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 and_o contrariwise_o a_o circle_n with_o in_o or_o about_o a_o rectiline_a figure_n in_o the_o five_o book_n he_o search_v out_o the_o nature_n of_o proportion_n a_o matter_n of_o wonderful_a use_n and_o deep_a consideration_n boâ—â—e_n for_o that_o otherwise_o he_o can_v not_o compare_v â—igure_n with_o figure_n or_o the_o side_n of_o figure_n together_o for_o whatsoever_o be_v compare_v to_o any_o other_o thing_n be_v compare_v unto_o it_o undoubted_o under_o some_o kind_n of_o proportion_n wherefore_o in_o the_o six_o book_n he_o compare_v figure_n together_o booâ—e_n one_o to_o a_o other_o likewise_o their_o side_n and_o for_o that_o the_o nature_n of_o proportion_n can_v not_o be_v full_o and_o clear_o see_v without_o the_o knowledge_n of_o number_n wherein_o it_o be_v first_o and_o chief_o find_v in_o the_o seven_o bookâ—_n eight_o booâ—â—_n and_o nine_o book_n book_n he_o entreatâ—th_v of_o number_n &_o of_o the_o kind_n and_o property_n thereof_o and_o because_o that_o the_o side_n of_o solid_a body_n for_o the_o most_o part_n be_v of_o such_o sort_n that_o compare_v together_o they_o have_v such_o proportion_n the_o one_o to_o the_o other_o booâ—e_n which_o can_v not_o be_v express_v by_o any_o number_n certain_a and_o therefore_o be_v call_v irrational_a line_n he_o in_o the_o ten_o book_n have_v write_v &_o teach_v which_o lineâ—_z be_v commensurable_a or_o incommensurable_a the_o one_o to_o the_o other_o and_o of_o the_o diversity_n of_o kind_n of_o irrational_a line_n with_o all_o the_o condition_n &_o propriety_n of_o they_o and_o thus_o have_v euclid_n in_o these_o ten_o foresay_a book_n full_o &_o most_o plenteous_o in_o a_o marvelous_a order_n teach_v whatsoever_o seem_v necessary_a and_o requisite_a to_o the_o knowledge_n of_o all_o superficial_a figure_n of_o what_o sort_n &_o form_n so_o ever_o they_o be_v now_o in_o these_o book_n follow_v he_o entreat_v of_o figure_n of_o a_o other_o kind_n namely_o of_o bodily_a figure_n follow_v as_o of_o cube_n pyramid_n cones_n column_n cilinder_n parallelipipedon_n sphere_n and_o such_o othersâ—_n and_o show_v the_o diversity_n of_o they_o the_o generation_n and_o production_n of_o they_o and_o demonstrate_v with_o great_a and_o wonderful_a art_n their_o propriety_n and_o passion_n with_o all_o their_o nature_n and_o condition_n he_o also_o compare_v one_o oâ—_n they_o to_o a_o other_o whereby_o to_o know_v the_o reason_n and_o proportion_n of_o the_o one_o to_o the_o other_o chief_o of_o the_o five_o body_n which_o be_v call_v regular_a body_n element_n and_o these_o be_v the_o thing_n of_o all_o other_o entreat_v of_o in_o geometry_n most_o worthy_a and_o of_o great_a dignity_n and_o as_o it_o be_v the_o end_n and_o final_a intent_n of_o the_o whole_a be_v of_o geometry_n and_o for_o who_o cause_n have_v be_v write_v and_o speak_v whatsoever_o have_v hitherto_o in_o the_o former_a book_n be_v say_v or_o write_v as_o the_o first_o book_n be_v a_o ground_n and_o a_o necessary_a entry_n to_o all_o the_o râ—st_o â—ollowing_v so_o be_v this_o eleven_o book_n a_o necessary_a entry_n and_o ground_n to_o the_o rest_n which_o follow_v 〈◊〉_d and_o as_o that_o contain_v the_o declaration_n of_o word_n and_o definition_n of_o thinger_n requisite_a to_o the_o knowledge_n of_o superficial_a figure_n and_o entreat_v of_o line_n and_o of_o their_o division_n and_o section_n which_o be_v the_o term_n and_o limit_n of_o superficial_a figure_n so_o in_o this_o book_n be_v set_v forth_o the_o declaration_n of_o word_n and_o definition_n of_o thing_n pertain_v to_o solid_a and_o corporal_a figure_n and_o also_o of_o superficiece_n which_o be_v the_o term_n &_o limit_n of_o solid_n moreover_o of_o the_o division_n and_o intersection_n of_o they_o and_o diverse_a other_o thing_n without_o which_o the_o knowledge_n of_o bodily_a and_o solid_a form_n can_v not_o be_v attain_a unto_o and_o first_o be_v set_v the_o definition_n as_o follow_v definition_n a_o solid_a or_o body_n be_v that_o which_o have_v length_n breadth_n and_o thickness_n definition_n and_o the_o term_n or_o limit_v of_o a_o solid_a be_v a_o superficies_n there_o be_v three_o kind_n of_o continual_a quantity_n a_o line_n a_o superficies_n and_o a_o solid_a or_o body_n the_o beginning_n of_o all_o which_o as_o before_o have_v be_v say_v be_v a_o point_n which_o be_v indivisible_a two_o of_o these_o quantity_n namely_o a_o line_n and_o a_o superficies_n be_v define_v of_o euclid_n before_o in_o his_o first_o book_n but_o the_o three_o kind_n namely_o a_o solid_a or_o body_n he_o there_o define_v not_o as_o a_o thing_n which_o pertain_v not_o then_o to_o his_o purpose_n but_o here_o in_o this_o place_n he_o set_v the_o definition_n thereof_o as_o that_o which_o chief_o now_o pertain_v to_o his_o purpose_n and_o without_o which_o nothing_o in_o these_o thing_n can_v profitable_o be_v teach_v a_o solid_a say_v he_o be_v that_o which_o have_v length_n breadth_n and_o thickness_n or_o depth_n there_o be_v as_o before_o have_v be_v teach_v three_o reason_n or_o mean_n of_o measure_v which_o be_v call_v common_o dimension_n namely_o length_n breadth_n and_o thickness_n these_o dimension_n be_v ascribe_v unto_o quantity_n only_o by_o these_o be_v all_o kind_n of_o quantity_n define_v â—â—_o be_v count_v perfect_a or_o imperfect_a according_a as_o they_o be_v partaker_n of_o few_o or_o more_o of_o they_o as_o euclid_n define_v a_o line_n ascribe_v unto_o it_o only_o one_o of_o these_o dimension_n namely_o length_n wherefore_o a_o line_n be_v the_o imperfect_a kind_n of_o quantity_n in_o define_v of_o a_o superficies_n he_o ascribe_v unto_o it_o two_o dimension_n namely_o length_n and_o breadth_n whereby_o a_o superficies_n be_v a_o quantity_n of_o
same_o be_v treble_a it_o be_v manifest_a by_o the_o 12._o of_o the_o five_o that_o all_o the_o prism_n be_v to_o all_o the_o pyramid_n treble_a wherefore_o parallelipipedon_n be_v treble_a to_o pyramid_n set_v upon_o the_o self_n same_o base_a with_o they_o and_o under_o the_o same_o altitude_n they_o for_o that_o they_o contain_v two_o prism_n three_o corollary_n if_o two_o prism_n be_v under_o one_o and_o the_o self_n same_o altitude_n have_v to_o their_o base_n either_o both_o triangle_n or_o both_o parallelogram_n the_o prism_n be_v the_o one_o to_o the_o other_o as_o their_o base_n be_v for_o forasmuch_o as_o those_o prism_n be_v equemultiqlice_n unto_o the_o pyramid_n upon_o the_o self_n same_o base_n and_o under_o the_o same_o altitude_n which_o pyramid_n be_v in_o proportion_n as_o their_o base_n it_o be_v manifest_a by_o the_o 15._o of_o the_o five_o that_o the_o prism_n be_v in_o the_o proportion_n of_o the_o base_n for_o by_o the_o former_a corollary_n the_o prism_n be_v treble_a to_o the_o pyramid_n sâ—t_v upon_o the_o triangular_a base_n four_o corollary_n prism_n be_v in_o sesquealtera_fw-la proportion_n to_o pyramid_n which_o have_v the_o self_n same_o quadrangled_a base_a that_o the_o prism_n have_v and_o be_v under_o the_o self_n same_o altitude_n for_o that_o pyramid_n contain_v two_o pyramid_n set_v upon_o a_o triangular_a base_a of_o the_o same_o prism_n for_o it_o be_v prove_v that_o that_o prism_n be_v treble_a to_o the_o pyramid_n which_o be_v set_v upon_o the_o half_a of_o his_o quadrangled_a base_a unto_o which_o the_o other_o which_o be_v set_v upon_o the_o whole_a base_a be_v double_a by_o the_o six_o of_o this_o book_n five_v corollary_n wherefore_o we_o may_v in_o like_a sort_n conclude_v that_o solid_n mention_v in_o the_o second_o corollary_n which_o solid_n campane_n call_v side_v column_n be_v under_o one_o and_o the_o self_n same_o altitude_n be_v in_o proportion_n the_o one_o to_o the_o other_o as_o their_o base_n which_o be_v poligonon_o figure_n for_o they_o be_v in_o the_o proportion_n of_o the_o pyramid_n or_o prism_n set_v upon_o the_o self_n same_o base_n and_o under_o the_o self_n same_o altitude_n that_o be_v they_o be_v in_o the_o proportion_n of_o the_o base_n of_o the_o say_a pyramid_n or_o prism_n for_o those_o solid_n may_v be_v divide_v into_o prism_n have_v the_o self_n same_o altitude_n when_o as_o their_o opposite_a base_n may_v be_v divide_v into_o triangle_n by_o the_o 20_o of_o the_o six_o upon_o which_o triangle_n prism_n be_v set_v be_v in_o proportion_n as_o their_o base_n by_o this_o 7._o proposition_n it_o plain_o appear_v that_o â—uâ—lide_a as_o it_o be_v before_o note_v in_o the_o diffinitionâ—â—_n under_o the_o definition_n of_o a_o prism_n comprehend_v also_o those_o kind_n of_o solid_n which_o campane_n call_v side_v column_n for_o in_o that_o he_o say_v every_o prism_n have_v a_o triangle_n to_o his_o base_a may_v be_v devidedâ—_n etc_n etc_n he_o need_v not_o take_v a_o prism_n in_o that_o sense_n which_o campane_n and_o most_o man_n take_v it_o to_o have_v add_v that_o particle_n have_v to_o his_o base_a a_o triangle_n for_o by_o their_o sense_n there_o be_v no_o prism_n but_o it_o may_v have_v to_o his_o base_a a_o triangleâ—_n and_o so_o it_o may_v seem_v that_o euclid_n ought_v without_o exception_n have_v say_v that_o every_o prism_n whatsoever_o may_v be_v divide_v into_o three_o pyramid_n equal_a the_o one_o to_o the_o other_o have_v also_o triangle_n to_o â—heir_a base_n for_o so_o do_v campane_n and_o flussas_n put_v the_o proposition_n leave_v out_o the_o former_a particle_n have_v to_o his_o base_a a_o triangle_n which_o yet_o be_v red_a in_o the_o greek_a copy_n &_o not_o leâ—t_v out_o by_o any_o other_o interpreter_n know_v abroad_o except_o by_o campane_n and_o flussas_n yea_o and_o the_o corollary_n follow_v of_o this_o proposition_n add_v by_o theon_n or_o euclid_n and_o amend_v by_o m._n dee_n seem_v to_o confirm_v this_o sense_n of_o this_o â—s_a 〈◊〉_d make_v manifest_a that_o every_o pyramid_n be_v the_o three_o part_n of_o the_o prism_n have_v the_o same_o base_a with_o it_o and_o equal_a altitude_n for_o and_o if_o the_o base_a of_o the_o prism_n have_v any_o other_o right_o line_v figure_n than_o a_o triangle_n and_o also_o the_o superficies_n opposite_a to_o the_o base_a the_o same_o figure_n that_o prism_n may_v be_v divide_v into_o prism_n have_v triangle_v base_n and_o the_o superficiece_n to_o those_o base_n opposite_a also_o triangle_v a_o â—â—ike_a and_o equal_o for_o there_o as_o we_o see_v be_v put_v these_o word_n â—or_a and_o if_o the_o base_a of_o the_o prism_n be_v any_o other_o right_o line_v figureâ—_n etc_n etc_n whereof_o a_o man_n may_v well_o infer_v that_o the_o base_a may_v be_v any_o other_o rectiline_a figure_n whatsoever_o &_o not_o only_o a_o triangle_n or_o a_o parallelogram_n and_o it_o be_v true_a also_o in_o that_o sense_n as_o it_o be_v plain_a to_o see_v by_o the_o second_o corollary_n add_v out_o of_o flussas_n which_o corollary_n as_o also_o the_o first_o of_o his_o corollary_n be_v in_o a_o manner_n all_o one_o with_o the_o corollary_n add_v by_o theon_n or_o euclid_n far_o theon_n in_o the_o demonstration_n of_o the_o 10._o proposition_n of_o this_o book_n as_o we_o shall_v afterwards_o see_v most_o plain_o call_v not_o only_o side_v column_n prism_n but_o also_o parallelipipedon_n and_o although_o the_o 40._o proposition_n of_o the_o eleven_o book_n may_v seem_v hereunto_o to_o be_v a_o lâ—t_n for_o that_o it_o can_v be_v understand_v of_o those_o prism_n only_o which_o have_v triangle_n to_o their_o like_a equal_a opposite_a and_o parallel_v side_n or_o but_o of_o some_o side_v column_n and_o not_o of_o all_o yet_o may_v that_o let_v be_v thus_o remove_v away_o to_o say_v that_o euclid_n in_o that_o proposition_n use_v genus_fw-la pro_fw-la specie_fw-la that_o be_v the_o general_a word_n for_o some_o special_a kind_n thereof_o which_o thing_n also_o be_v not_o rare_a not_o only_o with_o he_o but_o also_o with_o other_o learned_a philosopher_n thus_o much_o i_o think_v good_a by_o the_o way_n to_o note_v in_o far_a defence_n of_o euclid_n definition_n of_o a_o prism_n the_o 8._o theorem_a the_o 8._o proposition_n pyramid_n be_v like_o &_o have_a triangle_n to_o their_o base_n be_v in_o treble_a proportion_n the_o one_o to_o the_o other_o of_o that_o in_o which_o their_o side_n of_o like_a proportion_n be_v svppose_v that_o these_o pyramid_n who_o base_n be_v the_o triangle_n gbc_n and_o hef_n and_o top_n the_o point_n a_o and_o d_o be_v like_o and_o in_o like_a sort_n describe_v and_o let_v ab_fw-la and_o de_fw-fr be_v side_n of_o like_a proportion_n then_o i_o say_v that_o the_o pyramid_n abcg_o be_v to_o the_o pyramid_n defh_o in_o treble_a proportion_n of_o that_o in_o which_o the_o side_n ab_fw-la be_v to_o the_o side_n de._n make_v perfect_a the_o parallelipipedon_n namely_o the_o solid_n bckl_n &_o efxo_n and_o forasmuch_o as_o the_o pyramid_n abcg_o be_v like_a to_o the_o pyramid_n defh_o construction_n therefore_o the_o angle_n abc_n be_v equal_a to_o the_o angle_n def_n demonstration_n &_o the_o angle_n gbc_n to_o the_o angle_n hef_fw-fr and_o moreover_o the_o angle_n abg_o to_o the_o angle_n deh_fw-it and_o as_o the_o line_n ab_fw-la be_v to_o the_o line_n de_fw-fr so_o be_v the_o line_n bc_o to_o the_o line_n of_o and_o the_o line_n bg_o to_o the_o line_n eh_o and_o for_o that_o as_o the_o line_n ab_fw-la be_v to_o the_o line_n de_fw-fr so_o be_v the_o line_n bc_o to_o the_o line_n of_o and_o the_o side_n about_o the_o equal_a angle_n be_v proportional_a therefore_o the_o parallelogram_n bm_n be_v like_a to_o the_o parallelogram_n ep_n and_o by_o the_o same_o reason_n the_o parallelogram_n bn_o be_v like_a to_o the_o parallelogram_n er_fw-mi and_o the_o parellelogramme_n bk_o be_v like_a unto_o the_o parallelogram_n exit_fw-la wherefore_o the_o three_o parallelogram_n bm_n kb_o and_o bn_o be_v like_a to_o the_o three_o parallelogram_n ep_n exit_fw-la and_o er._n but_o the_o three_o parallelogram_n bm_n kb_o and_o bn_o be_v equal_a and_o like_a to_o the_o three_o opposite_a parallelogram_n and_o the_o three_o parallelogram_n ep_n exit_fw-la and_o ere_o be_v equal_a and_o like_a to_o the_o three_o opposite_a parallelogram_n wherefore_o the_o parallelipipedon_n bckl_n and_o efxo_n be_v comprehend_v under_o plain_a superficiece_n like_a and_o equal_a in_o multitude_n wherefore_o the_o solid_a bckl_n be_v like_a to_o the_o solid_a efxo_n but_o like_o parallelipipedon_n be_v by_o the_o 33._o of_o the_o eleven_o in_o treble_a proportion_n the_o one_o to_o the_o other_o of_o that_o in_o which_o side_n of_o like_a proportion_n be_v to_o side_n of_o like_a proportion_n wherefore_o the_o solid_a bckl_n be_v to_o the_o solid_a efxo_n in_o treble_a
line_n which_o subtend_v the_o angle_n zob_n to_o the_o three_o line_n which_o subtend_v the_o angle_n zkb_n but_o by_o construction_n boy_n be_v equal_a to_o bk_o therefore_o oz_n be_v equal_a to_o kz_o and_o the_o three_o alâ—o_o be_v equal_a to_o the_o three_o wherefore_o the_o point_n z_o in_o respect_n of_o the_o two_o triangle_n rectangle_v ozb_n and_o kzb_n determine_v one_o and_o the_o same_o magnitude_n iâ—_z the_o line_n bz_o which_o can_v not_o be_v if_o any_o other_o point_n in_o the_o line_n bz_o be_v assign_v near_o or_o far_o of_o from_o the_o point_n b._n one_o only_a point_n therefore_o be_v that_o at_o which_o the_o two_o perpendicular_n kz_o and_o oz_n fall_v but_o by_o construction_n oz_n fall_v at_o z_o the_o point_n and_o therefore_o at_o the_o same_o z_o do_v the_o perpendicular_a draw_v from_o king_n fall_v likewise_o which_o be_v require_v to_o be_v demonstrate_v although_o a_o brief_a monition_n may_v herein_o have_v serve_v for_o the_o pregnant_a or_o the_o humble_a learner_n yet_o for_o they_o that_o be_v well_o please_v to_o have_v thing_n make_v plain_a with_o many_o word_n and_o for_o the_o stiffnecked_a busy_a body_n it_o be_v necessary_a with_o my_o controlment_n of_o other_o to_o annex_v the_o cause_n &_o reason_n thereof_o both_o invincible_a and_o also_o evident_a a_o corollary_n 1._o hereby_o it_o be_v manifest_a that_o two_o equal_a circle_n cut_v one_o the_o other_o by_o the_o whole_a diameter_n if_o from_o one_o and_o the_o same_o end_n of_o their_o common_a diameter_n equal_a portion_n of_o their_o circumference_n be_v take_v and_o from_o the_o point_n end_v those_o equal_a portion_n two_o perpendicular_n be_v let_v down_o to_o their_o common_a diameter_n those_o perpendicular_n shall_v fall_v upon_o one_o and_o the_o same_o point_n of_o their_o common_a diameter_n 2._o second_o it_o follow_v that_o those_o perpendicular_n be_v equal_a ¶_o note_n from_o circle_n in_o our_o first_o supposition_n each_o to_o other_o perpendicular_o erect_v we_o proceed_v and_o infer_v now_o these_o corollary_n whether_o they_o be_v perpendicular_o erect_v or_o no_o by_o reasou_v the_o demonstration_n have_v a_o like_a force_n upon_o our_o supposition_n here_o use_v ¶_o the_o 16._o theorem_a the_o 18._o proposition_n sphere_n be_v in_o treble_a proportion_n the_o one_o to_o the_o other_o of_o that_o in_o which_o their_o diameter_n be_v svppose_v that_o there_o be_v two_o sphere_n abc_n and_o def_n and_o let_v their_o diameter_n be_v bc_o and_o ef._n then_o i_o say_v that_o the_o sphere_n abc_n be_v to_o the_o sphere_n def_n in_o treble_a proportion_n of_o that_o in_o which_o the_o diameter_n bc_o be_v to_o the_o diameter_n ef._n proposition_n for_o if_o not_o than_o the_o sphere_n abc_n be_v in_o treble_a proportion_n of_o that_o in_o which_o bc_o be_v to_o of_o either_o to_o some_o sphere_n less_o than_o the_o sphere_n def_n or_o to_o some_o sphere_n great_a first_o let_v it_o be_v unto_o a_o less_o namely_o to_o ghk_v case_n and_o imagine_v that_o the_o sphere_n def_n and_o ghk_o be_v both_o about_o one_o and_o the_o self_n same_o centre_n impossibility_n and_o by_o the_o proposition_n next_o go_v before_o describe_v in_o the_o great_a sphere_n def_n a_o polihedron_n or_o a_o solid_a of_o many_o side_n not_o touch_v the_o superficies_n of_o the_o less_o sphere_n ghk_v and_o suppose_v also_o that_o in_o the_o sphere_n abc_n be_v inscribe_v a_o polihedron_n like_a to_o the_o polihedron_n which_o be_v in_o the_o sphere_n def_n wherefore_o by_o the_o corollary_n of_o the_o same_o the_o polihedron_n which_o be_v in_o the_o sphere_n abc_n be_v to_o the_o polihedron_n which_o be_v in_o the_o sphere_n def_n in_o treble_a proportion_n of_o that_o in_o which_o the_o diameter_n bc_o be_v to_o the_o diameter_n ef._n but_o by_o supposition_n the_o sphere_n abc_n be_v to_o the_o sphere_n ghk_v in_o treble_a proportion_n of_o that_o in_o which_o the_o diameter_n bc_o be_v to_o the_o diameter_n ef._n wherefore_o as_o the_o sphere_n abc_n be_v to_o the_o sphere_n ghk_o so_o be_v the_o polihedron_n which_o be_v describe_v in_o the_o sphere_n abc_n to_o the_o polihedron_n which_o be_v describe_v in_o the_o sphere_n def_n by_o the_o 11._o of_o the_o five_o wherefore_o alternate_o by_o the_o 16._o of_o the_o five_o as_o the_o sphere_n abc_n be_v to_o the_o polihedron_n which_o be_v describe_v in_o it_o so_o be_v the_o sphere_n ghk_v to_o the_o polihedron_n which_o be_v in_o the_o sphere_n def_n but_o the_o sphere_n abc_n be_v great_a than_o the_o polihedron_n which_o be_v describe_v in_o it_o wherefore_o also_o the_o sphere_n ghk_o be_v great_a than_o the_o polihedron_n which_o be_v in_o the_o sphere_n def_n by_o the_o 14._o of_o the_o five_o but_o it_o be_v also_o less_o for_o it_o be_v contain_v in_o it_o which_o impossible_a wherefore_o the_o sphere_n abc_n be_v not_o in_o treble_a proportion_n of_o that_o in_o which_o the_o diameter_n bc_o be_v to_o the_o diameter_n of_o to_o any_o sphere_n less_o than_o the_o sphere_n def_n in_o like_a sort_n also_o may_v we_o prove_v that_o the_o sphere_n def_n be_v not_o in_o treble_a proportion_n of_o that_o in_o which_o the_o diameter_n of_o be_v to_o the_o diameter_n bc_o to_o any_o sphere_n less_o than_o the_o sphere_n abc_n case_n now_o i_o say_v that_o the_o sphere_n abc_n be_v not_o in_o treble_a proportion_n of_o that_o in_o which_o the_o diameter_n bc_o be_v to_o the_o diameter_n of_o to_o any_o sphere_n great_a they_o the_o sphere_n def_n for_o if_o it_o be_v possible_a let_v it_o be_v to_o a_o great_a namely_o to_o lmn_o wherefore_o by_o conversion_n the_o sphere_n lmn_o be_v to_o the_o sphere_n abc_n in_o treble_a proportion_n of_o that_o in_o which_o the_o diameter_n of_o be_v to_o the_o diameter_n bc._n but_o as_o the_o sphere_n lmn_o be_v to_o the_o sphere_n abc_n so_o be_v the_o sphere_n def_n to_o some_o sphere_n less_o they_o the_o sphere_n abc_n booâ—â—_n as_o it_o have_v before_o be_v prove_v for_o the_o sphere_n lmn_o be_v great_a than_o the_o sphere_n def_n wherefore_o the_o sphere_n def_n be_v in_o treble_a proportion_n of_o that_o in_o which_o the_o diameter_n of_o be_v to_o the_o diameter_n bc_o to_o some_o sphere_n less_o they_o the_o sphere_n abc_n which_o be_v prove_v to_o be_v impossible_a wherefore_o the_o sphere_n abc_n be_v not_o in_o treble_a proportion_n of_o that_o in_o which_o be_v be_v to_o of_o to_o any_o sphere_n great_a they_o the_o sphere_n def_n and_o it_o be_v also_o prove_v that_o it_o be_v not_o to_o any_o less_o wherefore_o the_o sphere_n abc_n be_v to_o the_o sphere_n def_n in_o treble_a proportion_n of_o that_o in_o which_o the_o diameter_n bc_o be_v to_o the_o diameter_n of_o which_o be_v require_v to_o be_v demonstrate_v a_o corollary_n add_v by_o flussas_n hereby_o it_o be_v manifest_a that_o sphere_n be_v the_o one_o to_o the_o other_o as_o like_o polihedrons_n and_o in_o like_a sort_n describe_v in_o they_o be_v namely_o each_o be_v in_o triple_a proportion_n of_o that_o in_o which_o the_o diameter_n a_o corollary_n add_v by_o mâ—_n dee_n it_o be_v then_o evident_a how_o to_o geve_v two_o right_a line_n have_v that_o proportion_n between_o they_o which_o any_o two_o sphere_n give_v have_v the_o one_o to_o the_o other_o for_o if_o to_o their_o diameter_n as_o to_o the_o first_o and_o second_o line_n of_o four_o in_o continual_a proportion_n you_o adjoin_v a_o three_o and_o a_o four_o line_n in_o continual_a proportion_n as_o i_o have_v teach_v before_o the_o first_o and_o four_o line_n shall_v answer_v the_o problem_n rule_n how_o general_a this_o rule_n be_v in_o any_o two_o like_a solid_n with_o their_o correspondent_a or_o omologall_n line_n i_o need_v not_o with_o more_o word_n declare_v ¶_o certain_a theorem_n and_o problem_n who_o use_n be_v manifold_a in_o sphere_n cones_n cylinder_n and_o other_o solid_n add_v by_o joh._n dee_n a_o theorem_a 1._o the_o whole_a superficies_n of_o any_o sphere_n be_v quadrupla_fw-la to_o the_o great_a circle_n in_o the_o same_o sphere_n contain_v it_o be_v needless_a to_o bring_v archimedes_n demonstration_n hereof_o into_o this_o place_n see_v his_o book_n of_o the_o sphere_n and_o cylinder_n with_o other_o his_o woâ—kes_n be_v every_o where_o to_o be_v have_v and_o the_o demonstration_n thereof_o easy_a a_o theorem_a 2._o every_o sphere_n be_v quadruplâ—_n to_o that_o cone_n who_o base_a be_v the_o great_a circle_n &_o height_n the_o semidiameter_n of_o the_o same_o sphere_n this_o be_v the_o 32._o proposition_n of_o archimedes_n fiâ—st_n book_n of_o the_o sphere_n and_o cylinder_n a_o problem_n 1._o a_o sphere_n be_v give_v to_o make_v a_o upright_a cone_n equal_a to_o the_o same_o or_o in_o any_o other_o proportion_n give_v between_o two_o right_a line_n and_o as_o concern_v the_o other_o part_n of_o
line_n add_v together_o shall_v be_v quintuple_a to_o the_o square_n make_v of_o the_o half_a of_o the_o whole_a line_n svppose_v that_o the_o right_a line_n ab_fw-la be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n in_o the_o point_n c._n and_o let_v the_o great_a segment_n thereof_o be_v ac_fw-la and_o unto_o ac_fw-la add_v direct_o a_o right_a line_n ad_fw-la and_o let_v ad_fw-la be_v equal_a to_o the_o half_a of_o the_o line_n ab_fw-la construction_n then_o i_o say_v that_o the_o square_a of_o the_o line_n cd_o be_v quintuple_a to_o the_o square_n of_o the_o line_n da._n describe_v by_o the_o 46._o of_o the_o first_o upon_o the_o line_n ab_fw-la and_o dc_o square_n namely_o ae_n &_o df._n and_o in_o the_o square_a df_o describe_v and_o make_v complete_a the_o figure_n and_o extend_v the_o line_n fc_o to_o the_o point_n g._n demonstration_n and_o forasmuch_o as_o the_o line_n ab_fw-la be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n in_o the_o point_n c_o therefore_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o be_v equal_a to_o the_o square_n of_o the_o line_n ac_fw-la but_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o be_v the_o parallelogram_n ce_fw-fr and_o the_o square_a of_o the_o line_n ac_fw-la be_v the_o square_a hf._n wherefore_o the_o parallelogram_n ce_fw-fr be_v equal_a to_o the_o square_a hf._n and_o forasmuch_o as_o the_o line_n basilius_n be_v double_a to_o the_o line_n ad_fw-la by_o construction_n 〈◊〉_d the_o line_n basilius_n be_v equal_a to_o the_o line_n ka_o and_o the_o line_n ad_fw-la to_o the_o line_n ah_o therefore_o also_o the_o line_n ka_o be_v double_a to_o the_o line_n ah_o but_o as_o the_o line_n ka_o be_v to_o the_o line_n ah_o so_o be_v the_o parallelogram_n ck_o to_o the_o parallelogram_n change_z wherefore_o the_o parallelogram_n ck_o be_v double_a to_o the_o parallelogram_n ch._n and_o the_o parallelogram_n lh_o and_o ch_n be_v double_a to_o the_o parallelogram_n change_z for_o supplement_n of_o parallelogram_n be_v bâ—_z the_o 4â—_o of_o the_o first_o equal_v the_o one_o to_o the_o other_o wherefore_o the_o parallelogram_n ck_o be_v equal_a to_o the_o parallelogram_n lh_o &_o ch._n and_o it_o be_v prove_v that_o the_o parallelogram_n ce_fw-fr be_v equal_a to_o the_o square_a fh_o wherefore_o the_o whole_a square_a ae_n be_v equal_a to_o the_o gnomon_n mxn_n and_o forasmuch_o as_o the_o line_n basilius_n iâ—_z double_a to_o the_o line_n ad_fw-la therefore_o the_o square_a of_o the_o line_n basilius_n be_v by_o the_o 20._o of_o the_o six_o quadruple_a to_o the_o square_n of_o the_o line_n dam_fw-ge that_o be_v the_o square_a ae_n to_o the_o square_a dh_o but_o the_o square_a ae_n be_v equal_a to_o the_o gnomon_n mxn_n wherefore_o the_o gnomon_n mxn_n be_v also_o quadruple_a to_o the_o square_a dh_o wherefore_o the_o whole_a square_a df_o be_v quintuple_a to_o the_o square_a dh_o but_o the_o square_a df_o iâ—_z the_o square_a of_o the_o line_n cd_o and_o the_o square_a dh_o be_v the_o square_a of_o the_o line_n da._n wherefore_o the_o square_a of_o the_o line_n cd_o be_v quintuple_a to_o the_o square_n of_o the_o line_n da._n if_o therefore_o a_o right_a line_n be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n and_o to_o the_o great_a segment_n be_v add_v the_o half_a of_o the_o whole_a line_n the_o square_n make_v of_o those_o two_o line_n add_v together_o shall_v be_v quintuple_a to_o the_o square_n make_v of_o the_o half_a of_o the_o whole_a line_n which_o be_v require_v to_o be_v demonstrate_v this_o proposition_n be_v a_o other_o way_n demonstrate_v after_o the_o five_o proposition_n of_o this_o book_n the_o 2._o theorem_a the_o â—_o proposition_n if_o a_o right_a line_n be_v in_o power_n quintuple_a to_o a_o segment_n of_o the_o same_o line_n the_o double_a of_o the_o say_a segment_n be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n and_o the_o great_a segment_n thereof_o be_v the_o other_o part_n of_o the_o line_n give_v at_o the_o beginning_n prove_v now_o that_o the_o double_a of_o the_o line_n ad_fw-la that_o be_v ab_fw-la be_v great_a than_o the_o line_n ac_fw-la may_v thus_o be_v prove_v for_o if_o not_o then_o if_o if_o it_o be_v possible_a let_v the_o line_n ac_fw-la be_v double_a to_o the_o line_n ad_fw-la wherefore_o the_o square_a of_o the_o line_n ac_fw-la be_v quadruple_a to_o the_o square_n of_o the_o line_n ad._n wherefore_o the_o square_n of_o the_o line_n ac_fw-la and_o ad_fw-la be_v quintuple_a to_o the_o square_n of_o the_o line_n ad._n and_o it_o be_v suppose_v that_o the_o square_a of_o the_o line_n dc_o be_v quintuple_a to_o the_o square_n of_o the_o line_n ad_fw-la wherefore_o the_o square_a of_o the_o line_n dc_o be_v equal_a to_o the_o square_n of_o the_o line_n ac_fw-la and_o ad_fw-la which_o be_v impossible_a by_o the_o 4._o of_o the_o second_o wherefore_o the_o line_n ac_fw-la be_v not_o double_a to_o the_o line_n ad._n in_o like_a sort_n also_o may_v we_o prove_v that_o the_o double_a of_o the_o line_n ad_fw-la be_v not_o less_o than_o the_o line_n ac_fw-la for_o this_o be_v much_o more_o absurd_a wherefore_o the_o double_a of_o the_o line_n ad_fw-la be_v great_a they_o the_o line_n acâ—_n which_o be_v require_v to_o be_v prove_v this_o proposition_n also_o be_v a_o other_o way_n demonstrate_v after_o the_o five_o proposition_n of_o this_o book_n two_o theorem_n in_o euclides_n method_n necessary_a add_v by_o m._n dee_n a_o theorem_a 1._o a_o right_a line_n can_v be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n but_o in_o one_o only_a point_n suppose_v a_o line_n divide_v by_o extreme_a and_o mean_a proportion_n to_o be_v ab_fw-la and_o let_v the_o great_a segment_n be_v ac_fw-la i_o say_v that_o ab_fw-la can_v not_o be_v divide_v by_o the_o say_a proportion_n in_o any_o other_o point_n then_o in_o the_o point_n c._n if_o a_o adversary_n will_v contend_v that_o it_o may_v in_o like_a sort_n be_v divide_v in_o a_o other_o point_n let_v his_o other_o point_n be_v suppose_v to_o be_v d_o make_v ad_fw-la the_o great_a segment_n of_o his_o imagine_a division_n which_o ad_fw-la also_o let_v be_v less_o than_o our_o ac_fw-la for_o the_o first_o discourse_n now_o forasmuch_o as_o by_o our_o adversary_n opinion_n ad_fw-la be_v the_o great_a segment_n of_o his_o divide_a lineâ—_z the_o parallelogram_n contain_v under_o ab_fw-la and_o db_o be_v equal_a to_o the_o square_n of_o ad_fw-la by_o the_o three_o definition_n and_o 17._o proposition_n of_o the_o six_o book_n and_o by_o the_o same_o definition_n and_o proposition_n the_o parallelogram_n under_o ab_fw-la and_o cb_o contain_v be_v equal_a to_o the_o square_n of_o our_o great_a segment_n ac_fw-la wherefore_o as_o the_o parallelogram_n under_o ab_fw-la and_o dâ—_n be_v to_o the_o square_n of_o ad_fw-la so_o iâ—_z 〈◊〉_d parallelogram_n under_o ab_fw-la and_o cb_o to_o the_o square_n of_o ac_fw-la for_o proportion_n of_o equality_n be_v conclude_v in_o they_o both_o but_o forasmuch_o as_o dâ—â—_n iâ—_z byâ—_n â—c_z supposition_n great_a them_z cb_o the_o parallelogram_n under_o ab_fw-la and_o db_o be_v great_a than_o the_o parallelogram_n under_o ac_fw-la and_o cb_o by_o the_o first_o of_o the_o six_o for_o ab_fw-la be_v their_o equal_a heith_n wherefore_o the_o square_a of_o ad_fw-la shall_v be_v great_a than_o the_o square_a of_o ac_fw-la by_o the_o 14._o of_o the_o five_o but_o the_o line_n ad_fw-la be_v less_o than_o the_o line_n ac_fw-la by_o supposition_n wherefore_o the_o square_a of_o ad_fw-la be_v less_o than_o the_o square_a of_o ac_fw-la and_o it_o be_v conclude_v also_o to_o be_v great_a than_o the_o square_a of_o ac_fw-la wherefore_o the_o square_a of_o ad_fw-la be_v both_o great_a than_o the_o square_a of_o acâ—_n and_o also_o less_o which_o be_v a_o thing_n impossible_a the_o square_a therefore_o of_o ad_fw-la be_v not_o equal_a to_o the_o parallelogram_n under_o ab_fw-la and_o db._n and_o therefore_o by_o the_o three_o definition_n of_o the_o six_o ab_fw-la be_v not_o divide_v by_o a_o extreme_a and_o mean_a proportion_n in_o the_o point_n d_o as_o our_o adversary_n imagine_v and_o second_o in_o like_a sort_n will_v the_o inconueniency_n fall_v out_o if_o we_o assign_v ad_fw-la our_o adversary_n great_a segment_n to_o be_v great_a than_o our_o ac_fw-la therefore_o see_v neither_o on_o the_o one_o side_n of_o our_o point_n c_o neither_o on_o the_o other_o side_n of_o the_o same_o point_n c_o any_o point_n can_v be_v have_v at_o which_o the_o line_n ab_fw-la can_v be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n it_o follow_v of_o necessity_n that_o ab_fw-la can_v be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n in_o the_o point_n c_o only_a therefore_o a_o right_a line_n can_v be_v divide_v by_o a_o extreme_a
and_o mean_a proportion_n but_o in_o one_o only_a point_n which_o be_v requisite_a to_o be_v demonstrate_v a_o theorem_a 2._o what_o right_a line_n so_o ever_o be_v divide_v into_o two_o part_n have_v those_o his_o two_o part_n proportional_a to_o the_o two_o segment_n of_o a_o line_n divide_v by_o extreme_a and_o mean_a proportion_n be_v also_o itself_o divide_v by_o a_o extreme_a and_o mean_a proportion_n and_o those_o his_o two_o part_n be_v his_o two_o segment_n of_o the_o say_a proportion_n suppose_v ab_fw-la to_o be_v a_o line_n divide_v by_o a_o extreme_a and_o mean_a proportion_n in_o the_o point_n c_o and_o ac_fw-la to_o be_v the_o great_a segment_n suppose_v also_o the_o right_a line_n de_fw-fr to_o be_v divide_v into_o two_o part_n in_o the_o point_n f_o and_o that_o the_o part_n df_o be_v to_o fe_o as_o the_o segment_n ac_fw-la be_v to_o cb_o or_o df_o to_o be_v to_o ac_fw-la as_o fe_o be_v to_o cb._n for_o so_o these_o payte_n be_v proportional_a to_o the_o say_a segment_n i_o say_v now_o that_o de_fw-fr be_v also_o divide_v by_o a_o extreme_a and_o mean_a proportion_n in_o the_o point_n f._n and_o that_o df_o fe_o be_v his_o segment_n of_o the_o say_a proportion_n for_o see_v as_o ac_fw-la be_v to_o cb_o so_o be_v df_o to_o fe_o by_o supposition_n therefore_o as_o ac_fw-la and_o cb_o which_o be_v ab_fw-la be_v to_o cb_o so_o be_v df_o and_o fe_o which_o be_v de_fw-fr to_o fe_o by_o the_o 18._o of_o the_o five_o wherefore_o alternate_o as_o ab_fw-la be_v to_o de_fw-fr so_o be_v cb_o to_o fe_o and_o therefore_o the_o residue_n ac_fw-la be_v to_o the_o residue_n df_o as_o ab_fw-la be_v to_o de_fw-fr by_o the_o five_o of_o the_o five_o and_o then_o alternate_o ac_fw-la be_v to_o ab_fw-la as_o de_fw-fr be_v to_o df._n now_o therefore_o backward_o ab_fw-la be_v to_o ac_fw-la as_o de_fw-fr be_v to_o df._n but_o as_o ab_fw-la be_v to_o ac_fw-la so_o be_v ac_fw-la to_o cb_o by_o the_o three_o definition_n of_o the_o six_o book_n wherefore_o de_fw-fr be_v to_o df_o as_o ac_fw-la be_v to_o cb_o by_o the_o 11._o of_o the_o five_o and_o by_o supposition_n as_o ac_fw-la be_v to_o cb_o so_o be_v df_o to_o fe_o wherefore_o by_o the_o 11._o of_o the_o five_o as_z de_fw-fr be_v to_o df_o so_o be_v df_o to_o fe_o wherefore_o by_o the_o 3._o definition_n of_o the_o six_o de_fw-fr be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n in_o the_o point_n f._n wherefore_o df_o and_o fe_o be_v the_o segment_n of_o the_o say_a proportion_n therefore_o what_o right_a line_n so_o ever_o be_v divide_v into_o two_o part_n have_v those_o his_o two_o part_n proportional_a to_o the_o two_o segment_n of_o a_o line_n divide_v by_o extreme_a and_o mean_a proportioned_a be_v also_o itself_o divide_v by_o a_o extreme_a and_o mean_a proportion_n and_o those_o his_o two_o part_n be_v his_o two_o segment_n of_o the_o say_v proportion_v which_o be_v requisite_a to_o be_v demonstrate_v note_n many_o way_n these_o two_o theorem_n may_v be_v demonstrate_v which_o i_o leave_v to_o the_o exercise_n of_o young_a student_n but_o utter_o to_o want_v these_o two_o theorem_n and_o their_o demonstration_n in_o so_o principal_a a_o line_n or_o rather_o the_o chief_a pillar_n of_o euclides_n geometrical_a palace_n geometry_n be_v hitherto_o and_o so_o will_v remain_v a_o great_a disgrace_n also_o i_o think_v it_o good_a to_o note_v unto_o you_o what_o we_o mean_v by_o one_o only_a point_n we_o mâ—â—â—â—_n that_o the_o quantity_n of_o the_o two_o segment_n can_v not_o be_v alter_v the_o whole_a line_n be_v once_o give_v and_o though_o from_o either_o end_n of_o the_o whole_a line_n the_o great_a segment_n may_v begin_v poiâ—t_n and_o so_o as_o it_o be_v the_o point_n of_o section_n may_v seem_v to_o be_v alter_v yet_o with_o we_o that_o be_v no_o alteration_n forasmuch_o as_o the_o quantity_n of_o the_o segment_n remain_v all_o one_o i_o mean_v the_o quantity_n of_o the_o great_a segment_n be_v all_o one_o at_o which_o end_n so_o ever_o it_o be_v take_v and_o therefore_o likewise_o the_o quantity_n of_o the_o less_o segment_n be_v all_o one_o etc_n etc_n the_o like_a consideration_n may_v be_v have_v in_o euclides_n ten_o book_n in_o the_o binomial_a line_n etc_n etc_n joâ—n_n dee_n 1569._o decemb._n 18._o the_o 3._o theorem_a the_o 3._o proposition_n if_o a_o right_a line_n be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n and_o to_o the_o less_o segment_n be_v add_v the_o half_a of_o the_o gerater_n segment_n the_o square_n make_v of_o those_o two_o line_n add_v together_o be_v quintuple_a to_o the_o square_n make_v of_o the_o half_a line_n of_o the_o great_a segment_n svppose_v that_o the_o right_a line_n ab_fw-la be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n in_o the_o point_n c._n and_o let_v the_o great_a segment_n thereof_o be_v ac_fw-la and_o divide_v ac_fw-la into_o two_o equal_a part_n in_o the_o point_n d._n then_o i_o say_v that_o the_o square_a of_o the_o line_n bd_o it_o quintuple_a to_o the_o square_n of_o the_o line_n dc_o construction_n describe_v by_o the_o 46._o of_o the_o first_o upon_o the_o line_n ab_fw-la a_o square_a ae_n and_o describe_v and_o make_v perfect_a the_o figure_n that_o be_v divide_v the_o line_n at_o like_a unto_o the_o division_n of_o the_o line_n ab_fw-la by_o the_o 10._o of_o the_o six_o in_o the_o point_n r_o h_o by_o which_o point_n draw_v by_o the_o 31._o of_o the_o first_o unto_o the_o line_n ab_fw-la parallel_a line_n rm_n and_o hn._n so_o likewise_o draw_v by_o the_o point_n d_o c_o unto_o the_o line_n be_v these_o parallel_a line_n dl_o and_o c_n &_o draw_v the_o diameter_n bt_n demonstration_n and_o forasmuch_o as_o the_o line_n ac_fw-la be_v double_a to_o the_o line_n dc_o therefore_o the_o square_a of_o ac_fw-la be_v quadruple_a to_o the_o square_n of_o dc_o by_o the_o 20._o of_o the_o six_o that_o be_v the_o square_a r_n to_o the_o square_a fg_o and_o forasmuch_o as_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o be_v equal_a to_o the_o square_n of_o the_o line_n ac_fw-la and_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o be_v equal_a to_o the_o parallelogram_n ce_fw-fr &_o the_o square_a of_o the_o line_n ac_fw-la be_v the_o square_a r_n wherefore_o the_o parallelogram_n ce_fw-fr be_v equal_a to_o the_o square_a rs._n but_o the_o square_a r_n be_v quadruple_a to_o the_o square_a fg_o wherefore_o the_o parallelogram_n ce_fw-fr also_o be_v quadruple_a to_o the_o square_a fg._n again_o forasmuch_o as_o the_o line_n ad_fw-la be_v equal_a to_o the_o line_n dc_o therefore_o the_o line_n hk_o be_v equal_a the_o line_n kf_o wherefore_o also_o the_o square_a gf_o be_v equal_a to_o the_o square_a hl_o wherefore_o the_o line_n gk_o be_v equal_a to_o the_o line_n kl_o that_o be_v the_o line_n mn_v to_o the_o line_n ne_fw-fr wherefore_o the_o parallelogram_n mf_n be_v equal_a to_o the_o parallelogram_n fe_o but_o the_o parallelogram_n mf_n be_v equal_a to_o the_o parallelogram_n cg_o wherefore_o the_o parallelogram_n cg_o be_v also_o equal_a to_o the_o parallelogram_n fe_o put_v the_o parallelogram_n cn_fw-la common_a to_o they_o both_o after_o wherefore_o the_o gnomon_n xop_n be_v equal_a to_o the_o parallelogram_n ck_o but_o the_o parallelogram_n ce_fw-fr be_v prove_v to_o be_v quadruple_a to_o gâ—_n the_o square_a wherefore_o the_o gnomon_n xop_n be_v quadruple_a to_o the_o square_a gf_o wherefore_o the_o square_a dn_o be_v quintuple_a to_o the_o square_a fg_o and_o dn_o be_v the_o square_a of_o the_o line_n db_o and_o gf_o the_o square_a of_o the_o line_n dc_o wherefore_o the_o square_a of_o the_o line_n db_o be_v quintuple_a to_o the_o square_n of_o the_o line_n dc_o if_o therefore_o a_o right_a line_n be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n and_o to_o the_o less_o segment_n be_v add_v the_o half_a of_o the_o great_a segment_n the_o square_n make_v of_o those_o two_o line_n add_v together_o be_v quintuple_a to_o the_o square_n make_v of_o the_o half_a line_n of_o the_o great_a segment_n which_o be_v require_v to_o be_v demonstrate_v you_o shall_v find_v this_o proposition_n a_o other_o way_n demonstrate_v after_o the_o five_o proposition_n of_o this_o book_n here_o follow_v m._n dee_n his_o addition_n ¶_o a_o theorem_a 1._o if_o a_o right_a line_n give_v be_v quintuple_a in_o power_n to_o the_o power_n of_o a_o segment_n of_o himself_o the_o double_a of_o that_o segment_n former_a and_o the_o other_o part_n remain_v of_o the_o first_o give_v line_n make_v a_o line_n divide_v by_o extreme_a and_o mean_a proportion_n and_o that_o double_a of_o the_o segment_n be_v the_o great_a part_n thereof_o forasmuch_o as_o this_o be_v the_o converse_n of_o euclides_n
namely_o the_o part_n great_a than_o the_o whole_a which_o be_v impossible_a wherefore_o the_o circumference_n shall_v not_o cut_v the_o line_n cd_o now_o i_o say_v that_o it_o shall_v not_o pass_v above_o the_o line_n cd_o and_o not_o touch_v it_o in_o the_o point_n d._n for_o if_o it_o be_v possible_a let_v it_o pass_v above_o it_o and_o extend_v the_o line_n cd_o to_o the_o circumference_n and_o let_v it_o cut_v it_o in_o the_o point_n â—_o and_o draw_v the_o line_n fb_o and_o favorina_n and_o it_o shall_v follow_v as_o before_o that_o the_o line_n cd_o be_v great_a than_o the_o line_n cf_o which_o be_v impossible_a wherefore_o that_o be_v manifest_v which_o be_v require_v to_o be_v prove_v ¶_o second_v assumpt_n if_o there_o be_v a_o right_a angle_n unto_o which_o a_o base_a be_v subtend_v and_o if_o upon_o the_o same_o be_v describe_v a_o semicircle_n the_o circumference_n thereof_o shall_v pass_v by_o the_o point_n of_o the_o right_a angle_n the_o converse_n of_o this_o be_v add_v after_o the_o demonstration_n of_o the_o 31._o of_o the_o three_o out_o of_o pelitarius_n and_o these_o two_o assumpte_n of_o campane_n be_v necessary_a for_o the_o better_a understanding_n of_o the_o demonstration_n of_o the_o secoâ—d_a part_n of_o this_o 13._o proposition_n wherein_o be_v prove_v that_o the_o pyramid_n be_v contain_v in_o the_o sphere_n give_v ¶_o certain_a corollarye_n add_v by_o flussas_n first_o corollary_n the_o diameter_n of_o the_o sphere_n be_v in_o power_n quadruple_a sesquialtera_fw-la to_o the_o line_n which_o be_v draw_v from_o the_o centre_n to_o the_o circumference_n of_o the_o circle_n which_o contain_v the_o base_a of_o the_o pyramid_n for_o forasmuch_o as_o it_o have_v be_v prove_v that_o the_o diameter_n kl_o be_v in_o power_n sesquialter_fw-la to_o the_o side_n of_o and_o it_o be_v prove_v also_o by_o the_o 12._o of_o this_o book_n that_o the_o side_n of_o be_v in_o power_n triple_a to_o the_o line_n eh_o which_o be_v draw_v from_o the_o centre_n of_o the_o circle_n contain_v the_o triangle_n efg_o but_o the_o proportion_n of_o the_o extreme_n namely_o of_o the_o diameter_n to_o the_o line_n eh_o consist_v of_o the_o proportion_n of_o the_o mean_n namely_o of_o the_o proportion_n of_o the_o diameter_n to_o the_o line_n of_o and_o of_o the_o proportion_n of_o the_o line_n of_o to_o the_o line_n eh_o by_o the_o 5._o definition_n of_o the_o six_o which_o proportion_n namely_o triple_a and_o sesquialter_fw-la add_v together_o make_v quadruple_a sesquialter_fw-la as_o it_o be_v easy_a to_o prove_v by_o that_o which_o be_v teach_v in_o the_o declaration_n of_o the_o 5._o definition_n of_o the_o six_o book_n wherefore_o the_o corollary_n be_v manifest_a ¶_o second_v corollary_n only_o the_o line_n which_o be_v draw_v from_o the_o angle_n of_o the_o pyramid_n to_o the_o base_a opposite_a unto_o it_o campane_n &_o pass_v by_o the_o centre_n of_o the_o sphere_n be_v perpendicular_a to_o the_o base_a and_o fall_v upon_o the_o centre_n of_o the_o circle_n which_o contain_v the_o base_a for_o if_o any_o other_o line_n than_o the_o line_n kmh_o which_o be_v draw_v by_o the_o centre_n of_o the_o sphere_n to_o the_o centre_n of_o the_o circle_n shall_v fall_v perpendicular_o upon_o the_o plain_a of_o the_o base_a then_o from_o one_o and_o the_o self_n same_o point_n shall_v be_v draw_v to_o one_o and_o the_o self_n same_o plain_a two_o perpendicular_a line_n contrary_a to_o the_o 13._o of_o the_o eleven_o which_o be_v impossible_a far_o if_o from_o the_o top_n king_n shall_v be_v draw_v to_o the_o centre_n of_o the_o base_a namely_o to_o the_o point_n h_o any_o other_o right_a line_n not_o pass_v by_o the_o centre_n m_o two_o right_a life_n shall_v include_v a_o superficies_n contrary_n to_o the_o last_o common_a sentence_n which_o be_v absurd_a wherefore_o only_o the_o line_n which_o be_v draw_v by_o the_o centre_n of_o the_o sphere_n to_o the_o centre_n of_o the_o base_a be_v perpendicular_a to_o the_o say_v base_a and_o the_o line_n which_o be_v draw_v from_o the_o angle_n perpendicular_o to_o the_o base_a shall_v pass_v by_o the_o centre_n of_o the_o sphere_n three_o corollary_n the_o perpendicular_a line_n which_o be_v draw_v from_o the_o centre_n of_o the_o sphere_n to_o the_o base_a of_o the_o pyramid_n book_n be_v equal_a to_o the_o six_o part_n of_o the_o diameter_n of_o the_o sphere_n for_o it_o be_v before_o prove_v that_o the_o line_n mh_o which_o be_v draw_v from_o the_o centre_n of_o the_o sphere_n to_o the_o centre_n of_o the_o base_a be_v equal_a to_o the_o line_n nc_n which_o line_n nc_n be_v the_o six_o part_n of_o the_o diameter_n ab_fw-la and_o therefore_o the_o line_n mh_o be_v the_o six_o part_n of_o the_o diameter_n of_o the_o sphere_n for_o the_o diameter_n ab_fw-la be_v equal_a to_o the_o diameter_n of_o the_o sphere_n as_o have_v also_o before_o be_v prove_v ¶_o the_o 2._o problem_n the_o 14._o proposition_n to_o make_v a_o octohedron_n and_o to_o comprehend_v it_o in_o the_o sphere_n give_v namely_o that_o wherein_o the_o pyramid_n be_v comprehend_v and_o to_o prove_v that_o the_o diameter_n of_o the_o sphere_n be_v in_o power_n double_a to_o the_o side_n of_o the_o octohedron_n construction_n take_v the_o diameter_n of_o the_o former_a sphere_n give_v which_o let_v be_v the_o line_n ab_fw-la and_o divide_v it_o by_o the_o 10._o of_o the_o first_o into_o two_o equal_a part_n in_o the_o point_n c._n and_o describe_v upon_o the_o line_n ab_fw-la a_o semicircle_n adb_o and_o by_o the_o 11._o of_o the_o first_o from_o the_o point_n c_o raise_v up_o unto_o the_o line_n ab_fw-la a_o perpendicular_a line_n cd_o and_o draw_v a_o right_a line_n from_o the_o point_n d_o to_o the_o point_n b._n and_o describe_v a_o square_a efgh_o have_v every_o one_o of_o his_o side_n equal_a to_o the_o line_n bd._n and_o draw_v the_o diagonal_a line_n fh_a &_o eglantine_n cut_v the_o one_o the_o other_o in_o the_o point_n k._n and_o by_o the_o 12._o of_o the_o eleven_o from_o the_o point_n king_n namely_o the_o point_n where_o the_o line_n fh_o and_o eglantine_n cut_v the_o one_o the_o other_o raise_v up_o to_o the_o plain_a superficies_n wherein_o the_o square_a efgh_o be_v a_o perpendicular_a line_n kl_o and_o extend_v the_o line_n kl_o on_o the_o other_o side_n of_o the_o plain_a superficies_n to_o the_o point_n m._n and_o let_v each_o of_o the_o line_n kl_o and_o km_o be_v put_v equal_a to_o one_o of_o these_o line_n ke_o kf_o kh_o or_o kg_n and_o draw_v these_o right_a line_n le_fw-fr lf_n lg_n lh_o menander_n mf_n mg_o and_o mh_o demonstration_n now_o forasmuch_o as_o the_o line_n ke_o be_v by_o the_o corollary_n of_o the_o 34._o of_o the_o first_o equal_a to_o the_o line_n kh_o and_o the_o angle_n angle_n ekh_n be_v a_o right_a angle_n therefore_o the_o square_a of_o he_o be_v double_a to_o the_o square_n of_o eke_o by_o the_o 47._o of_o the_o first_o again_o forasmuch_o as_o the_o line_n lk_n be_v equal_a to_o the_o line_n ke_o by_o position_n and_o the_o angle_n lke_n be_v by_o the_o second_o definition_n of_o the_o eleven_o a_o right_a angle_n therefore_o the_o square_n of_o the_o line_n el_n be_v double_a to_o the_o square_n of_o the_o line_n eke_o and_o it_o be_v prove_v that_o the_o square_a of_o the_o line_n he_o be_v double_a to_o the_o square_n of_o the_o line_n eke_o wherefore_o the_o square_a of_o the_o line_n le_fw-fr be_v equal_a to_o the_o square_n of_o the_o line_n eh_o wherefore_o also_o the_o line_n le_fw-fr be_v equal_a to_o the_o line_n eh_o and_o by_o the_o same_o reason_n the_o line_n lh_o be_v also_o equal_a to_o the_o line_n he._n wherefore_o the_o triangle_n lhe_n be_v equilater_n in_o like_a sort_n may_v we_o prove_v that_o every_o one_o of_o the_o rest_n of_o the_o triangle_n who_o base_n be_v the_o side_n of_o the_o square_a efgh_o and_o top_n the_o point_n l_o and_o m_o be_v equilater_n the_o say_v eight_o triangle_n also_o be_v equal_a the_o one_o to_o the_o other_o for_o every_o side_n of_o each_o be_v equal_a to_o the_o side_n of_o the_o square_a efgh_o wherefore_o there_o be_v make_v a_o octohedron_n contain_v under_o eight_o triangle_n who_o side_n be_v equal_a now_o it_o be_v require_v to_o comprehend_v it_o in_o the_o sphere_n give_v and_o to_o prove_v that_o the_o diameter_n of_o the_o sphere_n be_v in_o power_n double_a to_o the_o side_n of_o the_o octohedron_n demonstration_n forasmuch_o as_o these_o three_o line_n lk_v km_o and_o ke_o be_v equal_a the_o one_o to_o the_o other_o therefore_o a_o semicircle_n describe_v upon_o the_o line_n lm_o shall_v pass_v also_o by_o the_o point_n e._n and_o by_o the_o same_o reason_n if_o the_o semicircle_n be_v turn_v round_o about_o until_o it_o return_v unto_o the_o self_n same_o
give_v wherein_o be_v contain_v the_o former_a solid_n and_o to_o prove_v that_o the_o side_n of_o the_o icosahedron_n be_v a_o irrational_a line_n of_o that_o kind_n which_o be_v call_v a_o less_o line_n demonstration_n now_o forasmuch_o as_o the_o line_n qp_v qv_n qt_n q_n and_o qr_o do_v each_o subtend_v right_a angle_n contain_v under_o the_o side_n of_o a_o equilater_n hexagon_n &_o of_o a_o equilater_n decagon_fw-mi inscribe_v in_o the_o circle_n prstv_n or_o in_o the_o circle_n efghk_n which_o two_a circle_n be_v equal_a therefore_o the_o say_a line_n be_v each_o equal_a to_o the_o side_n of_o the_o pentagon_n inscribe_v in_o the_o foresay_a circle_n by_o the_o 10._o of_o this_o book_n and_o be_v equal_a the_o one_o to_o the_o other_o by_o the_o 4._o of_o the_o first_o for_o all_o the_o angle_n at_o the_o point_n w_n which_o they_o subtend_v be_v right_a angle_n wherefore_o the_o five_o triangle_n qpv_o qpr_fw-la qr_n qst_fw-la and_o qtv_n which_o be_v contain_v under_o the_o say_a line_n qv_n qp_n qr_o q_n qt_n and_o under_o the_o side_n of_o the_o pentagon_n vpr_v be_v equilater_n and_o equal_a to_o the_o ten_o former_a triangle_n and_o by_o the_o same_o reason_n the_o five_o triangle_n opposite_a unto_o they_o namely_o the_o triangle_n yml_n ymn_n ynx_n yxo_n and_o yol_n be_v equilater_n and_o equal_a to_o the_o say_v ten_o triangle_n for_o the_o line_n ill_o ym_fw-fr yn_n yx_n and_o you_o do_v subtend_v right_a angle_n contain_v under_o the_o side_n of_o a_o equilater_n hexagon_n and_o of_o a_o equilater_n decagon_fw-mi inscribe_v in_o the_o circle_n efghk_n which_o be_v equal_a to_o the_o circle_n prstv_n wherefore_o there_o be_v describe_v a_o solid_a contain_v under_o 20._o equilater_n triangle_n wherefore_o by_o the_o last_o definition_n of_o the_o eleven_o there_o be_v describe_v a_o icosahedron_n now_o it_o be_v require_v to_o comprehend_v it_o in_o the_o sphere_n give_v and_o to_o prove_v that_o the_o side_n of_o the_o icosahedron_n be_v a_o irrational_a line_n of_o that_o kind_n which_o be_v call_v a_o less_o line_n forasmuch_o as_o the_o line_n zw_n be_v the_o side_n of_o a_o hexagon_n &_o the_o line_n wq_n be_v the_o side_n of_o a_o decagon_n therefore_o the_o line_n zq_n be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n in_o the_o point_n w_n and_o his_o great_a segment_n be_v zw_n by_o the_o 9_o of_o the_o thirteen_o wherefore_o as_o the_o line_n qz_n be_v to_o the_o line_n zw_n so_o be_v the_o line_n zw_n to_o the_o line_n wq_n but_o the_o zw_n be_v equal_a to_o the_o line_n zl_n by_o construction_n and_o the_o line_n wq_n to_o the_o line_n zy_n by_o construction_n also_o wherefore_o as_o the_o line_n qz_n be_v to_o the_o line_n zl_n so_o be_v the_o line_n zl_n to_o the_o line_n zy_n and_o the_o angle_n qzlâ—_n and_o lzy_n be_v right_a angle_n by_o the_o 2._o definition_n of_o the_o eleven_o if_o therefore_o we_o draw_v a_o right_a line_n from_o the_o point_n l_o to_o the_o point_n qui_fw-fr the_o angle_n ylq_n shall_v be_v a_o right_a angle_n by_o reason_n of_o the_o likeness_n of_o the_o triangle_n ylq_n and_o zlq_n by_o the_o 8._o of_o the_o six_o wherefore_o a_o semicircle_n describe_v upon_o the_o line_n qy_n shall_v pass_v also_o by_o the_o point_n l_o by_o the_o assumpt_n add_v by_o campane_n after_o the_o 13._o of_o this_o book_n and_o by_o the_o same_o reason_n also_o for_o that_o as_o the_o line_n qz_n be_v the_o line_n zw_n so_o be_v the_o line_n zw_n to_o the_o line_n wq_n flussas_n but_o the_o line_n zq_n be_v equal_a to_o the_o line_n yw_n and_o the_o line_n zw_n to_o the_o line_n pw_o wherefore_o as_o the_o line_n yw_n be_v to_o the_o line_n wp_n so_o be_v the_o line_n pw_o to_o the_o line_n wq_n and_o therefore_o again_o if_o we_o draw_v a_o right_a line_n from_o the_o point_n p_o to_o the_o point_n y_fw-fr the_o angle_n ypq_n shall_v be_v a_o right_a angle_n wherefore_o a_o semicircle_n describe_v upon_o the_o line_n qy_n shall_v pass_v also_o by_o the_o point_n p_o by_o the_o former_a assumpt_n &_o if_o the_o diameter_n qy_o abide_v fix_v the_o semicircle_n be_v turn_v round_o about_o until_o it_o come_v to_o the_o self_n same_o place_n from_o whence_o it_o begin_v first_o to_o be_v move_v it_o shall_v pass_v both_o by_o the_o point_n p_o and_o also_o by_o the_o rest_n of_o the_o point_n of_o the_o angle_n of_o the_o icosahedron_n and_o the_o icosahedron_n shall_v be_v comprehend_v in_o a_o sphere_n i_o say_v also_o that_o it_o be_v contain_v in_o the_o sphere_n give_v divide_v by_o the_o 10._o of_o the_o first_o the_o line_n zw_n into_o two_o equal_a part_n in_o the_o point_n a._n and_o forasmuch_o as_o the_o right_a line_n zq_n be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n in_o the_o point_n w_n and_o his_o less_o segment_n be_v qw_o therefore_o the_o segment_n qw_o have_v add_v unto_o it_o the_o half_a of_o the_o great_a segment_n namely_o the_o line_n wa_n be_v by_o the_o 3._o of_o this_o book_n in_o power_n quintuple_a to_o the_o square_n make_v of_o the_o half_a of_o the_o great_a segment_n wherefore_o the_o square_a of_o the_o line_n qa_n be_v quintuple_a to_o the_o square_n of_o the_o line_n â—w_o but_o unto_o the_o square_n of_o the_o qa_n the_o square_a of_o the_o line_n qy_n be_v quadruple_a by_o the_o corollary_n of_o the_o 20._o of_o the_o six_o for_o the_o line_n qy_n be_v double_a to_o the_o line_n qa_n and_o by_o the_o same_o reason_n unto_o the_o square_n of_o the_o wa_n the_o square_a of_o the_o line_n zw_n be_v quadruple_a wherefore_o the_o square_a of_o the_o line_n qy_n be_v quintuple_a to_o the_o square_n of_o the_o line_n zw_n by_o the_o 15._o of_o the_o five_o and_o forasmuch_o as_o the_o line_n ac_fw-la be_v quadruple_a to_o the_o line_n cb_o therefore_o the_o line_n ab_fw-la be_v quintuple_a to_o the_o line_n cb._n but_o as_o the_o line_n ab_fw-la be_v to_o the_o line_n bc_o so_o be_v the_o square_a of_o the_o line_n ab_fw-la to_o the_o square_n of_o the_o line_n bd_o by_o the_o 8_o of_o the_o six_o and_o corollary_n of_o the_o 20._o of_o the_o same_o wherefore_o the_o square_a of_o the_o line_n ab_fw-la be_v quintuple_a to_o the_o square_n of_o the_o line_n bd_o and_o it_o be_v be_v prove_v that_o the_o square_a of_o the_o line_n qy_n be_v quintuple_a to_o the_o square_n of_o the_o line_n zw_n and_o the_o line_n bd_o be_v equal_a to_o the_o line_n zw_n for_o either_o of_o they_o be_v by_o position_n equal_a to_o the_o line_n which_o be_v draw_v from_o the_o centre_n of_o the_o circle_n efghk_v to_o the_o circumference_n wherefore_o the_o line_n ab_fw-la be_v equal_a to_o the_o yq_n but_o the_o line_n ab_fw-la be_v the_o diameter_n of_o the_o sphere_n give_v wherefore_o the_o line_n yq_n which_o be_v prove_v to_o be_v the_o diameter_n of_o the_o sphere_n contain_v the_o icosahedron_n be_v equal_a to_o the_o diameter_n of_o the_o sphere_n give_v wherefore_o the_o icosahedron_n be_v contain_v in_o the_o sphere_n give_v now_o i_o say_v that_o the_o side_n of_o the_o icosahedron_n be_v a_o irrational_a line_n of_o that_o kind_n which_o be_v call_v a_o less_o line_n for_o forasmuch_o as_o the_o diameter_n of_o the_o sphere_n be_v rational_a and_o be_v in_o power_n quintuple_a to_o the_o square_n of_o the_o line_n draw_v from_o the_o centre_n of_o the_o circle_n olmnx_n wherefore_o also_o the_o line_n which_o be_v draw_v from_o the_o centre_n of_o the_o circle_n olmnx_n be_v rational_a wherefore_o the_o diameter_n also_o be_v commensurable_a to_o the_o same_o line_n by_o the_o 6._o of_o the_o ten_o be_v rational_a but_o if_o in_o a_o circle_n have_v a_o rational_a line_n to_o his_o diameter_n be_v describe_v a_o equilater_n pentagon_n the_o side_n of_o the_o pentagon_n be_v by_o the_o 11._o of_o this_o book_n a_o irrational_a line_n of_o that_o kind_n which_o be_v call_v a_o less_o line_n but_o the_o side_n of_o the_o pentagon_n olmnx_n be_v also_o the_o side_n of_o the_o icosahedron_n describe_v as_o have_v before_o be_v prove_v wherefore_o the_o side_n of_o the_o icosahedron_n be_v a_o irrational_a line_n of_o that_o kind_n which_o be_v call_v a_o less_o line_n wherefore_o there_o be_v describe_v a_o icosahedron_n and_o it_o be_v contain_v in_o the_o sphere_n give_v and_o it_o be_v prove_v that_o the_o side_n of_o thou_o icosahedron_n be_v a_o irrational_a line_n of_o that_o kind_n which_o be_v call_v a_o less_o line_n which_o be_v require_v to_o be_v do_v and_o to_o be_v prove_v a_o corollary_n hereby_o it_o be_v manifest_a that_o the_o diameter_n of_o the_o sphere_n be_v in_o power_n quintuple_a to_o the_o line_n which_o be_v draw_v from_o the_o centre_n of_o the_o circle_n to_o
the_o circumference_n on_o which_o the_o icosahedron_n be_v describe_v and_o that_o the_o diameter_n of_o the_o sphere_n be_v compose_v of_o the_o side_n of_o a_o hexagon_n and_o of_o two_o side_n of_o a_o decagon_n describe_v in_o one_o and_o the_o self_n same_o circle_n flussas_n prove_v the_o icosahedron_n describe_v to_o be_v contain_v in_o a_o sphere_n by_o draw_v right_a line_n from_o the_o point_n a_o to_o the_o point_n p_o and_o g_o after_o this_o manner_n forasmuch_o as_o the_o line_n zw_n wp_n be_v put_v equal_a to_o the_o line_n draw_v from_o the_o centre_n to_o the_o circumferenceâ—_n and_o the_o line_n draw_v from_o the_o centre_n to_o the_o circumference_n be_v double_a to_o the_o line_n awe_n by_o construction_n therefore_o the_o line_n wp_n be_v also_o double_a to_o the_o same_o line_n awe_n wherefore_o the_o square_a of_o the_o line_n wp_n be_v quadruple_a to_o the_o square_n of_o the_o line_n awe_n by_o the_o corollary_n of_o the_o 20._o of_o the_o six_o and_o those_o line_n pw_o and_o wa_n contain_v a_o right_a angle_n pwa_n as_o have_v before_o be_v prove_v be_v subtend_v of_o the_o right_a line_n ap_fw-mi wherefore_o by_o the_o 47._o of_o the_o first_o the_o line_n ap_fw-mi contain_v in_o power_n the_o line_n pw_o and_o wa._n wherefore_o the_o right_a line_n ap_fw-mi be_v in_o power_n quintuple_a to_o the_o line_n wa._n wherefore_o the_o right_a line_n ap_fw-mi and_o aq_n be_v quintuple_a to_o one_o and_o the_o same_o line_n wa_n be_v by_o the_o 9_o of_o the_o flu_v equal_a in_o like_a sort_n also_o may_v we_o prove_v that_o unto_o those_o line_n ap_fw-mi and_o aq_n be_v equal_a the_o rest_n of_o the_o line_n draw_v from_o the_o point_n a_o to_o the_o rest_n of_o the_o angle_n r_o s_o t_o v._o for_o they_o subtend_v right_a angle_n contain_v of_o the_o line_n wa_n and_o of_o the_o line_n draw_v from_o the_o centre_n to_o the_o circumference_n and_o forasmuch_o as_o unto_o the_o line_n wa_n be_v equal_a the_o line_n valerio_n which_o be_v likewise_o erect_v perpendicular_o unto_o the_o other_o plain_a superficies_n olmnx_n therefore_o line_n draw_v from_o the_o point_n a_o to_o the_o angle_n oh_o l_o m_o n_o x_o and_o subtend_v right_a angle_n at_o the_o point_n z_o contain_v under_o line_n draw_v from_o the_o centre_n to_o the_o circumference_n and_o under_o the_o line_n az_n be_v equal_a not_o only_o the_o one_o to_o the_o other_o but_o also_o to_o the_o line_n draw_v from_o the_o say_a point_n a_o to_o the_o former_a angle_n at_o the_o point_n p_o r_o s_o t_o vâ—_n for_o the_o line_n draw_v from_o the_o centre_n to_o the_o circumference_n of_o each_o circle_n be_v equal_a &_o the_o line_n awe_n be_v equal_a to_o the_o line_n az._n but_o the_o line_n ap_fw-mi be_v prove_v equal_a to_o the_o line_n aq_n which_o be_v the_o half_a of_o the_o whole_a qy_n therefore_o the_o residue_n ay_o be_v equal_a to_o the_o foresay_a line_n ap_fw-mi aq_n etc_n etc_n wherefore_o make_v the_o centre_n the_o point_n a_o and_o the_o space_n one_o of_o those_o line_n aq_n ap_fw-mi etc_n etc_n extend_v the_o superficies_n of_o a_o sphere_n &_o it_o shall_v touch_v the_o 12._o angle_n of_o the_o icosahedron_n which_o be_v at_o the_o point_n oh_o l_o m._n n_o x_o pea_n king_n s_o t_o five_o q_o y_fw-fr which_o sphere_n be_v describe_v if_o upon_o the_o diameter_n qy_n be_v draw_v a_o semicircle_n and_o the_o say_a semicircle_n be_v move_v about_o till_o it_o return_v unto_o the_o same_o place_n from_o whence_o it_o begin_v first_o to_o be_v move_v ¶_o a_o corollary_n add_v by_o flussas_n the_o opposite_a side_n of_o a_o icosahedron_n be_v parallel_n for_o the_o diameter_n of_o the_o sphere_n do_v fall_v upon_o the_o opposite_a angle_n of_o the_o icosahedron_n as_o it_o be_v manifest_a by_o the_o right_a line_n qy_n if_o therefore_o there_o be_v imagine_v to_o be_v draw_v the_o two_o diameter_n pn_n and_o om_o they_o shall_v concur_v in_o the_o point_n f_o wherefore_o the_o right_a line_n which_o join_v they_o together_o pv_n and_o ln_o be_v in_o one_o and_o the_o self_n same_o plain_a superficies_n by_o the_o 2._o of_o the_o eleven_o and_o forasmuch_o as_o the_o alternate_a angle_n at_o the_o end_n of_o the_o diameter_n be_v equal_a by_o the_o 8._o of_o the_o first_o for_o the_o triangle_n contain_v under_o equal_a semidiameteâ—s_n and_o the_o side_n of_o the_o icosahedron_n be_v equiangle_n therefore_o by_o the_o 28._o of_o the_o first_o the_o line_n pv_n and_o ln_o be_v paralle_n ¶_o the_o 5._o problem_n the_o 17._o proposition_n to_o make_v a_o dodecahedron_n and_o to_o comprehend_v it_o in_o the_o sphere_n give_v wherein_o be_v comprehend_v the_o foresay_a solid_n and_o to_o prove_v that_o the_o side_n of_o the_o dodecahedron_n be_v a_o irrational_a line_n of_o that_o kind_n which_o be_v call_v a_o residual_a line_n superficies_n now_o i_o say_v that_o it_o be_v in_o one_o and_o the_o self_n same_o plain_a superficies_n forasmuch_o as_o the_o line_n zv_n be_v a_o parallel_n to_o the_o line_n sir_n as_o be_v before_o prove_v but_o unto_o the_o same_o line_n sir_n be_v the_o line_n cb_o a_o parallel_n by_o the_o 28._o of_o the_o first_o wherefore_o by_o the_o 9_o of_o the_o eleven_o the_o line_n we_o be_v a_o parallel_n to_o the_o line_n cb._n wherefore_o by_o the_o seven_o of_o the_o eleven_o the_o right_a line_n which_o join_v they_o together_o be_v in_o the_o self_n same_o plain_n wherein_o be_v the_o parallel_a line_n wherefore_o the_o trapesium_n buzc_n be_v in_o one_o plain_n and_o the_o triangle_n bwc_n be_v in_o one_o plain_n by_o the_o 2._o of_o the_o eleven_o now_o to_o prove_v that_o the_o trapesium_n buzc_n &_o the_o triangle_n bwc_n be_v in_o one_o and_o the_o self_n same_o plain_a we_o must_v prove_v that_o the_o right_a line_n yh_n and_o hw_o be_v make_v direct_o one_o right_a line_n which_o thing_n be_v thus_o prove_v forasmuch_o as_o the_o line_n hp_n be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n in_o the_o point_n t_o and_o his_o great_a segment_n be_v the_o line_n pt_v therefore_o as_o the_o line_n hp_n be_v to_o the_o line_n pt_v so_o be_v the_o line_n pt_v to_o the_o line_n th._n but_o the_o line_n hp_n be_v equal_a to_o the_o line_n ho_o and_o the_o line_n pt_v to_o either_o of_o these_o line_n tw_n and_o oy_o wherefore_o as_o the_o line_n ho_o be_v to_o the_o line_n oy_fw-it so_o be_v the_o line_n with_o to_o the_o line_n th._n but_o the_o line_n ho_o and_o tw_n be_v side_n of_o like_a proportion_n be_v parallel_n by_o the_o 6._o of_o the_o eleven_o for_o either_o of_o they_o be_v erect_v perpendicular_o to_o the_o plain_a superficies_n bd_o â—_o and_o the_o line_n th._n and_o oy_o be_v parallel_n which_o be_v also_o side_n of_o like_a proportion_n by_o the_o same_o 6._o of_o the_o eleven_o for_o either_o of_o they_o be_v also_o erect_v perpendicular_o to_o the_o plain_a superficies_n bf_o but_o when_o there_o be_v two_o triangle_n have_v two_o side_n proportional_a to_o two_o side_n so_o set_v upon_o one_o angle_n that_o their_o side_n of_o like_a proportion_n be_v also_o parallel_n as_o the_o triangle_n yoh_n and_o htw_n be_v â—_o who_o two_o side_n oh_o &_o ht_o be_v in_o the_o two_o base_n of_o the_o cube_fw-la make_v a_o angle_n at_o the_o point_n h_o the_o side_n remain_v of_o those_o triangle_n shall_v by_o the_o 32._o of_o the_o six_o be_v in_o one_o right_a line_n wherefore_o the_o line_n yh_n &_o hw_o make_v both_o one_o right_a line_n but_o every_o right_a line_n be_v by_o the_o 3._o of_o the_o eleven_o in_o one_o &_o the_o self_n same_o plain_a superficies_n wherefore_o if_o you_o draw_v a_o right_a line_n from_o b_o to_o a'n_z there_o shall_v be_v make_v a_o triangle_n bye_o which_o shall_v be_v in_o one_o and_o the_o self_n same_o plain_a by_o the_o 2._o of_o the_o eleven_o and_o therefore_o the_o whole_a pentagon_n figure_n vbwcz_n be_v in_o one_o and_o the_o self_n same_o plain_a superficies_n now_o also_o i_o say_v that_o it_o be_v equiangle_n equiangle_n for_o forasmuch_o as_o the_o right_a line_n no_o be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n in_o the_o point_n r_o and_o his_o great_a segment_n be_v or_o therefore_o as_o both_o the_o line_n no_o and_o or_o add_v together_o be_v to_o the_o line_n on_o so_o by_o the_o 5._o of_o this_o book_n be_v the_o line_n on_o to_o the_o line_n or_o but_o the_o line_n or_o be_v equal_a to_o the_o line_n os_fw-mi wherefore_o as_o the_o line_n sn_a be_v to_o the_o line_n no_o so_o be_v the_o line_n no_o to_o the_o line_n os_fw-mi wherefore_o the_o line_n sn_a be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n in_o the_o point_n oh_o
double_a to_o the_o side_n of_o the_o octohedron_n &_o the_o side_n be_v in_o power_n sequitertia_fw-la to_o the_o perpendiclar_a line_n by_o the_o 12._o of_o this_o book_n wherefore_o the_o diameter_n thereof_o be_v in_o power_n duple_fw-fr superbipartiens_fw-la tertias_fw-la to_o the_o perpendicular_a line_n wherefore_o also_o the_o diameter_n and_o the_o perpendicular_a line_n be_v rational_a and_o commensurable_a by_o the_o 6._o of_o the_o ten_o as_o touch_v a_o icosahedron_n it_o be_v prove_v in_o the_o 16._o of_o this_o book_n that_o the_o side_n thereof_o be_v a_o less_o line_n when_o the_o diameter_n of_o the_o sphere_n be_v rational_a and_o forasmuch_o as_o the_o angle_n of_o the_o inclination_n of_o the_o base_n thereof_o be_v contain_v of_o the_o perpendicular_a line_n of_o the_o triangle_n and_o subtend_v of_o the_o right_a line_n which_o subtend_v the_o angle_n of_o the_o pentagon_n which_o contain_v five_o side_n of_o the_o icosahedron_n and_o unto_o the_o perpendicular_a line_n the_o side_n be_v commensurable_a namely_o be_v in_o power_n sesquitertia_fw-la unto_o they_o by_o the_o corollary_n of_o the_o 12._o of_o this_o book_n therefore_o the_o perpendicular_a line_n which_o contain_v the_o angle_n be_v irrational_a line_n namely_o less_o line_n by_o the_o 105._o of_o the_o ten_o book_n and_o forasmuch_o as_o the_o diameter_n contain_v in_o power_n both_o the_o side_n of_o the_o icosahedron_n and_o the_o line_n which_o subtend_v the_o foresay_a angle_n if_o from_o the_o power_n of_o the_o diameter_n which_o be_v rational_a be_v take_v away_o the_o power_n of_o the_o side_n of_o the_o icosahedron_n which_o be_v irrational_a it_o be_v manifest_a that_o the_o residue_n which_o be_v the_o power_n of_o the_o subtend_a line_n shall_v be_v irrational_a for_o if_o it_o shall_v be_v rational_a the_o number_n which_o measure_v the_o whole_a power_n of_o the_o diameter_n and_o the_o part_n take_v away_o of_o the_o subtend_a line_n shall_v also_o by_o the_o 4._o common_a sentence_n of_o the_o seven_o measure_n the_o residue_n namely_o the_o power_n of_o the_o side_n irrational_a which_o be_v irrational_a for_o that_o it_o be_v a_o less_o line_n which_o be_v absurd_a wherefore_o it_o be_v manifest_a that_o the_o right_a line_n which_o compose_v the_o angle_n of_o the_o inclination_n of_o the_o base_n of_o the_o icosahedron_n be_v irrational_a line_n for_o the_o subtend_a line_n have_v to_o the_o line_n contain_v a_o great_a proportion_n than_o the_o whole_a have_v to_o the_o great_a segment_n the_o angle_n of_o the_o inclination_n of_o the_o base_n of_o a_o dodecahedron_n be_v contain_v under_o two_o perpendicular_n of_o the_o base_n of_o the_o dodecahedron_n and_o be_v subtend_v of_o that_o right_a line_n who_o great_a segment_n be_v the_o side_n of_o a_o cube_n inscribe_v in_o the_o dodecahedron_n which_o right_a line_n be_v equal_a to_o the_o line_n which_o couple_v the_o section_n into_o two_o equal_a part_n of_o the_o opposite_a side_n of_o the_o dodecahedron_n and_o this_o couple_a line_n we_o say_v be_v a_o irrational_a line_n for_o that_o the_o diameter_n of_o the_o sphere_n contain_v in_o power_n both_o the_o couple_a line_n and_o the_o side_n of_o the_o dodecahedron_n but_o the_o side_n of_o the_o dodecahedron_n be_v a_o irrational_a line_n namely_o a_o residual_a line_n by_o the_o 17._o of_o this_o book_n wherefore_o the_o residue_n namely_o the_o couple_a line_n be_v a_o irrational_a line_n as_o it_o be_v â—asy_a to_o prove_v by_o the_o 4._o common_a sentence_n of_o the_o seven_o and_o that_o the_o perpendicular_a line_n which_o contain_v the_o angle_n of_o the_o inclination_n be_v irrational_a be_v thus_o prove_v by_o the_o proportion_n of_o the_o subtend_a line_n of_o the_o foresay_a angle_n of_o inclination_n to_o the_o line_n which_o contain_v the_o angle_n be_v find_v out_o the_o obliquity_n of_o the_o angle_n angle_n for_o if_o the_o subtend_a line_n be_v in_o power_n double_a to_o the_o line_n which_o contain_v the_o angle_n then_o be_v the_o angle_n a_o right_a angle_n by_o the_o 48._o of_o the_o first_o but_o if_o it_o be_v in_o power_n less_o than_o the_o double_a it_o be_v a_o acute_a angle_n by_o the_o 23._o of_o the_o second_o but_o if_o it_o be_v in_o power_n more_o than_o the_o double_a or_o have_v a_o great_a proportion_n than_o the_o whole_a have_v to_o the_o great_a segmentâ—_n the_o angle_n shall_v be_v a_o obtuse_a angle_n by_o the_o 12._o of_o the_o second_o and_o 4._o of_o the_o thirteen_o by_o which_o may_v be_v prove_v that_o the_o square_a of_o the_o whole_a be_v great_a than_o the_o double_a of_o the_o square_n of_o the_o great_a segment_n this_o be_v to_o be_v note_v that_o that_o which_o flussas_n have_v here_o teach_v touch_v the_o inclination_n of_o the_o base_n of_o the_o â—ive_a regular_a body_n hypsicles_n teach_v after_o the_o 5_o proposition_n of_o the_o 15._o book_n where_o he_o confess_v that_o he_o receive_v it_o of_o one_o isidorus_n and_o seek_v to_o make_v the_o matter_n more_o clear_a he_o endeavour_v himself_o to_o declare_v that_o the_o angle_n of_o the_o inclination_n of_o the_o solid_n be_v give_v and_o that_o they_o be_v either_o acute_a or_o obtuse_a according_a to_o the_o nature_n of_o the_o solid_a although_o euclid_v in_o all_o his_o 15._o book_n have_v not_o yet_o show_v what_o a_o thing_n give_v be_v wherefore_o flussas_n frame_v his_o demonstration_n upon_o a_o other_o ground_n proceed_v after_o a_o other_o manner_n which_o seem_v more_o plain_a and_o more_o apt_o hereto_o be_v place_v then_o there_o albeit_o the_o reader_n in_o that_o place_n shall_v not_o be_v frustrate_a of_o his_o also_o the_o end_n of_o the_o thirteen_o book_n of_o euclides_n element_n ¶_o the_o fourteen_o book_n of_o euclides_n element_n in_o this_o book_n which_o be_v common_o account_v the_o 14._o book_n of_o euclid_n be_v more_o at_o large_a entreat_v of_o our_o principal_a purpose_n book_n namely_o of_o the_o comparison_n and_o proportion_n of_o the_o five_o regular_a body_n customable_o call_v the_o 5._o figure_n or_o form_n of_o pythagoras_n the_o one_o to_o the_o other_o and_o also_o of_o their_o side_n together_o each_o to_o other_o which_o thing_n be_v of_o most_o secret_a use_n and_o inestimable_a pleasure_n and_o commodity_n to_o such_o as_o diligent_o search_v for_o they_o and_o attain_v unto_o they_o which_o thing_n also_o undoubted_o for_o the_o worthiness_n and_o hardness_n thereof_o for_o thing_n of_o most_o price_n be_v most_o hard_a be_v first_o search_v and_o find_v out_o of_o philosopher_n not_o of_o the_o inferior_a or_o mean_a sort_n but_o of_o the_o deep_a and_o most_o ground_a philosopher_n and_o best_a exercise_v in_o geometry_n and_o albeit_o this_o book_n with_o the_o book_n follow_v namely_o the_o 15._o book_n have_v be_v hitherto_o of_o all_o man_n for_o the_o most_o part_n and_o be_v also_o at_o this_o day_n number_v and_o account_v among_o euclides_n book_n and_o suppose_v to_o be_v two_o of_o he_o namely_o the_o 14._o and_o 15._o in_o order_n as_o all_o exemplar_n not_o only_o new_a and_o late_o set_v abroad_o but_o also_o old_a monument_n write_v by_o hand_n do_v manifest_o witness_v yet_o it_o be_v think_v by_o the_o best_a learned_a in_o these_o day_n that_o these_o two_o book_n be_v none_o of_o euclides_n but_o of_o some_o other_o author_n no_o less_o worthy_a nor_o of_o less_o estimation_n and_o authority_n notwithstanding_o than_o euclid_n apollonius_n a_o man_n of_o deep_a knowledge_n a_o great_a philosopher_n and_o in_o geometry_n marvellous_a who_o wonderful_a book_n write_v of_o the_o section_n of_o cones_fw-la which_o exercise_n &_o occupy_v thewitte_v of_o the_o wise_a and_o best_a learned_a be_v yet_o remain_v be_v think_v and_o that_o not_o without_o just_a cause_n to_o be_v the_o author_n of_o they_o or_o as_o some_o think_v hypsicles_n himself_o for_o what_o can_v be_v more_o plain_o then_o that_o which_o he_o himself_o witness_v in_o the_o preface_n of_o this_o book_n basilides_n of_o tire_n say_v hypsicles_n and_o my_o father_n together_o scan_v and_o peyse_v a_o writing_n or_o book_n of_o apollonius_n which_o be_v of_o the_o comparison_n of_o a_o dodecahedron_n to_o a_o icosahedron_n inscribe_v in_o one_o and_o the_o self_n same_o sphere_n and_o what_o proportion_n these_o figure_n have_v the_o one_o to_o the_o other_o find_v that_o apollonius_n have_v fail_v in_o this_o matter_n but_o afterward_o say_v he_o i_o find_v a_o other_o copy_n or_o book_n of_o apollonius_n wherein_o the_o demonstration_n of_o that_o matter_n be_v full_a and_o perfect_a and_o show_v it_o unto_o they_o whereat_o they_o much_o rejoice_v by_o which_o word_n it_o seem_v to_o be_v manifest_a that_o apollonius_n be_v the_o first_o author_n of_o this_o book_n which_o be_v afterward_o set_v forth_o by_o hypsicles_n for_o so_o his_o own_o word_n after_o in_o
the_o same_o preface_n seem_v to_o import_v the_o preface_n of_o hypsicles_n before_o the_o fourteen_o book_n friend_n protarchus_n when_o that_o basilides_n of_o tire_n come_v into_o alexandria_n have_v familiar_a friendship_n with_o my_o father_n by_o reason_n of_o his_o knowledge_n in_o the_o mathematical_a science_n he_o remain_v with_o he_o a_o long_a time_n yea_o even_o all_o the_o time_n of_o the_o pestilence_n and_o sometime_o reason_v between_o themselves_o of_o that_o which_o apollonius_n have_v write_v touch_v the_o comparison_n of_o a_o dodecahedron_n and_o of_o a_o icosahedron_n inscribe_v in_o one_o and_o the_o self_n same_o sphere_n what_o proportion_n such_o body_n have_v the_o one_o to_o the_o other_o they_o judge_v that_o apollonius_n have_v somewhat_o err_v therein_o wherefore_o they_o as_o my_o father_n declare_v unto_o i_o diligent_o weigh_v it_o write_v it_o perfect_o howbeit_o afterward_o i_o happen_v to_o find_v a_o other_o book_n write_v of_o apollonius_n which_o contain_v in_o it_o the_o right_a demonstration_n of_o that_o which_o they_o seek_v for_o which_o when_o they_o see_v they_o much_o rejoice_v as_o for_o that_o which_o apollonius_n write_v may_v be_v see_v of_o all_o man_n for_o that_o it_o be_v in_o â—uery_n man_n hand_n and_o that_o which_o be_v of_o we_o more_o diligent_o afterward_o write_v again_o i_o think_v good_a to_o send_v and_o dedicate_v unto_o you_o as_z to_o one_o who_o i_o think_v worthy_a commendation_n both_o for_o that_o deep_a knowledge_n which_o i_o know_v you_o have_v in_o all_o kind_n of_o learning_n and_o chief_o in_o geometry_n so_o that_o you_o be_v able_a ready_o to_o judge_v of_o those_o thing_n which_o be_v speak_v and_o also_o for_o the_o great_a love_n and_o good_a will_n which_o you_o bear_v towards_o my_o father_n and_o i_o wherefore_o vouchsafe_v gentle_o to_o accept_v this_o which_o i_o send_v unto_o you_o but_o now_o be_v it_o time_n to_o end_v our_o preface_n and_o to_o begin_v the_o matter_n ¶_o the_o 1._o theorem_a the_o 1._o proposition_n flussas_n a_o perpendicular_a line_n draw_v from_o the_o centre_n of_o a_o circle_n to_o the_o side_n of_o a_o pentagon_n describe_v in_o the_o same_o circle_n be_v the_o half_a of_o these_o two_o line_n namely_o of_o the_o side_n of_o a_o hexagon_n figure_n and_o of_o the_o side_n of_o a_o decagon_n figure_n be_v both_o describe_v in_o the_o self_n same_o circle_n svppose_v that_o the_o circle_n be_v abc_n construction_n and_o let_v the_o side_n of_o a_o equilater_n pentagon_n describe_v in_o the_o circle_n abc_n be_v bc._n and_o by_o the_o 1._o of_o the_o three_o take_v the_o centre_n of_o the_o circle_n and_o let_v the_o same_o be_v d._n and_o by_o the_o 12._o of_o the_o first_o from_o the_o point_n d_o draw_v unto_o the_o line_n bc_o a_o perpendicular_a line_n de._n and_o extend_v the_o right_a line_n de_fw-fr direct_o to_o the_o point_n f._n then_o i_o say_v that_o the_o line_n de_fw-fr which_o be_v draw_v from_o the_o centre_n to_o bc_o the_o side_n of_o the_o pentagon_n be_v the_o half_a of_o the_o side_n of_o a_o hexagon_n and_o of_o a_o decagon_n take_v together_o and_o describe_v in_o the_o same_o circle_n draw_v these_o right_a line_n dc_o and_o cf._n and_o unto_o the_o line_n of_o put_v a_o equal_a line_n ge._n and_o draw_v a_o right_a line_n from_o the_o point_n g_o to_o the_o point_n c._n demonstration_n now_o forasmuch_o as_o the_o circumference_n of_o the_o whole_a circle_n be_v quintuple_a to_o the_o circumference_n bfc_n which_o be_v subtend_v of_o the_o side_n of_o the_o pentagon_n and_o the_o circumference_n acf_n be_v the_o half_a of_o the_o circumference_n of_o the_o whole_a circle_n and_o the_o circumference_n cf_o which_o be_v subtend_v of_o the_o side_n of_o the_o decagon_n be_v the_o half_a of_o the_o circumference_n bcf_n therefore_o the_o circumference_n acf_n be_v quintuple_a to_o the_o circumference_n cf_o by_o the_o 15._o of_o the_o â—iâ—t_n wherefore_o the_o circumference_n ac_fw-la be_v qradruple_a to_o the_o circumference_n fc_o but_o as_o the_o circumference_n ac_fw-la be_v to_o the_o circumference_n fc_o so_o be_v the_o angle_n adc_o to_o the_o angle_n fdc_n by_o the_o last_o of_o the_o six_o wherefore_o the_o angle_n adc_o be_v quadruple_a to_o the_o angle_n fdc_n but_o the_o angle_n adc_o be_v double_a to_o the_o angle_n efc_n by_o the_o 20._o of_o the_o three_o wherefore_o the_o angle_n efc_n be_v double_a to_o the_o angle_n gdc_n but_o the_o angle_n efc_n be_v equal_a to_o the_o angle_n egc_n by_o the_o 4._o of_o the_o first_o wherefore_o the_o angle_n egc_n be_v double_a to_o the_o angle_n edc_n wherefore_o the_o line_n dg_o be_v equal_a to_o the_o line_n gc_o by_o the_o 32._o and_o 6._o of_o the_o first_o but_o the_o line_n gc_o be_v equal_a to_o the_o line_n cf_o by_o the_o 4._o of_o the_o first_o wherefore_o the_o line_n dg_o be_v equal_a to_o the_o line_n cf._n and_o the_o line_n ge_z be_v equal_a to_o the_o line_n of_o by_o construction_n wherefore_o the_o line_n de_fw-fr be_v equal_a to_o the_o line_n of_o and_o fc_n add_v together_o unto_o the_o line_n of_o and_o fc_n add_v the_o line_n de._n wherefore_o the_o line_n df_o and_o fc_o add_v together_o be_v double_a to_o the_o line_n de._n but_o the_o line_n df_o be_v equal_a to_o the_o side_n of_o the_o hexagon_n and_o fc_a to_o the_o side_n of_o the_o decagon_n wherefore_o the_o line_n de_fw-fr be_v the_o half_a of_o the_o side_n of_o the_o hexagon_n and_o of_o the_o side_n of_o the_o decagon_n be_v both_o add_v together_o and_o describe_v in_o one_o and_o the_o self_n same_o circle_n it_o be_v manifest_a same_o by_o the_o proposition_n of_o the_o thirteen_o book_n that_o a_o perpendicular_a line_n draw_v from_o the_o centre_n of_o a_o circle_n to_o the_o side_n of_o a_o equilater_n triangle_n describe_v in_o the_o same_o circle_n be_v half_a of_o the_o semidiameter_n of_o the_o circle_n wherefore_o by_o this_o proposition_n a_o perpendicular_a draw_v from_o the_o câ—ntre_n of_o a_o circle_n to_o the_o side_n of_o a_o pentagon_n be_v equal_a to_o the_o perpendicular_a draw_v from_o the_o centre_n to_o the_o side_n of_o the_o triangle_n â—nd_v to_o half_a of_o the_o side_n of_o the_o decagon_n describe_v in_o the_o same_o circle_n ¶_o the_o 2._o theorem_a the_o 2._o proposition_n one_o and_o the_o self_n same_o circle_n comprehend_v both_o the_o pentagon_n of_o a_o dodecahedron_n and_o the_o triangle_n of_o a_o icosahedron_n flussas_n describe_v in_o one_o and_o the_o self_n same_o sphere_n this_o theorem_a be_v describe_v of_o aristeus_n in_o that_o book_n who_o title_n be_v the_o comparison_n of_o the_o five_o figure_n and_o be_v describe_v of_o apollonius_n in_o his_o second_o edition_n of_o the_o comparison_n of_o a_o dodecahedron_n to_o a_o icosahedron_n which_o be_v proposition_n that_o as_o the_o superficies_n of_o a_o dodecahedron_n be_v to_o the_o superficies_n of_o a_o icosahedron_n so_o be_v the_o dodecahedron_n to_o a_o icosahedron_n for_o that_o a_o perpendicular_a line_n draw_v from_o the_o centre_n of_o a_o sphere_n to_o the_o pentagon_n of_o a_o dodecahedron_n and_o to_o the_o triangle_n of_o a_o icosahedron_n be_v one_o and_o the_o self_n same_o now_o must_v we_o also_o prove_v that_o one_o and_o the_o self_n same_o circle_n comprehend_v both_o the_o pentagon_n of_o a_o dodecahedron_n and_o also_o the_o triangle_n of_o a_o icosahedron_n describe_v in_o one_o and_o the_o self_n same_o sphere_n first_o this_o be_v prove_v flussas_n if_o in_o a_o circle_n be_v describe_v a_o equilater_n pentagon_n the_o square_n which_o be_v make_v of_o the_o side_n of_o the_o pentagon_n and_o of_o that_o right_a line_n which_o be_v subtend_v under_o two_o side_n of_o the_o pentagon_n be_v quintuple_a to_o the_o square_n of_o the_o semidiameter_n oâ—_n the_o circle_n suppose_v that_o abc_n be_v a_o circle_n assumpt_n and_o let_v the_o side_n of_o a_o pentagon_n in_o the_o circle_n abc_n be_v ac_fw-la and_o take_v by_o the_o 1._o of_o the_o three_o the_o centre_n of_o the_o circle_n and_o let_v the_o same_o be_v d._n and_o by_o the_o 12._o of_o the_o first_o from_o the_o point_n d_o draw_v unto_o the_o line_n ac_fw-la a_o perpendicular_a line_n df._n and_o extend_v the_o line_n df_o on_o either_o side_n to_o the_o point_n b_o and_o e._n and_o draw_v a_o right_a line_n from_o the_o point_n a_o to_o the_o point_n b._n now_o i_o say_v that_o the_o square_n of_o the_o line_n basilius_n and_o ac_fw-la be_v quintuple_a to_o the_o square_n of_o the_o line_n de._n draw_v a_o right_a line_n from_o the_o point_n a_o to_o the_o point_n e._n wherefore_o the_o line_n ae_n be_v the_o side_n of_o a_o decagon_n figure_n and_o forasmuch_o as_o the_o line_n be_v be_v double_a to_o thâ—_z line_n de_fw-fr assumpt_n therefore_o the_o square_n
22._o of_o the_o six_o wherefore_o by_o composition_n by_o the_o 18._o of_o the_o five_o as_o both_o the_o line_n ab_fw-la &_o bc_o add_v the_o one_o to_o the_o other_o together_o with_o the_o line_n ac_fw-la that_o be_v as_z two_z such_o line_n as_o ab_fw-la be_v be_v to_o the_o line_n ac_fw-la so_o be_v both_o the_o line_n de_fw-fr and_o of_o add_v the_o one_o to_o the_o other_o together_o with_o the_o line_n df_o that_o be_v two_o such_o line_n as_o de_fw-fr be_v to_o the_o line_n df._n and_o in_o the_o same_o proportion_n be_v the_o half_n of_o the_o antecedent_n by_o the_o 15._o of_o the_o five_o wherefore_o as_o the_o line_n ab_fw-la be_v to_o the_o line_n ac_fw-la so_o be_v the_o line_n de_fw-fr to_o the_o line_n df._n and_o therefore_o by_o the_o 19_o of_o the_o five_o as_o the_o line_n ab_fw-la be_v to_o the_o line_n bc_o so_o be_v the_o line_n df_o to_o the_o line_n fe_o wherefore_o also_o by_o division_n by_o the_o 17._o of_o the_o five_o as_o the_o line_n ac_fw-la be_v to_o the_o line_n cb_o so_o be_v the_o line_n df_o to_o the_o line_n de_fw-fr now_o that_o we_o have_v prove_v proposition_n that_o any_o right_a line_n whatsoever_o be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n what_o proportion_n the_o line_n contain_v in_o power_n the_o square_n make_v of_o the_o whole_a line_n and_o of_o the_o great_a segment_n add_v together_o have_v to_o the_o line_n contain_v in_o power_n the_o square_n make_v of_o the_o whole_a line_n and_o of_o the_o less_o segment_n add_v together_o the_o same_o proportion_n have_v the_o side_n of_o the_o cube_fw-la to_o the_o side_n of_o the_o icosahedron_n now_o also_o that_o we_o have_v prove_v proposition_n that_o as_o the_o side_n of_o the_o cube_fw-la be_v to_o the_o side_n of_o the_o icosahedron_n so_o be_v the_o superficies_n of_o the_o dodecahedron_n to_o the_o superficies_n of_o the_o icosahedron_n be_v both_o describe_v in_o one_o and_o the_o self_n same_o sphere_n and_o moreover_o see_v that_o we_o have_v prove_v proposition_n that_o as_o the_o superficies_n of_o the_o dodecahedron_n be_v to_o the_o superficies_n of_o the_o icosahedron_n so_o be_v the_o dodecahedron_n to_o the_o icosahedron_n for_o that_o both_o the_o pentagon_n of_o the_o dodecahedron_n and_o the_o triangle_n of_o the_o icosahedron_n be_v comprehend_v in_o one_o and_o the_o self_n same_o circle_n corollary_n all_o these_o thing_n i_o say_v be_v prove_v it_o be_v manifest_a that_o if_o in_o one_o and_o the_o self_n same_o sphere_n be_v describe_v a_o dodecahedron_n and_o a_o icosahedron_n they_o shall_v be_v in_o proportion_n the_o one_o to_o the_o other_o as_z a_o right_a line_n whatsoever_o be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n the_o line_n contain_v in_o power_n the_o square_n of_o the_o whole_a line_n and_o of_o the_o great_a segment_n add_v together_o be_v to_o the_o line_n contain_v in_o power_n the_o square_n of_o the_o whole_a line_n and_o of_o the_o less_o segment_n add_v together_o for_o for_o that_o as_o the_o dodecahedron_n be_v to_o the_o icosahedron_n so_o be_v the_o superficies_n of_o the_o dodecahedron_n to_o the_o superficies_n of_o the_o icosahedron_n that_o be_v the_o side_n of_o the_o cube_fw-la to_o the_o side_n of_o the_o icosahedron_n but_o as_o the_o side_n of_o the_o cube_fw-la be_v to_o the_o side_n of_o the_o icosahedron_n so_o any_o right_a line_n what_o so_o ever_o be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n be_v the_o line_n contain_v in_o power_n the_o square_n of_o the_o whole_a line_n and_o of_o the_o great_a segment_n add_v together_o to_o the_o line_n contain_v in_o power_n the_o square_n of_o the_o whole_a line_n and_o of_o the_o less_o segment_n add_v together_o wherefore_o as_o a_o dodecahedron_n be_v to_o a_o icosahedron_n describe_v in_o one_o and_o the_o self_n same_o sphere_n so_o any_o right_a line_n what_o so_o ever_o be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n be_v the_o line_n contain_v in_o power_n the_o square_n of_o the_o whole_a line_n &_o of_o the_o great_a segment_n add_v together_o to_o the_o line_n contain_v in_o power_n the_o square_n of_o the_o whole_a line_n and_o of_o the_o less_o segment_n add_v together_o the_o end_n of_o the_o fourteen_o book_n of_o euclides_n element_n after_o hypsicles_n ¶_o the_o fourteen_o book_n of_o euclides_n element_n after_o flussas_n for_o that_o the_o fouretenth_fw-mi book_n as_o it_o be_v set_v forth_o by_o flussas_n contain_v in_o it_o more_o proposition_n than_o be_v find_v in_o hypsicles_n &_o also_o some_o of_o those_o proposition_n which_o hypsicles_n have_v be_v by_o he_o somewhat_o otherwise_o demonstrate_v i_o think_v my_o labour_n well_o bestow_v for_o the_o reader_n sake_n to_o turn_v it_o also_o all_o whole_a notwithstanding_o my_o travail_n before_o take_v in_o turn_v the_o same_o book_n after_o hypsicles_n where_o note_v you_o that_o here_o in_o this_o 14._o book_n after_o flussas_n and_o in_o the_o other_o book_n follow_v namely_o the_o 15._o and_o 16._o i_o have_v in_o allege_v of_o the_o proposition_n of_o the_o same_o 14._o book_n follow_v the_o order_n and_o number_n of_o the_o proposition_n as_o flussas_n have_v place_v they_o ¶_o the_o first_o proposition_n a_o perpendicular_a line_n draw_v from_o the_o centre_n of_o a_o circle_n to_o the_o side_n of_o a_o pentagon_n inscribe_v in_o the_o same_o circle_n campane_n be_v the_o half_a of_o these_o two_o line_n take_v together_o namely_o of_o the_o side_n of_o the_o hexagon_n and_o of_o the_o side_n of_o the_o decagon_n inscribe_v in_o the_o same_o circle_n take_v a_o circle_n abc_n and_o inscribe_v in_o it_o the_o side_n of_o a_o pentagon_n which_o let_v be_v bc_o and_o take_v the_o centre_n of_o the_o circle_n which_o let_v be_v the_o point_n d_o construction_n and_o from_o it_o draw_v unto_o the_o side_n bc_o a_o perpendicular_a line_n de_fw-fr which_o produce_v to_o the_o point_n â—_o and_o unto_o the_o line_n e_o fletcher_n put_v the_o line_n eglantine_n equal_a and_o draw_v these_o right_a line_n cg_o cd_o and_o cf._n then_o i_o say_v that_o the_o right_a line_n de_fw-fr which_o be_v draw_v from_o the_o centre_n to_o bc_o the_o side_n of_o the_o pentagon_n be_v the_o half_a of_o â—he_n sideâ—_n of_o the_o decagon_n and_o hexagon_n take_v together_o demonstration_n forasmuch_o as_o the_o line_n de_fw-fr be_v a_o perpendicular_a â—nto_n the_o line_n bc_o therefore_o the_o section_n be_v and_o ec_o shall_v be_v equal_a by_o the_o 3._o of_o the_o three_o and_o the_o line_n of_o be_v common_a unto_o they_o both_o and_o the_o angle_n fec_n and_o feb_n be_v right_a angle_n by_o supposition_n wherefore_o the_o base_n bf_o and_o fc_o be_v equal_a by_o the_o 4._o of_o the_o first_o but_o the_o line_n bc_o be_v the_o side_n of_o a_o pentagon_n by_o construction_n wherefore_o fc_o which_o subtend_v the_o half_a of_o the_o side_n of_o the_o pentagon_n be_v the_o side_n of_o the_o decagon_n inscribe_v in_o the_o circle_n abc_n but_o unto_o the_o line_n fc_o be_v by_o the_o 4._o of_o the_o first_o equal_a the_o line_n cg_o for_o they_o subtend_v right_a angle_n ceg_v and_o cef_n which_o be_v contain_v under_o equal_a side_n wherefore_o also_o the_o angle_n cge_n and_o cfe_n of_o the_o triangle_n cfg_n be_v equal_a by_o the_o 5._o of_o the_o first_o and_o forasmuch_o as_o the_o ark_n fc_o be_v subtend_v of_o the_o side_n of_o a_o decagon_n the_o ark_n ca_n shall_v be_v quadruple_a to_o the_o ark_n cf_o wherefore_o also_o the_o angle_n cda_n shall_v be_v quadruple_a to_o the_o angle_n cdf_n by_o the_o last_o of_o the_o sixâ—_o and_o forasmuch_o as_o the_o same_o angle_n cda_n which_o be_v set_v at_o the_o centre_n be_v double_a to_o the_o angle_n cfa_n which_o be_v set_v at_o the_o circumference_n by_o the_o 20._o of_o the_o three_o therefore_o the_o angle_n cfa_n or_o cfd_n be_v double_a to_o the_o angle_n cdâ—_n namely_o the_o half_a of_o quadruple_a but_o unto_o the_o angle_n cfd_n or_o cfg_n be_v prove_v equal_a the_o angle_n cgf_n wherefore_o the_o outward_a angle_n cgf_n be_v double_a to_o the_o angle_n cdf_n wherefore_o the_o angle_n cdg_n and_o dcg_n shall_v be_v equal_a for_o unto_o those_o two_o angle_n the_o angle_n cgf_n be_v equal_a by_o the_o 32._o of_o the_o first_o wherefore_o the_o side_n gc_o and_o gd_o be_v equal_a by_o the_o 6._o of_o the_o first_o wherefore_o also_o the_o line_n gd_o be_v equal_a to_o the_o line_n fc_o which_o be_v the_o side_n of_o the_o decagon_n but_o unto_o the_o right_a line_n fe_o be_v equal_a the_o line_n eglantine_n by_o construction_n wherefore_o the_o whole_a line_n de_fw-fr be_v equal_a to_o the_o two_o line_n câ—_n and_o fe_o wherefore_o those_o line_n take_v together_o namely_o the_o line_n df_o and_o fc_o shall_v
together_o and_o a_o perpendicular_a line_n draw_v from_o the_o centre_n of_o the_o sphere_n to_o any_o base_a of_o the_o cube_fw-la be_v equal_a to_o half_a the_o side_n of_o the_o cube_fw-la which_o be_v require_v to_o be_v prouâ—d_v ¶_o a_o corollary_n if_o two_o three_o of_o the_o power_n of_o the_o diameter_n of_o the_o sphere_n be_v multiply_v into_o the_o perpendicular_a line_n equal_a to_o half_a the_o side_n of_o the_o cube_fw-la campane_n there_o shall_v be_v produce_v a_o solid_a equal_a to_o the_o solid_a of_o the_o cube_fw-la for_o it_o be_v before_o manifest_v that_o two_o three_o part_n of_o the_o power_n of_o the_o diameter_n of_o the_o sphere_n be_v equal_a to_o two_o base_n of_o the_o cube_fw-la if_o therefore_o unto_o each_o of_o those_o two_o three_o be_v apply_v half_o the_o altitude_n of_o the_o cube_fw-la they_o shall_v make_v each_o of_o those_o solid_n equal_a to_o half_o of_o the_o cube_fw-la by_o the_o 31._o of_o the_o eleven_o for_o they_o have_v equal_a base_n wherefore_o two_o of_o those_o solid_n be_v equal_a to_o the_o whole_a cube_fw-la you_o shall_v understand_v gentle_a reader_n that_o campane_n in_o his_o 14._o book_n of_o euclides_n element_n have_v 18._o proposition_n with_o diverse_a corollary_n follow_v of_o they_o some_o of_o which_o proposition_n and_o corollary_n i_o have_v before_o in_o the_o twelve_o and_o thirteen_o book_n add_v out_o of_o flussas_n as_o corollary_n which_o thing_n also_o i_o have_v note_v on_o the_o side_n of_o those_o corollary_n namely_o with_o what_o proposition_n or_o corollary_n of_o campane_n 14._o book_n they_o do_v agree_v the_o rest_n of_o his_o 18._o proposition_n and_o corollary_n be_v contain_v in_o the_o twelve_o former_a proposition_n and_o corollary_n of_o this_o 14._o book_n after_o flussas_n where_o you_o may_v see_v on_o the_o side_n of_o each_o proposition_n and_o corollary_n with_o what_o proposition_n and_o corollary_n of_o campane_n they_o agree_v but_o the_o eight_o proposition_n follow_v together_o with_o their_o corollary_n flussas_n have_v add_v of_o himself_o as_o he_o himself_o affirm_v the_o 13._o proposition_n one_o and_o the_o self_n same_o circle_n contain_v both_o the_o square_n of_o a_o cube_fw-la and_o the_o triangle_n of_o a_o octohedron_n describe_v in_o one_o and_o the_o self_n same_o sphere_n svppose_v that_o there_o be_v a_o cube_fw-la abg_o and_o a_o octohedron_n def_n describe_v in_o one_o and_o the_o self_n same_o sphere_n who_o diameter_n let_v be_v ab_fw-la or_o dh_o and_o let_v the_o line_n draw_v from_o the_o centre_n that_o be_v the_o semidiameter_n of_o the_o circle_n which_o ctonaine_v the_o base_n of_o those_o solid_n â—_o be_v ca_n and_o id_fw-la then_o i_o say_v that_o the_o line_n ca_n and_o id_fw-la be_v equal_a forasmuch_o as_o ab_fw-la the_o diameter_n of_o the_o sphere_n which_o contain_v the_o cube_fw-la demonstration_n be_v in_o power_n triple_a to_o bg_o the_o side_n of_o the_o cube_fw-la by_o the_o 15._o of_o the_o thirteen_o unto_o which_o side_n ag_z the_o diameter_n of_o the_o base_a of_o the_o cube_fw-la be_v in_o power_n double_a by_o the_o 47._o of_o the_o first_o which_o line_n agnostus_n be_v also_o the_o diameter_n of_o the_o circle_n which_o contain_v the_o base_a by_o the_o 9_o of_o the_o four_o therefore_o ab_fw-la the_o diameter_n of_o the_o sphere_n be_v in_o power_n sesquialter_fw-la to_o the_o line_n agnostus_n namely_o of_o what_o part_n the_o line_n ab_fw-la contain_v in_o power_n 12._o of_o the_o same_o the_o line_n agnostus_n shall_v contain_v in_o power_n 8._o and_o therefore_o the_o right_a line_n ac_fw-la which_o be_v draw_v from_o the_o centre_n of_o the_o circle_n to_o the_o circumference_n contain_v in_o power_n of_o the_o same_o part_n 2._o wherefore_o the_o diameter_n of_o the_o sphere_n be_v in_o power_n sextuple_a to_o the_o line_n which_o be_v draw_v from_o the_o centre_n to_o the_o circumference_n of_o the_o circle_n which_o contain_v the_o square_a of_o the_o cube_fw-la but_o the_o diameter_n of_o the_o self_n same_o sphere_n which_o contain_v the_o octohedron_n be_v one_o and_o the_o self_n same_o with_o the_o diameter_n of_o the_o cube_fw-la namely_o dh_o be_v equal_a to_o ab_fw-la and_o the_o same_o diameter_n be_v also_o the_o diameter_n of_o the_o square_n which_o be_v make_v of_o the_o side_n of_o the_o octohedron_n wherefore_o the_o say_a diameter_n be_v in_o power_n double_a to_o the_o side_n of_o the_o same_o octohedron_n by_o the_o 14._o of_o the_o thirteen_o but_o the_o side_n df_o be_v in_o power_n triple_a to_o the_o line_n draw_v from_o the_o centre_n to_o the_o circumference_n of_o the_o circle_n which_o contain_v the_o triangle_n of_o the_o octohedron_n namely_o to_o the_o line_n id_fw-la by_o the_o 12._o of_o the_o thirteen_o wherefore_o the_o self_n same_o diameter_n ab_fw-la or_o dh_o which_o be_v in_o power_n sextuple_a to_o the_o line_n draw_v from_o the_o centre_n to_o the_o circumference_n of_o the_o circle_n which_o contain_v the_o square_a of_o the_o cube_fw-la be_v also_o sextuple_a to_o the_o line_n id_fw-la draw_v from_o the_o centre_n to_o the_o circumference_n of_o the_o circle_n which_o contain_v the_o triangle_n of_o the_o octohedron_n wherefore_o the_o line_n draw_v from_o the_o centre_n of_o the_o circle_n to_o the_o circumference_n which_o contain_v the_o base_n of_o the_o cube_fw-la and_o of_o the_o octohedron_n be_v equal_a and_o therefore_o the_o circle_n be_v equal_a by_o the_o first_o definition_n of_o the_o three_o wherefore_o one_o and_o the_o self_n same_o circle_n contain_v &c._a &c._a as_o in_o the_o proposition_n which_o be_v require_v to_o be_v prove_v a_o corollary_n hereby_o it_o be_v manifest_a that_o perpendicular_n couple_v together_o in_o a_o sphere_n the_o centre_n of_o the_o circle_n which_o contain_v the_o opposite_a base_n of_o the_o cube_fw-la and_o of_o the_o octohedron_n be_v equal_a for_o the_o circle_n be_v equal_a by_o the_o second_o corollary_n of_o the_o assumpt_n of_o the_o 16._o of_o the_o twelve_o and_o the_o line_n which_o pass_v by_o the_o centre_n of_o the_o sphere_n couple_v together_o the_o centre_n of_o the_o base_n be_v also_o equal_a by_o the_o first_o corollary_n of_o the_o same_o wherefore_o the_o perpendicular_a which_o couple_v together_o the_o opposite_a base_n of_o the_o octohedron_n be_v equal_a to_o the_o side_n of_o the_o cube_fw-la for_o either_o of_o they_o be_v the_o altitude_n erect_v the_o 14._o proposition_n a_o octohedron_n be_v to_o the_o triple_a of_o a_o tetrahedron_n contain_v in_o one_o and_o the_o self_n same_o sphere_n in_o that_o proportion_n that_o their_o side_n be_v svppose_v that_o there_o be_v a_o octohedron_n abcd_o and_o a_o tetrahedron_n efgh_o upon_o who_o base_a fgh_n erect_v a_o prism_n construction_n which_o be_v do_v by_o erect_v from_o the_o angle_n of_o the_o base_a perpendicular_a line_n equal_a to_o the_o altitude_n of_o the_o tetrahedron_n which_o prism_n shall_v be_v triple_a to_o the_o tetrahedron_n efgh_o by_o the_o first_o corollary_n of_o the_o 7._o of_o the_o twelve_o then_o i_o say_v that_o the_o octohedron_n abcd_o be_v to_o the_o prism_n which_o be_v triple_a to_o the_o tetrahedron_n efgh_o as_o the_o side_n bc_o be_v to_o the_o side_n fg._n for_o forasmuch_o as_o the_o side_n of_o the_o opposite_a base_n of_o the_o octohedron_n demonstration_n be_v right_a line_n touch_v the_o one_o the_o other_o and_o be_v parellel_n to_o other_o right_a line_n touch_v the_o one_o the_o other_o for_o the_o side_n of_o the_o square_n which_o be_v compose_v of_o the_o side_n of_o the_o octohedron_n be_v opposite_a wherefore_o the_o opposite_a plain_a triangle_n namely_o abc_n &_o kid_n shall_v be_v parallel_n and_o so_o the_o rest_n by_o the_o 15._o of_o the_o eleven_o let_v the_o diameter_n of_o the_o octohedron_n be_v the_o line_n ad._n now_o then_o the_o whole_a octohedron_n be_v cut_v into_o four_o equal_a and_o like_a pyramid_n set_v upon_o the_o base_n of_o the_o octohedron_n and_o have_v the_o same_o altitude_n with_o it_o &_o be_v about_o the_o diameter_n ad_fw-la namely_o the_o pyramid_n set_v upon_o the_o base_a bid_v and_o have_v his_o top_n the_o point_n a_o and_o also_o the_o pyramid_n set_v upon_o the_o base_a bcd_o have_v his_o top_n the_o same_o point_n a._n likewise_o the_o pyramid_n set_v upon_o the_o base_a ikd_v &_o have_v his_o top_n the_o same_o point_n a_o and_o moreover_o the_o pyramid_n set_v upon_o the_o base_a ckd_v and_o have_v his_o top_n the_o former_a point_n a_o which_o pyramid_n shall_v be_v equal_a by_o the_o 8._o definition_n of_o the_o eleven_o for_o they_o each_o consist_v of_o two_o base_n of_o the_o octohedron_n and_o of_o two_o triangle_n contain_v under_o the_o diameter_n ad_fw-la and_o two_o side_n of_o the_o octohedron_n wherefore_o the_o prism_n which_o be_v set_v upon_o the_o base_a of_o the_o octohedron_n
and_o have_v the_o same_o altitude_n with_o it_o name_o the_o altitude_n of_o the_o parallel_a base_n as_o it_o be_v manifest_a by_o the_o former_a be_v equal_a to_o three_o of_o those_o pyramid_n of_o the_o octohedron_n by_o the_o first_o corollary_n of_o the_o seven_o of_o the_o twelve_o wherefore_o that_o prism_n shall_v have_v to_o the_o other_o prism_n under_o the_o same_o altitude_n compose_v of_o the_o 4._o pyramid_n of_o the_o whole_a octohedron_n the_o proportion_n of_o the_o triangular_a base_n by_o the_o 3._o corollary_n of_o the_o same_o and_o forasmuch_o as_o 4._o pyramid_n be_v unto_o 3._o pyramid_n in_o sesquitercia_fw-la proportion_n therefore_o the_o triangule_a base_a of_o the_o prism_n which_o contain_v 4._o pyramid_n be_v in_o sesquitertia_fw-la proportion_n to_o the_o base_a of_o the_o prism_n which_o contain_v three_o pyramid_n of_o the_o same_o octohedron_n and_o be_v set_v upon_o the_o base_a of_o the_o octohedron_n and_o under_o the_o altitude_n thereof_o that_o be_v in_o sesquitercia_fw-la proportion_n to_o the_o base_a of_o the_o octohedron_n but_o the_o base_a of_o the_o same_o octohedron_n be_v in_o sesquitertia_fw-la proportion_n to_o the_o base_a of_o the_o pyramid_n by_o the_o â—enth_n of_o this_o book_n wherefore_o the_o triangular_a base_n namely_o of_o the_o prism_n which_o contain_v four_o pyramid_n of_o the_o octohedron_n and_o be_v under_o the_o altitude_n thereof_o be_v equal_a to_o the_o triangular_a base_n of_o the_o prism_n which_o contain_v three_o pyramid_n under_o the_o altitude_n of_o the_o pyramid_n efgh_o but_o the_o prism_n of_o the_o octohedron_n be_v equal_a to_o the_o octohedron_n and_o the_o prism_n of_o the_o pyramid_n efgh_o be_v prove_v triple_a to_o the_o same_o pyramid_n efgh_o now_o than_o the_o prism_n set_v upon_o equal_a base_n be_v the_o one_o to_o the_o other_o as_o their_o altitude_n be_v by_o the_o corollary_n of_o the_o 25._o of_o the_o eleven_o namely_o as_o be_v the_o parallelipidedon_n their_o double_n by_o the_o corollary_n of_o the_o 31._o of_o the_o eleven_o but_o the_o altitude_n of_o the_o octohedron_n be_v equal_a to_o the_o side_n of_o the_o cube_fw-la contain_v in_o the_o same_o sphere_n by_o the_o corollary_n of_o the_o 13._o of_o this_o book_n and_o the_o side_n of_o the_o cube_fw-la be_v in_o power_n to_o the_o altitude_n of_o the_o tetrahedon_n in_o that_o proportion_n that_o 12._o be_v to_o 16_o by_o the_o 18._o of_o the_o thirteen_o and_o the_o side_n of_o the_o octohedron_n be_v to_o the_o side_n of_o the_o pyramid_n in_o that_o proportion_n that_o 18._o be_v to_o 24._o by_o the_o same_o 18._o of_o the_o thirteen_o which_o proportion_n be_v one_o &_o the_o self_n same_o with_o the_o proportion_n of_o 12._o to_o 16._o wherefore_o that_o prism_n which_o be_v equal_a to_o the_o octohedron_n be_v to_o the_o prism_n which_o be_v triple_a to_o the_o tetrahedron_n in_o that_o proportion_n that_o the_o altitude_n or_o that_o the_o side_n be_v wherefore_o a_o octohedron_n be_v to_o the_o triple_a of_o a_o tetrahedron_n contain_v in_o one_o and_o the_o self_n same_o sphere_n in_o that_o proportion_n that_o their_o side_n be_v which_o be_v require_v to_o be_v demonstrate_v a_o corollary_n the_o side_n of_o a_o tetrahedron_n &_o of_o a_o octohedron_n be_v proportional_a with_o their_o altitude_n for_o the_o side_n &_o altitude_n be_v in_o power_n sesquitercia_fw-la moreover_o the_o diameter_n of_o the_o sphere_n be_v to_o the_o side_n of_o the_o tetrahedron_n as_o the_o side_n of_o the_o octohedron_n be_v to_o the_o â—â—de_n of_o the_o cubeâ—_n namely_o the_o power_n of_o each_o be_v in_o sesquialter_fw-la proportion_n by_o the_o 18._o of_o the_o thirteen_o the_o 15._o proposition_n if_o a_o rational_a line_n contain_v in_o power_n two_o line_n make_v the_o whole_a and_o the_o great_a segment_n and_o again_o contain_v in_o power_n two_o line_n make_v the_o whole_a and_o the_o less_o segment_n the_o great_a segment_n shall_v be_v the_o side_n of_o the_o icosahedron_n and_o the_o less_o segment_n shall_v be_v the_o side_n of_o the_o dodecahedron_n contain_v in_o one_o and_o the_o self_n same_o sphere_n svppose_v that_o agnostus_n be_v the_o diameter_n of_o the_o sphere_n which_o contain_v the_o icosahedron_n abgc_n and_o let_v bg_o subtend_v the_o side_n of_o the_o pentagon_n describe_v of_o the_o side_n of_o the_o icosahedron_n by_o the_o 16._o of_o the_o thirteen_o moreover_o upon_o the_o same_o diameter_n agnostus_n or_o df_o equal_a unto_o it_o construction_n let_v there_o be_v describe_v a_o dodecahedron_n defh_o by_o the_o 1â—_o of_o the_o thirteen_o who_o opposite_n side_n ed_z and_o fh_o let_v be_v cut_v into_o two_o equal_a part_n in_o the_o point_n i_o and_o king_n and_o draw_v a_o line_n from_o i_o to_o k._n and_o let_v the_o line_n of_o couple_n two_o of_o the_o opposite_a angle_n of_o the_o base_n which_o be_v join_v together_o then_o i_o say_v that_o ab_fw-la the_o side_n of_o the_o icosahedron_n be_v the_o great_a segment_n which_o the_o diameter_n agnostus_n contain_v in_o power_n together_o with_o the_o whole_a line_n and_o line_n ed_z be_v the_o less_o segment_n which_o the_o same_o diameter_n agnostus_n or_o df_o contain_v in_o power_n together_o with_o the_o whole_a demonstration_n for_o forasmuch_o as_o the_o opposite_a side_n ab_fw-la and_o gc_o of_o the_o icosahedron_n be_v couple_v by_o the_o diameter_n agnostus_n and_o bc_o be_v equal_a &_o parallel_n by_o the_o 2._o corollary_n of_o the_o 16._o of_o the_o thirteen_o the_o right_a line_n bg_o &_o ac_fw-la which_o couple_v they_o together_o be_v equal_a &_o parallel_n by_o the_o 33._o of_o the_o first_o moreover_o the_o angle_n bac_n &_o abg_o be_v subtend_v of_o equal_a diamâ—ters_n shall_v by_o the_o 8._o of_o the_o first_o be_v equal_a &_o by_o the_o 29_o of_o the_o 〈◊〉_d they_o shall_v be_v right_a angle_n wherefore_o the_o right_a line_n agnostus_n 〈◊〉_d in_o power_n the_o too_o line_n ab_fw-la and_o bg_o by_o the_o 47._o of_o 〈…〉_o and_o forasmuch_o as_o the_o line_n bg_o subtend_v the_o angle_n of_o the_o pentagon_n compose_v of_o the_o side_n of_o the_o icosahedron_n the_o great_a segment_n of_o the_o right_a line_n bg_o shall_v be_v the_o right_a line_n ab_fw-la by_o the_o â—_o of_o the_o thirteen_o which_o line_n ab_fw-la together_o with_o the_o whole_a line_n bg_o the_o line_n agnostus_n contain_v in_o power_n and_o forasmuch_o aâ—_z the_o line_n ik_fw-mi couple_v the_o opposite_a and_o parallel_a side_n ed_z and_o fh_o of_o the_o dodecahedron_n make_v at_o those_o point_n right_a angle_n by_o the_o 3._o corollary_n of_o the_o 17._o of_o tâ—e_z thirteen_o the_o right_a line_n of_o which_o couple_v together_o equal_a and_o parallel_a line_n ei_o &_o fk_v shall_v be_v equal_a to_o the_o same_o line_n ik_fw-mi by_o the_o 33._o of_o the_o first_o wherefore_o the_o angle_n def_n shall_v be_v â—_o right_n angle_n by_o the_o 29._o of_o the_o first_o wherefore_o the_o diameter_n df_o contain_v in_o power_n the_o two_o line_n ed_z and_o ef._n but_o the_o less_o segment_n of_o the_o line_n ik_fw-mi be_v ed_z the_o side_n of_o the_o dodecahedron_n by_o the_o 4._o corollary_n of_o the_o 17â—_n of_o the_o thirteen_o wherefore_o the_o same_o line_n ed_z be_v also_o the_o less_o segment_n of_o the_o line_n of_o which_o be_v equal_a unto_o the_o line_n ik_fw-mi wherefore_o the_o diameter_n df_o contain_v in_o power_n the_o two_o line_n ed_z and_o of_o by_o the_o 47._o of_o the_o first_o contain_v in_o powâ—râ—_n ed_z the_o side_n of_o the_o dodecahedron_n the_o less_o segment_n together_o with_o the_o whole_a if_o therefore_o a_o rational_a line_n agnostus_n or_o df_o contain_v in_o power_n two_o line_n ab_fw-la and_o bg_o do_v make_v the_o whole_a line_n and_o the_o great_a segment_n and_o again_o contain_v in_o power_n two_o line_n of_o and_o ed_z do_v make_v the_o whole_a line_n and_o the_o less_o segment_n the_o great_a segment_n ab_fw-la shall_v be_v the_o side_n of_o the_o icosahedron_n and_o the_o less_o segment_n ed_z shall_v be_v the_o side_n of_o the_o dodecahedron_n contain_v in_o one_o and_o the_o self_n same_o sphere_n the_o 16._o proposition_n if_o the_o power_n of_o the_o side_n of_o a_o octohedron_n be_v express_v by_o two_o right_a lineâ—_z join_v together_o by_o a_o extreme_a and_o meâ—ne_a proportion_n the_o side_n of_o the_o icosahedron_n contain_v in_o the_o same_o sphere_n shall_v be_v duple_v to_o the_o less_o segment_n let_v ab_fw-la the_o side_n of_o the_o octohedron_n abg_o contain_v in_o power_n the_o two_o line_n c_o and_o h_o which_o let_v have_v that_o proportion_n that_o the_o whole_a have_v to_o the_o great_a segment_n by_o the_o corollarye_a of_o the_o first_o proposition_n add_v by_o flussas_n after_o the_o last_o proposition_n of_o the_o six_o book_n construction_n and_o let_v the_o icosahedron_n contain_v in_o the_o same_o sphere_n be_v
overthrow_v and_o overwhelm_v the_o whole_a world_n he_o be_v utter_o rude_a and_o ignorant_a in_o the_o greek_a tongue_n so_o that_o certain_o he_o never_o read_v euclid_n in_o the_o greek_a nor_o of_o like_a translate_v out_o of_o the_o greek_a but_o have_v it_o translate_v out_o of_o the_o arabike_a tongue_n the_o arabian_n be_v man_n of_o great_a study_n and_o industry_n and_o common_o great_a philosopher_n notable_a physician_n and_o in_o mathematical_a art_n most_o expert_a so_o that_o all_o kind_n of_o good_a learning_n flourish_v and_o reign_v amongst_o they_o in_o a_o manner_n only_o these_o man_n turn_v whatsoever_o good_a author_n be_v in_o the_o greek_a tongue_n of_o what_o art_n and_o knowledge_n so_o ever_o it_o be_v into_o the_o arabike_a tongue_n and_o from_o thence_o be_v many_o of_o they_o turn_v into_o the_o latin_a and_o by_o that_o mean_v many_o greek_a author_n come_v to_o the_o hand_n of_o the_o latin_n and_o not_o from_o the_o first_o fountain_n the_o greek_a tongue_n wherein_o they_o be_v first_o write_v as_o appear_v by_o many_o word_n of_o the_o arabike_a tongue_n yet_o remain_v in_o such_o book_n as_o be_v zenith_n nadir_n helmuayn_fw-mi helmuariphe_n and_o infinite_a such_o other_o which_o arabian_n also_o in_o translate_n such_o greek_a work_n be_v accustom_v to_o add_v as_o they_o think_v good_a &_o for_o the_o full_a understanding_n of_o the_o author_n many_o thing_n as_o be_v to_o be_v see_v in_o diverse_a author_n as_o namely_o in_o theodosius_n de_fw-la sphera_fw-la where_o you_o see_v in_o the_o old_a translation_n which_o be_v undoubteldy_a out_o of_o the_o arabike_a many_o proposition_n almost_o every_o three_o or_o four_o leaf_n some_o such_o copy_n of_o euclid_n most_o likely_a do_v campanus_n follow_v wherein_o he_o find_v those_o proposition_n which_o he_o have_v more_o &_o above_o those_o which_o be_v find_v in_o the_o greek_a set_v out_o by_o hypsicles_n and_o that_o not_o only_o in_o this_o 15._o book_n but_o also_o in_o the_o 14._o book_n wherein_o also_o you_o find_v many_o proposition_n more_o they_o be_v find_v in_o the_o greek_a set_v out_o also_o by_o hypsicles_n likewise_o in_o the_o book_n before_o you_o shall_v find_v many_o proposition_n add_v and_o many_o invert_v and_o set_v out_o of_o order_n far_o otherwise_o than_o they_o be_v place_v in_o the_o greek_a examplar_n flussas_n also_o a_o diligent_a restorer_n of_o euclid_n a_o man_n also_o which_o have_v well_o deserve_v of_o the_o whole_a art_n of_o geometry_n have_v add_v moreover_o in_o this_o book_n as_o also_o in_o the_o former_a 14._o book_n he_o add_v 8._o proposition_n 9_o proposition_n of_o his_o own_o touch_v the_o inscription_n and_o circumscription_n 〈…〉_o body_n very_o singular_a â—ndoubtedly_o and_o witty_a all_o which_o for_o that_o nothing_o shall_v want_v to_o the_o desirous_a lover_n of_o knowledge_n i_o have_v faithful_o with_o no_o small_a pain_n turn_v and_o whereas_o flâ—ssâ—â—_n in_o the_o begin_n of_o the_o eleven_o book_n namely_o in_o the_o end_n of_o the_o definition_n there_o â—eâ—_n put_v two_o definition_n of_o the_o inscription_n and_o circumscription_n of_o solid_n or_o corporal_a figure_n within_o or_o about_o the_o one_o the_o other_o which_o certain_o be_v not_o to_o be_v reject_v yet_o for_o that_o until_o this_o present_a 15._o book_n there_o be_v no_o mention_n make_v of_o the_o inscription_n or_o circumscription_n of_o these_o body_n i_o think_v it_o not_o so_o convenient_a thâ—râ—_n to_o place_v they_o but_o to_o refer_v they_o to_o the_o begin_n of_o this_o 15._o book_n where_o they_o be_v in_o manner_n of_o necessity_n require_v to_o the_o elucidation_n of_o the_o proposition_n and_o dâ—monstrationâ—_n of_o the_o same_o the_o definition_n be_v these_o definition_n 1._o a_o solid_a figure_n be_v then_o â—aid_v to_o be_v inscribe_v in_o a_o solid_a figure_n when_o the_o angle_n of_o the_o figure_n inscribe_v touch_n together_o at_o one_o time_n either_o the_o angle_n of_o the_o figure_n circumscribe_v or_o the_o superficiece_n or_o the_o side_n definition_n 2._o a_o solid_a figure_n be_v then_o say_v to_o be_v circumscribe_v about_o a_o solid_a figure_n when_o together_o at_o one_o time_n either_o the_o angle_n or_o the_o superficiece_n or_o the_o side_n of_o the_o figure_n circumscribe_v â—ouch_v the_o angle_n of_o the_o figure_n inscribe_v in_o the_o fourâ—â—_n book_n in_o the_o definition_n of_o the_o inscription_n or_o circumscription_n of_o plain_a rectiline_n figure_v one_o with_o in_o or_o about_o a_o other_o be_v require_v that_o all_o the_o angle_n of_o the_o figuâ—â—_n inscribe_v shall_v at_o one_o time_n touch_v all_o the_o side_n of_o the_o figure_n circumscribe_v but_o in_o the_o five_o regular_a solid_n â—o_o who_o chief_o these_o two_o definition_n pertain_v for_o that_o the_o number_n of_o their_o angle_n superficiece_n &_o side_n be_v not_o equal_a one_o compare_v to_o a_o other_o it_o be_v not_o of_o necessity_n that_o all_o the_o angle_n of_o the_o solid_a inscribe_v shall_v together_o at_o one_o time_n touch_v either_o all_o the_o angle_n or_o all_o the_o superficiece_n or_o all_o the_o side_n of_o the_o solid_a circumscribe_v but_o it_o be_v sufficient_a that_o those_o angle_n of_o the_o inscribe_v solid_a which_o touch_n do_v at_o one_o time_n together_o each_o touch_n some_o one_o angle_n of_o the_o figure_n circumscribe_v or_o some_o one_o base_a or_o some_o one_o side_n so_o that_o if_o the_o angle_n of_o the_o inscribe_v figure_n do_v at_o one_o time_n touch_v the_o angle_n of_o the_o figure_n circumscribe_v none_o of_o they_o may_v at_o the_o same_o time_n touch_v either_o the_o base_n or_o the_o side_n of_o the_o same_o circumscribe_v figure_n and_o so_o if_o they_o touch_v the_o base_n they_o may_v touch_v neither_o angle_n nor_o side_n and_o likewise_o if_o they_o touch_v the_o side_n they_o may_v touch_v neither_o angle_n nor_o base_n and_o although_o sometime_o all_o the_o angle_n of_o the_o figure_n inscribe_v can_v not_o touch_v either_o the_o angle_n or_o the_o base_n or_o the_o side_n of_o the_o figure_n circumscribe_v by_o reason_n the_o number_n of_o the_o angle_n base_n or_o side_n of_o the_o say_a figure_n circumscribe_v want_v of_o the_o number_n of_o the_o angle_n of_o the_o â—igure_n inscribe_v yet_o shall_v those_o angle_n of_o the_o inscribe_v figure_n which_o touch_n so_o touch_v that_o the_o void_a place_n leave_v between_o the_o inscribe_v and_o circumscribe_v figure_n shall_v on_o every_o side_n be_v equal_a and_o like_a as_o you_o may_v afterward_o in_o this_o fifteen_o book_n most_o plain_o perceive_v ¶_o the_o 1._o proposition_n the_o 1._o problem_n in_o a_o cube_n give_v to_o describe_v admonish_v a_o trilater_n equilater_n pyramid_n svppose_v that_o the_o cube_fw-la give_v be_v abcdefgh_o in_o the_o same_o cube_fw-la it_o be_v require_v to_o inscribe_v a_o tetrahedron_n draw_v these_o right_a line_n ac_fw-la construction_n ce_fw-fr ae_n ah_o eh_o hc_n demonstration_n now_o it_o be_v manifest_a that_o the_o triangle_n aec_o ahe_o ahc_n and_o i_o be_v equilater_n for_o their_o side_n be_v the_o diameter_n of_o equal_a square_n wherefore_o aech_n be_v a_o trilater_n equilater_n pyramid_n or_o tetrahedron_n &_o it_o be_v inscribe_v in_o the_o cube_fw-la give_v by_o the_o first_o definition_n of_o this_o book_n which_o be_v require_v to_o be_v do_v ¶_o the_o 2._o proposition_n the_o 2._o problem_n in_o a_o trilater_n equilater_n pyramid_n give_v to_o describe_v a_o octohedron_n svppose_v that_o the_o trilater_n equilater_n pyramid_n give_v be_fw-mi abcd_o who_o side_n let_v be_v divide_v into_o two_o equal_a part_n in_o the_o point_n e_o z_o i_o king_n l_o t._n construction_n and_o draw_v these_o 12._o right_a line_n ez_o zi_n je_n kl_o lt_n tk_n eke_o kz_o zl_n li_n it_o and_o te_fw-la which_o 12._o right_a line_n be_v demonstration_n by_o the_o 4._o of_o the_o first_o equal_a for_o they_o subtend_v equal_a plain_a angle_n of_o the_o base_n of_o the_o pyramid_n and_o those_o equal_a angle_n be_v contain_v under_o equal_a side_n namely_o under_o the_o half_n of_o the_o side_n of_o the_o pyramid_n wherefore_o the_o triangle_n tkl_n tli_n tie_v tek_v zkl_n zli_n zib_n zek_n be_v equilater_n and_o they_o limitate_v and_o contain_v the_o solid_a tklezi_n wherefore_o the_o solid_a tklezi_n be_v a_o octohedron_n by_o the_o 23._o definition_n of_o the_o eleven_o and_o the_o angle_n of_o the_o same_o octohedron_n do_v touch_v the_o side_n of_o the_o pyramid_n abcd_o in_o the_o point_n e_o z_o i_o t_o edward_n l._n wherefore_o the_o octohedron_n be_v inscribe_v in_o the_o pyramid_n by_o the_o 1._o definition_n of_o this_o book_n wherefore_o in_o the_o trilater_n equilater_n pyramid_n give_v be_v inscribe_v a_o octohedron_n which_o be_v require_v to_o be_v do_v a_o corollary_n add_v by_o flussas_n hereby_o it_o be_v manifest_a that_o a_o pyramid_n be_v cut_v into_o two_o
subtend_v angle_n of_o triangle_n like_v unto_o the_o triangle_n who_o angle_n the_o line_n ac_fw-la bf_o and_o pl_z subtend_n be_v cut_v into_o two_o equal_a part_n in_o the_o point_n z_o an_o i_o by_o the_o 4._o of_o the_o six_o so_o also_o be_v the_o other_o line_n nv_n xm_o d_n qt_fw-la which_o be_v equal_a unto_o the_o line_n hk_o &_o ge_z cut_v in_o like_a sort_n and_o they_o shall_v cut_v the_o line_n ac_fw-la bf_o and_o pl_z like_z wherefore_o the_o line_n ko_o which_o be_v equal_a to_o rz_n shall_v make_v the_o great_a segment_n the_o line_n ro_n which_o be_v equal_a to_o the_o line_n zk_n for_o the_o great_a segment_n of_o the_o rz_n be_v the_o line_n zk_n and_o therefore_o the_o line_n oi_o shall_v be_v the_o less_o segment_n when_o as_o the_o whole_a line_n ri_n be_v equal_a to_o the_o whole_a line_n rz_n wherefore_o the_o square_n of_o the_o whole_a line_n ko_o and_o of_o the_o less_o segment_n oi_o be_v triple_a to_o the_o square_n of_o the_o great_a segment_n ro_n by_o the_o 4._o of_o the_o thirteen_o wherefore_o the_o line_n ki_v which_o contain_v in_o power_n the_o two_o line_n ko_o and_o oi_o be_v in_o power_n triple_a to_o the_o line_n ro_n by_o the_o 47._o of_o the_o first_o for_o the_o angle_n koi_n be_v a_o right_a angle_n and_o forasmuch_o as_o the_o line_n fe_o and_o fg_o which_o be_v the_o less_o segment_n of_o the_o side_n of_o the_o octohedron_n be_v equal_a and_o the_o line_n fk_o be_v common_a to_o they_o both_o and_o the_o angle_n kfg_n and_o kfe_n of_o the_o triangle_n of_o the_o octohedron_n be_v equal_a the_o base_n kg_v and_o ke_o shall_v by_o the_o 4_o of_o the_o first_o be_v equal_a and_o therefore_o the_o angle_n kie_fw-mi and_o kig_n which_o they_o subtend_v be_v equal_a by_o the_o 8._o of_o the_o first_o wherefore_o they_o be_v right_a angle_n by_o the_o 13._o of_o the_o first_o wherefore_o the_o right_a line_n ke_o which_o contain_v in_o power_n the_o two_o line_n ki_v and_o â—e_n by_o the_o 47._o of_o the_o first_o be_v in_o power_n quadruple_a to_o the_o line_n ro_n or_o je_n for_o the_o line_n ri_n be_v prove_v to_o be_v in_o power_n triple_a to_o the_o same_o line_n ro_n but_o the_o line_n ge_z be_v double_a to_o the_o line_n je_n wherefore_o the_o line_n ge_z be_v also_o in_o power_n 〈…〉_o pf_n and_o by_o the_o same_o reason_n may_v be_v prove_v that_o the_o â—est_n of_o the_o eleven_o solid_a angle_n of_o the_o 〈◊〉_d be_v 〈…〉_o the_o section_n of_o every_o one_o of_o the_o side_n of_o the_o octohedron_n namely_o in_o the_o point_n e_o n_o five_o h_o â—_o m_o â—_o d_o s_o q_o t._n wherefore_o there_o be_v 12._o angle_n of_o the_o icosahedron_n moreover_o forasmuch_o as_o every_o one_o of_o the_o base_n of_o the_o octohedron_n do_v each_o contain_v triangle_n of_o the_o icosahedron_n 〈…〉_o pyramid_n abcâ—fp_n which_o be_v the_o half_a of_o the_o octohedron_n the_o triangle_n fcp_n receive_v in_o thâ—_z section_n of_o his_o side_n the_o â—_o triangle_n gm_n and_o the_o triangle_n cpb_n contain_v the_o triangle_n nxs_n and_o thâ—_z triangle_n â—ap_n contain_v the_o triangle_n hnd_n and_o moreover_o the_o triangle_n apf_n contain_v the_o triangle_n edge_n and_o the_o same_o may_v be_v prove_v in_o the_o opposite_a pyramid_n abcfl_fw-mi wherefore_o there_o shall_v be_v eight_o triangleâ—_n and_o forasmuch_o as_o beside_o these_o triangle_n to_o every_o one_o of_o the_o solid_a angle_n of_o the_o octohedron_n 〈◊〉_d subtend_v two_o triangle_n as_o the_o triangle_n keg_n and_o meg_n to_o the_o angle_n f_o and_o the_o triangle_n hnv_n and_o xnv_n to_o the_o angle_n b_o also_o the_o triangle_n nd_n and_o â—ds_n to_o the_o angle_n p_o likewise_o the_o triangleâ—_n dhk_n and_o qhk_v to_o the_o angle_n a_o moreover_o the_o triangle_n eqt_a and_o vqt_a to_o the_o angle_n l_o and_o final_o the_o triangle_n sxm_n and_o txm_n to_o the_o angle_n c_o these_o 12._o triangle_n be_v add_v to_o thâ—â—_n for_o 〈◊〉_d triangle_n shall_v produce_v _o triangle_n equal_a and_o equilater_v couple_v together_o which_o shall_v male_a a_o icosahedron_n by_o the_o 25._o definition_n of_o the_o eleven_o and_o it_o shall_v be_v inscribe_v in_o the_o octohedron_n give_v abcâ—â—l_o by_o the_o first_o definition_n of_o this_o book_n for_o the_o 1â—_o angle_n thereof_o be_v set_v in_o 1â—_o like_a section_n of_o the_o side_n of_o the_o octohedron_n wherefore_o in_o a_o octohedron_n give_v be_v inscribe_v a_o icosahedron_n ¶_o first_n corollary_n the_o side_n of_o a_o equilater_n triangle_n be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n a_o right_a line_n subtend_v within_o the_o triangle_n the_o angle_n which_o be_v contain_v under_o the_o great_a segment_n and_o the_o less_o be_v in_o power_n duple_n to_o the_o less_o segment_n of_o the_o same_o side_n for_o the_o line_n ke_o which_o subtend_v the_o angle_n kfe_n of_o the_o triangle_n afl_fw-mi which_o angle_n kfe_n be_v contain_v under_o the_o two_o segment_n kf_a &_o fe_o be_v prove_v equal_a 〈◊〉_d the_o line_n hk_o which_o contain_v in_o power_n the_o two_o less_o segment_n ha_o and_o ak_o by_o the_o 47._o of_o the_o â—â—rst_a foâ—_n 〈◊〉_d angle_n hak_n be_v 〈…〉_o second_o corollary_n the_o base_n of_o the_o icosahedron_n be_v concentrical_a that_o be_v have_v one_o and_o the_o self_n same_o centre_n with_o the_o base_n of_o the_o octohedron_n which_o contain_v it_o for_o suppose_v that_o 〈…〉_o octohedron_n 〈◊〉_d ecd_v the_o base_a of_o a_o icosahedron_n and_o let_v the_o centre_n of_o the_o base_a abg_o be_v the_o point_n f._n and_o draw_v these_o right_a line_n favorina_n fb_o fc_o and_o fe_o now_o than_o the_o 〈…〉_o to_o the_o two_o line_n fb_o and_o bc_o for_o they_o be_v line_n draw_v from_o the_o centre_n and_o be_v also_o less_o segment_n and_o they_o contain_v the_o 〈…〉_o ¶_o the_o 17._o problem_n the_o 17._o proposition_n in_o a_o octohedron_n give_v to_o inscribe_v a_o dodecahedron_n construction_n svppose_v that_o the_o octohedron_n give_v be_v abgdec_n who_o 12._o â—ides_n let_v be_v cut_v by_o a_o extreme_a and_o mean_a proportion_n as_o in_o the_o former_a proposition_n it_o be_v manifest_a that_o of_o the_o right_a line_n which_o couple_v thâ—se_a section_n be_v make_v 20._o triangle_n of_o which_o 8._o be_v concentrical_a with_o the_o base_n of_o the_o octohedron_n by_o the_o second_o corollary_n of_o the_o former_a proposition_n if_o therefore_o in_o every_o one_o of_o the_o centre_n of_o the_o 20._o triangle_n be_v inscribe_v by_o the_o 1._o of_o this_o book_n every_o one_o of_o the_o â—â—_o â—â—gles_n of_o the_o dodecahedron_n demonstration_n we_o shall_v find_v that_o â—_o angle_n of_o the_o dodecahedron_n be_v set_v in_o the_o 8._o centre_n of_o the_o base_n of_o the_o octohedron_n namely_o these_o angle_n i_o u._fw-mi ct_n oh_o m_o a_o p_o and_o x_o and_o of_o the_o other_o 12._o solid_a angle_n there_o be_v two_o in_o the_o centre_n of_o the_o two_o triangle_n which_o have_v one_o side_n common_a under_o every_o one_o of_o the_o solid_a angle_n of_o the_o octohedron_n namely_o under_o the_o solid_a angle_n a_o the_o two_o solid_a angle_n king_n z_o under_o the_o solid_a angle_n b_o the_o two_o solid_a angle_n h_o t_o under_o the_o solid_a angle_n g_o the_o two_o solid_a angle_n in_o v_z under_o the_o solid_a angle_n d_o the_o two_o solid_a angle_n f_o l_o under_o the_o solid_a angle_n e_o the_o two_o solid_a angle_n s_o n_o under_o the_o solid_a angle_n c_o the_o two_o solid_a angle_n queen_n r_o and_o forasmuch_o as_o in_o the_o octohedron_n be_v six_o solid_a angle_n under_o they_o shall_v be_v subtend_v 12._o solid_a angle_n of_o the_o dodâ—cahedron_n and_o so_o be_v mâ—de_v 20._o solid_a angle_n compose_v of_o 12._o equal_a and_o equilater_v superficial_a pentagon_n as_o it_o be_v 〈◊〉_d by_o the_o 5._o of_o this_o book_n which_o therefore_o contain_v a_o dodecahedron_n by_o the_o 24._o definition_n of_o the_o eleven_o and_o it_o be_v inscribe_v in_o the_o octohedron_n by_o the_o 1._o definition_n of_o this_o book_n for_o that_o every_o one_o of_o the_o base_n of_o the_o octohedron_n do_v receive_v angle_n thereof_o wherefore_o in_o a_o octohedron_n give_v be_v inscribe_v a_o dodecahedron_n ¶_o the_o 18._o problem_n the_o 18._o proposition_n in_o a_o trilater_n and_o equilater_n pyramid_n to_o inscribe_v a_o cube_n construction_n svppose_v that_o there_o be_v a_o trilater_n equilater_n pyramid_n who_o base_a let_v be_v abc_n and_o â—oppe_v the_o point_n d._n and_o let_v it_o be_v comprehend_v in_o a_o sphereâ—_n by_o the_o 13._o of_o the_o 〈◊〉_d and_o lâ—â—_n the_o centre_n of_o that_o sphere_n be_v the_o point_n e._n and_o from_o the_o solid_a angle_n a_o b_o c_o d_o draw_v right_a line_n pass_v by_o the_o centre_n e_o unto_o the_o opposite_a base_n of_o
side_n gd_v the_o angle_n m_o n_o under_o the_o side_n ab_fw-la the_o angle_n t_o s_o under_o the_o side_n bg_o the_o angle_n p_o oh_o and_o under_o the_o side_n agnostus_n the_o angle_n r_o q_o so_o there_o rest_v 4._o angle_n who_o true_a place_n we_o will_v now_o appoint_v forasmuch_o as_o a_o cube_fw-la contain_v in_o one_o and_o the_o self_n same_o sphere_n with_o a_o dodecahedron_n be_v inscribe_v in_o the_o same_o dodecahedron_n as_o it_o be_v manifest_a by_o the_o 17._o of_o the_o thirteen_o and_o 8._o of_o this_o book_n it_o follow_v that_o a_o cube_fw-la and_o a_o dodecahedron_n circumscribe_v about_o it_o be_v contain_v in_o one_o and_o the_o self_n same_o body_n for_o that_o their_o angle_n concur_v in_o one_o and_o the_o self_n same_o point_n and_o it_o be_v prove_v in_o the_o 18._o of_o this_o book_n that_o 4._o angle_n of_o the_o cube_fw-la inscribe_v in_o the_o pyramid_n be_v set_v in_o the_o middle_a section_n of_o the_o perpendicularâ—_n which_o be_v draw_v from_o the_o solid_a angle_n of_o the_o pyramid_n to_o the_o opposite_a base_n wherefore_o the_o other_o 4._o angle_n of_o the_o dodecahedron_n be_v also_o as_o the_o angle_n of_o the_o cube_fw-la set_v in_o those_o middle_a section_n of_o the_o perpendicular_n namely_o the_o angle_n v_o be_v set_v in_o the_o midst_n of_o the_o perpendicular_a ahâ—_n the_o angle_n y_fw-fr in_o the_o midst_n of_o the_o perpendicular_a bf_o the_o angle_n x_o in_o the_o midst_n of_o the_o perpendicular_a ge_z and_o last_o the_o angle_n d_o in_o the_o midst_n of_o the_o perpendicular_a d_o which_o be_v draw_v from_o the_o top_n of_o the_o pyramid_n to_o the_o opposite_a base_a wherefore_o those_o 4._o angle_n of_o the_o dodecahedron_n may_v be_v say_v to_o be_v direct_o under_o the_o solid_a angle_n of_o the_o pyramid_n or_o they_o may_v be_v say_v to_o be_v set_v at_o the_o perpendicular_n wherefore_o the_o dodecahedron_n after_o this_o manner_n set_v be_v inscribe_v in_o the_o pyramid_n give_v by_o the_o first_o definition_n of_o this_o book_n for_o that_o upon_o every_o one_o of_o the_o base_n of_o the_o pyramid_n be_v set_v a_o angle_n of_o the_o dodecahedron_n inscribe_v wherefore_o in_o a_o trilater_n equilater_n pyramid_n be_v inscribe_v a_o dodecahedron_n the_o 21._o problem_n the_o 21._o proposition_n in_o every_o one_o of_o the_o regular_a solid_n to_o inscribe_v a_o sphere_n in_o the_o 13._o of_o thâ—_z thirteen_o and_o thâ—_z other_o 4._o proposition_n follow_v he_o iâ—_z be_v declare_v that_o â—he_n â—â—_o regular_a solidesâ—â—re_o so_o contain_v in_o a_o sphere_n that_o â—ight_a linâ—â—_n draw_v from_o the_o cenâ—â—â—_n oâ—_n the_o 〈…〉_o of_o 〈◊〉_d solid_a inscribe_v be_v equal_a which_o right_a line_n therefore_o make_v pyramid_n who_o â—oppes_n be_v the_o centre_n of_o the_o sphere_n or_o of_o the_o solid_a and_o the_o basâ—â—â—â—e_a cuâ—â—â—_n one_o of_o the_o base_n of_o those_o solid_n and_o 〈…〉_o solid_a squall_n and_o like_v the_o one_o to_o the_o other_o and_o describe_v in_o equal_a circle_n those_o circle_n shall_v cut_v the_o sphere_n for_o the_o angle_n which_o touch_v the_o circumference_n of_o the_o circle_n touch_v also_o the_o superficies_n of_o the_o sphere_n wherefore_o perpendiculars_n draw_v from_o the_o centre_n of_o the_o sphere_n to_o the_o base_n or_o to_o the_o plain_a superficiece_n of_o the_o equal_a circle_n be_v equal_a by_o the_o corollary_n of_o the_o assumpt_n of_o the_o 1â—_o of_o the_o twelve_o wherefore_o make_v the_o centre_n the_o 〈◊〉_d of_o the_o sphere_n which_o 〈◊〉_d the_o solid_a and_o thâ—_z space_n some_o one_o of_o the_o equal_a perpendicularâ—_n dâ—scribâ—_n a_o sphere_n and_o it_o shall_v touch_v every_o one_o of_o the_o base_n of_o 〈◊〉_d solid_a 〈…〉_o perficies_fw-la of_o the_o sphere_n pass_v beyond_o those_o base_n when_o as_o those_o pâ—â—peâ—diculars_n 〈…〉_o be_v draw_v from_o the_o centre_n to_o the_o base_n by_o the_o 3._o corollary_n of_o the_o saâ—â—â—â—â—umpt_n wherâ—fore_o â—e_v have_v iâ—_z every_o one_o of_o the_o regular_a body_n inscribe_v a_o sphere_n which_o regular_a boâ—â—â—_n be_v in_o number_n one_o iâ—_z 〈◊〉_d by_o the_o corollary_n of_o the_o 1â—_o of_o the_o 〈◊〉_d a_o corollary_n the_o regular_a figure_n inscribe_v in_o sphere_n and_o also_o the_o sphere_n circumscribe_v about_o they_o or_o contain_v they_o have_v one_o and_o the_o self_n same_o centre_n namely_o their_o pyramid_n the_o â—ngles_n of_o who_o base_n touch_v the_o superâ—â—â—â—â—â—_n of_o thâ—â—â—here_o do_v from_o those_o angle_n cause_v equal_a right_a line_n to_o be_v drawâ—â—_n to_o one_o and_o â—he_n self_n 〈◊〉_d poynâ—_n make_v the_o topâ—â—â—_n of_o the_o pyramid_n in_o the_o same_o point_n and_o therefore_o they_o 〈…〉_o thâ—_z câ—â—tres_n of_o the_o sphere_n in_o the_o self_n same_o top_n when_o 〈◊〉_d the_o right_a line_n draw_v from_o those_o angle_n to_o the_o croâ—â—ed_a superficies_n wherein_o be_v 〈◊〉_d the_o angle_n of_o the_o base_n of_o the_o pyramid_n be_v equal_o a_o advertisement_n of_o flussasâ—_n â—_z of_o these_o solid_n only_o the_o octohedron_n receive_v the_o other_o solid_n inscribe_v one_o with_o 〈…〉_o other_o for_o the_o octohedron_n contain_v the_o icosahedron_n inscribe_v in_o it_o and_o the_o same_o icosahedron_n contain_v the_o dodecahedron_n inscribe_v in_o the_o same_o icosahedron_n and_o the_o same_o dodecahedron_n contain_v the_o cube_fw-la inscribe_v in_o the_o same_o octohedron_n and_o 〈…〉_o â—â—râ—â—mscribeth_v the_o pyramid_n inscribe_v in_o the_o say_v octohedron_n but_o this_o happen_v not_o in_o the_o other_o solid_n the_o end_n of_o the_o fivetenth_fw-mi book_n of_o euclides_n elemenâ—â—â—_n after_o caâ—paâ—â—_n and_o 〈◊〉_d ¶_o the_o sixteen_o book_n of_o the_o element_n of_o geometry_n add_v by_o flussas_n in_o the_o former_a fivetenth_fw-mi book_n have_v be_v teach_v how_o to_o inscribe_v the_o five_o regular_a solid_n one_o with_o in_o a_o other_o now_o seem_v to_o rest_n to_o compare_v those_o solid_a so_o inscribe_v one_o to_o a_o other_o and_o to_o set_v forth_o their_o passion_n and_o propriety_n which_o thing_n flussas_n consider_v in_o this_o sixteen_o book_n add_v by_o he_o book_n have_v excellent_o well_o and_o most_o cunning_o perform_v for_o which_o undoubted_o he_o have_v of_o all_o they_o which_o have_v a_o love_n to_o the_o mathematicals_n deserve_v much_o praise_n and_o commendation_n both_o for_o the_o great_a traâ—ailes_n and_o paynâ—s_n which_o it_o be_v most_o likely_a he_o have_v taâ—â—n_v in_o invent_v such_o strange_a and_o wonderful_a proposition_n with_o their_o demonstration_n in_o this_o book_n contain_v as_o also_o for_o participate_v and_o communicate_v abroad_o the_o same_o to_o other_o which_o book_n also_o that_o the_o reader_n shall_v want_v nothing_o conduce_v to_o the_o perfection_n of_o euclides_n element_n i_o have_v with_o some_o travail_n translate_v &_o for_o the_o worthiness_n â—hereof_o have_v add_v it_o aâ—_z a_o sixteen_o book_n to_o the_o 15._o book_n of_o euclid_n vouchsafe_v therefore_o gentle_a reader_n diligent_o to_o read_v and_o poise_v it_o for_o in_o it_o shall_v you_o find_v noâ—_n only_a matter_n strange_a and_o delectable_a but_o also_o occasion_n of_o invention_n of_o great_a thing_n pertain_v to_o the_o nature_n of_o the_o five_o regular_a solidâ—sâ—_n ¶_o the_o 1._o proposition_n a_o dodecahedron_n and_o a_o cube_fw-la inscribe_v in_o it_o and_o a_o pyramid_n inscribe_v in_o the_o same_o cube_fw-la be_v contain_v in_o one_o and_o the_o self_n same_o sphere_n for_o the_o angle_n of_o the_o pyramiâ—_n be_v seâ—_z in_o the_o angel_n of_o the_o cube_fw-la wherein_o it_o be_v inscribe_v by_o the_o first_o of_o the_o fivetenthâ—_n and_o all_o the_o angle_n of_o the_o cube_fw-la be_v set_v in_o the_o angle_n of_o the_o dodecahedâ—â—â—_n circumscribe_v 〈…〉_o 〈◊〉_d the_o 8._o of_o the_o fifteen_o and_o all_o the_o angle_n of_o the_o dodecahedron_n be_v set_v in_o the_o superficies_n of_o the_o sphere_n by_o the_o 17._o of_o the_o thirteen_o wherefore_o those_o three_o solid_n inscribe_v one_o within_o a_o other_o be_v contain_v in_o one_o and_o the_o self_n same_o sphere_n by_o the_o first_o definition_n of_o the_o fifteen_o a_o dodecahedron_n therefore_o and_o a_o cube_fw-la inscribe_v in_o it_o and_o a_o pyramid_n inscribe_v in_o the_o same_o cube_fw-la be_v contain_v 〈…〉_o â—â—lfe_n same_o sphere_n 〈…〉_o these_o three_o solid_n livre_n 〈…〉_o elf_n same_o icosahedron_n or_o octohedron_n or_o pyramid_n 〈…〉_o i_o icosahedron_n by_o the_o 5.11_o &_o 12._o of_o the_o fifteen_o and_o they_o be_v 〈…〉_o ctohedron_n by_o the_o 4._o 6._o and_o 16._o of_o the_o same_o last_o they_o be_v inscribe_v in_o 〈…〉_o the_o first_o 18._o and_o 19_o of_o the_o same_o for_o the_o angle_n of_o all_o these_o solid_a 〈…〉_o the_o circumscribe_v icosahedron_n or_o octohedron_n or_o pyramid_n ¶_o the_o 〈…〉_o the_o proportion_n of_o a_o dodecahedron_n circumscribe_v about_o a_o cube_fw-la to_o a_o dodecahedron_n inscribe_v in_o the_o same_o cube_fw-la be_v
line_n a_fw-la be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n in_o the_o point_n h_o by_o the_o â—_o of_o the_o thirteen_o but_o as_o the_o line_n a_o be_v to_o the_o line_n ah_o so_o be_v the_o line_n ad_fw-la to_o the_o line_n ae_n by_o the_o 2._o of_o six_o for_o the_o lineâ—_z fh_o and_o on_o be_v parallel_n and_o again_o as_o the_o line_n ad_fw-la be_v to_o the_o line_n ae_n so_o by_o the_o same_o be_v the_o line_n agnostus_n to_o the_o line_n aq_n and_o the_o line_n ai_fw-fr to_o the_o line_n ap_n for_o the_o line_n pq_n and_o give_v be_v parallel_n wherefore_o the_o line_n agnostus_n and_o a_z be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n in_o the_o point_v q_n &_o p_o &_o the_o line_n aq_n shall_v be_v the_o great_a segment_n of_o the_o line_n agnostus_n or_o ab_fw-la and_o forasmuch_o as_o the_o wholâ—_n line_n agnostus_n be_v to_o the_o great_a segment_n aq_n as_o the_o great_a segment_n a_z be_v to_o the_o residue_n ap_n the_o line_n aâ—_z shall_v be_v the_o less_o segment_n of_o the_o whole_a line_n aâ—_z or_o ag._n wherefore_o the_o liâ—â—_n peq_n which_o by_o the_o point_n e_o pass_v parallelwise_o to_o the_o line_n give_v cut_v the_o line_n agnostus_n and_o basilius_n by_o a_o extreme_a and_o mean_a proportion_n in_o the_o point_n queen_n and_o p._n and_o by_o the_o same_o reason_n the_o line_n â—r_n which_o by_o the_o point_n c_o pass_v parallelwise_o to_o the_o line_n be_o shall_v fall_v upon_o the_o section_n p_o and_o r_o so_o also_o shall_v the_o line_n rq_n which_o by_o the_o point_n d_z pass_v parallelwise_o to_o the_o line_n bl_o fall_v upoâ—_n the_o section_n rq_n wherefore_o either_o of_o the_o line_n pe_n and_o eq_n shall_v be_v equal_a to_o the_o line_n cd_o in_o the_o parallelogram_n pd_v and_o qc_n by_o the_o 34._o of_o the_o first_o and_o forasmuch_o as_o the_o line_n pe_n and_o eq_n be_v equal_a the_o line_n pc_n cr_n rd_o and_o dq_n shall_v be_v likewise_o equal_a wherefore_o the_o triangle_n prq_n iâ—â—quilater_n and_o cut_v the_o side_n of_o the_o base_a of_o the_o pyramiâ—_n in_o the_o point_n p_o q_o r_o by_o a_o extreme_a and_o mean_a proportion_n and_o in_o it_o be_v inscribe_v the_o base_a ecd_v of_o the_o icosahedron_n contain_v in_o the_o foresaid_a pyramid_n if_o therefore_o from_o the_o angle_n of_o the_o base_a of_o a_o pyramid_n be_v draw_v to_o the_o opposite_a side_n right_a line_n cut_v the_o say_a side_n by_o a_o extreme_a and_o mean_a proportion_n they_o shall_v contain_v the_o base_a of_o the_o icosahedron_n inscribe_v in_o the_o pyramid_n which_o base_a shall_v be_v inscribe_v in_o a_o equilater_n triangle_n who_o angle_n cut_v the_o side_n of_o the_o base_a of_o the_o pyramid_n by_o a_o extreme_a &_o mean_a proportion_n ¶_o a_o corollary_n the_o side_n of_o a_o icosahedron_n inscribe_v in_o a_o octohedron_n be_v the_o great_a segment_n of_o the_o line_n which_o be_v draw_v from_o the_o angle_n of_o the_o base_a of_o the_o octohedron_n cut_v the_o opposite_a side_n by_o a_o extreme_a and_o mean_a proportion_n for_o by_o the_o 16._o of_o the_o fifteen_o fkh_n be_v the_o base_a of_o the_o octohedron_n which_o contain_v the_o base_a of_o the_o icosahedron_n cde_v unto_o which_o triangle_n fkh_n the_o triangle_n hkg_n be_v equal_a as_o have_v be_v prove_v by_o the_o point_n h_o draw_v unto_o the_o line_n menander_n a_o parallel_a line_n ht_o cut_v the_o line_n dn_o in_o the_o point_n s._n wherefore_o es_fw-ge dt_fw-ge and_o et_fw-la be_v parallelogram_n and_o therefore_o the_o line_n eh_o and_o mt_n be_v equal_a and_o the_o line_n em_n and_o ht_o be_v like_o cut_n in_o the_o point_n d_o and_o s_o by_o the_o 34._o of_o the_o first_o wherefore_o the_o great_a segment_n of_o the_o line_n ht_o be_v the_o line_n his_o which_o be_v equal_a to_o ed_z the_o side_n of_o the_o icosahedron_n but_o by_o the_o 2._o of_o the_o six_o the_o line_n tk_n be_v cut_v like_o to_o the_o line_n hk_o by_o the_o parallel_n dm_o and_o therefore_o by_o the_o 2._o of_o the_o fourteen_o it_o be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n but_o the_o line_n tm_n be_v equal_a to_o the_o line_n eh_o wherefore_o also_o the_o line_n tk_n be_v equal_a to_o the_o line_n of_o or_o dh_o wherefore_o the_o residue_v eh_o and_o tg_n be_v equal_a for_o the_o whole_a line_n fh_a and_o kg_o be_v equal_a wherefore_o kg_o the_o side_n of_o the_o triangle_n hkg_n be_v in_o the_o point_n t_o divide_v by_o a_o extreme_a and_o mean_a proportion_n in_o the_o point_n t_o by_o the_o right_a line_n ht_o and_o the_o great_a segment_n thereof_o be_v the_o line_n ed_z the_o side_n of_o the_o icosahedron_n inscribe_v in_o the_o octohedron_n who_o base_a be_v the_o triangle_n hkg_v or_o the_o triangle_n fkh_n which_o be_v equal_a to_o the_o triangle_n hkg_v by_o the_o 16._o of_o the_o fifteen_o ¶_o the_o 5._o proposition_n the_o side_n of_o a_o pyramid_n divide_v by_o a_o extreme_a and_o mean_a proportion_n make_v the_o less_o segment_n in_o power_n double_a to_o the_o side_n of_o the_o icosahedron_n inscribe_v in_o it_o svppose_v that_o abg_o be_v the_o base_a of_o a_o pyramid_n construction_n and_o let_v the_o base_a of_o the_o icosahedron_n inscribe_v in_o it_o be_v cde_o describe_v of_o three_o right_a line_n which_o be_v draw_v from_o the_o angle_n of_o the_o base_a abg_o cut_v the_o opposite_a side_n by_o a_o extreme_a and_o mean_a proportion_n by_o the_o former_a proposition_n namely_o of_o these_o three_o line_n be_o by_o and_o give_v then_o i_o say_v that_z a_z the_o less_o segment_n of_o the_o side_n aâ—_z be_v in_o power_n duple_n to_o ce_fw-fr the_o side_n of_o the_o icosahedron_n for_o forasmuch_o as_o by_o the_o former_a proposition_n it_o be_v prove_v that_o the_o triangle_n cde_o be_v inscribe_v in_o a_o equilater_n triangle_n demonstration_n who_o angle_n cut_v the_o side_n of_o abg_o the_o base_a of_o the_o pyramid_n by_o a_o extreme_a and_o mean_a proportion_n let_v that_o triangle_n be_v fhk_v cut_v the_o line_n ab_fw-la in_o the_o point_n f._n wherefore_o the_o less_o segment_n favorina_n be_v equal_a to_o the_o segment_n ai_fw-fr by_o the_o 2._o of_o the_o fourteen_o for_o the_o line_n ab_fw-la and_o ag_z be_v cut_v like_o moreover_o the_o side_n fh_o of_o the_o triangle_n fhk_n be_v in_o the_o point_n d_o cut_n into_o two_o equal_a part_n as_o in_o the_o former_a proposition_n it_o be_v prove_v and_o fce_v also_o by_o the_o same_o be_v a_o parallelogram_n wherefore_o the_o line_n ce_fw-fr and_o fd_o be_v equal_a by_o the_o 33_o of_o the_o first_o and_o forasmuch_o as_o the_o line_n fh_o subtend_v the_o angle_n bag_n of_o a_o equilater_n triangle_n which_o angle_n be_v contain_v under_o the_o great_a segment_n ah_o and_o the_o less_o segment_n afâ—_n therefore_o the_o line_n fh_o be_v in_o power_n double_a to_o the_o line_n of_o or_o to_o the_o line_n ai_fw-fr the_o less_o segment_n by_o the_o corollary_n of_o the_o 16._o of_o the_o fifteen_o but_o the_o same_o line_n fh_o be_v in_o power_n quadruple_a to_o the_o line_n ce_fw-fr by_o the_o 4._o of_o the_o second_o for_o the_o line_n fh_o be_v double_a to_o the_o line_n ce_fw-fr wherefore_o the_o line_n ai_fw-fr be_v the_o half_a of_o the_o square_n of_o the_o line_n fh_o be_v in_o power_n duple_n to_o the_o line_n ce_fw-fr to_o which_o the_o line_n fh_o be_v in_o power_n quadruple_a wherefore_o the_o side_n agnostus_n of_o the_o pyramid_n be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n make_v thâ—_z less_o segment_n ai_fw-fr in_o power_n duple_n to_o the_o side_n ce_fw-fr of_o the_o icosahedron_n inscribe_v in_o it_o ¶_o a_o corollary_n the_o side_n of_o a_o icosahedron_n inscribe_v in_o a_o pyramid_n be_v a_o residual_a line_n for_o the_o diameter_n of_o the_o sphere_n which_o contain_v the_o five_o regular_a body_n be_v rational_a be_v in_o power_n sesquialtera_fw-la to_o the_o side_n of_o the_o pyramid_n by_o the_o 13._o of_o the_o thirteen_o and_o therefore_o the_o side_n of_o the_o pyramid_n be_v rational_a by_o the_o definition_n which_o side_n be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n make_v the_o less_o segment_n a_o residual_a line_n by_o the_o 6._o of_o the_o thirteen_o wherefore_o the_o side_n of_o the_o icosahedron_n be_v commensurable_a to_o the_o same_o less_o segment_n for_o the_o square_n of_o the_o side_n of_o the_o icosahedron_n be_v the_o half_a of_o the_o square_n of_o the_o say_v less_o segment_n be_v a_o residual_a line_n by_o that_o which_o be_v add_v after_o the_o 103._o of_o the_o ten_o book_n ¶_o the_o 6._o proposition_n the_o side_n of_o a_o cube_n contain_v in_o power_n half_o the_o side_n of_o a_o equilater_n triangular_a pyramid_n inscribe_v in_o the_o say_a
proposition_n after_o prâ—â—lus_n a_o corollary_n take_v out_o of_o flussates_n demonstration_n lead_v to_o 〈◊〉_d absurdity_n a_o addition_n oâ—_n pelitarius_n demonstration_n three_o case_n in_o this_o proposition_n the_o first_o case_n construction_n demonstration_n three_o case_n in_o this_o proposition_n the_o first_o case_n every_o case_n may_v happen_v seven_o diverse_a way_n the_o like_a variety_n in_o each_o of_o the_o other_o two_o case_n euclides_n construction_n and_o demostration_n serve_v in_o all_o these_o case_n and_o in_o their_o varity_n also_o construction_n demonstration_n how_o triangle_n be_v say_v to_o be_v in_o the_o self_n same_o parallel_a line_n comparison_n of_o two_o triangle_n who_o side_n be_v equal_a their_o base_n and_o angle_n at_o the_o top_n be_v unequal_a when_o they_o be_v less_o than_o two_o right_a angle_n construction_n demonstration_n three_o case_n in_o this_o proposition_n each_o of_o these_o case_n also_o may_v be_v diverse_o note_n an_o other_o addition_n of_o pelitarius_n construction_n demonstration_n this_o theorem_a the_o converse_n of_o the_o 37._o proposition_n a_o addition_n of_o flâ—ssases_n a_o addition_n of_o campanus_n construction_n demonstration_n lead_v to_o a_o absurdity_n this_o proposition_n be_v the_o converse_n of_o the_o 38._o proposition_n demonstration_n two_o case_n in_o this_o proposition_n a_o corollary_n the_o self_n same_o demonstration_n will_v serve_v if_o the_o triangle_n &_o the_o parallelogram_n be_v upon_o equal_a base_n the_o converse_n of_o this_o proposition_n an_o other_o converse_n of_o the_o same_o proposition_n comparison_n of_o a_o triangle_n and_o a_o trapesium_n be_v upon_o one_o &_o the_o self_n same_o base_a and_o in_o the_o self_n same_o parallel_a line_n construction_n demonstration_n supplement_n &_o complement_n three_o case_n in_o this_o theorem_a the_o first_o case_n this_o proposition_n call_v gnomical_a and_o mystical_a the_o converse_n of_o this_o proposition_n construction_n demonstration_n application_n of_o space_n with_o excess_n or_o want_v a_o ancient_a invention_n of_o pythagoras_n how_o a_o figure_n be_v say_v to_o be_v apply_v to_o a_o line_n three_o thing_n give_v in_o this_o proposition_n the_o converse_n of_o this_o proposition_n construction_n demonstration_n a_o addition_n of_o pelitarius_n to_o describe_v a_o square_n mechanical_o a_o addition_n of_o procâ—â—â—_n the_o converse_n thereof_o construction_n demonstration_n pythagoras_n the_o first_o inventor_n of_o this_o proposition_n a_o addition_n of_o pâ—lâ—tariâ—â—_n an_o other_o addition_n of_o pelitarius_n an_o other_o addition_n of_o pelitarius_n an_o other_o addition_n of_o pelitarius_n a_o corollary_n this_o proposition_n be_v the_o converse_n of_o the_o former_a the_o argument_n of_o the_o second_o book_n what_o be_v the_o power_n of_o a_o line_n many_o compendious_a rule_n of_o reckon_v gather_v one_o of_o this_o book_n and_o also_o many_o rule_n of_o algebra_n two_o wonderful_a proposition_n in_o this_o book_n first_o definition_n what_o a_o parallelogram_n be_v four_o kind_n of_o parallelogram_n second_o definition_n a_o proposition_n add_v by_o campane_n after_o the_o last_o proposition_n of_o the_o first_o book_n construction_n demonstration_n barlaam_n barlaam_n construction_n demonstration_n barlaam_n construction_n demonstration_n barlaam_n construction_n demonstration_n a_o corollary_n barlaam_n construction_n demonstration_n constrâ—ction_n demonstration_n construction_n demonstration_n construction_n demonstration_n many_o and_o singular_a use_n of_o this_o proposition_n this_o proposition_n can_v not_o be_v reduce_v unto_o number_n demonstration_n demonstration_n a_o corollary_n this_o proposition_n true_a in_o all_o kind_n of_o triangle_n construction_n demonstration_n the_o argument_n of_o this_o book_n the_o first_o definition_n definition_n of_o unequal_a circle_n second_o definition_n a_o contigent_a line_n three_o definition_n the_o touch_n of_o circle_n be_v 〈◊〉_d in_o one_o poâ—â—â—_n only_o circle_n may_v touch_v together_o two_o maâ—â—â—_n of_o way_n four_o definition_n five_o definition_n six_o definition_n mix_v angle_n arke_n chord_n seven_o definition_n difference_n of_o a_o angle_n of_o a_o section_n and_o of_o a_o angle_n in_o a_o section_n eight_o definition_n nine_o definition_n ten_o definition_n two_o definition_n first_o second_o why_o euclid_n define_v not_o equal_a section_n constuction_n demonstration_n lead_v to_o a_o impossibility_n correlary_a demonstration_n lead_v to_o a_o impossibility_n the_o first_o para_fw-it of_o this_o proposition_n construction_n demonstration_n the_o second_o part_n converse_v of_o the_o first_o demonstration_n demonstration_n lead_v to_o a_o impossibility_n two_o case_n in_o this_o proposition_n construction_n demonstration_n lead_v to_o a_o impossibility_n demonstration_n lead_v to_o a_o impossibility_n two_o caseâ—_n in_o this_o proposition_n construction_n the_o first_o part_n of_o this_o proposition_n demonstration_n second_o part_n three_o part_n this_o demonstrate_v by_o a_o argument_n lead_v to_o a_o impossibilie_n an_o other_o demonstration_n of_o the_o latter_a part_n of_o the_o proposition_n lead_v also_o to_o a_o impossibility_n a_o corollary_n three_o part_n an_o other_o demonstration_n of_o the_o latter_a part_n lead_v also_o to_o a_o impossibility_n this_o proposion_n be_v common_o call_v caâ—dâ—_n panonis_fw-la a_o corollary_n construction_n demonstration_n an_o other_o demonstration_n of_o the_o same_o lead_v also_o to_o a_o impossibility_n demonstration_n lead_v to_o a_o impossibility_n an_o other_o demonstration_n of_o the_o same_o lead_v also_o to_o a_o impossibility_n construction_n demonstration_n lead_v to_o a_o impossibility_n an_o other_o demonstration_n of_o the_o same_o lead_v also_o to_o a_o impossibility_n the_o same_o again_o demonstrate_v by_o a_o argument_n lead_v to_o a_o absurdititie_n demonstratiâ—_n lead_v to_o a_o impossibility_n an_o other_o demonstration_n after_o pelitarius_n lead_v also_o to_o a_o absurdity_n of_o circle_n which_o touch_v the_o one_o the_o other_o inward_o of_o circle_n which_o touch_v the_o one_o the_o other_o outward_o an_o other_o demonstration_n after_o pelitarius_n &_o flussates_n of_o circle_n which_o touch_n the_o one_o the_o other_o outward_o of_o circle_n which_o touch_n the_o one_o the_o other_o inward_o the_o first_o part_n of_o this_o theorem_a construction_n demonstration_n demonstration_n the_o second_o part_n which_o be_v the_o converse_n of_o the_o first_o an_o other_o demonstration_n of_o the_o first_o part_n after_o campane_n construction_n demonstration_n an_o other_o demonstration_n after_o campane_n the_o first_o part_n of_o this_o theorem_a demonstration_n lead_v to_o a_o absurdity_n second_o part_n three_o part_n construction_n demonstration_n a_o addition_n of_o pelitarius_n this_o problem_n commodious_a for_o the_o inscribe_v and_o circumscribe_v of_o figure_n in_o or_o abouâ—_n circle_n demonstration_n lead_v to_o a_o impossibility_n an_o other_o demonstration_n after_o orontius_n demonstration_n lead_v to_o a_o impossibility_n two_o case_n in_o this_o proposition_n the_o one_o when_o the_o angle_n set_v at_o the_o circumference_n include_v the_o centre_n demonstration_n the_o other_o when_o the_o same_o angle_n set_v at_o the_o circumference_n include_v not_o the_o centre_n construction_n demonstration_n three_o case_n in_o this_o proposition_n the_o first_o case_n the_o second_o case_n the_o three_o case_n construction_n demonstration_n demonstration_n lead_v to_o a_o impossibility_n a_o addition_n of_o campane_n demonstrate_v by_o pelitariâ—s_n demonstration_n lead_v to_o a_o impossibility_n an_o other_o demonstration_n construction_n three_o case_n in_o this_o proposition_n the_o first_o case_n demonstration_n the_o second_o case_n the_o three_o case_n a_o addition_n construction_n demonstration_n demonstration_n lead_v to_o a_o impossibility_n construction_n demonstration_n the_o converse_n of_o the_o former_a proposition_n construction_n demonstration_n construction_n demonstration_n second_o part_n thirâ—_n part_n the_o five_o and_o last_o part_n an_o other_o demonstration_n to_o prove_v that_o the_o angâ—e_n in_o a_o semicircle_n be_v a_o right_a angle_n a_o corollary_n a_o addition_n of_o pâ—litarius_n demonstration_n leaâ—ing_v to_o a_o absurdity_n a_o addition_n of_o campane_n construction_n demonstration_n two_o case_n in_o this_o proposition_n three_o case_n in_o this_o proposition_n the_o first_o case_n construction_n demonstration_n the_o second_o case_n construction_n demonstration_n the_o three_o case_n construction_n demonstration_n construction_n demonstration_n two_o case_n in_o this_o proposition_n first_o case_n demonstration_n the_o second_o câ—se_n construction_n demonstration_n three_o case_n in_o this_o proposition_n construction_n two_o case_n in_o this_o proposition_n the_o first_o case_n demonstration_n the_o second_o case_n construction_n demonstration_n first_o corollary_n second_o corollary_n three_o corollary_n this_o proposition_n be_v the_o converse_n of_o the_o former_a construction_n demonstration_n an_o other_o demonstration_n after_o pelitarius_n the_o argument_n of_o this_o book_n first_o definition_n second_o definition_n the_o inscriptition_n and_o circumscription_n of_o rectiline_a figure_n pertain_v only_o to_o regular_a figure_n the_o three_o definition_n the_o four_o definition_n the_o five_o definition_n the_o six_o devition_n seven_o definition_n
casâ—_n demonstration_n construction_n demonstration_n demonstration_n lead_v to_o a_o impossibility_n note_v this_o manner_n of_o imagination_n mathematical_a demonstration_n lead_v to_o a_o impossibility_n construction_n demonstration_n demonstration_n lead_v to_o a_o absurdity_n in_o tâ—is_fw-la â—ronoâ—â—â—oâ—â—â—_n must_v understand_v the_o proportional_a â—artes_n or_o sâ—â—â—ions_n to_o be_v thâ—se_a which_o be_v contain_v 〈◊〉_d the_o parallel_a superficies_n construction_n demonstration_n construction_n demonstration_n demonstration_n lead_v to_o a_o impossibility_n demonstration_n construction_n demonstration_n two_o case_n in_o this_o proposition_n the_o first_o case_n second_o case_n construction_n demonstration_n an_o other_o demonstration_n construction_n demonstration_n construction_n three_o case_n in_o this_o proposition_n the_o first_o case_n a_o necessary_a thing_n to_o be_v prove_v before_o he_o procee_v any_o â—arther_a in_o the_o construction_n of_o the_o problem_n problem_n which_o how_o to_o find_v out_o be_v teach_v at_o the_o end_n of_o this_o demonstration_n and_o also_o be_v teach_v in_o the_o assumpt_n put_v before_o the_o 14._o proposition_n of_o the_o ten_o book_n demonstration_n of_o the_o first_o case_n second_o case_n three_o case_n an_o other_o demonstration_n to_o prove_v that_o thâ—_z line_n ab_fw-la be_v not_o less_o they_o the_o line_n lx._n this_o be_v before_o taâ—ght_n in_o the_o ten_o book_n in_o the_o assumpt_n put_v before_o the_o 14._o proposition_n proposition_n m._n dee_n to_o avoid_v cavillation_n add_v to_o euclides_n proposition_n this_o word_n six_o who_o i_o have_v follow_v according_o and_o not_o zamberts_n in_o this_o this_o kind_n of_o body_n mention_v in_o the_o proposition_n be_v call_v a_o parallelipipedom_n according_a to_o the_o definition_n before_o give_v thereof_o demonstration_n that_o the_o opposite_a side_n be_v parallelogram_n demonstration_n that_o the_o opposite_a superficies_n be_v equal_a equal_a ab_fw-la be_v equal_a to_o dc_o because_o the_o superficies_n ac_fw-la be_v prove_v a_o parallelogram_n and_o by_o the_o same_o reason_n be_v bh_o equal_a to_o cf_o because_o the_o superficies_n fb_o be_v prove_v a_o parallelogram_n therefore_o the_o 34._o of_o the_o first_o be_v our_o proof_n fâ—â—â—â—_n corollary_n second_o corolry_n these_o solid_n which_o he_o speak_v of_o in_o this_o corollary_n be_v of_o some_o call_v side_v column_n three_o corollary_n constrâ—ction_n demonstration_n demonstration_n look_v at_o the_o end_n of_o the_o demonstration_n what_o be_v understand_v by_o stand_a line_n john_n do_v you_o his_o figure_n by_o this_o figure_n it_o appear_v why_o â—uch_v prism_n be_v call_v pledge_n of_o 〈◊〉_d uâ—ry_a shape_n of_o a_o wedge_n as_o be_v the_o solid_a defgac_n etc_n etc_n stand_v line_n construction_n demonstration_n i_o dees_n figure_n two_o case_n in_o this_o proposition_n thâ—_z first_o case_n construction_n we_o be_v behold_v to_o m._n do_v you_o for_o invent_v this_o figure_n with_o other_o which_o till_o his_o reform_n be_v as_o much_o misshapen_a as_o this_o be_v and_o so_o both_o in_o the_o greek_a and_o latin_a copy_n remain_v demonstration_n demonstration_n note_v now_o how_o the_o base_a respective_o be_v takend_v for_o so_o may_v alteration_n of_o respect_n alter_v the_o name_n of_o the_o bound_n either_o of_o solid_n or_o plain_n second_o case_n case_n there_o you_o perceive_v how_o the_o base_a be_v diverse_o consider_v &_o choose_v as_o before_o we_o advertise_v you_o construction_n demonstration_n construction_n demonstration_n *_o *_o note_v this_o famous_a lemma_n the_o double_a oâ—â—he_n cube_n cube_n note_n note_n 〈…〉_o lemma_n note_v what_o iâ—_z yet_o lack_v requisite_a to_o the_o double_n of_o the_o cube_n the_o converse_n of_o both_o the_o part_n of_o the_o first_o case_n the_o converse_n of_o the_o second_o case_n the_o general_a conclusion_n construction_n demonstration_n of_o the_o first_o part_n demonstration_n of_o the_o second_o part_n to_o find_v two_o middle_a proportional_n between_o two_o number_n give_v note_v the_o practice_n of_o approach_v to_o precieâ—es_n in_o cubik_fw-mi root_n root_n this_o be_v the_o way_n to_o apply_v any_o square_n give_v to_o a_o line_n also_o give_v sufficienty_fw-mi extend_v a_o problem_n worth_a the_o search_n for_o construction_n demonstration_n construction_n demonstration_n demonstration_n it_o be_v evident_a that_o those_o perpendicular_n be_v all_o one_o with_o the_o stand_a line_n of_o the_o solid_n if_o their_o solid_a angle_n be_v make_v of_o superâ—icâ—â—il_n right_a angle_n only_o double_v of_o the_o 〈◊〉_d etc_n etc_n etc_n dâ—mâ—â—â—â—ratioâ—â—f_a possibility_n in_o the_o problem_n an_o other_o argument_n to_o comâ—oâ—t_v the_o studious_a demonstration_n of_o the_o first_o part_n demonstration_n of_o the_o second_o part_n which_o be_v the_o converse_n of_o the_o first_o part_n demonstration_n lead_v to_o a_o impossibility_n construction_n demonstration_n construction_n demonstration_n demonstration_n which_o of_o some_o be_v call_v side_v column_n side_v column_n construction_n demonstration_n the_o square_n of_o the_o circle_n demonstration_n â—eading_v to_o a_o impossibility_n two_o case_n in_o this_o proposition_n the_o first_o case_n that_o a_o square_a within_o any_o circle_n describe_v be_v big_a than_o half_a the_o circle_n that_o the_o isosceles_a triangle_n without_o the_o square_n be_v great_a than_o half_a the_o segment_n wherein_o they_o be_v second_o case_n case_n this_o as_o 〈…〉_o afterwards_o at_o the_o end_n of_o the_o demâ—straâ—ion_n prove_v construction_n demonstration_n construction_n demonstration_n diâ—â—reence_n between_o the_o first_o problem_n and_o the_o second_o construction_n demonstration_n construction_n demonstration_n construction_n demonstration_n construction_n note_v this_o well_o for_o it_o iâ—_z of_o great_a use_n an_o other_o way_n of_o demonstration_n of_o the_o fâ—â—st_a problem_n of_o thâ—â—_n addition_n note_v this_o properâ—ie_n of_o a_o triangle_n rectangle_n construction_n demonstration_n demonstration_n though_o i_o say_v without_o the_o square_n yet_o you_o must_v think_v that_o it_o may_v be_v also_o within_o the_o square_n &_o that_o diverse_o wherefore_o this_o problem_n may_v have_v diverse_a case_n so_o but_o brief_o to_o aside_o all_o may_v thus_o be_v say_v cut_v any_o side_n of_o that_o square_n into_o 3_n parte_v in_o the_o proportion_n of_o x_o to_o y._n note_v the_o manner_n of_o the_o drift_n in_o this_o demonstration_n and_o construction_n mix_o and_o with_o no_o determination_n to_o the_o construction_n as_o common_o iâ—_z in_o problemeâ—â—_n which_o be_v here_o of_o i_o so_o usedâ—_n for_o a_o example_n to_o young_a student_n of_o variety_n in_o art_n construction_n demonstration_n demonstration_n note_v and_o remember_v one_o â—eâ—th_n in_o these_o solid_n the_o conclusion_n of_o the_o first_o part_n demonstration_n of_o the_o second_o part_n namely_o that_o it_o be_v divide_v moreover_o into_o two_o equal_a prism_n conclusion_n of_o the_o second_o part_n demonstration_n of_o the_o last_o part_n that_o the_o two_o prism_n be_v great_a than_o the_o half_a of_o the_o whole_a pyramid_n conclusion_n of_o the_o last_o part_n conclusion_n of_o the_o whole_a proposition_n proposition_n a_o assumpt_n a_o assumpt_n conclusion_n of_o the_o whole_a demonstration_n lead_v to_o a_o impossibility_n impossibility_n in_o the_o assuâ—pâ—â—â—llowinâ—_n the_o second_o proposition_n of_o this_o bâ—â—ke_n construction_n demonstration_n demonstration_n note_n side_v column_n sometime_o call_v prism_n be_v triple_a to_o pyramid_n have_v one_o base_a and_o equal_a heâ—th_n with_o they_o note_v â—arallelipipedons_n treble_v to_o pyramid_n of_o one_o base_a and_o heith_n with_o they_o construction_n demonstration_n a_o addition_n by_o campane_n and_o flussas_n demonstration_n of_o the_o first_o part_n demonstration_n â—f_n the_o second_o part_n which_o iâ—_z the_o converse_n of_o the_o first_o constr_n 〈◊〉_d 〈◊〉_d parallelipipedon_n call_v prism_n prism_n by_o this_o it_o be_v manifest_a that_o euclid_n comprehend_v side_v column_n also_o under_o the_o name_n of_o a_o prism_n prism_n a_o prism_n have_v for_o his_o base_a a_o polygonon_n figure_n as_o we_o have_v often_o before_o note_v unto_o you_o note_v m._n do_v you_o his_o chief_a purpose_n in_o his_o addition_n demonstration_n touch_v cylinder_n second_o case_n second_o parâ—_n which_o concern_v cillinder_n construction_n demonstration_n construction_n demonstration_n touch_v cylinder_n demonstration_n touch_v cones_n first_o part_n of_o the_o proposition_n demonstrate_v touch_v cones_n two_o case_n in_o this_o proposition_n the_o first_o case_n second_o case_n construction_n demonstration_n touch_v cylinder_n second_o part_n demonstrate_v construction_n construction_n note_v this_o lm_o because_o of_o kz_o in_o the_o next_o proposition_n and_o here_o the_o point_n m_o for_o the_o point_n z_o in_o the_o next_o demonstration_n i_o dee_n dee_n for_o that_o the_o section_n be_v make_v by_o the_o number_n two_o that_o be_v by_o take_v half_n and_o of_o the_o residue_n the_o halâ—eâ—_n and_o so_o to_o ld_n be_v a_o half_a and_o a_o residue_n which_o shall_v be_v a_o common_a measure_n back_o again_o to_o make_v side_n of_o the_o poligonon_n figure_n construction_n
the_o 17._o proposition_n of_o the_o 14._o book_n construction_n â—rist_n part_n of_o the_o demonstration_n demonstration_n for_o the_o 4._o angle_n at_o the_o point_n king_n be_v equal_a to_o four_o right_a angle_n by_o the_o corollary_n of_o tâ—e_z 15._o of_o the_o first_o and_o those_o 4._o angle_n be_v equal_a the_o one_o to_o the_o other_o by_o the_o â—_o of_o the_o â—irst_n and_o therefore_o each_o be_v a_o right_a angle_n second_o part_n of_o the_o demonstration_n demonstration_n for_o the_o square_n of_o the_o line_n ab_fw-la which_o be_v prove_v equal_a to_o the_o square_n of_o the_o line_n lm_o be_v double_a to_o the_o square_n of_o the_o line_n bd_o which_o be_v also_o equal_a to_o the_o square_n of_o the_o line_n le._n this_o corollary_n be_v the_o 16._o proposition_n of_o the_o 14._o book_n after_o campane_n first_o part_n of_o the_o demonstration_n second_o part_n of_o the_o construction_n second_o part_n of_o the_o demonstration_n demonstration_n by_o the_o 2._o assumpt_v of_o the_o 13._o of_o this_o book_n three_o part_n of_o the_o demonstration_n second_o part_n of_o the_o demonstration_n for_o the_o line_n qw_o be_v equal_a to_o the_o line_n iz_n &_o the_o line_n zw_n be_v common_a to_o they_o both_o this_o part_n be_v again_o afterward_o demonstrate_v by_o flussas_n the_o pentagon_n vbwcr_n prove_v to_o be_v in_o one_o and_o the_o self_n same_o plain_a superficies_n the_o pentagon_n vbwcz_n it_o prove_v equiangle_n equiangle_n look_v for_o a_o far_a construction_n after_o flussas_n at_o the_o end_n of_o the_o demonstration_n that_o the_o side_n of_o the_o dodecahedron_n be_v a_o residual_a line_n draw_v in_o the_o former_a figure_n these_o line_n cta_n ctl_n câ—d_v the_o side_n of_o a_o pyramid_n the_o side_n of_o a_o cube_fw-la the_o side_n of_o a_o dodecahedron_n comparison_n of_o the_o five_o side_n of_o the_o foresay_a body_n an_o other_o demommonstration_n to_o prove_v that_o the_o side_n of_o the_o icosahedron_n be_v great_a than_o the_o side_n of_o the_o dodecahedron_n that_o 3._o square_n of_o the_o line_n fb_o be_v great_a they_o 6._o square_n of_o the_o line_n nb._n that_o there_o can_v be_v no_o other_o solid_a beside_o these_o five_o contain_v under_o equilater_n and_o equiangle_n base_n that_o the_o angle_n of_o a_o equilater_n and_o equiangle_n pentagon_n be_v one_o right_a angle_n and_o a_o 〈◊〉_d part_n 〈◊〉_d which_o thing_n be_v also_o before_o prove_v in_o the_o 〈◊〉_d of_o the_o 32._o of_o the_o â—irst_n the_o side_n of_o the_o angle_n of_o the_o inclâ—â—â—tion_n of_o the_o 〈◊〉_d of_o the_o 〈◊〉_d be_v 〈◊〉_d rational_a the_o side_n of_o the_o angle_n of_o the_o inclination_n of_o the_o 〈◊〉_d â—f_n tâ—e_z 〈…〉_o that_o the_o plain_n of_o a_o octohedron_n be_v in_o liâ—e_z sort_n incline_v that_o the_o plain_n of_o a_o icosahedron_n be_v in_o like_a sort_n incline_v that_o the_o plain_n of_o a_o dâ—â—â—â—â—hedron_n be_v 〈◊〉_d like_o sort_n incline_v the_o side_n of_o the_o angle_n of_o the_o inclination_n of_o the_o supeâ—ficieces_n of_o the_o tetrahedron_n be_v prove_v rational_a the_o side_n of_o the_o angle_n of_o the_o inclination_n of_o the_o superficiece_n of_o the_o cube_fw-la prove_v rational_a the_o side_n of_o thâ—_z angle_n etc_n etc_n of_o the_o octohedron_n prove_v rational_a the_o side_n of_o the_o angle_n etc_n etc_n of_o the_o icosahedron_n prove_v irrational_a how_o to_o know_v whether_o the_o angle_n of_o the_o inclination_n be_v a_o right_a angle_n a_o acute_a angle_n or_o a_o oblique_a angle_n the_o argument_n of_o the_o fourteen_o book_n first_o proposition_n after_o flussas_n construction_n demonstration_n demonstration_n this_o be_v manifest_a by_o the_o 12._o proposition_n of_o the_o thirtenh_o book_n as_o campane_n well_o gather_v in_o a_o corollary_n of_o the_o same_o the_o 4._o pâ—posâ—tion_n after_o flussas_n flussas_n this_o be_v afterward_o prove_v in_o the_o 4._o proposition_n this_o assumpt_n be_v the_o 3._o proposition_n after_o flussas_n construction_n of_o the_o assumpt_n demonstration_n of_o the_o assumpt_n construction_n of_o the_o proposition_n demonstration_n of_o the_o proposition_n proposition_n thâ—_z 5._o proposition_n aâ—tâ—r_a 〈◊〉_d construction_n demonstration_n the_o 5._o proposition_n aâ—ter_a fâ—ussas_n demonstration_n demonstration_n this_o be_v the_o reason_n of_o the_o corollary_n follow_v a_o corollary_n which_o also_o flussas_n put_v as_o a_o corollary_n after_o the_o 5._o proposition_n in_o his_o order_n the_o 6._o pâ—â—positionâ—â—ter_a flussas_n construction_n demonstration_n demonstration_n this_o be_v not_o hard_a to_o prove_v by_o the_o 15._o 16._o and_o 19_o of_o the_o â—â—â—eth_v â—â—â—eth_v in_o the_o corollary_n of_o the_o 17._o of_o the_o tâ—irtenth_o tâ—irtenth_o 〈◊〉_d again_o be_v require_v the_o assumpt_n which_o be_v afterward_o prove_v in_o this_o 4_o proposition_n proposition_n but_o first_o the_o assumpt_n follow_v the_o construction_n whereof_o here_o begin_v be_v to_o be_v prove_v the_o assumpt_n which_o also_o flussas_n put_v as_o a_o assumpt_n aâ—ter_v the_o 6._o proposition_n demonstration_n of_o the_o assumpt_n construction_n pertain_v to_o the_o second_o demonstration_n of_o the_o 4._o proposition_n second_o demonstration_n oâ—_n the_o 4._o proposition_n the_o 7â—_n proposition_n after_o flussas_n construction_n demonstration_n demonstration_n here_o again_o be_v require_v the_o assumpt_n afterward_o prove_v in_o this_o 4._o proposition_n proposition_n as_o may_v by_o the_o assumpt_n afterward_o in_o this_o proposition_n be_v plain_o prove_v the_o 8._o prodition_n aâ—ter_a flussas_n flussas_n by_o the_o corollary_n add_v by_o flussas_n after_o have_v assumpt_v put_v after_o the_o 17._o proposition_n of_o the_o 12._o book_n corollary_n of_o the_o 8._o after_o flussas_n this_o assumpt_n be_v the_o 3._o proposition_n aâ—ter_v â—lussas_n and_o be_v it_o which_o 〈◊〉_d time_n have_v be_v take_v aâ—_z grant_v in_o this_o book_n and_o oâ—ce_n also_o in_o the_o last_o proposition_n of_o the_o 13._o book_n as_o we_o have_v beâ—ore_o note_v demonstration_n demonstration_n in_o the_o 4._o section_n â—f_o this_o proposition_n proposition_n in_o the_o 1._o and_o 3_o section_n of_o the_o same_o proposition_n proposition_n in_o the_o 5._o section_n of_o the_o same_o proposition_n a_o corollary_n the_o first_o proposition_n after_o campane_n construction_n demonstration_n the_o 2._o proposition_n after_o campane_n demonstration_n lead_v to_o a_o impossibility_n the_o 4._o proposition_n after_o campane_n construction_n demonstration_n this_o corollary_n campane_n also_o â—utteth_v after_o the_o 4._o proposition_n in_o his_o order_n the_o 5._o proposition_n after_o campane_n construction_n demonstration_n this_o be_v the_o 6._o and_o 7._o proposition_n after_o campane_n construction_n demonstration_n this_o corollary_n campane_n also_o add_v after_o the_o 7._o proposition_n iâ—_z his_o order_n the_o 5._o proposition_n aâ—ter_v campane_n construction_n demonstration_n this_o assumpt_a campane_n also_o have_v after_o the_o 8._o proposition_n in_o his_o order_n construction_n demonstration_n the_o 9_o proposition_n after_o campane_n construction_n demonstration_n this_o campane_n put_v aâ—_z a_o corollary_n in_o the_o 9_o proposition_n after_o his_o order_n this_o corollary_n be_v the_o 9_o proposition_n after_o campane_n the_o 12._o proposition_n after_o campane_n construction_n demonstration_n the_o 13._o proposition_n after_o campane_n the_o 14._o proposition_n after_o campane_n demonstration_n of_o the_o first_o part_n demonstration_n of_o the_o second_o part_n the_o 17._o proposition_n after_o campane_n firâ—t_o part_n of_o the_o construction_n first_o part_n of_o the_o demonstration_n second_o parâ—_n of_o the_o construction_n second_o part_n of_o the_o demonstration_n the_o 18._o proposition_n after_o campane_n demonstration_n of_o the_o first_o part_n demonstration_n of_o the_o second_o part_n the_o corollary_n of_o the_o 8._o proposition_n after_o campane_n demonstration_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 construction_n demonstration_n the_o argument_n of_o the_o 15._o book_n book_n in_o this_o proposition_n as_o also_o in_o all_o the_o other_o follow_v by_o the_o name_n of_o a_o pyramid_n understand_v a_o tetrahedron_n as_o i_o have_v before_o admonish_v construction_n demonstration_n construction_n demonstration_n construction_n demonstration_n construction_n demonstration_n construction_n demonstration_n that_o which_o here_o follow_v concern_v the_o inclination_n of_o the_o plain_n of_o the_o five_o solid_n be_v before_o teach_v â—hough_o not_o altogether_o after_o the_o same_o manner_n out_o of_o flussas_n in_o the_o latter_a â—nde_n of_o the_o 13_o book_n construction_n demonstration_n construction_n demonstration_n construction_n demonstration_n construction_n demonstration_n construction_n demonstration_n first_o part_n of_o the_o construction_n first_o part_n of_o the_o demonstration_n second_o part_n of_o the_o construction_n second_o part_n of_o the_o demonstration_n three_o part_n of_o the_o construction_n three_o part_n of_o the_o demonstration_n construction_n demonstration_n construction_n demonstration_n construction_n demonstration_n construction_n demonstration_n construction_n demonstration_n produce_v in_o the_o figure_n the_o line_n tf_n to_o the_o point_n b._n construction_n demonstration_n this_o proposition_n campane_n have_v &_o be_v the_o last_o also_o in_o order_n of_o the_o 15._o book_n with_o he_o the_o argument_n of_o the_o 16._o book_n construction_n demonstration_n demonstration_n by_o a_o pyramid_n understand_v a_o tetrahedron_n throughout_o all_o this_o book_n construction_n demonstration_n construction_n demonstration_n construction_n demonstration_n construction_n demonstration_n demonstration_n that_o be_v aâ—_z 18._o to_o 1._o demonstration_n demonstration_n that_o iâ—_z as_o 9_o to_o 2._o 2._o that_o be_v as_o 18._o to_o 2._o or_o 9_o to_o 1._o draw_v in_o the_o figure_n a_o line_n from_o b_o to_o h._n h._n what_o the_o duple_n of_o a_o extreme_a and_o mean_a proportion_n be_v construction_n demonstration_n demonstration_n constrution_n demonstration_n construction_n demonstration_n demonstration_n demonstration_n that_o be_v at_o 13._o 1_o â—_o be_v toâ—_n â—_z demonstration_n demonstration_n construction_n demonstration_n construction_n demonstration_n extend_v in_o the_o figure_n a_o line_n â—rom_o the_o point_n e_o to_o the_o point_n b._n extend_v in_o the_o figure_n a_o line_n from_o the_o point_n e_o to_o the_o point_n b._n demonstration_n construction_n demonstration_n second_o part_n of_o the_o demonstration_n icosidodecahedron_n exoctohedron_n that_o the_o exoctohedron_n be_v contain_v in_o a_o sphere_n that_o the_o exoctohedron_n be_v contain_v in_o the_o sphere_n give_v that_o the_o diaâ—â—â—ter_n of_o the_o sphere_n be_v doâ—ble_a to_o the_o side_n â—f_o the_o exoctohedron_n that_o the_o icosidodecahedron_n be_v contain_v in_o the_o sphere_n give_v give_v that_o be_v as_o 8._o 103â—_n â—_z fault_n escape_v â—cl_n â—ag_n line_n faultesâ—_n co_n 〈◊〉_d 〈◊〉_d  _o  _o  _o errata_fw-la lib._n 1._o  _o 1_o 2_o 41_o point_n b._n at_o campane_n point_n c_o aâ—_z campane_n 3_o 1_o 22_o aâ—l_a line_n draw_v all_o righâ—_n 〈…〉_o 3_o 1_o 28_o line_n draw_v to_o the_o superficies_n right_a line_n draw_v to_o the_o circumference_n 9_o 1_o 42_o liâ—es_n ab_fw-la and_o ac_fw-la line_n ab_fw-la and_o bc_o 15_o 1_o 35_o be_v equal_a be_v prove_v equal_a 20_o 2_o 28_o by_o the_o first_o by_o the_o four_o 21_o 1_o 39_o tâ—e_z centre_n c._n the_o centre_n e_o 2d_o 2_o â—_o iâ—â—ower_a right_n if_o two_o right_a 25_o 2_o 3_o fâ—â—â—_n petition_n five_o petition_n 49_o 2_o 7_o 14._o â—â—_o 32.64_o etc_n etc_n 4.8.16.32.64_o &_o 53_o 1_o 39_o the_o triangle_n ng_z the_o triangle_n kâ—_n 54_o 2_o 25_o by_o the_o 44_o by_o the_o 42_o 57_o 2_o 23_o and_o câ—g_v in_o the_o and_o cgb_n be_v thâ—_z  _o  _o  _o in_o stead_n of_o â—lussates_n through_o out_o 〈◊〉_d whole_a book_n read_v â—lusâ—as_v  _o  _o  _o errata_fw-la lib._n 2._o  _o 60_o 2_o 29_o gnomon_n fgeh_a gnomon_n ahkd_v  _o  _o 30_o gnomon_n ehfg_n gnomon_n â—ckd_v 69_o 1_o 18_o the_o whole_a line_n the_o whole_a â—igure_n 76_o 2_o 9_o the_o diameter_n cd_o the_o diameter_n ahf_n  _o  _o  _o errata_fw-la lib._n 3._o  _o 82_o 2_o 36_o angle_n equal_a to_o the_o angle_n 92_o 1_o last_o the_o line_n ac_fw-la be_v the_o line_n of_o 〈◊〉_d  _o  _o  _o errata_fw-la lib._n 4._o  _o 110_o 2_o 10_o cd_o touch_v the_o ed_z touch_v the_o  _o  _o 12_o side_n of_o the_o other_o angle_n of_o the_o other_o 115_o 1_o 21_o and_o hb_o and_o he_o 117_o 2_o 44_o the_o angle_n acd_o the_o angle_n acb_o 118_o 1_o 2_o into_o ten_o equal_a into_o two_o equal_a 121_o 1_o 3â—_o cd_o and_o ea_fw-la cd_o de_fw-fr &_o ea_fw-la â—â—â—_o 1_o 29_o the_o first_o the_o three_o  _o  _o  _o errata_fw-la lib._n 5._o  _o 126_o 1_o 43_o it_o make_v 12._o more_o than_o 17._o by_o 5._o it_o make_v 24._o more_o than_o 17._o by_o 7._o 129_o 1_n  _o in_o stead_n of_o the_o figure_n of_o the_o 6._o definition_n draw_v in_o the_o magâ—â—_n a_o figure_n like_o unto_o thâ—s_n 134_o 2_o 4_o as_o ab_fw-la be_v to_o a_o so_o be_v cd_o to_o c_o as_z ab_fw-la be_v to_o b_o so_o be_v cd_o to_o d_o 141_o 2_o last_v but_o if_o king_n exceed_v m_o but_o if_o h_o exceed_v m_o lief_a be_v death_n and_o death_n be_v lief_a aetatis_fw-la svae_fw-la xxxx_n at_o london_n print_v by_o john_n day_n dwell_v over_o aldersgate_n beneath_o saint_n martins_n ¶_o these_o book_n be_v to_o be_v sell_v at_o his_o shop_n under_o the_o gate_n 1570._o
measure_v the_o square_a number_n produce_v of_o ef._n wherefore_o also_o by_o the_o 14._o of_o the_o eight_o the_o number_n g_z measure_v the_o number_n of_o and_o the_o number_n g_o also_o measure_v itself_o wherefore_o the_o number_n g_z measure_v these_o number_n of_o and_o g_z when_o yet_o they_o be_v prime_a the_o one_o to_o the_o other_o which_o be_v impossible_a wherefore_o the_o diameter_n a_o be_v not_o commensurable_a in_o length_n to_o the_o side_n b._n wherefore_o it_o be_v incommensurable_a which_o be_v require_v to_o be_v demonstrate_v an_o other_o demonstration_n after_o flussas_n suppose_v that_o upon_o the_o line_n ab_fw-la be_v describe_v a_o square_n who_o diameter_n let_v be_v the_o line_n ac_fw-la then_o i_o say_v that_o the_o side_n ab_fw-la be_v incommensurable_a in_o length_n unto_o the_o diameter_n ac_fw-la forasmuch_o as_o the_o line_n ab_fw-la and_o bc_o be_v equal_a therefore_o the_o square_a of_o the_o line_n ac_fw-la be_v double_a to_o the_o square_n of_o the_o line_n ab_fw-la by_o the_o 47._o of_o the_o first_o take_v by_o the_o 2._o of_o the_o eight_o number_n how_o many_o soever_o in_o continual_a proportion_n from_o unity_n and_o in_o the_o proportion_n of_o the_o square_n of_o the_o line_n ab_fw-la and_o ac_fw-la which_o let_v be_v the_o number_n d_o e_o f_o g._n and_o forasmuch_o as_o the_o first_o from_o unity_n namely_o e_o be_v no_o square_a number_n for_o that_o it_o be_v a_o prime_a number_n neither_o be_v also_o any_o other_o of_o the_o say_a number_n a_o square_a number_n except_o the_o three_o from_o unity_n and_o so_o all_o the_o rest_n leving_fw-mi one_o between_o by_o the_o 10._o of_o the_o nine_o wherefore_o d_z be_v to_o e_o or_o e_o to_o fletcher_n or_o fletcher_n to_o g_z in_o that_o proportion_n that_o a_o square_a number_n be_v to_o a_o number_n not_o square_a wherefore_o by_o the_o corollary_n of_o the_o 25._o of_o the_o eight_o they_o be_v not_o in_o that_o proportion_n the_o one_o to_o the_o other_o that_o a_o square_a number_n be_v to_o a_o square_a number_n wherefore_o neither_o also_o have_v the_o square_n of_o the_o line_n ab_fw-la and_o ac_fw-la which_o be_v in_o the_o same_o proportion_n that_o porportion_n that_o a_o square_a number_n have_v to_o a_o square_a number_n wherefore_o by_o the_o 9_o of_o this_o book_n their_o side_n namely_o the_o side_n ab_fw-la and_o the_o diameter_n ac_fw-la be_v incommensurable_a in_o length_n the_o one_o to_o the_o other_o which_o be_v require_v to_o be_v prove_v this_o demonstration_n i_o think_v good_a to_o add_v for_o that_o the_o former_a demonstration_n seem_v not_o so_o full_a and_o they_o be_v think_v of_o some_o to_o be_v none_o of_o theon_n as_o also_o the_o proposition_n to_o be_v none_o of_o euclides_n here_o follow_v a_o instruction_n by_o some_o studious_a and_o skilful_a grecian_a perchance_o theon_n which_o teach_v we_o of_o far_a use_n and_o fruit_n of_o these_o irrational_a line_n see_v that_o there_o be_v find_v out_o right_a line_n incommensurable_a in_o length_n the_o one_o to_o the_o â—ther_n as_o the_o line_n a_o and_o b_o there_o may_v also_o be_v find_v out_o many_o other_o magnitude_n have_v length_n and_o breadth_n such_o as_o be_v plain_a superficiece_n which_o shall_v be_v incommensurable_a the_o one_o to_o the_o other_o for_o if_o by_o the_o 13._o of_o the_o six_o between_o the_o line_n a_o and_o b_o there_o be_v take_v the_o mean_a proportional_a line_n namely_o c_z then_o by_o the_o second_o corollary_n of_o the_o 20._o of_o the_o six_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 figure_n describe_v upon_o the_o line_n a_o to_o the_o figure_n describe_v upon_o the_o line_n c_o be_v both_o like_a and_o in_o like_a sort_n describe_v that_o be_v whether_o they_o be_v square_n which_o be_v always_o like_o the_o one_o to_o the_o other_o or_o whether_o they_o be_v any_o other_o like_o rectiline_a figure_n or_o whether_o they_o be_v circle_n about_o the_o diameter_n a_o and_o c._n for_o circle_n have_v that_o proportion_n the_o one_o to_o the_o other_o that_o the_o square_n of_o their_o diameter_n have_v by_o the_o 2._o of_o the_o twelve_o wherefore_o by_o the_o second_o part_n of_o the_o 10._o of_o the_o ten_o the_o â—_z describe_v upon_o the_o line_n a_o and_o c_o being_n like_o and_o in_o like_a sort_n describe_v be_v incommensurable_a the_o one_o to_o the_o other_o wherefore_o by_o this_o mean_n there_o be_v find_v out_o superficieces_o incommensurable_a the_o one_o to_o the_o other_o in_o like_a sort_n there_o may_v be_v find_v out_o figure_n commensurable_a the_o one_o to_o the_o other_o if_o you_o put_v the_o line_n a_o and_o b_o to_o be_v commensurable_a in_o length_n the_o one_o to_o the_o other_o and_o see_v that_o it_o be_v so_o now_o let_v we_o also_o prove_v that_o even_o in_o soliâ—es_n also_o or_o body_n there_o be_v some_o commensurable_a the_o one_o to_o the_o other_o and_o other_o some_o incommensurable_a the_o one_o to_o the_o other_o for_o if_o from_o each_o of_o the_o square_n of_o the_o line_n a_o and_o b_o or_o from_o any_o other_o rectiline_a figure_n equal_a to_o these_o square_n be_v erect_v solid_n of_o equal_a altitude_n whether_o those_o solid_n be_v compose_v of_o equidistant_a supersiciece_n or_o whether_o they_o be_v pyramid_n or_o prism_n thosâ—_n solid_n sâ—_n erected_a shall_v be_v in_o that_o proportion_n the_o one_o to_o the_o other_o that_o their_o base_n be_v by_o the_o 32._o oâ—_n the_o eleven_o and_o 5._o and_o 6._o of_o the_o twelve_o howbeit_o there_o be_v no_o such_o proposition_n concern_v prism_n and_o so_o if_o the_o base_n of_o the_o solid_n bâ—_z commensurable_a the_o one_o to_o the_o other_o the_o solid_n also_o shall_v be_v commensurable_a the_o one_o to_o the_o other_o and_o if_o the_o base_n be_v incommensurable_a the_o one_o to_o the_o other_o the_o solid_n also_o shall_v be_v incommensurable_a the_o one_o to_o the_o other_o by_o the_o 10._o of_o the_o ten_o and_o if_o there_o be_v two_o circle_n a_o and_o b_o and_o upon_o each_o of_o the_o circle_n be_v erect_v cones_n or_o cilinder_n of_o equal_a altitude_n those_o cones_n &_o cilinder_n sâ—all_v be_v in_o that_o proportion_n the_o one_o to_o the_o other_o that_o the_o circle_n be_v which_o be_v their_o base_n by_o the_o 11._o of_o the_o twelve_o and_o so_o if_o the_o circle_n be_v commensurable_a the_o one_o to_o the_o other_o the_o cones_n and_o cilinder_n also_o shall_v be_v commensurable_a the_o one_o to_o the_o other_o but_o if_o the_o circle_n be_v incommensurable_a the_o one_o to_o the_o other_o the_o cones_n also_o and_o cilinder_n shall_v be_v incommensurable_a the_o one_o to_o the_o other_o by_o the_o 10._o of_o the_o ten_o wherefore_o it_o be_v manifest_a that_o not_o only_o in_o line_n and_o superâ—icieces_n but_o also_o in_o solid_n or_o body_n be_v find_v commensurabilitie_n or_o incommensurability_n a_o advertisement_n by_o john_n dee_n although_o this_o proposition_n be_v by_o euclid_n to_o this_o book_n allot_v as_o by_o the_o ancient_a graecian_n publish_v under_o the_o name_n of_o aristoteles_n 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d it_o will_v seem_v to_o be_v and_o also_o the_o property_n of_o the_o same_o agreeable_a to_o the_o matter_n of_o this_o book_n and_o the_o proposition_n itself_o so_o famous_a in_o philosophy_n and_o logic_n as_o it_o be_v will_v in_o manner_n crave_v his_o elemental_a place_n in_o this_o ten_o book_n yet_o the_o dignity_n &_o perfectionâ—_n of_o mathematical_a method_n can_v not_o allow_v it_o here_o as_o in_o due_a order_n follow_v but_o most_o apt_o after_o the_o 9_o proposition_n of_o this_o book_n as_o a_o corollary_n of_o the_o last_o part_n thereof_o and_o undoubted_o the_o proposition_n have_v for_o this_o 2000_o year_n be_v notable_o regard_v among_o the_o greek_n philosopher_n and_o before_o aristotle_n time_n be_v conclude_v with_o the_o very_a same_o inconvenience_n to_o the_o gaynesayer_n that_o the_o first_o demonstration_n here_o induce_v namely_o odd_a number_n to_o be_v equal_a to_o even_o as_o may_v appearâ—_n in_o aristotle_n work_n name_v analitica_fw-la prima_fw-la the_o first_o book_n and_o 40._o chapter_n but_o else_o in_o very_a many_o place_n of_o his_o work_n he_o make_v mention_n of_o the_o proposition_n evident_a also_o it_o be_v that_o euclid_n be_v about_o aristotle_n time_n and_o in_o that_o age_n the_o most_o excellent_a geometrician_n among_o the_o greek_n wherefore_o see_v it_o be_v so_o public_a in_o his_o time_n so_o famous_a and_o so_o appertain_v to_o the_o property_n of_o this_o book_n it_o be_v most_o likely_a both_o to_o be_v know_v to_o euclid_n and_o also_o to_o have_v be_v by_o he_o in_o apt_a order_n place_v but_o of_o the_o disorder_v of_o it_o can_v remain_v no_o doubt_n if_o you_o consider_v in_o zamberts_n translation_n