o be_v commensurable_a in_o length_n to_o a_o rational_a line_n put_v the_o line_n fk_o also_o be_v commensurable_a in_o length_n to_o the_o same_o wherefore_o the_o line_n bc_o be_v a_o four_o binomial_a line_n and_o the_o line_n fe_o be_v a_o four_o residual_a line_n and_o if_o the_o line_n bd_o be_v commensurable_a in_o length_n to_o the_o rational_a line_n the_o line_n eke_o be_v likewise_o commensurable_a in_o length_n to_o the_o same_o and_o then_o the_o line_n bc_o be_v a_o five_o binomial_a line_n and_o the_o line_n of_o a_o five_o residual_a line_n and_o if_o neither_o of_o the_o line_n cd_o nor_o db_o be_v commensurable_a in_o length_n to_o the_o rational_a line_n neither_o also_o of_o the_o line_n fk_v nor_o eke_o be_v commensurable_a in_o length_n to_o the_o same_o and_o then_o the_o line_n bc_o be_v a_o six_o binomial_a line_n and_o the_o line_n fe_o a_o six_o residual_a line_n wherefore_o the_o line_n fe_o be_v a_o residual_a line_n who_o name_n namely_o fk_o and_o eke_o be_v commensurable_a to_o the_o name_n of_o the_o binomial_a line_n namely_o to_o the_o name_n cd_o and_o db_o and_o be_v in_o the_o self_n same_o proportion_n and_o the_o residual_a line_n of_o be_v in_o the_o self_n same_o order_n of_o residual_a line_n that_o the_o binomial_a line_n bc_o be_v of_o binomial_a line_n wherefore_o the_o square_a of_o a_o rational_a line_n apply_v unto_o a_o binomial_a line_n make_v the_o breadth_n or_o other_o side_n a_o residual_a line_n who_o name_n be_v commensurable_a to_o the_o name_n of_o the_o binomial_a line_n and_o in_o the_o self_n same_o proportion_n and_o moreover_o that_o residual_a line_n be_v in_o the_o self_n same_o order_n of_o residual_a line_n that_o the_o binomial_a line_n be_v of_o binomial_a line_n which_o be_v require_v to_o be_v demonstrate_v here_o be_v the_o assumpt_n of_o the_o forego_n proposition_n confirm_v now_o let_v we_o declare_v how_o as_o the_o line_n hf_o be_v to_o the_o line_n fe_o so_o to_o make_v the_o line_n fk_o to_o the_o line_n eke_o the_o line_n cd_o be_v great_a than_o the_o line_n bd_o by_o supposition_n wherefore_o also_o the_o line_n hf_o be_v great_a than_o the_o line_n fe_o by_o alternate_a proportion_n and_o the_o 14._o of_o the_o five_o construction_n from_o the_o line_n hf_o take_v away_o the_o line_n fl_fw-mi equal_a to_o the_o line_n fe_o wherefore_o the_o line_n remain_v namely_o hl_o be_v less_o than_o the_o line_n hf_o for_o the_o line_n hf_o be_v equal_a to_o the_o line_n hl_o &_o lf_n as_o hl_o be_v to_o hf_o so_o by_o the_o 12._o of_o the_o six_o let_v fe_o be_v to_o fk_v demonstration_n wherefore_o by_o contrary_a proportion_n by_o the_o corollary_n of_o the_o 4._o of_o the_o five_o as_o hf_o be_v to_o hl_o so_o be_v fk_v to_o fe_o wherefore_o by_o conversion_n of_o proportion_n by_o the_o corollary_n of_o the_o 19_o of_o the_o five_o as_o hf_o be_v to_o lf_n that_o be_v to_o the_o line_n equal_a unto_o it_o namely_o to_o fe_o so_o be_v the_o line_n fk_o to_o the_o line_n eke_o m._n do_v you_o of_o this_o assumpt_n make_v 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d that_o be_v acquisive_o a_o problem_n universal_a thus_o two_o unequal_a right_a line_n be_v propound_v to_o adjoin_v unto_o the_o less_o a_o right_a line_n which_o take_v with_o the_o less_o as_o one_o right_a line_n shall_v have_v the_o same_o proportion_n to_o the_o line_n adjoin_v which_o the_o great_a of_o the_o two_o propound_v have_v to_o the_o less_o the_o
of_o the_o one_o be_v commensurable_a in_o length_n to_o the_o rational_a line_n put_v neither_o name_n of_o the_o other_o also_o shall_v be_v commensurable_a in_o length_n to_o the_o same_o rational_a line_n put_v by_o the_o 13._o of_o the_o same_o wherefore_o the_o residual_a line_n gz_n shall_v be_v in_o the_o self_n same_o order_n of_o residual_a line_n that_o the_o binomial_a line_n gb_o be_v of_o binomial_a line_n by_o the_o definition_n of_o residual_a and_o binomial_a line_n the_o square_a therefore_o of_o a_o rational_a line_n apply_v to_o a_o binomial_a lineâ—_z &_o â—_o which_o be_v require_v 〈◊〉_d be_v prove_v ¶_o the_o 89._o theorem_a the_o 113._o proposition_n the_o square_a of_o a_o rational_a line_n apply_v unto_o a_o residual_a make_v the_o breadth_n or_o other_o side_n a_o binomial_a line_n who_o name_n be_v commensurable_a to_o the_o name_n of_o the_o residual_a line_n and_o in_o the_o self_n same_o proportion_n and_o moreover_o that_o binomial_a line_n be_v in_o the_o self_n same_o order_n of_o binomial_a line_n that_o the_o residual_a line_n be_v of_o residual_a line_n svppose_v that_o a_o be_v a_o rational_a line_n and_o bd_o a_o residual_a line_n and_o unto_o the_o square_n of_o the_o line_n a_o let_v that_o which_o be_v contain_v under_o the_o line_n bd_o and_o kh_o be_v equal_a wherefore_o the_o square_a of_o the_o rational_a line_n a_o apply_v unto_o the_o residual_a line_n bd_o make_v the_o breadth_n or_o other_o side_n kh_o then_o i_o say_v that_o the_o line_n kh_o be_v a_o binomial_a line_n who_o name_n be_v commensurable_a to_o the_o name_n of_o the_o residual_a line_n bd_o and_o in_o the_o self_n same_o proportion_n and_o that_o the_o line_n kh_o be_v in_o the_o self_n same_o order_n of_o binomial_a line_n that_o the_o line_n bd_o be_v of_o residual_a line_n unto_o the_o line_n bd_o let_v the_o line_n convenient_o join_v be_v dc_o construction_n wherefore_o the_o line_n bc_o and_o dc_o be_v rational_a commensurable_a in_o power_n only_o and_o unto_o the_o square_n of_o the_o line_n a_o let_v the_o parallelogram_n contain_v under_o the_o line_n bc_o and_o g_o be_v equal_a but_o the_o square_n of_o the_o line_n a_o be_v rational_a demonstration_n wherefore_o the_o parallelogram_n contain_v under_o the_o line_n bc_o and_o g_o be_v also_o rational_a wherefore_o also_o the_o line_n g_o be_v rational_a and_o commensurable_a in_o length_n to_o the_o line_n bc_o by_o the_o 20._o of_o the_o ten_o now_o forasmuch_o as_o the_o parallelogram_n contain_v under_o the_o line_n bc_o and_o g_o be_v equal_a to_o that_o which_o be_v contain_v under_o the_o line_n bd_o and_o kh_o therefore_o by_o the_o 16._o of_o the_o six_o as_o the_o line_n bc_o be_v to_o the_o line_n bd_o so_o iâ—_z the_o line_n kh_o to_o the_o line_n g._n but_o the_o line_n bc_o be_v great_a than_o the_o line_n bd._n wherefore_o also_o the_o line_n kh_o be_v great_a than_o the_o line_n g._n unto_o the_o line_n g_o lâ—t_fw-fr the_o line_n ke_o be_v equal_a wherefore_o the_o line_n ke_o be_v rational_a and_o commensurable_a in_o length_n to_o the_o line_n bc_o as_o also_o the_o line_n g_o be_v by_o the_o 12._o of_o the_o ten_o and_o for_o that_o as_o bc_o be_v to_o bd_o so_o be_v kh_o to_o ke_o wherefore_o by_o â—duersion_n of_o proportion_n by_o the_o corollary_n of_o the_o 19_o of_o the_o five_o as_o bc_o be_v to_o dc_o so_o be_v kh_o to_o eh_o assumpt_n â—kh_n into_o eh_o so_o let_v the_o line_n fh_o be_v to_o the_o line_n of_o how_o this_o be_v to_o be_v do_v we_o will_v decare_v at_o the_o end_n of_o this_o demonstration_n wherefore_o the_o residue_n kf_o be_v to_o the_o residue_n fh_o as_o the_o whole_a kh_o be_v to_o the_o whole_a he_o by_o the_o 19_o of_o the_o five_o that_o be_v as_o the_o line_n bc_o be_v to_o the_o line_n cd_o but_o the_o line_n bc_o and_o cd_o be_v commensurable_a in_o power_n only_o wherefore_o also_o the_o line_n kf_a and_o fh_o be_v commensurable_a in_o power_n only_o and_o for_o that_o as_o kh_o be_v to_o he_o so_o be_v kf_a to_o fh_v but_o as_o kh_o be_v to_o he_o so_o be_v also_o hf_o to_o fe_o therefore_o as_o kf_n be_v to_o fh_v so_o be_v fh_o to_o fe_o wherefore_o by_o the_o corollary_n of_o the_o 19_o of_o the_o six_o as_o the_o first_o be_v to_o the_o three_o so_o be_v the_o square_a of_o the_o first_o to_o the_o square_n of_o the_o second_o wherefore_o as_o kf_n be_v to_o fe_o so_o be_v the_o square_a of_o the_o line_n kf_n to_o the_o square_n of_o the_o line_n fh_o but_o these_o square_n be_v commensurable_a for_o the_o line_n kf_a and_o fh_o be_v commensurable_a in_o power_n wherefore_o the_o line_n kf_a and_o fe_o be_v commensurable_a in_o length_n wherefore_o by_o the_o second_o part_n of_o the_o 15._o of_o the_o ten_o the_o line_n ke_o and_o of_o be_v commensurable_a in_o length_n wherefore_o by_o the_o same_o the_o line_n kf_a and_o fe_o be_v commensurable_a in_o length_n but_o the_o line_n ke_o be_v rational_a and_o commensurable_a in_o length_n to_o the_o line_n bc_o wherefore_o the_o line_n kf_o be_v also_o rational_a and_o commensurable_a in_o length_n to_o the_o line_n bc._n and_o for_o that_o as_o the_o line_n bc_o be_v to_o the_o line_n cd_o so_o it_o kf_n to_o eh_o therefore_o alternate_o by_o the_o 16._o of_o the_o five_o as_o bc_o be_v to_o kf_n so_o be_v cd_o to_o fh_o but_o the_o line_n bc_o be_v commensurable_a in_o length_n to_o the_o line_n kf_o wherefore_o the_o line_n cd_o be_v commensurable_a in_o length_n to_o the_o line_n fh_o but_o the_o line_n cd_o be_v rational_a wherefore_o also_o the_o line_n fh_o be_v rational_a and_o the_o line_n bc_o and_o cd_o be_v rational_a commensurable_a in_o power_n only_o wherefore_o the_o line_n kf_a and_o fh_o be_v rational_a commensurable_a in_o power_n only_o wherefore_o the_o line_n kh_o be_v a_o binomial_a line_n who_o name_n be_v commensurable_a to_o the_o name_n of_o the_o residual_a line_n and_o in_o the_o same_o proportion_n i_o say_v moreover_o that_o it_o be_v a_o binomial_a of_o the_o self_n same_o order_n of_o binomial_a line_n that_o the_o line_n bd_o be_v of_o residual_a line_n for_o if_o the_o line_n bc_o be_v in_o power_n more_o than_o the_o line_n cd_o by_o the_o square_n of_o a_o line_n commensurable_a in_o length_n to_o the_o line_n bc_o the_o line_n kf_o be_v also_o in_o power_n more_o than_o the_o line_n fh_o by_o the_o square_n of_o a_o line_n commensurable_a in_o length_n to_o the_o line_n kf_o by_o the_o 14._o of_o the_o ten_o and_o if_o the_o line_n bc_o be_v commensurable_a in_o length_n to_o the_o rational_a line_n put_v the_o line_n kf_o be_v also_o by_o the_o 12._o of_o the_o ten_o commensurable_a in_o length_n to_o the_o rational_a line_n and_o so_o the_o line_n bd_o be_v a_o first_o residual_a line_n and_o the_o line_n kh_o be_v in_o like_a sort_n a_o first_o binomial_a line_n if_o the_o line_n cd_o be_v commensurable_a in_o length_n to_o the_o rational_a line_n the_o line_n fh_o be_v also_o commensurable_a in_o length_n to_o the_o same_o line_n and_o so_o the_o line_n bd_o be_v a_o second_o residual_a line_n and_o the_o line_n kh_o a_o second_o binomial_a line_n and_o if_o neither_o of_o the_o line_n bc_o nor_o cd_o be_v commensurable_a in_o length_n to_o the_o rational_a line_n neither_o also_o of_o the_o line_n kf_n nor_o fh_o be_v commensurable_a in_o length_n to_o the_o same_o and_o so_o the_o line_n bd_o be_v a_o three_o residual_a line_n and_o the_o line_n kh_o a_o three_o binomial_a line_n but_o if_o the_o line_n bc_o be_v in_o power_n more_o than_o the_o line_n cd_o by_o the_o square_n of_o a_o line_n incommensurable_a in_o length_n to_o the_o line_n bc_o the_o line_n kf_o be_v in_o power_n more_o they_o the_o line_n fh_o by_o the_o square_n of_o a_o line_n incommensurable_a in_o length_n to_o the_o line_n kf_o by_o the_o 14._o of_o the_o ten_o and_o if_o the_o line_n bc_o be_v commensurable_a in_o length_n to_o the_o rational_a line_n put_v the_o line_n kf_o be_v also_o commensurable_a in_o length_n to_o the_o same_o line_n and_o so_o the_o line_n bd_o be_v a_o four_o residual_a line_n and_o the_o line_n kh_o a_o four_o binomial_a line_n and_o if_o the_o line_n cd_o be_v commensurable_a in_o length_n to_o the_o rational_a line_n the_o line_n fh_o be_v also_o commensurable_a in_o length_n to_o the_o same_o &_o so_o the_o line_n bd_o be_v a_o five_o residual_a line_n &_o the_o line_n kh_o a_o five_o binomial_a line_n and_o if_o neither_o of_o the_o line_n bc_o nor_o cd_o be_v commensurable_a in_o length_n to_o the_o rational_a line_n neither_o also_o of_o the_o line_n kf_n nor_o fh_o be_v commensurable_a in_o length_n
be_v equal_a to_o the_o angle_n lxn._n wherefore_o the_o whole_a angle_n mxn_n be_v equal_a to_o these_o two_o angle_n abc_n and_o ghk_n but_o the_o angle_n abc_n and_o ghk_n be_v great_a than_o the_o angle_n def_n wherefore_o the_o angle_n mxn_n be_v great_a than_o the_o angle_n def_n and_o forasmuch_o as_o these_o two_o line_n de_fw-fr and_o of_o be_v equal_a to_o these_o two_o line_n mx_o and_o xn_o and_o the_o base_a df_o be_v equal_a to_o the_o base_a mn_n therefore_o by_o the_o 8._o of_o the_o first_o the_o angle_n mxn_n be_v equal_a to_o the_o angle_n def_n and_o it_o be_v prove_v that_o it_o be_v also_o great_a which_o be_v impossible_a wherefore_o the_o line_n ab_fw-la be_v not_o equal_a to_o the_o line_n lx._n in_o like_a sort_n also_o may_v we_o prove_v that_o it_o be_v not_o less_o wherefore_o the_o line_n ab_fw-la be_v great_a than_o the_o line_n lx._n and_o again_o if_o unto_o the_o plain_a superficies_n of_o the_o circle_n be_v erect_v perpendicular_o from_o the_o point_n x_o a_o line_n xr_n who_o square_a be_v equal_a to_o that_o which_o the_o square_a of_o the_o line_n ab_fw-la exceed_v the_o square_a of_o the_o line_n lx_o and_o the_o rest_n of_o the_o construction_n be_v do_v in_o this_o that_o be_v in_o the_o former_a case_n then_o shall_v the_o problem_n be_v finish_v i_o say_v moreover_o that_o the_o line_n ab_fw-la be_v not_o less_o than_o the_o line_n lx._n lx._n for_o if_o it_o be_v possiblâ—_n let_v it_o be_v less_o and_o unto_o the_o line_n ab_fw-la put_v by_o the_o 2._o of_o the_o first_o the_o line_n xo_o equal_a and_o unto_o the_o line_n bc_o put_v the_o line_n xp_n equal_a and_o draw_v a_o right_a line_n from_o the_o point_n oh_o to_o the_o point_n p._n and_o forasmuch_o as_o the_o line_n ab_fw-la be_v equal_a to_o the_o line_n bc_o therefore_o the_o line_n xo_o be_v equal_a to_o the_o line_n xp_n wherefore_o the_o residue_n ol_n be_v equal_a to_o the_o residue_n mp_n wherefore_o by_o the_o 2._o of_o the_o six_o the_o line_n lm_o be_v a_o parallel_n to_o the_o line_n po._n and_o the_o triangle_n lxm_o be_v equiangle_n to_o the_o triangle_n pxo._n wherefore_o by_o the_o 6._o of_o the_o six_o as_o the_o line_n lx_o be_v to_o the_o line_n lm_o so_o be_v the_o line_n xo_o to_o the_o line_n po._n wherefore_o alternate_o by_o the_o 16._o of_o the_o five_o as_o the_o line_n lx_o be_v to_o the_o line_n xo_o so_o be_v the_o line_n lm_o to_o the_o line_n open_v but_o the_o line_n lx_o be_v great_a than_o the_o line_n xo_o wherefore_o also_o the_o line_n lm_o be_v great_a than_o the_o line_n open_v but_o the_o line_n lm_o be_v equal_a to_o the_o line_n ac_fw-la wherefore_o also_o the_o line_n ac_fw-la be_v great_a than_o the_o line_n open_v now_o forasmuch_o as_o these_o two_o line_n ab_fw-la and_o bc_o be_v equal_a to_o these_o two_o line_n ox_n &_o xp_n the_o one_o to_o the_o other_o and_o the_o base_a ac_fw-la be_v great_a than_o the_o base_a open_a therefore_o by_o the_o 25._o of_o the_o first_o the_o angle_n abc_n be_v great_a than_o the_o angle_n oxp._n and_o in_o like_a sort_n if_o we_o put_v the_o line_n xr_v equal_a to_o either_o of_o these_o line_n xo_o or_o xp_n and_o draw_v a_o right_a line_n from_o the_o point_n oh_o to_o the_o point_n r_o we_o may_v prove_v that_o the_o angle_n ghk_o be_v great_a than_o the_o angle_n oxr._n unto_o the_o right_a line_n lx_o and_o unto_o the_o point_n in_o it_o x_o make_v by_o the_o 23._o of_o the_o first_o unto_o the_o angle_n abc_n a_o equal_a angle_n lxs_n and_o unto_o the_o angle_n ghk_v make_v a_o equal_a angle_n lxt._n and_o by_o the_o second_o of_o the_o first_o let_v either_o of_o these_o line_n sx_n and_o xt_v be_v equal_a to_o the_o line_n ox_n and_o draw_v these_o line_n os_fw-mi ot_fw-mi and_o n-ab_n and_o forasmuch_o as_o these_o two_o line_n ab_fw-la and_o bc_o be_v equal_a to_o these_o two_o line_n ox_n and_o xs_n and_o the_o angle_n abc_n be_v equal_a to_o the_o angle_n oxs_n therefore_o by_o the_o 4._o of_o the_o first_o the_o base_a ac_fw-la that_o be_v lm_o be_v equal_a to_o the_o base_a os_fw-mi and_o by_o the_o same_o reason_n also_o the_o line_n ln_o be_v equal_a to_o the_o line_n ot_fw-mi and_o forasmuch_o as_o these_o two_o line_n lm_o and_o ln_o be_v equal_a to_o these_o two_o line_n so_o and_o ot_fw-mi and_o the_o angle_n mln_n be_v great_a than_o the_o angle_n sot_n therefore_o by_o the_o 25._o of_o the_o first_o the_o base_a mn_n be_v great_a than_o the_o base_a n-ab_n but_o the_o line_n mn_o be_v equal_a to_o the_o line_n df._n wherefore_o the_o line_n df_o be_v great_a than_o the_o line_n n-ab_n now_o forasmuch_o as_o these_o two_o line_n de_fw-fr and_o of_o be_v equal_a to_o these_o two_o line_n sx_n and_o xt_v and_o the_o base_a df_o be_v great_a than_o the_o base_a n-ab_n therefore_o by_o the_o 25._o of_o the_o first_o the_o angle_n def_n be_v great_a than_o the_o angle_n sxt_v but_o the_o angle_n sxt_v be_v equal_a to_o the_o angle_n abc_n and_o ghk_n wherâ—fore_o also_o the_o angle_n def_n be_v great_a than_o the_o angle_n abc_n and_o ghk_n but_o it_o be_v also_o less_o which_o be_v impossible_a proposition_n but_o now_o let_v we_o declare_v how_o to_o find_v out_o the_o line_n xr_n who_o square_a shall_v be_v equal_a â—o_o that_o which_o the_o square_a of_o the_o line_n ab_fw-la exceed_v the_o square_a of_o the_o line_n lx._n take_v the_o two_o right_a line_n ab_fw-la and_o lx_o and_o let_v ab_fw-la be_v the_o great_a and_o upon_o ab_fw-la describe_v a_o semicircle_n acb_o and_o from_o the_o point_n a_o apply_v into_o the_o semicircle_n a_o right_a line_n ac_fw-la equal_a to_o the_o right_a line_n lx_o and_o draw_v a_o right_a line_n from_o the_o point_n c_o to_o the_o point_n b._n and_o forasmuch_o as_o in_o the_o semicircle_n acb_o be_v a_o angle_n acb_o therefore_o by_o the_o 31._o of_o the_o three_o the_o angle_n acb_o be_v a_o right_a angle_n wherefore_o by_o the_o 47._o of_o the_o first_o the_o square_a of_o the_o line_n ab_fw-la be_v equal_a to_o the_o square_n of_o the_o line_n ac_fw-la and_o cb_o wherefore_o the_o square_a of_o the_o line_n ab_fw-la be_v in_o power_n more_o than_o the_o square_a of_o the_o line_n ac_fw-la by_o the_o square_n of_o the_o line_n cb_o but_o the_o line_n ac_fw-la be_v equal_a to_o the_o line_n lx_o wherefore_o the_o square_a of_o the_o line_n ab_fw-la be_v in_o power_n more_o than_o the_o square_a of_o the_o line_n lx_o by_o the_o square_n of_o the_o line_n cb._n if_o therefore_o unto_o the_o line_n cb_o we_o make_v the_o line_n xr_v equal_a then_o be_v the_o square_a of_o the_o line_n ab_fw-la great_a than_o the_o square_a of_o the_o line_n lx_o by_o the_o square_n of_o the_o line_n xr_n which_o be_v require_v to_o be_v do_v in_o this_o figure_n may_v you_o more_o full_o sâ—â—_n the_o construction_n and_o demonstration_n of_o the_o â—â—rst_a case_n of_o the_o former_a 23._o proposition_n if_o you_o erect_v perpendicular_o the_o triangle_n â—_o rn_v and_o unto_o it_o bend_v the_o triangle_n lmr_n that_o the_o angle_n r_o of_o each_o may_v join_v together_o in_o the_o point_n r._n and_o so_o full_o understand_v this_o case_n the_o other_o case_n will_v not_o be_v hard_a to_o conceive_v ¶_o the_o 21._o theorem_a the_o 24._o proposition_n if_o a_o solid_a or_o body_n be_v contain_v under_o thereof_o six_o parallel_n plain_a superficiece_n the_o opposite_a plain_a superficiece_n of_o the_o same_o body_n be_v equal_a and_o parallelogram_n svppose_v that_o this_o solid_a body_n cdhg_n be_v contain_v under_o these_o 6._o parallel_n plain_a superficiece_n namely_o ac_fw-la gf_o bg_o ce_fw-fr fb_o and_o ae_n then_o i_o say_v that_o the_o opposite_a superficiece_n of_o the_o same_o body_n be_v equal_a and_o parallelogram_n it_o be_v to_o weet_v the_o two_o opposite_n ac_fw-la and_o gf_o and_o the_o two_o opposite_n bg_o and_o ce_fw-fr and_o the_o two_o opposite_n fb_o and_o ae_n to_o be_v equal_a and_o all_o to_o be_v parallelogram_n parallelogram_n for_o forasmuch_o as_o two_o parallel_n plain_a superficiece_n that_o be_v bg_o and_o ce_fw-fr be_v divide_v by_o the_o plain_a superficies_n ac_fw-la their_o common_a section_n be_v by_o the_o 16._o of_o the_o eleven_o parallel_v wherefore_o the_o line_n ab_fw-la be_v a_o parallel_n to_o the_o line_n cd_o against_o forasmuch_o as_o two_o parallel_n plain_a superficiece_n fb_o and_o ae_n be_v divide_v by_o the_o plain_a superficies_n ac_fw-la their_o common_a section_n be_v by_o the_o same_o proposition_n parallel_n wherefore_o the_o line_n ad_fw-la be_v a_o parallel_n to_o the_o line_n bc._n and_o it_o be_v also_o prove_v that_o the_o line_n ab_fw-la be_v a_o parallel_n to_o
one_o of_o the_o circle_n ¶_o a_o assumpt_n add_v by_o flussas_n if_o a_o sphere_n be_v cut_v of_o a_o plain_a superficies_n the_o common_a section_n of_o the_o superficiece_n shall_v be_v the_o circumference_n of_o a_o circle_n suppose_v that_o the_o sphere_n abc_n be_v cut_v by_o the_o plain_a superficies_n aeb_n and_o let_v the_o centre_n of_o the_o sphere_n be_v the_o point_n d._n construction_n and_o from_o the_o point_n d_o let_v there_o be_v draw_v unto_o the_o plain_a superficies_n aeb_n a_o perpendicular_a line_n by_o the_o 11._o of_o the_o eleven_o which_o let_v be_v the_o line_n de._n and_o from_o the_o point_n e_o draw_v in_o the_o plain_a superficies_n aeb_n unto_o the_o common_a section_n of_o the_o say_a superficies_n and_o the_o sphere_n line_n how_o many_o so_o ever_o namely_o ea_fw-la and_o ebb_n and_o draw_v these_o line_n dam_fw-ge and_o db._n now_o forasmuch_o as_o the_o right_a angle_n dea_n and_o deb_n be_v equal_a for_o the_o line_n de_fw-fr be_v erect_v perpendicular_o to_o the_o plain_a superficies_n demonstration_n and_o the_o right_a line_n dam_n &_o db_o which_o subtend_v those_o angle_n be_v by_o the_o 12._o definition_n of_o the_o eleven_o equal_a which_o right_a line_n moreover_o by_o the_o 47._o of_o the_o first_o do_v contain_v in_o power_n the_o square_n of_o the_o line_n de_fw-fr ea_fw-la and_o de_fw-fr ebb_n if_o therefore_o from_o the_o square_n of_o the_o line_n de_fw-fr ea_fw-la and_o de_fw-fr ebb_v you_o take_v away_o the_o square_a of_o the_o line_n de_fw-fr which_o be_v common_a unto_o they_o the_o residue_n namely_o the_o square_n of_o the_o line_n ea_fw-la &_o ebb_n shall_v be_v equal_a wherefore_o also_o the_o line_n ea_fw-la and_o ebb_n be_v equal_a and_o by_o the_o same_o reason_n may_v we_o prove_v that_o all_o the_o right_a line_n draw_v from_o the_o point_n e_o to_o the_o line_n which_o be_v the_o common_a section_n of_o the_o superficies_n of_o sphere_n and_o of_o the_o plain_a superficies_n be_v equal_a wherefore_o that_o line_n shall_v be_v the_o circumference_n of_o a_o circle_n by_o the_o 15._o definition_n of_o the_o first_o circle_n but_o if_o it_o happen_v the_o plain_a superficies_n which_o cut_v the_o sphere_n to_o pass_v by_o the_o centre_n of_o the_o sphere_n the_o right_a line_n draw_v from_o the_o centre_n of_o the_o sphere_n to_o their_o common_a section_n shalâ—_n be_v equal_a by_o the_o 12._o definition_n of_o the_o eleven_o for_o that_o common_a section_n be_v in_o the_o superficies_n of_o the_o sphere_n wherefore_o of_o necessity_n the_o plain_a superficies_n comprehend_v under_o that_o line_n of_o the_o common_a section_n shall_v be_v a_o circle_n and_o his_o centre_n shall_v be_v one_o and_o the_o same_o with_o the_o centre_n of_o the_o sphere_n john_n dee_n euclid_n have_v among_o the_o definition_n of_o solid_n omit_v certain_a which_o be_v easy_a to_o conceive_v by_o a_o kind_n of_o analogy_n as_o a_o segment_n of_o a_o sphere_n a_o sector_n of_o a_o sphere_n the_o vertex_fw-la or_o top_n of_o the_o segment_n of_o a_o sphere_n with_o such_o like_a but_o that_o if_o need_n be_v some_o fartheâ—_n light_n may_v be_v give_v in_fw-ge this_o figure_n next_o before_o description_n understand_v a_o segment_n of_o the_o sphere_n abc_n to_o be_v that_o part_n of_o the_o sphere_n contain_v between_o the_o circle_n ab_fw-la who_o centre_n be_v e_o and_o the_o spherical_a superficies_n afb_o to_o which_o be_v a_o less_o segment_n add_v the_o cone_n adb_o who_o base_a be_v the_o former_a circle_n and_o top_n the_o centre_n of_o the_o sphere_n and_o you_o have_v dafb_n a_o sector_n of_o a_o sphere_n or_o solid_a sector_n as_o i_o call_v it_o de_fw-fr extend_a to_o fletcher_n show_v the_o top_n or_o vertex_fw-la of_o the_o segment_n to_o be_v the_o point_n f_o and_o of_o be_v the_o altitude_n of_o the_o segment_n spherical_a of_o segment_n some_o be_v great_a they_o the_o half_a sphere_n some_o be_v less_o as_o before_o abf_n be_v less_o the_o remanent_fw-la abc_n be_v a_o segment_n great_a than_o the_o half_a sphere_n ¶_o a_o corollary_n add_v by_o the_o same_o flussas_n by_o the_o foresay_a assumpt_n it_o be_v manifest_a that_o if_o from_o the_o centre_n of_o a_o sphere_n the_o line_n draw_v perpendicular_o unto_o the_o circle_n which_o cut_v the_o sphere_n be_v equal_a those_o circle_n be_v equal_a and_o the_o perpendicular_a line_n so_o draw_v fall_n upon_o the_o centre_n of_o the_o same_o circle_n for_o the_o line_n which_o be_v draw_v from_o the_o centre_n of_o the_o sphere_n to_o the_o circumference_n contain_v in_o power_n the_o power_n of_o the_o perpendicular_a line_n and_o the_o power_n of_o the_o line_n which_o join_v together_o the_o end_n of_o those_o line_n wherefore_o from_o that_o square_a or_o power_n of_o the_o line_n from_o the_o centre_n of_o the_o sphere_n to_o the_o circumference_n or_o common_a section_n draw_v which_o be_v the_o semidiameter_n of_o the_o sphere_n take_v away_o the_o power_n of_o the_o perpendicular_a which_o be_v common_a to_o they_o â—ound_v about_o it_o follow_v that_o the_o residue_v how_o many_o so_o ever_o they_o be_v be_v equal_a power_n and_o therefore_o the_o line_n be_v equal_a the_o one_o to_o the_o other_o wherefore_o they_o will_v describe_v equal_a circle_n by_o the_o first_o definition_n of_o the_o three_o and_o upon_o their_o centre_n fall_v the_o perpendicular_a line_n by_o the_o 9_o of_o the_o three_o corollary_n and_o those_o circle_n upon_o which_o fall_v the_o great_a perpendicular_a line_n be_v the_o less_o circle_n for_o the_o power_n of_o the_o line_n draw_v from_o the_o centre_n of_o the_o sphere_n to_o the_o circumference_n be_v always_o one_o and_o equal_a to_o the_o power_n of_o the_o perpendicular_a line_n and_o also_o to_o the_o power_n of_o the_o line_n draw_v from_o the_o centre_n of_o the_o circle_n to_o their_o circumference_n the_o great_a that_o the_o power_n of_o the_o perpendicular_a line_n take_v away_o from_o the_o power_n contain_v they_o both_o be_v the_o less_o be_v the_o power_n and_o therefore_o the_o line_n remain_v which_o be_v the_o semidiameter_n of_o the_o circle_n and_o therefore_o the_o less_o be_v the_o circle_n which_o they_o describe_v wherefore_o if_o the_o circle_n be_v equal_a the_o perpendicular_a line_n fall_v from_o the_o centre_n of_o the_o sphere_n upon_o they_o shall_v also_o be_v equal_a for_o if_o they_o shall_v be_v great_a or_o less_o the_o circle_n shall_v be_v unequal_a as_o it_o be_v before_o manifest_a but_o we_o suppose_v the_o perpendicular_n to_o be_v equal_a corollary_n also_o the_o perpendicular_a line_n fall_v upon_o those_o base_n be_v the_o least_o of_o all_o that_o be_v draw_v from_o the_o centre_n of_o the_o sphere_n for_o the_o other_o draw_v from_o the_o centre_n of_o the_o sphere_n to_o the_o circumference_n of_o the_o circle_n be_v in_o power_n equal_a both_o to_o the_o power_n of_o the_o perpendicular_n and_o to_o the_o power_n of_o the_o line_n join_v these_o perpendicular_n and_o these_o subtendent_fw-la line_n together_o make_v triangle_n rectangleround_v about_o as_o most_o easy_o you_o may_v conceive_v of_o the_o figure_n here_o annex_v a_o the_o centre_n of_o the_o sphere_n ab_fw-la the_o line_n from_o the_o centre_n of_o the_o sphere_n to_o the_o circumference_n of_o the_o circle_n make_v by_o the_o section_n bcb_n the_o diameter_n of_o circle_n make_v by_o the_o section_n ac_fw-la the_o perpendicular_n from_o the_o centre_n of_o the_o sphere_n to_o the_o circlesâ—_n who_o diameter_n bcb_n be_v one_o both_o side_n or_o in_o any_o situation_n else_o cb_o the_o semidiameter_n of_o the_o circle_n make_v by_o the_o section_n ao_o a_o perpendicular_a long_o than_o ac_fw-la and_o therefore_o the_o semidiameter_n ob_fw-la be_v less_o acb_o &_o aob_n triangle_n rectangle_n ¶_o the_o 2._o problem_n the_o 17._o proposition_n two_o sphere_n consist_v both_o about_o one_o &_o the_o self_n same_o centre_n be_v give_v to_o inscribe_v in_o the_o great_a sphere_n a_o solid_a of_o many_o side_n which_o be_v call_v a_o polyhedron_n which_o shall_v not_o touch_v the_o superficies_n of_o the_o less_o sphere_n svppose_v that_o there_o be_v two_o sphere_n about_o one_o &_o the_o self_n same_o centre_n namely_o about_o a._n it_o be_v require_v in_o the_o great_a sphere_n to_o inscribe_v a_o polyhedron_n or_o a_o solid_a of_o many_o side_n which_o shall_v not_o with_o his_o superficies_n touch_v the_o superficies_n of_o the_o less_o sphere_n let_v the_o sphere_n be_v cut_v by_o some_o one_o plain_a superficies_n pass_v by_o the_o centre_n a._n construction_n then_o shall_v their_o section_n be_v circle_n flussas_n for_o by_o the_o 12._o definition_n of_o the_o eleven_o the_o diameter_n remain_v fix_v and_o the_o semicircle_n be_v turn_v round_o about_o make_v a_o sphere_n wherefore_o in_o what_o position_n so_o ever_o you_o imagine_v the_o
three_o proposition_n we_o will_v use_v the_o same_o supposition_n and_o construction_n there_o specify_v so_o far_o as_o they_o shall_v serve_v our_o purpose_n begin_v therefore_o at_o the_o conclusion_n we_o must_v infer_v the_o part_n of_o the_o proposition_n before_o grant_v it_o be_v conclude_v that_o the_o square_n of_o the_o line_n db_o be_v quintuple_a to_o the_o square_n of_o the_o line_n dc_o his_o own_o segment_n therefore_o dn_o the_o square_n of_o db_o be_v quintuple_a to_o gf_o the_o square_n of_o dc_o but_o the_o squaâ—e_n of_o ac_fw-la the_o double_a of_o dc_o which_o be_v r_n be_v quadruple_a to_o gf_o by_o the_o second_o corollary_n of_o the_o 20._o of_o the_o six_o and_o therefore_o r_n with_o gf_o be_v quintuple_a to_o gf_o and_o so_o it_o be_v evident_a that_o the_o square_a dn_o be_v equal_a to_o the_o square_a r_n together_o with_o the_o square_a gf_o wherefore_o from_o those_o two_o equal_n take_v the_o square_a gf_o common_a to_o they_o both_o remain_v the_o square_a r_n equal_a to_o the_o gnomon_n xop_n but_o to_o the_o gnomon_n xop_n the_o parallelogram_n 3._o ce_fw-fr be_v equal_a wherefore_o the_o square_a of_o the_o line_n ac_fw-la which_o be_v r_n be_v equal_a to_o the_o parallelogram_n câ—_n which_o parallelogamme_n be_v contain_v under_o be_v equal_a to_o ab_fw-la and_o cb_o the_o part_n remain_v of_o the_o first_o line_n gruen_o which_o be_v db._n and_o the_o line_n ab_fw-la be_v make_v of_o the_o double_a of_o the_o segment_n dc_o and_o of_o cbâ—_n the_o other_o part_n of_o the_o line_n db_o first_o goven_v wherefore_o the_o double_a of_o the_o segment_n dc_o with_o cb_o the_o part_n remain_v which_o altogether_o be_v the_o whole_a line_n ab_fw-la be_v to_o ac_fw-la the_o double_a of_o the_o segment_n dc_o as_o that_o same_o ac_fw-la be_v to_o cb_o by_o the_o second_o part_n of_o the_o 16._o of_o the_o six_o therefore_o by_o the_o 3._o definition_n of_o the_o six_o book_n the_o whole_a line_n ab_fw-la be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n &_o ac_fw-la the_o double_a of_o the_o segment_n dc_o be_v middle_a proportional_a be_v the_o great_a part_n thereof_o wherefore_o if_o a_o right_a line_n be_v quintuple_a in_o power_n etc_n etc_n as_o in_o the_o proposition_n which_o be_v to_o be_v demonstrate_v or_o thus_o it_o may_v be_v demonstrate_v forasmuch_o as_o the_o square_n dn_o be_v quintuple_a to_o the_o square_a gf_o i_o mean_v the_o square_n of_o db_o the_o line_n give_v too_o the_o square_a oâ—_n dc_o the_o segment_n and_o the_o same_o square_a dn_o be_v equal_a to_o the_o parallelogram_n under_o ab_fw-la cb_o with_o the_o square_n make_v of_o the_o line_n dc_o by_o the_o six_o of_o the_o second_o for_o unto_o the_o line_n ac_fw-la equal_o divide_v the_o line_n cb_o be_v as_o it_o be_v adjoin_v wherefore_o the_o parallelogram_n under_o ab_fw-la cb_o together_o with_o the_o square_n of_o dc_o which_o be_v gf_o be_v quintuple_a to_o the_o square_a gf_o make_v oâ—_n thâ—_z line_n dc_o take_v then_o that_o square_a gf_o â—rom_o the_o parallelogram_n under_o ab_fw-la cb_o that_o parallelogram_n under_o ab_fw-la cb_o remain_v alone_o be_v but_o quadruple_a to_o the_o say_v square_a of_o the_o line_n dc_o but_o by_o the_o 4._o of_o the_o second_o or_o the_o second_o corollary_n of_o the_o 20._o of_o the_o six_o r_n â—he_z square_a of_o the_o line_n ac_fw-la be_v quadrupla_fw-la to_o the_o same_o square_a gfâ—_n wherefore_o by_o the_o 7._o of_o the_o five_o the_o square_a of_o the_o line_n ac_fw-la be_v equal_a to_o the_o parallelogram_n under_o ab_fw-la cb_o and_o so_o by_o the_o second_o part_n of_o the_o 16._o of_o the_o six_o ab_fw-la ac_fw-la and_o cb_o be_v three_o line_n in_o continual_a proportion_n and_o see_v ab_fw-la be_v great_a they_o ac_fw-la the_o same_o ac_fw-la the_o double_a of_o the_o line_n dc_o shall_v be_v great_a than_o the_o part_n bc_o remain_v wherefore_o by_o the_o 3._o definition_n of_o the_o six_o ab_fw-la compose_v or_o make_v of_o the_o double_a of_o dc_o and_o the_o other_o part_n of_o db_o remain_v be_v divide_v by_o a_o extreme_a and_o middle_n proportion_n and_o also_o his_o great_a segment_n be_v ac_fw-la the_o double_a of_o the_o segment_n dc_o wherefore_o if_o a_o right_a line_n be_v quintuple_a in_o power_n etc_n etc_n as_o in_o the_o proposition_n which_o be_v to_o be_v demonstrate_v a_o theorem_a 2._o if_o a_o right_a line_n divide_v by_o a_o extreme_a and_o mean_a proportion_n be_v give_v and_o to_o the_o great_a segment_n â—herof_o he_o direct_o adjoin_v a_o line_n equal_a to_o the_o whole_a line_n give_v that_o adjoin_v line_n and_o the_o say_v great_a segment_n do_v make_v a_o line_n divide_v by_o extreme_a and_o mean_a proportion_n who_o great_a segment_n be_v the_o line_n adjoin_v suppose_v the_o line_n give_v divide_v by_o extreme_a and_o mean_a proportion_n to_o be_v ab_fw-la divide_v in_o the_o point_n c_o and_o his_o great_a segment_n let_v be_v ac_fw-la unto_fw-la ac_fw-la direct_o adjoin_v a_o line_n equal_a to_o ab_fw-la let_v that_o be_v ad_fw-la i_o say_v that_o ad_fw-la together_o with_o ac_fw-la that_o be_v dc_o be_v a_o divide_v by_o extreme_a and_o middle_n proportion_n who_o great_a segment_n be_v ad_fw-la the_o line_n adjoin_v divide_v ad_fw-la equal_o in_o the_o point_n e._n now_o forasmuch_o as_o ae_n be_v the_o half_a of_o ad_fw-la by_o construction_n it_o be_v also_o the_o half_a of_o ab_fw-la equal_a to_o ad_fw-la by_o construction_n wherefore_o by_o the_o 1._o of_o the_o thirteen_o the_o square_a of_o the_o line_n compose_v of_o ac_fw-la and_o ae_n which_o â—ne_n be_v ec_o be_v quintuple_a to_o the_o square_n of_o the_o line_n ae_n wherefore_o the_o double_a of_o ae_n and_o the_o line_n ac_fw-la compose_v as_o in_o one_o right_a line_n be_v a_o line_n divide_v by_o extreme_a and_o mean_a proportion_n by_o the_o converse_n of_o this_o three_o by_o i_o demonstrate_v and_o the_o double_a of_o ae_n be_v the_o great_a segment_n but_o dc_o be_v the_o line_n compose_v of_o the_o double_a of_o ae_n &_o the_o line_n ac_fw-la and_o with_o all_o ad_fw-la be_v the_o double_a of_o ae_n wherefore_o dc_o be_v a_o line_n divide_v by_o extreme_a and_o mean_a proportion_n and_o ad_fw-la iâ—_z hiâ—_z great_a segment_n if_o a_o right_a line_n therefore_o divide_v by_o extreme_a and_o mean_a proportion_n be_v give_v and_o to_o the_o great_a segment_n thereof_o be_v direct_o adjoin_v a_o line_n equal_a to_o the_o whole_a line_n give_v that_o adjoin_v line_n and_o the_o say_v great_a segment_n do_v make_v a_o line_n divide_v by_o extreme_a and_o mean_a proportion_n who_o great_a segment_n be_v the_o line_n adjoin_v which_o be_v require_v to_o be_v demonstrate_v two_o other_o brief_a demonstration_n of_o the_o same_o forasmuch_o as_o ad_fw-la be_v to_o ac_fw-la as_o ab_fw-la be_v to_o ac_fw-la because_o ad_fw-la be_v equal_a to_o ab_fw-la by_o construction_n but_o as_o ab_fw-la be_v to_o ac_fw-la so_o be_v ac_fw-la to_o cb_o by_o supposition_n therefore_o by_o the_o 11._o of_o the_o five_o as_z ac_fw-la be_v to_o cb_o so_o be_v ad_fw-la to_o ac_fw-la demonstrate_v wherefore_o as_o ac_fw-la and_o cb_o which_o be_v ab_fw-la be_v to_o cb_o so_o be_v ad_fw-la and_o ac_fw-la which_o be_v dc_o to_o ac_fw-la therefore_o everse_o as_o ab_fw-la be_v to_o ac_fw-la so_o be_v dc_o to_o ad._n and_o it_o be_v prove_v ad_fw-la to_o be_v to_o ac_fw-la as_o ac_fw-la be_v to_o cb._n wherefore_o as_o ab_fw-la be_v to_o ac_fw-la and_o ac_fw-la to_o cb_o so_o be_v dc_o to_o ad_fw-la and_o ad_fw-la to_o ac_fw-la but_o ab_fw-la ac_fw-la and_o cb_o be_v in_o continual_a proportion_n by_o supposition_n wherefore_o dc_o ad_fw-la and_o ac_fw-la be_v in_o continual_a proportion_n wherefore_o by_o the_o 3._o definition_n of_o the_o six_o book_n dc_o be_v divide_v by_o extreme_a and_o middle_a proportion_n and_o his_o great_a segment_n be_v ad._n which_o be_v to_o be_v demonstrate_v note_v from_o the_o mark_n demonstrate_v how_o this_o have_v two_o demonstration_n one_o i_o have_v set_v in_o the_o margin_n by_o ¶_o a_o corollary_n 1._o upon_o euclides_n three_o proposition_n demonstrate_v it_o be_v make_v evident_a that_o of_o a_o line_n divide_v by_o extreme_a and_o mean_a proportion_n if_o you_o produce_v the_o less_o segment_n equal_o to_o the_o length_n of_o the_o great_a the_o line_n thereby_o adjoin_v together_o with_o the_o say_v less_o segment_n make_v a_o new_a line_n divide_v by_o extreme_a and_o middle_a proportion_n who_o less_o segment_n be_v the_o line_n adjoin_v for_o if_o ab_fw-la be_v divide_v by_o extreme_a and_o middle_a proportion_n in_o the_o point_n c_o ac_fw-la be_v the_o great_a segment_n and_o cb_o be_v produce_v from_o the_o point_n b_o make_v a_o line_n with_o cb_o equal_a to_o ac_fw-la which_o let_v be_v cq_n and_o the_o
as_o the_o line_n ab_fw-la be_v to_o the_o line_n bc_o so_o be_v the_o square_a of_o the_o line_n ab_fw-la to_o the_o square_n of_o the_o line_n bd_o by_o the_o corollary_n of_o the_o 8._o and_o 20._o of_o the_o six_o wherefore_o the_o square_a of_o the_o line_n ab_fw-la be_v treble_a to_o the_o square_n of_o the_o line_n bd._n and_o it_o be_v prove_v that_o the_o square_a of_o the_o line_n gk_o be_v treble_a to_o the_o square_n of_o the_o line_n ke_o and_o the_o line_n ke_o be_v put_v equal_a to_o the_o line_n bd._n wherefore_o the_o line_n kg_n be_v also_o equal_a to_o the_o line_n ab_fw-la and_o the_o line_n ab_fw-la be_v the_o diameter_n of_o the_o sphere_n give_v wherefore_o the_o line_n kg_n be_v equal_a to_o the_o diameter_n of_o the_o sphere_n give_v wherefore_o the_o cube_fw-la be_v comprehend_v in_o the_o sphere_n give_v and_o it_o be_v also_o prove_v that_o the_o diameter_n of_o the_o sphere_n be_v in_o power_n treble_a to_o the_o side_n of_o the_o cube_fw-la which_o be_v require_v tâ—â—e_o do_v and_o to_o be_v prove_v an_o other_o demonstration_n after_o flussas_n suppose_v that_o the_o diameter_n of_o the_o sphere_n give_v in_o the_o former_a proposition_n be_v the_o line_n aâ—_z and_o let_v the_o centre_n be_v the_o point_n c_o upon_o which_o describe_v a_o semicircle_n adb_o and_o from_o the_o diameter_n ab_fw-la cut_v of_o a_o three_o part_n bg_o by_o the_o 9_o of_o the_o six_o and_o from_o the_o point_n g_o raise_v up_o unto_o the_o line_n ab_fw-la a_o perpendicular_a line_n dg_o by_o the_o 11._o of_o the_o first_o and_o draw_v these_o right_a line_n dam_n dc_o and_o db._n and_o unto_o the_o right_a line_n db_o put_v a_o equal_a right_a line_n zi_n and_o upon_o the_o line_n zi_n describe_v a_o square_a ezit_n and_o from_o the_o point_n e_o z_o i_o t_o erect_v unto_o the_o superficies_n ezit_n perpendicular_a line_n eke_o zh_n in_o tn_n by_o the_o 12._o of_o the_o eleven_o and_o put_v every_o one_o of_o those_o perpendicular_a line_n equal_a to_o the_o line_n zi_n and_o draw_v these_o right_a line_n kh_o hm_n mn_o and_o nk_v each_o of_o which_o shall_v be_v equal_a and_o parallel_n to_o the_o line_n zi_n and_o to_o the_o rest_n of_o the_o line_n of_o the_o square_n by_o the_o 33._o of_o the_o first_o and_o moreover_o they_o shall_v contain_v equal_a angle_n by_o the_o 10._o of_o the_o eleven_o and_o therefore_o the_o angle_n be_v right_a angle_n for_o that_o ezit_n be_v a_o square_n wherefore_o the_o rest_n of_o the_o base_n shall_v be_v square_n wherefore_o the_o solid_a ezitkhmn_n be_v contain_v under_o 6._o equal_a square_n be_v a_o cube_fw-la by_o the_o 21._o definition_n of_o the_o eleven_o extend_v by_o the_o opposite_a side_n ke_o and_o mi_n of_o the_o cube_fw-la a_o plain_a keim_n and_o again_o by_o the_o other_o opposite_a side_n nt_v and_o hz_n extend_v a_o other_o plain_a hztn_n now_o forasmuch_o as_o each_o of_o these_o plain_n divide_v the_o solid_a into_o two_o equal_a part_n namely_o into_o two_o prism_n equal_a and_o like_a by_o the_o 8._o definition_n of_o the_o eleven_o therefore_o those_o plain_n shall_v cut_v the_o cube_fw-la by_o the_o centre_n by_o the_o corollary_n of_o the_o 39_o of_o the_o eleven_o wherefore_o the_o common_a section_n of_o those_o plain_n shall_v pass_v by_o the_o centre_n let_v that_o common_a section_n be_v the_o line_n lf_n and_o forasmuch_o as_o the_o side_n hn_o and_o km_o of_o the_o superficiece_n keim_n and_o hztn_n do_v divide_v the_o one_o the_o other_o into_o two_o equal_a part_n by_o the_o corollary_n of_o the_o 34._o of_o the_o first_o and_o so_o likewise_o do_v the_o side_n zt_v and_o ei_o therefore_o the_o common_a section_n lf_o be_v draw_v by_o these_o section_n and_o divide_v the_o plain_n keim_n and_o hztn_n into_o two_o equal_a part_n by_o the_o first_o of_o the_o six_o for_o their_o base_n be_v equal_a and_o the_o altitude_n be_v one_o and_o the_o â—ame_n namely_o the_o altitude_n of_o the_o cube_fw-la wherefore_o the_o line_n lf_n shall_v divide_v into_o two_o equal_a part_n the_o diameter_n of_o his_o plain_n namely_o the_o right_a line_n ki_v em_n zn_n and_o nt_v which_o be_v the_o diameter_n of_o the_o cube_fw-la wherefore_o those_o diameter_n shall_v concur_v and_o cut_v one_o the_o other_o in_o one_o and_o the_o self_n same_o point_n let_v the_o same_o be_v o._n wherefore_o the_o right_a line_n ok_n oe_n oi_o om_o oh_o oz_n ot_fw-mi and_o on_o shall_v be_v equal_a the_o onâ—_n to_o the_o other_o for_o that_o they_o be_v the_o half_n of_o the_o diameter_n of_o equal_a and_o like_a rectangle_n parallelogram_n wherefore_o make_v the_o centre_n the_o point_n oh_o and_o the_o space_n any_o of_o these_o line_n oe_z or_o ok_n etc_n etc_n a_o sphere_n describe_v shall_v pass_v by_o every_o one_o of_o the_o angle_n of_o the_o cube_fw-la namely_o which_o be_v at_o the_o point_n e_o z_o i_o t_o edward_n h_o m_o n_o by_o the_o 12._o definition_n of_o the_o eleven_o for_o that_o all_o the_o line_n draw_v from_o the_o point_n oh_o to_o the_o angle_n of_o the_o cube_fw-la be_v equal_a but_o the_o right_a line_n ei_o contain_v in_o power_n the_o two_o equal_a right_a line_n ez_o and_o zi_n by_o the_o 47._o of_o the_o first_o wherefore_o the_o square_a of_o the_o line_n ei_o be_v double_a to_o the_o square_n of_o the_o line_n zi_n and_o forasmuch_o as_o the_o right_a line_n ki_v subtend_v the_o right_a angle_n kei_n for_o that_o the_o right_a line_n keâ—_n be_v erect_v perpendicular_o to_o the_o plaiâ—e_a superficies_n of_o the_o right_a line_n ez_o and_o zt_n by_o the_o 4._o of_o the_o eleven_o â—_o therefore_o the_o square_a of_o the_o line_n ki_v be_v equal_a to_o the_o square_n of_o the_o line_n ei_o and_o eke_o but_o the_o square_n of_o the_o line_n ei_o be_v double_a to_o the_o square_n of_o the_o line_n eke_o for_o it_o be_v double_a to_o the_o square_n of_o the_o line_n zi_n as_o have_v be_v prove_v and_o the_o base_n of_o the_o cube_fw-la be_v equal_a square_n wherefore_o the_o square_a of_o the_o line_n ki_v be_v triple_a to_o the_o square_n of_o the_o line_n ke_o that_o be_v to_o the_o square_n of_o the_o line_n zi_n but_o the_o right_a line_n zi_n be_v equal_a to_o thâ—_z right_a line_n db_o by_o position_n unto_o who_o square_a the_o square_n of_o the_o diameter_n ab_fw-la be_v triple_a by_o that_o which_o be_v demonstrate_v in_o the_o 13._o proposition_n of_o this_o book_n wherefore_o the_o diameter_n ki_v &_o db_o be_v equal_a wherefore_o there_o be_v describe_v a_o cube_fw-la king_n and_o it_o be_v comprehend_v in_o the_o sphere_n give_v wherein_o the_o other_o solid_n be_v contain_v the_o diameter_n of_o which_o sphere_n be_v the_o line_n ab_fw-la and_o the_o diameter_n ki_v or_o ab_fw-la of_o the_o same_o sphere_n be_v prove_v to_o be_v in_o power_n triple_a to_o the_o side_n of_o the_o cube_fw-la namely_o to_o the_o line_n db_o or_o zi_n ¶_o corollaryes_fw-mi add_v by_o flussas_n first_o corollary_n hereby_o it_o be_v manifest_a that_o the_o diameter_n of_o a_o sphere_n contain_v in_o power_n the_o side_n both_o of_o a_o pyramid_n and_o of_o a_o cube_fw-la inscribe_v in_o it_o for_o the_o power_n of_o the_o side_n of_o the_o pyramid_n be_v two_o three_o of_o the_o power_n of_o the_o diameter_n by_o the_o 13._o of_o this_o book_n and_o the_o power_n of_o the_o side_n of_o the_o cube_fw-la be_v by_o this_o proposition_n one_o three_o of_o the_o power_n of_o the_o say_a diameter_n wherefore_o the_o diameter_n of_o the_o sphere_n contain_v in_o power_n the_o side_n of_o the_o pyramid_n and_o of_o the_o cube_fw-la .._o ¶_o second_v corollary_n all_o the_o diameter_n of_o a_o cube_fw-la cut_v the_o one_o the_o other_o into_o two_o equal_a part_n in_o the_o centre_n of_o the_o sphere_n which_o contain_v the_o cube_fw-la and_o moreover_o those_o diameter_n do_v in_o the_o self_n same_o point_n cut_v into_o two_o equal_a part_n the_o right_a line_n which_o join_v together_o the_o centre_n of_o the_o opposite_a base_n as_o it_o be_v manifest_a to_o see_v by_o the_o right_a line_n lof_o for_o the_o angle_n lko_n and_o fio_n be_v equal_a by_o the_o 29._o of_o the_o first_o and_o it_o be_v prove_v that_o they_o be_v contain_v under_o equal_a side_n wherefore_o by_o the_o 4._o of_o the_o first_o the_o base_n lo_o and_o foe_n be_v equal_a in_o like_a sort_n may_v be_v prove_v that_o the_o rest_n of_o the_o right_a line_n which_o join_v together_o the_o centre_n of_o the_o opposite_a base_n do_v cut_v the_o one_o the_o other_o into_o two_o equal_a part_n in_o the_o centre_n o._n ¶_o the_o 4._o problem_n the_o 16._o proposition_n to_o make_v a_o icosahedron_n and_o to_o comprehend_v it_o in_o the_o sphere_n
and_o his_o great_a segment_n be_v the_o line_n no_o wherefore_o the_o square_n of_o the_o line_n n_n and_o so_o be_v treble_a to_o the_o square_n of_o the_o line_n no_o by_o the_o 4._o of_o this_o book_n but_o the_o line_n no_o be_v equal_a to_o the_o nb_n and_o the_o line_n so_o to_o the_o line_n sz_n wherefore_o the_o square_n of_o the_o line_n n_n and_o zs_n be_v treble_a to_o the_o square_n of_o the_o line_n nb_n wherefore_o the_o square_n of_o the_o line_n zs_n sn_a and_o nb_n be_v quadruple_a to_o the_o square_n of_o the_o line_n nb._n but_o unto_o the_o square_n of_o the_o line_n sn_a &_o nb_n by_o the_o 47._o of_o the_o first_o be_v equal_a the_o square_n of_o the_o line_n sb_n wherefore_o the_o square_n of_o the_o line_n b_n and_o sz_n that_o be_v the_o square_a of_o the_o line_n bz_o by_o the_o 47._o of_o the_o first_o for_o the_o angle_n zsb_n be_v a_o right_a angle_n by_o position_n be_v quadruple_a to_o the_o square_n of_o the_o line_n nb._n wherefore_o the_o line_n bz_o be_v double_a to_o the_o line_n bn_o by_o the_o corollary_n of_o the_o 20._o of_o the_o six_o but_o the_o line_n bc_o be_v also_o double_a to_o the_o line_n bn_o wherefore_o the_o line_n bz_o be_v equal_a to_o the_o line_n bc._n now_o forasmuch_o as_o these_o two_o line_n bv_n and_o we_o be_v equal_a to_o these_o two_o line_n bw_o and_o wc_n and_o the_o base_a bz_o be_v equal_a to_o the_o base_a bc_o therefore_o by_o the_o 8._o of_o the_o first_o the_o angle_n buz_n be_v equal_a to_o the_o angle_n bwc_n and_o in_o like_a sort_n by_o the_o 8._o of_o the_o first_o may_v we_o prove_v that_o the_o angle_n vzc_n be_v equal_a to_o the_o angle_n bwc_n prove_v first_o that_o the_o line_n cb_o and_o cv_o be_v equal_a which_o be_v prove_v equal_a by_o this_o that_o the_o line_n ns_n be_v equal_a to_o the_o line_n xr_n and_o therefore_o the_o line_n cr_n be_v equal_a to_o the_o line_n bs_n by_o the_o 47._o of_o the_o first_o wherefore_o also_o by_o the_o same_o the_o line_n cv_o be_v equal_a to_o the_o line_n bz_o that_o be_v to_o the_o line_n bc_o for_o the_o line_n bc_o &_o bz_n be_v prove_v equal_a wherefore_o the_o three_o angle_n bwc_n buz_n and_o vzc_n be_v equal_a the_o one_o to_o the_o other_o but_o if_o in_o a_o equilater_n pentagon_n figure_n there_o be_v three_o angle_n equal_a the_o one_o to_o the_o other_o the_o pentagon_n be_v by_o the_o 7._o of_o the_o thirteen_o equiangle_n wherefore_o the_o pentagon_n buzcw_o be_v equiangle_n and_o it_o be_v also_o prove_v that_o it_o be_v equilater_n wherefore_o the_o pentagon_n buzcw_o be_v both_o equilater_n &_o equiangle_n and_o it_o be_v make_v upon_o one_o of_o the_o side_n of_o the_o cube_fw-la namely_o upon_o bc._n demonstration_n if_o therefore_o upon_o every_o one_o of_o the_o twelve_o side_n of_o the_o cube_fw-la be_v use_v the_o like_a construction_n there_o shall_v then_o be_v make_v a_o dodecahedron_n contain_v under_o twelve_o pentagons_n equilater_n and_o equiangle_n now_o i_o say_v that_o the_o side_n of_o the_o dodecahedron_n be_v a_o irrational_a line_n of_o that_o kind_n which_o be_v call_v a_o residual_a line_n line_n for_o forasmuch_o as_o the_o line_n no_o be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n in_o the_o point_n r_o and_o his_o great_a segment_n be_v the_o line_n or_o and_o the_o line_n ox_n be_v also_o divide_v by_o a_o extreme_a and_o mean_a proportion_n in_o the_o point_n s_o and_o his_o great_a segment_n be_v the_o line_n os_fw-mi wherefore_o the_o whole_a line_n nx_o be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n and_o his_o great_a segment_n be_v the_o line_n rs._n for_o for_o that_o as_o the_o line_n on_o be_v to_o the_o line_n or_o so_o be_v the_o line_n or_o to_o the_o line_n nr_n and_o in_o the_o same_o proportion_n also_o be_v their_o double_n for_o the_o part_n of_o equemultiplices_fw-la have_v one_o and_o the_o self_n same_o proportion_n with_o the_o whole_a by_o the_o 15._o of_o the_o five_o wherefore_o as_o the_o line_n nx_o be_v to_o the_o line_n r_n so_o be_v the_o line_n r_n to_o both_o the_o line_n nr_n and_o sx_n add_v together_o but_o the_o line_n nx_o be_v great_a than_o the_o line_n r_n by_o both_o the_o line_n nr_n and_o sx_n add_v together_o wherefore_o the_o line_n nx_o be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n and_o his_o great_a segment_n be_v the_o line_n rs._n but_o the_o line_n rs_n be_v equal_a to_o the_o line_n we_o as_o have_v before_o be_v prove_v wherefore_o the_o line_n nx_o be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n and_o his_o great_a segment_n be_v the_o line_n we_o and_o forasmuch_o as_o the_o diameter_n of_o the_o sphere_n be_v rational_a and_o be_v in_o power_n treble_a to_o the_o side_n of_o the_o cube_fw-la by_o the_o 15._o of_o this_o book_n therefore_o the_o line_n nx_o be_v the_o side_n of_o the_o cube_fw-la be_v rational_a but_o if_o a_o rational_a line_n be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n either_o of_o the_o segment_n be_v by_o the_o 6._o of_o this_o book_n a_o irrational_a line_n of_o that_o kind_n which_o be_v call_v a_o residual_a line_n wherefore_o the_o line_n we_o be_v the_o side_n of_o the_o dodecahedron_n be_v a_o irrational_a line_n of_o that_o kind_n which_o be_v call_v a_o residual_a line_n wherefore_o there_o be_v make_v a_o dodecahedron_n and_o it_o be_v comprehend_v in_o the_o sphere_n give_v wherein_o the_o other_o solid_n be_v contain_v and_o it_o be_v prove_v that_o the_o side_n of_o the_o dodecahedron_n be_v a_o residual_a line_n which_o be_v require_v to_o be_v do_v and_o also_o to_o be_v prove_v ¶_o corollary_n hereby_o it_o be_v manifest_a that_o the_o side_n of_o a_o cube_fw-la be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n the_o great_a segment_n thereof_o be_v the_o side_n of_o the_o dodecahedron_n as_o it_o be_v manifest_a by_o the_o line_n we_o which_o be_v prove_v to_o be_v the_o great_a segment_n of_o the_o right_a line_n nx_o namely_o of_o the_o side_n of_o the_o cube_fw-la a_o further_a construction_n of_o the_o dodecahedron_n after_o flussas_n forasmuch_o as_o it_o have_v be_v prove_v that_o the_o pentagon_n buzcw_o be_v equilater_n and_o equiangle_n and_o touch_v one_o of_o the_o side_n of_o the_o cube_fw-la câ—d_v let_v we_o show_v also_o by_o what_o mean_v upon_o each_o of_o the_o 12._o side_n of_o the_o cube_fw-la may_v in_o like_a sort_n be_v apply_v pentagon_n join_v one_o to_o the_o other_o and_o compose_v the_o 12._o base_n of_o the_o dodecahedron_n draw_v in_o the_o former_a figure_n these_o right_a line_n a_z id_fw-la il_fw-fr ctk_n now_o forasmuch_o as_o the_o line_n pl_z be_v in_o the_o point_n ct_v divide_v like_o unto_o the_o line_n ph_n on_o or_o ox_n and_o upon_o the_o point_n t_o p_o ct_n be_v erect_v perpendicular_a line_n equal_a unto_o the_o line_n oy_fw-it and_o the_o rest_n namely_o unto_o the_o great_a segment_n and_o the_o line_n t_o w_n and_o ct_n i_o be_v prove_v parallel_n therefore_o the_o line_n wi_n and_o tct_n be_v parallel_n by_o the_o 7._o of_o the_o eleven_o and_o 33._o of_o the_o first_o wherefore_o also_o by_o the_o 9_o of_o the_o eleven_o the_o line_n wi_n and_o dc_o be_v parallel_n wherefore_o by_o the_o 7._o of_o the_o eleven_o cwid_v be_v a_o plain_a superficies_n and_o the_o triangle_n aid_n be_v a_o plain_a superficies_n by_o the_o 2._o of_o the_o eleven_o now_o it_o be_v manifest_a that_o the_o right_a line_n id_fw-la &_o ja_z be_v equal_a to_o the_o right_a line_n wc_n for_o the_o right_a line_n all_o &_o â—ct_v which_o be_v equal_a to_o the_o right_a line_n bh_o &_o ht_o do_v make_v the_o subtended_a line_n a_o ct_n and_o bt_v equal_a by_o the_o 4._o of_o the_o first_o and_o again_o forasmuch_o as_o the_o line_n bt_v and_o tw_n contain_v a_o right_a angle_n btw_n as_o also_o do_v the_o right_a line_n act_n and_o cti_n contain_v the_o right_a angle_n acti_fw-la for_o the_o right_a line_n with_o and_o ict_n be_v erect_v perpendicular_o unto_o one_o and_o the_o self_n same_o plain_n abcd_o by_o supposition_n and_o the_o square_n of_o the_o line_n bt_v and_o tw_n be_v equal_a to_o the_o square_n of_o the_o line_n act_n and_o cti_fw-la for_o it_o be_v prove_v that_o the_o line_n bt_n be_v equal_a to_o the_o line_n act_n and_o the_o line_n tw_n to_o the_o line_n cti_n and_o unto_o the_o square_n of_o the_o line_n bt_v and_o tw_n be_v equal_a the_o square_n of_o the_o line_n bw_o by_o the_o 47._o of_o the_o first_o likewise_o by_o the_o same_o unto_o the_o square_n of_o the_o line_n act_n and_o cti_n be_v equal_a the_o square_n of_o the_o
f._n and_o draw_v these_o right_a line_n favorina_n fb_o fc_o fd_o fe_o wherefore_o those_o line_n do_v divide_v the_o angle_n of_o the_o pentagon_n into_o two_o equal_a part_n in_o the_o point_n a_o b_o c_o d_o e_o by_o the_o 4._o of_o the_o first_o and_o forasmuch_o as_o the_o five_o angle_n that_o be_v at_o the_o point_n f_o aâ—e_fw-fr equal_a to_o four_o right_a angle_n by_o the_o corollary_n of_o the_o 15._o of_o the_o first_o and_o they_o be_v equal_v the_o one_o to_o the_o other_o by_o the_o 8._o of_o the_o first_o therefore_o one_o of_o those_o angle_n as_o â—or_a example_n sake_n the_o angle_n afb_o be_v a_o fiâ—th_v part_n less_o than_o a_o right_a angle_n wherefore_o the_o angle_n remain_v namely_o fab._n &_o abf_n be_v one_o right_a angle_n and_o a_o five_o part_n over_o but_o the_o angle_n fab._n be_v equal_a to_o the_o angle_n fbc_n wherefore_o the_o whole_a angle_n abc_n be_v one_o of_o the_o angle_n of_o the_o pentagon_n be_v a_o right_a angle_n and_o a_o five_o part_v more_o than_o a_o right_a angle_n which_o be_v require_v to_o be_v prove_v ¶_o a_o corollary_n add_v by_o flussas_n now_o let_v we_o teach_v how_o those_o five_o solid_n have_v each_o like_a inclination_n of_o their_o base_n â—iâ—st_v let_v we_o take_v a_o pyramid_n and_o divide_v one_o of_o the_o side_n thereof_o into_o two_o equal_a part_n and_o from_o the_o two_o angle_n opposite_a unto_o that_o side_n dâ—aw_v perpendiculars_n which_o shall_v fall_v upon_o the_o section_n by_o the_o corollary_n of_o the_o 12._o of_o the_o thirteen_o rational_a and_o at_o the_o say_a point_n of_o division_n as_o may_v easy_o be_v prove_v wherefore_o they_o shall_v contain_v the_o angâ—e_n of_o the_o inclination_n of_o the_o plain_n by_o the_o 4._o definition_n of_o the_o eleven_o which_o angle_n be_v subtend_v of_o the_o opposite_a side_n of_o the_o pyramid_n now_o forasmuch_o as_o the_o rest_n of_o the_o angle_n of_o the_o inclination_n of_o the_o plain_n of_o the_o pyramid_n be_v contain_v under_o two_o perpendicular_a line_n of_o the_o triangle_n and_o be_v subtend_v of_o the_o side_n of_o the_o pyramid_n it_o follow_v by_o the_o 8._o of_o the_o firâ—t_o that_o those_o angle_n be_v equal_a wherâ—foâ—e_o by_o the_o 5._o definition_n of_o the_o eleven_o the_o superficiece_n be_v in_o like_a sort_n incline_v the_o one_o to_o the_o other_o 〈…〉_o one_o of_o the_o side_n of_o a_o cube_n be_v divide_v into_o two_o equal_a part_n if_o from_o the_o say_a section_n be_v draw_v in_o two_o of_o the_o base_n thereof_o two_o perpendicular_a line_n they_o shall_v be_v parallel_n and_o equal_a to_o the_o side_n of_o the_o square_n which_o contain_v a_o right_a angle_n and_o forasmuch_o as_o all_o the_o angle_n of_o the_o base_n of_o the_o cube_n be_v right_a angle_n therefore_o those_o perpendicular_n fall_v upon_o the_o section_n of_o the_o side_n common_a to_o the_o two_o base_n shall_v contâ—yne_v a_o right_a angle_n by_o the_o 10._o of_o the_o eleven_o which_o self_n angle_n be_v the_o angle_n of_o inclination_n by_o the_o 4._o definition_n of_o the_o eleven_o and_o be_v subtend_v of_o the_o diameter_n of_o the_o base_a of_o the_o cube_n and_o by_o the_o same_o reason_n may_v we_o prove_v that_o the_o rest_n of_o the_o angle_n of_o the_o inclination_n of_o the_o base_n of_o the_o cube_fw-la be_v right_a angle_n wherefore_o the_o inclination_n of_o the_o superficiece_n of_o the_o cube_fw-la the_o one_o to_o the_o other_o be_v equal_a by_o the_o 5._o definition_n of_o the_o eleven_o in_o a_o octohedron_n take_v the_o diameter_n which_o couple_v the_o two_o opposite_a angle_n incline_v and_o from_o those_o opposite_a angle_n draw_v to_o one_o and_o the_o selâ—e_a same_o side_n of_o the_o octohedron_n in_o two_o base_n thereof_o two_o perpendicular_a line_n which_o shall_v divide_v that_o side_n into_o two_o equal_a part_n and_o perpendicular_o by_o the_o corollary_n of_o the_o 12._o of_o the_o thirteen_o wherefore_o those_o perpendicular_n shall_v contain_v the_o angle_n of_o the_o inclination_n of_o the_o base_n by_o the_o 4._o definition_n of_o the_o eleven_o and_o the_o same_o angle_n be_v subtend_v of_o the_o diameter_n of_o the_o octohedron_n wherefore_o the_o rest_n of_o the_o angle_n after_o the_o same_o manner_n describe_v in_o the_o rest_n of_o the_o base_n be_v comprehend_v and_o subtend_v of_o equal_a side_n shall_v by_o the_o 8._o of_o the_o first_o be_v equal_a the_o one_o to_o the_o other_o and_o therefore_o the_o inclination_n of_o the_o plain_n in_o the_o octohedron_n shall_v by_o the_o 5._o definition_n of_o the_o eleven_o be_v equal_a in_o a_o icosahedron_n let_v there_o be_v draw_v from_o the_o angle_n of_o two_o of_o the_o base_n incline_v to_o one_o side_n common_a to_o both_o the_o say_v base_n perpendicular_n which_o shall_v contain_v the_o angle_n of_o the_o inclination_n of_o the_o base_n by_o the_o 4._o definition_n of_o the_o eleven_o which_o angle_n be_v subtend_v of_o the_o right_a line_n which_o subtend_v the_o angle_n of_o the_o pentagon_n which_o contain_v five_o side_n of_o the_o icosahedron_n by_o the_o 16._o of_o this_o book_n for_o it_o couple_v the_o two_o opposite_a angle_n of_o the_o triangle_n which_o be_v join_v together_o wherefore_o the_o rest_n of_o the_o angle_n of_o the_o inclination_n of_o the_o base_n be_v after_o the_o same_o manner_n find_v out_o they_o shall_v be_v contain_v under_o equal_a side_n and_o subtend_v of_o equal_a base_n and_o therefore_o by_o the_o 8._o of_o the_o fiâ—st_n those_o angle_n shall_v be_v equal_a wherefore_o also_o all_o the_o inclination_n of_o the_o base_n of_o the_o icosahedron_n the_o one_o to_o the_o other_o shalbâ—_n equal_a by_o the_o 5._o definition_n of_o the_o eleven_o in_o a_o dodecahedron_n incline_v from_o the_o two_o opposite_a angle_n of_o two_o next_o pentagon_n draw_v to_o their_o common_a side_n perpendicular_a line_n pass_v by_o the_o centre_n of_o the_o say_a pentagon_n which_o shall_v where_o they_o fall_v divide_v the_o side_n into_o two_o equal_a part_n by_o the_o 3._o of_o the_o three_o for_o the_o base_n of_o a_o dodecahedron_n be_v contain_v in_o a_o circle_n and_o the_o angle_n contaynâ—d_v under_o those_o perpendicular_a line_n be_v the_o inclination_n of_o those_o base_n by_o the_o 4._o definition_n of_o the_o eleven_o and_o the_o foresay_a opposite_a angle_n be_v couple_v by_o a_o right_a line_n equal_a to_o the_o right_a line_n which_o couple_v the_o opposite_a section_n into_o two_o equal_a part_n of_o the_o side_n of_o the_o dodecahedron_n by_o the_o 33._o of_o the_o first_o for_o they_o couple_v together_o the_o half_a side_n of_o the_o dodecahedron_n which_o half_n be_v parallel_n and_o equal_a by_o the_o 3._o corollary_n of_o the_o 17._o of_o this_o book_n which_o couple_v line_n also_o be_v equal_a by_o the_o same_o corollary_n wherefore_o the_o angle_n be_v contain_v of_o equal_a perpendicular_a line_n and_o subtend_v of_o equal_a couple_a line_n shall_v by_o the_o 8._o of_o the_o first_o be_v equal_a and_o they_o be_v the_o angle_n of_o the_o inclination_n wherefore_o the_o base_n of_o the_o dodecahedron_n be_v in_o like_a sort_n incline_v the_o one_o to_o the_o other_o by_o the_o 5._o definition_n of_o the_o eleven_o flussas_n after_o this_o teach_v how_o to_o know_v the_o rationality_n or_o irrationality_n of_o the_o side_n of_o the_o triangle_n which_o contain_v the_o angle_n of_o the_o inclination_n of_o the_o superficiece_n of_o the_o foresay_a body_n in_o a_o pyramid_n the_o angle_n of_o the_o inclination_n be_v contain_v under_o two_o perpâ—dicular_a line_n of_o the_o triangle_n rational_a and_o be_v subtend_v of_o the_o side_n of_o the_o pyramid_n now_o the_o side_n of_o the_o pyramid_n be_v in_o power_n sesquitertia_fw-la to_o the_o perpendicular_a line_n by_o the_o corollary_n of_o the_o 12._o of_o this_o book_n and_o therefore_o the_o triangle_n contain_v of_o those_o perpendicular_a line_n and_o the_o side_n of_o pyramid_n have_v his_o side_n rational_a &_o commensurable_a in_o power_n the_o one_o to_o the_o other_o forasmuch_o as_o the_o two_o side_n of_o a_o cube_n or_o right_a line_n equal_a to_o they_o subtend_v under_o the_o diameter_n of_o one_o of_o the_o base_n rational_a do_v make_v the_o angle_n of_o the_o inclination_n and_o the_o diameter_n of_o the_o cube_fw-la be_v in_o power_n sesquialter_fw-la to_o the_o diameter_n of_o the_o base_a which_o diameter_n of_o the_o base_a be_v in_o power_n double_a to_o the_o side_n by_o the_o 47._o of_o the_o first_o therefore_o those_o line_n be_v rational_a and_o commensurable_a in_o power_n in_o a_o octohedron_n rational_a who_o two_o perpendicular_n of_o the_o base_n contain_v the_o angle_n of_o the_o inclination_n of_o the_o octohedron_n which_o angle_n also_o be_v subtend_v of_o the_o diameter_n of_o the_o octohedron_n the_o diameter_n be_v in_o power_n
or_o tetrahedron_n adc_o be_v to_o the_o pyramid_n akg_v as_o 64._o be_v to_o â—7_o which_o be_v triple_a to_o the_o proportion_n of_o 4._o to_o 3._o and_o forasmuch_o as_o the_o line_n ac_fw-la be_v unto_o the_o line_n agnostus_n in_o length_n sesquitertia_fw-la of_o what_o part_n the_o line_n ac_fw-la contain_v in_o power_n 64_o of_o the_o same_o part_n do_v the_o line_n agnostus_n contain_v in_o power_n 36._o for_o by_o the_o 2._o of_o the_o six_o the_o proportion_n of_o the_o power_n or_o square_n be_v duple_n to_o the_o porportion_n of_o the_o side_n which_o be_v as_o 64._o be_v to_o 48._o now_o then_o upon_o the_o line_n r_n which_o let_v be_v equal_a to_o the_o line_n agnostus_n let_v there_o be_v a_o equilater_n triangle_n qr_v describe_v by_o the_o first_o of_o the_o first_o construction_n and_o from_o the_o angle_n qui_fw-fr draw_v to_o the_o base_a r_n a_o perpendicular_a line_n qt_o and_o extend_v the_o line_n r_n to_o the_o point_n x._o and_o as_o 27._o be_v to_o 64._o so_o by_o the_o corollary_n of_o the_o 6._o of_o the_o ten_o let_v the_o line_n r_v be_v to_o the_o line_n rx_n and_o divide_v â—he_z line_n rx_n into_o two_o equal_a part_n in_o the_o point_n v_o and_o draw_v the_o line_n qu._n and_o forasmuch_o as_o the_o line_n rs_n be_v equal_a to_o the_o line_n agnostus_n of_o what_o part_n the_o line_n ac_fw-la contain_v in_o power_n 64._o of_o the_o same_o part_n the_o line_n rs_n contain_v in_o power_n 36._o for_o it_o be_v prove_v that_o the_o line_n agnostus_n contain_v in_o power_n 36._o of_o those_o part_n demonstration_n and_o of_o what_o part_n the_o line_n rs_n contain_v in_o power_n 36_o of_o the_o same_o part_n the_o 〈◊〉_d qt_o contain_v in_o power_n â—7_o by_o the_o corollary_n of_o the_o 12._o of_o the_o thiâ—tenth_o wherefore_o of_o what_o part_n the_o line_n az_o contain_v in_o power_n 64._o of_o the_o same_o part_n the_o line_n qt_o contain_v in_o power_n 27._o wherefore_o the_o right_a line_n qt_o shall_v be_v equal_a to_o the_o right_a line_n ln_o by_o supposition_n again_o forasmuch_o as_o the_o line_n rs_n be_v put_v equal_a to_o the_o line_n agnostus_n and_o of_o what_o part_n the_o line_n rs_n contain_v in_o length_n 27._o of_o the_o same_o part_n be_v the_o line_n rx_n put_v to_o contain_v in_o length_n 64._o and_o of_o what_o part_n the_o line_n rx_n contain_v in_o length_n 64._o of_o the_o same_o the_o line_n ac_fw-la which_o be_v in_o length_n sesquitertia_fw-la to_o the_o line_n agnostus_n or_o rs_n contain_v 36._o wherefore_o the_o line_n rv_n which_o be_v the_o half_a of_o the_o line_n rx_n contain_v in_o length_n of_o the_o same_o part_n 32._o of_o which_o the_o line_n ac_fw-la contain_v in_o length_n 36._o wherefore_o the_o line_n rv_n be_v to_o the_o line_n ac_fw-la subsesquioctava_fw-la and_o therefore_o the_o line_n rv_n be_v equal_a to_o the_o line_n lm_o which_o be_v also_o subsesquioctava_fw-la to_o the_o same_o line_n ac_fw-la and_o forasmuch_o as_o the_o line_n nl_o be_v equal_a to_o the_o line_n qt_n and_o the_o line_n lm_o to_o the_o line_n rv_n as_o before_o have_v be_v prove_v the_o rectangle_n parallelogram_n contain_v under_o the_o line_n qt_a and_o rv_n shall_v be_v equal_a to_o the_o rectangle_n parallelogram_n contain_v under_o the_o line_n nl_o which_o be_v in_o power_n â—7_o 64_o to_o the_o side_n ac_fw-la and_o under_o the_o line_n lm_o which_o be_v in_o length_n subsesquioctava_fw-la to_o the_o same_o side_n ac_fw-la but_o that_o which_o be_v contain_v under_o the_o line_n qt_a and_o rv_n be_v double_a to_o the_o triangle_n qur_n by_o the_o 41._o of_o the_o first_o and_o to_o the_o same_o triangle_n qur_n be_v the_o triangle_n qxr_fw-fr duple_n by_o the_o first_o of_o the_o six_o wherefore_o the_o whole_a triangle_n qxr_n be_v equal_a to_o that_o which_o be_v contain_v under_o the_o line_n qt_a and_o rv_n and_o therefore_o be_v equal_a to_o the_o parallelogram_n mn_v and_o forasmuch_o as_o the_o line_n rx_n by_o supposition_n contain_v in_o length_n 64._o of_o those_o part_n of_o which_o the_o line_n rs_n contain_v 27_o and_o the_o triangle_n qrx_n and_o qr_n be_v by_o the_o first_o of_o the_o six_o in_o the_o proportion_n of_o their_o base_n that_o be_v as_o 64._o be_v to_o 27_o but_o as_o 64._o be_v to_o 27._o so_o be_v the_o pyramid_n or_o tetrahedron_n adc_o to_o the_o pyramid_n akg_v wherefore_o as_o the_o parallelogram_n nm_o or_o the_o triangle_n qrx_n be_v to_o the_o triangle_n qr_n so_o iâ—_z the_o pyramid_n adc_o to_o the_o pyramid_n akg_v and_o forasmuch_o as_o the_o semidiameter_n ah_o be_v the_o altitude_n of_o the_o pyramid_n akg_v and_o also_o of_o the_o two_o equal_a and_o like_a pyramid_n of_o the_o octohedron_n which_o have_v their_o common_a base_a in_o the_o square_a of_o the_o octohedron_n by_o the_o corollary_n of_o the_o 14._o of_o the_o thirteen_o therefore_o as_o the_o base_a of_o the_o pyramid_n akg_v which_o be_v the_o triangle_n qr_n be_v to_o two_o square_n of_o the_o octohâ—dron_n that_o be_v to_o the_o square_n of_o the_o diameter_n ab_fw-la which_o be_v equal_a to_o those_o square_n by_o the_o 47._o of_o the_o first_o so_o be_v the_o pyramid_n akg_v to_o the_o octohedron_n aeb_fw-mi by_o the_o 6._o of_o the_o twelve_o and_o forasmuch_o as_o the_o parallelogram_n mn_o be_v to_o the_o base_a qr_n as_o the_o pyramid_n adc_o be_v to_o the_o pyramid_n akg_v and_o the_o base_a qr_n be_v to_o the_o square_n of_o the_o line_n be_v as_o the_o pyramid_n akg_n be_v to_o the_o octohedron_n aeb_fw-mi therefore_o by_o proportion_n of_o equality_n taâ—ing_v away_o thâ—_z meaneâ—_n by_o the_o 22._o of_o the_o five_o as_o the_o parallelogram_n nm_o be_v to_o the_o square_n of_o the_o line_n be_v so_o be_v the_o pyramid_n adc_o to_o the_o octohedron_n aeb_fw-mi inscribe_v in_o one_o and_o the_o self_n same_o sphere_n but_o the_o parallelogram_n nm_o be_v contain_v under_o the_o line_n nl_o which_o by_o supposition_n be_v in_o power_n â—7_o â—â—_o to_o ac_fw-la the_o side_n of_o the_o tetrahedron_n adc_o and_o under_o the_o line_n lm_o which_o be_v also_o by_o supposition_n in_o length_n subsesquioctava_fw-la to_o the_o same_o line_n ac_fw-la wherefore_o a_o tetrahedron_n &_o a_o octohedron_n inscribe_v in_o one_o and_o the_o self_n same_o sphere_n be_v in_o proportion_n as_o the_o rectangle_n parallelogram_n contain_v under_o the_o line_n which_o contain_v in_o power_n 27._o sixty_o four_o part_n of_o the_o side_n of_o the_o tetrahedron_n and_o under_o the_o line_n which_o be_v subsesquioctava_fw-la to_o the_o same_o side_n of_o the_o tetrahedron_n be_v to_o the_o square_n of_o the_o diameter_n of_o the_o sphere_n which_o be_v require_v to_o be_v prove_v ¶_o the_o 12._o proposition_n campane_n if_o a_o cube_fw-la be_v contain_v in_o a_o sphere_n the_o square_a of_o the_o diameter_n double_v be_v equal_a to_o all_o the_o superficiece_n of_o the_o cube_fw-la take_v together_o and_o a_o perpendicular_a line_n draw_v from_o the_o centre_n of_o the_o sphere_n to_o any_o base_a of_o the_o cube_fw-la be_v equal_a to_o half_a the_o side_n of_o the_o cube_fw-la part_n for_o forasmuch_o as_o by_o the_o 15._o of_o the_o thirteen_o the_o diameter_n of_o the_o sphere_n be_v in_o power_n triple_a to_o the_o side_n of_o thâ—_z cube_fw-la therefore_o the_o square_n of_o the_o diameter_n double_v be_v sextuple_a to_o the_o base_a of_o the_o same_o cube_fw-la but_o the_o sextuple_a of_o the_o power_n of_o one_o of_o the_o side_n contain_v the_o whole_a superficies_n of_o the_o cube_fw-la 〈◊〉_d or_o the_o cube_fw-la be_v compose_v of_o six_o square_a superficiece_n by_o the_o 2d_o definition_n of_o thâ—_z eleven_o who_o side_n therefore_o be_v equal_a wherefore_o the_o square_a of_o the_o diameter_n double_v be_v equal_a to_o the_o whole_a superficies_n of_o the_o cube_fw-la and_o forasmuch_o as_o the_o diameter_n of_o the_o cube_fw-la and_o the_o line_n which_o fall_v perpendicular_o upon_o the_o opposite_a base_n of_o the_o cube_fw-la part_n do_v cut_v the_o one_o the_o other_o into_o two_o equal_a part_n in_o the_o centre_n of_o the_o sphere_n which_o contain_v the_o cube_fw-la by_o the_o 2._o corollary_n of_o the_o 15._o of_o the_o thirteen_o and_o the_o whole_a right_a line_n which_o couple_v the_o centre_n of_o the_o opposite_a base_n be_v equal_a to_o the_o side_n of_o the_o cube_fw-la by_o the_o 33._o of_o the_o first_o for_o it_o couple_v the_o equal_a and_o parallel_a semidiameter_n of_o the_o base_n therefore_o the_o half_a thereof_o shall_v be_v equal_a to_o the_o half_a of_o the_o side_n of_o the_o cube_fw-la by_o the_o 15._o of_o the_o five_o if_o therefore_o a_o cube_fw-la be_v contain_v in_o a_o sphere_n the_o square_a of_o the_o diameter_n double_v be_v equal_a to_o all_o the_o superficiece_n of_o the_o cube_fw-la take_v
together_o and_o a_o perpendicular_a line_n draw_v from_o the_o centre_n of_o the_o sphere_n to_o any_o base_a of_o the_o cube_fw-la be_v equal_a to_o half_a the_o side_n of_o the_o cube_fw-la which_o be_v require_v to_o be_v prouâ—d_v ¶_o a_o corollary_n if_o two_o three_o of_o the_o power_n of_o the_o diameter_n of_o the_o sphere_n be_v multiply_v into_o the_o perpendicular_a line_n equal_a to_o half_a the_o side_n of_o the_o cube_fw-la campane_n there_o shall_v be_v produce_v a_o solid_a equal_a to_o the_o solid_a of_o the_o cube_fw-la for_o it_o be_v before_o manifest_v that_o two_o three_o part_n of_o the_o power_n of_o the_o diameter_n of_o the_o sphere_n be_v equal_a to_o two_o base_n of_o the_o cube_fw-la if_o therefore_o unto_o each_o of_o those_o two_o three_o be_v apply_v half_o the_o altitude_n of_o the_o cube_fw-la they_o shall_v make_v each_o of_o those_o solid_n equal_a to_o half_o of_o the_o cube_fw-la by_o the_o 31._o of_o the_o eleven_o for_o they_o have_v equal_a base_n wherefore_o two_o of_o those_o solid_n be_v equal_a to_o the_o whole_a cube_fw-la you_o shall_v understand_v gentle_a reader_n that_o campane_n in_o his_o 14._o book_n of_o euclides_n element_n have_v 18._o proposition_n with_o diverse_a corollary_n follow_v of_o they_o some_o of_o which_o proposition_n and_o corollary_n i_o have_v before_o in_o the_o twelve_o and_o thirteen_o book_n add_v out_o of_o flussas_n as_o corollary_n which_o thing_n also_o i_o have_v note_v on_o the_o side_n of_o those_o corollary_n namely_o with_o what_o proposition_n or_o corollary_n of_o campane_n 14._o book_n they_o do_v agree_v the_o rest_n of_o his_o 18._o proposition_n and_o corollary_n be_v contain_v in_o the_o twelve_o former_a proposition_n and_o corollary_n of_o this_o 14._o book_n after_o flussas_n where_o you_o may_v see_v on_o the_o side_n of_o each_o proposition_n and_o corollary_n with_o what_o proposition_n and_o corollary_n of_o campane_n they_o agree_v but_o the_o eight_o proposition_n follow_v together_o with_o their_o corollary_n flussas_n have_v add_v of_o himself_o as_o he_o himself_o affirm_v the_o 13._o proposition_n one_o and_o the_o self_n same_o circle_n contain_v both_o the_o square_n of_o a_o cube_fw-la and_o the_o triangle_n of_o a_o octohedron_n describe_v in_o one_o and_o the_o self_n same_o sphere_n svppose_v that_o there_o be_v a_o cube_fw-la abg_o and_o a_o octohedron_n def_n describe_v in_o one_o and_o the_o self_n same_o sphere_n who_o diameter_n let_v be_v ab_fw-la or_o dh_o and_o let_v the_o line_n draw_v from_o the_o centre_n that_o be_v the_o semidiameter_n of_o the_o circle_n which_o ctonaine_v the_o base_n of_o those_o solid_n â—_o be_v ca_n and_o id_fw-la then_o i_o say_v that_o the_o line_n ca_n and_o id_fw-la be_v equal_a forasmuch_o as_o ab_fw-la the_o diameter_n of_o the_o sphere_n which_o contain_v the_o cube_fw-la demonstration_n be_v in_o power_n triple_a to_o bg_o the_o side_n of_o the_o cube_fw-la by_o the_o 15._o of_o the_o thirteen_o unto_o which_o side_n ag_z the_o diameter_n of_o the_o base_a of_o the_o cube_fw-la be_v in_o power_n double_a by_o the_o 47._o of_o the_o first_o which_o line_n agnostus_n be_v also_o the_o diameter_n of_o the_o circle_n which_o contain_v the_o base_a by_o the_o 9_o of_o the_o four_o therefore_o ab_fw-la the_o diameter_n of_o the_o sphere_n be_v in_o power_n sesquialter_fw-la to_o the_o line_n agnostus_n namely_o of_o what_o part_n the_o line_n ab_fw-la contain_v in_o power_n 12._o of_o the_o same_o the_o line_n agnostus_n shall_v contain_v in_o power_n 8._o and_o therefore_o the_o right_a line_n ac_fw-la which_o be_v draw_v from_o the_o centre_n of_o the_o circle_n to_o the_o circumference_n contain_v in_o power_n of_o the_o same_o part_n 2._o wherefore_o the_o diameter_n of_o the_o sphere_n be_v in_o power_n sextuple_a to_o the_o line_n which_o be_v draw_v from_o the_o centre_n to_o the_o circumference_n of_o the_o circle_n which_o contain_v the_o square_a of_o the_o cube_fw-la but_o the_o diameter_n of_o the_o self_n same_o sphere_n which_o contain_v the_o octohedron_n be_v one_o and_o the_o self_n same_o with_o the_o diameter_n of_o the_o cube_fw-la namely_o dh_o be_v equal_a to_o ab_fw-la and_o the_o same_o diameter_n be_v also_o the_o diameter_n of_o the_o square_n which_o be_v make_v of_o the_o side_n of_o the_o octohedron_n wherefore_o the_o say_a diameter_n be_v in_o power_n double_a to_o the_o side_n of_o the_o same_o octohedron_n by_o the_o 14._o of_o the_o thirteen_o but_o the_o side_n df_o be_v in_o power_n triple_a to_o the_o line_n draw_v from_o the_o centre_n to_o the_o circumference_n of_o the_o circle_n which_o contain_v the_o triangle_n of_o the_o octohedron_n namely_o to_o the_o line_n id_fw-la by_o the_o 12._o of_o the_o thirteen_o wherefore_o the_o self_n same_o diameter_n ab_fw-la or_o dh_o which_o be_v in_o power_n sextuple_a to_o the_o line_n draw_v from_o the_o centre_n to_o the_o circumference_n of_o the_o circle_n which_o contain_v the_o square_a of_o the_o cube_fw-la be_v also_o sextuple_a to_o the_o line_n id_fw-la draw_v from_o the_o centre_n to_o the_o circumference_n of_o the_o circle_n which_o contain_v the_o triangle_n of_o the_o octohedron_n wherefore_o the_o line_n draw_v from_o the_o centre_n of_o the_o circle_n to_o the_o circumference_n which_o contain_v the_o base_n of_o the_o cube_fw-la and_o of_o the_o octohedron_n be_v equal_a and_o therefore_o the_o circle_n be_v equal_a by_o the_o first_o definition_n of_o the_o three_o wherefore_o one_o and_o the_o self_n same_o circle_n contain_v &c._a &c._a as_o in_o the_o proposition_n which_o be_v require_v to_o be_v prove_v a_o corollary_n hereby_o it_o be_v manifest_a that_o perpendicular_n couple_v together_o in_o a_o sphere_n the_o centre_n of_o the_o circle_n which_o contain_v the_o opposite_a base_n of_o the_o cube_fw-la and_o of_o the_o octohedron_n be_v equal_a for_o the_o circle_n be_v equal_a by_o the_o second_o corollary_n of_o the_o assumpt_n of_o the_o 16._o of_o the_o twelve_o and_o the_o line_n which_o pass_v by_o the_o centre_n of_o the_o sphere_n couple_v together_o the_o centre_n of_o the_o base_n be_v also_o equal_a by_o the_o first_o corollary_n of_o the_o same_o wherefore_o the_o perpendicular_a which_o couple_v together_o the_o opposite_a base_n of_o the_o octohedron_n be_v equal_a to_o the_o side_n of_o the_o cube_fw-la for_o either_o of_o they_o be_v the_o altitude_n erect_v the_o 14._o proposition_n a_o octohedron_n be_v to_o the_o triple_a of_o a_o tetrahedron_n contain_v in_o one_o and_o the_o self_n same_o sphere_n in_o that_o proportion_n that_o their_o side_n be_v svppose_v that_o there_o be_v a_o octohedron_n abcd_o and_o a_o tetrahedron_n efgh_o upon_o who_o base_a fgh_n erect_v a_o prism_n construction_n which_o be_v do_v by_o erect_v from_o the_o angle_n of_o the_o base_a perpendicular_a line_n equal_a to_o the_o altitude_n of_o the_o tetrahedron_n which_o prism_n shall_v be_v triple_a to_o the_o tetrahedron_n efgh_o by_o the_o first_o corollary_n of_o the_o 7._o of_o the_o twelve_o then_o i_o say_v that_o the_o octohedron_n abcd_o be_v to_o the_o prism_n which_o be_v triple_a to_o the_o tetrahedron_n efgh_o as_o the_o side_n bc_o be_v to_o the_o side_n fg._n for_o forasmuch_o as_o the_o side_n of_o the_o opposite_a base_n of_o the_o octohedron_n demonstration_n be_v right_a line_n touch_v the_o one_o the_o other_o and_o be_v parellel_n to_o other_o right_a line_n touch_v the_o one_o the_o other_o for_o the_o side_n of_o the_o square_n which_o be_v compose_v of_o the_o side_n of_o the_o octohedron_n be_v opposite_a wherefore_o the_o opposite_a plain_a triangle_n namely_o abc_n &_o kid_n shall_v be_v parallel_n and_o so_o the_o rest_n by_o the_o 15._o of_o the_o eleven_o let_v the_o diameter_n of_o the_o octohedron_n be_v the_o line_n ad._n now_o then_o the_o whole_a octohedron_n be_v cut_v into_o four_o equal_a and_o like_a pyramid_n set_v upon_o the_o base_n of_o the_o octohedron_n and_o have_v the_o same_o altitude_n with_o it_o &_o be_v about_o the_o diameter_n ad_fw-la namely_o the_o pyramid_n set_v upon_o the_o base_a bid_v and_o have_v his_o top_n the_o point_n a_o and_o also_o the_o pyramid_n set_v upon_o the_o base_a bcd_o have_v his_o top_n the_o same_o point_n a._n likewise_o the_o pyramid_n set_v upon_o the_o base_a ikd_v &_o have_v his_o top_n the_o same_o point_n a_o and_o moreover_o the_o pyramid_n set_v upon_o the_o base_a ckd_v and_o have_v his_o top_n the_o former_a point_n a_o which_o pyramid_n shall_v be_v equal_a by_o the_o 8._o definition_n of_o the_o eleven_o for_o they_o each_o consist_v of_o two_o base_n of_o the_o octohedron_n and_o of_o two_o triangle_n contain_v under_o the_o diameter_n ad_fw-la and_o two_o side_n of_o the_o octohedron_n wherefore_o the_o prism_n which_o be_v set_v upon_o the_o base_a of_o the_o octohedron_n
and_o have_v the_o same_o altitude_n with_o it_o name_o the_o altitude_n of_o the_o parallel_a base_n as_o it_o be_v manifest_a by_o the_o former_a be_v equal_a to_o three_o of_o those_o pyramid_n of_o the_o octohedron_n by_o the_o first_o corollary_n of_o the_o seven_o of_o the_o twelve_o wherefore_o that_o prism_n shall_v have_v to_o the_o other_o prism_n under_o the_o same_o altitude_n compose_v of_o the_o 4._o pyramid_n of_o the_o whole_a octohedron_n the_o proportion_n of_o the_o triangular_a base_n by_o the_o 3._o corollary_n of_o the_o same_o and_o forasmuch_o as_o 4._o pyramid_n be_v unto_o 3._o pyramid_n in_o sesquitercia_fw-la proportion_n therefore_o the_o triangule_a base_a of_o the_o prism_n which_o contain_v 4._o pyramid_n be_v in_o sesquitertia_fw-la proportion_n to_o the_o base_a of_o the_o prism_n which_o contain_v three_o pyramid_n of_o the_o same_o octohedron_n and_o be_v set_v upon_o the_o base_a of_o the_o octohedron_n and_o under_o the_o altitude_n thereof_o that_o be_v in_o sesquitercia_fw-la proportion_n to_o the_o base_a of_o the_o octohedron_n but_o the_o base_a of_o the_o same_o octohedron_n be_v in_o sesquitertia_fw-la proportion_n to_o the_o base_a of_o the_o pyramid_n by_o the_o â—enth_n of_o this_o book_n wherefore_o the_o triangular_a base_n namely_o of_o the_o prism_n which_o contain_v four_o pyramid_n of_o the_o octohedron_n and_o be_v under_o the_o altitude_n thereof_o be_v equal_a to_o the_o triangular_a base_n of_o the_o prism_n which_o contain_v three_o pyramid_n under_o the_o altitude_n of_o the_o pyramid_n efgh_o but_o the_o prism_n of_o the_o octohedron_n be_v equal_a to_o the_o octohedron_n and_o the_o prism_n of_o the_o pyramid_n efgh_o be_v prove_v triple_a to_o the_o same_o pyramid_n efgh_o now_o than_o the_o prism_n set_v upon_o equal_a base_n be_v the_o one_o to_o the_o other_o as_o their_o altitude_n be_v by_o the_o corollary_n of_o the_o 25._o of_o the_o eleven_o namely_o as_o be_v the_o parallelipidedon_n their_o double_n by_o the_o corollary_n of_o the_o 31._o of_o the_o eleven_o but_o the_o altitude_n of_o the_o octohedron_n be_v equal_a to_o the_o side_n of_o the_o cube_fw-la contain_v in_o the_o same_o sphere_n by_o the_o corollary_n of_o the_o 13._o of_o this_o book_n and_o the_o side_n of_o the_o cube_fw-la be_v in_o power_n to_o the_o altitude_n of_o the_o tetrahedon_n in_o that_o proportion_n that_o 12._o be_v to_o 16_o by_o the_o 18._o of_o the_o thirteen_o and_o the_o side_n of_o the_o octohedron_n be_v to_o the_o side_n of_o the_o pyramid_n in_o that_o proportion_n that_o 18._o be_v to_o 24._o by_o the_o same_o 18._o of_o the_o thirteen_o which_o proportion_n be_v one_o &_o the_o self_n same_o with_o the_o proportion_n of_o 12._o to_o 16._o wherefore_o that_o prism_n which_o be_v equal_a to_o the_o octohedron_n be_v to_o the_o prism_n which_o be_v triple_a to_o the_o tetrahedron_n in_o that_o proportion_n that_o the_o altitude_n or_o that_o the_o side_n be_v wherefore_o a_o octohedron_n be_v to_o the_o triple_a of_o a_o tetrahedron_n contain_v in_o one_o and_o the_o self_n same_o sphere_n in_o that_o proportion_n that_o their_o side_n be_v which_o be_v require_v to_o be_v demonstrate_v a_o corollary_n the_o side_n of_o a_o tetrahedron_n &_o of_o a_o octohedron_n be_v proportional_a with_o their_o altitude_n for_o the_o side_n &_o altitude_n be_v in_o power_n sesquitercia_fw-la moreover_o the_o diameter_n of_o the_o sphere_n be_v to_o the_o side_n of_o the_o tetrahedron_n as_o the_o side_n of_o the_o octohedron_n be_v to_o the_o â—â—de_n of_o the_o cubeâ—_n namely_o the_o power_n of_o each_o be_v in_o sesquialter_fw-la proportion_n by_o the_o 18._o of_o the_o thirteen_o the_o 15._o proposition_n if_o a_o rational_a line_n contain_v in_o power_n two_o line_n make_v the_o whole_a and_o the_o great_a segment_n and_o again_o contain_v in_o power_n two_o line_n make_v the_o whole_a and_o the_o less_o segment_n the_o great_a segment_n shall_v be_v the_o side_n of_o the_o icosahedron_n and_o the_o less_o segment_n shall_v be_v the_o side_n of_o the_o dodecahedron_n contain_v in_o one_o and_o the_o self_n same_o sphere_n svppose_v that_o agnostus_n be_v the_o diameter_n of_o the_o sphere_n which_o contain_v the_o icosahedron_n abgc_n and_o let_v bg_o subtend_v the_o side_n of_o the_o pentagon_n describe_v of_o the_o side_n of_o the_o icosahedron_n by_o the_o 16._o of_o the_o thirteen_o moreover_o upon_o the_o same_o diameter_n agnostus_n or_o df_o equal_a unto_o it_o construction_n let_v there_o be_v describe_v a_o dodecahedron_n defh_o by_o the_o 1â—_o of_o the_o thirteen_o who_o opposite_n side_n ed_z and_o fh_o let_v be_v cut_v into_o two_o equal_a part_n in_o the_o point_n i_o and_o king_n and_o draw_v a_o line_n from_o i_o to_o k._n and_o let_v the_o line_n of_o couple_n two_o of_o the_o opposite_a angle_n of_o the_o base_n which_o be_v join_v together_o then_o i_o say_v that_o ab_fw-la the_o side_n of_o the_o icosahedron_n be_v the_o great_a segment_n which_o the_o diameter_n agnostus_n contain_v in_o power_n together_o with_o the_o whole_a line_n and_o line_n ed_z be_v the_o less_o segment_n which_o the_o same_o diameter_n agnostus_n or_o df_o contain_v in_o power_n together_o with_o the_o whole_a demonstration_n for_o forasmuch_o as_o the_o opposite_a side_n ab_fw-la and_o gc_o of_o the_o icosahedron_n be_v couple_v by_o the_o diameter_n agnostus_n and_o bc_o be_v equal_a &_o parallel_n by_o the_o 2._o corollary_n of_o the_o 16._o of_o the_o thirteen_o the_o right_a line_n bg_o &_o ac_fw-la which_o couple_v they_o together_o be_v equal_a &_o parallel_n by_o the_o 33._o of_o the_o first_o moreover_o the_o angle_n bac_n &_o abg_o be_v subtend_v of_o equal_a diamâ—ters_n shall_v by_o the_o 8._o of_o the_o first_o be_v equal_a &_o by_o the_o 29_o of_o the_o 〈◊〉_d they_o shall_v be_v right_a angle_n wherefore_o the_o right_a line_n agnostus_n 〈◊〉_d in_o power_n the_o too_o line_n ab_fw-la and_o bg_o by_o the_o 47._o of_o 〈…〉_o and_o forasmuch_o as_o the_o line_n bg_o subtend_v the_o angle_n of_o the_o pentagon_n compose_v of_o the_o side_n of_o the_o icosahedron_n the_o great_a segment_n of_o the_o right_a line_n bg_o shall_v be_v the_o right_a line_n ab_fw-la by_o the_o â—_o of_o the_o thirteen_o which_o line_n ab_fw-la together_o with_o the_o whole_a line_n bg_o the_o line_n agnostus_n contain_v in_o power_n and_o forasmuch_o aâ—_z the_o line_n ik_fw-mi couple_v the_o opposite_a and_o parallel_a side_n ed_z and_o fh_o of_o the_o dodecahedron_n make_v at_o those_o point_n right_a angle_n by_o the_o 3._o corollary_n of_o the_o 17._o of_o tâ—e_z thirteen_o the_o right_a line_n of_o which_o couple_v together_o equal_a and_o parallel_a line_n ei_o &_o fk_v shall_v be_v equal_a to_o the_o same_o line_n ik_fw-mi by_o the_o 33._o of_o the_o first_o wherefore_o the_o angle_n def_n shall_v be_v â—_o right_n angle_n by_o the_o 29._o of_o the_o first_o wherefore_o the_o diameter_n df_o contain_v in_o power_n the_o two_o line_n ed_z and_o ef._n but_o the_o less_o segment_n of_o the_o line_n ik_fw-mi be_v ed_z the_o side_n of_o the_o dodecahedron_n by_o the_o 4._o corollary_n of_o the_o 17â—_n of_o the_o thirteen_o wherefore_o the_o same_o line_n ed_z be_v also_o the_o less_o segment_n of_o the_o line_n of_o which_o be_v equal_a unto_o the_o line_n ik_fw-mi wherefore_o the_o diameter_n df_o contain_v in_o power_n the_o two_o line_n ed_z and_o of_o by_o the_o 47._o of_o the_o first_o contain_v in_o powâ—râ—_n ed_z the_o side_n of_o the_o dodecahedron_n the_o less_o segment_n together_o with_o the_o whole_a if_o therefore_o a_o rational_a line_n agnostus_n or_o df_o contain_v in_o power_n two_o line_n ab_fw-la and_o bg_o do_v make_v the_o whole_a line_n and_o the_o great_a segment_n and_o again_o contain_v in_o power_n two_o line_n of_o and_o ed_z do_v make_v the_o whole_a line_n and_o the_o less_o segment_n the_o great_a segment_n ab_fw-la shall_v be_v the_o side_n of_o the_o icosahedron_n and_o the_o less_o segment_n ed_z shall_v be_v the_o side_n of_o the_o dodecahedron_n contain_v in_o one_o and_o the_o self_n same_o sphere_n the_o 16._o proposition_n if_o the_o power_n of_o the_o side_n of_o a_o octohedron_n be_v express_v by_o two_o right_a lineâ—_z join_v together_o by_o a_o extreme_a and_o meâ—ne_a proportion_n the_o side_n of_o the_o icosahedron_n contain_v in_o the_o same_o sphere_n shall_v be_v duple_v to_o the_o less_o segment_n let_v ab_fw-la the_o side_n of_o the_o octohedron_n abg_o contain_v in_o power_n the_o two_o line_n c_o and_o h_o which_o let_v have_v that_o proportion_n that_o the_o whole_a have_v to_o the_o great_a segment_n by_o the_o corollarye_a of_o the_o first_o proposition_n add_v by_o flussas_n after_o the_o last_o proposition_n of_o the_o six_o book_n construction_n and_o let_v the_o icosahedron_n contain_v in_o the_o same_o sphere_n be_v
def_n who_o side_n let_v be_v de_fw-fr and_o let_v the_o right_a line_n subtend_v the_o angle_n of_o the_o pentagon_n make_v of_o the_o side_n of_o the_o icosahedron_n be_v the_o line_n ef._n then_o i_o say_v that_o the_o side_n ed_z be_v in_o power_n double_a to_o the_o line_n h_o the_o less_o of_o those_o segment_n forasmuch_o as_o by_o that_o which_o be_v demonstrate_v in_o the_o 15._o of_o this_o book_n demonstration_n it_o be_v manifest_a that_o ed_z the_o side_n of_o the_o icosahedron_n be_v the_o great_a segment_n of_o the_o line_n efâ—_n and_o that_o the_o diameter_n df_o contain_v in_o power_n the_o two_o line_n ed_z and_o of_o namely_o the_o whole_a and_o the_o great_a segment_n but_o by_o supposition_n the_o side_n ab_fw-la contain_v in_o power_n the_o two_o line_n c_o &_o h_o join_v together_o in_o the_o self_n same_o proportion_n wherefore_o the_o line_n of_o be_v to_o the_o line_n ed_z as_z the_o line_n c_o be_v to_o the_o line_n h_o by_o the_o â—_o oâ—_n this_o bokeâ—_n and_o alternally_n by_o the_o 16._o of_o the_o five_o the_o line_n of_o be_v to_o the_o line_n c_o as_o the_o line_n ed_z be_v to_o the_o line_n h._n and_o forasmuch_o as_o the_o line_n df_o contain_v in_o power_n the_o two_o line_n ed_z and_o of_o and_o the_o line_n ab_fw-la contain_v in_o power_n the_o two_o line_n c_o and_o h_o therefore_o the_o square_n of_o the_o line_n of_o and_o ed_z be_v to_o the_o square_n of_o the_o line_n df_o as_o the_o square_n of_o the_o line_n c_o and_o h_o to_o the_o square_a ab_fw-la and_o alternate_o the_o square_n of_o the_o line_n of_o and_o â—d_a be_v to_o the_o square_n of_o the_o line_n c_o and_o h_o as_o the_o square_n of_o the_o line_n df_o be_v to_o the_o square_n of_o the_o line_n abâ—_n but_o df_o the_o diameter_n be_v by_o the_o 14._o of_o the_o thirtenâ—h_o iâ—_z power_n double_a to_o ab_fw-la the_o side_n of_o the_o octohedron_n inscribe_v by_o supposition_n in_o the_o same_o sphere_n wherefore_o the_o square_n of_o the_o line_n of_o and_o ed_z be_v double_a to_o the_o square_n of_o the_o line_n c_o and_o h._n and_o therefore_o one_o square_a of_o the_o line_n ed_z be_v double_a to_o one_o square_a of_o the_o line_n h_o by_o the_o 12._o of_o the_o five_o wherefore_o ed_z the_o side_n of_o the_o icosahedron_n be_v in_o power_n duple_n to_o the_o line_n h_o which_o be_v the_o less_o segment_n if_o therefore_o the_o poweâ—_n of_o the_o side_n of_o a_o octohedron_n be_v express_v by_o two_o right_a line_n join_v together_o by_o a_o extreme_a and_o mean_a proportion_n the_o side_n of_o the_o icosahedron_n contain_v in_o the_o same_o sphere_n shall_v be_v duple_v to_o the_o less_o segment_n the_o 17._o proposition_n if_o the_o side_n of_o a_o dodecahedron_n and_o the_o right_a line_n of_o who_o the_o say_a side_n be_v the_o less_o segment_n be_v so_o set_v that_o they_o make_v a_o right_a angle_n the_o right_a line_n which_o contain_v in_o power_n half_o the_o line_n subtend_v the_o angle_n be_v the_o side_n of_o a_o octohedron_n contain_v in_o the_o self_n same_o sphere_n svppose_v that_o ab_fw-la be_v the_o side_n of_o a_o dodecahedron_n and_o let_v the_o right_a line_n of_o which_o that_o side_n be_v the_o less_o segment_n be_v agnostus_n namely_o which_o couple_v the_o opposite_a side_n of_o the_o dodecahedron_n by_o the_o 4._o corollary_n of_o the_o 17._o of_o the_o thirteen_o construction_n and_o let_v those_o line_n be_v so_o set_v that_o they_o make_v a_o right_a angle_n at_o the_o point_n a._n and_o draw_v the_o right_a line_n bg_o and_o let_v the_o line_n d_o contain_n in_o power_n half_o the_o line_n bg_o by_o the_o first_o proposition_n add_v by_o flussas_n after_o the_o last_o of_o the_o six_o then_o i_o say_v that_o the_o line_n d_o be_v the_o side_n of_o a_o octohedron_n contain_v in_o the_o same_o sphere_n forasmuch_o as_o the_o line_n agnostus_n make_v the_o great_a segment_n gc_o the_o side_n of_o the_o cube_fw-la contain_v in_o the_o same_o sphere_n by_o the_o same_o 4._o corollary_n of_o the_o 17._o of_o the_o thirteen_o demonstration_n and_o the_o square_n of_o the_o whole_a line_n ag._n and_o of_o the_o less_o segment_n ab_fw-la be_v triple_a to_o the_o square_n of_o the_o great_a segment_n gc_o by_o the_o 4._o of_o the_o thirteen_o moreover_o the_o diameter_n of_o the_o sphere_n be_v in_o power_n triple_a to_o the_o same_o line_n gc_o the_o side_n of_o the_o cube_fw-la by_o the_o 15._o of_o the_o thirteen_o wherefore_o the_o line_n bg_o be_v equal_a to_o the_o 〈◊〉_d for_o it_o contain_v in_o power_n the_o two_o line_n ab_fw-la and_o ag_z by_o the_o 47._o of_o the_o first_o and_o therefore_o it_o contain_v in_o power_n the_o triple_a of_o the_o line_n gc_o but_o the_o side_n of_o the_o octohedron_n contain_v in_o the_o same_o sphere_n be_v in_o power_n triple_a to_o half_a the_o diameter_n of_o the_o sphere_n by_o the_o 14._o of_o the_o thirteen_o and_o by_o supposition_n the_o line_n d_o contaiâ—â—â—â—_n in_o powâ—â—_n the_o half_a of_o the_o line_n bg_o wherefore_o the_o line_n d_o contain_v in_o power_n the_o half_a of_o the_o same_o diameter_n be_v the_o side_n of_o a_o octohedron_n if_o therefore_o the_o side_n of_o a_o dodecahedron_n and_o the_o right_a line_n of_o who_o the_o say_a side_n be_v the_o less_o segment_n be_v so_o set_v that_o they_o make_v a_o right_a angle_n the_o right_a line_n which_o contain_v in_o power_n half_o the_o line_n subtend_v the_o angle_n be_v the_o side_n of_o a_o ocâ—â—â—edron_n contain_v in_o the_o self_n same_o sphere_n which_o be_v require_v to_o be_v prove_v a_o corollary_n unto_o what_o right_a line_n the_o side_n of_o the_o octoâ—edron_n be_v in_o power_n sesquialter_fw-la unto_o the_o same_o line_n the_o side_n of_o the_o dodecahedron_n inscribe_v in_o the_o same_o sphere_n be_v the_o great_a segment_n for_o the_o side_n of_o the_o dodecahedron_n be_v the_o great_a segment_n of_o the_o segment_n cg_o unto_o which_o d_z the_o side_n of_o the_o octohedron_n be_v in_o power_n sesquialter_n that_o be_v be_v half_a of_o the_o power_n of_o the_o line_n bg_o which_o be_v triple_a unto_o the_o line_n cg_o ¶_o the_o 18._o proposition_n if_o the_o side_n of_o a_o tetrahedron_n contain_v in_o power_n two_o right_a line_n join_v together_o by_o a_o extreme_a and_o mean_a proportion_n the_o side_n of_o a_o icosahedron_n describe_v in_o the_o self_n same_o sphere_n be_v in_o power_n sesquialter_fw-la to_o the_o less_o right_a line_n svppose_v that_o abc_n be_v a_o tetrahedron_n and_o let_v his_o side_n be_v ab_fw-la construction_n who_o power_n let_v be_v divide_v into_o the_o line_n agnostus_n and_o gb_o join_v together_o by_o a_o extreme_a and_o mean_a proportion_n namely_o let_v it_o be_v divide_v into_o agnostus_n the_o whole_a line_n and_o gb_o the_o great_a seâ—ment_n by_o the_o corollary_n of_o the_o first_o proposition_n add_v by_o flussas_n after_o the_o last_o of_o the_o six_o and_o let_v ed_z be_v the_o side_n of_o the_o icosahedron_n edf_o contain_v in_o the_o self_n same_o sphere_n and_o let_v the_o line_n which_o subtend_v the_o angle_n of_o the_o pentagon_n describe_v of_o the_o side_n of_o the_o icosahedron_n be_v ef._n then_o i_o say_v that_z ed_z the_o side_n of_o the_o icosahedron_n be_v in_o power_n sesquialter_fw-la to_o the_o less_o line_n gb_o demonstration_n forasmuch_o as_o by_o that_o which_o be_v demonstrate_v in_o the_o 15._o of_o this_o book_n the_o side_n ed_z be_v the_o greâ—ter_a segment_n of_o the_o line_n of_o which_o subtend_v the_o angle_n of_o the_o pentagon_n but_o as_o the_o whole_a line_n of_o be_v to_o the_o great_a segment_n ed_z so_o be_v the_o same_o grâ—â—ter_a segment_n to_o the_o less_o by_o the_o 30._o of_o the_o six_o and_o by_o supposition_n agnostus_n be_v the_o whole_a line_n and_o gâ—_n the_o great_a segment_n wherefore_o as_o of_o be_v to_o ed_z so_o be_v agnostus_n to_o gâ—_n by_o the_o second_o of_o the_o fourteen_o and_o alternate_o the_o line_n of_o be_v to_o the_o line_n agnostus_n as_o the_o line_n ed_z be_v to_o the_o line_n gb_o and_o forasmuch_o as_o by_o supposition_n the_o line_n ab_fw-la contain_v in_o power_n the_o two_o line_n agnostus_n and_o gb_o therefore_o by_o the_o 4â—_o of_o the_o first_o the_o angle_n agb_n be_v a_o right_a angle_n but_o the_o angle_n def_n be_v a_o right_a angle_n by_o that_o which_o be_v demonstrate_v in_o the_o 15._o of_o this_o book_n wherefore_o the_o triangle_n agâ—_n and_o feed_v be_v equiangle_n by_o the_o â—_o of_o the_o six_o wherefore_o their_o side_n be_v proportional_a namely_o as_o the_o line_n ed_z be_v to_o the_o line_n gb_o so_o be_v the_o line_n fd_o to_o the_o line_n ab_fw-la by_o the_o 4._o of_o the_o six_o but_o by_o that_o which_o have_v before_o
be_v demonstrate_v fd_o be_v the_o diameter_n of_o the_o sphere_n which_o contain_v the_o icosahedron_n which_o diameter_n be_v in_o power_n sesquialter_fw-la to_o ab_fw-la the_o side_n of_o the_o tetrahedron_n inscribe_v in_o theâ—_z same_o sphere_n by_o the_o 13._o of_o the_o thirteen_o wherefore_o the_o line_n ed_z the_o side_n of_o the_o icosahedron_n be_v in_o power_n sesquialter_fw-la to_o gâ—_n the_o great_a segment_n or_o less_o line_n if_o therefore_o the_o side_n of_o a_o tetrahedron_n contain_v in_o power_n two_o right_a line_n join_v together_o a_o extreme_a and_o mean_a proportion_n the_o side_n of_o a_o icosahedron_n describe_v in_o the_o self_n same_o sphere_n be_v in_o power_n sesquialter_fw-la to_o the_o less_o right_a line_n ¶_o the_o 19_o proposition_n the_o superficies_n of_o a_o cube_n be_v to_o the_o superficies_n of_o a_o octohedron_n inscribe_v in_o one_o and_o the_o self_n same_o sphere_n in_o that_o proportion_n that_o the_o solid_n be_v construction_n svppose_v that_o abcde_v be_v a_o cube_n who_o four_o diameter_n let_v be_v the_o line_n ac_fw-la bc_o dc_o and_o ec_o produce_v on_o each_o side_n let_v also_o the_o octohedron_n inscribe_v in_o the_o self_n same_o sphere_n be_v fghk_v who_o three_o diameter_n let_v be_v fh_o gk_o and_o on_o then_o i_o say_v that_o the_o cube_fw-la abdella_n be_v to_o the_o octohedron_n fgh_n as_o the_o superficies_n of_o the_o cube_fw-la be_v to_o the_o superficies_n of_o the_o octohedron_n draw_v from_o the_o centre_n of_o the_o cube_fw-la to_o the_o base_a abed_o a_o perpendicular_a line_n cr._n and_o from_o the_o centre_n of_o the_o octohedron_n draw_v to_o the_o base_a gnh_o a_o perpendicular_a line_n â—l_o demonstration_n and_o forasmuch_o as_o the_o three_o diameter_n of_o the_o cube_fw-la do_v pass_n by_o the_o 〈◊〉_d c_o therefore_o by_o the_o 2._o corollary_n of_o the_o 15._o of_o the_o thirteen_o â—here_o shall_v be_v make_v of_o the_o cube_fw-la six_o pyramid_n as_o this_o pyramid_n abdec_n equal_a to_o the_o whole_a cube_fw-la for_o there_o be_v in_o the_o cube_fw-la â—ixe_v base_n upon_o which_o fall_v equal_a perpendicular_n from_o the_o cenâ—â—â—_n by_o the_o corollary_n of_o the_o assumpt_n of_o the_o 16._o of_o the_o twelve_o for_o the_o base_n be_v contain_v in_o equal_a circle_v of_o the_o sphere_n but_o in_o the_o octohedron_n the_o three_o diameter_n do_v make_v upon_o the_o 8._o base_n 8._o pyramid_n have_v their_o top_n in_o the_o centre_n by_o the_o 3._o corollary_n of_o the_o 14â—_o of_o the_o thirteen_o now_o the_o base_n of_o the_o cube_fw-la and_o of_o the_o octohedron_n be_v contain_v in_o equal_a circle_n of_o the_o sphere_n by_o the_o 13._o of_o this_o book_n wherefore_o they_o shall_v be_v equal_o distant_a from_o the_o centre_n and_o the_o perpendicular_a line_n cr_n and_o â—_o shall_v be_v equal_a by_o the_o corollary_n of_o the_o assumpt_n of_o the_o 16._o of_o the_o twelve_o wherefore_o the_o pyramid_n of_o the_o cube_fw-la shall_v be_v under_o one_o and_o the_o self_n same_o altitude_n with_o the_o pyramid_n of_o the_o octohedron_n namely_o under_o the_o perpendicular_a line_n draw_v from_o the_o centre_n to_o the_o base_n wherefore_o six_o pyramid_n of_o the_o cube_fw-la be_v to_o 8._o pyramid_n of_o the_o octohedron_n be_v under_o one_o and_o the_o same_o altitude_n in_o that_o proportion_n that_o their_o base_n be_v by_o the_o 6._o of_o the_o twelve_o that_o be_v one_o pyramid_n set_v upon_o six_o base_n of_o the_o cube_fw-la and_o have_v to_o his_o altitude_n the_o perpendicular_a line_n which_o pyramid_n be_v equal_a to_o the_o six_o pyramid_n by_o the_o same_o 6._o of_o the_o twelve_o be_v to_o one_o pyramid_n set_v upon_o the_o 8._o base_n of_o the_o octohedron_n be_v equal_a to_o the_o octohedron_n and_o also_o under_o onâ—_n and_o the_o self_n same_o altitude_n in_o that_o proportion_n that_o six_o base_n of_o the_o cube_fw-la which_o contain_v the_o whole_a superficies_n of_o the_o cube_fw-la be_v to_o 8._o base_n of_o the_o octohedronâ—_n which_o contain_v the_o whole_a superficies_n of_o the_o octohedron_n for_o the_o solid_n of_o those_o pyramid_n be_v in_o proportion_n the_o one_o to_o the_o other_o as_o their_o base_n be_v by_o the_o self_n same_o 6._o of_o the_o twelve_o wherefore_o â—he_z superficies_n of_o the_o cube_fw-la be_v to_o the_o superficies_n of_o the_o octohedron_n inscribe_v in_o one_o and_o the_o self_n same_o sphere_n in_o that_o proportion_n that_o the_o solid_n be_v which_o be_v require_v to_o be_v prove_v ¶_o the_o 20._o proposition_n if_o a_o cube_n and_o a_o octohedron_n be_v contain_v in_o one_o &_o the_o self_n same_o sphere_n they_o shall_v be_v in_o proportion_n the_o one_o to_o the_o other_o as_o the_o side_n of_o the_o cube_n be_v to_o the_o semidiameter_n of_o the_o sphere_n svppose_v that_o the_o octohedron_n aecdb_n be_v inscribe_v in_o the_o sphere_n abcd_o and_o let_v the_o cube_fw-la inscribe_v in_o the_o same_o sphere_n be_v fghim_n who_o diameter_n let_v be_v high_a construction_n which_o be_v equal_a to_o the_o diameter_n ac_fw-la by_o the_o 15._o of_o the_o thirteen_o let_v the_o half_a of_o the_o diameter_n be_v ae_n then_o i_o say_v that_o the_o cube_fw-la fghim_n be_v to_o the_o octohedron_n aecdb_n as_o the_o side_n mg_o be_v to_o the_o semidiameter_n ae_n forasmuch_o as_o the_o diameter_n ac_fw-la be_v in_o power_n double_a to_o bk_o the_o side_n of_o the_o octohedron_n by_o the_o 14._o of_o the_o thirteen_o and_o be_v in_o power_n triple_a to_o mg_o the_o side_n of_o the_o cube_fw-la by_o the_o 15._o of_o the_o same_o therefore_o the_o square_a bkdl_n shall_v be_v sesquialter_n to_o fm_o the_o square_a of_o the_o cube_fw-la from_o the_o line_n ae_n cut_v of_o a_o three_o part_n a_o and_o from_o the_o line_n mg_o cut_v of_o likewise_o a_o three_o part_n go_v by_o the_o 9_o of_o the_o six_o demonstration_n now_o than_o the_o line_n en_fw-fr shall_v be_v two_o three_o part_n of_o the_o line_n ae_n and_o so_o also_o shall_v the_o line_n more_n be_v of_o the_o line_n mg_o wherefore_o the_o parallelipipedon_n set_v upon_o the_o base_a bkdl_n and_o have_v his_o altitude_n the_o line_n ea_fw-la be_v triple_a to_o the_o parallelipipedon_n set_v upon_o the_o same_o base_a and_o have_v his_o altitude_n the_o line_n a_fw-la by_o the_o corollary_n of_o the_o 31._o of_o the_o eleven_o but_o it_o be_v also_o triple_a to_o the_o pyramid_n abkdl_n which_o be_v set_v upon_o the_o same_o base_a and_o be_v under_o the_o same_o altitude_n by_o the_o second_o corollary_n of_o the_o 7._o of_o the_o twelve_o wherefore_o the_o pyramid_n abkdl_n be_v equal_a to_o the_o parallelipipedon_n which_o be_v set_v upon_o the_o base_a bkdl_n and_o have_v to_o his_o altitude_n the_o line_n an._n but_o unto_o that_o parallelipipedom_n be_v double_a the_o parallelipipedon_n which_o be_v set_v upon_o the_o same_o base_a bkdl_n and_o have_v to_o his_o altitude_n a_o line_n double_a to_o the_o line_n en_fw-fr by_o the_o corollary_n of_o the_o 31._o of_o the_o first_o and_o unto_o the_o pyramid_n be_v double_a the_o octohedron_n abkldc_n by_o the_o 2._o corollary_n of_o the_o 14._o of_o the_o thirteen_o wherefore_o the_o octohedron_n abkdlc_n be_v equal_a to_o the_o parallelipipedon_n set_v upon_o the_o base_a bkld_a &_o have_v his_o altitude_n the_o line_n en_fw-fr by_o the_o 15._o of_o the_o five_o but_o the_o parallelipipedon_n set_v upon_o the_o base_a bkdl_n which_o be_v sesquialter_fw-la to_o the_o base_a fm_o and_o have_v to_o his_o altitude_n the_o line_n more_n which_o be_v two_o three_o part_n of_o the_o side_n of_o the_o cube_fw-la mg_o be_v equal_a to_o the_o cube_fw-la fg_o by_o the_o 2._o part_n of_o the_o 34._o of_o the_o eleven_o for_o it_o be_v before_o prove_v that_o the_o base_a bkdl_n be_v sesquialter_fw-la to_o the_o base_a fm_o now_o than_o these_o two_o parallelipipedon_n namely_o the_o parallelipipedon_n which_o be_v set_v upon_o the_o base_a bkdl_n which_o be_v sesquialter_fw-la to_o the_o base_a of_o the_o cube_fw-la and_o have_v to_o his_o altitude_n the_o line_n more_n which_o be_v two_o three_o part_n of_o mg_o the_o side_n of_o the_o cube_fw-la which_o parallelipipedon_n be_v prove_v equal_a to_o the_o cube_fw-la and_o the_o parallelipipedon_n set_v upon_o the_o same_o base_a bkdl_n and_o have_v his_o altitude_n the_o line_n en_fw-fr which_o parallelipipedon_n be_v prove_v equal_a to_o the_o octohedron_n these_o two_o parallelipipedon_n i_o say_v be_v the_o one_o to_o the_o other_o as_o the_o altitude_n more_n be_v to_o the_o altitude_n en_fw-fr by_o the_o corollary_n of_o the_o 31._o of_o the_o eleven_o wherefore_o also_o as_o the_o altitude_n more_n be_v to_o the_o altitude_n en_fw-fr so_o be_v the_o cube_fw-la fghim_n to_o the_o octohedron_n abkdlc_n by_o the_o 7._o of_o the_o five_o but_o as_o the_o line_n more_n be_v to_o the_o line_n en_fw-fr so_o be_v
wherefore_o the_o line_n ce_fw-fr be_v to_o the_o line_n egg_n as_o the_o great_a segment_n to_o the_o less_o and_o therefore_o their_o proportion_n be_v as_o the_o whole_a line_n i_fw-mi be_v to_o the_o great_a segment_n ce_fw-fr and_o as_o the_o great_a segment_n ce_fw-fr be_v to_o the_o less_o segment_n egg_n wherefore_o the_o whole_a line_n ceg_n which_o make_v the_o great_a segment_n and_o the_o less_o be_v equal_a to_o the_o whole_a line_n i_fw-mi or_o je_n and_o forasmuch_o as_o two_o parallel_n plain_a superficiece_n namely_o that_o which_o be_v extend_v by_o job_n and_o that_o which_o be_v extend_v by_o the_o line_n ag_fw-mi be_v cut_v by_o the_o plain_n of_o the_o triangle_n be_v which_o pass_v by_o the_o line_n ag_fw-mi and_o ib_n their_o common_a section_n ag_v and_o ib_n shall_v be_v parallel_n by_o the_o 16._o of_o the_o eleven_o but_o the_o angle_n buy_v or_o bic_n be_v a_o right_a angle_n wherefore_o the_o angle_n agc_n be_v also_o a_o right_a angle_n by_o the_o 29._o of_o the_o first_o and_o those_o right_a angle_n be_v contain_v under_o equal_a side_n namely_o the_o line_n gc_n be_v equal_a to_o the_o line_n ci_o and_o the_o line_n ag_fw-mi to_o the_o line_n by_o by_o the_o 33._o of_o the_o first_o wherefore_o the_o base_n ca_o and_o cb_o be_v equal_a by_o the_o 4._o of_o the_o first_o but_o of_o the_o line_n cb_o the_o line_n ce_fw-fr be_v prove_v to_o be_v the_o great_a segment_n wherefore_o the_o same_o line_n ce_fw-fr be_v also_o the_o great_a segment_n of_o the_o line_n ca_o but_o cn_fw-la be_v also_o the_o great_a segment_n of_o the_o same_o line_n ca._n wherefore_o unto_o the_o line_n ce_fw-fr the_o line_n cn_fw-la which_o be_v the_o side_n of_o the_o dodecahedron_n and_o be_v set_v at_o the_o diameter_n be_v equal_a and_o by_o the_o same_o reason_n the_o rest_n of_o the_o side_n which_o be_v set_v at_o the_o diameter_n may_v be_v prove_v eguall_a to_o line_n equal_a to_o the_o line_n ce._n wherefore_o the_o pentagon_n inscribe_v in_o the_o circle_n where_o in_o be_v contain_v the_o triangle_n be_v be_v by_o the_o 11._o of_o the_o four_o equiangle_n and_o equilater_n and_o forasmch_v as_o two_o pentagon_n set_v upon_o every_o one_o of_o the_o base_n of_o the_o cube_fw-la do_v make_v a_o dodecahedron_n and_o six_o base_n of_o the_o cube_fw-la do_v receive_v twelve_o angle_n of_o the_o dodecahedron_n and_o the_o 8._o semidiameter_n do_v in_o the_o point_n where_o they_o be_v cut_v by_o a_o extreme_a and_o mean_a proportion_n receive_v the_o rest_n therefore_o the_o 12._o pentagon_n base_n contain_v 20._o solid_a angle_n do_v inscribe_v the_o dodecahedron_n in_o the_o cube_fw-la by_o the_o 1._o definition_n of_o this_o book_n wherefore_o in_o a_o cube_fw-la give_v be_v inscribe_v a_o dodecahedron_n which_o be_v require_v to_o be_v do_v first_o corollary_n the_o diameter_n of_o the_o sphere_n which_o contain_v the_o dodecahedron_n contain_v in_o power_n these_o two_o side_n namely_o the_o side_n of_o the_o dodecahedron_n and_o the_o side_n of_o the_o cube_fw-la wherein_o the_o dodecahedron_n be_v inscribe_v for_o in_o the_o first_o figure_n a_o line_n draw_v from_o the_o centre_n oh_o to_o the_o point_n b_o the_o angle_n of_o the_o dodecahedron_n namely_o the_o line_n ob_fw-la contain_v in_o power_n these_o two_o line_n ov_v the_o half_a side_n of_o the_o cube_fw-la and_o ub_z the_o half_a side_n of_o the_o dodecahedron_n by_o the_o 47._o of_o the_o first_o wherefore_o by_o the_o 15._o of_o the_o five_o the_o double_a of_o the_o line_n ob_fw-la which_o be_v the_o diameter_n of_o the_o sphere_n contain_v the_o dodecahedron_n contain_v in_o power_n the_o double_a of_o the_o other_o line_n ov_n and_o ub_z which_o be_v the_o side_n of_o the_o cube_fw-la and_o of_o the_o dodecahedron_n ¶_o second_v corollary_n the_o side_n of_o a_o cube_fw-la divide_v by_o a_o extreme_a and_o mean_a proportion_n make_v the_o less_o segment_n the_o side_n of_o the_o dodecahedron_n inscribe_v in_o it_o and_o the_o great_a segment_n the_o side_n of_o the_o cube_fw-la inscribe_v in_o the_o same_o dodecahedron_n for_o it_o be_v before_o prove_v that_o the_o side_n of_o the_o dodecahedron_n be_v the_o great_a segment_n of_o be_v the_o side_n of_o the_o triangle_n becâ—_n but_o the_o side_n be_v which_o be_v equal_a to_o the_o lineâ—_z gb_o and_o sf_n be_v the_o great_a segment_n of_o gf_o the_o side_n of_o the_o cube_fw-la which_o line_n â—e_v subtend_v thâ—_z angle_n of_o the_o pentagon_n be_v by_o the_o â—_o of_o this_o book_n the_o side_n of_o the_o cube_fw-la inscribe_v in_o the_o dodecahedron_n three_o corollary_n the_o side_n of_o a_o cube_fw-la be_v equal_a to_o the_o side_n of_o a_o dodecahedron_n inscribe_v in_o it_o and_o circumscribe_v about_o it_o for_o it_o be_v manifest_a by_o this_o proposition_n that_o the_o side_n of_o a_o cube_fw-la make_v the_o less_o segment_n the_o side_n of_o a_o dodecahedron_n inscribe_v in_o it_o namely_o as_o in_o the_o first_o figure_n the_o line_n b_v the_o side_n of_o the_o dodecahedron_n inscribe_v be_v the_o less_o segment_n of_o the_o line_n gf_o the_o side_n of_o the_o cube_fw-la and_o it_o be_v prove_v in_o the_o 17._o of_o the_o thirteen_o that_o the_o same_o side_n of_o the_o cube_fw-la subtend_v the_o angle_n of_o the_o pentagon_n of_o the_o dodecahedron_n circumscribe_v and_o therefore_o it_o make_v the_o great_a segment_n the_o side_n of_o the_o dodecahedron_n or_o of_o the_o pentagon_n by_o the_o first_o corollary_n of_o the_o same_o wherefore_o it_o be_v equal_a to_o both_o those_o segment_n the_o 14._o problem_n the_o 14._o proposition_n in_o a_o cube_fw-la give_v to_o inscribe_v a_o icosahedron_n svppose_v that_o the_o cube_fw-la give_v by_o abc_n construction_n the_o centre_n of_o who_o base_n let_v be_v the_o point_n d_o e_o g_o h_o i_o king_n by_o which_o point_n draw_v in_o the_o base_n unto_o the_o other_o side_n parallel_v not_o touch_v the_o one_o the_o other_o and_o divide_v the_o line_n draw_v from_o the_o centre_n as_o the_o line_n dt_n etc_n etc_n by_o a_o extreme_a and_o mean_a proportion_n in_o the_o point_n a_o f_o l_o m_o n_o b_o p_o q_o r_o s_o c_o oh_o by_o the_o 30._o of_o the_o six_o and_o let_v the_o great_a segment_n be_v about_o the_o centre_n and_o draw_v these_o right_a line_n all_o agnostus_n be_o and_o tg_n and_o forasmuch_o as_o the_o line_n cut_v be_v parallel_n to_o the_o side_n of_o the_o cube_fw-la demonstration_n they_o shall_v make_v right_a angle_n the_o one_o with_o the_o other_o by_o the_o 29._o of_o the_o first_o and_o forasmuch_o as_o they_o be_v equal_a their_o section_n shall_v be_v equal_a for_o that_o the_o section_n be_v like_a by_o the_o 2._o of_o the_o fourteen_o wherefore_o the_o line_n tg_n be_v equal_a to_o the_o line_n dt_n for_o they_o be_v each_o half_a side_n of_o the_o cube_fw-la wherefore_o the_o square_a of_o the_o whole_a line_n tg_n and_o of_o the_o less_o segment_n ta_n be_v triple_a to_o the_o square_n of_o the_o line_n ad_fw-la the_o great_a segment_n by_o the_o 4._o of_o the_o thirteen_o but_o the_o line_n agnostus_n contain_v in_o power_n the_o line_n at_o &_o tg_n for_o the_o angle_n atg_n be_v a_o right_a angle_n wherefore_o the_o square_a of_o the_o line_n agnostus_n be_v triple_a to_o the_o square_n of_o the_o line_n ad._n and_o forasmuch_o as_o the_o line_n mgl_n be_v erect_v perpendicular_o to_o the_o plain_a pass_v by_o the_o line_n at_o &_o which_o be_v parallel_n to_o the_o base_n of_o the_o cube_fw-la by_o the_o corollary_n of_o the_o 14._o of_o the_o eleven_o therefore_o the_o angle_n agl_n be_v a_o right_a angle_n but_o the_o line_n lg_n be_v equal_a to_o the_o line_n ad_fw-la for_o they_o be_v the_o great_a segment_n of_o equal_a line_n wherefore_o the_o line_n agnostus_n which_o be_v in_o power_n triple_a to_o the_o line_n ad_fw-la be_v in_o power_n triple_a to_o the_o line_n lg_n wherefore_o add_v unto_o the_o same_o square_n of_o the_o line_n agnostus_n the_o square_a of_o the_o line_n lg_n the_o square_a of_o the_o line_n all_o which_o by_o the_o 47._o of_o the_o first_o contain_v in_o power_n the_o two_o line_n agnostus_n and_o gl_n shall_v be_v quadruple_a to_o the_o line_n ad_fw-la or_o lg_n wherefore_o the_o line_n all_o be_v double_a to_o the_o line_n ad_fw-la by_o the_o 20._o of_o the_o six_o and_o therefore_o be_v equal_a to_o the_o line_n of_o or_o to_o the_o line_n lm_o and_o by_o the_o same_o reason_n may_v we_o prove_v that_o every_o one_o of_o the_o other_o line_n which_o couple_v the_o next_o section_n of_o the_o line_n cut_v as_o the_o line_n be_o pf_n pm_n mq_n and_o the_o rest_n be_v equal_a wherefore_o the_o triangle_n alm_n apf_n ample_n pmq_n and_o the_o rest_n such_o like_a be_v equal_a equiangle_n and_o equilater_n
line_n a_fw-la be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n in_o the_o point_n h_o by_o the_o â—_o of_o the_o thirteen_o but_o as_o the_o line_n a_o be_v to_o the_o line_n ah_o so_o be_v the_o line_n ad_fw-la to_o the_o line_n ae_n by_o the_o 2._o of_o six_o for_o the_o lineâ—_z fh_o and_o on_o be_v parallel_n and_o again_o as_o the_o line_n ad_fw-la be_v to_o the_o line_n ae_n so_o by_o the_o same_o be_v the_o line_n agnostus_n to_o the_o line_n aq_n and_o the_o line_n ai_fw-fr to_o the_o line_n ap_n for_o the_o line_n pq_n and_o give_v be_v parallel_n wherefore_o the_o line_n agnostus_n and_o a_z be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n in_o the_o point_v q_n &_o p_o &_o the_o line_n aq_n shall_v be_v the_o great_a segment_n of_o the_o line_n agnostus_n or_o ab_fw-la and_o forasmuch_o as_o the_o wholâ—_n line_n agnostus_n be_v to_o the_o great_a segment_n aq_n as_o the_o great_a segment_n a_z be_v to_o the_o residue_n ap_n the_o line_n aâ—_z shall_v be_v the_o less_o segment_n of_o the_o whole_a line_n aâ—_z or_o ag._n wherefore_o the_o liâ—â—_n peq_n which_o by_o the_o point_n e_o pass_v parallelwise_o to_o the_o line_n give_v cut_v the_o line_n agnostus_n and_o basilius_n by_o a_o extreme_a and_o mean_a proportion_n in_o the_o point_n queen_n and_o p._n and_o by_o the_o same_o reason_n the_o line_n â—r_n which_o by_o the_o point_n c_o pass_v parallelwise_o to_o the_o line_n be_o shall_v fall_v upon_o the_o section_n p_o and_o r_o so_o also_o shall_v the_o line_n rq_n which_o by_o the_o point_n d_z pass_v parallelwise_o to_o the_o line_n bl_o fall_v upoâ—_n the_o section_n rq_n wherefore_o either_o of_o the_o line_n pe_n and_o eq_n shall_v be_v equal_a to_o the_o line_n cd_o in_o the_o parallelogram_n pd_v and_o qc_n by_o the_o 34._o of_o the_o first_o and_o forasmuch_o as_o the_o line_n pe_n and_o eq_n be_v equal_a the_o line_n pc_n cr_n rd_o and_o dq_n shall_v be_v likewise_o equal_a wherefore_o the_o triangle_n prq_n iâ—â—quilater_n and_o cut_v the_o side_n of_o the_o base_a of_o the_o pyramiâ—_n in_o the_o point_n p_o q_o r_o by_o a_o extreme_a and_o mean_a proportion_n and_o in_o it_o be_v inscribe_v the_o base_a ecd_v of_o the_o icosahedron_n contain_v in_o the_o foresaid_a pyramid_n if_o therefore_o from_o the_o angle_n of_o the_o base_a of_o a_o pyramid_n be_v draw_v to_o the_o opposite_a side_n right_a line_n cut_v the_o say_a side_n by_o a_o extreme_a and_o mean_a proportion_n they_o shall_v contain_v the_o base_a of_o the_o icosahedron_n inscribe_v in_o the_o pyramid_n which_o base_a shall_v be_v inscribe_v in_o a_o equilater_n triangle_n who_o angle_n cut_v the_o side_n of_o the_o base_a of_o the_o pyramid_n by_o a_o extreme_a &_o mean_a proportion_n ¶_o a_o corollary_n the_o side_n of_o a_o icosahedron_n inscribe_v in_o a_o octohedron_n be_v the_o great_a segment_n of_o the_o line_n which_o be_v draw_v from_o the_o angle_n of_o the_o base_a of_o the_o octohedron_n cut_v the_o opposite_a side_n by_o a_o extreme_a and_o mean_a proportion_n for_o by_o the_o 16._o of_o the_o fifteen_o fkh_n be_v the_o base_a of_o the_o octohedron_n which_o contain_v the_o base_a of_o the_o icosahedron_n cde_v unto_o which_o triangle_n fkh_n the_o triangle_n hkg_n be_v equal_a as_o have_v be_v prove_v by_o the_o point_n h_o draw_v unto_o the_o line_n menander_n a_o parallel_a line_n ht_o cut_v the_o line_n dn_o in_o the_o point_n s._n wherefore_o es_fw-ge dt_fw-ge and_o et_fw-la be_v parallelogram_n and_o therefore_o the_o line_n eh_o and_o mt_n be_v equal_a and_o the_o line_n em_n and_o ht_o be_v like_o cut_n in_o the_o point_n d_o and_o s_o by_o the_o 34._o of_o the_o first_o wherefore_o the_o great_a segment_n of_o the_o line_n ht_o be_v the_o line_n his_o which_o be_v equal_a to_o ed_z the_o side_n of_o the_o icosahedron_n but_o by_o the_o 2._o of_o the_o six_o the_o line_n tk_n be_v cut_v like_o to_o the_o line_n hk_o by_o the_o parallel_n dm_o and_o therefore_o by_o the_o 2._o of_o the_o fourteen_o it_o be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n but_o the_o line_n tm_n be_v equal_a to_o the_o line_n eh_o wherefore_o also_o the_o line_n tk_n be_v equal_a to_o the_o line_n of_o or_o dh_o wherefore_o the_o residue_v eh_o and_o tg_n be_v equal_a for_o the_o whole_a line_n fh_a and_o kg_o be_v equal_a wherefore_o kg_o the_o side_n of_o the_o triangle_n hkg_n be_v in_o the_o point_n t_o divide_v by_o a_o extreme_a and_o mean_a proportion_n in_o the_o point_n t_o by_o the_o right_a line_n ht_o and_o the_o great_a segment_n thereof_o be_v the_o line_n ed_z the_o side_n of_o the_o icosahedron_n inscribe_v in_o the_o octohedron_n who_o base_a be_v the_o triangle_n hkg_v or_o the_o triangle_n fkh_n which_o be_v equal_a to_o the_o triangle_n hkg_v by_o the_o 16._o of_o the_o fifteen_o ¶_o the_o 5._o proposition_n the_o side_n of_o a_o pyramid_n divide_v by_o a_o extreme_a and_o mean_a proportion_n make_v the_o less_o segment_n in_o power_n double_a to_o the_o side_n of_o the_o icosahedron_n inscribe_v in_o it_o svppose_v that_o abg_o be_v the_o base_a of_o a_o pyramid_n construction_n and_o let_v the_o base_a of_o the_o icosahedron_n inscribe_v in_o it_o be_v cde_o describe_v of_o three_o right_a line_n which_o be_v draw_v from_o the_o angle_n of_o the_o base_a abg_o cut_v the_o opposite_a side_n by_o a_o extreme_a and_o mean_a proportion_n by_o the_o former_a proposition_n namely_o of_o these_o three_o line_n be_o by_o and_o give_v then_o i_o say_v that_z a_z the_o less_o segment_n of_o the_o side_n aâ—_z be_v in_o power_n duple_n to_o ce_fw-fr the_o side_n of_o the_o icosahedron_n for_o forasmuch_o as_o by_o the_o former_a proposition_n it_o be_v prove_v that_o the_o triangle_n cde_o be_v inscribe_v in_o a_o equilater_n triangle_n demonstration_n who_o angle_n cut_v the_o side_n of_o abg_o the_o base_a of_o the_o pyramid_n by_o a_o extreme_a and_o mean_a proportion_n let_v that_o triangle_n be_v fhk_v cut_v the_o line_n ab_fw-la in_o the_o point_n f._n wherefore_o the_o less_o segment_n favorina_n be_v equal_a to_o the_o segment_n ai_fw-fr by_o the_o 2._o of_o the_o fourteen_o for_o the_o line_n ab_fw-la and_o ag_z be_v cut_v like_o moreover_o the_o side_n fh_o of_o the_o triangle_n fhk_n be_v in_o the_o point_n d_o cut_n into_o two_o equal_a part_n as_o in_o the_o former_a proposition_n it_o be_v prove_v and_o fce_v also_o by_o the_o same_o be_v a_o parallelogram_n wherefore_o the_o line_n ce_fw-fr and_o fd_o be_v equal_a by_o the_o 33_o of_o the_o first_o and_o forasmuch_o as_o the_o line_n fh_o subtend_v the_o angle_n bag_n of_o a_o equilater_n triangle_n which_o angle_n be_v contain_v under_o the_o great_a segment_n ah_o and_o the_o less_o segment_n afâ—_n therefore_o the_o line_n fh_o be_v in_o power_n double_a to_o the_o line_n of_o or_o to_o the_o line_n ai_fw-fr the_o less_o segment_n by_o the_o corollary_n of_o the_o 16._o of_o the_o fifteen_o but_o the_o same_o line_n fh_o be_v in_o power_n quadruple_a to_o the_o line_n ce_fw-fr by_o the_o 4._o of_o the_o second_o for_o the_o line_n fh_o be_v double_a to_o the_o line_n ce_fw-fr wherefore_o the_o line_n ai_fw-fr be_v the_o half_a of_o the_o square_n of_o the_o line_n fh_o be_v in_o power_n duple_n to_o the_o line_n ce_fw-fr to_o which_o the_o line_n fh_o be_v in_o power_n quadruple_a wherefore_o the_o side_n agnostus_n of_o the_o pyramid_n be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n make_v thâ—_z less_o segment_n ai_fw-fr in_o power_n duple_n to_o the_o side_n ce_fw-fr of_o the_o icosahedron_n inscribe_v in_o it_o ¶_o a_o corollary_n the_o side_n of_o a_o icosahedron_n inscribe_v in_o a_o pyramid_n be_v a_o residual_a line_n for_o the_o diameter_n of_o the_o sphere_n which_o contain_v the_o five_o regular_a body_n be_v rational_a be_v in_o power_n sesquialtera_fw-la to_o the_o side_n of_o the_o pyramid_n by_o the_o 13._o of_o the_o thirteen_o and_o therefore_o the_o side_n of_o the_o pyramid_n be_v rational_a by_o the_o definition_n which_o side_n be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n make_v the_o less_o segment_n a_o residual_a line_n by_o the_o 6._o of_o the_o thirteen_o wherefore_o the_o side_n of_o the_o icosahedron_n be_v commensurable_a to_o the_o same_o less_o segment_n for_o the_o square_n of_o the_o side_n of_o the_o icosahedron_n be_v the_o half_a of_o the_o square_n of_o the_o say_v less_o segment_n be_v a_o residual_a line_n by_o that_o which_o be_v add_v after_o the_o 103._o of_o the_o ten_o book_n ¶_o the_o 6._o proposition_n the_o side_n of_o a_o cube_n contain_v in_o power_n half_o the_o side_n of_o a_o equilater_n triangular_a pyramid_n inscribe_v in_o the_o say_a
cube_n for_o forasmuch_o as_o the_o side_n of_o the_o pyramid_n inscribe_v in_o the_o cube_fw-la subtend_v two_o side_n of_o the_o cube_fw-la which_o contain_v a_o right_a angle_n by_o the_o 1._o of_o the_o fifteen_o it_o be_v manifest_a by_o the_o 47._o of_o the_o first_o that_o the_o side_n of_o the_o pyramid_n subtend_v the_o say_a side_n be_v in_o power_n duple_n to_o the_o side_n of_o the_o cube_fw-la wherefore_o also_o the_o square_a of_o the_o side_n of_o the_o cube_fw-la be_v the_o half_a of_o the_o square_n of_o the_o side_n of_o the_o pyramid_n the_o side_n therefore_o of_o a_o cube_fw-la contain_v in_o power_n half_o the_o side_n of_o a_o equilater_n triangular_a pyramid_n inscribe_v in_o the_o say_v cube_fw-la ¶_o the_o 7._o proposition_n the_o side_n of_o a_o pyramid_n be_v duple_n to_o the_o side_n of_o a_o octohedron_n inscribe_v in_o it_o forasmuch_o as_o by_o the_o 2._o of_o the_o fivetenth_fw-mi it_o be_v prove_v that_o the_o side_n of_o the_o octohedron_n inscribe_v in_o a_o pyramid_n couple_v the_o middle_a section_n of_o the_o side_n of_o the_o pyramid_n wherefore_o the_o side_n of_o the_o pyramid_n and_o of_o the_o octohedron_n be_v parallel_n by_o the_o corollary_n of_o the_o 39_o of_o the_o first_o and_o therefore_o by_o the_o corollary_n of_o the_o 2._o of_o the_o six_o they_o subtend_v like_o triangle_n wherefore_o by_o the_o 4._o of_o the_o six_o the_o side_n of_o the_o pyramid_n be_v double_a to_o the_o side_n of_o the_o octohedron_n namely_o in_o the_o proportion_n of_o the_o side_n the_o side_n therefore_o of_o a_o pyramid_n be_v duple_n to_o the_o side_n of_o a_o octohedron_n inscribe_v in_o it_o ¶_o the_o 8._o proposition_n the_o side_n of_o a_o cube_n be_v in_o power_n duple_n to_o the_o side_n of_o a_o octohedron_n inscribe_v in_o it_o it_o be_v prove_v in_o the_o 3._o of_o the_o fifteen_o that_o the_o diameter_n of_o the_o octohedron_n inscribe_v in_o the_o cube_fw-la couple_v the_o centre_n of_o the_o opposite_a base_n of_o the_o cube_fw-la wherefore_o the_o say_a diameter_n be_v equal_a to_o the_o side_n of_o the_o cube_fw-la but_o the_o same_o be_v also_o the_o diameter_n of_o the_o square_n make_v of_o the_o side_n of_o the_o octohedron_n namely_o be_v the_o diameter_n of_o the_o sphere_n which_o contain_v it_o by_o the_o 14._o of_o the_o thirteen_o wherefore_o that_o diameter_n be_v equal_a to_o the_o side_n of_o the_o cube_fw-la be_v in_o power_n double_a to_o the_o side_n of_o that_o square_n or_o to_o the_o side_n of_o the_o octohedron_n inscribe_v in_o it_o by_o the_o 47._o of_o the_o first_o the_o side_n therefore_o of_o a_o cube_n be_v in_o power_n duple_n to_o the_o side_n of_o a_o octohedron_n inscribe_v in_o it_o which_o be_v require_v to_o be_v prove_v ¶_o the_o 9_o proposition_n the_o side_n of_o a_o dodecahedron_n be_v the_o great_a segment_n of_o the_o line_n which_o contain_v in_o power_n half_o the_o side_n of_o the_o pyramid_n inscribe_v in_o the_o say_v dodecahedron_n svppose_v that_o of_o the_o dodecahedron_n abgd_v the_o side_n be_v ab_fw-la and_o let_v the_o base_a of_o the_o cube_fw-la inscribe_v in_o the_o dodecahedron_n be_v ecfh_n by_o the_o â—â—_o of_o the_o fifteen_o and_o let_v the_o side_n of_o the_o pyramid_n inscribe_v in_o the_o cube_fw-la be_fw-mi change_z by_o the_o 1._o of_o the_o fifteen_o construction_n wherefore_o the_o same_o pyramid_n be_v inscribe_v in_o the_o dodecahedron_n by_o the_o 10._o of_o the_o fifteen_o then_o i_o say_v that_o ab_fw-la the_o side_n of_o the_o dodecahedron_n be_v the_o great_a segment_n of_o the_o line_n which_o contain_v in_o power_n half_o the_o line_n change_z which_o be_v the_o side_n of_o the_o pyramid_n inscribe_v in_o the_o dodecahedron_n demonstration_n for_o forasmuch_o as_o ec_o the_o side_n of_o the_o cube_fw-la be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n make_v the_o great_a segment_n the_o line_n ab_fw-la the_o side_n of_o the_o dodecahedron_n by_o the_o â—â—rst_a corollary_n of_o the_o 17._o of_o the_o thirteen_o for_o they_o be_v contain_v in_o one_o and_o the_o self_n same_o sphere_n by_o the_o first_o of_o this_o book_n and_o the_o line_n ec_o the_o side_n of_o the_o cube_fw-la contain_v in_o power_n the_o half_a of_o the_o side_n change_z by_o the_o 6._o of_o this_o book_n wherefore_o ab_fw-la the_o side_n of_o the_o dodecahedron_n be_v the_o great_a segment_n of_o the_o line_n ec_o which_o contain_v in_o power_n the_o half_a of_o the_o line_n change_z which_o be_v the_o side_n of_o the_o dodecahedron_n inscribe_v in_o the_o pyramid_n the_o side_n therefore_o of_o a_o dodecahedron_n be_v the_o great_a segment_n of_o the_o line_n which_o contain_v in_o power_n half_o the_o side_n of_o the_o pyramid_n inscribe_v in_o the_o say_v dodecahedron_n ¶_o the_o 10._o proposition_n the_o side_n of_o a_o icosahedron_n be_v the_o mean_a proportional_a between_o the_o side_n of_o the_o cube_n circumscribe_v about_o the_o icosahedron_n and_o the_o side_n of_o the_o dodecahedron_n inscribe_v in_o the_o same_o cube_n svppose_v that_o there_o be_v a_o cube_fw-la abfd_n in_o which_o let_v there_o be_v inscribe_v a_o icosahedron_n cligor_fw-la by_o the_o 14._o of_o the_o fifteen_o construction_n let_v also_o the_o dodecahedron_n inscribe_v in_o the_o same_o be_v edmnp_n by_o the_o 13._o of_o the_o same_o now_o forasmuch_o as_o cl_n the_o side_n of_o the_o icosahedron_n be_v the_o great_a segment_n of_o ab_fw-la the_o side_n of_o the_o cube_fw-la circumscribe_v about_o it_o by_o the_o 3._o corollary_n of_o the_o 14._o of_o the_o fifteen_o demonstration_n and_o the_o side_n ed_z of_o the_o dodecahedron_n inscribe_v in_o thesame_o cube_fw-la be_v the_o less_o segment_n of_o the_o same_o side_n ab_fw-la of_o the_o cube_fw-la by_o the_o 2._o corollary_n of_o the_o 13._o of_o the_o fifteen_o it_o follow_v that_o ab_fw-la the_o side_n of_o the_o cube_fw-la be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n make_v the_o great_a segment_n cl_n the_o side_n of_o the_o icosahedron_n inscribe_v in_o it_o and_o the_o less_o segment_n ed_z the_o side_n of_o the_o dodecahedron_n likewise_o inscribe_v in_o it_o wherefore_o as_o the_o whole_a line_n ab_fw-la the_o side_n of_o the_o cube_fw-la be_v to_o the_o great_a segment_n cl_n the_o side_n of_o the_o icosahedron_n so_o be_v the_o great_a segment_n cl_n the_o side_n of_o the_o icosahedron_n to_o the_o less_o segment_n edâ—_n the_o side_n of_o the_o dodecahedron_n by_o the_o three_o definition_n of_o the_o six_o wherefore_o the_o side_n of_o a_o icosahedron_n be_v the_o mean_a proportional_a between_o the_o side_n of_o the_o cube_fw-la circumscribe_v about_o the_o icosahedron_n and_o the_o side_n of_o the_o dodecahedron_n inscribe_v in_o the_o same_o cube_fw-la ¶_o the_o 11._o proposition_n the_o side_n of_o a_o pyramid_n be_v in_o power_n 1._o octodecuple_n to_o the_o side_n of_o the_o cube_fw-la inscribe_v in_o it_o for_o by_o that_o which_o be_v demonstrate_v in_o the_o 18._o of_o the_o fifteen_o the_o side_n of_o the_o pyramid_n be_v triple_a to_o the_o diameter_n of_o the_o base_a of_o the_o cube_fw-la inscribe_v in_o it_o demonstration_n and_o therefore_o it_o be_v in_o power_n nonecuple_n to_o the_o same_o diameter_n by_o the_o 20._o of_o the_o six_o but_o the_o diamer_n be_v in_o power_n double_a to_o the_o side_n of_o the_o cube_fw-la by_o the_o 47._o of_o the_o first_o and_o the_o double_a of_o nonecuple_n make_v octodecuple_n wherefore_o the_o side_n of_o the_o pyramid_n be_v in_o power_n octodecuple_n to_o the_o side_n of_o the_o cube_fw-la inscribe_v in_o it_o ¶_o the_o 12._o proposition_n the_o side_n of_o a_o pyramid_n be_v in_o power_n octodecuple_n to_o that_o right_a line_n who_o great_a segment_n be_v the_o side_n of_o the_o dodecahedron_n inscribe_v in_o the_o pyramid_n forasmuch_o as_o the_o dodecahedron_n and_o the_o cube_fw-la inscribe_v in_o it_o be_v set_v in_o one_o and_o the_o sâ—lfâ—_n same_o pyramid_n by_o the_o corollary_n of_o the_o first_o of_o this_o book_n and_o the_o side_n of_o the_o pyramid_n circumscribe_v about_o the_o cube_fw-la be_v in_o power_n octodecuple_n to_o the_o side_n of_o the_o cube_fw-la inscribe_v by_o the_o former_a proposition_n but_o the_o great_a segment_n of_o the_o self_n same_o side_n of_o the_o cube_fw-la be_v the_o side_n of_o the_o dodecahedron_n which_o contain_v the_o cube_fw-la by_o the_o corollary_n of_o the_o 17._o of_o the_o thirteen_o wherefore_o the_o side_n of_o the_o pyramid_n be_v in_o power_n octodecuple_n to_o that_o right_a line_n namely_o to_o the_o side_n of_o the_o cube_fw-la who_o great_a segment_n be_v the_o side_n of_o the_o dodecahedron_n inscribe_v in_o the_o pyramid_n ¶_o the_o 13._o proposition_n the_o side_n of_o a_o icosahedron_n inscribe_v in_o a_o octohedron_n be_v in_o power_n duple_n to_o the_o less_o segment_n of_o the_o side_n of_o the_o same_o
octohedron_n forasmuch_o as_o in_o the_o 17._o of_o the_o fifteen_o it_o be_v prove_v that_o the_o side_n of_o a_o icosahedron_n inscribe_v in_o a_o pyramid_n couple_v together_o the_o two_o section_n which_o be_v produce_v by_o a_o extreme_a and_o mean_a proportion_n of_o the_o side_n of_o the_o octohedron_n which_o make_v a_o right_a angle_n and_o that_o right_a angle_n be_v contain_v under_o the_o less_o segment_n of_o the_o side_n of_o the_o octohedron_n and_o be_v subtend_v of_o the_o side_n of_o the_o icosahedron_n inscribe_v it_o follow_v therefore_o that_o the_o side_n of_o the_o icosahedron_n which_o subtend_v the_o right_a angle_n be_v in_o power_n equal_a to_o the_o two_o line_n which_o contain_v the_o say_a angle_n by_o the_o 47._o of_o the_o first_o be_v in_o power_n duple_n to_o every_o one_o of_o the_o less_o segmente_n of_o the_o side_n of_o the_o octohedron_n which_o contain_v a_o right_a angle_n wherefore_o the_o side_n of_o a_o icosahedron_n inscribe_v in_o a_o octohedron_n be_v in_o power_n duple_n to_o the_o less_o segment_n of_o the_o â—ide_v of_o the_o same_o octohedron_n ¶_o the_o 14._o proposition_n the_o side_n of_o the_o octohedron_n and_o of_o the_o cube_n inscribe_v in_o it_o be_v in_o power_n the_o one_o to_o the_o other_o 2._o in_o quadrupla_fw-la sesquialter_fw-la proportion_n svppose_v that_o abgde_v be_v a_o octohedron_n and_o let_v the_o cube_fw-la inscribe_v in_o it_o be_v fchi_n then_o i_o say_v that_o ab_fw-la the_o side_n of_o the_o octohedron_n be_v in_o power_n quadruple_a sesquialter_fw-la to_o fi_n the_o â—ide_v of_o the_o cube_fw-la let_v there_o be_v draw_v to_o be_v the_o base_a of_o the_o triangle_n abe_n a_o perpendicular_a a_fw-la and_o again_o let_v there_o be_v draw_v to_o the_o same_o base_a in_o the_o triangle_n gâ—e_v the_o perpendicular_a gn_n which_o a_o &_o gn_n shall_v pass_v by_o the_o centre_n fletcher_n and_o i_o and_o the_o line_n of_o be_v duple_n to_o the_o line_n fn_o by_o the_o corollary_n of_o the_o 12._o of_o the_o thirteen_o wherefore_o the_o line_n ao_o be_v duple_n to_o the_o line_n oe_z by_o the_o 2._o of_o the_o six_o for_o the_o line_n foe_n and_o ne_fw-fr be_v parallel_n and_o therefore_o the_o diameter_n agnostus_n be_v triple_a to_o the_o line_n fi._n wherefore_o the_o power_n of_o agnostus_n be_v 1._o noncuple_n to_o the_o power_n of_o fi._n but_o the_o line_n agnostus_n be_v in_o power_n duple_n to_o the_o side_n ab_fw-la by_o the_o 14._o of_o the_o thirteen_o wherefore_o the_o square_a of_o the_o line_n ab_fw-la be_v ing_o the_o half_a of_o the_o square_n of_o the_o line_n agnostus_n which_o be_v noncuple_n to_o the_o square_n of_o the_o line_n fi_n iâ—_z quadruple_a sesquialter_fw-la to_o the_o square_n of_o the_o line_n fi._n the_o side_n therefore_o of_o the_o octohedâ—â—â—â—nd_n of_o the_o cube_fw-la inscribe_v in_o itâ—_n be_v in_o power_n the_o one_o to_o the_o other_o in_o quadruple_a sesquialter_fw-la proportion_n ¶_o the_o 1â—_o proposition_n the_o side_n of_o the_o octohedron_n be_v in_o power_n quadruple_a sesquialter_fw-la to_o that_o right_a line_n who_o great_a segment_n be_v the_o side_n of_o the_o dodecahedron_n inscribe_v in_o the_o same_o octohedron_n forasmuch_o as_o in_o the_o 14._o of_o this_o book_n it_o be_v prove_v that_o the_o side_n of_o the_o octohedron_n be_v in_o power_n quadruple_a sesquialter_fw-la to_o the_o side_n of_o the_o cube_fw-la inscribe_v in_o it_o but_o the_o side_n of_o the_o cube_fw-la be_v cut_v by_o a_o extreme_a and_o mean_a proportion_n make_v the_o great_a segment_n the_o side_n of_o the_o dodecahedron_n circumscribe_v about_o it_o by_o the_o 3._o corollary_n of_o the_o 13._o of_o the_o fifteen_o therefore_o the_o side_n of_o the_o octohedron_n be_v in_o power_n quadruple_a sesquialter_fw-la to_o that_o right_a line_n namely_o to_o the_o side_n of_o the_o cube_fw-la who_o great_a segment_n be_v the_o side_n of_o the_o dodecahedron_n inscribe_v in_o the_o cube_fw-la but_o the_o dodecahedron_n and_o the_o cube_fw-la inscribe_v one_o within_o a_o other_o arâ—_n inscribe_v in_o one_o and_o the_o self_n same_o octohedron_n by_o the_o corollary_n of_o the_o first_o of_o this_o book_n the_o side_n therefore_o of_o the_o octohedron_n be_v in_o power_n quadruple_a sesquialter_fw-la to_o that_o right_a line_n who_o great_a segment_n be_v the_o side_n of_o the_o dodecahedron_n inscribe_v in_o the_o same_o octohedron_n ¶_o the_o 16._o proposition_n the_o side_n of_o a_o icosahedron_n be_v the_o great_a segment_n of_o that_o right_a line_n which_o be_v in_o power_n duple_n to_o the_o side_n of_o the_o octohedron_n inscribe_v in_o the_o same_o icosahedron_n svppose_v that_o there_o be_v a_o icosahedron_n abgdfhec_n who_o side_n let_v be_v bg_o or_o â—câ—_n and_o let_v the_o octohedron_n inscribe_v in_o it_o be_v akdâ—_n and_o let_v the_o side_n thereof_o be_v al._n then_o i_o say_v that_o the_o side_n â—c_z be_v the_o great_a segment_n of_o that_o right_a line_n which_o be_v in_o power_n duple_n to_o the_o side_n al._n for_o forasmuch_o as_o figure_n inscribe_v and_o circumscribe_v have_v oâ—e_a &_o the_o selfsame_o centre_n by_o the_o corollary_n of_o the_o â—1_n of_o the_o fifteen_o let_v the_o same_o be_v the_o point_n i_o now_o right_a lineâ—_z draw_v by_o thâ—â—_n 〈◊〉_d to_o the_o middle_a section_n of_o the_o opposite_a side_n namely_o the_o line_n aid_n and_o kil_n do_v in_o the_o point_n i_o â—ut_z 〈…〉_o the_o other_o inâ—â—_n two_o squall_n 〈◊〉_d and_o perpendicular_o by_o the_o corollary_n of_o the_o 14._o of_o the_o fifteen_o and_o forasmuch_o as_o they_o couple_v the_o middle_a section_n of_o the_o opposite_a line_n bg_o and_o hf_o therefore_o they_o cut_v they_o perpendiular_o h._n wherefore_o also_o the_o line_n bg_o 〈…〉_o be_v parallel_n by_o the_o 4._o corollary_n of_o the_o 14._o of_o the_o 〈…〉_o now_o then_o draw_v a_o line_n from_o b_o to_o h_o and_o the_o say_a â—â—ne_n bh_o shall_v be_v equal_a and_o parallel_v to_o the_o line_n kl_o by_o the_o 33._o of_o the_o first_o but_o the_o line_n bh_o subtend_v â—wâ—_n side_n of_o the_o pentagon_n which_o be_v compose_v of_o the_o side_n of_o the_o icosahedron_n namely_o the_o side_n basilius_n and_o ah_o wherefore_o the_o line_n bh_o be_v cut_v by_o a_o extreme_a and_o mean_a proportion_n make_v the_o great_a segment_n the_o side_n of_o the_o pentagon_n by_o the_o 8._o of_o the_o thirteen_o which_o side_n be_v also_o the_o side_n of_o the_o icosahedron_n namely_o ec_o and_o unto_o the_o line_n bh_o the_o line_n klâ—_n be_v equal_a and_o the_o line_n kl_o be_v in_o power_n duple_n to_o all_o the_o side_n of_o the_o octohedron_n by_o the_o 47._o of_o the_o first_o for_o in_o the_o square_a akdl_n the_o angle_n kal_n be_v a_o right_a angle_n wherefore_o ec_o the_o side_n of_o the_o icosahedron_n be_v the_o great_a segment_n of_o the_o line_n bh_o or_o kl_o which_o be_v in_o power_n duple_n to_o all_o â—he_z side_n of_o the_o octohedron_n inscribe_v in_o the_o icosahedron_n wherefore_o the_o side_n of_o a_o icosahedron_n be_v the_o great_a segment_n of_o that_o right_a line_n which_o be_v in_o power_n duple_n to_o the_o side_n of_o the_o octohedron_n inscribe_v in_o the_o same_o icosahedron_n ¶_o the_o 17._o proposition_n the_o side_n of_o a_o cube_n be_v to_o the_o side_n of_o a_o dodecahedron_n inscribe_v in_o it_o in_o duple_n proportion_n of_o a_o extreme_a and_o mean_a proportion_n for_o it_o be_v manife_a by_o the_o â—_o corollary_n of_o the_o 13._o of_o the_o fifteen_o that_o the_o side_n of_o a_o cube_fw-la divide_v by_o a_o extreme_a and_o mean_a proportion_n make_v the_o less_o segment_n the_o side_n of_o the_o dodecahedron_n inscribe_v in_o it_o but_o the_o whole_a be_v to_o the_o less_o segment_n in_o duple_n proportion_n of_o that_o in_o which_o it_o be_v to_o the_o great_a by_o the_o 10._o definition_n of_o the_o five_o for_o the_o whole_a the_o great_a segment_n and_o the_o less_o be_v line_n in_o continual_a proportion_n by_o the_o 3._o definition_n of_o the_o six_o wherefore_o the_o whole_a namely_o the_o side_n of_o the_o cube_fw-la be_v to_o the_o side_n of_o the_o dodecahedron_n inscribe_v in_o it_o namely_o to_o his_o less_o segment_n in_o duple_n proportion_n of_o a_o extreme_a and_o mean_a proportion_n '_o namely_o be_v of_o that_o which_o the_o whole_a have_v â—o_o the_o great_a segment_n by_o the_o 2._o of_o the_o fourteen_o ¶_o the_o 18._o proposition_n the_o side_n of_o a_o dodecahedron_n be_v to_o the_o side_n of_o a_o cube_n inscribe_v in_o it_o in_o converse_n proportion_n of_o a_o extreme_a and_o mean_a proportion_n it_o be_v prove_v in_o the_o 3._o corollary_n of_o the_o 13._o of_o the_o fifteen_o that_o the_o side_n of_o a_o dodecahedron_n circumscribe_v about_o a_o cube_n be_v the_o great_a segment_n of_o the_o side_n of_o the_o same_o cube_n wherefore_o the_o whole_a
proportion_n which_o be_v in_o power_n duple_n to_o of_o the_o side_n of_o the_o octohedron_n inscribe_v in_o the_o dodecahedron_n draw_v the_o diameter_n el_n and_o fk_o of_o the_o octohedron_n now_o they_o couple_v the_o middle_a section_n of_o the_o opposite_a side_n of_o the_o dodecahedron_n ab_fw-la and_o gd_o by_o the_o 9_o of_o the_o fifteen_o &_o 3._o corollary_n of_o the_o 17._o of_o the_o thirteen_o &_o every_o one_o of_o those_o diameter_n be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n do_v make_v the_o less_o segment_n the_o side_n of_o the_o dodecahedron_n by_o the_o 4._o corollary_n of_o the_o same_o wherefore_o the_o side_n ab_fw-la be_v the_o less_o segment_n of_o the_o line_n fk_o but_o the_o line_n fk_o contain_v in_o power_n the_o two_o equal_a line_n of_o &_o eke_o by_o the_o 47._o of_o the_o first_o for_o the_o angle_n fek_n be_v a_o right_a angle_n of_o the_o square_a fekl_n of_o the_o octohedron_n wherefore_o the_o line_n fk_o be_v in_o power_n duple_n to_o the_o line_n ef._n wherefore_o the_o line_n ab_fw-la the_o side_n of_o the_o dodecahedron_n be_v the_o less_o segment_n of_o the_o line_n fk_o which_o be_v in_o power_n duple_n to_o of_o the_o sidâ—_n of_o the_o octohedron_n the_o side_n therefore_o of_o a_o dodecahedron_n iâ—_z the_o less_o segment_n of_o that_o right_a line_n which_o be_v in_o power_n duple_n to_o the_o side_n of_o the_o octohedron_n inscribe_v in_o the_o same_o dodâ—cahedron_n ¶_o the_o 22._o proposition_n the_o diameter_n of_o a_o icosahedron_n be_v in_o power_n sesquitertia_fw-la to_o the_o side_n of_o the_o same_o icosahedron_n and_o also_o be_v in_o power_n sesquialter_fw-la to_o the_o side_n of_o the_o pyramid_n inscribe_v in_o the_o icosahedron_n for_o forasmuch_o as_o it_o have_v be_v prove_v by_o the_o 10._o of_o this_o book_n that_o if_o from_o the_o power_n of_o the_o diameter_n of_o the_o icosahedron_n be_v take_v away_o the_o triple_a of_o the_o power_n of_o the_o side_n of_o the_o cube_fw-la inscribe_v in_o it_o there_o shall_v be_v leave_v a_o square_a sesquitertia_fw-la to_o the_o square_n of_o the_o side_n of_o the_o icosahedron_n but_o the_o power_n of_o the_o side_n of_o the_o cube_fw-la triple_a be_v the_o diameter_n of_o the_o same_o cube_fw-la by_o the_o 15._o of_o the_o thirteen_o and_o the_o cube_fw-la &_o the_o pyramid_n inscribe_v in_o it_o be_v contain_v in_o one_o &_o the_o self_n same_o sphere_n by_o the_o first_o of_o this_o book_n and_o in_o one_o &_o the_o self_n same_o icosahedron_n by_o the_o corollary_n of_o the_o same_o wherefore_o one_o and_o the_o self_n same_o diameter_n of_o the_o cube_fw-la or_o of_o the_o sphere_n which_o contain_v the_o cube_fw-la and_o the_o pyramid_n be_v in_o power_n sesquialter_fw-la to_o the_o side_n of_o the_o pyramid_n by_o the_o 13._o of_o the_o thirteen_o wherefore_o it_o follow_v that_o if_o from_o the_o diameter_n of_o the_o icosahedron_n be_v take_v away_o the_o triple_a power_n of_o the_o side_n of_o the_o cube_fw-la or_o the_o sesquialter_fw-la power_n of_o the_o side_n of_o the_o pyramid_n which_o be_v the_o power_n of_o one_o and_o the_o self_n same_o diameter_n there_o shall_v be_v leave_v the_o sesquitertia_fw-la power_n of_o the_o side_n of_o the_o icosahedron_n the_o diameter_n therefore_o of_o a_o icosahedron_n be_v in_o power_n sesquitertia_fw-la to_o the_o side_n of_o the_o same_o icosahedron_n and_o also_o be_v in_o power_n sesquialter_fw-la to_o the_o side_n of_o the_o pyramid_n inscribe_v in_o the_o icosahedron_n the_o 23._o proposition_n the_o side_n of_o a_o dodecahedron_n be_v to_o the_o side_n of_o a_o icosahedron_n inscribe_v in_o it_o as_o the_o less_o segment_n of_o the_o perpendicular_a of_o the_o pentagon_n be_v to_o that_o line_n which_o be_v draw_v from_o the_o centre_n to_o the_o side_n of_o the_o same_o pentagon_n constrution_n let_v there_o be_v take_v a_o dodecahedron_n abgdfso_n who_o side_n let_v be_v as_o or_o so_o and_o let_v the_o icosahedron_n inscribe_v in_o it_o be_v klnmne_a who_o side_n let_v be_v kl_o from_o the_o two_o angle_n of_o the_o pentagonâ—_n bas_n and_o fas_fw-la of_o the_o dodecahedron_n namely_o from_o the_o angleâ—â—_n and_o fletcher_n let_v there_o be_v draw_v to_o the_o common_a base_a as_o perpendicular_a line_n bc_o &_o fc_n which_o shall_v pass_v by_o the_o centre_n king_n &_o l_o of_o the_o say_v pentagon_n by_o the_o corollary_n of_o the_o 10._o of_o the_o thirteen_o draw_v the_o line_n bf_o and_o ro._n now_o forasmuch_o as_o the_o line_n ro_n subtend_v the_o angle_n ofr_n of_o thâ—_z pentagon_n of_o the_o dodecahedron_n it_o shall_v cut_v the_o line_n fc_o by_o a_o extreme_a and_o mean_a proportion_n by_o the_o 3._o of_o this_o book_n let_v it_o cut_v it_o in_o the_o point_n i_o and_o forasmuch_o as_o the_o line_n kl_o be_v the_o side_n of_o the_o icosahedron_n inscribe_v in_o the_o dodecahedron_n it_o couple_v the_o centre_n of_o the_o base_n of_o the_o dodecahedron_n for_o the_o angle_n of_o the_o icosahedron_n be_v set_v in_o the_o centre_n of_o the_o base_n of_o the_o dodecahedron_n by_o the_o 7._o of_o the_o fifteen_o now_o i_o say_v that_o so_o the_o side_n of_o the_o dodecahedron_n be_v to_o kl_o the_o side_n of_o the_o icosahedron_n as_o the_o less_o segment_n if_o of_o the_o perpendicular_a line_n cf_o be_v to_o the_o line_n lc_n which_o be_v draw_v from_o the_o centre_n l_o to_o as_o the_o side_n of_o the_o pentagon_n for_o forasmuch_o as_o in_o the_o triangle_n bcf_n the_o two_o side_n cb_o and_o cf_o be_v in_o the_o centre_n l_o and_o king_n cut_v like_o proportional_o demonstration_n the_o line_n bf_o and_o kl_o shall_v be_v parellel_n by_o the_o 2._o of_o the_o six_o wherefore_o the_o triangle_n bcf_n and_o kcl_n shall_v be_v equiangle_n by_o the_o corollary_n of_o the_o same_o wherefore_o as_o the_o line_n cl_n be_v to_o the_o line_n klâ—_n so_o be_v the_o line_n cf_o to_o the_o line_n bf_o by_o the_o 4._o of_o the_o six_o but_o cf_o make_v the_o less_o segment_n the_o line_n if_o by_o the_o 3._o of_o this_o book_n and_o the_o linâ—_n bf_o make_v the_o less_o segment_n the_o line_n so_o namely_o the_o side_n of_o the_o dodecahedron_n by_o the_o 2._o corollary_n of_o the_o 13._o of_o the_o fifteen_o for_o the_o line_n bf_o which_o couple_v the_o angle_n b_o and_o f_o of_o the_o base_n of_o the_o dodecahedron_n be_v equal_a to_o the_o side_n of_o the_o cube_fw-la which_o contain_v the_o dodecahedron_n by_o the_o .13_o of_o the_o fifteen_o wherefore_o as_o the_o whole_a line_n câ—_n be_v to_o the_o whole_a line_n bf_o so_o be_v the_o less_o segment_n if_o to_o the_o less_o segment_n so_o by_o the_o 2._o of_o the_o 14_o but_o as_o the_o line_n cf_o be_v to_o the_o line_n bf_o so_o be_v the_o line_n cl_n prove_v to_o be_v to_o the_o line_n kl_o wherefore_o as_o the_o line_n if_o be_v to_o the_o line_n so_o so_o be_v the_o line_n cl_n to_o the_o line_n kl_o wherefore_o alternate_o by_o the_o 16._o of_o the_o five_o as_o the_o line_n if_o the_o less_o segment_n of_o the_o perpendicular_a of_o the_o pentagon_n fas_fw-la be_v to_o the_o line_n lc_n which_o be_v draw_v from_o the_o centre_n of_o the_o pentagon_n to_o the_o base_a so_o be_v the_o line_n so_o the_o side_n of_o the_o dodecahedron_n to_o thâ—_z line_n kl_o the_o side_n of_o the_o icosahedron_n inscribe_v in_o it_o the_o side_n therefore_o of_o a_o dodecahedron_n be_v to_o the_o side_n of_o a_o icosahedron_n inscribe_v in_o it_o as_o the_o less_o segment_n of_o the_o perpendicular_a of_o the_o pentagon_n be_v to_o that_o line_n which_o be_v draw_v from_o the_o cenâ—re_n to_o the_o side_n of_o the_o same_o pentagon_n ¶_o the_o 24._o proposition_n if_o half_a of_o the_o side_n of_o a_o icosahedron_n be_v divide_v by_o a_o extreme_a &_o mean_a proportion_n and_o if_o the_o less_o segment_n thereof_o be_v take_v away_o from_o the_o whole_a side_n and_o again_o from_o the_o residue_n be_v take_v away_o the_o three_o part_n that_o which_o remain_v shall_v be_v equal_a to_o the_o side_n of_o the_o dodecahedron_n inscribe_v in_o the_o same_o icosahedron_n svppose_v that_o abgdf_n be_v a_o pentagon_n construction_n contain_v five_o side_n of_o the_o icosahedron_n by_o the_o 16._o of_o the_o thirteen_o and_o let_v it_o be_v inscribe_v in_o a_o circle_n who_o centre_n let_v be_v the_o point_n e._n and_o upon_o the_o side_n of_o the_o pentagon_n let_v there_o be_v rear_v up_o triangle_n make_v a_o solid_a angle_n of_o the_o icosahedron_n at_o the_o point_n i_o by_o the_o 16._o of_o the_o thirteen_o and_o in_o the_o circle_n abdella_n inscribe_v a_o equilater_n triangle_n ahk_n from_o the_o centre_n e_o draw_v to_o hk_v the_o side_n of_o the_o triangle_n and_o gd_o the_o side_n of_o the_o pentagon_n a_o perpendicular_a line_n which_o
power_n of_o the_o other_o side_n and_o if_o the_o half_a of_o the_o power_n remain_v be_v apply_v upon_o the_o whole_a base_a it_o shall_v make_v the_o breadth_n that_o section_n of_o the_o base_a which_o be_v couple_v to_o the_o first_o side_n for_o from_o the_o power_n of_o the_o base_a bg_o and_o of_o one_o of_o the_o side_n agnostus_n that_o be_v from_o the_o square_n bd_o and_o gl_n the_o power_n of_o the_o other_o side_n ab_fw-la namely_o the_o square_a bk_o or_o the_o parallelogram_n em_n be_v take_v away_o and_o of_o the_o residue_n namely_o of_o the_o square_n bd_o and_o of_o the_o parallelogram_n odd_a or_o doctor_n which_o by_o supposition_n be_v equal_a unto_o odd_a the_o half_a name_o of_o the_o whole_a fr_z which_o be_v pd_v for_o the_o line_n graccus_n and_o pb_n be_v equal_a to_o the_o line_n go_v and_o po_n be_v apply_v to_o the_o whole_a line_n bg_o or_o gd_o and_o make_v the_o bredthe_a the_o line_n pg_n the_o section_n of_o the_o base_a bg_o which_o section_n be_v couple_v to_o the_o first_o side_n ag._n and_o by_o the_o same_o reason_n in_o the_o other_o side_n if_o from_o the_o square_n bd_o and_o bk_o be_v take_v away_o the_o square_a gl_n there_o shall_v remain_v the_o rectangle_n parallelogram_n foyes_n for_o the_o parallalelogramme_n em_n be_v equal_a to_o the_o square_a bk_o and_o the_o parallelogram_n gm_n to_o the_o parallelogram_n odd_a wherefore_o fp_n the_o half_a of_o the_o residue_n foyes_n make_v the_o breadth_n bp_o which_o be_v couple_v to_o the_o first_o side_n take_v ab_fw-la a_o corollary_n 2._o if_o a_o perpendicular_a draw_v from_o a_o angle_n of_o a_o triangle_n do_v cut_v the_o base_a the_o section_n be_v to_o the_o other_o side_n in_o power_n proportional_a by_o a_o arithmetical_a proportion_n for_o it_o be_v prove_v that_o the_o excess_n of_o the_o power_n of_o the_o line_n agnostus_n and_o ab_fw-la be_v one_o and_o the_o same_o with_o the_o excess_n of_o the_o power_n of_o the_o line_n pg_n and_o pb_n if_o therefore_o the_o power_n do_v equal_o exceed_v the_o one_o the_o other_o they_o shall_v by_o a_o arithmetical_a proportion_n be_v proportional_a the_o end_n of_o the_o sixteen_o book_n of_o the_o element_n of_o geometry_n add_v 〈…〉_o a_o brief_a treatise_n add_v by_o flussas_n of_o mix_a and_o compose_v regular_a solid_n regular_a solid_n be_v say_v to_o be_v compose_v and_o mix_v when_o each_o of_o they_o be_v transform_v into_o other_o solid_n keep_v still_o the_o form_n number_n and_o inclination_n of_o the_o base_n which_o they_o before_o have_v one_o to_o the_o other_o some_o of_o which_o yet_o be_v transform_v into_o mix_a solid_n and_o other_o some_o into_o simple_a into_o mix_v as_o a_o dodecahedron_n and_o a_o icosahedron_n which_o be_v transform_v or_o alter_v if_o you_o divide_v their_o side_n into_o two_o equal_a part_n and_o take_v away_o the_o solid_a angle_n subtend_v of_o plain_a superficial_a figure_n make_v by_o the_o line_n couple_v those_o middle_a section_n for_o the_o solid_a remain_v after_o the_o take_v away_o of_o those_o solid_a angle_n be_v call_v a_o icosidodecahedron_n icosidodecahedron_n if_o you_o divide_v the_o side_n of_o a_o cube_fw-la and_o of_o a_o octohedron_n into_o two_o equal_a part_n and_o couple_v the_o section_n the_o solid_a angle_n subtend_v of_o the_o plain_a superficiece_n make_v by_o the_o couple_a line_n be_v take_v away_o there_o shall_v be_v leave_v a_o solid_a which_o be_v call_v a_o exoctohedron_n exoctohedron_n so_o that_o both_o of_o a_o dodecahedron_n and_o also_o of_o a_o icosahedron_n the_o solid_a which_o be_v make_v shall_v be_v call_v a_o icosidodecahedron_n and_o likewise_o the_o solid_a make_v of_o a_o cube_n &_o also_o of_o a_o octohedron_n shall_v be_v call_v a_o exoctohedron_n but_o the_o other_o solid_a namely_o a_o pyramid_n or_o tetrahedron_n be_v transform_v into_o a_o simple_a solid_a for_o if_o you_o divide_v into_o two_o equal_a part_n every_o one_o of_o the_o side_n of_o the_o pyramid_n triangle_n describe_v of_o the_o line_n which_o couple_v the_o section_n and_o subtend_v and_o take_v away_o solid_a angle_n of_o the_o pyramid_n be_v equal_a and_o like_a unto_o the_o equilater_n triangle_n leave_v in_o every_o one_o of_o the_o base_n of_o all_o which_o triangle_n be_v produce_v a_o octohedron_n namely_o a_o simple_a and_o not_o a_o compose_a solid_a for_o the_o octohedron_n have_v four_o base_n like_v in_o number_n fonne_z and_o mutual_a inclination_n with_o the_o base_n of_o the_o pyramid_n and_o have_v the_o other_o four_o base_n with_o like_a situation_n opposite_a and_o parallel_a to_o the_o former_a wherefore_o the_o application_n of_o the_o pyramid_n take_v twice_o make_v a_o simple_a octohedron_n as_o the_o other_o solid_n make_v a_o mix_a compound_n solid_a ¶_o first_n definition_n a_o exoctohedron_n be_v a_o solid_a figure_n contain_v of_o six_o equal_a square_n and_o eight_o equilater_n and_o equal_a triangle_n ¶_o second_v definition_n a_o icosidodecahedron_n be_v a_o solid_a figure_n contain_v under_o twelve_o equilater_n equal_a and_o equiangle_n pentagons_n and_o twenty_o equal_a and_o equilater_n triangle_n for_o the_o better_a understanding_n of_o the_o two_o former_a definition_n and_o also_o of_o the_o two_o proposition_n follow_v i_o have_v here_o set_v two_o figure_n who_o form_n if_o you_o first_o describe_v upon_o past_v paper_n or_o such_o like_a matter_n and_o then_o cut_v they_o and_o fold_v they_o according_o they_o will_v represent_v unto_o you_o the_o perfect_a form_n of_o a_o exoctohedron_n and_o of_o a_o icosidodecahedron_n ¶_o the_o first_o problem_n to_o describe_v a_o equilater_n and_o equiangle_n exoctohedron_n and_o to_o contain_v it_o in_o a_o sphere_n give_v and_o to_o prove_v that_o the_o diameter_n of_o the_o sphere_n be_v double_a to_o the_o side_n of_o the_o say_a exoctohedron_n and_o now_o forasmuch_o as_o the_o opposite_a side_n and_o diameter_n of_o the_o base_n of_o the_o cube_fw-la be_v parallel_n sphere_n the_o plain_n extend_v by_o the_o right_a line_n qt_o ur_fw-la shall_v be_v a_o parallelogram_n and_o for_o that_o also_o in_o that_o plain_n lie_v qr_o the_o diameter_n of_o the_o cube_fw-la and_o in_o the_o same_o plain_n also_o be_v the_o line_n mh_o which_o divide_v the_o say_a plain_n into_o two_o equal_a part_n and_o also_o couple_v the_o opposite_a angle_n of_o the_o exoctohedron_n this_o line_n mh_o therefore_o divide_v the_o diameter_n into_o two_o equal_a part_n by_o the_o corollary_n of_o the_o 34._o of_o the_o first_o and_o also_o divide_v itself_o in_o the_o same_o point_n which_o let_v be_v s_o into_o two_o equal_a part_n by_o the_o 4_o of_o the_o first_o and_o by_o the_o same_o reason_n may_v we_o prove_v that_o the_o rest_n of_o the_o line_n which_o couple_n the_o opposite_a angle_n of_o the_o exoctohedron_n do_v in_o saint_n the_o centre_n of_o the_o cube_fw-la divide_v thâ—_z one_o the_o other_o into_o two_o equal_a part_n for_o every_o one_o of_o the_o angle_n of_o the_o exoctohedron_n be_v set_v in_o every_o one_o of_o the_o base_n of_o the_o cube_fw-la wheâ—â—â—ore_o make_v the_o centre_n the_o point_n s_o and_o the_o space_n shall_v or_o sm_n describe_v a_o sphere_n and_o it_o shall_v touch_v every_o one_o of_o the_o angle_n eqâ—edistant_a from_o the_o point_n s_o and_o forasmuch_o as_o ab_fw-la the_o diameter_n of_o the_o sphere_n give_v be_v put_v equal_a to_o the_o diameter_n of_o the_o base_a of_o the_o cube_fw-la give_v namely_o to_o the_o line_n rt_n and_o the_o same_o line_n rt_n be_v equal_a to_o the_o line_n mh_o by_o the_o 33._o of_o the_o first_o which_o line_n mh_o couple_a the_o opposite_a angle_n of_o the_o exoctohedron_n be_v draw_v by_o the_o centre_n wherefore_o it_o be_v the_o diameter_n of_o the_o sphere_n give_v which_o contain_v the_o exoctohedron_n final_o forasmuch_o as_o in_o the_o triangle_n rft_v the_o line_n po_n do_v cut_v the_o side_n into_o two_o equal_a part_n exoctohedron_n it_o shall_v cut_v they_o proportional_o with_o the_o base_n namely_o as_o fr_z be_v to_o fp_n so_o shall_v rt_n be_v to_o po_n by_o the_o 2._o of_o the_o six_o but_o fr_z be_v double_a to_o fp_n by_o supposition_n wherefore_o rt_n or_o the_o diameter_n he_o be_v also_o double_a to_o the_o line_n po_n the_o side_n of_o the_o exoctohedron_n wherefore_o we_o have_v describe_v a_o equilater_n &_o equiangle_n exoctohedron_n and_o comprehend_v it_o in_o a_o sphere_n give_v and_o have_v prove_v that_o the_o diameter_n of_o the_o sphere_n be_v double_a to_o the_o side_n of_o the_o exoctohedron_n ¶_o the_o 2._o problem_n to_o describe_v a_o equilater_n &_o equiangle_n icosidodecahedron_n &_o to_o comprehend_v it_o in_o a_o sphere_n give_v and_o to_o prove_v that_o the_o diameter_n be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n make_v the_o great_a segment_n double_a to_o the_o side_n
to_o be_v prâ—â—â—d_v beâ—oâ—e_n 〈◊〉_d â—all_n to_o the_o demonstration_n construction_n demonstration_n lead_v to_o a_o absurdity_n a_o corollary_n this_o problem_n reduce_v to_o a_o theorem_a construction_n demonstration_n how_o magnitude_n be_v say_v to_o be_v in_o proportion_n the_o onâ—_n to_o the_o other_o as_o number_n be_v to_o number_v this_o proposition_n be_v the_o converse_n of_o forme_a construction_n demonstration_n a_o corollary_n construction_n demonstration_n construction_n demonstration_n demonstration_n lead_v to_o a_o absurdity_n this_o be_v the_o 〈…〉_o demonstration_n the_o first_o part_n demonstrate_v an_o other_o demonstration_n of_o the_o first_o part_n an_o other_o demonstration_n oâ—_n the_o same_o first_o part_n after_o montaureus_n demonstration_n of_o the_o second_o part_n which_o be_v the_o converse_n of_o the_o former_a an_o other_o demonstration_n of_o the_o second_o part_n this_o assumpt_n follow_v as_o a_o corollary_n of_o the_o 25_o but_o so_o as_o it_o may_v also_o be_v here_o in_o method_n place_v you_o shall_v â—inde_v it_o after_o the_o 53._o of_o this_o book_n absolute_o demonstrate_v for_o there_o it_o serve_v to_o the_o 54._o his_o demonstration_n demonstration_n of_o the_o three_o part_n demonstration_n of_o the_o four_o part_n which_o be_v the_o converse_n of_o the_o â—_o conclusion_n of_o the_o whole_a proposition_n a_o corollary_n proâ—e_v of_o the_o first_o part_n of_o the_o corollary_n proof_n of_o the_o second_o part_n proof_n of_o the_o three_o pâ—rt_n proâ—e_a oâ—_n the_o four_o part_n certain_a annotation_n â—ut_z of_o montauâ—â—us_fw-la rule_v to_o know_v whether_o two_o superficial_a number_n be_v like_a or_o no._n this_o assumpt_n be_v the_o converse_n of_o the_o 26._o of_o the_o eight_o demonstration_n oâ—_n the_o first_o part_n demonstration_n of_o the_o second_o partâ—_n a_o corollary_n to_o find_v out_o the_o first_o line_n incommensurable_a in_o length_n only_o to_o the_o line_n give_v to_o find_v out_o the_o second_o line_n incommensurable_a both_o in_o length_n and_o in_o power_n to_o the_o line_n give_v construction_n demonstration_n tâ—is_n be_v wiâ—h_n zambert_n a_o aâ—â—â—mpt_n but_o uâ—â—eâ—ly_o improper_o â—lâ—ssateâ—_n mane_v iâ—_z a_o corollary_n but_o the_o greeâ—e_n and_o montaureus_n maâ—e_n it_o a_o proposition_n but_o every_o way_n a_o â—nfallible_a truth_n 〈…〉_o demonstration_n lead_v to_o a_o absurdity_n demonstration_n lead_v to_o a_o absurdity_n a_o corollary_n a_o corollary_n demonstration_n an_o other_o way_n to_o prove_v that_o the_o line_n a_o e_o c_o f_o be_v proportional_a demonstration_n of_o the_o first_o part_n demonstration_n of_o the_o second_o paâ—t_n which_o be_v the_o converse_n of_o the_o first_o a_o corollary_n demonstration_n of_o the_o first_o part_n by_o a_o argument_n leadindg_fw-mi to_o a_o absurdity_n demonstration_n of_o the_o second_o paâ—t_n lead_v also_o to_o a_o impossibility_n and_o this_o second_o part_n be_v the_o converse_n of_o the_o first_o demonstration_n of_o the_o second_o part_n which_o be_v the_o converse_n of_o the_o first_o how_o to_o divide_v the_o line_n bc_o ready_o in_o such_o sort_n as_o iâ—_z require_v in_o the_o proposition_n demonstration_n of_o the_o second_o part_n which_o be_v the_o converse_n of_o tâ—e_z former_a a_o other_o demonstrationâ—y_v a_o argument_n lead_v to_o a_o absurdity_n a_o assumpt_n a_o corollary_n add_v by_o montaureuâ—_n cause_n cause_o of_o increase_v the_o difficulty_n of_o this_o book_n note_n construction_n demonstration_n diverse_a caâ—es_n in_o this_o proposition_n the_o second_o case_n the_o first_o kind_n of_o rational_a line_n commensurable_a in_o length_n this_o particle_n in_o the_o proposition_n according_a to_o any_o of_o the_o foresay_a way_n be_v not_o in_o vain_a put_v the_o second_o kind_n of_o rational_a line_n commensurable_a in_o length_n the_o three_o case_n the_o three_o kind_n of_o rational_a line_n commensurable_a in_o length_n the_o four_o case_n this_o proposition_n be_v the_o converse_n of_o the_o former_a proposition_n construction_n demonstration_n a_o assumpt_n construction_n demonstration_n definition_n of_o a_o medial_a line_n a_o corollary_n this_o assumpt_n be_v nothing_o else_o but_o a_o part_n of_o the_o first_o proposition_n of_o the_o six_o book_n 〈◊〉_d how_o a_o square_n be_v say_v to_o be_v apply_v upon_o a_o line_n construction_n demonstration_n construction_n demonstration_n note_n a_o corollary_n construction_n demonstration_n lead_v to_o a_o absurdity_n construction_n demonstration_n construction_n demonstration_n demonstration_n a_o corollary_n to_o find_v out_o two_o square_a number_n exceed_v the_o one_o the_o other_o by_o a_o square_a â—umber_n a_o assumpt_n construction_n demonstration_n montaureus_n make_v this_o a_o assumpt_n as_o the_o grecke_n text_n seem_v to_o do_v likewise_o but_o without_o a_o cause_n construction_n demonstration_n this_o assumpt_n set_v foâ—th_o nothing_o â—_z but_o that_o which_o the_o first_o oâ—_n the_o sâ—â—t_n jet_v â—orth_o and_o therefore_o in_o sâ—me_a examplar_n it_o be_v not_o find_v construction_n demonstration_n construction_n demonstration_n construction_n demonstration_n a_o corollary_n i_o dee_n dee_n the_o second_o corollary_n corollary_n therefore_o if_o you_o divide_v the_o square_n of_o the_o side_n ac_fw-la by_o the_o side_n bc_o the_o portion_n dc_o will_v be_v the_o product_n etc_n etc_n as_o in_o the_o former_a corollary_n i_o dâ—e_n dâ—e_n the_o third_o corollary_n corollary_n therefore_o if_o the_o parallelogram_n of_o basilius_n and_o ac_fw-la be_v divide_v by_o bc_o the_o product_n will_v geve_v the_o perpendicular_a d_o a._n these_o three_o corollarye_n in_o practice_n logistical_a and_o geometrical_a be_v profitable_a an_o other_o demonstration_n of_o this_o four_o part_n of_o the_o determination_n a_o assumpt_n construction_n demonstration_n the_o first_o part_n of_o the_o determination_n conclude_v the_o second_o part_n conclude_v the_o total_a conclusion_n construction_n demonstration_n the_o first_o part_n of_o the_o determination_n conclude_v the_o second_o part_n conclude_v the_o total_a conclusion_n construction_n demonstration_n the_o first_o part_n conclude_v the_o second_o part_n conclude_v the_o three_o part_n conclude_v the_o total_a conclusion_n the_o first_o senary_a by_o composition_n definition_n of_o a_o binomial_a line_n six_o kind_n of_o binomial_a line_n demonstration_n definition_n of_o a_o first_o bimedial_a line_n construction_n demonstration_n definition_n of_o a_o second_o bimedial_a line_n demonstration_n definition_n of_o a_o great_a line_n definition_n of_o a_o line_n who_o power_n be_v rational_a and_o medial_a definition_n of_o a_o liâ—e_z contain_v in_o power_n two_o medial_n a_o assumpt_n the_o second_o senary_a by_o composition_n demonstration_n lead_v to_o a_o impossibility_n a_o corollary_n demonstration_n lead_v to_o a_o impossibilâ—â—e_n demonstration_n lead_v to_o a_o impossibility_n demonstration_n lead_v to_o a_o impossibility_n demonstration_n lead_v to_o a_o impossibiâ—â—â—e_n demonstration_n lead_v to_o a_o impossibility_n construction_n demonstration_n lead_v to_o a_o absurdity_n six_o kind_n of_o binomial_a line_n a_o binomial_a line_n consi_v of_o two_o paâ—tâ—s_n firsâ—_o definition_n seconâ—_o definition_n three_o â—â—â—â—â—â—ition_n four_o definition_n five_o definition_n six_o definition_n the_o three_o senary_a by_o composition_n construction_n demonstration_n construction_n demonstration_n construction_n demonstration_n construction_n demonstration_n construction_n demonstration_n construction_n demonstration_n a_o corollary_n add_v by_o flussates_n m._n do_v you_o his_o book_n call_v tyâ—â—câ—niâ—m_fw-la mathematicum_fw-la this_o assumpt_n as_o be_v before_o note_v follow_v most_o â—riâ—fly_o without_o far_a demonstration_n of_o the_o 25._o of_o this_o book_n demonstration_n a_o assumpt_n the_o four_o senary_a by_o composition_n construction_n demonstration_n the_o first_o part_n of_o this_o demonstration_n conclude_v the_o second_o part_n of_o the_o demonstration_n conclude_v the_o three_o part_n conclude_v the_o total_a conclusion_n demonstration_n the_o first_o part_n of_o this_o demonstration_n conclude_v the_o three_o part_n conclude_v the_o four_o part_n concludeâ—_n the_o five_o part_n conclude_v the_o total_a conclusion_n demonstration_n construction_n demonstration_n demonstration_n demonstration_n demonstration_n look_v after_o the_o assumpt_n conclude_v at_o this_o mark_n for_o plain_a open_n of_o this_o place_n the_o use_n of_o this_o assumpt_n be_v in_o the_o next_o proposition_n &_o other_o follow_v the_o five_o senary_a by_o composition_n construction_n demonstration_n conclude_v that_o dg_o be_v a_o binomial_a line_n construction_n demonstration_n conclude_v that_o dg_o be_v a_o binomial_a line_n construction_n demonstration_n †_o *_o ‡_o dg_o conclude_v a_o binomial_a line_n a_o corollary_n add_v by_o m._n dee_n construction_n demonstration_n construction_n demonstration_n construction_n demonstration_n the_o six_o senary_n construction_n demonstration_n construction_n demonstration_n a_o corollary_n add_v by_o flussetes_n note_n construction_n demonstration_n an_o other_o demonstration_n after_o p._n montaureus_n an_o other_o demonstration_n after_o campane_n construction_n demonstration_n an_o other_o demonstration_n afâ—â—r_v campane_n construction_n demonstration_n a_o assumpt_n an_o other_o demonstration_n after_o campanâ—_n note_n
magnitude_n a_o be_v commensurable_a unto_o the_o magnitude_n c_o therefore_o by_o the_o 5._o of_o the_o ten_o a_o have_v unto_o c_o such_o proportion_n as_o number_n have_v to_o number_v let_v a_o have_v unto_o c_o that_o proportion_n that_o the_o number_n d_o have_v to_o the_o number_n e._n again_o forasmuch_o as_o b_o be_v commensurable_a unto_o c_o therefore_o by_o the_o self_n same_o c_z have_v unto_o b_o that_o proportion_n that_o number_n have_v to_o number_v let_v c_o have_v unto_o b_o that_o proportion_n that_o the_o number_n fletcher_n have_v unto_o the_o number_n g._n now_o then_o take_v the_o least_o number_n in_o continual_a proportion_n and_o in_o these_o proportion_n give_v namely_o that_o the_o number_n d_o have_v to_o the_o number_n e_o and_o that_o the_o number_n fletcher_n have_v to_o the_o number_n g_z by_o the_o 4._o of_o the_o eight_o which_o let_v be_v the_o number_n â—_o king_n l._n demonstration_n so_o that_o as_o the_o number_n d_o be_v to_o the_o number_n e_o so_fw-mi let_v the_o number_n h_o be_v to_o the_o number_n king_n and_o as_o the_o number_n fletcher_n be_v to_o the_o number_n g_o so_o let_v the_o number_n king_n be_v to_o the_o number_n l._n now_o for_o that_o as_o a_o be_v to_o c_o so_o be_v d_z to_o e_o but_o as_z d_z be_v to_o e_o so_o be_v h_o to_o king_n therefore_o as_o a_o be_v to_o c_o so_o be_v h_o to_o k._n again_o for_o that_o as_o c_o be_v to_o b_o so_o be_v fletcher_n to_o g_z but_o as_o fletcher_n be_v to_o g_z so_o be_v king_n to_o l_o therefore_o as_o c_z be_v to_o b_o so_o be_v king_n to_o l._n but_o it_o be_v now_o prove_v that_o as_o a_o be_v to_o c_o so_o be_v h_o to_o k._n wherefore_o of_o equality_n by_o the_o 22._o of_o the_o five_o as_o a_o be_v to_o b_o so_o be_v the_o number_n h_o to_o the_o number_n l._n wherefore_o a_o have_v unto_o b_o such_o proportion_n as_o number_n have_v to_o number_v wherefore_o by_o the_o six_o of_o the_o ten_o the_o magnitude_n a_o be_v commensurable_a unto_o the_o magnitude_n b._n magnitude_n therefore_o commensurable_a to_o one_o and_o the_o self_n same_o magnitude_n 〈…〉_o be_v also_o commensurable_a the_o one_o to_o the_o other_o which_o be_v require_v to_o be_v prove_v ¶_o a_o assumpt_n if_o there_o be_v two_o magnitude_n compare_v to_o one_o and_o the_o self_n same_o magnitude_n and_o if_o the_o one_o of_o they_o be_v commensurable_a unto_o it_o and_o the_o other_o incommensurable_a those_o magnitude_n be_v incommensurable_a the_o one_o to_o the_o other_o svppose_v that_o there_o be_v two_o magnitude_n namely_o absurdity_n a_o and_o b_o and_o let_v c_o be_v a_o certain_a other_o magnitude_n and_o let_v a_o â—e_n commensurable_a unto_o c_z and_o let_v b_o be_v commensurable_a unto_o the_o self_n same_o c._n then_o i_o say_v that_o the_o magnitude_n a_o be_v incommensurable_a unto_o b._n for_o if_o a_o be_v commensurable_a unto_o b_o forasmuch_o as_o a_o be_v also_o commensurable_a unto_o c_z therefore_o by_o the_o 12._o of_o the_o ten_o b_o be_v also_o commensurable_a unto_o c_z which_o be_v contrary_a to_o the_o supposition_n ¶_o the_o 10._o theorem_a the_o 13._o proposition_n if_o there_o be_v two_o magnitude_n commensurable_a and_o if_o the_o one_o of_o they_o be_v incommensurable_a to_o any_o other_o magnitude_n the_o other_o also_o shall_v be_v incommensurable_a unto_o the_o same_o svppose_v that_o these_o two_o magnitude_n a_o b_o be_v commensurable_a the_o one_o to_o the_o other_o and_o let_v the_o one_o of_o they_o namely_o a_o be_v incommensurable_a unto_o a_o other_o magnitude_n absurdity_n namely_o unto_o c._n then_o i_o say_v that_o the_o other_o magnitude_n also_o namely_o b_o be_v incommensurable_a unto_o c._n for_o if_o b_o be_v commensurable_a unto_o c_z then_o forasmuch_o as_o a_o be_v commensurable_a unto_o b_o therefore_o by_o the_o 12._o of_o the_o ten_o the_o magnitude_n a_o also_o be_v commensurable_a unto_o the_o magnitude_n c._n but_o it_o be_v suppose_v to_o be_v incommensurable_a unto_o it_o which_o be_v impossible_a wherefore_o the_o magnitude_n b_o and_o c_o be_v not_o commensurable_a wherefore_o they_o be_v incommensurable_a if_o therefore_o there_o be_v two_o magnitude_n commensurable_a and_o if_o the_o one_o of_o they_o be_v incommensurable_a to_o any_o other_o magnitude_n the_o other_o also_o shall_v be_v incommensurable_a unto_o the_o same_o which_o be_v require_v to_o be_v prove_v ¶_o a_o corollary_n add_v by_o montaureus_n magnitude_n commensurable_a to_o magnitude_n incommensurable_a be_v also_o incommensurable_a the_o one_o to_o the_o other_o corollary_n suppose_v that_o the_o magnitude_n a_o and_o b_o be_v incommensurable_a the_o one_o to_o the_o other_o and_o let_v the_o magnitude_n c_o be_v commensurable_a to_o a_o and_o let_v the_o magnitude_n d_o be_v commensurable_a unto_o b._n then_o i_o say_v that_o the_o magnituâ—â—s_n c_o and_o d_o be_v incommensurable_a the_o one_o to_o the_o other_o for_o a_o and_o c_o be_v commensurable_a of_o which_o the_o magnitude_n a_o be_v incommensurable_a unto_o b_o wherefore_o by_o this_o 13._o proposition_n the_o magnitude_n c_o and_o b_o be_v also_o incommensurable_a but_o the_o magnitudeâ—_n b_o and_o d_z be_v commensurable_a wherefore_o by_o the_o same_o or_o by_o the_o former_a assumpt_n the_o magnitude_n c_o and_o d_o be_v incommensurable_a the_o one_o to_o the_o other_o this_o corollary_n theon_n use_v often_o time_n as_o in_o the_o 22._o 26._o and_o 36_o proposition_n of_o this_o book_n and_o in_o other_o proposition_n also_o ¶_o a_o assumpt_n two_o unequal_a right_a line_n be_v give_v to_o fiâ—de_v out_o how_o much_o the_o great_a be_v in_o power_n more_o than_o the_o less_o corollary_n and_o like_v in_o sort_n two_o right_a line_n be_v give_v by_o this_o mean_n may_v be_v find_v out_o a_o right_a line_n which_o contain_v they_o both_o in_o power_n suppose_v that_o the_o two_o right_a line_n give_v be_v ad_fw-la and_o db._n it_o be_v require_v to_o â—inde_v out_o a_o right_a line_n that_o contain_v they_o both_o in_o power_n let_v the_o line_n ab_fw-la and_o db_o be_v so_o put_v that_o they_o comprehend_v a_o right_a angle_n adb_o and_o draw_v a_o right_a line_n from_o a_o to_o b._n now_o again_o it_o be_v manifest_a by_o the_o 47._o of_o the_o â—irst_n that_o the_o line_n ab_fw-la contain_v in_o power_n the_o line_n ad_fw-la and_o db._n ¶_o the_o 11._o theorem_a the_o 14._o proposition_n if_o there_o be_v sour_a right_a line_n proportional_a and_o if_o the_o first_o be_v in_o power_n more_o than_o the_o second_o by_o the_o square_n of_o a_o right_a line_n commensurable_a in_o length_n unto_o the_o first_o the_o three_o also_o shall_v be_v in_o power_n more_o than_o the_o four_o by_o the_o square_n of_o a_o right_a line_n commensurable_a unto_o the_o three_o and_o if_o the_o first_o be_v in_o power_n more_o than_o the_o second_o by_o the_o square_n of_o a_o right_a line_n incommensurable_a in_o length_n unto_o the_o first_o the_o three_o also_o shall_v be_v in_o power_n more_o than_o the_o four_o by_o the_o square_n of_o a_o right_a line_n incommensurable_a in_o length_n to_o the_o three_o svppose_v that_o these_o four_o right_a line_n a_o b_o c_o d_o be_v proportional_a as_o a_o be_v to_o b_o so_o let_v c_o be_v to_o d._n and_o let_v a_o be_v in_o power_n more_o than_o b_o by_o the_o square_n of_o the_o line_n e._n and_o likewise_o let_v c_o be_v in_o power_n more_o than_o d_o by_o the_o square_n of_o the_o line_n f._n demonstration_n then_o i_o say_v that_o if_o a_o be_v commensurable_a in_o length_n unto_o the_o line_n e_o c_o also_o shall_v be_v commensurable_a in_o length_n unto_o the_o line_n f._n and_o if_o a_o be_v incommensurable_a in_o length_n to_o the_o line_n e_o c_o also_o shall_v be_v incommensurable_a in_o length_n to_o the_o line_n f._n for_o for_o that_o as_o a_o be_v to_o b_o so_o be_v c_z to_o d_z therefore_o as_o the_o square_n of_o the_o line_n a_o be_v to_o the_o square_n of_o the_o line_n b_o so_o be_v the_o square_a of_o the_o line_n c_o to_o the_o square_n of_o the_o line_n d_o by_o the_o 22._o of_o the_o six_o but_o by_o supposition_n unto_o the_o square_n of_o the_o line_n a_o be_v equal_a the_o square_n oâ—_n the_o line_n e_o and_o b_o and_o unto_o the_o square_n of_o the_o line_n c_o be_v equal_a the_o square_n of_o the_o of_o the_o line_n d_o and_o f_o wherefore_o as_o the_o square_n of_o the_o line_n e_o and_o b_o which_o be_v equal_a to_o the_o square_n of_o the_o line_n a_o be_v to_o the_o square_n of_o the_o line_n b_o so_fw-mi be_v the_o square_n of_o the_o line_n d_o and_o f_o which_o be_v equal_a to_o the_o square_n of_o the_o line_n c_o to_o the_o square_n of_o the_o line_n d_o by_o the_o seven_o of_o the_o five_o wherefore_o
by_o the_o 17._o of_o the_o five_o as_o the_o square_n of_o the_o line_n â—_o iâ—_z to_o the_o square_n of_o the_o line_n b_o so_o be_v the_o square_a of_o the_o line_n f_o to_o the_o square_n of_o the_o line_n d._n wherefore_o also_o as_o the_o line_n e_o be_v to_o the_o line_n â—_o so_o be_v the_o line_n f_o to_o the_o line_n d_o by_o the_o second_o part_n of_o the_o 22._o of_o the_o six_o wherefore_o contrariwise_o by_o the_o corollary_n of_o the_o four_o of_o the_o five_o as_o b_o be_v to_o e_o so_o be_v d_z to_o f._n but_o by_o supposition_n â—s_v a_o be_v to_o b_o so_o be_v c_z to_o d_z wherefore_o of_o equality_n by_o the_o 22._o of_o the_o five_o aâ—_z a_o be_v to_o â—_o so_o be_v c_z be_v f._n if_o therefore_o a_o be_v commensurable_a in_o length_n unto_o e_o c_o also_o shall_v be_v commensurable_a in_o length_n unto_o fletcher_n and_o if_o it_o ba_o incommensurable_a in_o length_n unto_o e_o c_o also_o shall_v be_v incommensurablel_v in_o length_n unto_o fletcher_n by_o the_o 10._o of_o this_o book_n if_o therefore_o there_o be_v four_o right_a tlines_n proportional_a and_o if_o the_o first_o be_v in_o power_n more_o than_o the_o secondby_o the_o square_n of_o a_o right_a line_n commensurable_a in_o length_n unto_o the_o first_o the_o three_o also_o shall_v be_v in_o power_n more_o than_o the_o four_o by_o the_o square_n of_o a_o right_a line_n commensurable_a in_o length_n unto_o the_o three_o and_o if_o the_o first_o be_v in_o power_n more_o they_o the_o second_o by_o the_o square_n of_o a_o right_a line_n incommensurable_a in_o length_n unto_o the_o first_o the_o three_o also_o shall_v be_v in_o power_n more_o the_o the_o four_o by_o the_o square_n of_o a_o right_a line_n incommensurable_a in_o length_n to_o the_o three_o which_o be_v require_v to_o be_v prove_v note_v that_o the_o line_n a_o may_v be_v prove_v to_o be_v in_o proportion_n to_o the_o line_n e_o as_o the_o line_n c_o be_v to_o the_o line_n f_o by_o a_o other_o way_n namely_o by_o conversion_n of_o proportion_n of_o someâ—_n as_o we_o have_v before_o note_v proportional_a call_v inverse_v proportion_n by_o the_o 19_o of_o the_o five_o for_o forasmuch_o as_o the_o four_o line_n a_o b_o c_o d_o be_v proportional_a therefore_o by_o the_o 22._o of_o the_o six_o their_o square_n also_o be_v proportional_a and_o forasmuch_o as_o the_o antecedent_n namely_o the_o square_a of_o the_o line_n a_o exceed_v the_o consequent_a namely_o the_o square_a of_o the_o line_n b_o by_o the_o square_n of_o the_o line_n e_o and_o the_o other_o antecedent_n namely_o the_o square_a of_o the_o line_n c_o exceed_v the_o other_o consequent_a namely_o the_o square_a of_o the_o line_n d_o by_o the_o square_n of_o the_o line_n f_o therefore_o as_o the_o square_n of_o the_o line_n a_o be_v to_o the_o excess_n namely_o to_o the_o square_n of_o the_o line_n e_o so_o be_v â—he_z square_a of_o the_o line_n c_o to_o the_o excess_n namely_o to_o the_o square_n of_o the_o line_n f._n wherefore_o by_o the_o second_o part_n of_o the_o 22._o of_o the_o six_o as_o the_o line_n a_o be_v to_o the_o line_n e_o so_o be_v the_o line_n c_o to_o the_o line_n f._n ¶_o the_o 12._o theorem_a the_o 15._o proposition_n if_o two_o magnitude_n commensurable_a be_v compose_v the_o whole_a magnitude_n compose_v also_o shall_v be_v commensurable_a to_o either_o of_o the_o two_o part_n and_o if_o the_o whole_a magnitude_n compose_v be_v commensurable_a to_o any_o one_o of_o the_o two_o part_n those_o two_o part_n shall_v also_o be_v commensurable_a let_v these_o two_o commensurable_a magnitude_n ab_fw-la and_o bc_o be_v compose_v or_o add_v together_o then_o i_o say_v that_o the_o whole_a magnitude_n ac_fw-la be_v commensurable_a to_o either_o of_o these_o part_n ab_fw-la and_o bc_o for_o forasmuch_o as_o ab_fw-la and_o bc_o be_v commensurable_a therefore_o by_o the_o first_o definition_n of_o the_o ten_o some_o one_o magnitude_n measure_v they_o both_o part_n let_v there_o be_v a_o magnitude_n that_o measure_v they_o and_o let_v the_o same_o be_v d._n now_o forasmuch_o as_o d_z measure_v ab_fw-la and_o bc_o it_o shall_v also_o measure_v the_o whole_a magnitude_n compose_v ac_fw-la by_o this_o common_a sentence_n what_o soever_o magnitude_n measure_v two_o other_o magnitude_n shall_v also_o measure_v the_o magnitude_n compose_v of_o they_o but_o the_o same_o d_o measure_v ab_fw-la and_o bc_o by_o supposition_n wherefore_o d_z measure_v ab_fw-la bc_o and_o ac_fw-la wherefore_o ac_fw-la be_v commensurable_a to_o either_o of_o these_o magnitude_n ab_fw-la and_o bc._n but_o now_o suppose_v that_o the_o whole_a compose_a magnitude_n ac_fw-la be_v commensurable_a to_o any_o one_o of_o these_o two_o magnitude_n ab_fw-la or_o bc_o let_v it_o be_v commensurable_a i_o say_v unto_o ab_fw-la first_o then_o i_o say_v that_o the_o two_o magnitude_n ab_fw-la and_o bc_o be_v commensurable_a for_o forasmuch_o as_o ab_fw-la and_o ac_fw-la be_v commonsurable_a some_o one_o magnitude_n measure_v they_o by_o the_o first_o definition_n of_o the_o ten_o let_v some_o magnitude_n measure_v they_o and_o let_v the_o same_o be_v d._n now_o forasmuch_o as_o d_z measure_v ab_fw-la and_o ac_fw-la it_o also_o measure_v the_o residue_n bc_o by_o this_o common_a sentence_n what_o soever_o measure_v the_o whole_a and_o the_o part_n take_v away_o shall_v also_o measure_v the_o residue_n but_o the_o same_o d_o measure_v the_o magnitude_n ab_fw-la by_o supposition_n wherefore_o d_z measure_v either_o of_o these_o magnitude_n ab_fw-la and_o bc._n wherefore_o the_o magnitude_n ab_fw-la and_o bc_o be_v commensurable_a if_o therefore_o two_o magnitude_n commensurable_a be_v compose_v the_o whole_a magnitude_n compose_v also_o shall_v be_v commensurable_a to_o either_o of_o the_o two_o part_n and_o if_o the_o whole_a magnitude_n compose_v be_v commensurable_a to_o any_o one_o of_o the_o two_o part_n those_o two_o part_n shall_v also_o be_v commensurable_a which_o be_v require_v to_o be_v demonstrate_v ¶_o a_o corollary_n add_v by_o montaureus_n corollary_n if_o a_o whole_a magnitude_n be_v commensurable_a to_o one_o of_o the_o two_o magnitude_n which_o make_v the_o whole_a magnitude_n it_o shall_v also_o be_v commensurable_a to_o the_o other_o of_o the_o two_o magnitude_n for_o if_o the_o whole_a magnitude_n ac_fw-la be_v commensurable_a unto_o the_o magnitude_n bc_o then_o by_o the_o 2._o part_n of_o this_o 15_o proposition_n the_o magnitude_n ab_fw-la and_o bc_o be_v commensurable_a wherefore_o by_o the_o first_o part_n of_o the_o same_o the_o magnitude_n ac_fw-la shall_v be_v commensurable_a to_o either_o of_o these_o magnitude_n ab_fw-la and_o bc._n this_o corollary_n theon_n use_v in_o the_o demonstration_n of_o the_o 17._o proposition_n and_o also_o of_o other_o proposition_n howbeit_o euclid_n leave_v it_o out_o for_o that_o it_o seem_v easy_a as_o in_o a_o manner_n do_v all_o other_o corollaryes_fw-mi ¶_o the_o 13._o theorem_a the_o 16._o proposition_n if_o two_o magnitude_n incommensurable_a be_v compose_v the_o whole_a magnitude_n also_o shall_v be_v incommensurable_a unto_o either_o of_o the_o two_o part_n componentes_fw-la and_o if_o the_o whole_a be_v incommensurable_a to_o one_o of_o the_o part_n componentes_fw-la those_o first_o magnitude_n also_o shall_v be_v incommensurable_a absurdity_n let_v these_o two_o incommensurable_a magnitude_n ab_fw-la &_o bc_o be_v compose_v or_o add_v together_o then_o i_o say_v that_o the_o whole_a magnitude_n ac_fw-la be_v incommensurable_a to_o either_o of_o these_o magnitude_n ab_fw-la and_o bc._n for_o if_o ac_fw-la and_o ab_fw-la be_v not_o incommensurable_a than_o some_o one_o magnitude_n measure_v they_o by_o the_o first_o definition_n of_o the_o ten_o let_v there_o be_v such_o a_o magnitude_n if_o it_o be_v possible_a and_o let_v the_o same_o be_v d._n now_o forasmuch_o as_o d_z measure_v ca_n and_o ab_fw-la it_o also_o measure_v the_o residue_n bc_o &_o it_o likewise_o measure_v ab_fw-la wherefore_o d_z measure_v ab_fw-la and_o bc._n wherefore_o by_o the_o first_o definition_n of_o the_o ten_o the_o magnitude_n ab_fw-la and_o bc_o be_v commensurable_a but_o it_o be_v suppose_v that_o they_o be_v incommensurable_a which_o be_v impossible_a wherefore_o no_o magnitude_n do_v measure_v the_o magnitude_n ab_fw-la and_o ac_fw-la wherefore_o the_o magnitude_n ca_n and_o ab_fw-la be_v incommensurable_a in_o like_a sort_n also_o may_v we_o prove_v that_o the_o magnitude_n ac_fw-la and_o cb_o be_v incommensurable_a first_o but_o now_o suppose_v that_o the_o magnitude_n ac_fw-la be_v incommensurable_a to_o one_o of_o these_o magnitude_n ab_fw-la or_o bc_o and_o first_o let_v it_o be_v incommensurable_a unto_o ab_fw-la then_o i_o say_v that_o the_o magnitude_n ab_fw-la and_o bc_o be_v incommensurable_a for_o if_o they_o be_v commensurable_a some_o one_o magnitude_n measure_v they_o let_v some_o one_o magnitude_n measure_v they_o &_o let_v the_o same_o be_v d._n
shall_v in_o like_a sort_n by_o the_o 14._o of_o the_o ten_o be_v in_o power_n more_o than_o the_o line_n df_o by_o the_o square_n of_o a_o line_n incommensurable_a in_o length_n to_o the_o line_n cf_o and_o then_o if_o the_o line_n ae_n be_v commensurable_a in_o length_n to_o the_o rational_a line_n the_o line_n cf_o shall_v also_o in_o like_a sort_n be_v commensurable_a in_o length_n to_o the_o same_o rational_a line_n and_o so_o either_o of_o the_o line_n ab_fw-la and_o cd_o be_v a_o four_o residual_a line_n and_o if_o the_o line_n be_v be_v commensurable_a in_o length_n to_o the_o rational_a line_n the_o line_n df_o shall_v also_o be_v commensurable_a in_o length_n to_o the_o same_o line_n and_o so_o either_o of_o the_o line_n ab_fw-la &_o cd_o be_v a_o â—iâ—t_o residual_a line_n and_o if_o neither_o of_o the_o line_n ae_n nor_o be_v be_v commensurable_a in_o length_n to_o the_o rational_a line_n in_o like_a sort_n neither_o of_o the_o line_n cf_o nor_o df_o shall_v be_v commensurable_a in_o length_n to_o the_o same_o rational_a line_n and_o so_o either_o of_o the_o line_n ab_fw-la &_o cd_o be_v a_o six_o residual_a line_n wherefore_o the_o line_n cd_o be_v a_o residual_a line_n of_o the_o self_n same_o order_n that_o the_o line_n ab_fw-la be_v a_o line_n therefore_o commensurable_a in_o length_n to_o a_o residual_a line_n be_v itself_o also_o a_o residual_a line_n of_o the_o self_n same_o order_n which_o be_v require_v to_o be_v prove_v as_o before_o touch_v binomial_a line_n so_o also_o touch_v residual_a line_n this_o be_v to_o be_v note_v that_o a_o line_n commensurable_a in_o length_n to_o a_o residual_a line_n be_v always_o a_o residual_a line_n of_o the_o self_n same_o order_n that_o the_o residual_a line_n be_v unto_o who_o it_o be_v commensurable_a note_n as_o have_v before_o in_o this_o 103._o proposition_n be_v prove_v but_o if_o a_o line_n be_v commensurable_a in_o power_n only_o to_o a_o residual_a lineâ—_z than_o follow_v it_o not_o yea_o it_o be_v impossible_a that_o that_o line_n shall_v be_v a_o residual_a of_o the_o self_n same_o order_n that_o the_o residual_a line_n be_v unto_o who_o it_o be_v commensurable_a in_o power_n only_o howbeit_o those_o two_o line_n shall_v of_o necessity_n be_v both_o either_o of_o the_o three_o first_o order_n of_o residâ—â—ll_a line_n or_o of_o the_o three_o last_o order_n which_o be_v not_o hard_a to_o prove_v if_o you_o mark_v diligent_o the_o former_a demonstration_n and_o that_o which_o be_v speak_v of_o binomial_a line_n as_o touch_v this_o matter_n ¶_o the_o 80._o theorem_a the_o 104._o proposition_n a_o line_n commensurable_a to_o a_o medial_a residual_a line_n be_v itself_o also_o a_o medial_a residual_a line_n and_o of_o the_o self_n same_o order_n svppose_v that_o ab_fw-la be_v a_o medial_a residual_a line_n unto_o who_o let_v the_o line_n cd_o be_v commensurable_a in_o length_n and_o in_o power_n or_o in_o power_n only_o then_o i_o say_v that_o cd_o be_v also_o a_o medial_a residual_a line_n and_o of_o the_o self_n same_o order_n for_o forasmuch_o as_o the_o line_n ab_fw-la be_v a_o medial_a residual_a line_n construction_n let_v the_o line_n convenient_o join_v unto_o iâ—_z 〈◊〉_d be_v wherefore_o the_o line_n ae_n and_o be_v be_v medial_a commensurable_a in_o power_n only_o as_o ab_fw-la be_v to_o cd_o so_o by_o the_o 22._o of_o the_o six_o let_v be_v be_v to_o df._n and_o in_o like_a sort_n as_o in_o the_o former_a so_o also_o in_o this_o may_v we_o prove_v demonstration_n that_o the_o line_n ae_n be_v commensurable_a in_o length_n and_o in_o power_n or_o in_o power_n only_o unto_o the_o line_n cf_o &_o the_o line_n be_v 〈◊〉_d the_o line_n df._n wherefore_o by_o the_o 23._o of_o the_o ten_o 〈◊〉_d line_n cf_o be_v a_o medial_a line_n and_o the_o line_n df_o be_v also_o a_o medial_a line_n for_o that_o it_o be_v commensurable_a to_o the_o medial_a line_n be._n and_o in_o like_a sort_n the_o line_n cf_o and_o df_o be_v commensurable_a in_o power_n only_o for_o that_o they_o have_v the_o self_n same_o proportion_n the_o one_o to_o the_o other_o that_o the_o line_n ae_n and_o ebb_n have_v which_o be_v commensurable_a in_o power_n only_o wherefore_o the_o line_n cd_o be_v a_o medial_a residual_a line_n medial_a i_o say_v moreover_o that_o it_o be_v of_o the_o self_n same_o order_n that_o the_o line_n ab_fw-la be_v for_o for_o that_o as_o the_o line_n ae_n be_v to_o the_o line_n be_v so_o be_v the_o line_n cf_o to_o the_o line_n df._n but_o as_o the_o line_n ae_n be_v to_o the_o line_n be_v so_o be_v the_o square_a of_o the_o line_n ae_n to_o the_o parallelogram_n contain_v under_o the_o line_n ae_n and_o be_v by_o the_o first_o of_o the_o six_o and_o as_o the_o line_n cf_o be_v to_o the_o line_n df_o so_o be_v the_o square_a of_o the_o line_n cf_o to_o the_o parallelogram_n contain_v under_o the_o line_n cf_o and_o df._n wherefore_o as_o the_o square_n of_o the_o line_n ae_n be_v to_o the_o parallelogram_n contain_v under_o the_o line_n ae_n and_o be_v so_o be_v the_o square_a of_o the_o line_n cf_o to_o the_o parallelogram_n contain_v under_o the_o line_n cf_o and_o df._n wherefore_o alternate_o as_o the_o square_n of_o the_o line_n ae_n be_v to_o the_o square_n of_o the_o line_n cf_o so_o be_v the_o parallelogram_n contain_v under_o the_o line_n ae_n and_o be_v to_o the_o parallelogram_n contain_v under_o the_o â—ines_n cf_o and_o df._n but_o the_o square_n of_o the_o line_n ae_n be_v commensurable_a to_o the_o square_n of_o the_o line_n cf_o for_o the_o line_n ae_n be_v commensurable_a to_o the_o line_n cf_o wherefore_o also_o the_o parallelogram_n contain_v under_o the_o line_n ae_n and_o be_v be_v commensurable_a to_o the_o parallelogram_n contain_v under_o the_o line_n cf_o and_o df._n wherefore_o if_o the_o parallelogram_n contain_v under_o the_o line_n ae_n and_o ebb_n be_v rational_a the_o parallelogram_n also_o contain_v under_o the_o line_n cf_o and_o fd_o shall_v be_v rational_a and_o then_o either_o of_o the_o line_n ab_fw-la and_o cd_o be_v a_o first_o medial_a residual_a line_n but_o if_o the_o parallelogram_n contain_v under_o the_o line_n ae_n and_o be_v be_v medial_a the_o parallelogram_n also_o contain_v under_o the_o line_n cf_o and_o fd_o shall_v be_v also_o medial_a by_o the_o corollary_n of_o the_o 23._o of_o the_o ten_o and_o so_o either_o of_o the_o line_n ab_fw-la and_o cd_o be_v a_o second_o medial_a residual_a line_n wherefore_o the_o line_n cd_o be_v a_o medial_a residual_a line_n of_o the_o self_n same_o order_n that_o the_o line_n ab_fw-la be_v a_o line_n therefore_o commensurable_a to_o a_o medial_a residual_a line_n be_v itself_o also_o a_o medial_a residual_a line_n of_o the_o self_n same_o order_n which_o be_v require_v to_o be_v demonstrate_v this_o theorem_a be_v understand_v general_o that_o whether_o a_o line_n be_v commensurable_a in_o length_n &_o in_o power_n or_o in_o power_n only_o to_o a_o medial_a residual_a line_n it_o be_v itself_o also_o a_o medial_a residual_a line_n and_o of_o the_o self_n same_o order_n which_o thing_n also_o be_v to_o be_v understand_v of_o the_o three_o theorem_n which_o follow_v an_o other_o demonstration_n after_o campane_n suppose_v that_o a_o be_v a_o medial_a residual_a line_n unto_o who_o let_v the_o line_n b_o be_v commensurable_a in_o length_n or_o in_o power_n only_o and_o take_v a_o rational_a line_n cd_o unto_o which_o apply_v the_o parallelogram_n ce_fw-fr equal_a to_o the_o square_n of_o the_o line_n a_o construction_n and_o unto_o the_o line_n fe_o which_o be_v equal_a to_o the_o line_n cd_o apply_v the_o parallelogram_n fâ—_n equal_a to_o the_o square_n of_o the_o line_n b._n now_o then_o the_o parallelogram_n ce_fw-fr and_o fg_o shall_v be_v commensurable_a demonstration_n for_o that_o the_o line_n a_o b_o be_v commensurable_a in_o power_n wherefore_o by_o the_o 1._o of_o the_o six_o and_o 10._o of_o this_o book_n thâ—_z line_n de_fw-fr and_o fg_o be_v commensurable_a in_o length_n now_o than_o if_o a_o be_v a_o first_o medial_a residual_a line_n then_o be_v the_o line_n de_fw-fr a_fw-fr second_v residual_a line_n by_o the_o 98._o of_o this_o book_n and_o if_o the_o line_n a_o be_v a_o sâ—cond_o medial_a residual_a line_n then_o be_v the_o line_n â—â—_o â—_z a_o three_o residual_a line_n by_o the_o 99_o of_o this_o book_n but_o if_o de_fw-fr be_fw-mi a_o second_o residual_a line_n gâ—_n also_o shall_v be_v a_o second_o residual_a line_n by_o the_o â—03_n of_o this_o book_n and_o if_o de_fw-fr be_fw-mi a_o three_o residual_a line_n ge_z also_o shall_v by_o the_o same_o be_v also_o a_o three_o residual_a line_n wherefore_o it_o follow_v by_o the_o 9â—_n and_o 93._o of_o this_o book_n that_o b_o be_v either_o a_o first_o
line_n add_v together_o shall_v be_v quintuple_a to_o the_o square_n make_v of_o the_o half_a of_o the_o whole_a line_n svppose_v that_o the_o right_a line_n ab_fw-la be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n in_o the_o point_n c._n and_o let_v the_o great_a segment_n thereof_o be_v ac_fw-la and_o unto_o ac_fw-la add_v direct_o a_o right_a line_n ad_fw-la and_o let_v ad_fw-la be_v equal_a to_o the_o half_a of_o the_o line_n ab_fw-la construction_n then_o i_o say_v that_o the_o square_a of_o the_o line_n cd_o be_v quintuple_a to_o the_o square_n of_o the_o line_n da._n describe_v by_o the_o 46._o of_o the_o first_o upon_o the_o line_n ab_fw-la and_o dc_o square_n namely_o ae_n &_o df._n and_o in_o the_o square_a df_o describe_v and_o make_v complete_a the_o figure_n and_o extend_v the_o line_n fc_o to_o the_o point_n g._n demonstration_n and_o forasmuch_o as_o the_o line_n ab_fw-la be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n in_o the_o point_n c_o therefore_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o be_v equal_a to_o the_o square_n of_o the_o line_n ac_fw-la but_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o be_v the_o parallelogram_n ce_fw-fr and_o the_o square_a of_o the_o line_n ac_fw-la be_v the_o square_a hf._n wherefore_o the_o parallelogram_n ce_fw-fr be_v equal_a to_o the_o square_a hf._n and_o forasmuch_o as_o the_o line_n basilius_n be_v double_a to_o the_o line_n ad_fw-la by_o construction_n 〈◊〉_d the_o line_n basilius_n be_v equal_a to_o the_o line_n ka_o and_o the_o line_n ad_fw-la to_o the_o line_n ah_o therefore_o also_o the_o line_n ka_o be_v double_a to_o the_o line_n ah_o but_o as_o the_o line_n ka_o be_v to_o the_o line_n ah_o so_o be_v the_o parallelogram_n ck_o to_o the_o parallelogram_n change_z wherefore_o the_o parallelogram_n ck_o be_v double_a to_o the_o parallelogram_n ch._n and_o the_o parallelogram_n lh_o and_o ch_n be_v double_a to_o the_o parallelogram_n change_z for_o supplement_n of_o parallelogram_n be_v bâ—_z the_o 4â—_o of_o the_o first_o equal_v the_o one_o to_o the_o other_o wherefore_o the_o parallelogram_n ck_o be_v equal_a to_o the_o parallelogram_n lh_o &_o ch._n and_o it_o be_v prove_v that_o the_o parallelogram_n ce_fw-fr be_v equal_a to_o the_o square_a fh_o wherefore_o the_o whole_a square_a ae_n be_v equal_a to_o the_o gnomon_n mxn_n and_o forasmuch_o as_o the_o line_n basilius_n iâ—_z double_a to_o the_o line_n ad_fw-la therefore_o the_o square_a of_o the_o line_n basilius_n be_v by_o the_o 20._o of_o the_o six_o quadruple_a to_o the_o square_n of_o the_o line_n dam_fw-ge that_o be_v the_o square_a ae_n to_o the_o square_a dh_o but_o the_o square_a ae_n be_v equal_a to_o the_o gnomon_n mxn_n wherefore_o the_o gnomon_n mxn_n be_v also_o quadruple_a to_o the_o square_a dh_o wherefore_o the_o whole_a square_a df_o be_v quintuple_a to_o the_o square_a dh_o but_o the_o square_a df_o iâ—_z the_o square_a of_o the_o line_n cd_o and_o the_o square_a dh_o be_v the_o square_a of_o the_o line_n da._n wherefore_o the_o square_a of_o the_o line_n cd_o be_v quintuple_a to_o the_o square_n of_o the_o line_n da._n if_o therefore_o a_o right_a line_n be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n and_o to_o the_o great_a segment_n be_v add_v the_o half_a of_o the_o whole_a line_n the_o square_n make_v of_o those_o two_o line_n add_v together_o shall_v be_v quintuple_a to_o the_o square_n make_v of_o the_o half_a of_o the_o whole_a line_n which_o be_v require_v to_o be_v demonstrate_v this_o proposition_n be_v a_o other_o way_n demonstrate_v after_o the_o five_o proposition_n of_o this_o book_n the_o 2._o theorem_a the_o â—_o proposition_n if_o a_o right_a line_n be_v in_o power_n quintuple_a to_o a_o segment_n of_o the_o same_o line_n the_o double_a of_o the_o say_a segment_n be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n and_o the_o great_a segment_n thereof_o be_v the_o other_o part_n of_o the_o line_n give_v at_o the_o beginning_n prove_v now_o that_o the_o double_a of_o the_o line_n ad_fw-la that_o be_v ab_fw-la be_v great_a than_o the_o line_n ac_fw-la may_v thus_o be_v prove_v for_o if_o not_o then_o if_o if_o it_o be_v possible_a let_v the_o line_n ac_fw-la be_v double_a to_o the_o line_n ad_fw-la wherefore_o the_o square_a of_o the_o line_n ac_fw-la be_v quadruple_a to_o the_o square_n of_o the_o line_n ad._n wherefore_o the_o square_n of_o the_o line_n ac_fw-la and_o ad_fw-la be_v quintuple_a to_o the_o square_n of_o the_o line_n ad._n and_o it_o be_v suppose_v that_o the_o square_a of_o the_o line_n dc_o be_v quintuple_a to_o the_o square_n of_o the_o line_n ad_fw-la wherefore_o the_o square_a of_o the_o line_n dc_o be_v equal_a to_o the_o square_n of_o the_o line_n ac_fw-la and_o ad_fw-la which_o be_v impossible_a by_o the_o 4._o of_o the_o second_o wherefore_o the_o line_n ac_fw-la be_v not_o double_a to_o the_o line_n ad._n in_o like_a sort_n also_o may_v we_o prove_v that_o the_o double_a of_o the_o line_n ad_fw-la be_v not_o less_o than_o the_o line_n ac_fw-la for_o this_o be_v much_o more_o absurd_a wherefore_o the_o double_a of_o the_o line_n ad_fw-la be_v great_a they_o the_o line_n acâ—_n which_o be_v require_v to_o be_v prove_v this_o proposition_n also_o be_v a_o other_o way_n demonstrate_v after_o the_o five_o proposition_n of_o this_o book_n two_o theorem_n in_o euclides_n method_n necessary_a add_v by_o m._n dee_n a_o theorem_a 1._o a_o right_a line_n can_v be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n but_o in_o one_o only_a point_n suppose_v a_o line_n divide_v by_o extreme_a and_o mean_a proportion_n to_o be_v ab_fw-la and_o let_v the_o great_a segment_n be_v ac_fw-la i_o say_v that_o ab_fw-la can_v not_o be_v divide_v by_o the_o say_a proportion_n in_o any_o other_o point_n then_o in_o the_o point_n c._n if_o a_o adversary_n will_v contend_v that_o it_o may_v in_o like_a sort_n be_v divide_v in_o a_o other_o point_n let_v his_o other_o point_n be_v suppose_v to_o be_v d_o make_v ad_fw-la the_o great_a segment_n of_o his_o imagine_a division_n which_o ad_fw-la also_o let_v be_v less_o than_o our_o ac_fw-la for_o the_o first_o discourse_n now_o forasmuch_o as_o by_o our_o adversary_n opinion_n ad_fw-la be_v the_o great_a segment_n of_o his_o divide_a lineâ—_z the_o parallelogram_n contain_v under_o ab_fw-la and_o db_o be_v equal_a to_o the_o square_n of_o ad_fw-la by_o the_o three_o definition_n and_o 17._o proposition_n of_o the_o six_o book_n and_o by_o the_o same_o definition_n and_o proposition_n the_o parallelogram_n under_o ab_fw-la and_o cb_o contain_v be_v equal_a to_o the_o square_n of_o our_o great_a segment_n ac_fw-la wherefore_o as_o the_o parallelogram_n under_o ab_fw-la and_o dâ—_n be_v to_o the_o square_n of_o ad_fw-la so_o iâ—_z 〈◊〉_d parallelogram_n under_o ab_fw-la and_o cb_o to_o the_o square_n of_o ac_fw-la for_o proportion_n of_o equality_n be_v conclude_v in_o they_o both_o but_o forasmuch_o as_o dâ—â—_n iâ—_z byâ—_n â—c_z supposition_n great_a them_z cb_o the_o parallelogram_n under_o ab_fw-la and_o db_o be_v great_a than_o the_o parallelogram_n under_o ac_fw-la and_o cb_o by_o the_o first_o of_o the_o six_o for_o ab_fw-la be_v their_o equal_a heith_n wherefore_o the_o square_a of_o ad_fw-la shall_v be_v great_a than_o the_o square_a of_o ac_fw-la by_o the_o 14._o of_o the_o five_o but_o the_o line_n ad_fw-la be_v less_o than_o the_o line_n ac_fw-la by_o supposition_n wherefore_o the_o square_a of_o ad_fw-la be_v less_o than_o the_o square_a of_o ac_fw-la and_o it_o be_v conclude_v also_o to_o be_v great_a than_o the_o square_a of_o ac_fw-la wherefore_o the_o square_a of_o ad_fw-la be_v both_o great_a than_o the_o square_a of_o acâ—_n and_o also_o less_o which_o be_v a_o thing_n impossible_a the_o square_a therefore_o of_o ad_fw-la be_v not_o equal_a to_o the_o parallelogram_n under_o ab_fw-la and_o db._n and_o therefore_o by_o the_o three_o definition_n of_o the_o six_o ab_fw-la be_v not_o divide_v by_o a_o extreme_a and_o mean_a proportion_n in_o the_o point_n d_o as_o our_o adversary_n imagine_v and_o second_o in_o like_a sort_n will_v the_o inconueniency_n fall_v out_o if_o we_o assign_v ad_fw-la our_o adversary_n great_a segment_n to_o be_v great_a than_o our_o ac_fw-la therefore_o see_v neither_o on_o the_o one_o side_n of_o our_o point_n c_o neither_o on_o the_o other_o side_n of_o the_o same_o point_n c_o any_o point_n can_v be_v have_v at_o which_o the_o line_n ab_fw-la can_v be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n it_o follow_v of_o necessity_n that_o ab_fw-la can_v be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n in_o the_o point_n c_o only_a therefore_o a_o right_a line_n can_v be_v divide_v by_o a_o extreme_a
and_o mean_a proportion_n but_o in_o one_o only_a point_n which_o be_v requisite_a to_o be_v demonstrate_v a_o theorem_a 2._o what_o right_a line_n so_o ever_o be_v divide_v into_o two_o part_n have_v those_o his_o two_o part_n proportional_a to_o the_o two_o segment_n of_o a_o line_n divide_v by_o extreme_a and_o mean_a proportion_n be_v also_o itself_o divide_v by_o a_o extreme_a and_o mean_a proportion_n and_o those_o his_o two_o part_n be_v his_o two_o segment_n of_o the_o say_a proportion_n suppose_v ab_fw-la to_o be_v a_o line_n divide_v by_o a_o extreme_a and_o mean_a proportion_n in_o the_o point_n c_o and_o ac_fw-la to_o be_v the_o great_a segment_n suppose_v also_o the_o right_a line_n de_fw-fr to_o be_v divide_v into_o two_o part_n in_o the_o point_n f_o and_o that_o the_o part_n df_o be_v to_o fe_o as_o the_o segment_n ac_fw-la be_v to_o cb_o or_o df_o to_o be_v to_o ac_fw-la as_o fe_o be_v to_o cb._n for_o so_o these_o payte_n be_v proportional_a to_o the_o say_a segment_n i_o say_v now_o that_o de_fw-fr be_v also_o divide_v by_o a_o extreme_a and_o mean_a proportion_n in_o the_o point_n f._n and_o that_o df_o fe_o be_v his_o segment_n of_o the_o say_a proportion_n for_o see_v as_o ac_fw-la be_v to_o cb_o so_o be_v df_o to_o fe_o by_o supposition_n therefore_o as_o ac_fw-la and_o cb_o which_o be_v ab_fw-la be_v to_o cb_o so_o be_v df_o and_o fe_o which_o be_v de_fw-fr to_o fe_o by_o the_o 18._o of_o the_o five_o wherefore_o alternate_o as_o ab_fw-la be_v to_o de_fw-fr so_o be_v cb_o to_o fe_o and_o therefore_o the_o residue_n ac_fw-la be_v to_o the_o residue_n df_o as_o ab_fw-la be_v to_o de_fw-fr by_o the_o five_o of_o the_o five_o and_o then_o alternate_o ac_fw-la be_v to_o ab_fw-la as_o de_fw-fr be_v to_o df._n now_o therefore_o backward_o ab_fw-la be_v to_o ac_fw-la as_o de_fw-fr be_v to_o df._n but_o as_o ab_fw-la be_v to_o ac_fw-la so_o be_v ac_fw-la to_o cb_o by_o the_o three_o definition_n of_o the_o six_o book_n wherefore_o de_fw-fr be_v to_o df_o as_o ac_fw-la be_v to_o cb_o by_o the_o 11._o of_o the_o five_o and_o by_o supposition_n as_o ac_fw-la be_v to_o cb_o so_o be_v df_o to_o fe_o wherefore_o by_o the_o 11._o of_o the_o five_o as_z de_fw-fr be_v to_o df_o so_o be_v df_o to_o fe_o wherefore_o by_o the_o 3._o definition_n of_o the_o six_o de_fw-fr be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n in_o the_o point_n f._n wherefore_o df_o and_o fe_o be_v the_o segment_n of_o the_o say_a proportion_n therefore_o what_o right_a line_n so_o ever_o be_v divide_v into_o two_o part_n have_v those_o his_o two_o part_n proportional_a to_o the_o two_o segment_n of_o a_o line_n divide_v by_o extreme_a and_o mean_a proportioned_a be_v also_o itself_o divide_v by_o a_o extreme_a and_o mean_a proportion_n and_o those_o his_o two_o part_n be_v his_o two_o segment_n of_o the_o say_v proportion_v which_o be_v requisite_a to_o be_v demonstrate_v note_n many_o way_n these_o two_o theorem_n may_v be_v demonstrate_v which_o i_o leave_v to_o the_o exercise_n of_o young_a student_n but_o utter_o to_o want_v these_o two_o theorem_n and_o their_o demonstration_n in_o so_o principal_a a_o line_n or_o rather_o the_o chief_a pillar_n of_o euclides_n geometrical_a palace_n geometry_n be_v hitherto_o and_o so_o will_v remain_v a_o great_a disgrace_n also_o i_o think_v it_o good_a to_o note_v unto_o you_o what_o we_o mean_v by_o one_o only_a point_n we_o mâ—â—â—â—_n that_o the_o quantity_n of_o the_o two_o segment_n can_v not_o be_v alter_v the_o whole_a line_n be_v once_o give_v and_o though_o from_o either_o end_n of_o the_o whole_a line_n the_o great_a segment_n may_v begin_v poiâ—t_n and_o so_o as_o it_o be_v the_o point_n of_o section_n may_v seem_v to_o be_v alter_v yet_o with_o we_o that_o be_v no_o alteration_n forasmuch_o as_o the_o quantity_n of_o the_o segment_n remain_v all_o one_o i_o mean_v the_o quantity_n of_o the_o great_a segment_n be_v all_o one_o at_o which_o end_n so_o ever_o it_o be_v take_v and_o therefore_o likewise_o the_o quantity_n of_o the_o less_o segment_n be_v all_o one_o etc_n etc_n the_o like_a consideration_n may_v be_v have_v in_o euclides_n ten_o book_n in_o the_o binomial_a line_n etc_n etc_n joâ—n_n dee_n 1569._o decemb._n 18._o the_o 3._o theorem_a the_o 3._o proposition_n if_o a_o right_a line_n be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n and_o to_o the_o less_o segment_n be_v add_v the_o half_a of_o the_o gerater_n segment_n the_o square_n make_v of_o those_o two_o line_n add_v together_o be_v quintuple_a to_o the_o square_n make_v of_o the_o half_a line_n of_o the_o great_a segment_n svppose_v that_o the_o right_a line_n ab_fw-la be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n in_o the_o point_n c._n and_o let_v the_o great_a segment_n thereof_o be_v ac_fw-la and_o divide_v ac_fw-la into_o two_o equal_a part_n in_o the_o point_n d._n then_o i_o say_v that_o the_o square_a of_o the_o line_n bd_o it_o quintuple_a to_o the_o square_n of_o the_o line_n dc_o construction_n describe_v by_o the_o 46._o of_o the_o first_o upon_o the_o line_n ab_fw-la a_o square_a ae_n and_o describe_v and_o make_v perfect_a the_o figure_n that_o be_v divide_v the_o line_n at_o like_a unto_o the_o division_n of_o the_o line_n ab_fw-la by_o the_o 10._o of_o the_o six_o in_o the_o point_n r_o h_o by_o which_o point_n draw_v by_o the_o 31._o of_o the_o first_o unto_o the_o line_n ab_fw-la parallel_a line_n rm_n and_o hn._n so_o likewise_o draw_v by_o the_o point_n d_o c_o unto_o the_o line_n be_v these_o parallel_a line_n dl_o and_o c_n &_o draw_v the_o diameter_n bt_n demonstration_n and_o forasmuch_o as_o the_o line_n ac_fw-la be_v double_a to_o the_o line_n dc_o therefore_o the_o square_a of_o ac_fw-la be_v quadruple_a to_o the_o square_n of_o dc_o by_o the_o 20._o of_o the_o six_o that_o be_v the_o square_a r_n to_o the_o square_a fg_o and_o forasmuch_o as_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o be_v equal_a to_o the_o square_n of_o the_o line_n ac_fw-la and_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o be_v equal_a to_o the_o parallelogram_n ce_fw-fr &_o the_o square_a of_o the_o line_n ac_fw-la be_v the_o square_a r_n wherefore_o the_o parallelogram_n ce_fw-fr be_v equal_a to_o the_o square_a rs._n but_o the_o square_a r_n be_v quadruple_a to_o the_o square_a fg_o wherefore_o the_o parallelogram_n ce_fw-fr also_o be_v quadruple_a to_o the_o square_a fg._n again_o forasmuch_o as_o the_o line_n ad_fw-la be_v equal_a to_o the_o line_n dc_o therefore_o the_o line_n hk_o be_v equal_a the_o line_n kf_o wherefore_o also_o the_o square_a gf_o be_v equal_a to_o the_o square_a hl_o wherefore_o the_o line_n gk_o be_v equal_a to_o the_o line_n kl_o that_o be_v the_o line_n mn_v to_o the_o line_n ne_fw-fr wherefore_o the_o parallelogram_n mf_n be_v equal_a to_o the_o parallelogram_n fe_o but_o the_o parallelogram_n mf_n be_v equal_a to_o the_o parallelogram_n cg_o wherefore_o the_o parallelogram_n cg_o be_v also_o equal_a to_o the_o parallelogram_n fe_o put_v the_o parallelogram_n cn_fw-la common_a to_o they_o both_o after_o wherefore_o the_o gnomon_n xop_n be_v equal_a to_o the_o parallelogram_n ck_o but_o the_o parallelogram_n ce_fw-fr be_v prove_v to_o be_v quadruple_a to_o gâ—_n the_o square_a wherefore_o the_o gnomon_n xop_n be_v quadruple_a to_o the_o square_a gf_o wherefore_o the_o square_a dn_o be_v quintuple_a to_o the_o square_a fg_o and_o dn_o be_v the_o square_a of_o the_o line_n db_o and_o gf_o the_o square_a of_o the_o line_n dc_o wherefore_o the_o square_a of_o the_o line_n db_o be_v quintuple_a to_o the_o square_n of_o the_o line_n dc_o if_o therefore_o a_o right_a line_n be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n and_o to_o the_o less_o segment_n be_v add_v the_o half_a of_o the_o great_a segment_n the_o square_n make_v of_o those_o two_o line_n add_v together_o be_v quintuple_a to_o the_o square_n make_v of_o the_o half_a line_n of_o the_o great_a segment_n which_o be_v require_v to_o be_v demonstrate_v you_o shall_v find_v this_o proposition_n a_o other_o way_n demonstrate_v after_o the_o five_o proposition_n of_o this_o book_n here_o follow_v m._n dee_n his_o addition_n ¶_o a_o theorem_a 1._o if_o a_o right_a line_n give_v be_v quintuple_a in_o power_n to_o the_o power_n of_o a_o segment_n of_o himself_o the_o double_a of_o that_o segment_n former_a and_o the_o other_o part_n remain_v of_o the_o first_o give_v line_n make_v a_o line_n divide_v by_o extreme_a and_o mean_a proportion_n and_o that_o double_a of_o the_o segment_n be_v the_o great_a part_n thereof_o forasmuch_o as_o this_o be_v the_o converse_n of_o euclides_n
line_n thereby_o adjoin_v let_v be_v bq_n i_o say_v that_o cq_n be_v a_o line_n also_o divide_v by_o a_o extreme_a and_o mean_a proportion_n in_o the_o point_n b_o and_o that_o bq_n the_o line_n adjoin_v be_v the_o less_o segment_n for_o by_o the_o third_o it_o be_v prove_v that_o half_a ac_fw-la which_o let_v be_v cd_o with_o cb_o as_o one_o line_n compose_v have_v his_o power_n or_o square_v quintuple_a to_o the_o power_n of_o the_o segment_n cd_o wherefore_o by_o the_o second_o of_o this_o book_n the_o double_a of_o c_o d_o be_v divide_v by_o extreme_a and_o middle_a proportioned_a and_o the_o great_a segment_n thereof_o shall_v be_v cb._n but_o by_o construction_n cq_n be_v the_o double_a of_o cd_o for_o it_o be_v equal_a to_o ac_fw-la wherefore_o cq_n be_v divide_v by_o extreme_a and_o middle_a proportion_n in_o the_o point_n b_o and_o the_o great_a segment_n thereof_o shall_v be_v cb._n wherefore_o bq_n be_v the_o less_o segment_n which_o be_v the_o line_n adjoin_v therefore_o a_o line_n be_v divide_v by_o extreme_a and_o middle_a proportion_n if_o the_o less_o segment_n be_v produce_v equal_o to_o the_o length_n of_o the_o great_a segment_n the_o line_n thereby_o adjoin_v together_o with_o the_o say_v less_o segment_n make_v a_o new_a line_n divide_v by_o extreme_a &_o mean_a proportion_n whoâ—e_v less_o segment_n be_v the_o line_n adjoin_v which_o be_v to_o be_v demonstrate_v ¶_o a_o corollary_n 2._o ifâ—_n from_o the_o great_a segment_n of_o a_o line_n divide_v by_o extreme_a and_o middle_a proportion_n a_o line_n equal_a to_o the_o less_o segment_n be_v cut_v of_o the_o great_a segment_n thereby_o be_v also_o divide_v by_o extreme_a and_o mean_a proportion_n who_o great_a segmentâ—_n shall_v be_v 〈◊〉_d that_o part_n of_o it_o which_o be_v cut_v of_o for_o take_v from_o ac_fw-la a_o line_n equal_a to_o cb_o let_v be_v remain_v i_o say_v that_o ac_fw-la be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n in_o the_o point_n r_o and_o that_o cr_n the_o line_n cut_v of_o be_v the_o great_a segment_n for_o it_o be_v prove_v in_o the_o former_a corollary_n that_o cq_n be_v divide_v by_o extreme_a and_o mean_a proportion_n in_o the_o point_n b._n but_o ac_fw-la be_v equal_a to_o cq_n by_o construction_n and_o cr_n be_v equal_a to_o cb_o by_o construction_n wherefore_o the_o residue_n ar_n be_v equal_a to_o bq_n the_o residue_n see_v therefore_o the_o whole_a ac_fw-la be_v equal_a to_o the_o whole_a cq_n and_o the_o great_a part_n of_o ac_fw-la which_o be_v cr_n be_v equal_a to_o cb_o the_o great_a part_n of_o cq_n and_o the_o less_o segment_n also_o equal_a to_o the_o less_o and_o withal_o see_v cq_n be_v prove_v to_o be_v divide_v by_o extreme_a &_o mean_a proportion_n in_o the_o point_n b_o it_o follow_v of_o necessity_n that_o ac_fw-la be_v divide_v by_o extreme_a and_o mean_a proportion_n in_o the_o point_n r._n and_o see_v cb_o be_v the_o great_a segment_n of_o cq_n cr_n shall_v be_v the_o great_a segment_n of_o ac_fw-la which_o be_v to_o be_v demonstrate_v a_o corollary_n 3._o it_o be_v evident_a thereby_o a_o line_n be_v divide_v by_o extreme_a and_o mean_a proportion_n that_o the_o line_n whereby_o the_o great_a segment_n exceed_v the_o less_o together_o with_o the_o less_o segment_n do_v make_v a_o line_n divide_v by_o extreme_a and_o mean_a proportion_n who_o less_o segment_n be_v the_o say_a line_n of_o exceesse_n or_o difference_n between_o the_o segment_n john_n dee_n ¶_o two_o new_a way_n to_o divide_v any_o right_a line_n give_v by_o a_o extreme_a and_o mean_a proportion_n demonstrate_v and_o add_v by_o m._n dee_n a_o problem_n to_o divide_v by_o a_o extreme_a and_o mean_a proportion_n any_o right_a line_n give_v in_o length_n and_o position_n suppose_v a_o line_n give_v in_o length_n and_o position_n to_o be_v ab_fw-la i_o say_v that_o ab_fw-la be_v to_o be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n construction_n divide_v ab_fw-la into_o two_o equal_a part_n as_o in_o the_o point_n c._n produce_v ab_fw-la direct_o from_o the_o point_n b_o to_o the_o point_n d_o make_v bd_o equal_a to_o bc._n to_o the_o line_n ad_fw-la and_o at_o the_o point_n d_o let_v a_o line_n be_v draw_v etc_n perpendicular_a by_o the_o 11._o of_o the_o first_o which_o let_v be_v df_o of_o what_o length_n you_o will_v from_o df_o and_o at_o the_o point_n d_o cut_v of_o the_o six_o part_n of_o df_o by_o the_o 9_o of_o the_o six_o and_o let_v that_o six_o part_n be_v the_o line_n dg_o upon_o df_o as_o a_o diameter_n describe_v a_o semicircle_n which_o let_v be_v dhf_n from_o the_o point_n g_o rear_v a_o line_n perpendicular_a to_o df_o which_o suppose_v to_o be_v gh_o and_o let_v it_o come_v to_o the_o circumference_n of_o dhf_n in_o the_o point_n h._n draw_v right_a line_n hd_v and_o hf._n produce_v dh_o from_o the_o point_n h_o so_o long_o till_o a_o line_n adjoin_v with_o dh_o be_v equal_a to_o hf_o which_o let_v be_v di_fw-it equal_a to_o hf._n from_o the_o point_n h_o to_o the_o point_n b_o the_o one_o end_n of_o our_o line_n give_v let_v a_o right_a line_n be_v draw_v as_o hb_o from_o the_o point_n i_o let_v a_o line_n be_v draw_v to_o the_o line_n ab_fw-la so_o that_o it_o be_v also_o parallel_a to_o the_o line_n hb_o which_o parallel_n line_n suppose_v to_o be_v ik_fw-mi cut_v the_o line_n ab_fw-la at_o the_o point_n k._n i_o say_v that_o ab_fw-la be_v divide_v by_o a_o extreme_a &_o mean_a proportion_n in_o the_o point_n k._n for_o the_o triangle_n dki_n have_v hb_o parallel_n to_o ik_fw-mi have_v his_o side_n dk_o and_o diego_n demonstration_n cut_v proportional_o by_o the_o 2._o of_o the_o six_o wherefore_o as_o ih_v be_v to_o hd_v so_o be_v kb_o to_o bd._n and_o therefore_o compound_o by_o the_o 18._o of_o the_o five_o as_o diego_n be_v to_o dh_o so_o be_v dk_o to_o db._n but_o by_o construction_n diego_n be_v equal_a to_o hf_o wherefore_o by_o the_o 7._o of_o the_o five_o di_z be_v to_o dh_o as_o hf_o be_v to_o dh_o wherefore_o by_o the_o 11._o of_o the_o five_o dk_o be_v to_o db_o as_o hf_o be_v to_o dh_o wherefore_o the_o square_a of_o dk_o be_v to_o the_o square_n of_o db_o as_o the_o square_n of_o hf_o be_v to_o the_o square_n of_o dh_o by_o the_o 22._o of_o the_o six_o but_o the_o square_n of_o hf_o be_v to_o the_o square_n of_o dh_o as_o the_o line_n gf_o be_v to_o the_o line_n gdâ—_n by_o my_o corollary_n upon_o the_o 5_o problem_n of_o my_o addition_n to_o the_o second_o proposition_n of_o the_o twelve_o wherefore_o by_o the_o 11._o of_o the_o five_o the_o square_a of_o dk_o be_v to_o the_o square_n of_o db_o as_o the_o line_n gf_o be_v to_o the_o line_n gd_o but_o by_o construction_n gf_o be_v quintuple_a to_o gd_o wherefore_o the_o square_a of_o dk_o be_v quintuple_a to_o the_o square_n of_o db_o and_o therefore_o the_o double_a of_o db_o be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n and_o bk_o be_v the_o great_a segment_n thereof_o by_o the_o 2._o of_o this_o thirteen_o wherefore_o see_v ab_fw-la be_v the_o double_a of_o db_o by_o construction_n the_o line_n ab_fw-la be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n and_o his_o great_a segment_n be_v the_o line_n bk_o wherefore_o ab_fw-la be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n in_o the_o point_n k._n we_o have_v therefore_o divide_v by_o extreme_a and_o mean_a proportion_n any_o line_n give_v in_o length_n and_o position_n which_o be_v requisite_a to_o be_v do_v the_o second_o way_n to_o execute_v this_o problem_n suppose_v the_o line_n give_v to_o be_v ab_fw-la divide_v aâ—_z into_o two_o equal_a part_n as_o suppose_v it_o to_o be_v do_v in_o the_o point_n c._n produce_v ab_fw-la from_o the_o point_n b_o adjoin_v a_o line_n equal_a to_o bc_o which_o let_v be_v bd._n to_o the_o right_a line_n ad_fw-la and_o at_o the_o point_n d_o erect_v a_o perpendicular_a line_n equal_a to_o bd_o let_v that_o be_v de._n produce_v ed_z from_o the_o point_n d_o to_o the_o point_n f_o make_v df_o to_o contain_v five_o such_o equal_a part_n as_o de_fw-fr be_v one_o now_o upon_o of_o as_o a_o diameter_n describe_v a_o semicircle_n which_o let_v 〈◊〉_d ekf_n and_o let_v the_o point_n where_o the_o circumference_n of_o ekf_n do_v cut_v the_o line_n ab_fw-la be_v the_o point_n k._n i_o say_v that_o ab_fw-la be_v divide_v in_o the_o point_n king_n by_o a_o extreme_a and_o mean_a proportion_n for_o by_o the_o 13._o of_o the_o six_o ed_z dk_o &_o df_o be_v three_o line_n in_o continual_a proportion_n dk_o be_v the_o middle_n proportional_a â—_o wherefore_o by_o the_o corollary_n of_o the_o 20._o of_o the_o six_o as_z ed_z be_v to_o df_o so_o be_v
circumference_n km_o and_o forasmuch_o as_o the_o circumference_n ab_fw-la be_v double_a to_o the_o circumference_n bk_o but_o the_o circumference_n cd_o be_v equal_a to_o the_o circumference_n ab_fw-la wherefore_o the_o circumference_n cd_o be_v double_a to_o the_o circumference_n bk_o but_o the_o circumference_n cd_o be_v double_a to_o the_o circumference_n cg_o wherefore_o the_o circumference_n cg_o be_v equal_a to_o the_o circumference_n bk_o but_o the_o circumference_n bk_o be_v double_a to_o the_o circumference_n km_o for_o that_o the_o circumference_n ka_o be_v double_a thereunto_o â—_o wherefore_o also_o the_o circumference_n cg_o be_v double_a to_o the_o circumference_n km_o but_o the_o circumference_n cb_o be_v also_o double_a to_o the_o circumference_n bk_o for_o the_o circumference_n cb_o be_v equal_a to_o the_o circumference_n basilius_n wherefore_o the_o whole_a circumference_n gb_o be_v double_a to_o the_o whole_a circumference_n bm_n by_o the_o 12._o of_o the_o five_o wherefore_o also_o the_o angle_n gfb_n be_v double_a to_o the_o angle_n bfm_n by_o the_o last_o of_o the_o six_o but_o the_o angle_n gfb_n be_v double_a to_o the_o angle_n fab._n by_o the_o 32._o of_o the_o first_o or_o 20._o of_o the_o three_o for_o the_o angle_n fab._n be_v equal_a to_o the_o angle_n abf_n wherefore_o the_o angle_n bfl_fw-mi be_v equal_a â—o_o the_o angle_n fab._n by_o the_o 15._o of_o the_o five_o and_o the_o angle_n abf_n be_v common_a to_o the_o two_o triangle_n abf_a and_o bfl_fw-mi wherefore_o by_o the_o corollary_n of_o the_o â—â—_o of_o the_o first_o the_o angle_n remain_v afb_o be_v equal_a to_o the_o angle_n remain_v blf_n wherefore_o the_o triangle_n abf_n be_v equiangle_n to_o the_o triangle_n bfl_fw-mi wherefore_o by_o the_o 4._o of_o the_o six_o proportional_o as_o the_o right_a line_n ab_fw-la be_v to_o the_o right_a line_n bf_o so_o be_v the_o same_o right_a line_n fb_o to_o the_o right_a line_n bl_o wherefore_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bl_o be_v equal_a to_o the_o square_n of_o the_o line_n bf_o by_o the_o 17._o of_o the_o six_o again_o forasmuch_o as_o the_o line_n all_o be_v equal_a to_o the_o line_n lk_v for_o by_o the_o last_o of_o the_o six_o the_o angle_n kfl_fw-mi be_v equal_a to_o the_o angle_n afl_fw-mi which_o equal_a angle_n be_v contain_v under_o the_o line_n fk_o fl_fw-mi and_o favorina_n fl_fw-mi &_o the_o line_n fk_o be_v equal_a to_o the_o line_n favorina_n and_o the_o line_n fl_fw-mi be_v common_a to_o they_o both_o wherefore_o by_o the_o 4._o of_o the_o first_o the_o line_n all_o be_v equal_a to_o the_o line_n lk_n and_o the_o line_n ln_o be_v common_a to_o they_o both_o &_o make_v right_a angle_n at_o the_o point_n n_o and_o by_o the_o 3._o of_o the_o three_o the_o base_a a_o be_v equal_a to_o the_o base_a kn._n wherefore_o also_o the_o angle_n lkn_n be_v equal_a to_o the_o angle_n lan._n but_o the_o angle_n lanthorn_n be_v equal_a to_o the_o angle_n kbl_n by_o the_o 5._o of_o the_o first_o wherefore_o the_o angle_n lkn_n be_v equal_a to_o the_o angle_n kbl_n and_o the_o angle_n kal_n be_v common_a to_o both_o the_o triangle_n akb_n and_o akl_n wherefore_o the_o angle_n remain_v akb_n be_v equal_a to_o the_o angle_n remain_v alk_n by_o the_o corollary_n of_o the_o 32._o of_o the_o first_o wherefore_o the_o triangle_n kba_n be_v equiangle_n to_o the_o triangle_n kla._n wherefore_o by_o the_o 4._o of_o the_o six_o proportional_o as_o the_o right_a line_n basilius_n be_v to_o the_o right_a line_n ak_o so_o be_v the_o same_o right_a line_n ka_o to_o the_o right_a line_n al._n wherefore_o that_o which_o be_v contain_v under_o the_o line_n basilius_n and_o all_o be_v equal_a to_o the_o square_n of_o the_o line_n ak_o by_o the_o 17._o of_o the_o six_o and_o it_o be_v prove_v that_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bl_o be_v equal_a to_o the_o square_n of_o the_o line_n bf_o wherefore_o that_o which_o be_v contain_v under_o ab_fw-la &_o bl_o together_o with_o that_o which_o be_v conâ—â—ined_v under_o basilius_n and_o all_o which_o by_o the_o 2._o of_o the_o second_o be_v the_o square_a of_o the_o line_n basilius_n be_v equal_a to_o the_o square_n of_o the_o line_n of_o and_o ak_o but_o the_o line_n basilius_n be_v the_o side_n of_o the_o pentagon_n figure_n and_o of_o the_o side_n of_o the_o hexagon_n figure_n by_o the_o corollary_n of_o the_o 15._o of_o the_o four_o and_o ak_o the_o side_n of_o the_o decagon_n figure_n wherefore_o the_o side_n of_o a_o pentagon_n figure_n contain_v in_o power_n both_o the_o side_n of_o a_o hexagon_n figure_n and_o of_o a_o decagon_n figure_n be_v describe_v all_o in_o one_o and_o the_o self_n same_o circle_n which_o be_v require_v to_o be_v demonstrate_v ¶_o a_o corollary_n add_v by_o flussas_n a_o perpendicular_a line_n from_o any_o angle_n draw_v to_o the_o base_a of_o a_o pentagon_n pass_v by_o the_o centre_n for_o if_o we_o draw_v a_o right_a line_n from_o the_o point_n a_o to_o the_o point_n c_o and_o a_o other_o from_o the_o point_n a_o to_o the_o point_n d_o those_o right_a line_n shâ—â—l_v be_v equal_a by_o the_o 4._o of_o the_o first_o and_o therefore_o in_o the_o triangle_n acd_v the_o angle_n at_o the_o point_n c_o and_o d_o be_v by_o the_o 5._o of_o the_o firstâ—_o equal_a but_o the_o angle_n make_v at_o the_o point_n where_o the_o line_n agnostus_n cut_v the_o line_n cd_o be_v by_o supposition_n right_a angle_n wherefore_o by_o the_o 26._o of_o the_o first_o the_o line_n cd_o be_v by_o the_o line_n agnostus_n divide_v into_o two_o equal_a part_n and_o it_o be_v also_o divide_v perpendicular_o wherefore_o by_o the_o corollary_n of_o the_o first_o of_o the_o three_o in_o the_o line_n agnostus_n be_v the_o centre_n of_o the_o circle_n and_o therefore_o the_o line_n agnostus_n pass_v by_o the_o centre_n ¶_o the_o 11._o theorem_a the_o 11._o proposition_n if_o in_o a_o circle_n have_v a_o rational_a line_n to_o his_o diameter_n be_v inscribe_v a_o equilater_n pentagon_n the_o side_n of_o the_o pentagon_n be_v a_o irrational_a line_n and_o be_v of_o that_o kind_n which_o be_v call_v a_o less_o line_n svppose_v that_o in_o the_o circle_n abcde_v have_v a_o rational_a line_n to_o his_o diameter_n be_v inscribe_v a_o pentagon_n figure_n abcde_o then_o i_o say_v that_o the_o side_n of_o the_o pentagon_n figure_n abcde_o namely_o the_o side_n ab_fw-la be_v a_o irrational_a line_n of_o that_o kind_n which_o be_v call_v a_o less_o line_n take_v by_o the_o 1._o of_o the_o three_o the_o centre_n of_o the_o circle_n and_o let_v the_o same_o be_v the_o point_n fletcher_n and_o draw_v a_o right_a line_n from_o the_o point_n a_o to_o the_o point_n fletcher_n construstion_n and_o a_o other_o from_o the_o point_n f_o to_o the_o point_n b_o and_o extend_v those_o line_n to_o the_o point_n g_o and_o h._n and_o draw_v a_o right_a line_n from_o the_o point_n a_o to_o the_o point_n c._n and_o from_o the_o semidiameter_n fh_o take_v the_o four_o part_n by_o the_o 9_o of_o the_o six_o and_o let_v the_o same_o be_v fk_v but_o the_o line_n fh_o be_v rational_a for_o that_o it_o be_v the_o half_a of_o the_o diameter_n which_o be_v suppose_v to_o be_v rational_a wherefore_o also_o the_o line_n fk_o be_v rational_a and_o the_o line_n or_o semidiameter_n bf_o be_v rational_a wherefore_o the_o whole_a line_n bk_o be_v rational_a demonstration_n and_o forasmuch_o as_o the_o circumference_n acg_n be_v equal_a to_o the_o circumference_n adg_n of_o which_o the_o circumference_n abc_n be_v equal_a to_o the_o circumference_n aed_n wherefore_o the_o residue_n cg_o be_v equal_a to_o the_o residue_n gd_o now_o if_o we_o draw_v a_o right_a line_n from_o the_o point_n a_o to_o the_o point_n d_o it_o be_v manifest_a that_o the_o angle_n alc_n and_o ald_n be_v right_a angle_n for_o forasmuch_o as_o the_o circumference_n cg_o be_v equal_a to_o the_o circumference_n gd_o therefore_o by_o the_o last_o of_o the_o six_o the_o angle_n cag_n be_v equal_a to_o the_o angle_n dag_n and_o the_o line_n ac_fw-la be_v equal_a to_o the_o line_n ad_fw-la for_o that_o the_o circumference_n which_o they_o subtend_v be_v equal_a and_o the_o line_n all_o be_v câ—mmon_a to_o they_o both_o therefore_o there_o be_v two_o line_n ac_fw-la and_o all_o equal_a to_o two_o line_n ad_fw-la and_o all_o and_o the_o angle_n cal_n be_v equal_a to_o the_o angle_n dalinea_n wherefore_o by_o the_o 4._o of_o the_o first_o the_o base_a cl_n be_v equal_a to_o the_o base_a ld_n and_o the_o rest_n of_o the_o angle_n to_o the_o rest_n of_o the_o angle_n and_o the_o line_n cd_o be_v double_a to_o the_o line_n cl._n and_o by_o the_o same_o reason_n may_v it_o be_v prove_v that_o the_o angle_n at_o the_o point_n m_o
same_o parallelogram_n be_v also_o irrational_a and_o be_v call_v a_o less_o line_n by_o the_o 94._o of_o the_o ten_o but_o the_o line_n ab_fw-la contain_v in_o power_n the_o parallelogram_n contain_v under_o the_o line_n hb_o and_o bm_n for_o if_o we_o draw_v a_o right_a line_n from_o the_o point_n a_o to_o the_o point_n h_o the_o triangle_n abh_n shall_v be_v like_a to_o the_o triangle_n abm_n by_o the_o 8._o of_o the_o six_o for_o from_o the_o right_a angle_n bah_o be_v draw_v to_o the_o base_a bh_o a_o perpendicular_a line_n and_o therefore_o as_o the_o line_n bh_o be_v to_o the_o line_n basilius_n so_o be_v the_o line_n ab_fw-la to_o the_o line_n bm_n this_o follow_v also_o of_o the_o corollary_n of_o the_o say_v 8._o of_o the_o six_o wherefore_o the_o line_n ab_fw-la which_o be_v the_o side_n of_o the_o pentagon_n figure_n be_v a_o irrational_a line_n of_o that_o kind_n which_o be_v call_v a_o less_o line_n if_o therefore_o in_o a_o circle_n have_v a_o rational_a line_n to_o his_o diameter_n be_v inscribe_v a_o equilater_n pentagon_n the_o side_n of_o the_o pentagon_n be_v a_o irrational_a line_n and_o be_v of_o that_o kind_n which_o be_v call_v a_o less_o line_n which_o be_v require_v to_o be_v demonstrate_v ¶_o the_o 12._o theorem_a the_o 12._o proposition_n if_o in_o a_o circle_n be_v describe_v a_o equilater_n triangle_n the_o square_n make_v of_o the_o side_n of_o the_o triangle_n be_v treble_a to_o the_o square_n make_v of_o the_o line_n which_o be_v draw_v from_o the_o centre_n of_o the_o circle_n to_o the_o circumference_n svppose_v that_o abc_n be_v a_o circle_n and_o in_o it_o describe_v a_o equilater_n triangle_n abc_n then_o i_o say_v that_o the_o square_n make_v of_o the_o side_n of_o the_o triangle_n abc_n be_v treble_a to_o the_o square_n make_v of_o the_o line_n draw_v from_o the_o centre_n of_o the_o circle_n abc_n to_o the_o circumference_n constrâ—yction_n take_v by_o the_o 1._o of_o the_o three_o the_o centre_n of_o the_o circle_n and_o let_v the_o same_o be_v d._n aâ—d_v draw_v a_o right_a line_n from_o the_o point_n a_o to_o the_o point_n d_o and_o extend_v it_o to_o the_o point_n e._n and_o draw_v a_o right_a line_n from_o the_o point_n b_o to_o the_o point_n e._n demonstration_n now_o forasmuch_o as_o the_o triangle_n abc_n be_v equilater_n therefore_o each_o of_o these_o three_o circumference_n ab_fw-la ac_fw-la &_o bec_n be_v the_o three_o part_n of_o the_o whole_a circumference_n of_o the_o circle_n abc_n wherefore_o the_o circumference_n be_v be_v the_o six_o part_n of_o the_o circumference_n of_o the_o circle_n for_o the_o circumference_n of_o the_o semicircle_n abe_n be_v equal_a to_o the_o circumference_n of_o the_o semicircle_n ace_n from_o which_o take_v away_o equal_a circumference_n ab_fw-la and_o ac_fw-la the_o circumference_n remain_v be_v shall_v be_v equal_a to_o the_o circumference_n remain_v ec_o wherefore_o the_o right_a line_n be_v be_v the_o side_n of_o a_o equilater_n hexagon_fw-mi figure_n describe_v in_o the_o circle_n wherefore_o it_o be_v equal_a to_o the_o line_n draw_v from_o the_o centre_n of_o the_o circle_n to_o the_o circumference_n that_o be_v unto_o the_o line_n de_fw-fr by_o the_o corollary_n of_o the_o 15._o of_o the_o six_o and_o forasmuch_o as_o the_o line_n ae_n be_v double_a to_o the_o line_n de_fw-fr therefore_o the_o square_a of_o the_o line_n ae_n be_v quadruple_a to_o the_o square_n of_o the_o line_n de_fw-fr by_o the_o 4._o of_o the_o second_o that_o be_v to_o the_o square_n of_o the_o line_n be._n but_o the_o square_n of_o the_o line_n ae_n be_v equal_a to_o the_o square_n of_o the_o line_n ab_fw-la and_o be_v by_o the_o 47._o of_o the_o first_o for_o the_o angle_n abe_n be_v by_o the_o 31._o of_o the_o three_o a_o right_a angle_n wherefore_o the_o square_n of_o the_o line_n ab_fw-la &_o be_v be_v quadruple_a to_o the_o square_n of_o the_o line_n be._n wherefore_o take_v away_o the_o square_a of_o the_o line_n be_v the_o squaâ—e_n of_o the_o line_n ab_fw-la shâ—lbe_n treble_n to_o the_o square_n of_o be_v but_o the_o line_n be_v be_v equal_a to_o the_o line_n de._n whâ—râ—fore_o the_o square_a of_o the_o line_n ab_fw-la be_v treble_a to_o the_o square_n of_o the_o line_n de._n wherefore_o the_o square_a make_v of_o the_o side_n of_o the_o triangle_n be_v treble_a to_o the_o square_n make_v of_o the_o line_n draw_v from_o the_o centre_n of_o the_o circle_n to_o the_o circumference_n which_o be_v require_v to_o be_v prove_v a_o corollary_n add_v by_o campane_n hereby_o it_o be_v manifest_a that_o the_o line_n bc_o which_o be_v the_o side_n of_o the_o equilater_n triangle_n divide_v the_o semidiameter_n de_fw-fr into_o two_o equal_a part_n for_o let_v the_o point_n of_o the_o division_n be_v f._n and_o suppose_v a_o line_n to_o be_v draw_v from_o the_o point_n d_o to_o the_o b_o and_o a_o other_o from_o the_o point_n d_o to_o the_o point_n c._n now_o it_o be_v manifest_a by_o the_o 4._o of_o the_o first_o that_o the_o line_n bf_o be_v equal_a to_o the_o line_n fc_o and_o therefore_o by_o the_o 3._o of_o the_o three_o all_o the_o angle_n at_o the_o point_n f_o be_v â—ight_a angle_n wherefore_o by_o the_o 47._o of_o the_o first_o the_o square_a of_o the_o line_n bd_o be_v equal_a to_o the_o square_n oâ—_n the_o lineâ—â—f_n and_o fd_o and_o by_o the_o same_o the_o square_n of_o the_o line_n be_v be_v equal_a to_o the_o square_n of_o the_o line_n bf_o and_o fe_o bâ—t_v the_o line_n bd_o be_v equal_a to_o the_o line_n be_v as_o have_v before_o be_v prove_v wherefore_o by_o the_o common_a sentence_n the_o two_o square_n of_o the_o two_o line_n bf_o and_o fd_o be_v equal_a to_o the_o two_o square_n of_o the_o lineâ—_z bf_o and_o fe_o wherefore_o take_v away_o the_o square_a of_o the_o line_n bf_o which_o be_v common_a to_o they_o both_o the_o residue_n namely_o the_o square_a of_o the_o line_n df_o shall_v be_v equal_v to_o the_o residue_n namely_o to_o the_o square_n of_o the_o line_n fe_o wherefore_o also_o the_o line_n fd_o be_v equal_a to_o the_o line_n fe_o wherefore_o hereby_o it_o be_v manifest_a that_o a_o perpendicular_a line_n draw_v from_o the_o centre_n of_o a_o circle_n to_o the_o side_n of_o a_o equilater_n triangle_n inscribe_v in_o it_o be_v equal_a to_o half_n of_o the_o line_n draw_v from_o the_o centre_n of_o the_o same_o circle_n to_o the_o circumference_n thereof_o a_o corollary_n add_v by_o flussas_n the_o side_n of_o a_o equilater_n triangle_n be_v in_o power_n sesquitertia_fw-la to_o the_o perpendicular_a line_n which_o be_v draw_v from_o one_o of_o the_o angle_n to_o the_o opposite_a side_n for_o of_o what_o part_v the_o line_n ab_fw-la contain_v in_o power_n 12._o of_o such_o part_n the_o line_n bf_o which_o be_v the_o half_a of_o ab_fw-la contain_v in_o power_n 3._o campane_n wherefore_o the_o residue_n namely_o the_o perpendicular_a line_n of_o contain_v in_o power_n of_o such_o part_n 9_o for_o the_o square_n of_o the_o line_n of_o and_o bf_o be_v by_o the_o 47._o of_o the_o first_o equal_a to_o the_o square_n of_o the_o line_n ab_fw-la now_o 1â—_o to_o 9_o be_v sesquitertiaâ—_n wherefore_o the_o power_n of_o the_o line_n ab_fw-la be_v to_o the_o power_n of_o the_o line_n of_o in_o sesquitertia_fw-la proportion_n moreover_o the_o side_n of_o the_o triangle_n be_v the_o mean_a proportional_a between_o the_o diameter_n and_o the_o perpendicular_a line_n for_o by_o the_o corollary_n of_o the_o 8._o of_o the_o six_o the_o line_n ae_n be_v to_o the_o line_n ab_fw-la as_o the_o line_n ab_fw-la be_v to_o the_o line_n af._n far_o the_o perpendicular_a line_n draw_v from_o the_o angle_n divide_v the_o base_a into_o two_o equal_a part_n and_o pass_v by_o the_o centre_n campane_n for_o if_o there_o shall_v be_v draw_v any_o other_o right_a line_n from_o the_o point_n a_o to_o the_o point_n fletcher_n they_o that_o which_o be_v draw_v by_o the_o point_n d_o two_o right_a line_n shall_v include_v a_o superficies_n which_o be_v impossible_a wherefore_o the_o contrary_a follow_v namely_o that_o the_o line_n which_o be_v draw_v from_o the_o angle_n pass_v by_o the_o centre_n be_v a_o perpendicular_a line_n to_o the_o base_a by_o the_o 3._o of_o the_o three_o the_o 1._o problem_n the_o 13._o proposition_n to_o make_v a_o tetrahedron_n pyramid_n and_o to_o comprehend_v it_o in_o a_o sphere_n give_v and_o to_o prove_v that_o the_o diameter_n of_o the_o sphere_n be_v in_o power_n sesquialtera_fw-la to_o the_o side_n of_o the_o pyramid_n in_o the_o semicircle_n adb_o of_o the_o former_a figure_n draw_v the_o line_n dn_o and_o divide_v the_o
line_n kg_v into_o two_o equal_a part_n in_o the_o point_n m._n fâ—ussas_n and_o draw_v a_o line_n from_o m_o to_o g._n and_o forasmuch_o as_o by_o construction_n the_o line_n kh_o be_v equal_a to_o the_o line_n ac_fw-la and_o the_o line_n hl_o to_o the_o line_n cb_o therefore_o the_o whole_a line_n ab_fw-la be_v equal_a to_o the_o whole_a line_n kl_o wherefore_o also_o the_o half_a of_o the_o line_n kl_o namely_o the_o line_n lm_o be_v equal_a to_o the_o semidiameter_n bn_o wherefore_o take_v away_o from_o those_o equal_a line_n equal_a part_n bc_o and_o lh_o the_o residue_v nc_n and_o mh_o shall_v be_v equal_a wherefore_o in_o the_o two_o triangle_n mhg_n and_o ncd_a the_o two_o side_n about_o the_o equal_a right_n angle_n dcn_n and_o ghm_n namely_o the_o side_n dc_o cn_fw-la and_o gh_o he_o be_v equal_a wherefore_o the_o base_n mg_o and_o nd_o be_v equal_a by_o the_o 4._o of_o the_o first_o and_o by_o the_o same_o reason_n may_v it_o be_v prove_v that_o right_a line_n draw_v from_o the_o point_n m_o to_o the_o point_n e_o and_o f_o be_v equal_a to_o the_o line_n nd_o but_o the_o right_a line_n nd_o be_v equal_a to_o the_o line_n a_fw-la which_o be_v draw_v from_o the_o centre_n to_o the_o circumference_n wherefore_o the_o line_n mg_o be_v equal_a to_o the_o line_n mk_o &_o also_o to_o the_o line_n i_o mf_n and_o ml_o wherefore_o make_v the_o centre_n the_o point_n m_o and_o the_o space_n mk_o or_o mg_o describe_v a_o semicircle_n kgl_n and_o the_o diameter_n kl_o abide_v fix_v let_v the_o say_a semicircle_n kgl_n be_v move_v round_o about_o until_o it_o return_v to_o the_o same_o place_n from_o whence_o it_o begin_v to_o be_v move_v and_o there_o shall_v be_v describe_v a_o sphere_n about_o the_o centre_n m_o by_o the_o 12._o definition_n of_o the_o eleven_o touch_v every_o one_o of_o the_o angle_n of_o the_o pyramid_n which_o be_v at_o the_o point_n king_n e_o f_o g_o for_o those_o angle_n be_v equal_o distant_a from_o the_o centre_n of_o the_o sphere_n namely_o by_o the_o semidiameter_n of_o the_o say_a sphere_n as_o have_v before_o be_v prove_v wherefore_o in_o the_o sphere_n give_v who_o diameter_n be_v the_o line_n kl_o or_o the_o line_n ab_fw-la be_v inscribe_v a_o tetrahedron_n efgk_n now_o i_o say_v that_o the_o diameter_n of_o the_o sphere_n be_v in_o power_n sesquialtera_fw-la to_o the_o side_n of_o the_o pyramid_n demonstration_n for_o forasmuch_o as_o the_o line_n ac_fw-la be_v double_a to_o the_o line_n cb_o by_o construction_n therefore_o the_o line_n ab_fw-la be_v treble_a to_o the_o line_n bc._n wherefore_o by_o conversion_n by_o the_o corollary_n of_o the_o 19_o of_o the_o five_o the_o line_n ab_fw-la be_v sesquialtera_fw-la to_o the_o line_n ac_fw-la but_o as_o the_o line_n basilius_n be_v to_o the_o line_n ac_fw-la so_o be_v the_o square_a of_o the_o line_n basilius_n to_o the_o square_n of_o the_o line_n ad._n for_o if_o we_o draw_v a_o right_a line_n from_o the_o point_n b_o to_o the_o point_n d_o as_o the_o line_n bd_o be_v to_o the_o line_n ad_fw-la so_o be_v the_o same_o ad_fw-la to_o the_o line_n ac_fw-la by_o reason_n of_o the_o likeness_n of_o the_o ttriangle_n dab_n &_o dac_o by_o the_o 8._o of_o the_o six_o &_o by_o reason_n also_o that_o as_o the_o first_o be_v to_o the_o three_o so_o be_v the_o square_a of_o the_o first_o to_o the_o square_n of_o the_o second_o by_o the_o corollary_n of_o the_o 20._o of_o the_o six_o wherefore_o the_o square_a of_o the_o line_n basilius_n be_v sesquialter_fw-la to_o the_o square_n of_o the_o line_n ad._n but_o the_o line_n basilius_n be_v equal_a to_o the_o diameter_n of_o the_o sphere_n give_v namely_o to_o the_o line_n kl_o as_o have_v be_v prove_v &_o the_o line_n ad_fw-la be_v equal_a to_o the_o side_n of_o the_o pyramid_n inscribe_v in_o the_o sphere_n wherefore_o the_o diameter_n of_o the_o sphere_n be_v in_o power_n sesquialter_fw-la to_o the_o side_n of_o the_o pyramid_n wherefore_o there_o be_v make_v a_o pyramid_n comprehend_v in_o a_o sphere_n give_v and_o the_o diameter_n of_o the_o sphere_n be_v sesquialtera_fw-la to_o the_o side_n of_o the_o pyramid_n which_o be_v require_v to_o be_v do_v and_o prove_v ¶_o a_o other_o demonstration_n to_o prove_v that_o as_o the_o line_n ab_fw-la be_v to_o the_o line_n bc_o so_o be_v the_o square_a of_o the_o line_n ad_fw-la to_o the_o square_n of_o the_o line_n dc_o let_v the_o description_n of_o the_o semicircle_n adb_o be_v as_o in_o the_o first_o description_n and_o upon_o the_o line_n ac_fw-la describe_v by_o the_o 46._o of_o the_o first_o a_o square_a ec_o and_o make_v perfect_v the_o parallelogram_n fb_o now_o forasmuch_o as_o the_o triangle_n dab_n be_v equiangle_n to_o the_o triangle_n dac_o by_o the_o 32._o of_o the_o six_o therefore_o as_o the_o line_n basilius_n be_v to_o the_o line_n ad_fw-la so_o be_v the_o line_n dam_fw-ge to_o the_o line_n ac_fw-la by_o the_o 4._o of_o the_o six_o wherefore_o that_o which_o be_v contain_v under_o the_o line_n basilius_n and_o ac_fw-la be_v equal_a to_o the_o square_n of_o the_o line_n ad_fw-la by_o the_o 17._o of_o the_o six_o and_o for_o that_o as_o the_o line_n ab_fw-la be_v to_o the_o line_n bc_o so_o be_v the_o parallelogram_n ebb_v to_o the_o parallelogram_n fb_o by_o the_o 1._o of_o the_o six_o and_o the_o parallelogram_n ebb_n be_v that_o which_o be_v contain_v under_o the_o line_n basilius_n and_o ac_fw-la for_o the_o line_n ea_fw-la be_v equal_a to_o the_o line_n ac_fw-la and_o the_o parallelogram_n bf_o be_v that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o bc._n wherefore_o as_o the_o line_n ab_fw-la be_v to_o the_o line_n bc_o so_o be_v that_o which_o be_v contain_v under_o the_o line_n basilius_n and_o ac_fw-la to_o that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb._n but_o that_o which_o be_v contain_v under_o the_o line_n basilius_n and_o ac_fw-la be_v equal_a to_o the_o square_n of_o the_o line_n ad_fw-la by_o the_o corollary_n of_o the_o 8._o of_o the_o six_o and_o that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o be_v equal_a to_o the_o square_n of_o the_o line_n cd_o for_o the_o perpendicular_a line_n dc_o be_v the_o mean_a proportional_a between_o the_o segment_n of_o the_o base_a namely_o ac_fw-la and_o cb_o by_o the_o former_a corollary_n of_o the_o 8._o of_o the_o six_o for_o that_o the_o angle_n adb_o be_v a_o right_a angle_n wherefore_o as_o the_o line_n ab_fw-la be_v to_o the_o line_n bc_o so_o be_v the_o square_a of_o the_o line_n ad_fw-la to_o the_o square_n of_o the_o line_n dc_o by_o the_o 11._o of_o the_o five_o which_o be_v require_v to_o be_v prove_v ¶_o two_o assumpte_n add_v by_o campane_n first_o assumpt_n suppose_v that_o upon_o the_o line_n ab_fw-la be_v erect_v perpendicular_o the_o line_n dc_o which_o line_n dc_o let_v be_v the_o mean_a proportional_a between_o the_o part_n of_o the_o line_n ab_fw-la namely_o ac_fw-la &_o cb_o so_o that_o as_o the_o line_n ac_fw-la be_v to_o the_o line_n cd_o so_o let_v the_o line_n cd_o be_v to_o the_o line_n cb._n and_o upon_o the_o line_n ab_fw-la describe_v a_o semicircle_n then_o i_o say_v that_o the_o circumference_n of_o that_o semicircle_n shall_v pass_v by_o the_o point_n d_o which_o be_v the_o end_n of_o the_o perpendicular_a line_n but_o if_o not_o than_o it_o shall_v either_o cut_v the_o line_n cd_o or_o it_o shall_v pass_v above_o it_o and_o include_v it_o not_o touch_v it_o first_o let_v it_o cut_v it_o in_o the_o point_n e._n and_o draw_v these_o right_a line_n ebb_v and_o ea_fw-la wherefore_o by_o the_o 31._o of_o the_o three_o the_o whole_a angle_n aeb_n be_v a_o right_a angle_n wherefore_o by_o the_o first_o part_n of_o the_o corollary_n of_o the_o 8._o of_o the_o six_o the_o line_n ac_fw-la be_v to_o the_o line_n â—c_z as_z the_o line_n ec_o be_v to_o the_o line_n cb._n but_o by_o the_o 8._o of_o the_o five_o the_o proportion_n of_o the_o line_n ac_fw-la to_o the_o line_n ec_o be_v great_a than_o the_o proportion_n of_o the_o same_o line_n ac_fw-la to_o the_o line_n cd_o for_o the_o line_n ce_fw-fr be_v less_o than_o the_o line_n cd_o now_o for_o that_o the_o line_n ce_fw-fr be_v to_o thâ—_z line_n cb_o as_o the_o line_n ac_fw-la be_v to_o the_o line_n ce_fw-fr and_o the_o line_n cd_o be_v to_o the_o line_n cb_o as_o the_o line_n ac_fw-la be_v to_o the_o line_n cd_o therefore_o by_o the_o 13._o of_o the_o five_o the_o proportion_n of_o the_o line_n ec_o to_o the_o line_n cb_o be_v great_a than_o the_o proportion_n of_o the_o line_n cd_o to_o the_o line_n cb._n wherefore_o by_o the_o 10._o of_o the_o five_o the_o line_n ec_o be_v great_a than_o the_o line_n dc_o
namely_o the_o part_n great_a than_o the_o whole_a which_o be_v impossible_a wherefore_o the_o circumference_n shall_v not_o cut_v the_o line_n cd_o now_o i_o say_v that_o it_o shall_v not_o pass_v above_o the_o line_n cd_o and_o not_o touch_v it_o in_o the_o point_n d._n for_o if_o it_o be_v possible_a let_v it_o pass_v above_o it_o and_o extend_v the_o line_n cd_o to_o the_o circumference_n and_o let_v it_o cut_v it_o in_o the_o point_n â—_o and_o draw_v the_o line_n fb_o and_o favorina_n and_o it_o shall_v follow_v as_o before_o that_o the_o line_n cd_o be_v great_a than_o the_o line_n cf_o which_o be_v impossible_a wherefore_o that_o be_v manifest_v which_o be_v require_v to_o be_v prove_v ¶_o second_v assumpt_n if_o there_o be_v a_o right_a angle_n unto_o which_o a_o base_a be_v subtend_v and_o if_o upon_o the_o same_o be_v describe_v a_o semicircle_n the_o circumference_n thereof_o shall_v pass_v by_o the_o point_n of_o the_o right_a angle_n the_o converse_n of_o this_o be_v add_v after_o the_o demonstration_n of_o the_o 31._o of_o the_o three_o out_o of_o pelitarius_n and_o these_o two_o assumpte_n of_o campane_n be_v necessary_a for_o the_o better_a understanding_n of_o the_o demonstration_n of_o the_o secoâ—d_a part_n of_o this_o 13._o proposition_n wherein_o be_v prove_v that_o the_o pyramid_n be_v contain_v in_o the_o sphere_n give_v ¶_o certain_a corollarye_n add_v by_o flussas_n first_o corollary_n the_o diameter_n of_o the_o sphere_n be_v in_o power_n quadruple_a sesquialtera_fw-la to_o the_o line_n which_o be_v draw_v from_o the_o centre_n to_o the_o circumference_n of_o the_o circle_n which_o contain_v the_o base_a of_o the_o pyramid_n for_o forasmuch_o as_o it_o have_v be_v prove_v that_o the_o diameter_n kl_o be_v in_o power_n sesquialter_fw-la to_o the_o side_n of_o and_o it_o be_v prove_v also_o by_o the_o 12._o of_o this_o book_n that_o the_o side_n of_o be_v in_o power_n triple_a to_o the_o line_n eh_o which_o be_v draw_v from_o the_o centre_n of_o the_o circle_n contain_v the_o triangle_n efg_o but_o the_o proportion_n of_o the_o extreme_n namely_o of_o the_o diameter_n to_o the_o line_n eh_o consist_v of_o the_o proportion_n of_o the_o mean_n namely_o of_o the_o proportion_n of_o the_o diameter_n to_o the_o line_n of_o and_o of_o the_o proportion_n of_o the_o line_n of_o to_o the_o line_n eh_o by_o the_o 5._o definition_n of_o the_o six_o which_o proportion_n namely_o triple_a and_o sesquialter_fw-la add_v together_o make_v quadruple_a sesquialter_fw-la as_o it_o be_v easy_a to_o prove_v by_o that_o which_o be_v teach_v in_o the_o declaration_n of_o the_o 5._o definition_n of_o the_o six_o book_n wherefore_o the_o corollary_n be_v manifest_a ¶_o second_v corollary_n only_o the_o line_n which_o be_v draw_v from_o the_o angle_n of_o the_o pyramid_n to_o the_o base_a opposite_a unto_o it_o campane_n &_o pass_v by_o the_o centre_n of_o the_o sphere_n be_v perpendicular_a to_o the_o base_a and_o fall_v upon_o the_o centre_n of_o the_o circle_n which_o contain_v the_o base_a for_o if_o any_o other_o line_n than_o the_o line_n kmh_o which_o be_v draw_v by_o the_o centre_n of_o the_o sphere_n to_o the_o centre_n of_o the_o circle_n shall_v fall_v perpendicular_o upon_o the_o plain_a of_o the_o base_a then_o from_o one_o and_o the_o self_n same_o point_n shall_v be_v draw_v to_o one_o and_o the_o self_n same_o plain_a two_o perpendicular_a line_n contrary_a to_o the_o 13._o of_o the_o eleven_o which_o be_v impossible_a far_o if_o from_o the_o top_n king_n shall_v be_v draw_v to_o the_o centre_n of_o the_o base_a namely_o to_o the_o point_n h_o any_o other_o right_a line_n not_o pass_v by_o the_o centre_n m_o two_o right_a life_n shall_v include_v a_o superficies_n contrary_n to_o the_o last_o common_a sentence_n which_o be_v absurd_a wherefore_o only_o the_o line_n which_o be_v draw_v by_o the_o centre_n of_o the_o sphere_n to_o the_o centre_n of_o the_o base_a be_v perpendicular_a to_o the_o say_v base_a and_o the_o line_n which_o be_v draw_v from_o the_o angle_n perpendicular_o to_o the_o base_a shall_v pass_v by_o the_o centre_n of_o the_o sphere_n three_o corollary_n the_o perpendicular_a line_n which_o be_v draw_v from_o the_o centre_n of_o the_o sphere_n to_o the_o base_a of_o the_o pyramid_n book_n be_v equal_a to_o the_o six_o part_n of_o the_o diameter_n of_o the_o sphere_n for_o it_o be_v before_o prove_v that_o the_o line_n mh_o which_o be_v draw_v from_o the_o centre_n of_o the_o sphere_n to_o the_o centre_n of_o the_o base_a be_v equal_a to_o the_o line_n nc_n which_o line_n nc_n be_v the_o six_o part_n of_o the_o diameter_n ab_fw-la and_o therefore_o the_o line_n mh_o be_v the_o six_o part_n of_o the_o diameter_n of_o the_o sphere_n for_o the_o diameter_n ab_fw-la be_v equal_a to_o the_o diameter_n of_o the_o sphere_n as_o have_v also_o before_o be_v prove_v ¶_o the_o 2._o problem_n the_o 14._o proposition_n to_o make_v a_o octohedron_n and_o to_o comprehend_v it_o in_o the_o sphere_n give_v namely_o that_o wherein_o the_o pyramid_n be_v comprehend_v and_o to_o prove_v that_o the_o diameter_n of_o the_o sphere_n be_v in_o power_n double_a to_o the_o side_n of_o the_o octohedron_n construction_n take_v the_o diameter_n of_o the_o former_a sphere_n give_v which_o let_v be_v the_o line_n ab_fw-la and_o divide_v it_o by_o the_o 10._o of_o the_o first_o into_o two_o equal_a part_n in_o the_o point_n c._n and_o describe_v upon_o the_o line_n ab_fw-la a_o semicircle_n adb_o and_o by_o the_o 11._o of_o the_o first_o from_o the_o point_n c_o raise_v up_o unto_o the_o line_n ab_fw-la a_o perpendicular_a line_n cd_o and_o draw_v a_o right_a line_n from_o the_o point_n d_o to_o the_o point_n b._n and_o describe_v a_o square_a efgh_o have_v every_o one_o of_o his_o side_n equal_a to_o the_o line_n bd._n and_o draw_v the_o diagonal_a line_n fh_a &_o eglantine_n cut_v the_o one_o the_o other_o in_o the_o point_n k._n and_o by_o the_o 12._o of_o the_o eleven_o from_o the_o point_n king_n namely_o the_o point_n where_o the_o line_n fh_o and_o eglantine_n cut_v the_o one_o the_o other_o raise_v up_o to_o the_o plain_a superficies_n wherein_o the_o square_a efgh_o be_v a_o perpendicular_a line_n kl_o and_o extend_v the_o line_n kl_o on_o the_o other_o side_n of_o the_o plain_a superficies_n to_o the_o point_n m._n and_o let_v each_o of_o the_o line_n kl_o and_o km_o be_v put_v equal_a to_o one_o of_o these_o line_n ke_o kf_o kh_o or_o kg_n and_o draw_v these_o right_a line_n le_fw-fr lf_n lg_n lh_o menander_n mf_n mg_o and_o mh_o demonstration_n now_o forasmuch_o as_o the_o line_n ke_o be_v by_o the_o corollary_n of_o the_o 34._o of_o the_o first_o equal_a to_o the_o line_n kh_o and_o the_o angle_n angle_n ekh_n be_v a_o right_a angle_n therefore_o the_o square_a of_o he_o be_v double_a to_o the_o square_n of_o eke_o by_o the_o 47._o of_o the_o first_o again_o forasmuch_o as_o the_o line_n lk_n be_v equal_a to_o the_o line_n ke_o by_o position_n and_o the_o angle_n lke_n be_v by_o the_o second_o definition_n of_o the_o eleven_o a_o right_a angle_n therefore_o the_o square_n of_o the_o line_n el_n be_v double_a to_o the_o square_n of_o the_o line_n eke_o and_o it_o be_v prove_v that_o the_o square_a of_o the_o line_n he_o be_v double_a to_o the_o square_n of_o the_o line_n eke_o wherefore_o the_o square_a of_o the_o line_n le_fw-fr be_v equal_a to_o the_o square_n of_o the_o line_n eh_o wherefore_o also_o the_o line_n le_fw-fr be_v equal_a to_o the_o line_n eh_o and_o by_o the_o same_o reason_n the_o line_n lh_o be_v also_o equal_a to_o the_o line_n he._n wherefore_o the_o triangle_n lhe_n be_v equilater_n in_o like_a sort_n may_v we_o prove_v that_o every_o one_o of_o the_o rest_n of_o the_o triangle_n who_o base_n be_v the_o side_n of_o the_o square_a efgh_o and_o top_n the_o point_n l_o and_o m_o be_v equilater_n the_o say_v eight_o triangle_n also_o be_v equal_a the_o one_o to_o the_o other_o for_o every_o side_n of_o each_o be_v equal_a to_o the_o side_n of_o the_o square_a efgh_o wherefore_o there_o be_v make_v a_o octohedron_n contain_v under_o eight_o triangle_n who_o side_n be_v equal_a now_o it_o be_v require_v to_o comprehend_v it_o in_o the_o sphere_n give_v and_o to_o prove_v that_o the_o diameter_n of_o the_o sphere_n be_v in_o power_n double_a to_o the_o side_n of_o the_o octohedron_n demonstration_n forasmuch_o as_o these_o three_o line_n lk_v km_o and_o ke_o be_v equal_a the_o one_o to_o the_o other_o therefore_o a_o semicircle_n describe_v upon_o the_o line_n lm_o shall_v pass_v also_o by_o the_o point_n e._n and_o by_o the_o same_o reason_n if_o the_o semicircle_n be_v turn_v round_o about_o until_o it_o return_v unto_o the_o self_n same_o
place_n from_o whence_o first_o it_o begin_v to_o be_v move_v it_o shall_v pass_v by_o the_o point_n f_o g_o h_o and_o the_o octohedron_n shall_v be_v comprehend_v in_o a_o sphere_n i_o say_v also_o that_o it_o be_v comprehend_v in_o the_o sphere_n give_v for_o forasmuch_o as_o the_o line_n lk_n be_v equal_a to_o the_o line_n km_o by_o position_n and_o the_o line_n ke_o be_v common_a to_o they_o both_o and_o they_o contain_v right_a angle_n by_o the_o 3._o definition_n of_o the_o eleven_o therefore_o by_o the_o 4._o of_o the_o first_o the_o base_a le_fw-fr be_v equal_a to_o the_o base_a em_n and_o forasmuch_o as_o the_o angle_n lem_n be_v a_o right_a angle_n by_o the_o 31._o of_o the_o three_o for_o it_o be_v in_o a_o semicircle_n as_o have_v be_v prove_v therefore_o the_o square_a of_o the_o line_n lm_o be_v double_a to_o the_o square_n of_o the_o line_n le_fw-fr by_o the_o 47._o of_o the_o first_o again_o forasmuch_o as_o the_o line_n ac_fw-la be_v equal_a to_o the_o line_n bc_o therefore_o the_o line_n ab_fw-la be_v double_a to_o the_o line_n bc_o by_o the_o definition_n of_o a_o circle_n but_o as_o the_o line_n ab_fw-la be_v to_o the_o line_n bc_o so_o be_v the_o square_a of_o the_o line_n ab_fw-la to_o the_o square_n of_o the_o line_n bd_o by_o the_o corollary_n of_o the_o 8._o and_o _o of_o the_o six_o wherefore_o the_o square_a of_o the_o line_n ab_fw-la be_v double_a to_o the_o square_n of_o the_o line_n bd._n and_o it_o be_v prove_v that_o the_o square_a of_o the_o line_n lm_o be_v double_a to_o the_o square_n of_o the_o line_n le._n wherefore_o the_o square_a of_o the_o line_n bd_o be_v equal_a to_o the_o square_n of_o the_o line_n le._n for_o the_o line_n eh_o which_o be_v equal_a to_o the_o line_n lf_n be_v put_v to_o be_v equal_a to_o the_o line_n db._n wherefore_o the_o square_a of_o the_o line_n ab_fw-la be_v equal_a to_o the_o square_n of_o the_o line_n lm_o wherefore_o the_o line_n ab_fw-la be_v equal_a to_o the_o line_n lm_o and_o the_o line_n ab_fw-la be_v the_o diameter_n of_o the_o sphere_n give_v wherefore_o the_o line_n lm_o be_v equal_a to_o the_o diameter_n of_o the_o sphere_n give_v wherefore_o the_o octoedron_n be_v contain_v in_o the_o sphere_n give_v le._n and_o it_o be_v also_o prove_v that_o the_o diameter_n of_o the_o sphere_n be_v in_o power_n double_a to_o the_o side_n of_o the_o octohedron_n wherefore_o there_o be_v make_v a_o octohedron_n and_o it_o be_v comprehend_v in_o the_o sphere_n give_v wherein_o be_v comprehend_v the_o pyramid_n and_o it_o be_v prove_v that_o the_o diameter_n of_o the_o sphere_n be_v in_o power_n double_a to_o the_o side_n of_o the_o octohedrn_n which_o be_v require_v to_o be_v do_v and_o to_o be_v prove_v certain_a corollary_n add_v by_o flussas_n first_o corollary_n the_o side_n of_o a_o pyramid_n be_v in_o power_n sesquitertia_fw-la to_o the_o side_n of_o a_o octahedron_n inscribe_v in_o the_o same_o sphere_n for_o forasmch_v as_o the_o diameter_n be_v in_o power_n double_a to_o the_o side_n of_o the_o octohedron_n therefore_o of_o what_o part_n the_o diameter_n contain_v in_o power_n 6._o of_o the_o same_o the_o side_n of_o the_o octohedron_n contain_v in_o power_n 3._o but_o of_o what_o part_n the_o diameter_n contain_v 6._o of_o the_o same_o the_o side_n of_o the_o pyramid_n contain_v 4._o by_o the_o 13._o of_o this_o book_n wherefore_o of_o what_o part_n the_o side_n of_o the_o pyramid_n contain_v 4._o of_o the_o same_o the_o side_n of_o the_o octohedron_n contain_v 3._o second_o corollary_n a_o octohedron_n be_v divide_v into_o two_o equal_a and_o like_a pyramid_n the_o common_a base_n of_o these_o pyramid_n be_v set_v upon_o every_o square_n contain_v of_o the_o side_n of_o the_o octohedron_n upon_o which_o square_a be_v set_v the_o â—â—_o triangle_n of_o the_o octohedron_n campane_n which_o pyramid_n be_v by_o the_o â—_o definition_n of_o the_o eleven_o equal_a and_o like_a and_o the_o foresay_a square_n common_a to_o those_o pyramid_n be_v the_o half_a of_o the_o square_n of_o the_o diameter_n of_o the_o sphere_n for_o it_o be_v the_o square_a of_o the_o side_n of_o the_o octohedron_n three_o corollary_n the_o three_o diameter_n of_o the_o octohedron_n do_v cut_v the_o one_o the_o other_o perpendicular_o into_o two_o equal_a part_n in_o the_o centre_n of_o the_o sphere_n which_o contain_v the_o say_a octohedron_n as_o it_o be_v manifest_a by_o the_o three_o diameter_n eglantine_n fh_o and_o lm_o which_o cut_v the_o one_o the_o other_o in_o the_o centre_n king_n equal_o and_o perpendicular_o ¶_o the_o 3._o problem_n the_o 15._o proposition_n to_o make_v a_o solid_a call_v a_o cube_fw-la and_o to_o comprehend_v it_o in_o the_o sphere_n give_v namely_o that_o sphere_n wherein_o the_o former_a two_o solid_n be_v comprehendâ—dâ—_n and_o to_o prove_v that_o the_o diameter_n of_o the_o sphere_n be_v in_o power_n treble_a to_o the_o side_n of_o the_o cube_fw-la take_v the_o diameter_n of_o the_o sphere_n give_v namely_o ab_fw-la and_o divide_v it_o in_o the_o point_n câ—_n so_o that_o let_v the_o line_n ac_fw-la be_v double_a to_o the_o line_n bc_o by_o the_o 9_o of_o the_o six_o and_o upon_o the_o line_n ab_fw-la describe_v a_o semicircle_n adb_o and_o by_o the_o 11._o of_o the_o first_o from_o the_o pâ—ynt_n c_o râ—yse_v up_o unto_o the_o line_n ab_fw-la a_o perpendicular_a line_n cd_o and_o draw_v a_o right_a linâ—_n db._n and_o describe_v a_o square_a efgh_o have_v every_o one_o of_o his_o side_n equal_a to_o the_o line_n db_o and_o from_o the_o point_n e_o f_o g_o h_o raise_v up_o by_o the_o 12._o of_o the_o eleven_o unto_o the_o plain_a superficies_n of_o the_o square_a efgh_o perpendicular_a line_n eke_o fl_fw-mi gm_n and_o hn_o and_o let_v every_o one_o of_o the_o line_n eke_o fl_fw-mi gm_n and_o hn_o be_v put_v equal_a to_o one_o of_o the_o line_n of_o fg_o gh_o or_o he_o which_o be_v the_o side_n of_o the_o square_n and_o draw_v these_o right_a line_n kl_o lm_o mn_a and_o nk_v demonstration_n wherefore_o there_o be_v make_v a_o cube_fw-la namely_o fn_o which_o be_v contain_v under_o six_o equal_a square_n construction_n now_o it_o be_v require_v to_o comprehend_v the_o same_o cube_fw-la in_o the_o sphere_n give_v and_o to_o prove_v that_o the_o diameter_n of_o the_o sphere_n be_v in_o power_n ble_o to_o the_o side_n of_o the_o cube_fw-la demonstration_n draw_v these_o right_a line_n kg_v and_o eglantine_n and_o forasmuch_o as_o the_o angle_n keg_n be_v a_o right_a angle_n for_o that_o the_o line_n ke_o be_v erect_v perpendicular_o to_o the_o plain_a superficies_n eâ—_n and_o therefore_o also_o to_o the_o right_a line_n eglantine_n by_o the_o 2._o definition_n of_o the_o eleven_o wherefore_o a_o semicircle_n describe_v upon_o the_o line_n kg_v shall_v book_n pass_v by_o the_o point_n e._n again_o forasmuch_o as_o the_o line_n fg_o be_v erect_v perpendicular_o to_o either_o of_o these_o line_n fl_fw-mi and_o fe_o by_o the_o definition_n of_o a_o square_a &_o by_o the_o 2._o definition_n of_o the_o eleven_o therefore_o the_o line_n fg_o be_v erect_v perpendicular_o to_o the_o plain_a superficies_n fk_o by_o the_o 4._o of_o the_o eleven_o wherefore_o if_o we_o draw_v a_o right_a line_n from_o the_o point_n f_o to_o the_o point_n king_n the_o line_n gf_o shall_v be_v erect_v perpendicular_o to_o the_o line_n kf_o by_o the_o 2._o definition_n of_o the_o eleven_o and_o by_o the_o same_o reason_n again_o a_o semicircle_n describe_v upon_o the_o line_n gk_o shall_v pass_v also_o by_o the_o point_n f._n and_o likewise_o shall_v it_o pass_v by_o the_o rest_n of_o the_o point_n of_o the_o angle_n of_o that_o cube_fw-la if_o now_o the_o diameter_n kg_v abide_v fix_v the_o semicircle_n be_v turn_v round_o about_o until_o it_o return_v into_o the_o self_n same_o place_n from_o whence_o it_o begin_v first_o to_o be_v move_v the_o cube_fw-la shall_v be_v comprehend_v in_o a_o sphere_n i_o say_v also_o that_o it_o be_v comprehend_v in_o the_o sphere_n give_v demonstration_n for_o forasmuch_o as_o the_o line_n gf_o be_v equal_a to_o the_o linâ—â—e_n and_o the_o angle_n fletcher_n be_v a_o right_a angle_n therefore_o the_o square_a of_o the_o line_n eglantine_n be_v by_o the_o 47._o of_o the_o first_o double_a to_o the_o square_n of_o the_o line_n â—f_n but_o the_o line_n of_o be_v equal_a to_o the_o line_n eke_o wherefore_o the_o square_a of_o the_o line_n eglantine_n be_v double_a to_o the_o square_n of_o the_o line_n eke_o wherefore_o the_o square_n of_o eglantine_n and_o eke_o that_o be_v the_o square_a of_o the_o line_n gk_o by_o the_o 47._o of_o the_o first_o be_v treble_a to_o the_o square_n of_o the_o line_n eke_o and_o forasmuch_o as_o the_o line_n ab_fw-la be_v treble_a to_o the_o line_n bc_o but_o
the_o circumference_n on_o which_o the_o icosahedron_n be_v describe_v and_o that_o the_o diameter_n of_o the_o sphere_n be_v compose_v of_o the_o side_n of_o a_o hexagon_n and_o of_o two_o side_n of_o a_o decagon_n describe_v in_o one_o and_o the_o self_n same_o circle_n flussas_n prove_v the_o icosahedron_n describe_v to_o be_v contain_v in_o a_o sphere_n by_o draw_v right_a line_n from_o the_o point_n a_o to_o the_o point_n p_o and_o g_o after_o this_o manner_n forasmuch_o as_o the_o line_n zw_n wp_n be_v put_v equal_a to_o the_o line_n draw_v from_o the_o centre_n to_o the_o circumferenceâ—_n and_o the_o line_n draw_v from_o the_o centre_n to_o the_o circumference_n be_v double_a to_o the_o line_n awe_n by_o construction_n therefore_o the_o line_n wp_n be_v also_o double_a to_o the_o same_o line_n awe_n wherefore_o the_o square_a of_o the_o line_n wp_n be_v quadruple_a to_o the_o square_n of_o the_o line_n awe_n by_o the_o corollary_n of_o the_o 20._o of_o the_o six_o and_o those_o line_n pw_o and_o wa_n contain_v a_o right_a angle_n pwa_n as_o have_v before_o be_v prove_v be_v subtend_v of_o the_o right_a line_n ap_fw-mi wherefore_o by_o the_o 47._o of_o the_o first_o the_o line_n ap_fw-mi contain_v in_o power_n the_o line_n pw_o and_o wa._n wherefore_o the_o right_a line_n ap_fw-mi be_v in_o power_n quintuple_a to_o the_o line_n wa._n wherefore_o the_o right_a line_n ap_fw-mi and_o aq_n be_v quintuple_a to_o one_o and_o the_o same_o line_n wa_n be_v by_o the_o 9_o of_o the_o flu_v equal_a in_o like_a sort_n also_o may_v we_o prove_v that_o unto_o those_o line_n ap_fw-mi and_o aq_n be_v equal_a the_o rest_n of_o the_o line_n draw_v from_o the_o point_n a_o to_o the_o rest_n of_o the_o angle_n r_o s_o t_o v._o for_o they_o subtend_v right_a angle_n contain_v of_o the_o line_n wa_n and_o of_o the_o line_n draw_v from_o the_o centre_n to_o the_o circumference_n and_o forasmuch_o as_o unto_o the_o line_n wa_n be_v equal_a the_o line_n valerio_n which_o be_v likewise_o erect_v perpendicular_o unto_o the_o other_o plain_a superficies_n olmnx_n therefore_o line_n draw_v from_o the_o point_n a_o to_o the_o angle_n oh_o l_o m_o n_o x_o and_o subtend_v right_a angle_n at_o the_o point_n z_o contain_v under_o line_n draw_v from_o the_o centre_n to_o the_o circumference_n and_o under_o the_o line_n az_n be_v equal_a not_o only_o the_o one_o to_o the_o other_o but_o also_o to_o the_o line_n draw_v from_o the_o say_a point_n a_o to_o the_o former_a angle_n at_o the_o point_n p_o r_o s_o t_o vâ—_n for_o the_o line_n draw_v from_o the_o centre_n to_o the_o circumference_n of_o each_o circle_n be_v equal_a &_o the_o line_n awe_n be_v equal_a to_o the_o line_n az._n but_o the_o line_n ap_fw-mi be_v prove_v equal_a to_o the_o line_n aq_n which_o be_v the_o half_a of_o the_o whole_a qy_n therefore_o the_o residue_n ay_o be_v equal_a to_o the_o foresay_a line_n ap_fw-mi aq_n etc_n etc_n wherefore_o make_v the_o centre_n the_o point_n a_o and_o the_o space_n one_o of_o those_o line_n aq_n ap_fw-mi etc_n etc_n extend_v the_o superficies_n of_o a_o sphere_n &_o it_o shall_v touch_v the_o 12._o angle_n of_o the_o icosahedron_n which_o be_v at_o the_o point_n oh_o l_o m._n n_o x_o pea_n king_n s_o t_o five_o q_o y_fw-fr which_o sphere_n be_v describe_v if_o upon_o the_o diameter_n qy_n be_v draw_v a_o semicircle_n and_o the_o say_a semicircle_n be_v move_v about_o till_o it_o return_v unto_o the_o same_o place_n from_o whence_o it_o begin_v first_o to_o be_v move_v ¶_o a_o corollary_n add_v by_o flussas_n the_o opposite_a side_n of_o a_o icosahedron_n be_v parallel_n for_o the_o diameter_n of_o the_o sphere_n do_v fall_v upon_o the_o opposite_a angle_n of_o the_o icosahedron_n as_o it_o be_v manifest_a by_o the_o right_a line_n qy_n if_o therefore_o there_o be_v imagine_v to_o be_v draw_v the_o two_o diameter_n pn_n and_o om_o they_o shall_v concur_v in_o the_o point_n f_o wherefore_o the_o right_a line_n which_o join_v they_o together_o pv_n and_o ln_o be_v in_o one_o and_o the_o self_n same_o plain_a superficies_n by_o the_o 2._o of_o the_o eleven_o and_o forasmuch_o as_o the_o alternate_a angle_n at_o the_o end_n of_o the_o diameter_n be_v equal_a by_o the_o 8._o of_o the_o first_o for_o the_o triangle_n contain_v under_o equal_a semidiameteâ—s_n and_o the_o side_n of_o the_o icosahedron_n be_v equiangle_n therefore_o by_o the_o 28._o of_o the_o first_o the_o line_n pv_n and_o ln_o be_v paralle_n ¶_o the_o 5._o problem_n the_o 17._o proposition_n to_o make_v a_o dodecahedron_n and_o to_o comprehend_v it_o in_o the_o sphere_n give_v wherein_o be_v comprehend_v the_o foresay_a solid_n and_o to_o prove_v that_o the_o side_n of_o the_o dodecahedron_n be_v a_o irrational_a line_n of_o that_o kind_n which_o be_v call_v a_o residual_a line_n superficies_n now_o i_o say_v that_o it_o be_v in_o one_o and_o the_o self_n same_o plain_a superficies_n forasmuch_o as_o the_o line_n zv_n be_v a_o parallel_n to_o the_o line_n sir_n as_o be_v before_o prove_v but_o unto_o the_o same_o line_n sir_n be_v the_o line_n cb_o a_o parallel_n by_o the_o 28._o of_o the_o first_o wherefore_o by_o the_o 9_o of_o the_o eleven_o the_o line_n we_o be_v a_o parallel_n to_o the_o line_n cb._n wherefore_o by_o the_o seven_o of_o the_o eleven_o the_o right_a line_n which_o join_v they_o together_o be_v in_o the_o self_n same_o plain_n wherein_o be_v the_o parallel_a line_n wherefore_o the_o trapesium_n buzc_n be_v in_o one_o plain_n and_o the_o triangle_n bwc_n be_v in_o one_o plain_n by_o the_o 2._o of_o the_o eleven_o now_o to_o prove_v that_o the_o trapesium_n buzc_n &_o the_o triangle_n bwc_n be_v in_o one_o and_o the_o self_n same_o plain_a we_o must_v prove_v that_o the_o right_a line_n yh_n and_o hw_o be_v make_v direct_o one_o right_a line_n which_o thing_n be_v thus_o prove_v forasmuch_o as_o the_o line_n hp_n be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n in_o the_o point_n t_o and_o his_o great_a segment_n be_v the_o line_n pt_v therefore_o as_o the_o line_n hp_n be_v to_o the_o line_n pt_v so_o be_v the_o line_n pt_v to_o the_o line_n th._n but_o the_o line_n hp_n be_v equal_a to_o the_o line_n ho_o and_o the_o line_n pt_v to_o either_o of_o these_o line_n tw_n and_o oy_o wherefore_o as_o the_o line_n ho_o be_v to_o the_o line_n oy_fw-it so_o be_v the_o line_n with_o to_o the_o line_n th._n but_o the_o line_n ho_o and_o tw_n be_v side_n of_o like_a proportion_n be_v parallel_n by_o the_o 6._o of_o the_o eleven_o for_o either_o of_o they_o be_v erect_v perpendicular_o to_o the_o plain_a superficies_n bd_o â—_o and_o the_o line_n th._n and_o oy_o be_v parallel_n which_o be_v also_o side_n of_o like_a proportion_n by_o the_o same_o 6._o of_o the_o eleven_o for_o either_o of_o they_o be_v also_o erect_v perpendicular_o to_o the_o plain_a superficies_n bf_o but_o when_o there_o be_v two_o triangle_n have_v two_o side_n proportional_a to_o two_o side_n so_o set_v upon_o one_o angle_n that_o their_o side_n of_o like_a proportion_n be_v also_o parallel_n as_o the_o triangle_n yoh_n and_o htw_n be_v â—_o who_o two_o side_n oh_o &_o ht_o be_v in_o the_o two_o base_n of_o the_o cube_fw-la make_v a_o angle_n at_o the_o point_n h_o the_o side_n remain_v of_o those_o triangle_n shall_v by_o the_o 32._o of_o the_o six_o be_v in_o one_o right_a line_n wherefore_o the_o line_n yh_n &_o hw_o make_v both_o one_o right_a line_n but_o every_o right_a line_n be_v by_o the_o 3._o of_o the_o eleven_o in_o one_o &_o the_o self_n same_o plain_a superficies_n wherefore_o if_o you_o draw_v a_o right_a line_n from_o b_o to_o a'n_z there_o shall_v be_v make_v a_o triangle_n bye_o which_o shall_v be_v in_o one_o and_o the_o self_n same_o plain_a by_o the_o 2._o of_o the_o eleven_o and_o therefore_o the_o whole_a pentagon_n figure_n vbwcz_n be_v in_o one_o and_o the_o self_n same_o plain_a superficies_n now_o also_o i_o say_v that_o it_o be_v equiangle_n equiangle_n for_o forasmuch_o as_o the_o right_a line_n no_o be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n in_o the_o point_n r_o and_o his_o great_a segment_n be_v or_o therefore_o as_o both_o the_o line_n no_o and_o or_o add_v together_o be_v to_o the_o line_n on_o so_o by_o the_o 5._o of_o this_o book_n be_v the_o line_n on_o to_o the_o line_n or_o but_o the_o line_n or_o be_v equal_a to_o the_o line_n os_fw-mi wherefore_o as_o the_o line_n sn_a be_v to_o the_o line_n no_o so_o be_v the_o line_n no_o to_o the_o line_n os_fw-mi wherefore_o the_o line_n sn_a be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n in_o the_o point_n oh_o
line_n ai._n wherefore_o the_o square_a of_o the_o line_n bw_o be_v equal_a to_o the_o square_n of_o the_o line_n ai_fw-fr wherefore_o also_o the_o line_n bt_n be_v equal_a to_o the_o line_n ai._n and_o by_o the_o same_o reason_n be_v the_o line_n id_fw-la and_o wc_n equal_a to_o the_o same_o line_n now_o forasmuch_o as_o the_o line_n ai_fw-fr and_o id_fw-la and_o the_o line_n all_o and_o ld_n be_v equal_a and_o the_o base_a il_fw-fr be_v common_a to_o they_o both_o the_o angle_n ali_n and_o dli_o shall_v be_v equal_a by_o the_o 8._o of_o the_o first_o and_o therefore_o they_o be_v right_a angle_n by_o the_o 10._o definition_n of_o the_o first_o and_o by_o the_o same_o reason_n be_v the_o angle_n whb_n and_o whc_n right_a angle_n and_o forasmuch_o as_o the_o two_o line_n ht_v and_o tw_n be_v equal_a to_o the_o two_o line_n lct_n and_o cti_n and_o they_o contain_v equal_a angle_n that_o be_v right_a angle_n by_o supposition_n therefore_o the_o angle_n wht_v and_o ilct_n be_v equal_a by_o the_o 4._o of_o the_o first_o wherefore_o the_o plain_a superficies_n aid_n be_v in_o like_a sort_n incline_v to_o the_o plain_a superficies_n abcd_o as_o the_o plain_a superficies_n bwc_n be_v incline_v to_o the_o same_o plain_n abcd_o by_o the_o 4._o definition_n of_o the_o eleven_o in_o like_a sort_n may_v we_o prove_v that_o the_o plain_n wcdi_n be_v in_o like_a sort_n incline_v to_o the_o plain_n abcd_o as_o the_o plain_a buzc_n be_v to_o the_o plain_n ebcf._n for_o that_o in_o the_o triangle_n yoh_n and_o ctpk_n which_o consist_v of_o equal_a side_n each_o to_o his_o correspondent_a side_n the_o angle_n yho_n and_o ctkp_n which_o be_v the_o angle_n of_o the_o inclination_n be_v equal_a and_o now_o if_o the_o right_a line_n ctk_n be_v extend_v to_o the_o point_n a_o and_o the_o pentagon_n cwida_n be_v make_v perfect_a we_o may_v by_o the_o same_o reason_n prove_v that_o that_o plain_n be_v equiangle_n and_o equilater_n that_o we_o prove_v the_o pentagon_n buzcw_o to_o be_v equaliter_fw-la and_o equiangle_n and_o likewise_o if_o the_o other_o plain_n bwia_fw-la and_o aid_v be_v make_v perfect_a they_o may_v be_v prove_v to_o be_v equal_a and_o like_a pentagon_n and_o in_o like_a sort_n situate_a and_o they_o be_v set_v upon_o these_o common_a right_a line_n bw_o wc_n wi_n ai_fw-fr and_o id_fw-la and_o observe_v this_o method_n there_o shall_v upon_o every_o one_o of_o the_o 12._o â—idâ—s_n of_o the_o cube_fw-la be_v set_v every_o one_o of_o the_o 12._o pentagon_n which_o compose_v the_o dodecahedron_n ¶_o certain_a corollarye_n add_v by_o flussas_n first_o corollary_n the_o side_n of_o a_o cube_fw-la be_v equal_a to_o the_o right_a line_n which_o subtend_v the_o angle_n of_o the_o pentagon_n of_o a_o dodecahedron_n contain_v in_o one_o and_o the_o self_n same_o sphere_n with_o the_o cube_fw-la for_o the_o angle_n bwc_n and_o aid_n be_v subtend_v of_o the_o line_n bc_o and_o ad._n which_o be_v side_n of_o the_o cubeâ—_n ¶_o second_v corollary_n in_o a_o dodecahedron_n there_o be_v six_o side_n every_o two_o of_o which_o be_v parallel_n and_o opposite_a who_o section_n into_o two_o equal_a part_n be_v couple_v by_o three_o right_a line_n which_o in_o the_o centre_n of_o the_o sphere_n which_o contain_v the_o dodecahedron_n divide_v into_o two_o equal_a part_n and_o perpendicular_o both_o themselves_o and_o also_o the_o side_n for_o upon_o the_o six_o base_n of_o the_o cube_fw-la be_v set_v six_o side_n of_o the_o dodecahedron_n as_o it_o have_v be_v prove_v by_o the_o line_n zv_o wi_n &c._a &c._a which_o be_v cut_v into_o two_o equal_a part_n by_o right_a line_n which_o join_v together_o the_o centre_n of_o the_o base_n of_o the_o cube_fw-la as_o the_o line_n the_fw-mi produce_v and_o the_o other_o like_a which_o line_n couple_v together_o the_o centre_n of_o the_o base_n be_v three_o in_o number_n cut_v the_o one_o the_o other_o perpendicular_o for_o they_o be_v parallel_n to_o the_o side_n of_o the_o cube_fw-la and_o they_o cut_v the_o one_o the_o other_o into_o two_o equal_a part_n in_o the_o centre_n of_o the_o sphere_n which_o contain_v the_o cube_fw-la by_o that_o which_o be_v demonstrate_v in_o the_o 15._o of_o this_o book_n and_o unto_o these_o equal_a line_n join_v together_o the_o centre_n of_o the_o base_n of_o the_o cube_fw-la be_v without_o the_o base_n add_v equal_a part_n oy_o p_o ct_v and_o the_o other_o like_a which_o by_o supposition_n be_v equal_a to_o half_n of_o the_o side_n of_o the_o dodecahedron_n wherefore_o the_o whole_a line_n which_o join_v together_o the_o sectoin_n of_o the_o opposite_a side_n of_o the_o dodecahedron_n be_v equal_a and_o they_o cut_v those_o side_n into_o two_o equal_a part_n and_o perpendicular_o three_o corollary_n a_o right_a line_n join_v together_o the_o point_n of_o the_o section_n of_o the_o opposite_a side_n of_o the_o dodecahedron_n into_o two_o equal_a part_n be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n the_o great_a segment_n thereof_o shall_v be_v the_o side_n of_o the_o cube_fw-la and_o the_o less_o segment_n the_o side_n of_o the_o dodecahedron_n contain_v in_o the_o self_n same_o sphere_n for_o it_o be_v prove_v that_o the_o right_a line_n yq_n be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n in_o the_o point_n oh_o and_o that_o his_o great_a segment_n oq_fw-la be_v half_a the_o side_n of_o the_o cube_fw-la and_o his_o less_o segment_n oy_o be_v half_o of_o the_o side_n we_o which_o be_v the_o side_n of_o the_o dodecahedron_n wherefore_o it_o follow_v by_o the_o 15._o of_o the_o five_o that_o their_o double_n be_v in_o the_o same_o proportion_n wherefore_o the_o double_a of_o the_o line_n yq_n which_o join_v the_o point_n opposite_a unto_o the_o line_n y_fw-fr be_v the_o whole_a and_o the_o great_a segment_n be_v the_o double_a of_o the_o line_n oq_fw-la which_o be_v the_o side_n of_o the_o cube_fw-la &_o the_o less_o segment_n be_v the_o double_a of_o the_o line_n the_fw-mi which_o be_v equal_a to_o the_o side_n of_o the_o dodecahedron_n namely_o to_o the_o side_n we_o ¶_o the_o 6._o problem_n the_o 18._o proposition_n to_o find_v out_o the_o side_n of_o the_o foresay_a five_o body_n and_o to_o compare_v they_o together_o take_v the_o diameter_n of_o the_o sphere_n give_v and_o let_v the_o same_o be_v ab_fw-la and_o divide_v it_o in_o the_o point_n c_o so_fw-mi that_o let_v the_o line_n ac_fw-la be_v equal_a to_o cb_o by_o the_o 10._o of_o the_o first_o and_o in_o the_o point_n d_o so_fw-mi that_o let_v ad_fw-la be_v double_a to_o db_o by_o the_o 9_o of_o the_o six_o and_o upon_o the_o line_n ab_fw-la describe_v a_o semicircle_n aeb_fw-mi and_o from_o the_o point_n c_o and_o d_o raise_v up_o by_o the_o 11._o of_o the_o first_o unto_o the_o line_n ab_fw-la perpendicular_a line_n ce_fw-fr and_o df._n and_o draw_v these_o right_a line_n of_o fb_o and_o be._n now_o forasmuch_o as_o the_o line_n ad_fw-la be_v double_a to_o the_o line_n db_o therefore_o the_o line_n ab_fw-la be_v treble_a to_o the_o line_n db._n wherefore_o the_o line_n basilius_n be_v sesquialter_fw-la to_o the_o line_n ad_fw-la for_o it_o be_v as_o 3._o to_o 2._o but_o as_o the_o line_n basilius_n be_v to_o the_o line_n ad_fw-la so_o be_v the_o square_a of_o the_o line_n basilius_n to_o the_o square_n of_o the_o line_n of_o by_o the_o 6._o of_o the_o six_o or_o by_o the_o corollary_n of_o the_o same_o and_o by_o the_o corollary_n of_o the_o 20._o of_o the_o same_o for_o the_o triangle_n afb_o be_v equiangle_n to_o the_o triangle_n afd_v wherefore_o the_o square_a of_o the_o line_n basilius_n be_v sesquialter_fw-la to_o the_o square_n of_o the_o line_n af._n but_o the_o diameter_n of_o a_o sphere_n be_v in_o power_n sesquialter_fw-la to_o the_o sidâ—_n of_o the_o pyramid_n pyramid_n by_o the_o 13._o of_o this_o book_n and_o the_o line_n ab_fw-la be_v the_o diameter_n of_o the_o sphere_n wherefore_o the_o line_n of_o be_v equal_a to_o the_o side_n of_o the_o pyramid_n again_o forasmuch_o as_o the_o line_n ab_fw-la be_v treble_a to_o the_o line_n bd_o but_o as_o the_o line_n ab_fw-la be_v to_o the_o line_n bd_o so_o be_v the_o square_a of_o the_o line_n ab_fw-la to_o the_o square_n of_o the_o line_n fb_o by_o the_o corollary_n of_o the_o 8._o and_o 20._o of_o the_o six_o wherefore_o the_o square_a of_o the_o line_n ab_fw-la be_v treble_a to_o the_o square_n of_o the_o line_n fb_o but_o the_o diameter_n oâ—_n a_o sphere_n be_v in_o power_n treble_a to_o the_o side_n of_o the_o cube_fw-la by_o the_o 15._o of_o this_o book_n and_o the_o diameter_n of_o the_o sphere_n be_v the_o line_n ab_fw-la wherefore_o the_o line_n bf_o be_v the_o side_n of_o the_o cube_fw-la cube_fw-la and_o forasmuch_o as_o the_o line_n fb_o be_v the_o
side_n of_o a_o cube_fw-la let_v it_o be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n in_o the_o point_n n_o and_o let_v the_o great_a segment_n thereof_o be_v nb._n wherefore_o the_o line_n nb_n be_v the_o side_n of_o a_o dodecahedron_n dodecahedron_n by_o the_o corollary_n of_o the_o 17._o of_o this_o book_n body_n and_o forasmuch_o as_o it_o have_v be_v prove_v by_o the_o 13._o of_o this_o book_n that_o the_o diameter_n of_o the_o sphere_n be_v in_o power_n sesquialter_fw-la to_o of_o the_o side_n of_o the_o pyramid_n and_o be_v in_o power_n double_a to_o be_v the_o side_n of_o the_o octohedron_n by_o the_o 14._o of_o the_o same_o and_o be_v in_o power_n treble_a to_o fb_o the_o side_n of_o the_o cube_fw-la by_o the_o 15._o of_o the_o same_o wherefore_o it_o follow_v that_o of_o what_o part_n the_o diameter_n of_o the_o sphere_n contain_v six_o of_o such_o part_n the_o side_n of_o the_o pyramid_n contain_v four_o and_o the_o side_n of_o the_o octohedron_n three_o and_o the_o side_n of_o the_o cube_fw-la two_o wherefore_o the_o side_n of_o the_o pyramid_n be_v in_o power_n to_o the_o side_n of_o the_o octohedron_n in_o sesquitertia_fw-la proportion_n and_o be_v in_o power_n to_o the_o side_n of_o the_o cube_fw-la in_o double_a proportion_n and_o the_o side_n of_o the_o octohedron_n be_v in_o power_n to_o the_o side_n of_o the_o cube_fw-la in_o sesquialtera_fw-la proportion_n wherefore_o the_o foresaid_a side_n of_o the_o three_o figure_n that_o be_v of_o the_o pyramid_n of_o the_o octohedron_n and_o oâ—_n the_o cube_fw-la be_v the_o one_o to_o the_o other_o in_o rational_a proportion_n wherefore_o they_o be_v rational_a but_o the_o other_o two_o side_n namely_o the_o side_n of_o the_o icosahedron_n and_o of_o the_o dodecahedron_n be_v neither_o the_o one_o to_o the_o other_o nor_o also_o to_o the_o foresay_a side_n in_o rational_a proportion_n â—or_a they_o be_v irrational_a line_n namely_o a_o less_o line_n and_o a_o residual_a line_n an_o other_o way_n to_o prove_v that_o the_o line_n mb_n be_v great_a than_o the_o line_n nb._n dodecahedron_n forasmuch_o as_o the_o line_n ad_fw-la be_v double_a to_o the_o line_n db_o therefore_o the_o line_n ab_fw-la be_v treble_a to_o the_o line_n db._n but_o as_o ab_fw-la be_v to_o bd_o so_o be_v the_o square_a of_o the_o line_n ab_fw-la to_o the_o square_n of_o the_o line_n bf_o by_o the_o 8._o of_o the_o six_o for_o the_o triangle_n fab._n be_v equiangle_n to_o the_o triangle_n fdb_n wherefore_o the_o square_a of_o the_o line_n ab_fw-la be_v treble_a to_o the_o square_n of_o the_o line_n bf_o and_o it_o be_v before_o prove_v that_o the_o square_n of_o the_o line_n ab_fw-la be_v quintuple_a to_o the_o square_n of_o the_o line_n kl_o wherefore_o five_o square_n make_v of_o the_o line_n kl_o be_v equal_a to_o three_o square_n make_v of_o the_o line_n fb_o but_o three_o square_n make_v of_o the_o line_n fb_o be_v great_a than_o six_o square_n make_v of_o the_o line_n nb_n as_o be_v straight_o way_n prove_v wherefore_o five_o square_n make_v of_o the_o line_n kl_o be_v great_a than_o six_o square_n make_v of_o the_o line_n nb._n wherefore_o also_o one_o square_a make_v of_o the_o line_n kl_o be_v great_a than_o one_o square_a make_v of_o the_o line_n nb._n wherefore_o the_o line_n kl_o be_v great_a than_o the_o line_n nb._n but_o the_o line_n kl_o be_v equal_a to_o the_o line_n lm_o wherefore_o the_o line_n lm_o be_v great_a than_o the_o line_n nb._n wherefore_o the_o line_n mb_n be_v much_o great_a than_o the_o line_n nb_n which_o be_v require_v to_o be_v prove_v but_o now_o let_v we_o prove_v that_o three_o square_n make_v of_o the_o line_n fb_o nb._n be_v great_a than_o six_o square_n make_v of_o the_o line_n nb._n forasmuch_o as_o the_o line_n bn_o be_v great_a than_o the_o line_n nf_n for_o it_o be_v the_o great_a segment_n of_o the_o line_n bf_o divide_v by_o a_o extreme_a and_o mean_a proportion_n therefore_o that_o which_o be_v contain_v under_o the_o line_n bf_o and_o bn_o be_v great_a than_o that_o which_o be_v contain_v under_o the_o line_n bf_o and_o fn_o by_o the_o 1._o of_o the_o six_o wherefore_o that_o which_o be_v contain_v under_o the_o line_n bf_o and_o bn_o together_o with_o that_o which_o be_v contain_v under_o the_o line_n bf_o and_o fn_o be_v great_a than_o that_o which_o be_v contain_v under_o the_o line_n bf_o and_o fn_o twice_o but_o that_o which_o be_v contain_v under_o the_o line_n bf_o and_o fn_o together_o with_o that_o which_o be_v contain_v under_o the_o line_n bf_o and_o bn_o be_v the_o square_a of_o the_o line_n bf_o by_o the_o 2._o of_o the_o second_o and_o that_o which_o be_v contain_v under_o the_o line_n bf_o and_o fn_o once_o be_v equal_a to_o the_o square_n of_o nb._n for_o the_o line_n fb_o be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n in_o the_o point_n n_o and_o by_o the_o 17._o of_o the_o six_o that_o which_o be_v contain_v under_o the_o extreme_n be_v equal_a to_o the_o square_n make_v of_o the_o middle_a line_n wherefore_o the_o square_a of_o the_o line_n fb_o be_v great_a than_o the_o double_a of_o the_o square_n of_o the_o line_n bn_o wherefore_o one_o of_o the_o square_n make_v of_o the_o line_n bf_o be_v great_a than_o two_o square_n make_v of_o the_o line_n bn_o wherefore_o also_o three_o square_n make_v of_o the_o line_n fb_o be_v great_a than_o six_o square_n make_v of_o the_o line_n bn_o which_o be_v require_v to_o be_v prove_v a_o corollary_n now_o also_o i_o say_v that_o beside_o the_o five_o foresay_a solid_n there_o can_v not_o be_v describe_v any_o other_o solid_a comprehend_v under_o figure_n equilater_n &_o equiangle_n the_o one_o to_o the_o other_o base_n for_o of_o two_o triangle_n or_o of_o any_o two_o other_o plain_a superficiece_n can_v not_o be_v make_v a_o solid_a angle_n for_o that_o be_v contrary_a to_o the_o definition_n of_o a_o solid_a angle_n under_o three_o triangle_n be_v contain_v the_o solid_a angle_n of_o a_o pyramid_n under_o four_o the_o solid_a angle_n of_o a_o octohedron_n under_o five_o the_o solid_a angle_n of_o a_o icosahedron_n of_o six_o equilater_n &_o equiangle_n triangle_n set_v to_o one_o point_n can_v not_o be_v make_v a_o solid_a angle_n for_o forasmuch_o as_o the_o angle_n of_o a_o equilater_n triangle_n be_v two_o three_o part_n of_o a_o right_a angle_n the_o six_o angle_n of_o the_o solid_a shall_v be_v equal_a to_o four_o right_a angle_n which_o be_v impossible_a for_o every_o solid_a angle_n be_v by_o the_o 21._o of_o the_o eleven_o contain_v under_o plain_a angle_n less_o they_o four_o right_a angle_n and_o by_o the_o same_o reason_n can_v not_o be_v make_v a_o solid_a angle_n contain_v under_o more_o them_z six_o plain_a superficial_a angle_n of_o equilater_n triangle_n under_o three_o square_n be_v contain_v the_o angle_n of_o a_o cube_fw-la under_o four_o square_n it_o be_v impossible_a that_o a_o solid_a angle_n shall_v be_v contain_v for_o then_o again_o it_o shall_v be_v contain_v under_o four_o right_a angle_n wherefore_o much_o less_o can_v any_o solid_a angle_n be_v contain_v under_o more_o square_n than_o four_o under_o three_o equilater_n and_o equiangle_n pentagon_n be_v contain_v the_o solid_a angle_n of_o a_o dodecahedron_n but_o under_o four_o it_o be_v impossible_a for_o forasmuch_o as_o the_o angle_n of_o a_o pentagon_n be_v a_o right_a angle_n and_o the_o five_o part_v more_o of_o a_o right_a angle_n the_o four_o angle_n shall_v be_v great_a than_o four_o right_a angle_n which_o be_v impossible_a and_o therefore_o much_o less_o can_v a_o solid_a angle_n be_v compose_v of_o more_o pentagon_n than_o four_o neither_o can_v a_o solid_a angle_n be_v contain_v under_o any_o other_o equilater_n and_o equiangle_n figure_n of_o many_o angle_n for_o that_o that_o also_o shall_v be_v absurd_a for_o the_o more_o the_o side_n increase_v the_o great_a be_v the_o angle_n which_o they_o contain_v and_o therefore_o the_o far_a of_o be_v the_o superficial_a angle_n contain_v of_o those_o side_n from_o compose_v of_o a_o solid_a angle_n wherefore_o beside_o the_o foresay_a five_o figure_n there_o can_v not_o be_v make_v any_o solid_a figure_n contain_v under_o equal_a side_n and_o equal_a angle_n which_o be_v require_v to_o be_v prove_v a_o assumpt_n but_o now_o that_o the_o angle_n of_o a_o equilater_n and_o equiangle_n pentagon_n be_v a_o right_a angle_n and_o a_o fiâ—th_v parâ—_n more_o of_o a_o right_a angle_n may_v thus_o be_v prove_v suppose_v that_o abcde_v be_v a_o equilater_n and_o equiangle_n pentagon_n â—irst_n and_o by_o the_o 14._o of_o the_o four_o describe_v about_o it_o a_o circle_n abcde_o and_o take_v by_o the_o 1._o of_o the_o three_o the_o centre_n thereof_o and_o let_v the_o same_o be_v
of_o the_o line_n be_v be_v quadruple_a to_o the_o square_n of_o de_fw-fr by_o the_o 20._o of_o the_o six_o but_o unto_o the_o square_n of_o the_o line_n be_v be_v equal_a the_o square_n of_o the_o line_n basilius_n and_o ae_n by_o the_o 47._o of_o the_o first_o for_o the_o angle_n bae_o be_v a_o right_a angle_n by_o the_o 31._o of_o the_o three_o wherefore_o the_o square_n of_o the_o line_n basilius_n and_o ae_n be_v quadruple_a to_o the_o square_n of_o the_o line_n de._n wherefore_o the_o square_n of_o the_o line_n ab_fw-la ae_n and_o de_fw-fr be_v quintuple_a to_o the_o square_n of_o the_o line_n de._n but_o the_o square_n of_o the_o line_n de_fw-fr and_o ae_n be_v equal_a to_o the_o square_n of_o the_o line_n ac_fw-la by_o the_o 10._o of_o the_o thirteen_o wherefore_o the_o square_n of_o the_o line_n basilius_n and_o ac_fw-la be_v quintuple_a to_o the_o square_n of_o the_o line_n de._n this_o be_v thus_o prove_v now_o be_v to_o be_v demonstrate_v that_o one_o and_o the_o self_n same_o circle_n comprehend_v both_o the_o pentagon_n of_o a_o dodecahedron_n &_o the_o triangle_n of_o a_o icosahedron_n describe_v in_o one_o &_o the_o self_n same_o circled_a proposition_n take_v the_o diameter_n of_o the_o sphere_n &_o â—_o let_v the_o same_o be_v ab_fw-la and_o in_o the_o same_o sphere_n describe_v a_o dodecahedron_n &_o also_o a_o icosahedron_n and_o let_v one_o of_o the_o pentagon_n of_o the_o dodecahedron_n be_v cdefg_n &_o let_v one_o of_o the_o triangle_n of_o the_o icosahedron_n be_v klh_n now_o i_o say_v that_o the_o semidiameter_n of_o the_o circle_n which_o be_v describe_v about_o they_o be_v equal_a that_o be_v that_o one_o and_o the_o self_n same_o circle_n contain_v both_o the_o pentagon_n cdefg_n and_o the_o triangle_n klh_n draw_v a_o right_a line_n from_o the_o point_n d_o to_o the_o point_n g._n wherefore_o the_o line_n dg_o be_v the_o side_n of_o a_o cube_fw-la by_o the_o corollary_n of_o the_o 17._o of_o the_o thirteen_o take_v a_o certain_a right_a line_n mn_o and_o let_v the_o square_n of_o the_o line_n ab_fw-la be_v quintuple_a to_o the_o square_n of_o the_o line_n mn_o by_o the_o assumpt_n put_v after_o the_o 6._o proposition_n of_o the_o ten_o but_o the_o diameter_n of_o a_o sphere_n be_v in_o power_n quintuple_a to_o the_o square_n of_o the_o semidiameter_n of_o the_o circle_n on_o which_o be_v describe_v the_o icosahedron_n by_o the_o corollary_n of_o the_o 16._o of_o the_o thirteen_o wherefore_o the_o line_n mn_o be_v the_o semidiameter_n of_o the_o circle_n on_o which_o be_v describe_v the_o icosahedron_n divide_v by_o the_o 30._o of_o the_o six_o the_o line_n mn_v by_o a_o extreme_a and_o mean_a proportion_n in_o the_o point_n x._o and_o let_v the_o great_a segment_n thereof_o be_v mx_o wherefore_o the_o line_n mx_o be_v the_o side_n of_o a_o decagon_n describe_v in_o the_o same_o circle_n by_o the_o corollary_n of_o the_o 9_o of_o the_o thirteen_o proposition_n and_o forasmuch_o as_o the_o square_n of_o the_o line_n ab_fw-la be_v quintuple_a to_o the_o square_n of_o the_o line_n mn_v but_o the_o square_n of_o the_o line_n basilius_n be_v treble_a to_o the_o square_n of_o the_o line_n dg_o by_o the_o corollary_n of_o the_o 15_o of_o the_o thirteen_o wherefore_o three_o square_n of_o the_o line_n dg_o be_v equal_a to_o five_o square_n of_o the_o line_n mn_v proposition_n but_o as_o three_o square_n of_o the_o line_n dg_o be_v to_o â—ive_a square_n of_o the_o line_n mn_o so_o be_v three_o square_n of_o the_o line_n cg_o to_o five_o square_n of_o the_o line_n mx_o wherefore_o three_o square_n of_o the_o line_n cg_o be_v equll_a to_o five_o square_n of_o the_o line_n mx_o but_o five_o square_n of_o the_o line_n cg_o be_v equal_a to_o â—ive_a square_n of_o the_o line_n mn_a &_o to_o five_o square_n of_o the_o mx_o for_o by_o the_o 10._o of_o the_o thirteen_o one_o square_a of_o the_o line_n cg_o be_v equal_a to_o one_o square_a of_o the_o line_n mn_a &_o to_o one_o square_a of_o the_o line_n mx_o wherefore_o five_o square_n of_o the_o line_n cg_o be_v equal_a to_o three_o square_n of_o the_o line_n dg_o and_o to_o three_o square_n of_o the_o line_n cg_o as_o it_o be_v not_o hard_a to_o prove_v mark_v what_o have_v before_o be_v prove_v but_o three_o square_n of_o the_o line_n dg_o together_o with_o three_o square_n of_o the_o line_n cg_o be_v equal_a to_o fifteen_o square_n of_o the_o semidiameter_n of_o the_o circle_n describe_v about_o the_o pentagon_n cdefg_n for_o it_o be_v before_o prove_v in_o the_o assumpt_n put_v in_o this_o proposition_n that_o the_o squareâ—_n of_o dg_o and_o g_o c_o take_v once_o be_v quintuple_a to_o the_o square_n of_o the_o semidiameter_n of_o the_o circle_n describe_v about_o the_o pentagon_n cdefg_n and_o five_o square_n of_o the_o line_n kl_o be_v equal_a to_o fifteen_o square_n of_o the_o semidiameter_n of_o the_o circle_n describe_v about_o the_o triangle_n klh_n for_o by_o the_o 12._o of_o the_o thirteen_o one_o square_a of_o the_o line_n lk_n be_v triple_a to_o one_o square_a of_o the_o line_n draw_v from_o the_o centre_n to_o the_o circumference_n wherefore_o fifteen_o square_n of_o the_o line_n draw_v from_o the_o centre_n to_o the_o circumference_n of_o the_o circle_n which_o contain_v the_o pentagon_n cdefg_n be_v equal_a to_o fifteen_o square_n of_o the_o line_n draw_v from_o the_o centre_n to_o the_o circumference_n of_o the_o circle_n which_o contain_v the_o triangle_n klh_n wherefore_o one_o of_o the_o square_n which_o be_v draw_v from_o the_o centre_n to_o the_o circumference_n of_o the_o one_o circle_n be_v equal_a to_o one_o of_o the_o square_n which_o be_v draw_v from_o the_o centre_n to_o the_o circumference_n of_o the_o other_o circle_n wherefore_o the_o diameter_n be_v equal_a to_o the_o diameter_n wherefore_o one_o and_o the_o self_n same_o circle_n comprehend_v both_o the_o pentagon_n of_o a_o dodecahedron_n and_o the_o triangle_n of_o a_o icosahedron_n describe_v in_o one_o and_o the_o self_n same_o circle_n which_o be_v require_v to_o be_v prove_v ¶_o the_o 3._o theorem_a the_o .3_o proposition_n 〈◊〉_d if_o there_o be_v a_o equilater_n and_o equiangle_n pentagon_n and_o about_o it_o be_v describe_v a_o circle_n and_o from_o the_o centre_n to_o one_o of_o the_o side_n be_v draw_v a_o perpendicular_a line_n that_o which_o be_v contain_v under_o one_o of_o the_o side_n and_o the_o perpendicular_a line_n thirty_o time_n be_v equal_a to_o the_o superficies_n of_o the_o dodecahedron_n svppose_v that_o abcd_o be_v a_o equilater_n and_o equiangle_n pentagon_n and_o about_o the_o same_o pentagon_n describe_v by_o the_o 14._o of_o the_o four_o a_o circle_n construction_n and_o let_v the_o centre_n thereof_o be_v the_o point_n f._n and_o from_o the_o point_n f_o draw_v by_o the_o 12._o of_o the_o first_o unto_o the_o line_n cd_o a_o perpendicular_a line_n fg._n now_o i_o say_v that_o that_o which_o be_v contain_v under_o the_o line_n cd_o and_o gf_o thirty_o time_n be_v equal_a to_o 12._o pentagon_n of_o the_o same_o quantity_n that_o the_o pentagon_n abcd_o be_v draw_v these_o right_a line_n cf_o and_o fd._n demonstration_n now_o forasmuch_o as_o that_o which_o be_v contain_v under_o the_o line_n cd_o and_o fg_o be_v double_a to_o the_o triangle_n cdf_n by_o the_o 41._o of_o the_o first_o therefore_o that_o which_o be_v contain_v under_o the_o line_n cd_o and_o fg_o five_o time_n be_v equal_a to_o ten_o of_o those_o triangle_n but_o ten_o of_o those_o triangle_n be_v two_o pentagon_n and_o six_o time_n ten_o of_o those_o triangle_n be_v all_o the_o pentagon_n wherefore_o that_o which_o be_v contain_v under_o the_o line_n cd_o and_o fg_o thirty_o time_n be_v equal_a to_o 12._o pentagon_n but_o 12._o pentagon_n be_v the_o superficies_n of_o dodecahedron_n wherefore_o that_o which_o be_v contain_v under_o the_o line_n cd_o and_o fg_o thirty_o time_n be_v equal_a to_o the_o superficies_n of_o the_o dodecahedron_n in_o like_a sort_n also_o may_v we_o prove_v that_o if_o there_o be_v a_o equilater_n triangle_n as_o for_o example_n fâ—ussas_n the_o triangle_n abc_n and_o about_o it_o be_v describe_v a_o circle_n and_o the_o centre_n of_o the_o circle_n be_v the_o point_n d_o and_o the_o perpendicular_a line_n be_v the_o line_n de_fw-fr that_o which_o be_v contain_v under_o the_o line_n bc_o and_o de_fw-fr thirty_o time_n be_v equal_a to_o the_o superficies_n of_o the_o icosahedron_n demonstration_n for_o again_o forasmuch_o as_o that_o which_o be_v contain_v under_o the_o line_n de_fw-fr and_o bc_o be_v double_a to_o the_o triangle_n dbc_n by_o the_o 41._o of_o the_o first_o therefore_o two_o triangle_n be_v equal_a to_o that_o which_o be_v contain_v under_o the_o line_n de_fw-fr and_o bc_o and_o three_o of_o those_o triangle_n contain_v the_o whole_a
of_o and_o gh_o once_o take_v each_o of_o those_o parallelogram_n five_o time_n wherefore_o that_o which_o be_v contain_v under_o the_o line_n of_o and_o hc_n ten_o time_n be_v equal_a to_o that_o which_o be_v contain_v under_o the_o line_n of_o &_o gh_o five_o time_n that_o be_v to_o two_o pentagon_n wherefore_o that_o which_o be_v contain_v under_o the_o line_n of_o and_o hc_n five_a time_n be_v equal_a to_o one_o pentagon_n but_o that_o which_o be_v contain_v under_o the_o line_n of_o and_o hc_n five_a time_n be_v equal_a by_o the_o 1._o of_o the_o six_o to_o that_o which_o be_v contain_v under_o the_o line_n of_o and_o hb_o for_o the_o line_n hb_o be_v quintuple_a to_o the_o line_n hc_n as_o it_o be_v easy_a to_o see_v by_o the_o construction_n and_o they_o be_v both_o under_o one_o &_o the_o self_n same_o altitude_n namely_o under_o af._n wherefore_o that_o which_o be_v contain_v under_o the_o line_n of_o and_o bh_o be_v equal_a to_o one_o pentagon_n this_o be_v prove_v now_o let_v there_o be_v draw_v a_o circle_n comprehend_v both_o the_o pentagon_n of_o a_o dodecahedron_n and_o the_o triangle_n of_o a_o icosahedron_n be_v both_o describe_v in_o one_o and_o the_o self_n same_o sphere_n proposition_n let_v the_o circle_n be_v abc_n and_o in_o it_o describe_v as_o before_o two_o side_n of_o a_o equilater_n pentagon_n namely_o basilius_n and_o acâ—_n and_o draw_v a_o right_a line_n from_o the_o point_n b_o to_o the_o point_n c_o and_o take_v the_o centre_n of_o the_o circle_n and_o let_v the_o same_o be_v e._n and_o from_o the_o point_n a_o to_o the_o point_n e_o draw_v a_o right_a line_n ae_n and_o extend_v the_o line_n ae_n to_o the_o point_n f._n and_o let_v it_o cut_v the_o line_n bc_o in_o the_o point_n k._n and_o let_v the_o line_n ae_n be_v double_a to_o the_o line_n eglantine_n &_o let_v the_o line_n ck_o be_v treble_a to_o the_o line_n change_z by_o the_o .9_o of_o the_o six_o and_o from_o the_o point_n g_o raise_v up_o by_o the_o .11_o of_o the_o first_o unto_o the_o line_n of_o a_o perpendicular_a line_n gm_n and_o extend_v the_o line_n gm_n direct_o to_o the_o point_n d._n wherefore_o the_o line_n md_o be_v the_o side_n of_o a_o equilite_a triangle_n by_o the_o corollary_n of_o the_o 1â—_o of_o the_o thirteen_o draw_v these_o right_a line_n ad_fw-la and_o am._n wherefore_o adm_n be_v a_o equilater_n triangle_n proposition_n and_o for_o as_o much_o as_o that_o which_o be_v contain_v under_o the_o line_n agnostus_n and_o bh_o be_v equal_a to_o the_o pentagon_n by_o the_o former_a assumpt_n and_o that_o which_o be_v contain_v under_o the_o line_n agnostus_n and_o gd_o be_v equal_a to_o the_o triangle_n adm_n therefore_o as_o that_o which_o be_v contain_v under_o the_o line_n agnostus_n and_o hb_o be_v to_o that_o which_o be_v contain_v under_o the_o line_n agnostus_n and_o gd_o so_o be_v the_o pentagon_n to_o the_o triangle_n but_o as_o that_o which_o be_v contain_v under_o the_o line_n bh_o &_o agnostus_n be_v to_o that_o which_o be_v contain_v under_o the_o line_n agnostus_n and_o gd_o so_o be_v the_o line_n bh_o to_o the_o line_n dg_o by_o the_o .1_o of_o the_o six_o wherefore_o by_o the_o .15_o of_o the_o five_o as_o 12._o such_o line_n as_o bh_o be_v be_v to_o .20_o such_o line_n as_o dg_o be_v so_o be_v 12._o pentagon_n to_o 20._o triangle_n that_o be_v the_o superficies_n of_o the_o dodecahedron_n to_o the_o superficies_n of_o the_o icosahedron_n and_o 12._o such_o line_n as_o bh_o be_v be_v equal_a to_o ten_o such_o line_n as_o bc_o be_v for_o the_o line_n hb_o be_v quintuple_a to_o the_o line_n hc_n and_o the_o line_n bc_o be_v sextuple_a to_o the_o line_n châ—_n wherefore_o six_o such_o line_n as_o bh_o be_v be_v equal_a to_o five_o such_o line_n as_o bc_o be_v and_o in_o the_o same_o proportion_n be_v their_o double_n and_o 20._o such_o line_n as_o the_o line_n dg_o be_v be_v equal_a to_o .10_o such_o line_n as_o the_o line_n dm_o be_v for_o the_o line_n dm_o be_v double_a to_o the_o line_n dg_o wherefore_o as_o 10._o such_o line_n as_o bc_o be_v be_v to_o 10._o such_o line_n as_o dm_o be_v that_o be_v as_o the_o line_n bc_o be_v to_o the_o line_n dm_o so_o be_v the_o superficies_n of_o the_o dodecahedron_n to_o the_o superficies_n of_o the_o icosahedron_n but_o the_o line_n bc_o be_v the_o side_n of_o the_o cube_fw-la and_o the_o line_n dm_o the_o side_n of_o the_o icosahedron_n wherefore_o by_o the_o 11._o of_o the_o five_o as_o the_o superficies_n of_o the_o dodecahedron_n be_v to_o the_o superficies_n of_o the_o icosahedron_n so_o be_v the_o line_n bc_o to_o the_o line_n dm_o that_o be_v the_o side_n of_o the_o cube_fw-la to_o the_o side_n of_o the_o icosahedron_n flussas_n now_o will_v we_o prove_v that_o a_o right_a line_n be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n what_o proportion_n the_o line_n contain_v in_o power_n the_o square_n of_o the_o whole_a line_n and_o of_o the_o great_a segment_n have_v to_o the_o line_n contain_v in_o power_n the_o square_n of_o the_o whole_a line_n and_o of_o the_o less_o segment_n the_o same_o proportion_n have_v the_o side_n of_o the_o cube_fw-la to_o the_o side_n of_o the_o icosahedron_n be_v both_o describe_v in_o one_o and_o the_o self_n same_o sphere_n construction_n svppose_v that_o ab_fw-la be_v a_o circle_n contain_v both_o the_o pentagon_n of_o a_o dodecahedron_n &_o the_o triangle_n of_o a_o icosahedron_n describe_v both_o in_o one_o and_o the_o self_n same_o sphere_n take_v the_o centre_n of_o the_o circle_n and_o let_v the_o same_o be_v c._n and_o from_o the_o point_n c_o extend_v to_o the_o circumference_n a_o right_a line_n at_o all_o aventure_n and_o let_v the_o same_o bc._n and_o by_o the_o 30._o of_o the_o six_o divide_v the_o line_n bc_o by_o a_o extreme_a and_o mean_a proportion_n in_o the_o point_n d_o and_o let_v the_o great_a segment_n thereof_o be_v cd_o wherefore_o the_o line_n cd_o be_v the_o side_n of_o a_o decagon_n describe_v in_o the_o same_o circle_n by_o the_o corollary_n of_o the_o 9_o of_o the_o thirteen_o take_v the_o side_n of_o a_o icosahedron_n and_o let_v the_o same_o be_v the_o line_n e_o and_o the_o side_n of_o a_o dodecahedron_n and_o let_v the_o same_o be_v the_o line_n f_o and_o the_o side_n of_o a_o cube_fw-la &_o let_v the_o same_o be_v the_o line_n g._n demonstration_n wherefore_o the_o line_n e_o be_v the_o side_n of_o a_o equilater_n triangle_n and_o fletcher_n of_o a_o equaliter_fw-la pentagon_n describe_v in_o one_o and_o the_o self_n same_o circle_n and_o the_o line_n g_z be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n his_o great_a segment_n be_v the_o line_n f_o by_o the_o corollary_n of_o the_o 17._o of_o the_o thirteen_o now_o forasmuch_o as_o the_o line_n e_o be_v the_o side_n of_o a_o equilater_n triangle_n but_o by_o the_o 12._o of_o the_o thirteen_o the_o side_n of_o a_o equilater_n triangle_n be_v in_o power_n treble_a to_o the_o line_n bc_o which_o be_v draw_v from_o the_o centre_n to_o the_o circumference_n therefore_o the_o square_n of_o the_o line_n e_o be_v treble_a to_o the_o square_n of_o the_o line_n bc_o but_o the_o square_n of_o the_o line_n bc_o and_o bd_o be_v by_o the_o 4._o of_o the_o thirteen_o treble_v to_o the_o square_n of_o the_o line_n cd_o wherefore_o as_o the_o square_n of_o the_o line_n e_o be_v to_o the_o square_n of_o the_o line_n cb_o so_o be_v the_o square_n of_o the_o line_n cb_o and_o bd_o to_o the_o square_n of_o the_o line_n cd_o wherefore_o alternate_o by_o the_o 16._o of_o the_o five_o as_o the_o square_n of_o the_o line_n e_o be_v to_o the_o square_n of_o the_o line_n cb_o and_o bd_o so_o be_v the_o square_a of_o the_o line_n cb_o to_o the_o square_n of_o the_o line_n cd_o proposition_n but_o as_o the_o square_n of_o the_o line_n bc_o be_v to_o the_o square_n of_o the_o line_n cd_o so_o be_v the_o square_a of_o the_o line_n g_z the_o side_z of_o the_o cube_fw-la to_o the_o square_n of_o the_o line_n f_o the_o side_n of_o dodecahedron_n for_o the_o line_n f_o be_v the_o great_a segment_n of_o the_o line_n g_z as_o be_v before_o prove_v wherefore_o by_o the_o .11_o of_o the_o five_o as_o the_o square_n of_o the_o line_n e_o be_v to_o the_o square_n cb_o and_o bd_o so_o be_v the_o square_a of_o the_o line_n g_o to_o the_o square_n of_o the_o line_n f._n wherefore_o alternate_o by_o the_o 16._o of_o the_o five_o &_o also_o by_o conversion_n by_o the_o corollary_n of_o the_o 4._o of_o the_o five_o as_o the_o square_n of_o the_o line_n g_o be_v to_o the_o square_n of_o the_o line_n e_o so_o be_v the_o square_a of_o the_o line_n f_o
to_o the_o square_n of_o the_o line_n cb_o &_o bd._n but_o unto_o the_o square_n of_o the_o line_n f_o be_v equal_a the_o square_n of_o the_o line_n bc_o &_o cd_o for_o the_o side_n of_o a_o pentagon_n contain_v in_o power_n both_o the_o side_n of_o a_o six_o angle_a figure_n and_o the_o side_n of_o a_o ten_o angle_a figure_n by_o the_o 10._o of_o the_o thirteen_o wherefore_o as_o the_o square_n of_o the_o line_n g_o be_v to_o the_o square_n of_o the_o line_n e_o so_fw-mi be_v the_o square_n of_o the_o line_n bc_o and_o cd_o to_o the_o square_n of_o the_o line_n cb_o and_o bd._n but_o as_o the_o square_n of_o the_o line_n cb_o and_o cd_o be_v to_o the_o square_n of_o the_o line_n cb_o &_o bd_o prove_v so_o any_o right_a line_n what_o so_o ever_o it_o be_v be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n be_v the_o line_n contain_v in_o power_n the_o square_n of_o the_o whole_a line_n and_o of_o the_o great_a segment_n to_o the_o line_n contain_v in_o power_n the_o square_n of_o the_o whole_a line_n and_o of_o the_o less_o segment_n wherefore_o by_o the_o 11._o of_o the_o five_o as_o the_o square_n of_o the_o line_n g_z the_o side_z of_o the_o cube_fw-la be_v to_o the_o square_n of_o the_o line_n e_o so_o any_o right_a line_n be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n be_v the_o line_n contain_v in_o power_n the_o square_n make_v of_o the_o whole_a line_n and_o of_o the_o great_a segment_n to_o the_o line_n contain_v in_o power_n the_o square_n make_v of_o the_o whole_a line_n and_o of_o the_o less_o segment_n but_o the_o line_n g_o be_v the_o side_n of_o the_o cube_n and_o the_o line_n e_o of_o the_o icosahedron_n by_o supposition_n if_o therefore_o a_o right_a line_n be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n as_o the_o line_n contain_v in_o power_n the_o square_n of_o the_o whole_a line_n and_o of_o the_o great_a segment_n be_v to_o the_o line_n contain_v in_o power_n the_o square_n of_o the_o whole_a line_n and_o of_o the_o less_o segment_n so_o be_v the_o side_n of_o the_o cube_fw-la to_o the_o side_n of_o the_o icosahedron_n be_v both_o describe_v in_o one_o and_o the_o self_n same_o sphere_n now_o will_v we_o prove_v that_o as_o the_o side_n of_o the_o cube_n be_v to_o the_o side_n of_o the_o icosahedron_n flussas_n so_o be_v the_o solid_a of_o the_o dodecahedron_n to_o the_o solid_a of_o the_o icosahedron_n forasmuch_o as_o equal_a circle_n comprehend_v both_o the_o pentagon_n of_o a_o dodecahedron_n and_o the_o triangle_n of_o a_o icosahedron_n be_v both_o describe_v in_o one_o and_o the_o self_n same_o sphere_n by_o the_o 2._o of_o this_o book_n but_o in_o a_o sphere_n equal_a circle_n be_v equal_o distant_a from_o the_o centre_n for_o the_o perpendicular_a line_n draw_v from_o the_o centre_n of_o the_o sphere_n to_o the_o plain_a superficiece_n of_o the_o circle_n be_v equal_a and_o do_v fall_v upon_o the_o centre_n of_o the_o circle_n book_n wherefore_o perpendicular_a line_n draw_v from_o the_o centre_n of_o the_o sphere_n to_o the_o centre_n of_o the_o circle_n comprehend_v both_o the_o triangle_n of_o a_o icosahedron_n and_o the_o pentagon_n of_o a_o dodecahedron_n be_v equal_a wherefore_o the_o pyramid_n who_o base_n be_v the_o pentagon_n of_o the_o dodecahedron_n be_v of_o equal_a altitude_n with_o the_o pyramid_n who_o base_n be_v the_o triangle_n of_o the_o icosahedron_n but_o pyramid_n of_o equal_a altitude_n be_v in_o that_o proportion_n the_o one_o to_o the_o other_o that_o their_o base_n be_v by_o the_o 5._o of_o the_o twelve_o wherefore_o as_o the_o pentagon_n be_v to_o the_o triangle_n so_o be_v the_o pyramid_n who_o base_a be_v the_o pentagon_n of_o the_o dodecahedron_n and_o top_n the_o centre_n of_o the_o sphere_n to_o the_o pyramid_n who_o base_a be_v the_o triangle_n and_o top_n the_o centre_n also_o of_o the_o sphere_n wherefore_o by_o the_o 15._o of_o the_o five_o as_o 12._o pentagon_n be_v to_o 20._o triangle_n so_o be_v 12._o pyramid_n have_v pentagon_n to_o their_o base_n to_o 20._o pyramid_n have_v triangle_n to_o their_o base_n but_o 12._o pentagon_n be_v the_o superficies_n of_o the_o dâ—decahedron_n and_o 20._o triangle_n be_v the_o superficies_n of_o the_o icosahedron_n wherefore_o as_o the_o superficies_n of_o the_o dodecahedron_n be_v to_o the_o superficies_n of_o the_o icosahedron_n so_o be_v 12._o pyramid_n have_v pentagon_n to_o their_o base_n to_o 20._o pyramid_n have_v triangle_n to_o their_o base_n but_o 12._o pyramid_n have_v pentagon_n to_o their_o base_n be_v the_o solid_a of_o the_o dodecahedron_n and_o 20._o pyramid_n have_v triangle_n to_o their_o base_n be_v the_o solid_a oâ—_n the_o icosahedron_n flussas_n wherefore_o by_o the_o 11._o of_o the_o fifthe_o as_o the_o superficies_n of_o the_o dodecahedron_n be_v to_o the_o superficies_n of_o the_o icosahedron_n â—o_o be_v the_o solid_a of_o the_o dodecahedron_n to_o the_o solid_a of_o the_o icosahedron_n but_o as_o the_o superficies_n of_o the_o dodecahedron_n be_v to_o the_o superficies_n of_o the_o icosahedron_n so_o have_v we_o prove_v that_o the_o side_n oâ—_n the_o cube_fw-la be_v to_o the_o side_n of_o the_o icosahedron_n wherefore_o by_o the_o 11._o of_o the_o five_o as_o the_o side_n of_o the_o cube_fw-la be_v to_o the_o side_n of_o the_o icosahedron_n so_o be_v the_o solid_a of_o the_o dodecahedron_n to_o the_o solid_a of_o the_o icosahedron_n note_v if_o two_o right_a line_n be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n they_o shall_v every_o way_n be_v in_o like_a proportion_n which_o thing_n be_v thus_o demonstrate_v let_v the_o line_n ab_fw-la be_v by_o the_o 30._o of_o the_o six_o divide_a by_o a_o extreme_a and_o mean_a proportion_n in_o the_o point_n c_o and_o let_v the_o great_a segment_n thereof_o be_v the_o line_n ca._n and_o likewise_o also_o let_v the_o line_n de_fw-fr be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n in_o the_o point_n fletcher_n and_o let_v the_o great_a segment_n thereof_o be_v the_o line_n df._n then_o i_o say_v that_o as_o the_o whole_a line_n ab_fw-la be_v to_o the_o great_a segment_n thereof_o ac_fw-la so_o be_v the_o whole_a line_n de_fw-fr to_o the_o great_a segment_n thereof_o df._n for_o forasmuch_o as_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o be_v equal_a to_o the_o square_n of_o the_o line_n ac_fw-la by_o the_o definition_n of_o a_o line_n divide_v be_v a_o extreme_a and_o mean_a proportion_n demonstration_n and_o that_o which_o be_v contain_v under_o the_o line_n de_fw-fr and_o of_o be_v also_o equal_a to_o the_o square_n of_o the_o line_n df_o by_o the_o same_o definition_n therefore_o as_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o be_v to_o the_o square_n of_o the_o line_n ac_fw-la so_o be_v that_o which_o be_v contain_v under_o the_o line_n de_fw-fr and_o of_o to_o the_o square_n of_o the_o line_n df._n for_o in_o each_o be_v the_o proportion_n of_o equality_n wherefore_o as_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o four_o time_n be_v to_o the_o square_n of_o the_o line_n ac_fw-la so_o be_v that_o which_o be_v contain_v under_o the_o line_n de_fw-fr and_o of_o four_o time_n to_o the_o square_n of_o the_o line_n df_o by_o the_o 15._o of_o the_o five_o wherefore_o by_o composition_n by_o the_o 18._o of_o the_o â—iâ—th_n as_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o four_o time_n together_o with_o the_o square_n of_o the_o line_n ac_fw-la be_v to_o the_o square_n of_o the_o line_n ac_fw-la so_o be_v that_o which_o be_v contain_v under_o the_o line_n de_fw-fr and_o of_o four_o time_n together_o with_o the_o square_n of_o the_o line_n df_o to_o the_o square_n of_o the_o line_n df._n wherefore_o as_o the_o square_n which_o be_v make_v of_o the_o line_n ab_fw-la and_o bc_o add_v together_o and_o make_v one_o line_n which_o square_a by_o the_o 8._o of_o the_o second_o be_v equal_a to_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o four_o time_n together_o with_o square_n of_o the_o line_n ac_fw-la be_v to_o the_o square_n of_o the_o line_n ac_fw-la so_o be_v the_o square_n make_v of_o the_o line_n de_fw-fr &_o of_o add_v together_o and_o make_v one_o line_n which_o square_a be_v also_o by_o the_o same_o equal_a to_o that_o which_o be_v contain_v under_o the_o line_n de_fw-fr and_o of_o four_o time_n together_o with_o the_o square_n of_o the_o line_n df_o to_o the_o square_n of_o the_o line_n df._n wherefore_o also_o as_o the_o line_n ab_fw-la &_o bc_o add_v together_o be_v to_o the_o line_n ac_fw-la so_o be_v the_o line_n de_fw-fr &_o of_o add_v together_o to_o the_o line_n df_o by_o the_o
22._o of_o the_o six_o wherefore_o by_o composition_n by_o the_o 18._o of_o the_o five_o as_o both_o the_o line_n ab_fw-la &_o bc_o add_v the_o one_o to_o the_o other_o together_o with_o the_o line_n ac_fw-la that_o be_v as_z two_z such_o line_n as_o ab_fw-la be_v be_v to_o the_o line_n ac_fw-la so_o be_v both_o the_o line_n de_fw-fr and_o of_o add_v the_o one_o to_o the_o other_o together_o with_o the_o line_n df_o that_o be_v two_o such_o line_n as_o de_fw-fr be_v to_o the_o line_n df._n and_o in_o the_o same_o proportion_n be_v the_o half_n of_o the_o antecedent_n by_o the_o 15._o of_o the_o five_o wherefore_o as_o the_o line_n ab_fw-la be_v to_o the_o line_n ac_fw-la so_o be_v the_o line_n de_fw-fr to_o the_o line_n df._n and_o therefore_o by_o the_o 19_o of_o the_o five_o as_o the_o line_n ab_fw-la be_v to_o the_o line_n bc_o so_o be_v the_o line_n df_o to_o the_o line_n fe_o wherefore_o also_o by_o division_n by_o the_o 17._o of_o the_o five_o as_o the_o line_n ac_fw-la be_v to_o the_o line_n cb_o so_o be_v the_o line_n df_o to_o the_o line_n de_fw-fr now_o that_o we_o have_v prove_v proposition_n that_o any_o right_a line_n whatsoever_o be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n what_o proportion_n the_o line_n contain_v in_o power_n the_o square_n make_v of_o the_o whole_a line_n and_o of_o the_o great_a segment_n add_v together_o have_v to_o the_o line_n contain_v in_o power_n the_o square_n make_v of_o the_o whole_a line_n and_o of_o the_o less_o segment_n add_v together_o the_o same_o proportion_n have_v the_o side_n of_o the_o cube_fw-la to_o the_o side_n of_o the_o icosahedron_n now_o also_o that_o we_o have_v prove_v proposition_n that_o as_o the_o side_n of_o the_o cube_fw-la be_v to_o the_o side_n of_o the_o icosahedron_n so_o be_v the_o superficies_n of_o the_o dodecahedron_n to_o the_o superficies_n of_o the_o icosahedron_n be_v both_o describe_v in_o one_o and_o the_o self_n same_o sphere_n and_o moreover_o see_v that_o we_o have_v prove_v proposition_n that_o as_o the_o superficies_n of_o the_o dodecahedron_n be_v to_o the_o superficies_n of_o the_o icosahedron_n so_o be_v the_o dodecahedron_n to_o the_o icosahedron_n for_o that_o both_o the_o pentagon_n of_o the_o dodecahedron_n and_o the_o triangle_n of_o the_o icosahedron_n be_v comprehend_v in_o one_o and_o the_o self_n same_o circle_n corollary_n all_o these_o thing_n i_o say_v be_v prove_v it_o be_v manifest_a that_o if_o in_o one_o and_o the_o self_n same_o sphere_n be_v describe_v a_o dodecahedron_n and_o a_o icosahedron_n they_o shall_v be_v in_o proportion_n the_o one_o to_o the_o other_o as_z a_o right_a line_n whatsoever_o be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n the_o line_n contain_v in_o power_n the_o square_n of_o the_o whole_a line_n and_o of_o the_o great_a segment_n add_v together_o be_v to_o the_o line_n contain_v in_o power_n the_o square_n of_o the_o whole_a line_n and_o of_o the_o less_o segment_n add_v together_o for_o for_o that_o as_o the_o dodecahedron_n be_v to_o the_o icosahedron_n so_o be_v the_o superficies_n of_o the_o dodecahedron_n to_o the_o superficies_n of_o the_o icosahedron_n that_o be_v the_o side_n of_o the_o cube_fw-la to_o the_o side_n of_o the_o icosahedron_n but_o as_o the_o side_n of_o the_o cube_fw-la be_v to_o the_o side_n of_o the_o icosahedron_n so_o any_o right_a line_n what_o so_o ever_o be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n be_v the_o line_n contain_v in_o power_n the_o square_n of_o the_o whole_a line_n and_o of_o the_o great_a segment_n add_v together_o to_o the_o line_n contain_v in_o power_n the_o square_n of_o the_o whole_a line_n and_o of_o the_o less_o segment_n add_v together_o wherefore_o as_o a_o dodecahedron_n be_v to_o a_o icosahedron_n describe_v in_o one_o and_o the_o self_n same_o sphere_n so_o any_o right_a line_n what_o so_o ever_o be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n be_v the_o line_n contain_v in_o power_n the_o square_n of_o the_o whole_a line_n &_o of_o the_o great_a segment_n add_v together_o to_o the_o line_n contain_v in_o power_n the_o square_n of_o the_o whole_a line_n and_o of_o the_o less_o segment_n add_v together_o the_o end_n of_o the_o fourteen_o book_n of_o euclides_n element_n after_o hypsicles_n ¶_o the_o fourteen_o book_n of_o euclides_n element_n after_o flussas_n for_o that_o the_o fouretenth_fw-mi book_n as_o it_o be_v set_v forth_o by_o flussas_n contain_v in_o it_o more_o proposition_n than_o be_v find_v in_o hypsicles_n &_o also_o some_o of_o those_o proposition_n which_o hypsicles_n have_v be_v by_o he_o somewhat_o otherwise_o demonstrate_v i_o think_v my_o labour_n well_o bestow_v for_o the_o reader_n sake_n to_o turn_v it_o also_o all_o whole_a notwithstanding_o my_o travail_n before_o take_v in_o turn_v the_o same_o book_n after_o hypsicles_n where_o note_v you_o that_o here_o in_o this_o 14._o book_n after_o flussas_n and_o in_o the_o other_o book_n follow_v namely_o the_o 15._o and_o 16._o i_o have_v in_o allege_v of_o the_o proposition_n of_o the_o same_o 14._o book_n follow_v the_o order_n and_o number_n of_o the_o proposition_n as_o flussas_n have_v place_v they_o ¶_o the_o first_o proposition_n a_o perpendicular_a line_n draw_v from_o the_o centre_n of_o a_o circle_n to_o the_o side_n of_o a_o pentagon_n inscribe_v in_o the_o same_o circle_n campane_n be_v the_o half_a of_o these_o two_o line_n take_v together_o namely_o of_o the_o side_n of_o the_o hexagon_n and_o of_o the_o side_n of_o the_o decagon_n inscribe_v in_o the_o same_o circle_n take_v a_o circle_n abc_n and_o inscribe_v in_o it_o the_o side_n of_o a_o pentagon_n which_o let_v be_v bc_o and_o take_v the_o centre_n of_o the_o circle_n which_o let_v be_v the_o point_n d_o construction_n and_o from_o it_o draw_v unto_o the_o side_n bc_o a_o perpendicular_a line_n de_fw-fr which_o produce_v to_o the_o point_n â—_o and_o unto_o the_o line_n e_o fletcher_n put_v the_o line_n eglantine_n equal_a and_o draw_v these_o right_a line_n cg_o cd_o and_o cf._n then_o i_o say_v that_o the_o right_a line_n de_fw-fr which_o be_v draw_v from_o the_o centre_n to_o bc_o the_o side_n of_o the_o pentagon_n be_v the_o half_a of_o â—he_n sideâ—_n of_o the_o decagon_n and_o hexagon_n take_v together_o demonstration_n forasmuch_o as_o the_o line_n de_fw-fr be_v a_o perpendicular_a â—nto_n the_o line_n bc_o therefore_o the_o section_n be_v and_o ec_o shall_v be_v equal_a by_o the_o 3._o of_o the_o three_o and_o the_o line_n of_o be_v common_a unto_o they_o both_o and_o the_o angle_n fec_n and_o feb_n be_v right_a angle_n by_o supposition_n wherefore_o the_o base_n bf_o and_o fc_o be_v equal_a by_o the_o 4._o of_o the_o first_o but_o the_o line_n bc_o be_v the_o side_n of_o a_o pentagon_n by_o construction_n wherefore_o fc_o which_o subtend_v the_o half_a of_o the_o side_n of_o the_o pentagon_n be_v the_o side_n of_o the_o decagon_n inscribe_v in_o the_o circle_n abc_n but_o unto_o the_o line_n fc_o be_v by_o the_o 4._o of_o the_o first_o equal_a the_o line_n cg_o for_o they_o subtend_v right_a angle_n ceg_v and_o cef_n which_o be_v contain_v under_o equal_a side_n wherefore_o also_o the_o angle_n cge_n and_o cfe_n of_o the_o triangle_n cfg_n be_v equal_a by_o the_o 5._o of_o the_o first_o and_o forasmuch_o as_o the_o ark_n fc_o be_v subtend_v of_o the_o side_n of_o a_o decagon_n the_o ark_n ca_n shall_v be_v quadruple_a to_o the_o ark_n cf_o wherefore_o also_o the_o angle_n cda_n shall_v be_v quadruple_a to_o the_o angle_n cdf_n by_o the_o last_o of_o the_o sixâ—_o and_o forasmuch_o as_o the_o same_o angle_n cda_n which_o be_v set_v at_o the_o centre_n be_v double_a to_o the_o angle_n cfa_n which_o be_v set_v at_o the_o circumference_n by_o the_o 20._o of_o the_o three_o therefore_o the_o angle_n cfa_n or_o cfd_n be_v double_a to_o the_o angle_n cdâ—_n namely_o the_o half_a of_o quadruple_a but_o unto_o the_o angle_n cfd_n or_o cfg_n be_v prove_v equal_a the_o angle_n cgf_n wherefore_o the_o outward_a angle_n cgf_n be_v double_a to_o the_o angle_n cdf_n wherefore_o the_o angle_n cdg_n and_o dcg_n shall_v be_v equal_a for_o unto_o those_o two_o angle_n the_o angle_n cgf_n be_v equal_a by_o the_o 32._o of_o the_o first_o wherefore_o the_o side_n gc_o and_o gd_o be_v equal_a by_o the_o 6._o of_o the_o first_o wherefore_o also_o the_o line_n gd_o be_v equal_a to_o the_o line_n fc_o which_o be_v the_o side_n of_o the_o decagon_n but_o unto_o the_o right_a line_n fe_o be_v equal_a the_o line_n eglantine_n by_o construction_n wherefore_o the_o whole_a line_n de_fw-fr be_v equal_a to_o the_o two_o line_n câ—_n and_o fe_o wherefore_o those_o line_n take_v together_o namely_o the_o line_n df_o and_o fc_o shall_v
be_v double_a to_o the_o line_n de._n wherefore_o the_o line_n de_fw-fr which_o be_v draw_v from_o the_o centre_n perpendicular_o to_o the_o side_n of_o the_o pentagon_n shall_v be_v the_o half_a of_o both_o these_o line_n take_v together_o namely_o of_o df_o the_o side_n of_o the_o hexagon_n and_o cf_o the_o side_n of_o the_o decagon_n for_o the_o line_n df_o which_o be_v draw_v from_o the_o centre_n be_v equal_a to_o the_o side_n of_o the_o hexagon_n by_o the_o corollary_n of_o the_o 15._o of_o the_o four_o wherefore_o a_o perpendicular_a line_n draw_v from_o the_o centre_n of_o a_o circle_n to_o the_o side_n of_o a_o pentagon_n inscribe_v in_o the_o same_o circle_n be_v the_o half_a of_o these_o two_o line_n take_v together_o namely_o of_o the_o side_n of_o the_o hexagon_n and_o of_o the_o side_n of_o the_o decagon_n inscribe_v in_o the_o same_o circle_n which_o be_v require_v to_o be_v prove_v a_o corollary_n if_o a_o right_a line_n draw_v perpendicular_o from_o the_o centre_n of_o a_o circle_n to_o the_o side_n of_o a_o pentagon_n be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n the_o great_a segment_n shall_v be_v the_o line_n which_o be_v draw_v from_o the_o same_o câ—â—tre_n to_o the_o side_n of_o a_o equilater_n triangle_n inscribe_v in_o the_o same_o circle_n for_o that_o liâ—â—_n draw_v to_o the_o side_n of_o the_o triangle_n be_v by_o the_o corollary_n of_o the_o 12._o of_o the_o thirteen_o the_o half_a of_o the_o line_n draw_v from_o the_o centre_n to_o the_o circumference_n that_o be_v of_o the_o side_n of_o the_o hexagon_n wherefore_o the_o residue_n shall_v be_v the_o half_a of_o the_o side_n of_o the_o decagon_n for_o the_o whole_a line_n be_v the_o half_a of_o the_o two_o side_n namely_o of_o the_o side_n of_o the_o hexagon_n and_o of_o the_o side_n of_o the_o decagon_n but_o of_o the_o side_n of_o a_o decagon_n and_o of_o a_o hexagon_n take_v together_o the_o great_a segment_n be_v the_o side_n of_o the_o hexagon_n by_o the_o 9_o of_o the_o thirteen_o wherefore_o the_o great_a segment_n of_o their_o half_n shall_v be_v the_o half_a of_o the_o hexagon_n by_o the_o 15._o of_o the_o five_o which_o half_o be_v the_o perpendicular_a line_n draw_v from_o the_o centre_n to_o the_o side_n of_o the_o triangle_n by_o the_o corollary_n of_o the_o 12._o of_o the_o thirteen_o ¶_o the_o second_o proposition_n if_o two_o right_a line_n be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n campane_n they_o shall_v be_v divide_v into_o the_o self_n same_o proportion_n svppose_v that_o these_o two_o right_a line_n ab_fw-la and_o de_fw-fr be_v each_o cut_n by_o a_o extreme_a and_o mean_a proportion_n in_o the_o point_n fletcher_n and_o z._n then_o i_o say_v that_o these_o two_o line_n be_v divide_v into_o the_o self_n same_o proportion_n that_o be_v that_o the_o line_n ab_fw-la be_v in_o the_o point_n fletcher_n divide_v in_o like_a sort_n as_o the_o line_n de_fw-fr be_v in_o the_o point_n z._n for_o if_o they_o be_v not_o in_o like_a sort_n cut_v let_v one_o of_o they_o namely_o de_fw-fr be_v cut_v like_o unto_o the_o line_n ab_fw-la in_o the_o point_n c._n impossibility_n so_o that_o let_v the_o line_n de_fw-fr be_v to_o dc_o the_o great_a part_n as_o the_o great_a part_v dc_o be_v to_o ce_fw-fr the_o less_o part_n by_o the_o 3._o definition_n of_o the_o six_o but_o by_o suppositition_n the_o line_n de_fw-fr be_v to_o the_o line_n dz_o as_o the_o line_n dz_o be_v to_o the_o line_n ze._n wherefore_o the_o right_a line_n de_fw-fr be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n in_o two_o point_n c_o and_o z._n but_o the_o proportion_n of_o de_fw-fr to_o dc_o the_o less_o line_n be_v great_a than_o the_o proportion_n of_o the_o same_o de_fw-fr to_o dz_o the_o great_a line_n by_o the_o 2._o part_n of_o the_o 8._o of_o the_o five_o but_o as_o de_fw-fr be_v to_o dc_o so_o be_v dc_o to_o ce_fw-fr wherefore_o the_o proportion_n of_o dc_o to_o ce_fw-fr be_v great_a than_o the_o proportion_n of_o dz_o to_o ze._n and_o forasmuch_o as_o dz_o be_v great_a than_o dc_o the_o proportion_n of_o dz_o to_o ce_fw-fr shall_v be_v great_a than_o the_o proportion_n of_o dc_o to_o ce_fw-fr by_o the_o 8._o of_o the_o five_o wherefore_o the_o proportion_n of_o dz_o to_o ce_fw-fr be_v much_o great_a than_o the_o proportion_n of_o dz_o to_o ze._n wherefore_o one_o and_o the_o self_n same_o magnitude_n namely_o dz_o have_v to_o ce_fw-fr the_o great_a line_n a_o great_a proportion_n than_o it_o have_v to_o see_fw-mi the_o less_o line_n contrary_a to_o the_o second_o part_n of_o the_o 8._o of_o the_o five_o which_o be_v impossible_a wherefore_o the_o right_a line_n ab_fw-la &_o de_fw-fr be_v not_o cut_v unlike_a wherefore_o they_o be_v cut_v like_o and_o into_o the_o self_n same_o proportion_n and_o the_o same_o demonstration_n also_o will_v serve_v if_o the_o point_n c_o fall_n in_o any_o other_o place_n for_o always_o some_o one_o of_o they_o shall_v be_v the_o great_a if_o therefore_o two_o right_a line_n be_v cut_v by_o a_o extreme_a and_o mean_a proportion_n they_o shall_v be_v cut_v into_o the_o self_n same_o proportion_n which_o be_v require_v to_o be_v prove_v ¶_o the_o three_o proposition_n if_o in_o a_o circle_n be_v describe_v a_o equilater_n pentagon_n campane_n the_o square_n make_v of_o the_o side_n of_o the_o pentagon_n and_o of_o the_o line_n which_o subtend_v two_o side_n of_o the_o pentagon_n these_o two_o square_n i_o say_v take_v together_o be_v quintuple_a to_o the_o square_n of_o the_o line_n draw_v from_o the_o centre_n of_o the_o circle_n to_o the_o circumference_n svppose_v that_o in_o the_o circle_n bcg_n the_o side_n of_o a_o pentagon_n be_v bg_o and_o let_v the_o line_n bc_o subtend_v two_o side_n thereof_o construction_n and_o let_v the_o line_n bg_o be_v divide_v into_o two_o equal_a part_n by_o a_o right_a line_n draw_v from_o the_o centre_n d_o namely_o by_o the_o diameter_n cde_v produce_v to_o the_o point_n z._n and_o draw_v the_o right_a line_n bz_o then_o i_o say_v that_o the_o right_a line_n bc_o and_o bg_o be_v in_o power_n quintuple_a to_o the_o right_a line_n dz_o which_o be_v draw_v from_o the_o centre_n to_o the_o circumference_n for_o forasmuch_o as_o by_o the_o 47._o of_o the_o first_o the_o square_n of_o the_o line_n cb_o and_o bz_o demonstration_n be_v equal_a to_o the_o square_n of_o the_o diameter_n cz_n therefore_o they_o be_v quadruple_a to_o the_o square_n of_o the_o line_n dz_o by_o the_o 20._o of_o the_o six_o for_o the_o line_n cz_n be_v double_a to_o the_o line_n dz_o wherefore_o the_o right_a line_n cb_o bz_o and_o zd_v be_v in_o power_n quintuple_a to_o the_o line_n zd_fw-mi but_o the_o right_a line_n bg_o contain_v in_o power_n the_o two_o line_n bz_o and_o zd_v by_o the_o 10._o of_o the_o thirteen_o for_o dz_o be_v the_o side_n of_o a_o hexagon_n &_o bz_o the_o side_n of_o a_o decagon_n wherefore_o the_o line_n bc_o and_o bg_o who_o power_n be_v equal_a to_o the_o power_n of_o the_o line_n cb_o bz_o zd_v be_v in_o power_n quintuple_a to_o the_o line_n dz_o if_o therefore_o in_o a_o circle_n be_v describe_v a_o equilater_n pentagon_n the_o square_n make_v of_o the_o side_n of_o the_o pentagon_n and_o of_o the_o line_n which_o subtend_v two_o side_n of_o the_o pentagon_n thâ—se_a two_a square_n i_o say_v take_v together_o be_v quintuple_a to_o the_o square_n of_o the_o line_n draw_v from_o the_o centre_n of_o the_o circle_n to_o the_o circumference_n ¶_o a_o corollary_n if_o a_o cube_n and_o a_o doderahedron_n be_v contain_v in_o one_o and_o the_o self_n same_o sphere_n order_n the_o side_n of_o the_o cube_n and_o the_o side_n of_o the_o dodecahedron_n be_v in_o power_n quintuple_a to_o the_o line_n which_o be_v draw_v from_o the_o centre_n of_o the_o circle_n which_o contain_v the_o pentagon_n of_o the_o dodecahedron_n for_o it_o be_v prove_v in_o the_o 17._o of_o the_o thirteen_o that_o the_o side_n of_o the_o cube_n subtend_v two_o side_n of_o the_o pentagon_n of_o the_o dodecahedron_n where_o the_o say_v solid_n be_v contain_v in_o one_o and_o the_o self_n same_o sphere_n wherefore_o the_o side_n of_o the_o cube_n subtend_v two_o side_n of_o the_o pentagon_n and_o the_o side_n of_o the_o same_o pentagon_n be_v contain_v in_o one_o and_o the_o self_n same_o circle_n wherefore_o by_o this_o proposition_n they_o be_v in_o power_n quintuple_a to_o the_o line_n which_o be_v draw_v from_o the_o centre_n of_o the_o same_o circle_n which_o contain_v the_o pentagon_n of_o the_o dodecahedron_n the_o 4._o proposition_n one_o and_o the_o self_n same_o circle_n contain_v both_o the_o pentagon_n of_o a_o dodecahedron_n campane_n and_o the_o triangle_n of_o a_o icosahedron_n describe_v in_o
one_o and_o the_o self_n same_o sphere_n let_v the_o diameter_n of_o the_o sphere_n give_v by_o ab_fw-la and_o let_v the_o base_n of_o the_o icosahedron_n and_o dodecahedron_n describe_v in_o it_o construction_n be_v the_o triangle_n mnr_n and_o the_o pentagon_n fkh_n and_o about_o they_o let_v there_o be_v describe_v circle_n by_o the_o 5._o and_o 14._o of_o the_o four_o and_o let_v the_o line_n draw_v from_o the_o centre_n of_o those_o circle_n to_o the_o circumference_n be_v ln_o and_o ok_n then_o i_o say_v that_o the_o line_n ln_o and_o ok_n be_v equal_a and_o therefore_o one_o and_o the_o self_n same_o circle_n contain_v both_o those_o figure_n let_v the_o right_a line_n ab_fw-la be_v in_o power_n quintuple_a to_o some_o one_o right_a line_n as_o to_o the_o line_n cg_o by_o the_o corollary_n of_o the_o 6._o of_o the_o ten_o and_o make_v the_o centre_n the_o point_n c_o &_o the_o space_n cg_o describe_v a_o circle_n dzg_v and_o let_v the_o side_n of_o a_o pentagon_n inscribe_v in_o that_o circle_n by_o the_o 11._o of_o the_o four_o be_v the_o line_n zg_v and_o let_v eglantine_n subtend_v half_o of_o the_o ark_n zg_v be_v the_o side_n of_o a_o decagon_n inscribe_v in_o that_o circle_n and_o by_o the_o 30._o of_o the_o six_o divide_v the_o line_n cg_o by_o a_o extreme_a &_o mean_a proportion_n in_o the_o point_n i_o demonstration_n now_o forasmuch_o as_o in_o the_o 16._o of_o the_o thirteen_o it_o be_v prove_v that_o this_o line_n cg_o unto_o who_o the_o diameter_n ab_fw-la of_o the_o sphere_n be_v in_o power_n quintuple_a be_v the_o line_n which_o be_v draw_v from_o the_o centre_n of_o the_o circle_n which_o contain_v five_o angle_n of_o the_o icosahedron_n and_o the_o side_n of_o the_o pentagon_n describe_v in_o that_o circle_n dzg_v namely_o the_o line_n zg_n be_v side_n of_o the_o icosahedron_n describe_v in_o the_o sphere_n who_o diameter_n be_v the_o line_n ab_fw-la therefore_o the_o right_a line_n zg_v be_v equal_a to_o the_o line_n mn_o which_o be_v put_v to_o be_v the_o side_n of_o the_o icosahedron_n or_o of_o his_o triangular_a base_a moreover_o by_o the_o 17._o of_o the_o thirteen_o it_o be_v manifest_a that_o the_o right_a line_n â—h_a which_o subtend_v the_o angle_n of_o the_o pentagon_n of_o the_o dodecahedron_n inscribe_v in_o the_o foresay_a sphere_n be_v the_o side_n of_o the_o cube_n inscribe_v in_o the_o self_n same_o sphere_n for_o upon_o the_o angle_n of_o the_o cube_fw-la be_v make_v the_o angle_n of_o the_o dodecahedron_n wherefore_o the_o diameter_n ab_fw-la be_v in_o power_n triple_a to_o fh_v the_o side_n of_o the_o cube_n by_o the_o 15._o of_o the_o thirteen_o but_o the_o same_o line_n ab_fw-la be_v by_o supposition_n in_o power_n quintuple_a to_o the_o line_n cg_o wherefore_o five_o square_n of_o the_o line_n cg_o be_v equal_a to_o three_o square_n of_o the_o line_n fh_o for_o each_o be_v equal_a to_o one_o and_o the_o self_n same_o square_n of_o the_o line_n ab_fw-la and_o forasmuch_o as_o eglantine_n the_o side_n of_o the_o decagon_n cut_v the_o right_a line_n cg_o by_o a_o extreme_a and_o mean_a proportion_n by_o the_o corollary_n of_o the_o 9_o of_o the_o thirteen_o likewise_o the_o line_n hk_o cut_v the_o line_n fh_o the_o side_n of_o the_o cube_n by_o a_o extreme_a and_o mean_a proportion_n by_o the_o corollary_n of_o the_o 17._o of_o the_o thirteen_o therefore_o the_o line_n cg_o and_o fh_o be_v divide_v into_o the_o self_n same_o proportion_n by_o the_o second_o of_o this_o book_n and_o the_o right_a line_n ci_o and_o eglantine_n which_o be_v the_o great_a segment_n of_o one_o and_o the_o self_n same_o line_n cg_o be_v equal_a and_o forasmuch_o as_o five_o square_n of_o the_o line_n cg_o be_v equal_a to_o three_o square_n of_o the_o line_n fh_o therefore_o five_o square_n of_o the_o line_n ge_z be_v equal_a to_o three_o square_n of_o the_o line_n hk_o for_o the_o line_n ge_z and_o hk_o be_v the_o great_a segment_n of_o the_o line_n cg_o and_o fh_o wherefore_o five_o squâ—reâ—_n of_o the_o lineâ—_z cg_o &_o ge_z be_v equal_a to_o the_o square_n of_o the_o 〈◊〉_d â—h_a &_o hk_a by_o the_o 1â—_o of_o the_o â—ift_n but_o unto_o the_o square_n of_o the_o line_n cg_o and_o geâ—_n be_v â—qual_a the_o squâ—re_n of_o thetine_a zg_n by_o the_o 10._o of_o the_o thirteen_o and_o unto_o the_o line_n zg_v the_o line_n mn_o be_v equal_a wherefore_o five_o square_n of_o the_o line_n mn_o be_v equal_a to_o three_o square_n of_o the_o line_n fh_o hk_o but_o the_o square_n of_o the_o line_n â—â—_o and_o hk_o 〈◊〉_d quintuple_a to_o the_o square_n of_o the_o line_n ok_n which_o be_v draw_v from_o the_o centre_n by_o the_o three_o of_o this_o book_n wherefore_o three_o square_n of_o the_o line_n fh_o and_o hk_o make_v 15._o square_n of_o the_o line_n ok_n and_o forasmuch_o as_o the_o square_n of_o the_o line_n mn_o be_v triple_a to_o the_o square_n of_o the_o line_n ln_o which_o be_v draw_v from_o the_o centre_n by_o the_o 12._o of_o the_o thirteen_o therefore_o five_o square_n of_o the_o line_n mn_o be_v equal_a to_o 15._o square_n of_o the_o line_n ln_o but_o five_o square_n of_o the_o line_n mn_o be_v equal_a unto_o three_o square_n of_o the_o line_n fh_o and_o hk_o wherefore_o one_o square_a of_o the_o line_n ln_o be_v equal_a to_o one_o square_a of_o the_o line_n ok_n be_v each_o the_o fifteen_o part_n of_o equal_a magnitude_n by_o the_o 15._o of_o the_o fifâ—â—_n wherefore_o the_o line_n ln_o and_o ok_n which_o be_v draw_v from_o the_o centre_n be_v equal_a wherefore_o also_o the_o circle_n nrm_n and_o fkh_n which_o be_v describe_v of_o those_o line_n be_v equal_a and_o those_o circle_n contain_v by_o supposition_n the_o base_n of_o the_o dodecahedron_n and_o of_o the_o icosahedron_n describe_v in_o one_o and_o the_o self_n same_o sphere_n wherefore_o one_o and_o the_o self_n same_o circle_n etc_n etc_n aâ—_z in_o thâ—_z proposition_n which_o be_v require_v to_o be_v prove_v the_o 5._o proposition_n if_o in_o a_o circle_n be_v inscribe_v the_o pentagon_n of_o a_o dodecahedron_n and_o the_o triangle_n of_o a_o icosahedron_n campane_n and_o from_o the_o centre_n to_o one_o of_o their_o side_n be_v draw_v a_o perpendicular_a line_n that_o which_o be_v contain_v 30._o time_n under_o the_o side_n &_o the_o perpendicular_a line_n fall_v upon_o it_o be_v equal_a to_o the_o superficies_n of_o that_o solid_a upon_o who_o side_n the_o perpendicular_a line_n fall_v svppose_v that_o in_o the_o circle_n age_n be_v describe_v the_o pentagon_n of_o a_o dodecahedron_n which_o let_v be_v abgde_v and_o the_o triangle_n of_o a_o icosahedron_n describe_v in_o the_o same_o sphere_n which_o let_v be_v afh_n construction_n and_o let_v the_o centre_n be_v the_o point_n c._n â—â—on_n which_o draw_v perpendicular_o the_o line_n ci_o to_o the_o side_n of_o the_o pentagon_n and_o the_o line_n cl_n to_o the_o side_n of_o the_o triangle_n then_o i_o say_v that_o the_o rectangle_n figure_n contain_v under_o the_o line_n ci_o and_o gd_o 30._o time_n be_v equal_a to_o the_o superficies_n of_o the_o dodecahedron_n and_o that_o that_o which_o be_v contain_v under_o the_o line_n cl_n &_o of_o 30._o time_n be_v equal_a to_o the_o superficies_n of_o the_o icosahedron_n describe_v in_o the_o same_o sphere_n draw_v these_o right_a line_n ca_n cf_o cg_o and_o cd_o demonstration_n now_o forasmuch_o as_o that_o which_o be_v contain_v under_o the_o base_a gd_o &_o the_o altitude_n i_fw-mi be_v double_a to_o the_o triangle_n gcd_n by_o the_o 41._o of_o the_o first_o and_o five_o triangle_n like_a and_o equal_a to_o the_o triangle_n gcd_n do_v make_v the_o pentagon_n abgde_v of_o the_o dodecahedron_n wherefore_o that_o which_o be_v contain_v under_o the_o line_n gd_o and_o i_fw-mi five_o time_n be_v equal_a to_o two_o pentagon_n wherefore_o that_o which_o be_v contain_v under_o the_o line_n gd_o and_o i_fw-mi _o time_n be_v equal_a to_o the_o 12._o pentagon_n which_o contain_v the_o superficies_n of_o the_o dodecahedron_n again_o that_o which_o be_v contain_v under_o the_o line_n cl_n and_o of_o be_v double_a to_o the_o triangle_n acf_n wherefore_o that_o which_o be_v contain_v under_o the_o line_n cl_n and_o of_o three_o time_n be_v equal_a to_o two_o such_o triangle_n as_o afh_n be_v which_o be_v one_o of_o the_o base_n of_o the_o icosahedron_n for_o the_o triangle_n acf_n be_v the_o three_o part_n of_o the_o triangle_n afh_v as_o it_o be_v easy_a to_o prove_v by_o the_o 8._o &_o 4._o of_o the_o first_o wherefore_o that_o which_o be_v contain_v under_o the_o line_n cl_n and_o af._n 30_o time_n time_n be_v equal_a to_o 10._o such_o triangle_n as_o afh_v iâ—_z which_o contain_v the_o superficies_n of_o the_o icosahedron_n and_o forasmuch_o as_o one_o and_o the_o self_n same_o
that_o be_v the_o right_a line_n agnostus_n gb_o bd_o dam_fw-ge ca_n cg_o cb_o cd_o and_o ja_z ig_n ib_n id_fw-la be_v equal_a the_o one_o to_o the_o other_o wherefore_o also_o the_o 8._o triangle_n cag_n cgb_n cbd_v cda_n jag_n igb_n ibd_v &_o ida_n be_v equal_a and_o equilater_n and_o therefore_o agbdci_n be_v a_o octohedron_n by_o the_o 23._o definition_n of_o the_o eleven_o and_o the_o say_v octahedron_n be_v include_v in_o the_o dodecahedron_n by_o the_o first_o definition_n of_o this_o book_n for_o that_o all_o the_o angle_n thereof_o do_v at_o one_o time_n touch_v the_o side_n of_o the_o dodecahedron_n wherefore_o in_o the_o dodecahedron_n give_v be_v include_v a_o octohedron_n which_o be_v require_v to_o be_v do_v ¶_o the_o 10._o proposition_n the_o 10._o problem_n in_o a_o dodecahedron_n give_v to_o inscribe_v a_o equilater_n trilater_n pyramid_n svppose_v that_o the_o dodecahedron_n give_v be_v abcd_o of_o which_o dodecahedron_n take_v three_o base_n meet_v at_o the_o point_n s_o namely_o these_o three_o base_n alsik_n dnsle_n and_o sibrn_n construction_n and_o of_o those_o three_o base_n take_v the_o three_o angle_n at_o the_o point_n a_o b_o d_o and_o draw_v these_o right_a line_n ab_fw-la bd_o and_o dam_fw-ge and_o let_v the_o diameter_n of_o the_o sphere_n contain_v the_o dodecahedron_n be_v so_o and_o then_o draw_v these_o right_a line_n ao_o boy_n and_o do_v now_o forasmuch_o as_o by_o the_o 17._o of_o the_o thirteen_o the_o angle_n of_o the_o dodecahedron_n be_v set_v in_o the_o superficies_n of_o the_o sphere_n describe_v about_o the_o dodecahedronâ—_n demonstration_n therefore_o if_o upon_o the_o diameter_n so_o and_o by_o the_o point_n a_o be_v describe_v a_o semicircle_n it_o shall_v make_v the_o angle_n sao_n a_o right_a angle_n by_o the_o 31._o of_o the_o three_o and_o likewise_o if_o the_o same_o semicircle_n be_v draw_v by_o the_o point_n d_o and_o b_o it_o shall_v also_o make_v the_o angle_n sbo_n and_o sdo_v right_a angle_n wherefore_o the_o diameter_n so_o contain_v in_o power_n both_o the_o line_n sa_o ao_o or_o the_o line_n sb_n boy_n or_o else_o sd_z do_v but_o the_o line_n sa_o sd_z sb_n be_v equal_a the_o one_o to_o the_o other_o for_o they_o each_o subtend_v one_o of_o the_o angle_n of_o equal_a pentagon_n wherefore_o the_o other_o line_n remain_v namely_o ao_o boy_n do_v be_v equal_a the_o one_o to_o the_o other_o and_o by_o the_o same_o reason_n may_v be_v prove_v that_o the_o diameter_n hd_v which_o subtend_v the_o two_o right_a line_n ha_o ad_fw-la contain_v in_o power_n both_o the_o say_v two_o right_a line_n and_o also_o contain_v in_o power_n both_o the_o right_a line_n hb_o and_o bd_o which_o two_o right_a line_n it_o also_o suhtend_v and_o moreover_o by_o the_o same_o reason_n the_o diameter_n ac_fw-la which_o subtend_v the_o right_a line_n cb_o and_o basilius_n contain_v in_o power_n both_o the_o say_v right_a line_n câ—_n and_o basilius_n but_o the_o right_a line_n ha_o hb_o and_o cb_o be_v equal_a the_o one_o to_o the_o other_o for_o that_o each_o of_o they_o also_o subtend_v one_o of_o the_o angle_n of_o equal_a pentagonsâ—_n wherefore_o the_o right_a line_n remain_v namely_o ad_fw-la bd_o and_o basilius_n be_v equal_a the_o one_o to_o the_o other_o and_o by_o the_o same_o reason_n may_v be_v prove_v that_o each_o of_o those_o right_a line_n ad_fw-la bd_o and_o basilius_n be_v equal_a to_o each_o of_o the_o right_a line_n ao_o boy_n and_o do_v wherefore_o the_o six_o right_a line_n ab_fw-la bd_o dam_fw-ge ao_o boy_n &_o do_v be_v equal_a the_o one_o to_o the_o other_o and_o therefore_o the_o triangle_n which_o be_v make_v of_o they_o namely_o the_o triangle_n abdella_n aob_n aod_n and_o bod_fw-mi be_v equal_a and_o equilater_n which_o triangle_n therefore_o do_v make_v a_o pyramid_n abdo_n who_o base_a be_v abdella_n and_o top_n the_o point_n o._n each_o of_o the_o angle_n of_o which_o pyramid_n namely_o the_o angle_n at_o the_o point_n a_o b_o d_o oh_o do_v in_o the_o self_n same_o point_n touch_v the_o angle_n of_o the_o dodecahedron_n wherefore_o the_o say_a pyramid_n be_v inscribe_v in_o the_o dodecahedron_n by_o the_o first_o definition_n of_o this_o book_n wherefore_o in_o a_o dodecahedron_n give_v be_v inscribe_v a_o trilater_n equilater_n pyramid_n which_o be_v require_v to_o be_v do_v ¶_o the_o 11._o proposition_n the_o 11._o problem_n in_o a_o icosahedron_n give_v to_o inscribe_v a_o cube_fw-la it_o be_v manifest_a by_o the_o 7._o of_o this_o book_n that_o the_o angle_n of_o a_o dodecahedron_n be_v set_v in_o the_o centre_n of_o the_o base_n of_o the_o icosahedron_n and_o by_o the_o 8._o of_o this_o book_n it_o be_v prove_v that_o the_o angle_n of_o a_o cube_fw-la be_v set_v in_o the_o angle_n of_o a_o dodecahedron_n wherefore_o the_o self_n same_o angle_n of_o the_o cube_fw-la shall_v of_o necessity_n be_v set_v in_o the_o centre_n of_o the_o base_n of_o icosahedron_n wherefore_o the_o cube_fw-la shall_v be_v inscribe_v in_o the_o icosahedron_n by_o the_o first_o definition_n of_o this_o book_n wherefore_o in_o a_o icosahedron_n give_v be_v include_v a_o cube_fw-la which_o be_v require_v to_o be_v do_v ¶_o the_o 12._o proposition_n the_o 12._o problem_n in_o a_o icosahedron_n give_v to_o inscribe_v a_o trilater_n equilater_n pyramid_n by_o the_o former_a proposition_n it_o be_v manifest_a that_o the_o angle_n of_o a_o cube_fw-la be_v set_v in_o the_o centre_n of_o the_o base_n of_o the_o icosâ—hedron_n and_o by_o the_o first_o of_o this_o book_n it_o be_v plain_a that_o the_o four_o angle_n of_o a_o pyramid_n be_v set_v in_o four_o angle_n of_o a_o cube_fw-la wherefore_o it_o be_v evident_a by_o the_o first_o definition_n of_o this_o book_n that_o a_o pyramid_n describe_v of_o right_a line_n join_v together_o these_o four_o centre_n of_o the_o base_n of_o the_o icosahedron_n shall_v be_v inscribe_v in_o the_o same_o icosahedron_n wherefore_o in_o a_o icosadron_n give_v be_v inscribe_v a_o equilater_n trilater_n pyramid_n which_o be_v require_v to_o be_v do_v ¶_o the_o 13._o problem_n the_o 13._o proposition_n in_o a_o cube_n give_v to_o inscribe_v a_o dodecahedron_n take_v a_o cube_n adfl_fw-mi construction_n and_o divide_v every_o one_o of_o the_o side_n thereof_o into_o two_o equal_a part_n in_o the_o point_n t_o h_o edward_n p_o g_o l_o m_o f_o and_o pkq_n and_o draw_v these_o right_a line_n tk_n gf_o pq_n hk_n ps_n and_o lm_o which_o line_n again_o divide_v into_o two_o equal_a part_n in_o the_o point_n n_o five_o a'n_z ay_o z_o x._o and_o draw_v these_o right_a line_n nigh_o vx_n and_o iz_n now_o the_o three_o line_n nigh_o vx_n and_o iz_n together_o with_o the_o diameter_n of_o the_o cube_fw-la shall_v cut_v the_o one_o the_o other_o into_o two_o equal_a part_n in_o the_o centre_n of_o the_o cube_fw-la by_o the_o 3â—_o of_o the_o eleven_o let_v that_o centre_n be_v the_o point_n o._n and_o not_o to_o stand_v long_o about_o the_o demonstration_n understand_v all_o these_o right_a line_n to_o be_v equal_a and_o parallel_n to_o the_o side_n of_o the_o cube_fw-la and_o to_o cut_v the_o one_o the_o other_o right_o angle_a wise_a by_o the_o 29._o of_o the_o first_o let_v their_o half_n namely_o fv_n gv_n high_a and_o king_n and_o the_o rest_n such_o like_a be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n by_o the_o 30._o of_o the_o six_o who_o great_a segment_n let_v be_v the_o line_n f_n gb_o hc_n and_o ke_o etc_n etc_n and_o draw_v these_o right_a line_n give_v ge_z bc_o and_o be._n now_o forasmuch_o as_o the_o line_n give_v be_v equal_a to_o the_o whole_a line_n gv_n which_o be_v the_o half_a of_o the_o side_n of_o the_o cube_fw-la demonstration_n and_o the_o line_n je_n be_v equal_a to_o the_o line_n bv_n that_o be_v to_o the_o less_o segment_n therefore_o the_o square_n of_o the_o line_n give_v and_o je_n be_v triple_a to_o the_o square_n of_o the_o line_n gb_o by_o the_o 4._o of_o the_o thirteen_o but_o unto_o the_o square_n of_o the_o line_n give_v and_o je_n the_o square_a of_o the_o line_n ge_z be_v equal_a by_o the_o 47._o of_o the_o firstâ—_o for_o the_o angle_n gie_n be_v a_o right_a angle_n wherefore_o the_o square_a of_o the_o line_n ge_z be_v triple_a to_o the_o square_n of_o the_o line_n gb_o and_o forasmuch_o as_o the_o line_n fg_o be_v erect_v perpendicular_o to_o the_o plain_a agkl_n by_o the_o 4._o of_o the_o eleven_o for_o it_o be_v erect_v perpendicular_o to_o the_o two_o line_n agnostus_n and_o give_v therefore_o the_o angle_n bge_n be_v a_o right_a angle_n for_o the_o line_n ge_z be_v draw_v in_o the_o plain_a agkl_n wherefore_o the_o line_n be_v contain_v in_o power_n the_o two_o line_n bg_o and_o ge_z by_o the_o 47._o of_o the_o first_o be_v in_o power_n quadruple_a to_o the_o
line_n gb_o for_o the_o line_n ge_z be_v prove_v to_o be_v in_o power_n triple_a to_o the_o same_o line_n gb_o wherefore_o the_o line_n be_v be_v in_o length_n double_a to_o the_o line_n bg_o by_o the_o _o of_o the_o six_o but_o by_o construction_n the_o line_n ce_fw-fr be_v double_a the_o line_n je_n wherefore_o the_o half_n gb_o and_o je_n be_v in_o proportion_n the_o one_o to_o the_o other_o as_o their_o double_n be_v and_o ce_fw-fr by_o the_o 15._o of_o the_o five_o wherefore_o the_o line_n ce_fw-fr be_v the_o great_a segment_n of_o the_o line_n be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n and_o forasmuch_o as_o the_o self_n same_o thing_n may_v be_v prove_v touch_v the_o line_n bc_o therefore_o the_o line_n be_v and_o bc_o be_v equal_a make_v a_o isosceles_a triangle_n now_o let_v we_o prove_v that_o three_o angle_n of_o the_o pentagon_n of_o the_o dodecahedron_n be_v set_v at_o the_o point_n b_o c_o e_o and_o the_o other_o two_o angle_n be_v set_v between_o the_o line_n bc_o and_o be._n forasmuch_o as_o the_o circle_n which_o contain_v the_o triangle_n be_v circumscribe_v the_o pentagon_n who_o side_n be_v the_o line_n ce_fw-fr by_o the_o 11._o of_o the_o four_o construction_n extend_v the_o plain_a of_o the_o triangle_n be_v by_o the_o parallel_a line_n db_n and_o he_o cut_v the_o line_n ad_fw-la namely_o the_o diameter_n of_o ad_fw-la the_o base_a of_o the_o cube_fw-la in_o the_o point_n i_o and_o let_v it_o cut_v the_o line_n ah_o the_o diameter_n of_o the_o cube_fw-la in_o the_o point_n m._n and_o by_o the_o point_n i_o draw_v in_o the_o base_a ad_fw-la a_o parallel_a line_n unto_o the_o line_n ad_fw-la which_o let_v be_v il._n and_o forasmuch_o as_o from_o the_o triangle_n ahn_n be_v by_o the_o parallel_n line_n li_o take_v away_o the_o triangle_n all_o like_v unto_o the_o whole_a triangle_n ahn_n demonstration_n by_o the_o corollary_n of_o the_o 2._o of_o the_o six_o the_o line_n all_o and_o li_o shall_v be_v equal_a but_o as_o the_o line_n ha_o be_v to_o the_o line_n ad_fw-la so_o by_o the_o 2._o of_o the_o six_o be_v the_o line_n hl_n to_o the_o line_n li_o or_o to_o the_o line_n la_n which_o be_v equal_a to_o the_o line_n li_o and_o the_o great_a segment_n of_o the_o line_n ha._n which_o be_v half_a the_o side_n of_o the_o cube_fw-la be_v as_o before_o have_v be_v prove_v the_o line_n ad_fw-la that_o be_v the_o line_n gb_o which_o be_v equal_a to_o the_o line_n ad_fw-la by_o the_o 33._o of_o the_o first_o wherefore_o the_o 〈◊〉_d segment_n of_o the_o line_n hl_n be_v the_o line_n â—a_o and_o as_o the_o whole_a line_n hl_n be_v to_o the_o great_a segment_n so_o shall_v the_o same_o great_a segment_n hl_n be_v to_o the_o less_o segment_n la_n by_o the_o 5._o of_o the_o thirteen_o wherefore_o the_o line_n ha_o be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n in_o the_o point_n l._n but_o in_o the_o triangle_n ahn_n the_o line_n na_fw-mi which_o be_v draw_v from_o the_o centre_n of_o the_o base_a ad_fw-la be_v in_o the_o point_n i_o cut_v like_o unto_o the_o line_n all_o by_o thu_o parallel_a line_n li_o by_o the_o same_o second_o of_o the_o six_o for_o the_o line_n hn_o and_o ill_n be_v parallel_n by_o construction_n wherefore_o the_o line_n na_fw-mi be_v in_o the_o point_n i_o divide_v by_o a_o extreme_a and_o mean_a proportion_n by_o the_o superficies_n dbeh_n and_o forasmuch_o as_o the_o line_n yon_n which_o couple_v the_o centre_n of_o the_o opposite_a base_n be_v a_o parallel_n to_o the_o line_n he_o a_o plain_a superficies_n extend_v by_o the_o line_n yond_o parallel_n wise_a to_o the_o plain_a dbeh_n the_o two_o plain_n shall_v cut_v the_o line_n ao_o and_o a_o the_o semidiameter_n of_o the_o cube_fw-la and_o the_o semidiameter_n of_o the_o base_a ad_fw-la into_o the_o self_n same_o proportion_n in_o the_o point_n m_o an_o i_o by_o the_o 17._o of_o the_o eleven_o but_o the_o line_n a_o be_v in_o the_o point_n i_o divide_v by_o a_o extreme_a &_o mean_a proportion_n wherefore_o the_o semidiameter_n of_o the_o cube_fw-la be_v in_o the_o point_n m_z divide_v by_o a_o extreme_a and_o mean_a proportion_n by_o the_o plain_a of_o the_o triangle_n be_v and_o forasmuch_o as_o the_o rest_n of_o the_o triangle_n describe_v in_o the_o cube_fw-la after_o the_o like_a manner_n may_v by_o the_o same_o reason_n be_v prove_v to_o be_v in_o a_o plain_a which_o cut_v the_o semidiameter_n of_o the_o cube_fw-la by_o a_o extreme_a and_o mean_v proportion_n it_o be_v manifest_v that_o three_o plain_n of_o the_o dodecahedron_n shall_v under_o every_o angle_n of_o the_o cube_fw-la concur_v in_o one_o &_o the_o self_n same_o point_n oâ—_n the_o semidiameter_n be_v cut_v by_o a_o extreme_a and_o mean_a proportion_n now_o rest_v to_o prove_v that_o the_o right_a line_n which_o couple_v that_o point_n of_o the_o semidiameter_n with_o the_o angle_n of_o the_o triangle_n bec_n be_v equal_a whereby_o may_v be_v prove_v that_o the_o pentagons_n be_v equilater_n and_o equiangle_n take_v the_o two_o base_n of_o the_o cube_fw-la whereon_o be_v set_v the_o triangle_n be_v construction_n namely_o the_o base_n of_o and_o ak_o take_v also_o the_o same_o diameter_n of_o the_o cube_fw-la that_o be_v before_o namely_o ah_o and_o let_v the_o side_n set_v at_o the_o point_n n_o of_o the_o section_n of_o the_o diameter_n by_o a_o extreme_a &_o mean_a proportion_n be_v the_o line_n cn_fw-la or_o bn_n and_o let_v the_o centre_n of_o the_o cube_fw-la be_v as_o before_o the_o point_n o._n and_o extend_v the_o line_n cn_fw-la to_o the_o line_n bd_o and_o let_v it_o concur_v with_o it_o in_o the_o point_n a._n and_o forasmuch_o as_o the_o plain_a which_o pass_v by_o the_o line_n hce_n and_o the_o centre_n oh_o cut_v the_o cube_fw-la into_o two_o equal_a part_n be_v parallel_n to_o of_o the_o base_a of_o the_o cube_fw-la by_o construction_n imagine_v that_o by_o the_o point_n n_o be_v extend_v a_o plain_a superficies_n parallel_v to_o the_o former_a parallel_n plain_n demonstration_n which_o shall_v cut_v the_o semidiameter_n oa_n &_o the_o line_n ca_o proportional_o in_o the_o point_n n_o by_o the_o 17._o of_o the_o eleven_o for_o those_o line_n do_v touch_v the_o extreme_a parallel_n plain_n extend_v by_o the_o line_n he_o and_o eo_fw-la and_o by_o the_o line_n add_v and_o db_n but_o it_o be_v prove_v that_o the_o line_n oa_n be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n in_o the_o point_n n_o wherefore_o the_o line_n ca_o be_v also_o 〈◊〉_d by_o a_o extreme_a and_o mean_a proportion_n in_o the_o point_n n._n again_o forasmuch_o as_o be_v be_v a_o isoscel_v triangle_n and_o it_o be_v prove_v that_o the_o line_n by_o cut_v the_o base_a ce_fw-fr into_o two_o equal_a part_n in_o the_o point_n i_o the_o angle_n bic_a and_o bieâ—_n shall_v be_v right_a angle_n imagine_v by_o the_o line_n by_o and_o the_o centre_n oh_o a_o plain_a to_o pass_v cut_v the_o cube_fw-la into_o two_o equal_a part_n parallel_v to_o the_o base_a ad._n and_o unto_o those_o plain_n let_v there_o be_v imagine_v a_o other_o parallel_n plain_o pass_v by_o the_o point_n n_o which_o let_v be_v ne_fw-fr which_o shall_v cut_v the_o semidiameter_n ao_o and_o the_o half_a side_n of_o the_o cube_fw-la namely_o the_o line_n lh_n like_v in_o the_o point_n n_o and_o c_o by_o the_o 17._o of_o the_o eleven_o wherefore_o the_o line_n ih_v be_v in_o the_o point_n e_o divide_v by_o a_o extreme_a 〈◊〉_d mean_a proportion_n wherefore_o the_o line_n he_o be_v equal_a to_o the_o line_n ci_o or_o je_n namely_o each_o be_v less_o segment_n and_o forasmuch_o as_o the_o line_n je_fw-fr be_v to_o the_o line_n i_fw-mi which_o be_v equal_a â—o_o the_o â—ine_o eh_o as_o the_o whole_a be_v to_o the_o great_a segment_n take_v away_o from_o the_o whole_a line_n je_fw-fr thâ—_z great_a segmâ—â—_n i_fw-mi there_o shall_v remain_v the_o less_o segment_n ce_fw-fr by_o the_o 5._o of_o the_o thirteen_o wherefore_o the_o line_n iâ—_z be_v divide_v by_o a_o extreme_a &_o mean_a proportion_n in_o the_o point_n c._n again_o unto_o the_o same_o plain_n imagine_v a_o other_o plain_n to_o pass_v by_o the_o point_n a_o parallel_n wise_a and_o let_v the_o same_o be_v ag_n now_o then_o by_o the_o same_o 17._o of_o the_o eleven_o the_o line_n ca_o and_o cg_n be_v in_o like_a sort_n cut_v in_o the_o point_n n_o and_o e._n but_o the_o line_n ca_o be_v in_o the_o point_n n_o cut_v by_o a_o extreme_a and_o mean_a proportion_n wherefore_o the_o line_n cg_n shall_v be_v cut_v in_o the_o point_n e_o by_o a_o extreme_a &_o mean_a proportion_n but_o the_o line_n i_fw-mi be_v to_o the_o line_n ce_fw-fr as_o the_o great_a segment_n be_v to_o the_o less_o
by_o the_o 4._o and_o eight_o of_o the_o first_o and_o forasmuch_o as_o upon_o every_o one_o of_o the_o line_n cut_v of_o the_o cube_fw-la be_v set_v two_o triangle_n as_o the_o triangle_n alm_n and_o blmâ—_n there_o shall_v be_v make_v 12._o triangle_n and_o forasmuch_o as_o under_o every_o one_o of_o the_o â—_o angle_n of_o the_o cube_fw-la be_v subtend_v the_o other_o 8._o triangle_n as_o the_o triangle_n amp._n etc_n etc_n of_o 1â—_o and_o 8._o triangle_n shall_v be_v produce_v 20._o triangle_n equal_a and_o equilater_n contain_v the_o solid_a of_o a_o icosahedron_n by_o the_o 25._o definition_n of_o the_o eleven_o which_o shall_v be_v inscribe_v in_o the_o cube_fw-la give_v abc_n by_o the_o first_o definition_n of_o this_o book_n the_o invention_n of_o the_o demonstration_n of_o this_o depend_v of_o the_o ground_n of_o the_o former_a wherefore_o in_o a_o cube_fw-la give_v we_o have_v describe_v a_o icosahedron_n which_o be_v require_v to_o be_v do_v first_o corollary_n the_o diameter_n of_o a_o sphere_n which_o contain_v a_o icosahedron_n contain_v two_o side_n namely_o the_o side_n of_o the_o icosahedron_n and_o the_o side_n of_o the_o cube_fw-la which_o contain_v the_o icosahedron_n for_o if_o we_o draw_v the_o line_n ab_fw-la it_o shall_v make_v the_o angle_n at_o the_o point_n a_o right_a angle_n for_o that_o it_o be_v a_o parallel_n to_o the_o side_n of_o the_o cube_fw-la wherefore_o the_o linâ—_n which_o couple_v the_o opposite_a angleâ—_n of_o the_o icosahedron_n at_o the_o point_v fletcher_n and_o b_o contain_v in_o power_n the_o line_n ab_fw-la the_o sidâ—_n of_o thâ—_z cube_fw-la and_o the_o line_n of_o the_o side_n of_o the_o icosahedron_n by_o the_o 47._o of_o the_o first_o which_o line_n fb_o be_v equal_a to_o the_o â—iameter_n of_o the_o sphere_n which_o contain_v the_o icosahedron_n by_o the_o demonstration_n of_o the_o â—â—_o of_o the_o thirteen_o second_o corollary_n the_o six_o opposite_a side_n of_o the_o icosahedron_n divide_v into_o two_o equal_a part_n their_o section_n be_v couple_v by_o three_o equal_a right_a line_n cut_v the_o one_o the_o other_o into_o two_o equal_a part_n and_o perpendicular_o in_o the_o centre_n of_o the_o sphere_n which_o contain_v the_o icosahedron_n for_o those_o three_o line_n be_v the_o three_o line_n which_o couple_v the_o centre_n of_o the_o base_n of_o the_o cube_fw-la which_o do_v in_o such_o sort_n in_o the_o centre_n of_o the_o cube_fw-la cut_v the_o one_o the_o other_o by_o the_o corollary_n of_o the_o three_o of_o this_o book_n and_o therefore_o be_v equal_a to_o the_o side_n of_o the_o cube_fw-la but_o right_a line_n draw_v from_o the_o centre_n of_o the_o cube_fw-la to_o the_o angle_n of_o the_o icosahedron_n every_o one_o of_o they_o shall_v subtend_v the_o half_a side_n of_o the_o cube_fw-la and_o the_o half_a side_n of_o the_o icosahedron_n which_o half_a side_n contain_v a_o right_a angle_n wherefore_o those_o line_n be_v equal_a whereby_o it_o be_v manifest_a that_o the_o foresay_a centre_n be_v the_o centre_n of_o the_o sphere_n which_o contain_v the_o icosahedron_n three_o corollary_n the_o side_n of_o a_o cube_fw-la divide_v by_o a_o extreme_a and_o mean_a proportion_n make_v the_o great_a segment_n the_o side_n of_o a_o icosahedron_n describe_v in_o it_o for_o the_o half_a side_n of_o the_o cube_fw-la make_v the_o half_a of_o the_o side_n of_o the_o icosahedron_n the_o great_a segment_n wherefore_o also_o the_o whole_a side_n of_o the_o cube_fw-la make_v the_o whole_a side_n of_o the_o icosahedron_n the_o great_a segment_n by_o the_o 15._o of_o the_o fifthe_o for_o the_o section_n be_v like_a by_o the_o â—_o of_o the_o fourteen_o ¶_o four_o corollary_n the_o side_n and_o base_n of_o the_o icosahedron_n which_o be_v opposite_a the_o one_o to_o the_o other_o be_v parallel_n forasmuch_o as_o every_o one_o of_o the_o opposite_a side_n of_o the_o icosahedron_n may_v be_v in_o the_o parallel_a line_n of_o the_o cube_fw-la namely_o in_o those_o parallel_n which_o be_v opposite_a in_o the_o cube_fw-la and_o the_o triangle_n which_o be_v make_v of_o parallel_a line_n be_v parallel_n by_o the_o 15._o of_o the_o eleven_o therefore_o the_o opposite_a triâ—ngleâ—_n of_o the_o icosahedron_n as_o also_o the_o side_n be_v parallel_n the_o one_o to_o the_o other_o ¶_o the_o 15._o problem_n the_o 15._o proposition_n in_o a_o icosahedron_n give_v to_o inscribe_v a_o octohedron_n svppose_v that_o the_o icosahedron_n give_v be_v acdf_n and_o by_o the_o former_a second_o corollary_n construction_n let_v there_o be_v take_v the_o three_o right_a line_n which_o cut_v the_o one_o the_o other_o into_o two_o equal_a part_n perpendicular_o and_o which_o couple_n the_o section_n into_o two_o equal_a part_n of_o the_o side_n of_o the_o icosahedron_n which_o let_v be_v be_v gh_o and_o kl_o cut_v the_o one_o the_o other_o in_o the_o point_n i_o and_o drawâ—_n these_o riâ—ht_a line_n â—g_z ge_z eh_o and_o hb_o and_o forasmuch_o as_o the_o angle_n at_o the_o point_n i_o be_v by_o construction_n right_a angle_n demonstration_n â—nd_v be_v conâ—â—ined_v under_o equal_a linesâ—_n the_o 〈◊〉_d gâ—_n and_o â—â—_o shall_v 〈…〉_o squâ—re_o by_o the_o â—_o of_o the_o â—irst_n likewise_o â—nto_o thoâ—e_a 〈◊〉_d shall_v be_v squall_v the_o line_n drâ—wâ—n_v from_o 〈◊〉_d point_n king_n and_o â—_o to_o every_o one_o of_o the_o poinâ—â—s_n â—_o g._n â—_o h_o and_o therefore_o the_o triangle_n which_o 〈◊〉_d the_o â—â—ramis_n dgenk_n shall_v be_v equal_a 〈…〉_o and_o by_o 〈…〉_o ¶_o the_o 16._o problem_n the_o 16._o proposition_n in_o a_o octohedron_n give_v to_o inscribe_v a_o icosahedron_n let_v there_o be_v take_v a_o octohedron_n who_o 6._o angle_n let_v be_v a_o b_o c_o f_o p_o l._n and_o draw_v the_o line_n ac_fw-la bf_o pl_z construction_n cut_v the_o one_o the_o other_o perpendicular_o in_o the_o point_n r_o by_o the_o 2._o corollary_n of_o the_o 14._o of_o the_o thirteen_o and_o let_v every_o one_o of_o the_o 12._o side_n of_o the_o octohedron_n be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n in_o the_o point_n h_o x_o m_o king_n d_o s_o n_o g_o five_o e_o q_o t._n and_o let_v the_o great_a segment_n be_v the_o line_n bh_o bx_n fm_o fk_o ad_fw-la aq_n cs_n et_fw-la pn_n pg_n lv_o le_fw-fr and_o draw_v these_o line_n hk_o xm_o ge_z nv_n d_n qt_n now_o forasmuch_o as_o in_o the_o triangle_n abf_n the_o side_n be_v cut_v proportional_o namely_o as_o the_o line_n bh_o be_v to_o the_o line_n ha_o so_o be_v the_o line_n fk_o to_o the_o line_n ka_o by_o the_o 2._o of_o the_o fourteen_o demonstration_n therefore_o the_o line_n hk_o shall_v be_v a_o parallel_n to_o the_o line_n bf_o by_o the_o 2._o of_o the_o six_o and_o forasmuch_o as_o the_o line_n ac_fw-la cut_v the_o line_n hk_o in_o the_o point_n z_o and_o the_o line_n zk_n be_v a_o parallel_n unto_o the_o line_n rf_n the_o line_n ramires_n shall_v be_v cut_v by_o a_o extreme_a and_o mean_a proportion_n in_o the_o point_n z_o by_o the_o 2._o of_o the_o six_o namely_o shall_v be_v cut_v like_o unto_o the_o line_n favorina_n and_o the_o great_a segment_n thereof_o shall_v be_v the_o line_n zr_n unto_o the_o line_n zk_v put_v the_o line_n ro_n equal_a by_o the_o 3._o of_o the_o first_o and_o draw_v the_o line_n ko_o now_o then_o the_o line_n ko_o shall_v be_v equal_a to_o the_o line_n zr_n by_o the_o 33._o of_o the_o â—irst_n draw_v the_o line_n kg_v ke_o and_o ki_v and_o forasmuch_o as_o the_o triangle_n arf_n and_o azk_v be_v equiangle_n by_o the_o 6._o of_o the_o six_o the_o side_n az_o and_o zk_v shall_v be_v equal_a the_o one_o to_o the_o other_o by_o the_o 4._o of_o the_o six_o for_o the_o side_n be_v and_o rf_n be_v equal_a wherefore_o the_o line_n zk_n shall_v be_v the_o less_o segment_n of_o the_o line_n ra._n but_o if_o the_o great_a segment_n rz_n be_v divide_v by_o a_o extreme_a &_o mean_a proportion_n the_o great_a segment_n thereof_o shall_v be_v the_o line_n zk_n which_o be_v the_o less_o segment_n of_o the_o whole_a line_n ramires_n by_o the_o 5._o of_o the_o thirâ—enth_o and_o forasmuch_o as_o the_o two_o line_n fe_o and_o fg_o be_v equal_a to_o the_o two_o line_n ah_o and_o ak_o namely_o each_o be_v less_o segment_n of_o equal_a side_n of_o the_o octohedron_n and_o the_o angle_n hak_n and_o efg_o be_v equal_a namely_o be_v right_a angle_n by_o the_o 14._o of_o the_o thirteen_o the_o base_n hk_o and_o gf_o shall_v be_v equal_a by_o the_o 4._o of_o the_o first_o and_o by_o the_o same_o reason_n unto_o they_o may_v be_v prove_v equal_a the_o line_n xm_o nv_n d_n and_o qt_o and_o forasmuch_o as_o the_o line_n ac_fw-la bf_o and_o pl_z do_v cut_v the_o one_o the_o other_o into_o two_o equal_a part_n and_o perpendicular_o by_o construction_n the_o line_n hk_o and_o ge_z which_o
subtend_v angle_n of_o triangle_n like_v unto_o the_o triangle_n who_o angle_n the_o line_n ac_fw-la bf_o and_o pl_z subtend_n be_v cut_v into_o two_o equal_a part_n in_o the_o point_n z_o an_o i_o by_o the_o 4._o of_o the_o six_o so_o also_o be_v the_o other_o line_n nv_n xm_o d_n qt_fw-la which_o be_v equal_a unto_o the_o line_n hk_o &_o ge_z cut_v in_o like_a sort_n and_o they_o shall_v cut_v the_o line_n ac_fw-la bf_o and_o pl_z like_z wherefore_o the_o line_n ko_o which_o be_v equal_a to_o rz_n shall_v make_v the_o great_a segment_n the_o line_n ro_n which_o be_v equal_a to_o the_o line_n zk_n for_o the_o great_a segment_n of_o the_o rz_n be_v the_o line_n zk_n and_o therefore_o the_o line_n oi_o shall_v be_v the_o less_o segment_n when_o as_o the_o whole_a line_n ri_n be_v equal_a to_o the_o whole_a line_n rz_n wherefore_o the_o square_n of_o the_o whole_a line_n ko_o and_o of_o the_o less_o segment_n oi_o be_v triple_a to_o the_o square_n of_o the_o great_a segment_n ro_n by_o the_o 4._o of_o the_o thirteen_o wherefore_o the_o line_n ki_v which_o contain_v in_o power_n the_o two_o line_n ko_o and_o oi_o be_v in_o power_n triple_a to_o the_o line_n ro_n by_o the_o 47._o of_o the_o first_o for_o the_o angle_n koi_n be_v a_o right_a angle_n and_o forasmuch_o as_o the_o line_n fe_o and_o fg_o which_o be_v the_o less_o segment_n of_o the_o side_n of_o the_o octohedron_n be_v equal_a and_o the_o line_n fk_o be_v common_a to_o they_o both_o and_o the_o angle_n kfg_n and_o kfe_n of_o the_o triangle_n of_o the_o octohedron_n be_v equal_a the_o base_n kg_v and_o ke_o shall_v by_o the_o 4_o of_o the_o first_o be_v equal_a and_o therefore_o the_o angle_n kie_fw-mi and_o kig_n which_o they_o subtend_v be_v equal_a by_o the_o 8._o of_o the_o first_o wherefore_o they_o be_v right_a angle_n by_o the_o 13._o of_o the_o first_o wherefore_o the_o right_a line_n ke_o which_o contain_v in_o power_n the_o two_o line_n ki_v and_o â—e_n by_o the_o 47._o of_o the_o first_o be_v in_o power_n quadruple_a to_o the_o line_n ro_n or_o je_n for_o the_o line_n ri_n be_v prove_v to_o be_v in_o power_n triple_a to_o the_o same_o line_n ro_n but_o the_o line_n ge_z be_v double_a to_o the_o line_n je_n wherefore_o the_o line_n ge_z be_v also_o in_o power_n 〈…〉_o pf_n and_o by_o the_o same_o reason_n may_v be_v prove_v that_o the_o â—est_n of_o the_o eleven_o solid_a angle_n of_o the_o 〈◊〉_d be_v 〈…〉_o the_o section_n of_o every_o one_o of_o the_o side_n of_o the_o octohedron_n namely_o in_o the_o point_n e_o n_o five_o h_o â—_o m_o â—_o d_o s_o q_o t._n wherefore_o there_o be_v 12._o angle_n of_o the_o icosahedron_n moreover_o forasmuch_o as_o every_o one_o of_o the_o base_n of_o the_o octohedron_n do_v each_o contain_v triangle_n of_o the_o icosahedron_n 〈…〉_o pyramid_n abcâ—fp_n which_o be_v the_o half_a of_o the_o octohedron_n the_o triangle_n fcp_n receive_v in_o thâ—_z section_n of_o his_o side_n the_o â—_o triangle_n gm_n and_o the_o triangle_n cpb_n contain_v the_o triangle_n nxs_n and_o thâ—_z triangle_n â—ap_n contain_v the_o triangle_n hnd_n and_o moreover_o the_o triangle_n apf_n contain_v the_o triangle_n edge_n and_o the_o same_o may_v be_v prove_v in_o the_o opposite_a pyramid_n abcfl_fw-mi wherefore_o there_o shall_v be_v eight_o triangleâ—_n and_o forasmuch_o as_o beside_o these_o triangle_n to_o every_o one_o of_o the_o solid_a angle_n of_o the_o octohedron_n 〈◊〉_d subtend_v two_o triangle_n as_o the_o triangle_n keg_n and_o meg_n to_o the_o angle_n f_o and_o the_o triangle_n hnv_n and_o xnv_n to_o the_o angle_n b_o also_o the_o triangle_n nd_n and_o â—ds_n to_o the_o angle_n p_o likewise_o the_o triangleâ—_n dhk_n and_o qhk_v to_o the_o angle_n a_o moreover_o the_o triangle_n eqt_a and_o vqt_a to_o the_o angle_n l_o and_o final_o the_o triangle_n sxm_n and_o txm_n to_o the_o angle_n c_o these_o 12._o triangle_n be_v add_v to_o thâ—â—_n for_o 〈◊〉_d triangle_n shall_v produce_v _o triangle_n equal_a and_o equilater_v couple_v together_o which_o shall_v male_a a_o icosahedron_n by_o the_o 25._o definition_n of_o the_o eleven_o and_o it_o shall_v be_v inscribe_v in_o the_o octohedron_n give_v abcâ—â—l_o by_o the_o first_o definition_n of_o this_o book_n for_o the_o 1â—_o angle_n thereof_o be_v set_v in_o 1â—_o like_a section_n of_o the_o side_n of_o the_o octohedron_n wherefore_o in_o a_o octohedron_n give_v be_v inscribe_v a_o icosahedron_n ¶_o first_n corollary_n the_o side_n of_o a_o equilater_n triangle_n be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n a_o right_a line_n subtend_v within_o the_o triangle_n the_o angle_n which_o be_v contain_v under_o the_o great_a segment_n and_o the_o less_o be_v in_o power_n duple_n to_o the_o less_o segment_n of_o the_o same_o side_n for_o the_o line_n ke_o which_o subtend_v the_o angle_n kfe_n of_o the_o triangle_n afl_fw-mi which_o angle_n kfe_n be_v contain_v under_o the_o two_o segment_n kf_a &_o fe_o be_v prove_v equal_a 〈◊〉_d the_o line_n hk_o which_o contain_v in_o power_n the_o two_o less_o segment_n ha_o and_o ak_o by_o the_o 47._o of_o the_o â—â—rst_a foâ—_n 〈◊〉_d angle_n hak_n be_v 〈…〉_o second_o corollary_n the_o base_n of_o the_o icosahedron_n be_v concentrical_a that_o be_v have_v one_o and_o the_o self_n same_o centre_n with_o the_o base_n of_o the_o octohedron_n which_o contain_v it_o for_o suppose_v that_o 〈…〉_o octohedron_n 〈◊〉_d ecd_v the_o base_a of_o a_o icosahedron_n and_o let_v the_o centre_n of_o the_o base_a abg_o be_v the_o point_n f._n and_o draw_v these_o right_a line_n favorina_n fb_o fc_o and_o fe_o now_o than_o the_o 〈…〉_o to_o the_o two_o line_n fb_o and_o bc_o for_o they_o be_v line_n draw_v from_o the_o centre_n and_o be_v also_o less_o segment_n and_o they_o contain_v the_o 〈…〉_o ¶_o the_o 17._o problem_n the_o 17._o proposition_n in_o a_o octohedron_n give_v to_o inscribe_v a_o dodecahedron_n construction_n svppose_v that_o the_o octohedron_n give_v be_v abgdec_n who_o 12._o â—ides_n let_v be_v cut_v by_o a_o extreme_a and_o mean_a proportion_n as_o in_o the_o former_a proposition_n it_o be_v manifest_a that_o of_o the_o right_a line_n which_o couple_v thâ—se_a section_n be_v make_v 20._o triangle_n of_o which_o 8._o be_v concentrical_a with_o the_o base_n of_o the_o octohedron_n by_o the_o second_o corollary_n of_o the_o former_a proposition_n if_o therefore_o in_o every_o one_o of_o the_o centre_n of_o the_o 20._o triangle_n be_v inscribe_v by_o the_o 1._o of_o this_o book_n every_o one_o of_o the_o â—â—_o â—â—gles_n of_o the_o dodecahedron_n demonstration_n we_o shall_v find_v that_o â—_o angle_n of_o the_o dodecahedron_n be_v set_v in_o the_o 8._o centre_n of_o the_o base_n of_o the_o octohedron_n namely_o these_o angle_n i_o u._fw-mi ct_n oh_o m_o a_o p_o and_o x_o and_o of_o the_o other_o 12._o solid_a angle_n there_o be_v two_o in_o the_o centre_n of_o the_o two_o triangle_n which_o have_v one_o side_n common_a under_o every_o one_o of_o the_o solid_a angle_n of_o the_o octohedron_n namely_o under_o the_o solid_a angle_n a_o the_o two_o solid_a angle_n king_n z_o under_o the_o solid_a angle_n b_o the_o two_o solid_a angle_n h_o t_o under_o the_o solid_a angle_n g_o the_o two_o solid_a angle_n in_o v_z under_o the_o solid_a angle_n d_o the_o two_o solid_a angle_n f_o l_o under_o the_o solid_a angle_n e_o the_o two_o solid_a angle_n s_o n_o under_o the_o solid_a angle_n c_o the_o two_o solid_a angle_n queen_n r_o and_o forasmuch_o as_o in_o the_o octohedron_n be_v six_o solid_a angle_n under_o they_o shall_v be_v subtend_v 12._o solid_a angle_n of_o the_o dodâ—cahedron_n and_o so_o be_v mâ—de_v 20._o solid_a angle_n compose_v of_o 12._o equal_a and_o equilater_v superficial_a pentagon_n as_o it_o be_v 〈◊〉_d by_o the_o 5._o of_o this_o book_n which_o therefore_o contain_v a_o dodecahedron_n by_o the_o 24._o definition_n of_o the_o eleven_o and_o it_o be_v inscribe_v in_o the_o octohedron_n by_o the_o 1._o definition_n of_o this_o book_n for_o that_o every_o one_o of_o the_o base_n of_o the_o octohedron_n do_v receive_v angle_n thereof_o wherefore_o in_o a_o octohedron_n give_v be_v inscribe_v a_o dodecahedron_n ¶_o the_o 18._o problem_n the_o 18._o proposition_n in_o a_o trilater_n and_o equilater_n pyramid_n to_o inscribe_v a_o cube_n construction_n svppose_v that_o there_o be_v a_o trilater_n equilater_n pyramid_n who_o base_a let_v be_v abc_n and_o â—oppe_v the_o point_n d._n and_o let_v it_o be_v comprehend_v in_o a_o sphereâ—_n by_o the_o 13._o of_o the_o 〈◊〉_d and_o lâ—â—_n the_o centre_n of_o that_o sphere_n be_v the_o point_n e._n and_o from_o the_o solid_a angle_n a_o b_o c_o d_o draw_v right_a line_n pass_v by_o the_o centre_n e_o unto_o the_o opposite_a base_n of_o
two_o line_n hif_n and_o tio_n cut_v the_o one_o the_o other_o be_v in_o one_o and_o the_o self_n same_o '_o plain_n by_o the_o 2._o of_o the_o eleven_o and_o therefore_o the_o point_n h_o t_o f_o oh_o be_v in_o one_o &_o the_o self_n same_o plain_a wherforeâ—_n the_o rectangle_n figure_n hoff_z be_v quadrilater_n and_o equilater_n and_o in_o one_o and_o the_o self_n same_o plain_n be_v a_o square_a by_o the_o definition_n of_o a_o square_a and_o by_o the_o same_o reason_n may_v the_o rest_n of_o the_o base_n of_o the_o solid_a be_v prove_v to_o be_v square_n equal_a and_o plain_a or_o superficial_a now_o than_o the_o solid_a be_v comprehend_v of_o 6._o equal_a square_n which_o be_v contain_v of_o 12._o equal_a side_n which_o square_n make_v 8._o solid_a angle_n of_o which_o four_o be_v in_o the_o centre_n of_o the_o base_n oâ—_n the_o pyramid_n and_o the_o other_o 4._o be_v in_o the_o middle_a section_n of_o the_o four_o perdendiculars_n wherefore_o the_o solid_a hoftpgrn_n be_v a_o cube_fw-la by_o the_o 21._o definition_n of_o the_o eleven_o and_o be_v inscribe_v in_o the_o pyramid_n by_o the_o first_o definition_n of_o this_o book_n wherefore_o in_o a_o trilater_n equilater_n pyramid_n give_v be_v inscribe_v a_o cube_fw-la ¶_o a_o corollary_n the_o line_n which_o cut_v into_o two_o equal_a part_n the_o opposite_a side_n of_o the_o pyramid_n be_v triple_a to_o the_o side_n of_o the_o cube_fw-la inscribe_v in_o the_o pyramid_n and_o pass_v by_o the_o centre_n of_o the_o cube_fw-la for_o the_o line_n sev_n who_o three_o part_n the_o line_n si_fw-it be_v cut_v the_o opposite_a side_n cd_o and_o ab_fw-la into_o two_o equll_a part_n but_o the_o line_n ei_o which_o be_v draw_v from_o the_o centre_n of_o the_o cube_fw-la to_o the_o base_a be_v prove_v to_o be_v a_o three_o part_n of_o the_o line_n es_fw-ge wherefore_o the_o side_n of_o the_o cube_fw-la which_o be_v double_a to_o the_o line_n ei_o shall_v be_v a_o three_o part_n of_o the_o whole_a line_n we_o which_o be_v as_o have_v be_v prove_v double_a to_o the_o line_n es._n the_o 19_o problem_n the_o 19_o proposition_n in_o a_o trilater_n equilater_n pyramid_n give_v to_o inscribe_v a_o icosahedron_n svppose_v that_o the_o pyramid_n be_v give_v 〈◊〉_d abâ—dâ—_n every_o one_o of_o who_o sâ—des_n 〈◊〉_d be_v divide_v into_o two_o equal_a part_n in_o the_o poyâ—â—â—â—â—_n m_o king_n l_o p_o n._n construction_n and_o iâ—_z every_o one_o of_o the_o base_n of_o that_o pyramid_n descry_v the_o trianglâ—â—_n lâ—â—_n pmn_n nkl_n and_o 〈…〉_o which_o triangle_n shall_v be_v equilater_n by_o the_o 4._o of_o the_o firâ—t_o â—or_a the_o side_n subtend_v equal_a angle_n of_o the_o pyramid_n contain_v under_o the_o half_n of_o the_o side_n of_o the_o same_o pyramisâ—_n wherefore_o the_o side_n of_o the_o say_a triangle_n be_v equal_a let_v those_o side_n be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n by_o the_o 30._o of_o the_o six_o in_o the_o point_n c_o e_o q_o r_o s_o t_o h_o i_o oh_o five_o a'n_z x._o now_o then_o those_o side_n be_v cut_v into_o the_o self_n same_o proportion_n by_o the_o 2._o of_o the_o fourtenth_n and_o therefore_o they_o make_v the_o liâ—e_z section_n equal_a by_o the_o â—_o part_n of_o the_o nine_o of_o the_o five_o now_o i_o say_v that_o the_o foresay_a point_n doâ—_n receive_v the_o angle_n of_o the_o icosahedron_n inscribe_v in_o the_o pyramid_n abâ—d_n in_o the_o foresay_a triangle_n let_v there_o again_o be_v make_v other_o triangle_n by_o couple_v the_o section_n and_o let_v those_o triangle_n be_v tr_n joh_n ceq_n and_o vxy_n which_o shall_v be_v equilater_n for_o every_o one_o of_o their_o side_n do_v subâ—â—â—d_v equal_a angle_n of_o equilater_n triangle_n and_o those_o say_v equal_a angle_n be_v contain_v under_o equal_a sideâ—_n namely_o under_o the_o great_a segment_n and_o the_o less_o â—_o and_o therefore_o the_o side_n which_o subtend_v those_o angle_n be_v equal_a by_o the_o 4._o of_o the_o first_o now_o let_v we_o prove_v that_o at_o each_o of_o the_o foresay_a point_n as_o for_o example_n at_o t_o be_v set_v the_o solid_a angle_n of_o a_o icosahâ—dronâ—_n demonstration_n forasmuch_o as_o the_o triangle_n trs_n and_o tqo_n be_v equilater_n and_o equal_a the_o 4._o right_a line_n tr_z it_o be_v tq_n and_o to_o shall_v be_v equal_a and_o forasmuch_o as_o â—pnk_v be_v a_o square_n cut_v the_o pyramid_n abâ—d_v into_o two_o equal_a paâ—â—â—â—_n by_o the_o corollay_n of_o the_o second_o of_o this_o bookeâ—_n the_o line_n th._n shall_v be_v in_o power_n duple_n to_o the_o line_n tn_n or_o nh_v by_o the_o 47._o of_o the_o first_o for_o the_o line_n tn_n or_o nh_v be_v equal_a for_o that_o by_o construction_n they_o be_v each_o less_o segment_n and_o the_o line_n rt_n or_o it_o be_v be_v in_o power_n duple_n to_o the_o same_o line_n tn_n or_o nh_v by_o the_o corollary_n of_o the_o 16._o of_o this_o book_n for_o it_o subtend_v the_o angle_n of_o the_o triangle_n contain_v under_o the_o two_o segment_n wherefore_o the_o line_n th._n it_o be_v tr_z tq_n and_o to_o be_v equal_a and_o so_o also_o be_v the_o line_n his_o sir_n rq_n qo_n and_o oh_o which_o subtend_v the_o angle_n at_o the_o point_n t_o equal_a for_o the_o line_n qr_o contain_v in_o power_n the_o two_o line_n pq_n and_o pr._n the_o less_o segment_n which_o two_o line_n the_o line_n th._n also_o contain_v in_o power_n and_o the_o rest_n of_o the_o line_n do_v subtend_v angle_n of_o equilater_n triangle_n contain_v under_o the_o great_a segment_n and_o the_o less_o wherefore_o the_o five_o triangle_n trs_n tsh_o tho_o toq_fw-la tqr_n be_v equilater_n and_o equal_a make_v the_o solid_a angle_n of_o a_o icosahedron_n at_o the_o point_n t_o by_o the_o 16._o of_o the_o thirteen_o in_o the_o side_n pn_n of_o the_o triangle_n p_o nm_o and_o by_o the_o same_o reason_n in_o the_o other_o side_n of_o the_o 4._o triangle_n pnm_n nkl_n fmk_n &_o lfp_n which_o be_v inscribe_v in_o the_o base_n of_o the_o pyramid_n which_o side_n be_v 12â—_n in_o number_n shall_v be_v set_v 12._o angle_n of_o the_o icosahedron_n contain_v under_o 20._o equal_a &_o equilater_n triangle_n of_o which_o fowere_n be_v set_v in_o the_o 4._o base_n of_o the_o pyramid_n namely_o these_o four_o triangle_n trs_n hoi_n ceq_n vxy_n 4._o triangle_n be_v under_o 4._o angle_n of_o the_o pyramid_n that_o be_v the_o four_o triangle_n cix_o ysh_a erv_n tqo_n and_o under_o every_o one_o of_o the_o six_o side_n of_o the_o pyramid_n be_v set_v two_o triangle_n namely_o under_o the_o side_n of_o the_o triangle_n this_o and_o thoâ—_n under_o the_o side_n db_o the_o triangle_n rqe_n and_o rqt_n under_o the_o side_n da_z the_o triangle_n coq_fw-la and_o coi_fw-fr under_o the_o side_n ab_fw-la the_o triangle_n exc_a and_o exuâ—_n under_o the_o side_n bg_o the_o triangle_n sur_n and_o svy_n and_o under_o the_o side_n agnostus_n the_o triangle_n jyh_n and_o jyx._n wherefore_o the_o solid_a be_v contain_v under_o 20._o equilater_n and_o equal_a triangle_n shall_v be_v a_o icosahedron_n by_o the_o 23._o definition_n of_o the_o eleven_o and_o shall_v be_v inscribe_v in_o the_o pyramid_n abâ—d_v by_o the_o first_o definition_n of_o this_o book_n for_o all_o his_o angle_n do_v at_o one_o time_n touch_v the_o base_n of_o the_o pyramid_n wherefore_o in_o a_o trilater_n equilater_n pyramid_n give_v we_o have_v inscribe_v a_o icosahedron_n ¶_o the_o 20._o proposition_n the_o 20._o problem_n in_o a_o trilater_n equilater_n pyramid_n give_v to_o inscribe_v a_o dodecahedron_n svppose_v that_o the_o pyramid_n give_v be_v abgd_v â—che_fw-mi of_o who_o side_n let_v be_v cut_v into_o two_o equal_a part_n and_o draw_v the_o line_n which_o couple_v the_o section_n which_o be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n and_o right_a line_n be_v draw_v by_o the_o section_n shall_v receive_v 20._o triangle_n make_v a_o icosahedron_n as_o in_o the_o former_a proposition_n it_o be_v manifest_a now_o than_o if_o we_o take_v the_o centre_n of_o those_o triangle_n we_o shall_v there_o find_v the_o 20._o angle_n of_o the_o dodecahedron_n inscribe_v in_o it_o by_o the_o 5._o of_o this_o book_n and_o forasmuch_o as_o 4._o base_n of_o the_o foresay_a icosahedron_n be_v concentricall_a with_o the_o base_n of_o the_o pyramid_n as_o it_o be_v prove_v in_o the_o 2._o corollary_n of_o the_o 6._o of_o this_o book_n there_o shall_v be_v place_v 4â—_o angle_n of_o the_o dodecahedron_n namely_o the_o 4._o angle_n e_o f_o h_o d_o in_o the_o 4._o centre_n of_o the_o base_n and_o of_o the_o other_o 16._o angle_n under_o every_o one_o of_o the_o 6._o side_n of_o the_o pyramid_n be_v subtend_v two_o namely_o under_o the_o side_n ad_fw-la the_o angle_n ck_o under_o the_o side_n bd_o the_o angle_n li_n under_o the_o
side_n of_o the_o cube_n inscribe_v be_v to_o the_o great_a segment_n namely_o to_o the_o side_n of_o the_o dodecahedron_n circumscribe_v in_o a_o extreme_a and_o mean_a proportion_n wherefore_o by_o conversion_n the_o great_a segment_n that_o be_v the_o side_n of_o the_o dodecahedron_n be_v to_o the_o whole_a namely_o to_o the_o side_n of_o the_o cube_n inscribe_v in_o the_o converse_n proportion_n of_o a_o extreme_a and_o mean_a proportion_n by_o the_o 13._o definition_n of_o the_o five_o ¶_o the_o 19_o proposition_n the_o side_n of_o a_o octohedron_n be_v sesquialter_fw-la to_o the_o side_n of_o a_o pyramid_n inscribe_v in_o it_o for_o by_o the_o corollary_n of_o the_o 14._o of_o the_o thirteen_o the_o octohedron_n be_v custe_a into_o two_o quadrilater_n pyramid_n one_o of_o which_o let_v be_v abgdf_n construction_n and_o let_v the_o centre_n of_o the_o circle_n which_o contain_v the_o 4._o base_n of_o the_o octohedron_n be_v king_n e_o i_o c._n and_o drâ—w_o these_o right_a line_n ke_o â—i_n i_fw-mi ck_fw-mi and_o ec_o whereforâ—_n kâ—ic_n be_v a_o square_a and_o one_o of_o the_o base_n of_o the_o cube_fw-la inscribe_v in_o the_o octohedron_n by_o the_o 4._o of_o the_o fifteen_o and_o forasmuch_o as_o the_o angle_n of_o a_o cube_fw-la and_o of_o the_o pyramid_n inscribe_v in_o it_o be_v for_o in_o the_o centre_n of_o the_o base_n of_o the_o octohedron_n circumscribe_v about_o the_o cube_fw-la by_o the_o 6â—_o of_o the_o fifteen_o and_o the_o side_n of_o the_o pyramid_n couple_v the_o opposite_a angleâ—_n of_o the_o base_a of_o thâ—_z cube_fw-la by_o the_o 1._o of_o the_o fifteen_o it_o be_v manifest_a that_o the_o line_n ec_o be_v the_o side_n of_o the_o pyramid_n inscribe_v in_o the_o octohedron_n abgdf_n then_o i_o say_v that_o gd_v the_o side_n of_o the_o octohedron_n be_v sesquialter_fw-la to_o ec_o the_o side_n of_o the_o pyramid_n inscribe_v in_o it_o from_o the_o point_n a_o draw_v to_o the_o base_n bg_o and_o fd_o perpendicular_n a_o and_o amâ—_n which_o by_o the_o corollary_n of_o the_o 12._o of_o the_o thirteen_o shall_v pass_v by_o the_o centre_n e_o and_o c._n demonstration_n and_o draw_v the_o line_n nm_o now_o forasmuch_o as_o bgdf_n be_v a_o square_a by_o the_o 14._o of_o the_o thirteen_o the_o line_n ng_z and_o md_o shall_v be_v parallel_n and_o equal_a for_o the_o line_n bg_o and_o fd_o be_v by_o the_o perpendicular_n cut_v into_o two_o equal_a part_n in_o the_o point_n n_o and_o m_o by_o the_o 3._o of_o the_o three_o wherefore_o the_o line_n nm_o and_o gd_o shall_v be_v parallel_n and_o equal_a by_o the_o 33._o of_o the_o first_o and_o forasmuch_o as_o the_o line_n a_o and_o be_o which_o be_v the_o perpendicular_n of_o equal_a and_o likeâ—_n triangle_n be_v câ—t_v a_o like_a in_o the_o point_n â—_o and_o c_o the_o line_n ec_o and_o nmâ—_n shall_v be_v parallel_n by_o the_o 2._o of_o the_o six_o and_o therefore_o by_o the_o corollary_n of_o the_o same_o the_o triangle_n aec_o and_o anm_n shall_v be_v like_a wherefore_o as_o the_o line_n a_o be_v to_o the_o line_n ae_n so_o be_v the_o line_n nm_o to_o the_o line_n ec_o by_o the_o 4._o of_o the_o six_o but_o the_o line_n a_o be_v sesquialter_fw-la to_o the_o line_n ae_n for_o the_o line_n ae_n be_v duple_n to_o the_o line_n en_fw-fr by_o the_o corollary_n of_o the_o 12â—_n of_o the_o thirtenthâ—_n wherefore_o the_o line_n nm_o or_o the_o line_n gd_o which_o be_v equal_a unto_o it_o be_v sesquialter_fw-la to_o the_o line_n ec_o wherefore_o gd_v the_o side_n of_o the_o octohedron_n be_v sesquialter_fw-la to_o ec_o the_o sidâ—_n of_o the_o pyramid_n inscribe_v in_o it_o ¶_o the_o _o proposition_n if_o from_o the_o power_n of_o the_o diameter_n of_o a_o icosahedron_n be_v take_v away_o the_o power_n triple_a of_o the_o side_n of_o the_o cube_fw-la inscribe_v in_o the_o icosahedron_n the_o power_n remain_v shall_v be_v sesquitertia_fw-la to_o the_o power_n of_o the_o side_n of_o the_o icosahedron_n let_v there_o be_v take_v a_o icosahedron_n abgd_v and_o lâ—â—_n two_o base_n of_o the_o cube_fw-la inscribe_v in_o it_o join_v together_o be_v ehkl_n and_o lkfc_n and_o let_v the_o diameter_n of_o the_o cube_fw-la be_v fh_o and_o the_o side_n be_v eh_o and_o let_v the_o diameter_n of_o the_o icosahedron_n be_v â—g_z and_o the_o side_n be_v ab_fw-la then_o i_o say_v that_o if_o from_o the_o power_n of_o the_o diameter_n gb_o be_v take_v away_o the_o power_n triple_a of_o eh_o the_o side_n of_o the_o cube_fw-la demonstration_n the_o power_n remain_v shall_v be_v sesquetertia_fw-la to_o the_o power_n of_o ab_fw-la the_o side_n of_o the_o icosahedron_n for_o forasmuch_o as_o the_o centre_n of_o inscribe_v and_o circumscribe_v figure_n be_v in_o one_o &_o the_o self_n same_o point_n by_o the_o corollary_n of_o the_o 21._o of_o the_o 〈◊〉_d the_o diameter_n bg_o and_o fh_o shall_v in_o one_o and_o the_o self_n same_o point_n cut_v the_o one_o the_o other_o into_o two_o equal_a part_n for_o we_o have_v before_o by_o the_o same_o corollary_n teach_v that_o the_o top_n of_o equal_a and_o like_a pyramid_n do_v in_o that_o point_n concur_v let_v that_o point_n be_v the_o centre_n i_o now_o the_o angle_n of_o the_o cube_fw-la which_o be_v at_o the_o point_n fletcher_n and_o h_o be_v set_v at_o the_o centre_n of_o the_o base_n of_o the_o icosahedron_n by_o the_o 11._o of_o the_o fivetenthâ—_n wherefore_o the_o line_n fh_o shall_v be_v perpendicular_a to_o both_o the_o base_n of_o the_o icosahedron_n by_o the_o corollary_n of_o the_o assumpt_n of_o the_o 16._o of_o the_o twelve_o wherefore_o the_o line_n ib_n contain_v in_o power_n the_o two_o line_n ih_o and_o hb_o by_o the_o 47._o of_o the_o first_o but_o the_o line_n hb_o be_v draw_v from_o the_o centre_n of_o the_o circle_n which_o contain_v the_o base_a of_o the_o icosahedron_n namely_o the_o angle_n b_o be_v place_v in_o the_o circumference_n and_o the_o point_n h_o be_v the_o centre_n wherefore_o the_o whole_a line_n bg_o contain_v in_o power_n the_o whole_a line_n fh_a and_o the_o diameter_n of_o the_o circle_n namely_o the_o double_a of_o the_o line_n bh_o by_o the_o 15â—_n of_o the_o five_o but_o the_o diameter_n which_o be_v double_a to_o the_o line_n hb_o be_v in_o power_n sesquitertiâ—_fw-la to_o the_o side_n of_o the_o equilater_n triangle_n inscribe_v in_o the_o same_o circleâ—_n by_o the_o corollary_n of_o the_o â—â—_o of_o the_o thirteen_o for_o it_o be_v in_o proportion_n to_o the_o sideâ—_n as_o the_o side_n be_v to_o the_o perpendicular_a by_o the_o corollary_n of_o the_o 8._o of_o thâ—â—ixth_n and_o fh_o the_o diameter_n of_o the_o cube_fw-la be_v in_o power_n triple_a to_o eh_o the_o side_n of_o the_o same_o cube_fw-la by_o the_o 15â—_n of_o the_o thirteen_o if_o therefore_o from_o the_o power_n of_o the_o diameter_n bg_o be_v take_v away_o the_o power_n triple_a of_o eh_o the_o side_n of_o the_o cube_fw-la inscribedâ—_n that_o isâ—_n the_o power_n of_o the_o line_n fh_o the_o residue_n namely_o the_o power_n of_o the_o diameter_n of_o the_o circle_n which_o be_v duple_n to_o the_o line_n hb_o shall_v be_v sesquitertiâ—_fw-la to_o the_o side_n of_o the_o triangle_n inscribe_v in_o that_o circle_n which_o self_n side_n be_v ab_fw-la the_o side_n of_o the_o icosahedron_n if_o therefore_o from_o the_o power_n of_o the_o diameter_n of_o a_o icosahedron_n be_v take_v away_o the_o power_n triple_a of_o the_o side_n of_o the_o cube_fw-la inscribe_v in_o the_o icosahedron_n the_o power_n remain_v shall_v be_v sâ—squitertia_fw-la â—o_o the_o power_n of_o the_o side_n of_o the_o icosahedron_n a_o corollary_n the_o diameter_n of_o the_o icosahedron_n contain_v in_o power_n two_o line_n namely_o the_o diameter_n of_o the_o cube_fw-la inscribe_v which_o couple_v the_o centre_n of_o the_o opposite_a base_n and_o the_o diameter_n of_o the_o circle_n which_o contain_v the_o base_a of_o the_o icosahedron_n for_o it_o be_v manifest_a that_o bg_o the_o diameter_n contain_v to_o power_n the_o line_n fh_o which_o douple_v the_o centre_n and_o the_o double_a of_o the_o line_n bh_o that_o be_v the_o diameter_n of_o the_o circle_n contain_v the_o basâ—_n wherein_o iâ—_z the_o centre_n hâ—_n ¶_o the_o 21._o proposition_n the_o side_n of_o a_o dodecaâ—edron_n be_v the_o less_o segment_n of_o that_o right_a line_n which_o be_v in_o power_n duple_n to_o the_o side_n of_o the_o octohedron_n inscribe_v in_o the_o same_o dodecahedron_n let_v there_o be_v take_v a_o dodecahedron_n abâ—dct_n one_o of_o who_o side_n let_v be_v ab_fw-la and_o let_v the_o octohedron_n inscribe_v in_o the_o dodecahedron_n be_v eflki_n one_o of_o who_o side_n let_v be_v ef._n then_o i_o say_v that_o ab_fw-la the_o side_n of_o the_o dodecahedron_n be_v the_o less_o segment_n of_o a_o certain_a right_n â—ine_o â—_o cut_v by_o a_o extreme_a and_o mean_a
same_o the_o cube_fw-la contain_v 12_o namely_o be_v sesquialter_fw-la to_o the_o pyramid_n wherefore_o of_o what_o part_n the_o cube_fw-la contain_v 12_o of_o the_o same_o the_o whole_a octohedron_n which_o be_v double_a to_o the_o pyramid_n abdfc_n contain_v 54._o which_o 54._o have_v to_o 12._o quadruple_a sesquialter_fw-la proportion_n wherefore_o the_o whole_a octohedron_n be_v to_o the_o cube_fw-la inscribe_v in_o it_o in_o quadruple_a sesquialter_fw-la proportion_n wherefore_o we_o have_v prove_v that_o a_o octohedron_n give_v be_v quadruple_a sesquialter_fw-la to_o a_o cube_fw-la inscribe_v in_o it_o ¶_o a_o corollary_n a_o octohedron_n be_v to_o a_o cube_fw-la inscribe_v in_o it_o in_o that_o proportion_n that_o the_o square_n of_o their_o side_n be_v for_o by_o the_o 14._o of_o this_o book_n the_o side_n of_o the_o octohedron_n be_v in_o power_n quadruple_a sesquialter_fw-la to_o the_o side_n of_o the_o cube_fw-la inscribe_v in_o it_o ¶_o the_o 29._o proposition_n to_o prove_v that_o a_o octohedron_n give_v be_v â—_z tredecuple_n sesquialter_fw-la to_o a_o trilater_n equilater_n pyramid_n inscribe_v in_o it_o let_v the_o octohedron_n geâ—en_o be_v ab_fw-la in_o which_o let_v there_o be_v inscribe_v a_o cube_fw-la fced_fw-la by_o the_o 4._o of_o the_o fifteen_o and_o in_o the_o cube_fw-la let_v there_o be_v inscribe_v a_o pyramid_n fegd_v by_o the_o â—_o of_o the_o fifteen_o and_o forasmuch_o as_o the_o angle_n of_o the_o pyramid_n be_v by_o the_o same_o first_o of_o the_o fifteen_o set_v in_o the_o angle_n of_o the_o cube_fw-la and_o the_o angle_n of_o the_o cube_fw-la be_v set_v in_o the_o centre_n of_o the_o base_n of_o the_o octohedron_n namely_o in_o the_o point_n f_o e_o c_o d_o g_o by_o the_o 4._o of_o the_o fifteen_o wherefore_o the_o angle_n of_o the_o pyramid_n be_v set_v in_o the_o centre_n f_o c_o e_o d_o of_o the_o octohedron_n wherefore_o the_o pyramid_n fedg_n be_v inscribe_v in_o the_o octohedron_n by_o the_o 6._o of_o the_o fifteen_o and_o forasmuch_o as_o the_o octohedron_n ab_fw-la be_v to_o the_o cube_fw-la fced_fw-la inscribe_v in_o it_o quadruple_a sesquialter_fw-la by_o the_o former_a proposition_n and_o the_o cube_fw-la cdef_n be_v to_o the_o pyramid_n fedg_v inscribe_v in_o it_o triple_a by_o the_o 25._o of_o book_n wherefore_o three_o magnitude_n be_v give_v namely_o the_o octohedron_n the_o cube_fw-la and_o the_o pyramid_n the_o proportion_n of_o the_o extreme_n namely_o of_o the_o octohedron_n to_o the_o pyramid_n be_v make_v of_o the_o proportion_n of_o the_o mean_n namely_o of_o the_o octohedron_n to_o the_o cube_fw-la and_o of_o the_o cube_fw-la to_o the_o pyramid_n as_o it_o be_v easy_a to_o see_v by_o the_o declaration_n upon_o the_o 10._o definition_n of_o the_o five_o now_o then_o multiply_v the_o quantity_n or_o denomination_n of_o the_o proportion_n namely_o of_o the_o octohedron_n to_o the_o cube_fw-la which_o be_v 4_o 1_o â—_o and_o of_o the_o cube_fw-la to_o the_o pyramid_n which_o be_v 3_o as_o be_v teach_v in_o the_o definition_n of_o the_o six_o there_o shall_v be_v produce_v 13_o 1_o â—_o namely_o the_o proportion_n of_o the_o octohedron_n to_o the_o pyramid_n inscribe_v in_o it_o for_o 4_o ½_n multiply_v by_o 3._o produce_v 13_o ½_n wherefore_o the_o octohedron_n be_v to_o the_o pyramid_n inscribe_v in_o it_o in_o tredecuple_n sesquialter_fw-la proportion_n wherefore_o we_o have_v prove_v that_o a_o octohedron_n be_v to_o a_o trilater_n equilater_n pyramid_n inscribe_v in_o it_o in_o tredecuple_n sesquialter_fw-la proportion_n ¶_o the_o 30._o proposition_n to_o prove_v that_o a_o trilater_n equilater_n pyramid_n be_v noncuple_n to_o a_o cube_fw-la inscribe_v in_o it_o svppose_v that_o the_o pyramid_n give_v be_v abcd_o who_o two_o base_n let_v be_v abc_n and_o dbc_n and_o let_v their_o centre_n be_v the_o point_n g_o and_o i._n and_o from_o the_o angle_n a_o draw_v unto_o the_o base_a bc_o a_o perpendicular_a ae_n likewise_o from_o the_o angle_n d_o draw_v unto_o the_o same_o base_a bc_o a_o perpendicular_a de_fw-fr and_o they_o shall_v concur_v in_o the_o section_n e_o by_o the_o 3._o of_o the_o three_o and_o in_o they_o shall_v be_v the_o centre_n g_o an_o i_o by_o the_o corollary_n of_o the_o first_o of_o the_o three_o and_o forasmuch_o as_o the_o line_n ad_fw-la be_v the_o side_n of_o the_o pyramid_n the_o same_o ad_fw-la shall_v be_v the_o diameter_n of_o the_o base_a of_o the_o cube_fw-la which_o contain_v the_o pyramid_n by_o the_o 1_o of_o the_o fivetenth_n demonstration_n draw_v the_o line_n give_v and_o forasmuch_o as_o the_o line_n give_v couple_v the_o centreâ—_n of_o the_o base_n of_o the_o pyramid_n the_o say_a line_n give_v shall_v be_v the_o diameter_n of_o the_o base_a of_o the_o cube_fw-la inscribe_v in_o the_o pyramid_n by_o the_o 18._o of_o the_o fifteen_o and_o forasmuch_o as_o the_o line_n agnostus_n be_v double_a to_o the_o line_n ge_z by_o the_o corollarye_a of_o the_o twelve_o of_o the_o thirteen_o the_o whole_a line_n ae_n shall_v be_v triple_a to_o the_o line_n ge_z and_o so_o be_v also_o the_o line_n de_fw-fr to_o the_o line_n je_n wherefore_o the_o line_n ad_fw-la and_o give_v be_v parallel_n by_o the_o 2._o of_o the_o six_o and_o therefore_o the_o triangle_n aed_n and_o gei_n be_v likeâ—_n by_o the_o corollary_n of_o the_o same_o and_o forasmuch_o as_o the_o triangle_n aed_n and_o gei_n be_v like_a the_o line_n adâ—_n shall_v be_v treble_v to_o the_o line_n give_v by_o the_o 4._o of_o the_o six_o but_o the_o line_n ad_fw-la be_v the_o diameter_n of_o the_o base_a of_o the_o cube_fw-la circumscribe_v about_o the_o pyramid_n abcd_o and_o the_o line_n give_v be_v the_o diameter_n of_o the_o base_a of_o the_o cube_fw-la inscribe_v in_o the_o pyramid_n abcd_o but_o the_o diameter_n of_o the_o base_n be_v equemultiplices_fw-la to_o the_o side_n namely_o be_v in_o power_n duple_n wherefore_o the_o side_n of_o the_o cube_fw-la circumscribe_v about_o the_o pyramid_n abcd_o be_v triple_a to_o the_o side_n of_o the_o cube_fw-la inscribe_v in_o the_o same_o pyramid_n by_o the_o 15._o of_o the_o five_o but_o like_o cube_n be_v in_o triple_a proportion_n the_o one_o to_o the_o other_o of_o that_o in_o which_o their_o side_n be_v by_o the_o 33._o of_o the_o eleven_o and_o the_o side_n be_v in_o triple_a proportion_n the_o one_o to_o the_o other_o wherefore_o triple_a take_v three_o time_n bring_v forth_o twenty_o sevencuple_n which_o be_v 27._o to_o 1_o for_o the_o 4._o term_n 27.9.3.1_o be_v set_v in_o triple_a proportion_n the_o proportion_n of_o the_o first_o to_o the_o four_o namely_o of_o 27._o to_o 1._o shall_v be_v treble_v to_o the_o proportion_n of_o the_o first_o to_o the_o second_o namely_o of_o 27._o to_o 9_o by_o the_o 10._o definition_n of_o the_o five_o which_o proportion_n of_o 27._o to_o 1._o be_v the_o proportion_n of_o the_o side_n triple_a which_o proportion_n also_o be_v find_v in_o like_o solides_fw-la wherefore_o of_o what_o part_n the_o cube_fw-la circumscribe_v contain_v 27._o of_o the_o same_o the_o cube_fw-la inscribe_v contain_v one_o but_o of_o what_o part_n the_o cube_fw-la circumscribe_v contain_v 27._o of_o the_o same_o the_o pyramid_n inscribe_v in_o it_o contain_v 9_o by_o the_o 25._o of_o this_o book_n wherefore_o of_o what_o part_n the_o pyramid_n ab_fw-la cd_o contain_v 9_o of_o the_o same_o the_o cube_fw-la inscribe_v in_o the_o pyramid_n contain_v one_o wherefore_o we_o have_v prove_v that_o a_o trilater_n and_o equilater_n pyramid_n be_v nonâ—cuple_n to_o a_o cube_fw-la inscribe_v in_o it_o ¶_o the_o 31._o proposition_n a_o octohedron_n have_v to_o a_o icosohedron_n inscribe_v in_o it_o that_o proportion_n which_o two_o base_n of_o the_o octohedron_n have_v to_o five_o base_n of_o the_o icosahedron_n svppose_v that_o the_o octohedron_n give_v be_v abcd_o and_o let_v the_o icosahedron_n inscribe_v in_o it_o be_v fghmklio_n then_o i_o say_v that_o the_o octohedron_n be_v to_o the_o icosahedron_n as_o two_o base_n of_o the_o octohedron_n be_v to_o five_o base_n of_o the_o icosahedron_n for_o forasmuch_o as_o the_o solid_a of_o the_o octohedron_n consist_v of_o eight_o pyramid_n set_v upon_o the_o base_n of_o the_o octohedron_n demonstration_n and_o have_v to_o their_o altitude_n a_o perpendicular_a line_n draw_v from_o the_o centre_n to_o the_o base_a let_v that_o perpendicular_a be_v er_fw-mi or_o es_fw-ge be_v draw_v from_o the_o centre_n e_o which_o centre_n be_v common_a to_o either_o of_o the_o solid_n by_o the_o corollary_n of_o the_o 21._o of_o the_o fifteen_o to_o the_o centre_n of_o the_o base_n namely_o to_o the_o point_n r_o and_o s._n wherefore_o for_o that_o three_o pyramid_n be_v equal_a and_o like_a they_o shall_v be_v equal_a to_o a_o prism_n set_v upon_o the_o self_n same_o base_a and_o under_o the_o self_n same_o altitude_n by_o the_o corollary_n of_o the_o seven_o of_o the_o twelve_o but_o
from_o a_o cube_fw-la and_o a_o octohedron_n compose_v into_o one_o solid_a there_o shall_v be_v leave_v a_o exocthedron_n moreover_o the_o solid_a angle_n take_v away_o from_o two_o pyramid_n compose_v into_o one_o solid_a there_o shall_v be_v leave_v a_o octohedron_n flussas_n after_o this_o set_v forth_o certain_a passion_n and_o property_n of_o the_o five_o simple_a regular_a body_n which_o although_o he_o demonstrate_v not_o yet_o be_v they_o not_o hard_a to_o be_v demonstrate_v if_o we_o well_o pease_n and_o conceive_v that_o which_o in_o the_o former_a book_n have_v be_v teach_v touch_v those_o solid_n of_o the_o nature_n of_o a_o trilater_n and_o equilater_n pyramid_n a_o trilater_n equilater_n pyramid_n be_v divide_v into_o two_o equal_a part_n by_o three_o equal_a square_n which_o in_o the_o centre_n of_o the_o pyramid_n cut_v the_o one_o the_o other_o into_o two_o equal_a part_n and_o perpendicular_o and_o who_o angle_n be_v set_v in_o the_o middle_a section_n of_o the_o side_n of_o the_o pyramid_n from_o a_o pyramid_n be_v take_v away_o 4._o pyramid_n like_a unto_o the_o whole_a which_o utter_o take_v away_o the_o side_n of_o the_o pyramid_n and_o that_o which_o be_v leave_v be_v a_o octohedron_n inscribe_v in_o the_o pyramy_n in_o which_o all_o the_o solid_n inscribe_v in_o the_o pyramid_n be_v contain_v a_o perpendicular_a draw_v from_o the_o angle_n of_o the_o pyramid_n to_o the_o base_a be_v double_a to_o the_o diameter_n of_o the_o cube_fw-la inscribe_v in_o it_o and_o a_o right_a line_n couple_v the_o middle_a section_n of_o the_o opposite_a side_n of_o the_o pyramid_n be_v triple_a to_o the_o side_n of_o the_o self_n same_o cube_fw-la the_o side_n also_o of_o the_o pyramid_n be_v triple_a to_o the_o diameter_n of_o the_o base_a of_o the_o cube_fw-la wherefore_o the_o same_o side_n of_o the_o pyramid_n be_v in_o power_n duple_n to_o the_o right_a line_n which_o couple_v the_o middle_a section_n of_o the_o opposite_a side_n and_o it_o be_v in_o power_n sesquialter_fw-la to_o the_o perpendicular_a which_o be_v draw_v from_o the_o angle_n to_o the_o base_a wherefore_o the_o perpendicular_a be_v in_o power_n sesquitertia_fw-la to_o the_o line_n which_o couple_v the_o middle_a section_n of_o the_o opposite_a side_n a_o pyramid_n and_o a_o octohedron_n inscribe_v in_o it_o also_o a_o icosahedron_n inscribe_v in_o the_o same_o octohedron_n do_v contain_v one_o and_o the_o self_n same_o sphere_n of_o the_o nature_n of_o a_o octohedron_n four_o perpendicular_n of_o a_o octohedron_n draw_v in_o 4._o base_n thereof_o from_o two_o opposite_a angle_n of_o the_o say_v octohedron_n and_o couple_v together_o by_o those_o 4._o base_n describe_v a_o rhombus_fw-la or_o diamond_n figure_n one_o of_o who_o diameter_n be_v in_o power_n duple_n to_o the_o other_o diameter_n for_o it_o have_v the_o same_o proportion_n that_o the_o diameter_n of_o the_o octohedron_n have_v to_o the_o side_n of_o the_o octohedron_n a_o octohedron_n &_o a_o icosahedron_n inscribe_v in_o it_o do_v contain_v one_o and_o the_o self_n same_o sphere_n the_o diameter_n of_o the_o solid_a of_o the_o octohedron_n be_v in_o power_n sesquialter_fw-la to_o the_o diameter_n of_o the_o circle_n which_o contain_v the_o base_a and_o be_v in_o power_n triple_a to_o the_o right_a line_n which_o couple_v the_o centre_n of_o the_o opposite_a base_n and_o be_v in_o power_n â—_z duple_fw-fr superbipartiens_fw-la tercias_fw-la to_o the_o perpendicular_a or_o side_n of_o the_o foresay_a rhombus_fw-la and_o moreover_o be_v in_o length_n triple_a to_o the_o line_n which_o couple_v the_o centre_n of_o the_o next_o base_n the_o angle_n of_o the_o inclination_n of_o the_o base_n of_o the_o octohedron_n do_v with_o the_o angle_n of_o the_o inclination_n of_o the_o base_n of_o the_o pyramid_n make_v angle_n equal_a to_o two_o right_a angle_n of_o the_o nature_n of_o a_o cube_n the_o diameter_n of_o a_o cube_fw-la be_v in_o power_n sesquialter_fw-la to_o the_o diameter_n of_o his_o base_a and_o be_v in_o power_n triple_a to_o his_o side_n and_o unto_o the_o line_n which_o couple_v the_o centre_n of_o the_o next_o base_n it_o be_v in_o power_n sextuple_a moreover_o the_o side_n of_o the_o cube_fw-la be_v to_o the_o side_n of_o the_o icosahedron_n inscribe_v in_o it_o as_o the_o whole_a be_v to_o the_o great_a segment_n unto_o the_o side_n of_o the_o dodecahedron_n it_o be_v as_o the_o whole_a be_v to_o the_o less_o segment_n unto_o the_o side_n of_o the_o octohedron_n it_o be_v in_o power_n duple_n and_o unto_o the_o side_n of_o the_o pyramid_n it_o be_v in_o power_n subduple_n moreover_o the_o cube_fw-la be_v triple_a to_o the_o pyramid_n but_o to_o the_o cube_fw-la the_o dodecahedron_n be_v in_o a_o manner_n duple_n wherefore_o the_o same_o dodecahedron_n be_v in_o a_o manner_n sextuple_a to_o the_o say_a pyramid_n of_o the_o nature_n of_o a_o icosahedron_n five_o triangle_n of_o a_o icosahedron_n do_v make_v a_o solid_a angle_n the_o base_n of_o which_o triangle_n make_v a_o pentagon_n if_o therefore_o from_o the_o opposite_a base_n of_o the_o icosahedron_n be_v take_v the_o other_o pentagon_n by_o they_o describe_v these_o pentagon_n shall_v in_o such_o sort_n cut_v the_o diameter_n of_o the_o icosahedron_n which_o couple_v the_o foresaid_a opposite_a angle_n that_o that_o part_n which_o be_v contain_v between_o the_o plain_n of_o those_o two_o pentagon_n shall_v be_v the_o great_a segment_n and_o the_o residue_n which_o be_v draw_v from_o the_o plain_a to_o the_o angle_n shall_v be_v the_o less_o segment_n if_o the_o opposite_a angle_n of_o two_o base_n join_v together_o be_v couple_v by_o a_o right_a line_n the_o great_a segment_n of_o that_o right_a line_n be_v the_o side_n of_o the_o icosahedron_n a_o line_n draw_v from_o the_o centre_n of_o the_o icosahedron_n to_o the_o angle_n be_v in_o power_n quintuple_a to_o half_a that_o line_n which_o be_v take_v between_o the_o pentagon_n or_o of_o the_o half_a of_o 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