angle_n bac_n god_n &_o dab_n be_v equal_a the_o one_o to_o the_o other_o then_o be_v it_o manifest_v that_o two_o of_o they_o which_o two_z so_o ever_o be_v take_v be_v great_a than_o the_o three_o but_o if_o not_o let_v the_o angle_n bac_n be_v the_o great_a of_o the_o three_o angle_n and_o unto_o the_o right_a line_n ab_fw-la and_o from_o the_o point_n a_o make_v in_o the_o plain_a superficies_n bac_n unto_o the_o angle_n dab_n a_o equal_a angle_n bae_o and_o by_o the_o 2._o of_o the_o first_o make_v the_o line_n ae_n equal_a to_o the_o line_n ad._n now_o a_o right_a line_n bec_n draw_v by_o the_o point_n e_o shall_v cut_v the_o right_a line_n ab_fw-la and_o ac_fw-la in_o the_o point_n b_o and_o c_o draw_v a_o right_a line_n from_o d_z to_o b_o and_o a_o outstretch_o from_o d_z to_o c._n and_o forasmuch_o as_o the_o line_n dam_n be_v equal_a to_o the_o line_n ae_n demonstration_n and_o the_o line_n ab_fw-la be_v common_a to_o they_o both_o therefore_o these_o two_o line_n dam_n and_o ab_fw-la be_v equal_a to_o these_o two_o line_n ab_fw-la and_o ae_n and_o the_o angle_n dab_n be_v equal_a to_o the_o angle_n bae_o wherefore_o by_o the_o 4._o of_o the_o first_o the_o base_a db_o be_v equal_a to_o the_o base_a be._n and_o forasmuch_o as_o these_o two_o line_n db_o and_o dc_o be_v great_a than_o the_o line_n bc_o of_o which_o the_o line_n db_o be_v prove_v to_o be_v equal_a to_o the_o line_n be._n wherefore_o the_o residue_n namely_o the_o line_n dc_o be_v great_a than_o the_o residue_n namely_o than_o the_o line_n ec_o and_o forasmuch_o as_o the_o line_n dam_n be_v equal_a to_o the_o line_n ae_n and_o the_o line_n ac_fw-la be_v common_a to_o they_o both_o and_o the_o base_a dc_o be_v great_a than_o the_o base_a ec_o therefore_o the_o angle_n dac_o be_v great_a than_o the_o angle_n eac_n and_o it_o be_v prove_v that_o the_o angle_n dab_n be_v equal_a to_o the_o angle_n bae_o wherefore_o the_o angle_n dab_n and_o dac_o be_v great_a than_o the_o angle_n bac_n if_o therefore_o a_o solid_a angle_n be_v contain_v under_o three_o plain_a superficial_a angle_n every_o two_o of_o those_o three_o angle_n which_o then_o so_o ever_o be_v take_v be_v great_a than_o the_o three_o which_o be_v require_v to_o be_v prove_v in_o this_o figure_n you_o may_v plain_o behold_v the_o former_a demonstration_n if_o you_o elevate_v the_o three_o triangle_n abdella_n aâ—c_n and_o acd_o in_o such_o â—orâ—that_o they_o may_v all_o meet_v together_o in_o the_o point_n a._n the_o 19_o theorem_a the_o 21._o proposition_n every_o solid_a angle_n be_v comprehend_v under_o plain_a angle_n less_o than_o four_o right_a angle_n svppose_v that_o a_o be_v a_o solid_a angle_n contain_v under_o these_o superficial_a angle_n bac_n dac_o and_o dab_n then_o i_o say_v that_o the_o angle_n bac_n dac_o and_o dab_n be_v less_o than_o four_o right_a angle_n construction_n take_v in_o every_o one_o of_o these_o right_a line_n acab_n and_o ad_fw-la a_o point_n at_o all_o adventure_n and_o let_v the_o same_o be_v b_o c_o d._n and_o draw_v these_o right_a line_n bc_o cd_o and_o db._n demonstration_n and_o forasmuch_o as_o the_o angle_n b_o be_v a_o solid_a angle_n for_o it_o be_v contain_v under_o three_o superficial_a angle_n that_o be_v under_o cba_o abdella_n and_o cbd_n therefore_o by_o the_o 20._o of_o the_o eleven_o two_o of_o they_o which_o two_z so_o ever_o be_v take_v be_v great_a than_o the_o three_o wherefore_o the_o angle_n cba_o and_o abdella_n be_v great_a than_o the_o angle_n cbd_v and_o by_o the_o same_o reason_n the_o angle_n bca_o and_o acd_o be_v great_a than_o the_o angle_n bcdâ—_n and_o moreover_o the_o angle_n cda_n and_o adb_o be_v great_a than_o the_o angle_n cdb_n wherefore_o these_o six_o angle_n cba_o abdella_n bca_o acd_o cda_n and_o adb_o be_v great_a they_o these_o three_o angle_n namely_o cbd_v bcd_o &_o cdb_n but_o the_o three_o angle_n cbd_v bdc_o and_o bcd_o be_v equal_a to_o two_o right_a angle_n wherefore_o the_o six_o angle_n cba_o abdella_n bca_o acd_o cda_n and_o adb_o be_v great_a they_o two_o right_a angle_n and_o forasmuch_o as_o in_o every_o one_o of_o these_o triangle_n abc_n and_o abdella_n and_o acd_o three_o angle_n be_v equal_a two_o right_a angle_n by_o the_o 32._o of_o the_o first_o wherefore_o the_o nine_o angle_n of_o the_o three_o triangle_n that_o be_v the_o angle_n cba_o acb_o bac_n acd_o dac_o cda_n adb_o dba_n and_o bad_a be_v equal_a to_o six_o right_a angle_n of_o which_o angle_n the_o six_o angle_n abc_n bca_o acd_o cda_n adb_o and_o dba_n be_v great_a than_o two_o right_a angle_n wherefore_o the_o angle_n remain_v namely_o the_o angle_n bac_n god_n and_o dab_n which_o contain_v the_o solid_a angle_n be_v less_o than_o sour_a right_a angle_n wherefore_o every_o solid_a angle_n be_v comprehend_v under_o plain_a angle_n less_o than_o four_o right_a angle_n which_o be_v require_v to_o be_v prove_v if_o you_o will_v more_o full_o see_v this_o demonstration_n compare_v it_o with_o the_o figure_n which_o i_o put_v for_o the_o better_a sight_n of_o the_o demonstration_n of_o the_o proposition_n next_o go_v before_o only_o here_o be_v not_o require_v the_o draught_n of_o the_o line_n ae_n although_o this_o demonstration_n of_o euclid_n be_v here_o put_v for_o solid_a angle_n contain_v under_o three_o superficial_a angle_n yet_o after_o the_o like_a manner_n may_v you_o proceed_v if_o the_o solid_a angle_n be_v contain_v under_o superficial_a angle_n how_o many_o so_o ever_o as_o for_o example_n if_o it_o be_v contain_v under_o four_o superficial_a angle_n if_o you_o follow_v the_o former_a construction_n the_o base_a will_v be_v a_o quadrangled_a figure_n who_o four_o angle_n be_v equal_a to_o four_o right_a angle_n but_o the_o 8._o angle_n at_o the_o base_n of_o the_o 4._o triangle_n set_v upon_o this_o quadrangled_a figure_n may_v by_o the_o 20._o proposition_n of_o this_o book_n be_v prove_v to_o be_v great_a than_o those_o 4._o angle_n of_o the_o quadrangled_a figure_n as_o we_o see_v by_o the_o discourse_n of_o the_o former_a demonstration_n wherefore_o those_o 8._o angle_n be_v great_a than_o four_o right_a angle_n but_o the_o 12._o angle_n of_o those_o four_o triangle_n be_v equal_a to_o 8._o right_a angle_n wherefore_o the_o four_o angle_n remain_v at_o the_o top_n which_o make_v the_o solid_a angle_n be_v less_o than_o four_o right_a angle_n and_o observe_v this_o course_n you_o may_v proceed_v infinite_o ¶_o the_o 20._o theorem_a the_o 22._o proposition_n if_o there_o be_v three_o superficial_a plain_a angle_n of_o which_o two_o how_o soever_o they_o be_v take_v be_v great_a than_o the_o three_o and_o if_o the_o right_a line_n also_o which_o contain_v those_o angle_n be_v equal_a then_o of_o the_o line_n couple_v those_o equal_a right_a line_n together_o it_o be_v possible_a to_o make_v a_o triangle_n svppose_v that_o there_o be_v three_o superficial_a angle_n abc_n def_n and_o ghk_v of_o which_o let_v two_o which_o two_o soever_o be_v take_v be_v great_a than_o the_o three_o that_o be_v let_v the_o angle_n abc_n and_o def_n be_v great_a than_o the_o angle_n ghk_o and_o let_v the_o angle_n def_n and_o ghk_o be_v great_a than_o the_o angle_n abc_n and_o moreover_o let_v the_o angle_n ghk_v and_o abc_n be_v great_a than_o the_o angle_n def_n and_o let_v the_o right_a line_n ab_fw-la bc_o de_fw-fr of_o gh_o and_o hk_o be_v equal_a the_o one_o to_o the_o other_o and_o draw_v a_o right_a line_n from_o the_o point_n a_o to_o the_o point_n c_o and_o a_o other_o from_o the_o point_n d_o to_o the_o point_n fletcher_n and_o moreover_o a_o other_o from_o the_o point_n g_o to_o the_o point_n k._n proposition_n then_o i_o say_v that_o it_o be_v possible_a of_o three_o right_a line_n equal_a to_o the_o line_n ac_fw-la df_o and_o gk_o to_o make_v a_o triangle_n that_o be_v that_o two_o of_o the_o right_a line_n ac_fw-la df_o and_o gk_o which_o then_o soever_o be_v take_v be_v great_a than_o the_o three_o now_o if_o the_o angle_n abc_n def_n case_n and_o ghk_v be_v equal_a the_o one_o to_o the_o other_o it_o be_v manifest_a that_o these_o right_a line_n ac_fw-la df_o and_o gk_o be_v also_o by_o the_o 4._o of_o the_o first_o equal_a the_o one_o to_o the_o other_o it_o be_v possible_a of_o three_o right_a line_n equal_a to_o the_o line_n ac_fw-la df_o and_o gk_o to_o make_v a_o triangle_n case_n but_o if_o they_o be_v not_o equal_a let_v they_o be_v unequal_a and_o by_o the_o 23._o of_o the_o first_o unto_o the_o right_a line_n hk_o construction_n and_o at_o the_o point_n in_o it_o h_o make_v unto_o the_o angle_n abc_n a_o equal_a angle_n khl._n and_o by_o the_o â—_o of_o the_o first_o to_o one_o of_o the_o line_n
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
cube_n for_o forasmuch_o as_o the_o side_n of_o the_o pyramid_n inscribe_v in_o the_o cube_fw-la subtend_v two_o side_n of_o the_o cube_fw-la which_o contain_v a_o right_a angle_n by_o the_o 1._o of_o the_o fifteen_o it_o be_v manifest_a by_o the_o 47._o of_o the_o first_o that_o the_o side_n of_o the_o pyramid_n subtend_v the_o say_a side_n be_v in_o power_n duple_n to_o the_o side_n of_o the_o cube_fw-la wherefore_o also_o the_o square_a of_o the_o side_n of_o the_o cube_fw-la be_v the_o half_a of_o the_o square_n of_o the_o side_n of_o the_o pyramid_n the_o side_n therefore_o of_o a_o cube_fw-la contain_v in_o power_n half_o the_o side_n of_o a_o equilater_n triangular_a pyramid_n inscribe_v in_o the_o say_v cube_fw-la ¶_o the_o 7._o proposition_n the_o side_n of_o a_o pyramid_n be_v duple_n to_o the_o side_n of_o a_o octohedron_n inscribe_v in_o it_o forasmuch_o as_o by_o the_o 2._o of_o the_o fivetenth_fw-mi it_o be_v prove_v that_o the_o side_n of_o the_o octohedron_n inscribe_v in_o a_o pyramid_n couple_v the_o middle_a section_n of_o the_o side_n of_o the_o pyramid_n wherefore_o the_o side_n of_o the_o pyramid_n and_o of_o the_o octohedron_n be_v parallel_n by_o the_o corollary_n of_o the_o 39_o of_o the_o first_o and_o therefore_o by_o the_o corollary_n of_o the_o 2._o of_o the_o six_o they_o subtend_v like_o triangle_n wherefore_o by_o the_o 4._o of_o the_o six_o the_o side_n of_o the_o pyramid_n be_v double_a to_o the_o side_n of_o the_o octohedron_n namely_o in_o the_o proportion_n of_o the_o side_n the_o side_n therefore_o of_o a_o pyramid_n be_v duple_n to_o the_o side_n of_o a_o octohedron_n inscribe_v in_o it_o ¶_o the_o 8._o proposition_n the_o side_n of_o a_o cube_n be_v in_o power_n duple_n to_o the_o side_n of_o a_o octohedron_n inscribe_v in_o it_o it_o be_v prove_v in_o the_o 3._o of_o the_o fifteen_o that_o the_o diameter_n of_o the_o octohedron_n inscribe_v in_o the_o cube_fw-la couple_v the_o centre_n of_o the_o opposite_a base_n of_o the_o cube_fw-la wherefore_o the_o say_a diameter_n be_v equal_a to_o the_o side_n of_o the_o cube_fw-la but_o the_o same_o be_v also_o the_o diameter_n of_o the_o square_n make_v of_o the_o side_n of_o the_o octohedron_n namely_o be_v the_o diameter_n of_o the_o sphere_n which_o contain_v it_o by_o the_o 14._o of_o the_o thirteen_o wherefore_o that_o diameter_n be_v equal_a to_o the_o side_n of_o the_o cube_fw-la be_v in_o power_n double_a to_o the_o side_n of_o that_o square_n or_o to_o the_o side_n of_o the_o octohedron_n inscribe_v in_o it_o by_o the_o 47._o of_o the_o first_o the_o side_n therefore_o of_o a_o cube_n be_v in_o power_n duple_n to_o the_o side_n of_o a_o octohedron_n inscribe_v in_o it_o which_o be_v require_v to_o be_v prove_v ¶_o the_o 9_o proposition_n the_o side_n of_o a_o dodecahedron_n be_v the_o great_a segment_n of_o the_o line_n which_o contain_v in_o power_n half_o the_o side_n of_o the_o pyramid_n inscribe_v in_o the_o say_v dodecahedron_n svppose_v that_o of_o the_o dodecahedron_n abgd_v the_o side_n be_v ab_fw-la and_o let_v the_o base_a of_o the_o cube_fw-la inscribe_v in_o the_o dodecahedron_n be_v ecfh_n by_o the_o â—â—_o of_o the_o fifteen_o and_o let_v the_o side_n of_o the_o pyramid_n inscribe_v in_o the_o cube_fw-la be_fw-mi change_z by_o the_o 1._o of_o the_o fifteen_o construction_n wherefore_o the_o same_o pyramid_n be_v inscribe_v in_o the_o dodecahedron_n by_o the_o 10._o of_o the_o fifteen_o then_o i_o say_v that_o ab_fw-la the_o side_n of_o the_o dodecahedron_n be_v the_o great_a segment_n of_o the_o line_n which_o contain_v in_o power_n half_o the_o line_n change_z which_o be_v the_o side_n of_o the_o pyramid_n inscribe_v in_o the_o dodecahedron_n demonstration_n for_o forasmuch_o as_o ec_o the_o side_n of_o the_o cube_fw-la be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n make_v the_o great_a segment_n the_o line_n ab_fw-la the_o side_n of_o the_o dodecahedron_n by_o the_o â—â—rst_a corollary_n of_o the_o 17._o of_o the_o thirteen_o for_o they_o be_v contain_v in_o one_o and_o the_o self_n same_o sphere_n by_o the_o first_o of_o this_o book_n and_o the_o line_n ec_o the_o side_n of_o the_o cube_fw-la contain_v in_o power_n the_o half_a of_o the_o side_n change_z by_o the_o 6._o of_o this_o book_n wherefore_o ab_fw-la the_o side_n of_o the_o dodecahedron_n be_v the_o great_a segment_n of_o the_o line_n ec_o which_o contain_v in_o power_n the_o half_a of_o the_o line_n change_z which_o be_v the_o side_n of_o the_o dodecahedron_n inscribe_v in_o the_o pyramid_n the_o side_n therefore_o of_o a_o dodecahedron_n be_v the_o great_a segment_n of_o the_o line_n which_o contain_v in_o power_n half_o the_o side_n of_o the_o pyramid_n inscribe_v in_o the_o say_v dodecahedron_n ¶_o the_o 10._o proposition_n the_o side_n of_o a_o icosahedron_n be_v the_o mean_a proportional_a between_o the_o side_n of_o the_o cube_n circumscribe_v about_o the_o icosahedron_n and_o the_o side_n of_o the_o dodecahedron_n inscribe_v in_o the_o same_o cube_n svppose_v that_o there_o be_v a_o cube_fw-la abfd_n in_o which_o let_v there_o be_v inscribe_v a_o icosahedron_n cligor_fw-la by_o the_o 14._o of_o the_o fifteen_o construction_n let_v also_o the_o dodecahedron_n inscribe_v in_o the_o same_o be_v edmnp_n by_o the_o 13._o of_o the_o same_o now_o forasmuch_o as_o cl_n the_o side_n of_o the_o icosahedron_n be_v the_o great_a segment_n of_o ab_fw-la the_o side_n of_o the_o cube_fw-la circumscribe_v about_o it_o by_o the_o 3._o corollary_n of_o the_o 14._o of_o the_o fifteen_o demonstration_n and_o the_o side_n ed_z of_o the_o dodecahedron_n inscribe_v in_o thesame_o cube_fw-la be_v the_o less_o segment_n of_o the_o same_o side_n ab_fw-la of_o the_o cube_fw-la by_o the_o 2._o corollary_n of_o the_o 13._o of_o the_o fifteen_o it_o follow_v that_o ab_fw-la the_o side_n of_o the_o cube_fw-la be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n make_v the_o great_a segment_n cl_n the_o side_n of_o the_o icosahedron_n inscribe_v in_o it_o and_o the_o less_o segment_n ed_z the_o side_n of_o the_o dodecahedron_n likewise_o inscribe_v in_o it_o wherefore_o as_o the_o whole_a line_n ab_fw-la the_o side_n of_o the_o cube_fw-la be_v to_o the_o great_a segment_n cl_n the_o side_n of_o the_o icosahedron_n so_o be_v the_o great_a segment_n cl_n the_o side_n of_o the_o icosahedron_n to_o the_o less_o segment_n edâ—_n the_o side_n of_o the_o dodecahedron_n by_o the_o three_o definition_n of_o the_o six_o wherefore_o the_o side_n of_o a_o icosahedron_n be_v the_o mean_a proportional_a between_o the_o side_n of_o the_o cube_fw-la circumscribe_v about_o the_o icosahedron_n and_o the_o side_n of_o the_o dodecahedron_n inscribe_v in_o the_o same_o cube_fw-la ¶_o the_o 11._o proposition_n the_o side_n of_o a_o pyramid_n be_v in_o power_n 1._o octodecuple_n to_o the_o side_n of_o the_o cube_fw-la inscribe_v in_o it_o for_o by_o that_o which_o be_v demonstrate_v in_o the_o 18._o of_o the_o fifteen_o the_o side_n of_o the_o pyramid_n be_v triple_a to_o the_o diameter_n of_o the_o base_a of_o the_o cube_fw-la inscribe_v in_o it_o demonstration_n and_o therefore_o it_o be_v in_o power_n nonecuple_n to_o the_o same_o diameter_n by_o the_o 20._o of_o the_o six_o but_o the_o diamer_n be_v in_o power_n double_a to_o the_o side_n of_o the_o cube_fw-la by_o the_o 47._o of_o the_o first_o and_o the_o double_a of_o nonecuple_n make_v octodecuple_n wherefore_o the_o side_n of_o the_o pyramid_n be_v in_o power_n octodecuple_n to_o the_o side_n of_o the_o cube_fw-la inscribe_v in_o it_o ¶_o the_o 12._o proposition_n the_o side_n of_o a_o pyramid_n be_v in_o power_n octodecuple_n to_o that_o right_a line_n who_o great_a segment_n be_v the_o side_n of_o the_o dodecahedron_n inscribe_v in_o the_o pyramid_n forasmuch_o as_o the_o dodecahedron_n and_o the_o cube_fw-la inscribe_v in_o it_o be_v set_v in_o one_o and_o the_o sâ—lfâ—_n same_o pyramid_n by_o the_o corollary_n of_o the_o first_o of_o this_o book_n and_o the_o side_n of_o the_o pyramid_n circumscribe_v about_o the_o cube_fw-la be_v in_o power_n octodecuple_n to_o the_o side_n of_o the_o cube_fw-la inscribe_v by_o the_o former_a proposition_n but_o the_o great_a segment_n of_o the_o self_n same_o side_n of_o the_o cube_fw-la be_v the_o side_n of_o the_o dodecahedron_n which_o contain_v the_o cube_fw-la by_o the_o corollary_n of_o the_o 17._o of_o the_o thirteen_o wherefore_o the_o side_n of_o the_o pyramid_n be_v in_o power_n octodecuple_n to_o that_o right_a line_n namely_o to_o the_o side_n of_o the_o cube_fw-la who_o great_a segment_n be_v the_o side_n of_o the_o dodecahedron_n inscribe_v in_o the_o pyramid_n ¶_o the_o 13._o proposition_n the_o side_n of_o a_o icosahedron_n inscribe_v in_o a_o octohedron_n be_v in_o power_n duple_n to_o the_o less_o segment_n of_o the_o side_n of_o the_o same_o
how_o often_o therefore_o note_n these_o five_o sundry_a sort_n of_o operation_n do_v for_o the_o most_o part_n of_o their_o execution_n differre_fw-la from_o the_o five_o operation_n of_o like_a general_a property_n and_o name_n in_o our_o whole_a number_n practisable_v so_o often_o for_o a_o more_o distinct_a doctrine_n we_o vulgar_o account_v and_o name_v it_o a_o other_o kind_n of_o arithmetic_n and_o by_o this_o reason_n the_o consideration_n doctrine_n and_o work_v in_o whole_a number_n only_o where_o of_o a_o unit_fw-la be_v no_o less_o part_n to_o be_v allow_v be_v name_v as_o it_o be_v a_o arithmetic_n by_o itself_o and_o so_o of_o the_o arithmetic_n of_o fraction_n in_o like_o sort_n the_o necessary_a wonderful_a and_o secret_a doctrine_n of_o proportion_n and_o proportionalytie_n have_v purchase_v unto_o itself_o a_o peculiar_a manner_n of_o handle_v and_o work_n and_o so_o may_v seem_v a_o other_o form_n of_o arithmetic_n moreover_o the_o astronomer_n for_o speed_n and_o more_o commodious_a calculation_n have_v devise_v a_o peculiar_a manner_n of_o order_v number_n about_o their_o circular_a motion_n by_o sexagenes_n and_o sexagesmes_n by_o sign_n degree_n and_o minute_n etc_n etc_n which_o common_o be_v call_v the_o arithmetic_n of_o astronomical_a or_o physical_a fraction_n that_o have_v i_o brief_o note_v by_o the_o name_n of_o arithmetic_n circular_a by_o cause_n it_o be_v also_o use_v in_o circle_n not_o astronomical_a etc_n etc_n practice_n have_v lead_v number_n far_a and_o have_v frame_v they_o to_o take_v upon_o they_o the_o show_n of_o magnitude_n property_n which_o be_v incommensurabilitie_n and_o irrationalitie_n for_o in_o puâ—e_a arithmetic_n a_o unit_fw-la be_v the_o common_a measure_n of_o all_o number_n and_o here_o number_n be_v become_v as_o lynes_n plain_n and_o solides_n some_o time_n rational_a some_o time_n irrational_o â—_z and_o have_v proper_a and_o peculiar_a character_n as_o √_o â—_o √_o â—_o and_o so_o of_o other_o which_o be_v to_o signify_v roâ—e_n square_n rote_n cubik_n and_o so_o forth_o &_o proper_a and_o peculiar_a fashion_n in_o the_o five_o principal_a part_n wherefore_o the_o practiser_n esteem_v this_o a_o diverse_a arithmetic_n from_o the_o other_o practice_n bring_v in_o here_o diverse_a compound_a of_o number_n as_o some_o time_n two_o three_o four_o or_o more_o radical_a number_n diverse_o knit_v by_o sign_n oâ—_n more_o &_o less_o as_o thus_o √_o â—_o 12_o +_o √_o â—_o 15._o or_o â—hus_fw-la √_o â—â—_o 19_o +_o √_o â—_o 12_o √_o â—_o 2d_o etc_n etc_n and_o some_o time_n with_o whole_a number_n or_o fraction_n of_o whole_a number_n among_o they_o as_o 20_o +_o √_o â—â—4â—_n √_o â—_o +_o 33_o √_o â—_o 10_o √_o â—â—_o 44_o +_o 12_o +_o √_o â—9_o and_o so_o infinite_o may_v hap_v the_o variety_n after_o this_o both_o the_o one_o and_o the_o other_o have_v fraction_n incident_a and_o so_o be_v this_o arithmetic_n great_o enlarge_v by_o diverse_a exhibiting_n and_o use_n of_o composition_n and_o mixtynge_n consider_v howâ—_n i_o be_v desirous_a to_o deliver_v the_o student_n from_o error_n and_o cavillation_n do_v give_v to_o this_o practice_n the_o name_n of_o the_o arithmetic_n of_o radical_a number_n not_o of_o irrational_a or_o fur_v numbersâ—_n which_o other_o while_n be_v rational_a though_o they_o have_v the_o sign_n of_o a_o rote_n before_o they_o which_o arithmetic_n of_o whole_a number_n most_o usual_a will_v say_v they_o have_v no_o such_o root_n and_o so_o account_v they_o fur_v number_n which_o general_o speak_v be_v untrue_a as_o euclides_n ten_o book_n may_v teach_v you_o therefore_o to_o call_v they_o general_o radical_a number_n by_o reason_n of_o the_o sign_n √_o prefix_a be_v a_o sure_a way_n and_o a_o sufficient_a general_a distinction_n from_o all_o other_o order_v and_o use_v of_o number_n and_o yet_o beside_o all_o this_o consider_v the_o infinite_a desire_n of_o knowledge_n and_o incredible_a power_n of_o man_n search_n and_o capacity_n how_o they_o joint_o have_v wade_v far_o by_o mixt_v of_o speculation_n and_o practice_n and_o have_v find_v out_o and_o attain_v to_o the_o very_a chief_a perfection_n almost_o of_o number_n practical_a use_n which_o thing_n be_v well_o to_o be_v perceive_v in_o that_o great_a arithmetical_a art_n of_o aequation_n common_o call_v the_o rule_n of_o coss._n or_o algebra_n the_o latin_n term_v it_o regulam_fw-la rei_fw-la &_o census_n that_o be_v the_o rule_n of_o the_o thing_n and_o his_o value_n with_o a_o apt_a name_n comprehend_v the_o first_o and_o last_o point_n of_o the_o work_n and_o the_o vulgar_a name_n both_o in_o italian_a french_a and_o spanish_a depend_v in_o name_a it_o upon_o the_o signification_n of_o the_o latin_a word_n res_n a_o thing_n unleast_o they_o use_v the_o name_n of_o algebra_n and_o therein_o common_o be_v a_o double_a error_n the_o one_o of_o they_o which_o think_v it_o to_o be_v of_o geber_n his_o invent_v the_o other_o of_o such_o as_o call_v it_o algebra_n for_o first_o though_o geber_n for_o his_o great_a skill_n in_o number_n geometry_n astronomy_n and_o other_o marvellous_a art_n may_v have_v seem_v able_a to_o have_v first_o devise_v the_o say_a rule_n and_o also_o the_o name_n carry_v with_o it_o a_o very_a near_a likeness_n of_o geber_n his_o name_n yet_o true_a it_o be_v that_o a_o greek_a philosopher_n and_o mathematicien_n name_v diophantus_n before_o geber_n his_o time_n write_v 13._o book_n thereof_o of_o which_o six_o be_v yet_o extant_a and_o i_o have_v they_o to_o 1550._o use_v of_o the_o famous_a mathematicien_fw-fr and_o my_o great_a friend_n petrus_n monâ—aureus_n and_o second_o the_o very_a name_n be_v algiebar_v and_o not_o algebra_n as_o by_o the_o arabien_fw-fr avicen_n may_v be_v prove_v who_o have_v these_o precise_a word_n in_o latin_a by_o andrea_n alpagus_n most_o perfect_a in_o the_o arabik_a tongue_n so_o translate_v scientia_fw-la faciendi_fw-la algiebar_n &_o almachabel_n i._o scientia_fw-la inveniendi_fw-la numerum_fw-la ignotum_fw-la per_fw-la additionem_fw-la numeri_fw-la &_o divisionem_fw-la &_o aequationem_fw-la which_o be_v to_o say_v the_o science_n of_o work_n algiebar_n and_o almachabel_n that_o be_v the_o science_n of_o find_v a_o unknown_a number_n by_o add_v of_o a_o number_n &_o division_n &_o aequation_n here_o have_v you_o the_o name_n and_o also_o the_o principal_a part_n of_o the_o rule_n touch_v to_o name_v it_o the_o rule_n or_o art_n of_o aequation_n do_v signify_v the_o middle_a part_n and_o the_o state_n of_o the_o rule_n this_o rule_n have_v his_o peculiar_a character_n and_o the_o principal_a part_n of_o arithmetic_n to_o it_o appertain_v do_v differe_o from_o the_o other_o arithmetical_a operation_n this_o arithmetic_n have_v number_n simple_a compound_v mix_v and_o fraction_n accordingly_z this_o rule_n and_o arithmetic_n of_o algiebar_n be_v so_o profound_a so_o general_a and_o so_o in_o manner_n contain_v the_o whole_a power_n of_o number_n application_n 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&_o omnium_fw-la primum_fw-la circulus_fw-la ille_fw-la est_fw-la in_fw-la quo_fw-la manifest_a solis_fw-la latio_fw-la etc_n etc_n and_o so_o forth_o his_o meteorological_a book_n be_v full_a of_o argument_n and_o effectual_a demonstration_n of_o the_o virtue_n operation_n and_o power_n of_o the_o heavenly_a body_n in_o and_o upon_o the_o four_o element_n and_o other_o body_n of_o they_o either_o perfect_o or_o unperfect_o compose_v and_o in_o his_o second_o book_n de_fw-fr generatione_fw-la &_o corruption_n in_o the_o ten_o chapter_n quocirca_fw-la &_o prima_fw-la latiâ—_n orâ—us_n &_o interitus_fw-la causa_fw-la non_fw-la est_fw-la sed_fw-la obliqui_fw-la circuli_fw-la latio_fw-la ea_fw-la namque_fw-la &_o continua_fw-la est_fw-la &_o dvobus_fw-la motibus_fw-la fit_a in_o english_a thus_o wherefore_o the_o uppermost_a motion_n be_v not_o the_o cause_n of_o generation_n and_o corruption_n but_o the_o motion_n of_o the_o zodiac_n for_o that_o both_o be_v continual_a and_o be_v cause_v 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the_o elephant_n do_v as_o the_o cynocephalus_n as_o the_o porpentine_n doâ—h_v nor_o will_v allow_v these_o perfect_a and_o incorruptible_a mighty_a body_n so_o much_o virtual_a radiation_n &_o force_n as_o they_o see_v in_o a_o little_a piece_n of_o a_o magnes_n stone_n which_o at_o great_a distance_n show_v his_o operation_n and_o perchance_o they_o think_v the_o sea_n &_o rivers_n as_o the_o thames_n to_o be_v some_o quick_a thing_n and_o sâ—_n to_o ebb_v aâ—d_z slow_a run_v in_o and_o out_o of_o themselves_o at_o â—heiâ—_n own_o fantasy_n god_n help_v god_n help_v sure_o these_o man_n come_v to_o short_a and_o either_o be_v to_o dull_a or_o wilful_o blind_a or_o perhaps_o to_o malicious_a the_o three_o man_n be_v the_o common_a and_o vulgar_a astrologien_n or_o practiser_n who_o be_v not_o due_o artificial_o and_o perfect_o furnish_v yet_o either_o for_o vain_a glory_n or_o gain_v or_o like_o a_o simple_a dolt_n &_o blind_a bayardâ—_n both_o in_o matter_n and_o manner_n err_v to_o the_o discredit_n of_o the_o wary_a and_o modest_a astrologien_n and_o to_o the_o rob_n of_o those_o most_o noble_a corporal_a
mark_v in_o your_o line_n what_o number_n the_o water_n cut_v take_v the_o weight_n of_o the_o same_o cube_n against_o in_o the_o same_o kind_n of_o water_n which_o you_o have_v before_o put_v that_o 〈◊〉_d also_o into_o the_o pyramid_n or_o cone_n where_o you_o do_v put_v the_o first_o mark_v now_o again_o in_o what_o number_n or_o place_n of_o the_o line_n the_o water_n cut_v they_o two_o way_n you_o may_v conclude_v your_o purpose_n it_o be_v to_o weet_v either_o by_o number_n or_o line_n by_o number_n as_o if_o you_o divide_v the_o side_n of_o your_o fundamental_a cube_n into_o so_o many_o equal_a part_n as_o it_o be_v capable_a of_o convenient_o with_o your_o ease_n and_o preciseness_n of_o the_o division_n for_o as_o the_o number_n of_o your_o first_o and_o less_o line_n in_o your_o hollow_a pyramid_n or_o cone_n be_v to_o the_o second_o or_o great_a both_o be_v count_v from_o the_o vertex_fw-la so_o shall_v the_o number_n of_o the_o side_n of_o your_o fundamental_a cube_n be_v 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the_o four_o find_v to_o it_o proportional_a by_o the_o 12._o of_o the_o six_o of_o euâ—lide_n water_n note_v that_o all_o this_o while_n i_o forget_v not_o my_o first_o proposition_n statical_a here_o rehearse_v that_o the_o supersicy_n of_o the_o water_n be_v spherical_a wherein_o use_v your_o discretion_n to_o the_o first_o line_n add_v a_o small_a hear_v breadth_n more_o and_o to_o the_o second_o half_o a_o hear_v breadth_n more_o to_o his_o length_n for_o you_o will_v easy_o perceive_v that_o the_o difference_n can_v be_v no_o great_a in_o any_o pyramid_n or_o cone_n of_o you_o to_o be_v handle_v which_o you_o shall_v thus_o try_v for_o â—inding_v the_o swell_n of_o the_o water_n above_o level_a square_n the_o semidiameter_n ☞_o from_o the_o centre_n of_o the_o earth_n to_o your_o first_o water_n superficies_n square_n then_o half_o the_o subtendent_fw-la of_o that_o watery_a superficies_n which_o subtendent_fw-la must_v have_v the_o equal_a part_n of_o his_o measure_n all_o one_o 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experience_n will_v teach_v you_o abundant_o with_o great_a ease_n pleasure_n and_o commodity_n â—â—_o thus_o may_v you_o double_v the_o cube_n mechanical_o treble_a it_o and_o so_o forth_o in_o any_o proportion_n now_o will_v i_o abridge_v your_o pain_n cost_n and_o care_v herein_o without_o all_o prepare_n of_o your_o fundamental_a cube_n you_o may_v alike_o work_v this_o conclusion_n for_o that_o be_v rather_o a_o kind_n of_o experimental_a demonstration_n than_o the_o short_a way_n and_o all_o upon_o one_o mathematical_a demonstration_n depend_v take_v water_n as_o much_o as_o convenient_o will_v serve_v your_o turn_n as_o i_o warn_v before_o of_o your_o fundamental_a cube_n bigness_n way_n it_o precise_o put_v that_o water_n into_o your_o pyramid_n or_o cone_n of_o the_o same_o kind_n of_o water_n then_o take_v again_o the_o same_o weight_n you_o have_v before_o put_v that_o likewise_o into_o the_o pyramid_n or_o cone_n for_o in_o each_o time_n your_o mark_v of_o the_o line_n how_o the_o water_n do_v cut_v they_o shall_v geve_v you_o the_o proportion_n between_o the_o radical_a side_n of_o any_o two_o cube_n whereof_o the_o one_o be_v double_a to_o the_o other_o work_v as_o before_o i_o have_v teach_v you_o ☜_o save_v that_o for_o you_o fundamental_a cube_n his_o radical_a side_n here_o you_o may_v take_v a_o right_a line_n at_o pleasure_n yet_o far_a proceed_v with_o our_o drop_n of_o natural_a truth_n you_o may_v now_o geve_v cube_n one_o to_o the_o other_o in_o any_o proportion_n give_v irrational_a rational_a or_o irrational_a on_o this_o manner_n make_v a_o hollow_a parallelipipedon_n of_o copper_n or_o tin_n with_o one_o base_a want_v or_o open_v as_o in_o our_o cubike_a coffin_n fron_n the_o bottom_n of_o that_o parallelipipedon_n raise_v up_o many_o perpendicular_n in_o every_o of_o his_o four_o side_n now_o if_o any_o proportion_n be_v assign_v you_o in_o right_a line_n cut_v one_o of_o your_o perpendicular_n or_o a_o line_n equal_a to_o it_o or_o less_o than_o it_o likewise_o by_o the_o 10._o of_o the_o six_o of_o euclid_n and_o those_o two_o part_n set_v in_o two_o sundry_a line_n of_o those_o perpendicular_n or_o you_o may_v set_v they_o both_o in_o one_o line_n make_v their_o beginning_n to_o be_v at_o the_o base_a and_o so_o their_o lengthe_n to_o extend_v upward_o now_o set_v your_o hollow_a parallelipipedon_n upright_o perpendicular_o steady_a pour_v in_o water_n handsome_o to_o the_o heith_n of_o your_o short_a line_n pour_v that_o water_n into_o the_o hollow_a pyramid_n or_o cone_n mark_v the_o place_n of_o the_o rise_n settle_v your_o hollow_a parallelipipedon_n again_o pour_v water_n into_o it_o unto_o the_o heith_n of_o the_o second_o line_n exact_o pour_v that_o water_n first_o due_o into_o the_o hollow_a pyramid_n or_o cone_n mark_v now_o again_o where_o the_o water_n cut_v the_o same_o line_n which_o you_o mark_v before_o for_o there_o as_o the_o first_o mark_v line_n be_v to_o the_o second_o so_o shall_v the_o two_o radical_a side_n be_v one_o to_o the_o other_o of_o any_o two_o cube_n which_o in_o their_o solidity_n shall_v have_v the_o same_o proportion_n which_o be_v at_o the_o first_o assign_v be_v it_o rational_a or_o irrational_a thus_o in_o sundry_a way_n you_o may_v furnish_v yourself_o with_o such_o strange_a and_o profitable_a matter_n which_o long_o have_v be_v wish_v for_o and_o though_o it_o be_v natural_o do_v and_o mechanical_o yet_o have_v it_o a_o good_a demonstration_n mathematical_a rest_n which_o be_v thisâ—_n always_o you_o have_v two_o like_a pyramid_n or_o two_o like_o cones_n in_o the_o proportion_n assign_v and_o like_o pyramid_n or_o cones_n be_v in_o proportion_n one_o to_o the_o other_o in_o the_o proportion_n of_o their_o homologal_a side_n or_o line_n triple_a wherefore_o if_o to_o the_o first_o and_o second_o line_n find_v
compare_v they_o all_o with_o triangle_n &_o also_o together_o the_o one_o with_o the_o other_o in_o it_o also_o be_v teach_v how_o a_o figure_n of_o any_o form_n may_v be_v change_v into_o a_o figure_n of_o a_o other_o form_n and_o for_o that_o it_o entreat_v of_o these_o most_o common_a and_o general_a thing_n this_o book_n be_v more_o universal_a than_o be_v the_o second_o three_o or_o any_o other_o and_o therefore_o just_o occupy_v the_o first_o place_n in_o order_n as_o that_o without_o which_o the_o other_o book_n of_o eâ—clide_n which_o follow_v and_o also_o the_o work_n of_o other_o which_o have_v write_v in_o geometry_n can_v be_v perceive_v nor_o understand_v and_o forasmuch_o â—s_v all_o the_o demonstration_n and_o proof_n of_o all_o the_o proposition_n in_o this_o whole_a book_n depend_v of_o these_o ground_n and_o principle_n follow_v which_o by_o reason_n of_o their_o plainness_n need_v no_o great_a declaration_n yet_o to_o remove_v all_o be_v it_o never_o so_o little_a obscurity_n there_o be_v here_o set_v certain_a short_a and_o manife_a exposition_n of_o they_o definition_n 1._o a_o sign_n or_o point_n be_v that_o which_o have_v no_o part_n point_n the_o better_a to_o understand_v what_o manâ—r_n of_o thing_n a_o sign_n or_o point_n be_v you_o must_v note_v that_o the_o nature_n and_o property_n of_o quantity_n whereof_o geometry_n entreat_v be_v to_o be_v divide_v so_o that_o whatsoever_o may_v be_v divide_v into_o sundâ—y_a part_n be_v call_v quantity_n but_o a_o point_n although_o it_o pertain_v to_o quantity_n and_o have_v his_o be_v in_o quantity_n yet_o be_v it_o no_o quantity_n for_o that_o it_o can_v be_v divide_v because_o as_o the_o definition_n say_v it_o have_v no_o part_n into_o which_o it_o shall_v be_v divide_v so_o that_o a_o point_n be_v the_o least_o thing_n that_o by_o mind_n and_o understanding_n can_v be_v imagine_v and_o conceive_v than_o which_o there_o can_v be_v nothing_o less_o as_o the_o point_n a_o in_o the_o margin_n a_o sign_n or_o point_n be_v of_o pythagoras_n scholar_n after_o this_o manner_n define_v pythagoras_n a_o point_n be_v a_o unity_n which_o have_v position_n number_n be_v conceive_v in_o mind_n without_o any_o form_n &_o figure_n and_o therefore_o without_o matter_n whereon_o to_o 〈◊〉_d figure_n &_o consequent_o without_o place_n and_o position_n wherefore_o unity_n be_v a_o part_n of_o number_n have_v no_o position_n or_o determinate_a place_n whereby_o it_o be_v manifest_a that_o â—umbâ—â—_n iâ—_z more_o simple_a and_o pure_a then_o be_v magnitude_n and_o also_o immaterial_a and_o so_o unity_n which_o iâ—_z the_o beginning_n of_o number_n be_v less_o material_a than_o a_o â—igne_n or_o poyâ—â—_n which_o be_v the_o begin_n of_o magnitude_n for_o a_o point_n be_v material_a and_o require_v position_n and_o place_n and_o â—â—â—rby_o differ_v from_o unity_n â—_o a_o line_n be_v length_n without_o breadth_n liâ—â—_n there_o pertain_v to_o quantiâ—â—e_v three_o dimension_n length_n breadth_n &_o thickness_n or_o depth_n and_o by_o these_o three_o be_v all_o quamtitieâ—_n measure_v &_o make_v know_v there_o be_v also_o according_a to_o these_o three_o dimension_n three_o kind_n of_o continual_a quantity_n a_o line_n a_o superficies_n or_o plain_a and_o a_o body_n the_o first_o kind_n namely_o a_o line_n be_v here_o define_v in_o these_o word_n a_o line_n be_v length_n without_o breadth_n a_o point_n for_o that_o it_o be_v no_o quantity_n nor_o have_v any_o part_n into_o which_o it_o may_v be_v divide_v but_o remain_v indivisible_a have_v not_o nor_o can_v have_v any_o of_o these_o three_o dimension_n it_o neither_o have_v length_n breadth_n nor_o thickness_n but_o to_o a_o line_n which_o be_v the_o first_o kind_n of_o quantity_n be_v attribute_v the_o first_o dimension_n namely_o length_n and_o only_o that_o for_o it_o have_v neither_o breadth_n nor_o thickness_n but_o be_v conceive_v to_o be_v draw_v in_o length_n only_o and_o by_o it_o it_o may_v be_v divide_v into_o part_n as_o many_o as_o you_o listen_v equal_a or_o unequal_a but_o as_o touch_v breadth_n it_o remain_v indivisible_a as_o the_o line_n ab_fw-la which_o be_v only_o draw_v in_o length_n may_v be_v divide_v in_o the_o point_n c_o equal_o or_o in_o the_o point_n d_o unequal_o and_o so_o into_o as_o many_o part_n as_o you_o listen_v there_o be_v also_o of_o diverse_a other_o give_v other_o definition_n of_o a_o line_n as_o these_o which_o follow_v line_n a_o line_n be_v the_o move_n of_o a_o point_n as_o the_o motion_n or_o draught_n of_o a_o pin_n or_o a_o pen_n to_o your_o sense_n make_v a_o line_n other_o again_o a_o line_n be_v a_o magnitude_n have_v one_o only_a space_n or_o dimension_n namely_o length_n want_v breadth_n and_o thickness_n line_n 3_o the_o end_n or_o limit_n of_o a_o line_n be_v point_n for_o a_o line_n have_v his_o beginning_n from_o a_o point_n and_o likewise_o end_v in_o a_o point_n so_o that_o by_o this_o also_o it_o be_v manifest_a that_o point_n for_o their_o simplicity_n and_o lack_n of_o composition_n be_v neither_o quantity_n nor_o part_n of_o quantity_n but_o only_o the_o term_n and_o end_n of_o quantity_n as_o the_o point_n a_o b_o be_v only_o the_o end_n of_o the_o line_n ab_fw-la and_o no_o part_n thereof_o and_o herein_o differ_v a_o point_n in_o quantity_n frâ—â—nity_n from_o unity_n in_o numberâ—_n for_o that_o although_o unity_n be_v the_o beginning_n of_o number_n and_o no_o number_n as_o a_o point_n be_v the_o begin_n of_o quantity_n and_o no_o quantity_n yet_o be_v unity_n a_o part_n of_o number_n number_n for_o number_n be_v nothing_o else_o but_o a_o collection_n of_o unity_n and_o therefore_o may_v be_v divide_v into_o they_o as_o into_o his_o part_n but_o a_o point_n be_v no_o part_n of_o quantity_n quantity_n or_o of_o a_o lyneâ—_n neither_o be_v a_o line_n compose_v of_o point_n as_o number_n be_v of_o unity_n for_o thing_n indivisible_a being_n never_o so_o many_o add_v together_o can_v never_o make_v a_o thing_n divisible_a as_o a_o instant_n in_o time_n be_v neither_o time_n nor_o part_n of_o time_n but_o only_o the_o beginning_n and_o end_n of_o time_n and_o couple_v &_o join_v part_n of_o time_n together_o line_n 4_o a_o right_a line_n be_v that_o which_o lie_v equal_o between_o his_o point_n as_o the_o whole_a line_n ab_fw-la lie_v straight_a and_o equal_o between_o the_o point_n ab_fw-la without_o any_o go_n up_o or_o come_v down_o on_o either_o side_n a_o right_a line_n be_v the_o short_a extension_n or_o draught_n that_o be_v or_o may_v be_v from_o one_o point_n to_o a_o other_o archimedes_n define_v it_o thus_o plato_n plato_n define_v a_o right_a line_n after_o this_o manner_n a_o right_a line_n be_v that_o who_o middle_a part_n shadow_v the_o exâ—remeâ—_n as_o if_o you_o put_v any_o thing_n in_o the_o middle_n of_o a_o right_a line_n you_o shall_v not_o see_v from_o the_o one_o end_n to_o the_o other_o which_o thing_n happen_v not_o in_o a_o crooked_a line_n the_o eclipse_n of_o the_o sun_n say_v astronomer_n then_o happen_v when_o the_o sun_n the_o moon_n &_o our_o eye_n be_v in_o one_o right_a line_n for_o the_o moon_n then_o be_v in_o the_o midst_n between_o we_o and_o the_o sun_n cause_v it_o to_o be_v darken_v diverse_a other_o define_v a_o right_a line_n diverse_o definition_n as_o follow_v a_o right_a line_n be_v that_o which_o stand_v firm_a between_o his_o extreme_n other_o again_o a_o right_a line_n be_v that_o which_o with_o a_o other_o line_n of_o like_a form_n can_v make_v a_o figure_n again_o other_o a_o right_a line_n be_v that_o which_o have_v not_o one_o part_n in_o a_o plain_a superficies_n and_o a_o other_o erect_v on_o high_a again_o other_o a_o right_a line_n be_v that_o all_o who_o part_n agree_v together_o with_o all_o his_o other_o part_n again_o other_o a_o right_a line_n be_v that_o who_o extreme_n abide_v can_v be_v alter_v euclid_n do_v not_o here_o define_v a_o crooked_a line_n line_n for_o it_o need_v not_o it_o may_v easy_o be_v understand_v by_o the_o definition_n of_o a_o right_a line_n for_o every_o contrary_n be_v well_o manifest_v &_o set_v forth_o by_o his_o contrary_n one_o crooked_a line_n may_v be_v more_o crooked_a than_o a_o other_o and_o from_o one_o point_n to_o a_o other_o may_v be_v draw_v infinite_a crooked_a line_n but_o one_o right_a line_n can_v be_v right_a than_o a_o other_o and_o therefore_o from_o one_o point_n to_o a_o other_o there_o may_v be_v draw_v but_o one_o tight_a line_n as_o by_o figure_n above_o set_n you_o may_v see_v 5_o a_o superficies_n be_v that_o which_o have_v only_a length_n and_o breadth_n superficies_n a_o superficies_n be_v the_o second_o kind_n of_o quantity_n and_o to_o it_o be_v attribute_v two_o dimension_n namely_o length_n and_o breadth_n way_n as_o in_o the_o
diameter_n be_v double_a to_o that_o square_n who_o diameter_n it_o be_v corollary_n the_o 34._o theorem_a the_o 48._o proposition_n if_o the_o square_n which_o be_v make_v of_o one_o of_o the_o side_n of_o a_o triangle_n be_v equal_a to_o the_o square_n which_o be_v make_v of_o the_o two_o other_o side_n of_o the_o same_o triangle_n the_o angle_n comprehend_v under_o those_o two_o other_o side_n be_v a_o right_a angle_n svppose_v that_o abc_n be_v a_o triangle_n and_o let_v the_o square_n which_o be_v make_v of_o one_o of_o the_o side_n there_o namely_o of_o the_o side_n bc_o be_v equal_a to_o the_o square_n which_o be_v make_v of_o the_o side_n basilius_n and_o ac_fw-la then_o i_o say_v that_o the_o angle_n bac_n be_v a_o right_a angle_n raise_v up_o by_o the_o 11._o proposition_n from_o the_o point_n a_o unto_o the_o right_a line_n ac_fw-la a_o perpendicular_a line_n ad._n and_o by_o the_o third_o proposition_n unto_o the_o line_n ab_fw-la put_v a_o equal_a line_n ad._n and_o by_o the_o first_o petition_n draw_v a_o right_a line_n from_o the_o point_n d_o to_o the_o poinâ—_n c._n and_o forasmuch_o as_o the_o line_n dam_n be_v equal_a to_o the_o line_n ab_fw-la the_o square_n which_o be_v make_v of_o the_o line_n dam_n be_v equal_a to_o the_o square_n which_o be_v make_v of_o the_o line_n ab_fw-la put_v the_o square_n of_o the_o line_n ac_fw-la common_a to_o they_o both_o wherefore_o the_o square_n of_o the_o line_n dam_n and_o ac_fw-la be_v equal_a to_o the_o square_n of_o the_o line_n basilius_n and_o ac_fw-la but_o by_o the_o proposition_n go_v before_o the_o square_a of_o the_o line_n dc_o be_v equal_a to_o the_o square_n of_o the_o line_n ad_fw-la and_o ac_fw-la for_o the_o angle_n dac_o be_v a_o right_a angle_n and_o the_o square_a of_o bc_o be_v by_o supposition_n equal_a to_o the_o square_n of_o ab_fw-la and_o ac_fw-la wherefore_o the_o square_a of_o dc_o be_v equal_a to_o the_o square_n of_o bc_o wherefore_o the_o side_n dc_o be_v equal_a to_o the_o side_n bc._n and_o forasmuch_o as_o ab_fw-la be_v equal_a to_o ad_fw-la â—nd_n ac_fw-la be_v common_a to_o they_o both_o therefore_o these_o two_o side_n dam_n and_o ac_fw-la be_v equal_a to_o these_o two_o side_n basilius_n and_o ac_fw-la the_o one_o to_o the_o other_o and_o the_o base_a dc_o be_v equal_a to_o the_o base_a be_v wherefore_o by_o the_o 8._o proposition_n the_o angle_n dac_o be_v equal_a to_o the_o angle_n bac_n but_o the_o angle_n dac_o be_v a_o right_a angle_n wherefore_o also_o the_o angle_n bac_n be_v a_o right_a angle_n if_o therefore_o the_o square_n which_o be_v make_v of_o one_o of_o the_o side_n of_o a_o triangle_n be_v equal_a to_o the_o square_n which_o be_v make_v of_o the_o two_o other_o side_n of_o the_o same_o triangle_n the_o angle_n comprehend_v under_o those_o two_o other_o side_n be_v a_o right_a angle_n which_o be_v require_v to_o be_v prove_v former_a this_o proposition_n be_v the_o converse_n of_o the_o former_a and_o be_v of_o pelitarius_n demonstrate_v by_o a_o argument_n lead_v to_o a_o impossibility_n after_o this_o manner_n the_o end_n of_o the_o first_o book_n of_o euclides_n element_n ¶_o the_o second_o book_n of_o euclides_n element_n in_o this_o second_o book_n euclid_n show_v book_n what_o be_v a_o gnomon_n and_o a_o right_a angle_a parallelogram_n also_o in_o this_o book_n be_v set_v forth_o the_o power_n of_o line_n divide_v even_o and_o uneven_o and_o of_o line_n add_v one_o to_o a_o other_o the_o power_n of_o a_o line_n line_n be_v the_o square_a of_o the_o same_o line_n that_o be_v a_o square_a every_o side_n of_o which_o be_v equal_a to_o the_o line_n so_o that_o here_o be_v set_v forth_o the_o quality_n and_o propriety_n of_o the_o square_n and_o right_n line_v figure_n which_o be_v make_v of_o line_n &_o of_o their_o part_n algebra_n the_o arithmetician_n also_o our_o of_o this_o book_n gather_v many_o compendious_a rule_n of_o reckon_n and_o many_o rule_v also_o of_o algebra_n with_o the_o equation_n therein_o use_v the_o ground_n also_o of_o those_o rule_n be_v for_o the_o most_o part_n by_o this_o second_o book_n demonstrate_v book_n this_o book_n moreover_o contain_v two_o wonderful_a proposition_n one_o of_o a_o obtuse_a angle_a triangle_n and_o the_o other_o of_o a_o acute_a which_o with_o the_o aid_n of_o the_o 47._o proposition_n of_o the_o first_o book_n of_o euclid_n which_o be_v of_o a_o rectangle_n triangle_n of_o how_o great_a force_n and_o profit_n they_o be_v in_o matter_n of_o astronomy_n they_o know_v which_o have_v travail_v in_o that_o art_n wherefore_o if_o this_o book_n have_v none_o other_o profit_n be_v side_n only_o for_o these_o 2._o proposition_n sake_n it_o be_v diligent_o to_o be_v embrace_v and_o study_v the_o definition_n 1._o every_o rectangled_a parallelogram_n definition_n be_v say_v to_o be_v contain_v under_o two_o right_a line_n comprehend_v a_o right_a angle_n a_o parallelogram_n be_v a_o figure_n of_o four_o side_n be_v who_o two_o opposite_a or_o contrary_a side_n be_v equal_a the_o one_o to_o the_o other_o there_o be_v of_o parallelogram_n four_o kind_n parallelogram_n a_o square_a a_o figure_n of_o one_o side_n long_o a_o rombus_n or_o diamond_n and_o a_o romboide_n or_o diamond_n like_o figure_n as_o before_o be_v say_v in_o the_o 33._o definition_n of_o the_o first_o book_n of_o these_o four_o sort_n the_o square_a and_o the_o figure_n of_o one_o side_n long_o be_v only_o right_a angle_a parallelogram_n for_o that_o all_o their_o angle_n be_v right_a angle_n and_o either_o of_o they_o be_v contain_v according_a to_o this_o definition_n under_o two_o right_a line_n whiâ—h_o concur_v together_o and_o cause_v the_o right_a angle_n and_o contain_v the_o same_o of_o which_o two_o line_n the_o one_o be_v the_o length_n of_o the_o figure_n &_o the_o other_o the_o breadth_n the_o parallelogram_n be_v imagine_v to_o be_v make_v by_o the_o draught_n or_o motion_n of_o one_o of_o the_o line_n into_o the_o length_n of_o the_o other_o as_o if_o two_o number_n shall_v be_v multiply_v the_o one_o into_o the_o other_o as_o the_o figure_n abcd_o be_v a_o parallelogram_n and_o be_v say_v to_o be_v contain_v under_o the_o two_o right_a line_n ab_fw-la and_o ac_fw-la which_o contain_v the_o right_a angle_n bac_n or_o under_o the_o two_o right_a line_n ac_fw-la and_o cd_o for_o they_o likewise_o contain_v the_o right_a angle_n acd_o of_o which_o 2._o line_n the_o one_o namely_o ab_fw-la be_v the_o length_n and_o the_o other_o namely_o ac_fw-la be_v the_o breadth_n and_o if_o we_o imagine_v the_o line_n ac_fw-la to_o be_v draw_v or_o move_v direct_o according_a to_o the_o length_n of_o the_o line_n ab_fw-la or_o contrary_a wise_a the_o line_n ab_fw-la to_o be_v move_v direct_o according_a to_o the_o length_n of_o the_o line_n ac_fw-la you_o shall_v produce_v the_o whole_a rectangle_n parallelogram_n abcd_o which_o be_v say_v to_o be_v contain_v of_o they_o even_o as_o one_o number_n multiply_v by_o a_o other_o produce_v a_o plain_a and_o right_a angle_a superficial_a number_n as_o you_o see_v in_o the_o figure_n here_o set_v where_o the_o number_n of_o six_o or_o six_o unity_n be_v multiply_v by_o the_o number_n of_o five_o or_o by_o five_o unity_n of_o which_o multiplication_n be_v produce_v 30._o which_o number_n be_v set_v down_o and_o describe_v by_o his_o unity_n represent_v a_o plain_n and_o a_o right_a angle_a number_n wherefore_o even_o as_o equal_a number_n multiple_v by_o equal_a number_n produce_v number_n equal_v the_o one_o to_o the_o other_o so_o rectangle_n parallelogram_n which_o be_v comprehend_v under_o equal_a line_n be_v equal_a the_o one_o to_o the_o other_o definition_n 2._o in_o every_o parallelogram_n one_o of_o those_o parallelogram_n which_o soever_o it_o be_v which_o be_v about_o the_o diameter_n together_o with_o the_o two_o supplement_n be_v call_v a_o gnomon_n those_o particular_a parallelogram_n be_v say_v to_o be_v about_o the_o diameter_n of_o the_o parallelogram_n which_o have_v the_o same_o diameter_n which_o the_o whole_a parallelogram_n have_v and_o supplement_n be_v such_o which_o be_v without_o the_o diameter_n of_o the_o whole_a parallelogram_n as_o of_o the_o parallelogram_n abcd_o the_o partial_a or_o particular_a parallelogram_n gkcf_n and_o ebkh_n be_v parallelogram_n about_o the_o diameter_n for_o that_o each_o of_o they_o have_v for_o his_o diameter_n a_o part_n of_o the_o diameter_n of_o the_o whole_a parallelogram_n as_o ck_v and_o kb_v the_o particular_a diameter_n be_v part_n of_o the_o line_n cb_o which_o be_v the_o diameter_n of_o the_o whole_a parallelogram_n and_o the_o two_o parallelogram_n aegk_n and_o khfd_n be_v supplement_n because_o they_o be_v without_o the_o diameter_n of_o the_o whole_a parallelogram_n namely_o cb._n now_o any_o one_o of_o those_o partial_a parallelogram_n
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
namely_o the_o first_o ten_o proposition_n as_o they_o follow_v in_o order_n which_o be_v undoubted_o great_a pleasure_n to_o consider_v also_o great_a increase_n &_o furniture_n of_o knowledge_n who_o proposition_n be_v set_v orderly_a after_o the_o proposition_n of_o euclid_n every_o one_o of_o barlaam_n correspondent_a to_o the_o same_o of_o euclid_n and_o doubtless_o it_o be_v wonderful_a to_o see_v how_o these_o two_o contrary_a kind_n of_o quantity_n quantity_n discrete_a or_o number_n and_o quantity_n continual_a or_o magnitude_n which_o be_v the_o subject_n or_o matterâ—_n of_o arithmitique_a and_o geometry_n shall_v have_v in_o they_o one_o and_o the_o same_o propriety_n common_a to_o they_o both_o in_o very_a many_o point_n and_o affection_n although_o not_o in_o all_o for_o a_o line_n may_v in_o such_o sort_n be_v divide_v that_o what_o proportion_n the_o whole_a have_v to_o the_o great_a part_n the_o same_o shall_v the_o great_a part_n have_v to_o the_o less_o but_o that_o can_v not_o be_v in_o number_n for_o a_o number_n can_v not_o so_o be_v divide_v that_o the_o whole_a number_n to_o the_o great_a part_n thereof_o shall_v have_v that_o proportion_n which_o the_o great_a part_n have_v to_o the_o less_o as_o jordane_n very_o plain_o prove_v in_o his_o book_n of_o arithmetic_n which_o thing_n campane_n also_o as_o we_o shall_v afterward_o in_o the_o 9_o book_n after_o the_o 15._o proposition_n see_v prove_v and_o as_o touch_v these_o ten_o first_o proposition_n of_o the_o second_o book_n of_o euclid_n demonstrate_v by_o barlaam_n in_o number_n they_o be_v also_o demonstrate_v of_o campane_n after_o the_o 15._o proposition_n of_o the_o 9_o book_n who_o demonstration_n i_o mind_n by_o god_n help_n to_o set_v forth_o when_o i_o shall_v come_v to_o the_o place_n they_o be_v also_o demonstrate_v of_o jordane_n that_o excellet_fw-la learned_a author_n in_o the_o first_o book_n of_o his_o arithmetic_n in_o the_o mean_a time_n i_o think_v it_o not_o amiss_o here_o to_o set_v forth_o the_o demonstration_n of_o barlaam_n for_o that_o they_o geve_v great_a light_n to_o the_o second_o book_n of_o euclid_n beside_o the_o inestimable_a pleasure_n which_o they_o bring_v to_o the_o studious_a considerer_n and_o now_o to_o declare_v the_o first_o proposition_n by_o number_n i_o have_v put_v this_o example_n follow_v take_v two_o number_n the_o one_o undivided_a as_o 74._o the_o other_o divide_v into_o what_o part_n and_o how_o many_o you_o listen_v as_o 37._o divide_v into_o 20._o 10._o 5._o and_o 2d_o which_o altogether_o make_v the_o whole_a 37._o then_o if_o you_o multiply_v the_o number_n undivided_a namely_o 74_o into_o all_o the_o part_n of_o the_o number_n divide_v as_o into_o 20._o 10._o 5._o and_o 2._o you_o shall_v produce_v 1480._o 740._o 370._o 148._o which_o add_v together_o make_v 2738_o which_o self_n number_n be_v also_o produce_v if_o you_o multiply_v the_o two_o number_n first_o give_v the_o one_o into_o the_o other_o as_o you_o see_v in_o the_o example_n on_o the_o other_o side_n set_v so_o by_o the_o aid_n of_o this_o proposition_n be_v get_v a_o compendious_a way_n of_o multiplication_n by_o break_v of_o one_o of_o the_o number_n into_o his_o part_n which_o oftentimes_o serve_v to_o great_a use_n in_o working_n chiâ—â—ly_o in_o the_o rule_n of_o proportion_n the_o demonstration_n of_o which_o proposition_n follow_v in_o barlaam_n but_o â—irst_n be_v put_v of_o the_o author_n these_o principle_n follow_v barlaam_n ¶_o principles_n 1._o a_o number_n be_v sâ—yd_v to_o multiply_v a_o other_o number_n when_o the_o number_n multiply_v be_v so_o oftentimes_o add_v to_o itself_o as_o there_o be_v unity_n in_o the_o number_n which_o multiply_v whereby_o be_v produce_v a_o certain_a number_n which_o the_o number_n multiply_v measure_v by_o the_o unity_n which_o be_v in_o the_o number_n which_o multipli_v 2._o and_o the_o number_n produce_v of_o that_o a_o multiplication_n be_v call_v a_o plain_a or_o superficial_a number_n 3._o a_o square_a number_n be_v that_o which_o be_v produce_v of_o the_o multiplicatian_n of_o any_o number_n into_o itself_o 4._o every_o less_o number_n compare_v to_o a_o great_a be_v say_v to_o be_v a_o part_n of_o the_o great_a whether_o the_o less_o measure_n the_o great_a or_o measure_v it_o not_o 5._o number_n who_o one_o and_o the_o self_n same_o number_n measure_v equal_o that_o be_v by_o one_o and_o the_o self_n same_o number_n be_v equal_a the_o one_o to_o the_o other_o 6._o number_n that_o be_v equemultiplâ—ces_n to_o one_o and_o the_o self_n same_o number_n that_o be_v which_o contain_v one_o and_o the_o same_o number_n equal_o and_o alike_o be_v equal_a the_o one_o to_o the_o other_o the_o first_o proposition_n two_o number_n be_v give_v if_o the_o one_o of_o they_o be_v divide_v into_o any_o number_n how_o many_o soever_o barlaam_n the_o plain_n or_o superficial_a number_n which_o be_v produce_v of_o the_o multiplication_n of_o the_o two_o number_n first_o give_v the_o one_o into_o the_o other_o shall_v be_v equal_a to_o the_o superficial_a number_n which_o be_v produce_v of_o the_o multiplication_n of_o the_o number_n not_o divide_v into_o every_o part_n of_o the_o number_n divide_v suppose_v that_o there_o be_v two_o number_n ab_fw-la and_o c._n and_o divide_v the_o number_n ab_fw-la into_o certain_a other_o number_n how_o many_o soever_o as_o into_o ad_fw-la de_fw-fr and_o ebb_v then_o i_o say_v that_o the_o superficial_a number_n which_o be_v produce_v of_o the_o multiplication_n of_o the_o number_n c_o into_o the_o number_n ab_fw-la be_v equal_a to_o the_o superficial_a number_n which_o be_v produce_v of_o the_o multiplication_n of_o the_o number_n c_o into_o the_o number_n ad_fw-la and_o of_o c_o into_o de_fw-fr and_o of_o c_o into_o ebb_n for_o let_v fletcher_n be_v the_o superficial_a number_n produce_v of_o the_o multiplication_n of_o the_o number_n c_o into_o the_o number_n ab_fw-la and_o let_v gh_o be_v the_o superficial_a number_n produce_v of_o the_o multiplication_n of_o c_o into_o ad_fw-la and_o let_v high_a be_v produce_v of_o the_o multiplication_n of_o c_o into_o de_fw-fr aâ—d_v final_o of_o the_o multiplication_n of_o c_o into_o ebb_n let_v there_o be_v produce_v the_o number_n ik_fw-mi now_o forasmuch_o as_o ab_fw-la multiply_v the_o number_n c_o produce_v the_o number_n f_o therefore_o the_o number_n c_o measure_v the_o number_n f_o by_o the_o unity_n which_o be_v in_o the_o number_n ab_fw-la and_o by_o the_o same_o reason_n may_v be_v prove_v that_o the_o number_n c_o do_v also_o measure_v the_o number_n gh_o by_o the_o unity_n which_o be_v in_o the_o number_n ad_fw-la and_o that_o it_o do_v measure_v the_o number_n high_a by_o the_o unity_n which_o be_v in_o the_o number_n df_o and_o final_o that_o it_o measure_v the_o number_n ik_fw-mi by_o the_o unity_n which_o be_v in_o the_o number_n ebb_n wherefore_o the_o number_n c_o measure_v the_o whole_a number_n gk_o by_o the_o unity_n which_o be_v in_o the_o number_n ab_fw-la but_o it_o before_o measure_v the_o number_n f_o by_o the_o unity_n which_o be_v in_o the_o number_n ab_fw-la wherefore_o either_o of_o these_o number_n fletcher_n and_o gk_o be_v equemultiplex_n to_o the_o number_n c._n but_o number_n which_o be_v equemultiplices_fw-la to_o one_o &_o the_o self_n same_o number_n be_v equal_a the_o one_o to_o the_o other_o by_o the_o 6._o definition_n wherefore_o the_o number_n fletcher_n be_v equal_a to_o the_o number_n gk_o but_o the_o number_n fletcher_n be_v the_o superficial_a number_n produce_v of_o the_o multiplication_n of_o the_o number_n c_o into_o the_o number_n ab_fw-la and_o the_o number_n gk_o be_v compose_v of_o the_o superficial_a number_n produce_v of_o the_o multiplication_n of_o the_o number_n c_o not_o divide_v into_o every_o one_o of_o the_o number_n ad_fw-la de_fw-fr and_o ebb_v if_o therefore_o there_o be_v two_o number_n give_v and_o the_o one_o of_o they_o be_v divide_v etc_n etc_n which_o be_v require_v to_o be_v prove_v the_o 2._o theorem_a the_o 2._o proposition_n if_o a_o right_a line_n be_v divide_v by_o chance_n the_o rectangle_v figure_n which_o be_v comprehend_v under_o the_o whole_a and_o every_o one_o of_o the_o part_n be_v equal_a to_o the_o square_n which_o be_v make_v of_o the_o whole_a svppose_v that_o the_o right_a line_n ab_fw-la be_v by_o chance_n deny_v in_o the_o point_n c._n then_o i_o say_v that_o the_o rectangle_n figure_n comprehend_v under_o ab_fw-la and_o bc_o together_o with_o the_o rectangle_n comprehend_v under_o ab_fw-la and_o ac_fw-la be_v equal_a unto_o the_o square_n make_v of_o ab_fw-la construction_n describe_v by_o the_o 46._o of_o the_o first_o upon_o ab_fw-la a_o square_a adeb_n and_o by_o the_o 31_o of_o the_o first_o by_o the_o point_n c_o draw_v a_o line_n parallel_n unto_o either_o of_o these_o line_n ad_fw-la aâ—d_fw-la be_v and_o let_v the_o same_o
definition_n of_o the_o first_o the_o line_n mb_n be_v equal_a unto_o the_o line_n mk_o put_v the_o line_n md_o common_a to_o the_o both_o wherefore_o these_o two_o line_n mk_o and_o md_o be_v equal_a to_o these_o two_o line_n bm_n and_o md_o the_o one_o to_o the_o other_o and_o the_o angle_n kmd_v be_v by_o the_o 23._o of_o the_o first_o equal_a to_o the_o angle_n bmd_v wherefore_o by_o the_o 4._o of_o the_o first_o the_o base_a dk_o be_v equal_a to_o the_o base_a db._n or_o it_o may_v thus_o be_v demonstrate_v impossibility_n draw_v by_o the_o first_o petition_n a_o line_n from_o m_o to_o n._n and_o for_o asmuch_o as_o by_o the_o 15._o definition_n of_o the_o first_o the_o line_n km_o be_v equal_a unto_o the_o line_n mn_o and_o the_o line_n md_o be_v common_a to_o they_o both_o and_o the_o base_a kd_v be_v equal_a to_o the_o base_a dn_o by_o supposition_n therefore_o by_o the_o 8._o of_o the_o first_o the_o angle_n kmd_v be_v equal_a to_o the_o angle_n dmn_n but_o the_o angle_n kmd_v be_v equal_a to_o the_o angle_n bmd_v wherefore_o the_o angle_n bmd_v be_v equal_a to_o the_o angle_n nmd_v the_o less_o unto_o the_o great_a which_o be_v impossible_a wherefore_o from_o the_o point_n d_o can_v not_o be_v draw_v unto_o the_o circumference_n abc_n on_o each_o side_n of_o dg_o the_o jest_n more_o than_o two_o equal_a right_a line_n if_o therefore_o without_o a_o circle_n be_v take_v any_o point_n and_o from_o that_o point_n be_v draw_v into_o the_o circle_n unto_o the_o circumference_n certain_a right_a line_n of_o which_o let_v one_o be_v draw_v by_o the_o centre_n and_o let_v the_o rest_n be_v draw_v at_o all_o adventure_n the_o great_a of_o those_o right_a line_n which_o fall_v in_o the_o concavity_n or_o hollowness_n of_o the_o circumference_n of_o the_o circle_n be_v that_o which_o pass_v by_o the_o centre_n and_o of_o all_o the_o other_o line_n that_o line_n which_o be_v nigh_o to_o the_o line_n which_o pass_v by_o the_o centre_n be_v great_a than_o that_o which_o be_v more_o distant_a but_o of_o those_o right_a line_n which_o end_n in_o the_o convexe_n part_n of_o the_o circumference_n that_o line_n be_v the_o lest_o which_o be_v draw_v from_o the_o point_n to_o the_o dimetient_fw-la and_o of_o the_o other_o line_n that_o which_o be_v nigh_o to_o the_o least_o be_v always_o less_o than_o that_o which_o be_v more_o distant_a and_o from_o that_o point_n can_v be_v draw_v unto_o the_o circumference_n on_o each_o side_n of_o the_o lest_o only_a two_o equal_a right_a line_n which_o be_v require_v to_o be_v prove_v this_o proposition_n be_v call_v common_o in_o old_a book_n amongst_o the_o barbarous_a caâ—dâ—_n panonis_n panonis_fw-la that_o be_v the_o peacock_n tail_n ¶_o a_o corollary_n hereby_o it_o be_v manifest_a corollary_n that_o the_o right_a line_n which_o be_v draw_v from_o the_o point_n give_v without_o the_o circle_n and_o fall_v within_o the_o circle_n be_v equal_o distant_a from_o the_o least_o or_o from_o the_o great_a which_o be_v draw_v by_o the_o centre_n be_v equal_a the_o one_o to_o the_o other_o but_o contrariwise_o if_o they_o be_v unequal_o distant_a whether_o they_o light_v upon_o the_o concave_n or_o convexe_a circumference_n of_o the_o circle_n they_o be_v unequal_a the_o 8._o theorem_a the_o 9_o proposition_n if_o within_o a_o circle_n be_v take_v a_o point_n and_o from_o that_o point_n be_v draw_v unto_o the_o circumference_n more_o than_o two_o equal_a right_a line_n the_o point_n take_v be_v the_o centre_n of_o the_o circle_n svppose_v that_o the_o circle_n be_v abc_n and_o within_o it_o let_v there_o be_v take_v the_o point_n d._n and_o from_o d_z let_v there_o be_v draw_v unto_o the_o circumference_n abc_n more_o than_o two_o equal_a right_a line_n construction_n that_o be_v da_z db_o and_o dc_o then_o i_o say_v that_o the_o point_n d_o be_v the_o centre_n of_o the_o circle_n abc_n draw_v by_o the_o first_o petition_n these_o right_a line_n ab_fw-la and_o bc_o demonstration_n and_o by_o the_o 10._o of_o the_o first_o divide_v they_o into_o two_o equal_a part_n in_o the_o point_n e_o and_o f_o namely_o the_o line_n ab_fw-la in_o the_o point_n e_o and_o the_o line_n bc_o in_o the_o point_n f._n and_o draw_v the_o line_n ed_z and_o fd_o and_o by_o the_o second_o petition_n extend_v the_o line_n ed_z and_o fd_v on_o each_o side_n to_o the_o point_n king_n g_o and_o h_o l._n and_o for_o asmuch_o as_o the_o line_n ae_n be_v equal_a unto_o the_o line_n ebb_n and_o the_o line_n ed_z be_v common_a to_o they_o both_o therefore_o these_o two_o side_n ae_n and_o ed_z be_v equal_a unto_o these_o two_o side_n be_v and_o ed_z and_o by_o supposition_n the_o base_a dam_n be_v equal_a to_o the_o base_a db._n wherefore_o by_o the_o 8._o of_o the_o first_o the_o angle_n aed_n be_v equal_a to_o the_o angle_n bed_n wherefore_o either_o of_o these_o angle_n aed_n and_o bed_n be_v a_o right_a angle_n wherefore_o the_o line_n gk_o divide_v the_o line_n ab_fw-la into_o two_o equal_a part_n and_o make_v right_a angle_n and_o for_o asmuch_o as_z if_o in_o a_o circle_n a_o right_a line_n divide_v a_o other_o right_a line_n into_o two_o equal_a part_n in_o such_o sort_n that_o it_o make_v also_o right_a angle_n in_o the_o line_n that_o divide_v be_v the_o centre_n of_o the_o circle_n by_o the_o corollary_n of_o the_o first_o of_o the_o three_o therefore_o by_o the_o same_o corollary_n in_o the_o line_n gk_o be_v the_o centre_n of_o the_o circle_n abc_n and_o by_o the_o same_o reason_n may_v we_o prove_v that_o in_o the_o line_n hl_o be_v the_o centre_n of_o the_o circle_n abc_n and_o the_o right_a line_n gk_o and_o hl_o have_v no_o other_o point_n common_a to_o they_o both_o beside_o the_o point_n d._n wherefore_o the_o point_n d_o be_v the_o centre_n of_o the_o circle_n abc_n if_o therefore_o within_o a_o circle_n be_v take_v a_o point_n and_o from_o that_o point_n be_v draw_v unto_o the_o circumference_n more_o than_o two_o equal_a right_a line_n the_o point_n take_v be_v the_o centre_n of_o the_o circle_n which_o be_v require_v to_o be_v prove_v ¶_o a_o other_o demonstration_n let_v there_o be_v take_v within_o the_o circle_n abc_n the_o point_n d._n impossibility_n and_o from_o the_o point_n d_o let_v there_o be_v draw_v unto_o the_o circumference_n more_o than_o two_o equal_a right_a line_n namely_o dam_n db_o and_o dc_o then_o i_o say_v that_o the_o point_n d_o be_v the_o centre_n of_o the_o circle_n for_o if_o not_o then_o if_o it_o be_v possible_a let_v the_o point_n e_o be_v the_o centre_n and_o draw_v a_o line_n from_o d_z to_o e_o and_o extend_v de_fw-fr to_o the_o point_n fletcher_n and_o g._n wherefore_o the_o line_n fg_o be_v the_o diameter_n of_o the_o circle_n abc_n and_o for_o asmuch_o as_o in_o fg_o the_o diameter_n of_o the_o circle_n abc_n be_v take_v a_o point_n namely_o d_z which_o be_v not_o the_o centre_n of_o that_o circle_n therefore_o by_o the_o 7._o of_o the_o three_o the_o line_n dg_o be_v the_o great_a and_o the_o line_n dc_o be_v great_a than_o the_o line_n db_o and_o the_o line_n db_o be_v greateâ—_n than_o the_o line_n da._n but_o the_o line_n dc_o db_o dam_n be_v also_o equal_a by_o supposition_n which_o be_v impossible_a wherefore_o the_o point_n e_o be_v not_o the_o centre_n of_o the_o circle_n abc_n and_o in_o like_a sort_n may_v we_o prove_v that_o no_o other_o point_n beside_o d._n wherefore_o the_o point_n d_o be_v the_o centre_n of_o the_o circle_n abc_n which_o be_v require_v to_o be_v prove_v the_o 9_o theorem_a the_o 10._o proposition_n a_o circle_n cut_v not_o a_o circle_n in_o more_o point_n than_o two_o for_o if_o it_o be_v possible_a let_v the_o circle_n abc_n cut_v the_o circle_n def_n in_o more_o point_n than_o two_o impossibility_n that_o be_v in_o b_o g_o h_o &_o f._n and_o draw_v line_n from_o b_o to_o g_z and_o from_o b_o to_o h._n and_o by_o the_o 10._o of_o the_o first_o divide_v either_o of_o the_o line_n bg_o &_o bh_o into_o two_o equal_a part_n in_o the_o point_n king_n and_o l._n and_o by_o the_o 11._o of_o the_o first_o from_o the_o point_n king_n raise_v up_o unto_o the_o line_n bh_o a_o perpendicular_a line_n kc_o and_o likewise_o from_o the_o point_n l_o raise_v up_o unto_o the_o line_n bg_o a_o perpendicular_a line_n lm_o and_o extend_v the_o line_n ck_o to_o the_o point_n a_o and_o lnm_n to_o the_o point_n x_o and_o e._n and_o for_o asmuch_o as_o in_o the_o circle_n abc_n the_o right_a line_n ac_fw-la divide_v the_o right_a line_n bh_o into_o two_o equal_a part_n and_o make_v right_a angle_n therefore_o by_o the_o 3._o of_o the_o three_o in_o the_o line_n ac_fw-la be_v the_o centre_n of_o
the_o circle_n abc_n again_o for_o asmuch_o as_o in_o the_o self_n same_o circle_n abc_n the_o right_a line_n nx_o that_o be_v the_o line_n menander_n divide_v the_o right_a line_n bg_o into_o two_o equal_a part_n and_o make_v right_a angle_n therefore_o by_o the_o three_o of_o the_o three_o in_o the_o line_n nx_o be_v the_o centre_n of_o the_o circle_n abc_n and_o it_o be_v prove_v that_o it_o be_v also_o in_o the_o line_n ac_fw-la and_o these_o two_o right_a line_n ac_fw-la and_o nx_o meet_v together_o in_o no_o other_o point_n beside_o o._n wherefore_o the_o point_n oh_o be_v the_o centre_n of_o the_o circle_n abc_n and_o in_o like_a sort_n may_v we_o prove_v that_o the_o point_n oh_o be_v the_o centre_n of_o the_o circle_n def_n wherefore_o the_o two_o circle_n abc_n and_o def_n devide_v the_o one_o the_o other_o have_v one_o and_o the_o same_o centre_n which_o by_o the_o 5._o of_o the_o three_o be_v impossible_a a_o circle_n therefore_o cut_v not_o a_o circle_n in_o more_o point_n than_o two_o which_o be_v require_v to_o be_v prove_v an_o other_o demonstration_n to_o prove_v the_o same_o suppose_v that_o the_o circle_n abc_n do_v cut_v the_o circle_n dgf_n in_o more_o point_n than_o two_o impossibility_n that_o be_v in_o b_o g_o f_o and_o h._n and_o by_o the_o first_o of_o the_o three_o take_v the_o centre_n of_o the_o circle_n abc_n and_o let_v the_o same_o be_v the_o point_n k._n and_o draw_v these_o right_a line_n kb_v kg_a and_o kf_n now_o for_o asmuch_o as_o within_o the_o circle_n def_n be_v take_v a_o certain_a point_n king_n and_o from_o that_o point_n be_v draw_v unto_o the_o circumference_n more_o than_o two_o equal_a right_a line_n namely_o kb_o kg_a and_o kf_n therefore_o by_o the_o 9_o of_o the_o three_o king_n be_v the_o centre_n of_o the_o circle_n def_n and_o the_o point_n king_n be_v the_o centre_n of_o the_o circle_n abc_n wherefore_o two_o circle_n cut_v the_o one_o the_o other_o have_v one_o and_o the_o same_o centre_n which_o by_o the_o 5._o of_o the_o three_o be_v impossible_a a_o circle_n therefore_o cut_v not_o a_o circle_n in_o more_o point_n than_o two_o which_o be_v require_v to_o be_v demonstrate_v the_o 10._o theorem_a the_o 11._o proposition_n if_o two_o circle_n touch_v the_o one_o the_o other_o inward_o their_o centre_n be_v give_v a_o right_a line_n join_v together_o their_o centre_n and_o produce_v will_v fall_v upon_o the_o touch_n of_o the_o circle_n svppose_v that_o these_o two_o circle_n abc_n and_o ade_n do_v touch_v the_o one_o the_o other_o in_o the_o point_n a._n construction_n and_o by_o the_o first_o of_o the_o three_o take_v the_o centre_n of_o the_o circle_n abc_n and_o let_v the_o same_o be_v f_o and_o likewise_o the_o centre_n of_o the_o circle_n ade_n and_o let_v the_o same_o be_v g._n then_o i_o say_v that_o a_o right_a line_n draw_v from_o fletcher_n to_o g_z and_o be_v produce_v will_v fall_v upon_o the_o point_n a._n for_o if_o not_o impossibility_n then_o if_o it_o be_v possible_a let_v it_o fall_v as_o the_o line_n fgdh_n do_v and_o draw_v these_o right_a line_n of_o &_o ag._n now_o for_o asmuch_o as_o the_o line_n agnostus_n and_o gf_o be_v by_o the_o 20._o of_o the_o first_o great_a then_o the_o line_n favorina_n that_o be_v than_o the_o line_n fh_o take_v away_o the_o line_n gf_o which_o be_v common_a to_o they_o both_o wherefore_o the_o residue_n agnostus_n be_v great_a than_o the_o residue_n gh_o but_o the_o line_n dg_o be_v equal_a unto_o the_o line_n ga_o by_o the_o 15._o definition_n of_o the_o first_o wherefore_o the_o line_n gd_o be_v great_a than_o the_o line_n gh_o the_o less_o than_o the_o great_a which_o be_v impossible_a wherefore_o a_o right_a line_n draw_v from_o the_o point_n f_o to_o the_o point_n g_o and_o produce_v fall_v not_o beside_o the_o point_n a_o which_o be_v the_o point_n of_o the_o touch_n wherefore_o it_o fall_v upon_o the_o touch_n if_o therefore_o two_o circle_n touch_v the_o one_o the_o other_o inward_o their_o centre_n be_v give_v a_o right_a line_n join_v together_o their_o centre_n and_o produce_v will_v fall_v upon_o the_o touch_n of_o the_o circle_n which_o be_v require_v to_o be_v prove_v an_o other_o demonstration_n to_o prove_v the_o same_o but_o now_o let_v it_o fall_v as_o gfc_n fall_v and_o extend_v the_o line_n gfc_fw-la to_o the_o point_n h_o impossibility_n and_o draw_v these_o right_a line_n agnostus_n and_o af._n and_o for_o asmuch_o as_o the_o line_n agnostus_n and_o gf_o be_v by_o the_o 20._o of_o the_o first_o great_a then_o the_o line_n af._n but_o the_o line_n of_o be_v equal_a unto_o the_o line_n cf_o that_o be_v unto_o the_o line_n fh_o take_v away_o the_o line_n fg_o common_a to_o they_o both_o wherefore_o the_o residue_n agnostus_n be_v great_a than_o the_o residue_n gh_o that_o be_v the_o line_n gd_o be_v great_a than_o the_o line_n gh_o the_o less_o great_a than_o the_o great_a which_o be_v impossible_a which_o thing_n may_v also_o be_v prove_v by_o the_o 7._o proposition_n of_o this_o book_n for_o for_o asmuch_o as_o the_o line_n hc_n be_v the_o diameter_n of_o the_o circle_n abc_n absurdititie_n &_o in_o it_o be_v take_v a_o point_n which_o be_v not_o the_o centre_n namely_o the_o point_n g_o therefore_o the_o line_n ga_o be_v great_a than_o the_o line_n gh_o by_o the_o say_v 7._o proposition_n but_o the_o line_n gd_o be_v equal_a to_o the_o line_n ga_o by_o the_o definition_n of_o a_o circle_n wherefore_o the_o line_n gd_o be_v great_a than_o the_o line_n gh_o namely_o the_o part_n great_a than_o the_o whole_a which_o be_v impossible_a the_o 11._o theorem_a the_o 12._o proposition_n if_o two_o circle_n touch_v the_o one_o the_o other_o outward_o a_o right_a line_n draw_v by_o their_o centre_n shall_v pass_v by_o the_o touch_n svppose_v that_o these_o two_o circle_n abc_n and_o ade_n do_v touch_v the_o one_o the_o other_o outward_o in_o the_o point_n a._n and_o by_o the_o three_o of_o the_o three_o take_v the_o centre_n of_o the_o circle_n abc_n impossibility_n and_o let_v the_o same_o be_v the_o point_n f_o and_o likewise_o the_o centre_n of_o the_o circle_n ade_n and_o let_v the_o same_o be_v the_o point_n g._n then_o i_o say_v that_o a_o right_a line_n draw_v from_o the_o point_n f_o to_o the_o point_n g_o shall_v pass_v by_o the_o point_n of_o the_o touch_n namely_o by_o the_o point_n a._n for_o if_o not_o then_o if_o it_o be_v possible_a let_v it_o pass_v as_o the_o right_a line_n fcdg_n do_v and_o draw_v these_o right_a line_n of_o &_o ag._n and_o for_o asmuch_o as_o the_o point_n fletcher_n be_v the_o centre_n of_o the_o circle_n abc_n therefore_o the_o line_n favorina_n be_v equal_a unto_o the_o line_n fc_o again_o for_o asmuch_o as_o the_o point_n g_z be_v the_o centre_n of_o the_o circle_n ade_n therefore_o the_o line_n ga_o be_v equal_a to_o the_o line_n gd_o and_o an_o it_o be_v prove_v that_o the_o line_n favorina_n be_v equal_a to_o the_o line_n fc_o wherefore_o the_o line_n favorina_n and_o agnostus_n be_v equal_a unto_o the_o line_n fc_a and_o gd_a wherefore_o the_o whole_a line_n fg_o be_v great_a than_o the_o line_n favorina_n and_o ag._n but_o it_o be_v also_o less_o by_o the_o 20._o of_o the_o first_o which_o be_v impossible_a wherefore_o a_o right_a line_n draw_v from_o the_o point_n f_o to_o the_o point_n g_o shall_v pass_v by_o the_o point_n of_o the_o touch_n namely_o by_o the_o point_n a._n if_o therefore_o two_o circle_n touch_v the_o one_o the_o other_o outward_o a_o right_a line_n draw_v by_o their_o centre_n shall_v pass_v by_o the_o touch_n which_o be_v require_v to_o be_v demonstrate_v ¶_o a_o other_o demonstration_n after_o pelitarius_n absurdity_n suppose_v that_o the_o two_o circle_n abc_n and_o def_n do_v touch_v the_o one_o the_o other_o outward_o in_o the_o point_n a_o and_o let_v g_o be_v the_o centre_n of_o the_o circle_n abc_n from_o which_o point_n produce_v by_o the_o touch_n of_o the_o circle_n the_o line_n ga_o to_o the_o point_n fletcher_n of_o the_o circumference_n def_n which_o for_o asmuch_o as_o it_o pass_v not_o by_o the_o centre_n of_o the_o circle_n def_n as_o the_o adversary_n affirm_v draw_v from_o the_o same_o centre_n g_o a_o other_o right_a line_n gk_o which_o if_o it_o be_v possible_a let_v pass_v by_o the_o centre_n of_o the_o circle_n def_n namely_o by_o the_o point_n h_o cut_v the_o circumference_n abc_n in_o the_o point_n b_o &_o the_o circumference_n def_n in_o the_o point_n d_o &_o let_v the_o opposite_a point_n thereof_o be_v in_o the_o point_n k._n and_o for_o asmuch_o as_o from_o the_o point_n g_z take_v without_o the_o circle_n def_n be_v draw_v the_o line_n gk_o pass_v by_o the_o centre_n h_o and_o
when_o perpendicular_a line_n draw_v from_o the_o centre_n to_o those_o line_n be_v equal_a by_o the_o 4._o definition_n of_o the_o three_o wherefore_o the_o line_n ab_fw-la and_o cd_o be_v equal_o distant_a from_o the_o centre_n but_o now_o suppose_v that_o the_o right_a line_n ab_fw-la and_o cd_o be_v equal_o distant_a from_o the_o centre_n first_o that_o be_v let_v the_o perpendicular_a line_n of_o be_v equal_a to_o the_o perpendicular_a line_n eglantine_n then_o i_o say_v that_o the_o line_n ab_fw-la be_v equal_a to_o the_o line_n cd_o for_o the_o same_o order_n of_o construction_n remain_v we_o may_v in_o like_a sort_n prove_v that_o the_o line_n ab_fw-la be_v double_a to_o the_o line_n of_o and_o that_o the_o line_n cd_o be_v double_a to_o the_o line_n cg_o and_o for_o asmuch_o as_o the_o line_n ae_n be_v equal_a to_o the_o line_n ce_fw-fr for_o they_o be_v draw_v from_o the_o centre_n to_o the_o circumference_n therefore_o the_o square_a of_o the_o line_n ae_n be_v equal_a to_o the_o square_n of_o the_o line_n ce._n but_o by_o the_o 47._o of_o the_o first_o to_o the_o square_n of_o the_o line_n ae_n be_v equal_a the_o square_n of_o the_o line_n of_o and_o fa._n and_o by_o the_o self_n same_o to_o the_o square_n of_o the_o line_n ce_fw-fr be_v equal_v the_o square_n of_o the_o line_n eglantine_n and_o gc_o wherefore_o the_o square_n of_o the_o line_n of_o and_o favorina_n be_v equal_a to_o the_o square_n of_o the_o line_n eglantine_n and_o gc_o of_o which_o the_o square_a of_o the_o line_n eglantine_n be_v equal_a to_o the_o square_n of_o the_o line_n of_o for_o the_o line_n of_o be_v equal_a to_o the_o line_n eglantine_n wherefore_o by_o the_o three_o common_a sentence_n the_o square_a remain_v namely_o the_o square_a of_o the_o line_n of_o be_v equal_a to_o the_o square_n of_o the_o line_n cg_o wherefore_o the_o line_n ac_fw-la be_v equal_a unto_o the_o line_n cg_o but_o the_o line_n ab_fw-la be_v double_a to_o the_o line_n of_o and_o the_o line_n cd_o be_v double_a to_o the_o line_n cg_o wherefore_o the_o line_n ab_fw-la be_v equal_a to_o the_o line_n cd_o wherefore_o in_o a_o circle_n equal_a right_a line_n be_v equal_o distant_a from_o the_o centre_n and_o line_n equal_o distant_a from_o the_o centre_n be_v equal_a the_o one_o to_o the_o outstretch_o which_o be_v require_v to_o be_v prove_v ¶_o a_o other_o demonstration_n for_o the_o first_o part_n after_o campane_n campane_n suppose_v that_o there_o be_v a_o circle_n abdc_n who_o centre_n let_v be_v the_o point_n e._n and_o draw_v in_o it_o two_o equal_a line_n ab_fw-la and_o cd_o then_o i_o say_v that_o they_o be_v equal_o distant_a from_o the_o centre_n draw_v from_o the_o centre_n unto_o the_o line_n ab_fw-la and_o cd_o these_o perpendicular_a line_n of_o and_o eglantine_n and_o by_o the_o 2._o part_n of_o the_o 3._o of_o this_o book_n the_o line_n ab_fw-la shall_v be_v equal_o divide_v in_o the_o point_n f._n and_o the_o line_n cd_o shall_v be_v equal_o divide_v in_o the_o point_n g._n and_o draw_v these_o right_a line_n ea_fw-la ebb_n ec_o and_o ed._n and_o for_o asmuch_o as_o in_o the_o triangle_n aeb_fw-mi the_o two_o side_n ab_fw-la and_o ae_n be_v equal_a to_o the_o two_o side_n cd_o and_o ce_fw-fr of_o the_o triangle_n ce_z &_o the_o base_a ebb_n be_v equal_a to_o the_o base_a ed._n therefore_o by_o the_o 8._o of_o the_o first_o the_o angle_n at_o the_o point_n a_o shall_v be_v equal_a to_o the_o angle_n at_o the_o point_n c._n and_o for_o asmuch_o as_o in_o the_o triangle_n aef_n the_o two_o side_n ae_n and_o of_o be_v equal_a to_o the_o two_o side_n ce_fw-fr and_o cg_o of_o the_o triangle_n ceg_n and_o the_o angle_n eaf_n be_v equal_a to_o the_o angle_n ceg_n therefore_o by_o the_o 4._o of_o the_o first_o the_o base_a of_o iâ—_z equal_a to_o the_o base_a eglantine_n which_o for_o asmuch_o as_o they_o be_v perpendicular_a line_n therefore_o the_o line_n ab_fw-la &_o cd_o be_v equal_o distant_a from_o the_o centre_n by_o the_o 4._o definition_n of_o this_o book_n the_o 14._o theorem_a the_o 15._o proposition_n in_o a_o circle_n the_o great_a line_n be_v the_o diameter_n and_o of_o all_o other_o line_n that_o line_n which_o be_v nigh_o to_o the_o centre_n be_v always_o great_a than_o that_o line_n which_o be_v more_o distant_a svppose_v that_o there_o be_v a_o circle_n abcd_o and_o let_v the_o diameter_n thereof_o be_v the_o line_n ad_fw-la and_o let_v the_o centre_n thereof_o be_v the_o point_n e._n and_o unto_o the_o diameter_n ad_fw-la let_v the_o line_n bc_o be_v nigh_a than_o the_o line_n fg._n then_o i_o say_v that_o the_o line_n ad_fw-la be_v the_o great_a and_o the_o line_n bc_o be_v great_a than_o the_o line_n fg._n draw_v by_o the_o 12._o of_o the_o first_o from_o the_o centre_n e_o to_o the_o line_n bc_o and_o fg_o perpendicular_a line_n eh_o and_o eke_o construction_n and_o for_o asmuch_o as_o the_o line_n bc_o be_v nigh_a unto_o the_o centre_n than_o the_o line_n fg_o therefore_o by_o the_o 4._o definition_n of_o the_o three_o the_o line_n eke_o be_v great_a than_o the_o line_n eh_o and_o by_o the_o three_o of_o the_o first_o put_v unto_o the_o line_n eh_o a_o equal_a line_n el._n and_o by_o the_o 11._o of_o the_o first_o from_o the_o point_n l_o raise_v up_o unto_o the_o line_n eke_o a_o perpendicular_a line_n lm_o and_o extend_v the_o line_n lm_o to_o the_o point_n n._n and_o by_o the_o first_o petition_n draw_v these_o right_a line_n em_n en_fw-fr of_o and_o eglantine_n and_o for_o asmuch_o as_o the_o line_n eh_o be_v equal_a to_o the_o line_n el_fw-es therefore_o by_o the_o 14._o of_o the_o three_o demonstration_n and_o by_o the_o 4._o definition_n of_o the_o same_o the_o line_n bc_o be_v equal_a to_o the_o line_n mn_v again_o for_o asmuch_o as_o the_o line_n ae_n be_v equal_a to_o the_o line_n em_n and_o the_o line_n ed_z to_o the_o line_n en_fw-fr therefore_o the_o line_n ad_fw-la be_v equal_a to_o the_o line_n menander_n and_o en_fw-fr but_o the_o line_n i_o and_o en_fw-fr be_v by_o the_o 20._o of_o the_o first_o great_a then_o the_o line_n mn_v wherefore_o the_o line_n ad_fw-la be_v great_a than_o the_o line_n mn_v and_o for_o asmuch_o as_o these_o two_o line_n menander_n and_o en_fw-fr be_v equal_v to_o these_o two_o line_n fe_o and_o eglantine_n by_o the_o 15._o definition_n of_o the_o first_o for_o they_o be_v draw_v from_o the_o centre_n to_o the_o circumference_n and_o the_o angle_n men_n be_v great_a than_o the_o angle_n feg_n therefore_o by_o the_o 24._o of_o the_o first_o the_o base_a mn_n be_v great_a than_o the_o base_a fg._n but_o it_o be_v prove_v that_o the_o line_n mn_o be_v equal_a to_o the_o line_n bc_o wherefore_o the_o line_n bc_o also_o be_v great_a than_o the_o line_n fg._n wherefore_o the_o diameter_n ad_fw-la be_v the_o great_a and_o the_o line_n bc_o be_v great_a than_o the_o line_n fg._n wherefore_o in_o a_o circle_n the_o great_a line_n be_v the_o diameter_n and_o of_o all_o the_o other_o line_n that_o line_n which_o be_v nigh_o to_o the_o centre_n be_v always_o great_a than_o that_o line_n which_o be_v more_o distant_a which_o be_v require_v to_o be_v prove_v ¶_o a_o other_o demonstration_n after_o campane_n in_o the_o circle_n abcd_o who_o centre_n let_v be_v the_o point_n e_o draw_v these_o line_n ab_fw-la ac_fw-la ad_fw-la fg_o and_o hk_o of_o which_o let_v the_o line_n ad_fw-la be_v the_o diameter_n of_o the_o circle_n campane_n then_o i_o say_v that_o the_o line_n ad_fw-la be_v the_o great_a of_o all_o the_o line_n and_o the_o other_o line_n each_o of_o the_o one_o be_v so_o much_o great_a than_o each_o of_o the_o other_o how_o much_o nigh_o it_o be_v unto_o the_o centre_n join_v together_o the_o end_n of_o all_o these_o line_n with_o the_o centre_n by_o draw_v these_o right_a line_n ebb_v ec_o eglantine_n eke_o eh_o and_o ef._n and_o by_o the_o 20._o of_o the_o first_o the_o two_o side_n of_o and_o eglantine_n of_o the_o triangle_n efg_o shall_v be_v great_a than_o the_o three_o side_n fg._n and_o for_o asmuch_o as_o the_o say_a side_n of_o &_o eglantine_n be_v equal_a to_o the_o line_n ad_fw-la by_o the_o definition_n of_o a_o circle_n therefore_o the_o line_n ad_fw-la be_v great_a than_o the_o line_n fg._n and_o by_o the_o same_o reason_n it_o be_v great_a than_o every_o one_o of_o the_o rest_n of_o the_o line_n if_o they_o be_v put_v to_o be_v base_n of_o triangle_n for_o that_o every_o two_o side_n draw_v from_o the_o centre_n be_v equal_a to_o the_o line_n ad._n which_o be_v the_o first_o part_n of_o the_o proposition_n again_o for_o asmuch_o as_o the_o two_o side_n of_o and_o eglantine_n of_o the_o triangle_n efg_o be_v equal_a to_o the_o
say_v that_o the_o line_n gfh_v which_o by_o the_o corollary_n of_o the_o 16._o of_o this_o book_n touch_v the_o circle_n be_v a_o parallel_n unto_o the_o line_n ab_fw-la circle_n for_o forasmuch_o as_o the_o right_a line_n cf_o fall_a upon_o either_o of_o these_o line_n ab_fw-la &_o gh_o make_v all_o the_o angle_n at_o the_o point_n â—_o right_n angle_n by_o the_o 3._o of_o this_o book_n and_o the_o two_o angle_n at_o the_o point_n fare_v suppose_v to_o be_v right_a angle_n therefore_o by_o the_o 29._o of_o the_o first_o the_o line_n ab_fw-la and_o gh_o be_v parallel_n which_o be_v require_v to_o be_v do_v and_o this_o problem_n be_v very_o commodious_a for_o the_o inscribe_v or_o circumscribe_v of_o figure_n in_o or_o about_o circle_n the_o 16._o theorem_a the_o 18._o proposition_n if_o a_o right_a line_n touch_v a_o circle_n and_o from_o the_o centre_n to_o the_o touch_n be_v draw_v a_o right_a line_n that_o right_a line_n so_o draw_v shall_v be_v a_o perpendicular_a line_n to_o the_o touch_n line_n svppose_v that_o the_o right_a line_n de_fw-fr do_v touch_v the_o circle_n abc_n in_o the_o point_n c._n and_o take_v the_o centre_n of_o the_o circle_n abc_n and_o let_v the_o same_o be_v f._n impossibility_n and_o by_o the_o first_o petition_n from_o the_o point_n fletcher_n to_o the_o point_n c_o draw_v a_o right_a line_n fc_o then_o i_o say_v that_o cf_o be_v a_o perpendicular_a line_n to_o de._n for_o if_o not_o draw_v by_o the_o 12._o of_o the_o first_o from_o the_o point_n fletcher_n to_o the_o line_n de_fw-fr a_o perpendicular_a line_n fg._n and_o for_o asmuch_o as_o the_o angle_n fgc_n be_v a_o right_a angle_n therefore_o the_o angle_n gcf_n be_v a_o acute_a angle_n wherefore_o the_o angle_n fgc_n be_v great_a than_o the_o angle_n fcg_n but_o unto_o the_o great_a angle_n be_v subtend_v the_o great_a side_n by_o the_o 19_o of_o the_o first_o wherefore_o the_o line_n fc_o be_v great_a than_o the_o line_n fg._n but_o the_o line_n fc_o be_v equal_a to_o the_o line_n fb_o for_o they_o be_v draw_v from_o the_o centre_n to_o the_o circumference_n wherefore_o the_o line_n fb_o also_o be_v great_a than_o the_o line_n fg_o namely_o the_o less_o than_o the_o great_a which_o be_v impossible_a wherefore_o the_o line_n fg_o be_v not_o a_o perpendicular_a line_n unto_o the_o line_n de._n and_o in_o like_a sort_n may_v we_o prove_v that_o no_o other_o line_n be_v a_o perpendicular_a line_n unto_o the_o line_n de_fw-fr beside_o the_o line_n fc_o wherefore_o the_o line_n fc_o be_v a_o perpendicular_a line_n to_o de._n if_o therefore_o a_o right_a line_n touch_v a_o circle_n &_o from_o you_o centre_n to_o the_o touch_n be_v draw_v a_o right_a line_n that_o right_a line_n so_o draw_v shall_v be_v a_o perpendicular_a line_n to_o the_o touch_n line_n which_o be_v require_v to_o be_v prove_v ¶_o a_o other_o demonstration_n after_o orontius_n suppose_v that_o the_o circle_n give_v be_v abc_n which_o let_v the_o right_a line_n de_fw-fr touch_n in_o the_o point_n c._n orontius_n and_o let_v the_o centre_n of_o the_o circle_n be_v the_o point_n f._n and_o draw_v a_o right_a line_n from_o fletcher_n to_o c._n then_o i_o say_v that_o the_o line_n fc_o be_v perpendicular_a unto_o the_o line_n de._n for_o if_o the_o line_n fc_o be_v not_o a_o perpendicule_a unto_o the_o line_n de_fw-fr then_o by_o the_o converse_n of_o the_o x._o definition_n of_o the_o first_o book_n the_o angle_n dcf_n &_o fce_n shall_v be_v unequal_a &_o therefore_o the_o one_o be_v great_a than_o a_o right_a angle_n and_o the_o other_o be_v less_o than_o a_o right_a angle_n for_o the_o angle_n dcf_n and_o fce_n be_v by_o the_o 13._o of_o the_o first_o equal_a to_o two_o right_a angle_n let_v the_o angle_n fce_n if_o it_o be_v possible_a be_v great_a than_o a_o right_a angle_n that_o be_v let_v it_o be_v a_o obtuse_a angle_n wherefore_o the_o angle_n dcf_n â—hal_v be_v a_o acute_a angle_n and_o forasmuch_o as_o by_o supposition_n the_o right_a line_n de_fw-fr touch_v the_o circle_n abc_n therefore_o it_o cut_v not_o the_o circle_n wherefore_o the_o circumference_n bc_o fall_v between_o the_o right_a line_n dc_o &_o cf_o &_o therefore_o the_o acute_a and_o rectiline_a angle_n dcf_n shall_v be_v great_a than_o the_o angle_n of_o the_o semicircle_n bcf_n which_o be_v contain_v under_o the_o circumference_n bc_o &_o the_o right_a line_n cf._n and_o so_o shall_v there_o be_v give_v a_o rectiline_n &_o acute_a angle_n great_a than_o the_o angle_n of_o a_o semicircle_n which_o be_v contrary_a to_o the_o 16._o proposition_n of_o this_o book_n wherefore_o the_o angle_n dcf_n be_v not_o less_o than_o a_o right_a angle_n in_o like_a sort_n also_o may_v we_o prove_v that_o it_o be_v not_o great_a than_o a_o right_a angle_n wherefore_o it_o be_v a_o right_a angle_n and_o therefore_o also_o the_o angle_n fce_n be_v a_o right_a angle_n wherefore_o the_o right_a line_n fc_o be_v a_o perpendicular_a unto_o the_o right_a line_n de_fw-fr by_o the_o 10._o definition_n of_o the_o firstâ—_o which_o be_v require_v to_o be_v prove_v the_o 17._o theorem_a the_o 19_o proposition_n if_o a_o right_a line_n do_v touch_v a_o circle_n and_o from_o the_o point_n of_o the_o touch_n be_v raise_v up_o unto_o the_o touch_n line_n a_o perpendicular_a line_n in_o that_o line_n so_o raise_v up_o be_v the_o centre_n of_o the_o circle_n svppose_v that_o the_o right_a line_n de_fw-fr do_v touch_v the_o circle_n abc_n in_o the_o point_n c._n and_o from_o c_o raise_v up_o by_o the_o 11._o of_o the_o first_o unto_o the_o line_n de_fw-fr a_o perpendicular_a line_n ca._n then_o i_o say_v that_o in_o the_o line_n ca_n be_v the_o centre_n of_o the_o circle_n for_o if_o not_o then_o if_o it_o be_v possible_a let_v the_o centre_n be_v without_o the_o line_n ca_n as_o in_o the_o point_n f._n and_o by_o the_o first_o petition_n draw_v a_o right_a line_n from_o c_o to_o f._n impossibility_n and_o for_o asmuch_o as_o a_o certain_a right_a line_n de_fw-fr touch_v the_o circle_n abc_n and_o from_o the_o centre_n to_o the_o touch_n be_v draw_v a_o right_a line_n cf_o therefore_o by_o the_o 18._o of_o the_o three_o fc_o be_v a_o perpendicular_a line_n to_o de._n wherefore_o the_o angle_n fce_n be_v a_o right_a angle_n but_o the_o angle_n ace_n be_v also_o a_o right_a angle_n wherefore_o the_o angle_n fce_n be_v equal_a to_o the_o angle_n ace_n namely_o the_o less_o unto_o the_o great_a which_o be_v impossibleâ—_n wherefore_o the_o point_n fletcher_n be_v not_o the_o centre_n of_o the_o circle_n abc_n and_o in_o like_a sort_n may_v we_o prove_v that_o it_o be_v no_o other_o where_o but_o in_o the_o line_n ac_fw-la if_o therefore_o a_o right_a line_n do_v touch_v a_o circle_n and_o from_o the_o point_n of_o the_o touch_n be_v raise_v up_o unto_o the_o touch_n line_v a_o perpendicular_a line_n in_o that_o line_n so_o raise_v up_o be_v the_o centre_n of_o the_o circle_n which_o be_v require_v to_o be_v prove_v the_o 18._o theorem_a the_o 20._o proposition_n in_o a_o circle_n a_o angle_n set_v at_o the_o centre_n be_v double_a to_o a_o angle_n set_v at_o the_o circumference_n so_o that_o both_o the_o angle_n have_v to_o their_o base_a one_o and_o the_o same_o circumference_n svppose_v that_o there_o be_v a_o circle_n abc_n and_o at_o the_o centre_n thereof_o centre_n namely_o the_o point_n e_o let_v the_o angle_n bec_n be_v set_v &_o at_o the_o circumference_n let_v there_o be_v set_v the_o angle_n bac_n and_o let_v they_o both_o have_v one_o and_o the_o same_o base_a namely_o the_o circumference_n bc._n then_o i_o say_v that_o the_o angle_n bec_n be_v double_a to_o the_o angle_n bac_n draw_v the_o right_a line_n ae_n and_o by_o the_o second_o petition_n extend_v it_o to_o the_o point_n f._n now_o for_o asmuch_o as_o the_o line_n ae_n be_v equal_a to_o the_o line_n ebb_n demonstration_n for_o they_o be_v draw_v from_o the_o centre_n unto_o the_o circumference_n the_o angle_n eab_n be_v equal_a to_o the_o angle_n eba_n by_o the_o 5._o of_o the_o first_o wherefore_o the_o angle_n eab_n and_o eba_n be_v double_a to_o the_o angle_n eab_n but_o by_o the_o 32._o of_o the_o same_o the_o angle_n bef_n be_v equal_a to_o the_o angle_n eab_n and_o eba_n wherefore_o the_o angle_n bef_n be_v double_a to_o the_o angle_n eab_n and_o by_o the_o same_o reason_n the_o angle_n fec_n be_v double_a to_o the_o angle_n eac_n wherefore_o the_o whole_a angle_n bec_n be_v double_a to_o the_o whole_a angle_n bac_n again_o suppose_v that_o there_o be_v set_v a_o other_o angle_n at_o the_o circumference_n centre_n and_o let_v the_o same_o be_v bdc_o and_o by_o the_o â—irst_n petition_n draw_v a_o line_n from_o d_z to_o e._n and_o by_o the_o second_o petition_n extend_v
and_o it_o be_v in_o the_o segment_n abc_n which_o be_v great_a than_o the_o semicircle_n part_n and_o forasmuch_o as_o in_o the_o circle_n there_o be_v a_o figure_n of_o four_o side_n namely_o abcd._n but_o if_o within_o a_o circle_n be_v describe_v a_o figure_n of_o four_o side_n the_o angle_n thereof_o which_o be_v opposite_a the_o one_o to_o the_o other_o be_v equal_a to_o two_o right_a angle_n by_o the_o 22._o of_o the_o three_o wherefore_o by_o the_o same_o the_o angle_n abc_n and_o adc_o be_v equal_a to_o two_o right_a angle_n but_o the_o angle_n abc_n be_v less_o than_o a_o right_a angle_n wherefore_o the_o angle_n remain_v adc_o be_v great_a than_o a_o right_a angle_n and_o it_o be_v in_o a_o segment_n which_o be_v less_o than_o the_o semicircle_n again_o forasmuch_o as_o the_o angle_n comprehend_v under_o the_o right_a line_n ac_fw-la and_o of_o be_v a_o right_a angle_n part_n therefore_o the_o angle_n comprehend_v under_o the_o right_a line_n ca_n and_o the_o circumference_n adc_o be_v less_o than_o a_o right_a angle_n wherefore_o in_o a_o circle_n a_o angle_n make_v in_o the_o semicircle_n be_v a_o right_a angle_n but_o a_o angle_n make_v in_o the_o segment_n great_a than_o the_o semicircle_n be_v less_o than_o a_o right_a angle_n and_o a_o angle_n make_v in_o the_o segment_n less_o than_o the_o semicircle_n be_v great_a than_o a_o right_a angle_n and_o moreover_o the_o angle_n of_o the_o great_a segment_n be_v great_a than_o a_o right_a angle_n &_o the_o angle_n of_o the_o less_o segment_n be_v less_o than_o a_o right_a angle_n which_o be_v require_v to_o be_v demonstrate_v an_o other_o demonstration_n to_o prove_v that_o the_o angle_n bac_n be_v a_o right_a angle_n angle_n forasmuch_o as_o the_o angle_n aec_o be_v double_a to_o the_o angle_n bae_o by_o the_o 32._o of_o the_o first_o for_o it_o be_v equal_a to_o the_o two_o inward_a angle_n which_o be_v opposite_a but_o the_o inward_a angle_n be_v by_o the_o 5._o of_o the_o first_o equal_a the_o one_o to_o the_o other_o and_o the_o angle_n aeb_fw-mi be_v double_a to_o the_o angle_n eac_n wherefore_o the_o angle_n aeb_fw-mi and_o aec_o be_v double_a to_o the_o angle_n bac_n but_o the_o angle_n aeb_fw-mi and_o aec_o be_v equal_a to_o two_o right_a angle_n wherefore_o the_o angle_n bac_n be_v a_o right_a angle_n which_o be_v require_v to_o be_v demonstrate_v correlary_a hereby_o it_o be_v manifest_a corollary_n that_o if_o in_o a_o triangle_n one_o angle_n be_v equal_a to_o the_o two_o other_o angle_n remain_v the_o same_o angle_n be_v a_o right_a angle_n for_o that_o the_o side_n angle_n to_o that_o one_o angle_n namely_o the_o angle_n which_o be_v make_v of_o the_o side_n produce_v without_o the_o triangle_n be_v equal_a to_o the_o same_o angle_n but_o when_o the_o side_n angle_n be_v equal_a the_o one_o to_o the_o other_o they_o be_v also_o right_a angle_n ¶_o a_o addition_n of_o pelitarius_n if_o in_o a_o circle_n be_v inscribe_v a_o rectangle_n triangle_n the_o side_n opposite_a unto_o the_o right_a angle_n shall_v be_v the_o diameter_n of_o the_o circle_n pâ—litarius_n suppose_v that_o in_o the_o circle_n abc_n be_v inscribe_v a_o rectangle_n triangle_n abc_n who_o angle_n at_o the_o point_n b_o let_v be_v a_o right_a angle_n then_o i_o say_v that_o the_o side_n ac_fw-la be_v the_o diameter_n of_o the_o circle_n for_o if_o not_o then_o shall_v the_o centre_n be_v without_o the_o line_n ac_fw-la as_o in_o the_o point_n e._n absurdity_n and_o draw_v a_o line_n from_o the_o point_n a_o to_o the_o point_n e_o &_o produce_v it_o to_o the_o circumference_n to_o the_o point_n d_o and_o let_v aed_n be_v the_o diameter_n and_o draw_v a_o line_n from_o the_o point_n b_o to_o the_o point_n d._n now_o by_o this_o 31._o proposition_n the_o angle_n abdella_n shall_v be_v a_o right_a angle_n and_o therefore_o shall_v be_v equal_a to_o the_o right_a angle_n abc_n namely_o the_o part_n to_o the_o whole_a which_o be_v absurd_a even_o so_o may_v we_o prove_v that_o the_o centre_n be_v in_o no_o other_o where_o but_o in_o the_o line_n ac_fw-la wherefore_o ac_fw-la be_v the_o diameter_n of_o the_o circle_n which_o be_v require_v to_o be_v prove_v ¶_o a_o addition_n of_o campane_n campane_n by_o this_o 31._o proposition_n and_o by_o the_o 16._o proposition_n of_o this_o book_n it_o be_v manifest_a that_o although_o in_o mix_a angle_n which_o be_v contain_v under_o a_o right_a line_n and_o the_o circumference_n of_o a_o circle_n there_o may_v be_v give_v a_o angle_n less_o &_o great_a than_o a_o right_a angle_n yet_o can_v there_o never_o be_v give_v a_o angle_n equal_a to_o a_o right_a angle_n for_o every_o section_n of_o a_o circle_n be_v either_o a_o semicircle_n or_o great_a than_o a_o semicircle_n or_o less_o but_o the_o angle_n of_o a_o semicircle_n be_v by_o the_o 16._o of_o this_o book_n less_o than_o a_o right_a angle_n and_o so_o also_o be_v the_o angle_n of_o a_o less_o section_n by_o this_o 31._o proposition_n likewise_o the_o angle_n of_o a_o great_a section_n be_v great_a than_o a_o right_a angle_n as_o it_o have_v in_o this_o proposition_n be_v prove_v the_o 28._o theorem_a the_o 32._o proposition_n if_o a_o right_a line_n touch_v a_o circle_n and_o from_o the_o touch_n be_v draw_v a_o right_a line_n cut_v the_o circle_n the_o angle_n which_o that_o line_n and_o the_o touch_n line_n make_v be_v equal_a to_o the_o angle_n which_o consist_v in_o the_o alternate_a segment_n of_o the_o circle_n svppose_v that_o the_o right_a line_n of_o do_v touch_v the_o circle_n abcd_o in_o the_o point_n b_o and_o from_o the_o point_n b_o let_v there_o be_v draw_v into_o the_o circle_n abcd_o a_o right_a line_n cut_v the_o circle_n and_o let_v the_o same_o be_v bd._n then_o i_o say_v that_o the_o angle_n which_o the_o line_n bd_o together_o with_o the_o touch_n line_n of_o do_v make_v be_v equal_a to_o the_o angle_n which_o be_v in_o the_o alternate_a segment_n of_o the_o circle_n that_o be_v the_o angle_n fbd_v be_v equal_a to_o the_o 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aliquote_v part_n and_o this_o kind_n of_o part_n be_v common_o use_v in_o arithmetic_n arithmetic_n the_o other_o kind_n of_o a_o part_n part_n be_v any_o less_o quantity_n in_o comparison_n of_o a_o great_a whether_o it_o be_v in_o number_n or_o magnitude_n and_o whether_o it_o measure_v or_o no._n as_o suppose_v the_o line_n ab_fw-la to_o be_v 17._o and_o let_v it_o be_v divide_v into_o two_o part_n in_o the_o point_n c_o namely_o into_o the_o line_n ac_fw-la &_o the_o line_n cb_o and_o let_v the_o line_n ac_fw-la the_o great_a part_n contain_v 12._o and_o let_v the_o line_n bc_o the_o less_o part_n contain_v 5._o now_o either_o of_o these_o line_n by_o this_o definition_n be_v a_o part_n of_o the_o whole_a line_n ab_fw-la for_o either_o of_o they_o be_v a_o less_o magnitude_n or_o quantity_n in_o comparison_n of_o the_o whole_a line_n ab_fw-la but_o neither_o of_o they_o measure_v the_o whole_a line_n ab_fw-la for_o the_o less_o line_n cb_o contain_v 5._o take_v as_o often_o as_o you_o listen_v will_v never_o make_v precise_o ab_fw-la which_o contain_v 17._o if_o you_o take_v it_o 3._o time_n it_o make_v only_o 15._o so_o lack_v it_o 2._o of_o 17._o which_o be_v to_o little_a if_o you_o take_v it_o 4._o time_n so_o make_v it_o 20._o them_z be_v there_o three_o to_o much_o so_o it_o never_o make_v precise_o 17._o but_o either_o to_o much_o or_o to_o little_a likewise_o the_o other_o part_n ac_fw-la measure_v not_o the_o whole_a line_n ab_fw-la for_o take_v once_o it_o make_v but_o 12._o which_o be_v less_o than_o 17._o and_o take_v twice_o it_o make_v 24._o which_o be_v more_o than_o 17._o by_o â—_o so_o it_o never_o precise_o make_v by_o take_v thereof_o the_o whole_a ab_fw-la but_o either_o more_o or_o less_o and_o this_o kind_n of_o part_n they_o common_o call_v pars_fw-la
constituens_fw-la or_o componens_fw-la componens_fw-la because_o that_o it_o with_o some_o other_o part_n or_o part_n make_v the_o whole_a as_o the_o line_n cb_o together_o with_o the_o line_n ac_fw-la make_v the_o whole_a line_n ab_fw-la of_o the_o barbarous_a it_o be_v call_v pars_fw-la aliquanta_fw-la aliquanta_fw-la in_o this_o signification_n it_o be_v take_v in_o bâ—rlaâ—â—_n in_o the_o begin_n of_o his_o book_n in_o the_o definition_n of_o a_o part_n when_o he_o say_v every_o less_o number_n compare_v to_o a_o great_a be_v say_v to_o be_v a_o part_n of_o the_o great_a whether_o the_o less_o measure_n the_o great_a or_o measure_v it_o not_o multiplex_fw-la be_v a_o great_a magnitude_n in_o respect_n of_o the_o less_o when_o the_o less_o measure_v the_o great_a definition_n as_o the_o line_n cd_o before_o set_v in_o the_o first_o example_n be_v multiplex_n to_o the_o line_n ab_fw-la for_o that_o cd_o a_o line_n contain_v 9_o be_v the_o great_a magnitude_n and_o be_v compare_v to_o 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multiplex_fw-la superperticular_a and_o multiplex_fw-la superpartiens_fw-la multiplex_fw-la be_v when_o the_o antecedent_n contain_v in_o itself_o the_o consequent_a certain_a time_n without_o more_o or_o less_o multiplex_fw-la as_o twice_o thrice_o four_o time_n and_o so_o far_o and_o this_o proportion_n have_v under_o it_o infinite_a kind_n for_o if_o the_o antecedent_n contain_v the_o consequent_a just_o twice_o it_o be_v call_v dupla_fw-la proportion_n proportion_n as_o 4_o to_o 2._o if_o thrice_o tripla_fw-la quintuple_a as_o 9_o to_o 3._o if_o 4._o time_n quadrupla_fw-la as_o 12._o to_o 3._o if_o 5._o time_n quintupla_fw-la as_o 15._o to_o 3._o and_o so_o infinite_o after_o the_o same_o manner_n superperticular_a superperticular_a be_v when_o the_o antecedent_n contain_v the_o consequent_a only_o once_o &_o moreover_o some_o one_o part_n thereof_o as_o a_o half_a a_o three_o or_o four_o etc_n etc_n this_o kind_n also_o have_v under_o it_o infinite_a kind_n for_o if_o the_o antecedent_n contain_v the_o consequent_a once_o and_o a_o half_a thereof_o it_o be_v call_v sesquialtera_fw-la sesquialtera_fw-la as_o 6._o to_o 4_o if_o once_o and_o a_o three_o part_n sesquitertia_fw-la sesquitertia_fw-la as_o 4._o to_o 3_o if_o once_o and_o a_o four_o part_n sesquiquarta_fw-la sesquiquarta_fw-la as_o 5._o to_o 4._o and_o so_o in_o like_a manner_n infinite_o superpartiens_fw-la superpartiens_fw-la be_v when_o the_o antecedent_n contain_v the_o consequent_a only_o once_o &_o moreover_o more_o part_n than_o one_o of_o the_o same_o as_o two_o three_o three_o fourthe_n four_o fifthe_n and_o so_o forth_o this_o also_o have_v infinite_a kind_n under_o it_o for_o if_o the_o antecedent_n contain_v above_o the_o consequent_a two_o part_n it_o be_v call_v superbipartiens_fw-la superbipartiens_fw-la as_o 7._o to_o 5._o if_o 3._o part_n supertripartiens_fw-la as_o 7._o to_o 4._o supertripartiens_fw-la if_o 4._o part_n superquadripartiens_n superquadripartiens_fw-la as_o 9_o to_o 5._o if_o 5._o part_n superquintipartiens_n as_o 11._o to_o 6._o superquintipartiens_fw-la and_o so_o forth_o infinite_o superperticular_a multiplex_fw-la superperticular_a be_v when_o the_o antecedent_n contain_v the_o consequent_a more_o than_o once_o and_o moreover_o only_o one_o part_n of_o the_o same_o this_o kind_n likewise_o have_v infinite_a kind_n under_o it_o for_o if_o the_o antecedent_n contain_v the_o consequent_a twice_o and_o half_o thereof_o it_o be_v call_v dupla_fw-la sesquialtera_fw-la sesquialtera_fw-la as_o 5._o to_o 2._o if_o twice_o and_o a_o three_o dupla_fw-la sesquitertia_fw-la as_o 7._o to_o 3._o sesquitertia_fw-la if_o thrice_o and_o a_o half_a tripla_fw-la sesquialtera_fw-la as_o 7._o to_o 2._o sesquialtera_fw-la if_o four_o time_n and_o a_o half_a quadrâ—pla_fw-la sesquialtera_fw-la as_o 9_o to_o 2._o and_o so_o go_v on_o infinite_o superpartiens_fw-la multiplex_fw-la superpartient_fw-fr be_v when_o the_o antecedent_n contain_v the_o consequent_a more_o than_o once_o and_o also_o more_o part_n than_o one_o of_o the_o consequent_a and_o this_o kind_n also_o have_v infinite_a kind_n under_o it_o for_o if_o the_o antecedent_n contain_v the_o consequent_a twice_o and_o two_o part_n over_o it_o be_v call_v dupla_fw-la superbipartiens_fw-la as_o 8._o to_o 3._o superbipartiens_fw-la if_o twice_o and_o three_o part_n dupla_fw-la supertripartiens_fw-la as_o 11._o to_o 4._o supertripartiens_fw-la if_o thrice_o and_o two_o part_n it_o be_v name_v tripla_fw-la superbipartiens_fw-la as_o 11._o to_o 3._o superbipartiens_fw-la if_o three_o time_n and_o four_o part_n treble_a superquadripartiens_n as_o 31._o to_o 9_o superquadâ—ipartiens_fw-la and_o so_o forth_o infinite_o here_o be_v to_o be_v note_v that_o the_o denomination_n of_o the_o proportion_n between_o any_o two_o number_n be_v have_v by_o devide_v of_o the_o great_a by_o the_o less_o for_o the_o quotient_a oâ—_n number_v produce_v of_o that_o division_n be_v even_o the_o denomination_n of_o the_o proportion_n proportion_n which_o in_o the_o first_o kind_n of_o proportion_n namely_o multiplex_n be_v ever_o a_o whole_a number_n and_o in_o all_o other_o kind_n of_o proportion_n it_o be_v a_o break_a number_n as_o if_o you_o will_v know_v the_o denomination_n of_o the_o proportion_n between_o 9_o and_o 3._o divide_v 9_o by_o 3._o so_o shall_v you_o have_v in_o the_o quotient_a 3._o which_o be_v a_o whole_a number_n and_o be_v the_o denomination_n of_o the_o proportion_n and_o show_v that_o the_o proportion_n between_o 9_o &_o 3._o be_v tripla_fw-la so_o the_o proportion_n between_o 12._o and_o 3._o be_v quadrupla_fw-la for_o that_o 12._o be_v divide_v by_o 3._o the_o quotient_n be_v 4._o and_o so_o of_o other_o in_o the_o kind_n of_o multiplex_n and_o although_o in_o this_o kind_a the_o quotient_n be_v ever_o a_o whole_a number_n yet_o proper_o it_o be_v refer_v to_o unity_n and_o so_o be_v represent_v in_o manner_n of_o a_o break_a number_n as_o 5_o â—_o and_o 4_o â—_o for_o unity_n be_v the_o denomination_n to_o a_o whole_a number_n likewise_o the_o denomination_n of_o the_o proportion_n between_o 4_o and_o 3_o be_v 1._o 1_o â—_o for_o that_o 4_o divide_v by_o 3._o have_v in_o the_o quotient_a 1_o 1_o â—_o one_o and_o a_o three_o part_n of_o which_o three_o part_n it_o be_v call_v sesquitercia_fw-la so_o the_o proportion_n between_o 7_o and_o 6._o be_v 1_o â…™_n one_o and_o a_o six_o of_o which_o fix_v part_v it_o be_v call_v sesquisexta_fw-la and_o so_o of_o other_o of_o that_o kind_n also_o between_o 7_o and_o 5_o the_o denomination_n of_o the_o proportion_n be_v 1_o â—_o â—_o one_o and_o two_o fifthe_n which_o denomination_n consist_v of_o two_o part_n namely_o of_o the_o munerator_n and_o denominator_fw-la of_o the_o quotient_a of_o 2._o and_o 5_o of_o which_o two_o fifthe_n it_o be_v call_v superbipartiens_fw-la quintas_fw-la for_o 2_o the_o numerator_n show_v the_o denomination_n of_o the_o number_n of_o the_o part_n and_o 5._o the_o denominator_fw-la show_v the_o denomination_n what_o part_n they_o be_v &_o so_o of_o other_o also_o the_o denomination_n between_o
of_o the_o one_o also_o to_o the_o residue_n of_o the_o other_o shall_v be_v equemultiplex_n as_o the_o whole_a be_v to_o the_o whole_a which_o be_v require_v to_o be_v prove_v the_o 6._o theorem_a the_o 6._o proposition_n if_o two_o magnitude_n be_v â—quemultiplices_n to_o two_o magnitude_n &_o any_o parâ—es_n take_v away_o of_o they_o also_o be_v aequemultiplice_n to_o the_o same_o magnitude_n the_o residue_v also_o of_o they_o shall_v unto_o the_o same_o magnitude_n be_v either_o equal_a or_o equemultiplices_fw-la svppose_v that_o there_o be_v two_o magnitude_n ab_fw-la and_o cd_o equemultiplices_fw-la to_o two_o magnitude_n e_o and_o f_o and_o let_v the_o part_n take_v away_o of_o the_o magnitude_n ab_fw-la and_o cd_o namely_o ag_z and_o change_z be_v equemultiplices_fw-la to_o the_o same_o magnitude_n e_o and_o f._n propotion_n then_o i_o say_v that_o the_o residue_v gb_o and_o hd_n be_v unto_o the_o self_n same_o magnitude_n e_o and_o f_o either_o equal_a or_o else_o equemultiplices_fw-la and_o in_o like_a sort_n may_v we_o prove_v that_o if_o gb_o be_v multiplex_n to_o e_o hd_v also_o shall_v be_v so_o multiplex_n unto_o f._n if_o therefore_o there_o be_v two_o magnitude_n equemultiplices_fw-la to_o two_o magnitude_n second_o and_o any_o part_n take_v away_o of_o they_o be_v also_o equemultiplices_fw-la to_o the_o same_o magnitude_n the_o residue_v also_o of_o they_o shall_v unto_o the_o same_o magnitude_n be_v either_o equal_a or_o equemultiplices_fw-la which_o be_v require_v to_o be_v prove_v the_o 7._o theorem_a the_o 7._o proposition_n equal_a magnitude_n have_v to_o one_o &_o the_o self_n same_o magnitude_n one_o and_o the_o same_o proportion_n and_o one_o and_o the_o same_o magnitude_n have_v to_o equal_a magnitude_n one_o and_o the_o self_n same_o proportion_n svppose_v that_o a_o and_o b_o be_v equal_a magnitude_n and_o take_v any_o other_o magnitude_n namely_o c._n then_o i_o say_v that_o either_o of_o these_o a_o and_o b_o have_v unto_o c_o one_o and_o the_o same_o proportion_n and_o that_o c_o also_o have_v to_o either_o of_o these_o a_o and_o b_o one_o and_o the_o same_o proportion_n i_o say_v moreover_o that_o c_z have_v to_o either_o of_o these_o a_o and_o b_o one_o and_o the_o same_o proportion_n demonstrate_v for_o the_o same_o order_n of_o construction_n remain_v we_o may_v in_o like_a sort_n prove_v that_o d_z be_v equal_a unto_o e_o &_o there_o be_v take_v a_o other_o multiplex_n to_o c_o namely_o f._n wherefore_o if_o fletcher_n exceed_v d_o it_o also_o exceed_v e_o and_o if_o it_o be_v equal_a it_o be_v equal_a and_o if_o it_o be_v less_o it_o be_v less_o but_o fletcher_n be_v multiplex_n to_o c_o and_o d_z &_o e_o be_v other_o equemultiplices_fw-la to_o a_o and_o b._n wherefore_o as_o c_z be_v to_o a_o so_o be_v c_o to_o b._n wherefore_o equal_a magnitude_n have_v to_o one_o and_o the_o same_o magnitude_n one_o and_o the_o same_o proportion_n and_o one_o and_o the_o same_o magnitude_n have_v to_o equal_a magnitude_n one_o and_o the_o self_n same_o proportion_n which_o be_v require_v to_o be_v demonstrate_v the_o 8._o theorem_a the_o 8._o proposition_n unequal_a magnitude_n be_v take_v the_o great_a have_v to_o one_o and_o the_o same_o magnitude_n a_o great_a proportion_n than_o have_v the_o less_o and_o that_o one_o and_o the_o same_o magnitude_n have_v to_o the_o less_o a_o great_a proportion_n than_o it_o have_v to_o the_o great_a svppose_v that_o ab_fw-la and_o c_o be_v unequal_a magnitude_n of_o which_o let_v ab_fw-la be_v the_o great_a and_o c_o the_o less_o and_o let_v there_o be_v a_o other_o magnitude_n whatsoever_o namely_o d._n then_o i_o say_v that_o ab_fw-la have_v unto_o d_z a_o great_a proportion_n than_o have_v c_z to_o d_z and_o also_o that_o d_z have_v to_o c_o a_o great_a proportion_n than_o it_o have_v to_o ab_fw-la for_o forasmuch_o as_o ab_fw-la be_v great_a than_o c_z let_v there_o be_v take_v a_o magnitude_n equal_a unto_o c_o namely_o be._n now_o than_o the_o less_o of_o these_o two_o magnitude_n ae_n and_o ebb_n be_v multiply_v will_v at_o the_o length_n be_v great_a than_o d._n demonstrate_v demonstrate_v i_o say_v moreover_o that_o d_z have_v to_o c_o a_o great_a proportion_n than_o d_z have_v to_o ab_fw-la for_o the_o same_o order_n of_o construction_n still_o remain_v we_o may_v in_o like_a sort_n prove_v that_o n_o be_v great_a than_o king_n and_o that_o it_o be_v not_o great_a than_o fh_o and_o n_o be_v multiplex_n to_o d_o and_o fh_o and_o king_n be_v certain_a other_o equemultiplices_fw-la to_o ab_fw-la and_o c._n wherefore_o d_z have_v to_o c_o a_o great_a proportion_n than_o d_z have_v to_o ab_fw-la ¶_o for_o that_o orontius_n seem_v to_o demonstrate_v this_o more_o plain_o therefore_o i_o think_v it_o not_o amiss_o here_o to_o set_v it_o suppose_v that_o there_o be_v two_o unequal_a magnitude_n of_o which_o leâ—_z aâ—_z bâ—_z the_o greâ—ter_n and_o c_o the_o less_o and_o let_v there_o be_v a_o certain_a other_o magnitude_n namely_o d._n then_o i_o say_v first_o that_o ab_fw-la have_v to_o d_o a_o great_a proportion_n than_o have_v c_z to_o d._n for_o forasmuch_o as_o by_o supposition_n ab_fw-la be_v great_a than_o the_o magnitude_n c_o therefore_o the_o magnitude_n ab_fw-la contain_v the_o same_o magnitude_n c_o and_o a_o other_o magnitude_n beside_o let_v eâ—_n be_v equal_a unto_o c_o and_o let_v ae_n be_v the_o part_n remain_v of_o the_o same_o magnitude_n part_n now_o ae_n and_o ebb_n be_v either_o unequal_a or_o equal_a the_o one_o to_o the_o other_o first_o let_v they_o be_v unequally_n and_o leâ—_z ae_n be_v less_o than_o ebb_n and_o unto_o ae_n the_o less_o take_v any_o multiplex_n whatsoâ—uâ—r_n so_o that_o it_o be_v great_a they_o the_o magnitude_n d_o and_o let_v the_o same_o be_v fg._n and_o how_o multiplex_n fg_o be_v to_o ae_n so_o multiplex_n let_v gh_o be_v to_o eâ—â—_n â—_z and_o king_n to_o c._n again_o take_v the_o duple_n of_o dâ—_n â—_z which_o let_v be_v l_o and_o then_o the_o triple_a and_o leâ—_z the_o same_o be_v m._n and_o so_o forward_o always_o add_v one_o until_o there_o be_v produce_v suâ—h_v a_o multiplex_n to_o d_o which_o shall_v be_v nâ—xt_v great_a than_o gh_o that_o be_v which_o amongst_o the_o 〈◊〉_d oâ—_n dâ—_n â—_z by_o the_o continual_a addition_n of_o one_o do_v first_o begin_v to_o exceade_v gh_o and_o let_v thâ—â—â—me_z be_v nâ—_n â—_z which_o let_v â—e_z quadruple_a to_o d._n now_o then_o the_o multiplex_n gh_o be_v the_o next_o multiplex_n less_o than_o nâ—_n â—_z and_o therefore_o â—s_v not_o less_o than_o m_o that_o be_v be_v either_o equal_a unto_o it_o or_o great_a than_o it_o difference_n and_o forasmuch_o â—s_v fg_o be_v equemultiplex_n to_o ae_n as_o gh_o be_v to_o eâ—_n therefore_o how_o multiplex_n fâ—_n be_v to_o ae_n so_o multiplex_n be_v fh_o to_o ab_fw-la by_o the_o first_o of_o the_o five_o but_o how_o multiplex_n fg_o be_v to_o ae_n so_o multiplex_n be_v king_n to_o c_o therefore_o how_o multiplex_n fh_o be_v to_o ab_fw-la so_o multiplex_n be_v king_n to_o c._n moreover_o forasmuch_o as_o gh_o and_o king_n be_v equemultiplices_fw-la unto_o ebb_n and_o câ—_n â—_z and_o ebb_n be_v by_o construction_n equal_a unto_o c_o therefore_o by_o the_o common_a sentence_n gh_o be_v equal_a unto_o k._n but_o gh_o be_v not_o less_o than_o m_o as_o have_v before_o be_v show_v and_o fgâ—_n â—_z be_v put_v to_o be_v great_a than_o d._n wherefore_o the_o whole_a fh_o be_v great_a than_o these_o two_o d_z and_o m._n but_o d_z and_o m_o be_v equal_a unto_o n._n for_o n_o be_v quadruple_a to_o d._n and_o m_o be_v triple_a to_o d_z do_v together_o with_o d_z make_v quadruple_a unto_o d._n wherefore_o fh_o be_v great_a than_o n._n farther_n edward_n be_v prove_v to_o be_v equal_a to_o gh_o wherefore_o king_n be_v less_o than_o n._n but_o fh_o and_o king_n be_v equemultiplices_fw-la unto_o ab_fw-la and_o c_o unto_o the_o first_o magnitude_n i_o say_v and_o the_o three_o and_o n_o be_v a_o certain_a other_o multiplex_n unto_o d_o which_o represent_v the_o second_o &_o the_o four_o magnitude_n and_o the_o multiplex_n of_o the_o first_o exceed_v the_o multiplex_n of_o the_o second_o but_o the_o multiplex_n of_o the_o three_o exceed_v not_o the_o multiplex_n of_o the_o four_o wherefore_o ab_fw-la the_o first_o have_v unto_o d_z the_o second_o a_o great_a proportion_n then_o have_v c_z the_o three_o to_o d_o the_o four_o by_o the_o 8._o definition_n of_o this_o book_n but_o if_o ae_n be_v great_a than_o ebb_n let_v ebb_v the_o less_o be_v multiply_v until_o there_o be_v produce_v a_o multiplex_n great_a than_o the_o magnitude_n d_o which_o let_v be_v gh_o and_o how_o multiplex_n gh_o be_v to_o ebb_v difference_n so_o multiplex_n let_v fg_o be_v to_o ae_n and_o king_n also_o to_o c._n then_o take_v unto_o d_z
such_o a_o multiplex_n as_o be_v next_o great_a than_o fg_o and_o again_o let_v the_o same_o be_v n_o which_o let_v be_v quadruple_a to_o d._n and_o in_o like_a sort_n as_o before_o may_v we_o prove_v that_o the_o whole_a fh_o be_v unto_o ab_fw-la equemultiplex_fw-la as_o gh_o be_v to_o ebb_v and_o also_o that_o fh_o &_o king_n be_v equemultiplices_fw-la unto_o abâ—_n â—_z &_o c_o and_o final_o that_o gh_o be_v equal_a unto_o k._n and_o forasmuch_o as_o the_o multiplex_n n_o be_v next_o great_a than_o fg_o therefore_o fg_o be_v not_o less_o than_o m._n but_o gh_o be_v great_a than_o d_z by_o construction_n wherefore_o the_o whole_a fh_o be_v great_a than_o d_z and_o m_o and_o so_o consequent_o be_v great_a than_o n._n but_o king_n exceed_v not_o n_o for_o king_n be_v equal_a to_o gh_o for_o how_o multiplex_n king_n be_v to_o ebb_v the_o less_o so_o multiplex_n be_v fg_o to_o aâ—_z the_o great_a bâ—t_v those_o magnitude_n which_o be_v equemultipliceâ—_n unto_o unequal_a magnitudâ—s_n be_v according_a to_o the_o same_o proportion_n unequal_a wherefore_o king_n be_v less_o than_o fg_o and_o therefore_o iâ—_z much_o less_o than_o n._n wherefore_o again_o the_o multiplex_n of_o the_o first_o exceed_v the_o multiplex_n of_o the_o second_o but_o the_o multiplex_n of_o the_o three_o exceed_v not_o thâ—_z multiplex_n of_o the_o four_o wherefore_o by_o the_o 8._o definition_n of_o the_o five_o aâ—_z the_o first_o have_v to_o d_z the_o second_o a_o great_a proportion_n then_o have_v c_z the_o three_o to_o d_o the_o four_o difference_n but_o now_o if_o ae_n be_v equal_a unto_o ebb_n either_o of_o they_o shall_v be_v equal_a unto_o c._n wherefore_o unto_o either_o of_o thosâ—_n three_o magnitude_n take_v equemultiplices_fw-la great_a than_o d._n so_o that_o let_v fg_o be_v multiplex_n to_o ae_n and_o gh_o unto_o ebb_n and_o king_n again_o to_o c_z which_o by_o the_o 6._o common_a sentence_n shall_v be_v equal_a the_o one_o to_o the_o other_o let_v n_o also_o be_v multiplex_n to_o d_o and_o be_v next_o great_a than_o every_o one_o of_o they_o namely_o let_v it_o be_v qâ—adruplâ—_n to_o d._n this_o construction_n finish_v we_o may_v again_o prove_v that_o fh_o and_o king_n be_v equemultiplices_fw-la to_o ab_fw-la and_o c_o and_o that_o fh_o the_o multiplex_n of_o thâ—_z first_o magnitude_n exceed_v no_o the_o multiplex_n of_o the_o second_o magnitudeâ—â—nd_n thaâ—_z king_n tâ—â—â—ultiplex_n of_o the_o three_o exceed_v not_o the_o multiplex_n of_o the_o four_o wherefore_o we_o may_v conclude_v that_o ab_fw-la have_v unto_o d_z a_o great_a proportion_n then_o have_v c_z to_o d._n now_o also_o i_o say_v that_o the_o self_n same_o magnitude_n d_z have_v unto_o the_o less_o magnitude_n c_o a_o great_a proportion_n they_o it_o have_v to_o the_o great_a ab_fw-la proposition_n and_o this_o may_v plain_o be_v gather_v by_o the_o foresay_a discourse_n without_o change_v the_o order_n of_o the_o magnitude_n &_o of_o the_o equemultiplices_fw-la for_o see_v that_o every_o way_n it_o be_v before_o prove_v that_o fh_o exceed_v n_o and_o king_n be_v exceed_v of_o the_o self_n same_o n_o therefore_o converse_o n_o exceed_v king_n but_o do_v not_o exceed_v fh_o but_o n_o be_v multiplex_n to_o d_o that_o be_v to_o the_o first_o and_o three_o magnitude_n and_o king_n be_v multiplex_n to_o the_o second_o namely_o to_o câ—_n â—_z and_o fh_o be_v multiplex_fw-la to_o the_o four_o namely_o to_o abâ—_n â—_z wherefore_o the_o multiplex_n of_o the_o first_o exceed_v the_o multiplex_n of_o the_o second_o but_o the_o multiplex_n of_o the_o three_o exceed_v not_o the_o multiplex_n of_o the_o four_o wherefore_o by_o the_o 8._o definition_n of_o this_o five_o book_n d_z the_o first_o have_v unto_o c_o the_fw-fr second_o a_o great_a proportion_n then_o have_v d_z the_o three_o to_o ab_fw-la the_o four_o which_o be_v require_v to_o be_v prove_v the_o 9_o theorem_a the_o 9_o proposition_n magnitude_n which_o have_v to_o one_o and_o the_o same_o magnitude_n one_o and_o the_o same_o proportion_n be_v equal_a the_o one_o to_o the_o other_o and_o those_o magnitude_n unto_o who_o one_o and_o the_o same_o magnitude_n have_v one_o and_o the_o same_o proportion_n be_v also_o equal_a svppose_v that_o either_o of_o these_o two_o magnitude_n a_o and_o b_o have_v to_o c_o one_o and_o the_o same_o proportion_n demonstrate_v then_o i_o say_v that_o a_o be_v equal_a unto_o b._n for_o if_o it_o be_v not_o then_o either_o of_o these_o a_o and_o b_o shall_v not_o have_v to_o c_o one_o &_o the_o same_o proportion_n by_o the_o 8._o of_o the_o five_o but_o by_o supposition_n they_o have_v wherefore_o a_o be_v equal_a unto_o b._n again_o prove_v suppose_v that_o the_o magnitude_n c_o have_v to_o either_o of_o these_o magnitude_n a_o and_o b_o one_o and_o the_o same_o proportion_n then_o i_o say_v that_o a_o be_v equal_a unto_o b._n for_o if_o it_o be_v not_o c_z shall_v not_o have_v to_o either_o of_o these_o a_o and_o b_o one_o and_o the_o same_o proportion_n by_o the_o former_a proposition_n but_o by_o supposition_n it_o have_v wherefore_o a_o be_v equal_a unto_o b._n wherefore_o magnitude_n which_o have_v to_o one_o and_o the_o same_o magnitude_n one_o and_o the_o same_o proportion_n be_v equal_a the_o one_o to_o the_o other_o and_o thosâ—_n magnitude_n unto_o who_o one_o and_o the_o same_o magnitude_n have_v one_o and_o the_o same_o proportion_n be_v also_o equal_a which_o be_v require_v to_o be_v prove_v the_o 10._o theorem_a the_o 10._o proposition_n of_o magnitude_n compare_v to_o one_o and_o the_o same_o magnitude_n that_o which_o have_v the_o great_a proportion_n be_v the_o great_a and_o that_o magnitude_n whereunto_o one_o and_o the_o same_o magnitude_n have_v the_o great_a proportion_n be_v the_o less_o svppose_v that_o a_o have_v to_o c_o a_o great_a proportion_n than_o b_o have_v to_o c._n then_o i_o say_v that_o a_o be_v great_a than_o b._n prove_v for_o if_o it_o be_v not_o then_o either_o a_o be_v equal_a unto_o b_o or_o less_o than_o it_o but_o a_o can_v be_v equal_a unto_o b_o for_o then_o either_o of_o these_o a_o and_o b_o shall_v have_v unto_o c_o one_o and_o the_o same_o proportion_n by_o the_o 7_o of_o the_o five_o but_o by_o supposition_n they_o have_v not_o wherefore_o a_o be_v not_o equal_a unto_o b._n neither_o also_o be_v a_o less_o than_o b_o for_o they_o shall_v a_o have_v to_o c_o a_o less_o proportion_n then_o have_v b_o to_o c_z by_o the_o 8._o of_o the_o five_o but_o by_o supposition_n it_o have_v not_o wherefore_o a_o be_v not_o less_o than_o b._n and_o it_o be_v also_o prove_v that_o it_o be_v not_o equal_a wherefore_o a_o be_v great_a than_o b._n demonstrate_v again_o suppose_v that_o c_o have_v to_o b_o a_o great_a proportion_n than_o c_z have_v to_o a._n then_o i_o say_v that_o b_o be_v less_o then_o a._n for_o if_o it_o be_v not_o then_o be_v it_o either_o equal_a unto_o it_o or_o else_o great_a but_o b_o can_v be_v equal_a unto_o a_o for_o than_o shall_v c_o have_v to_o either_o of_o these_o a_o and_o b_o one_o and_o the_o same_o proportion_n by_o the_o 7._o of_o the_o five_o but_o by_o supposition_n it_o have_v not_o wherefore_o b_o be_v not_o equal_a unto_o a._n neither_o also_o be_v b_o great_a than_o a_o for_o than_o shall_v c_o have_v to_o b_o a_o less_o proportion_n than_o it_o have_v to_o a_o by_o the_o 8._o of_o the_o five_o but_o by_o supposition_n it_o have_v not_o wherefore_o b_o be_v not_o great_a than_o a._n and_o it_o be_v prove_v that_o it_o be_v not_o equal_a unto_o a_o wherefore_o b_o be_v less_o than_o a._n wherefore_o of_o magnitude_n compare_v to_o one_o and_o the_o same_o magnitude_n that_o which_o have_v y_fw-fr great_a proportion_n be_v the_o great_a and_o that_o magnitude_n whereunto_o one_o and_o the_o same_o magnitude_n have_v the_o great_a proportion_n be_v the_o less_o which_o be_v require_v to_o be_v prove_v the_o 11._o theorem_a the_o 11._o proposition_n proportion_n which_o be_v one_o and_o the_o self_n same_o to_o any_o one_o proportion_n be_v also_o the_o self_n same_o the_o one_o to_o the_o other_o svpppose_v that_o as_o a_o be_v to_o b_o so_o be_v c_z to_o d_z and_o as_z c_z be_v to_o d_z so_o be_v e_o to_o f._n then_o i_o say_v that_o as_o a_o be_v to_o b_o so_o be_v e_o to_o f._n construction_n take_v equemultiplices_fw-la to_o a_o c_o and_o e_o which_o let_v be_v g_o h_o k._n and_o likewise_o to_o b_o d_o and_o f_o take_v any_o other_o equemultiplices_fw-la which_o let_v be_v l_o m_o and_o n._n and_o because_o as_o a_o be_v to_o b_o so_o be_v c_z to_o d_z and_o to_o a_o and_o g_o be_v take_v equemultiplices_fw-la g_z &_o h_o &_o to_o b_o and_o d_o be_v take_v certain_a other_o equemultiplices_fw-la l_o
multiplex_n be_v mp_n compose_v of_o the_o three_o and_o six_o to_o fd._n and_o for_o that_o as_o ab_fw-la be_v to_o be_v so_o be_v cd_o to_o df_o and_o to_o ab_fw-la &_o cd_o be_v take_v equemultiplices_fw-la gk_o and_o ln_o and_o likewise_o to_o ebb_v and_o fd_o be_v take_v certain_a other_o equemultiplices_fw-la that_o be_v ho_o and_o mp_n if_o therefore_o gk_o exceed_v ho_o then_o ln_o also_o exceed_v mp_n and_o if_o it_o be_v equal_a it_o be_v equal_a and_o if_o it_o be_v less_o it_o be_v less_o by_o the_o conversion_n of_o the_o six_o definition_n of_o the_o five_o let_v gk_o exceed_v ho_o wherefore_o kh_o common_a to_o they_o both_o be_v take_v away_o the_o residue_n gh_o shall_v exceed_v the_o residue_n ko_o but_o if_o gk_o exceed_v ho_o then_o do_v ln_o exceed_v mp_n wherefore_o let_v ln_o exceed_v mp_n and_o mn_v which_o be_v common_a to_o the_o both_o be_v take_v away_o a'n_z residue_n lm_o shall_v exceed_v the_o residue_n np._n wherefore_o if_o gh_o exceed_v ko_o then_o shall_v lm_o exceed_v np._n and_o in_o like_a sort_n may_v we_o prove_v that_o if_o gh_v be_v equal_a unto_o ko_o than_o lm_o shall_v be_v equal_a unto_o np_n and_o if_o it_o be_v less_o it_o shall_v be_v less_o but_o gh_o and_o lm_o be_v equemultiplices_fw-la to_o ae_n and_o gf_o and_o likewise_o ko_o and_o np_n be_v certain_a other_o equemultiplices_fw-la to_o ebb_v and_o fd._n wherefore_o as_o ae_n be_v to_o ebb_v so_o â—cf_n to_o fd_o by_o the_o sixâ—_o definition_n of_o the_o five_o if_o compose_a magnitude_n therefore_o be_v proportional_a then_o also_o divide_v they_o shall_v be_v proportional_a which_o be_v require_v to_o be_v demonstrate_v the_o 18._o theorem_a the_o 18._o proposition_n if_o magnitude_n divide_v be_v proportional_a then_o also_o compose_v they_o shall_v be_v proportional_a svppose_v that_o the_o magnitude_n divide_v be_v proportional_a be_v ae_n ebb_n cf_o &_o fd_o so_o that_o as_o ae_n be_v to_o ebb_v so_o let_v cf_o be_v to_o fd._n composition_n then_o i_o say_v that_o compose_v also_o they_o shall_v be_v proportional_a that_o be_v as_z ab_fw-la be_v to_o be_v so_o be_v cd_o to_o df._n for_o if_o ab_fw-la be_v not_o unto_o be_v as_o cd_o be_v to_o fd_o then_o shall_v ab_fw-la be_v unto_o be_v former_a as_o cd_o be_v either_o unto_o a_o magnitude_n less_o than_o fd_o on_o unto_o a_o magnitude_n great_a let_v it_o first_o be_v unto_o a_o less_o namely_o to_o dg_v impossibility_n and_o forasmuch_o as_o as_z ab_fw-la be_v to_o be_v so_o be_v cd_o to_o dg_o the_o compose_a magnitude_n therefore_o be_v proportional_a wherefore_o divide_v also_o they_o shall_v be_v proportional_a by_o the_o 17._o of_o the_o first_o wherefore_o as_o ae_n be_v to_o ebb_v so_o be_v cg_o to_o gd_o but_o by_o supposition_n as_o ae_n be_v to_o ebb_v so_o be_v cf_o to_o fd._n wherefore_o by_o the_o 11._o of_o the_o five_o as_o cg_o be_v to_o gd_o so_o be_v cf_o to_o fd._n now_o then_o there_o be_v four_o magnitude_n cg_o gd_o cf_o and_o fd_o of_o which_o the_o first_o cg_o be_v great_a than_o the_o three_o cf._n wherefore_o by_o the_o 14._o of_o the_o five_o the_o second_o gd_o be_v great_a than_o the_o four_o fd._n but_o it_o be_v also_o put_v to_o be_v less_o than_o it_o which_o be_v impossible_a wherefore_o it_o can_v not_o be_v that_o as_o ab_fw-la be_v to_o be_v so_o be_v cd_o to_o a_o magnitude_n less_o than_o fd._n in_o like_a sort_n may_v we_o prove_v that_o it_o can_v not_o be_v so_o to_o a_o magnitude_n great_a than_o fd._n for_o by_o the_o same_o order_n of_o demonstration_n it_o will_v follow_v that_o fd_o be_v great_a than_o the_o say_v great_a magnitude_n which_o be_v impossible_a wherefore_o it_o must_v be_v to_o the_o self_n same_o if_o therefore_o magnitude_n divide_v be_v proportional_a then_o also_o compose_v they_o shall_v be_v proportional_a which_o be_v require_v to_o be_v prove_v the_o 19_o theorem_a the_o 19_o proposition_n if_o the_o whole_a be_v to_o the_o whole_a as_o the_o part_n take_v away_o be_v to_o the_o part_n take_v away_o then_o shall_v the_o residue_n be_v unto_o the_o residue_n as_o the_o whole_a be_v to_o the_o whole_a svppose_v that_o as_o the_o whole_a ab_fw-la be_v to_o the_o whole_a cd_o magnitude_n so_o be_v the_o part_n take_v away_o ae_n to_o the_o part_n take_v away_o cf._n then_o i_o say_v that_o the_o residue_n ebb_n shall_v be_v unto_o the_o residue_n fd_o as_o the_o whole_a ab_fw-la be_v to_o the_o whole_a cd_o for_o for_o that_o as_o the_o whole_a ab_fw-la be_v to_o the_o whole_a cd_o so_o be_v ae_n to_o ce_fw-fr therefore_o alternate_o also_o by_o the_o 16._o of_o the_o five_o as_o ab_fw-la be_v to_o ae_n so_o be_v cd_o to_o cf._n aâ—d_z for_o that_o when_o magnitude_n compose_v be_v proportional_a the_o same_o divide_v also_o be_v proportional_a by_o the_o 17._o of_o the_o five_o â—_o therefore_o as_o be_v be_v to_o ea_fw-la so_o be_v df_o to_o fc_o wherefore_o alternate_o also_o by_o the_o 16._o of_o the_o five_o as_o be_v be_v to_o df_o so_o be_v ea_fw-la to_o fc_fw-la but_o as_o ae_n be_v to_o cf_o so_o by_o supposition_n be_v the_o whole_a ab_fw-la to_o the_o whole_a cd_o wherefore_o the_o residue_n ebb_n shall_v be_v unto_o the_o residue_n fd_o as_o the_o whole_a ab_fw-la be_v to_o the_o whole_a cd_o if_o therefore_o the_o whole_a be_v to_o the_o whole_a as_o the_o part_n take_v away_o be_v to_o the_o part_n take_v away_o then_o shall_v the_o residue_n be_v unto_o the_o residue_n as_o the_o whole_a be_v to_o the_o whole_a which_o be_v require_v to_o be_v prove_v ¶_o alemma_n or_o assumpt_n and_o forasmuch_o as_o by_o supposition_n as_o ab_fw-la be_v to_o cd_o alemma_n so_o be_v ae_n to_o cf_o and_o alternate_o as_o ab_fw-la be_v to_o ae_n so_o be_v cd_o to_o cf._n and_o now_o it_o be_v prove_v that_o as_o ab_fw-la be_v to_o cd_o so_o be_v ebb_v to_o fd._n wherefore_o again_o alternate_o as_o ab_fw-la be_v to_o â—b_n so_o be_v cd_o to_o fd._n wherefore_o it_o follow_v that_o as_o ab_fw-la be_v to_o ae_n so_o be_v cd_o to_o cf_o and_o again_o as_o the_o same_o ab_fw-la be_v to_o ebb_n so_o be_v the_o same_o cd_o to_o df._n corollary_n and_o hereby_o it_o be_v manifest_a corollary_n that_o if_o magnitude_n compose_v be_v proportional_a then_o also_o by_o conversion_n of_o proportion_n which_o of_o some_o be_v call_v proportion_n by_o eversion_n proportion_n and_o which_o be_v as_o before_o it_o be_v define_v when_o the_o antecedent_n be_v compare_v to_o the_o excess_n wherein_o the_o antecedent_n exceed_v the_o consequent_a they_o shall_v be_v proportional_a the_o 20._o theorem_a the_o 20._o proposition_n if_o there_o be_v three_o magnitude_n in_o one_o order_n and_o as_o many_o other_o magnitude_n in_o a_o other_o order_n which_o be_v take_v two_o and_o two_o in_o each_o order_n be_v in_o one_o and_o the_o same_o proportion_n and_o if_o of_o equality_n in_o the_o first_o order_n the_o first_o be_v great_a than_o the_o three_o then_o in_o the_o second_o order_n the_o first_o also_o shall_v be_v great_a than_o the_o three_o and_o if_o it_o be_v equal_a it_o shall_v be_v equal_a and_o if_o it_o be_v less_o it_o shall_v be_v less_o svppose_v that_o there_o be_v three_o magnitude_n in_o one_o order_n proportionality_n namely_o a_o b_o c_o &_o let_v there_o be_v as_o many_o magnitude_n in_o a_o other_o order_n which_o let_v be_v d_o e_o f_o which_o be_v take_v two_o and_o two_o in_o each_o order_n let_v be_v in_o one_o and_o the_o same_o proportion_n that_o be_v as_o a_o be_v to_o b_o so_o let_v d_o be_v to_o e_o and_o as_z b_o be_v to_o c_o so_o let_v e_o be_v to_o f._n but_o now_o if_o a_o be_v equal_a unto_o c_o d_o also_o shall_v be_v equal_a unto_o f._n for_o then_o a_o and_o c_o have_v unto_o b_o one_o and_o the_o same_o proportion_n by_o the_o first_o part_n of_o the_o seven_o of_o this_o book_n difference_n and_o for_o that_o as_o a_o be_v to_o b_o so_o be_v d_z to_o e_o and_o as_z c_z be_v to_o b_o so_o be_v fletcher_n to_o e_o therefore_o d_z and_o f_o have_v unto_o e_o one_o and_o the_o same_o proportion_n wherefore_o by_o the_o first_o part_n of_o the_o 9_o of_o this_o book_n d_z be_v equal_a unto_o f._n but_o now_o suppose_v that_o a_o be_v less_o then_o c._n then_o also_o shall_v d_o be_v less_o then_o f._n for_o by_o the_o 8._o of_o this_o book_n c_o shall_v have_v unto_o b_o a_o great_a proportion_n than_o have_v a_o to_o b._n difference_n but_o as_o a_o be_v to_o b_o so_o be_v d_z to_o e_o by_o supposition_n and_o as_o c_o be_v to_o b_o so_o have_v we_o prove_v be_v fletcher_n to_o e._n wherefore_o fletcher_n have_v unto_o e_o a_o great_a
proportion_n than_o have_v d_z to_o e._n wherefore_o by_o the_o first_o part_n of_o the_o 10._o of_o this_o book_n fletcher_n be_v great_a than_o d._n if_o therefore_o there_o be_v three_o magnitude_n in_o one_o order_n and_o as_o many_o other_o magnâ—tudes_n in_o a_o other_o order_n which_o be_v take_v two_o and_o two_o in_o each_o order_n be_v in_o one_o and_o the_o same_o proportion_n and_o if_o of_o equality_n in_o the_o first_o order_n the_o first_o be_v great_a than_o the_o three_o then_o in_o the_o second_o order_n also_o the_o first_o shall_v be_v great_a than_o the_o three_o and_o if_o it_o be_v equal_a it_o shall_v be_v equal_a and_o if_o it_o be_v less_o it_o shall_v be_v less_o which_o be_v require_v to_o be_v prove_v the_o 21._o theorem_a the_o 21._o proposition_n if_o there_o be_v three_o magnitude_n in_o one_o order_n and_o as_o many_o other_o magnitude_n in_o a_o other_o order_n which_o be_v take_v two_o and_o two_o in_o each_o order_n be_v in_o one_o and_o the_o same_o proportion_n and_o their_o proportion_n be_v perturbate_n if_o of_o equality_n in_o the_o first_o order_n the_o first_o be_v great_a than_o the_o three_o they_o in_o the_o second_o order_n the_o first_o also_o shall_v be_v great_a than_o the_o three_o and_o if_o it_o be_v equal_a it_o shall_v be_v equal_a and_o if_o it_o be_v less_o it_o shall_v be_v less_o svppose_v that_o there_o be_v three_o magnitude_n in_o one_o order_n proportionality_n namely_o a_o b_o c_o and_o let_v there_o be_v as_o many_o other_o magnitude_n in_o a_o other_o which_o let_v be_v d_o e_o f_o which_o be_v take_v two_o &_o two_o in_o each_o order_n let_v be_v in_o one_o and_o the_o same_o proportion_n and_o let_v their_o proportion_n be_v perturbate_n so_o that_o as_o a_o be_v to_o b_o so_o let_v e_o be_v to_o fletcher_n &_o as_o b_o be_v to_o c_o so_o let_v d_o be_v to_o e_o and_o of_o equality_n let_v a_o be_v great_a than_o g._n then_o i_o say_v that_z d_z also_o be_v great_a thâ—n_n e_o and_o if_o it_o be_v equal_a it_o be_v equal_a and_o if_o it_o be_v less_o it_o be_v less_o difference_n likewise_o if_o a_o be_v less_o than_o c_o d_o also_o be_v less_o than_o f._n for_o then_o c_o shall_v have_v unto_o b_o a_o great_a proportion_n then_o have_v a_o to_o b_o by_o the_o 8._o of_o the_o five_o wherefore_o e_o also_o â—ath_z unto_o d_z a_o great_a proportion_n than_o it_o â—ath_z to_o f._n wherefore_o by_o the_o second_o part_n of_o the_o 10._o of_o this_o book_n d_o be_v less_o than_o f._n if_o therefore_o there_o be_v three_o magnitude_n in_o one_o order_n &_o as_o many_o other_o magnitude_n in_o a_o other_o order_n which_o be_v take_v two_o and_o two_o in_o each_o order_n be_v in_o one_o &_o the_o same_o proportion_n &_o their_o proportion_n be_v perturbate_n and_o if_o of_o equality_n in_o the_o first_o order_n the_o first_o be_v great_a than_o the_o three_o then_o in_o the_o second_o order_n the_o first_o also_o shall_v be_v great_a than_o the_o thirdâ—_o and_o if_o it_o be_v equal_a it_o shall_v be_v equal_a and_o if_o it_o be_v less_o it_o shall_v be_v less_o which_o be_v require_v to_o be_v prove_v the_o 22._o theorem_a the_o 22._o proposition_n if_o there_o be_v a_o number_n of_o magnitude_n how_o many_o soever_o in_o one_o order_n and_o as_o many_o other_o magnitude_n in_o a_o other_o order_n which_o be_v take_v two_o and_o two_o in_o each_o order_n be_v in_o one_o and_o the_o same_o proportion_n they_o shall_v also_o of_o equality_n be_v in_o one_o and_o the_o same_o proportion_n proportionality_n svppose_v that_o there_o be_v a_o certain_a number_n of_o magnitude_n in_o one_o order_n as_o for_o example_n a_o b_o c_o and_o let_v there_o be_v as_o many_o other_o magnitude_n in_o a_o other_o order_n which_o let_v be_v d_o e_o f_o which_o be_v take_v two_o and_o two_o let_v be_v in_o one_o and_o the_o same_o proportion_n so_o that_o as_o a_o be_v to_o b_o so_o let_v d_o be_v to_o e_o and_o as_z b_o be_v to_o c_o so_o let_v e_o be_v to_o f._n then_o i_o say_v that_o of_o equality_n they_o shall_v be_v in_o the_o same_o proportion_n that_o be_v as_o a_o be_v to_o câ—_n so_o be_v d_z to_o f._n construction_n take_v unto_o a_o and_o d_o equemultiplices_fw-la g_z &_o h_o and_o likewise_o to_o b_o &_o e_o take_v any_o other_o equemultiplices_fw-la whatsoever_o namely_o king_n and_o lâ—_n and_o moreover_o unto_o c_z and_o f_o take_v any_o other_o equemultiplices_fw-la also_o what_o soever_o namely_o m_o and_o n._n and_o forasmuch_o as_o demonstration_n as_o a_o be_v to_o b_o so_o be_v d_z to_o e_o and_o unto_o a_o and_o d_o be_v take_v equemultiplices_fw-la g_z and_o h_o and_o likewise_o unto_o b_o and_o e_o be_v take_v certain_a other_o equemâ—ltiplices_n king_n and_o l_o therefore_o by_o the_o 4._o of_o the_o five_o as_z g_z be_v to_o kâ—_n so_o be_v h_o to_o l._n and_o by_o the_o same_o reason_n as_o king_n be_v to_o m_o so_o be_v l_o to_o n._n see_v therefore_o that_o there_o be_v in_o order_n three_o magnitude_n g_o king_n m_o &_o as_o many_o other_o magnitude_n in_o a_o other_o order_n namely_o h_o l_o n_o which_o be_v compare_v two_o to_o two_o be_v in_o one_o and_o the_o same_o proportion_n therefore_o of_o equality_n by_o the_o 20._o of_o the_o five_o if_o n_o exceed_v m_o then_o shall_v h_o exceed_v g_z and_o if_o it_o be_v equal_a it_o shall_v be_v equal_a and_o if_o it_o be_v less_o it_o shall_v be_v less_o but_o g_z and_o h_o be_v equemultiplices_fw-la unto_o a_o and_o d_o and_o m_o and_o n_o be_v certain_a other_o equemultiplices_fw-la unto_o c_z and_o f_o therefore_o by_o the_o 6._o definition_n of_o the_o five_o as_o a_o be_v to_o c_o so_o be_v d_z to_o f._n so_o also_o if_o there_o be_v more_o magnitude_n than_o three_o in_o â—ither_o order_n order_n the_o first_o of_o the_o one_o order_n shall_v be_v to_o the_o last_o as_o the_o first_o of_o the_o other_o order_n be_v to_o the_o last_o as_o if_o there_o be_v four_o in_o one_o order_n namely_o abcd_o and_o other_o four_o in_o the_o other_o order_n namely_o efgh_o aâ—cdeâ—gh_n we_o may_v with_o three_o magnitude_n a_o b_o c_o and_o e_o f_o g_o prove_v that_o as_o a_o be_v to_o c_o so_o be_v e_o to_o g_z and_o then_o leave_v out_o in_o either_o order_n the_o second_o and_o take_v the_o four_o as_o leave_v out_o b_o and_o f_o and_o take_v d_z and_o h_o we_o may_v prove_v by_o these_o three_o and_o three_o a_o c_o d_o and_o e_o g_o h_o that_o as_o a_o be_v to_o d_o so_o be_v e_o to_o h._n and_o observe_v this_o order_n this_o demonstration_n will_v serve_v how_o many_o soever_o the_o magnitude_n be_v in_o either_o order_n if_o therefore_o there_o be_v a_o number_n of_o magnitude_n how_o many_o soever_o in_o one_o order_n and_o as_o many_o other_o magnitude_n in_o a_o other_o order_n which_o be_v take_v two_o and_o two_o in_o each_o order_n be_v in_o one_o and_o the_o same_o proportion_n they_o shall_v also_o of_o equality_n be_v in_o one_o and_o the_o same_o proportion_n which_o be_v require_v to_o be_v demonstrate_v the_o 23._o theorem_a the_o 23._o proposition_n if_o there_o be_v three_o magnitude_n in_o one_o order_n and_o as_o many_o other_o magnitude_n in_o a_o other_o order_n which_o be_v take_v two_o &_o two_o in_o each_o order_n be_v in_o one_o and_o the_o same_o proportion_n and_o if_o also_o their_o proportion_n be_v perturbate_n then_o of_o equality_n they_o shall_v be_v in_o one_o and_o the_o same_o proportion_n proprotionalitie_n svppose_v that_o there_o be_v in_o one_o order_n three_o magnitude_n namely_o a_o b_o c_o &_o let_v be_v take_v in_o a_o other_o order_n as_o many_o other_o magnitud_n which_o let_v be_v d_o e_o f_o which_o be_v take_v two_o and_o two_o in_o each_o order_n let_v be_v in_o one_o and_o the_o same_o proportion_n and_o suppose_v that_o their_o proportion_n be_v perturbate_n so_o that_o as_o a_o be_v to_o b_o so_o let_v e_o be_v f_o and_o as_z b_o be_v to_o c_o so_o let_v d_o be_v to_o e._n then_o i_o say_v that_o as_o a_o be_v to_o c_o so_o be_v d_z to_o f._n take_v unto_o a_o b_o d_o construction_n equemultiplices_fw-la and_o let_v the_o same_o be_v ghk_v and_o likewise_o unto_o c_z e_o f_o take_v any_o other_o equemultiplices_fw-la whatsoever_o and_o let_v the_o same_o be_v lmn_o and_o forasmuch_o as_o g_z and_o h_o be_v equemultiplices_fw-la unto_o a_o and_o b_o but_o the_o part_n of_o equemultiplices_fw-la be_v in_o the_o same_o proportion_n that_o their_o equemultiplices_fw-la be_v by_o the_o 15._o of_o the_o five_o wherefore_o as_o a_o be_v to_o b_o so_o be_v g_z to_o h._n demonstration_n and_o by_o the_o
therefore_o alternate_o as_o ab_fw-la be_v to_o e_o so_o be_v cd_o to_o fletcher_n by_o the_o 16._o of_o the_o five_o but_o ab_fw-la be_v great_a than_o e_o wherefore_o also_o cd_o be_v great_a than_o fletcher_n which_o thing_n may_v also_o be_v prove_v by_o the_o 14._o of_o the_o same_o now_o for_o that_o as_o ab_fw-la be_v to_o cd_o so_o be_v e_o to_o fletcher_n but_o e_o be_v equal_a unto_o agnostus_n and_o fletcher_n be_v equal_a unto_o change_z therefore_o as_o ab_fw-la be_v to_o cd_o so_o be_v ag_z to_o change_z and_o forasmuch_o as_o the_o whole_a ab_fw-la be_v to_o the_o whole_a cd_o so_o be_v the_o part_n take_v away_o agnostus_n to_o the_o part_n take_v away_o change_z therefore_o the_o residue_n gb_o by_o the_o 1â—_o of_o the_o five_o be_v unto_o the_o residue_n hd_v as_o the_o whole_a ab_fw-la be_v to_o the_o whole_a cd_o but_o ab_fw-la the_o first_o be_v great_a than_o cd_o the_o three_o wherefore_o gb_o the_o second_o be_v great_a than_o hd_v the_o four_o by_o the_o 14._o of_o the_o five_o and_o forasmuch_o as_o agnostus_n be_v equal_a unto_o e_o &_o change_z be_v equal_a unto_o fletcher_n therefore_o agnostus_n and_o fletcher_n be_v equal_a unto_o ch_n and_o e._n and_o forasmuch_o as_o if_o unto_o thing_n unequal_a be_v add_v thing_n equal_a all_o shall_v be_v unequal_a by_o the_o four_o common_a sentence_n therefore_o see_v that_o gb_o and_o dh_o be_v unequal_a and_o gb_o be_v the_o great_a if_o unto_o gb_o be_v add_v agnostus_n and_o fletcher_n and_o likewise_o if_o unto_o hd_n be_v add_v change_z &_o e_o there_o shall_v be_v produce_v ab_fw-la and_o fletcher_n great_a than_o cd_o &_o e._n if_o therefore_o there_o be_v four_o magnitude_n proportional_a the_o great_a and_o the_o least_o of_o they_o shall_v be_v great_a they_o the_o other_o remain_v which_o be_v require_v to_o be_v demonstrate_v here_o follow_v certain_a proposition_n add_v by_o campane_n which_o be_v not_o to_o be_v contemn_v and_o be_v cite_v even_o of_o the_o best_a learned_a namely_o of_o johannes_n regio_fw-la montanus_fw-la in_o the_o epitome_n which_o he_o write_v upon_o ptolemy_n ¶_o the_o first_o proposition_n if_o there_o be_v four_o quantity_n and_o if_o the_o proportion_n of_o the_o first_o to_o the_o second_o be_v great_a than_o the_o proportion_n of_o the_o three_o to_o the_o four_o then_o contrariwise_o by_o conversion_n the_o proportion_n of_o the_o second_o to_o the_o first_o shall_v be_v less_o than_o the_o proportion_n of_o the_o four_o to_o the_o three_o affirmative_o it_o may_v also_o be_v demonstrate_v direct_o for_o let_v e_o be_v unto_o b_o as_z c_z be_v to_o d._n then_o converse_o b_o be_v to_o e_o as_z d_z be_v to_o c._n and_o forasmuch_o as_o a_o be_v great_a than_o e_o by_o the_o first_o part_n of_o the_o ten_o of_o this_o book_n therefore_o by_o the_o second_o part_n of_o the_o 8_o of_o the_o same_o b_o have_v unto_o a_o a_o less_o proportion_n than_o have_v b_o to_o e._n wherefore_o by_o the_o 13._o of_o the_o same_o b_o have_v unto_o aâ—_n â—_z less_o proportion_n than_o have_v d_z to_o c_z which_o be_v require_v to_o be_v prove_v ¶_o the_o second_o proposition_n if_o there_o be_v four_o quantity_n and_o if_o the_o proportion_n of_o the_o first_o to_o the_o second_o be_v great_a than_o the_o proportion_n of_o the_o three_o to_o the_o four_o then_o alternate_o the_o proportion_n of_o the_o first_o to_o the_o three_o shall_v be_v great_a than_o the_o proportion_n of_o the_o second_o to_o the_o four_o affirmative_o this_o may_v also_o be_v demonstrate_v affirmative_o let_v e_o be_v unto_o b_o as_z c_z be_v to_o d._n now_o they_o by_o the_o first_o part_n of_o the_o ten_o of_o this_o book_n e_o be_v less_o than_o a_o wherefore_o by_o the_o first_o part_n of_o the_o 8._o of_o the_o same_o the_o proportion_n of_o a_o to_o c_o be_v great_a than_o the_o proportion_n of_o e_o to_o c._n but_o alternate_o e_o be_v to_o c_o as_z b_o be_v to_o d._n wherefore_o by_o the_o 13._o of_o the_o same_o a_o have_v to_o c_o a_o 〈…〉_o ¶_o the_o three_o proposition_n if_o there_o be_v four_o quantity_n and_o if_o the_o proportion_n of_o the_o first_o tâ—_n the_o second_o be_v great_a than_o the_o proportion_n of_o the_o three_o to_o the_o four_o then_o by_o composition_n also_o the_o proportion_n of_o the_o fâ—â—th_n and_o second_v to_o the_o second_o shall_v be_v great_a than_o the_o proportion_n of_o the_o three_o and_o fourthâ—_o to_o the_o four_o this_o may_v also_o be_v demonstrate_v affirmative_o same_o forasmuch_o as_o the_o proportion_n of_o a_o to_o b_o be_v great_a than_o the_o proportion_n of_o c_o to_o d_z let_v e_o be_v unto_o b_o as_z c_z be_v to_o d._n and_o so_o by_o the_o first_o part_n of_o the_o 10._o of_o this_o book_n e_o shall_v be_v less_o than_o a._n and_o therefore_o by_o the_o common_a sentence_n ebb_n shall_v be_v less_o than_o ab_fw-la wherefore_o by_o the_o first_o part_n of_o the_o 8._o of_o the_o same_o ab_fw-la have_v unto_o b_o a_o great_a proportion_n than_o have_v ebb_n to_o b._n but_o by_o composition_n ebb_n be_v to_o b_o as_o cd_o be_v to_o d._n for_o by_o supposition_n e_o be_v unto_o b_o as_o be_v to_o d._n wherefore_o by_o the_o 12._o of_o this_o book_n ab_fw-la have_v to_o b_o a_o great_a proportion_n than_o have_v cd_o to_o d_o which_o be_v require_v to_o be_v prove_v ¶_o the_o four_o proposition_n if_o there_o be_v four_o quantity_n and_o if_o the_o proportion_n of_o the_o first_o and_o the_o second_o to_o the_o second_o be_v great_a than_o the_o proportion_n of_o the_o three_o and_o four_o to_o the_o four_o then_o by_o division_n also_o the_o proportion_n of_o the_o first_o to_o the_o second_o shall_v be_v great_a thâ—n_n the_o proportion_n of_o the_o third_o to_o the_o four_o suppose_v that_o the_o proportion_n of_o ab_fw-la to_o b_o be_v great_a than_o the_o proportion_n of_o cd_o to_o d._n impossibility_n then_o i_o say_v that_o by_o division_n also_o the_o proportion_n of_o a_o to_o b_o be_v great_a than_o the_o proportion_n of_o c_o to_o d._n for_o it_o can_v be_v the_o same_o for_o then_o by_o composition_n ab_fw-la shall_v be_v to_o b_o as_o cd_o be_v to_o d._n neither_o also_o can_v it_o be_v less_o for_o if_o the_o proportion_n of_o c_o to_o d_z be_v great_a than_o the_o proportion_n of_o a_o to_o b_o then_o by_o the_o former_a proposition_n the_o proportion_n of_o cd_o to_o d_z shall_v be_v great_a than_o the_o proportion_n of_o ab_fw-la to_o b_o which_o be_v contrary_a also_o to_o the_o supposition_n wherefore_o the_o proportion_n of_o a_o to_o b_o be_v neither_o one_o and_o the_o same_o with_o the_o proportion_n of_o c_o to_o d_z 〈…〉_o it_o wherefore_o it_o be_v great_a than_o it_o which_o be_v require_v to_o be_v prove_v affirmative_o the_o same_o may_v also_o be_v prove_v affirmative_o suppose_v that_o ebb_n be_v unto_o b_o as_o cd_o be_v to_o d._n now_o then_o by_o the_o first_o part_n of_o the_o 10._o of_o the_o five_o ebb_n shall_v be_v less_o than_o ab_fw-la and_o therefore_o by_o the_o common_a sentence_n e_o be_v less_o than_o a_o wherefore_o by_o the_o first_o part_n of_o the_o 8._o of_o this_o book_n the_o proportion_n of_o e_o to_o b_o be_v less_o than_o the_o proportion_n of_o a_o to_o b_o but_o as_z e_o be_v to_o b_o so_o be_v c_z to_o d_z wherefore_o the_o proportion_n of_o c_o to_o d_z be_v less_o than_o the_o proportion_n of_o a_o to_o b._n wherefore_o the_o proportion_n of_o a_o to_o b_o be_v great_a than_o the_o proportion_n of_o c_o to_o d_z which_o be_v require_v to_o be_v prove_v ¶_o the_o five_o proposition_n if_o there_o be_v four_o quantity_n and_o if_o the_o proportion_n of_o the_o first_o and_o the_o second_o to_o the_o second_o be_v great_a than_o the_o proportion_n of_o the_o three_o and_o the_o four_o to_o the_o four_o then_o by_o eversion_n the_o proportion_n of_o the_o first_o and_o second_o to_o the_o first_o shall_v be_v less_o than_o the_o proportion_n of_o the_o three_o and_o four_o to_o the_o three_o suppose_v that_o the_o proportion_n of_o ab_fw-la to_o b_o be_v great_a than_o the_o proportion_n of_o cd_o to_o d._n then_o i_o say_v that_o by_o eversion_n the_o proportion_n of_o ab_fw-la to_o a_o be_v less_o than_o the_o proportion_n of_o cd_o to_o hundred_o demonstration_n for_o by_o division_n by_o the_o former_a proposition_n the_o proportion_n of_o a_o to_o b_o be_v great_a than_o the_o proportion_n of_o c_o to_o d._n wherefore_o by_o the_o first_o of_o these_o proposition_n converse_o b_o have_v unto_o a_o a_o less_o proportion_n they_o have_v d_z to_o c._n wherefore_o by_o the_o 3._o of_o the_o same_o by_o composition_n the_o proportion_n of_o ab_fw-la to_o a_o be_v less_o they_o the_o
proportion_n of_o cd_o to_o c_o which_o be_v require_v to_o be_v prove_v ¶_o the_o six_o proposition_n if_o there_o be_v take_v three_o quantity_n in_o one_o order_n and_o as_o many_o in_o a_o other_o order_n and_o if_o the_o proportion_n of_o the_o first_o to_o the_o second_o in_o the_o first_o order_n be_v great_a than_o the_o proportion_n of_o the_o first_o to_o the_o second_o in_o the_o latter_a order_n then_o also_o the_o proportion_n of_o the_o first_o to_o the_o three_o in_o the_o first_o order_n shall_v be_v great_a than_o the_o proportion_n of_o the_o first_o to_o the_o three_o in_o the_o latter_a order_n suppose_v that_o there_o be_v three_o quantity_n in_o one_o order_n a_o b_o c_o &_o as_o many_o other_o quantity_n in_o a_o other_o order_n d_o e_o f._n and_o let_v the_o proportion_n of_o a_o to_o b_o in_o the_o first_o order_n be_v great_a than_o the_o proportion_n of_o d_z to_o e_o in_o the_o second_o order_n and_o let_v also_o the_o proportion_n of_o b_o to_o c_z in_o the_o first_o order_n be_v great_a than_o the_o proportion_n of_o e_o to_o fletcher_n in_o the_o second_o order_n then_o i_o say_v that_o the_o proportion_n of_o a_o to_o c_o in_o the_o first_o order_n be_v great_a they_o the_o proportion_n of_o d_o to_o fletcher_n in_o the_o second_o order_n demonstrationâ—_n for_o let_v g_o be_v unto_o c_o as_z e_o be_v to_o f._n now_o then_o by_o the_o first_o part_n of_o the_o 10_o of_o this_o book_n g_o shall_v be_v less_o then_o b._n and_o therefore_o by_o the_o second_o part_n of_o the_o 8._o of_o the_o â—â—me_n the_o proportion_n of_o a_o to_o g_o iâ—_z great_a thâ—n_o the_o proportion_n of_o â—_o to_o â—_o wherefore_o the_o proportion_n of_o a_o to_o g_z be_v much_o great_a thâ—n_n the_o proportion_n of_o d_o to_o e._n now_o then_o let_v â—_o be_v 〈…〉_o d_o be_v to_o e._n wherefore_o by_o the_o first_o part_n of_o the_o 100l_n of_o the_o same_o a_o be_v great_a them_z h._n and_o therefore_o by_o the_o first_o part_n of_o the_o 8._o of_o the_o same_o the_o proportion_n of_o a_o to_o c_o be_v great_a than_o the_o proportion_n of_o h_o to_o c._n but_o by_o proportion_n of_o equality_n h_o be_v to_o c_o as_z d_z be_v to_o fletcher_n for_o h_o be_v to_o g_z as_z d_z iâ—_z to_o e_o and_o g_z be_v to_o c_z as_z e_o be_v to_o f._n wherefore_o by_o the_o 12._o of_o the_o same_o a_o have_v to_o c_o a_o great_a proportion_n than_o have_v d_z to_o fletcher_n which_o be_v require_v to_o be_v prove_v ¶_o the_o seven_o proposition_n if_o there_o be_v take_v three_o quantity_n in_o one_o order_n and_o as_o many_o other_o in_o a_o other_o order_n and_o if_o the_o proportion_n of_o the_o second_o to_o the_o three_o in_o the_o first_o order_n be_v great_a than_o the_o proportion_n of_o the_o first_o to_o the_o second_o in_o the_o latter_a order_n if_o also_o the_o proportion_n of_o the_o first_o to_o the_o second_o in_o the_o first_o order_n be_v great_a than_o the_o proportion_n of_o the_o second_o to_o the_o three_o in_o the_o latter_a order_n then_o shall_v the_o proportion_n of_o the_o first_o to_o the_o three_o in_o the_o first_o order_n be_v great_a than_o the_o proportion_n of_o the_o first_o to_o the_o three_o in_o the_o latter_a order_n suppose_v that_o there_o be_v three_o quantity_n in_o one_o order_n a_o b_o c_o and_o as_o many_o other_o in_o a_o other_o order_n d_o e_o f._n and_o let_v the_o proportion_n of_o b_o to_o c_z in_o the_o first_o order_n be_v great_a than_o the_o proportion_n of_o d_z to_o e_o in_o the_o second_o order_n and_o let_v also_o the_o proportion_n of_o a_o to_o b_o in_o the_o first_o order_n be_v great_a than_o the_o proportion_n of_o e_o to_o fletcher_n in_o the_o second_o order_n then_o i_o say_v that_o a_o have_v to_o c_o a_o great_a proportion_n than_o have_v d_z to_o f._n this_o pertain_v to_o proportion_n of_o equality_n for_o let_v g_o be_v unto_o c_o as_z d_z be_v to_o e._n and_o by_o the_o first_o part_n of_o the_o 10._o of_o this_o bokeâ—_n g_o shall_v be_v less_o they_o â—_o and_o therefore_o by_o the_o second_o part_n of_o the_o 8._o of_o the_o same_o the_o proportion_n of_o a_o to_o g_z be_v great_a than_o the_o proportion_n of_o a_o to_o b._n wherefore_o a_o have_v unto_o g_z a_o much_o great_a proportion_n than_o have_v â—_o to_o f._n now_o then_o let_v h_o be_v unto_o g_o as_z e_o be_v to_o f._n and_o by_o the_o first_o part_n of_o the_o 10._o of_o the_o same_o a_o shall_v be_v great_a than_o h._n and_o by_o the_o first_o part_n of_o the_o 8._o of_o the_o same_o the_o proportion_n of_o a_o to_o c_o be_v great_a than_o the_o proportion_n of_o h_o to_o c._n but_o by_o the_o 23._o of_o the_o same_o the_o proportion_n of_o h_o to_o c_z be_v as_o the_o proportion_n of_o d_o to_o fletcher_n for_o g_o be_v to_o c_z as_z d_z be_v to_o e_o and_o h_z be_v to_o g_z as_z e_o be_v to_o f._n wherefore_o by_o the_o 12._o of_o the_o same_o the_o proportion_n of_o a_o to_o c_o be_v great_a than_o the_o proportion_n of_o d_o to_o fletcher_n which_o be_v require_v to_o be_v prove_v ¶_o the_o eight_o proposition_n if_o the_o proportion_n of_o the_o whole_a to_o the_o whole_a be_v great_a thâ—n_n the_o proportion_n of_o a_o part_n take_v away_o to_o a_o part_n take_v away_o they_o shall_v the_o proportion_n of_o the_o residue_n unto_o the_o residue_n be_v great_a than_o the_o proportion_n of_o the_o whole_a to_o the_o whole_a suppose_v that_o there_o be_v two_o quantity_n ab_fw-la &_o c_o d_o from_o which_o let_v there_o be_v cut_v of_o these_o magnitude_n ae_n and_o cf_o and_o let_v the_o residue_n be_v ebb_n and_o fd._n and_o let_v the_o proportion_n of_o ab_fw-la to_o cd_o be_v great_a than_o the_o proportion_n of_o ae_n to_o cf._n then_o i_o say_v that_o the_o proportion_n of_o ebb_n to_o fd_o be_v great_a than_o the_o proportion_n of_o ab_fw-la to_o cd_o for_o by_o the_o second_o of_o these_o proposition_n now_o add_v alternate_o the_o proportion_n of_o aâ—_z to_o aâ—_z 〈◊〉_d great_a than_o the_o proportion_n of_o cd_o to_o cf._n and_o therefore_o by_o eversion_n of_o proportion_n by_o the_o 5._o of_o the_o same_o the_o proportion_n of_o ab_fw-la to_o eâ—_n be_v less_o than_o the_o proportion_n of_o cd_o to_o fd._n wherefore_o again_o alternate_o the_o proportion_n of_o ab_fw-la to_o cd_o be_n less_o than_o the_o proportion_n of_o ebb_n to_o fd_o which_o be_v require_v to_o be_v prove_v ¶_o the_o nine_o proposition_n if_o quantity_n how_o many_o soever_o in_o one_o order_n be_v compare_v to_o as_o many_o other_o in_o a_o other_o order_n and_o if_o there_o be_v a_o great_a proportion_n of_o every_o one_o that_o go_v before_o to_o that_o whereunto_o it_o be_v refer_v then_o of_o any_o that_o follow_v to_o that_o whereunto_o it_o be_v refer_v the_o proportion_n of_o they_o all_o take_v together_o unto_o all_o the_o other_o take_v together_o shall_v be_v great_a than_o the_o proportion_n of_o any_o that_o follow_v to_o that_o whereunto_o it_o be_v compare_v and_o also_o then_o the_o proportion_n of_o all_o they_o take_v together_o to_o all_o the_o other_o take_v together_o but_o shall_v be_v less_o than_o the_o proportion_n of_o the_o first_o to_o the_o first_o suppose_v that_o there_o be_v three_o quantity_n in_o one_o order_n a_o b_o c_o &_o as_o many_o other_o in_o a_o other_o order_n d_o e_o f._n and_o let_v the_o proportion_n of_o a_o to_o d_o be_v great_a they_o the_o proportion_n of_o b_o to_o e_o let_v also_o the_o proportion_n of_o b_o to_o e_o be_v great_a than_o the_o proportion_n of_o c_o to_o f._n then_o i_o say_v that_o the_o proportion_n of_o abc_n take_v all_o together_o to_o def_n take_v altogether_o be_v great_a they_o the_o proportion_n of_o b_o to_o e_o and_o also_o then_o the_o proportion_n of_o c_o to_o fletcher_n &_o more_o over_o they_o the_o proportion_n of_o b_o &_o c_o take_v together_o to_o of_o take_v together_o but_o be_v less_o than_o the_o proportion_n of_o a_o to_o d_o demonstration_n for_o forasmuch_o as_o a_o have_v to_o d_o a_o great_a proportion_n they_o have_v b_o to_o e_o therefore_o alternate_o a_o have_v to_o b_o a_o great_a proportion_n than_o have_v d_z to_o e_o wherefore_o by_o composition_n ab_fw-la have_v to_o b_o a_o great_a proportion_n they_o have_v de_fw-fr to_o e._n and_o again_o alternate_o ab_fw-la have_v to_o de_fw-fr a_o great_a proportion_n than_o have_v b_o to_o e._n wherefore_o by_o the_o former_a proposition_n a_o have_v to_o â—_o a_o great_a proportion_n than_o have_v ab_fw-la to_o de._n and_o by_o the_o same_o reason_n may_v it_o be_v prove_v that_o have_v to_o e_o a_o great_a
triangle_n unto_o the_o section_n divide_v the_o angle_n of_o the_o triangle_n into_o two_o equal_a part_n this_o construction_n be_v the_o half_a part_n of_o that_o gnomical_a figure_n describe_v in_o the_o 43._o proposition_n of_o the_o first_o book_n which_o gnomical_a figure_n be_v of_o great_a use_n in_o a_o manner_n in_o all_o geometrical_a demonstration_n the_o 4._o theorem_a the_o 4._o proposition_n in_o equiangle_n triangle_n the_o side_n which_o contain_v the_o equal_a angle_n be_v proportional_a and_o the_o side_n which_o be_v subtend_v under_o the_o equal_a angle_n be_v of_o like_a proportion_n svppose_v that_o there_o be_v two_o equiangle_n triangle_n abc_n and_o dce_n and_o let_v the_o angle_n abc_n of_o the_o one_o triangle_n be_v equal_a unto_o the_o angle_v dce_n of_o the_o other_o triangle_n and_o the_o angle_n bac_n equal_a unto_o the_o angle_v cde_o and_o moreover_o the_o angle_n acb_o equal_a unto_o the_o angle_n dec_n then_o i_o say_v that_o those_o side_n of_o the_o triangle_n abc_n &_o dce_n which_o include_v the_o equal_a angle_n be_v proportional_a and_o the_o side_n which_o be_v subtend_v under_o the_o equal_a angle_n be_v of_o like_a proportion_n construction_n for_o let_v two_o side_n of_o the_o say_a triangle_n namely_o two_o of_o those_o side_n which_o be_v subtend_v under_o equal_a angle_n as_o for_o example_n the_o side_n bc_o and_o ce_fw-fr be_v so_o set_v that_o they_o both_o make_v one_o right_a line_n and_o because_o the_o angle_n abc_n &_o acb_o be_v less_o than_o two_o right_a angle_n by_o the_o 17._o of_o the_o first_o but_o the_o angle_n acb_o be_v equal_a unto_o the_o angle_n dec_n therefore_o the_o angle_n abc_n &_o dec_n be_v less_o they_o two_o right_a angle_n wherefore_o the_o line_n basilius_n &_o ed_z be_v produce_v will_v at_o the_o length_n meet_v together_o let_v they_o meet_v and_o join_v together_o in_o the_o point_n f._n demonstration_n and_o because_o by_o supposition_n the_o angle_n dce_n be_v equal_a unto_o the_o angle_n abc_n therefore_o the_o line_n bf_o be_v by_o the_o 28._o of_o the_o first_o a_o parallel_n unto_o tâ—e_z line_n cd_o and_o forasmuch_o as_o by_o supposition_n the_o angle_n acb_o be_v equal_a unto_o the_o angle_n dec_n therefore_o again_o by_o the_o 28._o of_o the_o first_o the_o line_n ac_fw-la be_v a_o parallel_n unto_o the_o line_n fe_o wherefore_o fadc_n be_v a_o parallelogram_n wherefore_o the_o side_n favorina_n be_v equal_a unto_o the_o side_n dc_o and_o the_o side_n ac_fw-la unto_o the_o side_n fd_o by_o the_o 34._o of_o the_o first_o and_o because_o unto_o one_o of_o the_o side_n of_o the_o triangle_n bfe_n namely_o to_o fe_o be_v draw_v a_o parallel_a line_n ac_fw-la therefore_o as_o basilius_n be_v to_o of_o so_o be_v bc_o to_o ce_fw-fr by_o the_o 2._o of_o the_o six_o but_o of_o be_v equal_a unto_o cd_o wherefore_o by_o the_o 11._o of_o the_o five_o as_o basilius_n be_v to_o cd_o so_o be_v bc_o to_o ce_fw-fr which_o be_v side_n subtend_v under_o equal_a angle_n wherefore_o alternate_o by_o the_o 16._o of_o the_o five_o as_o ab_fw-la be_v to_o bc_o so_o be_v dc_o to_o ce._n again_o forasmuch_o as_o cd_o be_v a_o parallel_n unto_o bf_o therefore_o again_o by_o the_o 2._o of_o the_o six_o as_o bc_o be_v to_o ce_fw-fr so_o be_v fd_v to_o de._n but_o fd_o be_v equal_a unto_o ac_fw-la wherefore_o as_o bc_o be_v to_o ce_fw-fr so_o be_v ac_fw-la to_o de_fw-fr which_o be_v also_o side_n subtend_v under_o equal_a angle_n wherefore_o alternate_o by_o the_o 16._o of_o the_o five_o â—s_v bc_o be_v to_o ca_n so_o be_v ce_fw-fr to_o edâ—_n wherefore_o forasmuch_o as_o it_o have_v be_v demonstrate_v that_o as_o ab_fw-la be_v unto_o be_v so_o be_v dc_o unto_o ceâ—_n but_o as_o dc_o be_v unto_o ca_n so_o be_v ce_fw-fr unto_o edâ—_n it_o follow_v of_o equality_n by_o the_o 22._o of_o the_o five_o that_o â—s_a basilius_n be_v unto_o ac_fw-la so_o be_v cd_o unto_o deâ—_n wherefore_o in_o equiangle_n triangleâ—_n the_o side_n which_o include_v the_o equal_a angle_n be_v proportional_a and_o the_o side_n which_o be_v subtend_v under_o the_o equal_a angle_n be_v of_o like_a proportion_n â—hich_z be_v require_v to_o be_v demonstrate_v the_o 5._o theorem_a the_o 5._o proposition_n if_o two_o triangle_n have_v their_o side_n proportional_a the_o triangâ—â—s_n be_v equiangle_n and_o those_o angle_n in_o they_o be_v equal_a under_o which_o be_v subtend_v side_n of_o like_a proportion_n svppose_v that_o there_o be_v two_o triangle_n abc_n &_o def_n have_v their_o side_n proportional_a as_o ab_fw-la be_v to_o bc_o so_o let_v de_fw-fr be_v to_o of_o proposition_n &_o as_o bc_o be_v to_o ac_fw-la so_o let_v of_o be_v to_o df_o and_o moreover_o as_o basilius_n be_v to_o ac_fw-la so_o let_v ed_z be_v to_o df._n then_o i_o say_v that_o the_o triangle_n abc_n be_v equiangle_n unto_o the_o triangle_n def_n and_o those_o angle_n in_o they_o be_v equal_a under_o which_o be_v subtend_v side_n of_o like_a proportion_n that_o be_v the_o angle_n abc_n be_v equal_a unto_o the_o angle_n def_n and_o the_o angle_n bca_o unto_o the_o angle_n efd_n and_o moreover_o the_o angle_n bac_n to_o the_o angle_n edf_o upon_o the_o right_a line_n of_o construction_n and_o unto_o the_o point_n in_o it_o e_o &_o f_o describe_v by_o the_o 23._o of_o the_o first_o angle_n equal_a unto_o the_o angle_n abc_n &_o acb_o which_o let_v be_v feg_fw-mi and_o efg_o namely_o let_v the_o angle_n feg_fw-mi be_v equal_a unto_o the_o angle_n abc_n and_o let_v the_o angle_n efg_o be_v equal_a to_o the_o angle_n acb_o demonstration_n and_o forasmuch_o as_o the_o angle_n abc_n and_o acb_o be_v less_o than_o two_o right_a angle_n by_o the_o 17._o of_o the_o first_o therefore_o also_o the_o angle_n feg_v and_o efg_o be_v less_o than_o two_o right_a angle_n wherefore_o by_o the_o 5._o petition_n of_o the_o first_o the_o right_a line_n eglantine_n &_o fg_o shall_v at_o the_o length_n concur_v let_v they_o concur_v in_o the_o point_n g._n wherefore_o efg_o be_v a_o triangle_n wherefore_o the_o angle_n remain_v bac_n be_v equal_a unto_o the_o angle_n remain_v egf_n by_o the_o first_o corollary_n of_o the_o 32._o of_o the_o first_o wherefore_o the_o triangle_n abc_n be_v equiangle_n unto_o the_o triangle_n gef_n wherefore_o in_o the_o triangle_n abc_n and_o egf_n the_o side_n which_o include_v the_o equal_a angle_n by_o the_o 4._o of_o the_o six_o be_v proportional_a and_o the_o side_n which_o be_v subtend_v under_o the_o equal_a angle_n be_v of_o like_a proportion_n wherefore_o as_o ab_fw-la be_v to_o bc_o so_o be_v ge_z to_o ef._n but_o as_o ab_fw-la be_v to_o bc_o so_o by_o supposition_n be_v de_fw-fr to_o ef._n wherefore_o as_o de_fw-fr be_v to_o of_o so_o be_v ge_z to_o of_o by_o the_o 11._o of_o the_o five_o wherefore_o either_o of_o these_o de_fw-fr and_o eglantine_n have_v to_o of_o one_o and_o the_o same_o proportion_n wherefore_o by_o the_o 9_o of_o the_o five_o de_fw-fr be_v equal_a unto_o eglantine_n and_o by_o the_o same_o reason_n also_o df_o be_v equal_a unto_o fg._n now_o forasmuch_o as_o de_fw-fr be_v equal_a to_o eglantine_n and_o of_o be_v common_a unto_o they_o both_o therefore_o these_o two_o side_n de_fw-fr &_o of_o be_v equal_a unto_o these_o two_o side_n ge_z and_o of_o and_o the_o base_a df_o be_v equal_a unto_o the_o base_a fg._n wherefore_o the_o angle_n def_n by_o the_o 8._o of_o the_o first_o be_v equal_a unto_o the_o angle_n gef_n and_o the_o triangle_n def_n by_o the_o 4._o of_o the_o first_o be_v equal_a unto_o the_o triangle_n gef_n and_o the_o rest_n of_o the_o angle_n of_o the_o one_o triangle_n be_v equal_a unto_o the_o rest_n of_o the_o angle_n of_o the_o other_o triangle_n the_o one_o to_o the_o outstretch_o under_o which_o be_v subtend_v equal_a side_n wherefore_o the_o angle_n dfe_n be_v equal_a unto_o the_o angle_n gfe_n and_o the_o angle_n edf_o unto_o the_o angle_n egf_n and_o becauseâ—_n the_o angle_n feed_v be_v equal_a unto_o the_o angle_n gef_n but_o the_o angle_n gef_n be_v equal_a unto_o the_o angle_n abc_n therefore_o the_o angle_n abc_n be_v also_o equal_a unto_o the_o angle_n feed_v and_o by_o the_o same_o reason_n the_o angle_n acb_o be_v equal_a unto_o the_o angle_v dfeâ—_n and_o moreover_o the_o angle_n bac_n unto_o the_o angle_n edf_o wherefore_o the_o triangle_n abc_n be_v equiangle_n unto_o the_o triangle_n def_n if_o two_o triangle_n therefore_o have_v their_o side_n proportional_a the_o triangle_n shall_v be_v equiangle_n &_o those_o angle_n in_o they_o shall_v be_v equal_a under_o which_o be_v subtend_v side_n of_o like_a proportion_n which_o be_v require_v to_o be_v demonstrate_v the_o 6._o theorem_a the_o 6._o proposition_n if_o there_o be_v two_o triangle_n whereof_o the_o one_o have_v one_o angle_n equal_a to_o one_o angle_n of_o
uno_fw-la vnum_fw-la quodque_fw-la idea_n est_fw-la quia_fw-la unum_fw-la numero_fw-la est_fw-la that_o be_v every_o thing_n therefore_o be_v that_o be_v therefore_o have_v his_o be_v in_o nature_n and_o be_v that_o it_o be_v for_o that_o it_o be_v on_o in_o number_n according_a whereunto_o jordane_n in_o that_o most_o excellent_a and_o absolute_a work_n of_o arithmetic_n which_o he_o write_v define_v unity_n after_o this_o manner_n vnitas_fw-la est_fw-la res_fw-la per_fw-la se_fw-la discretio_fw-la that_o be_v unity_n be_v proper_o and_o of_o itself_o the_o difference_n of_o any_o thing_n that_o be_v unity_n unity_n be_v that_o whereby_o every_o thing_n do_v proper_o and_o essential_o differ_v and_o be_v a_o other_o thing_n from_o all_o other_o certain_o a_o very_a apt_a definition_n and_o it_o make_v plain_a the_o definition_n here_o set_v of_o euclid_n definition_n 2_o number_n be_v a_o multitude_n compose_v of_o unity_n as_o the_o number_n of_o three_o be_v a_o multitude_n compose_v and_o make_v of_o three_o unity_n likewise_o the_o number_n of_o five_o be_v nothing_o else_o but_o the_o composition_n &_o put_v together_o of_o five_o unity_n although_o as_o be_v before_o say_v between_o a_o point_n in_o magnitude_n and_o unity_n in_o multitude_n there_o be_v great_a agreement_n and_o many_o thiâ—â—â—â—_n be_v common_a to_o they_o both_o for_o as_o a_o point_n be_v the_o begin_n of_o magnitude_n so_o be_v unity_n the_o beginning_n of_o number_n and_o as_o a_o point_n in_o magnitude_n be_v indivisible_a so_o be_v also_o unity_n in_o number_n indivisible_a yet_o in_o this_o they_o differ_v and_o disagree_v unity_n there_o be_v no_o line_n or_o magnitude_n make_v of_o point_n as_o of_o his_o part_n so_o that_o although_o a_o point_n be_v the_o beginning_n of_o a_o line_n yet_o be_v it_o no_o part_n thereof_o but_o unity_n as_o it_o be_v the_o begin_n of_o number_n so_o be_v it_o also_o a_o part_n thereof_o which_o be_v somewhat_o more_o manifest_o set_v of_o boetius_fw-la in_o a_o other_o definition_n of_o number_n which_o he_o give_v in_o his_o arithmetic_n boetius_fw-la which_o be_v thus_o numerus_fw-la est_fw-la quantitatâ—_fw-la acernus_fw-la ex_fw-la unitatibus_fw-la profusus_fw-la that_o be_v number_n be_v a_o mass_n or_o heap_n of_o quantity_n produce_v of_o unity_n which_o definition_n in_o substance_n be_v all_o one_o with_o the_o first_o wherein_o be_v say_v most_o plain_o that_o the_o heap_n or_o mass_n number_n that_o be_v the_o whole_a substance_n of_o the_o quantity_n of_o number_n be_v produce_v &_o make_v of_o unity_n so_o that_o unity_n be_v as_o it_o be_v the_o very_a matter_n of_o number_n as_o four_o unity_n add_v together_o be_v the_o matter_n whereof_o the_o number_n 4._o be_v make_v &_o each_o of_o these_o unity_n be_v a_o part_n of_o the_o number_n four_o namely_o a_o four_o part_n or_o a_o quarter_n unto_o this_o definition_n agree_v also_o the_o definition_n give_v of_o jordane_n jordane_n which_o be_v thus_o number_n be_v a_o quantity_n which_o gather_v together_o thing_n sever_v a_o sunder_o number_n as_o five_o man_n be_v in_o themselves_o sever_v and_o distinct_a be_v by_o the_o number_n five_o bring_v together_o as_o it_o be_v into_o one_o mass_n and_o so_o of_o other_o and_o although_o unity_n be_v no_o number_n yet_o it_o contain_v in_o it_o the_o virtue_n and_o power_n of_o all_o number_n and_o be_v set_v and_o take_v for_o they_o number_n in_o this_o place_n for_o the_o far_o elucidation_n of_o thing_n partly_o before_o set_v and_o chief_o hereafter_o to_o be_v set_v because_o euclid_n here_o do_v make_v mention_n of_o diverse_a kind_n of_o number_n and_o also_o define_v the_o same_o be_v to_o be_v note_v that_o number_n may_v be_v consider_v three_o manner_n of_o way_n first_o number_n may_v be_v consider_v absolute_o without_o compare_v it_o to_o any_o other_o number_n wayâ—_n or_o without_o apply_v it_o to_o any_o other_o thing_n only_o view_v â—nd_n payse_v what_o it_o be_v in_o itself_o and_o in_o his_o own_o nature_n only_o and_o what_o part_n it_o have_v and_o what_o propriety_n and_o passion_n as_o this_o number_n six_o may_v be_v consider_v absolute_o in_o his_o own_o nature_n that_o it_o be_v a_o even_a number_n and_o that_o it_o be_v a_o perfect_a number_n and_o have_v many_o more_o condition_n and_o propriety_n and_o so_o conceive_v you_o of_o all_o other_o number_n whatsoever_o of_o 9_o 12._o and_o so_o forth_o an_o other_o way_n number_n may_v be_v consider_v by_o way_n of_o comparison_n and_o in_o respect_n of_o some_o other_o number_n either_o as_o equal_a to_o itself_o or_o as_o great_a they_o itself_o or_o as_o less_o they_o itself_o as_o 12._o may_v be_v consider_v as_o compare_v to_o 12._o which_o be_v equal_a unto_o it_o or_o as_o to_o 24._o which_o be_v great_a than_o it_o for_o 12_o be_v the_o half_a thereof_o of_o as_o to_o 6._o which_o be_v less_o than_o it_o as_o be_v the_o double_a thereof_o and_o of_o this_o consideration_n of_o number_n arise_v and_o spring_v all_o kind_n and_o variety_n of_o proportion_n as_o have_v before_o be_v declare_v in_o the_o explanation_n of_o the_o principle_n of_o the_o five_o book_n so_o that_o of_o that_o matter_n it_o be_v needless_a any_o more_o to_o be_v say_v in_o this_o place_n thus_o much_o of_o this_o for_o the_o declaration_n of_o the_o thing_n follow_v 3_o a_o part_n be_v a_o less_o number_n in_o comparison_n to_o the_o great_a when_o the_o less_o measure_v the_o great_a definition_n as_o the_o number_n 3_o compare_v to_o the_o number_n 12._o be_v a_o part_n for_o 3_o be_v a_o less_o number_n than_o be_v 12._o and_o moreover_o it_o measure_v 12_o the_o great_a number_n for_o 3_o take_v or_o add_v to_o itself_o certain_a time_n namely_o 4_o time_n make_v 12._o for_o 3_o four_o time_n be_v 12._o likewise_o be_v â—_o a_o part_n of_o 8_o 2_o be_n less_o than_o 8_o and_o take_v 4_o time_n it_o make_v 8._o for_o the_o better_a understanding_n of_o this_o definition_n and_o how_o this_o word_n parte_fw-la be_v diverse_o take_v in_o arithmetic_n and_o in_o geometry_n read_v the_o declaration_n of_o the_o first_o definition_n of_o the_o 5._o book_n 4_o part_n be_v a_o less_o number_n in_o respect_n of_o the_o great_a when_o the_o less_o measure_v not_o the_o great_a definition_n as_o the_o number_n 3_o compare_v to_o 5_o be_v part_n of_o 5_o and_o not_o a_o part_n for_o the_o number_n 3_o be_v less_o than_o the_o number_n 5_o and_o do_v not_o measure_v 5._o for_o take_v once_o it_o make_v but_o 3._o once_o 3_o be_v 3_o which_o be_v less_o than_o 5._o and_o 3_o take_v twice_o make_v 6_o which_o be_v more_o than_o 5._o wherefore_o it_o be_v no_o part_n of_o 5_o but_o part_n namely_o three_o five_o part_n of_o 5._o for_o in_o the_o number_n 3_o there_o be_v 3_o unity_n and_o every_o unity_n be_v the_o five_o part_n of_o 5._o wherefore_o 3_o be_v three_o five_o part_n of_o 5_o and_o so_o of_o other_o 5_o multiplex_n be_v a_o great_a number_n in_o comparison_n of_o the_o less_o when_o the_o less_o measure_v the_o great_a definition_n as_o 9_o compare_v to_o 3_o be_v multiplex_n the_o number_n 9_o be_v great_a than_o the_o number_n 3_n and_o moreover_o 3_o the_o less_o number_n measure_v 9_o the_o great_a number_n for_o 3_o take_v certain_a time_n namely_o 3_o time_n make_v 9_o three_o time_n three_o be_v 9_o for_o the_o more_o ample_a and_o full_a knowledge_n of_o this_o definition_n read_v what_o be_v say_v in_o the_o explanation_n of_o the_o second_o definition_n of_o the_o 5_o book_n where_o multiplex_n be_v sufficient_o entreat_v of_o with_o all_o his_o kind_n 6_o 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 definition_n as_o the_o number_n 6_o may_v be_v divide_v into_o 3_o and_o 3_o which_o be_v his_o part_n and_o they_o be_v equal_a the_o one_o not_o exceed_v the_o other_o this_o definition_n of_o euclid_n be_v to_o be_v understand_v of_o two_o such_o equal_a part_n which_o join_v together_o make_v the_o whole_a number_n as_o 3_o and_o 3_o the_o equal_a part_n of_o 6_o join_v together_o make_v 6_o for_o otherwise_o many_o number_n both_o even_o and_o odd_a may_v be_v divide_v into_o many_o equal_a part_n as_o into_o 4._o 5._o 6â—_o or_o more_o and_o therefore_o into_o 2._o as_o 9_o may_v be_v divide_v into_o 3_o and_o 3_o which_o be_v his_o part_n and_o be_v also_o equal_a for_o the_o one_o of_o they_o exceed_v not_o the_o other_o yet_o be_v not_o therefore_o this_o number_n â—_o a_o even_o number_n for_o that_o 3_o and_o 3_o these_o equal_a part_n of_o 9_o add_v together_o make_v not_o 9_o but_o only_o 6._o likewise_o take_v the_o definition_n so_o general_o every_o number_n whatsoever_o shall_v be_v a_o even_a numberâ—_n for_o
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 want_v number_n for_o all_o his_o part_n name_o 3._o and_o 1._o more_o part_n he_o have_v not_o add_v together_o make_v only_o 4_o which_o be_v less_o than_o 9_o also_o 26._o be_v a_o diminute_a number_n all_o his_o part_n namely_o 13.2.1_o add_v together_o make_v only_o 16_o which_o be_v a_o number_n much_o less_o than_o 26._o and_o so_o of_o such_o like_a sentence_n campane_n and_o flussates_n here_o add_v certain_a common_a sentence_n some_o of_o which_o for_o that_o they_o be_v in_o these_o three_o book_n follow_v sometime_o allege_v i_o think_v good_a here_o to_o annex_v sentence_n 1_o the_o less_o part_n be_v that_o which_o have_v the_o great_a denomination_n and_o the_o great_a part_n be_v that_o which_o have_v the_o less_o denomination_n as_o the_o number_n 6._o and_o 8._o be_v either_o of_o they_o a_o part_n of_o the_o number_n 24_o 6._o be_v a_o four_o part_n 4._o time_n 6._o be_v 24_o and_o 8._o be_v a_o three_o part_n 3._o time_n 8._o be_v 24._o now_o forasmuch_o as_o 4_o 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
be_v require_v to_o be_v demonstrate_v after_o this_o proposition_n campane_n demonstrate_v in_o number_n these_o four_o kind_n of_o proportionality_n campane_n namely_o proportion_n converse_n compose_v divide_a and_o everse_v which_o be_v in_o continual_a quantity_n demonstrate_v in_o the_o 4._o 17._o 18._o and_o 19_o proposition_n of_o the_o five_o book_n and_o first_o he_o demonstrate_v converse_v proportion_n in_o this_o manner_n case_n but_o if_o a_o be_v great_a than_o b_o c_o also_o be_v great_a than_o d_z and_o what_o part_n or_o part_n b_o be_v of_o a_o the_o self_n same_o part_n or_o part_n be_v d_z of_o c._n wherefore_o by_o the_o same_o definition_n as_o b_o be_v to_o a_o so_o iâ—_z d_o to_o c_z which_o be_v require_v to_o be_v prove_v proportionality_n divide_v be_v thus_o demonstrate_v suppose_v that_o the_o number_n ab_fw-la be_v to_o the_o number_n b_o as_o the_o number_n cd_o be_v to_o the_o number_n d._n then_o i_o say_v divide_v that_o divide_v also_o as_o a_o be_v to_o b_o so_o be_v c_o to_o d._n for_o for_o that_o as_o ab_fw-la be_v to_o b_o so_o be_v cd_o to_o d_o therefore_o alternate_o by_o the_o 14._o of_o this_o book_n as_o ab_fw-la be_v to_o cd_o so_o be_v b_o to_o d._n wherefore_o by_o the_o 11._o of_o this_o book_n as_o ab_fw-la be_v to_o cd_o so_o be_v a_o to_o c._n wherefore_o as_o b_o be_v to_o d_z so_o be_v a_o to_o c_o and_o for_o that_o as_o a_o be_v to_o c_o so_o be_v b_o to_o d_z therefore_o alternate_o as_o a_o be_v to_o b_o so_o be_v c_o to_o d._n proportionality_n compose_v be_v thus_o demonstrate_v compose_v if_o a_o be_v unto_o b_o as_z c_z be_v to_o d_z then_o shall_v ab_fw-la be_v to_o b_o as_z cd_o be_v to_o d._n for_o alternate_o a_o be_v to_o c_o as_z b_o be_v to_o d._n wherefore_o by_o the_o 13._o of_o this_o book_n as_o ab_fw-la namely_o all_o the_o antecedentes_fw-la be_v to_o cd_o namely_o to_o all_o the_o consequentes_fw-la so_o be_v b_o to_o d_z namely_o one_o of_o the_o antecedentes_fw-la to_o one_o of_o the_o consequentes_fw-la wherefore_o alternate_o as_o ab_fw-la be_v to_o b_o so_o be_v cd_o to_o d._n euerse_a proportionality_n be_v thus_o prove_v proportionality_n supposâ—_n that_o ab_fw-la be_v to_o b_o as_z cd_o be_v to_o d_o then_o shall_v ab_fw-la be_v to_o a_o as_o cd_o be_v to_o c._n for_o alternate_o ab_fw-la be_v to_o cd_o aâ—_z b_o be_v to_o d._n wherefâ—râ—_n by_o the_o 13._o of_o this_o booâ—â—_n aâ—_z be_v â—_o cd_o as_o a_o be_v to_o c._n wherefore_o alternate_o ab_fw-la iâ—_z to_o a_o aâ—_z cd_o iâ—_z to_o c_o whiâ—h_o be_v require_v to_o be_v prove_v ¶_o a_o proportion_n here_o add_v by_o campane_n if_o the_o proportion_n of_o the_o first_o to_o the_o second_o be_v as_o the_o proportion_n of_o the_o three_o to_o the_o foâ—rth_n and_o if_o the_o proportion_n of_o â—he_n five_o to_o the_o second_o be_v as_o the_o proportion_n of_o the_o six_o to_o the_o four_o then_o the_o proportion_n of_o the_o first_o and_o the_o five_o take_v together_o shall_v be_v to_o the_o second_o as_o the_o proportion_n â—f_o the_o three_o and_o the_o six_o take_v together_o to_o the_o four_o and_o after_o the_o same_o manner_n may_v you_o prove_v the_o converse_n of_o this_o proposition_n if_o b_o be_v to_o a_o as_z d_z be_v to_o câ—_n and_o if_o also_o b_o be_v unto_o e_o as_z d_z be_v to_o fletcher_n then_o shall_v b_o be_v to_o ae_n as_o d_z be_v to_o cf._n for_o by_o converse_v proportionality_n prâ—position_n a_o be_v to_o b_o as_z c_z be_v to_o d._n wherefore_o of_o equality_n a_o be_v to_o e_o as_z c_z be_v to_o f._n wherefore_o by_o composition_n a_o and_o e_o be_v to_o e_o as_z c_z and_o f_o be_v to_o f._n wherefore_o converse_o e_o be_v to_o a_o and_o e_o as_z fletcher_n be_v to_o c_o and_o f._n but_o by_o supposition_n b_o be_v to_o e_o as_z d_z be_v to_o f._n wherefore_o again_o by_o proportion_n of_o equality_n b_o be_v to_o a_o and_o e_o as_z d_z be_v to_o c_z and_o f_o demonstration_n which_o be_v require_v to_o be_v prove_v a_o corollary_n by_o this_o also_o it_o be_v manifest_a that_o if_o the_o proportion_n of_o number_n how_o many_o soever_o unto_o the_o first_o be_v as_o the_o proportion_n of_o as_o many_o other_o number_n unto_o the_o second_o campane_n then_o shall_v the_o proportion_n of_o the_o number_n compose_v of_o all_o the_o number_n that_o be_v antecedentes_fw-la to_o the_o first_o be_v to_o the_o first_o as_o the_o number_n compose_v of_o all_o the_o number_n that_o be_v antecedentes_fw-la to_o the_o second_o be_v to_o the_o second_o and_o also_o converse_o if_o the_o proportion_n of_o the_o first_o to_o number_n how_o many_o soever_o be_v as_o the_o proportion_n of_o the_o second_o to_o as_o many_o other_o number_n then_o shall_v the_o proportion_n of_o the_o first_o to_o the_o number_n compose_v of_o all_o the_o number_n that_o be_v consequentes_fw-la to_o itself_o be_v as_o the_o proportion_n of_o the_o second_o to_o the_o number_n compose_v of_o all_o the_o number_n that_o be_v consequence_n to_o itself_o ¶_o the_o 13._o theorem_a the_o 15._o proposition_n if_o unity_n measure_v any_o number_n and_o a_o other_o number_n do_v so_o many_o time_n measure_v a_o other_o number_n unity_n also_o shall_v alternate_o so_o many_o time_n measure_v the_o three_o number_n as_o the_o second_o do_v the_o four_o svppose_v that_o unity_n a_o do_v measure_v the_o number_n bc_o and_o let_v a_o other_o number_n d_o so_o many_o time_n measure_v some_o other_o number_n namely_o ef._n then_o i_o say_v that_o alternate_o unity_n a_o shall_v so_o many_o time_n measure_v the_o number_n d_o as_o the_o number_n bc_o do_v measure_v the_o number_n ef._n coâ—strâ—ctioâ—_n for_o forasmuch_o as_o unity_n a_o do_v so_o many_o time_n measure_v bc_o as_o d_z do_v of_o therefore_o how_o many_o unity_n there_o be_v in_o bc_o so_o many_o number_n be_v there_o in_o of_o equal_a unto_o d._n divide_v i_o say_v bc_o into_o the_o unity_n which_o be_v in_o it_o that_o be_v into_o bg_o gh_o and_o hc_n and_o divide_v likewise_o of_o into_o the_o number_n equal_a unto_o d_o that_o be_v into_o eke_o kl_o and_o lf_a now_o than_o the_o multitude_n of_o these_o bg_o gh_o and_o hc_n be_v equal_a unto_o the_o multitude_n of_o these_o eke_o kl_o lf_n demonstration_n and_o forasmuch_o as_o these_o unity_n bg_o gh_o and_o hc_n be_v equal_a the_o one_o to_o the_o other_o and_o these_o number_n eke_o kl_o &_o lf_a be_v also_o equal_a the_o one_o to_o the_o other_o and_o the_o multitude_n of_o the_o unity_n bg_o gh_o and_o hc_n be_v equal_a unto_o the_o multitude_n of_o the_o number_n eke_o kl_o &_o lf_a therefore_o as_o unity_n bg_o be_v to_o the_o number_n eke_o so_o be_v unity_n gh_o to_o the_o number_n kl_o and_o also_o unity_n hc_n to_o the_o number_n lf_n wherefore_o by_o the_o 12â—_n of_o the_o seventh_n as_o one_o of_o the_o antecedeâ—tâ—â—â—s_n to_o one_o of_o the_o consequentes_fw-la so_o be_v all_o the_o antecedenâ—es_n to_o all_o the_o consequentes_fw-la wherefore_o as_o unity_n bg_o be_v to_o the_o number_n eke_o so_o be_v the_o number_n bc_o to_o the_o number_n ef._n but_o unity_n bg_o be_v equal_a unto_o unity_n a_o and_o the_o number_n eke_o to_o the_o number_n d._n wherefore_o by_o the_o 7._o common_a sentence_n as_o unity_n a_o be_v to_o the_o number_n d_o so_o be_v the_o number_n bc_o to_o the_o number_n ef._n wherefore_o unity_n a_o measure_v the_o number_n d_o so_fw-mi many_o time_n as_o bc_o measure_v of_o by_o the_o 21_o definition_n of_o this_o book_n which_o be_v require_v to_o be_v prove_v ¶_o the_o 14._o theorem_a the_o 16._o proposition_n if_o two_o number_n multiply_v themselves_o the_o one_o into_o the_o other_o produce_v any_o number_n the_o number_n produce_v shall_v be_v equal_a the_o one_o into_o the_o other_o svppose_v that_o there_o be_v two_o number_n a_o and_o b_o and_o let_v a_o multiply_v b_o produce_v c_o and_o let_v b_o multiply_v a_o produce_v d._n thâ—n_o i_o say_v that_o the_o number_n câ—_n equal_a unto_o the_o nâ—mber_n d._n demonstration_n take_v any_o unity_n name_o e._n and_o forasmuch_o as_o a_o multiply_a b_o produce_v c_z therefore_o b_o measure_v c_o by_o the_o unity_n which_o be_v in_o a._n and_o unity_n e_o measure_v the_o number_n a_o by_o those_o unity_n which_o be_v in_o the_o number_n a._n wherefore_o unity_n e_o so_o many_o time_n measure_v a_o as_z b_o measure_v c._o wherefore_o alternate_o by_o the_o 15._o of_o the_o seven_o unity_n e_o measure_v the_o number_n b_o so_o many_o time_n as_o a_o measure_v c._n again_o for_o that_o b_o multiply_v a_o produce_v d_z therefore_o a_o measure_v d_z by_o thâ—_z unity_n which_o be_v in_o b._n and_o unity_n e_o
denomination_n of_o the_o number_n b._n for_o how_o often_o b_o measure_v a_o so_o many_o unity_n let_v there_o be_v in_o c._n and_o let_v d_o be_v unity_n and_o forasmuch_o as_o b_o measure_v a_o by_o those_o unity_n which_o be_v in_o c_o and_o unity_n d_z measure_v c_o by_o those_o unity_n which_o be_v in_o c_o therefore_o unity_n d_z so_o many_o time_n measure_v the_o number_n c_o as_z b_o do_v measure_n a._n demonstration_n wherefore_o alternate_o by_o the_o 15._o of_o the_o seven_o unity_n d_o so_o many_o time_n measure_v b_o as_z c_z do_v measure_n a._n wherefore_o what_o part_n unity_n d_z be_v of_o the_o number_n b_o the_o same_o part_n be_v c_o of_o a._n but_o unity_n d_o be_v a_o part_n of_o b_o have_v his_o denomination_n of_o b._n wherefore_o c_o also_o be_v a_o part_n of_o a_o have_v his_o denomination_n of_o b._n wherefore_o a_o have_v c_z as_o a_o part_n take_v his_o denomination_n of_o b_o which_o be_v require_v to_o be_v prove_v the_o meaning_n of_o this_o proposition_n be_v that_o if_o three_o measure_n any_o number_n that_o number_n have_v a_o three_o part_n and_o if_o four_o measure_n any_o number_n the_o say_a number_n have_v a_o four_o part_n and_o so_o forth_o ¶_o the_o 35._o theorem_a the_o 40._o proposition_n if_o a_o number_n have_v any_o part_n the_o number_n whereof_o the_o part_n take_v his_o denomination_n shall_v measure_v it_o svppose_v that_o the_o number_n a_o have_v a_o part_n namely_o b_o and_o let_v the_o part_n b_o have_v his_o denomination_n of_o the_o number_n c._n then_o i_o say_v that_z c_z measure_v a._n let_v d_o be_v unity_n and_o forasmuch_o as_o b_o be_v a_o part_n of_o a_o have_v his_o denomination_n of_o c_o proposition_n and_o d_z be_v unity_n be_v also_o a_o part_n of_o the_o number_n c_o have_v his_o denomination_n of_o c_o therefore_o what_o part_n unity_n d_z be_v of_o the_o number_n c_o the_o same_o part_n be_v also_o b_o of_o a_o wherefore_o unity_n d_z so_o many_o â—â—â—es_n measure_v the_o number_n c_o demonstration_n as_z b_o measurâ—â—â—_n a._n wherefore_o alternate_o by_o â—â—e_z 15._o â—f_n the_o sâ—â—â—nth_n unity_n d_z so_o many_o â—â—mes_n mâ—â—â—â—reth_v the_o nuâ—â—beâ—_n b_o 〈◊〉_d c_o measâ—â—eth_fw-mi a._n wherefâ—â—â—_n c_o measure_v a_o which_o be_v requiâ—â—d_v to_o bâ—_z prove_v this_o proposition_n be_v the_o converse_n of_o the_o former_a and_o the_o meaning_n thereof_o be_v that_o every_o number_n have_v a_o three_o part_n be_v measure_v of_o three_o and_o have_v a_o four_o part_n be_v measure_v of_o four_o and_o so_o forth_o ¶_o the_o 6._o problem_n the_o 41._o proposition_n to_o find_v out_o the_o least_o number_n that_o contain_v the_o part_n give_v svppose_v that_o the_o part_n give_v be_v a_o b_o c_o namely_o let_v a_o be_v a_o half_a part_n b_o a_o three_o part_n &_o c_o a_fw-fr four_o part_n construction_n now_o it_o be_v require_v to_o find_v out_o the_o least_o number_n which_o contain_v the_o part_n a_o b_o c._n let_v the_o say_a part_n a_o b_o c_o have_v their_o denomination_n of_o the_o number_n d_o e_o f._n and_o take_v by_o the_o 38._o of_o the_o seven_o â—he_z least_o number_n which_o the_o number_n d_o e_o f_o measure_n and_o let_v the_o same_o be_v g._n and_o forasmuch_o as_o the_o number_n d_o e_o f_o measure_v the_o number_n g_o therefore_o the_o number_n g_z have_v payte_n denominate_v of_o the_o number_n d_o e_o f_o by_o the_o 39_o of_o the_o seven_o but_o the_o part_n a_o b_o c_o have_v their_o denomination_n of_o the_o number_n d_o e_o f._n wherefore_o g_z have_v those_o part_n a_o b_o c._n i_o say_v also_o that_o it_o be_v the_o least_o number_n which_o have_v these_o part_n absurdity_n for_o if_o g_z be_v not_o the_o least_o number_n which_o contain_v those_o part_n a_o b_o c_o then_o let_v there_o be_v some_o number_n less_o than_o g_o which_o contain_v the_o say_a part_n a_o b_o c._n and_o suppose_v the_o same_o to_o be_v the_o number_n h._n and_o forasmuch_o as_o h_o have_v the_o say_a part_n a_o b_o c_o therefore_o the_o number_n that_o the_o part_n a_o b_o c_o take_v their_o denomination_n of_o shall_v measure_v h_o by_o the_o 40._o of_o the_o seven_o but_o the_o number_n wheroâ—_n the_o part_n a_o b_o c_o take_v their_o denomination_n of_o be_v d_z e_o f._n wherefore_o the_o number_n d_o e_o f_o measure_n thâ—_z number_n h_o which_o be_v less_o than_o g_o which_o be_v impossible_a for_o g_z be_v suppose_v to_o be_v tâ—â—_n lâ—â—sâ—_n number_n that_o the_o number_n d_o e_o f_o do_v measure_n wherefore_o there_o be_v no_o number_n less_o than_o g_o which_o contain_v these_o part_n a_o b_o c_o which_o be_v require_v to_o be_v do_v corrolary_n campane_n hereby_o it_o be_v manifest_a that_o if_o there_o be_v take_v the_o least_o number_n that_o number_n how_o many_o soever_o do_v measure_n the_o say_a number_n shall_v be_v the_o least_o which_o have_v the_o part_n denominate_v of_o the_o say_a number_n how_o many_o soever_o campane_n after_o he_o have_v teach_v to_o find_v out_o the_o first_o lest_o number_n that_o contain_v the_o part_n give_v teach_v also_o to_o find_v out_o the_o second_o lest_o number_n infinite_o that_o be_v which_o except_o the_o least_o of_o all_o be_v less_o than_o all_o other_o and_o also_o the_o three_o least_o and_o the_o four_o etc_n etc_n the_o second_o be_v find_v out_o by_o double_v the_o number_n g._n for_o the_o number_n which_o measure_v the_o number_n gâ—hall_v â—hall_z also_o measure_v the_o double_a thereof_o by_o the_o 5._o commoâ—_n sentence_n of_o the_o seven_o but_o there_o can_v be_v give_v a_o number_n great_a than_o the_o number_n g_o &_o less_o than_o the_o double_a thereof_o who_o the_o part_n give_v shall_v â—â—asureâ—_n for_o forasmuch_o as_o the_o part_n give_v do_v â—easurâ—_n the_o whole_a namely_o which_o be_v less_o than_o the_o double_a and_o they_o also_o measure_v the_o part_n take_v away_o namely_o the_o number_n g_o they_o shall_v also_o measure_v the_o residue_n namely_o a_o number_n less_o than_o g_o which_o be_v prove_v to_o be_v the_o least_o number_n that_o they_o do_v measure_n which_o be_v impossibleâ—_n wherefore_o the_o second_o number_n which_o the_o say_v part_n give_v do_v measureâ—_n must_v exceed_v g_z needs_o reach_v to_o the_o double_a of_o g_o and_o the_o three_o to_o the_o treble_a and_o the_o four_o to_o the_o quadruple_a and_o so_o infinite_o for_o those_o part_n can_v never_o measure_v any_o number_n less_o than_o the_o number_n g._n by_o this_o proposition_n also_o it_o be_v easy_a to_o find_v out_o the_o least_o number_n contain_v the_o part_n give_v of_o part_n as_o if_o we_o will_v find_v out_o the_o least_o number_n which_o contain_v one_o three_o 〈◊〉_d â—â—_o a_o halâ—e_a part_n part_n and_o one_o four_o part_n of_o a_o three_o part_n reduce_v the_o say_v dâ—â—ers_n fraction_n into_o simple_a fraction_n by_o the_o common_a 〈◊〉_d of_o reduce_v of_o frâ—â—â—ions_n namely_o the_o 〈◊〉_d of_o a_o halâ—e_n into_o a_o 〈◊〉_d part_n of_o a_o wholey_n and_o â—he_n four_o of_o thiâ—d_o into_o a_o twelve_o part_n of_o anâ—_z whole_a and_o then_o by_o this_o problem_n search_v out_o the_o least_o number_n which_o contain_v a_o sixâ—_o part_n and_o a_o twelve_o part_n and_o so_o have_v you_o do_v the_o end_n of_o the_o seven_o book_n of_o euclides_n element_n ¶_o the_o eighthe_a book_n of_o euclides_n element_n after_o that_o euclid_n have_v in_o the_o seven_o book_n entreat_v of_o the_o propriety_n of_o number_n in_o general_a and_o of_o certain_a kind_n thereof_o more_o special_o and_o of_o prime_n and_o compose_a number_n with_o other_o now_o in_o this_o eight_o book_n he_o prosecute_v far_a and_o find_v out_o and_o demonstrate_v the_o property_n and_o passion_n of_o certain_a other_o kind_n of_o number_n book_n as_o of_o the_o least_o number_n in_o proportion_n and_o how_o such_o may_v be_v find_v out_o infinite_o in_o whatsoever_o proportion_n which_o thing_n be_v both_o delectable_a and_o to_o great_a use_n also_o here_o be_v entreat_v of_o plain_a number_n and_o solid_a and_o of_o their_o side_n and_o proportion_n of_o they_o likewise_o of_o the_o passion_n of_o number_n square_a and_o cube_fw-la and_o of_o the_o nature_n and_o condition_n of_o their_o side_n and_o of_o the_o mean_a proportional_a number_n of_o plain_n solid_a square_a and_o cube_fw-la number_n with_o many_o other_o thing_n very_o requisite_a and_o necessary_a to_o be_v know_v ¶_o the_o first_o theorem_a the_o first_o proposition_n if_o there_o be_v number_n in_o continual_a proportion_n howmanysoever_o and_o if_o their_o extreme_n be_v prime_a the_o one_o to_o the_o other_o they_o be_v the_o least_o of_o all_o number_n that_o have_v one_o and_o the_o same_o proportion_n with_o they_o svppose_v that_o the_o number_n
number_n be_v make_v of_o unity_n and_o therefore_o can_v a_o point_n be_v a_o common_a part_n of_o all_o line_n and_o measure_v they_o as_o unity_n be_v a_o common_a part_n of_o all_o number_n and_o measure_v they_o unity_n take_v certain_a time_n make_v any_o number_n for_o there_o be_v not_o in_o any_o number_n infinite_a unity_n but_o a_o point_n take_v certain_a time_n yea_o as_o often_o as_o you_o listen_v never_o make_v any_o line_n for_o that_o in_o every_o line_n there_o be_v infinite_a point_n wherefore_o line_n figure_n and_o body_n in_o geometry_n be_v oftentimes_o incommensurable_a and_o irrational_a now_o which_o be_v rational_a and_o which_o irrational_a which_o commensurable_a and_o which_o incommensurable_a how_o many_o and_o how_o sundry_a sort_n and_o kind_n there_o be_v of_o they_o what_o be_v their_o nature_n passion_n and_o property_n do_v euclid_n most_o manifest_o show_v in_o this_o book_n and_o demonstrate_v they_o most_o exact_o this_o ten_o book_n have_v ever_o hitherto_o of_o all_o man_n and_o be_v yet_o think_v &_o account_v to_o be_v the_o hard_a book_n to_o understand_v of_o all_o the_o book_n of_o euclid_n euclid_n which_o common_a receive_a opinion_n have_v cause_v many_o to_o shrink_v and_o have_v as_o it_o be_v deter_v they_o from_o the_o handle_n and_o treaty_n thereof_o there_o have_v be_v in_o deed_n in_o time_n past_a and_o be_v present_o in_o these_o our_o day_n many_o which_o have_v deal_v and_o have_v take_v great_a and_o good_a diligence_n in_o comment_v amend_v and_o restore_a of_o the_o six_o first_o book_n of_o euclid_n and_o there_o have_v stay_v themselves_o and_o go_v no_o far_o be_v deter_v and_o make_v afraid_a as_o it_o seem_v by_o the_o opinion_n of_o the_o hardness_n of_o this_o book_n to_o pass_v forth_o to_o the_o book_n follow_v truth_n it_o be_v that_o this_o book_n have_v in_o it_o somewhat_o a_o other_o &_o strange_a manner_n of_o matter_n entreat_v of_o former_a them_z the_o other_o book_n before_o have_v and_o the_o demonstration_n also_o thereof_o &_o the_o order_n seem_v likewise_o at_o the_o first_o somewhat_o strange_a and_o unaccustomed_a which_o thing_n may_v seem_v also_o to_o cause_v the_o obscurity_n thereof_o and_o to_o fear_v away_o many_o from_o the_o read_n and_o diligent_a study_n of_o the_o same_o so_o much_o that_o many_o of_o the_o well_o learned_a have_v much_o complain_v of_o the_o darkness_n and_o difficulty_n thereof_o and_o have_v think_v it_o a_o very_a hard_a thing_n and_o in_o manner_n impossible_a to_o attain_v to_o the_o right_n and_o full_a understanding_n of_o this_o book_n algebra_n without_o the_o aid_n and_o help_v of_o some_o other_o knowledge_n and_o learning_n and_o chief_o without_o the_o knowledge_n of_o that_o more_o secret_a and_o subtle_a part_n of_o arithmetic_n common_o call_v algebra_n which_o undoubted_o first_o well_o have_v and_o know_v will_v geve_v great_a light_n thereunto_o yet_o certain_o may_v this_o book_n very_o well_o be_v enter_v into_o and_o full_o understand_v without_o any_o strange_a help_n or_o succour_n only_o by_o diligent_a observation_n of_o the_o order_n and_o course_n of_o euclides_n write_n so_o that_o he_o which_o diligent_o have_v peruse_v and_o full_o understand_v the_o 9_o book_n go_v before_o and_o mark_v also_o earnest_o the_o principle_n and_o definition_n of_o this_o â—enth_fw-mi book_n he_o shall_v well_o perceive_v that_o euclid_n be_v of_o himself_o a_o sufficient_a teacher_n and_o instructor_n understand_v and_o need_v not_o the_o help_n of_o any_o other_o and_o shall_v soon_o see_v that_o this_o ten_o book_n be_v not_o of_o such_o hardness_n and_o obscurity_n as_o it_o have_v be_v hitherto_o think_v yea_o i_o doubt_v not_o but_o that_o by_o the_o travel_n and_o industry_n take_v in_o this_o translation_n and_o by_o addition_n and_o emendation_n get_v of_o other_o there_o shall_v appear_v in_o it_o no_o hardness_n at_o all_o but_o shall_v be_v as_o easy_a as_o the_o rest_n of_o his_o book_n be_v definition_n definition_n 1_o magnitude_n commensurable_a be_v suchwhich_v one_o and_o the_o self_n same_o measure_n do_v measure_n first_o he_o show_v what_o magnitude_n be_v commensurable_a one_o to_o a_o other_o to_o the_o better_a and_o more_o clear_a understanding_n of_o this_o definition_n note_v that_o that_o measure_v whereby_o any_o magnitude_n be_v measure_v be_v less_o than_o the_o magnitude_n which_o it_o measure_v or_o at_o least_o equal_a unto_o it_o for_o the_o great_a can_v by_o no_o mean_n measure_v the_o less_o far_o it_o behove_v that_o that_o measure_n if_o it_o be_v equal_a to_o that_o which_o be_v measure_v take_v once_o make_v the_o magnitude_n which_o be_v measure_v if_o it_o be_v less_o than_o oftentimes_o take_v and_o repeat_v it_o must_v precise_o render_v and_o make_v the_o magnitude_n which_o it_o measure_v which_o thing_n in_o number_n be_v easy_o see_v for_o that_o as_o be_v before_o say_v all_o number_n be_v commensurable_a one_o to_o a_o other_o and_o although_o euclid_n in_o this_o definition_n comprehend_v purpose_o only_o magnitude_n which_o be_v continual_a quantity_n as_o be_v line_n superficiece_n and_o body_n yet_o undoubted_o the_o explication_n of_o this_o and_o such_o like_a place_n be_v apt_o to_o be_v seek_v of_o number_n as_o well_o rational_a as_o irrational_a for_o that_o all_o quantity_n commensurable_a have_v that_o proportion_n the_o one_o to_o the_o other_o which_o number_n have_v to_o number_n in_o number_n therefore_o 9_o and_o 12_o be_v commensurable_a because_o there_o be_v one_o common_a measure_n which_o measure_v they_o both_o namely_o the_o number_n 3_n first_o it_o measure_v 12_o for_o it_o be_v less_o than_o 12._o and_o be_v take_v certain_a time_n namely_o 4_o time_n it_o make_v exact_o 12_o 3_o time_n 4_o be_v 12_o it_o also_o measure_v 9_o for_o it_o be_v less_o than_o 9_o and_o also_o take_v certain_a time_n namely_o 3_o time_n it_o make_v precise_o 9_o 3_o time_n 3_o be_v 9_o likewise_o be_v it_o in_o magnitude_n if_o one_o magnitude_n measure_v two_o other_o magnitude_n those_o two_o magnitude_n so_o measure_v be_v say_v to_o be_v commensurable_a as_o for_o example_n if_o the_o line_n c_o be_v double_v do_v make_v the_o line_n b_o and_o the_o same_o line_n c_o triple_a do_v make_v the_o line_n a_o then_o be_v the_o two_o line_n a_o and_o b_o line_n or_o magnitude_n commensurable_a for_o that_o one_o measure_n namely_o the_o line_n c_o measure_v they_o both_o first_o the_o line_n c_o be_v less_o they_o the_o line_n a_o and_o alsolesse_v they_o the_o line_n b_o also_o the_o line_n c_o take_v or_o repeat_v certain_a time_n namely_o 3_o time_n make_v precise_o the_o line_n a_o and_o the_o same_o line_n c_o take_v also_o certain_a time_n namely_o two_o time_n make_v precise_o the_o line_n b._n so_o that_o the_o line_n c_o be_v a_o common_a measure_n to_o they_o both_o and_o do_v measure_v they_o both_o and_o therefore_o be_v the_o two_o line_n a_o and_o b_o line_n commensurable_a and_o so_o imagine_v you_o of_o magnitude_n of_o other_o kind_n as_o of_o superficial_a figure_n and_o also_o of_o body_n 2_o incommensurable_a magnitude_n be_v such_o which_o no_o one_o common_a measure_n do_v measure_n definition_n this_o definition_n nead_v no_o explanation_n at_o all_o other_o it_o be_v easy_o understand_v by_o the_o definition_n go_v before_o of_o line_n commensurable_a for_o contrary_n be_v make_v manifest_a by_o compare_v of_o the_o one_o to_o the_o other_o as_o if_o the_o line_n c_o or_o any_o other_o line_n oftentimes_o iterate_v do_v not_o render_v precise_o the_o line_n a_o nor_o the_o line_n b_o them_z be_v the_o line_n a_o and_o b_o incommensurable_a also_o if_o the_o line_n c_o or_o any_o other_o line_n certain_a time_n repeat_v do_v exact_o render_v the_o line_n a_o and_o do_v not_o measure_v the_o line_n b_o or_o if_o it_o measure_v the_o line_n b_o and_o measure_v not_o also_o the_o line_n a_o the_o line_n a_o and_o b_o be_v yet_o line_n incommensurable_a &_o so_o of_o other_o magnitude_n as_o of_o superficiece_n and_o body_n 3_o right_a line_n commensurable_a in_o power_n be_v such_o who_o square_n one_o and_o the_o self_n same_o superficies_n area_n or_o plat_n do_v measure_n definition_n to_o the_o declaration_n of_o this_o definition_n we_o must_v first_o call_v to_o mind_n what_o be_v understand_v &_o mean_v by_o the_o power_n of_o a_o line_n which_o as_o we_o have_v before_o in_o the_o former_a book_n note_v be_v nothing_o else_o but_o the_o square_a thereof_o or_o any_o other_o plain_a figure_n equal_a to_o the_o square_a thereof_o and_o so_o great_a power_n &_o hability_n â—s_v a_o line_n say_v to_o have_v as_o be_v the_o quantity_n of_o the_o square_n which_o it_o be_v able_a to_o describe_v or_o a_o figure_n superficial_a equal_a to_o the_o square_a
thereof_o be_v 〈◊〉_d this_o iâ—_z also_o to_o be_v note_v that_o of_o line_n some_o be_v commensurable_a in_o length_n the_o one_o to_o the_o other_o and_o some_o be_v commensurable_a the_o one_o to_o the_o other_o in_o power_n of_o line_n commensurable_a in_o length_n the_o one_o to_o the_o other_o be_v give_v a_o example_n in_o the_o declaration_n of_o the_o first_o definition_n namely_o the_o line_n a_o and_o b_o which_o be_v commensurable_a in_o length_n one_o and_o the_o self_n measure_n namely_o the_o line_n c_o measure_v the_o length_n of_o either_o of_o they_o of_o the_o other_o kind_n be_v give_v this_o definition_n here_o set_v for_o the_o open_n of_o which_o take_v this_o example_n let_v there_o be_v a_o certain_a line_n namely_o the_o line_n bc_o and_o let_v the_o square_n of_o that_o line_n be_v the_o square_n bcde_v suppose_v also_o a_o other_o line_n namely_o the_o line_n fh_o &_o let_v the_o square_v thereof_o be_v the_o square_a fhik_n and_o let_v a_o certain_a superficies_n namely_o the_o superficies_n a_o measure_v the_o square_n bcde_v take_v 16._o time_n which_o be_v the_o number_n of_o the_o little_a area_n square_n plait_n or_o superficiece_n contain_v and_o describe_v within_o the_o say_v square_n each_o of_o which_o be_v equal_a to_o the_o superficie_fw-la a._n again_o let_v the_o same_o superficies_n a_o measure_n the_o square_a fhik_n 9_o time_n take_v according_a to_o the_o number_n of_o the_o fieldâ—s_n or_o superficiece_n contain_v and_o describe_v in_o the_o same_o you_o see_v they_o that_o one_o and_o the_o self_n same_o superficies_n namely_o the_o superficies_n a_o be_v a_o common_a measure_n to_o both_o these_o square_n and_o by_o certain_a repetition_n thereof_o measure_v they_o both_o wherefore_o the_o two_o line_n bc_o and_o fh_o which_o be_v the_o side_n or_o line_n produce_v these_o square_n and_o who_o power_n these_o square_n be_v be_v by_o this_o definition_n line_n commensurable_a in_o power_n 4_o line_n incommensurable_a be_v such_o who_o square_n no_o one_o plat_n or_o superficies_n do_v measure_n definition_n this_o definition_n be_v easy_a to_o be_v understand_v by_o that_o which_o be_v say_v in_o the_o definition_n last_o set_v before_o this_o and_o nead_v no_o far_a declaration_n and_o thereof_o take_v this_o example_n if_o neither_o the_o superficies_n a_o nor_o any_o other_o superficies_n do_v measure_v the_o two_o square_n b_o cde_o and_o fhik_n or_o if_o it_o measure_v the_o one_o â—â—rely_o bcde_v and_o not_o the_o other_o fhik_n or_o if_o it_o measure_v the_o square_a fhik_n and_o not_o the_o square_n bcde_v the_o two_o line_n bc_o and_o fh_o be_v in_o power_n incommensurable_a and_o therefore_o also_o incommensurable_a in_o length_n for_o whatsoever_o line_n be_v incommensurable_a in_o power_n the_o same_o be_v also_o incommensurable_a in_o length_n as_o shall_v afterward_o in_o the_o 9_o proposition_n of_o this_o book_n be_v prove_v and_o therefore_o such_o line_n be_v here_o define_v to_o be_v absolute_o incommensurable_a these_o thing_n thus_o stand_v it_o may_v easy_o appear_v that_o if_o a_o line_n be_v assign_v and_o lay_v before_o we_o there_o may_v be_v innumerable_a other_o line_n commensurable_a unto_o it_o and_o other_o incommensurable_a unto_o it_o of_o commensurable_a line_n some_o be_v commensurable_a in_o length_n and_o power_n and_o some_o in_o power_n only_o 5_o and_o that_o right_a line_n so_o set_v forth_o be_v call_v a_o rational_a line_n thus_o may_v you_o see_v how_o to_o the_o suppose_a line_n first_o set_v may_v be_v compare_v infinite_a line_n line_n some_o commensurable_a both_o in_o length_n &_o power_n and_o some_o commensurable_a in_o power_n only_o and_o incommensurable_a in_o length_n and_o some_o incommensurable_a both_o in_o power_n &_o in_o length_n and_o this_o first_o line_n so_o set_v whereunto_o and_o to_o who_o square_n the_o other_o line_n and_o their_o square_n be_v compare_v be_v call_v a_o rational_a line_n common_o of_o the_o most_o part_n of_o writer_n line_n but_o some_o there_o be_v which_o mislike_n that_o it_o shall_v be_v call_v a_o rational_a line_n &_o that_o not_o without_o just_a cause_n in_o the_o greek_a copy_n it_o be_v call_v 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d rete_fw-la which_o signify_v a_o thing_n that_o may_v be_v speak_v &_o express_v by_o word_n a_o thing_n certain_a grant_v and_o appoint_v wherefore_o flussates_n a_o man_n which_o bestow_v great_a travel_n and_o diligence_n in_o restore_v of_o these_o element_n of_o euclid_n leave_v this_o word_n rational_a call_v this_o line_n suppose_v and_o first_o set_v a_o line_n certain_a because_o the_o part_n thereof_o into_o which_o it_o be_v divide_v be_v certain_a certain_a and_o know_v and_o may_v be_v express_v by_o voice_n and_o also_o be_v count_v by_o number_n other_o line_n be_v to_o this_o line_n incommensurable_a who_o part_n be_v not_o distinct_o know_v but_o be_v uncertain_a nor_o can_v be_v express_v by_o name_n nor_o assign_v by_o number_n which_o be_v of_o other_o man_n call_v irrational_a he_o call_v uncertain_a and_o surd_a line_n petrus_n montaureus_n although_o he_o do_v not_o very_o well_o like_a of_o the_o name_n yet_o he_o alter_v it_o not_o but_o use_v it_o in_o all_o his_o book_n likewise_o will_v we_o do_v here_o for_o that_o the_o word_n have_v be_v and_o be_v so_o universal_o receive_v and_o therefore_o will_v we_o use_v the_o same_o name_n and_o call_v it_o a_o rational_a line_n for_o it_o be_v not_o so_o great_a a_o matter_n what_o name_n we_o geve_v to_o thing_n so_o that_o we_o full_o understand_v the_o thing_n which_o the_o name_n signify_v this_o rational_a line_n thus_o here_o define_v be_v the_o ground_n and_o foundation_n of_o all_o the_o proposition_n almost_o of_o this_o whole_a ten_o book_n book_n and_o chief_o from_o the_o ten_o proposition_n forward_o so_o that_o unless_o you_o first_o place_n this_o rational_a line_n and_o have_v a_o special_a and_o continual_a regard_n unto_o it_o before_o you_o begin_v any_o demonstration_n you_o shall_v not_o easy_o understand_v it_o for_o it_o be_v as_o it_o be_v the_o touch_n and_o trial_n of_o all_o other_o line_n by_o which_o it_o be_v know_v whether_o any_o of_o they_o be_v rational_a or_o not_o note_n and_o this_o may_v be_v call_v the_o first_o rational_a line_n purpose_n the_o line_n rational_a of_o purpose_n or_o a_o rational_a line_n set_v in_o the_o first_o place_n and_o so_o make_v distinct_a and_o sever_v from_o other_o rational_a line_n of_o which_o shall_v be_v speak_v afterward_o and_o this_o must_v you_o well_o commit_v to_o memory_n definition_n 6_o line_n which_o be_v commensurable_a to_o this_o line_n whether_o in_o length_n and_o power_n or_o in_o power_n only_o be_v also_o call_v rational_a this_o definition_n need_v no_o declaration_n at_o all_o but_o be_v easy_o perceive_v if_o the_o first_o definition_n be_v remember_v which_o show_v what_o magnitude_n be_v commensurable_a and_o the_o three_o which_o show_v what_o line_n be_v commensurable_a in_o power_n here_o notâ—_n how_o apt_o &_o natural_o euclid_n in_o this_o place_n use_v these_o word_n commensurable_a either_o in_o length_n and_o power_n or_o in_o power_n only_o because_o that_o all_o line_n which_o be_v commensurable_a in_o length_n be_v also_o commensurable_a in_o pour_v when_o he_o speak_v of_o line_n commensurable_a in_o length_n he_o ever_o add_v and_o in_o power_n but_o when_o he_o speak_v of_o line_n commensurable_a in_o power_n he_o add_v this_o word_n only_o and_o add_v not_o this_o word_n in_o length_n as_o he_o in_o the_o other_o add_v this_o word_n in_o power_n for_o not_o all_o line_n which_o be_v commensurable_a in_o power_n be_v straight_o way_n commensurable_a also_o in_o length_n of_o this_o definition_n take_v this_o example_n let_v the_o first_o line_n rational_a of_o purpose_n which_o be_v suppose_v and_o lay_v forth_o who_o part_n be_v certain_a &_o know_v and_o may_v be_v express_v name_v and_o number_v be_v ab_fw-la the_o quadrate_n whereof_o let_v be_v abcd_o then_o suppose_v again_o a_o other_o line_n namely_o the_o line_n of_o which_o let_v be_v commensurable_a both_o in_o length_n and_o in_o power_n to_o the_o first_o rational_a line_n that_o be_v as_o before_o be_v teach_v let_v one_o line_n measure_v the_o length_n of_o each_o line_n and_o also_o lâ—t_v one_o superficies_n measure_v the_o two_o square_n of_o the_o say_v two_o line_n as_o here_o in_o the_o example_n be_v suppose_v and_o also_o appear_v to_o the_o eye_n then_o be_v the_o line_n e_o fletcher_n also_o a_o rational_a line_n moreover_o if_o the_o line_n of_o be_v commensurable_a in_o power_n only_o to_o the_o rational_a line_n ab_fw-la first_o set_v and_o suppose_v so_o that_o no_o one_o line_n do_v measure_v the_o two_o line_n ab_fw-la and_o of_o as_o in_o example_n yâ—_n see_v to_o be_v for_o
that_o the_o line_n of_o be_v make_v equal_a to_o the_o line_n ad_fw-la which_o be_v the_o diameter_n of_o the_o square_n abcd_o of_o which_o square_v the_o line_n ab_fw-la be_v a_o side_n it_o be_v certain_a that_o the_o â—ide_v of_o a_o square_n be_v incommensurable_a in_o length_n to_o the_o diameter_n of_o the_o same_o square_n if_o there_o be_v yet_o find_v any_o one_o superficies_n which_o measure_v the_o two_o square_n abcd_o and_o efgh_a as_o here_o do_v the_o triangle_n abdella_n or_o the_o triangle_n acd_v note_v in_o the_o square_n abcd_o or_o any_o of_o the_o four_o triangle_n note_v in_o the_o square_a efgh_o as_o appear_v somewhat_o more_o manifest_o in_o the_o second_o example_n in_o the_o declaration_n of_o the_o last_o definition_n go_v before_o the_o line_n of_o be_v also_o a_o rational_a line_n note_v that_o these_o line_n which_o here_o be_v call_v rational_a line_n be_v not_o rational_a line_n of_o purpose_n or_o by_o supposition_n as_o be_v the_o first_o rational_a line_n but_o be_v rational_a only_o by_o reason_n of_o relation_n and_o comparison_n which_o they_o have_v unto_o it_o because_o they_o be_v commensurable_a unto_o it_o either_o in_o length_n and_o power_n or_o in_o power_n only_o far_o here_o be_v to_o be_v note_v that_o these_o word_n length_n and_o power_n and_o power_n only_o be_v join_v only_o with_o these_o wordeâ—_n commensurable_a or_o incommensurable_a and_o be_v never_o join_v with_o these_o word_n rational_a or_o irrational_a so_o that_o no_o line_n can_v be_v call_v rational_a in_o length_n or_o in_o power_n nor_o like_o wise_a can_v they_o be_v call_v irrational_a in_o length_n or_o in_o power_n wherein_o undoubted_o campanus_n be_v deceive_v book_n who_o use_v those_o word_n &_o speech_n indifferent_o cause_v &_o bring_v in_o great_a obscurity_n to_o the_o proposition_n and_o demonstration_n of_o this_o book_n which_o he_o shall_v easy_o see_v which_o mark_v with_o diligence_n the_o demonstration_n of_o campanus_n in_o this_o book_n 7_o line_n which_o be_v incommensurable_a to_o the_o rational_a line_n be_v call_v irrational_a definition_n by_o line_n incommensurable_a to_o the_o rational_a line_n suppose_v in_o this_o place_n he_o understand_v such_o as_o be_v incommensurable_a unto_o it_o both_o in_o length_n and_o in_o power_n for_o there_o be_v no_o line_n incommensurable_a in_o power_n only_o for_o it_o can_v be_v that_o any_o line_n shall_v so_o be_v incommensurable_a in_o power_n only_o that_o they_o be_v not_o also_o incommensurable_a in_o length_n what_o so_o ever_o line_n be_v incommensurable_a in_o power_n the_o same_o be_v also_o incommensurable_a in_o length_n neither_o can_v euclid_n here_o in_o this_o place_n mean_a line_n incommensurable_a in_o length_n only_o for_o in_o the_o definition_n before_o he_o call_v they_o rational_a line_n neither_o may_v they_o be_v place_v amongst_o irrational_a line_n wherefore_o it_o remain_v that_o in_o this_o diffintion_n he_o speak_v only_o of_o those_o line_n which_o be_v incommensurable_a to_o the_o rational_a line_n first_o give_v and_o suppose_v both_o in_o length_n and_o in_o power_n which_o by_o all_o mean_n be_v incommensurable_a to_o the_o rational_a line_n &_o therefore_o most_o apt_o be_v they_o call_v irrational_a line_n this_o definition_n be_v easy_a to_o be_v understand_v by_o that_o which_o have_v be_v say_v before_o yet_o for_o the_o more_o plainness_n see_v this_o example_n let_v the_o â—â—rst_a rational_a line_n suppose_v be_v the_o line_n ab_fw-la who_o square_a or_o quadrate_n let_v be_v abcd._n and_o let_v there_o be_v give_v a_o other_o line_n of_o which_o lâ—t_v be_v to_o the_o rational_a line_n incommensurable_a in_o length_n and_o power_n so_o that_o let_v no_o one_o line_n measure_v the_o length_n of_o the_o two_o line_n ab_fw-la and_o of_o and_o let_v the_o square_n of_o the_o line_n of_o be_v efgh_a now_o if_o also_o there_o be_v no_o one_o superficies_n which_o measure_v the_o two_o square_n abcd_o and_o efgh_a as_o be_v suppose_v to_o be_v in_o this_o example_n they_o be_v the_o line_n of_o a_o irrational_a line_n which_o word_n irrational_a as_o before_o do_v this_o word_n rational_a mislike_v many_o learned_a in_o this_o knowledge_n of_o geometry_n flussates_n uncertain_a as_o he_o leave_v the_o word_n rational_a and_o in_o stead_n thereof_o use_v this_o word_n certain_a so_o here_o he_o leave_v the_o word_n irrational_a and_o use_v in_o place_n thereof_o this_o word_n uncertain_a and_o ever_o name_v these_o line_n uncertain_a line_n petrus_n montaureus_n also_o mislike_v the_o word_n irrational_a will_v rather_o have_v they_o to_o be_v call_v surd_a line_n yet_o because_o this_o word_n irrational_a have_v ever_o by_o custom_n and_o long_a use_n so_o general_o be_v receive_n he_o use_v continual_o the_o same_o in_o greek_a such_o line_n be_v call_v 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d alogoi_n which_o signify_v nameless_a unspeakable_a uncertain_a in_o determinate_a line_n and_o with_o out_o proportion_n not_o that_o these_o irrational_a line_n have_v no_o proportion_n at_o all_o either_o to_o the_o first_o rational_a line_n or_o between_o themselves_o but_o be_v so_o name_v for_o that_o their_o proportion_n to_o the_o rational_a line_n can_v be_v express_v in_o number_n that_o be_v undoubted_o very_o untrue_a which_o many_o write_v that_o their_o proportion_n be_v unknown_a both_o to_o we_o and_o to_o nature_n be_v it_o not_o think_v you_o a_o thing_n very_o absurd_a to_o say_v that_o there_o be_v any_o thing_n in_o nature_n and_o produce_v by_o nature_n to_o be_v hide_v from_o nature_n and_o not_o to_o be_v know_v of_o nature_n it_o can_v not_o be_v say_v that_o their_o proportion_n be_v utter_o hide_v and_o unknown_a to_o we_o much_o less_o unto_o nature_n although_o we_o can_v geve_v they_o their_o name_n and_o distinct_o express_v they_o by_o number_n otherwise_o shall_v euclid_n have_v take_v all_o this_o travel_n and_o wonderful_a diligence_n bestow_v in_o this_o bookeâ—_n in_o vain_a and_o to_o no_o useâ—_n in_o which_o he_o do_v nothing_o ellâ—_n but_o teach_v the_o propriety_n and_o passion_n of_o these_o irrational_a linesâ—_n and_o show_v the_o proportion_n which_o they_o have_v the_o one_o to_o the_o other_o here_o be_v also_o to_o be_v note_v which_o thing_n also_o tartalea_n have_v before_o diligent_o notedâ—_n that_o campanus_n and_o many_o other_o writer_n of_o geometryâ—_n over_o much_o â—â—â—ed_a and_o be_v deceive_v in_o that_o they_o write_v and_o teach_v that_o all_o these_o line_n who_o square_n be_v not_o signify_v and_o may_v be_v express_v by_o a_o square_a number_n although_o they_o may_v by_o any_o other_o number_n as_o by_o 11._o 12._o 14._o and_o such_o other_o not_o square_a number_n be_v irrational_a line_n which_o be_v manifest_o repugnant_a to_o the_o ground_n and_o principle_n of_o euclid_n who_o will_v that_o all_o line_n which_o be_v commensurable_a to_o the_o rational_a line_n whether_o it_o be_v in_o length_n and_o power_n or_o in_o power_n only_o shall_v be_v rational_a undoubted_o this_o have_v be_v one_o of_o the_o chief_a and_o great_a cause_n of_o the_o wonderful_a confusion_n and_o darkness_n of_o this_o book_n book_n which_o so_o have_v toss_v and_o turmoil_v the_o wit_n of_o all_o both_o writer_n and_o reader_n master_n and_o scholar_n and_o so_o overwhelm_v they_o that_o they_o can_v not_o with_o out_o infinite_a travel_n and_o sweat_v attain_v to_o the_o truth_n and_o perfect_a understanding_n thereof_o definition_n 8_o the_o square_n which_o be_v describe_v of_o the_o rational_a right_a line_n suppose_v be_v rational_a until_o this_o definition_n have_v euclid_n set_v forth_o the_o nature_n and_o propriety_n of_o the_o first_o kind_n of_o magnitude_n namely_o of_o line_n how_o they_o be_v rational_a or_o irrational_a now_o he_o begin_v to_o â—hew_v how_o the_o second_o kind_n of_o magnitude_n namely_o superficies_n be_v one_o to_o the_o other_o rational_a or_o irrational_a this_o definition_n be_v very_a plain_n suppose_v the_o line_n ab_fw-la to_o be_v the_o rational_a line_n have_v his_o part_n and_o division_n certain_o know_v the_o square_a of_o which_o line_n let_v be_v the_o square_a abcd._n now_o because_o it_o be_v the_o square_a of_o the_o rational_a line_n ab_fw-la it_o be_v also_o call_v rational_a and_o as_o the_o line_n ab_fw-la be_v the_o first_o rational_a line_n unto_o which_o other_o line_n compare_v be_v count_v rational_a or_o irrational_a so_o be_v the_o quadrat_fw-la or_o square_v thereof_o the_o â—irst_n rational_a superficies_n unto_o which_o all_o other_o square_n or_o figure_n compare_v be_v count_v and_o name_v rational_a or_o irrational_a 9_o such_o which_o be_v commensurable_a unto_o it_o be_v rational_a definition_n in_o this_o definition_n where_o it_o be_v say_v such_o as_o be_v commensurable_a to_o the_o square_n of_o the_o rational_a line_n be_v not_o understand_v only_o other_o square_n or_o
quadrates_n but_o all_o other_o kind_n of_o rectiline_a figure_n plain_a plait_n &_o superficiese_n what_o so_o ever_o so_o that_o if_o any_o such_o figure_n be_v commensurable_a unto_o that_o rational_a squareâ—_n it_o be_v also_o rational_a as_o suppose_v that_o the_o square_a of_o the_o rational_a line_n which_o be_v also_o rational_a be_v abcd_o suppose_v 〈◊〉_d so_o some_o other_o square_a as_o the_o square_a efgh_o to_o be_v commensurable_a to_o the_o same_o they_o be_v the_o square_a efgh_o also_o rational_a so_o also_o if_o the_o rectiline_a figure_n klmn_n which_o be_v a_o figure_n on_o the_o one_o side_n long_o be_v commensurable_a unto_o the_o say_a square_a as_o be_v suppose_v in_o this_o exampleâ—_n it_o be_v also_o a_o rational_a superficies_n and_o so_o of_o all_o other_o superficiese_n 10_o such_o which_o be_v incommensurable_a unto_o it_o be_v irrational_a definition_n where_o it_o be_v say_v in_o this_o definition_n such_o which_o be_v incommensurable_a it_o be_v general_o to_o be_v take_v as_o be_v this_o word_n commensurable_a in_o the_o definition_n before_o for_o all_o superficiese_n whether_o they_o be_v square_n or_o figure_n on_o the_o one_o side_n long_o or_o otherwise_o what_o manner_n of_o right_n line_v figure_n so_o ever_o it_o be_v if_o they_o be_v incommensurable_a unto_o the_o rational_a square_n suppose_v they_o be_v they_o irrational_a as_o let_v thâ—_z square_v abcd_o be_v the_o square_n of_o the_o suppose_a rational_a line_n which_o square_n therefore_o be_v also_o rational_a suppose_v also_o also_o a_o other_o square_a namely_o the_o square_a e_o suppose_v also_o any_o other_o figure_n as_o for_o example_n sake_n a_o figure_n of_o one_o side_n long_o which_o let_v be_v f_o now_o if_o the_o square_a e_o and_o the_o figure_n fletcher_n be_v both_o incommensurable_a to_o the_o rational_a square_n abcd_o then_o be_v 〈◊〉_d of_o these_o figure_n e_o &_o fletcher_n irrational_a and_o so_o of_o other_o 11_o and_o these_o line_n who_o powere_n they_o be_v be_v irrational_a if_o they_o be_v square_n then_o be_v their_o side_n irrational_a if_o they_o be_v not_o square_n definition_n but_o some_o other_o rectiline_a figure_n then_o shall_v the_o line_n who_o square_n be_v equal_a to_o these_o rectiline_a figure_n be_v irrational_a suppose_v that_o the_o rational_a square_n be_v abcd._n suppose_v also_o a_o other_o square_a namely_o the_o square_a e_o which_o let_v be_v incommensurable_a to_o the_o rational_a square_n &_o therefore_o be_v it_o irrational_a and_o let_v the_o side_n or_o line_n which_o produce_v this_o square_n be_v the_o line_n fg_o then_o shall_v the_o line_n fg_o by_o this_o definition_n be_v a_o irrational_a line_n because_o it_o be_v the_o side_n of_o a_o irrational_a square_n let_v also_o the_o figure_n h_o be_v a_o figure_n on_o the_o one_o side_n long_o which_o may_v be_v any_o other_o rectiline_a figure_n rectangle_v or_o not_o rectangle_v triangle_n pentagone_fw-mi trapezite_fw-mi or_o what_o so_o ever_o else_o be_v incommensurable_a to_o the_o rational_a square_n abcd_o then_o because_o the_o figure_n h_o be_v not_o a_o square_a it_o have_v no_o side_n or_o root_n to_o produce_v it_o yet_o may_v there_o be_v a_o square_n make_v equal_a unto_o it_o for_o that_o all_o such_o figure_n may_v be_v reduce_v into_o triangle_n and_o so_o into_o square_n by_o the_o 14._o of_o the_o second_o suppose_v that_o the_o square_a queen_n be_v equal_a to_o the_o irrational_a figure_n h._n the_o side_n of_o which_o figure_n q_n let_v be_v the_o line_n kl_o then_o shall_v the_o line_n kl_o be_v also_o a_o irrational_a line_n because_o the_o power_n or_o square_v thereof_o be_v equal_a to_o the_o irrational_a figure_n h_o and_o thus_o conceive_v of_o other_o the_o like_a these_o irrational_a line_n and_o figure_n be_v the_o chief_a matter_n and_o subject_n which_o be_v entreat_v of_o in_o all_o this_o ten_o book_n the_o knowledge_n of_o which_o be_v deep_a and_o secret_a and_o pertain_v to_o the_o high_a and_o most_o worthy_a part_n of_o geometry_n wherein_o stand_v the_o pith_n and_o marry_v of_o the_o hole_n science_n the_o knowledge_n hereof_o bring_v light_n to_o all_o the_o book_n follow_v with_o out_o which_o they_o be_v hard_a and_o can_v be_v at_o all_o understand_v and_o for_o the_o more_o plainness_n you_o shall_v note_v that_o of_o irrational_a line_n there_o be_v diâ—ers_n sort_n and_o kind_n but_o they_o who_o name_n be_v set_v in_o a_o table_n here_o follow_v and_o be_v in_o number_n 13._o be_v the_o chief_a and_o in_o this_o ten_o book_n sufficient_o for_o euclides_n principal_a purpose_n discourse_v on_o a_o medial_a line_n a_o binomial_a line_n a_o first_o bimedial_a line_n a_o second_o bimedial_a line_n a_o great_a line_n a_o line_n contain_v in_o power_n a_o rational_a superficies_n and_o a_o medial_a superficies_n a_o line_n contain_v in_o power_n two_o medial_a superficiece_n a_o residual_a line_n a_o first_o medial_a residual_a line_n a_o second_o medial_a residual_a line_n a_o less_o line_n a_o line_n make_v with_o a_o rational_a superficies_n the_o whole_a superficies_n medial_a a_o line_n make_v with_o a_o medial_a superficies_n the_o whole_a superficies_n medial_a of_o all_o which_o kind_n the_o definition_n together_o with_o there_o declaration_n shall_v be_v set_v here_o after_o in_o their_o due_a place_n ¶_o the_o 1._o theorem_a the_o 1._o proposition_n two_o unequal_a magnitude_n be_v give_v if_o from_o the_o great_a be_v take_v away_o more_o than_o the_o half_a and_o from_o the_o residue_n be_v again_o take_v away_o more_o than_o the_o half_a and_o so_o be_v do_v still_o continual_o there_o shall_v at_o length_n be_v leave_v a_o certain_a magnitude_n lesser_a than_o the_o less_o of_o the_o magnitude_n first_o give_v svppose_v that_o there_o be_v two_o unequal_a magnitude_n ab_fw-la and_o c_o of_o which_o let_v ab_fw-la be_v the_o great_a then_o i_o say_v that_o if_o from_o ab_fw-la be_v take_v away_o more_o than_o the_o half_a construction_n and_o from_o the_o residue_n be_v take_v again_o more_o than_o the_o half_a and_o so_o still_o continual_o there_o shall_v at_o the_o length_n be_v leave_v a_o certain_a magnitude_n lesser_a than_o the_o less_o magnitude_n give_v namely_o then_o c._n for_o forasmuch_o as_o c_z be_v the_o less_o magnitude_n therefore_o c_z may_v be_v so_o multiply_v that_o at_o the_o length_n it_o will_v be_v great_a than_o the_o magnitude_n ab_fw-la by_o the_o 5._o definition_n of_o the_o five_o book_n demonstration_n let_v it_o be_v so_o multiply_v and_o let_v the_o multiplex_n of_o c_z great_a than_o ab_fw-la be_v de._n and_o divide_v de_fw-fr into_o the_o part_n equal_a unto_o c_o which_o let_v be_v df_o fg_o and_o ge._n and_o from_o the_o magnitude_n ab_fw-la take_v away_o more_o than_o the_o half_a which_o let_v be_v bh_o and_o again_o from_o ah_o take_v away_o more_o than_o the_o half_a which_o let_v be_v hk_a and_o so_o do_v continual_o until_o the_o division_n which_o be_v in_o the_o magnitude_n ab_fw-la be_v equal_a in_o multitude_n unto_o the_o division_n which_o be_v in_o the_o magnitude_n de._n so_o that_o let_v the_o division_n ak_o kh_o and_o hb_o be_v equal_a in_o multitude_n unto_o the_o division_n df_o fg_o and_o ge._n and_o forasmuch_o as_o the_o magnitude_n de_fw-fr be_v great_a than_o the_o magnitude_n ab_fw-la and_o from_o de_fw-fr be_v take_v away_o less_o than_o the_o half_a that_o be_v eglantine_n which_o detraction_n or_o take_v away_o be_v understand_v to_o be_v do_v by_o the_o former_a division_n of_o the_o magnitude_n de_fw-fr into_o the_o part_n equal_a unto_o c_o for_o as_o a_o magnitude_n be_v by_o multiplication_n increase_v so_o be_v it_o by_o division_n diminish_v and_o from_o ab_fw-la be_v take_v away_o more_o than_o the_o half_a that_o be_v bh_o therefore_o the_o residue_n gd_o be_v great_a than_o the_o residue_n ha_o which_o thing_n be_v most_o true_a and_o most_o easy_a to_o conceive_v if_o we_o remember_v this_o principle_n that_o the_o residue_n of_o a_o great_a magnitude_n after_o the_o take_v away_o of_o the_o half_a or_o less_o than_o the_o half_a be_v ever_o great_a than_o the_o residue_n of_o a_o less_o magnitude_n after_o the_o take_v away_o of_o more_o than_o the_o half_a and_o forasmuch_o as_o the_o magnitude_n gd_o be_v great_a than_o the_o magnitude_n ha_o and_o from_o gd_o be_v take_v away_o the_o half_a that_o be_v gf_o and_o from_o ah_o be_v take_v away_o more_o than_o the_o half_a that_o be_v hk_o therefore_o the_o residue_n df_o be_v great_a than_o the_o residue_n ak_o by_o the_o foresay_a principle_n but_o the_o magnitude_n df_o be_v equal_a unto_o the_o magnitude_n c_o by_o supposition_n wherefore_o also_o the_o magnitude_n c_o be_v great_a than_o the_o magnitude_n ak_o wherefore_o the_o magnitude_n ak_o be_v less_o than_o the_o magnitude_n c._n wherefore_o of_o the_o magnitude_n
magnitude_n a_o be_v commensurable_a unto_o the_o magnitude_n c_o therefore_o by_o the_o 5._o of_o the_o ten_o a_o have_v unto_o c_o such_o proportion_n as_o number_n have_v to_o number_v let_v a_o have_v unto_o c_o that_o proportion_n that_o the_o number_n d_o have_v to_o the_o number_n e._n again_o forasmuch_o as_o b_o be_v commensurable_a unto_o c_o therefore_o by_o the_o self_n same_o c_z have_v unto_o b_o that_o proportion_n that_o number_n have_v to_o number_v let_v c_o have_v unto_o b_o that_o proportion_n that_o the_o number_n fletcher_n have_v unto_o the_o number_n g._n now_o then_o take_v the_o least_o number_n in_o continual_a proportion_n and_o in_o these_o proportion_n give_v namely_o that_o the_o number_n d_o have_v to_o the_o number_n e_o and_o that_o the_o number_n fletcher_n have_v to_o the_o number_n g_z by_o the_o 4._o of_o the_o eight_o which_o let_v be_v the_o number_n â—_o king_n l._n demonstration_n so_o that_o as_o the_o number_n d_o be_v to_o the_o number_n e_o so_fw-mi let_v the_o number_n h_o be_v to_o the_o number_n king_n and_o as_o the_o number_n fletcher_n be_v to_o the_o number_n g_o so_o let_v the_o number_n king_n be_v to_o the_o number_n l._n now_o for_o that_o as_o a_o be_v to_o c_o so_o be_v d_z to_o e_o but_o as_z d_z be_v to_o e_o so_o be_v h_o to_o king_n therefore_o as_o a_o be_v to_o c_o so_o be_v h_o to_o k._n again_o for_o that_o as_o c_o be_v to_o b_o so_o be_v fletcher_n to_o g_z but_o as_o fletcher_n be_v to_o g_z so_o be_v king_n to_o l_o therefore_o as_o c_z be_v to_o b_o so_o be_v king_n to_o l._n but_o it_o be_v now_o prove_v that_o as_o a_o be_v to_o c_o so_o be_v h_o to_o k._n wherefore_o of_o equality_n by_o the_o 22._o of_o the_o five_o as_o a_o be_v to_o b_o so_o be_v the_o number_n h_o to_o the_o number_n l._n wherefore_o a_o have_v unto_o b_o such_o proportion_n as_o number_n have_v to_o number_v wherefore_o by_o the_o six_o of_o the_o ten_o the_o magnitude_n a_o be_v commensurable_a unto_o the_o magnitude_n b._n magnitude_n therefore_o commensurable_a to_o one_o and_o the_o self_n same_o magnitude_n 〈…〉_o be_v also_o commensurable_a the_o one_o to_o the_o other_o which_o be_v require_v to_o be_v prove_v ¶_o a_o assumpt_n if_o there_o be_v two_o magnitude_n compare_v to_o one_o and_o the_o self_n same_o magnitude_n and_o if_o the_o one_o of_o they_o be_v commensurable_a unto_o it_o and_o the_o other_o incommensurable_a those_o magnitude_n be_v incommensurable_a the_o one_o to_o the_o other_o svppose_v that_o there_o be_v two_o magnitude_n namely_o absurdity_n a_o and_o b_o and_o let_v c_o be_v a_o certain_a other_o magnitude_n and_o let_v a_o â—e_n commensurable_a unto_o c_z and_o let_v b_o be_v commensurable_a unto_o the_o self_n same_o c._n then_o i_o say_v that_o the_o magnitude_n a_o be_v incommensurable_a unto_o b._n for_o if_o a_o be_v commensurable_a unto_o b_o forasmuch_o as_o a_o be_v also_o commensurable_a unto_o c_z therefore_o by_o the_o 12._o of_o the_o ten_o b_o be_v also_o commensurable_a unto_o c_z which_o be_v contrary_a to_o the_o supposition_n ¶_o the_o 10._o theorem_a the_o 13._o proposition_n if_o there_o be_v two_o magnitude_n commensurable_a and_o if_o the_o one_o of_o they_o be_v incommensurable_a to_o any_o other_o magnitude_n the_o other_o also_o shall_v be_v incommensurable_a unto_o the_o same_o svppose_v that_o these_o two_o magnitude_n a_o b_o be_v commensurable_a the_o one_o to_o the_o other_o and_o let_v the_o one_o of_o they_o namely_o a_o be_v incommensurable_a unto_o a_o other_o magnitude_n absurdity_n namely_o unto_o c._n then_o i_o say_v that_o the_o other_o magnitude_n also_o namely_o b_o be_v incommensurable_a unto_o c._n for_o if_o b_o be_v commensurable_a unto_o c_z then_o forasmuch_o as_o a_o be_v commensurable_a unto_o b_o therefore_o by_o the_o 12._o of_o the_o ten_o the_o magnitude_n a_o also_o be_v commensurable_a unto_o the_o magnitude_n c._n but_o it_o be_v suppose_v to_o be_v incommensurable_a unto_o it_o which_o be_v impossible_a wherefore_o the_o magnitude_n b_o and_o c_o be_v not_o commensurable_a wherefore_o they_o be_v incommensurable_a if_o therefore_o there_o be_v two_o magnitude_n commensurable_a and_o if_o the_o one_o of_o they_o be_v incommensurable_a to_o any_o other_o magnitude_n the_o other_o also_o shall_v be_v incommensurable_a unto_o the_o same_o which_o be_v require_v to_o be_v prove_v ¶_o a_o corollary_n add_v by_o montaureus_n magnitude_n commensurable_a to_o magnitude_n incommensurable_a be_v also_o incommensurable_a the_o one_o to_o the_o other_o corollary_n suppose_v that_o the_o magnitude_n a_o and_o b_o be_v incommensurable_a the_o one_o to_o the_o other_o and_o let_v the_o magnitude_n c_o be_v commensurable_a to_o a_o and_o let_v the_o magnitude_n d_o be_v commensurable_a unto_o b._n then_o i_o say_v that_o the_o magnituâ—â—s_n c_o and_o d_o be_v incommensurable_a the_o one_o to_o the_o other_o for_o a_o and_o c_o be_v commensurable_a of_o which_o the_o magnitude_n a_o be_v incommensurable_a unto_o b_o wherefore_o by_o this_o 13._o proposition_n the_o magnitude_n c_o and_o b_o be_v also_o incommensurable_a but_o the_o magnitudeâ—_n b_o and_o d_z be_v commensurable_a wherefore_o by_o the_o same_o or_o by_o the_o former_a assumpt_n the_o magnitude_n c_o and_o d_o be_v incommensurable_a the_o one_o to_o the_o other_o this_o corollary_n theon_n use_v often_o time_n as_o in_o the_o 22._o 26._o and_o 36_o proposition_n of_o this_o book_n and_o in_o other_o proposition_n also_o ¶_o a_o assumpt_n two_o unequal_a right_a line_n be_v give_v to_o fiâ—de_v out_o how_o much_o the_o great_a be_v in_o power_n more_o than_o the_o less_o corollary_n and_o like_v in_o sort_n two_o right_a line_n be_v give_v by_o this_o mean_n may_v be_v find_v out_o a_o right_a line_n which_o contain_v they_o both_o in_o power_n suppose_v that_o the_o two_o right_a line_n give_v be_v ad_fw-la and_o db._n it_o be_v require_v to_o â—inde_v out_o a_o right_a line_n that_o contain_v they_o both_o in_o power_n let_v the_o line_n ab_fw-la and_o db_o be_v so_o put_v that_o they_o comprehend_v a_o right_a angle_n adb_o and_o draw_v a_o right_a line_n from_o a_o to_o b._n now_o again_o it_o be_v manifest_a by_o the_o 47._o of_o the_o â—irst_n that_o the_o line_n ab_fw-la contain_v in_o power_n the_o line_n ad_fw-la and_o db._n ¶_o the_o 11._o theorem_a the_o 14._o proposition_n if_o there_o be_v sour_a right_a line_n proportional_a and_o if_o the_o first_o be_v in_o power_n more_o than_o the_o second_o by_o the_o square_n of_o a_o right_a line_n commensurable_a in_o length_n unto_o the_o first_o the_o three_o also_o shall_v be_v in_o power_n more_o than_o the_o four_o by_o the_o square_n of_o a_o right_a line_n commensurable_a unto_o the_o three_o and_o if_o the_o first_o be_v in_o power_n more_o than_o the_o second_o by_o the_o square_n of_o a_o right_a line_n incommensurable_a in_o length_n unto_o the_o first_o the_o three_o also_o shall_v be_v in_o power_n more_o than_o the_o four_o by_o the_o square_n of_o a_o right_a line_n incommensurable_a in_o length_n to_o the_o three_o svppose_v that_o these_o four_o right_a line_n a_o b_o c_o d_o be_v proportional_a as_o a_o be_v to_o b_o so_o let_v c_o be_v to_o d._n and_o let_v a_o be_v in_o power_n more_o than_o b_o by_o the_o square_n of_o the_o line_n e._n and_o likewise_o let_v c_o be_v in_o power_n more_o than_o d_o by_o the_o square_n of_o the_o line_n f._n demonstration_n then_o i_o say_v that_o if_o a_o be_v commensurable_a in_o length_n unto_o the_o line_n e_o c_o also_o shall_v be_v commensurable_a in_o length_n unto_o the_o line_n f._n and_o if_o a_o be_v incommensurable_a in_o length_n to_o the_o line_n e_o c_o also_o shall_v be_v incommensurable_a in_o length_n to_o the_o line_n f._n for_o for_o that_o as_o a_o be_v to_o b_o so_o be_v c_z to_o d_z therefore_o as_o the_o square_n of_o the_o line_n a_o be_v to_o the_o square_n of_o the_o line_n b_o so_o be_v the_o square_a of_o the_o line_n c_o to_o the_o square_n of_o the_o line_n d_o by_o the_o 22._o of_o the_o six_o but_o by_o supposition_n unto_o the_o square_n of_o the_o line_n a_o be_v equal_a the_o square_n oâ—_n the_o line_n e_o and_o b_o and_o unto_o the_o square_n of_o the_o line_n c_o be_v equal_a the_o square_n of_o the_o of_o the_o line_n d_o and_o f_o wherefore_o as_o the_o square_n of_o the_o line_n e_o and_o b_o which_o be_v equal_a to_o the_o square_n of_o the_o line_n a_o be_v to_o the_o square_n of_o the_o line_n b_o so_fw-mi be_v the_o square_n of_o the_o line_n d_o and_o f_o which_o be_v equal_a to_o the_o square_n of_o the_o line_n c_o to_o the_o square_n of_o the_o line_n d_o by_o the_o seven_o of_o the_o five_o wherefore_o
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
contain_v under_o the_o line_n bd_o and_o dc_o be_v equal_a to_o the_o square_n of_o the_o line_n da._n as_o touch_v the_o four_o that_o the_o parallelogram_n contain_v under_o the_o line_n bc_o and_o ad_fw-la be_v equal_a to_o the_o parallelogram_n contain_v under_o the_o line_n basilius_n and_o ac_fw-la be_v thus_o prove_v for_o forasmuch_o as_o as_o we_o have_v already_o declare_v the_o triangle_n abc_n be_v like_a and_o therefore_o equiangle_n to_o the_o triangle_n abdella_n therefore_o as_o the_o line_n bc_o be_v to_o the_o line_n ac_fw-la dâ—e_n so_o be_v the_o line_n basilius_n to_o the_o line_n ad_fw-la by_o the_o 4._o of_o the_o six_o corollary_n but_o if_o there_o be_v four_o right_a line_n proportional_a that_o which_o be_v contain_v under_o the_o first_o and_o the_o last_o be_v equal_a to_o that_o which_o be_v contain_v under_o the_o two_o mean_n by_o the_o 16._o of_o the_o six_o wherefore_o that_o which_o be_v contain_v under_o the_o line_n bc_o and_o ad_fw-la be_v equal_a to_o that_o which_o be_v contain_v under_o the_o line_n basilius_n and_o ac_fw-la profitable_a i_o say_v moreover_o that_o if_o there_o be_v make_v a_o parallelogram_n complete_a determination_n contain_v under_o the_o line_n bc_o and_o ad_fw-la which_o let_v be_v ec_o and_o if_o likewise_o be_v make_v complete_a the_o parallelogram_n contain_v under_o the_o line_n basilius_n and_o ac_fw-la which_o let_v be_v of_o it_o may_v by_o a_o other_o way_n be_v prove_v that_o the_o parallelogram_n ec_o be_v equal_a to_o the_o parallelogram_n af._n for_o forasmuch_o as_o either_o of_o they_o be_v double_a to_o the_o triangle_n acb_o by_o the_o 41._o of_o the_o first_o and_o thing_n which_o be_v double_a to_o one_o and_o the_o self_n same_o thing_n be_v equal_a the_o one_o to_o the_o other_o wherefore_o that_o which_o be_v contain_v under_o the_o line_n bc_o and_o ad_fw-la be_v equal_a to_o that_o which_o be_v contain_v under_o the_o line_n basilius_n and_o ac_fw-la 2._o ¶_o a_o assumpt_n if_o a_o right_a line_n be_v divide_v into_o two_o unequal_a part_n as_o the_o great_a part_n be_v to_o the_o less_o assumpt_n so_o be_v the_o parallelogram_n contain_v under_o the_o whole_a line_n and_o the_o great_a part_n to_o the_o parallelogram_n contain_v under_o the_o whole_a line_n and_o the_o less_o part_n this_o assumpt_n differ_v little_a from_o the_o first_o proposition_n of_o the_o six_o book_n 3._o ¶_o a_o assumpt_n if_o there_o be_v two_o unequal_a right_a line_n and_o if_o the_o less_o be_v divide_v into_o two_o equal_a part_n the_o parallelogram_n contain_v under_o the_o two_o unequal_a line_n be_v double_a to_o the_o parallelogram_n contain_v under_o the_o great_a line_n &_o half_a of_o the_o less_o line_n suppose_v that_o there_o be_v two_o unequal_a right_a line_n ab_fw-la and_o bc_o of_o which_o leâ—_z ab_fw-la be_v the_o great_a and_o divide_v the_o line_n bc_o into_o two_o equal_a part_n in_o the_o point_n d._n thâ—n_o i_o say_v that_o the_o parallelogram_n contain_v under_o the_o line_n ab_fw-la &_o bc_o be_v double_a to_o the_o parallelogram_n contain_v under_o the_o line_n ab_fw-la and_o bd._n from_o the_o point_n b_o raise_v up_o upon_o the_o right_a line_n bc_o a_o perpendicular_a line_n be_v and_o let_v be_v be_v equal_a to_o the_o line_n basilius_n and_o draw_v from_o the_o point_n c_o and_o d_o the_o line_n cg_o and_o df_o parallel_v and_o equal_a to_o be_v and_o then_o draw_v the_o right_a line_n gfe_n the_o figure_n be_v complete_a nâ—â—_n for_o that_o aâ—â—he_a line_n db_o be_v to_o the_o line_n dc_o so_o be_v the_o parallelogram_n bf_o to_o the_o parallelogram_n dg_o by_o the_o 1._o of_o the_o six_o therefore_o by_o composition_n of_o proportion_n as_o the_o whole_a line_n bc_o be_v to_o the_o line_n dc_o so_o be_v the_o parallelogram_n bg_o to_o the_o parallelogram_n dg_o by_o the_o 18._o of_o the_o five_o but_o the_o line_n bc_o be_v double_a to_o the_o line_n dc_o wherefore_o the_o parallelogram_n bg_o be_v double_a to_o the_o parallelogram_n dg_o but_o the_o parallâ—logramme_n bg_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o for_o the_o line_n ab_fw-la be_v equal_a to_o the_o line_n be_v and_o the_o parallelogram_n dg_o be_v contain_v under_o the_o line_n ab_fw-la and_o bd_o for_o the_o line_n bd_o be_v equal_a to_o the_o line_n dc_o and_o the_o line_n ab_fw-la to_o the_o line_n df_o which_o be_v require_v to_o be_v demonstrate_v ¶_o the_o 10._o problem_n the_o 33._o proposition_n to_o â—inde_v out_o two_o right_a line_n incommensurable_a in_o power_n who_o square_n add_v together_o make_v a_o rational_a superficies_n and_o the_o parallelogram_n contain_v under_o they_o make_v a_o medial_a superficies_n take_v by_o the_o 30._o of_o the_o ten_o two_o rational_a right_a line_n commensurable_a in_o power_n only_o namely_o ab_fw-la and_o bc_o so_o that_o let_v the_o line_n ab_fw-la be_v the_o great_a be_v in_o power_n more_o than_o the_o line_n bc_o be_v the_o less_o construction_n by_o the_o square_n of_o a_o line_n incommensurable_a in_o length_n unto_o the_o line_n ab_fw-la and_o by_o the_o 10._o of_o the_o first_o divide_v the_o line_n bc_o into_o two_o equal_a part_n in_o the_o point_n d._n and_o upon_o the_o line_n ab_fw-la apply_v a_o parallelogram_n equal_a to_o the_o square_a either_o of_o the_o line_n bd_o or_o of_o the_o line_n dc_o and_o want_v in_o figure_n by_o a_o square_a by_o the_o 28._o of_o the_o six_o and_o let_v that_o parallelogram_n be_v that_o which_o be_v contain_v under_o the_o line_n ae_n and_o ebb_n and_o upon_o the_o line_n ab_fw-la describe_v a_o semicircle_n afb_o and_o by_o the_o 11._o of_o the_o first_o from_o the_o point_n e_o raise_v up_o unto_o the_o line_n ab_fw-la a_o perpendicular_a line_n of_o cut_v the_o circumference_n in_o the_o point_n f._n and_o draw_v line_n from_o a_o to_o fletcher_n demonstration_n and_o from_o fletcher_n to_o b._n and_o forasmuch_o as_o there_o be_v two_o unequal_a right_a line_n ab_fw-la and_o bc_o and_o the_o line_n ab_fw-la be_v in_o power_n more_o than_o the_o line_n bc_o by_o the_o square_n of_o a_o line_n incommensurable_a in_o length_n unto_o ab_fw-la and_o upon_o the_o line_n ab_fw-la be_v apply_v a_o parallelogram_n equal_a to_o the_o four_o part_n oâ—_n the_o square_a of_o the_o line_n bc_o that_o be_v to_o the_o square_n of_o the_o half_a of_o the_o line_n bc_o and_o want_v in_o â—igure_n by_o a_o square_a and_o the_o say_a parallelogram_n be_v that_o which_o be_v contain_v under_o the_o line_n ae_n and_o ebb_n wherefore_o by_o the_o 2._o part_n of_o the_o 18._o of_o the_o ten_o the_o line_n ae_n be_v incommensurable_a in_o length_n unto_o the_o line_n ebb_n but_o as_o the_o line_n ae_n be_v to_o the_o line_n ebb_n so_o be_v the_o parallelogram_n contain_v under_o the_o line_n basilius_n and_o ae_n to_o the_o parallelogram_n contain_v under_o the_o line_n ab_fw-la and_o be_v by_o the_o second_o assumpt_n before_o put_v and_o that_o which_o be_v contain_v under_o the_o line_n basilius_n and_o ae_n be_v equal_a to_o the_o square_n of_o the_o line_n of_o by_o the_o second_o part_n of_o the_o first_o assumpt_v before_o put_v and_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o be_v be_v by_o the_o first_o part_n of_o the_o same_o assumpt_n equal_a to_o the_o square_n of_o the_o line_n bf_o wherefore_o the_o square_a of_o the_o line_n of_o be_v incommensurable_a to_o the_o square_n of_o the_o line_n bf_o wherefore_o the_o line_n of_o and_o bf_o be_v incommensurable_a in_o power_n and_o forasmuch_o as_o ab_fw-la be_v a_o rational_a line_n by_o supposition_n therefore_o by_o the_o 7_o definition_n of_o the_o ten_o the_o square_a of_o the_o line_n ab_fw-la be_v rational_a conclude_v wherefore_o also_o the_o square_n of_o the_o line_n of_o and_o fb_o add_v together_o make_v a_o rational_a superficies_n for_o by_o the_o 47._o of_o the_o first_o they_o be_v equal_a to_o the_o square_n of_o the_o line_n ab_fw-la again_o forasmuch_o as_o by_o the_o three_o part_n of_o the_o first_o assumpt_n go_v before_o that_o which_o be_v contain_v under_o the_o line_n ae_n and_o ebb_n be_v equal_a to_o the_o square_n of_o the_o line_n ef._n but_o by_o supposition_n that_o which_o be_v contain_v under_o the_o line_n ae_n and_o ebb_n be_v equal_a to_o the_o square_n of_o the_o line_n bd._n wherefore_o the_o line_n fe_o be_v equal_a to_o the_o line_n bd._n wherefore_o the_o linâ—_n bg_o be_v double_a to_o the_o line_n â—_o e._n wherefore_o by_o the_o three_o assumpt_v go_v before_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o be_v double_a to_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o ef._n but_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o be_v by_o supposition_n medial_a
length_n and_o so_o likewise_o shall_v the_o line_n ad_fw-la and_o db_o be_v for_o every_o line_n measure_v itself_o and_o any_o other_o line_n equal_a to_o itself_o moreover_o the_o line_n db_o be_v either_o one_o and_o the_o same_o with_o the_o line_n acâ—_n that_o be_v be_v equal_a to_o the_o line_n ac_fw-la oâ—_n else_o it_o be_v greater-then_a the_o line_n ac_fw-la either_o else_o it_o be_v less_o than_o it_o if_o db_o be_v equal_a to_o the_o line_n ac_fw-la then_o put_v the_o line_n db_o upon_o the_o line_n ac_fw-la each_o end_n of_o the_o one_o shall_v agree_v with_o each_o end_n of_o the_o other_o wherefore_o put_v the_o point_n b_o upon_o the_o point_n a_o the_o point_n d_o also_o shall_v fall_v upon_o the_o point_n c_o and_o the_o line_n ad_fw-la which_o be_v the_o rest_n of_o the_o line_n ac_fw-la shall_v also_o be_v equal_a to_o the_o line_n cb_o which_o be_v the_o rest_n of_o the_o line_n db._n wherefore_o the_o line_n ab_fw-la be_v divide_v into_o his_o name_n in_o the_o point_n c._n and_o so_o also_o shall_v the_o line_n ab_fw-la be_v divide_v in_o the_o point_n d_o be_v divide_v in_o the_o self_n â—ame_n point_n that_o the_o self_n same_o line_n ab_fw-la be_v before_o divide_v in_o the_o point_n c_o which_o be_v contrary_a to_o the_o supposition_n for_o by_o supposition_n it_o be_v divide_v in_o sundry_a point_n namely_o in_o c_o &_o d._n but_o if_o the_o line_n db_o be_v greaterâ—_n the_o the_o line_n ac_fw-la let_v the_o line_n ab_fw-la be_v deâ—ided_v into_o two_o equal_a part_n in_o the_o point_n e._n wherefore_o the_o point_n c_o &_o d_o shall_v not_o equal_o be_v distant_a from_o the_o point_n e_o now_o by_o the_o first_o assupt_v go_v before_o this_o proposition_n that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ad_fw-la &_o db_o be_v great_a they_o that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ac_fw-la &_o cbâ—_n but_o that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ad_fw-la &_o db_o together_o with_o that_o which_o be_v contain_v under_o the_o line_n ad_fw-la &_o db_o twice_o be_v equal_a to_o that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ac_fw-la &_o cb_o together_o with_o that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o twice_o for_o either_o of_o they_o be_v equal_a to_o the_o square_n of_o the_o whole_a line_n ab_fw-la by_o the_o 4._o of_o the_o second_o wherefore_o how_o much_o that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ad_fw-la and_o db_o add_v together_o be_v great_a than_o that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ac_fw-la and_o cb_o add_v together_o so_o much_o be_v that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o twice_o great_a than_o that_o which_o be_v contain_v under_o the_o line_n ad_fw-la and_o db_o twice_o but_o that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ad_fw-la and_o db_o exceed_v that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ac_fw-la and_o cb_o by_o a_o rational_a superficies_n by_o the_o 2._o assumpt_n go_v before_o this_o proposition_n for_o that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ad_fw-la and_o db_o be_v rational_a and_o so_o also_o be_v that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ac_fw-la and_o cb_o for_o the_o line_n ad_fw-la and_o db_o be_v put_v to_o be_v rational_a commensurable_a in_o power_n only_o and_o so_o likewise_o be_v the_o line_n ac_fw-la and_o cb._n wherefore_o also_o that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o twice_o exceed_v that_o which_o be_v contain_v under_o the_o line_n ad_fw-la &_o db_o twice_o by_o a_o rational_a superficies_n when_o yet_o notwithstanding_o they_o be_v both_o medial_a superficiece_n by_o the_o 21._o of_o the_o ten_o which_o by_o the_o 26._o of_o the_o same_o be_v impossible_a and_o if_o the_o line_n db_o be_v less_o than_o the_o line_n ac_fw-la we_o may_v by_o the_o like_a demonstration_n prove_v the_o self_n same_o impossibility_n wherefore_o a_o binomial_a line_n be_v in_o one_o point_n only_o divide_v into_o his_o name_n which_o be_v require_v to_o be_v demonstrate_v 〈…〉_o ollary_n add_v by_o flussates_n two_o ration_n 〈…〉_o surable_a in_o power_n only_o be_v add_v together_o can_v be_v equal_a to_o two_o other_o rational_a line_n 〈…〉_o in_o power_n only_o add_v together_o corollary_n for_o either_o of_o they_o shall_v make_v a_o binomia_fw-la 〈…〉_o so_o shall_v a_o binomial_a line_n be_v divide_v into_o his_o name_n in_o more_o point_n than_o onâ—â—â—ch_v by_o this_o proposition_n be_v prove_v to_o be_v impossible_a the_o like_a shall_v follow_v in_o the_o five_o 〈◊〉_d irrational_a line_n as_o touch_v their_o two_o name_n ¶_o the_o 31._o problem_n the_o 43._o proposition_n a_o first_o bimedial_a line_n be_v in_o one_o point_n only_o divide_v into_o his_o name_n svppose_v that_o ab_fw-la be_v a_o first_o bimedial_a line_n and_o let_v it_o be_v divide_v into_o his_o part_n in_o the_o point_n c_o so_fw-mi that_o let_v the_o line_n ac_fw-la and_o cb_o be_v medial_a commensurable_a in_o power_n only_o and_o contain_v a_o rational_a superficies_n then_o i_o say_v that_o the_o line_n ab_fw-la can_v not_o be_v divide_v into_o his_o name_n in_o any_o other_o point_n then_o in_o c._n for_o if_o it_o be_v possible_a let_v it_o be_v divide_v into_o his_o name_n in_o the_o point_n d_o impossibilâ—â—e_n so_fw-mi that_o let_v ad_fw-la &_o db_o be_v medial_a line_n commensurable_a in_o power_n only_o comprehend_v a_o rational_a superficies_n now_o forasmuch_o as_o how_o much_o that_o which_o be_v contain_v under_o the_o line_n ad_fw-la and_o db_o twice_o diâ—ferreth_v from_o that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o twice_o so_o much_o differ_v that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ad_fw-la and_o db_o from_o that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ac_fw-la and_o cb_o but_o that_o which_o be_v contain_v under_o the_o line_n ad_fw-la and_o db_o twice_o differ_v from_o that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o twice_o by_o a_o rational_a superficies_n by_o the_o second_o assumpt_v go_v before_o the_o 41._o of_o the_o ten_o for_o either_o of_o those_o superficiece_n be_v rational_a wherefore_o that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ac_fw-la and_o cb_o differ_v from_o that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ad_fw-la and_o db_o by_o a_o rational_a superficies_n when_o yet_o they_o be_v both_o medial_a superficiece_n which_o be_v impossible_a wherefore_o a_o first_o bimedial_a line_n be_v in_o one_o point_n only_o divide_v into_o his_o name_n which_o be_v require_v to_o be_v prove_v ¶_o the_o 32._o theorem_a the_o 44._o proposition_n a_o second_o bimedial_a line_n be_v in_o one_o point_n only_o divide_v into_o his_o name_n svppose_v that_o the_o line_n ab_fw-la be_v a_o second_o bimedial_a line_n be_v divide_v into_o his_o name_n in_o the_o point_n c_o so_o that_o let_v the_o line_n ac_fw-la and_o cb_o be_v medial_a line_n commensurable_a in_o power_n only_o comprehend_v a_o medial_a superficies_n it_o be_v manifest_a that_o the_o point_n c_o divide_v not_o the_o whole_a line_n ab_fw-la into_o two_o equal_a part_n impossibility_n for_o the_o line_n ac_fw-la and_o cb_o be_v not_o commensurable_a in_o length_n the_o one_o to_o the_o other_o now_o i_o say_v that_o the_o line_n ab_fw-la can_v be_v divide_v into_o his_o name_n in_o any_o other_o point_n but_o only_o in_o c._n for_o if_o it_o be_v possible_a let_v it_o be_v divide_v into_o his_o name_n in_o the_o point_n d_o so_fw-mi that_o let_v not_o the_o line_n ac_fw-la be_v one_o and_o the_o same_o that_o be_v let_v it_o not_o be_v equal_a with_o the_o line_n db._n but_o let_v it_o be_v great_a than_o it_o now_o it_o be_v manifest_a by_o the_o first_o assumpt_n go_v before_o the_o 42._o proposition_n of_o this_o book_n that_o the_o square_n of_o the_o line_n ac_fw-la and_o cb_o be_v great_a than_o the_o square_n of_o the_o line_n ad_fw-la and_o db._n and_o also_o that_o the_o line_n ad_fw-la and_o db_o be_v medial_a line_n commensurable_a in_o power_n only_o comprehend_v a_o medial_a supersicy_n take_v a_o rational_a line_n ef._n and_o by_o the_o 44._o of_o the_o first_o upon_o the_o line_n of_o apply_v a_o rectangle_n parallelogram_n eke_o equal_a to_o the_o square_n of_o the_o line_n ab_fw-la from_o which_o parallelogram_n take_v away_o the_o parallelogram_n eglantine_n equal_a to_o the_o square_n of_o the_o line_n ac_fw-la and_o cb_o wherefore_o the_o residue_n namely_o the_o parallelogram_n hk_o
the_o definition_n of_o a_o first_o binomial_a line_n seâ—_z before_o the_o 48._o proposition_n of_o this_o book_n the_o line_n dg_o be_v a_o first_o binomial_a line_n which_o be_v require_v to_o be_v prove_v this_o proposition_n and_o the_o five_o follow_v be_v the_o converse_n of_o the_o six_o former_a proposition_n ¶_o the_o 43._o theorem_a the_o 61._o proposition_n the_o square_a of_o a_o first_o bimedial_a line_n apply_v to_o a_o rational_a line_n make_v the_o breadth_n or_o other_o side_n a_o second_o binomial_a line_n svppose_v that_o the_o line_n ab_fw-la be_v a_o first_o bimedial_a line_n and_o let_v it_o be_v suppose_v to_o be_v divide_v into_o his_o part_n in_o the_o point_n c_o of_o which_o let_v ac_fw-la be_v the_o great_a part_n take_v also_o a_o rational_a line_n de_fw-fr and_o by_o the_o 44._o of_o the_o first_o apply_v to_o the_o line_n de_fw-fr the_o parallelogram_n df_o equal_a to_o the_o square_n of_o the_o line_n ab_fw-la &_o make_v in_o breadth_n the_o line_n dg_o construction_n then_o i_o say_v that_o the_o line_n dg_o be_v a_o second_o binomial_a line_n let_v the_o same_o construction_n be_v in_o this_o that_o be_v in_o the_o proposition_n go_v before_o and_o forasmuch_o as_o the_o line_n ab_fw-la be_v a_o first_o bimedial_a line_n demonstration_n and_o be_v divide_v into_o his_o part_n in_o the_o point_n c_o therefore_o by_o the_o 37._o of_o the_o ten_o the_o line_n ac_fw-la and_o cb_o be_v medial_a commensurable_a in_o power_n only_o comprehend_v a_o rational_a superficies_n wherefore_o also_o the_o square_n of_o the_o line_n ac_fw-la and_o cb_o be_v medial_a wherefore_o the_o parallelogram_n dl_o be_v medial_a by_o the_o corollary_n of_o the_o 23._o of_o the_o ten_o and_o it_o be_v apply_v upon_o the_o rational_a line_n de._n wherefore_o by_o the_o 22._o of_o the_o ten_o the_o line_n md_o be_v rational_a and_o incommensurable_a in_o length_n to_o the_o line_n de._n again_o forasmuch_o as_o that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o twice_o be_v rational_a therefore_o also_o the_o parallelogram_n mf_n be_v rational_a and_o it_o be_v apply_v unto_o the_o rational_a line_n ml_o wherefore_o the_o line_n mg_o be_v rational_a and_o commensurable_a in_o length_n to_o the_o line_n ml_o that_o be_v to_o the_o line_n de_fw-fr by_o the_o 20._o of_o the_o ten_o wherefore_o the_o line_n dm_o be_v incommensurable_a in_o length_n to_o the_o line_n mg_o and_o they_o be_v both_o rational_a wherefore_o the_o line_n dm_o and_o mg_o be_v rational_a commensurable_a in_o power_n only_o wherefore_o the_o whole_a line_n dg_o be_v a_o binomial_a line_n line_n now_o rest_v to_o prove_v that_o it_o be_v a_o second_o binomial_a line_n forasmuch_o as_o the_o square_n of_o the_o line_n ac_fw-la and_o cb_o be_v great_a than_o that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o twice_o by_o the_o assumpt_n before_o the_o 60._o of_o this_o book_n therefore_o the_o parallelogram_n dl_o be_v great_a than_o the_o parallelogrrmme_a mf_n wherefore_o also_o by_o the_o first_o of_o the_o six_o the_o line_n dm_o be_v great_a than_o the_o line_n mg_o and_o forasmuch_o as_o the_o square_n of_o the_o line_n ac_fw-la be_v commensurable_a to_o the_o square_n of_o the_o line_n cb_o therefore_o the_o parallelogram_n dh_o be_v commensurable_a to_o the_o parallelogram_n kl_o wherefore_o also_o the_o line_n dk_o be_v commensurable_a in_o length_n to_o the_o line_n km_o and_o that_o which_o be_v contain_v under_o the_o line_n dk_o and_o km_o be_v equal_a to_o the_o square_n of_o the_o line_n mn_o that_o be_v to_o the_o four_o part_n of_o the_o square_n of_o the_o line_n mg_o wherefore_o by_o the_o 17._o of_o the_o ten_o the_o line_n dm_o be_v in_o power_n more_o than_o the_o line_n mg_o by_o the_o square_n of_o a_o line_n commensurable_a in_o length_n unto_o the_o line_n dm_o and_o the_o line_n mg_o be_v commensurable_a in_o length_n to_o the_o rational_a line_n put_v namely_o to_o de._n wherefore_o the_o line_n dg_o be_v a_o second_o binomial_a line_n which_o be_v require_v to_o be_v prove_v ¶_o the_o 44._o theorem_a the_o 62._o proposition_n the_o square_a of_o a_o second_o bimedial_a line_n apply_v unto_o a_o rational_a line_n make_v the_o breadth_n or_o other_o side_n thereof_o a_o three_o binomial_a line_n svppose_v that_o ab_fw-la be_v a_o second_o bimedial_a line_n and_o let_v ab_fw-la be_v suppose_v to_o be_v divide_v into_o his_o part_n in_o the_o point_n c_o so_fw-mi that_o let_v ac_fw-la be_v the_o great_a part_n and_o take_v a_o rational_a line_n de._n and_o by_o the_o 44._o of_o the_o first_o unto_o the_o line_n de_fw-fr apply_v the_o parallelogram_n df_o equal_a to_o the_o square_n of_o the_o line_n ab_fw-la and_o make_v in_o breadth_n the_o line_n dg_o then_o i_o say_v that_o the_o line_n dg_o be_v a_o three_o binomial_a line_n let_v the_o self_n same_o construction_n be_v in_o this_o that_o be_v in_o the_o proposition_n next_o go_v before_o and_o forasmuch_o as_o the_o line_n ab_fw-la be_v a_o second_o bimedial_a line_n construction_n and_o be_v divide_v into_o his_o part_n in_o the_o point_n c_o demonstration_n therefore_o by_o the_o 38._o of_o the_o ten_o the_o line_n ac_fw-la and_o cb_o be_v medial_n commensurable_a in_o power_n only_o comprehend_v a_o medial_a superficies_n wherefore_o †_o that_o which_o be_v make_v of_o the_o square_n of_o the_o line_n ac_fw-la and_o cb_o add_v together_o be_v medial_a and_o it_o be_v equal_a to_o the_o parallelogram_n dl_o by_o construction_n wherefore_o the_o parallelogram_n dl_o be_v medial_a and_o be_v apply_v unto_o the_o rational_a line_n de_fw-fr wherefore_o by_o the_o 22._o of_o the_o ten_o the_o line_n md_o be_v rational_a and_o incommensurable_a in_o length_n to_o the_o line_n de._n and_o by_o the_o like_a reason_n also_o *_o the_o line_n mg_o be_v rational_a and_o incommensurable_a in_o length_n to_o the_o line_n ml_o that_o be_v to_o the_o line_n de._n wherefore_o either_o of_o these_o line_n dm_o and_o mg_o be_v rational_a and_o incommensurable_a in_o length_n to_o the_o line_n de._n and_o forasmuch_o as_o the_o line_n ac_fw-la be_v incommensurable_a in_o length_n to_o the_o line_n cb_o but_o as_o the_o line_n ac_fw-la be_v to_o the_o line_n cb_o so_o by_o the_o assumpt_n go_v before_o the_o 22._o of_o the_o ten_o be_v the_o square_a of_o the_o line_n ac_fw-la to_o that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb._n wherefore_o the_o square_a of_o the_o line_n ac_fw-la be_v incâ—mmmensurable_a to_o that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb._n wherefore_o that_o ‡_o that_o which_o be_v make_v of_o the_o square_n of_o the_o line_n ac_fw-la and_o cb_o add_v together_o be_v incommensurable_a to_o that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o twice_o that_o be_v the_o parallelogram_n dl_o to_o the_o parallelogram_n mf_n wherefore_o by_o the_o first_o of_o the_o six_o and_o 10._o of_o the_o ten_o the_o line_n dm_o be_v incommensurable_a in_o length_n to_o the_o line_n mg_o and_o they_o be_v prove_v both_o rational_a line_n wherefore_o the_o whole_a line_n dg_o be_v a_o binomial_a line_n by_o the_o definition_n in_o the_o 36._o of_o the_o ten_o now_o rest_v to_o prove_v that_o it_o be_v a_o three_o binomial_a line_n as_o in_o the_o former_a proposition_n so_o also_o in_o this_o may_v we_o conclude_v that_o the_o line_n dm_o be_v great_a than_o the_o line_n mg_o and_o that_o the_o line_n dk_o be_v commensurable_a in_o length_n to_o the_o line_n km_o and_o that_o that_o which_o be_v contain_v under_o the_o line_n dk_o and_o km_o be_v equal_a to_o the_o square_n of_o the_o line_n mn_v wherefore_o the_o line_n dm_o be_v in_o power_n more_o than_o the_o line_n mg_o by_o the_o square_n of_o a_o line_n commensurable_a in_o length_n unto_o the_o line_n dm_o and_o neither_o of_o the_o line_n dm_o nor_o mg_o be_v commensurable_a in_o length_n to_o the_o rational_a line_n de._n wherefore_o by_o the_o definition_n of_o a_o three_o binomiâ—ll_v line_v the_o line_n dg_o be_v a_o three_o binomial_a line_n which_o be_v require_v to_o be_v prove_v ¶_o here_o follow_v certain_a annotation_n by_o m._n dee_n make_v upon_o three_o place_n in_o the_o demonstration_n which_o be_v not_o very_o evident_a to_o young_a beginner_n †_o the_o square_n of_o the_o line_n ac_fw-la and_o câ—_n be_v medial_n 〈◊〉_d iâ—_z teach_v after_o the_o 21â—_n of_o this_o ten_o and_o therefore_o forasmuch_o as_o they_o be_v by_o supposition_n commensurable_a the_o one_o to_o the_o other_o by_o the_o 15._o of_o the_o ten_o the_o compound_n of_o they_o both_o be_v commensurable_a to_o each_o part_n but_o the_o part_n be_v medial_n therefore_o by_o the_o corollary_n of_o the_o 23._o of_o the_o ten_o the_o compound_n shall_v be_v
in_o like_a sort_n be_v divide_v as_o the_o line_n ab_fw-la be_v by_o that_o which_o have_v be_v demonstrate_v in_o the_o 66._o proposition_n of_o this_o bookeâ—_n let_v it_o be_v so_o divide_v in_o the_o point_n e._n wherefore_o it_o can_v not_o be_v so_o divide_v in_o any_o other_o point_n by_o the_o 42â—_n of_o this_o book_n and_o for_o that_o the_o line_n ab_fw-la â—â—_o to_o the_o line_n dz_o as_o the_o line_n agnostus_n be_v to_o the_o line_n de_fw-fr but_o the_o line_n agnostus_n &_o de_fw-fr namely_o the_o great_a name_n be_v commensurable_a in_o length_n the_o one_o to_o the_o other_o by_o the_o 10._o of_o this_o book_n for_o that_o they_o be_v commensurable_a in_o length_n to_o 〈◊〉_d and_o the_o self_n same_o rational_a line_n by_o the_o first_o definition_n of_o binomial_a line_n wherefore_o the_o line_n ab_fw-la and_o dz_o be_v commensurable_a in_o length_n by_o the_o 13._o of_o this_o book_n but_o by_o supposition_n they_o be_v commensurable_a in_o power_n only_o which_o be_v impossible_a the_o self_n same_o demonstration_n also_o will_v serve_v if_o we_o suppose_v the_o line_n ab_fw-la to_o be_v a_o second_o binomial_a line_n for_o the_o less_o name_n gb_o and_o ez_o be_v commensurable_a in_o length_n to_o one_o and_o the_o self_n same_o rational_a line_n shall_v also_o be_v commensurable_a in_o length_n the_o one_o to_o the_o other_o and_o therefore_o the_o line_n ab_fw-la and_o dz_o which_o be_v in_o the_o self_n same_o proportion_n with_o they_o shall_v also_o be_v commensurable_a in_o length_n the_o one_o to_o the_o other_o which_o be_v contrary_a to_o the_o supposition_n far_o if_o the_o square_n of_o the_o line_n ab_fw-la and_o dz_o be_v apply_v unto_o the_o rational_a line_n cf_o namely_o the_o parallelogram_n et_fw-la and_o hl_o they_o shall_v make_v the_o breadthes_fw-mi change_z and_o hk_o first_o binomial_a line_n of_o what_o order_n soever_o the_o line_n ab_fw-la &_o dz_o who_o square_n be_v apply_v unto_o the_o rational_a line_n be_v by_o the_o 60._o of_o this_o book_n wherefore_o it_o be_v manifest_a that_o under_o a_o rational_a line_n and_o a_o first_o binomial_a line_n be_v confuse_o contain_v all_o the_o power_n of_o binomial_a line_n by_o the_o 54._o of_o this_o book_n wherefore_o the_o only_a commensuration_n of_o the_o power_n do_v not_o of_o necessity_n bring_v forth_o one_o and_o the_o self_n same_o order_n of_o binomial_a line_n the_o self_n same_o thing_n also_o may_v be_v prove_v if_o the_o line_n ab_fw-la and_o dz_o be_v suppose_v to_o be_v a_o four_o or_o five_o binomial_a line_n who_o power_n only_o be_v conmmensurable_a namely_o that_o they_o shall_v as_o the_o first_o bring_v forth_o binomial_a line_n of_o diverse_a order_n now_o forasmuch_o as_o the_o power_n of_o the_o line_n agnostus_n and_o gb_o and_o de_fw-fr and_o ez_o be_v commensurable_a &_o proportional_a it_o be_v manifest_a that_o if_o the_o line_n agnostus_n be_v in_o power_n more_o than_o the_o line_n gb_o by_o the_o square_n of_o a_o line_n commensurable_a in_o length_n unto_o agnostus_n the_o line_n de_fw-fr also_o shall_v be_v in_o power_n more_o than_o the_o line_n ez_o by_o the_o square_n of_o a_o line_n commensurable_a in_o length_n unto_o the_o line_n de_fw-fr by_o the_o 16._o of_o this_o book_n and_o so_o shall_v the_o two_o line_n ab_fw-la and_o dz_o be_v each_o of_o the_o three_o first_o binomial_a line_n but_o if_o the_o line_n agnostus_n be_v in_o power_n more_o than_o the_o line_n gb_o by_o the_o square_n of_o a_o line_n incommensurable_a in_o length_n unto_o the_o line_n agnostus_n the_o line_n de_fw-fr shall_v also_o be_v in_o powâ—r_a 〈◊〉_d than_o the_o line_n ez_o by_o the_o square_n of_o a_o line_n incomensurable_a in_o length_n unto_o the_o line_n de_fw-fr by_o the_o self_n same_o proposition_n and_o so_o shall_v eke_v of_o the_o line_n ab_fw-la and_o dz_o be_v of_o the_o three_o last_o binomial_a line_n but_o why_o it_o be_v not_o so_o in_o the_o three_o and_o six_o binomial_a line_n the_o reason_n be_v for_o that_o in_o they_o neither_o of_o the_o nameâ—_n be_v commensurable_a in_o length_n to_o the_o rational_a line_n put_v fc_n ¶_o the_o 50._o theorem_a the_o 68_o proposition_n a_o line_n commensurable_a to_o a_o great_a line_n be_v also_o a_o great_a line_n svppose_v that_o the_o line_n ab_fw-la be_v a_o great_a line_n and_o unto_o the_o line_n ab_fw-la let_v the_o line_n cd_o be_v commensurable_a construction_n then_o i_o say_v that_o the_o line_n cd_o also_o be_v a_o great_a line_n divide_v the_o line_n ab_fw-la into_o his_o part_n in_o the_o point_n e._n wherefore_o by_o the_o 39_o of_o the_o ten_o the_o line_n ae_n and_o ebb_n be_v incommensurable_a in_o power_n have_v that_o which_o be_v make_v of_o the_o square_n of_o they_o add_v together_o rational_a and_o that_o which_o be_v contain_v under_o they_o medial_a and_o let_v the_o rest_n of_o the_o construction_n be_v in_o this_o as_o it_o be_v in_o the_o former_a and_o for_o that_o as_o the_o line_n ab_fw-la be_v to_o the_o line_n cd_o demonstration_n so_o be_v the_o line_n ae_n to_o the_o line_n cf_o &_o thâ—_z line_n ebb_n to_o the_o line_n fd_o but_o the_o line_n ab_fw-la be_v commensurable_a to_o the_o line_n cd_o by_o supposition_n wherefore_o the_o line_n ae_n be_v commensurable_a to_o the_o line_n cf_o and_o the_o line_n ebb_n to_o the_o line_n fd._n and_o for_o that_o as_o the_o line_n ae_n be_v to_o the_o line_n cf_o so_o be_v the_o line_n ebb_n to_o the_o line_n fd._n therefore_o alternate_o by_o the_o 16._o of_o the_o five_o as_o the_o line_n ae_n be_v to_o the_o line_n ebb_n so_o be_v the_o line_n cf_o to_o the_o line_n fd._n wherefore_o by_o composition_n also_o by_o the_o 18._o of_o the_o five_o as_o the_o line_n ab_fw-la be_v to_o the_o line_n ebb_n so_o be_v the_o line_n cd_o to_o the_o line_n fd._n wherefore_o by_o the_o 22._o of_o the_o six_o as_o the_o square_n of_o the_o line_n ab_fw-la be_v to_o the_o square_n of_o the_o line_n ebb_n so_o be_v the_o square_a of_o the_o line_n cd_o to_o the_o square_n of_o the_o line_n fd._n and_o in_o like_a sort_n may_v we_o prove_v that_o as_o the_o square_n of_o the_o line_n ab_fw-la be_v to_o the_o square_n of_o the_o line_n ae_n so_o be_v the_o square_a of_o the_o line_n cd_o to_o the_o square_n of_o the_o line_n cf._n wherefore_o by_o the_o 11._o of_o the_o five_o as_o the_o square_n of_o the_o line_n ab_fw-la be_v to_o the_o square_n of_o the_o line_n ae_n and_o ebb_n so_o be_v the_o square_a of_o the_o line_n cd_o to_o the_o square_n of_o the_o line_n cf_o and_o fd._n wherefore_o alternate_o by_o the_o 16._o of_o the_o five_o as_o the_o square_n of_o the_o line_n ab_fw-la be_v to_o the_o square_n of_o the_o line_n cd_o so_o be_v the_o square_n of_o the_o line_n ae_n and_o ebb_n to_o the_o square_n of_o the_o line_n cf_o and_o fd._n but_o the_o square_n of_o the_o line_n ab_fw-la be_v commensurable_a to_o the_o square_n of_o the_o line_n cd_o for_o the_o line_n ab_fw-la be_v commensurable_a to_o the_o line_n cd_o by_o supposition_n wherefore_o also_o the_o square_n of_o the_o line_n ae_n and_o ebb_n be_v commensurable_a to_o the_o square_n of_o the_o line_n cf_o and_o fd._n but_o the_o square_n of_o the_o line_n ae_n and_o ebb_n be_v incommensurable_a and_o be_v add_v together_o be_v rational_a wherefore_o the_o square_n of_o the_o line_n cf_o and_o fd_o be_v incommensurable_a &_o be_v add_v together_o be_v also_o rational_a and_o in_o like_a sort_n may_v we_o prove_v that_o that_o which_o be_v contain_v under_o the_o line_n ae_n and_o ebb_n twice_o be_v commensurable_a to_o that_o which_o be_v contain_v under_o the_o line_n cf_o and_o fd_o twice_o but_o that_o which_o be_v contain_v under_o the_o line_n ae_n and_o ebb_n twice_o be_v medial_a wherefore_o also_o that_o which_o be_v contain_v under_o the_o line_n cf_o and_o fd_o twice_o be_v medial_a wherefore_o the_o line_n cf_o and_o fd_o be_v incommensurable_a in_o power_n have_v that_o which_o be_v make_v of_o the_o square_n of_o they_o add_v together_o rational_a and_o that_o which_o be_v contain_v under_o they_o medial_a wherefore_o by_o the_o 39_o of_o the_o ten_o the_o whole_a line_n cd_o be_v irrational_a &_o be_v call_v a_o great_a line_n a_o line_n therefore_o commensurable_a to_o a_o great_a line_n be_v also_o a_o great_a line_n an_o other_o demonstration_n of_o peter_n montaureus_n to_o prove_v the_o same_o suppose_v that_o the_o line_n ab_fw-la be_v a_o great_a line_n and_o unto_o it_o let_v the_o line_n cd_o be_v commensurable_a any_o way_n that_o be_v either_o both_o in_o length_n and_o in_o power_n or_o else_o in_o power_n only_o montaureus_n then_o i_o say_v that_o the_o line_n cd_o also_o be_v a_o great_a
line_n divide_v the_o line_n ab_fw-la into_o his_o part_n in_o the_o point_n e._n and_o let_v the_o rest_n of_o the_o construction_n be_v in_o this_o as_o it_o be_v in_o the_o former_a and_o for_o that_o as_o the_o line_n ab_fw-la be_v to_o the_o line_n cd_o so_o be_v the_o line_n ae_n to_o the_o line_n cf_o and_o the_o line_n ebb_n to_o the_o line_n fd_o therefore_o as_o the_o line_n ae_n be_v to_o the_o line_n cf_o so_o be_v the_o line_n ebb_n to_o the_o line_n fd_o but_o the_o line_n ab_fw-la be_v commensurable_a to_o the_o line_n cd_o wherefore_o also_o the_o line_n ae_n be_v commensurable_a to_o the_o line_n cf_o and_o likewise_o the_o line_n ebb_n to_o the_o line_n fd._n and_o for_o thâ—â—_n as_o the_o line_n ae_n be_v to_o the_o line_n cf_o so_o be_v the_o line_n ebb_n to_o the_o line_n fd_o therefore_o alternate_o as_o the_o line_n ae_n be_v to_o the_o line_n ebb_n so_o be_v the_o line_n cf_o to_o the_o line_n fd._n wherefore_o by_o the_o 22._o of_o the_o six_o as_o the_o square_n of_o the_o line_n ae_n be_v to_o the_o square_n of_o the_o line_n ebb_n so_o be_v the_o square_a of_o the_o line_n cf_o to_o the_o square_n of_o the_o line_n fd._n wherefore_o by_o composition_n by_o the_o 18._o of_o the_o five_o as_o that_o which_o be_v make_v of_o the_o square_n of_o the_o line_n aâ—_z and_o eâ—_n add_v together_o be_v to_o the_o square_n of_o the_o line_n ebb_n so_o be_v that_o which_o be_v make_v of_o the_o squareâ—_n of_o the_o lyneâ—_n câ—_n and_o fd_o add_v together_o to_o the_o square_n of_o the_o line_n fd._n wherefore_o by_o contrary_a proportion_n as_o the_o square_n of_o the_o line_n ebb_n be_v to_o that_o which_o be_v make_v of_o the_o square_n of_o the_o line_n ae_n and_o eâ—_n add_v together_o so_o be_v the_o square_a of_o the_o line_n fd_v to_o that_o which_o be_v make_v of_o the_o square_n of_o the_o line_n cf_o and_o fd_o add_v together_o wherefore_o alternate_o as_o the_o square_n of_o the_o line_n ebb_n be_v to_o the_o square_n of_o the_o line_n fd_o so_o be_v that_o which_o be_v make_v of_o the_o square_n 〈◊〉_d the_o lâ—nes_n ae_n and_o ebb_v add_v together_o to_o that_o which_o be_v make_v of_o the_o square_n of_o the_o line_n cf_o and_o fd_o add_v together_o but_o the_o square_n of_o the_o line_n ebb_n be_v commensurable_a to_o the_o square_n of_o the_o line_n fd_o for_o it_o have_v already_o be_v prove_v that_o the_o line_n ebb_v and_o fd_o be_v commensurable_a wherefore_o that_o which_o be_v make_v of_o the_o square_n of_o the_o line_n ae_n &_o ebb_n add_v together_o be_v commensurable_a to_o that_o which_o be_v make_v of_o the_o square_n of_o câ—_n &_o fd_o add_v together_o but_o that_o which_o be_v make_v of_o the_o square_n of_o the_o line_n ae_n and_o ebb_v add_v together_o be_v rational_a by_o supposition_n wherefore_o that_o which_o be_v make_v of_o the_o square_n of_o the_o line_n cf_o and_o fd_o add_v together_o be_v also_o rational_a and_o as_o the_o line_n ae_n be_v to_o the_o line_n ebb_n so_o be_v the_o line_n cf_o to_o the_o line_n fd_o but_o as_o the_o line_n ae_n be_v to_o the_o line_n ebb_n so_o be_v the_o square_a of_o the_o line_n a_o 〈…〉_o contain_v under_o the_o line_n ae_n and_o ebb_n therefore_o at_o the_o line_n cf_o be_v to_o the_o line_n fd_o so_o be_v the_o square_a of_o the_o line_n ae_n to_o the_o parallelogram_n contain_v under_o the_o line_n ae_n and_o ebb_n &_o as_o the_o line_n cf_o be_v to_o the_o line_n fd_o so_o be_v the_o square_a of_o the_o line_n cf_o to_o the_o parallelogram_n contain_v under_o the_o line_n â—f_a &_o fd._n wherefore_o as_o the_o square_n of_o the_o line_n ae_n be_v to_o the_o parallelogram_n conâ—â—â—â—ed_v under_o the_o line_n ae_n and_o ebb_n so_o be_v the_o square_a of_o the_o line_n cf_o to_o the_o parallelogram_n contain_v under_o the_o line_n cf_o and_o fd._n wherefore_o 〈◊〉_d â—s_v the_o square_a of_o the_o line_n ae_n be_v to_o the_o square_n of_o the_o line_n cf_o so_o be_v the_o parallelogram_n contain_v under_o the_o line_n ae_n and_o ebb_n to_o the_o parallelogram_n 〈◊〉_d undeâ—_n the_o line_n â—â—_o and_o â—â—_o but_o the_o square_n of_o the_o line_n ae_n be_v commensurable_a to_o the_o square_n of_o the_o line_n cf_o for_o it_o be_v already_o prâ—â—â—d_v that_o the_o line_n ae_n and_o cf_o be_v commensurable_a wherefore_o the_o parallelogram_n contain_v under_o the_o line_n ae_n and_o ebb_n be_v commensurable_a to_o the_o parallelogram_n contain_v under_o the_o line_n cf_o and_o fd._n but_o the_o parallelogram_n contain_v under_o the_o line_n ae_n and_o ebb_n be_v medial_a by_o supposition_n wherefore_o the_o parallelogram_n contain_v under_o the_o line_n cf_o and_o â—d_a also_o be_v medial_a and_o as_o it_o have_v already_o be_v prove_v as_o the_o line_n ae_n be_v to_o the_o line_n ebb_n so_o be_v the_o line_n cf_o to_o the_o line_n fd._n but_o the_o line_n ae_n be_v by_o supposition_n incommensurable_a in_o power_n to_o the_o line_n ebb_n wherefore_o by_o the_o 10._o of_o the_o ten_o the_o line_n cf_o be_v incommensurable_a in_o power_n to_o the_o line_n fd._n wherefore_o the_o line_n cf_o and_o fd_o be_v incommensurable_a in_o power_n have_v that_o which_o be_v make_v of_o the_o square_n of_o they_o add_v together_o rational_a and_o that_o which_o be_v contain_v under_o they_o medial_a wherefore_o the_o whole_a line_n cd_o be_v by_o the_o 39_o of_o the_o ten_o a_o great_a line_n wherefore_o a_o line_n commensurable_a to_o a_o great_a line_n be_v also_o a_o great_a line_n which_o be_v require_v to_o be_v demonstrate_v a_o other_o more_o brief_a demonstration_n of_o the_o same_o after_o campane_n suppose_v that_o a_o be_v a_o great_a line_n unto_o which_o let_v the_o line_n b_o be_v commensurable_a either_o in_o length_n and_o power_n or_o in_o power_n only_o campane_n and_o take_v a_o rational_a line_n cd_o and_o upon_o it_o apply_v the_o superficies_n câ—_n equal_a to_o the_o square_n of_o the_o line_n a_o and_o also_o upon_o the_o line_n fe_o which_o be_v equal_a to_o the_o rational_a line_n cd_o apply_v the_o parallelogram_n fg_o equal_a to_o the_o square_n of_o the_o line_n b._n and_o forasmuch_o as_o the_o square_n of_o the_o two_o line_n a_o and_o â—_o be_v commensurable_a by_o supposition_n the_o superficies_n câ—_n shall_v be_v commensurable_a unto_o the_o superficies_n fg_o and_o therefore_o by_o the_o first_o of_o the_o six_o and_o ten_o of_o this_o book_n the_o line_n de_fw-fr be_v commensurable_a in_o length_n to_o the_o line_n gb_o and_o forasmuch_o as_o by_o the_o â—3_o of_o this_o book_n the_o line_n de_fw-fr be_v a_o four_o binomial_a line_n therefore_o by_o the_o â—6_o of_o this_o book_n the_o line_n ge_z be_v also_o a_o four_o binomial_a line_n wherefore_o by_o the_o 57_o of_o this_o book_n the_o line_n b_o which_o contain_v in_o power_n the_o superficies_n fg_o be_v a_o great_a line_n ¶_o the_o 51._o theorem_a the_o 69._o proposition_n a_o line_n commensurable_a to_o a_o line_n contain_v in_o power_n a_o rational_a and_o a_o medial_a be_v also_o a_o line_n contain_v in_o power_n a_o rational_a and_o a_o medial_a svppose_v that_o ab_fw-la be_v a_o line_n contain_v in_o power_n a_o rational_a and_o a_o medial_a and_o unto_o the_o line_n ab_fw-la let_v the_o line_n cd_o be_v commensurable_a whether_o in_o length_n and_o power_n or_o in_o power_n only_o then_o i_o say_v that_o the_o line_n cd_o be_v a_o line_n contain_v in_o power_n a_o rational_a &_o a_o medial_a duide_v the_o line_n ab_fw-la into_o his_o part_n in_o the_o point_n e._n construction_n wherefore_o by_o the_o 40._o of_o the_o ten_o the_o line_n ae_n and_o ebb_n be_v incommensurable_a in_o power_n have_v that_o which_o be_v make_v of_o the_o square_n of_o they_o add_v together_o medial_a and_o that_o which_o be_v contain_v under_o they_o national_a let_v the_o same_o construction_n be_v in_o this_o that_o be_v in_o the_o former_a demonstration_n and_o in_o like_a sort_n we_o may_v prove_v that_o the_o line_n cf_o and_o fd_o be_v incommensurable_a in_o power_n and_o that_o that_o which_o be_v make_v of_o the_o square_n of_o the_o line_n ae_n and_o ebb_n be_v commensurable_a to_o that_o which_o be_v make_v of_o the_o square_n of_o the_o line_n cf_o and_o fd_o and_o that_o that_o also_o which_o be_v contain_v under_o the_o line_n ae_n and_o ebb_n be_v commensurable_a to_o that_o which_o be_v contain_v under_o the_o line_n cf_o and_o fd._n wherefore_o that_o which_o be_v make_v of_o the_o square_n of_o the_o line_n cf_o and_o fd_o be_v medial_a and_o that_o which_o be_v contain_v under_o the_o line_n cf_o and_o fd_o be_v rational_a wherefore_o the_o whole_a line_n cd_o be_v a_o line_n contain_v in_o
to_o the_o same_o and_o so_o the_o line_n bd_o be_v a_o six_o residual_a line_n and_o the_o line_n kh_o be_v a_o six_o binomial_a line_n wherefore_o kh_o be_v a_o binomial_a line_n who_o name_n kf_a and_o fh_o be_v commensurable_a to_o the_o name_n of_o the_o residual_a line_n bd_o namely_o to_o bc_o and_o cd_o and_o in_o the_o self_n same_o proportion_n and_o the_o binomial_a line_n kh_o be_v in_o the_o self_n same_o order_n of_o binomial_a line_n that_o the_o residual_a bd_o be_v of_o residual_a line_n wherefore_o the_o square_a of_o a_o rational_a line_n apply_v unto_o a_o residual_a line_n make_v the_o breadth_n or_o other_o side_n a_o binomial_a line_n who_o name_n be_v commensurable_a to_o the_o name_n of_o the_o residual_a line_n and_o in_o the_o self_n same_o proportion_n and_o moreover_o the_o binomial_a line_n be_v in_o the_o self_n same_o order_n of_o binomial_a line_n that_o the_o residual_a line_n be_v of_o residual_a line_n which_o be_v require_v to_o be_v demonstrate_v the_o assumpt_v confirm_v now_o let_v we_o declare_v how_o as_o the_o line_n kh_o be_v to_o the_o line_n eh_o so_o to_o make_v the_o line_n hf_o to_o the_o line_n fe_o add_v unto_o the_o line_n kh_o direct_o a_o line_n equal_a to_o he_o and_o let_v the_o whole_a line_n be_v kl_o and_o by_o the_o ten_o of_o the_o six_o let_v the_o line_n he_o be_v divide_v as_o the_o whole_a line_n kl_o be_v divide_v in_o the_o point_n h_o let_v the_o line_n he_o be_v so_o divide_v in_o the_o point_n f._n wherefore_o as_o the_o line_n kh_o be_v to_o the_o line_n hl_o that_o be_v to_o the_o line_n he_o so_o be_v the_o line_n hf_o to_o the_o line_n fe_o an_o other_o demonstration_n after_o flussas_n suppose_v that_o a_o be_v a_o rational_a line_n and_o let_v bd_o be_v a_o residual_a line_n and_o upon_o the_o line_n bd_o apply_v the_o parallelogram_n dt_n equal_a to_o the_o square_n of_o the_o line_n a_o by_o the_o 45._o of_o the_o first_o make_v in_o breadth_n the_o line_n bt_n flussas_n then_o i_o say_v that_o bt_n be_v a_o binominall_a line_n such_o a_o one_o as_o be_v require_v in_o the_o proposition_n forasmuch_o as_o bd_o be_v a_o residual_a line_n let_v the_o line_n convenient_o join_v unto_o it_o be_v gd_o wherefore_o the_o line_n bg_o and_o gd_o be_v rational_a commensurable_a in_o power_n only_o construction_n upon_o the_o rational_a line_n bg_o apply_v the_o parallelogram_n by_o equal_a to_o the_o square_n of_o the_o line_n a_o and_o make_v in_o breadth_n the_o line_n be._n wherefore_o the_o line_n be_v be_v rational_a and_o commensurable_a in_o length_n to_o the_o line_n bg_o by_o the_o 20._o of_o the_o ten_o now_o forasmuch_o as_o the_o parallelogram_n by_o and_o td_v be_v equal_a for_o that_o they_o be_v each_o equal_a to_o the_o square_n of_o the_o line_n a_o demonstration_n therefore_o reciprocal_a by_o the_o 14._o of_o the_o six_o as_o the_o line_n bt_n be_v to_o the_o line_n be_v so_o be_v the_o line_n bg_o to_o the_o line_n bd._n wherefore_o by_o conversion_n of_o proportion_n by_o the_o corollary_n of_o the_o 19_o of_o the_o five_o as_o the_o line_n bt_n be_v to_o the_o line_n te_fw-la so_o be_v the_o line_n bg_o to_o the_o line_n gd_o as_o the_o line_n bg_o be_v to_o the_o line_n gd_o so_o let_v the_o line_n tz_n be_v to_o the_o line_n see_fw-mi by_o the_o corollary_n of_o the_o 10._o of_o the_o six_o wherefore_o by_o the_o 11._o of_o the_o five_o the_o line_n bt_n be_v to_o the_o line_n te_fw-la as_o the_o line_n tz_n be_v to_o the_o line_n ze._n for_o either_o of_o they_o be_v as_o the_o line_n bg_o be_v to_o the_o line_n gd_o wherefore_o the_o residue_n bz_o be_v to_o the_o residue_n zt_n as_o the_o whole_a bt_n be_v to_o the_o whole_a te_fw-la by_o the_o 19_o of_o the_o five_o wherefore_o by_o the_o 11._o of_o the_o five_o the_o line_n bz_o be_v to_o the_o line_n zt_n as_o the_o line_n zt_n be_v to_o the_o line_n ze._n wherefore_o the_o line_n tz_n be_v the_o mean_a proportional_a between_o the_o line_n bz_o and_o ze._n wherefore_o the_o square_a of_o the_o first_o namely_o of_o the_o line_n bz_o be_v to_o the_o square_n of_o the_o second_o namely_o of_o the_o line_n zt_n as_o the_o first_o namely_o the_o line_n bz_o be_v to_o the_o three_o namely_o to_o the_o line_n see_fw-mi by_o the_o corollary_n of_o the_o 20._o of_o the_o six_o and_o for_o that_o as_o the_o line_n bg_o be_v to_o the_o line_n gd_o so_o be_v the_o line_n tz_n to_o the_o line_n see_fw-mi but_o as_o the_o line_n tz_n be_v to_o the_o line_n see_fw-mi so_o be_v the_o line_n bz_o to_o the_o line_n zt_n wherefore_o as_o the_o line_n bg_o be_v to_o the_o line_n gd_o so_o be_v the_o line_n bz_o to_o the_o line_n zt_n by_o the_o 11._o of_o the_o five_o wherefore_o the_o line_n bz_o and_o zt_n be_v commensurable_a in_o power_n only_o as_o also_o be_v the_o line_n bg_o and_o gd_o which_o be_v the_o name_n of_o the_o residual_a line_n bd_o by_o the_o 10._o of_o this_o book_n wherefore_o the_o right_a line_n bz_o and_o see_fw-mi be_v commensurable_a in_o length_n for_o we_o have_v prove_v that_o they_o be_v in_o the_o same_o proportion_n that_o the_o square_n of_o the_o line_n bz_o and_o zt_n be_v and_o therefore_o by_o the_o corollary_n of_o the_o 15._o of_o this_o book_n the_o residue_n be_v which_o be_v a_o rational_a line_n be_v commensurable_a in_o length_n unto_o the_o same_o line_n bz_o wherefore_o also_o the_o line_n bg_o which_o be_v commensurable_a in_o length_n unto_o the_o line_n be_v shall_v also_o be_v commensurable_a in_o length_n unto_o the_o same_o line_n ez_o by_o the_o 12._o of_o the_o ten_o and_o it_o be_v prove_v that_o the_o line_n rz_n be_v to_o the_o line_n zt_n commensurable_a in_o power_n only_o wherefore_o the_o right_a line_n bz_o and_o zt_n be_v rational_a commensurable_a in_o power_n only_o wherefore_o the_o whole_a line_n bt_n be_v a_o binomial_a line_n by_o the_o 36._o of_o this_o book_n and_o for_o that_o as_o the_o line_n bg_o be_v to_o the_o line_n gd_o so_o be_v the_o line_n bz_o to_o the_o line_n zt_n therefore_o alternate_o by_o the_o 16._o of_o the_o five_o the_o line_n bg_o be_v to_o the_o line_n bz_o as_o the_o line_n gd_o be_v to_o the_o line_n zt_n but_o the_o line_n bg_o be_v commensurable_a in_o length_n unto_o the_o line_n bz_o wherefore_o by_o the_o 10._o of_o this_o book_n the_o line_n gd_o be_v commensurable_a in_o length_n unto_o the_o line_n zt_n wherefore_o the_o name_n bg_o and_o gd_o of_o the_o residual_a line_n bd_o be_v commensurable_a in_o length_n unto_o the_o name_n bz_o and_o zt_v of_o the_o binomial_a line_n bt_n and_o the_o line_n bz_o be_v to_o the_o line_n zt_n in_o the_o same_o proportion_n that_o the_o line_n bg_o be_v to_o the_o line_n gd_o as_o before_o it_o be_v more_o manifest_a and_o that_o they_o be_v of_o one_o and_o the_o self_n same_o order_n be_v thus_o prove_v if_o the_o great_a or_o less_o name_n of_o the_o residual_a line_n namely_o the_o right_a line_n bg_o or_o gd_o be_v commensurable_a in_o length_n to_o any_o rational_a line_n put_v the_o great_a name_n also_o or_o less_o namely_o bz_o or_o zt_n shall_v be_v commensurable_a in_o length_n to_o the_o same_o rational_a line_n put_v by_o the_o 12._o of_o this_o book_n and_o if_o neither_o of_o the_o name_n of_o the_o residual_a line_n be_v commensurable_a in_o length_n unto_o the_o rational_a line_n put_v neither_o of_o the_o name_n of_o the_o binomial_a line_n shall_v be_v commensurable_a in_o length_n unto_o the_o same_o rational_a line_n put_v by_o the_o 13._o of_o the_o ten_o and_o if_o the_o great_a name_n bg_o be_v in_o power_n more_o than_o the_o less_o name_n by_o the_o square_n of_o a_o line_n commensurable_a in_o length_n unto_o the_o line_n bg_o the_o great_a name_n also_o bz_o shall_v be_v in_o power_n more_o than_o the_o less_o by_o the_o square_n of_o a_o line_n commensurable_a in_o length_n unto_o the_o line_n bz_o and_o if_o the_o one_o be_v in_o power_n more_o by_o the_o square_n of_o a_o line_n incommensurable_a in_o length_n the_o other_o also_o shall_v be_v in_o power_n more_o by_o the_o square_n of_o a_o line_n incommensurable_a in_o length_n by_o the_o 14._o of_o this_o book_n the_o square_a therefore_o of_o a_o rational_a line_n etc_n etc_n which_o be_v require_v to_o be_v prove_v ¶_o the_o 90._o theorem_a the_o 114._o proposition_n joint_o if_o a_o parallelogram_n be_v contain_v under_o a_o residual_a line_n &_o a_o binomial_a line_n who_o name_n be_v commensurable_a to_o the_o name_n of_o the_o residual_a line_n and_o in_o the_o selâ—e_a same_o proportion_n the_o line_n which_o contain_v in_o power_n
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 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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
four_o the_o second_o and_o three_o be_v to_o be_v find_v which_o may_v between_o a_o &_o b_o be_v two_o mean_n in_o continual_a proportion_n as_o now_o lemma_n suppose_v such_o two_o line_n find_v and_o let_v they_o be_v c_z and_o d._n wherefore_o by_o euclides_n corollary_n as_o a_o be_v to_o b_o if_o a_o be_v take_v as_o first_o so_o shall_v the_o parallelipipedon_n describe_v of_o a_o be_v to_o the_o like_a parallelipipedon_n and_o in_o like_a sort_n describe_v of_o c_o be_v the_o second_o of_o the_o four_o line_n in_o continual_a proportion_n it_o be_v to_o weâ—e_v a_o c_o d_o and_o b._n or_o if_o b_o shall_v be_v take_v as_o first_o and_o that_o thus_o they_o be_v orderly_a in_o continual_a proportion_n b_o d_o c_o a_o then_o by_o the_o say_a corollary_n as_o b_o be_v to_o a_o so_o shall_v the_o parallelipipedon_n describe_v of_o b_o be_v unto_o the_o like_o parallelipipedon_n and_o in_o like_a sort_n describe_v oâ—_n d._n and_o unto_o a_o parallelipipedon_n of_o a_o or_o b_o at_o pleasure_n describe_v may_v a_o other_o of_o c_z or_o d_o be_v make_v like_o and_o in_o like_a sort_n situate_v or_o describe_v by_o the_o 27._o of_o this_o eleven_o book_n wherefore_o any_o two_o right_a line_n be_v give_v etc_o etc_o which_o be_v require_v to_o be_v do_v thus_o have_v i_o most_o brief_o bring_v to_o your_o understanding_n if_o first_o b_o be_v double_a to_o a_o than_o what_o parallelipipedon_n soever_o be_v describe_v of_o a_o the_o like_a parallelipipedon_n and_o in_o like_a sort_n describe_v of_o c_o shall_v be_v double_a to_o the_o parallelipipedon_n describe_v of_o a._n and_o so_o likewise_o second_o if_o a_o be_v double_a to_o b_o the_o parallelipipedon_n of_o d_z shall_v be_v double_a to_o the_o like_a of_o b_o describe_v both_o be_v like_o situate_v wherefore_o if_o of_o a_o or_o b_o be_v cube_n make_v the_o cube_n of_o c_o and_o d_o be_v prove_v double_a to_o they_o cube_n as_o that_o of_o c_o to_o the_o cube_n of_o a_o and_o the_o cube_n of_o d_o to_o the_o cube_n of_o b_o in_o the_o second_o case_n note_n and_o so_o of_o any_o proportion_n else_o between_o a_o and_o b._n now_o also_o do_v you_o most_o clear_o perceive_v the_o mathematical_a occasion_n whereby_o first_o of_o all_o man_n hypocrates_n to_o double_v any_o cube_n give_v be_v lead_v to_o the_o former_a lemma_n between_o any_o two_o right_a line_n give_v to_o find_v two_o other_o right_a line_n which_o shall_v be_v with_o the_o two_o first_o line_n in_o continual_a proportion_n after_o who_o time_n many_o year_n divine_a plato_n heron_n philo_n appollonius_n diâ—â—lâ—â—_n pappus_n sporus_n menechâ—us_n archytas_n tarentinus_n who_o make_v the_o wooden_a dove_n to_o sly_a eratoâ—â—hene_n nicomedes_n with_o many_o other_o to_o their_o immortal_a fame_n and_o renoun_n publish_v lemma_n diverse_a their_o witty_a device_n method_n and_o engine_n which_o yet_o be_v extant_a whereby_o to_o execute_v this_o problematical_a lemma_n but_o not_o withstand_v all_o the_o travail_n of_o the_o foresaid_a philosopher_n and_o mathematician_n yea_o and_o all_o other_o do_n and_o contrivinge_n unto_o this_o day_n about_o the_o say_a lemma_n yet_o there_o remain_v sufficient_a matter_n mathematical_o so_o to_o demonstrate_v the_o same_o that_o most_o exact_o &_o ready_o it_o may_v also_o be_v mechanical_o practise_v that_o who_o soever_o shall_v achieve_v that_o feat_n shall_v not_o be_v count_v a_o second_o archimedes_n but_o rather_o a_o perale_n mathematicien_fw-fr and_o mathematicorum_fw-la princeps_fw-la i_o will_v sundry_a way_n in_o my_o brief_a addition_n and_o annotation_n upon_o euclid_n excite_v you_o thereto_o cube_n yea_o and_o bring_v before_o your_o eye_n sundry_a new_a way_n by_o meinuented_a and_o in_o this_o book_n so_o place_v as_o matter_n thereof_o to_o my_o invention_n appertain_v may_v geve_v occasion_n leave_v the_o far_o full_a &_o absolute_a my_o conclude_v of_o the_o lemma_n to_o a_o other_o place_n and_o time_n which_o will_v now_o more_o compendious_o be_v do_v so_o great_a a_o part_n thereof_o be_v before_o hand_n in_o this_o book_n publish_v ¶_o a_o corollary_n add_v by_o flussas_n parallelipipedon_n consist_v upon_o equal_a base_n be_v in_o proportion_n the_o one_o to_o the_o other_o as_o their_o altitude_n be_v for_o if_o those_o altitude_n be_v cut_v by_o a_o plain_a superficies_n parallel_v to_o the_o base_n the_o section_n shall_v be_v in_o proportion_n the_o one_o to_o the_o other_o as_o the_o section_n of_o the_o base_n cut_v by_o the_o 25._o of_o this_o book_n which_o section_n of_o the_o base_n be_v the_o one_o to_o the_o other_o in_o that_o proportion_n that_o their_o side_n or_o the_o altitude_n of_o the_o solid_n be_v by_o the_o â—irst_n of_o the_o six_o wherefore_o the_o solid_n be_v the_o one_o to_o the_o other_o as_o their_o altitude_n be_v but_o if_o the_o base_n be_v unlike_a the_o self_n same_o thing_n may_v be_v prove_v by_o the_o corollary_n of_o the_o 25._o of_o this_o book_n which_o by_o the_o 25._o proposition_n be_v prove_v in_o like_a base_n ¶_o the_o 29._o theorem_a the_o 34._o proposition_n in_o equal_a parallelipipedon_n the_o base_n be_v reciprocal_a to_o their_o altitude_n and_o parallelipipedon_n who_o base_n be_v reciprocal_a to_o their_o altitude_n be_v equal_a the_o one_o to_o the_o other_o but_o now_o again_o suppose_v that_o the_o base_n of_o the_o parallelipipedon_n ab_fw-la and_o cd_o be_v reciprocal_a to_o their_o altitude_n case_n that_o be_v as_o the_o base_a eh_o be_v to_o the_o base_a np_n so_o let_v the_o altitude_n of_o the_o solid_a cd_o be_v to_o the_o altitude_n of_o the_o solid_a ab_fw-la then_o i_o say_v that_o the_o solid_a ab_fw-la be_v equal_a to_o the_o solid_a cd_o for_o again_o let_v the_o stand_a line_n be_v erect_v perpendicular_o to_o their_o base_n and_o now_o if_o the_o base_a eh_o be_v equal_a to_o the_o base_a np_n but_o as_o the_o base_a eh_o be_v to_o the_o base_a np_n so_o be_v the_o altitude_n of_o the_o solid_a cd_o to_o the_o altitude_n of_o the_o solid_a ab_fw-la wherefore_o the_o altitude_n oâ—_n the_o solid_a cd_o be_v equal_a to_o the_o altitude_n of_o the_o solid_a ab_fw-la but_o parallelipipedon_n consist_v upon_o equal_a base_n and_o under_o one_o and_o the_o self_n same_o altitude_n be_v by_o the_o 31._o of_o the_o eleven_o equal_v the_o one_o to_o the_o other_o wherefore_o the_o solid_a ab_fw-la be_v equal_a to_o the_o solid_a cd_o but_o now_o suppose_v that_o the_o base_a eh_o be_v not_o equal_a to_o the_o base_a np_n but_o let_v the_o base_a eh_o be_v the_o great_a wherefore_o also_o the_o altitude_n of_o the_o solid_a cd_o that_o be_v the_o line_n cm_o be_v great_a than_o the_o altitude_n of_o the_o solid_a ab_fw-la that_o be_v than_o the_o line_n ag._n put_v again_o by_o the_o 3._o of_o the_o first_o the_o line_n et_fw-la equal_a to_o the_o line_n agnostus_n and_o make_v perfect_v the_o solid_a cz_n now_o for_o that_o as_o the_o base_a eh_o be_v to_o the_o base_a np_n so_o be_v the_o line_n mc_n to_o the_o line_n ag._n but_o the_o line_n agnostus_n be_v equal_a to_o the_o line_n et_fw-la wherefore_o as_o the_o base_a eh_o be_v to_o the_o base_a np_n so_o be_v the_o line_n cm_o to_o the_o line_n et_fw-la but_o as_o the_o base_a eh_o be_v to_o the_o base_a np_n so_o by_o the_o 32._o of_o the_o eleven_o be_v the_o solid_a ab_fw-la to_o the_o solid_a cz_n â—or_a the_o solides_fw-la ab_fw-la and_o cz_n be_v under_o equal_a altitude_n and_o as_o the_o line_n cm_o be_v to_o the_o line_n et_fw-la so_o by_o the_o 1._o of_o the_o six_o be_v the_o base_a mp_n to_o the_o base_a pâ—t_n and_o by_o the_o 32._o of_o the_o eleven_o the_o solid_a cd_o to_o the_o solid_a cz_n wherefore_o also_o by_o the_o 11._o and_o 9_o of_o the_o five_o as_o the_o solid_a ab_fw-la be_v to_o the_o solid_a cz_n so_o be_v the_o solid_a cd_o to_o the_o solid_a cz_n wherefore_o either_o of_o these_o solid_n ab_fw-la and_o cd_o have_v to_o the_o solid_a cz_n one_o and_o the_o same_o proportion_n wherefore_o by_o the_o 7._o of_o the_o five_o the_o solid_a ab_fw-la be_v equal_a to_o the_o solid_a cd_o which_o be_v require_v to_o be_v demonstrate_v again_o suppose_v that_o the_o base_n of_o the_o parallelipipedon_n ab_fw-la and_o cd_o be_v reciprocal_a to_o their_o altitude_n case_n that_o be_v as_o the_o base_a eh_o be_v to_o the_o base_a np_n so_o let_v the_o altitude_n of_o the_o solid_a cd_o be_v to_o the_o altitude_n of_o the_o solid_a ab_fw-la then_o i_o say_v that_o the_o solid_a ab_fw-la be_v equal_a to_o the_o solid_a cd_o for_o the_o same_o order_n of_o construction_n remain_v for_o that_o as_o the_o base_a eh_o be_v to_o the_o base_a np_n so_o be_v the_o altitude_n of_o the_o solid_a cd_o to_o the_o altitude_n
and_o so_o of_o such_o like_a who_o can_v not_o ready_o fall_v into_o archimedes_n reckon_v and_o account_n by_o his_o method_n to_o find_v the_o proportion_n of_o the_o circumference_n of_o any_o circle_n to_o his_o diameter_n to_o be_v almost_o triple_a and_o one_o seven_o of_o the_o diameter_n but_o to_o be_v more_o than_o triple_a and_o ten_o one_o &_o seventithe_n that_o be_v to_o be_v less_o than_o 3_o 1_o â—_o and_o more_o than_o 3_o 10_o 71._o and_o where_o archimedes_n use_v a_o poligonon_n figure_n of_o 96._o side_n he_o that_o for_o exercise_v sake_n or_o for_o earnest_a desire_n of_o a_o more_o nearness_n will_v use_v polygonon_n figure_n of_o 384._o side_n or_o more_o may_v well_o travail_v therein_o till_o either_o weariness_n cause_v he_o stay_v or_o else_o he_o find_v his_o labour_n fruitless_a in_o deed_n archimedes_n conclude_v proportion_n of_o the_o circumference_n to_o the_o diameter_n have_v hitherto_o serve_v the_o vulgar_a andâ—_n mechanical_a worâ—_n man_n wherewith_o who_o so_o be_v not_o concent_v let_v his_o own_o methodical_a travail_n satisfy_v his_o desire_n or_o let_v he_o procure_v other_o thereto_o for_o narrow_a term_n of_o great_a and_o less_o find_v and_o appoint_v to_o the_o circumference_n will_v also_o win_v to_o the_o area_n of_o the_o circle_n a_o near_a quantity_n see_v it_o be_v well_o demonstrate_v of_o archimedâ—s_n that_o a_o triangle_n rectangle_n of_o who_o two_o side_n contain_v the_o right_a angle_n one_o be_v equal_a to_o the_o semidiameter_n of_o the_o circle_n and_o the_o other_o to_o the_o circumference_n of_o the_o same_o be_v equal_a to_o the_o area_n of_o that_o circle_n upon_o which_o two_o theorem_n it_o follow_v that_o the_o square_n make_v of_o the_o diameter_n be_v in_o that_o proportion_n to_o the_o circle_n very_o near_o in_o which_o 14_o be_v to_o 11â—_n wherefore_o every_o circle_n be_v eleven_o fowertenthe_n well_o near_o of_o the_o square_n about_o he_o describe_v the_o one_o side_n then_o of_o that_o square_n divide_v into_o 14._o equal_a part_n circle_n and_o from_o that_o point_n which_o end_v the_o eleven_o part_n draw_v to_o the_o opposite_a side_n a_o line_n parallel_n to_o the_o other_o side_n and_o so_o make_v perfect_a the_o parallelogram_n then_o by_o the_o last_o proposition_n of_o the_o second_o book_n unto_o that_o parallelogram_n who_o one_o side_n have_v those_o 11._o equal_a part_n make_v a_o square_a equal_a then_o be_v it_o evident_a that_o square_a to_o be_v equal_a to_o the_o circle_n about_o which_o the_o first_o square_a be_v describe_v as_o you_o may_v here_o behold_v in_o these_o figure_n gentle_a friend_n the_o great_a desire_n which_o i_o have_v that_o both_o with_o pleasure_n and_o also_o profit_n thou_o may_v spend_v thy_o time_n in_o these_o excellent_a study_n do_v cause_v i_o here_o to_o furnish_v thou_o somewhat_o extraordinary_o about_o the_o circle_n not_o only_o by_o point_v unto_o thou_o the_o wellspring_n of_o archimedes_n his_o so_o much_o wonder_v at_o and_o just_o commend_v travail_n in_o the_o former_a 3._o theorem_n here_o repeat_v but_o also_o to_o make_v thou_o more_o apt_a to_o understand_v and_o practice_v this_o and_o other_o book_n follow_v where_o use_v of_o the_o circle_n may_v be_v have_v in_o any_o consideration_n as_o in_o cones_n cylinder_n and_o sphere_n etc_n etc_n ¶_o a_o corollary_n 1._o by_o archimedes_n second_v theorem_a as_o i_o have_v here_o allege_v they_o it_o be_v manifest_a that_o a_o parallelogram_n contain_v either_o under_o the_o semidiameter_n and_o half_a the_o circumference_n or_o under_o the_o half_a semidiameter_n and_o whole_a circumference_n of_o any_o circle_n be_v equal_a to_o the_o circle_n by_o the_o 41._o of_o the_o first_o and_o first_o of_o the_o sixâ—_o ¶_o a_o corollary_n 2._o likewise_o it_o be_v evident_a that_o the_o parallelogram_n contain_v under_o the_o semidiameter_n and_o half_a of_o any_o portion_n of_o the_o circumference_n of_o a_o circle_n give_v be_v equal_a to_o that_o sector_n of_o the_o same_o circle_n to_o which_o the_o whole_a portion_n of_o the_o circumference_n give_v do_v belong_v or_o you_o may_v use_v the_o half_a semidiameter_n and_o the_o whole_a portion_n of_o the_o circumference_n as_o side_n of_o the_o say_a parallelogram_n the_o far_a win_n and_o infer_v i_o commiâ—â—â—_n to_o your_o skill_n care_n and_o â—â—udyâ—_n but_o in_o a_o other_o sort_n will_v i_o geve_v you_o new_a aid_n and_o instruction_n here_o ¶_o a_o theorem_a of_o all_o circle_n the_o circumference_n to_o their_o own_o diameter_n have_v one_o and_o the_o same_o proportion_n in_o what_o one_o circle_n soever_o they_o be_v assign_v that_o be_v as_o archimedes_n have_v demonstrate_v almost_o as_o 22._o to_o 7_o or_o near_o if_o near_o be_v fouâ—d_v until_o the_o very_a precise_a proportion_n be_v demonstrate_v which_o what_o soever_o it_o be_v in_o all_o circumference_n to_o their_o proper_a diameter_n will_v be_v demonstrate_v one_o and_o the_o same_o a_o corollary_n 1._o wherefore_o if_o two_o circle_n be_v propound_v which_o suppose_v to_o be_v a_o and_o b_o as_z the_o circumference_n of_o a_o be_v to_o the_o circumference_n of_o b_o so_o be_v the_o diameter_n of_o a_o to_o the_o diameter_n of_o b._n for_o by_o the_o former_a theorem_a as_o the_o circumference_n of_o a_o be_v to_o his_o own_o diameter_n so_o be_v the_o circumference_n of_o b_o to_o his_o own_o diameter_n wherefore_o alternate_o as_o the_o circumference_n of_o a_o be_v to_o the_o circumference_n of_o b_o so_o be_v the_o diameter_n of_o a_o to_o the_o diameter_n of_o b_o which_o be_v require_v to_o be_v demonstrate_v a_o corollary_n 2._o it_o be_v now_o then_o evident_a that_o we_o can_v geve_v two_o circle_n who_o circumference_n one_o to_o the_o other_o shall_v have_v any_o proportion_n give_v in_o two_o right_a line_n the_o great_a mechanical_a use_n beside_o mathematical_a consideration_n which_o these_o two_o corollaryes_fw-mi may_v have_v in_o wheel_n of_o milles_n clock_n crane_n and_o other_o engine_n for_o water_n work_v and_o for_o war_n and_o many_o other_o purpose_n the_o earnest_n and_o witty_a mechanicien_n will_v soon_o boult_v out_o &_o glad_o practice_v â—_o john_n dee_fw-mi ¶_o the_o 2._o theorem_a the_o 2._o proposition_n circle_n be_v in_o that_o proportion_n the_o one_o to_o the_o other_o that_o the_o square_n of_o their_o diameter_n be_v svppose_v that_o there_o be_v two_o circle_n abcd_o and_o efgh_a and_o let_v their_o diameter_n be_v bd_o and_o fh_o then_o i_o say_v that_o as_o the_o square_n of_o the_o line_n db_o be_v to_o the_o square_n of_o the_o line_n fh_o case_n so_o be_v the_o circle_n abcd_o to_o the_o circle_n efgh_o for_o if_o the_o circle_n abcd_o be_v not_o unto_o the_o circle_n efgh_o as_o the_o square_n of_o the_o line_n bd_o be_v to_o the_o square_n of_o the_o line_n fh_o then_o the_o square_a of_o the_o line_n bd_o shall_v be_v to_o the_o square_n of_o the_o line_n fh_o as_o the_o circle_n abcd_o be_v to_o a_o superficies_n either_o less_o than_o the_o circle_n efgh_o or_o great_a first_o let_v the_o square_n of_o the_o line_n bd_o be_v to_o the_o square_n of_o the_o line_n fh_o as_o the_o circle_n abcd_o be_v to_o a_o superficies_n less_o than_o the_o circle_n efgh_o namely_o to_o the_o superficies_n s._n describe_v by_o the_o 6._o of_o the_o four_o in_o the_o circle_n efgh_v a_o square_a efgh_o now_o this_o square_a thus_o describe_v be_v great_a than_o the_o half_a of_o the_o circle_n efgh_o for_o if_o by_o the_o point_n e_o f_o g_o h_o we_o draw_v right_a line_n touch_v the_o circle_n be_v the_o square_a efgh_o be_v the_o half_a of_o the_o square_n describe_v about_o the_o circle_n but_o the_o square_n describe_v about_o the_o circle_n be_v great_a than_o the_o circle_n wherefore_o the_o square_a efgh_o which_o be_v inscribe_v in_o the_o circle_n be_v great_a than_o the_o half_a of_o the_o circle_n efgh_o divide_v the_o circumference_n of_o fg_o gh_o and_o he_o into_o two_o equal_a part_n in_o the_o point_n king_n l_o m_o n._n and_o draw_v these_o right_a line_n eke_o kf_n fl_fw-mi lg_n gm_n mh_o hn_o and_o ne._n wherefore_o every_o one_o of_o these_o triangle_n ekf_a flg_n gmh_o and_o hne_n be_v great_a than_o the_o half_a of_o the_o segment_n of_o the_o circle_n which_o be_v describe_v about_o it_o for_o if_o by_o the_o point_n king_n l_o m_o n_o be_v draw_v line_n touch_v the_o circle_n and_o then_o be_v make_v perfect_v the_o parallelogram_n make_v of_o the_o right_a line_n of_o fg_o gh_o &_o he_o every_o one_o of_o the_o triangle_n ekf_n flg_n gmh_o &_o hne_n be_v the_o half_a of_o the_o parallelogragramme_n which_o be_v describe_v about_o it_o by_o the_o 41._o of_o the_o first_o but_o the_o segment_n describe_v about_o it_o be_v less_o than_o the_o parallelogram_n wherefore_o every_o one_o
of_o these_o triangle_n ekf_a flg_n gmh_o and_o hne_n be_v great_a than_o the_o half_a of_o the_o segment_n of_o the_o circle_n which_o be_v describe_v about_o it_o now_o then_o devide_v the_o circumference_n remain_v into_o two_o equal_a part_n and_o draw_v right_a line_n from_o the_o point_n where_o those_o division_n be_v make_v &_o so_o continual_o do_v this_o we_o shall_v at_o the_o length_n by_o the_o 1._o of_o the_o ten_o leave_v certain_a segment_n of_o the_o circle_n which_o shall_v be_v less_o than_o the_o excess_n whereby_o the_o circle_n efgh_o exceed_v the_o superficies_n s._n for_o it_o have_v be_v prove_v in_o the_o first_o proposition_n of_o the_o ten_o book_n that_o two_o unequal_a magnitude_n be_v give_v if_o from_o the_o great_a be_v take_v away_o more_o than_o the_o half_a and_o likewise_o again_o from_o the_o residue_n more_o than_o the_o halâ—e_n and_o so_o continual_o there_o shall_v at_o the_o length_n be_v leave_v a_o certain_a magnitude_n which_o shall_v be_v less_o than_o the_o less_o magnitude_n give_v let_v there_o be_v such_o segment_n leave_v &_o let_v the_o segment_n of_o the_o circle_n efgh_o namely_o which_o be_v make_v by_o the_o line_n eke_o kf_n fl_fw-mi lg_n gm_n mh_o hn_o and_o ne_fw-fr be_v less_o than_o the_o excess_n whereby_o the_o circle_n efgh_o exceed_v the_o superficies_n s._n wherefore_o the_o residue_n namely_o the_o poligonon_n figure_n ekflgmhn_n be_v great_a than_o the_o superficies_n s._n inscribe_v in_o the_o circle_n abcd_o a_o poligonon_n figure_n like_o to_o the_o poligonon_n figure_n ekflgmhn_n and_o let_v the_o same_o be_v axbocpdr_n wherefore_o by_o the_o proposition_n next_o go_v before_o as_o the_o square_n of_o the_o line_n bd_o be_v to_o the_o square_n of_o the_o line_n fh_o so_o be_v the_o poligonon_n figure_n axbocpdr_n to_o the_o poligonon_n figure_n ekflgmhn_n but_o as_o the_o square_n of_o the_o line_n bd_o be_v to_o the_o square_n of_o the_o line_n fg_o so_o be_v the_o circle_n abcd_o suppose_v to_o be_v to_o the_o superficies_n s._n wherefore_o by_o the_o 11._o of_o the_o five_o as_o the_o circle_n abcd_o be_v to_o the_o superficies_n s_o so_o be_v the_o poligonon_n figure_n axbocpdr_n to_o the_o poligonon_n figure_n ekflgmhn_n wherefore_o alternate_o by_o the_o 16._o of_o the_o five_o as_o the_o circle_n abcd_o be_v to_o the_o poligonon_n figure_n describe_v in_o it_o so_o be_v the_o superficies_n s_o to_o the_o poligonon_n figure_n ekflgmhn_n but_o the_o circle_n abcd_o be_v great_a than_o the_o poligonon_n figure_n describe_v in_o it_o wherefore_o also_o the_o superficies_n saint_n be_v great_a than_o the_o poligonon_n figure_n ekflghmn_a but_o it_o be_v also_o less_o which_o be_v impossible_a wherefore_o as_o the_o square_n of_o the_o line_n bd_o be_v to_o the_o square_n of_o the_o line_n fh_o so_o be_v not_o the_o circle_n abcd_o to_o any_o superficies_n less_o than_o the_o circle_n efgh_o case_n in_o like_a sort_n also_o may_v wprove_v that_o as_o the_o square_n of_o the_o line_n fh_o be_v to_o the_o square_n of_o the_o line_n bd_o so_o be_v not_o the_o circle_n efgh_o to_o any_o superficies_n less_o than_o the_o circle_n abcd._n i_o say_v namely_o that_o as_o the_o square_n of_o the_o line_n bd_o be_v to_o the_o square_n of_o the_o line_n fh_o so_o be_v not_o the_o circle_n abcd_o to_o any_o superficies_n great_a they_o than_o the_o circle_n efgh_o for_o if_o it_o be_v possible_a let_v it_o be_v to_o a_o great_a namely_o to_o the_o superficies_n s._n wherefore_o by_o conversion_n as_o the_o square_n of_o the_o line_n fh_o be_v to_o the_o square_n of_o the_o line_n bd_o so_o be_v the_o superficies_n s_o to_o the_o circle_n abcd._n prove_v but_o as_o the_o superficies_n saint_n be_v to_o the_o circle_n abcd_o so_o be_v the_o circle_n efgh_o to_o some_o superficies_n lâ—sse_v they_o the_o circle_n abcd._n wherefore_o by_o the_o 11._o of_o the_o five_o as_o the_o square_n of_o the_o line_n fh_o be_v to_o the_o square_n of_o the_o line_n bd_o so_o be_v the_o circle_n efgh_o to_o some_o superficies_n less_o than_o the_o circle_n abcd_o which_o be_v in_o the_o first_o case_n prove_v to_o be_v impossible_a wherefore_o as_o the_o square_n of_o the_o line_n bd_o be_v to_o the_o square_n of_o the_o line_n fh_o so_o be_v not_o the_o circle_n abcd_o to_o any_o superficies_n great_a than_o the_o circle_n efgh_o and_o it_o be_v also_o prove_v that_o it_o be_v not_o to_o any_o less_o wherefore_o as_o the_o square_n of_o the_o lâ—ne_n bd_o be_v to_o the_o square_n of_o the_o line_n fh_o so_o be_v the_o circle_n abcd_o to_o the_o circle_n efgh_o wherefore_o circle_n be_v in_o that_o proportion_n the_o one_o to_o the_o other_o that_o the_o square_n of_o their_o diameter_n be_v which_o be_v require_v to_o be_v prove_v ¶_o a_o assumpt_n i_o say_v now_o that_o the_o superficies_n saint_n be_v great_a than_o the_o circle_n efgh_o as_o the_o superficies_n saint_n be_v to_o the_o circle_n abcd_o so_o be_v the_o circle_n efgh_o to_o some_o superficies_n less_o than_o the_o circle_n abcd._n for_o as_o the_o superficies_n saint_n be_v to_o the_o circle_n abcd_o so_o let_v the_o circle_n efgh_o be_v to_o the_o superficies_n t._n now_o i_o say_v that_o the_o superficies_n t_n be_v less_o than_o the_o circle_n abcd._n for_o for_o that_o as_o the_o superficies_n saint_n be_v to_o the_o circle_n abcd_o so_o be_v the_o circle_n efgh_o to_o the_o superficies_n t_o therefore_o alternate_o by_o the_o 16._o of_o the_o five_o as_o the_o superficies_n saint_n be_v to_o the_o circle_n efgh_o so_o be_v the_o circle_n abcd_o to_o the_o superficies_n t._n but_o the_o superficies_n saint_n be_v great_a than_o the_o circle_n efgh_o by_o supposition_n wherefore_o also_o the_o circle_n abcd_o be_v great_a than_o the_o superficies_n t_n by_o the_o 14._o of_o the_o five_o wherefore_o as_o the_o superficies_n saint_n be_v to_o the_o circle_n abcd_o so_o be_v the_o circle_n efgh_o to_o some_o superficies_n less_o than_o the_o circle_n abcd_o which_o be_v require_v to_o be_v demonstrate_v ¶_o a_o corollary_n add_v by_o flussas_n circle_n have_v the_o one_o to_o the_o other_o that_o proportion_n that_o like_o poligonon_n figure_n and_o in_o like_a sort_n describe_v in_o they_o have_v for_o it_o be_v by_o the_o first_o proposition_n prove_v that_o the_o poligonon_n figure_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 the_o diameter_n have_v which_o proportion_n likewise_o by_o this_o proposition_n the_o circle_n have_v ¶_o very_o needful_a problem_n and_o corollarye_n by_o master_n john_n dee_n invent_v who_o wonderful_a use_n also_o be_v partly_o declare_v a_o problem_n 1._o two_o circle_n be_v give_v to_o find_v two_o right_a line_n which_o have_v the_o same_o proportion_n one_o to_o the_o other_o that_o the_o give_v circle_n have_v oâ—e_v to_o the_o otherâ—_n suppose_v a_o and_o b_o to_o be_v the_o diameter_n of_o two_o circle_n give_v i_o say_v that_o two_o right_a line_n be_v to_o be_v find_v have_v that_o proportion_n that_o the_o circle_n of_o a_o have_v to_o the_o circle_n of_o b._n let_v to_o a_o &_o b_o by_o the_o 11_o of_o the_o six_o a_o three_o proportional_a line_n be_v find_v which_o suppose_v to_o be_v c._n construction_n i_o say_v now_o that_o a_o have_v to_o c_o that_o proportion_n which_o the_o circle_n of_o a_o have_v to_o the_o circle_n of_o b._n for_o forasmuch_o as_o a_o b_o and_o c_o be_v by_o construction_n three_o proportional_a line_n demonstration_n the_o square_a of_o a_o be_v to_o the_o square_n of_o b_o as_z a_o be_v to_o c_o by_o the_o corollary_n of_o the_o 20._o of_o the_o six_o â—_o but_o as_o the_o square_n of_o the_o line_n a_o be_v to_o the_o square_n of_o the_o line_n b_o so_o be_v the_o circle_n who_o diameter_n be_v the_o line_n a_o to_o the_o circle_n who_o diameter_n be_v the_o line_n b_o by_o this_o second_o of_o the_o eleven_o wherefore_o the_o circle_n of_o the_o lineâ—_z a_o and_o b_o be_v in_o the_o proportion_n of_o the_o right_a line_n a_o and_o c._n therefore_o two_o circle_n be_v give_v we_o have_v find_v two_o right_a line_n have_v the_o same_o proportion_n between_o they_o that_o the_o circle_n give_v have_v one_o to_o the_o other_o which_o ought_v to_o be_v do_v a_o problem_n 2._o two_o circle_n be_v give_v and_o a_o right_a line_n to_o find_v a_o other_o right_a line_n to_o which_o the_o line_n give_v shall_v have_v that_o proportion_n which_o the_o one_o circle_n have_v to_o the_o other_o suppose_v two_o circle_n give_v which_o let_v be_v a_o &_o b_o &_o a_o right_a line_n give_v which_o let_v be_v c_o i_o say_v that_o a_o other_o right_a line_n be_v to_o be_v â—ounde_v to_o which_o the_o line_n c_o shall_v have_v that_o proportion_n that_o
last_o proposition_n in_o the_o second_o book_n and_o also_o of_o the_o 31._o in_o the_o six_o book_n ¶_o a_o problem_n 5._o two_o unequal_a circle_n be_v give_v to_o find_v a_o circle_n equal_a to_o the_o excess_n of_o the_o great_a to_o the_o less_o suppose_v the_o two_o unequal_a circle_n give_v to_o be_v abc_n &_o def_n &_o let_v abc_n be_v the_o great_a construction_n who_o diameter_n suppose_v to_o be_v ac_fw-la &_o the_o diameter_n of_o def_n suppose_v to_o be_v df._n i_o say_v a_o circle_n must_v be_v find_v equal_a to_o that_o excess_n in_o magnitude_n by_o which_o abc_n be_v great_a thâ—_z def_n by_o the_o first_o of_o the_o four_o in_o the_o circle_n abc_n apply_v a_o right_a line_n equal_a to_o df_o who_o one_o end_n let_v be_v at_o c_o and_o the_o other_o let_v be_v at_o b._n fron_n b_o to_o a_o draw_v a_o right_a line_n by_o the_o 30._o of_o the_o three_o it_o may_v appear_v demonstration_n that_o abc_n be_v a_o right_a angle_n and_o thereby_o abc_n the_o triangle_n be_v rectangle_v wherefore_o by_o the_o first_o of_o the_o two_o corollary_n here_o before_o the_o circle_z abc_n be_v equal_a to_o the_o circle_n def_n for_o bc_o by_o construction_n be_v equal_a to_o df_o and_o more_o over_o to_o the_o circle_n who_o diameter_n be_v ab_fw-la that_o circle_n therefore_o who_o diameter_n be_v ab_fw-la be_v the_o circle_n contain_v the_o magnitude_n by_o which_o abc_n be_v great_a than_o def_n wherefore_o two_o unequal_a circle_n be_v give_v we_o have_v find_v a_o circle_n equal_a to_o the_o excess_n of_o the_o great_a to_o the_o less_o which_o ought_v to_o be_v do_v a_o problem_n 6._o a_o circle_n be_v give_v to_o find_v two_o circle_n equal_a to_o the_o same_o which_o find_v circle_n shall_v have_v the_o one_o to_o the_o other_o any_o proportion_n give_v in_o two_o right_a line_n suppose_v abc_n a_o circle_n give_v and_o the_o proportion_n give_v let_v it_o be_v that_o which_o be_v between_o the_o two_o right_a line_n d_z and_o e._n i_o say_v that_o two_o circle_n be_v to_o be_v find_v equal_a to_o abc_n and_o with_o all_o one_o to_o the_o other_o construction_n in_o the_o proportion_n of_o d_o to_o e._n let_v the_o diameter_n of_o abc_n be_v ac_fw-la as_z d_z be_v to_o e_o so_o let_v ac_fw-la be_v divide_v by_o the_o 10._o of_o the_o six_o in_o the_o point_n f._n at_o fletcher_n to_o the_o line_n ac_fw-la let_v a_o perpendicular_a be_v draw_v fb_o and_o let_v it_o mete_v the_o circumference_n at_o the_o point_n b._n from_o the_o point_n b_o to_o the_o point_v a_o and_o c_o let_v right_a line_n be_v draw_v ba_o and_o bc._n i_o say_v that_o the_o circle_n who_o diameter_n be_v the_o line_n basilius_n and_o bc_o be_v equal_a to_o the_o circle_n abc_n and_o that_o those_o circle_n have_v to_o their_o diameter_n the_o line_n basilius_n and_o bc_o be_v one_o to_o the_o other_o in_o the_o proportion_n of_o the_o line_n d_o to_o the_o line_n e._n for_o first_o that_o they_o be_v equal_a it_o be_v evident_a by_o reason_n that_o abc_n be_v a_o triangle_n rectangle_n wherefore_o by_o the_o 47._o of_o the_o first_o the_o square_n of_o basilius_n and_o bc_o be_v equal_a to_o the_o square_n of_o ac_fw-la and_o so_o by_o this_o second_o it_o be_v manife_a the_o two_o circle_n to_o be_v equal_a to_o the_o circle_n abc_n second_o as_o d_o be_v to_o â—_o so_o be_v of_o to_o fc_n by_o construction_n and_o as_o the_o line_n of_o be_v to_o the_o line_n fc_o so_o be_v the_o square_a of_o the_o line_n â—a_o to_o the_o square_n of_o the_o line_n bc._n which_o thing_n we_o will_v brief_o prove_v thus_o the_o parallelogram_n contain_v under_o ac_fw-la and_o of_o use_n be_v equal_a to_o the_o square_n of_o basilius_n by_o the_o lemma_n after_o the_o 32._o of_o the_o ten_o book_n and_o by_o the_o same_o lemma_n or_o assumpt_n the_o parallelogram_n contain_v under_o ac_fw-la and_o â—c_z be_v equal_a to_o the_o square_n of_o the_o line_n bc._n wherefore_o as_o the_o first_o parallelogram_n have_v itself_o to_o the_o secondâ—_n so_o have_v the_o square_a of_o basilius_n equal_a to_o the_o first_o parallelogram_n itself_o to_o the_o square_n of_o bc_o equal_a to_o the_o second_o parallelogram_n but_o both_o the_o parallelogram_n have_v one_o height_n namely_o the_o line_n ac_fw-la and_o base_n the_o line_n of_o and_o fc_n wherefore_o as_o of_o be_v to_o fc_n so_o be_v the_o parallelogâ—amme_n contain_v under_o ac_fw-la of_o to_o the_o parallelogram_n contain_v under_o ac_fw-la fc_fw-la by_o the_o fiâ—st_n of_o the_o six_o and_o therefore_o as_o of_o be_v to_o fc_n so_o be_v the_o square_a of_o basilius_n to_o the_o square_n of_o bc._n and_o as_o the_o square_n of_o basilius_n be_v to_o the_o square_n of_o bc_o so_o be_v the_o circle_n who_o diameter_n be_v basilius_n to_o the_o circle_n who_o diameter_n be_v bc_o by_o this_o second_o of_o the_o twelve_o wherefore_o the_o circle_n who_o diameter_n be_v basilius_n be_v to_o the_o circle_n who_o diameter_n be_v bc_o as_o d_z be_v to_o e._n and_o before_o we_o prove_v they_o equal_a to_o the_o circle_n abc_n wherâ—fore_o a_o circle_n be_v give_v we_o have_v find_v two_o circle_n equal_a to_o the_o same_o which_o have_v the_o one_o to_o the_o other_o any_o proportion_n give_v in_o two_o right_a line_n which_o ought_v to_o be_v do_v note_n addition_n heâ—e_o may_v you_o perceive_v a_o other_o way_n how_o to_o execute_v my_o first_o problem_n for_o if_o you_o make_v a_o right_a angle_n contain_v of_o the_o diameter_n give_v as_o in_o this_o figure_n suppose_v they_o basilius_n and_o bc_o and_o then_o subtend_v the_o right_a angle_n with_o the_o line_n ac_fw-la and_o from_o the_o right_a angle_n let_v fall_v a_o line_n perpendicular_a to_o the_o base_a ac_fw-la that_o perpendicular_a at_o the_o point_n of_o his_o fall_n divide_v ac_fw-la into_o of_o and_o fc_n of_o the_o proportion_n require_v a_o corollary_n it_o follow_v of_o thing_n manifest_o prove_v in_o the_o demonstration_n of_o this_o problem_n that_o in_o a_o triangle_n rectangle_n if_o from_o the_o right_a angle_n to_o the_o base_a a_o perpendicular_a be_v let_v fall_v the_o same_o perpendicular_a cut_v the_o base_a into_o two_o part_n rectangle_n in_o that_o proportion_n one_o to_o the_o other_o that_o the_o square_n of_o the_o righâ—_n line_n contain_v the_o right_a angle_n be_v in_o one_o to_o the_o other_o those_o on_o the_o one_o side_n the_o perpendicular_a be_v compare_v to_o those_o on_o the_o other_o both_o square_a and_o segment_n a_o problem_n 7._o between_o two_o circle_n give_v to_o find_v a_o circle_n middle_a proportional_a let_v the_o two_o circle_n give_v be_v acd_o and_o bef_a i_o say_v that_o a_o circle_n be_v to_o be_v find_v which_o between_o acd_a and_o bef_n be_v middle_a proportional_a construction_n let_v the_o diameter_n of_o acd_a be_v ad_fw-la and_o of_o bef_n let_v bâ—_n be_v the_o diameter_n between_o ad_fw-la and_o bf_o find_v a_o line_n middle_a proportional_a by_o the_o 13._o of_o the_o six_o which_o let_v be_v hk_a i_o say_v that_o a_o circle_n who_o diameter_n be_v hk_n be_v middle_a proportional_a between_o acd_a and_o bef_a to_o ad_fw-la hk_o and_o bf_o three_o right_a line_n in_o continual_a proportion_n by_o construction_n let_v a_o four_o line_n be_v find_v to_o which_o bf_o shall_v have_v that_o proportion_n that_o ad_fw-la have_v to_o hk_n by_o the_o 12._o of_o the_o six_o &_o let_v that_o line_n be_v â—_o demonstration_n it_o be_v manifest_a that_o the_o â—ower_a line_n ad_fw-la hk_o bf_o and_o l_o be_v in_o continual_a proportion_n for_o by_o construction_n as_o ad_fw-la be_v to_o hk_n so_o be_v bâ—_n to_o l._n and_o by_o construction_n on_o as_z ad_fw-la be_v to_o hk_n so_o be_v hk_v to_o bf_o wherefore_o hk_n be_v to_o bf_o as_o bf_o be_v to_o l_o by_o the_o 11._o of_o the_o five_o wherefore_o the_o 4._o line_n be_v in_o continual_a proportion_n wherefore_o as_o the_o first_o be_v to_o the_o three_o that_o be_v ad_fw-la to_o bf_o so_o be_v the_o square_a of_o the_o first_o to_o the_o square_n of_o the_o second_o that_o be_v the_o square_a of_o ad_fw-la to_o the_o square_n of_o hk_n by_o the_o corollary_n of_o the_o 20._o of_o the_o six_o and_o by_o the_o same_o corollary_n as_o hk_n be_v to_o l_o so_o be_v the_o square_a of_o hk_n to_o the_o square_n of_o bf_o but_o by_o alternate_a proportion_n the_o line_n ad_fw-la be_v to_o bf_o as_o hk_n be_v to_o l_o wherefore_o the_o square_a of_o ad_fw-la be_v to_o the_o square_n of_o hk_n as_o the_o square_n of_o hk_n be_v to_o the_o square_n of_o bf_o wherefore_o the_o square_a of_o hk_n be_v middle_a proportional_a between_o the_o square_n of_o ad_fw-la and_o the_o square_n of_o bf_o but_o as_o the_o square_n be_v
with_o the_o altitude_n of_o the_o say_a pyramid_n a_o and_o b_o shall_v be_v equal_a by_o the_o 6._o of_o this_o book_n wherefore_o by_o the_o first_o part_n of_o this_o proposition_n the_o base_n of_o the_o pyramid_n c_o to_o d_z be_v reciprocal_a with_o the_o altitude_n of_o d_o to_o c._n but_o in_o what_o proportion_n be_v the_o base_n c_o to_o d_z in_o the_o same_o be_v the_o base_n a_o to_o b_o forasmuch_o as_o they_o be_v equal_a and_o in_o what_o proportion_n be_v the_o altitude_n of_o d_z to_o c_z in_o the_o same_o be_v the_o altitude_n of_o b_o to_o a_o which_o altitude_n be_v likewise_o equal_a wherefore_o by_o the_o 11._o of_o the_o five_o in_o what_o proportion_n the_o base_n a_o to_o b_o be_v in_o the_o same_o reciprocal_a be_v the_o altitude_n of_o the_o pyramid_n b_o to_o a._n in_o like_a sort_n by_o the_o second_o part_n of_o this_o proposition_n may_v be_v prove_v the_o converse_n of_o this_o corollary_n the_o same_o thing_n follow_v also_o in_o a_o prism_n and_o in_o a_o side_v column_n as_o have_v before_o at_o large_a be_v declare_v in_o the_o corollary_n of_o the_o 40._o proposition_n of_o the_o 11._o book_n for_o those_o solid_n be_v in_o proportion_n the_o one_o to_o the_o other_o as_o the_o pyramid_n or_o parallelipipedon_n for_o they_o be_v either_o part_n of_o equemultiplices_fw-la or_o equemultiplices_fw-la to_o part_n the_o 10._o theorem_a the_o 10._o proposition_n every_o cone_n be_v the_o three_o part_n of_o a_o cilind_a have_v one_o and_o the_o self_n same_o base_a and_o one_o and_o the_o self_n same_o altitude_n with_o it_o svppose_v that_o there_o be_v a_o cone_fw-mi have_v to_o his_o base_a the_o circle_n abcd_o and_o let_v there_o be_v a_o cilind_a have_v the_o self_n same_o base_a and_o also_o the_o same_o altitude_n that_o the_o cone_fw-mi have_v then_o i_o say_v that_o the_o cone_fw-mi be_v the_o three_o part_n of_o the_o cilind_a that_o be_v that_o the_o cilind_a be_v in_o treble_a proportion_n to_o the_o cone_n for_o if_o the_o cilind_a be_v not_o in_o treble_a proportion_n to_o the_o cone_n than_o the_o cilind_a be_v either_o in_o great_a proportion_n then_o triple_a to_o the_o cone_n or_o else_o in_o less_o first_o let_v it_o be_v in_o great_a than_o triple_a 〈◊〉_d and_o describe_v by_o the_o 6._o of_o the_o four_o in_o the_o circle_n abcd_o a_o square_a abcd._n now_o the_o square_n abcd_o be_v great_a than_o the_o half_a of_o the_o circle_n abcd_o for_o if_o about_o the_o circle_n abcd_o we_o describe_v a_o square_a the_o square_n describe_v in_o the_o circle_n abcd_o be_v the_o half_a of_o the_o square_n describe_v about_o the_o circle_n and_o let_v there_o be_v parallelipipedon_n prism_n describe_v upon_o those_o square_n prism_n equal_a in_o altitude_n with_o the_o cilind_a but_o prism_n be_v in_o that_o proportion_n the_o one_o to_o the_o other_o that_o their_o base_n be_v by_o the_o 32._o of_o the_o eleven_o and_o 5._o corollary_n of_o the_o 7._o of_o this_o book_n wherefore_o the_o prism_n describe_v upon_o the_o square_a abcd_o be_v the_o half_a of_o the_o prism_n describe_v upon_o the_o square_a that_o be_v describe_v about_o the_o circle_n now_o the_o clind_a be_v less_o than_o the_o prism_n which_o be_v make_v of_o the_o square_n describe_v abouâ—_n the_o circle_n abcd_o be_v equal_a in_o altitude_n with_o it_o for_o it_o contain_v it_o wherefore_o the_o prism_n describe_v upon_o the_o square_a abcd_o and_o be_v equal_a in_o altitude_n with_o the_o cylinder_n be_v great_a than_o half_a the_o cylinder_n divide_v by_o the_o 30._o of_o the_o three_o the_o circumference_n ab_fw-la bc_o cd_o and_o da_z into_o two_o equal_a part_n in_o the_o point_n e_o f_o g_o h_o and_o draw_v these_o right_a line_n ae_n ebb_n bf_o fc_o cg_o gd_o dh_o &_o ha._n wherefore_o every_o one_o of_o these_o triangle_n aeb_fw-mi bfc_n cgd_v and_o dha_n be_v great_a than_o half_a of_o that_o segment_n of_o the_o circle_n abcd_o which_o be_v describe_v about_o it_o as_o we_o have_v before_o in_o the_o 2._o proposition_n declare_v describe_v upon_o every_o one_o of_o these_o triangle_n aeb_fw-mi bfc_n cgd_v and_o dha_n a_o prism_n of_o equal_a altitude_n with_o the_o cylinder_n wherefore_o every_o one_o of_o these_o prism_n so_o describe_v be_v great_a than_o the_o half_a part_n of_o the_o segment_n of_o the_o cylinder_n that_o be_v set_v upon_o the_o say_a segment_n of_o the_o circle_n for_o if_o by_o the_o point_n e_o f_o g_o h_o be_v draw_v parallel_a line_n to_o the_o line_n ab_fw-la bc_o cd_o and_o da_z and_o then_o be_v make_v perfect_a the_o parallelogram_n make_v by_o those_o parallel_a line_n and_o moreover_o upon_o those_o parallelogram_n be_v erect_v parallelipipedon_n equal_v in_o altitude_n with_o the_o cylinder_n the_o prism_n which_o be_v describe_v upon_o each_o of_o the_o triangle_n aeb_fw-mi bfc_n cgd_v and_o dha_n be_v the_o half_n of_o every_o one_o of_o those_o parallelipipedon_n and_o the_o segment_n of_o the_o cylinder_n be_v less_o than_o those_o parallelipipedon_n so_o describe_v wherefore_o also_o every_o one_o of_o the_o prism_n which_o be_v describe_v upon_o the_o triangle_n aeb_fw-mi bfc_n cgd_v and_o dha_n be_v great_a than_o the_o half_a of_o the_o segment_n of_o the_o cylinder_n set_n upon_o the_o say_a segment_n now_o therefore_o devide_v every_o one_o of_o the_o circumference_n remain_v into_o two_o equal_a part_n and_o draw_v right_a line_n and_o raise_v up_o upon_o every_o one_o of_o these_o triangle_n prism_n equal_a in_o altitude_n with_o the_o cylinder_n and_o do_v this_o continual_o we_o shall_v at_o the_o length_n by_o the_o first_o of_o the_o ten_o leave_v certain_a segment_n of_o the_o cylinder_n which_o shall_v be_v less_o then_o the_o excess_n whereby_o the_o cylinder_n exceed_v the_o cone_n more_o than_o thrice_o let_v those_o segment_n be_v ae_n ebb_n bf_o fc_o cg_o gd_o dh_o and_o ha._n wherefore_o the_o prism_n remain_v who_o base_a be_v the_o polygonon_n â—igure_n aebfcgdh_n and_o altitude_n the_o self_n same_o that_o the_o cylinder_n have_v be_v great_a than_o the_o cone_n take_v three_o time_n prism_n but_o the_o prism_n who_o base_a be_v the_o polygonon_n figure_n aebfcgdh_n and_o altitude_n the_o self_n same_o that_o the_o cylinder_n have_v be_v treble_a to_o the_o pyramid_n who_o base_a be_v the_o polygonon_n figure_n aebfcgda_n and_o altitude_n the_o self_n same_o that_o the_o cone_fw-mi have_v by_o the_o corollary_n of_o the_o 3._o of_o this_o book_n wherefore_o also_o the_o pyramid_n who_o base_a be_v the_o polygonon_n figure_n aebfcgdh_n and_o top_n the_o self_n same_o that_o the_o cone_fw-mi have_v be_v great_a than_o the_o cone_n which_o have_v to_o his_o base_a the_o circle_n abcd._n but_o it_o be_v also_o less_o for_o it_o be_v contain_v of_o it_o which_o be_v impossible_a wherefore_o the_o cylinder_n be_v not_o in_o great_a proportion_n then_o triple_a to_o the_o cone_n i_o say_v moreover_o that_o the_o cylinder_n be_v not_o in_o less_o proportion_n then_o triple_a to_o the_o coneâ—_n for_o if_o it_o be_v possible_a let_v the_o cylinder_n be_v in_o less_o proportion_n then_o triple_a to_o the_o cone_n wherefore_o by_o conversion_n the_o cone_n be_v great_a than_o the_o three_o part_n of_o the_o cylinder_n describe_v now_o by_o the_o six_o of_o the_o four_o in_o the_o circle_n abcd_o a_o square_a abcd._n wherefore_o the_o square_a abcd_o be_v great_a than_o the_o half_a of_o the_o circle_n abcd_o upon_o the_o square_n abcd_o describe_v a_o pyramid_n have_v one_o &_o the_o self_n same_o altitude_n with_o the_o cone_n wherefore_o the_o pyramid_n so_o describe_v be_v great_a they_o half_n of_o the_o cone_fw-it for_o if_o as_o we_o have_v before_o declare_v we_o describe_v a_o square_a about_o the_o circle_n the_o square_a abcd_o be_v the_o half_a of_o the_o square_n describe_v about_o the_o circle_n and_o if_o upon_o the_o square_n be_v describe_v parallelipipedon_n equal_v in_o altitude_n with_o the_o cone_n which_o solid_n be_v also_o call_v prism_n the_o prism_n or_o parallelipipedon_n describe_v upon_o the_o square_a abcd_o be_v the_o half_a of_o the_o prism_n which_o be_v describe_v upon_o the_o square_n describe_v about_o the_o circle_n for_o they_o be_v the_o one_o to_o the_o other_o in_o that_o proportion_n that_o their_o base_n be_v by_o the_o 32._o of_o the_o eleven_o &_o 5._o corollary_n of_o the_o 7._o of_o this_o book_n wherefore_o also_o their_o three_o part_n be_v in_o the_o self_n same_o proportion_n by_o the_o 15._o of_o the_o five_o wherefore_o the_o pyramid_n who_o base_a be_v the_o square_a abcd_o be_v the_o half_a of_o the_o pyramid_n set_v upon_o the_o square_n describe_v about_o the_o circle_n but_o the_o pyramid_n set_v upon_o the_o square_n describe_v about_o the_o circle_n be_v great_a than_o the_o cone_n who_o
fd._n wherefore_o cones_fw-la &_o cylinder_n consist_v upon_o equal_a base_n be_v in_o proportion_n the_o one_o to_o the_o other_o as_o their_o altitude_n which_o be_v require_v to_o be_v demonstrate_v ¶_o the_o 15._o theorem_a the_o 15._o proposition_n in_o equal_a cones_n and_o cylinder_n the_o base_n be_v reciprocal_a to_o their_o altitude_n and_o cones_fw-la and_o cylinder_n who_o base_n be_v reciprocal_a to_o their_o altitude_n be_v equal_a the_o one_o to_o the_o other_o svppose_v that_o these_o cones_fw-la acl_n egn_n or_o these_o cylinder_n axe_n eo_fw-la who_o base_n be_v the_o circle_n abcd_o efgh_o and_o axe_n kl_o and_o mn_v which_o axe_n be_v also_o the_o altitude_n of_o the_o cones_fw-la &_o cylinder_n be_v equal_a the_o one_o to_o the_o other_o cones_n then_o i_o say_v that_o the_o base_n of_o the_o cylinder_n xa_n &_o eo_fw-la be_v reciprokal_n to_o their_o altitude_n that_o be_v that_o as_o the_o base_a abcd_o be_v to_o the_o base_a efgh_o proposition_n so_o the_o altitude_n mn_v to_o the_o altitude_n kl_o for_o the_o altitude_n kl_o be_v either_o equal_a to_o the_o altitude_n mn_a or_o not_o first_o let_v it_o be_v equal_a case_n but_o the_o cylinder_n axe_n be_v equal_a to_o the_o cylinder_n eq_n but_o cones_fw-la and_o cylinder_n consist_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 be_v by_o the_o 11._o of_o the_o twelve_o wherefore_o the_o base_a abcd_o be_v equal_a to_o the_o base_a efgh_o wherefore_o also_o they_o be_v reciprokal_a as_o the_o base_a abcd_o be_v to_o the_o base_a efgh_o so_o be_v the_o altitude_n mn_v to_o the_o altitude_n kl_o but_o now_o suppose_v that_o the_o altitude_n lk_n be_v not_o equal_a to_o the_o altitude_n m._n n_o construction_n but_o let_v the_o altitude_n mn_v be_v great_a and_o by_o the_o 3._o of_o the_o first_o from_o the_o altitude_n mn_v take_v away_o pm_v equal_a to_o the_o altitude_n kl_o so_o that_o let_v the_o line_n pm_n be_v put_v equal_a to_o the_o line_n kl_o and_o by_o the_o point_n p_o let_v there_o be_v extend_v a_o plain_a superficies_n tus_z which_o let_v cut_v the_o cylinder_n eo_fw-la and_o be_v a_o parallel_n to_o the_o two_o opposite_a plain_a superâ—icieces_n that_o be_v to_o the_o circle_n efgh_o and_o ro._n cylinder_n and_o make_v the_o base_a the_o circle_n efgh_o &_o the_o altitude_n mp_n imagine_v a_o cylinder_n es._n and_o for_o that_o the_o cylinder_n axe_n be_v equal_a to_o the_o cylinder_n eo_fw-la and_o there_o be_v a_o other_o cylinder_n es_fw-ge therefore_o by_o the_o 7._o of_o the_o five_o as_o the_o cylinder_n axe_n be_v to_o the_o cylinder_n es_fw-ge so_o be_v the_o cylinder_n eo_fw-la to_o the_o cylinder_n es._n but_o as_o the_o cylinder_n axe_n be_v to_o the_o cylinder_n es_fw-ge so_o be_v the_o base_a abcd_o to_o the_o base_a efgh_o for_o the_o cylinder_n axe_n and_o es_fw-ge be_v under_o one_o and_o the_o self_n same_o altitude_n and_o as_o the_o cylinder_n eo_fw-la be_v to_o the_o cylinder_n es_fw-ge so_o be_v the_o altitude_n mn_v to_o the_o altitude_n mp_n for_o cylinder_n consist_v upon_o equal_a base_n be_v in_o proportion_n the_o one_o to_o the_o other_o as_o their_o altitude_n wherefore_o as_o the_o base_a abcd_o be_v â—o_o the_o base_a efgh_o so_o be_v the_o altitude_n mn_v to_o the_o altitude_n mp_n but_o the_o altitude_n pm_n be_v equal_a to_o the_o altitude_n kl_o wherefore_o as_o the_o base_a abcd_o be_v to_o the_o base_a efgh_o so_o be_v the_o altitude_n mn_v to_o the_o altitude_n kl_o wherefore_o in_o the_o equal_a cylinder_n axe_n and_o eo_fw-la the_o base_n be_v reciprocal_a to_o their_o altitude_n but_o now_o suppose_v that_o the_o base_n of_o the_o cylinder_n axe_n and_o eo_fw-la be_v reciprokal_n to_o their_o altitude_n that_o be_v as_o the_o base_a abcd_o be_v to_o the_o base_a efgh_o so_o be_v the_o altitude_n mn_v to_o the_o altitude_n kl_o demonstrate_v then_o i_o say_v that_o the_o cylinder_n axe_n be_v equal_a to_o the_o cylinder_n eo_fw-la for_o the_o self_n same_o order_n of_o construction_n remain_v for_o that_o as_o the_o base_a abcd_o be_v to_o the_o base_a efgh_o so_o be_v the_o altitude_n mn_v to_o the_o altitude_n kl_o but_o the_o altitude_n kl_o be_v equal_a to_o the_o altitude_n pm_n wherefore_o as_o the_o base_a abcd_o be_v to_o the_o base_a efgh_o so_o be_v the_o altitude_n mn_v to_o the_o altitude_n pm_n but_o as_o the_o base_a abcd_o be_v to_o the_o base_a efgh_o so_o be_v the_o cylinder_n axe_n to_o the_o cylinder_n es_fw-ge for_o they_o be_v under_o equal_a altitude_n and_o as_o the_o altitude_n mn_o be_v to_o the_o altitude_n pm_n so_o be_v the_o cylinder_n eo_fw-la to_o the_o cylinder_n es_fw-ge by_o the_o 14._o of_o the_o twelve_o wherefore_o also_o as_o the_o cylinder_n axe_n be_v to_o the_o cylinder_n es_fw-ge so_o be_v the_o cylinder_n eo_fw-la to_o the_o cylinder_n es._n wherefore_o the_o cylinder_n axe_n be_v equal_a to_o the_o cylinder_n eo_fw-la by_o the_o 9_o of_o the_o five_o and_o so_o also_o be_v it_o in_o the_o cones_fw-la which_o haâ—â—_n the_o self_n same_o base_n and_o altitude_n with_o the_o cylinder_n wherefore_o in_o equal_a cones_fw-la and_o cylinder_n the_o base_n be_v reciprocal_a to_o their_o altitude_n etc_n etc_n which_o be_v require_v to_o be_v demonstrate_v a_o corollary_n add_v by_o campane_n and_o flussas_n hitherto_o have_v be_v show_v the_o passion_n and_o propriety_n of_o cones_fw-la and_o cylinder_n who_o altitude_n fall_v perpendicular_o upon_o the_o base_n now_o will_v we_o declare_v that_o cones_fw-la and_o cilinder_n who_o altitude_n fall_v oblique_o upon_o their_o base_n have_v also_o the_o self_n same_o passion_n and_o propriety_n which_o the_o foresay_a cones_fw-la and_o cilinder_n have_v forasmuch_o as_o in_o the_o ten_o of_o this_o book_n it_o be_v say_v that_o every_o cone_n be_v the_o three_o part_n of_o a_o cilind_a have_v one_o and_o the_o self_n same_o base_a &_o one_o &_o the_o self_n same_o altitude_n with_o it_o which_o thing_n be_v demonstrate_v by_o a_o cilind_a give_v who_o base_a be_v cut_v by_o a_o square_n inscribe_v in_o it_o and_o upon_o the_o side_n of_o the_o square_n be_v describe_v isosceles_a triangle_n make_v a_o polygonon_n figure_n and_o again_o upon_o the_o side_n of_o this_o polygonon_n figure_n be_v infinite_o after_o the_o same_o manner_n describe_v other_o isosceles_a triangle_n take_v away_o more_o they_o the_o half_a as_o have_v oftentimes_o be_v declare_v therefore_o it_o be_v manifest_a that_o the_o solid_n set_v upon_o these_o base_n be_v under_o the_o same_o altitude_n that_o the_o cilind_a incline_v be_v and_o be_v also_o include_v in_o the_o same_o cilind_a do_v take_v away_o more_o than_o the_o half_a of_o the_o cilind_a and_o also_o more_o they_o the_o half_a of_o the_o residue_n as_o it_o have_v be_v prove_v in_o erect_a cylinder_n for_o these_o incline_n solid_n be_v under_o equal_a altitude_n and_o upon_o equal_a base_n with_o the_o erect_a solid_n be_v equal_a to_o the_o erect_a solid_n by_o the_o corollary_n of_o the_o _o of_o the_o eleven_o wherefore_o they_o also_o in_o like_a sort_n as_o the_o erect_a take_v away_o more_o than_o the_o half_a if_o therefore_o we_o compare_v the_o incline_v cilind_a to_o a_o cone_n set_v upon_o the_o self_n same_o base_a and_o have_v his_o altitude_n erect_v and_o reason_n by_o a_o argument_n lead_v to_o a_o impossibility_n by_o the_o demonstration_n of_o the_o ten_o of_o this_o book_n we_o may_v prove_v that_o the_o side_v solid_a include_v in_o the_o incline_v cylinder_n be_v great_a than_o the_o triple_a of_o his_o pyramid_n and_o it_o be_v also_o equal_a to_o the_o same_o which_o be_v impossible_a and_o this_o be_v the_o first_o case_n wherein_o it_o be_v prove_v that_o the_o cilind_a not_o be_v equal_a to_o the_o triple_a of_o the_o cone_n be_v not_o great_a than_o the_o triple_a of_o the_o same_o and_o as_o touch_v the_o second_o case_n we_o may_v after_o the_o same_o manner_n conclude_v that_o that_o â—ided_v solid_a contain_v in_o the_o cylinder_n be_v great_a than_o the_o cylinder_n which_o be_v very_o absurd_a wherefore_o if_o the_o cylinder_n be_v neither_o great_a than_o the_o triple_a of_o the_o cone_n nor_o less_o it_o must_v needs_o be_v equal_a to_o the_o same_o the_o demonstration_n of_o these_o incline_v cylinder_n most_v plain_o follow_v the_o demonstration_n of_o the_o erect_a cylinder_n for_o it_o have_v already_o be_v prove_v that_o pyramid_n and_o side_v solid_n which_o be_v also_o call_v general_o prism_n be_v set_v upon_o equal_a base_n and_o under_o one_o and_o the_o self_n same_o altitude_n whether_o the_o altitude_n be_v erect_v or_o incline_v be_v equal_a the_o one_o to_o the_o other_o namely_o be_v in_o proportion_n as_o their_o base_n be_v by_o
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
also_o divide_v into_o two_o equal_a part_n be_v cylinder_n which_o two_o equal_a cylinder_n let_v be_v ig_o and_o fk_v the_o axe_n of_o ig_n suppose_v to_o be_v hn_o and_o of_o fk_v the_o axe_n to_o be_v nm_o and_o for_o that_o fg_o be_v a_o upright_o cylinder_n and_o at_o the_o point_n n_o cut_v by_o a_o plain_a superficies_n parallel_v to_o his_o opposite_a base_n the_o common_a section_n of_o that_o plain_a superficies_n and_o the_o cylinder_n fg_o must_v be_v a_o circle_n circle_n equal_a to_o his_o base_a flb_n and_o have_v his_o centre_n the_o point_n n._n which_o circle_n let_v be_v jok_v and_o see_v that_o flb_n be_v by_o supposition_n equal_a to_o the_o great_a circle_n in_o a_o jok_n also_o shall_v be_v equal_a to_o the_o great_a circle_n in_o a_o contain_v also_o by_o reason_n mh_o be_v by_o supposition_n equal_a to_o the_o diameter_n of_o a_o and_o nh_o by_o construction_n half_a of_o mh_o it_o be_v manifest_a that_o nh_v be_v equal_a to_o the_o semidiameter_n of_o a._n if_o therefore_o you_o suppose_v a_o cone_n to_o have_v the_o circle_n jok_v to_o hiâ—_z base_a and_o nh_v to_o his_o heith_n the_o sphere_n a_o shall_v be_v to_o that_o cone_n quadrupla_fw-la by_o the_o 2._o theorem_a let_v that_o cone_n be_v hiok_v wherefore_o a_o be_v quadrupla_fw-la to_o hiok_n and_o the_o cylinder_n ig_n have_v the_o same_o base_a with_o hiok_n the_o circle_n jok_v and_o the_o same_o heith_n the_o right_a line_n nh_v be_v triple_a to_o the_o cone_n hiok_v by_o the_o 10._o of_o this_o twelve_o book_n but_o to_o ig_v the_o whole_a cylinder_n fg_o be_v double_a as_o be_v prove_v wherefore_o fg_o be_v triple_a and_o triple_a to_o the_o cone_n hiok_v that_o be_v sextuple_a and_o a_o be_v prove_v quadrupla_fw-la to_o the_o same_o hiok_n wherefore_o fg_o be_v to_o hiok_v as_o 6._o to_o 1_o and_o a_o be_v to_o hiok_v as_o 4._o to_o 1_o demonstrate_v therefore_o fg_o be_v to_o a_o as_z 6_o to_o 4_o which_o in_o the_o least_o term_n be_v as_z 3_o to_o 2._o but_o 3_o to_o 2_o be_v the_o term_n of_o sesquialtera_fw-la proportion_n wherefore_o the_o cylinder_n fg_o be_v to_o a_o sesquialtera_fw-la in_o proportion_n second_o forasmuch_o as_o the_o superficies_n of_o a_o cylinder_n his_o two_o opposite_a base_n except_v be_v equal_a to_o that_o circle_n who_o semidiameter_n be_v middle_a proportional_a between_o the_o side_n of_o the_o cylinder_n and_o the_o diameter_n of_o his_o base_a as_o unto_o the_o 10._o of_o this_o book_n i_o have_v add_v but_o of_o fg_o the_o side_n bg_o be_v parallel_n and_o equal_a to_o the_o axe_n mh_o must_v also_o be_v equal_a to_o the_o diameter_n of_o a._n and_o the_o base_a flb_n be_v by_o supposition_n equal_a to_o the_o great_a circle_n in_o a_o contain_v must_v have_v his_o diameter_n fb_o equal_a to_o the_o say_a diameter_n of_o a._n the_o middle_a proportional_a therefore_o between_o bg_o and_o fb_o be_v equal_a each_o to_o other_o shalâ—_n be_v a_o line_n equal_a to_o either_o of_o they_o as_o iâ—_z 〈◊〉_d set_n bg_o and_o fb_o together_o as_o one_o line_n and_o upon_o that_o line_n compose_v as_o a_o diameter_n make_v a_o semicircle_n and_o from_o the_o centre_n to_o the_o circumference_n draw_v a_o linâ—_n perpendicular_a to_o the_o say_a diameter_n demonstrate_v by_o the_o 〈◊〉_d of_o the_o six_o that_o perpendicular_a be_v middle_n proportional_a between_o fb_o and_o bg_o the_o semidiametersâ—_n and_o he_o himself_o also_o a_o semidiameter_n and_o therefore_o by_o the_o definition_n of_o a_o circle_n equal_a to_o fb_o and_o likewise_o to_o bg_o and_o a_o circle_n have_v his_o semidiameter_n equal_a to_o the_o diameter_n fb_o be_v quadruple_a to_o the_o circle_n flb_n for_o the_o square_n of_o every_o whole_a line_n be_v quadruple_a to_o the_o squâ—re_n of_o his_o half_a line_n as_o may_v be_v prove_v by_o the_o 4._o of_o the_o second_o and_o by_o the_o second_o of_o this_o twelve_o circle_n be_v one_o to_o the_o other_o as_o the_o square_n of_o their_o diameter_n be_v wherefore_o the_o superficies_n cylindricall_fw-mi of_o fg_o alone_o be_v quadrupla_fw-la to_o his_o base_a flb_n but_o if_o a_o certain_a quantity_n be_v dupla_fw-la to_o one_o thing_n and_o a_o other_o quadrupla_fw-la to_o the_o same_o one_o thing_n those_o two_o quantity_n together_o be_v sextupla_fw-la to_o the_o same_o one_o thing_n therefore_o see_v the_o base_a opposite_a to_o flb_n be_v equal_a to_o to_o flb_n add_v to_o flb_n make_v that_o compound_n double_a to_o flb_n that_o double_a add_v to_o the_o cylindrical_a superficies_n of_o fg_o do_v make_v a_o superficies_n sextupla_fw-la to_o flb_n and_o the_o superficies_n of_o a_o be_v quadrupla_fw-la to_o the_o same_o flb_n by_o the_o first_o theorem_a therefore_o the_o cylindrical_a superficies_n of_o fg_o with_o the_o superficiece_n of_o his_o two_o base_n be_v to_o the_o superficies_n flb_n as_o 6_o to_o 1_o and_o the_o superficies_n of_o a_o to_o flb_n be_v as_o 4_o to_o 1._o wherefore_o the_o cylindrical_a superficies_n of_o fg_o &_o his_o two_o base_n together_o be_v to_o the_o superficies_n of_o a_o as_z 6_o to_o 4_o that_o be_v in_o the_o small_a term_n as_o 3_o to_o 2._o which_o be_v proper_a to_o sesâ—uialtera_fw-la proportion_n three_o it_o be_v already_o make_v evident_a that_o the_o superficies_n cylindrical_a of_o fg_o only_o by_o itself_o be_v quadrupla_fw-la to_o flb_n and_o also_o it_o be_v prove_v that_o the_o superficies_n of_o the_o sphere_n a_o be_v quadrupla_fw-la to_o the_o same_o flb_n wherefore_o by_o the_o 7._o of_o the_o five_o the_o cylindrical_a superficies_n of_o fg_o be_v equal_a to_o the_o superficies_n of_o a._n therefore_o every_o cylinder_n which_o have_v his_o base_a the_o great_a circle_n in_o a_o sphere_n and_o heith_n equal_a to_o the_o diameter_n of_o that_o sphere_n be_v sesquialtera_fw-la to_o that_o spear_n also_o the_o superficies_n of_o that_o cylinder_n with_o his_o two_o base_n be_v sesquialtera_fw-la to_o the_o superficies_n of_o the_o sphere_n and_o without_o his_o two_o base_n be_v equal_a to_o the_o superficies_n of_o the_o sphere_n which_o be_v to_o be_v demonstrate_v the_o lemma_n if_o a_o be_v to_o c_o as_z 6_o to_o 1_o and_o b_o to_o c_o as_z 4_o to_o 1_o a_o be_v to_o b_o as_z 6_o to_o 4._o for_o see_v b_o be_v to_o c_o as_z 4_o to_o 1_o by_o supposition_n therefore_o backward_o by_o the_o 4._o of_o the_o five_o c_z be_v to_o b_o as_z 1_o to_o 4._o imagine_v now_o two_o order_n of_o qnantity_n the_o first_o a_o c_o and_o b_o the_o second_o 6_o 1_o and_o 4._o forasmuch_o as_o a_o be_v to_o c_o as_z 6_o to_o 1_o by_o supposition_n and_o c_z be_v to_o b_o as_z 1_o to_o 4_o as_o we_o have_v prove_v wherefore_o a_o be_v to_o b_o as_z 6_o to_o 4_o by_o the_o 22_o of_o the_o five_o therefore_o if_o a_o be_v to_o c_o as_z 6_o to_o 1_o and_o b_o to_o c_z as_z 4_o to_o 1_o a_o be_v to_o b_o as_z 6_o to_o 4._o which_o be_v to_o be_v prove_v note_n slight_a thing_n some_o time_n lack_v evident_a proof_n breed_n doubt_n or_o ignorance_n and_o i_o need_n not_o warnâ—_n you_o how_o genâ—rall_a this_o demonstration_n be_v for_o if_o you_o put_v in_o the_o place_n of_o 6_o and_o 4_o any_o other_o number_n the_o like_a manner_n of_o conclusion_n will_v follow_v so_o likewise_o in_o place_n of_o 1._o any_o other_o one_o number_n may_v be_v as_z if_o a_o be_v to_o c_o as_o â—_o to_o 5_o and_o b_o unto_o c_o be_v as_z 7_o to_o 5_o a_o shall_v be_v to_o b_o as_z 6_o to_o 7._o etc_n etc_n a_o problem_n 4._o to_o a_o sphere_n give_v to_o make_v a_o cylinder_n equal_a or_o in_o any_o proportion_n give_v between_o two_o right_a line_n suppose_v the_o give_v sphere_n to_o be_v a_o and_o the_o proportion_n give_v to_o be_v that_o between_o x_o and_o y._n i_o say_v that_o a_o cylinder_n be_v to_o be_v make_v equal_a to_o aâ—_z or_o else_o in_o the_o same_o proportion_n to_o a_o that_o be_v between_o x_o to_o y._n let_v a_o cylinder_n be_v make_v such_o one_o as_o the_o theorem_a next_o before_o suppose_a that_o shall_v have_v his_o base_a equal_a to_o the_o great_a circle_n in_o a_o construction_n and_o height_n equal_a to_o the_o diameter_n of_o a_o let_v that_o cylinder_n bâ—_z the_o upright_o cylinder_n bc._n leâ—_n the_o one_o side_n of_o bc_o be_v the_o right_a line_n qc_n divide_v qc_n into_o three_o equal_a partâ—_n of_o which_o let_v qe_n contain_v two_o and_o let_v the_o three_o part_n be_v ce._n by_o the_o point_n e_o suppose_v a_o plain_a parallel_n to_o the_o base_n of_o bc_o to_o pass_v through_o the_o cylinder_n bc_o cut_v the_o same_o by_o the_o circle_n de._n i_o say_v that_o the_o cylinder_n be_v be_v equal_a to_o the_o sphere_n a._n for_o see_v bc_o be_v a_o
contain_v in_o a_o the_o sphere_n be_v the_o circle_n bcd_o construction_n and_o by_o the_o problem_n of_o my_o addition_n upon_o the_o second_o proposition_n of_o this_o book_n as_o x_o be_v to_o you_o so_o let_v the_o circle_n bcd_v be_v to_o a_o other_o circle_n find_v let_v that_o other_o circle_n be_v efg_o and_o his_o diameter_n eglantine_n i_o say_v that_o the_o spherical_a superficies_n of_o the_o sphere_n a_o have_v to_o the_o spherical_a superficies_n of_o the_o sphere_n who_o great_a circle_n be_v efg_o or_o his_o equal_a that_o proportion_n which_o x_o have_v to_o y._n demonstration_n for_o by_o construction_n bcd_o be_v to_o efg_o as_o x_o be_v to_o you_o and_o by_o the_o theorem_a next_o beforeâ—_n as_o bcd_o be_v to_o â—fg_n so_o be_v the_o spherical_a superficies_n of_o a_o who_o great_a circle_n be_v bcd_o by_o supposition_n to_o the_o spherical_a superficies_n of_o the_o sphere_n who_o great_a circle_n be_v efg_o wherefore_o by_o the_o 11._o of_o the_o five_o as_o x_o be_v to_o you_o so_o be_v the_o spherical_a superficies_n of_o a_o to_o the_o spherical_a superficies_n of_o the_o sphere_n who_o great_a circle_n be_v efg_o wherefore_o the_o sphere_n who_o diameter_n be_v eglantine_n the_o diameter_n also_o of_o efg_o be_v the_o sphere_n to_o who_o spherical_a superficies_n the_o spherical_a superficies_n of_o the_o sphere_n a_o have_v that_o proportion_n which_o x_o have_v to_o y._n a_o sphere_n be_v give_v therefore_o we_o have_v give_v a_o other_o sphere_n to_o who_o spherical_a superficies_n the_o superficies_n spherical_a of_o the_o sphere_n geum_z have_v any_o proportion_n give_v between_o two_o right_a line_n which_o ought_v to_o be_v do_v a_o problem_n 10._o a_o sphere_n be_v give_v and_o a_o circle_n less_o than_o the_o great_a circle_n in_o the_o same_o sphere_n contain_v to_o cope_v in_o the_o sphere_n give_v a_o circle_n equal_a to_o the_o circle_n give_v suppose_v a_o to_o be_v the_o sphere_n give_v and_o the_o circle_n give_v less_o than_o the_o great_a circle_n in_o a_o contain_v to_o be_v fkg_v spâ—erâ—_n i_o say_v that_o in_o the_o sphere_n a_o a_o circle_n equal_a to_o the_o circle_n fkg_v be_v to_o be_v coapt_v first_o understand_v what_o we_o mean_v here_o by_o coapt_a of_o a_o circle_n in_o a_o sphere_n we_o say_v that_o circle_n to_o be_v coapt_v in_o a_o sphere_n who_o whole_a circumference_n be_v in_o the_o superficies_n of_o the_o same_o sphere_n let_v the_o great_a circle_n in_o the_o sphere_n a_o contain_v be_v the_o circle_n bcd_o who_o diameter_n suppose_v to_o be_v bd_o and_o of_o the_o circle_n fkg_v let_v fg_o be_v the_o diameter_n construction_n by_o the_o 1._o of_o the_o four_o let_v a_o line_n equal_a to_o fg_o be_v coapt_v in_o the_o circle_n bcd_o which_o line_n coapt_v let_v be_v be._n and_o by_o the_o line_n be_v suppose_v a_o plain_a to_o pass_v cut_v the_o sphere_n a_o and_o to_o be_v perpendicular_o erect_v to_o the_o superficies_n of_o bcd_o demonstration_n see_v that_o the_o portion_n of_o the_o plain_n remain_v in_o the_o sphere_n be_v call_v their_o common_a section_n the_o say_a section_n shall_v be_v a_o circle_n as_o before_o be_v prove_v and_o the_o common_a section_n of_o the_o say_a plain_n and_o the_o great_a circle_n bcd_o which_o be_v be_v by_o supposition_n shall_v be_v the_o diameter_n of_o the_o same_o circle_n as_o we_o will_v prove_v for_o let_v that_o circle_n be_v blem_n let_v the_o centre_n of_o the_o sphere_n a_o be_v the_o point_n h_o which_o h_o be_v also_o the_o centre_n of_o the_o circle_n bcd_o because_o bcd_o be_v the_o great_a circle_n in_o a_o contain_v from_o h_o the_o centre_n of_o the_o sphere_n a_o let_v a_o line_n perpendicular_o be_v let_v fall_n to_o the_o circle_n blem_n let_v that_o line_n be_v ho_o and_o it_o be_v evident_a that_o ho_o shall_v fall_v upon_o the_o common_a section_n be_v by_o the_o 38._o of_o the_o eleven_o and_o it_o divide_v be_v into_o two_o equal_a part_n by_o the_o second_o part_n of_o the_o three_o proposition_n of_o the_o three_o book_n by_o which_o point_n oh_o all_o other_o line_n draw_v in_o the_o circle_n blem_n be_v at_o the_o same_o point_n oh_o divide_v into_o two_o equal_a part_n as_o if_o from_o the_o point_n m_o by_o the_o point_n oh_o a_o right_a line_n be_v draw_v one_o the_o other_o side_n come_v to_o the_o circumference_n at_o the_o point_n n_o it_o be_v manifest_a that_o nom_fw-la be_v divide_v into_o two_o equal_a part_n at_o the_o point_n oh_o euclid_n by_o reason_n if_o from_o the_o centre_n h_o to_o the_o point_n n_o and_o m_o right_a line_n be_v draw_v hn_o and_o he_o the_o square_n of_o he_o and_o hn_o be_v equal_a for_o that_o all_o the_o semidiameter_n of_o the_o sphere_n be_v equal_a and_o therefore_o their_o square_n be_v equal_a one_o to_o the_o other_o and_o the_o square_a of_o the_o perpendicular_a ho_o be_v common_a wherefore_o the_o square_a of_o the_o three_o line_n more_n be_v equal_a to_o the_o square_n of_o the_o three_o line_n no_o and_o therefore_o the_o line_n more_n to_o the_o line_n no_o so_o therefore_o be_v nm_o equal_o divide_v at_o the_o point_n o._n and_o so_o may_v be_v prove_v of_o all_o other_o right_a line_n draw_v in_o the_o circle_n blem_n pass_v by_o the_o point_n oh_o to_o the_o circumference_n one_o both_o side_n wherefore_o o_n be_v the_o centre_n of_o the_o circle_n blem_n and_o therefore_o be_v pass_v by_o the_o point_n oh_o be_v the_o diameter_n of_o the_o circle_n blem_n which_o circle_n i_o say_v be_v equal_a to_o fkg_v for_o by_o construction_n be_v be_v equal_a to_o fg_o and_o be_v be_v prove_v the_o diameter_n of_o blem_n and_o fg_o be_v by_o supposition_n the_o diameter_n of_o the_o circle_n fkg_v wherefore_o blem_n be_v equal_a to_o fkg_v the_o circle_n give_v and_o blem_n be_v in_o a_o the_o sphere_n give_v wherefore_o we_o have_v in_o a_o sphere_n give_v coapt_v a_o circle_n equal_a to_o a_o circle_n give_v which_o be_v to_o be_v do_v a_o corollary_n beside_o our_o principal_a purpose_n in_o this_o problem_n evident_o demonstrate_v this_o be_v also_o make_v manifest_a that_o if_o the_o great_a circle_n in_o a_o sphere_n be_v cut_v by_o a_o other_o circle_n erect_v upon_o he_o at_o right_a angle_n that_o the_o other_o circle_n be_v cut_v by_o the_o centre_n and_o that_o their_o common_a section_n be_v the_o diameter_n of_o that_o other_o circle_n and_o therefore_o that_o other_o circle_n divide_v be_v into_o two_o equal_a part_n a_o problem_n 11._o a_o sphere_n be_v give_v and_o a_o circle_n less_o than_o double_a the_o great_a circle_n in_o the_o same_o sphere_n contain_v to_o cut_v of_o a_o segment_n of_o the_o same_o sphere_n who_o spherical_a superficies_n shall_v be_v equal_a to_o the_o circle_n give_v suppose_v king_n to_o be_v a_o sphere_n give_v who_o great_a circle_n let_v be_v abc_n and_o the_o circle_n give_v suppose_v to_o be_v def_n construction_n i_o say_v that_o a_o segment_n of_o the_o sphere_n king_n be_v to_o be_v cut_v of_o so_o great_a that_o his_o spherical_a superficies_n shall_v be_v equal_a to_o the_o circle_n def_n let_v the_o diameter_n of_o the_o circle_n abc_n be_v the_o line_n ab_fw-la at_o the_o point_n a_o in_o the_o circle_n abc_n cope_v a_o right_a line_n equal_a to_o the_o semidiameter_n of_o the_o circle_n def_n by_o the_o first_o of_o the_o four_o which_o line_n suppose_v to_o be_v ah_o from_o the_o point_n h_o to_o the_o diameter_n ab_fw-la let_v a_o perpendicular_a line_n be_v draw_v which_o suppose_v to_o be_v high_a produce_v high_a to_o the_o other_o side_n of_o the_o circumference_n and_o let_v it_o come_v to_o the_o circumference_n at_o the_o point_n l._n by_o the_o right_a line_n hil_n perpendicular_a to_o ab_fw-la suppose_v a_o plain_a superficies_n to_o pass_v perpendicular_o erect_v upon_o the_o circle_n abc_n and_o by_o this_o plain_a superficies_n the_o sphere_n to_o be_v cut_v into_o two_o segment_n one_o less_o than_o the_o half_a sphere_n namely_o hali_n and_o the_o other_o great_a than_o the_o half_a sphere_n namely_o hbli_n i_o say_v that_o the_o spherical_a superficies_n of_o the_o segment_n of_o the_o sphere_n king_n in_o which_o the_o segment_n of_o the_o great_a circle_n hali_n be_v contain_v who_o base_a be_v the_o circle_n pass_v by_o hil_n and_o top_n the_o point_n a_o be_v equal_a to_o the_o circle_n def_n demonstration_n for_o the_o circle_n who_o semidiameter_n be_v equal_a to_o the_o line_n ah_o be_v equal_a to_o the_o spherical_a superficies_n of_o the_o segment_n hal_n by_o the_o 4._o theorem_a here_o add_v and_o by_o construction_n ah_o be_v equal_a to_o the_o semidiameter_n of_o the_o circle_n def_n therefore_o the_o spherical_a superficies_n of_o the_o segment_n of_o the_o sphere_n king_n cut_v of_o by_o the_o
superficies_n or_o solidity_n in_o the_o hole_n or_o in_o partâ—_n such_o certain_a knowledge_n demonstrative_a may_v arise_v and_o such_o mechanical_a exercise_n thereby_o be_v devise_v that_o sure_a i_o be_o to_o the_o sincere_a &_o true_a student_n great_a light_n aid_n and_o comfortable_a courage_n far_a to_o wade_v will_v enter_v into_o his_o heart_n and_o to_o the_o mechanical_a witty_a and_o industrous_a deviser_n new_a manner_n of_o invention_n &_o execution_n in_o his_o work_n will_v with_o small_a travail_n for_o foot_n application_n come_v to_o his_o perceiveraunce_n and_o understanding_n therefore_o even_o a_o manifold_a speculation_n &_o practice_n may_v be_v have_v with_o the_o circle_n his_o quantity_n be_v not_o know_v in_o any_o kind_n of_o small_a certain_a measure_n so_o likewise_o of_o the_o sphere_n many_o problem_n may_v be_v execute_v and_o his_o precise_a quantity_n in_o certain_a measure_n not_o determine_v or_o know_v yet_o because_o both_o one_o of_o the_o first_o humane_a occasion_n of_o invent_v and_o stablish_v this_o art_n be_v measure_v of_o the_o earth_n and_o therefore_o call_v geometria_n that_o be_v earthmeasure_v and_o also_o the_o chief_a and_o general_a end_n in_o deed_n be_v measure_n and_o measure_n require_v a_o determination_n of_o quantity_n in_o a_o certain_a measure_n by_o number_n express_v it_o be_v needful_a for_o mechanical_a earthmeasure_n not_o to_o be_v ignorant_a of_o the_o measure_n and_o content_n of_o the_o circle_n neither_o of_o the_o sphere_n his_o measure_n and_o quantity_n as_o near_o as_o sense_n can_v imagine_v or_o wish_v and_o in_o very_a deed_n the_o quantity_n and_o measure_n of_o the_o circle_n be_v know_v make_v not_o only_o the_o cone_n and_o cylinder_n but_o also_o the_o sphere_n his_o quantity_n to_o be_v as_o precise_o know_v and_o certain_a therefore_o see_v in_o respect_n of_o the_o circle_n quantity_n by_o archimedes_n specify_v this_o theorem_a be_v note_v unto_o you_o i_o will_v by_o order_n upon_o that_o as_o a_o supposition_n infer_v the_o conclusion_n of_o this_o our_o theorem_n note_n 1._o wherefore_o if_o you_o divide_v the_o one_o side_n as_o tq_n of_o the_o cube_fw-la tx_n into_o 21._o equal_a part_n and_o where_o 11._o part_n do_v end_n reckon_v from_o t_o suppose_v the_o point_n p_o and_o by_o that_o point_n p_o imagine_v a_o plain_a pass_v parallel_n to_o the_o opposite_a base_n to_o cut_v the_o cube_fw-la tx_n and_o thereby_o the_o cube_fw-la tx_n to_o be_v divide_v into_o two_o rectangle_n parallelipipedon_n namely_o tn_n and_o px_n it_o be_v manifest_a give_v tn_n to_o be_v equal_a to_o the_o sphere_n a_o by_o construction_n and_o the_o 7._o of_o the_o five_o note_n 2._o second_o the_o whole_a quantity_n of_o the_o sphere_n a_o assign_v be_v contain_v in_o the_o rectangle_n parallelipipedon_n tn_n you_o may_v easy_o transform_v the_o same_o quantity_n into_o other_o parallelipipedon_n rectangle_v of_o what_o height_n and_o of_o what_o parallelogram_n base_a you_o listen_v by_o my_o first_o and_o second_o problem_n upon_o the_o 34._o of_o this_o book_n and_o the_o like_a may_v you_o do_v to_o any_o assign_a part_n of_o the_o sphere_n a_o by_o the_o like_a mean_n devide_v the_o parallelipipedon_n tn_n as_o the_o part_n assign_v do_v require_v as_o if_o a_o three_o four_o five_o or_o six_o part_n of_o the_o sphere_n a_o be_v to_o be_v have_v in_o a_o parallelipipedon_n of_o any_o parallelograâ—â—e_n base_a assign_v or_o of_o any_o heith_n assign_v then_o devide_v tp_n into_o so_o many_o part_n as_o into_o 4._o if_o a_o four_o part_n be_v to_o be_v transform_v or_o into_o five_o if_o a_o five_o part_n be_v to_o be_v transform_v etc_n etc_n and_o then_o proceed_v â—s_v you_o do_v with_o cut_v of_o tn_n from_o tx_n and_o that_o i_o say_v of_o parallelipipedon_n may_v in_o like_a sort_n by_o my_o â—â—yd_n two_o problem_n add_v to_o the_o 34._o of_o this_o book_n be_v do_v in_o any_o side_v column_n pyramid_n and_o prismeâ—_n so_o thâ—â—_n in_o pyramid_n and_o some_o prism_n you_o use_v the_o caution_n necessary_a in_o respect_n of_o their_o quan_fw-mi 〈…〉_o odyes_n have_v parallel_n equal_a and_o opposite_a base_n who_o part_n 〈…〉_o re_fw-mi in_o their_o proposition_n be_v by_o euclid_n demonstrate_v and_o final_o 〈…〉_o addition_n you_o have_v the_o way_n and_o order_n how_o to_o geve_v to_o a_o sphere_n or_o any_o segment_n oâ—_n the_o same_o cones_n or_o cylinder_n equal_a or_o in_o any_o proportion_n between_o two_o right_a line_n give_v with_o many_o other_o most_o necessary_a speculation_n and_o practice_n about_o the_o sphere_n i_o trust_v that_o i_o have_v sufficient_o â—raughted_v your_o imagination_n for_o your_o honest_a and_o profitable_a study_n herein_o and_o also_o give_v you_o reaâ—â—_n â—â—tter_n wheâ—â—_n with_o to_o sâ—â—p_v the_o mouth_n of_o the_o malicious_a ignorant_a and_o arrogant_a despiser_n of_o the_o most_o excellent_a discourse_n travail_n and_o invention_n mathematical_a sting_v aswell_o the_o heavenly_a sphere_n &_o star_n their_o spherical_a solidity_n geometry_n with_o their_o convene_n spherical_a superficies_n to_o the_o earth_n at_o all_o time_n respect_v and_o their_o distance_n from_o the_o earth_n as_o also_o the_o whole_a earthly_a sphere_n and_o globe_n itself_o and_o infinite_a other_o case_n concern_v sphere_n or_o globe_n may_v hereby_o with_o as_o much_o ease_n and_o certainty_n be_v determine_v of_o as_o of_o the_o quantity_n of_o any_o bowl_n ball_n or_o bullet_n which_o we_o may_v gripe_v in_o our_o hand_n reason_n and_o experience_n be_v our_o witness_n and_o without_o these_o aid_n such_o thing_n of_o importance_n never_o able_a of_o we_o certain_o to_o be_v know_v or_o attain_a unto_o here_o end_n m._n john_n do_v you_o his_o addition_n upon_o the_o last_o proposition_n of_o the_o twelve_o book_n a_o proposition_n add_v by_o flussas_n if_o a_o sphere_n touch_v a_o plain_a superficiesâ—_n a_o right_a line_n draw_v from_o the_o centre_n to_o the_o touch_n shall_v be_v erect_v perpendicular_o to_o the_o plain_a superficies_n suppose_v that_o there_o be_v a_o sphere_n bcdl_n who_o centre_n let_v be_v the_o point_n a._n and_o let_v the_o plain_a superficies_n gci_n touch_v the_o spear_n in_o the_o point_n c_o and_o extend_v a_o right_a line_n from_o the_o centre_n a_o to_o the_o point_n c._n then_o i_o say_v that_o the_o line_n ac_fw-la be_v erect_v perpendicular_o to_o tâ—e_z plain_a gic._n let_v the_o sphere_n be_v cut_v by_o plain_a superficiece_n pass_v by_o the_o right_a line_n lac_n which_o plain_n let_v be_v abcdl_n and_o acel_n which_o let_v cut_v the_o plain_n gci_n by_o the_o right_a line_n gch_o and_o kci_fw-la now_o it_o be_v manifest_a by_o the_o assumpt_n put_v before_o the_o 17._o of_o this_o book_n that_o the_o two_o section_n of_o the_o sphere_n shall_v be_v circle_n have_v to_o their_o diameter_n the_o line_n lac_n which_o be_v also_o the_o diameter_n of_o the_o sphere_n wherefore_o the_o right_a line_n gch_o and_o kci_fw-la which_o be_v draw_v in_o the_o plain_n gci_n do_v at_o the_o point_n c_o fall_n without_o the_o circle_n bcdl_n and_o ecl._n wherefore_o they_o touch_v the_o circle_n in_o the_o point_n c_o by_o the_o second_o definition_n of_o the_o three_o wherefore_o the_o right_a line_n lac_n make_v right_a angle_n with_o the_o line_n gch_v and_o kci_fw-la by_o the_o 16._o of_o the_o three_o wherefore_o by_o the_o 4._o of_o the_o eleven_o the_o right_a line_n ac_fw-la be_v erect_v perpendicular_o to_o to_o the_o plain_a superficies_n gci_n wherein_o be_v draw_v the_o line_n gch_v and_o kci_fw-la if_o therefore_o a_o sphere_n touch_v a_o plain_a superficies_n a_o right_a line_n draw_v from_o the_o centre_n to_o the_o touch_n shall_v be_v erect_v perpendicular_o to_o the_o plain_a superficies_n which_o be_v require_v to_o be_v prove_v the_o end_n of_o the_o twelve_o book_n of_o euclides_n element_n ¶_o the_o thirteen_o book_n of_o euclides_n element_n in_o this_o thirteen_o book_n be_v set_v forth_o certain_a most_o wonderful_a and_o excellent_a passion_n of_o a_o line_n divide_v by_o a_o extreme_a and_o mean_a proportion_n book_n a_o matter_n undoubted_o of_o great_a and_o infinite_a use_n in_o geometry_n as_o you_o shall_v both_o in_o this_o book_n and_o in_o the_o other_o book_n follow_v most_o evident_o perceive_v it_o teach_v moreover_o the_o composition_n of_o the_o five_o regular_a solid_n and_o how_o to_o inscribe_v they_o in_o a_o sphere_n give_v and_o also_o set_v forth_o certain_a comparison_n of_o the_o say_a body_n both_o the_o one_o to_o the_o other_o and_o also_o to_o the_o sphere_n wherein_o they_o be_v describe_v the_o 1._o theorem_a the_o 1._o proposition_n if_o a_o right_a line_n be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n and_o to_o the_o great_a segment_n be_v add_v the_o half_a of_o the_o whole_a line_n the_o square_n make_v of_o those_o two_o
three_o proposition_n we_o will_v use_v the_o same_o supposition_n and_o construction_n there_o specify_v so_o far_o as_o they_o shall_v serve_v our_o purpose_n begin_v therefore_o at_o the_o conclusion_n we_o must_v infer_v the_o part_n of_o the_o proposition_n before_o grant_v it_o be_v conclude_v that_o the_o square_n of_o the_o line_n db_o be_v quintuple_a to_o the_o square_n of_o the_o line_n dc_o his_o own_o segment_n therefore_o dn_o the_o square_n of_o db_o be_v quintuple_a to_o gf_o the_o square_n of_o dc_o but_o the_o squaâ—e_n of_o ac_fw-la the_o double_a of_o dc_o which_o be_v r_n be_v quadruple_a to_o gf_o by_o the_o second_o corollary_n of_o the_o 20._o of_o the_o six_o and_o therefore_o r_n with_o gf_o be_v quintuple_a to_o gf_o and_o so_o it_o be_v evident_a that_o the_o square_a dn_o be_v equal_a to_o the_o square_a r_n together_o with_o the_o square_a gf_o wherefore_o from_o those_o two_o equal_n take_v the_o square_a gf_o common_a to_o they_o both_o remain_v the_o square_a r_n equal_a to_o the_o gnomon_n xop_n but_o to_o the_o gnomon_n xop_n the_o parallelogram_n 3._o ce_fw-fr be_v equal_a wherefore_o the_o square_a of_o the_o line_n ac_fw-la which_o be_v r_n be_v equal_a to_o the_o parallelogram_n câ—_n which_o parallelogamme_n be_v contain_v under_o be_v equal_a to_o ab_fw-la and_o cb_o the_o part_n remain_v of_o the_o first_o line_n gruen_o which_o be_v db._n and_o the_o line_n ab_fw-la be_v make_v of_o the_o double_a of_o the_o segment_n dc_o and_o of_o cbâ—_n the_o other_o part_n of_o the_o line_n db_o first_o goven_v wherefore_o the_o double_a of_o the_o segment_n dc_o with_o cb_o the_o part_n remain_v which_o altogether_o be_v the_o whole_a line_n ab_fw-la be_v to_o ac_fw-la the_o double_a of_o the_o segment_n dc_o as_o that_o same_o ac_fw-la be_v to_o cb_o by_o the_o second_o part_n of_o the_o 16._o of_o the_o six_o therefore_o by_o the_o 3._o definition_n of_o the_o six_o book_n the_o whole_a line_n ab_fw-la be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n &_o ac_fw-la the_o double_a of_o the_o segment_n dc_o be_v middle_a proportional_a be_v the_o great_a part_n thereof_o wherefore_o if_o a_o right_a line_n be_v quintuple_a in_o power_n etc_n etc_n as_o in_o the_o proposition_n which_o be_v to_o be_v demonstrate_v or_o thus_o it_o may_v be_v demonstrate_v forasmuch_o as_o the_o square_n dn_o be_v quintuple_a to_o the_o square_a gf_o i_o mean_v the_o square_n of_o db_o the_o line_n give_v too_o the_o square_a oâ—_n dc_o the_o segment_n and_o the_o same_o square_a dn_o be_v equal_a to_o the_o parallelogram_n under_o ab_fw-la cb_o with_o the_o square_n make_v of_o the_o line_n dc_o by_o the_o six_o of_o the_o second_o for_o unto_o the_o line_n ac_fw-la equal_o divide_v the_o line_n cb_o be_v as_o it_o be_v adjoin_v wherefore_o the_o parallelogram_n under_o ab_fw-la cb_o together_o with_o the_o square_n of_o dc_o which_o be_v gf_o be_v quintuple_a to_o the_o square_a gf_o make_v oâ—_n thâ—_z line_n dc_o take_v then_o that_o square_a gf_o â—rom_o the_o parallelogram_n under_o ab_fw-la cb_o that_o parallelogram_n under_o ab_fw-la cb_o remain_v alone_o be_v but_o quadruple_a to_o the_o say_v square_a of_o the_o line_n dc_o but_o by_o the_o 4._o of_o the_o second_o or_o the_o second_o corollary_n of_o the_o 20._o of_o the_o six_o r_n â—he_z square_a of_o the_o line_n ac_fw-la be_v quadrupla_fw-la to_o the_o same_o square_a gfâ—_n wherefore_o by_o the_o 7._o of_o the_o five_o the_o square_a of_o the_o line_n ac_fw-la be_v equal_a to_o the_o parallelogram_n under_o ab_fw-la cb_o and_o so_o by_o the_o second_o part_n of_o the_o 16._o of_o the_o six_o ab_fw-la ac_fw-la and_o cb_o be_v three_o line_n in_o continual_a proportion_n and_o see_v ab_fw-la be_v great_a they_o ac_fw-la the_o same_o ac_fw-la the_o double_a of_o the_o line_n dc_o shall_v be_v great_a than_o the_o part_n bc_o remain_v wherefore_o by_o the_o 3._o definition_n of_o the_o six_o ab_fw-la compose_v or_o make_v of_o the_o double_a of_o dc_o and_o the_o other_o part_n of_o db_o remain_v be_v divide_v by_o a_o extreme_a and_o middle_n proportion_n and_o also_o his_o great_a segment_n be_v ac_fw-la the_o double_a of_o the_o segment_n dc_o wherefore_o if_o a_o right_a line_n be_v quintuple_a in_o power_n etc_n etc_n as_o in_o the_o proposition_n which_o be_v to_o be_v demonstrate_v a_o theorem_a 2._o if_o a_o right_a line_n divide_v by_o a_o extreme_a and_o mean_a proportion_n be_v give_v and_o to_o the_o great_a segment_n â—herof_o he_o direct_o adjoin_v a_o line_n equal_a to_o the_o whole_a line_n give_v that_o adjoin_v line_n and_o the_o say_v great_a segment_n do_v make_v a_o line_n divide_v by_o extreme_a and_o mean_a proportion_n who_o great_a segment_n be_v the_o line_n adjoin_v suppose_v the_o line_n give_v divide_v by_o extreme_a and_o mean_a proportion_n to_o be_v ab_fw-la divide_v in_o the_o point_n c_o and_o his_o great_a segment_n let_v be_v ac_fw-la unto_fw-la ac_fw-la direct_o adjoin_v a_o line_n equal_a to_o ab_fw-la let_v that_o be_v ad_fw-la i_o say_v that_o ad_fw-la together_o with_o ac_fw-la that_o be_v dc_o be_v a_o divide_v by_o extreme_a and_o middle_n proportion_n who_o great_a segment_n be_v ad_fw-la the_o line_n adjoin_v divide_v ad_fw-la equal_o in_o the_o point_n e._n now_o forasmuch_o as_o ae_n be_v the_o half_a of_o ad_fw-la by_o construction_n it_o be_v also_o the_o half_a of_o ab_fw-la equal_a to_o ad_fw-la by_o construction_n wherefore_o by_o the_o 1._o of_o the_o thirteen_o the_o square_a of_o the_o line_n compose_v of_o ac_fw-la and_o ae_n which_o â—ne_n be_v ec_o be_v quintuple_a to_o the_o square_n of_o the_o line_n ae_n wherefore_o the_o double_a of_o ae_n and_o the_o line_n ac_fw-la compose_v as_o in_o one_o right_a line_n be_v a_o line_n divide_v by_o extreme_a and_o mean_a proportion_n by_o the_o converse_n of_o this_o three_o by_o i_o demonstrate_v and_o the_o double_a of_o ae_n be_v the_o great_a segment_n but_o dc_o be_v the_o line_n compose_v of_o the_o double_a of_o ae_n &_o the_o line_n ac_fw-la and_o with_o all_o ad_fw-la be_v the_o double_a of_o ae_n wherefore_o dc_o be_v a_o line_n divide_v by_o extreme_a and_o mean_a proportion_n and_o ad_fw-la iâ—_z hiâ—_z great_a segment_n if_o a_o right_a line_n therefore_o divide_v by_o extreme_a and_o mean_a proportion_n be_v give_v and_o to_o the_o great_a segment_n thereof_o be_v direct_o adjoin_v a_o line_n equal_a to_o the_o whole_a line_n give_v that_o adjoin_v line_n and_o the_o say_v great_a segment_n do_v make_v a_o line_n divide_v by_o extreme_a and_o mean_a proportion_n who_o great_a segment_n be_v the_o line_n adjoin_v which_o be_v require_v to_o be_v demonstrate_v two_o other_o brief_a demonstration_n of_o the_o same_o forasmuch_o as_o ad_fw-la be_v to_o ac_fw-la as_o ab_fw-la be_v to_o ac_fw-la because_o ad_fw-la be_v equal_a to_o ab_fw-la by_o construction_n but_o as_o ab_fw-la be_v to_o ac_fw-la so_o be_v ac_fw-la to_o cb_o by_o supposition_n therefore_o by_o the_o 11._o of_o the_o five_o as_z ac_fw-la be_v to_o cb_o so_o be_v ad_fw-la to_o ac_fw-la demonstrate_v wherefore_o as_o ac_fw-la and_o cb_o which_o be_v ab_fw-la be_v to_o cb_o so_o be_v ad_fw-la and_o ac_fw-la which_o be_v dc_o to_o ac_fw-la therefore_o everse_o as_o ab_fw-la be_v to_o ac_fw-la so_o be_v dc_o to_o ad._n and_o it_o be_v prove_v ad_fw-la to_o be_v to_o ac_fw-la as_o ac_fw-la be_v to_o cb._n wherefore_o as_o ab_fw-la be_v to_o ac_fw-la and_o ac_fw-la to_o cb_o so_o be_v dc_o to_o ad_fw-la and_o ad_fw-la to_o ac_fw-la but_o ab_fw-la ac_fw-la and_o cb_o be_v in_o continual_a proportion_n by_o supposition_n wherefore_o dc_o ad_fw-la and_o ac_fw-la be_v in_o continual_a proportion_n wherefore_o by_o the_o 3._o definition_n of_o the_o six_o book_n dc_o be_v divide_v by_o extreme_a and_o middle_a proportion_n and_o his_o great_a segment_n be_v ad._n which_o be_v to_o be_v demonstrate_v note_v from_o the_o mark_n demonstrate_v how_o this_o have_v two_o demonstration_n one_o i_o have_v set_v in_o the_o margin_n by_o ¶_o a_o corollary_n 1._o upon_o euclides_n three_o proposition_n demonstrate_v it_o be_v make_v evident_a that_o of_o a_o line_n divide_v by_o extreme_a and_o mean_a proportion_n if_o you_o produce_v the_o less_o segment_n equal_o to_o the_o length_n of_o the_o great_a the_o line_n thereby_o adjoin_v together_o with_o the_o say_v less_o segment_n make_v a_o new_a line_n divide_v by_o extreme_a and_o middle_a proportion_n who_o less_o segment_n be_v the_o line_n adjoin_v for_o if_o ab_fw-la be_v divide_v by_o extreme_a and_o middle_a proportion_n in_o the_o point_n c_o ac_fw-la be_v the_o great_a segment_n and_o cb_o be_v produce_v from_o the_o point_n b_o make_v a_o line_n with_o cb_o equal_a to_o ac_fw-la which_o let_v be_v cq_n and_o the_o
line_n thereby_o adjoin_v let_v be_v bq_n i_o say_v that_o cq_n be_v a_o line_n also_o divide_v by_o a_o extreme_a and_o mean_a proportion_n in_o the_o point_n b_o and_o that_o bq_n the_o line_n adjoin_v be_v the_o less_o segment_n for_o by_o the_o third_o it_o be_v prove_v that_o half_a ac_fw-la which_o let_v be_v cd_o with_o cb_o as_o one_o line_n compose_v have_v his_o power_n or_o square_v quintuple_a to_o the_o power_n of_o the_o segment_n cd_o wherefore_o by_o the_o second_o of_o this_o book_n the_o double_a of_o c_o d_o be_v divide_v by_o extreme_a and_o middle_a proportioned_a and_o the_o great_a segment_n thereof_o shall_v be_v cb._n but_o by_o construction_n cq_n be_v the_o double_a of_o cd_o for_o it_o be_v equal_a to_o ac_fw-la wherefore_o cq_n be_v divide_v by_o extreme_a and_o middle_a proportion_n in_o the_o point_n b_o and_o the_o great_a segment_n thereof_o shall_v be_v cb._n wherefore_o bq_n be_v the_o less_o segment_n which_o be_v the_o line_n adjoin_v therefore_o a_o line_n be_v divide_v by_o extreme_a and_o middle_a proportion_n if_o the_o less_o segment_n be_v produce_v equal_o to_o the_o length_n of_o the_o great_a segment_n the_o line_n thereby_o adjoin_v together_o with_o the_o say_v less_o segment_n make_v a_o new_a line_n divide_v by_o extreme_a &_o mean_a proportion_n whoâ—e_v less_o segment_n be_v the_o line_n adjoin_v which_o be_v to_o be_v demonstrate_v ¶_o a_o corollary_n 2._o ifâ—_n from_o the_o great_a segment_n of_o a_o line_n divide_v by_o extreme_a and_o middle_a proportion_n a_o line_n equal_a to_o the_o less_o segment_n be_v cut_v of_o the_o great_a segment_n thereby_o be_v also_o divide_v by_o extreme_a and_o mean_a proportion_n who_o great_a segmentâ—_n shall_v be_v 〈◊〉_d that_o part_n of_o it_o which_o be_v cut_v of_o for_o take_v from_o ac_fw-la a_o line_n equal_a to_o cb_o let_v be_v remain_v i_o say_v that_o ac_fw-la be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n in_o the_o point_n r_o and_o that_o cr_n the_o line_n cut_v of_o be_v the_o great_a segment_n for_o it_o be_v prove_v in_o the_o former_a corollary_n that_o cq_n be_v divide_v by_o extreme_a and_o mean_a proportion_n in_o the_o point_n b._n but_o ac_fw-la be_v equal_a to_o cq_n by_o construction_n and_o cr_n be_v equal_a to_o cb_o by_o construction_n wherefore_o the_o residue_n ar_n be_v equal_a to_o bq_n the_o residue_n see_v therefore_o the_o whole_a ac_fw-la be_v equal_a to_o the_o whole_a cq_n and_o the_o great_a part_n of_o ac_fw-la which_o be_v cr_n be_v equal_a to_o cb_o the_o great_a part_n of_o cq_n and_o the_o less_o segment_n also_o equal_a to_o the_o less_o and_o withal_o see_v cq_n be_v prove_v to_o be_v divide_v by_o extreme_a &_o mean_a proportion_n in_o the_o point_n b_o it_o follow_v of_o necessity_n that_o ac_fw-la be_v divide_v by_o extreme_a and_o mean_a proportion_n in_o the_o point_n r._n and_o see_v cb_o be_v the_o great_a segment_n of_o cq_n cr_n shall_v be_v the_o great_a segment_n of_o ac_fw-la which_o be_v to_o be_v demonstrate_v a_o corollary_n 3._o it_o be_v evident_a thereby_o a_o line_n be_v divide_v by_o extreme_a and_o mean_a proportion_n that_o the_o line_n whereby_o the_o great_a segment_n exceed_v the_o less_o together_o with_o the_o less_o segment_n do_v make_v a_o line_n divide_v by_o extreme_a and_o mean_a proportion_n who_o less_o segment_n be_v the_o say_a line_n of_o exceesse_n or_o difference_n between_o the_o segment_n john_n dee_n ¶_o two_o new_a way_n to_o divide_v any_o right_a line_n give_v by_o a_o extreme_a and_o mean_a proportion_n demonstrate_v and_o add_v by_o m._n dee_n a_o problem_n to_o divide_v by_o a_o extreme_a and_o mean_a proportion_n any_o right_a line_n give_v in_o length_n and_o position_n suppose_v a_o line_n give_v in_o length_n and_o position_n to_o be_v ab_fw-la i_o say_v that_o ab_fw-la be_v to_o be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n construction_n divide_v ab_fw-la into_o two_o equal_a part_n as_o in_o the_o point_n c._n produce_v ab_fw-la direct_o from_o the_o point_n b_o to_o the_o point_n d_o make_v bd_o equal_a to_o bc._n to_o the_o line_n ad_fw-la and_o at_o the_o point_n d_o let_v a_o line_n be_v draw_v etc_n perpendicular_a by_o the_o 11._o of_o the_o first_o which_o let_v be_v df_o of_o what_o length_n you_o will_v from_o df_o and_o at_o the_o point_n d_o cut_v of_o the_o six_o part_n of_o df_o by_o the_o 9_o of_o the_o six_o and_o let_v that_o six_o part_n be_v the_o line_n dg_o upon_o df_o as_o a_o diameter_n describe_v a_o semicircle_n which_o let_v be_v dhf_n from_o the_o point_n g_o rear_v a_o line_n perpendicular_a to_o df_o which_o suppose_v to_o be_v gh_o and_o let_v it_o come_v to_o the_o circumference_n of_o dhf_n in_o the_o point_n h._n draw_v right_a line_n hd_v and_o hf._n produce_v dh_o from_o the_o point_n h_o so_o long_o till_o a_o line_n adjoin_v with_o dh_o be_v equal_a to_o hf_o which_o let_v be_v di_fw-it equal_a to_o hf._n from_o the_o point_n h_o to_o the_o point_n b_o the_o one_o end_n of_o our_o line_n give_v let_v a_o right_a line_n be_v draw_v as_o hb_o from_o the_o point_n i_o let_v a_o line_n be_v draw_v to_o the_o line_n ab_fw-la so_o that_o it_o be_v also_o parallel_a to_o the_o line_n hb_o which_o parallel_n line_n suppose_v to_o be_v ik_fw-mi cut_v the_o line_n ab_fw-la at_o the_o point_n k._n i_o say_v that_o ab_fw-la be_v divide_v by_o a_o extreme_a &_o mean_a proportion_n in_o the_o point_n k._n for_o the_o triangle_n dki_n have_v hb_o parallel_n to_o ik_fw-mi have_v his_o side_n dk_o and_o diego_n demonstration_n cut_v proportional_o by_o the_o 2._o of_o the_o six_o wherefore_o as_o ih_v be_v to_o hd_v so_o be_v kb_o to_o bd._n and_o therefore_o compound_o by_o the_o 18._o of_o the_o five_o as_o diego_n be_v to_o dh_o so_o be_v dk_o to_o db._n but_o by_o construction_n diego_n be_v equal_a to_o hf_o wherefore_o by_o the_o 7._o of_o the_o five_o di_z be_v to_o dh_o as_o hf_o be_v to_o dh_o wherefore_o by_o the_o 11._o of_o the_o five_o dk_o be_v to_o db_o as_o hf_o be_v to_o dh_o wherefore_o the_o square_a of_o dk_o be_v to_o the_o square_n of_o db_o as_o the_o square_n of_o hf_o be_v to_o the_o square_n of_o dh_o by_o the_o 22._o of_o the_o six_o but_o the_o square_n of_o hf_o be_v to_o the_o square_n of_o dh_o as_o the_o line_n gf_o be_v to_o the_o line_n gdâ—_n by_o my_o corollary_n upon_o the_o 5_o problem_n of_o my_o addition_n to_o the_o second_o proposition_n of_o the_o twelve_o wherefore_o by_o the_o 11._o of_o the_o five_o the_o square_a of_o dk_o be_v to_o the_o square_n of_o db_o as_o the_o line_n gf_o be_v to_o the_o line_n gd_o but_o by_o construction_n gf_o be_v quintuple_a to_o gd_o wherefore_o the_o square_a of_o dk_o be_v quintuple_a to_o the_o square_n of_o db_o and_o therefore_o the_o double_a of_o db_o be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n and_o bk_o be_v the_o great_a segment_n thereof_o by_o the_o 2._o of_o this_o thirteen_o wherefore_o see_v ab_fw-la be_v the_o double_a of_o db_o by_o construction_n the_o line_n ab_fw-la be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n and_o his_o great_a segment_n be_v the_o line_n bk_o wherefore_o ab_fw-la be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n in_o the_o point_n k._n we_o have_v therefore_o divide_v by_o extreme_a and_o mean_a proportion_n any_o line_n give_v in_o length_n and_o position_n which_o be_v requisite_a to_o be_v do_v the_o second_o way_n to_o execute_v this_o problem_n suppose_v the_o line_n give_v to_o be_v ab_fw-la divide_v aâ—_z into_o two_o equal_a part_n as_o suppose_v it_o to_o be_v do_v in_o the_o point_n c._n produce_v ab_fw-la from_o the_o point_n b_o adjoin_v a_o line_n equal_a to_o bc_o which_o let_v be_v bd._n to_o the_o right_a line_n ad_fw-la and_o at_o the_o point_n d_o erect_v a_o perpendicular_a line_n equal_a to_o bd_o let_v that_o be_v de._n produce_v ed_z from_o the_o point_n d_o to_o the_o point_n f_o make_v df_o to_o contain_v five_o such_o equal_a part_n as_o de_fw-fr be_v one_o now_o upon_o of_o as_o a_o diameter_n describe_v a_o semicircle_n which_o let_v 〈◊〉_d ekf_n and_o let_v the_o point_n where_o the_o circumference_n of_o ekf_n do_v cut_v the_o line_n ab_fw-la be_v the_o point_n k._n i_o say_v that_o ab_fw-la be_v divide_v in_o the_o point_n king_n by_o a_o extreme_a and_o mean_a proportion_n for_o by_o the_o 13._o of_o the_o six_o ed_z dk_o &_o df_o be_v three_o line_n in_o continual_a proportion_n dk_o be_v the_o middle_n proportional_a â—_o wherefore_o by_o the_o corollary_n of_o the_o 20._o of_o the_o six_o as_z ed_z be_v to_o df_o so_o be_v
the_o square_a of_o ed_z to_o the_o square_n of_o dk_o but_o by_o construction_n ed_z be_v subquintuple_a to_o df._n wherefore_o the_o square_a of_o ed_z be_v subquintuple_a to_o the_o square_n of_o dk_o and_o therefore_o the_o square_a of_o dk_o be_v quintuple_a to_o the_o square_n of_o ed._n and_o ed_z be_v equal_a to_o ed_z by_o construction_n therefore_o the_o square_a of_o dk_o be_v quintuple_a to_o the_o square_n of_o e_o d._n wherefore_o the_o double_a of_o bd_o be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n who_o great_a segment_n be_v bkâ—_n â—_z by_o the_o second_o of_o this_o thirteâ—th_o but_o by_o construction_n ab_fw-la be_v the_o double_a of_o â—dâ—_n â—_z wherefore_o ab_fw-la be_v divide_v by_o extreme_a and_o mean_a proportion_n and_o his_o great_a segment_n be_v bk_o and_o thereby_o kâ—_n â—_z the_o point_n of_o the_o division_n we_o have_v therefore_o divide_v by_o extreme_a and_o mean_a proportion_n any_o right_a line_n give_v in_o length_n and_o position_n which_o be_v to_o be_v do_v noteâ—_n each_o of_o these_o way_n may_v well_o be_v execute_v but_o in_o the_o first_o you_o have_v this_o advantage_n that_o the_o diameter_n be_v take_v at_o pleasure_n which_o â—n_v the_o second_o way_n be_v ever_o just_a thrice_o so_o long_o as_o the_o line_n give_v to_o be_v divide_v john_n dee_n ¶_o the_o 4._o theorem_a the_o 4._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 the_o square_n make_v of_o the_o whole_a line_n and_o of_o the_o less_o segment_n be_v treble_a to_o the_o square_n make_v 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 &_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 then_o i_o say_v that_o the_o square_n make_v of_o the_o line_n ab_fw-la and_o bc_o be_v treble_a to_o the_o square_n of_o the_o line_n ac_fw-la 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 adeb_n and_o make_v perfect_a the_o figure_n now_o forasmuch_o as_o the_o line_n ab_fw-la demonstration_n be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n in_o the_o point_n c_o and_o the_o great_a segment_n thereof_o be_v the_o line_n ac_fw-la 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 cb_o be_v the_o parallelogram_n ak_o and_o the_o square_a of_o the_o line_n ac_fw-la be_v the_o square_a fd._n wherefore_o the_o parallelogram_n ak_o be_v equal_a to_o the_o square_a fd._n and_o the_o parallelogram_n of_o be_v equal_a to_o the_o parallelogram_n fe_o put_v the_o square_a ck_n common_a to_o they_o both_o wherefore_o the_o whole_a parallelogram_n ak_o be_v equal_a to_o the_o whole_a parallelogram_n ce_fw-fr wherefore_o the_o parallelogram_n ce_fw-fr and_o ak_o be_v double_a to_o the_o parallelogram_n ak_o bâ—t_v the_o parallelogram_n ak_o and_o ce_fw-fr be_v the_o gnomon_n lmn_o and_o the_o square_a ck_n wherefore_o the_o gnomon_n lmn_o and_o the_o square_a ck_n be_v double_a to_o the_o parallelogram_n ak_o but_o it_o be_v prove_v that_o the_o parallelogram_n ak_o be_v equal_a to_o the_o square_a df._n wherefore_o the_o gnomon_n lmn_o and_o the_o square_a ck_n be_v double_a to_o the_o square_a df._n wherefore_o the_o gnomon_n lmn_o and_o the_o square_n ck_a and_o df_o be_v treble_a to_o the_o square_a df._n but_o the_o gnomon_n lmn_o and_o the_o square_n ck_a and_o df_o be_v the_o whole_a square_a ae_n together_o with_o the_o square_a ck_n which_o be_v the_o square_n of_o the_o line_n ab_fw-la and_o bc._n and_o df_o be_v the_o square_a of_o the_o line_n ac_fw-la wherefore_o the_o square_n of_o the_o line_n ab_fw-la and_o bc_o be_v treble_a to_o the_o square_n of_o the_o line_n ac_fw-la 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 the_o square_n make_v of_o the_o whole_a line_n and_o of_o the_o less_o segment_n be_v treble_a to_o the_o square_n make_v of_o the_o great_a segment_n which_o be_v require_v to_o be_v prove_v look_v for_o a_o other_o demonstration_n of_o this_o proposition_n after_o the_o five_o proposition_n of_o this_o book_n ¶_o the_o 5._o theorem_a the_o 5._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 unto_o it_o be_v add_v a_o right_a â—ine_o proportion_n equal_a to_o the_o great_a segment_n the_o whole_a right_a line_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 right_a line_n give_v at_o the_o beginning_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_o and_o let_v the_o great_a segment_n thereof_o be_v ac_fw-la and_o unto_o the_o line_n ab_fw-la add_v the_o line_n ad_fw-la equal_a to_o the_o line_n ac_fw-la then_o i_o say_v that_o the_o line_n dâ—_n be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n in_o the_o point_n a_o and_o the_o great_a segment_n thereof_o be_v the_o right_a line_n pâ—t_n at_o the_o beginning_n namely_o ab_fw-la describe_v by_o the_o 46._o of_o the_o first_o upon_o on_o the_o line_n ab_fw-la a_o square_a ae_n and_o make_v perfect_a the_o figure_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 ceâ—_n and_o the_o square_a of_o the_o â—â—ne_n ac_fw-la be_v the_o square_a ch._n wherefore_o the_o parallelogram_n ce_fw-fr be_v equal_a to_o the_o square_a ch._n but_o unto_o the_o square_a change_z be_v equal_a the_o square_a dh_o by_o the_o first_o of_o the_o six_o and_o unto_o the_o parallelogram_n ce_fw-fr be_v equal_a the_o parallelogram_n he._n wherefore_o the_o parallelogram_n dh_o be_v equal_a to_o the_o parallelogram_n he._n add_v the_o parallelogram_n hb_o common_a to_o they_o both_o wherefore_o the_o whole_a parallelogram_n dk_o be_v equal_a to_o the_o whole_a square_a ae_n and_o the_o parallelogram_n dk_o be_v that_o which_o be_v contain_v under_o the_o line_n bd_o and_o dam_fw-ge for_o the_o line_n ad_fw-la be_v equal_a to_o the_o line_n dl_o &_o the_o square_a ae_n be_v the_o square_a of_o the_o line_n ab_fw-la wherefore_o that_o which_o be_v contain_v under_o the_o line_n ad_fw-la and_o db_o be_v equal_a to_o the_o square_n of_o the_o line_n ab_fw-la wherefore_o as_o the_o line_n db_o be_v to_o the_o line_n basilius_n so_o be_v the_o line_n basilius_n to_o the_o line_n ad_fw-la by_o the_o 17._o of_o the_o six_o but_o the_o line_n db_o be_v great_a than_o the_o line_n basilius_n wherefore_o the_o line_n basilius_n be_v great_a than_o the_o line_n ad._n wherefore_o the_o line_n bd_o be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n in_o the_o point_n a_o and_o his_o great_a segment_n be_v the_o line_n ab_fw-la 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 unto_o â—t_a be_v add_v a_o right_a line_n equal_a to_o the_o great_a segment_n the_o whole_a right_a line_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 right_a line_n give_v at_o the_o beginning_n which_o be_v require_v to_o be_v demonstrate_v this_o proposition_n be_v again_o afterward_o demonstrate_v a_o corollary_n add_v by_o campane_n hereby_o it_o be_v 〈…〉_o from_o the_o greaâ—â—â—_n 〈◊〉_d of_o a_o line_n divide_v by_o a_o extreme_a &_o mean_a proportion_n proposition_n be_v 〈◊〉_d away_o 〈◊〉_d segment_n the_o say_v great_a a_o segment_n shall_v 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 shall_v be_v the_o line_n take_v away_o as_o let_v the_o line_n â—â—_o be_v divide_v by_o a_o extreme_a and_o meanâ—_z proportion_n in_o the_o point_n c._n and_o leâ—_z the_o 〈…〉_o line_n 〈…〉_o a._n d._n i_o say_v that_o ac_fw-la be_v also_o divide_v by_o a_o extreme_a and_o meanâ—_z proportion_n in_o the_o point_n d_o and_o that_o his_o great_a portion_n be_v dc_o for_o by_o the_o definitino_fw-it of_o a_o line_n so_o divide_v ab_fw-la be_v to_o ac_fw-la as_o ac_fw-la be_v to_o cb._n but_o as_o ac_fw-la be_v to_o cb_o so_o be_v ac_fw-la to_o dc_o by_o the_o 7._o of_o the_o 〈…〉_o be_v equal_a to_o cb_o wherefore_o by_o the_o 11._o of_o the_o five_o as_z ab_fw-la be_v to_o ac_fw-la so_o be_v ac_fw-la to_o cd_o
the_o line_n ab_fw-la be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n in_o the_o point_n c_o and_o the_o great_a segment_n thereof_o be_v the_o line_n acâ—_n therefore_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o be_v equal_a to_o the_o square_n of_o the_o line_n ac_fw-la wherefore_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o twice_o be_v double_a to_o the_o square_n of_o ac_fw-la wherefore_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o twice_o together_o with_o the_o square_n of_o the_o line_n ac_fw-la be_v treble_a to_o the_o square_n of_o the_o line_n ac_fw-la but_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o twice_o together_o with_o the_o square_n of_o the_o line_n ac_fw-la be_v the_o square_n of_o the_o line_n ab_fw-la and_o bc_o by_o the_o 7._o of_o the_o second_o wherefore_o the_o square_n of_o the_o line_n ab_fw-la and_o bc_o be_v treble_a to_o the_o square_n of_o the_o line_n ac_fw-la which_o be_v require_v to_o be_v demonstrate_v resolution_n of_o the_o 5._o theorem_a suppose_v that_o a_o certain_a right_a line_n ab_fw-la be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n in_o the_o point_n c._n and_o let_v the_o great_a segment_n thereof_o be_v the_o line_n ac_fw-la and_o unto_o the_o line_n ab_fw-la add_v a_o line_n equal_a to_o the_o line_n ac_fw-la and_o let_v the_o same_o be_v ad._n theâ—_n i_o say_v that_o the_o line_n db_o be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n in_o the_o point_n a._n and_o the_o great_a segment_n thereof_o be_v the_o line_n ab_fw-la for_o forasmuch_o as_o the_o line_n db_o be_v divide_v by_o a_o extreme_a &_o mean_a proportion_n in_o the_o point_n a_o and_o the_o great_a segment_n thereof_o be_v the_o line_n ab_fw-la therefore_o as_o the_o line_n db_o be_v to_o the_o line_n basilius_n so_o be_v the_o line_n basilius_n to_o the_o line_n ad_fw-la but_o the_o line_n ad_fw-la be_v equal_a to_o the_o line_n ac_fw-la wherefore_o as_o the_o line_n db_o be_v to_o the_o line_n basilius_n so_o be_v the_o line_n basilius_n to_o the_o line_n ac_fw-la wherefore_o by_o conversion_n as_o the_o line_n bd_o be_v to_o the_o line_n dam_fw-ge so_o be_v the_o line_n ab_fw-la to_o the_o line_n bc_o by_o the_o corollary_n of_o the_o 19_o of_o the_o five_o wherefore_o by_o division_n by_o the_o 17._o of_o the_o five_o as_o the_o line_n basilius_n be_v to_o the_o line_n ad_fw-la â—o_o be_v the_o line_n ac_fw-la to_o the_o line_n cb._n but_o the_o line_n ad_fw-la be_v equal_a to_o the_o line_n ac_fw-la wherefore_o as_o the_o line_n basilius_n be_v to_o the_o line_n ac_fw-la so_o be_v the_o line_n ac_fw-la to_o the_o line_n cb._n and_o so_o it_o be_v indeed_o for_o the_o line_n ab_fw-la be_v by_o supposition_n divide_v by_o a_o extreme_a and_o mean_a proportion_n in_o the_o point_n c._n composition_n of_o the_o 5._o theorem_a now_o forasmuch_o as_o the_o line_n ab_fw-la be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n in_o the_o point_n c_o therefore_o as_o the_o line_n basilius_n be_v to_o the_o line_n ac_fw-la so_o be_v the_o line_n ac_fw-la to_o the_o line_n cb_o but_o the_o line_n ac_fw-la be_v equal_a to_o the_o line_n ad._n wherefore_o as_o the_o line_n basilius_n be_v to_o the_o line_n ad_fw-la so_o be_v the_o line_n ac_fw-la to_o the_o line_n cb._n wherefore_o by_o composition_n by_o the_o 18._o of_o the_o five_o as_o the_o line_n bd_o be_v to_o the_o line_n dam_fw-ge so_o be_v the_o line_n ab_fw-la to_o the_o line_n bc._n wherefore_o by_o conversion_n by_o the_o corollary_n of_o the_o 19_o of_o the_o five_o as_o the_o line_n db_o be_v to_o the_o line_n basilius_n so_o be_v the_o line_n basilius_n to_o the_o line_n ac_fw-la but_o the_o line_n ac_fw-la be_v equal_a to_o the_o line_n ad._n wherefore_o as_o the_o line_n db_o be_v to_o the_o line_n basilius_n so_o be_v the_o line_n basilius_n to_o the_o line_n ac_fw-la wherefore_o the_o line_n db_o be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n in_o the_o point_n a_o and_o his_o great_a segment_n be_v the_o line_n ab_fw-la which_o be_v require_v to_o be_v demonstrate_v a_o advice_n by_o john_n dee_n add_v sing_v it_o be_v doubtless_a that_o this_o parcel_n of_o resolution_n and_o composition_n be_v not_o of_o euclides_n do_v it_o can_v not_o â—ustly_o be_v impute_v to_o euclid_n that_o he_o have_v thereby_o either_o superfluity_n or_o any_o part_n disproportion_v in_o his_o whole_a composition_n elemental_a and_o though_o for_o one_o thing_n one_o good_a demonstration_n well_o suffice_v for_o stablish_v of_o the_o verity_n yet_o oâ—_n one_o thing_n diverse_o demonstrate_v to_o the_o diligent_a examiner_n of_o the_o diverse_a mean_n by_o which_o that_o variety_n arise_v do_v grow_v good_a occasion_n of_o invent_v demonstration_n where_o matter_n be_v more_o strange_a hard_a and_o barren_a also_o though_o resolution_n be_v not_o in_o all_o euclid_n before_o use_v yet_o thanks_n be_v to_o be_v give_v to_o the_o greek_a scholic_a writer_n who_o do_v leave_v both_o the_o definition_n and_o also_o so_o short_a and_o easy_a example_n of_o a_o method_n so_o ancient_a and_o so_o profitable_a the_o antiquity_n of_o it_o be_v above_o 2000_o year_n it_o be_v to_o weâ—e_v ever_o since_o plato_n his_o time_n and_o the_o profit_n thereof_o so_o great_a that_o thus_o i_o find_v in_o the_o greek_a record_v page_n 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d proclus_n have_v speak_v of_o some_o by_o nature_n excellent_a in_o invent_v demonstration_n pithy_a and_o brief_a say_v yet_o be_v there_o method_n give_v for_o that_o purpose_n and_o in_o deed_n that_o the_o best_a which_o by_o resolution_n reduce_v the_o thing_n inquire_v of_o to_o a_o undoubted_a principle_n which_o method_n plato_n teach_v leodamas_n as_o iâ—_z report_v and_o he_o be_v register_v thereby_o to_o have_v be_v the_o inventor_n of_o many_o thing_n in_o geometry_n and_o very_o in_o problem_n it_o be_v the_o chief_a aid_n for_o win_v and_o order_v a_o demonstration_n first_o by_o supposition_n of_o the_o thing_n inquire_v of_o to_o be_v do_v by_o due_a and_o orderly_a resolution_n to_o bring_v it_o to_o a_o stay_n at_o a_o undoubted_a verity_n in_o which_o point_n of_o art_n great_a abundance_n of_o example_n be_v to_o be_v see_v in_o that_o excellent_a and_o mighty_a mathematician_n archimedes_n &_o in_o his_o expositor_n eutocius_n in_o menaechmus_n likewise_o and_o in_o diocles_n book_n the_o pytiâ—s_n and_o in_o many_o other_o and_o now_o for_o as_o much_o as_o our_o euclid_n in_o the_o last_o six_o proposition_n of_o this_o thirteen_o book_n propound_v and_o conclude_v those_o problem_n which_o be_v the_o end_n scope_n and_o principal_a purpose_n to_o which_o all_o the_o premise_n of_o the_o 12._o book_n and_o the_o rest_n of_o this_o thirteen_o be_v direct_v and_o order_v it_o shall_v be_v artificial_o do_v and_o to_o a_o great_a commodity_n by_o resolution_n backward_o from_o these_o 6._o problem_n to_o return_v to_o the_o first_o definition_n of_o the_o first_o book_n i_o mean_v to_o the_o definition_n of_o a_o point_n which_o be_v nothing_o hard_a to_o do_v and_o i_o do_v counsel_n all_o such_o as_o desire_v to_o attain_v to_o the_o profound_a knowledge_n of_o geometry_n arithmetic_n or_o any_o branch_n of_o the_o science_n mathematical_a so_o by_o resolution_n discrete_o and_o advise_o to_o resolve_v unlose_v unjoint_n and_o disseaver_v every_o part_n of_o any_o work_n mathematical_a that_o therbyâ—_n aswell_o the_o due_a place_n of_o every_o verity_n and_o his_o proof_n as_o also_o what_o be_v either_o superfluous_a or_o want_v may_v evident_o appear_v for_o so_o to_o invent_v &_o there_o with_o to_o order_v their_o write_n be_v the_o custom_n of_o they_o who_o in_o the_o old_a time_n be_v most_o excellent_a and_o i_o for_o my_o part_n in_o write_v any_o mathematical_a conclusion_n which_o require_v great_a discourse_n at_o length_n have_v find_v by_o experience_n the_o commodity_n of_o it_o such_o that_o to_o do_v other_o way_n be_v to_o i_o a_o confusion_n and_o a_o unmethodicall_a heap_v of_o matter_n together_o beside_o the_o difficulty_n of_o invent_v the_o matter_n to_o be_v dispose_v and_o order_v i_o have_v occasion_n thus_o to_o geve_v you_o friendly_a advice_n for_o your_o beâ—ofeâ—_n because_o some_o of_o late_a have_v inveigh_v against_o euclid_n or_o theon_n in_o this_o place_n otherwise_o than_o i_o will_v wish_v they_o have_v the_o 6._o theorem_a the_o 6._o proposition_n if_o a_o rational_a right_a line_n be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n either_o of_o the_o segment_n be_v a_o irrational_a line_n of_o that_o kind_n which_o be_v call_v a_o residual_a line_n svppose_v that_o ab_fw-la be_v a_o rational_a line_n be_v devide_o 〈◊〉_d extreme_a
give_v wherein_o be_v contain_v the_o former_a solid_n and_o to_o prove_v that_o the_o side_n of_o the_o icosahedron_n be_v a_o irrational_a line_n of_o that_o kind_n which_o be_v call_v a_o less_o line_n demonstration_n now_o forasmuch_o as_o the_o line_n qp_v qv_n qt_n q_n and_o qr_o do_v each_o subtend_v right_a angle_n contain_v under_o the_o side_n of_o a_o equilater_n hexagon_n &_o of_o a_o equilater_n decagon_fw-mi inscribe_v in_o the_o circle_n prstv_n or_o in_o the_o circle_n efghk_n which_o two_a circle_n be_v equal_a therefore_o the_o say_a line_n be_v each_o equal_a to_o the_o side_n of_o the_o pentagon_n inscribe_v in_o the_o foresay_a circle_n by_o the_o 10._o of_o this_o book_n and_o be_v equal_a the_o one_o to_o the_o other_o by_o the_o 4._o of_o the_o first_o for_o all_o the_o angle_n at_o the_o point_n w_n which_o they_o subtend_v be_v right_a angle_n wherefore_o the_o five_o triangle_n qpv_o qpr_fw-la qr_n qst_fw-la and_o qtv_n which_o be_v contain_v under_o the_o say_a line_n qv_n qp_n qr_o q_n qt_n and_o under_o the_o side_n of_o the_o pentagon_n vpr_v be_v equilater_n and_o equal_a to_o the_o ten_o former_a triangle_n and_o by_o the_o same_o reason_n the_o five_o triangle_n opposite_a unto_o they_o namely_o the_o triangle_n yml_n ymn_n ynx_n yxo_n and_o yol_n be_v equilater_n and_o equal_a to_o the_o say_v ten_o triangle_n for_o the_o line_n ill_o ym_fw-fr yn_n yx_n and_o you_o do_v subtend_v right_a angle_n contain_v under_o the_o side_n of_o a_o equilater_n hexagon_n and_o of_o a_o equilater_n decagon_fw-mi inscribe_v in_o the_o circle_n efghk_n which_o be_v equal_a to_o the_o circle_n prstv_n wherefore_o there_o be_v describe_v a_o solid_a contain_v under_o 20._o equilater_n triangle_n wherefore_o by_o the_o last_o definition_n of_o the_o eleven_o there_o be_v describe_v a_o icosahedron_n now_o it_o be_v require_v to_o comprehend_v it_o in_o the_o sphere_n give_v and_o to_o prove_v that_o the_o side_n of_o the_o icosahedron_n be_v a_o irrational_a line_n of_o that_o kind_n which_o be_v call_v a_o less_o line_n forasmuch_o as_o the_o line_n zw_n be_v the_o side_n of_o a_o hexagon_n &_o the_o line_n wq_n be_v the_o side_n of_o a_o decagon_n therefore_o the_o line_n zq_n be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n in_o the_o point_n w_n and_o his_o great_a segment_n be_v zw_n by_o the_o 9_o of_o the_o thirteen_o wherefore_o as_o the_o line_n qz_n be_v to_o the_o line_n zw_n so_o be_v the_o line_n zw_n to_o the_o line_n wq_n but_o the_o zw_n be_v equal_a to_o the_o line_n zl_n by_o construction_n and_o the_o line_n wq_n to_o the_o line_n zy_n by_o construction_n also_o wherefore_o as_o the_o line_n qz_n be_v to_o the_o line_n zl_n so_o be_v the_o line_n zl_n to_o the_o line_n zy_n and_o the_o angle_n qzlâ—_n and_o lzy_n be_v right_a angle_n by_o the_o 2._o definition_n of_o the_o eleven_o if_o therefore_o we_o draw_v a_o right_a line_n from_o the_o point_n l_o to_o the_o point_n qui_fw-fr the_o angle_n ylq_n shall_v be_v a_o right_a angle_n by_o reason_n of_o the_o likeness_n of_o the_o triangle_n ylq_n and_o zlq_n by_o the_o 8._o of_o the_o six_o wherefore_o a_o semicircle_n describe_v upon_o the_o line_n qy_n shall_v pass_v also_o by_o the_o point_n l_o by_o the_o assumpt_n add_v by_o campane_n after_o the_o 13._o of_o this_o book_n and_o by_o the_o same_o reason_n also_o for_o that_o as_o the_o line_n qz_n be_v the_o line_n zw_n so_o be_v the_o line_n zw_n to_o the_o line_n wq_n flussas_n but_o the_o line_n zq_n be_v equal_a to_o the_o line_n yw_n and_o the_o line_n zw_n to_o the_o line_n pw_o wherefore_o as_o the_o line_n yw_n be_v to_o the_o line_n wp_n so_o be_v the_o line_n pw_o to_o the_o line_n wq_n and_o therefore_o again_o if_o we_o draw_v a_o right_a line_n from_o the_o point_n p_o to_o the_o point_n y_fw-fr the_o angle_n ypq_n shall_v be_v a_o right_a angle_n wherefore_o a_o semicircle_n describe_v upon_o the_o line_n qy_n shall_v pass_v also_o by_o the_o point_n p_o by_o the_o former_a assumpt_n &_o if_o the_o diameter_n qy_o abide_v fix_v the_o semicircle_n be_v turn_v round_o about_o until_o it_o come_v to_o the_o self_n same_o place_n from_o whence_o it_o begin_v first_o to_o be_v move_v it_o shall_v pass_v both_o by_o the_o point_n p_o and_o also_o by_o the_o rest_n of_o the_o point_n of_o the_o angle_n of_o the_o icosahedron_n and_o the_o icosahedron_n shall_v be_v comprehend_v in_o a_o sphere_n i_o say_v also_o that_o it_o be_v contain_v in_o the_o sphere_n give_v divide_v by_o the_o 10._o of_o the_o first_o the_o line_n zw_n into_o two_o equal_a part_n in_o the_o point_n a._n and_o forasmuch_o as_o the_o right_a line_n zq_n be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n in_o the_o point_n w_n and_o his_o less_o segment_n be_v qw_o therefore_o the_o segment_n qw_o have_v add_v unto_o it_o the_o half_a of_o the_o great_a segment_n namely_o the_o line_n wa_n be_v by_o the_o 3._o of_o this_o book_n in_o power_n quintuple_a to_o the_o square_n make_v of_o the_o half_a of_o the_o great_a segment_n wherefore_o the_o square_a of_o the_o line_n qa_n be_v quintuple_a to_o the_o square_n of_o the_o line_n â—w_o but_o unto_o the_o square_n of_o the_o qa_n the_o square_a of_o the_o line_n qy_n be_v quadruple_a by_o the_o corollary_n of_o the_o 20._o of_o the_o six_o for_o the_o line_n qy_n be_v double_a to_o the_o line_n qa_n and_o by_o the_o same_o reason_n unto_o the_o square_n of_o the_o wa_n the_o square_a of_o the_o line_n zw_n be_v quadruple_a wherefore_o the_o square_a of_o the_o line_n qy_n be_v quintuple_a to_o the_o square_n of_o the_o line_n zw_n by_o the_o 15._o of_o the_o five_o and_o forasmuch_o as_o the_o line_n ac_fw-la be_v quadruple_a to_o the_o line_n cb_o therefore_o the_o line_n ab_fw-la be_v quintuple_a to_o the_o line_n cb._n but_o as_o the_o line_n ab_fw-la be_v to_o the_o line_n bc_o so_o be_v the_o square_a of_o the_o line_n ab_fw-la to_o the_o square_n of_o the_o line_n bd_o by_o the_o 8_o of_o the_o six_o and_o corollary_n of_o the_o 20._o of_o the_o same_o wherefore_o the_o square_a of_o the_o line_n ab_fw-la be_v quintuple_a to_o the_o square_n of_o the_o line_n bd_o and_o it_o be_v be_v prove_v that_o the_o square_a of_o the_o line_n qy_n be_v quintuple_a to_o the_o square_n of_o the_o line_n zw_n and_o the_o line_n bd_o be_v equal_a to_o the_o line_n zw_n for_o either_o of_o they_o be_v by_o position_n equal_a to_o the_o line_n which_o be_v draw_v from_o the_o centre_n of_o the_o circle_n efghk_v to_o the_o circumference_n wherefore_o the_o line_n ab_fw-la be_v equal_a to_o the_o yq_n but_o the_o line_n ab_fw-la be_v the_o diameter_n of_o the_o sphere_n give_v wherefore_o the_o line_n yq_n which_o be_v prove_v to_o be_v the_o diameter_n of_o the_o sphere_n contain_v the_o icosahedron_n be_v equal_a to_o the_o diameter_n of_o the_o sphere_n give_v wherefore_o the_o icosahedron_n be_v contain_v in_o the_o sphere_n give_v now_o i_o say_v that_o the_o side_n of_o the_o icosahedron_n be_v a_o irrational_a line_n of_o that_o kind_n which_o be_v call_v a_o less_o line_n for_o forasmuch_o as_o the_o diameter_n of_o the_o sphere_n be_v rational_a and_o be_v in_o power_n quintuple_a to_o the_o square_n of_o the_o line_n draw_v from_o the_o centre_n of_o the_o circle_n olmnx_n wherefore_o also_o the_o line_n which_o be_v draw_v from_o the_o centre_n of_o the_o circle_n olmnx_n be_v rational_a wherefore_o the_o diameter_n also_o be_v commensurable_a to_o the_o same_o line_n by_o the_o 6._o of_o the_o ten_o be_v rational_a but_o if_o in_o a_o circle_n have_v a_o rational_a line_n to_o his_o diameter_n be_v describe_v a_o equilater_n pentagon_n the_o side_n of_o the_o pentagon_n be_v by_o the_o 11._o of_o this_o book_n a_o irrational_a line_n of_o that_o kind_n which_o be_v call_v a_o less_o line_n but_o the_o side_n of_o the_o pentagon_n olmnx_n be_v also_o the_o side_n of_o the_o icosahedron_n describe_v as_o have_v before_o be_v prove_v wherefore_o the_o side_n of_o the_o icosahedron_n be_v a_o irrational_a line_n of_o that_o kind_n which_o be_v call_v a_o less_o line_n wherefore_o there_o be_v describe_v a_o icosahedron_n and_o it_o be_v contain_v in_o the_o sphere_n give_v and_o it_o be_v prove_v that_o the_o side_n of_o thou_o icosahedron_n be_v a_o irrational_a line_n of_o that_o kind_n which_o be_v call_v a_o less_o line_n which_o be_v require_v to_o be_v do_v and_o to_o be_v prove_v a_o corollary_n hereby_o it_o be_v manifest_a that_o the_o diameter_n of_o the_o sphere_n be_v in_o power_n quintuple_a to_o the_o line_n which_o be_v draw_v from_o the_o centre_n of_o the_o circle_n to_o
double_a to_o the_o side_n of_o the_o octohedron_n &_o the_o side_n be_v in_o power_n sequitertia_fw-la to_o the_o perpendiclar_a line_n by_o the_o 12._o of_o this_o book_n wherefore_o the_o diameter_n thereof_o be_v in_o power_n duple_fw-fr superbipartiens_fw-la tertias_fw-la to_o the_o perpendicular_a line_n wherefore_o also_o the_o diameter_n and_o the_o perpendicular_a line_n be_v rational_a and_o commensurable_a by_o the_o 6._o of_o the_o ten_o as_o touch_v a_o icosahedron_n it_o be_v prove_v in_o the_o 16._o of_o this_o book_n that_o the_o side_n thereof_o be_v a_o less_o line_n when_o the_o diameter_n of_o the_o sphere_n be_v rational_a and_o forasmuch_o as_o the_o angle_n of_o the_o inclination_n of_o the_o base_n thereof_o be_v contain_v of_o the_o perpendicular_a line_n of_o the_o triangle_n and_o subtend_v of_o the_o right_a line_n which_o subtend_v the_o angle_n of_o the_o pentagon_n which_o contain_v five_o side_n of_o the_o icosahedron_n and_o unto_o the_o perpendicular_a line_n the_o side_n be_v commensurable_a namely_o be_v in_o power_n sesquitertia_fw-la unto_o they_o by_o the_o corollary_n of_o the_o 12._o of_o this_o book_n therefore_o the_o perpendicular_a line_n which_o contain_v the_o angle_n be_v irrational_a line_n namely_o less_o line_n by_o the_o 105._o of_o the_o ten_o book_n and_o forasmuch_o as_o the_o diameter_n contain_v in_o power_n both_o the_o side_n of_o the_o icosahedron_n and_o the_o line_n which_o subtend_v the_o foresay_a angle_n if_o from_o the_o power_n of_o the_o diameter_n which_o be_v rational_a be_v take_v away_o the_o power_n of_o the_o side_n of_o the_o icosahedron_n which_o be_v irrational_a it_o be_v manifest_a that_o the_o residue_n which_o be_v the_o power_n of_o the_o subtend_a line_n shall_v be_v irrational_a for_o if_o it_o shall_v be_v rational_a the_o number_n which_o measure_v the_o whole_a power_n of_o the_o diameter_n and_o the_o part_n take_v away_o of_o the_o subtend_a line_n shall_v also_o by_o the_o 4._o common_a sentence_n of_o the_o seven_o measure_n the_o residue_n namely_o the_o power_n of_o the_o side_n irrational_a which_o be_v irrational_a for_o that_o it_o be_v a_o less_o line_n which_o be_v absurd_a wherefore_o it_o be_v manifest_a that_o the_o right_a line_n which_o compose_v the_o angle_n of_o the_o inclination_n of_o the_o base_n of_o the_o icosahedron_n be_v irrational_a line_n for_o the_o subtend_a line_n have_v to_o the_o line_n contain_v a_o great_a proportion_n than_o the_o whole_a have_v to_o the_o great_a segment_n the_o angle_n of_o the_o inclination_n of_o the_o base_n of_o a_o dodecahedron_n be_v contain_v under_o two_o perpendicular_n of_o the_o base_n of_o the_o dodecahedron_n and_o be_v subtend_v of_o that_o right_a line_n who_o great_a segment_n be_v the_o side_n of_o a_o cube_n inscribe_v in_o the_o dodecahedron_n which_o right_a line_n be_v equal_a to_o the_o line_n which_o couple_v the_o section_n into_o two_o equal_a part_n of_o the_o opposite_a side_n of_o the_o dodecahedron_n and_o this_o couple_a line_n we_o say_v be_v a_o irrational_a line_n for_o that_o the_o diameter_n of_o the_o sphere_n contain_v in_o power_n both_o the_o couple_a line_n and_o the_o side_n of_o the_o dodecahedron_n but_o the_o side_n of_o the_o dodecahedron_n be_v a_o irrational_a line_n namely_o a_o residual_a line_n by_o the_o 17._o of_o this_o book_n wherefore_o the_o residue_n namely_o the_o couple_a line_n be_v a_o irrational_a line_n as_o it_o be_v â—asy_a to_o prove_v by_o the_o 4._o common_a sentence_n of_o the_o seven_o and_o that_o the_o perpendicular_a line_n which_o contain_v the_o angle_n of_o the_o inclination_n be_v irrational_a be_v thus_o prove_v by_o the_o proportion_n of_o the_o subtend_a line_n of_o the_o foresay_a angle_n of_o inclination_n to_o the_o line_n which_o contain_v the_o angle_n be_v find_v out_o the_o obliquity_n of_o the_o angle_n angle_n for_o if_o the_o subtend_a line_n be_v in_o power_n double_a to_o the_o line_n which_o contain_v the_o angle_n then_o be_v the_o angle_n a_o right_a angle_n by_o the_o 48._o of_o the_o first_o but_o if_o it_o be_v in_o power_n less_o than_o the_o double_a it_o be_v a_o acute_a angle_n by_o the_o 23._o of_o the_o second_o but_o if_o it_o be_v in_o power_n more_o than_o the_o double_a or_o have_v a_o great_a proportion_n than_o the_o whole_a have_v to_o the_o great_a segmentâ—_n the_o angle_n shall_v be_v a_o obtuse_a angle_n by_o the_o 12._o of_o the_o second_o and_o 4._o of_o the_o thirteen_o by_o which_o may_v be_v prove_v that_o the_o square_a of_o the_o whole_a be_v great_a than_o the_o double_a of_o the_o square_n of_o the_o great_a segment_n this_o be_v to_o be_v note_v that_o that_o which_o flussas_n have_v here_o teach_v touch_v the_o inclination_n of_o the_o base_n of_o the_o â—ive_a regular_a body_n hypsicles_n teach_v after_o the_o 5_o proposition_n of_o the_o 15._o book_n where_o he_o confess_v that_o he_o receive_v it_o of_o one_o isidorus_n and_o seek_v to_o make_v the_o matter_n more_o clear_a he_o endeavour_v himself_o to_o declare_v that_o the_o angle_n of_o the_o inclination_n of_o the_o solid_n be_v give_v and_o that_o they_o be_v either_o acute_a or_o obtuse_a according_a to_o the_o nature_n of_o the_o solid_a although_o euclid_v in_o all_o his_o 15._o book_n have_v not_o yet_o show_v what_o a_o thing_n give_v be_v wherefore_o flussas_n frame_v his_o demonstration_n upon_o a_o other_o ground_n proceed_v after_o a_o other_o manner_n which_o seem_v more_o plain_a and_o more_o apt_o hereto_o be_v place_v then_o there_o albeit_o the_o reader_n in_o that_o place_n shall_v not_o be_v frustrate_a of_o his_o also_o the_o end_n of_o the_o thirteen_o book_n of_o euclides_n element_n ¶_o the_o fourteen_o book_n of_o euclides_n element_n in_o this_o book_n which_o be_v common_o account_v the_o 14._o book_n of_o euclid_n be_v more_o at_o large_a entreat_v of_o our_o principal_a purpose_n book_n namely_o of_o the_o comparison_n and_o proportion_n of_o the_o five_o regular_a body_n customable_o call_v the_o 5._o figure_n or_o form_n of_o pythagoras_n the_o one_o to_o the_o other_o and_o also_o of_o their_o side_n together_o each_o to_o other_o which_o thing_n be_v of_o most_o secret_a use_n and_o inestimable_a pleasure_n and_o commodity_n to_o such_o as_o diligent_o search_v for_o they_o and_o attain_v unto_o they_o which_o thing_n also_o undoubted_o for_o the_o worthiness_n and_o hardness_n thereof_o for_o thing_n of_o most_o price_n be_v most_o hard_a be_v first_o search_v and_o find_v out_o of_o philosopher_n not_o of_o the_o inferior_a or_o mean_a sort_n but_o of_o the_o deep_a and_o most_o ground_a philosopher_n and_o best_a exercise_v in_o geometry_n and_o albeit_o this_o book_n with_o the_o book_n follow_v namely_o the_o 15._o book_n have_v be_v hitherto_o of_o all_o man_n for_o the_o most_o part_n and_o be_v also_o at_o this_o day_n number_v and_o account_v among_o euclides_n book_n and_o suppose_v to_o be_v two_o of_o he_o namely_o the_o 14._o and_o 15._o in_o order_n as_o all_o exemplar_n not_o only_o new_a and_o late_o set_v abroad_o but_o also_o old_a monument_n write_v by_o hand_n do_v manifest_o witness_v yet_o it_o be_v think_v by_o the_o best_a learned_a in_o these_o day_n that_o these_o two_o book_n be_v none_o of_o euclides_n but_o of_o some_o other_o author_n no_o less_o worthy_a nor_o of_o less_o estimation_n and_o authority_n notwithstanding_o than_o euclid_n apollonius_n a_o man_n of_o deep_a knowledge_n a_o great_a philosopher_n and_o in_o geometry_n marvellous_a who_o wonderful_a book_n write_v of_o the_o section_n of_o cones_fw-la which_o exercise_n &_o occupy_v thewitte_v of_o the_o wise_a and_o best_a learned_a be_v yet_o remain_v be_v think_v and_o that_o not_o without_o just_a cause_n to_o be_v the_o author_n of_o they_o or_o as_o some_o think_v hypsicles_n himself_o for_o what_o can_v be_v more_o plain_o then_o that_o which_o he_o himself_o witness_v in_o the_o preface_n of_o this_o book_n basilides_n of_o tire_n say_v hypsicles_n and_o my_o father_n together_o scan_v and_o peyse_v a_o writing_n or_o book_n of_o apollonius_n which_o be_v of_o the_o comparison_n of_o a_o dodecahedron_n to_o a_o icosahedron_n inscribe_v in_o one_o and_o the_o self_n same_o sphere_n and_o what_o proportion_n these_o figure_n have_v the_o one_o to_o the_o other_o find_v that_o apollonius_n have_v fail_v in_o this_o matter_n but_o afterward_o say_v he_o i_o find_v a_o other_o copy_n or_o book_n of_o apollonius_n wherein_o the_o demonstration_n of_o that_o matter_n be_v full_a and_o perfect_a and_o show_v it_o unto_o they_o whereat_o they_o much_o rejoice_v by_o which_o word_n it_o seem_v to_o be_v manifest_a that_o apollonius_n be_v the_o first_o author_n of_o this_o book_n which_o be_v afterward_o set_v forth_o by_o hypsicles_n for_o so_o his_o own_o word_n after_o in_o
one_o and_o the_o self_n same_o sphere_n let_v the_o diameter_n of_o the_o sphere_n give_v by_o ab_fw-la and_o let_v the_o base_n of_o the_o icosahedron_n and_o dodecahedron_n describe_v in_o it_o construction_n be_v the_o triangle_n mnr_n and_o the_o pentagon_n fkh_n and_o about_o they_o let_v there_o be_v describe_v circle_n by_o the_o 5._o and_o 14._o of_o the_o four_o and_o let_v the_o line_n draw_v from_o the_o centre_n of_o those_o circle_n to_o the_o circumference_n be_v ln_o and_o ok_n then_o i_o say_v that_o the_o line_n ln_o and_o ok_n be_v equal_a and_o therefore_o one_o and_o the_o self_n same_o circle_n contain_v both_o those_o figure_n let_v the_o right_a line_n ab_fw-la be_v in_o power_n quintuple_a to_o some_o one_o right_a line_n as_o to_o the_o line_n cg_o by_o the_o corollary_n of_o the_o 6._o of_o the_o ten_o and_o make_v the_o centre_n the_o point_n c_o &_o the_o space_n cg_o describe_v a_o circle_n dzg_v and_o let_v the_o side_n of_o a_o pentagon_n inscribe_v in_o that_o circle_n by_o the_o 11._o of_o the_o four_o be_v the_o line_n zg_v and_o let_v eglantine_n subtend_v half_o of_o the_o ark_n zg_v be_v the_o side_n of_o a_o decagon_n inscribe_v in_o that_o circle_n and_o by_o the_o 30._o of_o the_o six_o divide_v the_o line_n cg_o by_o a_o extreme_a &_o mean_a proportion_n in_o the_o point_n i_o demonstration_n now_o forasmuch_o as_o in_o the_o 16._o of_o the_o thirteen_o it_o be_v prove_v that_o this_o line_n cg_o unto_o who_o the_o diameter_n ab_fw-la of_o the_o sphere_n be_v in_o power_n quintuple_a be_v the_o line_n which_o be_v draw_v from_o the_o centre_n of_o the_o circle_n which_o contain_v five_o angle_n of_o the_o icosahedron_n and_o the_o side_n of_o the_o pentagon_n describe_v in_o that_o circle_n dzg_v namely_o the_o line_n zg_n be_v side_n of_o the_o icosahedron_n describe_v in_o the_o sphere_n who_o diameter_n be_v the_o line_n ab_fw-la therefore_o the_o right_a line_n zg_v be_v equal_a to_o the_o line_n mn_o which_o be_v put_v to_o be_v the_o side_n of_o the_o icosahedron_n or_o of_o his_o triangular_a base_a moreover_o by_o the_o 17._o of_o the_o thirteen_o it_o be_v manifest_a that_o the_o right_a line_n â—h_a which_o subtend_v the_o angle_n of_o the_o pentagon_n of_o the_o dodecahedron_n inscribe_v in_o the_o foresay_a sphere_n be_v the_o side_n of_o the_o cube_n inscribe_v in_o the_o self_n same_o sphere_n for_o upon_o the_o angle_n of_o the_o cube_fw-la be_v make_v the_o angle_n of_o the_o dodecahedron_n wherefore_o the_o diameter_n ab_fw-la be_v in_o power_n triple_a to_o fh_v the_o side_n of_o the_o cube_n by_o the_o 15._o of_o the_o thirteen_o but_o the_o same_o line_n ab_fw-la be_v by_o supposition_n in_o power_n quintuple_a to_o the_o line_n cg_o wherefore_o five_o square_n of_o the_o line_n cg_o be_v equal_a to_o three_o square_n of_o the_o line_n fh_o for_o each_o be_v equal_a to_o one_o and_o the_o self_n same_o square_n of_o the_o line_n ab_fw-la and_o forasmuch_o as_o eglantine_n the_o side_n of_o the_o decagon_n cut_v the_o right_a line_n cg_o by_o a_o extreme_a and_o mean_a proportion_n by_o the_o corollary_n of_o the_o 9_o of_o the_o thirteen_o likewise_o the_o line_n hk_o cut_v the_o line_n fh_o the_o side_n of_o the_o cube_n by_o a_o extreme_a and_o mean_a proportion_n by_o the_o corollary_n of_o the_o 17._o of_o the_o thirteen_o therefore_o the_o line_n cg_o and_o fh_o be_v divide_v into_o the_o self_n same_o proportion_n by_o the_o second_o of_o this_o book_n and_o the_o right_a line_n ci_o and_o eglantine_n which_o be_v the_o great_a segment_n of_o one_o and_o the_o self_n same_o line_n cg_o be_v equal_a and_o forasmuch_o as_o five_o square_n of_o the_o line_n cg_o be_v equal_a to_o three_o square_n of_o the_o line_n fh_o therefore_o five_o square_n of_o the_o line_n ge_z be_v equal_a to_o three_o square_n of_o the_o line_n hk_o for_o the_o line_n ge_z and_o hk_o be_v the_o great_a segment_n of_o the_o line_n cg_o and_o fh_o wherefore_o five_o squâ—reâ—_n of_o the_o lineâ—_z cg_o &_o ge_z be_v equal_a to_o the_o square_n of_o the_o 〈◊〉_d â—h_a &_o hk_a by_o the_o 1â—_o of_o the_o â—ift_n but_o unto_o the_o square_n of_o the_o line_n cg_o and_o geâ—_n be_v â—qual_a the_o squâ—re_n of_o thetine_a zg_n by_o the_o 10._o of_o the_o thirteen_o and_o unto_o the_o line_n zg_v the_o line_n mn_o be_v equal_a wherefore_o five_o square_n of_o the_o line_n mn_o be_v equal_a to_o three_o square_n of_o the_o line_n fh_o hk_o but_o the_o square_n of_o the_o line_n â—â—_o and_o hk_o 〈◊〉_d quintuple_a to_o the_o square_n of_o the_o line_n ok_n which_o be_v draw_v from_o the_o centre_n by_o the_o three_o of_o this_o book_n wherefore_o three_o square_n of_o the_o line_n fh_o and_o hk_o make_v 15._o square_n of_o the_o line_n ok_n and_o forasmuch_o as_o the_o square_n of_o the_o line_n mn_o be_v triple_a to_o the_o square_n of_o the_o line_n ln_o which_o be_v draw_v from_o the_o centre_n by_o the_o 12._o of_o the_o thirteen_o therefore_o five_o square_n of_o the_o line_n mn_o be_v equal_a to_o 15._o square_n of_o the_o line_n ln_o but_o five_o square_n of_o the_o line_n mn_o be_v equal_a unto_o three_o square_n of_o the_o line_n fh_o and_o hk_o wherefore_o one_o square_a of_o the_o line_n ln_o be_v equal_a to_o one_o square_a of_o the_o line_n ok_n be_v each_o the_o fifteen_o part_n of_o equal_a magnitude_n by_o the_o 15._o of_o the_o fifâ—â—_n wherefore_o the_o line_n ln_o and_o ok_n which_o be_v draw_v from_o the_o centre_n be_v equal_a wherefore_o also_o the_o circle_n nrm_n and_o fkh_n which_o be_v describe_v of_o those_o line_n be_v equal_a and_o those_o circle_n contain_v by_o supposition_n the_o base_n of_o the_o dodecahedron_n and_o of_o the_o icosahedron_n describe_v in_o one_o and_o the_o self_n same_o sphere_n wherefore_o one_o and_o the_o self_n same_o circle_n etc_n etc_n aâ—_z in_o thâ—_z proposition_n which_o be_v require_v to_o be_v prove_v the_o 5._o proposition_n if_o in_o a_o circle_n be_v inscribe_v the_o pentagon_n of_o a_o dodecahedron_n and_o the_o triangle_n of_o a_o icosahedron_n campane_n and_o from_o the_o centre_n to_o one_o of_o their_o side_n be_v draw_v a_o perpendicular_a line_n that_o which_o be_v contain_v 30._o time_n under_o the_o side_n &_o the_o perpendicular_a line_n fall_v upon_o it_o be_v equal_a to_o the_o superficies_n of_o that_o solid_a upon_o who_o side_n the_o perpendicular_a line_n fall_v svppose_v that_o in_o the_o circle_n age_n be_v describe_v the_o pentagon_n of_o a_o dodecahedron_n which_o let_v be_v abgde_v and_o the_o triangle_n of_o a_o icosahedron_n describe_v in_o the_o same_o sphere_n which_o let_v be_v afh_n construction_n and_o let_v the_o centre_n be_v the_o point_n c._n â—â—on_n which_o draw_v perpendicular_o the_o line_n ci_o to_o the_o side_n of_o the_o pentagon_n and_o the_o line_n cl_n to_o the_o side_n of_o the_o triangle_n then_o i_o say_v that_o the_o rectangle_n figure_n contain_v under_o the_o line_n ci_o and_o gd_o 30._o time_n be_v equal_a to_o the_o superficies_n of_o the_o dodecahedron_n and_o that_o that_o which_o be_v contain_v under_o the_o line_n cl_n &_o of_o 30._o time_n be_v equal_a to_o the_o superficies_n of_o the_o icosahedron_n describe_v in_o the_o same_o sphere_n draw_v these_o right_a line_n ca_n cf_o cg_o and_o cd_o demonstration_n now_o forasmuch_o as_o that_o which_o be_v contain_v under_o the_o base_a gd_o &_o the_o altitude_n i_fw-mi be_v double_a to_o the_o triangle_n gcd_n by_o the_o 41._o of_o the_o first_o and_o five_o triangle_n like_a and_o equal_a to_o the_o triangle_n gcd_n do_v make_v the_o pentagon_n abgde_v of_o the_o dodecahedron_n wherefore_o that_o which_o be_v contain_v under_o the_o line_n gd_o and_o i_fw-mi five_o time_n be_v equal_a to_o two_o pentagon_n wherefore_o that_o which_o be_v contain_v under_o the_o line_n gd_o and_o i_fw-mi _o time_n be_v equal_a to_o the_o 12._o pentagon_n which_o contain_v the_o superficies_n of_o the_o dodecahedron_n again_o that_o which_o be_v contain_v under_o the_o line_n cl_n and_o of_o be_v double_a to_o the_o triangle_n acf_n wherefore_o that_o which_o be_v contain_v under_o the_o line_n cl_n and_o of_o three_o time_n be_v equal_a to_o two_o such_o triangle_n as_o afh_n be_v which_o be_v one_o of_o the_o base_n of_o the_o icosahedron_n for_o the_o triangle_n acf_n be_v the_o three_o part_n of_o the_o triangle_n afh_v as_o it_o be_v easy_a to_o prove_v by_o the_o 8._o &_o 4._o of_o the_o first_o wherefore_o that_o which_o be_v contain_v under_o the_o line_n cl_n and_o af._n 30_o time_n time_n be_v equal_a to_o 10._o such_o triangle_n as_o afh_v iâ—_z which_o contain_v the_o superficies_n of_o the_o icosahedron_n and_o forasmuch_o as_o one_o and_o the_o self_n same_o
spherâ—_n contain_v the_o dodecahedron_n of_o this_o pentagon_n and_o the_o icosahedron_n of_o this_o triangle_n by_o the_o 4._o of_o this_o book_n â—_o and_o the_o line_n cl_n fall_v perpendicular_o upon_o the_o side_n of_o the_o icosahedron_n and_o the_o line_n ci_o upon_o the_o side_n of_o the_o dodecahedron_n that_o which_o be_v 30._o time_n contain_v under_o the_o side_n and_o the_o perpendicular_a line_n fall_v upon_o it_o be_v equal_a to_o the_o superficies_n of_o that_o solid_a upon_o who_o side_n the_o perpendicular_a fall_v if_o therefore_o in_o a_o circle_n etc_n etc_n as_o in_o the_o proposition_n which_o be_v require_v to_o be_v demonstrate_v a_o corollary_n order_n the_o superficiece_n of_o a_o dodecahedron_n and_o of_o a_o icosahedron_n describe_v in_o one_o and_o the_o self_n same_o sphere_n be_v the_o one_o to_o the_o other_o as_o that_o which_o be_v contain_v under_o the_o side_n of_o the_o one_o and_o the_o perpendicular_a line_n draw_v unto_o it_o from_o the_o centre_n of_o his_o base_a to_o that_o which_o be_v contain_v under_o the_o side_n of_o the_o other_o and_o the_o perpendicular_a line_n draw_v to_o it_o from_o the_o centre_n of_o his_o base_a for_o aâ—_z thirtyâ—_n timâ—s_o be_v to_o thirty_o time_n so_o be_v once_o to_o once_o by_o the_o 15._o of_o thâ—_z five_o the_o 6._o proposition_n the_o superficies_n of_o a_o dodecahedron_n be_v to_o the_o superficies_n of_o a_o icosahedron_n describe_v in_o one_o and_o the_o self_n same_o sphere_n campane_n in_o that_o proportion_n that_o the_o side_n of_o the_o cube_n be_v to_o the_o side_n of_o the_o icosahedron_n contain_v in_o the_o self_n same_o sphere_n construction_n svppose_v that_o there_o be_v a_o circle_n abg_o &_o in_o it_o by_o the_o 4._o of_o this_o book_n let_v there_o be_v inscribe_v the_o sideâ—_n of_o a_o dodecahedron_n and_o of_o a_o icosahedron_n contain_v in_o onâ—_n and_o the_o self_n same_o sphere_n and_o let_v the_o side_n oâ—_n the_o dodecahedron_n be_v agnostus_n and_o the_o side_n of_o the_o icosahedron_n be_v dg_o and_o let_v the_o centre_n be_v the_o point_n e_o from_o which_o draw_v unto_o those_o sâ—des_n perpendicular_a line_n ei_o and_o ez_o and_o produce_v the_o line_n ei_o to_o the_o point_n b_o and_o draw_v the_o linâ—_n bg_o and_o let_v the_o side_n of_o the_o cube_fw-la contain_v in_o the_o self_n same_o sphere_n be_v gc_o then_o i_o say_v that_o the_o superficies_n of_o the_o dodecahedron_n iâ—_z to_o the_o superficies_n of_o the_o icosahedron_n as_o the_o line_n â—g_z iâ—_z to_o the_o liâ—â—_n gd_o for_o forasmuch_o as_o the_o line_n ei_o beinâ—_n divide_v by_o a_o extreme_a and_o mean_a proportion_n the_o great_a segment_n thereof_o shall_v be_v the_o linâ—_n ez_o by_o the_o corollary_n of_o the_o first_o of_o this_o book_n demonstration_n and_o the_o line_n cg_o be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n his_o great_a segment_n be_v the_o line_n agnostus_n by_o the_o corollary_n of_o the_o 17._o of_o the_o thirteen_o wherefore_o the_o right_a line_n ei_o and_o cg_o â—râ—_n cut_v proportional_o by_o the_o second_o of_o this_o bâ—oke_n whâ—râ—fore_o as_o the_o line_n cg_o be_v to_o the_o line_n agnostus_n so_o be_v the_o line_n ei_o to_o the_o line_n ez_fw-fr wherâ—fore_o that_o which_o it_o contain_v under_o the_o extreme_n cg_o and_o ez_o be_v squall_n to_o that_o which_o iâ—_z contaynâ—d_v under_o the_o mean_n agnostus_n and_o ei._n by_o the_o 16._o of_o the_o six_o but_o as_o that_o which_o iâ—_z contain_v under_o the_o linâ—â—_n cg_o and_o â—z_n be_v to_o that_o which_o be_v contain_v under_o the_o line_n dg_o and_o ez_o so_o by_o the_o first_o of_o the_o six_o iâ—_z the_o linâ—_n cg_o to_o the_o line_n dg_o for_o both_o those_o parallelogram_n have_v oâ—â—_n and_o the_o self_n same_o altitude_n namely_o the_o line_n ez_fw-fr wherefore_o as_o that_o which_o be_v contain_v under_o the_o line_n ei_o and_o agnostus_n which_o iâ—_z prove_v equal_a to_o that_o which_o be_v contain_v under_o the_o lineâ—_z cg_o and_o ez_o be_v to_o that_o which_o be_v contain_v under_o the_o line_n dg_o and_o ez_o so_o be_v the_o line_n cg_o to_o the_o liâ—â—_n dg_o but_o as_o that_o which_o be_v contain_v under_o the_o line_n ei_o and_o agnostus_n be_v to_o that_o which_o be_v contain_v under_o the_o line_n dg_o and_o ez_o so_o by_o the_o corollary_n of_o the_o former_a proposition_n be_v the_o superficies_n of_o the_o dodecahedron_n to_o the_o superficies_n of_o the_o icosahedron_n wherefore_o as_o the_o superficies_n â—â—_o the_o dodecahedron_n be_v to_o the_o superficies_n of_o the_o icosahedron_n so_o be_v cg_o the_o side_n of_o the_o cube_fw-la to_o gd_v the_o side_n of_o the_o icosahedron_n the_o superficies_n therefore_o of_o a_o dodecahedron_n be_v to_o the_o superficiesâ—_n etc_n etc_n as_o in_o the_o proposition_n which_o be_v require_v to_o be_v prove_v a_o assumpt_n the_o pentagon_n of_o a_o dodecahedron_n order_n be_v equal_a to_o that_o which_o be_v contain_v under_o the_o perpendicular_a line_n which_o fall_v upon_o the_o base_a of_o the_o triangle_n of_o the_o icosahedron_n and_o five_o six_o part_n of_o the_o side_n of_o the_o cube_fw-la the_o say_v three_o solid_n be_v describe_v in_o one_o and_o the_o self_n same_o sphere_n suppose_v that_o in_o the_o circle_n abeg_n the_o pentagon_n of_o a_o dodecahedron_n be_v aâ—cig_a and_o let_v two_o side_n thereof_o ab_fw-la and_o agnostus_n be_v subtend_v of_o the_o right_a line_n bg_o and_o let_v the_o triangle_n of_o the_o icosahedron_n inscribe_v in_o the_o self_n same_o sphere_n construction_n by_o the_o 4._o of_o this_o book_n be_v afh_n and_o let_v the_o centre_n of_o the_o circle_n be_v the_o point_n d_o and_o let_v the_o diameter_n be_v ade_n cut_v fh_o the_o side_n of_o the_o triangle_n in_o the_o point_n z_o and_o cut_v the_o line_n bg_o in_o the_o point_n k._n and_o draw_v the_o right_a line_n bd._n and_o from_o the_o right_a line_n kg_v cut_v of_o a_o three_o part_n tg_n by_o the_o 9_o of_o the_o six_o now_o than_o the_o line_n bg_o subtend_v two_o side_n of_o the_o dodecahedron_n shall_v be_v the_o side_n of_o the_o cube_fw-la inscribe_v in_o the_o same_o sphere_n demonstration_n by_o the_o 17._o of_o the_o thirteen_o and_o the_o triangle_n of_o the_o icosahedron_n of_o the_o same_o sphere_n shall_v be_v aâ—h_a by_o the_o 4._o of_o this_o book_n and_o the_o line_n az_o which_o pass_v by_o the_o centre_n d_o shall_v fall_v perpendicular_o upon_o the_o side_n of_o the_o triangle_n for_o forasmuch_o as_o the_o angle_n gay_a &_o bae_o be_v equal_a by_o the_o 27._o of_o the_o thirdâ—_o for_o they_o be_v see_v upon_o equal_a circumference_n therefore_o the_o â—ases_v bk_o and_o kg_n be_v by_o the_o â—_o of_o the_o first_o equal_a wherefore_o the_o line_n bt_n contain_v 5._o six_o part_n of_o the_o line_n bg_o then_o i_o say_v that_o that_o which_o be_v contain_v under_o the_o line_n az_o and_o bt_n be_v equal_a to_o the_o pentagon_n aâ—câ—g_n for_o forasmuch_o as_o the_o line_n â—z_n be_v sesquialter_n to_o the_o line_n ad_fw-la for_o the_o line_n dâ—_n be_v divide_v into_o two_o equal_a part_n in_o the_o point_n z_o by_o the_o corollary_n of_o the_o â—2â—_n of_o the_o thirteen_o likewise_o by_o construction_n the_o line_n kg_n be_v sesquialter_fw-la to_o the_o line_n kt_n therefore_o as_o the_o line_n az_o be_v to_o the_o line_n ad_fw-la so_o be_v the_o line_n kg_v to_o the_o 〈◊〉_d â—t_a wherefore_o that_o which_o be_v contain_v undeâ—_n the_o 〈◊〉_d az_o and_o kt_n be_v equal_a to_o that_o which_o be_v contain_v under_o the_o mean_n ad_fw-la and_o kg_n by_o the_o 16._o of_o the_o six_o but_o unto_o the_o line_n kg_n be_v the_o line_n â—k_v â—roved_v equal_a wherefore_o that_o which_o be_v contain_v under_o the_o line_n az_o and_o kt_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 bk_o but_o that_o which_o be_v contain_v under_o the_o line_n ad_fw-la and_o bk_o be_v by_o the_o 41._o of_o the_o first_o double_a to_o the_o triangle_n abdella_n wherefore_o that_o which_o be_v contain_v under_o the_o line_n az_o and_o kt_n be_v double_a to_o the_o same_o triangle_n abdella_n and_o forasmuch_o as_o the_o pentagon_n abcig_n contaynethâ—_n 〈…〉_o equal_a â—o_o the_o triangle_n abdella_n and_o that_o which_o be_v contain_v under_o the_o line_n az_o and_o kt_n contain_v two_o such_o triangle_n therefore_o the_o pentagon_n abcig_n be_v duple_fw-fr sesquialter_fw-la to_o the_o rectangle_n parallelogram_n contain_v under_o the_o line_n az_o and_o kt_n and_o 〈…〉_o 1._o of_o the_o six_o that_o which_o be_v contain_v under_o the_o line_n az_o and_o bt_n be_v to_o that_o which_o be_v contain_v under_o the_o line_n az_o and_o kt_n as_o the_o base_a bt_n be_v to_o the_o base_a â—â—tâ—_n therefore_o that_o which_o be_v contain_v under_o the_o line_n az_o
and_o bt_n be_v duple_fw-fr sesquialter_fw-la to_o that_o which_o be_v contain_v unâ—â—r_o the_o line_n az_o &_o kt_n but_o unto_o that_o which_o be_v contain_v under_o the_o line_n az_o and_o kt_n the_o pentagon_n abcig_n be_v prove_v duple_n sesquialter_fw-la wherefore_o the_o pentagon_n abcig_n of_o the_o dodecahedron_n be_v equal_a to_o that_o which_o be_v contain_v under_o the_o perpendicular_a line_n az_o and_o under_o the_o line_n bt_v which_o be_v five_o six_o part_n of_o the_o line_n bg_o ¶_o the_o 7._o proposition_n campane_n a_o right_a line_n divide_v by_o a_o extreme_a and_o mean_a proportion_n what_o proportion_n the_o line_n contain_v in_o power_n the_o whole_a line_n and_o the_o great_a segment_n have_v to_o the_o line_n contain_v in_o power_n the_o whole_a and_o the_o less_o segment_n the_o same_o have_v the_o side_n of_o the_o cube_fw-la to_o the_o side_n of_o the_o icosahedron_n contain_v in_o one_o and_o the_o same_o sphere_n construction_n take_v a_o circle_n abe_n and_o in_o it_o by_o the_o 11._o of_o the_o four_o inscribe_v a_o equilater_n pentagon_n bzech_n and_o by_o the_o second_o of_o the_o same_o a_o equilater_n triangle_n abi_n and_o let_v the_o centre_n thereof_o be_v the_o point_n g._n and_o draw_v a_o line_n from_o g_z to_o b._n and_o divide_v the_o line_n gb_o by_o a_o extreme_a and_o mean_a proportion_n in_o the_o point_n d_o by_o the_o 30._o of_o the_o six_o and_o let_v the_o line_n ml_o contain_v in_o power_n both_o the_o whole_a line_n gb_o and_o his_o less_o segment_n bd_o by_o the_o corollary_n of_o the_o 13._o of_o the_o ten_o and_o draw_v the_o right_a line_n bâ—_n subtend_v the_o angle_n of_o the_o pentagon_n which_o shall_v be_v the_o side_n of_o the_o cube_fw-la by_o the_o corollary_n of_o the_o 17._o of_o the_o thirteen_o â—_o and_o the_o line_n by_o shall_v be_v the_o side_n of_o the_o icosahedron_n and_o the_o line_n â—z_n the_o side_n of_o the_o dodecahedron_n by_o the_o 4._o of_o this_o book_n then_o i_o say_v that_o be_v the_o side_n of_o the_o cube_fw-la be_v to_o by_o the_o side_n of_o the_o icosahedron_n as_o the_o line_n contain_v in_o power_n the_o line_n bg_o &_o gd_o be_v to_o the_o line_n contain_v in_o power_n the_o line_n gb_o and_o bd._n demonstration_n for_o forasmuch_o as_o by_o the_o 12._o of_o the_o thirteen_o the_o line_n by_o be_v in_o power_n triple_a to_o the_o line_n bg_o and_o by_o the_o 4._o of_o the_o same_o the_o square_n of_o the_o line_n gb_o &_o bd_o be_v triple_a to_o the_o square_n of_o the_o line_n gd_o wherefore_o by_o the_o 15._o of_o the_o five_o the_o square_a of_o the_o line_n by_o be_v to_o the_o square_n of_o the_o line_n gb_o &_o bd_o namely_o triple_a to_o treble_v as_o the_o square_n of_o the_o line_n bâ—_n be_v to_o the_o square_n of_o the_o line_n gd_v namely_o as_o one_o be_v to_o one_o but_o as_o the_o square_n of_o the_o line_n bg_o be_v to_o the_o square_n of_o the_o gd_o so_o be_v the_o square_a of_o the_o line_n be_v to_o the_o square_n of_o the_o line_n bz_o for_o the_o line_n bg_o gd_o and_o be_v bz_o be_v in_o one_o and_o the_o same_o proportion_n by_o the_o second_o of_o this_o book_n for_o bz_o be_v the_o great_a segment_n of_o the_o line_n be_v by_o the_o corollary_n of_o the_o 17._o of_o the_o thirteen_o wherefore_o the_o square_a of_o the_o line_n be_v be_v to_o the_o square_n of_o the_o line_n bz_o as_o the_o square_n of_o the_o line_n by_o be_v to_o the_o square_n of_o the_o line_n bg_o and_o bd._n wherefore_o alternate_o the_o square_a of_o the_o line_n be_v be_v to_o the_o square_n of_o the_o line_n by_o as_o the_o square_n of_o the_o line_n bz_o be_v to_o the_o square_n of_o the_o line_n gb_o and_o bd._n but_o the_o square_n of_o the_o line_n bz_o be_v equal_a to_o the_o square_n of_o the_o line_n bg_o and_o gd_o by_o the_o 10._o of_o the_o thirteen_o for_o the_o line_n bg_o be_v equal_a to_o the_o side_n of_o the_o hexagon_n and_o the_o line_n gd_o to_o the_o side_n of_o the_o decagon_n by_o the_o corollary_n of_o the_o 9_o of_o the_o same_o wherefore_o the_o square_n of_o the_o line_n bg_o and_o gd_o be_v to_o the_o square_n of_o the_o line_n gâ—_n and_o bd_o as_o the_o square_n of_o the_o line_n be_v be_v to_o the_o square_n of_o the_o line_n by_o but_o the_o line_n zb_n contain_v in_o power_n the_o line_n bg_o and_o gd_o and_o the_o line_n ml_o contain_v in_o power_n the_o line_n gb_o and_o bd_o by_o construction_n wherefore_o as_o the_o line_n zb_n which_o contain_v in_o power_n the_o whole_a line_n bg_o and_o the_o great_a segment_n gd_o be_v to_o the_o line_n ml_o which_o contain_v in_o in_o power_n the_o whole_a line_n gb_o and_o the_o less_o segment_n bd_o so_o be_v be_v the_o side_n of_o the_o cube_fw-la to_o by_o the_o side_n of_o the_o icosahedron_n by_o the_o 22._o of_o the_o six_o wherefore_o a_o right_a line_n divide_v by_o a_o extreme_a and_o mean_a proportion_n what_o proportion_n the_o line_n contain_v in_o power_n the_o whole_a line_n and_o the_o great_a segment_n have_v to_o the_o line_n contain_v in_o power_n the_o whole_a line_n and_o the_o less_o segment_n the_o same_o have_v the_o side_n of_o the_o cube_fw-la to_o the_o side_n of_o the_o icosahedron_n contain_v in_o one_o and_o the_o same_o sphere_n which_o be_v require_v to_o be_v prove_v ¶_o the_o 8._o proposition_n order_n the_o solid_a of_o a_o dodecahedron_n be_v to_o the_o solid_a of_o a_o icosahedron_n as_o the_o side_n of_o a_o cube_n be_v to_o the_o side_n of_o a_o icosahedron_n all_o those_o solid_n be_v describe_v in_o one_o and_o the_o self_n same_o sphere_n forasmuch_o as_o in_o the_o 4._o of_o this_o book_n it_o have_v be_v prove_v that_o one_o and_o the_o self_n same_o circle_n contain_v both_o the_o triangle_n of_o a_o icosahedron_n and_o the_o pentagon_n of_o a_o dodecahedron_n describe_v in_o one_o and_o the_o self_n same_o sphere_n wherefore_o the_o circle_n which_o contain_v those_o base_n be_v equal_a the_o perpendicular_n also_o which_o be_v draw_v from_o the_o centre_n of_o the_o sphere_n to_o those_o circle_n shall_v be_v equal_a by_o the_o corollary_n of_o the_o assumpt_n of_o the_o 16_o of_o the_o twelve_o and_o therefore_o the_o pyramid_n set_v upon_o the_o base_n of_o those_o solid_n have_v one_o and_o the_o self_n same_o altitude_n for_o the_o altitude_n of_o those_o pyramid_n concurrâ—_n in_o the_o centre_n wherefore_o they_o be_v in_o proportion_n as_o their_o base_n be_v by_o the_o 5._o and_o 6._o of_o the_o twelve_o and_o therefore_o the_o pyramid_n which_o compose_v the_o dodecahedron_n arâ—_n to_o the_o pyramid_n which_o compose_v the_o icosahedron_n as_o the_o base_n be_v which_o base_n be_v the_o superficiece_n of_o those_o solid_n wherefore_o their_o solid_n be_v the_o one_o to_o the_o other_o as_o their_o superficiece_n be_v but_o the_o superficies_n of_o the_o dodecahedron_n be_v to_o the_o superficies_n of_o the_o icosahedron_n as_o the_o side_n of_o the_o cube_fw-la be_v to_o the_o side_n of_o the_o icosahedron_n by_o the_o 6._o of_o this_o book_n wherefore_o by_o the_o 11._o of_o the_o five_o as_o the_o solid_a of_o the_o dodecahedron_n be_v to_o the_o solid_a of_o the_o icosahedron_n so_o be_v the_o side_n of_o the_o cube_fw-la to_o the_o side_n of_o the_o icosahedron_n all_o the_o say_v solid_n be_v inscribe_v in_o one_o and_o the_o self_n same_o sphere_n wherefore_o the_o solid_a of_o a_o dodecahedron_n be_v to_o the_o solid_a of_o a_o icosahedron_n as_o the_o side_n of_o a_o cube_fw-la be_v to_o the_o side_n of_o a_o icosahedron_n all_o those_o solid_n be_v describe_v in_o one_o and_o the_o self_n same_o sphere_n which_o be_v require_v to_o be_v prove_v a_o corollary_n the_o solid_a of_o a_o dodecahedron_n be_v to_o the_o solid_a of_o a_o icosahedron_n campane_n as_o the_o superficiece_n of_o the_o one_o be_v to_o the_o superficiece_n of_o the_o other_o be_v describe_v in_o one_o and_o the_o self_n same_o sphere_n namely_o as_o the_o side_n of_o the_o cube_fw-la be_v to_o the_o side_n of_o the_o icosahedron_n as_o be_v before_o manifest_v for_o they_o be_v resolve_v into_o pyramid_n of_o one_o and_o the_o self_n same_o altitude_n ¶_o the_o 9_o proposition_n if_o the_o side_n of_o a_o equilater_n triangle_n be_v rational_a the_o superficies_n shall_v be_v irrational_a campane_n of_o that_o kind_n which_o be_v call_v medial_a svppose_v that_o abg_o be_v a_o equilater_n triangle_n and_o from_o the_o point_n a_o draw_v unto_o the_o side_n bg_o a_o perpendicular_a line_n ad_fw-la and_o let_v the_o line_n ab_fw-la be_v rational_a then_o i_o say_v that_o the_o superficies_n abg_o be_v medial_a construction_n forasmuch_o as_o the_o line_n ab_fw-la be_v in_o power_n
def_n who_o side_n let_v be_v de_fw-fr and_o let_v the_o right_a line_n subtend_v the_o angle_n of_o the_o pentagon_n make_v of_o the_o side_n of_o the_o icosahedron_n be_v the_o line_n ef._n then_o i_o say_v that_o the_o side_n ed_z be_v in_o power_n double_a to_o the_o line_n h_o the_o less_o of_o those_o segment_n forasmuch_o as_o by_o that_o which_o be_v demonstrate_v in_o the_o 15._o of_o this_o book_n demonstration_n it_o be_v manifest_a that_o ed_z the_o side_n of_o the_o icosahedron_n be_v the_o great_a segment_n of_o the_o line_n efâ—_n and_o that_o the_o diameter_n df_o contain_v in_o power_n the_o two_o line_n ed_z and_o of_o namely_o the_o whole_a and_o the_o great_a segment_n but_o by_o supposition_n the_o side_n ab_fw-la contain_v in_o power_n the_o two_o line_n c_o &_o h_o join_v together_o in_o the_o self_n same_o proportion_n wherefore_o the_o line_n of_o be_v to_o the_o line_n ed_z as_z the_o line_n c_o be_v to_o the_o line_n h_o by_o the_o â—_o oâ—_n this_o bokeâ—_n and_o alternally_n by_o the_o 16._o of_o the_o five_o the_o line_n of_o be_v to_o the_o line_n c_o as_o the_o line_n ed_z be_v to_o the_o line_n h._n and_o forasmuch_o as_o the_o line_n df_o contain_v in_o power_n the_o two_o line_n ed_z and_o of_o and_o the_o line_n ab_fw-la contain_v in_o power_n the_o two_o line_n c_o and_o h_o therefore_o the_o square_n of_o the_o line_n of_o and_o ed_z be_v to_o the_o square_n of_o the_o line_n df_o as_o the_o square_n of_o the_o line_n c_o and_o h_o to_o the_o square_a ab_fw-la and_o alternate_o the_o square_n of_o the_o line_n of_o and_o â—d_a be_v to_o the_o square_n of_o the_o line_n c_o and_o h_o as_o the_o square_n of_o the_o line_n df_o be_v to_o the_o square_n of_o the_o line_n abâ—_n but_o df_o the_o diameter_n be_v by_o the_o 14._o of_o the_o thirtenâ—h_o iâ—_z power_n double_a to_o ab_fw-la the_o side_n of_o the_o octohedron_n inscribe_v by_o supposition_n in_o the_o same_o sphere_n wherefore_o the_o square_n of_o the_o line_n of_o and_o ed_z be_v double_a to_o the_o square_n of_o the_o line_n c_o and_o h._n and_o therefore_o one_o square_a of_o the_o line_n ed_z be_v double_a to_o one_o square_a of_o the_o line_n h_o by_o the_o 12._o of_o the_o five_o wherefore_o ed_z the_o side_n of_o the_o icosahedron_n be_v in_o power_n duple_n to_o the_o line_n h_o which_o be_v the_o less_o segment_n if_o therefore_o the_o poweâ—_n of_o the_o side_n of_o a_o octohedron_n be_v express_v by_o two_o right_a line_n join_v together_o by_o a_o extreme_a and_o mean_a proportion_n the_o side_n of_o the_o icosahedron_n contain_v in_o the_o same_o sphere_n shall_v be_v duple_v to_o the_o less_o segment_n the_o 17._o proposition_n if_o the_o side_n of_o a_o dodecahedron_n and_o the_o right_a line_n of_o who_o the_o say_a side_n be_v the_o less_o segment_n be_v so_o set_v that_o they_o make_v a_o right_a angle_n the_o right_a line_n which_o contain_v in_o power_n half_o the_o line_n subtend_v the_o angle_n be_v the_o side_n of_o a_o octohedron_n contain_v in_o the_o self_n same_o sphere_n svppose_v that_o ab_fw-la be_v the_o side_n of_o a_o dodecahedron_n and_o let_v the_o right_a line_n of_o which_o that_o side_n be_v the_o less_o segment_n be_v agnostus_n namely_o which_o couple_v the_o opposite_a side_n of_o the_o dodecahedron_n by_o the_o 4._o corollary_n of_o the_o 17._o of_o the_o thirteen_o construction_n and_o let_v those_o line_n be_v so_o set_v that_o they_o make_v a_o right_a angle_n at_o the_o point_n a._n and_o draw_v the_o right_a line_n bg_o and_o let_v the_o line_n d_o contain_n in_o power_n half_o the_o line_n bg_o by_o the_o first_o proposition_n add_v by_o flussas_n after_o the_o last_o of_o the_o six_o then_o i_o say_v that_o the_o line_n d_o be_v the_o side_n of_o a_o octohedron_n contain_v in_o the_o same_o sphere_n forasmuch_o as_o the_o line_n agnostus_n make_v the_o great_a segment_n gc_o the_o side_n of_o the_o cube_fw-la contain_v in_o the_o same_o sphere_n by_o the_o same_o 4._o corollary_n of_o the_o 17._o of_o the_o thirteen_o demonstration_n and_o the_o square_n of_o the_o whole_a line_n ag._n and_o of_o the_o less_o segment_n ab_fw-la be_v triple_a to_o the_o square_n of_o the_o great_a segment_n gc_o by_o the_o 4._o of_o the_o thirteen_o moreover_o the_o diameter_n of_o the_o sphere_n be_v in_o power_n triple_a to_o the_o same_o line_n gc_o the_o side_n of_o the_o cube_fw-la by_o the_o 15._o of_o the_o thirteen_o wherefore_o the_o line_n bg_o be_v equal_a to_o the_o 〈◊〉_d for_o it_o contain_v in_o power_n the_o two_o line_n ab_fw-la and_o ag_z by_o the_o 47._o of_o the_o first_o and_o therefore_o it_o contain_v in_o power_n the_o triple_a of_o the_o line_n gc_o but_o the_o side_n of_o the_o octohedron_n contain_v in_o the_o same_o sphere_n be_v in_o power_n triple_a to_o half_a the_o diameter_n of_o the_o sphere_n by_o the_o 14._o of_o the_o thirteen_o and_o by_o supposition_n the_o line_n d_o contaiâ—â—â—â—_n in_o powâ—â—_n the_o half_a of_o the_o line_n bg_o wherefore_o the_o line_n d_o contain_v in_o power_n the_o half_a of_o the_o same_o diameter_n be_v the_o side_n of_o a_o octohedron_n if_o therefore_o the_o side_n of_o a_o dodecahedron_n and_o the_o right_a line_n of_o who_o the_o say_a side_n be_v the_o less_o segment_n be_v so_o set_v that_o they_o make_v a_o right_a angle_n the_o right_a line_n which_o contain_v in_o power_n half_o the_o line_n subtend_v the_o angle_n be_v the_o side_n of_o a_o ocâ—â—â—edron_n contain_v in_o the_o self_n same_o sphere_n which_o be_v require_v to_o be_v prove_v a_o corollary_n unto_o what_o right_a line_n the_o side_n of_o the_o octoâ—edron_n be_v in_o power_n sesquialter_fw-la unto_o the_o same_o line_n the_o side_n of_o the_o dodecahedron_n inscribe_v in_o the_o same_o sphere_n be_v the_o great_a segment_n for_o the_o side_n of_o the_o dodecahedron_n be_v the_o great_a segment_n of_o the_o segment_n cg_o unto_o which_o d_z the_o side_n of_o the_o octohedron_n be_v in_o power_n sesquialter_n that_o be_v be_v half_a of_o the_o power_n of_o the_o line_n bg_o which_o be_v triple_a unto_o the_o line_n cg_o ¶_o the_o 18._o proposition_n if_o the_o side_n of_o a_o tetrahedron_n contain_v in_o power_n two_o right_a line_n join_v together_o by_o a_o extreme_a and_o mean_a proportion_n the_o side_n of_o a_o icosahedron_n describe_v in_o the_o self_n same_o sphere_n be_v in_o power_n sesquialter_fw-la to_o the_o less_o right_a line_n svppose_v that_o abc_n be_v a_o tetrahedron_n and_o let_v his_o side_n be_v ab_fw-la construction_n who_o power_n let_v be_v divide_v into_o the_o line_n agnostus_n and_o gb_o join_v together_o by_o a_o extreme_a and_o mean_a proportion_n namely_o let_v it_o be_v divide_v into_o agnostus_n the_o whole_a line_n and_o gb_o the_o great_a seâ—ment_n by_o the_o corollary_n of_o the_o first_o proposition_n add_v by_o flussas_n after_o the_o last_o of_o the_o six_o and_o let_v ed_z be_v the_o side_n of_o the_o icosahedron_n edf_o contain_v in_o the_o self_n same_o sphere_n and_o let_v the_o line_n which_o subtend_v the_o angle_n of_o the_o pentagon_n describe_v of_o the_o side_n of_o the_o icosahedron_n be_v ef._n then_o i_o say_v that_z ed_z the_o side_n of_o the_o icosahedron_n be_v in_o power_n sesquialter_fw-la to_o the_o less_o line_n gb_o demonstration_n forasmuch_o as_o by_o that_o which_o be_v demonstrate_v in_o the_o 15._o of_o this_o book_n the_o side_n ed_z be_v the_o greâ—ter_a segment_n of_o the_o line_n of_o which_o subtend_v the_o angle_n of_o the_o pentagon_n but_o as_o the_o whole_a line_n of_o be_v to_o the_o great_a segment_n ed_z so_o be_v the_o same_o grâ—â—ter_a segment_n to_o the_o less_o by_o the_o 30._o of_o the_o six_o and_o by_o supposition_n agnostus_n be_v the_o whole_a line_n and_o gâ—_n the_o great_a segment_n wherefore_o as_o of_o be_v to_o ed_z so_o be_v agnostus_n to_o gâ—_n by_o the_o second_o of_o the_o fourteen_o and_o alternate_o the_o line_n of_o be_v to_o the_o line_n agnostus_n as_o the_o line_n ed_z be_v to_o the_o line_n gb_o and_o forasmuch_o as_o by_o supposition_n the_o line_n ab_fw-la contain_v in_o power_n the_o two_o line_n agnostus_n and_o gb_o therefore_o by_o the_o 4â—_o of_o the_o first_o the_o angle_n agb_n be_v a_o right_a angle_n but_o the_o angle_n def_n be_v a_o right_a angle_n by_o that_o which_o be_v demonstrate_v in_o the_o 15._o of_o this_o book_n wherefore_o the_o triangle_n agâ—_n and_o feed_v be_v equiangle_n by_o the_o â—_o of_o the_o six_o wherefore_o their_o side_n be_v proportional_a namely_o as_o the_o line_n ed_z be_v to_o the_o line_n gb_o so_o be_v the_o line_n fd_o to_o the_o line_n ab_fw-la by_o the_o 4._o of_o the_o six_o but_o by_o that_o which_o have_v before_o
the_o whole_a line_n mg_o to_o the_o whole_a line_n ea_fw-la by_o the_o 18._o of_o the_o five_o wherefore_o as_o mg_o the_o side_n of_o the_o cube_fw-la be_v to_o ea_fw-la the_o semidiameter_n so_o be_v the_o line_n fghim_n to_o the_o octohedron_n abkdlc_n inscribe_v in_o one_o &_o the_o self_n same_o sphere_n if_o therefore_o a_o cube_fw-la and_o a_o octohedron_n be_v contain_v in_o one_o and_o the_o self_n same_o sphere_n they_o shall_v be_v in_o proportion_n the_o one_o to_o the_o other_o as_o the_o side_n of_o the_o cube_fw-la be_v to_o the_o semidiameter_n of_o the_o sphere_n which_o be_v require_v to_o be_v demonstrate_v a_o corollary_n distinct_o to_o notify_v the_o power_n of_o the_o side_n of_o the_o five_o solid_n by_o the_o power_n of_o the_o diameter_n of_o the_o sphere_n the_o side_n of_o the_o tetrahedron_n and_o of_o the_o cube_fw-la do_v cut_v the_o power_n of_o the_o diameter_n of_o the_o sphere_n into_o two_o square_n which_o be_v in_o proportion_n double_a the_o one_o to_o the_o other_o the_o octohedron_n cut_v the_o power_n of_o the_o diameter_n into_o two_o equal_a square_n the_o icosahedron_n into_o two_o square_n who_o proportion_n be_v duple_n to_o the_o proportion_n of_o a_o line_n divide_v by_o a_o extreme_a and_o mean_a proportion_n who_o less_o segment_n be_v the_o side_n of_o the_o icosahedron_n and_o the_o dodecahedron_n into_o two_o square_n who_o proportion_n be_v quadruple_a to_o the_o proportion_n of_o a_o line_n divide_v by_o a_o extreme_a and_o mean_a proportion_n who_o less_o segment_n be_v the_o side_n of_o the_o dodecahedron_n for_o ad_fw-la the_o diameter_n of_o the_o sphere_n contain_v in_o power_n ab_fw-la the_o side_n of_o the_o tetrahedron_n and_o bd_o the_o side_n of_o the_o cube_fw-la which_o bd_o be_v in_o power_n half_a of_o the_o side_n ab_fw-la the_o diameter_n also_o of_o the_o sphere_n contain_v in_o power_n ac_fw-la and_o cd_o two_o equal_a side_n of_o the_o octohedron_n but_o the_o diameter_n contain_v in_o power_n the_o whole_a line_n ae_n and_o the_o great_a segment_n thereof_o ed_z which_o be_v the_o side_n of_o the_o icosahedron_n by_o the_o 15._o of_o this_o book_n wherefore_o their_o power_n be_v in_o duple_n proportion_n of_o that_o in_o which_o the_o side_n be_v by_o the_o first_o corollary_n of_o the_o 20._o of_o the_o six_o have_v their_o proportion_n duple_n to_o the_o proportion_n of_o a_o extreme_a &_o mean_a proportion_n far_o the_o diameter_n contain_v in_o power_n the_o whole_a line_n of_o and_o his_o less_o segment_n fd_o which_o be_v the_o side_n of_o the_o dodecahedron_n by_o the_o same_o 15._o of_o this_o book_n wherefore_o the_o whole_a have_v to_o the_o less_o â—_o double_a proportion_n of_o that_o which_o the_o extreme_a have_v to_o the_o mean_a namely_o of_o the_o whole_a to_o the_o great_a segment_n by_o the_o 10._o definition_n of_o the_o five_o it_o follow_v that_o the_o proportion_n of_o the_o power_n be_v double_a to_o the_o double_a proportion_n of_o the_o side_n by_o the_o same_o first_o corollary_n of_o the_o 20._o of_o the_o six_o that_o be_v be_v quadruple_a to_o the_o proportion_n of_o the_o extreme_a and_o of_o the_o mean_a by_o the_o definition_n of_o the_o six_o a_o advertisement_n add_v by_o flussas_n by_o this_o mean_n therefore_o the_o diameter_n of_o a_o sphere_n be_v give_v there_o shall_v be_v give_v the_o side_n of_o every_o one_o of_o the_o body_n inscribe_v and_o forasmuch_o as_o three_o of_o those_o body_n have_v their_o side_n commensurable_a in_o power_n only_o and_o not_o in_o length_n unto_o the_o diameter_n give_v for_o their_o power_n be_v in_o the_o proportion_n of_o a_o square_a number_n to_o a_o number_n not_o square_a wherefore_o they_o have_v not_o the_o proportion_n of_o a_o square_a number_n to_o a_o square_a number_n by_o the_o corollary_n of_o the_o 25._o of_o the_o eight_o wherefore_o also_o their_o side_n be_v incommensurabe_n in_o length_n by_o the_o 9_o of_o the_o ten_o therefore_o it_o be_v sufficient_a to_o compare_v the_o power_n and_o not_o the_o length_n of_o those_o side_n the_o one_o to_o the_o otherâ—_n which_o power_n be_v contain_v in_o the_o power_n of_o the_o diameter_n namely_o from_o the_o power_n of_o the_o diameter_n let_v there_o ble_v take_v away_o the_o power_n of_o the_o cube_fw-la and_o there_o shall_v remain_v the_o power_n of_o the_o tetrahedron_n and_o take_v away_o the_o power_n of_o the_o tetrahedron_n there_o remain_v the_o power_n of_o the_o cube_fw-la and_o take_v away_o from_o the_o power_n of_o the_o diameter_n half_o the_o power_n thereof_o there_o shall_v be_v leave_v the_o power_n of_o the_o side_n of_o the_o octohedron_n but_o forasmuch_o as_o the_o side_n of_o the_o dodecahedron_n and_o of_o the_o icosahedron_n be_v prove_v to_o be_v irrational_a for_o the_o side_n of_o the_o icosahedron_n be_v a_o less_o line_n by_o the_o 16._o of_o the_o thirteen_o and_o the_o side_n of_o the_o dedocahedron_n be_v a_o residual_a line_n by_o the_o 17._o of_o the_o same_o therefore_o those_o side_n be_v unto_o the_o diameter_n which_o be_v a_o rational_a line_n set_v incommensurable_a both_o in_o length_n and_o in_o power_n wherefore_o their_o comparison_n can_v not_o be_v define_v or_o describe_v by_o any_o proportion_n express_v by_o number_n by_o the_o 8._o of_o the_o ten_o neither_o can_v they_o be_v compare_v the_o one_o to_o the_o other_o for_o irrational_a line_n of_o diverse_a kind_n be_v incommensurable_a the_o one_o to_o the_o other_o for_o if_o they_o shall_v be_v commensurable_a they_o shall_v be_v of_o one_o and_o the_o self_n same_o kind_n by_o the_o 103._o and_o 105._o of_o the_o ten_o which_o be_v impossible_a wherefore_o we_o seek_v to_o compare_v they_o to_o the_o power_n of_o the_o diameter_n think_v they_o can_v not_o be_v more_o apt_o express_v then_o by_o such_o proportion_n which_o cut_v that_o rational_a power_n of_o the_o diameter_n according_a to_o their_o side_n namely_o divide_v the_o power_n of_o the_o diameter_n by_o line_n which_o have_v that_o proportion_n that_o the_o great_a segment_n have_v to_o the_o less_o to_o put_v the_o less_o segment_n to_o be_v the_o side_n of_o the_o icosahedron_n &_o devide_v the_o say_a power_n of_o the_o diameter_n by_o line_n have_v the_o proportion_n of_o the_o whole_a to_o the_o less_o segment_n to_o express_v the_o side_n of_o the_o dodecahedron_n by_o the_o less_o segment_n which_o thing_n may_v well_o be_v do_v between_o magnitude_n incommensurable_a the_o end_n of_o the_o fourteen_o book_n of_o euclides_n element_n after_o flussas_n ¶_o the_o fifteen_o book_n of_o euclides_n element_n this_o finetenth_n and_o last_o book_n of_o euclid_n or_o rather_o the_o second_o book_n of_o appollonius_n or_o hypsicles_n book_n teach_v the_o inscription_n and_o circumscription_n of_o the_o five_o regular_a body_n one_o within_o and_o about_o a_o other_o a_o thing_n undouted_o pleasant_a and_o delectable_a in_o mind_n to_o contemplate_v and_o also_o profitable_a and_o necessary_a in_o act_n to_o practice_v for_o without_o practice_n in_o act_n it_o be_v very_o hard_a to_o see_v and_o conceive_v the_o construction_n and_o demonstration_n of_o the_o proposition_n of_o this_o book_n unless_o a_o man_n have_v a_o very_a deep_a sharp_a &_o fine_a imagination_n wherefore_o i_o will_v wish_v the_o diligent_a student_n in_o this_o book_n to_o make_v the_o study_n thereof_o more_o pleasant_a unto_o he_o to_o have_v present_o before_o his_o eye_n the_o body_n form_v &_o frame_v of_o past_v paper_n as_o i_o teach_v after_o the_o definition_n of_o the_o eleven_o book_n and_o then_o to_o draw_v and_o describe_v the_o line_n and_o division_n and_o superficiece_n according_a to_o the_o construction_n of_o the_o proposition_n in_o which_o description_n if_o he_o be_v wary_a and_o diligent_a he_o shall_v find_v all_o thing_n in_o these_o solid_a matter_n as_o clear_v and_o as_o manifest_v unto_o the_o eye_n as_o be_v thing_n before_o teach_v only_o in_o plain_a or_o superficial_a figure_n and_o although_o i_o have_v before_o in_o the_o twelve_o book_n admonish_v the_o reader_n hereof_o yet_o because_o in_o this_o book_n chief_o that_o thing_n be_v require_v i_o think_v it_o shall_v not_o be_v irksome_a unto_o he_o again_o to_o be_v put_v in_o mind_n thereof_o far_o this_o be_v to_o be_v note_v that_o in_o the_o greek_a exemplar_n be_v find_v in_o this_o 15._o book_n only_o 5._o proposition_n which_o 5._o be_v also_o only_o touch_v and_o set_v forth_o by_o hypsicy_n unto_o which_o campane_n add_v 8._o and_o so_o make_v up_o the_o number_n of_o 13._o campane_n undoubted_o although_o he_o be_v very_o well_o learn_v and_o that_o general_o in_o all_o kind_n of_o learning_n yet_o assure_o be_v bring_v up_o in_o a_o time_n of_o rudeness_n when_o all_o good_a letter_n be_v darken_v &_o barberousnes_n have_v
right_a linâ—s_n now_o then_o multiply_v the_o 20._o triangle_n into_o the_o side_n of_o one_o of_o the_o triangle_n and_o so_o shall_v there_o be_v produce_v 6â—_o â—he_z half_a of_o which_o be_v 30._o and_o so_o many_o side_n have_v a_o icosahedron_n and_o in_o like_a sort_n in_o a_o dodecahedron_n forasmuch_o as_o 12._o pentagon_n make_v a_o dodecahedron_n and_o every_o pentagon_n contain_v â—_o right_a linesâ—_n multiply_v â—â—_o into_o 12._o and_o there_o shall_v be_v produce_v 60._o the_o half_a of_o which_o be_v 30._o and_o so_o many_o be_v the_o side_n of_o a_o dodecahedron_n and_o the_o reason_n why_o we_o take_v the_o half_a iâ—_z for_o that_o every_o side_n whether_o it_o be_v of_o a_o triangle_n or_o of_o a_o pentagon_n or_o of_o a_o square_a as_o in_o a_o cube_fw-la â—s_v take_v twice_o and_o by_o the_o same_o reason_n may_v you_o find_v out_o how_o many_o side_n be_v in_o a_o cube_fw-la and_o in_o a_o pyramid_n and_o in_o a_o octohedron_n but_o now_o again_o if_o you_o will_v find_v out_o the_o number_n of_o the_o angle_n of_o every_o one_o of_o the_o solid_a figure_n when_o you_o have_v do_v the_o same_o multiplication_n that_o you_o do_v before_o divide_v the_o same_o side_n by_o the_o number_n of_o the_o plain_a superficiece_n which_o comprehend_v one_o of_o the_o angle_n of_o the_o solid_n as_o for_o example_n forasmuch_o as_o 5._o triangle_n contain_v the_o solid_a angle_n of_o a_o icosahedron_n divide_v 60._o by_o 5._o and_o there_o will_v come_v forth_o 12._o and_o so_o many_o solid_a angle_n have_v a_o icosahedron_n in_o a_o dodecahedron_n forasmuch_o as_o three_o pentagon_n comprehend_v a_o angle_n divide_v 60._o by_o 3._o and_o there_o will_v come_v forth_o 20_o and_o so_o many_o be_v the_o angle_n of_o a_o dodecahedron_n and_o by_o the_o same_o reason_n may_v you_o find_v out_o how_o many_o angle_n be_v in_o each_o of_o the_o rest_n of_o the_o solid_a figure_n book_n if_o it_o be_v require_v to_o be_v know_v how_o one_o of_o the_o plain_n of_o any_o of_o the_o five_o solid_n be_v give_v there_o may_v be_v find_v out_o the_o inclination_n of_o the_o say_a plain_n the_o one_o to_o the_o other_o which_o contain_v each_o of_o the_o solid_n this_o as_o say_v isidorus_n our_o great_a master_n be_v foâ—â—d_v out_o after_o this_o manner_n it_o be_v manifest_a that_o in_o a_o cube_fw-la the_o plain_n which_o contain_v iâ—_z doâ—_n 〈◊〉_d the_o one_o the_o other_o by_o a_o right_a angle_n but_o in_o a_o tetrahedron_n one_o of_o the_o triangle_n be_v give_v let_v the_o end_n of_o one_o of_o the_o side_n of_o the_o say_a triangle_n be_v the_o centre_n and_o let_v the_o space_n be_v the_o perpendicular_a line_n draw_v from_o the_o top_n of_o the_o triangle_n to_o the_o base_a and_o describe_v circumferâ—nces_n of_o a_o circle_n which_o shall_v cut_v the_o one_o the_o other_o and_o from_o the_o intersection_n to_o the_o centre_n draw_v right_a line_n which_o shall_v contain_v the_o inclination_n of_o the_o plain_n contain_v the_o tetrahedron_n in_o a_o octoâ—edron_n take_v one_o of_o the_o side_n of_o the_o triangle_n thereof_o and_o upon_o it_o describe_v a_o square_a and_o draw_v the_o diagonal_a line_n and_o make_v the_o centre_n the_o end_n of_o the_o diagonal_a line_n and_o the_o space_n likewise_o the_o perpendicular_a line_n draw_v from_o the_o top_n of_o the_o triangle_n to_o the_o base_a describe_v circumference_n and_o again_o from_o the_o common_a section_n to_o the_o centre_n draw_v right_a line_n and_o they_o shall_v contain_v the_o inclination_n seek_v for_o in_o a_o icosahedron_n upon_o the_o side_n of_o one_o of_o the_o triangle_n thereof_o describe_v a_o pentagon_n and_o draw_v the_o line_n which_o subtend_v one_o of_o the_o angle_n of_o the_o say_a pentagon_n and_o make_v the_o centre_n the_o end_n of_o that_o line_n and_o the_o space_n the_o perpendicular_a line_n of_o the_o triangle_n describe_v circumference_n and_o draw_v from_o the_o common_a intersection_n of_o the_o circumference_n unto_o the_o centre_n right_a line_n and_o they_o shall_v contain_v likewise_o the_o inclination_n of_o the_o plain_n of_o the_o icosahedron_n in_o a_o dodecahedron_n take_v one_o of_o the_o pentagon_n and_o draw_v likewise_o the_o line_n which_o subtend_v one_o of_o the_o angle_n of_o the_o pentagon_n and_o make_v the_o centre_n the_o end_n of_o that_o line_n and_o the_o space_n the_o perpendicular_a line_n draw_v from_o the_o section_n into_o two_o equal_a part_n of_o that_o line_n to_o the_o side_n of_o the_o pentagon_n which_o be_v parallel_n unto_o it_o describe_v circumference_n and_o from_o the_o point_n of_o the_o intersection_n of_o the_o circumference_n draw_v unto_o the_o centre_n right_a line_n and_o they_o shall_v also_o contain_v the_o inclination_n of_o the_o plain_n of_o the_o dodecahedron_n thus_o do_v this_o most_o singular_a learned_a man_n reason_n think_v the_o demonstration_n in_o every_o one_o of_o they_o to_o be_v plain_a and_o clear_a but_o to_o make_v the_o demonstration_n of_o they_o manifest_a i_o think_v it_o good_a to_o declare_v and_o make_v open_a his_o wordesâ—_n and_o first_o in_o a_o tâ—trahedronâ—_n the_o end_n of_o the_o fivetenth_fw-mi book_n of_o euclides_n element_n after_o 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d ¶_o the_o 6._o proposition_n the_o 6._o problem_n in_o a_o octohedron_n give_v to_o inscribe_v a_o trilater_n equilater_n pyramid_n svppose_v thaâ—_z the_o octohedron_n whereâ—â—_n the_o tetrahedron_n be_v require_v to_o be_v insâ—riâ—â—â—_n be_v abgdei_n take_v 〈…〉_o base_n of_o the_o octoâ—â—dron_n that_o be_v construction_n 〈…〉_o close_o in_o the_o loweâ—â—_n triangle_n bgd_v namely_o aeâ—_n head_n igd_v and_o let_v the_o four_o be_v aib_n which_o be_v opposite_a to_o the_o low_a triangle_n before_o put_v namely_o to_o egg_v and_o take_v the_o centre_n of_o those_o four_o base_n which_o let_v be_v the_o point_n h_o c_o n_o â—_o and_o upon_o the_o triangle_n hcn_n erect_v a_o pyramid_n hcnl_n now_o forasmuch_o as_o these_o two_o base_n of_o the_o octohedron_n namely_o age_n and_o abi_n be_v set_v upon_o the_o right_a line_n eglantine_n and_o by_o which_o be_v opposite_a the_o one_o to_o the_o otherâ—_n in_o the_o square_a gebi_n of_o the_o octohedron_n from_o the_o poinâ—_n a_o draâ—e_v by_o the_o centre_n of_o the_o base_n namely_o by_o the_o centre_n h_o l_o perpendicular_a line_n ahf_n alk_v cut_v the_o line_n eglantine_n and_o by_o 〈◊〉_d two_o equal_a part_n in_o the_o point_n f_o edward_n by_o the_o corollary_n of_o the_o 1â—_o of_o the_o thirteen_o wherefore_o a_o right_a line_n draw_v from_o the_o point_n f_o to_o the_o point_n king_n demonstration_n shall_v be_v a_o parallel_n and_o equal_a to_o the_o side_n of_o the_o octohedron_n namely_o to_o â—â—_o and_o give_v by_o the_o 33._o of_o the_o first_o and_o the_o right_a line_n hl_o which_o cut_v the_o 〈…〉_o of_o ak_o proportional_o for_o ah_o and_o all_o be_v draw_v from_o the_o centre_n of_o equal_a circle_n to_o the_o circumference_n be_v a_o parallel_n to_o the_o right_a line_n fk_o by_o the_o 2._o of_o the_o six_o and_o also_o to_o the_o side_n of_o the_o octohedron_n namely_o to_o eâ—_n and_o ig_n by_o the_o 9_o of_o the_o eleven_o wherefore_o as_o the_o line_n of_o be_v to_o the_o line_n ah_o so_o be_v the_o line_n fk_o to_o the_o line_n hl_o by_o the_o 4._o of_o the_o six_o for_o the_o triangle_n afk_v and_o ahl_n be_v like_o by_o thâ—_z corollary_n of_o the_o 2._o of_o the_o six_o but_o the_o line_n of_o be_v in_o sesquialter_fw-la proportion_n to_o the_o line_n ah_o for_o the_o side_n eglantine_n make_v hf_o the_o half_a of_o the_o right_a line_n ah_o by_o the_o corollary_n of_o the_o 12._o of_o the_o thirteen_o wherefore_o fk_n or_o give_v the_o side_n of_o the_o octohedron_n be_v sesquialter_fw-la to_o the_o righâ—line_n hl._n and_o by_o the_o same_o reason_n may_v we_o prove_v that_o the_o side_n of_o the_o octohedron_n be_v sesquialter_fw-la to_o the_o rest_n of_o the_o right_a line_n which_o make_v the_o pyramid_n hnci_n namely_o to_o the_o right_a lineâ—â—_n n_o nc_n ci_o ln_o and_o change_z wherefore_o those_o right_a line_n be_v equal_a and_o therefore_o the_o triangleâ—_n which_o be_v describe_v of_o they_o namely_o the_o triangle_n hcn_n hnl_n ncl_n and_o chl._n which_o make_v the_o pyramid_n hncl_n be_v equal_a and_o equilater_n and_o forasmuch_o as_o the_o angle_n of_o the_o same_o pyramid_n namely_o the_o angle_n h_o c_o n_o l_o do_v end_n in_o the_o centre_n of_o the_o base_n of_o the_o octohedron_n therefore_o it_o be_v inscribe_v â—o_o the_o same_o octohedron_n by_o the_o first_o definition_n of_o this_o book_n wherefore_o in_o a_o octohedron_n â—even_v be_v inscribe_v a_o trilate_a equilaâ—â—â—â—â—â—amisâ—_n which_o be_v require_v to_o â—e_a donâ—_n a_o corollary_n the_o base_n of_o a_o pyramid_n inscribe_v in_o a_o octohedron_n be_v parallel_n
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
octohedron_n forasmuch_o as_o in_o the_o 17._o of_o the_o fifteen_o it_o be_v prove_v that_o the_o side_n of_o a_o icosahedron_n inscribe_v in_o a_o pyramid_n couple_v together_o the_o two_o section_n which_o be_v produce_v by_o a_o extreme_a and_o mean_a proportion_n of_o the_o side_n of_o the_o octohedron_n which_o make_v a_o right_a angle_n and_o that_o right_a angle_n be_v contain_v under_o the_o less_o segment_n of_o the_o side_n of_o the_o octohedron_n and_o be_v subtend_v of_o the_o side_n of_o the_o icosahedron_n inscribe_v it_o follow_v therefore_o that_o the_o side_n of_o the_o icosahedron_n which_o subtend_v the_o right_a angle_n be_v in_o power_n equal_a to_o the_o two_o line_n which_o contain_v the_o say_a angle_n by_o the_o 47._o of_o the_o first_o be_v in_o power_n duple_n to_o every_o one_o of_o the_o less_o segmente_n of_o the_o side_n of_o the_o octohedron_n which_o contain_v a_o right_a angle_n wherefore_o the_o side_n of_o a_o icosahedron_n inscribe_v in_o a_o octohedron_n be_v in_o power_n duple_n to_o the_o less_o segment_n of_o the_o â—ide_v of_o the_o same_o octohedron_n ¶_o the_o 14._o proposition_n the_o side_n of_o the_o octohedron_n and_o of_o the_o cube_n inscribe_v in_o it_o be_v in_o power_n the_o one_o to_o the_o other_o 2._o in_o quadrupla_fw-la sesquialter_fw-la proportion_n svppose_v that_o abgde_v be_v a_o octohedron_n and_o let_v the_o cube_fw-la inscribe_v in_o it_o be_v fchi_n then_o i_o say_v that_o ab_fw-la the_o side_n of_o the_o octohedron_n be_v in_o power_n quadruple_a sesquialter_fw-la to_o fi_n the_o â—ide_v of_o the_o cube_fw-la let_v there_o be_v draw_v to_o be_v the_o base_a of_o the_o triangle_n abe_n a_o perpendicular_a a_fw-la and_o again_o let_v there_o be_v draw_v to_o the_o same_o base_a in_o the_o triangle_n gâ—e_v the_o perpendicular_a gn_n which_o a_o &_o gn_n shall_v pass_v by_o the_o centre_n fletcher_n and_o i_o and_o the_o line_n of_o be_v duple_n to_o the_o line_n fn_o by_o the_o corollary_n of_o the_o 12._o of_o the_o thirteen_o wherefore_o the_o line_n ao_o be_v duple_n to_o the_o line_n oe_z by_o the_o 2._o of_o the_o six_o for_o the_o line_n foe_n and_o ne_fw-fr be_v parallel_n and_o therefore_o the_o diameter_n agnostus_n be_v triple_a to_o the_o line_n fi._n wherefore_o the_o power_n of_o agnostus_n be_v 1._o noncuple_n to_o the_o power_n of_o fi._n but_o the_o line_n agnostus_n be_v in_o power_n duple_n to_o the_o side_n ab_fw-la by_o the_o 14._o of_o the_o thirteen_o wherefore_o the_o square_a of_o the_o line_n ab_fw-la be_v ing_o the_o half_a of_o the_o square_n of_o the_o line_n agnostus_n which_o be_v noncuple_n to_o the_o square_n of_o the_o line_n fi_n iâ—_z quadruple_a sesquialter_fw-la to_o the_o square_n of_o the_o line_n fi._n the_o side_n therefore_o of_o the_o octohedâ—â—â—â—nd_n of_o the_o cube_fw-la inscribe_v in_o itâ—_n be_v in_o power_n the_o one_o to_o the_o other_o in_o quadruple_a sesquialter_fw-la proportion_n ¶_o the_o 1â—_o proposition_n the_o side_n of_o the_o octohedron_n be_v in_o power_n quadruple_a sesquialter_fw-la to_o that_o right_a line_n who_o great_a segment_n be_v the_o side_n of_o the_o dodecahedron_n inscribe_v in_o the_o same_o octohedron_n forasmuch_o as_o in_o the_o 14._o of_o this_o book_n it_o be_v prove_v that_o the_o side_n of_o the_o octohedron_n be_v in_o power_n quadruple_a sesquialter_fw-la to_o the_o side_n of_o the_o cube_fw-la inscribe_v in_o it_o but_o the_o side_n of_o the_o cube_fw-la be_v cut_v by_o a_o extreme_a and_o mean_a proportion_n make_v the_o great_a segment_n the_o side_n of_o the_o dodecahedron_n circumscribe_v about_o it_o by_o the_o 3._o corollary_n of_o the_o 13._o of_o the_o fifteen_o therefore_o the_o side_n of_o the_o octohedron_n be_v in_o power_n quadruple_a sesquialter_fw-la to_o that_o right_a line_n namely_o to_o the_o side_n of_o the_o cube_fw-la who_o great_a segment_n be_v the_o side_n of_o the_o dodecahedron_n inscribe_v in_o the_o cube_fw-la but_o the_o dodecahedron_n and_o the_o cube_fw-la inscribe_v one_o within_o a_o other_o arâ—_n inscribe_v in_o one_o and_o the_o self_n same_o octohedron_n by_o the_o corollary_n of_o the_o first_o of_o this_o book_n the_o side_n therefore_o of_o the_o octohedron_n be_v in_o power_n quadruple_a sesquialter_fw-la to_o that_o right_a line_n who_o great_a segment_n be_v the_o side_n of_o the_o dodecahedron_n inscribe_v in_o the_o same_o octohedron_n ¶_o the_o 16._o proposition_n the_o side_n of_o a_o icosahedron_n be_v the_o great_a segment_n of_o that_o right_a line_n which_o be_v in_o power_n duple_n to_o the_o side_n of_o the_o octohedron_n inscribe_v in_o the_o same_o icosahedron_n svppose_v that_o there_o be_v a_o icosahedron_n abgdfhec_n who_o side_n let_v be_v bg_o or_o â—câ—_n and_o let_v the_o octohedron_n inscribe_v in_o it_o be_v akdâ—_n and_o let_v the_o side_n thereof_o be_v al._n then_o i_o say_v that_o the_o side_n â—c_z be_v the_o great_a segment_n of_o that_o right_a line_n which_o be_v in_o power_n duple_n to_o the_o side_n al._n for_o forasmuch_o as_o figure_n inscribe_v and_o circumscribe_v have_v oâ—e_a &_o the_o selfsame_o centre_n by_o the_o corollary_n of_o the_o â—1_n of_o the_o fifteen_o let_v the_o same_o be_v the_o point_n i_o now_o right_a lineâ—_z draw_v by_o thâ—â—_n 〈◊〉_d to_o the_o middle_a section_n of_o the_o opposite_a side_n namely_o the_o line_n aid_n and_o kil_n do_v in_o the_o point_n i_o â—ut_z 〈…〉_o the_o other_o inâ—â—_n two_o squall_n 〈◊〉_d and_o perpendicular_o by_o the_o corollary_n of_o the_o 14._o of_o the_o fifteen_o and_o forasmuch_o as_o they_o couple_v the_o middle_a section_n of_o the_o opposite_a line_n bg_o and_o hf_o therefore_o they_o cut_v they_o perpendiular_o h._n wherefore_o also_o the_o line_n bg_o 〈…〉_o be_v parallel_n by_o the_o 4._o corollary_n of_o the_o 14._o of_o the_o 〈…〉_o now_o then_o draw_v a_o line_n from_o b_o to_o h_o and_o the_o say_a â—â—ne_n bh_o shall_v be_v equal_a and_o parallel_v to_o the_o line_n kl_o by_o the_o 33._o of_o the_o first_o but_o the_o line_n bh_o subtend_v â—wâ—_n side_n of_o the_o pentagon_n which_o be_v compose_v of_o the_o side_n of_o the_o icosahedron_n namely_o the_o side_n basilius_n and_o ah_o wherefore_o the_o line_n bh_o be_v cut_v by_o a_o extreme_a and_o mean_a proportion_n make_v the_o great_a segment_n the_o side_n of_o the_o pentagon_n by_o the_o 8._o of_o the_o thirteen_o which_o side_n be_v also_o the_o side_n of_o the_o icosahedron_n namely_o ec_o and_o unto_o the_o line_n bh_o the_o line_n klâ—_n be_v equal_a and_o the_o line_n kl_o be_v in_o power_n duple_n to_o all_o the_o side_n of_o the_o octohedron_n by_o the_o 47._o of_o the_o first_o for_o in_o the_o square_a akdl_n the_o angle_n kal_n be_v a_o right_a angle_n wherefore_o ec_o the_o side_n of_o the_o icosahedron_n be_v the_o great_a segment_n of_o the_o line_n bh_o or_o kl_o which_o be_v in_o power_n duple_n to_o all_o â—he_z side_n of_o the_o octohedron_n inscribe_v in_o the_o icosahedron_n wherefore_o the_o side_n of_o a_o icosahedron_n be_v the_o great_a segment_n of_o that_o right_a line_n which_o be_v in_o power_n duple_n to_o the_o side_n of_o the_o octohedron_n inscribe_v in_o the_o same_o icosahedron_n ¶_o the_o 17._o proposition_n the_o side_n of_o a_o cube_n be_v to_o the_o side_n of_o a_o dodecahedron_n inscribe_v in_o it_o in_o duple_n proportion_n of_o a_o extreme_a and_o mean_a proportion_n for_o it_o be_v manife_a by_o the_o â—_o corollary_n of_o the_o 13._o of_o the_o fifteen_o that_o the_o side_n of_o a_o cube_fw-la divide_v by_o a_o extreme_a and_o mean_a proportion_n make_v the_o less_o segment_n the_o side_n of_o the_o dodecahedron_n inscribe_v in_o it_o but_o the_o whole_a be_v to_o the_o less_o segment_n in_o duple_n proportion_n of_o that_o in_o which_o it_o be_v to_o the_o great_a by_o the_o 10._o definition_n of_o the_o five_o for_o the_o whole_a the_o great_a segment_n and_o the_o less_o be_v line_n in_o continual_a proportion_n by_o the_o 3._o definition_n of_o the_o six_o wherefore_o the_o whole_a namely_o the_o side_n of_o the_o cube_fw-la be_v to_o the_o side_n of_o the_o dodecahedron_n inscribe_v in_o it_o namely_o to_o his_o less_o segment_n in_o duple_n proportion_n of_o a_o extreme_a and_o mean_a proportion_n '_o namely_o be_v of_o that_o which_o the_o whole_a have_v â—o_o the_o great_a segment_n by_o the_o 2._o of_o the_o fourteen_o ¶_o the_o 18._o proposition_n the_o side_n of_o a_o dodecahedron_n be_v to_o the_o side_n of_o a_o cube_n inscribe_v in_o it_o in_o converse_n proportion_n of_o a_o extreme_a and_o mean_a proportion_n it_o be_v prove_v in_o the_o 3._o corollary_n of_o the_o 13._o of_o the_o fifteen_o that_o the_o side_n of_o a_o dodecahedron_n circumscribe_v about_o a_o cube_n be_v the_o great_a segment_n of_o the_o side_n of_o the_o same_o cube_n wherefore_o the_o whole_a
a_o right_a line_n couple_v their_o centre_n be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n make_v the_o great_a segment_n the_o right_a line_n which_o couple_v the_o centre_n of_o the_o next_o base_n if_o by_o the_o centre_n of_o five_o base_n set_v upon_o one_o base_a be_v draw_v a_o plain_a superficies_n and_o by_o the_o centre_n of_o the_o base_n which_o be_v set_v upon_o the_o opposite_a base_a be_v draw_v also_o a_o plain_a superficies_n and_o then_o be_v draw_v a_o right_a line_n couple_v the_o centre_n of_o the_o opposite_a base_n that_o right_a line_n be_v so_o cut_v that_o each_o of_o his_o part_n set_v without_o the_o plain_a superficies_n be_v the_o great_a segment_n of_o that_o part_n which_o be_v contain_v between_o the_o plain_n the_o side_n of_o the_o dodecahedron_n be_v the_o great_a segment_n of_o the_o line_n which_o subtend_v the_o angle_n of_o the_o pentagon_n a_o perpendicular_a line_n draw_v from_o the_o centre_n of_o the_o dodecahedron_n to_o one_o of_o the_o base_n be_v in_o power_n quintuple_a to_o half_a the_o line_n which_o be_v between_o the_o plain_n and_o therefore_o the_o whole_a line_n which_o couple_v the_o centre_n of_o the_o opposite_a base_n be_v in_o power_n quintuple_a to_o the_o whole_a line_n which_o be_v between_o the_o say_a plain_n the_o line_n which_o subtâ—deth_v the_o angle_n of_o the_o base_a of_o the_o dodecahedron_n together_o with_o the_o side_n of_o the_o base_a be_v in_o power_n quintuple_a to_o the_o line_n which_o be_v draw_v from_o the_o centre_n of_o the_o circle_n which_o contain_v the_o base_a to_o the_o circumference_n a_o section_n of_o a_o sphere_n contain_v three_o base_n of_o the_o dodecahedron_n take_v a_o three_o part_n of_o the_o diameter_n of_o the_o say_a sphere_n the_o side_n of_o the_o dodecahedron_n and_o the_o line_n which_o subtend_v the_o angle_n of_o the_o pentagon_n be_v equal_a to_o the_o right_a line_n which_o couple_v the_o middle_a section_n of_o the_o opposite_a side_n of_o the_o dodecahedron_n ¶_o the_o end_n of_o the_o element_n of_o geometry_n of_o the_o most_o ancient_a philosopher_n 〈◊〉_d of_o megara_n the_o intent_n of_o this_o preface_n number_n note_v the_o word_n unit_fw-la to_o express_v the_o greek_a monaâ—_n &_o not_o unity_n as_o we_o hauâ—_n all_o common_o till_o now_o use_v magnitude_n a_o point_n a_o line_n magnitude_n ano._n 1488._o ☞_o arithmetic_n note_n note_n anno._n 1550._o r._n b._n note_n 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d ☞_o this_o noble_a earl_n die_v anno._n 1554._o scarce_o of_o 24._o year_n of_o ageâ—_n have_v no_o issue_n by_o his_o wife_n daughter_n to_o the_o duke_n of_o somerset_n justice._n ☞_o ☞_o plato_n 7._o de_fw-fr rep._n ☞_o ☞_o note_n the_o difference_n between_o strataruhmetrie_n and_o tacticie_n i.d._n fâ—endâ—â—_n you_o will_v find_v it_o hard_o to_o perform_v my_o description_n of_o â—his_n fâ—ate_n but_o by_o châ—râ—graphieâ—_n you_o may_v help_v yourself_o some_o â—hat_n wherâ—_n thâ—_z figure_v know_v in_o sidâ—â—â—nd_a angle_n be_v not_o regular_a and_o â—_z resolution_n into_o triangle_n can_v sâ—â—uâ—_n etc_n etc_n and_o yet_o you_o will_v find_v it_o strange_a to_o deal_v thus_o general_o with_o arithmetical_a figure_n and_o that_o for_o battle_n â—ay_n their_o coâ—tentâ—â—â—_n differ_v so_o much_o from_o like_o geometrâ—call_a â—_z a_o marvelous_a glass_n ☞_o s.w.p._n ☞_o note_n 1._o 2._o 3._o 5._o 6._o 7._o 8._o i.d._n read_v in_o aristotle_n his_o 8._o book_n of_o politic_n the_o 5_o 6_o and_o 7._o chapter_n where_o you_o shall_v have_v some_o occasion_n far_o to_o think_v of_o music_n than_o common_o be_v think_v ☜_o ☜_o anno._n 1548_o and_o 1549._o in_o lovayn_n note_n i.d._n the_o cut_v of_o a_o sphere_n according_a to_o any_o proportion_n assign_v may_v by_o this_o proposition_n be_v do_v mechanical_o by_o temper_v liquour_n to_o a_o certain_a weight_n in_o respect_n of_o the_o weight_n of_o the_o sphere_n 〈◊〉_d swyâ—â—â—ng_n a_o common_a error_n note_v a_o paradox_n n._n t._n 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de_fw-fr his_fw-la quae_fw-la mundo_fw-la mirabiliter_fw-la eveniunt_fw-la cap._n 8._o tusc._n â—_o â—_o ☞_o a_o digression_n apologetical_a apologetical_a a_o proverb_n fair_a fish_v and_o catch_v a_o frog_n ☞_o psal._n 140._o act._n 7._o c._n lib._n 30._o cap._n 1._o ☜_o r._n b._n ☞_o university_n ☜_o ☞_o the_o ground_n plat_z of_o this_o preface_n in_o a_o table_n the_o argumâ—â—â—_n of_o the_o first_o book_n definition_n of_o a_o point_n definition_n of_o a_o point_n after_o pythagoras_n definition_n of_o a_o liâ—â—_n an_o other_o definition_n of_o a_o line_n a_o other_o the_o end_n of_o a_o line_n difference_n of_o a_o point_n frâ—â—nity_n unity_n be_v a_o part_n of_o number_n a_o point_n be_v no_o part_n of_o quantity_n definition_n of_o a_o right_a line_n definition_n thereof_o after_o plato_n an_o other_o definition_n a_o other_o a_o other_o a_o other_o a_o other_o why_o euclid_n here_o define_v not_o a_o crooked_a line_n definition_n of_o a_o superficies_n a_o superficies_n 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the_o same_o affirmative_o an_o other_o demonstration_n of_o the_o same_o affirmative_o an_o other_o demonstration_n of_o the_o same_o demonstration_n lead_v to_o a_o impossibility_n an_o other_o demonstration_n of_o the_o same_o affirmative_o demonstration_n demonstrationâ—_n demonstration_n the_o argument_n of_o this_o six_o book_n this_o book_n necessary_a for_o the_o use_n of_o instrument_n of_o geometry_n the_o first_o definition_n the_o second_o definition_n reciprocal_a figure_n call_v mutual_a figure_n the_o three_o definition_n the_o four_o definition_n the_o five_o definition_n an_o other_o example_n of_o substraction_n of_o proportion_n the_o six_o definition_n demonstration_n of_o the_o first_o part_n demonstration_n of_o the_o second_o part_n a_o corollary_n add_v by_o flussates_n the_o first_o part_n of_o this_o theorem_a demonstration_n of_o the_o second_o part_n a_o corollary_n add_v by_o flussates_n construction_n demonstration_n of_o the_o first_o part_n 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demonstration_n demonstration_n the_o circle_n so_o make_v or_o so_o consider_v in_o the_o sphere_n be_v call_v the_o great_a circle_n all_o other_o not_o have_v the_o centre_n of_o the_o sphere_n to_o be_v their_o centre_n alsoâ—_n be_v call_v less_o circle_n note_v these_o description_n description_n an_o other_o corollary_n corollary_n an_o other_o corollary_n construction_n construction_n this_o be_v also_o prove_v in_o the_o assumpt_n before_o add_v out_o oâ—_n flussas_n note_v what_o a_o great_a or_o great_a circle_n in_o a_o spear_n be_v first_o part_n of_o the_o construction_n noteâ—_n noteâ—_n you_o know_v full_a well_o that_o in_o the_o superficies_n of_o the_o sphere_n â—â—â—ly_o the_o circumference_n of_o the_o circle_n be_v but_o by_o thâ—se_a circumference_n the_o limitation_n and_o assign_v of_o circle_n be_v use_v and_o so_o the_o circumference_n of_o a_o circle_n usual_o call_v a_o circle_n which_o in_o this_o place_n can_v not_o offend_v this_o figure_n be_v restore_v by_o m._n dee_n his_o diligence_n for_o in_o the_o greek_a and_o latin_a euclides_n the_o line_n gl_n the_o line_n agnostus_n and_o the_o line_n kz_o in_o which_o three_o line_n the_o chief_a pinch_n of_o both_o the_o demonstration_n do_v stand_v be_v untrue_o draw_v as_o by_o compare_v the_o studious_a may_v perceive_v note_n you_o must_v imagine_v 〈◊〉_d right_a line_n axe_n to_o be_v perpendicular_a upon_o the_o diameter_n bd_o and_o ce_fw-fr though_o here_o ac_fw-la the_o semidiater_n seem_v to_o be_v part_n of_o ax._n and_o so_o in_o other_o point_n in_o this_o figure_n and_o many_o other_o strengthen_v your_o imagination_n according_a to_o the_o tenor_n of_o construction_n though_o in_o the_o delineation_n in_o plain_a sense_n be_v not_o satisfy_v note_n boy_n equal_a to_o bk_o in_o respect_n of_o m._n dee_n his_o demonstration_n follow_v follow_v note_v â—his_fw-la point_n z_o that_o you_o may_v the_o better_a understand_v m._n dee_n his_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 which_o of_o necessity_n shall_v fall_v upon_o z_o as_o m._n dee_n prove_v it_o and_o his_o proof_n be_v set_v after_o at_o this_o mark_n ✚_o follow_v i_o dee_n dee_n but_o az_o be_v great_a them_z agnostus_n as_o in_o the_o former_a proposition_n km_o be_v evident_a to_o be_v great_a than_o kg_v so_o may_v it_o also_o be_v make_v manifest_a that_o kz_o do_v neither_o touch_n nor_o cut_v the_o circle_n fgâ—h_n a_o other_o prove_v that_o the_o line_n ay_o be_v great_a they_o the_o line_n ag._n ag._n this_o as_o a_o assumpt_n be_v present_o prove_v two_o case_n in_o this_o proposition_n the_o first_o case_n demonstration_n lead_v to_o a_o impossibility_n second_o case_n case_n as_o it_o be_v â—asiâ—_n to_o gather_v by_o the_o assumpt_n put_v after_o the_o secoâ—â—_n of_o this_o booâ—â—_n note_v a_o general_a rule_n the_o second_o part_n of_o the_o problem_n two_o way_n execute_v a_o upright_a cone_n the_o second_o part_n of_o the_o problem_n the_o second_o â—aâ—â—_n oâ—_n the_o problem_n ☜_o ☜_o this_o may_v easy_o be_v demonstrate_v as_o in_o thâ—_z 17._o proposition_n the_o section_n of_o a_o sphere_n be_v prove_v to_o be_v a_o circle_n circle_n for_o take_v away_o all_o doubt_n this_o aâ—_z a_o lemma_n afterward_o be_v demonstrate_v a_o lemma_n as_o it_o be_v present_o demonstrate_v construction_n demonstration_n the_o second_o part_n of_o the_o problem_n *_o construction_n demonstration_n an_o other_o way_n of_o execute_v this_o problem_n the_o converse_n of_o the_o assumpt_n a_o great_a error_n common_o maintain_v between_o straight_o and_o crooked_a all_o manner_n of_o proportion_n may_v be_v give_v construction_n demonstration_n the_o definition_n of_o a_o circled_a â—â—apâ—â—d_n in_o a_o spâ—erâ—_n construction_n demonstration_n this_o be_v manifest_a if_o you_o consider_v the_o two_o triangle_n rectangle_v hom_o and_o hon_n and_o then_o with_o all_o use_v the_o 47._o of_o the_o first_o of_o euclid_n construction_n demonstration_n construction_n demonstration_n this_o in_o manner_n of_o a_o lemmâ—_n be_v present_o prove_v note_v here_o of_o axe_n base_a &_o solidity_n more_o than_o i_o need_n to_o bring_v any_o far_a proof_n for_o note_n note_n i_o say_v half_a a_o circular_a revolution_n for_o that_o suffice_v in_o the_o whole_a diameter_n n-ab_n to_o describe_v a_o circle_n by_o iâ—_z it_o be_v move_v â—â—out_v his_o centre_n q_o etc_n etc_n lib_fw-la 2_o prop_n 2._o de_fw-fr spheâ—a_n &_o cylindrâ—_n note_n note_n a_o rectangle_n parallelipipedon_n give_v equal_a to_o a_o sphere_n give_v to_o a_o sphere_n or_o to_o any_o part_n of_o a_o sphere_n assign_v as_o a_o three_o four_o five_o etc_o etc_o to_o geve_v a_o parallelipipedon_n equal_a side_v colume_n pyramid_n and_o prism_n to_o be_v give_v equal_a to_o a_o sphere_n or_o to_o any_o certain_a part_n thereof_o to_o a_o sphere_n or_o any_o segment_n or_o sector_n of_o the_o same_o to_o geve_v a_o cone_n or_o cylinder_n equal_a or_o in_o any_o proportion_n assign_v far_o use_v of_o spherical_a geometry_n the_o argument_n of_o the_o thirteen_o book_n construction_n demonstration_n demonstration_n the_o assumpt_v prove_v prove_v because_o ac_fw-la be_v suppose_v great_a than_o ad_fw-la therefore_o his_o residue_n be_v less_o than_o the_o residue_n of_o ad_fw-la by_o the_o common_a sentence_n wherefore_o by_o the_o supposition_n db_o be_v great_a than_o â—c_z the_o chieâ—e_a line_n in_o all_o euclides_n geometry_n what_o be_v mean_v here_o by_o a_o section_n in_o one_o only_a poiâ—t_n construction_n demonstration_n demonstration_n note_v how_o ce_fw-fr and_o the_o gnonom_n xop_n be_v prove_v equal_a for_o it_o serve_v in_o the_o converse_n demonstrate_v by_o m._n dee_n here_o next_o after_o this_o proposition_n â—the_v converse_v of_o the_o former_a former_a as_o we_o haâ—e_z note_v the_o place_n of_o the_o peculiar_a proâ—e_n there_o â—in_a the_o demonstration_n of_o the_o 3._o 3._o therefore_o by_o my_o second_o theorem_a add_v upon_o the_o second_o proposition_n dc_o be_v divide_v by_o extreme_a and_o mean_a proportion_n in_o the_o point_n a._n and_o because_o ac_fw-la be_v big_a than_o cb_o therefore_o dam_n be_v great_a than_o ac_fw-la wherefore_o if_o a_o right_a line_n etc_n etc_n as_o in_o the_o proposition_n which_o be_v to_o be_v demonstrate_v demonstrate_v therefore_o by_o my_o second_o theorem_a add_v upon_o the_o second_o proposition_n dc_o be_v divide_v by_o extreme_a and_o mean_a proportion_n in_o the_o point_n a._n and_o because_o ac_fw-la be_v big_a than_o cb_o therefore_o dam_n be_v great_a than_o ac_fw-la wherefore_o if_o a_o right_a line_n etc_n etc_n as_o in_o the_o proposition_n which_o be_v to_o be_v demonstrate_v construction_n construction_n though_o i_o say_v perpendicular_a yes_o you_o may_v perceive_v how_o infinite_a other_o position_n will_v serve_v so_o that_o diego_n and_o ad_fw-la make_v a_o angle_n for_o a_o triangle_n to_o have_v his_o side_n proportional_o cut_v etc_n etc_n demonstration_n demonstration_n i_o dee_n this_o be_v most_o evident_a of_o my_o second_o theorem_a add_v to_o the_o three_o proposition_n for_o to_o add_v to_o a_o whole_a line_n a_o line_n equal_a to_o the_o great_a segment_n &_o to_o add_v to_o the_o great_a segment_n a_o line_n equal_a to_o the_o whole_a line_n be_v all_o one_o thing_n in_o the_o line_n produce_v by_o the_o whole_a line_n i_o mean_v the_o line_n divide_v by_o extreme_a and_o mean_a proportion_n this_o be_v before_o demonstrate_v most_o evident_o and_o brief_o by_o m._n dee_n after_o the_o 3._o proposition_n note_n note_v 4._o proportional_a line_n note_v two_o middle_a proportional_n note_v 4._o way_n of_o progression_n in_o the_o proportion_n of_o a_o line_n divide_v by_o extreme_a and_o middle_a proportion_n what_o resolution_n and_o composition_n be_v have_v before_o be_v teach_v in_o the_o begin_n of_o the_o first_o book_n book_n proclus_n in_o the_o greek_a in_o the_o 58._o page_n construction_n demonstration_n two_o case_n in_o this_o proposition_n construction_n thâ—_z first_o case_n demonstration_n the_o second_o case_n construction_n demonstration_n construction_n demonstration_n this_o corollary_n be_v the_o 3._o proposition_n of_o theâ—4_n â—4_o book_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 construction_n demonstration_n construstion_n demonstration_n constrâ—yction_n demonstration_n this_o corollary_n be_v the_o 11._o proposition_n of_o the_o 14._o book_n after_o campane_n this_o corollary_n be_v the_o 3._o corollary_n after_o the_o 17._o proposition_n of_o the_o 14_o book_n after_o campane_n campane_n by_o the_o name_n oâ—_n a_o pyramid_n both_o here_o &_o iâ—_z this_o book_n follow_v understand_v a_o tetrahedron_n an_o other_o construction_n and_o demonstration_n of_o the_o second_o part_n after_o fâ—ussas_n three_o part_n of_o the_o demonstration_n this_o corollary_n be_v the_o 15._o proposition_n of_o the_o 14._o book_n after_o campane_n this_o corollary_n campane_n put_v as_o a_o corollary_n after_o
the_o circle_n a_o construction_n have_v to_o the_o circle_n b._n as_o the_o diameter_n of_o the_o circle_n a_o be_v to_o the_o diameter_n of_o the_o circle_n b_o so_fw-mi let_v the_o line_n c_o be_v to_o a_o four_o line_n by_o the_o 12._o of_o the_o 〈…〉_o line_n be_v d._n and_o by_o the_o 11._o of_o the_o six_o let_v a_o third_o line_n proportional_a be_v find_v to_o the_o line_n c_o &_o d_o which_o let_v be_v eâ—_n i_o say_v now_o that_o the_o line_n c_o have_v too_o the_o line_n e_o that_o proportion_n which_o the_o circle_n a_o have_v to_o the_o circle_n b._n for_o by_o construction_n the_o lineâ—_z c_o d_o and_o e_o demonstration_n be_v proportional_a therefore_o the_o square_n of_o câ—_n be_v to_o the_o square_n of_o d_o as_z c_z be_v to_o e_o by_o the_o corollary_n of_o the_o 10._o of_o the_o six_o but_o by_o construction_n as_o the_o diameter_n of_o the_o circle_n a_o be_v to_o the_o diameter_n of_o the_o circle_n b_o so_o be_v c_o to_o d_z wherefore_o as_o the_o square_n of_o the_o diameter_n of_o the_o circle_n a_o be_v to_o the_o square_n of_o the_o diameter_n of_o the_o circle_n b_o so_o be_v the_o square_a of_o the_o line_n c_o to_o the_o square_n of_o the_o line_n d_o by_o the_o 22._o of_o the_o six_o but_o as_o the_o square_n of_o the_o diameter_n of_o a_o the_o circle_n be_v to_o the_o square_n of_o the_o diameter_n of_o the_o circle_n d_o so_o be_v the_o circle_n a_o to_o the_o circle_n b_o by_o the_o second_o of_o the_o twelve_o wherefore_o by_o 11._o of_o the_o five_o as_o the_o circle_n a_o be_v to_o the_o circle_n b_o so_o be_v the_o square_a of_o the_o line_n c_o to_o the_o square_n of_o the_o liâ—e_z d._n but_o it_o be_v prove_v that_o â—s_v the_o square_a of_o the_o line_n c_o be_v to_o the_o square_n of_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 wherefore_o by_o the_o 11._o of_o the_o five_o as_o the_o circle_n a_o be_v to_o the_o circle_n b_o so_o be_v the_o line_n c_o to_o the_o line_n e._n two_o circle_n be_v give_v therefore_o and_o a_o right_a line_n we_o have_v find_v a_o right_a line_n to_o which_o the_o right_a line_n give_v have_v that_o proportion_n which_o the_o one_o circle_n have_v to_o the_o other_o which_o ought_v to_o be_v do_v note_n the_o difference_n between_o this_o problem_n and_o that_o next_o before_o be_v this_o there_o although_o we_o have_v two_o circle_n give_v second_o and_o two_o line_n be_v find_v in_o that_o proportion_n the_o one_o to_o the_o other_o in_o which_o the_o give_v circle_n be_v and_o here_o likewise_o be_v two_o circle_n give_v and_o two_o lineâ—_z also_o be_v have_v in_o the_o same_o proportion_n that_o the_o give_v circle_n be_v yet_o there_o we_o take_v at_o pleasure_n the_o first_o of_o the_o two_o line_n whereunto_o we_o frame_v the_o second_o proportional_o to_o the_o circle_n give_v but_o here_o the_o first_o of_o the_o two_o line_n be_v assign_v point_a aâ—d_z determine_v to_o we_o and_o not_o our_o choice_n to_o be_v have_v therein_o as_o be_v in_o the_o former_a problem_n a_o problem_n 3._o a_o circle_n be_v give_v to_o find_v a_o other_o circle_n to_o which_o the_o give_v câ—rcle_n be_v in_o any_o proportion_n give_v in_fw-ge two_o right_a line_n suppose_v the_o circle_n aâ—c_n give_v and_o therefore_o his_o semidiameter_n be_v give_v whereby_o his_o diameter_n also_o be_v give_v which_o diameter_n let_v be_v ac_fw-la let_v the_o proportion_n give_v be_v that_o which_o be_v between_o â—_o f_o two_o right_a line_n i_o say_v a_o circle_n be_v to_o be_v find_v unto_o which_o aâ—c_n have_v that_o proportion_n that_o â—_o have_v to_o â—_o as_o â—_o be_v to_o â—_o construction_n so_o let_v ac_fw-la the_o diameter_n be_v to_o a_o other_o right_a line_n by_o the_o 12._o of_o the_o six_o which_o line_n suppose_v to_o be_v 11._o between_o ac_fw-la and_o â—â—_o find_v a_o middle_n proportional_a line_n by_o the_o 13._o of_o the_o six_o which_o let_v be_v ln_o by_o the_o 10._o of_o the_o first_o divide_v â—n_n into_o two_o equal_a payte_n and_o let_v that_o be_v do_v in_o the_o point_n â—_o now_o upon_o â—â—_o o_o being_n make_v the_o centre_n describe_v a_o circle_n which_o let_v be_v â—nâ—_n demonstration_n i_o say_v that_o abc_n be_v to_o lmn_o as_o â—_o be_v to_o f._n for_o see_v that_o ac_fw-la ln_o and_o h_o be_v three_o right_a line_n in_o continual_a proportion_n by_o construction_n therefore_o by_o the_o corollary_n of_o the_o 20._o of_o the_o six_o as_o ac_fw-la be_v to_o h_o so_o be_v the_o square_a of_o aâ—_z to_o the_o square_n of_o ln_o but_o ac_fw-la be_v to_o h_o as_z â—_o be_v to_o â—_o by_o construction_n wherefore_o the_o square_a of_o ac_fw-la be_v to_o the_o square_n of_o ln_o as_o e_o be_v fletcher_n but_o as_o the_o square_n of_o the_o diameter_n ac_fw-la be_v to_o the_o square_n of_o the_o diameter_n ln_o so_o be_v the_o circle_n abc_n to_o the_o circle_n â—mn_v by_o this_o 2._o of_o the_o twelve_o wherefore_o by_o the_o 11._o of_o the_o five_o the_o circle_n abc_n be_v to_o the_o circle_n lmn_o as_o â—_o be_v to_o f._n a_o circle_n be_v give_v therefore_o a_o other_o circle_n be_v find_v to_o which_o the_o give_v circle_n be_v in_o any_o proportion_n give_v between_o two_o right_a line_n which_o ought_v to_o be_v do_v a_o problem_n 4._o two_o circle_n be_v give_v to_o find_v one_o circle_n equal_a to_o they_o both_o suppose_v the_o two_o circle_n give_v have_v their_o diameter_n aâ—_z &_o cd_o i_o say_v that_o a_o circle_n must_v be_v â—ound_v equal_a to_o the_o two_o circle_n who_o diameter_n be_v aâ—_z and_o cd_o unto_o the_o line_n aâ—_z construction_n at_o the_o point_n a_o erect_v a_o perpendicular_a line_n aâ—_z from_o which_o sufficient_o produce_v cut_v a_o line_n equal_a to_o cd_o which_o let_v be_v af._n by_o the_o first_o petition_n draw_v from_o fletcher_n to_o â—_o a_o right_n line_n so_o be_v faâ—_n make_v a_o triangle_n rectangle_n i_o say_v now_o that_o a_o circle_n who_o diameter_n be_v fâ—_n be_v equal_a to_o the_o two_o circle_n who_o diameter_n be_v aâ—_z and_o â—d_a demonstration_n for_o by_o the_o 47._o of_o the_o first_o the_o square_a of_o fâ—_n be_v equal_a to_o the_o square_n of_o aâ—_z &_o af._n which_o of_o be_v by_o construction_n equal_a to_o cdâ—_n wherefore_o the_o square_a of_o fâ—_n be_v equal_a to_o the_o square_n of_o ab_fw-la and_o cd_o but_o circle_n be_v one_o to_o a_o other_o as_o the_o square_n of_o their_o diameter_n be_v one_o to_o the_o other_o by_o this_o second_o of_o the_o twelve_o therefore_o the_o circle_n who_o diameter_n be_v â—â—_o be_v equal_a to_o the_o circle_n who_o diameter_n be_v aâ—_z and_o cd_o therefore_o two_o circle_n be_v give_v we_o have_v find_v a_o circle_n equal_a to_o they_o both_o which_o be_v require_v to_o be_v do_v a_o corollary_n 1._o hereby_o it_o be_v make_v evident_a that_o in_o all_o triangle_n rectangle_n the_o circle_n semicircle_n quadrant_n oâ—_n any_o other_o portion_n of_o circle_n describe_v upon_o the_o subtendent_fw-la line_n be_v equal_a to_o the_o two_o circle_n semicircle_n quadrant_n or_o any_o two_o other_o like_a portion_n of_o circle_n describe_v on_o the_o two_o line_n comprehend_v the_o right_a angle_n like_v to_o like_v be_v compare_v for_o like_a part_n have_v that_o proportion_n between_o themselves_o that_o their_o whole_a magnitude_n have_v of_o which_o they_o be_v like_o part_n by_o the_o 15._o of_o the_o five_o but_o of_o the_o whole_a circle_n in_o the_o former_a problem_n it_o be_v evident_a and_o therefore_o in_o the_o forname_v like_o portion_n of_o circle_n it_o be_v a_o true_a consequent_a a_o corollary_n 2._o by_o the_o former_a problem_n it_o be_v also_o manifest_a unto_o circle_n three_o four_o five_o or_o to_o how_o many_o so_o ever_o one_o will_v geve_v one_o circle_n may_v be_v give_v equal_a for_o if_o first_o to_o any_o two_o by_o the_o former_a problem_n you_o find_v one_o equal_a and_o then_o unto_o your_o find_a circle_n and_o the_o three_o of_o the_o give_v circle_n as_o two_o give_v circle_n find_v one_o other_o circle_n equal_a and_o then_o to_o that_o second_o find_v circle_n and_o to_o the_o four_o of_o the_o first_o give_v circlesâ—_n as_o two_o circle_n one_o new_a circle_n be_v find_v equal_a and_o so_o proceed_v till_o you_o have_v once_o couple_v orderly_a every_o one_o of_o your_o propound_a circle_n except_o the_o first_o and_o second_o already_o do_v with_o the_o new_a circle_n thus_o find_v for_o so_o the_o last_o find_v circle_n be_v equal_a to_o all_o the_o first_o give_v circle_n if_o you_o doubt_v or_o sufficient_o understand_v i_o not_o help_v yourself_o by_o the_o discourse_n and_o demonstration_n of_o the_o