first_o demonstration_n demonstration_n lead_v to_o a_o impossibility_n this_o proposition_n in_o discreet_a quantity_n answer_v to_o the_o 23._o proposition_n of_o the_o five_o book_n in_o continual_a quantity_n this_o and_o the_o eleven_o proposition_n follow_v declare_v the_o passion_n and_o property_n ofâ—_n prime_a number_n demonstration_n lead_v to_o a_o impossibility_n this_o be_v the_o converse_n of_o the_o former_a proposition_n demonstration_n lead_v to_o a_o absurdity_n demonstration_n lead_v to_o a_o absurdity_n demonstration_n lead_v to_o a_o absurdity_n demonstration_n demonstration_n demonstration_n demonstration_n of_o the_o first_o part_n lead_v to_o a_o absurdity_n demonstration_n of_o the_o second_o part_n which_o be_v the_o conâ—câ—se_n of_o the_o first_o lean_v also_o to_o a_o absurdity_n demonstrasion_n lead_v to_o a_o absurdity_n demonstrasion_n a_o corollary_n â—â—ded_v by_o campave_n demonstration_n lâ—ading_v to_o a_o impossibility_n an_o other_o demonstration_n demonstration_n two_o case_n in_o this_o proposition_n the_o first_o case_n the_o second_o case_n demonstration_n demonstration_n lead_v to_o a_o absurdity_n a_o corollary_n add_v by_o campane_n two_o case_n in_o this_o proposition_n the_o first_o case_n demonstration_n lead_v to_o a_o absurdity_n the_o second_o caseâ—_n demonstration_n lead_v to_o a_o absurdity_n demonstration_n lead_v to_o a_o impossibility_n two_o case_n in_o this_o proposition_n the_o first_o case_n demonstration_n leaâ—iâ—g_v â—o_o a_o absurdity_n the_o second_o case_n demonstration_n lead_v to_o a_o absurdity_n a_o corollary_n demonstration_n the_o converse_n of_o the_o former_a proposition_n demonstration_n construction_n demonstration_n leâ—ding_v to_o a_o absurdity_n a_o corollary_n adâ—ed_v by_o campane_n how_o to_o â—inde_v out_o the_o second_o least_o number_n and_o the_o three_o and_o so_o â—orth_o infinite_o how_o to_o siâ—â—_n out_o the_o least_o â—â—mâ—_n a_o conâ—ayâ—â—g_n â—â—e_z paâ—â—s_n of_o part_n the_o arguâ—â—â—â—_n of_o the_o eight_o book_n demonstration_n lead_v to_o a_o absurdity_n construction_n demonstration_n this_o proposition_n be_v the_o â—â—uerse_a of_o the_o first_o demonstrationâ—_n two_o case_n in_o this_o proposition_n the_o first_o case_n demonstration_n lead_v to_o a_o absurdity_n the_o second_o case_n demonstration_n this_o proposition_n in_o number_n answer_v to_o the_o of_o the_o six_o touch_v parellelogramme_n construction_n demonstration_n an_o other_o demonstration_n after_o campane_n demonstration_n demonstration_n lead_v to_o a_o impossibility_n demonstration_n a_o corollary_n add_v by_o flussates_n construction_n demonstration_n this_o proposition_n be_v the_o converse_n of_o the_o former_a construction_n demonstration_n the_o first_o part_n of_o this_o proposition_n demonstrate_v the_o second_o part_n demonstrate_v construction_n the_o first_o part_n of_o this_o prâ—position_n demonstrate_v the_o second_o part_n demonstrate_v construction_n demonstration_n the_o first_o part_n of_o this_o proposition_n the_o second_o part_n be_v the_o converse_n of_o the_o first_o the_o first_o part_n of_o this_o proposition_n the_o second_o part_n be_v the_o converse_n of_o the_o first_o a_o negative_a proportion_n the_o first_o part_n of_o this_o proposition_n the_o second_o part_n be_v the_o converse_n of_o the_o first_o a_o negative_a proposition_n the_o first_o part_n of_o this_o proposition_n the_o second_o part_n be_v the_o converse_n of_o the_o first_o demonstration_n of_o the_o fiâ—st_a part_n of_o this_o proposition_n demonstration_n of_o the_o second_o part_n demonstration_n of_o the_o first_o part_n of_o this_o proposition_n the_o second_o part_n this_o proposition_n be_v the_o converse_n of_o the_o 18._o proposition_n construction_n demonstration_n this_o proposition_n be_v the_o converse_n of_o the_o 19_o proposition_n construction_n demonstration_n demonstration_n demonstration_n demonstration_n demonstration_n a_o corollary_n add_v by_o flussates_n construction_n construction_n demonstration_n a_o corollary_n add_v by_o flussates_n another_o corollary_n add_v by_o flussates_n the_o argument_n of_o the_o niâ—th_o book_n demonstration_n this_o proposition_n be_v the_o converse_n oâ—_n tâ—e_z formâ—â—_n demonstration_n a_o corollary_n aâ—ded_v by_o campane_n demonstration_n demonstration_n demonstration_n a_o corollary_n add_v by_o campane_n demonstration_n demonstration_n demonstration_n of_o the_o first_o part_n the_o second_o part_n demonstrate_v demostration_n of_o the_o three_o part_n demostration_n of_o the_o first_o part_n of_o this_o proposition_n the_o second_o pâ—rt_n demonstrate_v demonstration_n of_o the_o first_o part_n leave_v to_o a_o absurdity_n demonstration_n of_o the_o â—â—cond_a pâ—â—â—_n lead_v alâ—o_o to_o a_o absurdity_n demonstration_n demonstration_n lead_v to_o a_o absurdity_n an_o other_o demonstration_n aâ—ter_n flussates_n demonstration_n lead_v to_o a_o absurdity_n an_o other_o demonstration_n after_o campane_n demonstration_n lead_v to_o a_o absurdity_n a_o proposition_n add_v by_o campane_n construction_n demonstration_n demonstration_n to_o prove_v that_o the_o number_n a_o and_o c_o be_v prime_a to_o b._n demonstratiou_n this_o proposition_n be_v the_o converse_n of_o the_o former_a demonstration_n this_o answer_v to_o the_o 2._o of_o the_o second_o demonstration_n this_o answer_v to_o the_o 3._o of_o the_o three_o demonstration_n this_o answer_v to_o thâ—_z 4._o of_o the_o second_o demonstration_n this_o answer_v to_o the_o 5._o of_o the_o second_o demonstration_n this_o answer_v to_o the_o 6._o of_o the_o second_o demonstration_n this_o answer_v to_o the_o 7._o of_o the_o second_o demonstration_n this_o answer_v to_o the_o 8._o of_o the_o second_o demonstratition_n this_o answer_v to_o thâ—_z 9_o of_o the_o second_o demonstration_n this_o answer_v to_o the_o 10._o oâ—_n the_o second_o demonstration_n a_o negative_a proposition_n demonstration_n leaâ—ing_v to_o a_o impossibility_n demonstration_n lead_v to_o a_o absurdity_n demonstration_n lead_v to_o a_o abjurditie_n three_o case_n in_o this_o proposition_n the_o first_o case_n the_o second_o case_n the_o three_o case_n divert_v case_n â—n_v this_o proposition_n the_o first_o case_n two_o case_n in_o this_o proposition_n the_o first_o case_n the_o second_o case_n demonstration_n demonstration_n demonstration_n demonstration_n demonstration_n demonstration_n demonstration_n demonstration_n demonstration_n a_o proposition_n add_v by_o campane_n a_o other_o add_v by_o he_o demonstration_n lead_v to_o a_o absurdity_n demonstration_n demonstration_n demonstration_n lead_v to_o a_o absurdity_n an_o other_o demonstration_n demonstration_n demonstration_n this_o proposition_n teachâ—th_v how_o to_o find_v out_o a_o perfect_a number_n construction_n demonstration_n demonstration_n lead_v to_o a_o absurdity_n the_o argument_n of_o the_o ten_o book_n difference_n between_o number_n and_o magnitude_n a_o line_n be_v not_o make_v of_o point_n as_o number_n be_v make_v of_o unity_n this_o book_n the_o hard_a to_o understand_v of_o all_o the_o book_n of_o euclid_n in_o this_o book_n be_v entreat_v of_o a_o strange_a manner_n of_o matter_n then_o in_o the_o former_a many_o even_o of_o the_o well_o learn_v have_v think_v that_o this_o book_n can_v not_o well_o be_v understand_v without_o algebra_n the_o nine_o former_a book_n &_o the_o principle_n of_o this_o â—ooke_n well_o understand_v this_o book_n will_v not_o be_v hard_a to_o understand_v the_o fâ—rst_a definition_n the_o second_o definition_n contraries_n make_v manifest_a by_o the_o compare_v of_o the_o one_o to_o the_o other_o the_o third_o definition_n what_o the_o power_n of_o a_o line_n be_v the_o four_o definition_n unto_o the_o suppose_a line_n first_o set_v may_v be_v compare_v infinite_a line_n why_o some_o mislike_n that_o the_o line_n first_o set_v shall_v be_v call_v a_o rational_a line_n flussates_n call_v this_o line_n a_o line_n certain_a this_o rational_a line_n the_o ground_n in_o a_o manner_n of_o all_o the_o proposition_n in_o this_o ten_o book_n note_n the_o line_n rational_a of_o purpose_n the_o six_o definition_n campâ—nus_n â—ath_z cause_v much_o obscurity_n in_o this_o ten_o book_n the_o seven_o definition_n flussates_n in_o steed_n of_o this_o word_n irrational_a use_v this_o word_n uncertain_a why_o they_o be_v call_v irrational_a line_n the_o cause_n of_o the_o obscurity_n and_o confusedness_n in_o this_o book_n the_o eight_o definition_n the_o nine_o definition_n the_o ten_o definition_n the_o eleven_o definition_n construction_n demonstration_n a_o corollary_n construction_n demonstration_n this_o proposition_n teach_v that_o incontinuall_a quantity_n which_o the_o first_o of_o the_o seven_o teach_v in_o discrete_a quantity_n construction_n demonstration_n lead_v to_o a_o absurdity_n two_o case_n in_o this_o proposition_n the_o first_o case_n this_o proposition_n teach_v that_o in_o continual_a quantity_n which_o the_o 2._o of_o the_o sâ—â—ith_v teach_v in_o number_n the_o second_o case_n demonstration_n lead_v to_o a_o absurdity_n a_o corollary_n this_o problem_n reduce_v to_o a_o theorem_a this_o proposition_n teach_v that_o in_o continual_a quantity_n which_o the_o 3._o of_o the_o second_o teach_v in_o number_n construction_n two_o case_n in_o this_o proposition_n the_o first_o case_n demonstration_n lead_v to_o a_o absurdity_n the_o second_o case_n a_o leâ—ma_n necessary_a
diameter_n be_v double_a to_o that_o square_n who_o diameter_n it_o be_v corollary_n the_o 34._o theorem_a the_o 48._o proposition_n if_o the_o square_n which_o be_v make_v of_o one_o of_o the_o side_n of_o a_o triangle_n be_v equal_a to_o the_o square_n which_o be_v make_v of_o the_o two_o other_o side_n of_o the_o same_o triangle_n the_o angle_n comprehend_v under_o those_o two_o other_o side_n be_v a_o right_a angle_n svppose_v that_o abc_n be_v a_o triangle_n and_o let_v the_o square_n which_o be_v make_v of_o one_o of_o the_o side_n there_o namely_o of_o the_o side_n bc_o be_v equal_a to_o the_o square_n which_o be_v make_v of_o the_o side_n basilius_n and_o ac_fw-la then_o i_o say_v that_o the_o angle_n bac_n be_v a_o right_a angle_n raise_v up_o by_o the_o 11._o proposition_n from_o the_o point_n a_o unto_o the_o right_a line_n ac_fw-la a_o perpendicular_a line_n ad._n and_o by_o the_o third_o proposition_n unto_o the_o line_n ab_fw-la put_v a_o equal_a line_n ad._n and_o by_o the_o first_o petition_n draw_v a_o right_a line_n from_o the_o point_n d_o to_o the_o poinâ—_n c._n and_o forasmuch_o as_o the_o line_n dam_n be_v equal_a to_o the_o line_n ab_fw-la the_o square_n which_o be_v make_v of_o the_o line_n dam_n be_v equal_a to_o the_o square_n which_o be_v make_v of_o the_o line_n ab_fw-la put_v the_o square_n of_o the_o line_n ac_fw-la common_a to_o they_o both_o wherefore_o the_o square_n of_o the_o line_n dam_n and_o ac_fw-la be_v equal_a to_o the_o square_n of_o the_o line_n basilius_n and_o ac_fw-la but_o by_o the_o proposition_n go_v before_o the_o square_a of_o the_o line_n dc_o be_v equal_a to_o the_o square_n of_o the_o line_n ad_fw-la and_o ac_fw-la for_o the_o angle_n dac_o be_v a_o right_a angle_n and_o the_o square_a of_o bc_o be_v by_o supposition_n equal_a to_o the_o square_n of_o ab_fw-la and_o ac_fw-la wherefore_o the_o square_a of_o dc_o be_v equal_a to_o the_o square_n of_o bc_o wherefore_o the_o side_n dc_o be_v equal_a to_o the_o side_n bc._n and_o forasmuch_o as_o ab_fw-la be_v equal_a to_o ad_fw-la â—nd_n ac_fw-la be_v common_a to_o they_o both_o therefore_o these_o two_o side_n dam_n and_o ac_fw-la be_v equal_a to_o these_o two_o side_n basilius_n and_o ac_fw-la the_o one_o to_o the_o other_o and_o the_o base_a dc_o be_v equal_a to_o the_o base_a be_v wherefore_o by_o the_o 8._o proposition_n the_o angle_n dac_o be_v equal_a to_o the_o angle_n bac_n but_o the_o angle_n dac_o be_v a_o right_a angle_n wherefore_o also_o the_o angle_n bac_n be_v a_o right_a angle_n if_o therefore_o the_o square_n which_o be_v make_v of_o one_o of_o the_o side_n of_o a_o triangle_n be_v equal_a to_o the_o square_n which_o be_v make_v of_o the_o two_o other_o side_n of_o the_o same_o triangle_n the_o angle_n comprehend_v under_o those_o two_o other_o side_n be_v a_o right_a angle_n which_o be_v require_v to_o be_v prove_v former_a this_o proposition_n be_v the_o converse_n of_o the_o former_a and_o be_v of_o pelitarius_n demonstrate_v by_o a_o argument_n lead_v to_o a_o impossibility_n after_o this_o manner_n the_o end_n of_o the_o first_o book_n of_o euclides_n element_n ¶_o the_o second_o book_n of_o euclides_n element_n in_o this_o second_o book_n euclid_n show_v book_n what_o be_v a_o gnomon_n and_o a_o right_a angle_a parallelogram_n also_o in_o this_o book_n be_v set_v forth_o the_o power_n of_o line_n divide_v even_o and_o uneven_o and_o of_o line_n add_v one_o to_o a_o other_o the_o power_n of_o a_o line_n line_n be_v the_o square_a of_o the_o same_o line_n that_o be_v a_o square_a every_o side_n of_o which_o be_v equal_a to_o the_o line_n so_o that_o here_o be_v set_v forth_o the_o quality_n and_o propriety_n of_o the_o square_n and_o right_n line_v figure_n which_o be_v make_v of_o line_n &_o of_o their_o part_n algebra_n the_o arithmetician_n also_o our_o of_o this_o book_n gather_v many_o compendious_a rule_n of_o reckon_n and_o many_o rule_v also_o of_o algebra_n with_o the_o equation_n therein_o use_v the_o ground_n also_o of_o those_o rule_n be_v for_o the_o most_o part_n by_o this_o second_o book_n demonstrate_v book_n this_o book_n moreover_o contain_v two_o wonderful_a proposition_n one_o of_o a_o obtuse_a angle_a triangle_n and_o the_o other_o of_o a_o acute_a which_o with_o the_o aid_n of_o the_o 47._o proposition_n of_o the_o first_o book_n of_o euclid_n which_o be_v of_o a_o rectangle_n triangle_n of_o how_o great_a force_n and_o profit_n they_o be_v in_o matter_n of_o astronomy_n they_o know_v which_o have_v travail_v in_o that_o art_n wherefore_o if_o this_o book_n have_v none_o other_o profit_n be_v side_n only_o for_o these_o 2._o proposition_n sake_n it_o be_v diligent_o to_o be_v embrace_v and_o study_v the_o definition_n 1._o every_o rectangled_a parallelogram_n definition_n be_v say_v to_o be_v contain_v under_o two_o right_a line_n comprehend_v a_o right_a angle_n a_o parallelogram_n be_v a_o figure_n of_o four_o side_n be_v who_o two_o opposite_a or_o contrary_a side_n be_v equal_a the_o one_o to_o the_o other_o there_o be_v of_o parallelogram_n four_o kind_n parallelogram_n a_o square_a a_o figure_n of_o one_o side_n long_o a_o rombus_n or_o diamond_n and_o a_o romboide_n or_o diamond_n like_o figure_n as_o before_o be_v say_v in_o the_o 33._o definition_n of_o the_o first_o book_n of_o these_o four_o sort_n the_o square_a and_o the_o figure_n of_o one_o side_n long_o be_v only_o right_a angle_a parallelogram_n for_o that_o all_o their_o angle_n be_v right_a angle_n and_o either_o of_o they_o be_v contain_v according_a to_o this_o definition_n under_o two_o right_a line_n whiâ—h_o concur_v together_o and_o cause_v the_o right_a angle_n and_o contain_v the_o same_o of_o which_o two_o line_n the_o one_o be_v the_o length_n of_o the_o figure_n &_o the_o other_o the_o breadth_n the_o parallelogram_n be_v imagine_v to_o be_v make_v by_o the_o draught_n or_o motion_n of_o one_o of_o the_o line_n into_o the_o length_n of_o the_o other_o as_o if_o two_o number_n shall_v be_v multiply_v the_o one_o into_o the_o other_o as_o the_o figure_n abcd_o be_v a_o parallelogram_n and_o be_v say_v to_o be_v contain_v under_o the_o two_o right_a line_n ab_fw-la and_o ac_fw-la which_o contain_v the_o right_a angle_n bac_n or_o under_o the_o two_o right_a line_n ac_fw-la and_o cd_o for_o they_o likewise_o contain_v the_o right_a angle_n acd_o of_o which_o 2._o line_n the_o one_o namely_o ab_fw-la be_v the_o length_n and_o the_o other_o namely_o ac_fw-la be_v the_o breadth_n and_o if_o we_o imagine_v the_o line_n ac_fw-la to_o be_v draw_v or_o move_v direct_o according_a to_o the_o length_n of_o the_o line_n ab_fw-la or_o contrary_a wise_a the_o line_n ab_fw-la to_o be_v move_v direct_o according_a to_o the_o length_n of_o the_o line_n ac_fw-la you_o shall_v produce_v the_o whole_a rectangle_n parallelogram_n abcd_o which_o be_v say_v to_o be_v contain_v of_o they_o even_o as_o one_o number_n multiply_v by_o a_o other_o produce_v a_o plain_a and_o right_a angle_a superficial_a number_n as_o you_o see_v in_o the_o figure_n here_o set_v where_o the_o number_n of_o six_o or_o six_o unity_n be_v multiply_v by_o the_o number_n of_o five_o or_o by_o five_o unity_n of_o which_o multiplication_n be_v produce_v 30._o which_o number_n be_v set_v down_o and_o describe_v by_o his_o unity_n represent_v a_o plain_n and_o a_o right_a angle_a number_n wherefore_o even_o as_o equal_a number_n multiple_v by_o equal_a number_n produce_v number_n equal_v the_o one_o to_o the_o other_o so_o rectangle_n parallelogram_n which_o be_v comprehend_v under_o equal_a line_n be_v equal_a the_o one_o to_o the_o other_o definition_n 2._o in_o every_o parallelogram_n one_o of_o those_o parallelogram_n which_o soever_o it_o be_v which_o be_v about_o the_o diameter_n together_o with_o the_o two_o supplement_n be_v call_v a_o gnomon_n those_o particular_a parallelogram_n be_v say_v to_o be_v about_o the_o diameter_n of_o the_o parallelogram_n which_o have_v the_o same_o diameter_n which_o the_o whole_a parallelogram_n have_v and_o supplement_n be_v such_o which_o be_v without_o the_o diameter_n of_o the_o whole_a parallelogram_n as_o of_o the_o parallelogram_n abcd_o the_o partial_a or_o particular_a parallelogram_n gkcf_n and_o ebkh_n be_v parallelogram_n about_o the_o diameter_n for_o that_o each_o of_o they_o have_v for_o his_o diameter_n a_o part_n of_o the_o diameter_n of_o the_o whole_a parallelogram_n as_o ck_v and_o kb_v the_o particular_a diameter_n be_v part_n of_o the_o line_n cb_o which_o be_v the_o diameter_n of_o the_o whole_a parallelogram_n and_o the_o two_o parallelogram_n aegk_n and_o khfd_n be_v supplement_n because_o they_o be_v without_o the_o diameter_n of_o the_o whole_a parallelogram_n namely_o cb._n now_o any_o one_o of_o those_o partial_a parallelogram_n
poligonon_n figure_n of_o 24._o side_n likewise_o of_o the_o hexagon_n ab_fw-la and_o of_o the_o pentagon_n ac_fw-la shall_v be_v make_v a_o poligonon_n figure_n of_o 30._o side_n one_o of_o who_o side_n shall_v subtend_v the_o ark_n bc._n for_o the_o denomination_n of_o ab_fw-la which_o be_v 6._o exceed_v the_o denomination_n of_o ac_fw-la which_o be_v 5._o only_a by_o unity_n so_o also_o forasmuch_o as_o the_o denomination_n of_o ab_fw-la which_o be_v 6._o exceed_v the_o denomination_n of_o ae_n which_o be_v 3._o by_o 3._o therefore_o the_o ark_n be_v shall_v contain_v 3._o side_n of_o a_o poligonon_n figure_n of_o .18_o side_n and_o observe_v this_o self_n same_o method_n and_o order_n a_o man_n may_v find_v out_o infinite_a side_n of_o a_o poligonon_n figure_n the_o end_n of_o the_o four_o book_n of_o euclides_n element_n ¶_o the_o five_o book_n of_o euclides_n element_n this_o five_o book_n of_o euclid_n be_v of_o very_o great_a commodity_n and_o use_n in_o all_o geometry_n book_n 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in_o that_o sort_n of_o understanding_n there_o be_v no_o number_n but_o that_o it_o may_v be_v divide_v into_o two_o equal_a part_n as_o this_o number_n 7_o may_v be_v divide_v into_o 3_o part_n namely_o 3._o 1._o and_o 3._o of_o which_o two_o namely_o 3_o and_o 3_o be_v equal_a yet_o iâ—_z not_o 7_o a_o even_a number_n because_o 3_o and_o 3_o add_v together_o make_v not_o 7._o boetius_fw-la therefore_o in_o the_o first_o book_n of_o his_o arithmetic_n boetius_fw-la for_o the_o more_o plain_n after_o this_o manner_n define_v a_o even_a number_n numerun_v paâ—_n est_fw-la qui_fw-la potest_fw-la in_o â—quâ—lia_fw-la due_a dividi_fw-la una_fw-la mediâ—â—â—n_fw-la intercedenta_fw-la that_o be_v a_o even_a number_n be_v that_o which_o may_v be_v divide_v into_o two_o equal_a part_n without_o a_o unity_n come_v between_o they_o number_n as_o 8_o be_v divide_v into_o 4_o and_o 4_o two_o equal_a part_n without_o anâ—_z unity_n come_v between_o they_o which_o add_v together_o make_v 8_o so_o 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the_o part_n add_v together_o be_v not_o equal_a to_o the_o whole_a nor_o make_v the_o whole_a but_o make_v either_o more_o or_o less_o wherefore_o of_o imperfect_a number_n there_o be_v two_o kind_n number_n the_o one_o be_v call_v abundant_a or_o a_o building_n the_o other_o 〈◊〉_d or_o want_v a_o number_n abund_v be_v that_o who_o part_n be_v all_o add_v together_o make_v more_o than_o the_o wâ—â—ds_n number_n who_o part_n they_o be_v as_o 12._o be_v a_o abundant_a number_n for_o all_o the_o parteâ—_n of_o 12._o namely_o 6.4.3.2_o and_o 1._o add_v together_o make_v 16_o which_o be_v more_o than_o 12._o likewise_o 18._o be_v a_o number_n abund_v all_o his_o partâ—_n name_o 9.6.3.2_o and_o 1._o add_v together_o make_v 20._o which_o be_v more_o than_o 18_o and_o so_o of_o other_o wanâ—â—ngâ—_n a_o number_n diminute_a or_o want_v be_v that_o who_o part_n be_v all_o add_v together_o make_v less_o than_o the_o whole_a or_o number_n who_o part_n they_o be_v as_o 9_o be_v a_o diminute_a or_o 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which_o denominate_v what_o part_n 6._o be_v of_o 24_o be_v great_a than_o 3._o which_o denominate_v what_o part_n 8._o be_v of_o 24._o therefore_o be_v 6._o a_o less_o part_n of_o 24â—_n than_o be_v 8._o and_o so_o be_v 8._o a_o great_a part_n of_o 24._o they_o 6._o be_v and_o so_o in_o other_o sentence_n 2_o whatsoever_o number_n be_v equemultiplices_fw-la to_o one_o &_o the_o self_n same_o number_n or_o to_o equal_a number_n be_v also_o equal_a the_o one_o to_o the_o other_o as_o if_o unto_o the_o number_n 3_o be_v take_v two_o number_n contain_v the_o same_o number_n four_o time_n that_o be_v be_v equemultiplices_fw-la to_o the_o same_o number_n three_o the_o say_v two_o number_n shall_v be_v equal_a for_o 4._o time_n 3._o will_v ever_o be_v 12._o so_o also_o will_v it_o be_v if_o unto_o the_o two_o equal_a number_n 3._o &_o 3._o be_v take_v two_o number_n the_o one_o contain_v the_o one_o number_n 3._o four_o time_n the_o other_o contain_v the_o other_o number_n 3._o also_o four_o time_n that_o be_v be_v equemultiplices_fw-la to_o the_o equal_a number_n 3._o and_o 3._o sentence_n 3_o those_o number_n to_o who_o one_o and_o the_o self_n same_o number_n be_v equmultiplex_n or_o who_o equemultiplices_fw-la be_v equal_a be_v also_o equal_a the_o on_o to_o the_o other_o as_o if_o the_o number_n 18._o be_v equemultiplex_n to_o any_o two_o number_n that_o be_v contain_v any_o two_o number_n twice_o thrice_o four_o time_n &_o c_o as_o for_o example_n 3._o time_n then_o be_v the_o say_v two_o number_n equal_a for_o 18._o divide_v by_o 3._o will_v ever_o bring_v forth_o 6._o so_o that_o that_o division_n make_v twice_o will_v bring_v forth_o 6._o and_o 6._o two_o equal_a number_n so_o also_o will_v it_o follow_v if_o the_o two_o number_n have_v equal_a equemultiplices_fw-la namely_o if_o 18._o and_o 18._o which_o be_v equal_a number_n contain_v any_o two_o number_n 3._o time_n 4_o if_o a_o number_n measure_v the_o whole_a and_o a_o part_n take_v away_o it_o shall_v also_o measure_v the_o residue_n sentence_n as_o if_o from_o 24._o be_v take_v away_o 9_o there_o remain_v 15._o and_o for_o as_o much_o as_o the_o number_n 3_o measure_v the_o whole_a number_n 24_o &_o also_o the_o number_n take_v away_o namely_o 9_o it_o shall_v also_o measure_v the_o residue_n which_o be_v 15â—_n for_o 3._o measure_v 15_o by_o five_o five_o time_n 3._o be_v 15._o and_o so_o of_o other_o sentence_n 5_o if_o a_o number_n measure_v any_o number_n it_o also_o measure_v every_o number_n that_o the_o say_a number_n measure_v as_o the_o number_n 6._o measure_v the_o number_n 12._o shall_v also_o measure_v all_o the_o number_n that_o 1â—_o measurethâ—_n as_o the_o number_n 24.36.48.60_o and_o so_o forth_o which_o the_o number_n 12._o do_v measure_n by_o the_o numberâ—_n 2.3.4_o and_o 5._o and_o for_o as_o much_o as_o the_o number_n 12._o do_v measure_v the_o number_n 24.36.48_o and_o 60._o and_o the_o number_n 6_o do_v measure_v the_o number_n 12._o namely_o by_o 2._o it_o follow_v by_o this_o common_a sentence_n that_o the_o number_n 6._o measure_v each_o of_o thâ—se_a number_n 24._o 36.48_o and_o 60._o and_o so_o of_o other_o 6_o if_o a_o number_n measure_v two_o number_n sentence_n it_o shall_v also_o measure_v the_o number_n compose_v of_o they_o as_o the_o number_n 3_o measure_v these_o two_o number_n 6._o and_o 9â—_n it_o measure_v 6._o by_o 2d_o and_o 9_o by_o 3._o and_o therefore_o by_o this_o common_a sentence_n it_o measure_v the_o number_n 15._o which_o be_v compose_v of_o the_o number_n 6._o and_o 9_o namely_o it_o measure_v it_o by_o 5._o 7_o if_o in_o number_n there_o be_v proportion_n how_o manysoever_o equal_a or_o the_o self_n same_o to_o one_o proportion_n sentence_n they_o shall_v alâ—o_o be_v equal_a or_o the_o self_n same_o the_o one_o to_o the_o other_o as_o if_o the_o proportion_n of_o the_o number_n 6._o to_o the_o number_n 3._o be_v as_o the_o proportion_n of_o the_o number_n 8._o to_o the_o number_n 4_o if_o also_o the_o proportion_n of_o the_o number_n 10._o to_o the_o number_n 5._o be_v as_o the_o proportion_n of_o the_o number_n 8._o to_o the_o number_n 4_o then_o shall_v the_o proportion_n of_o the_o number_n 6._o to_o the_o number_n 3._o be_v as_o the_o proportion_n of_o the_o number_n 10._o be_v to_o the_o number_n 5_o namely_o each_o proportion_n be_v duple_n and_o so_o of_o other_o euclid_n in_o his_o â—_o book_n the_o 11._o proposition_n demonstrate_v this_o also_o in_o continual_a quantity_n which_o although_o as_o touch_v that_o kind_n of_o quantity_n it_o may_v have_v be_v put_v also_o as_o a_o principle_n as_o in_o number_n he_o take_v it_o yet_o for_o that_o in_o all_o magnitude_n their_o proportion_n can_v not_o be_v express_v as_o have_v before_o be_v note_v &_o shall_v be_v afterward_o in_o the_o ten_o book_n more_o at_o large_a make_v manifest_v therefore_o he_o demonstrate_v it_o there_o in_o that_o place_n and_o prove_v that_o it_o be_v true_a as_o touch_v all_o proportion_n general_o whither_o they_o be_v rational_a or_o irrational_a ¶_o the_o first_o proposition_n the_o first_o theorem_a if_o there_o be_v give_v two_o unequal_a number_n and_o if_o in_o take_v the_o less_o continual_o from_o the_o great_a the_o number_n remain_v do_v not_o measure_v the_o number_n go_v before_o until_o it_o shall_v come_v to_o unity_n then_o be_v those_o number_n which_o be_v at_o the_o beginning_n give_v prime_a the_o one_o to_o the_o other_o svppose_v that_o there_o be_v two_o unequal_a number_n ab_fw-la the_o great_a and_o cd_o the_o less_o and_o from_o ab_fw-la the_o great_a take_v away_o cd_o the_o less_o as_o oâ—ten_v as_o you_o can_v leave_v favorina_n constrâ—ctioâ—_n and_o from_o cd_o take_v away_o favorina_n as_o often_o as_o you_o can_v leave_v the_o number_n gc_o and_o from_o favorina_n take_v away_o gc_o as_o often_o as_o you_o can_v and_o so_o do_v continual_o till_o there_o remain_v only_a unity_n which_o let_v be_v ha._n absurdity_n then_o i_o say_v that_o no_o number_n measure_v the_o number_n ab_fw-la and_o cd_o for_o if_o it_o be_v possible_a let_v some_o number_n measure_v they_o and_o let_v the_o same_o be_v e._n now_o cd_o measure_n ab_fw-la leave_v a_o less_o number_n than_o itself_o which_o let_v be_v faâ—_n â—_z and_o favorina_n measure_v dc_o leave_v also_o a_o less_o than_o itself_o namely_o gc_o and_o gc_o measure_v favorina_n leave_v unity_n ha._n and_o forasmuch_o as_o the_o number_n e_o measure_v dc_o and_o the_o number_n cd_o measure_v the_o number_n bf_o therefore_o the_o number_n e_o also_o measure_v bf_o and_o it_o measure_v the_o whole_a number_n basilius_n wherefore_o it_o also_o measure_v that_o which_o remain_v namely_o the_o number_n favorina_n by_o the_o
that_o what_o part_n ae_n be_v of_o cf_o the_o same_o part_n be_v ebb_v of_o cg_o therefore_o what_o part_n ae_n be_v of_o cf_o the_o same_o part_n by_o the_o 5._o of_o the_o seven_o be_v ab_fw-la of_o fg._n but_o what_o part_n ae_n be_v of_o cf_o the_o same_o part_n by_o supposition_n be_v ab_fw-la of_o cd_o wherefore_o what_o part_v ab_fw-la be_v of_o fg_o the_o self_n same_o part_n be_v ab_fw-la of_o cd_o construction_n wherefore_o ab_fw-la be_v one_o &_o the_o self_n same_o part_n of_o both_o these_o number_n gf_o and_o cd_o wherefore_o gf_o be_v equal_a unto_o cd_o by_o the_o second_o common_a sentence_n of_o the_o seven_o take_v away_o cf_o which_o be_v common_a to_o they_o both_o wherefore_o the_o residue_n gc_o be_v equal_a unto_o the_o residue_n fd._n demonstration_n and_o forasmuch_o as_o what_o part_n ae_n be_v of_o cf_o the_o same_o part_n be_v ebb_v of_o gc_o but_o gc_o be_v equal_a unto_o fd_o therefore_o what_o part_n ae_n be_v of_o fc_n the_o self_n same_o part_n be_v ebb_v of_o fd._n but_o what_o part_n ae_n be_v of_o cf_o the_o same_o part_n be_v ab_fw-la of_o cd_o wherefore_o what_o part_n ebb_n be_v of_o fd_o the_o same_o part_n be_v ab_fw-la of_o cd_o wherefore_o the_o residue_n ebb_n be_v of_o the_o residue_n fd_v the_o self_n same_o part_n that_o the_o whole_a ab_fw-la be_v of_o the_o whole_a cd_o which_o be_v require_v to_o be_v demonstrate_v ¶_o the_o 6._o theorem_a the_o 8._o proposition_n if_o a_o number_n be_v of_o a_o number_n the_o self_n same_o part_n that_o a_o part_n take_v away_o be_v of_o a_o part_n take_v away_o the_o residue_n also_o shall_v be_v of_o the_o residue_n the_o self_n same_o part_n that_o the_o whole_a be_v of_o the_o whole_a svppose_v that_o the_o number_n ab_fw-la be_v of_o the_o number_n cd_o the_o self_n same_o part_n that_o the_o part_n take_v away_o ae_n be_v of_o the_o part_n take_v cf._n then_o i_o say_v that_o the_o residue_n ebb_n be_v of_o the_o residue_n fd_v the_o self_n same_o part_n that_o the_o whole_a ab_fw-la be_v of_o the_o whole_a cd_o unto_o ab_fw-la put_v a_o equal_a number_n gh_o constuâ—ction_n wherefore_o what_o part_v gh_o be_v of_o cd_o the_o self_n same_o part_n be_v ae_n of_o cf._n divide_v gh_o into_o the_o part_n of_o cd_o that_o be_v gk_o and_o kh_o and_o likewise_o ae_n into_o the_o part_n of_o cf_o that_o be_v into_o all_o and_o le._n now_o then_o the_o multitude_n of_o these_o gk_o and_o kh_o be_v equal_a unto_o the_o multitude_n of_o these_o all_o and_o le._n demonstration_n and_o forasmuch_o as_o what_o part_n gk_o be_v of_o cd_o the_o self_n same_o part_n be_v all_o of_o cf_o but_o cd_o be_v great_a than_o cf._n wherefore_o gk_o be_v great_a than_o al._n put_v unto_o all_o a_o equal_a number_n mg_o wherefore_o what_o part_n gk_o be_v of_o cd_o the_o same_o part_n be_v gm_n of_o cf._n wherefore_o the_o residue_n mk_o be_v by_o the_o 7._o of_o the_o seven_o of_o the_o residue_n fd_o the_o self_n same_o part_n that_o the_o whole_a gk_o be_v of_o the_o whole_a cd_o again_o forasmuch_o as_o what_o part_n kh_o be_v of_o cd_o the_o self_n same_o part_n be_v el_n of_o cf_o but_o cd_o be_v great_a than_o cf._n wherefore_o hk_n be_v great_a than_o el._n put_v unto_o el_n a_o equal_a number_n kn._n wherefore_o what_o part_n kh_o be_v of_o cd_o the_o self_n same_o part_n be_v kn_n of_o cf._n wherefore_o the_o residue_n also_o nh_v be_v by_o the_o 7._o of_o the_o seven_o of_o the_o residue_n fd_o the_o self_n same_o part_n that_o the_o whole_a kh_o be_v of_o the_o whole_a dc_o wherefore_o both_o these_o mk_o and_o nh_v add_v together_o be_v by_o the_o 5._o of_o the_o seven_o of_o df_o the_o self_n same_o part_n that_o the_o whole_a hg_o be_v of_o the_o whole_a cd_o but_o both_o these_o mk_o and_o nh_v add_v together_o be_v equal_a unto_o ebb_n and_o hg_o be_v equal_a unto_o basilius_n wherefore_o the_o residue_n ebb_n be_v of_o the_o residue_n fd_v the_o self_n same_o part_n that_o the_o whole_a ab_fw-la be_v of_o the_o whole_a cd_o which_o be_v require_v to_o be_v prove_v ¶_o a_o other_o demonstration_n after_o flussates_n suppose_v that_o the_o number_n ab_fw-la be_v of_o the_o number_n cd_o the_o self_n same_o part_n that_o the_o part_n take_v away_o ae_n be_v of_o the_o part_n take_v away_o cf._n flussates_n then_o i_o say_v that_o the_o residue_n ebb_n be_v of_o the_o residue_n fd_v the_o self_n same_o part_n that_o the_o whole_a ab_fw-la be_v of_o the_o whole_a cd_o let_v ebb_n be_v of_o ci_o the_o self_n same_o part_n that_o ab_fw-la be_v of_o cd_o or_o ae_n of_o câ—_n now_o forasmuch_o as_o ebb_n be_v of_o ci_o the_o self_n same_o part_n that_o ae_n be_v of_o cf_o therefore_o both_o these_o ae_n and_o ebb_v add_v together_o be_v of_o both_o these_o cf_o and_o ci_o add_v together_o that_o be_v the_o whole_a ab_fw-la be_v of_o the_o whole_a fi_n the_o self_n same_o part_n that_o ae_n be_v of_o cf_o by_o the_o six_o of_o this_o book_n but_o what_o part_v ae_n be_v of_o cf_o the_o self_n same_o part_n be_v the_o number_n ab_fw-la of_o the_o number_n cd_o by_o supposition_n wherefore_o what_o part_v the_o number_n ab_fw-la be_v of_o the_o number_n fi_n though_o self_n same_o part_n be_v the_o same_o number_n ab_fw-la of_o the_o number_n cd_o wherefore_o the_o number_n fi_n and_o cd_o be_v equal_a take_v away_o the_o number_n cf_o which_o be_v common_a to_o they_o both_o wherefore_o the_o number_n remain_v ci_o and_o â—d_a be_v equal_a wherefore_o what_o part_v the_o number_n ebb_n be_v of_o the_o number_n ci_o the_o self_n same_o part_n be_v the_o same_o number_n ebb_n of_o the_o number_n fd._n but_o what_o part_n ebb_n be_v of_o ci_o the_o self_n same_o part_n by_o construction_n be_v ab_fw-la of_o cd_o wherefore_o what_o part_v the_o residue_n ebb_n be_v of_o the_o residue_n fd_o the_o self_n same_o part_n be_v the_o whole_a ab_fw-la of_o the_o whole_a cd_o which_o be_v require_v to_o be_v prove_v ¶_o the_o 7._o theorem_a the_o 9_o proposition_n if_o a_o number_n be_v a_o part_n of_o a_o number_n and_o if_o a_o other_o number_n be_v the_o self_n same_o part_n of_o a_o other_o number_n then_o alternate_o what_o part_n or_o part_n the_o first_o be_v of_o the_o three_o the_o self_n same_o part_n or_o part_n shall_v the_o second_o be_v of_o the_o four_o svppose_v that_o the_o number_n a_o be_v of_o the_o number_n bc_o the_o self_n same_o part_n that_o a_o other_o number_n d_z be_v of_o a_o other_o number_n ef._n and_o let_v a_o be_v less_o then_o d._n then_o i_o say_v that_o alternate_o what_o part_n or_o part_n a_o be_v of_o d_o the_o self_n same_o part_n or_o part_n be_v bc_o of_o ef._n construction_n for_o forasmuch_o as_o what_o part_n a_o be_v of_o bc_o the_o self_n same_o part_n be_v d_o of_o of_o therefore_o how_o many_o number_n there_o be_v in_o bc_o equal_a unto_o a_o so_o many_o be_v there_o in_o of_o equal_a unto_o d._n divide_v bc_o into_o the_o number_n equal_a unto_o a_o that_o be_v into_o bg_o &_o gc_o and_o likewise_o of_o into_o the_o number_n equal_a unto_o d_o that_o be_v into_o eh_o and_o hf._n now_o then_o the_o multitude_n of_o these_o bg_o and_o gc_o be_v equal_a unto_o the_o multitude_n of_o these_o eh_o &_o hf._n demonstration_n and_o forasmuch_o as_o the_o number_n bg_o and_o gc_o be_v equal_a the_o one_o to_o the_o other_o the_o number_n also_o eh_o and_o hf_o be_v equal_a the_o one_o to_o the_o other_o and_o the_o multitude_n of_o these_o bg_o &_o gc_o be_v equal_a unto_o the_o multitude_n of_o these_o eh_o and_o hf._n wherefore_o what_o part_n or_o part_n bg_o be_v of_o eh_o the_o self_n same_o part_n or_o part_n be_v gc_o of_o hf._n wherefore_o what_o part_n or_o part_n bg_o be_v of_o eh_o the_o self_n same_o part_n or_o part_n by_o the_o five_o &_o six_o of_o the_o seven_o be_v bg_o and_o gc_o add_v together_o of_o eh_o and_o hf_o add_v together_o but_o bg_o be_v equal_a unto_o a_o and_o eh_o unto_o d._n wherefore_o what_o part_n or_o part_n a_o be_v of_o d_o the_o self_n same_o part_n or_o part_n be_v bg_o of_o of_o which_o be_v require_v to_o be_v demonstrate_v ¶_o the_o 8._o theorem_a the_o 10._o proposition_n if_o a_o number_n be_v part_n of_o a_o number_n and_o a_o other_o number_n the_o self_n same_o part_n of_o a_o other_o number_n then_o alternate_o what_o part_n or_o part_v the_o first_o be_v of_o the_o three_o the_o self_n same_o part_n or_o part_n be_v the_o second_o of_o the_o four_o svppose_v that_o the_o number_n ab_fw-la be_v of_o the_o number_n c_o the_o self_n same_o part_n that_o a_o
which_o be_v require_v to_o be_v prove_v ¶_o the_o 34._o theorem_a the_o 34._o proposition_n if_o a_o number_n be_v neither_o double_v from_o two_o nor_o have_v to_o his_o half_a part_n a_o odd_a number_n it_o shall_v be_v a_o number_n both_o even_o even_o and_o even_o odd_a svppose_v that_o the_o number_n a_o be_v a_o number_n neither_o double_v from_o the_o number_n two_o neither_o also_o let_v it_o have_v to_o his_o half_a part_n a_o odd_a number_n then_o i_o say_v that_o a_o be_v a_o number_n both_o even_o even_o and_o even_o odd_a demonstration_n that_o a_o be_v even_o even_o it_o be_v manifest_a for_o the_o half_a thereof_o be_v not_o odd_a and_o be_v measure_v by_o the_o number_n 2._o which_o be_v a_o even_a number_n now_o i_o say_v that_o it_o be_v even_o odd_a also_o for_o if_o we_o divide_v a_o into_o two_o equal_a part_n and_o so_o continue_v still_o we_o shall_v at_o the_o length_n light_n upon_o a_o certain_a odd_a number_n which_o shall_v measure_v a_o by_o a_o even_a number_n for_o if_o we_o shall_v not_o light_v upon_o such_o a_o odd_a number_n which_o measure_v a_o by_o a_o even_a number_n we_o shall_v at_o the_o length_n come_v unto_o the_o number_n two_o and_o so_o shall_v a_o be_v one_o of_o those_o number_n which_o be_v double_v from_o two_o upward_a which_o be_v contrary_a to_o the_o supposition_n wherefore_o a_o be_v even_o odd_a and_o it_o be_v prove_v that_o it_o be_v even_o even_o wherefore_o a_o be_v a_o number_n both_o even_o even_o and_o even_o odd_a which_o be_v require_v to_o be_v demonstrate_v this_o proposition_n and_o the_o two_o former_a manifest_o declare_v that_o which_o we_o note_v upon_o the_o ten_o definition_n of_o the_o seven_o book_n namely_o that_o campane_n and_o flussates_n and_o diverse_a other_o interpreter_n of_o euclid_n only_a theon_n except_v do_v not_o right_o understand_v the_o 8._o and_o 9_o definition_n of_o the_o same_o book_n concern_v a_o number_n even_o even_o and_o a_o number_n even_o odd_a for_o in_o the_o one_o definition_n they_o add_v unto_o euclides_n word_n extant_a in_o the_o greek_a this_o word_n only_o as_o we_o there_o note_v and_o in_o the_o other_o this_o word_n all_o so_o that_o after_o their_o definition_n a_o number_n can_v not_o be_v even_o even_o unless_o it_o be_v measure_v only_o by_o even_a number_n likewise_o a_o number_n can_v not_o be_v even_o odd_a unless_o all_o the_o even_a number_n which_o do_v measure_v it_o do_v measure_v it_o by_o a_o odd_a number_n the_o contrary_a whereof_o in_o this_o proposition_n we_o manifest_o see_v for_o here_o euclid_n prove_v that_o one_o number_n may_v be_v both_o even_o even_o and_o even_o odd_a and_o in_o the_o two_o former_a proposition_n he_o prove_v that_o some_o number_n be_v even_o even_o only_o and_o some_o even_o odd_a only_o which_o word_n only_o have_v be_v in_o vain_a of_o he_o add_v if_o no_o number_n even_o even_o can_v be_v measure_v by_o a_o odd_a number_n or_o if_o all_o the_o number_n that_o measure_v a_o number_n even_o odd_a must_v needs_o measure_v it_o by_o a_o odd_a number_n although_o campane_n and_o flussates_n to_o avoid_v this_o absurdity_n have_v wrest_v the_o 32._o proposition_n of_o this_o book_n from_o the_o true_a sense_n of_o the_o greek_a and_o as_o it_o be_v interpret_v of_o theon_n so_o also_o have_v flussates_n wrest_v the_o 33._o proposition_n for_o whereas_o euclid_n say_v every_o number_n produce_v by_o the_o double_n of_o two_o upward_a be_v even_o even_o only_o they_o say_v only_o the_o number_n produce_v by_o the_o double_n of_o two_o be_v even_o even_o likewise_o whereas_o euclid_n say_v a_o number_n who_o hafle_n part_n be_v odd_a be_v even_o odd_a only_o flussates_n say_v only_o a_o number_n who_o half_a part_n be_v odd_a be_v even_o odd_a which_o their_o interpretation_n be_v not_o true_a neither_o can_v be_v apply_v to_o the_o proposition_n as_o they_o be_v extant_a in_o the_o greek_a in_o deed_n the_o say_v 32._o and_o 33._o proposition_n as_o they_o put_v they_o be_v true_a touch_v those_o number_n which_o be_v even_o even_o only_o or_o even_o odd_a only_o for_o no_o number_n be_v even_o even_o only_o but_o those_o only_o which_o be_v double_v from_o two_o upward_o likewise_o no_o number_n be_v even_o odd_a only_o but_o those_o only_o who_o half_o be_v a_o odd_a number_n but_o this_o let_v not_o but_o that_o a_o number_n may_v be_v even_o even_o although_o it_o be_v not_o double_v from_o two_o upward_a &_o also_o that_o a_o number_n may_v be_v even_o odd_a although_o it_o have_v not_o to_o his_o half_n a_o odd_a number_n as_o in_o this_o 34._o proposition_n euclid_n have_v plain_o prove_v which_o thing_n can_v by_o no_o mean_n be_v true_a if_o the_o foresay_a 32._o &_o 33._o propositon_n of_o this_o book_n shall_v have_v that_o sense_n and_o meaning_n wherein_o they_o take_v it_o ¶_o the_o 35._o theorem_a the_o 35._o proposition_n if_o there_o be_v number_n in_o continual_a proportion_n how_o many_o soever_o and_o if_o from_o the_o second_o and_o last_o be_v take_v away_o number_n equal_a unto_o the_o first_o as_o the_o excess_n of_o the_o second_o be_v to_o the_o first_o so_o be_v the_o excess_n of_o the_o last_o to_o all_o the_o number_n go_v before_o the_o last_o svppose_v that_o these_o number_n a_o bc_o d_o and_o of_o be_v in_o continual_a proportion_n beginning_n at_o a_o the_o least_o and_o from_o bc_o which_o be_v the_o second_o take_v away_o cg_o equal_a unto_o the_o first_o namely_o to_o a_o and_o likewise_o from_o of_o the_o last_o take_v away_o fh_o equal_a also_o unto_o the_o first_o namely_o to_o a._n then_o i_o say_v that_o as_o the_o excess_n bg_o be_v to_o a_o the_o first_o so_o be_v he_o the_o excess_n to_o all_o the_o number_n d_o bc_o and_o a_o which_o go_v before_o the_o last_o number_n namely_o ef._n demonstration_n forasmuch_o as_o of_o be_v the_o great_a for_o the_o second_o be_v suppose_v great_a than_o the_o first_o put_v the_o number_n fl_fw-mi equal_v to_o the_o number_n d_o and_o likewise_o the_o number_n fk_o equal_a to_o the_o number_n bc._n and_o forasmuch_o as_o fk_n be_v equal_a unto_o cb_o of_o which_o fh_o be_v equal_a unto_o gc_o therefore_o the_o residue_n hk_o be_v equal_a unto_o the_o residue_n gb_o and_o for_o that_o as_o the_o whole_a fâ—_n be_v to_o the_o whole_a fl_fw-mi so_o be_v the_o part_n take_v away_o fl_fw-mi to_o the_o part_n take_v away_o fk_a therefore_o the_o residue_n le_fw-fr be_v to_o the_o residue_n kl_o as_o the_o whole_a â—e_n be_v to_o the_o whole_a fl_fw-mi by_o the_o 11._o of_o the_o seven_o so_o likewise_o for_o that_o fl_fw-mi be_v to_o fk_v as_o fk_n be_v to_o fh_v kl_o shall_v be_v to_o hk_v as_o the_o whole_a fl_fw-mi be_v to_o the_o whole_a fk_n by_o the_o same_o proposition_n but_o as_o fe_o be_v to_o fl_fw-mi and_o as_o fl_fw-mi be_v to_o fk_v and_o fk_a to_o fh_v so_o be_v fe_o to_o d_z and_o d_z to_o bc_o and_o bc_o 〈◊〉_d a._n wherefore_o as_o le_fw-fr be_v to_o kl_o and_o as_o kl_o be_v to_o hk_v so_o be_v d_z to_o bc._n wherefore_o alternate_o by_o the_o 23._o of_o the_o seven_o as_z le_fw-fr be_v to_o d_z so_o be_v kl_o to_o be_v bc_o and_o as_o kl_o be_v to_o bc_o so_o be_v hk_v to_o a._n wherefore_o also_o as_o one_o of_o the_o antecedentes_fw-la be_v to_o one_o of_o the_o consequentes_fw-la so_o be_v all_o the_o antecedentes_fw-la to_o all_o the_o consequentes_fw-la wherefore_o as_o kh_o be_v to_o a_o so_o be_v hk_v kl_o and_o le_fw-fr to_o d_z bc_o and_o a_o by_o the_o 12._o of_o the_o seven_o but_o it_o be_v prove_v that_o kh_o be_v equal_a unto_o bg_o wherefore_o as_o bg_o which_o be_v the_o excess_n of_o the_o second_o be_v to_o a_o so_o be_v eh_o the_o excess_n of_o the_o last_o unto_o the_o number_n go_v before_o d_o bc_o and_o a._n wherefore_o as_o the_o excess_n of_o the_o second_o be_v unto_o the_o first_o so_o be_v the_o excess_n of_o the_o last_o to_o all_o the_o number_n go_v before_o the_o last_o which_o be_v require_v to_o be_v prove_v ¶_o the_o 36._o theorem_a the_o 36._o proposition_n if_o from_o unity_n be_v take_v number_n how_o many_o soever_o in_o double_a proportion_n continual_o until_o the_o whole_a add_v together_o be_v a_o prime_a number_n and_o if_o the_o whole_a multiply_v the_o last_o produce_v any_o number_n that_o which_o be_v produce_v be_v a_o perfect_v number_n number_n svppose_v that_o from_o unity_n be_v take_v these_o number_n a_o b_o c_o d_o in_o double_a proportion_n continual_o so_o that_o all_o those_o number_n a_o b_o c_o d_o &_o unity_n add_v together_o make_v a_o prime_a number_n and_o let_v e_o be_v the_o number_n compose_v of_o
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
let_v the_o same_o be_v e._n and_o as_o the_o number_n d_o be_v to_o the_o number_n ab_fw-la so_o let_v the_o square_n of_o the_o line_n e_o be_v to_o the_o square_n of_o fg._n wherefore_o by_o the_o 6._o of_o the_o ten_o the_o line_n e_o be_v commensurable_a in_o power_n to_o the_o line_n fg_o &_o the_o line_n e_o be_v rational_a wherefore_o also_o the_o line_n fg_o be_v rational_a and_o for_o that_o the_o number_n d_o have_v not_o â—o_o the_o number_n ab_fw-la that_o proportion_n that_o a_o square_a number_n have_v to_o a_o square_a number_n therefore_o neither_o also_o shall_v the_o square_a of_o the_o line_n e_o have_v to_o the_o square_n of_o the_o line_n fg_o that_o proportion_n that_o a_o square_a number_n have_v to_o a_o square_a number_n wherefore_o the_o line_n fg_o be_v incommensurable_a in_o length_n to_o the_o line_n e._n again_o as_o the_o number_n basilius_n be_v to_o the_o number_n ac_fw-la so_o let_v the_o square_n of_o the_o line_n fg_o be_v to_o the_o square_n of_o the_o line_n gh_o demonstration_n wherefore_o by_o the_o 6._o of_o the_o ten_o the_o square_a of_o the_o line_n fg_o be_v commensurable_a to_o the_o square_n of_o the_o line_n gh_o and_o the_o square_a of_o the_o line_n fg_o be_v rational_a wherefore_o the_o square_a of_o the_o line_n gh_o be_v also_o rational_a wherefore_o also_o the_o line_n gh_o be_v rational_a and_o for_o that_o the_o number_n ab_fw-la have_v not_o to_o the_o number_n ac_fw-la that_o proportion_n that_o a_o square_a number_n have_v to_o a_o square_a number_n therefore_o neither_o also_o have_v the_o square_a of_o the_o line_n fg_o to_o the_o square_n of_o the_o line_n gh_o that_o proportion_n that_o a_o square_a number_n have_v to_o a_o square_a number_n wherefore_o the_o line_n fg_o be_v incommensurable_a in_o length_n to_o the_o line_n gh_o wherefore_o the_o line_n fg_o and_o gh_o be_v rational_a commensurable_a in_o power_n only_o wherefore_o the_o whole_a line_n fh_o be_v a_o binomial_a line_n i_o say_v moreover_o that_o it_o be_v a_o six_o binomial_a line_n for_o for_o that_o as_o the_o number_n d_o be_v to_o the_o number_n ab_fw-la so_o be_v the_o square_a of_o the_o line_n e_o to_o the_o square_n of_o the_o line_n fg._n and_o as_o the_o number_n basilius_n be_v to_o the_o number_n ac_fw-la so_o be_v the_o square_a of_o the_o line_n fg_o to_o the_o square_n of_o the_o line_n gh_o wherefore_o of_o equality_n by_o the_o 22._o of_o the_o five_o as_o the_o number_n d_z be_v to_o the_o number_n ac_fw-la so_o be_v the_o square_a of_o the_o line_n e_o to_o the_o square_n of_o the_o line_n gh_o but_o the_o number_n d_o have_v not_o to_o the_o number_n ac_fw-la that_o proportion_n that_o a_o square_a number_n have_v to_o a_o square_a number_n wherefore_o neither_o also_o have_v the_o square_a of_o the_o line_n e_o to_o the_o square_n of_o the_o line_n gh_o that_o proportion_n that_o a_o square_a number_n have_v to_o a_o square_a number_n wherefore_o the_o line_n e_o be_v incommensurable_a in_o length_n to_o the_o line_n gh_o and_o it_o be_v already_o prove_v that_o the_o line_n fg_o be_v also_o incommensurable_a in_o length_n to_o the_o line_n e._n wherefore_o either_o of_o these_o line_n fg_o and_o gh_o be_v incommensurable_a in_o length_n to_o the_o line_n e._n and_o for_o that_o as_o the_o number_n â—a_o be_v to_o the_o number_n ac_fw-la so_o be_v the_o square_a of_o the_o line_n fg_o to_o the_o square_n of_o the_o line_n gh_o therefore_o the_o square_n of_o the_o line_n fg_o be_v great_a than_o the_o square_a of_o the_o line_n gh_o unto_o the_o square_n of_o the_o line_n fg_o let_v the_o square_n of_o the_o line_n gh_o and_o king_n be_v equal_a wherefore_o by_o eversion_n of_o proportion_n as_o the_o number_n ab_fw-la be_v to_o the_o number_n bc_o so_o be_v the_o square_a of_o the_o line_n fg_o to_o the_o square_n of_o the_o line_n k._n but_o the_o number_n ab_fw-la have_v not_o to_o the_o number_n bc_o that_o proportion_n that_o a_o square_a number_n have_v to_o a_o square_a number_n wherefore_o neither_o also_o have_v the_o square_a of_o the_o line_n fg_o to_o the_o square_n of_o the_o line_n king_n that_o proportion_n that_o a_o square_a number_n have_v to_o a_o square_a number_n wherefore_o the_o line_n fg_o be_v incommensurable_a in_o length_n unto_o the_o line_n k._n wherefore_o the_o line_n fg_o be_v in_o power_n more_o than_o the_o line_n gh_o by_o the_o square_n of_o a_o line_n incommensurable_a in_o length_n to_o it_o and_o the_o line_n fg_o and_o gh_o be_v rational_a commensurable_a in_o power_n only_o and_o neither_o of_o the_o line_n fg_o &_o gh_o be_v commensurable_a in_o length_n to_o the_o rational_a line_n give_v namely_o to_o e._n wherefore_o the_o line_n fh_o be_v a_o six_o binomial_a line_n which_o be_v require_v to_o be_v find_v out_o ¶_o a_o corollary_n add_v out_o of_o flussates_n by_o the_o 6._o forme_a proposition_n it_o iâ—_z manifest_a hoâ—_n 〈◊〉_d divide_v any_o right_a line_n give_v into_o the_o name_n of_o every_o one_o of_o the_o sixâ—_o foresay_a binomiall_n line_n flussates_n for_o if_o it_o be_v require_v to_o divide_v a_o right_a line_n give_v into_o a_o first_o binomial_a line_n then_o by_o the_o 48â—_n of_o this_o book_n find_v out_o a_o first_o binomial_a line_n and_o this_o right_a line_n be_v so_o find_v out_o divide_v into_o his_o name_n you_o may_v by_o the_o 10._o of_o the_o six_o divide_v the_o right_a line_n give_v in_o like_a sort_n and_o so_o in_o the_o other_o five_o follow_v although_o i_o here_o note_n unto_o you_o this_o corollary_n out_o of_o 〈…〉_o in_o very_a conscience_n and_o of_o grateful_a â—indeâ—_n i_o be_o enforce_v to_o certify_v you_o that_o iâ—_z any_o year_v before_o the_o travail_n of_o flussas_n upon_o euâ—liâ—â—â—_n geometrical_a element_n be_v publish_v the_o order_n how_o to_o divide_v not_o only_o the_o 6._o binomial_a line_n into_o their_o name_n but_o also_o to_o add_v to_o the_o 6._o residâ—â—ls_v their_o due_a part_n â—nd_v fâ—rthermore_o to_o divide_v all_o the_o other_o irrational_a line_n of_o this_o ten_o book_n into_o the_o part_n distinct_a of_o which_o they_o be_v compose_v with_o many_o other_o strange_a conclusion_n mathematical_a to_o the_o better_a understanding_n of_o this_o ten_o book_n and_o other_o mathematical_a book_n most_o necessary_a be_v by_o m._n john_n dee_fw-mi invent_v and_o demonstrate_v mathematicum_fw-la as_o in_o his_o book_n who_o title_n be_v tyrocinium_fw-la mathematicum_fw-la dedicate_v to_o petruâ—_n nonnius_n an._n 1559._o may_v at_o large_a appear_v where_o also_o be_v one_o new_a art_n with_o sundry_a particular_a point_n whereby_o the_o mathematical_a science_n great_o may_v be_v enrich_v which_o his_o book_n i_o hope_v god_n will_v one_o day_n allow_v he_o opportunity_n to_o publish_v with_o diverse_a other_o his_o mathematical_a and_o metaphysical_a labour_n and_o invention_n ¶_o a_o assumpt_n be_v a_o right_a line_n be_v divide_v into_o two_o part_n how_o soever_o the_o rectangle_n parallelogram_n contain_v under_o both_o the_o part_n be_v the_o mean_a proportional_a between_o the_o square_n of_o the_o same_o part_n and_o the_o rectangle_n parallelogram_n contain_v under_o the_o whole_a line_n and_o one_o of_o the_o part_n be_v the_o mean_a proportional_a between_o the_o square_n of_o the_o whole_a line_n and_o the_o square_a of_o the_o say_a part_n book_n suppose_v that_o there_o be_v two_o square_n ab_fw-la and_o bc_o and_o let_v the_o line_n db_o and_o be_v so_o be_v put_v that_o they_o both_o make_v one_o right_a line_n wherefore_o by_o the_o 14._o of_o the_o first_o the_o line_n fb_o and_o bg_o make_v also_o both_o one_o right_a line_n and_o make_v perfect_a the_o parallelogram_n ac_fw-la then_o i_o say_v that_o the_o rectangle_n parallelogram_n dg_o be_v the_o mean_a proportional_a between_o the_o square_n ab_fw-la and_o bc_o and_o moreover_o that_o the_o parallelogram_n dc_o be_v the_o mean_a proportional_a between_o the_o square_n ac_fw-la and_o cb._n first_o the_o parallelogram_n agnostus_n be_v a_o square_a for_o forasmuch_o as_o the_o line_n db_o be_v equal_a to_o the_o line_n bf_o and_o the_o line_n be_v unto_o the_o line_n bg_o therefore_o the_o whole_a line_n de_fw-fr be_v equal_a to_o the_o whole_a line_n fg._n but_o the_o line_n de_fw-fr be_v equal_a to_o either_o of_o these_o line_n ah_o &_o kc_o and_o the_o line_n fg_o be_v equal_a to_o either_o of_o these_o line_n ak_o and_o hc_n by_o the_o 34._o of_o the_o first_o demonstration_n wherefore_o the_o parallelogram_n ac_fw-la be_v equilater_n it_o be_v also_o rectangle_n by_o the_o 29._o of_o the_o first_o wherefore_o by_o the_o 46._o of_o the_o first_o the_o parallelogram_n ac_fw-la be_v a_o square_a now_o for_o that_o as_o the_o line_n fb_o be_v to_o the_o line_n bg_o so_o be_v the_o line_n db_o
superficies_n or_o solidity_n in_o the_o hole_n or_o in_o partâ—_n such_o certain_a knowledge_n demonstrative_a may_v arise_v and_o such_o mechanical_a exercise_n thereby_o be_v devise_v that_o sure_a i_o be_o to_o the_o sincere_a &_o true_a student_n great_a light_n aid_n and_o comfortable_a courage_n far_a to_o wade_v will_v enter_v into_o his_o heart_n and_o to_o the_o mechanical_a witty_a and_o industrous_a deviser_n new_a manner_n of_o invention_n &_o execution_n in_o his_o work_n will_v with_o small_a travail_n for_o foot_n application_n come_v to_o his_o perceiveraunce_n and_o understanding_n therefore_o even_o a_o manifold_a speculation_n &_o practice_n may_v be_v have_v with_o the_o circle_n his_o quantity_n be_v not_o know_v in_o any_o kind_n of_o small_a certain_a measure_n so_o likewise_o of_o the_o sphere_n many_o problem_n may_v be_v execute_v and_o his_o precise_a quantity_n in_o certain_a measure_n not_o determine_v or_o know_v yet_o because_o both_o one_o of_o the_o first_o humane_a occasion_n of_o invent_v and_o stablish_v this_o art_n be_v measure_v of_o the_o earth_n and_o therefore_o call_v geometria_n that_o be_v earthmeasure_v and_o also_o the_o chief_a and_o general_a end_n in_o deed_n be_v measure_n and_o measure_n require_v a_o determination_n of_o quantity_n in_o a_o certain_a measure_n by_o number_n express_v it_o be_v needful_a for_o mechanical_a earthmeasure_n not_o to_o be_v ignorant_a of_o the_o measure_n and_o content_n of_o the_o circle_n neither_o of_o the_o sphere_n his_o measure_n and_o quantity_n as_o near_o as_o sense_n can_v imagine_v or_o wish_v and_o in_o very_a deed_n the_o quantity_n and_o measure_n of_o the_o circle_n be_v know_v make_v not_o only_o the_o cone_n and_o cylinder_n but_o also_o the_o sphere_n his_o quantity_n to_o be_v as_o precise_o know_v and_o certain_a therefore_o see_v in_o respect_n of_o the_o circle_n quantity_n by_o archimedes_n specify_v this_o theorem_a be_v note_v unto_o you_o i_o will_v by_o order_n upon_o that_o as_o a_o supposition_n infer_v the_o conclusion_n of_o this_o our_o theorem_n note_n 1._o wherefore_o if_o you_o divide_v the_o one_o side_n as_o tq_n of_o the_o cube_fw-la tx_n into_o 21._o equal_a part_n and_o where_o 11._o part_n do_v end_n reckon_v from_o t_o suppose_v the_o point_n p_o and_o by_o that_o point_n p_o imagine_v a_o plain_a pass_v parallel_n to_o the_o opposite_a base_n to_o cut_v the_o cube_fw-la tx_n and_o thereby_o the_o cube_fw-la tx_n to_o be_v divide_v into_o two_o rectangle_n parallelipipedon_n namely_o tn_n and_o px_n it_o be_v manifest_a give_v tn_n to_o be_v equal_a to_o the_o sphere_n a_o by_o construction_n and_o the_o 7._o of_o the_o five_o note_n 2._o second_o the_o whole_a quantity_n of_o the_o sphere_n a_o assign_v be_v contain_v in_o the_o rectangle_n parallelipipedon_n tn_n you_o may_v easy_o transform_v the_o same_o quantity_n into_o other_o parallelipipedon_n rectangle_v of_o what_o height_n and_o of_o what_o parallelogram_n base_a you_o listen_v by_o my_o first_o and_o second_o problem_n upon_o the_o 34._o of_o this_o book_n and_o the_o like_a may_v you_o do_v to_o any_o assign_a part_n of_o the_o sphere_n a_o by_o the_o like_a mean_n devide_v the_o parallelipipedon_n tn_n as_o the_o part_n assign_v do_v require_v as_o if_o a_o three_o four_o five_o or_o six_o part_n of_o the_o sphere_n a_o be_v to_o be_v have_v in_o a_o parallelipipedon_n of_o any_o parallelograâ—â—e_n base_a assign_v or_o of_o any_o heith_n assign_v then_o devide_v tp_n into_o so_o many_o part_n as_o into_o 4._o if_o a_o four_o part_n be_v to_o be_v transform_v or_o into_o five_o if_o a_o five_o part_n be_v to_o be_v transform_v etc_n etc_n and_o then_o proceed_v â—s_v you_o do_v with_o cut_v of_o tn_n from_o tx_n and_o that_o i_o say_v of_o parallelipipedon_n may_v in_o like_a sort_n by_o my_o â—â—yd_n two_o problem_n add_v to_o the_o 34._o of_o this_o book_n be_v do_v in_o any_o side_v column_n pyramid_n and_o prismeâ—_n so_o thâ—â—_n in_o pyramid_n and_o some_o prism_n you_o use_v the_o caution_n necessary_a in_o respect_n of_o their_o quan_fw-mi 〈…〉_o odyes_n have_v parallel_n equal_a and_o opposite_a base_n who_o part_n 〈…〉_o re_fw-mi in_o their_o proposition_n be_v by_o euclid_n demonstrate_v and_o final_o 〈…〉_o addition_n you_o have_v the_o way_n and_o order_n how_o to_o geve_v to_o a_o sphere_n or_o any_o segment_n oâ—_n the_o same_o cones_n or_o cylinder_n equal_a or_o in_o any_o proportion_n between_o two_o right_a line_n give_v with_o many_o other_o most_o necessary_a speculation_n and_o practice_n about_o the_o sphere_n i_o trust_v that_o i_o have_v sufficient_o â—raughted_v your_o imagination_n for_o your_o honest_a and_o profitable_a study_n herein_o and_o also_o give_v you_o reaâ—â—_n â—â—tter_n wheâ—â—_n with_o to_o sâ—â—p_v the_o mouth_n of_o the_o malicious_a ignorant_a and_o arrogant_a despiser_n of_o the_o most_o excellent_a discourse_n travail_n and_o invention_n mathematical_a sting_v aswell_o the_o heavenly_a sphere_n &_o star_n their_o spherical_a solidity_n geometry_n with_o their_o convene_n spherical_a superficies_n to_o the_o earth_n at_o all_o time_n respect_v and_o their_o distance_n from_o the_o earth_n as_o also_o the_o whole_a earthly_a sphere_n and_o globe_n itself_o and_o infinite_a other_o case_n concern_v sphere_n or_o globe_n may_v hereby_o with_o as_o much_o ease_n and_o certainty_n be_v determine_v of_o as_o of_o the_o quantity_n of_o any_o bowl_n ball_n or_o bullet_n which_o we_o may_v gripe_v in_o our_o hand_n reason_n and_o experience_n be_v our_o witness_n and_o without_o these_o aid_n such_o thing_n of_o importance_n never_o able_a of_o we_o certain_o to_o be_v know_v or_o attain_a unto_o here_o end_n m._n john_n do_v you_o his_o addition_n upon_o the_o last_o proposition_n of_o the_o twelve_o book_n a_o proposition_n add_v by_o flussas_n if_o a_o sphere_n touch_v a_o plain_a superficiesâ—_n a_o right_a line_n draw_v from_o the_o centre_n to_o the_o touch_n shall_v be_v erect_v perpendicular_o to_o the_o plain_a superficies_n suppose_v that_o there_o be_v a_o sphere_n bcdl_n who_o centre_n let_v be_v the_o point_n a._n and_o let_v the_o plain_a superficies_n gci_n touch_v the_o spear_n in_o the_o point_n c_o and_o extend_v a_o right_a line_n from_o the_o centre_n a_o to_o the_o point_n c._n then_o i_o say_v that_o the_o line_n ac_fw-la be_v erect_v perpendicular_o to_o tâ—e_z plain_a gic._n let_v the_o sphere_n be_v cut_v by_o plain_a superficiece_n pass_v by_o the_o right_a line_n lac_n which_o plain_n let_v be_v abcdl_n and_o acel_n which_o let_v cut_v the_o plain_n gci_n by_o the_o right_a line_n gch_o and_o kci_fw-la now_o it_o be_v manifest_a by_o the_o assumpt_n put_v before_o the_o 17._o of_o this_o book_n that_o the_o two_o section_n of_o the_o sphere_n shall_v be_v circle_n have_v to_o their_o diameter_n the_o line_n lac_n which_o be_v also_o the_o diameter_n of_o the_o sphere_n wherefore_o the_o right_a line_n gch_o and_o kci_fw-la which_o be_v draw_v in_o the_o plain_n gci_n do_v at_o the_o point_n c_o fall_n without_o the_o circle_n bcdl_n and_o ecl._n wherefore_o they_o touch_v the_o circle_n in_o the_o point_n c_o by_o the_o second_o definition_n of_o the_o three_o wherefore_o the_o right_a line_n lac_n make_v right_a angle_n with_o the_o line_n gch_v and_o kci_fw-la by_o the_o 16._o of_o the_o three_o wherefore_o by_o the_o 4._o of_o the_o eleven_o the_o right_a line_n ac_fw-la be_v erect_v perpendicular_o to_o to_o the_o plain_a superficies_n gci_n wherein_o be_v draw_v the_o line_n gch_v and_o kci_fw-la if_o therefore_o a_o sphere_n touch_v a_o plain_a superficies_n a_o right_a line_n draw_v from_o the_o centre_n to_o the_o touch_n shall_v be_v erect_v perpendicular_o to_o the_o plain_a superficies_n which_o be_v require_v to_o be_v prove_v the_o end_n of_o the_o twelve_o book_n of_o euclides_n element_n ¶_o the_o thirteen_o book_n of_o euclides_n element_n in_o this_o thirteen_o book_n be_v set_v forth_o certain_a most_o wonderful_a and_o excellent_a passion_n of_o a_o line_n divide_v by_o a_o extreme_a and_o mean_a proportion_n book_n a_o matter_n undoubted_o of_o great_a and_o infinite_a use_n in_o geometry_n as_o you_o shall_v both_o in_o this_o book_n and_o in_o the_o other_o book_n follow_v most_o evident_o perceive_v it_o teach_v moreover_o the_o composition_n of_o the_o five_o regular_a solid_n and_o how_o to_o inscribe_v they_o in_o a_o sphere_n give_v and_o also_o set_v forth_o certain_a comparison_n of_o the_o say_a body_n both_o the_o one_o to_o the_o other_o and_o also_o to_o the_o sphere_n wherein_o they_o be_v describe_v the_o 1._o theorem_a the_o 1._o proposition_n if_o a_o right_a line_n be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n and_o to_o the_o great_a segment_n be_v add_v the_o half_a of_o the_o whole_a line_n the_o square_n make_v of_o those_o two_o
place_n from_o whence_o first_o it_o begin_v to_o be_v move_v it_o shall_v pass_v by_o the_o point_n f_o g_o h_o and_o the_o octohedron_n shall_v be_v comprehend_v in_o a_o sphere_n i_o say_v also_o that_o it_o be_v comprehend_v in_o the_o sphere_n give_v for_o forasmuch_o as_o the_o line_n lk_n be_v equal_a to_o the_o line_n km_o by_o position_n and_o the_o line_n ke_o be_v common_a to_o they_o both_o and_o they_o contain_v right_a angle_n by_o the_o 3._o definition_n of_o the_o eleven_o therefore_o by_o the_o 4._o of_o the_o first_o the_o base_a le_fw-fr be_v equal_a to_o the_o base_a em_n and_o forasmuch_o as_o the_o angle_n lem_n be_v a_o right_a angle_n by_o the_o 31._o of_o the_o three_o for_o it_o be_v in_o a_o semicircle_n as_o have_v be_v prove_v therefore_o the_o square_a of_o the_o line_n lm_o be_v double_a to_o the_o square_n of_o the_o line_n le_fw-fr by_o the_o 47._o of_o the_o first_o again_o forasmuch_o as_o the_o line_n ac_fw-la be_v equal_a to_o the_o line_n bc_o therefore_o the_o line_n ab_fw-la be_v double_a to_o the_o line_n bc_o by_o the_o definition_n of_o a_o circle_n but_o as_o the_o line_n ab_fw-la be_v to_o the_o line_n bc_o so_o be_v the_o square_a of_o the_o line_n ab_fw-la to_o the_o square_n of_o the_o line_n bd_o by_o the_o corollary_n of_o the_o 8._o and_o _o of_o the_o six_o wherefore_o the_o square_a of_o the_o line_n ab_fw-la be_v double_a to_o the_o square_n of_o the_o line_n bd._n and_o it_o be_v prove_v that_o the_o square_a of_o the_o line_n lm_o be_v double_a to_o the_o square_n of_o the_o line_n le._n wherefore_o the_o square_a of_o the_o line_n bd_o be_v equal_a to_o the_o square_n of_o the_o line_n le._n for_o the_o line_n eh_o which_o be_v equal_a to_o the_o line_n lf_n be_v put_v to_o be_v equal_a to_o the_o line_n db._n wherefore_o the_o square_a of_o the_o line_n ab_fw-la be_v equal_a to_o the_o square_n of_o the_o line_n lm_o wherefore_o the_o line_n ab_fw-la be_v equal_a to_o the_o line_n lm_o and_o the_o line_n ab_fw-la be_v the_o diameter_n of_o the_o sphere_n give_v wherefore_o the_o line_n lm_o be_v equal_a to_o the_o diameter_n of_o the_o sphere_n give_v wherefore_o the_o octoedron_n be_v contain_v in_o the_o sphere_n give_v le._n and_o it_o be_v also_o prove_v that_o the_o diameter_n of_o the_o sphere_n be_v in_o power_n double_a to_o the_o side_n of_o the_o octohedron_n wherefore_o there_o be_v make_v a_o octohedron_n and_o it_o be_v comprehend_v in_o the_o sphere_n give_v wherein_o be_v comprehend_v the_o pyramid_n and_o it_o be_v prove_v that_o the_o diameter_n of_o the_o sphere_n be_v in_o power_n double_a to_o the_o side_n of_o the_o octohedrn_n which_o be_v require_v to_o be_v do_v and_o to_o be_v prove_v certain_a corollary_n add_v by_o flussas_n first_o corollary_n the_o side_n of_o a_o pyramid_n be_v in_o power_n sesquitertia_fw-la to_o the_o side_n of_o a_o octahedron_n inscribe_v in_o the_o same_o sphere_n for_o forasmch_v as_o the_o diameter_n be_v in_o power_n double_a to_o the_o side_n of_o the_o octohedron_n therefore_o of_o what_o part_n the_o diameter_n contain_v in_o power_n 6._o of_o the_o same_o the_o side_n of_o the_o octohedron_n contain_v in_o power_n 3._o but_o of_o what_o part_n the_o diameter_n contain_v 6._o of_o the_o same_o the_o side_n of_o the_o pyramid_n contain_v 4._o by_o the_o 13._o of_o this_o book_n wherefore_o of_o what_o part_n the_o side_n of_o the_o pyramid_n contain_v 4._o of_o the_o same_o the_o side_n of_o the_o octohedron_n contain_v 3._o second_o corollary_n a_o octohedron_n be_v divide_v into_o two_o equal_a and_o like_a pyramid_n the_o common_a base_n of_o these_o pyramid_n be_v set_v upon_o every_o square_n contain_v of_o the_o side_n of_o the_o octohedron_n upon_o which_o square_a be_v set_v the_o â—â—_o triangle_n of_o the_o octohedron_n campane_n which_o pyramid_n be_v by_o the_o â—_o definition_n of_o the_o eleven_o equal_a and_o like_a and_o the_o foresay_a square_n common_a to_o those_o pyramid_n be_v the_o half_a of_o the_o square_n of_o the_o diameter_n of_o the_o sphere_n for_o it_o be_v the_o square_a of_o the_o side_n of_o the_o octohedron_n three_o corollary_n the_o three_o diameter_n of_o the_o octohedron_n do_v cut_v the_o one_o the_o other_o perpendicular_o into_o two_o equal_a part_n in_o the_o centre_n of_o the_o sphere_n which_o contain_v the_o say_a octohedron_n as_o it_o be_v manifest_a by_o the_o three_o diameter_n eglantine_n fh_o and_o lm_o which_o cut_v the_o one_o the_o other_o in_o the_o centre_n king_n equal_o and_o perpendicular_o ¶_o the_o 3._o problem_n the_o 15._o proposition_n to_o make_v a_o solid_a call_v a_o cube_fw-la and_o to_o comprehend_v it_o in_o the_o sphere_n give_v namely_o that_o sphere_n wherein_o the_o former_a two_o solid_n be_v comprehendâ—dâ—_n and_o to_o prove_v that_o the_o diameter_n of_o the_o sphere_n be_v in_o power_n treble_a to_o the_o side_n of_o the_o cube_fw-la take_v the_o diameter_n of_o the_o sphere_n give_v namely_o ab_fw-la and_o divide_v it_o in_o the_o point_n câ—_n so_o that_o let_v the_o line_n ac_fw-la be_v double_a to_o the_o line_n bc_o by_o the_o 9_o of_o the_o six_o and_o upon_o the_o line_n ab_fw-la describe_v a_o semicircle_n adb_o and_o by_o the_o 11._o of_o the_o first_o from_o the_o pâ—ynt_n c_o râ—yse_v up_o unto_o the_o line_n ab_fw-la a_o perpendicular_a line_n cd_o and_o draw_v a_o right_a linâ—_n db._n and_o describe_v a_o square_a efgh_o have_v every_o one_o of_o his_o side_n equal_a to_o the_o line_n db_o and_o from_o the_o point_n e_o f_o g_o h_o raise_v up_o by_o the_o 12._o of_o the_o eleven_o unto_o the_o plain_a superficies_n of_o the_o square_a efgh_o perpendicular_a line_n eke_o fl_fw-mi gm_n and_o hn_o and_o let_v every_o one_o of_o the_o line_n eke_o fl_fw-mi gm_n and_o hn_o be_v put_v equal_a to_o one_o of_o the_o line_n of_o fg_o gh_o or_o he_o which_o be_v the_o side_n of_o the_o square_n and_o draw_v these_o right_a line_n kl_o lm_o mn_a and_o nk_v demonstration_n wherefore_o there_o be_v make_v a_o cube_fw-la namely_o fn_o which_o be_v contain_v under_o six_o equal_a square_n construction_n now_o it_o be_v require_v to_o comprehend_v the_o same_o cube_fw-la in_o the_o sphere_n give_v and_o to_o prove_v that_o the_o diameter_n of_o the_o sphere_n be_v in_o power_n ble_o to_o the_o side_n of_o the_o cube_fw-la demonstration_n draw_v these_o right_a line_n kg_v and_o eglantine_n and_o forasmuch_o as_o the_o angle_n keg_n be_v a_o right_a angle_n for_o that_o the_o line_n ke_o be_v erect_v perpendicular_o to_o the_o plain_a superficies_n eâ—_n and_o therefore_o also_o to_o the_o right_a line_n eglantine_n by_o the_o 2._o definition_n of_o the_o eleven_o wherefore_o a_o semicircle_n describe_v upon_o the_o line_n kg_v shall_v book_n pass_v by_o the_o point_n e._n again_o forasmuch_o as_o the_o line_n fg_o be_v erect_v perpendicular_o to_o either_o of_o these_o line_n fl_fw-mi and_o fe_o by_o the_o definition_n of_o a_o square_a &_o by_o the_o 2._o definition_n of_o the_o eleven_o therefore_o the_o line_n fg_o be_v erect_v perpendicular_o to_o the_o plain_a superficies_n fk_o by_o the_o 4._o of_o the_o eleven_o wherefore_o if_o we_o draw_v a_o right_a line_n from_o the_o point_n f_o to_o the_o point_n king_n the_o line_n gf_o shall_v be_v erect_v perpendicular_o to_o the_o line_n kf_o by_o the_o 2._o definition_n of_o the_o eleven_o and_o by_o the_o same_o reason_n again_o a_o semicircle_n describe_v upon_o the_o line_n gk_o shall_v pass_v also_o by_o the_o point_n f._n and_o likewise_o shall_v it_o pass_v by_o the_o rest_n of_o the_o point_n of_o the_o angle_n of_o that_o cube_fw-la if_o now_o the_o diameter_n kg_v abide_v fix_v the_o semicircle_n be_v turn_v round_o about_o until_o it_o return_v into_o the_o self_n same_o place_n from_o whence_o it_o begin_v first_o to_o be_v move_v the_o cube_fw-la shall_v be_v comprehend_v in_o a_o sphere_n i_o say_v also_o that_o it_o be_v comprehend_v in_o the_o sphere_n give_v demonstration_n for_o forasmuch_o as_o the_o line_n gf_o be_v equal_a to_o the_o linâ—â—e_n and_o the_o angle_n fletcher_n be_v a_o right_a angle_n therefore_o the_o square_a of_o the_o line_n eglantine_n be_v by_o the_o 47._o of_o the_o first_o double_a to_o the_o square_n of_o the_o line_n â—f_n but_o the_o line_n of_o be_v equal_a to_o the_o line_n eke_o wherefore_o the_o square_a of_o the_o line_n eglantine_n be_v double_a to_o the_o square_n of_o the_o line_n eke_o wherefore_o the_o square_n of_o eglantine_n and_o eke_o that_o be_v the_o square_a of_o the_o line_n gk_o by_o the_o 47._o of_o the_o first_o be_v treble_a to_o the_o square_n of_o the_o line_n eke_o and_o forasmuch_o as_o the_o line_n ab_fw-la be_v treble_a to_o the_o line_n bc_o but_o
first_o part_n of_o this_o proposition_n demonstration_n of_o the_o of_o the_o same_o the_o second_o part_n which_o be_v the_o converse_n of_o the_o first_o the_o first_o parâ—_n of_o this_o proposition_n demonstration_n of_o the_o same_o the_o second_o part_n which_o be_v the_o converse_n of_o the_o first_o demonstration_n of_o the_o first_o part_n the_o second_o part_n which_o be_v the_o converse_n of_o the_o first_o the_o first_o part_n of_o thâ—â—_n theorem_a the_o second_o part_n which_o be_v the_o converse_n of_o the_o first_o a_o corollary_n description_n of_o the_o rectiline_a figure_n require_v demonstration_n demonstration_n a_o corollary_n the_o first_o parâ—_n of_o this_o theorem_a the_o second_o part_n demonstrate_v the_o three_o part_n the_o first_o corollary_n the_o second_o corollary_n demonstration_n the_o first_o part_n of_o this_o proposition_n the_o second_o part_n which_o be_v the_o converse_n of_o the_o first_o first_o note_v that_o this_o be_v prove_v in_o the_o assumpt_n follow_v a_o assumpt_n an_o other_o demonstration_n of_o the_o second_o part_n after_o flussates_n an_o other_o demonstration_n after_o flussate_n demonstration_n of_o this_o proposition_n wherein_o be_v first_o prove_v that_o the_o parallegramme_n eglantine_n be_v like_a to_o the_o whole_a parallelogram_n abcd._n that_o the_o parallelogram_n kh_o be_v like_a to_o the_o whole_a parallelogram_n abcd_o that_o the_o parallelogram_n eglantine_n and_o kh_o be_v like_o the_o one_o to_o the_o other_o an_o other_o demonstration_n after_o flussates_n a_o addition_n of_o pelitarius_n another_o addition_n of_o pelitarius_n construction_n demonstration_n demonstration_n demonstration_n by_o the_o dimetient_fw-la be_v understand_v here_o the_o dimetient_fw-la which_o be_v draw_v from_o the_o angle_n which_o be_v common_a to_o they_o both_o to_o the_o opposite_a angle_n demonstration_n lead_v to_o a_o absurdity_n an_o other_o way_n after_o flussates_n in_o this_o proposition_n be_v two_o case_n in_o the_o first_o the_o parallelogram_n compare_v to_o the_o parallelogram_n describe_v of_o the_o half_a line_n be_v describe_v upon_o a_o line_n great_a they_o the_o half_a line_n in_o the_o second_o upon_o a_o line_n less_o the_o first_o case_n where_o the_o parellelogramme_n compare_v namely_o of_o be_v describe_v upon_o the_o line_n ak_o which_o be_v great_a than_o the_o half_a line_n ac_fw-la demonstration_n of_o this_o case_n the_o second_o case_n where_o the_o parallelogram_n compare_v namely_o ae_n be_v describe_v upon_o the_o line_n ad_fw-la which_o be_v less_o than_o the_o line_n ac_fw-la demonstration_n of_o the_o second_o case_n construction_n two_o case_n in_o this_o proposition_n the_o first_o case_n the_o second_o case_n a_o corollary_n add_v by_o flussates_n and_o be_v put_v of_o theon_n as_o a_o assumpt_n beâ—ore_o the_o 17._o proposition_n of_o the_o ten_o book_n which_o â—or_a that_o it_o follow_v of_o this_o proposition_n i_o think_v it_o not_o amiss_o here_o to_o place_n construction_n demonstration_n construction_n demonstration_n an_o other_o way_n construction_n demonstration_n the_o converse_n of_o the_o former_a proposition_n demonstration_n that_o the_o angle_n at_o thâ—_z centre_n be_v in_o proportion_n the_o one_o to_o the_o other_o as_o the_o circumference_n whereon_o they_o be_v that_o the_o angle_n at_o the_o circumference_n be_v so_o also_o that_o the_o sector_n be_v so_o also_o construction_n of_o the_o problem_n demonstartion_n of_o the_o same_o the_o first_o corollary_n the_o second_o corollary_n the_o three_o corollary_n demonstration_n of_o this_o proposition_n demonstration_n of_o this_o proposition_n demonstration_n of_o this_o proposition_n demonstration_n of_o the_o first_o part_n of_o this_o proposition_n demonstration_n of_o the_o second_o part_n why_o euclid_n in_o the_o midst_n of_o his_o work_n be_v compel_v to_o add_v these_o three_o book_n of_o number_n arithmetic_n of_o more_o excellency_n than_o geometry_n thing_n intellectual_a of_o more_o worthiness_n theâ—_z thing_n sensible_a arithmetic_n minister_v prinâ—ciples_n and_o ground_n in_o a_o manner_n to_o all_o science_n boetius_fw-la cap._n 2._o lib._n prim_fw-la arithmeti_n timaus_n the_o argument_n of_o the_o seven_o book_n the_o first_o definition_n without_o unity_n shall_v be_v confusion_n of_o thing_n â—oetius_fw-la in_o his_o book_n dâ—_z unitate_fw-la &_o uno_fw-la an_o other_o desinition_n of_o unity_n the_o second_o definition_n difference_n between_o a_o point_n and_o unity_n boetius_fw-la an_o other_o desinition_n of_o number_n jordane_n an_o other_o definition_n of_o number_n unity_n have_v in_o it_o the_o virtue_n and_o power_n of_o all_o number_n number_n consider_v three_o manner_n of_o wayâ—_n the_o three_o definition_n the_o four_o definition_n the_o five_o definition_n the_o six_o definition_n boetius_fw-la an_o other_o definition_n of_o a_o even_a number_n note_n pithagoraâ—_n an_o other_o definition_n an_o other_o definition_n an_o other_o definition_n the_o seven_o definition_n an_o other_o definition_n of_o a_o odd_a number_n an_o other_o definition_n the_o eight_o definition_n campane_n an_o other_o definition_n of_o a_o even_o even_o number_n flussates_n an_o other_o definition_n boetius_fw-la an_o other_o definition_n the_o nine_o definition_n campane_n an_o other_o definition_n flussates_n an_o other_o definition_n an_o other_o definition_n the_o ten_o definition_n this_o definition_n not_o find_v in_o the_o greek_n an_o other_o definition_n boeâ—ius_n definition_n of_o a_o number_n even_o even_o and_o even_o â—d_a the_o eleven_o definition_n flusâ—ates_n an_o other_o definition_n the_o twelve_o definition_n prime_n number_n call_v incomposed_a number_n the_o thirteen_o definition_n the_o fourteen_o definition_n the_o fifteen_o definition_n the_o sixteen_o definition_n two_o number_n require_v in_o multiplication_n the_o seventeen_o definition_n why_o they_o be_v call_v superficial_a number_n the_o eighteen_o definition_n why_o they_o be_v call_v solid_a number_n the_o nineteen_o definition_n why_o it_o be_v call_v a_o square_a number_n the_o twenty_o definition_n why_o it_o be_v call_v a_o cube_fw-la number_n the_o twenty_o one_o definition_n why_o the_o definition_n of_o proportional_a magnitude_n be_v unlike_a to_o the_o definitio_fw-la of_o proportional_a number_n the_o twenty_o two_o definition_n the_o twenty_o three_o definition_n perfect_a number_n rare_a &_o of_o great_a use_n in_o magic_a &_o in_o secret_a philosophy_n in_o what_o respect_n a_o number_n be_v perfect_a two_o kind_n of_o imperfect_a number_n a_o â—â—mber_n wanâ—â—ngâ—_n common_a sentence_n â—irst_n common_a sentence_n â—â—cond_v â—ommon_a sentence_n three_o common_a sentence_n forth_o common_a sentence_n â—iâ—th_o common_a sentence_n six_o common_a sentence_n seven_o comâ—mon_a sentence_n constrâ—ctioâ—_n demonstratiâ—â—_n lead_v to_o a_o absurdity_n the_o converse_n of_o â—his_n proposition_n how_o to_o â—now_v whether_o two_o number_n give_v be_v prime_a the_o one_o to_o the_o other_o two_o case_n in_o this_o problem_n the_o first_o case_n the_o second_o case_n demonstration_n of_o the_o second_o case_n that_o cf_o be_v a_o common_a measure_n to_o the_o number_n ab_fw-la and_o cd_o that_o cf_o be_v the_o great_a common_a measure_n to_o ab_fw-la and_o cd_o the_o second_o case_n two_o case_n in_o this_o proposition_n the_o first_o case_n the_o second_o case_n this_o proposition_n and_o the_o 6._o proposition_n in_o discrete_a quantity_n answer_v to_o the_o first_o of_o the_o five_o in_o continual_a quantity_n demonstration_n construction_n demonstration_n thiâ—_n proposition_n and_o the_o next_o follow_v in_o discreet_a quantity_n answer_v to_o the_o five_o proposition_n of_o the_o five_o book_n in_o continual_a quantity_n construction_n demonstration_n constuâ—ction_n demonstration_n an_o other_o demonstration_n after_o flussates_n construction_n demonstration_n construction_n demonstration_n this_o proposition_n iâ—_z discreet_a quantity_n answer_v to_o the_o nine_o proposition_n of_o the_o five_o book_n in_o continual_a quantity_n demonstration_n this_o in_o discreet_a quantity_n answer_v to_o the_o twelve_o proposition_n of_o the_o five_o in_o continual_a quantity_n demonstration_n this_o in_o discrete_a quantity_n answer_v to_o the_o sixteen_o proposition_n of_o the_o five_o book_n in_o continual_a quantity_n note_n this_o in_o discrete_a quantity_n answer_v to_o tâ—â—_n twenty_o one_o proposition_n oâ—_n the_o five_o book_n in_o continual_a quantity_n demonstration_n certain_a addition_n of_o campane_n the_o second_o case_n proportionality_n divide_v proportionality_o compose_v euerse_a proportionality_n the_o conversâ—_n of_o the_o same_o prâ—position_n demonstration_n a_o corollary_n follow_v thâ—se_a proposition_n adâ—ed_v by_o campane_n coâ—strâ—ctioâ—_n demonstration_n demonstration_n â—emonstraâ—ion_n a_o corollary_n add_v by_o flussâ—tes_n demonstration_n this_o proposition_n and_o the_o former_a may_v be_v extend_v to_o number_n how_o many_o soever_o the_o second_o part_n of_o this_o proposition_n which_o be_v the_o converse_n of_o the_o first_o demonstration_n a_o assumpt_v add_v by_o campane_n this_o proposition_n in_o number_n demonstrate_v that_o which_o the_o 17._o of_o the_o six_o demonstrate_v in_o line_n demonstration_n the_o second_o part_n which_o be_v the_o converse_n of_o the_o
to_o be_v prâ—â—â—d_v beâ—oâ—e_n 〈◊〉_d â—all_n to_o the_o demonstration_n construction_n demonstration_n lead_v to_o a_o absurdity_n a_o corollary_n this_o problem_n reduce_v to_o a_o theorem_a construction_n demonstration_n how_o magnitude_n be_v say_v to_o be_v in_o proportion_n the_o onâ—_n to_o the_o other_o as_o number_n be_v to_o number_v this_o proposition_n be_v the_o converse_n of_o forme_a construction_n demonstration_n a_o corollary_n construction_n demonstration_n construction_n demonstration_n demonstration_n lead_v to_o a_o absurdity_n this_o be_v the_o 〈…〉_o demonstration_n the_o first_o part_n demonstrate_v an_o other_o demonstration_n of_o the_o first_o part_n an_o other_o demonstration_n oâ—_n the_o same_o first_o part_n after_o montaureus_n demonstration_n of_o the_o second_o part_n which_o be_v the_o converse_n of_o the_o former_a an_o other_o demonstration_n of_o the_o second_o part_n this_o assumpt_n follow_v as_o a_o corollary_n of_o the_o 25_o but_o so_o as_o it_o may_v also_o be_v here_o in_o method_n place_v you_o shall_v â—inde_v it_o after_o the_o 53._o of_o this_o book_n absolute_o demonstrate_v for_o there_o it_o serve_v to_o the_o 54._o his_o demonstration_n demonstration_n of_o the_o three_o part_n demonstration_n of_o the_o four_o part_n which_o be_v the_o converse_n of_o the_o â—_o conclusion_n of_o the_o whole_a proposition_n a_o corollary_n proâ—e_v of_o the_o first_o part_n of_o the_o corollary_n proof_n of_o the_o second_o part_n proof_n of_o the_o three_o pâ—rt_n proâ—e_a oâ—_n the_o four_o part_n certain_a annotation_n â—ut_z of_o montauâ—â—us_fw-la rule_v to_o know_v whether_o two_o superficial_a number_n be_v like_a or_o no._n this_o assumpt_n be_v the_o converse_n of_o the_o 26._o of_o the_o eight_o demonstration_n oâ—_n the_o first_o part_n demonstration_n of_o the_o second_o partâ—_n a_o corollary_n to_o find_v out_o the_o first_o line_n incommensurable_a in_o length_n only_o to_o the_o line_n give_v to_o find_v out_o the_o second_o line_n incommensurable_a both_o in_o length_n and_o in_o power_n to_o the_o line_n give_v construction_n demonstration_n tâ—is_n be_v wiâ—h_n zambert_n a_o aâ—â—â—mpt_n but_o uâ—â—eâ—ly_o improper_o â—lâ—ssateâ—_n mane_v iâ—_z a_o corollary_n but_o the_o greeâ—e_n and_o montaureus_n maâ—e_n it_o a_o proposition_n but_o every_o way_n a_o â—nfallible_a truth_n 〈…〉_o demonstration_n lead_v to_o a_o absurdity_n demonstration_n lead_v to_o a_o absurdity_n a_o corollary_n a_o corollary_n demonstration_n an_o other_o way_n to_o prove_v that_o the_o line_n a_o e_o c_o f_o be_v proportional_a demonstration_n of_o the_o first_o part_n demonstration_n of_o the_o second_o paâ—t_n which_o be_v the_o converse_n of_o the_o first_o a_o corollary_n demonstration_n of_o the_o first_o part_n by_o a_o argument_n leadindg_fw-mi to_o a_o absurdity_n demonstration_n of_o the_o second_o paâ—t_n lead_v also_o to_o a_o impossibility_n and_o this_o second_o part_n be_v the_o converse_n of_o the_o first_o demonstration_n of_o the_o second_o part_n which_o be_v the_o converse_n of_o the_o first_o how_o to_o divide_v the_o line_n bc_o ready_o in_o such_o sort_n as_o iâ—_z require_v in_o the_o proposition_n demonstration_n of_o the_o second_o part_n which_o be_v the_o converse_n of_o tâ—e_z former_a a_o other_o demonstrationâ—y_v a_o argument_n lead_v to_o a_o absurdity_n a_o assumpt_n a_o corollary_n add_v by_o montaureuâ—_n cause_n cause_o of_o increase_v the_o difficulty_n of_o this_o book_n note_n construction_n demonstration_n diverse_a caâ—es_n in_o this_o proposition_n the_o second_o case_n the_o first_o kind_n of_o rational_a line_n commensurable_a in_o length_n this_o particle_n in_o the_o proposition_n according_a to_o any_o of_o the_o foresay_a way_n be_v not_o in_o vain_a put_v the_o second_o kind_n of_o rational_a line_n commensurable_a in_o length_n the_o three_o case_n the_o three_o kind_n of_o rational_a line_n commensurable_a in_o length_n the_o four_o case_n this_o proposition_n be_v the_o converse_n of_o the_o former_a proposition_n construction_n demonstration_n a_o assumpt_n construction_n demonstration_n definition_n of_o a_o medial_a line_n a_o corollary_n this_o assumpt_n be_v nothing_o else_o but_o a_o part_n of_o the_o first_o proposition_n of_o the_o six_o book_n 〈◊〉_d how_o a_o square_n be_v say_v to_o be_v apply_v upon_o a_o line_n construction_n demonstration_n construction_n demonstration_n note_n a_o corollary_n construction_n demonstration_n lead_v to_o a_o absurdity_n construction_n demonstration_n construction_n demonstration_n demonstration_n a_o corollary_n to_o find_v out_o two_o square_a number_n exceed_v the_o one_o the_o other_o by_o a_o square_a â—umber_n a_o assumpt_n construction_n demonstration_n montaureus_n make_v this_o a_o assumpt_n as_o the_o grecke_n text_n seem_v to_o do_v likewise_o but_o without_o a_o cause_n construction_n demonstration_n this_o assumpt_n set_v foâ—th_o nothing_o â—_z but_o that_o which_o the_o first_o oâ—_n the_o sâ—â—t_n jet_v â—orth_o and_o therefore_o in_o sâ—me_a examplar_n it_o be_v not_o find_v construction_n demonstration_n construction_n demonstration_n construction_n demonstration_n a_o corollary_n i_o dee_n dee_n the_o second_o corollary_n corollary_n therefore_o if_o you_o divide_v the_o square_n of_o the_o side_n ac_fw-la by_o the_o side_n bc_o the_o portion_n dc_o will_v be_v the_o product_n etc_n etc_n as_o in_o the_o former_a corollary_n i_o dâ—e_n dâ—e_n the_o third_o corollary_n corollary_n therefore_o if_o the_o parallelogram_n of_o basilius_n and_o ac_fw-la be_v divide_v by_o bc_o the_o product_n will_v geve_v the_o perpendicular_a d_o a._n these_o three_o corollarye_n in_o practice_n logistical_a and_o geometrical_a be_v profitable_a an_o other_o demonstration_n of_o this_o four_o part_n of_o the_o determination_n a_o assumpt_n construction_n demonstration_n the_o first_o part_n of_o the_o determination_n conclude_v the_o second_o part_n 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a_o impossibility_n demonstration_n lead_v to_o a_o impossibility_n demonstration_n lead_v to_o a_o impossibiâ—â—â—e_n demonstration_n lead_v to_o a_o impossibility_n construction_n demonstration_n lead_v to_o a_o absurdity_n six_o kind_n of_o binomial_a line_n a_o binomial_a line_n consi_v of_o two_o paâ—tâ—s_n firsâ—_o definition_n seconâ—_o definition_n three_o â—â—â—â—â—â—ition_n four_o definition_n five_o definition_n six_o definition_n the_o three_o senary_a by_o composition_n construction_n demonstration_n construction_n demonstration_n construction_n demonstration_n construction_n demonstration_n construction_n demonstration_n construction_n demonstration_n a_o corollary_n add_v by_o flussates_n m._n do_v you_o his_o book_n call_v tyâ—â—câ—niâ—m_fw-la mathematicum_fw-la this_o assumpt_n as_o be_v before_o note_v follow_v most_o â—riâ—fly_o without_o far_a demonstration_n of_o the_o 25._o of_o this_o book_n demonstration_n a_o assumpt_n the_o four_o senary_a by_o composition_n construction_n demonstration_n the_o first_o part_n of_o this_o demonstration_n conclude_v the_o second_o part_n of_o the_o demonstration_n conclude_v the_o three_o part_n conclude_v the_o total_a conclusion_n demonstration_n the_o first_o part_n of_o this_o demonstration_n conclude_v the_o three_o part_n conclude_v the_o four_o part_n concludeâ—_n the_o five_o part_n conclude_v the_o total_a conclusion_n demonstration_n construction_n demonstration_n demonstration_n demonstration_n demonstration_n look_v after_o the_o assumpt_n conclude_v at_o this_o mark_n for_o plain_a open_n of_o this_o place_n the_o use_n of_o this_o assumpt_n be_v in_o the_o next_o proposition_n &_o other_o follow_v the_o five_o senary_a by_o composition_n construction_n demonstration_n conclude_v that_o dg_o be_v a_o binomial_a line_n construction_n demonstration_n conclude_v that_o dg_o be_v a_o binomial_a line_n construction_n demonstration_n †_o *_o ‡_o dg_o conclude_v a_o binomial_a line_n a_o corollary_n add_v by_o m._n dee_n construction_n demonstration_n construction_n demonstration_n construction_n demonstration_n the_o six_o senary_n construction_n demonstration_n construction_n demonstration_n a_o corollary_n add_v by_o flussetes_n note_n construction_n demonstration_n an_o other_o demonstration_n after_o p._n montaureus_n an_o other_o demonstration_n after_o campane_n construction_n demonstration_n an_o other_o demonstration_n afâ—â—r_v campane_n construction_n demonstration_n a_o assumpt_n an_o other_o demonstration_n after_o campanâ—_n note_n
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and_o power_n of_o number_n for_o proportion_n arithmetical_a and_o geometrical_a plato_n in_o his_o book_n call_v epinomis_n which_o book_n be_v the_o threasury_n of_o all_o his_o doctrine_n where_o his_o purpose_n be_v to_o seek_v a_o science_n which_o when_o a_o man_n have_v it_o perfect_o he_o may_v seem_v and_o so_o be_v in_o deed_n wise._n he_o brief_o of_o other_o science_n discourse_v find_v they_o not_o able_a to_o bring_v it_o to_o pass_v but_o of_o the_o science_n of_o number_n he_o say_v illa_fw-la qua_fw-la numerum_fw-la mortalium_fw-la generi_fw-la dâ—â—n_fw-la id_fw-la profecto_fw-la efficietâ—_n deum_fw-la antem_fw-la aliquem_fw-la magis_fw-la quam_fw-la fortunam_fw-la â—d_a saâ—â—tem_fw-la nostram_fw-la hoc_fw-la mâ—nus_fw-la nobis_fw-la arbitror_fw-la contulisse_fw-la etc_n etc_n nam_fw-la ipsum_fw-la â—onorum_fw-la omnium_fw-la authorem_fw-la cur_n non_fw-la maximi_fw-la boni_fw-la prudentiae_fw-la dico_fw-la causam_fw-la arbitramur_fw-la that_o science_n very_o which_o have_v teach_v mankind_n number_n shall_v be_v able_a to_o bring_v it_o to_o pass_v and_o i_o think_v a_o certain_a god_n rather_o than_o fortune_n to_o have_v give_v we_o this_o gift_n for_o our_o bliss_n for_o why_o shall_v we_o not_o judge_v he_o who_o be_v the_o author_n of_o all_o good_a thing_n to_o be_v also_o the_o cause_n of_o thâ—_z great_a good_a thing_n namely_o wisdom_n there_o at_o length_n he_o prove_v wisdom_n to_o be_v attain_v by_o good_a skill_n of_o number_n with_o which_o great_a testimony_n and_o the_o manifold_a profess_v and_o reason_n before_o expressedâ—_n you_o may_v be_v sufficient_o and_o full_o persuade_v of_o the_o perfect_a science_n of_o arithmetic_n to_o make_v this_o account_v that_o of_o all_o science_n ☞_o next_o to_o theologie_n it_o be_v most_o divine_a most_o pure_a most_o ample_a and_o general_a most_o profound_a most_o subtle_a most_o commodious_a and_o most_o necessary_a who_o be_v next_o sister_n be_v the_o absolute_a science_n of_o magnitude_n of_o which_o by_o the_o direction_n and_o aid_n of_o he_o who_o magnitude_n be_v infinite_a and_o of_o we_o incomprehensible_a i_o now_o
the_o line_n ab_fw-la be_v long_o than_o the_o line_n cd_o &_o if_o you_o take_v from_o they_o two_o equal_a line_n as_o ebb_v and_o fd_o the_o part_n remain_v which_o be_v the_o line_n ae_n and_o cf_o shall_v be_v unequal_a the_o one_o to_o the_o other_o namely_o the_o line_n ae_n shall_v be_v great_a than_o the_o line_n cf_o which_o be_v ever_o true_a in_o all_o quantity_n whatsoever_o 5_o and_o if_o to_o unequal_a thing_n you_o add_v equal_a thing_n the_o whole_a shall_v be_v unequal_a as_o if_o you_o have_v two_o unequal_a line_n namely_o ae_n the_o great_a and_o cf_o the_o less_o &_o if_o you_o add_v unto_o they_o two_o equal_a line_n ebb_n &_o fd_o then_o may_v you_o conclude_v that_o the_o whole_a line_n compose_v be_v unequal_a namely_o that_o the_o whole_a line_n ab_fw-la be_v great_a than_o the_o whole_a line_n cd_o and_o so_o of_o all_o other_o quantity_n 6_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 as_o if_o the_o line_n ab_fw-la be_v double_a to_o the_o line_n of_o and_o if_o also_o the_o line_n cd_o be_v double_a to_o the_o same_o line_n efâ—_n â—_z they_o may_v you_o by_o this_o common_a sentence_n conclude_v that_o the_o two_o line_n ab_fw-la &_o cd_o be_v equal_a the_o one_o to_o the_o other_o and_o this_o be_v true_a in_o all_o quantity_n and_o that_o not_o only_o when_o they_o be_v double_a but_o also_o if_o they_o be_v triple_a or_o quadruple_a or_o in_o what_o proportion_n soever_o it_o be_v of_o the_o great_a inequality_n iâ—_z which_o be_v when_o the_o great_a quantity_n be_v compare_v to_o the_o less_o 7_o thing_n which_o be_v the_o half_a of_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 as_o if_o the_o line_n ab_fw-la be_v the_o half_a of_o the_o line_n of_o and_o if_o the_o line_n cd_o be_v the_o half_a also_o of_o the_o same_o line_n of_o then_o may_v you_o conclude_v by_o this_o common_a sentence_n that_o the_o two_o line_n ab_fw-la and_o cd_o be_v equal_a the_o one_o to_o the_o other_o this_o be_v also_o true_a in_o all_o kind_n of_o quantity_n and_o that_o not_o only_a when_o it_o be_v a_o half_a but_o also_o if_o it_o be_v a_o three_o a_o quarter_n or_o in_o what_o proportion_n soever_o it_o be_v of_o the_o less_o in_o equality_n which_o be_v when_o the_o less_o quantity_n be_v compare_v to_o the_o great_a iâ—_z 8_o thing_n which_o agree_v together_o be_v equal_v the_o one_o to_o the_o other_o such_o thing_n be_v say_v to_o agree_v together_o which_o when_o they_o be_v apply_v the_o one_o to_o the_o other_o or_o set_v the_o one_o upon_o the_o other_o the_o one_o â—â—â—â—deth_v not_o the_o other_o in_o any_o thing_n as_o if_o the_o two_o triangle_n abc_n and_o def_n be_v apply_v the_o one_o to_o the_o other_o and_o the_o triangle_n abc_n be_v set_v upon_o the_o triangle_n def_n if_o then_o the_o angle_n a_o do_v just_o agree_v with_o the_o angle_n d_o and_o the_o angle_n b_o with_o the_o angle_n e_o and_o also_o the_o angle_n c_o with_o the_o angle_n f_o and_o moreover_o â—f_a the_o line_n ab_fw-la do_v just_o fall_v upon_o the_o line_n de_fw-fr and_o the_o line_n ac_fw-la upon_o the_o line_n df_o and_o also_o the_o line_n bc_o upon_o the_o line_n of_o so_o that_o on_o every_o part_n of_o these_o two_o triangle_n there_o be_v just_a agreement_n then_o may_v you_o conclude_v that_o the_o two_o triangle_n be_v equal_a 9_o every_o whole_a be_v great_a than_o his_o part_n be_v the_o principle_n thus_o place_v &_o end_v now_o follow_v the_o proposition_n which_o be_v sentence_n set_v forth_o to_o be_v prove_v by_o reason_v and_o demonstration_n and_o therefore_o they_o be_v again_o repeat_v in_o the_o end_n of_o the_o demonstrationâ—_n for_o the_o proposition_n be_v ever_o the_o conclusion_n and_o that_o which_o ought_v to_o be_v prove_v sort_n proposition_n be_v of_o two_o sort_n the_o one_o be_v call_v a_o problem_n the_o other_o a_o theorem_a be_v a_o problem_n be_v a_o proposition_n which_o require_v some_o action_n or_o do_v as_o the_o make_v of_o some_o â—igure_n or_o to_o divide_v a_o figure_n or_o line_n to_o apply_v figure_n to_o â—igure_n to_o add_v figure_n together_o or_o to_o subtrah_n one_o from_o a_o other_o to_o describe_v to_o inscribe_v to_o circumscribe_v one_o figure_n within_o or_o without_o another_o and_o such_o like_a as_o of_o the_o first_o proposition_n of_o the_o first_o book_n be_v a_o problem_n which_o be_v thuss_v upon_o a_o right_a line_n give_v not_o be_v infinite_a to_o describe_v a_o equilater_n triangle_n or_o a_o triangle_n of_o three_o equal_a side_n for_o in_o it_o beside_o the_o demonstration_n and_o contemplation_n of_o the_o mind_n be_v require_v somewhat_o to_o be_v do_v name_o to_o make_v a_o equilater_n triangle_n upon_o a_o line_n give_v and_o in_o the_o end_n of_o every_o problem_n after_o the_o demonstration_n be_v conclude_v aâ—ter_v this_o manâ—er_n which_o be_v the_o thing_n which_o be_v require_v to_o be_v do_v be_v a_o theorem_a be_v a_o proposition_n which_o require_v the_o search_a ouâ—_n and_o demonstration_n of_o some_o property_n or_o passion_n of_o some_o figure_n wherein_o be_v only_a speculation_n and_o contemplation_n of_o mind_n without_o do_v or_o work_v of_o any_o thing_n as_o the_o five_o proposition_n of_o the_o first_o book_n which_o be_v thus_o a_o isosceles_a or_o triangle_n of_o two_o equal_a side_n have_v his_o angle_n at_o thâ—_z base_a equal_a the_o one_o to_o the_o other_o etc_n etc_n be_v a_o theorem_a for_o in_o it_o be_v require_v only_o to_o be_v prove_v and_o make_v plain_a by_o reason_n and_o demonstration_n that_o these_o two_o angle_n be_v equal_a without_o further_a work_n or_o do_v and_o in_o the_o endâ—_n of_o â—uâ—ry_a theorem_a after_o the_o demonstration_n be_v conclude_v after_o this_o manner_n which_o thing_n be_v require_v to_o be_v demonstrate_v or_o prove_v the_o first_o problem_n the_o first_o proposition_n upon_o a_o right_a line_n give_v not_o be_v infinite_a to_o describe_v a_o equilater_n triangle_n or_o a_o triangle_n of_o three_o equal_a side_n svppose_v that_o the_o right_a line_n give_v be_v ab_fw-la it_o be_v require_v upon_o the_o line_n ab_fw-la to_o describe_v a_o equilater_n triangle_n namely_o a_o triangle_n of_o three_o equal_a side_n construction_n now_o therefore_o make_v the_o centre_n the_o point_n a_o and_o the_o space_n ab_fw-la describe_v by_o the_o three_o petition_n a_o circle_n bcd_o and_o again_o by_o the_o same_o make_v the_o centre_n the_o point_n b_o and_o the_o space_n bâ—_n describe_v a_o other_o circle_n ace_n and_o by_o the_o first_o petition_n from_o the_o point_n c_o wherein_o the_o circle_n cut_v the_o one_o the_o other_o draw_v one_o right_a line_n to_o the_o point_n a_o and_o a_o other_o right_a line_n to_o the_o point_n b._n and_o forasmuch_o as_o the_o point_n a_o be_v the_o centre_n of_o the_o circle_n cbd_v demonstration_n therefore_o by_o the_o 15._o definition_n the_o line_n ac_fw-la be_v equal_a to_o the_o line_n ab_fw-la again_o forasmuch_o as_o the_o point_n b_o be_v the_o centre_n of_o the_o circle_n caesar_n therefore_o by_o the_o same_o definition_n the_o line_n bc_o be_v equal_a to_o the_o line_n basilius_n and_o it_o be_v prove_v that_o the_o line_n ac_fw-la be_v equal_a to_o the_o line_n ab_fw-la wherefore_o either_o of_o these_o line_n ca_n and_o cb_o be_v equal_a to_o the_o line_n ab_fw-la but_o thing_n which_o be_v equal_a to_o one_o and_o the_o same_o thing_n be_v also_o equal_a the_o one_o to_o the_o other_o by_o the_o first_o common_a sentence_n wherefore_o the_o line_n ca_n also_o be_v equal_a to_o the_o line_n cb._n wherefore_o these_o three_o right_a line_n ca_n ab_fw-la and_o bc_o be_v equal_a the_o one_o to_o the_o other_o wherefore_o the_o triangle_n abc_n be_v equilater_n wherefore_o upon_o the_o line_n ab_fw-la be_v describe_v a_o equilater_n triangle_n abc_n wherefore_o upon_o a_o line_n give_v not_o be_v infinite_a there_o be_v describe_v a_o equilater_n triangle_n which_o be_v the_o thing_n which_o be_v require_v to_o be_v do_v a_o triangle_n or_o any_o other_o rectiline_v figure_n be_v then_o say_v to_o be_v set_v or_o describe_v upon_o a_o line_n when_o the_o line_n be_v one_o of_o the_o side_n of_o the_o figure_n this_o first_o proposition_n be_v a_o problem_n because_o it_o require_v act_n or_o do_v namely_o to_o describe_v a_o triangle_n and_o this_o be_v to_o be_v note_v that_o every_o proposition_n whether_o it_o be_v a_o problem_n or_o a_o theorem_a common_o contain_v in_o it_o a_o thing_n give_v and_o a_o thing_n require_v to_o be_v search_v out_o although_o it_o be_v not_o always_o so_o and_o the_o thing_n give_v give_v be_v ever_o
about_o the_o diameter_n together_o with_o the_o two_o supplement_n make_v a_o gnomon_n as_o the_o parallelogram_n ebkh_n with_o the_o two_o supplement_n aegk_n and_o khfd_n make_v the_o gnomon_n fgeh_a likewise_o the_o parallelogram_n gkcf_n with_o the_o same_o two_o supplement_n make_v the_o gnomon_n ehfg_n and_o this_o definition_n of_o a_o gnomon_n extend_v itself_o and_o be_v general_a to_o all_o kind_n of_o parallelogram_n whether_o they_o be_v square_n or_o figure_n of_o one_o side_n long_o or_o rhombus_fw-la or_o romboide_n to_o be_v short_a if_o you_o take_v away_o from_o the_o whole_a parallelogram_n one_o of_o the_o partial_a parallelogram_n which_o be_v about_o the_o diameter_n whether_o you_o will_v the_o rest_n of_o the_o figure_n be_v a_o gnomon_n campane_v after_o the_o last_o proposition_n of_o the_o first_o book_n add_v this_o proposition_n book_n two_o square_n be_v give_v to_o adjoin_v to_o one_o of_o they_o a_o gnomon_n equal_a to_o the_o other_o square_n which_o for_o that_o as_o than_o it_o be_v not_o teach_v what_o a_o gnomon_n be_v i_o there_o omit_v think_v that_o it_o may_v more_o apt_o be_v place_v here_o the_o do_v and_o demonstration_n whereof_o be_v thus_o suppose_v that_o there_o be_v two_o square_n ab_fw-la and_o cd_o unto_o one_o of_o which_o namely_o unto_o ab_fw-la it_o be_v require_v to_o add_v a_o gnomon_n equal_a to_o the_o other_o square_n namely_o to_o cd_o produce_v the_o side_n bf_o of_o the_o square_a ab_fw-la direct_o to_o the_o point_n e._n and_o put_v the_o line_n fe_o equal_a to_o the_o side_n of_o the_o square_a cd_o and_o draw_v a_o line_n from_o e_o to_o a._n now_o then_o forasmuch_o as_o efa_n be_v a_o rectangle_n triangle_n therefore_o by_o the_o 47._o of_o the_o first_o the_o square_a of_o the_o line_n ea_fw-la be_v equal_a to_o the_o square_n of_o the_o line_n of_o &_o fa._n but_o the_o square_n of_o the_o line_n of_o be_v equal_a to_o the_o square_a cd_o &_o the_o square_a of_o the_o side_n favorina_n be_v the_o square_a ab_fw-la wherefore_o the_o square_a of_o the_o line_n ae_n be_v equal_a to_o the_o two_o square_n cd_o and_o ab_fw-la but_o the_o side_n of_o and_o favorina_n be_v by_o the_o 21._o of_o the_o first_o long_o than_o the_o side_n ae_n and_o the_o side_n favorina_n be_v equal_a to_o the_o side_n fb_o wherefore_o the_o side_n of_o and_o fb_o be_v long_o they_o the_o side_n ae_n wherefore_o the_o whole_a line_n be_v be_v long_o than_o the_o line_n ae_n from_o the_o line_n be_v cut_v of_o a_o line_n equal_a to_o the_o line_n ae_n which_o let_v be_v bc._n and_o by_o the_o 46._o proposition_n upon_o the_o line_n bc_o describe_v a_o square_a which_o let_v be_v bcgh_n which_o shall_v be_v equal_a to_o the_o square_n of_o the_o line_n ae_n but_o the_o square_n of_o the_o line_n ae_n be_v equal_a to_o the_o two_o square_n ab_fw-la and_o dc_o wherefore_o the_o square_a bcgh_n be_v equal_a to_o the_o same_o square_n wherefore_o forasmuch_o as_o the_o square_a bcgh_n be_v compose_v of_o the_o square_a ab_fw-la and_o of_o the_o gnomon_n fgah_n the_o say_a gnomon_n shall_v be_v equal_a unto_o the_o square_a cd_o which_o be_v require_v to_o be_v do_v a_o other_o more_o ready_a way_n after_o pelitarius_n suppose_v that_o there_o be_v two_o square_n who_o side_n let_v be_v ab_fw-la and_o bc._n it_o be_v require_v unto_o the_o square_n of_o the_o line_n ab_fw-la to_o add_v a_o gnomon_n equal_a to_o the_o square_n of_o the_o line_n bc._n set_v the_o line_n ab_fw-la and_o bc_o in_o such_o sort_n that_o they_o make_v a_o right_a angle_n abc_n and_o draw_v a_o line_n from_o a_o to_o c._n and_o upon_o the_o line_n ab_fw-la describe_v a_o square_n which_o let_v be_v abde_n and_o produce_v the_o line_n basilius_n to_o the_o point_n fletcher_n and_o put_v the_o line_n bf_o equal_a to_o the_o line_n ac_fw-la and_o upon_o the_o line_n bf_o describe_v a_o square_n which_o let_v be_v bfgh_o which_o shall_v be_v equal_a to_o the_o square_n of_o the_o line_n ac_fw-la when_o as_o the_o line_n bf_o and_o ac_fw-la be_v equal_a and_o therefore_o it_o be_v equal_a to_o the_o square_n of_o the_o two_o line_n ab_fw-la and_o bc._n now_o forasmuch_o as_o the_o square_a bfgh_o be_v make_v complete_a by_o the_o square_a abde_n and_o by_o the_o gnomon_n fegd_v the_o gnomon_n fegd_v shall_v be_v equal_a to_o the_o square_n of_o the_o line_n bc_o which_o be_v require_v to_o be_v do_v the_o 1._o theorem_a the_o 1._o proposition_n if_o there_o be_v two_o right_a line_n and_o if_o the_o one_o of_o they_o be_v divide_v into_o part_n how_o many_o soever_o the_o rectangle_n figure_n comprehend_v under_o the_o two_o right_a line_n be_v equal_a to_o the_o rectangle_n figure_n which_o be_v comprehend_v under_o the_o line_n undivided_a and_o under_o every_o one_o of_o the_o part_n of_o the_o other_o line_n svppose_v that_o there_o be_v two_o right_a line_n a_o and_o bc_o and_o let_v one_o of_o they_o namely_o bc_o be_v divide_v at_o all_o adventure_n in_o the_o point_n d_o and_o e._n then_o i_o say_v that_o the_o rectangle_n figure_n comprehend_v under_o the_o line_n a_o and_o bc_o be_v equal_a unto_o the_o rectangle_n figure_n comprehend_v under_o the_o line_n a_o and_o bd_o &_o unto_o the_o rectangle_n figure_n which_o be_v comprehend_v under_o the_o line_n a_o and_o de_fw-fr and_o also_o unto_o the_o rectangle_n figure_n which_o be_v comprehend_v under_o the_o line_n a_o and_o ec_o construction_n for_o from_o the_o point_n brayse_v up_o by_o the_o 11._o of_o the_o first_o unto_o the_o right_a line_n bc_o a_o perpendicular_a line_n bf_o &_o unto_o the_o line_n a_o by_o the_o three_o of_o the_o first_o put_v the_o line_n bg_o equal_a and_o by_o the_o point_n g_z by_o the_o 31._o of_o the_o first_o draw_v a_o parallel_n line_n unto_o the_o right_a line_n bc_o and_o let_v the_o same_o be_v gm_n and_o by_o the_o self_n same_o by_o the_o poor_n d_z e_o and_o c_o draw_v unto_o the_o line_n bg_o these_o parallel_a line_n dk_o demonstration_n el_n and_o ch._n now_o then_o the_o parallelogram_n bh_o be_v equal_a to_o these_o parallelogram_n bk_o dl_o and_o eh_o but_o the_o parallelogram_n bh_o be_v equal_a unto_o that_o which_o be_v contain_v under_o the_o line_n a_o and_o bc._n for_o it_o be_v comprehend_v under_o the_o line_n gb_o &_o bc_o and_o the_o line_n gb_o be_v equal_a unto_o the_o line_n a_o and_o the_o parallelogram_n bk_o be_v equal_a to_o that_o which_o be_v contain_v under_o the_o line_n a_o and_o bd_o for_o it_o be_v comprehend_v under_o the_o line_n gb_o and_o bd_o and_o bg_o be_v equal_a unto_o a_o and_o the_o parallelogram_n dl_o be_v equal_a to_o that_o which_o be_v contain_v under_o the_o line_n a_o and_o de_fw-fr for_o the_o line_n dk_o that_o be_v bg_o be_v equal_a unto_o a_o and_o moreover_o likewise_o the_o parallelogram_n eh_o be_v equal_a to_o that_o which_o be_v contain_v under_o the_o line_n a_o &_o ec_o wherefore_o that_o which_o be_v comprehend_v under_o the_o line_n a_o &_o bc_o be_v equal_a to_o that_o which_o be_v comprehend_v under_o the_o line_n a_o &_o bd_o &_o unto_o that_o which_o be_v comprehend_v under_o the_o line_n a_o and_o de_fw-fr and_o moreover_o unto_o that_o which_o be_v comprehend_v under_o the_o line_n a_o and_o ec_o if_o therefore_o there_o be_v two_o right_a line_n and_o if_o the_o one_o of_o they_o be_v divide_v into_o part_n how_o many_o soever_o the_o rectangle_n figure_n comprehend_v under_o the_o two_o right_a line_n be_v equal_a to_o the_o rectangle_n figure_n which_o be_v comprehend_v under_o the_o line_n undivided_a and_o under_o every_o one_o of_o the_o part_n of_o the_o other_o line_n which_o be_v require_v to_o be_v demonstrate_v because_o that_o all_o the_o proposition_n of_o this_o second_o book_n for_o the_o most_o part_n be_v true_a both_o in_o line_n and_o in_o number_n and_o may_v be_v declare_v by_o both_o therefore_o have_v i_o have_v add_v to_o every_o proposition_n convenient_a number_n for_o the_o manifestation_n of_o the_o same_o and_o to_o the_o end_n the_o studious_a and_o diligent_a reader_n may_v the_o more_o full_o perceive_v and_o understand_v the_o agreement_n of_o this_o art_n of_o geometry_n with_o the_o science_n of_o arithmetic_n and_o how_o never_o &_o dear_a sister_n they_o be_v together_o so_o that_o the_o one_o can_v without_o great_a blemish_n be_v without_o the_o other_o i_o have_v here_o also_o join_v a_o little_a book_n of_o arithmetic_n write_v by_o one_o barlaam_n a_o greek_a author_n a_o man_n of_o great_a knowledge_n in_o which_o book_n be_v by_o the_o author_n demonstrate_v many_o of_o the_o self_n same_o propriety_n and_o passion_n in_o number_n which_o euclid_n in_o this_o his_o second_o book_n have_v demonstrate_v in_o magnitude_n
touch_v the_o circle_n abc_n and_o let_v the_o same_o be_v de._n construction_n and_o by_o the_o first_o of_o the_o same_o let_v the_o point_n fletcher_n be_v the_o centre_n of_o the_o circle_n abc_n and_o draw_v these_o right_a line_n fe_o fb_o and_o fd._n wherefore_o the_o angle_n feed_v be_v a_o right_a angle_n demonstration_n and_o for_o asmuch_o as_o the_o right_a line_n de_fw-fr touch_v the_o circle_n abc_n and_o the_o right_a line_n dca_n cut_v the_o same_o therefore_o by_o the_o proposition_n go_v before_o that_o which_o be_v contain_v under_o the_o line_n ad_fw-la and_o dc_o be_v equal_a to_o the_o square_n of_o the_o line_n de._n but_o that_o which_o be_v contain_v under_o the_o line_n ad_fw-la and_o dc_o be_v suppose_v to_o be_v equal_a to_o the_o square_n of_o the_o line_n db._n wherefore_o the_o square_a of_o the_o line_n de_fw-fr be_v equal_a to_o the_o square_n of_o the_o line_n db._n wherefore_o also_o the_o line_n de_fw-fr be_v equal_a to_o the_o line_n db._n and_o the_o line_n fe_o be_v equal_a to_o the_o line_n fb_o for_o they_o be_v draw_v from_o the_o centre_n to_o the_o circumference_n now_o therefore_o these_o two_o line_n de_fw-fr and_o of_o be_v equal_a to_o these_o two_o line_n db_o and_o bf_o and_o fd_o be_v a_o common_a base_a to_o they_o both_o wherefore_o by_o the_o 8._o of_o the_o first_o the_o angle_n def_n be_v equal_a to_o the_o angle_n dbf_n but_o the_o angle_n def_n be_v a_o right_a angle_n wherefore_o also_o the_o angle_n dbf_n be_v a_o right_a angle_n and_o the_o line_n fb_o be_v produce_v shall_v be_v the_o diameter_n of_o the_o circle_n but_o if_o from_o the_o end_n of_o the_o diameter_n of_o a_o circle_n be_v draw_v a_o right_a line_n make_v right_a angle_n the_o right_a line_n so_o draw_v touch_v the_o circle_n by_o the_o corollary_n of_o the_o 16._o of_o the_o three_o wherefore_o the_o right_a line_n db_o touch_v the_o circle_n abc_n and_o the_o like_a demonstration_n will_v serve_v if_o the_o centre_n be_v in_o the_o line_n ac_fw-la if_o therefore_o without_o a_o circle_n be_v take_v a_o certain_a point_n and_o from_o that_o point_n be_v draw_v to_o the_o circle_n two_o right_a line_n of_o which_o the_o one_o do_v cut_v the_o circle_n and_o the_o other_o fall_v upon_o the_o circle_n and_o that_o in_o such_o sort_n that_o the_o rectangle_n parallelogram_n which_o be_v contain_v under_o the_o whole_a right_a line_n which_o cut_v the_o circle_n and_o that_o portion_n of_o the_o same_o line_n that_o lie_v between_o the_o point_n and_o the_o utter_a circumference_n of_o the_o circle_n be_v equal_a to_o the_o square_n make_v of_o the_o line_n that_o fall_v upon_o the_o circle_n then_o the_o line_n that_o so_o fall_v upon_o the_o circle_n shall_v touch_v the_o circle_n which_o be_v require_v to_o be_v prove_v ¶_o a_o other_o demonstration_n after_o pelitarius_n suppose_v that_o there_o be_v a_o circle_n bcd_o who_o centre_n let_v be_v e_o pelitarius_n and_o take_v a_o point_n without_o it_o namely_o a_o and_o from_o the_o point_n a_o draw_v two_o right_a line_n abdella_n and_o ac_fw-la of_o which_o let_v abdella_n cut_v the_o circle_n in_o the_o point_n b_o &_o let_v the_o other_o fall_n upon_o it_o and_o let_v that_o which_o be_v contain_v under_o the_o line_n ad_fw-la and_o ab_fw-la be_v equal_a to_o the_o square_n of_o the_o line_n ac_fw-la then_o i_o say_v that_o the_o line_n ac_fw-la touch_v the_o circle_n for_o first_o if_o the_o line_n abdella_n do_v pass_v by_o the_o centre_n draw_v the_o right_a line_n ce._n and_o by_o the_o 6._o of_o the_o second_o that_o which_o be_v contain_v under_o the_o line_n ad_fw-la and_o ab_fw-la together_o with_o the_o square_n of_o the_o line_n ebb_n that_o be_v with_o the_o square_n of_o the_o line_n ec_o for_o the_o line_n ebb_v and_o ec_o be_v equal_a be_v equal_a to_o the_o square_n of_o the_o line_n ae_n but_o that_o which_o be_v contain_v under_o the_o line_n ad_fw-la and_o ab_fw-la be_v suppose_v to_o be_v equal_a to_o the_o square_n of_o the_o line_n ac_fw-la wherefore_o the_o square_a of_o the_o line_n ac_fw-la together_o with_o the_o square_n of_o the_o line_n ce_fw-fr be_v equal_a to_o the_o square_n of_o the_o line_n ae_n wherefore_o by_o the_o last_o of_o the_o first_o the_o angle_n at_o the_o point_n c_o be_v a_o right_a angle_n wherefore_o by_o the_o 18._o of_o this_o book_n the_o line_n ac_fw-la touch_v the_o circle_n but_o if_o the_o line_n abdella_n do_v not_o pass_v by_o the_o centre_n draw_v from_o the_o point_n a_o the_o line_n ad_fw-la in_o which_o let_v be_v the_o centre_n e._n and_o forasmuch_o as_o that_o which_o be_v contain_v under_o this_o whole_a line_n and_o his_o outward_a part_n be_v equal_a to_o that_o which_o be_v contain_v under_o the_o line_n ad_fw-la and_o ab_fw-la by_o the_o first_o corollary_n before_o put_v therefore_o the_o same_o be_v equal_a to_o the_o square_n of_o the_o line_n ac_fw-la wherefore_o the_o angle_n eca_n be_v a_o right_a angle_n as_o have_v before_o be_v prove_v in_o the_o first_o part_n of_o this_o proposition_n and_o therefore_o the_o line_n ac_fw-la touch_v the_o circle_n which_o be_v require_v to_o be_v prove_v the_o end_n of_o the_o three_o book_n of_o euclides_n element_n ¶_o the_o four_o book_n of_o euclides_n element_n this_o four_o book_n intreat_v of_o the_o inscription_n &_o circumscription_n of_o rectiline_a figure_n book_n how_o one_o right_a line_v figure_n may_v be_v inscribe_v within_o a_o other_o right_o line_v figure_n and_o how_o a_o right_a line_v figure_n may_v be_v circumscribe_v about_o a_o other_o right_o line_v figure_n in_o such_o as_o may_v be_v inscribe_v and_o circumscribe_v within_o or_o about_o the_o other_o for_o all_o right_n line_v figure_n can_v so_o be_v inscribe_v or_o circumscribe_v within_o or_o about_o the_o other_o also_o it_o teach_v how_o a_o triangle_n a_o square_a and_o certain_a other_o rectiline_a figure_n be_v regular_a may_v be_v inscribe_v within_o a_o circle_n also_o how_o they_o may_v be_v circumscribe_v about_o a_o circle_n likewise_o how_o a_o circle_n may_v be_v inscribe_v within_o they_o and_o how_o it_o may_v be_v circumscribe_v about_o they_o and_o because_o the_o manner_n of_o entreaty_n in_o this_o book_n be_v diverse_a from_o the_o entreaty_n of_o the_o former_a book_n he_o use_v in_o this_o other_o word_n and_o term_n than_o he_o use_v in_o they_o the_o definition_n of_o which_o in_o order_n here_o after_o follow_v definition_n a_o rectiline_a figure_n be_v say_v to_o be_v inscribe_v in_o a_o rectiline_a figure_n definition_n when_o every_o one_o of_o the_o angle_n of_o the_o inscribe_v figure_n touch_v every_o one_o of_o the_o side_n of_o the_o figure_n wherein_o it_o be_v inscribe_v as_o the_o triangle_n abc_n be_v inscribe_v in_o the_o triangle_n def_n because_o that_o every_o angle_n of_o the_o triangle_n inscribe_v namely_o the_o triangle_n abc_n touch_v every_o side_n of_o the_o triangle_n within_o which_o it_o be_v describe_v namely_o of_o the_o triangle_n def_n as_o the_o angle_n cab_n touch_v the_o side_n ed_z the_o angle_n abc_n touch_v the_o side_n df_o and_o the_o angle_n acb_o touch_v the_o side_n ef._n so_o likewise_o the_o square_a abcd_o be_v say_v to_o be_v inscribe_v within_o the_o square_a efgh_o for_o every_o angle_n of_o it_o touch_v some_o one_o side_n of_o the_o other_o so_o also_o the_o pentagon_n or_o five_o angle_a figure_n abcde_o be_v inscribe_v within_o the_o pentagon_n or_o five_o angle_a figure_n fghikâ—_n â—_z as_o you_o see_v in_o the_o figureâ—_n likewise_o a_o rectiline_a figure_n be_v say_v to_o be_v circumscribe_v about_o a_o rectiline_a figure_n definition_n when_o every_o one_o of_o the_o side_n of_o the_o figure_n circumscribe_v touch_v every_o one_o of_o the_o angle_n of_o the_o figure_n about_o which_o it_o be_v circumscribe_v as_o in_o the_o former_a description_n the_o triangle_n def_n be_v say_v to_o be_v circumscribe_v about_o the_o triangle_n abc_n for_o that_o every_o side_n of_o the_o figure_n circumscribe_v namely_o of_o the_o triangle_n def_n touch_v every_o angle_n of_o the_o figure_n wherabout_o it_o be_v circumscribe_v as_o the_o side_n df_o of_o the_o triangle_n def_n circumscribe_v touch_v the_o angle_n abc_n of_o the_o triangle_n abc_n about_o which_o it_o be_v circumscribe_v and_o the_o side_n of_o touch_v the_o angle_n bca_o and_o the_o side_n cd_o touch_v the_o angle_n cab_n likewise_o understand_v you_o of_o the_o square_a efgh_o which_o be_v circumscribe_v about_o the_o square_n abcd_o for_o every_o side_n of_o the_o one_o touch_v some_o one_o side_n of_o the_o other_o even_o so_o by_o the_o same_o reason_n the_o pentagon_n fghik_n be_v circumscribe_v about_o the_o pentagon_n abcde_o as_o you_o see_v in_o the_o figure_n on_o the_o other_o side_n and_o thus_o may_v you_o
the_o denomination_n of_o they_o both_o the_o denomination_n of_o sextupla_fw-la proportion_n be_v 6_o the_o denomination_n of_o dupla_fw-la proportion_n be_v 2._o now_o divide_v 6._o the_o denomination_n of_o the_o one_o by_o 2._o the_o denomination_n of_o the_o other_o the_o quotient_n shall_v be_v 3_o which_o be_v the_o denomination_n of_o a_o new_a proportion_n namely_o tripla_fw-la so_o that_o when_o dupla_fw-la proportion_n be_v subtrahe_v from_o sextupla_fw-la there_o shall_v remain_v tripla_fw-la proportion_n and_o thus_o may_v you_o do_v in_o all_o other_o 6._o a_o parallelogram_n apply_v to_o a_o right_a line_n be_v say_v to_o want_v in_o form_n by_o a_o parallelogram_n like_a to_o one_o give_v when_o the_o parallelogram_n apply_v want_v to_o the_o fill_n of_o the_o whole_a line_n by_o a_o parallelogram_n like_a to_o one_o give_v definition_n and_o then_o be_v it_o say_v to_o exceed_v when_o it_o exceed_v the_o line_n by_o a_o parallelogram_n like_a to_o that_o which_o be_v give_v as_o let_v e_o be_v a_o parallelogramme_n give_v and_o let_v ab_fw-la be_v a_o right_a line_n to_o who_o be_v apply_v the_o parallelogram_n acdf_n now_o if_o it_o want_v of_o the_o fill_n of_o the_o line_n ab_fw-la by_o the_o parallelogram_n dfgb_n be_v like_a to_o the_o parallelogram_n give_v e_o then_o be_v the_o parallelogram_n say_v to_o want_v in_o form_n by_o a_o parallelogram_n like_a unto_o a_o parallelogram_n give_v likewise_o if_o it_o exceed_v as_o the_o parallelogram_n acgd_v apply_v to_o the_o linâ—_n abâ—_n if_o it_o exceed_v it_o by_o the_o parallelogram_n fgbd_v be_v like_o to_o the_o parallelogram_n fletcher_n which_o be_v give_v then_o be_v the_o parallelogram_n abgd_v say_v to_o exceed_v in_o form_n by_o a_o parallelogram_n like_a to_o a_o parallelogram_n give_v this_o definition_n be_v add_v by_o flussates_n as_o it_o seem_v it_o be_v not_o in_o any_o common_a greek_a book_n abroad_o nor_o in_o any_o commentary_n it_o be_v for_o many_o theorem_n follow_v very_o necessary_a the_o 1._o theorem_a the_o 1._o proposition_n triangle_n &_o parallelogram_n which_o be_v under_o one_o &_o the_o self_n same_o altitude_n be_v in_o proportion_n as_o the_o base_a of_o the_o one_o be_v to_o the_o base_a of_o the_o other_o and_o forasmuch_o as_o the_o line_n cb_o bg_o and_o gh_o be_v equal_a the_o one_o to_o the_o other_o part_n therefore_o the_o triangle_n also_o ahg_n agb_n and_o abc_n be_v by_o the_o 38._o of_o the_o first_o equal_a the_o one_o to_o the_o other_o wherefore_o how_o multiplex_n the_o base_a hc_n be_v to_o the_o base_a bc_o so_o multiplex_n also_o be_v the_o triangle_n ahc_n to_o the_o triangle_n abc_n and_o by_o the_o same_o reason_n also_o how_o multiplex_n the_o base_a lc_n be_v to_o the_o base_a dc_o so_o multiplex_n also_o be_v the_o triangle_n alc_n to_o the_o triangle_n adc_o wherefore_o if_o the_o base_a hc_n be_v equal_a unto_o the_o base_a cl_n then_o by_o the_o 38._o of_o the_o first_o the_o triangle_n ahc_n be_v equal_a unto_o the_o triangle_n acl_n and_o if_o the_o base_a hc_n exceed_v the_o base_a cl_n then_o also_o the_o triangle_n ahc_n exceed_v the_o triangle_n acl_n and_o if_o the_o base_a be_v less_o the_o triangle_n also_o shall_v be_v less_o now_o then_o there_o be_v four_o magnitude_n namely_o the_o two_o basâ—s_n bc_o and_o cd_o and_o the_o two_o triangle_n abc_n and_o acd_o and_o to_o the_o base_a bc_o and_o to_o the_o triangle_n abc_n namely_o to_o the_o first_o and_o the_o three_o be_v take_v equemulâ—iplices_n namely_o the_o base_a hc_n and_o the_o triangle_n ahc_n and_o likewise_o to_o the_o base_a cd_o and_o to_o the_o triangle_n adc_o namely_o to_o the_o second_o and_o the_o four_o be_v take_v certain_a other_o equemultiplices_fw-la that_o be_v the_o base_a cl_n and_o the_o triangle_n alc_n and_o it_o have_v be_v prove_v that_o if_o the_o multiplex_n of_o the_o first_o magnitude_n that_o be_v the_o base_a hc_n do_v exceed_v the_o multiplex_n of_o the_o second_o that_o be_v the_o base_a cl_n the_o multiplex_n also_o of_o the_o three_o that_o be_v the_o triangle_n ahc_n exceed_v the_o multiplex_n of_o the_o â—_z that_o be_v the_o triangle_n alc_n and_o if_o the_o say_v base_a hc_n be_v equal_a to_o the_o say_a baâ—â—_n cl_n the_o triangle_n also_o ahc_n be_v equal_a to_o the_o triangle_n alc_n and_o if_o it_o be_v less_o it_o iâ—_z less_o wherefore_o by_o the_o six_o definition_n of_o the_o five_o as_o the_o first_o of_o the_o foresay_a magnitude_n be_v to_o the_o second_o so_o be_v the_o three_o to_o the_o four_o wherefore_o as_o the_o base_a bc_o be_v to_o the_o base_a cd_o so_o be_v the_o triangle_n abc_n to_o the_o triangle_n acd_o and_o because_o by_o the_o 41._o of_o the_o first_o the_o parallelogram_n ec_o be_v double_a to_o the_o triangle_n abc_n part_n and_o by_o the_o same_o the_o parallelogram_n fc_o be_v double_a to_o the_o triangle_n acd_o therefore_o the_o parallelogram_n ec_o and_o fc_fw-la be_v equemultiplices_fw-la unto_o the_o triangle_n abc_n and_o acd_o but_o the_o part_n of_o equemultiplices_fw-la by_o the_o 15._o of_o the_o five_o have_v one_o and_o the_o same_o proportion_n with_o theiâ—_z equemultiplices_fw-la wherefore_o as_o the_o triangle_n abc_n be_v to_o the_o triangle_n acd_o so_o be_v the_o parallelogram_n ec_o to_o the_o parallelogram_n fc_o and_o forasmuch_o as_o it_o have_v be_v demonstrate_v that_o as_o the_o base_a bc_o be_v to_o the_o base_a cd_o so_o be_v the_o triangle_n abc_n to_o the_o triangle_n acd_o and_o as_o the_o triangle_n abc_n be_v to_o the_o triangle_n acd_v so_o be_v the_o parallelogram_n ec_o to_o the_o parallelogram_n fc_o wherefore_o by_o the_o 11._o of_o the_o five_o as_o the_o base_a bc_o be_v to_o the_o base_a cd_o so_o be_v the_o parallelogram_n ec_o to_o the_o parallelogram_n fc_o the_o parallelogram_n may_v also_o be_v demonstrate_v a_o part_n by_o themselves_o as_o the_o triangle_n be_v if_o we_o describe_v upon_o the_o base_n bg_o gh_o and_o dk_o &_o kl_o parallelogram_n under_o the_o self_n same_o altitude_n that_o the_o parallelogrammeâ—_n give_v be_v wherefore_o triangle_n and_o parallelogram_n which_o be_v under_o one_o and_o the_o self_n same_o altitude_n be_v in_o proportion_n as_o the_o base_a of_o the_o one_o be_v to_o the_o base_a of_o the_o other_o which_o be_v require_v to_o be_v demonstrate_v here_o flussates_n add_v this_o corollary_n if_o two_o right_a line_n be_v give_v the_o one_o of_o they_o be_v divide_v how_o so_o ever_o flussates_n the_o rectangle_n figure_v contain_v under_o the_o whole_a line_n undivided_a and_o each_o of_o the_o segment_n of_o the_o line_n divide_v be_v in_o proportion_n the_o one_o to_o the_o other_o as_o the_o segment_n be_v the_o one_o to_o the_o other_o for_o imagine_v the_o figure_n basilius_n and_o ad_fw-la in_o the_o former_a description_n to_o be_v rectangle_v the_o rectangle_n figure_v contain_v under_o the_o whole_a right_a line_n ac_fw-la and_o the_o segment_n of_o the_o right_a line_n bd_o which_o be_v cuâ—_n in_o the_o point_n c_o namely_o the_o parallelogram_n basilius_n and_o ad_fw-la be_v in_o proportion_n the_o one_o to_o the_o other_o as_o the_o segmente_n bc_o and_o cd_o be_v the_o 2._o theorem_a the_o 2._o proposition_n if_o to_o any_o one_o of_o the_o side_n of_o a_o triangle_n be_v draw_v a_o parallel_n right_a line_n it_o shall_v cut_v the_o side_n of_o the_o same_o triangle_n proportional_o and_o if_o the_o side_n of_o a_o triangle_n be_v cut_v proportional_o a_o right_a line_n draw_v from_o section_n to_o section_n be_v a_o parallel_n to_o the_o other_o side_n of_o the_o triangle_n svppose_v that_o there_o be_v a_o triangle_n abc_n unto_o one_o of_o the_o side_n whereof_o namely_o unto_o bc_o let_v there_o be_v draw_v a_o parallel_a line_n de_fw-fr cut_v the_o side_n ac_fw-la and_o ab_fw-la in_o the_o point_n e_o and_o d._n then_o i_o say_v first_o that_o as_o bd_o be_v to_o da_z so_o be_v ce_fw-fr to_o ea_fw-la theorem_a draw_v a_o line_n from_o b_o to_o e_o &_o also_o from_o c_z to_o d._n wherefore_o by_o the_o 37._o of_o the_o first_o the_o triangle_n bde_v be_v equal_a unto_o the_o triangle_n cde_o for_o they_o be_v set_v upon_o one_o and_o the_o same_o base_a de_fw-fr and_o be_v contain_v within_o the_o self_n same_o parallel_n de_fw-fr and_o bc._n consider_v also_o a_o certain_a other_o triangle_n ade_n now_o thing_n equal_a by_o the_o 7._o of_o the_o five_o have_v to_o oneself_o thing_n one_o and_o the_o same_o proportion_n wherefore_o as_o the_o triangle_n bde_n be_v to_o the_o triangle_n ade_n so_o be_v the_o triangle_n cde_o to_o the_o triangle_n ade_n but_o as_o the_o triangle_n bde_n be_v to_o the_o triangle_n ade_n so_o be_v the_o base_a bd_o to_o the_o base_a dam_fw-la by_o the_o first_o of_o this_o book_n for_o they_o be_v under_o one_o
seven_o but_o the_o lest_o number_n in_o any_o proportion_n measure_v those_o number_n that_o have_v one_o and_o the_o same_o proportion_n with_o they_o equal_o the_o great_a the_o great_a and_o the_o less_o the_o less_o by_o the_o 21_o of_o the_o seven_o wherefore_o b_o measure_v e_o namely_o the_o consequent_a the_o consequent_a and_o forasmuch_o as_o a_o multiply_a b_o and_o e_o produce_v c_z and_o d_o therefore_o by_o the_o 17._o of_o the_o seven_o as_o â—_o be_v to_o e_o so_o be_v c_o to_o d._n but_o b_o measure_v e._n wherefore_o c_z also_o measure_v d_z the_o great_a the_o less_o which_o be_v impossible_a wherefore_o if_o those_o number_n a_o and_o b_o be_v prime_a they_o shall_v measure_v no_o number_n less_o then_o c._n wherefore_o c_o be_v the_o lest_o number_n which_o a_o and_o b_o measure_n but_o now_o suppose_v that_o a_o and_o b_o be_v not_o prime_a the_o one_o to_o the_o other_o caseâ—_n and_o take_v by_o the_o 35._o of_o the_o seven_o the_o lest_o number_n that_o have_v one_o and_o the_o same_o proportion_n with_o a_o and_o b_o and_o let_v the_o same_o be_v fletcher_n and_o e._n wherefore_o that_o which_o be_v produce_v of_o a_o into_o e_o be_v equal_a to_o that_o which_o be_v produce_v of_o b_o into_o fletcher_n by_o the_o 19_o of_o the_o seven_o let_v a_o multiply_v e_o produce_v câ—_n wherefore_o b_o multiply_a fletcher_n produce_v also_o c._n wherefore_o a_o and_o b_o measure_n c._n then_o i_o say_v that_z c_z be_v the_o lest_o number_n that_o they_o measure_v for_o if_o it_o be_v not_o those_o number_n a_o and_o b_o shall_v measure_v some_o number_n less_o than_o c_o let_v they_o measure_v a_o number_n less_o than_o c_o and_o let_v the_o same_o be_v dâ—_n and_o how_o often_o a_o measure_v d_z so_o many_o unity_n let_v there_o be_v in_o g_o and_o how_o often_o â—_o measure_v dâ—_n so_o many_o unity_n let_v there_o be_v in_o h._n absurdity_n now_o then_o a_o multiply_v g_o produce_v d._n and_o b_o multiply_a h_o produce_v also_o d._n wherefore_o that_o which_o be_v produce_v of_o a_o into_o g_z be_v equal_a to_o that_o which_o be_v produce_v of_o b_o into_o h._n wherefore_o by_o the_o 19_o of_o the_o seven_o as_o a_o be_v to_o b_o so_o be_v h_o to_o gâ—_n but_o as_o a_o be_v to_o b_o so_o be_v fletcher_n to_o e._n wherefore_o as_o fletcher_n be_v to_o e_o so_o be_v h_o to_o g_z but_o the_o lest_o number_n in_o any_o proportion_n measure_v the_o number_n that_o have_v the_o same_o proportion_n with_o they_o equal_o the_o great_a the_o great_a &_o the_o less_o the_o less_o by_o the_o 21._o of_o the_o seven_o wherefore_o e_o measure_v g._n and_o forasmuch_o as_o a_o multiply_a g_z and_o e_o produce_v c_z and_o dâ—_n therefore_o by_o the_o 17._o of_o the_o five_o as_o e_o be_v to_o g_z so_o be_v c_o to_o d._n but_o e_o measure_v g._n wherefore_o c_z also_o measure_v d_z the_o great_a the_o less_o which_o be_v impossible_a wherefore_o those_o number_n a_o and_o b_o do_v not_o measure_v any_o number_n less_o than_o â—_o wherefore_o â—_o be_v the_o lest_o number_n that_o be_v measure_v by_o a_o and_o bâ—_n whiâ—h_n be_v require_v to_o be_v do_v the_o 33._o theorem_a the_o 37._o proposition_n if_o two_o number_n measure_v any_o number_n the_o least_o number_n also_o which_o they_o measure_v measure_v the_o self_n same_o number_n svppose_v that_o there_o be_v two_o number_n give_v a_o and_o bâ—_n â—_z and_o let_v they_o measure_v the_o number_n cd_o impossibility_n and_o let_v the_o least_o number_n that_o they_o measure_v be_v e._n then_o i_o say_v that_o e_o also_o measure_v the_o number_n cd_o for_o if_o e_o do_v not_o measure_v cd_o let_v e_o measure_v cd_o that_o be_v subtrahe_v out_o of_o cd_o as_o often_o as_o you_o can_v as_o for_o example_n once_o leave_v a_o less_o than_o itself_o namely_o cf._n and_o let_v the_o number_n subtrahe_v which_o e_o measure_v be_v fd_o and_o forasmuch_o as_o a_o and_o b_o measure_v e_o and_o e_o measure_v df_o therefore_o a_o and_o b_o also_o measure_n df._n and_o they_o measure_v the_o whole_a cd_o wherefore_o by_o the_o 4._o common_a sentence_n of_o the_o seven_o they_o measure_v also_o that_o which_o remain_v cf_o be_v less_o then_o e_o which_o be_v impossible_a wherefore_o e_o of_o necessity_n measure_v cd_o which_o be_v require_v to_o be_v prove_v ¶_o the_o 5._o problem_n the_o 38._o proposition_n three_o number_n be_v give_v to_o find_v out_o the_o least_o number_n which_o they_o measure_v svppose_v that_o there_o be_v three_o number_n give_v a_o b_o c._n it_o be_v require_v to_o find_v out_o the_o least_o number_n which_o they_o measure_v take_v by_o the_o 36._o of_o the_o seven_o the_o least_o number_n which_o a_o and_o b_o measure_n and_o let_v the_o same_o be_v d._n proposition_n now_o then_o c_o either_o measure_v d_z or_o else_o measure_v it_o not_o case_n first_o let_v it_o measure_v it_o and_o the_o number_n also_o a_o and_o b_o measure_v d_o wherefore_o a_o b_o c_o measure_n d._n now_o then_o i_o say_v that_o d_z be_v the_o least_o number_n which_o they_o measure_v for_o if_o not_o let_v the_o number_n a_o b_o c_o measure_v some_o number_n less_o than_o d_o and_o let_v the_o same_o be_v e._n absurdity_n and_o forasmuch_o as_o a_o b_o c_o measure_v e_o therefore_o also_o a_o and_o b_o measure_v e_o wherefore_o by_o the_o 37._o of_o the_o seven_o the_o least_o number_n which_o those_o number_n a_o and_o b_o measure_n shall_v also_o measure_v e._n but_o the_o least_o number_n which_o a_o and_o b_o measure_n be_v d._n wherefore_o d_o measure_v e_o the_o great_a the_o less_o which_o be_v impossible_a wherefore_o these_o number_n a_o b_o c_o shall_v not_o measure_v any_o number_n less_o then_o d._n wherefore_o d_o be_v the_o least_o number_n that_o a_o b_o c_o do_v measure_n but_o now_o suppose_v that_o c_o measure_v not_o d._n and_o take_v by_o the_o 36._o of_o the_o seven_o the_o least_o number_n which_o those_o two_o number_n c_o and_o d_o do_v measure_n and_o let_v the_o same_o be_v e._n case_n and_o forasmuch_o as_o a_o and_o b_o measure_v d_o and_o d_z measure_v e_o therefore_o a_o and_o b_o also_o measure_v e_o and_o c_o also_o measure_v e_o wherefore_o a_o b_o c_o also_o measure_n e._n i_o say_v moreover_o that_o e_o be_v the_o least_o number_n which_o a_o b_o c_o measure_n for_o if_o it_o be_v not_o let_v there_o be_v some_o less_o number_n than_o e_o which_o they_o measure_v absurdity_n and_o let_v the_o same_o be_v f._n and_o forasmuch_o as_o a_o b_o c_o measure_n f._n therefore_o a_o and_o b_o also_o measure_n fletcher_n wherefore_o the_o least_o number_n which_o these_o number_n a_o and_o b_o do_v measure_n do_v also_o measure_n fletcher_n by_o the_o 37._o of_o the_o seven_o but_o the_o least_o number_n which_o a_o and_o b_o do_v measure_n be_v d._n wherefore_o d_o measure_v f._n and_o c_o also_o measure_v f._n wherefore_o d_z and_o c_z measure_n f._n wherefore_o the_o least_o number_n which_o c_z and_o d_z do_v measure_n shall_v also_o by_o the_o self_n same_o measure_n f._n but_o the_o least_o â—â—mbeâ—_n which_o c_o &_o d_z measure_n be_v e._n wherefore_o e_o measure_v fletcher_n namely_o the_o great_a the_o less_o which_o be_v impossible_a wherefore_o these_o number_n a_o b_o c_o do_v not_o measure_v any_o number_n less_o then_o e._n wherefore_o e_o be_v the_o least_o number_n which_o a_o b_o c_o do_v measure_n which_o be_v require_v to_o be_v demonstrate_v in_o like_a manner_n also_o how_o many_o number_n soever_o be_v give_v may_v be_v find_v out_o the_o least_o number_n which_o they_o â—easure_v for_o if_o unto_o the_o three_o number_n a_o b_o c_o be_v add_v a_o forth_o then_o if_o the_o say_v forth_o number_n measure_v the_o number_n e_o then_z be_v e_o the_o least_o number_n which_o the_o four_o number_n give_v measure_n but_o if_o it_o do_v not_o measure_v e_o them_z by_o the_o 37._o of_o this_o book_n must_v you_o find_v but_o the_o least_o number_n which_o e_o and_o the_o forth_o number_n measure_n which_o shall_v be_v the_o number_n seek_v for_o and_o so_o likewise_o if_o there_o be_v five_o six_o or_o how_o manysoever_o give_v corollary_n hereby_o it_o be_v manifest_a that_o the_o least_o common_a measure_n to_o number_n howmanysoever_o corollary_n measure_v every_o number_n which_o the_o say_a number_n how_o many_o soever_o measure_n ¶_o the_o 34._o theorem_a the_o 39_o proposition_n if_o a_o number_n measure_v any_o number_n the_o number_n measure_v shall_v have_v a_o part_n after_o the_o denomination_n of_o the_o number_n measure_v svppose_v that_o there_o be_v a_o number_n b_o which_o let_v measure_v the_o number_n a._n then_o i_o say_v that_o a_o have_v a_o part_n take_v his_o
the_o eight_o between_o a_o and_o b_o there_o fall_v a_o mean_a proportional_a number_n but_o if_o between_o two_o number_n fall_v number_n in_o continual_a proportion_n how_o many_o number_n fall_v between_o they_o so_o many_o also_o by_o the_o 8._o of_o the_o eight_o shall_v there_o fall_v between_o the_o number_n that_o have_v the_o same_o proportion_n with_o they_o wherefore_o between_o c_z and_o d_o also_o there_o fall_v a_o mean_a proportional_a number_n but_o d_z be_v a_o square_a number_n wherefore_o by_o the_o 22._o of_o the_o eight_o c_o also_o be_v a_o square_a number_n which_o be_v require_v to_o be_v prove_v ¶_o the_o 2._o theorem_a the_o 2._o proposition_n if_o two_o number_n multiply_v the_o one_o the_o other_o produce_v a_o square_a number_n those_o number_n be_v like_o plain_a number_n formâ—â—_n svppose_v that_o two_o numberâ—_n a_o and_o b_o multiply_v the_o one_o the_o other_o do_v produce_v c_o a_o square_a number_n then_o i_o say_v that_o a_o and_o b_o be_v like_o plain_a number_n for_o let_v a_o multiply_v himself_o produce_v d._n wherefore_o d_z be_v a_o square_a number_n and_o forasmuch_o as_o a_o multiply_v himself_o produce_v d_z demonstration_n and_o multiply_v b_o produce_v c_z therefore_o by_o the_o 17._o of_o the_o seâ—enth_o as_o a_o be_v to_o b_o so_o be_v d_z to_o c._n and_o forasmuch_o as_o d_z be_v a_o square_a number_n and_o so_o likewise_o be_v c_o therefore_o d_z and_o c_o be_v like_o plain_a number_n wherefore_o between_o d_z and_o c_o there_o be_v by_o the_o 18._o of_o the_o eight_o one_o mean_v proportional_a number_n but_o as_o d_z be_v to_o c_o so_o be_v a_o to_o b._n wherefore_o by_o the_o 8._o of_o the_o eight_o between_o a_o and_o b_o there_o be_v one_o mean_v proportional_a number_n but_o if_o between_o two_o number_n there_o be_v one_o mean_v proportional_a number_n those_o number_n be_v by_o the_o 20._o of_o the_o eight_o like_o plain_a number_n wherefore_o a_o and_o b_o be_v like_o plain_a number_n which_o be_v require_v to_o be_v prove_v a_o corollary_n add_v by_o campane_n hâ—â—â—â—_n it_o be_v manifest_a thâ—t_v two_o square_a number_n multiply_v the_o one_o into_o the_o other_o do_v always_o produce_v a_o squaâ—â—_n numâ—â—r_n campane_n for_o they_o be_v like_o superficial_a number_n and_o therefore_o the_o number_n produce_v of_o they_o be_v by_o the_o first_o of_o this_o book_n a_o square_a number_n but_o a_o square_a number_n mulâ—â—plyeâ—_n into_o a_o number_n not_o square_a produce_v a_o number_n not_o square_a for_o if_o they_o shall_v produce_v a_o square_a number_n they_o shall_v be_v like_o superficial_a number_n by_o this_o proposition_n but_o they_o be_v not_o wherefore_o they_o produce_v a_o number_n not_o square_a but_o if_o a_o square_a number_n multiply_v into_o a_o other_o number_n produce_v a_o square_a number_n that_o other_o number_n shall_v be_v a_o square_a number_n for_o by_o this_o proposition_n that_o other_o number_n be_v like_a unto_o the_o square_a number_n which_o multiply_v it_o and_o therefore_o be_v a_o square_a number_n but_o if_o a_o square_a number_n multiply_v into_o a_o other_o number_n produce_v a_o number_n not_o square_a neither_o shall_v that_o other_o number_n also_o be_v a_o square_a number_n for_o if_o it_o shall_v be_v a_o square_a number_n then_o be_v multiply_v into_o the_o square_a number_n it_o shall_v produce_v a_o square_a number_n by_o the_o first_o part_n of_o this_o corollary_n the_o 3._o theorem_a the_o 3._o proposition_n if_o a_o cube_fw-la number_n multiply_v himself_o produce_v a_o number_n the_o number_n produce_v shall_v be_v a_o cube_fw-la number_n svppose_v that_o a_o be_v a_o cube_fw-la number_n multiply_v himself_o do_v produce_v the_o number_n b._n then_o i_o say_v that_o b_o be_v a_o cube_fw-la number_n take_v the_o side_n of_o a_o and_o let_v the_o same_o be_v the_o number_n c_o and_o let_v c_o multiply_v himself_o produce_v the_o number_n d._n now_o it_o be_v manifest_a that_o c_z multiply_v d_o produce_v a_o by_o the_o 20._o definition_n of_o the_o seven_o demonstration_n and_o forasmuch_o as_o c_z multiply_v himself_o produce_v d_z therefore_o c_z measure_v d_z by_o those_o unity_n which_o be_v in_o c._n but_o unity_n also_o measure_v c_z by_o those_o unity_n which_o be_v in_o c._n wherefore_o as_o unity_n be_v to_o c_o so_o be_v c_z to_o d._n again_o forasmuch_o as_o c_z multiply_v d_o produce_v a_o therefore_o d_z measure_v a_o by_o those_o unity_n which_o be_v in_o c._n but_o unity_n measure_v c_z by_o those_o unity_n which_o be_v in_o c_o wherefore_o as_o unity_n be_v to_o c_o so_o be_v d_z to_o a._n but_o as_o unity_n be_v to_o c_o so_o be_v c_z to_o d_z wherefore_o as_o unity_n be_v to_o c_o so_o be_v c_z to_o d_z &_o d_z to_o a._n wherefore_o between_o unity_n &_o a_o there_o be_v two_o mean_v proportional_a number_n namely_o c_o d._n again_o forasmuch_o as_o a_o multiply_v himself_o produce_v b_o therefore_o a_o measure_v b_o by_o those_o unity_n which_o be_v in_o a._n but_o unity_n also_o measure_v a_o by_o those_o unity_n which_o be_v in_o a._n wherefore_o as_o unity_n be_v to_o a_o so_o be_v a_o to_o b._n but_o between_o a_o and_o unity_n there_o be_v two_o mean_v proportional_a number_n wherefore_o between_o a_o and_o b_o also_o there_o be_v two_o mean_v proportional_a number_n by_o the_o 8._o of_o the_o eight_o but_o if_o between_o two_o number_n there_o be_v two_o mean_v proportional_a number_n and_o if_o the_o first_o be_v a_o cube_fw-la number_n the_o four_o also_o shall_v be_v a_o cube_fw-la number_n by_o the_o 21._o of_o the_o eight_o but_o a_o be_v a_o cube_fw-la number_n wherefore_o b_o also_o be_v a_o cube_fw-la number_n which_o be_v require_v to_o be_v prove_v ¶_o the_o 4._o theorem_a the_o 4._o proposition_n if_o a_o cube_fw-la number_n multiply_v a_o cube_fw-la number_n produce_v any_o number_n the_o number_n produce_v shall_v be_v a_o cube_fw-la number_n svppose_v that_o the_o cube_fw-la number_n a_o multiply_v the_o cube_fw-la number_n b_o do_v produce_v the_o number_n c._n then_o i_o say_v that_o c_z be_v a_o cube_fw-la number_n for_o let_v a_o multiply_v himself_o produce_v d._n wherefore_o d_z be_v a_o cube_fw-la number_n by_o the_o proposition_n go_v before_o demonstration_n and_o forasmuch_o as_o a_o multiply_v himself_o produce_v d_z and_o multiply_v b_o it_o produce_v c_z therefore_o by_o the_o 17._o of_o the_o seven_o as_o a_o be_v to_o b_o so_o be_v d_z to_o c._n and_o forasmuch_o as_o a_o and_o b_o be_v cube_fw-la number_n therefore_o a_o and_o b_o be_v like_o solid_a number_n wherefore_o between_o a_o and_o b_o by_o the_o 19_o of_o the_o eight_o there_o be_v two_o mean_v proportional_a number_n wherefore_o also_o by_o the_o 8._o of_o the_o same_o between_o d_z and_o c_o there_o be_v two_o mean_v proportional_a number_n but_o d_z be_v a_o cube_fw-la number_n wherefore_o c_z also_o be_v a_o cube_fw-la number_n by_o the_o 23._o of_o the_o eight_o which_o be_v require_v to_o be_v demonstrate_v ¶_o the_o 5._o theorem_a the_o 5._o proposition_n if_o a_o cube_fw-la number_n multiply_v any_o number_n produce_v a_o cube_fw-la number_n the_o number_n multiply_v be_v a_o cube_fw-la number_n svppose_v that_o the_o cube_fw-la number_n a_o multiply_v the_o number_n b_o do_v produce_v a_o cube_fw-la number_n namely_o c._n then_o i_o say_v that_z b_o be_v a_o cube_fw-la number_n for_o let_v a_o multiply_v himself_o produce_v d._n wherefore_o by_o the_o 3._o of_o the_o nine_o d_o be_v a_o cube_fw-la number_n and_o forasmuch_o as_o a_o multiply_v himself_o produce_v d_z demonstration_n and_o multiply_v b_o it_o produce_v c_z therefore_o by_o the_o 17._o of_o the_o seven_o as_o a_o be_v to_o b_o so_o be_v d_z to_o c._n and_o forasmuch_o as_o d_z and_o c_z be_v cube_fw-la number_n they_o be_v also_o like_o solid_a number_n wherefore_o by_o the_o 19_o of_o the_o eight_o between_o d_z and_o c_o there_o be_v two_o mean_v proportional_a number_n but_o as_o d_z be_v to_o c_o so_o be_v a_o to_o b._n wherefore_o by_o the_o 8._o of_o the_o eight_o between_o a_o and_o b_o there_o be_v two_o mean_v proportional_a number_n but_o a_o be_v a_o cube_fw-la number_n wherefore_o b_o also_o be_v a_o cube_fw-la number_n by_o the_o 23._o of_o the_o eight_o which_o be_v require_v to_o be_v prove_v ¶_o a_o corollary_n add_v by_o campane_n hereby_o it_o be_v manifest_a that_o if_o a_o cube_fw-la number_n multiply_v a_o number_n not_o cube_fw-la it_o shall_v produce_v a_o number_n not_o cube_fw-la campane_n for_o if_o it_o shall_v produce_v a_o cube_fw-la number_n than_o the_o number_n multiply_v shall_v also_o be_v a_o cube_fw-la number_n by_o this_o proposition_n which_o be_v contrary_a to_o the_o supposition_n for_o it_o be_v suppose_v to_o be_v no_o cube_fw-la number_n
the_o double_a of_o b_o be_v prime_a unto_o the_o double_a of_o b_o then_o the_o two_o number_n whereof_o the_o number_n be_v compose_v namely_o the_o number_n compose_v of_o a_o and_o c_o and_o the_o double_a of_o b_o shall_v be_v prime_a the_o one_o to_o the_o other_o by_o the_o 30_o of_o the_o seven_o and_o therefore_o the_o number_n compose_v of_o a_o and_o c_o shall_v be_v prime_a to_o b_o take_v once_o for_o if_o any_o number_n shall_v measure_v the_o two_o number_n namely_o the_o number_n compose_v of_o a_o and_o c_o and_o the_o number_n b_o it_o shall_v also_o measure_v the_o number_n compose_v of_o a_o and_o c_o and_o the_o double_a of_o b_o by_o the_o 5._o common_a sentence_n of_o the_o seven_o which_o be_v not_o possible_a for_o that_o they_o be_v prove_v to_o be_v prime_a number_n here_o have_v i_o add_v a_o other_o demonstration_n of_o the_o former_a proposition_n after_o campane_n which_o prove_v that_o in_o number_n how_o many_o soever_o which_o be_v there_o prove_v only_o touch_v three_o number_n and_o the_o demonstration_n seement_n somewhat_o more_o perspicuous_a than_o theon_n demonstration_n and_o thus_o he_o put_v the_o proposition_n if_o number_n how_o many_o soever_o be_v in_o continual_a proportion_n be_v the_o least_o that_o have_v one_o &_o the_o same_o proportion_n with_o they_o every_o one_o of_o they_o shall_v be_v to_o the_o number_n compose_v of_o the_o rest_n prime_a second_o i_o say_v that_o this_o be_v so_o in_o every_o one_o of_o they_o namely_o that_o c_z be_v a_o prime_a number_n to_o the_o number_n compose_v of_o a_o b_o d._n for_o if_o not_o then_o as_o before_o let_v e_o measure_v c_o and_o the_o number_n compose_v of_o a_o b_o d_o which_o e_o shall_v be_v a_o number_n not_o prime_a either_o to_o fletcher_n or_o to_o g_z by_o the_o former_a proposition_n add_v by_o campane_n wherefore_o let_v h_o measure_v they_o and_o forasmuch_o as_o h_o measure_v e_o it_o shall_v also_o measure_v the_o whole_a a_o b_o c_o d_o who_o e_o measure_v and_o forasmuch_o as_o h_o measure_v one_o of_o these_o number_n fletcher_n or_o g_o it_o shall_v measure_v one_o of_o the_o extreme_n a_o or_o d_o which_o be_v produce_v of_o fletcher_n or_o g_o by_o the_o second_o of_o the_o eight_o if_o they_o be_v multiply_v into_o the_o mean_n l_o or_o k._n and_o moreover_o the_o same_o h_o shall_v measure_v the_o meame_n bc_o by_o the_o 5._o common_a sentence_n of_o the_o seven_o when_o as_o by_o supposition_n it_o measure_v either_o fletcher_n or_o g._n which_o measure_n b_o c_o by_o the_o second_o of_o the_o eight_o but_o the_o same_o h_o measure_v the_o whole_a a_o b_o c_o d_o as_o we_o have_v prove_v for_o that_o it_o measure_v e._n wherefore_o it_o shall_v also_o measure_v the_o residue_n namely_o the_o number_n compose_v of_o the_o extreme_n a_o and_o d_z by_o the_o 4._o common_a sentence_n of_o the_o seven_o and_o it_o measure_v one_o of_o these_o a_o or_o d_z for_o it_o measure_v one_o of_o these_o fletcher_n or_o g_o which_o produce_v a_o and_o d_z wherefore_o the_o same_o h_o shall_v measure_v one_o of_o these_o a_o or_o d_o and_o also_o the_o other_o of_o they_o by_o the_o former_a common_a sentence_n which_o number_n a_o and_o d_o be_v by_o the_o 3._o of_o the_o eight_o prime_n the_o one_o to_o the_o other_o which_o be_v absurd_a this_o may_v also_o be_v prove_v in_o every_o one_o of_o these_o number_n a_o b_o c_o d._n wherefore_o no_o number_n shall_v measure_v one_o of_o these_o number_n a_o b_o c_o d_o and_o the_o number_n compose_v of_o the_o rest_n wherefore_o they_o be_v prime_a the_o one_o to_o the_o other_o if_o therefore_o number_n how_o many_o soever_o etc_o etc_o which_o be_v require_v to_o be_v prove_v here_o as_o i_o promise_v i_o have_v add_v campane_n demonstration_n of_o those_o proposition_n in_o number_n which_o euclâ—de_n in_o the_o second_o book_n demonstrate_v in_o line_n and_o that_o in_o this_o place_n so_o much_o the_o rather_o for_o that_o theon_n as_o we_o see_v in_o the_o demonstration_n of_o the_o 15._o proposition_n seem_v to_o allege_v the_o 3._o &_o 4._o proposition_n of_o the_o second_o book_n which_o although_o they_o concern_v line_n only_o yet_o as_o we_o there_o declare_v and_o prove_v be_v they_o true_a also_o in_o number_n ¶_o the_o first_o proposition_n add_v by_o campane_n that_o number_n which_o be_v produce_v of_o the_o multiplication_n of_o one_o number_n into_o number_n how_o many_o soever_o be_v equal_a to_o that_o number_n which_o be_v produce_v of_o the_o multiplication_n of_o the_o same_o number_n into_o the_o number_n compose_v of_o they_o this_o prove_v that_o in_o number_n which_o the_o first_o of_o the_o second_o prove_v touch_v line_n suppoâ—â—_n that_o the_o number_n a_o be_v multiply_v into_o the_o number_n b_o and_o into_o the_o number_n c_o demonstratiou_n and_o into_o the_o number_n d_o do_v produce_v the_o number_n e_o f_o and_o g._n then_o i_o say_v that_o the_o number_n produce_v of_o a_o multiply_a into_o the_o number_n compose_v of_o b_o c_o and_o d_z be_v equal_a to_o the_o number_n compose_v of_o e_o f_o and_o g._n for_o by_o the_o converse_n of_o the_o definition_n of_o a_o number_n multiply_v what_o part_n unity_n be_v of_o a_o the_o self_n same_o part_n be_v b_o of_o e_o and_o c_o of_o fletcher_n and_o also_o d_z of_o g._n wherefore_o by_o the_o 5._o of_o the_o seven_o what_o part_n unity_n be_v of_o a_o the_o self_n same_o part_n be_v the_o number_n compose_v of_o b_o c_o and_o d_o of_o the_o number_n compose_v of_o e_o f_o and_o g._n wherefore_o by_o the_o definition_n that_o which_o be_v produce_v of_o a_o into_o the_o number_n compose_v of_o b_o c_o d_o be_v equal_a to_o the_o number_n compose_v of_o e_o f_o g_o which_o be_v require_v to_o be_v prove_v the_o second_o proposition_n that_o number_n which_o be_v produce_v of_o the_o multiplication_n of_o number_n how_o many_o soever_o into_o one_o number_n be_v equal_a to_o that_o number_n which_o be_v produce_v of_o the_o multiplication_n of_o the_o number_n compose_v of_o they_o into_o the_o same_o number_n this_o be_v the_o converse_n of_o the_o former_a as_o if_o the_o â—â—â—bers_n â—_o and_o g_z and_o d_o multiply_v into_o the_o number_n a_o do_v produce_v the_o number_n e_o and_o f_o and_o g._n former_a then_o the_o number_n compose_v of_o b_o c_o d._n multiply_v into_o the_o number_n a_o shall_v produce_v the_o number_n compose_v of_o the_o number_n e_o f_o g._n which_o thing_n be_v easy_o prove_v by_o the_o 16._o of_o the_o seven_o and_o by_o the_o former_a proposition_n ¶_o the_o three_o proposition_n that_o number_n which_o be_v produce_v of_o the_o multiplication_n of_o number_n how_o many_o soever_o into_o other_o number_n how_o many_o soever_o be_v equal_a to_o that_o number_n which_o be_v produce_v of_o the_o multiplication_n of_o the_o number_n compose_v of_o those_o first_o number_n into_o the_o number_n compose_v of_o these_o latter_a number_n as_o if_o the_o number_n a_o b_o c_o do_v multiply_v the_o number_n d_o e_o f_o each_o one_o each_o other_o and_o if_o the_o number_n produce_v be_v add_v together_o then_o i_o say_v that_o the_o number_n compose_v of_o the_o number_n produce_v be_v equal_a to_o the_o number_n produce_v of_o the_o number_n compose_v of_o the_o number_n a_o b_o demonstration_n c_o into_o the_o number_n compose_v of_o the_o number_n d_o e_o f._n for_o by_o the_o former_a proposition_n that_o which_o be_v produce_v of_o the_o number_n compose_v of_o a_o b_o c_o into_o d_z be_v equal_a to_o that_o which_o be_v produce_v of_o every_o one_o of_o the_o say_a number_n into_o d_o and_o by_o the_o same_o reason_n that_o which_o be_v produce_v of_o the_o number_n compose_v of_o a_o b_o c_o into_o e_o be_v equal_a to_o that_o which_o be_v produce_v of_o every_o one_o of_o the_o say_a number_n into_o e_o and_o so_o likewise_o that_o which_o be_v produce_v of_o the_o number_n compose_v of_o a_o b_o c_o into_o fletcher_n be_v equal_a to_o that_o which_o be_v produce_v of_o every_o one_o of_o the_o say_a number_n into_o f._n but_o by_o the_o first_o of_o these_o proposition_n thâ—â—_n which_o be_v produce_v of_o the_o number_n compose_v of_o these_o number_n a_o b_o c_o into_o every_o one_o of_o these_o number_n d_o e_o fletcher_n be_v equal_a to_o that_o which_o be_v produce_v of_o the_o number_n compose_v into_o the_o number_n compose_v wherefore_o that_o be_v manifest_v which_o be_v require_v to_o be_v prove_v ¶_o the_o four_o proposition_n if_o a_o number_n be_v deuâ—â—â—d_v into_o part_n how_o many_o soever_o that_o number_n which_o be_v produce_v of_o the_o whole_a into_o himself_o be_v equal_a to_o that_o number_n which_o be_v produce_v of_o
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
the_o square_a of_o the_o line_n a_o have_v not_o unto_o the_o square_n of_o the_o line_n b_o the_o same_o proportion_n that_o a_o square_a number_n have_v to_o a_o square_a number_n then_o i_o say_v that_o the_o line_n a_o and_o b_o be_v incommensurable_a in_o length_n for_o if_o the_o line_n a_o and_o b_o be_v commensurable_a in_o length_n than_o the_o square_a of_o the_o line_n a_o shall_v have_v unto_o the_o square_n of_o the_o line_n b_o the_o same_o proportion_n that_o a_o square_a number_n have_v to_o a_o square_a number_n by_o the_o first_o part_n of_o this_o proposition_n but_o by_o supposition_n it_o have_v not_o wherefore_o the_o line_n a_o and_o b_o be_v not_o commensurable_a in_o length_n proposition_n wherefore_o they_o be_v incomensurable_a in_o length_n wherefore_o square_n make_v of_o right_a line_n commensura_fw-la in_o length_n have_v that_o proportion_n the_o one_o to_o the_o other_o that_o a_o square_a number_n have_v to_o a_o square_a number_n and_o square_n which_o have_v that_o proportion_n the_o one_o to_o the_o other_o that_o a_o square_a number_n have_v to_o a_o squaâ—e_a number_n shall_v also_o have_v the_o side_n commensurable_a in_o length_n but_o square_n describe_v of_o right_a line_n incommensurable_a in_o length_n have_v not_o that_o proportion_n the_o one_o to_o the_o other_o that_o a_o square_a number_n have_v to_o a_o square_a number_n and_o square_n which_o have_v not_o that_o proportion_n the_o one_o to_o the_o other_o that_o a_o square_a number_n have_v to_o a_o square_a number_n have_v not_o also_o their_o side_n commensurable_a in_o length_n which_o be_v all_o that_o be_v require_v to_o be_v prove_v ¶_o corollary_n hereby_o it_o be_v manifest_a that_o right_a line_n commensurable_a in_o length_n be_v also_o ever_o commensurable_a in_o power_n but_o right_a line_n commensurable_a in_o power_n be_v not_o always_o commensurable_a in_o length_n corollary_n and_o right_a line_n incommensurable_a in_o length_n be_v not_o always_o incommensurable_a in_o power_n but_o right_a line_n incommensurable_a in_o power_n be_v ever_o also_o incommensurable_a in_o length_n for_o forasmuch_o as_o square_n make_v of_o right_a line_n commensurable_a in_o length_n have_v that_o proportion_n the_o one_o to_o the_o other_o corollary_n that_o a_o square_a number_n have_v to_o a_o square_a number_n by_o the_o first_o part_n of_o this_o proposition_n but_o magnitude_n which_o have_v that_o proportion_n the_o one_o to_o the_o other_o that_o number_n simple_o have_v to_o number_n be_v by_o the_o six_o of_o the_o ten_o commensurable_a wherefore_o right_a line_n commensurable_a in_o length_n be_v commensurable_a not_o only_o in_o length_n but_o also_o in_o power_n part_n again_o forasmuch_o as_o there_o be_v certain_a square_n which_o have_v not_o that_o proportion_n the_o one_o to_o the_o other_o that_o a_o square_a number_n have_v to_o a_o square_a number_n but_o yet_o have_v that_o proportion_n the_o one_o to_o the_o other_o which_o number_n simple_o have_v to_o number_v their_o side_n in_o deed_n be_v in_o power_n commensurable_a for_o that_o they_o describe_v square_n which_o have_v that_o proportion_n which_o number_n simple_o have_v to_o number_n which_o square_n be_v therefore_o commensurable_a by_o the_o 6._o of_o this_o book_n but_o the_o say_v side_n be_v incommensurable_a in_o length_n by_o the_o latter_a part_n of_o this_o proposition_n wherâ—fore_o it_o be_v true_a that_o line_n commensurable_a in_o power_n be_v not_o straight_o way_n commensurable_a in_o length_n also_o pâ—rt_n and_o by_o the_o selâ—e_a same_o reason_n be_v prove_v also_o that_o three_o part_n of_o the_o corollary_n that_o line_n incommensurable_a in_o length_n be_v not_o always_o incommensurable_a in_o power_n for_o they_o may_v be_v incommensurable_a in_o length_n but_o yet_o commensurable_a in_o power_n as_o in_o those_o square_n which_o be_v in_o proportion_n the_o one_o to_o the_o other_o as_o number_n be_v to_o number_n but_o not_o as_o a_o square_a number_n be_v to_o a_o square_a number_n part_n but_o right_a line_n incommensurable_a in_o power_n be_v always_o also_o incommensurable_a in_o length_n for_o iâ—_z they_o be_v commensurable_a in_o length_n they_o shall_v also_o be_v commensurable_a in_o power_n by_o the_o first_o part_n of_o this_o corollary_n but_o they_o be_v suppose_v to_o be_v incommensurable_a in_o length_n which_o be_v absurd_a wherefore_o right_a line_n incommensurable_a in_o power_n be_v ever_o incommensurable_a in_o lengthy_a for_o the_o better_a understanding_n of_o this_o proposition_n and_o the_o other_o follow_v i_o have_v here_o add_v certain_a annotation_n take_v out_o of_o montaureus_n montauâ—â—us_fw-la and_o first_o as_o touch_v the_o signification_n oâ—_n word_n and_o term_n herein_o use_v whâ—ch_a arâ—_n such_o that_o unless_o they_o be_v well_o mark_v and_o poise_v the_o matter_n will_v be_v obscure_a and_o hard_a and_o in_o a_o manner_n inexplicable_a first_o this_o you_o must_v note_v that_o line_n to_o be_v commensurable_a in_o length_n and_o line_n to_o be_v in_o proportion_n the_o one_o to_o the_o other_o as_o number_n be_v to_o number_n be_v all_o one_o so_o that_o whatsoever_o line_n be_v commensurable_a in_o length_n be_v also_o in_o proportion_n the_o one_o to_o the_o other_o as_o number_n be_v to_o number_v and_o converse_o what_o so_o ever_o line_n be_v in_o proportion_n the_o one_o to_o the_o other_o as_o number_n be_v to_o number_n be_v also_o commensurable_a in_o length_n as_o it_o be_v manifest_a by_o the_o 5_o and_o 6_o of_o this_o book_n likewise_o line_n to_o be_v incommensurable_a in_o length_n and_o not_o to_o be_v in_o proportion_n the_o one_o to_o the_o other_o as_o number_n be_v to_o number_n be_v all_o one_o as_o it_o be_v manifest_a by_o the_o 7._o and_o 8._o of_o this_o book_n wherefore_o that_o which_o be_v say_v in_o this_o theorem_a aught_o to_o be_v understand_v of_o line_n commensurable_a in_o length_n and_o incommensurable_a in_o length_n this_o moreover_o be_v to_o be_v note_v that_o it_o be_v not_o all_o one_o numbers_z to_o be_v square_a number_n and_o to_o be_v in_o proportion_n the_o one_o to_o the_o other_o as_o a_o square_a number_n be_v to_o a_o square_a number_n for_o although_o square_a number_n be_v in_o proportion_n the_o one_o to_o the_o other_o as_o a_o square_a number_n be_v to_o a_o square_a number_n yet_o be_v not_o all_o those_o number_n which_o be_v in_o proportion_n the_o one_o to_o the_o other_o as_o a_o square_a number_n be_v to_o a_o square_a number_n square_a number_n for_o they_o may_v be_v like_o superficial_a number_n and_o yet_o not_o square_a number_n which_o yet_o be_v in_o proportion_n the_o one_o to_o the_o other_o as_o a_o square_a number_n be_v to_o a_o square_a number_n although_o all_o square_a number_n be_v like_o superficial_a number_n for_o between_o two_o square_a number_n there_o fall_v one_o mean_a proportional_a number_n by_o the_o 11._o of_o the_o eight_o but_o if_o between_o two_o number_n there_o fall_v one_o mean_v proportional_a number_n those_o two_o number_n be_v like_o superficial_a number_n by_o the_o 20._o of_o the_o eight_o so_o also_o if_o two_o number_n be_v in_o proportion_n the_o one_o to_o the_o other_o as_o a_o square_a number_n be_v to_o a_o square_a number_n they_o shall_v be_v like_o superficial_a number_n by_o the_o first_o corollary_n add_v after_o the_o last_o proposition_n of_o the_o eight_o book_n and_o now_o to_o know_v whether_o two_o superficial_a number_n give_v be_v like_o superficial_a number_n or_o no_o no._n it_o be_v thus_o find_v out_o first_o if_o between_o the_o two_o number_n give_v there_o fall_v no_o mean_a proportional_a then_o be_v not_o these_o two_o number_n like_o superficial_a number_n by_o the_o 18._o of_o the_o eight_o but_o if_o there_o do_v fall_v between_o they_o a_o mean_v proportional_a then_o be_v they_o like_v superâ—iciall_a number_n by_o the_o 20._o of_o the_o eight_o moreover_o two_o like_o superficial_a number_n multiply_v the_o one_o into_o the_o other_o do_v produce_v a_o square_a number_n by_o the_o firsâ—_o of_o the_o nine_o wherefore_o if_o they_o do_v not_o produce_v a_o square_a number_n then_o be_v they_o not_o like_o superficial_a number_n and_o if_o the_o one_o be_v multiply_v into_o the_o other_o they_o produce_v a_o square_a number_n then_o be_v they_o like_v superficial_a by_o the_o 2._o of_o the_o nine_o moreover_o if_o the_o say_v two_o superficial_a number_n be_v in_o superperticular_a or_o superbipartient_fw-fr proportion_n then_o be_v they_o not_o like_o superficial_a number_n for_o if_o they_o shall_v be_v like_a then_o shall_v there_o be_v a_o mean_a proportional_a between_o they_o by_o the_o 20._o of_o the_o eight_o but_o that_o be_v contrary_a to_o the_o corollary_n of_o the_o 20._o of_o the_o eight_o and_o the_o easy_o to_o conceive_v the_o demonstration_n follow_v take_v this_o example_n of_o that_o which_o we_o have_v say_v ¶_o
a_o assumpt_n forasmuch_o as_o in_o the_o eight_o book_n in_o the_o 26._o proposition_n it_o be_v prove_v that_o like_o plain_a number_n have_v that_o proportion_n the_o one_o to_o the_o other_o that_o a_o square_a number_n have_v to_o a_o square_a number_n and_o likewise_o in_o the_o 24._o of_o the_o same_o book_n it_o be_v prove_v that_o if_o two_o number_n have_v that_o proportion_n the_o one_o to_o the_o other_o eight_o that_o a_o square_a number_n have_v to_o a_o square_a number_n those_o number_n be_v like_o plain_a number_n hereby_o it_o be_v manifest_a that_o unlike_o plain_a number_n that_o be_v who_o side_n be_v not_o proportional_a have_v not_o that_o proportion_n the_o one_o to_o the_o other_o that_o a_o square_a number_n have_v to_o a_o square_a number_n for_o if_o they_o have_v than_o shall_v they_o be_v like_o plain_a number_n which_o be_v contrary_a to_o the_o supposition_n wherefore_o unlike_o plain_a number_n have_v not_o that_o proportion_n the_o one_o to_o the_o other_o that_o a_o square_a number_n have_v to_o a_o square_a number_n and_o therefore_o square_n which_o have_v that_o proportion_n the_o one_o to_o the_o other_o that_o unlike_o plain_a number_n have_v shall_v have_v their_o side_n incommensurable_a in_o length_n by_o the_o last_o part_n of_o the_o former_a proposition_n for_o that_o those_o square_n have_v not_o that_o proportion_n the_o one_o to_o the_o other_o that_o a_o square_a number_n have_v to_o a_o square_a number_n ¶_o the_o 8._o theorem_a the_o 10._o proposition_n if_o four_o magnitude_n be_v proportional_a and_o if_o the_o first_o be_v commensurable_a unto_o the_o second_o the_o three_o also_o shall_v be_v commensurable_a unto_o the_o four_o and_o if_o the_o first_o be_v incommensurable_a unto_o the_o second_o the_o three_o shall_v also_o be_v incommensurable_a unto_o the_o four_o svppose_v that_o these_o four_o magnitude_n a_o b_o c_o d_o be_v proportional_a as_o a_o be_v to_o b_o so_o let_v c_o be_v to_o d_o and_o let_v a_o be_v commensurable_a unto_o b._n then_o i_o say_v that_o c_z be_v also_o commensurable_a unto_o d._n part_n for_o forasmuch_o as_o a_o be_v commensurable_a unto_o b_o it_o have_v by_o the_o five_o of_o the_o ten_o that_o proportion_n that_o number_n have_v to_o number_v but_o as_o a_o be_v to_o b_o so_o be_v c_o to_o d._n wherefore_o c_z also_o have_v unto_o d_z that_o proportion_n that_o number_n have_v to_o number_v wherefore_o c_o be_v commensurable_a unto_o d_z by_o the_o 6._o of_o the_o ten_o but_o now_o suppose_v that_o the_o magnitude_n a_o be_v incommensurable_a unto_o the_o magnitude_n b._n partâ—_n then_o i_o say_v that_o the_o magnitude_n c_o also_o be_v incommensurable_a unto_o the_o magnitude_n d._n for_o forasmuch_o as_o a_o be_v incommensurable_a unto_o b_o therefore_o by_o the_o 7._o of_o this_o book_n a_o have_v not_o unto_o b_o such_o proportion_n as_o number_n have_v to_o number_v but_o as_o a_o be_v to_o b_o so_o be_v c_o to_o d._n wherefore_o c_o have_v not_o unto_o d_z such_o proportion_n as_o number_n have_v to_o number_v wherefore_o by_o the_o 8._o of_o the_o ten_o c_o be_v incommensurable_a unto_o d._n if_o therefore_o there_o be_v four_o magnitude_n proportional_a and_o if_o the_o first_o be_v commensurable_a unto_o the_o second_o the_o three_o also_o shall_v be_v commensurable_a unto_o the_o four_o and_o if_o the_o first_o be_v incommensurable_a unto_o the_o second_o the_o three_o shall_v also_o be_v incommensurable_a unto_o the_o four_o which_o be_v require_v to_o be_v prove_v ¶_o a_o corollary_n add_v by_o montaureus_n if_o there_o be_v four_o line_n proportional_a and_o if_o the_o two_o first_o or_o the_o two_o last_o be_v commensurable_a in_o power_n only_o the_o other_o two_o also_o shall_v be_v commensurable_a in_o power_n only_o corollary_n this_o be_v prove_v by_o the_o 22._o of_o the_o six_o and_o by_o this_o ten_o proposition_n and_o this_o corollary_n euclid_n use_v in_o the_o 27._o and_o 28._o proposition_n of_o this_o book_n and_o in_o other_o proposition_n also_o ¶_o the_o 3._o problem_n the_o 11._o proposition_n unto_o a_o right_a line_n first_o set_v and_o give_v which_o be_v call_v a_o rational_a line_n to_o find_v out_o two_o right_a line_n incommensurable_a the_o one_o in_o length_n only_o and_o the_o other_o in_o length_n and_o also_o in_o power_n svppose_v that_o the_o right_a line_n first_o set_v and_o give_v which_o be_v call_v a_o rational_a line_n of_o purpose_n be_v a._n it_o be_v require_v unto_o the_o say_a line_n a_o to_o find_v out_o two_o right_a line_n incommensurable_a the_o one_o in_o length_n only_o the_o other_o both_o in_o length_n and_o in_o power_n give_v take_v by_o that_o which_o be_v add_v after_o the_o 9_o proposition_n of_o this_o book_n two_o number_n b_o and_o c_o not_o have_v that_o proportion_n the_o one_o to_o the_o other_o that_o a_o square_a number_n have_v to_o a_o square_a number_n that_o be_v let_v they_o not_o be_v like_o plain_a number_n for_o like_o plain_a number_n by_o the_o 26._o of_o the_o eight_o have_v that_o proportion_n the_o one_o to_o the_o other_o that_o a_o square_a number_n have_v to_o a_o square_a number_n and_o as_o the_o number_n b_o be_v to_o the_o number_n c_o so_fw-mi let_v the_o square_n of_o the_o line_n a_o be_v unto_o the_o square_n of_o a_o other_o line_n namely_o of_o d_z how_o to_o do_v this_o be_v teach_v in_o the_o assumpt_n put_v before_o the_o 6._o proposition_n of_o this_o book_n wherefore_o the_o square_a of_o the_o line_n a_o be_v unto_o the_o square_n of_o the_o line_n d_o commensurable_a by_o the_o six_o of_o the_o ten_o and_o forasmuch_o as_o the_o number_n b_o have_v not_o unto_o the_o number_n c_o that_o proportion_n that_o a_o square_a number_n have_v to_o a_o square_a number_n therefore_o the_o square_a of_o the_o line_n a_o have_v not_o unto_o the_o square_n of_o the_o line_n d_z that_o proportion_n that_o a_o square_a number_n have_v to_o a_o number_n wherefore_o by_o the_o 9_o of_o the_o ten_o the_o line_n a_o be_v unto_o the_o line_n d_o incommensurable_a in_o length_n only_o and_o so_o be_v find_v out_o the_o first_o line_n namely_o d_z incommensurable_a in_o length_n only_o to_o the_o line_n give_v a._n give_v again_o take_v by_o the_o 13._o of_o the_o six_o the_o mean_a proportional_a between_o the_o line_n a_o and_o d_o and_o let_v the_o same_o be_v e._n wherefore_o as_o the_o line_n a_o be_v to_o the_o line_n d_o so_o be_v the_o square_a of_o the_o line_n a_o to_o the_o square_n of_o the_o line_n e_o by_o the_o corollary_n of_o the_o 20._o of_o the_o six_o but_o the_o line_n a_o be_v unto_o the_o line_n d_o incommensurable_a in_o length_n wherefore_o also_o the_o square_a of_o the_o line_n a_o be_v unto_o the_o square_n of_o the_o line_n e_o incommensurable_a by_o the_o second_o part_n of_o the_o former_a proposition_n now_o forasmuch_o as_o the_o square_n of_o the_o line_n a_o be_v incommensurable_a to_o the_o square_n of_o the_o line_n e_o it_o follow_v by_o the_o definition_n of_o incommensurable_a line_n that_o the_o line_n a_o be_v incommensurable_a in_o power_n to_o the_o line_n e._n wherefore_o unto_o the_o right_a line_n give_v and_o first_o set_v a_o which_o be_v a_o rational_a line_n and_o which_o be_v suppose_v to_o have_v such_o division_n and_o so_o many_o part_n as_o you_o listen_v to_o conceive_v in_o mind_n as_o in_o this_o example_n 11_z whereunto_o as_o be_v declare_v in_o the_o 5._o definition_n of_o this_o book_n may_v be_v compare_v infinite_a other_o line_n either_o commensurable_a or_o incommensurable_a be_v find_v out_o the_o line_n d_o incommensurable_a in_o length_n only_o wherefore_o the_o line_n d_o be_v rational_a by_o the_o six_o definition_n of_o this_o book_n for_o that_o it_o be_v incommensurable_a in_o length_n only_o to_o the_o line_n a_o which_o be_v the_o first_o line_n set_v and_o be_v by_o supposition_n rational_a there_o be_v also_o find_v out_o the_o line_n e_o which_o be_v unto_o the_o same_o line_n a_o incommensurable_a not_o only_o in_o length_n but_o also_o in_o power_n which_o line_n e_o compare_v to_o the_o rational_a line_n a_o be_v by_o the_o definition_n irrational_a for_o euclid_n always_o call_v those_o line_n irrational_a which_o be_v incommensurable_a both_o in_o length_n and_o in_o power_n to_o the_o line_n first_o set_v and_o by_o supposition_n rational_a ¶_o the_o 9_o theorem_a the_o 12._o proposition_n magnitude_n commensurable_a to_o one_o and_o the_o self_n same_o magnitude_n be_v also_o commensurable_a the_o one_o to_o the_o other_o svppose_v that_o either_o of_o these_o magnitude_n a_o and_o b_o be_v commensurable_a unto_o the_o magnitude_n c_o then_o i_o say_v that_o the_o magnitude_n a_o be_v commensurable_a unto_o the_o magnitude_n b._n construction_n for_o forasmuch_o as_o the_o
now_o forasmuch_o as_o d_z measure_v ab_fw-la and_o bc_o it_o also_o measure_v the_o whole_a magnitude_n ac_fw-la and_o it_o measure_v ab_fw-la wherefore_o d_z measure_v these_o magnitude_n ca_n and_o ab_fw-la wherefore_o ca_n &_o ab_fw-la be_v commensurable_a and_o they_o be_v suppose_v to_o be_v incommensurableâ—_n which_o be_v impossible_a wherefore_o no_o magnitude_n measure_v ab_fw-la and_o bc._n wherefore_o the_o magnitude_n ab_fw-la and_o bc_o be_v incommensurable_a and_o in_o like_a sort_n may_v they_o be_v prove_v to_o be_v incommensurable_a if_o the_o magnitude_n ac_fw-la be_v suppose_v to_o be_v incommensurable_a unto_o bc._n if_o therefore_o there_o be_v two_o magnitude_n incommensurable_a compose_v the_o whole_a also_o shall_v be_v incommensurable_a unto_o either_o of_o the_o two_o part_n component_a and_o if_o the_o whole_a be_v incommensurable_a to_o one_o of_o the_o part_n component_n those_o first_o magnitude_n shall_v be_v incommensurable_a which_o be_v require_v to_o be_v prove_v ¶_o a_o corollary_n add_v by_o montaureus_n if_o a_o whole_a magnitude_n be_v incommensurable_a to_o one_o of_o the_o two_o magnitude_n which_o make_v the_o whole_a magnitude_n it_o shall_v also_o be_v incommensurable_a to_o the_o other_o of_o the_o two_o magnitude_n for_o if_o the_o whole_a magnitude_n ac_fw-la be_v incommensurable_a unto_o the_o magnitude_n bc_o then_o by_o the_o 2_o part_n of_o this_o 16._o theorem_o the_o magnitude_n ab_fw-la and_o bc_o shall_v be_v incommensurable_a wherefore_o by_o the_o first_o part_n of_o the_o same_o theorem_a the_o magnitude_n ac_fw-la shall_v be_v incommensurable_a to_o either_o of_o these_o magnitude_n ab_fw-la and_o bc._n this_o corollary_n 〈◊〉_d use_v in_o the_o demonstration_n of_o the_o â—3_o theorem_a &_o also_o of_o other_o proposition_n ¶_o a_o assumpt_n if_o upon_o a_o right_a line_n be_v apply_v a_o parallelogram_n want_v in_o figure_n by_o a_o square_n the_o parallelogram_n so_o apply_v be_v equal_a to_o that_o parallelogram_n which_o be_v contain_v under_o the_o segment_n of_o the_o right_a line_n which_o segment_n be_v make_v by_o reason_n of_o that_o application_n this_o assumpt_n i_o before_o add_v as_o a_o corollary_n out_o of_o flussates_n after_o the_o 28._o proposition_n of_o the_o six_o book_n ¶_o the_o 14._o theorem_a the_o 17._o proposition_n if_o there_o be_v two_o right_a line_n unequal_a and_o if_o upon_o the_o great_a be_v apply_v a_o parallelogram_n equal_a unto_o the_o four_o part_n of_o the_o square_n of_o the_o less_o line_n and_o want_v in_o figure_n by_o a_o square_a if_o also_o the_o parallelogram_n thus_o apply_v divide_v the_o line_n where_o upon_o it_o be_v apply_v into_o part_n commensurable_a in_o length_n then_o shall_v the_o great_a line_n be_v in_o power_n more_o than_o the_o less_o by_o the_o square_n of_o a_o line_n commensurable_a in_o length_n unto_o the_o great_a and_o if_o the_o great_a be_v in_o power_n more_o than_o the_o less_o by_o the_o square_n of_o a_o right_a line_n commensurable_a in_o length_n unto_o the_o great_a and_o if_o also_o upon_o the_o great_a be_v apply_v a_o parallelogram_n equal_a unto_o the_o four_o part_n of_o the_o square_n of_o the_o less_o line_n and_o want_v in_o figure_n by_o a_o square_n then_o shall_v it_o divide_v the_o great_a line_n into_o part_n commensurable_a but_o now_o suppose_v that_o the_o line_n bc_o be_v in_o power_n more_o than_o the_o line_n a_o by_o the_o square_n of_o a_o line_n commensurable_a in_o length_n unto_o the_o line_n bc._n first_o and_o upon_o the_o line_n bc_o let_v there_o be_v apply_v a_o rectangle_n parallelogram_n equal_a unto_o the_o four_o part_n of_o the_o square_n of_o the_o line_n a_o and_o want_v in_o figure_n by_o a_o square_a and_o let_v the_o say_a parallelogram_n be_v that_o which_o be_v contain_v under_o the_o line_n bd_o and_o dc_o then_o must_v we_o prove_v that_o the_o line_n bd_o be_v unto_o the_o line_n dc_o commensurable_a in_o length_n the_o same_o construction_n and_o supposition_n that_o be_v before_o remain_v we_o may_v in_o like_a sort_n prove_v that_o the_o line_n bc_o be_v in_o power_n more_o than_o the_o line_n a_o by_o the_o square_n of_o the_o line_n fd._n but_o by_o supposition_n the_o line_n bc_o be_v in_o power_n more_o they_o the_o line_n a_o by_o the_o square_n of_o a_o line_n commensurable_a unto_o it_o in_o length_n wherefore_o the_o line_n bc_o be_v unto_o the_o line_n fd_v commensurable_a in_o length_n wherefore_o the_o line_n compose_v of_o the_o two_o line_n bf_o and_o dc_o be_v commensurable_a in_o length_n unto_o the_o line_n fd_o by_o the_o second_o part_n of_o the_o 15._o of_o the_o ten_o wherefore_o by_o the_o 12._o of_o the_o ten_o or_o by_o the_o first_o part_n of_o the_o 15._o of_o the_o ten_o the_o line_n bc_o be_v commensurable_a in_o length_n to_o the_o line_n compose_v of_o bf_o and_o dc_o but_o the_o whole_a line_n conpose_v bf_o and_o dc_o be_v commensurable_a in_o length_n unto_o dc_o for_o bf_o as_o before_o have_v be_v prove_v be_v equal_a to_o dc_o wherefore_o the_o line_n bc_o be_v commensurable_a in_o length_n unto_o the_o line_n dc_o by_o the_o 12._o of_o the_o ten_o whâ—â—â—fore_o also_o the_o line_n bd_o be_v commensurable_a in_o length_n unto_o the_o line_n dc_o by_o the_o second_o part_n of_o thâ—_z 15._o of_o the_o teâ—th_n if_o therefore_o there_o be_v two_o right_a line_n unequal_a and_o if_o upon_o the_o great_a be_v apply_v a_o parallelogram_n equal_a unto_o the_o four_o part_n of_o the_o square_n of_o the_o less_o and_o want_v in_o figure_n by_o a_o square_a if_o also_o the_o parallelogram_n thus_o apply_v divide_v the_o line_n whereupon_o it_o be_v apply_v into_o part_n commensurable_a in_o length_n then_o shall_v the_o great_a line_n be_v in_o power_n more_o than_o the_o less_o by_o the_o square_n of_o a_o line_n commensurable_a in_o length_n unto_o the_o great_a and_o if_o the_o great_a be_v in_o power_n more_o than_o the_o less_o by_o the_o square_n of_o a_o line_n commensurable_a in_o length_n unto_o the_o great_a and_o if_o also_o upon_o the_o great_a be_v apply_v a_o parallelogram_n equal_a unto_o the_o four_o part_n of_o the_o square_n make_v of_o the_o less_o and_o want_v in_o figure_n by_o a_o square_n then_o shall_v it_o divide_v the_o great_a line_n into_o part_n commensurable_a in_o length_n which_o be_v require_v to_o be_v prove_v campanâ—_n after_o this_o proposition_n reach_v how_o we_o may_v ready_o apply_v upon_o the_o line_n bc_o a_o parallelogram_n equal_a to_o the_o four_o part_n of_o the_o square_n of_o half_n of_o the_o line_n a_o and_o want_v in_o figure_n by_o a_o square_a after_o this_o manner_n divide_v the_o line_n bc_o into_o two_o line_n in_o such_o sort_n that_o half_a of_o the_o line_n a_o shall_v be_v the_o mean_a proportional_a between_o those_o two_o line_n which_o be_v possible_a when_o as_o the_o line_n bc_o be_v suppose_v to_o be_v great_a than_o the_o line_n a_o and_o may_v thus_o be_v do_v proposition_n divide_v the_o line_n bc_o into_o two_o equal_a part_n in_o the_o point_n e_o and_o describe_v upon_o the_o line_n bc_o a_o semicircle_n bhc_n and_o unto_o the_o line_n bc_o and_o from_o the_o point_n c_o erect_v a_o perpâ—dicular_a line_n ck_o and_o put_v the_o line_n ck_o equal_a to_o half_o of_o the_o line_n aâ—_z and_o by_o the_o point_n king_n draw_v unto_o the_o line_n ec_o a_o parallel_n line_n kh_o cut_v the_o semicircle_n in_o the_o point_n h_o which_o it_o must_v needs_o cut_v forasmuch_o as_o the_o line_n bc_o be_v great_a than_o the_o line_n a_o and_o from_o the_o point_n h_o draw_v unto_o the_o line_n bc_o a_o perpendicular_a liâ—e_z hd_v which_o line_n hdâ—_n forasmuch_o as_o by_o the_o 34_o of_o the_o first_o it_o be_v equal_a unto_o the_o line_n kc_o shall_v also_o be_v equal_a to_o half_o of_o the_o line_n a_o draw_v the_o line_n bh_o and_o hc_n now_o then_o by_o the_o â—â—_o of_o the_o three_o the_o angle_n bhc_n be_v a_o right_a aâ—gle_n wherefore_o by_o the_o corollary_n of_o the_o eight_o of_o the_o six_o book_n the_o line_n hd_v be_v the_o mean_a proportional_a between_o the_o line_n bd_o and_o dc_o wherefore_o the_o half_a of_o the_o line_n a_o which_o be_v equal_a unto_o the_o line_n hd_v be_v the_o mean_a proportional_a between_o the_o line_n bd_o and_o dc_o wherefore_o that_o which_o be_v contain_v under_o the_o line_n bd_o and_o dc_o be_v equal_a to_o the_o four_o part_n of_o the_o square_n of_o the_o line_n a._n and_o so_o if_o upon_o the_o line_n bd_o be_v describe_v a_o rectangle_n parallelogram_n have_v his_o other_o side_n equal_a to_o the_o line_n dc_o there_o shall_v be_v apply_v upon_o the_o line_n bc_o a_o rectangle_n parallelogram_n equal_a unto_o the_o square_n of_o half_n of_o the_o line_n a_o and_o want_v in_o figure_n by_o
28._o proposition_n to_o find_v out_o medial_a right_a line_n commensurable_a in_o power_n only_o contain_v a_o medial_a parallelogram_n let_v there_o be_v put_v three_o rational_a right_a line_n commensurable_a in_o power_n only_o namely_o a_o b_o and_o c_o construction_n and_o by_o the_o 13._o of_o the_o six_o take_v the_o mean_a proportional_a between_o the_o line_n a_o and_o b_o &_o let_v thâ—_z same_o be_v d._n and_o as_o the_o line_n b_o be_v to_o the_o line_n c_o so_o by_o the_o 12._o of_o the_o six_o let_v the_o line_n d_o be_v to_o the_o line_n e._n and_o forasmuch_o as_o the_o line_n a_o and_o b_o be_v rational_a commensurable_a in_o power_n only_o demonstration_n therefore_o by_o the_o 21._o of_o the_o ten_o that_o which_o be_v contain_v under_o the_o line_n a_o and_o b_o that_o be_v the_o square_a of_o the_o line_n d_o be_v medial_a wherefore_o d_z be_v a_o medial_a line_n and_o forasmuch_o as_o the_o line_n b_o and_o c_o be_v commensurable_a in_o power_n only_o and_o as_o the_o line_n b_o be_v to_o the_o line_n c_o so_o be_v the_o line_n d_o to_o the_o line_n e_o wherefore_o the_o line_n d_z and_o e_o be_v commensurable_a in_o power_n only_o by_o the_o corollary_n of_o the_o ten_o of_o this_o book_n but_o d_o be_v a_o medial_a line_n wherefore_o e_o also_o be_v a_o medial_a line_n by_o the_o 23._o of_o this_o book_n wherefore_o d_z &_o e_o be_v medial_a line_n commensurable_a in_o power_n only_o i_o say_v also_o that_o they_o contain_v a_o medial_a parallelogram_n for_o for_o that_o as_o the_o line_n b_o be_v to_o the_o line_n c_o so_o be_v the_o line_n d_o to_o the_o line_n e_o therefore_o alternate_o by_o the_o 16_o of_o the_o five_o as_o the_o line_n b_o be_v to_o the_o line_n d_o so_o be_v the_o line_n c_o to_o the_o line_n e._n but_o as_o the_o line_n b_o be_v to_o the_o line_n d_o so_o be_v the_o line_n d_o to_o the_o line_n aâ—_z by_o converse_n proportion_n which_o be_v prove_v by_o the_o corollary_n of_o the_o four_o of_o the_o five_o wherefore_o as_o the_o line_n d_o be_v to_o the_o line_n a_o so_o be_v the_o line_n c_o to_o the_o line_n e._n wherefore_o that_o which_o be_v contain_v under_o the_o line_n a_o &_o c_o be_v by_o the_o 16._o of_o the_o sixâ—_o equal_a to_o that_o which_o be_v contain_v under_o the_o line_n d_o &_o e._n but_o that_o which_o be_v contain_v under_o the_o line_n a_o and_o c_o be_v medial_a by_o the_o 21._o of_o the_o ten_o wherefore_o that_o which_o be_v contain_v under_o the_o line_n d_o and_o e_o be_v medial_a wherefore_o there_o be_v find_v out_o medial_a line_n commensurable_a in_o power_n only_o contain_v a_o medial_a superficies_n which_o be_v require_v to_o be_v do_v a_o assumpt_n to_o find_v out_o two_o square_a number_n which_o add_v together_o make_v a_o square_a number_n let_v there_o be_v put_v two_o like_o superficial_a number_n ab_fw-la and_o bc_o which_o how_o to_o find_v out_o have_v be_v teach_v after_o the_o 9_o proposition_n of_o this_o book_n and_o let_v they_o both_o be_v either_o even_a number_n or_o odd_a and_o let_v the_o great_a number_n be_v ab_fw-la and_o forasmuch_o as_o if_o from_o any_o even_a number_n be_v take_v away_o a_o even_a number_n or_o from_o a_o odd_a number_n be_v take_v away_o a_o odd_a number_n the_o residue_n shall_v be_v even_o by_o the_o 24._o and_o 26_o of_o the_o nine_o if_o therefore_o from_o ab_fw-la be_v a_o even_a number_n be_v take_v away_o bc_o a_o even_a number_n or_o from_o ab_fw-la be_v a_o odd_a number_n be_v take_v away_o bc_o be_v also_o odd_a the_o residue_n ac_fw-la shall_v be_v even_o divide_v the_o number_n ac_fw-la into_o two_o equal_a part_n in_o d_o wherefore_o the_o number_n which_o be_v produce_v of_o ab_fw-la into_o bc_o together_o with_o the_o square_a number_n of_o cd_o be_v by_o the_o six_o of_o the_o second_o as_o barlaam_n demonstrate_v it_o in_o number_n equal_a to_o the_o square_a number_n of_o bd._n but_o that_o which_o be_v produce_v of_o ab_fw-la into_o bc_o be_v a_o square_a number_n for_o it_o be_v prove_v by_o the_o first_o of_o the_o nine_o that_o if_o two_o like_o plain_a number_n multiply_v the_o one_o the_o other_o produce_v any_o number_n the_o number_n produce_v shall_v be_v a_o square_a number_n wherefore_o there_o be_v find_v out_o two_o square_a number_n the_o one_o be_v the_o square_a number_n which_o be_v produce_v of_o ab_fw-la into_o bc_o and_o the_o other_o the_o square_a number_n produce_v of_o cd_o which_o add_v together_o make_v a_o square_a number_n namely_o the_o square_a number_n produce_v of_o bd_o multiply_v into_o himself_o forasmuch_o as_o they_o be_v demonstrate_v equal_a to_o it_o corollary_n a_o corollary_n â—umber_n and_o hereby_o it_o be_v manifest_a that_o there_o be_v find_v out_o two_o square_a number_n namely_o the_o 〈◊〉_d the_o square_a number_n of_o bd_o and_o the_o other_o the_o square_a number_n of_o cd_o so_o that_o that_o number_n wherein_o the_o one_o exceed_v the_o other_o the_o number_n i_o say_v which_o be_v produce_v of_o ab_fw-la into_o bc_o be_v also_o a_o square_a number_n namely_o when_o aâ—_z &_o bc_o be_v like_o plain_a number_n but_o when_o they_o be_v not_o like_o plain_a number_n then_o be_v there_o find_v out_o two_o square_a number_n the_o square_a number_n of_o bd_o and_o the_o square_a number_n of_o dc_o who_o excess_n that_o be_v the_o number_n whereby_o the_o great_a exceed_v the_o less_o namely_o that_o which_o be_v produce_v of_o ab_fw-la into_o bc_o be_v not_o a_o square_a number_n ¶_o a_o assumpt_n to_o find_v out_o two_o square_a number_n which_o add_v together_o make_v not_o a_o square_a number_n assumpt_n let_v ab_fw-la and_o bc_o be_v like_o plain_a number_n so_o that_o by_o the_o first_o of_o the_o nine_o that_o which_o be_v produce_v of_o ab_fw-la into_o bc_o be_v a_o square_a number_n and_o let_v ac_fw-la be_v a_o even_a number_n and_o divide_v câ—_n into_o two_o equal_a parâ—es_n in_o d._n now_o by_o that_o which_o have_v before_o be_v say_v in_o the_o former_a assumpt_n it_o be_v manifest_a that_o the_o square_a number_n produce_v of_o ab_fw-la into_o bc_o together_o with_o the_o square_a number_n of_o cd_o be_v equal_a to_o the_o square_a number_n of_o bd._n take_v away_o from_o cd_o unity_n de._n wherefore_o that_o which_o be_v produce_v of_o ab_fw-la into_o bc_o together_o with_o the_o square_n of_o ce_fw-fr be_v less_o than_o the_o square_a number_n of_o bd._n now_o then_o i_o say_v that_o the_o square_a numâ—er_n produce_v of_o ab_fw-la into_o bc_o add_v to_o the_o square_a number_n of_o ce_fw-fr make_v not_o a_o square_a number_n for_o if_o they_o do_v make_v a_o square_a number_n than_o that_o square_a number_n which_o they_o make_v be_v either_o great_a they_o the_o square_a number_n of_o be_v or_o equal_a unto_o it_o or_o less_o than_o it_o first_o great_a it_n can_v be_v for_o it_o be_v already_o prove_v that_o the_o square_a number_n produce_v of_o ab_fw-la into_o bc_o together_o with_o the_o square_a number_n of_o ce_fw-fr be_v less_o than_o the_o square_a number_n of_o bd._n but_o between_o the_o square_a number_n of_o bd_o and_o the_o square_a number_n of_o be_v there_o be_v no_o mean_a square_a number_n for_o the_o number_n bd_o exceed_v the_o number_n be_v only_o by_o unity_n which_o unity_n can_v by_o no_o mean_n be_v divide_v into_o number_n or_o if_o the_o number_n produce_v of_o ab_fw-la into_o bc_o together_o with_o the_o square_n of_o the_o number_n ce_fw-fr shall_v be_v great_a than_o the_o square_a of_o the_o number_n be_v then_o shall_v the_o self_n same_o number_n produce_v of_o ab_fw-la into_o bc_o together_o with_o the_o square_n of_o the_o number_n ce_fw-fr be_v equal_a to_o the_o square_n of_o the_o number_n bd_o the_o contrary_a whereof_o be_v already_o prove_v wherefore_o if_o it_o be_v possible_a let_v that_o which_o be_v produce_v of_o ab_fw-la into_o bc_o together_o with_o the_o square_a number_n of_o the_o number_n ce_fw-fr be_v equal_a to_o the_o square_a number_n of_o be._n and_o let_v ga_o be_v double_a to_o unity_n de_fw-fr that_o be_v let_v it_o be_v the_o number_n two_o now_o forasmuch_o as_o the_o whole_a number_n ac_fw-la be_v by_o supposition_n double_a to_o the_o whole_a number_n cd_o of_o which_o the_o number_n agnostus_n be_v double_a to_o unity_n de_fw-fr therefore_o by_o the_o 7._o of_o the_o seven_o the_o residue_n namely_o the_o number_n gc_o be_v double_a to_o the_o residue_n namely_o to_o the_o number_n ec_o wherefore_o the_o number_n gc_o be_v divide_v into_o two_o equal_a part_n in_o e._n wherefore_o that_o which_o be_v produce_v of_o gb_o into_o bc_o together_o with_o the_o square_a number_n of_o ce_fw-fr be_v equal_a to_o the_o square_a number_n
two_o other_o proposition_n go_v next_o before_o it_o so_o far_o misplace_v that_o where_o they_o be_v word_n for_o word_n before_o duâ—ly_o place_v be_v the_o 105._o and_o 106._o yet_o here_o after_o the_o book_n end_v they_o be_v repeat_v with_o the_o number_n of_o 116._o and_o 117._o proposition_n zambert_n therein_o be_v more_o faithful_a to_o follow_v as_o he_o find_v in_o his_o greek_n example_n than_o he_o be_v skilful_a or_o careful_a to_o do_v what_o be_v necessary_a yea_o and_o some_o greek_n write_v ancient_a copy_n have_v they_o not_o so_o though_o in_o deed_n they_o be_v well_o demonstrate_v yet_o truth_n disord_v be_v half_a disgracedâ—_n especial_o where_o the_o pattern_n of_o good_a order_n by_o profession_n be_v avouch_v to_o be_v but_o through_o ignorance_n arrogancy_n and_o â—emerltie_n of_o unskilful_a method_n master_n many_o thing_n remain_v yet_o in_o these_o geometrical_a element_n undue_o tumble_v in_o though_o true_a yet_o with_o disgrace_n which_o by_o help_n of_o so_o many_o wit_n and_o hability_n of_o such_o as_o now_o may_v have_v good_a cause_n to_o be_v skilful_a herein_o will_v i_o hope_v ere_o long_o be_v take_v away_o and_o thing_n of_o importance_n want_v supply_v the_o end_n of_o the_o ten_o book_n of_o euclides_n element_n ¶_o the_o eleven_o book_n of_o euclides_n element_n book_n hitherto_o have_v â—vclidâ—_n in_o thâ—sâ—_n former_a book_n with_o a_o wonderful_a method_n and_o order_n entreat_v of_o such_o kind_n of_o figure_n superficial_a which_o be_v or_o may_v be_v describe_v in_o a_o superficies_n or_o plain_a and_o have_v teach_v and_o set_v forth_o their_o property_n nature_n generation_n and_o production_n even_o from_o the_o first_o root_n ground_n and_o beginning_n of_o they_o namely_o from_o a_o point_n which_o although_o it_o be_v indivisible_a yet_o be_v it_o the_o begin_n of_o all_o quantity_n continual_a and_o of_o it_o and_o of_o the_o motion_n and_o slow_v thereof_o be_v produce_v a_o line_n and_o consequent_o all_o quantity_n continual_a as_o all_o figure_n plain_a and_o solid_a what_o so_o ever_o euclid_n therefore_o in_o his_o first_o book_n begin_v with_o it_o boot_n and_o from_o thence_o go_v he_o to_o a_o line_n as_o to_o a_o thing_n most_o simple_a next_o unto_o a_o point_n then_o to_o a_o superficies_n and_o to_o angle_n and_o so_o through_o the_o whole_a first_o book_n boâ—â—e_n he_o entreat_v of_o these_o most_o simple_a and_o plain_a ground_n in_o the_o second_o book_n he_o entreat_v further_o â—â—oâ—e_n and_o go_v unto_o more_o hard_a matter_n and_o teach_v of_o division_n of_o line_n and_o of_o the_o multiplication_n of_o line_n and_o of_o their_o part_n and_o of_o their_o passion_n and_o property_n and_o for_o that_o rightline_v â—_z be_v far_o distant_a in_o nature_n and_o property_n from_o round_a and_o circular_a figure_n in_o the_o three_o book_n he_o instruct_v the_o reader_n of_o the_o nature_n and_o condition_n of_o circle_n booâ—e_n in_o the_o four_o book_n he_o compare_v figure_n of_o right_a line_n and_o circle_n together_o bâ—oâ—e_n and_o teach_v how_o to_o describe_v a_o figure_n of_o right_a line_n with_o in_o or_o about_o a_o circle_n