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 line_n commensurable_a and_o so_o imagine_v you_o of_o magnitude_n of_o other_o kind_n as_o of_o superficial_a figure_n and_o also_o of_o body_n 2_o incommensurable_a magnitude_n be_v such_o which_o no_o one_o common_a measure_n do_v measure_n definition_n this_o definition_n nead_v no_o explanation_n at_o all_o other_o it_o be_v easy_o understand_v by_o the_o definition_n go_v before_o of_o line_n commensurable_a for_o contrary_n be_v make_v manifest_a by_o compare_v of_o the_o one_o to_o the_o other_o as_o if_o the_o line_n c_o or_o any_o other_o line_n oftentimes_o iterate_v do_v not_o render_v precise_o the_o line_n a_o nor_o the_o line_n b_o them_z be_v the_o line_n a_o and_o b_o incommensurable_a also_o if_o the_o line_n c_o or_o any_o other_o line_n certain_a time_n repeat_v do_v exact_o render_v the_o line_n a_o and_o do_v not_o measure_v the_o line_n b_o or_o if_o it_o measure_v the_o line_n b_o and_o measure_v not_o also_o the_o line_n a_o the_o line_n a_o and_o b_o be_v yet_o line_n incommensurable_a &_o so_o of_o other_o magnitude_n as_o of_o superficiece_n and_o body_n 3_o right_a line_n commensurable_a in_o power_n be_v such_o who_o square_n one_o and_o the_o self_n same_o superficies_n area_n or_o plat_n do_v measure_n definition_n to_o the_o declaration_n of_o this_o definition_n we_o must_v first_o call_v to_o mind_n what_o be_v understand_v &_o mean_v by_o the_o power_n of_o a_o line_n which_o as_o we_o have_v before_o in_o the_o former_a book_n note_v be_v nothing_o else_o but_o the_o square_a thereof_o or_o any_o other_o plain_a figure_n equal_a to_o the_o square_a thereof_o and_o so_o great_a power_n &_o hability_n â—s_v a_o line_n say_v to_o have_v as_o be_v the_o quantity_n of_o the_o square_n which_o it_o be_v able_a to_o describe_v or_o a_o figure_n superficial_a equal_a to_o the_o square_a
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 superficies_n namely_o the_o superficies_n a_o measure_v the_o square_n bcde_v take_v 16._o time_n which_o be_v the_o number_n of_o the_o little_a area_n square_n plait_n or_o superficiece_n contain_v and_o describe_v within_o the_o say_v square_n each_o of_o which_o be_v equal_a to_o the_o superficie_fw-la a._n again_o let_v the_o same_o superficies_n a_o measure_n the_o square_a fhik_n 9_o time_n take_v according_a to_o the_o number_n of_o the_o fieldâ—s_n or_o superficiece_n contain_v and_o describe_v in_o the_o same_o you_o see_v they_o that_o one_o and_o the_o self_n same_o superficies_n namely_o the_o superficies_n a_o be_v a_o common_a measure_n to_o both_o these_o square_n and_o by_o certain_a repetition_n thereof_o measure_v they_o both_o wherefore_o the_o two_o line_n bc_o and_o fh_o which_o be_v the_o side_n or_o line_n produce_v these_o square_n and_o who_o power_n these_o square_n be_v be_v by_o this_o definition_n line_n commensurable_a in_o power_n 4_o line_n incommensurable_a be_v such_o who_o square_n no_o one_o plat_n or_o superficies_n do_v measure_n definition_n this_o definition_n be_v easy_a to_o be_v understand_v by_o that_o which_o be_v say_v in_o the_o definition_n last_o set_v before_o this_o and_o nead_v no_o far_a declaration_n and_o thereof_o take_v this_o example_n if_o neither_o the_o superficies_n a_o nor_o any_o other_o superficies_n do_v measure_v the_o two_o square_n b_o cde_o and_o fhik_n or_o if_o it_o measure_v the_o one_o â—â—rely_o bcde_v and_o not_o the_o other_o fhik_n or_o if_o it_o measure_v the_o square_a fhik_n and_o not_o the_o square_n bcde_v the_o two_o line_n bc_o and_o fh_o be_v in_o power_n incommensurable_a and_o therefore_o also_o incommensurable_a in_o length_n for_o whatsoever_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 as_o shall_v afterward_o in_o the_o 9_o proposition_n of_o this_o book_n be_v prove_v and_o therefore_o such_o line_n be_v here_o define_v to_o be_v absolute_o incommensurable_a these_o thing_n thus_o stand_v it_o may_v easy_o appear_v that_o if_o a_o line_n be_v assign_v and_o lay_v before_o we_o there_o may_v be_v innumerable_a other_o line_n commensurable_a unto_o it_o and_o other_o incommensurable_a unto_o it_o of_o commensurable_a line_n some_o be_v commensurable_a in_o length_n and_o power_n and_o some_o in_o power_n only_o 5_o and_o that_o right_a line_n so_o set_v forth_o be_v call_v a_o rational_a line_n thus_o may_v you_o see_v how_o to_o the_o suppose_a line_n first_o set_v may_v be_v compare_v infinite_a line_n line_n some_o commensurable_a both_o in_o length_n &_o power_n and_o some_o commensurable_a in_o power_n only_o and_o incommensurable_a in_o length_n and_o some_o incommensurable_a both_o in_o power_n &_o in_o length_n and_o this_o first_o line_n so_o set_v whereunto_o and_o to_o who_o square_n the_o other_o line_n and_o their_o square_n be_v compare_v be_v call_v a_o rational_a line_n common_o of_o the_o most_o part_n of_o writer_n line_n but_o some_o there_o be_v which_o mislike_n that_o it_o shall_v be_v call_v a_o rational_a line_n &_o that_o not_o without_o just_a cause_n in_o the_o greek_a copy_n it_o be_v call_v 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d rete_fw-la which_o signify_v a_o thing_n that_o may_v be_v speak_v &_o express_v by_o word_n a_o thing_n certain_a grant_v and_o appoint_v wherefore_o flussates_n a_o man_n which_o bestow_v great_a travel_n and_o diligence_n in_o restore_v of_o these_o element_n of_o euclid_n leave_v this_o word_n rational_a call_v this_o line_n suppose_v and_o first_o set_v a_o line_n certain_a because_o the_o part_n thereof_o into_o which_o it_o be_v divide_v be_v certain_a certain_a and_o know_v and_o may_v be_v express_v by_o voice_n and_o also_o be_v count_v by_o number_n other_o line_n be_v to_o this_o line_n incommensurable_a who_o part_n be_v not_o distinct_o know_v but_o be_v uncertain_a nor_o can_v be_v express_v by_o name_n nor_o assign_v by_o number_n which_o be_v of_o other_o man_n call_v irrational_a he_o call_v uncertain_a and_o surd_a line_n petrus_n montaureus_n although_o he_o do_v not_o very_o well_o like_a of_o the_o name_n yet_o he_o alter_v it_o not_o but_o use_v it_o in_o all_o his_o book_n likewise_o will_v we_o do_v here_o for_o that_o the_o word_n have_v be_v and_o be_v so_o universal_o receive_v and_o therefore_o will_v we_o use_v the_o same_o name_n and_o call_v it_o a_o rational_a line_n for_o it_o be_v not_o so_o great_a a_o matter_n what_o name_n we_o geve_v to_o thing_n so_o that_o we_o full_o understand_v the_o thing_n which_o the_o name_n signify_v this_o rational_a line_n thus_o here_o define_v be_v the_o ground_n and_o foundation_n of_o all_o the_o proposition_n almost_o of_o this_o whole_a ten_o book_n book_n and_o chief_o from_o the_o ten_o proposition_n forward_o so_o that_o unless_o you_o first_o place_n this_o rational_a line_n and_o have_v a_o special_a and_o continual_a regard_n unto_o it_o before_o you_o begin_v any_o demonstration_n you_o shall_v not_o easy_o understand_v it_o for_o it_o be_v as_o it_o be_v the_o touch_n and_o trial_n of_o all_o other_o line_n by_o which_o it_o be_v know_v whether_o any_o of_o they_o be_v rational_a or_o not_o note_n and_o this_o may_v be_v call_v the_o first_o rational_a line_n purpose_n the_o line_n rational_a of_o purpose_n or_o a_o rational_a line_n set_v in_o the_o first_o place_n and_o so_o make_v distinct_a and_o sever_v from_o other_o rational_a line_n of_o which_o shall_v be_v speak_v afterward_o and_o this_o must_v you_o well_o commit_v to_o memory_n definition_n 6_o line_n which_o be_v commensurable_a to_o this_o line_n whether_o in_o length_n and_o power_n or_o in_o power_n only_o be_v also_o call_v rational_a this_o definition_n need_v no_o declaration_n at_o all_o but_o be_v easy_o perceive_v if_o the_o first_o definition_n be_v remember_v which_o show_v what_o magnitude_n be_v commensurable_a and_o the_o three_o which_o show_v what_o line_n be_v commensurable_a in_o power_n here_o notâ—_n how_o apt_o &_o natural_o euclid_n in_o this_o place_n use_v these_o word_n commensurable_a either_o in_o length_n and_o power_n or_o in_o power_n only_o because_o that_o all_o line_n which_o be_v commensurable_a in_o length_n be_v also_o commensurable_a in_o pour_v when_o he_o speak_v of_o line_n commensurable_a in_o length_n he_o ever_o add_v and_o in_o power_n but_o when_o he_o speak_v of_o line_n commensurable_a in_o power_n he_o add_v this_o word_n only_o and_o add_v not_o this_o word_n in_o length_n as_o he_o in_o the_o other_o add_v this_o word_n in_o power_n for_o not_o all_o line_n which_o be_v commensurable_a in_o power_n be_v straight_o way_n commensurable_a also_o in_o length_n of_o this_o definition_n take_v this_o example_n let_v the_o first_o line_n rational_a of_o purpose_n which_o be_v suppose_v and_o lay_v forth_o who_o part_n be_v certain_a &_o know_v and_o may_v be_v express_v name_v and_o number_v be_v ab_fw-la the_o quadrate_n whereof_o let_v be_v abcd_o then_o suppose_v again_o a_o other_o line_n namely_o the_o line_n of_o which_o let_v be_v commensurable_a both_o in_o length_n and_o in_o power_n to_o the_o first_o rational_a line_n that_o be_v as_o before_o be_v teach_v let_v one_o line_n measure_v the_o length_n of_o each_o line_n and_o also_o lâ—t_v one_o superficies_n measure_v the_o two_o square_n of_o the_o say_v two_o line_n as_o here_o in_o the_o example_n be_v suppose_v and_o also_o appear_v to_o the_o eye_n then_o be_v the_o line_n e_o fletcher_n also_o a_o rational_a line_n moreover_o if_o the_o line_n of_o be_v commensurable_a in_o power_n only_o to_o the_o rational_a line_n ab_fw-la first_o set_v and_o suppose_v so_o that_o no_o one_o line_n do_v measure_v the_o two_o line_n ab_fw-la and_o of_o as_o in_o example_n yâ—_n see_v to_o be_v for_o
that_o the_o line_n of_o be_v make_v equal_a to_o the_o line_n ad_fw-la which_o be_v the_o diameter_n of_o the_o square_n abcd_o of_o which_o square_v the_o line_n ab_fw-la be_v a_o side_n it_o be_v certain_a that_o the_o â—ide_v of_o a_o square_n be_v incommensurable_a in_o length_n to_o the_o diameter_n of_o the_o same_o square_n if_o there_o be_v yet_o find_v any_o one_o superficies_n which_o measure_v the_o two_o square_n abcd_o and_o efgh_a as_o here_o do_v the_o triangle_n abdella_n or_o the_o triangle_n acd_v note_v in_o the_o square_n abcd_o or_o any_o of_o the_o four_o triangle_n note_v in_o the_o square_a efgh_o as_o appear_v somewhat_o more_o manifest_o in_o the_o second_o example_n in_o the_o declaration_n of_o the_o last_o definition_n go_v before_o the_o line_n of_o be_v also_o a_o rational_a line_n note_v that_o these_o line_n which_o here_o be_v call_v rational_a line_n be_v not_o rational_a line_n of_o purpose_n or_o by_o supposition_n as_o be_v the_o first_o rational_a line_n but_o be_v rational_a only_o by_o reason_n of_o relation_n and_o comparison_n which_o they_o have_v unto_o it_o because_o they_o be_v commensurable_a unto_o it_o either_o in_o length_n and_o power_n or_o in_o power_n only_o far_o here_o be_v to_o be_v note_v that_o these_o word_n length_n and_o power_n and_o power_n only_o be_v join_v only_o with_o these_o wordeâ—_n commensurable_a or_o incommensurable_a and_o be_v never_o join_v with_o these_o word_n rational_a or_o irrational_a so_o that_o no_o line_n can_v be_v call_v rational_a in_o length_n or_o in_o power_n nor_o like_o wise_a can_v they_o be_v call_v irrational_a in_o length_n or_o in_o power_n wherein_o undoubted_o campanus_n be_v deceive_v book_n who_o use_v those_o word_n &_o speech_n indifferent_o cause_v &_o bring_v in_o great_a obscurity_n to_o the_o proposition_n and_o demonstration_n of_o this_o book_n which_o he_o shall_v easy_o see_v which_o mark_v with_o diligence_n the_o demonstration_n of_o campanus_n in_o this_o book_n 7_o line_n which_o be_v incommensurable_a to_o the_o rational_a line_n be_v call_v irrational_a definition_n by_o line_n incommensurable_a to_o the_o rational_a line_n suppose_v in_o this_o place_n he_o understand_v such_o as_o be_v incommensurable_a unto_o it_o both_o in_o length_n and_o in_o power_n for_o there_o be_v no_o line_n incommensurable_a in_o power_n only_o for_o it_o can_v be_v that_o any_o line_n shall_v so_o be_v incommensurable_a in_o power_n only_o that_o they_o be_v not_o also_o incommensurable_a in_o length_n what_o so_o ever_o line_n be_v incommensurable_a in_o power_n the_o same_o be_v also_o incommensurable_a in_o length_n neither_o can_v euclid_n here_o in_o this_o place_n mean_a line_n incommensurable_a in_o length_n only_o for_o in_o the_o definition_n before_o he_o call_v they_o rational_a line_n neither_o may_v they_o be_v place_v amongst_o irrational_a line_n wherefore_o it_o remain_v that_o in_o this_o diffintion_n he_o speak_v only_o of_o those_o line_n which_o be_v incommensurable_a to_o the_o rational_a line_n first_o give_v and_o suppose_v both_o in_o length_n and_o in_o power_n which_o by_o all_o mean_n be_v incommensurable_a to_o the_o rational_a line_n &_o therefore_o most_o apt_o be_v they_o call_v irrational_a line_n this_o definition_n be_v easy_a to_o be_v understand_v by_o that_o which_o have_v be_v say_v before_o yet_o for_o the_o more_o plainness_n see_v this_o example_n let_v the_o â—â—rst_a rational_a line_n suppose_v be_v the_o line_n ab_fw-la who_o square_a or_o quadrate_n let_v be_v abcd._n and_o let_v there_o be_v give_v a_o other_o line_n of_o which_o lâ—t_v be_v to_o the_o rational_a line_n incommensurable_a in_o length_n and_o power_n so_o that_o let_v no_o one_o line_n measure_v the_o length_n of_o the_o two_o line_n ab_fw-la and_o of_o and_o let_v the_o square_n of_o the_o line_n of_o be_v efgh_a now_o if_o also_o there_o be_v no_o one_o superficies_n which_o measure_v the_o two_o square_n abcd_o and_o efgh_a as_o be_v suppose_v to_o be_v in_o this_o example_n they_o be_v the_o line_n of_o a_o irrational_a line_n which_o word_n irrational_a as_o before_o do_v this_o word_n rational_a mislike_v many_o learned_a in_o this_o knowledge_n of_o geometry_n flussates_n uncertain_a as_o he_o leave_v the_o word_n rational_a and_o in_o stead_n thereof_o use_v this_o word_n certain_a so_o here_o he_o leave_v the_o word_n irrational_a and_o use_v in_o place_n thereof_o this_o word_n uncertain_a and_o ever_o name_v these_o line_n uncertain_a line_n petrus_n montaureus_n also_o mislike_v the_o word_n irrational_a will_v rather_o have_v they_o to_o be_v call_v surd_a line_n yet_o because_o this_o word_n irrational_a have_v ever_o by_o custom_n and_o long_a use_n so_o general_o be_v receive_n he_o use_v continual_o the_o same_o in_o greek_a such_o line_n be_v call_v 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d alogoi_n which_o signify_v nameless_a unspeakable_a uncertain_a in_o determinate_a line_n and_o with_o out_o proportion_n not_o that_o these_o irrational_a line_n have_v no_o proportion_n at_o all_o either_o to_o the_o first_o rational_a line_n or_o between_o themselves_o but_o be_v so_o name_v for_o that_o their_o proportion_n to_o the_o rational_a line_n can_v be_v express_v in_o number_n that_o be_v undoubted_o very_o untrue_a which_o many_o write_v that_o their_o proportion_n be_v unknown_a both_o to_o we_o and_o to_o nature_n be_v it_o not_o think_v you_o a_o thing_n very_o absurd_a to_o say_v that_o there_o be_v any_o thing_n in_o nature_n and_o produce_v by_o nature_n to_o be_v hide_v from_o nature_n and_o not_o to_o be_v know_v of_o nature_n it_o can_v not_o be_v say_v that_o their_o proportion_n be_v utter_o hide_v and_o unknown_a to_o we_o much_o less_o unto_o nature_n although_o we_o can_v geve_v they_o their_o name_n and_o distinct_o express_v they_o by_o number_n otherwise_o shall_v euclid_n have_v take_v all_o this_o travel_n and_o wonderful_a diligence_n bestow_v in_o this_o bookeâ—_n in_o vain_a and_o to_o no_o useâ—_n in_o which_o he_o do_v nothing_o ellâ—_n but_o teach_v the_o propriety_n and_o passion_n of_o these_o irrational_a linesâ—_n and_o show_v the_o proportion_n which_o they_o have_v the_o one_o to_o the_o other_o here_o be_v also_o to_o be_v note_v which_o thing_n also_o tartalea_n have_v before_o diligent_o notedâ—_n that_o campanus_n and_o many_o other_o writer_n of_o geometryâ—_n over_o much_o â—â—â—ed_a and_o be_v deceive_v in_o that_o they_o write_v and_o teach_v that_o all_o these_o line_n who_o square_n be_v not_o signify_v and_o may_v be_v express_v by_o a_o square_a number_n although_o they_o may_v by_o any_o other_o number_n as_o by_o 11._o 12._o 14._o and_o such_o other_o not_o square_a number_n be_v irrational_a line_n which_o be_v manifest_o repugnant_a to_o the_o ground_n and_o principle_n of_o euclid_n who_o will_v that_o all_o line_n which_o be_v commensurable_a to_o the_o rational_a line_n whether_o it_o be_v in_o length_n and_o power_n or_o in_o power_n only_o shall_v be_v rational_a undoubted_o this_o have_v be_v one_o of_o the_o chief_a and_o great_a cause_n of_o the_o wonderful_a confusion_n and_o darkness_n of_o this_o book_n book_n which_o so_o have_v toss_v and_o turmoil_v the_o wit_n of_o all_o both_o writer_n and_o reader_n master_n and_o scholar_n and_o so_o overwhelm_v they_o that_o they_o can_v not_o with_o out_o infinite_a travel_n and_o sweat_v attain_v to_o the_o truth_n and_o perfect_a understanding_n thereof_o definition_n 8_o the_o square_n which_o be_v describe_v of_o the_o rational_a right_a line_n suppose_v be_v rational_a until_o this_o definition_n have_v euclid_n set_v forth_o the_o nature_n and_o propriety_n of_o the_o first_o kind_n of_o magnitude_n namely_o of_o line_n how_o they_o be_v rational_a or_o irrational_a now_o he_o begin_v to_o â—hew_v how_o the_o second_o kind_n of_o magnitude_n namely_o superficies_n be_v one_o to_o the_o other_o rational_a or_o irrational_a this_o definition_n be_v very_a plain_n suppose_v the_o line_n ab_fw-la to_o be_v the_o rational_a line_n have_v his_o part_n and_o division_n certain_o know_v the_o square_a of_o which_o line_n let_v be_v the_o square_a abcd._n now_o because_o it_o be_v the_o square_a of_o the_o rational_a line_n ab_fw-la it_o be_v also_o call_v rational_a and_o as_o the_o line_n ab_fw-la be_v the_o first_o rational_a line_n unto_o which_o other_o line_n compare_v be_v count_v rational_a or_o irrational_a so_o be_v the_o quadrat_fw-la or_o square_v thereof_o the_o â—irst_n rational_a superficies_n unto_o which_o all_o other_o square_n or_o figure_n compare_v be_v count_v and_o name_v rational_a or_o irrational_a 9_o such_o which_o be_v commensurable_a unto_o it_o be_v rational_a definition_n in_o this_o definition_n where_o it_o be_v say_v such_o as_o be_v commensurable_a to_o the_o square_n of_o the_o rational_a line_n be_v not_o understand_v only_o other_o square_n or_o
quadrates_n but_o all_o other_o kind_n of_o rectiline_a figure_n plain_a plait_n &_o superficiese_n what_o so_o ever_o so_o that_o if_o any_o such_o figure_n be_v commensurable_a unto_o that_o rational_a squareâ—_n it_o be_v also_o rational_a as_o suppose_v that_o the_o square_a of_o the_o rational_a line_n which_o be_v also_o rational_a be_v abcd_o suppose_v 〈◊〉_d so_o some_o other_o square_a as_o the_o square_a efgh_o to_o be_v commensurable_a to_o the_o same_o they_o be_v the_o square_a efgh_o also_o rational_a so_o also_o if_o the_o rectiline_a figure_n klmn_n which_o be_v a_o figure_n on_o the_o one_o side_n long_o be_v commensurable_a unto_o the_o say_a square_a as_o be_v suppose_v in_o this_o exampleâ—_n it_o be_v also_o a_o rational_a superficies_n and_o so_o of_o all_o other_o superficiese_n 10_o such_o which_o be_v incommensurable_a unto_o it_o be_v irrational_a definition_n where_o it_o be_v say_v in_o this_o definition_n such_o which_o be_v incommensurable_a it_o be_v general_o to_o be_v take_v as_o be_v this_o word_n commensurable_a in_o the_o definition_n before_o for_o all_o superficiese_n whether_o they_o be_v square_n or_o figure_n on_o the_o one_o side_n long_o or_o otherwise_o what_o manner_n of_o right_n line_v figure_n so_o ever_o it_o be_v if_o they_o be_v incommensurable_a unto_o the_o rational_a square_n suppose_v they_o be_v they_o irrational_a as_o let_v thâ—_z square_v abcd_o be_v the_o square_n of_o the_o suppose_a rational_a line_n which_o square_n therefore_o be_v also_o rational_a suppose_v also_o also_o a_o other_o square_a namely_o the_o square_a e_o suppose_v also_o any_o other_o figure_n as_o for_o example_n sake_n a_o figure_n of_o one_o side_n long_o which_o let_v be_v f_o now_o if_o the_o square_a e_o and_o the_o figure_n fletcher_n be_v both_o incommensurable_a to_o the_o rational_a square_n abcd_o then_o be_v 〈◊〉_d of_o these_o figure_n e_o &_o fletcher_n irrational_a and_o so_o of_o other_o 11_o and_o these_o line_n who_o powere_n they_o be_v be_v irrational_a if_o they_o be_v square_n then_o be_v their_o side_n irrational_a if_o they_o be_v not_o square_n definition_n but_o some_o other_o rectiline_a figure_n then_o shall_v the_o line_n who_o square_n be_v equal_a to_o these_o rectiline_a figure_n be_v irrational_a suppose_v that_o the_o rational_a square_n be_v abcd._n suppose_v also_o a_o other_o square_a namely_o the_o square_a e_o which_o let_v be_v incommensurable_a to_o the_o rational_a square_n &_o therefore_o be_v it_o irrational_a and_o let_v the_o side_n or_o line_n which_o produce_v this_o square_n be_v the_o line_n fg_o then_o shall_v the_o line_n fg_o by_o this_o definition_n be_v a_o irrational_a line_n because_o it_o be_v the_o side_n of_o a_o irrational_a square_n let_v also_o the_o figure_n h_o be_v a_o figure_n on_o the_o one_o side_n long_o which_o may_v be_v any_o other_o rectiline_a figure_n rectangle_v or_o not_o rectangle_v triangle_n pentagone_fw-mi trapezite_fw-mi or_o what_o so_o ever_o else_o be_v incommensurable_a to_o the_o rational_a square_n abcd_o then_o because_o the_o figure_n h_o be_v not_o a_o square_a it_o have_v no_o side_n or_o root_n to_o produce_v it_o yet_o may_v there_o be_v a_o square_n make_v equal_a unto_o it_o for_o that_o all_o such_o figure_n may_v be_v reduce_v into_o triangle_n and_o so_o into_o square_n by_o the_o 14._o of_o the_o second_o suppose_v that_o the_o square_a queen_n be_v equal_a to_o the_o irrational_a figure_n h._n the_o side_n of_o which_o figure_n q_n let_v be_v the_o line_n kl_o then_o shall_v the_o line_n kl_o be_v also_o a_o irrational_a line_n because_o the_o power_n or_o square_v thereof_o be_v equal_a to_o the_o irrational_a figure_n h_o and_o thus_o conceive_v of_o other_o the_o like_a these_o irrational_a line_n and_o figure_n be_v the_o chief_a matter_n and_o subject_n which_o be_v entreat_v of_o in_o all_o this_o ten_o book_n the_o knowledge_n of_o which_o be_v deep_a and_o secret_a and_o pertain_v to_o the_o high_a and_o most_o worthy_a part_n of_o geometry_n wherein_o stand_v the_o pith_n and_o marry_v of_o the_o hole_n science_n the_o knowledge_n hereof_o bring_v light_n to_o all_o the_o book_n follow_v with_o out_o which_o they_o be_v hard_a and_o can_v be_v at_o all_o understand_v and_o for_o the_o more_o plainness_n you_o shall_v note_v that_o of_o irrational_a line_n there_o be_v diâ—ers_n sort_n and_o kind_n but_o they_o who_o name_n be_v set_v in_o a_o table_n here_o follow_v and_o be_v in_o number_n 13._o be_v the_o chief_a and_o in_o this_o ten_o book_n sufficient_o for_o euclides_n principal_a purpose_n discourse_v on_o a_o medial_a line_n a_o binomial_a line_n a_o first_o bimedial_a line_n a_o second_o bimedial_a line_n a_o great_a line_n a_o line_n contain_v in_o power_n a_o rational_a superficies_n and_o a_o medial_a superficies_n a_o line_n contain_v in_o power_n two_o medial_a superficiece_n a_o residual_a line_n a_o first_o medial_a residual_a line_n a_o second_o medial_a residual_a line_n a_o less_o line_n a_o line_n make_v with_o a_o rational_a superficies_n the_o whole_a superficies_n medial_a a_o line_n make_v with_o a_o medial_a superficies_n the_o whole_a superficies_n medial_a of_o all_o which_o kind_n the_o definition_n together_o with_o there_o declaration_n shall_v be_v set_v here_o after_o in_o their_o due_a place_n ¶_o the_o 1._o theorem_a the_o 1._o proposition_n two_o unequal_a magnitude_n be_v give_v if_o from_o the_o great_a be_v take_v away_o more_o than_o the_o half_a and_o from_o the_o residue_n be_v again_o take_v away_o more_o than_o the_o half_a and_o so_o be_v do_v still_o continual_o there_o shall_v at_o length_n be_v leave_v a_o certain_a magnitude_n lesser_a than_o the_o less_o of_o the_o magnitude_n first_o give_v svppose_v that_o there_o be_v two_o unequal_a magnitude_n ab_fw-la and_o c_o of_o which_o let_v ab_fw-la be_v the_o great_a then_o i_o say_v that_o if_o from_o ab_fw-la be_v take_v away_o more_o than_o the_o half_a construction_n and_o from_o the_o residue_n be_v take_v again_o more_o than_o the_o half_a and_o so_o still_o continual_o there_o shall_v at_o the_o length_n be_v leave_v a_o certain_a magnitude_n lesser_a than_o the_o less_o magnitude_n give_v namely_o then_o c._n for_o forasmuch_o as_o c_z be_v the_o less_o magnitude_n therefore_o c_z may_v be_v so_o multiply_v that_o at_o the_o length_n it_o will_v be_v great_a than_o the_o magnitude_n ab_fw-la by_o the_o 5._o definition_n of_o the_o five_o book_n demonstration_n let_v it_o be_v so_o multiply_v and_o let_v the_o multiplex_n of_o c_z great_a than_o ab_fw-la be_v de._n and_o divide_v de_fw-fr into_o the_o part_n equal_a unto_o c_o which_o let_v be_v df_o fg_o and_o ge._n and_o from_o the_o magnitude_n ab_fw-la take_v away_o more_o than_o the_o half_a which_o let_v be_v bh_o and_o again_o from_o ah_o take_v away_o more_o than_o the_o half_a which_o let_v be_v hk_a and_o so_o do_v continual_o until_o the_o division_n which_o be_v in_o the_o magnitude_n ab_fw-la be_v equal_a in_o multitude_n unto_o the_o division_n which_o be_v in_o the_o magnitude_n de._n so_o that_o let_v the_o division_n ak_o kh_o and_o hb_o be_v equal_a in_o multitude_n unto_o the_o division_n df_o fg_o and_o ge._n and_o forasmuch_o as_o the_o magnitude_n de_fw-fr be_v great_a than_o the_o magnitude_n ab_fw-la and_o from_o de_fw-fr be_v take_v away_o less_o than_o the_o half_a that_o be_v eglantine_n which_o detraction_n or_o take_v away_o be_v understand_v to_o be_v do_v by_o the_o former_a division_n of_o the_o magnitude_n de_fw-fr into_o the_o part_n equal_a unto_o c_o for_o as_o a_o magnitude_n be_v by_o multiplication_n increase_v so_o be_v it_o by_o division_n diminish_v and_o from_o ab_fw-la be_v take_v away_o more_o than_o the_o half_a that_o be_v bh_o therefore_o the_o residue_n gd_o be_v great_a than_o the_o residue_n ha_o which_o thing_n be_v most_o true_a and_o most_o easy_a to_o conceive_v if_o we_o remember_v this_o principle_n that_o the_o residue_n of_o a_o great_a magnitude_n after_o the_o take_v away_o of_o the_o half_a or_o less_o than_o the_o half_a be_v ever_o great_a than_o the_o residue_n of_o a_o less_o magnitude_n after_o the_o take_v away_o of_o more_o than_o the_o half_a and_o forasmuch_o as_o the_o magnitude_n gd_o be_v great_a than_o the_o magnitude_n ha_o and_o from_o gd_o be_v take_v away_o the_o half_a that_o be_v gf_o and_o from_o ah_o be_v take_v away_o more_o than_o the_o half_a that_o be_v hk_o therefore_o the_o residue_n df_o be_v great_a than_o the_o residue_n ak_o by_o the_o foresay_a principle_n but_o the_o magnitude_n df_o be_v equal_a unto_o the_o magnitude_n c_o by_o supposition_n wherefore_o also_o the_o magnitude_n c_o be_v great_a than_o the_o magnitude_n ak_o wherefore_o the_o magnitude_n ak_o be_v less_o than_o the_o magnitude_n c._n wherefore_o of_o the_o magnitude_n
the_o line_n ab_fw-la be_v rational_a by_o the_o definition_n wherefore_o by_o the_o definition_n also_o of_o rational_a figure_n the_o parallelogram_n cd_o shall_v be_v rational_a now_o rest_v a_o other_o caâ—e_n of_o the_o third_o kind_n of_o rational_a line_n commensurable_a in_o length_n the_o one_o to_o the_o other_o which_o be_v to_o the_o rational_a line_n ab_fw-la first_o set_v commensurable_a in_o power_n only_o and_o yet_o be_v therefore_o rational_a line_n and_o let_v the_o line_n ce_fw-fr and_o ed_z be_v commensurable_a in_o length_n the_o one_o to_o the_o other_o length_n now_o then_o let_v the_o self_n same_o construction_n remain_v that_o be_v in_o the_o former_a so_o that_o let_v the_o line_n ce_fw-fr and_o ed_z be_v rational_a commensurable_a in_o power_n only_o unto_o the_o line_n ab_fw-la but_o let_v they_o be_v commensurable_a in_o length_n the_o one_o to_o the_o other_o then_o i_o say_v that_o in_o this_o case_n also_o the_o parallelogram_n cd_o be_v rational_a first_o it_o may_v be_v prove_v as_o before_o that_o the_o parallelogram_n cd_o be_v commensurable_a to_o the_o square_a df._n wherefore_o by_o the_o 12._o of_o this_o book_n the_o parallelogram_n cd_o shall_v be_v commensurable_a to_o the_o square_n of_o the_o line_n abâ—_n but_o the_o square_n of_o the_o line_n ab_fw-la be_v rational_a case_n wherefore_o by_o the_o definition_n the_o parallelogram_n cd_o shall_v be_v also_o rational_a this_o case_n be_v well_o to_o be_v note_v for_o it_o serve_v to_o the_o demonstration_n and_o understanding_n of_o the_o 25._o proposition_n of_o this_o book_n ¶_o the_o 17._o theorem_a the_o 20._o proposition_n if_o upon_o a_o rational_a line_n be_v apply_v a_o rational_a rectangle_n parallelogram_n the_o other_o side_n that_o make_v the_o breadth_n thereof_o shall_v be_v a_o rational_a line_n and_o commensurable_a in_o length_n unto_o that_o line_n whereupon_o the_o rational_a parallelogram_n be_v apply_v svppose_v that_o this_o rational_a rectangle_n parallelogram_n ac_fw-la be_v apply_v upon_o the_o line_n ab_fw-la which_o let_v be_v rational_a according_a to_o any_o one_o of_o the_o foresay_a way_n whether_o it_o be_v the_o first_o rational_a line_n set_v proposition_n or_o any_o other_o line_n commensurable_a to_o the_o rational_a line_n first_o set_v and_o that_o in_o length_n and_o in_o power_n or_o in_o power_n only_o for_o one_o of_o these_o three_o way_n as_o be_v declare_v in_o the_o assumpt_n put_v before_o the_o 19_o proposition_n of_o this_o book_n be_v a_o line_n call_v rational_a and_o make_v in_o breadth_n the_o line_n bc._n then_o i_o say_v that_o the_o line_n bc_o be_v rational_a and_o commensurable_a in_o length_n unto_o the_o line_n basilius_n construction_n desrcribe_v by_o the_o 46._o of_o the_o first_o upon_o the_o line_n basilius_n a_o square_a ad._n wherefore_o by_o the_o 9_o definition_n of_o the_o ten_o the_o square_a ad_fw-la be_v rational_a but_o the_o parallelogram_n ac_fw-la also_o be_v rational_a by_o supposition_n demonstration_n wherefore_o by_o the_o conversion_n of_o the_o definition_n of_o rational_a figure_n or_o by_o the_o 12._o of_o this_o book_n the_o square_a dam_n be_v commensurable_a unto_o the_o parallelogram_n ac_fw-la but_o as_o the_o square_a dam_n be_v to_o the_o parallelogram_n ac_fw-la so_o be_v the_o line_n db_o to_o the_o line_n bc_o by_o the_o first_o of_o the_o six_o wherefore_o by_o the_o 10._o of_o the_o ten_o the_o line_n db_o be_v commensurable_a unto_o the_o line_n bc._n but_o the_o line_n db_o be_v equal_a unto_o the_o line_n basilius_n wherefore_o the_o line_n ab_fw-la be_v commensurable_a unto_o the_o line_n bc._n but_o the_o line_n ab_fw-la be_v rational_a wherefore_o the_o line_n bc_o also_o be_v rational_a and_o commensurable_a in_o length_n unto_o the_o line_n basilius_n if_o therefore_o upon_o a_o rational_a line_n be_v apply_v a_o rational_a rectangle_n parallelogram_n the_o other_o side_n that_o make_v the_o breadth_n thereof_o shall_v be_v a_o rational_a line_n commensurable_a in_o length_n unto_o that_o line_n whereupon_o the_o rational_a parallelogram_n be_v apply_v which_o be_v require_v to_o be_v demonstrate_v ¶_o a_o assumpt_n a_o line_n contain_v in_o power_n a_o irrational_a superficies_n be_v irrational_a assumpt_n suppose_v that_o the_o line_n ab_fw-la contain_v in_o power_n a_o irrational_a superficies_n that_o be_v let_v the_o square_v describe_v upon_o the_o line_n ab_fw-la be_v equal_a unto_o a_o irrational_a superficies_n then_o i_o say_v that_o the_o line_n ab_fw-la be_v irrational_a for_o if_o the_o line_n ab_fw-la be_v rational_a they_o shall_v the_o square_a of_o the_o line_n ab_fw-la be_v also_o rational_a for_o so_o be_v it_o put_v in_o the_o definition_n but_o by_o supposition_n it_o be_v not_o wherefore_o the_o line_n ab_fw-la be_v irrational_a a_o line_n therefore_o contain_v in_o power_n a_o irrational_a superficies_n be_v irrational_a ¶_o the_o 18._o theorem_a the_o 21._o proposition_n a_o rectangle_n figure_n comprehend_v under_o two_o rational_a right_a line_n commensurable_a in_o power_n only_o be_v irrational_a and_o the_o line_n which_o in_o power_n contain_v that_o rectangle_n figure_n be_v irrational_a &_o be_v call_v a_o medial_a line_n svppose_v that_o this_o rectangle_n figure_n ac_fw-la be_v comprehend_v under_o these_o rational_a right_a line_n ab_fw-la and_o bc_o commensurable_a in_o power_n only_o then_o i_o say_v that_o the_o superficies_n ac_fw-la be_v irrational_a and_o the_o line_n which_o contain_v it_o in_o power_n be_v irrational_a and_o be_v call_v a_o medial_a line_n construction_n describe_v by_o the_o 46._o of_o the_o first_o upon_o the_o line_n ab_fw-la a_o square_a ad._n demonstration_n wherefore_o the_o square_a ad_fw-la be_v rational_a and_o forasmuch_o as_o the_o line_n ab_fw-la be_v unto_o the_o line_n bc_o incommensurable_a in_o length_n for_o they_o be_v suppose_v to_o be_v commensurable_a in_o power_n only_o and_o the_o line_n ab_fw-la be_v equal_a unto_o the_o line_n bd_o therefore_o also_o the_o lineâ—_z bd_o be_v unto_o the_o line_n bc_o incommensurable_a in_o length_n and_o 〈◊〉_d â—hâ—_n linâ—_n 〈…〉_o be_v to_o the_o line_n â—_o c_o so_fw-mi 〈◊〉_d the_o square_a ad_fw-la to_o the_o parallelogram_n ac_fw-la by_o the_o first_o of_o the_o fiuâ—_o wherefore_o by_o the_o 10._o of_o the_o ten_o the_o square_a dam_n be_v unto_o the_o parallelogram_n ac_fw-la incommensurable_a but_o the_o square_a dam_n be_v rational_a wherefore_o the_o parallelogram_n ac_fw-la be_v irrational_a wherefore_o also_o the_o line_n that_o contain_v the_o superficies_n ac_fw-la in_o power_n that_o be_v who_o square_a be_v equal_a unto_o the_o parallelogram_n ac_fw-la be_v by_o the_o assumpt_n go_v before_o irrational_a and_o it_o be_v call_v a_o medial_a line_n for_o that_o the_o square_n which_o be_v make_v of_o it_o be_v equal_a to_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o and_o therefore_o it_o be_v by_o the_o second_o part_n of_o the_o 17._o of_o the_o six_o a_o mean_a proportional_a line_n between_o the_o line_n ab_fw-la and_o bc._n a_o rectangle_n figure_n therefore_o comprehend_v under_o rational_a right_a line_n which_o be_v commensurable_a in_o power_n only_o be_v irrational_a and_o the_o line_n which_o in_o power_n contain_v that_o rectangle_n figure_n be_v irrational_a and_o be_v call_v a_o medial_a line_n at_o this_o proposition_n do_v euclid_n first_o entreat_v of_o the_o generation_n and_o production_n of_o irrational_a line_n and_o here_o he_o search_v out_o the_o first_o kind_n of_o they_o which_o he_o call_v a_o medial_a line_n and_o the_o definition_n thereof_o be_v full_o gather_v and_o take_v out_o of_o this_o 21._o proposition_n which_o be_v this_o a_o medial_a line_n be_v a_o irrational_a line_n who_o square_a be_v equal_a to_o a_o rectangled_a figure_n contain_v of_o two_o rational_a line_n commensurable_a in_o power_n only_o line_n it_o be_v call_v a_o medial_a line_n as_o theon_n right_o say_v for_o two_o cause_n first_o for_o that_o the_o power_n or_o square_v which_o it_o producethâ—_n be_v equal_a to_o a_o medial_a superficies_n or_o parallelogram_n for_o as_o that_o line_n which_o produce_v a_o rational_a square_n be_v call_v a_o rational_a line_n and_o that_o line_n which_o produce_v a_o irrational_a square_n or_o a_o square_n equal_a to_o a_o irrational_a figure_n general_o be_v call_v a_o irrational_a line_n so_o iâ—_z thaâ—_z line_n which_o produce_v a_o medial_a square_n or_o a_o square_n equal_a to_o a_o medial_a superficies_n call_v by_o special_a name_n a_o medial_a line_n second_o it_o be_v call_v a_o medial_a line_n because_o it_o be_v a_o mean_a proportional_a between_o the_o two_o line_n commensurable_a in_o power_n only_o which_o comprehend_v the_o medial_a superficies_n ¶_o a_o corollary_n add_v by_o flussates_n a_o rectangle_n parallelogram_n contain_v under_o a_o rational_a line_n and_o a_o irrational_a line_n be_v irrational_a corollary_n for_o if_o the_o line_n ab_fw-la be_v rational_a and_o
if_o the_o line_n cb_o be_v irrational_a they_o shall_v be_v incommensurable_a but_o as_o the_o line_n bd_o which_o be_v equal_a to_o the_o line_n basilius_n be_v to_o the_o line_n bc_o so_o be_v the_o square_a ad_fw-la to_o the_o parallelogram_n ac_fw-la wherefore_o the_o parallelogram_n ac_fw-la shall_v be_v incommensurable_a to_o the_o square_a ad_fw-la which_o be_v rational_a for_o that_o the_o line_n ab_fw-la whereupon_o it_o be_v describe_v be_v suppose_v to_o be_v rational_a wherefore_o the_o parallelogram_n ac_fw-la which_o be_v contain_v under_o the_o rational_a line_n ab_fw-la and_o the_o irrational_a line_n bc_o be_v irrational_a ¶_o a_o assumpt_n if_o there_o be_v two_o right_a line_n as_o the_o first_o be_v to_o the_o second_o so_o be_v the_o square_n which_o be_v describe_v upon_o the_o first_o to_o the_o parallelogram_n which_o be_v contain_v under_o the_o two_o right_a line_n suppose_v that_o there_o be_v two_o right_a line_n ab_fw-la and_o bc._n book_n then_o i_o say_v that_o as_o the_o line_n ab_fw-la be_v to_o the_o line_n bc_o so_o be_v the_o square_a of_o the_o line_n ab_fw-la â—â—_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc._n describe_v by_o the_o 46._o of_o the_o first_o upon_o the_o line_n ab_fw-la a_o square_a ad._n and_o make_v perfect_a the_o parallelogram_n ac_fw-la now_o for_o that_o as_o the_o line_n ab_fw-la be_v to_o the_o line_n bc_o for_o the_o line_n ab_fw-la be_v equal_a to_o the_o line_n bd_o so_o be_v the_o square_a ad_fw-la to_o the_o parallelogram_n ca_n by_o the_o first_o of_o the_o sixâ—_o and_o ad_fw-la be_v the_o square_n which_o be_v make_v of_o the_o line_n ab_fw-la and_o ac_fw-la be_v that_o which_o be_v contain_v under_o the_o line_n bd_o and_o bc_o that_o be_v under_o the_o line_n ab_fw-la &_o bc_o therefore_o as_o the_o line_n ab_fw-la be_v to_o the_o line_n bc_o so_o be_v the_o square_n describe_v upon_o the_o the_o line_n ab_fw-la to_o the_o rectangle_n figure_n contain_v under_o the_o line_n ab_fw-la &_o bc._n and_o converse_o as_o the_o parallelogram_n which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o be_v to_o the_o square_n of_o the_o line_n ab_fw-la so_o be_v the_o line_n cb_o to_o the_o line_n basilius_n ¶_o the_o 19_o theorem_a the_o 22._o proposition_n if_o upon_o a_o rational_a line_n be_v apply_v the_o square_a of_o a_o medial_a line_n the_o other_o side_n that_o make_v the_o breadth_n thereof_o shall_v be_v rational_a and_o incommensurable_a in_o length_n to_o the_o line_n whereupon_o the_o parallelogram_n be_v apply_v svppose_v that_o a_o be_v a_o medial_a line_n and_o let_v bc_o be_v a_o line_n rational_a and_o upon_o the_o line_n bc_o describe_v a_o rectangle_n parallelogram_n equal_a unto_o the_o square_n of_o the_o line_n a_o and_o let_v the_o same_o be_v bd_o make_v in_o breadth_n the_o line_n cd_o then_o i_o say_v that_o the_o line_n cd_o be_v rational_a and_o incommensurable_a in_o length_n unto_o the_o line_n cb._n 〈◊〉_d for_o forasmuch_o as_o a_o be_v a_o medial_a line_n it_o contain_v in_o power_n by_o the_o 21._o of_o the_o ten_o a_o rectangle_n parallelogram_n comprehend_v under_o rational_a right_a line_n commensurable_a in_o power_n only_o suppose_v that_o be_v contain_v in_o power_n the_o parallelogram_n gf_o and_o by_o supposition_n it_o also_o contain_v in_o power_n the_o parallelogram_n bd._n wherefore_o the_o parallelogram_n bd_o be_v equal_a unto_o the_o parallelogram_n gf_o and_o it_o be_v also_o equiangle_n unto_o it_o for_o that_o they_o be_v each_o rectangle_n but_o in_o parallelogram_n equal_a and_o equiangle_n the_o side_n which_o contain_v the_o equal_a angle_n be_v reciprocal_a by_o the_o 14._o of_o the_o six_o wherefore_o what_o proportion_n the_o line_n bc_o have_v to_o the_o line_n eglantine_n the_o same_o have_v the_o line_n of_o to_o the_o line_n cd_o therefore_o by_o the_o 22._o of_o the_o six_o as_o the_o square_n of_o the_o line_n bc_o be_v to_o the_o square_n of_o the_o line_n eglantine_n so_o be_v the_o square_a of_o the_o line_n of_o to_o the_o square_n of_o the_o line_n cd_o but_o the_o square_n of_o the_o line_n bc_o be_v commensurable_a unto_o the_o square_n of_o the_o line_n eglantine_n by_o supposition_n for_o either_o of_o they_o be_v rational_a wherefore_o by_o the_o the_o 10._o of_o the_o ten_o the_o square_a of_o the_o line_n of_o be_v commensurable_a unto_o the_o square_n of_o the_o linâ—_n cd_o but_o the_o square_n of_o the_o line_n of_o be_v rational_a wherefore_o the_o square_a of_o the_o line_n cd_o be_v likewise_o rational_a wherefore_o the_o line_n cd_o be_v rational_a and_o forasmuch_o as_o the_o line_n of_o be_v incommensurable_a in_o length_n unto_o the_o line_n eglantine_n for_o they_o be_v suppose_v to_o be_v commensurable_a in_o power_n only_o but_o as_o the_o line_n of_o be_v to_o the_o line_n eglantine_n so_o by_o the_o assumpt_n go_v before_o be_v the_o square_a of_o the_o line_n of_o to_o the_o parallelogram_n which_o be_v contain_v under_o the_o line_n of_o and_o eglantine_n wherefore_o by_o the_o 10._o of_o the_o ten_o the_o square_a of_o the_o line_n of_o be_v incommensurable_a unto_o the_o parallelogram_n which_o be_v contain_v under_o the_o line_n fe_o and_o eglantine_n but_o unto_o the_o square_n of_o the_o line_n of_o the_o square_n of_o the_o line_n cd_o be_v commensurable_a for_o it_o be_v prove_v that_o â—ither_o of_o they_o be_v a_o rational_a linâ—_n and_o that_o which_o be_v contain_v under_o the_o line_n dc_o and_o cb_o be_v commensurable_a unto_o that_o which_o be_v contain_v under_o the_o line_n fe_o and_o eglantine_n for_o they_o be_v both_o equal_a to_o the_o square_n of_o the_o line_n a._n wherefore_o by_o the_o 13._o of_o the_o ten_o the_o square_a of_o the_o line_n cd_o be_v incommensurable_a to_o that_o which_o be_v contain_v under_o the_o line_n dc_o and_o cb._n but_o as_o the_o square_n of_o the_o line_n cd_o be_v to_o that_o which_o be_v contain_v under_o the_o line_n dc_o and_o cb_o so_o by_o the_o assumpt_n go_v before_o be_v the_o line_n dc_o to_o the_o line_n cb._n wherefore_o the_o line_n dc_o be_v incommensurable_a in_o length_n unto_o the_o line_n cb._n wherefore_o the_o line_n cd_o be_v rational_a and_o incommensurable_a in_o length_n unto_o the_o line_n cb._n if_o therefore_o upon_o a_o rational_a line_n be_v apply_v the_o square_a of_o a_o medial_a line_n the_o other_o side_n that_o make_v the_o breadth_n thereof_o shall_v be_v rational_a and_o incommensurable_a in_o length_n to_o the_o line_n whereupon_o the_o parallelogram_n be_v apply_v which_o be_v require_v to_o be_v prove_v a_o square_n be_v say_v to_o be_v apply_v upon_o a_o line_n when_o it_o or_o a_o parallelogram_n equal_a unto_o it_o be_v apply_v upon_o the_o say_a line_n line_n if_o upon_o a_o rational_a line_n give_v we_o will_v apply_v a_o rectangle_n parallelogram_n equal_a to_o the_o square_n of_o a_o medial_a line_n give_v and_o so_o of_o any_o line_n give_v we_o must_v by_o the_o 11._o of_o the_o six_o find_v out_o the_o three_o line_n proportional_a with_o the_o rational_a line_n and_o the_o medial_a line_n give_v so_o yet_o that_o the_o rational_a line_n be_v the_o first_o and_o the_o medial_a line_n give_v which_o contain_v in_o power_n the_o square_a to_o be_v apply_v be_v the_o second_o for_o then_o the_o superficies_n contain_v under_o the_o first_o and_o the_o three_o shall_v be_v equal_a to_o the_o square_n of_o the_o middle_a line_n by_o the_o 17._o of_o the_o six_o ¶_o the_o 20._o theorem_a the_o 23._o proposition_n a_o right_a line_n commensurable_a to_o a_o medial_a line_n be_v also_o a_o medial_a line_n svppose_v that_o a_o be_v a_o medial_a line_n and_o unto_o the_o line_n a_o let_v the_o line_n b_o be_v commensurable_a either_o in_o length_n &_o in_o power_n or_o in_o power_n only_o construction_n then_o i_o say_v that_o b_o also_o be_v a_o medial_a line_n let_v there_o be_v put_v a_o rational_a line_n cd_o and_o upon_o the_o line_n cd_o apply_v a_o rectangle_n parallelogram_n ce_fw-fr equal_a unto_o the_o square_n of_o the_o line_n a_o and_o make_v in_o breadth_n the_o line_n ed._n wherefore_o by_o the_o proposition_n go_v before_o the_o line_n ed_z be_v rational_a and_o incommensurable_a in_o length_n unto_o the_o line_n cd_o demonstration_n and_o again_o upon_o the_o line_n cd_o apply_v a_o rectangle_n parallelogram_n cf_o equal_a unto_o the_o square_n of_o the_o line_n b_o and_o make_v in_o breadth_n the_o line_n df._n and_o forasmuch_o as_o the_o line_n a_o be_v commensurable_a unto_o the_o line_n b_o therefore_o the_o square_a of_o the_o line_n a_o be_v commensurable_a to_o the_o square_n of_o the_o line_n b._n but_o the_o parallelogram_n ec_o be_v equal_a to_o the_o square_n of_o the_o linâ—_n a_o and_o the_o parallelogram_n cf_o be_v equal_a to_o
the_o square_n of_o the_o line_n b_o wherefore_o the_o parallelogram_n ec_o be_v commensurable_a unto_o the_o parallelogram_n cf._n but_o as_o the_o parallelogram_n ec_o be_v to_o the_o parallelogram_n cf_o so_o be_v the_o line_n ed_z to_o the_o line_n df_o by_o the_o first_o of_o the_o six_o wherefore_o by_o the_o 10._o of_o the_o ten_o the_o line_n ed_z be_v commensurable_a in_o length_n unto_o the_o line_n df._n but_o the_o line_n ed_z be_v rational_a and_o incommensurable_a in_o length_n unto_o the_o line_n dc_o wherefore_o the_o line_n df_o be_v rational_a and_o incommensurable_a in_o length_n unto_o the_o line_n dc_o by_o the_o 13._o of_o the_o ten_o wherefore_o the_o line_n cd_o and_o df_o be_v rational_a commensurable_a in_o power_n only_o but_o a_o rectangle_n figure_n comprehend_v under_o rational_a right_a line_n commensurable_a in_o power_n only_o be_v by_o the_o â—1_n of_o the_o ten_o irrational_a and_o the_o line_n that_o contain_v it_o in_o power_n be_v irrational_a and_o be_v call_v a_o medial_a line_n wherefore_o the_o line_n that_o contain_v in_o power_n that_o which_o be_v comprehend_v under_o the_o line_n cd_o and_o df_o be_v a_o medial_a line_n but_o the_o line_n b_o contain_v in_o power_n the_o parallelogram_n which_o be_v comprehend_v under_o the_o line_n cd_o and_o df_o wherefore_o the_o line_n b_o be_v a_o medial_a line_n a_o right_a line_n therefore_o commensurable_a to_o a_o medial_a line_n be_v also_o a_o medial_a line_n which_o be_v require_v to_o be_v prove_v ¶_o corollary_n hereby_o it_o be_v manifest_a that_o a_o superficies_n commensurable_a unto_o a_o medial_a superficies_n be_v also_o a_o medial_a superficies_n for_o the_o line_n which_o containâ—_n in_o power_n those_o superficiece_n be_v commensurable_a in_o power_n of_o which_o the_o one_o be_v a_o medial_a line_n by_o the_o definition_n of_o a_o medial_a line_n in_o the_o 21._o of_o this_o ten_o wherefore_o the_o other_o also_o be_v a_o medial_a line_n by_o this_o 23._o proposition_n and_o as_o it_o be_v say_v of_o rational_a line_n so_o also_o be_v it_o to_o be_v say_v oâ—_n medial_a line_n namely_o that_o a_o liâ—e_z commensurable_a to_o a_o medial_a line_n be_v also_o a_o medial_a line_n a_o line_n i_o say_v which_o be_v commensurable_a unto_o a_o medial_a line_n whether_o it_o be_v commensurable_a in_o length_n and_o also_o in_o power_n or_o else_o in_o power_n only_o for_o universal_o it_o be_v true_a that_o line_n commensurable_a in_o length_n be_v also_o commensurable_a in_o power_n now_o if_o unto_o a_o medial_a line_n there_o be_v a_o line_n commensurable_a in_o power_n if_o it_o be_v commensurable_a in_o length_n they_o be_v those_o line_n call_v medial_a line_n commensurable_a in_o length_n &_o in_o power_n but_o if_o they_o be_v commensurable_a in_o power_n only_o thâ—y_o be_v call_v medial_a line_n commensurable_a in_o power_n only_o there_o be_v also_o other_o right_a line_n incommensurable_a in_o length_n to_o the_o medial_a line_n and_o commensurable_a in_o power_n only_o to_o the_o same_o and_o these_o line_n be_v also_o call_v medial_a for_o that_o they_o be_v commensurable_a in_o power_n to_o the_o medial_a line_n and_o in_o aâ—_z muâ—h_o as_o they_o be_v medial_a line_n they_o be_v commensurable_a in_o power_n the_o one_o to_o the_o other_o but_o be_v compare_v the_o one_o to_o the_o other_o they_o may_v be_v commensurable_a either_o in_o length_n and_o therefore_o in_o power_n or_o else_o in_o power_n only_o and_o then_o if_o they_o be_v commensurable_a in_o length_n they_o be_v call_v also_o medial_a line_n commensurable_a in_o length_n and_o so_o consequent_o they_o be_v understand_v to_o be_v commensurable_a in_o power_n but_o iâ—_z they_o be_v commensurable_a in_o power_n only_o yet_o notwithstanding_o they_o also_o be_v call_v medial_a line_n commensurable_a in_o power_n only_o flussates_n after_o this_o proposition_n teach_v how_o to_o come_v to_o the_o understanding_n of_o medial_a superficiece_n and_o line_n by_o surd_a number_n after_o this_o manner_n namely_o to_o express_v the_o medial_a superficiece_n by_o the_o root_n of_o number_n which_o be_v not_o square_a number_n and_o the_o line_n contain_v in_o power_n such_o medial_a superficiece_n by_o the_o root_n of_o root_n of_o number_n not_o square_a medial_a line_n also_o commensurable_a be_v express_v by_o the_o root_n of_o root_n of_o like_a superficial_a number_n but_o yet_o not_o square_a but_o such_o as_o have_v that_o proportion_n that_o the_o square_n of_o square_a number_n have_v for_o the_o root_n of_o those_o number_n and_o the_o root_n of_o root_n be_v in_o proportion_n as_o number_n be_v namely_o if_o the_o square_n be_v proportional_a the_o side_n also_o shall_v be_v proportional_a by_o the_o 22._o of_o the_o six_o but_o medial_a line_n incommensurable_a in_o power_n be_v the_o root_n of_o root_n of_o number_n which_o have_v not_o that_o proportion_n that_o square_a number_n have_v for_o their_o root_n be_v the_o power_n of_o medial_a line_n which_o be_v incommensurable_a by_o the_o 9_o of_o the_o ten_o but_o medial_a line_n commensurable_a in_o power_n only_o be_v the_o root_n of_o root_n of_o number_n which_o have_v that_o proportion_n that_o simple_a square_a number_n have_v and_o not_o which_o the_o square_n of_o square_n have_v for_o the_o root_n which_o be_v the_o power_n of_o the_o medial_a line_n be_v commensurable_a but_o the_o root_n of_o root_n which_o express_v the_o say_v medial_a line_n be_v incommensurable_a wherefore_o there_o may_v be_v find_v out_o infinite_a medial_a line_n incommensurable_a in_o powâ—r_n by_o compare_v infinite_a unlike_o plain_a number_n the_o one_o to_o the_o other_o for_o unlike_o plain_a number_n which_o have_v not_o the_o proportion_n of_o square_a number_n do_v make_v the_o root_n which_o express_v the_o superficiece_n of_o medial_a line_n incommensurable_a by_o the_o 9_o of_o the_o ten_o and_o therefore_o the_o medial_a line_n contain_v in_o power_n those_o superficiece_n be_v incommensurable_a in_o length_n for_o line_n incommensurable_a in_o power_n be_v always_o incommensurable_a in_o length_n by_o the_o corollary_n of_o the_o 9_o of_o the_o ten_o ¶_o the_o 21._o theorem_a the_o 24._o proposition_n a_o rectangle_n parallelogram_n comprehend_v under_o medial_a line_n commensurable_a in_o length_n be_v a_o medial_a rectangle_n parallelogram_n svppose_v that_o the_o rectangle_n parallelogram_n agnostus_n be_v comprehend_v under_o these_o medial_a right_a line_n ab_fw-la and_o bc_o which_o let_v be_v commensurable_a in_o length_n construction_n then_o i_o say_v that_o ac_fw-la be_v a_o medial_a rectangle_n parallelogram_n describe_v by_o the_o 46._o of_o the_o first_o upon_o the_o line_n ab_fw-la a_o square_a ad._n demonstration_n wherefore_o the_o square_a ad_fw-la be_v a_o medial_a superficies_n and_o forasmuch_o as_o the_o line_n ab_fw-la be_v commensurablâ—_n in_o length_n unto_o the_o line_n bc_o and_o the_o line_n ab_fw-la be_v equal_a unto_o the_o line_n bd_o therefore_o the_o line_n bd_o be_v commensurable_a in_o length_n unto_o the_o line_n bc._n but_o 〈◊〉_d the_o line_n db_o be_v to_o the_o line_n bc_o so_o be_v the_o square_a dam_n to_o the_o parallelogram_n ac_fw-la by_o the_o first_o of_o the_o six_o wherefore_o by_o the_o 10._o of_o the_o ten_o the_o square_a dam_n be_v commensurable_a unto_o the_o parallelogram_n ac_fw-la but_o the_o square_a dam_n be_v medial_a for_o that_o it_o be_v describe_v upon_o a_o medial_a line_n wherefore_o ac_fw-la also_o be_v a_o medial_a parallelogram_n by_o the_o former_a corollary_n a_o rectangle_v etc_o etc_o which_o be_v require_v to_o be_v prove_v ¶_o the_o 22â—_n theorem_a the_o 25._o proposition_n a_o rectangle_n parallelogram_n comprehend_v under_o medial_a right_a line_n commensurable_a in_o power_n only_o be_v either_o rational_a or_o medial_a and_o now_o if_o the_o line_n hk_o be_v commensurable_a in_o length_n unto_o the_o line_n he_o that_o be_v unto_o the_o line_n fg_o note_n which_o be_v equal_a to_o the_o line_n he_o than_o by_o the_o 19_o of_o the_o ten_o the_o parallelogram_n nh_o be_v rational_a but_o if_o it_o be_v incommensurable_a in_o length_n unto_o the_o line_n fg_o than_o the_o line_n hk_o and_o he_o be_v rational_a commensurable_a in_o power_n only_o and_o so_o shall_v the_o parallelogram_n hn_o be_v medial_a wherefore_o the_o parallelogram_n hn_o be_v either_o rational_a or_o medial_a but_o the_o parallelogram_n hn_o be_v equal_a to_o the_o parallelogram_n ag._n wherefore_o the_o parallelogram_n ac_fw-la be_v either_o rational_a or_o medial_a a_o rectangle_n parallelogram_n therefore_o comprehend_v under_o medial_a right_a line_n commensurable_a in_o power_n only_o be_v either_o rational_a or_o medial_a which_o be_v require_v to_o be_v demonstrate_v how_o to_o find_v medial_a line_n commensurable_a in_o power_n only_o contain_v a_o rational_a parallelogram_n and_o also_o other_o medial_a line_n commensurable_a in_o power_n contain_v a_o medial_a
to_o the_o number_n ce_fw-fr so_o be_v the_o square_a of_o the_o line_n ab_fw-la to_o the_o square_n of_o the_o line_n of_o therefore_o by_o conversion_n by_o the_o corollary_n of_o the_o 19_o of_o the_o five_o as_o the_o number_n cd_o be_v to_o the_o number_n de_fw-fr so_o be_v the_o square_a of_o the_o line_n ab_fw-la to_o the_o square_n to_o the_o line_n fb_o but_o the_o number_n c_o d_o have_v not_o to_o the_o number_n de_fw-fr that_o proportion_n that_o a_o square_a nâ—mbeâ—_n hâ—th_v to_o a_o square_a number_n wherefore_o neither_o also_o the_o square_a of_o the_o line_n ab_fw-la have_v to_o the_o square_n of_o the_o line_n bf_o that_o proportion_n that_o a_o square_a number_n have_v to_o a_o square_a number_n wherefore_o the_o line_n ab_fw-la be_v by_o the_o 9_o of_o the_o ten_o incommensurable_a in_o length_n to_o the_o line_n bf_o and_o the_o line_n ab_fw-la be_v in_o power_n more_o than_o the_o line_n of_o by_o the_o square_n of_o the_o right_a line_n bf_o which_o be_v incommensurable_a in_o length_n unto_o the_o line_n ab_fw-la wherefore_o the_o line_n ab_fw-la and_o of_o be_v rational_a commensurable_a in_o power_n only_o and_o the_o line_n ab_fw-la be_v in_o power_n more_o than_o the_o line_n of_o by_o the_o square_n of_o the_o line_n fb_o which_o be_v commensurable_a in_o length_n unto_o the_o line_n abâ—_n â—_z which_o be_v require_v to_o be_v do_v ¶_o a_o assumpt_n if_o there_o be_v two_o right_a line_n have_v between_o themselves_o any_o proportion_n as_o the_o one_o right_a line_n be_v to_o the_o other_o so_o be_v the_o parallelogram_n contain_v under_o both_o the_o right_a line_n to_o the_o square_n of_o the_o less_o of_o those_o two_o line_n suppose_v that_o these_o two_o right_a ab_fw-la and_o bc_o be_v in_o some_o certain_a proportion_n find_v then_o i_o say_v that_o as_o the_o line_n ab_fw-la be_v to_o the_o line_n bc_o so_o be_v the_o parallelogram_n contain_v under_o ab_fw-la and_o bc_o to_o the_o square_n of_o bc._n describe_v the_o square_a of_o the_o line_n bc_o and_o let_v the_o same_o be_v cd_o and_o make_v perfect_a the_o parallelogram_n ad_fw-la now_o it_o be_v manifest_a that_o as_o the_o line_n ab_fw-la be_v to_o the_o line_n bc_o so_o be_v the_o parallelogram_n ad_fw-la to_o the_o parallelogram_n or_o square_n be_v by_o the_o first_o of_o the_o six_o but_o the_o parallelogram_n ad_fw-la be_v that_o which_o be_v bontain_v under_o the_o line_n ab_fw-la and_o bc_o for_o the_o line_n bc_o be_v equal_a to_o the_o line_n bd_o and_o the_o parallelogram_n be_v be_v the_o square_n of_o the_o line_n bc._n wherefore_o as_o the_o line_n ab_fw-la be_v to_o the_o line_n bc_o so_o be_v the_o parallelogram_n coutain_v under_o the_o line_n ab_fw-la and_o bc_o to_o the_o square_n of_o the_o line_n bc_o which_o be_v require_v to_o be_v prove_v ¶_o the_o 8._o problem_n the_o 31._o proposition_n to_o find_v out_o two_o medial_a line_n commensurable_a in_o power_n only_o comprehend_v a_o rational_a superficies_n so_o that_o the_o great_a shall_v be_v in_o power_n more_o than_o the_o less_o by_o the_o square_n of_o a_o line_n commensurable_a in_o length_n unto_o the_o great_a let_v there_o be_v take_v by_o the_o 29._o of_o the_o ten_o two_o rational_a line_n commensurable_a in_o power_n only_o a_o and_o b_o construction_n so_o that_o let_v the_o line_n a_o be_v the_o great_a be_v in_o power_n more_o than_o the_o line_n b_o be_v the_o less_o by_o the_o square_n of_o a_o line_n commensurable_a in_o length_n unto_o the_o line_n aâ—_n â—_z and_o let_v the_o square_n of_o the_o line_n c_o be_v equal_a to_o the_o parallelogram_n contain_v under_o the_o line_n a_o and_o b_o which_o be_v do_v by_o find_v out_o the_o mean_a proportional_a line_n namely_o the_o line_n c_o between_o the_o line_n a_o and_o b_o by_o the_o 13._o of_o the_o six_o now_o the_o parallelogram_n contain_v under_o the_o line_n a_o and_o b_o be_v medial_a by_o the_o 21._o of_o this_o book_n wherefore_o by_o the_o corollary_n of_o the_o 23._o of_o the_o ten_o the_o square_a also_o of_o the_o line_n c_o be_v medial_a wherefore_o the_o line_n c_o also_o be_v medial_a unto_o the_o square_n of_o the_o line_n b_o let_v the_o parallelogram_n contain_v under_o the_o line_n c_o and_o d_o be_v equal_a by_o find_v out_o a_o three_o line_n proportional_a namely_o the_o line_n d_o to_o the_o two_o line_n c_o and_o b_o by_o the_o 11._o of_o the_o six_o but_o the_o square_n of_o the_o line_n b_o be_v rational_a demonstration_n wherefore_o the_o parallelogram_n contain_v under_o the_o line_n c_o and_o d_o be_v rational_a and_o for_o that_o as_o the_o line_n a_o be_v to_o the_o line_n b_o so_o be_v the_o parallelogram_n contain_v under_o the_o line_n a_o and_o b_o to_o the_o square_n of_o the_o line_n b_o by_o the_o assumpt_n go_v before_o but_o unto_o the_o parallelogram_n contain_v under_o the_o line_n a_o and_o b_o be_v equal_v the_o square_n of_o the_o line_n c_o and_o unto_o the_o square_n of_o the_o line_n b_o be_v equal_a the_o parallelogram_n contain_v under_o the_o line_n c_o and_o d_o as_o it_o have_v now_o be_v prove_v therefore_o as_o the_o line_n a_o be_v to_o the_o line_n b_o so_o be_v the_o square_a of_o the_o line_n c_o to_o the_o parallelogram_n contain_v under_o the_o line_n c_o d._n but_o as_o the_o square_n of_o the_o line_n c_o be_v to_o that_o which_o be_v contain_v under_o the_o line_n c_o and_o d_o so_o be_v the_o line_n c_o to_o the_o line_n d._n wherefore_o as_o the_o line_n a_o be_v to_o the_o line_n b_o so_o be_v the_o line_n c_o to_o the_o line_n d._n but_o by_o supposition_n the_o line_n a_o be_v commensurable_a unto_o the_o line_n b_o in_o power_n only_o wherefore_o by_o the_o 11._o of_o the_o ten_o the_o line_n c_o also_o be_v unto_o the_o line_n d_o commensurable_a in_o power_n only_o but_o the_o line_n c_o be_v medial_a wherefore_o by_o the_o 23â—_n of_o the_o ten_o the_o line_n d_o also_o be_v medial_a and_o for_o that_o as_o the_o line_n a_o be_v to_o the_o line_n b_o so_o be_v the_o line_n c_o to_o the_o line_n d_o but_o the_o line_n a_o be_v in_o power_n more_o than_o the_o line_n b_o by_o the_o square_n of_o a_o line_n commensurable_a in_o length_n unto_o the_o line_n a_o by_o supposition_n wherefore_o the_o line_n c_o also_o be_v in_o power_n more_o than_o the_o line_n d_o by_o the_o square_n of_o a_o line_n commensurable_a in_o length_n unto_o the_o line_n c._n wherefore_o there_o be_v find_v out_o two_o medial_a line_n c_o and_o d_o commensurable_a in_o power_n only_o comprehend_v a_o rational_a superficies_n and_o the_o line_n c_o be_v in_o power_n more_o than_o the_o line_n d_o by_o the_o square_n of_o a_o line_n commensurable_a in_o length_n unto_o the_o line_n c._n and_o in_o like_a sort_n may_v be_v find_v out_o two_o medial_a line_n commensurable_a in_o power_n only_o contain_v a_o rational_a superficies_n so_o that_o the_o great_a shall_v be_v in_o power_n more_o they_o the_o less_o by_o the_o square_n of_o a_o line_n incommensurable_a in_o length_n to_o the_o great_a namely_o when_o the_o line_n a_o be_v in_o power_n more_o they_o the_o line_n b_o by_o the_o square_n of_o a_o line_n incommensurable_n in_o length_n unto_o the_o line_n a_o which_o to_o do_v be_v teach_v by_o the_o 30._o of_o this_o book_n the_o self_n same_o construction_n remain_v that_o part_n of_o this_o proposition_n from_o these_o word_n and_o for_o that_o as_o the_o line_n a_o be_v to_o the_o line_n b_o to_o these_o word_n but_o by_o supposition_n the_o line_n a_o be_v commensurable_a unto_o the_o line_n b_o may_v more_o easy_o be_v demonstrate_v after_o this_o manner_n the_o line_n c_o b_o d_o be_v in_o continual_a proportion_n by_o the_o second_o part_n of_o the_o 17._o of_o the_o six_o but_o the_o line_n a_o c_o d_o be_v also_o in_o continual_a proportion_n by_o the_o same_o wherefore_o by_o the_o 11._o of_o the_o five_o as_o the_o line_n a_o be_v to_o the_o line_n c_o so_o be_v the_o line_n b_o to_o the_o line_n d._n wherefore_o alternate_o as_o the_o line_n a_o be_v to_o the_o line_n b_o so_fw-mi be_v the_o line_n c_o to_o the_o line_n d._n etc_n etc_n which_o be_v require_v to_o be_v do_v ¶_o a_o assumpt_n if_o there_o be_v three_o right_a line_n have_v between_o themselves_o any_o proportion_n as_o the_o first_o be_v to_o the_o three_o so_o be_v the_o parallelogram_n contain_v under_o the_o first_o and_o the_o second_o to_o the_o parallelogram_n contain_v under_o the_o second_o and_o the_o three_o suppose_v that_o these_o three_o line_n ab_fw-la b_o c_o and_o cd_o be_v in_o some_o certain_a proportion_n then_o i_o say_v that_o as_o the_o line_n ab_fw-la be_v to_o
the_o line_n cd_o so_o be_v the_o parallelogram_n contain_v under_o the_o line_n ab_fw-la and_o bc_o to_o the_o parallelogram_n contain_v under_o the_o line_n bc_o and_o cd_o construction_n from_o the_o point_n a_o raise_v up_o unto_o the_o line_n ab_fw-la a_o perpendicular_a line_n ae_n and_o let_v ae_n be_v equal_a to_o the_o line_n bc_o and_o by_o the_o point_n e_o draw_v unto_o the_o line_n ad_fw-la a_o parallel_n line_n eke_o and_o by_o every_o one_o of_o the_o point_n b_o c_o and_o d_o draw_v unto_o the_o line_n ae_n parallel_a line_n bf_o change_z and_o dk_o and_o for_o that_o as_o the_o line_n ab_fw-la be_v to_o the_o line_n bc_o so_o be_v the_o parallelogram_n of_o to_o the_o parallelogram_n bh_o by_o the_o first_o of_o the_o six_o demonstration_n and_o as_o the_o line_n bc_o be_v to_o the_o linâ—_n cd_o so_o be_v the_o parallelogram_n bh_o to_o the_o parallelogram_n ck_o wherefore_o of_o equality_n as_o the_o line_n ab_fw-la be_v to_o the_o line_n cd_o so_o be_v the_o parallelogram_n of_o to_o the_o parallelogram_n ck_o but_o the_o parallelogram_n of_o be_v that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o for_o the_o line_n ae_n be_v put_v equal_a to_o the_o line_n bc._n and_o the_o parallelogram_n ck_o be_v that_o which_o be_v contain_v under_o the_o line_n bc_o and_o cd_o for_o the_o line_n bc_o be_v equal_a to_o the_o line_n change_z for_o that_o the_o line_n change_z be_v equal_a to_o the_o line_n ae_n by_o the_o 34._o of_o the_o first_o if_o therefore_o there_o be_v three_o right_a line_n have_v between_o themselves_o any_o proportion_n as_o the_o first_o be_v to_o the_o three_o so_o be_v the_o parallelogram_n contain_v under_o the_o first_o and_o the_o second_o to_o the_o parallelogram_n contain_v under_o the_o second_o and_o the_o three_o which_o be_v require_v to_o be_v demonstrate_v ¶_o the_o 9_o problem_n the_o 32._o proposition_n to_o find_v out_o two_o medial_a line_n commensurable_a in_o power_n only_o comprehend_v a_o medial_a superficies_n so_o that_o the_o great_a shall_v be_v in_o power_n more_o than_o the_o less_o by_o the_o square_n of_o a_o line_n commensurable_a in_o length_n unto_o the_o great_a let_v there_o be_v take_v three_o rational_a line_n commensurable_a in_o power_n only_o a_o b_o c_o so_fw-mi that_o by_o the_o 29._o of_o the_o ten_o let_v the_o line_n a_o be_v in_o power_n more_o than_o the_o line_n c_o by_o the_o square_n of_o a_o line_n commensurable_a in_o length_n unto_o the_o line_n a._n construction_n and_o unto_o the_o parallelogram_n contain_v under_o the_o line_n a_o &_o b_o let_v the_o square_n of_o the_o line_n d_o be_v equal_a but_o that_o which_o be_v contain_v under_o the_o line_n a_o and_o b_o be_v medial_a wherefore_o by_o the_o corollary_n of_o the_o 23._o of_o the_o ten_o the_o square_a of_o the_o line_n d_o also_o be_v medial_a wherefore_o the_o line_n d_o also_o be_v medial_a and_o unto_o that_o which_o be_v contain_v under_o the_o line_n b_o and_o c_o let_v be_v equal_a that_o which_o be_v contain_v under_o the_o line_n d_o and_o e_o which_o be_v do_v by_o â—inding_v out_o a_o four_o line_n proportional_a unto_o the_o line_n d_o b_o c_o which_o let_v be_v the_o line_n e_o demonstration_n and_o for_o that_o by_o the_o assumpt_n go_v before_o as_o that_o which_o be_v contain_v under_o the_o line_n a_o and_o b_o be_v to_o that_o which_o be_v contain_v under_o the_o line_n b_o and_o c_o so_o be_v the_o line_n a_o to_o the_o line_n c._n but_o unto_o that_o which_o be_v contain_v under_o the_o line_n a_o &_o b_o be_v equal_a the_o square_n of_o the_o line_n d_o and_o unto_o that_o which_o be_v contain_v under_o the_o line_n b_o &_o c_o be_v equal_a that_o which_o be_v contain_v under_o the_o line_n d_o and_o e._n wherefore_o as_o the_o line_n a_o be_v to_o the_o line_n c_o so_fw-mi iâ—_z the_o square_a of_o the_o line_n d_o to_o that_o which_o be_v contain_v under_o the_o line_n d_o and_o e._n but_o as_o the_o square_n of_o the_o line_n d_o be_v to_o that_o which_o be_v contain_v under_o the_o line_n d_o and_o e_o so_o be_v the_o line_n d_o to_o the_o line_n e_o by_o the_o assumpt_n put_v before_o the_o 22._o of_o the_o ten_o wherefore_o as_o the_o line_n a_o be_v to_o the_o line_n c_o so_o be_v the_o line_n d_o to_o the_o line_n e._n but_o the_o line_n a_o be_v unto_o the_o line_n c_o commensurable_a in_o power_n only_o wherefore_o the_o line_n d_o be_v unto_o the_o line_n e_o commensurable_a in_o power_n only_o but_o d_z be_v a_o medial_a line_n wherefore_o by_o the_o 23._o of_o the_o ten_o e_o also_o be_v a_o medial_a line_n and_o for_o that_o as_o the_o line_n a_o be_v to_o the_o line_n c_o so_o be_v the_o line_n d_o to_o the_o line_n e_o and_o the_o line_n a_o be_v in_o power_n more_o than_o the_o line_n c_o by_o the_o square_n of_o a_o line_n commensurable_a in_o length_n unto_o the_o line_n a._n wherefore_o by_o the_o 14._o of_o the_o ten_o d_o be_v in_o power_n more_o than_o e_o by_o the_o square_n of_o a_o line_n commensurable_a in_o length_n unto_o the_o line_n d._n i_o say_v moreover_o that_o that_o which_o be_v contain_v under_o the_o line_n d_o and_o e_o be_v medial_a for_o forasmuch_o as_o that_o which_o be_v contain_v under_o the_o line_n b_o &_o c_o be_v equal_a to_o that_o which_o be_v contain_v under_o the_o line_n d_o and_o e_o but_o that_o which_o be_v contain_v under_o the_o line_n b_o and_o c_o be_v medial_a wherefore_o that_o which_o be_v contain_v under_o the_o line_n d_o and_o e_o be_v also_o medial_a wherefore_o there_o be_v find_v out_o two_o medial_a line_n d_z and_o e_o commensurable_a in_o power_n only_o comprehend_v a_o medial_a superficies_n so_o that_o the_o great_a be_v in_o power_n more_o than_o the_o less_o by_o the_o square_n of_o a_o line_n commensurable_a in_o length_n to_o the_o great_a which_o be_v require_v to_o be_v do_v and_o thus_o iâ—_z iâ—_z euâ—deâ—t_n how_o in_o like_a sort_n may_v be_v find_v out_o two_o medial_a line_n commâ—â—surable_a in_o power_n only_o corollary_n contain_v a_o medial_a superficies_n so_o that_o the_o great_a shall_v be_v in_o power_n more_o theâ—_z the_o less_o by_o the_o square_n of_o a_o line_n incommensurable_a in_o length_n unto_o the_o great_a when_o the_o line_n a_o be_v in_o power_n more_o than_o the_o line_n c_o by_o the_o square_n of_o a_o line_n incommensurable_a in_o length_n unto_o the_o line_n a_o as_o the_o thirty_o teach_v us._n 1._o ¶_o a_o assumpt_n suppose_v that_o there_o be_v a_o rectangle_n triangle_n abc_n have_v the_o angle_n bac_n a_o right_a angle_n and_o â—y_a the_o 12._o of_o the_o first_o from_o the_o point_n a_o to_o the_o right_a line_n bc_o a_o perpendicular_a line_n be_v draw_v ad_fw-la then_o i_o say_v first_o that_o the_o parallelogram_n contain_v under_o the_o line_n câ—_n and_o bd_o be_v equal_a to_o the_o square_n of_o the_o line_n basilius_n second_o i_o say_v that_o the_o parallelogram_n contain_v under_o the_o line_n bc_o and_o cd_o be_v equal_a to_o the_o square_n of_o the_o line_n ca._n three_o i_o say_v that_o the_o parallelogram_n contain_v under_o the_o line_n bd_o and_o dc_o be_v equal_a to_o the_o square_n of_o the_o line_n ad._n and_o fourthly_a i_o say_v that_o the_o parallelogram_n contain_v under_o the_o line_n bc_o &_o ad_fw-la be_v equal_a to_o the_o parallelogram_n contain_v under_o the_o line_n basilius_n &_o ac_fw-la as_o touch_v the_o second_o that_o the_o parellelogramme_n contain_v under_o the_o line_n bâ—_n and_o cd_o be_v equal_a to_o the_o square_n of_o the_o line_n ac_fw-la be_v by_o the_o self_n same_o reason_n prove_v for_o the_o triangle_n abc_n be_v like_a to_o the_o triangle_n adc_o wherefore_o as_o the_o line_n bc_o be_v to_o the_o line_n ac_fw-la dee_n so_o be_v the_o line_n ac_fw-la to_o the_o line_n dc_o corollary_n wherefore_o the_o parallelogram_n contain_v under_o the_o line_n bc_o and_o cd_o be_v equal_a to_o the_o square_n of_o the_o line_n ac_fw-la corollary_n as_o touch_v the_o three_o that_o the_o parallelogram_n contain_v under_o the_o line_n bd_o and_o dc_o be_v equal_a to_o the_o square_n of_o the_o line_n dam_fw-ge be_v thus_o prove_v for_o forasmuch_o as_o if_o in_o a_o rectangle_n triangle_n be_v draw_v from_o the_o right_a angle_n to_o the_o base_a a_o perpendicular_a line_n the_o perpendicular_a so_o draw_v be_v the_o mean_a proportional_a between_o the_o segment_n of_o the_o base_a by_o the_o corollary_n of_o the_o 8._o of_o the_o six_o therefore_o as_o the_o line_n bd_o be_v to_o the_o line_n dam_fw-ge so_o be_v thâ—_z line_n ad_fw-la to_o the_o line_n dc_o wherefore_o by_o the_o 1â—_o of_o the_o six_o the_o parallelogram_n
contain_v under_o the_o line_n bd_o and_o dc_o be_v equal_a to_o the_o square_n of_o the_o line_n da._n as_o touch_v the_o four_o that_o the_o parallelogram_n contain_v under_o the_o line_n bc_o and_o ad_fw-la be_v equal_a to_o the_o parallelogram_n contain_v under_o the_o line_n basilius_n and_o ac_fw-la be_v thus_o prove_v for_o forasmuch_o as_o as_o we_o have_v already_o declare_v the_o triangle_n abc_n be_v like_a and_o therefore_o equiangle_n to_o the_o triangle_n abdella_n therefore_o as_o the_o line_n bc_o be_v to_o the_o line_n ac_fw-la dâ—e_n so_o be_v the_o line_n basilius_n to_o the_o line_n ad_fw-la by_o the_o 4._o of_o the_o six_o corollary_n but_o if_o there_o be_v four_o right_a line_n proportional_a that_o which_o be_v contain_v under_o the_o first_o and_o the_o last_o be_v equal_a to_o that_o which_o be_v contain_v under_o the_o two_o mean_n by_o the_o 16._o of_o the_o six_o wherefore_o that_o which_o be_v contain_v under_o the_o line_n bc_o and_o ad_fw-la be_v equal_a to_o that_o which_o be_v contain_v under_o the_o line_n basilius_n and_o ac_fw-la profitable_a i_o say_v moreover_o that_o if_o there_o be_v make_v a_o parallelogram_n complete_a determination_n contain_v under_o the_o line_n bc_o and_o ad_fw-la which_o let_v be_v ec_o and_o if_o likewise_o be_v make_v complete_a the_o parallelogram_n contain_v under_o the_o line_n basilius_n and_o ac_fw-la which_o let_v be_v of_o it_o may_v by_o a_o other_o way_n be_v prove_v that_o the_o parallelogram_n ec_o be_v equal_a to_o the_o parallelogram_n af._n for_o forasmuch_o as_o either_o of_o they_o be_v double_a to_o the_o triangle_n acb_o by_o the_o 41._o of_o the_o first_o and_o thing_n which_o be_v double_a to_o one_o and_o the_o self_n same_o thing_n be_v equal_a the_o one_o to_o the_o other_o wherefore_o that_o which_o be_v contain_v under_o the_o line_n bc_o and_o ad_fw-la be_v equal_a to_o that_o which_o be_v contain_v under_o the_o line_n basilius_n and_o ac_fw-la 2._o ¶_o a_o assumpt_n if_o a_o right_a line_n be_v divide_v into_o two_o unequal_a part_n as_o the_o great_a part_n be_v to_o the_o less_o assumpt_n so_o be_v the_o parallelogram_n contain_v under_o the_o whole_a line_n and_o the_o great_a part_n to_o the_o parallelogram_n contain_v under_o the_o whole_a line_n and_o the_o less_o part_n this_o assumpt_n differ_v little_a from_o the_o first_o proposition_n of_o the_o six_o book_n 3._o ¶_o a_o assumpt_n if_o there_o be_v two_o unequal_a right_a line_n and_o if_o the_o less_o be_v divide_v into_o two_o equal_a part_n the_o parallelogram_n contain_v under_o the_o two_o unequal_a line_n be_v double_a to_o the_o parallelogram_n contain_v under_o the_o great_a line_n &_o half_a of_o the_o less_o line_n suppose_v that_o there_o be_v two_o unequal_a right_a line_n ab_fw-la and_o bc_o of_o which_o leâ—_z ab_fw-la be_v the_o great_a and_o divide_v the_o line_n bc_o into_o two_o equal_a part_n in_o the_o point_n d._n thâ—n_o i_o say_v that_o the_o parallelogram_n contain_v under_o the_o line_n ab_fw-la &_o bc_o be_v double_a to_o the_o parallelogram_n contain_v under_o the_o line_n ab_fw-la and_o bd._n from_o the_o point_n b_o raise_v up_o upon_o the_o right_a line_n bc_o a_o perpendicular_a line_n be_v and_o let_v be_v be_v equal_a to_o the_o line_n basilius_n and_o draw_v from_o the_o point_n c_o and_o d_o the_o line_n cg_o and_o df_o parallel_v and_o equal_a to_o be_v and_o then_o draw_v the_o right_a line_n gfe_n the_o figure_n be_v complete_a nâ—â—_n for_o that_o aâ—â—he_a line_n db_o be_v to_o the_o line_n dc_o so_o be_v the_o parallelogram_n bf_o to_o the_o parallelogram_n dg_o by_o the_o 1._o of_o the_o six_o therefore_o by_o composition_n of_o proportion_n as_o the_o whole_a line_n bc_o be_v to_o the_o line_n dc_o so_o be_v the_o parallelogram_n bg_o to_o the_o parallelogram_n dg_o by_o the_o 18._o of_o the_o five_o but_o the_o line_n bc_o be_v double_a to_o the_o line_n dc_o wherefore_o the_o parallelogram_n bg_o be_v double_a to_o the_o parallelogram_n dg_o but_o the_o parallâ—logramme_n bg_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o for_o the_o line_n ab_fw-la be_v equal_a to_o the_o line_n be_v and_o the_o parallelogram_n dg_o be_v contain_v under_o the_o line_n ab_fw-la and_o bd_o for_o the_o line_n bd_o be_v equal_a to_o the_o line_n dc_o and_o the_o line_n ab_fw-la to_o the_o line_n df_o which_o be_v require_v to_o be_v demonstrate_v ¶_o the_o 10._o problem_n the_o 33._o proposition_n to_o â—inde_v out_o two_o right_a line_n incommensurable_a in_o power_n who_o square_n add_v together_o make_v a_o rational_a superficies_n and_o the_o parallelogram_n contain_v under_o they_o make_v a_o medial_a superficies_n take_v by_o the_o 30._o of_o the_o ten_o two_o rational_a right_a line_n commensurable_a in_o power_n only_o namely_o ab_fw-la and_o bc_o so_o that_o let_v the_o line_n ab_fw-la be_v the_o great_a be_v in_o power_n more_o than_o the_o line_n bc_o be_v the_o less_o construction_n by_o the_o square_n of_o a_o line_n incommensurable_a in_o length_n unto_o the_o line_n ab_fw-la and_o by_o the_o 10._o of_o the_o first_o divide_v the_o line_n bc_o into_o two_o equal_a part_n in_o the_o point_n d._n and_o upon_o the_o line_n ab_fw-la apply_v a_o parallelogram_n equal_a to_o the_o square_a either_o of_o the_o line_n bd_o or_o of_o the_o line_n dc_o and_o want_v in_o figure_n by_o a_o square_a by_o the_o 28._o of_o the_o six_o and_o let_v that_o parallelogram_n be_v that_o which_o be_v contain_v under_o the_o line_n ae_n and_o ebb_n and_o upon_o the_o line_n ab_fw-la describe_v a_o semicircle_n afb_o and_o by_o the_o 11._o of_o the_o first_o from_o the_o point_n e_o raise_v up_o unto_o the_o line_n ab_fw-la a_o perpendicular_a line_n of_o cut_v the_o circumference_n in_o the_o point_n f._n and_o draw_v line_n from_o a_o to_o fletcher_n demonstration_n and_o from_o fletcher_n to_o b._n and_o forasmuch_o as_o there_o be_v two_o unequal_a right_a line_n ab_fw-la and_o bc_o and_o the_o line_n ab_fw-la be_v in_o power_n more_o than_o the_o line_n bc_o by_o the_o square_n of_o a_o line_n incommensurable_a in_o length_n unto_o ab_fw-la and_o upon_o the_o line_n ab_fw-la be_v apply_v a_o parallelogram_n equal_a to_o the_o four_o part_n oâ—_n the_o square_a of_o the_o line_n bc_o that_o be_v to_o the_o square_n of_o the_o half_a of_o the_o line_n bc_o and_o want_v in_o â—igure_n by_o a_o square_a and_o the_o say_a parallelogram_n be_v that_o which_o be_v contain_v under_o the_o line_n ae_n and_o ebb_n wherefore_o by_o the_o 2._o part_n of_o the_o 18._o of_o the_o ten_o the_o line_n ae_n be_v incommensurable_a in_o length_n unto_o the_o line_n ebb_n but_o as_o the_o line_n ae_n be_v to_o the_o line_n ebb_n so_o be_v the_o parallelogram_n contain_v under_o the_o line_n basilius_n and_o ae_n to_o the_o parallelogram_n contain_v under_o the_o line_n ab_fw-la and_o be_v by_o the_o second_o assumpt_n before_o put_v and_o that_o which_o be_v contain_v under_o the_o line_n basilius_n and_o ae_n be_v equal_a to_o the_o square_n of_o the_o line_n of_o by_o the_o second_o part_n of_o the_o first_o assumpt_v before_o put_v and_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o be_v be_v by_o the_o first_o part_n of_o the_o same_o assumpt_n equal_a to_o the_o square_n of_o the_o line_n bf_o wherefore_o the_o square_a of_o the_o line_n of_o be_v incommensurable_a to_o the_o square_n of_o the_o line_n bf_o wherefore_o the_o line_n of_o and_o bf_o be_v incommensurable_a in_o power_n and_o forasmuch_o as_o ab_fw-la be_v a_o rational_a line_n by_o supposition_n therefore_o by_o the_o 7_o definition_n of_o the_o ten_o the_o square_a of_o the_o line_n ab_fw-la be_v rational_a conclude_v wherefore_o also_o the_o square_n of_o the_o line_n of_o and_o fb_o add_v together_o make_v a_o rational_a superficies_n for_o by_o the_o 47._o of_o the_o first_o they_o be_v equal_a to_o the_o square_n of_o the_o line_n ab_fw-la again_o forasmuch_o as_o by_o the_o three_o part_n of_o the_o first_o assumpt_n go_v before_o that_o which_o be_v contain_v under_o the_o line_n ae_n and_o ebb_n be_v equal_a to_o the_o square_n of_o the_o line_n ef._n but_o by_o supposition_n that_o which_o be_v contain_v under_o the_o line_n ae_n and_o ebb_n be_v equal_a to_o the_o square_n of_o the_o line_n bd._n wherefore_o the_o line_n fe_o be_v equal_a to_o the_o line_n bd._n wherefore_o the_o linâ—_n bg_o be_v double_a to_o the_o line_n â—_o e._n wherefore_o by_o the_o three_o assumpt_v go_v before_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o be_v double_a to_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o ef._n but_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o be_v by_o supposition_n medial_a
wherefore_o by_o the_o corollary_n of_o the_o 23._o of_o the_o ten_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o of_o be_v also_o medial_a but_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o of_o conclude_v be_v by_o the_o last_o part_n of_o the_o first_o assumpt_n go_v before_o equal_a to_o that_o which_o be_v contain_v under_o the_o line_n of_o and_o fb_o wherefore_o that_o which_o be_v contain_v under_o the_o line_n of_o &_o fb_o be_v a_o medial_a superficies_n and_o it_o be_v prove_v that_o that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n of_o and_o fb_o add_v together_o be_v rational_a conclusion_n wherefore_o there_o be_v find_v out_o two_o right_a line_n of_o and_o fb_o incommensurable_a in_o power_n who_o square_n add_v together_o make_v a_o rational_a superficies_n and_o the_o parallelogram_n contain_v under_o they_o be_v a_o medial_a superficiâ—sâ—_n which_o be_v require_v to_o be_v do_v ¶_o the_o 11._o problem_n the_o 34._o proposition_n to_o find_v out_o two_o right_a line_n incommensurable_a in_o power_n who_o square_n add_v together_o make_v a_o medial_a superficies_n and_o the_o parallelogram_n contain_v under_o they_o make_v a_o rational_a superficies_n take_v by_o the_o 31._o of_o the_o ten_o two_o medial_a line_n ab_fw-la and_o bc_o commensurable_a in_o power_n only_o comprehend_v a_o rational_a superficies_n construction_n so_o that_o let_v the_o line_n ab_fw-la be_v in_o power_n more_o than_o the_o line_n bc_o by_o the_o square_n of_o a_o line_n incommensurable_a in_o length_n unto_o the_o line_n ab_fw-la and_o describe_v upon_o the_o line_n ab_fw-la a_o semicircle_n adb_o and_o by_o the_o 10._o of_o the_o first_o divide_v the_o line_n bc_o unto_o two_o equal_a part_n in_o the_o point_n e._n and_o by_o the_o 28._o of_o the_o six_o upon_o the_o line_n ab_fw-la apply_v a_o parallelogram_n equal_a to_o the_o square_n of_o the_o line_n be_v and_o want_v in_o figure_n by_o a_o square_a and_o let_v that_o parallelogram_n be_v that_o which_o be_v contain_v under_o the_o line_n of_o and_o fb_o wherefore_o the_o line_n of_o be_v incommensurable_a in_o length_n unto_o the_o line_n fb_o by_o the_o 2._o part_n of_o the_o 18._o of_o the_o ten_o and_o from_o the_o point_n f_o unto_o the_o right_a line_n ab_fw-la raise_v up_o by_o the_o 11._o of_o the_o first_o a_o perpendicular_a line_n fd_o and_o draw_v line_n from_o a_o to_o d_o and_o from_o d_z to_o b._n demonstration_n and_o forasmuch_o as_o the_o line_n of_o be_v incommensurable_a unto_o the_o line_n fb_o but_o by_o the_o second_o assumpt_v go_v before_o the_o 33._o of_o the_o ten_o as_o the_o line_n of_o be_v to_o the_o line_n fb_o so_o be_v the_o parallelogram_n contain_v under_o the_o line_n basilius_n and_o of_o to_o the_o parallelogram_n contain_v under_o the_o line_n basilius_n and_o bf_o wherefore_o by_o the_o ten_o of_o the_o ten_o that_o which_o be_v contain_v under_o the_o line_n basilius_n and_o of_o be_v incommensurable_a to_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bf_o but_o that_o which_o be_v contain_v under_o the_o line_n basilius_n and_o of_o be_v equal_a to_o the_o square_n of_o the_o line_n ad_fw-la and_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bf_o be_v also_o equal_a to_o the_o square_n of_o the_o line_n db_o by_o the_o second_o part_n of_o the_o first_o assumpt_n go_v before_o the_o 33._o of_o the_o ten_o wherefore_o the_o square_a of_o the_o line_n ad_fw-la be_v incommensurable_a to_o the_o square_n of_o the_o line_n db._n wherefore_o the_o line_n ad_fw-la and_o db_o be_v incommensurable_a in_o power_n and_o forasmuch_o as_o the_o square_n of_o the_o line_n ab_fw-la be_v medial_a therefore_o also_o the_o superficies_n make_v of_o the_o square_n of_o the_o line_n ad_fw-la and_o db_o add_v together_o be_v medial_a conclude_v for_o the_o square_n of_o the_o line_n ad_fw-la and_o db_o be_v by_o the_o 47._o of_o the_o first_o equal_a to_o the_o square_n of_o the_o line_n ab_fw-la and_o forasmuch_o as_o the_o line_n bc_o be_v double_a to_o the_o line_n fd_o as_o it_o be_v prove_v in_o the_o proposition_n go_v before_o therefore_o the_o parallelogram_n contain_v under_o the_o line_n ab_fw-la and_o and_o bc_o be_v double_a to_o the_o parallelogram_n contain_v under_o the_o line_n ab_fw-la and_o fd_v by_o the_o three_o assumpt_v go_v before_o the_o 33._o proposition_n wherefore_o it_o be_v also_o commensurable_a unto_o it_o by_o the_o six_o of_o the_o ten_o but_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o be_v suppose_v to_o be_v rational_a wherefore_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o fd_o be_v also_o rational_a but_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o fd_v be_v equal_a to_o that_o which_o be_v contain_v under_o the_o line_n ad_fw-la and_o db_o by_o the_o last_o part_n of_o the_o first_o assumpt_n go_v before_o the_o 33._o of_o the_o ten_o wherefore_o that_o which_o be_v contain_v under_o the_o line_n ad_fw-la and_o db_o be_v also_o rational_a conclusion_n wherefore_o there_o be_v â—ound_v out_o two_o right_a line_n ad_fw-la and_o db_o incommensurable_a in_o power_n who_o square_n add_v together_o make_v a_o medial_a superficies_n and_o the_o parallelogram_n contain_v under_o they_o make_v a_o rational_a superficies_n which_o be_v require_v to_o be_v do_v ¶_o the_o 12._o problem_n the_o 35._o proposition_n to_o find_v out_o two_o right_a line_n incommensurable_a in_o power_n who_o square_n add_v together_o make_v a_o medial_a superficies_n and_o the_o parallelogram_n contain_v under_o they_o make_v also_o a_o medial_a superficies_n which_o parallelogram_n moreover_o shall_v be_v incommensurable_a to_o the_o superficies_n make_v of_o the_o square_n of_o those_o line_n add_v together_o take_v by_o the_o 32._o of_o the_o ten_o two_o medial_a line_n ab_fw-la and_o bc_o commensurable_a in_o power_n only_o construction_n comprehend_v a_o medial_a superficies_n so_o that_o let_v the_o line_n ab_fw-la be_v in_o power_n more_o than_o the_o line_n bc_o by_o the_o square_n of_o a_o line_n incommensurable_a in_o length_n unto_o the_o line_n ab_fw-la and_o upon_o the_o line_n ab_fw-la describe_v a_o semicircle_n adb_o and_o let_v the_o rest_n of_o the_o construction_n be_v as_o it_o be_v in_o the_o two_o former_a proposition_n and_o forasmuch_o as_o by_o the_o 2_o part_n of_o the_o 18._o of_o the_o ten_o the_o line_n of_o be_v incommensurable_a in_o length_n unto_o the_o line_n fb_o demonstration_n therefore_o the_o line_n ad_fw-la be_v incommensurable_a in_o power_n unto_o the_o line_n db_o by_o that_o which_o be_v demonstrate_v in_o the_o proposition_n go_v before_o and_o forasmuch_o as_o the_o square_n of_o the_o line_n ab_fw-la be_v medial_a therefore_o that_o also_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ad_fw-la and_o db_o which_o square_n be_v equal_a to_o the_o square_n of_o the_o line_n ab_fw-la by_o the_o 47._o of_o the_o first_o be_v medial_a conclude_v and_o forasmuch_o as_o that_o which_o be_v contain_v under_o the_o line_n of_o and_o fb_o be_v equal_a to_o either_o of_o the_o square_n of_o the_o line_n ebb_v and_o fd_o for_o by_o supposition_n the_o parallelogram_n contain_v under_o the_o line_n of_o and_o fb_o be_v equal_a to_o the_o square_n of_o the_o line_n ebb_n and_o the_o same_o parallelogram_n be_v equal_a to_o the_o square_n of_o the_o line_n df_o by_o the_o three_o part_n of_o the_o first_o assumpt_n go_v before_o the_o 33._o of_o the_o ten_o wherefore_o the_o line_n be_v be_v equal_a to_o the_o line_n df._n wherefore_o the_o line_n bc_o be_v double_a to_o the_o line_n fd._n wherefore_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o be_v double_a to_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o fd._n wherefore_o they_o be_v commensurable_a by_o the_o six_o of_o this_o book_n but_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o be_v medial_a by_o supposition_n wherefore_o also_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o fd_o be_v medial_a by_o the_o corollary_n of_o the_o 23_o of_o the_o ten_o but_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o fd_v be_v by_o the_o four_o part_n of_o the_o first_o assumpt_n go_v before_o the_o 33._o of_o the_o ten_o equal_v to_o that_o which_o be_v contain_v under_o the_o line_n ad_fw-la and_o db_o conclude_v wherefore_o that_o which_o be_v contain_v under_o the_o line_n ad_fw-la and_o db_o be_v also_o medial_a and_o forasmuch_o as_o the_o line_n ab_fw-la be_v incommensurable_a in_o length_n unto_o the_o line_n bc._n but_o the_o line_n bc_o be_v commensurable_a in_o length_n unto_o the_o line_n be._n wherefore_o by_o the_o 13â—_n of_o
the_o ten_o the_o line_n ab_fw-la be_v incommensurable_a in_o length_n unto_o the_o line_n be._n wherefore_o the_o square_a of_o the_o line_n ab_fw-la be_v incommensurable_a to_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o be_v by_o the_o first_o of_o the_o six_o and_o 10._o of_o this_o book_n but_o unto_o the_o square_n of_o the_o line_n ab_fw-la be_v equal_a the_o square_n of_o the_o line_n ad_fw-la and_o db_o add_v together_o by_o the_o 47._o of_o the_o first_o and_o unto_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o be_v be_v equal_a that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o fd_v that_o be_v which_o be_v contain_v under_o ad_fw-la and_o db._n for_o the_o parallelogram_n contain_v under_o the_o line_n ab_fw-la and_o fd_o be_v equal_a to_o the_o parallelogram_n contain_v under_o the_o line_n ad_fw-la and_o db_o by_o the_o last_o part_n of_o the_o first_o assumpt_n go_v before_o the_o 33._o of_o this_o ten_o book_n conclude_v wherefore_o that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ad_fw-la and_o db_o be_v incommensurable_a to_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o db._n wherefore_o there_o be_v find_v out_o two_o right_a line_n ad_fw-la and_o db_o incommensurable_a in_o power_n who_o square_n add_v together_o make_v a_o medial_a superficies_n conclusion_n and_o the_o parallelogram_n contain_v under_o they_o make_v also_o a_o medial_a superficies_n which_o parallelogram_n moreover_o be_v incommensurable_a to_o the_o superficies_n compose_v of_o the_o square_n of_o those_o line_n add_v together_o which_o be_v require_v to_o be_v do_v the_o beginning_n of_o the_o senaries_n by_o composition_n ¶_o the_o 2d_o theorem_a the_o 36._o proposition_n if_o two_o rational_a line_n commensurable_a in_o power_n only_o be_v add_v together_o composition_n the_o whole_a line_n be_v irrational_a and_o be_v call_v a_o binomium_n or_o a_o binomial_a line_n 〈…〉_o b_o and_o bc_o be_v incommensurable_a to_o the_o square_n of_o the_o line_n bc._n but_o unto_o the_o parallelogram_n contain_v under_o the_o line_n ab_fw-la and_o bc_o be_v commensurable_a the_o parallelogram_n contain_v under_o ab_fw-la and_o bc_o twice_o by_o the_o 6._o of_o the_o ten_o wherefore_o that_o which_o be_v contain_v under_o ab_fw-la and_o bc_o twice_o be_v incommensurable_a to_o the_o square_n of_o the_o line_n bc_o by_o the_o 13_o of_o the_o ten_o but_o unto_o the_o square_n of_o the_o line_n bc_o be_v commensurable_a that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ab_fw-la and_o bc_o by_o the_o 15._o of_o the_o ten_o for_o by_o supposition_n the_o line_n ab_fw-la and_o bc_o be_v commensurable_a in_o power_n only_o wherefore_o by_o the_o 13._o of_o the_o ten_o that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ab_fw-la and_o bc_o add_v together_o be_v incommensurable_a to_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o twice_o wherefore_o by_o the_o 16._o of_o the_o ten_o that_o which_o be_v contain_v under_o ab_fw-la and_o bc_o twice_o together_o with_o the_o square_n of_o the_o line_n ab_fw-la and_o bc_o which_o by_o the_o 4._o of_o the_o second_o be_v equal_a to_o the_o square_n of_o the_o whole_a line_n ac_fw-la be_v incommensurable_a to_o that_o which_o be_v compose_v of_o the_o square_n of_o ab_fw-la and_o bc_o add_v together_o but_o that_o which_o be_v compose_v of_o the_o square_n of_o ab_fw-la and_o bc_o add_v together_o be_v rational_a for_o it_o be_v commensurable_a to_o either_o of_o the_o square_n of_o the_o line_n ab_fw-la and_o bg_o of_o which_o either_o of_o they_o be_v rational_a by_o supposition_n wherefore_o the_o square_a of_o the_o line_n ac_fw-la be_v by_o the_o 10._o definition_n of_o the_o ten_o irrational_a wherefore_o the_o line_n ac_fw-la also_o be_v irrational_a and_o be_v call_v a_o binomial_a line_n line_n this_o proposition_n show_v the_o generation_n and_o production_n of_o the_o second_o kind_n of_o irrational_a line_n which_o be_v call_v a_o binomium_n or_o a_o binomial_a line_n the_o definition_n whereof_o be_v full_o gather_v out_o of_o this_o proposition_n and_o that_o thus_o a_o binomium_n or_o a_o binomial_a line_n be_v a_o irrational_a line_n compose_v of_o two_o rational_a line_n commensurable_a the_o one_o to_o the_o other_o in_o power_n only_o and_o it_o be_v call_v a_o binomium_n that_o be_v have_v two_o name_n because_o it_o be_v make_v of_o two_o such_o line_n as_o of_o his_o part_n which_o be_v only_o commensurable_a in_o power_n and_o not_o in_o length_n and_o therefore_o each_o part_n or_o line_n or_o at_o the_o least_o the_o one_o of_o they_o as_o touch_v length_n be_v uncertain_a and_o unknown_a wherefore_o be_v join_v together_o their_o quantity_n can_v be_v express_v by_o any_o one_o number_n or_o name_n but_o each_o part_n remain_v to_o be_v several_o name_v in_o such_o sort_n as_o it_o may_v and_o of_o these_o binomial_a line_n there_o be_v six_o several_a kind_n line_n the_o first_o binomiall_n the_o second_o the_o three_o the_o four_o the_o five_o and_o the_o six_o of_o what_o nature_n and_o condition_n each_o of_o these_o be_v shall_v be_v know_v by_o their_o definitious_a which_o be_v afterward_o set_v in_o their_o due_a place_n ¶_o the_o 25._o theorem_a the_o 37._o proposition_n if_o two_o medial_a line_n commensurable_a in_o power_n only_o contain_v a_o rational_a superficies_n be_v add_v together_o the_o whole_a line_n be_v irrational_a and_o be_v call_v a_o first_o bimedial_a line_n let_v these_o two_o medial_a line_n ab_fw-la and_o bc_o be_v commensurable_a in_o power_n only_o and_o contain_v a_o rational_a superficies_n the_o 27._o of_o the_o ten_o teach_v to_o find_v out_o two_o such_o line_n be_v compose_v then_o i_o say_v that_o the_o whole_a line_n ac_fw-la be_v irrational_a for_o as_o 〈◊〉_d say_v in_o the_o proposition_n next_o go_v before_o that_o which_o be_v compose_v of_o the_o square_n of_o the_o 〈◊〉_d ab_fw-la and_o bc_o be_v incommensurable_a to_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o twisâ—_n wherefore_o by_o the_o 16._o of_o the_o ten_o that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ab_fw-la and_o bc_o together_o with_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o twice_o demonstration_n that_o be_v the_o square_a of_o the_o line_n ac_fw-la be_v incommensurable_a to_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o twice_o but_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o twice_o iâ—_z commensurable_a to_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o once_o by_o the_o 6._o of_o the_o ten_o wherefore_o the_o square_a of_o the_o whole_a line_n ac_fw-la be_v by_o the_o 13â—_n of_o the_o ten_o incommensurable_a â—o_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o once_o but_o by_o supposition_n the_o line_n ab_fw-la and_o bc_o comprehend_v a_o rational_a superficies_n wherefore_o the_o square_a of_o the_o whole_a line_n ac_fw-la be_v irrational_a wherefore_o also_o the_o line_n ac_fw-la be_v irrational_a and_o it_o be_v call_v a_o first_o bimedial_a line_n the_o three_o irrational_a line_n which_o be_v call_v a_o first_o bimedial_a line_n be_v show_v by_o this_o proposition_n and_o the_o definition_n thereof_o be_v by_o it_o make_v manifest_a line_n which_o be_v this_o a_o first_o bimedial_a line_n be_v a_o irrational_a line_n which_o be_v compose_v of_o two_o medial_a line_n commensurable_a in_o power_n only_o contain_v a_o rational_a parallelogram_n it_o be_v call_v a_o first_o bimedial_a line_n by_o cause_n the_o two_o medial_a line_n or_o part_n whereof_o it_o be_v compose_v contain_v a_o rational_a superficies_n which_o be_v prefer_v before_o a_o irrational_a ¶_o the_o 26._o theorem_a the_o 38._o proposition_n if_o two_o medial_a line_n commensurable_a in_o power_n only_o contain_v a_o medial_a superficies_n be_v add_v together_o the_o whole_a line_n be_v irrational_a and_o be_v call_v a_o second_o bimedial_a line_n let_v these_o two_o medial_a line_n ab_fw-la and_o bc_o be_v commensurable_a in_o power_n only_o and_o contain_v a_o medial_a superficies_n the_o 28._o of_o the_o ten_o teach_v to_o findâ—_n out_o two_o such_o line_n be_v add_v together_o construction_n then_o i_o say_v that_o the_o whole_a line_n ac_fw-la be_v irrational_a take_v a_o rational_a line_n de._n and_o by_o the_o 44._o of_o the_o first_o upon_o the_o line_n de_fw-fr apply_v the_o parallelogram_n df_o equal_a to_o the_o square_n of_o the_o line_n ac_fw-la who_o be_v other_o side_n let_v be_v the_o line_n dg_o and_o forasmuch_o as_o the_o square_n of_o the_o line_n ac_fw-la be_v by_o the_o 4._o of_o the_o second_o equal_a to_o that_o which_o be_v compose_v of_o
the_o square_n of_o the_o line_n ab_fw-la and_o bc_o demonstration_n together_o with_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o twice_o but_o the_o square_n of_o the_o line_n ac_fw-la be_v equal_a to_o the_o parallelogram_n df._n wherefore_o the_o parallelogram_n df_o be_v equal_a to_o that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ab_fw-la and_o bc_o together_o with_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o twice_o now_o then_o again_o by_o the_o 44_o of_o the_o first_o upon_o the_o line_n de_fw-fr apply_v the_o parallelogram_n eh_o equal_a to_o the_o square_n of_o the_o line_n ab_fw-la and_o bc._n wherefore_o the_o parallelogram_n remain_v namely_o hf_o be_v equal_a to_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o twice_o and_o forasmuch_o as_o either_o of_o these_o line_n ab_fw-la and_o bc_o be_v medial_a therefore_o the_o square_n of_o the_o line_n ab_fw-la and_o bc_o be_v also_o medial_a and_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o twice_o be_v by_o the_o corollary_n of_o the_o â—4_o of_o the_o ten_o medial_a for_o by_o the_o 6._o of_o this_o book_n it_o be_v commensurable_a â—â—_o that_o 〈◊〉_d be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o once_o which_o be_v by_o supposition_n medial_a 〈…〉_o square_n of_o the_o line_n ab_fw-la and_o bc_o be_v equal_a the_o parallelogram_n eh_o and_o unto_o that_o 〈◊〉_d contain_v under_o the_o line_n ab_fw-la and_o bc_o twice_o be_v equal_a the_o parallelogram_n he_o 〈…〉_o either_o of_o these_o parallelogram_n he_o and_o of_o be_v medial_a and_o they_o be_v apply_v upon_o the_o rational_a line_n ed._n wherefore_o by_o the_o 22._o of_o the_o ten_o either_o of_o these_o line_n dh_o and_o hg_o be_v a_o rational_a line_n and_o incommensurable_a in_o length_n unto_o the_o line_n de._n and_o forasmuch_o as_o by_o supposition_n the_o line_n ab_fw-la be_v incommensurable_a in_o length_n unto_o the_o line_n bc._n but_o as_o the_o line_n ab_fw-la be_v to_o the_o line_n be_v â—_z so_o be_v the_o square_a of_o the_o line_n ab_fw-la to_o the_o parallelogram_n which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o by_o the_o first_o of_o the_o six_o wherefore_o by_o the_o 10_o of_o this_o bookâ—_n the_o square_a of_o the_o line_n ab_fw-la be_v incommensurable_a to_o the_o parallelogram_n contain_v under_o the_o line_n ab_fw-la and_o bc._n but_o to_o the_o square_n of_o the_o line_n ab_fw-la be_v commensâ—able_a that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ab_fw-la and_o bc_o by_o the_o 15._o of_o the_o ten_o for_o the_o square_n of_o the_o line_n ab_fw-la and_o bc_o be_v commensurable_a when_o as_o the_o line_n ab_fw-la and_o bc_o be_v put_v to_o be_v commensurable_a in_o power_n only_o and_o to_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o be_v commensurable_a that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o twice_o by_o the_o 6_o of_o the_o ten_o wherefore_o that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ab_fw-la and_o bc_o be_v incommensurable_a to_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o twice_o but_o to_o the_o square_n of_o the_o line_n ab_fw-la and_o bc_o be_v equal_a the_o parallelogram_n eh_o and_o to_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o twice_o be_v equal_a the_o parallelogram_n fh_o wherefore_o the_o parallelogram_n fh_o be_v incommensurable_a to_o the_o parallelogram_n he._n wherefore_o the_o line_n dh_o be_v incommensurable_a in_o length_n to_o the_o line_n hg_o by_o the_o 1_o of_o the_o six_o and_o 10_o of_o this_o book_n and_o it_o be_v prove_v that_o they_o be_v rational_a line_n wherefore_o the_o line_n dh_o &_o hg_o be_v rational_a commensurable_a in_o power_n only_o wherefore_o by_o the_o 36._o of_o the_o ten_o the_o whole_a line_n dg_o be_v irrational_o and_o the_o line_n de_fw-fr be_v rational_a but_o a_o rectangle_n superficies_n comprehend_v under_o a_o rational_a line_n and_o a_o irrational_a line_n be_v by_o the_o corollary_n add_v after_o the_o 21_o of_o the_o ten_o irrational_a wherefore_o the_o superficies_n df_o be_v irrational_a and_o the_o line_n also_o which_o contain_v it_o in_o power_n be_v irrational_a but_o the_o line_n ac_fw-la contain_v in_o power_n the_o superficies_n df._n wherefore_o the_o line_n ac_fw-la be_v irrational_a and_o it_o be_v call_v a_o second_o bimedial_a line_n this_o proposition_n show_v the_o generation_n of_o the_o four_o irrational_a line_n call_v a_o second_o bimedial_a line_n line_n the_o definition_n whereof_o be_v evident_a by_o this_o proposition_n which_o be_v thus_o a_o second_o bimedial_a line_n be_v a_o irrational_a line_n which_o be_v make_v of_o two_o medial_a line_n commensurable_a in_o power_n only_o join_v together_o which_o comprehend_v a_o medial_a superficies_n and_o it_o be_v call_v a_o second_o bimediall_n because_o the_o two_o medial_a line_n of_o which_o it_o be_v compose_v contain_v a_o medial_a superficies_n and_o not_o a_o rational_a now_o a_o medial_a be_v by_o nature_n &_o in_o knowledge_n after_o a_o rational_a ¶_o the_o 27._o theorem_a the_o 39_o proposition_n if_o two_o right_a line_n incommensurable_a in_o power_n be_v add_v together_o have_v that_o which_o be_v compose_v of_o the_o square_n of_o they_o rational_a and_o the_o parallelogram_n contain_v under_o they_o medial_a the_o whole_a right_a line_n be_v irrational_a and_o be_v call_v a_o great_a line_n let_v tâ—ese_n two_o right_a line_n ab_fw-la and_o bc_o be_v incommensurable_a in_o power_n only_o and_o make_v that_o which_o be_v require_v in_o the_o proposition_n the_o 33._o of_o the_o ten_o teach_v to_o find_v out_o two_o such_o line_n be_v add_v together_o then_o i_o say_v that_o the_o whole_a line_n ac_fw-la be_v irrational_a demonstration_n for_o forasmuch_o as_o by_o supposition_n the_o parallelogram_n contain_v under_o the_o line_n ab_fw-la and_o bc_o be_v medial_a therefore_o the_o parallelogram_n contain_v twice_o under_o the_o line_n ab_fw-la and_o bc_o be_v medial_a for_o that_o which_o be_v contain_v under_o ab_fw-la and_o bc_o twice_o be_v commensurable_a to_o that_o which_o be_v contain_v under_o ab_fw-la and_o bc_o once_o by_o the_o 6._o of_o the_o ten_o wherefore_o by_o the_o corollary_n of_o the_o 23._o of_o the_o ten_o that_o which_o be_v contain_v under_o ab_fw-la &_o bc_o twice_o be_v medial_a but_o by_o supposition_n that_o which_o iâ—_z compose_v of_o the_o square_n of_o the_o line_n ab_fw-la and_o bc_o be_v rational_a wherefore_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o twice_o be_v incommensurable_a to_o that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ab_fw-la and_o bc._n wherefore_o by_o the_o 16._o of_o the_o ten_o that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ab_fw-la and_o bc_o together_o with_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la &_o bc_o twice_o which_o be_v by_o the_o 4._o of_o the_o second_o equal_a to_o the_o square_n of_o the_o line_n ac_fw-la be_v incommensurable_a to_o that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ab_fw-la and_o bc._n but_o that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ab_fw-la and_o bc_o be_v rational_a wherefore_o the_o square_a of_o the_o whole_a line_n ac_fw-la be_v irrational_a wherefore_o the_o line_n ac_fw-la also_o be_v irrational_a and_o be_v call_v a_o great_a line_n and_o it_o be_v call_v a_o great_a line_n for_o that_o that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ab_fw-la &_o bc_o which_o be_v rational_a be_v great_a than_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o twice_o which_o be_v medial_a now_o it_o be_v meet_v that_o the_o name_n shall_v be_v give_v according_a to_o the_o property_n of_o the_o rational_a a_o assumpt_n this_o proposition_n teach_v the_o production_n of_o the_o five_o irrational_a line_n which_o be_v call_v a_o great_a line_n which_o be_v by_o the_o sense_n of_o this_o proposition_n thus_o define_v a_o great_a line_n be_v a_o irrational_a line_n which_o be_v compose_v of_o two_o right_a line_n which_o be_v incommensurable_a in_o power_n line_n the_o square_n of_o which_o add_v together_o make_v a_o rational_a superficies_n and_o the_o parallelogram_n which_o they_o contain_v be_v medial_a it_o be_v therefore_o call_v a_o great_a line_n as_o theon_n say_v because_o the_o square_n of_o the_o two_o line_n of_o which_o it_o be_v compose_v add_v together_o be_v rational_a be_v great_a than_o the_o medial_a superficies_n contain_v under_o they_o twice_o and_o it_o be_v convenient_a that_o the_o denomination_n be_v take_v of_o the_o propriety_n of_o the_o
rational_a part_n rather_o than_o of_o the_o medial_a part_n ¶_o the_o 28._o theorem_a the_o 40._o proposition_n if_o two_o right_a line_n incommensurable_a in_o power_n be_v add_v together_o have_v that_o which_o be_v make_v of_o the_o square_n of_o they_o add_v together_o medial_a and_o the_o parallelogram_n contain_v under_o they_o rational_a the_o whole_a right_a line_n be_v irrational_a and_o be_v call_v a_o line_n contain_v in_o power_n a_o rational_a and_o a_o medial_a superficies_n in_o this_o proposition_n be_v teach_v the_o generation_n of_o the_o six_o irrational_a line_n which_o be_v call_v a_o line_n who_o power_n be_v rational_a and_o medial_a the_o definition_n of_o which_o be_v gather_v of_o this_o proposition_n after_o this_o manner_n a_o line_n who_o power_n be_v rational_a and_o medial_a be_v a_o irrational_a line_n which_o be_v make_v of_o two_o right_a line_n incommensurable_a in_o power_n add_v together_o medial_a who_o square_n add_v together_o make_v a_o medial_a superficies_n but_o that_o supersicy_n which_o they_o contain_v be_v rational_a the_o reason_n of_o the_o name_n be_v before_o set_v forth_o in_o the_o proposition_n ¶_o the_o 29._o theorem_a the_o 41._o proposition_n if_o two_o right_a line_n incommensurable_a in_o power_n be_v add_v together_o have_v that_o which_o be_v compose_v of_o the_o square_n of_o they_o add_v together_o medial_a and_o the_o parallelogram_n contain_v under_o they_o medial_a and_o also_o incommensurable_a to_o that_o which_o be_v compose_v of_o the_o square_n of_o they_o add_v togethers_o the_o whole_a right_a line_n be_v irrational_a and_o be_v call_v a_o line_n contain_v in_o power_n two_o medial_n in_o this_o proposition_n be_v teach_v the_o nature_n of_o the_o 7._o kind_n of_o irrational_a line_n which_o be_v call_v a_o line_n who_o power_n be_v two_o medial_n the_o definition_n whereof_o be_v take_v of_o this_o proposition_n after_o this_o manner_n a_o line_n who_o power_n be_v two_o medial_n be_v a_o irrational_a line_n which_o be_v compose_v of_o two_o right_a line_n incommensurable_a in_o power_n medial_n the_o square_n of_o which_o add_v together_o make_v a_o medial_a superficies_n and_o that_o which_o be_v contain_v under_o they_o be_v also_o medial_a and_o moreover_o it_o be_v incommensurable_a to_o that_o which_o be_v compose_v of_o the_o two_o square_n add_v together_o the_o reason_n why_o this_o line_n be_v call_v a_o line_n who_o power_n be_v two_o medial_n be_v before_o in_o the_o end_n of_o the_o demonstration_n declare_v and_o that_o the_o say_v irrational_a line_n be_v divide_v one_o way_n only_o that_o be_v in_o one_o point_n only_o into_o the_o right_a line_n of_o which_o they_o be_v compose_v and_o which_o make_v every_o one_o of_o the_o kind_n of_o those_o irrational_a line_n shall_v straight_o way_n be_v demonstrate_v but_o first_o will_v we_o demonstrate_v two_o assumpte_n here_o follow_v ¶_o a_o assumpt_n take_v a_o right_a line_n and_o let_v the_o same_o be_v ab_fw-la and_o divide_v it_o into_o two_o unequal_a part_n in_o the_o point_n c_o assumpt_n and_o again_o divide_v the_o same_o line_n ab_fw-la into_o two_o other_o unequal_a part_n in_o a_o other_o point_n namely_o in_o d_o and_o let_v the_o line_n ac_fw-la by_o supposition_n be_v great_a than_o the_o line_n db._n then_o i_o say_v that_o the_o square_n of_o the_o line_n ac_fw-la and_o bc_o add_v together_o be_v great_a than_o the_o square_n of_o the_o line_n ad_fw-la and_o db_o add_v together_o divide_v the_o line_n ab_fw-la by_o the_o 10._o of_o the_o first_o into_o two_o equal_a part_n in_o the_o point_n e._n and_o forasmuch_o as_o the_o line_n ac_fw-la be_v great_a than_o the_o line_n db_o take_v away_o the_o line_n dc_o which_o be_v common_a to_o they_o both_o wherefore_o the_o residue_n ad_fw-la be_v great_a than_o the_o residue_n cb_o but_o the_o line_n ae_n be_v equal_a to_o the_o line_n ebb_n wherefore_o the_o line_n de_fw-fr be_v less_o than_o the_o line_n ec_o wherefore_o the_o point_n c_o and_o d_o be_v not_o equal_o distant_a from_o the_o point_n e_o which_o be_v the_o point_n of_o the_o section_n into_o two_o equal_a part_n and_o forasmuch_o as_o by_o the_o 5._o of_o the_o second_o that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o together_o with_o the_o square_n of_o the_o line_n ec_o be_v equal_a to_o the_o square_n of_o the_o line_n ebb_n and_o by_o the_o same_o reason_n that_o which_o be_v contain_v under_o the_o line_n ad_fw-la and_o db_o together_o with_o the_o square_n of_o the_o line_n de_fw-fr be_v also_o equal_a to_o the_o self_n same_o square_n of_o the_o line_n ebb_v wherefore_o that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o together_o with_o the_o square_n of_o the_o line_n ec_o be_v equal_a to_o that_o which_o be_v contain_v under_o the_o line_n ad_fw-la and_o db_o together_o with_o the_o square_n of_o the_o line_n de_fw-fr of_o which_o the_o square_a of_o the_o line_n de_fw-fr be_v less_o than_o the_o square_a of_o the_o line_n ec_o for_o it_o be_v prove_v that_o the_o line_n de_fw-fr be_v less_o than_o the_o line_n ec_o wherefore_o the_o parallelogram_n remain_v contain_v under_o the_o line_n ac_fw-la and_o cb_o be_v less_o they_o the_o parallelogram_n remain_v contain_v under_o the_o line_n ad_fw-la and_o db._n wherefore_o also_o that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o twice_o be_v less_o than_o that_o which_o be_v contain_v under_o the_o line_n ad_fw-la and_o db_o twice_o but_o by_o the_o four_o of_o the_o second_o the_o square_a of_o the_o whole_a line_n ab_fw-la be_v equal_a to_o that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ac_fw-la and_o cb_o together_o with_o that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o twice_o and_o by_o the_o same_o reason_n the_o square_n of_o the_o whole_a line_n ab_fw-la be_v equal_a to_o that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ad_fw-la and_o db_o together_o with_o that_o which_o be_v contain_v under_o the_o line_n ad_fw-la and_o db_o twice_o wherefore_o that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ac_fw-la and_o cb_o together_o with_o that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o twice_o be_v equal_a to_o that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ad_fw-la and_o db_o together_o with_o that_o which_o be_v contain_v under_o the_o line_n ad_fw-la and_o db_o twice_o but_o it_o be_v already_o prove_v that_o that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o twice_o be_v less_o than_o that_o which_o be_v contain_v under_o the_o line_n ad_fw-la &_o db_o twice_o wherefore_o the_o residue_n namely_o that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ac_fw-la and_o cb_o be_v great_a than_o the_o residue_n namely_o then_o that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ad_fw-la and_o db_o which_o be_v require_v to_o be_v demonstrate_v ¶_o a_o assumpt_n a_o rational_a superficies_n exceed_v a_o rational_a superficies_n by_o a_o rational_a superficies_n let_v ad_fw-la be_v a_o rational_a superficies_n and_o let_v it_o exceed_v of_o be_v also_o a_o rational_a superficies_n by_o the_o superficies_n ed._n then_o i_o say_v that_o the_o superficies_n ed_z be_v also_o rational_a for_o the_o parallelogram_n ad_fw-la be_v commensurable_a to_o the_o parallelogram_n of_o for_o that_o either_o of_o they_o be_v rational_a wherefore_o by_o the_o second_o part_n of_o the_o 15._o of_o the_o ten_o the_o parallelogram_n of_o be_v commensurable_a to_o the_o parallelogram_n ed._n but_o the_o the_o parallelogram_n of_o be_v rational_a wherefore_o also_o the_o parallelogram_n ed_z be_v rational_a ¶_o the_o 30._o theorem_a the_o 42._o proposition_n a_o binomial_a line_n be_v in_o one_o point_n only_o divide_v into_o his_o name_n composition_n svppose_v that_o ab_fw-la be_v a_o binomial_a line_n and_o in_o the_o point_n g_o let_v it_o be_v divide_v into_o his_o name_n that_o be_v into_o the_o line_n whereof_o the_o whole_a line_n ab_fw-la be_v compose_v wherefore_o these_o line_n ac_fw-la and_o cb_o be_v rational_a commensurable_a in_o power_n only_o now_o i_o say_v that_o the_o line_n ab_fw-la can_v in_o any_o other_o point_n beside_o c_o be_v divide_v into_o two_o rational_a line_n commensurable_a in_o power_n only_o impossibility_n for_o if_o it_o be_v possible_a let_v it_o be_v divide_v in_o the_o point_n d_o so_fw-mi that_o let_v the_o line_n ad_fw-la and_o de_fw-fr bâ—_z rational_a commensurable_a in_o power_n only_o first_o this_o manifest_a that_o neither_o of_o these_o point_n c_o and_o d_o divide_v the_o right_a line_n ab_fw-la into_o two_o equal_a part_n otherwise_o the_o line_n ac_fw-la and_o cb_o shall_v be_v rational_a commensurable_a in_o
length_n and_o so_o likewise_o shall_v the_o line_n ad_fw-la and_o db_o be_v for_o every_o line_n measure_v itself_o and_o any_o other_o line_n equal_a to_o itself_o moreover_o the_o line_n db_o be_v either_o one_o and_o the_o same_o with_o the_o line_n acâ—_n that_o be_v be_v equal_a to_o the_o line_n ac_fw-la oâ—_n else_o it_o be_v greater-then_a the_o line_n ac_fw-la either_o else_o it_o be_v less_o than_o it_o if_o db_o be_v equal_a to_o the_o line_n ac_fw-la then_o put_v the_o line_n db_o upon_o the_o line_n ac_fw-la each_o end_n of_o the_o one_o shall_v agree_v with_o each_o end_n of_o the_o other_o wherefore_o put_v the_o point_n b_o upon_o the_o point_n a_o the_o point_n d_o also_o shall_v fall_v upon_o the_o point_n c_o and_o the_o line_n ad_fw-la which_o be_v the_o rest_n of_o the_o line_n ac_fw-la shall_v also_o be_v equal_a to_o the_o line_n cb_o which_o be_v the_o rest_n of_o the_o line_n db._n wherefore_o the_o line_n ab_fw-la be_v divide_v into_o his_o name_n in_o the_o point_n c._n and_o so_o also_o shall_v the_o line_n ab_fw-la be_v divide_v in_o the_o point_n d_o be_v divide_v in_o the_o self_n â—ame_n point_n that_o the_o self_n same_o line_n ab_fw-la be_v before_o divide_v in_o the_o point_n c_o which_o be_v contrary_a to_o the_o supposition_n for_o by_o supposition_n it_o be_v divide_v in_o sundry_a point_n namely_o in_o c_o &_o d._n but_o if_o the_o line_n db_o be_v greaterâ—_n the_o the_o line_n ac_fw-la let_v the_o line_n ab_fw-la be_v deâ—ided_v into_o two_o equal_a part_n in_o the_o point_n e._n wherefore_o the_o point_n c_o &_o d_o shall_v not_o equal_o be_v distant_a from_o the_o point_n e_o now_o by_o the_o first_o assupt_v go_v before_o this_o proposition_n that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ad_fw-la &_o db_o be_v great_a they_o that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ac_fw-la &_o cbâ—_n but_o that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ad_fw-la &_o db_o together_o with_o that_o which_o be_v contain_v under_o the_o line_n ad_fw-la &_o db_o twice_o be_v equal_a to_o that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ac_fw-la &_o cb_o together_o with_o that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o twice_o for_o either_o of_o they_o be_v equal_a to_o the_o square_n of_o the_o whole_a line_n ab_fw-la by_o the_o 4._o of_o the_o second_o wherefore_o how_o much_o that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ad_fw-la and_o db_o add_v together_o be_v great_a than_o that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ac_fw-la and_o cb_o add_v together_o so_o much_o be_v that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o twice_o great_a than_o that_o which_o be_v contain_v under_o the_o line_n ad_fw-la and_o db_o twice_o but_o that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ad_fw-la and_o db_o exceed_v that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ac_fw-la and_o cb_o by_o a_o rational_a superficies_n by_o the_o 2._o assumpt_n go_v before_o this_o proposition_n for_o that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ad_fw-la and_o db_o be_v rational_a and_o so_o also_o be_v that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ac_fw-la and_o cb_o for_o the_o line_n ad_fw-la and_o db_o be_v put_v to_o be_v rational_a commensurable_a in_o power_n only_o and_o so_o likewise_o be_v the_o line_n ac_fw-la and_o cb._n wherefore_o also_o that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o twice_o exceed_v that_o which_o be_v contain_v under_o the_o line_n ad_fw-la &_o db_o twice_o by_o a_o rational_a superficies_n when_o yet_o notwithstanding_o they_o be_v both_o medial_a superficiece_n by_o the_o 21._o of_o the_o ten_o which_o by_o the_o 26._o of_o the_o same_o be_v impossible_a and_o if_o the_o line_n db_o be_v less_o than_o the_o line_n ac_fw-la we_o may_v by_o the_o like_a demonstration_n prove_v the_o self_n same_o impossibility_n wherefore_o a_o binomial_a line_n be_v in_o one_o point_n only_o divide_v into_o his_o name_n which_o be_v require_v to_o be_v demonstrate_v 〈…〉_o ollary_n add_v by_o flussates_n two_o ration_n 〈…〉_o surable_a in_o power_n only_o be_v add_v together_o can_v be_v equal_a to_o two_o other_o rational_a line_n 〈…〉_o in_o power_n only_o add_v together_o corollary_n for_o either_o of_o they_o shall_v make_v a_o binomia_fw-la 〈…〉_o so_o shall_v a_o binomial_a line_n be_v divide_v into_o his_o name_n in_o more_o point_n than_o onâ—â—â—ch_v by_o this_o proposition_n be_v prove_v to_o be_v impossible_a the_o like_a shall_v follow_v in_o the_o five_o 〈◊〉_d irrational_a line_n as_o touch_v their_o two_o name_n ¶_o the_o 31._o problem_n the_o 43._o proposition_n a_o first_o bimedial_a line_n be_v in_o one_o point_n only_o divide_v into_o his_o name_n svppose_v that_o ab_fw-la be_v a_o first_o bimedial_a line_n and_o let_v it_o be_v divide_v into_o his_o part_n in_o the_o point_n c_o so_fw-mi that_o let_v the_o line_n ac_fw-la and_o cb_o be_v medial_a commensurable_a in_o power_n only_o and_o contain_v a_o rational_a superficies_n then_o i_o say_v that_o the_o line_n ab_fw-la can_v not_o be_v divide_v into_o his_o name_n in_o any_o other_o point_n then_o in_o c._n for_o if_o it_o be_v possible_a let_v it_o be_v divide_v into_o his_o name_n in_o the_o point_n d_o impossibilâ—â—e_n so_fw-mi that_o let_v ad_fw-la &_o db_o be_v medial_a line_n commensurable_a in_o power_n only_o comprehend_v a_o rational_a superficies_n now_o forasmuch_o as_o how_o much_o that_o which_o be_v contain_v under_o the_o line_n ad_fw-la and_o db_o twice_o diâ—ferreth_v from_o that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o twice_o so_o much_o differ_v that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ad_fw-la and_o db_o from_o that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ac_fw-la and_o cb_o but_o that_o which_o be_v contain_v under_o the_o line_n ad_fw-la and_o db_o twice_o differ_v from_o that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o twice_o by_o a_o rational_a superficies_n by_o the_o second_o assumpt_v go_v before_o the_o 41._o of_o the_o ten_o for_o either_o of_o those_o superficiece_n be_v rational_a wherefore_o that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ac_fw-la and_o cb_o differ_v from_o that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ad_fw-la and_o db_o by_o a_o rational_a superficies_n when_o yet_o they_o be_v both_o medial_a superficiece_n which_o be_v impossible_a wherefore_o a_o first_o bimedial_a line_n be_v in_o one_o point_n only_o divide_v into_o his_o name_n which_o be_v require_v to_o be_v prove_v ¶_o the_o 32._o theorem_a the_o 44._o proposition_n a_o second_o bimedial_a line_n be_v in_o one_o point_n only_o divide_v into_o his_o name_n svppose_v that_o the_o line_n ab_fw-la be_v a_o second_o bimedial_a line_n be_v divide_v into_o his_o name_n in_o the_o point_n c_o so_o that_o let_v the_o line_n ac_fw-la and_o cb_o be_v medial_a line_n commensurable_a in_o power_n only_o comprehend_v a_o medial_a superficies_n it_o be_v manifest_a that_o the_o point_n c_o divide_v not_o the_o whole_a line_n ab_fw-la into_o two_o equal_a part_n impossibility_n for_o the_o line_n ac_fw-la and_o cb_o be_v not_o commensurable_a in_o length_n the_o one_o to_o the_o other_o now_o i_o say_v that_o the_o line_n ab_fw-la can_v be_v divide_v into_o his_o name_n in_o any_o other_o point_n but_o only_o in_o c._n for_o if_o it_o be_v possible_a let_v it_o be_v divide_v into_o his_o name_n in_o the_o point_n d_o so_fw-mi that_o let_v not_o the_o line_n ac_fw-la be_v one_o and_o the_o same_o that_o be_v let_v it_o not_o be_v equal_a with_o the_o line_n db._n but_o let_v it_o be_v great_a than_o it_o now_o it_o be_v manifest_a by_o the_o first_o assumpt_n go_v before_o the_o 42._o proposition_n of_o this_o book_n that_o the_o square_n of_o the_o line_n ac_fw-la and_o cb_o be_v great_a than_o the_o square_n of_o the_o line_n ad_fw-la and_o db._n and_o also_o that_o the_o line_n ad_fw-la and_o db_o be_v medial_a line_n commensurable_a in_o power_n only_o comprehend_v a_o medial_a supersicy_n take_v a_o rational_a line_n ef._n and_o by_o the_o 44._o of_o the_o first_o upon_o the_o line_n of_o apply_v a_o rectangle_n parallelogram_n eke_o equal_a to_o the_o square_n of_o the_o line_n ab_fw-la from_o which_o parallelogram_n take_v away_o the_o parallelogram_n eglantine_n equal_a to_o the_o square_n of_o the_o line_n ac_fw-la and_o cb_o wherefore_o the_o residue_n namely_o the_o parallelogram_n hk_o
which_o be_v contain_v under_o the_o line_n ad_fw-la and_o db_o twice_o by_o a_o rational_a superficies_n for_o either_o of_o they_o be_v rational_a wherefore_o also_o the_o square_n of_o the_o line_n ac_fw-la and_o cb_o add_v together_o exceed_v the_o square_n of_o the_o line_n ad_fw-la and_o db_o add_v together_o by_o a_o rational_a superficies_n when_o yet_o each_o of_o they_o be_v a_o medial_a superficies_n which_o be_v impossible_a wherefore_o a_o line_n contain_v in_o power_n a_o rational_a and_o a_o medial_a be_v in_o one_o point_n only_o divide_v into_o his_o name_n which_o be_v require_v to_o be_v demonstrate_v ¶_o the_o 35._o theorem_a the_o 47._o proposition_n a_o line_n contain_v in_o power_n two_o medial_n be_v in_o one_o point_n only_o divide_v into_o his_o name_n svppose_v that_o ab_fw-la be_v a_o line_n contain_v in_o power_n two_o medial_n be_v divide_v into_o his_o name_n in_o the_o point_n c_o so_fw-mi that_o let_v the_o line_n ac_fw-la and_o cb_o be_v incommensurable_a in_o power_n have_v that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ac_fw-la &_o cb_o medial_a and_o that_o also_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o medial_a and_o moreover_o incommensurable_a â—o_o that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ac_fw-la and_o cb._n then_o i_o say_v construction_n that_o the_o line_n ab_fw-la can_v in_o no_o other_o point_n be_v divide_v into_o his_o name_n but_o only_o in_o the_o point_n c._n for_o if_o it_o be_v possible_a let_v it_o be_v divide_v into_o his_o name_n in_o the_o point_n d_o so_fw-mi that_o let_v not_o the_o line_n ac_fw-la be_v one_o and_o the_o same_o that_o be_v equal_a with_o the_o line_n db_o but_o by_o supposition_n let_v the_o line_n ac_fw-la be_v the_o great_a and_o take_v a_o rational_a line_n ef._n and_o by_o the_o 43._o of_o the_o first_o upon_o the_o line_n of_o apply_v a_o rectangle_n parallelogram_n eglantine_n equal_a to_o that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ac_fw-la and_o cb_o and_o likewise_o upon_o the_o line_n hc_n which_o be_v equal_a to_o the_o line_n of_o apply_v the_o parallelogram_n hk_o equal_a to_o that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o twice_o wherefore_o the_o whole_a parallelogram_n eke_o be_v equal_a to_o the_o square_n of_o the_o line_n ab_fw-la absurdity_n again_o upon_o the_o same_o line_n of_o describe_v the_o parallelogram_n el_n equal_a to_o the_o square_n of_o the_o line_n ad_fw-la and_o db._n wherefore_o the_o residue_n namely_o that_o which_o be_v contain_v under_o the_o line_n ad_fw-la and_o db_o twice_o be_v equal_a to_o the_o parallelogram_n remain_v namely_o to_o mk_o and_o forasmuch_o as_o that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ac_fw-la and_o cb_o be_v by_o supposition_n medial_a therefore_o the_o parallelogram_n eglantine_n which_o be_v equal_a unto_o it_o be_v also_o medial_a and_o it_o be_v apply_v upon_o the_o rational_a line_n ef._n wherefore_o by_o the_o 22._o of_o the_o ten_o the_o line_n he_o be_v rational_a and_o incommensurable_a in_o length_n unto_o the_o line_n ef._n and_o by_o the_o same_o reason_n also_o the_o line_n hn_o be_v rational_a and_o incommensurable_a in_o length_n to_o the_o same_o line_n ef._n and_o forasmuch_o as_o that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ac_fw-la and_o cb_o be_v incommensurable_a to_o that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o twice_o for_o it_o be_v suppose_v to_o be_v incommensurable_a to_o that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o once_o therefore_o the_o parallelogram_n eglantine_n be_v incommensurable_a to_o the_o parallelogram_n h_o â—_o wherefore_o the_o line_n eh_o also_o be_v incommensurable_a in_o length_n to_o the_o line_n hn_o and_o they_o be_v rational_a line_n wherefore_o the_o line_n eh_o and_o hn_o be_v rational_a commensurable_a in_o power_n only_o wherefore_o the_o whole_a line_n en_fw-fr be_v a_o binomial_a line_n and_o be_v divide_v into_o his_o name_n in_o the_o point_n h._n and_o in_o like_a sort_n may_v we_o prove_v that_o the_o same_o binomial_a line_n en_fw-fr be_v divide_v into_o his_o name_n in_o the_o point_n m_o and_o that_o the_o line_n eh_o be_v not_o one_o and_o the_o same_o that_o be_v equal_a with_o the_o line_n mn_o as_o it_o be_v prove_v in_o the_o end_n of_o the_o demonstration_n of_o the_o 44._o of_o this_o book_n wherefore_o a_o binomial_a line_n be_v divide_v into_o his_o name_n in_o two_o sundry_a point_n which_o be_v impossible_a by_o the_o 42._o of_o the_o ten_o wherefore_o a_o line_n contain_v in_o power_n two_o medial_n be_v not_o in_o sundry_a point_n divide_v into_o his_o name_n wherefore_o it_o be_v divide_v in_o one_o point_n only_o which_o be_v require_v to_o be_v demonstrate_v ¶_o second_a definition_n it_o be_v show_v before_o that_o of_o binomial_a line_n there_o be_v six_o kind_n line_n the_o definition_n of_o all_o which_o be_v here_o now_o set_v and_o be_v call_v second_o definition_n all_o binomial_a line_n as_o all_o other_o kind_n of_o irrational_a line_n be_v conceive_v consider_v and_o perfect_o understand_v only_o in_o respect_n of_o a_o rational_a line_n who_o part_n as_o before_o be_v teach_v be_v certain_a and_o know_v and_o may_v be_v distinct_o express_v by_o number_n unto_o which_o line_n they_o be_v compare_v this_o rationalâ—_fw-la line_n must_v you_o ever_o have_v before_o your_o eye_n in_o all_o these_o definition_n so_o shall_v they_o all_o be_v â—asie_a enough_o paâ—tâ—s_n a_o binomial_a line_n you_o know_v be_v make_v of_o two_o part_n or_o name_n whereof_o the_o one_o be_v great_a than_o the_o other_o wherefore_o the_o power_n or_o square_v also_o of_o the_o one_o be_v great_a than_o the_o power_n or_o square_a oâ—_n the_o other_o the_o three_o first_o kind_n of_o binomial_a line_n namely_o the_o first_o the_o seconâ—_o &_o the_o three_o be_v produce_v when_o the_o square_a of_o the_o great_a name_n or_o part_n of_o a_o binomâ—all_n eâ—cedeth_v the_o square_a of_o the_o less_o name_n or_o part_n by_o the_o square_n of_o a_o line_n which_o be_v commensurable_a in_o length_n to_o it_o namely_o to_o the_o great_a the_o three_o last_o kind_n namely_o the_o four_o the_o â—iâ—t_n and_o the_o six_o be_v produce_v when_o the_o square_a of_o the_o great_a name_n or_o part_n â—â—â—â—edeth_v the_o square_a of_o the_o less_o name_n or_o part_n by_o the_o square_n of_o a_o line_n incommensurable_a in_o length_n unto_o it_o that_o be_v to_o the_o great_a part_n definition_n a_o first_o binomial_a line_n be_v who_o square_a of_o the_o great_a part_n exceed_v the_o square_a of_o tâ—e_z less_o part_n â—y_a the_o square_n of_o a_o line_n commensurable_a in_o length_n to_o the_o great_a part_n and_o the_o great_a part_n be_v also_o commensurable_a in_o length_n to_o tâ—e_z rational_a line_n first_o set_v as_o lâ—t_v the_o rational_a line_n first_o set_v be_v ab_fw-la who_o part_n be_v distinct_o know_v suppose_v also_o that_o the_o line_n ce_fw-fr be_v a_o binomial_a line_n who_o name_n or_o part_n let_v be_v cd_o and_o de._n and_o let_v the_o square_n of_o the_o line_n cd_o the_o great_a part_n exceed_v the_o square_n of_o the_o line_n de_fw-fr the_o less_o part_n by_o the_o square_n of_o the_o line_n fg_o which_o line_n fg_o let_v bâ—e_o commensurable_a in_o length_n to_o the_o line_n cd_o which_o be_v the_o great_a part_n of_o the_o binomial_a line_n and_o moreover_o let_v the_o line_n cd_o the_o great_a paâ—t_n be_v commensurble_n in_o length_n to_o the_o rational_a line_n first_o set_v namely_o to_o ab_fw-la so_o by_o this_o dâ—â—inition_n the_o binomial_a line_n ce_fw-fr be_v a_o first_o binomial_a line_n definition_n a_o second_o binomial_a line_n be_v when_o the_o square_a of_o the_o great_a part_n exceed_v the_o square_a of_o the_o less_o part_n by_o the_o square_n of_o a_o line_n commensurable_a in_o length_n unto_o it_o and_o the_o less_o part_n be_v commensurable_a in_o length_n to_o the_o rational_a line_n first_o set_v as_o suppose_v ever_o the_o rational_a line_n let_v ce_fw-fr be_v a_o binomial_a line_n divide_v in_o the_o point_n d._n the_o square_n of_o who_o great_a part_n cd_o let_v exceed_v the_o square_n of_o the_o less_o part_n de_fw-fr by_o the_o square_n of_o the_o line_n fg_o which_o line_n â—g_z let_v be_v commensurable_a in_o length_n unto_o the_o line_n cd_o tâ—e_z great_a pâ—â—â—_n oâ—_n the_o binomial_a line_n and_o let_v also_o the_o line_n de_fw-fr the_o less_o part_n of_o the_o binomial_a line_n be_v commensurable_a in_o length_n to_o the_o rational_a line_n first_o set_v ab_fw-la so_o by_o this_o definition_n the_o binomial_a line_n ce_fw-fr be_v a_o second_o binomial_a line_n â—â—â—â—â—â—ition_n a_o three_o binomial_a
to_o the_o line_n be._n but_o as_o the_o line_n fb_o be_v to_o the_o line_n bg_o so_o by_o the_o 1._o of_o the_o six_o be_v the_o parallelogram_n ab_fw-la which_o be_v the_o square_a of_o the_o line_n db_o to_o the_o parallelogram_n dg_o and_o as_o the_o line_n db_o be_v to_o the_o line_n be_v so_o be_v the_o same_o parallelogram_n dg_o to_o the_o parallelogram_n bc_o which_o be_v the_o square_a of_o the_o line_n be._n wherefore_o as_o the_o square_a ab_fw-la be_v to_o the_o parallelogram_n dg_o so_o be_v the_o same_o parallelogram_n dg_o to_o the_o square_a bc._n wherefore_o the_o parallelogram_n dg_o be_v the_o mean_a proportional_a between_o the_o square_n ab_fw-la and_o bc._n i_o say_v moreover_o that_o the_o parallelogram_n dc_o be_v the_o mean_a proportional_a between_o the_o square_n ac_fw-la and_o cb._n for_o for_o that_o as_o the_o line_n ad_fw-la be_v to_o the_o line_n dk_o so_o be_v the_o line_n kg_v to_o the_o line_n gc_o for_o they_o be_v each_o equal_a to_o each_o wherefore_o by_o composition_n by_o the_o 18._o of_o the_o five_o as_o the_o line_n ak_o be_v to_o the_o line_n kd_v so_o be_v the_o line_n kc_o to_o the_o line_n cg_o but_o as_o the_o line_n ak_o be_v to_o the_o line_n kd_v so_o be_v the_o square_a of_o the_o line_n ak_o which_o be_v the_o square_a ac_fw-la to_o the_o parallelogram_n contain_v under_o the_o line_n ak_o and_o kd_v which_o be_v the_o parallelogram_n cd_o and_o as_o the_o line_n kc_o be_v to_o the_o line_n cg_o so_o also_o be_v the_o parallelogram_n dc_o to_o the_o square_n of_o the_o line_n gc_o which_o be_v the_o square_a bc._n wherefore_o as_o the_o square_a ac_fw-la be_v to_o the_o parallelogram_n dc_o so_o be_v the_o parallelogram_n dc_o to_o the_o square_a bc._n wherefore_o the_o parallelogram_n dc_o be_v the_o mean_a proportional_a between_o the_o square_n ac_fw-la and_o bc_o which_o be_v require_v to_o be_v demonstrate_v ¶_o a_o assumpt_n magnitude_n that_o be_v mean_a proportionalls_n between_o the_o self_n same_o or_o equal_a magnitude_n be_v also_o equal_a the_o one_o to_o the_o other_o suppose_v that_o there_o be_v three_o magnitude_n a_o b_o c._n assumpt_n and_o as_o a_o be_v to_o b_o so_o let_v b_o be_v to_o c._n and_o likewise_o as_o the_o same_o magnitude_n a_o be_v to_o d_o so_o let_v d_o be_v to_o the_o same_o magnitude_n c._n then_o i_o say_v that_o b_o and_o d_o be_v equal_a the_o one_o the_o other_o for_o the_o proportion_n of_o a_o unto_o c_o be_v double_a to_o that_o proportion_n which_o a_o have_v to_o b_o by_o the_o 10._o definition_n of_o the_o five_o and_o likewise_o the_o self_n same_o proportion_n of_o a_o to_o c_o be_v by_o the_o same_o definition_n double_a to_o that_o proportion_n which_o a_o have_v to_o d._n but_o magnitude_n who_o equemultiplices_fw-la be_v either_o equal_a or_o the_o self_n same_o be_v also_o equal_a wherefore_o as_o a_o be_v to_o b_o so_o be_v a_o to_o d._n wherefore_o by_o the_o 9_o of_o the_o five_o b_o and_o d_o be_v equal_a the_o one_o to_o the_o other_o so_o shall_v if_o also_o be_v if_o there_o be_v other_o magnitude_n equal_a to_o a_o and_o c_o namely_o e_o and_o f_o between_o which_o let_v the_o magnitude_n d_o be_v the_o mean_a proportional_a ¶_o the_o 36._o theorem_a the_o 54._o proposition_n if_o a_o superficies_n be_v contain_v under_o a_o rational_a line_n &_o a_o first_o binomial_a line_n the_o line_n which_o contain_v in_o power_n that_o superficies_n be_v a_o irrational_a line_n &_o a_o binomial_a line_n composition_n svppose_v that_o the_o superficies_n abcd_o be_v contain_v under_o the_o rational_a line_n ab_fw-la and_o under_o a_o first_o binomial_a line_n ad._n then_o i_o say_v that_o the_o line_n which_o contain_v in_o power_n the_o superficies_n ac_fw-la be_v a_o irrational_a line_n and_o a_o binomial_a line_n construction_n for_o forasmuch_o as_o the_o line_n ad_fw-la be_v a_o first_o binomial_a line_n it_o be_v in_o one_o only_a point_n divide_v into_o his_o name_n by_o the_o 42._o of_o this_o ten_o let_v it_o be_v divide_v into_o his_o name_n in_o the_o point_n e._n and_o let_v ae_n be_v the_o great_a name_n now_o it_o be_v manifest_a that_o the_o line_n ae_n and_o ed_z be_v rational_a commensurable_a in_o power_n only_o and_o that_o the_o line_n ae_n be_v in_o power_n more_o than_o the_o line_n ed_z by_o the_o square_n of_o a_o line_n commensurable_a in_o length_n to_o the_o line_n ae_n and_o moreover_o that_o the_o line_n ae_n be_v commensurable_a in_o length_n to_o the_o rational_a line_n give_v ab_fw-la by_o the_o definition_n of_o a_o first_o binomial_a line_n set_v before_o the_o 48._o proposition_n of_o this_o ten_o divide_v by_o the_o 10._o of_o the_o first_o the_o line_n ed_z into_o two_o equal_a part_n in_o the_o point_n f._n and_o forasmuch_o as_o the_o line_n ae_n be_v in_o power_n more_o than_o the_o line_n ed_z by_o the_o square_n of_o a_o line_n commensurable_a in_o length_n unto_o the_o line_n ae_n therefore_o if_o upon_o the_o great_a line_n namely_o upon_o the_o line_n ae_n be_v apply_v a_o parallelogram_n equal_a to_o the_o four_o part_n of_o the_o square_n of_o the_o less_o line_n that_o be_v to_o the_o square_n of_o the_o line_n of_o &_o want_v in_o form_n by_o a_o square_a it_o shall_v divide_v the_o great_a line_n namely_o ae_n into_o two_o part_n còmmensurable_a in_o length_n the_o one_o to_o the_o other_o by_o the_o second_o part_n of_o the_o 17._o of_o the_o ten_o apply_v therefore_o upon_o the_o line_n ae_n a_o parallelogram_n equal_a to_o the_o square_n of_o the_o line_n of_o and_o want_v in_o form_n by_o a_o square_a by_o the_o 28._o of_o the_o six_o and_o let_v the_o same_o be_v that_o which_o be_v contain_v under_o the_o line_n agnostus_n and_o ge._n wherefore_o the_o line_n agnostus_n be_v commensurable_a in_o length_n to_o the_o line_n ge._n draw_v by_o the_o point_n g_o e_o and_o fletcher_n to_o either_o of_o these_o line_n ab_fw-la and_o dg_v these_o parallel_a line_n gh_o eke_o and_o fl_fw-mi by_o the_o 31._o of_o the_o first_o and_o by_o the_o 14._o of_o the_o second_o unto_o the_o parallelogram_n ah_o describe_v a_o equal_a square_a sn_a and_o unto_o the_o parallelogram_n gk_o describe_v by_o the_o same_o a_o equal_a square_a np._n and_o let_v these_o line_n mn_a &_o nx_o be_v so_o put_v that_o they_o both_o make_v one_o right_a line_n wherefore_o by_o the_o 14._o of_o the_o first_o the_o line_n rn_v and_o no_o make_v also_o both_o one_o right_a line_n demonstration_n make_v perfect_a the_o parallelogram_n sp._n wherefore_o the_o parallelogram_n sparke_n be_v a_o square_a by_o those_o thing_n which_o be_v demonstrate_v after_o the_o determination_n in_o the_o first_o assumpt_n go_v before_o and_o forasmuch_o as_o that_o which_o be_v contain_v under_o the_o line_n agnostus_n and_o ge_z be_v equal_a to_o the_o square_n of_o the_o line_n of_o by_o construction_n therefore_o as_o the_o line_n agnostus_n be_v to_o the_o of_o so_o be_v the_o line_n of_o to_o the_o line_n eglantine_n by_o the_o 14._o or_o 17._o of_o the_o six_o wherefore_o also_o by_o the_o 1._o of_o the_o six_o as_o the_o parallelogram_n ah_o be_v to_o the_o parallelogram_n el_fw-es so_o be_v the_o parallelogram_n el_fw-es to_o the_o parallelogram_n gk_o wherefore_o the_o parallelogram_n el_n be_v the_o mean_a proportional_a between_o the_o parallelogram_n ah_o and_o gk_o but_o the_o parallelogram_n ah_o be_v equal_a to_o the_o square_a sn_a and_o the_o parallelogram_n gk_o be_v equal_a to_o the_o square_a np_n by_o construction_n wherefore_o the_o parallelogram_n el_n be_v the_o mean_a proportional_a between_o the_o square_n sn_a and_o np_a by_o the_o 7._o of_o the_o five_o but_o by_o the_o first_o assumpt_n go_v before_o the_o parallelogram_n mr._n be_v the_o mean_a proportional_a between_o the_o square_n sn_a and_o np._n wherefore_o the_o parallelogram_n mr._n be_v equal_a to_o the_o parallelogram_n el_fw-es by_o the_o last_o assumpt_v go_v before_o but_o the_o parallelogram_n mr._n be_v equal_a to_o the_o parallelogram_n ox_n by_o the_o 43._o of_o the_o first_o and_o the_o parallelogram_n el_n be_v equal_a to_o the_o parallelogram_n fc_o by_o construction_n and_o by_o the_o first_o of_o the_o six_o wherefore_o the_o whole_a parallelogram_n ec_o be_v equal_a to_o the_o two_o parallelogram_n mr._n &_o ox_n and_o the_o parallelogram_n ah_o and_o gk_o be_v equal_a to_o the_o square_n sn_a and_o np_a by_o construction_n wherefore_o the_o whole_a parallelogram_n ac_fw-la be_v equal_a to_o the_o whole_a square_a sparke_n that_o be_v to_o the_o square_n of_o the_o line_n mx_o conclude_v wherefore_o the_o line_n mx_o contain_v in_o power_n the_o parallelogram_n ac_fw-la i_o say_v moreover_o that_o the_o line_n mx_o be_v a_o binomial_a line_n for_o forasmuch_o as_o by_o
the_o 17._o of_o the_o ten_o the_o line_n agnostus_n be_v commensurable_a in_o length_n to_o the_o line_n eglantine_n therefore_o by_o the_o 15._o of_o the_o ten_o the_o whole_a line_n ae_n be_v commensurable_a in_o length_n to_o either_o of_o thâ—se_a line_n agnostus_n and_o ge._n but_o by_o supposition_n the_o line_n ae_n be_v commensurable_a in_o length_n to_o the_o line_n ab_fw-la wherefore_o by_o the_o 12._o of_o the_o ten_o either_o of_o the_o line_n agnostus_n &_o ge_z be_v commensurable_a in_o length_n to_o the_o line_n ab_fw-la but_o the_o line_n ab_fw-la be_v rational_a wherefore_o either_o of_o these_o line_n agnostus_n and_o ge_z be_v rational_a wherefore_o by_o the_o 19_o of_o the_o ten_o either_o of_o these_o parallelogram_n ah_o and_o gk_o be_v rational_a wherefore_o by_o the_o first_o of_o the_o six_o and_o 10._o of_o the_o ten_o the_o parallelogram_n ah_o be_v commensurable_a to_o the_o parallelogram_n gk_o but_o the_o parallelogram_n ah_o be_v equal_a to_o the_o square_a sn_a conclude_v and_o the_o parallelogram_n gk_o be_v equal_a to_o the_o square_a np_n wherefore_o the_o square_n sn_a and_o np_a which_o be_v the_o square_n of_o the_o line_n mn_a and_o nx_o be_v rational_a and_o commensurable_a and_o forasmuch_o as_o by_o supposition_n the_o line_n ae_n be_v incommensurable_a in_o length_n to_o the_o line_n ed._n but_o the_o line_n ae_n be_v commensurable_a in_o length_n to_o the_o line_n ag._n and_o the_o line_n de_fw-fr be_v commensurable_a in_o length_n to_o the_o line_n of_o for_o it_o be_v double_a to_o it_o by_o construction_n wherefore_o by_o the_o 13._o of_o the_o ten_o the_o line_n agnostus_n be_v incommensurable_a in_o length_n to_o the_o line_n ef._n wherefore_o the_o parallelogram_n ah_o be_v incommensurable_a to_o the_o parallelogram_n el._n but_o the_o parallelogram_n ah_o be_v equal_a to_o the_o square_a sn_a and_o the_o parallelogram_n el_n be_v equal_a to_o the_o parallelogram_n mr._n wherefore_o the_o square_a sn_a be_v incommensurable_a to_o the_o parallelogram_n mr._n but_o as_o the_o square_a sn_a be_v to_o the_o parallelogram_n mr._n so_o be_v the_o line_n on_o to_o the_o line_n nr_n by_o the_o 1._o of_o the_o six_o conclude_v wherefore_o the_o line_n on_o be_v incommensurable_a to_o the_o line_n nr_n but_o the_o line_n on_o be_v equal_a to_o the_o line_n mn_o and_o the_o line_n nr_n to_o the_o line_n nx_o wherefore_o the_o line_n mn_o be_v incommensurable_a to_o the_o line_n nx_o and_o it_o be_v already_o prove_v that_o the_o square_n of_o the_o line_n mn_a and_o nx_o be_v rational_a and_o commensurable_a wherefore_o the_o line_n mn_a and_o nx_o be_v rational_a commensurable_a in_o power_n only_o conclusion_n wherefore_o the_o whole_a line_n mx_o be_v a_o binomial_a line_n and_o it_o contain_v in_o power_n the_o parallelogram_n ac_fw-la which_o be_v require_v to_o be_v prove_v ¶_o the_o 37._o theorem_a the_o 55._o proposition_n if_o a_o superficies_n be_v comprehend_v under_o a_o rational_a line_n and_o a_o second_o binomial_a line_n the_o line_n that_o contain_v in_o power_n that_o superficies_n be_v irrational_a and_o be_v a_o first_o bimedial_a line_n svppose_v that_o the_o superficies_n abcd_o be_v contain_v under_o a_o rational_a line_n ab_fw-la and_o under_o a_o second_o binomial_a line_n ad._n then_o i_o say_v that_o the_o line_n that_o contain_v in_o power_n the_o superficies_n ac_fw-la be_v a_o first_o bimedial_a line_n for_o forasmuch_o as_o ad_fw-la be_v a_o second_o binomial_a line_n it_o can_v in_o one_o only_a point_n be_v divide_v into_o his_o name_n by_o the_o 43._o of_o this_o ten_o let_v it_o therefore_o by_o supposition_n be_v divide_v into_o his_o name_n in_o the_o point_n e_o so_fw-mi that_o let_v ae_n be_v the_o great_a name_n wherefore_o the_o line_n ae_n and_o ed_z be_v rational_a commensurable_a in_o power_n only_o and_o the_o line_n ae_n be_v in_o power_n more_o than_o the_o line_n ed_z by_o the_o square_n of_o a_o line_n commensurable_a in_o length_n to_o ae_n and_o the_o less_o name_n namely_o ed_z be_v commensurable_a in_o length_n to_o the_o line_n ab_fw-la by_o the_o definition_n of_o a_o second_o binomial_a line_n set_v before_o the_o 48._o proposition_n of_o this_o ten_o divide_v the_o line_n ed_z by_o the_o ten_o of_o the_o first_o into_o two_o equal_a part_n in_o the_o point_n f._n and_o by_o the_o 28._o of_o the_o six_o upon_o the_o line_n ae_n apply_v a_o parallelogram_n equal_a to_o the_o square_n of_o the_o line_n of_o and_o want_v in_o figure_n by_o a_o square_a and_o let_v the_o same_o parallelogram_n be_v that_o which_o be_v contain_v under_o the_o line_n agnostus_n and_o ge._n wherefore_o by_o the_o second_o part_n of_o the_o 17._o of_o this_o ten_o the_o line_n agnostus_n be_v commensurable_a in_o length_n to_o the_o line_n ge._n and_o by_o the_o 31._o of_o the_o first_o by_o the_o point_n g_o e_o f_o draw_v unto_o the_o line_n ab_fw-la and_o cd_o these_o parallel_a line_n gh_o eke_o &_o fl._n and_o by_o the_o 14._o of_o the_o second_o unto_o the_o parallelogram_n ah_o describe_v a_o equal_a square_a sn_a and_o to_o the_o parallelogram_n gk_o describe_v a_o equal_a square_a np_n and_o let_v the_o line_n mn_a &_o nx_o be_v so_o put_v that_o they_o both_o make_v one_o right_a line_n wherefore_o by_o the_o 14._o of_o the_o first_o the_o line_n also_o rn_v and_o no_o make_v both_o one_o right_a line_n demonstration_n make_v perfect_a the_o parallelogram_n sp._n now_o it_o be_v manifest_a by_o that_o which_o have_v be_v demonstrate_v in_o the_o proposition_n next_o go_v before_o that_o the_o parallelogâ—ame_n mr._n be_v the_o mean_a proportional_a between_o the_o square_n sn_a and_o np_a and_o be_v equal_a to_o the_o parallelogram_n el_fw-es and_o that_o the_o line_n mx_o contain_v in_o power_n the_o superficies_n ac_fw-la now_o rest_v to_o prove_v that_o the_o line_n mx_o be_v a_o first_o bimedial_a line_n forasmuch_o as_o the_o line_n ae_n be_v incommensurable_a in_o length_n to_o the_o line_n ed_z and_o the_o line_n ed_z be_v commensurable_a in_o length_n to_o the_o line_n ab_fw-la therefore_o by_o the_o 13._o of_o the_o ten_o the_o line_n ae_n be_v incommensurable_a in_o length_n to_o the_o line_n ab_fw-la and_o forasmuch_o as_o the_o line_n agnostus_n be_v commensurable_a in_o length_n to_o the_o line_n ge_z therefore_o the_o whole_a line_n ae_n be_v by_o the_o 15._o of_o the_o ten_o commensurable_a in_o length_n to_o either_o of_o these_o line_n agnostus_n and_o ge._n but_o the_o line_n ae_n be_v rational_a wherefore_o either_o of_o these_o line_n agnostus_n and_o ge_z be_v rational_a and_o forasmuch_o as_o the_o line_n ae_n be_v incommensurable_a in_o length_n to_o the_o line_n ab_fw-la but_o the_o line_n ae_n be_v commensurable_a in_o length_n to_o either_o of_o these_o line_n agnostus_n and_o ge_z wherefore_o either_o of_o the_o line_n agnostus_n and_o ge_z be_v incommensurable_a in_o length_n to_o the_o line_n ab_fw-la by_o the_o 13._o of_o the_o ten_o wherefore_o the_o line_n ab_fw-la agnostus_n and_o ge_z be_v rational_a commensurable_a in_o power_n only_o wherefore_o by_o the_o 21._o of_o the_o ten_o either_o of_o these_o parallelogram_n ah_o and_o gk_o be_v a_o medial_a superficies_n wherefore_o also_o either_o of_o these_o square_n sn_a and_o np_n be_v a_o medial_a superficies_n by_o the_o corollary_n of_o the_o 23._o of_o the_o ten_o wherefore_o the_o line_n mn_a &_o nx_o be_v medial_a line_n by_o the_o 21._o of_o this_o ten_o conclude_v and_o forasmuch_o as_o the_o line_n agnostus_n be_v commensurable_a in_o length_n to_o the_o line_n ge_z therefore_o by_o the_o 1._o of_o the_o six_o and_o 11._o of_o the_o ten_o the_o parallelogram_n ah_o be_v commensurable_a to_o the_o parallelogram_n gk_o that_o be_v the_o square_a sn_a to_o the_o square_a np_n that_o be_v the_o square_a of_o the_o line_n mn_v to_o the_o square_n of_o the_o line_n nx_o wherefore_o the_o line_n mn_a and_o nx_o be_v medial_n commensurable_a in_o power_n conclude_v and_o forasmuch_o as_o the_o line_n ae_n be_v incommensurable_a in_o length_n to_o the_o line_n ed_z but_o the_o line_n ae_n be_v commensurable_a in_o length_n to_o the_o line_n agnostus_n and_o the_o line_n ed_z be_v commensurable_a in_o length_n to_o the_o line_n eâ—_n therefore_o by_o the_o 13._o of_o the_o ten_o the_o line_n agnostus_n be_v incommensurable_a in_o length_n to_o the_o line_n eâ—_n wherefore_o by_o the_o â—_o of_o the_o six_o and_o 11._o of_o the_o ten_o the_o parallelogram_n ah_o be_v incommensurable_a to_o the_o parallelogram_n el_fw-es that_o be_v thâ—_z square_v sn_a to_o the_o parallelogram_n mr._n that_o be_v the_o line_n on_o be_v incommensurable_a to_o the_o line_n nr_n that_o be_v the_o line_n mn_v to_o the_o line_n nx_o and_o it_o be_v prove_v that_o the_o line_n mn_a and_o nx_o be_v medial_a line_n commensurablâ—_n in_o power_n concludeâ—_n
wherefore_o the_o line_n mn_a and_o nx_o be_v medial_a line_n commensurable_a in_o power_n only_o now_o i_o say_v moreover_o that_o they_o comprehend_v a_o rational_a superficies_n for_o forasmuch_o as_o by_o supposition_n the_o line_n de_fw-fr be_v commensurable_a in_o length_n to_o either_o of_o these_o line_n ab_fw-la and_o of_o therefore_o the_o line_n fe_o be_v commensurable_a in_o length_n to_o the_o line_n eke_o which_o be_v equal_a to_o the_o line_n ab_fw-la by_o thâ—_z 12._o of_o the_o ten_o and_o either_o of_o these_o line_n of_o and_o eke_o be_v a_o rational_a line_n wherefore_o the_o parallelogram_n el_fw-es that_o be_v the_o parallelogram_n mr._n be_v a_o rational_a superficies_n by_o the_o 19_o of_o the_o ten_o conclude_v but_o the_o parallelogram_n mr._n be_v that_o which_o be_v contain_v under_o the_o line_n mn_a and_o nx_o but_o if_o two_o medial_a line_n commensurable_a in_o power_n only_o and_o comprehend_v a_o rational_a superficies_n be_v add_v together_o the_o whole_a line_n be_v irrational_a and_o be_v call_v a_o first_o bimediall_n by_o the_o 37._o of_o the_o ten_o conclusion_n wherefore_o the_o line_n mx_o be_v a_o first_o bimedial_a line_n which_o be_v require_v to_o be_v demonstrate_v ¶_o the_o 38._o theorem_a the_o 56._o proposition_n if_o a_o superficies_n be_v contain_v under_o a_o rational_a line_n and_o a_o three_o binomial_a line_n the_o line_n that_o contain_v in_o power_n that_o superficies_n be_v irrational_a and_o be_v a_o second_o bimedial_a line_n svppose_v that_o the_o superficies_n abcd_o be_v comprehend_v under_o the_o rational_a line_n ab_fw-la and_o a_o three_o binomial_a line_n ad_fw-la and_o let_v the_o line_n ad_fw-la be_v suppose_v to_o be_v divide_v into_o his_o name_n in_o thâ—_z point_n e_o of_o which_o let_v ae_n be_v the_o great_a name_n then_o i_o say_v that_o the_o line_n that_o contain_v in_o power_n the_o superficies_n ac_fw-la be_v irrational_a and_o be_v a_o second_o bimedial_a line_n let_v the_o same_o construction_n of_o the_o figure_n be_v in_o this_o that_o be_v in_o the_o two_o proposition_n nexâ—_n go_v before_o and_o now_o forasmuch_o as_o the_o line_n ad_fw-la be_v a_o three_o binomial_a line_n therefore_o these_o line_n ae_n and_o ed_z be_v rational_a commensurable_a in_o power_n only_o demonstration_n and_o the_o line_n ae_n be_v in_o power_n more_o they_o the_o line_n ed_z by_o the_o square_n of_o a_o line_n commensurable_a in_o length_n to_o the_o line_n ae_n and_o neither_o of_o the_o line_n ae_n nor_o ed_z be_v commensurable_a in_o length_n to_o the_o line_n ab_fw-la by_o the_o definition_n of_o a_o three_o binomial_a line_n set_v before_o the_o 48._o proposition_n as_o in_o the_o former_a proposition_n it_o be_v demonstrate_v so_o also_o may_v it_o in_o this_o proposition_n be_v prove_v that_o the_o line_n mx_o contain_v in_o power_n the_o superficies_n ac_fw-la and_o that_o the_o line_n mn_a and_o nx_o be_v medial_a line_n commensurable_a in_o power_n only_o wherefore_o the_o line_n mx_o be_v a_o bimedial_a line_n now_o rest_v to_o prove_v that_o it_o be_v a_o second_o bimedial_a line_n forasmuch_o as_o the_o line_n de_fw-fr be_v by_o supposition_n incommensurable_a in_o length_n to_o the_o line_n ab_fw-la that_o be_v to_o the_o line_n eke_o but_o the_o line_n ed_z be_v commensurable_a in_o length_n to_o the_o line_n ef._n wherefore_o by_o the_o 13._o of_o the_o ten_o the_o line_n of_o be_v incommensurable_a in_o length_n to_o the_o line_n eke_o and_o the_o line_n fe_o and_o eke_o be_v rational_a for_o by_o supposition_n the_o line_n ed_z be_v rational_a unto_o which_o the_o line_n fe_o be_v commensurable_a wherefore_o the_o line_n fe_o and_o eke_o be_v rational_a line_n commensurable_a in_o power_n only_o wherefore_o by_o the_o 21._o of_o the_o ten_o the_o parallelogram_n el_fw-es that_o be_v the_o parallelogram_n mr._n which_o be_v contain_v under_o the_o line_n mn_a and_o nx_o be_v a_o medial_a superficies_n wherefore_o that_o which_o be_v contain_v under_o the_o line_n mn_a and_o nx_o be_v a_o medial_a superficies_n wherefore_o the_o line_n mx_o be_v a_o second_o bimedial_a line_n by_o the_o 38._o proposition_n and_o definition_n annex_v thereto_o which_o be_v require_v to_o be_v prove_v ¶_o the_o 39_o theorem_a the_o 57_o proposition_n if_o a_o superficies_n be_v contain_v under_o a_o rational_a line_n and_o a_o four_o binomial_a line_n the_o line_n which_o contain_v in_o power_n that_o superficies_n be_v irrational_a and_o be_v a_o great_a line_n svppose_v that_o the_o superficies_n ac_fw-la be_v comprehend_v under_o a_o rational_a line_n ab_fw-la and_o a_o four_o binomial_a line_n ad_fw-la &_o let_v the_o binomial_a line_n ad_fw-la be_v suppose_v to_o be_v divide_v into_o his_o name_n in_o the_o point_n e_o so_fw-mi that_o let_v the_o line_n ae_n be_v the_o great_a name_n then_o i_o say_v that_o the_o line_n which_o contain_v in_o power_n the_o superficies_n ac_fw-la be_v irrational_a and_o be_v a_o great_a line_n construction_n for_o forasmuch_o as_o the_o line_n ad_fw-la be_v a_o four_o binomial_a line_n therefore_o the_o line_n ae_n and_o ed_z be_v rational_a commensurable_a in_o power_n only_o and_o the_o line_n ae_n be_v in_o power_n more_o than_o the_o line_n ed_z by_o the_o square_n of_o a_o line_n incommensurable_a in_o length_n to_o ae_n and_o the_o line_n ae_n be_v commensurable_a in_o length_n to_o the_o line_n ab_fw-la divide_v by_o the_o 10._o of_o the_o first_o the_o line_n de_fw-fr into_o two_o equal_a part_n in_o the_o point_n f._n and_o upon_o the_o line_n ae_n apply_v a_o parallelogram_n equal_a to_o the_o square_n of_o of_o and_o want_v in_o figure_n by_o a_o square_n and_o let_v the_o same_o parallelogram_n be_v that_o which_o be_v contain_v under_o the_o line_n agnostus_n &_o ge._n wherefore_o by_o the_o second_o part_n of_o the_o 18._o of_o the_o ten_o the_o line_n agnostus_n be_v incommensurable_a in_o length_n to_o the_o line_n eglantine_n draw_v unto_o the_o line_n ab_fw-la by_o the_o point_n g_o e_o f_o parallel_a line_n gh_o eke_o and_o pl_z and_o let_v the_o rest_n of_o the_o construction_n be_v as_o it_o be_v in_o the_o three_o former_a proposition_n demonstration_n now_o it_o be_v manifest_a that_o the_o line_n mx_o contain_v in_o power_n the_o superficies_n ac_fw-la now_o rest_v to_o prove_v that_o the_o line_n mx_o be_v a_o irrational_a line_n and_o a_o great_a line_n forasmuch_o as_o the_o line_n agnostus_n be_v incommensurable_a in_o length_n to_o the_o line_n eglantine_n therefore_o by_o the_o 1._o of_o the_o six_o and_o 11._o of_o the_o ten_o the_o parallelogram_n ah_o be_v incommensurable_a to_o the_o parallelogram_n gk_o that_o be_v the_o square_a sn_a to_o the_o square_a np._n wherefore_o the_o line_n mn_a and_o nk_n be_v incommensurable_a in_o power_n and_o forasmuch_o 〈◊〉_d the_o line_n ae_n be_v commensurable_a in_o length_n to_o the_o rational_a line_n aâ—_z therefore_o the_o parallelogram_n ak_o be_v rational_a and_o it_o be_v equal_a to_o the_o square_n of_o the_o line_n mn_a and_o nx_o wherefore_o that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n mn_a and_o nx_o add_v together_o be_v rational_a and_o forasmuch_o as_o the_o line_n ed_z be_v incommensurable_a in_o length_n to_o the_o line_n ab_fw-la that_o be_v to_o the_o line_n eke_o but_o the_o line_n ed_z be_v commensurable_a in_o length_n to_o the_o line_n of_o therefore_o the_o line_n of_o be_v incommensurable_a in_o length_n to_o the_o line_n eke_o wherefore_o the_o line_n eke_o and_o of_o be_v rational_a commensurable_a in_o power_n only_o wherefore_o by_o the_o 21._o of_o the_o ten_o the_o parallelogram_n le_fw-fr that_o be_v the_o parallelogram_n mr._n be_v medial_a and_o the_o parallelogram_n mr._n be_v that_o which_o be_v contain_v under_o the_o line_n mn_v and_o nx_o wherefore_o that_o which_o be_v contain_v under_o the_o line_n mn_a and_o nx_o be_v medial_a and_o that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n mn_a &_o nx_o be_v prove_v to_o be_v rational_a &_o the_o line_n mn_o be_v demonstrate_v to_o be_v incommensurable_a in_o power_n to_o the_o line_n nx_o but_o if_o two_o line_n incommensurable_a in_o power_n be_v add_v together_o have_v that_o which_o be_v make_v of_o the_o square_n of_o they_o add_v together_o rational_a &_o that_o which_o be_v under_o they_o medial_a the_o whole_a line_n be_v irrational_a and_o be_v call_v a_o great_a line_n by_o the_o 39_o of_o the_o ten_o wherefore_o the_o line_n mx_o be_v irrational_a and_o be_v a_o great_a line_n and_o it_o contain_v in_o power_n the_o superficies_n ac_fw-la which_o be_v require_v to_o be_v demonstrate_v ¶_o the_o 40._o theorem_a the_o 58._o proposition_n if_o a_o superficies_n be_v contain_v under_o a_o rational_a line_n and_o a_o five_o binomial_a line_n the_o line_n which_o contain_v in_o power_n that_o superficies_n be_v
irrational_a and_o be_v a_o line_n contain_v in_o power_n a_o rational_a and_o a_o medial_a superficies_n svppose_v that_o the_o superficies_n agnostus_n be_v contain_v under_o the_o rational_a line_n aâ—_z and_o under_o a_o five_o binomial_a line_n addend_n let_v the_o same_o linâ—_n ad_fw-la be_v suppose_v to_o be_v divide_v into_o his_o name_n in_o the_o point_n e_o so_fw-mi that_o let_v the_o line_n ae_n be_v the_o great_a name_n then_o i_o say_v that_o the_o line_n which_o contain_v in_o power_n the_o superficies_n ac_fw-la be_v irrational_a and_o be_v a_o line_n contain_v in_o power_n a_o rational_a and_o a_o medial_a superficies_n demonstration_n let_v the_o self_n same_o construction_n be_v in_o this_o that_o be_v in_o the_o four_o proposition_n next_o go_v before_o and_o it_o be_v manifest_a that_o the_o line_n mx_o contain_v in_o power_n the_o superficiâ—â—_n ag._n now_o test_v to_o prove_v that_o the_o line_n mx_o be_v a_o line_n contain_v in_o power_n a_o rational_a &_o a_o medial_a superficies_n forasmuch_o as_o the_o line_n agnostus_n be_v incommensurable_a in_o length_n to_o the_o line_n ge_z therefore_o by_o the_o 1._o of_o the_o six_o and_o 10._o of_o the_o ten_o the_o parallelogram_n ah_o be_v incommensurable_a to_o the_o parallelogram_n he_o that_o be_v the_o square_a of_o the_o line_n mn_v to_o the_o square_n of_o the_o line_n nx_o wherefore_o the_o line_n mn_a and_o nx_o be_v incommensurable_a in_o power_n and_o forasmuch_o as_o the_o line_n ad_fw-la iâ—_z a_o fifâ—_n binomial_a line_n and_o his_o less_o name_n or_o part_n be_v the_o line_n ed_z therefore_o the_o line_n ed_z be_v commensurable_a in_o length_n to_o the_o line_n ab_fw-la but_o the_o line_n ae_n be_v incommensurable_a in_o length_n to_o the_o line_n ed._n wherefore_o by_o the_o 13._o of_o the_o ten_o the_o line_n ab_fw-la be_v incommensurable_a in_o length_n to_o the_o line_n ae_n wherefore_o the_o line_n ab_fw-la and_o ae_n be_v rational_a commensurable_a in_o power_n only_o wherefore_o by_o the_o 21._o of_o the_o ten_o the_o parallelogram_n ak_o be_v medial_a that_o be_v that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n mn_a &_o nx_o add_v together_o and_o forasmuch_o as_o the_o line_n de_fw-fr be_v commensurable_a in_o length_n to_o the_o line_n ab_fw-la that_o be_v to_o the_o line_n eke_o but_o the_o line_n de_fw-fr be_v commensurable_a in_o length_n to_o the_o line_n of_o wherefore_o by_o the_o 12._o of_o the_o ten_o the_o line_n of_o be_v also_o commensurable_a in_o length_n to_o the_o line_n eke_o and_o the_o line_n eke_o be_v rational_a wherefore_o by_o the_o 19_o of_o the_o ten_o the_o parallelogram_n el_fw-es that_o be_v the_o parallelogram_n mr._n which_o be_v contain_v under_o the_o line_n mn_a and_o nx_o be_v rational_a wherefore_o the_o line_n mn_a and_o nx_o be_v incommensurable_a in_o power_n have_v that_o which_o be_v compose_v of_o the_o square_n of_o they_o add_v together_o medial_a and_o that_o which_o be_v contain_v under_o they_o rational_a wherefore_o by_o the_o 40._o of_o the_o ten_o the_o whole_a line_n mx_o be_v a_o line_n contain_v in_o power_n a_o rational_a and_o a_o medial_a superficies_n and_o it_o contain_v in_o power_n the_o superficies_n ac_fw-la which_o be_v require_v to_o be_v prove_v ¶_o the_o 41._o theorem_a the_o 59_o proposition_n if_o a_o superficies_n be_v contain_v under_o a_o rational_a line_n and_o a_o six_o binomial_a line_n the_o line_n which_o contain_v in_o power_n that_o superficies_n be_v irrational_a &_o be_v call_v a_o line_n contain_v in_o power_n two_o medial_n svppose_v that_o the_o superficies_n abcd_o be_v contain_v under_o the_o rational_a line_n ab_fw-la and_o under_o a_o six_o binomial_a line_n ad_fw-la and_o let_v the_o line_n ad_fw-la be_v suppose_v to_o be_v divide_v â—â—to_o his_o name_n in_o the_o point_n e_o so_fw-mi that_o let_v the_o line_n ae_n be_v the_o great_a name_n then_o i_o say_v that_o the_o line_n that_o contain_v in_o power_n the_o superficies_n ac_fw-la be_v irrational_a and_o be_v a_o line_n contain_v in_o power_n two_o medial_n demonstration_n let_v the_o self_n same_o construction_n be_v in_o this_o that_o be_v in_o the_o former_a proposition_n now_o it_o be_v manifest_a that_o the_o line_n mx_o contain_v in_o power_n the_o superficies_n ac_fw-la and_o that_o the_o line_n mn_o be_v incommensurable_a in_o power_n to_o the_o line_n nx_o and_o forasmuch_o as_o the_o line_n ae_n be_v incommensurable_a in_o length_n to_o the_o line_n ab_fw-la therefore_o the_o line_n ae_n and_o ab_fw-la be_v rational_a commensurable_a in_o power_n only_o wherefore_o by_o the_o 〈◊〉_d of_o the_o ten_o the_o parallelogrammâ—_n ak_o that_o be_v that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n mn_a and_o nx_o add_v together_o be_v medial_a again_o forasmuch_o as_o the_o line_n ed_z be_v incommensurable_a in_o length_n to_o the_o line_n ab_fw-la therefore_o also_o the_o line_n of_o be_v incommensurable_a in_o length_n to_o the_o line_n eke_o wherefore_o the_o line_n of_o and_o eke_o be_v rational_a commensurable_a in_o power_n only_o wherefore_o the_o parallelogram_n el_fw-es that_o be_v the_o parallelogram_n mr._n which_o be_v contain_v under_o the_o line_n mn_a and_o nx_o be_v medial_a and_o forasmuch_o as_o the_o line_n ae_n be_v incommensurable_a in_o length_n to_o the_o line_n of_o therefore_o the_o parallelogram_n ak_o be_v also_o incommensurable_a to_o the_o parallelogram_n el_fw-es by_o the_o first_o of_o the_o six_o and_o 100l_n of_o the_o ten_o but_o the_o parallelogram_n ak_o be_v equal_a to_o that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n mn_a and_o nx_o add_v together_o and_o the_o parallelogram_n el_n be_v equal_a to_o that_o which_o be_v contain_v under_o the_o line_n mn_a and_o nx_o wherefore_o that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n mn_a and_o nigh_o add_v together_o be_v incommensurable_a to_o that_o which_o be_v contain_v under_o the_o lâ—nes_n mn_n and_o nxâ—_n and_o eâ—â—her_n of_o they_o name_o that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n mn_a and_o nx_o add_v together_o and_o that_o which_o be_v contain_v uâ—der_v the_o line_n mn_a and_o nâ—_z be_v prove_v medial_a and_o the_o line_n mn_a and_o nx_o be_v prove_v incommensurable_a in_o power_n wherefore_o by_o the_o 41._o of_o the_o ten_o the_o whole_a line_n mx_o be_v a_o line_n contain_v in_o power_n two_o medial_n and_o it_o contain_v in_o power_n the_o superfice_n ac_fw-la which_o be_v require_v to_o be_v demonstrate_v a_o assumpt_n if_o a_o right_a line_n be_v divide_v into_o two_o unequal_a part_n the_o square_n which_o be_v make_v of_o the_o unequal_a part_n be_v great_a than_o the_o rectangle_n parallelogram_n contain_v under_o the_o unequal_a part_n twice_o suppose_v that_o ab_fw-la be_v a_o right_a line_n and_o let_v it_o be_v 〈…〉_o point_fw-fr c._n and_o let_v the_o line_n ac_fw-la be_v the_o great_a part_n 〈…〉_o and_o â—b_n be_v great_a they_o that_o which_o be_v contain_v under_o the_o line_n aâ—_z and_o cb_o twice_o dâ—â—idâ—_n by_o the_o 10._o of_o the_o first_o the_o line_n ab_fw-la into_o two_o equal_a part_n in_o the_o point_n d._n now_o forasmuch_o as_o the_o right_a line_n ab_fw-la be_v divide_v into_o two_o equal_a parteâ—_n in_o the_o point_n d_o and_o into_o two_o unequal_a parteâ—_n in_o the_o point_n c_o therefore_o by_o the_o 5._o of_o the_o second_o that_o which_o be_v contain_v under_o the_o line_n 〈…〉_o line_n cd_o be_v equal_a to_o the_o square_n of_o the_o line_n ad._n 〈…〉_o the_o line_n ac_fw-la and_o cb_o omit_v the_o square_a of_o the_o line_n cd_o be_v less_o than_o the_o square_a of_o the_o ad_fw-la by_o the_o 9_o common_a sentence_n and_o the_o seven_o of_o the_o five_o wherefore_o that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o twice_o be_v less_o than_o the_o double_a of_o the_o square_n of_o the_o line_n ad_fw-la that_o be_v them_z twice_o the_o square_a of_o the_o line_n ad_fw-la by_o place_n alternate_a proportion_n and_o the_o 14._o of_o the_o five_o but_o the_o square_n of_o the_o line_n ac_fw-la and_o cb_o be_v double_a to_o the_o square_n of_o the_o line_n ad_fw-la and_o dc_o by_o the_o 9_o of_o the_o second_o therefore_o the_o square_n of_o ac_fw-la and_o cb_o be_v more_o than_o double_a to_o the_o square_n of_o ad_fw-la alone_o leave_v out_o the_o square_n of_o dc_o by_o the_o 8._o of_o the_o five_o but_o the_o parallelogram_n contain_v under_o the_o line_n ac_fw-la and_o â—b_v twice_o be_v prove_v less_o they_o the_o double_a of_o the_o square_n of_o the_o line_n ad._n therefore_o the_o same_o parallelogram_n contain_v under_o the_o line_n ac_fw-la and_o cb_o twice_o be_v much_o less_o than_o the_o square_n of_o the_o line_n ac_fw-la and_o cb._n if_o a_o right_a line_n
the_o definition_n of_o a_o first_o binomial_a line_n seâ—_z before_o the_o 48._o proposition_n of_o this_o book_n the_o line_n dg_o be_v a_o first_o binomial_a line_n which_o be_v require_v to_o be_v prove_v this_o proposition_n and_o the_o five_o follow_v be_v the_o converse_n of_o the_o six_o former_a proposition_n ¶_o the_o 43._o theorem_a the_o 61._o proposition_n the_o square_a of_o a_o first_o bimedial_a line_n apply_v to_o a_o rational_a line_n make_v the_o breadth_n or_o other_o side_n a_o second_o binomial_a line_n svppose_v that_o the_o line_n ab_fw-la be_v a_o first_o bimedial_a line_n and_o let_v it_o be_v suppose_v to_o be_v divide_v into_o his_o part_n in_o the_o point_n c_o of_o which_o let_v ac_fw-la be_v the_o great_a part_n take_v also_o a_o rational_a line_n de_fw-fr and_o by_o the_o 44._o of_o the_o first_o apply_v to_o the_o line_n de_fw-fr the_o parallelogram_n df_o equal_a to_o the_o square_n of_o the_o line_n ab_fw-la &_o make_v in_o breadth_n the_o line_n dg_o construction_n then_o i_o say_v that_o the_o line_n dg_o be_v a_o second_o binomial_a line_n let_v the_o same_o construction_n be_v in_o this_o that_o be_v in_o the_o proposition_n go_v before_o and_o forasmuch_o as_o the_o line_n ab_fw-la be_v a_o first_o bimedial_a line_n demonstration_n and_o be_v divide_v into_o his_o part_n in_o the_o point_n c_o therefore_o by_o the_o 37._o of_o the_o ten_o the_o line_n ac_fw-la and_o cb_o be_v medial_a commensurable_a in_o power_n only_o comprehend_v a_o rational_a superficies_n wherefore_o also_o the_o square_n of_o the_o line_n ac_fw-la and_o cb_o be_v medial_a wherefore_o the_o parallelogram_n dl_o be_v medial_a by_o the_o corollary_n of_o the_o 23._o of_o the_o ten_o and_o it_o be_v apply_v upon_o the_o rational_a line_n de._n wherefore_o by_o the_o 22._o of_o the_o ten_o the_o line_n md_o be_v rational_a and_o incommensurable_a in_o length_n to_o the_o line_n de._n again_o forasmuch_o as_o that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o twice_o be_v rational_a therefore_o also_o the_o parallelogram_n mf_n be_v rational_a and_o it_o be_v apply_v unto_o the_o rational_a line_n ml_o wherefore_o the_o line_n mg_o be_v rational_a and_o commensurable_a in_o length_n to_o the_o line_n ml_o that_o be_v to_o the_o line_n de_fw-fr by_o the_o 20._o of_o the_o ten_o wherefore_o the_o line_n dm_o be_v incommensurable_a in_o length_n to_o the_o line_n mg_o and_o they_o be_v both_o rational_a wherefore_o the_o line_n dm_o and_o mg_o be_v rational_a commensurable_a in_o power_n only_o wherefore_o the_o whole_a line_n dg_o be_v a_o binomial_a line_n line_n now_o rest_v to_o prove_v that_o it_o be_v a_o second_o binomial_a line_n forasmuch_o as_o the_o square_n of_o the_o line_n ac_fw-la and_o cb_o be_v great_a than_o that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o twice_o by_o the_o assumpt_n before_o the_o 60._o of_o this_o book_n therefore_o the_o parallelogram_n dl_o be_v great_a than_o the_o parallelogrrmme_a mf_n wherefore_o also_o by_o the_o first_o of_o the_o six_o the_o line_n dm_o be_v great_a than_o the_o line_n mg_o and_o forasmuch_o as_o the_o square_n of_o the_o line_n ac_fw-la be_v commensurable_a to_o the_o square_n of_o the_o line_n cb_o therefore_o the_o parallelogram_n dh_o be_v commensurable_a to_o the_o parallelogram_n kl_o wherefore_o also_o the_o line_n dk_o be_v commensurable_a in_o length_n to_o the_o line_n km_o and_o that_o which_o be_v contain_v under_o the_o line_n dk_o and_o km_o be_v equal_a to_o the_o square_n of_o the_o line_n mn_o that_o be_v to_o the_o four_o part_n of_o the_o square_n of_o the_o line_n mg_o wherefore_o by_o the_o 17._o of_o the_o ten_o the_o line_n dm_o be_v in_o power_n more_o than_o the_o line_n mg_o by_o the_o square_n of_o a_o line_n commensurable_a in_o length_n unto_o the_o line_n dm_o and_o the_o line_n mg_o be_v commensurable_a in_o length_n to_o the_o rational_a line_n put_v namely_o to_o de._n wherefore_o the_o line_n dg_o be_v a_o second_o binomial_a line_n which_o be_v require_v to_o be_v prove_v ¶_o the_o 44._o theorem_a the_o 62._o proposition_n the_o square_a of_o a_o second_o bimedial_a line_n apply_v unto_o a_o rational_a line_n make_v the_o breadth_n or_o other_o side_n thereof_o a_o three_o binomial_a line_n svppose_v that_o ab_fw-la be_v a_o second_o bimedial_a line_n and_o let_v ab_fw-la be_v suppose_v to_o be_v divide_v into_o his_o part_n in_o the_o point_n c_o so_fw-mi that_o let_v ac_fw-la be_v the_o great_a part_n and_o take_v a_o rational_a line_n de._n and_o by_o the_o 44._o of_o the_o first_o unto_o the_o line_n de_fw-fr apply_v the_o parallelogram_n df_o equal_a to_o the_o square_n of_o the_o line_n ab_fw-la and_o make_v in_o breadth_n the_o line_n dg_o then_o i_o say_v that_o the_o line_n dg_o be_v a_o three_o binomial_a line_n let_v the_o self_n same_o construction_n be_v in_o this_o that_o be_v in_o the_o proposition_n next_o go_v before_o and_o forasmuch_o as_o the_o line_n ab_fw-la be_v a_o second_o bimedial_a line_n construction_n and_o be_v divide_v into_o his_o part_n in_o the_o point_n c_o demonstration_n therefore_o by_o the_o 38._o of_o the_o ten_o the_o line_n ac_fw-la and_o cb_o be_v medial_n commensurable_a in_o power_n only_o comprehend_v a_o medial_a superficies_n wherefore_o †_o that_o which_o be_v make_v of_o the_o square_n of_o the_o line_n ac_fw-la and_o cb_o add_v together_o be_v medial_a and_o it_o be_v equal_a to_o the_o parallelogram_n dl_o by_o construction_n wherefore_o the_o parallelogram_n dl_o be_v medial_a and_o be_v apply_v unto_o the_o rational_a line_n de_fw-fr wherefore_o by_o the_o 22._o of_o the_o ten_o the_o line_n md_o be_v rational_a and_o incommensurable_a in_o length_n to_o the_o line_n de._n and_o by_o the_o like_a reason_n also_o *_o the_o line_n mg_o be_v rational_a and_o incommensurable_a in_o length_n to_o the_o line_n ml_o that_o be_v to_o the_o line_n de._n wherefore_o either_o of_o these_o line_n dm_o and_o mg_o be_v rational_a and_o incommensurable_a in_o length_n to_o the_o line_n de._n and_o forasmuch_o as_o the_o line_n ac_fw-la be_v incommensurable_a in_o length_n to_o the_o line_n cb_o but_o as_o the_o line_n ac_fw-la be_v to_o the_o line_n cb_o so_o by_o the_o assumpt_n go_v before_o the_o 22._o of_o the_o ten_o be_v the_o square_a of_o the_o line_n ac_fw-la to_o that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb._n wherefore_o the_o square_a of_o the_o line_n ac_fw-la be_v incâ—mmmensurable_a to_o that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb._n wherefore_o that_o ‡_o that_o which_o be_v make_v of_o the_o square_n of_o the_o line_n ac_fw-la and_o cb_o add_v together_o be_v incommensurable_a to_o that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o twice_o that_o be_v the_o parallelogram_n dl_o to_o the_o parallelogram_n mf_n wherefore_o by_o the_o first_o of_o the_o six_o and_o 10._o of_o the_o ten_o the_o line_n dm_o be_v incommensurable_a in_o length_n to_o the_o line_n mg_o and_o they_o be_v prove_v both_o rational_a line_n wherefore_o the_o whole_a line_n dg_o be_v a_o binomial_a line_n by_o the_o definition_n in_o the_o 36._o of_o the_o ten_o now_o rest_v to_o prove_v that_o it_o be_v a_o three_o binomial_a line_n as_o in_o the_o former_a proposition_n so_o also_o in_o this_o may_v we_o conclude_v that_o the_o line_n dm_o be_v great_a than_o the_o line_n mg_o and_o that_o the_o line_n dk_o be_v commensurable_a in_o length_n to_o the_o line_n km_o and_o that_o that_o which_o be_v contain_v under_o the_o line_n dk_o and_o km_o be_v equal_a to_o the_o square_n of_o the_o line_n mn_v wherefore_o the_o line_n dm_o be_v in_o power_n more_o than_o the_o line_n mg_o by_o the_o square_n of_o a_o line_n commensurable_a in_o length_n unto_o the_o line_n dm_o and_o neither_o of_o the_o line_n dm_o nor_o mg_o be_v commensurable_a in_o length_n to_o the_o rational_a line_n de._n wherefore_o by_o the_o definition_n of_o a_o three_o binomiâ—ll_v line_v the_o line_n dg_o be_v a_o three_o binomial_a line_n which_o be_v require_v to_o be_v prove_v ¶_o here_o follow_v certain_a annotation_n by_o m._n dee_n make_v upon_o three_o place_n in_o the_o demonstration_n which_o be_v not_o very_o evident_a to_o young_a beginner_n †_o the_o square_n of_o the_o line_n ac_fw-la and_o câ—_n be_v medial_n 〈◊〉_d iâ—_z teach_v after_o the_o 21â—_n of_o this_o ten_o and_o therefore_o forasmuch_o as_o they_o be_v by_o supposition_n commensurable_a the_o one_o to_o the_o other_o by_o the_o 15._o of_o the_o ten_o the_o compound_n of_o they_o both_o be_v commensurable_a to_o each_o part_n but_o the_o part_n be_v medial_n therefore_o by_o the_o corollary_n of_o the_o 23._o of_o the_o ten_o the_o compound_n shall_v be_v
medial_a â—_o for_o that_o mx_o be_v equal_a by_o construction_n to_o that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o which_o be_v prove_v medial_a therefore_o by_o the_o corollary_n of_o the_o 23._o of_o this_o ten_o mx_o be_v medial_a and_o therefore_o by_o the_o same_o corollary_n his_o double_a mf_n be_v medial_a and_o it_o be_v apply_v to_o a_o rational_a line_n ml_o be_v equal_a to_o dâ—_n therefore_o by_o the_o 22._o of_o the_o ten_o the_o line_n mg_o be_v rational_a and_o incommensurable_a in_o length_n to_o ml_o that_o be_v to_o de._n ‡_o because_o the_o compound_n of_o the_o two_o square_n of_o the_o line_n ac_fw-la and_o câ—_n be_v commensurable_a one_o to_o the_o other_o be_v also_o to_o either_o square_n by_o the_o 15._o commensurable_a therefore_o to_o the_o square_n of_o ac_fw-la but_o the_o square_n of_o ac_fw-la be_v prove_v incommensurable_a to_o that_o which_o be_v contain_v under_o ac_fw-la &_o cb_o once_o wherefore_o by_o the_o 13._o of_o the_o ten_o the_o compound_n of_o the_o two_o square_n of_o the_o line_n ac_fw-la and_o cb_o be_v incommensurable_a to_o that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o câ—_n once_o but_o to_o that_o which_o be_v twice_o contain_v under_o the_o same_o line_n ac_fw-la and_o cb_o the_o paralleloganrme_n once_o contain_v be_v commensurable_a for_o it_o be_v as_o 1._o be_v to_o 2._o therefore_o that_o which_o be_v make_v of_o the_o square_n of_o the_o line_n ac_fw-la and_o cb_o be_v incommensurable_a to_o the_o parallelogram_n contain_v under_o ac_fw-la and_o cb_o twice_o by_o the_o say_v 13._o of_o this_o ten_o ¶_o a_o corollary_n dee_n hereby_o it_o be_v evident_a that_o the_o square_n make_v of_o the_o two_o part_n of_o a_o second_o bimedial_a line_n compose_v be_v a_o compound_n medial_a and_o that_o the_o same_o compound_n be_v incommensurable_a to_o the_o parallelogram_n contain_v under_o the_o two_o part_n of_o the_o second_o bimedial_a line_n the_o proof_n hereof_o be_v in_o the_o first_o and_o three_o annotation_n here_o before_o annex_v ¶_o the_o 45._o theorem_a the_o 63._o proposition_n the_o square_a of_o a_o great_a line_n apply_v unto_o a_o rational_a line_n make_v the_o breadth_n or_o other_o side_n a_o four_o binomial_a line_n svppose_v that_o the_o line_n ab_fw-la be_v a_o great_a line_n and_o let_v it_o be_v suppose_v to_o be_v divide_v into_o his_o part_n in_o the_o point_n c_o so_fw-mi that_o let_v ac_fw-la be_v the_o great_a part_n and_o take_v a_o rational_a line_n de._n and_o by_o the_o 44._o of_o the_o first_o unto_o the_o line_n de_fw-fr apply_v the_o parallelogram_n df_o equal_a to_o the_o square_n of_o the_o line_n ab_fw-la and_o make_v in_o breadth_n the_o line_n dg_o then_o i_o say_v that_o the_o line_n dg_o be_v a_o four_o binomial_a line_n let_v the_o self_n same_o construction_n be_v in_o this_o that_o be_v in_o the_o former_a proposition_n construction_n and_o forasmuch_o as_o the_o line_n ab_fw-la be_v a_o great_a line_n &_o be_v divide_v into_o his_o part_n in_o the_o point_n c_o demonstration_n therefore_o the_o line_n ac_fw-la and_o cb_o be_v incommensurable_a in_o power_n have_v that_o which_o be_v make_v of_o the_o square_n of_o they_o add_v together_o rational_a and_o the_o parallelogram_n which_o be_v contain_v under_o they_o medial_a now_o forasmuch_o as_o that_o which_o be_v make_v of_o the_o square_n of_o the_o line_n ac_fw-la and_o cb_o add_v together_o be_v rational_a therefore_o the_o parallelogram_n dl_o be_v rational_a wherefore_o also_o the_o line_n md_o be_v rational_a and_o commensurable_a in_o length_n to_o the_o line_n de_fw-fr by_o the_o 20._o of_o this_o ten_o again_o forasmuch_o as_o that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o twice_o be_v medial_a that_o be_v the_o parallelogram_n mf_n and_o it_o be_v apply_v unto_o the_o rational_a line_n ml_o therefore_o by_o the_o â—2_o of_o the_o ten_o the_o line_n mg_o be_v rational_a and_o incommensurable_a in_o length_n to_o the_o line_n de_fw-fr therefore_o by_o the_o 13._o of_o the_o ten_o the_o line_n dm_o be_v incommensurable_a in_o length_n to_o the_o line_n mg_o wherefore_o the_o line_n dm_o and_o mg_o be_v rational_a commensurable_a in_o power_n only_o wherefore_o the_o whole_a line_n dg_o be_v a_o binomial_a line_n now_o rest_v to_o prove_v that_o it_o be_v also_o a_o four_o binomial_a line_n even_o as_o in_o the_o former_a proposition_n so_o also_o in_o this_o may_v we_o conclude_v that_o the_o line_n dm_o be_v great_a than_o the_o line_n mc_n and_o that_o that_o which_o be_v contain_v under_o the_o line_n dk_o and_o km_o be_v equal_a to_o the_o square_n of_o the_o line_n mn_v now_o forasmuch_o as_o the_o square_n of_o the_o line_n ac_fw-la be_v incommensurable_a to_o the_o square_n of_o the_o line_n cb_o therefore_o the_o parallelogram_n dh_o be_v incommensurable_a to_o the_o parallelogram_n kl_o wherefore_o by_o the_o 1._o of_o the_o six_o and_o 10._o of_o the_o ten_o the_o line_n dk_o be_v incommensurable_a in_o length_n to_o the_o line_n km_o but_o if_o there_o be_v two_o unequal_a right_a line_n and_o if_o upon_o the_o great_a be_v apply_v a_o parallelogram_n equal_a to_o the_o four_o part_n of_o the_o square_n make_v of_o the_o less_o and_o want_v in_o figure_n by_o a_o square_a and_o if_o also_o the_o parallelogram_n thus_o apply_v divide_v the_o line_n whereupon_o it_o be_v apply_v into_o part_n incommensurable_a in_o length_n the_o great_a line_n sâ—all_v be_v in_o power_n more_o than_o the_o less_o by_o the_o square_n of_o a_o line_n incommensurable_a in_o length_n to_o the_o great_a by_o the_o 18._o of_o the_o ten_o wherefore_o the_o line_n dm_o be_v in_o power_n more_o than_o the_o line_n mg_o by_o the_o square_n of_o a_o line_n incommensurable_a in_o length_n to_o dm_o and_o the_o line_n dm_o and_o mg_o be_v prove_v to_o be_v rational_a commensurable_a in_o power_n only_o and_o the_o line_n dm_o be_v commensurable_a in_o length_n to_o the_o rational_a line_n give_v de._n wherefore_o the_o line_n dg_o be_v a_o four_o binomial_a line_n which_o be_v require_v to_o be_v prove_v ¶_o the_o 46._o theorem_a the_o 64._o proposition_n the_o square_a of_o a_o line_n contain_v in_o power_n a_o rational_a and_o a_o medial_a superficies_n apply_v to_o a_o rational_a line_n make_v the_o breadth_n or_o other_o side_n a_o five_o binomial_a line_n svppose_v that_o the_o line_n ab_fw-la be_v a_o line_n contain_v in_o power_n a_o rational_a and_o a_o medial_a superficies_n and_o let_v it_o be_v suppose_v to_o be_v divide_v into_o his_o part_n in_o the_o point_n c_o so_fw-mi that_o let_v ac_fw-la be_v the_o great_a part_n and_o take_v a_o rational_a line_n de._n and_o by_o the_o 44._o of_o the_o first_o unto_o the_o line_n de_fw-fr apply_v the_o parallelogram_n df_o equal_a to_o the_o square_n of_o the_o line_n ab_fw-la and_o make_v in_o breadth_n the_o line_n dg_o then_o i_o say_v that_o the_o line_n dg_o be_v a_o five_o binomial_a line_n construction_n let_v the_o self_n same_o construction_n be_v in_o this_o that_o be_v in_o the_o former_a and_o forasmuch_o as_o ab_fw-la be_v a_o line_n contain_v in_o power_n a_o rational_a and_o a_o medial_a superficies_n demonstration_n and_o be_v divide_v into_o his_o part_n in_o the_o point_n c_o therefore_o the_o line_n ac_fw-la &_o cb_o be_v incommensurable_a in_o power_n have_v that_o which_o be_v make_v of_o the_o square_n of_o they_o add_v together_o medial_a and_o that_o which_o be_v contain_v under_o then_o rational_a now_o forasmuch_o as_o that_o which_o be_v make_v of_o the_o square_n of_o the_o line_n ac_fw-la and_o cb_o add_v together_o be_v medial_a therefore_o also_o the_o parallelogram_n dl_o be_v medial_a wherefore_o by_o the_o 22._o of_o the_o ten_o the_o line_n dm_o be_v rational_a and_o incommensurable_a in_o length_n to_o the_o line_n de._n again_o forasmuch_o as_o that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o twice_o that_o be_v the_o parallelogram_n mf_n be_v rational_a therefore_o by_o the_o 20._o the_o line_n mg_o be_v rational_a &_o commensurable_a in_o length_n to_o the_o line_n de._n wherefore_o by_o the_o 13._o of_o the_o ten_o the_o line_n dm_o be_v incommensurable_a in_o length_n to_o the_o line_n mg_o wherefore_o the_o line_n dm_o and_o mg_o be_v rational_a commensurable_a in_o power_n only_o wherefore_o the_o whole_a line_n dg_o be_v a_o binomial_a line_n i_o say_v moreover_o that_o it_o be_v a_o five_o binomiall_n for_o as_o in_o the_o former_a so_o also_o in_o this_o may_v it_o be_v prove_v that_o that_o which_o be_v contain_v under_o the_o line_n dk_o and_o km_o be_v equal_a to_o the_o square_n of_o mn_a the_o half_a of_o the_o less_o and_o that_o the_o line_n dk_o be_v incommensurable_a in_o length_n
line_n divide_v the_o line_n ab_fw-la into_o his_o part_n in_o the_o point_n e._n and_o let_v the_o rest_n of_o the_o construction_n be_v in_o this_o as_o it_o be_v in_o the_o former_a and_o for_o that_o as_o the_o line_n ab_fw-la be_v to_o the_o line_n cd_o so_o be_v the_o line_n ae_n to_o the_o line_n cf_o and_o the_o line_n ebb_n to_o the_o line_n fd_o therefore_o as_o the_o line_n ae_n be_v to_o the_o line_n cf_o so_o be_v the_o line_n ebb_n to_o the_o line_n fd_o but_o the_o line_n ab_fw-la be_v commensurable_a to_o the_o line_n cd_o wherefore_o also_o the_o line_n ae_n be_v commensurable_a to_o the_o line_n cf_o and_o likewise_o the_o line_n ebb_n to_o the_o line_n fd._n and_o for_o thâ—â—_n as_o the_o line_n ae_n be_v to_o the_o line_n cf_o so_o be_v the_o line_n ebb_n to_o the_o line_n fd_o therefore_o alternate_o as_o the_o line_n ae_n be_v to_o the_o line_n ebb_n so_o be_v the_o line_n cf_o to_o the_o line_n fd._n wherefore_o by_o the_o 22._o of_o the_o six_o as_o the_o square_n of_o the_o line_n ae_n be_v to_o the_o square_n of_o the_o line_n ebb_n so_o be_v the_o square_a of_o the_o line_n cf_o to_o the_o square_n of_o the_o line_n fd._n wherefore_o by_o composition_n by_o the_o 18._o of_o the_o five_o as_o that_o which_o be_v make_v of_o the_o square_n of_o the_o line_n aâ—_z and_o eâ—_n add_v together_o be_v to_o the_o square_n of_o the_o line_n ebb_n so_o be_v that_o which_o be_v make_v of_o the_o squareâ—_n of_o the_o lyneâ—_n câ—_n and_o fd_o add_v together_o to_o the_o square_n of_o the_o line_n fd._n wherefore_o by_o contrary_a proportion_n as_o the_o square_n of_o the_o line_n ebb_n be_v to_o that_o which_o be_v make_v of_o the_o square_n of_o the_o line_n ae_n and_o eâ—_n add_v together_o so_o be_v the_o square_a of_o the_o line_n fd_v to_o that_o which_o be_v make_v of_o the_o square_n of_o the_o line_n cf_o and_o fd_o add_v together_o wherefore_o alternate_o as_o the_o square_n of_o the_o line_n ebb_n be_v to_o the_o square_n of_o the_o line_n fd_o so_o be_v that_o which_o be_v make_v of_o the_o square_n 〈◊〉_d the_o lâ—nes_n ae_n and_o ebb_v add_v together_o to_o that_o which_o be_v make_v of_o the_o square_n of_o the_o line_n cf_o and_o fd_o add_v together_o but_o the_o square_n of_o the_o line_n ebb_n be_v commensurable_a to_o the_o square_n of_o the_o line_n fd_o for_o it_o have_v already_o be_v prove_v that_o the_o line_n ebb_v and_o fd_o be_v commensurable_a wherefore_o that_o which_o be_v make_v of_o the_o square_n of_o the_o line_n ae_n &_o ebb_n add_v together_o be_v commensurable_a to_o that_o which_o be_v make_v of_o the_o square_n of_o câ—_n &_o fd_o add_v together_o but_o that_o which_o be_v make_v of_o the_o square_n of_o the_o line_n ae_n and_o ebb_v add_v together_o be_v rational_a by_o supposition_n wherefore_o that_o which_o be_v make_v of_o the_o square_n of_o the_o line_n cf_o and_o fd_o add_v together_o be_v also_o rational_a and_o as_o the_o line_n ae_n be_v to_o the_o line_n ebb_n so_o be_v the_o line_n cf_o to_o the_o line_n fd_o but_o as_o the_o line_n ae_n be_v to_o the_o line_n ebb_n so_o be_v the_o square_a of_o the_o line_n a_o 〈…〉_o contain_v under_o the_o line_n ae_n and_o ebb_n therefore_o at_o the_o line_n cf_o be_v to_o the_o line_n fd_o so_o be_v the_o square_a of_o the_o line_n ae_n to_o the_o parallelogram_n contain_v under_o the_o line_n ae_n and_o ebb_n &_o as_o the_o line_n cf_o be_v to_o the_o line_n fd_o so_o be_v the_o square_a of_o the_o line_n cf_o to_o the_o parallelogram_n contain_v under_o the_o line_n â—f_a &_o fd._n wherefore_o as_o the_o square_n of_o the_o line_n ae_n be_v to_o the_o parallelogram_n conâ—â—â—â—ed_v under_o the_o line_n ae_n and_o ebb_n so_o be_v the_o square_a of_o the_o line_n cf_o to_o the_o parallelogram_n contain_v under_o the_o line_n cf_o and_o fd._n wherefore_o 〈◊〉_d â—s_v the_o square_a of_o the_o line_n ae_n be_v to_o the_o square_n of_o the_o line_n cf_o so_o be_v the_o parallelogram_n contain_v under_o the_o line_n ae_n and_o ebb_n to_o the_o parallelogram_n 〈◊〉_d undeâ—_n the_o line_n â—â—_o and_o â—â—_o but_o the_o square_n of_o the_o line_n ae_n be_v commensurable_a to_o the_o square_n of_o the_o line_n cf_o for_o it_o be_v already_o prâ—â—â—d_v that_o the_o line_n ae_n and_o cf_o be_v commensurable_a wherefore_o the_o parallelogram_n contain_v under_o the_o line_n ae_n and_o ebb_n be_v commensurable_a to_o the_o parallelogram_n contain_v under_o the_o line_n cf_o and_o fd._n but_o the_o parallelogram_n contain_v under_o the_o line_n ae_n and_o ebb_n be_v medial_a by_o supposition_n wherefore_o the_o parallelogram_n contain_v under_o the_o line_n cf_o and_o â—d_a also_o be_v medial_a and_o as_o it_o have_v already_o be_v prove_v as_o the_o line_n ae_n be_v to_o the_o line_n ebb_n so_o be_v the_o line_n cf_o to_o the_o line_n fd._n but_o the_o line_n ae_n be_v by_o supposition_n incommensurable_a in_o power_n to_o the_o line_n ebb_n wherefore_o by_o the_o 10._o of_o the_o ten_o the_o line_n cf_o be_v incommensurable_a in_o power_n to_o the_o line_n fd._n wherefore_o the_o line_n cf_o and_o fd_o be_v incommensurable_a in_o power_n have_v that_o which_o be_v make_v of_o the_o square_n of_o they_o add_v together_o rational_a and_o that_o which_o be_v contain_v under_o they_o medial_a wherefore_o the_o whole_a line_n cd_o be_v by_o the_o 39_o of_o the_o ten_o a_o great_a line_n wherefore_o a_o line_n commensurable_a to_o a_o great_a line_n be_v also_o a_o great_a line_n which_o be_v require_v to_o be_v demonstrate_v a_o other_o more_o brief_a demonstration_n of_o the_o same_o after_o campane_n suppose_v that_o a_o be_v a_o great_a line_n unto_o which_o let_v the_o line_n b_o be_v commensurable_a either_o in_o length_n and_o power_n or_o in_o power_n only_o campane_n and_o take_v a_o rational_a line_n cd_o and_o upon_o it_o apply_v the_o superficies_n câ—_n equal_a to_o the_o square_n of_o the_o line_n a_o and_o also_o upon_o the_o line_n fe_o which_o be_v equal_a to_o the_o rational_a line_n cd_o apply_v the_o parallelogram_n fg_o equal_a to_o the_o square_n of_o the_o line_n b._n and_o forasmuch_o as_o the_o square_n of_o the_o two_o line_n a_o and_o â—_o be_v commensurable_a by_o supposition_n the_o superficies_n câ—_n shall_v be_v commensurable_a unto_o the_o superficies_n fg_o and_o therefore_o by_o the_o first_o of_o the_o six_o and_o ten_o of_o this_o book_n the_o line_n de_fw-fr be_v commensurable_a in_o length_n to_o the_o line_n gb_o and_o forasmuch_o as_o by_o the_o â—3_o of_o this_o book_n the_o line_n de_fw-fr be_v a_o four_o binomial_a line_n therefore_o by_o the_o â—6_o of_o this_o book_n the_o line_n ge_z be_v also_o a_o four_o binomial_a line_n wherefore_o by_o the_o 57_o of_o this_o book_n the_o line_n b_o which_o contain_v in_o power_n the_o superficies_n fg_o be_v a_o great_a line_n ¶_o the_o 51._o theorem_a the_o 69._o proposition_n a_o line_n commensurable_a to_o a_o line_n contain_v in_o power_n a_o rational_a and_o a_o medial_a be_v also_o a_o line_n contain_v in_o power_n a_o rational_a and_o a_o medial_a svppose_v that_o ab_fw-la be_v a_o line_n contain_v in_o power_n a_o rational_a and_o a_o medial_a and_o unto_o the_o line_n ab_fw-la let_v the_o line_n cd_o be_v commensurable_a whether_o in_o length_n and_o power_n or_o in_o power_n only_o then_o i_o say_v that_o the_o line_n cd_o be_v a_o line_n contain_v in_o power_n a_o rational_a &_o a_o medial_a duide_v the_o line_n ab_fw-la into_o his_o part_n in_o the_o point_n e._n construction_n wherefore_o by_o the_o 40._o of_o the_o ten_o the_o line_n ae_n and_o ebb_n be_v incommensurable_a in_o power_n have_v that_o which_o be_v make_v of_o the_o square_n of_o they_o add_v together_o medial_a and_o that_o which_o be_v contain_v under_o they_o national_a let_v the_o same_o construction_n be_v in_o this_o that_o be_v in_o the_o former_a demonstration_n and_o in_o like_a sort_n we_o may_v prove_v that_o the_o line_n cf_o and_o fd_o be_v incommensurable_a in_o power_n and_o that_o that_o which_o be_v make_v of_o the_o square_n of_o the_o line_n ae_n and_o ebb_n be_v commensurable_a to_o that_o which_o be_v make_v of_o the_o square_n of_o the_o line_n cf_o and_o fd_o and_o that_o that_o also_o which_o be_v contain_v under_o the_o line_n ae_n and_o ebb_n be_v commensurable_a to_o that_o which_o be_v contain_v under_o the_o line_n cf_o and_o fd._n wherefore_o that_o which_o be_v make_v of_o the_o square_n of_o the_o line_n cf_o and_o fd_o be_v medial_a and_o that_o which_o be_v contain_v under_o the_o line_n cf_o and_o fd_o be_v rational_a wherefore_o the_o whole_a line_n cd_o be_v a_o line_n contain_v in_o
power_n a_o rational_a and_o a_o medial_a which_o be_v require_v to_o be_v demonstrate_v an_o other_o demonstration_n of_o the_o same_o after_o campane_n suppose_v that_o ab_fw-la be_v a_o line_n contain_v in_o power_n a_o rational_a and_o a_o medial_a campane_n whereunto_o let_v the_o line_n gd_v be_v commensurable_a either_o in_o length_n and_o power_n or_o in_o power_n only_o then_o i_o say_v that_o the_o line_n gd_o be_v a_o line_n contain_v in_o power_n a_o rational_a and_o a_o medial_a take_v a_o rational_a line_n ez_o upon_o which_o by_o the_o 45._o of_o the_o first_o apply_v a_o rectangle_n parallelogram_n ezfc_n equal_a to_o the_o square_n of_o the_o line_n ab_fw-la and_o upon_o the_o line_n cf_o which_o be_v equal_a to_o the_o line_n ez_o apply_v the_o parallelogram_n fchi_n equal_a to_o the_o square_n of_o the_o line_n gdâ—_n and_o let_v the_o breadth_n of_o the_o say_a parallelogram_n be_v the_o line_n eglantine_n and_o ch._n and_o forasmuch_o as_o the_o line_n ab_fw-la be_v commensurable_a to_o the_o line_n gd_o at_o the_o least_o in_o power_n only_o therefore_o the_o parallelogram_n of_o and_o fh_o which_o be_v equal_a to_o their_o square_n shall_v be_v commensurable_a wherefore_o by_o the_o 1._o of_o the_o six_o the_o right_a line_n ec_o and_o change_z be_v commensurable_a in_o length_n and_o forasmuch_o as_o the_o parallelogram_n of_o which_o be_v equal_a to_o the_o square_n of_o the_o line_n aâ—_z which_o contain_v in_o power_n â—_o rational_a and_o a_o medial_a be_v apply_v upon_o the_o rational_a ez_o make_v in_o breadth_n the_o line_n ec_o therefore_o the_o line_n ec_o be_v a_o five_o binomial_a line_n by_o the_o 64._o of_o this_o book_n unto_o which_o line_n ec_o the_o line_n change_z be_v commensurable_a in_o length_n wherefore_o by_o the_o 66._o of_o this_o book_n the_o line_n change_z be_v also_o a_o five_o binomial_a line_n and_o forasmuch_o as_o the_o superficies_n ci_o be_v contain_v under_o the_o rational_a line_n ez_o that_o be_v cf_o and_o a_o five_o binomall_a line_n change_z therefore_o the_o line_n which_o contain_v in_o power_n the_o superficies_n ci_o which_o by_o supposition_n be_v the_o line_n gd_o be_v a_o line_n contain_v in_o power_n a_o rational_a and_o a_o medial_a by_o the_o 58._o of_o this_o book_n a_o line_n therefore_o commensurable_a to_o a_o line_n contain_v in_o power_n a_o rational_a and_o a_o medial_a etc_n etc_n ¶_o the_o 52._o theorem_a the_o 70._o proposition_n a_o line_n commensurable_a to_o a_o line_n contain_v in_o power_n two_o medial_n be_v also_o a_o line_n contain_v in_o power_n two_o medial_n svppose_v that_o ab_fw-la be_v a_o line_n contain_v in_o power_n two_o medial_n and_o unto_o the_o line_n ab_fw-la let_v the_o line_n cd_o be_v commensurable_a whether_o in_o length_n &_o power_n or_o in_o power_n only_o then_o i_o say_v that_o the_o line_n cd_o be_v a_o line_n contain_v in_o power_n two_o medial_n forasmuch_o as_o the_o line_n ab_fw-la be_v a_o line_n contain_v in_o power_n two_o medial_n construction_n let_v it_o be_v divide_v into_o his_o part_n in_o the_o point_n e._n wherefore_o by_o the_o 41._o of_o the_o ten_o the_o line_n ae_n and_o ebb_n be_v incommensurable_a in_o power_n have_v that_o which_o be_v make_v of_o the_o square_n of_o they_o add_v together_o medial_a and_o that_o also_o which_o be_v contain_v under_o they_o medial_a and_o that_o which_o be_v make_v of_o the_o square_n of_o the_o line_n ae_n &_o ebb_n be_v incommensurable_a to_o that_o which_o be_v contain_v under_o the_o line_n ae_n and_o ebb_n let_v the_o self_n same_o construction_n be_v in_o this_o that_o be_v in_o the_o former_a demonstration_n and_o in_o like_a sort_n may_v we_o prove_v that_o the_o line_n cf_o &_o fd_o be_v incommensurable_a in_o power_n and_o that_o that_o which_o be_v make_v of_o the_o square_n of_o the_o line_n ae_n and_o ebb_v add_v together_o be_v commensurable_a to_o that_o which_o be_v make_v of_o the_o square_n of_o the_o line_n cf_o and_o fd_o add_v together_o and_o that_o that_o also_o which_o be_v contain_v under_o the_o line_n ae_n and_o ebb_n be_v commensurable_a to_o that_o which_o be_v contain_v under_o the_o line_n cf_o and_o fd._n wherefore_o that_o which_o be_v make_v of_o the_o square_n of_o the_o line_n cf_o and_o fd_o be_v medial_a by_o the_o corollary_n of_o the_o 23._o of_o the_o ten_o and_o that_o which_o be_v contain_v under_o the_o line_n cf_o and_o fd_o be_v medial_a by_o the_o same_o corollary_n â—_o and_o moreover_o that_o which_o be_v make_v of_o the_o square_n of_o the_o line_n cf_o &_o fd_o be_v incommensurable_a to_o that_o which_o be_v contain_v under_o the_o line_n cf_o and_o fd._n wherefore_o the_o line_n cd_o be_v a_o line_n contain_v in_o power_n two_o medial_n which_o be_v require_v to_o be_v prove_v ¶_o a_o assumpt_n add_v by_o montaureus_n assumpt_n that_o that_o which_o be_v make_v of_o the_o square_n of_o the_o line_n cf_o and_o fd_o add_v together_o be_v incommensurable_a to_o that_o which_o be_v contain_v under_o the_o line_n cf_o and_o fd_o be_v thus_o prove_v for_o because_o as_o that_o which_o be_v make_v of_o the_o square_n of_o the_o line_n ae_n and_o ebb_v add_v together_o be_v to_o the_o square_n of_o the_o line_n ae_n so_o be_v that_o which_o be_v make_v of_o the_o square_n of_o the_o line_n cf_o and_o fd_o add_v together_o to_o the_o square_n of_o the_o line_n cf_o as_o it_o be_v prove_v in_o the_o proposition_n go_v before_o therefore_o alternate_o as_o that_o which_o be_v make_v of_o the_o square_n of_o ae_n and_o ebb_v add_v together_o be_v to_o that_o which_o be_v make_v of_o the_o square_n of_o cf_o and_o fd_o add_v together_o so_o be_v the_o square_a of_o the_o line_n ae_n to_o the_o square_n of_o the_o line_n cf._n but_o before_o namely_o in_o the_o 68_o proposition_n it_o be_v prove_v that_o as_o the_o square_n of_o the_o line_n ae_n be_v to_o the_o square_n of_o the_o line_n cf_o so_o be_v the_o parallelogram_n contain_v under_o the_o line_n ae_n and_o ebb_n to_o the_o parallelogram_n contain_v under_o the_o line_n cf_o and_o fd._n wherefore_o as_o that_o which_o be_v make_v of_o the_o square_n of_o the_o line_n ae_n and_o ebb_n be_v to_o that_o which_o be_v make_v of_o the_o square_n of_o the_o line_n cf_o and_o fd_o so_o be_v the_o parallelogram_n contain_v under_o the_o line_n ae_n and_o ebb_n to_o the_o parallelogram_n contain_v under_o the_o line_n cf_o and_o fd._n wherefore_o alternate_o as_o that_o which_o be_v make_v of_o the_o square_n of_o the_o line_n ae_n and_o ebb_n be_v to_o the_o parallelogram_n contain_v under_o the_o line_n ae_n and_o ebb_n so_o be_v that_o which_o be_v make_v of_o the_o square_n of_o the_o line_n cf_o and_o fd_o to_o the_o parallelogram_n contain_v under_o the_o line_n cf_o and_o fd._n but_o by_o supposition_n that_o which_o be_v make_v of_o the_o square_n of_o the_o line_n ae_n and_o ebb_n be_v incommensurable_a to_o the_o parallelogram_n contain_v under_o the_o line_n ae_n &_o ebb_n wherefore_o that_o which_o be_v make_v of_o the_o square_n of_o the_o line_n cf_o and_o fd_o add_v together_o be_v incommensurable_a to_o the_o parallelogram_n contain_v under_o the_o line_n cf_o and_o fd_o which_o be_v require_v to_o be_v prove_v an_o other_o demonstration_n after_o campane_n suppose_v that_o ab_fw-la be_v a_o line_n contain_v in_o power_n two_o medial_n whereunto_o let_v the_o line_n gd_v be_v commensurable_a either_o in_o length_n and_o in_o power_n or_o in_o power_n only_o campanâ—_n then_o i_o say_v that_o the_o line_n gd_o be_v a_o line_n contain_v in_o power_n two_o medial_n let_v the_o same_o construction_n be_v in_o this_o that_o be_v in_o the_o former_a and_o forasmuch_o as_o the_o parallelogram_n of_o be_v equal_a to_o the_o square_n of_o the_o line_n ab_fw-la and_o be_v apply_v upon_o a_o rational_a line_n ez_o it_o make_v the_o breadth_n ec_o a_o six_o binomial_a line_n by_o the_o 65._o of_o this_o book_n and_o forasmuch_o as_o the_o parallelogram_n of_o &_o ci_o which_o be_v equal_a unto_o the_o square_n of_o the_o line_n ab_fw-la and_o gd_o which_o be_v suppose_v to_o be_v commensurable_a be_v commensurable_a therefore_o the_o line_n ec_o and_o change_z be_v commensurable_a in_o length_n by_o the_o first_o of_o the_o six_o but_o ec_o be_v a_o six_o binomial_a line_n wherefore_o change_z also_o be_v a_o six_o binomial_a line_n by_o the_o 66._o of_o this_o book_n and_o forasmuch_o as_o the_o superficies_n ci_o be_v contain_v under_o the_o rational_a line_n cf_o and_o a_o six_o binomial_a line_n change_z therefore_o the_o line_n which_o contain_v in_o power_n the_o superficies_n ci_o namely_o the_o line_n gd_o be_v a_o line_n contain_v in_o power_n two_o medial_n by_o the_o 59_o of_o
this_o book_n wherefore_o a_o line_n commensurable_a to_o a_o line_n contain_v in_o power_n two_o medial_n etc_n etc_n a_o annotation_n if_o other_o to_o have_v be_v speak_v of_o six_o senary_n of_o which_o the_o first_o senary_n contain_v the_o production_n of_o irrational_a line_n by_o composition_n the_o second_o the_o division_n of_o they_o namely_o that_o those_o line_n be_v in_o one_o poinâ—_n only_a devideâ—_n the_o three_o the_o find_v out_o of_o binomial_a line_n of_o the_o first_o i_o say_v the_o second_o the_o three_o the_o four_o the_o five_o and_o the_o six_o after_o that_o begin_v the_o forth_o senary_a contain_v the_o difference_n of_o irrational_a line_n between_o themselves_o for_o by_o the_o nature_n of_o every_o one_o of_o the_o binomial_a line_n be_v demonstrate_v the_o difference_n of_o irrational_a line_n the_o five_o entreâ—teth_v of_o the_o application_n of_o the_o square_n of_o every_o irrational_a line_n namely_o what_o irrational_a line_n be_v the_o breadthes_fw-mi of_o every_o superficies_n so_o apply_v in_o the_o six_o senary_a be_v prove_v that_o any_o line_n commensurable_a to_o any_o irrational_a line_n be_v also_o a_o irrational_a line_n of_o the_o same_o nature_n and_o now_o shall_v be_v speak_v of_o the_o seven_o senary_a wherein_o again_o be_v plain_o set_v forth_o the_o rest_n of_o the_o difference_n of_o the_o say_a line_n between_o themselves_o and_o theâ—e_n be_v even_o in_o those_o irrational_a line_n a_o arithmetical_a proportionality_n note_n and_o that_o line_n which_o be_v the_o arithmetical_a mean_a proportional_a between_o the_o part_n of_o any_o irrational_a line_n be_v also_o a_o irrational_a line_n of_o the_o self_n same_o kind_n first_o it_o be_v certain_a that_o there_o be_v a_o arithmetical_a proportion_n between_o those_o part_n for_o suppose_v that_o the_o line_n ab_fw-la be_v any_o of_o the_o foresay_a irrational_a line_n as_o for_o example_n let_v it_o be_v a_o binomial_a line_n &_o let_v it_o be_v divide_v into_o his_o name_n in_o the_o point_n c._n and_o let_v ac_fw-la be_v the_o great_a name_n from_o which_o take_v away_o the_o line_n ad_fw-la equal_a to_o the_o less_o name_n namely_o to_o cb._n and_o divide_v the_o line_n cd_o into_o two_o equal_a part_n in_o the_o point_n e._n it_o be_v manifest_a that_o the_o line_n ae_n be_v equal_a to_o the_o line_n ebb_n let_v the_o line_n fg_o be_v equal_a to_o either_o of_o they_o it_o be_v plain_a that_o how_o much_o the_o line_n ac_fw-la differ_v from_o the_o line_n fg_o so_o much_o the_o same_o line_n fg_o differ_v from_o the_o line_n cb_o for_o in_o each_o be_v the_o difference_n of_o the_o line_n de_fw-fr or_o ec_o which_o be_v the_o property_n of_o arithmetical_a proportionality_n and_o it_o be_v manifest_a that_o the_o line_n fg_o be_v commensurable_a in_o length_n to_o the_o line_n ab_fw-la for_o it_o be_v the_o half_a thereof_o wherefore_o by_o the_o 66._o of_o the_o ten_o the_o line_n fg_o be_v a_o binomial_a line_n and_o after_o the_o self_n same_o manner_n may_v it_o be_v prove_v touch_v the_o rest_n of_o the_o irrational_a line_n ¶_o the_o 53._o theorem_a the_o 71._o proposition_n if_o two_o superficiece_n namely_o a_o rational_a and_o a_o medial_a superficies_n be_v compose_v together_o the_o line_n which_o contain_v in_o power_n the_o whole_a superficies_n be_v one_o of_o these_o four_o irrational_a line_n either_o a_o binomial_a line_n or_o a_o first_o bimedial_a line_n or_o a_o great_a line_n or_o a_o line_n contain_v in_o power_n a_o rational_a and_o a_o medial_a superficies_n but_o now_o let_v the_o line_n eh_o be_v in_o power_n more_o than_o the_o line_n hk_o by_o the_o square_n of_o a_o line_n incommensurable_a in_o length_n to_o the_o line_n eh_o case_n now_o the_o great_a name_n that_o be_v eh_o be_v commensurable_a in_o length_n to_o the_o rational_a line_n give_v ef._n wherefore_o the_o line_n eke_o be_v afourth_o binomial_a line_n and_o the_o line_n of_o be_v rational_a but_o if_o a_o superficies_n be_v contain_v under_o a_o rational_a line_n and_o afourth_o binomial_a line_n the_o line_n that_o contain_v in_o power_n the_o same_o superficies_n be_v by_o the_o 57_o of_o the_o ten_o irrational_a and_o be_v a_o great_a line_n wherefore_o the_o line_n which_o contain_v in_o power_n the_o parallelogram_n ei_o be_v a_o great_a line_n wherefore_o also_o the_o line_n contain_v in_o power_n the_o superficies_n ad_fw-la be_v a_o great_a line_n case_n but_o now_o suppose_v that_o the_o superficies_n ab_fw-la which_o be_v rational_a be_v less_o than_o the_o superficies_n cd_o which_o be_v medial_a wherefore_o also_o the_o parallelogram_n eglantine_n be_v less_o than_o the_o parallelogram_n high_a wherefore_o also_o the_o line_n eh_o be_v less_o than_o the_o line_n hk_o now_o the_o line_n hk_o be_v in_o power_n more_o than_o the_o line_n eh_o either_o by_o the_o square_n of_o a_o line_n commensurable_a in_o length_n to_o the_o line_n hk_o or_o by_o the_o square_n of_o a_o line_n incommensurable_a in_o length_n unto_o the_o line_n hk_o first_o let_v it_o be_v in_o power_n more_o by_o the_o square_n of_o a_o line_n commensurable_a in_o length_n unto_o hk_n case_n now_o the_o less_o name_n that_o be_v eh_o be_v commensurable_a in_o length_n to_o the_o rational_a line_n give_v of_o as_o it_o be_v before_o prove_v wherefore_o the_o whole_a line_n eke_o be_v a_o second_o binomial_a line_n and_o the_o line_n of_o be_v a_o rational_a line_n but_o if_o a_o superficies_n be_v contain_v under_o a_o rational_a line_n and_o a_o second_o binomial_a line_n the_o line_n that_o contain_v in_o power_n the_o same_o superficies_n be_v by_o the_o 55._o of_o the_o ten_o a_o first_o bimedial_a line_n wherefore_o the_o line_n which_o contain_v in_o power_n the_o parallelogram_n ei_o be_v a_o first_o bimedial_a line_n wherefore_o also_o the_o line_n that_o contain_v in_o power_n the_o superficies_n ad_fw-la be_v a_o first_o bimedial_a line_n case_n but_o now_o let_v the_o line_n hk_o be_v in_o power_n more_o than_o the_o line_n eh_o by_o the_o square_n of_o a_o line_n incommensurable_a in_o length_n to_o the_o line_n hk_o now_o the_o less_o name_n that_o be_v eh_o be_v commensurable_a in_o length_n to_o the_o rational_a line_n give_v ef._n wherefore_o the_o whole_a line_n eke_o be_v a_o five_o binomial_a line_n and_o the_o line_n of_o be_v rational_a but_o if_o a_o superficies_n be_v contain_v under_o a_o rational_a line_n and_o a_o five_o binomial_a line_n the_o line_n that_o contain_v in_o power_n the_o same_o superficies_n be_v by_o the_o 58._o of_o the_o ten_o a_o line_n contain_v in_o power_n a_o rational_a and_o a_o medial_a wherefore_o the_o line_n that_o contain_v in_o power_n the_o parallelogram_n ei_o be_v a_o line_n contain_v in_o power_n a_o rational_a and_o a_o medial_a wherefore_o also_o the_o line_n that_o contain_v in_o power_n the_o superficies_n ad_fw-la be_v a_o line_n contain_v in_o power_n a_o rational_a and_o a_o medial_a if_o therefore_o a_o rational_a and_o a_o medial_a superficies_n be_v add_v together_o the_o line_n which_o contain_v in_o power_n the_o whole_a superficies_n be_v one_o of_o these_o four_o irrational_a line_n namely_o either_o a_o binomial_a line_n or_o a_o first_o bimedial_a line_n or_o a_o great_a line_n or_o a_o line_n contain_v in_o power_n a_o rational_a and_o a_o medial_a which_o be_v require_v to_o be_v demonstrate_v ¶_o the_o 54._o theorem_a the_o 72._o proposition_n if_o two_o medial_a superficiece_n incommensurable_a the_o one_o to_o the_o other_o be_v compose_v together_o the_o line_n contain_v in_o power_n the_o whole_a superficies_n be_v one_o of_o the_o two_o irrational_a line_n remain_v namely_o either_o a_o second_o bimedial_a line_n or_o a_o line_n contain_v in_o power_n two_o medial_n let_v these_o two_o medial_a superficiece_n ab_fw-la and_o cd_o be_v incommensurable_a the_o one_o to_o the_o other_o be_v add_v together_o then_o i_o say_v that_o the_o line_n which_o contain_v in_o power_n the_o superficies_n ad_fw-la be_v either_o a_o second_o bimedial_a line_n or_o a_o line_n contain_v in_o power_n two_o medial_n construction_n for_o the_o superficies_n ab_fw-la be_v either_o great_a or_o less_o than_o the_o superficies_n cd_o for_o they_o can_v by_o no_o mean_n be_v equal_a when_o as_o they_o be_v incommensurable_a case_n first_o let_v the_o superficies_n ab_fw-la be_v great_a than_o the_o superficies_n cd_o and_o take_v a_o rational_a line_n ef._n and_o by_o the_o 44._o of_o the_o first_o unto_o the_o line_n of_o apply_v the_o parallelogram_n eglantine_n equal_a to_o the_o superficies_n ab_fw-la and_o make_v in_o breadth_n the_o line_n eh_o and_o unto_o the_o same_o line_n of_o that_o be_v to_o the_o line_n hg_o apply_v the_o parallelogram_n high_a equal_a to_o the_o superficies_n cd_o &_o make_v in_o breadth_n the_o line_n hk_o and_o forasmuch_o as_o
either_o of_o these_o superficiece_n ab_fw-la &_o cd_o be_v medial_a therefore_o also_o either_o of_o these_o parallelogram_n eglantine_n and_o high_a be_v medial_a and_o they_o be_v each_o apply_v to_o the_o rational_a line_n of_o make_v in_o breadth_n the_o line_n eh_o and_o hk_o wherefore_o by_o the_o 22._o of_o the_o ten_o either_o of_o these_o line_n eh_o and_o hk_o be_v rational_a and_o incommensurable_a in_o length_n to_o the_o line_n ef._n and_o forasmuch_o as_o the_o superficies_n ab_fw-la be_v incommensurable_a to_o the_o superficies_n cd_o and_o the_o superficies_n ab_fw-la be_v equal_a to_o the_o parallelogram_n eâ—_n and_o the_o superficies_n cd_o to_o the_o parallelogram_n high_a therefore_o the_o parallelogram_n eglantine_n be_v incommensurable_a to_o the_o parallelogram_n high_a but_o by_o the_o 1._o of_o the_o six_o as_o the_o parallelogram_n eglantine_n be_v to_o the_o parallelogram_n high_a so_o be_v the_o line_n ehâ—_n to_o the_o line_n hk_o wherefore_o by_o the_o 10._o of_o the_o ten_o the_o line_n eh_o iâ—â—spansâ—_n hkâ—_n wherefore_o the_o â—â—nes_n eh_o â—nd_v hk_o be_v rational_a commensurable_a in_o power_n only_o wherefore_o the_o whole_a line_n eke_o be_v a_o binomial_a line_n and_o as_o in_o the_o former_a proposition_n so_o alsâ—_n in_o this_o may_v it_o be_v prove_v that_o the_o line_n eh_o be_v great_a than_o the_o line_n hk_o wherefore_o the_o line_n eh_o be_v in_o power_n more_o than_o the_o line_n hk_o either_o by_o the_o square_n of_o a_o line_n commensurable_a in_o length_n to_o the_o line_n eh_o or_o by_o the_o square_n of_o a_o line_n incommensurable_a in_o length_n to_o the_o line_n eh_o first_o let_v it_o be_v great_a by_o the_o square_n of_o a_o line_n commensurable_a in_o length_n unto_o the_o line_n eh_o case_n now_o neither_o of_o these_o line_n eh_o and_o hk_o be_v commensurable_a in_o length_n to_o the_o rational_a line_n give_v ef._n wherefore_o the_o whole_a line_n eke_o be_v a_o three_o binomial_a line_n and_o the_o line_n of_o be_v a_o rational_a line_n but_o if_o a_o superficies_n be_v contain_v under_o a_o rational_a line_n &_o a_o three_o binomial_a line_n the_o line_n that_o contain_v in_o power_n the_o same_o superficies_n be_v by_o the_o 56._o of_o the_o ten_o a_o second_o bimedial_a line_n wherefore_o the_o line_n that_o contain_v in_o power_n the_o superficies_n ei_o that_o be_v the_o superficies_n ad_fw-la be_v a_o second_o bimedial_a line_n case_n but_o now_o suppose_v that_o the_o line_n eh_o be_v in_o power_n more_o than_o the_o line_n hk_o by_o the_o square_n of_o a_o line_n incommensurable_a in_o length_n to_o the_o line_n eh_o and_o forasmuch_o as_o either_o of_o these_o line_n eh_o and_o hk_o be_v incommensurable_a in_o length_n to_o the_o rational_a line_n give_v of_o therefore_o the_o line_n eke_o be_v a_o six_o binomial_a line_n but_o if_o a_o superficies_n be_v contain_v under_o a_o rational_a line_n and_o a_o six_o binomial_a line_n the_o line_n that_o contain_v in_o power_n the_o same_o superficies_n be_v by_o the_o 59_o of_o the_o ten_o a_o line_n contain_v in_o power_n two_o medial_n wherefore_o the_o line_n that_o contain_v in_o power_n the_o superficies_n ad_fw-la be_v a_o line_n contain_v in_o power_n two_o medial_n case_n and_o after_o the_o self_n same_o manner_n if_o the_o superficies_n ab_fw-la be_v less_o than_o the_o superficies_n cd_o may_v we_o prove_v that_o the_o line_n that_o contain_v in_o power_n the_o superficies_n ad_fw-la be_v either_o a_o second_o bimedial_a line_n or_o a_o line_n contain_v in_o power_n two_o medial_n if_o therefore_o two_o medial_a superficiece_n incommensurable_a the_o one_o to_o the_o other_o be_v add_v together_o the_o line_n contain_v in_o power_n the_o whole_a superficies_n be_v one_o of_o the_o two_o irrational_a line_n remain_v namely_o either_o a_o second_o bimedial_a line_n or_o a_o line_n contain_v in_o power_n two_o medial_n which_o be_v require_v to_o be_v prove_v ¶_o a_o corollary_n follow_v of_o the_o former_a proposition_n a_o binomial_a line_n and_o the_o other_o irrational_a line_n follow_v it_o be_v neither_o medial_a line_n nor_o one_o and_o the_o same_o between_o themselves_o for_o the_o square_n of_o a_o medial_a line_n apply_v to_o a_o rational_a line_n corollary_n make_v the_o breadth_n rational_a and_o incommensurale_fw-la in_o length_n to_o the_o rational_a line_n whereunto_o it_o be_v apply_v by_o the_o 22._o of_o the_o ten_o the_o square_a of_o a_o binomial_a line_n apply_v to_o â—_o rational_a line_n make_v the_o breadth_n a_o first_o binomial_a line_n by_o the_o 60._o of_o the_o ten_o the_o square_a of_o a_o first_o bimedial_a line_n apply_v unto_o a_o rational_a line_n make_v the_o breadth_n a_o second_o binomial_a line_n by_o the_o 61._o of_o the_o ten_o the_o square_a of_o a_o second_o bimedial_a line_n apply_v unto_o a_o rational_a line_n make_v the_o breadth_n a_o three_o binomial_a line_n by_o the_o 62._o of_o the_o ten_o the_o square_a of_o a_o great_a line_n apply_v to_o a_o rational_a line_n make_v the_o breadth_n a_o four_o binomial_a line_n by_o the_o 63._o of_o the_o ten_o the_o square_a of_o a_o line_n contain_v in_o power_n a_o rational_a &_o a_o medial_a superficies_n make_v the_o breadth_n a_o five_o binomial_a line_n by_o the_o 64._o of_o the_o ten_o and_o the_o square_a of_o a_o line_n contain_v in_o power_n two_o medial_n apply_v unto_o a_o rational_a line_n make_v the_o breadth_n a_o six_o binomial_a line_n by_o the_o 65._o of_o the_o ten_o see_v therefore_o that_o these_o aforesaid_a breadthes_fw-mi differ_v both_o from_o the_o first_o breadth_n for_o that_o it_o be_v rational_a and_o differ_v also_o the_o one_o from_o the_o other_o for_o that_o they_o be_v binomial_n of_o diverse_a order_n it_o be_v manifest_a that_o those_o irrational_a line_n differ_v also_o the_o one_o from_o the_o other_o here_o begin_v the_o senaries_n by_o substraction_n ¶_o the_o 55._o theorem_a the_o 73._o proposition_n if_o from_o a_o rational_a line_n be_v take_v away_o a_o rational_a line_n commensurable_a in_o power_n only_o to_o the_o whole_a line_n the_o residue_n be_v a_o irrational_a line_n and_o be_v call_v a_o residual_a line_n svppose_v that_o ab_fw-la be_v a_o rational_a line_n and_o from_o ab_fw-la take_v away_o a_o rational_a line_n bc_o commensurable_a in_o power_n only_o to_o the_o whole_a line_n ab_fw-la substraction_n then_o i_o say_v that_o the_o line_n remain_v namely_o ac_fw-la be_v irrational_a and_o be_v call_v a_o residual_a line_n for_o forasmuch_o as_o the_o line_n ab_fw-la be_v incommensurable_a in_o length_n unto_o the_o line_n bc_o and_o by_o the_o assumpt_n go_v before_o the_o 22._o of_o the_o ten_o as_o the_o line_n ab_fw-la be_v to_o the_o line_n bc_o demonstration_n so_o iâ—_z the_o square_a of_o the_o line_n ab_fw-la to_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o wherefore_o by_o the_o 10._o of_o the_o ten_o the_o square_a of_o the_o line_n ab_fw-la be_v incommensurable_a to_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc._n but_o unto_o the_o square_n of_o the_o line_n ab_fw-la be_v commensurable_a the_o square_n of_o the_o line_n ab_fw-la and_o bc_o by_o the_o 15._o of_o the_o ten_o wherefore_o the_o square_n of_o the_o line_n ab_fw-la and_o bc_o be_v incommensurable_a to_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc._n but_o unto_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o be_v commensurable_a that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o twice_o wherefore_o the_o square_n of_o the_o line_n ab_fw-la and_o bc_o be_v incommensurable_a to_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o twice_o but_o the_o square_n of_o the_o line_n ab_fw-la and_o bc_o be_v equal_a to_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o twice_o and_o to_o the_o square_n of_o the_o line_n ac_fw-la by_o the_o 7._o of_o the_o second_o wherefore_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o twice_o together_o with_o the_o square_n of_o the_o line_n ac_fw-la be_v incommensurable_a to_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o twice_o wherefore_o by_o the_o 2_o part_n of_o the_o 16._o of_o the_o ten_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o twice_o be_v incommensurable_a to_o the_o square_n of_o the_o line_n ac_fw-la wherefore_o by_o the_o first_o part_n of_o the_o same_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o twice_o together_o with_o the_o square_n of_o the_o line_n ac_fw-la that_o be_v the_o square_n of_o the_o line_n ab_fw-la and_o bc_o be_v incommensurable_a to_o the_o square_n of_o the_o line_n ac_fw-la but_o the_o square_n of_o the_o line_n ab_fw-la and_o
bc_o be_v rational_a for_o the_o line_n ab_fw-la and_o bc_o be_v put_v to_o be_v rational_a wherefore_o the_o line_n ac_fw-la be_v irrational_a and_o be_v call_v a_o residual_a line_n which_o be_v require_v to_o be_v prove_v an_o other_o demonstration_n after_o campane_n campane_n demonstrate_v this_o proposition_n by_o a_o figure_n more_o brief_o after_o this_o mâ—ner_n campane_n let_v the_o superficies_n eglantine_n be_v equal_a to_o the_o square_n of_o the_o line_n ab_fw-la and_o bc_o add_v together_o which_o shall_v be_v rational_a for_o that_o the_o line_n ab_fw-la and_o bc_o be_v suppose_v to_o be_v rational_a commensurable_a in_o power_n only_o fron_n which_o superficies_n take_v away_o the_o superficies_n df_o equal_a to_o that_o which_o be_v conâ—â—yaâ—d_v under_o the_o line_n ab_fw-la &_o dc_o twice_o which_o shall_v be_v medial_a by_o the_o 21._o of_o this_o book_n now_o by_o the_o 7._o of_o the_o second_o the_o superficies_n fg_o be_v equal_a to_o the_o square_n of_o the_o line_n ac_fw-la and_o forasmuch_o as_o the_o superficies_n eglantine_n be_v incommensurable_a to_o the_o superficies_n df_o for_o that_o the_o one_o be_v rational_a and_o the_o other_o medial_a therefore_o by_o the_o 16._o of_o this_o book_n the_o 〈◊〉_d superficies_n eglantine_n be_v incommensurable_a to_o the_o superficies_n fg._n wherefore_o the_o superficies_n fg_o be_v irrational_a and_o therefore_o the_o line_n ac_fw-la which_o contain_v it_o in_o power_n be_v irrational_a which_o be_v require_v to_o be_v prove_v a_o annotation_n of_o p._n montaâ—reâ—s_n this_o theorem_a teach_v nothing_o else_o but_o that_o that_o portion_n of_o the_o great_a name_n of_o a_o binomial_a line_n which_o remain_v after_o the_o take_n away_o of_o the_o less_o name_n from_o the_o great_a name_n be_v irrational_a which_o be_v call_v a_o residual_a line_n that_o be_v to_o say_v if_o from_o the_o great_a name_n of_o a_o binomial_a line_n which_o great_a name_n be_v a_o rational_a line_n commensurable_a in_o power_n only_o to_o the_o less_o name_n be_v take_v away_o the_o less_o name_n which_o self_n less_o name_n be_v also_o commensurable_a in_o power_n only_o to_o the_o great_a name_n which_o great_a name_n this_o theorem_a call_v the_o whole_a line_n the_o rest_n of_o the_o line_n which_o remain_v be_v irrational_a which_o he_o call_v a_o residual_a line_n wherefore_o all_o the_o line_n which_o be_v entreat_v in_o this_o theorem_a and_o in_o the_o five_o other_o which_o follow_v be_v the_o portion_n remain_v of_o the_o great_a part_n of_o the_o whole_a line_n which_o be_v entreat_v of_o in_o the_o 36.37.38.39.40.41_o proposition_n after_o the_o take_v away_o the_o less_o part_n from_o the_o great_a in_o this_o proposition_n be_v set_v forth_o the_o nature_n of_o the_o eight_o kind_n of_o irrational_a line_n which_o be_v call_v a_o residual_a line_n the_o definition_n whereof_o by_o this_o proposition_n be_v thus_o line_n a_o residual_a line_n be_v a_o irrational_a line_n which_o remain_v when_o from_o a_o rational_a line_n give_v be_v take_v away_o a_o rational_a line_n commensurable_a to_o the_o whole_a line_n in_o power_n only_o ¶_o the_o 56._o theorem_a the_o 74._o proposition_n if_o from_o a_o medial_a line_n be_v take_v away_o a_o medial_a line_n commensurable_a in_o power_n only_o to_o the_o whole_a line_n and_o comprehend_v together_o with_o the_o whole_a line_n a_o rational_a superficies_n the_o residue_n be_v a_o irrational_a line_n and_o be_v call_v a_o first_o medial_a residual_a line_n out_o of_o this_o proposition_n be_v take_v the_o definition_n of_o the_o nine_o kind_n of_o irrational_a line_n which_o be_v call_v a_o first_o residual_a medial_a line_n the_o definition_n whereof_o be_v thus_o line_n a_o first_o residual_a medial_a line_n be_v a_o irrational_a line_n which_o remain_v when_o from_o a_o medial_a line_n be_v take_v away_o a_o medial_a line_n commensurable_a to_o the_o whole_a in_o power_n only_o and_o the_o part_n take_v away_o and_o the_o whole_a line_n contain_v a_o medial_a superficies_n an_o other_o demonstration_n after_o campane_n let_v the_o line_n de_fw-fr be_v rational_a upon_o which_o apply_v the_o superficies_n df_o equal_a to_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o twice_o and_o let_v the_o superficies_n ge_z be_v equal_a to_o that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ab_fw-la and_o bc_o campane_n wherefore_o by_o the_o 7._o of_o the_o second_o the_o superficies_n fg_o be_v equal_a to_o the_o square_n of_o the_o line_n ac_fw-la and_o forasmuch_o as_o by_o supposition_n the_o superficies_n eglantine_n be_v medial_a therefore_o by_o the_o 22._o of_o the_o ten_o the_o line_n dg_o be_v rational_a commensurable_a in_o power_n only_o to_o the_o rational_a line_n de._n and_o forasmuch_o as_o by_o supposition_n the_o superficies_n eh_o be_v rational_a therefore_o by_o the_o 20._o of_o the_o ten_o the_o line_n dh_o be_v rational_a commensurable_a in_o length_n unto_o the_o rational_a line_n de._n wherefore_o the_o line_n dg_o and_o dh_o be_v rational_a commensurable_a in_o power_n only_o by_o the_o assumpt_n put_v before_o the_o 13._o of_o this_o book_n wherefore_o by_o the_o 73_o of_o this_o book_n the_o line_n gh_o be_v a_o residual_a line_n and_o be_v therefore_o irrational_a wherefore_o by_o the_o corollary_n of_o the_o 21._o of_o this_o book_n the_o superficies_n fg_o be_v irrational_a and_o therefore_o the_o line_n ac_fw-la which_o contain_v it_o in_o power_n be_v irrational_a and_o be_v call_v a_o first_o medial_n residual_a line_n ¶_o the_o 57_o theorem_a the_o 75._o proposition_n if_o from_o a_o medial_a line_n be_v take_v away_o a_o medial_a line_n commensurable_a in_o power_n only_o to_o the_o whole_a line_n and_o comprehend_v together_o with_o the_o whole_a line_n a_o medial_a superficies_n the_o residue_n be_v a_o irrational_a line_n and_o be_v call_v a_o second_o medial_a residual_a line_n svppose_v that_o ab_fw-la be_v a_o medial_a line_n and_o from_o ab_fw-la take_v away_o a_o medial_a line_n cb_o commensurable_a in_o power_n only_o to_o the_o whole_a line_n ab_fw-la and_o comprehend_v together_o with_o the_o whole_a line_n ab_fw-la a_o medial_a superficies_n namely_o the_o parallelogram_n contain_v under_o the_o line_n ab_fw-la and_o bc._n construction_n then_o i_o say_v that_o the_o residue_n namely_o the_o line_n ac_fw-la be_v irrational_a and_o be_v call_v a_o second_o medial_a residual_a line_n take_v a_o rational_a line_n diego_n and_o by_o the_o 44._o of_o the_o first_o unto_o the_o line_n diego_n apply_v the_o parallelogram_n de_fw-fr equal_a to_o the_o square_n of_o the_o line_n ab_fw-la &_o bc_o and_o make_v in_o breadth_n the_o line_n dg_o demonstration_n and_o unto_o the_o same_o line_n diego_n apply_v the_o parallelogram_n dh_fw-mi equal_a to_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la &_o bc_o twice_o and_o make_v in_o breadth_n the_o line_n df._n now_o the_o parallelogram_n dh_o be_v less_o than_o the_o parallelogram_n de_fw-fr for_o that_o also_o the_o square_a of_o the_o line_n ab_fw-la and_o bc_o be_v great_a than_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o twice_o by_o the_o square_n of_o the_o line_n ac_fw-la by_o the_o 7._o of_o the_o second_o wherefore_o the_o parallelogram_n remain_v namely_o fe_o be_v equal_a to_o the_o square_n of_o the_o line_n ac_fw-la and_o forasmuch_o as_o the_o square_n of_o the_o line_n ab_fw-la and_o bc_o be_v medial_a therefore_o also_o the_o parallelogram_n de_fw-fr be_v medial_a and_o be_v apply_v to_o the_o rational_a line_n diego_n make_v in_o breadth_n the_o line_n dg_o wherefore_o by_o the_o 22._o of_o the_o ten_o the_o line_n dg_o be_v rational_a and_o incommensurable_a in_o length_n to_o the_o line_n di._n again_o forasmuch_o as_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o be_v medial_a therefore_o also_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o twice_o be_v medial_a but_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o twice_o be_v equal_a to_o the_o parallelogram_n dh_fw-mi wherefore_o the_o parallelogram_n dh_o be_v medial_a and_o be_v apply_v to_o the_o rational_a line_n diego_n make_v in_o breadth_n the_o line_n df._n wherefore_o the_o line_n df_o be_v rational_a and_o incommensurable_a in_o length_n to_o the_o line_n di._n and_o forasmuch_o as_o the_o line_n ab_fw-la and_o bc_o be_v commensurable_a in_o power_n only_o therefore_o the_o line_n ab_fw-la be_v incommensurable_a in_o length_n to_o the_o line_n bc._n wherefore_o by_o the_o assumpt_n go_v before_o the_o 22._o of_o the_o ten_o and_o by_o the_o 10._o of_o the_o ten_o the_o square_a of_o the_o line_n ab_fw-la be_v incommensurable_a to_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc._n but_o unto_o the_o square_n of_o the_o line_n ab_fw-la be_v commensurable_a
the_o square_n of_o ab_fw-la and_o bc_o by_o the_o 15._o of_o the_o ten_o and_o unto_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o be_v commensurable_a to_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o twice_o wherefore_o the_o square_n of_o the_o line_n ab_fw-la and_o bc_o be_v incommensurable_a to_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o twice_o but_o unto_o the_o square_n of_o the_o line_n ab_fw-la and_o bc_o be_v equal_a the_o parallelogram_n de_fw-fr and_o to_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o twice_o be_v equal_a the_o parallelogram_n dh_fw-mi wherefore_o the_o parallelogram_n de_fw-fr be_v incommensurable_a to_o the_o parallelogram_n dh_fw-mi but_o as_o the_o parallelogram_n de_fw-fr be_v to_o the_o parallelogram_n dh_o so_o iâ—_z the_o line_n gd_o to_o the_o line_n df._n wherefore_o the_o line_n gd_o be_v incommensurable_a in_o length_n to_o the_o line_n df._n and_o either_o of_o they_o be_v rational_a wherefore_o the_o line_n gd_o and_o df_o be_v rational_a commensurable_a in_o power_n only_o wherefore_o the_o line_n fg_o be_v a_o residual_a line_n by_o the_o 73._o proposition_n of_o the_o ten_o and_o the_o line_n de_fw-fr be_v a_o rational_a line_n but_o a_o superficies_n comprehend_v under_o a_o rational_a line_n and_o a_o irrational_a line_n be_v irrational_a by_o the_o 21_o of_o the_o teâ—â—â—â—_n and_o the_o line_n which_o contain_v in_o power_n the_o same_o superficies_n be_v irrational_a by_o the_o assumpt_n go_v before_o the_o same_o wherefore_o the_o parallelogram_n fe_o be_v irrational_a but_o the_o line_n ac_fw-la contain_v in_o power_n the_o parallelogram_n fe_o wherefore_o the_o line_n ac_fw-la be_v a_o irrational_a line_n and_o be_v call_v a_o second_o medial_a residual_a line_n and_o this_o second_o medial_a residual_a line_n be_v that_o part_n of_o the_o great_a part_n of_o a_o bimedial_a line_n which_o remain_v after_o the_o take_n away_o of_o the_o less_o part_n from_o the_o great_a which_o be_v require_v to_o be_v prove_v an_o other_o demonstrtion_n more_o brief_a after_o campane_n this_o proposition_n set_v forth_o the_o nature_n of_o the_o ten_o kind_n of_o irrational_a line_n which_o be_v call_v a_o second_o residual_a medial_a line_n which_o be_v thus_o define_v line_n a_o second_o residualâ—_n medial_a line_n be_v a_o irrational_a line_n which_o remain_v when_o from_o a_o medial_a line_n be_v take_v away_o a_o medial_a line_n commensurable_a to_o the_o whole_a in_o power_n only_o and_o the_o part_n take_v away_o &_o the_o whole_a line_n contain_v a_o medial_a superficies_n ¶_o the_o 58._o theorem_a the_o 76._o proposition_n iâ—â—rom_o a_o right_a line_n be_v take_v away_o a_o right_a line_n incommensurable_a in_o power_n to_o the_o whole_a and_o if_o that_o which_o be_v make_v of_o the_o square_n of_o the_o whole_a line_n and_o of_o the_o line_n take_v away_o add_v together_o be_v rational_a and_o the_o parallelogram_n contain_v under_o the_o same_o line_n medial_a the_o line_n remain_v be_v irrational_a and_o be_v call_v a_o less_o line_n in_o this_o proposition_n be_v contain_v the_o definition_n of_o the_o eleven_o kind_n of_o irrational_a line_n which_o be_v call_v a_o less_o line_n who_o definition_n be_v thus_o a_o less_o line_n be_v a_o irrational_a line_n which_o remain_v line_n when_o from_o a_o right_a line_n be_v take_v away_o a_o right_a line_n incommensurable_a in_o power_n to_o the_o whole_a and_o the_o square_a of_o the_o whole_a line_n &_o the_o square_a of_o the_o part_n take_v away_o add_v together_o make_v a_o rational_a superficies_n and_o the_o parallelogram_n contain_v of_o they_o be_v medial_a this_o proposition_n may_v after_o campane_n way_n be_v demonstrate_v if_o you_o remember_v well_o the_o order_n &_o position_n which_o he_o in_o the_o three_o former_a proposition_n use_v ¶_o the_o 19_o theorem_a the_o 77._o proposition_n if_o from_o a_o right_a line_n be_v take_v away_o a_o right_a line_n incommensurable_a in_o power_n to_o the_o whole_a line_n and_o if_o that_o which_o be_v make_v of_o the_o square_n of_o the_o whole_a line_n and_o of_o the_o line_n take_v away_o add_v together_o be_v medial_a and_o the_o parallelogram_n contain_v under_o the_o same_o line_n rational_a the_o line_n remain_v be_v irrational_a and_o be_v call_v a_o line_n make_v with_o a_o rational_a superficies_n the_o whole_a superficies_n medial_a in_o this_o proposition_n be_v declare_v the_o nature_n of_o the_o twelve_o kind_a of_o irrational_a line_n which_o be_v call_v a_o line_n make_v with_o a_o rational_a superficies_n the_o whole_a superficies_n medial_a who_o definition_n be_v thus_o line_n a_o line_n make_v with_o a_o rational_a superficies_n the_o whole_a superficies_n medial_a be_v a_o irrational_a line_n which_o remain_v when_o from_o a_o right_a line_n be_v take_v away_o a_o right_a line_n incommensurable_a in_o power_n to_o the_o whole_a line_n and_o the_o square_a of_o the_o whole_a line_n &_o the_o square_a of_o the_o part_n take_v away_o add_v together_o make_v a_o medial_a superficies_n and_o the_o parallelogram_n contain_v of_o they_o be_v rational_a this_o proposition_n also_o may_v after_o campane_n way_n be_v demonstrate_v observe_v the_o former_a caution_n ¶_o the_o 60._o theorem_a the_o 78._o proposition_n if_o from_o a_o right_a line_n be_v take_v away_o a_o right_a line_n incommensurable_a in_o power_n to_o the_o whole_a line_n and_o if_o that_o which_o be_v make_v of_o the_o square_n of_o the_o whole_a line_n and_o of_o the_o line_n take_v away_o add_v together_o be_v medial_a and_o the_o parallelogram_n contain_v under_o the_o same_o line_n be_v also_o medial_a and_o incommensurable_a to_o that_o which_o be_v make_v of_o the_o square_n of_o the_o say_v line_n add_v together_o the_o line_n remain_v be_v irrational_a and_o be_v call_v a_o line_n make_v with_o a_o medial_a superficies_n the_o whole_a superficies_n medial_a this_o proposition_n may_v thus_o more_o brief_o be_v demonstrate_v forasmuch_o as_o that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ab_fw-la and_o bc_o be_v medial_a and_o that_o also_o which_o be_v contain_v under_o they_o be_v medial_a therefore_o the_o parallelogrammâ—s_n de_fw-fr and_o dh_o which_o be_v equal_a unto_o they_o be_v medial_a but_o a_o medial_a superficies_n exceed_v not_o a_o medial_a superficies_n by_o a_o rational_a superficies_n wherefore_o the_o superficies_n fe_o which_o be_v the_o excess_n of_o the_o medial_a superficies_n de_fw-fr above_o the_o medial_a superficies_n dh_o be_v irrational_a and_o therefore_o the_o line_n ac_fw-la which_o contain_v it_o in_o power_n be_v irrational_a etc_n etc_n in_o this_o proposition_n be_v show_v the_o condition_n and_o nature_n of_o the_o thirteen_o and_o last_o kind_n of_o irrational_a line_n which_o be_v call_v a_o line_n make_v with_o a_o medial_a superficies_n the_o whole_a superficies_n medial_a who_o definition_n be_v thus_o a_o line_n make_v with_o a_o medial_a superficies_n the_o whole_a superficies_n medial_a be_v a_o irrational_a line_n which_o remain_v line_n when_o from_o a_o right_a liâ—e_z be_v take_v away_o a_o right_a line_n incommensurable_a in_o power_n to_o the_o whole_a line_n and_o the_o square_n of_o the_o whole_a line_n and_o of_o the_o line_n take_v away_o add_v together_o make_v a_o medial_a superficies_n and_o the_o parallelogram_n contain_v of_o they_o be_v also_o a_o medial_a superficies_n moreover_o the_o square_n of_o they_o be_v incommensurable_a to_o the_o parallelogram_n contain_v of_o they_o a_o assumpt_n of_o campane_n if_o there_o be_v four_o quantity_n &_o if_o the_o difference_n of_o the_o first_o to_o the_o second_o be_v as_o the_o difference_n of_o the_o three_o to_o the_o four_o then_o alternate_o as_o the_o difference_n of_o the_o first_o be_v to_o the_o three_o so_o be_v the_o difference_n of_o the_o second_o to_o the_o four_o this_o be_v to_o be_v understand_v of_o quantity_n in_o like_a sort_n refer_v the_o one_o to_o the_o other_o that_o be_v if_o the_o first_o be_v great_a than_o the_o second_o campane_n the_o three_o ought_v to_o be_v great_a than_o the_o four_o and_o if_o the_o first_o be_v less_o than_o the_o second_o the_o three_o ought_v to_o be_v less_o than_o the_o four_o and_o be_v also_o to_o be_v understand_v in_o arithmeticiall_a proportionality_n as_o for_o example_n let_v the_o difference_n of_o a_o be_v unto_o b_o as_o the_o difference_n of_o c_o be_v to_o d._n artificial_a then_o i_o say_v that_o as_o the_o difference_n of_o a_o be_v to_o c_o so_o be_v the_o difference_n of_o b_o to_o d._n for_o by_o this_o common_a sentence_n the_o difference_n of_o the_o extreme_n be_v compose_v of_o the_o difference_n of_o the_o extreme_n to_o the_o mean_n the_o difference_n of_o a_o to_o c_z be_v compose_v of_o the_o difference_n of_o a_o
to_o b_o and_o of_o the_o difference_n of_o b_o to_o c._n and_o by_o the_o same_o common_a sentence_n the_o difference_n of_o b_o to_o d_z be_v compose_v of_o the_o difference_n of_o b_o to_o c_z and_o of_o â—he_n difference_n of_o c_o to_o d._n and_o forasmuch_o as_o by_o supposition_n the_o difference_n of_o a_o to_o b_o be_v as_o the_o difference_n of_o c_o to_o d_z and_o the_o difference_n of_o b_o to_o c_z be_v common_a to_o they_o both_o wherefore_o it_o follow_v that_o as_o the_o difference_n of_o a_o be_v to_o c_o so_o be_v the_o difference_n of_o b_o to_o d_z which_o be_v require_v to_o be_v prove_v ¶_o the_o 61._o theorem_a the_o 79._o proposition_n unto_o a_o residual_a line_n can_v be_v join_v one_o only_o right_a line_n rational_a and_o commensurable_a in_o power_n only_o to_o the_o whole_a line_n senary_n let_v ab_fw-la be_v a_o residual_a line_n and_o unto_o it_o let_v the_o line_n bc_o be_v suppose_v to_o be_v join_v so_o that_o let_v the_o line_n ac_fw-la and_o bc_o be_v rational_a commensurable_a in_o power_n only_o then_o i_o say_v that_o unto_o the_o line_n ab_fw-la can_v be_v join_v any_o other_o rational_a line_n commensurable_a in_o power_n only_o to_o the_o whole_a line_n impossibility_n for_o if_o it_o be_v possible_a let_v bd_o be_v such_o a_o line_n add_v unto_o it_o wherefore_o the_o line_n ad_fw-la and_o db_o be_v rational_a commensurable_a in_o power_n only_o and_o forasmuch_o as_o how_o much_o the_o square_n of_o the_o line_n ad_fw-la and_o db_o do_v exceed_v that_o which_o be_v contain_v under_o the_o line_n ad_fw-la and_o db_o twice_o so_o much_o also_o do_v the_o square_n of_o the_o line_n ac_fw-la and_o cb_o exceed_v that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o twice_o for_o the_o excess_n of_o each_o be_v one_o and_o the_o same_o namely_o the_o square_a of_o the_o line_n ab_fw-la by_o the_o 7._o of_o the_o second_o wherefore_o alternate_o by_o the_o â—ormer_a assumpt_n of_o campanus_n how_o much_o the_o square_n of_o the_o line_n ad_fw-la and_o db_o do_v exceed_v the_o square_n of_o the_o line_n ac_fw-la &_o cb_o so_o much_o also_o exceed_v that_o which_o be_v contain_v under_o the_o line_n ad_fw-la and_o db_o twice_o that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o twice_o but_o that_o which_o be_v make_v of_o the_o square_n of_o the_o line_n ad_fw-la and_o db_o add_v together_o exceed_v that_o which_o be_v make_v of_o the_o square_n of_o the_o line_n ac_fw-la and_o cb_o add_v together_o by_o a_o rational_a superficies_n for_o they_o be_v either_o of_o they_o rational_a wherefore_o that_o which_o be_v contain_v under_o the_o line_n ad_fw-la and_o db_o twice_o exceed_v that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o twice_o by_o a_o rational_a superficies_n but_o that_o which_o be_v contain_v under_o the_o line_n ad_fw-la and_o db_o twice_o be_v medial_a for_o it_o be_v commensurable_a to_o that_o which_o be_v contain_v under_o the_o line_n ad_fw-la and_o db_o once_o which_o superficies_n be_v medial_a by_o the_o 21._o of_o the_o ten_o and_o by_o the_o same_o reason_n also_o that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o twice_o be_v medial_a wherefore_o a_o medial_a superficies_n differ_v from_o a_o medial_a superficies_n by_o a_o rational_a superficies_n which_o by_o the_o 26._o of_o the_o ten_o be_v impossible_a wherefore_o unto_o the_o line_n ab_fw-la can_v be_v join_v any_o other_o rational_a line_n beside_o bc_o commensurable_a in_o power_n only_o to_o the_o whole_a line_n wherefore_o unto_o a_o residual_a line_n can_v be_v join_v one_o only_o right_a line_n rational_a and_o commensurable_a in_o power_n only_o to_o the_o whole_a line_n which_o be_v require_v to_o be_v demonstrate_v ¶_o the_o 62._o theorem_a the_o 80._o proposition_n unto_o a_o first_o medial_n residual_a line_n can_v be_v join_v one_o only_o medial_a right_a line_n commensurable_a in_o power_n only_o to_o the_o whole_a line_n and_o comprehend_v with_o the_o whole_a line_n a_o rational_a superficies_n svppose_v that_o ab_fw-la be_v a_o first_o medial_a residual_a line_n &_o unto_o ab_fw-la join_v the_o line_n bc_o so_o that_o let_v the_o line_n ac_fw-la and_o bc_o be_v medial_a commensurable_a in_o power_n only_o &_o let_v that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o bc_o be_v rational_a then_o i_o say_v that_o unto_o the_o line_n ab_fw-la can_v be_v join_v any_o other_o medial_a line_n commensurable_a in_o power_n only_o to_o the_o whole_a line_n and_o comprehend_v together_o with_o the_o whole_a line_n a_o rational_a superficies_n absurdity_n for_o if_o it_o be_v possible_a let_v the_o line_n bd_o be_v such_o a_o line_n wherefore_o the_o line_n ad_fw-la and_o db_o be_v medial_a commensurable_a in_o power_n only_o and_o that_o which_o be_v contain_v under_o the_o line_n ad_fw-la and_o db_o be_v rational_a and_o forasmuch_o as_o how_o much_o the_o square_n of_o the_o line_n ad_fw-la and_o db_o exceed_v that_o which_o be_v contain_v under_o the_o line_n ad_fw-la and_o db_o twice_o so_o much_o also_o exceed_v the_o square_n of_o the_o line_n ac_fw-la &_o bc_o that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o twice_o for_o the_o excess_n of_o each_o be_v one_o and_o the_o same_o namely_o the_o square_a of_o the_o line_n ab_fw-la wherefore_o alternate_o as_o it_o be_v say_v in_o the_o former_a proposition_n how_o much_o the_o square_n of_o the_o line_n ad_fw-la and_o db_o exceed_v the_o square_n of_o the_o line_n ac_fw-la and_o cb_o so_o much_o also_o that_o which_o be_v contain_v under_o the_o line_n ad_fw-la and_o db_o twice_o exceed_v that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o twice_o but_o that_o which_o be_v contain_v under_o the_o line_n ad_fw-la and_o db_o twice_o exceed_v that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o twice_o by_o a_o rational_a superficies_n for_o they_o be_v either_o of_o they_o a_o rational_a supersicy_n wherefore_o that_o which_o be_v make_v of_o the_o square_n of_o the_o line_n ad_fw-la &_o db_o exceed_v that_o which_o be_v make_v of_o the_o square_n of_o the_o line_n ac_fw-la &_o cb_o by_o a_o rational_a superficies_n which_o by_o the_o 26._o of_o the_o ten_o be_v impossible_a for_o they_o be_v either_o of_o they_o medial_a for_o those_o four_o line_n be_v put_v to_o be_v medial_a wherefore_o unto_o a_o first_o medial_a residual_a line_n can_v be_v join_v only_a one_o right_a medial_a line_n commensurable_a in_o power_n only_o to_o the_o whole_a line_n and_o comprehend_v with_o the_o whole_a line_n a_o rational_a superficies_n which_o be_v require_v to_o be_v prove_v ¶_o the_o 63._o theorem_a the_o 81._o proposition_n unto_o a_o second_o medial_a residual_a line_n can_v be_v join_v only_a one_o medial_a right_a line_n commensurable_a in_o power_n only_o to_o the_o whole_a line_n and_o comprehend_v with_o the_o whole_a line_n a_o medial_a superficies_n svppose_v that_o ab_fw-la be_v a_o second_o medial_a residual_a line_n &_o unto_o the_o line_n ab_fw-la join_v the_o line_n bc_o so_o that_o let_v the_o line_n ac_fw-la and_o cb_o be_v medial_a commensurable_a in_o power_n only_o and_o let_v that_o which_o be_v comprehend_v under_o the_o line_n ac_fw-la and_o cb_o be_v medial_a then_o i_o say_v that_o unto_o the_o line_n ab_fw-la can_v not_o be_v join_v any_o other_o medial_a right_a line_n commensurable_a in_o power_n only_o to_o the_o whole_a line_n and_o comprehend_v together_o with_o the_o whole_a line_n a_o medial_a superficies_n for_o if_o it_o be_v possible_a let_v the_o line_n bd_o be_v such_o a_o line_n wherefore_o the_o line_n ad_fw-la &_o db_o be_v medial_a commensurable_a in_o power_n only_o and_o that_o which_o be_v contain_v under_o the_o line_n ad_fw-la and_o db_o be_v also_o medial_a take_v a_o rational_a line_n ef._n construction_n and_o by_o the_o 44._o of_o the_o first_o unto_o the_o line_n of_o apply_v the_o parallelogram_n eglantine_n equal_a to_o the_o square_n of_o the_o line_n ac_fw-la and_o cb_o and_o make_v in_o breadth_n the_o line_n em_n and_o from_o that_o parallelogram_n eglantine_n take_v away_o the_o parallelogram_n hg_o equal_a to_o that_o which_o be_v contain_v under_o ac_fw-la and_o cb_o twice_o and_o make_v in_o breadth_n the_o line_n he_o wherefore_o the_o parallelogram_n remain_v namely_o el_n be_v by_o the_o 7._o of_o the_o second_o equal_a to_o the_o square_n of_o the_o line_n ab_fw-la wherefore_o the_o line_n ab_fw-la contain_v in_o power_n the_o parallelogram_n el._n again_o unto_o the_o line_n of_o apply_v by_o the_o 44._o of_o the_o first_o the_o parallelogram_n ei_o equal_a to_o the_o square_n of_o the_o line_n ad_fw-la and_o db_o and_o make_v in_o breadth_n the_o line_n en_fw
but_o the_o square_n of_o the_o line_n ad_fw-la and_o db_o be_v equal_a to_o that_o which_o be_v contain_v under_o the_o line_n ad_fw-la and_o db_o twice_o and_o to_o the_o square_n of_o the_o line_n ab_fw-la absurdity_n wherefore_o the_o parallelogram_n ei_o be_v equal_a to_o that_o which_o be_v contain_v under_o the_o line_n ad_fw-la &_o db_o twice_o and_o to_o the_o square_n of_o the_o line_n ab_fw-la but_o the_o parallelogram_n el_n be_v equal_a to_o the_o square_n of_o the_o line_n ab_fw-la wherefore_o the_o parallelogram_n remain_v namely_o high_a be_v equal_a to_o that_o which_o be_v contain_v under_o the_o line_n ad_fw-la and_o db_o twice_o and_o forasmuch_o as_o the_o line_n ac_fw-la and_o cb_o be_v â—â—â—iall_a therefore_o the_o square_n also_o of_o the_o line_n ac_fw-la and_o cb_o be_v medial_a and_o they_o be_v equal_a to_o the_o parallelogram_n eglantine_n wherefore_o the_o parallelogram_n eglantine_n be_v by_o that_o which_o be_v speak_v in_o the_o 75._o proposition_n medial_a and_o it_o be_v apply_v unto_o the_o rational_a line_n of_o make_v in_o breadth_n the_o line_n em_n wherefore_o by_o the_o 22._o of_o the_o ten_o the_o line_n em_n be_v rational_a and_o incommensurable_a in_o length_n to_o the_o line_n ef._n again_o forasmuch_o as_o that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o be_v medial_a therefore_o by_o the_o corollary_n of_o the_o 23._o of_o the_o ten_o that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o twice_o be_v also_o medial_a and_o it_o be_v equal_a to_o the_o parallelogram_n hg_o wherefore_o also_o the_o parallelogram_n hg_o be_v medial_a and_o be_v apply_v to_o the_o rational_a line_n of_o make_v in_o breadth_n the_o line_n he_o wherefore_o by_o the_o 22_o of_o the_o ten_o the_o line_n he_o be_v rational_a and_o incommensurable_a in_o length_n to_o the_o line_n ef._n and_o forasmuch_o as_o the_o line_n ac_fw-la and_o cb_o be_v commensurable_a in_o power_n only_o therefore_o the_o line_n ac_fw-la be_v incommensurable_a in_o length_n to_o the_o line_n cb._n but_o as_o the_o line_n ac_fw-la be_v to_o the_o line_n cb_o so_o by_o the_o assumpt_n go_v before_o the_o 22._o of_o the_o ten_o be_v the_o square_a of_o the_o line_n ac_fw-la to_o that_o which_o be_v contain_v under_o the_o line_n ac_fw-la &_o cb._n wherefore_o by_o the_o 10._o of_o the_o ten_o the_o square_a of_o the_o line_n ac_fw-la be_v incommensurable_a to_o that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb._n but_o unto_o the_o square_n of_o the_o line_n ac_fw-la be_v commensurable_a the_o square_n of_o ac_fw-la &_o cb_o and_o unto_o that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o be_v commensurable_a that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o twice_o wherefore_o the_o square_n of_o the_o line_n ac_fw-la &_o cb_o be_v incommensurable_a to_o that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o twice_o but_o unto_o the_o square_n of_o the_o line_n ac_fw-la and_o cb_o be_v equal_a the_o parallelogram_n eglantine_n and_o unto_o that_o which_o be_v contain_v under_o the_o line_n ac_fw-la &_o cb_o twice_o be_v equal_a the_o parallelogram_n gh_o wherefore_o the_o parallelogram_n eglantine_n be_v incommensurable_a to_o the_o parallelogram_n hg_o but_o as_o the_o parallelogram_n eglantine_n be_v to_o the_o parallelogrmme_n hg_o so_o be_v the_o line_n em_n to_o the_o line_n he_o wherefore_o the_o line_n em_n be_v incommensurable_a in_o length_n to_o the_o line_n he_o and_o they_o be_v both_o rational_a line_n wherefore_o the_o line_n em_n and_o mh_o be_v rational_a commensurable_a in_o power_n only_o wherefore_o the_o line_n eh_o be_v a_o residual_a line_n and_o unto_o it_o be_v join_v a_o rational_a line_n he_o commensurable_a in_o power_n only_o to_o the_o whole_a line_n em_n in_o like_a sort_n also_o may_v it_o be_v prove_v that_o unto_o the_o line_n eh_o be_v join_v the_o line_n hn_o be_v also_o rational_a and_o commensurable_a in_o power_n only_o to_o the_o whole_a line_n en_fw-fr wherefore_o unto_o a_o residual_a line_n be_v join_v â—â—re_o than_o one_o only_a line_n commensurable_a in_o power_n only_o to_o the_o whole_a line_n which_o by_o the_o 79._o of_o the_o ten_o be_v impossible_a wherefore_o unto_o a_o second_o medial_a residual_a line_n can_v be_v join_v only_a one_o medial_a right_a line_n commensurable_a in_o power_n only_o to_o the_o whole_a line_n and_o comprehend_v with_o the_o whole_a line_n a_o medial_a superficies_n which_o be_v require_v to_o be_v demonstrate_v ¶_o the_o 64._o theorem_a the_o 82._o proposition_n unto_o a_o less_o line_n can_v be_v join_v only_a one_o right_a line_n incommensurable_a in_o power_n to_o the_o whole_a line_n and_o make_v together_o with_o the_o whole_a line_n that_o which_o be_v make_v of_o their_o square_n add_v together_o rational_a and_o that_o which_o be_v contain_v under_o they_o medial_a svppose_v that_o ab_fw-la be_v a_o less_o line_n and_o to_o ab_fw-la join_v the_o line_n bc_o so_o that_o let_v bc_o be_v such_o a_o line_n as_o be_v require_v in_o the_o theorem_a wherefore_o the_o line_n ac_fw-la and_o cb_o be_v incommensurable_a in_o power_n have_v that_o which_o be_v make_v of_o the_o square_n of_o they_o add_v together_o rational_a and_o that_o which_o be_v contain_v under_o they_o medial_a then_o i_o say_v that_o unto_o ab_fw-la can_v be_v join_v any_o other_o such_o right_a line_n absurdity_n for_o if_o it_o be_v possible_a lâ—t_v the_o line_n bd_o be_v such_o a_o line_n wherefore_o the_o line_n ad_fw-la &_o db_o be_v incommensurable_a in_o power_n have_v that_o which_o be_v make_v of_o the_o square_n of_o they_o add_v together_o rational_a and_o that_o which_o be_v contain_v under_o they_o medial_a and_o for_o that_o how_o much_o the_o square_n of_o the_o line_n ad_fw-la and_o db_o exceed_v the_o square_n of_o the_o line_n ac_fw-la and_o cb_o so_o much_o that_o which_o be_v contain_v under_o the_o line_n ad_fw-la and_o db_o twice_o exceed_v that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o twice_o by_o those_o thing_n which_o be_v speak_v in_o the_o 79._o proposition_n but_o that_o which_o be_v make_v of_o the_o square_n of_o the_o line_n ad_fw-la and_o db_o add_v together_o exceed_v that_o which_o be_v make_v of_o the_o square_n of_o the_o line_n ac_fw-la and_o cb_o add_v together_o by_o a_o rational_a superficies_n for_o they_o be_v either_o of_o they_o rational_a by_o supposition_n wherefore_o that_o which_o be_v contain_v under_o the_o line_n ad_fw-la and_o db_o twice_o exceed_v that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o twice_o by_o a_o rational_a superficies_n which_o by_o the_o 26._o of_o the_o ten_o be_v impossible_a for_o either_o of_o they_o be_v medial_a by_o supposition_n wherefore_o unto_o a_o less_o line_n can_v be_v join_v only_a one_o right_a line_n incommensurable_a in_o power_n to_o the_o whole_a line_n and_o make_v together_o with_o the_o whole_a line_n that_o which_o be_v make_v of_o their_o square_n add_v together_o rational_a and_o that_o which_o be_v contain_v under_o they_o medial_a which_o be_v require_v to_o be_v demonstrate_v ¶_o the_o 65._o theorem_a the_o 83._o proposition_n unto_o a_o line_n make_v with_o a_o rational_a superficies_n the_o whole_a superficies_n medial_a can_v be_v join_v only_a one_o right_a line_n incommensurable_a in_o power_n to_o the_o whole_a line_n and_o make_v together_o with_o the_o whole_a line_n that_o which_o be_v make_v of_o their_o square_n add_v together_o medial_a and_o that_o which_o be_v contain_v under_o they_o rational_a svppose_v that_o ab_fw-la be_v a_o line_n make_v with_o a_o rational_a superficies_n the_o whole_a superficies_n medial_a and_o unto_o it_o let_v the_o line_n bc_o be_v join_v so_o that_o let_v bc_o be_v such_o a_o line_n as_o be_v require_v in_o the_o theorem_a wherefore_o the_o line_n ac_fw-la and_o cb_o be_v incommensurable_a in_o power_n haâ—ing_v that_o which_o be_v make_v of_o the_o square_n of_o the_o line_n ac_fw-la and_o cb_o add_v together_o medial_a and_o that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o rational_a impossibility_n then_o i_o say_v that_o unto_o the_o line_n ab_fw-la can_v be_v join_v any_o other_o such_o line_n for_o if_o it_o be_v possible_a let_v the_o line_n bd_o be_v such_o a_o line_n wherefore_o the_o line_n ad_fw-la and_o db_o be_v incommensurable_a in_o power_n have_v that_o which_o be_v make_v of_o the_o square_n of_o the_o line_n ad_fw-la and_o db_o add_v together_o medial_a and_o that_o which_o be_v contain_v under_o the_o line_n ad_fw-la and_o db_o rational_a now_o for_o that_o how_o much_o the_o square_n of_o the_o line_n ad_fw-la and_o db_o exceed_v the_o square_n of_o the_o line_n ac_fw-la and_o cb_o so_o much_o
that_o which_o be_v contain_v under_o the_o line_n ad_fw-la and_o db_o twice_o exceed_v that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o twice_o by_o that_o which_o be_v speak_v in_o the_o 79._o proposition_n but_o that_o which_o be_v contain_v under_o the_o line_n ad_fw-la and_o db_o twice_o exceed_v that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o twice_o by_o a_o rational_a superficies_n for_o they_o be_v either_o of_o they_o rational_a by_o supposition_n wherefore_o that_o which_o be_v make_v of_o the_o square_n of_o the_o line_n ad_fw-la and_o db_o add_v together_o exceed_v that_o which_o be_v make_v of_o the_o square_n of_o the_o line_n ac_fw-la and_o cb_o add_v together_o by_o a_o rational_a superficies_n which_o by_o the_o 26._o of_o the_o ten_o be_v impossible_a for_o they_o be_v either_o of_o they_o medial_a by_o supposition_n wherefore_o unto_o the_o line_n ab_fw-la can_v be_v join_v any_o other_o line_n beside_o bc_o make_v that_o which_o be_v require_v in_o the_o proposition_n wherefore_o unto_o a_o line_n make_v with_o a_o rational_a superficies_n the_o whole_a superficies_n medial_a can_v be_v join_v only_a one_o right_a line_n incommensurable_a in_o power_n to_o the_o whole_a line_n and_o make_v together_o with_o the_o whole_a line_n that_o which_o be_v make_v of_o their_o squaâ—es_n add_v together_o medial_a and_o that_o which_o be_v contain_v under_o they_o rational_a which_o be_v require_v to_o be_v prove_v ¶_o the_o 66._o theorem_a the_o 84._o proposition_n unto_o a_o line_n make_v with_o a_o medial_a superficies_n the_o whole_a superficies_n medial_a can_v be_v join_v only_a one_o right_a line_n incommensurable_a in_o power_n to_o the_o whole_a line_n and_o make_v together_o with_o the_o whole_a line_n that_o which_o be_v make_v of_o their_o square_n add_v together_o medial_a and_o that_o which_o be_v contain_v under_o they_o medial_a and_o moreover_o make_v that_o which_o be_v make_v of_o the_o square_n of_o they_o add_v together_o incommensurable_a to_o