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
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 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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
first_o demonstration_n demonstration_n lead_v to_o a_o impossibility_n this_o proposition_n in_o discreet_a quantity_n answer_v to_o the_o 23._o proposition_n of_o the_o five_o book_n in_o continual_a quantity_n this_o and_o the_o eleven_o proposition_n follow_v declare_v the_o passion_n and_o property_n ofâ—_n prime_a number_n demonstration_n lead_v to_o a_o impossibility_n this_o be_v the_o converse_n of_o the_o former_a proposition_n demonstration_n lead_v to_o a_o absurdity_n demonstration_n lead_v to_o a_o absurdity_n demonstration_n lead_v to_o a_o absurdity_n demonstration_n demonstration_n demonstration_n demonstration_n of_o the_o first_o part_n lead_v to_o a_o absurdity_n demonstration_n of_o the_o second_o part_n which_o be_v the_o conâ—câ—se_n of_o the_o first_o lean_v also_o to_o a_o absurdity_n demonstrasion_n lead_v to_o a_o absurdity_n demonstrasion_n a_o corollary_n â—â—ded_v by_o campave_n demonstration_n lâ—ading_v to_o a_o impossibility_n an_o other_o demonstration_n demonstration_n two_o case_n in_o this_o proposition_n the_o first_o case_n the_o second_o case_n demonstration_n demonstration_n lead_v to_o a_o absurdity_n a_o corollary_n add_v by_o campane_n two_o case_n in_o this_o proposition_n the_o first_o case_n demonstration_n lead_v to_o a_o absurdity_n the_o second_o caseâ—_n demonstration_n lead_v to_o a_o absurdity_n demonstration_n lead_v to_o a_o impossibility_n two_o case_n in_o this_o proposition_n the_o first_o case_n demonstration_n leaâ—iâ—g_v â—o_o a_o absurdity_n the_o second_o case_n demonstration_n lead_v to_o a_o absurdity_n a_o corollary_n demonstration_n the_o converse_n of_o the_o former_a proposition_n demonstration_n construction_n demonstration_n leâ—ding_v to_o a_o absurdity_n a_o corollary_n adâ—ed_v by_o campane_n how_o to_o â—inde_v out_o the_o second_o least_o number_n and_o the_o three_o and_o so_o â—orth_o infinite_o how_o to_o siâ—â—_n out_o the_o least_o â—â—mâ—_n a_o conâ—ayâ—â—g_n â—â—e_z paâ—â—s_n of_o part_n the_o arguâ—â—â—â—_n of_o the_o eight_o book_n demonstration_n lead_v to_o a_o absurdity_n construction_n demonstration_n this_o proposition_n be_v the_o â—â—uerse_a of_o the_o first_o demonstrationâ—_n two_o case_n in_o this_o proposition_n the_o first_o case_n demonstration_n lead_v to_o a_o absurdity_n the_o second_o case_n demonstration_n this_o proposition_n in_o number_n answer_v to_o the_o of_o the_o six_o touch_v parellelogramme_n construction_n demonstration_n an_o other_o demonstration_n after_o campane_n demonstration_n demonstration_n lead_v to_o a_o impossibility_n demonstration_n a_o corollary_n add_v by_o flussates_n construction_n demonstration_n this_o proposition_n be_v the_o converse_n of_o the_o former_a construction_n demonstration_n the_o first_o part_n of_o this_o proposition_n demonstrate_v the_o second_o part_n demonstrate_v construction_n the_o first_o part_n of_o this_o prâ—position_n demonstrate_v the_o second_o part_n demonstrate_v construction_n demonstration_n the_o first_o part_n of_o this_o proposition_n the_o second_o part_n be_v the_o converse_n of_o the_o first_o the_o first_o part_n of_o this_o proposition_n the_o second_o part_n be_v the_o converse_n of_o the_o first_o a_o negative_a proportion_n the_o first_o part_n of_o this_o proposition_n the_o second_o part_n be_v the_o converse_n of_o the_o first_o a_o negative_a proposition_n the_o first_o part_n of_o this_o proposition_n the_o second_o part_n be_v the_o converse_n of_o the_o first_o demonstration_n of_o the_o fiâ—st_a part_n of_o this_o proposition_n demonstration_n of_o the_o second_o part_n demonstration_n of_o the_o first_o part_n of_o this_o proposition_n the_o second_o part_n this_o proposition_n be_v the_o converse_n of_o the_o 18._o proposition_n construction_n demonstration_n this_o proposition_n be_v the_o converse_n of_o the_o 19_o proposition_n construction_n demonstration_n demonstration_n demonstration_n demonstration_n demonstration_n a_o corollary_n add_v by_o flussates_n construction_n construction_n demonstration_n a_o corollary_n add_v by_o flussates_n another_o corollary_n add_v by_o flussates_n the_o argument_n of_o the_o niâ—th_o book_n demonstration_n this_o proposition_n be_v the_o converse_n oâ—_n tâ—e_z formâ—â—_n demonstration_n a_o corollary_n aâ—ded_v by_o campane_n demonstration_n demonstration_n demonstration_n a_o corollary_n add_v by_o campane_n demonstration_n demonstration_n demonstration_n of_o the_o first_o part_n the_o second_o part_n demonstrate_v demostration_n of_o the_o three_o part_n demostration_n of_o the_o first_o part_n of_o this_o proposition_n the_o second_o pâ—rt_n demonstrate_v demonstration_n of_o the_o first_o part_n leave_v to_o a_o absurdity_n demonstration_n of_o the_o â—â—cond_a pâ—â—â—_n lead_v alâ—o_o to_o a_o absurdity_n demonstration_n demonstration_n lead_v to_o a_o absurdity_n an_o other_o demonstration_n aâ—ter_n flussates_n demonstration_n lead_v to_o a_o absurdity_n an_o other_o demonstration_n after_o campane_n demonstration_n lead_v to_o a_o absurdity_n a_o proposition_n add_v by_o campane_n construction_n demonstration_n demonstration_n to_o prove_v that_o the_o number_n a_o and_o c_o be_v prime_a to_o b._n demonstratiou_n this_o proposition_n be_v the_o converse_n of_o the_o former_a demonstration_n this_o answer_v to_o the_o 2._o of_o the_o second_o demonstration_n this_o answer_v to_o the_o 3._o of_o the_o three_o demonstration_n this_o answer_v to_o thâ—_z 4._o of_o the_o second_o demonstration_n this_o answer_v to_o the_o 5._o of_o the_o second_o demonstration_n this_o answer_v to_o the_o 6._o of_o the_o second_o demonstration_n this_o answer_v to_o the_o 7._o of_o the_o second_o demonstration_n this_o answer_v to_o the_o 8._o of_o the_o second_o demonstratition_n this_o answer_v to_o thâ—_z 9_o of_o the_o second_o demonstration_n this_o answer_v to_o the_o 10._o oâ—_n the_o second_o demonstration_n a_o negative_a proposition_n demonstration_n leaâ—ing_v to_o a_o impossibility_n demonstration_n lead_v to_o a_o absurdity_n demonstration_n lead_v to_o a_o abjurditie_n three_o case_n in_o this_o proposition_n the_o first_o case_n the_o second_o case_n the_o three_o case_n divert_v case_n â—n_v this_o proposition_n the_o first_o case_n two_o case_n in_o this_o proposition_n the_o first_o case_n the_o second_o case_n demonstration_n demonstration_n demonstration_n demonstration_n demonstration_n demonstration_n demonstration_n demonstration_n demonstration_n a_o proposition_n add_v by_o campane_n a_o other_o add_v by_o he_o demonstration_n lead_v to_o a_o absurdity_n demonstration_n demonstration_n demonstration_n lead_v to_o a_o absurdity_n an_o other_o demonstration_n demonstration_n demonstration_n this_o proposition_n teachâ—th_v how_o to_o find_v out_o a_o perfect_a number_n construction_n demonstration_n demonstration_n lead_v to_o a_o absurdity_n the_o argument_n of_o the_o ten_o book_n difference_n between_o number_n and_o magnitude_n a_o line_n be_v not_o make_v of_o point_n as_o number_n be_v make_v of_o unity_n this_o book_n the_o hard_a to_o understand_v of_o all_o the_o book_n of_o euclid_n in_o this_o book_n be_v entreat_v of_o a_o strange_a manner_n of_o matter_n then_o in_o the_o former_a many_o even_o of_o the_o well_o learn_v have_v think_v that_o this_o book_n can_v not_o well_o be_v understand_v without_o algebra_n the_o nine_o former_a book_n &_o the_o principle_n of_o this_o â—ooke_n well_o understand_v this_o book_n will_v not_o be_v hard_a to_o understand_v the_o fâ—rst_a definition_n the_o second_o definition_n contraries_n make_v manifest_a by_o the_o compare_v of_o the_o one_o to_o the_o other_o the_o third_o definition_n what_o the_o power_n of_o a_o line_n be_v the_o four_o definition_n unto_o the_o suppose_a line_n first_o set_v may_v be_v compare_v infinite_a line_n why_o some_o mislike_n that_o the_o line_n first_o set_v shall_v be_v call_v a_o rational_a line_n flussates_n call_v this_o line_n a_o line_n certain_a this_o rational_a line_n the_o ground_n in_o a_o manner_n of_o all_o the_o proposition_n in_o this_o ten_o book_n note_n the_o line_n rational_a of_o purpose_n the_o six_o definition_n campâ—nus_n â—ath_z cause_v much_o obscurity_n in_o this_o ten_o book_n the_o seven_o definition_n flussates_n in_o steed_n of_o this_o word_n irrational_a use_v this_o word_n uncertain_a why_o they_o be_v call_v irrational_a line_n the_o cause_n of_o the_o obscurity_n and_o confusedness_n in_o this_o book_n the_o eight_o definition_n the_o nine_o definition_n the_o ten_o definition_n the_o eleven_o definition_n construction_n demonstration_n a_o corollary_n construction_n demonstration_n this_o proposition_n teach_v that_o incontinuall_a quantity_n which_o the_o first_o of_o the_o seven_o teach_v in_o discrete_a quantity_n construction_n demonstration_n lead_v to_o a_o absurdity_n two_o case_n in_o this_o proposition_n the_o first_o case_n this_o proposition_n teach_v that_o in_o continual_a quantity_n which_o the_o 2._o of_o the_o sâ—â—ith_v teach_v in_o number_n the_o second_o case_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 this_o proposition_n teach_v that_o in_o continual_a quantity_n which_o the_o 3._o of_o the_o second_o teach_v in_o number_n construction_n two_o case_n in_o this_o proposition_n the_o first_o case_n demonstration_n lead_v to_o a_o absurdity_n the_o second_o case_n a_o leâ—ma_n necessary_a
about_o the_o diameter_n together_o with_o the_o two_o supplement_n make_v a_o gnomon_n as_o the_o parallelogram_n ebkh_n with_o the_o two_o supplement_n aegk_n and_o khfd_n make_v the_o gnomon_n fgeh_a likewise_o the_o parallelogram_n gkcf_n with_o the_o same_o two_o supplement_n make_v the_o gnomon_n ehfg_n and_o this_o definition_n of_o a_o gnomon_n extend_v itself_o and_o be_v general_a to_o all_o kind_n of_o parallelogram_n whether_o they_o be_v square_n or_o figure_n of_o one_o side_n long_o or_o rhombus_fw-la or_o romboide_n to_o be_v short_a if_o you_o take_v away_o from_o the_o whole_a parallelogram_n one_o of_o the_o partial_a parallelogram_n which_o be_v about_o the_o diameter_n whether_o you_o will_v the_o rest_n of_o the_o figure_n be_v a_o gnomon_n campane_v after_o the_o last_o proposition_n of_o the_o first_o book_n add_v this_o proposition_n book_n two_o square_n be_v give_v to_o adjoin_v to_o one_o of_o they_o a_o gnomon_n equal_a to_o the_o other_o square_n which_o for_o that_o as_o than_o it_o be_v not_o teach_v what_o a_o gnomon_n be_v i_o there_o omit_v think_v that_o it_o may_v more_o apt_o be_v place_v here_o the_o do_v and_o demonstration_n whereof_o be_v thus_o suppose_v that_o there_o be_v two_o square_n ab_fw-la and_o cd_o unto_o one_o of_o which_o namely_o unto_o ab_fw-la it_o be_v require_v to_o add_v a_o gnomon_n equal_a to_o the_o other_o square_n namely_o to_o cd_o produce_v the_o side_n bf_o of_o the_o square_a ab_fw-la direct_o to_o the_o point_n e._n and_o put_v the_o line_n fe_o equal_a to_o the_o side_n of_o the_o square_a cd_o and_o draw_v a_o line_n from_o e_o to_o a._n now_o then_o forasmuch_o as_o efa_n be_v a_o rectangle_n triangle_n therefore_o by_o the_o 47._o of_o the_o first_o the_o square_a of_o the_o line_n ea_fw-la be_v equal_a to_o the_o square_n of_o the_o line_n of_o &_o fa._n but_o the_o square_n of_o the_o line_n of_o be_v equal_a to_o the_o square_a cd_o &_o the_o square_a of_o the_o side_n favorina_n be_v the_o square_a ab_fw-la wherefore_o the_o square_a of_o the_o line_n ae_n be_v equal_a to_o the_o two_o square_n cd_o and_o ab_fw-la but_o the_o side_n of_o and_o favorina_n be_v by_o the_o 21._o of_o the_o first_o long_o than_o the_o side_n ae_n and_o the_o side_n favorina_n be_v equal_a to_o the_o side_n fb_o wherefore_o the_o side_n of_o and_o fb_o be_v long_o they_o the_o side_n ae_n wherefore_o the_o whole_a line_n be_v be_v long_o than_o the_o line_n ae_n from_o the_o line_n be_v cut_v of_o a_o line_n equal_a to_o the_o line_n ae_n which_o let_v be_v bc._n and_o by_o the_o 46._o proposition_n upon_o the_o line_n bc_o describe_v a_o square_a which_o let_v be_v bcgh_n which_o shall_v be_v equal_a to_o the_o square_n of_o the_o line_n ae_n but_o the_o square_n of_o the_o line_n ae_n be_v equal_a to_o the_o two_o square_n ab_fw-la and_o dc_o wherefore_o the_o square_a bcgh_n be_v equal_a to_o the_o same_o square_n wherefore_o forasmuch_o as_o the_o square_a bcgh_n be_v compose_v of_o the_o square_a ab_fw-la and_o of_o the_o gnomon_n fgah_n the_o say_a gnomon_n shall_v be_v equal_a unto_o the_o square_a cd_o which_o be_v require_v to_o be_v do_v a_o other_o more_o ready_a way_n after_o pelitarius_n suppose_v that_o there_o be_v two_o square_n who_o side_n let_v be_v ab_fw-la and_o bc._n it_o be_v require_v unto_o the_o square_n of_o the_o line_n ab_fw-la to_o add_v a_o gnomon_n equal_a to_o the_o square_n of_o the_o line_n bc._n set_v the_o line_n ab_fw-la and_o bc_o in_o such_o sort_n that_o they_o make_v a_o right_a angle_n abc_n and_o draw_v a_o line_n from_o a_o to_o c._n and_o upon_o the_o line_n ab_fw-la describe_v a_o square_n which_o let_v be_v abde_n and_o produce_v the_o line_n basilius_n to_o the_o point_n fletcher_n and_o put_v the_o line_n bf_o equal_a to_o the_o line_n ac_fw-la and_o upon_o the_o line_n bf_o describe_v a_o square_n which_o let_v be_v bfgh_o which_o shall_v be_v equal_a to_o the_o square_n of_o the_o line_n ac_fw-la when_o as_o the_o line_n bf_o and_o ac_fw-la be_v equal_a and_o therefore_o it_o be_v equal_a to_o the_o square_n of_o the_o two_o line_n ab_fw-la and_o bc._n now_o forasmuch_o as_o the_o square_a bfgh_o be_v make_v complete_a by_o the_o square_a abde_n and_o by_o the_o gnomon_n fegd_v the_o gnomon_n fegd_v shall_v be_v equal_a to_o the_o square_n of_o the_o line_n bc_o which_o be_v require_v to_o be_v do_v the_o 1._o theorem_a the_o 1._o proposition_n if_o there_o be_v two_o right_a line_n and_o if_o the_o one_o of_o they_o be_v divide_v into_o part_n how_o many_o soever_o the_o rectangle_n figure_n comprehend_v under_o the_o two_o right_a line_n be_v equal_a to_o the_o rectangle_n figure_n which_o be_v comprehend_v under_o the_o line_n undivided_a and_o under_o every_o one_o of_o the_o part_n of_o the_o other_o line_n svppose_v that_o there_o be_v two_o right_a line_n a_o and_o bc_o and_o let_v one_o of_o they_o namely_o bc_o be_v divide_v at_o all_o adventure_n in_o the_o point_n d_o and_o e._n then_o i_o say_v that_o the_o rectangle_n figure_n comprehend_v under_o the_o line_n a_o and_o bc_o be_v equal_a unto_o the_o rectangle_n figure_n comprehend_v under_o the_o line_n a_o and_o bd_o &_o unto_o the_o rectangle_n figure_n which_o be_v comprehend_v under_o the_o line_n a_o and_o de_fw-fr and_o also_o unto_o the_o rectangle_n figure_n which_o be_v comprehend_v under_o the_o line_n a_o and_o ec_o construction_n for_o from_o the_o point_n brayse_v up_o by_o the_o 11._o of_o the_o first_o unto_o the_o right_a line_n bc_o a_o perpendicular_a line_n bf_o &_o unto_o the_o line_n a_o by_o the_o three_o of_o the_o first_o put_v the_o line_n bg_o equal_a and_o by_o the_o point_n g_z by_o the_o 31._o of_o the_o first_o draw_v a_o parallel_n line_n unto_o the_o right_a line_n bc_o and_o let_v the_o same_o be_v gm_n and_o by_o the_o self_n same_o by_o the_o poor_n d_z e_o and_o c_o draw_v unto_o the_o line_n bg_o these_o parallel_a line_n dk_o demonstration_n el_n and_o ch._n now_o then_o the_o parallelogram_n bh_o be_v equal_a to_o these_o parallelogram_n bk_o dl_o and_o eh_o but_o the_o parallelogram_n bh_o be_v equal_a unto_o that_o which_o be_v contain_v under_o the_o line_n a_o and_o bc._n for_o it_o be_v comprehend_v under_o the_o line_n gb_o &_o bc_o and_o the_o line_n gb_o be_v equal_a unto_o the_o line_n a_o and_o the_o parallelogram_n bk_o be_v equal_a to_o that_o which_o be_v contain_v under_o the_o line_n a_o and_o bd_o for_o it_o be_v comprehend_v under_o the_o line_n gb_o and_o bd_o and_o bg_o be_v equal_a unto_o a_o and_o the_o parallelogram_n dl_o be_v equal_a to_o that_o which_o be_v contain_v under_o the_o line_n a_o and_o de_fw-fr for_o the_o line_n dk_o that_o be_v bg_o be_v equal_a unto_o a_o and_o moreover_o likewise_o the_o parallelogram_n eh_o be_v equal_a to_o that_o which_o be_v contain_v under_o the_o line_n a_o &_o ec_o wherefore_o that_o which_o be_v comprehend_v under_o the_o line_n a_o &_o bc_o be_v equal_a to_o that_o which_o be_v comprehend_v under_o the_o line_n a_o &_o bd_o &_o unto_o that_o which_o be_v comprehend_v under_o the_o line_n a_o and_o de_fw-fr and_o moreover_o unto_o that_o which_o be_v comprehend_v under_o the_o line_n a_o and_o ec_o if_o therefore_o there_o be_v two_o right_a line_n and_o if_o the_o one_o of_o they_o be_v divide_v into_o part_n how_o many_o soever_o the_o rectangle_n figure_n comprehend_v under_o the_o two_o right_a line_n be_v equal_a to_o the_o rectangle_n figure_n which_o be_v comprehend_v under_o the_o line_n undivided_a and_o under_o every_o one_o of_o the_o part_n of_o the_o other_o line_n which_o be_v require_v to_o be_v demonstrate_v because_o that_o all_o the_o proposition_n of_o this_o second_o book_n for_o the_o most_o part_n be_v true_a both_o in_o line_n and_o in_o number_n and_o may_v be_v declare_v by_o both_o therefore_o have_v i_o have_v add_v to_o every_o proposition_n convenient_a number_n for_o the_o manifestation_n of_o the_o same_o and_o to_o the_o end_n the_o studious_a and_o diligent_a reader_n may_v the_o more_o full_o perceive_v and_o understand_v the_o agreement_n of_o this_o art_n of_o geometry_n with_o the_o science_n of_o arithmetic_n and_o how_o never_o &_o dear_a sister_n they_o be_v together_o so_o that_o the_o one_o can_v without_o great_a blemish_n be_v without_o the_o other_o i_o have_v here_o also_o join_v a_o little_a book_n of_o arithmetic_n write_v by_o one_o barlaam_n a_o greek_a author_n a_o man_n of_o great_a knowledge_n in_o which_o book_n be_v by_o the_o author_n demonstrate_v many_o of_o the_o self_n same_o propriety_n and_o passion_n in_o number_n which_o euclid_n in_o this_o his_o second_o book_n have_v demonstrate_v in_o magnitude_n
&_o m._n demonstrationâ—_n if_o therefore_o g_z exceed_v l_o then_z also_o h_o exceed_v m_o and_o if_o it_o be_v equal_a it_o be_v equal_a and_o if_o it_o be_v less_o it_o be_v less_o by_o the_o converse_n of_o the_o 6â—_o definition_n of_o the_o five_o again_o because_o that_o as_o c_o be_v to_o d_z so_o be_v e_o to_o fletcher_n and_o to_o c_o and_o e_o be_v take_v â—â—â—emâ—ltiplices_n h_o â—â—d_z king_n and_o likewise_o to_o d_z &_o f_o be_v take_v certain_a other_o equemultiplices_fw-la m_o &_o n._n if_o therefore_o h_o exceed_v m_o than_o also_o king_n exceed_v n_o and_o if_o it_o be_v equal_a it_o be_v equal_a and_o if_o it_o be_v less_o it_o be_v less_o by_o the_o same_o converse_n but_o if_o king_n exceed_v m_o than_o also_o g_z exceed_v l_o and_o if_o it_o be_v equal_a it_o be_v equal_a and_o if_o it_o be_v less_o it_o be_v less_o by_o the_o same_o converse_n wherefore_o if_o g_z exceed_v l_o then_z king_n also_o exceed_v n_o and_o if_o it_o be_v equal_a it_o be_v equal_a and_o if_o it_o be_v less_o it_o be_v less_o but_o g_z &_o king_n be_v equemultiplices_fw-la of_o a_o &_o e._n and_o l_o &_o n_o be_v certain_a other_o equemultiplices_fw-la of_o b_o &_o f._n wherefore_o by_o the_o 6._o definition_n as_o a_o be_v to_o b_o so_o be_v e_o to_o f._n proportion_n therefore_o which_o be_v one_o and_o the_o self_n same_o to_o any_o one_o proportion_n be_v also_o the_o self_n same_o one_o to_o the_o other_o which_o be_v require_v to_o be_v prove_v the_o 12._o theorem_a the_o 12._o proposition_n if_o there_o be_v a_o number_n of_o magnitude_n how_o many_o soeâ—â—r_n proportional_a 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 svppose_v that_o there_o be_v a_o number_n of_o magnitude_n how_o many_o soever_o namely_o a_o b_o c_o d_o e_o f_o in_o proportion_n so_o that_o as_o a_o be_v to_o b_o so_o let_v c_o be_v to_o d_o and_o e_o to_o f._n then_o i_o say_v that_o as_o a_o be_v to_o b_o so_o 〈◊〉_d a_o c_o e_o to_o b_o d_o f._n take_v equemultiplices_fw-la to_o a_o c_o and_o e._n constrâ—ction_n and_o let_v the_o same_o be_v g_o h_o k._n and_o likewise_o to_o b_o d_o and_o fletcher_n â—ake_z any_o other_o equemultiplices_fw-la which_o to_o be_v l_o m_o n._n and_o because_o that_o 〈◊〉_d a_o be_v to_o b_o so_o iâ—_z c_o to_o d_z and_o e_o to_o f._n demonstrationâ—_n and_o to_o a_o c_o e_o be_v take_v â—quemultiplices_n g_o h_o king_n and_o likewise_o to_o â—_o d_o f_o be_v take_v certain_a other_o equemâ—â—tipliâ—â—s_n l_o m_o n._n if_o therefore_o g_z exceed_v l_o h_o also_o exceed_v m_o and_o kn_n and_o if_o it_o be_v equal_a it_o be_v equal_a and_o if_o it_o be_v less_o it_o be_v less_o â—y_a the_o converse_n of_o the_o sixâ—_o definition_n of_o the_o five_o wherefore_o if_o g_z exceed_v l_o then_z g_z h_o edward_n also_o exceed_v l_o m_o n_o and_o if_o they_o be_v equal_a they_o be_v equal_a and_o if_o they_o be_v less_o they_o be_v less_o by_o the_o same_o but_o g_z and_o g_z h_o edward_n be_v equemultiplices_fw-la to_o the_o magnitude_n a_o and_o to_o the_o magnitude_n a_o c_o e._n for_o by_o the_o first_o of_o the_o five_o if_o there_o be_v a_o number_n of_o magnitude_n equemultiplices_fw-la to_o a_o like_a number_n of_o magnitude_n each_o to_o each_o how_o multiplex_n one_o magnitude_n be_v to_o one_o so_o multiplices_fw-la be_v all_o the_o magnitude_n to_o all_o and_o by_o the_o same_o reason_n also_o l_o and_o l_o m_o n_o be_v equemultiplices_fw-la to_o the_o magnitude_n b_o and_o to_o the_o magnitude_n b_o d_o f_o wherefore_o as_o a_o be_v to_o b_o so_o be_v a_o c_o e_o to_o b_o d_o f_o by_o the_o six_o definition_n of_o the_o five_o if_o therefore_o there_o be_v a_o number_n of_o magnitude_n how_o many_o soever_o proportional_a 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 which_o be_v require_v to_o be_v prove_v the_o 13._o theorem_a the_o 13._o proposition_n if_o the_o first_o have_v unto_o the_o second_o the_o self_n same_o proportion_n that_o the_o three_o have_v to_o the_o four_o and_o if_o the_o three_o have_v unto_o the_o four_o a_o great_a proportion_n they_o the_o five_o have_v to_o the_o six_o they_o shall_v the_o first_o also_o have_v unto_o the_o second_o a_o great_a proportion_n than_o have_v the_o five_o to_o the_o six_o svppose_v that_o there_o be_v six_o magnitude_n of_o which_o let_v a_o be_v the_o first_o b_o the_o second_o c_o the_o three_o d_z the_o four_o e_o the_o five_o and_o fletcher_n the_o six_o suppose_v that_o a_o the_o first_o have_v unto_o b_o the_o second_o the_o self_n same_o proportion_n that_o c_o the_o three_o have_v to_o d_o the_o four_o and_o let_v c_o the_fw-fr three_o have_v unto_o d_z the_o four_o a_o great_a proportion_n than_o have_v e_o the_o five_o to_o f_o the_o six_o then_o i_o say_v that_o a_o the_o first_o have_v to_o b_o the_o second_o a_o great_a proportion_n then_o have_v e_o the_o five_o to_o f_o the_o six_o construction_n for_o forasmuch_o as_o c_z have_v to_o d_z a_o great_a proportion_n than_o have_v e_o to_o fletcher_n therefore_o there_o be_v certain_a equemultiplices_fw-la to_o c_o and_o e_o and_o likewise_o any_o other_o equemultiplices_fw-la whatsoever_o to_o d_z and_o f_o which_o be_v compare_v together_o the_o multiplex_n to_o c_o shall_v exceed_v the_o multiplex_n to_o d_o but_o the_o multiplex_n to_o e_o shall_v not_o exceed_v the_o multiplex_n to_o fletcher_n by_o the_o converse_n of_o the_o eight_o definition_n of_o this_o book_n let_v those_o multiplices_fw-la be_v take_v and_o suppose_v that_o the_o equemultiplices_fw-la to_o c_o and_o e_o be_v g_z and_o h_o and_o likewise_o to_o d_z and_o f_o take_v any_o other_o equemultiplices_fw-la whatsoever_o and_o let_v the_o same_o be_v king_n and_o l_o so_fw-mi that_o let_v g_z exceed_v king_n but_o let_v not_o h_o exceed_v l._n and_o how_o multiplex_n g_z be_v to_o c_o so_o multiplex_n let_v m_o be_v to_o a._n and_o how_o multiplex_n king_n be_v to_o d_o so_o multiplex_n also_o let_v n_o be_v to_o b._n and_o because_o that_o as_o a_o be_v to_o b_o so_o be_v c_z to_o d_z and_o to_o a_o and_o c_o be_v take_v equemultiplices_fw-la m_o and_o g._n demonstration_n and_o likewise_o to_o b_o and_o d_o be_v take_v certain_a other_o equemultiplices_fw-la n_o &_o king_n if_o therefore_o m_o exceed_v n_o g_o also_o exceed_v king_n and_o if_o it_o be_v equal_a it_o be_v equal_a and_o if_o it_o be_v less_o it_o be_v less_o by_o the_o conversion_n of_o the_o six_o definition_n of_o the_o five_o but_o by_o construction_n g_o excedetâ—_n edward_n wherefore_o m_o also_o exceed_v n_o but_o h_o exceed_v not_o l._n but_o m_o &_o h_o be_v equemultiplices_fw-la to_o a_o &_o e_o and_o n_o &_o l_o be_v certain_a other_o equemultiplicesâ—_n whatsoever_o to_o b_o and_o f._n wherefore_o a_o have_v unto_o b_o a_o great_a proportion_n than_o e_o have_v to_o fletcher_n by_o the_o 8._o definition_n if_o therefore_o the_o first_o have_v unto_o the_o second_o the_o self_n same_o proportion_n that_o the_o three_o have_v to_o the_o four_o and_o if_o the_o three_o have_v unto_o the_o four_o a_o great_a proportion_n than_o the_o five_o have_v to_o the_o six_o then_o shall_v the_o firsâ—_o also_o have_v unto_o the_o second_o a_o great_a proportion_n than_o have_v the_o 〈◊〉_d to_o the_o sixths_n which_o be_v require_v to_o be_v prove_v ¶_o a_o addition_n of_o campane_n if_o there_o be_v four_o quantity_n campane_n and_o if_o the_o first_o have_v unto_o the_o second_o a_o great_a proportion_n they_o have_v the_o three_o to_o the_o four_o then_o shall_v there_o be_v some_o equemultiplices_fw-la of_o the_o first_o and_o the_o three_o which_o be_v compare_v to_o some_o equemultiplices_fw-la of_o the_o second_o and_o the_o four_o the_o multiplex_n of_o the_o first_o shall_v be_v great_a than_o the_o multiplex_n of_o the_o second_o but_o the_o multiplex_n of_o the_o three_o shall_v not_o be_v great_a than_o the_o multiplex_n of_o the_o four_o although_o this_o proposition_n here_o put_v by_o campane_n need_v no_o demonstration_n for_o that_o it_o be_v but_o the_o converse_n of_o the_o 8._o definition_n of_o this_o book_n yet_o think_v i_o it_o not_o worthy_a to_o be_v omit_v for_o that_o it_o reach_v the_o way_n to_o find_v out_o such_o equemultiplices_fw-la that_o the_o multiplex_n of_o the_o first_o shall_v exceed_v the_o multiplex_n of_o the_o second_o but_o the_o multiplex_n of_o the_o three_o shall_v not_o exceed_v the_o multiplex_n of_o the_o four_o the_o 14._o theorem_a the_o 14
and_o the_o same_o proportion_n wherefore_o by_o the_o 9_o of_o the_o five_o the_o figure_n nh_o be_v equal_a unto_o the_o figure_n sir_n and_o it_o be_v unto_o it_o like_v and_o in_o like_a sort_n situate_a follow_v but_o in_o like_a and_o equal_a rectiline_a figure_n be_v in_o like_a sort_n situate_a the_o side_n of_o like_a proportion_n on_o which_o they_o be_v describe_v be_v equal_a wherefore_o the_o line_n gh_o be_v equal_a unto_o the_o line_n qr_o and_o because_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 of_o to_o the_o line_n qr_o but_o the_o line_n qr_o be_v equal_a unto_o the_o line_n gh_o therefore_o as_o the_o line_n ab_fw-la be_v to_o he_o line_n cd_o so_o be_v the_o line_n of_o to_o the_o line_n gh_o if_o therefore_o there_o be_v four_o right_a line_n proportional_a the_o rectiline_a figure_n also_o describe_v upon_o they_o be_v like_a and_o in_o like_a sort_n situate_a shall_v be_v proportional_a and_o if_o the_o rectiline_a figure_n upon_o they_o describe_v be_v like_o and_o in_o like_a sort_n situate_v be_v proportional_a those_o right_a line_n also_o shall_v be_v proportional_a which_o be_v require_v to_o be_v prove_v a_o assumpt_n and_o now_o that_o in_o like_a and_o equal_a figure_n be_v in_o like_a sort_n situate_a the_o side_n of_o like_a proportion_n be_v also_o equal_a which_o thing_n be_v before_o in_o this_o proposition_n take_v as_o grant_v may_v thus_o be_v prove_v assumpt_n suppose_v that_o the_o rectiline_a figure_n nh_v and_o sir_n be_v equal_a and_o like_a and_o as_o hg_o be_v to_o gn_v so_o let_v rq_n be_v to_o q_n and_o let_v gh_o and_o qr_o be_v side_n of_o like_a proportion_n then_o i_o say_v that_o the_o side_n rq_n be_v equal_a unto_o the_o side_n gh_o for_o if_o they_o be_v unequal_a the_o one_o of_o they_o be_v great_a than_o the_o other_o let_v the_o side_n rq_n be_v great_a than_o the_o side_n hg_o and_o for_o that_o as_o the_o line_n rq_n be_v to_o the_o line_n q_n so_o be_v the_o line_n hg_o to_o the_o line_n gn_n and_o alternate_o also_o by_o the_o 16._o of_o the_o five_o as_o the_o line_n rq_n be_v to_o the_o line_n hg_o so_o be_v the_o line_n q_n to_o the_o line_n gn_n but_o the_o line_n rq_n be_v great_a than_o the_o line_n hg_o wherefore_o also_o the_o line_n q_n be_v great_a than_o the_o line_n gn_n wherefore_o also_o the_o figure_n rs_n be_v great_a than_o the_o figure_n hn_o but_o by_o supposition_n it_o be_v equal_a unto_o it_o which_o be_v impossible_a wherefore_o the_o line_n qr_o be_v not_o great_a than_o the_o line_n gh_o in_o like_a sort_n also_o may_v we_o prove_v that_o it_o be_v not_o less_o than_o it_o wherefore_o it_o be_v equal_a unto_o it_o which_o be_v require_v to_o be_v prove_v flussates_n demonstrate_v this_o second_o part_n more_o brief_o flussates_n by_o the_o first_o corollary_n of_o the_o _o of_o this_o book_n thus_o forasmuch_o as_o the_o rectiline_a figure_n be_v by_o supposition_n in_o one_o and_o the_o same_o proportion_n and_o the_o same_o proportion_n be_v double_a to_o the_o proportion_n of_o the_o side_n ab_fw-la to_o cd_o and_o of_o to_o gh_o by_o the_o foresay_a corollary_n the_o proportion_n also_o of_o the_o side_n shall_v be_v one_o and_o the_o self_n same_o by_o the_o 7._o common_a sentence_n namely_o the_o line_n ab_fw-la shall_v be_v unto_o the_o line_n cd_o as_o the_o line_n of_o be_v to_o the_o line_n gh_o the_o 17._o theorem_a the_o 23._o proposition_n equiangle_n parallelogrammes_n have_v the_o one_o to_o the_o other_o that_o proportion_n which_o be_v compose_v of_o the_o side_n flussates_n demonstrate_v this_o theorem_a without_o take_v of_o these_o three_o line_n king_n l_o m_o after_o this_o manner_n flussate_n forasmuch_o as_o say_v he_o it_o have_v be_v declare_v upon_o the_o 10._o definition_n of_o the_o five_o book_n and_o â—ift_a definition_n of_o this_o book_n that_o the_o proportion_n of_o the_o extreme_n consist_v of_o the_o proportion_n of_o the_o mean_n let_v we_o suppose_v two_o equiangle_n parallelogram_n abgd_v and_o gezi_n and_o let_v the_o angle_n at_o the_o point_n g_o in_o either_o be_v equal_a and_o let_v the_o line_n bg_o and_o give_v be_v set_v direct_o that_o they_o both_o make_v one_o right_a line_n namely_o bgi._n wherefore_o egg_v also_o shall_v be_v one_o right_a line_n by_o the_o converse_n of_o the_o 15._o of_o the_o first_o make_v complete_a the_o parallelogram_n gt_fw-mi then_o i_o say_v that_o the_o proportion_n of_o the_o parallelogram_n agnostus_n &_o gz_n be_v compose_v of_o the_o proportion_n of_o the_o side_n bg_o to_o give_v and_o dg_o to_o ge._n for_o forasmuch_o as_o that_o there_o be_v three_o magnitude_n agnostus_n gt_n and_o gz_n and_o gt_n be_v the_o mean_a of_o the_o say_a magnitude_n and_o the_o proportion_n of_o the_o extreme_n agnostus_n to_o gz_n consist_v of_o the_o mean_a proportion_n by_o the_o 5._o definition_n of_o this_o book_n namely_o of_o the_o proportion_n of_o agnostus_n to_o gt_n and_o of_o the_o proportion_n gt_fw-mi to_o gz_n but_o the_o proportion_n of_o agnostus_n to_o gt_n be_v one_o and_o the_o self_n same_o with_o the_o proportion_n of_o the_o side_n bg_o to_o give_v by_o the_o first_o of_o this_o book_n and_o the_o proportion_n also_o of_o gt_n to_o gz_n be_v one_o and_o the_o self_n same_o with_o the_o proportion_n of_o the_o other_o side_n namely_o dg_o to_o ge_z by_o the_o same_o proposition_n wherefore_o the_o proportion_n of_o the_o parallelogram_n agnostus_n to_o gz_n consist_v of_o the_o proportion_n of_o the_o side_n bg_o to_o give_v and_o dc_o to_o ge._n wherefore_o equiangle_n parallelogram_n be_v the_o one_o to_o the_o other_o in_o that_o proportion_n which_o be_v compose_v of_o their_o side_n which_o be_v require_v to_o be_v prove_v the_o 18._o theorem_a the_o 24._o proposition_n in_o every_o parallelogram_n the_o parallelogram_n about_o the_o dimecient_a be_v like_a unto_o the_o whole_a and_o also_o like_o the_o one_o to_o the_o other_o abcd._n svppose_v that_o there_o be_v a_o parallelogram_n abcd_o and_o let_v the_o dimecient_a thereof_o be_v ac_fw-la and_o let_v the_o parallelogram_n about_o the_o dimecient_a ac_fw-la be_v eglantine_n and_o hk_o then_o i_o say_v that_o either_o of_o these_o parallelogram_n eglantine_n and_o hk_o be_v like_a unto_o the_o whole_a parallelogram_n abcd_o and_o also_o be_v like_o the_o one_o to_o the_o other_o for_o forasmuch_o as_o to_o one_o of_o the_o side_n of_o the_o triangle_n abc_n namely_o to_o bc_o be_v draw_v a_o parallel_a line_n of_o therefore_o as_o be_v be_v to_o ea_fw-la so_o by_o the_o 2._o of_o the_o six_o be_v cf_o to_o fa._n again_o forasmuch_o as_o to_o one_o of_o the_o side_n of_o the_o triangle_n adc_o namely_o to_o cd_o be_v draw_v a_o parallel_a line_n fâ—_n therefore_o by_o the_o same_o as_o cf_o be_v to_o favorina_n so_o be_v dg_o to_o ga._n but_o as_o cf_o be_v to_o favorina_n so_o be_v it_o prove_v that_o be_v be_v to_o ea_fw-la wherefore_o as_o be_v be_v to_o ea_fw-la so_o by_o the_o 11._o of_o the_o five_o â—_o be_v dg_fw-mi to_o ga._n wherefore_o by_o composition_n by_o the_o 18._o of_o the_o five_o as_o basilius_n be_v to_o aeâ—_n so_o be_v dam_n to_o ag._n and_o alternate_o by_o the_o 16._o of_o the_o five_o as_o basilius_n be_v to_o ad_fw-la so_o be_v ea_fw-la to_o ag._n wherefore_o in_o the_o parallelogrammesâ—_n abcd_o and_o eglantine_n the_o side_n which_o be_v about_o the_o common_a angle_n bad_a be_v proportional_a and_o because_o the_o line_n gf_o be_v a_o parallel_n unto_o the_o line_n dcâ—_n therefore_o the_o angle_n agf_n by_o the_o 29â—_n of_o theâ—_z first_o be_v equal_a unto_o the_o angle_v adc_o â—_o the_o angle_n gfa_n equal_a unto_o the_o angle_v dca_n and_o the_o angle_n dac_o be_v common_a to_o the_o two_o triangle_n adc_o and_o afg_n wherefore_o the_o triangle_n dac_o be_v equiangle_n unto_o the_o triangle_n agf_n and_o by_o the_o same_o reason_n the_o triangle_n abc_n be_v equiangle_n unto_o the_o triangle_n aef_n wherefore_o the_o whole_a parallelogram_n abcd_o be_v equiangle_n unto_o the_o parallelogram_n eglantine_n wherefore_o as_o ad_fw-la be_v in_o proportion_n to_o dc_o so_o by_o the_o 4._o of_o the_o six_o be_v agnostus_n to_o gf_o and_o as_o dc_o be_v to_o ca_n so_o be_v gf_o to_o fa._n and_o as_o ac_fw-la be_v to_o cb_o so_o be_v of_o to_o fe_o and_o moreover_o as_o cb_o be_v to_o ba_o so_o be_v fe_o to_o ea_fw-la and_o forasmuch_o as_o it_o be_v prove_v that_o as_o dâ—_n be_v to_o ca_n so_o be_v gf_o to_o favorina_n but_o as_o ac_fw-la be_v to_o câ—_n so_o be_v of_o to_o fe_o wherefore_o of_o equality_n by_o the_o 22._o of_o the_o five_o as_o dc_o be_v to_o cb_o so_o be_v gf_o to_o fe_o wherefore_o in_o the_o parallelogram_n abcd_o and_o
which_o be_v require_v to_o be_v prove_v ¶_o the_o 34._o theorem_a the_o 34._o proposition_n if_o a_o number_n be_v neither_o double_v from_o two_o nor_o have_v to_o his_o half_a part_n a_o odd_a number_n it_o shall_v be_v a_o number_n both_o even_o even_o and_o even_o odd_a svppose_v that_o the_o number_n a_o be_v a_o number_n neither_o double_v from_o the_o number_n two_o neither_o also_o let_v it_o have_v to_o his_o half_a part_n a_o odd_a number_n then_o i_o say_v that_o a_o be_v a_o number_n both_o even_o even_o and_o even_o odd_a demonstration_n that_o a_o be_v even_o even_o it_o be_v manifest_a for_o the_o half_a thereof_o be_v not_o odd_a and_o be_v measure_v by_o the_o number_n 2._o which_o be_v a_o even_a number_n now_o i_o say_v that_o it_o be_v even_o odd_a also_o for_o if_o we_o divide_v a_o into_o two_o equal_a part_n and_o so_o continue_v still_o we_o shall_v at_o the_o length_n light_n upon_o a_o certain_a odd_a number_n which_o shall_v measure_v a_o by_o a_o even_a number_n for_o if_o we_o shall_v not_o light_v upon_o such_o a_o odd_a number_n which_o measure_v a_o by_o a_o even_a number_n we_o shall_v at_o the_o length_n come_v unto_o the_o number_n two_o and_o so_o shall_v a_o be_v one_o of_o those_o number_n which_o be_v double_v from_o two_o upward_a which_o be_v contrary_a to_o the_o supposition_n wherefore_o a_o be_v even_o odd_a and_o it_o be_v prove_v that_o it_o be_v even_o even_o wherefore_o a_o be_v a_o number_n both_o even_o even_o and_o even_o odd_a which_o be_v require_v to_o be_v demonstrate_v this_o proposition_n and_o the_o two_o former_a manifest_o declare_v that_o which_o we_o note_v upon_o the_o ten_o definition_n of_o the_o seven_o book_n namely_o that_o campane_n and_o flussates_n and_o diverse_a other_o interpreter_n of_o euclid_n only_a theon_n except_v do_v not_o right_o understand_v the_o 8._o and_o 9_o definition_n of_o the_o same_o book_n concern_v a_o number_n even_o even_o and_o a_o number_n even_o odd_a for_o in_o the_o one_o definition_n they_o add_v unto_o euclides_n word_n extant_a in_o the_o greek_a this_o word_n only_o as_o we_o there_o note_v and_o in_o the_o other_o this_o word_n all_o so_o that_o after_o their_o definition_n a_o number_n can_v not_o be_v even_o even_o unless_o it_o be_v measure_v only_o by_o even_a number_n likewise_o a_o number_n can_v not_o be_v even_o odd_a unless_o all_o the_o even_a number_n which_o do_v measure_v it_o do_v measure_v it_o by_o a_o odd_a number_n the_o contrary_a whereof_o in_o this_o proposition_n we_o manifest_o see_v for_o here_o euclid_n prove_v that_o one_o number_n may_v be_v both_o even_o even_o and_o even_o odd_a and_o in_o the_o two_o former_a proposition_n he_o prove_v that_o some_o number_n be_v even_o even_o only_o and_o some_o even_o odd_a only_o which_o word_n only_o have_v be_v in_o vain_a of_o he_o add_v if_o no_o number_n even_o even_o can_v be_v measure_v by_o a_o odd_a number_n or_o if_o all_o the_o number_n that_o measure_v a_o number_n even_o odd_a must_v needs_o measure_v it_o by_o a_o odd_a number_n although_o campane_n and_o flussates_n to_o avoid_v this_o absurdity_n have_v wrest_v the_o 32._o proposition_n of_o this_o book_n from_o the_o true_a sense_n of_o the_o greek_a and_o as_o it_o be_v interpret_v of_o theon_n so_o also_o have_v flussates_n wrest_v the_o 33._o proposition_n for_o whereas_o euclid_n say_v every_o number_n produce_v by_o the_o double_n of_o two_o upward_a be_v even_o even_o only_o they_o say_v only_o the_o number_n produce_v by_o the_o double_n of_o two_o be_v even_o even_o likewise_o whereas_o euclid_n say_v a_o number_n who_o hafle_n part_n be_v odd_a be_v even_o odd_a only_o flussates_n say_v only_o a_o number_n who_o half_a part_n be_v odd_a be_v even_o odd_a which_o their_o interpretation_n be_v not_o true_a neither_o can_v be_v apply_v to_o the_o proposition_n as_o they_o be_v extant_a in_o the_o greek_a in_o deed_n the_o say_v 32._o and_o 33._o proposition_n as_o they_o put_v they_o be_v true_a touch_v those_o number_n which_o be_v even_o even_o only_o or_o even_o odd_a only_o for_o no_o number_n be_v even_o even_o only_o but_o those_o only_o which_o be_v double_v from_o two_o upward_o likewise_o no_o number_n be_v even_o odd_a only_o but_o those_o only_o who_o half_o be_v a_o odd_a number_n but_o this_o let_v not_o but_o that_o a_o number_n may_v be_v even_o even_o although_o it_o be_v not_o double_v from_o two_o upward_a &_o also_o that_o a_o number_n may_v be_v even_o odd_a although_o it_o have_v not_o to_o his_o half_n a_o odd_a number_n as_o in_o this_o 34._o proposition_n euclid_n have_v plain_o prove_v which_o thing_n can_v by_o no_o mean_n be_v true_a if_o the_o foresay_a 32._o &_o 33._o propositon_n of_o this_o book_n shall_v have_v that_o sense_n and_o meaning_n wherein_o they_o take_v it_o ¶_o the_o 35._o theorem_a the_o 35._o proposition_n if_o there_o be_v number_n in_o continual_a proportion_n how_o many_o soever_o and_o if_o from_o the_o second_o and_o last_o be_v take_v away_o number_n equal_a unto_o the_o first_o as_o the_o excess_n of_o the_o second_o be_v to_o the_o first_o so_o be_v the_o excess_n of_o the_o last_o to_o all_o the_o number_n go_v before_o the_o last_o svppose_v that_o these_o number_n a_o bc_o d_o and_o of_o be_v in_o continual_a proportion_n beginning_n at_o a_o the_o least_o and_o from_o bc_o which_o be_v the_o second_o take_v away_o cg_o equal_a unto_o the_o first_o namely_o to_o a_o and_o likewise_o from_o of_o the_o last_o take_v away_o fh_o equal_a also_o unto_o the_o first_o namely_o to_o a._n then_o i_o say_v that_o as_o the_o excess_n bg_o be_v to_o a_o the_o first_o so_o be_v he_o the_o excess_n to_o all_o the_o number_n d_o bc_o and_o a_o which_o go_v before_o the_o last_o number_n namely_o ef._n demonstration_n forasmuch_o as_o of_o be_v the_o great_a for_o the_o second_o be_v suppose_v great_a than_o the_o first_o put_v the_o number_n fl_fw-mi equal_v to_o the_o number_n d_o and_o likewise_o the_o number_n fk_o equal_a to_o the_o number_n bc._n and_o forasmuch_o as_o fk_n be_v equal_a unto_o cb_o of_o which_o fh_o be_v equal_a unto_o gc_o therefore_o the_o residue_n hk_o be_v equal_a unto_o the_o residue_n gb_o and_o for_o that_o as_o the_o whole_a fâ—_n be_v to_o the_o whole_a fl_fw-mi so_o be_v the_o part_n take_v away_o fl_fw-mi to_o the_o part_n take_v away_o fk_a therefore_o the_o residue_n le_fw-fr be_v to_o the_o residue_n kl_o as_o the_o whole_a â—e_n be_v to_o the_o whole_a fl_fw-mi by_o the_o 11._o of_o the_o seven_o so_o likewise_o for_o that_o fl_fw-mi be_v to_o fk_v as_o fk_n be_v to_o fh_v kl_o shall_v be_v to_o hk_v as_o the_o whole_a fl_fw-mi be_v to_o the_o whole_a fk_n by_o the_o same_o proposition_n but_o as_o fe_o be_v to_o fl_fw-mi and_o as_o fl_fw-mi be_v to_o fk_v and_o fk_a to_o fh_v so_o be_v fe_o to_o d_z and_o d_z to_o bc_o and_o bc_o 〈◊〉_d a._n wherefore_o as_o le_fw-fr be_v to_o kl_o and_o as_o kl_o be_v to_o hk_v so_o be_v d_z to_o bc._n wherefore_o alternate_o by_o the_o 23._o of_o the_o seven_o as_z le_fw-fr be_v to_o d_z so_o be_v kl_o to_o be_v bc_o and_o as_o kl_o be_v to_o bc_o so_o be_v hk_v to_o a._n wherefore_o also_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 kh_o be_v to_o a_o so_o be_v hk_v kl_o and_o le_fw-fr to_o d_z bc_o and_o a_o by_o the_o 12._o of_o the_o seven_o but_o it_o be_v prove_v that_o kh_o be_v equal_a unto_o bg_o wherefore_o as_o bg_o which_o be_v the_o excess_n of_o the_o second_o be_v to_o a_o so_o be_v eh_o the_o excess_n of_o the_o last_o unto_o the_o number_n go_v before_o d_o bc_o and_o a._n wherefore_o as_o the_o excess_n of_o the_o second_o be_v unto_o the_o first_o so_o be_v the_o excess_n of_o the_o last_o to_o all_o the_o number_n go_v before_o the_o last_o which_o be_v require_v to_o be_v prove_v ¶_o the_o 36._o theorem_a the_o 36._o proposition_n if_o from_o unity_n be_v take_v number_n how_o many_o soever_o in_o double_a proportion_n continual_o until_o the_o whole_a add_v together_o be_v a_o prime_a number_n and_o if_o the_o whole_a multiply_v the_o last_o produce_v any_o number_n that_o which_o be_v produce_v be_v a_o perfect_v number_n number_n svppose_v that_o from_o unity_n be_v take_v these_o number_n a_o b_o c_o d_o in_o double_a proportion_n continual_o so_o that_o all_o those_o number_n a_o b_o c_o d_o &_o unity_n add_v together_o make_v a_o prime_a number_n and_o let_v e_o be_v the_o number_n compose_v 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
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
reform_v by_o m._n dee_n describe_v for_o it_o in_o the_o plain_n especial_o if_o you_o remember_v the_o form_n of_o the_o figure_n of_o the_o 29._o proposition_n of_o this_o book_n only_o that_o which_o there_o you_o conceive_v to_o be_v the_o base_a imagine_v here_o in_o both_o the_o figure_n of_o this_o second_o case_n to_o be_v the_o upper_a superficies_n opposite_a to_o the_o base_a and_o that_o which_o be_v there_o suppose_v to_o be_v the_o upper_a superficies_n conceive_v here_o to_o be_v the_o base_a you_o may_v describe_v they_o upon_o past_v paper_n for_o your_o better_a sight_n take_v heed_n you_o note_v the_o letter_n right_o according_a as_o the_o construction_n require_v flussas_n demonstrate_v this_o proposition_n a_o otherway_o take_v only_o the_o base_n of_o the_o solid_n and_o that_o after_o this_o manner_n take_v equal_a base_n which_o yet_o for_o the_o sure_a understanding_n let_v be_v utter_o unlike_a namely_o aebf_n and_o adch_n and_o let_v one_o of_o the_o side_n of_o each_o concur_v in_o one_o &_o the_o same_o right_a line_n aed_n &_o the_o base_n be_v upon_o one_o and_o the_o self_n same_o plain_n let_v there_o be_v suppose_v to_o be_v set_v upon_o they_o parallelipipedon_n under_o one_o &_o the_o self_n same_o altitude_n then_o i_o say_v that_o the_o solid_a set_v upon_o the_o base_a ab_fw-la be_v equal_a to_o the_o solid_a set_v upon_o the_o base_a ah_o by_o the_o point_n e_o draw_v unto_o the_o line_n ac_fw-la a_o parallel_n line_n eglantine_n which_o if_o it_o fall_v without_o the_o base_a ab_fw-la produce_v the_o right_a line_n hc_n to_o the_o point_n i_o now_o forasmuch_o as_o ab_fw-la and_o ah_o be_v parallelogrmae_n therefore_o by_o the_o 24._o of_o this_o book_n the_o triangle_n aci_n and_o egl_n shall_v be_v equaliter_fw-la the_o one_o to_o the_o other_o and_o by_o the_o 4._o of_o the_o first_o they_o shall_v be_v equiangle_n and_o equal_a and_o by_o the_o first_o definition_n of_o the_o six_o and_o four_o proposition_n of_o the_o same_o they_o shall_v be_v like_a wherefore_o prism_n erect_v upon_o those_o triangle_n and_o under_o the_o same_o altitude_n that_o the_o solid_n ab_fw-la and_o ah_o aâ—e_n shall_v be_v equal_a and_o like_a by_o the_o 8._o definition_n of_o this_o book_n for_o they_o be_v contain_v under_o like_o plain_a superficiece_n equal_a both_o in_o multitude_n and_o magnitude_n add_v the_o solid_a set_v upon_o the_o base_a acle_n common_a to_o they_o both_o wherefore_o the_o solid_a set_v upon_o the_o base_a aegc_n be_v equal_a to_o the_o solid_a set_v upon_o the_o base_a aeli_n and_o forasmuch_o as_o the_o superficiece_n aebf_n and_o adhc_n be_v equal_a by_o supposition_n and_o the_o part_n take_v away_o agnostus_n be_v equal_a to_o the_o part_n take_v away_o all_o therefore_o the_o residue_n by_o shall_v be_v equal_a to_o the_o residue_n gd_o wherefore_o as_o agnostus_n be_v to_o gd_a as_o all_o be_v to_o by_o namely_o equal_n to_o equal_n but_o as_o agnostus_n be_v to_o gd_o so_o iâ—_z the_o solid_a set_v upon_o agnostus_n to_o the_o solid_a set_v upon_o gd_o by_o the_o 25._o of_o this_o book_n for_o it_o be_v cut_v by_o a_o plain_a superficies_n set_v upon_o the_o line_n ge_z which_o superficies_n be_v parallel_n to_o the_o opposite_a superficiece_n wherefore_o as_o all_o be_v to_o by_o so_o be_v the_o solid_a set_v upon_o all_o to_o the_o solid_a set_v upon_o by_o wherefore_o by_o the_o 11._o of_o the_o five_o as_o the_o solid_a set_v upon_o agnostus_n or_o upon_o all_o which_o be_v equal_a unto_o it_o be_v to_o the_o solid_a set_v upon_o gd_o so_o be_v the_o same_o solid_a set_v upon_o agnostus_n or_o all_o to_o the_o solid_a set_v upon_o by_o wherefore_o by_o the_o 2._o part_n of_o the_o 9_o of_o the_o five_o the_o solid_n set_v upon_o gd_o and_o by_o shall_v be_v equal_a unto_o which_o solid_n if_o you_o add_v equal_a solid_n namely_o the_o solid_a set_v upon_o agnostus_n to_o the_o solid_a set_v upon_o gd_o and_o the_o solid_a set_v upon_o all_o to_o the_o solid_a set_v upon_o by_o the_o whole_a solid_n set_v upon_o the_o base_a ah_o and_o upon_o the_o base_a ab_fw-la â—hall_v be_v equal_a wherefore_o parallelipedon_n consist_v upon_o equal_a base_n and_o be_v under_o one_o and_o the_o self_n same_o altitude_n be_v equal_a the_o one_o to_o the_o other_o which_o be_v require_v to_o be_v prove_v ¶_o the_o 27._o theorem_a the_o 32._o proposition_n parallelipipedon_n be_v under_o one_o and_o the_o self_n same_o 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 svppose_v that_o these_o parallelipipedon_n ab_fw-la and_o cd_o be_v under_o one_o &_o the_o self_n same_o altitude_n then_o i_o say_v that_o those_o parallelipipedon_n ab_fw-la and_o cd_o 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 that_o be_v that_o as_o the_o base_a ae_n be_v to_o the_o base_a cf_o so_o be_v the_o parallelipipedon_n ab_fw-la to_o the_o parallelipipedon_n cd_o construction_n upon_o the_o line_n fg_o describe_v by_o the_o 45._o of_o the_o first_o the_o parallelogram_n fh_o equal_a to_o the_o parallelogram_n ae_n and_o equiangle_n with_o the_o parallelogram_n cf._n and_o upon_o the_o base_a fh_o describe_v a_o parallelipipedom_n of_o the_o self_n same_o altitude_n that_o the_o parallelipipedom_n cd_o be_v &_o let_v the_o same_o be_v gk_o demonstration_n now_o by_o the_o 31._o of_o the_o eleven_o the_o parallelipipedon_n ab_fw-la be_v equal_a to_o the_o parallelipipedon_n gk_o for_o they_o consist_v upon_o equal_a base_n namely_o ae_n and_o fh_o and_o be_v under_o one_o and_o the_o self_n same_o altitude_n and_o forasmuch_o as_o the_o parallelipipedon_n ck_o be_v cut_v by_o a_o plain_a superficies_n dg_o be_v parallel_n to_o either_o of_o the_o opposite_a plain_a superâ—icieces_n therefore_o by_o the_o 25._o of_o the_o eleven_o as_o the_o base_a hf_o be_v to_o the_o base_a fc_n so_o be_v the_o parallelipipedon_n gk_o to_o parallelipipedon_v cd_o but_o the_o base_a hf_o be_v equal_a to_o the_o base_a ae_n and_o the_o parallelipipedon_n gk_o be_v prove_v equal_a to_o the_o parallelipipedon_n ab_fw-la wherefore_o as_o the_o base_a aâ—e_n be_v to_o the_o base_a cf_o so_o be_v the_o parallelipedon_n ab_fw-la to_o the_o parallelipipedon_n cd_o wherefore_o parallelipipedon_n be_v under_o one_o and_o the_o self_n same_o 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 which_o be_v require_v to_o be_v demonstrate_v i_o need_v not_o to_o put_v any_o other_o figure_n for_o the_o declaration_n of_o this_o demonstration_n for_o it_o be_v easy_a to_o see_v by_o the_o figure_n there_o describe_v howbeit_o you_o may_v for_o the_o more_o full_a sight_n thereof_o describe_v solid_n of_o past_v paper_n according_a to_o the_o construction_n there_o set_v forth_o which_o will_v not_o be_v hard_o for_o you_o to_o do_v if_o you_o remember_v the_o description_n of_o such_o body_n before_o teach_v a_o corollary_n add_v by_o flussas_n equal_a parallelipipedon_n contain_v under_o one_o and_o the_o self_n same_o altitude_n have_v also_o their_o base_n equal_a for_o if_o the_o base_n shall_v be_v unequal_a the_o parallelipipedon_n also_o shall_v be_v unequal_a by_o this_o 32_o proposition_n and_o equal_a parallelipipedon_n have_v equal_a base_n have_v also_o one_o and_o the_o self_n same_o altitude_n for_o if_o they_o shall_v have_v a_o great_a altitude_n they_o shall_v exceed_v the_o equal_a parallelipipedon_n which_o have_v the_o self_n same_o altitude_n but_o if_o they_o shall_v have_v a_o less_o they_o shall_v want_v so_o much_o of_o those_o self_n same_o equal_a parallelipipedon_n the_o 28._o theorem_a the_o 33._o proposition_n like_a parallelipipedon_n be_v in_o treble_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 of_o like_a proportion_n be_v svppose_v that_o these_o parallelipipedon_n ab_fw-la and_o cd_o be_v like_a &_o let_v the_o side_n ae_n and_o cf_o be_v side_n of_o like_a proportion_n then_o i_o say_v the_o parallelipipedon_n ab_fw-la be_v unto_o the_o parallelipipedon_n cd_o in_o treble_a proportion_n of_o that_o in_o which_o the_o side_n ae_n be_v to_o the_o side_n cf._n extend_v the_o right_a line_n ae_n ge_z and_o he_o to_o the_o point_n king_n l_o m._n construction_n and_o by_o the_o 2._o of_o the_o first_o unto_o the_o line_n cf_o put_v the_o line_n eke_o equal_a and_o unto_o the_o line_n fn_o put_v the_o line_n el_n equal_a and_o moreover_o unto_o the_o line_n fr_z put_v the_o line_n em_n equal_a and_o make_v perfect_a the_o parallelogram_n kl_o and_o the_o parallelipipedon_n ko_o demonstration_n now_o forasmuch_o as_o these_o two_o line_n eke_o and_o el_n be_v equal_a to_o these_o two_o line_n cf_o and_o fn_o but_o the_o angle_n kel_n be_v equal_a to_o the_o angle_n cfn_n for_o the_o angle_n
a_o triangle_n and_o if_o the_o parallelogram_n be_v double_a to_o the_o triangle_n those_o prism_n be_v by_o the_o 40._o of_o the_o eleven_o equal_v the_o one_o to_o the_o other_o therefore_o the_o prism_n contain_v under_o the_o two_o triangle_n bkf_a and_o ehg_n and_o under_o the_o three_o parallelogram_n ebfg_n ebkh_n and_o khfg_n part_n be_v equal_a to_o the_o prism_n contain_v under_o the_o two_o triangle_n gfc_a and_o hkl_n and_o under_o the_o three_o parallelogram_n kfcl_n lcgh_n and_o hkfg_n and_o it_o be_v manifest_a that_o both_o these_o prism_n of_o which_o the_o base_a of_o one_o be_v the_o parallelogram_n ebfg_n pyramid_n and_o the_o opposite_a unto_o it_o the_o line_n kh_o and_o the_o base_a of_o the_o other_o be_v the_o triangle_n gfc_n and_o the_o opposite_a side_n unto_o it_o the_o triangle_n klh_n be_v great_a than_o both_o these_o pyramid_n who_o base_n be_v the_o triangle_n age_n and_o hkl_n and_o top_n the_o point_n h_o &_o d._n for_o if_o we_o draw_v these_o right_a line_n of_o and_o eke_o the_o prism_n who_o base_a be_v the_o parallelogram_n ebfg_n and_o the_o opposite_a unto_o it_o the_o right_a line_n hk_o be_v great_a than_o the_o pyramid_n who_o base_a be_v the_o triangle_n ebf_n &_o top_n the_o point_n k._n but_o the_o pyramid_n who_o base_a be_v the_o triangle_n ebf_n and_o top_n the_o point_n king_n be_v equal_a to_o the_o pyramid_n who_o base_a be_v the_o triangle_n aeg_n and_o top_n the_o point_n h_o for_o they_o be_v contain_v under_o equal_a and_o like_a plain_a superficiece_n wherefore_o also_o the_o prism_n who_o base_a be_v the_o parallelogram_n ebfg_n and_o the_o opposite_a unto_o it_o the_o right_a line_n hk_o be_v great_a than_o the_o pyramid_n who_o base_a be_v the_o triangle_n aeg_n and_o top_n the_o point_n h._n but_o the_o prism_n who_o base_a be_v the_o parallelogram_n ebfg_n and_o the_o opposite_a unto_o it_o the_o right_a line_n hk_o be_v equal_a to_o the_o prism_n who_o base_a be_v the_o triangle_n gfc_n and_o the_o opposite_a side_n unto_o it_o the_o triangle_n hkl_n and_o the_o pyramid_n who_o base_a be_v the_o triangle_n aeg_n and_o top_n the_o point_n h_o be_v equal_a to_o the_o pyramid_n who_o base_a be_v the_o triangle_n hkl_n and_o top_n the_o point_n d._n part_n wherefore_o the_o foresay_a two_o prism_n be_v great_a than_o the_o foresay_a two_o pyramid_n who_o base_n be_v the_o triangle_n aeg_n hkl_n and_o top_n the_o point_n h_o and_o d._n wherefore_o the_o whole_a pyramid_n who_o base_a be_v the_o triangle_n abc_n proposition_n and_o top_n the_o point_n d_o be_v divide_v into_o two_o pyramid_n equal_a and_o like_v the_o one_o to_o the_o other_o and_o like_v also_o unto_o the_o whole_a pyramid_n have_v also_o triangle_n to_o their_o base_n and_o into_o two_o equal_a prism_n and_o the_o two_o prism_n be_v great_a than_o half_a of_o the_o whole_a pyramid_n which_o be_v require_v to_o be_v demonstrate_v if_o you_o will_v with_o diligence_n read_v these_o four_o book_n follow_v of_o euclid_n which_o concern_v body_n and_o clear_o see_v the_o demonstration_n in_o they_o contain_v it_o shall_v be_v requisite_a for_o you_o when_o you_o come_v to_o any_o proposition_n which_o concern_v a_o body_n or_o body_n whether_o they_o be_v regular_a or_o not_o first_o to_o describe_v of_o pâ—sâ—ed_a paper_n according_a as_o i_o teach_v you_o in_o the_o end_n of_o the_o definition_n of_o the_o eleven_o book_n such_o a_o body_n or_o body_n as_o be_v there_o require_v and_o have_v your_o body_n or_o body_n thus_o describe_v when_o you_o have_v note_v it_o with_o letter_n according_a to_o the_o figure_n set_v forth_o upon_o a_o plain_a in_o the_o proposition_n follow_v the_o construction_n require_v in_o the_o proposition_n as_o for_o example_n in_o this_o three_o proposition_n it_o be_v say_v that_o every_o pyramid_n have_v a_o triangle_n to_o â—is_n base_a may_v be_v divide_v into_o two_o pyramid_n etc_n etc_n here_o first_o describe_v a_o pyramid_n of_o past_v paper_n haâ—ing_v his_o base_a triangle_v it_o skill_v not_o whether_o it_o be_v equilater_n or_o equiangle_v or_o not_o only_o in_o this_o proposition_n be_v require_v that_o the_o base_a be_v a_o triangle_n then_o for_o that_o the_o proposition_n suppose_v the_o base_a of_o the_o pyramid_n to_o be_v the_o triangle_n abc_n note_v the_o base_a of_o your_o pyramid_n which_o you_o have_v describe_v with_o the_o letter_n abc_n and_o the_o top_n of_o your_o pyramid_n with_o the_o letter_n d_o for_o so_o be_v require_v in_o the_o proposition_n and_o thus_o have_v you_o your_o body_n order_v ready_a to_o the_o construction_n now_o in_o the_o construction_n it_o be_v require_v that_o you_o divide_v the_o line_n ab_fw-la bc_o ca._n etc_o etc_o namely_o the_o six_o line_n which_o be_v the_o side_n of_o the_o four_o triangle_n contain_v the_o pyramid_n into_o two_o equal_a part_n in_o the_o poyntet_fw-la â—_o f_o g_o etc_n etc_n that_o be_v you_o must_v divide_v the_o line_n ab_fw-la of_o your_o pyramid_n into_o two_o equal_a part_n and_o note_v the_o point_n of_o the_o division_n with_o the_o letter_n e_o and_o so_o the_o line_n bc_o be_v divide_v into_o two_o equal_a part_n note_v the_o point_n of_o the_o division_n with_o the_o letter_n f._n and_o so_o the_o rest_n and_o this_o order_n follow_v you_o as_o touch_v the_o rest_n of_o the_o construction_n there_o put_v and_o when_o you_o have_v finish_v the_o construction_n compare_v your_o body_n thus_o describe_v with_o the_o demonstration_n and_o it_o will_v make_v it_o very_o plain_a and_o easy_a to_o be_v understand_v whereas_o without_o such_o a_o body_n describe_v of_o matter_n it_o be_v hard_o for_o young_a beginner_n unless_o they_o have_v a_o very_a deep_a imagination_n full_o to_o conceive_v the_o demonstration_n by_o the_o sigâ—e_n as_o it_o be_v describe_v in_o a_o plain_a here_o for_o the_o better_a declaration_n of_o that_o which_o i_o have_v say_v have_v i_o set_v a_o figure_n who_o form_n if_o you_o describe_v upon_o past_v paper_n note_v with_o the_o like_a letter_n and_o cut_v the_o line_n â—a_o dam_n dc_o and_o fold_v it_o according_o it_o will_v make_v a_o pyramid_n describe_v according_a to_o the_o construction_n require_v in_o the_o proposition_n and_o this_o order_n follow_v you_o as_o touch_v all_o other_o proposition_n which_o concern_v body_n ¶_o a_o other_o demonstration_n after_o campane_n of_o the_o 3._o proposition_n suppose_v that_o there_o be_v a_o pyramid_n abcd_o have_v to_o his_o base_a the_o triangle_n bcd_o and_o let_v his_o top_n be_v the_o solid_a angle_n a_o from_o which_o let_v there_o be_v draw_v three_o subtended_a line_n ab_fw-la ac_fw-la and_o ad_fw-la to_o the_o three_o angle_n of_o the_o base_a and_o divide_v all_o the_o side_n of_o the_o base_a into_o two_o equal_a part_n in_o the_o three_o point_n e_o f_o g_o divide_v also_o the_o three_o subtended_a line_n ab_fw-la ac_fw-la and_o ad_fw-la in_o two_o equal_a part_n in_o the_o three_o point_n h_o edward_n l._n and_o draw_v in_o the_o base_a these_o two_o line_n of_o and_o eglantine_n so_o shall_v the_o base_a of_o the_o pyramid_n be_v divide_v into_o three_o superficiece_n whereof_o two_o be_v the_o two_o triangle_n bef_n and_o egg_v which_o be_v like_o both_o the_o one_o to_o the_o other_o and_o also_o to_o the_o whole_a base_a by_o the_o 2_o part_n of_o the_o second_o of_o the_o six_o &_o by_o the_o definition_n of_o like_a superâ—iciecâ—s_n &_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 the_o three_o superficies_n be_v a_o quadrangled_a parallelogram_n namely_o efgc_n which_o be_v double_a to_o the_o triangle_n egg_v by_o the_o 40._o and_o 41._o of_o the_o first_o now_o then_o again_o from_o the_o point_n h_o draw_v unto_o the_o point_n e_o and_o f_o these_o two_o subtendent_fw-la line_n he_o and_o hf_o draw_v also_o a_o subtended_a line_n from_o the_o point_n king_n to_o the_o point_n g._n and_o draw_v these_o line_n hk_o kl_o and_o lh_o wherefore_o the_o whole_a pyramid_n abcd_o be_v divide_v into_o two_o pyramid_n which_o be_v hbef_n and_o ahkl_n and_o into_o two_o prism_n of_o which_o the_o one_o be_v ehfgkc_n and_o be_v set_v upon_o the_o quadrangled_a base_a cfge_n the_o other_o be_v egdhkl_n and_o have_v to_o his_o base_a the_o triangle_n egg_v now_o as_o touch_v the_o two_o pyramid_n hbef_a and_o ahkl_n that_o they_o be_v equal_v the_o one_o to_o the_o other_o and_o also_o that_o they_o be_v like_o both_o the_o one_o to_o the_o other_o and_o also_o to_o the_o whole_a it_o be_v manifest_a by_o the_o definition_n of_o equal_a and_o like_a body_n and_o by_o the_o 10._o of_o the_o eleven_o and_o by_o 2._o part_n of_o the_o second_o of_o the_o six_o and_o that_o the_o two_o prism_n be_v equal_a it_o
it_o comprehend_v wherefore_o the_o pyramid_n who_o base_a be_v the_o square_n abcd_o and_o altitude_n the_o self_n same_o that_o the_o cone_fw-mi have_v be_v great_a than_o the_o half_a of_o the_o cone_fw-it divide_v by_o the_o 30._o of_o the_o three_o every_o one_o of_o the_o circumference_n ab_fw-la bc_o cd_o and_o dam_n into_o two_o equal_a part_n in_o the_o point_n e_o f_o g_o and_o h_o and_o draw_v these_o right_a line_n ae_n ebb_n bf_o fc_o cg_o gd_o dh_o and_o ha._n wherefore_o every_o one_o of_o these_o triangle_n aeb_fw-mi bfc_n cgd_v and_o dha_n be_v great_a than_o the_o half_a part_n of_o the_o segment_n of_o the_o circle_n describe_v about_o it_o upon_o every_o one_o of_o these_o triangle_n aeb_fw-mi bfc_n cgd_v and_o dha_n describe_v a_o pyramid_n of_o equal_a altitude_n with_o the_o cone_n and_o after_o the_o same_o manner_n every_o one_o of_o those_o pyramid_n so_o describe_v be_v great_a than_o the_o half_a part_n of_o the_o segment_n of_o the_o cone_fw-it set_v upon_o the_o segment_n of_o the_o circle_n now_o therefore_o divide_v by_o the_o 30_o of_o the_o three_o the_o circumference_n remain_v into_o two_o equal_a part_n &_o draw_v right_a line_n &_o raise_v up_o upon_o every_o one_o of_o those_o triangle_n a_o pyramid_n of_o equal_a altitude_n with_o the_o cone_n and_o do_v this_o continual_o we_o shall_v at_o the_o length_n by_o the_o first_o of_o the_o ten_o leave_v certain_a segment_n of_o the_o cone_n which_o shall_v be_v less_o then_o the_o excess_n whereby_o the_o cone_n exceed_v the_o three_o part_n of_o the_o cylinder_n let_v those_o segment_n be_v ae_n ebb_n bf_o fc_o cg_o gd_o dh_o and_o ha._n wherefore_o the_o pyramid_n remain_v who_o base_a be_v the_o poligonon_n figure_n aebfcgdh_n and_o altitude_n the_o self_n same_o with_o the_o cone_n be_v great_a than_o the_o three_o part_n of_o the_o cylinder_n but_o the_o pyramid_n who_o base_a be_v the_o polygonon_n figure_n aebfcgdh_n and_o altitude_n the_o self_n same_o with_o the_o cone_n be_v the_o three_o part_n of_o the_o prism_n who_o base_a be_v the_o poligonon_n figure_n aebfcgdh_n and_o altitude_n the_o self_n same_o with_o the_o cylinder_n wherefore_o you_o the_o prism_n who_o base_a be_v the_o polygonon_n figure_n aebfcgdh_n and_o altitude_n the_o self_n same_o with_o the_o cylinder_n be_v great_a than_o the_o cylinder_n who_o base_a be_v the_o circle_n abcd._n but_o it_o be_v also_o less_o for_o it_o be_v contain_v of_o it_o which_o be_v impossible_a wherefore_o the_o cylinder_n be_v not_o in_o less_o proportion_n to_o the_o cone_n then_o in_o treble_a proportion_n and_o it_o be_v prove_v that_o it_o be_v not_o in_o great_a proportion_n to_o the_o cone_n then_o in_o treble_a proportion_n wherefore_o the_o cone_n be_v the_o three_o part_n of_o the_o cylinder_n wherefore_o every_o cone_n be_v the_o three_o part_n of_o a_o cylinder_n have_v one_o &_o the_o self_n same_o base_a and_o one_o and_o the_o self_n same_o altitude_n with_o it_o which_o be_v require_v to_o be_v demonstrate_v ¶_o add_v by_o m._n john_n dee_n ¶_o a_o theorem_a 1._o the_o superficies_n of_o every_o upright_o cylinder_n except_o his_o base_n be_v equal_a to_o that_o circle_n who_o semidiameter_n be_v middle_a proportional_a between_o the_o side_n of_o the_o cylinder_n and_o the_o diameter_n of_o his_o base_a ¶_o a_o theorem_a 2._o the_o superficies_n of_o every_o upright_o or_o isosceles_a cone_n except_o the_o base_a be_v equal_a to_o that_o circle_n who_o semidiameter_n be_v middle_a proportional_a between_o the_o side_n of_o that_o cone_n and_o the_o semidiameter_n of_o the_o circle_n which_o be_v the_o base_a of_o the_o cone_n my_o intent_n in_o addition_n be_v not_o to_o amend_v euclideâ—_n method_n which_o need_v little_a add_v or_o none_o at_o all_o but_o my_o desire_n be_v somewhat_o to_o furnish_v you_o towards_o a_o more_o general_a art_n mathematical_a them_z euclides_n element_n addition_n remain_v in_o the_o term_n in_o which_o they_o be_v write_v can_v sufficient_o help_v you_o unto_o and_o though_o euclides_n element_n with_o my_o addition_n run_v not_o in_o one_o methodical_a race_n towards_o my_o mark_n yet_o in_o the_o mean_a space_n my_o addition_n either_o geve_v light_a where_o they_o be_v annex_v to_o euclides_n matter_n or_o geve_v some_o ready_a aid_n and_o show_v the_o way_n to_o dilate_v your_o discourse_n mathematical_a or_o to_o invent_v and_o practice_v thing_n mechanical_o and_o in_o deed_n if_o more_o leysor_n have_v happen_v many_o more_o strange_a matter_n mathematical_a have_v according_a to_o my_o purpose_n general_a be_v present_o publish_v to_o your_o knowledge_n but_o want_v of_o due_a leasour_n cause_v you_o to_o want_v that_o which_o my_o good_a will_n towards_o you_o most_o hearty_o do_v wish_v you_o as_o concern_v the_o two_o theorem_n here_o annex_v their_o verity_n be_v by_o archimedes_n in_o his_o book_n of_o the_o sphere_n and_o cylinder_n manifest_o demonstrate_v and_o at_o large_a you_o may_v therefore_o bold_o trust_v to_o they_o and_o use_v they_o as_o supposition_n in_o any_o your_o purpose_n till_o you_o have_v also_o their_o demonstration_n but_o if_o you_o well_o remember_v my_o instruction_n upon_o the_o first_o proposition_n of_o this_o book_n and_o my_o other_o addition_n upon_o the_o second_o with_o the_o supposition_n how_o a_o cylinder_n and_o a_o cone_n be_v mathematical_o produce_v you_o will_v not_o need_v archimedes_n demonstration_n nor_o yet_o be_v utter_o ignorant_a of_o the_o solid_a quantity_n of_o this_o cylinder_n and_o cone_n here_o compare_v the_o diameter_n of_o their_o base_a and_o heith_n be_v know_v in_o any_o measure_n neither_o can_v their_o crooked_a superficies_n remain_v unmeasured_a whereof_o undoubted_o great_a pleasure_n and_o commodity_n may_v grow_v to_o the_o sincere_a student_n and_o precise_a practiser_n ¶_o the_o 11._o theorem_a the_o 11._o proposition_n cones_n and_o cylinder_n be_v under_o one_o and_o the_o self_n same_o altitude_n be_v in_o that_o proportion_n the_o one_o other_o that_o their_o base_n be_v in_o like_a sort_n also_o may_v we_o proveâ—_v that_o as_o the_o circle_n efgh_o be_v to_o the_o circle_n abcd_o so_o be_v not_o the_o cone_n en_fw-fr to_o any_o solid_a less_o than_o the_o cone_n al._n now_o i_o say_v that_o as_o the_o circle_n abcd_o be_v to_o the_o circle_n efgh_o so_o be_v not_o the_o cone_n all_o to_o any_o solid_a great_a than_o the_o cone_fw-mi en_fw-fr for_o if_o it_o be_v possible_a let_v it_o be_v unto_o a_o great_a namely_o to_o the_o solid_a x._o wherefore_o by_o conversion_n as_o the_o circle_n efgh_o be_v to_o the_o circle_n abcd_o so_o be_v the_o solid_a x_o to_o the_o cone_n all_o but_o as_o the_o solid_a x_o be_v to_o the_o cone_n all_o so_o be_v the_o cone_n en_fw-fr to_o some_o solid_a less_o than_o the_o cone_n all_o as_o we_o may_v see_v by_o the_o assumpt_n put_v after_o thâ—_z second_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 circle_n efgh_o be_v to_o the_o circle_n abcgâ—_n so_o be_v the_o cone_n en_fw-fr to_o some_o solid_a less_o than_o the_o cone_n all_o which_o we_o have_v prove_v to_o be_v impossible_a wherefore_o as_o the_o circle_n abcd_o be_v to_o the_o circle_n efgh_o so_o be_v not_o the_o cone_n all_o to_o any_o solid_a great_a than_o the_o cone_fw-mi en_fw-fr and_o it_o be_v also_o prove_v that_o it_o be_v not_o to_o any_o less_o wherefore_o as_o the_o circle_n abcd_o be_v to_o the_o circle_n efgh_o so_o be_v the_o cone_n all_o to_o the_o cone_n en_fw-fr but_o as_o the_o cone_n be_v to_o the_o cone_n so_o be_v the_o cylinder_n to_o the_o cylinder_n by_o the_o 15._o of_o the_o five_o for_o the_o one_o be_v in_o treble_a proportion_n to_o the_o other_o cylinder_n wherefore_o by_o the_o 11._o of_o the_o five_o as_o the_o circle_n abcd_o be_v to_o the_o circle_n efgh_o so_o be_v the_o cylinder_n which_o be_v set_v upon_o they_o the_o one_o to_o the_o other_o the_o say_a cylinder_n being_n under_o equal_a altitude_n with_o the_o cones_fw-la cones_n therefore_o and_o cylinder_n be_v under_o one_o &_o the_o self_n same_o 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 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 like_a cones_n and_o cylinder_n be_v in_o treble_a proportion_n of_o that_o in_o which_o the_o diameter_n of_o their_o base_n be_v now_o also_o i_o say_v that_o the_o cone_fw-mi abcdl_n be_v not_o to_o any_o solid_a great_a than_o the_o cone_n efghn_n case_n in_o treble_a proportion_n of_o that_o in_o which_o the_o diameter_n bd_o be_v to_o the_o diameter_n fh_o for_o if_o it_o be_v possible_a let_v it_o be_v to_o a_o great_a namely_o to_o the_o solid_a x._o wherefore_o by_o conversion_n by_o the_o
superficies_n or_o solidity_n in_o the_o hole_n or_o in_o partâ—_n such_o certain_a knowledge_n demonstrative_a may_v arise_v and_o such_o mechanical_a exercise_n thereby_o be_v devise_v that_o sure_a i_o be_o to_o the_o sincere_a &_o true_a student_n great_a light_n aid_n and_o comfortable_a courage_n far_a to_o wade_v will_v enter_v into_o his_o heart_n and_o to_o the_o mechanical_a witty_a and_o industrous_a deviser_n new_a manner_n of_o invention_n &_o execution_n in_o his_o work_n will_v with_o small_a travail_n for_o foot_n application_n come_v to_o his_o perceiveraunce_n and_o understanding_n therefore_o even_o a_o manifold_a speculation_n &_o practice_n may_v be_v have_v with_o the_o circle_n his_o quantity_n be_v not_o know_v in_o any_o kind_n of_o small_a certain_a measure_n so_o likewise_o of_o the_o sphere_n many_o problem_n may_v be_v execute_v and_o his_o precise_a quantity_n in_o certain_a measure_n not_o determine_v or_o know_v yet_o because_o both_o one_o of_o the_o first_o humane_a occasion_n of_o invent_v and_o stablish_v this_o art_n be_v measure_v of_o the_o earth_n and_o therefore_o call_v geometria_n that_o be_v earthmeasure_v and_o also_o the_o chief_a and_o general_a end_n in_o deed_n be_v measure_n and_o measure_n require_v a_o determination_n of_o quantity_n in_o a_o certain_a measure_n by_o number_n express_v it_o be_v needful_a for_o mechanical_a earthmeasure_n not_o to_o be_v ignorant_a of_o the_o measure_n and_o content_n of_o the_o circle_n neither_o of_o the_o sphere_n his_o measure_n and_o quantity_n as_o near_o as_o sense_n can_v imagine_v or_o wish_v and_o in_o very_a deed_n the_o quantity_n and_o measure_n of_o the_o circle_n be_v know_v make_v not_o only_o the_o cone_n and_o cylinder_n but_o also_o the_o sphere_n his_o quantity_n to_o be_v as_o precise_o know_v and_o certain_a therefore_o see_v in_o respect_n of_o the_o circle_n quantity_n by_o archimedes_n specify_v this_o theorem_a be_v note_v unto_o you_o i_o will_v by_o order_n upon_o that_o as_o a_o supposition_n infer_v the_o conclusion_n of_o this_o our_o theorem_n note_n 1._o wherefore_o if_o you_o divide_v the_o one_o side_n as_o tq_n of_o the_o cube_fw-la tx_n into_o 21._o equal_a part_n and_o where_o 11._o part_n do_v end_n reckon_v from_o t_o suppose_v the_o point_n p_o and_o by_o that_o point_n p_o imagine_v a_o plain_a pass_v parallel_n to_o the_o opposite_a base_n to_o cut_v the_o cube_fw-la tx_n and_o thereby_o the_o cube_fw-la tx_n to_o be_v divide_v into_o two_o rectangle_n parallelipipedon_n namely_o tn_n and_o px_n it_o be_v manifest_a give_v tn_n to_o be_v equal_a to_o the_o sphere_n a_o by_o construction_n and_o the_o 7._o of_o the_o five_o note_n 2._o second_o the_o whole_a quantity_n of_o the_o sphere_n a_o assign_v be_v contain_v in_o the_o rectangle_n parallelipipedon_n tn_n you_o may_v easy_o transform_v the_o same_o quantity_n into_o other_o parallelipipedon_n rectangle_v of_o what_o height_n and_o of_o what_o parallelogram_n base_a you_o listen_v by_o my_o first_o and_o second_o problem_n upon_o the_o 34._o of_o this_o book_n and_o the_o like_a may_v you_o do_v to_o any_o assign_a part_n of_o the_o sphere_n a_o by_o the_o like_a mean_n devide_v the_o parallelipipedon_n tn_n as_o the_o part_n assign_v do_v require_v as_o if_o a_o three_o four_o five_o or_o six_o part_n of_o the_o sphere_n a_o be_v to_o be_v have_v in_o a_o parallelipipedon_n of_o any_o parallelograâ—â—e_n base_a assign_v or_o of_o any_o heith_n assign_v then_o devide_v tp_n into_o so_o many_o part_n as_o into_o 4._o if_o a_o four_o part_n be_v to_o be_v transform_v or_o into_o five_o if_o a_o five_o part_n be_v to_o be_v transform_v etc_n etc_n and_o then_o proceed_v â—s_v you_o do_v with_o cut_v of_o tn_n from_o tx_n and_o that_o i_o say_v of_o parallelipipedon_n may_v in_o like_a sort_n by_o my_o â—â—yd_n two_o problem_n add_v to_o the_o 34._o of_o this_o book_n be_v do_v in_o any_o side_v column_n pyramid_n and_o prismeâ—_n so_o thâ—â—_n in_o pyramid_n and_o some_o prism_n you_o use_v the_o caution_n necessary_a in_o respect_n of_o their_o quan_fw-mi 〈…〉_o odyes_n have_v parallel_n equal_a and_o opposite_a base_n who_o part_n 〈…〉_o re_fw-mi in_o their_o proposition_n be_v by_o euclid_n demonstrate_v and_o final_o 〈…〉_o addition_n you_o have_v the_o way_n and_o order_n how_o to_o geve_v to_o a_o sphere_n or_o any_o segment_n oâ—_n the_o same_o cones_n or_o cylinder_n equal_a or_o in_o any_o proportion_n between_o two_o right_a line_n give_v with_o many_o other_o most_o necessary_a speculation_n and_o practice_n about_o the_o sphere_n i_o trust_v that_o i_o have_v sufficient_o â—raughted_v your_o imagination_n for_o your_o honest_a and_o profitable_a study_n herein_o and_o also_o give_v you_o reaâ—â—_n â—â—tter_n wheâ—â—_n with_o to_o sâ—â—p_v the_o mouth_n of_o the_o malicious_a ignorant_a and_o arrogant_a despiser_n of_o the_o most_o excellent_a discourse_n travail_n and_o invention_n mathematical_a sting_v aswell_o the_o heavenly_a sphere_n &_o star_n their_o spherical_a solidity_n geometry_n with_o their_o convene_n spherical_a superficies_n to_o the_o earth_n at_o all_o time_n respect_v and_o their_o distance_n from_o the_o earth_n as_o also_o the_o whole_a earthly_a sphere_n and_o globe_n itself_o and_o infinite_a other_o case_n concern_v sphere_n or_o globe_n may_v hereby_o with_o as_o much_o ease_n and_o certainty_n be_v determine_v of_o as_o of_o the_o quantity_n of_o any_o bowl_n ball_n or_o bullet_n which_o we_o may_v gripe_v in_o our_o hand_n reason_n and_o experience_n be_v our_o witness_n and_o without_o these_o aid_n such_o thing_n of_o importance_n never_o able_a of_o we_o certain_o to_o be_v know_v or_o attain_a unto_o here_o end_n m._n john_n do_v you_o his_o addition_n upon_o the_o last_o proposition_n of_o the_o twelve_o book_n a_o proposition_n add_v by_o flussas_n if_o a_o sphere_n touch_v a_o plain_a superficiesâ—_n a_o right_a line_n draw_v from_o the_o centre_n to_o the_o touch_n shall_v be_v erect_v perpendicular_o to_o the_o plain_a superficies_n suppose_v that_o there_o be_v a_o sphere_n bcdl_n who_o centre_n let_v be_v the_o point_n a._n and_o let_v the_o plain_a superficies_n gci_n touch_v the_o spear_n in_o the_o point_n c_o and_o extend_v a_o right_a line_n from_o the_o centre_n a_o to_o the_o point_n c._n then_o i_o say_v that_o the_o line_n ac_fw-la be_v erect_v perpendicular_o to_o tâ—e_z plain_a gic._n let_v the_o sphere_n be_v cut_v by_o plain_a superficiece_n pass_v by_o the_o right_a line_n lac_n which_o plain_n let_v be_v abcdl_n and_o acel_n which_o let_v cut_v the_o plain_n gci_n by_o the_o right_a line_n gch_o and_o kci_fw-la now_o it_o be_v manifest_a by_o the_o assumpt_n put_v before_o the_o 17._o of_o this_o book_n that_o the_o two_o section_n of_o the_o sphere_n shall_v be_v circle_n have_v to_o their_o diameter_n the_o line_n lac_n which_o be_v also_o the_o diameter_n of_o the_o sphere_n wherefore_o the_o right_a line_n gch_o and_o kci_fw-la which_o be_v draw_v in_o the_o plain_n gci_n do_v at_o the_o point_n c_o fall_n without_o the_o circle_n bcdl_n and_o ecl._n wherefore_o they_o touch_v the_o circle_n in_o the_o point_n c_o by_o the_o second_o definition_n of_o the_o three_o wherefore_o the_o right_a line_n lac_n make_v right_a angle_n with_o the_o line_n gch_v and_o kci_fw-la by_o the_o 16._o of_o the_o three_o wherefore_o by_o the_o 4._o of_o the_o eleven_o the_o right_a line_n ac_fw-la be_v erect_v perpendicular_o to_o to_o the_o plain_a superficies_n gci_n wherein_o be_v draw_v the_o line_n gch_v and_o kci_fw-la if_o therefore_o a_o sphere_n touch_v a_o plain_a superficies_n a_o right_a line_n draw_v from_o the_o centre_n to_o the_o touch_n shall_v be_v erect_v perpendicular_o to_o the_o plain_a superficies_n which_o be_v require_v to_o be_v prove_v the_o end_n of_o the_o twelve_o book_n of_o euclides_n element_n ¶_o the_o thirteen_o book_n of_o euclides_n element_n in_o this_o thirteen_o book_n be_v set_v forth_o certain_a most_o wonderful_a and_o excellent_a passion_n of_o a_o line_n divide_v by_o a_o extreme_a and_o mean_a proportion_n book_n a_o matter_n undoubted_o of_o great_a and_o infinite_a use_n in_o geometry_n as_o you_o shall_v both_o in_o this_o book_n and_o in_o the_o other_o book_n follow_v most_o evident_o perceive_v it_o teach_v moreover_o the_o composition_n of_o the_o five_o regular_a solid_n and_o how_o to_o inscribe_v they_o in_o a_o sphere_n give_v and_o also_o set_v forth_o certain_a comparison_n of_o the_o say_a body_n both_o the_o one_o to_o the_o other_o and_o also_o to_o the_o sphere_n wherein_o they_o be_v describe_v the_o 1._o theorem_a the_o 1._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 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 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
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
proposition_n after_o prâ—â—lus_n a_o corollary_n take_v out_o of_o flussates_n demonstration_n lead_v to_o 〈◊〉_d absurdity_n a_o addition_n oâ—_n pelitarius_n demonstration_n three_o case_n in_o this_o proposition_n the_o first_o case_n construction_n demonstration_n three_o case_n in_o this_o proposition_n the_o first_o case_n every_o case_n may_v happen_v seven_o diverse_a way_n the_o like_a variety_n in_o each_o of_o the_o other_o two_o case_n euclides_n construction_n and_o demostration_n serve_v in_o all_o these_o case_n and_o in_o their_o varity_n also_o construction_n demonstration_n how_o triangle_n be_v say_v to_o be_v in_o the_o self_n same_o parallel_a line_n comparison_n of_o two_o triangle_n who_o side_n be_v equal_a their_o base_n and_o angle_n at_o the_o top_n be_v unequal_a when_o they_o be_v less_o than_o two_o right_a angle_n construction_n demonstration_n three_o case_n in_o this_o proposition_n each_o of_o these_o case_n also_o may_v be_v diverse_o note_n an_o other_o addition_n of_o pelitarius_n construction_n demonstration_n this_o theorem_a the_o converse_n of_o the_o 37._o proposition_n a_o addition_n of_o flâ—ssases_n a_o addition_n of_o campanus_n construction_n demonstration_n lead_v to_o a_o absurdity_n this_o proposition_n be_v the_o converse_n of_o the_o 38._o proposition_n demonstration_n two_o case_n in_o this_o proposition_n a_o corollary_n the_o self_n same_o demonstration_n will_v serve_v if_o the_o triangle_n &_o the_o parallelogram_n be_v upon_o equal_a base_n the_o converse_n of_o this_o proposition_n an_o other_o converse_n of_o the_o same_o proposition_n comparison_n of_o a_o triangle_n and_o a_o trapesium_n be_v upon_o one_o &_o the_o self_n same_o base_a and_o in_o the_o self_n same_o parallel_a line_n construction_n demonstration_n supplement_n &_o complement_n three_o case_n in_o this_o theorem_a the_o first_o case_n this_o proposition_n call_v gnomical_a and_o mystical_a the_o converse_n of_o this_o proposition_n construction_n demonstration_n application_n of_o space_n with_o excess_n or_o want_v a_o ancient_a invention_n of_o pythagoras_n how_o a_o figure_n be_v say_v to_o be_v apply_v to_o a_o line_n three_o thing_n give_v in_o this_o proposition_n the_o converse_n of_o this_o proposition_n construction_n demonstration_n a_o addition_n of_o pelitarius_n to_o describe_v a_o square_n mechanical_o a_o addition_n of_o procâ—â—â—_n the_o converse_n thereof_o construction_n demonstration_n pythagoras_n the_o first_o inventor_n of_o this_o proposition_n a_o addition_n of_o pâ—lâ—tariâ—â—_n an_o other_o addition_n of_o pelitarius_n an_o other_o addition_n of_o pelitarius_n an_o other_o addition_n of_o pelitarius_n a_o corollary_n this_o proposition_n be_v the_o converse_n of_o the_o former_a the_o argument_n of_o the_o second_o book_n what_o be_v the_o power_n of_o a_o line_n many_o compendious_a rule_n of_o reckon_v gather_v one_o of_o this_o book_n and_o also_o many_o rule_n of_o algebra_n two_o wonderful_a proposition_n in_o this_o book_n first_o definition_n what_o a_o parallelogram_n be_v four_o kind_n of_o parallelogram_n second_o definition_n a_o proposition_n add_v by_o campane_n after_o the_o last_o proposition_n of_o the_o first_o book_n construction_n demonstration_n barlaam_n barlaam_n construction_n demonstration_n barlaam_n construction_n demonstration_n barlaam_n construction_n demonstration_n a_o corollary_n barlaam_n construction_n demonstration_n constrâ—ction_n demonstration_n construction_n demonstration_n construction_n demonstration_n many_o and_o singular_a use_n of_o this_o proposition_n this_o proposition_n can_v not_o be_v reduce_v unto_o number_n demonstration_n demonstration_n a_o corollary_n this_o proposition_n true_a in_o all_o kind_n of_o triangle_n construction_n demonstration_n the_o argument_n of_o this_o book_n the_o first_o definition_n definition_n of_o unequal_a circle_n second_o definition_n a_o contigent_a line_n three_o definition_n the_o touch_n of_o circle_n be_v 〈◊〉_d in_o one_o poâ—â—â—_n only_o circle_n may_v touch_v together_o two_o maâ—â—â—_n of_o way_n four_o definition_n five_o definition_n six_o definition_n mix_v angle_n arke_n chord_n seven_o definition_n difference_n of_o a_o angle_n of_o a_o section_n and_o of_o a_o angle_n in_o a_o section_n eight_o definition_n nine_o definition_n ten_o definition_n two_o definition_n first_o second_o why_o euclid_n define_v not_o equal_a section_n constuction_n demonstration_n lead_v to_o a_o impossibility_n correlary_a demonstration_n lead_v to_o a_o impossibility_n the_o first_o para_fw-it of_o this_o proposition_n construction_n demonstration_n the_o second_o part_n converse_v of_o the_o first_o demonstration_n demonstration_n lead_v to_o a_o impossibility_n two_o case_n in_o this_o proposition_n construction_n demonstration_n lead_v to_o a_o impossibility_n demonstration_n lead_v to_o a_o impossibility_n two_o caseâ—_n in_o this_o proposition_n construction_n the_o first_o part_n of_o this_o proposition_n demonstration_n second_o part_n three_o part_n this_o demonstrate_v by_o a_o argument_n lead_v to_o a_o impossibilie_n an_o other_o demonstration_n of_o the_o latter_a part_n of_o the_o proposition_n lead_v also_o to_o a_o impossibility_n a_o corollary_n three_o part_n an_o other_o demonstration_n of_o the_o latter_a part_n lead_v also_o to_o a_o impossibility_n this_o proposion_n be_v common_o call_v caâ—dâ—_n panonis_fw-la a_o corollary_n construction_n demonstration_n an_o other_o demonstration_n of_o the_o same_o lead_v also_o to_o a_o impossibility_n demonstration_n lead_v to_o a_o impossibility_n an_o other_o demonstration_n of_o the_o same_o lead_v also_o to_o a_o impossibility_n construction_n demonstration_n lead_v to_o a_o impossibility_n an_o other_o demonstration_n of_o the_o same_o lead_v also_o to_o a_o impossibility_n the_o same_o again_o demonstrate_v by_o a_o argument_n lead_v to_o a_o absurdititie_n demonstratiâ—_n lead_v to_o a_o impossibility_n an_o other_o demonstration_n after_o pelitarius_n lead_v also_o to_o a_o absurdity_n of_o circle_n which_o touch_v the_o one_o the_o other_o inward_o of_o circle_n which_o touch_v the_o one_o the_o other_o outward_o an_o other_o demonstration_n after_o pelitarius_n &_o flussates_n of_o circle_n which_o touch_n the_o one_o the_o other_o outward_o of_o circle_n which_o touch_n the_o one_o the_o other_o inward_o the_o first_o part_n of_o this_o theorem_a construction_n demonstration_n demonstration_n the_o second_o part_n which_o be_v the_o converse_n of_o the_o first_o an_o other_o demonstration_n of_o the_o first_o part_n after_o campane_n construction_n demonstration_n an_o other_o demonstration_n after_o campane_n the_o first_o part_n of_o this_o theorem_a demonstration_n lead_v to_o a_o absurdity_n second_o part_n three_o part_n construction_n demonstration_n a_o addition_n of_o pelitarius_n this_o problem_n commodious_a for_o the_o inscribe_v and_o circumscribe_v of_o figure_n in_o or_o abouâ—_n circle_n demonstration_n lead_v to_o a_o impossibility_n an_o other_o demonstration_n after_o orontius_n demonstration_n lead_v to_o a_o impossibility_n two_o case_n in_o this_o proposition_n the_o one_o when_o the_o angle_n set_v at_o the_o circumference_n include_v the_o centre_n demonstration_n the_o other_o when_o the_o same_o angle_n set_v at_o the_o circumference_n include_v not_o the_o centre_n construction_n demonstration_n three_o case_n in_o this_o proposition_n the_o first_o case_n the_o second_o case_n the_o three_o case_n construction_n demonstration_n demonstration_n lead_v to_o a_o impossibility_n a_o addition_n of_o campane_n demonstrate_v by_o pelitariâ—s_n demonstration_n lead_v to_o a_o impossibility_n an_o other_o demonstration_n construction_n three_o case_n in_o this_o proposition_n the_o first_o case_n demonstration_n the_o second_o case_n the_o three_o case_n a_o addition_n construction_n demonstration_n demonstration_n lead_v to_o a_o impossibility_n construction_n demonstration_n the_o converse_n of_o the_o former_a proposition_n construction_n demonstration_n construction_n demonstration_n second_o part_n thirâ—_n part_n the_o five_o and_o last_o part_n an_o other_o demonstration_n to_o prove_v that_o the_o angâ—e_n in_o a_o semicircle_n be_v a_o right_a angle_n a_o corollary_n a_o addition_n of_o pâ—litarius_n demonstration_n leaâ—ing_v to_o a_o absurdity_n a_o addition_n of_o campane_n construction_n demonstration_n two_o case_n in_o this_o proposition_n three_o case_n in_o this_o proposition_n the_o first_o case_n construction_n demonstration_n the_o second_o case_n construction_n demonstration_n the_o three_o case_n construction_n demonstration_n construction_n demonstration_n two_o case_n in_o this_o proposition_n first_o case_n demonstration_n the_o second_o câ—se_n construction_n demonstration_n three_o case_n in_o this_o proposition_n construction_n two_o case_n in_o this_o proposition_n the_o first_o case_n demonstration_n the_o second_o case_n construction_n demonstration_n first_o corollary_n second_o corollary_n three_o corollary_n this_o proposition_n be_v the_o converse_n of_o the_o former_a construction_n demonstration_n an_o other_o demonstration_n after_o pelitarius_n the_o argument_n of_o this_o book_n first_o definition_n second_o definition_n the_o inscriptition_n and_o circumscription_n of_o rectiline_a figure_n pertain_v only_o to_o regular_a figure_n the_o three_o definition_n the_o four_o definition_n the_o five_o definition_n the_o six_o devition_n seven_o definition_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 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theorem_a the_o 27._o proposition_n if_o a_o right_a line_n fall_v upon_o two_o right_a line_n do_v make_v the_o alternate_a angle_n equal_v the_o one_o to_o the_o other_o those_o two_o right_a line_n be_v parallel_n the_o one_o to_o the_o other_o svppose_v that_o the_o right_a line_n of_o fall_v upon_o these_o two_o right_a line_n ab_fw-la and_o cd_o do_v make_v the_o alternate_a angle_n namely_o the_o angle_n aef_n &_o efd_n equal_v the_o one_o to_o the_o other_o then_o i_o say_v that_o ab_fw-la be_v a_o parallel_a line_n to_o cd_o for_o if_o not_o than_o these_o line_n produce_v shall_v mete_v together_o either_o on_o the_o side_n of_o b_o and_o d_o or_o on_o the_o side_n of_o a_o &_o c._n absurdity_n let_v they_o be_v produce_v therefore_o and_o let_v they_o mete_v if_o it_o be_v possible_a on_o the_o side_n of_o b_o and_o d_o in_o the_o point_n g._n wherefore_o in_o the_o triangle_n gef_n the_o outward_a angle_n aef_n be_v equal_a to_o the_o inward_a and_o opposite_a angle_n efg_o which_o by_o the_o 16._o proposition_n be_v impossible_a wherefore_o the_o line_n ab_fw-la and_o cd_o be_v produce_v on_o the_o side_n of_o b_o and_o d_o shall_v not_o meet_v in_o like_a sort_n also_o may_v it_o be_v prove_v that_o they_o shall_v not_o mete_v on_o the_o side_n of_o a_o and_o c._n but_o line_n which_o be_v produce_v on_o no_o side_n meet_v together_o be_v parrallell_a line_n by_o the_o last_o definition_n wherefore_o ab_fw-la be_v a_o parallel_v line_n to_o cd_o if_o therefore_o a_o right_a line_n fall_v upon_o two_o right_a line_n do_v make_v the_o alternate_a angle_n equal_v the_o one_o to_o the_o other_o those_o two_o right_a line_n be_v parallel_n the_o one_o to_o the_o other_o which_o be_v require_v to_o be_v demonstrate_v absurdity_n this_o word_n alternate_a be_v of_o euclid_n in_o diverse_a place_n diverse_o take_v sometime_o for_o a_o kind_n of_o situation_n in_o place_n and_o sometime_o for_o a_o order_n in_o proportion_n in_o which_o signification_n he_o use_v it_o in_o the_o v_o book_n and_o in_o his_o book_n of_o number_n and_o in_o the_o first_o signification_n he_o use_v it_o here_o in_o this_o place_n and_o general_o in_o all_o his_o other_o book_n absurdity_n have_v to_o do_v with_o line_n &_o figure_n and_o those_o two_o angle_n he_o call_v alternate_a which_o be_v both_o contain_v within_o two_o parallel_n or_o equidistant_a line_n be_v neither_o angle_n in_o order_n nor_o be_v on_o the_o one_o and_o self_n same_o side_n but_o be_v separate_v the_o one_o from_o the_o other_o by_o the_o line_n which_o fall_v on_o the_o two_o line_n the_o one_o angle_n be_v above_o and_o the_o other_o beneath_o the_o 19_o theorem_a the_o 28._o proposition_n if_o a_o right_a line_n fall_v upon_o two_o right_a line_n make_v the_o outward_a angle_n equal_a to_o the_o inward_a and_o opposite_a angle_n on_o one_o and_o the_o same_o side_n or_o the_o inward_a angle_n on_o one_o and_o the_o same_o side_n equal_a to_o two_o right_a angle_n those_o two_o right_a line_n shall_v be_v parallel_n the_o one_o to_o the_o other_o svppose_v that_o the_o right_a line_n of_o fall_a upon_o these_o two_o right_a line_n ab_fw-la and_o cd_o do_v make_v the_o outward_a angle_n egb_n equal_a to_o the_o inward_a and_o opposite_a angle_n ghd_v or_o do_v make_v the_o inward_a angle_n on_o one_o and_o the_o same_o side_n that_o be_v the_o angle_n bgh_a and_o ghd_v equal_a to_o two_o right_a angle_n then_o i_o say_v that_o the_o line_n ab_fw-la be_v a_o parallel_a line_n to_o the_o line_n cd_o for_o forasmuch_o as_o the_o angle_n egb_n be_v by_o supposition_n equal_a to_o the_o angle_n ghd_v demonstration_n and_o the_o angle_n egb_n be_v by_o the_o 15._o proposition_n equal_a to_o the_o angle_n agh_o therefore_o the_o angle_n agh_o be_v equal_a to_o the_o angle_n ghd_v and_o they_o be_v alternate_a angle_n wherefore_o by_o the_o 27._o proposition_n ab_fw-la be_v a_o parallel_a line_n to_o cd_o again_o forasmuch_o as_o the_o angle_n bgh_a and_o ghd_v be_v by_o supposition_n equal_a to_o two_o right_a angle_n &_o by_o the_o 13._o proposition_n the_o angle_n agh_o and_o bgh_o be_v also_o equal_a to_o two_o right_a angle_n wherefore_o the_o angle_n agh_o and_o bgh_o be_v equal_a to_o the_o angle_n bgh_a and_o ghd_v take_v away_o the_o angle_n bgh_n which_o be_v common_a to_o they_o both_o wherefore_o the_o angle_n remain_v namely_o agh_o be_v equal_a to_o the_o angle_n remain_v namely_o to_o ghd_v and_o they_o be_v alternate_a angle_n wherefore_o by_o the_o former_a proposition_n ab_fw-la be_v a_o parallel_a line_n to_o cd_o if_o therefore_o a_o right_a line_n fall_a upon_o two_o right_a line_n do_v make_v the_o outward_a angle_n equal_a to_o the_o inward_a and_o opposite_a angle_n on_o one_o and_o the_o same_o side_n or_o the_o inward_a angle_n on_o one_o and_o the_o same_o side_n equal_a to_o two_o right_a angle_n those_o two_o right_a line_n shall_v be_v parallel_n the_o one_o to_o the_o other_o which_o be_v require_v to_o be_v prove_v ptolomeus_n demonstrate_v the_o second_o part_n of_o this_o proposition_n namely_o that_o the_o two_o inward_a angle_n on_o one_o and_o the_o same_o side_n be_v equal_a the_o right_a line_n be_v parellel_n after_o this_o manner_n ptolemy_n the_o 20._o theorem_a the_o 29._o proposition_n a_o right_a line_n line_n fall_v upon_o two_o parallel_n right_a line_n make_v the_o alternate_a angle_n equal_v the_o one_o to_o the_o other_o and_o also_o the_o outward_a angle_n equal_a to_o the_o inward_a and_o opposite_a angle_n on_o one_o and_o the_o same_o side_n and_o moreover_o the_o inward_a angle_n on_o one_o and_o the_o same_o side_n equal_a to_o two_o right_a angle_n and_o the_o angle_n agh_o be_v by_o the_o 15._o proposition_n equal_a to_o
ae_n or_o ec_o and_o multiply_v 4._o into_o himself_o and_o there_o be_v produce_v 16_o which_o add_v unto_o 64_o and_o there_o shall_v be_v produce_v 80_o who_o root_n be_v √_o â—_o 80_o which_o be_v the_o line_n ebb_n or_o the_o line_n of_o by_o the_o 47._o of_o the_o first_o and_o forasmuch_o as_o the_o line_n of_o be_v √_o â—_o 80._o &_o the_o line_n ea_fw-la be_v 4._o therefore_o the_o line_n of_o be_v √_o â—_o 80_o 4._o and_o so_o much_o shall_v the_o line_n ah_o be_v and_o the_o line_n bh_o shall_v be_v 8_o √_o â—_o 80_o 4_o that_o be_v 12_o √_o â—_o 80._o now_o they_o 12_o √_o â—_o 80_o multiply_v into_o 8_o shall_v be_v as_o much_o as_o √_o â—_o 80_o 4._o multiply_v into_o itself_o for_o of_o either_o of_o they_o be_v produce_v 96_o √_o 5120._o the_o 11._o theorem_a the_o 12._o proposition_n in_o obtuseangle_v triangle_n the_o square_n which_o be_v make_v of_o the_o side_n subtend_v the_o obtuse_a angle_n be_v great_a than_o the_o square_n which_o be_v make_v of_o the_o side_n which_o comprehend_v the_o obtuse_a angle_n by_o the_o rectangle_n figure_n which_o be_v comprehend_v twice_o under_o one_o of_o those_o side_n which_o be_v about_o the_o obtuse_a angle_n upon_o which_o be_v produce_v fall_v a_o perpendicular_a line_n and_o that_o which_o be_v outward_o take_v between_o the_o perpendicular_a line_n and_o the_o obtuse_a angle_n svppose_v that_o abc_n be_v a_o obtuseangle_n triangle_n have_v the_o angle_n bac_n obtuse_a and_o from_o the_o point_n b_o by_o the_o 12._o of_o the_o first_o draw_v a_o perpendicular_a line_n unto_o ca_n produce_v and_o let_v the_o same_o be_v bd._n then_o i_o say_v that_o the_o square_n which_o be_v make_v of_o the_o side_n bc_o be_v great_a than_o the_o square_n which_o be_v make_v of_o the_o side_n basilius_n and_o ac_fw-la by_o the_o rectangle_n figure_n comprehend_v under_o the_o line_n ca_n and_o ad_fw-la twice_o demonstration_n for_o forasmuch_o as_o the_o right_a line_n cd_o be_v by_o chance_n divide_v in_o the_o point_n a_o therefore_o by_o the_o 4._o of_o the_o second_o the_o square_a which_o be_v make_v of_o cd_o be_v equal_a to_o the_o square_n which_o be_v make_v of_o ca_n and_o ad_fw-la and_o unto_o the_o rectangle_n figure_n contain_v under_o ca_n and_o ad_fw-la twice_o put_v the_o square_n which_o be_v make_v of_o db_o common_a unto_o they_o both_o wherefore_o the_o square_n which_o be_v make_v of_o cd_o and_o db_o be_v equal_a to_o the_o square_n which_o be_v make_v of_o the_o line_n ca_n ad_fw-la and_o db_o and_o unto_o the_o rectangle_n figure_n contain_v under_o the_o line_n ca_n and_o ad_fw-la twice_o but_o by_o the_o 47._o of_o the_o first_o the_o square_a which_o be_v make_v of_o cb_o be_v equal_a to_o the_o square_n which_o be_v make_v of_o the_o line_n cd_o and_o db._n for_o the_o angle_n at_o the_o point_n d_o be_v a_o right_a angle_n and_o unto_o the_o square_n which_o be_v make_v of_o ad_fw-la and_o db_o by_o the_o self_n same_o be_v equal_a the_o square_n which_o be_v mâ—de_v of_o ab_fw-la wherefore_o the_o square_n which_o be_v make_v of_o cb_o be_v equal_a to_o the_o square_n which_o be_v make_v of_o ca_n and_o ab_fw-la and_o unto_o the_o rectangle_n figure_n contain_v under_o the_o line_n ca_n and_o ad_fw-la twice_o wherefore_o the_o square_v which_o be_v make_v of_o cb_o be_v great_a than_o the_o square_n which_o be_v make_v of_o ca_n and_o ab_fw-la by_o the_o rectangle_n figure_n contain_v under_o the_o line_n ca_n and_o ad_fw-la twice_o in_o obtuseangle_v triangle_n therefore_o the_o square_n which_o be_v make_v of_o the_o side_n subtend_v the_o obtuse_a angle_n be_v great_a than_o the_o square_n which_o be_v make_v of_o the_o side_n which_o comprehend_v the_o obtuse_a angle_n by_o the_o rectangle_n figure_n which_o be_v comprehend_v twice_o under_o one_o of_o those_o side_n which_o be_v about_o the_o obtuse_a angle_n upon_o which_o be_v produce_v fall_v a_o perpendicular_a line_n and_o that_o which_o be_v outward_o take_v between_o the_o perpendicular_a line_n and_o the_o obtuse_a angle_n which_o be_v require_v to_o be_v demonstrate_v of_o what_o force_n this_o proposition_n and_o the_o proposition_n follow_v touch_v the_o measure_n of_o the_o obtuseangle_n triangle_n and_o the_o acuteangle_a triangle_n with_o the_o aid_n of_o the_o 47._o proposition_n of_o the_o first_o book_n touch_v the_o rightangle_n triangle_n he_o shall_v well_o perceive_v which_o shall_v at_o any_o time_n need_v the_o art_n of_o triangle_n in_o which_o by_o three_o thing_n know_v be_v ever_o search_v out_o three_o other_o thing_n unknown_a by_o help_n of_o the_o table_n of_o arke_n and_o cord_n the_o 12._o theorem_a the_o 13._o proposition_n in_o acuteangle_v triangle_n the_o square_n which_o be_v make_v of_o the_o side_n that_o subtend_v the_o acute_a angle_n be_v less_o than_o the_o square_n which_o be_v make_v of_o the_o side_n which_o comprehend_v the_o acute_a angle_n by_o the_o rectangle_n figure_n which_o be_v comprehend_v twice_o under_o one_o of_o those_o side_n which_o be_v about_o the_o acuteangle_n upon_o which_o fall_v a_o perpendicular_a line_n and_o that_o which_o be_v inward_o take_v between_o the_o perpendicular_a line_n and_o the_o acute_a angle_n svppose_v that_o abc_n be_v a_o acuteangle_n triangle_n have_v the_o angle_n at_o the_o point_fw-fr b_o acute_a &_o by_o the_o 12._o of_o the_o first_o from_o the_o point_n a_o draw_v unto_o the_o line_n bc_o a_o perpendicular_a line_n ad._n then_o i_o say_v that_o the_o square_n which_o be_v make_v of_o the_o line_n ac_fw-la be_v less_o than_o the_o square_n which_o be_v make_v of_o the_o line_n cb_o and_o basilius_n by_o the_o rectangle_n figure_n contain_v under_o the_o line_n cb_o and_o bd_o twice_o demonstration_n for_o forasmuch_o as_o the_o right_a line_n bc_o be_v by_o chance_n divide_v in_o the_o point_n d_o therefore_o by_o the_o 7._o of_o the_o second_o the_o square_n which_o be_v make_v of_o the_o line_n cb_o and_o bd_o be_v equal_a to_o the_o rectangle_n figure_n contain_v under_o the_o line_n cb_o and_o db_o twice_o and_o unto_o the_o square_n which_o be_v make_v of_o line_n cd_o put_v the_o square_n which_o be_v make_v of_o the_o line_n dam_fw-ge common_a unto_o they_o both_o wherefore_o the_o square_n which_o be_v make_v of_o the_o line_n cb_o bd_o and_o da_z be_v equal_a unto_o the_o rectangle_n figure_n contain_v under_o the_o line_n cb_o and_o bd_o twice_o and_o unto_o the_o square_n which_o be_v make_v of_o ad_fw-la and_o dc_o but_o to_o the_o square_n which_o be_v make_v of_o the_o line_n bd_o and_o da_z be_v equal_a the_o square_v which_o be_v make_v of_o the_o line_n ab_fw-la for_o th'angle_v at_o the_o point_fw-fr d_o be_v a_o right_a angle_n and_o unto_o the_o square_n which_o be_v make_v of_o the_o line_n ad_fw-la and_o dc_o be_v equal_a the_o square_n which_o be_v make_v of_o the_o line_n ac_fw-la by_o the_o 47._o of_o the_o first_o wherefore_o the_o square_n which_o be_v make_v of_o the_o line_n cb_o and_o basilius_n be_v equal_a to_o the_o square_n which_o be_v make_v of_o the_o line_n ac_fw-la and_o to_o that_o which_o be_v contain_v under_o the_o line_n cb_o and_o bd_o twice_o wherefore_o the_o square_n which_o be_v make_v of_o the_o line_n ac_fw-la be_v take_v alone_o be_v less_o than_o the_o square_n which_o be_v make_v of_o the_o line_n cb_o and_o basilius_n by_o the_o rectangle_n figure_n which_o be_v contain_v under_o the_o line_n cb_o and_o bd_o twice_o in_o rectangle_n triangle_n therefore_o the_o square_n which_o be_v make_v of_o the_o side_n that_o subtend_v the_o acute_a angle_n be_v less_o than_o the_o square_n which_o be_v make_v of_o the_o side_n which_o comprehend_v the_o acute_a angle_n by_o the_o rectangle_n figure_n which_o be_v comprehend_v twice_o under_o one_o of_o those_o side_n which_o be_v about_o the_o acute_a angle_n upon_o which_o fall_v a_o perpendicular_a line_n and_o that_o which_o be_v inward_o take_v between_o the_o perpendicular_a line_n and_o the_o acute_a angle_n which_o be_v require_v to_o be_v prove_v ¶_o a_o corollary_n add_v by_o orontius_n corollary_n hereby_o be_v easy_o gather_v that_o such_o a_o perpendicular_a line_n in_o rectangle_n triangle_n fall_v of_o necessity_n upon_o the_o side_n of_o the_o triangle_n that_o be_v neither_o within_o the_o triangle_n nor_o without_o but_o in_o obtuseangle_v triangle_n it_o fall_v without_o and_o in_o acuteangle_v triangle_n within_o for_o the_o perpendicular_a line_n in_o obtuseangle_a triangle_n and_o acuteangle_v triangle_n can_v not_o exact_o agree_v with_o the_o side_n of_o the_o triangle_n for_o then_o a_o obtuse_a &_o a_o acuteangle_n shall_v be_v equal_a to_o a_o right_a angle_n contrary_a to_o the_o eleven_o and_o twelve_o definition_n of_o the_o first_o book_n likewise_o in_o obtuseangle_a
triangle_n it_o can_v not_o fall_v within_o nor_o in_o acuteangle_a triangle_n without_o for_o then_o the_o outward_a angle_n of_o a_o triangle_n shall_v be_v less_o than_o the_o inward_a and_o opposite_a angle_n which_o be_v contrary_a to_o the_o 16._o of_o the_o first_o triangle_n and_o this_o be_v to_o be_v note_v that_o although_o proper_o a_o acuteangle_n triangle_n by_o the_o definition_n thereof_o give_v in_fw-ge the_o first_o book_n be_v that_o triangle_n who_o angle_n be_v all_o acute_a yet_o forasmuch_o as_o there_o be_v no_o triangle_n but_o that_o it_o have_v a_o acute_a angle_n this_o proposition_n be_v to_o be_v understand_v &_o be_v true_a general_o in_o all_o kind_n of_o triangle_n whatsoever_o and_o may_v be_v declare_v by_o they_o as_o you_o may_v easy_o prove_v the_o 2._o problem_n the_o 14._o proposition_n unto_o a_o rectiline_a figure_n give_v to_o make_v a_o square_a equal_a svppose_v that_o the_o rectiline_a figure_n give_v be_v a._n it_o be_v require_v to_o make_v a_o square_a equal_a unto_o the_o rectiline_a figure_n a._n construction_n make_v by_o the_o 45._o of_o the_o first_o unto_o the_o rectiline_a figure_n a_o a_o equal_a rectangle_n parallelogram_n bcde_v now_o if_o the_o line_n be_v be_v equal_a unto_o the_o line_n ed_z then_z be_v the_o thing_n do_v which_o be_v require_v for_o unto_o the_o rectiline_a figure_n a_o be_v make_v a_o equal_a square_a bd._n but_o if_o not_o one_o of_o these_o line_n be_v &_o be_v ed_z the_o great_a let_v be_v be_v the_o great_a and_o let_v it_o be_v produce_v unto_o the_o point_fw-fr f._n and_o by_o the_o 3._o of_o the_o first_o put_v unto_o ed_z a_o equal_a line_n ef._n and_o by_o the_o 10._o of_o the_o first_o divide_v the_o line_n bf_o into_o two_o equal_a part_n in_o the_o point_n g._n and_o make_v the_o centre_n the_o point_n g_o and_o the_o space_n gb_o or_o gf_o describe_v a_o semicircle_n bhf_n and_o by_o the_o 2._o petition_n extend_v the_o line_n de_fw-fr unto_o the_o point_n h._n demonstration_n and_o by_o the_o 1._o petition_n draw_v a_o line_n from_o g_z to_o h._n and_o forasmuch_o as_o the_o right_a line_n fb_o be_v divide_v into_o two_o equal_a part_n in_o the_o point_n g_o and_o into_o two_o unequal_a part_n in_o the_o point_n e_o therefore_o by_o the_o 5._o of_o the_o second_o the_o rectangle_n figure_n comprehend_v under_o the_o line_n be_v and_o of_o together_o with_o the_o square_n which_o be_v make_v of_o the_o line_n eglantine_n be_v equal_a to_o the_o square_n which_o be_v make_v of_o the_o line_n gf_o but_o the_o line_n gf_o be_v equal_a unto_o the_o line_n gh_o wherefore_o the_o rectangle_n figure_n comprehend_v under_o the_o line_n be_v and_o of_o together_o with_o the_o square_n which_o be_v make_v of_o the_o line_n ge_z be_v equal_a to_o square_v which_o be_v make_v of_o the_o line_n gh_o but_o unto_o the_o square_n which_o be_v make_v of_o the_o line_n gh_o be_v equal_a the_o square_n which_o be_v make_v of_o the_o line_n he_o and_o ge_z by_o 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the_o other_o in_o the_o point_n d_o and_o that_o only_a point_n be_v common_a to_o they_o both_o neither_o do_v the_o one_o enter_v into_o the_o other_o if_o any_o part_n of_o the_o one_o enter_v into_o any_o part_n of_o the_o other_o than_o the_o one_o cut_v and_o divide_v the_o other_o and_o touch_v the_o one_o the_o other_o not_o in_o one_o point_n only_o as_o in_o the_o other_o before_o but_o in_o two_o pointâ—s_n and_o have_v also_o a_o superficies_n common_a to_o they_o both_o as_o the_o circle_n ghk_v and_o hlk_v cut_v the_o one_o the_o other_o in_o two_o point_n h_o and_o edward_n and_o the_o one_o enter_v into_o the_o other_o also_o the_o superficies_n hk_o be_v common_a to_o they_o both_o for_o it_o be_v a_o part_n of_o the_o circle_n ghk_o and_o also_o it_o be_v a_o part_n of_o the_o circle_n hlk._n definition_n right_o line_n in_o a_o circle_n be_v say_v to_o be_v equal_o distant_a from_o the_o centre_n when_o perpendicular_a line_n draw_v from_o the_o centre_n unto_o 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centre_n of_o the_o circle_n and_o receive_v the_o circumference_n of_o the_o say_a section_n first_o the_o other_o pertain_v to_o the_o angle_n in_o the_o section_n which_o as_o before_o be_v say_v be_v ever_o in_o the_o circumference_n as_o if_o the_o angle_n bac_n be_v in_o the_o centre_n a_o and_o receive_v of_o the_o circumference_n blc_n be_v equal_a to_o the_o angle_n feg_n be_v also_o in_o the_o centre_n e_o and_o receive_v of_o the_o circumference_n fkg_v then_o be_v the_o two_o section_n bcl_n and_o fgk_n like_v by_o the_o first_o definition_n by_o the_o same_o definition_n also_o be_v the_o other_o two_o section_n like_a namely_o bcd_o and_o fgh_a for_o that_o the_o angle_n bac_n be_v equal_a to_o the_o
when_o perpendicular_a line_n draw_v from_o the_o centre_n to_o those_o line_n be_v equal_a by_o the_o 4._o definition_n of_o the_o three_o wherefore_o the_o line_n ab_fw-la and_o cd_o be_v equal_o distant_a from_o the_o centre_n but_o now_o suppose_v that_o the_o right_a line_n ab_fw-la and_o cd_o be_v equal_o distant_a from_o the_o centre_n first_o that_o be_v let_v the_o perpendicular_a line_n of_o be_v equal_a to_o the_o perpendicular_a line_n eglantine_n then_o i_o say_v that_o the_o line_n ab_fw-la be_v equal_a to_o the_o line_n cd_o for_o the_o same_o order_n of_o construction_n remain_v we_o may_v in_o like_a sort_n prove_v that_o the_o line_n ab_fw-la be_v double_a to_o the_o line_n of_o and_o that_o the_o line_n cd_o be_v double_a to_o the_o line_n cg_o and_o for_o asmuch_o as_o the_o line_n ae_n be_v equal_a to_o the_o line_n ce_fw-fr for_o they_o be_v draw_v from_o the_o centre_n to_o the_o circumference_n therefore_o the_o square_a of_o the_o line_n ae_n be_v equal_a to_o the_o square_n of_o the_o line_n ce._n but_o by_o the_o 47._o of_o the_o first_o to_o the_o square_n of_o the_o line_n ae_n be_v equal_a the_o square_n of_o the_o line_n of_o and_o fa._n and_o by_o the_o self_n same_o to_o the_o square_n of_o the_o line_n ce_fw-fr be_v equal_v the_o square_n of_o the_o line_n eglantine_n and_o gc_o wherefore_o the_o square_n of_o the_o line_n of_o and_o favorina_n be_v equal_a to_o the_o square_n of_o the_o line_n eglantine_n and_o gc_o of_o which_o the_o square_a of_o the_o line_n eglantine_n be_v equal_a to_o the_o square_n of_o the_o line_n of_o for_o the_o line_n of_o be_v equal_a to_o the_o line_n eglantine_n wherefore_o by_o the_o three_o common_a sentence_n the_o square_a remain_v namely_o the_o square_a of_o the_o line_n of_o be_v equal_a to_o the_o square_n of_o the_o line_n cg_o wherefore_o the_o line_n ac_fw-la be_v equal_a unto_o the_o line_n cg_o but_o the_o line_n ab_fw-la be_v double_a to_o the_o line_n of_o and_o the_o line_n cd_o be_v double_a to_o the_o line_n cg_o wherefore_o the_o line_n ab_fw-la be_v equal_a to_o the_o line_n cd_o wherefore_o in_o a_o circle_n equal_a right_a line_n be_v equal_o distant_a from_o the_o centre_n and_o line_n equal_o distant_a from_o the_o centre_n be_v equal_a the_o one_o to_o the_o outstretch_o which_o be_v require_v to_o be_v prove_v ¶_o a_o other_o demonstration_n for_o the_o first_o part_n after_o campane_n campane_n suppose_v that_o there_o be_v a_o circle_n abdc_n who_o centre_n let_v be_v the_o point_n e._n and_o draw_v in_o it_o two_o equal_a line_n ab_fw-la and_o cd_o then_o i_o say_v that_o they_o be_v equal_o distant_a from_o the_o centre_n draw_v from_o the_o centre_n unto_o the_o line_n ab_fw-la and_o cd_o these_o perpendicular_a line_n of_o and_o eglantine_n and_o by_o the_o 2._o part_n of_o the_o 3._o of_o this_o book_n the_o line_n ab_fw-la shall_v be_v equal_o divide_v in_o the_o point_n f._n and_o the_o line_n cd_o shall_v be_v equal_o divide_v in_o the_o point_n g._n and_o draw_v these_o right_a line_n ea_fw-la ebb_n ec_o and_o ed._n and_o for_o asmuch_o as_o in_o the_o triangle_n aeb_fw-mi the_o two_o side_n ab_fw-la and_o ae_n be_v equal_a to_o the_o two_o side_n cd_o and_o ce_fw-fr of_o the_o triangle_n ce_z &_o the_o base_a ebb_n be_v equal_a to_o the_o base_a ed._n therefore_o by_o the_o 8._o of_o the_o first_o the_o angle_n at_o the_o point_n a_o shall_v be_v equal_a to_o the_o angle_n at_o the_o point_n c._n and_o for_o asmuch_o as_o in_o the_o triangle_n aef_n the_o two_o side_n ae_n and_o of_o be_v equal_a to_o the_o two_o side_n ce_fw-fr and_o cg_o of_o the_o triangle_n ceg_n and_o the_o angle_n eaf_n be_v equal_a to_o the_o angle_n ceg_n therefore_o by_o the_o 4._o of_o the_o first_o the_o base_a of_o iâ—_z equal_a to_o the_o base_a eglantine_n which_o for_o asmuch_o as_o they_o be_v perpendicular_a line_n therefore_o the_o line_n ab_fw-la &_o cd_o be_v equal_o distant_a from_o the_o centre_n by_o the_o 4._o definition_n of_o this_o book_n the_o 14._o theorem_a the_o 15._o proposition_n in_o a_o circle_n the_o great_a line_n be_v the_o diameter_n and_o of_o all_o other_o line_n that_o line_n which_o be_v nigh_o to_o the_o centre_n be_v always_o great_a than_o that_o line_n which_o be_v more_o distant_a svppose_v that_o there_o be_v a_o circle_n abcd_o and_o let_v the_o diameter_n thereof_o be_v the_o line_n ad_fw-la and_o let_v the_o centre_n thereof_o be_v the_o point_n e._n and_o unto_o the_o diameter_n ad_fw-la let_v the_o line_n bc_o be_v nigh_a than_o the_o line_n fg._n then_o i_o say_v that_o the_o line_n ad_fw-la be_v the_o great_a and_o the_o line_n bc_o be_v great_a than_o the_o line_n fg._n draw_v by_o the_o 12._o of_o the_o first_o from_o the_o centre_n e_o to_o the_o line_n bc_o and_o fg_o perpendicular_a line_n eh_o and_o eke_o construction_n and_o for_o asmuch_o as_o the_o line_n bc_o be_v nigh_a unto_o the_o centre_n than_o the_o line_n fg_o therefore_o by_o the_o 4._o definition_n of_o the_o three_o the_o line_n eke_o be_v great_a than_o the_o line_n eh_o and_o by_o the_o three_o of_o the_o first_o put_v unto_o the_o line_n eh_o a_o equal_a line_n el._n and_o by_o the_o 11._o of_o the_o first_o from_o the_o point_n l_o raise_v up_o unto_o the_o line_n eke_o a_o perpendicular_a line_n lm_o and_o extend_v the_o line_n lm_o to_o the_o point_n n._n and_o by_o the_o first_o petition_n draw_v these_o right_a line_n em_n en_fw-fr of_o and_o eglantine_n and_o for_o asmuch_o as_o the_o line_n eh_o be_v equal_a to_o the_o line_n el_fw-es therefore_o by_o the_o 14._o of_o the_o three_o demonstration_n and_o by_o the_o 4._o definition_n of_o the_o same_o the_o line_n bc_o be_v equal_a to_o the_o line_n mn_v again_o for_o asmuch_o as_o the_o line_n ae_n be_v equal_a to_o the_o line_n em_n and_o the_o line_n ed_z to_o the_o line_n en_fw-fr therefore_o the_o line_n ad_fw-la be_v equal_a to_o the_o line_n menander_n and_o en_fw-fr but_o the_o line_n i_o and_o en_fw-fr be_v by_o the_o 20._o of_o the_o first_o great_a then_o the_o line_n mn_v wherefore_o the_o line_n ad_fw-la be_v great_a than_o the_o line_n mn_v and_o for_o asmuch_o as_o these_o two_o line_n menander_n and_o en_fw-fr be_v equal_v to_o these_o two_o line_n fe_o and_o eglantine_n by_o the_o 15._o definition_n of_o the_o first_o for_o they_o be_v draw_v from_o the_o centre_n to_o the_o circumference_n and_o the_o angle_n men_n be_v great_a than_o the_o angle_n feg_n therefore_o by_o the_o 24._o of_o the_o first_o the_o base_a mn_n be_v great_a than_o the_o base_a fg._n but_o it_o be_v prove_v that_o the_o line_n mn_o be_v equal_a to_o the_o line_n bc_o wherefore_o the_o line_n bc_o also_o be_v great_a than_o the_o line_n fg._n wherefore_o the_o diameter_n ad_fw-la be_v the_o great_a and_o the_o line_n bc_o be_v great_a than_o the_o line_n fg._n wherefore_o in_o a_o circle_n the_o great_a line_n be_v the_o diameter_n and_o of_o all_o the_o other_o line_n that_o line_n which_o be_v nigh_o to_o the_o centre_n be_v always_o great_a than_o that_o line_n which_o be_v more_o distant_a which_o be_v require_v to_o be_v prove_v ¶_o a_o other_o demonstration_n after_o campane_n in_o the_o circle_n abcd_o who_o centre_n let_v be_v the_o point_n e_o draw_v these_o line_n ab_fw-la ac_fw-la ad_fw-la fg_o and_o hk_o of_o which_o let_v the_o line_n ad_fw-la be_v the_o diameter_n of_o the_o circle_n campane_n then_o i_o say_v that_o the_o line_n ad_fw-la be_v the_o great_a of_o all_o the_o line_n and_o the_o other_o line_n each_o of_o the_o one_o be_v so_o much_o great_a than_o each_o of_o the_o other_o how_o much_o nigh_o it_o be_v unto_o the_o centre_n join_v together_o the_o end_n of_o all_o these_o line_n with_o the_o centre_n by_o draw_v these_o right_a line_n ebb_v ec_o eglantine_n eke_o eh_o and_o ef._n and_o by_o the_o 20._o of_o the_o first_o the_o two_o side_n of_o and_o eglantine_n of_o the_o triangle_n efg_o shall_v be_v great_a than_o the_o three_o side_n fg._n and_o for_o asmuch_o as_o the_o say_a side_n of_o &_o eglantine_n be_v equal_a to_o the_o line_n ad_fw-la by_o the_o definition_n of_o a_o circle_n therefore_o the_o line_n ad_fw-la be_v great_a than_o the_o line_n fg._n and_o by_o the_o same_o reason_n it_o be_v great_a than_o every_o one_o of_o the_o rest_n of_o the_o line_n if_o they_o be_v put_v to_o be_v base_n of_o triangle_n for_o that_o every_o two_o side_n draw_v from_o the_o centre_n be_v equal_a to_o the_o line_n ad._n which_o be_v the_o first_o part_n of_o the_o proposition_n again_o for_o asmuch_o as_o the_o two_o side_n of_o and_o eglantine_n of_o the_o triangle_n efg_o be_v equal_a to_o the_o
say_v that_o the_o line_n gfh_v which_o by_o the_o corollary_n of_o the_o 16._o of_o this_o book_n touch_v the_o circle_n be_v a_o parallel_n unto_o the_o line_n ab_fw-la circle_n for_o forasmuch_o as_o the_o right_a line_n cf_o fall_a upon_o either_o of_o these_o line_n ab_fw-la &_o gh_o make_v all_o the_o angle_n at_o the_o point_n â—_o right_n angle_n by_o the_o 3._o of_o this_o book_n and_o the_o two_o angle_n at_o the_o point_n fare_v suppose_v to_o be_v right_a angle_n therefore_o by_o the_o 29._o of_o the_o first_o the_o line_n ab_fw-la and_o gh_o be_v parallel_n which_o be_v require_v to_o be_v do_v and_o this_o problem_n be_v very_o commodious_a for_o the_o inscribe_v or_o circumscribe_v of_o figure_n in_o or_o about_o circle_n the_o 16._o theorem_a the_o 18._o proposition_n if_o a_o right_a line_n touch_v a_o circle_n and_o from_o the_o centre_n to_o the_o touch_n be_v draw_v a_o right_a line_n that_o right_a line_n so_o draw_v shall_v be_v a_o perpendicular_a line_n to_o the_o touch_n line_n svppose_v that_o the_o right_a line_n de_fw-fr do_v touch_v the_o circle_n abc_n in_o the_o point_n c._n and_o take_v the_o centre_n of_o the_o circle_n abc_n and_o let_v the_o same_o be_v f._n impossibility_n and_o by_o the_o first_o petition_n from_o the_o point_n fletcher_n to_o the_o point_n c_o draw_v a_o right_a line_n fc_o then_o i_o say_v that_o cf_o be_v a_o perpendicular_a line_n to_o de._n for_o if_o not_o draw_v by_o the_o 12._o of_o the_o first_o from_o the_o point_n fletcher_n to_o the_o line_n de_fw-fr a_o perpendicular_a line_n fg._n and_o for_o asmuch_o as_o the_o angle_n fgc_n be_v a_o right_a angle_n therefore_o the_o angle_n gcf_n be_v a_o acute_a angle_n wherefore_o the_o angle_n fgc_n be_v great_a than_o the_o angle_n fcg_n but_o unto_o the_o great_a angle_n be_v subtend_v the_o great_a side_n by_o the_o 19_o of_o the_o first_o wherefore_o the_o line_n fc_o be_v great_a than_o the_o line_n fg._n but_o the_o line_n fc_o be_v equal_a to_o the_o line_n fb_o for_o they_o be_v draw_v from_o the_o centre_n to_o the_o circumference_n wherefore_o the_o line_n fb_o also_o be_v great_a than_o the_o line_n fg_o namely_o the_o less_o than_o the_o great_a which_o be_v impossible_a wherefore_o the_o line_n fg_o be_v not_o a_o perpendicular_a line_n unto_o the_o line_n de._n and_o in_o like_a sort_n may_v we_o prove_v that_o no_o other_o line_n be_v a_o perpendicular_a line_n unto_o the_o line_n de_fw-fr beside_o the_o line_n fc_o wherefore_o the_o line_n fc_o be_v a_o perpendicular_a line_n to_o de._n if_o therefore_o a_o right_a line_n touch_v a_o circle_n &_o from_o you_o centre_n to_o the_o touch_n be_v draw_v a_o right_a line_n that_o right_a line_n so_o draw_v shall_v be_v a_o perpendicular_a line_n to_o the_o touch_n line_n which_o be_v require_v to_o be_v prove_v ¶_o a_o other_o demonstration_n after_o orontius_n suppose_v that_o the_o circle_n give_v be_v abc_n which_o let_v the_o right_a line_n de_fw-fr touch_n in_o the_o point_n c._n orontius_n and_o let_v the_o centre_n of_o the_o circle_n be_v the_o point_n f._n and_o draw_v a_o right_a line_n from_o fletcher_n to_o c._n then_o i_o say_v that_o the_o line_n fc_o be_v perpendicular_a unto_o the_o line_n de._n for_o if_o the_o line_n fc_o be_v not_o a_o perpendicule_a unto_o the_o line_n de_fw-fr then_o by_o the_o converse_n of_o the_o x._o definition_n of_o the_o first_o book_n the_o angle_n dcf_n &_o fce_n shall_v be_v unequal_a &_o therefore_o the_o one_o be_v great_a than_o a_o right_a angle_n and_o the_o other_o be_v less_o than_o a_o right_a angle_n for_o the_o angle_n dcf_n and_o fce_n be_v by_o the_o 13._o of_o the_o first_o equal_a to_o two_o right_a angle_n let_v the_o angle_n fce_n if_o it_o be_v possible_a be_v great_a than_o a_o right_a angle_n that_o be_v let_v it_o be_v a_o obtuse_a angle_n wherefore_o the_o angle_n dcf_n â—hal_v be_v a_o acute_a angle_n and_o forasmuch_o as_o by_o supposition_n the_o right_a line_n de_fw-fr touch_v the_o circle_n abc_n therefore_o it_o cut_v not_o the_o circle_n wherefore_o the_o circumference_n bc_o fall_v between_o the_o right_a line_n dc_o &_o cf_o &_o therefore_o the_o acute_a and_o rectiline_a angle_n dcf_n shall_v be_v great_a than_o the_o angle_n of_o the_o semicircle_n bcf_n which_o be_v contain_v under_o the_o circumference_n bc_o &_o the_o right_a line_n cf._n and_o so_o shall_v there_o be_v give_v a_o rectiline_n &_o acute_a angle_n great_a than_o the_o angle_n of_o a_o semicircle_n which_o be_v contrary_a to_o the_o 16._o proposition_n of_o this_o book_n wherefore_o the_o angle_n dcf_n be_v not_o less_o than_o a_o right_a angle_n in_o like_a sort_n also_o may_v we_o prove_v that_o it_o be_v not_o great_a than_o a_o right_a angle_n wherefore_o it_o be_v a_o right_a angle_n and_o therefore_o also_o the_o angle_n fce_n be_v a_o right_a angle_n wherefore_o the_o right_a line_n fc_o be_v a_o perpendicular_a unto_o the_o right_a line_n de_fw-fr by_o the_o 10._o definition_n of_o the_o firstâ—_o which_o be_v require_v to_o be_v prove_v the_o 17._o theorem_a the_o 19_o proposition_n if_o a_o right_a line_n do_v touch_v a_o circle_n and_o from_o the_o point_n of_o the_o touch_n be_v raise_v up_o unto_o the_o touch_n line_n a_o perpendicular_a line_n in_o that_o line_n so_o raise_v up_o be_v the_o centre_n of_o the_o circle_n svppose_v that_o the_o right_a line_n de_fw-fr do_v touch_v the_o circle_n abc_n in_o the_o point_n c._n and_o from_o c_o raise_v up_o by_o the_o 11._o of_o the_o first_o unto_o the_o line_n de_fw-fr a_o perpendicular_a line_n ca._n then_o i_o say_v that_o in_o the_o line_n ca_n be_v the_o centre_n of_o the_o circle_n for_o if_o not_o then_o if_o it_o be_v possible_a let_v the_o centre_n be_v without_o the_o line_n ca_n as_o in_o the_o point_n f._n and_o by_o the_o first_o petition_n draw_v a_o right_a line_n from_o c_o to_o f._n impossibility_n and_o for_o asmuch_o as_o a_o certain_a right_a line_n de_fw-fr touch_v the_o circle_n abc_n and_o from_o the_o centre_n to_o the_o touch_n be_v draw_v a_o right_a line_n cf_o therefore_o by_o the_o 18._o of_o the_o three_o fc_o be_v a_o perpendicular_a line_n to_o de._n wherefore_o the_o angle_n fce_n be_v a_o right_a angle_n but_o the_o angle_n ace_n be_v also_o a_o right_a angle_n wherefore_o the_o angle_n fce_n be_v equal_a to_o the_o angle_n ace_n namely_o the_o less_o unto_o the_o great_a which_o be_v impossibleâ—_n wherefore_o the_o point_n fletcher_n be_v not_o the_o centre_n of_o the_o circle_n abc_n and_o in_o like_a sort_n may_v we_o prove_v that_o it_o be_v no_o other_o where_o but_o in_o the_o line_n ac_fw-la if_o therefore_o a_o right_a line_n do_v touch_v a_o circle_n and_o from_o the_o point_n of_o the_o touch_n be_v raise_v up_o unto_o the_o touch_n line_v a_o perpendicular_a line_n in_o that_o line_n so_o raise_v up_o be_v the_o centre_n of_o the_o circle_n which_o be_v require_v to_o be_v prove_v the_o 18._o theorem_a the_o 20._o proposition_n in_o a_o circle_n a_o angle_n set_v at_o the_o centre_n be_v double_a to_o a_o angle_n set_v at_o the_o circumference_n so_o that_o both_o the_o angle_n have_v to_o their_o base_a one_o and_o the_o same_o circumference_n svppose_v that_o there_o be_v a_o circle_n abc_n and_o at_o the_o centre_n thereof_o centre_n namely_o the_o point_n e_o let_v the_o angle_n bec_n be_v set_v &_o at_o the_o circumference_n let_v there_o be_v set_v the_o angle_n bac_n and_o let_v they_o both_o have_v one_o and_o the_o same_o base_a namely_o the_o circumference_n bc._n then_o i_o say_v that_o the_o angle_n bec_n be_v double_a to_o the_o angle_n bac_n draw_v the_o right_a line_n ae_n and_o by_o the_o second_o petition_n extend_v it_o to_o the_o point_n f._n now_o for_o asmuch_o as_o the_o line_n ae_n be_v equal_a to_o the_o line_n ebb_n demonstration_n for_o they_o be_v draw_v from_o the_o centre_n unto_o the_o circumference_n the_o angle_n eab_n be_v equal_a to_o the_o angle_n eba_n by_o the_o 5._o of_o the_o first_o wherefore_o the_o angle_n eab_n and_o eba_n be_v double_a to_o the_o angle_n eab_n but_o by_o the_o 32._o of_o the_o same_o the_o angle_n bef_n be_v equal_a to_o the_o angle_n eab_n and_o eba_n wherefore_o the_o angle_n bef_n be_v double_a to_o the_o angle_n eab_n and_o by_o the_o same_o reason_n the_o angle_n fec_n be_v double_a to_o the_o angle_n eac_n wherefore_o the_o whole_a angle_n bec_n be_v double_a to_o the_o whole_a angle_n bac_n again_o suppose_v that_o there_o be_v set_v a_o other_o angle_n at_o the_o circumference_n centre_n and_o let_v the_o same_o be_v bdc_o and_o by_o the_o â—irst_n petition_n draw_v a_o line_n from_o d_z to_o e._n and_o by_o the_o second_o petition_n extend_v
touch_v the_o circle_n abc_n and_o let_v the_o same_o be_v de._n construction_n and_o by_o the_o first_o of_o the_o same_o let_v the_o point_n fletcher_n be_v the_o centre_n of_o the_o circle_n abc_n and_o draw_v these_o right_a line_n fe_o fb_o and_o fd._n wherefore_o the_o angle_n feed_v be_v a_o right_a angle_n demonstration_n and_o for_o asmuch_o as_o the_o right_a line_n de_fw-fr touch_v the_o circle_n abc_n and_o the_o right_a line_n dca_n cut_v the_o same_o therefore_o by_o the_o proposition_n go_v before_o that_o which_o be_v contain_v under_o the_o line_n ad_fw-la and_o dc_o be_v equal_a to_o the_o square_n of_o the_o line_n de._n but_o that_o which_o be_v contain_v under_o the_o line_n ad_fw-la and_o dc_o be_v suppose_v to_o be_v equal_a to_o the_o square_n of_o the_o line_n db._n wherefore_o the_o square_a of_o the_o line_n de_fw-fr be_v equal_a to_o the_o square_n of_o the_o line_n db._n wherefore_o also_o the_o line_n de_fw-fr be_v equal_a to_o the_o line_n db._n and_o the_o line_n fe_o be_v equal_a to_o the_o line_n fb_o for_o they_o be_v draw_v from_o the_o centre_n to_o the_o circumference_n now_o therefore_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 db_o and_o bf_o and_o fd_o be_v a_o common_a base_a to_o they_o both_o wherefore_o by_o the_o 8._o of_o the_o first_o the_o angle_n def_n be_v equal_a to_o the_o angle_n dbf_n but_o the_o angle_n def_n be_v a_o right_a angle_n wherefore_o also_o the_o angle_n dbf_n be_v a_o right_a angle_n and_o the_o line_n fb_o be_v produce_v shall_v be_v the_o diameter_n of_o the_o circle_n but_o if_o from_o the_o end_n of_o the_o diameter_n of_o a_o circle_n be_v draw_v a_o right_a line_n make_v right_a angle_n the_o right_a line_n so_o draw_v touch_v the_o circle_n by_o the_o corollary_n of_o the_o 16._o of_o the_o three_o wherefore_o the_o right_a line_n db_o touch_v the_o circle_n abc_n and_o the_o like_a demonstration_n will_v serve_v if_o the_o centre_n be_v in_o the_o line_n ac_fw-la if_o therefore_o without_o a_o circle_n be_v take_v a_o certain_a point_n and_o from_o that_o point_n be_v draw_v to_o the_o circle_n two_o right_a line_n of_o which_o the_o one_o do_v cut_v the_o circle_n and_o the_o other_o fall_v upon_o the_o circle_n and_o that_o in_o such_o sort_n that_o the_o rectangle_n parallelogram_n which_o be_v contain_v under_o the_o whole_a right_a line_n which_o cut_v the_o circle_n and_o that_o portion_n of_o the_o same_o line_n that_o lie_v between_o the_o point_n and_o the_o utter_a circumference_n of_o the_o circle_n be_v equal_a to_o the_o square_n make_v of_o the_o line_n that_o fall_v upon_o the_o circle_n then_o the_o line_n that_o so_o fall_v upon_o the_o circle_n shall_v touch_v the_o circle_n which_o be_v require_v to_o be_v prove_v ¶_o a_o other_o demonstration_n after_o pelitarius_n suppose_v that_o there_o be_v a_o circle_n bcd_o who_o centre_n let_v be_v e_o pelitarius_n and_o take_v a_o point_n without_o it_o namely_o a_o and_o from_o the_o point_n a_o draw_v two_o right_a line_n abdella_n and_o ac_fw-la of_o which_o let_v abdella_n cut_v the_o circle_n in_o the_o point_n b_o &_o let_v the_o other_o fall_n upon_o it_o and_o let_v that_o which_o be_v contain_v under_o the_o line_n ad_fw-la and_o ab_fw-la be_v equal_a to_o the_o square_n of_o the_o line_n ac_fw-la then_o i_o say_v that_o the_o line_n ac_fw-la touch_v the_o circle_n for_o first_o if_o the_o line_n abdella_n do_v pass_v by_o the_o centre_n draw_v the_o right_a line_n ce._n and_o by_o the_o 6._o of_o the_o second_o that_o which_o be_v contain_v under_o the_o line_n ad_fw-la and_o ab_fw-la together_o with_o the_o square_n of_o the_o line_n ebb_n that_o be_v with_o the_o square_n of_o the_o line_n ec_o for_o the_o line_n ebb_v and_o ec_o be_v equal_a be_v equal_a to_o the_o square_n of_o the_o line_n ae_n but_o that_o which_o be_v contain_v under_o the_o line_n ad_fw-la and_o ab_fw-la be_v suppose_v to_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 together_o with_o the_o square_n of_o the_o line_n ce_fw-fr be_v equal_a to_o the_o square_n of_o the_o line_n ae_n wherefore_o by_o the_o last_o of_o the_o first_o the_o angle_n at_o the_o point_n c_o be_v a_o right_a angle_n wherefore_o by_o the_o 18._o of_o this_o book_n the_o line_n ac_fw-la touch_v the_o circle_n but_o if_o the_o line_n abdella_n do_v not_o pass_v by_o the_o centre_n draw_v from_o the_o point_n a_o the_o line_n ad_fw-la in_o which_o let_v be_v the_o centre_n e._n and_o forasmuch_o as_o that_o which_o be_v contain_v under_o this_o whole_a line_n and_o his_o outward_a part_n 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 ab_fw-la by_o the_o first_o corollary_n before_o put_v therefore_o the_o same_o be_v equal_a to_o the_o square_n of_o the_o line_n ac_fw-la wherefore_o the_o angle_n eca_n be_v a_o right_a angle_n as_o have_v before_o be_v prove_v in_o the_o first_o part_n of_o this_o proposition_n and_o therefore_o the_o line_n ac_fw-la touch_v the_o circle_n which_o be_v require_v to_o be_v prove_v the_o end_n of_o the_o three_o book_n of_o euclides_n element_n ¶_o the_o four_o book_n of_o euclides_n element_n this_o four_o book_n intreat_v of_o the_o inscription_n &_o circumscription_n of_o rectiline_a figure_n book_n how_o one_o right_a line_v figure_n may_v be_v inscribe_v within_o a_o other_o right_o line_v figure_n and_o how_o a_o right_a line_v figure_n may_v be_v circumscribe_v about_o a_o other_o right_o line_v figure_n in_o such_o as_o may_v be_v inscribe_v and_o circumscribe_v within_o or_o about_o the_o other_o for_o all_o right_n line_v figure_n can_v so_o be_v inscribe_v or_o circumscribe_v within_o or_o about_o the_o other_o also_o it_o teach_v how_o a_o triangle_n a_o square_a and_o certain_a other_o rectiline_a figure_n be_v regular_a may_v be_v inscribe_v within_o a_o circle_n also_o how_o they_o may_v be_v circumscribe_v about_o a_o circle_n likewise_o how_o a_o circle_n may_v be_v inscribe_v within_o they_o and_o how_o it_o may_v be_v circumscribe_v about_o they_o and_o because_o the_o manner_n of_o entreaty_n in_o this_o book_n be_v diverse_a from_o the_o entreaty_n of_o the_o former_a book_n he_o use_v in_o this_o other_o word_n and_o term_n than_o he_o use_v in_o they_o the_o definition_n of_o which_o in_o order_n here_o after_o follow_v definition_n a_o rectiline_a figure_n be_v say_v to_o be_v inscribe_v in_o a_o rectiline_a figure_n definition_n when_o every_o one_o of_o the_o angle_n of_o the_o inscribe_v figure_n touch_v every_o one_o of_o the_o side_n of_o the_o figure_n wherein_o it_o be_v inscribe_v as_o the_o triangle_n abc_n be_v inscribe_v in_o the_o triangle_n def_n because_o that_o every_o angle_n of_o the_o triangle_n inscribe_v namely_o the_o triangle_n abc_n touch_v every_o side_n of_o the_o triangle_n within_o which_o it_o be_v describe_v namely_o of_o the_o triangle_n def_n as_o the_o angle_n cab_n touch_v the_o side_n ed_z the_o angle_n abc_n touch_v the_o side_n df_o and_o the_o angle_n acb_o touch_v the_o side_n ef._n so_o likewise_o the_o square_a abcd_o be_v say_v to_o be_v inscribe_v within_o the_o square_a efgh_o for_o every_o angle_n of_o it_o touch_v some_o one_o side_n of_o the_o other_o so_o also_o the_o pentagon_n or_o five_o angle_a figure_n abcde_o be_v inscribe_v within_o the_o pentagon_n or_o five_o angle_a figure_n fghikâ—_n â—_z as_o you_o see_v in_o the_o figureâ—_n likewise_o a_o rectiline_a figure_n be_v say_v to_o be_v circumscribe_v about_o a_o rectiline_a figure_n definition_n when_o every_o one_o of_o the_o side_n of_o the_o figure_n circumscribe_v touch_v every_o one_o of_o the_o angle_n of_o the_o figure_n about_o which_o it_o be_v circumscribe_v as_o in_o the_o former_a description_n the_o triangle_n def_n be_v say_v to_o be_v circumscribe_v about_o the_o triangle_n abc_n for_o that_o every_o side_n of_o the_o figure_n circumscribe_v namely_o of_o the_o triangle_n def_n touch_v every_o angle_n of_o the_o figure_n wherabout_o it_o be_v circumscribe_v as_o the_o side_n df_o of_o the_o triangle_n def_n circumscribe_v touch_v the_o angle_n abc_n of_o the_o triangle_n abc_n about_o which_o it_o be_v circumscribe_v and_o the_o side_n of_o touch_v the_o angle_n bca_o and_o the_o side_n cd_o touch_v the_o angle_n cab_n likewise_o understand_v you_o of_o the_o square_a efgh_o which_o be_v circumscribe_v about_o the_o square_n abcd_o for_o every_o side_n of_o the_o one_o touch_v some_o one_o side_n of_o the_o other_o even_o so_o by_o the_o same_o reason_n the_o pentagon_n fghik_n be_v circumscribe_v about_o the_o pentagon_n abcde_o as_o you_o see_v in_o the_o figure_n on_o the_o other_o side_n and_o thus_o may_v you_o
line_n fb_o favorina_n &_o fe_o and_o for_o asmuch_o as_o the_o angle_n bcd_o be_v equal_a to_o the_o angle_n cde_o &_o the_o half_a of_o the_o angle_n bcd_o be_v the_o angle_n fcd_n and_o likewise_o the_o half_a of_o the_o angle_n cde_o be_v the_o angle_n cdf_n wherefore_o the_o angle_n fcd_n be_v equal_a to_o the_o angle_n fdc_n wherefore_o the_o side_n fc_o be_v equal_a to_o the_o side_n fd._n in_o like_a sort_n also_o may_v it_o be_v prove_v that_o every_o one_o of_o these_o line_n fb_o favorina_n and_o fe_o be_v equal_a to_o every_o one_o of_o these_o line_n fc_a and_o fd._n wherefore_o these_o five_o right_a line_n favorina_n fb_o fc_o fd_o and_o fe_o be_v equal_a the_o one_o to_o the_o other_o wherefore_o make_v the_o centre_n f_o and_o the_o space_n favorina_n or_o fb_o or_o fc_n or_o fd_o or_o fe_o describe_v a_o circle_n and_o it_o will_v pass_v by_o the_o point_n a_o b_o c_o d_o e_o and_o shall_v be_v describe_v â—bout_o the_o five_o angle_a figure_n abcde_o which_o be_v equiangle_n and_o equilater_n let_v this_o circle_n be_v describe_v and_o let_v the_o same_o be_v abcde_o wherefore_o about_o the_o pentagon_n give_v beâ—â—g_v both_o equiangle_n and_o equilater_n be_v describe_v a_o circle_n which_o be_v require_v to_o be_v â—one_n the_o 15._o problem_n the_o 15._o proposition_n in_o a_o circle_n give_v to_o describe_v a_o hexagon_n or_o figure_n of_o six_o angle_n equilater_n and_o equiangle_n svppose_v that_o the_o circle_n give_v be_v abcdef_n it_o be_v require_v in_o the_o circle_n give_v abcde_o to_o describe_v a_o figure_n of_o six_o angle_n of_o equal_a side_n and_o of_o equal_a angle_n dâ—aw_v the_o diameter_n of_o the_o circle_n abcdef_n and_o let_v the_o same_o be_v ad._n and_o by_o the_o first_o of_o the_o three_o take_v the_o câ—nâ—re_n of_o the_o circled_a and_o let_v the_o same_o be_v g._n and_o make_v the_o centre_n d_o and_o the_o space_n dg_o describe_v by_o the_o three_o petition_n a_o circle_n cgeh_a and_o draw_v right_a line_n from_o e_o to_o g_z and_o from_o g_z to_o c_z extend_v they_o to_o the_o point_n b_o and_o f_o of_o the_o circumference_n of_o the_o circle_n give_v and_o draw_v these_o right_a line_n ab_fw-la bc_o cd_o de_fw-fr of_o and_o fa._n then_o i_o say_v that_o abcdef_n be_v a_o hexagon_n figure_n of_o equal_a side_n and_o of_o equal_a angle_n demonstration_n for_o forasmuch_o as_o the_o point_n g_z be_v the_o centre_n of_o the_o circle_n abcdef_n therefore_o by_o the_o 15._o definition_n of_o the_o first_o the_o line_n ge_z be_v equal_a unto_o the_o line_n gd_o again_o forasmuch_o as_o the_o point_n d_o be_v the_o centre_n of_o the_o circle_n cgeh_a therefore_o by_o the_o self_n same_o the_o line_n de_fw-fr be_v equal_a unto_o the_o line_n dg_o and_o it_o be_v prove_v that_o the_o line_n ge_z be_v equal_a unto_o the_o line_n gd_o wherefore_o the_o line_n ge_z be_v equal_a unto_o the_o line_n ed_z by_o the_o first_o common_a sentence_n wherefore_o the_o triangle_n egg_v be_v equilater_n and_o biâ—_n three_o angle_n namely_o egg_v gde_v deg_n be_v equal_a the_o one_o to_o the_o other_o and_o forasmuch_o as_o by_o the_o 5._o of_o the_o first_o in_o triangle_n of_o two_o equal_a side_n colmonell_n call_v isosceles_a the_o angle_n at_o the_o base_a be_v equal_a the_o one_o to_o the_o other_o and_o the_o three_o angle_n of_o a_o triangle_n be_v by_o the_o 30._o of_o the_o first_o equal_v â—nto_o two_o right_a angle_n therefore_o the_o angle_n egg_v be_v the_o three_o part_n of_o two_o right_a angle_n and_o in_o like_a sorâ—e_n may_v it_o be_v prove_v that_o the_o angle_n dgc_n be_v the_o three_o part_n of_o two_o right_a angle_n and_o forasmuch_o as_o the_o right_a line_n cg_o stand_v upon_o the_o right_a line_n ebb_n do_v by_o the_o 13._o of_o the_o fiâ—â—t_n make_v the_o two_o side_n angle_n â—gc_n and_o cgâ—_n equal_a to_o two_o right_a angle_n therefore_o the_o angle_n remain_v cgb_n be_v the_o three_o part_n of_o two_o right_a angle_n wherefore_o the_o angle_n egg_v dgc_n and_o cgb_n be_v equal_a the_o one_o to_o the_o other_o wherefore_o their_o head_n angle_n that_o be_v bga_n agf_n and_o fge_n be_v by_o the_o 15._o of_o the_o first_o equal_a to_o these_o angle_n egg_v dgc_n and_o cgb_n wherefore_o these_o six_o angle_n egg_v dgc_n cgb_n bga_n agf_n and_o fge_n be_v equal_a the_o one_o to_o the_o other_o but_o equal_a angle_n consist_v upon_o equal_a circumference_n by_o 〈…〉_o the_o three_o therefore_o these_o six_o circumference_n abâ—_n be_v cdâ—_n deâ—_n efâ—_n and_o â—a_o be_v equal_a the_o one_o to_o the_o other_o but_o under_o equal_a circumference_n be_v â—â—btâ—â—ded_v equal_a right_a line_n by_o the_o 29._o of_o the_o same_o wherefore_o these_o six_o right_a line_n ab_fw-la bc_o cd_o de_fw-fr of_o and_o favorina_n be_v equal_a the_o one_o to_o the_o other_o wherefore_o the_o hexagon_n abcdef_n be_v equilater_n i_o say_v also_o that_o it_o be_v equiangle_n for_o forasmuch_o as_o the_o circumference_n of_o be_v equal_a unto_o the_o circumference_n ed_z add_v the_o circumference_n abcd_o common_a to_o they_o both_o wherefore_o the_o whole_a circumference_n fabcd_n be_v equal_a to_o the_o whole_a circumference_n edcba_n and_o upon_o the_o circumference_n fabcd_n consist_v the_o angle_n feed_v and_o upon_o the_o circumference_n edcba_n consist_v the_o angle_n afe._n wherefore_o the_o angle_n afe_n be_v equal_a to_o the_o angle_n def_n in_o like_a sort_n also_o may_v it_o be_v prove_v that_o the_o rest_n of_o the_o angle_n of_o the_o hexagon_n abcdef_n that_o be_v every_o one_o of_o these_o angle_n fab._n abc_n bcd_o and_o cde_o be_v equal_a to_o every_o one_o of_o these_o angle_n afe_a and_o feed_v wherefore_o the_o hexagon_n figure_n abcdef_n be_v equiangle_n and_o it_o be_v prove_v that_o it_o be_v also_o equilater_n and_o it_o be_v describe_v in_o the_o circle_n abcdef_n wherefore_o in_o the_o circle_n give_v abcdef_n be_v describe_v a_o figure_n of_o six_o angle_n of_o equal_a side_n and_o of_o equal_a angle_n which_o be_v require_v to_o be_v doneâ—_n ¶_o a_o other_o way_n to_o do_v the_o same_o after_o orontius_n suppose_v that_o the_o circle_n give_v be_v abcdef_n in_o which_o first_o let_v there_o be_v describe_v a_o equilater_n and_o equiangle_n triangle_n ace_n by_o the_o second_o of_o this_o book_n orontius_n wherefore_o the_o arke_n abc_n cde_o efa_n be_v by_o the_o 28._o of_o the_o three_o equal_v the_o one_o to_o the_o other_o divide_v every_o one_o of_o those_o three_o arke_n into_o two_o equal_a part_n by_o the_o 30._o of_o the_o same_o in_o the_o point_n b_o d_o &_o f._n and_o draw_v these_o right_a line_n abâ—_n bc_o cd_o de_fw-fr of_o and_o fa._n now_o then_o by_o the_o 2._o definition_n of_o this_o book_n there_o shall_v be_v describe_v in_o the_o circle_n geve_v a_o hexagon_n figure_n abcdef_n which_o must_v needs_o be_v equilater_n for_o that_o every_o one_o of_o the_o arke_n which_o subtend_v the_o side_n thereof_o be_v equal_a the_o one_o to_o the_o other_o i_o say_v also_o that_o it_o be_v equiangle_n for_o every_o angle_n of_o the_o hexagon_n figure_n be_v set_v upon_o equal_a arke_n namely_o upon_o four_o such_o part_n of_o the_o circumference_n whereof_o the_o whole_a circumference_n contain_v six_o wherefore_o the_o angle_n of_o the_o hexagon_n figure_n be_v equal_a the_o one_o to_o the_o other_o by_o the_o 27._o of_o the_o three_o wherefore_o in_o the_o circle_n give_v abcdef_n be_v inscribe_v a_o equilater_n and_o equiangle_n hexagon_n figure_n which_o be_v require_v to_o be_v do_v ¶_o a_o other_o way_n to_o do_v the_o same_o after_o pelitarius_n suppose_v that_o the_o circle_n in_o which_o be_v to_o be_v inscribe_v 〈◊〉_d equilater_n &_o â—â—â—iangle_n hexagon_n figure_n be_v abcde_o pelitarius_n who_o câ—ntre_n let_v be_v f._n and_o from_o the_o centre_n draw_v the_o semidiameter_n fa._n and_o from_o the_o point_n a_o apply_v by_o the_o first_o oâ—â—hys_n book_n the_o line_n â—â—_o equal_a to_o the_o semidiameter_n which_o i_o say_v be_v the_o side_n of_o a_o equilater_n and_o equiangle_n hexagon_n figure_n to_o be_v inscribe_v in_o the_o circle_n abcde_o draw_v â—â—â—ght_n â—ine_o from_o e_o to_o bâ—_n â—_z and_o for_o asmuch_o as_o the_o line_n ab_fw-la be_v equal_a to_o the_o line_n faâ—_n â—_z &_o it_o be_v also_o equal_a to_o the_o line_n fb_o therefore_o the_o triangle_n afb_o be_v equilater_n and_o by_o the_o 5._o of_o the_o first_o equiangle_n now_o then_o upon_o the_o centre_n f_o describe_v the_o angle_n bfc_n equal_a to_o the_o angle_n afâ—_n or_o to_o the_o angle_n fba_n which_o be_v all_o one_o by_o draw_v the_o right_a line_n fc_o and_o draw_v a_o line_n from_o b_o to_o c._n and_o for_o asmuch_o as_o the_o angle_n afb_o be_v the_o three_o part_n of_o two_o right_a angle_n by_o the_o
poligonon_n figure_n of_o 24._o side_n likewise_o of_o the_o hexagon_n ab_fw-la and_o of_o the_o pentagon_n ac_fw-la shall_v be_v make_v a_o poligonon_n figure_n of_o 30._o side_n one_o of_o who_o side_n shall_v subtend_v the_o ark_n bc._n for_o the_o denomination_n of_o ab_fw-la which_o be_v 6._o exceed_v the_o denomination_n of_o ac_fw-la which_o be_v 5._o only_a by_o unity_n so_o also_o forasmuch_o as_o the_o denomination_n of_o ab_fw-la which_o be_v 6._o exceed_v the_o denomination_n of_o ae_n which_o be_v 3._o by_o 3._o therefore_o the_o ark_n be_v shall_v contain_v 3._o side_n of_o a_o poligonon_n figure_n of_o .18_o side_n and_o observe_v this_o self_n same_o method_n and_o order_n a_o man_n may_v find_v out_o infinite_a side_n of_o a_o poligonon_n figure_n the_o end_n of_o the_o four_o book_n of_o euclides_n element_n ¶_o the_o five_o book_n of_o euclides_n element_n this_o five_o book_n of_o euclid_n be_v of_o very_o great_a commodity_n and_o use_n in_o all_o geometry_n book_n and_o much_o diligence_n ought_v to_o be_v bestow_v therein_o it_o ought_v of_o all_o other_o to_o be_v thorough_o and_o most_o perfect_o and_o ready_o know_v for_o nothing_o in_o the_o book_n follow_v can_v be_v understand_v without_o it_o the_o knowledge_n of_o they_o all_o depend_v of_o it_o and_o not_o only_o they_o and_o other_o write_n of_o geometry_n but_o all_o other_o science_n also_o and_o art_n as_o music_n astronomy_n perspective_n arithmetic_n the_o art_n of_o account_n and_o reckon_n with_o other_o such_o like_a this_o book_n therefore_o be_v as_o it_o be_v a_o chief_a treasure_n and_o a_o peculiar_a jewel_n much_o to_o be_v account_v of_o it_o entreat_v of_o proportion_n and_o analogy_n or_o proportionality_n which_o pertain_v not_o only_o unto_o line_n figure_n and_o body_n in_o geometry_n but_o also_o unto_o sound_n &_o voice_n of_o which_o music_n entreat_v as_o witness_v boetius_fw-la and_o other_o which_o write_v of_o music_n also_o the_o whole_a art_n of_o astronomy_n teach_v to_o measure_v proportion_n of_o time_n and_o move_n archimedes_n and_o jordan_n with_o other_o write_v of_o weight_n affirm_v that_o there_o be_v proportion_n between_o weight_n and_o weight_n and_o also_o between_o place_n &_o place_n you_o see_v therefore_o how_o large_a be_v the_o use_n of_o this_o five_o book_n wherefore_o the_o definition_n also_o thereof_o be_v common_a although_o hereof_o euclid_n they_o be_v accommodate_v and_o apply_v only_o to_o geometry_n the_o first_o author_n of_o this_o book_n be_v as_o it_o be_v affirm_v of_o many_o one_o eudoxus_n who_o be_v plato_n scholar_n eudoxus_n but_o it_o be_v afterward_o frame_v and_o put_v in_o order_n by_o euclid_n definition_n definition_n a_o part_n be_v a_o less_o magnitude_n in_o respect_n of_o a_o great_a magnitude_n when_o the_o less_o measure_v the_o great_a as_o in_o the_o other_o book_n before_o so_o in_o this_o the_o author_n first_o set_v orderly_a the_o definition_n and_o declaration_n of_o such_o term_n and_o word_n which_o be_v necessary_o require_v to_o the_o entreaty_n of_o the_o subject_n and_o matter_n thereof_o which_o be_v proportion_n and_o comparison_n of_o proportion_n or_o proportionality_n and_o first_o he_o show_v what_o a_o part_n be_v here_o be_v to_o be_v consider_v that_o all_o the_o definition_n of_o this_o five_o book_n be_v general_a to_o geometry_n and_o arithmetic_n and_o be_v true_a in_o both_o art_n even_o as_o proportion_n and_o proportionality_n be_v common_a to_o they_o both_o and_o chief_o appertain_v to_o number_v neither_o can_v they_o apt_o be_v apply_v to_o matter_n of_o geometry_n but_o in_o respect_n of_o number_n and_o by_o number_n yet_o in_o this_o book_n and_o in_o these_o definition_n here_o set_v euclid_n seem_v to_o speak_v of_o they_o only_o geometrical_o as_o they_o be_v apply_v to_o quantity_n continual_a as_o to_o line_n superficiece_n and_o body_n for_o that_o he_o yet_o continue_v in_o geometry_n i_o will_v notwithstanding_o for_o facility_n and_o far_a help_n of_o the_o reader_n declare_v they_o both_o by_o example_n in_o number_n and_o also_o in_o line_n for_o the_o clear_a understanding_n of_o a_o part_n it_o be_v to_o be_v note_v way_n that_o a_o part_n be_v take_v in_o the_o mathematical_a science_n two_o manner_n of_o way_n way_n one_o way_n a_o part_n be_v a_o less_o quantity_n in_o respect_n of_o a_o great_a whether_o it_o measure_v the_o great_a oâ—_n no._n the_o second_o way_n way_n a_o part_n be_v only_o that_o less_o quantity_n in_o respect_n of_o the_o great_a which_o measure_v the_o great_a a_o less_o quantity_n be_v say_v to_o measure_n or_o number_v a_o great_a quantity_n great_a when_o it_o be_v oftentimes_o take_v make_v precise_o the_o great_a quantity_n without_o more_o or_o less_o or_o be_v as_o oftentimes_o take_v from_o the_o great_a as_o it_o may_v there_o remain_v nothing_o as_o suppose_v the_o line_n ab_fw-la to_o contain_v 3._o and_o the_o line_n cd_o to_o contain_v 9_o they_o do_v the_o line_n ab_fw-la measure_n the_o line_n cd_o for_o that_o if_o it_o be_v take_v certain_a time_n namely_o 3._o time_n it_o make_v precise_o the_o line_n cd_o that_o be_v 9_o without_o more_o or_o less_o again_o if_o the_o say_v less_o line_n ab_fw-la be_v take_v from_o the_o great_a cd_o as_o often_o as_o it_o may_v be_v namely_o 3._o time_n there_o shall_v remain_v nothing_o of_o the_o great_a so_o the_o number_n 3._o be_v say_v to_o measure_v 12._o for_o that_o be_v take_v certain_a time_n namely_o four_o time_n it_o make_v just_a 12._o the_o great_a quantity_n and_o also_o be_v take_v from_o 12._o as_o often_o as_o it_o may_v namely_o 4._o time_n there_o shall_v remain_v nothing_o and_o in_o this_o meaning_n and_o signification_n do_v euclid_n undoubted_o here_o in_o this_o define_v a_o part_n part_n say_v that_o it_o be_v a_o less_o magnitude_n in_o comparison_n of_o a_o great_a when_o the_o less_o measure_v the_o great_a as_o the_o line_n ab_fw-la before_o set_v contain_v 3_n be_v a_o less_o quantity_n in_o comparison_n of_o the_o line_n cd_o which_o contain_v 9_o and_o also_o measure_v it_o for_o it_o be_v certain_a time_n take_v namely_o 3._o time_n precise_o make_v it_o or_o take_v from_o it_o as_o often_o as_o it_o may_v there_o remain_v nothing_o wherefore_o by_o this_o definition_n the_o line_n ab_fw-la be_v a_o part_n of_o the_o line_n cd_o likewise_o in_o number_n the_o number_n 5._o be_v a_o part_n of_o the_o number_n 15._o for_o it_o be_v a_o less_o number_n or_o quantity_n compare_v to_o the_o great_a and_o also_o it_o measure_v the_o great_a for_o be_v take_v certain_a time_n namely_o 3._o time_n it_o make_v 15._o and_o this_o kind_n of_o part_n be_v call_v common_o pars_fw-la metiens_fw-la or_o mensurans_fw-la mensuranâ—_n that_o be_v a_o measure_a part_n some_o call_v it_o pars_fw-la multiplicatina_fw-la multiplicatiâ—a_fw-la and_o of_o the_o barbarous_a it_o be_v call_v pars_fw-la aliquota_fw-la aliquota_fw-la that_o be_v a_o aliquote_v part_n and_o this_o kind_n of_o part_n be_v common_o use_v in_o arithmetic_n arithmetic_n the_o other_o kind_n of_o a_o part_n part_n be_v any_o less_o quantity_n in_o comparison_n of_o a_o great_a whether_o it_o be_v in_o number_n or_o magnitude_n and_o whether_o it_o measure_v or_o no._n as_o suppose_v the_o line_n ab_fw-la to_o be_v 17._o and_o let_v it_o be_v divide_v into_o two_o part_n in_o the_o point_n c_o namely_o into_o the_o line_n ac_fw-la &_o the_o line_n cb_o and_o let_v the_o line_n ac_fw-la the_o great_a part_n contain_v 12._o and_o let_v the_o line_n bc_o the_o less_o part_n contain_v 5._o now_o either_o of_o these_o line_n by_o this_o definition_n be_v a_o part_n of_o the_o whole_a line_n ab_fw-la for_o either_o of_o they_o be_v a_o less_o magnitude_n or_o quantity_n in_o comparison_n of_o the_o whole_a line_n ab_fw-la but_o neither_o of_o they_o measure_v the_o whole_a line_n ab_fw-la for_o the_o less_o line_n cb_o contain_v 5._o take_v as_o often_o as_o you_o listen_v will_v never_o make_v precise_o ab_fw-la which_o contain_v 17._o if_o you_o take_v it_o 3._o time_n it_o make_v only_o 15._o so_o lack_v it_o 2._o of_o 17._o which_o be_v to_o little_a if_o you_o take_v it_o 4._o time_n so_o make_v it_o 20._o them_z be_v there_o three_o to_o much_o so_o it_o never_o make_v precise_o 17._o but_o either_o to_o much_o or_o to_o little_a likewise_o the_o other_o part_n ac_fw-la measure_v not_o the_o whole_a line_n ab_fw-la for_o take_v once_o it_o make_v but_o 12._o which_o be_v less_o than_o 17._o and_o take_v twice_o it_o make_v 24._o which_o be_v more_o than_o 17._o by_o â—_o so_o it_o never_o precise_o make_v by_o take_v thereof_o the_o whole_a ab_fw-la but_o either_o more_o or_o less_o and_o this_o kind_n of_o part_n they_o common_o call_v pars_fw-la
of_o the_o one_o also_o to_o the_o residue_n of_o the_o other_o shall_v be_v equemultiplex_n as_o the_o whole_a be_v to_o the_o whole_a which_o be_v require_v to_o be_v prove_v the_o 6._o theorem_a the_o 6._o proposition_n if_o two_o magnitude_n be_v â—quemultiplices_n to_o two_o magnitude_n &_o any_o parâ—es_n take_v away_o of_o they_o also_o be_v aequemultiplice_n to_o the_o same_o magnitude_n the_o residue_v also_o of_o they_o shall_v unto_o the_o same_o magnitude_n be_v either_o equal_a or_o equemultiplices_fw-la svppose_v that_o there_o be_v two_o magnitude_n ab_fw-la and_o cd_o equemultiplices_fw-la to_o two_o magnitude_n e_o and_o f_o and_o let_v the_o part_n take_v away_o of_o the_o magnitude_n ab_fw-la and_o cd_o namely_o ag_z and_o change_z be_v equemultiplices_fw-la to_o the_o same_o magnitude_n e_o and_o f._n propotion_n then_o i_o say_v that_o the_o residue_v gb_o and_o hd_n be_v unto_o the_o self_n same_o magnitude_n e_o and_o f_o either_o equal_a or_o else_o equemultiplices_fw-la and_o in_o like_a sort_n may_v we_o prove_v that_o if_o gb_o be_v multiplex_n to_o e_o hd_v also_o shall_v be_v so_o multiplex_n unto_o f._n if_o therefore_o there_o be_v two_o magnitude_n equemultiplices_fw-la to_o two_o magnitude_n second_o and_o any_o part_n take_v away_o of_o they_o be_v also_o equemultiplices_fw-la to_o the_o same_o magnitude_n the_o residue_v also_o of_o they_o shall_v unto_o the_o same_o magnitude_n be_v either_o equal_a or_o equemultiplices_fw-la which_o be_v require_v to_o be_v prove_v the_o 7._o theorem_a the_o 7._o proposition_n equal_a magnitude_n have_v to_o one_o &_o the_o self_n same_o magnitude_n one_o and_o the_o same_o proportion_n and_o one_o and_o the_o same_o magnitude_n have_v to_o equal_a magnitude_n one_o and_o the_o self_n same_o proportion_n svppose_v that_o a_o and_o b_o be_v equal_a magnitude_n and_o take_v any_o other_o magnitude_n namely_o c._n then_o i_o say_v that_o either_o of_o these_o a_o and_o b_o have_v unto_o c_o one_o and_o the_o same_o proportion_n and_o that_o c_o also_o have_v to_o either_o of_o these_o a_o and_o b_o one_o and_o the_o same_o proportion_n i_o say_v moreover_o that_o c_z have_v to_o either_o of_o these_o a_o and_o b_o one_o and_o the_o same_o proportion_n demonstrate_v for_o the_o same_o order_n of_o construction_n remain_v we_o may_v in_o like_a sort_n prove_v that_o d_z be_v equal_a unto_o e_o &_o there_o be_v take_v a_o other_o multiplex_n to_o c_o namely_o f._n wherefore_o if_o fletcher_n exceed_v d_o it_o also_o exceed_v e_o and_o if_o it_o be_v equal_a it_o be_v equal_a and_o if_o it_o be_v less_o it_o be_v less_o but_o fletcher_n be_v multiplex_n to_o c_o and_o d_z &_o e_o be_v other_o equemultiplices_fw-la to_o a_o and_o b._n wherefore_o as_o c_z be_v to_o a_o so_o be_v c_o to_o b._n wherefore_o equal_a magnitude_n have_v to_o one_o and_o the_o same_o magnitude_n one_o and_o the_o same_o proportion_n and_o one_o and_o the_o same_o magnitude_n have_v to_o equal_a magnitude_n one_o and_o the_o self_n same_o proportion_n which_o be_v require_v to_o be_v demonstrate_v the_o 8._o theorem_a the_o 8._o proposition_n unequal_a magnitude_n be_v take_v the_o great_a have_v to_o one_o and_o the_o same_o magnitude_n a_o great_a proportion_n than_o have_v the_o less_o and_o that_o one_o and_o the_o same_o magnitude_n have_v to_o the_o less_o a_o great_a proportion_n than_o it_o have_v to_o the_o great_a svppose_v that_o ab_fw-la and_o c_o be_v unequal_a magnitude_n of_o which_o let_v ab_fw-la be_v the_o great_a and_o c_o the_o less_o and_o let_v there_o be_v a_o other_o magnitude_n whatsoever_o namely_o d._n then_o i_o say_v that_o ab_fw-la have_v unto_o d_z a_o great_a proportion_n than_o have_v c_z to_o d_z and_o also_o that_o d_z have_v to_o c_o a_o great_a proportion_n than_o it_o have_v to_o ab_fw-la for_o forasmuch_o as_o ab_fw-la be_v great_a than_o c_z let_v there_o be_v take_v a_o magnitude_n equal_a unto_o c_o namely_o be._n now_o than_o the_o less_o of_o these_o two_o magnitude_n ae_n and_o ebb_n be_v multiply_v will_v at_o the_o length_n be_v great_a than_o d._n demonstrate_v demonstrate_v i_o say_v moreover_o that_o d_z have_v to_o c_o a_o great_a proportion_n than_o d_z have_v to_o ab_fw-la for_o the_o same_o order_n of_o construction_n still_o remain_v we_o may_v in_o like_a sort_n prove_v that_o n_o be_v great_a than_o king_n and_o that_o it_o be_v not_o great_a than_o fh_o and_o n_o be_v multiplex_n to_o d_o and_o fh_o and_o king_n be_v certain_a other_o equemultiplices_fw-la to_o ab_fw-la and_o c._n wherefore_o d_z have_v to_o c_o a_o great_a proportion_n than_o d_z have_v to_o ab_fw-la ¶_o for_o that_o orontius_n seem_v to_o demonstrate_v this_o more_o plain_o therefore_o i_o think_v it_o not_o amiss_o here_o to_o set_v it_o suppose_v that_o there_o be_v two_o unequal_a magnitude_n of_o which_o leâ—_z aâ—_z bâ—_z the_o greâ—ter_n and_o c_o the_o less_o and_o let_v there_o be_v a_o certain_a other_o magnitude_n namely_o d._n then_o i_o say_v first_o that_o ab_fw-la have_v to_o d_o a_o great_a proportion_n than_o have_v c_z to_o d._n for_o forasmuch_o as_o by_o supposition_n ab_fw-la be_v great_a than_o the_o magnitude_n c_o therefore_o the_o magnitude_n ab_fw-la contain_v the_o same_o magnitude_n c_o and_o a_o other_o magnitude_n beside_o let_v eâ—_n be_v equal_a unto_o c_o and_o let_v ae_n be_v the_o part_n remain_v of_o the_o same_o magnitude_n part_n now_o ae_n and_o ebb_n be_v either_o unequal_a or_o equal_a the_o one_o to_o the_o other_o first_o let_v they_o be_v unequally_n and_o leâ—_z ae_n be_v less_o than_o ebb_n and_o unto_o ae_n the_o less_o take_v any_o multiplex_n whatsoâ—uâ—r_n so_o that_o it_o be_v great_a they_o the_o magnitude_n d_o and_o let_v the_o same_o be_v fg._n and_o how_o multiplex_n fg_o be_v to_o ae_n so_o multiplex_n let_v gh_o be_v to_o eâ—â—_n â—_z and_o king_n to_o c._n again_o take_v the_o duple_n of_o dâ—_n â—_z which_o let_v be_v l_o and_o then_o the_o triple_a and_o leâ—_z the_o same_o be_v m._n and_o so_o forward_o always_o add_v one_o until_o there_o be_v produce_v suâ—h_v a_o multiplex_n to_o d_o which_o shall_v be_v nâ—xt_v great_a than_o gh_o that_o be_v which_o amongst_o the_o 〈◊〉_d oâ—_n dâ—_n â—_z by_o the_o continual_a addition_n of_o one_o do_v first_o begin_v to_o exceade_v gh_o and_o let_v thâ—â—â—me_z be_v nâ—_n â—_z which_o let_v â—e_z quadruple_a to_o d._n now_o then_o the_o multiplex_n gh_o be_v the_o next_o multiplex_n less_o than_o nâ—_n â—_z and_o therefore_o â—s_v not_o less_o than_o m_o that_o be_v be_v either_o equal_a unto_o it_o or_o great_a than_o it_o difference_n and_o forasmuch_o â—s_v fg_o be_v equemultiplex_n to_o ae_n as_o gh_o be_v to_o eâ—_n therefore_o how_o multiplex_n fâ—_n be_v to_o ae_n so_o multiplex_n be_v fh_o to_o ab_fw-la by_o the_o first_o of_o the_o five_o but_o how_o multiplex_n fg_o be_v to_o ae_n so_o multiplex_n be_v king_n to_o c_o therefore_o how_o multiplex_n fh_o be_v to_o ab_fw-la so_o multiplex_n be_v king_n to_o c._n moreover_o forasmuch_o as_o gh_o and_o king_n be_v equemultiplices_fw-la unto_o ebb_n and_o câ—_n â—_z and_o ebb_n be_v by_o construction_n equal_a unto_o c_o therefore_o by_o the_o common_a sentence_n gh_o be_v equal_a unto_o k._n but_o gh_o be_v not_o less_o than_o m_o as_o have_v before_o be_v show_v and_o fgâ—_n â—_z be_v put_v to_o be_v great_a than_o d._n wherefore_o the_o whole_a fh_o be_v great_a than_o these_o two_o d_z and_o m._n but_o d_z and_o m_o be_v equal_a unto_o n._n for_o n_o be_v quadruple_a to_o d._n and_o m_o be_v triple_a to_o d_z do_v together_o with_o d_z make_v quadruple_a unto_o d._n wherefore_o fh_o be_v great_a than_o n._n farther_n edward_n be_v prove_v to_o be_v equal_a to_o gh_o wherefore_o king_n be_v less_o than_o n._n but_o fh_o and_o king_n be_v equemultiplices_fw-la unto_o ab_fw-la and_o c_o unto_o the_o first_o magnitude_n i_o say_v and_o the_o three_o and_o n_o be_v a_o certain_a other_o multiplex_n unto_o d_o which_o represent_v the_o second_o &_o the_o four_o magnitude_n and_o the_o multiplex_n of_o the_o first_o exceed_v the_o multiplex_n of_o the_o second_o but_o the_o multiplex_n of_o the_o three_o exceed_v not_o the_o multiplex_n of_o the_o four_o wherefore_o ab_fw-la the_o first_o have_v unto_o d_z the_o second_o a_o great_a proportion_n then_o have_v c_z the_o three_o to_o d_o the_o four_o by_o the_o 8._o definition_n of_o this_o book_n but_o if_o ae_n be_v great_a than_o ebb_n let_v ebb_v the_o less_o be_v multiply_v until_o there_o be_v produce_v a_o multiplex_n great_a than_o the_o magnitude_n d_o which_o let_v be_v gh_o and_o how_o multiplex_n gh_o be_v to_o ebb_v difference_n so_o multiplex_n let_v fg_o be_v to_o ae_n and_o king_n also_o to_o c._n then_o take_v unto_o d_z
such_o a_o multiplex_n as_o be_v next_o great_a than_o fg_o and_o again_o let_v the_o same_o be_v n_o which_o let_v be_v quadruple_a to_o d._n and_o in_o like_a sort_n as_o before_o may_v we_o prove_v that_o the_o whole_a fh_o be_v unto_o ab_fw-la equemultiplex_fw-la as_o gh_o be_v to_o ebb_v and_o also_o that_o fh_o &_o king_n be_v equemultiplices_fw-la unto_o abâ—_n â—_z &_o c_o and_o final_o that_o gh_o be_v equal_a unto_o k._n and_o forasmuch_o as_o the_o multiplex_n n_o be_v next_o great_a than_o fg_o therefore_o fg_o be_v not_o less_o than_o m._n but_o gh_o be_v great_a than_o d_z by_o construction_n wherefore_o the_o whole_a fh_o be_v great_a than_o d_z and_o m_o and_o so_o consequent_o be_v great_a than_o n._n but_o king_n exceed_v not_o n_o for_o king_n be_v equal_a to_o gh_o for_o how_o multiplex_n king_n be_v to_o ebb_v the_o less_o so_o multiplex_n be_v fg_o to_o aâ—_z the_o great_a bâ—t_v those_o magnitude_n which_o be_v equemultipliceâ—_n unto_o unequal_a magnitudâ—s_n be_v according_a to_o the_o same_o proportion_n unequal_a wherefore_o king_n be_v less_o than_o fg_o and_o therefore_o iâ—_z much_o less_o than_o n._n wherefore_o again_o the_o multiplex_n of_o the_o first_o exceed_v the_o multiplex_n of_o the_o second_o but_o the_o multiplex_n of_o the_o three_o exceed_v not_o thâ—_z multiplex_n of_o the_o four_o wherefore_o by_o the_o 8._o definition_n of_o the_o five_o aâ—_z the_o first_o have_v to_o d_z the_o second_o a_o great_a proportion_n then_o have_v c_z the_o three_o to_o d_o the_o four_o difference_n but_o now_o if_o ae_n be_v equal_a unto_o ebb_n either_o of_o they_o shall_v be_v equal_a unto_o c._n wherefore_o unto_o either_o of_o thosâ—_n three_o magnitude_n take_v equemultiplices_fw-la great_a than_o d._n so_o that_o let_v fg_o be_v multiplex_n to_o ae_n and_o gh_o unto_o ebb_n and_o king_n again_o to_o c_z which_o by_o the_o 6._o common_a sentence_n shall_v be_v equal_a the_o one_o to_o the_o other_o let_v n_o also_o be_v multiplex_n to_o d_o and_o be_v next_o great_a than_o every_o one_o of_o they_o namely_o let_v it_o be_v qâ—adruplâ—_n to_o d._n this_o construction_n finish_v we_o may_v again_o prove_v that_o fh_o and_o king_n be_v equemultiplices_fw-la to_o ab_fw-la and_o c_o and_o that_o fh_o the_o multiplex_n of_o thâ—_z first_o magnitude_n exceed_v no_o the_o multiplex_n of_o the_o second_o magnitudeâ—â—nd_n thaâ—_z king_n tâ—â—â—ultiplex_n of_o the_o three_o exceed_v not_o the_o multiplex_n of_o the_o four_o wherefore_o we_o may_v conclude_v that_o ab_fw-la have_v unto_o d_z a_o great_a proportion_n then_o have_v c_z to_o d._n now_o also_o i_o say_v that_o the_o self_n same_o magnitude_n d_z have_v unto_o the_o less_o magnitude_n c_o a_o great_a proportion_n they_o it_o have_v to_o the_o great_a ab_fw-la proposition_n and_o this_o may_v plain_o be_v gather_v by_o the_o foresay_a discourse_n without_o change_v the_o order_n of_o the_o magnitude_n &_o of_o the_o equemultiplices_fw-la for_o see_v that_o every_o way_n it_o be_v before_o prove_v that_o fh_o exceed_v n_o and_o king_n be_v exceed_v of_o the_o self_n same_o n_o therefore_o converse_o n_o exceed_v king_n but_o do_v not_o exceed_v fh_o but_o n_o be_v multiplex_n to_o d_o that_o be_v to_o the_o first_o and_o three_o magnitude_n and_o king_n be_v multiplex_n to_o the_o second_o namely_o to_o câ—_n â—_z and_o fh_o be_v multiplex_fw-la to_o the_o four_o namely_o to_o abâ—_n â—_z wherefore_o the_o multiplex_n of_o the_o first_o exceed_v the_o multiplex_n of_o the_o second_o but_o the_o multiplex_n of_o the_o three_o exceed_v not_o the_o multiplex_n of_o the_o four_o wherefore_o by_o the_o 8._o definition_n of_o this_o five_o book_n d_z the_o first_o have_v unto_o c_o the_fw-fr second_o a_o great_a proportion_n then_o have_v d_z the_o three_o to_o ab_fw-la the_o four_o which_o be_v require_v to_o be_v prove_v the_o 9_o theorem_a the_o 9_o proposition_n magnitude_n which_o have_v to_o one_o and_o the_o same_o magnitude_n one_o and_o the_o same_o proportion_n be_v equal_a the_o one_o to_o the_o other_o and_o those_o magnitude_n unto_o who_o one_o and_o the_o same_o magnitude_n have_v one_o and_o the_o same_o proportion_n be_v also_o equal_a svppose_v that_o either_o of_o these_o two_o magnitude_n a_o and_o b_o have_v to_o c_o one_o and_o the_o same_o proportion_n demonstrate_v then_o i_o say_v that_o a_o be_v equal_a unto_o b._n for_o if_o it_o be_v not_o then_o either_o of_o these_o a_o and_o b_o shall_v not_o have_v to_o c_o one_o &_o the_o same_o proportion_n by_o the_o 8._o of_o the_o five_o but_o by_o supposition_n they_o have_v wherefore_o a_o be_v equal_a unto_o b._n again_o prove_v suppose_v that_o the_o magnitude_n c_o have_v to_o either_o of_o these_o magnitude_n a_o and_o b_o one_o and_o the_o same_o proportion_n then_o i_o say_v that_o a_o be_v equal_a unto_o b._n for_o if_o it_o be_v not_o c_z shall_v not_o have_v to_o either_o of_o these_o a_o and_o b_o one_o and_o the_o same_o proportion_n by_o the_o former_a proposition_n but_o by_o supposition_n it_o have_v wherefore_o a_o be_v equal_a unto_o b._n wherefore_o magnitude_n which_o have_v to_o one_o and_o the_o same_o magnitude_n one_o and_o the_o same_o proportion_n be_v equal_a the_o one_o to_o the_o other_o and_o thosâ—_n magnitude_n unto_o who_o one_o and_o the_o same_o magnitude_n have_v one_o and_o the_o same_o proportion_n be_v also_o equal_a which_o be_v require_v to_o be_v prove_v the_o 10._o theorem_a the_o 10._o proposition_n of_o magnitude_n compare_v to_o one_o and_o the_o same_o magnitude_n that_o which_o have_v the_o great_a proportion_n be_v the_o great_a and_o that_o magnitude_n whereunto_o one_o and_o the_o same_o magnitude_n have_v the_o great_a proportion_n be_v the_o less_o svppose_v that_o a_o have_v to_o c_o a_o great_a proportion_n than_o b_o have_v to_o c._n then_o i_o say_v that_o a_o be_v great_a than_o b._n prove_v for_o if_o it_o be_v not_o then_o either_o a_o be_v equal_a unto_o b_o or_o less_o than_o it_o but_o a_o can_v be_v equal_a unto_o b_o for_o then_o either_o of_o these_o a_o and_o b_o shall_v have_v unto_o c_o one_o and_o the_o same_o proportion_n by_o the_o 7_o of_o the_o five_o but_o by_o supposition_n they_o have_v not_o wherefore_o a_o be_v not_o equal_a unto_o b._n neither_o also_o be_v a_o less_o than_o b_o for_o they_o shall_v a_o have_v to_o c_o a_o less_o proportion_n then_o have_v b_o to_o c_z by_o the_o 8._o of_o the_o five_o but_o by_o supposition_n it_o have_v not_o wherefore_o a_o be_v not_o less_o than_o b._n and_o it_o be_v also_o prove_v that_o it_o be_v not_o equal_a wherefore_o a_o be_v great_a than_o b._n demonstrate_v again_o suppose_v that_o c_o have_v to_o b_o a_o great_a proportion_n than_o c_z have_v to_o a._n then_o i_o say_v that_o b_o be_v less_o then_o a._n for_o if_o it_o be_v not_o then_o be_v it_o either_o equal_a unto_o it_o or_o else_o great_a but_o b_o can_v be_v equal_a unto_o a_o for_o than_o shall_v c_o have_v to_o either_o of_o these_o a_o and_o b_o one_o and_o the_o same_o proportion_n by_o the_o 7._o of_o the_o five_o but_o by_o supposition_n it_o have_v not_o wherefore_o b_o be_v not_o equal_a unto_o a._n neither_o also_o be_v b_o great_a than_o a_o for_o than_o shall_v c_o have_v to_o b_o a_o less_o proportion_n than_o it_o have_v to_o a_o by_o the_o 8._o of_o the_o five_o but_o by_o supposition_n it_o have_v not_o wherefore_o b_o be_v not_o great_a than_o a._n and_o it_o be_v prove_v that_o it_o be_v not_o equal_a unto_o a_o wherefore_o b_o be_v less_o than_o a._n wherefore_o of_o magnitude_n compare_v to_o one_o and_o the_o same_o magnitude_n that_o which_o have_v y_fw-fr great_a proportion_n be_v the_o great_a and_o that_o magnitude_n whereunto_o one_o and_o the_o same_o magnitude_n have_v the_o great_a proportion_n be_v the_o less_o which_o be_v require_v to_o be_v prove_v the_o 11._o theorem_a the_o 11._o proposition_n proportion_n which_o be_v one_o and_o the_o self_n same_o to_o any_o one_o proportion_n be_v also_o the_o self_n same_o the_o one_o to_o the_o other_o svpppose_v that_o as_o a_o be_v to_o b_o so_o be_v c_z to_o d_z and_o as_z c_z be_v to_o d_z so_o be_v e_o to_o f._n then_o i_o say_v that_o as_o a_o be_v to_o b_o so_o be_v e_o to_o f._n construction_n take_v equemultiplices_fw-la to_o a_o c_o and_o e_o which_o let_v be_v g_o h_o k._n and_o likewise_o to_o b_o d_o and_o f_o take_v any_o other_o equemultiplices_fw-la which_o let_v be_v l_o m_o and_o n._n and_o because_o as_o a_o be_v to_o b_o so_o be_v c_z to_o d_z and_o to_o a_o and_o g_o be_v take_v equemultiplices_fw-la g_z &_o h_o &_o to_o b_o and_o d_o be_v take_v certain_a other_o equemultiplices_fw-la l_o
proportion_n than_o have_v bc_o to_o ef._n wherefore_o a_o have_v to_o d_o a_o great_a proportion_n than_o have_v bc_o to_o ef._n wherefore_o alternate_o a_o have_v to_o bc_o a_o great_a proportion_n than_o have_v d_z to_o of_o wherefore_o by_o composition_n abc_n have_v to_o bc_o a_o great_a proportion_n than_o have_v def_n to_o ef._n wherefore_o again_o alternate_o abc_n have_v to_o def_n a_o great_a proportion_n than_o have_v bc_o to_o ef._n wherefore_o by_o the_o former_a proposition_n the_o proportion_n of_o a_o to_o d_z be_v great_a than_o the_o proportion_n of_o abc_n to_o def_n which_o be_v require_v to_o be_v prove_v the_o end_n of_o the_o five_o book_n of_o euclides_n element_n ¶_o the_o six_o book_n of_o euclides_n element_n this_o spaniard_o book_n be_v for_o use_v and_o practice_v book_n a_o most_o special_a book_n in_o it_o be_v teach_v the_o proportion_n of_o one_o figure_n to_o a_o other_o figure_n &_o of_o their_o side_n the_o one_o to_o the_o other_o and_o of_o the_o side_n of_o one_o to_o the_o side_n of_o a_o other_o likewise_o of_o the_o angle_n of_o the_o one_o to_o the_o angle_n of_o the_o other_o moreover_o it_o teach_v the_o description_n of_o figure_n like_o to_o â—_z give_v and_o marvelous_a application_n of_o figure_n to_o line_n even_o or_o with_o decrease_n or_o excess_n with_o many_o other_o theorem_n not_o only_o of_o the_o propoâ—tions_n of_o right_n line_v figure_n but_o also_o of_o sector_n of_o circle_n with_o their_o angle_n on_o the_o theorem_n and_o problem_n of_o this_o book_n depend_v for_o the_o most_o part_n the_o composition_n of_o all_o instrument_n of_o measure_v length_n breadth_n or_o deepen_n and_o also_o the_o reason_n of_o the_o use_n of_o the_o same_o instrument_n as_o of_o the_o geometrical_a â—quarâ—_n geometry_n the_o scale_n of_o the_o astrolabe_n the_o quadrant_a the_o staff_n and_o such_o other_o the_o use_n of_o which_o instrument_n beside_o all_o other_o mechanical_a instrument_n of_o raise_v up_o of_o move_v and_o draw_v huge_a thing_n incredible_a to_o the_o ignorant_a and_o infinite_a other_o begin_v which_o likewise_o have_v their_o ground_n out_o of_o this_o book_n be_v of_o wonderful_a and_o unspeakable_a profit_n beside_o the_o inestimable_a pleasure_n which_o be_v in_o they_o definition_n 1._o like_o rectiline_a figure_n be_v such_o definition_n who_o angle_n be_v equal_a the_o one_o to_o the_o other_o and_o who_o side_n about_o the_o equal_a angle_n be_v proportional_a as_o if_o you_o take_v any_o two_o rectiline_a figure_n as_o for_o example_n two_o triangle_n abc_n and_o def_n 〈…〉_o of_o the_o one_o triangle_n be_v equal_a to_o the_o angle_n of_o the_o other_o namely_o if_o the_o angle_n a_o be_v equal_a to_o the_o angle_n d_o and_o the_o angle_n b_o equal_v to_o the_o angle_n e_o &_o also_o the_o angle_n c_o equal_v to_o the_o angle_n f._n and_o moreover_o iâ—_z the_o side_n which_o contain_v the_o equal_a angle_n be_v proportional_a as_o if_o the_o side_n ab_fw-la have_v that_o proportion_n to_o the_o side_n bc_o whâ—ch_v the_o side_n de_fw-fr have_v to_o the_o side_n of_o and_o also_o if_o the_o side_n bc_o be_v unto_o the_o side_n ca_n as_o â—he_n side_n of_o be_v to_o the_o side_n fd_o and_o moreover_o if_o the_o side_n ca_n be_v to_o the_o side_n ab_fw-la as_o the_o side_n fd_o be_v to_o the_o side_n de_fw-fr then_o be_v these_o two_o triangle_n say_v to_o be_v like_o and_o so_o judge_v you_o of_o any_o other_o kind_n of_o figure_n as_o if_o in_o the_o parallelogram_n abcd_o and_o efgh_a the_o angle_n a_o be_v equal_a to_o the_o angle_n e_o and_o the_o angle_n b_o equal_v to_o the_o angle_n fletcher_n and_o the_o angle_n c_o equal_v to_o the_o angle_n g_o and_o the_o angle_n d_o equal_v to_o the_o angle_n h._n and_o farthermore_o if_o the_o side_n ac_fw-la have_v that_o proportion_n to_o the_o side_n cd_o which_o the_o side_n eglantine_n have_v to_o the_o side_n gh_o and_o if_o also_o the_o side_n cd_o be_v to_o the_o side_n db_o as_o the_o side_n gh_o be_v to_o the_o side_n hf_o and_o moreover_o if_o the_o side_n db_o be_v to_o the_o side_n basilius_n as_o the_o side_n hf_o be_v to_o the_o side_n fe_o and_o final_o if_o the_o side_n basilius_n be_v to_o the_o side_n ac_fw-la as_o the_o side_n fe_o be_v to_o the_o side_n eglantine_n then_o be_v these_o parallelogram_n like_a definition_n 2._o reciprocal_a figure_n be_v those_o when_o the_o termeâ—_n of_o proportion_n be_v both_o antecedentes_fw-la and_o consequentes_fw-la in_o either_o figure_n as_o if_o you_o have_v two_o parallelogram_n abcd_o and_o efgh_o if_o the_o side_n ab_fw-la to_o the_o side_n of_o a_o antecedent_n of_o the_o first_o figure_n to_o a_o consequent_a of_o the_o second_o figure_n have_v mutual_o the_o same_o proportion_n which_o the_o side_n eglantine_n have_v to_o the_o side_n ac_fw-la a_o antecedent_n of_o the_o second_o figure_n to_o a_o consequent_a of_o the_o first_o figure_n then_o be_v these_o two_o figure_n reciprocal_a they_o be_v call_v of_o some_o figure_n of_o mutual_a side_n and_o that_o undoubted_o not_o amiss_o nor_o unapt_o figure_n and_o to_o make_v this_o definition_n more_o plain_a campane_n and_o pestitarius_n and_o othersâ—_n thus_o put_v it_o reciprocal_a figure_n be_v when_o the_o side_n of_o other_o 〈◊〉_d mutual_o proportional_a as_o in_o the_o example_n and_o declaration_n before_o give_v among_o the_o barbarous_a they_o be_v call_v mutekesia_n reserve_v still_o the_o arabike_a word_n definition_n 3._o a_o right_a line_n be_v say_v to_o be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n when_o the_o whole_a be_v to_o the_o great_a part_n as_o the_o great_a part_n be_v to_o the_o less_o as_o if_o the_o line_n ab_fw-la be_v so_o divide_v in_o the_o point_n c_o that_o the_o whole_a line_n ab_fw-la have_v the_o same_o proportion_n to_o the_o great_a part_n thereof_o namely_o to_o ac_fw-la which_o the_o same_o great_a part_n ac_fw-la have_v to_o the_o less_o part_n thereof_o namely_o to_o cb_o then_o be_v the_o line_n ab_fw-la divide_v by_o a_o extreme_a and_o mean_a proportion_n common_o it_o be_v call_v a_o line_n divide_v by_o proportion_n haâ—ing_v a_o mean_a and_o two_o extreme_n how_o to_o divide_v a_o line_n in_o such_o sort_n be_v teach_v in_o the_o 11._o proposition_n of_o the_o second_o book_n but_o not_o under_o this_o form_n of_o proportion_n 4._o the_o alitude_n of_o a_o figure_n be_v a_o perpendicular_a line_n draw_v from_o the_o top_n to_o the_o base_a definition_n as_o the_o altitude_n or_o height_n of_o the_o triangle_n abc_n be_v the_o line_n ad_fw-la be_v draw_v perpendicular_o from_o the_o point_n a_o be_v the_o top_n or_o high_a part_n of_o the_o triangle_n to_o the_o base_a thereof_o bc._n so_o likewise_o in_o other_o figure_n as_o you_o see_v in_o the_o example_n here_o set_v that_o which_o here_o â—ee_n call_v the_o altitude_n or_o height_n of_o a_o figure_n in_o the_o first_o book_n in_o the_o 35._o proposition_n and_o certain_a other_o follow_v he_o teach_v to_o be_v contain_v within_o two_o equidistant_a line_n so_o that_o figure_v to_o have_v one_o altitude_n and_o to_o be_v contain_v within_o two_o equidistant_a line_n be_v all_o one_o so_o in_o all_o these_o example_n if_o from_o the_o high_a point_n of_o the_o figure_n you_o draw_v a_o equidistant_a line_n to_o the_o base_a thereof_o and_o then_o from_o that_o point_n draw_v a_o perpendicular_a to_o the_o same_o base_a that_o perpendicular_a be_v the_o altitude_n of_o the_o figure_n 5._o a_o proportion_n be_v say_v to_o be_v make_v of_o two_o proportion_n or_o more_o when_o the_o quantity_n of_o the_o proportion_n multiply_v the_o one_o into_o the_o other_o produce_v a_o other_o quantity_n definition_n an_o other_o example_n where_o the_o great_a inequality_n and_o the_o less_o inequality_n be_v mix_v together_o 6._o 4._o 2._o 3._o the_o denomination_n of_o the_o proportion_n of_o 6._o to_o 4_o be_v 1_o â—_o â—_o example_n of_o 4._o to_o 2_o be_v â—_o â—_o and_o of_o 2._o to_o 3_o be_v â—_o â—_o now_o if_o you_o multiply_v as_o you_o ought_v all_o these_o denomination_n together_o you_o shall_v produce_v 12._o to_o 6_o namely_o dupla_fw-la proportion_n forasmuch_o as_o so_o much_o have_v hitherto_o be_v speak_v of_o addition_n of_o proportion_n it_o shall_v not_o be_v unnecessary_a somewhat_o also_o to_o say_v of_o substraction_n of_o they_o proportion_n where_o it_o be_v to_o be_v note_v that_o as_o addition_n of_o they_o be_v make_v by_o multiplication_n of_o their_o denomination_n the_o one_o into_o the_o other_o so_o be_v the_o substraction_n of_o the_o one_o from_o the_o other_o do_v by_o division_n of_o the_o denomination_n of_o the_o one_o by_o the_o denomination_n of_o the_o other_o as_o if_o you_o will_v from_o sextupla_fw-la proportion_n subtrahe_fw-la dupla_fw-la proportion_n take_v
the_o denomination_n of_o they_o both_o the_o denomination_n of_o sextupla_fw-la proportion_n be_v 6_o the_o denomination_n of_o dupla_fw-la proportion_n be_v 2._o now_o divide_v 6._o the_o denomination_n of_o the_o one_o by_o 2._o the_o denomination_n of_o the_o other_o the_o quotient_n shall_v be_v 3_o which_o be_v the_o denomination_n of_o a_o new_a proportion_n namely_o tripla_fw-la so_o that_o when_o dupla_fw-la proportion_n be_v subtrahe_v from_o sextupla_fw-la there_o shall_v remain_v tripla_fw-la proportion_n and_o thus_o may_v you_o do_v in_o all_o other_o 6._o a_o parallelogram_n apply_v to_o a_o right_a line_n be_v say_v to_o want_v in_o form_n by_o a_o parallelogram_n like_a to_o one_o give_v when_o the_o parallelogram_n apply_v want_v to_o the_o fill_n of_o the_o whole_a line_n by_o a_o parallelogram_n like_a to_o one_o give_v definition_n and_o then_o be_v it_o say_v to_o exceed_v when_o it_o exceed_v the_o line_n by_o a_o parallelogram_n like_a to_o that_o which_o be_v give_v as_o let_v e_o be_v a_o parallelogramme_n give_v and_o let_v ab_fw-la be_v a_o right_a line_n to_o who_o be_v apply_v the_o parallelogram_n acdf_n now_o if_o it_o want_v of_o the_o fill_n of_o the_o line_n ab_fw-la by_o the_o parallelogram_n dfgb_n be_v like_a to_o the_o parallelogram_n give_v e_o then_o be_v the_o parallelogram_n say_v to_o want_v in_o form_n by_o a_o parallelogram_n like_a unto_o a_o parallelogram_n give_v likewise_o if_o it_o exceed_v as_o the_o parallelogram_n acgd_v apply_v to_o the_o linâ—_n abâ—_n if_o it_o exceed_v it_o by_o the_o parallelogram_n fgbd_v be_v like_o to_o the_o parallelogram_n fletcher_n which_o be_v give_v then_o be_v the_o parallelogram_n abgd_v say_v to_o exceed_v in_o form_n by_o a_o parallelogram_n like_a to_o a_o parallelogram_n give_v this_o definition_n be_v add_v by_o flussates_n as_o it_o seem_v it_o be_v not_o in_o any_o common_a greek_a book_n abroad_o nor_o in_o any_o commentary_n it_o be_v for_o many_o theorem_n follow_v very_o necessary_a the_o 1._o theorem_a the_o 1._o proposition_n triangle_n &_o parallelogram_n which_o be_v under_o one_o &_o the_o self_n same_o altitude_n be_v in_o proportion_n as_o the_o base_a of_o the_o one_o be_v to_o the_o base_a of_o the_o other_o and_o forasmuch_o as_o the_o line_n cb_o bg_o and_o gh_o be_v equal_a the_o one_o to_o the_o other_o part_n therefore_o the_o triangle_n also_o ahg_n agb_n and_o abc_n be_v by_o the_o 38._o of_o the_o first_o equal_a the_o one_o to_o the_o other_o wherefore_o how_o multiplex_n the_o base_a hc_n be_v to_o the_o base_a bc_o so_o multiplex_n also_o be_v the_o triangle_n ahc_n to_o the_o triangle_n abc_n and_o by_o the_o same_o reason_n also_o how_o multiplex_n the_o base_a lc_n be_v to_o the_o base_a dc_o so_o multiplex_n also_o be_v the_o triangle_n alc_n to_o the_o triangle_n adc_o wherefore_o if_o the_o base_a hc_n be_v equal_a unto_o the_o base_a cl_n then_o by_o the_o 38._o of_o the_o first_o the_o triangle_n ahc_n be_v equal_a unto_o the_o triangle_n acl_n and_o if_o the_o base_a hc_n exceed_v the_o base_a cl_n then_o also_o the_o triangle_n ahc_n exceed_v the_o triangle_n acl_n and_o if_o the_o base_a be_v less_o the_o triangle_n also_o shall_v be_v less_o now_o then_o there_o be_v four_o magnitude_n namely_o the_o two_o basâ—s_n bc_o and_o cd_o and_o the_o two_o triangle_n abc_n and_o acd_o and_o to_o the_o base_a bc_o and_o to_o the_o triangle_n abc_n namely_o to_o the_o first_o and_o the_o three_o be_v take_v equemulâ—iplices_n namely_o the_o base_a hc_n and_o the_o triangle_n ahc_n and_o likewise_o to_o the_o base_a cd_o and_o to_o the_o triangle_n adc_o namely_o to_o the_o second_o and_o the_o four_o be_v take_v certain_a other_o equemultiplices_fw-la that_o be_v the_o base_a cl_n and_o the_o triangle_n alc_n and_o it_o have_v be_v prove_v that_o if_o the_o multiplex_n of_o the_o first_o magnitude_n that_o be_v the_o base_a hc_n do_v exceed_v the_o multiplex_n of_o the_o second_o that_o be_v the_o base_a cl_n the_o multiplex_n also_o of_o the_o three_o that_o be_v the_o triangle_n ahc_n exceed_v the_o multiplex_n of_o the_o â—_z that_o be_v the_o triangle_n alc_n and_o if_o the_o say_v base_a hc_n be_v equal_a to_o the_o say_a baâ—â—_n cl_n the_o triangle_n also_o ahc_n be_v equal_a to_o the_o triangle_n alc_n and_o if_o it_o be_v less_o it_o iâ—_z less_o wherefore_o by_o the_o six_o definition_n of_o the_o five_o as_o the_o first_o of_o the_o foresay_a magnitude_n be_v to_o the_o second_o so_o be_v the_o three_o to_o the_o four_o wherefore_o as_o the_o base_a bc_o be_v to_o the_o base_a cd_o so_o be_v the_o triangle_n abc_n to_o the_o triangle_n acd_o and_o because_o by_o the_o 41._o of_o the_o first_o the_o parallelogram_n ec_o be_v double_a to_o the_o triangle_n abc_n part_n and_o by_o the_o same_o the_o parallelogram_n fc_o be_v double_a to_o the_o triangle_n acd_o therefore_o the_o parallelogram_n ec_o and_o fc_fw-la be_v equemultiplices_fw-la unto_o the_o triangle_n abc_n and_o acd_o but_o the_o part_n of_o equemultiplices_fw-la by_o the_o 15._o of_o the_o five_o have_v one_o and_o the_o same_o proportion_n with_o theiâ—_z equemultiplices_fw-la wherefore_o as_o the_o triangle_n abc_n be_v to_o the_o triangle_n acd_o so_o be_v the_o parallelogram_n ec_o to_o the_o parallelogram_n fc_o and_o forasmuch_o as_o it_o have_v be_v demonstrate_v that_o as_o the_o base_a bc_o be_v to_o the_o base_a cd_o so_o be_v the_o triangle_n abc_n to_o the_o triangle_n acd_o and_o as_o the_o triangle_n abc_n be_v to_o the_o triangle_n acd_v so_o be_v the_o parallelogram_n ec_o to_o the_o parallelogram_n fc_o wherefore_o by_o the_o 11._o of_o the_o five_o as_o the_o base_a bc_o be_v to_o the_o base_a cd_o so_o be_v the_o parallelogram_n ec_o to_o the_o parallelogram_n fc_o the_o parallelogram_n may_v also_o be_v demonstrate_v a_o part_n by_o themselves_o as_o the_o triangle_n be_v if_o we_o describe_v upon_o the_o base_n bg_o gh_o and_o dk_o &_o kl_o parallelogram_n under_o the_o self_n same_o altitude_n that_o the_o parallelogrammeâ—_n give_v be_v wherefore_o triangle_n and_o parallelogram_n which_o be_v under_o one_o and_o the_o self_n same_o altitude_n be_v in_o proportion_n as_o the_o base_a of_o the_o one_o be_v to_o the_o base_a of_o the_o other_o which_o be_v require_v to_o be_v demonstrate_v here_o flussates_n add_v this_o corollary_n if_o two_o right_a line_n be_v give_v the_o one_o of_o they_o be_v divide_v how_o so_o ever_o flussates_n the_o rectangle_n figure_v contain_v under_o the_o whole_a line_n undivided_a and_o each_o of_o the_o segment_n of_o the_o line_n divide_v be_v in_o proportion_n the_o one_o to_o the_o other_o as_o the_o segment_n be_v the_o one_o to_o the_o other_o for_o imagine_v the_o figure_n basilius_n and_o ad_fw-la in_o the_o former_a description_n to_o be_v rectangle_v the_o rectangle_n figure_v contain_v under_o the_o whole_a right_a line_n ac_fw-la and_o the_o segment_n of_o the_o right_a line_n bd_o which_o be_v cuâ—_n in_o the_o point_n c_o namely_o the_o parallelogram_n basilius_n and_o ad_fw-la be_v in_o proportion_n the_o one_o to_o the_o other_o as_o the_o segmente_n bc_o and_o cd_o be_v the_o 2._o theorem_a the_o 2._o proposition_n if_o to_o any_o one_o of_o the_o side_n of_o a_o triangle_n be_v draw_v a_o parallel_n right_a line_n it_o shall_v cut_v the_o side_n of_o the_o same_o triangle_n proportional_o and_o if_o the_o side_n of_o a_o triangle_n be_v cut_v proportional_o a_o right_a line_n draw_v from_o section_n to_o section_n be_v a_o parallel_n to_o the_o other_o side_n of_o the_o triangle_n svppose_v that_o there_o be_v a_o triangle_n abc_n unto_o one_o of_o the_o side_n whereof_o namely_o unto_o bc_o let_v there_o be_v draw_v a_o parallel_a line_n de_fw-fr cut_v the_o side_n ac_fw-la and_o ab_fw-la in_o the_o point_n e_o and_o d._n then_o i_o say_v first_o that_o as_o bd_o be_v to_o da_z so_o be_v ce_fw-fr to_o ea_fw-la theorem_a draw_v a_o line_n from_o b_o to_o e_o &_o also_o from_o c_z to_o d._n wherefore_o by_o the_o 37._o of_o the_o first_o the_o triangle_n bde_v be_v equal_a unto_o the_o triangle_n cde_o for_o they_o be_v set_v upon_o one_o and_o the_o same_o base_a de_fw-fr and_o be_v contain_v within_o the_o self_n same_o parallel_n de_fw-fr and_o bc._n consider_v also_o a_o certain_a other_o triangle_n ade_n now_o thing_n equal_a by_o the_o 7._o of_o the_o five_o have_v to_o oneself_o thing_n one_o and_o the_o same_o proportion_n wherefore_o as_o the_o triangle_n bde_n be_v to_o the_o triangle_n ade_n so_o be_v the_o triangle_n cde_o to_o the_o triangle_n ade_n but_o as_o the_o triangle_n bde_n be_v to_o the_o triangle_n ade_n so_o be_v the_o base_a bd_o to_o the_o base_a dam_fw-la by_o the_o first_o of_o this_o book_n for_o they_o be_v under_o one_o
and_o the_o self_n same_o top_n namely_o e_o and_o therefore_o be_v under_o one_o and_o the_o same_o altitude_n and_o by_o the_o same_o reason_n as_o the_o triangle_n cde_o be_v to_o the_o triangle_n ade_n so_o be_v the_o line_n ce_fw-fr to_o the_o line_n ea_fw-la wherefore_o by_o the_o 11._o of_o the_o five_o as_o the_o line_n bd_o be_v to_o the_o line_n dam_fw-ge so_o be_v the_o line_n ce_fw-fr to_o the_o line_n ea_fw-la but_o now_o suppose_v that_o in_o the_o triangle_n abc_n the_o side_n ab_fw-la &_o ac_fw-la be_v cut_v proportional_o so_o that_o as_o bd_o be_v to_o da_z so_o let_v ce_fw-fr be_v to_o ea_fw-la &_o draw_v a_o line_n from_o d_z to_o e._n then_o second_o i_o say_v that_o the_o line_n de_fw-fr be_v a_o parallel_n to_o the_o line_n bc._n part_n for_o the_o same_o order_n of_o construction_n be_v keep_v for_o that_o as_z bd_o be_v to_o da_z so_o be_v ce_fw-fr to_o ea_fw-la but_o as_o bd_o be_v to_o da_z so_o be_v the_o triangle_n bde_v to_o the_o triangle_n ade_n by_o the_o 1._o of_o the_o six_o &_o as_o ce_fw-fr be_v to_o ea_fw-la so_o by_o the_o same_o be_v the_o triangle_n cde_o to_o the_o triangle_n ade_n therefore_o by_o the_o 11._o of_o the_o five_o as_o the_o triangle_n bde_v be_v to_o the_o triangle_n ade_n so_o be_v the_o triangle_n cde_o to_o the_o triangle_n ade_n wherefore_o either_o of_o these_o triangle_n bde_v and_o cde_o have_v to_o the_o triangle_n ade_n one_o and_o the_o same_o proportion_n wherefore_o by_o the_o 9_o of_o the_o five_o the_o triangle_n bde_v be_v equal_a unto_o the_o triangle_n cde_o and_o they_o be_v upon_o one_o and_o the_o self_n base_a namely_o de._n but_o triangle_n equal_a and_o set_v upon_o one_o base_a be_v also_o contain_v within_o the_o same_o parallel_a line_n by_o the_o 39_o of_o the_o first_o wherefore_o the_o line_n de_fw-fr be_v unto_o the_o line_n bc_o a_o parallel_n if_o therefore_o to_o any_o one_o of_o the_o side_n of_o a_o triangle_n be_v draw_v a_o parallel_a line_n it_o cut_v the_o other_o side_n of_o the_o same_o triangle_n proportional_o and_o if_o the_o side_n of_o a_o triangle_n be_v cut_v proportional_o a_o right_a line_n draw_v from_o section_n to_o section_n be_v parallel_n to_o the_o other_o side_n of_o the_o triangle_n which_o thing_n be_v require_v to_o be_v demonstrate_v ¶_o here_o also_o flussates_n add_v a_o corollary_n if_o a_o line_n parallel_n to_o one_o of_o the_o side_n of_o a_o triangle_n do_v cut_v the_o triangle_n it_o shall_v cut_v of_o from_o the_o whole_a triangle_n a_o triangle_n like_a to_o the_o whole_a triangle_n flussates_n for_o as_o it_o have_v be_v prove_v it_o divide_v the_o side_n proportional_o so_o that_o as_o ec_o be_v to_o ea_fw-la so_o be_v bd_o to_o dam_fw-ge wherefore_o by_o the_o 18._o of_o the_o five_o as_z ac_fw-la be_v to_o ae_n so_o be_v ab_fw-la to_o ad._n wherefore_o alternate_o by_o the_o 16._o of_o the_o five_o as_o ac_fw-la be_v to_o ab_fw-la so_o be_v ae_n to_o ad_fw-la wherefore_o in_o the_o two_o triangle_n ead_n and_o cab_n the_o side_n about_o the_o common_a angle_n a_o be_v proportional_a the_o say_v triangle_n also_o be_v equiangle_n for_o forasmuch_o as_o the_o right_a line_n aec_o and_o adb_o do_v fall_v upon_o the_o parallel_a line_n ed_z and_o cb_o therefore_o by_o the_o 29._o of_o the_o firsâ—_o they_o make_v the_o angle_n aed_n and_o ade_n in_o the_o triangle_n ade_n equal_a to_o the_o angle_n acb_o and_o abc_n in_o the_o triangle_n acb_o wherefore_o by_o the_o first_o definition_n of_o this_o book_n the_o whole_a triangle_n abc_n be_v like_a unto_o the_o triangle_n cut_v of_o ade_n the_o 3._o theorem_a the_o 3._o proposition_n if_o a_o angle_n of_o a_o triangle_n be_v divide_v into_o two_o equal_a part_n and_o if_o the_o right_a line_n which_o divide_v the_o angle_n divide_v also_o the_o base_a the_o segment_n of_o the_o base_a shall_v be_v in_o the_o same_o proportion_n the_o one_o to_o the_o other_o that_o the_o other_o side_n of_o the_o triangle_n be_v and_o if_o the_o segmente_n of_o the_o base_a be_v in_o the_o same_o proportion_n that_o the_o other_o side_n of_o the_o say_a triangle_n be_v a_o right_n draw_v from_o the_o top_n of_o the_o triangle_n unto_o the_o section_n shall_v divide_v the_o angle_n of_o the_o triangle_n into_o two_o equal_a part_n svppose_v that_o there_o be_v a_o triangle_n abc_n and_o by_o the_o 9_o of_o the_o first_o let_v the_o angle_n bac_n be_v divide_v into_o two_o equal_a part_n by_o the_o right_a line_n ad_fw-la which_o let_v cut_v also_o the_o base_a bc_o in_o the_o point_n d._n then_o i_o say_v that_o as_o the_o segment_n bd_o be_v to_o the_o segment_n dc_o so_o be_v the_o side_n basilius_n to_o the_o side_n ac_fw-la construction_n for_o by_o the_o point_n c_o by_o the_o 31._o of_o the_o first_o draw_v unto_o the_o line_n da_z a_o parallel_a line_n ce_fw-fr and_o extend_v the_o line_n basilius_n till_o it_o concur_v with_o the_o line_n ce_fw-fr in_o the_o point_n e_o and_o do_v make_v the_o triangle_n bec_n part_n but_o the_o line_n basilius_n shall_v concur_v with_o the_o line_n ce_fw-fr by_o the_o 5._o petition_n for_o that_o the_o angle_n ebc_n and_o be_v be_v less_o than_o two_o right_a angle_n for_o the_o angle_n ecb_n be_v equal_a to_o the_o outward_a and_o opposite_a angle_n adb_o by_o the_o 29._o of_o the_o first_o and_o the_o two_o angle_n adb_o and_o dba_n of_o the_o triangle_n bad_a be_v less_o than_o two_o right_a angle_n by_o the_o 17._o of_o the_o first_o now_o forasmuch_o as_o upon_o the_o parallel_n ad_fw-la and_o ec_o fall_v the_o right_a line_n ac_fw-la therefore_o by_o the_o 29._o of_o the_o first_o the_o angle_n ace_n be_v equal_a unto_o the_o angle_n god_n but_o unto_o the_o angle_n god_n be_v the_o angle_n bid_v suppose_v to_o be_v equal_a wherefore_o the_o angle_n bad_a be_v also_o equal_a unto_o the_o angle_n ace_n again_o because_o upon_o the_o parallel_n ad_fw-la and_o ec_o fall_v the_o right_a line_n bae_o the_o outward_a angle_n bad_a by_o the_o 28._o of_o the_o first_o be_v equal_a unto_o the_o inward_a angle_n aec_o but_o before_o it_o be_v provell_v that_o the_o angle_n ace_n be_v equal_a unto_o the_o angle_v bad_a wherefore_o the_o angle_n ace_n be_v equal_a unto_o the_o angle_v aec_o wherefore_o by_o the_o 6._o of_o the_o first_o the_o side_z ae_n be_v equal_a unto_o the_o side_n ac_fw-la and_o because_o to_o one_o of_o the_o side_n of_o the_o triangle_n be_v namely_o to_o ec_o be_v draw_v a_o parallel_a line_n ad_fw-la therefore_o by_o the_o 2._o of_o the_o six_o as_o bd_o be_v to_o dc_o so_o be_v basilius_n to_o ae_n but_o ae_n be_v equal_a unto_o ac_fw-la therefore_o as_o bd_o be_v to_o dc_o so_o be_v basilius_n to_o ac_fw-la but_o now_o suppose_v that_o as_o the_o segment_n bd_o be_v to_o the_o segment_n dc_o first_o so_o be_v the_o side_n basilius_n to_o the_o side_n ac_fw-la &_o draw_v a_o line_n from_o a_o to_o d._n then_o i_o say_v that_o the_o angle_n bac_n be_v by_o the_o right_a line_n ad_fw-la divide_v into_o two_o equal_a part_n for_o the_o same_o order_n of_o construction_n remain_v for_o that_o as_o bd_o be_v to_o dc_o so_o be_v basilius_n to_o ac_fw-la but_o as_o bd_o be_v to_o dc_o so_o be_v basilius_n to_o ae_n by_o the_o 2._o of_o the_o six_o for_o unto_o one_o of_o the_o side_n of_o the_o triangle_n be_v namely_o unto_o the_o side_n ec_o be_v draw_v a_o parallel_a line_n ad._n wherefore_o also_o as_o basilius_n be_v to_o ac_fw-la so_o be_v basilius_n to_o ae_n by_o the_o 11._o of_o the_o five_o wherefore_o by_o the_o 9_o of_o the_o five_o ac_fw-la be_v equal_a unto_o ae_n wherefore_o also_o by_o the_o 5._o of_o the_o first_o the_o angle_n aec_o be_v equal_a unto_o the_o angle_n ace_n but_o the_o angle_n aec_o by_o the_o 29._o of_o the_o first_o be_v equal_a unto_o the_o outward_a angle_n bad_a and_o the_o angle_n ace_n be_v equal_a unto_o the_o angle_n god_n which_o be_v alternate_a unto_o he_o wherefore_o the_o angle_n bad_a be_v equal_a unto_o the_o angle_n god_n wherefore_o the_o angle_n bac_n be_v by_o the_o right_a line_n ad_fw-la divide_v into_o two_o equal_a part_n wherefore_o if_o a_o angle_n of_o a_o triangle_n be_v divide_v into_o two_o equal_a part_n and_o if_o the_o right_a line_n which_o divide_v the_o angle_n cut_v also_o the_o base_a the_o segment_n of_o the_o base_a shall_v be_v in_o the_o same_o proportion_n the_o one_o to_o the_o other_o that_o the_o other_o side_n of_o the_o say_a triangle_n be_v and_o if_o the_o segment_n of_o the_o base_a be_v in_o the_o same_o proportion_n that_o the_o other_o side_n of_o the_o say_a triangle_n be_v a_o right_a line_n draw_v from_o the_o top_n of_o the_o
to_o bg_o and_o draw_v a_o line_n from_o a_o to_o g._n now_o forasmuch_o as_o ab_fw-la be_v to_o bc_o as_o de_fw-fr be_v to_o of_o therefore_o alternate_o by_o the_o 16._o of_o the_o five_o as_o ab_fw-la be_v to_o de_fw-fr so_o be_v bc_o to_o ef._n demonstration_n but_o as_o bc_o be_v to_o of_o so_o be_v of_o to_o bg_o wherefore_o also_o by_o the_o 11._o of_o the_o five_o as_o ab_fw-la be_v to_o de_fw-fr so_o be_v of_o to_o bg_o wherefore_o the_o side_n of_o the_o triangle_n abg_o &_o def_n which_o include_v the_o equal_a angle_n be_v reciprocal_a proportional_a but_o if_o in_o triangle_n have_v one_o angle_n of_o the_o one_o equal_a to_o one_o angle_n of_o the_o outstretch_o the_o side_n which_o include_v the_o equal_a angle_n be_v reciprokal_n the_o triangle_n also_o by_o the_o 15._o the_o six_o shall_v be_v equal_a wherefore_o the_o triangle_n abg_o be_v equal_a unto_o the_o triangle_n def_n and_o for_o that_o as_z a'n_z line_n bc_o be_v to_o the_o line_n of_o so_o be_v the_o line_n of_o to_o the_o line_n bg_o but_o if_o there_o be_v three_o line_n in_o proportion_n the_o first_o shall_v have_v to_o the_o three_o double_a proportion_n that_o it_o have_v to_o the_o second_o by_o the_o 10._o definition_n of_o the_o five_o therefore_o the_o line_n bc_o have_v unto_o the_o line_n bg_o double_a proportion_n that_o it_o have_v to_o the_o line_n ef._n but_o as_o bc_o be_v to_o bg_o so_o by_o the_o 1._o of_o the_o six_o be_v the_o triangle_n abc_n to_o the_o triangle_n abg_o wherefore_o the_o tiangle_n abc_n be_v unto_o the_o triangle_n abg_o in_o double_a proportion_n that_o the_o side_n bc_o be_v to_o the_o side_n ef._n but_o the_o triangle_n abg_o be_v equal_a to_o the_o triangle_n def_n wherefore_o also_o the_o triangle_n abc_n be_v unto_o the_o triangle_n def_n in_o double_a proportion_n that_o the_o side_n bc_o be_v to_o the_o side_n ef._n wherefore_o like_a triangle_n be_v one_o to_o the_o other_o in_o double_a proportion_n that_o the_o side_n of_o like_a proportion_n be_v which_o be_v require_v to_o be_v prove_v corollary_n corollary_n hereby_o it_o be_v manifest_a that_o if_o there_o be_v three_o right_a line_n in_o proportion_n as_o the_o first_o be_v to_o the_o three_o so_o be_v the_o triangle_n describe_v upon_o the_o first_o unto_o the_o triangle_n describe_v upon_o the_o second_o so_o that_o the_o say_v triangle_n be_v like_a and_o in_o like_a â—ort_n describe_v for_o it_o have_v be_v prove_v that_o as_o the_o line_n cb_o be_v to_o the_o line_n bg_o so_o be_v the_o triangle_n abc_n to_o the_o triangle_n def_n which_o be_v require_v to_o be_v demonstrate_v the_o 14._o theorem_a the_o 20._o proposition_n like_a poligonon_n figure_n be_v divide_v into_o like_a triangle_n and_o equal_a in_o number_n and_o of_o like_a proportion_n to_o the_o whole_a and_o the_o one_o poligonon_n figure_n be_v to_o the_o other_o poligonon_n figure_n in_o double_a proportion_n that_o one_o of_o the_o side_n of_o like_a proportion_n be_v to_o one_o of_o the_o side_n of_o like_a proportion_n svppose_v that_o the_o like_a poligonon_n figure_n be_v abcde_o &_o fghkl_o have_v the_o angle_n at_o the_o point_n f_o equal_a to_o the_o angle_n at_o the_o point_n a_o and_o the_o angle_n at_o the_o point_n â—_o equal_a to_o the_o angle_n at_o the_o point_n b_o and_o the_o angle_n at_o the_o point_n h_o equal_a to_o you_o angle_v at_o the_o point_n c_o and_o so_o of_o the_o rest_n and_o moreover_o as_o the_o side_n ab_fw-la be_v to_o the_o side_n bc_o so_o let_v the_o side_n fg_o be_v to_o the_o side_n gh_o and_o as_o the_o side_n bc_o be_v to_o the_o side_n cd_o so_o let_v the_o side_n gh_o be_v to_o the_o side_n hk_o and_o so_o forth_o and_o let_v the_o side_n ab_fw-la &_o fg_o be_v side_n of_o like_a proportion_n then_o i_o say_v first_o theorem_a that_o these_o poligonon_n figure_v abcde_o &_o fghkl_o be_v divide_v into_o like_a triangle_n and_o equal_a in_o number_n for_o draw_v these_o right_a line_n ac_fw-la ad_fw-la fh_o &_o fkâ—_n and_o forasmuch_o as_o by_o supposition_n that_o be_v by_o reason_n the_o figure_n abcde_o be_v like_a unto_o the_o figure_n fghkl_o the_o angle_n b_o be_v equal_a unto_o the_o angle_n g_o and_o as_z the_o side_n ab_fw-la be_v to_o the_o side_n bc_o so_o be_v the_o side_n fg_o to_o the_o side_n gh_o it_o follow_v that_o the_o two_o triangle_n abc_n and_o fgh_n have_v one_o angle_n of_o the_o one_o equal_a to_o one_o angle_n of_o the_o other_o and_o have_v also_o the_o side_n about_o the_o equal_a angle_n proportional_a wherefore_o by_o the_o 6._o of_o the_o six_o the_o triangle_n abc_n be_v equiangle_n unto_o the_o triangle_n fgh_n and_o those_o angle_n in_o they_o be_v equal_a under_o which_o be_v subtend_v side_n of_o like_a proportion_n namely_o the_o angle_n bac_n be_v equal_a to_o the_o angle_n gfh_v and_o the_o angle_n bca_o to_o the_o angle_n ghf_n wherefore_o by_o the_o 4._o of_o the_o six_o the_o side_n which_o be_v about_o the_o equal_a angle_n be_v proportional_a and_o the_o side_n which_o be_v subtend_v under_o the_o equal_a angle_n be_v of_o like_a proportion_n wherefore_o as_o ac_fw-la be_v to_o bc_o so_o be_v fh_o to_o gh_o but_o by_o supposition_n as_o bc_o be_v to_o cd_o so_o be_v gh_o to_o hk_v wherefore_o of_o equality_n by_o the_o 22._o of_o the_o five_o as_o ac_fw-la be_v to_o cd_o so_o be_v fh_o to_o hk_v and_o forasmuch_o as_o by_o supposition_n the_o whole_a angle_n bcd_o be_v equal_a to_o the_o whole_a angle_n ghk_o and_o it_o be_v prove_v that_o the_o angle_n bca_o be_v eâ—â—all_a to_o the_o angle_n ghf_n therefore_o the_o angle_n remain_v acd_o be_v equal_a to_o the_o angle_n remain_v fhk_n by_o the_o 3._o common_a sentence_n wherefore_o the_o â—riaâ—glâ—s_v acd_o and_o fhk_v have_v again_o one_o angle_n of_o the_o one_o equal_a to_o one_o angle_n of_o the_o other_o and_o the_o side_n which_o be_v about_o the_o equal_a side_n be_v proportional_a wherefore_o by_o the_o â—ame_n six_o of_o this_o book_n the_o triangle_n acd_o &_o fhk_v be_v equiangle_n and_o by_o the_o 4._o of_o this_o book_n the_o side_n which_o be_v about_o the_o equal_a angle_n 〈◊〉_d proportional_a and_o by_o the_o same_o reason_n may_v we_o prove_v that_o the_o triangle_n adâ—_n be_v equiangle_n unto_o the_o triangle_n fkl_n and_o that_o the_o side_n which_o be_v about_o the_o equal_a angle_n be_v proportional_a wherefore_o the_o triangle_n abc_n be_v like_o to_o the_o triangle_n fgh_n and_o the_o triangle_n acd_o to_o the_o triangle_n fhk_n and_o also_o the_o triangle_n ade_n to_o the_o triangle_n fkl_n by_o the_o first_o definition_n of_o this_o six_o book_n wherefore_o the_o poligonon_n figure_v give_v abcde_o and_o fghkl_o be_v divide_v into_o triangle_n like_a and_o equal_a in_o number_n demonstrate_v i_o say_v moreover_o that_o the_o triangle_n be_v the_o one_o to_o the_o other_o and_o to_o the_o whole_a poligonon_n figure_v proportional_a that_o be_v as_o the_o triangle_n abc_n be_v to_o the_o triangle_n fgh_o so_o be_v the_o triangle_n acd_o to_o the_o triangle_n fhk_n and_o the_o triangle_n ade_n to_o the_o triangle_n fkl_n and_o as_o the_o triangle_n abc_n be_v to_o the_o triangle_n fgh_o so_o be_v the_o poligonon_n figure_n abcde_o to_o the_o poligonon_n figure_n fghkl_o for_o forasmuch_o as_o the_o triangle_n abc_n be_v like_a to_o the_o triangle_n fgh_o and_o ac_fw-la and_o fh_o be_v side_n of_o like_a proportion_n therefore_o the_o proportion_n of_o the_o triangle_n abc_n to_o the_o triangle_n fgh_n be_v double_a to_o the_o proportion_n of_o the_o side_n ac_fw-la to_o the_o 〈◊〉_d fh_o by_o the_o former_a proposition_n and_o therefore_o also_o the_o proportion_n of_o the_o triangle_n acd_o to_o the_o triangle_n fkh_n be_v double_a to_o the_o proportion_n that_o the_o same_o side_n ac_fw-la have_v to_o the_o side_n fh_o wherefore_o by_o the_o 11._o of_o the_o five_o as_o the_o triangle_n abc_n be_v to_o the_o triangle_n fgh_o so_o be_v the_o triangle_n acd_o to_o the_o triangle_n fhk_n again_o forasmuch_o as_o the_o triangle_n acd_o be_v like_a to_o the_o triangle_n fhk_n and_o the_o side_n ad_fw-la &_o fk_o be_v of_o like_a proportion_n therefore_o the_o proportion_n of_o the_o triangle_n acd_o to_o the_o triangle_n fhk_n be_v double_a to_o the_o proportion_n of_o the_o side_n ad_fw-la to_o the_o side_n fk_o by_o the_o foresay_a 19_o of_o the_o six_o and_o by_o the_o same_o reason_n the_o proportion_n of_o the_o triangle_n ade_n to_o the_o triangle_n fkl_n be_v double_a to_o the_o proportion_n of_o the_o same_o side_n ad_fw-la to_o the_o side_n fk_o wherefore_o by_o the_o 11._o of_o the_o five_o as_o the_o triangle_n acd_o be_v to_o the_o triangle_n fhk_n so_o be_v the_o triangle_n ade_n to_o the_o triangle_n fkl_n but_o
eglantine_n the_o side_n which_o include_v the_o equal_a angle_n be_v proportional_a wherefore_o the_o parallelogram_n abcd_o be_v by_o the_o first_o definition_n of_o the_o six_o like_v unto_o the_o parallelogram_n eglantine_n and_o by_o the_o same_o reason_n also_o the_o parallelogram_n abcd_o be_v like_a to_o the_o parallelogram_n kh_o abcd_o wherefore_o either_o of_o these_o parallelogram_n eglantine_n and_o kh_o be_v like_a unto_o the_o parallelogram_n abcd._n but_o rectiline_a figure_n which_o be_v like_a to_o one_o and_o the_o same_o rectiline_a figure_n be_v also_o by_o the_o 21._o of_o the_o six_o like_o the_o one_o to_o the_o other_o wherefore_o the_o parallelogram_n eglantine_n be_v like_a to_o the_o parallelogram_n hk_o other_o wherefore_o in_o every_o parallelogram_n the_o parallelogram_n about_o the_o dimecient_a be_v like_a unto_o the_o whole_a and_o also_o like_o 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 other_o more_o brief_a demonstration_n after_o flussates_n suppose_v that_o there_o be_v a_o parallelogram_n abcd_o who_o dimetient_a let_v bâ—_z aâ—_z flussates_n about_o which_o let_v consist_v these_o parallelogram_n eke_o and_o ti_o have_v the_o angle_n at_o the_o point_n â—_o and_o 〈…〉_o with_o the_o whole_a parallelogram_n abcd._n then_o i_o say_v that_o those_o parallelogram_n eke_o and_o ti_fw-mi be_v like_a to_o the_o whole_a parallelogram_n db_o and_o also_o alâ—_z like_o the_o one_o to_o the_o other_o for_o forasmuch_o as_o bd_o eke_o and_o ti_fw-mi be_v parallelogram_n therefore_o the_o right_a line_n azg_v fall_v upon_o these_o parallel_a line_n aeb_fw-mi kzt_n and_o di_z g_z or_o upon_o these_o parallel_a line_n akd_v ezi_n and_o btg_n make_v these_o angle_n equal_a the_o one_o to_o the_o other_o namely_o the_o angle_n eaz_n to_o the_o angle_n kza_n &_o the_o angle_n eza_n to_o the_o angle_n kaz_n and_o the_o angle_n tzg_v to_o the_o angle_n zgi_n and_o the_o angle_n tgz_n to_o the_o angle_n izg_v and_o the_o angle_n bag_n to_o the_o angle_n agd_v and_o final_o the_o angle_n bga_n to_o the_o angle_n dag_n wherefore_o by_o the_o first_o corollary_n of_o the_o 32._o of_o the_o first_o and_o by_o the_o 34._o of_o the_o first_o the_o angle_n remain_v be_v equal_v the_o one_o to_o the_o other_o namely_o the_o angle_n b_o to_o the_o angle_n d_o and_o the_o angle_n e_o to_o the_o angle_n king_n and_o the_o angle_n t_o to_o the_o angle_n i_o wherefore_o these_o triangle_n be_v equiangle_n and_o therefore_o like_o the_o one_o to_o the_o other_o namely_o the_o triangle_n abg_o to_o the_o triangle_n gda_n and_o the_o triangle_n aez_n to_o the_o triangle_n zka_n &_o the_o triangle_n ztg_n to_o the_o triangle_n giz._n wherefore_o as_o the_o side_n ab_fw-la be_v to_o the_o side_n bg_o so_o be_v the_o side_n ae_n to_o the_o side_n ez_fw-fr and_o the_o side_n zt_n to_o the_o side_n tg_n wherefore_o the_o parallelogram_n contain_v under_o those_o right_a line_n namely_o the_o parallelogram_n abgd_v eke_o &_o ti_fw-mi be_v like_o the_o one_o to_o the_o other_o by_o the_o first_o definition_n of_o this_o book_n wherefore_o in_o every_o parallelogram_n the_o parallelogram_n etc_n etc_n as_o before_o which_o be_v require_v to_o be_v demonstrate_v ¶_o a_o problem_n add_v by_o pelitarius_n two_o equiangle_n parallelogrammes_n be_v give_v so_o that_o they_o be_v not_o like_a to_o cut_v of_o from_o one_o of_o they_o a_o parallelogram_n like_a unto_o the_o other_o pelitarius_n suppose_v that_o the_o two_o equiangle_n parallelogram_n be_v abcd_o and_o cefg_n which_o let_v not_o be_v like_o the_o one_o to_o the_o other_o it_o be_v require_v from_o the_o parallelogram_n abcd_o to_o cut_v of_o a_o parallelogram_n like_a unto_o the_o parallelogram_n cefg_n let_v the_o angle_n c_o of_o the_o one_o be_v equal_a to_o the_o angle_n c_o of_o the_o other_o and_o let_v the_o two_o parallelogram_n be_v so_o 〈◊〉_d that_o the_o line_n bc_o &_o cg_o may_v make_v both_o one_o right_a line_n namely_o bg_o wherefore_o also_o the_o right_a line_n dc_o and_o ce_fw-fr shall_v both_o make_v one_o right_a line_n namely_o de._n and_o draw_v a_o line_n from_o the_o point_n f_o to_o the_o point_n c_o and_o produce_v the_o line_n fc_o till_o it_o concur_v with_o the_o line_n ad_fw-la in_o the_o point_n h._n and_o draw_v the_o line_n hk_o parallel_v to_o the_o line_n cd_o by_o the_o 31._o of_o the_o first_o then_o i_o say_v that_o from_o the_o parallelogram_n ac_fw-la be_v cut_v of_o the_o parallelogram_n cdhk_n like_v unto_o the_o parallelogram_n eglantine_n which_o thing_n be_v manifest_a by_o this_o 24._o proposition_n for_o that_o both_o the_o say_v parallelogram_n be_v describe_v about_o one_o &_o the_o self_n same_o dimetient_fw-la and_o to_o the_o end_n it_o may_v the_o more_o plain_o be_v see_v i_o have_v make_v complete_a the_o parallelogram_n abgl_n ¶_o a_o other_o problem_n add_v by_o pelitarius_n between_o two_o rectiline_a superficiece_n to_o find_v out_o a_o mean_a superficies_n proportional_a pelitarius_n suppose_v that_o the_o two_o superficiece_n be_v a_o and_o b_o between_o which_o it_o be_v require_v to_o place_v a_o mean_a superficies_n proportional_a reduce_v the_o say_v two_o rectiline_a figure_n a_o and_o b_o unto_o two_o like_a parallelogram_n by_o the_o 18._o of_o this_o book_n or_o if_o you_o think_v good_a reduce_v either_o of_o they_o to_o a_o square_a by_o the_o last_o of_o the_o second_o and_o let_v the_o say_v two_o parallelogram_n like_o the_o one_o to_o the_o other_o and_o equal_a to_o the_o superficiece_n a_o and_o b_o be_v cdef_n and_o fghk_n and_o let_v the_o angle_n fletcher_n in_o either_o of_o they_o be_v equal_a which_o two_o angle_n let_v be_v place_v in_o such_o sort_n that_o the_o two_o parallelogram_n ed_z and_o hg_o may_v be_v about_o one_o and_o the_o self_n same_o dimetient_fw-la ck_o which_o be_v do_v by_o put_v the_o right_a line_n of_o and_o fg_o in_o such_o sort_n that_o they_o both_o make_v one_o right_a line_n namely_o eglantine_n and_o make_v complete_a the_o parallelogram_n clk_n m._n then_o i_o say_v that_o either_o of_o the_o supplement_n fl_fw-mi &_o fm_o be_v a_o mean_a proportional_a between_o the_o superficiece_n cf_o &_o fk_o that_o be_v between_o the_o superficiece_n a_o and_o b_o namely_o as_o the_o superficies_n hg_o be_v to_o the_o superficies_n fl_fw-mi so_o be_v the_o same_o superficies_n fl_fw-mi to_o the_o superficies_n ed._n for_o by_o this_o 24._o proposition_n the_o line_n hf_o be_v to_o the_o line_n fd_o as_o the_o line_n gf_o be_v to_o the_o line_n fe_o but_o by_o the_o first_o of_o this_o book_n as_o the_o line_n hf_o be_v to_o the_o line_n fd_o so_o be_v the_o superficies_n hg_o to_o the_o superficies_n fl_fw-mi and_o as_o the_o line_n gf_o be_v to_o the_o line_n fe_o so_o also_o by_o the_o same_o be_v the_o superficies_n fl_fw-mi to_o the_o superficies_n ed._n wherefore_o by_o the_o 11._o of_o the_o five_o as_o the_o superficies_n hg_o be_v to_o the_o superficies_n fl_fw-mi so_o be_v the_o same_o superficies_n fl_fw-mi to_o the_o superficies_n ed_z which_o be_v require_v to_o be_v do_v the_o 7._o problem_n the_o 25._o proposition_n unto_o a_o rectiline_a figure_n give_v to_o describe_v a_o other_o figure_n like_o which_o shall_v also_o be_v equal_a unto_o a_o other_o rectiline_a figure_n give_v svppose_v that_o the_o rectiline_a figure_n give_v whereunto_o be_v require_v a_o other_o to_o be_v make_v like_o be_v abc_n and_o let_v the_o other_o rectiline_a figure_n whereunto_o the_o same_o be_v require_v to_o be_v make_v equal_a be_v d._n now_o it_o be_v require_v to_o describe_v a_o rectiline_a figure_n like_o unto_o the_o figure_n abc_n and_o equal_a unto_o the_o figure_n d._n construction_n upon_o the_o line_n bc_o describe_v by_o the_o 44._o of_o the_o first_o a_o parallelogram_n be_v equal_a unto_o the_o triangle_n abc_n and_o by_o the_o same_o upon_o the_o line_n ce_fw-fr describe_v the_o parallelogram_n c_o m_o equal_a unto_o the_o rectiline_a figure_n d_o and_o in_o the_o say_a parallelogram_n let_v the_o angle_n fce_n be_v equal_a unto_o the_o angle_n cbl_n and_o forasmuch_o as_o the_o angle_n fce_n be_v by_o construction_n equal_a to_o the_o angle_n cbl_n demonstration_n add_v the_o angle_n be_v common_a to_o they_o both_o wherefore_o the_o angle_n lbc_n and_o be_v be_v equal_a unto_o the_o angle_n be_v and_o ecf_n but_o the_o angle_n lbc_n and_o be_v be_v equal_a to_o two_o right_a angle_n by_o the_o 29._o of_o the_o first_o wherefore_o also_o the_o angle_n be_v and_o ecf_n be_v equal_a to_o two_o right_a angle_n wherefore_o the_o line_n bc_o and_o cf_o by_o the_o 14._o of_o the_o first_o make_v both_o one_o right_a line_n namely_o bf_o and_o in_o like_a sort_n do_v the_o line_n le_fw-fr and_o em_n make_v both_o one_o right_a line_n namely_o lm_o then_o by_o the_o
figure_n comprehend_v under_o the_o line_n gb_o and_o ge_z be_v equal_a to_o the_o square_n make_v of_o the_o line_n agnostus_n by_o the_o 36._o of_o the_o three_o it_o follow_v that_o the_o touch_n line_n ga_o be_v a_o mean_a proportional_a between_o the_o extreme_n gb_o and_o ge_z by_o the_o second_o part_n of_o the_o 17._o of_o the_o six_o for_o that_o by_o that_o proposition_n the_o line_n gb_o ga_o and_o ge_z be_v proportional_a and_o by_o the_o same_o reason_n may_v it_o be_v prove_v that_o the_o line_n ga_o be_v a_o mean_a proportional_a between_o the_o line_n gd_o and_o gc_o and_o so_o of_o all_o other_o if_o therefore_o from_o a_o point_n gevenâ—_n &_o câ—_n which_o be_v require_v to_o be_v demonstrate_v the_o end_n of_o the_o six_o book_n of_o euclides_n element_n ¶_o the_o seven_o book_n of_o euclides_n element_n hitherto_o in_o the_o six_o book_n before_o have_v euclid_n pass_v through_o and_o entreat_v of_o the_o element_n of_o geometry_n without_o the_o aid_n and_o succour_v of_o number_n but_o the_o matter_n which_o remain_v to_o be_v teach_v and_o to_o be_v speak_v of_o in_o these_o his_o geometrical_a book_n which_o follow_v as_o in_o the_o ten_o eleven_o and_o so_o forth_o he_o can_v by_o no_o mean_n full_o and_o clear_o make_v plain_a &_o demonstrate_v without_o the_o help_n and_o aid_n of_o number_n in_o the_o ten_o be_v entreat_v of_o line_n irrational_a and_o uncertain_a and_o that_o of_o many_o &_o sundry_a kind_n and_o in_o the_o eleven_o &_o the_o other_o follow_v he_o teach_v the_o nature_n of_o body_n and_o compare_v their_o side_n and_o line_n together_o all_o which_o for_o the_o most_o part_n be_v also_o irrational_a and_o as_o rational_a quantity_n and_o the_o comparison_n and_o proportion_n of_o they_o can_v be_v know_v nor_o exact_o try_v but_o by_o the_o mean_a of_o number_n in_o which_o they_o be_v first_o see_v and_o perceive_v even_o so_o likewise_o can_v irrational_a quantity_n be_v know_v and_o find_v out_o without_o number_n as_o straightness_n be_v the_o trial_n of_o crookedness_n and_o inequality_n be_v try_v by_o equality_n so_o be_v quantity_n irrational_a perceive_v and_o know_v by_o quantity_n rational_a which_o be_v â—irst_n and_o chief_o find_v among_o number_n wherefore_o in_o these_o three_o book_n follow_v be_v as_o it_o be_v in_o the_o midst_n of_o his_o element_n he_o be_v compel_v of_o necessity_n to_o entreat_v of_o number_n number_n although_o not_o so_o full_o as_o the_o nature_n of_o number_n require_v yet_o so_o much_o as_o shall_v seem_v to_o be_v fit_a and_o sufficient_o to_o serve_v for_o his_o purpose_n whereby_o be_v see_v the_o necessity_n that_o the_o one_o art_n namely_o geometry_n have_v of_o the_o other_o namely_o of_o arithmetic_n and_o also_o of_o what_o excellency_n and_o worthiness_n arithmetic_n be_v above_o geometry_n geometry_n in_o that_o geometry_n borrow_v of_o it_o principle_n aid_n and_o succour_n and_o be_v as_o it_o be_v maim_v with_o out_o it_o whereas_o arithmetic_n be_v of_o itself_o sufficient_a and_o nead_v not_o at_o all_o any_o aid_n of_o geometry_n but_o be_v absolute_a and_o perfect_a in_o itself_o and_o may_v well_o be_v teach_v and_o attain_a unto_o without_o it_o again_o the_o matter_n or_o subject_n where_o about_o geometry_n be_v occupy_v which_o be_v line_n figure_n and_o body_n be_v such_o as_o offer_v themselves_o to_o the_o sense_n as_o triangle_n square_n circle_n cube_n and_o other_o be_v see_v &_o judge_v to_o be_v such_o as_o they_o be_v by_o the_o sight_n but_o number_n which_o be_v the_o subject_n and_o matter_n of_o arithmetic_n fall_v under_o no_o sense_n nor_o be_v represent_v by_o any_o shape_n form_n or_o figure_n and_o therefore_o can_v be_v judge_v by_o any_o sense_n but_o only_o by_o consideration_n of_o the_o mind_n and_o understanding_n now_o thing_n sensible_a be_v far_o under_o in_o degree_n then_o be_v thing_n intellectual_a sensible_a and_o be_v of_o nature_n much_o more_o gross_a than_o they_o wherefore_o number_n as_o be_v only_o intellectual_a be_v more_o pure_a more_o immaterial_a and_o more_o subtle_a far_o then_o be_v magnitude_n and_o extend_v itself_o far_a for_o arithmetic_n not_o only_o aid_v geometry_n but_o minister_v principle_n and_o ground_n to_o many_o other_o nay_o rather_o to_o all_o other_o science_n and_o art_n science_n as_o to_o music_n astronomy_n natural_a philosophy_n perspective_n with_o other_o what_o other_o thing_n be_v in_o music_n entreat_v of_o than_o number_n contract_v to_o sound_n and_o voice_n in_o astronomy_n who_o without_o the_o knowledge_n of_o number_n can_v do_v any_o thing_n either_o in_o search_v out_o of_o the_o motion_n of_o the_o heaven_n and_o their_o course_n either_o in_o judge_v and_o foreshow_v the_o effect_n of_o they_o in_o natural_a philosophy_n it_o be_v of_o no_o small_a force_n the_o wise_a and_o best_a learned_a philosopher_n that_o have_v be_v as_o pythagoras_n timeus_n plato_n and_o their_o follower_n find_v out_o 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timaus_n &_o also_o plato_n in_o his_o timâ—â—_n follow_v he_o show_v how_o the_o soul_n be_v compose_v of_o harmonical_a number_n and_o consonantes_fw-la of_o music_n number_n compass_v all_o thing_n and_o be_v after_o these_o man_n the_o be_v and_o very_a essence_n of_o all_o thing_n and_o minister_v aid_n and_o help_v as_o to_o all_o other_o knowledge_n so_o also_o no_o small_a to_o geometry_n which_o thing_n cause_v euclidâ—_n in_o the_o midst_n of_o his_o book_n of_o geometry_n to_o insert_v and_o place_n these_o three_o book_n of_o arithmetic_n as_o without_o the_o aid_n of_o which_o he_o can_v not_o well_o pass_v any_o father_n in_o this_o seven_o book_n he_o â—irst_n place_v the_o general_a principle_n and_o first_o ground_n of_o arithmetic_n and_o set_v the_o definition_n of_o number_n or_o multitude_n and_o the_o kind_n thereof_o as_o in_o the_o first_o book_n he_o do_v of_o magnitude_n and_o the_o kind_n and_o part_n thereof_o book_n after_o that_o he_o entreat_v of_o 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book_n for_o the_o like_a reason_n and_o cause_n be_v a_o point_n first_o define_v unity_n say_v euclid_n be_v that_o whereby_o every_o thing_n be_v say_v to_o be_v one_o that_o be_v unity_n be_v that_o whereby_o every_o thing_n be_v divide_v and_o separate_v from_o a_o other_o and_o remain_v on_o in_o itself_o pure_a and_o distinct_a from_o all_o other_o otherwise_o be_v not_o this_o unity_n thing_n whereby_o all_o thing_n be_v sejoin_v the_o on_o from_o the_o other_o all_o thing_n shall_v suffer_v mixtion_n and_o be_v in_o confusion_n and_o where_o confusion_n be_v there_o be_v no_o order_n nor_o any_o thing_n can_v be_v exact_o know_v either_o what_o it_o be_v or_o what_o be_v the_o nature_n and_o what_o be_v the_o property_n thereof_o unity_n therefore_o be_v that_o which_o make_v every_o thing_n to_o be_v that_o which_o it_o be_v boetius_fw-la say_v very_o apt_o
uno_fw-la vnum_fw-la quodque_fw-la idea_n est_fw-la quia_fw-la unum_fw-la numero_fw-la est_fw-la that_o be_v every_o thing_n therefore_o be_v that_o be_v therefore_o have_v his_o be_v in_o nature_n and_o be_v that_o it_o be_v for_o that_o it_o be_v on_o in_o number_n according_a whereunto_o jordane_n in_o that_o most_o excellent_a and_o absolute_a work_n of_o arithmetic_n which_o he_o write_v define_v unity_n after_o this_o manner_n vnitas_fw-la est_fw-la res_fw-la per_fw-la se_fw-la discretio_fw-la that_o be_v unity_n be_v proper_o and_o of_o itself_o the_o difference_n of_o any_o thing_n that_o be_v unity_n unity_n be_v that_o whereby_o every_o thing_n do_v proper_o and_o essential_o differ_v and_o be_v a_o other_o thing_n from_o all_o other_o certain_o a_o very_a apt_a definition_n and_o it_o make_v plain_a the_o definition_n here_o set_v of_o euclid_n definition_n 2_o number_n be_v a_o multitude_n compose_v of_o unity_n as_o the_o number_n of_o three_o be_v a_o multitude_n compose_v and_o make_v of_o three_o unity_n likewise_o the_o number_n of_o five_o be_v nothing_o else_o but_o the_o composition_n &_o put_v together_o of_o five_o unity_n although_o as_o be_v before_o say_v between_o a_o point_n in_o magnitude_n and_o unity_n in_o multitude_n there_o be_v great_a agreement_n and_o many_o thiâ—â—â—â—_n be_v common_a to_o they_o both_o for_o as_o a_o point_n be_v the_o begin_n of_o magnitude_n so_o be_v unity_n the_o beginning_n of_o number_n and_o as_o a_o point_n in_o magnitude_n be_v indivisible_a so_o be_v also_o unity_n in_o number_n indivisible_a yet_o in_o this_o they_o differ_v and_o disagree_v unity_n there_o be_v no_o line_n or_o magnitude_n make_v of_o point_n as_o of_o his_o part_n so_o that_o although_o a_o point_n be_v the_o beginning_n of_o a_o line_n yet_o be_v it_o no_o part_n thereof_o but_o unity_n as_o it_o be_v the_o begin_n of_o number_n so_o be_v it_o also_o a_o part_n thereof_o which_o be_v somewhat_o more_o manifest_o set_v of_o 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number_n arise_v and_o spring_v all_o kind_n and_o variety_n of_o proportion_n as_o have_v before_o be_v declare_v in_o the_o explanation_n of_o the_o principle_n of_o the_o five_o book_n so_o that_o of_o that_o matter_n it_o be_v needless_a any_o more_o to_o be_v say_v in_o this_o place_n thus_o much_o of_o this_o for_o the_o declaration_n of_o the_o thing_n follow_v 3_o a_o part_n be_v a_o less_o number_n in_o comparison_n to_o the_o great_a when_o the_o less_o measure_v the_o great_a definition_n as_o the_o number_n 3_o compare_v to_o the_o number_n 12._o be_v a_o part_n for_o 3_o be_v a_o less_o number_n than_o be_v 12._o and_o moreover_o it_o measure_v 12_o the_o great_a number_n for_o 3_o take_v or_o add_v to_o itself_o certain_a time_n namely_o 4_o time_n make_v 12._o for_o 3_o four_o time_n be_v 12._o likewise_o be_v â—_o a_o part_n of_o 8_o 2_o be_n less_o than_o 8_o and_o take_v 4_o time_n it_o make_v 8._o for_o the_o better_a understanding_n of_o this_o definition_n and_o 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number_n even_o even_o be_v that_o which_o may_v be_v divide_v into_o two_o even_a part_n and_o that_o part_n again_o into_o two_o even_a part_n and_o so_o continual_o devide_v without_o stay_n 〈◊〉_d come_v to_o unity_n as_o by_o exampleâ—_n 64._o may_v be_v divide_v into_o 31_o and._n 32._o and_o either_o of_o these_o part_n may_v be_v divide_v into_o two_o even_a part_n for_o 32_o may_v be_v divide_v into_o 16_o and_o 16._o again_o 16_o may_v be_v divide_v into_o 8_o and_o 8_o which_o be_v even_o part_n and_o 8_o into_o 4_o and_o 4._o again_o 4_o into_o â—_o and_o â—_o and_o last_o of_o all_o may_v â—_o be_v divide_v into_o one_o and_o one_o 9_o a_o number_n even_o odd_a call_v in_o latin_a pariter_fw-la impar_fw-la be_v that_o which_o a_o even_a number_n measure_v by_o a_o odd_a number_n definition_n as_o the_o number_n 6_o which_o 2_o a_o even_a number_n measure_v by_o 3_o a_o odd_a number_n three_o time_n 2_o be_v 6._o likewise_o 10._o which_o 2._o a_o even_a number_n measure_v by_o 5_o a_o odd_a number_n in_o this_o definition_n also_o be_v find_v by_o all_o the_o expositor_n of_o euclid_n the_o same_o want_n that_o be_v find_v in_o the_o definition_n next_o before_o and_o for_o that_o it_o extend_v itself_o to_o large_a for_o there_o be_v infinite_a number_n which_o even_a number_n do_v measure_n by_o odd_a number_n which_o yet_o after_o their_o mind_n be_v not_o even_o odd_a number_n as_o for_o example_n 12._o for_o 4_o a_o even_a number_n measure_v 12_o by_o â—_o a_o odd_a numberâ—_n three_o time_n 4_o be_v 12._o yet_o be_v not_o 12_o as_z they_o think_v a_o even_o odd_a number_n wherefore_o campane_n amend_v it_o after_o his_o think_n campane_n by_o add_v of_o this_o word_n all_o as_o he_o do_v in_o the_o first_o and_o define_v it_o after_o this_o manner_n a_o number_n even_o odd_a be_v when_o all_o the_o even_a number_n which_o measure_v it_o do_v measure_n it_o by_o uneven_a time_n that_o be_v by_o a_o odd_a number_n as_o 10._o be_v a_o number_n even_o 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same_o part_n here_o he_o define_v which_o number_n be_v call_v proportional_a that_o be_v what_o number_n have_v one_o and_o the_o self_n same_o proportion_n for_o example_n 6._o to_o 3_o and_o 4._o to_o 2_o be_v number_n proportional_a and_o have_v one_o and_o thâ—_z self_n same_o proportion_n for_o 6._o the_o first_o be_v to_o 3._o the_o second_o equemultiplex_n as_o 4._o the_o three_o be_v to_o 2._o the_o four_o â—_o be_v double_a to_o 3_o and_o so_o be_v 4._o double_a to_o 2._o likewise_o these_o four_o number_n be_v in_o like_a proportion_n 3_o 9.4.1â—_n for_o what_o part_n 3._o be_v of_o 9_o such_o part_n be_v 4._o of_o 12_o â—3_o of_o 9_o be_v a_o three_o part_n so_o be_v also_o 4._o of_o 12._o a_o three_o part_n so_o be_v these_o four_o number_n also_o in_o proportion_n â—_o 5_o 4._o 10_o what_o part_n 2._o be_v of_o 5_o such_o part_n be_v 4._o of_o 10_o 2_o of_o 5_o be_v two_o â—ift_a part_n likewise_o 4._o of_o 10_o be_v two_o five_o part_n moreover_o these_o number_n 8.6_o 1â—_o 9_o be_v in_o proportion_n for_o what_o and_o how_o many_o part_n 8._o be_v of_o 6_o such_o &_o so_o many_o part_n be_v 12._o of_o 9_o 8._o of_o 6_o be_v four_o three_o part_n for_o one_o three_o part_n of_o 6._o be_v 2_o which_o take_v four_o time_n make_v 8_o so_o 12._o of_o 9_o be_v also_o four_o thyrd_o part_n for_o one_o three_o part_n of_o 9_o be_v 3_o which_o take_v four_o time_n make_v 1â—_o and_o so_o conceive_v you_o of_o all_o other_o proportional_a number_n in_o the_o 〈◊〉_d definition_n of_o the_o v_o book_n euclid_n give_v a_o â—arre_n other_o definition_n of_o magnitude_n proportional_a and_o much_o unlike_a to_o this_o which_o he_o here_o give_v of_o number_n proportional_a number_n the_o reason_n be_v as_o there_o also_o be_v partly_o note_v for_o that_o there_o he_o give_v a_o definition_n common_a to_o all_o quantity_n discrete_a and_o continual_a rational_a and_o irrational_a and_o therefore_o be_v constrain_v to_o geve_v the_o definition_n by_o the_o excess_n equality_n or_o want_n of_o their_o equemultiplices_fw-la and_o that_o general_o only_a for_o that_o irrational_a quantity_n have_v no_o certain_a part_n or_o common_a measure_n to_o be_v measure_v by_o or_o know_v neither_o can_v they_o be_v express_v by_o any_o certain_a number_n but_o here_o in_o this_o place_n because_o in_o number_n there_o be_v no_o irrational_a quantity_n but_o all_o be_v certain_o know_v so_o that_o both_o they_o and_o the_o proportion_n between_o they_o may_v be_v express_v by_o number_n certain_a and_o know_v by_o reason_n of_o their_o part_n certain_a and_o for_o that_o they_o have_v some_o common_a measure_n to_o measure_v they_o at_o the_o least_o unity_n which_o be_v a_o common_a measure_n to_o all_o number_n he_o give_v here_o this_o definition_n of_o proportional_a number_n by_o that_o the_o one_o be_v like_o equemultiplex_n to_o the_o other_o or_o the_o same_o part_n or_o the_o same_o part_n which_o definition_n be_v much_o easy_a than_o be_v the_o other_o and_o be_v not_o so_o large_a as_o be_v the_o other_o neither_o extend_v itself_o general_o to_o all_o kind_n of_o quantity_n rational_a and_o irrational_a but_o contain_v itself_o within_o the_o limit_n and_o bond_n of_o rational_a quantity_n and_o number_n 22_o like_o plain_a number_n and_o like_o solid_a number_n be_v such_o definition_n which_o have_v their_o side_n proportional_a before_o he_o show_v that_o a_o plain_a number_n have_v two_o side_n and_o a_o solid_a number_n three_o side_n now_o he_o show_v by_o this_o definition_n which_o plain_a number_n and_o which_o solid_a number_n be_v like_o the_o one_o to_o the_o other_o the_o likeness_n of_o which_o number_n depend_v altogether_o of_o the_o proportion_n of_o the_o side_n of_o these_o number_n so_o that_o if_o the_o two_o side_n of_o one_o plain_a number_n have_v the_o same_o proportion_n the_o one_o to_o the_o other_o that_o the_o two_o side_n of_o the_o other_o plain_a number_n have_v the_o one_o to_o the_o other_o then_o be_v such_o two_o plain_a number_n like_o for_o a_o example_n 6_o and_o 24_o be_v two_o plain_a number_n the_o side_n of_o 6_o be_v 2_o and_o 3_o two_o time_n 3_o make_v 6_o the_o side_n of_o 24_o be_v 4_o and_o 6_o four_o time_n 6_o make_v 24._o again_o the_o same_o proportion_n that_o be_v between_o 3_o and_o 2_o the_o side_n of_o 6._o be_v also_o between_o 6_o and_o 4_o the_o side_n of_o 24._o wherefore_o 24_o and_o 6_o be_v two_o like_o plain_a and_o superficial_a number_n and_o so_o of_o other_o plain_a number_n after_o the_o same_o manner_n be_v it_o in_o solid_a number_n if_o three_o side_n of_o the_o one_o be_v in_o like_a proportion_n together_o as_o be_v the_o three_o side_n of_o the_o other_o then_o be_v the_o one_o solid_a number_n like_o to_o the_o other_o as_o 24_o and_o 192_o be_v solid_a number_n the_o side_n of_o 24._o be_v 2._o 3._o and_o 4_o two_o time_n there_o take_v 4_o time_n be_v 24._o the_o side_n of_o 192_o be_v 4.6_o and_o 8_o for_o four_o time_n 6._o 8_o time_n make_v 192._o again_o the_o proportion_n of_o 4_o to_o 3_o be_v sesquitercia_fw-la the_o proportion_n of_o 3_o to_o 2_o be_v sesquialtera_fw-la which_o be_v the_o proportion_n of_o the_o side_n of_o the_o one_o solid_a number_n namely_o of_o 24_o the_o proportion_n between_o 8_o and_o 6_o be_v sesquâ—â—ercia_n the_o proportion_n between_o 6_o and_o 4_o be_v sesquialtera_fw-la which_o be_v the_o proportion_n of_o the_o side_n of_o the_o other_o solid_a number_n namely_o of_o 191._o and_o they_o be_v one_o and_o the_o same_o with_o the_o proportione_n of_o the_o 〈◊〉_d of_o the_o other_o wherefore_o thâ—se_a two_z solid_a number_n 24_o and_o 192_o be_v like_a and_o so_o of_o other_o solid_a number_n 23_o a_o perfect_a number_n be_v that_o which_o be_v equal_a to_o all_o his_o part_n definition_n as_o the_o part_n of_o 6_o be_v 1._o 2._o 3._o three_o be_v the_o half_a of_o 6_o two_o the_o three_o part_n and_o 1._o the_o six_o part_n and_o more_o port_n 6_o have_v not_o which_o thrâ—â—_n payte_v 1._o â—_o 3_o add_v together_o make_v 6_o the_o whole_a number_n who_o part_n they_o be_v wherefore_o 6_o be_v a_o perfect_a number_n so_o likewise_o be_v 28_o a_o perfect_a number_n the_o part_n whereof_o be_v these_o number_n 14._o 7._o 2._o and_o 1_o 14_o be_v the_o half_a thereof_o 7_o be_v the_o quarter_n 4_o be_v the_o seven_o part_n 2_o be_v a_o fourteen_o part_n and_o 1_o a_o 28_o part_n and_o these_o be_v all_o the_o part_n of_o 28._o all_o which_o namely_o 1_o 2_o 3_o 4_o 7_o and_o 14_o add_v together_o make_v just_o without_o more_o or_o lâ—sse_v 28._o wherefore_o 〈…〉_o a_o perfect_a number_n and_o so_o of_o other_o the_o like_a this_o kind_n of_o number_n be_v very_o rare_a and_o seldom_o find_v philosophy_n from_o 1_o to_o 10_o there_o be_v but_o one_o perfect_a number_n namely_o 6._o from_o 10_o to_o a_o 100_o there_o be_v also_o but_o one_o that_o be_v 28._o also_o from_o 100_o to_o 1000_o there_o be_v but_o one_o which_o be_v 496._o from_o 1000_o to_o 10000_o likewise_o but_o one_o so_o that_o between_o every_o sâ—ay_n in_o number_v which_o be_v ever_o in_o the_o ten_o place_n there_o be_v find_v but_o one_o perfect_a number_n and_o for_o their_o rareness_n and_o great_a perfection_n they_o be_v of_o marvellous_a use_n in_o magic_a and_o in_o the_o secret_a part_n of_o philosophy_n this_o kind_n of_o number_n be_v call_v perfect_a perfect_a in_o respect_n 〈…〉_o number_n which_o be_v imperfect_a for_o as_o the_o nature_n of_o a_o perfect_a number_n stand_v in_o this_o that_o all_o part_n add_v together_o be_v equal_a to_o the_o whole_a and_o make_v the_o whole_a so_o in_o a_o imperfect_a number_n all_o
other_o number_n de_fw-fr be_v of_o a_o other_o number_n fletcher_n and_o let_v ab_fw-la be_v less_o than_o de._n then_o i_o say_v that_o alternate_o also_o what_o part_n or_o part_n ab_fw-la be_v of_o de_fw-fr the_o self_n same_o part_n or_o part_n be_v c_o of_o f._n forasmuch_o as_o what_o part_v ab_fw-la be_v of_o c_o the_o self_n same_o part_n be_v de_fw-fr of_o fletcher_n construction_n therefore_o how_o many_o part_n of_o c_o there_o be_v in_o ab_fw-la so_o many_o part_n of_o fletcher_n also_o be_v there_o in_o de._n divide_v ab_fw-la into_o the_o part_n of_o c_o that_o be_v into_o ag_z and_o gb_o and_o likewise_o de_fw-fr into_o the_o part_n of_o fletcher_n that_o be_v dh_o and_o he._n now_o then_o the_o multitude_n of_o these_o agnostus_n and_o gb_o be_v equal_a unto_o the_o multitude_n of_o these_o dh_o and_o he._n demonstration_n and_o forasmuch_o as_o what_o part_n agnostus_n be_v of_o c_o the_o self_n same_o part_n be_v dh_o of_o fletcher_n therefore_o alternate_o also_o by_o the_o former_a what_o part_n or_o part_n agnostus_n be_v of_o dh_o the_o self_n same_o part_n or_o part_n be_v c_o of_o f._n and_o by_o the_o same_o reason_n also_o what_o part_n or_o part_n gb_o be_v of_o he_o the_o same_o part_n or_o part_n be_v c_o of_o f._n wherefore_o what_o part_n or_o part_n agnostus_n be_v of_o dh_o the_o self_n same_o part_n or_o part_n be_v ab_fw-la of_o de_fw-fr by_o the_o 6._o of_o the_o seven_o but_o what_o part_n or_o part_n agnostus_n be_v of_o dh_o the_o self_n same_o part_n or_o part_n be_v it_o prove_v that_o c_z be_v of_o f._n wherefore_o what_o part_n or_o part_n ab_fw-la be_v of_o d_o e_o the_o self_n same_o part_n or_o part_n be_v c_o of_o fletcher_n which_o be_v require_v to_o be_v prove_v ¶_o the_o 9_o theorem_a the_o 11._o proposition_n if_o the_o whole_a be_v to_o the_o whole_a as_o a_o part_n take_v away_o be_v to_o a_o part_n take_v away_o then_o shall_v the_o residue_n be_v unto_o the_o residue_n as_o the_o whole_a be_v to_o the_o whole_a quantity_n svppose_v that_o the_o whole_a number_n ab_fw-la be_v unto_o the_o whole_a number_n cd_o as_o the_o part_n take_v away_o ae_n be_v to_o the_o part_n take_v away_o cf._n then_o i_o say_v that_o the_o residue_n ebb_n be_v to_o the_o residue_n fd_o as_o the_o whole_a ab_fw-la be_v to_o the_o whole_a cd_o for_o forasmuch_o as_o ab_fw-la be_v to_o cd_o as_o ae_n be_v to_o cf_o therefore_o what_o part_n or_o part_n ab_fw-la be_v of_o cd_o the_o self_n same_o part_n or_o part_n be_v ae_n of_o cf._n wherefore_o also_o the_o residue_n ebb_n be_v of_o the_o residue_n fd_o by_o the_o 8._o of_o the_o seven_o the_o self_n same_o part_n oâ—_n part_n that_o ab_fw-la be_v of_o cd_o demonstration_n wherefore_o also_o by_o the_o 21._o definition_n of_o this_o book_n as_o ebb_n be_v to_o fd_o so_o be_v ab_fw-la to_o cd_o which_o be_v require_v to_o be_v prove_v ¶_o the_o 10._o theorem_a the_o 12._o proposition_n if_o there_o be_v a_o multitude_n of_o number_n how_o many_o soever_o proportional_a 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 svppose_v that_o there_o be_v a_o multitude_n of_o number_n how_o many_o soever_o proportional_a namely_o a_o b_o c_o d_o so_fw-mi that_o as_o a_o be_v to_o b_o so_o let_v c_o be_v to_o d._n quantity_n then_o i_o say_v that_o as_o one_o of_o the_o antecedentes_fw-la namely_o a_o be_v to_o one_o of_o the_o consequentes_fw-la namely_o to_o b_o or_o as_z c_z be_v to_o d_z so_o be_v all_o the_o antecedentes_fw-la namely_o a_o and_o c_o to_o all_o the_o consequentes_fw-la namely_o to_o b_o and_o d._n for_o forasmuch_o as_o by_o supposition_n as_o a_o be_v to_o b_o so_o be_v c_z to_o d_z therefore_o what_o part_n or_o part_n a_o be_v of_o b_o the_o self_n same_o part_n or_o part_n be_v c_z of_o d_z by_o the_o 21._o definition_n of_o this_o book_n wherefore_o alternate_o what_o part_n or_o part_n a_o be_v of_o c_o the_o self_n same_o part_n or_o part_n be_v b_o of_o d_z by_o the_o nine_o and_o ten_o of_o the_o seven_o wherefore_o both_o these_o number_n add_v together_o demonstration_n a_o and_o c_o be_v of_o both_o these_o numbers_z b_o and_o d_z add_v together_o the_o self_n same_o part_n or_o part_n that_o a_o be_v of_o b_o by_o the_o 5._o and_o 6._o of_o the_o seven_o wherefore_o by_o the_o 21._o definition_n of_o the_o seven_o as_o one_o of_o the_o antecedent_n namely_o a_o be_v to_o one_o of_o the_o consequentes_fw-la namely_o to_o b_o so_o be_v all_o the_o antecedentes_fw-la a_fw-la and_o c_o to_o all_o the_o consequentes_fw-la b_o &_o d._n which_o be_v require_v to_o be_v prove_v ¶_o the_o 11._o theorem_a the_o 13._o proposition_n if_o there_o be_v four_o number_n proportional_a then_o alternate_o also_o they_o shall_v be_v proportional_a svppose_v that_o there_o be_v four_o number_n proportional_a quantity_n a_o b_o c_o d_o so_fw-mi that_o as_o a_o be_v to_o b_o so_o let_v c_o be_v to_o d._n then_o i_o say_v that_o alternate_o also_o they_o shall_v be_v proportional_a that_o be_v as_o a_o be_v to_o c_o so_o be_v b_o to_o d._n for_o forasmuch_o as_o by_o supposition_n as_o a_o be_v to_o b_o so_o be_v c_z to_o d_z therefore_o by_o the_o 21._o definition_n of_o this_o book_n what_o part_n or_o part_n a_o be_v of_o b_o the_o self_n same_o part_n or_o part_n be_v c_o of_o d._n therefore_o alternate_o what_o part_n or_o part_n a_o be_v of_o c_o the_o self_n same_o part_n or_o part_n be_v b_o of_o d_z by_o the_o 9_o of_o the_o seven_o &_o also_o by_o the_o 10._o of_o the_o same_o wherefore_o as_o a_o be_v to_o c_o so_o be_v b_o to_o d_z by_o the_o 21._o definition_n of_o this_o book_n which_o be_v require_v to_o be_v prove_v here_o be_v to_o be_v note_v that_o although_o in_o the_o foresay_a example_n and_o demonstration_n the_o number_n a_o be_v suppose_v to_o be_v less_o than_o the_o number_n b_o and_o so_o the_o number_n c_o be_v less_o than_o the_o number_n d_o note_n yet_o will_v the_o same_o serve_v also_o though_o a_o be_v suppose_v to_o be_v great_a than_o b_o whereby_o also_o c_o shall_v be_v great_a than_o d_z as_z in_o thâ—s_n example_n here_o put_v for_o for_o that_o by_o supposition_n as_o a_o be_v to_o b_o so_o be_v c_z to_o d_z and_o a_o be_v suppose_v to_o be_v great_a than_o b_o and_o c_z great_a than_o d_z therefore_o by_o the_o 21._o definition_n of_o this_o book_n how_o multiplex_n a_o be_v to_o b_o so_o multiplex_n be_v c_o to_o d_z and_o therefore_o what_o part_n or_o part_n b_o be_v of_o a_o the_o self_n same_o part_n or_o part_n be_v d_z of_o c._n wherefore_o alternate_o what_o part_n or_o part_n b_o be_v of_o d_o the_o self_n same_o part_n or_o part_n be_v a_o of_o c_o and_o therefore_o by_o the_o same_o definition_n b_o be_v to_o d_z as_z a_o be_v to_o c._n and_o so_o must_v you_o understand_v of_o the_o former_a proposition_n next_o go_v before_o ¶_o the_o 12._o theorem_a the_o 14._o proposition_n if_o there_o be_v a_o multitude_n of_o number_n how_o many_o soever_o and_o also_o other_o number_n equal_a unto_o they_o in_o multitude_n which_o be_v compare_v two_o and_o two_o be_v in_o one_o and_o the_o same_o proportion_n they_o shall_v also_o of_o equality_n be_v in_o one_o and_o the_o same_o proportion_n quantity_n svppose_v that_o there_o be_v a_o multitude_n of_o number_n how_o many_o soever_o namely_o a_o b_o c_o and_o let_v the_o other_o number_n equal_a unto_o they_o in_o multitude_n be_v d_o e_o f_o which_o be_v compare_v two_o and_o two_o let_v be_v in_o one_o and_o the_o same_o proportion_n that_o be_v as_z a_o to_o b_o so_o let_v d_o be_v to_o e_o and_o as_o b_o be_v to_o c_o so_o let_v e_o be_v to_o f._n then_o i_o say_v that_o of_o equality_n as_o a_o be_v to_o c_o so_o be_v d_z to_o f._n for_o forasmuch_o as_o by_o supposition_n as_o a_o be_v to_o b_o so_o be_v d_z to_o e_o therefore_o alternate_o also_o by_o the_o 13_o of_o the_o seven_o as_o a_o be_v to_o d_o so_o be_v b_o to_o e._n again_o for_o that_o as_z b_o be_v to_o c_o so_o be_v e_o to_o fletcher_n demonstration_n therefore_o alternate_o also_o by_o the_o self_n same_o as_z b_o be_v to_o e_o so_o be_v c_o to_o f._n but_o as_o b_o be_v to_o e_o so_o be_v a_o to_o d._n wherefore_o by_o the_o seven_o common_a sentence_n of_o the_o seven_o as_o a_o be_v to_o d_o so_o be_v c_o to_o f._n wherefore_o alternate_o by_o the_o 13._o of_o the_o seven_o as_o a_o be_v to_o c_o so_o be_v d_z to_o fletcher_n which_o
be_v require_v to_o be_v demonstrate_v after_o this_o proposition_n campane_n demonstrate_v in_o number_n these_o four_o kind_n of_o proportionality_n campane_n namely_o proportion_n converse_n compose_v divide_a and_o everse_v which_o be_v in_o continual_a quantity_n demonstrate_v in_o the_o 4._o 17._o 18._o and_o 19_o proposition_n of_o the_o five_o book_n and_o first_o he_o demonstrate_v converse_v proportion_n in_o this_o manner_n case_n but_o if_o a_o be_v great_a than_o b_o c_o also_o be_v great_a than_o d_z and_o what_o part_n or_o part_n b_o be_v of_o a_o the_o self_n same_o part_n or_o part_n be_v d_z of_o c._n wherefore_o by_o the_o same_o definition_n as_o b_o be_v to_o a_o so_o iâ—_z d_o to_o c_z which_o be_v require_v to_o be_v prove_v proportionality_n divide_v be_v thus_o demonstrate_v suppose_v that_o the_o number_n ab_fw-la be_v to_o the_o number_n b_o as_o the_o number_n cd_o be_v to_o the_o number_n d._n then_o i_o say_v divide_v that_o divide_v also_o as_o a_o be_v to_o b_o so_o be_v c_o to_o d._n for_o for_o that_o as_o ab_fw-la be_v to_o b_o so_o be_v cd_o to_o d_o therefore_o alternate_o by_o the_o 14._o of_o this_o book_n as_o ab_fw-la be_v to_o cd_o so_o be_v b_o to_o d._n wherefore_o by_o the_o 11._o of_o this_o book_n as_o ab_fw-la be_v to_o cd_o so_o be_v a_o to_o c._n wherefore_o as_o b_o be_v to_o d_z so_o be_v a_o to_o c_o and_o for_o that_o as_o a_o be_v to_o c_o so_o be_v b_o to_o d_z therefore_o alternate_o as_o a_o be_v to_o b_o so_o be_v c_o to_o d._n proportionality_n compose_v be_v thus_o demonstrate_v compose_v if_o a_o be_v unto_o b_o as_z c_z be_v to_o d_z then_o shall_v ab_fw-la be_v to_o b_o as_z cd_o be_v to_o d._n for_o alternate_o a_o be_v to_o c_o as_z b_o be_v to_o d._n wherefore_o by_o the_o 13._o of_o this_o book_n as_o ab_fw-la namely_o all_o the_o antecedentes_fw-la be_v to_o cd_o namely_o to_o all_o the_o consequentes_fw-la so_o be_v b_o to_o d_z namely_o one_o of_o the_o antecedentes_fw-la to_o one_o of_o the_o consequentes_fw-la wherefore_o alternate_o as_o ab_fw-la be_v to_o b_o so_o be_v cd_o to_o d._n euerse_a proportionality_n be_v thus_o prove_v proportionality_n supposâ—_n that_o ab_fw-la be_v to_o b_o as_z cd_o be_v to_o d_o then_o shall_v ab_fw-la be_v to_o a_o as_o cd_o be_v to_o c._n for_o alternate_o ab_fw-la be_v to_o cd_o aâ—_z b_o be_v to_o d._n wherefâ—râ—_n by_o the_o 13._o of_o this_o booâ—â—_n aâ—_z be_v â—_o cd_o as_o a_o be_v to_o c._n wherefore_o alternate_o ab_fw-la iâ—_z to_o a_o aâ—_z cd_o iâ—_z to_o c_o whiâ—h_o be_v require_v to_o be_v prove_v ¶_o a_o proportion_n here_o add_v by_o campane_n if_o the_o proportion_n of_o the_o first_o to_o the_o second_o be_v as_o the_o proportion_n of_o the_o three_o to_o the_o foâ—rth_n and_o if_o the_o proportion_n of_o â—he_n five_o to_o the_o second_o be_v as_o the_o proportion_n of_o the_o six_o to_o the_o four_o then_o the_o proportion_n of_o the_o first_o and_o the_o five_o take_v together_o shall_v be_v to_o the_o second_o as_o the_o proportion_n â—f_o the_o three_o and_o the_o six_o take_v together_o to_o the_o four_o and_o after_o the_o same_o manner_n may_v you_o prove_v the_o converse_n of_o this_o proposition_n if_o b_o be_v to_o a_o as_z d_z be_v to_o câ—_n and_o if_o also_o b_o be_v unto_o e_o as_z d_z be_v to_o fletcher_n then_o shall_v b_o be_v to_o ae_n as_o d_z be_v to_o cf._n for_o by_o converse_v proportionality_n prâ—position_n a_o be_v to_o b_o as_z c_z be_v to_o d._n wherefore_o of_o equality_n a_o be_v to_o e_o as_z c_z be_v to_o f._n wherefore_o by_o composition_n a_o and_o e_o be_v to_o e_o as_z c_z and_o f_o be_v to_o f._n wherefore_o converse_o e_o be_v to_o a_o and_o e_o as_z fletcher_n be_v to_o c_o and_o f._n but_o by_o supposition_n b_o be_v to_o e_o as_z d_z be_v to_o f._n wherefore_o again_o by_o proportion_n of_o equality_n b_o be_v to_o a_o and_o e_o as_z d_z be_v to_o c_z and_o f_o demonstration_n which_o be_v require_v to_o be_v prove_v a_o corollary_n by_o this_o also_o it_o be_v manifest_a that_o if_o the_o proportion_n of_o number_n how_o many_o soever_o unto_o the_o first_o be_v as_o the_o proportion_n of_o as_o many_o other_o number_n unto_o the_o second_o campane_n then_o shall_v the_o proportion_n of_o the_o number_n compose_v of_o all_o the_o number_n that_o be_v antecedentes_fw-la to_o the_o first_o be_v to_o the_o first_o as_o the_o number_n compose_v of_o all_o the_o number_n that_o be_v antecedentes_fw-la to_o the_o second_o be_v to_o the_o second_o and_o also_o converse_o if_o the_o proportion_n of_o the_o first_o to_o number_n how_o many_o soever_o be_v as_o the_o proportion_n of_o the_o second_o to_o as_o many_o other_o number_n then_o shall_v the_o proportion_n of_o the_o first_o to_o the_o number_n compose_v of_o all_o the_o number_n that_o be_v consequentes_fw-la to_o itself_o be_v as_o the_o proportion_n of_o the_o second_o to_o the_o number_n compose_v of_o all_o the_o number_n that_o be_v consequence_n to_o itself_o ¶_o the_o 13._o theorem_a the_o 15._o proposition_n if_o unity_n measure_v any_o number_n and_o a_o other_o number_n do_v so_o many_o time_n measure_v a_o other_o number_n unity_n also_o shall_v alternate_o so_o many_o time_n measure_v the_o three_o number_n as_o the_o second_o do_v the_o four_o svppose_v that_o unity_n a_o do_v measure_v the_o number_n bc_o and_o let_v a_o other_o number_n d_o so_o many_o time_n measure_v some_o other_o number_n namely_o ef._n then_o i_o say_v that_o alternate_o unity_n a_o shall_v so_o many_o time_n measure_v the_o number_n d_o as_o the_o number_n bc_o do_v measure_v the_o number_n ef._n coâ—strâ—ctioâ—_n for_o forasmuch_o as_o unity_n a_o do_v so_o many_o time_n measure_v bc_o as_o d_z do_v of_o therefore_o how_o many_o unity_n there_o be_v in_o bc_o so_o many_o number_n be_v there_o in_o of_o equal_a unto_o d._n divide_v i_o say_v bc_o into_o the_o unity_n which_o be_v in_o it_o that_o be_v into_o bg_o gh_o and_o hc_n and_o divide_v likewise_o of_o into_o the_o number_n equal_a unto_o d_o that_o be_v into_o eke_o kl_o and_o lf_a now_o than_o the_o multitude_n of_o these_o bg_o gh_o and_o hc_n be_v equal_a unto_o the_o multitude_n of_o these_o eke_o kl_o lf_n demonstration_n and_o forasmuch_o as_o these_o unity_n bg_o gh_o and_o hc_n be_v equal_a the_o one_o to_o the_o other_o and_o these_o number_n eke_o kl_o &_o lf_a be_v also_o equal_a the_o one_o to_o the_o other_o and_o the_o multitude_n of_o the_o unity_n bg_o gh_o and_o hc_n be_v equal_a unto_o the_o multitude_n of_o the_o number_n eke_o kl_o &_o lf_a therefore_o as_o unity_n bg_o be_v to_o the_o number_n eke_o so_o be_v unity_n gh_o to_o the_o number_n kl_o and_o also_o unity_n hc_n to_o the_o number_n lf_n wherefore_o by_o the_o 12â—_n of_o the_o seventh_n as_o one_o of_o the_o antecedeâ—tâ—â—â—s_n to_o one_o of_o the_o consequentes_fw-la so_o be_v all_o the_o antecedenâ—es_n to_o all_o the_o consequentes_fw-la wherefore_o as_o unity_n bg_o be_v to_o the_o number_n eke_o so_o be_v the_o number_n bc_o to_o the_o number_n ef._n but_o unity_n bg_o be_v equal_a unto_o unity_n a_o and_o the_o number_n eke_o to_o the_o number_n d._n wherefore_o by_o the_o 7._o common_a sentence_n as_o unity_n a_o be_v to_o the_o number_n d_o so_o be_v the_o number_n bc_o to_o the_o number_n ef._n wherefore_o unity_n a_o measure_v the_o number_n d_o so_fw-mi many_o time_n as_o bc_o measure_v of_o by_o the_o 21_o definition_n of_o this_o book_n which_o be_v require_v to_o be_v prove_v ¶_o the_o 14._o theorem_a the_o 16._o proposition_n if_o two_o number_n multiply_v themselves_o the_o one_o into_o the_o other_o produce_v any_o number_n the_o number_n produce_v shall_v be_v equal_a the_o one_o into_o the_o other_o svppose_v that_o there_o be_v two_o number_n a_o and_o b_o and_o let_v a_o multiply_v b_o produce_v c_o and_o let_v b_o multiply_v a_o produce_v d._n thâ—n_o i_o say_v that_o the_o number_n câ—_n equal_a unto_o the_o nâ—mber_n d._n demonstration_n take_v any_o unity_n name_o e._n and_o forasmuch_o as_o a_o multiply_a b_o produce_v c_z therefore_o b_o measure_v c_o by_o the_o unity_n which_o be_v in_o a._n and_o unity_n e_o measure_v the_o number_n a_o by_o those_o unity_n which_o be_v in_o the_o number_n a._n wherefore_o unity_n e_o so_o many_o time_n measure_v a_o as_z b_o measure_v c._o wherefore_o alternate_o by_o the_o 15._o of_o the_o seven_o unity_n e_o measure_v the_o number_n b_o so_o many_o time_n as_o a_o measure_v c._n again_o for_o that_o b_o multiply_v a_o produce_v d_z therefore_o a_o measure_v d_z by_o thâ—_z unity_n which_o be_v in_o b._n and_o unity_n e_o
measure_v b_o by_o the_o unity_n which_o be_v in_o b._n wherefore_o unity_n e_o so_o many_o time_n measure_v the_o number_n b_o as_o a_o measure_v d._n but_o unity_n e_o so_fw-mi many_o time_n measure_v the_o number_n b_o as_o a_o measure_v c._n wherefore_o a_o measure_v either_o of_o these_o number_n c_o and_o d_o a_o like_a wherefore_o by_o the_o 3._o common_a sentence_n of_o this_o book_n c_z be_v equal_a unto_o d_o which_o be_v require_v to_o be_v demonstrate_v the_o 15._o theorem_a the_o 17._o proposition_n if_o one_o number_n multiply_v two_o number_n and_o produce_v other_o number_n the_o number_n produce_v of_o they_o shall_v be_v in_o the_o self_n same_o proportion_n that_o the_o number_n multiply_v be_v svppose_v that_o the_o number_n a_o multiply_v two_o number_n b_o and_o c_o do_v produce_v the_o number_n d_o and_o e._n then_o i_o say_v that_o as_z b_o be_v to_o c_o so_o be_v d_z to_o e._n take_v unity_n namely_o f._n and_o forasmuch_o as_o a_o multiply_v b_o produce_v d_z therefore_o b_o measure_v d_z by_o those_o unity_n that_o be_v in_o a._n and_o unity_n fletcher_n measure_v a_o by_o thâ—sâ—_n uâ—itiâ—â—_fw-fr whih_o be_v in_o a._n â—emonstraâ—ion_n wherefore_o unity_n fletcher_n so_o many_o time_n measure_v the_o number_n a_o as_z b_o measure_v d._n wherefore_o as_o unity_n i_o be_v to_o the_o number_n a._n so_o be_v the_o number_n b_o to_o the_o number_n d_o by_o the_o 21_o definition_n of_o this_o book_n and_o by_o the_o same_o reason_n as_o unity_n fletcher_n be_v to_o the_o number_n a_o so_o be_v the_o number_n c_o to_o the_o number_n e_o wherefore_o also_o by_o the_o 7._o common_a sentence_n of_o this_o book_n as_o b_o be_v to_o d_z so_o be_v c_o to_o e._n wherefore_o alternate_o by_o the_o 15._o of_o the_o seven_o as_o b_o be_v to_o c_o so_o be_v d_z to_o e._n if_o therefore_o one_o number_n multiply_v two_o numbersâ—_n and_o produce_v other_o number_n the_o number_n produce_v of_o they_o shall_v be_v in_o the_o self_n same_o proportion_n that_o the_o number_n multiply_v be_v which_o be_v require_v to_o be_v prove_v here_o fluâ—â—tes_n add_v thiâ—_z corollary_n if_o two_o numberâ—_n have_v one_o and_o the_o samâ—_n proportion_n with_o two_o other_o number_n do_v multiply_v thâ—_z oâ—e_v the_o other_o alternate_o flussâ—tes_n and_o produce_v any_o number_n the_o number_n produce_v of_o they_o shall_v be_v equal_a the_o one_o to_o the_o other_o suppose_v that_o there_o be_v two_o numberâ—_n â—_o and_o b_o and_o also_o two_o other_o number_n c_o and_o d_o have_v thâ—_z same_o proportion_n that_o the_o number_n a_o and_o b_o have_v and_o let_v the_o number_n a_o and_o b_o multiply_v the_o numberâ—_n c_o &_o d_o alternate_o that_o be_v let_v a_o multiply_v d_o produce_v f_o and_o let_v b_o multiply_v c_o produce_v e._n then_o i_o say_v that_o the_o number_n produce_v namely_o e_o &_o f_o be_v equal_a let_v a_o and_o b_o multiply_v the_o one_o the_o other_o in_o such_o sort_n that_o let_v a_o multiply_v b_o produce_v g_o and_o let_v b_o multiply_v a_o produce_v h_o now_o then_o the_o number_n g_o and_o h_o be_v equal_a by_o the_o 16._o of_o this_o bookeâ—_n and_o forasmuch_o as_o a_o multipliâ—ng_a the_o two_o number_n b_o and_o d_z produce_v the_o number_n g_o and_o f_o therefore_o g_z be_v to_o â—_o as_z b_o be_v to_o d_z by_o this_o proposition_n so_o likewise_o b_o multiply_v the_o two_o number_n a_o and_o c_o produce_v the_o two_o number_n h_o and_o e._n wherefore_o by_o the_o same_o h_o be_v to_o e_o as_o a_o be_v to_o c._n but_o alternate_o by_o the_o 13._o of_o this_o book_n a_o be_v to_o c_o as_z b_o be_v to_o d_z but_o as_o a_o be_v to_o c_o so_o be_v h_z to_o e_o and_o as_z â—_o be_v to_o d_z so_o be_v g_z to_o â—_o wherefore_o by_o the_o seven_o common_a sentence_n as_o h_o be_v to_o eâ—_n â—_z so_o be_v g_z to_o f._n wherefore_o alternate_o by_o the_o 13._o of_o this_o book_n h_o be_v to_o g_z as_z e_o be_v to_o f._n but_o it_o be_v prove_v that_o g_z &_o h_o be_v equal_a wherefore_o e_o and_o fletcher_n which_o have_v the_o same_o proportion_n that_o a_o and_o b_o have_v be_v equal_a if_o therefore_o there_o be_v two_o number_n etc_n etc_n which_o be_v require_v to_o be_v prove_v ¶_o the_o 16._o theorem_a the_o 18._o proposition_n if_o two_o number_n multiply_v any_o number_n &_o produce_v other_o number_n the_o number_n of_o they_o produce_v shall_v be_v in_o the_o same_o proportion_n that_o the_o number_n multiply_v be_v svppose_v that_o two_o number_n a_o and_o b_o multiply_v the_o number_n c_o do_v produce_v the_o number_n d_o and_o e._n demonstration_n then_o i_o say_v that_o as_o a_o be_v to_o b_o so_o be_v d_z to_o e._n for_o forasmuch_o as_o a_o multiply_v c_o produce_v d_z therefore_o c_o multiply_v a_o produce_v also_o d_z by_o the_o 16._o of_o this_o book_n and_o by_o the_o same_o reason_n c_o multiply_v b_o produce_v e._n now_o then_o one_o number_n c_z multiply_v two_o number_n a_o and_o b_o produce_v the_o number_n d_z and_o e._n wherefore_o by_o the_o 17._o of_o the_o seven_o as_o a_o be_v to_o b_o so_o be_v d_z to_o e_o which_o be_v require_v to_o be_v demonstrate_v this_o proposition_n and_o the_o former_a touch_v two_o number_n may_v be_v extend_v to_o number_n how_o many_o soever_o soever_o so_o that_o if_o one_o number_n multiply_v number_n how_o many_o soever_o and_o produce_v any_o number_n the_o proportion_n of_o the_o number_n produce_v and_o of_o the_o number_n multiply_v shall_v be_v one_o and_o the_o self_n same_o likewise_o if_o number_n how_o many_o soever_o multiply_v one_o number_n and_o produce_v any_o number_n the_o proportion_n of_o the_o number_n producedâ—_n and_o of_o the_o number_n multiply_v shall_v be_v one_o and_o the_o self_n same_o which_o thing_n by_o this_o and_o the_o former_a proposition_n repeat_v as_o often_o as_o be_v needful_a be_v not_o hard_a to_o prove_v ¶_o the_o 17._o theorem_a the_o 19_o proposition_n if_o there_o be_v four_o number_n in_o proportion_n the_o number_n produce_v of_o the_o first_o and_o the_o four_o be_v equal_a to_o that_o number_n which_o be_v produce_v of_o the_o second_o and_o the_o three_o and_o if_o the_o number_n which_o be_v produce_v of_o the_o first_o and_o the_o four_o be_v equal_a to_o that_o which_o be_v produce_v of_o the_o second_o &_o the_o three_o those_o four_o number_n shall_v be_v in_o proportion_n but_o now_o again_o suppose_v that_o e_o be_v equal_a unto_o f._n then_o i_o say_v that_o as_o a_o be_v to_o b_o so_o be_v c_o to_o d._n first_o for_o the_o same_o order_n of_o construction_n remain_v still_o forasmuch_o as_o a_o multiply_v c_o &_o d_o produce_v g_z and_o e_o therefore_o by_o the_o 17._o of_o the_o seven_o as_o c_o be_v to_o d_z so_o be_v g_z to_o e_o but_o e_o be_v equal_a unto_o fletcher_n but_o if_o two_o number_n be_v equal_a one_o number_n shall_v have_v unto_o they_o onâ—_n and_o the_o same_o proportion_n wherefore_o as_o g_z be_v to_o e_o so_o be_v g_z to_o f._n but_o as_o g_z be_v to_o e_o so_o be_v c_o to_o d._n wherefore_o as_o c_o be_v to_o d_z so_o be_v g_z to_o fletcher_n but_o as_o g_z be_v to_o fletcher_n so_o be_v a_o to_o b_o by_o the_o 18._o of_o the_o seven_o wherefore_o as_o a_o be_v to_o b_o so_o be_v c_z to_o d_z demonstration_n which_o be_v require_v to_o be_v prove_v campane_n here_o campane_n add_v that_o it_o be_v needless_a to_o demonstrate_v that_o if_o one_o number_n have_v to_o two_o number_n one_o and_o the_o same_o proportion_n the_o say_v two_o number_n shall_v be_v equal_a or_o that_o if_o they_o be_v equal_a one_o number_n have_v to_o they_o one_o and_o the_o same_o proportion_n for_o say_v he_o if_o g_o have_v unto_o e_o and_o f_o one_o and_o the_o same_o proportion_n they_o either_o what_o part_n or_o part_n g_z be_v to_o e_o the_o same_o part_n or_o part_n be_v g_z also_o of_o fletcher_n or_o how_o multiplex_n g_z be_v to_o e_o so_o multiplex_n be_v g_z to_o fletcher_n by_o the_o 21._o definition_n and_o therefore_o by_o the_o 2_o and_o 3_o common_a sentence_n the_o say_a number_n shall_v be_v equal_a and_o so_o converse_o if_o the_o two_o number_n e_o and_o f_o be_v equal_a then_o shall_v the_o number_n e_o and_o f_o be_v either_o the_o self_n same_o part_n or_o part_n of_o the_o number_n g_o or_o they_o shall_v be_v equemultiplices_fw-la unto_o it_o and_o therefore_o by_o the_o same_o definition_n the_o number_n g_o shall_v have_v to_o the_o number_n e_o and_o f_o one_o and_o the_o same_o proportion_n ¶_o the_o 18._o theorem_a the_o 20._o proposition_n if_o there_o be_v three_o number_n in_o proportion_n the_o number_n
all_o those_o number_n a_o b_o c_o d_o &_o unity_n add_v together_o and_o let_v e_o multiply_v d_z which_o be_v the_o last_o number_n produce_v the_o number_n fg._n then_o i_o say_v that_o fg_o be_v a_o perfect_a number_n construction_n how_o many_o in_o multitude_n a_o b_o c_o d_o be_v so_o many_o in_o continual_a double_a proportion_n take_v beginning_n at_o e_o which_o let_v be_v the_o number_n e_o hk_o l_o and_o m._n wherefore_o of_o equality_n by_o the_o 13._o of_o the_o seven_o as_o a_o be_v to_o d_o so_o be_v e_o to_o m._n wherefore_o that_o which_o be_v produce_v of_o e_o into_o d_z be_v equal_a to_o that_o which_o be_v produce_v of_o a_o into_o m._n but_o that_o which_o be_v produce_v of_o e_o into_o d_z be_v the_o number_n fg._n wherefore_o that_o which_o be_v produce_v of_o a_o into_o m_o be_v equal_a unto_o fg._n wherefore_o a_o multiply_a m_o produce_v fg._n wherefore_o m_o measure_v fg_o by_o those_o unity_n which_o be_v in_o a._n but_o a_o be_v the_o number_n two_o wherefore_o fg_o be_v double_a to_o m._n and_o the_o number_n m_o l_o hk_o and_o e_o be_v also_o in_o continual_a double_a proportion_n wherefore_o all_o the_o number_n e_o hk_o l_o m_o and_o fg_o be_v continual_o proportional_a in_o double_a proportion_n take_v from_o the_o second_o number_n kh_o and_o from_o the_o last_o fg_o a_o number_n equal_a unto_o the_o first_o namely_o to_o e_o and_o let_v those_o number_n take_v be_v hn_o &_o fx_n demonstration_n wherefore_o by_o the_o proposition_n go_v before_o as_o the_o excess_n of_o the_o second_o number_n be_v to_o the_o first_o number_n so_o be_v the_o excess_n of_o the_o last_o to_o all_o the_o number_n go_v before_o it_o wherefore_o as_o nk_v be_v to_o e_o so_o be_v xg_n to_o these_o number_n m_o l_o kh_o and_o e._n but_o nk_n be_v equal_a unto_o e_o for_o it_o be_v the_o half_a of_o hk_n which_o be_v suppose_v to_o be_v double_a to_o e_o wherefore_o xg_n be_v equal_a unto_o these_o number_n m_o l_o hk_o and_o e._n but_o xf_n be_v equal_a unto_o e_o and_o e_o be_v equal_a unto_o these_o number_n a_o b_o c_o d_o and_o unto_o unity_n wherefore_o the_o whole_a number_n fg_o be_v equal_a unto_o these_o number_n e_o hk_o l_o m_o absurdity_n and_o also_o unto_o these_o number_n a_o b_o c_o d_o and_o unto_o unity_n moreover_o i_o say_v that_o unity_n and_o all_o the_o number_n a_o b_o c_o d_o e_o hk_o l_o and_o m_o do_v measure_n the_o number_n fg._n that_o unity_n measure_v it_o it_o need_v not_o proof_n and_o forasmuch_o as_o fg_o be_v produce_v of_o d_z into_o e_o therefore_o d_z and_o e_o do_v measure_n it_o and_o forasmuch_o as_o the_o double_a from_o unity_n namely_o the_o number_n a_o b_o c_o do_v measure_n the_o number_n d_o by_o the_o 13._o of_o this_o book_n therefore_o they_o shall_v also_o measure_v the_o number_n fg_o who_o d_o measure_v by_o the_o â—_o common_a sentence_n by_o the_o same_o reason_n forasmuch_o as_o the_o number_n e_o hk_o l_o and_o m_o be_v unto_o fg_o as_o unity_n and_o the_o number_n a_o â—_o c_o be_v unto_o d_z namely_o in_o subduple_n proportion_n and_o unity_n and_o the_o number_n a_o b_o c_o do_v 〈◊〉_d d_o therefore_o also_o the_o number_n e_o hk_o l_o and_o m_o shall_v measure_v the_o number_n fg_o noâ—_n i_o say_v also_o that_o no_o other_o number_n measure_v fg_o beside_o these_o number_n a_o b_o c_o d_o e_o hk_o l_o m_o and_o unity_n for_o if_o it_o be_v possible_a let_v oh_o measure_n fg._n and_o let_v oh_o not_o be_v any_o of_o these_o number_n a_o b_o c_o d_o e_o hk_o l_o and_o m._n and_o how_o often_o oh_o measure_v fg_o 〈◊〉_d unity_n let_v there_o be_v in_o p._n wherefore_o oh_o multiply_a pea_n produce_v fg._n but_o e_o also_o multiply_a d_z produce_v fg._n wherefore_o by_o the_o 19_o of_o the_o seven_o as_z e_o be_v to_o oh_o so_o be_v pea_n to_o d._n wherefore_o alternate_o by_o the_o 9_o of_o the_o seven_o as_o e_o be_v to_o p_o so_o be_v oh_o to_o d._n and_o forasmuch_o as_o from_o unity_n be_v these_o number_n in_o continual_a proportion_n a_o b_o c_o d_o and_o the_o number_n a_o which_o be_v next_o after_o unity_n be_v a_o prime_a number_n therefore_o by_o the_o 13._o of_o the_o nine_o no_v other_o number_n measure_v d_z beside_o the_o number_n a_o b_o c._n and_o it_o be_v suppose_v that_o oh_o be_v not_o one_o and_o the_o same_o with_o any_o of_o these_o number_n a_o b_o c._n wherefore_o o_n measure_v not_o d._n but_o as_o oh_o be_v to_o d_o so_o be_v e_o to_o p._n wherefore_o neither_o do_v e_o measure_n p._n and_o e_o be_v a_o prime_a number_n but_o by_o the_o 31._o of_o the_o seven_o every_o prime_a number_n be_v to_o every_o number_n that_o it_o measure_v not_o a_o prime_a number_n wherefore_o e_o and_o p_o be_v prime_a the_o one_o to_o the_o other_o yea_o they_o be_v prime_a and_o the_o least_o but_o by_o the_o 21._o of_o the_o seven_o the_o least_o measure_v the_o number_n that_o have_v one_o and_o the_o same_o proportion_n with_o they_o equal_o the_o antecedent_n the_o antecedent_n and_o the_o consequent_a the_o consequent_a and_o as_o e_o be_v to_o p_o so_o be_v oh_o to_o d._n wherefore_o how_o many_o time_n e_o measure_v oh_o so_o many_o time_n pea_n measure_v d._n but_o no_o other_o number_n measure_v d_z beside_o the_o number_n a_o b_o c_o by_o the_o 13._o of_o this_o book_n wherefore_o pea_n be_v one_o and_o the_o same_o with_o one_o of_o these_o number_n a_o b_o c._n suppose_v that_o p_o be_v one_o and_o the_o same_o with_o b_o &_o how_o many_o b_o c_o d_o be_v in_o multitude_n so_o many_o take_v from_o e_o upward_o namely_o e_o hk_o and_o l._n but_o e_o hk_o and_o l_o be_v in_o the_o same_o proportion_n that_o b_o c_o d_o be_v wherefore_o of_o equality_n as_o b_o be_v to_o d_z so_o be_v e_o to_o l._n wherefore_o that_o which_o be_v produce_v of_o b_o into_o l_o be_v equal_a to_o that_o which_o be_v produce_v of_o d_o into_o e._n but_o that_o which_o be_v produce_v of_o d_z into_o e_o be_v equal_a to_o that_o which_o be_v produce_v of_o pea_n into_o o._n wherefore_o that_o which_o be_v produce_v of_o pea_n into_o oh_o be_v equal_a to_o that_o which_o be_v produce_v of_o b_o into_o l._n wherefore_o as_o pea_n be_v to_o b_o so_o be_v l_o too_o oh_o and_o pea_n be_v one_o &_o the_o same_o with_o b_o wherefore_o l_o also_o be_v one_o and_o the_o same_o with_o oh_o which_o be_v impossible_a for_o oh_o be_v suppose_v not_o to_o be_v one_o and_o the_o same_o with_o any_o of_o the_o number_n give_v wherefore_o no_o number_n measure_v fg_o beside_o these_o number_n a_o b_o c_o d_o e_o hk_o l_o m_o and_o unity_n and_o it_o be_v prove_v that_o fg_o be_v equal_a unto_o these_o number_n a_o b_o c_o d_o e_o hk_o l_o m_o and_o unity_n which_o be_v the_o part_n thereof_o by_o the_o 39_o of_o the_o seven_o but_o a_o perfect_a number_n by_o the_o definition_n be_v that_o which_o be_v equal_a unto_o all_o his_o part_n wherefore_o fg_o be_v a_o perfect_a numberâ—_n which_o be_v require_v to_o be_v prove_v the_o end_n of_o the_o nine_o book_n of_o euclides_n element_n ¶_o the_o ten_o book_n of_o euclides_n element_n in_o this_o ten_o book_n do_v euclid_n entreat_v of_o line_n and_o other_o magnitude_n rational_a &_o irrational_a book_n but_o chief_o of_o irrational_a magnitude_n commensurable_a and_o incommensurable_a of_o which_o hitherto_o in_o all_o his_o former_a 9_o book_n he_o have_v make_v no_o mention_n at_o all_o and_o herein_o differ_v number_n from_o magnitude_n magnitude_n or_o arithmetic_n from_o geometry_n for_o that_o although_o in_o arithmetic_n certain_a number_n be_v call_v prime_a number_n in_o consideration_n of_o themselves_o or_o in_o respect_n of_o a_o other_o and_o so_o be_v call_v incommensurable_a for_o that_o no_o one_o number_n measure_v they_o but_o only_a unity_n yet_o in_o deed_n and_o to_o speak_v absolute_o and_o true_o there_o be_v no_o two_o number_n incommensurable_a but_o have_v one_o common_a measure_n which_o measure_v they_o both_o if_o none_o other_o yet_o have_v they_o unity_n which_o be_v a_o common_a part_n and_o measure_n to_o all_o number_n and_o all_o number_n be_v make_v of_o unity_n as_o of_o their_o part_n unity_n as_o have_v before_o be_v show_v in_o the_o declaration_n of_o the_o definition_n of_o the_o seven_o book_n but_o in_o magnitude_n it_o be_v far_o otherwise_o for_o although_o many_o line_n plain_a figure_n and_o body_n be_v commensurable_a and_o may_v have_v one_o measure_n to_o measure_v they_o yet_o all_o have_v not_o so_o nor_o can_v have_v for_o that_o a_o line_n be_v not_o make_v of_o point_n as_o
number_n be_v make_v of_o unity_n and_o therefore_o can_v a_o point_n be_v a_o common_a part_n of_o all_o line_n and_o measure_v they_o as_o unity_n be_v a_o common_a part_n of_o all_o number_n and_o measure_v they_o unity_n take_v certain_a time_n make_v any_o number_n for_o there_o be_v not_o in_o any_o number_n infinite_a unity_n but_o a_o point_n take_v certain_a time_n yea_o as_o often_o as_o you_o listen_v never_o make_v any_o line_n for_o that_o in_o every_o line_n there_o be_v infinite_a point_n wherefore_o line_n figure_n and_o body_n in_o geometry_n be_v oftentimes_o incommensurable_a and_o irrational_a now_o which_o be_v rational_a and_o which_o irrational_a which_o commensurable_a and_o which_o incommensurable_a how_o many_o and_o how_o sundry_a sort_n and_o kind_n there_o be_v of_o they_o what_o be_v their_o nature_n passion_n and_o property_n do_v euclid_n most_o manifest_o show_v in_o this_o book_n and_o demonstrate_v they_o most_o exact_o this_o ten_o book_n have_v ever_o hitherto_o of_o all_o man_n and_o be_v yet_o think_v &_o account_v to_o be_v the_o hard_a book_n to_o understand_v of_o all_o the_o book_n of_o euclid_n euclid_n which_o common_a receive_a opinion_n have_v cause_v many_o to_o shrink_v and_o have_v as_o it_o be_v deter_v they_o from_o the_o handle_n and_o treaty_n thereof_o there_o have_v be_v in_o deed_n in_o time_n past_a and_o be_v present_o in_o these_o our_o day_n many_o which_o have_v deal_v and_o have_v take_v great_a and_o good_a diligence_n in_o comment_v amend_v and_o restore_a of_o the_o six_o first_o book_n of_o euclid_n and_o there_o have_v stay_v themselves_o and_o go_v no_o far_o be_v deter_v and_o make_v afraid_a as_o it_o seem_v by_o the_o opinion_n of_o the_o hardness_n of_o this_o book_n to_o pass_v forth_o to_o the_o book_n follow_v truth_n it_o be_v that_o this_o book_n have_v in_o it_o somewhat_o a_o other_o &_o strange_a manner_n of_o matter_n entreat_v of_o former_a them_z the_o other_o book_n before_o have_v and_o the_o demonstration_n also_o thereof_o &_o the_o order_n seem_v likewise_o at_o the_o first_o somewhat_o strange_a and_o unaccustomed_a which_o thing_n may_v seem_v also_o to_o cause_v the_o obscurity_n thereof_o and_o to_o fear_v away_o many_o from_o the_o read_n and_o diligent_a study_n of_o the_o same_o so_o much_o that_o many_o of_o the_o well_o learned_a have_v much_o complain_v of_o the_o darkness_n and_o difficulty_n thereof_o and_o have_v think_v it_o a_o very_a hard_a thing_n and_o in_o manner_n impossible_a to_o attain_v to_o the_o right_n and_o full_a understanding_n of_o this_o book_n algebra_n without_o the_o aid_n and_o help_v of_o some_o other_o knowledge_n and_o learning_n and_o chief_o without_o the_o knowledge_n of_o that_o more_o secret_a and_o subtle_a part_n of_o arithmetic_n common_o call_v algebra_n which_o undoubted_o first_o well_o have_v and_o know_v will_v geve_v great_a light_n thereunto_o yet_o certain_o may_v this_o book_n very_o well_o be_v enter_v into_o and_o full_o understand_v without_o any_o strange_a help_n or_o succour_n only_o by_o diligent_a observation_n of_o the_o order_n and_o course_n of_o euclides_n write_n so_o that_o he_o which_o diligent_o have_v peruse_v and_o full_o understand_v the_o 9_o book_n go_v before_o and_o mark_v also_o earnest_o the_o principle_n and_o definition_n of_o this_o â—enth_fw-mi book_n he_o shall_v well_o perceive_v that_o euclid_n be_v of_o himself_o a_o sufficient_a teacher_n and_o instructor_n understand_v and_o need_v not_o the_o help_n of_o any_o other_o and_o shall_v soon_o see_v that_o this_o ten_o book_n be_v not_o of_o such_o hardness_n and_o obscurity_n as_o it_o have_v be_v hitherto_o think_v yea_o i_o doubt_v not_o but_o that_o by_o the_o travel_n and_o industry_n take_v in_o this_o translation_n and_o by_o addition_n and_o emendation_n get_v of_o other_o there_o shall_v appear_v in_o it_o no_o hardness_n at_o all_o but_o shall_v be_v as_o easy_a as_o the_o rest_n of_o his_o book_n be_v definition_n definition_n 1_o magnitude_n commensurable_a be_v suchwhich_v one_o and_o the_o self_n same_o measure_n do_v measure_n first_o he_o show_v what_o magnitude_n be_v commensurable_a one_o to_o a_o other_o to_o the_o better_a and_o more_o clear_a understanding_n of_o this_o definition_n note_v that_o that_o measure_v whereby_o any_o magnitude_n be_v measure_v be_v less_o than_o the_o magnitude_n which_o it_o measure_v or_o at_o least_o equal_a unto_o it_o for_o the_o great_a can_v by_o no_o mean_n measure_v the_o less_o far_o it_o behove_v that_o that_o measure_n if_o it_o be_v equal_a to_o that_o which_o be_v measure_v take_v once_o make_v the_o magnitude_n which_o be_v measure_v if_o it_o be_v less_o than_o oftentimes_o take_v and_o repeat_v it_o must_v precise_o render_v and_o make_v the_o magnitude_n which_o it_o measure_v which_o thing_n in_o number_n be_v easy_o see_v for_o that_o as_o be_v before_o say_v all_o number_n be_v commensurable_a one_o to_o a_o other_o and_o although_o euclid_n in_o this_o definition_n comprehend_v purpose_o only_o magnitude_n which_o be_v continual_a quantity_n as_o be_v line_n superficiece_n and_o body_n yet_o undoubted_o the_o explication_n of_o this_o and_o such_o like_a place_n be_v apt_o to_o be_v seek_v of_o number_n as_o well_o rational_a as_o irrational_a for_o that_o all_o quantity_n commensurable_a have_v that_o proportion_n the_o one_o to_o the_o other_o which_o number_n have_v to_o number_n in_o number_n therefore_o 9_o and_o 12_o be_v commensurable_a because_o there_o be_v one_o common_a measure_n which_o measure_v they_o both_o namely_o the_o number_n 3_n first_o it_o measure_v 12_o for_o it_o be_v less_o than_o 12._o and_o be_v take_v certain_a time_n namely_o 4_o time_n it_o make_v exact_o 12_o 3_o time_n 4_o be_v 12_o it_o also_o measure_v 9_o for_o it_o be_v less_o than_o 9_o and_o also_o take_v certain_a time_n namely_o 3_o time_n it_o make_v precise_o 9_o 3_o time_n 3_o be_v 9_o likewise_o be_v it_o in_o magnitude_n if_o one_o magnitude_n measure_v two_o other_o magnitude_n those_o two_o magnitude_n so_o measure_v be_v say_v to_o be_v commensurable_a as_o for_o example_n if_o the_o line_n c_o be_v double_v do_v make_v the_o line_n b_o and_o the_o same_o line_n c_o triple_a do_v make_v the_o line_n a_o then_o be_v the_o two_o line_n a_o and_o b_o line_n or_o magnitude_n commensurable_a for_o that_o one_o measure_n namely_o the_o line_n c_o measure_v they_o both_o first_o the_o line_n c_o be_v less_o they_o the_o line_n a_o and_o alsolesse_v they_o the_o line_n b_o also_o the_o line_n c_o take_v or_o repeat_v certain_a time_n namely_o 3_o time_n make_v precise_o the_o line_n a_o and_o the_o same_o line_n c_o take_v also_o certain_a time_n namely_o two_o time_n make_v precise_o the_o line_n b._n so_o that_o the_o line_n c_o be_v a_o common_a measure_n to_o they_o both_o and_o do_v measure_v they_o both_o and_o therefore_o be_v the_o two_o line_n a_o and_o b_o 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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 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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 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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
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
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
two_o other_o proposition_n go_v next_o before_o it_o so_o far_o misplace_v that_o where_o they_o be_v word_n for_o word_n before_o duâ—ly_o place_v be_v the_o 105._o and_o 106._o yet_o here_o after_o the_o book_n end_v they_o be_v repeat_v with_o the_o number_n of_o 116._o and_o 117._o proposition_n zambert_n therein_o be_v more_o faithful_a to_o follow_v as_o he_o find_v in_o his_o greek_n example_n than_o he_o be_v skilful_a or_o careful_a to_o do_v what_o be_v necessary_a yea_o and_o some_o greek_n write_v ancient_a copy_n have_v they_o not_o so_o though_o in_o deed_n they_o be_v well_o demonstrate_v yet_o truth_n disord_v be_v half_a disgracedâ—_n especial_o where_o the_o pattern_n of_o good_a order_n by_o profession_n be_v avouch_v to_o be_v but_o through_o ignorance_n arrogancy_n and_o â—emerltie_n of_o unskilful_a method_n master_n many_o thing_n remain_v yet_o in_o these_o geometrical_a element_n undue_o tumble_v in_o though_o true_a yet_o with_o disgrace_n which_o by_o help_n of_o so_o many_o wit_n and_o hability_n of_o such_o as_o now_o may_v have_v good_a cause_n to_o be_v skilful_a herein_o will_v i_o hope_v ere_o long_o be_v take_v away_o and_o thing_n of_o importance_n want_v supply_v the_o end_n of_o the_o ten_o book_n of_o euclides_n element_n ¶_o the_o eleven_o book_n of_o euclides_n element_n book_n hitherto_o have_v â—vclidâ—_n in_o thâ—sâ—_n former_a book_n with_o a_o wonderful_a method_n and_o order_n entreat_v of_o such_o kind_n of_o figure_n superficial_a which_o be_v or_o may_v be_v describe_v in_o a_o superficies_n or_o plain_a and_o have_v teach_v and_o set_v forth_o their_o property_n nature_n generation_n and_o production_n even_o from_o the_o first_o root_n ground_n and_o beginning_n of_o they_o namely_o from_o a_o point_n which_o although_o it_o be_v indivisible_a yet_o be_v it_o the_o begin_n of_o all_o quantity_n continual_a and_o of_o it_o and_o of_o the_o motion_n and_o slow_v thereof_o be_v produce_v a_o line_n and_o consequent_o all_o quantity_n continual_a as_o all_o figure_n plain_a and_o solid_a what_o so_o ever_o euclid_n therefore_o in_o his_o first_o book_n begin_v with_o it_o boot_n and_o from_o thence_o go_v he_o to_o a_o line_n as_o to_o a_o thing_n most_o simple_a next_o unto_o a_o point_n then_o to_o a_o superficies_n and_o to_o angle_n and_o so_o through_o the_o whole_a first_o book_n boâ—â—e_n he_o entreat_v of_o these_o most_o simple_a and_o plain_a ground_n in_o the_o second_o book_n he_o entreat_v further_o â—â—oâ—e_n and_o go_v unto_o more_o hard_a matter_n and_o teach_v of_o division_n of_o line_n and_o of_o the_o multiplication_n of_o line_n and_o of_o their_o part_n and_o of_o their_o passion_n and_o property_n and_o for_o that_o rightline_v â—_z be_v far_o distant_a in_o nature_n and_o property_n from_o round_a and_o circular_a figure_n in_o the_o three_o book_n he_o instruct_v the_o reader_n of_o the_o nature_n and_o condition_n of_o circle_n booâ—e_n in_o the_o four_o book_n he_o compare_v figure_n of_o right_a line_n and_o circle_n together_o bâ—oâ—e_n and_o teach_v how_o to_o describe_v a_o figure_n of_o right_a line_n with_o in_o or_o about_o a_o circle_n and_o contrariwise_o a_o circle_n with_o in_o or_o about_o a_o rectiline_a figure_n in_o the_o five_o book_n he_o search_v out_o the_o nature_n of_o proportion_n a_o matter_n of_o wonderful_a use_n and_o deep_a consideration_n boâ—â—e_n for_o that_o otherwise_o he_o can_v not_o compare_v â—igure_n with_o figure_n or_o the_o side_n of_o figure_n together_o for_o whatsoever_o be_v compare_v to_o any_o other_o thing_n be_v compare_v unto_o it_o undoubted_o under_o some_o kind_n of_o proportion_n wherefore_o in_o the_o six_o book_n he_o compare_v figure_n together_o booâ—e_n one_o to_o a_o other_o likewise_o their_o side_n and_o for_o that_o the_o nature_n of_o proportion_n can_v not_o be_v full_o and_o clear_o see_v without_o the_o knowledge_n of_o number_n wherein_o it_o be_v first_o and_o chief_o find_v in_o the_o seven_o bookâ—_n eight_o booâ—â—_n and_o nine_o book_n book_n he_o entreatâ—th_v of_o number_n &_o of_o the_o kind_n and_o property_n thereof_o and_o because_o that_o the_o side_n of_o solid_a body_n for_o the_o most_o part_n be_v of_o such_o sort_n that_o compare_v together_o they_o have_v such_o proportion_n the_o one_o to_o the_o other_o booâ—e_n which_o can_v not_o be_v express_v by_o any_o number_n certain_a and_o therefore_o be_v call_v irrational_a line_n he_o in_o the_o ten_o book_n have_v write_v &_o teach_v which_o lineâ—_z be_v commensurable_a or_o incommensurable_a the_o one_o to_o the_o other_o and_o of_o the_o diversity_n of_o kind_n of_o irrational_a line_n with_o all_o the_o condition_n &_o propriety_n of_o they_o and_o thus_o have_v euclid_n in_o these_o ten_o foresay_a book_n full_o &_o most_o plenteous_o in_o a_o marvelous_a order_n teach_v whatsoever_o seem_v necessary_a and_o requisite_a to_o the_o knowledge_n of_o all_o superficial_a figure_n of_o what_o sort_n &_o form_n so_o ever_o they_o be_v now_o in_o these_o book_n follow_v he_o entreat_v of_o figure_n of_o a_o other_o kind_n namely_o of_o bodily_a figure_n follow_v as_o of_o cube_n pyramid_n cones_n column_n cilinder_n parallelipipedon_n sphere_n and_o such_o othersâ—_n and_o show_v the_o diversity_n of_o they_o the_o generation_n and_o production_n of_o they_o and_o demonstrate_v with_o great_a and_o wonderful_a art_n their_o propriety_n and_o passion_n with_o all_o their_o nature_n and_o condition_n he_o also_o compare_v one_o oâ—_n they_o to_o a_o other_o whereby_o to_o know_v the_o reason_n and_o proportion_n of_o the_o one_o to_o the_o other_o chief_o of_o the_o five_o body_n which_o be_v call_v regular_a body_n element_n and_o these_o be_v the_o thing_n of_o all_o other_o entreat_v of_o in_o geometry_n most_o worthy_a and_o of_o great_a dignity_n and_o as_o it_o be_v the_o end_n and_o final_a intent_n of_o the_o whole_a be_v of_o geometry_n and_o for_o who_o cause_n have_v be_v write_v and_o speak_v whatsoever_o have_v hitherto_o in_o the_o former_a book_n be_v say_v or_o write_v as_o the_o first_o book_n be_v a_o ground_n and_o a_o necessary_a entry_n to_o all_o the_o râ—st_o â—ollowing_v so_o be_v this_o eleven_o book_n a_o necessary_a entry_n and_o ground_n to_o the_o rest_n which_o follow_v 〈◊〉_d and_o as_o that_o contain_v the_o declaration_n of_o word_n and_o definition_n of_o thinger_n requisite_a to_o the_o knowledge_n of_o superficial_a figure_n and_o entreat_v of_o line_n and_o of_o their_o division_n and_o section_n which_o be_v the_o term_n and_o limit_n of_o superficial_a figure_n so_o in_o this_o book_n be_v set_v forth_o the_o declaration_n of_o word_n and_o definition_n of_o thing_n pertain_v to_o solid_a and_o corporal_a figure_n and_o also_o of_o superficiece_n which_o be_v the_o term_n &_o limit_n of_o solid_n moreover_o of_o the_o division_n and_o intersection_n of_o they_o and_o diverse_a other_o thing_n without_o which_o the_o knowledge_n of_o bodily_a and_o solid_a form_n can_v not_o be_v attain_a unto_o and_o first_o be_v set_v the_o definition_n as_o follow_v definition_n a_o solid_a or_o body_n be_v that_o which_o have_v length_n breadth_n and_o thickness_n definition_n and_o the_o term_n or_o limit_v of_o a_o solid_a be_v a_o superficies_n there_o be_v three_o kind_n of_o continual_a quantity_n a_o line_n a_o superficies_n and_o a_o solid_a or_o body_n the_o beginning_n of_o all_o which_o as_o before_o have_v be_v say_v be_v a_o point_n which_o be_v indivisible_a two_o of_o these_o quantity_n namely_o a_o line_n and_o a_o superficies_n be_v define_v of_o euclid_n before_o in_o his_o first_o book_n but_o the_o three_o kind_n namely_o a_o solid_a or_o body_n he_o there_o define_v not_o as_o a_o thing_n which_o pertain_v not_o then_o to_o his_o purpose_n but_o here_o in_o this_o place_n he_o set_v the_o definition_n thereof_o as_o that_o which_o chief_o now_o pertain_v to_o his_o purpose_n and_o without_o which_o nothing_o in_o these_o thing_n can_v profitable_o be_v teach_v a_o solid_a say_v he_o be_v that_o which_o have_v length_n breadth_n and_o thickness_n or_o depth_n there_o be_v as_o before_o have_v be_v teach_v three_o reason_n or_o mean_n of_o measure_v which_o be_v call_v common_o dimension_n namely_o length_n breadth_n and_o thickness_n these_o dimension_n be_v ascribe_v unto_o quantity_n only_o by_o these_o be_v all_o kind_n of_o quantity_n define_v â—â—_o be_v count_v perfect_a or_o imperfect_a according_a as_o they_o be_v partaker_n of_o few_o or_o more_o of_o they_o as_o euclid_n define_v a_o line_n ascribe_v unto_o it_o only_o one_o of_o these_o dimension_n namely_o length_n wherefore_o a_o line_n be_v the_o imperfect_a kind_n of_o quantity_n in_o define_v of_o a_o superficies_n he_o ascribe_v unto_o it_o two_o dimension_n namely_o length_n and_o breadth_n whereby_o a_o superficies_n be_v a_o quantity_n of_o
if_o the_o angle_n lmk_n be_v a_o acute_a angle_n then_o be_v that_o angle_n the_o inclination_n of_o the_o superficies_n abcd_o unto_o the_o superficies_n efgh_o by_o this_o definition_n because_o it_o be_v contain_v of_o perpendicular_a line_n draw_v in_o either_o of_o the_o superficiece_n to_o one_o and_o the_o self_n same_o point_n be_v the_o common_a section_n of_o they_o both_o 5_o plain_a superficiece_n be_v in_o like_a sort_n incline_v the_o onâ—_n 〈…〉_o she_o definition_n when_o the_o say_v angle_n of_o inclination_n be_v equal_a the_o one_o to_o the_o o_o 〈…〉_o this_o definition_n need_v no_o declaration_n at_o all_o but_o be_v most_o manifest_a by_o the_o definition_n last_o go_v before_o for_o in_o consider_v the_o inclination_n of_o diverse_a superficiece_n to_o other_o if_o the_o acute_a angle_n contain_v under_o the_o perpendicular_a line_n draw_v in_o they_o from_o one_o point_n assign_v in_o each_o of_o their_o common_a section_n be_v equal_a as_o if_o to_o the_o 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certain_a right_a line_n draw_v by_o the_o centre_n and_o one_o each_o side_n end_v at_o the_o superficies_n of_o the_o same_o sphere_n definition_n this_o definition_n also_o be_v not_o hard_o but_o may_v easy_o be_v couceave_v by_o the_o definition_n of_o the_o diameter_n of_o a_o circle_n for_o as_o the_o diameter_n of_o a_o circle_n be_v a_o right_a line_n draw_v from_o one_o side_n of_o the_o circumfrence_n of_o a_o circle_n to_o the_o other_o pass_v by_o the_o centre_n of_o the_o circle_n so_o imagine_v you_o a_o right_a line_n to_o be_v draw_v from_o one_o side_n of_o the_o superficies_n of_o a_o sphere_n to_o the_o other_o pass_v by_o the_o centre_n of_o the_o sphere_n sphere_n and_o that_o line_n be_v the_o diameter_n of_o the_o sphere_n so_o it_o be_v not_o all_o one_o to_o say_v the_o axe_n of_o a_o sphere_n and_o the_o diameter_n of_o a_o sphere_n any_o line_n in_o a_o sphere_n draw_v from_o side_n to_o side_n by_o the_o centre_n be_v a_o diameter_n but_o not_o every_o line_n so_o draw_v by_o the_o centre_n be_v the_o axe_n of_o the_o sphere_n but_o only_o one_o right_a line_n about_o which_o the_o sphere_n be_v imagine_v to_o be_v movedâ—_n so_o that_o the_o name_n of_o a_o diameter_n of_o a_o sphere_n be_v more_o general_a then_o be_v the_o name_n of_o a_o axe_n for_o every_o axe_n in_o a_o sphere_n be_v a_o diameter_n of_o the_o same_o but_o not_o every_o diameter_n of_o a_o sphere_n be_v a_o axe_n of_o the_o same_o and_o therefore_o flussas_n set_v a_o diameter_n in_o the_o definition_n of_o a_o axe_n as_o a_o more_o general_a word_n â—n_o this_o manner_n the_o axe_n of_o a_o sphere_n be_v that_o fix_a diameter_n above_o which_o the_o sphere_n be_v move_v a_o sphere_n as_o also_o a_o circle_n may_v have_v infinite_a diameter_n but_o it_o can_v have_v but_o
signify_v last_o of_o all_o a_o dodecahedron_n heaven_n for_o that_o it_o be_v make_v of_o pâ—ntagoâ—_n who_o angle_n be_v more_o ample_a and_o large_a than_o the_o angle_n of_o the_o other_o body_n and_o by_o that_o â—eaâ—â—â—_n draw_v more_o â—â—_o roundness_n 〈◊〉_d &_o to_o the_o form_n and_o nature_n of_o a_o sphere_n they_o assign_v to_o a_o sphere_n namely_o 〈…〉_o who_o so_o will_v 〈…〉_o in_o his_o tineus_n shall_v â—ead_a of_o these_o figure_n and_o of_o their_o mutual_a proportionâ—â—â—raunge_a maâ—terâ—_n which_o hâ—re_o be_v not_o to_o be_v entreat_v of_o this_o which_o be_v say_v shall_v be_v sufficient_a for_o the_o 〈◊〉_d of_o they_o and_o for_o thâ—_z declaration_n of_o their_o definition_n after_o all_o these_o definition_n here_o set_v of_o euclid_n flussas_n have_v add_v a_o other_o definition_n which_o 〈◊〉_d of_o a_o parallelipipedon_n which_o because_o it_o have_v not_o hitherto_o of_o euclid_n in_o any_o place_n be_v define_v and_o because_o it_o be_v very_o good_a and_o necessary_a to_o be_v have_v i_o think_v good_a not_o to_o omit_v it_o thus_o it_o be_v a_o parallelipipedon_n be_v a_o solid_a figure_n comprehend_v under_o four_o plain_a quadrangle_n figure_n parallelipipedon_n of_o which_o those_o which_o be_v opposite_a be_v parallel_n because_o these_o five_o regular_a body_n here_o define_v be_v not_o by_o these_o figure_n here_o set_v so_o full_o and_o lively_o express_v that_o the_o studious_a beholder_n can_v thorough_o according_a to_o their_o definition_n conceive_v they_o i_o have_v here_o give_v of_o they_o other_o description_n draw_v in_o a_o plain_n by_o which_o you_o may_v easy_o attain_v to_o the_o knowledge_n of_o they_o for_o if_o you_o draw_v the_o like_a form_n in_o matter_n that_o will_v bow_v and_o geve_v place_n as_z most_o apt_o you_o may_v do_v in_o fine_a past_v paper_n such_o as_o pastwive_n make_v woman_n paste_n of_o &_o they_o with_o a_o knife_n cut_v every_o line_n fine_o not_o through_o but_o half_a way_n only_o if_o they_o you_o bow_v and_o bend_v they_o according_o you_o shall_v most_o plain_o and_o manifest_o see_v the_o form_n and_o shape_n of_o these_o body_n even_o as_o their_o definition_n show_v and_o it_o shall_v be_v very_o necessary_a for_o you_o to_o hadâ—â—tore_v of_o that_o past_v paper_n by_o you_o for_o so_o shall_v yoâ—_n upon_o it_o 〈…〉_o the_o form_n of_o other_o body_n as_o prism_n and_o parallelipopedon_n 〈…〉_o set_v forth_o in_o these_o five_o book_n follow_v and_o see_v the_o very_a 〈◊〉_d of_o thâ—se_a body_n there_o mention_v which_o will_v make_v these_o book_n concern_v body_n as_o easy_a unto_o you_o as_o be_v the_o other_o book_n who_o figure_n you_o may_v plain_o see_v upon_o a_o plain_a superficies_n describe_v thiâ—_z figurâ—_n which_o consi_v of_o twâ—luâ—â—quilâ—â—â—â—_n and_o equiangle_n pâ—ntâ—â—â—â—â—_n upoâ—_n the_o foresay_a matter_n and_o fine_o cut_v as_o before_o be_v â—â—ught_v tâ—â—â—lâ—uâ—n_v line_n contain_v within_o thâ—_z figurâ—_n and_o bow_n and_o fold_n the_o penâ—â—gonâ—_n according_o and_o they_o will_v so_o close_o together_o thaâ—_z thâ—y_n will_v â—â—kâ—_n thâ—_z very_a form_n of_o a_o dodecahedron_n dâ—dâ—â—â—â—edron_n icosaâ—edron_n if_o you_o describe_v this_o figure_n which_o consist_v of_o twenty_o equilater_n and_o equiangle_n triangle_n upon_o the_o foresay_a matter_n and_o fine_o cut_v as_o before_o be_v show_v the_o ninâ—tâ—ne_a line_n which_o be_v contain_v within_o the_o figure_n and_o then_o bow_n and_o fold_n they_o according_o they_o will_v in_o such_o sort_n close_o together_o that_o therâ—_n will_v be_v make_v a_o perfect_v form_n of_o a_o icosahedron_n because_o in_o these_o five_o book_n there_o be_v sometime_o require_v other_o body_n beside_o the_o foresay_a five_o regular_a body_n as_o pyramid_n of_o diverse_a form_n prism_n and_o other_o i_o have_v here_o set_v forth_o three_o figure_n of_o three_o sundry_a pyramid_n one_o have_v to_o his_o base_a a_o triangle_n a_o other_o a_o quadrangle_n figure_n the_o other_o â—_o pentagonâ—_n which_o if_o you_o describe_v upon_o the_o foresay_a matter_n &_o fine_o cut_v as_o it_o be_v before_o teach_v the_o line_n contain_v within_o each_o figure_n namely_o in_o the_o first_o three_o line_n in_o the_o second_o four_o line_n and_o in_o the_o three_o five_o line_n and_o so_o bend_v and_o fold_v they_o according_o they_o will_v so_o close_o together_o at_o the_o top_n that_o they_o will_v â—ake_v pyramid_n of_o that_o form_n that_o their_o base_n be_v of_o and_o if_o you_o conceive_v well_o the_o describe_v of_o these_o you_o may_v most_o easy_o describe_v the_o body_n of_o a_o pyramid_n of_o what_o form_n so_o ever_o you_o will._n because_o these_o five_o book_n follow_v be_v somewhat_o hard_o for_o young_a beginner_n by_o reason_n they_o must_v in_o the_o figure_n describe_v in_o a_o plain_a imagine_v line_n and_o superficiece_n to_o be_v elevate_v and_o erect_v the_o one_o to_o the_o other_o and_o also_o conceive_v solid_n or_o body_n which_o for_o that_o they_o have_v not_o hitherto_o be_v acquaint_v with_o will_v at_o the_o first_o sight_n be_v somewhat_o strange_a unto_o they_o i_o have_v for_o their_o more_o â—ase_a in_o this_o eleven_o book_n at_o the_o end_n of_o the_o demonstration_n of_o every_o proposition_n either_o set_v new_a figure_n if_o they_o concern_v the_o elevate_n or_o erect_v of_o line_n or_o superficiece_n or_o else_o if_o they_o concern_v body_n i_o have_v show_v how_o they_o shall_v describe_v body_n to_o be_v compare_v with_o the_o construction_n and_o demonstration_n of_o the_o proposition_n to_o they_o belong_v and_o if_o they_o diligent_o weigh_v the_o manner_n observe_v in_o this_o eleven_o book_n touch_v the_o description_n of_o new_a figure_n agree_v with_o the_o figure_n describe_v in_o the_o plain_a it_o shall_v not_o be_v hard_o for_o they_o of_o themselves_o to_o do_v the_o like_a in_o the_o other_o book_n follow_v when_o they_o come_v to_o a_o proposition_n which_o concern_v either_o the_o elevate_n or_o erect_v of_o line_n and_o superficiece_n or_o any_o kind_n of_o body_n to_o be_v imagine_v ¶_o the_o 1._o theorem_a the_o 1._o proposition_n that_o part_n of_o a_o right_a line_n shall_v be_v in_o a_o ground_n plain_a superficies_n &_o part_v elevate_v upward_o be_v impossible_a for_o if_o it_o be_v possible_a let_v part_n of_o the_o right_a line_n abc_n namely_o the_o part_n ab_fw-la be_v in_o a_o ground_n plain_a superficies_n and_o the_o other_o part_n thereof_o namely_o bc_o be_v elevate_v upward_o and_o produce_v direct_o upon_o the_o ground_n plain_a superficies_n the_o right_a line_n ab_fw-la beyond_o the_o point_n b_o unto_o the_o point_n d._n wherefore_o unto_o two_o right_a line_n give_v abc_n and_o abdella_n the_o line_n ab_fw-la be_v a_o common_a section_n or_o part_n which_o be_v impossible_a impossibility_n for_o a_o right_a line_n can_v not_o touch_v a_o right_a line_n in_o 〈◊〉_d point_n then_o one_o uâ—lesse_o those_o right_n be_v exact_o agree_v and_o lay_v the_o one_o upon_o the_o other_o wherefore_o that_o part_n of_o a_o right_a line_n shall_v be_v in_o a_o ground_n plain_a superficies_n and_o part_v elevate_v upward_o be_v impossible_a which_o be_v require_v to_o be_v prove_v this_o figure_n more_o plain_o set_v forth_o the_o foresay_a demonstration_n if_o you_o elevate_v the_o superficies_n wherein_o the_o line_n bc._n an_o other_o demonstration_n after_o flâ—sâ—â—s_n if_o it_o be_v possible_a let_v there_o be_v a_o right_a line_n abg_o who_o part_n ab_fw-la let_v be_v in_o the_o ground_n plain_a superficies_n aed_n flussas_n and_o let_v the_o rest_n thereof_o bg_o be_v elevate_v on_o high_a that_o be_v without_o the_o plain_n aed_n then_o i_o say_v that_o abg_o be_v not_o one_o right_a line_n for_o forasmuch_o as_o aed_n be_v a_o plain_a superficies_n produce_v direct_o &_o equal_o upon_o the_o say_a plain_n aed_n the_o right_a line_n ab_fw-la towards_o d_z which_o by_o the_o 4._o definition_n of_o the_o first_o shall_v be_v a_o right_a line_n and_o from_o some_o one_o point_n of_o the_o right_a line_n abdella_n namely_o from_o c_o draâ—_n unto_o the_o point_n g_o a_o right_a line_n cg_o wherefore_o in_o the_o triangle_n 〈…〉_o the_o outward_a angâ—â—_n abâ—_n be_v equal_a to_o the_o two_o inward_a and_o opposite_a angle_n by_o the_o 32._o of_o the_o first_o and_o therefore_o it_o be_v less_o than_o two_o right_a angle_n by_o the_o 17._o of_o the_o same_o wherefore_o the_o line_n abg_o forasmuch_o as_o it_o make_v a_o angle_n be_v not_o â—_o right_n line_n wherefore_o that_o part_n of_o a_o right_a line_n shall_v be_v in_o a_o ground_n plain_a superficies_n and_o part_v elevate_v upward_o be_v impossible_a if_o you_o mark_v well_o the_o figure_n before_o add_v for_o the_o plain_a declaration_n of_o euclides_n demonstration_n iâ—_z will_v not_o be_v hard_o for_o you_o to_o coâ—â—â—â—e_v this_o figure_n which_o ulyss_n put_v for_o his_o demonstration_n â—_o wherein_o
same_o superficies_n wherefore_o these_o right_a line_n ab_fw-la bd_o and_o dc_o be_v in_o one_o and_o the_o self_n same_o superficies_n and_o either_o of_o these_o angle_n abdella_n and_o bdc_o be_v a_o right_a angle_n by_o supposition_n wherefore_o by_o the_o 28._o of_o the_o first_o the_o line_n ab_fw-la be_v a_o parallel_n to_o the_o line_n cd_o if_o therefore_o two_o right_a line_n be_v erect_v perpendicular_o to_o one_o and_o the_o self_n same_o plain_a superficies_n those_o right_a line_n be_v parallel_n the_o one_o to_o the_o other_o which_o be_v require_v to_o be_v prove_v here_o for_o the_o better_a understanding_n of_o this_o 6._o proposition_n i_o have_v describe_v a_o other_o figure_n as_o touch_v which_o if_o you_o erect_v the_o superficies_n abdella_n perpendicular_o to_o the_o superficies_n bde_v and_o imagine_v only_o a_o line_n to_o be_v draw_v from_o the_o point_n a_o to_o the_o point_n e_o if_o you_o will_v you_o may_v extend_v a_o thread_n from_o the_o say_a point_n a_o to_o the_o point_n e_o and_o so_o compare_v it_o with_o the_o demonstration_n it_o will_v make_v both_o the_o proposition_n and_o also_o the_o demonstration_n most_o clear_a unto_o you_o ¶_o a_o other_o demonstration_n of_o the_o six_o proposition_n by_o m._n dee_n suppose_v that_o the_o two_o right_a line_n ab_fw-la &_o cd_o be_v perpendicular_o erect_v to_o one_o &_o the_o same_o plain_a superficies_n namely_o the_o plain_a superficies_n open_v then_o i_o say_v that_o â—â—_o and_o cd_o be_v parallel_n let_v the_o end_n point_v of_o the_o right_a line_n ab_fw-la and_o cd_o which_o touch_v the_o plain_a superficies_n oâ—_n be_v the_o point_n â—_o and_o d_o fronâ—_n to_o d_o let_v a_o straight_a line_n be_v draw_v by_o the_o first_o petition_n and_o by_o the_o second_o petition_n let_v the_o straight_a line_n â—d_a be_v extend_v as_o to_o the_o point_n m_o &_o n._n now_o forasmuch_o as_o the_o right_a line_n ab_fw-la from_o the_o point_n â—_o produce_v do_v cut_v the_o line_n mn_v by_o construction_n therefore_o by_o the_o second_o proposition_n of_o this_o eleven_o book_n the_o right_a line_n ab_fw-la &_o mn_o be_v in_o one_o plainâ—_n superficies_n which_o let_v be_v qr_o cut_v the_o superficies_n open_v in_o the_o right_a line_n mn_o by_o the_o same_o mean_n may_v we_o conclude_v the_o right_a line_n cd_o to_o be_v in_o one_o plain_a superficies_n with_o the_o right_a line_n mn_o but_o the_o right_a line_n mn_o by_o supposition_n be_v in_o the_o plain_a superficies_n qr_o wherefore_o cd_o be_v in_o the_o plain_a superficies_n qr_o and_o aâ—_z the_o right_a line_n be_v prove_v to_o be_v in_o the_o same_o plain_a superficies_n qr_o therefore_o ab_fw-la and_o cd_o be_v in_o one_o plain_a superficies_n namely_o qr_o and_o forasmuch_o as_o the_o line_n aâ—_z and_o cd_o by_o supposition_n be_v perpendicular_a upon_o the_o plain_a superficies_n open_v therefore_o by_o the_o second_o definition_n of_o this_o book_n with_o all_o the_o right_a line_n draw_v in_o the_o superficies_n open_v and_o touch_v ab_fw-la and_o cd_o the_o same_o perpendiculars_n aâ—_z and_o cd_o do_v make_v right_a angle_n but_o by_o construction_n mn_v be_v draw_v in_o the_o plain_a superficies_n open_a touch_v the_o perpendicular_n ab_fw-la and_o cd_o at_o the_o point_n â—_o and_o d._n therefore_o the_o perpendicular_n aâ—_z and_o cd_o make_v with_o the_o right_a line_n mn_v two_o right_a angle_n namely_o abn_n and_o cdm_o and_o mn_v the_o right_a line_n be_v prove_v to_o be_v in_o the_o one_o and_o the_o same_o plain_a superficies_n with_o the_o right_a line_n ab_fw-la &_o cd_o namely_o in_o the_o plain_a superficies_n qr_o wherefore_o by_o the_o second_o part_n of_o the_o 28._o proposition_n of_o the_o first_o book_n the_o right_a lineâ—_z ab_fw-la and_o cd_o be_v parallel_n if_o therefore_o two_o right_a line_n be_v erect_v perpendicular_o to_o one_o and_o the_o self_n same_o plain_a superficies_n those_o right_a line_n be_v parallel_n the_o one_o to_o the_o other_o which_o be_v require_v to_o be_v demonstrate_v a_o corollary_n add_v by_o m._n dee_n hereby_o it_o be_v evident_a that_o any_o two_o right_a line_n perpendicular_o erect_v to_o one_o and_o the_o self_n same_o plain_a superficies_n be_v also_o themselves_o in_o one_o and_o the_o same_o plain_a superficies_n which_o be_v likewise_o perpendicular_o erect_v to_o the_o same_o plain_a superficies_n unto_o which_o the_o two_o right_a line_n be_v perpendicular_a the_o first_o part_n hereof_o be_v prove_v by_o the_o former_a construction_n and_o demonstration_n that_o the_o right_a line_n ab_fw-la and_o cd_o be_v in_o one_o and_o the_o same_o plain_a superficies_n qâ—_n the_o second_o part_n be_v also_o manifest_a that_o be_v that_o the_o plain_a superficies_n qr_o be_v perpendicular_o erect_v upon_o the_o plain_a superficies_n open_v for_o that_o aâ—_z and_o cd_o be_v in_o the_o plain_a superficies_n qr_o be_v by_o supposition_n perpendicular_a to_o the_o plain_a superficies_n open_v wherefore_o by_o the_o three_o definition_n of_o this_o book_n qr_o be_v perpendicular_o erect_v to_o or_o upon_o open_a which_o be_v require_v to_o be_v prove_v io._n do_v you_o his_o advice_n upon_o the_o assumpt_n of_o the_o 6._o as_o concern_v the_o make_n of_o the_o line_n de_fw-fr equal_a to_o the_o right_a line_n ab_fw-la very_o the_o second_o of_o the_o first_o without_o some_o far_a consideration_n be_v not_o proper_o enough_o allege_v and_o no_o wonder_n it_o be_v for_o that_o in_o the_o former_a bookeâ—_n whatsoeâ—â—â—â—aâ—h_v of_o line_n be_v speak_v the_o same_o have_v always_o be_v imagine_v to_o be_v in_o one_o only_a plain_a superficies_n consider_v or_o execute_v but_o here_o the_o perpendicular_a line_n ab_fw-la be_v not_o in_o the_o same_o plaint_n superficies_n that_o the_o right_a line_n db_o be_v therefore_o some_o other_o help_n must_v be_v put_v into_o the_o hand_n of_o young_a beginner_n how_o to_o bring_v this_o problem_n to_o execution_n which_o be_v this_o most_o plain_a and_o brief_a understand_v that_o bd_o the_o right_n line_n be_v the_o common_a section_n of_o the_o plain_a superficies_n wherein_o the_o perpendicular_n ab_fw-la and_o cd_o be_v &_o of_o the_o other_o plain_a superficies_n to_o which_o they_o be_v perpendicular_n the_o first_o of_o these_o in_o my_o former_a demonstration_n of_o the_o 6_o â—_o i_o note_v by_o the_o plain_a superficies_n qr_o and_o the_o other_o i_o note_v by_o the_o plain_a superficies_n open_v wherefore_o bd_o be_v a_o right_a line_n common_a to_o both_o the_o plain_a supâ—rficieces_n qr_o &_o op_n thereby_o the_o ponit_n b_o and_o d_o be_v common_a to_o the_o plain_n qr_o and_o op_n now_o from_o bd_o sufficient_o extend_v cut_v a_o right_a line_n equal_a to_o ab_fw-la which_o suppose_v to_o be_v bf_o by_o the_o three_o of_o the_o first_o and_o orderly_a to_o bf_o make_v de_fw-fr equal_a by_o the_o 3._o oâ—_n the_o first_o if_o de_fw-fr be_fw-mi great_a than_o bf_o which_o always_o you_o may_v cause_v so_o to_o be_v by_o produce_v of_o de_fw-fr sufficient_o now_o forasmuch_o as_o bf_o by_o construction_n be_v cut_v equal_a to_o ab_fw-la and_o de_fw-fr also_o by_o construction_n put_v equal_a to_o bf_o therefore_o by_o the_o 1._o common_a sentence_n de_fw-fr be_v put_v equal_a to_o ab_fw-la which_o be_v require_v to_o be_v do_v in_o like_a sort_n if_o de_fw-fr be_v a_o line_n give_v to_o who_o ab_fw-la be_v to_o be_v cut_v and_o make_v equal_a first_o out_o of_o the_o line_n db_o sufficient_o produce_v cut_v of_o dg_o equal_a to_o de_fw-fr by_o the_o three_o of_o the_o first_o and_o by_o the_o same_o 3._o cut_v from_o basilius_n sufficient_o produce_v basilius_n equal_a to_o dg_v then_o be_v it_o evident_a that_o to_o the_o right_a line_n de_fw-fr the_o perpendicular_a line_n ab_fw-la be_v put_v equal_a and_o though_o this_o be_v easy_a to_o conceive_v yet_o i_o have_v design_v the_o figure_n according_o whereby_o you_o may_v instruct_v your_o imagination_n many_o such_o help_n be_v in_o this_o book_n requisite_a as_o well_o to_o inform_v the_o young_a student_n therewith_o as_o also_o to_o master_n the_o froward_a gaynesayer_n of_o our_o conclusion_n or_o interrupter_n of_o our_o demonstration_n course_n ¶_o the_o 7._o theorem_a the_o 7._o proposition_n if_o there_o be_v two_o parallel_n right_a line_n and_o in_o either_o of_o they_o be_v take_v a_o point_n at_o all_o adventure_n a_o right_a line_n draw_v by_o the_o say_a point_n be_v in_o the_o self_n same_o superficies_n with_o the_o parallel_n right_a line_n svppose_v that_o these_o two_o right_a line_n ab_fw-la and_o cd_o be_v parallel_n and_o in_o either_o of_o they_o take_v a_o point_n at_o all_o adventure_n namely_o e_o and_o f._n then_o i_o say_v that_o a_o right_a line_n draw_v from_o the_o point_n e_o to_o the_o point_n fletcher_n be_v in_o the_o self_n same_o plain_a superficies_n that_o the_o
a_o other_o from_o the_o point_n x_o to_o the_o point_n f._n and_o forasmuch_o as_o these_o two_o parallel_a superficiece_n kl_o and_o mn_o be_v cut_v by_o the_o superficies_n ebdx_n therefore_o their_o common_a section_n which_o be_v the_o line_n exit_fw-la and_o bd_o be_v by_o the_o 16._o of_o the_o eleven_o parallel_v the_o one_o to_o the_o other_o and_o by_o the_o same_o reason_n also_o forasmuch_o as_o the_o two_o parallel_a superficies_n gh_o and_o kl_o be_v cut_v by_o the_o superficies_n axfc_n their_o common_a section_n ac_fw-la and_o xf_n be_v by_o the_o 16._o of_o the_o eleven_o parallel_v and_o forasmuch_o as_o to_o one_o of_o the_o side_n of_o the_o triangle_n abdâ—_n namely_o to_o the_o side_n bd_o be_v draw_v a_o parallel_a line_n exit_fw-la therefore_o by_o the_o 2._o of_o the_o six_o proportional_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 axe_n to_o the_o line_n xd_v again_o forasmuch_o as_o to_o one_o of_o the_o side_n of_o the_o triangle_n adc_o namely_o to_o the_o side_n ac_fw-la be_v draw_v a_o parallel_a line_n xf_n therefore_o by_o the_o 2._o of_o the_o six_o proportional_o as_o the_o line_n axe_n be_v to_o the_o line_n xd_v so_o be_v the_o line_n cf_o to_o the_o line_n fd._n and_o it_o be_v prove_v that_o as_o the_o line_n axe_n be_v to_o the_o line_n xd_v so_o be_v the_o line_n ae_n to_o the_o line_n ebb_n therefore_o also_o by_o the_o 11._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 if_o therefore_o two_o right_a line_n â—e_z divide_v by_o plain_a superâ—icieces_n be_v parallel_n the_o part_n of_o the_o line_n divide_v shall_v be_v proportional_a which_o be_v require_v to_o be_v demonstrate_v in_o this_o figure_n it_o be_v more_o easy_a to_o see_v the_o former_a demonstration_n if_o you_o erect_v perpendicular_o unto_o the_o ground_n superficies_n acbd_v the_o three_o superficiece_n gh_o kl_o and_o mn_v or_o if_o you_o so_o â—râ—ct_v they_o that_o thâ—y_v be_v equedistant_a one_o to_o the_o other_o ¶_o the_o 16._o theorem_a the_o 18._o proposition_n if_o a_o right_a line_n be_v erect_v perpendicular_o to_o a_o plain_a superficies_n all_o the_o superficiece_n extend_v by_o that_o right_a line_n be_v erect_v perpendicular_o to_o the_o self_n same_o plain_a superficies_n svppose_v that_o a_o right_a line_n ab_fw-la be_v erect_v perpendicular_o to_o a_o ground_n superficies_n then_o i_o say_v that_o all_o the_o superficiece_n pass_v by_o the_o line_n ab_fw-la be_v erect_v perpendicular_o to_o the_o ground_n superficies_n extend_v a_o superficies_n by_o the_o line_n ab_fw-la and_o let_v the_o same_o be_v ed_z &_o let_v the_o common_a section_n of_o the_o plain_a superficies_n and_o of_o the_o ground_n superficies_n be_v the_o right_a line_n ce._n and_o take_v in_o the_o line_n ce_fw-fr a_o point_n at_o all_o adventure_n construction_n and_o let_v the_o same_o be_v f_o and_o by_o the_o 11._o of_o the_o first_o from_o the_o point_n fletcher_n draw_v unto_o the_o line_n ce_fw-fr a_o perpendicular_a line_n in_o the_o superficies_n de_fw-fr and_o let_v the_o same_o be_v fg._n and_o forasmuch_o as_o the_o line_n ab_fw-la be_v erect_v perpendicular_o to_o the_o ground_n superficies_n therefore_o by_o the_o 2._o definition_n of_o the_o eleven_o the_o line_n ab_fw-la be_v erect_v perpendicular_o to_o all_o the_o right_a line_n that_o be_v in_o the_o ground_n plain_a superficies_n demonstration_n and_o which_o touch_v it_o wherefore_o it_o be_v erect_v perpendicular_o to_o the_o line_n ce._n wherefore_o the_o angle_n abe_n be_v a_o right_a angle_n and_o the_o angle_n gfb_n be_v also_o a_o right_a angle_n by_o construction_n wherefore_o by_o the_o 28_o of_o the_o first_o the_o line_n ab_fw-la be_v a_o parallel_n to_o the_o line_n fg._n but_o the_o line_n ab_fw-la be_v erect_v perpendicular_o to_o the_o ground_n superficies_n wherefore_o by_o the_o 8._o of_o the_o eleven_o the_o line_n fg_o be_v also_o erect_v perpendicular_o to_o the_o ground_n superficies_n and_o forasmuch_o as_o by_o the_o 3._o definition_n of_o the_o eleven_o a_o plain_a superficies_n be_v then_o erect_v perpendicular_o to_o a_o plain_a superficies_n when_o all_o the_o right_a line_n draw_v in_o one_o of_o the_o plain_a superficiece_n unto_o the_o common_a section_n of_o those_o two_o plain_a superficiece_n make_v therewith_o right_a angle_n do_v also_o make_v right_a angle_n with_o the_o other_o plain_a superficies_n and_o it_o be_v prove_v that_o the_o line_n fg_o draw_v in_o one_o of_o the_o plain_a superficiece_n namely_o in_o de_fw-fr perpendicular_o to_o the_o common_a section_n of_o the_o plain_a superficiece_n namely_o to_o the_o line_n ce_fw-fr be_v erect_v perpendicular_o to_o the_o ground_n superficies_n wherefore_o the_o plain_a superficies_n de_fw-fr be_v erect_v perpendicular_o to_o the_o ground_n superficies_n in_o like_a sort_n also_o may_v we_o prove_v that_o all_o the_o plain_a superficiece_n which_o pass_v by_o the_o line_n ab_fw-la be_v erect_v perpendicular_o to_o the_o ground_n superficies_n if_o therefore_o a_o right_a line_n be_v erect_v perpendicular_o to_o a_o plain_a superficies_n all_o the_o superficiece_n pass_v by_o the_o right_a line_n be_v erect_v perpendicular_o to_o the_o self_n same_o plain_a superficies_n which_o be_v require_v to_o be_v demonstrate_v in_o this_o figure_n here_o set_v you_o may_v erect_v perpendicular_o at_o your_o pleasure_n the_o superficies_n wherein_o be_v draw_v the_o line_n dc_o gf_o ab_fw-la and_o he_o to_o the_o ground_n superficies_n wherein_o be_v draw_v the_o line_n cfbe_n and_o so_o plain_o compare_v it_o with_o the_o demonstration_n before_o put_v ¶_o the_o 17._o theorem_a the_o 19_o proposition_n if_o two_o plain_a superficiece_n cut_v the_o one_o the_o other_o be_v erect_v perpendicular_o to_o any_o plain_a superficies_n their_o common_a section_n be_v also_o erect_v perpendicular_o to_o the_o self_n same_o plain_a superficies_n svppose_v that_o these_o two_o plain_a superâ—icieces_n ab_fw-la &_o bc_o cut_v the_o one_o the_o other_o be_v erect_v perpendicular_o to_o a_o ground_n superficies_n and_o let_v their_o common_a section_n be_v the_o line_n bd._n then_o i_o say_v that_o the_o line_n bd_o be_v erect_v perpendicular_o to_o the_o ground_n superficies_n impossibility_n for_o if_o not_o then_o by_o the_o 11._o of_o the_o first_o from_o the_o point_n d_o draw_v in_o the_o superficies_n ab_fw-la unto_o the_o right_a line_n da_z a_o perpendicular_a line_n de._n and_o in_o the_o superficies_n cb_o draw_v unto_o the_o line_n dc_o a_o perpendicular_a line_n df._n and_o forasmuch_o as_o the_o superficies_n ab_fw-la be_v erect_v perpendicular_o to_o the_o ground_n superficies_n and_o in_o the_o plain_a superficies_n ab_fw-la unto_o the_o common_a section_n of_o the_o plain_a superficies_n and_o of_o the_o ground_n superficies_n namely_o to_o the_o line_n dam_n be_v erect_v a_o perpendicular_a line_n de_fw-fr therefore_o by_o the_o converse_n of_o the_o 3._o definition_n of_o this_o book_n the_o line_n de_fw-fr be_v erect_v perpendicular_o to_o the_o ground_n superficies_n and_o in_o like_a sort_n may_v we_o prove_v that_o the_o line_n df_o be_v erect_v perpendicular_o to_o the_o ground_n superficies_n wherefore_o from_o one_o and_o the_o self_n same_o point_n namely_o from_o d_o be_v erect_v perpendicular_o to_o the_o ground_n superficies_n two_o right_a line_n both_o on_o one_o and_o the_o self_n same_o side_n which_o be_v by_o the_o 15._o of_o the_o eleven_o impossible_a wherefore_o from_o the_o point_n d_o can_v not_o be_v erect_v perpendicular_o to_o the_o ground_n superficies_n any_o other_o right_a line_n beside_o bd_o which_o be_v the_o common_a section_n of_o the_o two_o superficiece_n ab_fw-la and_o bc._n if_o therefore_o two_o plain_a superâ—icieces_n cut_v the_o one_o the_o other_o be_v erect_v perpendicular_o to_o any_o plain_a superficies_n their_o common_a section_n be_v also_o erect_v perpendicular_o to_o the_o self_n same_o plain_a superficies_n which_o be_v require_v to_o be_v prove_v here_o have_v i_o set_v a_o other_o figure_n which_o will_v more_o plain_o show_v unto_o you_o the_o former_a demonstration_n if_o you_o erect_v perpendicular_o to_o the_o ground_n superficies_n ac_fw-la the_o two_o superficiece_n ab_fw-la and_o bc_o which_o cut_v the_o one_o the_o other_o in_o the_o line_n bd._n the_o 18._o theorem_a the_o 20._o proposition_n if_o a_o solid_a angle_n be_v contain_v under_o three_o plain_a superficial_a angle_n every_o two_o of_o those_o three_o angle_n which_o then_o so_o ever_o be_v take_v be_v great_a than_o the_o three_o svppose_v that_o the_o solid_a angle_n a_o be_v contain_v under_o three_o plain_a superficial_a angle_n that_o be_v under_o bac_n god_n and_o dab_n then_o i_o say_v that_o two_o of_o these_o superficial_a angle_n how_o so_o ever_o they_o be_v take_v be_v great_a than_o the_o three_o if_o the_o
the_o line_n dc_o wherefore_o the_o superficies_n ac_fw-la be_v a_o parallelogram_n in_o like_a sort_n also_o may_v we_o prove_v that_o every_o one_o of_o these_o superficice_n ce_fw-fr gf_o bg_o fb_o and_o ae_n be_v parallelogram_n draw_v a_o right_a line_n from_o the_o point_n a_o to_o the_o point_n h_o and_o a_o other_o from_o the_o point_n d_o to_o the_o point_n f._n equal_a and_o forasmuch_o as_o the_o line_n ab_fw-la be_v prove_v a_o parallel_n to_o the_o line_n cd_o and_o the_o line_n bh_o to_o the_o line_n cf_o therefore_o these_o two_o right_a line_n ab_fw-la and_o bh_o touch_v the_o one_o the_o other_o be_v parallel_n to_o these_o two_o right_a line_n dc_o and_o cf_o touch_v also_o the_o one_o the_o other_o and_o not_o be_v in_o one_o and_o the_o self_n same_o plain_a superficies_n wherefore_o by_o the_o 10._o of_o the_o eleven_o they_o comprehend_v equal_a angle_n wherefore_o the_o angle_n abh_n be_v equal_a to_o the_o angle_n dcf_n and_o forasmuch_o as_o these_o two_o line_n ab_fw-la and_o bh_o be_v proof_n equal_a to_o these_o two_o line_n dc_o and_o cf_o and_o the_o angle_n abh_n be_v prove_v equal_a to_o the_o angle_n dcfâ—_n therefore_o by_o the_o 4._o of_o the_o first_o the_o base_a ah_o be_v equal_a to_o the_o base_a df_o and_o the_o triangle_n abh_n be_v equal_a to_o the_o triangle_n dcf_n and_o forasmuch_o as_o by_o the_o 41._o of_o the_o first_o the_o parallelogram_n bg_o be_v double_a to_o the_o triangle_n abh_n and_o the_o parallelogram_n ce_fw-fr be_v also_o double_a to_o the_o triangle_n dcf_n therefore_o the_o parallelogram_n bg_o be_v equal_a to_o the_o parallelogram_n ce._n in_o like_a sort_n also_o may_v we_o prove_v that_o the_o parallelogram_n ac_fw-la be_v equal_a to_o the_o parallelogram_n gf_o and_o the_o parallelogram_n ae_n to_o the_o parallelogram_n fb_o if_o therefore_o a_o solid_a or_o body_n be_v contain_v under_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 &_o parallelogram_n which_o be_v require_v to_o be_v demonstrate_v i_o have_v for_o the_o better_a help_n of_o young_a beginner_n describe_v here_o a_o other_o figure_n who_o form_n if_o it_o be_v describe_v upon_o past_v paper_n with_o the_o letter_n place_v in_o the_o same_o order_n that_o it_o be_v here_o and_o then_o if_o you_o cut_v fine_o these_o line_n agnostus_n de_fw-fr and_o cf_o not_o through_o the_o paper_n and_o fold_v it_o according_o compare_v it_o with_o the_o demonstration_n and_o it_o will_v shake_v of_o all_o hardness_n from_o it_o the_o 22._o theorem_a the_o 25._o proposition_n if_o a_o parallelipipedom_n be_v cut_v of_o a_o plain_a superficies_n be_v a_o parallel_n to_o the_o two_o opposite_a plain_a superficiece_n of_o the_o same_o body_n then_o as_o the_o base_a be_v to_o the_o base_a so_o be_v the_o one_o solid_a to_o the_o other_o solid_a i_o have_v for_o the_o better_a sight_n of_o the_o construction_n &_o demonstration_n of_o the_o former_a 25._o proposition_n here_o set_v another_o figure_n who_o form_n if_o you_o describe_v upon_o past_v paper_n and_o fine_o cut_v the_o three_o line_n xi_o b_n and_o tie_v not_o through_o the_o paper_n but_o half_a way_n and_o then_o fold_v it_o according_o and_o compare_v it_o with_o the_o construction_n and_o demonstration_n you_o shall_v see_v that_o it_o will_v geve_v great_a light_n thereunto_o here_o flussas_n add_v three_o corollary_n first_o corollary_n if_o a_o prism_n be_v cut_v of_o a_o plain_a superficies_n parallel_v to_o the_o opposite_a superficiece_n corollary_n the_o session_n of_o the_o prism_n shall_v be_v the_o one_o to_o the_o other_o in_o that_o proportion_n that_o the_o section_n of_o the_o base_a be_v the_o one_o to_o the_o other_o for_o the_o section_n of_o the_o base_n which_o base_n by_o the_o 11._o definition_n of_o this_o book_n be_v parallelogram_n shall_v likewise_o be_v parallelogram_n by_o the_o 16._o of_o this_o book_n when_o as_o the_o superficies_n which_o cut_v be_v parallelel_n to_o the_o opposite_a superâ—icieces_n and_o shall_v also_o be_v equiangle_n wherefore_o if_o unto_o the_o base_n by_o produce_v the_o right_a line_n be_v add_v like_o and_o equal_a base_n as_o be_v before_o show_v in_o a_o parallelipipedon_n of_o those_o section_n shall_v be_v make_v as_o many_o like_a prism_n as_o you_o will._n and_o so_o by_o the_o same_o reason_n namely_o by_o the_o common_a excess_n equality_n or_o want_v of_o the_o multiplices_fw-la of_o the_o base_n &_o of_o the_o section_n by_o the_o 5._o definition_n of_o the_o five_o may_v be_v prove_v that_o every_o section_n of_o the_o prism_n multiply_v by_o any_o multiplication_n whatsoever_o shall_v have_v to_o any_o other_o section_n that_o proportion_n that_o the_o section_n of_o the_o base_n have_v second_o corollary_n solides_n who_o be_v two_o opposite_a superficieâ—es_n be_v poligonon_o figure_n like_o equal_a and_o parallel_n the_o other_o superficies_n which_o of_o necessity_n be_v parallelogram_n column_n be_v cut_v of_o a_o plain_a superficies_n parallel_v to_o the_o two_o opposite_a superficies_n the_o section_n of_o the_o base_a be_v the_o one_o to_o the_o other_o as_o the_o section_n of_o the_o solid_a be_v thâ—_z one_o to_o the_o other_o which_o thing_n be_v manifest_a for_o such_o solid_n be_v divide_v into_o prism_n which_o have_v one_o common_a side_n namely_o the_o axe_n or_o right_a line_n which_o be_v draw_v by_o the_o centre_n of_o the_o opposite_a base_n wherefore_o how_o many_o paâ—allelogrammes_n or_o base_n be_v set_v upon_o the_o opposite_a poligonon_fw-fr figure_n of_o so_o many_o prism_n shall_v the_o whole_a solid_a be_v compose_v for_o those_o polygonon_n figure_n be_v divide_v into_o so_o many_o like_a triangle_n by_o the_o 20._o of_o the_o six_o which_o describe_v prism_n which_o prism_n be_v cut_v by_o a_o superficies_n parallel_v to_o the_o opposite_a superficiece_n the_o section_n of_o the_o base_n shall_v by_o the_o former_a corollary_n be_v proportional_a with_o the_o section_n of_o the_o prism_n wherefore_o by_o the_o â—â—_o of_o the_o five_o as_o the_o section_n of_o the_o one_o be_v the_o one_o to_o the_o other_o so_o be_v the_o section_n of_o the_o whole_a the_o one_o to_o the_o other_o of_o these_o solid_n there_o be_v infinite_a kind_n according_a to_o the_o variety_n of_o the_o opposite_a and_o parallel_v poligonon_o figure_n which_o poligonon_o figure_n do_v alter_v the_o angle_n of_o the_o parallelogram_n set_v upon_o they_o according_a to_o the_o diversity_n oâ—_n their_o situation_n but_o this_o be_v no_o let_v at_o all_o to_o this_o corollary_n for_o that_o which_o we_o have_v prove_v will_v always_o follow_v when_o as_o the_o superficiece_n which_o be_v set_v upon_o the_o opposite_a like_a &_o equal_a polygonon_n and_o parallel_n superficiece_n be_v always_o parallelogram_n three_o corollary_n corollary_n tâ—e_z foresay_a solid_n compose_v of_o prism_n be_v cut_v by_o a_o plain_a superficies_n parallel_v to_o the_o opposite_a superficiece_n be_v the_o one_o to_o the_o other_o as_o the_o head_n or_o high_a part_n cut_v be_v for_o it_o be_v prove_v that_o they_o be_v the_o one_o to_o the_o other_o as_o the_o base_n be_v which_o base_n forasmuch_o as_o they_o be_v parâ—llelogrammes_n be_v the_o one_o to_o the_o other_o as_o the_o right_a line_n be_v upon_o which_o they_o be_v set_v by_o the_o first_o of_o the_o six_o which_o right_a line_n be_v the_o head_n or_o high_a part_n of_o the_o prism_n the_o 4._o problem_n the_o 26._o proposition_n upon_o a_o right_a line_n give_v and_o at_o a_o point_n in_o it_o give_v to_o make_v a_o solid_a angle_n equal_a to_o a_o solid_a angle_n give_v in_o these_o two_o 〈…〉_o here_o put_v you_o may_v in_o 〈◊〉_d clear_o concern_v the_o â—â—â—â—mer_a construction_n and_o dâ—â—monstration_n if_o you_o erect_v peâ—â—pendicularly_o unto_o the_o ground_n superficies_n the_o triangle_n alb_n and_o dce_n &_o elevate_v the_o triangle_n alh_n and_o dcf_n that_o the_o line_n la_fw-fr and_o cd_o of_o they_o may_v exact_o agree_v with_o the_o lineâ—_z la_fw-fr and_o cd_o of_o the_o â—riangles_n erecâ—edâ—_n for_o so_o order_v they_o if_o you_o compare_v the_o former_a construction_n and_o demonstration_n with_o they_o they_o will_v be_v plaint_n unto_o you_o although_o euclides_n demonstration_n be_v touch_v solid_a angle_n which_o be_v contain_v only_o under_o three_o superficial_a angle_n that_o be_v who_o base_n be_v triangle_n yet_o by_o it_o may_v you_o perform_v the_o problem_n touch_v solid_a angle_n contain_v under_o superficial_a angle_n how_o many_o soever_o that_o be_v have_v to_o their_o base_n any_o kind_n of_o poligonon_n figure_n for_o every_o poligonon_n figure_n may_v by_o the_o 20._o of_o the_o six_o be_v resolve_v into_o like_a tringles_n and_o so_o also_o shall_v the_o solid_a angle_n be_v divide_v into_o so_o many_o solid_a angle_n as_o there_o be_v
at_o all_o adventure_n namely_o d_o five_o g_o s_o and_o a_o right_a line_n be_v draw_v from_o the_o point_n d_o to_o the_o point_n g_o and_o a_o other_o from_o the_o point_n five_o to_o the_o point_n s._n wherefore_o by_o the_o 7._o of_o the_o eleven_o the_o line_n dg_o and_o we_o be_v in_o one_o and_o the_o self_n same_o plain_a superficies_n and_o forasmuch_o as_o the_o line_n de_fw-fr be_v a_o parallel_n to_o the_o line_n bg_o therefore_o by_o the_o 24._o of_o the_o first_o the_o angle_n edt_n be_v equal_a to_o the_o angle_n bgt_fw-mi for_o they_o be_v alternate_a angle_n and_o likewise_o the_o angle_n dtv_n be_v equal_a to_o the_o angle_n gts_fw-fr now_o then_o there_o be_v two_o triangle_n that_o be_v dtv_n and_o gts_fw-fr have_v two_o angle_n of_o the_o one_o equal_a to_o two_o angle_n of_o the_o other_o and_o one_o side_n of_o the_o one_o equal_a to_o one_o side_n of_o the_o other_o namely_o the_o side_n which_o subtend_v the_o equal_a angle_n that_o be_v the_o side_n dv_o to_o the_o side_n g_n for_o they_o be_v the_o half_n of_o the_o line_n de_fw-fr and_o bg_o wherefore_o the_o side_n remain_v be_v equal_a to_o the_o side_n remain_v wherefore_o the_o line_n dt_n be_v equal_a to_o the_o line_n tg_n and_o the_o line_n vt_fw-la to_o the_o line_n it_o be_v if_o therefore_o the_o opposite_a side_n of_o a_o parallelipipedon_n be_v divide_v into_o two_o equal_a part_n and_o by_o their_o section_n be_v extend_v plain_a superficiece_n the_o common_a section_n of_o those_o plain_a superficiece_n and_o the_o diameter_n of_o the_o parallelipipedon_n do_v divide_v the_o one_o the_o other_o into_o two_o equal_a part_n which_o be_v require_v to_o be_v demonstrate_v a_o corollary_n add_v by_o flussas_n every_o plain_a superficies_n extend_v by_o the_o centre_n of_o a_o parallelipipedon_n divide_v that_o solid_a into_o two_o equal_a part_n and_o so_o do_v not_o any_o other_o plain_a superficies_n not_o extend_v by_o the_o centre_n for_o every_o plain_n extend_v by_o the_o centre_n cut_v the_o diameter_n of_o the_o parallelipipedon_n in_o the_o centre_n into_o two_o equal_a part_n for_o it_o be_v prove_v that_o plain_a superficiece_n which_o cut_v the_o solid_a into_o two_o equal_a part_n do_v cut_v the_o dimetient_fw-la into_o two_o equal_a part_n in_o the_o centre_n wherefore_o all_o the_o line_n draw_v by_o the_o centre_n in_o that_o plain_a superficies_n shall_v make_v angle_n with_o the_o dimetient_fw-la and_o forasmuch_o as_o the_o diameter_n fall_v upon_o the_o parallel_n right_a line_n of_o the_o solid_a which_o describe_v the_o opposite_a side_n of_o the_o say_v solid_a or_o upon_o the_o parallel_n plain_a superficiece_n of_o the_o solid_a which_o make_v angel_n at_o the_o end_n of_o the_o diameter_n the_o triangle_n contain_v under_o the_o diameter_n and_o the_o right_a line_n extend_v in_o that_o plain_n by_o the_o centre_n and_o the_o right_a line_n which_o be_v draw_v in_o the_o opposite_a superficiece_n of_o the_o solid_a join_v together_o the_o end_n of_o the_o foresay_a right_a line_n namely_o the_o end_n of_o the_o diameter_n and_o the_o end_n of_o the_o line_n draw_v by_o the_o centre_n in_o the_o superficies_n extend_v by_o the_o centre_n shall_v always_o be_v equal_a and_o equiangle_n by_o the_o 26._o of_o the_o first_o for_o the_o opposite_a right_a line_n draw_v by_o the_o opposite_a plain_a superficiece_n of_o the_o solid_a do_v make_v equal_a angle_n with_o the_o diameter_n forasmuch_o as_o they_o be_v parallel_a line_n by_o the_o 16._o of_o this_o book_n but_o the_o angle_n at_o the_o centre_n be_v equal_a by_o the_o 15._o of_o the_o first_o for_o they_o be_v head_n angle_n &_o one_o side_n be_v equal_a to_o one_o side_n namely_o half_o the_o dimetient_fw-la wherefore_o the_o triangle_n contain_v under_o every_o right_a line_n draw_v by_o the_o centre_n of_o the_o parallelipipedon_n in_o the_o superficies_n which_o be_v extend_v also_o by_o the_o say_a centre_n and_o the_o diameter_n thereof_o who_o end_n be_v the_o angle_n of_o the_o solid_a be_v equal_a equilater_n &_o equiangle_n by_o the_o 26._o of_o the_o first_o wherefore_o it_o follow_v that_o the_o plain_a superficies_n which_o cut_v the_o parallelipipedon_n do_v make_v the_o part_n of_o the_o base_n on_o the_o opposite_a side_n equal_a and_o equiangle_n and_o therefore_o like_a and_o equal_a both_o in_o multitude_n and_o in_o magnitude_n wherefore_o the_o two_o solid_a section_n of_o that_o solid_a shall_v be_v equal_a and_o like_a by_o the_o 8._o definition_n of_o this_o book_n and_o now_o that_o no_o other_o plain_a superficies_n beside_o that_o which_o be_v extend_v by_o the_o centre_n divide_v the_o parallelipipedon_n into_o two_o equal_a part_n it_o be_v manifest_a if_o unto_o the_o plain_a superficies_n which_o be_v not_o extend_v by_o the_o centre_n we_o extend_v by_o the_o centre_n a_o parallel_n plain_a superficies_n by_o the_o corollary_n of_o the_o 15._o of_o this_o book_n for_o forasmuch_o as_o that_o superficies_n which_o be_v extend_v by_o the_o centre_n do_v divide_v the_o parallelipipedom_n into_o two_o equal_a parâ—â—_n it_o be_v manifest_a that_o the_o other_o plain_a superficies_n which_o be_v parallel_n to_o the_o superficies_n which_o divide_v the_o solid_a into_o two_o equal_a part_n be_v in_o one_o of_o the_o equal_a part_n of_o the_o solid_a wherefore_o see_v that_o the_o whole_a be_v ever_o great_a than_o his_o part_n it_o must_v needs_o be_v that_o one_o of_o these_o section_n be_v less_o than_o the_o half_a of_o the_o solid_a and_o therefore_o the_o other_o be_v great_a for_o the_o better_a understanding_n of_o this_o former_a proposition_n &_o also_o of_o this_o corollary_n add_v by_o flussas_n it_o shall_v be_v very_o needful_a for_o you_o to_o describe_v of_o past_v paper_n or_o such_o like_a matter_n a_o parallelipipedom_n or_o a_o cube_n and_o to_o divide_v all_o the_o parallelogram_n thereof_o into_o two_o equal_a part_n by_o draw_v by_o the_o centre_n of_o the_o say_a parallelogram_n which_o centre_n be_v the_o point_n make_v by_o the_o cut_n of_o diagonal_a line_n draw_v from_o thâ—_z opposite_a angle_n of_o the_o say_a parallelogram_n line_n parallel_n to_o the_o side_n of_o the_o parallelogram_n as_o in_o the_o former_a figure_n describe_v in_o a_o plain_a you_o may_v see_v be_v the_o six_o parallelogram_n de_fw-fr eh_o ha_o ad_fw-la dh_o and_o cg_o who_o these_o parallel_a line_n draw_v by_o the_o centre_n of_o the_o say_a parallelogram_n namely_o xo_o or_o pr._n and_o px_n do_v divide_v into_o two_o equal_a part_n by_o which_o four_o line_n you_o must_v imagine_v a_o plain_a superficies_n to_o be_v extend_v also_o these_o parallel_a line_n kl_o ln_o nm_o and_o mâ—_n by_o which_o four_o line_n likewise_o yâ—_n must_v imagine_v a_o plain_a superficies_n to_o be_v extend_v you_o may_v if_o you_o will_v put_v within_o your_o body_n make_v thus_o of_o past_v paper_n two_o superficiece_n make_v also_o of_o the_o say_a paper_n have_v to_o their_o limit_n line_n equal_a to_o the_o foresay_a parallel_n line_n which_o superficiece_n must_v also_o be_v divide_v into_o two_o equal_a part_n by_o parallel_a line_n draw_v by_o their_o centre_n and_o must_v cut_v the_o one_o the_o other_o by_o these_o parallel_a line_n and_o for_o the_o diameter_n of_o this_o body_n extend_v a_o thread_n from_o one_o angle_n in_o the_o base_a of_o the_o solid_a to_o his_o opposite_a angle_n which_o shall_v pass_v by_o the_o centre_n of_o the_o parallelipipedon_n as_o do_v the_o line_n dg_o in_o the_o figure_n before_o describe_v in_o the_o plain_n and_o draw_v in_o the_o base_a and_o the_o opposite_a superficies_n unto_o it_o diagonal_a line_n from_o the_o angle_n from_o which_o be_v extend_v the_o diameter_n of_o the_o solid_a as_o in_o the_o former_a description_n be_v the_o line_n bg_o and_o de._n and_o when_o you_o have_v thus_o describe_v this_o body_n compare_v it_o with_o the_o former_a demonstration_n and_o it_o will_v make_v it_o very_o plain_a unto_o you_o so_o your_o letter_n agree_v with_o the_o letter_n of_o the_o figure_n describe_v in_o the_o book_n and_o this_o description_n will_v plain_o set_v forth_o unto_o you_o the_o corollary_n follow_v that_o proposition_n for_o where_o as_o to_o the_o understanding_n of_o the_o demonstration_n of_o the_o proposition_n the_o superficiece_n put_v within_o the_o body_n be_v extend_v by_o parallel_a line_n draw_v by_o the_o centre_n of_o the_o base_n of_o the_o parallelipipedon_n to_o the_o understanding_n of_o the_o say_a corollary_n you_o may_v extend_v a_o superficies_n by_o any_o other_o line_n draw_v in_o the_o say_a base_n so_o that_o yet_o it_o pass_v through_o the_o midst_n of_o the_o thread_n which_o be_v suppose_v to_o be_v the_o centre_n of_o the_o parallelipipedon_n ¶_o the_o 35._o theorem_a the_o 40._o proposition_n if_o there_o be_v
and_o so_o of_o such_o like_a who_o can_v not_o ready_o fall_v into_o archimedes_n reckon_v and_o account_n by_o his_o method_n to_o find_v the_o proportion_n of_o the_o circumference_n of_o any_o circle_n to_o his_o diameter_n to_o be_v almost_o triple_a and_o one_o seven_o of_o the_o diameter_n but_o to_o be_v more_o than_o triple_a and_o ten_o one_o &_o seventithe_n that_o be_v to_o be_v less_o than_o 3_o 1_o â—_o and_o more_o than_o 3_o 10_o 71._o and_o where_o archimedes_n use_v a_o poligonon_n figure_n of_o 96._o side_n he_o that_o for_o exercise_v sake_n or_o for_o earnest_a desire_n of_o a_o more_o nearness_n will_v use_v polygonon_n figure_n of_o 384._o side_n or_o more_o may_v well_o travail_v therein_o till_o either_o weariness_n cause_v he_o stay_v or_o else_o he_o find_v his_o labour_n fruitless_a in_o deed_n archimedes_n conclude_v proportion_n of_o the_o circumference_n to_o the_o diameter_n have_v hitherto_o serve_v the_o vulgar_a andâ—_n mechanical_a worâ—_n man_n wherewith_o who_o so_o be_v not_o concent_v let_v his_o own_o methodical_a travail_n satisfy_v his_o desire_n or_o let_v he_o procure_v other_o thereto_o for_o narrow_a term_n of_o great_a and_o less_o find_v and_o appoint_v to_o the_o circumference_n will_v also_o win_v to_o the_o area_n of_o the_o circle_n a_o near_a quantity_n see_v it_o be_v well_o demonstrate_v of_o archimedâ—s_n that_o a_o triangle_n rectangle_n of_o who_o two_o side_n contain_v the_o right_a angle_n one_o be_v equal_a to_o the_o semidiameter_n of_o the_o circle_n and_o the_o other_o to_o the_o circumference_n of_o the_o same_o be_v equal_a to_o the_o area_n of_o that_o circle_n upon_o which_o two_o theorem_n it_o follow_v that_o the_o square_n make_v of_o the_o diameter_n be_v in_o that_o proportion_n to_o the_o circle_n very_o near_o in_o which_o 14_o be_v to_o 11â—_n wherefore_o every_o circle_n be_v eleven_o fowertenthe_n well_o near_o of_o the_o square_n about_o he_o describe_v the_o one_o side_n then_o of_o that_o square_n divide_v into_o 14._o equal_a part_n circle_n and_o from_o that_o point_n which_o end_v the_o eleven_o part_n draw_v to_o the_o opposite_a side_n a_o line_n parallel_n to_o the_o other_o side_n and_o so_o make_v perfect_a the_o parallelogram_n then_o by_o the_o last_o proposition_n of_o the_o second_o book_n unto_o that_o parallelogram_n who_o one_o side_n have_v those_o 11._o equal_a part_n make_v a_o square_a equal_a then_o be_v it_o evident_a that_o square_a to_o be_v equal_a to_o the_o circle_n about_o which_o the_o first_o square_a be_v describe_v as_o you_o may_v here_o behold_v in_o these_o figure_n gentle_a friend_n the_o great_a desire_n which_o i_o have_v that_o both_o with_o pleasure_n and_o also_o profit_n thou_o may_v spend_v thy_o time_n in_o these_o excellent_a study_n do_v cause_v i_o here_o to_o furnish_v thou_o somewhat_o extraordinary_o about_o the_o circle_n not_o only_o by_o point_v unto_o thou_o the_o wellspring_n of_o archimedes_n his_o so_o much_o wonder_v at_o and_o just_o commend_v travail_n in_o the_o former_a 3._o theorem_n here_o repeat_v but_o also_o to_o make_v thou_o more_o apt_a to_o understand_v and_o practice_v this_o and_o other_o book_n follow_v where_o use_v of_o the_o circle_n may_v be_v have_v in_o any_o consideration_n as_o in_o cones_n cylinder_n and_o sphere_n etc_n etc_n ¶_o a_o corollary_n 1._o by_o archimedes_n second_v theorem_a as_o i_o have_v here_o allege_v they_o it_o be_v manifest_a that_o a_o parallelogram_n contain_v either_o under_o the_o semidiameter_n and_o half_a the_o circumference_n or_o under_o the_o half_a semidiameter_n and_o whole_a circumference_n of_o any_o circle_n be_v equal_a to_o the_o circle_n by_o the_o 41._o of_o the_o first_o and_o first_o of_o the_o sixâ—_o ¶_o a_o corollary_n 2._o likewise_o it_o be_v evident_a that_o the_o parallelogram_n contain_v under_o the_o semidiameter_n and_o half_a of_o any_o portion_n of_o the_o circumference_n of_o a_o circle_n give_v be_v equal_a to_o that_o sector_n of_o the_o same_o circle_n to_o which_o the_o whole_a portion_n of_o the_o circumference_n give_v do_v belong_v or_o you_o may_v use_v the_o half_a semidiameter_n and_o the_o whole_a portion_n of_o the_o circumference_n as_o side_n of_o the_o say_a parallelogram_n the_o far_a win_n and_o infer_v i_o commiâ—â—â—_n to_o your_o skill_n care_n and_o â—â—udyâ—_n but_o in_o a_o other_o sort_n will_v i_o geve_v you_o new_a aid_n and_o instruction_n here_o ¶_o a_o theorem_a of_o all_o circle_n the_o circumference_n to_o their_o own_o diameter_n have_v one_o and_o the_o same_o proportion_n in_o what_o one_o circle_n soever_o they_o be_v assign_v that_o be_v as_o archimedes_n have_v demonstrate_v almost_o as_o 22._o to_o 7_o or_o near_o if_o near_o be_v fouâ—d_v until_o the_o very_a precise_a proportion_n be_v demonstrate_v which_o what_o soever_o it_o be_v in_o all_o circumference_n to_o their_o proper_a diameter_n will_v be_v demonstrate_v one_o and_o the_o same_o a_o corollary_n 1._o wherefore_o if_o two_o circle_n be_v propound_v which_o suppose_v to_o be_v a_o and_o b_o as_z the_o circumference_n of_o a_o be_v to_o the_o circumference_n of_o b_o so_o be_v the_o diameter_n of_o a_o to_o the_o diameter_n of_o b._n for_o by_o the_o former_a theorem_a as_o the_o circumference_n of_o a_o be_v to_o his_o own_o diameter_n so_o be_v the_o circumference_n of_o b_o to_o his_o own_o diameter_n wherefore_o alternate_o as_o the_o circumference_n of_o a_o be_v to_o the_o circumference_n of_o b_o so_o be_v the_o diameter_n of_o a_o to_o the_o diameter_n of_o b_o which_o be_v require_v to_o be_v demonstrate_v a_o corollary_n 2._o it_o be_v now_o then_o evident_a that_o we_o can_v geve_v two_o circle_n who_o circumference_n one_o to_o the_o other_o shall_v have_v any_o proportion_n give_v in_o two_o right_a line_n the_o great_a mechanical_a use_n beside_o mathematical_a consideration_n which_o these_o two_o corollaryes_fw-mi may_v have_v in_o wheel_n of_o milles_n clock_n crane_n and_o other_o engine_n for_o water_n work_v and_o for_o war_n and_o many_o other_o purpose_n the_o earnest_n and_o witty_a mechanicien_n will_v soon_o boult_v out_o &_o glad_o practice_v â—_o john_n dee_fw-mi ¶_o the_o 2._o theorem_a the_o 2._o proposition_n circle_n be_v in_o that_o proportion_n the_o one_o to_o the_o other_o that_o the_o square_n of_o their_o diameter_n be_v svppose_v that_o there_o be_v two_o circle_n abcd_o and_o efgh_a and_o let_v their_o diameter_n be_v bd_o and_o fh_o then_o i_o say_v that_o as_o the_o square_n of_o the_o line_n db_o be_v to_o the_o square_n of_o the_o line_n fh_o case_n so_o be_v the_o circle_n abcd_o to_o the_o circle_n efgh_o for_o if_o the_o circle_n abcd_o be_v not_o unto_o the_o circle_n efgh_o as_o the_o square_n of_o the_o line_n bd_o be_v to_o the_o square_n of_o the_o line_n fh_o then_o the_o square_a of_o the_o line_n bd_o shall_v be_v to_o the_o square_n of_o the_o line_n fh_o as_o the_o circle_n abcd_o be_v to_o a_o superficies_n either_o less_o than_o the_o circle_n efgh_o or_o great_a first_o let_v the_o square_n of_o the_o line_n bd_o be_v to_o the_o square_n of_o the_o line_n fh_o as_o the_o circle_n abcd_o be_v to_o a_o superficies_n less_o than_o the_o circle_n efgh_o namely_o to_o the_o superficies_n s._n describe_v by_o the_o 6._o of_o the_o four_o in_o the_o circle_n efgh_v a_o square_a efgh_o now_o this_o square_a thus_o describe_v be_v great_a than_o the_o half_a of_o the_o circle_n efgh_o for_o if_o by_o the_o point_n e_o f_o g_o h_o we_o draw_v right_a line_n touch_v the_o circle_n be_v the_o square_a efgh_o be_v the_o half_a of_o the_o square_n describe_v about_o the_o circle_n but_o the_o square_n describe_v about_o the_o circle_n be_v great_a than_o the_o circle_n wherefore_o the_o square_a efgh_o which_o be_v inscribe_v in_o the_o circle_n be_v great_a than_o the_o half_a of_o the_o circle_n efgh_o divide_v the_o circumference_n of_o fg_o gh_o and_o he_o into_o two_o equal_a part_n in_o the_o point_n king_n l_o m_o n._n and_o draw_v these_o right_a line_n eke_o kf_n fl_fw-mi lg_n gm_n mh_o hn_o and_o ne._n wherefore_o every_o one_o of_o these_o triangle_n ekf_a flg_n gmh_o and_o hne_n be_v great_a than_o the_o half_a of_o the_o segment_n of_o the_o circle_n which_o be_v describe_v about_o it_o for_o if_o by_o the_o point_n king_n l_o m_o n_o be_v draw_v line_n touch_v the_o circle_n and_o then_o be_v make_v perfect_v the_o parallelogram_n make_v of_o the_o right_a line_n of_o fg_o gh_o &_o he_o every_o one_o of_o the_o triangle_n ekf_n flg_n gmh_o &_o hne_n be_v the_o half_a of_o the_o parallelogragramme_n which_o be_v describe_v about_o it_o by_o the_o 41._o of_o the_o first_o but_o the_o segment_n describe_v about_o it_o be_v less_o than_o the_o parallelogram_n wherefore_o every_o one_o
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
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
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 and_o the_o great_a segment_n thereof_o be_v the_o line_n acâ—_n 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 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 double_a to_o the_o square_n of_o ac_fw-la 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 treble_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 twice_o together_o with_o the_o square_n of_o the_o line_n ac_fw-la be_v the_o square_n of_o the_o line_n ab_fw-la and_o bc_o by_o the_o 7._o of_o the_o second_o wherefore_o the_o square_n of_o the_o line_n ab_fw-la and_o bc_o be_v treble_a to_o the_o square_n of_o the_o line_n ac_fw-la which_o be_v require_v to_o be_v demonstrate_v resolution_n of_o the_o 5._o theorem_a suppose_v that_o a_o certain_a 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 the_o line_n ac_fw-la and_o unto_o the_o line_n ab_fw-la add_v a_o line_n equal_a to_o the_o line_n ac_fw-la and_o let_v the_o same_o be_v ad._n theâ—_n i_o say_v that_o the_o line_n db_o be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n in_o the_o point_n a._n and_o the_o great_a segment_n thereof_o be_v the_o line_n ab_fw-la for_o forasmuch_o as_o the_o line_n db_o be_v divide_v by_o a_o extreme_a &_o mean_a proportion_n in_o the_o point_n a_o and_o the_o great_a segment_n thereof_o be_v the_o line_n ab_fw-la therefore_o as_o the_o line_n db_o be_v to_o the_o line_n basilius_n so_o be_v the_o line_n basilius_n to_o the_o line_n ad_fw-la but_o the_o line_n ad_fw-la be_v equal_a to_o the_o line_n ac_fw-la wherefore_o as_o the_o line_n db_o be_v to_o the_o line_n basilius_n so_o be_v the_o line_n basilius_n to_o the_o line_n ac_fw-la wherefore_o by_o conversion_n as_o the_o line_n bd_o be_v to_o the_o line_n dam_fw-ge so_o be_v the_o line_n ab_fw-la to_o the_o line_n bc_o by_o the_o corollary_n of_o the_o 19_o of_o the_o five_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 basilius_n be_v to_o the_o line_n ad_fw-la â—o_o be_v the_o line_n ac_fw-la to_o the_o line_n cb._n but_o the_o line_n ad_fw-la be_v equal_a to_o the_o line_n ac_fw-la wherefore_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 line_n ac_fw-la to_o the_o line_n cb._n and_o so_o it_o be_v indeed_o for_o the_o line_n ab_fw-la be_v by_o supposition_n divide_v by_o a_o extreme_a and_o mean_a proportion_n in_o the_o point_n c._n composition_n of_o the_o 5._o theorem_a now_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 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 line_n ac_fw-la to_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 ad._n wherefore_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 ac_fw-la to_o the_o line_n cb._n wherefore_o by_o composition_n by_o the_o 18._o of_o the_o five_o as_o the_o line_n bd_o be_v to_o the_o line_n dam_fw-ge so_o be_v the_o line_n ab_fw-la 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 as_o the_o line_n db_o be_v to_o the_o line_n basilius_n so_o be_v the_o line_n basilius_n to_o the_o line_n ac_fw-la but_o the_o line_n ac_fw-la be_v equal_a to_o the_o line_n ad._n wherefore_o as_o the_o line_n db_o be_v to_o the_o line_n basilius_n so_o be_v the_o line_n basilius_n to_o the_o line_n ac_fw-la wherefore_o the_o line_n db_o be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n in_o the_o point_n a_o and_o his_o great_a segment_n be_v the_o line_n ab_fw-la which_o be_v require_v to_o be_v demonstrate_v a_o advice_n by_o john_n dee_n add_v sing_v it_o be_v doubtless_a that_o this_o parcel_n of_o resolution_n and_o composition_n be_v not_o of_o euclides_n do_v it_o can_v not_o â—ustly_o be_v impute_v to_o euclid_n that_o he_o have_v thereby_o either_o superfluity_n or_o any_o part_n disproportion_v in_o his_o whole_a composition_n elemental_a and_o though_o for_o one_o thing_n one_o good_a demonstration_n well_o suffice_v for_o stablish_v of_o the_o verity_n yet_o oâ—_n one_o thing_n diverse_o demonstrate_v to_o the_o diligent_a examiner_n of_o the_o diverse_a mean_n by_o which_o that_o variety_n arise_v do_v grow_v good_a occasion_n of_o invent_v demonstration_n where_o matter_n be_v more_o strange_a hard_a and_o barren_a also_o though_o resolution_n be_v not_o in_o all_o euclid_n before_o use_v yet_o thanks_n be_v to_o be_v give_v to_o the_o greek_a scholic_a writer_n who_o do_v leave_v both_o the_o definition_n and_o also_o so_o short_a and_o easy_a example_n of_o a_o method_n so_o ancient_a and_o so_o profitable_a the_o antiquity_n of_o it_o be_v above_o 2000_o year_n it_o be_v to_o weâ—e_v ever_o since_o plato_n his_o time_n and_o the_o profit_n thereof_o so_o great_a that_o thus_o i_o find_v in_o the_o greek_a record_v page_n 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d proclus_n have_v speak_v of_o some_o by_o nature_n excellent_a in_o invent_v demonstration_n pithy_a and_o brief_a say_v yet_o be_v there_o method_n give_v for_o that_o purpose_n and_o in_o deed_n that_o the_o best_a which_o by_o resolution_n reduce_v the_o thing_n inquire_v of_o to_o a_o undoubted_a principle_n which_o method_n plato_n teach_v leodamas_n as_o iâ—_z report_v and_o he_o be_v register_v thereby_o to_o have_v be_v the_o inventor_n of_o many_o thing_n in_o geometry_n and_o very_o in_o problem_n it_o be_v the_o chief_a aid_n for_o win_v and_o order_v a_o demonstration_n first_o by_o supposition_n of_o the_o thing_n inquire_v of_o to_o be_v do_v by_o due_a and_o orderly_a resolution_n to_o bring_v it_o to_o a_o stay_n at_o a_o undoubted_a verity_n in_o which_o point_n of_o art_n great_a abundance_n of_o example_n be_v to_o be_v see_v in_o that_o excellent_a and_o mighty_a mathematician_n archimedes_n &_o in_o his_o expositor_n eutocius_n in_o menaechmus_n likewise_o and_o in_o diocles_n book_n the_o pytiâ—s_n and_o in_o many_o other_o and_o now_o for_o as_o much_o as_o our_o euclid_n in_o the_o last_o six_o proposition_n of_o this_o thirteen_o book_n propound_v and_o conclude_v those_o problem_n which_o be_v the_o end_n scope_n and_o principal_a purpose_n to_o which_o all_o the_o premise_n of_o the_o 12._o book_n and_o the_o rest_n of_o this_o thirteen_o be_v direct_v and_o order_v it_o shall_v be_v artificial_o do_v and_o to_o a_o great_a commodity_n by_o resolution_n backward_o from_o these_o 6._o problem_n to_o return_v to_o the_o first_o definition_n of_o the_o first_o book_n i_o mean_v to_o the_o definition_n of_o a_o point_n which_o be_v nothing_o hard_a to_o do_v and_o i_o do_v counsel_n all_o such_o as_o desire_v to_o attain_v to_o the_o profound_a knowledge_n of_o geometry_n arithmetic_n or_o any_o branch_n of_o the_o science_n mathematical_a so_o by_o resolution_n discrete_o and_o advise_o to_o resolve_v unlose_v unjoint_n and_o disseaver_v every_o part_n of_o any_o work_n mathematical_a that_o therbyâ—_n aswell_o the_o due_a place_n of_o every_o verity_n and_o his_o proof_n as_o also_o what_o be_v either_o superfluous_a or_o want_v may_v evident_o appear_v for_o so_o to_o invent_v &_o there_o with_o to_o order_v their_o write_n be_v the_o custom_n of_o they_o who_o in_o the_o old_a time_n be_v most_o excellent_a and_o i_o for_o my_o part_n in_o write_v any_o mathematical_a conclusion_n which_o require_v great_a discourse_n at_o length_n have_v find_v by_o experience_n the_o commodity_n of_o it_o such_o that_o to_o do_v other_o way_n be_v to_o i_o a_o confusion_n and_o a_o unmethodicall_a heap_v of_o matter_n together_o beside_o the_o difficulty_n of_o invent_v the_o matter_n to_o be_v dispose_v and_o order_v i_o have_v occasion_n thus_o to_o geve_v you_o friendly_a advice_n for_o your_o beâ—ofeâ—_n because_o some_o of_o late_a have_v inveigh_v against_o euclid_n or_o theon_n in_o this_o place_n otherwise_o than_o i_o will_v wish_v they_o have_v the_o 6._o theorem_a the_o 6._o proposition_n if_o a_o rational_a right_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 a_o irrational_a line_n of_o that_o kind_n which_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 be_v devide_o 〈◊〉_d extreme_a
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
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
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
overthrow_v and_o overwhelm_v the_o whole_a world_n he_o be_v utter_o rude_a and_o ignorant_a in_o the_o greek_a tongue_n so_o that_o certain_o he_o never_o read_v euclid_n in_o the_o greek_a nor_o of_o like_a translate_v out_o of_o the_o greek_a but_o have_v it_o translate_v out_o of_o the_o arabike_a tongue_n the_o arabian_n be_v man_n of_o great_a study_n and_o industry_n and_o common_o great_a philosopher_n notable_a physician_n and_o in_o mathematical_a art_n most_o expert_a so_o that_o all_o kind_n of_o good_a learning_n flourish_v and_o reign_v amongst_o they_o in_o a_o manner_n only_o these_o man_n turn_v whatsoever_o good_a author_n be_v in_o the_o greek_a tongue_n of_o what_o art_n and_o knowledge_n so_o ever_o it_o be_v into_o the_o arabike_a tongue_n and_o from_o thence_o be_v many_o of_o they_o turn_v into_o the_o latin_a and_o by_o that_o mean_v many_o greek_a author_n come_v to_o the_o hand_n of_o the_o latin_n and_o not_o from_o the_o first_o fountain_n the_o greek_a tongue_n wherein_o they_o be_v first_o write_v as_o appear_v by_o many_o word_n of_o the_o arabike_a tongue_n yet_o remain_v in_o such_o book_n as_o be_v zenith_n nadir_n helmuayn_fw-mi helmuariphe_n and_o infinite_a such_o other_o which_o arabian_n also_o in_o translate_n such_o greek_a work_n be_v accustom_v to_o add_v as_o they_o think_v good_a &_o for_o the_o full_a understanding_n of_o the_o author_n many_o thing_n as_o be_v to_o be_v see_v in_o diverse_a author_n as_o namely_o in_o theodosius_n de_fw-la sphera_fw-la where_o you_o see_v in_o the_o old_a translation_n which_o be_v undoubteldy_a out_o of_o the_o arabike_a many_o proposition_n almost_o every_o three_o or_o four_o leaf_n some_o such_o copy_n of_o euclid_n most_o likely_a do_v campanus_n follow_v wherein_o he_o find_v those_o proposition_n which_o he_o have_v more_o &_o above_o those_o which_o be_v find_v in_o the_o greek_a set_v out_o by_o hypsicles_n and_o that_o not_o only_o in_o this_o 15._o book_n but_o also_o in_o the_o 14._o book_n wherein_o also_o you_o find_v many_o proposition_n more_o they_o be_v find_v in_o the_o greek_a set_v out_o also_o by_o hypsicles_n likewise_o in_o the_o book_n before_o you_o shall_v find_v many_o proposition_n add_v and_o many_o invert_v and_o set_v out_o of_o order_n far_o otherwise_o than_o they_o be_v place_v in_o the_o greek_a examplar_n flussas_n also_o a_o diligent_a restorer_n of_o euclid_n a_o man_n also_o which_o have_v well_o deserve_v of_o the_o whole_a art_n of_o geometry_n have_v add_v moreover_o in_o this_o book_n as_o also_o in_o the_o former_a 14._o book_n he_o add_v 8._o proposition_n 9_o proposition_n of_o his_o own_o touch_v the_o inscription_n and_o circumscription_n 〈…〉_o body_n very_o singular_a â—ndoubtedly_o and_o witty_a all_o which_o for_o that_o nothing_o shall_v want_v to_o the_o desirous_a lover_n of_o knowledge_n i_o have_v faithful_o with_o no_o small_a pain_n turn_v and_o whereas_o flâ—ssâ—â—_n in_o the_o begin_n of_o the_o eleven_o book_n namely_o in_o the_o end_n of_o the_o definition_n there_o â—eâ—_n put_v two_o definition_n of_o the_o inscription_n and_o circumscription_n of_o solid_n or_o corporal_a figure_n within_o or_o about_o the_o one_o the_o other_o which_o certain_o be_v not_o to_o be_v reject_v yet_o for_o that_o until_o this_o present_a 15._o book_n there_o be_v no_o mention_n make_v of_o the_o inscription_n or_o circumscription_n of_o these_o body_n i_o think_v it_o not_o so_o convenient_a thâ—râ—_n to_o place_v they_o but_o to_o refer_v they_o to_o the_o begin_n of_o this_o 15._o book_n where_o they_o be_v in_o manner_n of_o necessity_n require_v to_o the_o elucidation_n of_o the_o proposition_n and_o dâ—monstrationâ—_n of_o the_o same_o the_o definition_n be_v these_o definition_n 1._o a_o solid_a figure_n be_v then_o â—aid_v to_o be_v inscribe_v in_o a_o solid_a figure_n when_o the_o angle_n of_o the_o figure_n inscribe_v touch_n together_o at_o one_o time_n either_o the_o angle_n of_o the_o figure_n circumscribe_v or_o the_o superficiece_n or_o the_o side_n definition_n 2._o a_o solid_a figure_n be_v then_o say_v to_o be_v circumscribe_v about_o a_o solid_a figure_n when_o together_o at_o one_o time_n either_o the_o angle_n or_o the_o superficiece_n or_o the_o side_n of_o the_o figure_n circumscribe_v â—ouch_v the_o angle_n of_o the_o figure_n inscribe_v in_o the_o fourâ—â—_n book_n in_o the_o definition_n of_o the_o inscription_n or_o circumscription_n of_o plain_a rectiline_n figure_v one_o with_o in_o or_o about_o a_o other_o be_v require_v that_o all_o the_o angle_n of_o the_o figuâ—â—_n inscribe_v shall_v at_o one_o time_n touch_v all_o the_o side_n of_o the_o figure_n circumscribe_v but_o in_o the_o five_o regular_a solid_n â—o_o who_o chief_o these_o two_o definition_n pertain_v for_o that_o the_o number_n of_o their_o angle_n superficiece_n &_o side_n be_v not_o equal_a one_o compare_v to_o a_o other_o it_o be_v not_o of_o necessity_n that_o all_o the_o angle_n of_o the_o solid_a inscribe_v shall_v together_o at_o one_o time_n touch_v either_o all_o the_o angle_n or_o all_o the_o superficiece_n or_o all_o the_o side_n of_o the_o solid_a circumscribe_v but_o it_o be_v sufficient_a that_o those_o angle_n of_o the_o inscribe_v solid_a which_o touch_n do_v at_o one_o time_n together_o each_o touch_n some_o one_o angle_n of_o the_o figure_n circumscribe_v or_o some_o one_o base_a or_o some_o one_o side_n so_o that_o if_o the_o angle_n of_o the_o inscribe_v figure_n do_v at_o one_o time_n touch_v the_o angle_n of_o the_o figure_n circumscribe_v none_o of_o they_o may_v at_o the_o same_o time_n touch_v either_o the_o base_n or_o the_o side_n of_o the_o same_o circumscribe_v figure_n and_o so_o if_o they_o touch_v the_o base_n they_o may_v touch_v neither_o angle_n nor_o side_n and_o likewise_o if_o they_o touch_v the_o side_n they_o may_v touch_v neither_o angle_n nor_o base_n and_o although_o sometime_o all_o the_o angle_n of_o the_o figure_n inscribe_v can_v not_o touch_v either_o the_o angle_n or_o the_o base_n or_o the_o side_n of_o the_o figure_n circumscribe_v by_o reason_n the_o number_n of_o the_o angle_n base_n or_o side_n of_o the_o say_a figure_n circumscribe_v want_v of_o the_o number_n of_o the_o angle_n of_o the_o â—igure_n inscribe_v yet_o shall_v those_o angle_n of_o the_o inscribe_v figure_n which_o touch_n so_o touch_v that_o the_o void_a place_n leave_v between_o the_o inscribe_v and_o circumscribe_v figure_n shall_v on_o every_o side_n be_v equal_a and_o like_a as_o you_o may_v afterward_o in_o this_o fifteen_o book_n most_o plain_o perceive_v ¶_o the_o 1._o proposition_n the_o 1._o problem_n in_o a_o cube_n give_v to_o describe_v admonish_v a_o trilater_n equilater_n pyramid_n svppose_v that_o the_o cube_fw-la give_v be_v abcdefgh_o in_o the_o same_o cube_fw-la it_o be_v require_v to_o inscribe_v a_o tetrahedron_n draw_v these_o right_a line_n ac_fw-la construction_n ce_fw-fr ae_n ah_o eh_o hc_n demonstration_n now_o it_o be_v manifest_a that_o the_o triangle_n aec_o ahe_o ahc_n and_o i_o be_v equilater_n for_o their_o side_n be_v the_o diameter_n of_o equal_a square_n wherefore_o aech_n be_v a_o trilater_n equilater_n pyramid_n or_o tetrahedron_n &_o it_o be_v inscribe_v in_o the_o cube_fw-la give_v by_o the_o first_o definition_n of_o this_o book_n which_o be_v require_v to_o be_v do_v ¶_o the_o 2._o proposition_n the_o 2._o problem_n in_o a_o trilater_n equilater_n pyramid_n give_v to_o describe_v a_o octohedron_n svppose_v that_o the_o trilater_n equilater_n pyramid_n give_v be_fw-mi abcd_o who_o side_n let_v be_v divide_v into_o two_o equal_a part_n in_o the_o point_n e_o z_o i_o king_n l_o t._n construction_n and_o draw_v these_o 12._o right_a line_n ez_o zi_n je_n kl_o lt_n tk_n eke_o kz_o zl_n li_n it_o and_o te_fw-la which_o 12._o right_a line_n be_v demonstration_n by_o the_o 4._o of_o the_o first_o equal_a for_o they_o subtend_v equal_a plain_a angle_n of_o the_o base_n of_o the_o pyramid_n and_o those_o equal_a angle_n be_v contain_v under_o equal_a side_n namely_o under_o the_o half_n of_o the_o side_n of_o the_o pyramid_n wherefore_o the_o triangle_n tkl_n tli_n tie_v tek_v zkl_n zli_n zib_n zek_n be_v equilater_n and_o they_o limitate_v and_o contain_v the_o solid_a tklezi_n wherefore_o the_o solid_a tklezi_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 angle_n of_o the_o same_o octohedron_n do_v touch_v the_o side_n of_o the_o pyramid_n abcd_o in_o the_o point_n e_o z_o i_o t_o edward_n l._n wherefore_o the_o octohedron_n be_v inscribe_v in_o the_o pyramid_n by_o the_o 1._o definition_n of_o this_o book_n wherefore_o in_o the_o trilater_n equilater_n pyramid_n give_v be_v inscribe_v a_o octohedron_n which_o be_v require_v to_o be_v do_v a_o corollary_n add_v by_o flussas_n hereby_o it_o be_v manifest_a that_o a_o pyramid_n be_v cut_v into_o two_o
equal_a part_n by_o every_o one_o of_o the_o three_o equal_a square_n which_o divide_v the_o octohedron_n into_o two_o equal_a part_n and_o perpendicular_o for_o the_o three_o diameter_n of_o those_o square_n do_v in_o the_o centre_n cut_v the_o one_o the_o other_o into_o two_o equal_a part_n and_o perpendicular_o by_o the_o three_o corollary_n of_o the_o 1â—_o of_o the_o thirteen_o which_o square_n as_o for_o example_n the_o square_a ekli_n do_v divide_v in_o sunder_o the_o pyramid_n and_o the_o prism_n namely_o the_o pyramid_n kltd_v and_o the_o prism_n klteia_n from_o the_o pyramid_n ekzb_n and_o the_o prism_n ekzilg_n which_o pyramid_n be_v equal_a the_o one_o to_o the_o other_o and_o so_o also_o be_v the_o prism_n equal_v the_o one_o to_o the_o other_o by_o the_o 3._o of_o the_o twelve_o and_o in_o like_a sort_n do_v the_o rest_n of_o the_o square_n namely_o kzit_n and_o zlte_n which_o square_n by_o the_o second_o corollary_n of_o the_o 14._o of_o the_o thirteen_o do_v divide_v the_o octohedron_n into_o two_o equal_a part_n ¶_o the_o 3._o proposition_n the_o 3._o problem_n in_o a_o cube_fw-la give_v to_o describe_v a_o octohedron_n take_v a_o cube_n namely_o abcdefgh_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 construction_n and_o draw_v right_a line_n couple_v together_o the_o section_n as_o for_o example_n these_o right_a line_n pq_n and_o r_n which_o shall_v be_v equal_a unto_o the_o side_n of_o the_o cube_fw-la by_o the_o 33._o of_o the_o first_o and_o shall_v divide_v the_o one_o the_o other_o into_o two_o equal_a part_n in_o the_o midst_n of_o the_o diameter_n agnostus_n in_o the_o point_n i_o by_o the_o corollary_n of_o the_o 34._o of_o the_o first_o wherefore_o the_o point_n i_o be_v the_o centre_n of_o the_o base_a of_o the_o cube_fw-la and_o by_o the_o same_o reason_n may_v be_v find_v out_o the_o centre_n of_o the_o rest_n of_o the_o base_n which_o let_v be_v the_o point_n king_n l_o oh_o n_o m._n and_o draw_v these_o right_a line_n li_n in_o more_n ol_n king_n kl_o km_o ko_o ni_fw-fr nl_o nm_o &_o no_o demonstration_n and_o now_o forasmuch_o as_o the_o angle_n ipl_n be_v a_o right_a angle_n by_o the_o 10._o of_o the_o eleven_o for_o the_o line_n ip_n and_o pl_z be_v parallel_n to_o the_o line_n ramires_n and_o ab_fw-la and_o the_o right_a line_n il_fw-fr subtend_v the_o right_a angle_n ipl_n namely_o it_o subtend_v the_o half_a side_n of_o the_o cube_fw-la which_o contain_v the_o right_a angle_n ipl_n and_o likewise_o the_o right_a line_n in_o subtend_v the_o angle_n iqm_n which_o be_v equal_a to_o the_o same_o angle_n ipl_n and_o be_v contain_v under_o right_a line_n equal_a to_o the_o right_a line_n which_o contain_v the_o angle_n ipl_n wherefore_o the_o right_a line_n in_o be_v equal_a to_o the_o right_a line_n ill_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 every_o one_o of_o the_o right_a line_n more_o ol_n king_n kl_o km_o ko_o ni_fw-fr nl_o nm_o and_o no_o which_o subtend_v angle_n equal_a to_o the_o self_n same_o angle_n ipl_n and_o be_v contain_v under_o side_n equal_a to_o the_o side_n which_o contain_v the_o angle_n ipl_n be_v equal_a to_o the_o right_a line_n il._n wherefore_o the_o triangle_n kli_n klo_n kmi_n kmo_fw-la and_o nli_n nlo_n nmi_n nmo_fw-la be_v equilater_n and_o equal_a and_o they_o contain_v the_o solid_a iklonm_n wherefore_o iklonm_n be_v a_o octohedron_n by_o the_o 23._o definition_n of_o the_o eleven_o and_o forasmuch_o as_o the_o angle_n thereof_o do_v altogether_o in_o the_o point_n i_o king_n l_o oh_o n_o m_o touch_v the_o base_n of_o the_o cube_fw-la which_o contain_v it_o it_o follow_v that_o the_o octohedron_n be_v inscribe_v in_o the_o cube_fw-la by_o the_o first_o definition_n of_o this_o book_n wherefore_o in_o the_o cube_fw-la give_v be_v describe_v a_o octohedron_n which_o be_v require_v to_o be_v do_v ¶_o a_o corollary_n aâ—ded_v by_o â—lussâ—â—_n hereby_o it_o be_v manifest_a that_o right_a line_n join_v together_o the_o centre_n of_o the_o opposite_a base_n of_o the_o cube_fw-la do_v 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 cube_fw-la or_o in_o the_o centre_n of_o the_o sphere_n which_o contain_v the_o cube_fw-la for_o forasmuch_o as_o the_o right_a line_n lm_o and_o io_o which_o knâ—â—_n together_o the_o centre_n of_o the_o opposite_a base_n of_o the_o cube_fw-la do_v also_o knit_v together_o the_o opposite_a angel_n of_o the_o octahedron_n inscribe_v in_o the_o cube_fw-la it_o follow_v by_o the_o 3._o corollary_n of_o the_o 14._o of_o the_o thirteen_o that_o those_o line_n lm_o and_o io_o do_v cut_v the_o one_o the_o other_o into_o two_o equal_a part_n in_o a_o point_n but_o the_o diameter_n of_o the_o cube_fw-la do_v also_o cut_v the_o one_o the_o other_o into_o two_o equal_a part_n by_o the_o 39_o of_o the_o eleven_o wherefore_o that_o point_n shall_v be_v the_o centre_n of_o the_o sphere_n which_o contain_v the_o câ—â—â—_n for_o make_v that_o point_n the_o centre_n and_o the_o space_n some_o one_o of_o the_o semidiameter_n describe_v a_o sphere_n and_o it_o shall_v pass_v by_o the_o angle_n of_o the_o cube_fw-la and_o likewise_o make_v the_o same_o point_n the_o centre_n and_o the_o space_n half_o of_o the_o line_n lm_o describe_v a_o sphere_n and_o it_o shall_v also_o pass_v by_o the_o angle_n of_o the_o octohedron_n ¶_o the_o 4._o proposition_n the_o 4._o problem_n in_o a_o octohedron_n give_v to_o describe_v a_o cube_n svppose_v that_o the_o octohedron_n give_v be_v abgdez_n and_o let_v the_o two_o pyramid_n thereof_o be_v abgde_v and_o zbgde_v construction_n and_o take_v the_o centre_n of_o the_o triangle_n of_o the_o pyramid_n abgde_v that_o be_v take_v the_o centre_n of_o the_o circle_n which_o contain_v those_o triangle_n and_o let_v those_o centre_n be_v the_o pointâ—s_n t_o i_o king_n l._n and_o by_o these_o centre_n let_v there_o be_v draw_v parallel_n line_n â—o_o the_o side_n of_o the_o square_n bgde_v which_o parallel_a â—igâ—â—_n linâ—â—_n let_v be_v mtn_n nlx_n xko_n &_o oim._n demonstration_n and_o forasmuch_o as_o thâ—se_a parallel_n right_a line_n do_v by_o the_o 2._o of_o the_o six_o cut_v the_o equal_a right_a line_n ab_fw-la agnostus_n ad_fw-la and_o ae_n proportional_o therefore_o they_o concur_v in_o the_o point_n m_o n_o x_o o._n wherefore_o the_o right_a line_n mn_v nx_o xo_o and_o om_fw-la which_o subtend_v equal_a angle_n set_v at_o the_o point_n a_o &_o contain_v under_o squall_n right_a line_n be_v equal_a by_o the_o 4._o of_o the_o first_o and_o moreover_o see_v that_o they_o be_v parallel_n unto_o the_o line_n bg_o gd_o de_fw-fr eâ—_n which_o make_v a_o square_a therefore_o mnxo_n be_v also_o a_o square_a by_o the_o 10._o of_o the_o eleven_o wherefore_o also_o by_o the_o 15._o of_o the_o â—ame_n the_o square_a mnxo_n be_v parallel_v to_o the_o square_n bgde_v for_o all_o tâ—e_z right_a line_n touch_v the_o one_o the_o other_o in_o the_o point_n of_o their_o section_n from_o the_o centre_n t_o i_o king_n l_o draw_v these_o right_a line_n ti_o ik_fw-mi kl_o ltâ—_n and_o draw_v the_o right_a line_n aic_n and_o forasmuch_o as_o i_o be_v the_o centre_n of_o the_o equilater_n triangle_n abe_n therefore_o the_o right_a line_n ai_fw-fr be_v extend_v cut_v the_o right_a line_n be_v into_o two_o equal_a part_n by_o the_o corollary_n of_o the_o 12._o of_o the_o thirteen_o and_o forasmuch_o as_o more_n be_v a_o parallel_n to_o be_v therefore_o the_o triangle_n aio_fw-la be_v like_a to_o the_o whole_a triangle_n ace_n by_o the_o corollary_n of_o the_o 2._o of_o the_o six_o and_o the_o right_a line_n more_n be_v divide_v into_o two_o equal_a part_n in_o the_o point_n i_o by_o the_o 4._o of_o the_o six_o and_o by_o the_o same_o reason_n may_v we_o prove_v that_o the_o right_a line_n mn_v nx_o xo_o be_v divide_v into_o two_o equal_a part_n in_o the_o point_n t_o l_o k._n wherefore_o also_o again_o the_o base_n ti_fw-mi ik_fw-mi kl_o lt_n which_o subtend_v the_o angle_n set_v at_o the_o point_n m_o oh_o x_o n_o which_o angle_n be_v right_a angle_n and_o be_v contain_v under_o equal_a side_n those_o base_n i_o say_v be_v equal_a and_o forasmuch_o as_o tim_n be_v a_o isosceles_a triangle_n therefore_o the_o angle_n set_v at_o the_o base_a namely_o the_o angle_n mti_n and_o mit_fw-fr be_v equal_a by_o the_o â—â—_o of_o the_o first_o but_o the_o angle_n m_o be_v a_o right_a angle_n wherefore_o each_o of_o the_o angle_n mit_fw-fr and_o mti_n be_v the_o half_a of_o a_o right_a angle_n and_o by_o the_o same_o reason_n the_o angle_n oik_a &_o oki_n be_v equal_a wherefore_o the_o angle_n remain_v namely_o tik_fw-mi be_v a_o right_a angle_n
by_o the_o 13._o of_o the_o first_o for_o the_o right_a line_n ti_fw-mi and_o ik_fw-mi be_v set_v upon_o the_o line_n mo._n and_o by_o the_o same_o reason_n may_v the_o rest_n of_o the_o angle_n namely_o ikl_n klt_n lti_fw-la be_v prove_v right_a angle_n and_o they_o be_v in_o one_o and_o the_o self_n same_o plain_a superficies_n namely_o mnxo_n by_o the_o 7._o of_o the_o eleven_o wherefore_o the_o right_a line_n which_o join_v together_o the_o centre_n of_o the_o plain_a superficial_a triangle_n which_o make_v the_o solid_a angle_n a_o do_v make_v the_o square_a itkl_n and_o by_o the_o same_o reason_n may_v be_v prove_v that_o the_o plain_a superficial_a triangle_n of_o the_o rest_n of_o the_o five_o solid_a angle_n of_o the_o octohedron_n set_v at_o the_o point_n b_o g_o z_o d_o e_o do_v in_o the_o centre_n of_o their_o base_n receive_v square_n so_o that_o there_o be_v in_o number_n six_o square_n for_o every_o octohedron_n have_v six_o solid_a angle_n and_o those_o square_n be_v equal_a for_o their_o side_n do_v contain_v equal_a angle_n of_o inclination_n contain_v under_o equal_a side_n namely_o under_o those_o side_n which_o be_v draw_v from_o the_o centre_n to_o the_o side_n of_o the_o equal_a triangle_n by_o the_o 2._o corollary_n of_o the_o 18._o of_o the_o thirteen_o wherefore_o itklrpus_n be_v a_o cube_fw-la by_o the_o 21._o definition_n of_o the_o eleven_o and_o have_v his_o angle_n in_o the_o centre_n of_o the_o base_n of_o the_o octohedron_n and_o therefore_o be_v inscribe_v in_o it_o by_o the_o first_o definition_n of_o this_o book_n wherefore_o in_o a_o octohedron_n give_v be_v describe_v a_o cube_fw-la which_o be_v require_v to_o be_v do_v the_o 5._o proposition_n the_o 5._o problem_n in_o a_o icosahedron_n give_v to_o describe_v a_o dodecahedron_n take_v a_o icosahedron_n one_o of_o who_o solid_a angle_n let_v be_v z._n construction_n now_o forasmuch_o as_o by_o those_o thing_n which_o have_v be_v prove_v in_o the_o 16._o of_o the_o thirteen_o the_o base_n of_o the_o triangle_n which_o contain_v the_o angle_n of_o the_o icosahedron_n do_v make_v a_o pentagon_n inscribe_v in_o a_o circle_n let_v that_o pentagon_n be_v abgde_v which_o be_v make_v of_o the_o five_o base_n of_o the_o triangle_n who_o plain_a superficial_a angle_n remain_v make_v the_o solid_a angle_n give_v namely_o z._n and_o take_v the_o centre_n of_o the_o circle_n which_o contain_v the_o foresay_a triangle_n which_o centre_n let_v be_v the_o point_n i_o t_o edward_n m_o l_o and_o draw_v these_o right_a line_n it_o tk_n km_o ml_o li._n demonstration_n now_o then_o a_o perpendicular_a line_n draw_v from_o the_o point_n z_o to_o the_o plain_a superficies_n of_o the_o pentagon_n abgde_v shall_v fall_v upon_o the_o centre_n of_o the_o circle_n which_o contain_v the_o pentagon_n abgde_v by_o those_o thing_n which_o have_v be_v prove_v in_o the_o self_n same_o 16._o of_o the_o thirteen_o moreover_o perpendicular_a line_n draw_v from_o the_o centre_n to_o the_o side_n of_o the_o pentagon_n abgde_v shall_v in_o the_o point_n c_o n_o oh_o where_o they_o fall_v cut_v the_o right_a line_n ab_fw-la bg_o gd_o into_o two_o equal_a part_n by_o the_o 3._o of_o the_o three_o draw_v these_o right_a line_n cn_fw-la and_o no_o and_o forasmuch_o as_o the_o angle_n cbn_n and_o ngo_n be_v equal_a and_o be_v contain_v under_o equal_a side_n therefore_o the_o base_a cn_fw-la be_v equal_a to_o the_o base_a no_o by_o the_o 4._o of_o the_o first_o moreover_o perpendicular_a line_n drâ—â—â—e_v from_o the_o point_n z_o to_o the_o bâ—sâ—â—_n of_o the_o pentagon_n abgde_v shall_v likewise_o cut_v the_o base_n into_o two_o equal_a part_n by_o thâ—â—_n of_o the_o three_o for_o the_o perpendicular_n pass_v by_o the_o centre_n by_o the_o corollary_n of_o the_o 12._o of_o the_o thirteen_o wherefore_o thâ—se_a perpendicular_a line_n shall_v fall_v upon_o the_o point_n c_o n_o o._n and_o now_o forasmuch_o as_o the_o right_a line_n zi_n ig_n be_v equal_a to_o the_o right_a line_n zt_v tn_n &_o also_o to_o the_o right_a line_n zk_v ko_o by_o reason_n of_o the_o likeness_n of_o the_o equal_a triangle_n therefore_o the_o line_n it_o be_v a_o parallel_n to_o the_o line_n cn_fw-la and_o so_o also_o be_v the_o line_n tk_n to_o the_o line_n no_o by_o the_o 2._o of_o the_o six_o wherefore_o the_o angle_n itk_n and_o cno_n be_v equal_a by_o the_o 11._o of_o the_o eleven_o again_o forasmuch_o as_o the_o triangle_n cbn_n and_o ngo_n be_v isoscel_n triangle_n therefore_o the_o angle_n bcn_n and_o bnc_n be_v equal_a by_o the_o 5._o of_o the_o first_o and_o by_o the_o same_o reason_n the_o angle_n gno_fw-la and_o gon_n be_v equal_a and_o moreover_o the_o angle_n bcn_n and_o bnc_n be_v equal_a to_o the_o angle_n gno_fw-la and_o go_v for_o that_o the_o triangle_n cbn_n and_o ngo_n be_v equal_a and_o like_a bâ—â—_n the_o three_o angle_n bnc_fw-la cno_n ong_n be_v equal_a to_o two_o right_a angle_n by_o the_o 13._o of_o the_o first_o for_o that_o upon_o the_o right_a line_n bâ—_n be_v set_v the_o right_a line_n cn_fw-la &_o on_o and_o the_o three_o angle_n of_o the_o triangle_n cbn_n namely_o the_o angle_n bnc_fw-la bcn_n or_o gno_n for_o the_o angle_n gno_n be_v equal_a to_o the_o angle_n bcn_n as_o it_o have_v be_v prove_v and_o nbc_n be_v also_o equal_a to_o two_o right_a angle_n by_o the_o 32._o of_o the_o first_o wherefore_o take_v away_o the_o angle_n bnc_n &_o gno_fw-la the_o angle_n remain_v namely_o cno_n be_v equal_a to_o the_o angle_n remain_v namely_o to_o cbn_n wherefore_o also_o the_o angle_n itk_n which_o be_v prove_v to_o be_v equal_a to_o the_o angle_n cno_n be_v equal_a to_o the_o angle_n cbn_n wherefore_o itk_n be_v the_o angle_n of_o a_o pentagon_n and_o by_o the_o same_o reason_n may_v be_v proâ—ed_v that_o the_o rest_n of_o the_o angle_n name_o the_o angle_n till_o ilm_n lmk_n mkt_v be_v equal_a to_o the_o rest_n of_o the_o angle_n namely_o to_o bae_o aed_n edg_n dgb_n wherefore_o itkml_n be_v a_o equilater_n and_o equiangle_n pentagon_n by_o the_o 4._o of_o the_o first_o for_o the_o equal_a base_n of_o the_o pentagon_n itkml_n do_v subtend_v equal_a angle_n set_v at_o the_o point_n z_o and_o comprehend_v under_o equal_a side_n moreover_o it_o be_v manifest_a that_o the_o pentagon_n i_o tkml_n be_v in_o one_o and_o the_o self_n same_o plain_a superficies_n for_o forasmuch_o as_o the_o angle_n onc_n and_o ncp_n be_v in_o one_o and_o the_o self_n sâ—me_v plain_a superficies_n namely_o in_o the_o superficies_n abgde_v but_o unto_o the_o same_o plain_a superficies_n the_o plain_a superficiece_n of_o the_o angle_n kti_fw-la and_o until_o be_v parallel_n by_o the_o 15._o of_o the_o eleven_o and_o the_o triangle_n kti_n and_o until_o concur_v wherefore_o they_o be_v in_o one_o and_o the_o self_n same_o plain_a superficies_n by_o the_o corollary_n of_o the_o 16._o of_o the_o eleven_o and_o by_o the_o same_o reason_n so_o may_v we_o prove_v that_o the_o triangle_n ilm_n lmk_n mkt_v be_v in_o the_o self_n same_o plain_a superficies_n wherein_o be_v the_o triangle_n kti_n and_o till_o wherefore_o the_o pentagon_n itkml_n be_v in_o one_o and_o the_o self_n same_o plain_a superficies_n wherefore_o the_o solid_a angle_n of_o the_o icosahedron_n namely_o the_o solid_a angle_n at_o the_o point_n z_o subtend_v a_o equilater_n and_o equiangle_n pentagon_n plain_a superficies_n which_o pentagon_n have_v his_o plain_a superficial_a angle_n in_o the_o centre_n of_o the_o triangle_n which_o make_v the_o solid_a angle_n z._n and_o in_o like_a sort_n may_v we_o prove_v that_o the_o other_o eleven_o solid_a angle_n of_o the_o icosahedron_n each_o of_o which_o eleven_o solid_a angle_n be_v equal_a and_o like_a to_o the_o solid_a angle_n z_o by_o the_o 16._o of_o the_o thirteen_o be_v subtend_v unto_o pentagon_n equal_a and_o like_a and_o in_o like_a sort_n set_v to_o the_o pentagon_n itkml_n and_o forasmuch_o as_o in_o those_o pentagon_n the_o right_a line_n which_o join_v together_o the_o centre_n of_o the_o base_n be_v common_a side_n it_o follow_v that_o those_o 12._o pentagon_n include_v a_o solid_a which_o solid_a be_v therefore_o a_o dodecahedron_n by_o the_o 24._o definition_n of_o the_o eleven_o and_o be_v by_o the_o first_o definition_n of_o this_o book_n describe_v in_o the_o icosahedron_n five_o side_n whereof_o 〈◊〉_d set_v upon_o the_o pentagon_n abgde_v wherefore_o in_o a_o icosahedron_n give_v iâ—_z inscribe_v a_o dodecahedron_n which_o be_v require_v to_o be_v do_v a_o annotation_n of_o hypsiâ—les_n this_o be_v to_o be_v note_v that_o if_o a_o man_n shall_v demand_v 〈◊〉_d many_o side_n a_o icosahedron_n have_v we_o may_v thus_o answer_v it_o be_v manifest_a that_o a_o icosahâ—râ—n_n be_v contain_v under_o 20._o triangle_n and_o that_o every_o triangle_n consist_v of_o three_o
right_a linâ—s_n now_o then_o multiply_v the_o 20._o triangle_n into_o the_o side_n of_o one_o of_o the_o triangle_n and_o so_o shall_v there_o be_v produce_v 6â—_o â—he_z half_a of_o which_o be_v 30._o and_o so_o many_o side_n have_v a_o icosahedron_n and_o in_o like_a sort_n in_o a_o dodecahedron_n forasmuch_o as_o 12._o pentagon_n make_v a_o dodecahedron_n and_o every_o pentagon_n contain_v â—_o right_a linesâ—_n multiply_v â—â—_o into_o 12._o and_o there_o shall_v be_v produce_v 60._o the_o half_a of_o which_o be_v 30._o and_o so_o many_o be_v the_o side_n of_o a_o dodecahedron_n and_o the_o reason_n why_o we_o take_v the_o half_a iâ—_z for_o that_o every_o side_n whether_o it_o be_v of_o a_o triangle_n or_o of_o a_o pentagon_n or_o of_o a_o square_a as_o in_o a_o cube_fw-la â—s_v take_v twice_o and_o by_o the_o same_o reason_n may_v you_o find_v out_o how_o many_o side_n be_v in_o a_o cube_fw-la and_o in_o a_o pyramid_n and_o in_o a_o octohedron_n but_o now_o again_o if_o you_o will_v find_v out_o the_o number_n of_o the_o angle_n of_o every_o one_o of_o the_o solid_a figure_n when_o you_o have_v do_v the_o same_o multiplication_n that_o you_o do_v before_o divide_v the_o same_o side_n by_o the_o number_n of_o the_o plain_a superficiece_n which_o comprehend_v one_o of_o the_o angle_n of_o the_o solid_n as_o for_o example_n forasmuch_o as_o 5._o triangle_n contain_v the_o solid_a angle_n of_o a_o icosahedron_n divide_v 60._o by_o 5._o and_o there_o will_v come_v forth_o 12._o and_o so_o many_o solid_a angle_n have_v a_o icosahedron_n in_o a_o dodecahedron_n forasmuch_o as_o three_o pentagon_n comprehend_v a_o angle_n divide_v 60._o by_o 3._o and_o there_o will_v come_v forth_o 20_o and_o so_o many_o be_v the_o angle_n of_o a_o dodecahedron_n and_o by_o the_o same_o reason_n may_v you_o find_v out_o how_o many_o angle_n be_v in_o each_o of_o the_o rest_n of_o the_o solid_a figure_n book_n if_o it_o be_v require_v to_o be_v know_v how_o one_o of_o the_o plain_n of_o any_o of_o the_o five_o solid_n be_v give_v there_o may_v be_v find_v out_o the_o inclination_n of_o the_o say_a plain_n the_o one_o to_o the_o other_o which_o contain_v each_o of_o the_o solid_n this_o as_o say_v isidorus_n our_o great_a master_n be_v foâ—â—d_v out_o after_o this_o manner_n it_o be_v manifest_a that_o in_o a_o cube_fw-la the_o plain_n which_o contain_v iâ—_z doâ—_n 〈◊〉_d the_o one_o the_o other_o by_o a_o right_a angle_n but_o in_o a_o tetrahedron_n one_o of_o the_o triangle_n be_v give_v let_v the_o end_n of_o one_o of_o the_o side_n of_o the_o say_a triangle_n be_v the_o centre_n and_o let_v the_o space_n be_v the_o perpendicular_a line_n draw_v from_o the_o top_n of_o the_o triangle_n to_o the_o base_a and_o describe_v circumferâ—nces_n of_o a_o circle_n which_o shall_v cut_v the_o one_o the_o other_o and_o from_o the_o intersection_n to_o the_o centre_n draw_v right_a line_n which_o shall_v contain_v the_o inclination_n of_o the_o plain_n contain_v the_o tetrahedron_n in_o a_o octoâ—edron_n take_v one_o of_o the_o side_n of_o the_o triangle_n thereof_o and_o upon_o it_o describe_v a_o square_a and_o draw_v the_o diagonal_a line_n and_o make_v the_o centre_n the_o end_n of_o the_o diagonal_a line_n and_o the_o space_n likewise_o the_o perpendicular_a line_n draw_v from_o the_o top_n of_o the_o triangle_n to_o the_o base_a describe_v circumference_n and_o again_o from_o the_o common_a section_n to_o the_o centre_n draw_v right_a line_n and_o they_o shall_v contain_v the_o inclination_n seek_v for_o in_o a_o icosahedron_n upon_o the_o side_n of_o one_o of_o the_o triangle_n thereof_o describe_v a_o pentagon_n and_o draw_v the_o line_n which_o subtend_v one_o of_o the_o angle_n of_o the_o say_a pentagon_n and_o make_v the_o centre_n the_o end_n of_o that_o line_n and_o the_o space_n the_o perpendicular_a line_n of_o the_o triangle_n describe_v circumference_n and_o draw_v from_o the_o common_a intersection_n of_o the_o circumference_n unto_o the_o centre_n right_a line_n and_o they_o shall_v contain_v likewise_o the_o inclination_n of_o the_o plain_n of_o the_o icosahedron_n in_o a_o dodecahedron_n take_v one_o of_o the_o pentagon_n and_o draw_v likewise_o the_o line_n which_o subtend_v one_o of_o the_o angle_n of_o the_o pentagon_n and_o make_v the_o centre_n the_o end_n of_o that_o line_n and_o the_o space_n the_o perpendicular_a line_n draw_v from_o the_o section_n into_o two_o equal_a part_n of_o that_o line_n to_o the_o side_n of_o the_o pentagon_n which_o be_v parallel_n unto_o it_o describe_v circumference_n and_o from_o the_o point_n of_o the_o intersection_n of_o the_o circumference_n draw_v unto_o the_o centre_n right_a line_n and_o they_o shall_v also_o contain_v the_o inclination_n of_o the_o plain_n of_o the_o dodecahedron_n thus_o do_v this_o most_o singular_a learned_a man_n reason_n think_v the_o demonstration_n in_o every_o one_o of_o they_o to_o be_v plain_a and_o clear_a but_o to_o make_v the_o demonstration_n of_o they_o manifest_a i_o think_v it_o good_a to_o declare_v and_o make_v open_a his_o wordesâ—_n and_o first_o in_o a_o tâ—trahedronâ—_n the_o end_n of_o the_o fivetenth_fw-mi book_n of_o euclides_n element_n after_o 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d ¶_o the_o 6._o proposition_n the_o 6._o problem_n in_o a_o octohedron_n give_v to_o inscribe_v a_o trilater_n equilater_n pyramid_n svppose_v thaâ—_z the_o octohedron_n whereâ—â—_n the_o tetrahedron_n be_v require_v to_o be_v insâ—riâ—â—â—_n be_v abgdei_n take_v 〈…〉_o base_n of_o the_o octoâ—â—dron_n that_o be_v construction_n 〈…〉_o close_o in_o the_o loweâ—â—_n triangle_n bgd_v namely_o aeâ—_n head_n igd_v and_o let_v the_o four_o be_v aib_n which_o be_v opposite_a to_o the_o low_a triangle_n before_o put_v namely_o to_o egg_v and_o take_v the_o centre_n of_o those_o four_o base_n which_o let_v be_v the_o point_n h_o c_o n_o â—_o and_o upon_o the_o triangle_n hcn_n erect_v a_o pyramid_n hcnl_n now_o forasmuch_o as_o these_o two_o base_n of_o the_o octohedron_n namely_o age_n and_o abi_n be_v set_v upon_o the_o right_a line_n eglantine_n and_o by_o which_o be_v opposite_a the_o one_o to_o the_o otherâ—_n in_o the_o square_a gebi_n of_o the_o octohedron_n from_o the_o poinâ—_n a_o draâ—e_v by_o the_o centre_n of_o the_o base_n namely_o by_o the_o centre_n h_o l_o perpendicular_a line_n ahf_n alk_v cut_v the_o line_n eglantine_n and_o by_o 〈◊〉_d two_o equal_a part_n in_o the_o point_n f_o edward_n by_o the_o corollary_n of_o the_o 1â—_o of_o the_o thirteen_o wherefore_o a_o right_a line_n draw_v from_o the_o point_n f_o to_o the_o point_n king_n demonstration_n shall_v be_v a_o parallel_n and_o equal_a to_o the_o side_n of_o the_o octohedron_n namely_o to_o â—â—_o and_o give_v by_o the_o 33._o of_o the_o first_o and_o the_o right_a line_n hl_o which_o cut_v the_o 〈…〉_o of_o ak_o proportional_o for_o ah_o and_o all_o be_v draw_v from_o the_o centre_n of_o equal_a circle_n to_o the_o circumference_n be_v a_o parallel_n to_o the_o right_a line_n fk_o by_o the_o 2._o of_o the_o six_o and_o also_o to_o the_o side_n of_o the_o octohedron_n namely_o to_o eâ—_n and_o ig_n by_o the_o 9_o of_o the_o eleven_o wherefore_o as_o the_o line_n of_o be_v to_o the_o line_n ah_o so_o be_v the_o line_n fk_o to_o the_o line_n hl_o by_o the_o 4._o of_o the_o six_o for_o the_o triangle_n afk_v and_o ahl_n be_v like_o by_o thâ—_z corollary_n of_o the_o 2._o of_o the_o six_o but_o the_o line_n of_o be_v in_o sesquialter_fw-la proportion_n to_o the_o line_n ah_o for_o the_o side_n eglantine_n make_v hf_o the_o half_a of_o the_o right_a line_n ah_o by_o the_o corollary_n of_o the_o 12._o of_o the_o thirteen_o wherefore_o fk_n or_o give_v the_o side_n of_o the_o octohedron_n be_v sesquialter_fw-la to_o the_o righâ—line_n hl._n and_o by_o the_o same_o reason_n may_v we_o prove_v that_o the_o side_n of_o the_o octohedron_n be_v sesquialter_fw-la to_o the_o rest_n of_o the_o right_a line_n which_o make_v the_o pyramid_n hnci_n namely_o to_o the_o right_a lineâ—â—_n n_o nc_n ci_o ln_o and_o change_z wherefore_o those_o right_a line_n be_v equal_a and_o therefore_o the_o triangleâ—_n which_o be_v describe_v of_o they_o namely_o the_o triangle_n hcn_n hnl_n ncl_n and_o chl._n which_o make_v the_o pyramid_n hncl_n be_v equal_a and_o equilater_n and_o forasmuch_o as_o the_o angle_n of_o the_o same_o pyramid_n namely_o the_o angle_n h_o c_o n_o l_o do_v end_n in_o the_o centre_n of_o the_o base_n of_o the_o octohedron_n therefore_o it_o be_v inscribe_v â—o_o the_o same_o octohedron_n by_o the_o first_o definition_n of_o this_o book_n wherefore_o in_o a_o octohedron_n â—even_v be_v inscribe_v a_o trilate_a equilaâ—â—â—â—â—â—amisâ—_n which_o be_v require_v to_o â—e_a donâ—_n a_o corollary_n the_o base_n of_o a_o pyramid_n inscribe_v in_o a_o octohedron_n be_v parallel_n
to_o the_o base_n of_o the_o octohedron_n for_o forasmuch_o as_o the_o side_n of_o the_o base_n of_o the_o pyramid_n touch_v the_o one_o the_o other_o be_v parallel_n to_o the_o side_n of_o the_o octohedron_n which_o also_o touch_v the_o one_o the_o other_o as_o for_o example_n hl_o be_v prove_v to_o be_v a_o parallel_n to_o give_v and_o lc_a to_o diego_n therefore_o by_o the_o 15._o of_o the_o eleven_o the_o plain_a superficies_n which_o be_v draw_v by_o the_o line_n hl_o and_o lc_n be_v a_o parallel_n to_o the_o plain_a superficies_n draw_v by_o the_o line_n give_v and_o di._n and_o so_o likewise_o of_o the_o rest_n second_o corollary_n a_o right_a line_n join_v together_o the_o centre_n of_o the_o opposite_a base_n of_o the_o octohedron_n be_v sesquialter_fw-la to_o the_o perpendicular_a line_n draw_v from_o the_o angle_n of_o the_o inscribe_v pyramid_n to_o the_o base_a thereof_o for_o forasmuch_o as_o the_o pyramid_n and_o the_o cube_fw-la which_o contain_v it_o do_v in_o the_o self_n same_o point_n end_v their_o angle_n by_o the_o 1._o of_o this_o book_n therefore_o they_o shall_v both_o be_v enclose_v in_o one_o and_o the_o self_n same_o octohedron_n by_o the_o 4._o of_o this_o book_n but_o the_o diameter_n of_o the_o cube_fw-la join_v together_o the_o centre_n of_o the_o opposite_a base_n of_o the_o octohedron_n and_o therefore_o be_v the_o diameter_n of_o the_o sphere_n which_o contain_v the_o cube_fw-la and_o the_o pyramid_n inscribe_v in_o the_o cube_fw-la by_o the_o 13._o and_o 14._o of_o the_o thirteen_o which_o diameter_n be_v sesquialter_fw-la to_o the_o perpendicular_a which_o be_v draw_v from_o the_o angle_n of_o the_o pyramid_n to_o the_o base_a thereof_o 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 base_a of_o the_o pyramid_n be_v the_o six_o part_n of_o the_o diameter_n by_o the_o 3._o corollary_n of_o the_o 13._o of_o the_o thirteen_o wherefore_o of_o what_o part_n the_o diameter_n contain_v six_o of_o the_o same_o part_n the_o perpendicular_a contain_v four_o ¶_o the_o 7._o proposition_n the_o 7._o problem_n in_o a_o dodecahedron_n give_v to_o inscribe_v a_o icosahedron_n svppose_v that_o the_o dodecahedron_n give_v be_v abcde_o and_o let_v the_o centre_n of_o the_o circle_n which_o contain_v six_o base_n of_o the_o same_o dodecahedron_n be_v the_o polnes_n l_o m_o n_o p_o q_o o._n construction_n and_o draw_v these_o right_a line_n ol_n om_o on_o open_v oq_fw-la and_o moreover_o these_o right_a line_n lm_o mn_a np_n pq_n ql_n and_o now_o forasmuch_o as_o equal_v and_o equilater_n pentagon_n be_v contain_v in_o equal_a circle_n therefore_o perpendicular_a line_n draw_v from_o their_o centre_n to_o the_o side_n shall_v be_v equal_a by_o the_o 14._o of_o the_o three_o and_o shall_v divide_v the_o side_n of_o the_o dodecahedron_n into_o two_o equal_a part_n by_o the_o 3._o of_o the_o same_o wherefore_o the_o foresay_a perpendicular_a line_n shall_v coâ—outre_v in_o the_o point_n of_o the_o section_n demonstration_n wherein_o the_o side_n be_v divide_v into_o two_o equal_a part_n as_o lf_a and_o mf_o do_v and_o they_o also_o contain_v equal_a angle_n namely_o the_o inclination_n of_o the_o base_n of_o the_o dodecahedron_n by_o the_o 2._o corollary_n of_o the_o 18._o of_o the_o thirteen_o wherefore_o the_o right_a line_n lm_o mn_a np_n pq_n ql_n and_o the_o rest_n of_o the_o right_a line_n which_o join_v together_o two_o centre_n of_o the_o base_n and_o which_o subtende_v the_o equal_a angle_n contain_v under_o the_o say_a equal_a perpendicular_a line_n 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 wherefore_o the_o triangle_n olm_n omnes_n onp_n opq_n oql_n and_o the_o rest_n of_o the_o triangle_n which_o be_v set_v at_o the_o centre_n of_o the_o pentagon_n be_v equilater_n and_o equal_a now_o forasmuch_o as_o the_o 12._o pentagon_n of_o a_o dodecahedron_n contain_v 60._o plain_a superficial_a angle_n of_o which_o 60._o every_n â—hre_o make_v one_o solid_a angle_n of_o the_o dodecahedron_n it_o follow_v that_o a_o dodecahedron_n have_v 20._o solid_a angle_n but_o each_o of_o those_o solid_a angle_n be_v subtend_v of_o each_o of_o the_o triangle_n of_o the_o icosahedron_n namely_o of_o each_o of_o those_o triangle_n which_o join_v together_o the_o centre_n of_o the_o pentagon_n which_o make_v the_o solid_a angle_n as_o we_o have_v before_o prove_v wherefore_o the_o 20_o equal_a and_o equilater_n triangle_n which_o subtende_v the_o 20._o solid_a angle_n of_o the_o dodecahedron_n and_o have_v their_o side_n which_o be_v draw_v from_o the_o centre_n of_o the_o pentagon_n common_a do_v make_v a_o icosahedron_n by_o the_o 25._o definition_n of_o the_o eleven_o and_o it_o be_v inscribe_v in_o the_o dodecahedron_n give_v by_o the_o first_o definition_n of_o this_o book_n for_o that_o the_o angle_n thereof_o do_v all_o at_o one_o time_n touch_v the_o base_n of_o the_o dodecahedron_n wherefore_o in_o a_o dodecahedron_n gevenâ—_n iâ—_z inscribe_v a_o icosahedron_n which_o be_v require_v to_o be_v do_v ¶_o the_o 8._o proposition_n the_o 8._o problem_n in_o a_o dodecahedron_n give_v to_o include_v a_o cube_fw-la describe_v by_o the_o 17._o of_o the_o thirteen_o a_o dodecahedron_n construction_n and_o by_o the_o same_o take_v the_o 12._o side_n of_o the_o cube_fw-la each_o of_o which_o subtend_v one_o angle_n of_o each_o of_o the_o 12._o base_n of_o the_o dodecahedron_n for_o the_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 by_o the_o 2._o corollary_n of_o the_o 17._o of_o the_o thirteen_o if_o therefore_o in_o the_o dodecahedron_n describe_v by_o the_o self_n same_o 17._o proposition_n we_o draw_v the_o 12._o right_a line_n subtend_v under_o the_o foresay_a 12._o angle_n and_o end_v in_o 8._o angle_n of_o the_o dodecahedron_n and_o concur_v together_o in_o such_o sort_n that_o they_o be_v in_o like_a sort_n situate_a as_o it_o be_v plain_o prove_v in_o that_o proposition_n then_o shall_v it_o be_v manifest_a that_o the_o right_a line_n draw_v in_o this_o dodecahedron_n from_o the_o foresay_a 8._o angle_n thereof_o do_v make_v the_o foresay_a cube_fw-la which_o therefore_o be_v include_v in_o the_o dodecahedron_n for_o that_o the_o side_n of_o the_o cube_fw-la be_v draw_v in_o the_o side_n of_o the_o dodecahedron_n and_o the_o angle_n of_o the_o same_o cube_fw-la be_v set_v in_o the_o angle_n of_o the_o say_a dodecahedron_n as_o for_o example_n take_v 4._o pentagon_n of_o a_o dodecahedron_n demonstration_n namely_o agibo_n bhcno_n ckedn_n and_o dfaon_n and_o draw_v these_o right_a line_n ab_fw-la bc_o cd_o da._n which_o four_o right_a line_n make_v a_o square_n for_o that_o each_o of_o those_o right_a line_n do_v subtend_v equal_a angle_n of_o equal_a pentagon_n &_o the_o angle_n which_o those_o 4._o right_a line_n contain_v be_v right_a angle_n as_o we_o prove_v in_o the_o construction_n of_o the_o dodecahedron_n in_o the_o 17._o proposition_n before_o allege_a wherefore_o the_o six_o base_n be_v square_n do_v make_v a_o cube_fw-la by_o the_o 21._o definition_n of_o the_o eleven_o and_o for_o that_o the_o 8._o angle_n of_o the_o say_v cube_fw-la be_v set_v in_o 8._o angle_n of_o the_o dodecaheeron_n therefore_o be_v the_o say_v cube_fw-la 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 be_v inscribe_v a_o cube_fw-la which_o be_v require_v to_o be_v do_v ¶_o the_o 9_o proposition_n the_o 9_o problem_n in_o a_o dodecahedron_n give_v to_o include_v a_o octohedron_n svppose_v that_o the_o dodecahedron_n give_v be_v abgd_v construction_n now_o by_o the_o 3._o corollary_n of_o the_o 17._o of_o the_o thirteen_o take_v the_o 6._o side_n which_o be_v opposite_a the_o one_o to_o the_o other_o those_o 6._o side_n i_o say_v who_o section_n wherein_o they_o be_v divide_v 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 wherein_o the_o dodecahedron_n be_v contain_v do_v cut_v the_o one_o the_o other_o perpendicular_o and_o let_v the_o point_n wherein_o the_o foresay_a side_n be_v cut_v into_o two_o equal_a part_n be_v a_o b_o g_o d_o c_o i._n and_o let_v the_o foresay_a three_o right_a line_n join_v together_o the_o say_a section_n be_v ab_fw-la gd_o and_o ci._n and_o let_v the_o centre_n of_o the_o sphere_n be_v e._n demonstration_n now_o forasmuch_o as_o by_o the_o foresay_a corollary_n those_o three_o right_a line_n be_v equal_a it_o follow_v by_o the_o 4._o of_o the_o first_o that_o the_o right_a line_n subtend_v the_o right_a angle_n which_o they_o make_v at_o the_o centre_n of_o the_o sphere_n which_o right_a angle_n be_v contain_v under_o the_o half_n of_o the_o say_v three_o right_a line_n be_v equal_a the_o one_o to_o the_o other_o
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
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 gd_v the_o angle_n m_o n_o under_o the_o side_n ab_fw-la the_o angle_n t_o s_o under_o the_o side_n bg_o the_o angle_n p_o oh_o and_o under_o the_o side_n agnostus_n the_o angle_n r_o q_o so_o there_o rest_v 4._o angle_n who_o true_a place_n we_o will_v now_o appoint_v forasmuch_o as_o a_o cube_fw-la contain_v in_o one_o and_o the_o self_n same_o sphere_n with_o a_o dodecahedron_n be_v inscribe_v in_o the_o same_o dodecahedron_n as_o it_o be_v manifest_a by_o the_o 17._o of_o the_o thirteen_o and_o 8._o of_o this_o book_n it_o follow_v that_o a_o cube_fw-la and_o a_o dodecahedron_n circumscribe_v about_o it_o be_v contain_v in_o one_o and_o the_o self_n same_o body_n for_o that_o their_o angle_n concur_v in_o one_o and_o the_o self_n same_o point_n and_o it_o be_v prove_v in_o the_o 18._o of_o this_o book_n that_o 4._o angle_n of_o the_o cube_fw-la inscribe_v in_o the_o pyramid_n be_v set_v in_o the_o middle_a section_n of_o the_o perpendicularâ—_n which_o be_v draw_v from_o the_o solid_a angle_n of_o the_o pyramid_n to_o the_o opposite_a base_n wherefore_o the_o other_o 4._o angle_n of_o the_o dodecahedron_n be_v also_o as_o the_o angle_n of_o the_o cube_fw-la set_v in_o those_o middle_a section_n of_o the_o perpendicular_n namely_o the_o angle_n v_o be_v set_v in_o the_o midst_n of_o the_o perpendicular_a ahâ—_n the_o angle_n y_fw-fr in_o the_o midst_n of_o the_o perpendicular_a bf_o the_o angle_n x_o in_o the_o midst_n of_o the_o perpendicular_a ge_z and_o last_o the_o angle_n d_o in_o the_o midst_n of_o the_o perpendicular_a d_o which_o be_v draw_v from_o the_o top_n of_o the_o pyramid_n to_o the_o opposite_a base_a wherefore_o those_o 4._o angle_n of_o the_o dodecahedron_n may_v be_v say_v to_o be_v direct_o under_o the_o solid_a angle_n of_o the_o pyramid_n or_o they_o may_v be_v say_v to_o be_v set_v at_o the_o perpendicular_n wherefore_o the_o dodecahedron_n after_o this_o manner_n set_v be_v inscribe_v in_o the_o pyramid_n give_v by_o the_o first_o definition_n of_o this_o book_n for_o that_o upon_o every_o one_o of_o the_o base_n of_o the_o pyramid_n be_v set_v a_o angle_n of_o the_o dodecahedron_n inscribe_v wherefore_o in_o a_o trilater_n equilater_n pyramid_n be_v inscribe_v a_o dodecahedron_n the_o 21._o problem_n the_o 21._o proposition_n in_o every_o one_o of_o the_o regular_a solid_n to_o inscribe_v a_o sphere_n in_o the_o 13._o of_o thâ—_z thirteen_o and_o thâ—_z other_o 4._o proposition_n follow_v he_o iâ—_z be_v declare_v that_o â—he_n â—â—_o regular_a solidesâ—â—re_o so_o contain_v in_o a_o sphere_n that_o â—ight_a linâ—â—_n draw_v from_o the_o cenâ—â—â—_n oâ—_n the_o 〈…〉_o of_o 〈◊〉_d solid_a inscribe_v be_v equal_a which_o right_a line_n therefore_o make_v pyramid_n who_o â—oppes_n be_v the_o centre_n of_o the_o sphere_n or_o of_o the_o solid_a and_o the_o basâ—â—â—â—e_a cuâ—â—â—_n one_o of_o the_o base_n of_o those_o solid_n and_o 〈…〉_o solid_a squall_n and_o like_v the_o one_o to_o the_o other_o and_o describe_v in_o equal_a circle_n those_o circle_n shall_v cut_v the_o sphere_n for_o the_o angle_n which_o touch_v the_o circumference_n of_o the_o circle_n touch_v also_o the_o superficies_n of_o the_o sphere_n wherefore_o perpendiculars_n draw_v from_o the_o centre_n of_o the_o sphere_n to_o the_o base_n or_o to_o the_o plain_a superficiece_n of_o the_o equal_a circle_n be_v equal_a by_o the_o corollary_n of_o the_o assumpt_n of_o the_o 1â—_o of_o the_o twelve_o wherefore_o make_v the_o centre_n the_o 〈◊〉_d of_o the_o sphere_n which_o 〈◊〉_d the_o solid_a and_o thâ—_z space_n some_o one_o of_o the_o equal_a perpendicularâ—_n dâ—scribâ—_n a_o sphere_n and_o it_o shall_v touch_v every_o one_o of_o the_o base_n of_o 〈◊〉_d solid_a 〈…〉_o perficies_fw-la of_o the_o sphere_n pass_v beyond_o those_o base_n when_o as_o those_o pâ—â—peâ—diculars_n 〈…〉_o be_v draw_v from_o the_o centre_n to_o the_o base_n by_o the_o 3._o corollary_n of_o the_o saâ—â—â—â—â—umpt_n wherâ—fore_o â—e_v have_v iâ—_z every_o one_o of_o the_o regular_a body_n inscribe_v a_o sphere_n which_o regular_a boâ—â—â—_n be_v in_o number_n one_o iâ—_z 〈◊〉_d by_o the_o corollary_n of_o the_o 1â—_o of_o the_o 〈◊〉_d a_o corollary_n the_o regular_a figure_n inscribe_v in_o sphere_n and_o also_o the_o sphere_n circumscribe_v about_o they_o or_o contain_v they_o have_v one_o and_o the_o self_n same_o centre_n namely_o their_o pyramid_n the_o â—ngles_n of_o who_o base_n touch_v the_o superâ—â—â—â—â—â—_n of_o thâ—â—â—here_o do_v from_o those_o angle_n cause_v equal_a right_a line_n to_o be_v drawâ—â—_n to_o one_o and_o â—he_n self_n 〈◊〉_d poynâ—_n make_v the_o topâ—â—â—_n of_o the_o pyramid_n in_o the_o same_o point_n and_o therefore_o they_o 〈…〉_o thâ—_z câ—â—tres_n of_o the_o sphere_n in_o the_o self_n same_o top_n when_o 〈◊〉_d the_o right_a line_n draw_v from_o those_o angle_n to_o the_o croâ—â—ed_a superficies_n wherein_o be_v 〈◊〉_d the_o angle_n of_o the_o base_n of_o the_o pyramid_n be_v equal_o a_o advertisement_n of_o flussasâ—_n â—_z of_o these_o solid_n only_o the_o octohedron_n receive_v the_o other_o solid_n inscribe_v one_o with_o 〈…〉_o other_o for_o the_o octohedron_n contain_v the_o icosahedron_n inscribe_v in_o it_o and_o the_o same_o icosahedron_n contain_v the_o dodecahedron_n inscribe_v in_o the_o same_o icosahedron_n and_o the_o same_o dodecahedron_n contain_v the_o cube_fw-la inscribe_v in_o the_o same_o octohedron_n and_o 〈…〉_o â—â—râ—â—mscribeth_v the_o pyramid_n inscribe_v in_o the_o say_v octohedron_n but_o this_o happen_v not_o in_o the_o other_o solid_n the_o end_n of_o the_o fivetenth_fw-mi book_n of_o euclides_n elemenâ—â—â—_n after_o caâ—paâ—â—_n and_o 〈◊〉_d ¶_o the_o sixteen_o book_n of_o the_o element_n of_o geometry_n add_v by_o flussas_n in_o the_o former_a fivetenth_fw-mi book_n have_v be_v teach_v how_o to_o inscribe_v the_o five_o regular_a solid_n one_o with_o in_o a_o other_o now_o seem_v to_o rest_n to_o compare_v those_o solid_a so_o inscribe_v one_o to_o a_o other_o and_o to_o set_v forth_o their_o passion_n and_o propriety_n which_o thing_n flussas_n consider_v in_o this_o sixteen_o book_n add_v by_o he_o book_n have_v excellent_o well_o and_o most_o cunning_o perform_v for_o which_o undoubted_o he_o have_v of_o all_o they_o which_o have_v a_o love_n to_o the_o mathematicals_n deserve_v much_o praise_n and_o commendation_n both_o for_o the_o great_a traâ—ailes_n and_o paynâ—s_n which_o it_o be_v most_o likely_a he_o have_v taâ—â—n_v in_o invent_v such_o strange_a and_o wonderful_a proposition_n with_o their_o demonstration_n in_o this_o book_n contain_v as_o also_o for_o participate_v and_o communicate_v abroad_o the_o same_o to_o other_o which_o book_n also_o that_o the_o reader_n shall_v want_v nothing_o conduce_v to_o the_o perfection_n of_o euclides_n element_n i_o have_v with_o some_o travail_n translate_v &_o for_o the_o worthiness_n â—hereof_o have_v add_v it_o aâ—_z a_o sixteen_o book_n to_o the_o 15._o book_n of_o euclid_n vouchsafe_v therefore_o gentle_a reader_n diligent_o to_o read_v and_o poise_v it_o for_o in_o it_o shall_v you_o find_v noâ—_n only_a matter_n strange_a and_o delectable_a but_o also_o occasion_n of_o invention_n of_o great_a thing_n pertain_v to_o the_o nature_n of_o the_o five_o regular_a solidâ—sâ—_n ¶_o the_o 1._o proposition_n a_o dodecahedron_n and_o a_o cube_fw-la inscribe_v in_o it_o and_o a_o pyramid_n inscribe_v in_o the_o same_o cube_fw-la be_v contain_v in_o one_o and_o the_o self_n same_o sphere_n for_o the_o angle_n of_o the_o pyramiâ—_n be_v seâ—_z in_o the_o angel_n of_o the_o cube_fw-la wherein_o it_o be_v inscribe_v by_o the_o first_o of_o the_o fivetenthâ—_n and_o all_o the_o angle_n of_o the_o cube_fw-la be_v set_v in_o the_o angle_n of_o the_o dodecahedâ—â—â—_n circumscribe_v 〈…〉_o 〈◊〉_d the_o 8._o of_o the_o fifteen_o and_o all_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 by_o the_o 17._o of_o the_o thirteen_o wherefore_o those_o three_o solid_n inscribe_v one_o within_o a_o other_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 definition_n of_o the_o fifteen_o a_o dodecahedron_n therefore_o and_o a_o cube_fw-la inscribe_v in_o it_o and_o a_o pyramid_n inscribe_v in_o the_o same_o cube_fw-la be_v contain_v 〈…〉_o â—â—lfe_n same_o sphere_n 〈…〉_o these_o three_o solid_n livre_n 〈…〉_o elf_n same_o icosahedron_n or_o octohedron_n or_o pyramid_n 〈…〉_o i_o icosahedron_n by_o the_o 5.11_o &_o 12._o of_o the_o fifteen_o and_o they_o be_v 〈…〉_o ctohedron_n by_o the_o 4._o 6._o and_o 16._o of_o the_o same_o last_o they_o be_v inscribe_v in_o 〈…〉_o the_o first_o 18._o and_o 19_o of_o the_o same_o for_o the_o angle_n of_o all_o these_o solid_a 〈…〉_o the_o circumscribe_v icosahedron_n or_o octohedron_n or_o pyramid_n ¶_o the_o 〈…〉_o the_o proportion_n of_o a_o dodecahedron_n circumscribe_v about_o a_o cube_fw-la to_o a_o dodecahedron_n inscribe_v in_o the_o same_o cube_fw-la be_v
construction_n two_o case_n in_o this_o proposition_n first_o caseâ—_n second_o case_n demonstration_n construction_n demonstration_n construction_n demonstration_n an_o other_o way_n after_o peliâ—arius_n construction_n demonstration_n construction_n demonstration_n demonstration_n lead_v to_o a_o impossibility_n three_o case_n in_o this_o proposition_n the_o three_o case_n construction_n demonstration_n construction_n demonstration_n construction_n demonstration_n construction_n demonstration_n construction_n demonstration_n a_o proposition_n add_v by_o petarilius_n note_n construction_n demonstration_n an_o other_o way_n also_o after_o pelitarius_n construction_n demonstration_n an_o other_o way_n to_o do_v the_o samâ—_n after_o pelitarius_n demonstration_n demonstration_n demonstration_n lead_v to_o a_o absurdity_n a_o corollary_n construction_n demonstration_n demonstration_n a_o â—ther_n way_n to_o do_v the_o same_o after_o orontius_n an_o other_o way_n after_o pelitarius_n construction_n demonstration_n a_o addition_n of_o flussates_n flussates_n a_o poligonon_n figure_n be_v a_o figure_n consist_v of_o many_o side_n the_o argument_n of_o this_o five_o book_n the_o first_o author_n of_o this_o book_n eudoxus_n the_o first_o definition_n a_o part_n take_v two_o manner_n of_o way_n the_o fiâ—st_a way_n the_o second_o way_n how_o a_o less_o quantity_n be_v say_v to_o measure_v a_o great_a in_o what_o signification_n euclid_n here_o take_v a_o part_n parâ—_n metienâ—_n or_o mensuranâ—_n pars_fw-la multiplicatiâ—a_fw-la pars_fw-la aliquota_fw-la this_o kind_n of_o part_n common_o use_v in_o arithmetic_n the_o other_o kind_n of_o part_n pars_fw-la constitâ—ens_fw-la or_o componens_fw-la pars_fw-la aliquanta_fw-la the_o second_o definition_n number_n very_o necessary_a for_o the_o understanding_n of_o this_o book_n and_o the_o other_o book_n follow_v the_o tâ—ird_o definition_n rational_a proportion_n divide_v â—â—to_o two_o kind_n proportion_n of_o equality_n proportion_n of_o inequality_n proportion_n of_o the_o great_a to_o the_o less_o multiplex_fw-la duple_n proportion_n triple_a quadruple_a quintuple_a superperticular_a sesquialtera_fw-la sesquitertia_fw-la sesquiquarta_fw-la superpartiens_fw-la superbipartiens_fw-la supertripartiens_fw-la superquadripartiens_fw-la superquintipartiens_fw-la multiplex_fw-la superperticular_a dupla_fw-la sesquialtera_fw-la dupla_fw-la sesquitertia_fw-la tripla_fw-la sesquialtera_fw-la multiplex_fw-la superpartiens_fw-la dupla_fw-la superbipartiens_fw-la dupla_fw-la supertripartiens_fw-la tripla_fw-la superbipartiens_fw-la tripla_fw-la superquadâ—ipartiens_fw-la how_o to_o knoâ—_n the_o denomination_n of_o any_o proportion_n proportion_n of_o the_o less_o in_o the_o great_a submultiplex_fw-la subsuperparticular_a subsuperpertient_fw-fr etc_n etc_n the_o four_o definition_n example_n of_o this_o definition_n in_o magnitude_n example_n thereof_o in_o number_n note_n the_o five_o definition_n a_o example_n of_o this_o definition_n in_o magnitude_n why_o euclid_n in_o define_v of_o proportion_n use_v multiplication_n the_o six_o definition_n a_o example_n of_o this_o definitiin_n in_o magnitude_n a_o example_n in_o number_n an_o other_o example_n in_o number_n an_o other_o example_n in_o number_n note_v this_o particle_n according_a to_o any_o multiplication_n a_o example_n where_o the_o equimultiplice_n of_o the_o first_o and_o three_o excee_v the_o equimultiplice_n of_o the_o second_o and_o four_o and_o yet_o the_o quantity_n give_v be_v not_o in_o one_o and_o the_o self_n same_o proportion_n a_o rule_n to_o produce_v equimultiplice_n of_o the_o first_o and_o three_o equal_a to_o the_o equimultiplice_n of_o the_o secondâ—_n and_o forth_o example_n thereof_o the_o seven_o definition_n 9_z 12_o 3_o 4_o proportionality_n of_o two_o sort_n continual_a and_o discontinuall_a a_o example_n of_o continual_a proportionality_n in_o number_n 16.8.4.2.1_o in_o coutinnall_a proportionality_n the_o quantity_n can_v be_v of_o one_o kind_n discontinuall_a propâ—rtionalitie_n example_n of_o discontinual_a proportionality_n in_o number_n in_o discontinual_a proportionality_n the_o proportion_n may_v be_v of_o diverse_a kind_n the_o eight_o definition_n a_o example_n of_o this_o definition_n in_o magnitude_n a_o example_n in_o number_n note_n the_o nine_o definition_n a_o example_n of_o this_o definition_n in_o magnitude_n example_n â—n_a number_n the_o ten_o definition_n a_o rule_n to_o add_v proportion_n to_o proportion_n 8._o 4._o 2._o 1._o 2_o 2_o 2_o 1_o 1_o 1_o the_o eleven_o definition_n example_n of_o this_o definition_n in_o magnitude_n example_n in_o number_n the_o twelfâ—h_o definition_n example_n of_o this_o definition_n in_o magnitud_n example_n in_o number_n the_o thirteen_o definition_n example_n of_o this_o definition_n in_o magnitud_n example_n in_o number_n the_o fourteen_o definition_n example_n of_o this_o definition_n in_o magnitud_n example_n in_o number_n the_o fiâ—tâ—ne_a definition_n this_o be_v the_o converse_n of_o the_o former_a definition_n example_n in_o magnitude_n example_n in_o number_n the_o sixteen_o definition_n a_o example_n of_o this_o definition_n in_o magnitude_n a_o example_n in_o number_n the_o sevententh_n definition_n a_o example_n of_o this_o definition_n in_o magnitude_n a_o example_n in_o number_n note_n the_o eighttenth_n definition_n a_o example_n of_o this_o definition_n in_o magnitude_n example_n in_o number_n the_o nineteen_o definition_n a_o example_n of_o this_o definition_n in_o magnitude_n example_n in_o number_n the_o 20._o definition_n the_o 2d_o definition_n these_o two_o last_o definition_n not_o find_v in_o the_o greek_a exampler_n construction_n demonstration_n demonstrationâ—_n construction_n demonstration_n construction_n demonstration_n alemmae_fw-la or_o a_o assumpt_n a_o corollary_n converse_v proportion_n construction_n demonstration_n two_o case_n in_o this_o propotion_n the_o second_o the_o 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the_o same_o affirmative_o an_o other_o demonstration_n of_o the_o same_o affirmative_o an_o other_o demonstration_n of_o the_o same_o demonstration_n lead_v to_o a_o impossibility_n an_o other_o demonstration_n of_o the_o same_o affirmative_o demonstration_n demonstrationâ—_n demonstration_n the_o argument_n of_o this_o six_o book_n this_o book_n necessary_a for_o the_o use_n of_o instrument_n of_o geometry_n the_o first_o definition_n the_o second_o definition_n reciprocal_a figure_n call_v mutual_a figure_n the_o three_o definition_n the_o four_o definition_n the_o five_o definition_n an_o other_o example_n of_o substraction_n of_o proportion_n the_o six_o definition_n demonstration_n of_o the_o first_o part_n demonstration_n of_o the_o second_o part_n a_o corollary_n add_v by_o flussates_n the_o first_o part_n of_o this_o theorem_a demonstration_n of_o the_o second_o part_n a_o corollary_n add_v by_o flussates_n construction_n demonstration_n of_o the_o first_o part_n 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first_o part_n of_o this_o proposition_n demonstration_n of_o the_o of_o the_o same_o the_o second_o part_n which_o be_v the_o converse_n of_o the_o first_o the_o first_o parâ—_n of_o this_o proposition_n demonstration_n of_o the_o same_o the_o second_o part_n which_o be_v the_o converse_n of_o the_o first_o demonstration_n of_o the_o first_o part_n the_o second_o part_n which_o be_v the_o converse_n of_o the_o first_o the_o first_o part_n of_o thâ—â—_n theorem_a the_o second_o part_n which_o be_v the_o converse_n of_o the_o first_o a_o corollary_n description_n of_o the_o rectiline_a figure_n require_v demonstration_n demonstration_n a_o corollary_n the_o first_o parâ—_n of_o this_o theorem_a the_o second_o part_n demonstrate_v the_o three_o part_n the_o first_o corollary_n the_o second_o corollary_n demonstration_n the_o first_o part_n of_o this_o proposition_n the_o second_o part_n which_o be_v the_o converse_n of_o the_o first_o first_o note_v that_o this_o be_v prove_v in_o the_o 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superficial_a number_n the_o eighteen_o definition_n why_o they_o be_v call_v solid_a number_n the_o nineteen_o definition_n why_o it_o be_v call_v a_o square_a number_n the_o twenty_o definition_n why_o it_o be_v call_v a_o cube_fw-la number_n the_o twenty_o one_o definition_n why_o the_o definition_n of_o proportional_a magnitude_n be_v unlike_a to_o the_o definitio_fw-la of_o proportional_a number_n the_o twenty_o two_o definition_n the_o twenty_o three_o definition_n perfect_a number_n rare_a &_o of_o great_a use_n in_o magic_a &_o in_o secret_a philosophy_n in_o what_o respect_n a_o number_n be_v perfect_a two_o kind_n of_o imperfect_a number_n a_o â—â—mber_n wanâ—â—ngâ—_n common_a sentence_n â—irst_n common_a sentence_n â—â—cond_v â—ommon_a sentence_n three_o common_a sentence_n forth_o common_a sentence_n â—iâ—th_o common_a sentence_n six_o common_a sentence_n seven_o comâ—mon_a sentence_n constrâ—ctioâ—_n demonstratiâ—â—_n lead_v to_o a_o absurdity_n the_o converse_n of_o â—his_n proposition_n how_o to_o â—now_v whether_o two_o number_n give_v be_v prime_a the_o one_o to_o the_o other_o two_o case_n in_o this_o problem_n the_o first_o case_n the_o second_o case_n demonstration_n of_o the_o second_o case_n that_o cf_o be_v a_o common_a measure_n to_o the_o number_n ab_fw-la and_o cd_o that_o cf_o be_v the_o great_a common_a measure_n to_o ab_fw-la and_o cd_o the_o second_o case_n two_o case_n in_o this_o proposition_n the_o first_o case_n the_o second_o case_n this_o proposition_n and_o the_o 6._o proposition_n in_o discrete_a quantity_n answer_v to_o the_o first_o of_o the_o five_o in_o continual_a quantity_n demonstration_n construction_n demonstration_n thiâ—_n proposition_n and_o the_o next_o follow_v in_o discreet_a quantity_n answer_v to_o the_o five_o proposition_n of_o the_o five_o book_n in_o continual_a quantity_n construction_n demonstration_n constuâ—ction_n demonstration_n an_o other_o demonstration_n after_o flussates_n construction_n demonstration_n construction_n demonstration_n this_o proposition_n iâ—_z discreet_a quantity_n answer_v to_o the_o nine_o proposition_n of_o the_o five_o book_n in_o continual_a quantity_n demonstration_n this_o in_o discreet_a quantity_n answer_v to_o the_o twelve_o proposition_n of_o the_o five_o in_o continual_a quantity_n demonstration_n this_o in_o discrete_a quantity_n answer_v to_o the_o sixteen_o proposition_n of_o the_o five_o book_n in_o continual_a quantity_n note_n this_o in_o discrete_a quantity_n answer_v to_o tâ—â—_n twenty_o one_o proposition_n oâ—_n the_o five_o book_n in_o continual_a quantity_n demonstration_n certain_a addition_n of_o campane_n the_o second_o case_n proportionality_n divide_v proportionality_o compose_v euerse_a proportionality_n the_o conversâ—_n of_o the_o same_o prâ—position_n demonstration_n a_o corollary_n follow_v thâ—se_a proposition_n adâ—ed_v by_o campane_n coâ—strâ—ctioâ—_n demonstration_n demonstration_n â—emonstraâ—ion_n a_o corollary_n add_v by_o flussâ—tes_n demonstration_n this_o proposition_n and_o the_o former_a may_v be_v extend_v to_o number_n how_o many_o soever_o the_o second_o part_n of_o this_o proposition_n which_o be_v the_o converse_n of_o the_o first_o demonstration_n a_o assumpt_v add_v by_o campane_n this_o proposition_n in_o number_n demonstrate_v that_o which_o the_o 17._o of_o the_o six_o demonstrate_v in_o line_n demonstration_n the_o second_o part_n which_o be_v the_o converse_n of_o the_o
second_o part_n of_o the_o first_o case_n the_o second_o case_n first_o part_n of_o the_o second_o case_n second_o part_n of_o the_o second_o case_n construction_n two_o case_n in_o this_o proposition_n the_o first_o case_n the_o first_o part_n of_o the_o first_o case_n 〈◊〉_d second_o 〈◊〉_d of_o the_o 〈◊〉_d case_n the_o second_o case_n a_o corollary_n the_o first_o senary_a by_o substraction_n demonstration_n an_o other_o demonstration_n after_o campane_n definition_n of_o the_o eight_o irrational_a line_n definition_n of_o 〈◊〉_d â—inth_o irrational_a line_n an_o other_o demonstration_n after_o campane_n construction_n demonstration_n definition_n of_o the_o ten_o irrational_a line_n definition_n of_o the_o eleven_o irrational_a line_n â—â—â—â—iâ—ition_n of_o the_o twelve_o irraâ—ionall_a line_n definition_n of_o the_o thirteen_o and_o last_o irrational_a line_n a_o assumpt_n of_o campane_n i_o dee_n though_o campane_n lemma_n be_v true_a yeâ—_z the_o manner_n of_o demonstrate_v it_o narrow_o consider_v be_v not_o artificial_a second_o senary_n demonstration_n lead_v to_o a_o impossibility_n demonstration_n lead_v to_o a_o absurdity_n construction_n demonstration_n lead_v to_o a_o absurdity_n demonstration_n lead_v to_o a_o absurdity_n demonstration_n lead_v to_o a_o impossibility_n construction_n demonstration_n 〈◊〉_d a_o abjurdâ—tâ—â—â—_n six_o kind_n of_o reâ—iduall_a line_n first_o definition_n second_o definition_n three_o definition_n four_o definition_n five_o definition_n six_o definition_n three_o senary_n construction_n demonstration_n construction_n demoâ—stratiâ—â—_n construction_n demonstration_n construction_n demonstration_n construction_n demonstration_n construction_n demonstration_n a_o other_o more_o ready_a way_n to_o find_v out_o the_o six_o residual_a line_n four_o senary_n the_o â—irst_n parâ—_n of_o the_o construction_n the_o first_o part_n of_o the_o demonstration_n note_n a_z and_o fk_o conclude_v rational_a parallelogram_n note_n dh_o 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propound_v both_o in_o the_o proposition_n and_o also_o in_o the_o determination_n construction_n demonstration_n construction_n demonstration_n here_o be_v the_o â—ower_a part_n of_o the_o propositiâ—_n more_o orderly_a huddle_v theâ—_z in_o the_o former_a demöstration_n construction_n demonstration_n a_o assumpt_n an_o other_o demonstration_n after_o flussas_n construction_n demonstration_n this_o be_v in_o a_o manner_n the_o converse_n of_o both_o the_o former_a proposition_n joint_o construction_n demonstration_n construction_n demonstration_n demonstration_n an_o other_o demonstration_n demonstration_n lead_v to_o a_o impossibility_n an_o other_o demonstration_n lead_v to_o a_o impossibility_n the_o argument_n of_o the_o eleven_o book_n a_o point_n the_o beginning_n of_o all_o quantity_n continual_a the_o method_n use_v by_o euclid_n in_o the_o ten_o â—â—â—mer_a boot_n â—irst_n boâ—â—e_n second_o â—â—oâ—e_n three_o booâ—e_n forth_o bâ—oâ—e_n â—iveth_v boâ—â—e_n six_o booâ—e_n seven_o bookâ—_n â—ight_a booâ—â—_n nine_o book_n ten_o booâ—e_n what_o be_v entreaâ—ea_n of_o in_o the_o fiâ—e_a boot_n follow_v 〈◊〉_d â—â—â—ular_a bodiesâ—_n the_o â—â—all_a end_n 〈…〉_o oâ—_n i_o uâ—â—â—â—es_v â—eomeâ—â—â—all_a element_n comparison_n â—â—_o the_o 〈◊〉_d â—â—oâ—e_n and_o 〈◊〉_d book_n 〈◊〉_d first_o definition_n a_o solid_a the_o most_o perfect_a quantity_n no_o science_n of_o thing_n infinite_a second_o definition_n three_o definition_n two_o difâ—initions_n include_v in_o this_o definition_n declaration_n of_o the_o first_o part_n declaration_n of_o the_o second_o part_n four_o definition_n five_o definition_n six_o definition_n seven_o defâ—inition_n eight_o definition_n nine_o dissilition_n ten_o definition_n eleven_o definition_n an_o other_o definition_n of_o a_o prism_n which_o be_v a_o special_a definition_n of_o a_o prism_n as_o it_o be_v common_o call_v and_o use_v this_o body_n call_v figura_fw-la serratilis_fw-la psellus_n twelve_o definition_n what_o be_v to_o be_v taâ—â—n_v heed_n of_o in_o the_o definition_n of_o a_o sphere_n give_v by_o johannes_n de_fw-fr sacro_fw-la busco_n theodosiuâ—_n definition_n of_o a_o sphere_n the_o circumference_n of_o a_o sphere_n galens_n definition_n 〈◊〉_d a_o sphâ—râ—_n the_o dignity_n of_o a_o sphere_n a_o sphere_n call_v a_o globe_n thirteen_o definition_n theodosius_n definition_n of_o the_o axe_n of_o a_o sphere_n fourtenth_n definition_n theodosius_n definition_n of_o the_o centre_n of_o a_o sphere_n flussas_n definition_n of_o the_o centre_n of_o a_o sphere_n fivetenth_fw-mi definition_n difference_n between_o the_o diameter_n &_o axe_n of_o a_o sphere_n sevententh_n definition_n first_o kind_n of_o cones_n a_o cone_n call_v of_o campane_n a_o roâ—â—de_a pyramid_n sevententh_n definition_n a_o conical_a superficies_n eightenth_n definition_n ninetenth_fw-mi definition_n a_o cillindricall_a superficies_n corollary_n a_o roundâ—_n column_n or_o sphere_n a_o corollary_n add_v by_o campane_n twenty_o definition_n twenty_o one_o definition_n twenty_o two_o definition_n a_o tetrahedron_n one_o of_o the_o five_o regular_a body_n difference_n between_o a_o tetrahedron_n and_o a_o pyramid_n psellus_n call_v a_o tetrahedron_n a_o pyramid_n twenty_o three_o definition_n twenty_o â—oâ—er_v definition_n twenty_o five_o definition_n five_o regular_a body_n the_o dignity_n of_o these_o body_n a_o tetrahedron_n ascribe_v unto_o the_o fire_n a_o octohedron_n ascribe_v unto_o the_o air_n a_o ikosahedron_n assign_v unto_o the_o water_n a_o cube_fw-la assign_v unto_o the_o earth_n a_o dodecahedron_n assign_v to_o heaven_n definition_n of_o a_o parallelipipedon_n a_o dâ—dâ—â—â—â—edron_n a_o icosaâ—edron_n demonstration_n lead_v to_o a_o impossibility_n an_o other_o demonstration_n after_o flussas_n construction_n demonstration_n lead_v to_o a_o impossibility_n demonstration_n lead_v to_o a_o impossibility_n construction_n demonstration_n demonstration_n lead_v to_o a_o impossibility_n construction_n construction_n a_o assumpt_n as_o m._n dee_n prove_v it_o demonstration_n demonstration_n lead_v to_o a_o impossibility_n this_o proposition_n be_v as_o it_o be_v the_o converse_n of_o the_o six_o construction_n demonstration_n construction_n demonstration_n construction_n demonstration_n construction_n two_o case_n in_o this_o proposition_n the_o first_o case_n john_n dee_n dee_n this_o require_v the_o imagination_n of_o a_o plain_a superficies_n pass_v by_o the_o point_n a_o and_o the_o straight_a line_n bc._n and_o so_o help_v yourself_o in_o the_o like_a case_n either_o mathematical_o imagine_v or_o mechanical_o practise_v second_o