equal_a side_n be_v most_o simple_a and_o easy_a to_o be_v know_v and_o be_v here_o first_o define_v and_o be_v call_v a_o equilater_n triangle_n as_o the_o triangle_n a_o in_o the_o example_n all_o who_o side_n be_v of_o one_o length_n isosceles_a 25._o isosceles_a be_v a_o triangle_n which_o have_v only_o two_o side_n equal_a the_o second_o kind_n of_o triangle_n have_v two_o side_n of_o one_o length_n but_o the_o three_o side_n be_v either_o long_o or_o short_a they_o the_o other_o two_o as_o be_v the_o triangle_n here_o figure_v b_o c_o d_o in_o the_o triangle_n b_o the_o two_o side_n ae_n and_o of_o be_v equal_a the_o one_o to_o the_o other_o and_o the_o side_n ae_n be_v long_a than_o any_o of_o they_o both_o likewise_o in_o the_o triangle_n c_o the_o two_o side_n gh_o and_o hk_v be_v squall_n and_o the_o side_n gk_o be_v great_a also_o in_o the_o triangle_n d_o the_o side_n lm_o and_o mn_a be_v squall_n and_o the_o side_n ln_o be_v short_a scalenum_n 26._o scalenum_n be_v a_o triangle_n who_o three_o side_n be_v all_o unequal_a as_o be_v the_o triangle_n e_o f_o in_o which_o there_o be_v no_o one_o side_n equal_a to_o any_o of_o the_o other_o for_o in_o the_o triangle_n e_o the_o side_n ac_fw-la be_v great_a than_o the_o side_n bc_o and_o the_o side_n bc_o be_v great_a they_o the_o side_n ab_fw-la likewise_o in_o the_o triangle_n fletcher_n the_o side_n dh_o be_v great_a they_o the_o side_n dg_o and_o the_o side_n dg_o be_v great_a than_o the_o side_n gh_o triangle_n 27._o again_o of_o triangle_n a_o orthigonium_fw-la or_o a_o rightangled_a triangle_n be_v a_o triangle_n which_o have_v a_o right_a angle_n as_o there_o be_v three_o kind_n of_o triangle_n by_o reason_n of_o the_o diversity_n of_o the_o side_n so_o be_v there_o also_o three_o kind_n of_o triangle_n by_o reason_n of_o the_o variety_n of_o the_o angle_n for_o every_o triangle_n either_o contain_v one_o right_a angle_n &_o two_o acute_a angle_n or_o one_o obtuse_a angle_n &_o two_o acute_a or_o three_o acute_a angle_n for_o it_o be_v impossible_a that_o one_o triangle_n shall_v containâ—_n two_o obtuse_a angle_n or_o two_o right_a angle_n or_o one_o obtuse_a angle_n and_o the_o other_o a_o right_a angle_n all_o which_o kind_n be_v here_o define_v first_o a_o rightangled_a triangle_n which_o have_v in_o it_o a_o right_a angle_n as_o the_o triangle_n bcd_o of_o which_o the_o angle_n bcd_o be_v a_o right_a angle_n triangle_n 28._o an_fw-mi ambligonium_fw-la or_o a_o obtuse_a angle_a triangle_n be_v a_o triangle_n which_o have_v a_o obtuse_a angle_n as_o be_v the_o triangle_n b_o who_o angle_n ac_fw-la d_z be_v a_o obtuse_a angle_n and_o be_v also_o a_o scalenon_fw-mi have_v his_o three_o side_n unequal_a the_o triangle_n e_o be_v likewise_o a_o ambligonion_a who_o angle_n fgh_n be_v a_o obtuse_a angle_n &_o be_v a_o isosceles_a have_v two_o of_o his_o side_n equal_a namely_o fg_o and_o gh_o triangle_n 29._o a_o oxigonium_n or_o a_o acute_a angle_a triangle_n be_v a_o triangle_n which_o have_v all_o his_o three_o angle_n acute_a as_o the_o triangle_n a_o b_o c_o in_o the_o example_n all_o who_o angle_n be_v acute_a of_o which_o a_o be_v a_o equilater_n triangle_n b_o a_o isosceles_a and_o c_o a_fw-fr scalenon_fw-mi a_o equilater_n triangle_n be_v most_o simple_a and_o have_v one_o uniform_a construction_n and_o therefore_o all_o the_o angle_n of_o it_o be_v equal_a and_o never_o have_v in_o it_o either_o a_o right_a angle_n or_o a_o obtuse_a but_o the_o angle_n of_o a_o isosceles_a or_o a_o scalenon_fw-mi may_v diverse_o vary_v it_o be_v also_o to_o be_v note_v that_o in_o comparison_n of_o any_o two_o side_n of_o a_o triangle_n the_o three_o be_v call_v a_o base_a as_o of_o the_o triangle_n abc_n in_o respect_n of_o the_o two_o line_n ab_fw-la and_o ac_fw-la the_o line_n bc_o be_v the_o base_a and_o in_o respect_n of_o the_o two_o side_n ac_fw-la and_o cb_o the_o line_n ab_fw-la be_v the_o base_a and_o likewise_o in_o respect_n of_o the_o two_o side_n cb_o &_o basilius_n the_o line_n ac_fw-la be_v the_o base_a 30_o of_o four_o syded_a figure_n square_a a_o quadrate_n or_o square_n be_v that_o who_o side_n be_v equal_a and_o his_o angle_n right_a as_o triangle_n have_v their_o difference_n and_o variety_n by_o reason_n of_o their_o side_n and_o angle_n so_o likewise_o do_v figure_n of_o four_o side_n take_v their_o variety_n and_o difference_n partly_o by_o reason_n of_o their_o side_n &_o partly_o by_o reason_n of_o their_o angle_n as_o appear_v by_o their_o definition_n the_o four_o side_v figure_n abcd_o be_v a_o square_a or_o a_o quadrate_n because_o it_o be_v a_o right_a angle_a figure_n all_o his_o angle_n be_v right_a angle_n and_o also_o all_o his_o four_o side_n be_v equal_a 31_o a_o figure_n on_o the_o one_o side_n long_o square_n or_o squarelike_a or_o as_o some_o call_v it_o a_o long_a square_n be_v that_o which_o have_v right_a angle_n but_o have_v not_o equal_a side_n this_o figure_n agree_v with_o a_o square_n touch_v his_o angle_n in_o that_o either_o of_o they_o have_v right_a angle_n and_o differ_v from_o it_o only_o by_o reason_n of_o his_o side_n in_o that_o the_o side_n thereof_o be_v not_o equal_a as_o be_v the_o side_n of_o a_o square_a as_o in_o the_o example_n the_o angle_n of_o the_o figure_n abcd_o be_v right_a angle_n but_o the_o two_o side_n thereof_o ab_fw-la and_o cd_o be_v long_o than_o the_o other_o two_o side_n ac_fw-la &_o bd._n 32_o rhombus_fw-la or_o a_o diamond_n be_v a_o figure_n have_v four_o equal_a side_n figure_n but_o it_o be_v not_o rightangle_v this_o figure_n agree_v with_o a_o square_n as_o touch_v the_o equality_n of_o line_n but_o differ_v from_o it_o in_o that_o it_o have_v not_o right_a angle_n as_o have_v the_o square_a as_o of_o this_o figure_n the_o four_o line_n ab_fw-la bc_o cd_o dam_n be_v equal_a but_o the_o angle_n thereof_o be_v not_o right_a angle_n for_o the_o two_o angle_n abc_n and_o adc_o be_v obtuse_a angle_n great_a than_o right_a angle_n &_o the_o other_o two_o angle_n bad_a and_o bcd_o be_v two_o acute_a angle_n less_o than_o two_o right_a angle_n and_o these_o four_o angle_n be_v yet_o equal_a to_o four_o right_a angle_n for_o as_o much_o as_o the_o acute_a angle_n wait_v of_o a_o right_a angle_n so_o much_o the_o obtuse_a angle_n exceed_v a_o right_a angle_n figure_n 33_o rhomboide_n or_o a_o diamond_n like_o be_v a_o figure_n who_o opposite_a side_n be_v equal_a and_o who_o opposite_a angle_n be_v also_o equal_a but_o it_o have_v neither_o equal_a side_n nor_o right_a angle_n as_o in_o the_o figure_n abcd_o all_o the_o four_o side_n be_v not_o equal_a but_o the_o two_o side_n ab_fw-la and_o cd_o be_v opposite_a the_o one_o to_o the_o other_o also_o the_o other_o two_o side_n ac_fw-la and_o bd_o be_v also_o opposite_a be_v equal_a the_o one_o to_o the_o other_o likewise_o the_o angle_n be_v not_o right_a angle_n but_o the_o angle_n cab_n and_o cdb_n be_v obtuse_a angle_n and_o opposite_a and_o equal_a the_o one_o to_o the_o other_o likewise_o the_o angle_n abdella_n and_o acd_o be_v acute_a angle_n and_o opposite_a and_o also_o equal_a the_o one_o to_o the_o other_o table_n 34_o all_o other_o figure_n of_o four_o side_n beside_o these_o be_v call_v trapezia_fw-la or_o table_n such_o be_v all_o figure_n in_o which_o be_v observe_v no_o equality_n of_o side_n nor_o angle_n as_o the_o figure_n a_o and_o b_o in_o the_o margin_n which_o have_v neither_o equal_a side_n nor_o equal_a angle_n but_o be_v describe_v at_o all_o adventure_n without_o observation_n of_o order_n and_o therefore_o they_o be_v call_v irregular_a figure_n line_n 35_o parallel_n or_o equidistant_a right_a line_n be_v such_o which_o be_v in_o one_o and_o the_o self_n same_o superficies_n and_o produce_v infinite_o on_o both_o side_n do_v never_o in_o any_o part_n concur_v as_o be_v the_o line_n ab_fw-la and_o cd_o in_o the_o example_n petition_n or_o request_n 1_o from_o any_o point_n to_o any_o point_n to_o draw_v a_o right_a line_n after_o the_o definition_n which_o be_v the_o first_o kind_n of_o principle_n now_o follow_v petition_n which_o be_v the_o second_o kind_n of_o principle_n be_v which_o be_v certain_a general_a sentence_n so_o plain_a &_o so_o perspicuous_a that_o they_o be_v perceive_v to_o be_v true_a as_o soon_o as_o they_o be_v utter_v &_o no_o man_n that_o have_v but_o common_a sense_n can_v nor_o will_v deny_v they_o of_o which_o the_o first_o be_v that_o which_o be_v here_o set_v as_o from_o the_o point_n a_o to_o the_o point_n b_o who_o will_v deny_v but_o easy_o grant_v that_o a_o right_a line_n may_v be_v draw_v for_o two_o point_n howsoever_o they_o be_v set_v be_v imagine_v to_o
the_o line_n ab_fw-la be_v long_o than_o the_o line_n cd_o &_o if_o you_o take_v from_o they_o two_o equal_a line_n as_o ebb_v and_o fd_o the_o part_n remain_v which_o be_v the_o line_n ae_n and_o cf_o shall_v be_v unequal_a the_o one_o to_o the_o other_o namely_o the_o line_n ae_n shall_v be_v great_a than_o the_o line_n cf_o which_o be_v ever_o true_a in_o all_o quantity_n whatsoever_o 5_o and_o if_o to_o unequal_a thing_n you_o add_v equal_a thing_n the_o whole_a shall_v be_v unequal_a as_o if_o you_o have_v two_o unequal_a line_n namely_o ae_n the_o great_a and_o cf_o the_o less_o &_o if_o you_o add_v unto_o they_o two_o equal_a line_n ebb_n &_o fd_o then_o may_v you_o conclude_v that_o the_o whole_a line_n compose_v be_v unequal_a namely_o that_o the_o whole_a line_n ab_fw-la be_v great_a than_o the_o whole_a line_n cd_o and_o so_o of_o all_o other_o quantity_n 6_o thing_n which_o be_v double_a to_o one_o and_o the_o self_n same_o thing_n be_v equal_a the_o one_o to_o the_o other_o as_o if_o the_o line_n ab_fw-la be_v double_a to_o the_o line_n of_o and_o if_o also_o the_o line_n cd_o be_v double_a to_o the_o same_o line_n efâ—_n â—_z they_o may_v you_o by_o this_o common_a sentence_n conclude_v that_o the_o two_o line_n ab_fw-la &_o cd_o be_v equal_a the_o one_o to_o the_o other_o and_o this_o be_v true_a in_o all_o quantity_n and_o that_o not_o only_o when_o they_o be_v double_a but_o also_o if_o they_o be_v triple_a or_o quadruple_a or_o in_o what_o proportion_n soever_o it_o be_v of_o the_o great_a inequality_n iâ—_z which_o be_v when_o the_o great_a quantity_n be_v compare_v to_o the_o less_o 7_o thing_n which_o be_v the_o half_a of_o one_o and_o the_o self_n same_o thing_n be_v equal_a the_o one_o to_o the_o other_o as_o if_o the_o line_n ab_fw-la be_v the_o half_a of_o the_o line_n of_o and_o if_o the_o line_n cd_o be_v the_o half_a also_o of_o the_o same_o line_n of_o then_o may_v you_o conclude_v by_o this_o common_a sentence_n that_o the_o two_o line_n ab_fw-la and_o cd_o be_v equal_a the_o one_o to_o the_o other_o this_o be_v also_o true_a in_o all_o kind_n of_o quantity_n and_o that_o not_o only_a when_o it_o be_v a_o half_a but_o also_o if_o it_o be_v a_o three_o a_o quarter_n or_o in_o what_o proportion_n soever_o it_o be_v of_o the_o less_o in_o equality_n which_o be_v when_o the_o less_o quantity_n be_v compare_v to_o the_o great_a iâ—_z 8_o thing_n which_o agree_v together_o be_v equal_v the_o one_o to_o the_o other_o such_o thing_n be_v say_v to_o agree_v together_o which_o when_o they_o be_v apply_v the_o one_o to_o the_o other_o or_o set_v the_o one_o upon_o the_o other_o the_o one_o â—â—â—â—deth_v not_o the_o other_o in_o any_o thing_n as_o if_o the_o two_o triangle_n abc_n and_o def_n be_v apply_v the_o one_o to_o the_o other_o and_o the_o triangle_n abc_n be_v set_v upon_o the_o triangle_n def_n if_o then_o the_o angle_n a_o do_v just_o agree_v with_o the_o angle_n d_o and_o the_o angle_n b_o with_o the_o angle_n e_o and_o also_o the_o angle_n c_o with_o the_o angle_n f_o and_o moreover_o â—f_a the_o line_n ab_fw-la do_v just_o fall_v upon_o the_o line_n de_fw-fr and_o the_o line_n ac_fw-la upon_o the_o line_n df_o and_o also_o the_o line_n bc_o upon_o the_o line_n of_o so_o that_o on_o every_o part_n of_o these_o two_o triangle_n there_o be_v just_a agreement_n then_o may_v you_o conclude_v that_o the_o two_o triangle_n be_v equal_a 9_o every_o whole_a be_v great_a than_o his_o part_n be_v the_o principle_n thus_o place_v &_o end_v now_o follow_v the_o proposition_n which_o be_v sentence_n set_v forth_o to_o be_v prove_v by_o reason_v and_o demonstration_n and_o therefore_o they_o be_v again_o repeat_v in_o the_o end_n of_o the_o demonstrationâ—_n for_o the_o proposition_n be_v ever_o the_o conclusion_n and_o that_o which_o ought_v to_o be_v prove_v sort_n proposition_n be_v of_o two_o sort_n the_o one_o be_v call_v a_o problem_n the_o other_o a_o theorem_a be_v a_o problem_n be_v a_o proposition_n which_o require_v some_o action_n or_o do_v as_o the_o make_v of_o some_o â—igure_n or_o to_o divide_v a_o figure_n or_o line_n to_o apply_v figure_n to_o â—igure_n to_o add_v figure_n together_o or_o to_o subtrah_n one_o from_o a_o other_o to_o describe_v to_o inscribe_v to_o circumscribe_v one_o figure_n within_o or_o without_o another_o and_o such_o like_a as_o of_o the_o first_o proposition_n of_o the_o first_o book_n be_v a_o problem_n which_o be_v thuss_v upon_o a_o right_a line_n give_v not_o be_v infinite_a to_o describe_v a_o equilater_n triangle_n or_o a_o triangle_n of_o three_o equal_a side_n for_o in_o it_o beside_o the_o demonstration_n and_o contemplation_n of_o the_o mind_n be_v require_v somewhat_o to_o be_v do_v name_o to_o make_v a_o equilater_n triangle_n upon_o a_o line_n give_v and_o in_o the_o end_n of_o every_o problem_n after_o the_o demonstration_n be_v conclude_v aâ—ter_v this_o manâ—er_n which_o be_v the_o thing_n which_o be_v require_v to_o be_v do_v be_v a_o theorem_a be_v a_o proposition_n which_o require_v the_o search_a ouâ—_n and_o demonstration_n of_o some_o property_n or_o passion_n of_o some_o figure_n wherein_o be_v only_a speculation_n and_o contemplation_n of_o mind_n without_o do_v or_o work_v of_o any_o thing_n as_o the_o five_o proposition_n of_o the_o first_o book_n which_o be_v thus_o a_o isosceles_a or_o triangle_n of_o two_o equal_a side_n have_v his_o angle_n at_o thâ—_z base_a equal_a the_o one_o to_o the_o other_o etc_n etc_n be_v a_o theorem_a for_o in_o it_o be_v require_v only_o to_o be_v prove_v and_o make_v plain_a by_o reason_n and_o demonstration_n that_o these_o two_o angle_n be_v equal_a without_o further_a work_n or_o do_v and_o in_o the_o endâ—_n of_o â—uâ—ry_a theorem_a after_o the_o demonstration_n be_v conclude_v after_o this_o manner_n which_o thing_n be_v require_v to_o be_v demonstrate_v or_o prove_v the_o first_o problem_n the_o first_o proposition_n upon_o a_o right_a line_n give_v not_o be_v infinite_a to_o describe_v a_o equilater_n triangle_n or_o a_o triangle_n of_o three_o equal_a side_n svppose_v that_o the_o right_a line_n give_v be_v ab_fw-la it_o be_v require_v upon_o the_o line_n ab_fw-la to_o describe_v a_o equilater_n triangle_n namely_o a_o triangle_n of_o three_o equal_a side_n construction_n now_o therefore_o make_v the_o centre_n the_o point_n a_o and_o the_o space_n ab_fw-la describe_v by_o the_o three_o petition_n a_o circle_n bcd_o and_o again_o by_o the_o same_o make_v the_o centre_n the_o point_n b_o and_o the_o space_n bâ—_n describe_v a_o other_o circle_n ace_n and_o by_o the_o first_o petition_n from_o the_o point_n c_o wherein_o the_o circle_n cut_v the_o one_o the_o other_o draw_v one_o right_a line_n to_o the_o point_n a_o and_o a_o other_o right_a line_n to_o the_o point_n b._n and_o forasmuch_o as_o the_o point_n a_o be_v the_o centre_n of_o the_o circle_n cbd_v demonstration_n therefore_o by_o the_o 15._o definition_n the_o line_n ac_fw-la be_v equal_a to_o the_o line_n ab_fw-la again_o forasmuch_o as_o the_o point_n b_o be_v the_o centre_n of_o the_o circle_n caesar_n therefore_o by_o the_o same_o definition_n the_o line_n bc_o be_v equal_a to_o the_o line_n basilius_n and_o it_o be_v prove_v that_o the_o line_n ac_fw-la be_v equal_a to_o the_o line_n ab_fw-la wherefore_o either_o of_o these_o line_n ca_n and_o cb_o be_v equal_a to_o the_o line_n ab_fw-la but_o thing_n which_o be_v equal_a to_o one_o and_o the_o same_o thing_n be_v also_o equal_a the_o one_o to_o the_o other_o by_o the_o first_o common_a sentence_n wherefore_o the_o line_n ca_n also_o be_v equal_a to_o the_o line_n cb._n wherefore_o these_o three_o right_a line_n ca_n ab_fw-la and_o bc_o be_v equal_a the_o one_o to_o the_o other_o wherefore_o the_o triangle_n abc_n be_v equilater_n wherefore_o upon_o the_o line_n ab_fw-la be_v describe_v a_o equilater_n triangle_n abc_n wherefore_o upon_o a_o line_n give_v not_o be_v infinite_a there_o be_v describe_v a_o equilater_n triangle_n which_o be_v the_o thing_n which_o be_v require_v to_o be_v do_v a_o triangle_n or_o any_o other_o rectiline_v figure_n be_v then_o say_v to_o be_v set_v or_o describe_v upon_o a_o line_n when_o the_o line_n be_v one_o of_o the_o side_n of_o the_o figure_n this_o first_o proposition_n be_v a_o problem_n because_o it_o require_v act_n or_o do_v namely_o to_o describe_v a_o triangle_n and_o this_o be_v to_o be_v note_v that_o every_o proposition_n whether_o it_o be_v a_o problem_n or_o a_o theorem_a common_o contain_v in_o it_o a_o thing_n give_v and_o a_o thing_n require_v to_o be_v search_v out_o although_o it_o be_v not_o always_o so_o and_o the_o thing_n give_v give_v be_v ever_o
three_o right_a line_n be_v a_o b_o c._n and_o unto_o some_o one_o of_o they_o namely_o flussates_n to_o c_o put_v a_o equal_a line_n de_fw-fr and_o by_o the_o second_o proposition_n from_o the_o point_n e_o draw_v the_o line_n eglantine_n equal_a to_o the_o line_n b_o and_o by_o the_o same_o unto_o the_o point_n d_o put_v the_o line_n dh_fw-mi equal_a to_o the_o line_n a._n and_o make_v the_o centre_n the_o point_n e_o &_o the_o space_n eglantine_n describe_v a_o circle_n fg_o likewise_o make_v the_o centre_n the_o point_n d_o and_o the_o space_n dh_o describe_v a_o other_o circle_n hf_o which_o circle_n let_v cut_v the_o one_o the_o other_o in_o the_o point_n f._n and_o draw_v these_o line_n df_o and_o ef._n then_o i_o say_v that_o dfe_n be_v a_o triangle_n describe_v of_o 3._o right_a line_n equal_n to_o the_o right_a line_n a_o b_o c._n for_o forasmuch_o as_o the_o line_n dh_o be_v equal_a to_o the_o right_a line_n a_o the_o line_n df_o shall_v also_o be_v equal_a to_o the_o same_o right_a line_n a._n for_o that_o the_o line_n dh_o and_o df_o be_v draw_v from_o the_o centre_n to_o the_o circumference_n likewise_o forasmuch_o as_o eglantine_n be_v equal_a to_o of_o by_o the_o 15._o definition_n and_o the_o right_a line_n b_o be_v equal_a to_o the_o same_o right_a line_n eglantine_n therefore_o the_o right_a line_n of_o be_v equal_a to_o the_o right_a line_n b_o but_o the_o right_a line_n de_fw-fr be_v put_v to_o be_v equal_a to_o the_o right_a line_n c._n wherefore_o of_o three_o right_a line_n ed_z df_o and_o fe_o which_o be_v equal_a to_o three_o right_a line_n give_v a_o b_o c_o be_v describe_v a_o triangle_n which_o be_v require_v to_o be_v do_v problem_n in_o this_o proposition_n the_o adversary_n peradventure_o will_v cavil_n that_o the_o circle_n shall_v not_o cut_v the_o one_o the_o other_o which_o thing_n euclid_n put_v they_o to_o do_v but_o now_o if_o they_o cut_v not_o the_o one_o the_o other_o either_o they_o touch_v the_o one_o the_o other_o or_o they_o be_v distaâ—nte_v the_o one_o from_o the_o other_o first_o if_o it_o be_v possible_a let_v they_o touch_n the_o one_o the_o other_o as_o in_o the_o figure_n here_o put_v the_o construction_n whereof_o answer_v to_o the_o construction_n of_o euclid_n instance_n and_o forasmuch_o as_o fletcher_n be_v the_o centre_n of_o the_o circle_n dk_o therefore_o the_o line_n df_o be_v equal_a to_o the_o line_n fn_o and_o forasmuch_o as_o the_o point_n g_o be_v the_o centre_n of_o the_o circle_n hl_o therefore_o the_o line_n hg_o be_v equal_a to_o the_o line_n gm_n wherefore_o these_o two_o line_n df_o and_o gh_o be_v equal_a to_o one_o line_n namely_o to_o fg._n but_o they_o be_v put_v to_o be_v great_a than_o it_o for_o the_o line_n df_o fg_o and_o gh_o be_v put_v to_o be_v equal_a to_o the_o line_n a_o b_o c_o every_o two_o of_o which_o be_v suppose_v to_o be_v great_a than_o the_o third_o wherefore_o they_o be_v both_o great_a and_o also_o equal_a which_o be_v impossible_a instance_n again_o if_o it_o be_v possible_a let_v the_o circle_n be_v distant_a the_o one_o from_o the_o other_o as_o be_v the_o circle_n dk_o and_o hl._n and_o forasmuch_o as_o fletcher_n be_v the_o centre_n of_o the_o circle_n dk_o therefore_o the_o line_n df_o be_v equal_a to_o the_o line_n fn_o and_o forasmuch_o as_o g_z be_v the_o centre_n of_o the_o circle_n lh_o therefore_o the_o line_n hg_o be_v equal_a to_o the_o line_n gm_n wherefore_o the_o whole_a line_n fg_o be_v great_a than_o the_o two_o line_n df_o and_o gh_o for_o the_o line_n fg_o exceed_v the_o line_n df_o and_o gh_o by_o the_o line_n nm_o but_o it_o be_v suppose_v that_o the_o 〈◊〉_d â—_o df_o and_o hg_o 〈…〉_o theâ—_z the_o 〈◊〉_d fg_o aâ—_z also_o iâ—_z 〈◊〉_d supposed_a that_o the_o line_n a_o and_o c_o be_v 〈…〉_o line_n b_o for_o thâ—_z 〈◊〉_d dâ—_n be_v put_v to_o be_v â—quâ—ll_o to_o the_o line_n a_o and_o the_o line_n fâ—_n to_o the_o line_n bâ—_n â—_z and_o the_o line_n hg_o to_o the_o line_n gâ—_n â—_z whâ—â—â—foâ—â—_n they_o be_v both_o great_a and_o also_o equal_a which_o be_v impossible_a wherefore_o the_o circle_v neither_o touch_n the_o one_o the_o other_o nor_o be_v distant_a the_o one_o from_o the_o other_o wherefore_o of_o necessity_n they_o cut_v the_o one_o the_o other_o which_o be_v require_v to_o be_v prove_v the_o 9_o problem_n the_o 23._o proposition_n upon_o a_o right_a line_n give_v and_o to_o a_o point_n in_o it_o give_v to_o make_v a_o rectiline_a angle_n equal_a to_o a_o rectiline_a angle_n give_v svppose_v that_o the_o right_a line_n give_v be_v ab_fw-la &_o let_v the_o point_n in_o it_o give_v be_v a._n and_o let_v also_o the_o rectiline_a angle_n give_v be_v dch_fw-ge it_o be_v require_v upon_o the_o right_a line_n give_v ab_fw-la and_o to_o the_o point_n in_o it_o give_v a_o to_o make_v a_o rectiline_a angle_n equal_a to_o the_o rectiline_a angle_n give_v dch_o construction_n take_v in_o either_o of_o the_o line_n cd_o and_o change_z a_o point_n at_o all_o adventure_n &_o let_v the_o same_o be_v d_o and_o e._n and_o by_o thâ—_z first_o petition_n draw_v a_o right_a line_n from_o d_z to_o e._n and_o of_o three_o right_a line_n of_o fg_o and_o ga_o which_o let_v be_v equal_a to_o the_o three_o right_a line_n give_v that_o be_v to_o cd_o de_fw-fr and_o ec_o make_v by_o the_o proposition_n go_v before_o a_o triangle_n and_o let_v the_o same_o be_v afgâ—_n so_o that_o let_v the_o line_n cd_o be_v equal_a to_o the_o line_n of_o and_o the_o line_n ce_fw-fr to_o the_o line_n agnostus_n and_o moreover_o the_o line_n de_fw-fr to_o the_o line_n fg._n demonstration_n and_o forasmuch_o as_o these_o two_o line_n dc_o and_o ce_fw-fr be_v equal_v to_o these_o two_o line_n favorina_n and_o agnostus_n the_o one_o to_o the_o other_o and_o the_o base_a de_fw-fr be_v equal_a to_o the_o base_a fg_o therefore_o by_o the_o 8._o proposition_n the_o angle_n dce_n be_v equal_a to_o the_o angle_n fag_n wherefore_o upon_o the_o right_a line_n give_v ab_fw-la and_o to_o the_o poiâ—â—_n iâ—_z iâ—_z give_v namely_o a_o be_v make_v a_o rectiline_a angle_n fag_n equal_a to_o the_o rectiline_a angle_n give_v dch_o which_o be_v require_v to_o be_v do_v an_o other_o construction_n and_o demonstration_n after_o proclus_n suppose_v that_o the_o right_a line_n give_v be_v ab_fw-la proclus_n &_o let_v the_o point_n in_o it_o give_v be_v a_o &_o let_v the_o rectiline_a angle_n give_v be_v cde_o it_o be_v require_v upon_o the_o right_a line_n give_v ab_fw-la &_o to_o the_o point_n in_o it_o give_v a_o to_o make_v a_o rectiline_a angle_n equal_a to_o the_o rectiline_a angle_n give_v cde_v draw_v a_o line_n from_o c_o to_o e._n and_o produce_v the_o line_n ab_fw-la on_o either_o side_n to_o the_o point_v fletcher_n and_o g._n and_o unto_o the_o line_n cd_o put_v the_o line_n favorina_n equal_a &_o unto_o the_o line_n de_fw-fr let_v the_o line_n ab_fw-la be_v equal_a &_o unto_o the_o line_n ec_o put_v the_o line_n bg_o equal_a and_o make_v the_o centre_n the_o point_n a_o &_o the_o space_n of_o describe_v a_o circle_n kf_o and_o again_o make_v the_o centre_n the_o point_n b_o and_o thâ—_z space_n bg_o describe_v a_o other_o circle_n â—l_v which_o shall_v of_o necessity_n cut_v the_o one_o the_o other_o as_o we_o have_v beâ—ore_o prove_v let_v they_o cut_v the_o one_o the_o other_o in_o the_o point_n m_o &_o n._n and_o draw_v these_o right_n linâ—s_v a_o be_o bn_o and_o bm_n and_o forasmuch_o as_o favorina_n be_v equal_a to_o amâ—_n and_o also_o to_o a_o by_o the_o definition_n of_o a_o circle_n but_o cd_o be_v equal_a to_o favorina_n wherefore_o the_o line_n be_o and_o a_o be_v echâ—_n equal_a to_o the_o line_n dc_o again_o forasmuch_o as_o bg_o be_v equal_a to_o bm_v and_o to_o bn_o and_o bg_o be_v equal_a to_o ce_fw-fr therefore_o either_o of_o these_o line_n bm_n and_o bn_o be_v equal_a to_o the_o line_n ce._n but_o the_o line_n basilius_n be_v equal_a to_o the_o line_n de._n wherefore_o these_o two_o line_n basilius_n &_o be_o be_v equal_a to_o these_o two_o line_n d_z e_o and_o dc_o the_o one_o to_o the_o other_o and_o the_o base_a bm_n be_v equal_a to_o the_o base_a ce._n wherefore_o by_o the_o 8._o proposition_n the_o angle_n mab_n be_v equal_a to_o the_o angle_n at_o the_o point_n d._n and_o by_o the_o same_o reason_n the_o angle_n nab_n be_v equal_a to_o the_o same_o angle_n at_o the_o point_n d._n wherefore_o upon_o the_o right_a line_n give_v ab_fw-la and_o to_o the_o point_n in_o it_o give_v a_o be_v describe_v a_o rectiline_a angle_n on_o either_o side_n of_o the_o line_n ab_fw-la namely_o on_o one_o side_n the_o rectiline_a angle_n nab_n and_o on_o the_o other_o
square_n which_o be_v make_v of_o the_o line_n ab_fw-la and_o ac_fw-la construction_n describe_v by_o the_o 46._o proposition_n upon_o the_o line_n bc_o a_o square_a bdce_n and_o by_o the_o same_o upon_o the_o line_n basilius_n and_o ac_fw-la describe_v the_o square_n abfg_n and_o ackh_n and_o by_o the_o point_n a_o draw_v by_o the_o 31._o proposition_n to_o either_o of_o these_o line_n bd_o and_o ce_fw-fr a_o parallel_a line_n al._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 a_o to_o the_o point_n d_o and_o a_o other_o from_o the_o point_n c_o to_o the_o point_n e._n demonstration_n and_o forasmuch_o as_o the_o angle_n bac_n and_o bag_n be_v right_a angle_n therefore_o unto_o a_o right_a line_n basilius_n and_o to_o a_o point_n in_o it_o give_v a_o be_v draw_v two_o right_a line_n ac_fw-la and_o ag_z not_o both_o on_o one_o and_o the_o same_o side_n make_v the_o two_o side_n angle_n equal_a to_o two_o right_a angle_n wherefore_o by_o the_o 14._o proposition_n the_o line_n ac_fw-la and_o ag_z make_v direct_o one_o right_a line_n and_o by_o the_o same_o reason_n the_o line_n basilius_n and_o ah_o make_v also_o direct_o one_o right_a line_n and_o forasmuch_o as_o the_o angle_n dbc_n be_v equal_a to_o the_o angle_n fba_n for_o either_o of_o they_o be_v a_o right_a angle_n put_v the_o angle_n abc_n common_a to_o they_o both_o wherefore_o the_o whole_a angle_n dba_n be_v equal_a to_o the_o whole_a angle_n fbc_n and_o forasmuch_o as_o these_o two_o line_n ab_fw-la and_o bd_o be_v equal_a to_o these_o two_o line_n bf_o and_o bc_o the_o one_o to_o the_o other_o and_o the_o angle_n dba_n be_v equal_a to_o the_o angle_n fbc_n therefore_o by_o the_o 4._o proposition_n the_o base_a ad_fw-la be_v equal_a to_o the_o base_a fc_n and_o the_o triangle_n abdella_n be_v equal_a to_o the_o triangle_n fbc_n but_o by_o the_o 31._o proposition_n the_o parallelogram_n bl_o be_v double_a to_o the_o triangle_n abdella_n for_o they_o have_v both_o one_o and_o the_o same_o base_a namely_o bd_o and_o be_v in_o the_o self_n same_o parallel_a line_n that_o be_v bd_o and_o all_o and_o by_o the_o same_o the_o square_a gb_o be_v double_a to_o the_o triangle_n fbc_n for_o they_o have_v both_o one_o and_o the_o self_n same_o base_a that_o be_v bf_o and_o be_v in_o the_o self_n same_o parallel_a line_n that_o be_v fb_o and_o gc_o but_o the_o double_n of_o thing_n equal_a be_v by_o the_o six_o common_a sentence_n equal_v the_o one_o to_o the_o other_o wherefore_o the_o parallelogram_n bl_o be_v equal_a to_o the_o square_a gb_o and_o in_o like_a sort_n if_o by_o the_o first_o petition_n there_o be_v draw_v a_o right_a line_n from_o the_o point_n a_o to_o the_o point_n e_o and_o a_o other_o from_o the_o point_n b_o to_o the_o point_n king_n we_o may_v prove_v that_o the_o parallelogram_n cl_n be_v equal_a to_o the_o square_a hc_n wherefore_o the_o whole_a square_a bdec_n be_v equal_a to_o the_o two_o square_n gb_o and_o hc_n but_o the_o square_a bdec_n be_v describe_v upon_o the_o line_n bc_o and_o the_o square_n gb_o and_o hc_n be_v describe_v upon_o the_o line_n basilius_n &_o ac_fw-la wherefore_o the_o square_a of_o the_o side_n bc_o be_v equal_a to_o the_o square_n of_o the_o side_n basilius_n and_o ac_fw-la wherefore_o in_o rectangle_n triangle_n the_o square_n which_o be_v make_v of_o the_o side_n that_o subtend_v the_o right_a angle_n be_v equal_a to_o the_o square_n which_o be_v make_v of_o the_o side_n contain_v the_o right_a angle_n which_o be_v require_v to_o be_v demonstrate_v this_o most_o excellent_a and_o notable_a theorem_a be_v first_o invent_v of_o the_o great_a philosopher_n pythagoras_n proposition_n who_o for_o the_o exceed_a joy_n conceive_v of_o the_o invention_n thereof_o offer_v in_o sacrifice_n a_o ox_n as_o record_n hiâ—rone_n proclus_n lycius_n &_o vitruutus_fw-la and_o it_o have_v be_v common_o call_v of_o barbarous_a writer_n of_o the_o latter_a time_n dulcarnon_n pâ—lâ—tariâ—â—_n a_o addition_n of_o pelitarius_n to_o reduce_v two_o unequal_a square_n to_o two_o equal_a square_n suppose_v that_o the_o square_n of_o the_o line_n ab_fw-la and_o ac_fw-la be_v unequal_a it_o be_v require_v to_o reduce_v they_o to_o two_o equal_a square_n join_v the_o two_o line_n ab_fw-la and_o ac_fw-la at_o their_o end_n in_o such_o sort_n that_o they_o make_v a_o right_a angle_n bac_n and_o draw_v a_o line_n from_o b_o to_o c._n then_o upon_o the_o two_o end_n b_o and_o c_o make_v two_o angle_n each_o of_o which_o may_v be_v equal_a to_o half_a a_o right_a angle_n this_o be_v do_v by_o erect_v upon_o the_o line_n bc_o perpendicule_a line_n from_o the_o point_n b_o and_o c_o and_o so_o by_o the_o 9_o proposition_n devide_v â—che_fw-mi of_o the_o right_a angle_n into_o two_o equal_a part_n and_o let_v the_o angle_n bcd_o and_o cbd_v be_v either_o of_o they_o half_o of_o a_o right_a angle_n and_o let_v the_o line_n bd_o and_o cd_o concur_v in_o the_o point_n d._n then_o i_o say_v that_o the_o two_o square_n of_o the_o side_n bd_o and_o cd_o be_v equal_a to_o the_o two_o square_n of_o the_o side_n ab_fw-la and_o ac_fw-la for_o by_o the_o 6_o proposition_n the_o two_o side_n db_o and_o dc_o be_v equal_a and_o the_o angle_n at_o the_o point_n d_o be_v by_o the_o 32._o proposition_n a_o right_a angle_n wherefore_o the_o square_a of_o the_o side_n bc_o be_v equal_a to_o the_o square_n of_o the_o two_o side_n db_o and_o dc_o by_o the_o 47._o proposition_n but_o it_o be_v also_o equal_a to_o the_o square_n of_o the_o two_o side_n ab_fw-la and_o ac_fw-la by_o the_o self_n same_o proposition_n wherefore_o by_o the_o common_a sentence_n the_o square_n of_o the_o two_o side_n bd_o and_o dc_o be_v equal_a to_o the_o square_n of_o the_o two_o side_n ab_fw-la and_o ac_fw-la which_o be_v require_v to_o be_v do_v an_o other_o addition_n of_o pelitarius_n pelitarius_n if_o two_o right_a angle_a triangle_n have_v equal_a base_n the_o square_n of_o the_o two_o side_n of_o the_o one_o be_v equal_a to_o the_o square_n of_o the_o two_o side_n of_o the_o other_o this_o be_v manifest_a by_o the_o former_a construction_n and_o demonstration_n an_o other_o addition_n of_o pelitarius_n pelitarius_n two_o unequal_a line_n be_v give_v to_o know_v how_o much_o the_o square_n of_o the_o one_o be_v great_a than_o the_o square_a of_o the_o other_o suppose_v that_o there_o be_v two_o unequal_a line_n ab_fw-la and_o bc_o of_o which_o let_v ab_fw-la be_v the_o great_a it_o be_v require_v to_o search_v out_o how_o much_o the_o square_n of_o ab_fw-la exceed_v the_o square_a of_o bc._n that_o be_v i_o will_v find_v out_o the_o square_n which_o with_o the_o square_n of_o the_o line_n bc_o shall_v be_v equal_a to_o the_o square_n of_o the_o line_n ab_fw-la put_v the_o line_n a_o b_o and_o bc_o direct_o that_o they_o make_v both_o one_o right_a line_n and_o make_v the_o centre_n the_o point_n b_o and_o the_o space_n basilius_n describe_v a_o circle_n ade_n and_o produce_v the_o line_n ac_fw-la to_o the_o circumference_n and_o let_v it_o concur_v with_o it_o in_o the_o point_n e._n and_o upon_o the_o line_n ae_n and_o from_o the_o point_n c_o erect_v by_o the_o 11._o proposition_n a_o perpendicular_a line_n cd_o which_o produce_v till_o it_o concur_v with_o the_o circumference_n in_o the_o point_n d_o &_o draw_v a_o line_n from_o b_o to_o d._n then_o i_o say_v that_o the_o square_n of_o the_o line_n cd_o be_v the_o excess_n of_o the_o square_n of_o the_o line_n ab_fw-la above_o the_o square_n of_o the_o line_n bc._n for_o forasmuch_o as_o in_o the_o triangle_n bcd_o the_o angle_n at_o the_o point_n c_o be_v a_o right_a angle_n the_o square_a of_o the_o base_a bd_o be_v equal_a to_o the_o square_n of_o the_o two_o side_n bc_o and_o cd_o by_o this_o 47._o proposition_n wherefore_o also_o the_o square_a of_o the_o line_n ab_fw-la be_v equal_a to_o the_o self_n same_o square_n of_o the_o line_n bc_o and_o cd_o wherefore_o the_o square_a of_o the_o line_n bc_o be_v so_o much_o less_o than_o the_o square_a of_o the_o line_n ab_fw-la as_o be_v the_o square_a of_o the_o line_n cd_o which_o be_v require_v to_o search_v out_o an_o other_o addition_n of_o pelitarius_n pelitarius_n the_o diameter_n of_o a_o square_n be_v give_v to_o geve_v the_o square_a thereof_o this_o be_v easy_a to_o be_v do_v for_o if_o upon_o the_o two_o end_n of_o the_o line_n be_v draw_v two_o half_a right_a angle_n and_o so_o be_v make_v perfect_a the_o triangle_n then_o shall_v be_v describe_v half_a of_o the_o square_n the_o other_o half_o whereof_o also_o be_v after_o the_o same_o manner_n easy_a to_o be_v describe_v hereby_o it_o be_v manifest_a that_o the_o square_n of_o the_o
former_a part_n of_o the_o number_n amount_v also_o to_o 112_o which_o be_v equal_a to_o the_o former_a sum_n as_o you_o see_v in_o the_o example_n the_o demonstration_n hereof_o follow_v in_o barlaam_n the_o three_o proposition_n barlaam_n if_o a_o number_n give_v be_v divide_v into_o two_o number_n the_o superficial_a number_n which_o be_v produce_v of_o the_o multiplication_n of_o the_o whole_a into_o one_o of_o the_o part_n be_v equal_a to_o the_o superficial_a number_n which_o be_v produce_v of_o the_o part_n the_o one_o into_o the_o other_o and_o to_o the_o square_a number_n produce_v of_o the_o foresay_a part_n suppose_v that_o the_o number_n give_v by_o ab_fw-la which_o let_v be_v divide_v into_o two_o number_n ac_fw-la and_o cb._n 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 ab_fw-la into_o the_o number_n bc_o 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 ac_fw-la into_o the_o number_n cb_o and_o to_o the_o square_a number_n produce_v of_o the_o number_n cb._n for_o let_v the_o number_n ab_fw-la multiply_v the_o number_n cb_o produce_v the_o number_n d._n and_o let_v the_o number_n ac_fw-la multiply_v the_o number_n cb_o produce_v the_o number_n of_o and_o final_o let_v the_o number_n cb_o multiply_v himself_o produce_v the_o number_n fg._n now_o forasmuch_o as_o the_o number_n ab_fw-la multiply_v the_o 〈…〉_o 〈…〉_o be_v require_v to_o be_v prove_v 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 chance_n the_o square_n which_o be_v make_v of_o the_o whole_a line_n be_v equal_a to_o the_o square_n which_o be_v make_v of_o the_o part_n &_o unto_o that_o rectangle_n figure_n which_o be_v comprehend_v under_o the_o part_n twice_o svppose_v that_o the_o right_a line_n ab_fw-la be_v by_o chance_n divide_v in_o the_o point_n c._n then_o i_o say_v that_o the_o square_n make_v of_o the_o line_n ab_fw-la be_v equal_a unto_o the_o square_n which_o be_v make_v of_o the_o line_n ac_fw-la and_o cb_o and_o unto_o the_o rectangle_n figure_n contain_v under_o the_o lineâ—_z ac_fw-la and_o cb_o awise_o construction_n describe_v by_o the_o â—â—_z of_o the_o first_o upon_o the_o line_n ab_fw-la a_o square_a adeb_n and_o draw_v a_o line_n from_o b_o to_o d_z 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 and_o be_v câ—tting_v the_o diameter_n bd_o in_o the_o point_n g_o and_o let_v the_o same_o be_v cf._n and_o by_o the_o point_n g_z by_o the_o self_n same_o draw_v a_o line_n parallel_n unto_o either_o of_o these_o line_n ab_fw-la and_o de_fw-fr and_o let_v the_o same_o be_v hk_a demonstration_n and_o forasmuch_o as_o the_o line_n cf_o be_v a_o parallel_n unto_o the_o line_n ad_fw-la and_o upon_o they_o fall_v a_o right_a line_n bd_o therefore_o by_o the_o 29._o of_o the_o first_o the_o outward_a angle_n cgb_n be_v equal_a unto_o the_o inward_a and_o opposite_a angle_n adb_o but_o the_o angle_n adb_o be_v by_o the_o 5._o of_o the_o first_o equal_a unto_o the_o angle_n abdella_n for_o the_o side_n basilius_n be_v equal_a unto_o the_o side_n ad_fw-la by_o the_o definition_n of_o a_o square_a wherefore_o the_o angle_n cgb_n be_v equal_a unto_o the_o angle_n gbc_n wherefore_o by_o the_o 6._o of_o the_o first_o the_o side_n bc_o be_v equal_a unto_o the_o side_n cg_o but_o cb_o be_v equal_a unto_o gkâ—_n and_o cg_o be_v equal_a unto_o kb_n wherefore_o gk_o be_v equal_a unto_o kb_n wherefore_o the_o figure_n cgkb_n consist_v of_o four_o equal_a side_n i_o say_v also_o that_o it_o be_v a_o rectangle_n figure_n for_o forasmuch_o as_o cg_o be_v a_o parallel_n unto_o bk_o &_o upon_o they_o fall_v a_o right_a line_n cb_o therefore_o by_o the_o 29._o of_o the_o 1._o the_o angle_n kbc_n and_o gcb_n be_v equal_a unto_o two_o right_a angle_n but_o the_o angle_n kbc_n be_v a_o right_a angle_n wherefore_o the_o angle_n bcg_n be_v also_o a_o right_a angle_n wherefore_o by_o the_o 34._o of_o the_o first_o the_o angle_n opposite_a unto_o they_o namely_o cgk_n and_o gkb_n be_v right_a angle_n wherefore_o cgkb_n be_v a_o rectangle_n figure_n and_o it_o be_v before_o prove_v that_o the_o side_n be_v equal_a wherefore_o it_o be_v a_o square_a and_o it_o be_v describe_v upon_o the_o line_n b_o â—_o and_o by_o the_o same_o reason_n also_o hf_o be_v a_o square_a and_o be_v describe_v upon_o the_o line_n hg_o that_o be_v upon_o the_o line_n ac_fw-la wherefore_o the_o square_n hf_o and_o ck_o be_v make_v of_o the_o line_n ac_fw-la and_o cb._n and_o forasmuch_o as_o the_o parallelogram_n agnostus_n be_v by_o the_o 43._o of_o the_o first_o equal_a unto_o the_o parallelogram_n ge._n and_o agnostus_n be_v that_o which_o be_v contain_v under_o ac_fw-la and_o cb_o for_o cg_o be_v equal_a unto_o cb_o wherefore_o ge_z be_v equal_a to_o that_o which_o be_v contain_v under_o ac_fw-la and_o cb._n wherefore_o agnostus_n and_o ge_z be_v equal_a unto_o that_o which_o be_v comprehend_v under_o ac_fw-la and_o cb_o twice_o and_o the_o square_n hf_o and_o ck_o be_v make_v of_o the_o line_n ac_fw-la and_o cb._n wherefore_o these_o four_o rectangle_n figure_v hf_o ck_o agnostus_n and_o ge_z be_v equal_a unto_o the_o square_n which_o be_v make_v of_o the_o line_n ac_fw-la and_o cb_o and_o to_o the_o rectangle_n figure_n which_o be_v comprehend_v under_o the_o line_n ac_fw-la and_o cb_o twice_o but_o the_o rectangle_n figure_v hf_o ck_o agnostus_n and_o ge_z be_v the_o whole_a rectangle_n figure_n adeb_n which_o be_v the_o square_n make_v of_o the_o line_n ab_fw-la wherefore_o the_o square_n which_o be_v make_v of_o the_o line_n ab_fw-la be_v equal_a to_o the_o square_n which_o be_v make_v of_o the_o line_n ac_fw-la and_o cb_o and_o unto_o the_o rectangle_n figure_n which_o be_v comprehend_v under_o the_o line_n ac_fw-la and_o cb_o awise_o if_o therefore_o a_o right_a line_n be_v divide_v by_o chance_n the_o square_n which_o be_v make_v of_o the_o whole_a line_n be_v equal_a to_o the_o square_n which_o be_v make_v of_o the_o part_n &_o unto_o the_o rectangle_n figure_n which_o be_v comprehend_v under_o the_o part_n awise_o which_o be_v require_v to_o be_v prove_v an_o other_o demonstration_n hereby_o it_o be_v manifest_a that_o the_o parallelogram_n which_o consist_v about_o the_o diameter_n of_o a_o square_n must_v needs_o be_v square_n corollary_n this_o proposition_n be_v of_o infinite_a use_n chief_o in_o surde_fw-la number_n by_o help_v of_o it_o be_v make_v in_o they_o addition_n &_o substraction_n also_o multiplication_n in_o binomial_n &_o residual_n and_o by_o help_n hereofalso_fw-la be_v demoustrate_v that_o kind_n of_o equation_n which_o be_v when_o there_o be_v three_o denomination_n in_o natural_a order_n or_o equal_o distant_a and_o two_o of_o the_o great_a denomination_n be_v equal_a to_o the_o third_o be_v less_o on_o this_o proposition_n be_v ground_v the_o extraction_n of_o square_a root_n and_o many_o other_o thing_n be_v also_o by_o it_o demonstrate_v a_o example_n of_o this_o proposition_n in_o number_n suppose_v a_o number_n namely_o 17._o to_o be_v divide_v into_o two_o part_n 9_o and_o 8._o the_o whole_a number_n 17._o multiply_v into_o himself_o produce_v 289._o the_o square_a number_n of_o 9_o and_o 8._o be_v 81._o and_o 64_o the_o number_n produce_v of_o the_o multiplication_n of_o the_o part_n the_o one_o into_o the_o other_o twice_o be_v 72._o and_o 72_o which_o two_o number_n add_v to_o the_o square_a number_n of_o 9_o and_o 8._o namely_o to_o 81._o and_o 64._o make_v also_o 289._o which_o be_v equal_a to_o the_o square_a number_n of_o the_o whole_a number_n 17._o as_o you_o see_v in_o the_o example_n the_o demonstration_n whereof_o follow_v in_o barlaam_n the_o four_o proposition_n barlaam_n if_o a_o number_n give_v be_v divide_v into_o two_o number_n the_o square_a number_n of_o the_o whole_a be_v equal_a to_o the_o square_a number_n of_o the_o part_n and_o to_o the_o superficial_a number_n which_o be_v produce_v of_o the_o multiplication_n of_o the_o part_n the_o one_o into_o the_o other_o twice_o suppose_v that_o the_o number_n give_v by_o ab_fw-la which_o let_v be_v divide_v into_o two_o number_n ac_fw-la and_o cb._n then_o i_o say_v that_o the_o square_a number_n of_o the_o whole_a number_n ab_fw-la be_v equal_a to_o the_o square_n of_o the_o part_n that_o be_v to_o the_o square_n of_o the_o number_n ac_fw-la and_o cb_o and_o to_o the_o superficial_a number_n produce_v of_o the_o multiplication_n of_o the_o number_n ac_fw-la and_o cb_o the_o one_o into_o the_o other_o twice_o let_v the_o square_a number_n produce_v of_o the_o multiplication_n of_o the_o whole_a number_n ab_fw-la into_o himself_o
the_o line_n ebb_v and_o fd_o be_v produce_v on_o that_o side_n that_o the_o line_n bd_o be_v will_v at_o the_o length_n meet_v together_o produce_v they_o and_o let_v they_o meet_v together_o in_o the_o point_n g._n and_o by_o the_o first_o petition_n draw_v a_o line_n from_o a_o to_o g._n demonstration_n and_o forasmuch_o as_o the_o line_n ac_fw-la be_v equal_a unto_o the_o line_n ce_fw-fr the_o angle_n also_o aec_o be_v by_o the_o 5._o of_o the_o first_o equal_a unto_o the_o angle_n eac_n and_o the_o angle_n at_o the_o point_fw-fr c_o be_v a_o right_a angle_n wherefore_o each_o of_o these_o angle_n eac_n &_o and_o aec_o be_v the_o half_a of_o a_o right_a angle_n and_o by_o the_o same_o reason_n each_o of_o these_o angle_n ceb_fw-mi and_o ebc_n be_v the_o half_a of_o a_o right_a angle_n wherefore_o the_o angle_n aeb_fw-mi be_v a_o right_a angle_n and_o forasmuch_o as_o the_o angle_n ebc_n be_v the_o half_a of_o a_o right_a angle_n therefore_o by_o the_o 15._o of_o the_o first_o the_o angle_n dbc_n be_v the_o half_a of_o a_o right_a angle_n but_o the_o angle_v bdâ—_n be_v a_o right_a angle_n for_o it_o be_v equal_a unto_o the_o angle_n dce_n for_o they_o be_v alternate_a angle_n wherefore_o the_o angle_n remain_v dgb_n be_v the_o half_a of_o a_o right_a angle_n wherefore_o by_o the_o 6_o common_a sentence_n of_o the_o first_o the_o angle_n dgb_n be_v equal_a to_o the_o angle_n dbg_n wherefore_o by_o the_o 6._o of_o the_o first_o the_o side_n bd_o be_v equal_a unto_o the_o side_n gd_o again_o forasmuch_o as_o the_o angle_n egf_n be_v the_o half_a of_o a_o right_a angle_n and_o the_o angle_n at_o the_o point_n fletcher_n be_v a_o right_a angle_n for_o by_o the_o 34._o of_o the_o first_o it_o be_v equal_a unto_o the_o opposite_a angle_n ecd_v wherefore_o the_o angle_n remain_v feg_n be_v the_o half_a of_o a_o right_a angle_n wherefore_o the_o angle_n egf_n be_v equal_a to_o the_o angle_n feg_n wherefore_o by_o the_o 6._o of_o the_o first_o the_o side_n fe_o be_v equal_a unto_o the_o side_n fg._n and_o forasmuch_o as_o ec_o be_v equal_a unto_o ca_n the_o square_a also_o which_o be_v make_v of_o ec_o be_v equal_a to_o the_o square_n which_o be_v make_v of_o ca._n wherefore_o the_o square_n which_o be_v make_v of_o ce_fw-fr and_o ca_n be_v double_a to_o the_o square_n which_o be_v make_v of_o ac_fw-la but_o the_o square_n which_o be_v make_v of_o ea_fw-la be_v by_o the_o 47._o of_o the_o first_o equal_a unto_o the_o square_n which_o be_v make_v of_o ec_o and_o ca._n wherefore_o the_o square_a which_o be_v make_v of_o ea_fw-la be_v double_a to_o the_o square_n which_o be_v make_v of_o acâ—_n again_o forasmuch_o as_o gf_o be_v equal_a unto_o of_o the_o square_a also_o which_o be_v make_v of_o gf_o be_v equal_a to_o the_o square_n which_o be_v make_v of_o fe_o wherefore_o the_o square_n which_o be_v make_v of_o gf_o and_o of_o be_v double_a to_o the_o square_n which_o be_v make_v of_o ef._n butler_n by_o the_o 47._o of_o the_o first_o the_o square_a which_o be_v make_v of_o eglantine_n be_v equal_a to_o the_o square_n which_o be_v make_v of_o gf_o and_o ef._n wherefore_o the_o square_a which_o be_v make_v of_o eglantine_n be_v double_a to_o the_o square_n which_o be_v make_v of_o ef._n but_o of_o be_v equal_a unto_o cd_o wherefore_o the_o square_v which_o be_v make_v of_o eglantine_n be_v double_a to_o the_o square_n which_o be_v make_v of_o cd_o and_o it_o be_v prove_v the_o square_n which_o be_v make_v of_o ea_fw-la be_v double_a to_o the_o square_n which_o be_v make_v of_o ac_fw-la wherefore_o the_o square_n which_o be_v make_v of_o ae_n and_o eglantine_n be_v double_a to_o the_o square_n which_o be_v make_v of_o ac_fw-la and_o cd_o but_o by_o the_o 47._o of_o this_o first_o the_o square_a which_o be_v make_v of_o agnostus_n be_v equal_a to_o the_o square_n which_o be_v make_v of_o ae_n and_o eglantine_n wherefore_o the_o square_a which_o be_v make_v of_o agnostus_n be_v double_a to_o the_o square_n which_o be_v make_v of_o ac_fw-la and_o cd_o but_o unto_o the_o square_n which_o be_v make_v of_o agnostus_n be_v equal_a the_o square_n which_o be_v make_v of_o ad_fw-la and_o dg_o wherefore_o the_o square_n which_o be_v make_v of_o ad_fw-la and_o dg_o be_v double_a to_o the_o square_n which_o be_v make_v of_o ac_fw-la and_o dc_o but_o dg_o be_v equal_a unto_o db._n wherefore_o the_o square_n which_o be_v make_v of_o ad_fw-la and_o db_o be_v double_a to_o the_o square_n which_o be_v make_v of_o ac_fw-la and_o dc_o if_o therefore_o a_o right_a line_n be_v divide_v into_o two_o equal_a part_n and_o unto_o it_o be_v add_v a_o other_o line_n direct_o the_o square_n which_o be_v make_v of_o the_o whole_a and_o that_o which_o be_v add_v as_o of_o one_o line_n together_o with_o the_o square_n which_o be_v make_v of_o the_o line_n which_o be_v add_v these_o two_o square_n i_o say_v be_v double_a to_o these_o square_n namely_o to_o the_o square_n which_o be_v make_v of_o the_o half_a line_n and_o to_o the_o square_n which_o be_v make_v of_o the_o other_o half_a line_n and_o that_o which_o be_v add_v as_o of_o one_o line_n which_o be_v require_v to_o be_v prove_v ¶_o a_o other_o demonstration_n after_o pelitarius_n suppose_v that_o the_o line_n ab_fw-la be_v divide_v into_o two_o equal_a part_n in_o the_o point_n c._n and_o unto_o it_o let_v there_o be_v add_v a_o other_o right_a line_n direct_o namely_o bd._n then_o i_o say_v that_o the_o square_a of_o ad_fw-la together_o with_o the_o square_n of_o bd_o be_v double_a to_o the_o square_n of_o ac_fw-la and_o cd_o the_o supplement_n eh_o be_v equal_a to_o the_o parallelogram_n hp_n and_o the_o square_n ah_o together_o with_o the_o lesser_a supplement_n no_o be_v equal_a to_o the_o other_o supplement_n hd_v by_o the_o first_o common_a sentence_n so_o oftentimes_o repeat_v as_o be_v need_n wherefore_o the_o two_o supplement_n eh_o and_o hd_n be_v equal_a to_o the_o square_n ah_o and_o to_o the_o gnomon_n khlpqo_n if_o therefore_o unto_o either_o of_o they_o be_v add_v the_o square_a qf_n the_o two_o supplement_n eh_o and_o hd_v together_o with_o the_o square_n of_o qf_n shall_v be_v equal_a to_o the_o square_n ah_o &_o to_o the_o gnomon_n khlpqo_n and_o to_o the_o square_a qf_n but_o these_o three_o figure_n do_v make_v the_o two_o square_n ah_o and_o hf._n wherefore_o the_o two_o supplement_n eh_o and_o hd_v together_o with_o the_o square_a qf_n be_v equal_a to_o the_o two_o square_n ah_o and_o hf_o which_o be_v the_o second_o thing_n to_o be_v prove_v wherefore_o the_o two_o square_n ah_o and_o hf_o be_v take_v twice_o be_v equal_a to_o the_o whole_a square_a de_fw-fr together_a with_o the_o square_n of_o qf_n wherefore_o the_o square_a de_fw-fr together_a with_o the_o square_a qf_n be_v double_a to_o the_o square_n ah_o and_o hf_o which_o be_v require_v to_o be_v prove_v ¶_o a_o example_n of_o this_o proposition_n in_o number_n take_v any_o even_a number_n as_o 18_o and_o take_v the_o half_a of_o it_o which_o be_v 9_o and_o unto_o 18._o the_o whole_a add_v any_o other_o number_n as_o 3._o which_o make_v 21._o take_v the_o square_a number_n of_o 21._o the_o whole_a number_n and_o the_o number_n add_v which_o make_v 441._o take_v also_o the_o square_a of_o 3_o the_o number_n add_v which_o be_v 9_o which_o two_o square_n add_v together_o make_v 450._o then_o add_v the_o half_a number_n 9_o to_o the_o number_n add_v 3._o which_o make_v 12._o and_o take_v the_o square_a of_o 9_o the_o half_a number_n and_o of_o 12._o the_o half_a number_n and_o the_o number_n add_v which_o square_n be_v 81._o and_o 144._o and_o which_o two_o square_n also_o add_v together_o make_v 225_o unto_o which_o sum_n the_o foresay_a number_n 450._o be_v double_a as_o you_o see_v in_o the_o example_n the_o demonstration_n whereof_o follow_v in_o barlaam_n the_o ten_o proposition_n if_o a_o even_a number_n be_v divide_v into_o two_o equal_a number_n and_o unto_o it_o be_v add_v any_o other_o number_n the_o square_a number_n of_o the_o whole_a number_n compose_v of_o the_o number_n and_o of_o that_o which_o be_v add_v and_o the_o square_a number_n of_o the_o number_n add_v these_o two_o square_a number_n i_o say_v add_v together_o be_v double_a to_o these_o square_a number_n namely_o to_o the_o square_n of_o the_o half_a number_n and_o to_o the_o square_n of_o the_o number_n compose_v of_o the_o half_a number_n and_o of_o the_o number_n add_v suppose_v that_o the_o number_n ab_fw-la be_v a_o even_a number_n be_v divide_v into_o two_o equal_a number_n ac_fw-la and_o cb_o and_o unto_o it_o let_v be_v add_v a_o other_o number_n bd._n then_o i_o say_v that_o the_o square_a
a_o circle_n be_v take_v any_o point_n which_o be_v not_o the_o centre_n of_o the_o circle_n and_o from_o that_o point_n be_v draw_v unto_o the_o circumference_n certain_a right_a line_n the_o great_a of_o those_o line_n shall_v be_v that_o line_n wherein_o be_v the_o centre_n and_o the_o lest_o shall_v be_v the_o residue_n of_o the_o same_o line_n and_o of_o all_o the_o other_o line_n that_o 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 and_o from_o that_o point_n can_v fall_v within_o the_o circle_n on_o each_o side_n of_o the_o least_o line_n only_o two_o equal_a right_a line_n 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 ad._n and_o take_v in_o it_o any_o point_n beside_o the_o centre_n of_o the_o circle_n and_o let_v the_o same_o be_v f._n construction_n and_o let_v the_o centre_n of_o the_o circle_n by_o the_o 1._o of_o the_o three_o be_v the_o point_n e._n and_o from_o the_o point_n f_o let_v there_o be_v draw_v unto_o the_o circumference_n abcd_o these_o right_a line_n fd_v fc_o and_o fg._n then_o i_o say_v that_o the_o line_n favorina_n be_v the_o great_a and_o the_o line_n fd_o be_v the_o jest_n and_o of_o the_o other_o line_n the_o line_n fb_o be_v great_a than_o the_o line_n fc_o and_o the_o line_n fc_o be_v great_a than_o the_o line_n fg._n proposition_n draw_v by_o the_o first_o petition_n these_o right_a line_n be_v ce_fw-fr and_o ge._n and_o for_o asmuch_o as_o by_o the_o 20._o of_o the_o first_o in_o every_o triangle_n two_o side_n be_v great_a than_o the_o three_o demonstration_n therefore_o the_o line_n ebb_v and_o of_o be_v great_a than_o the_o residue_n namely_o then_o the_o line_n fb_o but_o the_o line_n ae_n be_v equal_a unto_o the_o line_n be_v by_o the_o 15._o definition_n of_o the_o first_o wherefore_o the_o line_n be_v and_o of_o be_v equal_a unto_o the_o line_n af._n wherefore_o the_o line_n of_o be_v great_a than_o then_o the_o line_n bf_o again_o for_o asmuch_o as_o the_o line_n be_v be_v equal_a unto_o ce_fw-fr by_o the_o 15._o definition_n of_o the_o first_o and_o the_o line_n fe_o be_v common_a unto_o they_o both_o therefore_o these_o two_o line_n be_v and_o of_o be_v equal_a unto_o these_o two_o ce_fw-fr and_o ef._n but_o the_o angle_n bef_n be_v great_a than_o the_o angle_n cef_n wherefore_o by_o the_o 24._o of_o the_o first_o the_o base_a bf_o be_v great_a than_o the_o base_a cf_o and_o by_o the_o same_o reason_n the_o line_n cf_o be_v great_a than_o the_o line_n fg._n again_o for_o asmuch_o as_o the_o line_n gf_o and_o fe_o be_v great_a than_o the_o line_n eglantine_n by_o the_o 20._o of_o the_o first_o part_n but_o by_o the_o 15._o definition_n of_o the_o first_o the_o line_n eglantine_n be_v equal_a unto_o the_o line_n ed_z wherefore_o the_o line_n gf_o and_o fe_o be_v great_a than_o the_o line_n ed_z take_v away_o of_o which_o be_v common_a to_o they_o both_o wherefore_o the_o residue_n gf_o be_v great_a than_o the_o residue_n fd._n wherefore_o the_o line_n favorina_n be_v the_o great_a and_o the_o line_n fd_o be_v the_o jest_n and_o the_o line_n fb_o be_v great_a than_o the_o line_n fc_o and_o the_o line_n fc_o be_v great_a than_o the_o line_n fg._n part_n now_o also_o i_o say_v that_o from_o the_o point_n f_o there_o can_v be_v draw_v only_o two_o equal_a right_a line_n into_o the_o circle_n abcd_o on_o each_o side_n of_o the_o least_o line_n namely_o fd._n for_n by_o the_o 23._o of_o the_o first_o upon_o the_o right_a line_n give_v of_o and_o to_o the_o point_n in_o it_o namely_o e_o make_v unto_o the_o angle_n gef_n a_o equal_a angle_n feh_n and_o by_o the_o first_o petition_n draw_v a_o line_n from_o fletcher_n to_o h._n now_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 eglantine_n be_v equal_a unto_o the_o line_n eh_o and_o the_o line_n of_o be_v common_a unto_o they_o both_o therefore_o these_o two_o line_n ge_z and_o of_o be_v equal_a unto_o these_o two_o line_n he_o and_o of_o and_o by_o construction_n the_o angle_n gef_n be_v equal_a unto_o the_o angle_n hef_fw-fr wherefore_o by_o the_o 4._o of_o the_o first_o the_o base_a fg_o be_v equal_a unto_o the_o base_a fh_o impossibilie_n i_o say_v moreover_o that_o from_o the_o point_n f_o can_v be_v draw_v into_o the_o circle_n no_o other_o right_a line_n equal_a unto_o the_o line_n fg._n for_o if_o it_o possible_a let_v the_o lineâ—_z fk_o be_v equal_a unto_o the_o line_n fg._n and_o for_o asmuch_o as_o fk_n be_v equal_a unto_o fg._n but_o the_o line_n fh_o be_v equal_a unto_o the_o line_n fg_o therefore_o the_o line_n fk_o be_v equal_a unto_o the_o line_n fh_o wherefore_o the_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 equal_a to_o that_o which_o be_v far_o of_o which_o we_o have_v before_o prove_v to_o be_v impossible_a impossibility_n or_o else_o it_o may_v thus_o be_v demonstrate_v draw_v by_o the_o first_o petition_n a_o line_n from_o e_o to_o king_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_v ge_z be_v equal_a unto_o the_o line_n eke_o and_o the_o line_n fe_o be_v common_a to_o they_o both_o and_o the_o base_a gf_o be_v equal_a unto_o the_o base_a fk_n therefore_o by_o the_o 8._o of_o the_o first_o the_o angle_n gef_n be_v equal_a to_o the_o angle_n kef_n but_o the_o angle_n gef_n be_v equal_a to_o the_o angle_n hef_fw-fr wherefore_o by_o the_o first_o common_a sentence_n the_o angle_n hef_fw-fr be_v equal_a to_o the_o angle_n kef_n the_o less_o unto_o the_o great_a which_o be_v impossible_a wherefore_o from_o the_o point_n fletcher_n there_o can_v be_v draw_v into_o the_o circle_n no_o other_o right_a line_n equal_a unto_o the_o line_n gf_o wherefore_o but_o one_o only_a if_o therefore_o in_o the_o diameter_n of_o a_o circle_n be_v take_v any_o point_n which_o be_v not_o the_o centre_n of_o the_o circle_n and_o from_o that_o point_n be_v draw_v unto_o the_o circumference_n certain_a right_a line_n the_o great_a of_o those_o right_a line_n shall_v be_v that_o wherein_o be_v the_o centre_n and_o the_o least_o shall_v be_v the_o residue_n and_o of_o all_o the_o other_o line_n that_o 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 and_o from_o that_o point_n can_v fall_v within_o the_o circle_n on_o each_o side_n of_o the_o least_o line_n only_o two_o equal_a right_a line_n which_o be_v require_v to_o be_v prove_v ¶_o a_o corollary_n corollary_n hereby_o it_o be_v manifest_a that_o two_o right_a line_n be_v draw_v from_o any_o one_o point_n of_o the_o diameter_n the_o one_o of_o one_o side_n and_o the_o other_o of_o the_o other_o side_n if_o with_o the_o diameter_n they_o make_v equal_a angle_n the_o say_v two_o right_a line_n be_v equal_a as_o in_o this_o place_n be_v the_o two_o line_n fg_o and_o fh_o the_o 7._o theorem_a the_o 8._o proposition_n if_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 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 be_v the_o least_o which_o be_v draw_v from_o the_o point_n to_o the_o diameter_n 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 least_o only_o two_o equal_a right_a line_n part_n now_o also_o i_o say_v that_o from_o the_o point_n d_o can_v be_v draw_v unto_o the_o circumference_n on_o each_o side_n of_o dg_o the_o least_o only_o two_o equal_a right_a line_n upon_o the_o right_a line_n md_o and_o unto_o the_o point_n in_o it_o m_o make_v by_o the_o 23._o of_o the_o first_o unto_o the_o angle_n kmd_v a_o equal_a angle_n dmb._n and_o by_o the_o first_o petition_n draw_v a_o line_n from_o d_z to_o b._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 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
from_o the_o same_o point_n isâ—_n draw_v also_o a_o other_o line_n not_o pass_v by_o the_o contre_fw-fr namely_o the_o line_n gf_o therefore_o by_o the_o 8._o of_o this_o book_n the_o outward_a part_n gd_o of_o the_o line_n gk_o shall_v be_v less_o than_o the_o outward_a part_n ga_o of_o the_o line_n gfâ—_n â—_z but_o the_o line_n ga_o be_v equal_a to_o the_o line_n gb_o wherefore_o the_o line_n gd_o be_v less_o than_o the_o line_n gbâ—_n â—_z namely_o the_o whole_a less_o than_o the_o partâ—_n which_o be_v absurd_a the_o 12._o theorem_a the_o 13._o proposition_n a_o circle_n can_v not_o touch_v a_o other_o circle_n in_o more_o point_n than_o one_o whether_o they_o touch_v within_o or_o without_o for_o if_o it_o be_v possible_a let_v the_o circle_n abcd_o touch_v the_o circle_n ebfd_n first_o inward_o in_o more_o point_n than_o one_o that_o be_v in_o d_z and_o b._n inward_o take_v by_o the_o first_o of_o the_o three_o the_o centre_n of_o the_o circle_n abcd_o and_o let_v the_o same_o be_v the_o point_fw-fr g_o and_o likewise_o the_o centre_n of_o the_o circle_n ebfd_n and_o let_v the_o same_o be_v the_o point_n h._n wherefore_o by_o the_o 11._o of_o the_o same_o a_o right_a line_n draw_v from_o the_o point_n g_o to_o the_o point_n h_o and_o produce_v will_v fall_v upon_o the_o point_n b_o and_o d_o let_v it_o so_o fall_v as_o the_o line_n bghd_v do_v and_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 abcd_o therefore_o by_o the_o 15._o definition_n of_o the_o first_o the_o line_n bg_o be_v equal_a to_o the_o line_n dg_o wherefore_o the_o line_n bg_o be_v great_a than_o then_o the_o line_n hd_v wherefore_o the_o line_n bh_o be_v much_o great_a than_o the_o line_n hd_v again_o for_o asmuch_o as_o the_o point_n h_o be_v the_o centre_n of_o the_o circle_n ebfd_n therefore_o by_o the_o same_o definition_n the_o line_n bh_o be_v equal_a to_o the_o line_n hd_v and_o it_o be_v prove_v that_o it_o be_v mâ—ch_o great_a than_o it_o which_o be_v impossible_a a_o circle_n therefore_o can_v not_o touch_v a_o circle_n inward_o in_o more_o point_n than_o one_o now_o i_o say_v that_o neither_o outward_o also_o a_o circle_n touch_v a_o circle_n in_o more_o point_n than_o one_o outward_o for_o if_o it_o be_v possible_a let_v the_o circle_n ack_z touch_v the_o circle_n abcd_o outward_o in_o more_o point_n they_o one_o that_o be_v in_o a_o and_o c_o and_o by_o the_o first_o petition_n draw_v a_o line_n from_o the_o point_n a_o to_o the_o point_n c._n now_o for_o asmuch_o as_o in_o the_o circumference_n of_o either_o of_o the_o circle_n abcd_o and_o ack_z be_v take_v two_o point_n at_o all_o adventure_n namely_o a_o and_o c_o therefore_o by_o the_o second_o of_o the_o three_o a_o right_a line_n join_v together_o those_o point_n shall_v fall_v within_o both_o the_o circle_n but_o it_o fall_v within_o the_o circle_n abcd_o &_o without_o the_o circle_n ack_z which_o be_v absurd_a wherefore_o a_o circle_n shall_v not_o touch_v a_o circle_n outward_o in_o more_o point_n than_o one_o and_o it_o be_v prove_v that_o neither_o also_o inward_o wherefore_o a_o circle_n can_v not_o touch_v a_o other_o circle_n in_o more_o point_n than_o one_o whether_o they_o touch_v within_o or_o without_o which_o be_v require_v to_o be_v demonstrate_v ¶_o a_o other_o demonstration_n after_o pelitarius_n and_o flussates_n outward_o suppose_v that_o there_o be_v two_o circle_n abg_o and_o adg_n which_o if_o it_o be_v possible_a let_v touch_v the_o one_o the_o other_o outward_o in_o more_o point_n than_o one_o namely_o in_o a_o and_o g._n let_v the_o centre_n of_o the_o circle_n abg_o be_v the_o point_n i_o and_o let_v the_o centre_n of_o the_o circle_n adg_n be_v the_o point_n k._n and_o draw_v a_o right_a line_n from_o the_o point_n i_o to_o the_o point_n king_n which_o by_o the_o 12._o of_o this_o book_n shall_v pass_v both_o by_o the_o point_n a_o and_o by_o the_o point_n g_o which_o be_v not_o possible_a for_o then_o two_o right_a line_n shall_v include_v a_o superficies_n contrary_a to_o the_o last_o common_a sentence_n it_o may_v also_o be_v thus_o demonstrate_v draw_v a_o line_n from_o the_o centre_n i_o to_o the_o centre_n king_n which_o shall_v pass_v by_o one_o of_o the_o touch_n as_o for_o example_n by_o the_o point_n a._n and_o draw_v these_o right_a line_n gk_o and_o give_v and_o so_o shall_v be_v make_v a_o triangle_n who_o two_o side_n gk_o and_o give_v shall_v not_o be_v great_a than_o the_o side_n ik_fw-mi which_o iâ—_z contrary_a to_o the_o 20._o of_o the_o first_o but_o now_o if_o it_o be_v possible_a let_v the_o foresay_a circle_n adg_n touch_v the_o circle_n abc_n inward_o in_o more_o point_n than_o one_o inward_o namely_o in_o the_o point_n a_o and_o g_o and_o let_v the_o centre_n of_o the_o circle_n abc_n be_v the_o point_n i_o as_o before_o and_o let_v the_o centre_n of_o the_o circle_n adg_n be_v the_o point_n king_n as_o also_o before_o and_o extend_v a_o line_n from_o the_o point_n i_o to_o the_o point_n king_n which_o shall_v fall_v upon_o the_o touch_n by_o the_o 11._o of_o this_o book_n draw_v also_o these_o line_n kg_v and_o ig_n and_o for_o asmuch_o as_o the_o line_n kg_n be_v equal_a to_o the_o line_n ka_o by_o the_o 15._o definition_n of_o the_o first_o add_v the_o line_n ki_v common_a to_o they_o both_o wherefore_o the_o whole_a line_n a_z be_v equal_a to_o the_o two_o line_n kg_v and_o ki_v but_o unto_o the_o line_n ai_fw-fr be_v equal_v the_o line_n ig_n by_o the_o definition_n of_o a_o circle_n wherefore_o in_o the_o triangle_n ikg_v the_o side_n ig_n be_v not_o less_o than_o the_o two_o side_n ik_fw-mi and_o kg_n which_o be_v contrary_a to_o the_o 20._o of_o the_o first_o the_o 13._o theorem_a the_o 14._o proposition_n 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 other_o theorem_a svppose_v that_o there_o be_v a_o circle_n abcd_o and_o let_v there_o be_v in_o it_o draw_v these_o equal_a right_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 construction_n take_v by_o the_o first_o of_o the_o three_o the_o centre_n of_o the_o circle_n abcd_o and_o let_v the_o same_o be_v the_o point_n e._n and_o by_o the_o 12._o of_o the_o first_o from_o the_o point_n e_o draw_v unto_o the_o line_n ab_fw-la &_o cd_o perpendicular_a line_n of_o and_o eglantine_n and_o by_o the_o first_o petition_n draw_v these_o right_a line_n ae_n and_o ce._n demonstration_n now_o for_o asmuch_o as_o a_o certain_a right_a line_n of_o draw_v by_o the_o centre_n cut_v a_o certain_a other_o right_a line_n ab_fw-la not_o draw_v by_o the_o centre_n in_o such_o sort_n that_o it_o make_v right_a angle_n therefore_o by_o the_o three_o of_o the_o three_o it_o divide_v it_o into_o two_o equal_a part_n wherefore_o the_o line_n of_o be_v equal_a to_o the_o line_n fb_o wherefore_o the_o line_n ab_fw-la be_v double_a to_o the_o line_n of_o and_o by_o the_o same_o reason_n also_o the_o line_n cd_o be_v double_a to_o the_o line_n cg_o but_o the_o line_n ab_fw-la be_v equal_a to_o the_o line_n cd_o wherefore_o the_o line_n of_o be_v also_o equal_a to_o the_o line_n cg_o 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 ae_n be_v equal_a to_o the_o line_n ec_o therefore_o the_o square_a of_o the_o line_n ec_o be_v equal_a to_o the_o square_n of_o the_o line_n ae_n but_o unto_o the_o square_n of_o the_o line_n ae_n be_v equal_a by_o the_o 47._o of_o the_o first_o the_o square_n of_o the_o line_n of_o &_o fe_o for_o the_o angle_n at_o the_o point_n fletcher_n be_v a_o right_a angle_n and_o by_o the_o self_n same_o to_o the_o square_n of_o the_o line_n ec_o be_v equal_a the_o square_n of_o the_o line_n eglantine_n and_o gc_o for_o the_o angle_n at_o the_o point_n g_z be_v a_o right_a angle_n wherefore_o the_o square_n of_o the_o line_n of_o and_o fe_o be_v equal_a to_o the_o square_n of_o the_o line_n cg_o and_o ge_z of_o which_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 for_o the_o line_n of_o be_v equal_a to_o the_o line_n cg_o 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 fe_o be_v equal_a to_o the_o square_n remain_v namely_o to_o the_o square_n of_o the_o line_n eglantine_n wherefore_o the_o line_n of_o be_v equal_a to_o the_o line_n eglantine_n demonstration_n but_o right_a line_n be_v say_v to_o be_v equal_o distant_a from_o the_o centre_n
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
but_o the_o line_n ab_fw-la be_v one_o line_n with_o the_o line_n cd_o so_o that_o thereby_o shall_v follow_v the_o contrary_a of_o the_o former_a proposition_n the_o 3._o problem_n the_o 25._o proposition_n a_o segment_n of_o a_o circle_n be_v give_v to_o describe_v the_o whole_a circle_n of_o the_o same_o segment_n svppose_v that_o the_o segment_n give_v be_v abc_n it_o be_v require_v to_o describe_v the_o whole_a circle_n of_o the_o same_o segment_n abc_n construction_n divide_v by_o the_o 10._o of_o the_o first_o the_o line_n ac_fw-la into_o two_o equal_a part_n in_o the_o point_n d._n and_o by_o the_o 11._o of_o the_o same_o from_o the_o point_n d_o raise_v up_o unto_o the_o line_n ac_fw-la a_o perpendicular_a line_n bd._n and_o by_o the_o first_o petition_n draw_v a_o right_a line_n from_o a_o to_o b._n now_o then_o the_o angle_n abdella_n be_v compare_v to_o the_o angle_n bad_a be_v either_o great_a than_o it_o or_o equal_a unto_o it_o or_o less_o than_o it_o case_n first_o let_v it_o be_v great_a and_o by_o the_o 23._o of_o the_o same_o upon_o the_o right_a line_n basilius_n and_o unto_o the_o point_n in_o it_o a_o make_v unto_o the_o angle_n abdella_n a_o equal_a angle_n bae_o and_o by_o the_o second_o petition_n extend_v the_o line_n bd_o unto_o the_o point_n e._n and_o by_o the_o first_o petition_n draw_v a_o line_n from_o e_o to_o c._n now_o for_o asmuch_o as_o the_o angle_n abe_n be_v equal_a to_o the_o angle_n bae_o demonstration_n therefore_o by_o the_o 6._o of_o the_o first_o the_o right_a line_n ebb_n be_v equal_a to_o the_o right_a line_n ae_n and_o for_o asmuch_o as_o the_o line_n ad_fw-la be_v equal_a to_o the_o line_n dc_o and_o the_o line_n de_fw-fr be_v common_a to_o they_o both_o therefore_o these_o two_o line_n ad_fw-la â—nd_fw-fr de_fw-fr be_v equal_a to_o these_o tâ—o_fw-la line_n cd_o and_o de_fw-fr the_o one_o to_o the_o other_o and_o the_o angle_n ade_n â—â—_o by_o the_o 4._o petition_n equal_a to_o the_o angle_n cde_o for_o either_o of_o they_o be_v a_o right_a angleâ—_n whereforâ—_n by_o the_o 4._o of_o the_o first_o the_o base_a ae_n be_v equal_a to_o the_o base_a ce._n but_o it_o be_v proâ—ed_v that_o the_o line_n ae_n be_v squall_n to_o the_o line_n be._n wherefore_o the_o line_n be_v also_o be_v equal_a to_o the_o line_n ce._n wherefore_o these_o three_o line_n ae_n ebb_n and_o ec_o be_v equal_a the_o one_o to_o the_o other_o wherefore_o make_v the_o centre_n e_o and_o the_o space_n either_o ae_n or_o ebb_n or_o ec_o describe_v by_o the_o three_o petition_n a_o circle_n and_o it_o shall_v pass_v by_o the_o point_n a_o b_o c._n wherefore_o there_o be_v describe_v the_o whole_a circle_n of_o the_o segment_n give_v and_o it_o be_v manifest_a that_o the_o segment_n abc_n be_v lesâ—e_n than_o a_o semicircle_n for_o the_o centre_n e_o fall_v without_o it_o the_o like_a demonstration_n also_o will_v serve_v if_o the_o angle_n abdella_n be_v equal_a to_o the_o angle_n bad_a case_n for_o the_o line_n ad_fw-la be_v equal_a to_o either_o of_o these_o line_n bd_o and_o dc_o there_o be_v three_o linesâ—_n dam_n db_o and_o dc_o equal_a the_o one_o to_o the_o other_o so_o that_o the_o point_n d_o shall_v be_v the_o centre_n of_o the_o circle_n be_v complete_a and_o abc_n shall_v be_v a_o semicircle_n but_o if_o the_o angle_n abdella_n be_v less_o than_o the_o angle_n bad_a case_n then_o by_o the_o 23._o of_o the_o first_o upon_o the_o right_a line_n basilius_n and_o unto_o the_o point_n in_o it_o a_o make_v unto_o the_o angle_n abdella_n a_o equal_a angle_n within_o y_fw-es segment_n abc_n and_o so_o the_o centre_n of_o the_o circle_n shall_v fall_v in_o the_o line_n db_o and_o it_o shall_v be_v the_o point_n e_o and_o the_o segment_n abc_n shall_v be_v great_a than_o a_o semicircle_n wherefore_o a_o segment_n be_v give_v there_o be_v describe_v the_o whole_a circle_n of_o the_o same_o segment_n â—hich_z be_v require_v to_o be_v do_v ¶_o a_o corollary_n hereby_o it_o be_v manifest_a that_o in_o a_o semicircle_n the_o angle_n bad_a be_v equal_a to_o the_o angle_n dba_n but_o in_o a_o section_n less_o than_o a_o semicircle_n it_o be_v less_o in_o a_o section_n great_a than_o a_o semicircle_n it_o be_v great_a and_o by_o this_o last_o general_a way_n addition_n if_o there_o be_v give_v three_o point_n set_v howsoever_o so_o that_o they_o be_v not_o all_o three_o in_o one_o right_a line_n a_o man_n may_v describe_v a_o circle_n which_o shall_v pass_v by_o all_o the_o say_v three_o point_n for_o as_o in_o the_o example_n before_o put_v if_o you_o suppose_v only_o the_o 3._o point_n a_o b_o c_o to_o be_v give_v and_o not_o the_o circumference_n abc_n to_o be_v draw_v yet_o follow_v the_o self_n â—ame_n order_v you_o do_v before_o that_o be_v draw_v a_o right_a line_n from_o a_o to_o b_o and_o a_o outstretch_o from_o b_o to_o c_z and_o divide_v the_o say_v right_a line_n into_o two_o equal_a part_n in_o the_o poor_n d_z and_o e_o and_o erect_v the_o perpendicular_a line_n df_o and_o of_o cut_v the_o one_o the_o other_o in_o the_o point_n fletcher_n and_o draw_v a_o â—ight_a line_n from_o fletcher_n to_o b_o and_o make_v the_o centre_n the_o point_n fletcher_n and_o the_o space_n fb_o describe_v a_o circle_n and_o it_o shall_v pass_v by_o the_o point_n a_o &_o c_o which_o may_v be_v prove_v by_o draw_v right_a line_n from_o a_o to_o fletcher_n and_o from_o fletcher_n to_o c._n for_o forasmuch_o as_o the_o two_o siâ—es_n ad_fw-la and_o df_o of_o the_o triangle_n adf_n be_v equal_a to_o the_o two_o side_n bd_o and_o df_o of_o the_o triangle_n bdf_n for_o by_o supposition_n the_o line_n ad_fw-la be_v equal_a to_o the_o line_n db_o and_o the_o line_n df_o be_v common_a to_o they_o botâ—_n and_o the_o angle_n adf_n be_v equal_a to_o the_o angle_n bdf_n for_o they_o be_v both_o right_a angle_n therefore_o by_o the_o 4._o of_o the_o first_o the_o base_a of_o be_v equal_a to_o the_o base_a bf_o and_o by_o the_o same_o reason_n the_o line_n fc_o be_v equal_a to_o the_o line_n fb_o wherefore_o these_o three_o line_n favorina_n fb_o and_o fc_o be_v equal_a the_o one_o to_o the_o other_o wherefore_o make_v the_o centre_n the_o point_n f_o and_o the_o space_n fb_o it_o shall_v also_o pass_v by_o the_o point_n a_o and_o c._n which_o be_v require_v to_o be_v do_v this_o proposition_n be_v very_o necessary_a for_o many_o thing_n as_o you_o shall_v afterward_o see_v pelitarius_n here_o add_v a_o brief_a way_n how_o to_o find_v out_o the_o centre_n of_o a_o circle_n which_o be_v common_o use_v of_o artificer_n the_o 23._o theorem_a the_o 26._o proposition_n equal_a angle_n in_o equal_a circle_n consist_v in_o equal_a circumference_n whether_o the_o angle_n be_v draw_v from_o the_o centre_n or_o from_o the_o circumference_n svppose_v that_o these_o circle_n abc_n and_o def_n be_v equal_a and_o from_o their_o centre_n namely_o the_o point_n g_o and_o h_o let_v there_o be_v draw_v these_o equal_a angle_n bgc_o and_o ehf_o and_o likewise_o from_o their_o circumference_n these_o equal_a angle_n bac_n and_o edf_o then_o i_o say_v that_o the_o circumference_n bkc_n be_v equal_a to_o the_o circumference_n elf_n draw_v by_o the_o first_o petition_n right_a line_n from_o b_o to_o c_z construction_n and_o from_o e_o to_o f._n and_o for_o asmuch_o as_o the_o circle_n abc_n and_o def_n be_v equal_a the_o right_a line_n also_o draw_v from_o their_o centre_n to_o their_o circumference_n demonstration_n be_v by_o the_o first_o definition_n of_o the_o three_o equal_v the_o one_o to_o the_o other_o wherefore_o these_o two_o line_n bg_o and_o gc_o be_v equal_a to_o these_o two_o line_n eh_o and_o hf._n and_o the_o angle_n at_o the_o point_n g_z be_v equal_a to_o the_o angle_n at_o the_o point_n h_o wherefore_o by_o the_o 4._o of_o the_o first_o the_o base_a bc_o be_v equal_a to_o the_o base_a ef._n and_o for_o asmuch_o as_o the_o angle_n at_o the_o point_n a_o be_v equal_a to_o the_o angle_n at_o the_o point_n d_o therefore_o the_o segment_n bac_n be_v like_a to_o the_o segment_n edf_o and_o they_o be_v describe_v upon_o equal_a right_a line_n bc_o and_o ef._n but_o like_a segment_n of_o circle_n describe_v upon_o equal_a right_a line_n be_v by_o the_o 24._o of_o the_o three_o equal_v the_o one_o to_o the_o other_o wherefore_o the_o segment_n bac_n be_v equal_a to_o the_o segment_n edf_o and_o the_o whole_a circle_n abc_n be_v equal_a to_o the_o whole_a circle_n def_n wherefore_o by_o the_o three_o common_a sentence_n the_o circumference_n remain_v bkc_n be_v equal_a to_o the_o circumference_n remain_v elf_n wherefore_o equal_a angle_n in_o equal_a circle_n consist_v in_o equal_a circumference_n whether_o the_o angle_n be_v draw_v
construction_n draw_v by_o the_o 17._o of_o the_o three_o a_o right_a line_n touch_v the_o circle_n abc_n and_o let_v the_o same_o be_v gah_n and_o let_v it_o touch_v in_o the_o point_n a._n and_o by_o the_o 23._o of_o the_o first_o unto_o the_o right_a line_n ah_o and_o unto_o the_o point_n in_o it_o a_o describe_v a_o angle_n hac_fw-la equal_a unto_o the_o angle_n def_n and_o by_o the_o self_n same_o unto_o the_o right_a line_n agnostus_n and_o unto_o the_o point_n in_o it_o a_o make_v a_o angle_n gab_n equal_a unto_o the_o angle_n dfe_n and_o draw_v a_o right_a line_n from_o b_o to_o c._n and_o for_o asmuch_o as_o a_o certain_a right_a line_n gah_n touch_v the_o circle_n abc_n demonstration_n and_o from_o the_o point_n where_o it_o touch_v namely_o a_o be_v draw_v into_o the_o circle_n a_o right_a line_n ac_fw-la therefore_o by_o the_o 31._o of_o the_o three_o the_o angle_n hac_fw-la be_v equal_a unto_o the_o angle_n abc_n which_o be_v in_o the_o alternate_a segment_n of_o the_o circle_n but_o the_o angle_n hac_fw-la be_v equal_a to_o the_o angle_n def_n wherefore_o the_o angle_n abc_n be_v equal_a to_o the_o angle_n def_n and_o by_o the_o same_o reason_n the_o angle_n acb_o be_v equal_a to_o the_o angle_n dfe_n wherefore_o the_o angle_n remain_v bac_n be_v equal_a unto_o the_o angle_n remain_v edf_o wherefore_o the_o triangle_n abc_n be_v equiangle_n unto_o the_o triangle_n def_n and_o it_o be_v describe_v in_o the_o circle_n give_v abc_n wherefore_o in_o a_o circle_n give_v be_v describe_v a_o triangle_n equiangle_n unto_o a_o triangle_n give_v which_o be_v require_v to_o be_v do_v the_o 3._o problem_n the_o 3._o proposition_n about_o a_o circle_n give_v to_o describe_v a_o triangle_n equiangle_n unto_o a_o triangle_n give_v svppose_v that_o the_o circle_n give_v be_v abc_n and_o let_v the_o triangle_n give_v be_v def_n it_o be_v require_v about_o the_o circle_n abc_n to_o describe_v a_o triangle_n equiangle_n unto_o the_o triangle_n def_n construction_n extend_v the_o line_n of_o on_o each_o side_n to_o the_o point_n g_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 then_o draw_v a_o right_a line_n kb_o and_o by_o the_o 23._o of_o the_o first_o unto_o the_o right_a line_n kb_o and_o unto_o the_o point_n in_o it_o king_n make_v a_o angle_n bka_fw-mi equal_a unto_o the_o angle_n deg_n and_o likewise_o make_v the_o angle_n bkc_n equal_a unto_o the_o angle_n dfh._n and_o by_o the_o 17._o of_o the_o three_o draw_v right_a line_n touch_v the_o circle_n a_o b_o c_o in_o the_o point_n a_o b_o c._n and_o let_v the_o same_o be_v lam_n mbn_n and_o ncl_n demonstration_n and_o for_o asmuch_o as_o the_o right_a line_n lm_o mn_a &_o nl_o do_v touch_v the_o circle_n abc_n in_o the_o point_n a_o b_o c_o and_o from_o the_o centre_n king_n unto_o the_o point_n a_o b_o c_o be_v draw_v right_a line_n ka_o kb_o and_o kc_o therefore_o the_o angle_n which_o be_v at_o the_o point_n a_o b_o c_o be_v right_a angle_n by_o the_o 18._o of_o the_o three_o and_o for_o asmuch_o as_o the_o four_o angle_n of_o the_o four_o side_v figure_n ambk_n be_v equal_a unto_o four_o right_a angle_n who_o angle_n kam_fw-mi &_o kbm_n be_v two_o right_a angle_n therefore_o the_o angle_n remain_v akb_n and_o ambiguity_n be_v equal_a to_o two_o right_a angle_n and_o the_o angle_n deg_n &_o def_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 wherefore_o the_o angle_n akb_n and_o ambiguity_n be_v equal_a unto_o the_o angle_n deg_n and_o def_n of_o which_o two_o angle_n the_o angle_n akb_n be_v equal_a unto_o the_o angle_n deg_n wherefore_o the_o angle_n remain_v amb_n be_v equal_a unto_o the_o angle_n remain_v def_n in_o like_a sort_n may_v it_o be_v prove_v that_o the_o angle_n lnm_n be_v equal_a to_o the_o angle_n dfe_n wherefore_o the_o angle_n remain_v mln_v be_v equal_a unto_o the_o angle_n remain_v edf_o wherefore_o the_o triangle_n lmn_o be_v equiangle_n unto_o the_o triangle_n def_n and_o it_o be_v describe_v about_o the_o circle_n abc_n wherefore_o about_o a_o circle_n give_v be_v describe_v a_o triangle_n equiangle_n unto_o a_o triangle_n give_v which_o be_v require_v to_o be_v do_v ¶_o a_o other_o way_n after_o pelitarius_n peliâ—arius_n in_o the_o circle_n abc_n inscribe_v a_o triangle_n ghk_v equiangle_n to_o the_o triangle_n edf_o by_o the_o former_a proposition_n so_o that_o let_v the_o angle_n at_o the_o point_n g_o be_v equal_a to_o the_o angle_n d_o and_o let_v the_o angle_n at_o the_o point_n h_o be_v equal_a to_o the_o angle_n e_o and_o let_v also_o the_o angle_n at_o the_o point_n king_n be_v equal_a to_o the_o angle_n f._n then_o draw_v the_o line_n lm_o parallel_n to_o the_o line_n gh_o which_o let_v touch_v the_o circle_n in_o the_o point_n a_o which_o may_v be_v do_v by_o the_o proposition_n add_v of_o the_o say_v pelitarius_n after_o the_o 17._o proposition_n construction_n draw_v likewise_o the_o line_n mn_a parallel_n unto_o the_o line_n hk_o and_o touch_v the_o circle_n in_o the_o point_n b_o and_o also_o draw_v the_o line_n ln_o parallel_n unto_o the_o line_n gk_o and_o touch_v the_o circle_n in_o the_o point_n c._n and_o these_o three_o line_n shall_v undoubted_o concur_v as_o in_o the_o point_n l_o m_o and_o n_o demonstration_n which_o may_v easy_o be_v prove_v if_o you_o produce_v on_o either_o side_n the_o line_n gh_o gk_o and_o hk_o until_o they_o cut_v the_o line_n lm_o ln_o and_o mn_v in_o the_o point_n oh_o p_o q_o r_o s_o t._n now_o i_o say_v that_o the_o triangle_n lmn_o circumscribe_v about_o the_o circle_n abc_n be_v equiangle_n to_o the_o triangle_n def_n for_o it_o be_v manifest_a that_o it_o be_v equiangle_n unto_o the_o triangle_n ghk_o by_o the_o propriety_n of_o parallel_a line_n for_o the_o angle_n mtq_n be_v equal_a to_o the_o angle_n at_o the_o point_n g_o of_o the_o triangle_n ghk_o by_o the_o 29._o of_o the_o first_o and_o therefore_o also_o the_o angle_n at_o the_o point_n l_o be_v equal_a to_o the_o self_n same_o angle_n at_o the_o point_n g_z for_o the_o angle_n at_o the_o point_n l_o be_v by_o the_o same_o 29._o proposition_n equal_a to_o the_o angle_n mtq_n and_o by_o the_o same_o reason_n the_o angle_n at_o the_o point_n m_o be_v equal_a to_o the_o angle_n at_o the_o point_n h_o of_o the_o self_n same_o triangle_n and_o the_o angle_n at_o the_o point_n n_o to_o the_o angle_n at_o the_o point_n k._n wherefore_o the_o whole_a triangle_n lmn_o be_v equiangle_n to_o the_o whole_a triangle_n ghk_o wherefore_o also_o it_o be_v equiangle_n to_o the_o triangle_n def_n which_o be_v require_v to_o be_v do_v the_o 4._o problem_n the_o 4._o proposition_n in_o a_o triangle_n give_v to_o describe_v a_o circle_n svppose_v that_o the_o triangle_n give_v be_v abc_n it_o be_v require_v to_o describe_v a_o circle_n in_o the_o triangle_n abc_n construction_n divide_v by_o the_o 9_o of_o the_o first_o the_o angle_n abc_n and_o acb_o into_o two_o equal_a parâ—es_n by_o two_o right_a line_n bd_o and_o cd_o and_o let_v these_o right_a line_n meet_v together_o in_o the_o point_n d._n and_o by_o the_o 12._o of_o the_o first_o from_o the_o point_n d_o draw_v unto_o the_o right_a line_n ab_fw-la bc_o and_o ca_n perpendicular_a line_n namely_o de_fw-fr df_o and_o dg_o demonstration_n and_o forasmuch_o as_o the_o angle_n abdella_n be_v equal_a to_o though_o angle_v cbd_v and_o the_o right_a angle_n bed_n be_v equal_a unto_o the_o right_a angle_n bfd_n now_o then_o there_o be_v two_o triangle_n ebb_v and_o fbd_v have_v two_o angle_n equal_a to_o two_o angle_n and_o one_o side_n equal_a to_o one_o side_n namely_o bd_o which_o be_v common_a to_o they_o both_o and_o subtend_v one_o of_o the_o equal_a angle_n wherefore_o by_o the_o 26._o of_o the_o first_o the_o rest_n of_o the_o side_n be_v equal_a unto_o the_o rest_n of_o the_o side_n wherefore_o the_o line_n de_fw-fr be_v equal_a unto_o the_o line_n df_o and_o by_o the_o same_o reason_n also_o the_o â—ne_a dg_o be_v equal_a unto_o the_o line_n df._n wherefore_o these_o three_o right_a line_n de_fw-fr df_o &_o dg_o be_v equal_a the_o one_o to_o the_o other_o by_o the_o first_o common_a sentence_n wherefore_o make_v the_o centre_n the_o point_n d_o and_o the_o space_n de_fw-fr or_o df_o or_o dg_o describe_v a_o circle_n and_o it_o will_v pass_v through_o the_o point_n e_o f_o g_o and_o will_v touch_v the_o right_a line_n ab_fw-la bc_o and_o ca._n for_o the_o angle_n make_v at_o the_o point_n e_o f_o g_o be_v right_a angle_n for_o if_o the_o circle_z cut_v those_o right_a line_n then_o from_o the_o end_n of_o
the_o diameter_n of_o the_o circle_n shall_v be_v draw_v a_o right_a line_n make_v two_o right_a angle_n impossibility_n &_o fall_v within_o the_o circle_n which_o be_v impossible_a as_o it_o be_v manifest_a by_o the_o 16._o of_o the_o three_o wherefore_o the_o circle_n describe_v d_z be_v the_o centre_n thereof_o and_o the_o space_n thereof_o be_v either_o de_fw-fr or_o df_o or_o dg_o cut_v not_o these_o right_a line_n ab_fw-la bc_o &_o ca._n wherefore_o by_o the_o corollary_n of_o the_o same_o it_o touch_v they_o and_o the_o circle_n be_v describe_v in_o the_o triangle_n abc_n wherefore_o in_o the_o triangle_n give_v abc_n be_v describe_v a_o circle_n efg_o which_o be_v require_v to_o be_v do_v the_o 5._o problem_n the_o 5._o proposition_n about_o a_o triangle_n give_v to_o describe_v a_o circle_n svppose_v that_o the_o triangle_n give_v be_v abc_n it_o be_v require_v about_o the_o triangle_n abc_n to_o describe_v a_o circle_n divide_v by_o the_o 10._o of_o the_o first_o the_o right_a lâ—nes_fw-la ab_fw-la and_o ac_fw-la into_o two_o equal_a part_n in_o the_o point_n d_o and_o e._n and_o from_o the_o point_n d_o and_o e_o by_o the_o 11._o of_o the_o first_o raise_v up_o unto_o the_o line_n ab_fw-la &_o ac_fw-la two_o perpendicular_a line_n df_o and_o ef._n proposition_n now_o these_o perpendicular_a line_n meet_v together_o either_o within_o the_o triangle_n abc_n or_o in_o the_o right_a line_n bc_o or_o else_o without_o the_o right_a line_n bc._n but_o now_o suppose_v that_o the_o right_a line_n df_o and_o of_o do_v meet_v together_o without_o the_o triangle_n abc_n in_o the_o point_n f._n case_n again_o as_o it_o be_v in_o the_o three_o description_n draw_v right_a line_n from_o fletcher_n to_o a_o from_o fletcher_n to_o b_o and_o from_o fletcher_n to_o c._n and_o forasmuch_o as_o the_o line_n ad_fw-la be_v equal_a unto_o the_o line_n db_o and_o the_o line_n df_o be_v common_a unto_o they_o both_o and_o make_v a_o right_a angle_n on_o each_o side_n of_o he_o wherefore_o by_o the_o 4._o of_o the_o first_o the_o base_a of_o be_v equal_a unto_o the_o base_a bf_o and_o in_o like_a sort_n may_v we_o prove_v that_o thâ—_z line_n cf_o be_v equal_a unto_o the_o line_n af._n wherefore_o again_o make_v fletcher_n the_o centre_n and_o the_o space_n favorina_n or_o fb_o or_o fc_n describe_v a_o circle_n and_o it_o shall_v pass_v by_o the_o point_n a_o b_o c_o and_o so_o be_v there_o a_o circle_n describe_v about_o the_o triangle_n abc_n as_o you_o see_v it_o iâ—_z in_o the_o three_o description_n wherefore_o about_o a_o triangle_n give_v be_v describe_v a_o circle_n which_o be_v require_v to_o be_v do_v correlary_a hereby_o it_o be_v manifest_a that_o when_o the_o centre_n of_o the_o circle_n fall_v within_o the_o triangle_n the_o angle_n bac_n be_v in_o a_o great_a segment_n of_o a_o circle_n be_v less_o they_o a_o right_a angle_n but_o when_o it_o fall_v upon_o the_o right_a line_n bc_o the_o angle_n bac_n be_v in_o a_o semicircle_n be_v a_o right_a angle_n but_o when_o the_o centre_n fall_v without_o the_o right_a line_n bc_o the_o angle_n bac_n be_v in_o a_o less_o segment_n of_o a_o circle_n be_v great_a than_o a_o right_a angle_n wherefore_o also_o when_o the_o angle_n give_v be_v less_o than_o a_o right_a angle_n the_o right_a line_n df_o and_o of_o will_v meet_v together_o within_o the_o say_a triangle_n but_o when_o it_o be_v a_o right_a angle_n they_o will_v meet_v together_o upon_o the_o line_n bc._n but_o when_o it_o be_v great_a than_o a_o right_a angle_n they_o will_v meet_v together_o without_o the_o right_a line_n bc._n the_o 6._o problem_n the_o 6._o proposition_n in_o a_o circle_n give_v to_o describe_v a_o square_n svppose_v that_o the_o circle_n give_v be_v abcd._n it_o be_v require_v in_o the_o circle_n abcd_o to_o describe_v a_o square_n draw_v in_o the_o circle_n abcd_o two_o diameter_n make_v right_a angle_n construction_n and_o let_v the_o same_o be_v ac_fw-la and_o bd_o and_o draw_v right_a line_n from_o a_o to_o b_o from_o b_o to_o c_z from_o c_z to_o d_z and_o from_o d_z to_o a._n and_o forasmuch_o as_o the_o line_n be_v be_v equal_a unto_o the_o line_n ed_z by_o the_o 15._o definition_n of_o the_o first_o for_o the_o point_n e_o be_v the_o centre_n demonstration_n and_o the_o line_n ea_fw-la be_v common_a to_o they_o both_o make_v on_o each_o side_n a_o right_a angle_n therefore_o by_o the_o 4._o of_o the_o first_o the_o base_a ab_fw-la be_v equal_a unto_o the_o base_a ad._n and_o by_o the_o same_o reason_n also_o either_o of_o these_o line_n bc_o and_o cd_o be_v equal_a to_o either_o of_o these_o line_n ab_fw-la and_o ad_fw-la wherefore_o abcd_o be_v a_o figure_n of_o four_o equal_a side_n i_o say_v also_o that_o it_o be_v a_o rectangle_n figure_n for_o forasmuch_o as_o the_o right_a line_n bd_o be_v the_o diameter_n of_o the_o circle_n abcd_o therefore_o the_o angle_n bid_v be_v in_o the_o semicircle_n be_v a_o right_a angle_n by_o the_o 31._o of_o the_o three_o and_o by_o the_o same_o reason_n every_o one_o of_o these_o angle_n abc_n bcd_o and_o cda_n be_v a_o right_a angle_n wherefore_o the_o four_o side_v figure_n abcd_o be_v a_o rectangle_n figure_n and_o it_o be_v prove_v that_o it_o consist_v of_o equal_a side_n wherefore_o by_o the_o 30._o definition_n of_o the_o first_o it_o be_v a_o square_a and_o it_o be_v describe_v in_o the_o circle_n abcd_o which_o be_v require_v to_o be_v do_v the_o 7._o problem_n the_o 7._o proposition_n about_o a_o circle_n give_v to_o describe_v a_o square_n svppose_v that_o the_o circle_n give_v be_v abcd._n it_o be_v require_v about_o the_o circle_n abcd_o to_o describe_v a_o square_n draw_v in_o the_o circle_n abcd_o two_o diameter_n make_v right_a angle_n construction_n where_o they_o cut_v the_o one_o the_o other_o and_o let_v the_o same_o be_v ac_fw-la and_o bd._n and_o by_o the_o point_n a_o b_o c_o d_o draw_v by_o the_o 17._o of_o the_o three_o right_a line_n touch_v the_o circle_n abcd_o and_o let_v the_o same_o be_v fg_o gh_o hk_o and_o kf_n demonstration_n now_o for_o asmuch_o as_o the_o right_a line_n fg_o touch_v the_o circle_n abcd_o in_o the_o point_n a_o and_o from_o the_o centre_n e_o to_o the_o point_n a_o where_o the_o touch_n be_v be_v draw_v a_o right_a line_n ea_fw-la therefore_o by_o the_o 18._o of_o the_o three_o the_o angle_n at_o the_o point_n a_o be_v right_a angle_n and_o by_o the_o same_o reason_n the_o angle_n which_o be_v at_o the_o point_n b_o c_o d_o be_v also_o right_a angle_n and_o forasmuch_o as_o the_o angle_n aeb_fw-mi be_v a_o right_a angle_n &_o the_o angle_n ebg_n be_v also_o a_o right_a angle_n therefore_o by_o the_o 28._o of_o the_o first_o the_o line_n gh_o be_v a_o parallel_n unto_o the_o line_n ac_fw-la and_o by_o the_o same_o reason_n the_o line_n ac_fw-la be_v a_o parallel_n unto_o the_o line_n fk_o in_o like_a sort_n also_o may_v we_o prove_v that_o either_o of_o these_o line_n gf_o and_o hk_o be_v a_o parallel_n unto_o the_o line_n bed_n wherefore_o these_o figure_n gk_o gc_o ak_o fb_o and_o bk_o be_v parallelogramesâ—_n wherefore_o by_o the_o 34._o of_o the_o first_o the_o line_n gf_o be_v equal_a unto_o the_o line_n hk_o and_o the_o line_n gh_o be_v equal_a unto_o the_o line_n fk_o and_o forasmuch_o as_o the_o line_n ac_fw-la be_v equal_a unto_o the_o line_n bd_o but_o the_o line_n ac_fw-la be_v equal_a unto_o either_o of_o these_o line_n gh_o and_o fk_v anâ—_z the_o line_n bd_o be_v equal_a to_o either_o of_o these_o line_n gf_o and_o hk_o wherefore_o either_o of_o these_o line_n gh_o and_o fk_o be_v equal_a to_o either_o of_o these_o line_n gf_o and_o hk_o wherefore_o the_o figure_n fghk_n consist_v of_o four_o equal_a side_n i_o say_v also_o that_o it_o be_v a_o rectangle_n figure_n for_o forasmuch_o as_o gbea_n be_v a_o parallelogram_n and_o the_o angle_n aeâ—_n be_v a_o right_a angle_n therefore_o by_o the_o 34._o of_o the_o first_o the_o angle_n agb_n be_v a_o right_a angle_n in_o like_a sort_n may_v we_o prove_v that_o the_o angle_n at_o the_o point_n h_o edward_n and_o fletcher_n be_v right_a angle_n wherefore_o fghk_n be_v a_o rectangle_n four_o side_v figure_n and_o it_o be_v prove_v that_o it_o consist_v of_o equal_a side_n wherefore_o it_o be_v a_o square_a and_o it_o be_v describe_v about_o the_o circle_n abcd._n wherefore_o about_o a_o circle_n give_v be_v describe_v a_o square_n which_o be_v require_v to_o be_v do_v the_o 8._o problem_n the_o 8._o proposition_n in_o a_o square_n give_v to_o describe_v a_o circle_n svppose_v that_o the_o square_n give_v be_v abcd._n it_o be_v require_v in_o the_o square_n abcd_o to_o describe_v a_o circle_n construction_n divide_v by_o the_o 10._o
problem_n the_o 12._o proposition_n about_o a_o circle_n give_v to_o describe_v a_o equilater_n and_o aquiangle_n pentagon_n svppose_v that_o the_o circle_n give_v be_v abcde_o it_o be_v require_v about_o the_o circle_n abcde_o to_o describe_v a_o figure_n of_o five_o angle_n consist_v of_o equal_a side_n and_o of_o equal_a angle_n construction_n take_v the_o point_n of_o the_o angle_n of_o a_o five_o angle_a figure_n describe_v by_o the_o 11._o of_o the_o four_o so_o that_o by_o the_o proposition_n go_v before_o let_v the_o circumference_n ab_fw-la bc_o cd_o de_fw-fr and_o ea_fw-la be_v equal_a the_o one_o to_o the_o other_o and_o by_o the_o point_n a_o b_o c_o d_o e_o draw_v by_o the_o 17._o of_o the_o three_o right_a line_n touch_v the_o circle_n and_o let_v the_o same_o be_v gh_o hk_o kl_o lm_o and_o mg_o and_o by_o the_o 1._o of_o the_o three_o take_v the_o centre_n of_o the_o circle_n &_o let_v the_o same_o be_v f._n and_o draw_v right_a line_n from_o fletcher_n to_o b_o from_o fletcher_n to_o king_n from_o fletcher_n to_o c_o from_o fletcher_n to_o l_o and_o from_o fletcher_n to_o d._n demonstration_n and_o forasmuch_o as_o the_o right_a line_n kl_o touch_v the_o circle_n abcde_o in_o the_o point_n c_o and_o from_o the_o centre_n f_o unto_o the_o point_n c_o where_o the_o touch_n be_v be_v draw_v a_o right_a line_n fc_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 unâ—o_v kl_o wherefore_o either_o of_o the_o angle_n which_o be_v at_o the_o point_n c_o be_v a_o right_a angle_n and_o by_o the_o same_o reason_n the_o angle_n which_o be_v at_o the_o point_n d_o and_o b_o be_v right_a angle_n and_o forasmuch_o as_o the_o angle_n fck_n be_v a_o right_a angle_n therefore_o the_o square_n which_o be_v make_v of_o fk_n be_v by_o the_o 47._o of_o the_o first_o equal_a to_o the_o square_n which_o be_v make_v of_o fc_o and_o ck_o and_o by_o the_o same_o reason_n also_o the_o square_n which_o be_v make_v of_o fk_n be_v equal_a to_o the_o square_n which_o be_v make_v of_o fb_o and_o bk_o wherefore_o the_o square_n which_o be_v make_v of_o fc_o and_o ck_o be_v equal_a to_o the_o square_n which_o be_v make_v of_o fb_o and_o bk_o of_o which_o the_o square_a which_o be_v make_v of_o fc_n be_v equal_a to_o the_o square_n which_o be_v make_v of_o fb_o wherefore_o the_o square_a which_o be_v make_v of_o ck_o be_v equal_a to_o the_o square_n which_o be_v make_v of_o bk_o wherefore_o the_o line_n bk_o be_v equal_a unto_o the_o line_n ck_o and_o forasmuch_o as_o fb_o be_v equal_a unto_o fc_n and_o fk_n be_v common_a to_o they_o both_o therefore_o these_o two_o bf_o and_o fk_o be_v equal_a to_o these_o two_o cf_o &_o fk_o and_o the_o base_a bk_o be_v equal_a unto_o the_o base_a ck_n wherefore_o by_o the_o 8._o of_o the_o first_o the_o angle_n bfk_n be_v equal_a unto_o the_o angle_n kfc_n and_o the_o angle_n bkf_n to_o the_o angle_n fkc_n wherefore_o the_o angle_n bfc_n be_v double_a to_o the_o angle_n kfc_n and_o the_o angle_n bkc_n be_v double_a to_o the_o angle_n fkc_n and_o by_o the_o same_o reason_n the_o angle_n cfd_n be_v double_a to_o the_o angle_n cfl_fw-mi and_o the_o angle_n dlc_n be_v double_a to_o the_o angle_n flc_n and_o forasmuch_o as_o the_o circumference_n bc_o be_v equal_a unto_o the_o circumference_n cd_o therefore_o by_o the_o 27._o of_o the_o three_o the_o angle_n bfc_n be_v equal_a to_o the_o angle_n cfd_n and_o the_o angle_n bfc_n be_v double_a to_o the_o angle_n kfc_n and_o the_o angle_n dfc_n be_v double_a to_o the_o angle_n lfc_n wherefore_o the_o angle_n kfc_n be_v equal_a unto_o the_o angle_n lfc_n now_o then_o there_o be_v two_o triangle_n fkc_n and_o flc_n have_v two_o angle_n equal_a to_o two_o angle_n and_o one_o side_n equal_a to_o one_o side_n namely_o fc_o which_o be_v common_a to_o they_o both_o wherefore_o by_o the_o 26._o of_o the_o three_o the_o other_o side_n remain_v be_v equal_a unto_o the_o side_n remain_v and_o the_o angle_n remain_v unto_o the_o angle_n remain_v wherefore_o the_o right_a line_n kc_o be_v equal_a to_o the_o right_a line_n cl_n and_o the_o angle_n fkc_n to_o the_o angle_n flc_n and_o forasmuch_o as_o kc_o be_v equal_a to_o cl_n therefore_o kl_o be_v double_a to_o kc_o and_o by_o the_o same_o reason_n also_o may_v it_o be_v prove_v that_o hk_n be_v double_a to_o bk_o and_o forasmuch_o as_o it_o be_v prove_v that_o bk_o be_v equal_a unto_o kc_o and_o kl_o be_v double_a to_o kc_o and_o h_o king_n double_a to_o b_o edward_n therefore_o h_o king_n be_v equal_a unto_o kl_o in_o like_a sort_n may_v we_o prove_v that_o every_o one_o of_o these_o line_n hg_o gm_n &_o ml_o be_v equal_a unto_o either_o of_o these_o line_n h_o edward_n and_o kl_o wherefore_o the_o five_o angle_a figure_n gh_o klm_a be_v of_o equal_a side_n i_o say_v also_o that_o it_o be_v of_o equal_a angle_n for_o forasmuch_o as_o the_o angle_n f_o kc_o be_v equal_a unto_o the_o angle_n flc_n and_o it_o be_v prove_v that_o the_o angle_n h_o kl_o be_v double_a to_o the_o angle_n f_o kc_o and_o the_o angle_n klm_n be_v double_a to_o the_o angle_n flc_n therefore_o the_o angle_n h_o kl_o be_v equal_a to_o the_o angle_n klm_n in_o like_a sort_n may_v it_o be_v prove_v that_o every_o one_o of_o these_o angle_n khg_n hgm_n and_o gml_n be_v equal_a to_o either_o of_o these_o angle_n h_o kl_o and_o klm_n wherefore_o the_o five_o angle_n gh_o king_n h_o kl_o klm_n lmg_n and_o mgh_n be_v equal_a the_o one_o to_o the_o other_o wherefore_o the_o five_o angle_a figure_n gh_o klm_a be_v equiangle_n and_o it_o be_v also_o prove_v that_o it_o be_v equilater_n and_o it_o be_v describe_v about_o the_o circle_n abcde_o which_o be_v require_v to_o be_v do_v ¶_o a_o other_o way_n to_o do_v the_o same_o after_o pelitarius_n by_o parallel_a line_n suppose_v that_o the_o circle_n give_v be_v abc_n pelitarius_n who_o centre_n let_v be_v the_o point_n f_o and_o in_o it_o by_o the_o former_a proposition_n inscribe_v a_o equilater_n and_o equiangle_n pentagon_n abcdeâ—_n by_o who_o five_o angle_n draw_v from_o the_o centre_n beyond_o the_o circumference_n five_o line_n fg_o fh_o fk_o fl_fw-mi and_o fm_o and_o it_o be_v manife_a that_o the_o five_o angle_n at_o the_o centre_n fletcher_n be_v equal_a when_o as_o the_o five_o side_n of_o the_o triangle_n within_o be_v equal_a and_o also_o their_o base_n it_o be_v manifest_a also_o that_o thâ—_z five_o angle_n of_o the_o pentagon_n which_o be_v at_o the_o circumference_n be_v divide_v into_o â—â—n_fw-la equal_a â—ngles_n by_o the_o 4._o of_o the_o firâ—â—_n now_o then_o between_o the_o two_o line_n fg_o and_o fh_o draw_v the_o line_n gh_o parallel_n to_o the_o side_n ab_fw-la and_o touch_v the_o circle_n abc_n which_o be_v do_v by_o a_o proposition_n add_v by_o pelitarius_n after_o the_o 17._o of_o the_o three_o and_o so_o likewise_o draw_v these_o line_n hk_o kl_o and_o lm_o parallel_n to_o each_o of_o these_o side_n bc_o cd_o and_o de_fw-fr and_o touch_v the_o circle_n and_o for_o asmuch_o as_o the_o line_n fg_o and_o fh_o fall_v upon_o the_o two_o parallel_a line_n ab_fw-la and_o gh_v demonstration_n the_o two_o angle_n fgh_o &_o fhg_n be_v equal_a to_o the_o two_o angle_n fab._n and_o fba_n the_o one_o to_o the_o other_o by_o the_o 29._o of_o the_o first_o wherefore_o by_o the_o six_o of_o the_o same_o the_o two_o line_n fg_o and_o fh_o be_v equal_a and_o by_o the_o same_o reason_n the_o two_o angle_n fhk_v &_o fkh_n be_v equal_a to_o the_o two_o angle_n fgh_a and_o fhg_n the_o one_o to_o the_o other_o and_o the_o line_n fk_o be_v equal_a to_o the_o line_n fh_o and_o therefore_o be_v equal_a to_o the_o line_n fg._n and_o forasmuch_o as_o the_o angle_n at_o the_o point_n f_o be_v equal_a therefore_o by_o the_o 4._o of_o the_o first_o the_o base_a hk_n be_v equal_a to_o the_o base_a gh_o in_o like_a sort_n may_v we_o prove_v that_o the_o three_o line_n fk_o fl_fw-mi and_o fm_o be_v equal_a to_o the_o two_o line_n fg_o and_o fh_o and_o also_o that_o the_o two_o base_n kl_o and_o lm_o be_v equal_a to_o the_o two_o base_n gh_o and_o hk_v and_o that_o the_o angle_n which_o they_o make_v with_o the_o line_n fk_o fl_fw-mi and_o fm_o be_v equal_a the_o one_o to_o the_o other_o now_o then_o draw_v the_o five_o line_n mg_o which_o shall_v be_v equal_a to_o the_o four_o former_a line_n by_o the_o 4._o of_o the_o first_o for_o that_o as_o we_o have_v prove_v the_o two_o line_n fg_o &_o fm_o be_v equal_a &_o the_o angle_n gfm_n be_v equal_a to_o every_o one_o of_o the_o angle_n at_o the_o point_n f._n this_o line_n also_o mg_o
5._o and_o 32._o of_o the_o first_o the_o angle_n bfc_n also_o shall_v be_v the_o three_o part_n of_o two_o right_a angle_n wherefore_o either_o of_o the_o two_o angle_n remain_v fbc_n and_o fcb_n for_o asmuch_o as_o they_o be_v equal_a by_o the_o 5._o of_o the_o first_o shall_v be_v two_o three_o part_n of_o two_o right_a angle_n by_o the_o 32._o of_o the_o same_o or_o by_o the_o 4._o of_o the_o first_o forasmuch_o as_o the_o angle_n bfc_n be_v equal_a to_o the_o angle_n fba_n and_o the_o two_o side_n fb_o and_o fc_o be_v equal_a to_o the_o two_o side_n ab_fw-la and_o bf_o the_o base_a bc_o shall_v be_v equal_a to_o the_o base_a bf_o and_o therefore_o be_v equal_a to_o the_o line_n fc_o wherefore_o the_o triangle_n fbc_n be_v equilater_n and_o equiangle_n last_o make_v the_o angle_n cfd_v equal_a to_o either_o of_o the_o angle_n at_o the_o point_n fletcher_n by_o draw_v the_o line_n fd._n and_o draw_v a_o line_n from_o c_o to_o d._n now_o then_o by_o the_o former_a reason_n the_o triangle_n fcd_n shall_v be_v equilater_n and_o equiangle_n and_o for_o asmuch_o as_o the_o three_o angle_n at_o the_o point_n f_o be_v equal_a to_o two_o right_a angle_n for_o each_o of_o they_o be_v the_o three_o part_n of_o two_o right_a angle_n therefore_o by_o the_o 14._o of_o the_o first_o ad_fw-la be_n one_o right_a line_n and_o for_o that_o cause_n be_v the_o diameter_n of_o the_o circle_n wherefore_o if_o the_o other_o semicircle_n afd_v be_v divide_v into_o so_o many_o equal_a part_n as_o the_o semicircle_n abcd_o be_v divide_v into_o it_o shall_v comprehend_v so_o many_o equal_a line_n subtend_v unto_o it_o wherefore_o the_o line_n ab_fw-la be_v the_o side_n of_o a_o equilater_n hexagon_n figure_n to_o be_v inscribe_v in_o the_o circle_n which_o hexagon_n figure_n also_o shall_v be_v equiangle_n for_o the_o half_a of_o the_o whole_a angle_n b_o be_v equal_a to_o the_o half_a of_o the_o whole_a angle_n c_o which_o be_v require_v to_o be_v do_v now_o than_o if_o we_o draw_v from_o the_o centre_n f_o a_o perpendicular_a line_n unto_o ad_fw-la which_o let_v be_v fe_o and_o draw_v also_o these_o right_a line_n be_v and_o ce_fw-fr there_o shall_v be_v describe_v a_o triangle_n bec_n who_o angle_n e_o which_o be_v at_o the_o top_n shall_v be_v the_o 6._o part_n of_o two_o right_a angle_n by_o the_o 20._o of_o the_o three_o for_o the_o angle_n bfc_n be_v double_a unto_o it_o and_o either_o of_o the_o two_o angle_n at_o the_o base_a namely_o the_o angle_n ebc_n and_o ecb_n be_v dupla_fw-la sesquialter_fw-la to_o the_o angle_n e_o that_o be_v either_o of_o they_o contain_v the_o angle_n e_o twice_o and_o half_o the_o angle_n e._n and_o by_o this_o reason_n be_v find_v out_o the_o side_n of_o a_o hexagon_n figure_n correlary_a hereby_o it_o be_v manifest_a that_o the_o side_n of_o a_o hexagon_n figure_n describe_v in_o a_o circle_n be_v equal_a to_o a_o right_a line_n draw_v from_o the_o centre_n of_o the_o say_a circle_n unto_o the_o circumference_n and_o if_o by_o the_o point_n a_o b_o c_o d_o e_o f_o be_v draw_v right_a line_n touch_v the_o circle_n then_o shall_v there_o be_v describe_v about_o the_o circle_n a_o hexagon_n figure_n equilater_n and_o equiangle_n which_o may_v be_v demonstrate_v by_o that_o which_o have_v be_v speak_v of_o the_o describe_v of_o a_o pentagon_n about_o a_o circle_n and_o moreover_o by_o those_o thing_n which_o have_v be_v speak_v of_o pentagons_n we_o may_v in_o a_o hexagon_n give_v either_o describe_v or_o circumscribe_v a_o circle_n which_o be_v require_v to_o be_v do_v the_o 16._o problem_n the_o 16._o proposition_n in_o a_o circle_n give_v to_o describe_v a_o quindecagon_n or_o figure_n of_o fifteen_o angle_n equilater_n and_o equiangle_n svppose_v that_o the_o circle_n give_v be_v abcd._n it_o be_v require_v in_o the_o circle_n abcd_o to_o describe_v a_o figure_n of_o fifteen_o angle_n consist_v of_o equal_a side_n and_o of_o equal_a angle_n describe_v in_o the_o circle_n abcd_o the_o side_n of_o a_o equilater_n triangle_n and_o let_v the_o same_o be_v ac_fw-la and_o in_o the_o ark_n ac_fw-la describe_v the_o side_n of_o a_o equilater_n pentagon_n and_o let_v the_o same_o be_v ab_fw-la construction_n now_o then_o of_o such_o equal_a part_n whereof_o the_o whole_a circle_n abcd_o contain_v fifteen_o of_o such_o part_n i_o say_v the_o circumference_n abc_n be_v the_o three_o part_n of_o the_o circle_n shall_v contain_v five_o and_o the_o circumference_n ab_fw-la be_v the_o five_o part_n of_o a_o circle_n shall_v contain_v three_o wherefore_o the_o residue_n bc_o shall_v contain_v two_o divide_v by_o the_o 30._o of_o the_o first_o the_o ark_n bc_o into_o two_o equal_a part_n in_o the_o point_n e._n demonstration_n wherefore_o either_o of_o these_o circumference_n be_v &_o ec_o be_v the_o fifteen_o part_n of_o the_o circle_n abcd._n if_o therefore_o there_o be_v draw_v right_a line_n from_o b_o to_o e_o and_o from_o e_o to_o c_z and_o then_o begin_v at_o the_o point_n b_o or_o at_o the_o point_n c_o there_o be_v apply_v into_o the_o circle_n abcd_o right_a line_n equal_a unto_o ebb_n or_o ec_o and_o so_o continue_v till_o you_o come_v to_o the_o point_n c_o if_o you_o begin_v at_o b_o or_o to_o the_o point_fw-fr b_o if_o you_o begin_v at_o c_o and_o there_o shall_v be_v describe_v in_o the_o circle_n abcd_o a_o figure_n of_o fifteen_o angle_n equilater_n and_o equiangle_n which_o be_v require_v to_o be_v do_v and_o in_o like_a sort_n as_o in_o a_o pentagon_n if_o by_o the_o point_n where_o the_o circle_n be_v divide_v be_v draw_v right_a line_n touch_v the_o circle_n in_o the_o say_a point_n there_o shall_v be_v describe_v about_o the_o circle_n a_o figure_n of_o fifteen_o angle_n equilater_n &_o equiangle_n and_o in_o like_a sort_n by_o the_o self_n same_o observation_n that_o be_v in_o pentagons_n we_o may_v in_o a_o figure_n of_o fifteen_o angle_n give_v be_v equilater_n and_o equiangle_n either_o inscribe_v or_o circumscribe_v a_o circle_n flussates_n ¶_o a_o addition_n of_o flussates_n to_o find_v out_o infinite_a figure_n of_o many_o angle_n if_o into_o a_o circle_n from_o one_o point_n be_v apply_v the_o side_n of_o two_o side_n poligonon_n figure_n the_o excess_n of_o the_o great_a ark_n above_o the_o less_o shall_v comprehend_v a_o ark_n contain_v so_o many_o side_n of_o the_o poligonon_n figure_n to_o be_v inscribe_v by_o how_o many_o unity_n the_o denomination_n of_o the_o poligonon_n figure_n of_o the_o less_o side_n exceed_v the_o denomination_n of_o the_o poligonon_n figure_n of_o the_o great_a side_n and_o the_o number_n of_o the_o side_n of_o the_o poligonon_n figure_n to_o be_v inscribe_v be_v produce_v of_o the_o multiplication_n of_o the_o denomination_n of_o the_o foresay_a poligonon_n figure_v the_o one_o into_o the_o other_o as_o for_o example_n suppose_v that_o into_o the_o circle_n abe_n be_v apply_v the_o side_n of_o a_o equilater_n and_o equiangle_n hexagon_n figure_n by_o the_o 15._o of_o this_o book_n which_o let_v be_v ab_fw-la and_o likewise_o the_o side_n of_o a_o pentagon_n by_o the_o 11._o of_o this_o book_n which_o let_v be_v ac_fw-la and_o the_o side_n of_o a_o square_n by_o the_o 6._o of_o this_o book_n which_o let_v be_v ad_fw-la and_o the_o side_n of_o a_o equilater_n triangle_n by_o the_o 2._o of_o this_o book_n which_o let_v be_v ae_n then_o i_o say_v that_o the_o excess_n of_o the_o ark_n ad_fw-la above_o the_o ark_n ab_fw-la which_o excess_n be_v the_o ark_n bd_o contain_v so_o many_o side_n of_o the_o poligonon_n figure_n to_o be_v inscribe_v of_o how_o many_o unity_n the_o denominator_fw-la of_o the_o hexagon_n ab_fw-la which_o be_v six_o exceed_v the_o denominator_fw-la of_o the_o square_a ad_fw-la which_o be_v four_o and_o forasmuch_o as_o that_o excess_n it_o two_o unity_n therefore_o in_o bd_o there_o shall_v be_v two_o side_n and_o the_o denominator_fw-la of_o the_o poligonon_n figure_n which_o be_v to_o be_v inscribe_v shall_v be_v produce_v of_o the_o multiplication_n of_o the_o denominator_n of_o the_o foresay_a poligonon_n figure_n namely_o of_o the_o multiplication_n of_o 6._o into_o 4._o which_o make_v 24._o which_o number_n be_v the_o denominator_fw-la of_o the_o poligonon_n figure_n who_o two_o side_n shall_v subtend_v the_o ark_n bd._n for_o of_o such_o equal_a part_n whereof_o the_o whole_a circumference_n contain_v 24_o of_o such_o part_n i_o say_v the_o circumference_n ab_fw-la contain_v 4_o and_o the_o circumference_n ad_fw-la contain_v 6._o wherefore_o if_o from_o ad_fw-la which_o subtend_v 6._o part_n be_v take_v away_o 4._o which_o ab_fw-la subtend_v there_o shall_v remain_v unto_o bd_o two_o of_o such_o part_n of_o which_o the_o whole_a contain_v 24._o wherefore_o of_o a_o hexagon_n and_o a_o square_n be_v make_v a_o
line_n in_o proportion_n the_o rectangle_n figure_n comprehend_v under_o the_o extreme_n be_v equal_a to_o the_o rectangle_n figure_n contain_v under_o the_o mean_n and_o if_o the_o rectangle_n figure_n which_o be_v contain_v under_o the_o extreme_n be_v equal_a unto_o the_o rectangle_n figure_n which_o be_v contain_v under_o the_o mean_n then_o be_v those_o four_o line_n in_o proportion_v which_o be_v require_v to_o be_v prove_v the_o 12._o theorem_a the_o 17._o proposition_n if_o there_o be_v three_o right_a line_n in_o proportion_n the_o rectangle_n figure_n comprehend_v under_o the_o extreme_n be_v equal_a unto_o the_o square_a that_o be_v make_v of_o the_o mean_a and_o if_o the_o rectangle_n figure_n which_o be_v make_v of_o the_o extreme_n be_v equal_a unto_o the_o square_n make_v of_o the_o mean_a then_o be_v those_o three_o right_a line_n proportional_a svppose_v that_o there_o be_v three_o line_n in_o proportion_n a_o b_o c_o so_fw-mi that_o as_o a_o be_v to_o b_o so_o let_v b_o be_v to_o c._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 c_o be_v equal_a unto_o the_o square_v make_v of_o the_o line_n b._n theorem_a unto_o the_o line_n b_o by_o the_o 2._o of_o the_o first_o put_v a_o equal_a line_n d._n and_o because_o by_o supposition_n as_o a_o be_v to_o b_o so_o be_v b_o to_o c_z but_o b_o be_v equal_a unto_o d_o wherefore_o by_o the_o 7._o of_o the_o five_o as_o a_o be_v to_o b_o so_o be_v d_z to_o c_z but_o if_o there_o be_v four_o right_a line_n proportional_a the_o rectangle_n figure_n comprehend_v under_o the_o extreme_n be_v equal_a unto_o the_o rectangle_n figure_n comprehend_v under_o the_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 a_o and_o c_o be_v equal_a unto_o that_o which_o be_v comprehend_v under_o the_o line_n b_o and_o d._n but_o that_o which_o be_v contain_v under_o the_o line_n b_o and_o d_o be_v the_o square_a of_o the_o line_n b_o for_o the_o line_n b_o be_v equal_a unto_o the_o line_n d._n wherefore_o the_o rectangle_n figure_n comprehend_v under_o the_o line_n a_o and_o c_o be_v equal_a unto_o the_o square_n make_v of_o the_o line_n b._n first_o but_o now_o suppose_v that_o that_o which_o be_v comprehend_v under_o the_o line_n a_o &_o c_o be_v equal_a unto_o the_o square_n make_v of_o the_o line_n b._n then_o also_o i_o say_v that_o as_o the_o line_n a_o be_v to_o the_o line_n b_o so_o be_v the_o line_n b_o to_o the_o line_n c._n the_o same_o order_n of_o construction_n that_o be_v before_o be_v keep_v forasmuch_o as_o that_o which_o be_v contain_v under_o the_o line_n a_o and_o c_o be_v equal_a unto_o the_o square_n which_o be_v make_v of_o the_o line_n b._n but_o the_o square_n which_o be_v make_v of_o the_o line_n b_o be_v that_o which_o be_v contain_v under_o the_o line_n b_o &_o d_o for_o the_o line_n b_o be_v put_v equal_a unto_o the_o line_n d._n wherefore_o that_o which_o be_v contain_v under_o the_o line_n a_o and_o c_o be_v equal_a unto_o that_o which_o be_v contain_v under_o the_o line_n b_o and_o d._n but_o if_o the_o rectangle_n figure_n comprehend_v under_o the_o extreme_n be_v equal_a unto_o the_o rectangle_n figure_n comprehend_v under_o the_o mean_a line_n the_o four_o right_a line_n shall_v be_v proportional_a by_o the_o 16._o of_o the_o six_o wherefore_o as_o the_o line_n a_o be_v to_o the_o line_n b_o so_o be_v the_o line_n d_o to_o the_o line_n c._n but_o the_o line_n b_o be_v equal_a unto_o the_o line_n d._n wherefore_o as_o the_o line_n a_o be_v to_o the_o line_n b_o so_o be_v b_o to_o the_o line_n c._n if_o therefore_o there_o be_v three_o right_a line_n in_o proportion_n the_o rectangle_n figure_n comprehend_v under_o the_o extreme_n be_v equal_a unto_o the_o square_a that_o be_v make_v of_o the_o mean_a and_o if_o the_o rectangle_n figure_n which_o be_v contain_v under_o the_o extreme_n be_v equal_a unto_o the_o square_n make_v of_o the_o mean_a then_o be_v those_o three_o right_a line_n proportional_a which_o be_v require_v to_o be_v demonstrate_v corollary_n ¶_o corollary_n add_v by_o flussates_n hereby_o we_o gather_v that_o every_o right_a line_n be_v a_o mean_a proportional_a between_o every_o two_o right_a line_n which_o make_v a_o rectangle_n figure_n equal_a to_o the_o square_n of_o the_o same_o right_a line_n the_o 6._o problem_n the_o 18._o proposition_n upon_o a_o right_a line_n give_v to_o describe_v a_o rectiline_a figure_n like_o and_o in_o like_a sort_n situate_a unto_o a_o rectiline_a figure_n give_v svppose_v that_o the_o riâ—ht_a line_n give_v be_v abâ—_n and_o let_v the_o rectiline_a figure_n give_v be_v eglantine_n it_o be_v require_v upon_o the_o right_a line_n give_v ab_fw-la to_o describe_v a_o rectiline_a figure_n like_o and_o in_o like_a sort_n situate_a unto_o the_o rectiline_a figure_n give_v ge._n draw_v a_o line_n from_o h_o to_o fletcher_n and_o unto_o the_o right_a line_n ab_fw-la and_o to_o the_o pâ—iâ—t_n in_o it_o a_o require_v make_v unto_o the_o angle_n e_o a_o equal_a angle_n dab_n by_o the_o 23._o of_o the_o firsâ—_o and_o unto_o the_o right_a line_n ab_fw-la and_o unto_o the_o point_n in_o it_o b_o by_o the_o same_o make_v unâ—o_o the_o angle_n efh_v a_o equal_a angle_n abdella_n wherefore_o the_o angle_n remain_v ehf_o be_v equal_a unto_o the_o angle_n remain_v adb_o wherefore_o the_o triangle_n hef_fw-fr be_v equiangle_n unto_o the_o triangle_n dab_n wherefore_o by_o the_o 4._o of_o the_o six_o as_o the_o side_n hf_o be_v in_o proportion_n to_o the_o side_n db_o so_o be_v the_o side_n he_o to_o the_o side_n dam_fw-ge and_o a'n_z side_n of_o to_o the_o side_n ab_fw-la again_o by_o the_o 23._o of_o the_o first_o unto_o the_o right_a line_n bd_o and_o unto_o the_o point_n in_o it_o d_o make_v unto_o the_o angle_n fhg_n a_o equal_a angle_n bdc_o and_o by_o the_o same_o unto_o the_o right_a line_n bd_o and_o unto_o the_o point_n in_o 〈◊〉_d b_o make_v unto_o the_o angle_n hfg_n a_o squall_n angle_n dbc_n demonstration_n wherefore_o the_o angle_n remain_v namely_o g_o be_v equal_a unto_o the_o angle_n remain_v namely_o to_o c._n wherefore_o the_o triangle_n hfg_n be_v equiangle_n unto_o the_o triangle_n dbc_n wherefore_o by_o the_o 4._o of_o the_o six_o as_o the_o side_n hf_o be_v in_o proportion_n to_o the_o side_n db_o so_o be_v the_o side_n hg_o to_o the_o side_n dc_o and_o the_o side_n gf_o to_o the_o side_n cb._n and_o it_o be_v already_o prove_v that_o as_o hf_o be_v to_o db_o so_o be_v he_o to_o dam_n and_o of_o to_o ab_fw-la wherefore_o by_o the_o 11._o of_o the_o five_o as_o eh_o be_v to_o ad_fw-la so_o be_v of_o to_o ab_fw-la and_o hg_o to_o dc_o and_o moreover_o gf_o to_o cb._n and_o forasmuch_o as_o the_o angle_n ehf_o be_v equal_a unto_o the_o angle_n adb_o and_o the_o angle_n fhg_n be_v equal_a unto_o the_o angle_n bdc_o therefore_o the_o whole_a angle_n ehg_n be_v equal_a unto_o the_o whole_a angle_n adc_o and_o by_o the_o same_o reason_n the_o angle_n efg_o be_v equal_a unto_o the_o angle_n abc_n but_o by_o construction_n the_o angle_n e_o be_v equal_a unto_o the_o angle_n a_o and_o the_o angle_n g_z be_v prove_v equal_a unto_o the_o angle_n c._n wherefore_o the_o figure_n ac_fw-la be_v equiangle_n unto_o the_o figure_n eglantine_n and_o those_o side_n which_o in_o it_o include_v the_o equal_a angle_n be_v proportional_a as_o we_o have_v before_o prove_v wherefore_o the_o rectiline_a figure_n ac_fw-la be_v by_o the_o first_o definition_n of_o the_o six_o like_o unto_o the_o rectiline_a figure_n give_v egâ—_n wherefore_o upon_o the_o right_a line_n give_v ab_fw-la be_v describe_v a_o rectiline_a figure_n ac_fw-la like_a &_o in_o like_a sort_n â—â—â—uatâ—_n uâ—to_o the_o rectiline_a figure_n give_v eglantine_n which_o be_v require_v to_o be_v do_v the_o 13._o theorem_a the_o 19_o proposition_n like_a triangle_n be_v one_o to_o the_o other_o in_o double_a proportion_n that_o the_o side_n of_o like_a proportion_n be_v svppose_v the_o triangle_n like_v to_o be_v abc_n and_o def_n have_v the_o angle_n b_o of_o the_o one_o triangle_n equal_a unto_o the_o angle_n e_o of_o the_o other_o triangle_n &_o as_o ab_fw-la be_v to_o bc_o so_o let_v de_fw-fr be_v to_o of_o so_o that_o let_v bc_o &_o of_o be_v side_n of_o like_a proportion_n then_o i_o say_v that_o the_o proportion_n of_o the_o triangle_n abc_n unto_o the_o triangle_n def_n be_v double_a to_o the_o proportion_n of_o the_o side_n bc_o to_o the_o side_n ef._n unto_o the_o two_o line_n bc_o and_o of_o by_o the_o 10._o of_o the_o six_o make_v a_o three_o line_n in_o proportion_n bg_o so_o that_o as_o bc_o be_v to_o of_o so_o let_v of_o be_v
13._o of_o the_o six_o take_v the_o mean_a proportional_a between_o the_o line_n bc_o and_o cf_o which_o let_v be_v gh_o and_o by_o the_o 18._o of_o the_o six_o upon_o the_o line_n gh_o let_v there_o be_v describe_v a_o rectiline_a figure_n khg_n like_a unto_o the_o rectiline_a figure_n abc_n and_o in_o like_a sort_n situate_a and_o for_o that_o as_o the_o line_n bc_o be_v to_o the_o line_n gh_o so_o be_v the_o line_n gh_o to_o the_o line_n cf_o demonstration_n but_o if_o there_o be_v three_o right_a line_n proportional_a as_o the_o first_o be_v to_o the_o three_o so_o be_v the_o figure_n which_o be_v describe_v of_o the_o first_o unto_o the_o figure_n which_o be_v describe_v of_o the_o second_o the_o say_a figure_n be_v like_o and_o in_o like_a sort_n situate_a by_o the_o second_o corollary_n of_o the_o 20._o of_o the_o six_o wherefore_o as_o the_o line_n bc_o be_v to_o the_o line_n cf_o so_o be_v the_o triangle_n abc_n to_o the_o triangle_n kgh_n but_o as_o the_o line_n bc_o be_v to_o the_o line_n cf_o so_o be_v the_o parallelogram_n be_v to_o the_o parallelogram_n of_o by_o the_o 1._o of_o the_o six_o wherefore_o as_o the_o triangle_n a_o bc_o be_v to_o the_o triangle_n kgh_o so_o be_v the_o parallelogram_n be_v to_o the_o parallelogram_n ef._n wherefore_o alternate_o also_o by_o the_o 16._o of_o the_o five_o as_o the_o triangle_n abc_n be_v to_o the_o parallelogram_n be_v so_o be_v the_o triangle_n kgh_o to_o the_o parallelogram_n of_o but_o the_o triangle_n abc_n be_v equal_a unto_o the_o parallelogram_n be_v wherefore_o also_o the_o triangle_n kgh_n be_v equal_a unto_o the_o parallelogram_n of_o but_o the_o parallelogram_n fe_o be_v equal_a unto_o the_o rectiline_a figure_n d._n wherefore_o also_o the_o rectiline_a figure_n kgh_o be_v equal_a unto_o the_o rectiline_a figure_n d_o and_o the_o rectiline_a figure_n kgh_o be_v by_o supposition_n like_o unto_o the_o rectiline_a figure_n abc_n wherefore_o there_o be_v describe_v a_o rectiline_a figure_n kgh_o like_o unto_o the_o rectiline_a figure_n give_v abc_n and_o equal_a unto_o the_o other_o rectiline_a figure_n give_v d_z which_o be_v require_v to_o be_v do_v the_o 19_o theorem_a the_o 26._o proposition_n if_o from_o a_o parallelogram_n be_v take_v away_o a_o parallelogram_n like_a unto_o the_o whole_a and_o in_o like_a sort_n set_v have_v also_o a_o angle_n common_a with_o it_o then_o be_v the_o parallelogram_n about_o one_o and_o the_o self_n same_o dimecient_a with_o the_o whole_a svppose_v that_o there_o be_v a_o parallelogram_n abcd_o and_o from_o the_o parallelogram_n abcd_o take_v away_o a_o parallelogram_n of_o like_a unto_o the_o parallelogram_n abcd_o and_o in_o like_a sort_n situate_a have_v also_o the_o angle_n dab_n common_a with_o it_o then_o i_o say_v that_o the_o parallelogram_n abcd_o and_o of_o be_v both_o about_o one_o and_o the_o self_n same_o absurdity_n dimecient_a afc_n that_o be_v that_o the_o dimecient_a afc_n of_o the_o whole_a parallelogram_n abcd_o pass_v by_o the_o angle_n f_o of_o the_o parallelogram_n of_o and_o be_v common_a to_o either_o of_o the_o parallelogram_n for_o if_o ac_fw-la do_v not_o pass_v by_o the_o point_n fletcher_n then_o if_o it_o be_v possible_a let_v it_o pass_v by_o some_o other_o point_n as_o ahc_n do_v now_o than_o the_o dimetient_fw-la ahc_n shall_v cut_v either_o the_o side_n gf_o or_o the_o side_n of_o of_o the_o parallelogram_n af._n let_v it_o cut_v the_o side_n gf_o in_o the_o point_n h._n and_o by_o the_o 31._o of_o the_o first_o by_o the_o point_n h_o let_v there_o be_v draw_v to_o either_o of_o these_o line_n ad_fw-la and_o bc_o a_o parallel_a line_n hk_o wherefore_o gk_o be_v a_o parallelogram_n and_o be_v about_o one_o and_o the_o self_n same_o dimetient_fw-la with_o the_o parallelogram_n abcd._n and_o forasmuch_o as_o the_o parallelogram_n abcd_o and_o gk_o be_v about_o one_o and_o the_o self_n same_o dimecient_a therefore_o by_o the_o 24._o of_o the_o six_o the_o parallelogram_n abcd_o be_v like_a unto_o the_o parallelogram_n gk_o wherefore_o as_o the_o line_n dam_n be_v to_o the_o line_n ab_fw-la so_o be_v the_o line_n ga_o to_o the_o line_n ak_o by_o the_o conversion_n of_o the_o first_o definition_n of_o the_o six_o and_o for_o that_o the_o parallelogram_n abcd_o and_o eglantine_n be_v by_o supposition_n like_v therefore_o as_o the_o line_n dam_n be_v to_o the_o line_n ab_fw-la so_o be_v the_o line_n ga_o to_o the_o line_n ae_n wherefore_o the_o line_n ga_o have_v one_o and_o the_o self_n proportion_n to_o either_o of_o these_o line_n ak_o and_o ae_n wherefore_o by_o the_o 9_o of_o the_o five_o the_o line_n ak_o be_v equal_a unto_o the_o line_n ae_n namely_o the_o less_o to_o the_o great_a which_o be_v impossible_a the_o self_n same_o inconvenience_n also_o will_v follow_v if_o you_o put_v the_o dimetient_fw-la ac_fw-la to_o cut_v the_o side_n fe_o wherefore_o ac_fw-la the_o dimetient_n of_o the_o whole_a parallelogram_n abcd_o pass_v by_o the_o angle_n and_o point_n f._n and_o therefore_o the_o parallelogram_n aefg_n be_v about_o one_o and_o the_o self_n same_o dimetient_fw-la with_o the_o whole_a parallelogram_n abcd._n wherefore_o if_o from_o a_o parallelogram_n be_v take_v away_o a_o parallelogram_n like_o unto_o the_o whole_a and_o in_o like_a sort_n situate_a have_v also_o a_o angle_n common_a with_o it_o then_o be_v that_o parallelogram_n about_o one_o and_o the_o self_n same_o dimetient_fw-la with_o the_o whole_a which_o be_v require_v to_o be_v prove_v ¶_o a_o other_o demonstration_n after_o flussates_n which_o prove_v this_o proposition_n affirmative_o from_o the_o parallelogram_n abgd_v let_v there_o be_v take_v away_o the_o parallelogram_n aezk_v like_a and_o in_o like_a sort_n situate_a with_o the_o whole_a parallelogram_n abgd_v and_o have_v also_o the_o angle_n a_o common_a with_o the_o whole_a parallelogram_n flussates_n then_o i_o say_v that_o both_o their_o diameter_n namely_o az_o and_o azg_n do_v make_v one_o and_o the_o self_n same_o right_a line_n divide_v the_o side_n ab_fw-la and_o bâ—_n into_o two_o equal_a part_n in_o the_o point_n c_o and_o f_o by_o the_o 10._o of_o the_o first_o and_o draw_v a_o line_n from_o c_o to_o f._n wherefore_o the_o line_n cf_o be_v a_o parallel_n to_o the_o right_a line_n agnostus_n by_o the_o corollary_n add_v by_o campane_n after_o the_o 29._o of_o the_o first_o wherefore_o the_o angle_n bag_n and_o bcf_n be_v equal_a by_o the_o 29._o of_o the_o first_o but_o the_o angle_n eaz_n be_v equal_a unto_o the_o angle_n bag_n by_o reason_n the_o parallelogram_n be_v suppose_v to_o be_v like_o wherefore_o the_o same_o angle_n eaz_n be_v equal_a to_o the_o angle_n bcf_n namely_o the_o outward_a angle_n to_o the_o inward_a and_o opposite_a angle_n wherefore_o by_o the_o 28._o of_o the_o first_o the_o line_n az_o and_o cf_o be_v parallel_a line_n now_o than_o the_o line_n az_o and_o agnostus_n be_v parallel_n to_o one_o and_o the_o self_n same_o line_n namely_o to_o cf_o do_v concur_v in_o the_o point_n a._n wherefore_o they_o be_v set_v direct_o the_o one_o to_o the_o other_o so_o that_o they_o both_o make_v one_o right_a line_n by_o that_o which_o be_v add_v in_o the_o end_n of_o the_o 30._o proposition_n of_o the_o first_o wherefore_o the_o parallelogram_n abgd_v and_o aezk_n be_v about_o one_o and_o the_o self_n fame_n dimetient_fw-la which_o be_v require_v to_o be_v prove_v the_o 20._o theorem_a the_o 27._o proposition_n of_o all_o parallelogram_n apply_v to_o a_o right_a line_n want_v in_o figure_n by_o parallelogram_n like_a and_o in_o like_a sort_n situate_a to_o that_o parallelogram_n which_o be_v describe_v of_o the_o half_a line_n the_o great_a parallelogram_n be_v that_o which_o be_v describe_v of_o the_o half_a line_n be_v like_a unto_o the_o want_n ac_fw-la let_v there_o be_v a_o right_a line_n ab_fw-la and_o by_o the_o 10._o of_o the_o first_o divide_v it_o in_o two_o equal_a part_n in_o the_o point_n c._n and_o unto_o the_o right_a line_n ab_fw-la apply_v a_o parallelogram_n ad_fw-la want_v in_o figure_n by_o the_o parallelogram_n db_o which_o let_v be_v like_a and_o in_o like_a sort_n describe_v unto_o the_o parallelogram_n describe_v of_o half_a the_o line_n ab_fw-la which_o be_v bc._n then_o i_o say_v that_o of_o all_o the_o parallelogram_n which_o may_v be_v apply_v unto_o the_o line_n ab_fw-la and_o which_o want_n in_o figure_n by_o parallelogram_n like_a and_o in_o like_a sort_n situate_a unto_o the_o parallelogram_n db_o the_o great_a be_v the_o parallelogram_n ad._n for_o unto_o the_o right_a line_n ab_fw-la let_v there_o be_v apply_v a_o parallelogram_n of_o want_v in_o figure_n by_o the_o parallelogram_n fb_o which_o let_v be_v like_a and_o in_o like_a sort_n situate_a unto_o the_o parallelogram_n db._n then_o i_o say_v that_o the_o parallelogram_n ad_fw-la be_v
mean_a proportion_n in_o the_o point_n c._n which_o be_v require_v to_o be_v do_v the_o 21._o theorem_a the_o 31._o proposition_n in_o rectangle_n triangle_n the_o figure_n make_v of_o the_o side_n subtend_v the_o right_a angle_n be_v equal_a unto_o the_o figure_n make_v of_o the_o side_n comprehend_v the_o right_a angle_n so_o that_o the_o say_v thrâ—e_a figure_n bâ—_z bâ—_z like_a and_o in_o like_a sort_n describe_v svppose_v that_o there_o be_v a_o triangle_n abc_n who_o angle_n bac_n let_v be_v a_o right_a angle_n then_o i_o say_v that_o the_o figure_n which_o be_v describe_v of_o the_o line_n bc_o be_v equal_a unto_o the_o two_o figure_n which_o be_v describe_v of_o the_o line_n basilius_n &_o ac_fw-la the_o say_v three_o figure_n be_v like_o the_o one_o to_o the_o other_o and_o in_o like_a sort_n describe_v from_o the_o point_n a_o by_o the_o 12._o of_o the_o first_o let_v there_o be_v draw_v unto_o the_o line_n bc_o a_o perpendicular_a line_n ad._n construction_n now_o forasmuch_o as_o in_o the_o rectangle_n triangle_n abc_n be_v draw_v from_o the_o right_a angle_n a_o unto_o the_o base_a bc_o a_o perpendicular_a line_n ad_fw-la demonstration_n therefore_o the_o triangle_n abdella_n and_o adc_o set_v upon_o the_o perpendicular_a line_n be_v like_a unto_o the_o whole_a triangle_n abc_n and_o also_o like_o the_o one_o to_o the_o other_o by_o the_o 8._o of_o the_o six_o and_o forasmuch_o as_o the_o triangle_n abc_n be_v like_a unto_o the_o triangle_n abdella_n therefore_o as_o the_o line_n cb_o be_v to_o the_o line_n basilius_n so_o be_v the_o line_n ab_fw-la to_o the_o line_n bd._n now_o for_o that_o there_o be_v three_o right_a line_n proportional_a therefore_o by_o the_o 2._o corollary_n of_o the_o 20._o of_o the_o six_o as_o the_o first_o be_v to_o the_o three_o so_o be_v the_o figure_n make_v of_o the_o first_o to_o the_o figure_n make_v of_o the_o second_o the_o say_a figure_n be_v like_o and_o in_o like_a sort_n describe_v wherefore_o as_o the_o line_n bc_o be_v to_o the_o line_n bd_o so_o be_v the_o figure_n make_v of_o the_o line_n bc_o to_o the_o figure_n make_v of_o the_o line_n basilius_n they_o be_v like_a and_o in_o like_a sort_n describe_v and_o by_o the_o same_o reason_n as_o the_o line_n bc_o be_v to_o the_o line_n cd_o so_o be_v the_o figure_n make_v of_o the_o line_n bc_o to_o the_o figure_n make_v of_o the_o line_n ca_n they_o be_v like_a and_o in_o like_a sort_n describe_v wherefore_o as_o the_o line_n bc_o be_v to_o the_o line_n bd_o and_o dc_o so_o be_v the_o figure_n make_v of_o the_o line_n bc_o to_o the_o figure_n make_v of_o the_o line_n basilius_n and_o ac_fw-la they_o be_v like_a and_o in_o like_a sort_n describe_v but_o the_o line_n bc_o be_v equal_a unto_o the_o line_n bd_o and_o dc_o wherefore_o the_o figure_n make_v of_o the_o line_n bc_o be_v equal_a unto_o the_o figure_n make_v of_o the_o line_n basilius_n and_o ac_fw-la they_o be_v like_a and_o in_o like_a sort_n describe_v wherefore_o in_o rectangle_n triangle_n the_o figure_n make_v of_o the_o side_n subtend_v the_o right_a angle_n be_v equal_a unto_o the_o figure_n make_v of_o the_o side_n comprehend_v the_o right_a angle_n so_o that_o the_o say_a thrâ—_n figure_n be_v like_a and_o in_o like_a sort_n describe_v which_o be_v require_v to_o be_v prove_v an_o other_o way_n forasmuch_o as_o by_o the_o first_o corollary_n of_o the_o 20._o of_o the_o six_o like_o rectiline_a figure_n be_v in_o double_a proportion_n to_o that_o that_o the_o side_n of_o like_a proportion_n be_v therefore_o the_o rectiline_a figure_n make_v of_o the_o line_n bc_o be_v unto_o tâ—e_z rectiline_a figure_n â—ade_v of_o the_o line_n basilius_n in_o double_a proportion_n to_o that_o that_o the_o line_n cb_o be_v to_o the_o line_n basilius_n and_o by_o the_o same_o the_o square_a also_o make_v of_o the_o line_n bc_o be_v unto_o the_o square_n make_v of_o the_o line_n basilius_n in_o double_a proportion_n to_o that_o that_o the_o line_n cb_o be_v unto_o the_o line_n basilius_n wherefore_o also_o as_o the_o rectiline_a figure_n make_v of_o the_o line_n cb_o be_v to_o the_o rectiline_a figure_n make_v of_o the_o line_n basilius_n so_o be_v the_o square_n make_v of_o the_o line_n cb_o to_o the_o square_n make_v of_o the_o line_n basilius_n and_o by_o the_o same_o reason_n also_o as_o the_o rectiline_a figure_n make_v of_o the_o line_n bc_o be_v to_o the_o rectiline_a figure_n make_v of_o the_o line_n ca_n so_o be_v the_o square_n make_v of_o the_o line_n bc_o to_o the_o square_n make_v of_o the_o line_n ca._n wherefore_o also_o as_o the_o rectiline_a figure_n make_v of_o the_o line_n bc_o be_v to_o the_o rectiline_a figure_n make_v of_o the_o line_n basilius_n and_o ac_fw-la so_o be_v the_o square_n make_v of_o the_o line_n bc_o to_o the_o square_n make_v of_o the_o line_n basilius_n and_o ac_fw-la but_o the_o square_n make_v of_o the_o line_n bc_o be_v equal_a unto_o the_o square_n make_v of_o the_o line_n basilius_n and_o ac_fw-la by_o the_o 47._o of_o the_o first_o wherefore_o also_o the_o rectiline_a figure_n make_v of_o the_o line_n bc_o be_v equal_a unto_o the_o rectiline_a figure_n make_v of_o the_o line_n basilius_n and_o ac_fw-la the_o say_v three_o figure_n be_v like_o and_o in_o like_a sort_n describe_v the_o converse_n of_o this_o proposition_n after_o campane_n proposition_n if_o the_o figure_n describe_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 figure_n which_o be_v describe_v of_o the_o two_o other_o side_n the_o say_a figure_n be_v like_o and_o in_o like_a sort_n describe_v the_o triangle_n shall_v be_v a_o rectangle_n triangle_n suppose_v that_o abc_n be_v a_o triangle_n and_o let_v the_o figure_n describe_v of_o the_o side_n bc_o be_v equal_a to_o the_o two_o figure_n describe_v of_o the_o side_n ab_fw-la and_o ac_fw-la the_o say_a figure_n be_v like_a and_o in_o like_a sort_n describe_v then_o i_o say_v that_o the_o angle_n a_o be_v a_o right_a angle_n let_v the_o angle_n god_n be_v a_o right_a angle_n and_o put_v the_o line_n ad_fw-la equal_a to_o the_o line_n ab_fw-la and_o draw_v a_o line_n from_o d_z to_o c._n now_o then_o by_o this_o 31._o proposition_n the_o figure_n make_v of_o the_o line_n cd_o be_v equal_a to_o the_o two_o figure_n make_v of_o the_o line_n ac_fw-la and_o ad_fw-la the_o say_a figure_n be_v like_o and_o in_o like_a sort_n describe_v wherefore_o also_o it_o be_v equal_a unto_o the_o figure_n make_v of_o the_o line_n bc_o which_o be_v by_o supposition_n equal_a to_o the_o two_o figure_n make_v of_o the_o line_n ac_fw-la and_o ad_fw-la for_o the_o line_n ad_fw-la be_v put_v equal_a to_o the_o line_n ab_fw-la wherefore_o the_o line_n dc_o be_v squall_n to_o the_o line_n bc._n wherefore_o by_o the_o 8._o of_o the_o first_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 prove_v the_o 22._o theorem_a the_o 32._o proposition_n if_o two_o triangle_n be_v set_v together_o at_o one_o angle_n have_v two_o side_n of_o the_o one_o proportional_a to_o two_o side_n of_o the_o other_o so_o that_o their_o side_n of_o like_a proportion_n be_v also_o parallel_v then_o the_o other_o side_n remain_v of_o those_o triangle_n shall_v be_v in_o one_o right_a line_n svppose_v the_o two_o triangle_n to_o be_v abc_n and_o dce_n and_o let_v two_o of_o their_o side_n ac_fw-la &_o dc_o make_v a_o angle_n acd_o and_o let_v the_o say_a triangle_n have_v two_o side_n of_o the_o one_o namely_o basilius_n and_o ac_fw-la proportional_a to_o two_o side_n of_o the_o other_o demonstration_n namely_o to_o dc_o and_o de_fw-fr so_o that_o as_z ab_fw-la be_v to_o ac_fw-la so_o let_v dc_o be_v to_o de._n and_o let_v ab_fw-la be_v a_o parallel_n unto_o dc_o and_o ac_fw-la a_o parallel_n unto_o de._n then_o i_o say_v that_o the_o line_n bc_o and_o ce_fw-fr be_v in_o one_o right_a line_n for_o forasmuch_o as_o the_o line_n ab_fw-la be_v a_o parallel_n unto_o the_o line_n dc_o and_o upon_o they_o light_v a_o right_a line_n ac_fw-la therefore_o by_o the_o 29._o of_o the_o first_o the_o alternate_a angle_n bac_n and_o acd_o be_v equal_a the_o one_o to_o the_o other_o and_o by_o the_o same_o reason_n the_o angle_n cde_o be_v equal_a unto_o the_o same_o angle_n acd_o wherefore_o the_o angle_n bac_n be_v equal_a unto_o the_o angle_n cde_o and_o forasmuch_o as_o there_o be_v two_o triangle_n abc_n and_o dce_n have_v the_o angle_n a_o of_o the_o one_o equal_a to_o the_o angle_n d_o of_o the_o other_o and_o the_o side_n about_o the_o equal_a angle_n be_v by_o supposition_n proportional_a that_o be_v 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 cd_o to_o the_o line_n de_fw-fr therefore_o the_o
be_v no_o difference_n but_o only_o the_o draught_n of_o the_o line_n gc_o ¶_o the_o 2._o theorem_a the_o 2._o proposition_n if_o two_o right_a line_n cut_v the_o ouâ—_n to_o the_o other_o they_o be_v â—â—â—ne_v and_o the_o self_n same_o plain_a superficies_n &_o every_o triangle_n be_v in_o one_o &_o the_o self_n same_o superficies_n svppose_v that_o these_o two_o right_a line_n ab_fw-la and_o cd_o do_v cut_v the_o one_o the_o other_o in_o the_o point_n e._n then_o i_o say_v that_o these_o line_n ab_fw-la and_o cd_o be_v in_o one_o and_o the_o self_n same_o superficies_n and_o that_o every_o triangle_n be_v in_o one_o &_o self_n same_o plain_a superficies_n construction_n take_v in_o the_o line_n ec_o and_o ebb_n point_v at_o all_o aventure_n and_o let_v the_o same_o be_v fletcher_n and_o g_o and_o draw_v a_o right_a line_n from_o the_o point_n b_o to_o the_o point_n c_o and_o a_o other_o from_o the_o point_n f_o to_o the_o point_n g._n and_o draw_v the_o line_n fh_o and_o gk_o first_o i_o say_v that_o the_o triangle_n ebc_n be_v in_o one_o and_o the_o same_o ground_n superficies_n impossibility_n for_o if_o part_n of_o the_o triangle_n ebc_n namely_o the_o triangle_n fch_o or_o the_o triangle_n gbk_n be_v in_o the_o ground_n superficies_n and_o the_o residue_n be_v in_o a_o other_o then_o also_o part_n of_o one_o of_o the_o right_a line_n ec_o or_o ebb_n shall_v be_v in_o the_o ground_n superficies_n and_o part_n in_o a_o other_o so_o also_o if_o part_n of_o the_o triangle_n ebc_n namely_o the_o part_n efg_o be_v in_o the_o ground_n superficies_n and_o the_o residue_n be_v in_o a_o other_o then_o also_o one_o part_n of_o each_o of_o the_o right_a line_n ec_o and_o ebb_n shall_v be_v in_o the_o ground_n superficies_n &_o a_o other_o part_n in_o a_o other_o superficies_n which_o by_o the_o first_o of_o the_o eleven_o be_v prove_v to_o be_v impossible_a wherefore_o the_o triangle_n ebc_n be_v in_o one_o and_o the_o self_n same_o plain_a superficies_n for_o in_o what_o superficies_n the_o triangle_n be_v be_v in_o the_o same_o also_o be_v either_o of_o the_o line_n ec_o and_o ebb_n and_o in_o what_o superficies_n either_o of_o the_o line_n ec_o and_o ebb_n be_v in_o the_o self_n same_o also_o be_v the_o line_n ab_fw-la and_o cd_o wherefore_o the_o right_a line_n line_n ab_fw-la and_o cd_o be_v in_o one_o &_o the_o self_n same_o plain_a superficies_n and_o every_o triangle_n be_v in_o one_o &_o the_o self_n same_o plain_a superficies_n which_o be_v require_v to_o be_v prove_v in_o this_o figure_n here_o set_v may_v you_o more_o plain_o conceive_v the_o demonstration_n of_o the_o former_a proposition_n where_o 〈◊〉_d may_v eleâ—â—â—â—_n what_o part_n of_o the_o triangle_n ecb_n you_o will_v namely_o the_o part_n fch_n or_o the_o part_n gbk_n or_o final_o the_o part_n fcgb_n as_o be_v require_v in_o the_o demonstration_n ¶_o the_o 3._o theorem_a the_o 3._o proposition_n if_o two_o plain_a superficiece_n cut_v the_o one_o the_o other_o their_o common_a section_n be_v a_o right_a line_n svppose_v that_o these_o two_o superficiece_n ab_fw-la &_o bc_o do_v cut_v the_o one_o the_o other_o and_o let_v their_o common_a sectiâ—â—â—e_n the_o line_n db._n then_o i_o say_v that_o db_o be_v a_o right_a line_n for_o if_o not_o draw_v from_o the_o point_n d_o to_o the_o point_n b_o a_o right_a line_n dfb_n in_o the_o plain_a superficies_n ab_fw-la impossibility_n and_o likewise_o from_o the_o same_o point_n draw_v a_o other_o right_a line_n deb_n in_o the_o plain_a superficies_n bc._n now_o therefore_o two_o right_a line_n deb_n and_o dfb_n shall_v â—ave_n the_o self_n saâ—â—_n eâ—deâ—_n and_o therefore_o do_v include_v a_o superficies_n which_o by_o the_o last_o common_a sentence_n be_v impossibleâ—_n wherefore_o the_o line_n deb_n and_o dfb_n be_v not_o right_a line_n in_o like_a sort_n also_o may_v we_o prove_v that_o no_o other_o right_a line_n can_v be_v draw_v from_o the_o point_n d_o to_o the_o point_n b_o beside_o the_o line_n db_o which_o be_v the_o common_a section_n of_o the_o two_o superficiece_n ab_fw-la and_o bc._n if_o therefore_o two_o plain_a superficiece_n cut_v the_o one_o the_o other_o their_o common_a section_n be_v a_o right_a line_n which_o be_v require_v to_o be_v demonstrate_v this_o figure_n here_o set_v show_v most_o plain_o not_o only_a this_o three_o proposition_n but_o also_o the_o demonstration_n thereof_o if_o you_o elevate_v the_o superficies_n ab_fw-la and_o so_o compare_v it_o with_o the_o demonstration_n ¶_o the_o 4._o theorem_a the_o 4._o proposition_n if_o from_o two_o right_a line_n cut_v the_o one_o the_o other_o at_o their_o common_a section_n a_o right_a line_n be_v perpendicular_o erect_v the_o same_o shall_v also_o be_v perpendicular_o erect_v from_o the_o plain_a superficies_n by_o the_o say_v two_o line_n pass_v svppose_v that_o there_o be_v two_o right_a line_n ab_fw-la and_o cd_o cut_v the_o one_o the_o other_o in_o the_o point_n e._n and_o from_o the_o point_n e_o let_v there_o be_v erect_v a_o right_a line_n of_o perpendicular_o to_o the_o say_v two_o right_a line_n ab_fw-la and_o cd_o then_o i_o say_v that_o the_o right_a line_n of_o construction_n be_v also_o erect_v perpendicular_a to_o the_o plain_a superficies_n which_o pass_v by_o the_o line_n a_o b_o and_o cd_o let_v these_o line_n ae_n ebb_n ec_o and_o ed_z be_v put_v equal_a the_o one_o to_o the_o other_o and_o by_o the_o point_n e_o extend_v a_o right_a line_n at_o all_o aventure_n and_o let_v the_o same_o be_v geh_a and_o draw_v these_o right_a line_n ad_fw-la cb_o favorina_n fg_o fd_o fc_o fh_o and_o fb_o demonstration_n and_o forasmuch_o as_o these_o two_o right_a line_n ae_n &_o ed_z be_v equal_a to_o these_o two_o line_n ce_fw-fr and_o ebb_n and_o they_o comprehend_v equal_a angle_n by_o the_o 15._o of_o the_o first_o therefore_o by_o the_o 4._o of_o the_o first_o the_o base_a ad_fw-la be_v equal_a to_o the_o base_a cb_o and_o the_o triangle_n aed_n be_v equal_a to_o the_o triangle_n ceb_fw-mi wherefore_o also_o the_o angle_n dae_n be_v equal_a to_o the_o angle_n ebc_n but_o the_o angle_n aeg_n be_v equal_a to_o the_o angle_n beh_n by_o the_o 15_o of_o the_o first_o wherefore_o there_o be_v two_o triangle_n age_n and_o beh_n have_v two_o angle_n of_o the_o one_o equal_a to_o two_o angle_n of_o the_o other_o each_o to_o his_o correspondent_a angle_n and_o one_o side_n of_o the_o one_o equal_a to_o one_o side_n of_o the_o other_o namely_o one_o of_o the_o side_n which_o lie_v between_o the_o equal_a angle_n namely_o the_o side_n ae_n be_v equal_a to_o the_o side_n ebb_n wherefore_o by_o the_o 26._o of_o the_o first_o the_o side_n remain_v be_v equal_a to_o the_o side_n remain_v wherefore_o the_o side_n ge_z be_v equal_a to_o the_o side_n eh_o and_o the_o side_n agnostus_n to_o the_o side_n bh_o and_o forasmuch_o as_o the_o line_n ae_n be_v equal_a to_o the_o line_n ebb_n and_o the_o line_n fe_o be_v common_a to_o they_o both_o and_o make_v with_o they_o right_a angle_n wherefore_o by_o the_o four_o of_o the_o first_o the_o base_a favorina_n it_o equal_a to_o the_o base_a fb_o and_o by_o the_o same_o reason_n the_o base_a fc_n be_v equal_a to_o the_o base_a fd._n and_o forasmuch_o as_o the_o line_n ad_fw-la be_v equal_a to_o the_o line_n bc_o and_o the_o line_n favorina_n be_v equal_a to_o the_o line_n fb_o as_o it_o have_v be_v prove_v therefore_o these_o two_o line_n favorina_n and_o ad_fw-la be_v equal_a to_o these_o two_o line_n fb_o &_o bc_o the_o one_o to_o the_o other_o &_o the_o base_a fd_o be_v equal_a to_o the_o base_a fc_n wherefore_o also_o the_o angle_n fad_n be_v equal_a to_o the_o angle_n fbc_n and_o again_o forasmuch_o as_o it_o have_v be_v prove_v that_o the_o line_n agnostus_n be_v equal_a to_o the_o line_n bh_o but_o the_o line_n favorina_n be_v equal_a to_o the_o line_n fb_o wherefore_o there_o be_v two_o line_n favorina_n and_o agnostus_n equal_a to_o two_o line_n fb_o and_o bh_o and_o it_o be_v prove_v that_o the_o angle_n fag_n be_v equal_a to_o the_o angle_n fbh_n wherefore_o by_o the_o 4._o of_o the_o first_o the_o base_a fg_o be_v equal_a to_o the_o base_a fh_o again_o forasmuch_o as_o it_o have_v be_v prove_v that_o the_o line_n ge_z be_v equal_a to_o the_o line_n eh_o and_o the_o line_n of_o be_v common_a to_o they_o both_o wherefore_o these_o two_o line_n ge_z and_o of_o be_v equal_a to_o these_o two_o line_n he_o and_o of_o and_o the_o base_a fh_o be_v equal_a to_o the_o base_a fg_o wherefore_o the_o angle_n gef_n be_v equal_a to_o the_o angle_n hef_fw-fr wherefore_o either_o of_o the_o angle_n gef_n and_o hef_n be_v a_o right_a angle_n wherefore_o the_o line_n of_o be_v erect_v from_o the_o point_n e_o
perpendicular_o to_o the_o line_n gh_o in_o like_a sort_n may_v we_o prove_v that_o the_o same_o line_n fe_o make_v right_a angle_n with_o all_o the_o right_a line_n which_o be_v draw_v upon_o the_o ground_n plain_a superficies_n and_o touch_v the_o point_n b._n but_o a_o right_a line_n be_v then_o erect_v perpendicular_o to_o a_o plain_a superficies_n when_o it_o make_v right_a angle_n with_o all_o the_o line_n which_o touch_v it_o and_o be_v draw_v upon_o the_o ground_n plain_a superficies_n by_o the_o 2._o definition_n of_o the_o eleven_o wherefore_o the_o right_a line_n fe_o be_v erect_v perpendicular_o to_o the_o ground_n plain_a superficies_n and_o the_o ground_n plain_a superficies_n be_v that_o which_o pass_v by_o these_o right_a line_n ab_fw-la and_o cd_o wherefore_o the_o right_a line_n fe_o be_v erect_v perpendicular_o to_o the_o plain_a superficies_n which_o pass_v by_o the_o right_a line_n ab_fw-la and_o cd_o if_o therefore_o from_o two_o right_a line_n cut_v the_o one_o the_o other_o and_o at_o their_o common_a section_n a_o right_a line_n be_v perpendicular_o erect_v it_o shall_v also_o be_v erect_v perpendicular_o to_o the_o plain_a superficies_n by_o the_o say_v two_o line_n pass_v which_o be_v require_v to_o be_v prove_v in_o this_o figure_n you_o may_v most_o evident_o conceive_v the_o former_a proposition_n and_o demonstration_n if_o you_o erect_v perpendicular_o unto_o the_o ground_n plain_a superficies_n acbd_v theâ—_z triangle_n afb_o and_o elevate_v the_o triangle_n afd_v &_o cfb_n in_o such_o sort_n that_o the_o line_n of_o of_o the_o triangle_n afb_o may_v join_v &_o make_v one_o line_n with_o the_o line_n of_o of_o the_o triangle_n afd_v and_o likewise_o that_o the_o line_n bf_o of_o the_o triangle_n afb_o may_v join_v &_o make_v one_o right_a line_n with_o the_o line_n bf_o of_o the_o triangle_n bfc_n ¶_o the_o 5._o theorem_a the_o 5._o proposition_n if_o unto_o three_o right_a line_n which_o touch_v the_o one_o the_o other_o be_v erect_v a_o perpendicular_a line_n from_o the_o common_a point_n where_o those_o three_o line_n touch_v those_o three_o right_a line_n be_v in_o one_o and_o the_o self_n same_o plain_a superficies_n svppose_v that_o unto_o these_o three_o right_a line_n bc_o bd_o and_o be_v touch_v the_o one_o the_o other_o in_o the_o point_n b_o be_v erect_v perpendicular_o from_o the_o point_n b_o the_o line_n ab_fw-la then_o i_o say_v that_o those_o three_o right_a line_n bc_o bd_o and_o be_v be_v in_o one_o &_o the_o self_n same_o plain_a superficies_n for_o if_o not_o then_o if_o it_o be_v possible_a let_v the_o line_n bd_o &_o be_v be_v in_o the_o ground_n superficies_n and_o let_v the_o line_n bc_o be_v erect_v upward_o now_o the_o line_n ab_fw-la and_o bc_o be_v in_o one_o and_o the_o same_o plain_a superficies_n by_o the_o 2._o of_o the_o eleven_o for_o they_o touch_v the_o one_o the_o other_o in_o the_o point_n b_o extend_v the_o plain_a superficies_n wherein_o the_o line_n ab_fw-la and_o bc_o be_v impossibility_n and_o it_o shall_v make_v at_o the_o length_n a_o common_a section_n with_o the_o ground_n superficies_n which_o common_a section_n shall_v be_v a_o right_a line_n by_o the_o 3._o of_o the_o eleven_o let_v that_o common_a section_n be_v the_o line_n bf_o wherefore_o the_o three_o right_a line_n ab_fw-la bc_o and_o bf_o be_v in_o one_o and_o the_o self_n same_o superficies_n namely_o in_o the_o superficies_n wherein_o the_o line_n ab_fw-la and_o bc_o be_v and_o forasmuch_o as_o the_o right_a line_n ab_fw-la be_v erect_v perpendicular_o to_o either_o of_o these_o line_n bd_o and_o be_v therefore_o the_o line_n ab_fw-la be_v also_o by_o the_o 4._o of_o the_o eleven_o erect_v perpendicular_o to_o the_o plain_a superficies_n wherein_o the_o line_n bd_o and_o be_v be_v but_o the_o superficies_n wherein_o the_o line_n bd_o and_o be_v be_v be_v the_o ground_n superficies_n wherefore_o the_o line_n ab_fw-la be_v erect_v perpendicular_o to_o the_o ground_n plain_a superficies_n wherefore_o by_o the_o 2._o definition_n of_o the_o eleven_o the_o line_n ab_fw-la make_v right_a angle_n with_o all_o the_o line_n which_o be_v draw_v upon_o the_o ground_n superficies_n and_o touch_v it_o but_o the_o line_n bf_o which_o be_v in_o the_o ground_n superficies_n do_v touch_v it_o wherefore_o the_o angle_n abf_n be_v a_o right_a angle_n and_o it_o be_v suppose_v that_o the_o angle_n abc_n be_v a_o right_a angle_n wherefore_o the_o angle_n abf_n be_v equal_a to_o the_o angle_n abc_n and_o they_o be_v in_o one_o and_o the_o self_n same_o plain_a superficies_n which_o be_v impossible_a wherefore_o the_o right_a line_n bc_o be_v not_o in_o a_o high_a superficies_n wherefore_o the_o right_a line_n bc_o bd_o be_v be_v in_o one_o and_o the_o self_n same_o plain_a superficies_n if_o therefore_o unto_o three_o right_a line_n touch_v the_o one_o the_o one_o the_o other_o be_v erect_v a_o perpendicular_a line_n from_o the_o common_a point_n where_o those_o three_o line_n touch_v those_o three_o right_a line_n be_v in_o one_o and_o the_o self_n same_o plain_a superficies_n which_o be_v require_v to_o be_v demonstrate_v this_o figure_n here_o set_v more_o plain_o declare_v the_o demonstration_n of_o the_o former_a proposition_n if_o you_o erect_v perpendicular_o unto_o the_o ground_n superficies_n the_o sâ—â—perficies_n wherein_o be_v draw_v the_o line_n 〈◊〉_d and_o so_o compare_v it_o with_o the_o say_a defenestration_n the_o 6._o theorem_a the_o 6._o proposition_n if_o two_o right_a line_n be_v erect_v perpendicular_o to_o one_o &_o the_o self_n same_o plain_a superficies_n those_o right_a line_n be_v parallel_n the_o one_o to_o the_o other_o svppose_v that_o these_o two_o right_a line_n ab_fw-la and_o cd_o be_v erect_v perpendicular_o to_o a_o ground_n plain_a superficies_n then_o i_o say_v that_o the_o line_n ab_fw-la be_v a_o parallel_n to_o the_o line_n cd_o let_v the_o point_n which_o those_o two_o right_a line_n touch_v in_o the_o plain_a superficies_n be_v b_o and_o d._n construction_n and_o draw_v a_o right_a line_n from_o the_o point_n b_o to_o the_o point_n d._n and_o by_o the_o 11._o of_o the_o first_o from_o the_o point_n d_o draw_v unto_o the_o line_n bd_o in_o the_o ground_n superficies_n a_o perpendicular_a line_n de._n and_o by_o the_o 2._o of_o the_o first_o demonstration_n put_v the_o line_n de_fw-fr equal_a to_o the_o line_n ab_fw-la and_o draw_v these_o right_a line_n be_v ae_n and_o ad._n and_o forasmuch_o as_o the_o line_n ab_fw-la be_v erect_v perpendicular_o to_o the_o ground_n superficies_n therefore_o by_o the_o 2._o definition_n of_o the_o eleven_o the_o line_n ab_fw-la make_v right_a angle_n with_o all_o the_o line_n which_o be_v draw_v upon_o the_o ground_n plain_a superficies_n and_o touch_v it_o but_o either_o of_o tâ—ese_n line_n bd_o and_o be_v which_o be_v in_o the_o ground_n superficies_n touch_v the_o line_n ab_fw-la wherefore_o either_o of_o these_o angle_n abdella_n and_o abe_n be_v a_o right_a angleâ—_n and_o by_o the_o same_o reason_n also_o either_o of_o the_o angle_n cdb_n &_o cde_o be_v a_o right_a angle_n and_o forasmuch_o as_o the_o line_n ab_fw-la be_v equal_a to_o the_o line_n de_fw-fr and_o the_o line_n bd_o be_v common_a to_o they_o both_o therefore_o these_o two_o line_n ab_fw-la and_o bd_o be_v equal_a to_o these_o two_o line_n ed_z and_o db_o and_o they_o contain_v right_a angle_n wherefore_o by_o the_o 4._o of_o the_o first_o the_o base_a ad_fw-la be_v equal_a to_o the_o base_a be._n and_o forasmuch_o as_o the_o line_n ab_fw-la be_v equal_a to_o the_o line_n de_fw-fr and_o the_o line_n ad_fw-la to_o the_o line_n be_v therefore_o these_o two_o line_n ab_fw-la and_o be_v be_v equal_a to_o these_o two_o line_n ed_z and_o da_z and_o the_o line_n ae_n be_v a_o common_a base_a to_o they_o both_o wherefore_o the_o angle_n abe_n be_v by_o the_o 8._o of_o the_o first_o equal_a to_o the_o angle_n eda_n but_o the_o angle_n abe_n be_v a_o right_a angle_n wherefore_o also_o the_o angle_n eda_n be_v a_o right_a angle_n wherefore_o the_o line_n ed_z be_v erect_v perpendicular_o to_o the_o line_n dam_fw-ge and_o it_o be_v also_o erect_v perpendicular_o to_o either_o of_o these_o line_n bd_o and_o dc_o wherefore_o the_o line_n ed_z be_v unto_o these_o three_o right_a line_n bd_o dam_fw-ge and_o dc_o erect_v perpendicular_o from_o the_o point_n where_o these_o three_o right_a line_n touch_v the_o one_o the_o other_o wherefore_o by_o the_o 5._o of_o the_o eleven_o these_o three_o right_a line_n bd_o dam_fw-ge and_o dc_o be_v in_o one_o and_o the_o self_n same_o superficies_n and_o in_o what_o superficies_n the_o line_n bd_o and_o da_z be_v in_o the_o self_n same_o also_o be_v the_o line_n basilius_n for_o every_o triangle_n be_v by_o the_o 2._o of_o the_o eleven_o in_o one_o and_o the_o self_n
the_o line_n of_o be_v erect_v perpendicular_o to_o the_o superficies_n wherein_o the_o line_n gh_o and_o gk_o be_v but_o the_o line_n of_o be_v a_o parallel_a line_n to_o the_o line_n ab_fw-la wherefore_o by_o tâ—e_z 8._o of_o the_o eleven_o the_o line_n ab_fw-la be_v erect_v perpendicular_o to_o the_o plain_a superficies_n wherein_o be_v the_o line_n gh_o and_o gk_o and_o by_o the_o same_o reason_n also_o the_o line_n cd_o be_v erect_v perpendicular_o to_o the_o plain_a superficies_n wherein_o be_v the_o line_n gh_o &_o gk_o wherefore_o either_o of_o these_o line_n ab_fw-la and_o cd_o be_v erect_v perpendicular_o to_o the_o plain_a superficies_n wherein_o the_o line_n gh_o and_o gk_o be_v but_o if_o two_o right_a line_n be_v erect_v perpendicular_o to_o one_o and_o the_o self_n same_o plain_a superficies_n those_o right_a line_n be_v parallel_n the_o one_o to_o the_o other_o by_o the_o 6._o of_o the_o eleven_o wherefore_o the_o line_n ab_fw-la be_v a_o parallel_n to_o the_o line_n cd_o wherefore_o right_a line_n which_o be_v parallel_n to_o one_o &_o the_o self_n same_o right_a line_n and_o be_v not_o in_o the_o self_n same_o superficies_n with_o it_o be_v also_o parallel_v the_o one_o to_o the_o other_o which_o be_v require_v to_o be_v prove_v this_o figure_n more_o clear_o manifest_v the_o former_a proposition_n and_o demonstration_n if_o you_o elevate_v the_o superficiece_n abef_n and_o cdef_n that_o they_o may_v incline_v and_o concur_v in_o the_o line_n ef._n ¶_o the_o 10._o theorem_a the_o 1_o 〈…〉_o if_o two_o right_a line_n touch_v the_o one_o the_o other_o 〈…〉_o she_o right_n line_n touch_v the_o one_o the_o other_o and_o no_o 〈…〉_o lfe_n same_o superficies_n with_o the_o two_o first_o those_o right_a line_n contain_v equal_a angle_n svppose_v that_o these_o two_o right_a line_n ab_fw-la and_o bc_o touch_v the_o one_o the_o other_o be_v parallel_n to_o these_o two_o line_n de_fw-fr and_o of_o touch_v also_o the_o one_o the_o other_o and_o not_o be_v in_o the_o self_n same_o superficies_n that_o the_o line_n ab_fw-la and_o bc_o be_v then_o i_o say_v that_o the_o angle_n abc_n be_v equal_a to_o the_o angle_n def_n construction_n for_o let_v the_o line_n ba_o bc_o ed_z of_o be_v put_v equal_a the_o one_o to_o the_o other_o and_o draw_v these_o right_a line_n ad_fw-la cf_o be_v ac_fw-la and_o df._n demonstration_n and_o forasmuch_o as_o the_o line_n basilius_n be_v equal_a to_o the_o line_n ed_z and_o also_o parallel_v unto_o it_o therefore_o by_o the_o 33._o of_o the_o first_o the_o line_n ad_fw-la be_v equal_a and_o parallel_v to_o the_o line_n be_v and_o by_o the_o same_o reason_n also_o the_o line_n cf_o be_v equal_a &_o parallel_v to_o the_o line_n be._n wherefore_o either_o of_o these_o line_n ad_fw-la and_o cf_o be_v equal_a &_o parallel_v to_o the_o line_n ebb_n but_o right_a line_n which_o be_v parallel_n to_o one_o and_o the_o self_n same_o right_a line_n and_o be_v not_o in_o the_o self_n same_o superficies_n with_o it_o be_v also_o by_o the_o 9_o of_o the_o eleven_o parallel_n the_o one_o to_o the_o other_o wherefore_o the_o line_n ad_fw-la be_v a_o parallel_a line_n to_o the_o line_n cf._n and_o the_o line_n ac_fw-la and_o df_o join_v they_o together_o wherefore_o by_o the_o 33._o of_o the_o first_o the_o line_n ac_fw-la be_v equal_a and_o parallel_v to_o the_o line_n df._n and_o forasmuch_o as_o these_o two_o right_a line_n ab_fw-la &_o bc_o be_v equal_a to_o these_o two_o right_a line_n de_fw-fr and_o of_o and_o the_o base_a ac_fw-la also_o be_v equal_a to_o the_o base_a df_o therefore_o by_o the_o 8._o of_o the_o first_o the_o angle_n abc_n be_v equal_a to_o the_o angle_n def_n if_o therefore_o two_o right_a line_n touch_v the_o one_o the_o other_o be_v parallel_n to_o two_o other_o right_a line_n touch_v the_o one_o the_o other_o and_o not_o be_v in_o one_o and_o the_o self_n same_o superficies_n with_o the_o two_o first_o those_o righâ—_n line_n contain_v equal_a angle_n which_o be_v require_v to_o be_v demonstrate_v this_o figure_n here_o set_v more_o plain_o declare_v the_o former_a proposition_n and_o demonstration_n if_o you_o elevate_v the_o superficiece_n dabe_n and_o fcbe_n till_o they_o concur_v in_o the_o line_n fe_o ¶_o the_o 1._o problem_n the_o 11._o proposition_n from_o a_o point_n give_v on_o high_a to_o draw_v unto_o a_o ground_n plain_a superficies_n a_o perpendicular_a right_a line_n case_n svppose_v that_o the_o point_n give_v on_o high_a be_v a_o and_o suppose_v a_o ground_n plain_a superficies_n namely_o bcgh_n it_o be_v require_v from_o the_o point_n a_o to_o draw_v unto_o the_o ground_n superficies_n a_o perpendicular_a line_n draw_v in_o the_o ground_n superficies_n a_o right_a line_n at_o adventure_n and_o let_v the_o same_o be_v bc._n dee_n and_o by_o the_o 12._o of_o the_o first_o from_o the_o point_n a_o draw_v unto_o the_o line_n bc_o a_o perpendicular_a line_n ad._n practise_v now_o if_o ad_fw-la be_v a_o perpendicular_a line_n to_o the_o ground_n superficies_n then_o be_v that_o do_v which_o be_v seek_v for_o but_o if_o not_o then_o by_o the_o 11._o of_o the_o first_o from_o the_o point_n d_o raise_v up_o in_o the_o ground_n superficies_n unto_o the_o line_n bc_o a_o perpendicular_a line_n de._n and_o by_o the_o 12._o of_o the_o first_o from_o the_o point_n a_o draw_v unto_o the_o line_n de_fw-fr a_o perpendicular_a line_n af._n and_o by_o the_o point_n f_o draw_v by_o the_o 31._o of_o the_o â—irst_n unto_o the_o line_n bc_o a_o parallel_a line_n fh_o and_o extend_v the_o line_n fh_o from_o the_o point_n f_o to_o the_o point_n g._n demonstration_n and_o forasmuch_o as_o the_o line_n bc_o be_v erect_v perpendicular_o to_o either_o of_o these_o line_n de_fw-fr and_o da_z therefore_o by_o the_o 4._o of_o the_o eleven_o the_o line_n bc_o be_v erect_v perpendicular_o to_o the_o superficies_n wherein_o the_o line_n ed_z and_o ad_fw-la be_v and_o to_o the_o line_n bc_o the_o line_n gh_o be_v a_o parallel_n but_o iâ—_z there_o be_v two_o parallel_n right_a line_n of_o which_o one_o be_v erect_v perpendicular_o to_o a_o certain_a plain_a superficies_n the_o other_o also_o by_o the_o 8._o of_o the_o eleven_o be_v erect_v perpendicular_o to_o the_o self_n same_o superficies_n wherefore_o the_o line_n gh_o be_v erect_v perpendicular_o to_o the_o plainâ—_n superficies_n wherein_o the_o line_n ed_z and_o da_z be_v wherefore_o also_o by_o the_o 2._o definition_n of_o the_o eleven_o the_o line_n gh_o be_v erect_v perpendicular_o to_o all_o the_o right_a line_n which_o touch_v it_o and_o be_v in_o the_o plain_a superficies_n wherein_o the_o line_n ed_z and_o ad_fw-la be_v but_o the_o line_n of_o touch_v it_o be_v in_o the_o superficies_n wherein_o the_o line_n ed_z and_o ad_fw-la be_v by_o the_o â—_o of_o this_o book_n wherefore_o the_o line_n gh_o be_v erect_v perpendicular_o to_o the_o line_n fa._n wherefore_o also_o the_o line_n favorina_n be_v erect_v perpendicular_o to_o the_o line_n gh_o and_o the_o line_n of_o be_v also_o erect_v perpendicular_o to_o the_o line_n de._n wherefore_o of_o be_v erect_v perpendicular_o to_o either_o of_o these_o line_n hg_o and_o de._n but_o if_o a_o right_a line_n be_v erect_v perpendicular_o from_o the_o common_a section_n of_o two_o right_a line_n cut_v the_o one_o the_o other_o it_o shall_v also_o be_v erect_v perpendicular_o to_o the_o plain_a superficies_n of_o the_o say_v two_o line_n by_o the_o 4._o of_o the_o eleven_o wherefore_o the_o line_n of_o be_v erect_v perpendicular_o to_o that_o superficies_n wherein_o the_o line_n ed_z and_o gh_o be_v but_o the_o superficies_n wherein_o the_o line_n ed_z and_o gh_o be_v be_v the_o ground_n superficies_n wherefore_o the_o line_n of_o be_v erect_v perpendicular_o to_o the_o ground_n superficies_n wherefore_o from_o a_o point_n give_v on_o high_a namely_o from_o the_o point_n a_o be_v draw_v to_o the_o ground_n superficies_n a_o perpendicular_a line_n which_o be_v require_v to_o be_v do_v in_o this_o figure_n shall_v you_o much_o more_o plain_o see_v both_o the_o case_n of_o this_o former_a demonstration_n for_o as_o touch_v the_o first_o case_n you_o must_v erect_v perpendicular_o to_o the_o ground_n superficies_n the_o superficies_n wherein_o be_v draw_v the_o line_n ad_fw-la and_o compare_v it_o with_o the_o demonstration_n and_o it_o will_v be_v clear_a unto_o you_o for_o the_o second_o case_n you_o must_v erect_v perpendicular_o unto_o the_o ground_n superficies_n the_o superficies_n wherein_o be_v draw_v the_o line_n of_o and_o unto_o it_o let_v the_o other_o superficies_n wherein_o be_v draw_v the_o line_n ad_fw-la incline_v so_o that_o the_o point_n a_o of_o the_o one_o may_v concur_v with_o the_o point_n a_o of_o the_o other_o and_o so_o with_o your_o figure_n thus_o order_v compare_v it_o with_o
the_o demonstration_n and_o there_o will_v be_v in_o it_o no_o hardness_n at_o all_o ¶_o the_o 2._o problem_n the_o 12._o proposition_n unto_o a_o plain_a superficies_n give_v and_o from_o a_o point_n in_o it_o give_v to_o raise_v up_o a_o perpendicular_a line_n svppose_v that_o there_o be_v a_o ground_n plain_a superficies_n give_v and_o let_v the_o point_n in_o it_o give_v be_v a._n it_o be_v require_v from_o the_o point_n a_o to_o raise_v up_o unto_o the_o ground_n plain_a superficies_n a_o perpendicular_a line_n understand_v some_o certain_a point_n on_o high_a and_o let_v the_o same_o be_v b._n construction_n and_o from_o the_o point_n b_o draw_v by_o the_o 11._o of_o the_o eleven_o a_o perpendicular_a line_n to_o the_o ground_n superficies_n and_o let_v the_o same_o be_v bc._n and_o by_o the_o 31._o of_o the_o first_o by_o the_o point_n a_o draw_v unto_o the_o line_n bc_o a_o parallel_a line_n da._n now_o forasmuch_o as_o there_o be_v two_o parallel_n right_a line_n ad_fw-la and_o cb_o the_o one_o of_o they_o namely_o demonstration_n cb_o be_v erect_v perpendicular_o to_o the_o 〈◊〉_d superficies_n wherefore_o the_o other_o line_n also_o namely_o ad_fw-la be_v 〈◊〉_d perpendicular_o to_o the_o same_o ground_n superficies_n by_o the_o eight_o of_o 〈…〉_o leventh_o wherefore_o unto_o a_o plain_a superficies_n give_v and_o 〈◊〉_d point_n in_o it_o give_v namely_o a_o be_v raise_v up_o a_o perpendicular_a lynâ—â—required_v to_o be_v do_v in_o this_o second_o figure_n you_o may_v consider_v plain_o the_o demonstration_n of_o the_o former_a proposition_n if_o you_o erect_v perpendicular_o the_o superficies_n wherein_o be_v draw_v the_o line_n ad_fw-la and_o cb._n ¶_o the_o 11._o theorem_a the_o 13._o prâ—position_n from_o one_o and_o the_o self_n point_n and_o to_o one_o and_o the_o self_n same_o plain_a superficies_n can_v not_o be_v erect_v two_o perpendicular_a right_a line_n on_o one_o and_o the_o self_n same_o side_n for_o if_o it_o be_v possible_a from_o the_o point_n a_o let_v there_o be_v erect_v perpendicular_o to_o one_o and_o the_o self_n same_o plain_a superficies_n two_o righâ—_n line_n ab_fw-la and_o ac_fw-la on_o one_o and_o the_o self_n same_o side_n impossibility_n and_o extend_v the_o superficies_n wherein_o be_v the_o line_n ab_fw-la and_o ac_fw-la mathematical_a and_o it_o shall_v make_v at_o length_n a_o common_a section_n in_o the_o ground_n superficies_n which_o common_a section_n shall_v be_v a_o right_a line_n and_o shall_v pass_v by_o the_o point_n a_o let_v that_o common_a section_n be_v the_o line_n dae_n wherefore_o by_o the_o 3._o of_o the_o eleven_o the_o line_n ab_fw-la ac_fw-la and_o dae_n be_v in_o one_o and_o the_o self_n same_o plain_a superficies_n and_o forasmuch_o as_o the_o line_n ca_n be_v erect_v perpendicular_o to_o the_o ground_n superficies_n therefore_o by_o the_o 2._o definition_n of_o the_o eleven_o it_o make_v right_a angle_n with_o all_o the_o right_a line_n that_o touch_v it_o and_o be_v in_o the_o ground_n superficies_n but_o the_o line_n dae_n touch_v it_o be_v in_o the_o ground_n superficies_n wherefore_o the_o angle_n caesar_n be_v a_o right_a angle_n and_o by_o the_o same_o reason_n also_o the_o angle_n bae_o be_v a_o right_a angle_n wherefore_o by_o the_o 4_o petition_n the_o angle_n caesar_n be_v equal_a to_o the_o angle_n bae_o the_o less_o to_o the_o more_o both_o angle_n be_v in_o one_o &_o the_o self_n same_o plain_a superficies_n which_o be_v impossible_a wherefore_o from_o one_o and_o the_o self_n same_o point_n and_o to_o one_o and_o the_o self_n same_o plain_a superficies_n can_v not_o be_v erect_v two_o perpendicular_a right_a line_n on_o one_o &_o the_o self_n same_o side_n which_o be_v require_v to_o be_v demonstrate_v in_o this_o figure_n if_o you_o erect_v perpendicular_o the_o superficies_n wherein_o be_v draw_v the_o line_n â—a_o and_o ca_n to_o the_o ground_n superficies_n wherein_o be_v draw_v the_o line_n dae_n and_o so_o compare_v it_o with_o the_o the_o demonstration_n of_o the_o former_a proposition_n it_o will_v be_v clear_a unto_o you_o m._n do_v you_o his_o annotation_n euclides_n word_n in_o this_o 13._o proposition_n admit_v two_o case_n one_o if_o thâ—_z 〈…〉_o in_o the_o plain_a superficies_n as_o common_o the_o demonstration_n suppose_v the_o other_o if_o the_o point_n assign_v be_v any_o where_o without_o the_o say_v plain_a superficies_n to_o which_o the_o perpendicular_n fall_v be_v consider_v contrary_n to_o either_o of_o which_o if_o the_o adversary_n affirm_v admit_v from_o one_o point_n two_o right_a line_n perpendicular_n to_o one_o and_o the_o self_n same_o plain_a superficies_n and_o on_o one_o and_o the_o same_o side_n thereof_o by_o the_o 6._o of_o the_o eleven_o he_o may_v be_v bridle_v which_o will_v â—ore_v he_o to_o confess_v his_o two_o perpendicular_n to_o be_v also_o parallel_v but_o by_o supposition_n agree_v one_o they_o concur_v at_o one_o and_o the_o same_o poynâ—_n which_o by_o the_o definition_n of_o parallel_n iâ—_z impossible_a therefore_o our_o adversary_n must_v recant_v aâ—d_n yield_v to_o out_o proposition_n ¶_o the_o 12._o theorem_a the_o 14._o proposition_n to_o whatsoever_o plain_a superficiece_n one_o and_o the_o self_n same_o right_a line_n be_v erect_v perpendicular_o those_o superficiece_n be_v parallel_n the_o one_o to_o the_o other_o svppose_v that_o a_o right_a line_n ab_fw-la be_v erect_v perpendicular_o to_o either_o of_o these_o plain_a superficiece_n cd_o and_o ef._n then_o i_o say_v that_o these_o superficiece_n cd_o and_o of_o be_v parallel_n the_o one_o to_o the_o other_o impossibility_n for_o if_o not_o then_o if_o they_o be_v extend_v they_o will_v at_o the_o length_n meet_v let_v they_o meet_v if_o it_o be_v possible_a now_o than_o their_o common_a section_n shall_v by_o the_o 3._o of_o the_o eleven_o be_v a_o right_a line_n let_v that_o common_a section_n be_v gh_o and_o in_o the_o line_n gh_o take_v a_o point_n at_o all_o adventure_n and_o let_v the_o same_o be_v k._n and_o draw_v a_o right_a line_n from_o the_o point_n a_o to_o the_o point_n king_n and_o a_o other_o from_o the_o point_n b_o to_o the_o point_n k._n and_o forasmuch_o as_o the_o line_n ab_fw-la be_v erect_v perpendicular_o to_o the_o plain_a superficies_n of_o therefore_o the_o line_n ab_fw-la be_v also_o erect_v perpendicular_o to_o the_o line_n bk_o which_o be_v in_o the_o extend_a superficies_n ef._n wherefore_o the_o angle_n abk_n be_v a_o right_a angle_n and_o by_o the_o same_o reason_n also_o the_o angle_n back_o be_v a_o right_a angle_n wherefore_o in_o the_o triangle_n abk_n these_o two_o angle_n abk_n &_o bak_n be_v equal_a to_o two_o right_a angle_n which_o by_o the_o 17._o of_o the_o first_o be_v impossible_a wherefore_o these_o superficiece_n cd_o and_o of_o be_v extend_v meet_v not_o together_o wherefore_o the_o superficiece_n cd_o and_o of_o be_v parallel_n wherefore_o to_o what_o soever_o plain_a superficiece_n one_o and_o the_o self_n same_o right_a line_n iâ—_z erect_v perpendicular_o those_o superficies_n be_v parallel_n the_o one_o to_o the_o other_o which_o be_v require_v to_o be_v prove_v in_o this_o figure_n may_v you_o plain_o see_v the_o former_a demonstration_n if_o you_o erect_v the_o three_o superficiece_n gd_o ge_z and_o klm_v perpendiculary_a to_o the_o ground_n plain_a superficies_n but_o yet_o in_o such_o sort_n that_o the_o two_o superficiâ—â—_n 〈…〉_o may_v concur_v in_o the_o common_a line_n g_o 〈…〉_o the_o demonstration_n a_o corollary_n add_v by_o campane_n if_o a_o right_a line_n be_v erect_v perpendicular_o to_o one_o of_o those_o superficies_n it_o 〈…〉_o erect_v perpendicular_o to_o the_o other_o for_o if_o it_o shall_v not_o be_v erect_v perpendicular_o to_o the_o other_o than_o it_o fall_v upon_o that_o other_o shall_v make_v with_o some_o one_o line_n thereof_o a_o angle_n less_o than_o a_o right_a angle_n which_o line_n shall_v by_o the_o 5._o petition_n of_o the_o first_o concur_v with_o some_o one_o line_n of_o that_o superficies_n whereunto_o it_o be_v perpendicular_a so_o that_o those_o superficiece_n shall_v not_o be_v parallel_n which_o be_v contrary_a to_o the_o supposition_n for_o they_o be_v suppse_v to_o be_v parallel_n ¶_o the_o 13._o theorem_a the_o 15._o proposition_n if_o two_o right_a line_n touch_v the_o one_o the_o other_o be_v parallel_n to_o two_o other_o right_a line_n touch_v also_o the_o one_o the_o other_o and_o not_o be_v in_o the_o self_n same_o plain_a superficies_n with_o the_o two_o first_o the_o plain_a superficiece_n extend_v by_o those_o right_a line_n be_v also_o parallel_n the_o one_o to_o the_o other_o svppose_v that_o these_o two_o right_a line_n ab_fw-la and_o bc_o touch_v the_o one_o the_o other_o be_v parallel_n to_o these_o two_o right_a line_n de_fw-fr &_o of_o touch_v also_o the_o one_o the_o other_o and_o not_o be_v in_o the_o self_n same_o plain_a superficies_n with_o
the_o right_a line_n ab_fw-la and_o bc._n then_o i_o say_v that_o the_o plain_a superficiece_n by_o the_o line_n ab_fw-la and_o bc_o and_o the_o line_n de_fw-fr and_o of_o be_v extend_v shall_v not_o meet_v together_o that_o be_v they_o be_v equedistant_a and_o parallel_n construction_n from_o the_o point_n b_o draw_v by_o the_o 11._o of_o the_o eleven_o a_o perpendicular_a line_n to_o the_o superficies_n wherein_o be_v the_o line_n de_fw-fr and_o of_o and_o let_v that_o perpendicular_a line_n be_v bg_o and_o by_o the_o point_n g_o in_o the_o plain_a superficies_n pass_v by_o de_fw-fr and_o of_o draw_v by_o the_o 31._o of_o the_o first_o unto_o the_o line_n ed_z a_o parallel_a line_n gh_o and_o likewise_o by_o that_o point_n g_o draw_v in_o the_o same_o superficies_n unto_o the_o line_n of_o a_o parallel_a line_n gk_o demonstration_n and_o forasmuch_o as_o the_o line_n bg_o be_v erect_v perpendicular_o to_o the_o superficies_n wherein_o be_v the_o line_n de_fw-fr and_o of_o therefore_o by_o the_o 2._o definition_n of_o the_o eleven_o it_o be_v also_o erect_v perpendicular_o to_o all_o the_o right_a line_n which_o touch_v it_o and_o be_v in_o the_o self_n same_o superficies_n wherein_o be_v the_o line_n de_fw-fr and_o ef._n but_o either_o of_o these_o line_n gh_o and_o gk_o touch_v it_o and_o be_v also_o in_o the_o superficies_n wherein_o be_v the_o line_n de_fw-fr and_o of_o therefore_o either_o of_o these_o angle_n bgh_o and_o bgk_n be_v a_o right_a angle_n and_o forasmuch_o as_o the_o line_n basilius_n be_v a_o parallel_n to_o the_o line_n gh_o that_o the_o line_n gh_o and_o gk_o be_v parallel_n unto_o the_o line_n ab_fw-la and_o bc_o it_o be_v manifest_a by_o the_o 9_o of_o this_o book_n therefore_o by_o the_o 29._o of_o the_o first_o the_o angle_n gba_n and_o bgh_n be_v equal_a to_o two_o right_a angle_n but_o the_o angle_n bgh_n be_v by_o construction_n a_o right_a angle_n therefore_o also_o the_o angle_n gba_n be_v a_o right_a angle_n therefore_o the_o line_n gb_o be_v erect_v perpendicular_o to_o the_o line_n basilius_n and_o by_o the_o same_o reason_n also_o may_v it_o be_v prove_v that_o the_o line_n bg_o be_v erect_v perpendicular_o to_o the_o line_n bc._n now_o forasmuch_o as_o the_o right_a line_n bg_o be_v erect_v perpendicular_o to_o these_o two_o right_a line_n basilius_n and_o bc_o touch_v the_o one_o the_o other_o therefore_o by_o the_o 4._o of_o the_o eleven_o the_o line_n bg_o be_v erect_v perpendicular_o to_o the_o superficies_n wherein_o be_v the_o line_n basilius_n and_o be_v and_o it_o be_v also_o erect_v perpendicular_o to_o the_o superficies_n wherein_o be_v the_o line_n gh_o and_o gk_o but_o the_o superficies_n wherein_o be_v the_o line_n gh_o and_o gk_o be_v that_o superficies_n wherein_o be_v the_o line_n de_fw-fr and_o of_o wherefore_o the_o line_n bg_o be_v erect_v perpendicular_o to_o the_o superficies_n wherein_o be_v the_o line_n de_fw-fr and_o ef._n wherefore_o the_o line_n bg_o be_v erect_v perpendicular_o to_o the_o superficies_n wherein_o be_v the_o line_n de_fw-fr and_o of_o and_o to_o the_o superficies_n wherein_o be_v the_o line_n ab_fw-la and_o bc._n but_o if_o one_o and_o the_o self_n same_o right_a line_n be_v erect_v perpendicular_o to_o plain_a superficiece_n those_o superficiece_n be_v by_o the_o 14._o of_o the_o eleven_o parallel_v the_o one_o to_o the_o other_o wherefore_o the_o superficies_n wherein_o be_v the_o line_n ab_fw-la and_o bc_o be_v a_o parallel_n to_o the_o superficies_n wherein_o be_v the_o line_n de_fw-fr and_o ef._n if_o therefore_o two_o right_a line_n touch_v the_o one_o the_o other_o be_v parallel_n to_o two_o other_o right_a line_n touch_v also_o the_o one_o the_o other_o and_o not_o be_v in_o the_o self_n same_o plain_a superficies_n with_o the_o two_o first_o the_o plain_a superficiece_n extend_v by_o those_o right_a line_n be_v also_o parallel_v the_o one_o to_o the_o other_o which_o be_v require_v to_o be_v demonstrate_v by_o this_o figure_n here_o put_v you_o may_v more_o clear_o see_v both_o the_o former_a 15._o proposition_n and_o also_o the_o demonstration_n thereof_o if_o you_o erect_v perpendicular_o unto_o the_o ground_n superficies_n the_o three_o superficiece_n abc_n khe_n and_o lhbm_n and_o so_o compare_v it_o with_o the_o demonstration_n ¶_o a_o corollary_n add_v by_o flussas_n unto_o a_o plain_a superficies_n be_v give_v to_o draw_v by_o a_o point_n give_v without_o it_o a_o parallel_n plain_a superficies_n suppose_v as_o in_o the_o former_a description_n that_o the_o superficies_n give_v be_fw-mi abc_n &_o let_v the_o point_n give_v without_o it_o be_v g._n now_o then_o by_o the_o point_n g_o draw_v by_o the_o 31._o of_o the_o first_o unto_o the_o line_n ab_fw-la and_o bc_o parallel_v line_n gh_o and_o hk_o and_o the_o superficies_n extend_v by_o the_o line_n gh_o and_o gk_o shall_v be_v parallel_n unto_o the_o superficies_n abc_n by_o this_o 15._o proposition_n the_o 14._o theorem_a the_o 16._o proposition_n if_o two_o parallel_n plain_a superficiece_n be_v cut_v by_o some_o one_o plain_a superficies_n their_o common_a section_n be_v parallel_a line_n svppose_v that_o these_o two_o plain_a superficiece_n ab_fw-la and_o cd_o be_v cut_v by_o this_o plain_a superficies_n efgh_o â—nd_v let_v their_o common_a section_n be_v the_o right_a line_n of_o and_o gh_o then_o i_o say_v that_o the_o line_n of_o be_v a_o parallel_n to_o the_o line_n gh_o for_o if_o not_o than_o the_o line_n of_o and_o gh_o be_v produce_v shall_v at_o the_o length_n meet_v together_o either_o on_o the_o side_n that_o the_o point_n fh_o be_v or_o on_o the_o side_n that_o the_o point_n e_o g_o be_v first_o let_v they_o be_v produce_v on_o that_o side_n that_o the_o point_n f_o h_o be_v and_o let_v they_o mete_v in_o the_o point_n k._n and_o forasmuch_o as_o the_o line_n efk_n be_v in_o the_o superficies_n ab_fw-la therefore_o all_o the_o point_n which_o be_v in_o the_o line_n of_o be_v in_o the_o superficies_n ab_fw-la by_o the_o first_o of_o this_o book_n but_o one_o of_o the_o point_n which_o be_v in_o the_o right_a line_n efk_n be_v the_o point_n king_n absurdity_n therefore_o the_o point_n king_n be_v in_o the_o superficies_n ab_fw-la and_o by_o the_o same_o reason_n also_o the_o point_n king_n be_v in_o the_o superficies_n cd_o wherefore_o the_o two_o superficiece_n ab_fw-la and_o cd_o be_v produce_v do_v mete_v together_o but_o by_o supposition_n they_o meet_v not_o together_o for_o they_o be_v suppose_v to_o be_v parallel_n wherefore_o the_o right_a line_n of_o and_o gh_o produce_v shall_v not_o meet_v together_o on_o that_o side_n that_o the_o point_n f_o h_o be_v in_o like_a sort_n also_o may_v we_o prove_v that_o the_o right_a line_n of_o and_o gh_o produce_v meet_v not_o together_o on_o that_o side_n that_o the_o point_n e_o g_o be_v but_o right_a line_n which_o be_v produce_v on_o no_o side_n mete_v together_o be_v parallel_n by_o the_o last_o definition_fw-la of_o the_o first_o wherefore_o the_o line_n of_o be_v a_o parallel_n to_o the_o line_n gh_o if_o therefore_o two_o parallel_n plain_a superficiece_n be_v cut_v by_o some_o one_o plain_a superficies_n their_o common_a section_n be_v parallel_a line_n which_o be_v require_v to_o be_v prove_v this_o figure_n here_o set_v more_o plain_o 〈…〉_o demonstration_n if_o you_o erect_v perpendicular_a 〈…〉_o superficies_n the_o three_o superficiece_n aâ—_z 〈…〉_o and_o so_o compare_v it_o with_o the_o demonstrâ—_n 〈…〉_o a_o corollary_n add_v by_o flussas_n if_o two_o plain_a superficiece_n be_v parallel_n to_o one_o and_o the_o sâ—lfe_a same_o plain_n 〈…〉_o also_o be_v parallel_n the_o one_o to_o the_o other_o or_o they_o shall_v make_v one_o and_o the_o self_n same_o plain_a sup_n 〈…〉_o in_o this_o figure_n here_o set_v you_o may_v more_o plain_o see_v the_o forme_a demonstration_n if_o you_o elevate_v to_o the_o ground_n superâ—icieces_v acdi_n the_o three_o superâ—icieces_n ab_fw-la dg_o &_o give_v and_o â—o_o compare_v it_o with_o the_o demonstration_n the_o 15._o theorem_a the_o 17._o proposition_n iâ—_z two_o right_a line_n be_v cut_v by_o plain_a superficiece_n be_v parallel_n the_o part_n oâ—_n the_o line_n divide_v shall_v be_v proportional_a demonstration_n suppose_v that_o these_o two_o right_a line_n ab_fw-la and_o cd_o be_v divide_v by_o these_o plain_a superfiâ—iâ—ces_n be_v parallel_n namely_o gh_o kl_o mn_v in_o the_o point_v a_o e_o b_o c_o f_o d._n then_o i_o say_v that_o as_o the_o right_a line_n ae_n be_v to_o the_o right_a line_n ebb_v so_o be_v the_o right_a line_n cf_o to_o the_o right_a line_n fd._n draw_v these_o right_a line_n ac_fw-la bd_o and_o ad._n and_o let_v the_o line_n ad_fw-la and_o the_o superficies_n kl_o concur_v in_o the_o point_n x._o and_o draw_v a_o right_a line_n from_o the_o point_n e_o to_o the_o point_n x_o and_o
ab_fw-la bg_o de_fw-fr of_o gh_o demonstration_n or_o hk_o make_v the_o line_n hl_o equal_a &_o draw_v these_o right_a line_n kl_o and_o gl_n and_o forasmuch_o as_o these_o two_o line_n ab_fw-la and_o dc_o be_v equal_a to_o these_o two_o line_n kh_o and_o hl_o and_o the_o angle_n b_o be_v equal_a to_o the_o angle_n khl_n therâ—â—â—â—_n by_o the_o 4._o of_o the_o first_o the_o base_a ac_fw-la be_v equal_a to_o the_o base_a kl_o and_o forasmuch_o as_o the_o angle_n abc_n and_o ghk_o be_v great_a than_o the_o angle_n def_n but_o the_o angle_n ghl_n be_v equal_a to_o the_o angle_n abc_n &_o ghkâ—_n therefore_o the_o angle_n ghl_n be_v great_a than_o the_o angle_n def_n and_o forasmuch_o as_o these_o two_o line_n gh_o and_o hl_o be_v equal_a to_o these_o two_o line_n de_fw-fr and_o of_o and_o the_o angle_n ghl_n be_v greaâ—er_a than_o the_o angle_n def_n therefore_o by_o the_o 25._o of_o the_o first_o the_o base_a gd_o be_v great_a they_o the_o base_a df._n but_o the_o line_n gk_o and_o kl_o be_v great_a than_o the_o line_n gl_n wherefore_o then_o line_n gk_o &_o kl_o be_v much_o great_a than_o the_o line_n df._n but_o the_o line_n kl_o be_v equal_a to_o the_o line_n ac_fw-la wherefore_o the_o line_n ac_fw-la and_o gk_o be_v great_a than_o the_o line_n df._n in_o like_a sort_n also_o may_v we_o proâ—â—â—_n thaâ—_z the_o line_n ac_fw-la and_o df_o be_v great_a than_o the_o line_n gk_o and_o that_o the_o line_n gk_o and_o df_o be_v great_a than_o the_o line_n ac_fw-la wherefore_o it_o be_v possible_a to_o make_v a_o triangle_n of_o three_o line_n equal_a to_o the_o line_n ac_fw-la df_o and_o gk_o which_o be_v require_v to_o be_v demonstrate_v an_o other_o demonstration_n suppose_v that_o the_o three_o superficial_a angle_n be_v abc_n def_n and_o ghk_v of_o which_o angle_n demonstration_n then_o howsoever_o they_o be_v take_v be_v great_a than_o the_o three_o and_o let_v they_o be_v contain_v under_o these_o equal_a right_a line_n ab_fw-la bc_o de_fw-fr of_o gh_o hk_o which_o equal_a right_a line_n let_v thâ—se_a line_n ac_fw-la df_o and_o gk_o join_v together_o 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 whiâ—h_v again_o be_v as_o much_o to_o say_v as_o that_o two_o of_o those_o line_n which_o then_o soever_o be_v take_v be_v great_a than_o the_o three_o now_o again_o if_o the_o angle_n b_o e_o h_o be_v equal_a the_o line_n also_o ac_fw-la df_o and_o gk_o be_v equal_a and_o so_o two_o of_o they_o shall_v be_v great_a than_o the_o three_o but_o if_o not_o lâ—t_v the_o angle_n b_o e_o h_o be_v unequal_a and_o let_v the_o angle_n b_o be_v great_a than_o â—ither_o of_o the_o angle_n e_o and_o h._n therefore_o by_o the_o 24._o of_o the_o first_o the_o right_a line_n ac_fw-la be_v great_a than_o either_o of_o the_o line_n df_o &_o gk_o and_o it_o be_v manifest_a that_o the_o line_n ac_fw-la with_o either_o of_o the_o line_n df_o or_o gk_o be_v great_a than_o the_o three_o i_o say_v also_o that_o the_o line_n df_o and_o gk_o be_v great_a than_o the_o line_n ac_fw-la unto_o the_o right_a line_n ab_fw-la construction_n and_o to_o the_o point_n in_o it_o b_o make_v by_o the_o 22._o of_o the_o first_o unto_o the_o angle_n ghk_v a_o equal_a angle_n ab_fw-la l_o and_o unto_o one_o of_o the_o line_n ab_fw-la bc_o de_fw-fr of_o gh_o or_o hk_v make_v by_o the_o 2._o of_o the_o first_o a_o equal_a line_n bl_o and_o draw_v a_o right_a line_n from_o the_o point_n a_o to_o the_o point_fw-fr l_o and_o a_o other_o from_o the_o point_n l_o to_o the_o point_n c._n demonstration_n and_o forasmuch_o as_o these_o two_o line_n ab_fw-la &_o bl_o be_v equal_a to_o these_o two_o line_n gh_o &_o hk_v the_o one_o to_o the_o other_o and_o they_o contain_v equal_a angle_n therefore_o by_o the_o 4._o of_o the_o first_o the_o base_a all_o be_v equal_a to_o the_o base_a gk_o and_o forasmuch_o as_o the_o angle_n e_o and_o h_o be_v great_a than_o the_o angle_n abc_n of_o which_o the_o angle_n ghk_o be_v equal_a to_o the_o angle_n abl_n therefore_o the_o angle_n remain_v namely_o the_o angle_n e_o be_v great_a than_o the_o angle_n lbc_n and_o forasmuch_o as_o these_o two_o line_n lb_n and_o bc_o be_v equal_a to_o these_o two_o line_n de_fw-fr and_o of_o the_o one_o to_o the_o other_o and_o the_o angle_n def_n be_v great_a than_o the_o angle_n lbc_n therefore_o by_o the_o 25._o of_o the_o first_o the_o base_a df_o be_v great_a than_o the_o base_a lc_n and_o it_o be_v prove_v that_o the_o line_n gk_o be_v equal_a to_o the_o line_n al._n wherefore_o the_o line_n df_o &_o gk_o be_v great_a than_o the_o linâ—s_v all_o &_o lc_n bâ—t_v the_o liâ—es_v all_o and_o lc_n be_v great_a than_o the_o line_n ac_fw-la wherefore_o the_o line_n df_o &_o gk_o be_v much_o great_a they_o thâ—_z line_n ac_fw-la wherefore_o two_o of_o these_o right_a line_n ac_fw-la df_o &_o gk_o which_o two_o soever_o be_v take_v be_v great_a than_o thâ—_z three_o wherefore_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 which_o be_v require_v to_o be_v demonstrate_v ¶_o the_o 3._o problem_n the_o 23._o proposition_n of_o three_o plain_a superficial_a angle_n two_o of_o which_o how_o soever_o they_o be_v take_v be_v great_a than_o the_o three_o to_o make_v a_o solid_a angle_n now_o it_o be_v necessary_a that_o those_o three_o superficial_a angle_n be_v less_o than_o four_o right_a angle_n svppose_v that_o the_o superficial_a angle_n give_v be_v abc_n def_n ghk_o of_o which_o let_v two_o how_o soever_o they_o be_v take_v be_v great_a than_o the_o three_o and_o moreover_o let_v those_o three_o angle_n be_v lâ—sse_o than_o four_o right_a angle_n it_o be_v require_v of_o three_o superficial_a angle_n equal_a to_o the_o angle_n abc_n def_n and_o ghk_v to_o make_v a_o solid_a or_o bodily_a angle_n let_v the_o line_n ab_fw-la bc_o de_fw-fr of_o gh_o and_o hk_o be_v make_v equal_a construction_n and_o draw_v a_o right_a line_n from_o the_o point_n a_o to_o the_o point_n c_o &_o a_o other_o from_o the_o point_n d_o to_o the_o point_n fletcher_n and_o a_o other_o from_o the_o point_n g_o to_o the_o point_n k._n now_o by_o the_o 22._o of_o the_o eleven_o it_o be_v possible_a of_o three_o right_a line_n equal_a to_o the_o right_a line_n ac_fw-la df_o and_o gk_o to_o make_v a_o triangle_n make_v such_o a_o triangle_n and_o let_v the_o same_o be_v lmn_o so_o that_o let_v the_o line_n ac_fw-la be_v equal_a to_o the_o line_n lm_o and_o the_o line_n df_o to_o the_o line_n mn_o and_o the_o line_n gk_o to_o the_o line_n ln_o and_o by_o the_o 5._o of_o the_o four_o about_o the_o triangle_n lmn_o describe_v a_o circle_n lmn_o and_o take_v by_o the_o 1._o of_o the_o three_o the_o centre_n of_o the_o same_o circle_n case_n which_o centre_n shall_v either_o be_v within_o the_o triangle_n lmn_o or_o in_o one_o of_o the_o side_n thereof_o or_o without_o it_o first_o let_v it_o be_v within_o the_o triangle_n and_o let_v the_o same_o be_v the_o point_n x_o &_o draw_v these_o right_a line_n lx_o mx_o and_o nx_o now_o i_o say_v that_o the_o line_n ab_fw-la be_v great_a than_o the_o line_n lx._n problem_n for_o if_o not_o than_o the_o line_n ab_fw-la be_v either_o equal_a to_o the_o line_n lx_o or_o else_o it_o be_v less_o than_o it_o first_o let_v it_o be_v equal_a and_o forasmuch_o as_o the_o line_n ab_fw-la be_v equal_a to_o the_o line_n lx_o but_o the_o line_n ab_fw-la be_v equal_a to_o the_o line_n bc_o therefore_o the_o line_n lx_o be_v equal_a to_o the_o line_n bc_o and_o unto_o the_o line_n lx_o the_o line_n xm_o be_v by_o the_o 15._o definition_n of_o the_o first_o equal_a wherefore_o these_o two_o line_n ab_fw-la and_o bc_o be_v equal_a to_o these_o two_o line_n lx_o and_o xm_o the_o one_o to_o the_o other_o and_o the_o base_a ac_fw-la be_v suppose_v to_o be_v equal_a to_o the_o base_a lm_o wherefore_o by_o the_o 8._o of_o the_o first_o the_o angle_n abc_n be_v equal_a to_o the_o angle_n lxm_o and_o by_o the_o same_o reason_n also_o the_o angle_n def_n be_v equal_a to_o the_o angle_n mxn_n and_o moreover_o the_o angle_n ghk_o to_o the_o angle_n nxl_n wherefore_o these_o three_o angle_n abc_n def_n and_o ghk_v be_v equal_a to_o these_o three_o angle_n lxm_o mxn_n &_o nxl_n but_o the_o three_o angle_n lxm_o mxn_n and_o nxl_o be_v equal_a to_o four_o right_a angle_n as_o it_o be_v manifest_a to_o see_v by_o the_o 13._o of_o the_o first_o if_o any_o one_o of_o these_o line_n mx_o lx_o or_o nx_o be_v
extend_v on_o the_o side_n that_o the_o point_n x_o be_v wherefore_o the_o three_o angle_n abc_n def_n and_o ghk_v be_v also_o equal_a to_o four_o right_a angle_n but_o they_o be_v suppose_v to_o be_v less_o than_o four_o right_a angle_n which_o be_v impossible_a wherefore_o the_o line_n ab_fw-la be_v not_o equal_a to_o the_o line_n lx._n i_o say_v also_o that_o the_o line_n ab_fw-la be_v not_o less_o than_o the_o line_n lx._n for_o if_o it_o be_v possible_a let_v it_o be_v less_o and_o by_o the_o 2._o of_o the_o first_o unto_o the_o line_n ab_fw-la put_v a_o equal_a line_n xo_o and_o to_o the_o line_n bc_o put_v a_o equal_a line_n xp_n and_o draw_v a_o right_a line_n from_o the_o point_n oh_o to_o the_o point_n p._n and_o forasmuch_o as_o the_o line_n ab_fw-la be_v equal_a to_o the_o line_n bc_o therefore_o also_o the_o line_n xo_o be_v equal_a to_o the_o line_n xp_n wherefore_o the_o residue_n ol_n be_v equal_a to_o the_o residue_n mp_n wherefore_o by_o the_o 2._o of_o the_o six_o the_o line_n lm_o be_v a_o parallel_n to_o the_o line_n opâ—_n and_o the_o triangle_n lmx_n be_v equiangle_n to_o the_o triangle_n opx._n wherefore_o as_o the_o line_n xl_o be_v to_o the_o line_n lm_o so_o be_v the_o line_n xo_o to_o the_o line_n open_v wherefore_o alternate_o by_o the_o 16._o of_o the_o five_o as_o the_o line_n lx_o be_v to_o the_o line_n xo_o so_o be_v the_o line_n lm_o to_o the_o line_n open_v but_o the_o line_n lx_o be_v great_a than_o the_o line_n xo_o wherefore_o also_o the_o line_n lm_o be_v great_a than_o the_o line_n open_v but_o the_o line_n lm_o be_v put_v to_o be_v equal_a to_o the_o line_n ac_fw-la wherefore_o also_o the_o line_n ac_fw-la be_v great_a than_o the_o line_n open_v now_o forasmuch_o as_o these_o two_o right_a line_n ab_fw-la and_o bc_o be_v equal_a to_o these_o two_o right_a line_n ox_n and_o xp_n and_o the_o base_a ac_fw-la be_v great_a than_o the_o base_a open_a therefore_o by_o the_o 25._o of_o the_o first_o the_o angle_n abc_n be_v great_a than_o the_o angle_n oxp._n in_o like_a sort_n also_o may_v we_o prove_v that_o the_o angle_n def_n be_v great_a than_o the_o angle_n mxn_n and_o that_o the_o angle_n ghk_o be_v great_a than_o the_o angle_n nxl_n wherefore_o the_o three_o angle_n abc_n def_n and_o ghk_v be_v great_a than_o the_o three_o angle_n lxm_o mxn_n and_o nxl_o but_o the_o angle_n abc_n def_n and_o ghk_v be_v suppose_v to_o be_v less_o than_o four_o right_a angle_n wherefore_o much_o more_o be_v the_o angle_n lxm_o mxn_n &_o nxl_o less_o than_o four_o right_a angle_n but_o they_o be_v also_o equal_a to_o four_o right_a angle_n which_o be_v impossible_a wherefore_o the_o line_n ab_fw-la be_v not_o less_o than_o the_o line_n lx_o and_o it_o be_v also_o prove_v that_o it_o be_v not_o equal_a unto_o it_o wherefore_o the_o line_n ab_fw-la be_v great_a than_o the_o line_n lx._n now_o from_o the_o point_n x_o raise_v up_o unto_o the_o plain_a superficies_n of_o the_o circle_n lmn_o a_o perpendicular_a line_n xr_n by_o the_o 12._o of_o the_o eleven_o and_o unto_o that_o which_o the_o square_a of_o the_o line_n ab_fw-la exceed_v the_o square_a of_o the_o line_n xl_o book_n let_v the_o square_n of_o the_o line_n xr_n be_v equal_a and_o draw_v a_o right_a line_n â—rom_o the_o point_n r_o to_o the_o point_n l_o and_o a_o other_o from_o the_o point_n r_o to_o the_o point_n m_o and_o a_o other_o from_o the_o point_n r_o to_o the_o point_n n._n and_o forasmuch_o as_o the_o line_n rx_n be_v erect_v perpendicular_o to_o the_o plain_a superficies_n of_o the_o circle_n lmn_o therefore_o by_o conversion_n of_o the_o second_o definition_n of_o the_o eleven_o the_o line_n rx_n be_v erect_v perpendicular_o to_o every_o one_o of_o these_o line_n lx_o mx_o and_o nx_o case_n and_o forasmuch_o as_o the_o line_n lx_o be_v equal_a to_o the_o line_n xm_o &_o the_o line_n xr_fw-fr iâ—_z common_a to_o they_o both_o and_o be_v also_o erect_v perpendicular_o to_o they_o both_o therefore_o by_o the_o 4._o of_o the_o first_o the_o base_a rl_n be_v equal_a to_o the_o base_a rm._n and_o by_o the_o same_o reason_n also_o the_o line_n rn_n be_v equal_a to_o either_o of_o these_o line_n rl_n and_o rm._n wherefore_o these_o three_o line_n rl_n rm_n and_o rn_v be_v equal_a the_o one_o to_o the_o other_o and_o forasmuch_o as_o unto_o that_o which_o the_o square_a of_o the_o line_n ab_fw-la exceed_v the_o square_a of_o the_o line_n lx_o the_o square_a of_o the_o line_n rx_n be_v suppose_v to_o be_v equal_a therefore_o the_o square_a of_o the_o line_n ab_fw-la be_v equal_a to_o the_o square_n of_o the_o line_n lx_o and_o rx_n but_o unto_o the_o square_n of_o the_o line_n lx_o and_o xr_n the_o square_a of_o the_o line_n lr_n be_v by_o the_o 47._o of_o the_o first_o equal_a for_o the_o angle_n lxr_n be_v a_o right_a angle_n wherefore_o the_o square_a of_o the_o line_n ab_fw-la be_v equal_a to_o the_o square_n of_o the_o line_n rl_n wherefore_o also_o the_o line_n ab_fw-la be_v equal_a to_o the_o line_n rl_n but_o unto_o the_o line_n ab_fw-la be_v equal_a every_o one_o of_o these_o line_n bc_o de_fw-fr of_o gh_o and_o hk_o and_o unto_o the_o line_n rl_n be_v equal_a either_o of_o these_o line_n rm_n and_o rn._n wherefore_o every_o one_o of_o these_o line_n ab_fw-la bc_o de_fw-fr of_o gh_o and_o hk_o be_v equal_a to_o every_o one_o of_o these_o line_n rl_n rm_n and_o rn._n and_o forasmuch_o as_o these_o two_o line_n rl_n and_o rm_n be_v equal_a to_o these_o two_o line_n ab_fw-la and_o bc_o and_o the_o base_a lm_o be_v suppose_v to_o be_v equal_a to_o the_o base_a ac_fw-la therefore_o by_o the_o 8._o of_o the_o first_o the_o angle_n lrm_n be_v equal_a to_o the_o angle_n abc_n and_o by_o the_o same_o reason_n also_o the_o mrn_n be_v equal_a to_o the_o angle_n def_n and_o the_o angle_n lrn_v to_o the_o angle_n ghk_o wherefore_o of_o three_o superficial_a angle_n lrm_v mrn_n and_o lrn_v which_o be_v equal_a to_o three_o superficial_a angle_n give_v namely_o to_o the_o angle_n abc_n def_n &_o ghk_v be_v make_v a_o solid_a angle_n r_o comprehend_v under_o the_o superficial_a angle_n lrm_v mrn_n and_o lrn_v which_o be_v require_v to_o be_v do_v case_n but_o now_o let_v the_o centre_n of_o the_o circle_n be_v in_o one_o of_o the_o side_n of_o the_o triangle_n let_v it_o be_v in_o thâ—_z side_n mn_n and_o let_v the_o centre_n be_v x._o and_o draw_v a_o right_a line_n from_o the_o point_n l_o to_o the_o point_n x._o i_o say_v again_o that_o the_o line_n ab_fw-la be_v great_a than_o the_o line_n lx._n for_o if_o not_o then_o ab_fw-la be_v either_o equal_a to_o lx_o or_o else_o it_o be_v less_o than_o it_o first_o let_v it_o be_v equal_a now_o they_o these_o two_o line_n ab_fw-la and_o bc_o that_o be_v de_fw-fr &_o of_o be_v equal_a to_o these_o two_o line_n mx_o and_o xl_o that_o be_v to_o the_o line_n mn_v but_o the_o line_n mn_o be_v suppose_v to_o be_v equal_a to_o the_o line_n df._n wherefore_o also_o the_o line_n de_fw-fr and_o of_o be_v equal_a to_o the_o line_n df_o which_o by_o the_o 20._o of_o the_o first_o be_v impossible_a wherefore_o the_o line_n ab_fw-la be_v not_o equal_a to_o the_o line_n lx._n in_o like_a sort_n also_o may_v we_o prove_v that_o it_o be_v not_o less_o wherefore_o the_o line_n ab_fw-la be_v great_a than_o the_o line_n lx._n and_o now_o if_o as_o before_o unto_o the_o plain_a superficies_n of_o the_o circle_n be_v erect_v from_o the_o point_n x_o a_o perpendicular_a line_n rx_n who_o square_a let_v be_v equal_a unto_o that_o which_o the_o square_a of_o the_o line_n ab_fw-la exceed_v the_o square_a of_o the_o line_n lx_o and_o if_o the_o rest_n of_o the_o construction_n and_o demonstration_n be_v observe_v in_o this_o that_o be_v in_o the_o formeâ—_n case_n then_o shall_v the_o problem_n be_v finish_v but_o now_o let_v the_o centre_n of_o the_o circle_n be_v without_o the_o triangle_n lmn_o and_o let_v it_o be_v in_o thâ—_z point_v x._o and_o draw_v these_o right_a line_n lx_o mx_o and_o nx_o case_n i_o say_v that_o in_o this_o case_n also_o the_o line_n ab_fw-la be_v great_a than_o the_o line_n lx._n for_o if_o not_o then_o be_v it_o either_o equal_a or_o less_o first_o let_v it_o be_v equal_a wherefore_o these_o two_o line_n ab_fw-la and_o bc_o be_v equal_a to_o these_o two_o line_n mx_o and_o xl_o the_o one_o to_o thâ—_z other_o and_o the_o base_a ac_fw-la be_v equal_a to_o the_o base_a ml_o wherefore_o by_o the_o 8._o of_o the_o first_o the_o angle_n abc_n be_v equal_a to_o the_o angle_n mxl_o and_o by_o the_o same_o reason_n also_o the_o angle_n ghk_o
be_v equal_a to_o the_o angle_n lxn._n wherefore_o the_o whole_a angle_n mxn_n be_v equal_a to_o these_o two_o angle_n abc_n and_o ghk_n but_o the_o angle_n abc_n and_o ghk_n be_v great_a than_o the_o angle_n def_n wherefore_o the_o angle_n mxn_n be_v great_a than_o the_o angle_n def_n and_o forasmuch_o as_o these_o two_o line_n de_fw-fr and_o of_o be_v equal_a to_o these_o two_o line_n mx_o and_o xn_o and_o the_o base_a df_o be_v equal_a to_o the_o base_a mn_n therefore_o by_o the_o 8._o of_o the_o first_o the_o angle_n mxn_n be_v equal_a to_o the_o angle_n def_n and_o it_o be_v prove_v that_o it_o be_v also_o great_a which_o be_v impossible_a wherefore_o the_o line_n ab_fw-la be_v not_o equal_a to_o the_o line_n lx._n in_o like_a sort_n also_o may_v we_o prove_v that_o it_o be_v not_o less_o wherefore_o the_o line_n ab_fw-la be_v great_a than_o the_o line_n lx._n and_o again_o if_o unto_o the_o plain_a superficies_n of_o the_o circle_n be_v erect_v perpendicular_o from_o the_o point_n x_o a_o line_n xr_n who_o square_a be_v equal_a to_o that_o which_o the_o square_a of_o the_o line_n ab_fw-la exceed_v the_o square_a of_o the_o line_n lx_o and_o the_o rest_n of_o the_o construction_n be_v do_v in_o this_o that_o be_v in_o the_o former_a case_n then_o shall_v the_o problem_n be_v finish_v i_o say_v moreover_o that_o the_o line_n ab_fw-la be_v not_o less_o than_o the_o line_n lx._n lx._n for_o if_o it_o be_v possiblâ—_n let_v it_o be_v less_o and_o unto_o the_o line_n ab_fw-la put_v by_o the_o 2._o of_o the_o first_o the_o line_n xo_o equal_a and_o unto_o the_o line_n bc_o put_v the_o line_n xp_n equal_a and_o draw_v a_o right_a line_n from_o the_o point_n oh_o to_o the_o point_n p._n and_o forasmuch_o as_o the_o line_n ab_fw-la be_v equal_a to_o the_o line_n bc_o therefore_o the_o line_n xo_o be_v equal_a to_o the_o line_n xp_n wherefore_o the_o residue_n ol_n be_v equal_a to_o the_o residue_n mp_n wherefore_o by_o the_o 2._o of_o the_o six_o the_o line_n lm_o be_v a_o parallel_n to_o the_o line_n po._n and_o the_o triangle_n lxm_o be_v equiangle_n to_o the_o triangle_n pxo._n wherefore_o by_o the_o 6._o of_o the_o six_o as_o the_o line_n lx_o be_v to_o the_o line_n lm_o so_o be_v the_o line_n xo_o to_o the_o line_n po._n wherefore_o alternate_o by_o the_o 16._o of_o the_o five_o as_o the_o line_n lx_o be_v to_o the_o line_n xo_o so_o be_v the_o line_n lm_o to_o the_o line_n open_v but_o the_o line_n lx_o be_v great_a than_o the_o line_n xo_o wherefore_o also_o the_o line_n lm_o be_v great_a than_o the_o line_n open_v but_o the_o line_n lm_o be_v equal_a to_o the_o line_n ac_fw-la wherefore_o also_o the_o line_n ac_fw-la be_v great_a than_o the_o line_n open_v now_o forasmuch_o as_o these_o two_o line_n ab_fw-la and_o bc_o be_v equal_a to_o these_o two_o line_n ox_n &_o xp_n the_o one_o to_o the_o other_o and_o the_o base_a ac_fw-la be_v great_a than_o the_o base_a open_a therefore_o by_o the_o 25._o of_o the_o first_o the_o angle_n abc_n be_v great_a than_o the_o angle_n oxp._n and_o in_o like_a sort_n if_o we_o put_v the_o line_n xr_v equal_a to_o either_o of_o these_o line_n xo_o or_o xp_n and_o draw_v a_o right_a line_n from_o the_o point_n oh_o to_o the_o point_n r_o we_o may_v prove_v that_o the_o angle_n ghk_o be_v great_a than_o the_o angle_n oxr._n unto_o the_o right_a line_n lx_o and_o unto_o the_o point_n in_o it_o x_o make_v by_o the_o 23._o of_o the_o first_o unto_o the_o angle_n abc_n a_o equal_a angle_n lxs_n and_o unto_o the_o angle_n ghk_v make_v a_o equal_a angle_n lxt._n and_o by_o the_o second_o of_o the_o first_o let_v either_o of_o these_o line_n sx_n and_o xt_v be_v equal_a to_o the_o line_n ox_n and_o draw_v these_o line_n os_fw-mi ot_fw-mi and_o n-ab_n and_o forasmuch_o as_o these_o two_o line_n ab_fw-la and_o bc_o be_v equal_a to_o these_o two_o line_n ox_n and_o xs_n and_o the_o angle_n abc_n be_v equal_a to_o the_o angle_n oxs_n therefore_o by_o the_o 4._o of_o the_o first_o the_o base_a ac_fw-la that_o be_v lm_o be_v equal_a to_o the_o base_a os_fw-mi and_o by_o the_o same_o reason_n also_o the_o line_n ln_o be_v equal_a to_o the_o line_n ot_fw-mi and_o forasmuch_o as_o these_o two_o line_n lm_o and_o ln_o be_v equal_a to_o these_o two_o line_n so_o and_o ot_fw-mi and_o the_o angle_n mln_n be_v great_a than_o the_o angle_n sot_n therefore_o by_o the_o 25._o of_o the_o first_o the_o base_a mn_n be_v great_a than_o the_o base_a n-ab_n but_o the_o line_n mn_o be_v equal_a to_o the_o line_n df._n wherefore_o the_o line_n df_o be_v great_a than_o the_o line_n n-ab_n now_o forasmuch_o as_o these_o two_o line_n de_fw-fr and_o of_o be_v equal_a to_o these_o two_o line_n sx_n and_o xt_v and_o the_o base_a df_o be_v great_a than_o the_o base_a n-ab_n therefore_o by_o the_o 25._o of_o the_o first_o the_o angle_n def_n be_v great_a than_o the_o angle_n sxt_v but_o the_o angle_n sxt_v be_v equal_a to_o the_o angle_n abc_n and_o ghk_n wherâ—fore_o also_o the_o angle_n def_n be_v great_a than_o the_o angle_n abc_n and_o ghk_n but_o it_o be_v also_o less_o which_o be_v impossible_a proposition_n but_o now_o let_v we_o declare_v how_o to_o find_v out_o the_o line_n xr_n who_o square_a shall_v be_v equal_a â—o_o that_o which_o the_o square_a of_o the_o line_n ab_fw-la exceed_v the_o square_a of_o the_o line_n lx._n take_v the_o two_o right_a line_n ab_fw-la and_o lx_o and_o let_v ab_fw-la be_v the_o great_a and_o upon_o ab_fw-la describe_v a_o semicircle_n acb_o and_o from_o the_o point_n a_o apply_v into_o the_o semicircle_n a_o right_a line_n ac_fw-la equal_a to_o the_o right_a line_n lx_o and_o draw_v a_o right_a line_n from_o the_o point_n c_o to_o the_o point_n b._n and_o forasmuch_o as_o in_o the_o semicircle_n acb_o be_v a_o angle_n acb_o therefore_o by_o the_o 31._o of_o the_o three_o the_o angle_n acb_o be_v a_o right_a angle_n wherefore_o by_o the_o 47._o of_o the_o first_o the_o square_a of_o the_o line_n ab_fw-la be_v equal_a to_o the_o square_n of_o the_o line_n ac_fw-la and_o cb_o wherefore_o the_o square_a of_o the_o line_n ab_fw-la be_v in_o power_n more_o than_o the_o square_a of_o the_o line_n ac_fw-la by_o the_o square_n of_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 lx_o wherefore_o the_o square_a of_o the_o line_n ab_fw-la be_v in_o power_n more_o than_o the_o square_a of_o the_o line_n lx_o by_o the_o square_n of_o the_o line_n cb._n if_o therefore_o unto_o the_o line_n cb_o we_o make_v the_o line_n xr_v equal_a then_o be_v the_o square_a of_o the_o line_n ab_fw-la great_a than_o the_o square_a of_o the_o line_n lx_o by_o the_o square_n of_o the_o line_n xr_n which_o be_v require_v to_o be_v do_v in_o this_o figure_n may_v you_o more_o full_o sâ—â—_n the_o construction_n and_o demonstration_n of_o the_o â—â—rst_a case_n of_o the_o former_a 23._o proposition_n if_o you_o erect_v perpendicular_o the_o triangle_n â—_o rn_v and_o unto_o it_o bend_v the_o triangle_n lmr_n that_o the_o angle_n r_o of_o each_o may_v join_v together_o in_o the_o point_n r._n and_o so_o full_o understand_v this_o case_n the_o other_o case_n will_v not_o be_v hard_a to_o conceive_v ¶_o the_o 21._o theorem_a the_o 24._o proposition_n if_o a_o solid_a or_o body_n be_v contain_v under_o thereof_o six_o parallel_n plain_a superficiece_n the_o opposite_a plain_a superficiece_n of_o the_o same_o body_n be_v equal_a and_o parallelogram_n svppose_v that_o this_o solid_a body_n cdhg_n be_v contain_v under_o these_o 6._o parallel_n plain_a superficiece_n namely_o ac_fw-la gf_o bg_o ce_fw-fr fb_o and_o ae_n then_o i_o say_v that_o the_o opposite_a superficiece_n of_o the_o same_o body_n be_v equal_a and_o parallelogram_n it_o be_v to_o weet_v the_o two_o opposite_n ac_fw-la and_o gf_o and_o the_o two_o opposite_n bg_o and_o ce_fw-fr and_o the_o two_o opposite_n fb_o and_o ae_n to_o be_v equal_a and_o all_o to_o be_v parallelogram_n parallelogram_n for_o forasmuch_o as_o two_o parallel_n plain_a superficiece_n that_o be_v bg_o and_o ce_fw-fr be_v divide_v by_o the_o plain_a superficies_n ac_fw-la their_o common_a section_n be_v by_o the_o 16._o of_o the_o eleven_o parallel_v wherefore_o the_o line_n ab_fw-la be_v a_o parallel_n to_o the_o line_n cd_o against_o forasmuch_o as_o two_o parallel_n plain_a superficiece_n fb_o and_o ae_n be_v divide_v by_o the_o plain_a superficies_n ac_fw-la their_o common_a section_n be_v by_o the_o same_o proposition_n parallel_n wherefore_o the_o line_n ad_fw-la be_v a_o parallel_n to_o the_o line_n bc._n and_o it_o be_v also_o prove_v that_o the_o line_n ab_fw-la be_v a_o parallel_n to_o
triangle_n in_o the_o base_a unto_o every_o one_o of_o which_o solid_a angle_n you_o may_v by_o this_o proposition_n describe_v 〈◊〉_d equal_a solid_a angle_n upon_o a_o line_n give_v and_o at_o a_o point_n in_o it_o give_v and_o so_o at_o the_o length_n the_o whole_a solid_a angle_n after_o this_o manner_n describe_v shall_v be_v equal_a to_o the_o solid_a angle_n give_v the_o 5._o theorem_a the_o 27._o proposition_n upon_o a_o right_a line_n give_v to_o describe_v a_o parallelipipedon_n like_o and_o in_o like_a sort_n situate_a to_o a_o parallelipipedon_n give_v â—vppose_v that_o the_o right_a line_n give_v be_v ab_fw-la and_o let_v the_o parallelipipedon_v give_v be_v cd_o it_o be_v require_v upon_o the_o right_a line_n give_v ab_fw-la to_o describe_v a_o parallelipipedon_n like_o and_o in_o like_a sort_n situate_a to_o the_o parallelipipedon_n give_v namely_o to_o cd_o constrâ—ction_n unto_o the_o right_a line_n ab_fw-la and_o at_o the_o point_n in_o it_o a_o describe_v by_o the_o 26._o of_o the_o eleâ—enth_o a_o solid_a angle_n equal_a to_o the_o solid_a angle_n c_o and_o let_v it_o be_v contain_v under_o these_o superficial_a angle_n bah_o hak_n and_o kab_n so_o that_o let_v the_o angle_n bah_o be_v equal_a to_o the_o angle_n ecf_n and_o the_o angle_n back_o to_o the_o angle_n ecg_n and_o moreover_o the_o angle_n kah_n to_o the_o angle_n gcf_n and_o as_o the_o line_n ec_o be_v to_o the_o line_n cg_o so_o let_v the_o ab_fw-la be_v to_o the_o line_n ak_o by_o the_o 12._o of_o the_o six_o and_o as_o the_o line_n gc_o be_v to_o the_o line_n cf_o so_o let_v the_o ka_o be_v 〈◊〉_d the_o line_n ah_o wherefore_o of_o equality_n by_o the_o 22._o of_o the_o five_o as_o the_o line_n ec_o be_v to_o the_o line_n cf_o so_o be_v the_o line_n basilius_n to_o the_o line_n ah_o demonstration_n make_v perfect_a the_o parallelogram_n bh_o and_o also_o the_o solid_a al._n now_o for_o that_o as_o the_o line_n ec_o be_v to_o the_o line_n cg_o so_o be_v the_o line_n basilius_n to_o the_o line_n ak_o therefore_o the_o side_n which_o contain_v the_o equal_a angle_n namely_o the_o angle_n ecg_n and_o back_o be_v proportional_a wherefore_o by_o the_o first_o definition_n of_o the_o six_o the_o parallelogram_n ge_z be_v like_a to_o the_o parallelogram_n kb_o and_o by_o the_o same_o reason_n the_o parallelogram_n kh_o be_v like_a to_o the_o parallelogram_n gf_o and_o moreover_o the_o parallelogram_n fe_o to_o the_o parallelogram_n hbâ—_n â—_z wherefore_o there_o be_v three_o parallelogram_n of_o the_o solid_a cd_o like_o to_o the_o three_o parallelogram_n of_o the_o solid_a al._n but_o the_o three_o other_o side_n in_o each_o of_o these_o solid_n be_v equal_a and_o like_a to_o their_o opposite_a side_n wherefore_o the_o whole_a solid_a cd_o be_v like_a to_o the_o whole_a solid_a al._n wherefore_o upon_o the_o right_a line_n give_v ab_fw-la be_v describe_v the_o solid_a all_o contain_v under_o parallel_n plain_a superficiece_n like_a and_o in_o like_a sort_n situate_a to_o the_o solid_a give_v cd_o contain_v also_o under_o parallel_n plain_a superficiece_n which_o be_v require_v to_o be_v do_v this_o demonstration_n be_v not_o hard_a to_o conceive_v by_o the_o former_a figure_n as_o it_o be_v describe_v in_o a_o plain_n if_o you_o that_o imagination_n of_o parallelipipedon_n describe_v in_o a_o plain_n which_o we_o before_o teach_v in_o the_o definition_n of_o a_o cube_fw-la howbeit_o i_o have_v here_o for_o the_o more_o ease_n of_o such_o as_o be_v not_o yet_o acquaint_v with_o solid_n describe_v two_o figure_n who_o form_n first_o describe_v upon_o past_v paper_n with_o the_o like_a letter_n note_v in_o they_o and_o then_o fine_o cut_v the_o three_o middle_a line_n of_o each_o figure_n and_o so_o fold_v they_o according_o and_o that_o do_v compare_v they_o with_o the_o construction_n and_o demonstration_n of_o this_o 27._o proposition_n and_o they_o will_v be_v very_o easy_a to_o conceive_v the_o 23._o theorem_a the_o 28._o proposition_n if_o a_o parallelipipedom_n be_v cut_v by_o a_o plain_a superficies_n draw_v by_o the_o diagonal_a line_n of_o those_o plain_a superficiece_n which_o be_v opposite_a that_o solid_a be_v by_o this_o plain_a superficies_n cut_v into_o two_o equal_a part_n a_o diagonal_a line_n be_v a_o right_a line_n which_o in_o angular_a figure_n be_v draw_v from_o one_o angle_n and_o extend_v to_o his_o contrary_a angle_n as_o you_o see_v in_o the_o figure_n ab_fw-la describe_v for_o the_o better_a sight_n of_o this_o demonstration_n a_o figure_n upon_o past_v paper_n like_o unto_o that_o which_o you_o describe_v for_o the_o 24._o proposition_n only_o alter_v the_o letter_n namely_o in_o stead_n of_o the_o letter_n a_o put_v the_o letter_n fletcher_n and_o in_o stead_n of_o the_o letter_n h_o the_o letter_n c_o moreover_o in_o stead_n of_o the_o letter_n c_o put_v the_o letter_n h_o and_o the_o letter_n e_o for_o the_o letter_n d_o and_o the_o letter_n a_o for_o the_o letter_n e_o and_o final_o put_v the_o letter_n d_o for_o the_o letter_n f._n and_o your_o figure_n thus_o order_v compare_v it_o with_o the_o demonstration_n only_o imagine_v a_o superficies_n to_o pass_v through_o the_o body_n by_o the_o diagonal_a line_n cf_o and_o de._n ¶_o the_o 24._o theorem_a the_o 29._o proposition_n parallelipipedon_n consist_v upon_o one_o and_o the_o self_n same_o base_a and_o under_o one_o and_o the_o self_n same_o altitude_n who_o line_n stand_a line_n be_v in_o the_o self_n same_o right_a line_n be_v equal_a the_o one_o to_o the_o other_o svppose_v that_o that_o these_o parallelipipedon_n cm_o and_o cn_fw-la do_v consist_v upon_o one_o and_o the_o self_n same_o base_a namely_o ab_fw-la and_o let_v they_o be_v under_o one_o and_o the_o self_n same_o altitude_n who_o stand_a line_n that_o be_v the_o four_o side_n of_o each_o solid_a which_o fall_v upon_o the_o base_a as_o the_o line_n of_o cd_o bh_o lm_o of_o the_o solid_a cm_o and_o the_o line_n ce_fw-fr bk_o agnostus_n and_o ln_o of_o the_o solid_a cn_fw-la let_v be_v in_o the_o self_n same_o right_a line_n or_o parallel_a line_n fn_o dk_o etc_n then_o i_o say_v that_o the_o solid_a cm_o be_v equal_a to_o the_o solid_a cn_fw-la for_o forasmuch_o as_o either_o of_o these_o superficiece_n cbdh_fw-mi cbek_n be_v a_o parallelogram_n therefore_o by_o the_o 34._o of_o the_o first_o the_o line_n cb_o be_v equal_a to_o either_o of_o these_o line_n dh_o and_o eke_o wherefore_o also_o the_o line_n dh_o be_v equal_a to_o the_o line_n eke_o take_v away_o eh_o which_o be_v common_a to_o they_o both_o wherefore_o the_o residue_n namely_o de_fw-fr be_v equal_a to_o the_o residue_n hk_o wherefore_o also_o the_o triangle_n dce_n be_v equal_a to_o the_o triangle_n hkb_n and_o the_o parallelogram_n dg_o be_v equal_a to_o the_o parallelogram_n hn._n and_o by_o the_o same_o reason_n the_o triangle_n agf_n be_v equal_a to_o the_o triangle_n mln_n and_o the_o parallelogram_n cf_o be_v equal_a to_o the_o parallelogram_n bm_n but_o the_o parallelogram_n cg_o be_v equal_a to_o the_o parallelogram_n bn_o by_o the_o 24._o of_o the_o ten_o for_o they_o be_v opposite_a the_o one_o to_o the_o other_o wherefore_o the_o prism_n contain_v under_o the_o two_o triangle_n fag_v and_o dce_n and_o under_o the_o three_o parrallelogramme_n ad_fw-la dg_o and_o cg_o be_v equal_a to_o the_o prism_n contain_v under_o the_o two_o triangle_n mln_v and_o hbk_n and_o under_o the_o three_o parallelogram_n that_o be_v bm_n nh_o and_o bn_o put_v that_o solid_a common_a to_o they_o both_o who_o base_a be_v the_o parallelogram_n ab_fw-la and_o the_o opposite_a side_n unto_o the_o base_a be_v the_o superficies_n gehm_n wherefore_o the_o whole_a parallelipipedon_n cm_o be_v equal_a to_o the_o whole_a parallelipipedon_n cn_fw-la wherefore_o parallelipipedon_n consist_v upon_o one_o and_o the_o self_n same_o base_a and_o under_o one_o and_o the_o self_n same_o altitude_n who_o stand_a line_n be_v in_o the_o self_n same_o right_a line_n be_v equal_a the_o one_o to_o the_o other_o which_o be_v requiâ—ed_v to_o be_v demonstrate_v although_o this_o demonstration_n be_v not_o hard_a to_o a_o good_a imagination_n to_o conceive_v by_o the_o former_a figure_n which_o yet_o by_o m._n deeâ—_n reform_v be_v much_o better_a than_o the_o figure_n of_o this_o proposition_n common_o describe_v in_o other_o copy_n both_o greek_a and_o latin_n yet_o for_o the_o ease_n of_o those_o which_o be_v young_a beginner_n in_o this_o matter_n of_o solid_n i_o have_v here_o set_v a_o other_o figure_n who_o form_n if_o it_o be_v describe_v upon_o past_v paper_n with_o the_o like_a letter_n to_o every_o line_n as_o they_o be_v here_o put_v and_o then_o if_o you_o fine_o cut_v not_o through_o but_o as_o it_o be_v half_a way_n the_o three_o line_n la_fw-fr nmgf_n and_o kh_v &_o so_o fold_v it_o according_o
&_o compare_v it_o with_o the_o demonstration_n it_o will_v geve_v great_a light_n thereunto_o stand_v line_n be_v call_v those_o four_o right_a line_n of_o every_o parallelipipedon_n which_o join_v together_o the_o angle_n of_o the_o upper_a and_o nether_a base_n of_o the_o same_o body_n line_n which_o according_a to_o the_o diversity_n of_o the_o angle_n of_o the_o solid_n may_v either_o be_v perpendicular_a upon_o the_o base_a or_o fall_v oblique_o and_o forasmuch_o as_o in_o this_o proposition_n and_o in_o the_o next_o proposition_n follow_v the_o solid_n compare_v together_o be_v suppose_v to_o have_v one_o and_o the_o self_n same_o base_a it_o be_v manifest_a that_o the_o stand_a line_n be_v in_o respect_n of_o the_o low_a base_a in_o the_o self_n same_o parallel_a line_n namely_o in_o the_o two_o side_n of_o the_o low_a base_a but_o because_o there_o be_v put_v two_o solid_n upon_o one_o and_o the_o self_n same_o base_a and_o under_o one_o and_o the_o self_n same_o altitude_n the_o two_o upper_a base_n of_o the_o solid_n may_v be_v diverse_o place_v for_o forasmuch_o as_o they_o be_v equal_a and_o like_a by_o the_o 24._o of_o this_o book_n either_o they_o may_v be_v place_v between_o the_o self_n same_o parallel_a line_n and_o they_o the_o stand_a line_n be_v in_o either_o solid_a say_v to_o be_v in_o the_o self_n same_o parallel_a line_n or_o right_a line_n namely_o when_o the_o two_o side_n of_o each_o of_o the_o upper_a base_n be_v contain_v in_o the_o self_n same_o parallel_a line_n but_o contrariwise_o if_o those_o two_o side_n of_o the_o upper_a base_n be_v not_o contain_v in_o the_o self_n same_o parallel_n or_o right_a line_n neither_o shall_v the_o stand_a line_n which_o be_v join_v to_o those_o side_n be_v say_v to_o be_v in_o the_o self_n same_o parallel_n or_o right_a line_n and_o therefore_o the_o stand_a line_n be_v say_v to_o be_v in_o the_o self_n same_o right_a line_n when_o the_o side_n of_o the_o upper_a base_n at_o the_o least_o two_o of_o the_o side_n be_v contain_v in_o the_o self_n same_o right_a line_n which_o thing_n be_v require_v in_o the_o supposition_n of_o this_o 29_o proposition_n but_o the_o stand_a line_n be_v say_v not_o to_o be_v in_o the_o self_n same_o right_a line_n when_o none_o of_o the_o two_o side_n of_o the_o upper_a base_n be_v contain_v in_o the_o self_n same_o right_a line_n which_o thing_n the_o next_o proposition_n follow_v suppose_v ¶_o the_o 25._o theorem_a the_o 30._o proposition_n parallelipipedon_n consist_v upon_o one_o and_o the_o self_n same_o base_a and_o under_o the_o self_n same_o altitude_n who_o stand_a line_n be_v not_o in_o the_o self_n same_o right_a line_n be_v equal_a the_o one_o to_o the_o other_o svppose_v that_o these_o parallelipipedon_n cm_o and_o cn_fw-la do_v consist_v upon_o one_o and_o the_o self_n same_o base_a namely_o ab_fw-la and_o under_o one_o and_o the_o self_n same_o altitude_n who_o stand_a line_n namely_o the_o line_n of_o cd_o bh_o and_o lm_o of_o the_o parallelipipedon_n cm_o and_o the_o stand_a line_n agnostus_n change_z bk_o and_o ln_o of_o the_o parallelipipedon_n cn_fw-la let_v not_o be_v in_o the_o self_n same_o right_a line_n then_o i_o say_v that_o the_o parallelipipedon_n cm_o be_v equal_a to_o the_o parallelipipedon_n cn_fw-la forasmuch_o as_o the_o upper_a superficiece_n fh_o &_o â—k_v of_o the_o former_a parallelipipedon_n be_v in_o one_o and_o the_o self_n same_o superficies_n by_o reason_n they_o be_v suppose_v to_o be_v of_o one_o and_o the_o self_n same_o altitude_n construction_n extend_v the_o line_n fd_v and_o mh_o till_o they_o concur_v with_o the_o line_n nâ—_z and_o ke_o sufficient_o both_o way_n extend_v for_o in_o diverse_a case_n their_o concourse_n be_v diverse_a let_v â—d_a extend_v meet_v with_o ng_z and_o cut_v it_o in_o the_o point_n x_o and_o with_o ke_o in_o the_o point_n p._n let_v likewise_o mh_o extend_v meet_v with_o ng_z sufficient_o produce_v in_o the_o point_n oh_o and_o with_o ke_o in_o the_o point_n r._n and_o draw_v these_o right_a line_n axe_n lo_o cp_n and_o br_n now_o by_o the_o 29._o of_o the_o eleven_o the_o solid_a cm_o who_o base_a be_v the_o parallelogram_n acbl_n and_o the_o parallelogram_n opposite_a unto_o it_o be_v fdhm_n be_v equal_a to_o the_o solid_a county_n who_o base_a be_v acbl_n and_o the_o opposite_a side_n the_o parallelogram_n xpro_n demonstration_n for_o they_o consist_v upon_o one_o and_o the_o self_n same_o base_a namely_o upon_o the_o parallelogram_n acbl_n who_o stand_a line_n of_o axe_n lm_o lo_o cd_o cp_n bh_o and_o â—k_v figure_n be_v in_o the_o self_n same_o right_a line_n fp_n and_o mr._n but_o the_o solid_a county_n who_o base_a be_v the_o parallelogram_n acbl_n and_o the_o opposite_a superficies_n unto_o it_o be_v xpro_n be_v equal_a to_o the_o solid_a cn_fw-la who_o base_a be_v the_o parallelogram_n acbl_n and_o the_o opposite_a superficies_n unto_o it_o be_v the_o superficies_n gâ—kn_n for_o they_o be_v upon_o one_o and_o the_o self_n same_o base_a namely_o acbl_n and_o their_o stand_a line_n agnostus_n axe_n cf_o cp_n ln_o lo_o bk_o and_o br_n be_v in_o the_o self_n same_o right_a line_n nx_n and_o pk_v wherefore_o also_o the_o solid_a cm_o be_v equal_a to_o the_o solid_a cn_fw-la wherefore_o parallelipipedon_n consist_v upon_o one_o and_o the_o self_n same_o base_a and_o under_o the_o self_n same_o altitude_n who_o stand_a line_n be_v not_o in_o the_o self_n same_o right_a line_n be_v equal_a the_o one_o to_o the_o other_o which_o be_v require_v to_o be_v prove_v this_o demonstration_n be_v somewhat_o hard_a than_o the_o former_a to_o conceive_v by_o the_o figure_n describe_v in_o the_o plain_a and_o yet_o before_o m._n do_v you_o invent_v that_o figure_n which_o be_v place_v for_o it_o it_o be_v much_o hard_o by_o reason_n one_o solid_a be_v contain_v in_o a_o other_o and_o therefore_o for_o the_o clear_a light_n thereof_o describe_v upon_o past_v paper_n this_o figure_n here_o put_v with_o the_o like_a letter_n and_o fine_o cut_v the_o line_n ac_fw-la cb_o eglantine_n bl_o epk_n rohm_n and_o fold_v it_o accordingly_z that_o every_o line_n may_v exact_o agree_v with_o his_o correspondent_a line_n which_o observe_v the_o letter_n of_o every_o line_n you_o shall_v easy_o do_v and_o so_o compare_v your_o figure_n with_o the_o demonstration_n and_o it_o will_v make_v it_o very_o plain_a unto_o you_o the_o 26._o theorem_a the_o 31._o proposit._fw-la parallelipipedon_n consist_v upon_o equal_a base_n and_o be_v under_o one_o and_o the_o self_n same_o altitude_n be_v equal_a the_o one_o to_o the_o other_o svppose_v that_o upon_o these_o equal_a base_n ab_fw-la and_o cd_o do_v consist_v these_o parallelipipedon_n ae_n and_o cf_o be_v under_o one_o &_o the_o self_n same_o altitude_n construction_n then_o i_o say_v that_o the_o solid_a ae_n be_v equal_a to_o the_o solid_a cf._n first_o let_v the_o stand_a line_n namely_o hk_o be_v agnostus_n lm_o open_v df_o cx_o and_o r_n be_v erect_v perpendicular_o to_o the_o base_n ab_fw-la and_o cd_o and_o let_v the_o angle_n alb_n not_o be_v equal_a to_o the_o angle_n crd._n extend_v the_o line_n cr_n to_o the_o point_n t._n and_o by_o the_o 23._o of_o the_o first_o upon_o the_o right_a line_n rt_n and_o at_o the_o point_n in_o it_o r_o describe_v unto_o remain_v the_o angle_n alb_n a_o equal_a angle_n tru._n and_o by_o the_o three_o of_o the_o first_o put_v the_o line_n rt_n equal_a to_o the_o line_n lb_n and_o the_o line_n rv_n equal_a to_o the_o line_n al._n and_o by_o the_o 31._o of_o the_o first_o by_o the_o point_n five_o draw_v unto_o the_o line_n rt_v a_o parallel_n line_n vw_n demonstration_n and_o make_v perfect_v the_o base_a rw_n and_o the_o solid_a yu._n now_o forasmuch_o as_o these_o two_o line_n tr_z and_o rv_n be_v equal_a to_o these_o two_o line_n bl_o and_o la_fw-fr and_o they_o contain_v equal_a angle_n therefore_o the_o parallelogram_n rw_n be_v equal_a and_o like_a to_o the_o parallelogram_n ab_fw-la again_o forasmuch_o as_o the_o line_n lb_n be_v equal_a to_o the_o line_n rt_n and_o the_o line_n lm_o to_o the_o line_n r_n for_o the_o line_n lm_o and_o r_n be_v the_o altitude_n of_o the_o parallelipipedon_n ae_n and_o cf_o which_o altitude_n be_v suppose_v to_o be_v equal_a and_o they_o contain_v right_a angle_n by_o supposition_n therefore_o the_o parallelogram_n ry_z be_v equal_a and_o like_a to_o the_o parallelogram_n bm_n and_o by_o the_o same_o reason_n also_o the_o parallelogram_n lg_n be_v equal_a and_o like_a to_o the_o parallelogram_n su._n wherefore_o three_o parallelogram_n of_o the_o solid_a ae_n be_v equal_a and_o like_a to_o the_o three_o parallelogram_n of_o the_o solid_a yu._n but_o these_o three_o parallelogram_n be_v equal_a and_o like_a to_o the_o three_o opposite_a
aeg_n be_v equal_a to_o the_o angle_n cfm_n by_o reason_n that_o the_o solid_n ab_fw-la and_o cd_o be_v like_a wherefore_o the_o parallelogram_n kl_o be_v equal_a and_o like_a to_o the_o parallelogram_n cn_fw-la and_o by_o the_o same_o reason_n also_o the_o parallelogram_n km_o be_v equal_a and_o like_a to_o the_o parallelogram_n cr_n and_o moreover_o the_o parallelogram_n oe_z to_o the_o parallelogram_n fd._n wherefore_o three_o parallelogram_n of_o the_o parallelipipedon_n ko_o be_v like_a and_o equal_a to_o three_o parallelogram_n of_o the_o parallelipipedon_n cd_o but_o those_o three_o side_n be_v equal_a and_o like_a to_o the_o three_o opposite_a side_n wherefore_o the_o whole_a parallelipipedon_n ko_o be_v equal_a and_o like_a to_o the_o whole_a parallelipipedon_n cd_o make_v perfect_a the_o parallelogram_n gk_o and_o upon_o the_o base_n gk_o and_o kl_o make_v perfect_a to_o the_o altitude_n of_o the_o parallelipipedon_n ab_fw-la the_o parallelipipedon_n exit_fw-la &_o lp_v and_o forasmuch_o as_o by_o reason_n that_o the_o parallelipipedon_n ab_fw-la &_o cd_o be_v like_a 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 eglantine_n to_o the_o line_n fn_o and_o the_o line_n eh_o to_o the_o line_n fr._n but_o the_o line_n cf_o be_v equal_a to_o the_o line_n eke_o and_o the_o line_n fn_o to_o the_o line_n el_fw-es and_o the_o line_n fr_z to_o the_o line_n em_n therefore_o as_o the_o line_n ae_n be_v to_o the_o line_n eke_o so_o be_v the_o line_n ge_z to_o the_o line_n el_fw-es and_o the_o line_n he_o to_o the_o line_n em_n by_o construction_n but_o as_o the_o line_n ae_n be_v to_o the_o line_n eke_o so_o be_v the_o parallelogram_n agnostus_n to_o the_o parallelogram_n gk_o by_o the_o first_o of_o the_o six_o and_o as_o the_o line_n ge_z be_v to_o the_o line_n el_fw-es so_o be_v the_o parallelogram_n gk_o to_o the_o parallelogram_n kl_o and_o moreover_o as_o the_o line_n he_o be_v to_o the_o line_n em_n so_o be_v the_o parallelogrammâ—_n pe_n to_o the_o parallelogram_n km_o wherefore_o by_o the_o 11._o of_o the_o five_o as_o the_o parallelogram_n agnostus_n be_v to_o the_o parallelogram_n gk_o so_o be_v the_o parallelogram_n gk_o to_o the_o parallelogram_n kl_o and_o the_o parallelogram_n pe_n to_o the_o parallelogram_n km_o but_o as_o the_o parallelogram_n agnostus_n be_v to_o the_o parallelogram_n gk_o so_o be_v the_o parallelipipedon_n ab_fw-la to_o the_o parallellpipedon_n exit_fw-la by_o the_o former_a proposition_n and_o as_o the_o parallelogram_n gk_o be_v to_o the_o parallelogram_n kl_o so_o by_o the_o same_o be_v the_o parallelipipedon_n xe_n to_o the_o parallelipipedon_n pl_z and_o again_o by_o the_o same_o as_o the_o parallelogram_n pe_n be_v to_o the_o parallelogram_n km_o so_o be_v the_o parallelipipedon_n pl_z to_o the_o parallelipipedon_n ko_o wherefore_o as_o the_o parallelipipedom_n ab_fw-la be_v to_o the_o parallelipipedon_n exit_fw-la so_o be_v the_o parallelipipedon_n exit_fw-la to_o the_o parallelipipedon_n pl_z and_o the_o parallelipipedon_n pl_z to_o the_o parallelipipedon_n ko_o but_o if_o there_o be_v four_o magnitude_n in*_n continual_a proportion_n the_o first_o shall_v be_v unto_o the_o four_o in_o treble_a proportion_n that_o it_o be_v to_o the_o second_o by_o the_o 10._o definition_n of_o the_o five_o wherefore_o the_o parallelipipedon_n ab_fw-la be_v unto_o the_o parallelipedon_n ko_o in_o treble_a proportion_n that_o the_o parallelipipedon_n ab_fw-la be_v to_o the_o parallelipipedon_n exit_fw-la but_o as_o the_o parallelipipedon_n ab_fw-la be_v to_o the_o parallelipipedom_n exit_fw-la so_o be_v the_o parallelogram_n agnostus_n to_o the_o parallelogram_n gk_o and_o the_o right_a line_n ae_n to_o the_o right_a line_n eke_o wherefore_o also_o the_o parallelipipedon_n ab_fw-la be_v to_o the_o parallelipipedon_n ko_o in_o treble_a proportion_n of_o that_o which_o the_o line_n ae_n have_v to_o the_o line_n eke_o but_o the_o parallelipipedon_n ko_o be_v equal_a to_o the_o parallelipipedon_n cd_o and_o the_o right_a line_n eke_o to_o the_o right_a line_n cf._n wherefore_o the_o parallelipipedon_n ab_fw-la be_v to_o the_o parallelipipedon_n cd_o in_o treble_a proportion_n that_o the_o side_n of_o like_a proportion_n namely_o ae_n be_v to_o the_o side_n of_o like_a proportion_n namely_o to_o cf._n wherefore_o like_a parallelipipedon_n be_v in_o treble_a proportion_n the_o one_o to_o the_o other_o of_o that_o in_o which_o their_o side_n of_o like_a proportion_n be_v which_o be_v require_v to_o be_v demonstrate_v ¶_o corellary_o hereby_o it_o be_v manifest_a that_o if_o there_o be_v four_o right_a line_n in_o *_o continual_a proportion_n as_o the_o first_o be_v to_o the_o four_o so_o shall_v the_o parallelipipedon_n describe_v of_o the_o first_o line_n be_v to_o the_o parallelipipedon_n describe_v of_o the_o second_o both_o the_o parallelipipedon_n be_v like_o and_o in_o like_a sort_n describe_v for_o the_o first_o line_n be_v to_o the_o four_o in_o triple_a proportion_n that_o it_o be_v to_o the_o second_o and_o it_o have_v before_o be_v prove_v that_o the_o parallelipipedon_n describe_v of_o the_o first_o be_v to_o the_o parallelipipedon_n describe_v of_o the_o second_o in_o the_o same_o proportion_n that_o the_o first_o line_n be_v to_o the_o four_o because_o the_o one_o of_o the_o figure_n before_o describe_v in_o a_o plain_a pertain_v to_o the_o demonstration_n of_o this_o 33._o proposition_n be_v not_o altogether_o so_o easy_a to_o a_o young_a beginner_n to_o conceive_v i_o have_v here_o for_o the_o same_o describe_v a_o other_o figure_n which_o if_o you_o first_o draw_v upon_o past_v paper_n and_o afterward_o cut_v the_o line_n &_o fold_v the_o side_n according_o will_v agree_v with_o the_o construction_n &_o demonstration_n of_o the_o say_a proposition_n howbeit_o this_o you_o must_v note_v that_o you_o must_v cut_v the_o line_n oq_fw-la &_o mr._n on_o the_o contrary_a side_n â—o_o that_o which_o you_o cut_v the_o other_o line_n for_o the_o solid_n which_o have_v to_o their_o base_a the_o parallelogram_n lk_n be_v set_v on_o upward_o and_o the_o other_o downward_o you_o may_v if_o you_o think_v good_a describe_v after_o the_o sâ—me_a manner_n of_o past_v paper_n a_o body_n equal_a to_o the_o solid_a cd_o though_o that_o be_v easy_a enough_o to_o conceive_v by_o the_o figure_n thereof_o describe_v in_o the_o plain_a ¶_o certain_a most_o profitable_a corollary_n annotation_n theorem_n and_o problem_n with_o other_o practice_n logistical_a and_o mechanical_a partly_o upon_o this_o 33._o and_o partly_o upon_o the_o 34._o 36._o and_o other_o follow_v add_v by_o master_n john_n dee_n ¶_o a_o corollary_n 1._o 1._o hereby_o it_o be_v manifest_a that_o two_o right_a line_n may_v be_v find_v which_o shall_v have_v that_o proportion_n the_o one_o to_o the_o other_o that_o any_o two_o like_a parallelipipedon_n and_o in_o like_a sort_n describe_v give_v have_v the_o one_o to_o the_o other_o suppose_v q_n and_o x_o to_o be_v two_o like_a parallelipipedon_n and_o in_o like_a sort_n describe_v of_o queen_n take_v any_o of_o the_o three_o line_n of_o which_o it_o be_v produce_v as_o namely_o rg_n of_o x_o take_v that_o right_a line_n of_o his_o production_n which_o line_n â—s_v answerable_a to_o r_o g_o in_o proportion_n which_o most_o apt_o after_o the_o greek_a name_n may_v be_v call_v omologall_n to_o rg_n &_o let_v that_o be_v tu._n by_o the_o 11._o of_o the_o six_o to_o rg_n and_o tu_fw-la let_v the_o three_o line_n in_o proportion_n with_o they_o be_v find_v and_o let_v that_o be_v y._n by_o the_o same_o 11._o of_o the_o six_o to_o tu_fw-la and_o y_fw-fr let_v the_o third_o right_a line_n be_v find_v in_o the_o say_a proportion_n of_o tu_fw-la to_o y_fw-fr &_o let_v that_o be_v z._n i_o say_v now_o that_o rg_n have_v that_o proportion_n to_o z_o which_o queen_n have_v to_o x._o for_o by_o construction_n we_o have_v four_o right_a line_n in_o continual_a propiotion_n namely_o rg_n tu_fw-la y_fw-fr and_o z._n wherefore_o by_o euclides_n corollary_n here_o before_o rg_n be_v to_o z_o as_o queen_n be_v to_o x._o wherefore_o we_o have_v find_v two_o right_a line_n have_v that_o proportion_n the_o one_o to_o the_o other_o which_o any_o two_o like_a parallelipipedon_n of_o like_a description_n give_v have_v the_o one_o to_o the_o other_o which_o be_v require_v to_o be_v do_v ¶_o a_o corollary_n as_o a_o converse_v of_o my_o â—ormer_a corollary_n do_v it_o follow_v to_o find_v two_o like_o parallelipipedon_n of_o like_a description_n which_o shall_v have_v that_o proportion_n the_o one_o to_o the_o other_o that_o any_o two_o right_a line_n give_v have_v the_o one_o to_o the_o other_o suppose_v the_o two_o right_a line_n give_v to_o be_v a_o and_o b_o imagine_v of_o four_o right_a line_n in_o continual_a proportion_n a._n to_o be_v the_o first_o and_o b_o to_o be_v the_o four_o or_o contrariwise_o b_o to_o be_v first_o &_o a_o to_o be_v the_o
superficies_n give_v to_o be_v â—_o and_o the_o rectangle_n parallelipipedon_n give_v to_o be_v am._n upon_o â—_o as_o a_o base_a must_n be_o be_v apply_v that_o be_v a_o rectangle_n parâ—llipipedon_n must_v be_v erect_v upon_o â—_o as_o a_o base_a which_o shall_v be_v equal_a to_o am._n by_o the_o last_o of_o the_o second_o to_o the_o right_n line_v figure_n â—_o let_v a_o equal_a square_n be_v make_v which_o suppose_v to_o be_v frx_n extend_v produce_v one_o side_n of_o the_o base_a of_o the_o parallelipipedom_n be_o which_o let_v be_v ac_fw-la extend_v to_o the_o point_n p._n let_v the_o other_o side_n of_o the_o say_v base_a concur_v with_o ac_fw-la be_v cg_o as_o cg_o be_v to_o fr_z the_o side_n of_o the_o square_a frx_n so_o let_v the_o same_o fr_z be_v to_o a_o line_n cut_v of_o from_o cp_n sufficient_o extend_v by_o the_o 11._o of_o the_o six_o and_o let_v that_o three_o proportional_a line_n be_v cp_n let_v the_o rectangle_n parallelogram_n be_v make_v perfect_a as_o cd_o it_o be_v evident_a that_o cd_o be_v equal_a to_o the_o square_a frx_n by_o the_o 17._o of_o the_o six_o and_o by_o construction_n frx_n be_v equal_a to_o b._n wherefore_o cd_o be_v equal_a to_o â—_o by_o the_o 12._o of_o the_o six_o as_o cp_n be_v to_o ac_fw-la so_o let_v a_o the_o heith_n of_o be_o be_v to_o the_o right_a line_n o._n i_o say_v that_o a_o solid_a perpendicular_o erect_v upon_o the_o base_a â—_o have_v the_o heith_n of_o the_o line_n oh_o be_v equal_a to_o the_o parallelipipedon_n am._n for_o cd_o be_v to_o agnostus_n as_o cp_n be_v to_o aâ—_z by_o the_o first_o of_o the_o six_o and_o â—_o be_v prove_v equal_a to_o cd_o wherefore_o by_o the_o 7._o of_o the_o five_o b_o be_v to_o ag_z as_o cp_n be_v to_o ac_fw-la but_o as_o cp_n be_v to_o ac_fw-la so_o be_v a_o to_o oh_o by_o construction_n wherefore_o b_o be_v to_o ag_z as_o a_o be_v to_o o._n so_o than_o the_o base_n â—_o and_o agnostus_n be_v reciprocal_o in_o proportion_n with_o the_o heithe_n a_o and_o o._n by_o this_o 34_o therefore_o a_o solid_a erect_v perpendicular_o upon_o â—_o as_o a_o base_a have_v the_o height_n oh_o be_v equal_a to_o am._n wherefore_o upon_o a_o right_n line_v plain_n superficies_n give_v we_o have_v apply_v a_o rectangle_n parallelipipedon_n give_v which_o be_v requisite_a to_o be_v do_v a_o problem_n 2._o a_o rectangle_n parallelipipedon_n be_v give_v to_o make_v a_o other_o equal_a to_o it_o of_o any_o heith_n assign_v suppose_v the_o rectangle_n parallelipipedon_n give_v to_o be_v a_o and_o the_o heith_n assign_v to_o be_v the_o right_a line_n â—_o now_o must_v we_o make_v a_o rectangle_n parallelipipedon_n equal_a to_o a_o who_o heith_n must_v be_v equal_a to_o â—_o according_a to_o the_o manner_n before_o use_v we_o must_v frame_v our_o construction_n to_o a_o reciprocal_a proportion_n between_o the_o base_n and_o heithe_n which_o will_v be_v do_v if_o as_o the_o heith_n assign_v bear_v itself_o in_o proportion_n to_o the_o heith_n of_o the_o parallelipipedon_n give_v so_o one_o of_o the_o side_n of_o the_o base_a of_o the_o parallelipipedon_n give_v be_v to_o a_o four_o line_n by_o the_o 12._o of_o the_o six_o find_v for_o that_o line_n find_v and_o the_o other_o side_n of_o the_o base_a of_o the_o give_v parallelipipedon_n contain_v a_o parallelogram_n which_o do_v serve_v for_o the_o base_a which_o only_o we_o want_v to_o use_v with_o our_o give_a heith_n and_o so_o be_v the_o problem_n to_o be_v execute_v note_n euclid_n in_o the_o 27._o of_o this_o eleven_o have_v teach_v how_o of_o a_o right_a line_n give_v to_o describe_v a_o parallepipedom_n like_a &_o likewise_o situate_v to_o a_o parallelipipedom_n give_v ay_o have_v also_o add_v how_o to_o a_o parallepipedon_n give_v a_o other_o may_v be_v make_v equal_a upon_o any_o right_n line_v base_a give_v or_o of_o any_o heith_n assign_v but_o if_o either_o euclid_n or_o any_o other_o before_o our_o time_n answerable_o to_o the_o 25._o of_o the_o six_o in_o plain_v have_v among_o solid_n invent_v this_o proposition_n for_o two_o unequal_a and_o unlike_a parallelipipedon_n be_v give_v to_o describe_v a_o parallelipipedon_n equal_a to_o the_o one_o and_o like_a to_o the_o other_o we_o will_v have_v give_v they_o their_o deserve_a praise_n and_o i_o will_v also_o have_v be_v right_o glad_a to_o have_v be_v ease_v of_o my_o great_a travail_n and_o discourse_n about_o the_o invent_v thereof_o here_o end_n i_o dee_n his_o addition_n upon_o this_o 34._o proposition_n the_o 30._o theorem_a the_o 35._o proposition_n if_o there_o be_v two_o superficial_a angle_n equal_a and_o from_o the_o point_n of_o those_o angle_n be_v elevate_v on_o high_a right_a line_n comprehend_v together_o with_o those_o right_a line_n which_o contain_v the_o superficial_a angle_n equal_a angle_n each_o to_o his_o corespondent_a angle_n and_o if_o in_o each_o of_o the_o elevate_v line_n be_v take_v a_o point_n at_o all_o aventure_n and_o from_o those_o point_n be_v draw_v perpendicular_a line_n to_o the_o ground_n plain_a superficiece_n in_o which_o be_v the_o angle_n give_v at_o the_o beginning_n and_o from_o the_o point_n which_o be_v by_o those_o perpendicular_a line_n make_v in_o the_o two_o plain_a superficiece_n be_v join_v to_o those_o angle_n which_o be_v put_v at_o the_o beginning_n right_a line_n those_o right_a line_n together_o with_o the_o line_n elevate_v on_o high_a shall_v contain_v equal_a angle_n svppose_v that_o these_o two_o rectiline_a superficial_a angle_n bac_n and_o edf_o be_v equal_a the_o one_o to_o the_o other_o construction_n and_o from_o the_o point_n a_o and_o d_o let_v there_o be_v elevate_v upward_o these_o right_a line_n agnostus_n and_o dm_o comprehend_v together_o with_o the_o line_n put_v at_o the_o begin_v equal_a angle_n each_o to_o his_o correspondent_a angle_n that_o be_v the_o angle_n mde_v to_o the_o angle_n gab_n and_o the_o angle_n mdf_n to_o the_o angle_n gac_n and_o take_v in_o the_o line_n agnostus_n and_o dm_o point_n at_o all_o adventure_n and_o let_v the_o same_o be_v g_z and_o m._n and_o by_o the_o 11._o of_o the_o eleven_o from_o the_o point_n g_z and_o m_o draw_v unto_o the_o ground_n plain_a superficiece_n wherein_o be_v the_o angle_n bac_n and_o e_o df_o perpendicular_a line_n gl_n and_o mn_o and_o let_v they_o fall_v in_o the_o say_v plain_a superâ—icieces_n in_o the_o point_n n_o and_o l_o and_o draw_v a_o right_a line_n from_o the_o point_n l_o to_o the_o point_n a_o and_o a_o other_o from_o the_o point_n n_o to_o the_o point_n d._n then_o i_o say_v that_o the_o angle_n galliard_n be_v equal_a to_o the_o angle_n mdn._n fron_n the_o great_a of_o the_o two_o line_n agnostus_n and_o dm_o which_o let_v be_v agnostus_n cut_v of_o by_o the_o 3._o of_o the_o first_o the_o line_n ah_o equal_a unto_o the_o line_n dm_o and_o by_o the_o 31._o of_o the_o first_o by_o the_o point_n h_o draw_v unto_o the_o line_n gl_n a_o parallel_a line_n and_o let_v the_o same_o be_v hk_a now_o the_o line_n gl_n be_v erect_v perpendicular_o to_o the_o ground_n plain_a superficies_n bal_n wherefore_o also_o by_o the_o 8._o of_o the_o eleven_o the_o line_n hk_o be_v erect_v perpendicular_o to_o the_o same_o ground_n plain_a superficies_n bac_n draw_v by_o the_o 12._o of_o the_o first_o from_o the_o point_n king_n and_o n_o unto_o the_o right_a line_n ab_fw-la ac_fw-la df_o &_o de_fw-fr perpendicular_a right_a line_n and_o let_v the_o same_o be_v kc_o nf_n kb_o ne._n and_o draw_v these_o right_a line_n hc_n cb_o mf_n fe_o now_o forasmuch_o aâ—_z by_o the_o 47._o of_o the_o first_o the_o square_a of_o the_o line_n ha_o be_v equal_a to_o the_o square_n of_o the_o line_n hk_o and_o ka_o demonstration_n but_o unto_o the_o square_n of_o the_o line_n ka_o be_v equal_a the_o square_n of_o the_o line_n kc_o and_o ca_n wherefore_o the_o square_a of_o the_o line_n ha_o be_v equal_a to_o the_o square_n of_o the_o line_n hk_o kc_o and_o ca._n but_o by_o the_o same_o unto_o the_o square_n of_o the_o line_n hk_o and_o kc_o be_v equal_a the_o square_n of_o the_o line_n hc_n wherefore_o the_o square_a of_o the_o line_n ha_o be_v equal_a to_o the_o square_n of_o the_o line_n hc_n and_o ca_n wherefore_o the_o angle_n hca_n be_v by_o the_o 48._o of_o the_o first_o a_o right_a angle_n and_o by_o the_o same_o reason_n also_o the_o angle_n mfd_n be_v a_o right_a angle_n wherefore_o the_o angle_n hca_n be_v equal_a to_o the_o angle_n mfd_n but_o the_o angle_n hac_fw-la be_v by_o supposition_n equal_a to_o the_o angle_n mdf_n wherefore_o there_o be_v two_o triangle_n mdf_n and_o hac_fw-la have_v two_o angle_n of_o the_o one_o equal_a to_o two_o angle_n of_o the_o other_o each_o to_o his_o correspondent_a angle_n
and_o one_o side_n of_o the_o one_o equal_a to_o one_o side_n of_o the_o other_o namely_o that_o side_n which_o subtend_v one_o of_o the_o equal_a angle_n that_o be_v the_o side_n ha_o be_v equal_a to_o the_o side_n dm_o by_o construction_n wherefore_o the_o side_n remain_v be_v by_o the_o 26._o of_o the_o first_o equal_a to_o the_o side_n remain_v wherefore_o the_o side_n ac_fw-la be_v equal_a to_o the_o side_n df._n in_o like_a sort_n may_v we_o prove_v that_o the_o side_n ab_fw-la be_v equal_a to_o the_o side_n de_fw-fr if_o you_o draw_v a_o right_a line_n from_o the_o point_n h_o to_o the_o point_n b_o and_o a_o other_o from_o the_o point_n m_o to_o the_o point_n e._n for_o forasmuch_o as_o the_o square_n of_o the_o line_n ah_o be_v by_o the_o 47._o of_o the_o first_o equal_v to_o the_o square_n of_o the_o line_n ak_o and_o kh_o and_o by_o the_o same_o unto_o the_o square_n of_o the_o line_n ak_o be_v equal_a the_o square_n of_o the_o line_n ab_fw-la and_o bk_o wherefore_o the_o square_n of_o the_o line_n ab_fw-la bk_o and_o kh_o be_v equal_a to_o the_o square_n of_o the_o line_n ah_o but_o unto_o the_o square_n of_o the_o line_n bk_o and_o kh_o be_v equal_a the_o square_n of_o the_o line_n bh_o by_o the_o 47._o of_o the_o first_o for_o the_o angle_n hkb_n be_v a_o right_a angle_n for_o that_o the_o line_n hk_o be_v erect_v perpendicular_o to_o the_o ground_n plain_a superficies_n wherefore_o the_o square_a of_o the_o line_n ah_o be_v equal_a to_o the_o square_n of_o the_o line_n ab_fw-la and_o bh_o wherefore_o by_o the_o 48._o of_o the_o first_o the_o angle_n abh_n be_v a_o right_a angle_n and_o by_o the_o same_o reason_n the_o angle_n dem_n be_v a_o right_a angle_n now_o the_o angle_n bah_o be_v equal_a to_o the_o angle_n edm_n for_o it_o be_v so_o suppose_a and_o the_o line_n ah_o be_v equal_a to_o the_o line_n dm_o wherefore_o by_o the_o 26._o of_o the_o first_o the_o line_n ab_fw-la be_v equal_a to_o the_o line_n de._n now_o forasmuch_o as_o the_o line_n ac_fw-la be_v equal_a to_o the_o line_n df_o and_o the_o line_n ab_fw-la to_o the_o line_n de_fw-fr therefore_o these_o two_o line_n ac_fw-la and_o ab_fw-la be_v equal_a to_o these_o two_o line_n fd_v and_o de._n but_o the_o angle_n also_o cab_n be_v by_o supposition_n equal_a to_o the_o angle_n fde_v wherefore_o by_o the_o 4._o of_o the_o first_o the_o base_a bc_o be_v equal_a to_o the_o base_a of_o and_o the_o triangle_n to_o the_o triangle_n and_o the_o rest_n of_o the_o angle_n to_o the_o rest_n of_o the_o angle_n wherefore_o the_o angle_n acb_o be_v equal_a to_o the_o angle_n dfe_n and_o the_o right_a angle_n ack_z be_v equal_a to_o the_o right_a angle_n dfn._n wherefore_o the_o angle_n remain_v namely_o bck_n be_v equal_a to_o the_o angle_n remain_v namely_o to_o efn_n and_o by_o the_o same_o reason_n also_o the_o angle_n cbk_n be_v equal_a to_o the_o angle_n fen_n wherefore_o there_o be_v two_o triangle_n bck_o &_o efn_n have_v two_o angle_n of_o the_o one_o equal_a to_o two_o angle_n of_o the_o other_o each_o to_o his_o correspondent_a angle_n and_o one_o side_n of_o the_o one_o equal_a to_o one_o side_n of_o the_o other_o namely_o that_o side_n that_o lie_v between_o the_o equal_a angle_n that_o be_v the_o side_n bc_o be_v equal_a to_o the_o side_n of_o wherefore_o by_o the_o 26._o of_o the_o first_o the_o side_n remain_v be_v equal_a to_o the_o side_n remain_v wherefore_o the_o side_n ck_o be_v equal_a to_o the_o side_n fn_o but_o the_o side_n ac_fw-la be_v equal_a to_o the_o side_n df._n wherefore_o these_o two_o side_n ac_fw-la and_o ck_o be_v equal_a to_o these_o two_o side_n df_o and_o fn_o and_o they_o contain_v equal_a angle_n wherefore_o by_o the_o 4._o of_o the_o first_o the_o base_a ak_n be_v equal_a to_o the_o base_a dn_o and_o forasmuch_o as_o the_o line_n ah_o be_v equal_a to_o the_o line_n dm_o therefore_o the_o square_a of_o the_o line_n ah_o be_v equal_a to_o the_o square_n of_o the_o line_n dm_o but_o unto_o the_o square_n of_o the_o line_n ah_o be_v equal_a the_o square_n of_o the_o line_n ak_o and_o kh_o by_o the_o 47._o of_o the_o first_o for_o the_o angle_n akh_n be_v a_o right_a angle_n and_o to_o the_o square_n of_o the_o line_n dm_o be_v equal_a the_o square_n of_o the_o line_n dn_o and_o nm_o for_o the_o angle_n dnm_n be_v a_o right_a angle_n wherefore_o the_o square_n of_o the_o line_n ak_o and_o kh_o be_v equal_a to_o the_o square_n of_o the_o line_n dn_o and_o nm_o of_o which_o two_o the_o square_a of_o the_o line_n ak_o be_v equal_a to_o the_o square_n of_o the_o line_n dn_o for_o the_o line_n ak_o be_v prove_v equal_a to_o the_o line_n a_fw-la wherefore_o the_o residue_n namely_o the_o square_a of_o the_o line_n kh_o be_v equal_a to_o the_o residue_n namely_o to_o the_o square_n of_o the_o line_n nm_o wherefore_o the_o line_n hk_o be_v equal_a to_o the_o line_n mn_v and_o forasmuch_o as_o these_o two_o line_n ha_o and_o ak_o be_v equal_a to_o these_o two_o line_n md_o and_o dn_o the_o one_o to_o the_o other_o and_o the_o base_a hk_n be_v equal_a to_o the_o base_a mn_n therefore_o by_o the_o 8._o of_o the_o first_o the_o angle_n hak_n be_v equal_a to_o the_o angle_n mdn._n if_o therefore_o there_o be_v two_o superficial_a angle_n equal_a and_o from_o the_o point_n of_o those_o angle_n be_v elevate_v on_o high_a right_a line_n comprehend_v together_o with_o those_o right_a line_n which_o be_v put_v at_o the_o beginning_n equal_a angle_n each_o to_o his_o corespondent_a angle_n and_o if_o in_o each_o of_o the_o erect_a line_n be_v take_v a_o point_n at_o all_o adventure_n and_o from_o those_o point_n be_v draw_v perpendicular_a line_n to_o the_o plain_a superficiece_n in_o which_o be_v the_o angle_n give_v at_o the_o beginning_n and_o frâ—â—_n the_o point_n which_o be_v by_o the_o perpendicular_a line_n make_v in_o the_o two_o plain_a superficiece_n be_v join_v right_a line_n to_o those_o angle_n which_o be_v put_v at_o the_o beginning_n those_o right_a line_n shall_v together_o with_o the_o line_n elevate_v on_o high_a make_v equal_a angle_n which_o be_v require_v to_o be_v prove_v because_o the_o figure_n of_o the_o former_a demonstration_n be_v somewhat_o hard_a to_o conceive_v as_o they_o be_v there_o draw_v in_o a_o plain_a by_o reason_n of_o the_o line_n that_o be_v imagine_v to_o be_v elevate_v on_o high_a i_o have_v here_o set_v other_o figure_n wherein_o you_o must_v erect_v perpendicular_o to_o the_o ground_n superficiece_v the_o two_o triangle_n bhk_v and_o emn_a and_o then_o elevate_v the_o triangle_n dfm_n &_o ache_n in_o such_o sort_n that_o the_o angle_n m_o and_o h_o of_o these_o triangle_n may_v concur_v with_o the_o angle_n m_o and_o h_o of_o the_o other_o erect_v triangle_n and_o then_o imagine_v only_o a_o line_n to_o be_v draw_v from_o the_o point_n g_o of_o the_o line_n agnostus_n to_o the_o point_n l_o in_o the_o ground_n superficies_n compare_v it_o with_o the_o former_a construction_n &_o demonstration_n and_o it_o will_v make_v it_o very_o easy_a to_o conceive_v ¶_o corollary_n by_o this_o it_o be_v manifest_a that_o if_o there_o be_v two_o rectiline_v superficial_a angle_n equal_a and_o upon_o those_o angle_n be_v elevate_v on_o high_a equal_a right_a line_n contain_v together_o with_o the_o right_a line_n put_v at_o the_o beginning_n equal_a angle_n perpendicular_a line_n draw_v from_o those_o elevate_v line_n to_o the_o ground_n plain_a superficiece_n wherein_o be_v the_o angle_n put_v at_o the_o beginning_n be_v equal_a the_o one_o to_o the_o other_o for_o it_o be_v manifest_a that_o the_o perpendicular_a line_n hk_o &_o mn_a which_o be_v draw_v from_o the_o end_n of_o the_o equal_a elevate_v line_n ah_o and_o dm_o to_o the_o ground_n superficiece_n be_v equal_a ¶_o the_o 31._o theorem_a the_o 36._o proposition_n if_o there_o be_v three_o right_a line_n proportional_a a_o parallelipipedon_n describe_v of_o those_o three_o right_a line_n be_v equal_a to_o the_o parallelipipedon_n describe_v of_o the_o middle_a line_n so_o that_o it_o consist_v of_o equal_a side_n and_o also_o be_v equiangle_n to_o the_o foresay_a parallelipipedon_n svppose_v that_o these_o three_o line_n a_o b_o c_o be_v proportional_a as_o a_o be_v to_o b_o so_o let_v b_o be_v to_o c._n then_o i_o say_v that_o the_o parallelipipedon_n make_v of_o the_o line_n a_o b_o c_o be_v equal_a to_o the_o parallelipipedon_n make_v of_o the_o line_n b_o so_fw-mi that_o the_o solid_a make_v of_o the_o line_n b_o consist_v of_o equal_a side_n and_o be_v also_o equiangle_n to_o the_o solid_a make_v of_o the_o line_n a_o b_o c._n describe_v by_o the_o 23._o of_o the_o
shall_v be_v proportional_a and_o iâ—_z the_o parallelipipedon_n describe_v of_o they_o be_v like_a and_o in_o like_a sort_n describe_v be_v proportional_a those_o right_a line_n also_o shall_v be_v proportional_a svppose_v that_o these_o four_o right_a line_n ab_fw-la cd_o of_o and_o gh_o be_v proportional_a as_o ab_fw-la be_v to_o cd_o so_o let_v of_o be_v to_o gh_o and_o upon_o the_o line_n ab_fw-la cd_o of_o and_o gh_o describe_v these_o parallelipipedon_n ka_o lc_n menander_n and_o ng_z be_v like_a and_o in_o like_a sort_n describe_v then_o i_o say_v that_o as_o the_o solid_a ka_o be_v â—o_o the_o solid_a lc_n so_o be_v the_o solid_a menander_n to_o the_o solid_a ng_z part_n for_o forasmuch_o as_o the_o parallelipipedon_n ka_o be_v like_a to_o the_o parallelipipedon_n lc_n therefore_o by_o the_o 33._o of_o the_o eleven_o the_o solid_a ka_o be_v to_o the_o solid_a lc_n in_o treble_a proportion_n of_o that_o which_o the_o side_n ab_fw-la be_v to_o the_o side_n cdâ—_n and_o by_o the_o same_o reason_n the_o parallelipipedon_n menander_n be_v to_o the_o parallelipipedon_n ng_z in_o treble_a proportion_n of_o that_o which_o the_o side_n of_o be_v to_o the_o side_n gh_o wherefore_o by_o the_o 11._o of_o the_o five_o as_o the_o parallelipipedon_n ka_o be_v to_o the_o parallelipipedon_n lc_n so_o be_v the_o parallelipipedon_n menander_n tâ—_n the_o parallelipipedon_n ng_z but_o now_o suppose_v that_o as_o the_o parallelipipedon_n ka_o be_v to_o the_o parallelipipedon_n lc_n so_o be_v the_o parallelipipedon_n menander_n to_o the_o parallelipipedon_n ng_z then_o i_o say_v part_n that_o as_o the_o right_a line_n ab_fw-la be_v to_o the_o right_a line_n cd_o so_o be_v the_o right_a liâ—â—_n of_o to_o the_o right_a line_n gh_o for_o again_o forasmuch_o as_o the_o solid_a ka_o be_v to_o the_o solid_a lc_n in_o treble_a proportion_n of_o that_o which_o the_o side_n ab_fw-la be_v to_o the_o side_n cd_o and_o the_o solid_a menander_n also_o be_v to_o the_o solid_a ng_z in_o treble_a proportion_n of_o that_o which_o the_o line_n of_o be_v to_o the_o line_n gh_o and_o as_o the_o solid_a ka_o be_v to_o the_o solid_a lc_n so_o be_v the_o solid_a menander_n to_o the_o solid_a ng_z wherefore_o also_o as_o the_o line_n ab_fw-la be_v to_o the_o line_n cd_o so_o be_v the_o line_n of_o to_o the_o line_n gh_o if_o therefore_o there_o be_v four_o right_a line_n proportional_a the_o parallelipipedon_n describe_v of_o those_o line_n be_v like_a &_o in_o like_a sort_n describe_v shall_v be_v proportional_a and_o if_o the_o parallelipipedon_n describe_v of_o they_o and_o be_v like_a and_o in_o like_a sort_n describe_v be_v proportional_a those_o right_a line_n also_o shall_v be_v proportional_o which_o be_v require_v to_o be_v prove_v ¶_o the_o 33._o theorem_a the_o 38._o proposition_n if_o a_o plain_a superficies_n be_v erect_v perpendicular_o to_o a_o plain_a superficies_n and_o from_o a_o point_n take_v in_o one_o of_o the_o plain_a superficiece_n be_v draw_v to_o the_o other_o plain_a superficies_n a_o perpendicular_a line_n that_o perpendicular_a line_n shall_v fall_v upon_o the_o common_a section_n of_o those_o plain_a superficiece_n svppose_v that_o the_o plain_a superficies_n cd_o be_v erect_v perpendicular_o to_o the_o plain_a superficies_n ab_fw-la and_o let_v their_o common_a section_n be_v the_o line_n dam_fw-ge and_o in_o the_o superficies_n cd_o take_v a_o point_n at_o all_o adventure_n and_o let_v the_o same_o be_v e._n then_o i_o say_v that_o a_o perpendicular_a line_n draw_v from_o the_o point_n e_o to_o the_o plain_a superficies_n ab_fw-la shall_v fall_v upon_o the_o right_a line_n da._n for_o if_o not_o then_o let_v it_o fall_v without_o the_o line_n dam_fw-ge as_o the_o line_n of_o do_v and_o let_v it_o fall_v upon_o the_o plain_a superficies_n ab_fw-la in_o the_o point_n f._n impossibility_n and_o by_o the_o 12._o of_o the_o first_o from_o the_o point_n fletcher_n draw_v unto_o the_o line_n dam_fw-ge be_v in_o the_o superficies_n ab_fw-la a_o perpendicular_a line_n fg_o which_o line_n also_o be_v erect_v perpendicular_o to_o the_o plain_a superficies_n cd_o by_o the_o three_o definition_n by_o reason_n we_o presuppose_v cd_o and_o ab_fw-la to_o be_v perpendicular_o erect_v each_o to_o other_o draw_v a_o right_a line_n from_o the_o point_n e_o to_o the_o point_n g._n and_o forasmuch_o as_o the_o line_n fg_o be_v erect_v perpendicular_o to_o the_o plain_a superficies_n cd_o and_o the_o line_n eglantine_n touch_v it_o be_v in_o the_o superficies_n cd_o wherefore_o the_o angle_n fge_n be_v by_o the_o 2._o definition_n of_o the_o eleven_o a_o right_a angle_n but_o the_o line_n of_o be_v also_o erect_v perpendicular_o to_o the_o superficies_n ab_fw-la wherefore_o the_o angle_n efg_o be_v a_o right_a angle_n now_o therefore_o two_o angle_n of_o the_o triangle_n efg_o be_v equal_a to_o two_o right_a angle_n which_o by_o the_o 17._o of_o the_o first_o be_v impossible_a wherefore_o a_o perpendicular_a line_n draw_v from_o the_o point_n e_o to_o the_o superficies_n ab_fw-la fall_v not_o without_o the_o line_n da._n wherefore_o it_o fall_v upon_o the_o line_n dam_fw-ge which_o be_v require_v to_o be_v prove_v ¶_o note_n campane_n make_v this_o as_o a_o corollary_n follow_v upon_o the_o 13_o and_o very_o well_o with_o small_a aid_n of_o other_o proposition_n he_o prove_v itâ—_n who_o demonstration_n there_o flussas_n have_v in_o this_o place_n and_o none_o other_o though_o he_o say_v that_o campane_n of_o such_o a_o proposition_n as_o of_o euclides_n make_v no_o mention_n in_o this_o figure_n you_o may_v more_o full_o see_v the_o former_a proposition_n and_o demonstration_n if_o you_o erect_v perpendicular_o unto_o the_o ground_n plain_a superficies_n ab_fw-la the_o superficies_n cd_o and_o imagine_v a_o line_n to_o be_v extend_v from_o the_o point_n e_o to_o the_o point_n fletcher_n instead_n whereof_o you_o may_v extend_v if_o you_o will_v a_o thread_n ¶_o the_o 34._o theorem_a the_o 39_o proposition_n if_o the_o opposite_a side_n of_o a_o parallelipipedon_n be_v divide_v into_o two_o equal_a part_n and_o by_o their_o common_a section_n be_v extend_v plain_a superficiece_n the_o common_a section_n of_o those_o plain_a superficiece_n and_o the_o diameter_n of_o the_o parallelipipedon_n shall_v divide_v the_o one_o the_o other_o into_o two_o equal_a part_n svppose_v that_o of_o be_v a_o parallelipipedon_n and_o let_v the_o opposite_a side_n thereof_o cf_o and_o ah_o be_v divide_v into_o two_o equal_a part_n in_o the_o point_n king_n l_o m_o n_o and_o likewise_o let_v the_o opposite_a side_n ad_fw-la and_o gf_o be_v divide_v into_o two_o equal_a part_n in_o the_o pointâ—s_v x_o pea_n oh_o r_o and_o by_o those_o section_n extend_v these_o two_o plain_a superficiece_n kn_n &_o xr_n and_o let_v the_o common_a section_n of_o those_o plain_a superficiece_n be_v the_o line_n we_o and_o let_v the_o diagonal_a line_n of_o the_o solid_a ab_fw-la be_v the_o line_n dg_o then_o i_o say_v that_o the_o line_n we_o and_o dg_o do_v divide_v the_o one_o the_o other_o into_o two_o equal_a part_n construction_n that_o be_v that_o the_o line_n vt_fw-la be_v equal_a to_o the_o line_n it_o be_v and_o the_o line_n dt_n to_o the_o line_n tg_n draw_v these_o right_a line_n dv_o we_o bs_n and_o sg_n demonstration_n now_o forasmuch_o as_o the_o line_n dx_o be_v a_o parallel_n to_o the_o line_n oe_z therefore_o by_o the_o 29._o of_o the_o first_o the_o angle_n dxv_o and_o who_n be_v alternate_a angle_n be_v equal_a the_o one_o to_o the_o other_o and_o forasmuch_o as_o the_o line_n dx_o be_v equal_a to_o the_o line_n oe_z and_o the_o line_n xv_o to_o the_o line_n vo_o and_o they_o comprehend_v equal_a angle_n wherefore_o the_o base_a dv_o be_v equal_a to_o the_o base_a we_o by_o the_o 4._o of_o the_o first_o and_o the_o triangle_n dxv_o be_v equal_a to_o the_o triangle_n who_n and_o the_o rest_n of_o the_o angle_n to_o the_o rest_n of_o the_o angle_n wherefore_o the_o angle_n xud_n be_v equal_a to_o the_o angle_n ove._n wherefore_o due_a be_v one_o right_a line_n and_o by_o the_o same_o reason_n bsg_n be_v also_o one_o right_a line_n and_o the_o line_n bs_n be_v equal_a to_o the_o line_n sg_n and_o forasmuch_o as_o the_o line_n ca_n be_v equal_a to_o the_o line_n db_o and_o be_v unto_o it_o a_o parallel_n but_o the_o line_n ca_n be_v equal_a to_o the_o line_n ge_z and_o be_v unto_o it_o also_o a_o parallel_n wherefore_o by_o the_o firsâ—_o common_a sentence_n the_o line_n db_o be_v equal_a to_o the_o line_n ge_z &_o be_v also_o a_o parallel_n unto_o it_o but_o the_o right_a line_n de_fw-fr and_o bg_o do_v join_v these_o parallel_a line_n together_o wherefore_o by_o the_o 33._o of_o the_o first_o the_o line_n de_fw-fr be_v a_o parallel_n unto_o the_o line_n bg_o and_o in_o either_o of_o these_o line_n be_v take_v point_n
other_o in_o one_o and_o the_o self_n same_o proportion_n be_v to_o the_o altitude_n in_o the_o same_o proportion_n that_o the_o base_n of_o the_o double_a solid_n namely_o of_o the_o parallelipipedon_n be_v for_o if_o the_o base_n of_o the_o equal_a parallelipipedon_n be_v reciprocal_a with_o their_o altitude_n than_o their_o half_n which_o be_v prism_n shall_v have_v their_o base_n reciprocal_a with_o their_o altitude_n by_o the_o 36._o proposition_n we_o may_v conclude_v that_o if_o there_o be_v three_o right_a line_n proportional_a the_o angle_n of_o a_o prism_n make_v of_o these_o three_o line_n be_v common_a with_o the_o angle_n of_o his_o parallelipipedon_n which_o be_v double_a do_v make_v a_o prism_n which_o be_v equal_a to_o the_o prism_n describe_v of_o the_o middle_a line_n and_o contain_v the_o like_a angle_n consist_v also_o of_o equal_a side_n for_o aâ—_z in_o the_o parallelipipedon_n so_o also_o in_o the_o prism_n this_o one_o thing_n be_v require_v namely_o that_o the_o three_o dimension_n of_o the_o proportional_a line_n do_v make_v a_o angle_n like_o unto_o the_o angle_n contain_v of_o the_o middle_a line_n take_v three_o time_n now_o than_o if_o the_o solid_a angle_n of_o the_o prism_n be_v make_v of_o those_o three_o right_a line_n there_o shall_v of_o they_o be_v make_v a_o angle_n like_o to_o the_o angle_n of_o the_o parallelipipedon_n which_o be_v double_a unto_o it_o wherefore_o it_o follow_v of_o necessity_n that_o the_o prism_n which_o be_v always_o the_o half_n of_o the_o parallelipipedon_n be_v equiangle_n the_o one_o to_o the_o other_o as_o also_o be_v their_o double_n although_o they_o be_v not_o equilater_n and_o therefore_o those_o half_n of_o equal_a solid_n be_v equal_a the_o one_o to_o the_o other_o namely_o that_o which_o be_v describe_v of_o the_o middle_n proportional_a line_n be_v equal_a to_o that_o which_o be_v describe_v of_o the_o three_o proportional_a line_n by_o the_o 37._o proposition_n also_o we_o may_v conclude_v the_o same_o touch_v prism_n which_o be_v conclude_v touch_a parallelipipedon_n for_o forasmuch_o as_o prism_n describe_v like_o &_o in_o like_a sort_n of_o the_o line_n give_v be_v the_o half_n of_o the_o parallelipipedon_n which_o be_v like_a and_o in_o like_a sort_n describe_v it_o follow_v that_o these_o prism_n have_v the_o one_o to_o the_o other_o the_o same_o proportion_n that_o the_o solid_n which_o be_v their_o double_n have_v and_o therefore_o if_o the_o line_n which_o describe_v they_o be_v porportionall_a they_o shall_v be_v proportional_a and_o so_o converse_o according_o to_o the_o rule_n of_o the_o say_v 37._o proposition_n but_o forasmuch_o as_o the_o 39_o proposition_n suppose_v the_o opposite_a superficial_a side_n of_o the_o solid_a to_o be_v parallelogram_n and_o the_o same_o solid_a to_o have_v one_o diameter_n which_o thing_n a_o prism_n can_v not_o have_v therefore_o this_o proposition_n can_v by_o â—o_o mean_v by_o apply_v to_o prism_n but_o as_o touch_v solid_n who_o base_n be_v two_o like_v equal_a and_o parallel_v poligonon_o figure_n and_o their_o side_n be_v parallelogram_n column_n forasmuch_o as_o by_o the_o second_o corollary_n of_o the_o 25._o of_o this_o book_n it_o have_v be_v declare_v that_o such_o solid_n be_v compose_v of_o prism_n it_o may_v easy_o be_v prove_v that_o their_o nature_n be_v such_o as_o be_v the_o nature_n of_o the_o prism_n whereof_o they_o be_v compose_v wherefore_o a_o parallelipipedon_n be_v by_o the_o 27._o proposition_n of_o this_o book_n describe_v there_o may_v also_o be_v describe_v the_o half_a thereof_o which_o be_v a_o prism_n and_o by_o the_o description_n of_o prism_n there_o may_v be_v compose_v a_o solid_a like_o unto_o a_o solid_a give_v compose_v of_o prism_n so_o that_o it_o be_v manifest_a that_o that_o which_o the_o 29._o 30._o 31._o 32._o 33._o 34._o and_o 37._o proposition_n set_v forth_o touch_v parallelipipedon_n may_v well_o be_v apply_v also_o to_o these_o kind_n of_o solid_n the_o end_n of_o the_o eleven_o book_n of_o euclides_n element_n ¶_o the_o twelve_o book_n of_o euclides_n element_n in_o this_o twelve_v book_n euclid_n set_v forth_o the_o passion_n and_o propriety_n of_o pyramid_n prism_n cones_n cylinder_n and_o sphere_n and_o compare_v pyramid_n first_o to_o pyramid_n then_o to_o prism_n so_o likewise_o do_v he_o cones_n and_o cylinder_n and_o last_o he_o compare_v sphere_n the_o one_o to_o the_o other_o but_o before_o he_o go_v to_o the_o treaty_n of_o those_o body_n he_o prove_v that_o like_o poligonon_n figure_v inscribe_v in_o circle_n and_o also_o the_o circle_n then selue_o be_v in_o proportion_n the_o one_o to_o the_o other_o as_o the_o square_n of_o the_o diameter_n of_o those_o circle_n be_v because_o that_o be_v necessary_a to_o be_v prove_v for_o the_o confirmation_n of_o certain_a passion_n and_o propriety_n of_o those_o body_n ¶_o the_o 1._o theorem_a the_o 1._o proposition_n like_a poligonon_n figure_v describe_v in_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 svppose_v that_o there_o be_v two_o circle_n abcde_o and_o fghkl_o and_o in_o they_o let_v there_o be_v describe_v like_o poligonon_n figure_n namely_o abcde_o and_o fghkl_o and_o let_v the_o diameter_n of_o the_o circle_n be_v bm_n and_o gn_n then_o i_o say_v that_o as_o the_o square_n of_o the_o line_n bm_n be_v to_o the_o square_n of_o the_o line_n gn_n so_o be_v the_o poligonon_n figure_n abcde_o to_o the_o poligonon_n figure_n fghkl_o construction_n draw_v these_o right_a line_n be_v be_o gl_n and_o fn_o and_o forasmuch_o as_o the_o poligonon_n figure_n abcde_o be_v like_a to_o the_o poligonon_n figure_n fghkl_o demonstration_n therefore_o the_o angle_n bae_o be_v equal_a to_o the_o angle_n gfl_fw-mi and_o as_o the_o line_n basilius_n be_v to_o the_o line_n ae_n so_o be_v the_o line_n gf_o to_o the_o line_n fl_fw-mi by_o the_o definition_n of_o like_a poligonon_n figure_v now_o therefore_o there_o be_v two_o triangle_n bae_o and_o gfl_fw-mi have_v one_o angle_n of_o the_o one_o equal_a to_o one_o angle_n of_o the_o other_o namely_o the_o angle_n bae_o equal_a to_o the_o angle_n gfl_fw-mi and_o the_o side_n about_o the_o equal_a angle_n be_v proportional_a wherefore_o by_o the_o first_o definition_n of_o the_o six_o the_o triangle_n abe_n be_v equiangle_n to_o the_o triangle_n fgl_n wherefore_o the_o angle_n aeb_fw-mi be_v equal_a to_o the_o angle_n flg_n but_o by_o the_o 21._o of_o the_o three_o the_o angle_n aeb_fw-mi be_v equal_a to_o the_o angle_n amb_n for_o they_o consist_v upon_o one_o and_o the_o self_n same_o circumference_n and_o by_o the_o same_o reason_n the_o angle_n flg_n be_v equal_a to_o the_o angle_n fng_v wherefore_o the_o angle_n amb_n be_v equal_a to_o the_o angle_n fng_v and_o the_o right_a angle_n bam_n be_v by_o the_o 4._o petition_n equal_a to_o the_o right_a angle_n gfn_n wherefore_o the_o angle_n remain_v be_v equal_a to_o the_o angle_n remain_v wherefore_o the_o triangle_n amb_n be_v equiangle_n to_o the_o triangle_n fng_v wherefore_o proportional_o as_o the_o line_n bm_n be_v to_o the_o line_n gn_n so_o be_v the_o line_n basilius_n to_o the_o line_n gf_o but_o the_o square_n of_o the_o line_n bm_n be_v to_o the_o square_n of_o the_o line_n gn_n in_o double_a proportion_n of_o that_o which_o the_o line_n bm_n be_v to_o the_o line_n gn_n by_o the_o corollary_n of_o the_o 20._o of_o the_o six_o and_o the_o poligonon_n figure_n abcde_o be_v to_o the_o poligonon_n figure_n fghkl_o in_o double_a proportion_n of_o that_o which_o the_o line_n basilius_n be_v to_o the_o line_n gf_o by_o the_o _o of_o the_o six_o 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 bm_n be_v to_o the_o square_n of_o the_o line_n gn_n so_o be_v the_o poligonon_n figure_n abcde_o to_o the_o poligonon_n figure_n fghkl_o wherefore_o like_a poligonon_n figure_v describe_v in_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 the_o diameter_n be_v which_o be_v require_v to_o be_v demonstrate_v ¶_o john_n dee_n his_o fruitful_a instruction_n with_o certain_a corollary_n and_o their_o great_a use_n who_o can_v not_o easy_o perceive_v what_o occasion_n and_o aid_n archimedes_n have_v by_o these_o first_o &_o second_o proposition_n to_o â—inde_v the_o near_a area_n or_o content_a of_o a_o circle_n between_o a_o poligonon_n figure_n within_o the_o circle_n and_o the_o like_a about_o the_o same_o circle_n describe_v who_o precise_a quantity_n be_v most_o easy_o know_v be_v comprehend_v of_o right_a line_n where_o also_o to_o avoid_v all_o occasion_n of_o error_n it_o be_v good_a in_o number_n not_o have_v precise_a square_a root_n to_o use_v the_o logistical_a process_n according_a to_o the_o rule_n with_o √_o â—_o 12_o √_o â—_o 19_o
same_o be_v treble_a it_o be_v manifest_a by_o the_o 12._o of_o the_o five_o that_o all_o the_o prism_n be_v to_o all_o the_o pyramid_n treble_a wherefore_o parallelipipedon_n be_v treble_a to_o pyramid_n set_v upon_o the_o self_n same_o base_a with_o they_o and_o under_o the_o same_o altitude_n they_o for_o that_o they_o contain_v two_o prism_n three_o corollary_n if_o two_o prism_n be_v under_o one_o and_o the_o self_n same_o altitude_n have_v to_o their_o base_n either_o both_o triangle_n or_o both_o parallelogram_n the_o prism_n be_v the_o one_o to_o the_o other_o as_o their_o base_n be_v for_o forasmuch_o as_o those_o prism_n be_v equemultiqlice_n unto_o the_o pyramid_n upon_o the_o self_n same_o base_n and_o under_o the_o same_o altitude_n which_o pyramid_n be_v in_o proportion_n as_o their_o base_n it_o be_v manifest_a by_o the_o 15._o of_o the_o five_o that_o the_o prism_n be_v in_o the_o proportion_n of_o the_o base_n for_o by_o the_o former_a corollary_n the_o prism_n be_v treble_a to_o the_o pyramid_n sâ—t_v upon_o the_o triangular_a base_n four_o corollary_n prism_n be_v in_o sesquealtera_fw-la proportion_n to_o pyramid_n which_o have_v the_o self_n same_o quadrangled_a base_a that_o the_o prism_n have_v and_o be_v under_o the_o self_n same_o altitude_n for_o that_o pyramid_n contain_v two_o pyramid_n set_v upon_o a_o triangular_a base_a of_o the_o same_o prism_n for_o it_o be_v prove_v that_o that_o prism_n be_v treble_a to_o the_o pyramid_n which_o be_v set_v upon_o the_o half_a of_o his_o quadrangled_a base_a unto_o which_o the_o other_o which_o be_v set_v upon_o the_o whole_a base_a be_v double_a by_o the_o six_o of_o this_o book_n five_v corollary_n wherefore_o we_o may_v in_o like_a sort_n conclude_v that_o solid_n mention_v in_o the_o second_o corollary_n which_o solid_n campane_n call_v side_v column_n be_v under_o one_o and_o the_o self_n same_o altitude_n be_v in_o proportion_n the_o one_o to_o the_o other_o as_o their_o base_n which_o be_v poligonon_o figure_n for_o they_o be_v in_o the_o proportion_n of_o the_o pyramid_n or_o prism_n set_v upon_o the_o self_n same_o base_n and_o under_o the_o self_n same_o altitude_n that_o be_v they_o be_v in_o the_o proportion_n of_o the_o base_n of_o the_o say_a pyramid_n or_o prism_n for_o those_o solid_n may_v be_v divide_v into_o prism_n have_v the_o self_n same_o altitude_n when_o as_o their_o opposite_a base_n may_v be_v divide_v into_o triangle_n by_o the_o 20_o of_o the_o six_o upon_o which_o triangle_n prism_n be_v set_v be_v in_o proportion_n as_o their_o base_n by_o this_o 7._o proposition_n it_o plain_o appear_v that_o â—uâ—lide_a as_o it_o be_v before_o note_v in_o the_o diffinitionâ—â—_n under_o the_o definition_n of_o a_o prism_n comprehend_v also_o those_o kind_n of_o solid_n which_o campane_n call_v side_v column_n for_o in_o that_o he_o say_v every_o prism_n have_v a_o triangle_n to_o his_o base_a may_v be_v devidedâ—_n etc_n etc_n he_o need_v not_o take_v a_o prism_n in_o that_o sense_n which_o campane_n and_o most_o man_n take_v it_o to_o have_v add_v that_o particle_n have_v to_o his_o base_a a_o triangle_n for_o by_o their_o sense_n there_o be_v no_o prism_n but_o it_o may_v have_v to_o his_o base_a a_o triangleâ—_n and_o so_o it_o may_v seem_v that_o euclid_n ought_v without_o exception_n have_v say_v that_o every_o prism_n whatsoever_o may_v be_v divide_v into_o three_o pyramid_n equal_a the_o one_o to_o the_o other_o have_v also_o triangle_n to_o â—heir_a base_n for_o so_o do_v campane_n and_o flussas_n put_v the_o proposition_n leave_v out_o the_o former_a particle_n have_v to_o his_o base_a a_o triangle_n which_o yet_o be_v red_a in_o the_o greek_a copy_n &_o not_o leâ—t_v out_o by_o any_o other_o interpreter_n know_v abroad_o except_o by_o campane_n and_o flussas_n yea_o and_o the_o corollary_n follow_v of_o this_o proposition_n add_v by_o theon_n or_o euclid_n and_o amend_v by_o m._n dee_n seem_v to_o confirm_v this_o sense_n of_o this_o â—s_a 〈◊〉_d make_v manifest_a that_o every_o pyramid_n be_v the_o three_o part_n of_o the_o prism_n have_v the_o same_o base_a with_o it_o and_o equal_a altitude_n for_o and_o if_o the_o base_a of_o the_o prism_n have_v any_o other_o right_o line_v figure_n than_o a_o triangle_n and_o also_o the_o superficies_n opposite_a to_o the_o base_a the_o same_o figure_n that_o prism_n may_v be_v divide_v into_o prism_n have_v triangle_v base_n and_o the_o superficiece_n to_o those_o base_n opposite_a also_o triangle_v a_o â—â—ike_a and_o equal_o for_o there_o as_o we_o see_v be_v put_v these_o word_n â—or_a and_o if_o the_o base_a of_o the_o prism_n be_v any_o other_o right_o line_v figureâ—_n etc_n etc_n whereof_o a_o man_n may_v well_o infer_v that_o the_o base_a may_v be_v any_o other_o rectiline_a figure_n whatsoever_o &_o not_o only_o a_o triangle_n or_o a_o parallelogram_n and_o it_o be_v true_a also_o in_o that_o sense_n as_o it_o be_v plain_a to_o see_v by_o the_o second_o corollary_n add_v out_o of_o flussas_n which_o corollary_n as_o also_o the_o first_o of_o his_o corollary_n be_v in_o a_o manner_n all_o one_o with_o the_o corollary_n add_v by_o theon_n or_o euclid_n far_o theon_n in_o the_o demonstration_n of_o the_o 10._o proposition_n of_o this_o book_n as_o we_o shall_v afterwards_o see_v most_o plain_o call_v not_o only_o side_v column_n prism_n but_o also_o parallelipipedon_n and_o although_o the_o 40._o proposition_n of_o the_o eleven_o book_n may_v seem_v hereunto_o to_o be_v a_o lâ—t_n for_o that_o it_o can_v be_v understand_v of_o those_o prism_n only_o which_o have_v triangle_n to_o their_o like_a equal_a opposite_a and_o parallel_v side_n or_o but_o of_o some_o side_v column_n and_o not_o of_o all_o yet_o may_v that_o let_v be_v thus_o remove_v away_o to_o say_v that_o euclid_n in_o that_o proposition_n use_v genus_fw-la pro_fw-la specie_fw-la that_o be_v the_o general_a word_n for_o some_o special_a kind_n thereof_o which_o thing_n also_o be_v not_o rare_a not_o only_o with_o he_o but_o also_o with_o other_o learned_a philosopher_n thus_o much_o i_o think_v good_a by_o the_o way_n to_o note_v in_o far_a defence_n of_o euclid_n definition_n of_o a_o prism_n the_o 8._o theorem_a the_o 8._o proposition_n pyramid_n be_v like_o &_o have_a triangle_n to_o their_o base_n be_v in_o treble_a proportion_n the_o one_o to_o the_o other_o of_o that_o in_o which_o their_o side_n of_o like_a proportion_n be_v svppose_v that_o these_o pyramid_n who_o base_n be_v the_o triangle_n gbc_n and_o hef_n and_o top_n the_o point_n a_o and_o d_o be_v like_o and_o in_o like_a sort_n describe_v and_o let_v ab_fw-la and_o de_fw-fr be_v side_n of_o like_a proportion_n then_o i_o say_v that_o the_o pyramid_n abcg_o be_v to_o the_o pyramid_n defh_o in_o treble_a proportion_n of_o that_o in_o which_o the_o side_n ab_fw-la be_v to_o the_o side_n de._n make_v perfect_a the_o parallelipipedon_n namely_o the_o solid_n bckl_n &_o efxo_n and_o forasmuch_o as_o the_o pyramid_n abcg_o be_v like_a to_o the_o pyramid_n defh_o construction_n therefore_o the_o angle_n abc_n be_v equal_a to_o the_o angle_n def_n demonstration_n &_o the_o angle_n gbc_n to_o the_o angle_n hef_fw-fr and_o moreover_o the_o angle_n abg_o to_o the_o angle_n deh_fw-it and_o as_o the_o line_n ab_fw-la be_v to_o the_o line_n de_fw-fr so_o be_v the_o line_n bc_o to_o the_o line_n of_o and_o the_o line_n bg_o to_o the_o line_n eh_o and_o for_o that_o as_o the_o line_n ab_fw-la be_v to_o the_o line_n de_fw-fr so_o be_v the_o line_n bc_o to_o the_o line_n of_o and_o the_o side_n about_o the_o equal_a angle_n be_v proportional_a therefore_o the_o parallelogram_n bm_n be_v like_a to_o the_o parallelogram_n ep_n and_o by_o the_o same_o reason_n the_o parallelogram_n bn_o be_v like_a to_o the_o parallelogram_n er_fw-mi and_o the_o parellelogramme_n bk_o be_v like_a unto_o the_o parallelogram_n exit_fw-la wherefore_o the_o three_o parallelogram_n bm_n kb_o and_o bn_o be_v like_a to_o the_o three_o parallelogram_n ep_n exit_fw-la and_o er._n but_o the_o three_o parallelogram_n bm_n kb_o and_o bn_o be_v equal_a and_o like_a to_o the_o three_o opposite_a parallelogram_n and_o the_o three_o parallelogram_n ep_n exit_fw-la and_o ere_o be_v equal_a and_o like_a to_o the_o three_o opposite_a parallelogram_n wherefore_o the_o parallelipipedon_n bckl_n and_o efxo_n be_v comprehend_v under_o plain_a superficiece_n like_a and_o equal_a in_o multitude_n wherefore_o the_o solid_a bckl_n be_v like_a to_o the_o solid_a efxo_n but_o like_o parallelipipedon_n be_v by_o the_o 33._o of_o the_o eleven_o in_o treble_a proportion_n the_o one_o to_o the_o other_o of_o that_o in_o which_o side_n of_o like_a proportion_n be_v to_o side_n of_o like_a proportion_n wherefore_o the_o solid_a bckl_n be_v to_o the_o solid_a efxo_n in_o treble_a
plain_a superficies_n of_o the_o four_o side_v figure_n kbos_n a_o perpendicular_a line_n ay_o and_o let_v it_o fall_v upon_o the_o plain_a superficies_n in_o the_o point_n y._n and_o draw_v these_o right_a line_n by_o &_o yk._n and_o forasmuch_o as_o the_o line_n ay_o be_v erect_v perpendicular_o to_o the_o plain_a superficies_n of_o bkos_n therefore_o the_o same_o ay_o be_v erect_v perpendicular_o to_o all_o the_o right_a line_n that_o touch_v it_o and_o be_v in_o the_o plain_a superficies_n of_o the_o four_o side_v figure_n by_o the_o â—_o definition_n of_o the_o eleven_o demonstration_n wherefore_o the_o line_n ay_o be_v erect_v perpendicular_o to_o either_o of_o these_o line_n by_o and_o yk._n and_o forasmuch_o as_o by_o the_o 15._o definition_n of_o the_o first_o the_o line_n aâ—_z be_v equal_a to_o the_o line_n ak_o therefore_o the_o square_a of_o the_o line_n ab_fw-la be_v equal_a to_o the_o square_n of_o the_o line_n ak_o and_o to_o the_o square_n of_o the_o line_n aâ—_z be_v equal_a the_o square_n of_o the_o line_n ay_o and_o yb_n by_o the_o 47._o of_o the_o first_o for_o the_o angle_n â—_o ya_fw-mi be_v a_o right_a angle_n and_o to_o the_o square_n of_o the_o line_n ak_o be_v equal_a the_o square_n of_o the_o line_n ay_o and_o yk._n wherefore_o the_o square_n of_o the_o line_n ay_o and_o yb_n be_v equal_a to_o the_o square_n of_o the_o line_n ay_o and_o tk_n take_v away_o the_o square_a of_o the_o line_n ay_o which_o be_v common_a to_o they_o both_o wherefore_o the_o residue_n namely_o the_o square_a of_o the_o by_o be_v equal_a to_o the_o residue_n namely_o to_o the_o square_n of_o the_o line_n yk._n wherefore_o the_o line_n by_o be_v equal_a to_o the_o line_n yk._n in_o like_a sort_n also_o may_v we_o prove_v that_o right_a line_n draw_v from_o the_o point_n y_fw-fr to_o the_o point_n oh_o and_o saint_n be_v equal_a to_o either_o of_o the_o line_n by_o and_o yk._n wherefore_o make_v the_o centre_n the_o point_n y_fw-fr and_o the_o space_n either_o the_o line_n by_o or_o the_o line_n yk_fw-mi describe_v a_o circle_n and_o it_o shall_v pass_v by_o the_o point_n oh_o and_o saint_n and_o the_o four_o side_v figure_n kbos_n shall_v be_v inscribe_v in_o the_o circle_n and_o forasmuch_o as_o the_o line_n kb_o be_v great_a than_o the_o line_n wz_n by_o the_o 2._o of_o the_o six_o because_o ak_o be_v great_a than_o awe_o but_o the_o line_n wz_n be_v equal_a to_o the_o line_n so._n wherefore_o the_o line_n bk_o be_v great_a they_o the_o line_n so._n but_o the_o line_n bk_o be_v equal_a to_o either_o of_o the_o line_n k_n and_o boy_n by_o construction_n wherefore_o either_o of_o the_o line_n k_n and_o â—o_n be_v great_a thâ—n_n the_o line_n so._n and_o forasmuch_o â—s_a in_o the_o circle_n be_v a_o four_o side_v figure_n kbos_n and_o the_o side_n bk_o boy_n and_o k_n be_v equal_a and_o the_o side_n os_fw-mi be_v less_o than_o any_o one_o of_o they_o and_o the_o line_n by_o be_v draw_v from_o the_o centre_n of_o the_o circle_n therefore_o the_o square_n of_o the_o line_n kb_o be_v great_a than_o the_o double_a of_o the_o square_n of_o the_o line_n by_o by_o the_o 12._o of_o the_o second_o afore_o that_o it_o subâ—endeth_v a_o angle_n great_a than_o a_o right_a angle_n contain_v of_o the_o two_o equal_a 〈◊〉_d bâ—_n and_o yk_n which_o angle_n byk_a be_v a_o obtuse_a angle_n for_o the_o 4._o angle_n at_o the_o centre_n y_fw-fr be_v equal_a to_o 4â—_o right_a angle_n of_o which_o three_o namely_o the_o angle_n byk_a kiss_v and_o byo_n be_v equal_a by_o the_o 4._o of_o the_o first_o and_o the_o four_o namely_o the_o angle_n syo_n be_v less_o than_o any_o of_o those_o three_o angle_n by_o the_o 25._o of_o the_o first_o draw_v by_o the_o 12._o of_o the_o first_o from_o the_o point_n king_n to_o the_o line_n bz_o a_o perpendicular_a line_n follow_v kz_o and_o forasmuch_o as_o the_o line_n bd_o be_v less_o than_o the_o double_a to_o the_o line_n dz_o for_o the_o line_n bd_o be_v double_a to_o the_o line_n dam_fw-ge which_o be_v less_o than_o the_o line_n dz_o but_o as_o the_o line_n bd_o be_v to_o the_o line_n dz_o so_o be_v the_o parallelogram_n contain_v under_o the_o line_n db_o and_o bz_o to_o the_o parallelogram_n contain_v under_o the_o line_n dz_o and_o zb_n by_o the_o 1._o of_o the_o six_o therefore_o if_o you_o describe_v upon_o the_o line_n bz_o a_o square_a and_o make_v perfect_v the_o parallelogram_n contain_v under_o the_o line_n zd_v and_o zb_n that_o which_o be_v contain_v under_o the_o line_n db_o &_o bz_o shall_v be_v less_o than_o the_o double_a to_o that_o which_o be_v contain_v under_o the_o line_n dz_o and_o zb_n and_o if_o you_o draw_v a_o right_a line_n from_o the_o point_n king_n to_o the_o point_n d_o that_o which_o be_v contain_v under_o the_o line_n db_o and_o bz_o equal_a to_o the_o square_n of_o the_o line_n bk_o by_o the_o corollary_n of_o the_o 8._o of_o the_o six_o for_o the_o angle_n bkd_v be_v a_o right_a angle_n by_o the_o 3â—_o of_o the_o three_o for_o it_o be_v in_o the_o semicircle_n bed_n and_o that_o which_o be_v contain_v under_o the_o line_n dz_o and_o zb_n be_v equal_a to_o the_o square_n of_o the_o line_n kz_o by_o the_o same_o corollary_n wherefore_o the_o square_a of_o the_o line_n kb_o be_v less_o than_o the_o double_a to_o the_o square_n of_o the_o line_n kz_o but_o the_o square_n of_o the_o line_n kb_o be_v great_a than_o the_o double_a to_o the_o square_n of_o the_o line_n by_o as_o before_o have_v be_v prove_v wherefore_o the_o square_a of_o the_o line_n kz_o be_v great_a than_o the_o square_a of_o the_o line_n by_o by_o the_o 10._o of_o the_o five_o and_o forasmuch_o as_o by_o the_o 15._o definition_n of_o the_o first_o the_o line_n basilius_n be_v equal_a to_o the_o line_n ka_o therefore_o also_o the_o square_a of_o the_o line_n basilius_n be_v equal_a to_o the_o square_n of_o the_o line_n ka_o but_o by_o the_o 47._o of_o the_o first_o unto_o the_o square_n of_o the_o line_n ab_fw-la be_v equal_a the_o square_n of_o the_o line_n by_o &_o ya_fw-mi for_o the_o angle_n bya_n be_v by_o construction_n a_o right_a angle_n and_o by_o the_o same_o reason_n to_o the_o square_n of_o the_o line_n ka_o be_v equal_a the_o square_n of_o the_o line_n kz_o and_o za_n for_o the_o angle_n kza_n be_v also_o by_o construction_n a_o right_a angle_n wherefore_o the_o square_n of_o the_o line_n by_o and_o ya_fw-mi be_v equal_a to_o the_o square_n of_o the_o line_n kz_o and_o â—aâ—_n of_o which_o the_o square_a of_o the_o line_n kz_o be_v great_a than_o the_o square_a of_o the_o line_n by_o as_o have_v before_o be_v prove_v wherefore_o the_o residue_n namely_o dee_n the_o square_a of_o the_o line_n za_n be_v less_o than_o the_o square_a of_o the_o line_n ya_fw-mi wherefore_o the_o line_n ya_fw-mi be_v great_a than_o the_o line_n az_o fgâ—h_n wherefore_o the_o line_n ay_o be_v much_o great_a theâ—_z the_o line_n ag._n but_o the_o line_n ay_o fall_v upon_o one_o of_o the_o base_n of_o the_o polihedron_n and_o the_o line_n agnostus_n fall_v upon_o the_o superficies_n of_o the_o less_o sphere_n wherefore_o the_o polihedron_n touch_v not_o the_o superficies_n of_o the_o less_o sphere_n a_o other_o and_o more_o ready_a demonstration_n to_o prove_v that_o the_o line_n ay_o be_v great_a than_o the_o line_n ag._n ag._n raise_v up_o by_o the_o 11._o of_o the_o first_o from_o the_o point_n g_o to_o the_o line_n agnostus_n a_o perpendicular_a line_n gl_n and_o draw_v a_o right_a line_n from_o the_o point_n a_o to_o the_o point_n l._n now_o they_o devide_v by_o the_o 30._o of_o the_o three_o the_o circumference_n eâ—_n into_o half_n &_o again_o that_o half_a into_o half_n &_o thus_o do_v continual_o we_o shall_v at_o the_o length_n by_o the_o corollary_n of_o the_o first_o of_o the_o ten_o leave_v a_o certain_a circumference_n which_o shall_v be_v less_o than_o the_o circumference_n of_o the_o circle_n bcd_v which_o be_v subtend_v of_o a_o line_n equal_a to_o the_o line_n gl_n let_v the_o circumference_n leave_v be_v kb_o wherefore_o also_o the_o right_a line_n kb_o be_v less_o than_o the_o right_a line_n gl_n and_o forasmuch_o as_o the_o four_o side_v figure_n bkos_n be_v in_o a_o circle_n and_o the_o line_n ob_fw-la bk_o and_o k_n be_v equal_a and_o the_o line_n os_fw-mi be_v less_o therefore_o the_o angle_n byk_a be_v a_o obtuse_a angle_n wherefore_o the_o line_n bk_o be_v great_a than_o the_o line_n by._n but_o the_o line_n gl_n be_v great_a than_o the_o line_n kb_o wherefore_o the_o line_n gl_n be_v much_o great_a than_o the_o line_n by._n wherefore_o also_o the_o square_a of_o the_o line_n gl_n be_v great_a than_o
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 then_o i_o say_v that_o either_o of_o the_o line_n ac_fw-la and_o cb_o be_v a_o irrational_a line_n of_o that_o kind_n which_o be_v call_v a_o residual_a extend_v the_o line_n ab_fw-la to_o the_o point_n d_o construction_n and_o let_v the_o line_n ad_fw-la be_v equal_a to_o half_o of_o the_o line_n ab_fw-la now_o forasmuch_o as_o the_o right_a line_n ab_fw-la demonstration_n be_v divide_v by_o a_o extreme_a &_o mean_a proportion_n in_o the_o point_n c_o and_o unto_o the_o great_a segment_n ac_fw-la be_v add_v a_o line_n ad_fw-la equal_a to_o the_o half_a of_o the_o right_a line_n ab_fw-la therefore_o by_o the_o 1._o of_o the_o thirteen_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 ad._n wherefore_o the_o square_a of_o the_o line_n cd_o have_v to_o the_o square_n of_o the_o line_n ad_fw-la that_o proportion_n that_o number_n have_v to_o number_v wherefore_o the_o square_a of_o the_o line_n cd_o be_v commensurable_a to_o the_o square_n of_o the_o line_n ad._n but_o the_o square_n of_o the_o line_n dam_n be_v rational_a for_o the_o line_n dam_n be_v rational_a forasmuch_o as_o it_o be_v the_o half_a of_o the_o rational_a line_n ab_fw-la wherefore_o the_o square_a of_o the_o line_n cd_o be_v rational_a wherefore_o also_o the_o linâ—_n cd_o be_v rational_a and_o forasmuch_o as_o the_o square_n of_o the_o line_n cd_o have_v not_o to_o the_o square_n of_o the_o line_n ad_fw-la that_o proportion_n that_o a_o square_a number_n have_v to_o a_o square_a number_n therefore_o by_o the_o 9_o of_o the_o ten_o the_o line_n cd_o be_v incommensurable_a in_o length_n to_o the_o line_n ad._n wherefore_o the_o line_n cd_o and_o dam_n be_v rational_a commensurable_a in_o power_n only_o wherefore_o the_o line_n ac_fw-la be_v a_o residual_a line_n by_o the_o 73._o of_o the_o ten_o again_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 and_o the_o great_a segment_n thereof_o be_v ac_fw-la therefore_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la &_o bc_o be_v equal_a to_o the_o square_n of_o the_o line_n ac_fw-la wherefore_o the_o square_a of_o the_o line_n ac_fw-la apply_v to_o the_o rational_a line_n ab_fw-la make_v the_o breadth_n bc._n but_o the_o square_n of_o a_o residual_a line_n apply_v to_o a_o rational_a line_n make_v the_o breadth_n a_o first_o residual_a line_n by_o the_o 97._o of_o the_o ten_o wherefore_o the_o line_n cb_o be_v a_o first_o residual_a line_n and_o it_o be_v prove_v that_o the_o line_n ac_fw-la be_v also_o a_o residual_a line_n if_o therefore_o a_o rational_a right_a line_n be_v divide_v by_o a_o extreme_a and_o meanâ—_z 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 which_o be_v require_v to_o be_v demonstrate_v ¶_o a_o corollary_n add_v by_o campane_n hereby_o it_o be_v manifest_a that_o if_o the_o great_a segment_n be_v a_o rational_a line_n the_o less_o segment_n shall_v be_v a_o residual_a line_n for_o if_o the_o great_a segment_n ac_fw-la of_o the_o right_a line_n acb_o be_v divide_v into_o two_o equal_a part_n in_o the_o point_n d_o the_o square_a of_o the_o line_n db_o shall_v be_v quintâ—plâ—_n to_o the_o square_n of_o the_o line_n dc_o by_o the_o 3._o of_o this_o book_n and_o forasmuch_o as_o the_o line_n cd_o be_v the_o â—alfe_a of_o the_o rational_a line_n suppose_v ac_fw-la be_v rational_a by_o the_o 6._o definition_n of_o the_o ten_o and_o unto_o the_o square_n of_o the_o line_n dc_o the_o square_a of_o the_o line_n db_o be_v commensurable_a for_o it_o be_v quintuple_a unto_o it_o wherefore_o the_o square_a of_o the_o line_n db_o be_v rational_a wherefore_o also_o the_o line_n db_o be_v rational_a and_o forasmuch_o as_o the_o square_n of_o the_o line_n db_o &_o dc_o be_v not_o in_o proportion_n as_o a_o square_a number_n be_v to_o a_o square_a number_n therefore_o the_o line_n db_o and_o dc_o be_v incommensurable_a in_o length_n by_o the_o 9_o of_o the_o ten_o wherefore_o they_o be_v commensurable_a in_o power_n only_o wherefore_o by_o the_o 73._o of_o the_o ten_o the_o line_n bc_o which_o be_v the_o less_o segment_n be_v a_o residual_a line_n the_o 7._o theorem_a the_o 7._o proposition_n if_o a_o equilater_n pentagon_n have_v three_o of_o his_o angle_n whether_o they_o follow_v in_o order_n or_o not_o in_o order_n equal_v the_o one_o to_o the_o other_o that_o pentagon_n shall_v be_v equiangle_n svppose_v that_o abcde_o be_v a_o equilater_n pentagon_n and_o let_v the_o angle_n of_o the_o say_a pentagon_n proposition_n namely_o first_o three_o angle_n follow_v in_o order_n which_o be_v at_o the_o point_v a_o b_o c_o be_v equal_a the_o one_o to_o the_o other_o then_o i_o say_v that_o the_o pentagon_n abcde_o be_v equiangle_n construction_n draw_v these_o right_a line_n ac_fw-la be_v and_o fd._n now_o forasmuch_o as_o these_o two_o line_n cb_o case_n and_o basilius_n be_v equal_a to_o these_o two_o line_n ba_o and_o ae_n the_o one_o to_o the_o other_o and_o the_o angle_n cba_o be_v equal_a to_o the_o angle_n bae_o demonstration_n therefore_o by_o the_o 4._o of_o the_o first_o the_o base_a ac_fw-la be_v equal_a to_o the_o base_a be_v and_o the_o triangle_n abc_n be_v equal_a to_o the_o triangle_n abe_n and_o the_o rest_n of_o the_o angle_n be_v equal_a to_o the_o rest_n of_o the_o angle_n under_o which_o be_v subtend_v equal_a side_n wherefore_o the_o angle_n bca_o be_v equal_a to_o the_o angle_n bea_fw-mi and_o the_o angle_n abe_n to_o the_o angle_n cab_n wherefore_o also_o the_o side_n of_o be_v equal_a to_o the_o side_n bf_o by_o the_o 6._o of_o the_o first_o and_o it_o be_v prove_v that_o the_o whole_a line_n ac_fw-la be_v equal_a to_o the_o whole_a line_n be._n wherefore_o the_o residue_n cf_o be_v equal_a to_o the_o residue_n fâ—_n and_o the_o line_n cd_o be_v equal_a to_o the_o line_n de._n wherefore_o these_o two_o line_n fc_a and_o cd_o be_v equal_a to_o these_o two_o line_n fe_o &_o ed_z and_o the_o base_a fd_o be_v common_a to_o they_o both_o wherefore_o the_o angle_n fcd_n be_v equal_a to_o the_o angle_n feed_v by_o the_o 8._o of_o the_o first_o and_o it_o be_v prove_v that_o the_o angle_n bca_o be_v equal_a to_o the_o angle_n aeb_fw-mi wherefore_o the_o whole_a angle_n bcd_o be_v equal_a to_o the_o whole_a angle_n aed_n but_o the_o angle_n bcd_o be_v suppose_v to_o be_v equal_a to_o the_o angle_n a_o and_o b._n wherefore_o the_o angle_n aed_n be_v equal_a to_o the_o angle_n a_o and_o b._n in_o like_a sort_n also_o may_v we_o prove_v that_o the_o angle_n cde_o be_v equal_a to_o the_o angle_n a_o and_o b._n wherefore_o the_o pentagon_n abcde_o be_v equiangle_n case_n but_o now_o suppose_v that_o three_o angle_n which_o follow_v not_o in_o order_n be_v equal_a the_o one_o to_o the_o other_o namely_o let_v the_o angle_n a_o c_o d_o be_v equal_a then_o i_o say_v that_o in_o this_o case_n also_o the_o pentagon_n abcde_o be_v equiangle_n draw_v a_o right_a line_n from_o the_o point_n b_o to_o the_o point_n d._n now_o forasmuch_o as_o these_o two_o line_n ba_o and_o aâ—_z be_v equal_a to_o these_o two_o line_n bc_o and_o cd_o and_o they_o comprehend_v equal_a angle_n therefore_o by_o the_o 4._o of_o the_o first_o the_o base_a be_v be_v equal_a to_o the_o base_a bd._n and_o the_o triangle_n abe_n be_v equal_a to_o the_o triangle_n bdc_o and_o the_o rest_n of_o the_o angle_n be_v equal_a to_o the_o rest_n of_o the_o angle_n under_o which_o be_v subtend_v equal_a side_n wherefore_o the_o angle_n aeb_fw-mi be_v equal_a to_o the_o angle_n cdb_n and_o the_o angle_n bed_n be_v equal_a to_o the_o angle_n bde_v by_o the_o 5._o of_o the_o first_o for_o the_o side_n be_v be_v equal_a to_o the_o side_n bd._n wherefore_o the_o whole_a angle_n aed_n be_v equal_a to_o the_o whole_a angle_n cde_o but_o the_o angle_n cde_o be_v suppose_v to_o be_v equal_a to_o the_o angle_n a_o and_o c._n wherefore_o the_o aed_n be_v equal_a to_o the_o angle_n a_o and_o c._n and_o by_o the_o same_o reason_n also_o the_o angle_n abc_n be_v equal_a to_o the_o angle_n a_o c_z and_o d._n wherefore_o the_o pentagon_n abcde_o be_v equiangle_n if_o therefore_o a_o equilater_n pentagon_n have_v three_o of_o his_o angle_n whither_o they_o follow_v in_o order_n or_o not_o in_o order_n equal_v the_o one_o to_o the_o other_o that_o pentagon_n shall_v be_v equiangle_n which_o be_v require_v to_o be_v prove_v the_o 8._o problem_n the_o 8._o proposition_n if_o in_o a_o equilater_n &_o equiangle_n pentagon_n two_o right_a line_n do_v subtend_v two_o of_o
be_v a_o line_n divide_v by_o a_o extreme_a and_o mean_a proportion_n and_o the_o great_a segment_n of_o the_o same_o be_v the_o side_n of_o the_o hexagon_n which_o be_v require_v to_o be_v prove_v a_o corollary_n add_v by_o flussas_n campane_n hereby_o it_o be_v manifest_a that_o the_o side_n of_o a_o exagon_n inscribe_v in_o a_o circle_n be_v cut_v by_o a_o extreme_a and_o mean_a proportion_n the_o great_a segment_n thereof_o be_v the_o side_n of_o the_o decagon_n inscribe_v in_o the_o same_o circle_n for_o if_o from_o the_o right_a line_n dc_o be_v cut_v of_o a_o right_a line_n equal_a to_o the_o line_n cb_o we_o may_v thus_o reason_n as_o the_o whole_a db_o be_v to_o the_o whole_a dc_o so_o be_v the_o part_n take_v away_o dc_o to_o the_o part_n take_v away_o cb_o wherefore_o by_o the_o 19_o of_o the_o five_o the_o residue_n be_v to_o the_o residue_n as_o the_o whole_a be_v to_o the_o whole_a wherefore_o the_o line_n dc_o be_v cut_v like_o unto_o the_o line_n db_o and_o therefore_o be_v cut_v by_o a_o extreme_a and_o mean_a proportion_n campane_n put_v the_o converse_n of_o this_o proposition_n after_o this_o manner_n if_o a_o line_n be_v divide_v by_o a_o 〈◊〉_d and_o â—â—ane_a proportion_n of_o 〈◊〉_d circle_n the_o great_a segment_n be_v the_o side_n of_o a_o equilater_n hexagon_n of_o the_o same_o shall_v the_o less_o segment_n be_v the_o side_n of_o a_o equilater_n decagon_n and_o of_o what_o circle_n the_o less_o segment_n be_v the_o side_n of_o a_o equilater_n decagon_n of_o the_o same_o be_v the_o great_a segment_n the_o side_n of_o a_o equilater_n hexagon_n for_o the_o former_a figure_n remain_v suppose_v that_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 c_o and_o let_v the_o great_a segment_n thereof_o be_v dc_o then_o i_o say_v that_o of_o what_o circle_n the_o line_n dc_o be_v the_o side_n of_o a_o equilater_n hexagon_n of_o the_o same_o circle_n be_v the_o line_n cb_o the_o side_n of_o a_o equilater_n decagon_n and_o of_o what_o circle_n the_o line_n bc_o be_v the_o side_n of_o a_o equilater_n decagon_n of_o the_o same_o be_v the_o line_n dc_o the_o side_n of_o a_o equilater_n hexagon_n part_n for_o if_o the_o line_n dc_o be_v the_o side_n of_o a_o hexagon_n inscribe_v in_o the_o circle_n then_o by_o the_o corollary_n of_o the_o 15._o of_o the_o four_o the_o line_n dc_o be_v equal_a to_o the_o line_n be._n and_o forasmuch_o as_o the_o proportion_n of_o the_o line_n bd_o to_o the_o line_n dc_o be_v as_o the_o proportion_n of_o the_o line_n dc_o to_o the_o line_n cb_o by_o supposition_n therefore_o by_o the_o 7._o of_o the_o five_o the_o proportion_n of_o the_o line_n bd_o to_o the_o line_n be_v be_v as_o the_o proportion_n of_o the_o line_n be_v to_o the_o line_n bc._n wherefore_o by_o the_o 6._o of_o the_o six_o the_o two_o triangle_n deb_n ebc_n be_v equiangle_n for_o the_o angle_n b_o be_v common_a to_o each_o triangle_n wherefore_o the_o angle_n d_o be_v equal_a to_o the_o angle_n ceb_fw-mi for_o they_o be_v subtend_v of_o side_n of_o like_a proportion_n and_o forasmuch_o as_o the_o angle_n aec_o be_v quadruple_a to_o the_o angle_n d_o by_o the_o 32._o of_o the_o first_o twice_o take_v and_o by_o the_o 5._o of_o the_o same_o therefore_o the_o same_o angle_n aec_o be_v quadruple_a to_o the_o angle_n ceb_fw-mi wherefore_o by_o the_o last_o of_o the_o six_o the_o circumference_n ac_fw-la be_v quadruple_a to_o to_o the_o circumference_n cb._n wherefore_o the_o line_n bc_o be_v the_o side_n of_o a_o decagon_n inscribe_v in_o the_o circle_n acb_o but_o now_o if_o the_o line_n bc_o be_v the_o side_n of_o a_o decagon_n inscribe_v in_o the_o circle_n abc_n the_o line_n cd_o shall_v be_v the_o side_n of_o a_o hexagon_n inscribe_v in_o the_o same_o circle_n part_n for_o let_v dc_o be_v the_o side_n of_o a_o hexagon_n inscribe_v in_o the_o circle_n h._n now_o by_o the_o first_o part_n of_o this_o proposition_n the_o line_n bc_o shall_v be_v the_o side_n of_o a_o decagon_n inscribe_v in_o the_o same_o circle_n suppose_v that_o in_o the_o two_o circle_n acb_o and_o h_o be_v inscribe_v equilater_n decagon_n all_o who_o side_n shall_v be_v equal_a to_o the_o line_n cb._n and_o forasmuch_o as_o every_o equilater_n figure_n inscribe_v in_o a_o circle_n be_v also_o equiangle_n therefore_o both_o the_o decagon_n be_v equiangle_n and_o forasmuch_o as_o all_o the_o angle_n of_o the_o one_o take_v together_o be_v equal_a to_o all_o the_o angle_n of_o the_o other_o take_v together_o as_o it_o be_v easy_a to_o be_v prove_v by_o that_o which_o be_v add_v after_o the_o 32._o of_o the_o first_o therefore_o one_o of_o these_o decagon_n be_v equiangle_n to_o the_o other_o and_o therefore_o the_o one_o be_v like_a to_o the_o other_o by_o the_o definition_n of_o like_a superficiece_n and_o for_o that_o if_o there_o be_v two_o like_o rectiline_a figure_n inscribe_v in_o two_o circle_n the_o proportion_n of_o the_o side_n of_o like_a proportion_n of_o those_o figure_n shall_v be_v as_o the_o proportion_n of_o the_o diameter_n of_o those_o circle_n as_o it_o be_v easy_a to_o prove_v by_o the_o corollary_n of_o the_o 20._o of_o the_o six_o and_o first_o of_o this_o bookâ—_n but_o the_o side_n of_o the_o like_a decaâ—ons_n inscribe_v in_o the_o two_o circle_n abc_n and_o h_o be_v equal_a therefore_o their_o diameter_n also_o be_v equal_a wherefore_o also_o their_o semidiameter_n be_v equal_a but_o the_o sâ—midiâ—meters_n and_o the_o side_n of_o the_o hexagon_n be_v equal_a by_o the_o corollary_n of_o the_o 1â—_o of_o the_o four_o wherefore_o the_o line_n dc_o be_v the_o side_n of_o a_o hexagon_n in_o the_o circle_n abc_n as_o also_o it_o be_v the_o side_n of_o a_o hexagon_n inscribe_v in_o the_o circle_n fletcher_n which_o be_v equal_a to_o the_o circled_a abc_n which_o be_v require_v to_o be_v prove_v ¶_o the_o 10._o theorem_a the_o 10._o proposition_n if_o in_o a_o circle_n be_v describe_v a_o equilater_n pentagon_n the_o side_n of_o the_o pentagon_n contain_v in_o power_n both_o the_o side_n of_o a_o hexagon_n and_o the_o side_n of_o a_o decagon_n be_v all_o describe_v in_o one_o and_o the_o self_n same_o circle_n svppose_v that_o abcde_v be_v a_o circle_n and_o in_o the_o circle_n abcde_v describe_v by_o the_o 11._o of_o the_o four_o a_o pentagon_n figure_n abcde_o then_o i_o say_v that_o the_o side_n of_o the_o pentagon_n â—igure_v abcde_o contain_v in_o power_n both_o the_o side_n of_o a_o hexagon_n figure_n and_o of_o a_o decagon_n figure_n be_v describe_v in_o the_o circle_n abcde_o construction_n take_v by_o the_o 1._o of_o the_o three_o the_o centre_n of_o the_o circle_n and_o let_v the_o same_o be_v f._n and_o draw_v a_o right_a line_n from_o the_o point_n a_o to_o the_o point_n fletcher_n extend_v it_o to_o the_o point_n g._n and_o draw_v a_o right_a line_n from_o the_o point_n f_o to_o the_o point_n b._n and_o from_o the_o point_n f_o draw_v by_o the_o 12._o of_o the_o first_o unto_o the_o line_n ab_fw-la a_o perpendicular_a line_n fh_o and_o extend_v it_o to_o the_o point_n k._n and_o draw_v the_o right_a line_n ak_o and_o kb_o and_o again_o from_o the_o point_n f_o draw_v by_o the_o same_o unto_o the_o line_n ak_o a_o perpendicular_a line_n fn_o and_o extend_v fn_o to_o the_o point_n m_o which_o line_n let_v cut_v the_o line_n ab_fw-la in_o the_o point_n l._n and_o draw_v a_o right_a line_n from_o the_o point_n king_n to_o the_o point_n l._n now_o forasmuch_o as_o the_o circumference_n abcg_o be_v equal_a to_o the_o circumference_n aedg_n demonstration_n of_o which_o the_o circumference_n abc_n be_v equal_a to_o the_o circumference_n aed_n therefore_o the_o rest_n of_o the_o circumference_n namely_o cg_o be_v equal_a to_o the_o rest_n of_o the_o circumference_n namely_o to_o dg_v but_o the_o circumference_n cd_o be_v subtend_v of_o the_o side_n of_o a_o pentagon_n wherefore_o the_o circumference_n cg_o be_v subtend_v of_o the_o side_n of_o a_o decagon_n â—igure_n and_o forasmuch_o as_o the_o line_n bh_o be_v equal_a to_o the_o line_n ha_o by_o the_o 3._o of_o the_o three_o and_o the_o line_n fh_o be_v common_a to_o they_o both_o &_o the_o angle_n at_o the_o point_n h_o be_v right_a angle_n wherefore_o by_o the_o 4._o of_o the_o first_o the_o angle_n afk_n be_v equal_a to_o the_o angle_n kfb._n wherefore_o also_o by_o the_o 26._o of_o the_o three_o the_o circumference_n ak_o be_v equal_a to_o the_o circumference_n kb_o wherefore_o the_o circumference_n ab_fw-la be_v double_a to_o the_o circumference_n bk_o wherefore_o the_o right_a line_n ak_o be_v the_o side_n of_o a_o decagon_n figure_n and_o by_o the_o same_o reason_n also_o the_o circumference_n ak_o be_v double_a to_o the_o
circumference_n km_o and_o forasmuch_o as_o the_o circumference_n ab_fw-la be_v double_a to_o the_o circumference_n bk_o but_o the_o circumference_n cd_o be_v equal_a to_o the_o circumference_n ab_fw-la wherefore_o the_o circumference_n cd_o be_v double_a to_o the_o circumference_n bk_o but_o the_o circumference_n cd_o be_v double_a to_o the_o circumference_n cg_o wherefore_o the_o circumference_n cg_o be_v equal_a to_o the_o circumference_n bk_o but_o the_o circumference_n bk_o be_v double_a to_o the_o circumference_n km_o for_o that_o the_o circumference_n ka_o be_v double_a thereunto_o â—_o wherefore_o also_o the_o circumference_n cg_o be_v double_a to_o the_o circumference_n km_o but_o the_o circumference_n cb_o be_v also_o double_a to_o the_o circumference_n bk_o for_o the_o circumference_n cb_o be_v equal_a to_o the_o circumference_n basilius_n wherefore_o the_o whole_a circumference_n gb_o be_v double_a to_o the_o whole_a circumference_n bm_n by_o the_o 12._o of_o the_o five_o wherefore_o also_o the_o angle_n gfb_n be_v double_a to_o the_o angle_n bfm_n by_o the_o last_o of_o the_o six_o but_o the_o angle_n gfb_n be_v double_a to_o the_o angle_n fab._n by_o the_o 32._o of_o the_o first_o or_o 20._o of_o the_o three_o for_o the_o angle_n fab._n be_v equal_a to_o the_o angle_n abf_n wherefore_o the_o angle_n bfl_fw-mi be_v equal_a â—o_o the_o angle_n fab._n by_o the_o 15._o of_o the_o five_o and_o the_o angle_n abf_n be_v common_a to_o the_o two_o triangle_n abf_a and_o bfl_fw-mi wherefore_o by_o the_o corollary_n of_o the_o â—â—_o of_o the_o first_o the_o angle_n remain_v afb_o be_v equal_a to_o the_o angle_n remain_v blf_n wherefore_o the_o triangle_n abf_n be_v equiangle_n to_o the_o triangle_n bfl_fw-mi wherefore_o by_o the_o 4._o of_o the_o six_o proportional_o as_o the_o right_a line_n ab_fw-la be_v to_o the_o right_a line_n bf_o so_o be_v the_o same_o right_a line_n fb_o to_o the_o right_a line_n bl_o wherefore_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bl_o be_v equal_a to_o the_o square_n of_o the_o line_n bf_o by_o the_o 17._o of_o the_o six_o again_o forasmuch_o as_o the_o line_n all_o be_v equal_a to_o the_o line_n lk_v for_o by_o the_o last_o of_o the_o six_o the_o angle_n kfl_fw-mi be_v equal_a to_o the_o angle_n afl_fw-mi which_o equal_a angle_n be_v contain_v under_o the_o line_n fk_o fl_fw-mi and_o favorina_n fl_fw-mi &_o the_o line_n fk_o be_v equal_a to_o the_o line_n favorina_n and_o the_o line_n fl_fw-mi be_v common_a to_o they_o both_o wherefore_o by_o the_o 4._o of_o the_o first_o the_o line_n all_o be_v equal_a to_o the_o line_n lk_n and_o the_o line_n ln_o be_v common_a to_o they_o both_o &_o make_v right_a angle_n at_o the_o point_n n_o and_o by_o the_o 3._o of_o the_o three_o the_o base_a a_o be_v equal_a to_o the_o base_a kn._n wherefore_o also_o the_o angle_n lkn_n be_v equal_a to_o the_o angle_n lan._n but_o the_o angle_n lanthorn_n be_v equal_a to_o the_o angle_n kbl_n by_o the_o 5._o of_o the_o first_o wherefore_o the_o angle_n lkn_n be_v equal_a to_o the_o angle_n kbl_n and_o the_o angle_n kal_n be_v common_a to_o both_o the_o triangle_n akb_n and_o akl_n wherefore_o the_o angle_n remain_v akb_n be_v equal_a to_o the_o angle_n remain_v alk_n by_o the_o corollary_n of_o the_o 32._o of_o the_o first_o wherefore_o the_o triangle_n kba_n be_v equiangle_n to_o the_o triangle_n kla._n wherefore_o by_o the_o 4._o of_o the_o six_o proportional_o as_o the_o right_a line_n basilius_n be_v to_o the_o right_a line_n ak_o so_o be_v the_o same_o right_a line_n ka_o to_o the_o right_a line_n al._n wherefore_o that_o which_o be_v contain_v under_o the_o line_n basilius_n and_o all_o be_v equal_a to_o the_o square_n of_o the_o line_n ak_o by_o the_o 17._o of_o the_o six_o and_o it_o be_v prove_v that_o that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bl_o be_v equal_a to_o the_o square_n of_o the_o line_n bf_o wherefore_o that_o which_o be_v contain_v under_o ab_fw-la &_o bl_o together_o with_o that_o which_o be_v conâ—â—ined_v under_o basilius_n and_o all_o which_o by_o the_o 2._o of_o the_o second_o be_v the_o square_a of_o the_o line_n basilius_n be_v equal_a to_o the_o square_n of_o the_o line_n of_o and_o ak_o but_o the_o line_n basilius_n be_v the_o side_n of_o the_o pentagon_n figure_n and_o of_o the_o side_n of_o the_o hexagon_n figure_n by_o the_o corollary_n of_o the_o 15._o of_o the_o four_o and_o ak_o the_o side_n of_o the_o decagon_n figure_n wherefore_o the_o side_n of_o a_o pentagon_n figure_n contain_v in_o power_n both_o the_o side_n of_o a_o hexagon_n figure_n and_o of_o a_o decagon_n figure_n be_v describe_v all_o in_o one_o and_o the_o self_n same_o circle_n which_o be_v require_v to_o be_v demonstrate_v ¶_o a_o corollary_n add_v by_o flussas_n a_o perpendicular_a line_n from_o any_o angle_n draw_v to_o the_o base_a of_o a_o pentagon_n pass_v by_o the_o centre_n for_o if_o we_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 a_o to_o the_o point_n d_o those_o right_a line_n shâ—â—l_v be_v equal_a by_o the_o 4._o of_o the_o first_o and_o therefore_o in_o the_o triangle_n acd_v the_o angle_n at_o the_o point_n c_o and_o d_o be_v by_o the_o 5._o of_o the_o firstâ—_o equal_a but_o the_o angle_n make_v at_o the_o point_n where_o the_o line_n agnostus_n cut_v the_o line_n cd_o be_v by_o supposition_n right_a angle_n wherefore_o by_o the_o 26._o of_o the_o first_o the_o line_n cd_o be_v by_o the_o line_n agnostus_n divide_v into_o two_o equal_a part_n and_o it_o be_v also_o divide_v perpendicular_o wherefore_o by_o the_o corollary_n of_o the_o first_o of_o the_o three_o in_o the_o line_n agnostus_n be_v the_o centre_n of_o the_o circle_n and_o therefore_o the_o line_n agnostus_n pass_v by_o the_o centre_n ¶_o the_o 11._o theorem_a the_o 11._o proposition_n if_o in_o a_o circle_n have_v a_o rational_a line_n to_o his_o diameter_n be_v inscribe_v a_o equilater_n pentagon_n the_o side_n of_o the_o pentagon_n be_v a_o irrational_a line_n and_o be_v of_o that_o kind_n which_o be_v call_v a_o less_o line_n svppose_v that_o in_o the_o circle_n abcde_v have_v a_o rational_a line_n to_o his_o diameter_n be_v inscribe_v a_o pentagon_n figure_n abcde_o then_o i_o say_v that_o the_o side_n of_o the_o pentagon_n figure_n abcde_o namely_o the_o side_n ab_fw-la 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 take_v by_o the_o 1._o of_o the_o three_o the_o centre_n of_o the_o circle_n and_o let_v the_o same_o be_v the_o point_n fletcher_n and_o draw_v a_o right_a line_n from_o the_o point_n a_o to_o the_o point_n fletcher_n construstion_n and_o a_o other_o from_o the_o point_n f_o to_o the_o point_n b_o and_o extend_v those_o line_n to_o the_o point_n g_o and_o h._n and_o draw_v a_o right_a line_n from_o the_o point_n a_o to_o the_o point_n c._n and_o from_o the_o semidiameter_n fh_o take_v the_o four_o part_n by_o the_o 9_o of_o the_o six_o and_o let_v the_o same_o be_v fk_v but_o the_o line_n fh_o be_v rational_a for_o that_o it_o be_v the_o half_a of_o the_o diameter_n which_o be_v suppose_v to_o be_v rational_a wherefore_o also_o the_o line_n fk_o be_v rational_a and_o the_o line_n or_o semidiameter_n bf_o be_v rational_a wherefore_o the_o whole_a line_n bk_o be_v rational_a demonstration_n and_o forasmuch_o as_o the_o circumference_n acg_n be_v equal_a to_o the_o circumference_n adg_n of_o which_o the_o circumference_n abc_n be_v equal_a to_o the_o circumference_n aed_n wherefore_o the_o residue_n cg_o be_v equal_a to_o the_o residue_n gd_o now_o if_o we_o draw_v a_o right_a line_n from_o the_o point_n a_o to_o the_o point_n d_o it_o be_v manifest_a that_o the_o angle_n alc_n and_o ald_n be_v right_a angle_n for_o forasmuch_o as_o the_o circumference_n cg_o be_v equal_a to_o the_o circumference_n gd_o therefore_o by_o the_o last_o of_o the_o six_o the_o angle_n cag_n be_v equal_a to_o the_o angle_n dag_n and_o the_o line_n ac_fw-la be_v equal_a to_o the_o line_n ad_fw-la for_o that_o the_o circumference_n which_o they_o subtend_v be_v equal_a and_o the_o line_n all_o be_v câ—mmon_a to_o they_o both_o therefore_o there_o be_v two_o line_n ac_fw-la and_o all_o equal_a to_o two_o line_n ad_fw-la and_o all_o and_o the_o angle_n cal_n be_v equal_a to_o the_o angle_n dalinea_n wherefore_o by_o the_o 4._o of_o the_o first_o the_o base_a cl_n be_v equal_a to_o the_o base_a ld_n and_o the_o rest_n of_o the_o angle_n to_o the_o rest_n of_o the_o angle_n and_o the_o line_n cd_o be_v double_a to_o the_o line_n cl._n and_o by_o the_o same_o reason_n may_v it_o be_v prove_v that_o the_o angle_n at_o the_o point_n m_o
line_n kg_v into_o two_o equal_a part_n in_o the_o point_n m._n fâ—ussas_n and_o draw_v a_o line_n from_o m_o to_o g._n and_o forasmuch_o as_o by_o construction_n the_o line_n kh_o be_v equal_a to_o the_o line_n ac_fw-la and_o the_o line_n hl_o to_o the_o line_n cb_o therefore_o the_o whole_a line_n ab_fw-la be_v equal_a to_o the_o whole_a line_n kl_o wherefore_o also_o the_o half_a of_o the_o line_n kl_o namely_o the_o line_n lm_o be_v equal_a to_o the_o semidiameter_n bn_o wherefore_o take_v away_o from_o those_o equal_a line_n equal_a part_n bc_o and_o lh_o the_o residue_v nc_n and_o mh_o shall_v be_v equal_a wherefore_o in_o the_o two_o triangle_n mhg_n and_o ncd_a the_o two_o side_n about_o the_o equal_a right_n angle_n dcn_n and_o ghm_n namely_o the_o side_n dc_o cn_fw-la and_o gh_o he_o be_v equal_a wherefore_o the_o base_n mg_o and_o nd_o be_v equal_a by_o the_o 4._o of_o the_o first_o and_o by_o the_o same_o reason_n may_v it_o be_v prove_v that_o right_a line_n draw_v from_o the_o point_n m_o to_o the_o point_n e_o and_o f_o be_v equal_a to_o the_o line_n nd_o but_o the_o right_a line_n nd_o be_v equal_a to_o the_o line_n a_fw-la which_o be_v draw_v from_o the_o centre_n to_o the_o circumference_n wherefore_o the_o line_n mg_o be_v equal_a to_o the_o line_n mk_o &_o also_o to_o the_o line_n i_o mf_n and_o ml_o wherefore_o make_v the_o centre_n the_o point_n m_o and_o the_o space_n mk_o or_o mg_o describe_v a_o semicircle_n kgl_n and_o the_o diameter_n kl_o abide_v fix_v let_v the_o say_a semicircle_n kgl_n be_v move_v round_o about_o until_o it_o return_v to_o the_o same_o place_n from_o whence_o it_o begin_v to_o be_v move_v and_o there_o shall_v be_v describe_v a_o sphere_n about_o the_o centre_n m_o by_o the_o 12._o definition_n of_o the_o eleven_o touch_v every_o one_o of_o the_o angle_n of_o the_o pyramid_n which_o be_v at_o the_o point_n king_n e_o f_o g_o for_o those_o angle_n be_v equal_o distant_a from_o the_o centre_n of_o the_o sphere_n namely_o by_o the_o semidiameter_n of_o the_o say_a sphere_n as_o have_v before_o be_v prove_v wherefore_o in_o the_o sphere_n give_v who_o diameter_n be_v the_o line_n kl_o or_o the_o line_n ab_fw-la be_v inscribe_v a_o tetrahedron_n efgk_n now_o i_o say_v that_o the_o diameter_n of_o the_o sphere_n be_v in_o power_n sesquialtera_fw-la to_o the_o side_n of_o the_o pyramid_n demonstration_n for_o forasmuch_o as_o the_o line_n ac_fw-la be_v double_a to_o the_o line_n cb_o by_o construction_n therefore_o the_o line_n ab_fw-la be_v treble_a 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 the_o line_n ab_fw-la be_v sesquialtera_fw-la to_o the_o line_n ac_fw-la but_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 square_a of_o the_o line_n basilius_n to_o the_o square_n of_o the_o line_n ad._n for_o if_o we_o draw_v a_o right_a line_n from_o the_o point_n b_o to_o the_o point_n d_o as_o the_o line_n bd_o be_v to_o the_o line_n ad_fw-la so_o be_v the_o same_o ad_fw-la to_o the_o line_n ac_fw-la by_o reason_n of_o the_o likeness_n of_o the_o ttriangle_n dab_n &_o dac_o by_o the_o 8._o of_o the_o six_o &_o by_o reason_n also_o that_o as_o the_o first_o be_v to_o the_o three_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 by_o the_o corollary_n of_o the_o 20._o of_o the_o six_o wherefore_o the_o square_a of_o the_o line_n basilius_n be_v sesquialter_fw-la to_o the_o square_n of_o the_o line_n ad._n but_o the_o line_n basilius_n be_v equal_a to_o the_o diameter_n of_o the_o sphere_n give_v namely_o to_o the_o line_n kl_o as_o have_v be_v prove_v &_o the_o line_n ad_fw-la be_v equal_a to_o the_o side_n of_o the_o pyramid_n inscribe_v in_o the_o sphere_n wherefore_o the_o diameter_n of_o the_o sphere_n be_v in_o power_n sesquialter_fw-la to_o the_o side_n of_o the_o pyramid_n wherefore_o there_o be_v make_v a_o pyramid_n comprehend_v in_o a_o sphere_n give_v and_o the_o diameter_n of_o the_o sphere_n be_v sesquialtera_fw-la to_o the_o side_n of_o the_o pyramid_n which_o be_v require_v to_o be_v do_v and_o prove_v ¶_o a_o other_o demonstration_n to_o prove_v that_o as_o the_o line_n ab_fw-la be_v to_o the_o line_n bc_o so_o be_v the_o square_a of_o the_o line_n ad_fw-la to_o the_o square_n of_o the_o line_n dc_o let_v the_o description_n of_o the_o semicircle_n adb_o be_v as_o in_o the_o first_o description_n and_o upon_o the_o line_n ac_fw-la describe_v by_o the_o 46._o of_o the_o first_o a_o square_a ec_o and_o make_v perfect_v the_o parallelogram_n fb_o now_o forasmuch_o as_o the_o triangle_n dab_n be_v equiangle_n to_o the_o triangle_n dac_o by_o the_o 32._o of_o the_o six_o therefore_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 dam_fw-ge to_o the_o line_n ac_fw-la by_o the_o 4._o of_o the_o six_o wherefore_o that_o which_o be_v contain_v under_o the_o line_n basilius_n and_o ac_fw-la be_v equal_a to_o the_o square_n of_o the_o line_n ad_fw-la by_o the_o 17._o of_o the_o six_o and_o for_o that_o as_o the_o line_n ab_fw-la be_v to_o the_o line_n bc_o so_o be_v the_o parallelogram_n ebb_v to_o the_o parallelogram_n fb_o by_o the_o 1._o of_o the_o six_o and_o the_o parallelogram_n ebb_n be_v that_o which_o be_v contain_v under_o the_o line_n basilius_n and_o ac_fw-la for_o the_o line_n ea_fw-la be_v equal_a to_o the_o line_n ac_fw-la and_o the_o parallelogram_n bf_o be_v that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o bc._n wherefore_o as_o the_o line_n ab_fw-la be_v to_o the_o line_n bc_o so_o be_v that_o which_o be_v contain_v under_o the_o line_n basilius_n and_o 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 but_o that_o which_o be_v contain_v under_o the_o line_n basilius_n and_o ac_fw-la be_v equal_a to_o the_o square_n of_o the_o line_n ad_fw-la by_o the_o corollary_n of_o the_o 8._o of_o the_o six_o and_o that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o be_v equal_a to_o the_o square_n of_o the_o line_n cd_o for_o the_o perpendicular_a line_n dc_o be_v the_o mean_a proportional_a between_o the_o segment_n of_o the_o base_a namely_o ac_fw-la and_o cb_o by_o the_o former_a corollary_n of_o the_o 8._o of_o the_o six_o for_o that_o the_o angle_n adb_o be_v a_o right_a angle_n wherefore_o as_o the_o line_n ab_fw-la be_v to_o the_o line_n bc_o so_o be_v the_o square_a of_o the_o line_n ad_fw-la to_o the_o square_n of_o the_o line_n dc_o by_o the_o 11._o of_o the_o five_o which_o be_v require_v to_o be_v prove_v ¶_o two_o assumpte_n add_v by_o campane_n first_o assumpt_n suppose_v that_o upon_o the_o line_n ab_fw-la be_v erect_v perpendicular_o the_o line_n dc_o which_o line_n dc_o let_v be_v the_o mean_a proportional_a between_o the_o part_n of_o the_o line_n ab_fw-la namely_o ac_fw-la &_o cb_o so_o that_o as_o the_o line_n ac_fw-la be_v to_o the_o line_n cd_o so_o let_v the_o line_n cd_o be_v to_o the_o line_n cb._n and_o upon_o the_o line_n ab_fw-la describe_v a_o semicircle_n then_o i_o say_v that_o the_o circumference_n of_o that_o semicircle_n shall_v pass_v by_o the_o point_n d_o which_o be_v the_o end_n of_o the_o perpendicular_a line_n but_o if_o not_o than_o it_o shall_v either_o cut_v the_o line_n cd_o or_o it_o shall_v pass_v above_o it_o and_o include_v it_o not_o touch_v it_o first_o let_v it_o cut_v it_o in_o the_o point_n e._n and_o draw_v these_o right_a line_n ebb_v and_o ea_fw-la wherefore_o by_o the_o 31._o of_o the_o three_o the_o whole_a angle_n aeb_n be_v a_o right_a angle_n wherefore_o by_o the_o first_o part_n of_o the_o corollary_n of_o the_o 8._o of_o the_o six_o the_o line_n ac_fw-la be_v to_o the_o line_n â—c_z as_z the_o line_n ec_o be_v to_o the_o line_n cb._n but_o by_o the_o 8._o of_o the_o five_o the_o proportion_n of_o the_o line_n ac_fw-la to_o the_o line_n ec_o be_v great_a than_o the_o proportion_n of_o the_o same_o line_n ac_fw-la to_o the_o line_n cd_o for_o the_o line_n ce_fw-fr be_v less_o than_o the_o line_n cd_o now_o for_o that_o the_o line_n ce_fw-fr be_v to_o thâ—_z line_n cb_o as_o the_o line_n ac_fw-la be_v to_o the_o line_n ce_fw-fr and_o the_o line_n cd_o be_v to_o the_o line_n cb_o as_o the_o line_n ac_fw-la be_v to_o the_o line_n cd_o therefore_o by_o the_o 13._o of_o the_o five_o the_o proportion_n of_o the_o line_n ec_o to_o the_o line_n cb_o be_v great_a than_o the_o proportion_n of_o the_o line_n cd_o to_o the_o line_n cb._n wherefore_o by_o the_o 10._o of_o the_o five_o the_o line_n ec_o be_v great_a than_o the_o line_n dc_o
namely_o the_o part_n great_a than_o the_o whole_a which_o be_v impossible_a wherefore_o the_o circumference_n shall_v not_o cut_v the_o line_n cd_o now_o i_o say_v that_o it_o shall_v not_o pass_v above_o the_o line_n cd_o and_o not_o touch_v it_o in_o the_o point_n d._n for_o if_o it_o be_v possible_a let_v it_o pass_v above_o it_o and_o extend_v the_o line_n cd_o to_o the_o circumference_n and_o let_v it_o cut_v it_o in_o the_o point_n â—_o and_o draw_v the_o line_n fb_o and_o favorina_n and_o it_o shall_v follow_v as_o before_o that_o the_o line_n cd_o be_v great_a than_o the_o line_n cf_o which_o be_v impossible_a wherefore_o that_o be_v manifest_v which_o be_v require_v to_o be_v prove_v ¶_o second_v assumpt_n if_o there_o be_v a_o right_a angle_n unto_o which_o a_o base_a be_v subtend_v and_o if_o upon_o the_o same_o be_v describe_v a_o semicircle_n the_o circumference_n thereof_o shall_v pass_v by_o the_o point_n of_o the_o right_a angle_n the_o converse_n of_o this_o be_v add_v after_o the_o demonstration_n of_o the_o 31._o of_o the_o three_o out_o of_o pelitarius_n and_o these_o two_o assumpte_n of_o campane_n be_v necessary_a for_o the_o better_a understanding_n of_o the_o demonstration_n of_o the_o secoâ—d_a part_n of_o this_o 13._o proposition_n wherein_o be_v prove_v that_o the_o pyramid_n be_v contain_v in_o the_o sphere_n give_v ¶_o certain_a corollarye_n add_v by_o flussas_n first_o corollary_n the_o diameter_n of_o the_o sphere_n be_v in_o power_n quadruple_a sesquialtera_fw-la to_o the_o line_n which_o be_v draw_v from_o the_o centre_n to_o the_o circumference_n of_o the_o circle_n which_o contain_v the_o base_a of_o the_o pyramid_n for_o forasmuch_o as_o it_o have_v be_v prove_v that_o the_o diameter_n kl_o be_v in_o power_n sesquialter_fw-la to_o the_o side_n of_o and_o it_o be_v prove_v also_o by_o the_o 12._o of_o this_o book_n that_o the_o side_n of_o be_v in_o power_n triple_a to_o the_o line_n eh_o which_o be_v draw_v from_o the_o centre_n of_o the_o circle_n contain_v the_o triangle_n efg_o but_o the_o proportion_n of_o the_o extreme_n namely_o of_o the_o diameter_n to_o the_o line_n eh_o consist_v of_o the_o proportion_n of_o the_o mean_n namely_o of_o the_o proportion_n of_o the_o diameter_n to_o the_o line_n of_o and_o of_o the_o proportion_n of_o the_o line_n of_o to_o the_o line_n eh_o by_o the_o 5._o definition_n of_o the_o six_o which_o proportion_n namely_o triple_a and_o sesquialter_fw-la add_v together_o make_v quadruple_a sesquialter_fw-la as_o it_o be_v easy_a to_o prove_v by_o that_o which_o be_v teach_v in_o the_o declaration_n of_o the_o 5._o definition_n of_o the_o six_o book_n wherefore_o the_o corollary_n be_v manifest_a ¶_o second_v corollary_n only_o the_o line_n which_o be_v draw_v from_o the_o angle_n of_o the_o pyramid_n to_o the_o base_a opposite_a unto_o it_o campane_n &_o pass_v by_o the_o centre_n of_o the_o sphere_n be_v perpendicular_a to_o the_o base_a and_o fall_v upon_o the_o centre_n of_o the_o circle_n which_o contain_v the_o base_a for_o if_o any_o other_o line_n than_o the_o line_n kmh_o which_o be_v draw_v by_o the_o centre_n of_o the_o sphere_n to_o the_o centre_n of_o the_o circle_n shall_v fall_v perpendicular_o upon_o the_o plain_a of_o the_o base_a then_o from_o one_o and_o the_o self_n same_o point_n shall_v be_v draw_v to_o one_o and_o the_o self_n same_o plain_a two_o perpendicular_a line_n contrary_a to_o the_o 13._o of_o the_o eleven_o which_o be_v impossible_a far_o if_o from_o the_o top_n king_n shall_v be_v draw_v to_o the_o centre_n of_o the_o base_a namely_o to_o the_o point_n h_o any_o other_o right_a line_n not_o pass_v by_o the_o centre_n m_o two_o right_a life_n shall_v include_v a_o superficies_n contrary_n to_o the_o last_o common_a sentence_n which_o be_v absurd_a wherefore_o only_o the_o line_n which_o be_v draw_v by_o the_o centre_n of_o the_o sphere_n to_o the_o centre_n of_o the_o base_a be_v perpendicular_a to_o the_o say_v base_a and_o the_o line_n which_o be_v draw_v from_o the_o angle_n perpendicular_o to_o the_o base_a shall_v pass_v by_o the_o centre_n of_o the_o sphere_n three_o corollary_n the_o perpendicular_a line_n which_o be_v draw_v from_o the_o centre_n of_o the_o sphere_n to_o the_o base_a of_o the_o pyramid_n book_n be_v equal_a to_o the_o six_o part_n of_o the_o diameter_n of_o the_o sphere_n for_o it_o be_v before_o prove_v that_o the_o line_n mh_o which_o be_v draw_v from_o the_o centre_n of_o the_o sphere_n to_o the_o centre_n of_o the_o base_a be_v equal_a to_o the_o line_n nc_n which_o line_n nc_n be_v the_o six_o part_n of_o the_o diameter_n ab_fw-la and_o therefore_o the_o line_n mh_o be_v the_o six_o part_n of_o the_o diameter_n of_o the_o sphere_n for_o the_o diameter_n ab_fw-la be_v equal_a to_o the_o diameter_n of_o the_o sphere_n as_o have_v also_o before_o be_v prove_v ¶_o the_o 2._o problem_n the_o 14._o proposition_n to_o make_v a_o octohedron_n and_o to_o comprehend_v it_o in_o the_o sphere_n give_v namely_o that_o wherein_o the_o pyramid_n be_v comprehend_v and_o to_o prove_v that_o the_o diameter_n of_o the_o sphere_n be_v in_o power_n double_a to_o the_o side_n of_o the_o octohedron_n construction_n take_v the_o diameter_n of_o the_o former_a sphere_n give_v which_o let_v be_v the_o line_n ab_fw-la and_o divide_v it_o by_o the_o 10._o of_o the_o first_o into_o two_o equal_a part_n in_o the_o point_n c._n and_o describe_v upon_o the_o line_n ab_fw-la a_o semicircle_n adb_o and_o by_o the_o 11._o of_o the_o first_o from_o the_o point_n c_o raise_v up_o unto_o the_o line_n ab_fw-la a_o perpendicular_a line_n cd_o and_o draw_v a_o right_a line_n from_o the_o point_n d_o to_o the_o point_n b._n and_o describe_v a_o square_a efgh_o have_v every_o one_o of_o his_o side_n equal_a to_o the_o line_n bd._n and_o draw_v the_o diagonal_a line_n fh_a &_o eglantine_n cut_v the_o one_o the_o other_o in_o the_o point_n k._n and_o by_o the_o 12._o of_o the_o eleven_o from_o the_o point_n king_n namely_o the_o point_n where_o the_o line_n fh_o and_o eglantine_n cut_v the_o one_o the_o other_o raise_v up_o to_o the_o plain_a superficies_n wherein_o the_o square_a efgh_o be_v a_o perpendicular_a line_n kl_o and_o extend_v the_o line_n kl_o on_o the_o other_o side_n of_o the_o plain_a superficies_n to_o the_o point_n m._n and_o let_v each_o of_o the_o line_n kl_o and_o km_o be_v put_v equal_a to_o one_o of_o these_o line_n ke_o kf_o kh_o or_o kg_n and_o draw_v these_o right_a line_n le_fw-fr lf_n lg_n lh_o menander_n mf_n mg_o and_o mh_o demonstration_n now_o forasmuch_o as_o the_o line_n ke_o be_v by_o the_o corollary_n of_o the_o 34._o of_o the_o first_o equal_a to_o the_o line_n kh_o and_o the_o angle_n angle_n ekh_n be_v a_o right_a angle_n therefore_o the_o square_a of_o he_o be_v double_a to_o the_o square_n of_o eke_o by_o the_o 47._o of_o the_o first_o again_o forasmuch_o as_o the_o line_n lk_n be_v equal_a to_o the_o line_n ke_o by_o position_n and_o the_o angle_n lke_n be_v by_o the_o second_o definition_n of_o the_o eleven_o a_o right_a angle_n therefore_o the_o square_n of_o the_o line_n el_n be_v double_a to_o the_o square_n of_o the_o line_n eke_o and_o it_o be_v prove_v that_o the_o square_a of_o the_o line_n he_o be_v double_a to_o the_o square_n of_o the_o line_n eke_o wherefore_o the_o square_a of_o the_o line_n le_fw-fr be_v equal_a to_o the_o square_n of_o the_o line_n eh_o wherefore_o also_o the_o line_n le_fw-fr be_v equal_a to_o the_o line_n eh_o and_o by_o the_o same_o reason_n the_o line_n lh_o be_v also_o equal_a to_o the_o line_n he._n wherefore_o the_o triangle_n lhe_n be_v equilater_n in_o like_a sort_n may_v we_o prove_v that_o every_o one_o of_o the_o rest_n of_o the_o triangle_n who_o base_n be_v the_o side_n of_o the_o square_a efgh_o and_o top_n the_o point_n l_o and_o m_o be_v equilater_n the_o say_v eight_o triangle_n also_o be_v equal_a the_o one_o to_o the_o other_o for_o every_o side_n of_o each_o be_v equal_a to_o the_o side_n of_o the_o square_a efgh_o wherefore_o there_o be_v make_v a_o octohedron_n contain_v under_o eight_o triangle_n who_o side_n be_v equal_a now_o it_o be_v require_v to_o comprehend_v it_o in_o the_o sphere_n give_v and_o to_o prove_v that_o the_o diameter_n of_o the_o sphere_n be_v in_o power_n double_a to_o the_o side_n of_o the_o octohedron_n demonstration_n forasmuch_o as_o these_o three_o line_n lk_v km_o and_o ke_o be_v equal_a the_o one_o to_o the_o other_o therefore_o a_o semicircle_n describe_v upon_o the_o line_n lm_o shall_v pass_v also_o by_o the_o point_n e._n and_o by_o the_o same_o reason_n if_o the_o semicircle_n be_v turn_v round_o about_o until_o it_o return_v unto_o the_o self_n same_o
give_v wherein_o be_v contain_v the_o former_a solid_n and_o to_o prove_v that_o the_o side_n of_o the_o icosahedron_n be_v a_o irrational_a line_n of_o that_o kind_n which_o be_v call_v a_o less_o line_n demonstration_n now_o forasmuch_o as_o the_o line_n qp_v qv_n qt_n q_n and_o qr_o do_v each_o subtend_v right_a angle_n contain_v under_o the_o side_n of_o a_o equilater_n hexagon_n &_o of_o a_o equilater_n decagon_fw-mi inscribe_v in_o the_o circle_n prstv_n or_o in_o the_o circle_n efghk_n which_o two_a circle_n be_v equal_a therefore_o the_o say_a line_n be_v each_o equal_a to_o the_o side_n of_o the_o pentagon_n inscribe_v in_o the_o foresay_a circle_n by_o the_o 10._o of_o this_o book_n and_o be_v equal_a the_o one_o to_o the_o other_o by_o the_o 4._o of_o the_o first_o for_o all_o the_o angle_n at_o the_o point_n w_n which_o they_o subtend_v be_v right_a angle_n wherefore_o the_o five_o triangle_n qpv_o qpr_fw-la qr_n qst_fw-la and_o qtv_n which_o be_v contain_v under_o the_o say_a line_n qv_n qp_n qr_o q_n qt_n and_o under_o the_o side_n of_o the_o pentagon_n vpr_v be_v equilater_n and_o equal_a to_o the_o ten_o former_a triangle_n and_o by_o the_o same_o reason_n the_o five_o triangle_n opposite_a unto_o they_o namely_o the_o triangle_n yml_n ymn_n ynx_n yxo_n and_o yol_n be_v equilater_n and_o equal_a to_o the_o say_v ten_o triangle_n for_o the_o line_n ill_o ym_fw-fr yn_n yx_n and_o you_o do_v subtend_v right_a angle_n contain_v under_o the_o side_n of_o a_o equilater_n hexagon_n and_o of_o a_o equilater_n decagon_fw-mi inscribe_v in_o the_o circle_n efghk_n which_o be_v equal_a to_o the_o circle_n prstv_n wherefore_o there_o be_v describe_v a_o solid_a contain_v under_o 20._o equilater_n triangle_n wherefore_o by_o the_o last_o definition_n of_o the_o eleven_o there_o be_v describe_v a_o icosahedron_n now_o it_o be_v require_v to_o comprehend_v it_o in_o the_o sphere_n give_v and_o to_o prove_v that_o the_o side_n of_o the_o icosahedron_n be_v a_o irrational_a line_n of_o that_o kind_n which_o be_v call_v a_o less_o line_n forasmuch_o as_o the_o line_n zw_n be_v the_o side_n of_o a_o hexagon_n &_o the_o line_n wq_n be_v the_o side_n of_o a_o decagon_n therefore_o the_o line_n zq_n be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n in_o the_o point_n w_n and_o his_o great_a segment_n be_v zw_n by_o the_o 9_o of_o the_o thirteen_o wherefore_o as_o the_o line_n qz_n be_v to_o the_o line_n zw_n so_o be_v the_o line_n zw_n to_o the_o line_n wq_n but_o the_o zw_n be_v equal_a to_o the_o line_n zl_n by_o construction_n and_o the_o line_n wq_n to_o the_o line_n zy_n by_o construction_n also_o wherefore_o as_o the_o line_n qz_n be_v to_o the_o line_n zl_n so_o be_v the_o line_n zl_n to_o the_o line_n zy_n and_o the_o angle_n qzlâ—_n and_o lzy_n be_v right_a angle_n by_o the_o 2._o definition_n of_o the_o eleven_o if_o therefore_o we_o draw_v a_o right_a line_n from_o the_o point_n l_o to_o the_o point_n qui_fw-fr the_o angle_n ylq_n shall_v be_v a_o right_a angle_n by_o reason_n of_o the_o likeness_n of_o the_o triangle_n ylq_n and_o zlq_n by_o the_o 8._o of_o the_o six_o wherefore_o a_o semicircle_n describe_v upon_o the_o line_n qy_n shall_v pass_v also_o by_o the_o point_n l_o by_o the_o assumpt_n add_v by_o campane_n after_o the_o 13._o of_o this_o book_n and_o by_o the_o same_o reason_n also_o for_o that_o as_o the_o line_n qz_n be_v the_o line_n zw_n so_o be_v the_o line_n zw_n to_o the_o line_n wq_n flussas_n but_o the_o line_n zq_n be_v equal_a to_o the_o line_n yw_n and_o the_o line_n zw_n to_o the_o line_n pw_o wherefore_o as_o the_o line_n yw_n be_v to_o the_o line_n wp_n so_o be_v the_o line_n pw_o to_o the_o line_n wq_n and_o therefore_o again_o if_o we_o draw_v a_o right_a line_n from_o the_o point_n p_o to_o the_o point_n y_fw-fr the_o angle_n ypq_n shall_v be_v a_o right_a angle_n wherefore_o a_o semicircle_n describe_v upon_o the_o line_n qy_n shall_v pass_v also_o by_o the_o point_n p_o by_o the_o former_a assumpt_n &_o if_o the_o diameter_n qy_o abide_v fix_v the_o semicircle_n be_v turn_v round_o about_o until_o it_o come_v to_o the_o self_n same_o place_n from_o whence_o it_o begin_v first_o to_o be_v move_v it_o shall_v pass_v both_o by_o the_o point_n p_o and_o also_o by_o the_o rest_n of_o the_o point_n of_o the_o angle_n of_o the_o icosahedron_n and_o the_o icosahedron_n shall_v be_v comprehend_v in_o a_o sphere_n i_o say_v also_o that_o it_o be_v contain_v in_o the_o sphere_n give_v divide_v by_o the_o 10._o of_o the_o first_o the_o line_n zw_n into_o two_o equal_a part_n in_o the_o point_n a._n and_o forasmuch_o as_o the_o right_a line_n zq_n be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n in_o the_o point_n w_n and_o his_o less_o segment_n be_v qw_o therefore_o the_o segment_n qw_o have_v add_v unto_o it_o the_o half_a of_o the_o great_a segment_n namely_o the_o line_n wa_n be_v by_o the_o 3._o of_o this_o book_n in_o power_n quintuple_a to_o the_o square_n make_v of_o the_o half_a of_o the_o great_a segment_n wherefore_o the_o square_a of_o the_o line_n qa_n be_v quintuple_a to_o the_o square_n of_o the_o line_n â—w_o but_o unto_o the_o square_n of_o the_o qa_n the_o square_a of_o the_o line_n qy_n be_v quadruple_a by_o the_o corollary_n of_o the_o 20._o of_o the_o six_o for_o the_o line_n qy_n be_v double_a to_o the_o line_n qa_n and_o by_o the_o same_o reason_n unto_o the_o square_n of_o the_o wa_n the_o square_a of_o the_o line_n zw_n be_v quadruple_a wherefore_o the_o square_a of_o the_o line_n qy_n be_v quintuple_a to_o the_o square_n of_o the_o line_n zw_n by_o the_o 15._o of_o the_o five_o and_o forasmuch_o as_o the_o line_n ac_fw-la be_v quadruple_a to_o the_o line_n cb_o therefore_o the_o line_n ab_fw-la be_v quintuple_a to_o the_o line_n cb._n but_o as_o the_o line_n ab_fw-la be_v to_o the_o line_n bc_o so_o be_v the_o square_a of_o the_o line_n ab_fw-la to_o the_o square_n of_o the_o line_n bd_o by_o the_o 8_o of_o the_o six_o and_o corollary_n of_o the_o 20._o of_o the_o same_o wherefore_o the_o square_a of_o the_o line_n ab_fw-la be_v quintuple_a to_o the_o square_n of_o the_o line_n bd_o and_o it_o be_v be_v prove_v that_o the_o square_a of_o the_o line_n qy_n be_v quintuple_a to_o the_o square_n of_o the_o line_n zw_n and_o the_o line_n bd_o be_v equal_a to_o the_o line_n zw_n for_o either_o of_o they_o be_v by_o position_n equal_a to_o the_o line_n which_o be_v draw_v from_o the_o centre_n of_o the_o circle_n efghk_v to_o the_o circumference_n wherefore_o the_o line_n ab_fw-la be_v equal_a to_o the_o yq_n but_o the_o line_n ab_fw-la be_v the_o diameter_n of_o the_o sphere_n give_v wherefore_o the_o line_n yq_n which_o be_v prove_v to_o be_v the_o diameter_n of_o the_o sphere_n contain_v the_o icosahedron_n be_v equal_a to_o the_o diameter_n of_o the_o sphere_n give_v wherefore_o the_o icosahedron_n be_v contain_v in_o the_o sphere_n give_v now_o i_o say_v that_o the_o side_n of_o the_o icosahedron_n be_v a_o irrational_a line_n of_o that_o kind_n which_o be_v call_v a_o less_o line_n for_o forasmuch_o as_o the_o diameter_n of_o the_o sphere_n be_v rational_a and_o be_v in_o power_n quintuple_a to_o the_o square_n of_o the_o line_n draw_v from_o the_o centre_n of_o the_o circle_n olmnx_n wherefore_o also_o the_o line_n which_o be_v draw_v from_o the_o centre_n of_o the_o circle_n olmnx_n be_v rational_a wherefore_o the_o diameter_n also_o be_v commensurable_a to_o the_o same_o line_n by_o the_o 6._o of_o the_o ten_o be_v rational_a but_o if_o in_o a_o circle_n have_v a_o rational_a line_n to_o his_o diameter_n be_v describe_v a_o equilater_n pentagon_n the_o side_n of_o the_o pentagon_n be_v by_o the_o 11._o of_o this_o book_n a_o irrational_a line_n of_o that_o kind_n which_o be_v call_v a_o less_o line_n but_o the_o side_n of_o the_o pentagon_n olmnx_n be_v also_o the_o side_n of_o the_o icosahedron_n describe_v as_o have_v before_o be_v prove_v wherefore_o the_o side_n of_o the_o icosahedron_n be_v a_o irrational_a line_n of_o that_o kind_n which_o be_v call_v a_o less_o line_n wherefore_o there_o be_v describe_v a_o icosahedron_n and_o it_o be_v contain_v in_o the_o sphere_n give_v and_o it_o be_v prove_v that_o the_o side_n of_o thou_o icosahedron_n be_v a_o irrational_a line_n of_o that_o kind_n which_o be_v call_v a_o less_o line_n which_o be_v require_v to_o be_v do_v and_o to_o be_v prove_v a_o corollary_n hereby_o it_o be_v manifest_a that_o the_o diameter_n of_o the_o sphere_n be_v in_o power_n quintuple_a to_o the_o line_n which_o be_v draw_v from_o the_o centre_n of_o the_o circle_n to_o
line_n ai._n wherefore_o the_o square_a of_o the_o line_n bw_o be_v equal_a to_o the_o square_n of_o the_o line_n ai_fw-fr wherefore_o also_o the_o line_n bt_n be_v equal_a to_o the_o line_n ai._n and_o by_o the_o same_o reason_n be_v the_o line_n id_fw-la and_o wc_n equal_a to_o the_o same_o line_n now_o forasmuch_o as_o the_o line_n ai_fw-fr and_o id_fw-la and_o the_o line_n all_o and_o ld_n be_v equal_a and_o the_o base_a il_fw-fr be_v common_a to_o they_o both_o the_o angle_n ali_n and_o dli_o shall_v be_v equal_a by_o the_o 8._o of_o the_o first_o and_o therefore_o they_o be_v right_a angle_n by_o the_o 10._o definition_n of_o the_o first_o and_o by_o the_o same_o reason_n be_v the_o angle_n whb_n and_o whc_n right_a angle_n and_o forasmuch_o as_o the_o two_o line_n ht_v and_o tw_n be_v equal_a to_o the_o two_o line_n lct_n and_o cti_n and_o they_o contain_v equal_a angle_n that_o be_v right_a angle_n by_o supposition_n therefore_o the_o angle_n wht_v and_o ilct_n be_v equal_a by_o the_o 4._o of_o the_o first_o wherefore_o the_o plain_a superficies_n aid_n be_v in_o like_a sort_n incline_v to_o the_o plain_a superficies_n abcd_o as_o the_o plain_a superficies_n bwc_n be_v incline_v to_o the_o same_o plain_n abcd_o by_o the_o 4._o definition_n of_o the_o eleven_o in_o like_a sort_n may_v we_o prove_v that_o the_o plain_n wcdi_n be_v in_o like_a sort_n incline_v to_o the_o plain_n abcd_o as_o the_o plain_a buzc_n be_v to_o the_o plain_n ebcf._n for_o that_o in_o the_o triangle_n yoh_n and_o ctpk_n which_o consist_v of_o equal_a side_n each_o to_o his_o correspondent_a side_n the_o angle_n yho_n and_o ctkp_n which_o be_v the_o angle_n of_o the_o inclination_n be_v equal_a and_o now_o if_o the_o right_a line_n ctk_n be_v extend_v to_o the_o point_n a_o and_o the_o pentagon_n cwida_n be_v make_v perfect_a we_o may_v by_o the_o same_o reason_n prove_v that_o that_o plain_n be_v equiangle_n and_o equilater_n that_o we_o prove_v the_o pentagon_n buzcw_o to_o be_v equaliter_fw-la and_o equiangle_n and_o likewise_o if_o the_o other_o plain_n bwia_fw-la and_o aid_v be_v make_v perfect_a they_o may_v be_v prove_v to_o be_v equal_a and_o like_a pentagon_n and_o in_o like_a sort_n situate_a and_o they_o be_v set_v upon_o these_o common_a right_a line_n bw_o wc_n wi_n ai_fw-fr and_o id_fw-la and_o observe_v this_o method_n there_o shall_v upon_o every_o one_o of_o the_o 12._o â—idâ—s_n of_o the_o cube_fw-la be_v set_v every_o one_o of_o the_o 12._o pentagon_n which_o compose_v the_o dodecahedron_n ¶_o certain_a corollarye_n add_v by_o flussas_n first_o corollary_n the_o side_n of_o a_o cube_fw-la be_v equal_a to_o the_o right_a line_n which_o subtend_v the_o angle_n of_o the_o pentagon_n of_o a_o dodecahedron_n contain_v in_o one_o and_o the_o self_n same_o sphere_n with_o the_o cube_fw-la for_o the_o angle_n bwc_n and_o aid_n be_v subtend_v of_o the_o line_n bc_o and_o ad._n which_o be_v side_n of_o the_o cubeâ—_n ¶_o second_v corollary_n in_o a_o dodecahedron_n there_o be_v six_o side_n every_o two_o of_o which_o be_v parallel_n and_o opposite_a who_o section_n into_o two_o equal_a part_n be_v couple_v by_o three_o right_a line_n which_o in_o the_o centre_n of_o the_o sphere_n which_o contain_v the_o dodecahedron_n divide_v into_o two_o equal_a part_n and_o perpendicular_o both_o themselves_o and_o also_o the_o side_n for_o upon_o the_o six_o base_n of_o the_o cube_fw-la be_v set_v six_o side_n of_o the_o dodecahedron_n as_o it_o have_v be_v prove_v by_o the_o line_n zv_o wi_n &c._a &c._a which_o be_v cut_v into_o two_o equal_a part_n by_o right_a line_n which_o join_v together_o the_o centre_n of_o the_o base_n of_o the_o cube_fw-la as_o the_o line_n the_fw-mi produce_v and_o the_o other_o like_a which_o line_n couple_v together_o the_o centre_n of_o the_o base_n be_v three_o in_o number_n cut_v the_o one_o the_o other_o perpendicular_o for_o they_o be_v parallel_n to_o the_o side_n of_o the_o cube_fw-la and_o they_o cut_v the_o one_o the_o other_o into_o two_o equal_a part_n in_o the_o centre_n of_o the_o sphere_n which_o contain_v the_o cube_fw-la by_o that_o which_o be_v demonstrate_v in_o the_o 15._o of_o this_o book_n and_o unto_o these_o equal_a line_n join_v together_o the_o centre_n of_o the_o base_n of_o the_o cube_fw-la be_v without_o the_o base_n add_v equal_a part_n oy_o p_o ct_v and_o the_o other_o like_a which_o by_o supposition_n be_v equal_a to_o half_n of_o the_o side_n of_o the_o dodecahedron_n wherefore_o the_o whole_a line_n which_o join_v together_o the_o sectoin_n of_o the_o opposite_a side_n of_o the_o dodecahedron_n be_v equal_a and_o they_o cut_v those_o side_n into_o two_o equal_a part_n and_o perpendicular_o three_o corollary_n a_o right_a line_n join_v together_o the_o point_n of_o the_o section_n of_o the_o opposite_a side_n of_o the_o dodecahedron_n into_o two_o equal_a part_n be_v 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 side_n of_o the_o cube_fw-la and_o the_o less_o segment_n the_o side_n of_o the_o dodecahedron_n contain_v in_o the_o self_n same_o sphere_n for_o it_o be_v prove_v that_o the_o right_a line_n yq_n be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n in_o the_o point_n oh_o and_o that_o his_o great_a segment_n oq_fw-la be_v half_a the_o side_n of_o the_o cube_fw-la and_o his_o less_o segment_n oy_o be_v half_o of_o the_o side_n we_o which_o be_v the_o side_n of_o the_o dodecahedron_n wherefore_o it_o follow_v by_o the_o 15._o of_o the_o five_o that_o their_o double_n be_v in_o the_o same_o proportion_n wherefore_o the_o double_a of_o the_o line_n yq_n which_o join_v the_o point_n opposite_a unto_o the_o line_n y_fw-fr be_v the_o whole_a and_o the_o great_a segment_n be_v the_o double_a of_o the_o line_n oq_fw-la which_o be_v the_o side_n of_o the_o cube_fw-la &_o the_o less_o segment_n be_v the_o double_a of_o the_o line_n the_fw-mi which_o be_v equal_a to_o the_o side_n of_o the_o dodecahedron_n namely_o to_o the_o side_n we_o ¶_o the_o 6._o problem_n the_o 18._o proposition_n to_o find_v out_o the_o side_n of_o the_o foresay_a five_o body_n and_o to_o compare_v they_o together_o take_v the_o diameter_n of_o the_o sphere_n give_v and_o let_v the_o same_o be_v ab_fw-la and_o divide_v it_o in_o the_o point_n c_o so_fw-mi that_o let_v the_o line_n ac_fw-la be_v equal_a to_o cb_o by_o the_o 10._o of_o the_o first_o and_o in_o the_o point_n d_o so_fw-mi that_o let_v ad_fw-la be_v double_a to_o db_o by_o the_o 9_o of_o the_o six_o and_o upon_o the_o line_n ab_fw-la describe_v a_o semicircle_n aeb_fw-mi and_o from_o the_o point_n c_o and_o d_o raise_v up_o by_o the_o 11._o of_o the_o first_o unto_o the_o line_n ab_fw-la perpendicular_a line_n ce_fw-fr and_o df._n and_o draw_v these_o right_a line_n of_o fb_o and_o be._n now_o forasmuch_o as_o the_o line_n ad_fw-la be_v double_a to_o the_o line_n db_o therefore_o the_o line_n ab_fw-la be_v treble_a to_o the_o line_n db._n wherefore_o the_o line_n basilius_n be_v sesquialter_fw-la to_o the_o line_n ad_fw-la for_o it_o be_v as_o 3._o to_o 2._o but_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 square_a of_o the_o line_n basilius_n to_o the_o square_n of_o the_o line_n of_o by_o the_o 6._o of_o the_o six_o or_o by_o the_o corollary_n of_o the_o same_o and_o by_o the_o corollary_n of_o the_o 20._o of_o the_o same_o for_o the_o triangle_n afb_o be_v equiangle_n to_o the_o triangle_n afd_v wherefore_o the_o square_a of_o the_o line_n basilius_n be_v sesquialter_fw-la to_o the_o square_n of_o the_o line_n af._n but_o the_o diameter_n of_o a_o sphere_n be_v in_o power_n sesquialter_fw-la to_o the_o sidâ—_n of_o the_o pyramid_n pyramid_n by_o the_o 13._o of_o this_o book_n and_o the_o line_n ab_fw-la be_v the_o diameter_n of_o the_o sphere_n wherefore_o the_o line_n of_o be_v equal_a to_o the_o side_n of_o the_o pyramid_n again_o forasmuch_o as_o the_o line_n ab_fw-la be_v treble_a to_o the_o line_n bd_o but_o as_o the_o line_n ab_fw-la be_v to_o the_o line_n bd_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 fb_o by_o the_o corollary_n of_o the_o 8._o and_o 20._o of_o the_o six_o wherefore_o the_o square_a of_o the_o line_n ab_fw-la be_v treble_a to_o the_o square_n of_o the_o line_n fb_o but_o the_o diameter_n oâ—_n a_o sphere_n be_v in_o power_n treble_a to_o the_o side_n of_o the_o cube_fw-la by_o the_o 15._o of_o this_o book_n and_o the_o diameter_n of_o the_o sphere_n be_v the_o line_n ab_fw-la wherefore_o the_o line_n bf_o be_v the_o side_n of_o the_o cube_fw-la cube_fw-la and_o forasmuch_o as_o the_o line_n fb_o be_v the_o
equal_a part_n by_o every_o one_o of_o the_o three_o equal_a square_n which_o divide_v the_o octohedron_n into_o two_o equal_a part_n and_o perpendicular_o for_o the_o three_o diameter_n of_o those_o square_n do_v in_o the_o centre_n cut_v the_o one_o the_o other_o into_o two_o equal_a part_n and_o perpendicular_o by_o the_o three_o corollary_n of_o the_o 1â—_o of_o the_o thirteen_o which_o square_n as_o for_o example_n the_o square_a ekli_n do_v divide_v in_o sunder_o the_o pyramid_n and_o the_o prism_n namely_o the_o pyramid_n kltd_v and_o the_o prism_n klteia_n from_o the_o pyramid_n ekzb_n and_o the_o prism_n ekzilg_n which_o pyramid_n be_v equal_a the_o one_o to_o the_o other_o and_o so_o also_o be_v the_o prism_n equal_v the_o one_o to_o the_o other_o by_o the_o 3._o of_o the_o twelve_o and_o in_o like_a sort_n do_v the_o rest_n of_o the_o square_n namely_o kzit_n and_o zlte_n which_o square_n by_o the_o second_o corollary_n of_o the_o 14._o of_o the_o thirteen_o do_v divide_v the_o octohedron_n into_o two_o equal_a part_n ¶_o the_o 3._o proposition_n the_o 3._o problem_n in_o a_o cube_fw-la give_v to_o describe_v a_o octohedron_n take_v a_o cube_n namely_o abcdefgh_n and_o divide_v every_o one_o of_o the_o side_n thereof_o into_o two_o equal_a part_n construction_n and_o draw_v right_a line_n couple_v together_o the_o section_n as_o for_o example_n these_o right_a line_n pq_n and_o r_n which_o shall_v be_v equal_a unto_o the_o side_n of_o the_o cube_fw-la by_o the_o 33._o of_o the_o first_o and_o shall_v divide_v the_o one_o the_o other_o into_o two_o equal_a part_n in_o the_o midst_n of_o the_o diameter_n agnostus_n in_o the_o point_n i_o by_o the_o corollary_n of_o the_o 34._o of_o the_o first_o wherefore_o the_o point_n i_o be_v the_o centre_n of_o the_o base_a of_o the_o cube_fw-la and_o by_o the_o same_o reason_n may_v be_v find_v out_o the_o centre_n of_o the_o rest_n of_o the_o base_n which_o let_v be_v the_o point_n king_n l_o oh_o n_o m._n and_o draw_v these_o right_a line_n li_n in_o more_n ol_n king_n kl_o km_o ko_o ni_fw-fr nl_o nm_o &_o no_o demonstration_n and_o now_o forasmuch_o as_o the_o angle_n ipl_n be_v a_o right_a angle_n by_o the_o 10._o of_o the_o eleven_o for_o the_o line_n ip_n and_o pl_z be_v parallel_n to_o the_o line_n ramires_n and_o ab_fw-la and_o the_o right_a line_n il_fw-fr subtend_v the_o right_a angle_n ipl_n namely_o it_o subtend_v the_o half_a side_n of_o the_o cube_fw-la which_o contain_v the_o right_a angle_n ipl_n and_o likewise_o the_o right_a line_n in_o subtend_v the_o angle_n iqm_n which_o be_v equal_a to_o the_o same_o angle_n ipl_n and_o be_v contain_v under_o right_a line_n equal_a to_o the_o right_a line_n which_o contain_v the_o angle_n ipl_n wherefore_o the_o right_a line_n in_o be_v equal_a to_o the_o right_a line_n ill_n by_o the_o 4._o of_o the_o first_o and_o by_o the_o same_o reason_n may_v we_o prove_v that_o every_o one_o of_o the_o right_a line_n more_o ol_n king_n kl_o km_o ko_o ni_fw-fr nl_o nm_o and_o no_o which_o subtend_v angle_n equal_a to_o the_o self_n same_o angle_n ipl_n and_o be_v contain_v under_o side_n equal_a to_o the_o side_n which_o contain_v the_o angle_n ipl_n be_v equal_a to_o the_o right_a line_n il._n wherefore_o the_o triangle_n kli_n klo_n kmi_n kmo_fw-la and_o nli_n nlo_n nmi_n nmo_fw-la be_v equilater_n and_o equal_a and_o they_o contain_v the_o solid_a iklonm_n wherefore_o iklonm_n be_v a_o octohedron_n by_o the_o 23._o definition_n of_o the_o eleven_o and_o forasmuch_o as_o the_o angle_n thereof_o do_v altogether_o in_o the_o point_n i_o king_n l_o oh_o n_o m_o touch_v the_o base_n of_o the_o cube_fw-la which_o contain_v it_o it_o follow_v that_o the_o octohedron_n be_v inscribe_v in_o the_o cube_fw-la by_o the_o first_o definition_n of_o this_o book_n wherefore_o in_o the_o cube_fw-la give_v be_v describe_v a_o octohedron_n which_o be_v require_v to_o be_v do_v ¶_o a_o corollary_n aâ—ded_v by_o â—lussâ—â—_n hereby_o it_o be_v manifest_a that_o right_a line_n join_v together_o the_o centre_n of_o the_o opposite_a base_n of_o the_o cube_fw-la do_v cut_v the_o one_o the_o other_o into_o two_o equal_a part_n and_o perpendicular_o in_o the_o centre_n of_o the_o cube_fw-la or_o in_o the_o centre_n of_o the_o sphere_n which_o contain_v the_o cube_fw-la for_o forasmuch_o as_o the_o right_a line_n lm_o and_o io_o which_o knâ—â—_n together_o the_o centre_n of_o the_o opposite_a base_n of_o the_o cube_fw-la do_v also_o knit_v together_o the_o opposite_a angel_n of_o the_o octahedron_n inscribe_v in_o the_o cube_fw-la it_o follow_v by_o the_o 3._o corollary_n of_o the_o 14._o of_o the_o thirteen_o that_o those_o line_n lm_o and_o io_o do_v cut_v the_o one_o the_o other_o into_o two_o equal_a part_n in_o a_o point_n but_o the_o diameter_n of_o the_o cube_fw-la do_v also_o cut_v the_o one_o the_o other_o into_o two_o equal_a part_n by_o the_o 39_o of_o the_o eleven_o wherefore_o that_o point_n shall_v be_v the_o centre_n of_o the_o sphere_n which_o contain_v the_o câ—â—â—_n for_o make_v that_o point_n the_o centre_n and_o the_o space_n some_o one_o of_o the_o semidiameter_n describe_v a_o sphere_n and_o it_o shall_v pass_v by_o the_o angle_n of_o the_o cube_fw-la and_o likewise_o make_v the_o same_o point_n the_o centre_n and_o the_o space_n half_o of_o the_o line_n lm_o describe_v a_o sphere_n and_o it_o shall_v also_o pass_v by_o the_o angle_n of_o the_o octohedron_n ¶_o the_o 4._o proposition_n the_o 4._o problem_n in_o a_o octohedron_n give_v to_o describe_v a_o cube_n svppose_v that_o the_o octohedron_n give_v be_v abgdez_n and_o let_v the_o two_o pyramid_n thereof_o be_v abgde_v and_o zbgde_v construction_n and_o take_v the_o centre_n of_o the_o triangle_n of_o the_o pyramid_n abgde_v that_o be_v take_v the_o centre_n of_o the_o circle_n which_o contain_v those_o triangle_n and_o let_v those_o centre_n be_v the_o pointâ—s_n t_o i_o king_n l._n and_o by_o these_o centre_n let_v there_o be_v draw_v parallel_n line_n â—o_o the_o side_n of_o the_o square_n bgde_v which_o parallel_a â—igâ—â—_n linâ—â—_n let_v be_v mtn_n nlx_n xko_n &_o oim._n demonstration_n and_o forasmuch_o as_o thâ—se_a parallel_n right_a line_n do_v by_o the_o 2._o of_o the_o six_o cut_v the_o equal_a right_a line_n ab_fw-la agnostus_n ad_fw-la and_o ae_n proportional_o therefore_o they_o concur_v in_o the_o point_n m_o n_o x_o o._n wherefore_o the_o right_a line_n mn_v nx_o xo_o and_o om_fw-la which_o subtend_v equal_a angle_n set_v at_o the_o point_n a_o &_o contain_v under_o squall_n right_a line_n be_v equal_a by_o the_o 4._o of_o the_o first_o and_o moreover_o see_v that_o they_o be_v parallel_n unto_o the_o line_n bg_o gd_o de_fw-fr eâ—_n which_o make_v a_o square_a therefore_o mnxo_n be_v also_o a_o square_a by_o the_o 10._o of_o the_o eleven_o wherefore_o also_o by_o the_o 15._o of_o the_o â—ame_n the_o square_a mnxo_n be_v parallel_v to_o the_o square_n bgde_v for_o all_o tâ—e_z right_a line_n touch_v the_o one_o the_o other_o in_o the_o point_n of_o their_o section_n from_o the_o centre_n t_o i_o king_n l_o draw_v these_o right_a line_n ti_o ik_fw-mi kl_o ltâ—_n and_o draw_v the_o right_a line_n aic_n and_o forasmuch_o as_o i_o be_v the_o centre_n of_o the_o equilater_n triangle_n abe_n therefore_o the_o right_a line_n ai_fw-fr be_v extend_v cut_v the_o right_a line_n be_v into_o two_o equal_a part_n by_o the_o corollary_n of_o the_o 12._o of_o the_o thirteen_o and_o forasmuch_o as_o more_n be_v a_o parallel_n to_o be_v therefore_o the_o triangle_n aio_fw-la be_v like_a to_o the_o whole_a triangle_n ace_n by_o the_o corollary_n of_o the_o 2._o of_o the_o six_o and_o the_o right_a line_n more_n be_v divide_v into_o two_o equal_a part_n in_o the_o point_n i_o by_o the_o 4._o of_o the_o six_o and_o by_o the_o same_o reason_n may_v we_o prove_v that_o the_o right_a line_n mn_v nx_o xo_o be_v divide_v into_o two_o equal_a part_n in_o the_o point_n t_o l_o k._n wherefore_o also_o again_o the_o base_n ti_fw-mi ik_fw-mi kl_o lt_n which_o subtend_v the_o angle_n set_v at_o the_o point_n m_o oh_o x_o n_o which_o angle_n be_v right_a angle_n and_o be_v contain_v under_o equal_a side_n those_o base_n i_o say_v be_v equal_a and_o forasmuch_o as_o tim_n be_v a_o isosceles_a triangle_n therefore_o the_o angle_n set_v at_o the_o base_a namely_o the_o angle_n mti_n and_o mit_fw-fr be_v equal_a by_o the_o â—â—_o of_o the_o first_o but_o the_o angle_n m_o be_v a_o right_a angle_n wherefore_o each_o of_o the_o angle_n mit_fw-fr and_o mti_n be_v the_o half_a of_o a_o right_a angle_n and_o by_o the_o same_o reason_n the_o angle_n oik_a &_o oki_n be_v equal_a wherefore_o the_o angle_n remain_v namely_o tik_fw-mi be_v a_o right_a angle_n
by_o the_o 13._o of_o the_o first_o for_o the_o right_a line_n ti_fw-mi and_o ik_fw-mi be_v set_v upon_o the_o line_n mo._n and_o by_o the_o same_o reason_n may_v the_o rest_n of_o the_o angle_n namely_o ikl_n klt_n lti_fw-la be_v prove_v right_a angle_n and_o they_o be_v in_o one_o and_o the_o self_n same_o plain_a superficies_n namely_o mnxo_n by_o the_o 7._o of_o the_o eleven_o wherefore_o the_o right_a line_n which_o join_v together_o the_o centre_n of_o the_o plain_a superficial_a triangle_n which_o make_v the_o solid_a angle_n a_o do_v make_v the_o square_a itkl_n and_o by_o the_o same_o reason_n may_v be_v prove_v that_o the_o plain_a superficial_a triangle_n of_o the_o rest_n of_o the_o five_o solid_a angle_n of_o the_o octohedron_n set_v at_o the_o point_n b_o g_o z_o d_o e_o do_v in_o the_o centre_n of_o their_o base_n receive_v square_n so_o that_o there_o be_v in_o number_n six_o square_n for_o every_o octohedron_n have_v six_o solid_a angle_n and_o those_o square_n be_v equal_a for_o their_o side_n do_v contain_v equal_a angle_n of_o inclination_n contain_v under_o equal_a side_n namely_o under_o those_o side_n which_o be_v draw_v from_o the_o centre_n to_o the_o side_n of_o the_o equal_a triangle_n by_o the_o 2._o corollary_n of_o the_o 18._o of_o the_o thirteen_o wherefore_o itklrpus_n be_v a_o cube_fw-la by_o the_o 21._o definition_n of_o the_o eleven_o and_o have_v his_o angle_n in_o the_o centre_n of_o the_o base_n of_o the_o octohedron_n and_o therefore_o be_v inscribe_v in_o it_o by_o the_o first_o definition_n of_o this_o book_n wherefore_o in_o a_o octohedron_n give_v be_v describe_v a_o cube_fw-la which_o be_v require_v to_o be_v do_v the_o 5._o proposition_n the_o 5._o problem_n in_o a_o icosahedron_n give_v to_o describe_v a_o dodecahedron_n take_v a_o icosahedron_n one_o of_o who_o solid_a angle_n let_v be_v z._n construction_n now_o forasmuch_o as_o by_o those_o thing_n which_o have_v be_v prove_v in_o the_o 16._o of_o the_o thirteen_o the_o base_n of_o the_o triangle_n which_o contain_v the_o angle_n of_o the_o icosahedron_n do_v make_v a_o pentagon_n inscribe_v in_o a_o circle_n let_v that_o pentagon_n be_v abgde_v which_o be_v make_v of_o the_o five_o base_n of_o the_o triangle_n who_o plain_a superficial_a angle_n remain_v make_v the_o solid_a angle_n give_v namely_o z._n and_o take_v the_o centre_n of_o the_o circle_n which_o contain_v the_o foresay_a triangle_n which_o centre_n let_v be_v the_o point_n i_o t_o edward_n m_o l_o and_o draw_v these_o right_a line_n it_o tk_n km_o ml_o li._n demonstration_n now_o then_o a_o perpendicular_a line_n draw_v from_o the_o point_n z_o to_o the_o plain_a superficies_n of_o the_o pentagon_n abgde_v shall_v fall_v upon_o the_o centre_n of_o the_o circle_n which_o contain_v the_o pentagon_n abgde_v by_o those_o thing_n which_o have_v be_v prove_v in_o the_o self_n same_o 16._o of_o the_o thirteen_o moreover_o perpendicular_a line_n draw_v from_o the_o centre_n to_o the_o side_n of_o the_o pentagon_n abgde_v shall_v in_o the_o point_n c_o n_o oh_o where_o they_o fall_v cut_v the_o right_a line_n ab_fw-la bg_o gd_o into_o two_o equal_a part_n by_o the_o 3._o of_o the_o three_o draw_v these_o right_a line_n cn_fw-la and_o no_o and_o forasmuch_o as_o the_o angle_n cbn_n and_o ngo_n be_v equal_a and_o be_v contain_v under_o equal_a side_n therefore_o the_o base_a cn_fw-la be_v equal_a to_o the_o base_a no_o by_o the_o 4._o of_o the_o first_o moreover_o perpendicular_a line_n drâ—â—â—e_v from_o the_o point_n z_o to_o the_o bâ—sâ—â—_n of_o the_o pentagon_n abgde_v shall_v likewise_o cut_v the_o base_n into_o two_o equal_a part_n by_o thâ—â—_n of_o the_o three_o for_o the_o perpendicular_n pass_v by_o the_o centre_n by_o the_o corollary_n of_o the_o 12._o of_o the_o thirteen_o wherefore_o thâ—se_a perpendicular_a line_n shall_v fall_v upon_o the_o point_n c_o n_o o._n and_o now_o forasmuch_o as_o the_o right_a line_n zi_n ig_n be_v equal_a to_o the_o right_a line_n zt_v tn_n &_o also_o to_o the_o right_a line_n zk_v ko_o by_o reason_n of_o the_o likeness_n of_o the_o equal_a triangle_n therefore_o the_o line_n it_o be_v a_o parallel_n to_o the_o line_n cn_fw-la and_o so_o also_o be_v the_o line_n tk_n to_o the_o line_n no_o by_o the_o 2._o of_o the_o six_o wherefore_o the_o angle_n itk_n and_o cno_n be_v equal_a by_o the_o 11._o of_o the_o eleven_o again_o forasmuch_o as_o the_o triangle_n cbn_n and_o ngo_n be_v isoscel_n triangle_n therefore_o the_o angle_n bcn_n and_o bnc_n be_v equal_a by_o the_o 5._o of_o the_o first_o and_o by_o the_o same_o reason_n the_o angle_n gno_fw-la and_o gon_n be_v equal_a and_o moreover_o the_o angle_n bcn_n and_o bnc_n be_v equal_a to_o the_o angle_n gno_fw-la and_o go_v for_o that_o the_o triangle_n cbn_n and_o ngo_n be_v equal_a and_o like_a bâ—â—_n the_o three_o angle_n bnc_fw-la cno_n ong_n be_v equal_a to_o two_o right_a angle_n by_o the_o 13._o of_o the_o first_o for_o that_o upon_o the_o right_a line_n bâ—_n be_v set_v the_o right_a line_n cn_fw-la &_o on_o and_o the_o three_o angle_n of_o the_o triangle_n cbn_n namely_o the_o angle_n bnc_fw-la bcn_n or_o gno_n for_o the_o angle_n gno_n be_v equal_a to_o the_o angle_n bcn_n as_o it_o have_v be_v prove_v and_o nbc_n be_v also_o equal_a to_o two_o right_a angle_n by_o the_o 32._o of_o the_o first_o wherefore_o take_v away_o the_o angle_n bnc_n &_o gno_fw-la the_o angle_n remain_v namely_o cno_n be_v equal_a to_o the_o angle_n remain_v namely_o to_o cbn_n wherefore_o also_o the_o angle_n itk_n which_o be_v prove_v to_o be_v equal_a to_o the_o angle_n cno_n be_v equal_a to_o the_o angle_n cbn_n wherefore_o itk_n be_v the_o angle_n of_o a_o pentagon_n and_o by_o the_o same_o reason_n may_v be_v proâ—ed_v that_o the_o rest_n of_o the_o angle_n name_o the_o angle_n till_o ilm_n lmk_n mkt_v be_v equal_a to_o the_o rest_n of_o the_o angle_n namely_o to_o bae_o aed_n edg_n dgb_n wherefore_o itkml_n be_v a_o equilater_n and_o equiangle_n pentagon_n by_o the_o 4._o of_o the_o first_o for_o the_o equal_a base_n of_o the_o pentagon_n itkml_n do_v subtend_v equal_a angle_n set_v at_o the_o point_n z_o and_o comprehend_v under_o equal_a side_n moreover_o it_o be_v manifest_a that_o the_o pentagon_n i_o tkml_n be_v in_o one_o and_o the_o self_n same_o plain_a superficies_n for_o forasmuch_o as_o the_o angle_n onc_n and_o ncp_n be_v in_o one_o and_o the_o self_n sâ—me_v plain_a superficies_n namely_o in_o the_o superficies_n abgde_v but_o unto_o the_o same_o plain_a superficies_n the_o plain_a superficiece_n of_o the_o angle_n kti_fw-la and_o until_o be_v parallel_n by_o the_o 15._o of_o the_o eleven_o and_o the_o triangle_n kti_n and_o until_o concur_v wherefore_o they_o be_v in_o one_o and_o the_o self_n same_o plain_a superficies_n by_o the_o corollary_n of_o the_o 16._o of_o the_o eleven_o and_o by_o the_o same_o reason_n so_o may_v we_o prove_v that_o the_o triangle_n ilm_n lmk_n mkt_v be_v in_o the_o self_n same_o plain_a superficies_n wherein_o be_v the_o triangle_n kti_n and_o till_o wherefore_o the_o pentagon_n itkml_n be_v in_o one_o and_o the_o self_n same_o plain_a superficies_n wherefore_o the_o solid_a angle_n of_o the_o icosahedron_n namely_o the_o solid_a angle_n at_o the_o point_n z_o subtend_v a_o equilater_n and_o equiangle_n pentagon_n plain_a superficies_n which_o pentagon_n have_v his_o plain_a superficial_a angle_n in_o the_o centre_n of_o the_o triangle_n which_o make_v the_o solid_a angle_n z._n and_o in_o like_a sort_n may_v we_o prove_v that_o the_o other_o eleven_o solid_a angle_n of_o the_o icosahedron_n each_o of_o which_o eleven_o solid_a angle_n be_v equal_a and_o like_a to_o the_o solid_a angle_n z_o by_o the_o 16._o of_o the_o thirteen_o be_v subtend_v unto_o pentagon_n equal_a and_o like_a and_o in_o like_a sort_n set_v to_o the_o pentagon_n itkml_n and_o forasmuch_o as_o in_o those_o pentagon_n the_o right_a line_n which_o join_v together_o the_o centre_n of_o the_o base_n be_v common_a side_n it_o follow_v that_o those_o 12._o pentagon_n include_v a_o solid_a which_o solid_a be_v therefore_o a_o dodecahedron_n by_o the_o 24._o definition_n of_o the_o eleven_o and_o be_v by_o the_o first_o definition_n of_o this_o book_n describe_v in_o the_o icosahedron_n five_o side_n whereof_o 〈◊〉_d set_v upon_o the_o pentagon_n abgde_v wherefore_o in_o a_o icosahedron_n give_v iâ—_z inscribe_v a_o dodecahedron_n which_o be_v require_v to_o be_v do_v a_o annotation_n of_o hypsiâ—les_n this_o be_v to_o be_v note_v that_o if_o a_o man_n shall_v demand_v 〈◊〉_d many_o side_n a_o icosahedron_n have_v we_o may_v thus_o answer_v it_o be_v manifest_a that_o a_o icosahâ—râ—n_n be_v contain_v under_o 20._o triangle_n and_o that_o every_o triangle_n consist_v of_o three_o
right_a linâ—s_n now_o then_o multiply_v the_o 20._o triangle_n into_o the_o side_n of_o one_o of_o the_o triangle_n and_o so_o shall_v there_o be_v produce_v 6â—_o â—he_z half_a of_o which_o be_v 30._o and_o so_o many_o side_n have_v a_o icosahedron_n and_o in_o like_a sort_n in_o a_o dodecahedron_n forasmuch_o as_o 12._o pentagon_n make_v a_o dodecahedron_n and_o every_o pentagon_n contain_v â—_o right_a linesâ—_n multiply_v â—â—_o into_o 12._o and_o there_o shall_v be_v produce_v 60._o the_o half_a of_o which_o be_v 30._o and_o so_o many_o be_v the_o side_n of_o a_o dodecahedron_n and_o the_o reason_n why_o we_o take_v the_o half_a iâ—_z for_o that_o every_o side_n whether_o it_o be_v of_o a_o triangle_n or_o of_o a_o pentagon_n or_o of_o a_o square_a as_o in_o a_o cube_fw-la â—s_v take_v twice_o and_o by_o the_o same_o reason_n may_v you_o find_v out_o how_o many_o side_n be_v in_o a_o cube_fw-la and_o in_o a_o pyramid_n and_o in_o a_o octohedron_n but_o now_o again_o if_o you_o will_v find_v out_o the_o number_n of_o the_o angle_n of_o every_o one_o of_o the_o solid_a figure_n when_o you_o have_v do_v the_o same_o multiplication_n that_o you_o do_v before_o divide_v the_o same_o side_n by_o the_o number_n of_o the_o plain_a superficiece_n which_o comprehend_v one_o of_o the_o angle_n of_o the_o solid_n as_o for_o example_n forasmuch_o as_o 5._o triangle_n contain_v the_o solid_a angle_n of_o a_o icosahedron_n divide_v 60._o by_o 5._o and_o there_o will_v come_v forth_o 12._o and_o so_o many_o solid_a angle_n have_v a_o icosahedron_n in_o a_o dodecahedron_n forasmuch_o as_o three_o pentagon_n comprehend_v a_o angle_n divide_v 60._o by_o 3._o and_o there_o will_v come_v forth_o 20_o and_o so_o many_o be_v the_o angle_n of_o a_o dodecahedron_n and_o by_o the_o same_o reason_n may_v you_o find_v out_o how_o many_o angle_n be_v in_o each_o of_o the_o rest_n of_o the_o solid_a figure_n book_n if_o it_o be_v require_v to_o be_v know_v how_o one_o of_o the_o plain_n of_o any_o of_o the_o five_o solid_n be_v give_v there_o may_v be_v find_v out_o the_o inclination_n of_o the_o say_a plain_n the_o one_o to_o the_o other_o which_o contain_v each_o of_o the_o solid_n this_o as_o say_v isidorus_n our_o great_a master_n be_v foâ—â—d_v out_o after_o this_o manner_n it_o be_v manifest_a that_o in_o a_o cube_fw-la the_o plain_n which_o contain_v iâ—_z doâ—_n 〈◊〉_d the_o one_o the_o other_o by_o a_o right_a angle_n but_o in_o a_o tetrahedron_n one_o of_o the_o triangle_n be_v give_v let_v the_o end_n of_o one_o of_o the_o side_n of_o the_o say_a triangle_n be_v the_o centre_n and_o let_v the_o space_n be_v the_o perpendicular_a line_n draw_v from_o the_o top_n of_o the_o triangle_n to_o the_o base_a and_o describe_v circumferâ—nces_n of_o a_o circle_n which_o shall_v cut_v the_o one_o the_o other_o and_o from_o the_o intersection_n to_o the_o centre_n draw_v right_a line_n which_o shall_v contain_v the_o inclination_n of_o the_o plain_n contain_v the_o tetrahedron_n in_o a_o octoâ—edron_n take_v one_o of_o the_o side_n of_o the_o triangle_n thereof_o and_o upon_o it_o describe_v a_o square_a and_o draw_v the_o diagonal_a line_n and_o make_v the_o centre_n the_o end_n of_o the_o diagonal_a line_n and_o the_o space_n likewise_o the_o perpendicular_a line_n draw_v from_o the_o top_n of_o the_o triangle_n to_o the_o base_a describe_v circumference_n and_o again_o from_o the_o common_a section_n to_o the_o centre_n draw_v right_a line_n and_o they_o shall_v contain_v the_o inclination_n seek_v for_o in_o a_o icosahedron_n upon_o the_o side_n of_o one_o of_o the_o triangle_n thereof_o describe_v a_o pentagon_n and_o draw_v the_o line_n which_o subtend_v one_o of_o the_o angle_n of_o the_o say_a pentagon_n and_o make_v the_o centre_n the_o end_n of_o that_o line_n and_o the_o space_n the_o perpendicular_a line_n of_o the_o triangle_n describe_v circumference_n and_o draw_v from_o the_o common_a intersection_n of_o the_o circumference_n unto_o the_o centre_n right_a line_n and_o they_o shall_v contain_v likewise_o the_o inclination_n of_o the_o plain_n of_o the_o icosahedron_n in_o a_o dodecahedron_n take_v one_o of_o the_o pentagon_n and_o draw_v likewise_o the_o line_n which_o subtend_v one_o of_o the_o angle_n of_o the_o pentagon_n and_o make_v the_o centre_n the_o end_n of_o that_o line_n and_o the_o space_n the_o perpendicular_a line_n draw_v from_o the_o section_n into_o two_o equal_a part_n of_o that_o line_n to_o the_o side_n of_o the_o pentagon_n which_o be_v parallel_n unto_o it_o describe_v circumference_n and_o from_o the_o point_n of_o the_o intersection_n of_o the_o circumference_n draw_v unto_o the_o centre_n right_a line_n and_o they_o shall_v also_o contain_v the_o inclination_n of_o the_o plain_n of_o the_o dodecahedron_n thus_o do_v this_o most_o singular_a learned_a man_n reason_n think_v the_o demonstration_n in_o every_o one_o of_o they_o to_o be_v plain_a and_o clear_a but_o to_o make_v the_o demonstration_n of_o they_o manifest_a i_o think_v it_o good_a to_o declare_v and_o make_v open_a his_o wordesâ—_n and_o first_o in_o a_o tâ—trahedronâ—_n the_o end_n of_o the_o fivetenth_fw-mi book_n of_o euclides_n element_n after_o 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d ¶_o the_o 6._o proposition_n the_o 6._o problem_n in_o a_o octohedron_n give_v to_o inscribe_v a_o trilater_n equilater_n pyramid_n svppose_v thaâ—_z the_o octohedron_n whereâ—â—_n the_o tetrahedron_n be_v require_v to_o be_v insâ—riâ—â—â—_n be_v abgdei_n take_v 〈…〉_o base_n of_o the_o octoâ—â—dron_n that_o be_v construction_n 〈…〉_o close_o in_o the_o loweâ—â—_n triangle_n bgd_v namely_o aeâ—_n head_n igd_v and_o let_v the_o four_o be_v aib_n which_o be_v opposite_a to_o the_o low_a triangle_n before_o put_v namely_o to_o egg_v and_o take_v the_o centre_n of_o those_o four_o base_n which_o let_v be_v the_o point_n h_o c_o n_o â—_o and_o upon_o the_o triangle_n hcn_n erect_v a_o pyramid_n hcnl_n now_o forasmuch_o as_o these_o two_o base_n of_o the_o octohedron_n namely_o age_n and_o abi_n be_v set_v upon_o the_o right_a line_n eglantine_n and_o by_o which_o be_v opposite_a the_o one_o to_o the_o otherâ—_n in_o the_o square_a gebi_n of_o the_o octohedron_n from_o the_o poinâ—_n a_o draâ—e_v by_o the_o centre_n of_o the_o base_n namely_o by_o the_o centre_n h_o l_o perpendicular_a line_n ahf_n alk_v cut_v the_o line_n eglantine_n and_o by_o 〈◊〉_d two_o equal_a part_n in_o the_o point_n f_o edward_n by_o the_o corollary_n of_o the_o 1â—_o of_o the_o thirteen_o wherefore_o a_o right_a line_n draw_v from_o the_o point_n f_o to_o the_o point_n king_n demonstration_n shall_v be_v a_o parallel_n and_o equal_a to_o the_o side_n of_o the_o octohedron_n namely_o to_o â—â—_o and_o give_v by_o the_o 33._o of_o the_o first_o and_o the_o right_a line_n hl_o which_o cut_v the_o 〈…〉_o of_o ak_o proportional_o for_o ah_o and_o all_o be_v draw_v from_o the_o centre_n of_o equal_a circle_n to_o the_o circumference_n be_v a_o parallel_n to_o the_o right_a line_n fk_o by_o the_o 2._o of_o the_o six_o and_o also_o to_o the_o side_n of_o the_o octohedron_n namely_o to_o eâ—_n and_o ig_n by_o the_o 9_o of_o the_o eleven_o wherefore_o as_o the_o line_n of_o be_v to_o the_o line_n ah_o so_o be_v the_o line_n fk_o to_o the_o line_n hl_o by_o the_o 4._o of_o the_o six_o for_o the_o triangle_n afk_v and_o ahl_n be_v like_o by_o thâ—_z corollary_n of_o the_o 2._o of_o the_o six_o but_o the_o line_n of_o be_v in_o sesquialter_fw-la proportion_n to_o the_o line_n ah_o for_o the_o side_n eglantine_n make_v hf_o the_o half_a of_o the_o right_a line_n ah_o by_o the_o corollary_n of_o the_o 12._o of_o the_o thirteen_o wherefore_o fk_n or_o give_v the_o side_n of_o the_o octohedron_n be_v sesquialter_fw-la to_o the_o righâ—line_n hl._n and_o by_o the_o same_o reason_n may_v we_o prove_v that_o the_o side_n of_o the_o octohedron_n be_v sesquialter_fw-la to_o the_o rest_n of_o the_o right_a line_n which_o make_v the_o pyramid_n hnci_n namely_o to_o the_o right_a lineâ—â—_n n_o nc_n ci_o ln_o and_o change_z wherefore_o those_o right_a line_n be_v equal_a and_o therefore_o the_o triangleâ—_n which_o be_v describe_v of_o they_o namely_o the_o triangle_n hcn_n hnl_n ncl_n and_o chl._n which_o make_v the_o pyramid_n hncl_n be_v equal_a and_o equilater_n and_o forasmuch_o as_o the_o angle_n of_o the_o same_o pyramid_n namely_o the_o angle_n h_o c_o n_o l_o do_v end_n in_o the_o centre_n of_o the_o base_n of_o the_o octohedron_n therefore_o it_o be_v inscribe_v â—o_o the_o same_o octohedron_n by_o the_o first_o definition_n of_o this_o book_n wherefore_o in_o a_o octohedron_n â—even_v be_v inscribe_v a_o trilate_a equilaâ—â—â—â—â—â—amisâ—_n which_o be_v require_v to_o â—e_a donâ—_n a_o corollary_n the_o base_n of_o a_o pyramid_n inscribe_v in_o a_o octohedron_n be_v parallel_n
that_o be_v the_o right_a line_n agnostus_n gb_o bd_o dam_fw-ge ca_n cg_o cb_o cd_o and_o ja_z ig_n ib_n id_fw-la be_v equal_a the_o one_o to_o the_o other_o wherefore_o also_o the_o 8._o triangle_n cag_n cgb_n cbd_v cda_n jag_n igb_n ibd_v &_o ida_n be_v equal_a and_o equilater_n and_o therefore_o agbdci_n be_v a_o octohedron_n by_o the_o 23._o definition_n of_o the_o eleven_o and_o the_o say_v octahedron_n be_v include_v in_o the_o dodecahedron_n by_o the_o first_o definition_n of_o this_o book_n for_o that_o all_o the_o angle_n thereof_o do_v at_o one_o time_n touch_v the_o side_n of_o the_o dodecahedron_n wherefore_o in_o the_o dodecahedron_n give_v be_v include_v a_o octohedron_n which_o be_v require_v to_o be_v do_v ¶_o the_o 10._o proposition_n the_o 10._o problem_n in_o a_o dodecahedron_n give_v to_o inscribe_v a_o equilater_n trilater_n pyramid_n svppose_v that_o the_o dodecahedron_n give_v be_v abcd_o of_o which_o dodecahedron_n take_v three_o base_n meet_v at_o the_o point_n s_o namely_o these_o three_o base_n alsik_n dnsle_n and_o sibrn_n construction_n and_o of_o those_o three_o base_n take_v the_o three_o angle_n at_o the_o point_n a_o b_o d_o and_o draw_v these_o right_a line_n ab_fw-la bd_o and_o dam_fw-ge and_o let_v the_o diameter_n of_o the_o sphere_n contain_v the_o dodecahedron_n be_v so_o and_o then_o draw_v these_o right_a line_n ao_o boy_n and_o do_v now_o forasmuch_o as_o by_o the_o 17._o of_o the_o thirteen_o the_o angle_n of_o the_o dodecahedron_n be_v set_v in_o the_o superficies_n of_o the_o sphere_n describe_v about_o the_o dodecahedronâ—_n demonstration_n therefore_o if_o upon_o the_o diameter_n so_o and_o by_o the_o point_n a_o be_v describe_v a_o semicircle_n it_o shall_v make_v the_o angle_n sao_n a_o right_a angle_n by_o the_o 31._o of_o the_o three_o and_o likewise_o if_o the_o same_o semicircle_n be_v draw_v by_o the_o point_n d_o and_o b_o it_o shall_v also_o make_v the_o angle_n sbo_n and_o sdo_v right_a angle_n wherefore_o the_o diameter_n so_o contain_v in_o power_n both_o the_o line_n sa_o ao_o or_o the_o line_n sb_n boy_n or_o else_o sd_z do_v but_o the_o line_n sa_o sd_z sb_n be_v equal_a the_o one_o to_o the_o other_o for_o they_o each_o subtend_v one_o of_o the_o angle_n of_o equal_a pentagon_n wherefore_o the_o other_o line_n remain_v namely_o ao_o boy_n do_v be_v equal_a the_o one_o to_o the_o other_o and_o by_o the_o same_o reason_n may_v be_v prove_v that_o the_o diameter_n hd_v which_o subtend_v the_o two_o right_a line_n ha_o ad_fw-la contain_v in_o power_n both_o the_o say_v two_o right_a line_n and_o also_o contain_v in_o power_n both_o the_o right_a line_n hb_o and_o bd_o which_o two_o right_a line_n it_o also_o suhtend_v and_o moreover_o by_o the_o same_o reason_n the_o diameter_n ac_fw-la which_o subtend_v the_o right_a line_n cb_o and_o basilius_n contain_v in_o power_n both_o the_o say_v right_a line_n câ—_n and_o basilius_n but_o the_o right_a line_n ha_o hb_o and_o cb_o be_v equal_a the_o one_o to_o the_o other_o for_o that_o each_o of_o they_o also_o subtend_v one_o of_o the_o angle_n of_o equal_a pentagonsâ—_n wherefore_o the_o right_a line_n remain_v namely_o ad_fw-la bd_o and_o basilius_n be_v equal_a the_o one_o to_o the_o other_o and_o by_o the_o same_o reason_n may_v be_v prove_v that_o each_o of_o those_o right_a line_n ad_fw-la bd_o and_o basilius_n be_v equal_a to_o each_o of_o the_o right_a line_n ao_o boy_n and_o do_v wherefore_o the_o six_o right_a line_n ab_fw-la bd_o dam_fw-ge ao_o boy_n &_o do_v be_v equal_a the_o one_o to_o the_o other_o and_o therefore_o the_o triangle_n which_o be_v make_v of_o they_o namely_o the_o triangle_n abdella_n aob_n aod_n and_o bod_fw-mi be_v equal_a and_o equilater_n which_o triangle_n therefore_o do_v make_v a_o pyramid_n abdo_n who_o base_a be_v abdella_n and_o top_n the_o point_n o._n each_o of_o the_o angle_n of_o which_o pyramid_n namely_o the_o angle_n at_o the_o point_n a_o b_o d_o oh_o do_v in_o the_o self_n same_o point_n touch_v the_o angle_n of_o the_o dodecahedron_n wherefore_o the_o say_a pyramid_n be_v inscribe_v in_o the_o dodecahedron_n by_o the_o first_o definition_n of_o this_o book_n wherefore_o in_o a_o dodecahedron_n give_v be_v inscribe_v a_o trilater_n equilater_n pyramid_n which_o be_v require_v to_o be_v do_v ¶_o the_o 11._o proposition_n the_o 11._o problem_n in_o a_o icosahedron_n give_v to_o inscribe_v a_o cube_fw-la it_o be_v manifest_a by_o the_o 7._o of_o this_o book_n that_o the_o angle_n of_o a_o dodecahedron_n be_v set_v in_o the_o centre_n of_o the_o base_n of_o the_o icosahedron_n and_o by_o the_o 8._o of_o this_o book_n it_o be_v prove_v that_o the_o angle_n of_o a_o cube_fw-la be_v set_v in_o the_o angle_n of_o a_o dodecahedron_n wherefore_o the_o self_n same_o angle_n of_o the_o cube_fw-la shall_v of_o necessity_n be_v set_v in_o the_o centre_n of_o the_o base_n of_o icosahedron_n wherefore_o the_o cube_fw-la shall_v be_v inscribe_v in_o the_o icosahedron_n by_o the_o first_o definition_n of_o this_o book_n wherefore_o in_o a_o icosahedron_n give_v be_v include_v a_o cube_fw-la which_o be_v require_v to_o be_v do_v ¶_o the_o 12._o proposition_n the_o 12._o problem_n in_o a_o icosahedron_n give_v to_o inscribe_v a_o trilater_n equilater_n pyramid_n by_o the_o former_a proposition_n it_o be_v manifest_a that_o the_o angle_n of_o a_o cube_fw-la be_v set_v in_o the_o centre_n of_o the_o base_n of_o the_o icosâ—hedron_n and_o by_o the_o first_o of_o this_o book_n it_o be_v plain_a that_o the_o four_o angle_n of_o a_o pyramid_n be_v set_v in_o four_o angle_n of_o a_o cube_fw-la wherefore_o it_o be_v evident_a by_o the_o first_o definition_n of_o this_o book_n that_o a_o pyramid_n describe_v of_o right_a line_n join_v together_o these_o four_o centre_n of_o the_o base_n of_o the_o icosahedron_n shall_v be_v inscribe_v in_o the_o same_o icosahedron_n wherefore_o in_o a_o icosadron_n give_v be_v inscribe_v a_o equilater_n trilater_n pyramid_n which_o be_v require_v to_o be_v do_v ¶_o the_o 13._o problem_n the_o 13._o proposition_n in_o a_o cube_n give_v to_o inscribe_v a_o dodecahedron_n take_v a_o cube_n adfl_fw-mi construction_n and_o divide_v every_o one_o of_o the_o side_n thereof_o into_o two_o equal_a part_n in_o the_o point_n t_o h_o edward_n p_o g_o l_o m_o f_o and_o pkq_n and_o draw_v these_o right_a line_n tk_n gf_o pq_n hk_n ps_n and_o lm_o which_o line_n again_o divide_v into_o two_o equal_a part_n in_o the_o point_n n_o five_o a'n_z ay_o z_o x._o and_o draw_v these_o right_a line_n nigh_o vx_n and_o iz_n now_o the_o three_o line_n nigh_o vx_n and_o iz_n together_o with_o the_o diameter_n of_o the_o cube_fw-la shall_v cut_v the_o one_o the_o other_o into_o two_o equal_a part_n in_o the_o centre_n of_o the_o cube_fw-la by_o the_o 3â—_o of_o the_o eleven_o let_v that_o centre_n be_v the_o point_n o._n and_o not_o to_o stand_v long_o about_o the_o demonstration_n understand_v all_o these_o right_a line_n to_o be_v equal_a and_o parallel_n to_o the_o side_n of_o the_o cube_fw-la and_o to_o cut_v the_o one_o the_o other_o right_o angle_a wise_a by_o the_o 29._o of_o the_o first_o let_v their_o half_n namely_o fv_n gv_n high_a and_o king_n and_o the_o rest_n such_o like_a be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n by_o the_o 30._o of_o the_o six_o who_o great_a segment_n let_v be_v the_o line_n f_n gb_o hc_n and_o ke_o etc_n etc_n and_o draw_v these_o right_a line_n give_v ge_z bc_o and_o be._n now_o forasmuch_o as_o the_o line_n give_v be_v equal_a to_o the_o whole_a line_n gv_n which_o be_v the_o half_a of_o the_o side_n of_o the_o cube_fw-la demonstration_n and_o the_o line_n je_n be_v equal_a to_o the_o line_n bv_n that_o be_v to_o the_o less_o segment_n therefore_o the_o square_n of_o the_o line_n give_v and_o je_n be_v triple_a to_o the_o square_n of_o the_o line_n gb_o by_o the_o 4._o of_o the_o thirteen_o but_o unto_o the_o square_n of_o the_o line_n give_v and_o je_n the_o square_a of_o the_o line_n ge_z be_v equal_a by_o the_o 47._o of_o the_o firstâ—_o for_o the_o angle_n gie_n be_v a_o right_a angle_n wherefore_o the_o square_a of_o the_o line_n ge_z be_v triple_a to_o the_o square_n of_o the_o line_n gb_o and_o forasmuch_o as_o the_o line_n fg_o be_v erect_v perpendicular_o to_o the_o plain_a agkl_n by_o the_o 4._o of_o the_o eleven_o for_o it_o be_v erect_v perpendicular_o to_o the_o two_o line_n agnostus_n and_o give_v therefore_o the_o angle_n bge_n be_v a_o right_a angle_n for_o the_o line_n ge_z be_v draw_v in_o the_o plain_a agkl_n wherefore_o the_o line_n be_v contain_v in_o power_n the_o two_o line_n bg_o and_o ge_z by_o the_o 47._o of_o the_o first_o be_v in_o power_n quadruple_a to_o the_o
by_o the_o 4._o and_o eight_o of_o the_o first_o and_o forasmuch_o as_o upon_o every_o one_o of_o the_o line_n cut_v of_o the_o cube_fw-la be_v set_v two_o triangle_n as_o the_o triangle_n alm_n and_o blmâ—_n there_o shall_v be_v make_v 12._o triangle_n and_o forasmuch_o as_o under_o every_o one_o of_o the_o â—_o angle_n of_o the_o cube_fw-la be_v subtend_v the_o other_o 8._o triangle_n as_o the_o triangle_n amp._n etc_n etc_n of_o 1â—_o and_o 8._o triangle_n shall_v be_v produce_v 20._o triangle_n equal_a and_o equilater_n contain_v the_o solid_a of_o a_o icosahedron_n by_o the_o 25._o definition_n of_o the_o eleven_o which_o shall_v be_v inscribe_v in_o the_o cube_fw-la give_v abc_n by_o the_o first_o definition_n of_o this_o book_n the_o invention_n of_o the_o demonstration_n of_o this_o depend_v of_o the_o ground_n of_o the_o former_a wherefore_o in_o a_o cube_fw-la give_v we_o have_v describe_v a_o icosahedron_n which_o be_v require_v to_o be_v do_v first_o corollary_n the_o diameter_n of_o a_o sphere_n which_o contain_v a_o icosahedron_n contain_v two_o side_n namely_o the_o side_n of_o the_o icosahedron_n and_o the_o side_n of_o the_o cube_fw-la which_o contain_v the_o icosahedron_n for_o if_o we_o draw_v the_o line_n ab_fw-la it_o shall_v make_v the_o angle_n at_o the_o point_n a_o right_a angle_n for_o that_o it_o be_v a_o parallel_n to_o the_o side_n of_o the_o cube_fw-la wherefore_o the_o linâ—_n which_o couple_v the_o opposite_a angleâ—_n of_o the_o icosahedron_n at_o the_o point_v fletcher_n and_o b_o contain_v in_o power_n the_o line_n ab_fw-la the_o sidâ—_n of_o thâ—_z cube_fw-la and_o the_o line_n of_o the_o side_n of_o the_o icosahedron_n by_o the_o 47._o of_o the_o first_o which_o line_n fb_o be_v equal_a to_o the_o â—iameter_n of_o the_o sphere_n which_o contain_v the_o icosahedron_n by_o the_o demonstration_n of_o the_o â—â—_o of_o the_o thirteen_o second_o corollary_n the_o six_o opposite_a side_n of_o the_o icosahedron_n divide_v into_o two_o equal_a part_n their_o section_n be_v couple_v by_o three_o equal_a right_a line_n cut_v the_o one_o the_o other_o into_o two_o equal_a part_n and_o perpendicular_o in_o the_o centre_n of_o the_o sphere_n which_o contain_v the_o icosahedron_n for_o those_o three_o line_n be_v the_o three_o line_n which_o couple_v the_o centre_n of_o the_o base_n of_o the_o cube_fw-la which_o do_v in_o such_o sort_n in_o the_o centre_n of_o the_o cube_fw-la cut_v the_o one_o the_o other_o by_o the_o corollary_n of_o the_o three_o of_o this_o book_n and_o therefore_o be_v equal_a to_o the_o side_n of_o the_o cube_fw-la but_o right_a line_n draw_v from_o the_o centre_n of_o the_o cube_fw-la to_o the_o angle_n of_o the_o icosahedron_n every_o one_o of_o they_o shall_v subtend_v the_o half_a side_n of_o the_o cube_fw-la and_o the_o half_a side_n of_o the_o icosahedron_n which_o half_a side_n contain_v a_o right_a angle_n wherefore_o those_o line_n be_v equal_a whereby_o it_o be_v manifest_a that_o the_o foresay_a centre_n be_v the_o centre_n of_o the_o sphere_n which_o contain_v the_o icosahedron_n three_o corollary_n the_o side_n of_o a_o cube_fw-la divide_v by_o a_o extreme_a and_o mean_a proportion_n make_v the_o great_a segment_n the_o side_n of_o a_o icosahedron_n describe_v in_o it_o for_o the_o half_a side_n of_o the_o cube_fw-la make_v the_o half_a of_o the_o side_n of_o the_o icosahedron_n the_o great_a segment_n wherefore_o also_o the_o whole_a side_n of_o the_o cube_fw-la make_v the_o whole_a side_n of_o the_o icosahedron_n the_o great_a segment_n by_o the_o 15._o of_o the_o fifthe_o for_o the_o section_n be_v like_a by_o the_o â—_o of_o the_o fourteen_o ¶_o four_o corollary_n the_o side_n and_o base_n of_o the_o icosahedron_n which_o be_v opposite_a the_o one_o to_o the_o other_o be_v parallel_n forasmuch_o as_o every_o one_o of_o the_o opposite_a side_n of_o the_o icosahedron_n may_v be_v in_o the_o parallel_a line_n of_o the_o cube_fw-la namely_o in_o those_o parallel_n which_o be_v opposite_a in_o the_o cube_fw-la and_o the_o triangle_n which_o be_v make_v of_o parallel_a line_n be_v parallel_n by_o the_o 15._o of_o the_o eleven_o therefore_o the_o opposite_a triâ—ngleâ—_n of_o the_o icosahedron_n as_o also_o the_o side_n be_v parallel_n the_o one_o to_o the_o other_o ¶_o the_o 15._o problem_n the_o 15._o proposition_n in_o a_o icosahedron_n give_v to_o inscribe_v a_o octohedron_n svppose_v that_o the_o icosahedron_n give_v be_v acdf_n and_o by_o the_o former_a second_o corollary_n construction_n let_v there_o be_v take_v the_o three_o right_a line_n which_o cut_v the_o one_o the_o other_o into_o two_o equal_a part_n perpendicular_o and_o which_o couple_n the_o section_n into_o two_o equal_a part_n of_o the_o side_n of_o the_o icosahedron_n which_o let_v be_v be_v gh_o and_o kl_o cut_v the_o one_o the_o other_o in_o the_o point_n i_o and_o drawâ—_n these_o riâ—ht_a line_n â—g_z ge_z eh_o and_o hb_o and_o forasmuch_o as_o the_o angle_n at_o the_o point_n i_o be_v by_o construction_n right_a angle_n demonstration_n â—nd_v be_v conâ—â—ined_v under_o equal_a linesâ—_n the_o 〈◊〉_d gâ—_n and_o â—â—_o shall_v 〈…〉_o squâ—re_o by_o the_o â—_o of_o the_o â—irst_n likewise_o â—nto_o thoâ—e_a 〈◊〉_d shall_v be_v squall_v the_o line_n drâ—wâ—n_v from_o 〈◊〉_d point_n king_n and_o â—_o to_o every_o one_o of_o the_o poinâ—â—s_n â—_o g._n â—_o h_o and_o therefore_o the_o triangle_n which_o 〈◊〉_d the_o â—â—ramis_n dgenk_n shall_v be_v equal_a 〈…〉_o and_o by_o 〈…〉_o ¶_o the_o 16._o problem_n the_o 16._o proposition_n in_o a_o octohedron_n give_v to_o inscribe_v a_o icosahedron_n let_v there_o be_v take_v a_o octohedron_n who_o 6._o angle_n let_v be_v a_o b_o c_o f_o p_o l._n and_o draw_v the_o line_n ac_fw-la bf_o pl_z construction_n cut_v the_o one_o the_o other_o perpendicular_o in_o the_o point_n r_o by_o the_o 2._o corollary_n of_o the_o 14._o of_o the_o thirteen_o and_o let_v every_o one_o of_o the_o 12._o side_n of_o the_o octohedron_n be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n in_o the_o point_n h_o x_o m_o king_n d_o s_o n_o g_o five_o e_o q_o t._n and_o let_v the_o great_a segment_n be_v the_o line_n bh_o bx_n fm_o fk_o ad_fw-la aq_n cs_n et_fw-la pn_n pg_n lv_o le_fw-fr and_o draw_v these_o line_n hk_o xm_o ge_z nv_n d_n qt_n now_o forasmuch_o as_o in_o the_o triangle_n abf_n the_o side_n be_v cut_v proportional_o namely_o as_o the_o line_n bh_o be_v to_o the_o line_n ha_o so_o be_v the_o line_n fk_o to_o the_o line_n ka_o by_o the_o 2._o of_o the_o fourteen_o demonstration_n therefore_o the_o line_n hk_o shall_v be_v a_o parallel_n to_o the_o line_n bf_o by_o the_o 2._o of_o the_o six_o and_o forasmuch_o as_o the_o line_n ac_fw-la cut_v the_o line_n hk_o in_o the_o point_n z_o and_o the_o line_n zk_n be_v a_o parallel_n unto_o the_o line_n rf_n the_o line_n ramires_n shall_v be_v cut_v by_o a_o extreme_a and_o mean_a proportion_n in_o the_o point_n z_o by_o the_o 2._o of_o the_o six_o namely_o shall_v be_v cut_v like_o unto_o the_o line_n favorina_n and_o the_o great_a segment_n thereof_o shall_v be_v the_o line_n zr_n unto_o the_o line_n zk_v put_v the_o line_n ro_n equal_a by_o the_o 3._o of_o the_o first_o and_o draw_v the_o line_n ko_o now_o then_o the_o line_n ko_o shall_v be_v equal_a to_o the_o line_n zr_n by_o the_o 33._o of_o the_o â—irst_n draw_v the_o line_n kg_v ke_o and_o ki_v and_o forasmuch_o as_o the_o triangle_n arf_n and_o azk_v be_v equiangle_n by_o the_o 6._o of_o the_o six_o the_o side_n az_o and_o zk_v shall_v be_v equal_a the_o one_o to_o the_o other_o by_o the_o 4._o of_o the_o six_o for_o the_o side_n be_v and_o rf_n be_v equal_a wherefore_o the_o line_n zk_n shall_v be_v the_o less_o segment_n of_o the_o line_n ra._n but_o if_o the_o great_a segment_n rz_n be_v divide_v by_o a_o extreme_a &_o mean_a proportion_n the_o great_a segment_n thereof_o shall_v be_v the_o line_n zk_n which_o be_v the_o less_o segment_n of_o the_o whole_a line_n ramires_n by_o the_o 5._o of_o the_o thirâ—enth_o and_o forasmuch_o as_o the_o two_o line_n fe_o and_o fg_o be_v equal_a to_o the_o two_o line_n ah_o and_o ak_o namely_o each_o be_v less_o segment_n of_o equal_a side_n of_o the_o octohedron_n and_o the_o angle_n hak_n and_o efg_o be_v equal_a namely_o be_v right_a angle_n by_o the_o 14._o of_o the_o thirteen_o the_o base_n hk_o and_o gf_o shall_v be_v equal_a by_o the_o 4._o of_o the_o first_o and_o by_o the_o same_o reason_n unto_o they_o may_v be_v prove_v equal_a the_o line_n xm_o nv_n d_n and_o qt_o and_o forasmuch_o as_o the_o line_n ac_fw-la bf_o and_o pl_z do_v cut_v the_o one_o the_o other_o into_o two_o equal_a part_n and_o perpendicular_o by_o construction_n the_o line_n hk_o and_o ge_z which_o
the_o pyramisâ—_n and_o they_o shall_v â—all_n perpendicular_o upon_o the_o base_n and_o shall_v also_o fall_v upon_o the_o centre_n of_o the_o circle_n which_o contain_v the_o base_n by_o the_o corollary_n of_o the_o 13._o of_o the_o thirteen_o let_v the_o cenâ—re_n of_o the_o triangle_n abc_n be_v the_o point_n g_o and_o let_v the_o centre_n of_o the_o triangle_n adc_o be_v the_o point_n hâ—_n and_o of_o the_o triangle_n adb_o let_v the_o point_n n_o be_v the_o centre_n and_o final_o let_v the_o point_n f_o be_v the_o centre_n of_o the_o other_o triangle_n dbc_n and_o let_v the_o right_a line_n fall_v upon_o those_o centre_n be_v deg_n beh_n cen_n &_o aef_n and_o by_o those_o centre_n g_o h_o n_o f_o let_v there_o be_v draw_v from_o the_o angle_n to_o the_o opposite_a side_n these_o right_a line_n agl_n dhk_n bnm_fw-la and_o dfl_fw-mi which_o shall_v fall_v perpendicular_o upon_o the_o side_n bc_o ca_n ad_fw-la and_o cb_o by_o the_o corollary_n of_o the_o 1â—_o of_o the_o thirteen_o and_o therefore_o they_o shall_v cuâ—_n they_o into_o two_o equal_a part_n in_o the_o point_n king_n l_o m_o by_o the_o 3._o of_o the_o three_o again_o let_v the_o line_n which_o be_v draw_v from_o the_o solid_a angle_n to_o the_o opposite_a base_n be_v divide_v into_o two_o equal_a part_n namely_o the_o line_n dg_o in_o the_o point_n t_o the_o line_n cn_fw-la in_o the_o point_n oh_o the_o line_n of_o in_o the_o point_n p_o and_o the_o line_n bh_o in_o â—he_n point_n r_o and_o draw_v the_o line_n ht_v ft_n ho_o and_o foyes_n now_o forasmuch_o as_o the_o line_n gk_o demonstration_n and_o gl_n which_o be_v draw_v from_o the_o centre_n of_o one_o and_o the_o self_n same_o triangle_n abc_n to_o the_o side_n be_v equal_a and_o the_o line_n dk_o and_o dl_o be_v equal_a for_o they_o be_v the_o perpendicular_n of_o equal_a &_o like_a triangle_n b._n and_o the_o line_n dg_o be_v common_a to_o they_o wherefore_o by_o the_o 8._o of_o the_o first_o the_o angle_n kdg_n &_o ldg_n be_v equal_a and_o forasmuch_o as_o the_o line_n hd_v &_o df_o be_v draw_v from_o the_o centre_n of_o equal_a circle_n which_o contain_v the_o equal_a triangle_n adc_o &_o dbc_n therefore_o they_o be_v equal_a &_o the_o line_n dt_n be_v common_a to_o they_o both_o and_o they_o contain_v equal_a angle_n as_o before_o have_v be_v prove_v wherefore_o the_o base_n ht_v and_o ft_v be_v equal_a by_o the_o 4._o of_o the_o first_o and_o by_o the_o same_o reason_n if_o we_o draw_v the_o line_n cf_o and_o change_z may_v we_o prove_v that_o the_o other_o line_n ho_o and_o foe_n be_v equal_a to_o the_o same_o line_n ht_v and_o ft_v and_o also_o the_o one_o to_o the_o other_o wherefore_o also_o after_o the_o same_o manner_n may_v be_v prove_v that_o the_o rest_n of_o the_o line_n which_o couple_n the_o centre_n of_o the_o triangle_n and_o the_o section_n of_o the_o perpendicular_n into_o two_o equal_a part_n as_o the_o line_n np_v graccus_n gp_n rn_n nt_v ph_n go_v and_o rf_n be_v equal_a and_o forasmuch_o as_o from_o every_o one_o of_o the_o centre_n of_o the_o base_n be_v draw_v three_o right_a line_n to_o the_o section_n into_o two_o equal_a part_n of_o the_o perpendiculers_n and_o there_o be_v four_o centre_n it_o follow_v that_o these_o equal_a right_a line_n so_o draw_v be_v twelve_o in_o number_n of_o which_o every_o three_o and_o three_o make_v a_o solid_a angle_n in_o the_o four_o centre_n of_o the_o base_n and_o in_o the_o four_o section_n into_o two_o equal_a part_n of_o the_o perpendicular_n wherefore_o that_o solid_a have_v 8._o angle_n contain_v under_o 12._o equal_a side_n which_o make_v six_o quadrangled_a figure_n namely_o hoff_z pgrn_n phog_n gofr_n frnt_v and_o tnph_n now_o let_v we_o prove_v that_o those_o quadrangled_a figure_n be_v rectangle_n forasmuch_o as_o upon_o dc_o the_o common_a base_a of_o the_o triangle_n adc_o and_o bdc_o fall_v the_o perpendicular_n as_o and_o b_n which_o be_v draw_v by_o the_o centre_n h_o and_o fletcher_n either_o of_o these_o line_n his_o and_o sf_n shall_v be_v the_o three_o part_n of_o either_o of_o these_o line_n as_o and_o sb_n for_o the_o line_n ah_o be_v duple_n to_o the_o line_n his_o and_o divide_v the_o base_a dc_o into_o two_o equal_a part_n by_o the_o corollary_n of_o the_o 12._o of_o the_o thirteen_o wherefore_o in_o the_o triangle_n abs_fw-la the_o side_n as_o and_o b_n be_v cut_v proportional_o in_o the_o point_v h_o and_o fletcher_n and_o therefore_o the_o line_n hf_o be_v a_o parallel_n to_o the_o side_n ab_fw-la by_o the_o 2._o of_o the_o six_o wherefore_o the_o triangle_n asb_n and_o hsf_n be_v equiangle_n by_o the_o 6._o of_o the_o six_o wherefore_o the_o base_a hf_o shall_v be_v the_o three_o part_n of_o the_o base_a ab_fw-la by_o the_o 4._o of_o the_o six_o we_o may_v also_o prove_v that_o the_o line_n to_o be_v the_o three_o part_n of_o the_o line_n dc_o for_o the_o line_n ec_o and_o ed_z which_o be_v draw_v from_o the_o centre_n of_o the_o sphere_n which_o contain_v the_o pyramid_n be_v equal_a and_o the_o line_n en_fw-fr which_o be_v draw_v from_o the_o centre_n to_o the_o base_a be_v the_o three_o part_n of_o the_o line_n ec_o so_o also_o be_v the_o line_n ge_z the_o three_o part_n of_o the_o line_n ed_z by_o the_o corollary_n of_o the_o 13._o of_o the_o thirteen_o for_o it_o be_v the_o six_o part_n of_o the_o diameter_n of_o the_o sphere_n which_o contain_v the_o pyramid_n and_o the_o line_n on_o be_v the_o half_a of_o the_o whole_a line_n nc_n wherefore_o the_o residue_n eo_fw-la be_v the_o three_o part_n of_o the_o line_n ecâ—_n and_o so_o also_o be_v the_o line_n et_fw-fr the_o three_o part_n of_o the_o line_n ed._n wherefore_o the_o line_n to_o in_o the_o triangle_n dec_n be_v a_o parallel_n to_o the_o line_n dc_o and_o be_v a_o three_o part_n of_o the_o same_o by_o the_o former_a 2._o and_o 4._o of_o the_o six_o as_o the_o line_n hf_o be_v prove_v the_o three_o part_n of_o the_o line_n ab_fw-la but_o ab_fw-la and_o dc_o be_v side_n of_o the_o pyramid_n be_v equal_a wherefore_o the_o line_n hf_o and_o to_o be_v the_o three_o part_n of_o equal_a line_n be_v equal_a by_o the_o 15._o of_o the_o five_o wherefore_o by_o the_o 8._o of_o the_o first_o the_o angle_n htf_n and_o tfo_n be_v equal_a and_o by_o the_o same_o reason_n the_o angle_n opposite_a unto_o they_o namely_o the_o angle_n foh_o and_o oht_fw-mi be_v equal_v the_o one_o to_o the_o other_o and_o also_o be_v equal_a to_o the_o say_a angle_n htâ—_n and_o tfo_n but_o these_o four_o angle_n be_v equal_a to_o 4._o right_a angle_n by_o the_o corollary_n of_o the_o 32._o of_o the_o first_o wherefore_o the_o angle_n of_o the_o quadrangle_n hoff_z be_v right_a angle_n and_o by_o the_o same_o reason_n may_v the_o angle_n of_o the_o other_o five_o quadrangled_a figure_n be_v prove_v right_a anglesâ—_n now_o rest_v to_o prove_v that_o the_o foresay_a quadrangle_v be_v each_o in_o one_o and_o the_o self_n same_o plain_a take_v the_o quadrangle_n hoff_z and_o forasmuch_o as_o in_o the_o triangle_n asb_n the_o line_n hf_o be_v prove_v a_o parallel_n to_o the_o line_n ab_fw-la therefore_o it_o cut_v the_o line_n sv_n and_o sb_n proportional_o in_o the_o point_n i_o and_o f._n by_o the_o 2._o of_o the_o six_o now_o then_o forasmuch_o as_o sf_n be_v prove_v the_o three_o part_n of_o the_o line_n sb_n the_o line_n si_fw-it shall_v also_o be_v the_o third_o part_n of_o the_o line_n su._n moreover_o forasmuch_o as_o the_o line_n we_o which_o couple_v the_o section_n into_o equal_a part_n of_o the_o opposite_a â—ides_n of_o the_o pyramid_n namely_o of_o the_o side_n ab_fw-la and_o dc_o be_v by_o the_o centre_n e_o divide_v into_o two_o equal_a part_n by_o the_o corollary_n of_o the_o second_o of_o this_o book_n for_o it_o be_v the_o diameter_n of_o the_o octohedron_n inscribe_v in_o the_o pyramid_n therefore_o the_o line_n si_fw-it be_v two_z three_o part_n of_o the_o half_a line_n se._n and_o by_o the_o same_o reason_n forasmuch_o as_o in_o the_o triangle_n dec_n the_o line_n to_o be_v prove_v to_o be_v a_o parallel_n to_o the_o side_n dc_o it_o shall_v in_o the_o self_n same_o triangle_n cut_v the_o line_n ce_fw-fr and_o see_fw-la proportional_o in_o the_o point_n oh_o and_o i_o by_o the_o same_o 2._o of_o the_o six_o but_o the_o line_n eo_fw-la be_v prove_v to_o be_v a_o three_o part_n of_o the_o line_n ec_o wherefore_o the_o line_n ei_o be_v also_o a_o three_o part_n of_o the_o line_n es._n wherefore_o the_o residue_n be_v shall_v be_v two_o three_o part_n of_o the_o whole_a line_n es._n wherefore_o the_o point_n i_o cut_v of_o the_o line_n to_o and_o hf._n wherefore_o the_o
power_n of_o the_o other_o side_n and_o if_o the_o half_a of_o the_o power_n remain_v be_v apply_v upon_o the_o whole_a base_a it_o shall_v make_v the_o breadth_n that_o section_n of_o the_o base_a which_o be_v couple_v to_o the_o first_o side_n for_o from_o the_o power_n of_o the_o base_a bg_o and_o of_o one_o of_o the_o side_n agnostus_n that_o be_v from_o the_o square_n bd_o and_o gl_n the_o power_n of_o the_o other_o side_n ab_fw-la namely_o the_o square_a bk_o or_o the_o parallelogram_n em_n be_v take_v away_o and_o of_o the_o residue_n namely_o of_o the_o square_n bd_o and_o of_o the_o parallelogram_n odd_a or_o doctor_n which_o by_o supposition_n be_v equal_a unto_o odd_a the_o half_a name_o of_o the_o whole_a fr_z which_o be_v pd_v for_o the_o line_n graccus_n and_o pb_n be_v equal_a to_o the_o line_n go_v and_o po_n be_v apply_v to_o the_o whole_a line_n bg_o or_o gd_o and_o make_v the_o bredthe_a the_o line_n pg_n the_o section_n of_o the_o base_a bg_o which_o section_n be_v couple_v to_o the_o first_o side_n ag._n and_o by_o the_o same_o reason_n in_o the_o other_o side_n if_o from_o the_o square_n bd_o and_o bk_o be_v take_v away_o the_o square_a gl_n there_o shall_v remain_v the_o rectangle_n parallelogram_n foyes_n for_o the_o parallalelogramme_n em_n be_v equal_a to_o the_o square_a bk_o and_o the_o parallelogram_n gm_n to_o the_o parallelogram_n odd_a wherefore_o fp_n the_o half_a of_o the_o residue_n foyes_n make_v the_o breadth_n bp_o which_o be_v couple_v to_o the_o first_o side_n take_v ab_fw-la a_o corollary_n 2._o if_o a_o perpendicular_a draw_v from_o a_o angle_n of_o a_o triangle_n do_v cut_v the_o base_a the_o section_n be_v to_o the_o other_o side_n in_o power_n proportional_a by_o a_o arithmetical_a proportion_n for_o it_o be_v prove_v that_o the_o excess_n of_o the_o power_n of_o the_o line_n agnostus_n and_o ab_fw-la be_v one_o and_o the_o same_o with_o the_o excess_n of_o the_o power_n of_o the_o line_n pg_n and_o pb_n if_o therefore_o the_o power_n do_v equal_o exceed_v the_o one_o the_o other_o they_o shall_v by_o a_o arithmetical_a proportion_n be_v proportional_a the_o end_n of_o the_o sixteen_o book_n of_o the_o element_n of_o geometry_n add_v 〈…〉_o a_o brief_a treatise_n add_v by_o flussas_n of_o mix_a and_o compose_v regular_a solid_n regular_a solid_n be_v say_v to_o be_v compose_v and_o mix_v when_o each_o of_o they_o be_v transform_v into_o other_o solid_n keep_v still_o the_o form_n number_n and_o inclination_n of_o the_o base_n which_o they_o before_o have_v one_o to_o the_o other_o some_o of_o which_o yet_o be_v transform_v into_o mix_a solid_n and_o other_o some_o into_o simple_a into_o mix_v as_o a_o dodecahedron_n and_o a_o icosahedron_n which_o be_v transform_v or_o alter_v if_o you_o divide_v their_o side_n into_o two_o equal_a part_n and_o take_v away_o the_o solid_a angle_n subtend_v of_o plain_a superficial_a figure_n make_v by_o the_o line_n couple_v those_o middle_a section_n for_o the_o solid_a remain_v after_o the_o take_v away_o of_o those_o solid_a angle_n be_v call_v a_o icosidodecahedron_n icosidodecahedron_n if_o you_o divide_v the_o side_n of_o a_o cube_fw-la and_o of_o a_o octohedron_n into_o two_o equal_a part_n and_o couple_v the_o section_n the_o solid_a angle_n subtend_v of_o the_o plain_a superficiece_n make_v by_o the_o couple_a line_n be_v take_v away_o there_o shall_v be_v leave_v a_o solid_a which_o be_v call_v a_o exoctohedron_n exoctohedron_n so_o that_o both_o of_o a_o dodecahedron_n and_o also_o of_o a_o icosahedron_n the_o solid_a which_o be_v make_v shall_v be_v call_v a_o icosidodecahedron_n and_o likewise_o the_o solid_a make_v of_o a_o cube_n &_o also_o of_o a_o octohedron_n shall_v be_v call_v a_o exoctohedron_n but_o the_o other_o solid_a namely_o a_o pyramid_n or_o tetrahedron_n be_v transform_v into_o a_o simple_a solid_a for_o if_o you_o divide_v into_o two_o equal_a part_n every_o one_o of_o the_o side_n of_o the_o pyramid_n triangle_n describe_v of_o the_o line_n which_o couple_v the_o section_n and_o subtend_v and_o take_v away_o solid_a angle_n of_o the_o pyramid_n be_v equal_a and_o like_a unto_o the_o equilater_n triangle_n leave_v in_o every_o one_o of_o the_o base_n of_o all_o which_o triangle_n be_v produce_v a_o octohedron_n namely_o a_o simple_a and_o not_o a_o compose_a solid_a for_o the_o octohedron_n have_v four_o base_n like_v in_o number_n fonne_z and_o mutual_a inclination_n with_o the_o base_n of_o the_o pyramid_n and_o have_v the_o other_o four_o base_n with_o like_a situation_n opposite_a and_o parallel_a to_o the_o former_a wherefore_o the_o application_n of_o the_o pyramid_n take_v twice_o make_v a_o simple_a octohedron_n as_o the_o other_o solid_n make_v a_o mix_a compound_n solid_a ¶_o first_n definition_n a_o exoctohedron_n be_v a_o solid_a figure_n contain_v of_o six_o equal_a square_n and_o eight_o equilater_n and_o equal_a triangle_n ¶_o second_v definition_n a_o icosidodecahedron_n be_v a_o solid_a figure_n contain_v under_o twelve_o equilater_n equal_a and_o equiangle_n pentagons_n and_o twenty_o equal_a and_o equilater_n triangle_n for_o the_o better_a understanding_n of_o the_o two_o former_a definition_n and_o also_o of_o the_o two_o proposition_n follow_v i_o have_v here_o set_v two_o figure_n who_o form_n if_o you_o first_o describe_v upon_o past_v paper_n or_o such_o like_a matter_n and_o then_o cut_v they_o and_o fold_v they_o according_o they_o will_v represent_v unto_o you_o the_o perfect_a form_n of_o a_o exoctohedron_n and_o of_o a_o icosidodecahedron_n ¶_o the_o first_o problem_n to_o describe_v a_o equilater_n and_o equiangle_n exoctohedron_n and_o to_o contain_v it_o in_o a_o sphere_n give_v and_o to_o prove_v that_o the_o diameter_n of_o the_o sphere_n be_v double_a to_o the_o side_n of_o the_o say_a exoctohedron_n and_o now_o forasmuch_o as_o the_o opposite_a side_n and_o diameter_n of_o the_o base_n of_o the_o cube_fw-la be_v parallel_n sphere_n the_o plain_n extend_v by_o the_o right_a line_n qt_o ur_fw-la shall_v be_v a_o parallelogram_n and_o for_o that_o also_o in_o that_o plain_n lie_v qr_o the_o diameter_n of_o the_o cube_fw-la and_o in_o the_o same_o plain_n also_o be_v the_o line_n mh_o which_o divide_v the_o say_a plain_n into_o two_o equal_a part_n and_o also_o couple_v the_o opposite_a angle_n of_o the_o exoctohedron_n this_o line_n mh_o therefore_o divide_v the_o diameter_n into_o two_o equal_a part_n by_o the_o corollary_n of_o the_o 34._o of_o the_o first_o and_o also_o divide_v itself_o in_o the_o same_o point_n which_o let_v be_v s_o into_o two_o equal_a part_n by_o the_o 4_o of_o the_o first_o and_o by_o the_o same_o reason_n may_v we_o prove_v that_o the_o rest_n of_o the_o line_n which_o couple_n the_o opposite_a angle_n of_o the_o exoctohedron_n do_v in_o saint_n the_o centre_n of_o the_o cube_fw-la divide_v thâ—_z one_o the_o other_o into_o two_o equal_a part_n for_o every_o one_o of_o the_o angle_n of_o the_o exoctohedron_n be_v set_v in_o every_o one_o of_o the_o base_n of_o the_o cube_fw-la wheâ—â—â—ore_o make_v the_o centre_n the_o point_n s_o and_o the_o space_n shall_v or_o sm_n describe_v a_o sphere_n and_o it_o shall_v touch_v every_o one_o of_o the_o angle_n eqâ—edistant_a from_o the_o point_n s_o and_o forasmuch_o as_o ab_fw-la the_o diameter_n of_o the_o sphere_n give_v be_v put_v equal_a to_o the_o diameter_n of_o the_o base_a of_o the_o cube_fw-la give_v namely_o to_o the_o line_n rt_n and_o the_o same_o line_n rt_n be_v equal_a to_o the_o line_n mh_o by_o the_o 33._o of_o the_o first_o which_o line_n mh_o couple_a the_o opposite_a angle_n of_o the_o exoctohedron_n be_v draw_v by_o the_o centre_n wherefore_o it_o be_v the_o diameter_n of_o the_o sphere_n give_v which_o contain_v the_o exoctohedron_n final_o forasmuch_o as_o in_o the_o triangle_n rft_v the_o line_n po_n do_v cut_v the_o side_n into_o two_o equal_a part_n exoctohedron_n it_o shall_v cut_v they_o proportional_o with_o the_o base_n namely_o as_o fr_z be_v to_o fp_n so_o shall_v rt_n be_v to_o po_n by_o the_o 2._o of_o the_o six_o but_o fr_z be_v double_a to_o fp_n by_o supposition_n wherefore_o rt_n or_o the_o diameter_n he_o be_v also_o double_a to_o the_o line_n po_n the_o side_n of_o the_o exoctohedron_n wherefore_o we_o have_v describe_v a_o equilater_n &_o equiangle_n exoctohedron_n and_o comprehend_v it_o in_o a_o sphere_n give_v and_o have_v prove_v that_o the_o diameter_n of_o the_o sphere_n be_v double_a to_o the_o side_n of_o the_o exoctohedron_n ¶_o the_o 2._o problem_n to_o describe_v a_o equilater_n &_o equiangle_n icosidodecahedron_n &_o to_o comprehend_v it_o in_o a_o sphere_n give_v and_o to_o prove_v that_o the_o diameter_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 double_a to_o the_o side_n
set_v before_o the_o thing_n require_v require_v in_o some_o proposition_n there_o be_v more_o thing_n give_v then_o one_o and_o more_o thing_n require_v then_o one_o in_o some_o there_o be_v nothing_o give_v at_o all_o moreover_o every_o problem_n &_o theorem_a be_v perfect_a and_o absolute_a aught_o to_o have_v all_o these_o part_n namely_o first_o the_o proposition_n proposition_n to_o be_v prove_v exposition_n then_o the_o exposition_n which_o be_v the_o explication_n of_o the_o thing_n give_v after_o that_o follow_v the_o determination_n determination_n which_o be_v the_o declaration_n of_o the_o thing_n require_v then_o be_v set_v the_o construction_n of_o such_o thing_n which_o be_v necessary_a either_o for_o the_o do_v of_o the_o proposition_n construction_n or_o for_o the_o demonstration_n afterwards_o follow_v the_o demonstration_n demonstration_n which_o be_v the_o reason_n and_o proof_n of_o the_o proposition_n and_o last_o of_o all_o be_v put_v the_o conclusion_n conclusion_n which_o be_v infer_v &_o prove_v by_o the_o demonstration_n and_o be_v ever_o the_o proposition_n but_o all_o those_o part_n be_v not_o of_o necessity_n require_v in_o every_o problem_n and_o theorem_a but_o the_o proposition_n demonstration_n and_o conclusion_n be_v necessary_a part_n &_o can_v never_o be_v absent_a the_o other_o part_n may_v sometime_o be_v away_o case_n further_o in_o diverse_a proposition_n there_o happen_v diverse_a case_n which_o be_v nothing_o else_o but_o variety_n of_o delineation_n and_o construction_n or_o change_v of_o position_n as_o when_o point_n line_n superâ—iciesses_n or_o body_n be_v change_v which_o thing_n happen_v in_o diverse_a proposition_n problem_n now_o then_o in_o this_o problem_n the_o thing_n give_v be_v the_o line_n give_v the_o thing_n require_v require_v to_o be_v search_v out_o be_v how_o upon_o that_o line_n to_o describe_v a_o equilater_n triangle_n the_o proposition_n of_o this_o problem_n be_v prâ—position_n upon_o a_o right_a line_n give_v not_o be_v infinite_a to_o describe_v a_o equilater_n triangle_n exposition_n the_o exposition_n be_v suppose_v that_o the_o right_a line_n give_v be_v ab_fw-la and_o this_o declare_v only_o the_o thing_n give_v the_o determination_n be_v determination_n it_o be_v require_v upon_o the_o line_n ab_fw-la to_o describe_v a_o equilater_n triangle_n for_o thereby_o as_o you_o see_v be_v declare_v only_o the_o thing_n require_v the_o construction_n begin_v at_o these_o woâ—ds_n construction_n now_o therefore_o make_v the_o centre_n the_o point_n a_o &_o the_o space_n ab_fw-la describe_v by_o the_o three_o peâ—ition_n a_o circle_n etc_o etc_o and_o continue_v until_o you_o come_v to_o these_o word_n and_o forasmuch_o at_o the_o point_n a_o etc_n etc_n for_o thetheâ—to_o be_v describe_v circle_n and_o line_n necessary_a both_o for_o the_o do_v of_o the_o proposition_n and_o also_o for_o the_o demonstration_n thereof_o which_o demonstration_n begin_v at_o these_o word_n demonstration_n and_o forasmuch_o as_o the_o point_n a_o be_v the_o centre_n of_o the_o circle_n cbd_v etc_o etc_o and_o so_o proceed_v till_o you_o come_v to_o these_o word_n wherefore_o upon_o the_o line_n ab_fw-la be_v describe_v a_o equilater_n triangle_n abc_n for_o until_o you_o come_v thither_o be_v by_o ground_n before_o set_v and_o construction_n have_v prove_v and_o make_v evident_a that_o the_o triangle_n make_v be_v equilater_n and_o then_o in_o these_o word_n conclusion_n wherefore_o upon_o the_o line_n ab_fw-la be_v describe_v a_o equilater_n triangle_n abc_n be_v put_v the_o first_o conclusion_n for_o there_o be_v common_o in_o every_o proposition_n two_o conclusion_n the_o one_o particular_a the_o other_o universal_a and_o from_o the_o first_o you_o go_v to_o the_o last_o and_o this_o be_v the_o first_o and_o particular_a conclusion_n â—or_a that_o it_o conclude_v that_o upon_o the_o line_n ab_fw-la be_v describe_v a_o equilater_n triangle_n which_o be_v according_a to_o the_o exposition_n after_o it_o câ—nclusâ—â—n_n follow_v the_o last_o and_o universal_a conclusion_n wherefore_o upon_o a_o right_a line_n give_v not_o be_v infinite_a be_v describe_v a_o equilater_n triangle_n for_o whether_o the_o line_n give_v be_v great_a or_o less_o than_o this_o line_n the_o â—ame_n construction_n and_o demonstration_n prove_v the_o same_o conclusion_n last_o of_o all_o be_v add_v this_o clause_n problem_n which_o be_v the_o thing_n which_o be_v require_v to_o be_v do_v whereby_o as_o we_o have_v before_o note_v be_v declare_v that_o this_o proposition_n be_v a_o problem_n and_o not_o a_o theorem_a as_o for_o variety_n of_o case_n in_o this_o proposition_n there_o be_v none_o proposition_n for_o that_o the_o line_n give_v can_v have_v no_o diversity_n of_o position_n as_o you_o have_v in_o this_o problem_n see_v plain_o set_v forth_o the_o thing_n give_v and_o the_o thing_n require_v moreover_o the_o proposition_n exposition_n determination_n construction_n demonstration_n and_o conclusion_n which_o be_v general_a also_o to_o many_o other_o both_o problem_n and_o theorem_n so_o may_v you_o by_o the_o example_n thereof_o distinct_a they_o and_o search_v they_o out_o in_o other_o problemesâ—_n â—_z and_o also_o theorem_n demonstration_n this_o also_o be_v to_o be_v note_v that_o there_o be_v three_o kind_n of_o demonstration_n the_o one_o be_v call_v demonstratio_fw-la a_o priori_fw-la or_o composition_n the_o other_o be_v call_v demonstratiâ—_n a_o posteriori_fw-la or_o resolution_n and_o the_o three_o be_v a_o demonstration_n lead_v to_o a_o impossibility_n a_o demonstration_n a_o priori_fw-la composition_n or_o composition_n be_v when_o in_o reason_v from_o the_o principle_n and_o first_o ground_n we_o pass_v descend_v continual_o till_o after_o many_o reason_n make_v we_o come_v at_o the_o length_n to_o conclude_v that_o which_o we_o first_o chief_o intend_v and_o this_o kind_n of_o demonstration_n use_v euclid_n in_o his_o bookeâ—_n for_o the_o most_o part_n a_o demonstration_n a_o posteriori_fw-la resolution_n or_o resolution_n be_v when_o contrariwise_o in_o reason_v we_o pass_v from_o the_o last_o conclusion_n make_v by_o the_o premise_n and_o by_o the_o premise_n of_o the_o premise_n continual_o ascend_v till_o we_o come_v to_o the_o first_o principle_n and_o ground_n which_o be_v indemonstrable_a and_o for_o their_o simplicity_n can_v suffer_v no_o far_o resolution_n a_o demonstration_n lead_v to_o a_o impossibility_n be_v that_o argument_n who_o conclusion_n be_v impossible_a impossibility_n that_o be_v when_o it_o conclude_v direct_o against_o any_o principle_n or_o against_o any_o proposition_n before_o prove_v by_o principle_n or_o proposition_n beâ—ore_o prove_v premise_n in_o a_o argument_n be_v proposition_n go_v before_o the_o conclusion_n by_o which_o the_o conclusion_n be_v prove_v be_v composition_n pass_v from_o the_o cause_n to_o the_o effect_n or_o from_o thing_n simple_a to_o thing_n more_o compound_v resolution_n contrariwise_o pass_v from_o thing_n compound_v to_o thing_n more_o simple_a or_o from_o the_o effect_n to_o â—he_n cause_n composition_n or_o the_o first_o kind_n of_o demonstration_n which_o pass_v from_o the_o principle_n may_v easy_o be_v see_v in_o this_o first_o proposition_n of_o euclid_n proposition_n the_o demonstration_n whereof_o begin_v thus_o and_o forasmuch_o as_o the_o point_n a_o be_v the_o centre_n of_o the_o circle_n cbd_v therefore_o the_o line_n ac_fw-la be_v equal_a to_o the_o line_n ab_fw-la this_o reason_n you_o see_v take_v his_o beginning_n of_o a_o principle_n namely_o of_o the_o definition_n of_o a_o circle_n and_o this_o be_v the_o first_o reason_n reason_n again_o forasmuch_o as_o b_o be_v the_o centre_n of_o the_o circle_n caesar_n therefore_o the_o line_n bc_o be_v equal_a to_o the_o line_n basilius_n which_o be_v the_o second_o reason_n reason_n and_o it_o be_v before_o prove_v that_o the_o line_n ac_fw-la be_v equal_a to_o the_o line_n ab_fw-la wherefore_o either_o of_o these_o line_n ca_n &_o cb_o be_v equal_a to_o the_o line_n ab_fw-la and_o this_o be_v the_o three_o reason_n reason_n but_o thing_n which_o be_v equal_a to_o one_o &_o the_o self_n same_o thing_n be_v also_o equal_a the_o one_o to_o the_o other_o wherefore_o the_o line_n ca_n be_v equal_a to_o the_o line_n cb._n and_o this_o be_v the_o four_o argument_n reason_n wherefore_o these_o three_o line_n ca_n ab_fw-la and_o bc_o be_v equal_a the_o one_o to_o the_o other_o which_o be_v the_o conclusion_n conclusion_n and_o the_o thing_n to_o be_v prove_v you_o may_v also_o in_o the_o same_o first_o proposition_n easy_o take_v a_o example_n of_o resolution_n proposition_n use_v a_o contrary_a order_n pass_a backward_o from_o the_o last_o conclusion_n of_o the_o former_a demonstration_n till_o you_o come_v to_o the_o first_o principle_n or_o ground_n whereon_o it_o begin_v for_o the_o last_o argument_n or_o reason_n in_o composition_n be_v the_o first_o in_o resolution_n &_o the_o first_o in_o composition_n be_v the_o last_o in_o resolution_n thus_o therefore_o must_v you_o proceed_v the_o triangle_n abc_n be_v contain_v of_o three_o equal_a right_a line_n
namely_o ab_fw-la ac_fw-la and_o bc_o and_o therefore_o it_o be_v a_o equilater_n triangle_n by_o the_o definition_n of_o a_o equilater_n triangle_n and_o this_o be_v the_o first_o reason_n reason_n that_o the_o three_o line_n be_v equal_a be_v thus_o prove_v the_o line_n ac_fw-la and_o cb_o be_v equal_a to_o the_o line_n ab_fw-la wherefore_o they_o be_v equal_v the_o one_o to_o the_o other_o and_o this_o be_v the_o second_o reason_n reason_n that_o the_o line_n abâ—_n and_o bc_o be_v equal_a be_v thus_o prove_v the_o line_n ab_fw-la and_o ac_fw-la be_v draw_v from_o the_o centre_n of_o the_o circle_n ace_n to_o the_o circumference_n of_o the_o same_o wherefore_o they_o be_v equal_a by_o the_o definition_n of_o a_o circle_n and_o this_o be_v the_o three_o reason_n reason_n likewise_o that_o the_o line_n ac_fw-la and_o ab_fw-la resolution_n be_v equal_a be_v prove_v by_o the_o same_o reason_n for_o the_o line_n ac_fw-la and_o ab_fw-la be_v draw_v from_o the_o centre_n of_o the_o circle_n bcd_v wherefore_o they_o be_v equal_a by_o the_o same_o definition_n of_o a_o circle_n this_o be_v the_o four_o reason_n or_o syllogism_n and_o thus_o be_v end_v the_o whole_a resolution_n for_o that_o you_o be_v come_v to_o a_o principle_n which_o be_v indemonstrable_a &_o can_v not_o be_v resolve_v of_o a_o â—emonstration_n lead_v to_o a_o impossibility_n or_o to_o a_o absurdity_n you_o may_v have_v a_o example_n in_o the_o forth_o proposition_n of_o this_o book_n a_o addition_n of_o campanus_n but_o now_o if_o upon_o the_o same_o line_n give_v namely_o ab_fw-la you_o will_v describe_v the_o other_o two_o kind_n of_o triangle_n namely_o a_o isosceles_a or_o a_o triangle_n of_o two_o equal_a side_n &_o a_o scalenon_fw-mi or_o a_o triangle_n of_o three_o unequal_a side_n first_o for_o the_o describe_v of_o a_o isosceles_a triangle_n produce_v the_o line_n ab_fw-la on_o either_o side_n triangle_n until_o it_o concur_v with_o the_o circumference_n of_o both_o the_o circle_n in_o the_o point_n d_o and_o f_o and_o make_v the_o centre_n the_o point_n a_o describe_v a_o circle_n hfg_n according_a to_o the_o quantity_n of_o the_o line_n af._n likewise_o make_v the_o centre_n the_o point_n b_o describe_v a_o circle_n hdg_n according_a to_o the_o quantity_n of_o the_o line_n bd._n now_o they_o these_o circle_n shall_v cut_v the_o one_o the_o other_o in_o two_o point_n which_o let_v be_v h_o and_o g_z and_o lâ—t_o the_o â—ndâ—s_n of_o the_o line_n give_v be_v join_v with_o one_o of_o the_o say_a section_n by_o two_o right_a line_n which_o let_v be_v agnostus_n and_o bg_o and_o forasmuch_o as_o these_o two_o line_n ab_fw-la and_o ad_fw-la be_v draw_v from_o the_o centre_n of_o the_o circle_n cde_o unto_o the_o circumference_n thereof_o therefore_o be_v they_o equal_a likewise_o the_o line_n basilius_n and_o bf_o for_o that_o they_o be_v draw_v from_o the_o centre_n of_o the_o circle_n eacf_n to_o the_o circumference_n thereof_o aâ—e_v equal_a and_o forasmuch_o as_o either_o of_o the_o line_n ad_fw-la and_o bf_o be_v equal_a to_o the_o line_n ab_fw-la therefore_o they_o be_v equal_v the_o one_o to_o the_o other_o wherefore_o put_v the_o line_n ab_fw-la common_a to_o they_o both_o the_o whole_a line_n bd_o shall_v be_v equal_a to_o the_o whole_a line_n af._n but_o bd_o be_v equal_a to_o bg_o for_o they_o be_v both_o draw_v from_o the_o centre_n of_o the_o circle_n hdg_n to_o the_o circumference_n thereof_o and_o likewise_o by_o the_o same_o reason_n the_o line_n of_o be_v equal_a to_o the_o line_n ag._n wherefore_o by_o the_o common_a sentence_n the_o line_n agnostus_n and_o bg_o be_v equal_a the_o one_o to_o the_o other_o and_o either_o of_o they_o be_v great_a than_o the_o line_n ab_fw-la for_o that_o either_o of_o the_o two_o line_n bd_o and_o of_o be_v great_a than_o the_o line_n ab_fw-la wherefore_o upon_o the_o line_n give_v be_v describe_v a_o isosceles_a or_o triangle_n of_o two_o equal_a side_n scalenum_n you_o may_v also_o describe_v upon_o the_o self_n same_o line_n a_o scaleâ—on_n or_o triangle_n of_o three_o unequal_a side_n if_o by_o two_o right_a line_n you_o join_v both_o the_o end_n of_o the_o line_n gruen_n to_o some_o one_o point_n that_o be_v in_o the_o circumference_n of_o one_o of_o the_o two_o great_a circle_n so_o that_o that_o point_n be_v not_o in_o one_o of_o the_o two_o section_n and_o that_o the_o line_n df_o do_v not_o concur_v with_o it_o when_o it_o be_v on_o either_o side_n produce_v continual_o and_o direct_o for_o let_v the_o point_n king_n be_v take_v in_o the_o circumference_n of_o the_o circle_n hdg_n and_o let_v it_o not_o be_v in_o any_o of_o the_o section_n neither_o let_v the_o line_n df_o concur_v with_o it_o when_o it_o be_v produce_v continual_o and_o direct_o unto_o the_o circumference_n thereof_o and_o draw_v these_o line_n ak_o and_o bk_o and_o the_o line_n ak_o shall_v cut_v the_o circumference_n of_o the_o circle_n hfg_n let_v it_o cut_v it_o in_o the_o point_n l_o now_o then_o by_o the_o common_a sentence_n the_o line_n bk_o shall_v be_v equal_v to_o the_o line_n all_o for_o by_o the_o definition_n of_o a_o circle_n the_o line_n bk_o be_v equal_a to_o the_o line_n bg_o and_o the_o line_n all_o be_v equal_a to_o the_o line_n agnostus_n which_o be_v equal_a to_o the_o line_n bg_o wherefore_o the_o line_n ak_o be_v great_a than_o the_o line_n bk_o and_o by_o the_o same_o reason_n may_v it_o be_v prove_v that_o the_o line_n bk_o be_v great_a than_o the_o line_n ab_fw-la wherefore_o the_o triangle_n abk_n consist_v of_o three_o unequal_a side_n and_o so_o have_v you_o upon_o the_o line_n give_v describe_v all_o the_o kind_n of_o triangle_n this_o be_v to_o be_v note_v that_o if_o a_o man_n will_v mechanical_o and_o ready_o not_o regard_v demonstration_n upon_o a_o line_n give_v describe_v a_o triangle_n of_o three_o equal_a side_n mechanical_o he_o need_v not_o to_o describe_v the_o whole_a foresay_a circle_n but_o only_o a_o little_a part_n of_o each_o namely_o where_o they_o cut_v the_o one_o the_o other_o and_o so_o from_o the_o point_n of_o the_o section_n to_o draw_v the_o line_n to_o the_o end_n of_o the_o line_n gevenâ—_n as_o in_o this_o figure_n here_o put_v and_o likewise_o if_o upon_o the_o say_a line_n he_o will_v describe_v a_o triangle_n of_o two_o equal_a side_n ready_o let_v he_o extend_v the_o compass_n according_a to_o the_o quantity_n that_o he_o will_v have_v the_o side_n to_o be_v whether_o long_o than_o the_o line_n give_v or_o short_a and_o so_o draw_v only_o a_o little_a part_n of_o each_o circle_n where_o they_o cut_v the_o one_o the_o other_o &_o from_o the_o point_n of_o the_o section_n draw_v the_o line_n to_o the_o end_n of_o the_o line_n give_v as_o in_o the_o figure_n here_o put_v note_v that_o in_o this_o the_o two_o side_n must_v be_v such_o that_o be_v join_v together_o they_o be_v long_o than_o the_o line_n give_v and_o so_o also_o if_o upon_o the_o say_a right_a line_n he_o will_v describe_v a_o triangle_n of_o three_o unequal_a side_n ready_o let_v he_o extend_v the_o compass_n first_o according_a to_o the_o quantity_n that_o he_o will_v have_v one_o of_o the_o unequal_a side_n to_o be_v and_o so_o draw_v a_o little_a part_n of_o the_o circle_n &_o then_o extend_v it_o according_a to_o the_o quantity_n that_o he_o will_v have_v the_o other_o unequal_a side_n to_o be_v and_o draw_v likewise_o a_o little_a part_n of_o the_o circle_n and_o that_o do_v from_o the_o point_n of_o the_o section_n draâ—_n the_o line_n to_o the_o end_n of_o the_o line_n give_v as_o in_o the_o figure_n here_o put_v note_v that_o in_o this_o the_o two_o side_n must_v be_v such_o that_o the_o circle_n describe_v according_a to_o their_o quantity_n may_v cut_v the_o one_o the_o other_o the_o second_o problem_n the_o second_o proposition_n fron_n a_o point_n give_v to_o draw_v a_o right_a line_n equal_a to_o a_o rightline_v give_v svppose_v that_o the_o point_n give_v be_v a_o &_o let_v the_o right_a line_n give_v be_v bc._n it_o be_v require_v from_o the_o point_n a_o to_o draw_v a_o right_a line_n equal_a to_o the_o line_n bc._n draw_v by_o the_o first_o petition_n from_o the_o point_n a_o to_o the_o point_n b_o a_o right_a line_n ab_fw-la construâ—tiâ—â—_n and_o upon_o the_o line_n ab_fw-la describe_v by_o the_o first_o proposition_n a_o equilater_n triangle_n and_o let_v the_o same_o be_v dab_n and_o extend_v by_o the_o second_o petition_n the_o right_a line_n dam_n &_o db_o to_o the_o point_n e_o and_o f_o &_o by_o the_o three_o petition_n make_v the_o centre_n b_o and_o the_o space_n bc_o describe_v a_o circle_n cgh_n &_o again_o by_o the_o same_o make_v the_o centre_n d_o and_o the_o space_n dg_o describe_v a_o circle_n gkl_n
the_o angle_n bac_n be_v not_o equal_a to_o the_o angle_n edf_o neither_o also_o be_v the_o angle_n bac_n less_o than_o the_o angle_n edf_o for_o than_o shall_v the_o base_a bc_o be_v less_o they_o the_o base_a of_o by_o the_o former_a proposition_n but_o by_o supposition_n it_o be_v not_o wherefore_o y_fw-fr angle_v bac_n be_v not_o less_o they_o the_o angle_n edf_o and_o it_o be_v already_o prove_v that_o it_o be_v not_o equal_a unto_o it_o wherefore_o the_o aâ—gle_v bac_n be_v great_a than_o the_o angle_n edf_o if_o therefore_o two_o triangle_n have_v two_o side_n of_o the_o one_o equal_a to_o two_o side_n of_o the_o other_o each_o to_o his_o correspondent_a side_n &_o if_o the_o base_a of_o the_o one_o be_v great_a than_o the_o base_a of_o the_o other_o the_o angle_n also_o of_o the_o same_o contain_v under_o the_o equal_a right_a line_n shall_v be_v great_a they_o the_o angle_n of_o the_o other_o which_o be_v require_v to_o be_v prove_v this_o proposition_n be_v plain_a opposite_a to_o the_o eight_o &_o be_v the_o converse_n of_o the_o four_o and_o twenty_o which_o go_v before_o demonstrate_v and_o it_o be_v prove_v as_o common_o all_o converse_n be_v by_o a_o reason_n lead_v to_o a_o absurdity_n but_o it_o may_v after_o menelaus_n alexandrinus_n be_v demonstrate_v direct_o alexandrinus_n after_o this_o manner_n suppose_v that_o there_o be_v two_o triangle_n abc_n &_o def_n have_v the_o two_o side_n ab_fw-la and_o ac_fw-la equal_a to_o the_o two_o side_n de_fw-fr and_o df_o the_o one_o to_o the_o other_o and_o leâ—_z the_o base_a bc_o be_v great_a than_o the_o base_a ef._n then_o â—_o say_v that_o the_o angle_n at_o the_o point_n a_o be_v great_a they_o the_o angle_n at_o the_o point_n d._n for_o from_o the_o base_a bc_o 〈◊〉_d of_o by_o the_o third_o a_o line_n eqnall_a to_o the_o base_a of_o and_o let_v the_o same_o be_v bg_o and_o upon_o the_o line_n gb_o and_o to_o the_o point_n b_o put_v by_o the_o 23_o proposition_n a_o angle_n equal_a to_o the_o angle_n def_n which_o let_v be_v gbh_n and_o let_v the_o line_n â—h_a be_v equal_a to_o the_o line_n de._n and_o draw_v a_o line_n from_o h_o to_o g_z and_o produce_v it_o beyond_o the_o point_n g_o which_o be_v produce_v shall_v fall_v either_o upon_o the_o angle_n a_o demonstration_n or_o upon_o the_o line_n ab_fw-la or_o upon_o the_o line_n ac_fw-la first_o let_v it_o fall_v upon_o the_o angle_n a._n case_n and_o forasmuch_o as_o these_o two_o line_n bg_o and_o bh_o be_v equal_a to_o these_o two_o line_n of_o and_o ed_z the_o one_o to_o the_o other_o and_o they_o contain_v equal_a angle_n by_o construction_n namely_o the_o angle_n gbh_n and_o def_n therefore_o by_o the_o 4._o proposition_n the_o base_a gh_o be_v equal_a to_o the_o df_o and_o the_o angle_n bhg_n to_o the_o angle_n edf_o again_o forasmuch_o as_o the_o line_n bh_o be_v equal_a to_o the_o line_n basilius_n for_o the_o line_n ab_fw-la be_v suppose_v to_o be_v equal_a to_o the_o line_n de_fw-fr unto_o which_o line_n the_o line_n bh_o be_v put_v equal_a therefore_o by_o the_o 5._o proposition_n the_o angle_n bha_n be_v equal_a to_o the_o angle_n bah_o wherefore_o also_o the_o angle_n edf_o be_v equal_a to_o the_o angle_n bah_o but_o the_o angle_n bac_n be_v great_a than_o the_o angle_n bah_o wherefore_o also_o the_o angle_n bac_n be_v great_a than_o the_o angle_n edf_o case_n hero_n mechanicus_fw-la also_o demonstrate_v it_o a_o other_o way_n and_o that_o by_o a_o direct_a demonstration_n the_o 17._o theorem_a the_o 26._o proposition_n if_o two_o triangle_n have_v two_o angle_n of_o the_o one_o equal_a to_o two_o angle_n of_o the_o other_o each_o to_o his_o correspondent_a angle_n and_o have_v also_o one_o side_n of_o the_o one_o equal_a to_o one_o side_n of_o the_o other_o either_o that_o side_n which_o lie_v between_o the_o equal_a angle_n or_o that_o which_o be_v subtend_v under_o one_o of_o the_o equal_a angle_n the_o other_o side_n also_o of_o the_o one_o shall_v be_v equal_a to_o the_o other_o side_n of_o the_o other_o each_o to_o his_o correspondent_a side_n and_o the_o other_o angle_n of_o the_o one_o shall_v be_v equal_a to_o the_o other_o angle_n of_o the_o other_o svppose_v that_o there_o be_v two_o triangle_n ab_fw-la c_z and_o def_n have_v two_o angle_n of_o the_o one_o that_o be_v the_o angle_n abc_n and_o bca_o equal_a to_o two_o angle_n of_o the_o other_o that_o be_v to_o the_o angle_n def_n and_o efd_n each_o to_o his_o correspondent_a angle_n that_o be_v the_o angle_n abc_n to_o the_o angle_n def_n and_o the_o angle_n bca_o to_o the_o angle_n efd_n and_o one_o side_n of_o the_o one_o equal_a to_o one_o side_n of_o the_o outstretch_o first_o that_o side_n which_o lie_v between_o the_o equal_a angle_n that_o be_v the_o side_n bc_o to_o the_o ef._n then_o i_o say_v that_o the_o other_o side_n also_o of_o the_o one_o shall_v be_v equal_a to_o the_o other_o side_n of_o the_o other_o each_o to_o his_o correspondent_a side_n that_o be_v that_o side_n ab_fw-la to_o the_o side_n de_fw-fr and_o the_o side_n ac_fw-la to_o the_o side_n df_o and_o the_o other_o angle_n of_o the_o one_o to_o the_o other_o angle_n of_o the_o other_o that_o be_v the_o angle_n bac_n to_o the_o angle_n edf_o for_o if_o the_o side_n ab_fw-la be_v not_o equal_a to_o the_o side_n de_fw-fr the_o one_o of_o they_o be_v great_a let_v the_o side_n ab_fw-la be_v great_a and_o by_o the_o 3._o proposition_n unto_o the_o line_n de_fw-fr put_v a_o equal_a line_n gb_o and_o draw_v a_o right_a line_n from_o the_o point_n g_o to_o the_o point_n c._n now_o forasmuch_o as_o the_o line_n gb_o absurditis_fw-la be_v equal_a to_o the_o line_n de_fw-fr and_o the_o line_n bc_o to_o the_o line_n of_o therefore_o these_o two_o line_n gb_o and_o bc_o be_v equal_a to_o these_o two_o line_n de_fw-fr and_o of_o the_o one_o to_o the_o other_o and_o the_o angle_n gbc_n be_v by_o supposition_n equal_a to_o the_o angle_n def_n wherefore_o by_o the_o 4._o proposytion_n the_o base_a gc_o be_v equal_a to_o the_o base_a df_o and_o the_o triangle_n gcb_n be_v equal_a to_o the_o triangle_n def_n and_o the_o angle_n remain_v be_v equal_a to_o the_o angle_n remain_v under_o which_o be_v subtend_v equal_a side_n wherefore_o the_o angle_n gcb_n be_v equal_a to_o the_o angle_n dfe_n but_o the_o angle_n dfe_n be_v suppose_v to_o be_v equal_a to_o the_o angle_n bca_o wherefore_o by_o the_o first_o common_a sentence_n the_o angle_n bcg_n be_v equal_a to_o the_o angle_n bca_o the_o less_o angle_n to_o the_o great_a which_o be_v impossible_a wherefore_o the_o line_n ab_fw-la be_v not_o unequal_a to_o the_o line_n de._n wherefore_o it_o be_v equal_a and_o the_o the_o line_n bc_o be_v equal_a to_o the_o line_n of_o now_o therefore_o there_o be_v two_o side_n ab_fw-la and_o bc_o equal_a to_o two_o side_n de_fw-fr and_o of_o the_o one_o to_o the_o other_o and_o the_o angle_n abc_n be_v equal_a to_o the_o angle_n def_n wherefore_o by_o the_o 4._o proposition_n the_o base_a ac_fw-la be_v equal_a to_o the_o base_a df_o and_o the_o angle_n remain_v bac_n be_v equal_a to_o the_o angle_n remain_v edf_o again_o suppose_v that_o the_o side_n subtend_v the_o equal_a angle_n be_v equal_a the_o one_o to_o the_o other_o let_v the_o side_n i_o say_v ab_fw-la be_v equal_a to_o the_o side_n de._n then_o again_o i_o say_v that_o the_o other_o side_n of_o the_o one_o be_v equal_a to_o the_o other_o side_n of_o the_o other_o each_o to_o his_o correspondent_a side_n that_o be_v the_o side_n ac_fw-la to_o the_o side_n df_o and_o the_o side_n bc_o to_o the_o side_n of_o and_o moreover_o the_o angle_n remain_v namely_o bac_n be_v equal_a to_o the_o angle_n remain_v that_o be_v to_o the_o angle_n edf_o for_o if_o the_o side_n bc_o be_v not_o equal_a to_o the_o side_n of_o the_o one_o of_o they_o be_v great_a let_v the_o side_n bc_o if_o it_o be_v possible_a be_v great_a and_o by_o the_o three_o proposytion_n unto_o the_o line_n of_o put_v a_o equal_a line_n bh_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 h._n and_o forasmuch_o as_o the_o line_n bh_o be_v equal_a to_o the_o line_n of_o and_o the_o line_n ab_fw-la to_o the_o line_n de_fw-fr therefore_o these_o two_o side_n ab_fw-la and_o bh_o be_v equal_a to_o these_o two_o side_n de_fw-fr and_o of_o the_o one_o to_o the_o other_o and_o they_o contain_v equal_a angle_n wherefore_o by_o the_o 4._o proposition_n the_o base_a ah_o be_v equal_a to_o the_o base_a df_o and_o the_o triangle_n abh_n be_v equal_a to_o the_o triangle_n def_n and_o the_o angle_n remain_v be_v equal_a to_o the_o angleâ—_n remain_v under_o which_o be_v subtend_v equal_a
side_n wherefore_o the_o angle_n bha_n be_v equal_a to_o the_o angle_n efd_n but_o the_o angle_n efd_n be_v equal_a to_o the_o angle_n bca_o wherefore_o the_o angle_n bha_n be_v equal_a to_o the_o angle_n bca_o wherefore_o the_o outward_a angle_n of_o the_o triangle_n ahc_n namely_o the_o angle_n bha_n be_v equal_a to_o the_o inward_a and_o opposite_a angle_n namely_o to_o the_o angle_n hca_n which_o by_o the_o 16_o proposition_n be_v impossible_a wherefore_o the_o side_n of_o be_v not_o unequal_a to_o the_o side_n bc_o wherefore_o it_o be_v equal_a and_o the_o side_n ab_fw-la be_v equal_a to_o the_o side_n de_fw-fr wherefore_o these_o two_o side_n ab_fw-la and_o bc_o be_v equal_a to_o these_o two_o side_n de_fw-fr and_o of_o the_o one_o to_o the_o other_o and_o they_o contain_v equal_a angle_n wherefore_o by_o the_o 4._o proposition_n the_o base_a ac_fw-la be_v equal_a to_o the_o base_a df_o and_o the_o triangle_n abc_n be_v equal_a to_o the_o triangle_n def_n and_o the_o angle_n remain_v namely_o the_o angle_n bac_n be_v equal_a to_o the_o angle_n remain_v that_o be_v to_o the_o angle_n edf_o if_o therefore_o two_o triangle_n have_v two_o angle_n of_o the_o one_o equal_a to_o two_o angle_n of_o the_o other_o each_o to_o his_o correspondent_a angle_n and_o have_v also_o one_o side_n of_o the_o one_o equal_a to_o oâ—e_v side_n of_o the_o other_o either_o that_o side_n which_o lie_v between_o the_o equal_a angle_n or_o that_o which_o be_v subtend_v under_o one_o of_o the_o equal_a angle_n the_o other_o side_n also_o of_o the_o one_o shall_v be_v equal_a to_o the_o other_o side_n of_o the_o other_o each_o to_o his_o correspondent_a side_n and_o the_o other_o angle_n of_o the_o one_o shall_v be_v equal_a to_o the_o other_o angle_n of_o the_o other_o which_o be_v require_v to_o be_v prove_v whereas_o in_o this_o proposition_n it_o be_v say_v that_o triangle_n be_v equal_a which_o have_v two_o angle_n of_o the_o one_o equal_a to_o two_o angle_n of_o the_o other_o the_o one_o to_o the_o other_o have_v also_o one_o side_n of_o the_o one_o equal_a to_o one_o side_n of_o the_o other_o either_o that_o side_n which_o lie_v between_o the_o equal_a angle_n or_o that_o side_n which_o subtend_v one_o of_o the_o equal_a angle_n this_o be_v to_o be_v note_v that_o without_o that_o caution_n touch_v the_o equal_a side_n the_o proposition_n shall_v not_o always_o be_v true_a as_o for_o example_n the_o reason_n whereof_o be_v for_o that_o the_o equal_a side_n in_o one_o triangle_n subtend_v one_o of_o the_o equal_a angle_n and_o in_o the_o other_o lie_v between_o the_o equal_a angle_n so_o that_o you_o see_v that_o it_o be_v of_o necessity_n that_o the_o equal_a side_n do_v in_o both_o triangle_n either_o subtend_v one_o of_o the_o equal_a angle_n or_o lie_v between_o the_o equal_a angle_n of_o this_o proposition_n be_v thales_n milesius_n the_o inventor_n proposition_n as_o witness_v eudemus_n in_o his_o book_n of_o geometrical_a enarration_n the_o 18._o theorem_a the_o 27._o proposition_n if_o a_o right_a line_n fall_v upon_o two_o right_a line_n do_v make_v the_o alternate_a angle_n equal_v the_o one_o to_o the_o other_o those_o two_o right_a line_n be_v parallel_n the_o one_o to_o the_o other_o svppose_v that_o the_o right_a line_n of_o fall_v upon_o these_o two_o right_a line_n ab_fw-la and_o cd_o do_v make_v the_o alternate_a angle_n namely_o the_o angle_n aef_n &_o efd_n equal_v the_o one_o to_o the_o other_o then_o i_o say_v that_o ab_fw-la be_v a_o parallel_a line_n to_o cd_o for_o if_o not_o than_o these_o line_n produce_v shall_v mete_v together_o either_o on_o the_o side_n of_o b_o and_o d_o or_o on_o the_o side_n of_o a_o &_o c._n absurdity_n let_v they_o be_v produce_v therefore_o and_o let_v they_o mete_v if_o it_o be_v possible_a on_o the_o side_n of_o b_o and_o d_o in_o the_o point_n g._n wherefore_o in_o the_o triangle_n gef_n the_o outward_a angle_n aef_n be_v equal_a to_o the_o inward_a and_o opposite_a angle_n efg_o which_o by_o the_o 16._o proposition_n be_v impossible_a wherefore_o the_o line_n ab_fw-la and_o cd_o be_v produce_v on_o the_o side_n of_o b_o and_o d_o shall_v not_o meet_v in_o like_a sort_n also_o may_v it_o be_v prove_v that_o they_o shall_v not_o mete_v on_o the_o side_n of_o a_o and_o c._n but_o line_n which_o be_v produce_v on_o no_o side_n meet_v together_o be_v parrallell_a line_n by_o the_o last_o definition_n wherefore_o ab_fw-la be_v a_o parallel_v line_n to_o cd_o if_o therefore_o a_o right_a line_n fall_v upon_o two_o right_a line_n do_v make_v the_o alternate_a angle_n equal_v the_o one_o to_o the_o other_o those_o two_o right_a line_n be_v parallel_n the_o one_o to_o the_o other_o which_o be_v require_v to_o be_v demonstrate_v absurdity_n this_o word_n alternate_a be_v of_o euclid_n in_o diverse_a place_n diverse_o take_v sometime_o for_o a_o kind_n of_o situation_n in_o place_n and_o sometime_o for_o a_o order_n in_o proportion_n in_o which_o signification_n he_o use_v it_o in_o the_o v_o book_n and_o in_o his_o book_n of_o number_n and_o in_o the_o first_o signification_n he_o use_v it_o here_o in_o this_o place_n and_o general_o in_o all_o his_o other_o book_n absurdity_n have_v to_o do_v with_o line_n &_o figure_n and_o those_o two_o angle_n he_o call_v alternate_a which_o be_v both_o contain_v within_o two_o parallel_n or_o equidistant_a line_n be_v neither_o angle_n in_o order_n nor_o be_v on_o the_o one_o and_o self_n same_o side_n but_o be_v separate_v the_o one_o from_o the_o other_o by_o the_o line_n which_o fall_v on_o the_o two_o line_n the_o one_o angle_n be_v above_o and_o the_o other_o beneath_o the_o 19_o theorem_a the_o 28._o proposition_n if_o a_o right_a line_n fall_v upon_o two_o right_a line_n make_v the_o outward_a angle_n equal_a to_o the_o inward_a and_o opposite_a angle_n on_o one_o and_o the_o same_o side_n or_o the_o inward_a angle_n on_o one_o and_o the_o same_o side_n equal_a to_o two_o right_a angle_n those_o two_o right_a line_n shall_v be_v parallel_n the_o one_o to_o the_o other_o svppose_v that_o the_o right_a line_n of_o fall_a upon_o these_o two_o right_a line_n ab_fw-la and_o cd_o do_v make_v the_o outward_a angle_n egb_n equal_a to_o the_o inward_a and_o opposite_a angle_n ghd_v or_o do_v make_v the_o inward_a angle_n on_o one_o and_o the_o same_o side_n that_o be_v the_o angle_n bgh_a and_o ghd_v equal_a to_o two_o right_a angle_n then_o i_o say_v that_o the_o line_n ab_fw-la be_v a_o parallel_a line_n to_o the_o line_n cd_o for_o forasmuch_o as_o the_o angle_n egb_n be_v by_o supposition_n equal_a to_o the_o angle_n ghd_v demonstration_n and_o the_o angle_n egb_n be_v by_o the_o 15._o proposition_n equal_a to_o the_o angle_n agh_o therefore_o the_o angle_n agh_o be_v equal_a to_o the_o angle_n ghd_v and_o they_o be_v alternate_a angle_n wherefore_o by_o the_o 27._o proposition_n ab_fw-la be_v a_o parallel_a line_n to_o cd_o again_o forasmuch_o as_o the_o angle_n bgh_a and_o ghd_v be_v by_o supposition_n equal_a to_o two_o right_a angle_n &_o by_o the_o 13._o proposition_n the_o angle_n agh_o and_o bgh_o be_v also_o equal_a to_o two_o right_a angle_n wherefore_o the_o angle_n agh_o and_o bgh_o be_v equal_a to_o the_o angle_n bgh_a and_o ghd_v take_v away_o the_o angle_n bgh_n which_o be_v common_a to_o they_o both_o wherefore_o the_o angle_n remain_v namely_o agh_o be_v equal_a to_o the_o angle_n remain_v namely_o to_o ghd_v and_o they_o be_v alternate_a angle_n wherefore_o by_o the_o former_a proposition_n ab_fw-la be_v a_o parallel_a line_n to_o cd_o if_o therefore_o a_o right_a line_n fall_a upon_o two_o right_a line_n do_v make_v the_o outward_a angle_n equal_a to_o the_o inward_a and_o opposite_a angle_n on_o one_o and_o the_o same_o side_n or_o the_o inward_a angle_n on_o one_o and_o the_o same_o side_n equal_a to_o two_o right_a angle_n those_o two_o right_a line_n shall_v be_v parallel_n the_o one_o to_o the_o other_o which_o be_v require_v to_o be_v prove_v ptolomeus_n demonstrate_v the_o second_o part_n of_o this_o proposition_n namely_o that_o the_o two_o inward_a angle_n on_o one_o and_o the_o same_o side_n be_v equal_a the_o right_a line_n be_v parellel_n after_o this_o manner_n ptolemy_n the_o 20._o theorem_a the_o 29._o proposition_n a_o right_a line_n line_n fall_v upon_o two_o parallel_n right_a line_n make_v the_o alternate_a angle_n equal_v the_o one_o to_o the_o other_o and_o also_o the_o outward_a angle_n equal_a to_o the_o inward_a and_o opposite_a angle_n on_o one_o and_o the_o same_o side_n and_o moreover_o the_o inward_a angle_n on_o one_o and_o the_o same_o side_n equal_a to_o two_o right_a angle_n and_o the_o angle_n agh_o be_v by_o the_o 15._o proposition_n equal_a to_o
into_o two_o equal_a part_n but_o also_o they_o divide_v the_o angle_n into_o two_o equal_a part_n but_o in_o parallogramme_n who_o side_n be_v not_o equal_a such_o as_o be_v a_o figure_n on_o the_o one_o side_n long_o and_o a_o rhomboide_n it_o be_v not_o so_o prâ—â—lus_n the_o converse_n of_o the_o first_o and_o second_o part_n of_o this_o proposition_n after_o proclus_n be_v 〈◊〉_d figure_n whatsoever_o have_v his_o opposite_a side_n and_o angle_n equal_a then_o be_v a_o parallellogram_n flussates_n a_o corollary_n take_v out_o of_o flussates_n a_o right_a line_n cut_v a_o parallelogram_n which_o way_n soever_o into_o two_o equal_a part_n shall_v also_o divide_v the_o diameter_n thereof_o into_o two_o equal_a part_n for_o if_o it_o be_v possible_a let_v the_o right_a line_n gc_o divide_v the_o parallelogram_n aebd_a into_o two_o equal_a part_n but_o let_v it_o divide_v the_o diameter_n de_fw-fr into_o two_o unequal_a part_n in_o the_o point_n i_o and_o lâ—t_v the_o part_n je_n be_v great_a than_o the_o part_n id_fw-la and_o unto_o the_o line_n id_fw-la put_v the_o line_n io_o equal_v by_o th3_n proposition_n absurdity_n and_o by_o the_o point_n oh_o draw_v unto_o the_o line_n ad_fw-la and_o be_v a_o parallel_a line_n of_o by_o the_o 31._o proposition_n wherefore_o in_o the_o triangle_n foi_fw-fr and_o cdi_o two_o angle_n of_o the_o one_o be_v equal_a to_o two_o angle_n of_o the_o other_o namely_o the_o angle_n jof_o and_o idc_n by_o the_o 29._o proposition_n &_o the_o angle_n fio_n &_o cid_n by_o the_o 15._o proposition_n &_o the_o side_n id_fw-la be_v equal_a to_o the_o side_n io._n wherefore_o by_o the_o 26._o proposition_n the_o triangle_n be_v equal_a wherefore_o the_o whole_a triangle_n eig_n be_v great_a than_o the_o triangle_n dic._n and_o forasmuch_o as_o the_o trapesium_n gbdc_n be_v suppose_v to_o be_v the_o half_a of_o the_o parrallelograme_n and_o the_o half_a of_o the_o same_o parallelogram_n be_v the_o triangle_n ebb_v by_o this_o proposition_n from_o the_o trapesium_n gbdc_n and_o the_o triangle_n ebb_v which_o be_v equal_a take_v away_o the_o trapesium_n gbdi_n which_o be_v common_a to_o they_o both_o and_o the_o residue_n namely_o the_o triangle_n dic_fw-la shall_v be_v equal_a to_o the_o residue_n namely_o to_o the_o triangle_n eig_n but_o it_o be_v also_o less_o as_o have_v before_o be_v prove_v which_o be_v impossible_a wherefore_o a_o right_a line_n devide_v a_o parallelogram_n into_o two_o equal_a part_n shall_v not_o divide_v the_o diameter_n thereof_o unequal_o wherefore_o it_o shall_v divide_v it_o equal_o which_o be_v require_v to_o be_v prove_v a_o addition_n of_o pelitarius_n between_o two_o right_a line_n be_v infinite_a and_o make_v a_o angle_n give_v pelitarius_n to_o place_v a_o line_n equal_a to_o a_o line_n give_v in_o such_o sort_n that_o it_o shall_v make_v with_o one_o of_o those_o line_n a_o angle_n equal_a to_o a_o other_o angle_n give_v now_o it_o behove_v that_o the_o two_o angle_n give_v be_v less_o than_o two_o right_a angle_n though_o this_o addition_n of_o pelitarius_n be_v not_o so_o much_o pertain_v to_o the_o proposition_n yet_o because_o it_o be_v witty_a and_o seem_v somewhat_o difficult_a i_o think_v it_o good_a here_o to_o anexe_v it_o the_o 25._o theorem_a the_o 35._o proposition_n parallelogram_n consist_v upon_o one_o and_o the_o same_o base_a and_o in_o the_o self_n same_o parallel_a line_n be_v equal_a the_o one_o to_o the_o other_o svppose_v that_o these_o parallelogram_n abcd_o and_o ebb_n cf_o do_v consist_v upon_o one_o and_o the_o same_o base_a that_o be_v upon_o bc_o and_o in_o the_o self_n same_o parallel_a line_n that_o be_v of_o and_o bc._n then_o i_o say_v that_o the_o parallelogram_n abcd_o be_v equal_a to_o the_o parallelogram_n ebcf._n demonstration_n for_o forasmuch_o as_o abcd_o be_v a_o parallelogram_n therefore_o by_o the_o 34._o proposition_n the_o side_n ad_fw-la be_v equal_a to_o the_o side_n bc_o and_o by_o the_o same_o reason_n also_o the_o side_n of_o be_v equal_a to_o the_o side_n bc_o wherefore_o ad_fw-la be_v equal_a to_o of_o and_o de_fw-fr be_v common_a to_o they_o both_o wherefore_o the_o whole_a line_n ae_n be_v equal_a to_o the_o whole_a line_n df._n and_o the_o side_n ab_fw-la be_v equal_a to_o the_o side_n dc_o wherefore_o these_o two_o ea_fw-la and_o ab_fw-la be_v equal_a to_o these_o two_o fd_a and_o dc_o the_o one_o to_o the_o other_o and_o the_o angle_n fdc_n be_v equal_a to_o the_o angle_n eab_n namely_o the_o outward_a angle_n to_o the_o inward_a angle_n by_o the_o â—9_o proposition_n wherefore_o by_o the_o 4_o proposition_n the_o base_a ebb_n be_v equal_a to_o the_o base_a fc_n and_o the_o triangle_n eab_n be_v equal_a to_o the_o triangle_n fdc_n take_v away_o the_o triangle_n dge_n which_o be_v common_a to_o they_o both_o wherefore_o the_o residue_n namely_o the_o trapesium_n abgd_v be_v equal_a to_o the_o residue_n that_o be_v to_o the_o trapesium_n egcf_n put_v the_o triangle_n gbc_n common_a to_o they_o both_o wherefore_o the_o whole_a parallelogram_n abcd_o be_v equal_a to_o the_o whole_a parallelogram_n ebcf._n wherefore_o parallelogram_n consist_v upon_o one_o and_o the_o same_o base_a and_o in_o the_o self_n same_o parallel_a line_n be_v equal_a the_o one_o to_o the_o other_o which_o be_v require_v to_o be_v demonstrate_v parallelogram_n be_v say_v to_o be_v in_o the_o self_n same_o parallel_a line_n when_o their_o base_n and_o the_o opposite_a side_n unto_o they_o be_v one_o and_o the_o self_n same_o line_n with_o the_o parallel_n case_n in_o this_o proposition_n be_v three_o case_n for_o the_o line_n be_v may_v cut_v the_o line_n of_o either_o beyond_o the_o point_n d_o or_o in_o the_o point_n d_o or_o on_o this_o side_n the_o point_n d._n when_o it_o cut_v the_o line_n of_o beyond_o the_o point_n d_o the_o demonstration_n before_o put_v serve_v the_o 26._o theorem_a the_o 36._o proposition_n parallelogram_n consist_v upon_o equal_a base_n and_o in_o the_o self_n same_o parallel_a line_n be_v equal_a the_o one_o to_o the_o other_o svppose_v that_o these_o parallelogram_n abcd_o and_o efgh_o do_v consist_v upon_o equal_a base_n that_o be_v upon_o bc_o and_o fg_o and_o in_o the_o self_n same_o parallel_a line_n that_o be_v ah_o and_o bg_o then_o i_o say_v that_o the_o parallelogram_n abcd_o be_v equal_a to_o the_o parallelogram_n efgh_o construction_n draw_v a_o right_a line_n from_o the_o point_n b_o to_o the_o point_n e_o and_o a_o other_o from_o the_o point_n c_o to_o the_o point_n h._n demonstration_n and_o forasmuch_o as_o bc_o be_v equal_a to_o fg_o but_o fg_o be_v equal_a to_o eh_o therefore_o bc_o also_o be_v equal_a to_o eh_o and_o they_o be_v parallel_a line_n and_o the_o line_n be_v and_o change_z join_v they_o together_o but_o two_o right_a line_n join_v together_o two_o equal_a right_a line_n be_v parallel_n be_v themselves_o also_o by_o the_o 33_o proposition_n equal_v the_o one_o to_o the_o other_o and_o parallel_n wherefore_o ebch_n be_v a_o parallelogram_n and_o be_v equal_a to_o the_o parallelogram_n abcd_o for_o they_o have_v both_o one_o and_o the_o same_o base_a that_o be_v bc._n and_o be_v in_o the_o self_n same_o parallel_a line_n that_o be_v bc_o &_o eh_o and_o by_o the_o same_o reason_n also_o the_o parallelogram_n efgh_o be_v equal_a to_o the_o parallelogram_n ebch_o wherefore_o the_o parallelogram_n abcd_o be_v equal_a to_o the_o parallelogram_n efgh_o wherefore_o parallelogram_n consist_v upon_o equal_a base_n and_o in_o the_o self_n same_o parallel_v line_n be_v equal_a the_o one_o to_o the_o other_o which_o be_v require_v to_o be_v prove_v proposition_n in_o this_o proposition_n also_o be_v three_o case_n for_o the_o equal_a base_n may_v either_o be_v utter_o separate_v a_o sunder_o or_o they_o may_v touch_v aâ—_z one_o of_o the_o end_n or_o they_o may_v have_v one_o part_n common_a to_o they_o both_o case_n euclides_n demonstration_n serve_v when_o the_o base_n be_v utter_o separate_v a_o sunder_o which_o yet_o may_v happen_v seven_o diverse_a way_n way_n for_o the_o base_n be_v separate_v a_o sunder_o their_o opposite_a side_n also_o may_v be_v utter_o separate_v a_o sunder_o beyond_o the_o point_n d_o as_o the_o side_n ad_fw-la and_o eh_o in_o the_o first_o figure_n or_o they_o may_v touch_v together_o in_o one_o of_o the_o end_n and_o the_o whole_a side_n may_v be_v beyond_o the_o point_n d_o as_o the_o side_n ad_fw-la and_o eh_o do_v in_o the_o second_o figure_n or_o they_o may_v just_o agree_v the_o one_o with_o the_o other_o that_o be_v the_o point_n a_o and_o d_o may_v fall_v upon_o the_o point_n e_o and_o h_o as_o in_o the_o four_o figure_n or_o the_o side_n ad_fw-la be_v produce_v on_o this_o side_n the_o point_n a_o part_n of_o the_o opposite_a side_n unto_o the_o base_a fg_o may_v be_v on_o this_o
to_o the_o angle_n bml_n but_o the_o angle_n fgb_n be_v equal_a to_o the_o angle_n d_o therefore_o the_o angle_n bml_n be_v equal_a to_o the_o angle_n d._n wherefore_o upon_o the_o right_a line_n give_v ab_fw-la be_v apply_v the_o parallelogram_n lb_n equal_a to_o the_o triangle_n give_v c_o and_o contain_v the_o angle_n bml_n equal_a to_o the_o rectiline_a angle_n give_v d_z which_o be_v require_v to_o be_v do_v application_n of_o space_n or_o figure_n to_o line_n with_o excess_n or_o want_n be_v say_v eudemus_n a_o ancient_a invention_n of_o pythagoras_n pythagoras_n when_o the_o space_n or_o figure_n be_v join_v to_o the_o whole_a line_n line_n they_o be_v the_o figure_n say_v to_o be_v apply_v to_o the_o line_n but_o if_o the_o length_n of_o the_o space_n be_v long_o than_o the_o line_n they_o it_o be_v say_v to_o exceed_v and_o if_o the_o length_n of_o the_o figure_n be_v short_a than_o the_o line_n so_o that_o part_n of_o the_o line_n remain_v without_o the_o figure_n describe_v then_o be_v it_o say_v to_o want_v in_o this_o problem_n be_v three_o thing_n give_v proposition_n a_o right_a line_n to_o which_o the_o application_n be_v make_v which_o here_o must_v be_v the_o one_o side_n of_o the_o parallelogram_n apply_v a_o triangle_n whereunto_o the_o parallelogram_n apply_v must_v be_v equal_a and_o a_o angle_n whereunto_o the_o angle_n of_o the_o parallelogram_n apply_v must_v be_v equal_a and_o if_o the_o angle_n give_v be_v a_o right_a angle_n they_o shall_v the_o parallelogram_n apply_v be_v either_o a_o square_a or_o a_o figure_n on_o the_o one_o side_n long_o but_o if_o the_o angle_n give_v be_v a_o obtuse_a or_o a_o acute_a angle_n then_o shall_v the_o parallelogram_n apply_v be_v a_o rhombus_fw-la or_o diamond_n figure_n or_o else_o a_o rhomboide_n or_o diamondlike_a figure_n the_o converse_n of_o this_o proposition_n after_o politarius_n upon_o a_o right_a line_n give_v to_o apply_v unto_o a_o parallelogram_n give_v a_o equal_a triangle_n have_v a_o angle_n equal_a to_o a_o angle_n give_v proposition_n suppose_v that_o the_o right_a line_n give_v be_v ab_fw-la and_o let_v the_o parallelogram_n give_v be_v cdef_n and_o let_v the_o angle_v give_v be_v g._n it_o be_v require_v upon_o the_o line_n ab_fw-la to_o describe_v a_o triangle_n equal_a to_o the_o parallelogram_n cdef_n have_v a_o angle_n equal_a to_o the_o angle_n g._n draw_v the_o diameter_n cf_o &_o produce_v cd_o beyond_o the_o point_n d_o to_o the_o point_n h._n and_o put_v the_o line_n dh_fw-mi equal_a to_o the_o line_n cd_o and_o draw_v a_o line_n from_o fletcher_n to_o h._n now_o they_o by_o the_o 41._o proposition_n the_o triangle_n chf_n be_v equal_a to_o the_o parallelogram_n cd_o ef._n and_o by_o this_o proposition_n upon_o the_o line_n ab_fw-la describe_v a_o parallelogram_n abkl_n equal_a to_o the_o triangle_n chf_n have_v the_o angle_n abl_n equal_a to_o the_o angle_n give_v g_z and_o produce_v the_o line_n bl_o beyond_o the_o point_n l_o to_o the_o point_n m._n and_o put_v the_o line_n lm_o equal_a to_o the_o line_n bl_o and_o draw_v a_o line_n from_o a_o to_o m._n then_o i_o say_v that_o upon_o the_o line_n ab_fw-la be_v describe_v the_o triangle_n abm_n which_o be_v such_o a_o triangle_n as_o be_v require_v for_o by_o the_o 41._o proposition_n the_o triangle_n abm_n be_v equal_a to_o the_o parallelogram_n abkl_n for_o that_o they_o be_v between_o two_o parallel_a line_n bm_n and_o ak_o &_o the_o base_a of_o the_o triangle_n be_v double_a to_o the_o base_a of_o the_o parallelogram_n but_o abkl_n be_v by_o construction_n equal_a to_o the_o triangle_n chf_n and_o the_o triangle_n chf_n be_v equal_a to_o the_o parallelogram_n cdef_n wherefore_o by_o the_o first_o common_a sentence_n the_o triangle_n abm_n be_v equal_a to_o the_o parallelogram_n give_v cd_o of_o and_o have_v his_o angle_n abm_n equal_a to_o the_o angle_n give_v g_z which_o be_v require_v to_o be_v do_v the_o 13._o problem_n the_o 45_o proposition_n to_o describe_v a_o parallelogram_n equal_a to_o any_o rectiline_a figure_n give_v and_o contain_v a_o angle_n equal_a to_o a_o rectiline_a angle_n give_v svppose_v that_o the_o rectiline_a figure_n give_v be_v abcd_o and_o let_v the_o rectiline_a angle_n give_v be_v e._n it_o be_v require_v to_o describe_v a_o parallelogram_n equal_a to_o the_o rectiline_a figure_n give_v abcd_o and_o contain_v a_o angle_n equal_a to_o the_o rectiline_a angle_n give_v e._n construction_n draw_v by_o the_o first_o petition_n a_o right_a line_n from_o the_o point_n d_o to_o the_o point_n b._n and_o by_o the_o 42._o proposition_n unto_o the_o triangle_n abdella_n describe_v a_o equal_a parallelogram_n fh_o have_v his_o angle_n fkh_n equal_a to_o the_o angle_n e._n and_o by_o the_o 4â—_o of_o the_o first_o upon_o the_o right_a line_n gh_o apply_v the_o parallelogram_n gm_n equal_a to_o the_o triangle_n dbc_n have_v his_o angle_n ghm_n equal_a to_o the_o angle_n e._n demonstration_n and_o forasmuch_o as_o either_o of_o those_o angle_n hkf_a and_o ghm_n be_v equal_a to_o the_o angle_n e_o therefore_o the_o angle_n hkf_n be_v equal_a to_o the_o angle_n ghm_n put_v the_o angle_n khg_n common_a to_o they_o both_o wherefore_o the_o angle_n fkh_n and_o khg_n be_v equal_a to_o the_o angle_n khg_n and_o ghm_n but_o the_o angle_n fkh_n and_o khg_n be_v by_o the_o 29._o proposition_n equal_a to_o two_o right_a angle_n wherefore_o the_o angle_n khg_n and_o ghm_n be_v equal_a to_o two_o right_a angle_n now_o then_o unto_o a_o right_a line_n gh_o and_o to_o a_o point_n in_o the_o same_o h_o be_v draw_v two_o right_a line_n kh_o and_o he_o not_o both_o on_o one_o and_o the_o same_o side_n make_v the_o side_n angle_n equal_a to_o two_o right_a angle_n wherefore_o by_o the_o 14._o proposition_n the_o line_n kh_o and_o he_o make_v direct_o one_o right_a line_n and_o forasmuch_o as_o upon_o the_o parallel_a line_n km_o and_o fg_o fall_v the_o right_a line_n hg_o therefore_o the_o alternate_a aagle_n mhg_n and_o hgf_n be_v by_o the_o 29_o proposition_n equal_v the_o one_o to_o the_o other_o put_v the_o angle_n hgl_n common_a to_o they_o both_o wherefore_o the_o angle_n mhg_n and_o hgl_n be_v equal_a to_o the_o angle_n hgf_n and_o hgl_n but_o the_o angle_n mhg_n &_o hgl_n be_v equal_a to_o two_o right_a angle_n by_o the_o 29._o proposition_n wherefore_o also_o the_o angle_n hgf_n and_o hgl_n be_v equal_a to_o two_o right_a angle_n wherefore_o by_o the_o 14._o proposition_n the_o line_n fg_o and_o gl_n make_v direct_o one_o right_a line_n and_o forasmuch_o as_o the_o line_n kf_o be_v by_o the_o 34._o proposition_n equal_a to_o the_o line_n hg_o and_o it_o be_v also_o parallel_a unto_o it_o and_o the_o line_n hg_o be_v by_o the_o same_o equal_a to_o the_o line_n ml_o therefore_o by_o the_o first_o common_a sentence_n the_o line_n fk_o be_v equal_a to_o the_o line_n ml_o and_o also_o a_o parallel_n unto_o it_o by_o the_o 30._o proposition_n but_o the_o right_a line_n km_o and_o fl_fw-mi join_v they_o together_o wherefore_o by_o the_o 33._o proposition_n the_o line_n km_o and_o fl_fw-mi be_v equal_v the_o on_o to_o the_o other_o and_o parallel_a line_n wherefore_o kflm_n be_v a_o parallelogram_n and_o forasmuch_o as_o the_o triangle_n abdella_n be_v eqnal_a to_o the_o parallelogram_n fh_o and_o the_o triangle_n dbc_n to_o the_o parallelogram_n gmâ—_n therefore_o the_o whole_a rectiline_a figure_n abcd_o be_v equal_a to_o the_o whole_a parallelogram_n kflm_n wherefore_o to_o the_o rectiline_a figure_n give_v abcd_o be_v make_v a_o equal_a parallegrame_n kflm_n who_o angle_n fkm_n be_v equal_a to_o the_o angle_n give_v namely_o to_o e_o which_o be_v require_v to_o be_v do_v the_o rectiline_a figure_n give_v 〈◊〉_d in_o the_o example_n of_o euclid_n be_v a_o parallelogram_n but_o if_o the_o rectiline_a figure_n be_v of_o many_o side_n as_o of_o 5.6_o or_o more_o they_o must_v you_o resolve_v the_o figure_n into_o his_o triangle_n as_o have_v be_v before_o teach_v in_o the_o 32._o proposition_n and_o they_o apply_v a_o parallelogram_n equal_a to_o every_o triangle_n upon_o a_o line_n give_v as_o before_o in_o the_o example_n of_o the_o author_n and_o the_o same_o kind_n of_o reason_v will_v serve_v that_o be_v before_o only_o by_o reason_n of_o the_o multitude_n of_o triangle_n you_o shall_v have_v need_n of_o often_o repetition_n of_o the_o 29._o and_o 14._o proposition_n to_o prove_v that_o the_o base_n of_o all_o the_o parallelogram_n make_v equal_a to_o all_o the_o triangle_n make_v one_o right_a line_n and_o so_o also_o of_o the_o top_n of_o the_o say_a parallelogram_n pelitarius_n add_v unto_o this_o proposition_n this_o problem_n follow_v two_o unequal_a rectiline_n superficiâ—câ—s_v be_v give_v to_o find_v out_o the_o excess_n of_o the_o great_a above_o the_o less_o pelitarius_n
by_o the_o point_n d_o draw_v unto_o the_o line_n ec_o a_o parallel_n line_n and_o let_v the_o same_o be_v df_o and_o by_o the_o self_n same_o by_o the_o point_n fletcher_n draw_v unto_o ab_fw-la a_o line_n parallel_n and_o let_v the_o same_o be_v fg._n and_o by_o the_o first_o petition_n draw_v a_o line_n from_o a_o to_o f._n demonstration_n and_o forasmuch_o as_o ac_fw-la be_v equal_a unto_o ce_fw-fr therefore_o by_o the_o 5._o of_o the_o first_o the_o angle_n eac_n be_v equal_a unto_o the_o angle_n cea_n and_o forasmuch_o as_o the_o angle_n at_o the_o point_n c_o be_v a_o right_a angle_n therefore_o the_o angle_n remain_v eac_n and_o aec_o be_v equal_a unto_o one_o right_a angle_n wherefore_o each_o of_o these_o angle_n eac_n and_o aec_o be_v the_o half_a of_o a_o right_a angle_n and_o by_o the_o same_o reason_n also_o each_o of_o these_o angle_n ebc_n and_o ceb_n be_v the_o half_a of_o a_o rigâ—t_a angle_n wherefore_o the_o whole_a angle_n aeb_n be_v a_o right_a angle_n and_o forasmuch_o as_o the_o angle_n gef_n be_v the_o half_a of_o a_o right_a angle_n but_o egf_n be_v a_o right_a angle_n for_o by_o the_o 29_o of_o the_o first_o it_o be_v equal_a unto_o the_o inward_a and_o opposite_a angle_n that_o be_v unto_o ecb_n wherefore_o the_o angle_n remain_v efg_o be_v the_o half_a of_o a_o right_a angle_n wherefore_o by_o the_o 6._o common_a sentence_n the_o angle_n gef_n be_v equal_a unto_o the_o angle_n efg_o wherefore_o also_o by_o the_o 6._o of_o the_o first_o the_o side_n eglantine_n be_v equal_a unto_o the_o side_n fg._n again_o forasmuch_o as_o the_o angle_n at_o the_o point_n b_o be_v the_o half_a of_o a_o right_a angle_n but_o the_o angle_n fdb_n be_v a_o right_a angle_n for_o it_o also_o by_o the_o 29._o of_o the_o first_o be_v equal_a unto_o the_o inward_a and_o opposite_a angle_n ecb_n wherefore_o the_o angle_n remain_v bfd_v be_v the_o half_a of_o a_o right_a angle_n wherefore_o the_o angle_n at_o the_o point_n b_o be_v equal_a unto_o the_o angle_n dfb_n wherefore_o by_o the_o 6._o of_o the_o first_o the_o side_n df_o be_v equal_a unto_o the_o side_n db._n and_o forasmuch_o as_o ac_fw-la be_v equal_a unto_o ce_fw-fr therefore_o the_o square_n which_o be_v make_v of_o ac_fw-la be_v equal_a unto_o the_o square_n which_o be_v make_v of_o ce._n wherefore_o the_o square_n which_o be_v make_v of_o ca_n and_o ce_fw-fr be_v double_v to_o the_o square_n which_o be_v make_v of_o ac_fw-la but_o by_o the_o 47._o of_o the_o first_o the_o square_a which_o be_v make_v of_o ea_fw-la be_v equal_a to_o the_o square_n which_o be_v make_v of_o ac_fw-la and_o ce_fw-fr for_o the_o angle_n ace_n be_v a_o right_a angle_n wherefore_o the_o square_a of_o ae_n be_v double_a to_o the_o square_n of_o ac_fw-la again_o forasmuch_o as_o eglantine_n be_v equal_a unto_o gf_o the_o square_a therefore_o which_o be_v make_v of_o eglantine_n be_v equal_a to_o the_o square_n which_o be_v make_v of_o gf_o wherefore_o the_o square_n which_o be_v make_v of_o ge_z and_o gf_o be_v double_a to_o the_o square_n which_o be_v make_v of_o gf_o but_o by_o the_o 47._o of_o the_o first_o the_o square_a which_o be_v make_v of_o of_o be_v equal_a to_o the_o square_n which_o be_v make_v of_o eglantine_n and_o gf_o wherefore_o the_o square_n which_o be_v make_v of_o of_o be_v double_a to_o the_o square_n which_o be_v make_v of_o gf_o but_o gf_o be_v equal_a unto_o cd_o wherefore_o the_o square_a which_o be_v make_v of_o of_o be_v double_a to_o the_o square_n which_o be_v make_v of_o cd_o and_o the_o square_n which_o be_v make_v of_o ae_n be_v double_a to_o the_o square_n which_o be_v make_v of_o ac_fw-la wherefore_o the_o square_n which_o be_v make_v of_o ae_n and_o of_o be_v double_a to_o the_o square_n which_o be_v make_v of_o ac_fw-la and_o cd_o but_o by_o the_o 47._o of_o the_o first_o the_o square_a which_o be_v make_v of_o of_o be_v equal_a to_o the_o square_n which_o be_v make_v of_o ae_n and_o of_o for_o the_o angle_n aef_n be_v a_o right_a angle_n wherefore_o the_o square_n which_o be_v make_v of_o of_o be_v double_a to_o the_o square_n which_o be_v make_v of_o ac_fw-la &_o cd_o but_o by_o the_o 47._o of_o the_o first_o the_o square_n which_o be_v make_v of_o ad_fw-la and_o df_o be_v equal_a to_o the_o square_v which_o be_v make_v of_o af._n for_o the_o angle_n ot_fw-mi the_o point_fw-fr d_z be_v a_o right_a angle_n wherefore_o the_o square_n which_o be_v make_v of_o ad_fw-la and_o df_o be_v double_a to_o the_o square_n which_o be_v make_v of_o ac_fw-la and_o cd_o but_o df_o be_v equal_a unto_o db._n wherefore_o the_o square_n which_o be_v make_v of_o ad_fw-la and_o db_o be_v double_a to_o the_o square_n which_o be_v make_v of_o ac_fw-la and_o cd_o if_o therefore_o a_o right_a line_n be_v divide_v into_o two_o equal_a part_n and_o into_o two_o unequal_a part_n the_o square_n which_o be_v make_v of_o the_o unequal_a part_n of_o the_o whole_a be_v double_a to_o the_o square_n which_o be_v make_v of_o the_o half_a line_n and_o of_o that_o line_n which_o be_v between_o the_o section_n which_o be_v require_v to_o be_v prove_v ¶_o a_o example_n of_o this_o proposition_n in_o number_n take_v any_o even_a number_n as_o 12._o and_o divide_v it_o first_o equal_o as_o into_o 6._o and_o 6._o &_o then_o unequal_o as_o into_o 8._o &_o 4._o and_o take_v the_o difference_n of_o the_o half_a to_o one_o of_o the_o unequal_a part_n which_o be_v 2._o and_o take_v the_o square_a number_n of_o the_o unequal_a part_n 8_o and_o 4_o which_o be_v sixpence_n and_o 16_o and_o add_v they_o together_o which_o make_v 80._o then_o take_v the_o square_n of_o the_o half_a 6._o and_o of_o the_o difference_n 2_o which_o be_v 36_o and_o 4_o which_o add_v together_o make_v 40._o unto_o which_o number_n the_o number_n compose_v of_o the_o square_n of_o the_o unequal_a part_n which_o iâ—_z 80_o iâ—_z double_a as_o you_o see_v in_o the_o example_n the_o demonstration_n whereof_o follow_v in_o barlaam_n the_o nine_o proposition_n if_o a_o number_n be_v divide_v into_o two_o equal_a number_n and_o again_o be_v divide_v into_o two_o inequal_a part_n the_o square_a number_n of_o the_o unequal_a number_n be_v double_a to_o the_o square_n which_o be_v make_v of_o the_o multiplication_n of_o the_o half_a number_n into_o itself_o together_o with_o the_o square_n which_o be_v make_v of_o the_o number_n set_v between_o they_o the_o 10._o theorem_a the_o 10._o proposition_n if_o a_o right_a line_n be_v divide_v into_o two_o equal_a part_n &_o unto_o it_o be_v add_v a_o other_o right_a line_n direct_o the_o square_n which_o be_v make_v of_o the_o whole_a &_o that_o which_o be_v add_v as_o of_o one_o line_n together_o with_o the_o square_n which_o be_v make_v of_o the_o line_n which_o be_v add_v these_o two_o square_n i_o say_v be_v double_a to_o these_o square_n namely_o to_o the_o square_n which_o be_v make_v of_o the_o half_a line_n &_o to_o the_o square_n which_o be_v make_v of_o the_o other_o half_a line_n and_o that_o which_o be_v add_v as_o of_o one_o line_n svppose_v that_o a_o certain_a right_a line_n ab_fw-la be_v divide_v into_o two_o equal_a part_n in_o the_o point_n c._n and_o unto_o it_o let_v there_o be_v add_v a_o other_o right_a line_n direct_o namely_o bd._n then_o i_o say_v that_o the_o square_n which_o be_v make_v of_o the_o line_n ad_fw-la and_o db_o be_v double_a to_o the_o square_n which_o be_v make_v of_o the_o line_n ac_fw-la and_o cd_o construction_n raise_v up_o by_o the_o 11._o of_o the_o first_o from_o the_o point_n c_o unto_o the_o right_a line_n acd_o a_o perpendicular_a line_n and_o let_v the_o same_o be_v ce._n and_o let_v ce_fw-fr by_o the_o 3._o of_o the_o first_o be_v make_v equal_a unto_o either_o of_o these_o line_n ac_fw-la and_o cb._n and_o by_o the_o first_o petition_n draw_v right_a line_n from_o e_o to_o a_o and_o from_o e_o to_o b._n and_o by_o the_o 31._o of_o the_o first_o by_o the_o point_n e_o draw_v a_o line_n parallel_n unto_o cd_o and_o let_v the_o same_o be_v ef._n and_o by_o the_o self_n same_o by_o the_o point_n d_o draw_v a_o line_n parallel_n unto_o ce_fw-fr and_o let_v the_o same_o be_v df._n and_o forasmuch_o as_o upon_o these_o parallel_a line_n ce_fw-fr &_o df_o light_v a_o certain_a right_a line_n of_o therefore_o by_o the_o 29._o of_o the_o first_o the_o angle_n cef_n and_o efd_n be_v equal_a unto_o two_o right_a angle_n wherefore_o the_o angle_n feb_n and_o efd_n be_v less_o than_o two_o right_a angle_n but_o line_n produce_v from_o angle_n less_o than_o two_o right_a angle_n by_o the_o five_o petition_n at_o the_o length_n meet_v together_o wherefore_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
two_o side_n eh_o and_o eke_o of_o the_o triangle_n ehk_n and_o the_o angle_n feg_n be_v great_a than_o the_o angle_n hek_n therefore_o by_o the_o 24._o of_o the_o first_o the_o base_a fg_o be_v great_a than_o the_o base_a hk_n and_o by_o the_o same_o reason_n may_v it_o be_v prove_v that_o the_o line_n ac_fw-la be_v great_a than_o the_o line_n ab_fw-la and_o so_o be_v manifest_v the_o whole_a proposition_n the_o 15._o theorem_a the_o 16._o proposition_n if_o from_o the_o end_n of_o the_o diameter_n of_o a_o circle_n be_v draw_v a_o right_a line_n make_v right_a angle_n it_o shall_v fall_v without_o the_o circle_n and_o between_o that_o right_a line_n and_o the_o circumference_n can_v not_o be_v draw_v a_o other_o right_a line_n and_o the_o angle_n of_o the_o semicircle_n be_v great_a than_o any_o acute_a angle_n make_v of_o right_a line_n but_o the_o other_o angle_n be_v less_o than_o any_o acute_a angle_n make_v of_o right_a line_n svppose_v that_o there_o be_v a_o circle_n abc_n who_o centre_n let_v be_v the_o point_n d_o and_o let_v the_o diameter_n thereof_o be_v ab_fw-la then_o i_o say_v that_o a_o right_a line_n draw_v from_o the_o point_n a_o make_v with_o the_o diameter_n ab_fw-la right_a angle_n shall_v fall_v without_o the_o circle_n theorem_a for_o if_o it_o do_v not_o then_o if_o it_o be_v possible_a let_v it_o fall_v within_o the_o circle_n as_o the_o line_n ac_fw-la do_v and_o draw_v a_o line_n from_o the_o point_n d_o to_o the_o point_n c._n absurdity_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 dam_n be_v equal_a to_o the_o line_n dc_o for_o they_o be_v draw_v from_o the_o centre_n to_o the_o circumference_n therefore_o the_o angle_n dac_o be_v equal_a to_o the_o angle_n acd_o but_o the_o angle_n dac_o be_v by_o supposition_n a_o right_a angle_n wherefore_o also_o the_o angle_n acd_o be_v a_o right_a angle_n wherefore_o the_o angle_n dac_o and_o acd_o be_v equal_a to_o two_o right_a angle_n which_o by_o the_o 17._o of_o the_o first_o be_v impossible_a wherefore_o a_o right_a line_n draw_v from_o the_o point_n a_o make_v with_o the_o diameter_n ab_fw-la right_a angle_n shall_v not_o fall_v within_o the_o circle_z in_o like_a sort_n also_o may_v we_o prove_v that_o it_o fall_v not_o in_o the_o circumference_n wherefore_o it_o fall_v without_o as_o the_o line_n ae_n do_v i_o say_v also_o part_n that_o between_o the_o right_a line_n ae_n and_o the_o circumference_n acb_o can_v not_o be_v draw_v a_o other_o right_a line_n for_o if_o it_o be_v possible_a let_v the_o line_n of_o so_o be_v draw_v and_o by_o the_o 12._o of_o the_o first_o from_o the_o point_n d_o draw_v unto_o the_o line_n favorina_n a_o perpendicular_a line_n dg_o and_o for_o asmuch_o as_o agd_n be_v a_o right_a angle_n but_o dag_n be_v less_o than_o a_o right_a angle_n therefore_o by_o the_o 19_o of_o the_o first_o the_o side_n ad_fw-la be_v great_a than_o the_o side_n dg_o but_o the_o line_n dam_n be_v equal_a to_o the_o line_n dh_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 dh_o be_v great_a than_o the_o line_n dg_o namely_o the_o less_o great_a than_o the_o great_a which_o be_v impossible_a wherefore_o between_o the_o right_a line_n ae_n and_o the_o circumference_n acb_o can_v not_o be_v draw_v a_o other_o right_a line_n i_o say_v moreover_o part_n that_o the_o angle_n of_o the_o semicircle_n contain_v under_o the_o right_a line_n ab_fw-la and_o the_o circumference_n cha_fw-mi be_v great_a than_o any_o acute_a angle_n make_v of_o right_a line_n and_o the_o angle_n remain_v contain_v under_o the_o circumference_n cha_fw-mi and_o the_o right_a line_n ae_n be_v less_o than_o any_o acute_a angle_n make_v of_o right_a line_n for_o if_o there_o be_v any_o angle_n make_v of_o right_a line_n great_a than_o that_o angle_n which_o be_v contain_v under_o the_o right_a line_n basilius_n and_o the_o circumference_n cha_fw-mi or_o less_o than_o that_o which_o be_v contain_v under_o the_o circumference_n cha_fw-mi and_o the_o right_a line_n ae_n then_o between_o the_o circumference_n cha_fw-mi and_o the_o right_a line_n ae_n there_o shall_v fall_v a_o right_a line_n which_o make_v the_o angle_n contain_v under_o the_o right_a line_n great_a than_o that_o angle_n which_o be_v contain_v under_o the_o right_a line_n basilius_n and_o the_o circumference_n cha_fw-mi and_o less_o than_o the_o angle_n which_o be_v contain_v under_o the_o circumference_n cha_fw-mi and_o the_o right_a line_n ae_n but_o there_o can_v fall_v no_o such_o line_n as_o it_o have_v before_o be_v prove_v wherefore_o no_o acute_a angle_n contain_v under_o right_a line_n be_v great_a than_o the_o angle_n contain_v under_o the_o right_a line_n basilius_n and_o the_o circumference_n cha_fw-mi nor_o also_o less_o than_o the_o angle_n contain_v under_o the_o circumference_n cha_fw-mi and_o the_o line_n ae_n correlary_a hereby_o it_o be_v manifest_a that_o a_o perpendicular_a line_n draw_v from_o the_o end_n of_o the_o diameter_n of_o a_o circle_n touch_v the_o circle_n and_o that_o a_o right_a line_n touch_v a_o circle_n in_o one_o point_n only_o for_o it_o be_v prove_v by_o the_o 2._o of_o the_o three_o that_o a_o right_a line_n draw_v from_o two_o point_n take_v in_o the_o circumference_n of_o a_o circle_n shall_v fall_v within_o the_o circle_n which_o be_v require_v to_o be_v demonstrate_v the_o 2._o problem_n the_o 17._o proposition_n from_o a_o point_n give_v to_o draw_v a_o right_a line_n which_o shall_v touch_v a_o circle_n give_v construction_n svppose_v that_o the_o point_n give_v be_v a_o and_o let_v the_o circle_n give_v be_v bcd_o it_o be_v require_v from_o the_o point_n a_o to_o draw_v a_o right_a line_n which_o shall_v touch_v the_o circle_n bcd_o take_v by_o the_o first_o of_o the_o three_o the_o centre_n of_o the_o circle_n and_o let_v the_o same_o be_v e._n and_o by_o the_o first_o petition_n draw_v the_o right_a line_n ade_n and_o make_v the_o centre_n e_o and_o the_o space_n ae_n describe_v by_o the_o three_o petition_n a_o circle_n afg_n and_o from_o the_o point_n d_o raise_v up_o by_o the_o 11._o of_o the_o first_o unto_o the_o line_n ea_fw-la a_o perpendicular_a line_n df._n and_o by_o the_o first_o petition_n draw_v these_o line_n ebf_a and_o ab_fw-la then_o i_o say_v that_o from_o the_o point_n a_o be_v draw_v to_o the_o circle_n bcd_v a_o touch_n line_n ab_fw-la demonstration_n for_o for_o asmuch_o as_o the_o point_n e_o be_v the_o centre_n of_o the_o circle_n bcd_o and_o also_o of_o the_o circle_n afg_n therefore_o the_o line_n ea_fw-la be_v equal_a to_o the_o line_n of_o and_o the_o line_n ed_z to_o the_o line_n ebb_n for_o they_o be_v draw_v from_o the_o centre_n to_o the_o circumference_n wherefore_o to_o these_o two_o line_n ae_n and_o ebb_n be_v equal_a these_o two_o line_n of_o &_o ed_z and_o the_o angle_n at_o the_o point_n e_o be_v common_a to_o they_o both_o wherefore_o by_o the_o 4._o of_o the_o first_o the_o base_a df_o be_v equal_a to_o the_o base_a ab_fw-la and_o the_o triangle_n def_n be_v equal_a to_o the_o triangle_n eba_n and_o the_o rest_n of_o the_o angle_n remain_v to_o the_o rest_n of_o the_o angle_n remain_v wherefore_o the_o angle_n edf_o be_v equal_a to_o the_o angle_n eba_n but_o the_o angle_n edf_o be_v a_o right_a angle_n wherefore_o also_o the_o angle_n eba_n be_v a_o right_a angle_n and_o the_o line_n ebb_n be_v draw_v from_o the_o centre_n but_o a_o perpendicular_a line_n draw_v from_o the_o end_n of_o the_o diameter_n of_o a_o circle_n touch_v the_o circle_n by_o the_o corellary_n of_o the_o 16._o of_o the_o three_o wherefore_o the_o line_n ab_fw-la touch_v the_o circle_n bcd_o wherefore_o from_o the_o point_n give_v namely_o a_o be_v draw_v unto_o the_o circle_n give_v bcd_o a_o touch_n line_n ab_fw-la which_o be_v require_v to_o be_v do_v ¶_o a_o addition_n of_o pelitarius_n pelitarius_n unto_o a_o right_a line_n which_o cut_v a_o circle_n to_o draw_v a_o parallel_n line_n which_o shall_v touch_v the_o circle_n suppose_v that_o the_o right_a line_n ab_fw-la do_v out_o the_o circle_n abc_n in_o the_o point_n a_o and_o b._n it_o be_v require_v to_o draw_v unto_o the_o line_n ab_fw-la a_o parallel_n line_n which_o shall_v touch_v the_o circle_n let_v the_o centre_n of_o the_o circle_n be_v the_o point_n d._n and_o divide_v the_o line_n ab_fw-la into_o two_o equal_a part_n in_o the_o point_n e._n and_o by_o the_o point_n e_o and_o by_o the_o centre_n d_o draw_v the_o diameter_n cdef_n and_o from_o the_o point_n fletcher_n which_o be_v the_o end_n of_o the_o diameter_n raise_v up_o by_o the_o 11._o of_o the_o first_o unto_o the_o diameter_n cf_o a_o perpendicular_a line_n gfh_v then_o i_o
equal_a unto_o the_o line_n ca._n two_o right_a angle_n and_o either_o of_o the_o angle_n at_o the_o baseâ—_n be_v two_o â—ift_a part_n of_o two_o right_a angle_n or_o four_o five_o part_n of_o one_o right_a angle_n which_o shall_v manifest_o appear_v if_o we_o divide_v two_o right_a angle_n into_o five_o part_n for_o then_o in_o this_o kind_n of_o triangle_n the_o angle_n at_o the_o top_n shall_v be_v one_o five_o part_n and_o either_o of_o the_o two_o angle_n at_o the_o base_a shall_v be_v two_o five_o part_n this_o also_o be_v to_o be_v note_v that_o the_o line_n ac_fw-la be_v the_o side_n of_o a_o equilater_n pentagon_n to_o be_v inscribe_v in_o the_o circle_n acd_o for_o by_o the_o latter_a construction_n it_o be_v manifest_a that_o the_o three_o arke_n ac_fw-la cd_o and_o de_fw-fr of_o the_o less_o circle_n be_v equal_a and_o forasmuch_o as_o by_o the_o same_o it_o be_v manifest_a that_o the_o two_o line_n ad_fw-la and_o ae_n be_v equal_a the_o ark_n also_o ae_n shall_v be_v equal_a to_o the_o ark_n ad_fw-la by_o the_o 20._o of_o the_o three_o wherefore_o their_o half_n also_o be_v equal_a if_o therefore_o the_o ark_n ae_n be_v by_o the_o 30._o of_o the_o three_o divide_v into_o two_o equal_a part_n the_o whole_a circumference_n acdea_n shall_v be_v divide_v into_o five_o equal_a arke_n and_o forasmuch_o as_o the_o line_n subtend_v the_o say_v equal_a arke_n be_v by_o the_o 2d_o of_o the_o same_o equal_a therefore_o every_o one_o of_o the_o say_a side_n shall_v be_v the_o side_n of_o a_o equilater_n pentagonâ—_n which_o be_v require_v to_o be_v prove_v and_o the_o same_o line_n ac_fw-la shall_v be_v the_o side_n of_o a_o equilater_n ten_o angle_a figure_n to_o be_v inscribe_v in_o the_o circle_n bde_v the_o demonstration_n whereof_o i_o omit_v for_o that_o it_o be_v demonstrate_v by_o proposition_n follow_v petarilius_n ¶_o a_o proposition_n add_v by_o pelitarius_n upon_o a_o right_a line_n give_v be_v finite_a to_o describe_v a_o equilater_n and_o equiangle_n pentagon_n figure_n if_o we_o consider_v well_o this_o demonstration_n of_o pelitarius_n it_o will_v not_o be_v hard_o for_o we_o upon_o a_o right_a line_n give_v to_o describe_v the_o rest_n of_o the_o figure_n who_o inscription_n hereafter_o follow_v note_n the_o 11._o problem_n the_o 11._o proposition_n in_o a_o circle_n give_v to_o describe_v a_o pentagon_n figure_n aequilater_n and_o equiangle_n svppose_v that_o the_o circle_n give_v be_v abcde_o construction_n it_o be_v require_v in_o the_o circle_n abcde_o to_o inscribe_v a_o figure_n of_o five_o angle_n of_o equal_a side_n and_o of_o equal_a angle_n take_v by_o the_o proposition_n go_v before_o a_o isosceles_a triangle_n fgh_o have_v either_o of_o the_o angle_n at_o the_o base_a gh_o double_a to_o the_o other_o angle_n namely_o unto_o the_o angle_n f._n and_o by_o the_o 2._o of_o the_o fourth_z in_o the_o circle_n abcde_v inscribe_v a_o triangle_n acd_a equiangle_n unto_o the_o triangle_n fgh_n so_o that_o let_v the_o angle_n god_n be_v equal_a to_o the_o angle_n fletcher_n &_o the_o angle_n acd_o unto_o the_o angle_n g_o and_o likewise_o the_o angle_n cda_n to_o the_o angle_n h._n wherefore_o either_o of_o these_o angle_n acd_o and_o cda_n be_v double_a to_o the_o angle_n god_n divide_v by_o the_o 9_o of_o the_o first_o either_o of_o these_o angle_n acd_o and_o cda_n into_o two_o equal_a part_n by_o the_o right_a line_n ce_fw-fr and_o db._n and_o draw_v right_a line_n from_o a_o to_o b_o from_o b_o to_o c_z from_o c_z to_o d_z from_o d_z to_o e_o and_o from_o e_o to_o a._n demonstration_n and_o forasmuch_o as_o either_o of_o these_o angle_n acd_o and_o cda_n be_v double_a to_o the_o angle_n god_n and_o they_o be_v divide_v into_o two_o equal_a part_n by_o the_o right_a line_n ce_fw-fr and_o db_o therefore_o the_o five_o angle_n dac_o ace_n ecd_n cdb_n and_o bda_n be_v equal_a the_o one_o to_o the_o other_o but_o equal_a angle_n by_o the_o 26._o of_o the_o three_o subtend_v equal_a circumference_n wherefore_o the_o five_o circumference_n ab_fw-la bc_o cd_o de_fw-fr and_o ea_fw-la be_v equal_a the_o one_o to_o the_o outstretch_o and_o by_o the_o 29_o of_o the_o same_o unto_o equal_a circumference_n be_v subtend_v equal_a right_a line_n wherefore_o the_o five_o right_a line_n ab_fw-la bc_o cd_o de_fw-fr &_o ea_fw-la be_v equal_a the_o one_o to_o the_o other_o wherefore_o the_o figure_v abcde_o have_v five_o angle_n be_v equilater_n now_o also_o i_o say_v that_o it_o be_v equiangle_n for_o forasmuch_o as_o the_o circumference_n ab_fw-la be_v equal_a to_o the_o circumference_n de_fw-fr put_v the_o circumference_n bcd_v common_a unto_o they_o both_o wherefore_o the_o whole_a circumference_n abcd_o be_v equal_a to_o the_o whole_a circumference_n edcb_n and_o upon_o the_o circumference_n abcd_o consist_v the_o angle_n aed_n and_o upon_o the_o circumference_n edcb_n consist_v the_o angle_n bae_o wherefore_o the_o angle_n bae_o be_v equal_a to_o the_o angle_n aed_n by_o the_o 27._o of_o the_o three_o and_o by_o the_o same_o reason_n every_o one_o of_o these_o angle_n abc_n and_o bcd_o and_o cde_o be_v equal_a to_o every_o one_o of_o these_o angle_n bae_o and_o aed_n wherefore_o the_o five_o angle_a figure_n abcde_o be_v equiangle_n and_o it_o be_v prove_v that_o be_v also_o equilater_n wherefore_o in_o a_o circle_n give_v be_v describe_v a_o figure_n of_o five_o angle_n equilater_n and_o equiangle_n which_o be_v require_v to_o be_v do_v ¶_o a_o other_o way_n to_o do_v the_o same_o after_o pelitarius_n here_o it_o be_v a_o pleasant_a thing_n to_o behold_v the_o variety_n of_o triangle_n for_o in_o the_o triangle_n abc_n either_o of_o the_o angle_n at_o the_o point_n a_o be_v one_o five_o part_n of_o a_o right_a angle_n whereby_o be_v produce_v the_o side_n of_o a_o ten_o angle_a figure_n to_o be_v inscribe_v in_o the_o selfsame_a circle_n which_o be_v manifest_a if_o we_o imagine_v the_o line_n bl_o and_o lc_a to_o be_v draw_v for_o the_o ark_n bc_o be_v divide_v into_o two_o equal_a part_n in_o the_o point_n l_o by_o the_o 26._o of_o the_o three_o so_o then_o by_o the_o inscription_n of_o a_o equilater_n triangle_n be_v know_v how_o to_o inscribe_v a_o hexagon_n figure_n namely_o by_o devide_v each_o of_o the_o arke_v subtend_v under_o the_o side_n of_o the_o triangle_n into_o two_o equal_a part_n and_o so_o always_o by_o the_o simple_a number_n of_o the_o side_n be_v know_v the_o double_a thereof_o as_o by_o a_o square_n be_v know_v a_o eight_o angle_a figure_n and_o by_o a_o eight_o angle_a figure_n a_o sixteen_o angle_a figure_n and_o so_o continual_o in_o the_o rest_n pelitarius_n teach_v yet_o a_o other_o way_n how_o to_o inscribe_v a_o pentagon_n pelitarius_n take_v the_o same_o circle_n that_o be_v before_o namely_o abc_n and_o the_o same_o triangle_n also_o defâ—_n â—_z and_o by_o the_o 17._o of_o the_o three_o draw_v the_o line_n man_n touch_v the_o circle_n in_o the_o point_n a._n and_o upon_o the_o line_n be_o and_o to_o the_o point_n a_o describe_v by_o the_o 23._o of_o the_o first_o the_o angle_n mab_n equal_a to_o one_o of_o these_o two_o angle_n e_o or_o fletcher_n either_o of_o which_o as_o it_o be_v manifest_a be_v less_o than_o a_o right_a angle_n by_o draw_v the_o right_a line_n ab_fw-la which_o let_v out_o the_o circumference_n in_o the_o point_n b._n again_o upon_o the_o line_n a_fw-la and_o to_o the_o point_n in_o it_o a_o describe_v the_o angle_n nac_n equal_a to_o the_o angle_n mab_n by_o draw_v the_o right_a line_n ac_fw-la which_o let_v cut_v the_o circumference_n in_o the_o point_n c._n and_o draw_v a_o right_a line_n from_o b_o to_o c._n then_o i_o say_v that_o bc_o be_v the_o side_n of_o a_o pentagon_n figure_n to_o be_v inscribe_v in_o the_o circle_n abc_n which_o be_v manifest_a if_o we_o divide_v the_o ark_n ab_fw-la into_o two_o equal_a part_n in_o the_o point_n h_o and_o draw_v these_o right_a line_n ah_o and_o bh_o and_o if_o also_o we_o divide_v the_o ark_n ac_fw-la into_o two_o equal_a part_n in_o the_o point_n g_o and_o draw_v these_o right_a line_n agnostus_n and_o cg_o for_o take_v the_o quadrangle_n figure_n abcg_o it_o be_v manifest_a by_o the_o 32._o of_o the_o three_o that_o the_o angle_n abc_n be_v equal_a to_o the_o alternate_a angle_n nac_n and_o therefore_o be_v equal_a to_o the_o angle_n e._n likewise_o take_v the_o quadrangle_n figure_n acbh_n the_o angle_n acb_o shall_v be_v equal_a to_o the_o alternate_a angle_n mab_n and_o therefore_o be_v equal_a to_o the_o angle_n f._n wherefore_o by_o the_o 32._o of_o the_o first_o as_o before_o the_o triangle_n abc_n be_v equiangle_n to_o the_o triangle_n def_n and_o now_o may_v you_o proceed_v in_o the_o demonstration_n as_o you_o do_v in_o the_o former_a the_o 12._o
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
to_o bg_o and_o draw_v a_o line_n from_o a_o to_o g._n now_o forasmuch_o as_o ab_fw-la be_v to_o bc_o as_o de_fw-fr be_v to_o of_o therefore_o alternate_o by_o the_o 16._o of_o the_o five_o as_o ab_fw-la be_v to_o de_fw-fr so_o be_v bc_o to_o ef._n demonstration_n but_o as_o bc_o be_v to_o of_o so_o be_v of_o to_o bg_o wherefore_o also_o by_o the_o 11._o of_o the_o five_o as_o ab_fw-la be_v to_o de_fw-fr so_o be_v of_o to_o bg_o wherefore_o the_o side_n of_o the_o triangle_n abg_o &_o def_n which_o include_v the_o equal_a angle_n be_v reciprocal_a proportional_a but_o if_o in_o triangle_n have_v one_o angle_n of_o the_o one_o equal_a to_o one_o angle_n of_o the_o outstretch_o the_o side_n which_o include_v the_o equal_a angle_n be_v reciprokal_n the_o triangle_n also_o by_o the_o 15._o the_o six_o shall_v be_v equal_a wherefore_o the_o triangle_n abg_o be_v equal_a unto_o the_o triangle_n def_n and_o for_o that_o as_z a'n_z line_n bc_o be_v to_o the_o line_n of_o so_o be_v the_o line_n of_o to_o the_o line_n bg_o but_o if_o there_o be_v three_o line_n in_o proportion_n the_o first_o shall_v have_v to_o the_o three_o double_a proportion_n that_o it_o have_v to_o the_o second_o by_o the_o 10._o definition_n of_o the_o five_o therefore_o the_o line_n bc_o have_v unto_o the_o line_n bg_o double_a proportion_n that_o it_o have_v to_o the_o line_n ef._n but_o as_o bc_o be_v to_o bg_o so_o by_o the_o 1._o of_o the_o six_o be_v the_o triangle_n abc_n to_o the_o triangle_n abg_o wherefore_o the_o tiangle_n abc_n be_v unto_o the_o triangle_n abg_o in_o double_a proportion_n that_o the_o side_n bc_o be_v to_o the_o side_n ef._n but_o the_o triangle_n abg_o be_v equal_a to_o the_o triangle_n def_n wherefore_o also_o the_o triangle_n abc_n be_v unto_o the_o triangle_n def_n in_o double_a proportion_n that_o the_o side_n bc_o be_v to_o the_o side_n ef._n wherefore_o like_a triangle_n be_v one_o to_o the_o other_o in_o double_a proportion_n that_o the_o side_n of_o like_a proportion_n be_v which_o be_v require_v to_o be_v prove_v corollary_n corollary_n hereby_o it_o be_v manifest_a that_o if_o there_o be_v three_o right_a line_n in_o proportion_n as_o the_o first_o be_v to_o the_o three_o so_o be_v the_o triangle_n describe_v upon_o the_o first_o unto_o the_o triangle_n describe_v upon_o the_o second_o so_o that_o the_o say_v triangle_n be_v like_a and_o in_o like_a â—ort_n describe_v for_o it_o have_v be_v prove_v that_o as_o the_o line_n cb_o be_v to_o the_o line_n bg_o so_o be_v the_o triangle_n abc_n to_o the_o triangle_n def_n which_o be_v require_v to_o be_v demonstrate_v the_o 14._o theorem_a the_o 20._o proposition_n like_a poligonon_n figure_n be_v divide_v into_o like_a triangle_n and_o equal_a in_o number_n and_o of_o like_a proportion_n to_o the_o whole_a and_o the_o one_o poligonon_n figure_n be_v to_o the_o other_o poligonon_n figure_n in_o double_a proportion_n that_o one_o of_o the_o side_n of_o like_a proportion_n be_v to_o one_o of_o the_o side_n of_o like_a proportion_n svppose_v that_o the_o like_a poligonon_n figure_n be_v abcde_o &_o fghkl_o have_v the_o angle_n at_o the_o point_n f_o equal_a to_o the_o angle_n at_o the_o point_n a_o and_o the_o angle_n at_o the_o point_n â—_o equal_a to_o the_o angle_n at_o the_o point_n b_o and_o the_o angle_n at_o the_o point_n h_o equal_a to_o you_o angle_v at_o the_o point_n c_o and_o so_o of_o the_o rest_n and_o moreover_o as_o the_o side_n ab_fw-la be_v to_o the_o side_n bc_o so_o let_v the_o side_n fg_o be_v to_o the_o side_n gh_o and_o as_o the_o side_n bc_o be_v to_o the_o side_n cd_o so_o let_v the_o side_n gh_o be_v to_o the_o side_n hk_o and_o so_o forth_o and_o let_v the_o side_n ab_fw-la &_o fg_o be_v side_n of_o like_a proportion_n then_o i_o say_v first_o theorem_a that_o these_o poligonon_n figure_v abcde_o &_o fghkl_o be_v divide_v into_o like_a triangle_n and_o equal_a in_o number_n for_o draw_v these_o right_a line_n ac_fw-la ad_fw-la fh_o &_o fkâ—_n and_o forasmuch_o as_o by_o supposition_n that_o be_v by_o reason_n the_o figure_n abcde_o be_v like_a unto_o the_o figure_n fghkl_o the_o angle_n b_o be_v equal_a unto_o the_o angle_n g_o and_o as_z the_o side_n ab_fw-la be_v to_o the_o side_n bc_o so_o be_v the_o side_n fg_o to_o the_o side_n gh_o it_o follow_v that_o the_o two_o triangle_n abc_n and_o fgh_n have_v one_o angle_n of_o the_o one_o equal_a to_o one_o angle_n of_o the_o other_o and_o have_v also_o the_o side_n about_o the_o equal_a angle_n proportional_a wherefore_o by_o the_o 6._o of_o the_o six_o the_o triangle_n abc_n be_v equiangle_n unto_o the_o triangle_n fgh_n and_o those_o angle_n in_o they_o be_v equal_a under_o which_o be_v subtend_v side_n of_o like_a proportion_n namely_o the_o angle_n bac_n be_v equal_a to_o the_o angle_n gfh_v and_o the_o angle_n bca_o to_o the_o angle_n ghf_n wherefore_o by_o the_o 4._o of_o the_o six_o the_o side_n which_o be_v about_o the_o equal_a angle_n be_v proportional_a and_o the_o side_n which_o be_v subtend_v under_o the_o equal_a angle_n be_v of_o like_a proportion_n wherefore_o as_o ac_fw-la be_v to_o bc_o so_o be_v fh_o to_o gh_o but_o by_o supposition_n as_o bc_o be_v to_o cd_o so_o be_v gh_o to_o hk_v wherefore_o of_o equality_n by_o the_o 22._o of_o the_o five_o as_o ac_fw-la be_v to_o cd_o so_o be_v fh_o to_o hk_v and_o forasmuch_o as_o by_o supposition_n the_o whole_a angle_n bcd_o be_v equal_a to_o the_o whole_a angle_n ghk_o and_o it_o be_v prove_v that_o the_o angle_n bca_o be_v eâ—â—all_a to_o the_o angle_n ghf_n therefore_o the_o angle_n remain_v acd_o be_v equal_a to_o the_o angle_n remain_v fhk_n by_o the_o 3._o common_a sentence_n wherefore_o the_o â—riaâ—glâ—s_v acd_o and_o fhk_v have_v again_o one_o angle_n of_o the_o one_o equal_a to_o one_o angle_n of_o the_o other_o and_o the_o side_n which_o be_v about_o the_o equal_a side_n be_v proportional_a wherefore_o by_o the_o â—ame_n six_o of_o this_o book_n the_o triangle_n acd_o &_o fhk_v be_v equiangle_n and_o by_o the_o 4._o of_o this_o book_n the_o side_n which_o be_v about_o the_o equal_a angle_n 〈◊〉_d proportional_a and_o by_o the_o same_o reason_n may_v we_o prove_v that_o the_o triangle_n adâ—_n be_v equiangle_n unto_o the_o triangle_n fkl_n and_o that_o the_o side_n which_o be_v about_o the_o equal_a angle_n be_v proportional_a wherefore_o the_o triangle_n abc_n be_v like_o to_o the_o triangle_n fgh_n and_o the_o triangle_n acd_o to_o the_o triangle_n fhk_n and_o also_o the_o triangle_n ade_n to_o the_o triangle_n fkl_n by_o the_o first_o definition_n of_o this_o six_o book_n wherefore_o the_o poligonon_n figure_v give_v abcde_o and_o fghkl_o be_v divide_v into_o triangle_n like_a and_o equal_a in_o number_n demonstrate_v i_o say_v moreover_o that_o the_o triangle_n be_v the_o one_o to_o the_o other_o and_o to_o the_o whole_a poligonon_n figure_v proportional_a that_o be_v as_o the_o triangle_n abc_n be_v to_o the_o triangle_n fgh_o so_o be_v the_o triangle_n acd_o to_o the_o triangle_n fhk_n and_o the_o triangle_n ade_n to_o the_o triangle_n fkl_n and_o as_o the_o triangle_n abc_n be_v to_o the_o triangle_n fgh_o so_o be_v the_o poligonon_n figure_n abcde_o to_o the_o poligonon_n figure_n fghkl_o for_o forasmuch_o as_o the_o triangle_n abc_n be_v like_a to_o the_o triangle_n fgh_o and_o ac_fw-la and_o fh_o be_v side_n of_o like_a proportion_n therefore_o the_o proportion_n of_o the_o triangle_n abc_n to_o the_o triangle_n fgh_n be_v double_a to_o the_o proportion_n of_o the_o side_n ac_fw-la to_o the_o 〈◊〉_d fh_o by_o the_o former_a proposition_n and_o therefore_o also_o the_o proportion_n of_o the_o triangle_n acd_o to_o the_o triangle_n fkh_n be_v double_a to_o the_o proportion_n that_o the_o same_o side_n ac_fw-la have_v to_o the_o side_n fh_o wherefore_o by_o the_o 11._o of_o the_o five_o as_o the_o triangle_n abc_n be_v to_o the_o triangle_n fgh_o so_o be_v the_o triangle_n acd_o to_o the_o triangle_n fhk_n again_o forasmuch_o as_o the_o triangle_n acd_o be_v like_a to_o the_o triangle_n fhk_n and_o the_o side_n ad_fw-la &_o fk_o be_v of_o like_a proportion_n therefore_o the_o proportion_n of_o the_o triangle_n acd_o to_o the_o triangle_n fhk_n be_v double_a to_o the_o proportion_n of_o the_o side_n ad_fw-la to_o the_o side_n fk_o by_o the_o foresay_a 19_o of_o the_o six_o and_o by_o the_o same_o reason_n the_o proportion_n of_o the_o triangle_n ade_n to_o the_o triangle_n fkl_n be_v double_a to_o the_o proportion_n of_o the_o same_o side_n ad_fw-la to_o the_o side_n fk_o wherefore_o by_o the_o 11._o of_o the_o five_o as_o the_o triangle_n acd_o be_v to_o the_o triangle_n fhk_n so_o be_v the_o triangle_n ade_n to_o the_o triangle_n fkl_n but_o
and_o the_o same_o proportion_n wherefore_o by_o the_o 9_o of_o the_o five_o the_o figure_n nh_o be_v equal_a unto_o the_o figure_n sir_n and_o it_o be_v unto_o it_o like_v and_o in_o like_a sort_n situate_a follow_v but_o in_o like_a and_o equal_a rectiline_a figure_n be_v in_o like_a sort_n situate_a the_o side_n of_o like_a proportion_n on_o which_o they_o be_v describe_v be_v equal_a wherefore_o the_o line_n gh_o be_v equal_a unto_o the_o line_n qr_o and_o because_o as_o the_o line_n ab_fw-la be_v to_o the_o line_n cd_o so_o be_v the_o line_n of_o to_o the_o line_n qr_o but_o the_o line_n qr_o be_v equal_a unto_o the_o line_n gh_o therefore_o as_o the_o line_n ab_fw-la be_v to_o he_o line_n cd_o so_o be_v the_o line_n of_o to_o the_o line_n gh_o if_o therefore_o there_o be_v four_o right_a line_n proportional_a the_o rectiline_a figure_n also_o describe_v upon_o they_o be_v like_a and_o in_o like_a sort_n situate_a shall_v be_v proportional_a and_o if_o the_o rectiline_a figure_n upon_o they_o describe_v be_v like_o and_o in_o like_a sort_n situate_v be_v proportional_a those_o right_a line_n also_o shall_v be_v proportional_a which_o be_v require_v to_o be_v prove_v a_o assumpt_n and_o now_o that_o in_o like_a and_o equal_a figure_n be_v in_o like_a sort_n situate_a the_o side_n of_o like_a proportion_n be_v also_o equal_a which_o thing_n be_v before_o in_o this_o proposition_n take_v as_o grant_v may_v thus_o be_v prove_v assumpt_n suppose_v that_o the_o rectiline_a figure_n nh_v and_o sir_n be_v equal_a and_o like_a and_o as_o hg_o be_v to_o gn_v so_o let_v rq_n be_v to_o q_n and_o let_v gh_o and_o qr_o be_v side_n of_o like_a proportion_n then_o i_o say_v that_o the_o side_n rq_n be_v equal_a unto_o the_o side_n gh_o for_o if_o they_o be_v unequal_a the_o one_o of_o they_o be_v great_a than_o the_o other_o let_v the_o side_n rq_n be_v great_a than_o the_o side_n hg_o and_o for_o that_o as_o the_o line_n rq_n be_v to_o the_o line_n q_n so_o be_v the_o line_n hg_o to_o the_o line_n gn_n and_o alternate_o also_o by_o the_o 16._o of_o the_o five_o as_o the_o line_n rq_n be_v to_o the_o line_n hg_o so_o be_v the_o line_n q_n to_o the_o line_n gn_n but_o the_o line_n rq_n be_v great_a than_o the_o line_n hg_o wherefore_o also_o the_o line_n q_n be_v great_a than_o the_o line_n gn_n wherefore_o also_o the_o figure_n rs_n be_v great_a than_o the_o figure_n hn_o but_o by_o supposition_n it_o be_v equal_a unto_o it_o which_o be_v impossible_a wherefore_o the_o line_n qr_o be_v not_o great_a than_o the_o line_n gh_o in_o like_a sort_n also_o may_v we_o prove_v that_o it_o be_v not_o less_o than_o it_o wherefore_o it_o be_v equal_a unto_o it_o which_o be_v require_v to_o be_v prove_v flussates_n demonstrate_v this_o second_o part_n more_o brief_o flussates_n by_o the_o first_o corollary_n of_o the_o _o of_o this_o book_n thus_o forasmuch_o as_o the_o rectiline_a figure_n be_v by_o supposition_n in_o one_o and_o the_o same_o proportion_n and_o the_o same_o proportion_n be_v double_a to_o the_o proportion_n of_o the_o side_n ab_fw-la to_o cd_o and_o of_o to_o gh_o by_o the_o foresay_a corollary_n the_o proportion_n also_o of_o the_o side_n shall_v be_v one_o and_o the_o self_n same_o by_o the_o 7._o common_a sentence_n namely_o the_o line_n ab_fw-la shall_v be_v unto_o the_o line_n cd_o as_o the_o line_n of_o be_v to_o the_o line_n gh_o the_o 17._o theorem_a the_o 23._o proposition_n equiangle_n parallelogrammes_n have_v the_o one_o to_o the_o other_o that_o proportion_n which_o be_v compose_v of_o the_o side_n flussates_n demonstrate_v this_o theorem_a without_o take_v of_o these_o three_o line_n king_n l_o m_o after_o this_o manner_n flussate_n forasmuch_o as_o say_v he_o it_o have_v be_v declare_v upon_o the_o 10._o definition_n of_o the_o five_o book_n and_o â—ift_a definition_n of_o this_o book_n that_o the_o proportion_n of_o the_o extreme_n consist_v of_o the_o proportion_n of_o the_o mean_n let_v we_o suppose_v two_o equiangle_n parallelogram_n abgd_v and_o gezi_n and_o let_v the_o angle_n at_o the_o point_n g_o in_o either_o be_v equal_a and_o let_v the_o line_n bg_o and_o give_v be_v set_v direct_o that_o they_o both_o make_v one_o right_a line_n namely_o bgi._n wherefore_o egg_v also_o shall_v be_v one_o right_a line_n by_o the_o converse_n of_o the_o 15._o of_o the_o first_o make_v complete_a the_o parallelogram_n gt_fw-mi then_o i_o say_v that_o the_o proportion_n of_o the_o parallelogram_n agnostus_n &_o gz_n be_v compose_v of_o the_o proportion_n of_o the_o side_n bg_o to_o give_v and_o dg_o to_o ge._n for_o forasmuch_o as_o that_o there_o be_v three_o magnitude_n agnostus_n gt_n and_o gz_n and_o gt_n be_v the_o mean_a of_o the_o say_a magnitude_n and_o the_o proportion_n of_o the_o extreme_n agnostus_n to_o gz_n consist_v of_o the_o mean_a proportion_n by_o the_o 5._o definition_n of_o this_o book_n namely_o of_o the_o proportion_n of_o agnostus_n to_o gt_n and_o of_o the_o proportion_n gt_fw-mi to_o gz_n but_o the_o proportion_n of_o agnostus_n to_o gt_n be_v one_o and_o the_o self_n same_o with_o the_o proportion_n of_o the_o side_n bg_o to_o give_v by_o the_o first_o of_o this_o book_n and_o the_o proportion_n also_o of_o gt_n to_o gz_n be_v one_o and_o the_o self_n same_o with_o the_o proportion_n of_o the_o other_o side_n namely_o dg_o to_o ge_z by_o the_o same_o proposition_n wherefore_o the_o proportion_n of_o the_o parallelogram_n agnostus_n to_o gz_n consist_v of_o the_o proportion_n of_o the_o side_n bg_o to_o give_v and_o dc_o to_o ge._n wherefore_o equiangle_n parallelogram_n be_v the_o one_o to_o the_o other_o in_o that_o proportion_n which_o be_v compose_v of_o their_o side_n which_o be_v require_v to_o be_v prove_v the_o 18._o theorem_a the_o 24._o proposition_n in_o every_o parallelogram_n the_o parallelogram_n about_o the_o dimecient_a be_v like_a unto_o the_o whole_a and_o also_o like_o the_o one_o to_o the_o other_o abcd._n svppose_v that_o there_o be_v a_o parallelogram_n abcd_o and_o let_v the_o dimecient_a thereof_o be_v ac_fw-la and_o let_v the_o parallelogram_n about_o the_o dimecient_a ac_fw-la be_v eglantine_n and_o hk_o then_o i_o say_v that_o either_o of_o these_o parallelogram_n eglantine_n and_o hk_o be_v like_a unto_o the_o whole_a parallelogram_n abcd_o and_o also_o be_v like_o the_o one_o to_o the_o other_o for_o forasmuch_o as_o to_o one_o of_o the_o side_n of_o the_o triangle_n abc_n namely_o to_o bc_o be_v draw_v a_o parallel_a line_n of_o therefore_o as_o be_v be_v to_o ea_fw-la so_o by_o the_o 2._o of_o the_o six_o be_v cf_o to_o fa._n again_o forasmuch_o as_o to_o one_o of_o the_o side_n of_o the_o triangle_n adc_o namely_o to_o cd_o be_v draw_v a_o parallel_a line_n fâ—_n therefore_o by_o the_o same_o as_o cf_o be_v to_o favorina_n so_o be_v dg_o to_o ga._n but_o as_o cf_o be_v to_o favorina_n so_o be_v it_o prove_v that_o be_v be_v to_o ea_fw-la wherefore_o as_o be_v be_v to_o ea_fw-la so_o by_o the_o 11._o of_o the_o five_o â—_o be_v dg_fw-mi to_o ga._n wherefore_o by_o composition_n by_o the_o 18._o of_o the_o five_o as_o basilius_n be_v to_o aeâ—_n so_o be_v dam_n to_o ag._n and_o alternate_o by_o the_o 16._o of_o the_o five_o as_o basilius_n be_v to_o ad_fw-la so_o be_v ea_fw-la to_o ag._n wherefore_o in_o the_o parallelogrammesâ—_n abcd_o and_o eglantine_n the_o side_n which_o be_v about_o the_o common_a angle_n bad_a be_v proportional_a and_o because_o the_o line_n gf_o be_v a_o parallel_n unto_o the_o line_n dcâ—_n therefore_o the_o angle_n agf_n by_o the_o 29â—_n of_o theâ—_z first_o be_v equal_a unto_o the_o angle_v adc_o â—_o the_o angle_n gfa_n equal_a unto_o the_o angle_v dca_n and_o the_o angle_n dac_o be_v common_a to_o the_o two_o triangle_n adc_o and_o afg_n wherefore_o the_o triangle_n dac_o be_v equiangle_n unto_o the_o triangle_n agf_n and_o by_o the_o same_o reason_n the_o triangle_n abc_n be_v equiangle_n unto_o the_o triangle_n aef_n wherefore_o the_o whole_a parallelogram_n abcd_o be_v equiangle_n unto_o the_o parallelogram_n eglantine_n wherefore_o as_o ad_fw-la be_v in_o proportion_n to_o dc_o so_o by_o the_o 4._o of_o the_o six_o be_v agnostus_n to_o gf_o and_o as_o dc_o be_v to_o ca_n so_o be_v gf_o to_o fa._n and_o as_o ac_fw-la be_v to_o cb_o so_o be_v of_o to_o fe_o and_o moreover_o as_o cb_o be_v to_o ba_o so_o be_v fe_o to_o ea_fw-la and_o forasmuch_o as_o it_o be_v prove_v that_o as_o dâ—_n be_v to_o ca_n so_o be_v gf_o to_o favorina_n but_o as_o ac_fw-la be_v to_o câ—_n so_o be_v of_o to_o fe_o wherefore_o of_o equality_n by_o the_o 22._o of_o the_o five_o as_o dc_o be_v to_o cb_o so_o be_v gf_o to_o fe_o wherefore_o in_o the_o parallelogram_n abcd_o and_o
eglantine_n the_o side_n which_o include_v the_o equal_a angle_n be_v proportional_a wherefore_o the_o parallelogram_n abcd_o be_v by_o the_o first_o definition_n of_o the_o six_o like_v unto_o the_o parallelogram_n eglantine_n and_o by_o the_o same_o reason_n also_o the_o parallelogram_n abcd_o be_v like_a to_o the_o parallelogram_n kh_o abcd_o wherefore_o either_o of_o these_o parallelogram_n eglantine_n and_o kh_o be_v like_a unto_o the_o parallelogram_n abcd._n but_o rectiline_a figure_n which_o be_v like_a to_o one_o and_o the_o same_o rectiline_a figure_n be_v also_o by_o the_o 21._o of_o the_o six_o like_o the_o one_o to_o the_o other_o wherefore_o the_o parallelogram_n eglantine_n be_v like_a to_o the_o parallelogram_n hk_o other_o wherefore_o in_o every_o parallelogram_n the_o parallelogram_n about_o the_o dimecient_a be_v like_a unto_o the_o whole_a and_o also_o like_o the_o one_o to_o the_o other_o which_o be_v require_v to_o be_v prove_v ¶_o a_o other_o more_o brief_a demonstration_n after_o flussates_n suppose_v that_o there_o be_v a_o parallelogram_n abcd_o who_o dimetient_a let_v bâ—_z aâ—_z flussates_n about_o which_o let_v consist_v these_o parallelogram_n eke_o and_o ti_o have_v the_o angle_n at_o the_o point_n â—_o and_o 〈…〉_o with_o the_o whole_a parallelogram_n abcd._n then_o i_o say_v that_o those_o parallelogram_n eke_o and_o ti_fw-mi be_v like_a to_o the_o whole_a parallelogram_n db_o and_o also_o alâ—_z like_o the_o one_o to_o the_o other_o for_o forasmuch_o as_o bd_o eke_o and_o ti_fw-mi be_v parallelogram_n therefore_o the_o right_a line_n azg_v fall_v upon_o these_o parallel_a line_n aeb_fw-mi kzt_n and_o di_z g_z or_o upon_o these_o parallel_a line_n akd_v ezi_n and_o btg_n make_v these_o angle_n equal_a the_o one_o to_o the_o other_o namely_o the_o angle_n eaz_n to_o the_o angle_n kza_n &_o the_o angle_n eza_n to_o the_o angle_n kaz_n and_o the_o angle_n tzg_v to_o the_o angle_n zgi_n and_o the_o angle_n tgz_n to_o the_o angle_n izg_v and_o the_o angle_n bag_n to_o the_o angle_n agd_v and_o final_o the_o angle_n bga_n to_o the_o angle_n dag_n wherefore_o by_o the_o first_o corollary_n of_o the_o 32._o of_o the_o first_o and_o by_o the_o 34._o of_o the_o first_o the_o angle_n remain_v be_v equal_v the_o one_o to_o the_o other_o namely_o the_o angle_n b_o to_o the_o angle_n d_o and_o the_o angle_n e_o to_o the_o angle_n king_n and_o the_o angle_n t_o to_o the_o angle_n i_o wherefore_o these_o triangle_n be_v equiangle_n and_o therefore_o like_o the_o one_o to_o the_o other_o namely_o the_o triangle_n abg_o to_o the_o triangle_n gda_n and_o the_o triangle_n aez_n to_o the_o triangle_n zka_n &_o the_o triangle_n ztg_n to_o the_o triangle_n giz._n wherefore_o as_o the_o side_n ab_fw-la be_v to_o the_o side_n bg_o so_o be_v the_o side_n ae_n to_o the_o side_n ez_fw-fr and_o the_o side_n zt_n to_o the_o side_n tg_n wherefore_o the_o parallelogram_n contain_v under_o those_o right_a line_n namely_o the_o parallelogram_n abgd_v eke_o &_o ti_fw-mi be_v like_o the_o one_o to_o the_o other_o by_o the_o first_o definition_n of_o this_o book_n wherefore_o in_o every_o parallelogram_n the_o parallelogram_n etc_n etc_n as_o before_o which_o be_v require_v to_o be_v demonstrate_v ¶_o a_o problem_n add_v by_o pelitarius_n two_o equiangle_n parallelogrammes_n be_v give_v so_o that_o they_o be_v not_o like_a to_o cut_v of_o from_o one_o of_o they_o a_o parallelogram_n like_a unto_o the_o other_o pelitarius_n suppose_v that_o the_o two_o equiangle_n parallelogram_n be_v abcd_o and_o cefg_n which_o let_v not_o be_v like_o the_o one_o to_o the_o other_o it_o be_v require_v from_o the_o parallelogram_n abcd_o to_o cut_v of_o a_o parallelogram_n like_a unto_o the_o parallelogram_n cefg_n let_v the_o angle_n c_o of_o the_o one_o be_v equal_a to_o the_o angle_n c_o of_o the_o other_o and_o let_v the_o two_o parallelogram_n be_v so_o 〈◊〉_d that_o the_o line_n bc_o &_o cg_o may_v make_v both_o one_o right_a line_n namely_o bg_o wherefore_o also_o the_o right_a line_n dc_o and_o ce_fw-fr shall_v both_o make_v one_o right_a line_n namely_o de._n and_o draw_v a_o line_n from_o the_o point_n f_o to_o the_o point_n c_o and_o produce_v the_o line_n fc_o till_o it_o concur_v with_o the_o line_n ad_fw-la in_o the_o point_n h._n and_o draw_v the_o line_n hk_o parallel_v to_o the_o line_n cd_o by_o the_o 31._o of_o the_o first_o then_o i_o say_v that_o from_o the_o parallelogram_n ac_fw-la be_v cut_v of_o the_o parallelogram_n cdhk_n like_v unto_o the_o parallelogram_n eglantine_n which_o thing_n be_v manifest_a by_o this_o 24._o proposition_n for_o that_o both_o the_o say_v parallelogram_n be_v describe_v about_o one_o &_o the_o self_n same_o dimetient_fw-la and_o to_o the_o end_n it_o may_v the_o more_o plain_o be_v see_v i_o have_v make_v complete_a the_o parallelogram_n abgl_n ¶_o a_o other_o problem_n add_v by_o pelitarius_n between_o two_o rectiline_a superficiece_n to_o find_v out_o a_o mean_a superficies_n proportional_a pelitarius_n suppose_v that_o the_o two_o superficiece_n be_v a_o and_o b_o between_o which_o it_o be_v require_v to_o place_v a_o mean_a superficies_n proportional_a reduce_v the_o say_v two_o rectiline_a figure_n a_o and_o b_o unto_o two_o like_a parallelogram_n by_o the_o 18._o of_o this_o book_n or_o if_o you_o think_v good_a reduce_v either_o of_o they_o to_o a_o square_a by_o the_o last_o of_o the_o second_o and_o let_v the_o say_v two_o parallelogram_n like_o the_o one_o to_o the_o other_o and_o equal_a to_o the_o superficiece_n a_o and_o b_o be_v cdef_n and_o fghk_n and_o let_v the_o angle_n fletcher_n in_o either_o of_o they_o be_v equal_a which_o two_o angle_n let_v be_v place_v in_o such_o sort_n that_o the_o two_o parallelogram_n ed_z and_o hg_o may_v be_v about_o one_o and_o the_o self_n same_o dimetient_fw-la ck_o which_o be_v do_v by_o put_v the_o right_a line_n of_o and_o fg_o in_o such_o sort_n that_o they_o both_o make_v one_o right_a line_n namely_o eglantine_n and_o make_v complete_a the_o parallelogram_n clk_n m._n then_o i_o say_v that_o either_o of_o the_o supplement_n fl_fw-mi &_o fm_o be_v a_o mean_a proportional_a between_o the_o superficiece_n cf_o &_o fk_o that_o be_v between_o the_o superficiece_n a_o and_o b_o namely_o as_o the_o superficies_n hg_o be_v to_o the_o superficies_n fl_fw-mi so_o be_v the_o same_o superficies_n fl_fw-mi to_o the_o superficies_n ed._n for_o by_o this_o 24._o proposition_n the_o line_n hf_o be_v to_o the_o line_n fd_o as_o the_o line_n gf_o be_v to_o the_o line_n fe_o but_o by_o the_o first_o of_o this_o book_n as_o the_o line_n hf_o be_v to_o the_o line_n fd_o so_o be_v the_o superficies_n hg_o to_o the_o superficies_n fl_fw-mi and_o as_o the_o line_n gf_o be_v to_o the_o line_n fe_o so_o also_o by_o the_o same_o be_v the_o superficies_n fl_fw-mi to_o the_o superficies_n ed._n wherefore_o by_o the_o 11._o of_o the_o five_o as_o the_o superficies_n hg_o be_v to_o the_o superficies_n fl_fw-mi so_o be_v the_o same_o superficies_n fl_fw-mi to_o the_o superficies_n ed_z which_o be_v require_v to_o be_v do_v the_o 7._o problem_n the_o 25._o proposition_n unto_o a_o rectiline_a figure_n give_v to_o describe_v a_o other_o figure_n like_o which_o shall_v also_o be_v equal_a unto_o a_o other_o rectiline_a figure_n give_v svppose_v that_o the_o rectiline_a figure_n give_v whereunto_o be_v require_v a_o other_o to_o be_v make_v like_o be_v abc_n and_o let_v the_o other_o rectiline_a figure_n whereunto_o the_o same_o be_v require_v to_o be_v make_v equal_a be_v d._n now_o it_o be_v require_v to_o describe_v a_o rectiline_a figure_n like_o unto_o the_o figure_n abc_n and_o equal_a unto_o the_o figure_n d._n construction_n upon_o the_o line_n bc_o describe_v by_o the_o 44._o of_o the_o first_o a_o parallelogram_n be_v equal_a unto_o the_o triangle_n abc_n and_o by_o the_o same_o upon_o the_o line_n ce_fw-fr describe_v the_o parallelogram_n c_o m_o equal_a unto_o the_o rectiline_a figure_n d_o and_o in_o the_o say_a parallelogram_n let_v the_o angle_n fce_n be_v equal_a unto_o the_o angle_n cbl_n and_o forasmuch_o as_o the_o angle_n fce_n be_v by_o construction_n equal_a to_o the_o angle_n cbl_n demonstration_n add_v the_o angle_n be_v common_a to_o they_o both_o wherefore_o the_o angle_n lbc_n and_o be_v be_v equal_a unto_o the_o angle_n be_v and_o ecf_n but_o the_o angle_n lbc_n and_o be_v be_v equal_a to_o two_o right_a angle_n by_o the_o 29._o of_o the_o first_o wherefore_o also_o the_o angle_n be_v and_o ecf_n be_v equal_a to_o two_o right_a angle_n wherefore_o the_o line_n bc_o and_o cf_o by_o the_o 14._o of_o the_o first_o make_v both_o one_o right_a line_n namely_o bf_o and_o in_o like_a sort_n do_v the_o line_n le_fw-fr and_o em_n make_v both_o one_o right_a line_n namely_o lm_o then_o by_o the_o
triangle_n abc_n be_v by_o the_o 6._o of_o the_o six_o equiangle_n unto_o the_o triangle_n dce_n wherefore_o the_o angle_n abc_n be_v equal_a unto_o the_o angle_n dce_n and_o it_o be_v prove_v that_o the_o angle_n acd_o be_v equal_a unto_o the_o angle_n bac_n wherefore_o the_o whole_a angle_n ace_n be_v equal_a unto_o the_o two_o angle_n abc_n and_o bac_n put_v the_o angle_n acb_o common_a to_o they_o both_o wherefore_o the_o angle_n ace_n and_o acb_o be_v equal_a unto_o the_o angle_n cab_n acb_o &_o cba_o but_o the_o angle_n cab_n acb_o and_o cba_o be_v by_o the_o 32._o of_o the_o first_o equal_a unto_o two_o right_a angle_n wherefore_o also_o the_o angle_n ace_n and_o acb_o be_v equal_a to_o two_o right_a angle_n now_o then_o unto_o the_o right_a line_n ac_fw-la and_o unto_o the_o point_n in_o it_o c_o be_v draw_v two_o right_a line_n bc_o and_o ce_fw-fr not_o on_o one_o and_o the_o same_o side_n make_v the_o side_n angle_n ace_n &_o acb_o equal_a to_o two_o right_a angle_n wherefore_o the_o line_n bc_o and_o ce_fw-fr by_o the_o 14._o of_o the_o first_o be_v set_v direct_o and_o do_v make_v one_o right_a line_n if_o therefore_o two_o triangle_n be_v set_v together_o at_o one_o angle_n have_v two_o side_n of_o the_o one_o proportional_a to_o two_o side_n of_o the_o outstretch_o so_o that_o their_o side_n of_o like_a proportion_n be_v also_o parallel_v then_o the_o side_n remain_v of_o those_o triangle_n shall_v be_v in_o one_o right_a line_n which_o be_v require_v to_o be_v prove_v although_o euclid_n do_v not_o distinct_o set_v forth_o the_o manner_n of_o proportion_n of_o like_a rectiline_a figure_n as_o he_o do_v of_o line_n in_o the_o 10._o proposition_n of_o this_o book_n and_o in_o the_o 3._o follow_v it_o yet_o as_o flussates_n note_v be_v that_o not_o hard_a to_o be_v do_v by_o the_o 22._o of_o this_o book_n â—or_a two_o like_o rectiline_a figure_n be_v give_v to_o find_v out_o a_o three_o proportional_o also_o between_o two_o rectiline_a superficiece_n give_v to_o find_v out_o a_o mean_a proportional_a which_o we_o before_o teach_v to_o do_v by_o pelitarius_n after_o the_o 24._o proposition_n of_o this_o book_n and_o moreover_o three_o like_o rectiline_a figure_n be_v give_v to_o find_v out_o a_o four_o proportional_a like_a and_o in_o like_a sort_n describe_v and_o such_o kind_n of_o proportion_n be_v easy_a to_o be_v find_v out_o by_o the_o proportion_n of_o line_n as_o thus_o if_o unto_o two_o side_n of_o like_a proportion_n we_o shall_v find_v out_o a_o three_o proportional_a by_o the_o 11._o of_o this_o bokeâ—_n the_o rectiline_a figure_n describe_v upon_o that_o line_n shall_v be_v the_o three_o rectiline_a figure_n proportional_a with_o the_o two_o first_o figure_n give_v by_o the_o 22._o of_o this_o book_n and_o if_o between_o two_o side_n of_o like_a proportion_n be_v take_v a_o mean_a proportional_a by_o the_o 13._o of_o this_o book_n the_o rectiline_a â—igure_n describe_v upon_o the_o say_a means_n shall_v likewise_o be_v a_o mean_a proportional_a between_o the_o two_o rectiline_a figure_n give_v by_o the_o same_o 22._o of_o the_o six_o and_o so_o if_o unto_o three_o side_n geâ—en_o be_v find_v out_o the_o four_o side_n proportional_a by_o the_o 12._o of_o this_o book_n the_o rectiline_a â—igure_n describe_v upon_o the_o say_v four_o line_n shall_v be_v the_o four_o rectiline_a figure_n proportional_a for_o if_o the_o right_a line_n be_v proportional_a the_o rectiline_a figure_n describe_v upon_o they_o shall_v also_o be_v proportional_a so_o that_o the_o say_v rectiline_a â—_z be_v like_o &_o in_o like_a sort_n describe_v by_o the_o say_v 22._o of_o the_o six_o the_o 23._o theorem_a the_o 33._o proposition_n in_o equal_a circle_n the_o angle_n have_v one_o and_o the_o self_n same_o proportion_n that_o the_o circumference_n have_v wherein_o they_o consist_v whether_o the_o angle_n be_v set_v at_o the_o centre_n or_o at_o the_o circumference_n and_o in_o like_a sort_n be_v the_o sector_n which_o be_v describe_v upon_o the_o centre_n svppose_v the_o equal_a circle_n to_o be_v abc_n and_o def_n who_o centre_n let_v be_v g_o and_o h_o and_o let_v the_o angle_n set_v at_o their_o centre_n g_z and_o h_o be_v bgc_o and_o ehf_o and_o let_v the_o angle_n set_v at_o their_o circumference_n be_v bac_n and_o edf_o then_o i_o say_v that_o as_o the_o circumference_n bc_o be_v to_o the_o circumference_n of_o so_o be_v the_o angle_n bgc_o to_o the_o angle_n ehf_o and_o the_o angle_n bac_n to_o the_o angle_n edf_o and_o moreover_o the_o sector_n gbc_n to_o the_o sector_n hef_fw-fr unto_o the_o circumference_n bc_o by_o the_o 38._o of_o the_o three_o put_v as_o many_o equal_a circumference_n in_o order_n as_o you_o will_v namely_o ck_n and_o kl_o and_o unto_o the_o circumference_n of_o put_v also_o as_o many_o equal_a circumference_n in_o number_n as_o you_o will_v namely_o fm_o and_o mn_v and_o draw_v these_o right_a line_n gk_o be_v gl_n hm_n and_o hn._n now_o forasmuch_o as_o the_o circumference_n bc_o ck_n and_o kl_o be_v equal_a the_o one_o to_o the_o other_o the_o angle_n also_o bgc_o and_o cgk_n and_o kgl_n be_v by_o the_o 27._o of_o the_o three_o equal_v the_o one_o to_o the_o other_o therefore_o how_o multiplex_n the_o circumference_n bl_o be_v to_o the_o circumference_n bc_o so_o multiplex_n be_v the_o angle_n bgl_n to_o the_o angle_n bgc_o and_o by_o the_o same_o reason_n also_o how_o multiplex_n the_o circumference_n ne_fw-fr be_v to_o the_o circumference_n of_o so_o multiplex_n be_v the_o angle_n the_o to_o the_o angle_n ehf_o wherefore_o if_o the_o circumference_n bl_o be_v equal_a unto_o the_o circumference_n en_fw-fr the_o angle_n bgl_n be_v equal_a unto_o the_o angle_n ehn_n and_o if_o the_o circumference_n bl_o be_v great_a than_o the_o circumference_n en_fw-fr the_o angle_n bgl_n be_v great_a than_o the_o angle_n the_o and_o if_o the_o circumference_n be_v less_o the_o angle_n be_v less_o now_o then_o there_o be_v four_o magnitude_n namely_o the_o two_o circumference_n bc_o and_o of_o and_o the_o two_o angle_n that_o be_v bgc_o and_o ehf_o and_o to_o the_o circumference_n bc_o and_o to_o the_o angle_n bgc_o that_o be_v to_o the_o first_o and_o three_o be_v take_v equemultiplices_fw-la namely_o the_o circumference_n bl_o and_o the_o angle_n bgl_n and_o likewise_o to_o the_o circumference_n of_o and_o to_o the_o angle_n ehf_o that_o be_v to_o the_o second_o and_o four_o be_v take_v certain_a other_o equemultiplices_fw-la namely_o the_o circumference_n en_fw-fr and_o the_o angle_n ehn._n and_o it_o be_v prove_v that_o if_o the_o circumference_n bl_o exceed_v the_o circumference_n en_fw-fr the_o angle_n also_o bgl_n exceed_v the_o angle_n ehn._n and_o if_o the_o circumference_n be_v equal_a the_o angle_n be_v equal_a and_o if_o the_o circumference_n be_v less_o the_o angle_n also_o be_v less_o wherefore_o by_o the_o 6._o definition_n of_o the_o five_o as_o the_o circumference_n bc_o be_v to_o the_o circumference_n of_o so_o be_v the_o angle_n bgc_o to_o the_o angle_n ehfâ—_n but_o as_o the_o angle_n bgc_o be_v to_o the_o angle_n ehf_o also_o so_o be_v the_o angle_n bac_n to_o the_o angle_n edf_o for_o the_o angle_n bgc_o be_v double_a to_o the_o angle_n bac_n and_o the_o angle_n ehf_o be_v also_o double_a to_o the_o angle_n edf_o by_o the_o 20._o of_o the_o three_o wherefore_o as_o the_o circumference_n bc_o be_v to_o the_o circumference_n of_o so_o be_v the_o angle_n bgc_o to_o the_o angle_n ehf_o and_o the_o angle_n bac_n to_o the_o angle_n edf_o wherefore_o in_o equal_a circle_n the_o angle_n be_v in_o one_o and_o the_o self_n same_o proportion_n that_o their_o circumference_n be_v whether_o the_o angle_n be_v set_v at_o the_o centre_n or_o at_o the_o circumference_n which_o be_v require_v to_o be_v prove_v i_o say_v moreover_o that_o as_o the_o circumference_n bc_o be_v to_o the_o circumference_n of_o also_o so_o be_v the_o sector_n gbc_n to_o the_o sector_n hef_fw-fr draw_v these_o line_n bc_o and_o ck_n and_o in_o the_o circumference_n bc_o and_o ck_n take_v point_n at_o all_o adventure_n namely_o pea_n and_o o._n and_o draw_v line_n from_o b_o to_o pea_n and_o from_o p_o to_o c_o from_o c_z to_o oh_o and_o from_o oh_o to_o k._n and_o forasmuch_o as_o by_o the_o 15._o definition_n of_o the_o first_o the_o two_o line_n bg_o and_o gc_o be_v equal_a unto_o the_o two_o line_n cg_o and_o gk_o and_o they_o also_o comprehend_v equal_a angle_n therefore_o by_o the_o 4._o of_o the_o first_o the_o base_a bc_o be_v equal_a unto_o the_o base_a ck_n &_o the_o triangle_n gbc_n be_v equal_a unto_o the_o triangle_n gck._n and_o see_v that_o the_o circumference_n bc_o be_v equal_a unto_o the_o circumference_n ck_o therefore_o the_o circumference_n remain_v of_o the_o whole_a circle_n abc_n
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
the_o line_n cd_o so_o be_v the_o parallelogram_n contain_v under_o the_o line_n ab_fw-la and_o bc_o to_o the_o parallelogram_n contain_v under_o the_o line_n bc_o and_o cd_o construction_n from_o the_o point_n a_o raise_v up_o unto_o the_o line_n ab_fw-la a_o perpendicular_a line_n ae_n and_o let_v ae_n be_v equal_a to_o the_o line_n bc_o and_o by_o the_o point_n e_o draw_v unto_o the_o line_n ad_fw-la a_o parallel_n line_n eke_o and_o by_o every_o one_o of_o the_o point_n b_o c_o and_o d_o draw_v unto_o the_o line_n ae_n parallel_a line_n bf_o change_z and_o dk_o and_o for_o that_o as_o the_o line_n ab_fw-la be_v to_o the_o line_n bc_o so_o be_v the_o parallelogram_n of_o to_o the_o parallelogram_n bh_o by_o the_o first_o of_o the_o six_o demonstration_n and_o as_o the_o line_n bc_o be_v to_o the_o linâ—_n cd_o so_o be_v the_o parallelogram_n bh_o to_o the_o parallelogram_n ck_o wherefore_o of_o equality_n as_o the_o line_n ab_fw-la be_v to_o the_o line_n cd_o so_o be_v the_o parallelogram_n of_o to_o the_o parallelogram_n ck_o but_o the_o parallelogram_n of_o be_v that_o which_o be_v contain_v under_o the_o line_n ab_fw-la and_o bc_o for_o the_o line_n ae_n be_v put_v equal_a to_o the_o line_n bc._n and_o the_o parallelogram_n ck_o be_v that_o which_o be_v contain_v under_o the_o line_n bc_o and_o cd_o for_o the_o line_n bc_o be_v equal_a to_o the_o line_n change_z for_o that_o the_o line_n change_z be_v equal_a to_o the_o line_n ae_n by_o the_o 34._o of_o the_o first_o if_o therefore_o there_o be_v three_o right_a line_n have_v between_o themselves_o any_o proportion_n as_o the_o first_o be_v to_o the_o three_o so_o be_v the_o parallelogram_n contain_v under_o the_o first_o and_o the_o second_o to_o the_o parallelogram_n contain_v under_o the_o second_o and_o the_o three_o which_o be_v require_v to_o be_v demonstrate_v ¶_o the_o 9_o problem_n the_o 32._o proposition_n to_o find_v out_o two_o medial_a line_n commensurable_a in_o power_n only_o comprehend_v a_o medial_a superficies_n so_o that_o the_o great_a shall_v be_v in_o power_n more_o than_o the_o less_o by_o the_o square_n of_o a_o line_n commensurable_a in_o length_n unto_o the_o great_a let_v there_o be_v take_v three_o rational_a line_n commensurable_a in_o power_n only_o a_o b_o c_o so_fw-mi that_o by_o the_o 29._o of_o the_o ten_o let_v the_o line_n a_o be_v in_o power_n more_o than_o the_o line_n c_o by_o the_o square_n of_o a_o line_n commensurable_a in_o length_n unto_o the_o line_n a._n construction_n and_o unto_o the_o parallelogram_n contain_v under_o the_o line_n a_o &_o b_o let_v the_o square_n of_o the_o line_n d_o be_v equal_a but_o that_o which_o be_v contain_v under_o the_o line_n a_o and_o b_o be_v medial_a wherefore_o by_o the_o corollary_n of_o the_o 23._o of_o the_o ten_o the_o square_a of_o the_o line_n d_o also_o be_v medial_a wherefore_o the_o line_n d_o also_o be_v medial_a and_o unto_o that_o which_o be_v contain_v under_o the_o line_n b_o and_o c_o let_v be_v equal_a that_o which_o be_v contain_v under_o the_o line_n d_o and_o e_o which_o be_v do_v by_o â—inding_v out_o a_o four_o line_n proportional_a unto_o the_o line_n d_o b_o c_o which_o let_v be_v the_o line_n e_o demonstration_n and_o for_o that_o by_o the_o assumpt_n go_v before_o as_o that_o which_o be_v contain_v under_o the_o line_n a_o and_o b_o be_v to_o that_o which_o be_v contain_v under_o the_o line_n b_o and_o c_o so_o be_v the_o line_n a_o to_o the_o line_n c._n but_o unto_o that_o which_o be_v contain_v under_o the_o line_n a_o &_o b_o be_v equal_a the_o square_n of_o the_o line_n d_o and_o unto_o that_o which_o be_v contain_v under_o the_o line_n b_o &_o c_o be_v equal_a that_o which_o be_v contain_v under_o the_o line_n d_o and_o e._n wherefore_o as_o the_o line_n a_o be_v to_o the_o line_n c_o so_fw-mi iâ—_z the_o square_a of_o the_o line_n d_o to_o that_o which_o be_v contain_v under_o the_o line_n d_o and_o e._n but_o as_o the_o square_n of_o the_o line_n d_o be_v to_o that_o which_o be_v contain_v under_o the_o line_n d_o and_o e_o so_o be_v the_o line_n d_o to_o the_o line_n e_o by_o the_o assumpt_n put_v before_o the_o 22._o of_o the_o ten_o wherefore_o as_o the_o line_n a_o be_v to_o the_o line_n c_o so_o be_v the_o line_n d_o to_o the_o line_n e._n but_o the_o line_n a_o be_v unto_o the_o line_n c_o commensurable_a in_o power_n only_o wherefore_o the_o line_n d_o be_v unto_o the_o line_n e_o commensurable_a in_o power_n only_o but_o d_z be_v a_o medial_a line_n wherefore_o by_o the_o 23._o of_o the_o ten_o e_o also_o be_v a_o medial_a line_n and_o for_o that_o as_o the_o line_n a_o be_v to_o the_o line_n c_o so_o be_v the_o line_n d_o to_o the_o line_n e_o and_o the_o line_n a_o be_v in_o power_n more_o than_o the_o line_n c_o by_o the_o square_n of_o a_o line_n commensurable_a in_o length_n unto_o the_o line_n a._n wherefore_o by_o the_o 14._o of_o the_o ten_o d_o be_v in_o power_n more_o than_o e_o by_o the_o square_n of_o a_o line_n commensurable_a in_o length_n unto_o the_o line_n d._n i_o say_v moreover_o that_o that_o which_o be_v contain_v under_o the_o line_n d_o and_o e_o be_v medial_a for_o forasmuch_o as_o that_o which_o be_v contain_v under_o the_o line_n b_o &_o c_o be_v equal_a to_o that_o which_o be_v contain_v under_o the_o line_n d_o and_o e_o but_o that_o which_o be_v contain_v under_o the_o line_n b_o and_o c_o be_v medial_a wherefore_o that_o which_o be_v contain_v under_o the_o line_n d_o and_o e_o be_v also_o medial_a wherefore_o there_o be_v find_v out_o two_o medial_a line_n d_z and_o e_o commensurable_a in_o power_n only_o comprehend_v a_o medial_a superficies_n so_o that_o the_o great_a be_v in_o power_n more_o than_o the_o less_o by_o the_o square_n of_o a_o line_n commensurable_a in_o length_n to_o the_o great_a which_o be_v require_v to_o be_v do_v and_o thus_o iâ—_z iâ—_z euâ—deâ—t_n how_o in_o like_a sort_n may_v be_v find_v out_o two_o medial_a line_n commâ—â—surable_a in_o power_n only_o corollary_n contain_v a_o medial_a superficies_n so_o that_o the_o great_a shall_v be_v in_o power_n more_o theâ—_z the_o less_o by_o the_o square_n of_o a_o line_n incommensurable_a in_o length_n unto_o the_o great_a when_o the_o line_n a_o be_v in_o power_n more_o than_o the_o line_n c_o by_o the_o square_n of_o a_o line_n incommensurable_a in_o length_n unto_o the_o line_n a_o as_o the_o thirty_o teach_v us._n 1._o ¶_o a_o assumpt_n suppose_v that_o there_o be_v a_o rectangle_n triangle_n abc_n have_v the_o angle_n bac_n a_o right_a angle_n and_o â—y_a the_o 12._o of_o the_o first_o from_o the_o point_n a_o to_o the_o right_a line_n bc_o a_o perpendicular_a line_n be_v draw_v ad_fw-la then_o i_o say_v first_o that_o the_o parallelogram_n contain_v under_o the_o line_n câ—_n and_o bd_o be_v equal_a to_o the_o square_n of_o the_o line_n basilius_n second_o i_o say_v that_o the_o parallelogram_n contain_v under_o the_o line_n bc_o and_o cd_o be_v equal_a to_o the_o square_n of_o the_o line_n ca._n three_o i_o say_v that_o the_o parallelogram_n contain_v under_o the_o line_n bd_o and_o dc_o be_v equal_a to_o the_o square_n of_o the_o line_n ad._n and_o fourthly_a i_o say_v that_o the_o parallelogram_n contain_v under_o the_o line_n bc_o &_o ad_fw-la be_v equal_a to_o the_o parallelogram_n contain_v under_o the_o line_n basilius_n &_o ac_fw-la as_o touch_v the_o second_o that_o the_o parellelogramme_n contain_v under_o the_o line_n bâ—_n and_o cd_o be_v equal_a to_o the_o square_n of_o the_o line_n ac_fw-la be_v by_o the_o self_n same_o reason_n prove_v for_o the_o triangle_n abc_n be_v like_a to_o the_o triangle_n adc_o wherefore_o as_o the_o line_n bc_o be_v to_o the_o line_n ac_fw-la dee_n so_o be_v the_o line_n ac_fw-la to_o the_o line_n dc_o corollary_n wherefore_o the_o parallelogram_n contain_v under_o the_o line_n bc_o and_o cd_o be_v equal_a to_o the_o square_n of_o the_o line_n ac_fw-la corollary_n as_o touch_v the_o three_o that_o the_o parallelogram_n contain_v under_o the_o line_n bd_o and_o dc_o be_v equal_a to_o the_o square_n of_o the_o line_n dam_fw-ge be_v thus_o prove_v for_o forasmuch_o as_o if_o in_o a_o rectangle_n triangle_n be_v draw_v from_o the_o right_a angle_n to_o the_o base_a a_o perpendicular_a line_n the_o perpendicular_a so_o draw_v be_v the_o mean_a proportional_a between_o the_o segment_n of_o the_o base_a by_o the_o corollary_n of_o the_o 8._o of_o the_o six_o therefore_o as_o the_o line_n bd_o be_v to_o the_o line_n dam_fw-ge so_o be_v thâ—_z line_n ad_fw-la to_o the_o line_n dc_o wherefore_o by_o the_o 1â—_o of_o the_o six_o the_o parallelogram_n
rational_a part_n rather_o than_o of_o the_o medial_a part_n ¶_o the_o 28._o theorem_a the_o 40._o proposition_n if_o two_o right_a line_n incommensurable_a in_o power_n be_v add_v together_o have_v that_o which_o be_v make_v of_o the_o square_n of_o they_o add_v together_o medial_a and_o the_o parallelogram_n contain_v under_o they_o rational_a the_o whole_a right_a line_n be_v irrational_a and_o be_v call_v a_o line_n contain_v in_o power_n a_o rational_a and_o a_o medial_a superficies_n in_o this_o proposition_n be_v teach_v the_o generation_n of_o the_o six_o irrational_a line_n which_o be_v call_v a_o line_n who_o power_n be_v rational_a and_o medial_a the_o definition_n of_o which_o be_v gather_v of_o this_o proposition_n after_o this_o manner_n a_o line_n who_o power_n be_v rational_a and_o medial_a be_v a_o irrational_a line_n which_o be_v make_v of_o two_o right_a line_n incommensurable_a in_o power_n add_v together_o medial_a who_o square_n add_v together_o make_v a_o medial_a superficies_n but_o that_o supersicy_n which_o they_o contain_v be_v rational_a the_o reason_n of_o the_o name_n be_v before_o set_v forth_o in_o the_o proposition_n ¶_o the_o 29._o theorem_a the_o 41._o proposition_n if_o two_o right_a line_n incommensurable_a in_o power_n be_v add_v together_o have_v that_o which_o be_v compose_v of_o the_o square_n of_o they_o add_v together_o medial_a and_o the_o parallelogram_n contain_v under_o they_o medial_a and_o also_o incommensurable_a to_o that_o which_o be_v compose_v of_o the_o square_n of_o they_o add_v togethers_o the_o whole_a right_a line_n be_v irrational_a and_o be_v call_v a_o line_n contain_v in_o power_n two_o medial_n in_o this_o proposition_n be_v teach_v the_o nature_n of_o the_o 7._o kind_n of_o irrational_a line_n which_o be_v call_v a_o line_n who_o power_n be_v two_o medial_n the_o definition_n whereof_o be_v take_v of_o this_o proposition_n after_o this_o manner_n a_o line_n who_o power_n be_v two_o medial_n be_v a_o irrational_a line_n which_o be_v compose_v of_o two_o right_a line_n incommensurable_a in_o power_n medial_n the_o square_n of_o which_o add_v together_o make_v a_o medial_a superficies_n and_o that_o which_o be_v contain_v under_o they_o be_v also_o medial_a and_o moreover_o it_o be_v incommensurable_a to_o that_o which_o be_v compose_v of_o the_o two_o square_n add_v together_o the_o reason_n why_o this_o line_n be_v call_v a_o line_n who_o power_n be_v two_o medial_n be_v before_o in_o the_o end_n of_o the_o demonstration_n declare_v and_o that_o the_o say_v irrational_a line_n be_v divide_v one_o way_n only_o that_o be_v in_o one_o point_n only_o into_o the_o right_a line_n of_o which_o they_o be_v compose_v and_o which_o make_v every_o one_o of_o the_o kind_n of_o those_o irrational_a line_n shall_v straight_o way_n be_v demonstrate_v but_o first_o will_v we_o demonstrate_v two_o assumpte_n here_o follow_v ¶_o a_o assumpt_n take_v a_o right_a line_n and_o let_v the_o same_o be_v ab_fw-la and_o divide_v it_o into_o two_o unequal_a part_n in_o the_o point_n c_o assumpt_n and_o again_o divide_v the_o same_o line_n ab_fw-la into_o two_o other_o unequal_a part_n in_o a_o other_o point_n namely_o in_o d_o and_o let_v the_o line_n ac_fw-la by_o supposition_n be_v great_a than_o the_o line_n db._n then_o i_o say_v that_o the_o square_n of_o the_o line_n ac_fw-la and_o bc_o add_v together_o be_v great_a than_o the_o square_n of_o the_o line_n ad_fw-la and_o db_o add_v together_o divide_v the_o line_n ab_fw-la by_o the_o 10._o of_o the_o first_o into_o two_o equal_a part_n in_o the_o point_n e._n and_o forasmuch_o as_o the_o line_n ac_fw-la be_v great_a than_o the_o line_n db_o take_v away_o the_o line_n dc_o which_o be_v common_a to_o they_o both_o wherefore_o the_o residue_n ad_fw-la be_v great_a than_o the_o residue_n cb_o but_o the_o line_n ae_n be_v equal_a to_o the_o line_n ebb_n wherefore_o the_o line_n de_fw-fr be_v less_o than_o the_o line_n ec_o wherefore_o the_o point_n c_o and_o d_o be_v not_o equal_o distant_a from_o the_o point_n e_o which_o be_v the_o point_n of_o the_o section_n into_o two_o equal_a part_n and_o forasmuch_o as_o by_o the_o 5._o of_o the_o second_o that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o together_o with_o the_o square_n of_o the_o line_n ec_o be_v equal_a to_o the_o square_n of_o the_o line_n ebb_n and_o by_o the_o same_o reason_n that_o which_o be_v contain_v under_o the_o line_n ad_fw-la and_o db_o together_o with_o the_o square_n of_o the_o line_n de_fw-fr be_v also_o equal_a to_o the_o self_n same_o square_n of_o the_o line_n ebb_v wherefore_o that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o together_o with_o the_o square_n of_o the_o line_n ec_o be_v equal_a to_o that_o which_o be_v contain_v under_o the_o line_n ad_fw-la and_o db_o together_o with_o the_o square_n of_o the_o line_n de_fw-fr of_o which_o the_o square_a of_o the_o line_n de_fw-fr be_v less_o than_o the_o square_a of_o the_o line_n ec_o for_o it_o be_v prove_v that_o the_o line_n de_fw-fr be_v less_o than_o the_o line_n ec_o wherefore_o the_o parallelogram_n remain_v contain_v under_o the_o line_n ac_fw-la and_o cb_o be_v less_o they_o the_o parallelogram_n remain_v contain_v under_o the_o line_n ad_fw-la and_o db._n wherefore_o also_o that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o twice_o be_v less_o than_o that_o which_o be_v contain_v under_o the_o line_n ad_fw-la and_o db_o twice_o but_o by_o the_o four_o of_o the_o second_o the_o square_a of_o the_o whole_a line_n ab_fw-la be_v equal_a to_o that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ac_fw-la and_o cb_o together_o with_o that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o twice_o and_o by_o the_o same_o reason_n the_o square_n of_o the_o whole_a line_n ab_fw-la be_v equal_a to_o that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ad_fw-la and_o db_o together_o with_o that_o which_o be_v contain_v under_o the_o line_n ad_fw-la and_o db_o twice_o wherefore_o that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ac_fw-la and_o cb_o together_o with_o that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o twice_o be_v equal_a to_o that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ad_fw-la and_o db_o together_o with_o that_o which_o be_v contain_v under_o the_o line_n ad_fw-la and_o db_o twice_o but_o it_o be_v already_o prove_v that_o that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o twice_o be_v less_o than_o that_o which_o be_v contain_v under_o the_o line_n ad_fw-la &_o db_o twice_o wherefore_o the_o residue_n namely_o that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ac_fw-la and_o cb_o be_v great_a than_o the_o residue_n namely_o then_o that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ad_fw-la and_o db_o which_o be_v require_v to_o be_v demonstrate_v ¶_o a_o assumpt_n a_o rational_a superficies_n exceed_v a_o rational_a superficies_n by_o a_o rational_a superficies_n let_v ad_fw-la be_v a_o rational_a superficies_n and_o let_v it_o exceed_v of_o be_v also_o a_o rational_a superficies_n by_o the_o superficies_n ed._n then_o i_o say_v that_o the_o superficies_n ed_z be_v also_o rational_a for_o the_o parallelogram_n ad_fw-la be_v commensurable_a to_o the_o parallelogram_n of_o for_o that_o either_o of_o they_o be_v rational_a wherefore_o by_o the_o second_o part_n of_o the_o 15._o of_o the_o ten_o the_o parallelogram_n of_o be_v commensurable_a to_o the_o parallelogram_n ed._n but_o the_o the_o parallelogram_n of_o be_v rational_a wherefore_o also_o the_o parallelogram_n ed_z be_v rational_a ¶_o the_o 30._o theorem_a the_o 42._o proposition_n a_o binomial_a line_n be_v in_o one_o point_n only_o divide_v into_o his_o name_n composition_n svppose_v that_o ab_fw-la be_v a_o binomial_a line_n and_o in_o the_o point_n g_o let_v it_o be_v divide_v into_o his_o name_n that_o be_v into_o the_o line_n whereof_o the_o whole_a line_n ab_fw-la be_v compose_v wherefore_o these_o line_n ac_fw-la and_o cb_o be_v rational_a commensurable_a in_o power_n only_o now_o i_o say_v that_o the_o line_n ab_fw-la can_v in_o any_o other_o point_n beside_o c_o be_v divide_v into_o two_o rational_a line_n commensurable_a in_o power_n only_o impossibility_n for_o if_o it_o be_v possible_a let_v it_o be_v divide_v in_o the_o point_n d_o so_fw-mi that_o let_v the_o line_n ad_fw-la and_o de_fw-fr bâ—_z rational_a commensurable_a in_o power_n only_o first_o this_o manifest_a that_o neither_o of_o these_o point_n c_o and_o d_o divide_v the_o right_a line_n ab_fw-la into_o two_o equal_a part_n otherwise_o the_o line_n ac_fw-la and_o cb_o shall_v be_v rational_a commensurable_a in_o
be_v equal_a to_o that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o twice_o again_o from_o the_o the_o parallelogram_n eke_o take_v away_o the_o parallelogram_n el_n equal_a to_o the_o square_n of_o the_o line_n ad_fw-la and_o db_o which_o be_v less_o than_o the_o square_n of_o the_o line_n ac_fw-la and_o cb._n wherefore_o the_o residue_n namely_o the_o parallelogram_n mk_o be_v equal_a to_o that_o which_o be_v contain_v under_o the_o line_n ad_fw-la and_o db_o twice_o and_o forasmuch_o as_o the_o square_n of_o the_o line_n ac_fw-la and_o cb_o be_v medial_a therefore_o the_o parallelogram_n eglantine_n also_o be_v medial_a impossibility_n and_o it_o be_v apply_v upon_o the_o rational_a line_n efâ—_n â—_z wherefore_o the_o line_n eh_o be_v rational_a and_o incommensurable_a in_o length_n to_o the_o line_n ef._n and_o by_o the_o same_o reason_n the_o parallelogram_n hk_o be_v medial_a for_o that_o which_o be_v equal_a unto_o it_o namely_o that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o twice_o be_v medial_a therefore_o the_o line_n hn_o be_v also_o rational_a and_o incommensurable_a in_o length_n unto_o the_o line_n ef._n and_o forasmuch_o as_o the_o line_n ac_fw-la and_o cb_o be_v medial_a line_n commensurable_a in_o power_n only_o therefore_o the_o line_n ac_fw-la be_v incommensurable_a in_o length_n unto_o the_o line_n cb._n but_o as_o the_o line_n ac_fw-la be_v to_o the_o line_n cb_o so_o 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_o by_o the_o 1._o of_o the_o six_o wherefore_o the_o square_a of_o the_o line_n ac_fw-la be_v incommensurable_a to_o that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb._n butt_n by_o the_o 16._o of_o the_o ten_o unto_o the_o square_n of_o the_o line_n ac_fw-la be_v commensurable_a the_o square_n of_o the_o line_n ac_fw-la and_o cb_o add_v together_o for_o the_o line_n ac_fw-la and_o cb_o be_v commensurable_a in_o power_n only_o and_o unto_o that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o be_v commensurable_a that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o twice_o wherefore_o that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ac_fw-la and_o cb_o be_v incommensurable_a to_o that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o twice_o but_o to_o the_o square_n of_o the_o line_n ac_fw-la and_o cb_o be_v equal_a the_o parallelogram_n eglantine_n and_o 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 be_v equal_a the_o paralelograme_n hk_o wherefore_o the_o parallelogram_n eglantine_n be_v incommensurable_a to_o the_o parallelogram_n hk_o wherefore_o also_o the_o line_n eh_o be_v incommensurable_a in_o length_n to_o the_o line_n hn._n and_o the_o line_n eh_o and_o hn_o be_v rational_a wherefore_o they_o be_v rational_a commensurable_a in_o power_n only_o but_o if_o two_o rational_a line_n commensurable_a in_o power_n only_o be_v add_v together_o the_o whole_a line_n be_v irrational_a and_o be_v call_v a_o binomial_a line_n by_o the_o 36._o of_o the_o ten_o wherefore_o the_o binomial_a line_n en_fw-fr be_v in_o the_o point_n h_o divide_v into_o his_o name_n and_o by_o the_o same_o reason_n also_o may_v it_o be_v prove_v that_o the_o line_n em_n and_o mn_o be_v rational_a line_n commensurable_a in_o power_n only_o wherefore_o en_fw-fr being_n a_o binomial_a line_n be_v divide_v into_o his_o name_n in_o sundry_a point_n namely_o in_o h_o and_o m_o neither_o be_v the_o line_n eh_o one_o and_o the_o same_o that_o be_v equal_a with_o mn_n for_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 bd_o and_o ad_fw-la by_o the_o 1._o assumpt_v put_v after_o the_o 41._o of_o the_o ten_o but_o the_o square_n of_o the_o line_n ad_fw-la and_o db_o be_v 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 by_o the_o assumpt_n put_v after_o the_o 39_o of_o the_o ten_o wherefore_o the_o square_n of_o the_o line_n ac_fw-la and_o cb_o that_o be_v the_o parallelogram_n eglantine_n be_v much_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 that_o be_v then_o the_o parallelogram_n mk_o wherefore_o by_o the_o first_o of_o the_o six_o the_o line_n eh_o be_v great_a than_o the_o line_n mn_v wherefore_o eh_o be_v not_o one_o and_o the_o same_o with_o mn_n wherefore_o a_o binomial_a line_n be_v in_o two_o sundry_a point_n divide_v into_o his_o name_n which_o be_v impossible_a the_o self_n same_o absurdity_n also_o will_v follow_v if_o the_o line_n ac_fw-la be_v suppose_v to_o be_v less_o than_o the_o line_n db._n a_o second_o binomial_a line_n therefore_o be_v not_o divide_v into_o his_o name_n in_o sundry_a point_n wherefore_o it_o be_v divide_v in_o one_o only_a which_o be_v require_v to_o be_v demonstrate_v ¶_o the_o 33._o theorem_a the_o 45._o proposition_n a_o great_a line_n be_v in_o one_o point_n only_o divide_v into_o his_o name_n let_v ab_fw-la be_v a_o great_a line_n be_v divide_v into_o his_o name_n in_o the_o point_n c_o so_fw-mi that_o let_v the_o line_n ac_fw-la and_o cb_o be_v rational_a incommensurable_a in_o power_n have_v that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ac_fw-la and_o cb_o rational_a and_o that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o medial_a impossibiâ—â—â—e_n then_o i_o say_v that_o the_o line_n ab_fw-la can_v not_o in_o any_o other_o point_n then_o in_o c_o be_v divide_v into_o his_o name_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 ad_fw-la and_o db_o be_v line_n incommensurable_a in_o power_n have_v that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ad_fw-la and_o db_o rational_a and_o that_o which_o be_v contain_v under_o the_o line_n ad_fw-la and_o db_o medial_a now_o forasmuch_o as_o how_o much_o the_o square_n of_o the_o line_n ac_fw-la and_o cb_o diâ—â—er_n from_o the_o square_n of_o the_o line_n ad_fw-la and_o db_o so_o much_o differ_v that_o which_o be_v contain_v under_o the_o line_n ad_fw-la and_o db_o twice_o 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 those_o thing_n which_o have_v beâ—e_v say_v in_o the_o demonstration_n of_o the_o 42._o proposition_n but_o the_o square_n of_o the_o line_n ac_fw-la and_o cb_o exceed_v 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 for_o they_o be_v either_o of_o they_o rational_a wherefore_o that_o which_o be_v containeâ—â—â—der_v the_o line_n ad_fw-la and_o db_o twice_o exceed_v that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o twice_o by_o a_o rational_a superficies_n when_o as_o either_o of_o they_o be_v a_o medial_a supersâ—â—â—es_n which_o be_v impossible_a by_o the_o 26._o of_o the_o ten_o wherefore_o a_o great_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 34._o theorem_a the_o 46._o proposition_n a_o line_n contain_v in_o power_n a_o rational_a and_o a_o medial_a be_v in_o one_o point_n only_o divide_v into_o his_o name_n let_v 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 be_v divide_v into_o his_o name_n in_o the_o point_n c_o so_fw-mi that_o let_v the_o line_n ac_fw-la &_o cb_o be_v incommensurable_a in_o power_n have_v that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ac_fw-la and_o cb_o medial_a and_o that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o rational_a impossibility_n then_o i_o say_v that_o the_o line_n ab_fw-la can_v not_o in_o any_o other_o point_n be_v divide_v into_o his_o name_n but_o only_o in_o the_o point_n c._n for_o if_o it_o be_v possible_a let_v it_o be_v divide_v into_o his_o name_n in_o the_o point_n d_o so_fw-mi that_o let_v the_o line_n ad_fw-la and_o db_o be_v incommensurable_a in_o power_n have_v that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ad_fw-la and_o db_o medial_a and_o that_o which_o be_v contain_v under_o the_o line_n ad_fw-la and_o db_o rational_a now_o 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 &_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 so_o much_o differ_v the_o square_n of_o the_o line_n ac_fw-la &_o cb_o add_v together_o from_o the_o square_n of_o the_o line_n ad_fw-la and_o db_o add_v together_o but_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
reform_v by_o m._n dee_n describe_v for_o it_o in_o the_o plain_n especial_o if_o you_o remember_v the_o form_n of_o the_o figure_n of_o the_o 29._o proposition_n of_o this_o book_n only_o that_o which_o there_o you_o conceive_v to_o be_v the_o base_a imagine_v here_o in_o both_o the_o figure_n of_o this_o second_o case_n to_o be_v the_o upper_a superficies_n opposite_a to_o the_o base_a and_o that_o which_o be_v there_o suppose_v to_o be_v the_o upper_a superficies_n conceive_v here_o to_o be_v the_o base_a you_o may_v describe_v they_o upon_o past_v paper_n for_o your_o better_a sight_n take_v heed_n you_o note_v the_o letter_n right_o according_a as_o the_o construction_n require_v flussas_n demonstrate_v this_o proposition_n a_o otherway_o take_v only_o the_o base_n of_o the_o solid_n and_o that_o after_o this_o manner_n take_v equal_a base_n which_o yet_o for_o the_o sure_a understanding_n let_v be_v utter_o unlike_a namely_o aebf_n and_o adch_n and_o let_v one_o of_o the_o side_n of_o each_o concur_v in_o one_o &_o the_o same_o right_a line_n aed_n &_o the_o base_n be_v upon_o one_o and_o the_o self_n same_o plain_n let_v there_o be_v suppose_v to_o be_v set_v upon_o they_o parallelipipedon_n under_o one_o &_o the_o self_n same_o altitude_n then_o i_o say_v that_o the_o solid_a set_v upon_o the_o base_a ab_fw-la be_v equal_a to_o the_o solid_a set_v upon_o the_o base_a ah_o by_o the_o point_n e_o draw_v unto_o the_o line_n ac_fw-la a_o parallel_n line_n eglantine_n which_o if_o it_o fall_v without_o the_o base_a ab_fw-la produce_v the_o right_a line_n hc_n to_o the_o point_n i_o now_o forasmuch_o as_o ab_fw-la and_o ah_o be_v parallelogrmae_n therefore_o by_o the_o 24._o of_o this_o book_n the_o triangle_n aci_n and_o egl_n shall_v be_v equaliter_fw-la the_o one_o to_o the_o other_o and_o by_o the_o 4._o of_o the_o first_o they_o shall_v be_v equiangle_n and_o equal_a and_o by_o the_o first_o definition_n of_o the_o six_o and_o four_o proposition_n of_o the_o same_o they_o shall_v be_v like_a wherefore_o prism_n erect_v upon_o those_o triangle_n and_o under_o the_o same_o altitude_n that_o the_o solid_n ab_fw-la and_o ah_o aâ—e_n shall_v be_v equal_a and_o like_a by_o the_o 8._o definition_n of_o this_o book_n for_o they_o be_v contain_v under_o like_o plain_a superficiece_n equal_a both_o in_o multitude_n and_o magnitude_n add_v the_o solid_a set_v upon_o the_o base_a acle_n common_a to_o they_o both_o wherefore_o the_o solid_a set_v upon_o the_o base_a aegc_n be_v equal_a to_o the_o solid_a set_v upon_o the_o base_a aeli_n and_o forasmuch_o as_o the_o superficiece_n aebf_n and_o adhc_n be_v equal_a by_o supposition_n and_o the_o part_n take_v away_o agnostus_n be_v equal_a to_o the_o part_n take_v away_o all_o therefore_o the_o residue_n by_o shall_v be_v equal_a to_o the_o residue_n gd_o wherefore_o as_o agnostus_n be_v to_o gd_a as_o all_o be_v to_o by_o namely_o equal_n to_o equal_n but_o as_o agnostus_n be_v to_o gd_o so_o iâ—_z the_o solid_a set_v upon_o agnostus_n to_o the_o solid_a set_v upon_o gd_o by_o the_o 25._o of_o this_o book_n for_o it_o be_v cut_v by_o a_o plain_a superficies_n set_v upon_o the_o line_n ge_z which_o superficies_n be_v parallel_n to_o the_o opposite_a superficiece_n wherefore_o as_o all_o be_v to_o by_o so_o be_v the_o solid_a set_v upon_o all_o to_o the_o solid_a set_v upon_o by_o wherefore_o by_o the_o 11._o of_o the_o five_o as_o the_o solid_a set_v upon_o agnostus_n or_o upon_o all_o which_o be_v equal_a unto_o it_o be_v to_o the_o solid_a set_v upon_o gd_o so_o be_v the_o same_o solid_a set_v upon_o agnostus_n or_o all_o to_o the_o solid_a set_v upon_o by_o wherefore_o by_o the_o 2._o part_n of_o the_o 9_o of_o the_o five_o the_o solid_n set_v upon_o gd_o and_o by_o shall_v be_v equal_a unto_o which_o solid_n if_o you_o add_v equal_a solid_n namely_o the_o solid_a set_v upon_o agnostus_n to_o the_o solid_a set_v upon_o gd_o and_o the_o solid_a set_v upon_o all_o to_o the_o solid_a set_v upon_o by_o the_o whole_a solid_n set_v upon_o the_o base_a ah_o and_o upon_o the_o base_a ab_fw-la â—hall_v be_v equal_a wherefore_o parallelipedon_n consist_v upon_o equal_a base_n and_o be_v under_o one_o and_o the_o self_n same_o altitude_n be_v equal_a the_o one_o to_o the_o other_o which_o be_v require_v to_o be_v prove_v ¶_o the_o 27._o theorem_a the_o 32._o proposition_n parallelipipedon_n be_v under_o one_o and_o the_o self_n same_o altitude_n be_v in_o that_o proportion_n the_o one_o to_o the_o other_o that_o their_o base_n be_v svppose_v that_o these_o parallelipipedon_n ab_fw-la and_o cd_o be_v under_o one_o &_o the_o self_n same_o altitude_n then_o i_o say_v that_o those_o parallelipipedon_n ab_fw-la and_o cd_o be_v in_o that_o proportion_n the_o one_o to_o the_o other_o that_o their_o base_n be_v that_o be_v that_o as_o the_o base_a ae_n be_v to_o the_o base_a cf_o so_o be_v the_o parallelipipedon_n ab_fw-la to_o the_o parallelipipedon_n cd_o construction_n upon_o the_o line_n fg_o describe_v by_o the_o 45._o of_o the_o first_o the_o parallelogram_n fh_o equal_a to_o the_o parallelogram_n ae_n and_o equiangle_n with_o the_o parallelogram_n cf._n and_o upon_o the_o base_a fh_o describe_v a_o parallelipipedom_n of_o the_o self_n same_o altitude_n that_o the_o parallelipipedom_n cd_o be_v &_o let_v the_o same_o be_v gk_o demonstration_n now_o by_o the_o 31._o of_o the_o eleven_o the_o parallelipipedon_n ab_fw-la be_v equal_a to_o the_o parallelipipedon_n gk_o for_o they_o consist_v upon_o equal_a base_n namely_o ae_n and_o fh_o and_o be_v under_o one_o and_o the_o self_n same_o altitude_n and_o forasmuch_o as_o the_o parallelipipedon_n ck_o be_v cut_v by_o a_o plain_a superficies_n dg_o be_v parallel_n to_o either_o of_o the_o opposite_a plain_a superâ—icieces_n therefore_o by_o the_o 25._o of_o the_o eleven_o as_o the_o base_a hf_o be_v to_o the_o base_a fc_n so_o be_v the_o parallelipipedon_n gk_o to_o parallelipipedon_v cd_o but_o the_o base_a hf_o be_v equal_a to_o the_o base_a ae_n and_o the_o parallelipipedon_n gk_o be_v prove_v equal_a to_o the_o parallelipipedon_n ab_fw-la wherefore_o as_o the_o base_a aâ—e_n be_v to_o the_o base_a cf_o so_o be_v the_o parallelipedon_n ab_fw-la to_o the_o parallelipipedon_n cd_o wherefore_o parallelipipedon_n be_v under_o one_o and_o the_o self_n same_o altitude_n be_v in_o that_o proportion_n the_o one_o to_o the_o other_o that_o their_o base_n be_v which_o be_v require_v to_o be_v demonstrate_v i_o need_v not_o to_o put_v any_o other_o figure_n for_o the_o declaration_n of_o this_o demonstration_n for_o it_o be_v easy_a to_o see_v by_o the_o figure_n there_o describe_v howbeit_o you_o may_v for_o the_o more_o full_a sight_n thereof_o describe_v solid_n of_o past_v paper_n according_a to_o the_o construction_n there_o set_v forth_o which_o will_v not_o be_v hard_o for_o you_o to_o do_v if_o you_o remember_v the_o description_n of_o such_o body_n before_o teach_v a_o corollary_n add_v by_o flussas_n equal_a parallelipipedon_n contain_v under_o one_o and_o the_o self_n same_o altitude_n have_v also_o their_o base_n equal_a for_o if_o the_o base_n shall_v be_v unequal_a the_o parallelipipedon_n also_o shall_v be_v unequal_a by_o this_o 32_o proposition_n and_o equal_a parallelipipedon_n have_v equal_a base_n have_v also_o one_o and_o the_o self_n same_o altitude_n for_o if_o they_o shall_v have_v a_o great_a altitude_n they_o shall_v exceed_v the_o equal_a parallelipipedon_n which_o have_v the_o self_n same_o altitude_n but_o if_o they_o shall_v have_v a_o less_o they_o shall_v want_v so_o much_o of_o those_o self_n same_o equal_a parallelipipedon_n the_o 28._o theorem_a the_o 33._o proposition_n like_a parallelipipedon_n be_v in_o treble_a proportion_n the_o one_o to_o the_o other_o of_o that_o in_o which_o their_o side_n of_o like_a proportion_n be_v svppose_v that_o these_o parallelipipedon_n ab_fw-la and_o cd_o be_v like_a &_o let_v the_o side_n ae_n and_o cf_o be_v side_n of_o like_a proportion_n then_o i_o say_v the_o parallelipipedon_n ab_fw-la be_v unto_o the_o parallelipipedon_n cd_o in_o treble_a proportion_n of_o that_o in_o which_o the_o side_n ae_n be_v to_o the_o side_n cf._n extend_v the_o right_a line_n ae_n ge_z and_o he_o to_o the_o point_n king_n l_o m._n construction_n and_o by_o the_o 2._o of_o the_o first_o unto_o the_o line_n cf_o put_v the_o line_n eke_o equal_a and_o unto_o the_o line_n fn_o put_v the_o line_n el_n equal_a and_o moreover_o unto_o the_o line_n fr_z put_v the_o line_n em_n equal_a and_o make_v perfect_a the_o parallelogram_n kl_o and_o the_o parallelipipedon_n ko_o demonstration_n now_o forasmuch_o as_o these_o two_o line_n eke_o and_o el_n be_v equal_a to_o these_o two_o line_n cf_o and_o fn_o but_o the_o angle_n kel_n be_v equal_a to_o the_o angle_n cfn_n for_o the_o angle_n
in_o continual_a proportion_n in_o the_o proportion_n of_o x_o to_o y._n three_o circle_n therefore_o be_v give_v we_o have_v find_v three_o circle_n equal_a to_o they_o and_o also_o in_o continual_a proportion_n in_o any_o proportion_n give_v between_o two_o right_a line_n which_o be_v requisite_a to_o be_v do_v ¶_o a_o corollary_n 1._o it_o be_v hereby_o very_o manifest_a that_o unto_o 4.5.6.10.20_o 100_o or_o how_o many_o circle_n soever_o shall_v be_v give_v we_o may_v find_v 3.4.5.8.10_o or_o how_o many_o soever_o shall_v be_v appoint_v which_o all_o together_o shall_v be_v equal_a to_o the_o circle_n give_v how_o many_o soever_o they_o be_v and_o with_o all_o our_o find_a circle_n to_o be_v in_o continual_a proportion_n in_o any_o proportion_n assign_v between_o two_o right_a line_n give_v for_o evermore_o by_o the_o corollary_n of_o the_o 4._o problem_n reduce_v all_o your_o circle_n to_o one_o and_o by_o the_o corollary_n of_o my_o 8._o problem_n make_v as_o many_o circle_n as_o you_o be_v appoint_v equal_a to_o the_o circle_n give_v and_o continual_a in_o proportion_n in_o the_o same_o wherein_o the_o two_o right_a line_n give_v areâ—_n and_o so_o have_v you_o perform_v the_o thing_n require_v note_n what_o incredible_a fruit_n in_o the_o science_n of_o proportion_n may_v hereby_o grow_v no_o man_n tongue_n can_v sufficient_o express_v and_o sorry_a i_o be_o that_o utter_o leisure_n be_v take_v from_o i_o somewhat_o to_o specify_v in_o particular_a hereof_o ¶_o the_o key_n of_o one_o of_o the_o chief_a treasure_n house_n belong_v to_o the_o state_n mathematical_a thatâ—_n which_o in_o these_o 9_o problem_n be_v say_v of_o circlesâ—_n be_v much_o more_o say_v of_o square_n by_o who_o mean_n circle_n be_v thus_o handle_v and_o therefore_o see_v to_o all_o polygonon_n right_o line_v figure_n equal_a square_n may_v be_v make_v by_o the_o fault_n of_o the_o second_o and_o contrariwise_o to_o any_o square_n a_o right_n line_v figure_n may_v be_v make_v equal_a and_o withal_o like_v to_o any_o right_n line_v figure_n give_v by_o the_o 25_o of_o the_o six_o and_o fourthly_a see_v upon_o the_o say_a plain_a figuresâ—_n as_o upon_o baseâ—_n may_v priâ—mes_n parallelipipedon_n pyramid_n side_v column_n cones_n and_o cylinder_n be_v rear_v which_o be_v solid_n all_o of_o one_o height_n shall_v have_v that_o proportion_n one_o to_o the_o other_o that_o their_o base_n have_v one_o to_o the_o other_o and_o five_o see_v sphere_n cones_n and_o cylinder_n be_v one_o to_o other_o in_o 〈◊〉_d know_v proportion_n and_o so_o may_v be_v make_v one_o to_o the_o other_o in_o any_o proportion_n assign_v and_o 〈◊〉_d see_v under_o every_o one_o of_o the_o kind_n of_o figure_n both_o plain_a and_o solid_a infinite_a case_n may_v chance_v by_o the_o aid_n of_o these_o problem_n to_o be_v solute_v and_o execute_v how_o infinite_a then_o upon_o infinite_a be_v the_o number_n of_o practice_n either_o mathematical_a oâ—_n mechanical_a to_o be_v perform_v of_o comparison_n between_o diverse_a kind_n of_o plain_n to_o plain_n and_o solid_n to_o solide_a farthermore_o to_o speak_v of_o plain_a superficial_a figure_n in_o respect_n of_o the_o conâ—â—r_n or_o area_n of_o the_o circle_n sundry_a mix_a line_n figure_n anular_a and_o lunular_a figure_n and_o also_o of_o circle_n to_o be_v give_v equal_a to_o the_o say_v â—â—â—sed_v figuresâ—_n and_o in_o all_o proportion_n else_o and_o evermore_o think_v of_o solid_n like_o high_a set_v upon_o any_o of_o those_o unused_a figure_n oh_o lord_n in_o consideration_n of_o all_o the_o premise_n how_o infinite_a how_o strange_a and_o â—â—credible_a â—â—â—â—â—â—ationâ—_n and_o practiseâ—_n may_v by_o the_o aid_n and_o direction_n of_o these_o few_o problem_n 〈◊〉_d ready_o into_o the_o imagination_n and_o hand_n of_o they_o that_o will_v bring_v their_o mind_n and_o intent_n whole_o and_o fix_o to_o such_o mathematical_a discourse_n in_o these_o element_n i_o intend_v but_o to_o geve_v to_o young_a beginner_n some_o 〈◊〉_d and_o courage_n to_o exercise_v â—heir_o own_o wit_n and_o talon_n in_o this_o most_o pleasant_a and_o profitable_a sciâ—ncâ—_n all_o â—hingeâ—_n 〈◊〉_d not_o neither_o yâ—â—â—an_n in_o every_o place_n be_v say_v opportunity_n and_o sufficienty_n best_o be_v to_o be_v allow_v ¶_o the_o 3._o theorem_a the_o 3._o proposition_n every_o pyramid_n have_v a_o triangle_n to_o his_o base_a may_v be_v divide_v into_o two_o pyramid_n equal_a and_o like_v the_o one_o to_o the_o other_o and_o also_o like_a to_o the_o whole_a have_v also_o triangle_n to_o their_o base_n and_o into_o two_o equal_a prism_n and_o those_o two_o prism_n be_v great_a than_o the_o half_a of_o the_o whole_a pyramid_n and_o forasmuch_o as_o to_o one_o of_o the_o side_n of_o the_o triangle_n adb_o namely_o to_o the_o side_n ab_fw-la be_v draw_v â—_o parallel_v line_n hk_o therefore_o the_o whole_a triangle_n aâ—db_n be_v equiangle_n to_o the_o triangle_n dhk_n and_o their_o side_n be_v proportional_a by_o the_o corollary_n of_o the_o â—_o of_o the_o six_o wherefore_o the_o triangle_n adb_o be_v like_a to_o the_o triangle_n dhk_n and_o by_o the_o same_o reason_n also_o the_o triangle_n dbc_n be_v like_a to_o the_o triangle_n dkl_n and_o the_o triangle_n adc_o to_o the_o triangle_n dhl_n and_o forasmuch_o as_o two_o right_a line_n basilius_n and_o ac_fw-la touch_v the_o one_o the_o other_o be_v parallel_n to_o two_o right_a line_n kh_o and_o hl_o touch_v also_o the_o one_o the_o other_o but_o not_o be_v in_o one_o and_o the_o self_n same_o superficies_n with_o the_o two_o first_o line_n therefore_o by_o the_o 10._o of_o the_o eleven_o they_o contain_v equal_a angle_n wherefore_o the_o angle_n bac_n be_v equal_a to_o the_o angle_n khl._n and_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 kh_o to_o the_o line_n he._n wherefore_o the_o triangle_n abc_n be_v like_a to_o the_o triangle_n khl._n wherefore_o the_o whole_a pyramid_n who_o base_a be_v the_o triangle_n abc_n &_o top_n the_o point_n d_o be_v like_a to_o the_o pyramid_n who_o base_a be_v the_o triangle_n hkl_n and_o top_n the_o point_n d._n but_o the_o pryamis_n who_o base_a be_v the_o triangle_n hkl_n and_o top_n the_o point_n d_o be_v prove_v to_o be_v like_a to_o the_o pyramid_n who_o base_a be_v the_o triangle_n aeg_n and_o top_n the_o point_n h._n wherefore_o also_o the_o pyramid_n who_o base_a be_v the_o triangle_n abc_n and_o top_n the_o point_n d_o be_v like_a to_o the_o pyramid_n who_o base_a be_v the_o triangle_n aeg_n and_o top_n the_o point_n h_o by_o the_o 21._o of_o the_o six_o part_n wherefore_o either_o of_o these_o pyramid_n aegh_n and_o hkld_a be_v like_a to_o the_o whole_a pyramid_n abcd._n prism_n andrea_n forasmuch_o as_o the_o line_n be_v kh_o be_v parallel_a line_n and_o equal_a as_o it_o have_v be_v prove_v therefore_o the_o right_a line_n bk_o and_o eh_o which_o join_v they_o together_o be_v equal_a and_o parallel_n by_o the_o 33._o of_o the_o first_o again_o forasmuch_o as_o the_o line_n be_v and_o fg_o be_v parallel_a line_n and_o equal_a therefore_o line_n eglantine_n and_o bf_o which_o join_v they_o together_o be_v also_o equal_a and_o parallel_n and_o by_o the_o same_o reason_n forasmuch_o as_o the_o line_n fg_o and_o kh_o be_v equal_a parallel_n the_o line_n fk_o and_o gh_o which_o join_v they_o together_o be_v also_o equal_a parallel_n wherefore_o behk_n degf_n and_o khgf_n be_v parallelogram_n and_o forasmuch_o as_o their_o opposite_a side_n be_v equal_a by_o the_o 34._o of_o the_o firstâ—_o therefore_o the_o triangle_n fhg_n &_o bkf_n be_v equiangle_n by_o the_o 8._o of_o the_o first_o and_o therefore_o by_o the_o 4._o of_o the_o same_o they_o be_v equal_a and_o moreover_o by_o the_o 15._o of_o the_o eleven_o their_o superficiecâ—s_n be_v parallel_n wherefore_o the_o solid_a bkfehg_n be_v a_o prism_n by_o the_o 11._o definition_n of_o the_o eleven_o likewise_o forasmuch_o as_o the_o side_n of_o the_o triangle_n hkl_n be_v equal_a and_o parallel_n to_o the_o side_n of_o the_o triangle_n gfc_n as_o it_o have_v before_o be_v prove_v it_o be_v manifest_a that_o cfkl_n fkhg_n and_o clhg_n be_v parallelogram_n by_o the_o 33._o of_o the_o first_o whereforâ—_n the_o whole_a solid_a klhfcg_n be_v a_o prism_n by_o the_o 11._o definition_n of_o the_o eleven_o and_o be_v contain_v under_o the_o say_a parallelogram_n cfkl_n fkhg_n and_o clhg_n and_o the_o two_o triangle_n hkl_n and_o gfc_n which_o be_v opposite_a and_o parallel_n and_o forasmuch_o as_o the_o line_n bf_o be_v equal_a to_o the_o line_n fc_o therefore_o by_o the_o 41._o of_o the_o first_o the_o parallelogram_n ebfg_n be_v double_a to_o the_o triangle_n gfc_n and_o forasmuch_o as_o if_o there_o be_v two_o prism_n of_o equal_a altitude_n and_o the_o one_o have_v to_o his_o base_a a_o parallelogram_n and_o the_o other_o
opposite_a side_n the_o triangle_n stv_n may_v thus_o be_v prove_v for_o in_o the_o self_n same_o construction_n imagine_v perpendicular_a line_n to_o be_v draw_v from_o the_o point_n g_o and_o h_o to_o the_o two_o plain_a superficiece_n wherein_o be_v the_o triangle_n abc_n and_o def_n now_o those_o perpendicular_a line_n shall_v be_v equal_a the_o one_o to_o the_o other_o for_o that_o the_o two_o pyramid_n be_v suppose_v to_o be_v of_o equal_a altitude_n and_o forasmuch_o as_o two_o right_a line_n namely_o gc_o and_o the_o perpendicular_a line_n draw_v from_o the_o point_n g_o be_v divide_v by_o two_o parallel_n plain_a superficiece_n namely_o abc_n and_o omn_n therefore_o by_o the_o 17._o of_o the_o eleven_o the_o part_n of_o the_o line_n divide_v be_v proportional_a but_o the_o line_n gc_o be_v by_o the_o plain_a superficies_n omnes_n divide_v into_o two_o equal_a part_n in_o the_o point_n n._n wherefore_o also_o the_o perpendicular_a line_n draw_v from_o the_o point_n g_o to_o the_o plain_a superficies_n wherein_o be_v the_o triangle_n abc_n be_v divide_v into_o two_o equal_a part_n by_o the_o superficies_n omnes_n &_o by_o the_o same_o reason_n also_o the_o perpendicular_a line_n which_o be_v draw_v from_o the_o point_n h_o to_o the_o plain_a superficies_n def_n be_v divide_v into_o two_o equal_a part_n by_o the_o plain_a superficies_n stv_n and_o the_o perpendicular_n draw_v from_o the_o point_n g_o and_o h_o to_o the_o plain_a superficiece_n abc_n and_o def_n be_v equal_a wherefore_o also_o the_o perpendicular_a line_n which_o be_v draw_v from_o the_o triangle_n omnes_n and_o stv_n to_o the_o plain_a superficiece_n abc_n and_o def_n be_v equal_a the_o one_o to_o the_o other_o wherefore_o also_o the_o prism_n who_o base_n be_v the_o triangle_n lxc_a and_o rwf_n and_o the_o opposite_a side_n the_o triangle_n omnes_n and_o stv_n be_v of_o equal_a altitude_n wherefore_o also_o the_o parallelipipedon_n which_o be_v describe_v of_o the_o foresaid_a prism_n and_o be_v equal_a in_o altitude_n with_o they_o be_v the_o one_o to_o the_o other_o as_o the_o base_a of_o the_o one_o be_v to_o the_o base_a of_o the_o other_o wherefore_o also_o as_o the_o half_a of_o the_o base_n of_o those_o parallelipipedon_n namely_o as_o the_o base_a lxc_n be_v to_o the_o base_a rwf_n so_o be_v the_o foresaid_a prism_n the_o one_o to_o the_o other_o if_o therefore_o there_o be_v two_o pyramid_n under_o equal_a altitude_n have_v triangle_n to_o their_o base_n and_o either_o of_o those_o pyramid_n be_v divide_v into_o two_o pyramid_n equal_v the_o one_o to_o the_o other_o and_o like_a unto_o the_o whole_a and_o into_o two_o equal_a prism_n whole_a and_o again_o if_o in_o either_o of_o the_o pyramid_n make_v of_o the_o two_o first_o pyramid_n be_v still_o observe_v the_o same_o order_n and_o manner_n then_o as_o the_o base_a of_o the_o one_o pyramid_n be_v to_o the_o base_a of_o the_o other_o pyramid_n so_o all_o the_o prism_n which_o be_v in_o the_o one_o pyramid_n to_o all_o the_o prism_n which_o be_v in_o the_o other_o be_v equal_a in_o multitude_n which_o be_v require_v to_o be_v prove_v ¶_o the_o 5._o theorem_a the_o 5._o proposition_n pyramid_n consist_v under_o one_o and_o the_o self_n same_o altitude_n have_v triangle_n to_o their_o base_n be_v in_o that_o proportion_n the_o one_o to_o the_o other_o that_o their_o base_n be_v svppose_v that_o these_o two_o pyramid_n who_o base_n be_v the_o triangle_n abc_n &_o def_n and_o top_n the_o point_n g_o and_o h_o be_v under_o equal_a altitude_n then_o i_o say_v that_o as_o the_o base_a abc_n be_v to_o the_o base_a def_n so_o be_v the_o pyramid_n abcg_o to_o the_o pyramid_n defh_o for_o if_o the_o pyramid_n abcg_o be_v not_o to_o the_o pyramid_n defh_o as_o the_o base_a abc_n be_v to_o the_o base_a def_n then_o as_o the_o base_a abc_n be_v to_o the_o base_a def_n impossibility_n so_o be_v the_o pyramid_n abcg_o to_o a_o solid_a either_o less_o than_o the_o pyramid_n defh_o or_o great_a first_o let_v it_o be_v to_o some_o less_o and_o let_v the_o same_o be_v x._o and_o by_o the_o 3._o of_o the_o twelve_o let_v the_o pyramid_n defh_o be_v divide_v into_o two_o pyramid_n equal_v the_o one_o to_o the_o other_o and_o like_v unto_o the_o whole_a and_o into_o two_o equal_a prism_n now_o the_o two_o prism_n be_v great_a than_o the_o half_a of_o the_o whole_a pyramid_n and_o again_o by_o the_o same_o let_v the_o pyramid_n which_o be_v make_v of_o the_o division_n be_v in_o like_a sort_n divide_v and_o do_v this_o continual_o until_o there_o remain_v some_o pyramid_n make_v of_o the_o pyramid_n defh_o which_o be_v less_o than_o the_o excess_n whereby_o the_o pyramid_n defh_n exceed_v the_o solid_a x._o let_v such_o pyramid_n be_v take_v and_o for_o example_n sake_n let_v those_o pyramid_n be_v dpr_n &_o stuh_o wherefore_o the_o prism_n remain_v which_o be_v in_o the_o pyramid_n defh_o be_v great_a than_o the_o solid_a x._o divide_v by_o the_o proposition_n next_o go_v before_o the_o pyramid_n abcg_o in_o like_a sort_n &_o as_o many_o time_n as_o the_o pyramid_n defh_n â—s_v divide_v wherefore_o by_o the_o same_o as_o the_o base_a abc_n be_v to_o the_o base_a def_n so_o be_v all_o the_o prism_n which_o be_v in_o the_o pyramid_n abcg_o to_o all_o the_o prism_n which_o be_v in_o the_o pyramid_n defh_o but_o as_o the_o base_a abc_n be_v to_o the_o base_a def_n so_o be_v the_o pyramid_n abcg_o to_o the_o solid_a x._o wherefore_o by_o the_o 11._o of_o the_o five_o as_o the_o pyramid_n abcg_o be_v to_o the_o solid_a x_o so_o be_v the_o prism_n which_o be_v in_o the_o pyramid_n abcg_o to_o the_o prism_n which_o be_v in_o the_o pyramid_n defh_o wherefore_o alternate_o by_o the_o 16._o of_o the_o five_o as_o the_o pyramid_n abcg_o be_v to_o the_o prism_n which_o be_v in_o it_o so_o be_v the_o solid_a x_o to_o the_o prism_n which_o be_v iâ—_z the_o pyramid_n defh_o but_o the_o pyramid_n abcg_o be_v great_a than_o the_o prism_n which_o be_v in_o it_o wherefore_o also_o the_o solid_a x_o be_v great_a than_o the_o prism_n which_o be_v in_o the_o pyramid_n defh_o by_o the_o 14._o of_o the_o five_o but_o it_o be_v suppose_v to_o be_v less_o which_o be_v impossible_a wherefore_o as_o the_o base_a abc_n be_v to_o the_o base_a def_n so_o be_v not_o the_o pyramid_n abcg_o to_o any_o solid_a less_o than_o the_o pyramid_n defh_o i_o say_v moreover_o that_o as_o the_o base_a abc_n be_v to_o the_o base_a def_n so_o be_v not_o the_o pyramid_n abcg_o to_o any_o solid_a great_a than_o the_o pyramid_n defh_o for_o if_o it_o be_v possible_a let_v it_o be_v unto_o some_o great_a namely_o to_o the_o solid_a x._o wherefore_o by_o conversion_n by_o the_o corollary_n of_o the_o 4._o of_o the_o â—iâ—â—_n as_o the_o base_a def_n be_v to_o the_o base_a abc_n so_o be_v the_o solid_a x_o to_o the_o pyramid_n abcg_o but_o as_o the_o solid_a x_o be_v to_o the_o pyramid_n abcg_o so_o be_v the_o pyramid_n defh_o to_o some_o solid_a less_o than_o the_o pyramid_n abcg_o bâ—â—ke_n as_o we_o have_v before_o prove_v wherefore_o also_o by_o the_o 11._o of_o the_o â—ift_n as_o thâ—_z base_a def_n be_v to_o the_o base_a abc_n so_o be_v the_o pyramid_n defh_o to_o some_o solid_a less_o than_o the_o pyramid_n abcg_o which_o thing_n we_o have_v prove_v to_o be_v impossible_a wherefore_o as_o the_o base_a abc_n be_v to_o the_o base_a def_n so_o be_v not_o the_o pyramid_n abcg_o to_o any_o solid_a great_a than_o the_o pyramid_n defh_o and_o it_o be_v alsâ—_n prove_v that_o it_o be_v not_o in_o that_o proportion_n to_o any_o less_o than_o the_o pyramid_n defh_o wherefore_o as_o the_o base_a abc_n be_v to_o the_o base_a def_n so_o be_v the_o pyramid_n abcg_o to_o the_o pyramid_n defh_o wherefore_o pyramid_n consist_v under_o one_o and_o the_o self_n same_o altitude_n and_o have_v triangle_n to_o their_o base_n be_v in_o that_o proportion_n the_o one_o to_o the_o other_o that_o their_o base_n be_v which_o be_v require_v to_o be_v demonstrate_v ¶_o the_o 6._o theorem_a the_o 6._o proposition_n pyramid_n consist_v under_o one_o and_o the_o self_n same_o altitude_n and_o have_v pâ—ligoâ—on_a figure_n to_o their_o base_n be_v in_o that_o proportion_n the_o one_o to_o the_o other_o that_o their_o base_n be_v svppose_v that_o there_o be_v two_o pyramid_n have_v to_o their_o base_n these_o poligâ—non_n figure_v abce_v and_o fghkl_o and_o let_v their_o â—oppes_n be_v the_o point_n m_o and_o n_o which_o let_v be_v of_o one_o and_o the_o self_n same_o altitude_n then_o i_o say_v that_o as_o the_o base_a abce_v be_v to_o the_o base_a fghkl_o so_o be_v the_o pyramid_n abcedm_n to_o the_o pyramid_n fghkln_n construction_n divide_v the_o base_a abce_v into_o these_o
one_o of_o the_o circle_n ¶_o a_o assumpt_n add_v by_o flussas_n if_o a_o sphere_n be_v cut_v of_o a_o plain_a superficies_n the_o common_a section_n of_o the_o superficiece_n shall_v be_v the_o circumference_n of_o a_o circle_n suppose_v that_o the_o sphere_n abc_n be_v cut_v by_o the_o plain_a superficies_n aeb_n and_o let_v the_o centre_n of_o the_o sphere_n be_v the_o point_n d._n construction_n and_o from_o the_o point_n d_o let_v there_o be_v draw_v unto_o the_o plain_a superficies_n aeb_n a_o perpendicular_a line_n by_o the_o 11._o of_o the_o eleven_o which_o let_v be_v the_o line_n de._n and_o from_o the_o point_n e_o draw_v in_o the_o plain_a superficies_n aeb_n unto_o the_o common_a section_n of_o the_o say_a superficies_n and_o the_o sphere_n line_n how_o many_o so_o ever_o namely_o ea_fw-la and_o ebb_n and_o draw_v these_o line_n dam_fw-ge and_o db._n now_o forasmuch_o as_o the_o right_a angle_n dea_n and_o deb_n be_v equal_a for_o the_o line_n de_fw-fr be_v erect_v perpendicular_o to_o the_o plain_a superficies_n demonstration_n and_o the_o right_a line_n dam_n &_o db_o which_o subtend_v those_o angle_n be_v by_o the_o 12._o definition_n of_o the_o eleven_o equal_a which_o right_a line_n moreover_o by_o the_o 47._o of_o the_o first_o do_v contain_v in_o power_n the_o square_n of_o the_o line_n de_fw-fr ea_fw-la and_o de_fw-fr ebb_n if_o therefore_o from_o the_o square_n of_o the_o line_n de_fw-fr ea_fw-la and_o de_fw-fr ebb_v you_o take_v away_o the_o square_a of_o the_o line_n de_fw-fr which_o be_v common_a unto_o they_o the_o residue_n namely_o the_o square_n of_o the_o line_n ea_fw-la &_o ebb_n shall_v be_v equal_a wherefore_o also_o the_o line_n ea_fw-la and_o ebb_n be_v equal_a and_o by_o the_o same_o reason_n may_v we_o prove_v that_o all_o the_o right_a line_n draw_v from_o the_o point_n e_o to_o the_o line_n which_o be_v the_o common_a section_n of_o the_o superficies_n of_o sphere_n and_o of_o the_o plain_a superficies_n be_v equal_a wherefore_o that_o line_n shall_v be_v the_o circumference_n of_o a_o circle_n by_o the_o 15._o definition_n of_o the_o first_o circle_n but_o if_o it_o happen_v the_o plain_a superficies_n which_o cut_v the_o sphere_n to_o pass_v by_o the_o centre_n of_o the_o sphere_n the_o right_a line_n draw_v from_o the_o centre_n of_o the_o sphere_n to_o their_o common_a section_n shalâ—_n be_v equal_a by_o the_o 12._o definition_n of_o the_o eleven_o for_o that_o common_a section_n be_v in_o the_o superficies_n of_o the_o sphere_n wherefore_o of_o necessity_n the_o plain_a superficies_n comprehend_v under_o that_o line_n of_o the_o common_a section_n shall_v be_v a_o circle_n and_o his_o centre_n shall_v be_v one_o and_o the_o same_o with_o the_o centre_n of_o the_o sphere_n john_n dee_n euclid_n have_v among_o the_o definition_n of_o solid_n omit_v certain_a which_o be_v easy_a to_o conceive_v by_o a_o kind_n of_o analogy_n as_o a_o segment_n of_o a_o sphere_n a_o sector_n of_o a_o sphere_n the_o vertex_fw-la or_o top_n of_o the_o segment_n of_o a_o sphere_n with_o such_o like_a but_o that_o if_o need_n be_v some_o fartheâ—_n light_n may_v be_v give_v in_fw-ge this_o figure_n next_o before_o description_n understand_v a_o segment_n of_o the_o sphere_n abc_n to_o be_v that_o part_n of_o the_o sphere_n contain_v between_o the_o circle_n ab_fw-la who_o centre_n be_v e_o and_o the_o spherical_a superficies_n afb_o to_o which_o be_v a_o less_o segment_n add_v the_o cone_n adb_o who_o base_a be_v the_o former_a circle_n and_o top_n the_o centre_n of_o the_o sphere_n and_o you_o have_v dafb_n a_o sector_n of_o a_o sphere_n or_o solid_a sector_n as_o i_o call_v it_o de_fw-fr extend_a to_o fletcher_n show_v the_o top_n or_o vertex_fw-la of_o the_o segment_n to_o be_v the_o point_n f_o and_o of_o be_v the_o altitude_n of_o the_o segment_n spherical_a of_o segment_n some_o be_v great_a they_o the_o half_a sphere_n some_o be_v less_o as_o before_o abf_n be_v less_o the_o remanent_fw-la abc_n be_v a_o segment_n great_a than_o the_o half_a sphere_n ¶_o a_o corollary_n add_v by_o the_o same_o flussas_n by_o the_o foresay_a assumpt_n it_o be_v manifest_a that_o if_o from_o the_o centre_n of_o a_o sphere_n the_o line_n draw_v perpendicular_o unto_o the_o circle_n which_o cut_v the_o sphere_n be_v equal_a those_o circle_n be_v equal_a and_o the_o perpendicular_a line_n so_o draw_v fall_n upon_o the_o centre_n of_o the_o same_o circle_n for_o the_o line_n which_o be_v draw_v from_o the_o centre_n of_o the_o sphere_n to_o the_o circumference_n contain_v in_o power_n the_o power_n of_o the_o perpendicular_a line_n and_o the_o power_n of_o the_o line_n which_o join_v together_o the_o end_n of_o those_o line_n wherefore_o from_o that_o square_a or_o power_n of_o the_o line_n from_o the_o centre_n of_o the_o sphere_n to_o the_o circumference_n or_o common_a section_n draw_v which_o be_v the_o semidiameter_n of_o the_o sphere_n take_v away_o the_o power_n of_o the_o perpendicular_a which_o be_v common_a to_o they_o â—ound_v about_o it_o follow_v that_o the_o residue_v how_o many_o so_o ever_o they_o be_v be_v equal_a power_n and_o therefore_o the_o line_n be_v equal_a the_o one_o to_o the_o other_o wherefore_o they_o will_v describe_v equal_a circle_n by_o the_o first_o definition_n of_o the_o three_o and_o upon_o their_o centre_n fall_v the_o perpendicular_a line_n by_o the_o 9_o of_o the_o three_o corollary_n and_o those_o circle_n upon_o which_o fall_v the_o great_a perpendicular_a line_n be_v the_o less_o circle_n for_o the_o power_n of_o the_o line_n draw_v from_o the_o centre_n of_o the_o sphere_n to_o the_o circumference_n be_v always_o one_o and_o equal_a to_o the_o power_n of_o the_o perpendicular_a line_n and_o also_o to_o the_o power_n of_o the_o line_n draw_v from_o the_o centre_n of_o the_o circle_n to_o their_o circumference_n the_o great_a that_o the_o power_n of_o the_o perpendicular_a line_n take_v away_o from_o the_o power_n contain_v they_o both_o be_v the_o less_o be_v the_o power_n and_o therefore_o the_o line_n remain_v which_o be_v the_o semidiameter_n of_o the_o circle_n and_o therefore_o the_o less_o be_v the_o circle_n which_o they_o describe_v wherefore_o if_o the_o circle_n be_v equal_a the_o perpendicular_a line_n fall_v from_o the_o centre_n of_o the_o sphere_n upon_o they_o shall_v also_o be_v equal_a for_o if_o they_o shall_v be_v great_a or_o less_o the_o circle_n shall_v be_v unequal_a as_o it_o be_v before_o manifest_a but_o we_o suppose_v the_o perpendicular_n to_o be_v equal_a corollary_n also_o the_o perpendicular_a line_n fall_v upon_o those_o base_n be_v the_o least_o of_o all_o that_o be_v draw_v from_o the_o centre_n of_o the_o sphere_n for_o the_o other_o draw_v from_o the_o centre_n of_o the_o sphere_n to_o the_o circumference_n of_o the_o circle_n be_v in_o power_n equal_a both_o to_o the_o power_n of_o the_o perpendicular_n and_o to_o the_o power_n of_o the_o line_n join_v these_o perpendicular_n and_o these_o subtendent_fw-la line_n together_o make_v triangle_n rectangleround_v about_o as_o most_o easy_o you_o may_v conceive_v of_o the_o figure_n here_o annex_v a_o the_o centre_n of_o the_o sphere_n ab_fw-la the_o line_n from_o the_o centre_n of_o the_o sphere_n to_o the_o circumference_n of_o the_o circle_n make_v by_o the_o section_n bcb_n the_o diameter_n of_o circle_n make_v by_o the_o section_n ac_fw-la the_o perpendicular_n from_o the_o centre_n of_o the_o sphere_n to_o the_o circlesâ—_n who_o diameter_n bcb_n be_v one_o both_o side_n or_o in_o any_o situation_n else_o cb_o the_o semidiameter_n of_o the_o circle_n make_v by_o the_o section_n ao_o a_o perpendicular_a long_o than_o ac_fw-la and_o therefore_o the_o semidiameter_n ob_fw-la be_v less_o acb_o &_o aob_n triangle_n rectangle_n ¶_o the_o 2._o problem_n the_o 17._o proposition_n two_o sphere_n consist_v both_o about_o one_o &_o the_o self_n same_o centre_n be_v give_v to_o inscribe_v in_o the_o great_a sphere_n a_o solid_a of_o many_o side_n which_o be_v call_v a_o polyhedron_n which_o shall_v not_o touch_v the_o superficies_n of_o the_o less_o sphere_n svppose_v that_o there_o be_v two_o sphere_n about_o one_o &_o the_o self_n same_o centre_n namely_o about_o a._n it_o be_v require_v in_o the_o great_a sphere_n to_o inscribe_v a_o polyhedron_n or_o a_o solid_a of_o many_o side_n which_o shall_v not_o with_o his_o superficies_n touch_v the_o superficies_n of_o the_o less_o sphere_n let_v the_o sphere_n be_v cut_v by_o some_o one_o plain_a superficies_n pass_v by_o the_o centre_n a._n construction_n then_o shall_v their_o section_n be_v circle_n flussas_n for_o by_o the_o 12._o definition_n of_o the_o eleven_o the_o diameter_n remain_v fix_v and_o the_o semicircle_n be_v turn_v round_o about_o make_v a_o sphere_n wherefore_o in_o what_o position_n so_o ever_o you_o imagine_v the_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
be_v right_a angle_n and_o that_o the_o line_n ac_fw-la be_v double_a to_o the_o line_n cm_o now_o forasmuch_o as_o the_o angle_n alc_n be_v equal_a to_o the_o angle_n amf_n for_o that_o they_o be_v both_o right_a angle_n and_o the_o angle_n lac_n be_v common_a to_o both_o the_o triangle_n alc_n and_o amf_n wherefore_o the_o angle_n remain_v namely_o acl_n be_v equal_a to_o the_o angle_n remain_v afm_n by_o the_o corollary_n of_o the_o 32._o of_o the_o first_o wherefore_o the_o triangle_n acl_n be_v equiangle_n to_o the_o triangle_n amf_n wherefore_o proportional_o by_o the_o 4._o of_o the_o six_o as_o the_o line_n lc_n be_v to_o the_o line_n ca_n so_o be_v the_o line_n mf_n to_o the_o line_n fa._n and_o in_o the_o same_o proportion_n also_o be_v the_o double_n of_o the_o antecedent_n lc_a and_o mf_n by_o the_o 15._o of_o the_o five_o wherefore_o as_o the_o double_a of_o the_o line_n lc_n be_v to_o the_o line_n ca_n so_o be_v the_o double_a of_o the_o line_n mf_n to_o the_o line_n fa._n but_o as_o the_o double_a of_o the_o line_n mf_n be_v to_o the_o line_n favorina_n so_o be_v the_o line_n mf_n to_o the_o half_a of_o the_o line_n favorina_n by_o the_o 15._o of_o the_o five_o wherefore_o as_o the_o double_a of_o the_o line_n lc_n be_v to_o the_o line_n ca_n so_o be_v the_o line_n mf_n to_o the_o half_a of_o the_o line_n favorina_n by_o the_o 11._o of_o the_o five_o and_o in_o the_o same_o proportion_n by_o the_o 15._o of_o the_o five_o be_v the_o half_n of_o the_o consequent_n namely_o of_o ca_n and_o of_o the_o halue_fw-mi of_o the_o line_n af._n wherefore_o as_o the_o double_a of_o the_o line_n lc_n be_v to_o the_o half_a of_o the_o line_n ac_fw-la so_o be_v the_o line_n mf_n to_o the_o four_o part_n of_o the_o line_n fa._n but_o the_o double_a of_o the_o line_n lc_n be_v the_o line_n dc_o and_o the_o half_a of_o the_o line_n ca_n be_v the_o line_n cm_o as_o have_v before_o be_v prove_v and_o the_o four_o part_n of_o the_o line_n favorina_n be_v the_o line_n fk_o for_o the_o line_n fk_o be_v the_o four_o part_n of_o the_o line_n fh_o by_o construction_n wherefore_o as_o the_o line_n dc_o be_v to_o the_o line_n cm_o so_o be_v the_o line_n mf_n to_o the_o line_n fk_o wherefore_o by_o composition_n by_o the_o 18._o of_o the_o five_o as_o both_o the_o line_n dc_o and_o cm_o be_v to_o the_o line_n cm_o so_o be_v the_o whole_a line_n mk_o to_o the_o line_n fk_o wherefore_o also_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 dc_o and_o cm_o be_v to_o the_o square_n of_o the_o line_n cm_o so_o be_v the_o square_a of_o the_o line_n mk_o to_o the_o square_n of_o the_o line_n fk_o and_o forasmuch_o as_o by_o the_o 8._o of_o the_o thirteen_o a_o line_n which_o be_v subtend_v under_o two_o side_n of_o a_o pentagon_n figure_n as_o be_v the_o line_n ac_fw-la be_v divide_v by_o a_o extreme_a &_o mean_a proportion_n the_o great_a segment_n be_v equal_a to_o the_o side_n of_o the_o pentagon_n figure_n that_o be_v unto_o the_o line_n dc_o and_o by_o the_o 1._o of_o the_o thirteen_o the_o great_a segment_n have_v add_v unto_o it_o the_o half_a of_o the_o whole_a be_v in_o power_n quintuple_a to_o the_o square_n make_v of_o the_o half_a of_o the_o whole_a and_o the_o half_a of_o the_o whole_a line_n ac_fw-la be_v the_o line_n cm_o wherefore_o the_o square_a that_o be_v make_v of_o the_o line_n dc_o and_o cm_o that_o be_v of_o the_o great_a segment_n and_o of_o the_o half_a of_o the_o whole_a as_o of_o one_o line_n be_v quintuple_a to_o the_o square_n of_o the_o line_n cm_o that_o be_v of_o the_o half_a of_o the_o whole_a but_o as_o the_o square_n make_v of_o the_o line_n dc_o and_o cm_o as_o of_o one_o line_n be_v to_o the_o square_n of_o the_o line_n cm_o so_o be_v it_o prove_v that_o the_o square_n of_o the_o line_n mk_o be_v to_o the_o square_n of_o the_o line_n fk_o wherefore_o the_o square_a of_o the_o line_n mk_o be_v quintuple_a to_o the_o square_n of_o the_o line_n fk_o but_o the_o square_n of_o the_o line_n kf_o be_v rational_a as_o have_v before_o be_v prove_v wherefore_o also_o the_o square_a of_o the_o line_n mk_o be_v rational_a by_o the_o 9_o definition_n of_o the_o ten_o for_o the_o square_n of_o the_o line_n mk_o have_v to_o the_o square_n of_o the_o line_n kf_o that_o proportion_n that_o number_n have_v to_o number_n namely_o that_o 5._o have_v to_o 1._o and_o therefore_o the_o say_a square_n be_v commensurable_a by_o the_o 6._o of_o the_o ten_o wherefore_o also_o the_o line_n mk_o be_v rational_a and_o forasmuch_o as_o the_o line_n bf_o be_v quadruple_a to_o the_o line_n fk_o for_o the_o semidiameter_n bf_o be_v equal_a to_o the_o semidiameter_n fh_o therefore_o the_o line_n bk_o be_v quintuple_a to_o the_o line_n fk_o wherefore_o the_o square_a of_o the_o line_n bk_o be_v 25._o time_n so_o much_o as_o the_o square_n of_o the_o line_n kf_o by_o the_o corollary_n of_o the_o 20_o of_o the_o six_o but_o the_o square_n of_o the_o line_n mk_o be_v quintuple_a to_o the_o square_n of_o the_o fk_n as_o be_v prove_v wherefore_o the_o square_a of_o the_o line_n bk_o be_v quintuple_a to_o the_o square_n of_o the_o line_n km_o wherefore_o the_o square_v of_o the_o line_n bk_o have_v not_o to_o the_o square_n of_o the_o line_n km_o that_o proportion_n that_o a_o square_a number_n have_v to_o a_o square_a number_n by_o the_o corollary_n of_o the_o 25._o of_o the_o eight_o wherefore_o by_o the_o 9_o of_o the_o ten_o the_o line_n bk_o be_v incommensurable_a in_o length_n to_o the_o line_n km_o and_o either_o of_o the_o line_n be_v rational_a wherefore_o the_o line_n bk_o and_o km_o be_v rational_a commensurable_a in_o power_n only_o but_o if_o from_o a_o rational_a line_n be_v take_v away_o a_o rational_a line_n be_v commensurable_a in_o power_n only_o to_o the_o whole_a that_o which_o remain_v be_v irrational_a and_o be_v by_o the_o 73._o of_o the_o ten_o call_v a_o residual_a line_n wherefore_o the_o line_n mb_n be_v a_o residual_a line_n and_o the_o line_n convenient_o join_v unto_o it_o be_v the_o line_n mk_o now_o i_o say_v that_o the_o line_n bm_n be_v a_o four_o residual_a line_n unto_o the_o excess_n of_o the_o square_n of_o the_o line_n bk_o above_o the_o square_n of_o the_o line_n km_o let_v the_o square_n of_o the_o line_n n_o be_v equal_a which_o excess_n how_o to_o find_v out_o be_v teach_v in_o the_o assumpt_n put_v after_o the_o 13._o proposition_n of_o the_o ten_o wherefore_o the_o line_n bk_o be_v in_o power_n more_o than_o the_o line_n km_o by_o the_o square_n of_o the_o line_n n._n and_o forasmuch_o as_o the_o line_n kf_o be_v commensurable_a in_o length_n to_o the_o line_n fb_o for_o it_o be_v the_o four_o part_n thereof_o therefore_o by_o the_o 16._o of_o the_o ten_o the_o whole_a line_n kb_o be_v commensurable_a in_o length_n to_o the_o line_n fb_o but_o the_o line_n fb_o be_v commensurable_a in_o length_n to_o the_o line_n bh_o namely_o the_o semidiameterâ—_n to_o the_o diameter_n wherefore_o the_o line_n bk_o be_v commensurable_a in_o length_n to_o the_o line_n bh_o by_o the_o 12._o of_o the_o ten_o and_o forasmuch_o as_o the_o square_n of_o the_o line_n bk_o be_v quintuple_a to_o the_o square_n of_o the_o line_n km_o therefore_o the_o square_a of_o the_o line_n bk_o have_v to_o the_o square_n of_o the_o line_n km_o that_o proportion_n that_o five_o have_v to_o one_o 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 the_o square_a of_o bk_o have_v to_o the_o square_n of_o the_o line_n n_o that_o proportion_n that_o five_o have_v to_o four_o &_o therefore_o it_o have_v not_o that_o proportion_n that_o a_o square_a number_n have_v to_o a_o square_a number_n by_o the_o corollary_n of_o the_o 25._o of_o the_o eight_o wherefore_o the_o line_n bk_o be_v incommensurable_a in_o length_n to_o the_o line_n n_o by_o the_o 9_o of_o the_o ten_o wherefore_o the_o line_n bk_o be_v in_o power_n more_o than_o the_o line_n km_o by_o the_o square_n of_o a_o line_n incommensurable_a in_o length_n to_o the_o line_n bk_o now_o then_o forasmuch_o as_o the_o whole_a line_n bk_o be_v in_o power_n more_o than_o the_o line_n convenient_o join_v namely_o then_o km_o by_o the_o square_n of_o a_o line_n incommensurable_a in_o length_n to_o the_o line_n bk_o and_o the_o whole_a line_n bk_o be_v commensurable_a in_o length_n to_o the_o rational_a line_n give_v bh_o therefore_o the_o line_n mb_n be_v a_o four_o residual_a line_n by_o the_o definition_n of_o a_o four_o residual_a line_n but_o a_o rectangle_n parallelogram_n
side_n of_o a_o cube_fw-la let_v it_o be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n in_o the_o point_n n_o and_o let_v the_o great_a segment_n thereof_o be_v nb._n wherefore_o the_o line_n nb_n be_v the_o side_n of_o a_o dodecahedron_n dodecahedron_n by_o the_o corollary_n of_o the_o 17._o of_o this_o book_n body_n and_o forasmuch_o as_o it_o have_v be_v prove_v by_o the_o 13._o of_o this_o book_n that_o the_o diameter_n of_o the_o sphere_n be_v in_o power_n sesquialter_fw-la to_o of_o the_o side_n of_o the_o pyramid_n and_o be_v in_o power_n double_a to_o be_v the_o side_n of_o the_o octohedron_n by_o the_o 14._o of_o the_o same_o and_o be_v in_o power_n treble_a to_o fb_o the_o side_n of_o the_o cube_fw-la by_o the_o 15._o of_o the_o same_o wherefore_o it_o follow_v that_o of_o what_o part_n the_o diameter_n of_o the_o sphere_n contain_v six_o of_o such_o part_n the_o side_n of_o the_o pyramid_n contain_v four_o and_o the_o side_n of_o the_o octohedron_n three_o and_o the_o side_n of_o the_o cube_fw-la two_o wherefore_o the_o side_n of_o the_o pyramid_n be_v in_o power_n to_o the_o side_n of_o the_o octohedron_n in_o sesquitertia_fw-la proportion_n and_o be_v in_o power_n to_o the_o side_n of_o the_o cube_fw-la in_o double_a proportion_n and_o the_o side_n of_o the_o octohedron_n be_v in_o power_n to_o the_o side_n of_o the_o cube_fw-la in_o sesquialtera_fw-la proportion_n wherefore_o the_o foresaid_a side_n of_o the_o three_o figure_n that_o be_v of_o the_o pyramid_n of_o the_o octohedron_n and_o oâ—_n the_o cube_fw-la be_v the_o one_o to_o the_o other_o in_o rational_a proportion_n wherefore_o they_o be_v rational_a but_o the_o other_o two_o side_n namely_o the_o side_n of_o the_o icosahedron_n and_o of_o the_o dodecahedron_n be_v neither_o the_o one_o to_o the_o other_o nor_o also_o to_o the_o foresay_a side_n in_o rational_a proportion_n â—or_a they_o be_v irrational_a line_n namely_o a_o less_o line_n and_o a_o residual_a line_n an_o other_o way_n to_o prove_v that_o the_o line_n mb_n be_v great_a than_o the_o line_n nb._n dodecahedron_n forasmuch_o as_o the_o line_n ad_fw-la be_v double_a to_o the_o line_n db_o therefore_o the_o line_n ab_fw-la be_v treble_a to_o the_o line_n db._n but_o as_o ab_fw-la be_v to_o bd_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 bf_o by_o the_o 8._o of_o the_o six_o for_o the_o triangle_n fab._n be_v equiangle_n to_o the_o triangle_n fdb_n wherefore_o the_o square_a of_o the_o line_n ab_fw-la be_v treble_a to_o the_o square_n of_o the_o line_n bf_o and_o it_o be_v before_o prove_v that_o the_o square_n 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 kl_o wherefore_o five_o square_n make_v of_o the_o line_n kl_o be_v equal_a to_o three_o square_n make_v of_o the_o line_n fb_o but_o three_o square_n make_v of_o the_o line_n fb_o be_v great_a than_o six_o square_n make_v of_o the_o line_n nb_n as_o be_v straight_o way_n prove_v wherefore_o five_o square_n make_v of_o the_o line_n kl_o be_v great_a than_o six_o square_n make_v of_o the_o line_n nb._n wherefore_o also_o one_o square_a make_v of_o the_o line_n kl_o be_v great_a than_o one_o square_a make_v of_o the_o line_n nb._n wherefore_o the_o line_n kl_o be_v great_a than_o the_o line_n nb._n but_o the_o line_n kl_o be_v equal_a to_o the_o line_n lm_o wherefore_o the_o line_n lm_o be_v great_a than_o the_o line_n nb._n wherefore_o the_o line_n mb_n be_v much_o great_a than_o the_o line_n nb_n which_o be_v require_v to_o be_v prove_v but_o now_o let_v we_o prove_v that_o three_o square_n make_v of_o the_o line_n fb_o nb._n be_v great_a than_o six_o square_n make_v of_o the_o line_n nb._n forasmuch_o as_o the_o line_n bn_o be_v great_a than_o the_o line_n nf_n for_o it_o be_v the_o great_a segment_n of_o the_o line_n bf_o divide_v by_o a_o extreme_a and_o mean_a proportion_n therefore_o that_o which_o be_v contain_v under_o the_o line_n bf_o and_o bn_o be_v great_a than_o that_o which_o be_v contain_v under_o the_o line_n bf_o and_o fn_o by_o the_o 1._o of_o the_o six_o wherefore_o that_o which_o be_v contain_v under_o the_o line_n bf_o and_o bn_o together_o with_o that_o which_o be_v contain_v under_o the_o line_n bf_o and_o fn_o be_v great_a than_o that_o which_o be_v contain_v under_o the_o line_n bf_o and_o fn_o twice_o but_o that_o which_o be_v contain_v under_o the_o line_n bf_o and_o fn_o together_o with_o that_o which_o be_v contain_v under_o the_o line_n bf_o and_o bn_o be_v the_o square_a of_o the_o line_n bf_o by_o the_o 2._o of_o the_o second_o and_o that_o which_o be_v contain_v under_o the_o line_n bf_o and_o fn_o once_o be_v equal_a to_o the_o square_n of_o nb._n for_o the_o line_n fb_o be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n in_o the_o point_n n_o and_o by_o the_o 17._o of_o the_o six_o that_o which_o be_v contain_v under_o the_o extreme_n be_v equal_a to_o the_o square_n make_v of_o the_o middle_a line_n wherefore_o the_o square_a of_o the_o line_n fb_o be_v great_a than_o the_o double_a of_o the_o square_n of_o the_o line_n bn_o wherefore_o one_o of_o the_o square_n make_v of_o the_o line_n bf_o be_v great_a than_o two_o square_n make_v of_o the_o line_n bn_o wherefore_o also_o three_o square_n make_v of_o the_o line_n fb_o be_v great_a than_o six_o square_n make_v of_o the_o line_n bn_o which_o be_v require_v to_o be_v prove_v a_o corollary_n now_o also_o i_o say_v that_o beside_o the_o five_o foresay_a solid_n there_o can_v not_o be_v describe_v any_o other_o solid_a comprehend_v under_o figure_n equilater_n &_o equiangle_n the_o one_o to_o the_o other_o base_n for_o of_o two_o triangle_n or_o of_o any_o two_o other_o plain_a superficiece_n can_v not_o be_v make_v a_o solid_a angle_n for_o that_o be_v contrary_a to_o the_o definition_n of_o a_o solid_a angle_n under_o three_o triangle_n be_v contain_v the_o solid_a angle_n of_o a_o pyramid_n under_o four_o the_o solid_a angle_n of_o a_o octohedron_n under_o five_o the_o solid_a angle_n of_o a_o icosahedron_n of_o six_o equilater_n &_o equiangle_n triangle_n set_v to_o one_o point_n can_v not_o be_v make_v a_o solid_a angle_n for_o forasmuch_o as_o the_o angle_n of_o a_o equilater_n triangle_n be_v two_o three_o part_n of_o a_o right_a angle_n the_o six_o angle_n of_o the_o solid_a shall_v be_v equal_a to_o four_o right_a angle_n which_o be_v impossible_a for_o every_o solid_a angle_n be_v by_o the_o 21._o of_o the_o eleven_o contain_v under_o plain_a angle_n less_o they_o four_o right_a angle_n and_o by_o the_o same_o reason_n can_v not_o be_v make_v a_o solid_a angle_n contain_v under_o more_o them_z six_o plain_a superficial_a angle_n of_o equilater_n triangle_n under_o three_o square_n be_v contain_v the_o angle_n of_o a_o cube_fw-la under_o four_o square_n it_o be_v impossible_a that_o a_o solid_a angle_n shall_v be_v contain_v for_o then_o again_o it_o shall_v be_v contain_v under_o four_o right_a angle_n wherefore_o much_o less_o can_v any_o solid_a angle_n be_v contain_v under_o more_o square_n than_o four_o under_o three_o equilater_n and_o equiangle_n pentagon_n be_v contain_v the_o solid_a angle_n of_o a_o dodecahedron_n but_o under_o four_o it_o be_v impossible_a for_o forasmuch_o as_o the_o angle_n of_o a_o pentagon_n be_v a_o right_a angle_n and_o the_o five_o part_v more_o of_o a_o right_a angle_n the_o four_o angle_n shall_v be_v great_a than_o four_o right_a angle_n which_o be_v impossible_a and_o therefore_o much_o less_o can_v a_o solid_a angle_n be_v compose_v of_o more_o pentagon_n than_o four_o neither_o can_v a_o solid_a angle_n be_v contain_v under_o any_o other_o equilater_n and_o equiangle_n figure_n of_o many_o angle_n for_o that_o that_o also_o shall_v be_v absurd_a for_o the_o more_o the_o side_n increase_v the_o great_a be_v the_o angle_n which_o they_o contain_v and_o therefore_o the_o far_a of_o be_v the_o superficial_a angle_n contain_v of_o those_o side_n from_o compose_v of_o a_o solid_a angle_n wherefore_o beside_o the_o foresay_a five_o figure_n there_o can_v not_o be_v make_v any_o solid_a figure_n contain_v under_o equal_a side_n and_o equal_a angle_n which_o be_v require_v to_o be_v prove_v a_o assumpt_n but_o now_o that_o the_o angle_n of_o a_o equilater_n and_o equiangle_n pentagon_n be_v a_o right_a angle_n and_o a_o fiâ—th_v parâ—_n more_o of_o a_o right_a angle_n may_v thus_o be_v prove_v suppose_v that_o abcde_v be_v a_o equilater_n and_o equiangle_n pentagon_n â—irst_n and_o by_o the_o 14._o of_o the_o four_o describe_v about_o it_o a_o circle_n abcde_o and_o take_v by_o the_o 1._o of_o the_o three_o the_o centre_n thereof_o and_o let_v the_o same_o be_v
wherefore_o the_o line_n ce_fw-fr be_v to_o the_o line_n egg_n as_o the_o great_a segment_n to_o the_o less_o and_o therefore_o their_o proportion_n be_v as_o the_o whole_a line_n i_fw-mi be_v to_o the_o great_a segment_n ce_fw-fr and_o as_o the_o great_a segment_n ce_fw-fr be_v to_o the_o less_o segment_n egg_n wherefore_o the_o whole_a line_n ceg_n which_o make_v the_o great_a segment_n and_o the_o less_o be_v equal_a to_o the_o whole_a line_n i_fw-mi or_o je_n and_o forasmuch_o as_o two_o parallel_n plain_a superficiece_n namely_o that_o which_o be_v extend_v by_o job_n and_o that_o which_o be_v extend_v by_o the_o line_n ag_fw-mi be_v cut_v by_o the_o plain_n of_o the_o triangle_n be_v which_o pass_v by_o the_o line_n ag_fw-mi and_o ib_n their_o common_a section_n ag_v and_o ib_n shall_v be_v parallel_n by_o the_o 16._o of_o the_o eleven_o but_o the_o angle_n buy_v or_o bic_n be_v a_o right_a angle_n wherefore_o the_o angle_n agc_n be_v also_o a_o right_a angle_n by_o the_o 29._o of_o the_o first_o and_o those_o right_a angle_n be_v contain_v under_o equal_a side_n namely_o the_o line_n gc_n be_v equal_a to_o the_o line_n ci_o and_o the_o line_n ag_fw-mi to_o the_o line_n by_o by_o the_o 33._o of_o the_o first_o wherefore_o the_o base_n ca_o and_o cb_o be_v equal_a by_o the_o 4._o of_o the_o first_o but_o of_o the_o line_n cb_o the_o line_n ce_fw-fr be_v prove_v to_o be_v the_o great_a segment_n wherefore_o the_o same_o line_n ce_fw-fr be_v also_o the_o great_a segment_n of_o the_o line_n ca_o but_o cn_fw-la be_v also_o the_o great_a segment_n of_o the_o same_o line_n ca._n wherefore_o unto_o the_o line_n ce_fw-fr the_o line_n cn_fw-la which_o be_v the_o side_n of_o the_o dodecahedron_n and_o be_v set_v at_o the_o diameter_n be_v equal_a and_o by_o the_o same_o reason_n the_o rest_n of_o the_o side_n which_o be_v set_v at_o the_o diameter_n may_v be_v prove_v eguall_a to_o line_n equal_a to_o the_o line_n ce._n wherefore_o the_o pentagon_n inscribe_v in_o the_o circle_n where_o in_o be_v contain_v the_o triangle_n be_v be_v by_o the_o 11._o of_o the_o four_o equiangle_n and_o equilater_n and_o forasmch_v as_o two_o pentagon_n set_v upon_o every_o one_o of_o the_o base_n of_o the_o cube_fw-la do_v make_v a_o dodecahedron_n and_o six_o base_n of_o the_o cube_fw-la do_v receive_v twelve_o angle_n of_o the_o dodecahedron_n and_o the_o 8._o semidiameter_n do_v in_o the_o point_n where_o they_o be_v cut_v by_o a_o extreme_a and_o mean_a proportion_n receive_v the_o rest_n therefore_o the_o 12._o pentagon_n base_n contain_v 20._o solid_a angle_n do_v inscribe_v the_o dodecahedron_n in_o the_o cube_fw-la by_o the_o 1._o definition_n of_o this_o book_n wherefore_o in_o a_o cube_fw-la give_v be_v inscribe_v a_o dodecahedron_n which_o be_v require_v to_o be_v do_v first_o corollary_n the_o diameter_n of_o the_o sphere_n which_o contain_v the_o dodecahedron_n contain_v in_o power_n these_o two_o side_n namely_o the_o side_n of_o the_o dodecahedron_n and_o the_o side_n of_o the_o cube_fw-la wherein_o the_o dodecahedron_n be_v inscribe_v for_o in_o the_o first_o figure_n a_o line_n draw_v from_o the_o centre_n oh_o to_o the_o point_n b_o the_o angle_n of_o the_o dodecahedron_n namely_o the_o line_n ob_fw-la contain_v in_o power_n these_o two_o line_n ov_v the_o half_a side_n of_o the_o cube_fw-la and_o ub_z the_o half_a side_n of_o the_o dodecahedron_n by_o the_o 47._o of_o the_o first_o wherefore_o by_o the_o 15._o of_o the_o five_o the_o double_a of_o the_o line_n ob_fw-la which_o be_v the_o diameter_n of_o the_o sphere_n contain_v the_o dodecahedron_n contain_v in_o power_n the_o double_a of_o the_o other_o line_n ov_n and_o ub_z which_o be_v the_o side_n of_o the_o cube_fw-la and_o of_o the_o dodecahedron_n ¶_o second_v corollary_n the_o side_n of_o a_o cube_fw-la divide_v by_o a_o extreme_a and_o mean_a proportion_n make_v the_o less_o segment_n the_o side_n of_o the_o dodecahedron_n inscribe_v in_o it_o and_o the_o great_a segment_n the_o side_n of_o the_o cube_fw-la inscribe_v in_o the_o same_o dodecahedron_n for_o it_o be_v before_o prove_v that_o the_o side_n of_o the_o dodecahedron_n be_v the_o great_a segment_n of_o be_v the_o side_n of_o the_o triangle_n becâ—_n but_o the_o side_n be_v which_o be_v equal_a to_o the_o lineâ—_z gb_o and_o sf_n be_v the_o great_a segment_n of_o gf_o the_o side_n of_o the_o cube_fw-la which_o line_n â—e_v subtend_v thâ—_z angle_n of_o the_o pentagon_n be_v by_o the_o â—_o of_o this_o book_n the_o side_n of_o the_o cube_fw-la inscribe_v in_o the_o dodecahedron_n three_o corollary_n the_o side_n of_o a_o cube_fw-la be_v equal_a to_o the_o side_n of_o a_o dodecahedron_n inscribe_v in_o it_o and_o circumscribe_v about_o it_o for_o it_o be_v manifest_a by_o this_o proposition_n that_o the_o side_n of_o a_o cube_fw-la make_v the_o less_o segment_n the_o side_n of_o a_o dodecahedron_n inscribe_v in_o it_o namely_o as_o in_o the_o first_o figure_n the_o line_n b_v the_o side_n of_o the_o dodecahedron_n inscribe_v be_v the_o less_o segment_n of_o the_o line_n gf_o the_o side_n of_o the_o cube_fw-la and_o it_o be_v prove_v in_o the_o 17._o of_o the_o thirteen_o that_o the_o same_o side_n of_o the_o cube_fw-la subtend_v the_o angle_n of_o the_o pentagon_n of_o the_o dodecahedron_n circumscribe_v and_o therefore_o it_o make_v the_o great_a segment_n the_o side_n of_o the_o dodecahedron_n or_o of_o the_o pentagon_n by_o the_o first_o corollary_n of_o the_o same_o wherefore_o it_o be_v equal_a to_o both_o those_o segment_n the_o 14._o problem_n the_o 14._o proposition_n in_o a_o cube_fw-la give_v to_o inscribe_v a_o icosahedron_n svppose_v that_o the_o cube_fw-la give_v by_o abc_n construction_n the_o centre_n of_o who_o base_n let_v be_v the_o point_n d_o e_o g_o h_o i_o king_n by_o which_o point_n draw_v in_o the_o base_n unto_o the_o other_o side_n parallel_v not_o touch_v the_o one_o the_o other_o and_o divide_v the_o line_n draw_v from_o the_o centre_n as_o the_o line_n dt_n etc_n etc_n by_o a_o extreme_a and_o mean_a proportion_n in_o the_o point_n a_o f_o l_o m_o n_o b_o p_o q_o r_o s_o c_o oh_o by_o the_o 30._o of_o the_o six_o and_o let_v the_o great_a segment_n be_v about_o the_o centre_n and_o draw_v these_o right_a line_n all_o agnostus_n be_o and_o tg_n and_o forasmuch_o as_o the_o line_n cut_v be_v parallel_n to_o the_o side_n of_o the_o cube_fw-la demonstration_n they_o shall_v make_v right_a angle_n the_o one_o with_o the_o other_o by_o the_o 29._o of_o the_o first_o and_o forasmuch_o as_o they_o be_v equal_a their_o section_n shall_v be_v equal_a for_o that_o the_o section_n be_v like_a by_o the_o 2._o of_o the_o fourteen_o wherefore_o the_o line_n tg_n be_v equal_a to_o the_o line_n dt_n for_o they_o be_v each_o half_a side_n of_o the_o cube_fw-la wherefore_o the_o square_a of_o the_o whole_a line_n tg_n and_o of_o the_o less_o segment_n ta_n be_v triple_a to_o the_o square_n of_o the_o line_n ad_fw-la the_o great_a segment_n by_o the_o 4._o of_o the_o thirteen_o but_o the_o line_n agnostus_n contain_v in_o power_n the_o line_n at_o &_o tg_n for_o the_o angle_n atg_n be_v a_o right_a angle_n wherefore_o the_o square_a of_o the_o line_n agnostus_n be_v triple_a to_o the_o square_n of_o the_o line_n ad._n and_o forasmuch_o as_o the_o line_n mgl_n be_v erect_v perpendicular_o to_o the_o plain_a pass_v by_o the_o line_n at_o &_o which_o be_v parallel_n to_o the_o base_n of_o the_o cube_fw-la by_o the_o corollary_n of_o the_o 14._o of_o the_o eleven_o therefore_o the_o angle_n agl_n be_v a_o right_a angle_n but_o the_o line_n lg_n be_v equal_a to_o the_o line_n ad_fw-la for_o they_o be_v the_o great_a segment_n of_o equal_a line_n wherefore_o the_o line_n agnostus_n which_o be_v in_o power_n triple_a to_o the_o line_n ad_fw-la be_v in_o power_n triple_a to_o the_o line_n lg_n wherefore_o add_v unto_o the_o same_o square_n of_o the_o line_n agnostus_n the_o square_a of_o the_o line_n lg_n the_o square_a of_o the_o line_n all_o which_o by_o the_o 47._o of_o the_o first_o contain_v in_o power_n the_o two_o line_n agnostus_n and_o gl_n shall_v be_v quadruple_a to_o the_o line_n ad_fw-la or_o lg_n wherefore_o the_o line_n all_o be_v double_a to_o the_o line_n ad_fw-la by_o the_o 20._o of_o the_o six_o and_o therefore_o be_v equal_a to_o the_o line_n of_o or_o to_o the_o line_n lm_o and_o by_o the_o same_o reason_n may_v we_o prove_v that_o every_o one_o of_o the_o other_o line_n which_o couple_v the_o next_o section_n of_o the_o line_n cut_v as_o the_o line_n be_o pf_n pm_n mq_n and_o the_o rest_n be_v equal_a wherefore_o the_o triangle_n alm_n apf_n ample_n pmq_n and_o the_o rest_n such_o like_a be_v equal_a equiangle_n and_o equilater_n
two_o line_n hif_n and_o tio_n cut_v the_o one_o the_o other_o be_v in_o one_o and_o the_o self_n same_o '_o plain_n by_o the_o 2._o of_o the_o eleven_o and_o therefore_o the_o point_n h_o t_o f_o oh_o be_v in_o one_o &_o the_o self_n same_o plain_a wherforeâ—_n the_o rectangle_n figure_n hoff_z be_v quadrilater_n and_o equilater_n and_o in_o one_o and_o the_o self_n same_o plain_n be_v a_o square_a by_o the_o definition_n of_o a_o square_a and_o by_o the_o same_o reason_n may_v the_o rest_n of_o the_o base_n of_o the_o solid_a be_v prove_v to_o be_v square_n equal_a and_o plain_a or_o superficial_a now_o than_o the_o solid_a be_v comprehend_v of_o 6._o equal_a square_n which_o be_v contain_v of_o 12._o equal_a side_n which_o square_n make_v 8._o solid_a angle_n of_o which_o four_o be_v in_o the_o centre_n of_o the_o base_n oâ—_n the_o pyramid_n and_o the_o other_o 4._o be_v in_o the_o middle_a section_n of_o the_o four_o perdendiculars_n wherefore_o the_o solid_a hoftpgrn_n be_v a_o cube_fw-la by_o the_o 21._o definition_n of_o the_o eleven_o and_o be_v inscribe_v in_o the_o pyramid_n by_o the_o first_o definition_n of_o this_o book_n wherefore_o in_o a_o trilater_n equilater_n pyramid_n give_v be_v inscribe_v a_o cube_fw-la ¶_o a_o corollary_n the_o line_n which_o cut_v into_o two_o equal_a part_n the_o opposite_a side_n of_o the_o pyramid_n be_v triple_a to_o the_o side_n of_o the_o cube_fw-la inscribe_v in_o the_o pyramid_n and_o pass_v by_o the_o centre_n of_o the_o cube_fw-la for_o the_o line_n sev_n who_o three_o part_n the_o line_n si_fw-it be_v cut_v the_o opposite_a side_n cd_o and_o ab_fw-la into_o two_o equll_a part_n but_o the_o line_n ei_o which_o be_v draw_v from_o the_o centre_n of_o the_o cube_fw-la to_o the_o base_a be_v prove_v to_o be_v a_o three_o part_n of_o the_o line_n es_fw-ge wherefore_o the_o side_n of_o the_o cube_fw-la which_o be_v double_a to_o the_o line_n ei_o shall_v be_v a_o three_o part_n of_o the_o whole_a line_n we_o which_o be_v as_o have_v be_v prove_v double_a to_o the_o line_n es._n the_o 19_o problem_n the_o 19_o proposition_n in_o a_o trilater_n equilater_n pyramid_n give_v to_o inscribe_v a_o icosahedron_n svppose_v that_o the_o pyramid_n be_v give_v 〈◊〉_d abâ—dâ—_n every_o one_o of_o who_o sâ—des_n 〈◊〉_d be_v divide_v into_o two_o equal_a part_n in_o the_o poyâ—â—â—â—â—_n m_o king_n l_o p_o n._n construction_n and_o iâ—_z every_o one_o of_o the_o base_n of_o that_o pyramid_n descry_v the_o trianglâ—â—_n lâ—â—_n pmn_n nkl_n and_o 〈…〉_o which_o triangle_n shall_v be_v equilater_n by_o the_o 4._o of_o the_o firâ—t_o â—or_a the_o side_n subtend_v equal_a angle_n of_o the_o pyramid_n contain_v under_o the_o half_n of_o the_o side_n of_o the_o same_o pyramisâ—_n wherefore_o the_o side_n of_o the_o say_a triangle_n be_v equal_a let_v those_o side_n be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n by_o the_o 30._o of_o the_o six_o in_o the_o point_n c_o e_o q_o r_o s_o t_o h_o i_o oh_o five_o a'n_z x._o now_o then_o those_o side_n be_v cut_v into_o the_o self_n same_o proportion_n by_o the_o 2._o of_o the_o fourtenth_n and_o therefore_o they_o make_v the_o liâ—e_z section_n equal_a by_o the_o â—_o part_n of_o the_o nine_o of_o the_o five_o now_o i_o say_v that_o the_o foresay_a point_n doâ—_n receive_v the_o angle_n of_o the_o icosahedron_n inscribe_v in_o the_o pyramid_n abâ—d_n in_o the_o foresay_a triangle_n let_v there_o again_o be_v make_v other_o triangle_n by_o couple_v the_o section_n and_o let_v those_o triangle_n be_v tr_n joh_n ceq_n and_o vxy_n which_o shall_v be_v equilater_n for_o every_o one_o of_o their_o side_n do_v subâ—â—â—d_v equal_a angle_n of_o equilater_n triangle_n and_o those_o say_v equal_a angle_n be_v contain_v under_o equal_a sideâ—_n namely_o under_o the_o great_a segment_n and_o the_o less_o â—_o and_o therefore_o the_o side_n which_o subtend_v those_o angle_n be_v equal_a by_o the_o 4._o of_o the_o first_o now_o let_v we_o prove_v that_o at_o each_o of_o the_o foresay_a point_n as_o for_o example_n at_o t_o be_v set_v the_o solid_a angle_n of_o a_o icosahâ—dronâ—_n demonstration_n forasmuch_o as_o the_o triangle_n trs_n and_o tqo_n be_v equilater_n and_o equal_a the_o 4._o right_a line_n tr_z it_o be_v tq_n and_o to_o shall_v be_v equal_a and_o forasmuch_o as_o â—pnk_v be_v a_o square_n cut_v the_o pyramid_n abâ—d_v into_o two_o equal_a paâ—â—â—â—_n by_o the_o corollay_n of_o the_o second_o of_o this_o bookeâ—_n the_o line_n th._n shall_v be_v in_o power_n duple_n to_o the_o line_n tn_n or_o nh_v by_o the_o 47._o of_o the_o first_o for_o the_o line_n tn_n or_o nh_v be_v equal_a for_o that_o by_o construction_n they_o be_v each_o less_o segment_n and_o the_o line_n rt_n or_o it_o be_v be_v in_o power_n duple_n to_o the_o same_o line_n tn_n or_o nh_v by_o the_o corollary_n of_o the_o 16._o of_o this_o book_n for_o it_o subtend_v the_o angle_n of_o the_o triangle_n contain_v under_o the_o two_o segment_n wherefore_o the_o line_n th._n it_o be_v tr_z tq_n and_o to_o be_v equal_a and_o so_o also_o be_v the_o line_n his_o sir_n rq_n qo_n and_o oh_o which_o subtend_v the_o angle_n at_o the_o point_n t_o equal_a for_o the_o line_n qr_o contain_v in_o power_n the_o two_o line_n pq_n and_o pr._n the_o less_o segment_n which_o two_o line_n the_o line_n th._n also_o contain_v in_o power_n and_o the_o rest_n of_o the_o line_n do_v subtend_v angle_n of_o equilater_n triangle_n contain_v under_o the_o great_a segment_n and_o the_o less_o wherefore_o the_o five_o triangle_n trs_n tsh_o tho_o toq_fw-la tqr_n be_v equilater_n and_o equal_a make_v the_o solid_a angle_n of_o a_o icosahedron_n at_o the_o point_n t_o by_o the_o 16._o of_o the_o thirteen_o in_o the_o side_n pn_n of_o the_o triangle_n p_o nm_o and_o by_o the_o same_o reason_n in_o the_o other_o side_n of_o the_o 4._o triangle_n pnm_n nkl_n fmk_n &_o lfp_n which_o be_v inscribe_v in_o the_o base_n of_o the_o pyramid_n which_o side_n be_v 12â—_n in_o number_n shall_v be_v set_v 12._o angle_n of_o the_o icosahedron_n contain_v under_o 20._o equal_a &_o equilater_n triangle_n of_o which_o fowere_n be_v set_v in_o the_o 4._o base_n of_o the_o pyramid_n namely_o these_o four_o triangle_n trs_n hoi_n ceq_n vxy_n 4._o triangle_n be_v under_o 4._o angle_n of_o the_o pyramid_n that_o be_v the_o four_o triangle_n cix_o ysh_a erv_n tqo_n and_o under_o every_o one_o of_o the_o six_o side_n of_o the_o pyramid_n be_v set_v two_o triangle_n namely_o under_o the_o side_n of_o the_o triangle_n this_o and_o thoâ—_n under_o the_o side_n db_o the_o triangle_n rqe_n and_o rqt_n under_o the_o side_n da_z the_o triangle_n coq_fw-la and_o coi_fw-fr under_o the_o side_n ab_fw-la the_o triangle_n exc_a and_o exuâ—_n under_o the_o side_n bg_o the_o triangle_n sur_n and_o svy_n and_o under_o the_o side_n agnostus_n the_o triangle_n jyh_n and_o jyx._n wherefore_o the_o solid_a be_v contain_v under_o 20._o equilater_n and_o equal_a triangle_n shall_v be_v a_o icosahedron_n by_o the_o 23._o definition_n of_o the_o eleven_o and_o shall_v be_v inscribe_v in_o the_o pyramid_n abâ—d_v by_o the_o first_o definition_n of_o this_o book_n for_o all_o his_o angle_n do_v at_o one_o time_n touch_v the_o base_n of_o the_o pyramid_n wherefore_o in_o a_o trilater_n equilater_n pyramid_n give_v we_o have_v inscribe_v a_o icosahedron_n ¶_o the_o 20._o proposition_n the_o 20._o problem_n in_o a_o trilater_n equilater_n pyramid_n give_v to_o inscribe_v a_o dodecahedron_n svppose_v that_o the_o pyramid_n give_v be_v abgd_v â—che_fw-mi of_o who_o side_n let_v be_v cut_v into_o two_o equal_a part_n and_o draw_v the_o line_n which_o couple_v the_o section_n which_o be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n and_o right_a line_n be_v draw_v by_o the_o section_n shall_v receive_v 20._o triangle_n make_v a_o icosahedron_n as_o in_o the_o former_a proposition_n it_o be_v manifest_a now_o than_o if_o we_o take_v the_o centre_n of_o those_o triangle_n we_o shall_v there_o find_v the_o 20._o angle_n of_o the_o dodecahedron_n inscribe_v in_o it_o by_o the_o 5._o of_o this_o book_n and_o forasmuch_o as_o 4._o base_n of_o the_o foresay_a icosahedron_n be_v concentricall_a with_o the_o base_n of_o the_o pyramid_n as_o it_o be_v prove_v in_o the_o 2._o corollary_n of_o the_o 6._o of_o this_o book_n there_o shall_v be_v place_v 4â—_o angle_n of_o the_o dodecahedron_n namely_o the_o 4._o angle_n e_o f_o h_o d_o in_o the_o 4._o centre_n of_o the_o base_n and_o of_o the_o other_o 16._o angle_n under_o every_o one_o of_o the_o 6._o side_n of_o the_o pyramid_n be_v subtend_v two_o namely_o under_o the_o side_n ad_fw-la the_o angle_n ck_o under_o the_o side_n bd_o the_o angle_n li_n under_o the_o
cube_fw-la inscribe_v in_o a_o dodecahedron_n do_v with_o his_o side_n subtend_v tâ—â—_n angle_n of_o 4._o pentagon_n concur_v at_o one_o and_o the_o self_n same_o side_n of_o the_o dodecahedron_n construction_n let_v that_o base_a of_o the_o cube_fw-la be_v abcd_o and_o let_v the_o side_n whereat_o 4._o base_n of_o the_o dodecahedron_n circumscribe_v concur_v be_v eglantine_n which_o shall_v contain_v a_o solid_a aebdgc_n set_v upon_o the_o base_a abcd._n divide_v the_o side_n ab_fw-la and_o dc_o into_o two_o equal_a part_n in_o the_o point_n l_o and_o n._n and_o draw_v the_o line_n ln_o which_o be_v a_o parallel_n to_o the_o side_n eglantine_n as_o it_o be_v manifest_a by_o the_o 17._o of_o the_o thirteen_o the_o perpendicular_n also_o ere_o and_o go_v which_o couple_n those_o parallel_n be_v each_o equal_a to_o half_a of_o the_o side_n eglantine_n and_o each_o be_v the_o great_a segment_n of_o half_a the_o side_n of_o the_o cube_fw-la and_o therefore_o the_o whole_a line_n eglantine_n be_v the_o great_a segment_n of_o the_o whole_a line_n ln_o the_o side_n of_o the_o cube_fw-la by_o the_o foresay_a 17._o of_o the_o thirteen_o by_o the_o point_n r_o and_o oh_o draw_v unto_o the_o side_n ab_fw-la and_o cd_o parallel_a line_n fh_a and_o ik_fw-mi and_o draw_v these_o right_a line_n of_o eh_o give_v and_o gk_o now_o forasmuch_o as_o the_o two_o line_n fh_a &_o er_o touch_v the_o one_o the_o other_o be_v parallel_n to_o the_o two_o line_n ik_fw-mi and_o go_v touch_v also_o the_o one_o the_o other_o demonstration_n &_o not_o be_v in_o the_o self_n same_o plain_n with_o the_o two_o first_o line_n therefore_o the_o plain_a superâ—icieces_n efh_n and_o gik_n pass_v by_o those_o line_n be_v parallel_n by_o the_o 15._o of_o the_o eleven_o which_o plain_n do_v cut_v the_o solid_a aebdgc_n whererefore_o there_o be_v make_v four_o quadrangled_a pyramid_n set_v upon_o the_o rectangle_n parallelogram_n lh_o lf_n nk_n and_o ni_fw-fr and_o have_v their_o top_n the_o point_n e_o and_o g._n and_o forasmuch_o as_o the_o triangle_n gok_v and_o erh_n be_v equal_a and_o like_a by_o the_o 4._o of_o the_o first_o namely_o b._n they_o contain_v equal_a angle_n comprehend_v under_o equal_a side_n and_o they_o be_v parallel_n by_o construction_n be_v set_v in_o the_o plain_n gik_fw-mi and_o efh_n the_o figure_n gkhe_v okhr_fw-ge and_o gore_n shall_v be_v parallelogram_n by_o the_o definition_n of_o a_o parallelogram_n and_o therefore_o the_o solid_a gokerh_n be_v a_o prism_n by_o the_o 11._o definition_n of_o the_o eleven_o and_o by_o the_o same_o reason_n may_v the_o solid_a goierf_n be_v prove_v to_o be_v a_o prism_n and_o forasmuch_o as_o upon_o equal_a base_n nokc_n and_o rlbh_n and_o under_o equal_a altitude_n og_n and_o re_n be_v set_v pyramid_n those_o pyramid_n shall_v be_v equal_a to_o â—hat_n pyramid_n which_o be_v set_v upon_o the_o â—ase_n ckid_v which_o be_v double_a to_o either_o of_o the_o base_n nokc_n and_o rlbh_n and_o under_o the_o same_o altitude_n og_n by_o the_o 6._o of_o the_o twelve_o and_o forasmuch_o as_o the_o side_n ge_z be_v the_o great_a segment_n of_o the_o line_n cb_o the_o line_n kh_o which_o by_o the_o 33._o of_o the_o first_o be_v equal_a to_o the_o line_n ge_z shall_v be_v the_o great_a segment_n of_o the_o same_o line_n cb_o by_o the_o 2._o of_o the_o fourteen_o wherefore_o the_o residue_v ck_a and_o hb_o shall_v make_v the_o less_o segment_n of_o the_o whole_a line_n cb_o but_o as_o the_o great_a segment_n kh_o be_v to_o the_o two_o line_n ck_a and_o hb_o the_o less_o segment_n so_o be_v the_o rectangle_n parallelogram_n oh_o to_o the_o two_o rectangle_n parallelogram_n oc_n and_o hl_o by_o the_o 1._o of_o the_o six_o wherefore_o the_o less_o segment_n of_o the_o parallelogram_n nb_n shall_v be_v the_o two_o parallelogram_n oc_n and_o hl._n put_v the_o line_n km_o double_a to_o the_o line_n kc_o and_o draw_v the_o line_n m_o parallel_n to_o the_o line_n cn_fw-la wherefore_o the_o parallelogram_n okms_n be_v equal_a to_o the_o parallelogram_n oc_n and_o hl_o by_o the_o 1._o of_o the_o six_o wherefore_o the_o pyramid_n set_v upon_o the_o base_a okm_n contain_v two_o three_o part_n of_o the_o prism_n set_v upon_o the_o self_n same_o base_a by_o the_o 4._o corollary_n of_o the_o 7._o of_o the_o twelve_o wherefore_o the_o prism_n which_o be_v set_v upon_o two_o three_o part_n of_o the_o base_a okm_n be_v equal_a to_o the_o two_o pyramid_n nokcg_n and_o rlbhe_n for_o the_o section_n of_o a_o prism_n be_v one_o to_o the_o other_o as_o the_o section_n of_o the_o base_a be_v by_o the_o first_o corollary_n of_o the_o 25._o of_o the_o eleven_o but_o the_o section_n of_o the_o base_a be_v as_o the_o section_n of_o the_o line_n cb_o or_o km_o by_o the_o 1._o of_o the_o six_o wherefore_o the_o two_o pyramid_n nokcg_n and_o rlbhe_n add_v unto_o the_o prism_n gokerh_n two_o three_o part_n of_o the_o prism_n set_v upon_o the_o base_a okm_n and_o forasmuch_o as_o the_o line_n km_o be_v the_o less_o segment_n of_o the_o whole_a line_n bc_o for_o it_o be_v equal_a to_o to_o the_o two_o line_n ck_a and_o hâ—_n and_o the_o prism_n set_v upon_o the_o base_a okhr_n be_v cut_v like_o unto_o the_o line_n km_o namely_o in_o each_o be_v take_v two_o three_o as_o have_v before_o be_v prove_v the_o prism_n equal_a to_o the_o two_o pyramid_n shall_v add_v unto_o the_o prism_n gokerh_n which_o be_v set_v upon_o the_o great_a segment_n kh_o two_o thâ—â—ds_o of_o the_o less_o segment_n wherefore_o in_o the_o line_n bc_o there_o shall_v remain_v one_o three_o part_n of_o the_o less_o segment_n and_o therefore_o in_o the_o rectangle_n parallelogram_n nb_n which_o be_v half_a the_o base_a of_o the_o cube_fw-la there_o shall_v remain_v the_o same_o three_o part_n of_o the_o less_o segment_n and_o by_o the_o same_o reason_n may_v we_o prove_v that_o in_o the_o other_o pyramid_n ondig_fw-ge and_o rlafe_a and_o in_o the_o prism_n goierf_n be_v leave_v the_o self_n same_o excess_n of_o the_o base_a land_n namely_o the_o three_o part_n of_o the_o less_o segment_n wherefore_o the_o whole_a prism_n contain_v between_o the_o triangle_n igk_n and_o feh_n and_o under_o the_o length_n of_o the_o great_a b._n segment_n and_o two_o three_o part_n of_o the_o less_o segment_n of_o bc_o the_o side_n of_o the_o cube_fw-la be_v equal_a to_o the_o solid_a compose_v of_o 4._o base_n of_o the_o dodecahedron_n and_o set_v upon_o the_o base_a of_o the_o cube_fw-la wherefore_o the_o base_a of_o that_o prism_n want_v of_o the_o whole_a base_a of_o the_o cube_fw-la only_o a_o three_o part_n of_o the_o less_o segment_n and_o the_o altitude_n of_o that_o prism_n be_v the_o line_n go_v which_o be_v the_o great_a segment_n of_o half_a the_o side_n of_o the_o cube_fw-la and_o forasmuch_o as_o unto_o the_o triangle_n igk_n be_v double_a the_o rctangle_n parallelogram_n set_v upon_o the_o same_o base_a ik_fw-mi the_o side_n of_o the_o cube_fw-la and_o under_o the_o altitude_n go_v by_o the_o 41._o of_o the_o first_o it_o follow_v that_o three_o rectangle_n parallelogram_n set_v upon_o the_o same_o base_a ik_fw-mi the_o side_n of_o the_o cube_fw-la and_o under_o the_o altitude_n og_n the_o great_a segment_n of_o half_a the_o side_n of_o the_o cube_fw-la be_v sextuple_a to_o the_o triangle_n igk_n wherefore_o those_o three_o rectangle_n parallelogram_n do_v make_v one_o rectangle_n parallelogram_n set_v upon_o the_o base_a ik_fw-mi and_o under_o the_o altitude_n of_o the_o line_n go_v triple_a but_o by_o the_o 7._o definition_n of_o the_o eleven_o there_o be_v six_o prism_n equal_a and_o like_a unto_o the_o foresay_a prism_n be_v set_v upon_o every_o one_o of_o the_o six_o base_n of_o the_o cube_fw-la which_o prism_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 3._o corollary_n of_o the_o 7._o of_o twelve_o wherefore_o the_o solid_a compose_v of_o these_o six_o prism_n shall_v want_v of_o the_o base_a abcd_o the_o three_o part_n of_o the_o less_o segment_n and_o take_v his_o altitude_n of_o the_o foresay_a rectangle_n parallelogram_n the_o say_a altitude_n shall_v be_v equal_a to_o three_o great_a segment_n one_o of_o which_o be_v go_v of_o half_a the_o side_n of_o the_o cube_fw-la now_o rest_v to_o prove_v that_o these_o three_o segment_n want_v of_o the_o side_n of_o the_o cube_fw-la by_o the_o less_o segment_n of_o the_o less_o segment_n of_o half_a the_o sidâ—_n of_o the_o cube_fw-la suppose_v that_o ab_fw-la the_o side_n of_o the_o cube_fw-la be_v divide_v into_o the_o great_a segment_n ac_fw-la and_o into_o the_o less_o segment_n cb_o by_o the_o 30._o of_o the_o six_o and_o divide_v into_o two_o equal_a part_n the_o line_n ac_fw-la in_o the_o point_n g_o and_o the_o line_n cb_o
the_o residue_n or_o of_o this_o excess_n but_o a_o pyramid_n be_v to_o the_o same_o cube_fw-la inscribe_v in_o it_o nonecuple_n by_o the_o 30._o of_o this_o book_n wherefore_o the_o dodecahedron_n inscribe_v in_o the_o pyramid_n and_o contain_v the_o same_o cube_fw-la twice_o take_v away_o the_o self_n same_o three_o of_o the_o less_o segment_n and_o moreover_o the_o less_o segment_n of_o the_o less_o segment_n of_o half_a the_o residue_n shall_v contain_v two_o nine_o part_n of_o the_o solid_a of_o the_o pyramid_n of_o which_o nine_o part_v each_o be_v equal_a unto_o the_o cube_fw-la take_v away_o this_o self_n same_o excess_n the_o solid_a therefore_o of_o a_o dodecahedron_n contain_v of_o a_o pyramid_n circumscribe_v about_o it_o two_o nine_o part_n take_v away_o a_o three_o part_n of_o one_o nine_o part_n of_o the_o less_o segment_n of_o a_o line_n divide_v by_o a_o extmere_n and_o mean_a proportion_n and_o moreover_o the_o less_o segment_n of_o the_o less_o segment_n of_o half_a the_o residue_n ¶_o the_o 36._o proposition_n a_o octohedron_n exceed_v a_o icosahedron_n inscribe_v in_o it_o by_o a_o parallelipipedon_n set_v upon_o the_o square_n of_o the_o side_n of_o the_o icosahedron_n and_o have_v to_o his_o altitude_n the_o line_n which_o be_v the_o great_a segment_n of_o half_a the_o semidiameter_n of_o the_o octohedron_n svppose_v that_o there_o be_v a_o octohedron_n abcfpl_n construction_n in_o which_o let_v there_o be_v inscribe_v a_o icosahedron_n hkegmxnudsqtâ—_n by_o the_o â—6_o of_o the_o fifteen_o and_o draw_v the_o diameter_n azrcbroif_n and_o the_o perpendicular_a ko_o parallel_n to_o the_o line_n azr_n then_o i_o say_v that_o the_o octohedron_n abcfpl_n be_v great_a thâ—n_n the_o icosahedron_n inscribe_v in_o it_o by_o a_o parallelipipedon_n set_v upon_o the_o square_n of_o the_o side_n hk_o or_o ge_z and_o have_v to_o his_o altitude_n the_o line_n ko_o or_o rz_n which_o be_v the_o great_a segment_n of_o the_o semidiameter_n ar._n forasmuch_o as_o in_o the_o same_o 16._o it_o have_v be_v prove_v that_o the_o triangle_n kdg_n and_o keq_n be_v describe_v in_o the_o base_n apf_n and_o alf_n of_o the_o octohedron_n demonstration_n therefore_o about_o the_o solid_a angle_n there_o remain_v upon_o the_o base_a feg_fw-mi three_o triangle_n keg_n kfe_n and_o kfg_n which_o contain_v a_o pyramid_n kefg_n unto_o which_o pyramid_n shall_v be_v equal_a and_o like_v the_o opposite_a pyramid_n mefg_n set_v upon_o the_o same_o base_a feg_n by_o the_o 8._o definition_n of_o the_o eleven_o and_o by_o the_o â—ame_n reason_n shall_v there_o at_o every_o solid_a angle_n of_o the_o octohedron_n remain_v two_o pyramid_n equal_a and_o like_a namely_o two_o upon_o the_o base_a ahk_n two_o upon_o the_o base_a bnv_n two_o upon_o the_o base_a dp_n and_o moreover_o two_o upon_o the_o base_a qlt._n now_o they_o there_o shall_v be_v make_v twelve_o pyramid_n set_v upon_o a_o base_a contain_v of_o the_o side_n of_o the_o icosahedron_n and_o under_o two_o leâ—â—e_a segment_n of_o the_o side_n of_o the_o octohedron_n contain_v a_o right_a angle_n as_o for_o example_n the_o base_a gef_n and_o forasmuch_o as_o the_o side_n ge_z subtend_v a_o right_a angle_n be_v by_o the_o 47._o of_o the_o â—irst_n in_o power_n duple_n to_o either_o of_o the_o line_n of_o and_o fg_o and_o so_o the_o â—â—deâ—_n kh_o be_v in_o power_n duple_n to_o either_o of_o the_o side_n ah_o and_o ak_o and_o either_o of_o the_o line_n ah_o ak_o or_o of_o fg_o be_v in_o power_n duple_n to_o either_o of_o the_o line_n az_o or_o zk_v which_o contain_v a_o right_a angle_n make_v in_o the_o triangle_n or_o base_a ahk_n by_o the_o perpendicular_a az_o wherefore_o it_o follow_v that_o the_o side_n ge_z or_o hk_o be_v in_o power_n quadruple_a to_o the_o triangle_n efg_o or_o ahk_n but_o the_o pyramid_n kefg_n have_v his_o base_a efg_o in_o the_o plain_a flbp_n of_o the_o octohedron_n shall_v have_v to_o his_o altitude_n the_o perpendicular_a ko_o by_o the_o 4._o definition_n of_o the_o six_o which_o be_v the_o great_a segment_n of_o the_o semidiameter_n of_o the_o octohedron_n by_o the_o 16._o of_o the_o fifteen_o wherefore_o three_o pyramid_n set_v under_o the_o same_o altitude_n and_o upon_o equal_a base_n shall_v be_v equal_a to_o one_o prism_n set_v upon_o the_o same_o base_a and_o under_o the_o same_o altitude_n by_o the_o 1._o corollary_n of_o the_o 7._o of_o the_o twelve_o wherefore_o 4._o prism_n set_v upon_o the_o base_a gef_n quadruple_v which_o be_v equal_a to_o the_o square_n of_o the_o side_n ge_z and_o under_o the_o altitude_n ko_o or_o rz_n the_o great_a segment_n which_o be_v equal_a to_o ko_o shall_v contain_v a_o solid_a equal_a to_o the_o twelve_o pyramid_n which_o twelve_o pyramid_n make_v the_o excess_n of_o the_o octohedron_n above_o the_o icosahedron_n inscribe_v in_o it_o a_o octohedron_n therefore_o exceed_v a_o icosahedron_n inscribe_v in_o it_o by_o a_o parallelipipedon_n set_v upon_o the_o square_n of_o the_o side_n of_o the_o icosahedron_n and_o have_v to_o his_o altitude_n the_o line_n which_o be_v the_o great_a segment_n of_o half_a the_o semidiameter_n of_o the_o octohedron_n ¶_o a_o corollary_n a_o pyramid_n exceed_v the_o double_a of_o a_o icosahedron_n inscribe_v in_o it_o by_o a_o solid_a set_v upon_o the_o square_n of_o the_o side_n of_o the_o icosahedron_n inscribe_v in_o it_o and_o have_v to_o his_o altitude_n that_o whole_a line_n of_o which_o the_o side_n of_o the_o icosahedron_n be_v the_o great_a segment_n for_o it_o be_v manifest_a by_o the_o 19_o of_o the_o fivetenth_n that_o a_o octohedron_n &_o a_o icosahedron_n inscribe_v in_o it_o be_v inscribe_v in_o one_o &_o the_o self_n same_o pyramid_n it_o have_v moreover_o be_v prove_v in_o the_o 26._o of_o this_o book_n that_o a_o pyramid_n be_v double_a to_o a_o octohedron_n inscribe_v in_o it_o wherefore_o the_o two_o excess_n of_o the_o two_o octohedron_n unto_o which_o the_o pyramid_n be_v equal_a above_o the_o two_o icosahedrons_n inscribe_v in_o the_o say_v two_o octohedron_n be_v bring_v into_o a_o solid_a the_o say_v solid_a shall_v be_v set_v upon_o the_o self_n same_o square_n of_o the_o side_n of_o the_o icosahedron_n and_o shall_v have_v to_o his_o altitude_n the_o perpendicular_a ko_o double_a who_o double_a couple_v the_o opposite_a side_n hk_o and_o xm_o make_v the_o great_a segment_n the_o same_o side_n of_o the_o icosahedron_n by_o the_o first_o and_o second_o corollary_n of_o the_o 14._o of_o the_o fiuâ—â—enâ—h_n the_o 37._o proposition_n if_o in_o a_o triangle_n have_v to_o his_o base_a a_o rational_a line_n set_v the_o side_n be_v commensurable_a in_o power_n to_o the_o base_a and_o from_o the_o top_n be_v draw_v to_o the_o base_a a_o perpendicular_a line_n cut_v the_o base_a the_o section_n of_o the_o base_a shall_v be_v commensurable_a in_o length_n to_o the_o whole_a base_a and_o the_o perpendicular_a shall_v be_v commensurable_a in_o power_n to_o the_o say_v whole_a base_a and_o now_o that_o the_o perpendicular_a ap_n be_v commensurable_a in_o power_n to_o the_o base_a bg_o demonstration_n iâ—_z thus_o prove_v forasmuch_o as_o the_o square_n of_o ab_fw-la be_v by_o supposition_n commensurable_a to_o the_o square_n of_o bg_o and_o unto_o the_o rational_a square_n of_o ab_fw-la be_v commensurable_a the_o rational_a square_n of_o bp_o by_o the_o 12._o of_o the_o eleven_o wherefore_o the_o residue_n namely_o the_o square_a of_o pa_o be_v commensurable_a to_o the_o same_o square_n of_o bp_o by_o the_o 2._o part_n of_o the_o 15._o of_o the_o eleven_o wherefore_o by_o the_o 12._o of_o the_o ten_o the_o square_a of_o pa_o be_v commensurable_a to_o the_o whole_a square_n of_o bg_o wherefore_o the_o perpendicular_a ap_n be_v commensurable_a in_o power_n to_o the_o base_a bg_o by_o the_o 3._o definition_n of_o the_o ten_o which_o be_v require_v to_o be_v prove_v in_o demonstrate_v of_o this_o we_o make_v no_o mention_n at_o all_o of_o the_o length_n of_o the_o side_n ab_fw-la and_o ag_z but_o only_o of_o the_o length_n of_o the_o base_a bg_o for_o that_o the_o line_n bg_o be_v the_o rational_a line_n first_o set_v and_o the_o other_o line_n ab_fw-la and_o ag_z be_v suppose_v to_o be_v commensurable_a in_o power_n only_o to_o the_o line_n bg_o wherefore_o if_o that_o be_v plain_o demonstrate_v when_o the_o side_n be_v commensurable_a in_o power_n only_o to_o the_o base_a much_o more_o easy_o will_v it_o follow_v if_o the_o same_o side_n be_v suppose_v to_o be_v commensurable_a both_o in_o length_n and_o in_o power_n to_o the_o base_a that_o be_v if_o their_o lengthe_n be_v express_v by_o the_o root_n of_o square_a number_n ¶_o a_o corollary_n 1._o by_o the_o former_a thing_n demonstrate_v it_o be_v manifest_a that_o if_o from_o the_o power_n of_o the_o base_a and_o of_o one_o of_o the_o side_n be_v take_v away_o the_o
consider_v the_o diverse_a trade_n of_o these_o two_o excellent_a philosopher_n and_o be_o most_o sure_a both_o that_o plato_n right_a well_o otherwise_o can_v teach_v and_o that_o aristotle_n may_v bold_o with_o his_o hearer_n have_v deal_v in_o like_a sort_n as_o plato_n do_v i_o be_o in_o no_o little_a pang_n of_o perplexity_n because_o that_o which_o i_o mislike_v be_v most_o easy_a for_o i_o to_o perform_v and_o to_o have_v plato_n for_o my_o example_n and_o that_o which_o i_o know_v to_o be_v most_o commendable_a and_o in_o this_o first_o bring_v into_o common_a handle_n the_o art_n mathematical_a to_o be_v most_o necessary_a be_v full_a of_o great_a difficulty_n and_o sundry_a danger_n yetâ—_n neither_o do_v i_o think_v it_o meâ—e_v for_o so_o strange_a matter_n as_o now_o be_v mean_v to_o be_v publish_v and_o to_o so_o strange_a a_o audience_n to_o be_v blunt_o at_o first_o put_v forth_o without_o a_o peculiar_a preface_n nor_o imitate_v aristotle_n well_o can_v i_o hope_v that_o accord_v to_o the_o ampleness_n and_o dignity_n of_o the_o state_n mathematical_a i_o be_o able_a either_o plain_o to_o prescribe_v the_o material_a bound_n or_o precise_o to_o express_v the_o chief_a purpose_n and_o most_o wonderful_a application_n thereof_o and_o though_o i_o be_o sure_a that_o such_o as_o do_v shrink_v from_o plato_n his_o sâ—hole_a after_o they_o have_v perceive_v his_o final_a conclusion_n will_v in_o these_o thing_n have_v be_v his_o most_o diligent_a hearer_n so_o infinite_o may_v their_o desire_n in_o fine_a and_o at_o length_n by_o our_o art_n mathematical_a be_v satisfy_v yet_o by_o this_o my_o preface_n &_o forewarn_n aswell_o all_o such_o may_v to_o their_o great_a behoof_n the_o soon_o hither_o be_v allure_v as_o also_o the_o pythagorical_a and_o platonical_a perfect_a scholar_n and_o the_o constant_a profound_a philosopher_n with_o more_o ease_n and_o speed_n may_v like_o the_o bee_n gather_v hereby_o both_o wax_n and_o honey_n wherefore_o sing_v i_o find_v great_a occasion_n for_o the_o cause_n allege_v and_o far_o in_o respect_n of_o my_o art_n mathematic_a general_n to_o use_v a_o certain_a forewarn_n and_o preface_n who_o content_n shall_v be_v that_o mighty_a most_o pleasant_a and_o fruitful_a mathematical_a tree_n with_o his_o chief_a arm_n and_o second_o grift_v branch_n preface_n both_o what_o every_o one_o be_v and_o also_o what_o commodity_n in_o general_a be_v to_o be_v look_v for_o aswell_o of_o griff_n as_o stock_n and_o forasmuch_o as_o this_o enterprise_n be_v so_o great_a that_o to_o this_o our_o time_n it_o never_o be_v to_o my_o knowledge_n by_o any_o achieve_v and_o also_o it_o be_v most_o hard_a in_o these_o our_o dreary_a day_n to_o such_o rare_a and_o strange_a art_n to_o win_v due_a and_o common_a credit_n nevertheless_o if_o for_o my_o sincere_a endeavour_n to_o satisfy_v your_o honest_a expectation_n you_o will_v but_o lend_v i_o your_o thankful_a mind_n a_o while_n and_o to_o such_o matter_n as_o for_o this_o time_n my_o pen_n with_o speed_n be_v able_a to_o deliver_v apply_v your_o eye_n or_o ear_n attentifely_a perchance_o at_o once_o and_o for_o the_o first_o saluting_n this_o preface_n you_o will_v find_v a_o lesson_n long_a enough_o and_o either_o you_o will_v for_o a_o second_o by_o this_o be_v make_v much_o the_o apt_a or_o short_o become_v well_o able_a yourselves_o of_o the_o lion_n claw_n to_o conjecture_v his_o royal_a symmetry_n and_o far_a property_n now_o then_o gentle_a my_o friend_n and_o country_n man_n turn_v your_o eye_n and_o bend_v your_o mind_n to_o that_o doctrine_n which_o for_o our_o present_a purpose_n my_o simple_a talon_n be_v able_a to_o yield_v you_o all_o thing_n which_o be_v &_o have_v be_v be_v find_v under_o a_o triple_a diversity_n general_a for_o either_o they_o be_v deem_v supernatural_a natural_a or_o of_o a_o three_o be_v thing_n supernatural_a be_v immaterial_a simple_a indivisible_a incorruptible_a &_o unchangeable_a thing_n natural_a be_v material_a compound_v divisible_a corruptible_a and_o changeable_a thing_n supernatural_a be_v of_o the_o mind_n only_o comprehend_v thing_n natural_a of_o the_o sense_n exterior_a be_v able_a to_o be_v perceive_v in_o thing_n natural_a probability_n and_o conjecture_n have_v place_n but_o in_o thing_n supernatural_a chief_a demonstration_n &_o most_o sure_a science_n be_v to_o be_v have_v by_o which_o property_n &_o comparason_n of_o these_o two_o more_o easy_o may_v be_v describe_v the_o state_n condition_n nature_n and_o property_n of_o those_o thing_n which_o we_o before_o term_v of_o a_o three_o be_v which_o by_o a_o peculiar_a name_n also_o be_v call_v thing_n mathematical_a for_o these_o be_v in_o a_o manner_n middle_a between_o thing_n supernatural_a and_o natural_a be_v not_o so_o absolute_a and_o excellent_a as_o thing_n supernatural_a nor_o yet_o so_o base_a and_o gross_a as_o thing_n natural_a but_o be_v thing_n immaterial_a and_o nevertheless_o by_o material_a thing_n able_a somewhat_o to_o be_v signify_v and_o though_o their_o particular_a image_n by_o art_n be_v aggâ—egable_a and_o divisible_a yet_o the_o general_a form_n notwithstanding_o be_v constant_a unchangeable_a untransformable_a and_o incorruptible_a neither_o of_o the_o sense_n can_v they_o at_o any_o time_n be_v perceive_v or_o judge_v nor_o yet_o for_o all_o that_o in_o the_o royal_a mind_n of_o man_n first_o conceive_v but_o surmount_v the_o imperfection_n of_o conjecture_n ween_v and_o opinion_n and_o come_n short_a of_o high_a intellectual_a conception_n be_v the_o mercurial_a fruit_n of_o dianoetical_a discourse_n in_o perfect_a imagination_n subsist_v a_o marvellous_a neutrality_n have_v these_o thing_n mathematical_a and_o also_o a_o strange_a participation_n between_o thing_n supernatural_a immortal_a intellectual_a simple_a and_o indivisible_a and_o thing_n natural_a mortal_a sensible_a compound_v and_o divisible_a probability_n and_o sensible_a proof_n may_v well_o serve_v in_o thing_n natural_a and_o be_v commendable_a in_o mathematical_a reason_n a_o probable_a argument_n be_v nothing_o regard_v nor_o yet_o the_o testimony_n of_o sense_n any_o whit_n credit_v but_o only_o a_o perfect_a demonstration_n of_o truth_n certain_a necessary_a and_o invincible_a universal_o and_o necessary_o conclude_v be_v allow_v as_o sufficient_a for_o a_o argument_n exact_o and_o pure_o mathematical_a of_o mathematical_a thing_n be_v two_o principal_a kindeâ—â—_n namely_o number_n number_n and_o magnitude_n number_n we_o define_v to_o be_v a_o certain_a mathematical_a sum_n of_o vnit_n use_v and_o a_o unit_fw-la be_v that_o thing_n mathematical_a indivisible_a by_o participation_n of_o some_o likeness_n of_o who_o property_n any_o thing_n which_o be_v in_o deâ—de_n or_o be_v count_v one_o may_v reasonable_o be_v call_v one._n we_o account_v a_o unit_fw-la a_o thing_n mathematical_a though_o it_o be_v no_o number_n and_o also_o indivisibleâ—_n because_o of_o it_o material_o number_n do_v consi_v which_o principal_o be_v a_o thing_n mathematical_a magnitude_n be_v a_o thing_n mathematical_a magnitude_n by_o participation_n of_o some_o likeness_n of_o who_o nature_n any_o thing_n be_v judge_v long_o broad_a or_o thick_a a_o thick_a magnitude_n we_o call_v a_o solid_a or_o a_o body_n what_o magnitude_n so_o ever_o be_v solid_a or_o thick_a be_v also_o broad_a &_o long_o a_o broad_a magnitude_n we_o call_v a_o superficies_n or_o a_o plain_a every_o plain_a magnitude_n have_v also_o length_n a_o long_a magnitude_n we_o term_v a_o line_n a_o line_n be_v neither_o thick_a nor_o broad_a but_o only_o long_a every_o certain_a line_n have_v two_o end_n the_o end_n of_o a_o line_n be_v point_n call_v point_n a_o point_n be_v a_o thing_n mathematical_a indivisible_a which_o may_v have_v a_o certain_a determine_a situation_n if_o a_o point_v move_v from_o a_o determine_v situation_n the_o way_n wherein_o it_o move_v be_v also_o a_o line_n mathematical_o produce_v whereupon_o of_o the_o ancient_a mathematician_n a_o line_n be_v call_v the_o race_n or_o course_n of_o a_o point_n line_n a_o point_v we_o define_v by_o the_o name_n of_o a_o thing_n mathematical_a though_o it_o be_v no_o magnitude_n and_o indivisible_a because_o it_o be_v the_o proper_a end_n and_o bind_v of_o a_o line_n which_o be_v a_o true_a magnitude_n magnitude_n and_o magnitude_n we_o may_v define_v to_o be_v that_o thing_n mathematical_a which_o be_v divisible_a for_o ever_o in_o part_n divisible_a long_a broad_a or_o thick_a therefore_o though_o a_o point_v be_v no_o magnitude_n yet_o terminative_o we_o reckon_v it_o a_o thing_n mathematical_a as_o i_o say_v by_o reason_n it_o be_v proper_o the_o end_n and_o bind_v of_o a_o line_n neither_o number_n nor_o magnitude_n have_v any_o materiality_n first_o we_o will_v consider_v of_o number_n and_o of_o the_o science_n mathematical_a to_o it_o appropriate_a call_v arithmetic_n and_o afterward_o of_o magnitude_n and_o his_o science_n call_v geometry_n but_o that_o name_n content_v i_o not_o whereof_o a_o
be_v term_v dial_v ancient_a be_v the_o use_n and_o more_o ancient_a be_v the_o invention_n the_o use_n do_v well_o appear_v to_o have_v be_v at_o the_o least_o above_o two_o thousand_o and_o three_o hundred_o year_n ago_o in_o 20._o king_n achaâ—_n dial_n then_o by_o the_o sun_n show_v the_o distinction_n of_o time_n by_o sun_n moon_n and_o star_n this_o dial_n may_v be_v perform_v and_o the_o precise_a time_n of_o day_n or_o night_n know_v but_o the_o demonstrative_a delineation_n of_o these_o dial_n of_o all_o sort_n require_v good_a skill_n both_o of_o astronomy_n and_o geometry_n elemental_a spherical_a phaenomenall_a and_o conikall_n then_o to_o use_v the_o ground_n of_o the_o art_n for_o any_o regular_a superficies_n in_o any_o place_n offer_v and_o in_o any_o possible_a apt_a position_n thereof_o thâ—ron_n to_o describe_v all_o manner_n of_o way_n how_o usual_a hour_n may_v be_v by_o the_o sun_n shadow_n true_o determine_v will_v be_v find_v no_o sleight_n painter_n work_v so_o to_o paint_n and_o prescribe_v the_o sun_n motion_n to_o the_o breadth_n of_o a_o hair_n in_o this_o feat_n in_o my_o youth_n i_o invent_v a_o way_n how_o in_o any_o horizontal_a mural_a or_o equinoctial_a dial_n etc_n etc_n at_o all_o hour_n the_o sun_n shine_v the_o sign_n and_o degree_n ascendent_a may_v be_v know_v which_o be_v a_o thing_n very_o necessary_a for_o the_o rise_n of_o those_o fix_a star_n who_o operation_n in_o the_o air_n be_v of_o great_a might_n evident_o i_o speak_v no_o further_o of_o the_o use_n hereof_o but_o forasmuch_o as_o man_n affair_n require_v knowledge_n of_o time_n &_o moment_n when_o neither_o sun_n moon_n or_o star_n can_v be_v see_v therefore_o by_o industry_n mechanical_a be_v invent_v first_o how_o by_o water_n run_v orderly_a the_o time_n and_o hour_n may_v be_v know_v whereof_o the_o famous_a ctesibius_n be_v inventor_n a_o man_n of_o vitrwius_fw-la to_o the_o sky_n just_o extol_v then_o after_o that_o by_o sand_n run_v be_v hour_n measure_v then_o by_o trochilic_a with_o weight_n and_o of_o late_a time_n by_o trochilic_a with_o spring_n without_o weight_n all_o these_o by_o sun_n or_o star_n direction_n in_o certain_a time_n require_v oversight_n and_o reformation_n according_a to_o the_o heavenly_a equinoctial_a motion_n beside_o the_o inequality_n of_o their_o own_o operation_n there_o remain_v without_o parabolical_a meaning_n herein_o among_o the_o philosopher_n motion_n a_o more_o excellent_a more_o commodious_a and_o more_o marvelous_a way_n than_o all_o these_o of_o have_v the_o motion_n of_o the_o primovant_n or_o first_o equinoctial_a motion_n by_o nature_n and_o arteâ—_n imitate_v which_o you_o shall_v by_o further_a search_n in_o weighty_a study_n hereafter_o understand_v more_o of_o and_o so_o it_o be_v time_n to_o finish_v this_o annotation_n of_o time_n distinction_n use_v in_o our_o common_a and_o private_a affair_n the_o commodity_n whereof_o no_o man_n will_v want_v that_o can_v tell_v how_o to_o bestow_v his_o tyme._n zographie_n be_v a_o art_n mathematical_a which_o teach_v and_o demonstrate_v how_o the_o intersection_n of_o all_o visual_a pyramid_n make_v by_o any_o plain_n assign_v the_o centre_n distance_n and_o light_n be_v determine_v may_v be_v by_o line_n and_o due_a proper_a colour_n represent_v a_o notable_a art_n be_v this_o and_o will_v require_v a_o whole_a volume_n to_o declare_v the_o property_n thereof_o and_o the_o commodity_n ensue_v great_a skill_n of_o geometry_n arithmetic_n perspective_n and_o anthropographie_n with_o many_o other_o particular_a artâ—s_n have_v the_o zographer_n need_n of_o for_o his_o perfection_n for_o the_o most_o excellent_a painter_n who_o be_v but_o the_o propre_fw-fr mechanicien_fw-fr &_o imitator_n sensible_a of_o the_o zographer_n have_v attein_v to_o such_o perfection_n that_o sense_n of_o man_n and_o beast_n have_v judge_v thing_n paint_v to_o be_v thing_n natural_a and_o not_o artificial_a alive_a and_o not_o dead_a this_o mechanical_a zographer_n common_o call_v the_o painter_n be_v marvellous_a in_o his_o skill_n and_o seem_v to_o have_v a_o certain_a divine_a power_n as_o of_o friend_n absent_a to_o make_v a_o friendly_a present_a comfort_n yea_o and_o of_o friend_n dead_a to_o give_v a_o continual_a silent_a presence_n not_o only_o with_o we_o but_o with_o our_o posterity_n for_o many_o age_n and_o so_o proceed_v consider_v how_o in_o winter_n he_o can_v show_v you_o the_o lively_a view_n of_o summer_n joy_n and_o riches_n and_o in_o summer_n exhibit_v the_o countenance_n of_o winter_n doleful_a state_n and_o nakedness_n city_n town_n fort_n wood_n army_n yea_o whole_a kingdom_n be_v they_o never_o so_o far_o or_o great_a can_v he_o with_o ease_n bring_v with_o he_o home_n to_o any_o man_n judgement_n as_o pattern_n lively_a of_o the_o thing_n rehearse_v in_o one_o little_a house_n can_v he_o enclose_v with_o great_a pleasure_n of_o the_o beholder_n the_o portraiture_n lively_a of_o all_o visible_a creature_n either_o on_o earth_n or_o in_o the_o earth_n live_v or_o in_o the_o water_n lie_v creep_v slide_v or_o swim_v or_o of_o any_o â—oule_n or_o fly_v in_o the_o air_n fly_v nay_o in_o respect_n of_o the_o star_n the_o sky_n the_o cloud_n yea_o in_o the_o show_n of_o the_o very_a light_n itself_o that_o divine_a creature_n can_v he_o match_v our_o eye_n judgement_n most_o near_o what_o a_o thing_n be_v this_o thing_n not_o yet_o be_v he_o can_v represent_v so_o as_z at_o their_o be_v the_o picture_n shall_v seam_n in_o manner_n to_o have_v create_v they_o to_o what_o artificer_n be_v not_o picture_n a_o great_a pleasure_n and_o commodity_n which_o of_o they_o all_o will_v refuse_v the_o direction_n and_o aid_n of_o picture_n the_o architect_n the_o goldsmith_n and_o the_o arras_n weaver_n of_o picture_n make_v great_a account_n our_o lively_a herbal_n our_o portraiture_n of_o bird_n beast_n and_o fish_n and_o our_o curious_a anatomy_n which_o way_n be_v they_o most_o perfect_o make_v or_o with_o most_o pleasure_n of_o we_o beholden_a be_v it_o not_o by_o picture_n only_o and_o if_o picture_n by_o the_o industry_n of_o the_o painter_n be_v thus_o commodious_a and_o marvellous_a what_o shall_v be_v think_v of_o zographie_n the_o schoolmaster_n of_o picture_n and_o chief_a governor_n though_o i_o mention_v not_o sculpture_n in_o my_o table_n of_o art_n mathematical_a yet_o may_v all_o man_n perceive_v how_o that_o picture_n and_o sculpture_n be_v sister_n germane_a and_o both_o right_a profitable_a in_o a_o common_a wealth_n and_o of_o sculpture_n aswell_o as_o of_o picture_n excellent_a artificer_n have_v write_v great_a book_n in_o commendation_n witness_v i_o take_v of_o georgio_n vasari_n pittore_fw-la aretino_n of_o pomponius_n gauricusâ—_n â—_z and_o other_o to_o these_o two_o art_n with_o other_o be_v a_o certain_a odd_a art_n call_v althalmasat_n much_o behold_v more_o than_o the_o common_a sculptor_n entayler_n carver_n cutâ—er_n graver_n founder_n or_o painter_n etc_o etc_o know_v their_o art_n to_o be_v commodious_a architecture_n to_o many_o may_v seem_v not_o worthy_a or_o not_o mete_v objection_n to_o be_v reckon_v among_o the_o art_n mathematical_o â—_z to_o who_o i_o think_v good_a to_o give_v some_o account_n of_o my_o so_o do_v not_o worthy_a will_v they_o say_v because_o it_o be_v but_o for_o building_n of_o a_o house_n palace_n church_n forte_n or_o such_o like_a gross_a work_n and_o you_o also_o define_v the_o art_n mathematical_a to_o be_v such_o as_o deal_v with_o no_o material_a or_o corruptible_a thing_n and_o alâ—o_o do_v demonstrative_o proceed_v in_o their_o faculty_n by_o number_n or_o magnitude_n first_o you_o see_v that_o i_o count_v here_o architecture_n among_o those_o art_n mathematical_a answer_n which_o be_v derive_v from_o the_o principal_n and_o you_o know_v that_o such_o may_v deal_v with_o natural_a thing_n and_o sensibâ—â—â—aâ—â—er_n of_o which_o some_o draw_v near_o to_o the_o simple_a and_o absolute_a mathematical_a speculation_n than_o other_o do_v and_o though_o the_o architect_n ☜_o procure_v inform_v &_o direct_v the_o mechanicien_n to_o handworke_n &_o the_o build_n actual_a of_o house_n castle_n or_o palace_n and_o be_v chief_a judge_n of_o the_o same_o yet_o with_o himself_o as_o chief_a master_n and_o architect_n remain_v the_o demonstrative_a reason_n and_o cause_n of_o the_o mechaniciens_fw-la work_n in_o line_n plain_a and_o solid_a by_o geometrical_a arithmetical_a optical_a musical_a astronomical_a cosmographical_a &_o to_o be_v brief_a by_o all_o the_o former_a derive_v art_n mathematical_a and_o other_o natural_a art_n able_a to_o be_v confirm_v and_o establish_v if_o this_o be_v soâ—then_a may_v you_o think_v that_o architecture_n have_v good_a and_o due_a allowance_n in_o this_o honest_a company_n of_o art_n mathematical_a derivative_n i_o will_v herein_o crave_v judgement_n of_o two_o most_o perfect_a architectâ—s_n the_o one_o be_v vitrwius_fw-la the_o roman_a who_o do_v write_v ten_o
superficies_n abcd._n who_o length_n be_v take_v by_o the_o line_n ab_fw-la or_o cd_o and_o breadth_n by_o the_o line_n ac_fw-la or_o bd_o and_o by_o reason_n of_o those_o two_o dimension_n a_o superficies_n may_v be_v divide_v two_o way_n namely_o by_o his_o length_n and_o by_o his_o breadth_n but_o not_o by_o thickness_n for_o it_o have_v none_o for_o that_o be_v attribute_v only_o to_o a_o body_n which_o be_v the_o three_o kind_n of_o quantity_n and_o have_v all_o three_o dimension_n length_n breadth_n and_o thickness_n and_o may_v be_v divide_v according_a to_o any_o of_o they_o other_o define_v a_o superficies_n thus_o superficies_n a_o superficies_n be_v the_o term_n or_o end_n of_o a_o body_n as_o a_o line_n be_v the_o end_n and_o term_v of_o a_o superficies_n 6_o extreme_n of_o a_o superficies_n be_v line_n superficies_n as_o the_o end_n limit_n or_o border_n of_o a_o line_n be_v point_n enclose_v the_o line_n so_o be_v line_n the_o limit_n border_n and_o end_n enclose_v a_o superficies_n as_o in_o the_o figure_n aforesaid_a you_o may_v see_v the_o superficies_n enclose_v with_o four_o line_n the_o extreme_n or_o limit_n of_o a_o body_n be_v superficiesles_fw-mi and_o therefore_o a_o superficies_n be_v of_o some_o thus_o define_v superficies_n a_o superficies_n be_v that_o which_o end_v or_o enclose_v a_o body_n as_o be_v to_o be_v see_v in_o the_o side_n of_o a_o die_n or_o of_o any_o other_o body_n 7_o a_o plain_a superficies_n be_v that_o which_o lie_v equal_o between_o his_o line_n superficies_n a_o plain_a superficies_n be_v the_o short_a extension_n or_o drâ—ught_n from_o one_o line_n to_o a_o other_o superficies_n like_v as_o a_o right_a line_n be_v the_o short_a extension_n or_o draught_n from_o one_o point_n to_o a_o other_o euclid_n also_o leave_v out_o here_o to_o speak_v of_o a_o crooked_a and_o hollow_a superficies_n because_o it_o may_v easy_o be_v understand_v by_o the_o definition_n of_o a_o plain_a superficies_n be_v his_o contrary_n and_o even_o as_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 &_o but_o one_o right_a line_n which_o be_v the_o short_a so_o from_o one_o line_n to_o a_o other_o may_v be_v draw_v infinite_a crooked_a superficiess_n &_o but_o one_o plain_a superficies_n which_o be_v the_o short_a here_o must_v you_o consider_v when_o there_o be_v in_o geometry_n mention_v make_v of_o point_n note_n line_n circle_n triangle_n or_o of_o any_o other_o figure_n you_o may_v not_o conceive_v of_o they_o as_o they_o be_v in_o matter_n as_o in_o wood_n in_o mettle_n in_o paper_n or_o in_o any_o such_o like_a for_o so_o be_v there_o no_o line_n but_o have_v some_o breadth_n and_o may_v be_v devidedâ—_n nor_o point_n but_o that_o shall_v have_v some_o part_n and_o may_v also_o be_v divide_v and_o so_o of_o other_o but_o you_o must_v conceive_v they_o in_o mind_n pluck_v they_o by_o imagination_n from_o all_o matter_n so_o shall_v you_o understand_v they_o true_o and_o perfect_o in_o their_o own_o nature_n as_o they_o be_v define_v as_o a_o line_n to_o be_v long_o and_o not_o broad_a and_o a_o point_n to_o be_v so_o little_a that_o it_o shall_v have_v no_o part_n at_o all_o superficies_n other_o otherwise_o define_v a_o plain_a superficies_n a_o plain_a superficies_n be_v that_o which_o be_v firm_o set_v between_o his_o extreme_n as_o before_o be_v say_v of_o a_o right_a line_n definition_n again_o a_o plain_a superficies_n be_v that_o unto_o all_o who_o part_n a_o right_a line_n may_v well_o be_v apply_v definition_n again_o a_o plain_a superficies_n be_v that_o which_o be_v the_o short_a of_o all_o superficies_n which_o have_v one_o &_o the_o self_n extreme_n as_o a_o rigâ—t_a line_n be_v the_o short_a line_n that_o can_v be_v draw_v between_o two_o point_n definition_n again_o a_o plain_a superficies_n be_v that_o who_o middle_a darken_v the_o extreme_n as_o be_v also_o say_v of_o a_o right_a line_n angle_n 8_o a_o plain_a angle_n be_v a_o inclination_n or_o bow_v of_o two_o line_n the_o one_o to_o the_o other_o and_o the_o one_o touch_v the_o other_o and_o not_o be_v direct_o join_v together_o as_o the_o two_o line_n ab_fw-la &_o bc_o incline_v the_o one_o to_o the_o other_o and_o touch_v the_o one_o the_o other_o in_o the_o point_n b_o in_o which_o point_n by_o reason_n of_o the_o inclination_n of_o the_o say_a line_n be_v make_v the_o angle_n abc_n but_o if_o the_o two_o line_n which_o touch_v the_o one_o the_o other_o be_v without_o all_o inclination_n of_o the_o one_o to_o the_o other_o and_o be_v draw_v direct_o the_o one_o to_o the_o other_o then_o make_v they_o not_o any_o angle_n at_o all_o as_o the_o line_n cd_o and_o de_fw-fr touch_v the_o one_o the_o other_o in_o the_o point_n d_o and_o yet_o as_o you_o see_v they_o make_v no_o angle_n angle_n 9_o and_o if_o the_o line_n which_o contain_v the_o angle_n be_v right_a line_n then_o be_v it_o call_v a_o rightlyne_v angle_n as_o the_o angle_n abc_n in_o the_o former_a figure_n be_v a_o rightlined_a angle_n because_o it_o be_v contain_v of_o right_a line_n where_o note_n that_o a_o angle_n be_v for_o the_o most_o part_n describe_v by_o three_o letter_n of_o which_o the_o second_o or_o middle_a letter_n represent_v the_o very_a angle_n and_o therefore_o be_v set_v at_o the_o angle_n by_o the_o contrary_a a_o crooked_a line_v angle_n be_v that_o which_o be_v contain_v of_o crooked_a line_n which_o may_v be_v diverse_o figure_v also_o a_o mix_a angle_n be_v that_o which_o be_v cause_v of_o they_o both_o namely_o of_o a_o right_a line_n and_o a_o crooked_a which_o may_v also_o be_v diverse_o figure_v as_o in_o the_o figure_n before_o set_v you_o may_v see_v angle_n there_o be_v of_o angle_n three_o kind_n a_o right_a angle_n a_o acute_a angle_n and_o a_o obtuse_a angle_n the_o definition_n of_o which_o now_o follow_v iâ—_z 10_o when_o a_o right_a line_n stand_v upon_o a_o right_a line_n make_v the_o angle_n on_o either_o side_n equal_o then_o either_o of_o those_o angle_n be_v a_o right_a angle_n and_o the_o right_a line_n which_o stand_v erect_v be_v call_v a_o perpendicular_a line_n to_o that_o upon_o which_o it_o stand_v as_o upon_o the_o right_a line_n cd_o suppose_v there_o do_v stand_v a_o other_o line_n ab_fw-la in_o such_o sort_n that_o it_o make_v the_o angle_n on_o either_o side_n thereof_o equal_a namely_o the_o angle_n abc_n on_o the_o one_o side_n equal_a to_o the_o angle_n abdella_n on_o the_o other_o side_n then_o be_v each_o of_o the_o two_o angle_n abc_n and_o abdella_n a_o right_a angle_n and_o the_o line_n ab_fw-la which_o stand_v erect_v upon_o the_o line_n cd_o without_o inclination_n to_o either_o part_n be_v a_o perpendicular_a line_n common_o call_v among_o artificer_n a_o plumb_n line_n â—â—_o 11_o a_o obtuse_a angle_n be_v that_o which_o be_v great_a than_o a_o right_a angle_n as_o the_o angle_n cbe_n in_o the_o example_n be_v a_o obtuse_a angle_n for_o it_o be_v great_a than_o the_o angle_n abc_n which_o be_v a_o right_a angle_n because_o it_o contain_v it_o and_o contain_v moreover_o the_o angle_n abe_n 12_o a_o acute_a angle_n be_v that_o which_o be_v less_o than_o a_o right_a angle_n be_v as_o the_o angle_n ebb_v in_o the_o figure_n before_o put_v be_v a_o acute_a angle_n for_o that_o it_o be_v less_o than_o the_o angle_n abdella_n which_o be_v a_o right_a angle_n for_o the_o right_a angle_n contain_v it_o and_o moreover_o the_o angle_n abe_n 13_o a_o limit_n or_o term_v be_v the_o end_n of_o every_o thing_n thing_n for_o as_o much_o as_o of_o thing_n infinite_a as_o plato_n say_v there_o be_v no_o science_n infinite_a therefore_o must_v magnitude_n or_o quantity_n whereof_o geometry_n entreat_v be_v finite_a and_o have_v border_n and_o limit_n to_o enclose_v it_o which_o be_v here_o define_v to_o be_v the_o end_n thereof_o as_o a_o point_n be_v the_o limit_n or_o term_v of_o a_o line_n because_o it_o be_v the_o end_n thereof_o a_o line_n likewise_o be_v the_o limit_n &_o term_v of_o a_o superficies_n and_o likewise_o a_o superficies_n be_v the_o limit_n and_o term_v of_o a_o body_n as_o be_v before_o declare_v 14_o a_o figure_n be_v that_o which_o be_v contain_v under_o one_o limit_n or_o term_v or_o many_o figure_n as_o the_o figure_n a_o be_v contain_v under_o one_o limit_n which_o be_v the_o round_a line_n also_o the_o figure_n b_o be_v contain_v under_o three_o right_a line_n and_o the_o figure_n c_o under_o four_o and_o so_o of_o other_o which_o be_v their_o limit_n or_o term_n 15_o a_o circle_n be_v a_o plain_a figure_n contain_v under_o one_o line_n which_o be_v call_v a_o circumference_n circle_n unto_o which_o all_o line_n draw_v from_o one_o point_n within_o the_o figure_n and_o fall_v upon_o
the_o circumference_n thereof_o be_v equal_a the_o one_o to_o the_o other_o as_o the_o figure_n here_o set_v be_v a_o plain_a figure_n that_o be_v a_o figure_n without_o grossness_n or_o thickness_n and_o be_v also_o contain_v under_o one_o line_n namely_o the_o crooked_a line_n b_o cd_o which_o be_v the_o circumference_n thereof_o it_o have_v moreover_o in_o the_o middle_n thereof_o a_o point_n namely_o the_o point_n a_o from_o which_o all_o the_o line_n draw_v to_o the_o superficies_n be_v equal_n as_o the_o line_n ab_fw-la ac_fw-la ad_fw-la and_o other_o how_o many_o soever_o of_o all_o figure_n a_o circle_n be_v the_o most_o perfect_a figure_n and_o therefore_o be_v it_o here_o first_o define_v 16_o and_o that_o point_n be_v call_v the_o centre_n of_o the_o circle_n as_o be_v the_o point_n a_o which_o be_v set_v in_o the_o mid_n of_o the_o former_a circle_n circle_n for_o the_o more_o easy_a declaration_n that_o all_o the_o line_n draw_v from_o the_o centre_n of_o the_o circle_n to_o the_o circumference_n be_v equal_a you_o must_v note_v that_o although_o a_o line_n be_v not_o make_v of_o point_n yet_o a_o point_n by_o his_o motion_n or_o draught_n describe_v a_o line_n likewise_o a_o line_n draw_v or_o move_v describe_v a_o superficiesâ—_n also_o a_o superficies_n be_v move_v make_v a_o solid_a or_o body_n now_o they_o imagine_v the_o line_n ab_fw-la the_o point_n a_o be_v fix_v to_o be_v move_v about_o in_o a_o plain_a superficies_n draw_v the_o point_n b_o continual_o about_o the_o point_n a_o till_o it_o return_v to_o the_o place_n where_o it_o begin_v first_o to_o move_v so_o shall_v the_o point_n b_o by_o this_o motion_n describe_v the_o circumference_n of_o the_o circle_n and_o the_o point_n a_o be_v fix_v be_v the_o centre_n of_o the_o circle_n which_o in_o all_o the_o time_n of_o the_o motion_n of_o the_o line_n have_v like_o distance_n from_o the_o circumference_n namely_o the_o length_n of_o the_o line_n ab_fw-la and_o for_o that_o all_o the_o line_n draw_v from_o the_o centre_n to_o the_o circumference_n be_v describe_v of_o that_o line_n they_o be_v also_o equal_a unto_o it_o &_o between_o themselves_o diameter_n 17_o a_o diameter_n of_o a_o circle_n be_v a_o right_a line_n which_o draw_v by_o the_o centre_n thereof_o and_o end_v at_o the_o circumference_n on_o either_o side_n divide_v the_o circle_n into_o two_o equal_a part_n as_o the_o line_n bac_n in_o this_o circle_n present_a be_v the_o diameter_n because_o it_o pass_v from_o the_o point_n b_o of_o the_o one_o side_n of_o the_o circumferenceâ—_n to_o the_o point_n c_o on_o the_o other_o side_n of_o the_o circumference_n &_o pass_v also_o by_o the_o point_n a_o be_v the_o centre_n of_o the_o circle_n and_o moreover_o it_o divide_v the_o circle_n into_o two_o equal_a part_n the_o one_o namely_o bdc_o be_v on_o the_o one_o side_n of_o the_o line_n &_o the_o other_o namely_o bec_n on_o the_o other_o side_n which_o thing_n do_v thales_n miletius_n which_o bring_v geometry_n out_o of_o egypt_n into_o grece_n first_o observe_v and_o prove_v for_o if_o a_o line_n draw_v by_o the_o centre_n do_v not_o divide_v a_o circle_n into_o two_o equal_a part_n all_o the_o line_n draw_v from_o the_o centre_n to_o the_o circumference_n shall_v not_o be_v equal_a semicircle_n 18_o a_o semicircle_n be_v a_o figure_n which_o be_v contain_v under_o the_o diameter_n and_o under_o that_o part_n of_o the_o circumference_n which_o be_v cut_v of_o by_o the_o diametre_fw-la as_o in_o the_o circle_n abcd_o the_o figure_n bac_n be_v a_o semicircle_n because_o it_o be_v contain_v of_o the_o right_a line_n bgc_o which_o be_v the_o diametre_fw-la and_o of_o the_o crooked_a line_n bac_n be_v that_o part_n of_o the_o circumference_n which_o be_v cut_v of_o by_o the_o diametre_fw-la bgc_o so_o likewise_o the_o other_o part_n of_o the_o circle_n namely_o bdcâ—_n â—_z be_v a_o semicircle_n as_o the_o other_o be_v circle_n 19_o a_o section_n or_o portion_n of_o a_o circle_n be_v a_o figure_n which_o be_v contain_v under_o a_o right_a line_n and_o a_o part_n of_o the_o circumference_n great_a or_o less_o than_o the_o semicircle_n as_o the_o figure_n abc_n in_o the_o example_n be_v a_o section_n of_o a_o circle_n &_o be_v great_a than_o half_a a_o circle_n and_o the_o figure_n adc_o be_v also_o a_o section_n of_o a_o circle_n and_o be_v less_o than_o a_o semicircle_n a_o section_n portion_n or_o part_n of_o a_o circle_n be_v all_o one_o and_o signify_v such_o a_o part_n which_o be_v either_o more_o or_o less_o than_o a_o semicircle_n so_o that_o a_o semicircle_n be_v not_o here_o call_v a_o section_n or_o portion_n of_o a_o circle_n a_o right_a line_n draw_v from_o one_o side_n of_o the_o circumference_n of_o a_o circle_n to_o the_o other_o not_o pass_a by_o the_o centre_n divide_v the_o circle_n into_o two_o unequal_a part_n which_o be_v two_o section_n of_o which_o that_o which_o contain_v the_o centre_n be_v call_v thâ—_z great_a section_n and_o the_o other_o be_v call_v the_o less_o section_n as_o in_o the_o example_n the_o part_n of_o the_o circle_n abc_n which_o contain_v in_o it_o the_o centre_n e_o be_v the_o great_a section_n be_v great_a than_o the_o half_a circle_n the_o other_o part_n namely_o adc_o which_o have_v not_o the_o centre_n in_o it_o be_v the_o less_o section_n of_o the_o circle_n be_v less_o than_o a_o semicircle_n figure_n 20_o rightlined_a figure_n be_v such_o which_o be_v contain_v under_o right_a line_n as_o be_v such_o as_o follow_v of_o which_o some_o be_v contain_v under_o three_o right_a line_n some_o under_o â—oure_n some_o under_o five_o and_o some_o under_o more_o figure_n 21_o three_o side_v figure_n or_o figure_n of_o three_o side_n be_v such_o which_o be_v contain_v under_o three_o right_a line_n as_o the_o figure_n in_o the_o example_n abc_n be_v a_o figure_n of_o three_o side_n because_o it_o be_v contain_v under_o three_o right_a line_n namely_o under_o the_o line_n ab_fw-la bc_o ca._n a_o figure_n of_o three_o side_n or_o a_o triangle_n be_v the_o first_o figure_n in_o order_n of_o all_o right_n line_v figure_n and_o therefore_o of_o all_o other_o it_o be_v first_o define_v for_o under_o less_o than_o three_o line_n can_v no_o figure_n be_v comprehend_v 22._o four_o side_v figure_n or_o figure_n of_o four_o side_n be_v such_o figure_n which_o be_v contain_v under_o four_o right_a line_n as_o the_o figure_n here_o set_v be_v a_o figure_n of_o four_o side_n for_o that_o it_o be_v comprehend_v under_o four_o right_a line_n namely_o ab_fw-la bd_o dc_o ca._n triangle_n and_o four_o side_v figure_n serve_v common_o to_o many_o use_n in_o demonstration_n of_o geometry_n wherefore_o the_o nature_n and_o property_n of_o they_o be_v much_o to_o be_v observe_v the_o use_n of_o other_o figure_n be_v more_o obscure_a 23._o many_o side_v figure_n be_v such_o which_o have_v more_o side_n than_o four_o figure_n right_o line_v figure_n have_v more_o side_n than_o four_o by_o continual_a add_v of_o side_n may_v be_v infinite_a wherefore_o to_o define_v they_o all_o several_o according_a to_o the_o number_n of_o their_o â—ides_n shall_v be_v very_o tedious_a or_o rather_o impossible_a therefore_o have_v euclid_n comprehend_v they_o under_o one_o name_n and_o under_o one_o definition_n call_v they_o many_o side_v figure_n as_o many_o as_o have_v more_o side_n than_o four_o as_o if_o they_o have_v five_o side_n six_o seven_o or_o moâ—_n here_o note_v you_o that_o every_o rightlined_a figure_n have_v as_o many_o angle_n as_o it_o have_v side_n &_o take_v his_o denomination_n aswell_o of_o the_o number_n of_o his_o angle_n as_o of_o the_o number_n of_o his_o side_n as_o a_o figure_n contain_v under_o three_o right_a line_n of_o the_o number_n of_o his_o three_o side_n be_v call_v a_o three_o side_v figure_n â—even_v so_o of_o the_o number_n of_o his_o three_o angle_n it_o be_v call_v a_o triangle_n likewise_o a_o figure_n contain_v under_o four_o right_a line_n by_o reason_n of_o the_o number_n of_o his_o side_n be_v call_v a_o four_o side_v figure_n and_o by_o reason_n of_o the_o number_n of_o his_o angle_n it_o be_v call_v a_o quadrangled_a figure_n and_o so_o of_o other_o 24._o of_o three_o side_v figure_n or_o triangle_n triangle_n a_o equilatre_n triangle_n be_v that_o which_o have_v three_o equal_a side_n triangle_n have_v their_o difference_n partly_o of_o their_o side_n and_o partly_o of_o their_o angle_n as_o touch_v the_o difference_n of_o their_o side_n there_o be_v three_o kind_n for_o either_o all_o three_o side_n of_o the_o triangle_n be_v equal_a or_o two_o only_o be_v equal_a &_o the_o three_o unequal_a 〈…〉_o three_o be_v unequal_a the_o one_o to_o the_o other_o the_o first_o kind_n of_o triangle_n namely_o that_o which_o have_v three_o
be_v require_v to_o be_v demonstrate_v this_o proposition_n may_v also_o be_v demonstrate_v without_o produce_v any_o of_o the_o side_n aâ—ter_v this_o manner_n the_o same_o may_v yet_o also_o be_v demonstrate_v a_o other_o way_n this_o proposition_n may_v yet_o moreover_o be_v demonstrate_v by_o a_o argument_n lead_v to_o a_o absurdity_n and_o that_o after_o this_o manner_n line_n a_o man_n may_v also_o more_o brief_o demonstrate_v this_o proposition_n by_o campanus_n definition_n of_o a_o right_a line_n which_o as_o we_o have_v before_o declare_v be_v thus_o a_o right_a line_n be_v the_o short_a extension_n or_o drawght_fw-mi that_fw-mi be_v or_o may_v be_v from_o one_o point_n to_o another_o wherefore_o any_o one_o side_n of_o a_o triangle_n for_o that_o it_o be_v a_o right_a line_n draw_v from_o some_o one_o point_n to_o some_o other_o one_o point_n be_v of_o necessity_n short_a than_o the_o other_o two_o side_n draw_v from_o and_o to_o the_o same_o point_n understanding_n epicurus_n and_o such_o as_o follow_v he_o deride_v this_o proposition_n not_o count_v it_o worthy_a to_o be_v add_v in_o the_o number_n of_o proposition_n of_o geometry_n for_o the_o easiness_n thereof_o for_o that_o it_o be_v manifest_a even_o to_o the_o sense_n but_o not_o all_o thing_n manifest_a to_o sense_n be_v straight_o way_n manifest_a to_o reason_n and_o understanding_n it_o pertain_v to_o one_o that_o be_v a_o teacher_n of_o science_n by_o proof_n and_o demonstration_n to_o render_v a_o certain_a and_o undoubted_a reason_n why_o it_o so_o appear_v to_o the_o senseâ—_n and_o in_o that_o only_o consist_v science_n the_o 14._o theorem_a the_o 21._o proposition_n if_o from_o the_o end_n of_o one_o of_o the_o side_n of_o a_o triangle_n be_v draw_v to_o any_o point_n within_o the_o say_a triangle_n two_o right_a line_n those_o right_a line_n so_o draw_v shall_v be_v less_o then_o the_o two_o other_o side_n of_o the_o triangle_n but_o shall_v contain_v the_o great_a angle_n svppose_v that_o abc_n be_v a_o triangle_n and_o from_o the_o end_n of_o the_o side_n bc_o namely_o from_o the_o point_n b_o and_o c_o let_v there_o be_v draw_v within_o the_o triangle_n two_o right_a line_n bd_o and_o cd_o to_o the_o point_n d._n then_o i_o say_v that_o the_o line_n bd_o and_o cd_o be_v less_o than_o the_o other_o side_n of_o the_o triangle_n namely_o than_o the_o side_n basilius_n and_o ac_fw-la and_o that_o the_o angle_n which_o they_o contain_v namely_o bdc_o be_v great_a than_o the_o angle_n bac_n extend_v by_o the_o second_o petition_n the_o line_n bd_o to_o the_o point_n e._n demonstration_n and_o forasmuch_o as_o by_o the_o 20._o proposition_n in_o every_o triangle_n the_o two_o side_n be_v great_a than_o the_o side_n remain_v therefore_o the_o two_o side_n of_o the_o triangle_n abe_n namely_o the_o side_n ab_fw-la and_o ae_n be_v great_a than_o the_o side_n ebb_n put_v the_o line_n ec_o common_a to_o they_o both_o wherefore_o the_o line_n basilius_n and_o ac_fw-la be_v great_a than_o the_o line_n be_v and_o ecâ—_n again_o forasmuch_o as_o by_o the_o same_o in_o the_o triangle_n ce_z the_o two_o side_n ce_fw-fr and_o ed_z be_v great_a than_o the_o side_n dc_o put_v the_o line_n db_o common_a to_o they_o bothâ—_n wherefore_o the_o line_n ce_fw-fr and_o ed_z be_v great_a than_o the_o line_n cd_o and_o db._n but_o it_o be_v prove_v that_o the_o line_n basilius_n and_o ac_fw-la be_v great_a than_o the_o line_n be_v and_o ec_o wherefore_o the_o line_n basilius_n and_o ac_fw-la be_v much_o great_a than_o the_o line_n bd_o and_o dc_o again_o forasmuch_o as_o by_o the_o 16._o proposition_n in_o every_o triangle_n the_o outward_a angle_n be_v great_a than_o the_o inward_a and_o opposite_a angle_n therefore_o the_o outward_a angle_n of_o the_o triangle_n cde_o namely_o bdc_o be_v great_a than_o the_o angle_n ce_z wherefore_o also_o by_o the_o same_o the_o outward_a angle_n of_o the_o triangle_n abe_n namely_o the_o angle_n ceb_n be_v great_a than_o the_o angle_n bac_n but_o it_o be_v prove_v that_o the_o angle_n bdc_o be_v great_a than_o the_o angle_n ceb_fw-mi wherefore_o the_o angle_n bdc_o be_v much_o great_a than_o the_o angle_n bac_n wherefore_o if_o from_o the_o end_n of_o one_o of_o the_o side_n of_o a_o triangle_n be_v draw_v to_o any_o point_n within_o the_o say_a triangle_n two_o right_a line_n those_o right_a line_n so_o draw_v shall_v be_v less_o than_o the_o two_o other_o side_n of_o the_o triangle_n but_o shall_v contain_v the_o great_a angle_n which_o be_v require_v to_o be_v demonstrate_v in_o this_o proposition_n be_v express_v that_o the_o two_o right_a line_n draw_v within_o the_o triangle_n have_v their_o beginning_n at_o the_o extreme_n of_o the_o side_n of_o the_o triangle_n for_o from_o the_o one_o extreme_a of_o the_o side_n of_o the_o triangle_n and_o from_o some_o one_o point_n of_o the_o same_o side_n may_v be_v draw_v two_o right_a line_n within_o the_o triangle_n which_o shall_v be_v long_o they_o the_o two_o outward_a line_n which_o be_v wonderful_a and_o seem_v strange_a that_o two_o right_a line_n draw_v upon_o a_o part_n of_o a_o line_n shall_v be_v great_a than_o two_o right_a line_n draw_v upon_o the_o whole_a line_n and_o again_o it_o be_v possible_a from_o the_o one_o extreme_a of_o the_o side_n of_o a_o triangle_n and_o from_o some_o one_o point_n of_o the_o same_o side_n to_o draw_v two_o right_a line_n within_o the_o triangle_n which_o shall_v contain_v a_o angle_n less_o than_o the_o angle_n contain_v under_o the_o two_o outward_a line_n as_o touch_v the_o first_o part_n the_o 8._o problem_n the_o 22._o proposition_n of_o three_o right_a line_n which_o be_v equal_a to_o three_o right_a line_n give_v to_o make_v a_o triangle_n but_o it_o behove_v two_o of_o those_o line_n which_o then_o soever_o be_v take_v to_o be_v great_a than_o the_o three_o for_o that_o in_o every_o triangle_n two_o side_n which_o two_o side_n soever_o be_v take_v be_v great_a than_o the_o side_n remain_v svppose_v that_o the_o three_o right_a line_n give_v be_v a_o b_o c_o of_o which_o let_v two_o of_o they_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 line_n a_o b_o be_v great_a than_o the_o line_n c_o and_o the_o line_n a_o c_o then_z the_o line_n b_o and_o the_o line_n b_o c_o then_z the_o line_n a._n it_o be_v require_v of_o three_o right_a line_n equal_a to_o the_o right_a line_n a_o b_o c_o to_o make_v a_o triangle_n construction_n take_v a_o right_a line_n have_v a_o appoint_a end_n on_o the_o side_n d_o and_o be_v infinite_a on_o the_o side_n e._n and_o by_o the_o 3._o proposition_n put_v unto_o the_o line_n a_o a_o equal_a line_n df_o and_o put_v unto_o the_o line_n b_o a_o equal_a line_n fg_o and_o unto_o the_o line_v c_o a_o equal_a line_n gh_o and_o make_v the_o centre_n fletcher_n and_o the_o space_n df_o describe_v by_o the_o 3._o petition_n a_o circle_n dkl_n again_o make_v the_o centre_n g_o and_o the_o space_n g_o h_o describe_v by_o the_o same_o a_o circle_n hkl_n and_o let_v the_o point_n of_o the_o intersection_n of_o the_o say_a circle_n be_v king_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 king_n to_o the_o point_fw-fr fletcher_n &_o a_o other_o from_o the_o point_n king_n to_o the_o point_n g._n then_o i_o say_v that_o of_o three_o right_a line_n equal_a to_o the_o line_n a_o b_o c_o be_v make_v a_o triangle_n kfg_n demonstration_n for_o forasmuch_o as_o the_o point_n fletcher_n be_v the_o centre_n of_o the_o circle_n dkl_n therefore_o by_o the_o 15._o definition_n the_o line_n fd_o be_v equal_a to_o the_o line_n fk_o but_o the_o line_n a_o be_v equal_a to_o the_o line_n fd_o wherefore_o by_o the_o first_o common_a sentence_n the_o line_n fk_o be_v equal_a to_o the_o line_n a._n again_o forasmuch_o as_o the_o point_n g_o be_v the_o centre_n of_o the_o circle_n lkh_n therefore_o by_o the_o same_o definition_n the_o line_n gk_o be_v equal_a to_o the_o line_n ghâ—_n but_o the_o line_n c_o be_v equal_a to_o the_o line_n gh_o wherefore_o by_o the_o first_o common_a sentence_n the_o line_n kg_n be_v equal_a to_o the_o line_n c._n but_o the_o line_n fg_o be_v by_o supposition_n equal_a to_o the_o line_n b_o wherefore_o these_o three_o right_a line_n gf_o fk_o and_o kg_v be_v equal_a to_o these_o three_o right_a line_n a_o b_o c._n wherefore_o of_o three_o right_a line_n that_o be_v kf_n fg_o and_o gk_o which_o be_v equal_a to_o the_o three_o right_a line_n give_v that_o be_v to_o a_o b_o c_o be_v make_v a_o triangle_n kfg_n which_o be_v require_v to_o be_v do_v an_o other_o construction_n and_o demonstration_n after_o flussates_n suppose_v that_o the_o
side_n the_o rectiline_a angle_n mab_n either_o of_o which_o be_v equal_a to_o the_o rectiline_a angle_n give_v cde_o which_o be_v require_v to_o be_v do_v an_o other_o construction_n also_o and_o demonstration_n after_o pelitarâ—us_n and_o if_o the_o perpendicular_a line_n chance_n to_o fall_v without_o the_o angle_n give_v namely_o if_o the_o angle_n give_v be_v a_o acute_a angle_n the_o self_n same_o manner_n of_o demonstration_n will_v serve_v but_o only_o that_o in_o stead_n of_o the_o second_o common_a sentence_n must_v be_v use_v the_o 3._o common_a sentence_n appollonius_n put_v another_o construction_n &_o demonstration_n of_o this_o proposition_n which_o though_o the_o demonstration_n thereof_o depend_v of_o proposition_n put_v in_o the_o three_o book_n yet_o for_o that_o the_o construction_n be_v very_o good_a for_o he_o that_o will_v ready_o and_o mechanical_o without_o demonstration_n describe_v upon_o a_o line_n give_v and_o to_o a_o point_n in_o it_o give_v a_o rectiline_a angle_n equal_a to_o a_o rectiline_a angle_n give_v i_o think_v not_o amiss_o here_o to_o place_v it_o and_o it_o be_v thus_o proposition_n oenopides_n be_v the_o first_o inventor_n of_o this_o proposition_n as_o witness_v eudemius_fw-la the_o 15._o theorem_a the_o 24._o proposition_n if_o two_o triangle_n have_v two_o side_n of_o the_o one_o equal_a to_o two_o side_n of_o the_o other_o each_o to_o his_o correspondent_a side_n and_o if_o the_o angle_n contain_v under_o the_o equal_a side_n of_o the_o one_o be_v great_a than_o the_o angle_n contain_v under_o the_o equal_a side_n of_o the_o other_o the_o base_a also_o of_o the_o same_o shall_v be_v great_a than_o the_o base_a of_o the_o other_o svppose_v that_o there_o be_v two_o triangle_n abc_n and_o def_n have_v two_o side_n of_o the_o one_o that_o be_v ab_fw-la and_o ac_fw-la equal_a to_o two_o side_n of_o the_o other_o that_o be_v to_o de_fw-fr and_o df_o each_o to_o his_o correspondent_a side_n that_o be_v the_o side_n ab_fw-la to_o the_o side_n de_fw-fr and_o the_o side_n ac_fw-la to_o the_o side_n df_o and_o suppose_v that_o the_o angle_n bac_n be_v great_a than_o the_o angle_n edf_o then_o i_o say_v that_o the_o base_a bc_o be_v great_a than_o the_o base_a ef._n for_o forasmuch_o as_o the_o angle_n bac_n be_v great_a than_o the_o angle_n edf_o construction_n make_v by_o the_o 23._o proposition_n upon_o the_o right_a line_n de_fw-fr and_o to_o the_o point_n in_o it_o give_v d_o a_o angle_n edge_v equal_a to_o the_o angle_n give_v bac_n and_o to_o one_o of_o these_o line_n that_o be_v either_o to_o ac_fw-la or_o df_o put_v a_o equal_a line_n dg_o and_o by_o the_o first_o petition_n draw_v a_o right_a line_n from_o the_o point_n g_o to_o the_o point_n e_o and_o a_o other_o from_o the_o point_n fletcher_n demonstration_n to_o the_o point_n g._n and_o forasmuch_o as_o the_o line_n ab_fw-la be_v equal_a to_o the_o line_n de_fw-fr and_o the_o line_n ac_fw-la to_o the_o line_n dg_o the_o one_o to_o the_o outstretch_o and_o the_o angle_n bac_n be_v by_o construction_n equal_a to_o the_o angle_n edg_n therefore_o by_o the_o 4._o proposition_n the_o base_a bc_o be_v equal_a to_o the_o base_a eglantine_n again_o for_o as_o much_o as_o the_o line_n dg_o be_v equal_a to_o the_o line_n df_o therâ—_n by_o the_o 5._o proposition_n the_o angle_n dgf_n be_v equal_a to_o the_o angle_n dfg_n wherefore_o the_o angle_n dfg_n be_v great_a than_o the_o angle_n egf_n wherefore_o the_o angle_n efg_o be_v much_o great_a than_o the_o angle_n egf_n and_o forasmuch_o as_o efg_o be_v a_o triangle_n have_v the_o angle_n efg_o great_a than_o the_o angle_n egf_n and_o by_o the_o 18._o prâ—position_n under_o the_o great_a angle_n be_v subtend_v the_o great_a side_n therefore_o the_o side_n eglantine_n be_v great_a than_o the_o side_n ef._n but_o the_o side_n eglantine_n be_v equal_a to_o the_o side_n bc_o wherefore_o the_o side_n bc_o be_v great_a than_o the_o side_n ef._n if_o therefore_o two_o triangle_n have_v two_o side_n of_o the_o one_o equal_a to_o two_o side_n of_o the_o other_o each_o to_o his_o correspondent_a side_n and_o if_o the_o angle_n contain_v under_o the_o equal_a side_n of_o the_o one_o be_v great_a than_o the_o angle_n contain_v under_o the_o equal_a side_n of_o the_o other_o the_o base_a also_o of_o the_o same_o shall_v be_v great_a than_o the_o base_a of_o the_o other_o which_o be_v require_v to_o be_v prove_v in_o this_o theorem_a may_v be_v three_o case_n prâ—position_n for_o the_o angle_n edg_n be_v put_v equal_a to_o the_o angle_n bac_n and_o the_o line_n dg_o be_v put_v equal_a to_o the_o line_n ac_fw-la and_o a_o line_n be_v draw_v from_o e_o to_o g_z the_o line_n eglantine_n shall_v either_o fall_v above_o the_o line_n gf_o or_o upon_o it_o or_o under_o it_o euclides_n demonstration_n serve_v 2._o when_o the_o line_n ge_z fall_v above_o the_o line_n gf_o as_o we_o have_v before_o manifest_o see_v but_o now_o let_v the_o line_n eglantine_n case_n fall_v under_o the_o line_n e_o fletcher_n as_o in_o the_o figure_n here_o put_v and_o forasmuch_o as_o these_o two_o line_n ab_fw-la and_o ac_fw-la be_v equal_a to_o these_o two_o line_n de_fw-fr and_o dg_o the_o one_o to_o the_o other_o and_o they_o contain_v equal_a angle_n therefore_o by_o the_o 4._o proposition_n the_o base_a bc_o be_v equal_a to_o the_o base_a eglantine_n and_o forasmuch_o as_o within_o the_o triangle_n deg_n the_o two_o linne_n df_o and_o fe_o be_v set_v upon_o the_o side_n de_fw-fr therefore_o by_o the_o 21._o proposition_n the_o line_n df_o and_o fâ—_n be_v less_o than_o the_o outward_a line_n dg_o and_o ge_z but_o the_o line_n dg_o be_v equal_a to_o the_o line_n df._n wherefore_o the_o line_n ge_z be_v great_a than_o the_o line_n fe_o but_o ge_z be_v equal_a to_o bc._n wherefore_o the_o line_n bc_o be_v great_a the_o the_o line_n ef._n which_o be_v require_v to_o be_v prove_v it_o may_v peradventure_o semeâ—_n that_o euclid_n shall_v here_o in_o this_o proposition_n have_v prove_v that_o not_o only_o the_o base_n of_o the_o triangle_n be_v unequal_a but_o also_o that_o the_o area_n of_o the_o same_o be_v unequal_a for_o so_o in_o the_o four_o proposition_n after_o he_o have_v prove_v the_o base_a to_o be_v equal_a he_o prove_v also_o the_o area_n to_o be_v equal_a but_o hereto_o may_v be_v answer_v &c._a that_o in_o equal_a angle_n and_o base_n and_o unequal_a angle_n and_o base_n the_o consideration_n be_v not_o like_a for_o the_o angle_n and_o base_n be_v equal_a the_o triangle_n also_o shall_v of_o necessity_n be_v equal_a but_o the_o angle_n and_o base_n be_v unequal_a the_o area_n shall_v not_o of_o necessity_n be_v equal_a for_o the_o triangle_n may_v both_o be_v equal_a and_o unequal_a and_o that_o may_v be_v the_o great_a which_o have_v the_o great_a angle_n and_o the_o great_a base_n and_o it_o may_v also_o be_v the_o less_o and_o for_o that_o cause_n euclid_n make_v not_o mention_v of_o the_o comparison_n of_o the_o triangle_n whereof_o this_o also_o may_v be_v a_o cause_n for_o that_o to_o the_o demonstration_n thereof_o be_v require_v certain_a proposition_n concern_v parallel_a line_n which_o we_o be_v not_o as_o yet_o come_v unto_o etc_n howbeit_o after_o the_o 37â—_n proposition_n of_o his_o book_n you_o shall_v find_v the_o comparison_n of_o the_o area_n of_o triangle_n which_o have_v their_o side_n equal_a and_o their_o base_n and_o angle_n at_o the_o top_n unequal_a the_o 16._o theorem_a the_o 25._o proposition_n if_o two_o triangle_n have_v two_o side_n of_o the_o one_o equal_a to_o two_o side_n of_o the_o other_o each_o to_o his_o correspondent_a side_n and_o if_o the_o base_a of_o the_o one_o be_v great_a than_o the_o base_a of_o the_o other_o the_o angle_n also_o of_o the_o same_o contain_v under_o the_o equal_a right_a linesâ—_n shall_v be_v great_a than_o the_o angle_n of_o the_o other_o svppose_v that_o there_o be_v two_o triangle_n a_o b_o c_o and_o def_n have_v two_o side_n of_o tb'one_n that_o be_v ab_fw-la and_o ac_fw-la equal_a to_o two_o side_n of_o the_o other_o that_o be_v to_o de_fw-fr and_o df_o each_o to_o his_o correspondent_a side_n namely_o the_o side_n ab_fw-la to_o the_o side_n df_o and_o the_o side_n ac_fw-la to_o the_o side_n df._n but_o let_v the_o base_a bc_o be_v great_a than_o the_o base_a ef._n then_o i_o say_v they_o the_o angle_n bac_n be_v great_a than_o the_o angle_n edf_o for_o if_o not_o absurdity_n then_o be_v it_o either_o equal_a unto_o it_o or_o less_o than_o it_o but_o the_o angle_n bac_n be_v not_o equal_a to_o the_o angle_n edf_o for_o if_o it_o be_v equal_a the_o base_a also_o bc_o shall_v by_o the_o 4._o proposition_n be_v equal_a to_o the_o base_a of_o but_o by_o supposition_n it_o be_v not_o wherefore_o
the_o line_n de_fw-fr unto_o the_o point_n g._n and_o in_o like_a sort_n may_v we_o prove_v that_o the_o angle_n gec_n be_v double_a to_o the_o angle_n edc_n of_o which_o the_o angle_n geb_fw-mi be_v double_a to_o the_o angle_n edb_n wherefore_o the_o angle_n remain_v bec_n be_v double_a to_o the_o angle_n remain_v bdc_o wherefore_o 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 which_o be_v require_v to_o be_v demonstrate_v the_o 19_o theorem_a the_o 21._o proposition_n in_o a_o circle_n the_o angle_n which_o consist_v in_o one_o and_o the_o self_n same_o section_n or_o segment_n be_v equal_a the_o one_o to_o the_o other_o svppose_v that_o there_o be_v a_o circle_n abcd_o &_o in_o the_o segment_n thereof_o baed_n let_v there_o consist_v these_o angle_n bad_a and_o bed_n then_o i_o say_v that_o the_o angle_n bad_a and_o bed_n be_v equal_a the_o one_o to_o the_o other_o take_v by_o the_o first_o of_o the_o three_o the_o centre_n of_o the_o circle_n abcd_o construction_n and_o let_v the_o same_o be_v the_o point_n f._n and_o by_o the_o first_o petition_n draw_v these_o line_n bf_o and_o fd._n demonstration_n now_o for_o asmuch_o as_o the_o angle_n bfd_n be_v set_v at_o the_o centre_n and_o the_o angle_n bad_a at_o the_o circumference_n and_o they_o have_v both_o one_o and_o the_o same_o base_a namely_o the_o circumference_n bcd_o therefore_o the_o angle_n bfd_n be_v by_o the_o proposition_n go_v before_o double_a to_o the_o angle_n bad_a and_o by_o the_o same_o reason_n the_o angle_n bfd_n be_v also_o double_a to_o the_o angle_n bed_n wherefore_o by_o the_o 7._o common_a sentence_n the_o angle_n bad_a be_v equal_a to_o the_o angle_n bed_n wherefore_o in_o a_o circle_n the_o angle_n which_o consist_v in_o one_o and_o the_o self_n same_o segment_n be_v equal_a the_o one_o to_o the_o other_o which_o be_v require_v to_o be_v prove_v case_n in_o this_o proposition_n be_v three_o case_n for_o the_o angle_n consist_v in_o one_o and_o the_o self_n same_o segment_n the_o segment_n may_v either_o be_v great_a they_o a_o semicircle_n or_o less_o than_o a_o semicircle_n or_o else_o just_a a_o semicircle_n for_o the_o first_o case_n the_o demonstration_n before_o put_v serve_v the_o self_n same_o construction_n and_o demonstration_n will_v also_o serve_v case_n if_o the_o angle_n be_v set_v in_o a_o semicircle_n as_o it_o be_v plaâ—ne_v to_o see_v in_o the_o figure_n here_o set_v the_o 20._o theorem_a the_o 22._o proposition_n 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 svppose_v that_o there_o be_v a_o circle_n abcd_o and_o let_v there_o be_v describe_v in_o it_o a_o figure_n of_o four_o side_n namely_o abcd._n then_o i_o say_v that_o 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 draw_v by_o the_o first_o petition_n these_o right_a line_n ac_fw-la and_o bd._n construction_n now_o for_o asmuch_o as_o by_o the_o 32._o of_o the_o first_o the_o three_o angle_n of_o every_o triangle_n be_v equal_a to_o two_o right_a angle_n therefore_o y_fw-fr three_o angle_n of_o the_o triangle_n abc_n namely_o demonstration_n cab_n abc_n and_o bca_o be_v equal_a to_o two_o right_a angle_n but_o by_o the_o 21._o of_o the_o three_o the_o angle_n cab_n be_v equal_a to_o the_o angle_n bdc_o for_o they_o consist_v in_o one_o and_o the_o self_n same_o segment_n namely_o badc_n and_o by_o the_o same_o proposition_n the_o angle_n acb_o be_v equal_a to_o the_o angle_n adb_o for_o they_o consist_v in_o one_o and_o the_o same_o segment_n adcb_n wherefore_o the_o wholâ—_n angle_n adc_o be_v equal_a to_o the_o angle_n bac_n and_o acb_o put_v the_o angle_n abc_n common_a to_o they_o both_o wherefore_o the_o angle_n abc_n bac_n and_o acb_o be_v equal_a to_o the_o angle_n abc_n and_o adc_o but_o the_o angle_n abc_n bac_n and_o acb_o be_v equal_a to_o two_o right_a angle_n wherefore_o the_o angle_n abc_n and_o adc_o be_v equal_a to_o two_o right_a angle_n and_o in_o like_a sort_n also_o may_v we_o prove_v that_o the_o angle_n bad_a and_o dcb_n be_v equal_a to_o two_o right_a angle_n if_o therefore_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 outstretch_o be_v equal_a to_o two_o right_a anglesâ—_n which_o be_v require_v to_o be_v prove_v the_o 21._o theorem_a the_o 23._o proposition_n upon_o one_o and_o the_o self_n same_o right_a line_n can_v not_o be_v describe_v two_o like_a and_o unequal_a segment_n of_o circle_n fall_v both_o on_o one_o and_o the_o self_n same_o side_n of_o the_o line_n impossibility_n for_o if_o it_o be_v possible_a let_v there_o be_v describe_v upon_o the_o right_a line_n ab_fw-la two_o like_v &_o unequal_a section_n of_o circle_n namely_o acb_o &_o adb_o fall_v both_o on_o one_o and_o the_o self_n same_o side_n of_o the_o line_n ab_fw-la and_o by_o the_o first_o petition_n draw_v the_o right_a line_n acd_o and_o by_o the_o three_o petition_n draw_v right_a line_n from_o c_o to_o b_o and_o from_o d_z to_o b._n and_o for_o asmuch_o as_o the_o segment_n acb_o be_v like_a to_o the_o segment_n adb_o and_o like_o segmente_n of_o circle_n be_v they_o which_o have_v equal_a angle_n by_o the_o 10._o definition_n of_o the_o three_o wherefore_o the_o angle_n acb_o be_v equal_a to_o the_o angle_n adb_o namely_o the_o outward_a angle_n of_o the_o triangle_n cdb_n to_o the_o inward_a angle_n which_o by_o the_o 16._o of_o the_o first_o be_v impossible_a wherefore_o upon_o one_o and_o the_o self_n same_o right_a line_n can_v not_o be_v describe_v two_o like_a &_o unequal_a segment_n of_o circle_n fall_v both_o on_o one_o &_o the_o self_n same_o side_n of_o the_o line_n which_o be_v require_v to_o be_v demonstrate_v pelitariâ—s_n here_o campane_n add_v that_o upon_o one_o and_o the_o self_n same_o right_a line_n can_v be_v describe_v two_o like_a and_o unequal_a section_n neither_o on_o one_o and_o the_o self_n same_o side_n of_o the_o line_n nor_o on_o the_o opposite_a side_n that_o they_o can_v not_o be_v describe_v on_o one_o and_o the_o self_n same_o side_n have_v be_v before_o demonstrate_v and_o that_o neither_o also_o on_o the_o opposite_a side_n pelitarius_n thus_o demonstrate_v the_o 22._o theorme_n the_o 24._o proposition_n like_a segment_n of_o circle_n describe_v upon_o equal_a right_a line_n be_v equal_a the_o one_o to_o the_o other_o svppose_v that_o upon_o these_o equal_a right_a line_n ab_fw-la and_o cd_o be_v describe_v these_o like_a segment_n of_o circle_n namely_o aeb_n and_o cfd_n then_o i_o say_v that_o the_o segment_n aeb_fw-mi be_v equal_a to_o the_o segment_n cfd_n for_o put_v the_o segment_n aeb_fw-mi upon_o the_o segment_n cfd_n and_o the_o point_n a_o upon_o the_o point_v c_o and_o the_o right_a line_n ab_fw-la upon_o the_o right_a line_n cd_o the_o point_n b_o also_o shall_v fall_v upon_o the_o point_n d_o for_o the_o line_n ab_fw-la be_v equal_a to_o the_o line_n cd_o and_o the_o right_a line_n ab_fw-la exact_o agree_v with_o the_o right_a line_n cd_o y_fw-mi segment_n also_o aeb_n shall_v exact_o agree_v with_o the_o segment_n cfd_n impossibility_n for_o if_o the_o right_a line_n ab_fw-la do_v exact_o agree_v with_o the_o right_a line_n cd_o and_o the_o segment_n aeb_fw-mi do_v not_o exact_o agree_v with_o the_o segment_n cfd_n but_o differ_v as_o the_o segment_n cgd_v do_v now_o by_o the_o 13._o of_o the_o three_o a_o circle_n cut_v not_o a_o circle_n in_o more_o point_n than_o two_o but_o the_o circle_n cgd_v cut_v the_o circle_n cfd_n in_o more_o point_n they_o two_o that_o be_v in_o the_o point_n c_o g_o and_o d_o which_o be_v by_o the_o same_o impossible_a wherefore_o the_o right_a line_n ab_fw-la exact_o agree_v with_o the_o right_a line_n cd_o the_o segment_n aeb_fw-mi shall_v not_o but_o exact_o agree_v with_o the_o segment_n cfd_n wherefore_o it_o exact_o agree_v with_o it_o and_o be_v equal_a unto_o it_o wherefore_o like_a segment_n of_o circle_n describe_v upon_o equal_a right_a line_n be_v equal_a the_o one_o to_o the_o other_o which_o be_v require_v to_o be_v prove_v this_o proposition_n may_v also_o be_v demonstrate_v by_o the_o former_a proposition_n for_o if_o the_o section_n aeb_fw-mi and_o cfd_n be_v like_o and_o set_v upon_o equal_a right_a line_n ab_fw-la and_o cd_o demonstration_n shall_v be_v unequal_a than_o the_o one_o be_v put_v upon_o the_o other_o the_o great_a shall_v exceed_v the_o less_o
touch_v the_o circle_n abc_n and_o let_v the_o same_o be_v de._n construction_n and_o by_o the_o first_o of_o the_o same_o let_v the_o point_n fletcher_n be_v the_o centre_n of_o the_o circle_n abc_n and_o draw_v these_o right_a line_n fe_o fb_o and_o fd._n wherefore_o the_o angle_n feed_v be_v a_o right_a angle_n demonstration_n and_o for_o asmuch_o as_o the_o right_a line_n de_fw-fr touch_v the_o circle_n abc_n and_o the_o right_a line_n dca_n cut_v the_o same_o therefore_o by_o the_o proposition_n go_v before_o that_o which_o be_v contain_v under_o the_o line_n ad_fw-la and_o dc_o be_v equal_a to_o the_o square_n of_o the_o line_n de._n but_o that_o which_o be_v contain_v under_o the_o line_n ad_fw-la and_o dc_o be_v suppose_v to_o be_v equal_a to_o the_o square_n of_o the_o line_n db._n wherefore_o the_o square_a of_o the_o line_n de_fw-fr be_v equal_a to_o the_o square_n of_o the_o line_n db._n wherefore_o also_o the_o line_n de_fw-fr be_v equal_a to_o the_o line_n db._n and_o the_o line_n fe_o be_v equal_a to_o the_o line_n fb_o for_o they_o be_v draw_v from_o the_o centre_n to_o the_o circumference_n now_o therefore_o these_o two_o line_n de_fw-fr and_o of_o be_v equal_a to_o these_o two_o line_n db_o and_o bf_o and_o fd_o be_v a_o common_a base_a to_o they_o both_o wherefore_o by_o the_o 8._o of_o the_o first_o the_o angle_n def_n be_v equal_a to_o the_o angle_n dbf_n but_o the_o angle_n def_n be_v a_o right_a angle_n wherefore_o also_o the_o angle_n dbf_n be_v a_o right_a angle_n and_o the_o line_n fb_o be_v produce_v shall_v be_v the_o diameter_n of_o the_o circle_n but_o if_o from_o the_o end_n of_o the_o diameter_n of_o a_o circle_n be_v draw_v a_o right_a line_n make_v right_a angle_n the_o right_a line_n so_o draw_v touch_v the_o circle_n by_o the_o corollary_n of_o the_o 16._o of_o the_o three_o wherefore_o the_o right_a line_n db_o touch_v the_o circle_n abc_n and_o the_o like_a demonstration_n will_v serve_v if_o the_o centre_n be_v in_o the_o line_n ac_fw-la if_o therefore_o without_o a_o circle_n be_v take_v a_o certain_a point_n and_o from_o that_o point_n be_v draw_v to_o the_o circle_n two_o right_a line_n of_o which_o the_o one_o do_v cut_v the_o circle_n and_o the_o other_o fall_v upon_o the_o circle_n and_o that_o in_o such_o sort_n that_o the_o rectangle_n parallelogram_n which_o be_v contain_v under_o the_o whole_a right_a line_n which_o cut_v the_o circle_n and_o that_o portion_n of_o the_o same_o line_n that_o lie_v between_o the_o point_n and_o the_o utter_a circumference_n of_o the_o circle_n be_v equal_a to_o the_o square_n make_v of_o the_o line_n that_o fall_v upon_o the_o circle_n then_o the_o line_n that_o so_o fall_v upon_o the_o circle_n shall_v touch_v the_o circle_n which_o be_v require_v to_o be_v prove_v ¶_o a_o other_o demonstration_n after_o pelitarius_n suppose_v that_o there_o be_v a_o circle_n bcd_o who_o centre_n let_v be_v e_o pelitarius_n and_o take_v a_o point_n without_o it_o namely_o a_o and_o from_o the_o point_n a_o draw_v two_o right_a line_n abdella_n and_o ac_fw-la of_o which_o let_v abdella_n cut_v the_o circle_n in_o the_o point_n b_o &_o let_v the_o other_o fall_n upon_o it_o and_o let_v that_o which_o be_v contain_v under_o the_o line_n ad_fw-la and_o ab_fw-la be_v equal_a to_o the_o square_n of_o the_o line_n ac_fw-la then_o i_o say_v that_o the_o line_n ac_fw-la touch_v the_o circle_n for_o first_o if_o the_o line_n abdella_n do_v pass_v by_o the_o centre_n draw_v the_o right_a line_n ce._n and_o by_o the_o 6._o of_o the_o second_o that_o which_o be_v contain_v under_o the_o line_n ad_fw-la and_o ab_fw-la together_o with_o the_o square_n of_o the_o line_n ebb_n that_o be_v with_o the_o square_n of_o the_o line_n ec_o for_o the_o line_n ebb_v and_o ec_o be_v equal_a be_v equal_a to_o the_o square_n of_o the_o line_n ae_n but_o that_o which_o be_v contain_v under_o the_o line_n ad_fw-la and_o ab_fw-la be_v suppose_v to_o be_v equal_a to_o the_o square_n of_o the_o line_n ac_fw-la wherefore_o the_o square_a of_o the_o line_n ac_fw-la together_o with_o the_o square_n of_o the_o line_n ce_fw-fr be_v equal_a to_o the_o square_n of_o the_o line_n ae_n wherefore_o by_o the_o last_o of_o the_o first_o the_o angle_n at_o the_o point_n c_o be_v a_o right_a angle_n wherefore_o by_o the_o 18._o of_o this_o book_n the_o line_n ac_fw-la touch_v the_o circle_n but_o if_o the_o line_n abdella_n do_v not_o pass_v by_o the_o centre_n draw_v from_o the_o point_n a_o the_o line_n ad_fw-la in_o which_o let_v be_v the_o centre_n e._n and_o forasmuch_o as_o that_o which_o be_v contain_v under_o this_o whole_a line_n and_o his_o outward_a part_n be_v equal_a to_o that_o which_o be_v contain_v under_o the_o line_n ad_fw-la and_o ab_fw-la by_o the_o first_o corollary_n before_o put_v therefore_o the_o same_o be_v equal_a to_o the_o square_n of_o the_o line_n ac_fw-la wherefore_o the_o angle_n eca_n be_v a_o right_a angle_n as_o have_v before_o be_v prove_v in_o the_o first_o part_n of_o this_o proposition_n and_o therefore_o the_o line_n ac_fw-la touch_v the_o circle_n which_o be_v require_v to_o be_v prove_v the_o end_n of_o the_o three_o book_n of_o euclides_n element_n ¶_o the_o four_o book_n of_o euclides_n element_n this_o four_o book_n intreat_v of_o the_o inscription_n &_o circumscription_n of_o rectiline_a figure_n book_n how_o one_o right_a line_v figure_n may_v be_v inscribe_v within_o a_o other_o right_o line_v figure_n and_o how_o a_o right_a line_v figure_n may_v be_v circumscribe_v about_o a_o other_o right_o line_v figure_n in_o such_o as_o may_v be_v inscribe_v and_o circumscribe_v within_o or_o about_o the_o other_o for_o all_o right_n line_v figure_n can_v so_o be_v inscribe_v or_o circumscribe_v within_o or_o about_o the_o other_o also_o it_o teach_v how_o a_o triangle_n a_o square_a and_o certain_a other_o rectiline_a figure_n be_v regular_a may_v be_v inscribe_v within_o a_o circle_n also_o how_o they_o may_v be_v circumscribe_v about_o a_o circle_n likewise_o how_o a_o circle_n may_v be_v inscribe_v within_o they_o and_o how_o it_o may_v be_v circumscribe_v about_o they_o and_o because_o the_o manner_n of_o entreaty_n in_o this_o book_n be_v diverse_a from_o the_o entreaty_n of_o the_o former_a book_n he_o use_v in_o this_o other_o word_n and_o term_n than_o he_o use_v in_o they_o the_o definition_n of_o which_o in_o order_n here_o after_o follow_v definition_n a_o rectiline_a figure_n be_v say_v to_o be_v inscribe_v in_o a_o rectiline_a figure_n definition_n when_o every_o one_o of_o the_o angle_n of_o the_o inscribe_v figure_n touch_v every_o one_o of_o the_o side_n of_o the_o figure_n wherein_o it_o be_v inscribe_v as_o the_o triangle_n abc_n be_v inscribe_v in_o the_o triangle_n def_n because_o that_o every_o angle_n of_o the_o triangle_n inscribe_v namely_o the_o triangle_n abc_n touch_v every_o side_n of_o the_o triangle_n within_o which_o it_o be_v describe_v namely_o of_o the_o triangle_n def_n as_o the_o angle_n cab_n touch_v the_o side_n ed_z the_o angle_n abc_n touch_v the_o side_n df_o and_o the_o angle_n acb_o touch_v the_o side_n ef._n so_o likewise_o the_o square_a abcd_o be_v say_v to_o be_v inscribe_v within_o the_o square_a efgh_o for_o every_o angle_n of_o it_o touch_v some_o one_o side_n of_o the_o other_o so_o also_o the_o pentagon_n or_o five_o angle_a figure_n abcde_o be_v inscribe_v within_o the_o pentagon_n or_o five_o angle_a figure_n fghikâ—_n â—_z as_o you_o see_v in_o the_o figureâ—_n likewise_o a_o rectiline_a figure_n be_v say_v to_o be_v circumscribe_v about_o a_o rectiline_a figure_n definition_n when_o every_o one_o of_o the_o side_n of_o the_o figure_n circumscribe_v touch_v every_o one_o of_o the_o angle_n of_o the_o figure_n about_o which_o it_o be_v circumscribe_v as_o in_o the_o former_a description_n the_o triangle_n def_n be_v say_v to_o be_v circumscribe_v about_o the_o triangle_n abc_n for_o that_o every_o side_n of_o the_o figure_n circumscribe_v namely_o of_o the_o triangle_n def_n touch_v every_o angle_n of_o the_o figure_n wherabout_o it_o be_v circumscribe_v as_o the_o side_n df_o of_o the_o triangle_n def_n circumscribe_v touch_v the_o angle_n abc_n of_o the_o triangle_n abc_n about_o which_o it_o be_v circumscribe_v and_o the_o side_n of_o touch_v the_o angle_n bca_o and_o the_o side_n cd_o touch_v the_o angle_n cab_n likewise_o understand_v you_o of_o the_o square_a efgh_o which_o be_v circumscribe_v about_o the_o square_n abcd_o for_o every_o side_n of_o the_o one_o touch_v some_o one_o side_n of_o the_o other_o even_o so_o by_o the_o same_o reason_n the_o pentagon_n fghik_n be_v circumscribe_v about_o the_o pentagon_n abcde_o as_o you_o see_v in_o the_o figure_n on_o the_o other_o side_n and_o thus_o may_v you_o
of_o other_o rectiline_a figure_n consider_v figure_n by_o these_o two_o definition_n it_o be_v manifest_a that_o the_o inscription_n and_o circumscription_n of_o rectiline_a figure_n here_o speak_v of_o pertain_v to_o such_o rectiline_a figure_n only_o which_o have_v equal_a side_n and_o equal_a angle_n which_o be_v common_o call_v regular_a it_o be_v also_o to_o be_v note_v that_o rectiline_a figure_n only_o of_o one_o kind_n or_o form_n can_v be_v inscribe_v or_o circumscribe_v the_o one_o within_o or_o about_o the_o other_o as_o a_o triangle_n within_o or_o about_o a_o triangle_n a_o square_a within_o or_o about_o a_o square_n and_o so_o a_o pentagon_n within_o or_o about_o a_o pentagon_n &_o likewise_o of_o other_o of_o one_o form_n but_o a_o triangle_n can_v not_o be_v inscribe_v or_o circumscribe_v within_o or_o about_o a_o square_n nor_o a_o square_a within_o or_o about_o a_o pentagon_n and_o so_o of_o other_o of_o diverse_a kind_n for_o every_o plain_a rectiline_a figure_n have_v so_o many_o angle_n as_o it_o have_v side_n wherefore_o the_o figure_n inscride_v must_v have_v so_o many_o angle_n as_o the_o figure_n in_o which_o it_o be_v inscribe_v have_v side_n and_o the_o angle_n of_o the_o one_o as_o be_v say_v must_v touch_v the_o side_n of_o the_o other_o and_o contrariwise_o in_o circumscription_n of_o figure_n the_o side_n of_o the_o figure_n circumscribe_v must_v touch_v the_o angle_n of_o the_o figure_n about_o which_o it_o be_v circumscribe_v definition_n a_o rectiline_a figure_n be_v say_v to_o be_v inscribe_v in_o a_o circle_n when_o every_o one_o of_o the_o angle_n of_o the_o inscribe_v figure_n touch_v the_o circumference_n of_o the_o circle_n a_o circle_n by_o reason_n of_o his_o uniform_a and_o regular_a distance_n which_o it_o have_v from_o the_o centre_n to_o the_o circumference_n may_v easy_o touch_v all_o the_o angle_n of_o any_o regular_a rectiline_a figure_n within_o it_o and_o also_o all_o the_o side_n of_o any_o figure_n without_o it_o and_o therefore_o any_o regular_a rectiline_a figure_n may_v be_v inscribe_v within_o it_o and_o also_o be_v circumscribe_v about_o it_o and_o again_o a_o circle_n may_v be_v both_o inscribe_v within_o any_o regular_a rectiline_a figure_n and_o also_o be_v circumscribe_v about_o it_o as_o the_o triangle_n abc_n be_v inscribe_v in_o the_o circle_n abcâ—_n â—_z for_o that_o every_o angle_n touch_v some_o one_o point_n of_o the_o circumference_n of_o the_o circle_n as_o the_o angle_n cab_n of_o the_o triangle_n abc_n touch_v the_o point_n a_o of_o the_o circumference_n of_o the_o circle_n and_o the_o angle_n abc_n of_o the_o triangle_n touch_v the_o point_n b_o of_o the_o circumference_n of_o the_o circle_n and_o also_o the_o angle_n acb_o of_o the_o triangle_n 〈…〉_o the_o point_n â—_o of_o the_o circumference_n of_o the_o circle_n in_o like_a manner_n the_o square_a adef_n be_v inscribe_v in_o the_o same_o circle_n abc_n for_o that_o every_o angle_n of_o the_o square_n inscribe_v touch_v some_o one_o point_n of_o the_o circle_n in_o which_o it_o be_v inscribe_v and_o so_o imagine_v you_o of_o rectiline_v figure_n a_o circle_n be_v say_v to_o be_v circumscribe_v about_o a_o rectiline_a figure_n definition_n when_o the_o circumference_n of_o the_o circle_n touch_v every_o one_o of_o the_o angle_n of_o the_o figure_n about_o which_o it_o be_v circumscribe_v as_o in_o the_o former_a example_n of_o the_o three_o definition_n the_o circle_n adef_n be_v circumscribe_v about_o the_o triangle_n abc_n because_o the_o circumference_n of_o the_o circle_n which_o be_v circumscribe_v touch_v every_o angle_n of_o the_o triangle_n about_o which_o it_o be_v circumscribedâ—_n namely_o the_o angle_n cab_n abc_n and_o bca_o likewise_o the_o same_o circle_n adef_n be_v circumscribe_v about_o the_o square_a adef_n by_o the_o same_o definition_n as_o you_o may_v see_v a_o circle_n be_v say_v to_o be_v inscribe_v in_o a_o rectiline_a figure_n definition_n when_o the_o circumference_n of_o the_o circle_n touch_v every_o one_o of_o the_o side_n of_o the_o figure_n within_o which_o it_o be_v inscribe_v as_o the_o circle_n abcd_o be_v inscribe_v within_o the_o triangle_n efg_o because_o the_o circumference_n of_o the_o circle_n touch_v every_o side_n of_o the_o triangle_n in_o which_o it_o be_v inscribedâ—_n namely_o the_o side_n of_o in_o the_o point_n b_o and_o the_o side_n gf_o in_o the_o point_n c_o and_o the_o side_n ge_z in_o the_o point_n d._n likewise_o by_o the_o same_o reason_n the_o same_o circle_n be_v inscribe_v within_o the_o square_a hikl_n and_o so_o may_v you_o judge_v of_o other_o rectiline_a figure_n a_o rectilined_a figure_n be_v say_v to_o be_v circumscribe_v about_o a_o circle_n devition_n when_o every_o one_o of_o the_o side_n of_o the_o figure_n circumscribe_v touch_v the_o circumference_n of_o the_o circle_n as_o in_o the_o former_a figure_n of_o the_o five_o definition_n the_o triangle_n efg_o be_v circumscribe_v about_o the_o circle_n abcd_o for_o that_o every_o side_n of_o the_o same_o triangle_n be_v circumscribe_v touch_v the_o circumference_n of_o the_o circle_n about_o which_o it_o be_v circumscribe_v as_o the_o side_n eglantine_n of_o the_o triangle_n efg_o touch_v the_o circumference_n of_o the_o circle_n in_o the_o point_n d_o and_o the_o side_n of_o touch_v it_o in_o the_o point_n b_o and_o the_o side_n gf_o in_o the_o point_n c._n likewise_o also_o the_o square_a hikl_n be_v circumscribe_v about_o the_o circle_n abcd_o for_o every_o one_o of_o his_o side_n touch_v the_o circumference_n of_o the_o circle_n namely_o in_o the_o point_n a_o b_o c_o d._n and_o thus_o consider_v of_o all_o other_o regular_a right_n line_v figure_n for_o of_o they_o only_o be_v understand_v these_o definition_n to_o be_v circumscribe_v about_o a_o circle_n or_o to_o be_v inscribe_v within_o a_o circle_n or_o of_o a_o circle_n to_o be_v circumscribe_v or_o inscribe_v about_o or_o within_o any_o of_o they_o definition_n a_o right_a line_n be_v say_v to_o be_v coapt_v or_o apply_v in_o a_o circle_n when_o the_o extreme_n or_o end_n thereof_o fall_v upon_o the_o circumference_n of_o the_o circle_n as_o the_o line_n bc_o be_v say_v to_o be_v coapt_v or_o to_o be_v apply_v to_o the_o circle_n abc_n for_o that_o both_o his_o extreme_n fall_v upon_o the_o circumference_n of_o the_o circle_n in_o the_o point_n b_o and_o c._n likewise_o the_o line_n de._n this_o definition_n be_v very_o necessary_a and_o be_v proper_o to_o be_v take_v of_o any_o line_n give_v to_o be_v coapt_v and_o apply_v into_o a_o circle_n so_o â—hat_n it_o exceed_v not_o the_o diameter_n of_o the_o circle_n give_v the_o 1._o problem_n the_o 1._o proposition_n in_o a_o circle_n give_v to_o apply_v a_o right_a line_n equal_a unto_o a_o right_a line_n give_v which_o exceed_v not_o the_o diameter_n of_o a_o circle_n svppose_v that_o the_o circle_n give_v be_v abc_n and_o let_v the_o right_a line_n give_v exceed_v not_o the_o diameter_n of_o the_o same_o circle_n be_v d._n now_o it_o be_v require_v in_o the_o circle_n give_v abc_n to_o apply_v a_o right_a line_n equal_a unto_o the_o right_a line_n d._n construction_n draw_v the_o diameter_n of_o the_o circle_n abc_n and_o let_v the_o same_o be_v bc._n now_o if_o the_o line_n bc_o be_v equal_a unto_o the_o line_n d_o then_o be_v that_o do_v which_o be_v require_v for_o in_o the_o circle_n give_v abc_n be_v apply_v a_o right_a line_n bc_o equal_a unto_o the_o right_a line_n d._n proposition_n but_o if_o not_o caseâ—_n then_o be_v the_o line_n bc_o great_a than_o the_o line_n d._n case_n and_o by_o the_o three_o of_o the_o first_o put_v unto_o the_o line_n d_o a_o equal_a line_n ce._n and_o make_v the_o centre_n c_o and_o the_o space_n ce_fw-fr describe_v by_o the_o three_o petition_n a_o circle_n egf_n cut_v the_o circle_n abc_n in_o the_o point_n fletcher_n &_o draw_v a_o line_n from_o c_o to_o f._n and_o for_o asmuch_o as_o the_o point_n c_o be_v the_o centre_n of_o the_o circle_n egf_n demonstration_n therefore_o by_o the_o 13._o definition_n of_o the_o first_o the_o line_n cf_o be_v equal_a unto_o the_o line_n ce._n but_o the_o line_n ce_fw-fr be_v equal_a unto_o the_o line_n d._n wherefore_o by_o the_o first_o common_a sentence_n the_o line_n cf_o also_o be_v equal_a unto_o the_o line_n d._n wherefore_o in_o the_o circle_n give_v abc_n be_v apply_v a_o right_a line_n ca_n equal_a unto_o the_o right_a line_n give_v d_z which_o be_v require_v to_o be_v do_v the_o 2._o problem_n the_o 2._o proposition_n in_o a_o circle_n give_v to_o describe_v a_o triangle_n equiangle_n unto_o a_o triangle_n give_v svppose_v that_o the_o circle_n give_v be_v abc_n and_o let_v the_o triangle_n give_v be_v def_n now_o it_o be_v require_v in_o the_o circle_n give_v abc_n to_o describe_v a_o triangle_n equiangle_n unto_o the_o triangle_n give_v def_n
proportion_n than_o have_v bc_o to_o ef._n wherefore_o a_o have_v to_o d_o a_o great_a proportion_n than_o have_v bc_o to_o ef._n wherefore_o alternate_o a_o have_v to_o bc_o a_o great_a proportion_n than_o have_v d_z to_o of_o wherefore_o by_o composition_n abc_n have_v to_o bc_o a_o great_a proportion_n than_o have_v def_n to_o ef._n wherefore_o again_o alternate_o abc_n have_v to_o def_n a_o great_a proportion_n than_o have_v bc_o to_o ef._n wherefore_o by_o the_o former_a proposition_n the_o proportion_n of_o a_o to_o d_z be_v great_a than_o the_o proportion_n of_o abc_n to_o def_n which_o be_v require_v to_o be_v prove_v the_o end_n of_o the_o five_o book_n of_o euclides_n element_n ¶_o the_o six_o book_n of_o euclides_n element_n this_o spaniard_o book_n be_v for_o use_v and_o practice_v book_n a_o most_o special_a book_n in_o it_o be_v teach_v the_o proportion_n of_o one_o figure_n to_o a_o other_o figure_n &_o of_o their_o side_n the_o one_o to_o the_o other_o and_o of_o the_o side_n of_o one_o to_o the_o side_n of_o a_o other_o likewise_o of_o the_o angle_n of_o the_o one_o to_o the_o angle_n of_o the_o other_o moreover_o it_o teach_v the_o description_n of_o figure_n like_o to_o â—_z give_v and_o marvelous_a application_n of_o figure_n to_o line_n even_o or_o with_o decrease_n or_o excess_n with_o many_o other_o theorem_n not_o only_o of_o the_o propoâ—tions_n of_o right_n line_v figure_n but_o also_o of_o sector_n of_o circle_n with_o their_o angle_n on_o the_o theorem_n and_o problem_n of_o this_o book_n depend_v for_o the_o most_o part_n the_o composition_n of_o all_o instrument_n of_o measure_v length_n breadth_n or_o deepen_n and_o also_o the_o reason_n of_o the_o use_n of_o the_o same_o instrument_n as_o of_o the_o geometrical_a â—quarâ—_n geometry_n the_o scale_n of_o the_o astrolabe_n the_o quadrant_a the_o staff_n and_o such_o other_o the_o use_n of_o which_o instrument_n beside_o all_o other_o mechanical_a instrument_n of_o raise_v up_o of_o move_v and_o draw_v huge_a thing_n incredible_a to_o the_o ignorant_a and_o infinite_a other_o begin_v which_o likewise_o have_v their_o ground_n out_o of_o this_o book_n be_v of_o wonderful_a and_o unspeakable_a profit_n beside_o the_o inestimable_a pleasure_n which_o be_v in_o they_o definition_n 1._o like_o rectiline_a figure_n be_v such_o definition_n who_o angle_n be_v equal_a the_o one_o to_o the_o other_o and_o who_o side_n about_o the_o equal_a angle_n be_v proportional_a as_o if_o you_o take_v any_o two_o rectiline_a figure_n as_o for_o example_n two_o triangle_n abc_n and_o def_n 〈…〉_o of_o the_o one_o triangle_n be_v equal_a to_o the_o angle_n of_o the_o other_o namely_o if_o the_o angle_n a_o be_v equal_a to_o the_o angle_n d_o and_o the_o angle_n b_o equal_v to_o the_o angle_n e_o &_o also_o the_o angle_n c_o equal_v to_o the_o angle_n f._n and_o moreover_o iâ—_z the_o side_n which_o contain_v the_o equal_a angle_n be_v proportional_a as_o if_o the_o side_n ab_fw-la have_v that_o proportion_n to_o the_o side_n bc_o whâ—ch_v the_o side_n de_fw-fr have_v to_o the_o side_n of_o and_o also_o if_o the_o side_n bc_o be_v unto_o the_o side_n ca_n as_o â—he_n side_n of_o be_v to_o the_o side_n fd_o and_o moreover_o if_o the_o side_n ca_n be_v to_o the_o side_n ab_fw-la as_o the_o side_n fd_o be_v to_o the_o side_n de_fw-fr then_o be_v these_o two_o triangle_n say_v to_o be_v like_o and_o so_o judge_v you_o of_o any_o other_o kind_n of_o figure_n as_o if_o in_o the_o parallelogram_n abcd_o and_o efgh_a the_o angle_n a_o be_v equal_a to_o the_o angle_n e_o and_o the_o angle_n b_o equal_v to_o the_o angle_n fletcher_n and_o the_o angle_n c_o equal_v to_o the_o angle_n g_o and_o the_o angle_n d_o equal_v to_o the_o angle_n h._n and_o farthermore_o if_o the_o side_n ac_fw-la have_v that_o proportion_n to_o the_o side_n cd_o which_o the_o side_n eglantine_n have_v to_o the_o side_n gh_o and_o if_o also_o the_o side_n cd_o be_v to_o the_o side_n db_o as_o the_o side_n gh_o be_v to_o the_o side_n hf_o and_o moreover_o if_o the_o side_n db_o be_v to_o the_o side_n basilius_n as_o the_o side_n hf_o be_v to_o the_o side_n fe_o and_o final_o if_o the_o side_n basilius_n be_v to_o the_o side_n ac_fw-la as_o the_o side_n fe_o be_v to_o the_o side_n eglantine_n then_o be_v these_o parallelogram_n like_a definition_n 2._o reciprocal_a figure_n be_v those_o when_o the_o termeâ—_n of_o proportion_n be_v both_o antecedentes_fw-la and_o consequentes_fw-la in_o either_o figure_n as_o if_o you_o have_v two_o parallelogram_n abcd_o and_o efgh_o if_o the_o side_n ab_fw-la to_o the_o side_n of_o a_o antecedent_n of_o the_o first_o figure_n to_o a_o consequent_a of_o the_o second_o figure_n have_v mutual_o the_o same_o proportion_n which_o the_o side_n eglantine_n have_v to_o the_o side_n ac_fw-la a_o antecedent_n of_o the_o second_o figure_n to_o a_o consequent_a of_o the_o first_o figure_n then_o be_v these_o two_o figure_n reciprocal_a they_o be_v call_v of_o some_o figure_n of_o mutual_a side_n and_o that_o undoubted_o not_o amiss_o nor_o unapt_o figure_n and_o to_o make_v this_o definition_n more_o plain_a campane_n and_o pestitarius_n and_o othersâ—_n thus_o put_v it_o reciprocal_a figure_n be_v when_o the_o side_n of_o other_o 〈◊〉_d mutual_o proportional_a as_o in_o the_o example_n and_o declaration_n before_o give_v among_o the_o barbarous_a they_o be_v call_v mutekesia_n reserve_v still_o the_o arabike_a word_n definition_n 3._o a_o right_a line_n be_v say_v to_o be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n when_o the_o whole_a be_v to_o the_o great_a part_n as_o the_o great_a part_n be_v to_o the_o less_o as_o if_o the_o line_n ab_fw-la be_v so_o divide_v in_o the_o point_n c_o that_o the_o whole_a line_n ab_fw-la have_v the_o same_o proportion_n to_o the_o great_a part_n thereof_o namely_o to_o ac_fw-la which_o the_o same_o great_a part_n ac_fw-la have_v to_o the_o less_o part_n thereof_o namely_o to_o cb_o then_o be_v the_o line_n ab_fw-la divide_v by_o a_o extreme_a and_o mean_a proportion_n common_o it_o be_v call_v a_o line_n divide_v by_o proportion_n haâ—ing_v a_o mean_a and_o two_o extreme_n how_o to_o divide_v a_o line_n in_o such_o sort_n be_v teach_v in_o the_o 11._o proposition_n of_o the_o second_o book_n but_o not_o under_o this_o form_n of_o proportion_n 4._o the_o alitude_n of_o a_o figure_n be_v a_o perpendicular_a line_n draw_v from_o the_o top_n to_o the_o base_a definition_n as_o the_o altitude_n or_o height_n of_o the_o triangle_n abc_n be_v the_o line_n ad_fw-la be_v draw_v perpendicular_o from_o the_o point_n a_o be_v the_o top_n or_o high_a part_n of_o the_o triangle_n to_o the_o base_a thereof_o bc._n so_o likewise_o in_o other_o figure_n as_o you_o see_v in_o the_o example_n here_o set_v that_o which_o here_o â—ee_n call_v the_o altitude_n or_o height_n of_o a_o figure_n in_o the_o first_o book_n in_o the_o 35._o proposition_n and_o certain_a other_o follow_v he_o teach_v to_o be_v contain_v within_o two_o equidistant_a line_n so_o that_o figure_v to_o have_v one_o altitude_n and_o to_o be_v contain_v within_o two_o equidistant_a line_n be_v all_o one_o so_o in_o all_o these_o example_n if_o from_o the_o high_a point_n of_o the_o figure_n you_o draw_v a_o equidistant_a line_n to_o the_o base_a thereof_o and_o then_o from_o that_o point_n draw_v a_o perpendicular_a to_o the_o same_o base_a that_o perpendicular_a be_v the_o altitude_n of_o the_o figure_n 5._o a_o proportion_n be_v say_v to_o be_v make_v of_o two_o proportion_n or_o more_o when_o the_o quantity_n of_o the_o proportion_n multiply_v the_o one_o into_o the_o other_o produce_v a_o other_o quantity_n definition_n an_o other_o example_n where_o the_o great_a inequality_n and_o the_o less_o inequality_n be_v mix_v together_o 6._o 4._o 2._o 3._o the_o denomination_n of_o the_o proportion_n of_o 6._o to_o 4_o be_v 1_o â—_o â—_o example_n of_o 4._o to_o 2_o be_v â—_o â—_o and_o of_o 2._o to_o 3_o be_v â—_o â—_o now_o if_o you_o multiply_v as_o you_o ought_v all_o these_o denomination_n together_o you_o shall_v produce_v 12._o to_o 6_o namely_o dupla_fw-la proportion_n forasmuch_o as_o so_o much_o have_v hitherto_o be_v speak_v of_o addition_n of_o proportion_n it_o shall_v not_o be_v unnecessary_a somewhat_o also_o to_o say_v of_o substraction_n of_o they_o proportion_n where_o it_o be_v to_o be_v note_v that_o as_o addition_n of_o they_o be_v make_v by_o multiplication_n of_o their_o denomination_n the_o one_o into_o the_o other_o so_o be_v the_o substraction_n of_o the_o one_o from_o the_o other_o do_v by_o division_n of_o the_o denomination_n of_o the_o one_o by_o the_o denomination_n of_o the_o other_o as_o if_o you_o will_v from_o sextupla_fw-la proportion_n subtrahe_fw-la dupla_fw-la proportion_n take_v
and_o the_o self_n same_o top_n namely_o e_o and_o therefore_o be_v under_o one_o and_o the_o same_o altitude_n and_o by_o the_o same_o reason_n as_o the_o triangle_n cde_o be_v to_o the_o triangle_n ade_n so_o be_v the_o line_n ce_fw-fr to_o the_o line_n ea_fw-la wherefore_o by_o the_o 11._o of_o the_o five_o as_o the_o line_n bd_o be_v to_o the_o line_n dam_fw-ge so_o be_v the_o line_n ce_fw-fr to_o the_o line_n ea_fw-la but_o now_o suppose_v that_o in_o the_o triangle_n abc_n the_o side_n ab_fw-la &_o ac_fw-la be_v cut_v proportional_o so_o that_o as_o bd_o be_v to_o da_z so_o let_v ce_fw-fr be_v to_o ea_fw-la &_o draw_v a_o line_n from_o d_z to_o e._n then_o second_o i_o say_v that_o the_o line_n de_fw-fr be_v a_o parallel_n to_o the_o line_n bc._n part_n for_o the_o same_o order_n of_o construction_n be_v keep_v for_o that_o as_z bd_o be_v to_o da_z so_o be_v ce_fw-fr to_o ea_fw-la but_o as_o bd_o be_v to_o da_z so_o be_v the_o triangle_n bde_v to_o the_o triangle_n ade_n by_o the_o 1._o of_o the_o six_o &_o as_o ce_fw-fr be_v to_o ea_fw-la so_o by_o the_o same_o be_v the_o triangle_n cde_o to_o the_o triangle_n ade_n therefore_o by_o the_o 11._o of_o the_o five_o as_o the_o triangle_n bde_v be_v to_o the_o triangle_n ade_n so_o be_v the_o triangle_n cde_o to_o the_o triangle_n ade_n wherefore_o either_o of_o these_o triangle_n bde_v and_o cde_o have_v to_o the_o triangle_n ade_n one_o and_o the_o same_o proportion_n wherefore_o by_o the_o 9_o of_o the_o five_o the_o triangle_n bde_v be_v equal_a unto_o the_o triangle_n cde_o and_o they_o be_v upon_o one_o and_o the_o self_n base_a namely_o de._n but_o triangle_n equal_a and_o set_v upon_o one_o base_a be_v also_o contain_v within_o the_o same_o parallel_a line_n by_o the_o 39_o of_o the_o first_o wherefore_o the_o line_n de_fw-fr be_v unto_o the_o line_n bc_o a_o parallel_n if_o therefore_o to_o any_o one_o of_o the_o side_n of_o a_o triangle_n be_v draw_v a_o parallel_a line_n it_o cut_v the_o other_o side_n of_o the_o same_o triangle_n proportional_o and_o if_o the_o side_n of_o a_o triangle_n be_v cut_v proportional_o a_o right_a line_n draw_v from_o section_n to_o section_n be_v parallel_n to_o the_o other_o side_n of_o the_o triangle_n which_o thing_n be_v require_v to_o be_v demonstrate_v ¶_o here_o also_o flussates_n add_v a_o corollary_n if_o a_o line_n parallel_n to_o one_o of_o the_o side_n of_o a_o triangle_n do_v cut_v the_o triangle_n it_o shall_v cut_v of_o from_o the_o whole_a triangle_n a_o triangle_n like_a to_o the_o whole_a triangle_n flussates_n for_o as_o it_o have_v be_v prove_v it_o divide_v the_o side_n proportional_o so_o that_o as_o ec_o be_v to_o ea_fw-la so_o be_v bd_o to_o dam_fw-ge wherefore_o by_o the_o 18._o of_o the_o five_o as_z ac_fw-la be_v to_o ae_n so_o be_v ab_fw-la to_o ad._n wherefore_o alternate_o by_o the_o 16._o of_o the_o five_o as_o ac_fw-la be_v to_o ab_fw-la so_o be_v ae_n to_o ad_fw-la wherefore_o in_o the_o two_o triangle_n ead_n and_o cab_n the_o side_n about_o the_o common_a angle_n a_o be_v proportional_a the_o say_v triangle_n also_o be_v equiangle_n for_o forasmuch_o as_o the_o right_a line_n aec_o and_o adb_o do_v fall_v upon_o the_o parallel_a line_n ed_z and_o cb_o therefore_o by_o the_o 29._o of_o the_o firsâ—_o they_o make_v the_o angle_n aed_n and_o ade_n in_o the_o triangle_n ade_n equal_a to_o the_o angle_n acb_o and_o abc_n in_o the_o triangle_n acb_o wherefore_o by_o the_o first_o definition_n of_o this_o book_n the_o whole_a triangle_n abc_n be_v like_a unto_o the_o triangle_n cut_v of_o ade_n the_o 3._o theorem_a the_o 3._o proposition_n if_o a_o angle_n of_o a_o triangle_n be_v divide_v into_o two_o equal_a part_n and_o if_o the_o right_a line_n which_o divide_v the_o angle_n divide_v also_o the_o base_a the_o segment_n of_o the_o base_a shall_v be_v in_o the_o same_o proportion_n the_o one_o to_o the_o other_o that_o the_o other_o side_n of_o the_o triangle_n be_v and_o if_o the_o segmente_n of_o the_o base_a be_v in_o the_o same_o proportion_n that_o the_o other_o side_n of_o the_o say_a triangle_n be_v a_o right_n draw_v from_o the_o top_n of_o the_o triangle_n unto_o the_o section_n shall_v divide_v the_o angle_n of_o the_o triangle_n into_o two_o equal_a part_n svppose_v that_o there_o be_v a_o triangle_n abc_n and_o by_o the_o 9_o of_o the_o first_o let_v the_o angle_n bac_n be_v divide_v into_o two_o equal_a part_n by_o the_o right_a line_n ad_fw-la which_o let_v cut_v also_o the_o base_a bc_o in_o the_o point_n d._n then_o i_o say_v that_o as_o the_o segment_n bd_o be_v to_o the_o segment_n dc_o so_o be_v the_o side_n basilius_n to_o the_o side_n ac_fw-la construction_n for_o by_o the_o point_n c_o by_o the_o 31._o of_o the_o first_o draw_v unto_o the_o line_n da_z a_o parallel_a line_n ce_fw-fr and_o extend_v the_o line_n basilius_n till_o it_o concur_v with_o the_o line_n ce_fw-fr in_o the_o point_n e_o and_o do_v make_v the_o triangle_n bec_n part_n but_o the_o line_n basilius_n shall_v concur_v with_o the_o line_n ce_fw-fr by_o the_o 5._o petition_n for_o that_o the_o angle_n ebc_n and_o be_v be_v less_o than_o two_o right_a angle_n for_o the_o angle_n ecb_n be_v equal_a to_o the_o outward_a and_o opposite_a angle_n adb_o by_o the_o 29._o of_o the_o first_o and_o the_o two_o angle_n adb_o and_o dba_n of_o the_o triangle_n bad_a be_v less_o than_o two_o right_a angle_n by_o the_o 17._o of_o the_o first_o now_o forasmuch_o as_o upon_o the_o parallel_n ad_fw-la and_o ec_o fall_v the_o right_a line_n ac_fw-la therefore_o by_o the_o 29._o of_o the_o first_o the_o angle_n ace_n be_v equal_a unto_o the_o angle_n god_n but_o unto_o the_o angle_n god_n be_v the_o angle_n bid_v suppose_v to_o be_v equal_a wherefore_o the_o angle_n bad_a be_v also_o equal_a unto_o the_o angle_n ace_n again_o because_o upon_o the_o parallel_n ad_fw-la and_o ec_o fall_v the_o right_a line_n bae_o the_o outward_a angle_n bad_a by_o the_o 28._o of_o the_o first_o be_v equal_a unto_o the_o inward_a angle_n aec_o but_o before_o it_o be_v provell_v that_o the_o angle_n ace_n be_v equal_a unto_o the_o angle_v bad_a wherefore_o the_o angle_n ace_n be_v equal_a unto_o the_o angle_v aec_o wherefore_o by_o the_o 6._o of_o the_o first_o the_o side_z ae_n be_v equal_a unto_o the_o side_n ac_fw-la and_o because_o to_o one_o of_o the_o side_n of_o the_o triangle_n be_v namely_o to_o ec_o be_v draw_v a_o parallel_a line_n ad_fw-la therefore_o by_o the_o 2._o of_o the_o six_o as_o bd_o be_v to_o dc_o so_o be_v basilius_n to_o ae_n but_o ae_n be_v equal_a unto_o ac_fw-la therefore_o as_o bd_o be_v to_o dc_o so_o be_v basilius_n to_o ac_fw-la but_o now_o suppose_v that_o as_o the_o segment_n bd_o be_v to_o the_o segment_n dc_o first_o so_o be_v the_o side_n basilius_n to_o the_o side_n ac_fw-la &_o draw_v a_o line_n from_o a_o to_o d._n then_o i_o say_v that_o the_o angle_n bac_n be_v by_o the_o right_a line_n ad_fw-la divide_v into_o two_o equal_a part_n for_o the_o same_o order_n of_o construction_n remain_v for_o that_o as_o bd_o be_v to_o dc_o so_o be_v basilius_n to_o ac_fw-la but_o as_o bd_o be_v to_o dc_o so_o be_v basilius_n to_o ae_n by_o the_o 2._o of_o the_o six_o for_o unto_o one_o of_o the_o side_n of_o the_o triangle_n be_v namely_o unto_o the_o side_n ec_o be_v draw_v a_o parallel_a line_n ad._n wherefore_o also_o as_o basilius_n be_v to_o ac_fw-la so_o be_v basilius_n to_o ae_n by_o the_o 11._o of_o the_o five_o wherefore_o by_o the_o 9_o of_o the_o five_o ac_fw-la be_v equal_a unto_o ae_n wherefore_o also_o by_o the_o 5._o of_o the_o first_o the_o angle_n aec_o be_v equal_a unto_o the_o angle_n ace_n but_o the_o angle_n aec_o by_o the_o 29._o of_o the_o first_o be_v equal_a unto_o the_o outward_a angle_n bad_a and_o the_o angle_n ace_n be_v equal_a unto_o the_o angle_n god_n which_o be_v alternate_a unto_o he_o wherefore_o the_o angle_n bad_a be_v equal_a unto_o the_o angle_n god_n wherefore_o the_o angle_n bac_n be_v by_o the_o right_a line_n ad_fw-la divide_v into_o two_o equal_a part_n wherefore_o if_o a_o angle_n of_o a_o triangle_n be_v divide_v into_o two_o equal_a part_n and_o if_o the_o right_a line_n which_o divide_v the_o angle_n cut_v also_o the_o base_a the_o segment_n of_o the_o base_a shall_v be_v in_o the_o same_o proportion_n the_o one_o to_o the_o other_o that_o the_o other_o side_n of_o the_o say_a triangle_n be_v and_o if_o the_o segment_n of_o the_o base_a be_v in_o the_o same_o proportion_n that_o the_o other_o side_n of_o the_o say_a triangle_n be_v a_o right_a line_n draw_v from_o the_o top_n of_o the_o
figure_n comprehend_v under_o the_o line_n gb_o and_o ge_z be_v equal_a to_o the_o square_n make_v of_o the_o line_n agnostus_n by_o the_o 36._o of_o the_o three_o it_o follow_v that_o the_o touch_n line_n ga_o be_v a_o mean_a proportional_a between_o the_o extreme_n gb_o and_o ge_z by_o the_o second_o part_n of_o the_o 17._o of_o the_o six_o for_o that_o by_o that_o proposition_n the_o line_n gb_o ga_o and_o ge_z be_v proportional_a and_o by_o the_o same_o reason_n may_v it_o be_v prove_v that_o the_o line_n ga_o be_v a_o mean_a proportional_a between_o the_o line_n gd_o and_o gc_o and_o so_o of_o all_o other_o if_o therefore_o from_o a_o point_n gevenâ—_n &_o câ—_n which_o be_v require_v to_o be_v demonstrate_v the_o end_n of_o the_o six_o book_n of_o euclides_n element_n ¶_o the_o seven_o book_n of_o euclides_n element_n hitherto_o in_o the_o six_o book_n before_o have_v euclid_n pass_v through_o and_o entreat_v of_o the_o element_n of_o geometry_n without_o the_o aid_n and_o succour_v of_o number_n but_o the_o matter_n which_o remain_v to_o be_v teach_v and_o to_o be_v speak_v of_o in_o these_o his_o geometrical_a book_n which_o follow_v as_o in_o the_o ten_o eleven_o and_o so_o forth_o he_o can_v by_o no_o mean_n full_o and_o clear_o make_v plain_a &_o demonstrate_v without_o the_o help_n and_o aid_n of_o number_n in_o the_o ten_o be_v entreat_v of_o line_n irrational_a and_o uncertain_a and_o that_o of_o many_o &_o sundry_a kind_n and_o in_o the_o eleven_o &_o the_o other_o follow_v he_o teach_v the_o nature_n of_o body_n and_o compare_v their_o side_n and_o line_n together_o all_o which_o for_o the_o most_o part_n be_v also_o irrational_a and_o as_o rational_a quantity_n and_o the_o comparison_n and_o proportion_n of_o they_o can_v be_v know_v nor_o exact_o try_v but_o by_o the_o mean_a of_o number_n in_o which_o they_o be_v first_o see_v and_o perceive_v even_o so_o likewise_o can_v irrational_a quantity_n be_v know_v and_o find_v out_o without_o number_n as_o straightness_n be_v the_o trial_n of_o crookedness_n and_o inequality_n be_v try_v by_o equality_n so_o be_v quantity_n irrational_a perceive_v and_o know_v by_o quantity_n rational_a which_o be_v â—irst_n and_o chief_o find_v among_o number_n wherefore_o in_o these_o three_o book_n follow_v be_v as_o it_o be_v in_o the_o midst_n of_o his_o element_n he_o be_v compel_v of_o necessity_n to_o entreat_v of_o number_n number_n although_o not_o so_o full_o as_o the_o nature_n of_o number_n require_v yet_o so_o much_o as_o shall_v seem_v to_o be_v fit_a and_o sufficient_o to_o serve_v for_o his_o purpose_n whereby_o be_v see_v the_o necessity_n that_o the_o one_o art_n namely_o geometry_n have_v of_o the_o other_o namely_o of_o arithmetic_n and_o also_o of_o what_o excellency_n and_o worthiness_n arithmetic_n be_v above_o geometry_n geometry_n in_o that_o geometry_n borrow_v of_o it_o principle_n aid_n and_o succour_n and_o be_v as_o it_o be_v maim_v with_o out_o it_o whereas_o arithmetic_n be_v of_o itself_o sufficient_a and_o nead_v not_o at_o all_o any_o aid_n of_o geometry_n but_o be_v absolute_a and_o perfect_a in_o itself_o and_o may_v well_o be_v teach_v and_o attain_a unto_o without_o it_o again_o the_o matter_n or_o subject_n where_o about_o geometry_n be_v occupy_v which_o be_v line_n figure_n and_o body_n be_v such_o as_o offer_v themselves_o to_o the_o sense_n as_o triangle_n square_n circle_n cube_n and_o other_o be_v see_v &_o judge_v to_o be_v such_o as_o they_o be_v by_o the_o sight_n but_o number_n which_o be_v the_o subject_n and_o matter_n of_o arithmetic_n fall_v under_o no_o sense_n nor_o be_v represent_v by_o any_o shape_n form_n or_o figure_n and_o therefore_o can_v be_v judge_v by_o any_o sense_n but_o only_o by_o consideration_n of_o the_o mind_n and_o understanding_n now_o thing_n sensible_a be_v far_o under_o in_o degree_n then_o be_v thing_n intellectual_a sensible_a and_o be_v of_o nature_n much_o more_o gross_a than_o they_o wherefore_o number_n as_o be_v only_o intellectual_a be_v more_o pure_a more_o immaterial_a and_o more_o subtle_a far_o then_o be_v magnitude_n and_o extend_v itself_o far_a for_o arithmetic_n not_o only_o aid_v geometry_n but_o minister_v principle_n and_o ground_n to_o many_o other_o nay_o rather_o to_o all_o other_o science_n and_o art_n science_n as_o to_o music_n astronomy_n natural_a philosophy_n perspective_n with_o other_o what_o other_o thing_n be_v in_o music_n entreat_v of_o than_o number_n contract_v to_o sound_n and_o voice_n in_o astronomy_n who_o without_o the_o knowledge_n of_o number_n can_v do_v any_o thing_n either_o in_o search_v out_o of_o the_o motion_n of_o the_o heaven_n and_o their_o course_n either_o in_o judge_v and_o foreshow_v the_o effect_n of_o they_o in_o natural_a philosophy_n it_o be_v of_o no_o small_a force_n the_o wise_a and_o best_a learned_a philosopher_n that_o have_v be_v as_o pythagoras_n timeus_n plato_n and_o their_o follower_n find_v out_o &_o teach_v most_o pithy_o and_o pure_o the_o secret_a and_o hide_a knowledge_n of_o the_o nature_n and_o condition_n of_o all_o thing_n by_o number_n and_o by_o the_o propriety_n and_o passion_n of_o they_o of_o what_o force_n number_n be_v in_o perspective_n let_v he_o declare_v and_o judge_v who_o have_v any_o thing_n travel_v therein_o yea_o to_o be_v short_a what_o can_v be_v worthy_o and_o with_o praise_n practise_v in_o common_a life_n of_o any_o man_n of_o any_o condition_n without_o the_o knowledge_n of_o number_n yea_o it_o have_v be_v teach_v of_o the_o chief_a amongst_o philosopher_n that_o all_o natural_a thing_n be_v frame_v and_o have_v their_o constitution_n of_o number_n boetius_fw-la say_v hoc_fw-la fuit_fw-la principal_n in_o animâ—_n câ—â—ditoris_fw-la exemplar_n arithmeti_n number_n say_v he_o be_v the_o principal_a example_n and_o patron_n in_o the_o mind_n of_o the_o creator_n of_o the_o world_n do_v not_o that_o great_a philosop_n timaus_n in_o his_o book_n timaus_n &_o also_o plato_n in_o his_o timâ—â—_n follow_v he_o show_v how_o the_o soul_n be_v compose_v of_o harmonical_a number_n and_o consonantes_fw-la of_o music_n number_n compass_v all_o thing_n and_o be_v after_o these_o man_n the_o be_v and_o very_a essence_n of_o all_o thing_n and_o minister_v aid_n and_o help_v as_o to_o all_o other_o knowledge_n so_o also_o no_o small_a to_o geometry_n which_o thing_n cause_v euclidâ—_n in_o the_o midst_n of_o his_o book_n of_o geometry_n to_o insert_v and_o place_n these_o three_o book_n of_o arithmetic_n as_o without_o the_o aid_n of_o which_o he_o can_v not_o well_o pass_v any_o father_n in_o this_o seven_o book_n he_o â—irst_n place_v the_o general_a principle_n and_o first_o ground_n of_o arithmetic_n and_o set_v the_o definition_n of_o number_n or_o multitude_n and_o the_o kind_n thereof_o as_o in_o the_o first_o book_n he_o do_v of_o magnitude_n and_o the_o kind_n and_o part_n thereof_o book_n after_o that_o he_o entreat_v of_o number_n general_o and_o of_o their_o part_n and_o search_v and_o demonstrate_v in_o general_a the_o most_o common_a passion_n and_o propriety_n of_o the_o same_o and_o chief_o of_o number_n prime_a or_o incommensurable_a and_o of_o number_n compose_v or_o commensurable_a and_o of_o their_o propriety_n and_o partly_o also_o of_o the_o comparison_n oâ—_n proportion_n of_o one_o number_n to_o a_o other_o ¶_o definition_n definition_n 1_o unity_n be_v that_o whereby_o every_o thing_n that_o be_v be_v say_v to_o be_v on_o as_o a_o point_n in_o magnitude_n be_v the_o least_o thing_n in_o magnitude_n and_o no_o magnitude_n at_o all_o &_o yet_o the_o ground_n and_o beginning_n of_o all_o magnitude_n even_o so_o be_v unity_n in_o multitude_n or_o number_n the_o least_o thing_n in_o number_n and_o no_o number_n at_o all_o and_o yet_o the_o ground_n and_o beginning_n of_o all_o number_n and_o therefore_o it_o be_v here_o in_o this_o place_n of_o euclid_n first_o define_v as_o in_o the_o first_o book_n for_o the_o like_a reason_n and_o cause_n be_v a_o point_n first_o define_v unity_n say_v euclid_n be_v that_o whereby_o every_o thing_n be_v say_v to_o be_v one_o that_o be_v unity_n be_v that_o whereby_o every_o thing_n be_v divide_v and_o separate_v from_o a_o other_o and_o remain_v on_o in_o itself_o pure_a and_o distinct_a from_o all_o other_o otherwise_o be_v not_o this_o unity_n thing_n whereby_o all_o thing_n be_v sejoin_v the_o on_o from_o the_o other_o all_o thing_n shall_v suffer_v mixtion_n and_o be_v in_o confusion_n and_o where_o confusion_n be_v there_o be_v no_o order_n nor_o any_o thing_n can_v be_v exact_o know_v either_o what_o it_o be_v or_o what_o be_v the_o nature_n and_o what_o be_v the_o property_n thereof_o unity_n therefore_o be_v that_o which_o make_v every_o thing_n to_o be_v that_o which_o it_o be_v boetius_fw-la say_v very_o apt_o
the_o square_a of_o the_o line_n a_o have_v not_o unto_o the_o square_n of_o the_o line_n b_o the_o same_o proportion_n that_o a_o square_a number_n have_v to_o a_o square_a number_n then_o i_o say_v that_o the_o line_n a_o and_o b_o be_v incommensurable_a in_o length_n for_o if_o the_o line_n a_o and_o b_o be_v commensurable_a in_o length_n than_o the_o square_a of_o the_o line_n a_o shall_v have_v unto_o the_o square_n of_o the_o line_n b_o the_o same_o proportion_n that_o a_o square_a number_n have_v to_o a_o square_a number_n by_o the_o first_o part_n of_o this_o proposition_n but_o by_o supposition_n it_o have_v not_o wherefore_o the_o line_n a_o and_o b_o be_v not_o commensurable_a in_o length_n proposition_n wherefore_o they_o be_v incomensurable_a in_o length_n wherefore_o square_n make_v of_o right_a line_n commensura_fw-la in_o length_n have_v that_o proportion_n the_o one_o to_o the_o other_o that_o a_o square_a number_n have_v to_o a_o square_a number_n and_o square_n which_o have_v that_o proportion_n the_o one_o to_o the_o other_o that_o a_o square_a number_n have_v to_o a_o squaâ—e_a number_n shall_v also_o have_v the_o side_n commensurable_a in_o length_n but_o square_n describe_v of_o right_a line_n incommensurable_a in_o length_n have_v not_o that_o proportion_n the_o one_o to_o the_o other_o that_o a_o square_a number_n have_v to_o a_o square_a number_n and_o square_n which_o have_v not_o that_o proportion_n the_o one_o to_o the_o other_o that_o a_o square_a number_n have_v to_o a_o square_a number_n have_v not_o also_o their_o side_n commensurable_a in_o length_n which_o be_v all_o that_o be_v require_v to_o be_v prove_v ¶_o corollary_n hereby_o it_o be_v manifest_a that_o right_a line_n commensurable_a in_o length_n be_v also_o ever_o commensurable_a in_o power_n but_o right_a line_n commensurable_a in_o power_n be_v not_o always_o commensurable_a in_o length_n corollary_n and_o right_a line_n incommensurable_a in_o length_n be_v not_o always_o incommensurable_a in_o power_n but_o right_a line_n incommensurable_a in_o power_n be_v ever_o also_o incommensurable_a in_o length_n for_o forasmuch_o as_o square_n make_v of_o right_a line_n commensurable_a in_o length_n have_v that_o proportion_n the_o one_o to_o the_o other_o corollary_n that_o a_o square_a number_n have_v to_o a_o square_a number_n by_o the_o first_o part_n of_o this_o proposition_n but_o magnitude_n which_o have_v that_o proportion_n the_o one_o to_o the_o other_o that_o number_n simple_o have_v to_o number_n be_v by_o the_o six_o of_o the_o ten_o commensurable_a wherefore_o right_a line_n commensurable_a in_o length_n be_v commensurable_a not_o only_o in_o length_n but_o also_o in_o power_n part_n again_o forasmuch_o as_o there_o be_v certain_a square_n which_o have_v not_o that_o proportion_n the_o one_o to_o the_o other_o that_o a_o square_a number_n have_v to_o a_o square_a number_n but_o yet_o have_v that_o proportion_n the_o one_o to_o the_o other_o which_o number_n simple_o have_v to_o number_v their_o side_n in_o deed_n be_v in_o power_n commensurable_a for_o that_o they_o describe_v square_n which_o have_v that_o proportion_n which_o number_n simple_o have_v to_o number_n which_o square_n be_v therefore_o commensurable_a by_o the_o 6._o of_o this_o book_n but_o the_o say_v side_n be_v incommensurable_a in_o length_n by_o the_o latter_a part_n of_o this_o proposition_n wherâ—fore_o it_o be_v true_a that_o line_n commensurable_a in_o power_n be_v not_o straight_o way_n commensurable_a in_o length_n also_o pâ—rt_n and_o by_o the_o selâ—e_a same_o reason_n be_v prove_v also_o that_o three_o part_n of_o the_o corollary_n that_o line_n incommensurable_a in_o length_n be_v not_o always_o incommensurable_a in_o power_n for_o they_o may_v be_v incommensurable_a in_o length_n but_o yet_o commensurable_a in_o power_n as_o in_o those_o square_n which_o be_v in_o proportion_n the_o one_o to_o the_o other_o as_o number_n be_v to_o number_n but_o not_o as_o a_o square_a number_n be_v to_o a_o square_a number_n part_n but_o right_a line_n incommensurable_a in_o power_n be_v always_o also_o incommensurable_a in_o length_n for_o iâ—_z they_o be_v commensurable_a in_o length_n they_o shall_v also_o be_v commensurable_a in_o power_n by_o the_o first_o part_n of_o this_o corollary_n but_o they_o be_v suppose_v to_o be_v incommensurable_a in_o length_n which_o be_v absurd_a wherefore_o right_a line_n incommensurable_a in_o power_n be_v ever_o incommensurable_a in_o lengthy_a for_o the_o better_a understanding_n of_o this_o proposition_n and_o the_o other_o follow_v i_o have_v here_o add_v certain_a annotation_n take_v out_o of_o montaureus_n montauâ—â—us_fw-la and_o first_o as_o touch_v the_o signification_n oâ—_n word_n and_o term_n herein_o use_v whâ—ch_a arâ—_n such_o that_o unless_o they_o be_v well_o mark_v and_o poise_v the_o matter_n will_v be_v obscure_a and_o hard_a and_o in_o a_o manner_n inexplicable_a first_o this_o you_o must_v note_v that_o line_n to_o be_v commensurable_a in_o length_n and_o line_n to_o be_v in_o proportion_n the_o one_o to_o the_o other_o as_o number_n be_v to_o number_n be_v all_o one_o so_o that_o whatsoever_o line_n be_v commensurable_a in_o length_n be_v also_o in_o proportion_n the_o one_o to_o the_o other_o as_o number_n be_v to_o number_v and_o converse_o what_o so_o ever_o line_n be_v in_o proportion_n the_o one_o to_o the_o other_o as_o number_n be_v to_o number_n be_v also_o commensurable_a in_o length_n as_o it_o be_v manifest_a by_o the_o 5_o and_o 6_o of_o this_o book_n likewise_o line_n to_o be_v incommensurable_a in_o length_n and_o not_o to_o be_v in_o proportion_n the_o one_o to_o the_o other_o as_o number_n be_v to_o number_n be_v all_o one_o as_o it_o be_v manifest_a by_o the_o 7._o and_o 8._o of_o this_o book_n wherefore_o that_o which_o be_v say_v in_o this_o theorem_a aught_o to_o be_v understand_v of_o line_n commensurable_a in_o length_n and_o incommensurable_a in_o length_n this_o moreover_o be_v to_o be_v note_v that_o it_o be_v not_o all_o one_o numbers_z to_o be_v square_a number_n and_o to_o be_v in_o proportion_n the_o one_o to_o the_o other_o as_o a_o square_a number_n be_v to_o a_o square_a number_n for_o although_o square_a number_n be_v in_o proportion_n the_o one_o to_o the_o other_o as_o a_o square_a number_n be_v to_o a_o square_a number_n yet_o be_v not_o all_o those_o number_n which_o be_v in_o proportion_n the_o one_o to_o the_o other_o as_o a_o square_a number_n be_v to_o a_o square_a number_n square_a number_n for_o they_o may_v be_v like_o superficial_a number_n and_o yet_o not_o square_a number_n which_o yet_o be_v in_o proportion_n the_o one_o to_o the_o other_o as_o a_o square_a number_n be_v to_o a_o square_a number_n although_o all_o square_a number_n be_v like_o superficial_a number_n for_o between_o two_o square_a number_n there_o fall_v one_o mean_a proportional_a number_n by_o the_o 11._o of_o the_o eight_o but_o if_o between_o two_o number_n there_o fall_v one_o mean_v proportional_a number_n those_o two_o number_n be_v like_o superficial_a number_n by_o the_o 20._o of_o the_o eight_o so_o also_o if_o two_o number_n be_v in_o proportion_n the_o one_o to_o the_o other_o as_o a_o square_a number_n be_v to_o a_o square_a number_n they_o shall_v be_v like_o superficial_a number_n by_o the_o first_o corollary_n add_v after_o the_o last_o proposition_n of_o the_o eight_o book_n and_o now_o to_o know_v whether_o two_o superficial_a number_n give_v be_v like_o superficial_a number_n or_o no_o no._n it_o be_v thus_o find_v out_o first_o if_o between_o the_o two_o number_n give_v there_o fall_v no_o mean_a proportional_a then_o be_v not_o these_o two_o number_n like_o superficial_a number_n by_o the_o 18._o of_o the_o eight_o but_o if_o there_o do_v fall_v between_o they_o a_o mean_v proportional_a then_o be_v they_o like_v superâ—iciall_a number_n by_o the_o 20._o of_o the_o eight_o moreover_o two_o like_o superficial_a number_n multiply_v the_o one_o into_o the_o other_o do_v produce_v a_o square_a number_n by_o the_o firsâ—_o of_o the_o nine_o wherefore_o if_o they_o do_v not_o produce_v a_o square_a number_n then_o be_v they_o not_o like_o superficial_a number_n and_o if_o the_o one_o be_v multiply_v into_o the_o other_o they_o produce_v a_o square_a number_n then_o be_v they_o like_v superficial_a by_o the_o 2._o of_o the_o nine_o moreover_o if_o the_o say_v two_o superficial_a number_n be_v in_o superperticular_a or_o superbipartient_fw-fr proportion_n then_o be_v they_o not_o like_o superficial_a number_n for_o if_o they_o shall_v be_v like_a then_o shall_v there_o be_v a_o mean_a proportional_a between_o they_o by_o the_o 20._o of_o the_o eight_o but_o that_o be_v contrary_a to_o the_o corollary_n of_o the_o 20._o of_o the_o eight_o and_o the_o easy_o to_o conceive_v the_o demonstration_n follow_v take_v this_o example_n of_o that_o which_o we_o have_v say_v ¶_o
which_o be_v contain_v under_o the_o line_n ad_fw-la and_o db_o twice_o by_o a_o rational_a superficies_n for_o either_o of_o they_o be_v rational_a wherefore_o also_o the_o square_n of_o the_o line_n ac_fw-la and_o cb_o add_v together_o exceed_v the_o square_n of_o the_o line_n ad_fw-la and_o db_o add_v together_o by_o a_o rational_a superficies_n when_o yet_o each_o of_o they_o be_v a_o medial_a superficies_n which_o be_v impossible_a wherefore_o a_o line_n contain_v in_o power_n a_o rational_a and_o a_o medial_a be_v in_o one_o point_n only_o divide_v into_o his_o name_n which_o be_v require_v to_o be_v demonstrate_v ¶_o the_o 35._o theorem_a the_o 47._o proposition_n a_o line_n contain_v in_o power_n two_o medial_n be_v in_o one_o point_n only_o divide_v into_o his_o name_n svppose_v that_o ab_fw-la be_v a_o line_n contain_v in_o power_n two_o medial_n be_v divide_v into_o his_o name_n in_o the_o point_n c_o so_fw-mi that_o let_v the_o line_n ac_fw-la and_o cb_o be_v incommensurable_a in_o power_n have_v that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ac_fw-la &_o cb_o medial_a and_o that_o also_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o medial_a and_o moreover_o incommensurable_a â—o_o that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ac_fw-la and_o cb._n then_o i_o say_v construction_n that_o the_o line_n ab_fw-la can_v in_o no_o other_o point_n be_v divide_v into_o his_o name_n but_o only_o in_o the_o point_n c._n for_o if_o it_o be_v possible_a let_v it_o be_v divide_v into_o his_o name_n in_o the_o point_n d_o so_fw-mi that_o let_v not_o the_o line_n ac_fw-la be_v one_o and_o the_o same_o that_o be_v equal_a with_o the_o line_n db_o but_o by_o supposition_n let_v the_o line_n ac_fw-la be_v the_o great_a and_o take_v a_o rational_a line_n ef._n and_o by_o the_o 43._o of_o the_o first_o upon_o the_o line_n of_o apply_v a_o rectangle_n parallelogram_n eglantine_n equal_a to_o that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ac_fw-la and_o cb_o and_o likewise_o upon_o the_o line_n hc_n which_o be_v equal_a to_o the_o line_n of_o apply_v the_o parallelogram_n hk_o equal_a to_o that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o twice_o wherefore_o the_o whole_a parallelogram_n eke_o be_v equal_a to_o the_o square_n of_o the_o line_n ab_fw-la absurdity_n again_o upon_o the_o same_o line_n of_o describe_v the_o parallelogram_n el_n equal_a to_o the_o square_n of_o the_o line_n ad_fw-la and_o db._n wherefore_o the_o residue_n namely_o that_o which_o be_v contain_v under_o the_o line_n ad_fw-la and_o db_o twice_o be_v equal_a to_o the_o parallelogram_n remain_v namely_o to_o mk_o and_o forasmuch_o as_o that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ac_fw-la and_o cb_o be_v by_o supposition_n medial_a therefore_o the_o parallelogram_n eglantine_n which_o be_v equal_a unto_o it_o be_v also_o medial_a and_o it_o be_v apply_v upon_o the_o rational_a line_n ef._n wherefore_o by_o the_o 22._o of_o the_o ten_o the_o line_n he_o be_v rational_a and_o incommensurable_a in_o length_n unto_o the_o line_n ef._n and_o by_o the_o same_o reason_n also_o the_o line_n hn_o be_v rational_a and_o incommensurable_a in_o length_n to_o the_o same_o line_n ef._n and_o forasmuch_o as_o that_o which_o be_v compose_v of_o the_o square_n of_o the_o line_n ac_fw-la and_o cb_o be_v incommensurable_a to_o that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o twice_o for_o it_o be_v suppose_v to_o be_v incommensurable_a to_o that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o once_o therefore_o the_o parallelogram_n eglantine_n be_v incommensurable_a to_o the_o parallelogram_n h_o â—_o wherefore_o the_o line_n eh_o also_o be_v incommensurable_a in_o length_n to_o the_o line_n hn_o and_o they_o be_v rational_a line_n wherefore_o the_o line_n eh_o and_o hn_o be_v rational_a commensurable_a in_o power_n only_o wherefore_o the_o whole_a line_n en_fw-fr be_v a_o binomial_a line_n and_o be_v divide_v into_o his_o name_n in_o the_o point_n h._n and_o in_o like_a sort_n may_v we_o prove_v that_o the_o same_o binomial_a line_n en_fw-fr be_v divide_v into_o his_o name_n in_o the_o point_n m_o and_o that_o the_o line_n eh_o be_v not_o one_o and_o the_o same_o that_o be_v equal_a with_o the_o line_n mn_o as_o it_o be_v prove_v in_o the_o end_n of_o the_o demonstration_n of_o the_o 44._o of_o this_o book_n wherefore_o a_o binomial_a line_n be_v divide_v into_o his_o name_n in_o two_o sundry_a point_n which_o be_v impossible_a by_o the_o 42._o of_o the_o ten_o wherefore_o a_o line_n contain_v in_o power_n two_o medial_n be_v not_o in_o sundry_a point_n divide_v into_o his_o name_n wherefore_o it_o be_v divide_v in_o one_o point_n only_o which_o be_v require_v to_o be_v demonstrate_v ¶_o second_a definition_n it_o be_v show_v before_o that_o of_o binomial_a line_n there_o be_v six_o kind_n line_n the_o definition_n of_o all_o which_o be_v here_o now_o set_v and_o be_v call_v second_o definition_n all_o binomial_a line_n as_o all_o other_o kind_n of_o irrational_a line_n be_v conceive_v consider_v and_o perfect_o understand_v only_o in_o respect_n of_o a_o rational_a line_n who_o part_n as_o before_o be_v teach_v be_v certain_a and_o know_v and_o may_v be_v distinct_o express_v by_o number_n unto_o which_o line_n they_o be_v compare_v this_o rationalâ—_fw-la line_n must_v you_o ever_o have_v before_o your_o eye_n in_o all_o these_o definition_n so_o shall_v they_o all_o be_v â—asie_a enough_o paâ—tâ—s_n a_o binomial_a line_n you_o know_v be_v make_v of_o two_o part_n or_o name_n whereof_o the_o one_o be_v great_a than_o the_o other_o wherefore_o the_o power_n or_o square_v also_o of_o the_o one_o be_v great_a than_o the_o power_n or_o square_a oâ—_n the_o other_o the_o three_o first_o kind_n of_o binomial_a line_n namely_o the_o first_o the_o seconâ—_o &_o the_o three_o be_v produce_v when_o the_o square_a of_o the_o great_a name_n or_o part_n of_o a_o binomâ—all_n eâ—cedeth_v the_o square_a of_o the_o less_o name_n or_o part_n by_o the_o square_n of_o a_o line_n which_o be_v commensurable_a in_o length_n to_o it_o namely_o to_o the_o great_a the_o three_o last_o kind_n namely_o the_o four_o the_o â—iâ—t_n and_o the_o six_o be_v produce_v when_o the_o square_a of_o the_o great_a name_n or_o part_n â—â—â—â—edeth_v the_o square_a of_o the_o less_o name_n or_o part_n by_o the_o square_n of_o a_o line_n incommensurable_a in_o length_n unto_o it_o that_o be_v to_o the_o great_a part_n definition_n a_o first_o binomial_a line_n be_v who_o square_a of_o the_o great_a part_n exceed_v the_o square_a of_o tâ—e_z less_o part_n â—y_a the_o square_n of_o a_o line_n commensurable_a in_o length_n to_o the_o great_a part_n and_o the_o great_a part_n be_v also_o commensurable_a in_o length_n to_o tâ—e_z rational_a line_n first_o set_v as_o lâ—t_v the_o rational_a line_n first_o set_v be_v ab_fw-la who_o part_n be_v distinct_o know_v suppose_v also_o that_o the_o line_n ce_fw-fr be_v a_o binomial_a line_n who_o name_n or_o part_n let_v be_v cd_o and_o de._n and_o let_v the_o square_n of_o the_o line_n cd_o the_o great_a part_n exceed_v the_o square_n of_o the_o line_n de_fw-fr the_o less_o part_n by_o the_o square_n of_o the_o line_n fg_o which_o line_n fg_o let_v bâ—e_o commensurable_a in_o length_n to_o the_o line_n cd_o which_o be_v the_o great_a part_n of_o the_o binomial_a line_n and_o moreover_o let_v the_o line_n cd_o the_o great_a paâ—t_n be_v commensurble_n in_o length_n to_o the_o rational_a line_n first_o set_v namely_o to_o ab_fw-la so_o by_o this_o dâ—â—inition_n the_o binomial_a line_n ce_fw-fr be_v a_o first_o binomial_a line_n definition_n a_o second_o binomial_a line_n be_v when_o the_o square_a of_o the_o great_a part_n exceed_v the_o square_a of_o the_o less_o part_n by_o the_o square_n of_o a_o line_n commensurable_a in_o length_n unto_o it_o and_o the_o less_o part_n be_v commensurable_a in_o length_n to_o the_o rational_a line_n first_o set_v as_o suppose_v ever_o the_o rational_a line_n let_v ce_fw-fr be_v a_o binomial_a line_n divide_v in_o the_o point_n d._n the_o square_n of_o who_o great_a part_n cd_o let_v exceed_v the_o square_n of_o the_o less_o part_n de_fw-fr by_o the_o square_n of_o the_o line_n fg_o which_o line_n â—g_z let_v be_v commensurable_a in_o length_n unto_o the_o line_n cd_o tâ—e_z great_a pâ—â—â—_n oâ—_n the_o binomial_a line_n and_o let_v also_o the_o line_n de_fw-fr the_o less_o part_n of_o the_o binomial_a line_n be_v commensurable_a in_o length_n to_o the_o rational_a line_n first_o set_v ab_fw-la so_o by_o this_o definition_n the_o binomial_a line_n ce_fw-fr be_v a_o second_o binomial_a line_n â—â—â—â—â—â—ition_n a_o three_o binomial_a
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
to_o b_o and_o of_o the_o difference_n of_o b_o to_o c._n and_o by_o the_o same_o common_a sentence_n the_o difference_n of_o b_o to_o d_z be_v compose_v of_o the_o difference_n of_o b_o to_o c_z and_o of_o â—he_n difference_n of_o c_o to_o d._n and_o forasmuch_o as_o by_o supposition_n the_o difference_n of_o a_o to_o b_o be_v as_o the_o difference_n of_o c_o to_o d_z and_o the_o difference_n of_o b_o to_o c_z be_v common_a to_o they_o both_o wherefore_o it_o follow_v that_o as_o the_o difference_n of_o a_o be_v to_o c_o so_o be_v the_o difference_n of_o b_o to_o d_z which_o be_v require_v to_o be_v prove_v ¶_o the_o 61._o theorem_a the_o 79._o proposition_n unto_o a_o residual_a line_n can_v be_v join_v one_o only_o right_a line_n rational_a and_o commensurable_a in_o power_n only_o to_o the_o whole_a line_n senary_n let_v ab_fw-la be_v a_o residual_a line_n and_o unto_o it_o let_v the_o line_n bc_o be_v suppose_v to_o be_v join_v so_o that_o let_v the_o line_n ac_fw-la and_o bc_o be_v rational_a commensurable_a in_o power_n only_o then_o i_o say_v that_o unto_o the_o line_n ab_fw-la can_v be_v join_v any_o other_o rational_a line_n commensurable_a in_o power_n only_o to_o the_o whole_a line_n impossibility_n for_o if_o it_o be_v possible_a let_v bd_o be_v such_o a_o line_n add_v unto_o it_o wherefore_o the_o line_n ad_fw-la and_o db_o be_v rational_a commensurable_a in_o power_n only_o and_o forasmuch_o as_o how_o much_o the_o square_n of_o the_o line_n ad_fw-la and_o db_o do_v exceed_v that_o which_o be_v contain_v under_o the_o line_n ad_fw-la and_o db_o twice_o so_o much_o also_o do_v the_o square_n of_o the_o line_n ac_fw-la and_o cb_o exceed_v that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o twice_o for_o the_o excess_n of_o each_o be_v one_o and_o the_o same_o namely_o the_o square_a of_o the_o line_n ab_fw-la by_o the_o 7._o of_o the_o second_o wherefore_o alternate_o by_o the_o â—ormer_a assumpt_n of_o campanus_n how_o much_o the_o square_n of_o the_o line_n ad_fw-la and_o db_o do_v exceed_v the_o square_n of_o the_o line_n ac_fw-la &_o cb_o so_o much_o also_o exceed_v that_o which_o be_v contain_v under_o the_o line_n ad_fw-la and_o db_o twice_o that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o twice_o but_o that_o which_o be_v make_v of_o the_o square_n of_o the_o line_n ad_fw-la and_o db_o add_v together_o exceed_v that_o which_o be_v make_v of_o the_o square_n of_o the_o line_n ac_fw-la and_o cb_o add_v together_o by_o a_o rational_a superficies_n for_o they_o be_v either_o of_o they_o rational_a wherefore_o that_o which_o be_v contain_v under_o the_o line_n ad_fw-la and_o db_o twice_o exceed_v that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o twice_o by_o a_o rational_a superficies_n but_o that_o which_o be_v contain_v under_o the_o line_n ad_fw-la and_o db_o twice_o be_v medial_a for_o it_o be_v commensurable_a to_o that_o which_o be_v contain_v under_o the_o line_n ad_fw-la and_o db_o once_o which_o superficies_n be_v medial_a by_o the_o 21._o of_o the_o ten_o and_o by_o the_o same_o reason_n also_o that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o twice_o be_v medial_a wherefore_o a_o medial_a superficies_n differ_v from_o a_o medial_a superficies_n by_o a_o rational_a superficies_n which_o by_o the_o 26._o of_o the_o ten_o be_v impossible_a wherefore_o unto_o the_o line_n ab_fw-la can_v be_v join_v any_o other_o rational_a line_n beside_o bc_o commensurable_a in_o power_n only_o to_o the_o whole_a line_n wherefore_o unto_o a_o residual_a line_n can_v be_v join_v one_o only_o right_a line_n rational_a and_o commensurable_a in_o power_n only_o to_o the_o whole_a line_n which_o be_v require_v to_o be_v demonstrate_v ¶_o the_o 62._o theorem_a the_o 80._o proposition_n unto_o a_o first_o medial_n residual_a line_n can_v be_v join_v one_o only_o medial_a right_a line_n commensurable_a in_o power_n only_o to_o the_o whole_a line_n and_o comprehend_v with_o the_o whole_a line_n a_o rational_a superficies_n svppose_v that_o ab_fw-la be_v a_o first_o medial_a residual_a line_n &_o unto_o ab_fw-la join_v the_o line_n bc_o so_o that_o let_v the_o line_n ac_fw-la and_o bc_o be_v medial_a commensurable_a in_o power_n only_o &_o let_v that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o bc_o be_v rational_a then_o i_o say_v that_o unto_o the_o line_n ab_fw-la can_v be_v join_v any_o other_o medial_a line_n commensurable_a in_o power_n only_o to_o the_o whole_a line_n and_o comprehend_v together_o with_o the_o whole_a line_n a_o rational_a superficies_n absurdity_n for_o if_o it_o be_v possible_a let_v the_o line_n bd_o be_v such_o a_o line_n wherefore_o the_o line_n ad_fw-la and_o db_o be_v medial_a commensurable_a in_o power_n only_o and_o that_o which_o be_v contain_v under_o the_o line_n ad_fw-la and_o db_o be_v rational_a and_o forasmuch_o as_o how_o much_o the_o square_n of_o the_o line_n ad_fw-la and_o db_o exceed_v that_o which_o be_v contain_v under_o the_o line_n ad_fw-la and_o db_o twice_o so_o much_o also_o exceed_v the_o square_n of_o the_o line_n ac_fw-la &_o bc_o that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o twice_o for_o the_o excess_n of_o each_o be_v one_o and_o the_o same_o namely_o the_o square_a of_o the_o line_n ab_fw-la wherefore_o alternate_o as_o it_o be_v say_v in_o the_o former_a proposition_n how_o much_o the_o square_n of_o the_o line_n ad_fw-la and_o db_o exceed_v the_o square_n of_o the_o line_n ac_fw-la and_o cb_o so_o much_o also_o that_o which_o be_v contain_v under_o the_o line_n ad_fw-la and_o db_o twice_o exceed_v that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o twice_o but_o that_o which_o be_v contain_v under_o the_o line_n ad_fw-la and_o db_o twice_o exceed_v that_o which_o be_v contain_v under_o the_o line_n ac_fw-la and_o cb_o twice_o by_o a_o rational_a superficies_n for_o they_o be_v either_o of_o they_o a_o rational_a supersicy_n wherefore_o that_o which_o be_v make_v of_o the_o square_n of_o the_o line_n ad_fw-la &_o db_o exceed_v that_o which_o be_v make_v of_o the_o square_n of_o the_o line_n ac_fw-la &_o cb_o by_o a_o rational_a superficies_n which_o by_o the_o 26._o of_o the_o ten_o be_v impossible_a for_o they_o be_v either_o of_o they_o medial_a for_o those_o four_o line_n be_v put_v to_o be_v medial_a wherefore_o unto_o a_o first_o medial_a residual_a line_n can_v be_v join_v only_a one_o right_a medial_a line_n commensurable_a in_o power_n only_o to_o the_o whole_a line_n and_o comprehend_v with_o the_o whole_a line_n a_o rational_a superficies_n which_o be_v require_v to_o be_v prove_v ¶_o the_o 63._o theorem_a the_o 81._o proposition_n unto_o a_o second_o medial_a residual_a line_n can_v be_v join_v only_a one_o medial_a right_a line_n commensurable_a in_o power_n only_o to_o the_o whole_a line_n and_o comprehend_v with_o the_o whole_a line_n a_o medial_a superficies_n svppose_v that_o ab_fw-la be_v a_o second_o medial_a residual_a line_n &_o unto_o the_o line_n ab_fw-la join_v the_o line_n bc_o so_o that_o let_v the_o line_n ac_fw-la and_o cb_o be_v medial_a commensurable_a in_o power_n only_o and_o let_v that_o which_o be_v comprehend_v under_o the_o line_n ac_fw-la and_o cb_o be_v medial_a then_o i_o say_v that_o unto_o the_o line_n ab_fw-la can_v not_o be_v join_v any_o other_o medial_a right_a line_n commensurable_a in_o power_n only_o to_o the_o whole_a line_n and_o comprehend_v together_o with_o the_o whole_a line_n a_o medial_a superficies_n for_o if_o it_o be_v possible_a let_v the_o line_n bd_o be_v such_o a_o line_n wherefore_o the_o line_n ad_fw-la &_o db_o be_v medial_a commensurable_a in_o power_n only_o and_o that_o which_o be_v contain_v under_o the_o line_n ad_fw-la and_o db_o be_v also_o medial_a take_v a_o rational_a line_n ef._n construction_n and_o by_o the_o 44._o of_o the_o first_o unto_o the_o line_n of_o apply_v the_o parallelogram_n eglantine_n equal_a to_o the_o square_n of_o the_o line_n ac_fw-la and_o cb_o and_o make_v in_o breadth_n the_o line_n em_n and_o from_o that_o parallelogram_n eglantine_n take_v away_o the_o parallelogram_n hg_o equal_a to_o that_o which_o be_v contain_v under_o ac_fw-la and_o cb_o twice_o and_o make_v in_o breadth_n the_o line_n he_o wherefore_o the_o parallelogram_n remain_v namely_o el_n be_v by_o the_o 7._o of_o the_o second_o equal_a to_o the_o square_n of_o the_o line_n ab_fw-la wherefore_o the_o line_n ab_fw-la contain_v in_o power_n the_o parallelogram_n el._n again_o unto_o the_o line_n of_o apply_v by_o the_o 44._o of_o the_o first_o the_o parallelogram_n ei_o equal_a to_o the_o square_n of_o the_o line_n ad_fw-la and_o db_o and_o make_v in_o breadth_n the_o line_n en_fw
two_o other_o proposition_n go_v next_o before_o it_o so_o far_o misplace_v that_o where_o they_o be_v word_n for_o word_n before_o duâ—ly_o place_v be_v the_o 105._o and_o 106._o yet_o here_o after_o the_o book_n end_v they_o be_v repeat_v with_o the_o number_n of_o 116._o and_o 117._o proposition_n zambert_n therein_o be_v more_o faithful_a to_o follow_v as_o he_o find_v in_o his_o greek_n example_n than_o he_o be_v skilful_a or_o careful_a to_o do_v what_o be_v necessary_a yea_o and_o some_o greek_n write_v ancient_a copy_n have_v they_o not_o so_o though_o in_o deed_n they_o be_v well_o demonstrate_v yet_o truth_n disord_v be_v half_a disgracedâ—_n especial_o where_o the_o pattern_n of_o good_a order_n by_o profession_n be_v avouch_v to_o be_v but_o through_o ignorance_n arrogancy_n and_o â—emerltie_n of_o unskilful_a method_n master_n many_o thing_n remain_v yet_o in_o these_o geometrical_a element_n undue_o tumble_v in_o though_o true_a yet_o with_o disgrace_n which_o by_o help_n of_o so_o many_o wit_n and_o hability_n of_o such_o as_o now_o may_v have_v good_a cause_n to_o be_v skilful_a herein_o will_v i_o hope_v ere_o long_o be_v take_v away_o and_o thing_n of_o importance_n want_v supply_v the_o end_n of_o the_o ten_o book_n of_o euclides_n element_n ¶_o the_o eleven_o book_n of_o euclides_n element_n book_n hitherto_o have_v â—vclidâ—_n in_o thâ—sâ—_n former_a book_n with_o a_o wonderful_a method_n and_o order_n entreat_v of_o such_o kind_n of_o figure_n superficial_a which_o be_v or_o may_v be_v describe_v in_o a_o superficies_n or_o plain_a and_o have_v teach_v and_o set_v forth_o their_o property_n nature_n generation_n and_o production_n even_o from_o the_o first_o root_n ground_n and_o beginning_n of_o they_o namely_o from_o a_o point_n which_o although_o it_o be_v indivisible_a yet_o be_v it_o the_o begin_n of_o all_o quantity_n continual_a and_o of_o it_o and_o of_o the_o motion_n and_o slow_v thereof_o be_v produce_v a_o line_n and_o consequent_o all_o quantity_n continual_a as_o all_o figure_n plain_a and_o solid_a what_o so_o ever_o euclid_n therefore_o in_o his_o first_o book_n begin_v with_o it_o boot_n and_o from_o thence_o go_v he_o to_o a_o line_n as_o to_o a_o thing_n most_o simple_a next_o unto_o a_o point_n then_o to_o a_o superficies_n and_o to_o angle_n and_o so_o through_o the_o whole_a first_o book_n boâ—â—e_n he_o entreat_v of_o these_o most_o simple_a and_o plain_a ground_n in_o the_o second_o book_n he_o entreat_v further_o â—â—oâ—e_n and_o go_v unto_o more_o hard_a matter_n and_o teach_v of_o division_n of_o line_n and_o of_o the_o multiplication_n of_o line_n and_o of_o their_o part_n and_o of_o their_o passion_n and_o property_n and_o for_o that_o rightline_v â—_z be_v far_o distant_a in_o nature_n and_o property_n from_o round_a and_o circular_a figure_n in_o the_o three_o book_n he_o instruct_v the_o reader_n of_o the_o nature_n and_o condition_n of_o circle_n booâ—e_n in_o the_o four_o book_n he_o compare_v figure_n of_o right_a line_n and_o circle_n together_o bâ—oâ—e_n and_o teach_v how_o to_o describe_v a_o figure_n of_o right_a line_n with_o in_o or_o about_o a_o circle_n and_o contrariwise_o a_o circle_n with_o in_o or_o about_o a_o rectiline_a figure_n in_o the_o five_o book_n he_o search_v out_o the_o nature_n of_o proportion_n a_o matter_n of_o wonderful_a use_n and_o deep_a consideration_n boâ—â—e_n for_o that_o otherwise_o he_o can_v not_o compare_v â—igure_n with_o figure_n or_o the_o side_n of_o figure_n together_o for_o whatsoever_o be_v compare_v to_o any_o other_o thing_n be_v compare_v unto_o it_o undoubted_o under_o some_o kind_n of_o proportion_n wherefore_o in_o the_o six_o book_n he_o compare_v figure_n together_o booâ—e_n one_o to_o a_o other_o likewise_o their_o side_n and_o for_o that_o the_o nature_n of_o proportion_n can_v not_o be_v full_o and_o clear_o see_v without_o the_o knowledge_n of_o number_n wherein_o it_o be_v first_o and_o chief_o find_v in_o the_o seven_o bookâ—_n eight_o booâ—â—_n and_o nine_o book_n book_n he_o entreatâ—th_v of_o number_n &_o of_o the_o kind_n and_o property_n thereof_o and_o because_o that_o the_o side_n of_o solid_a body_n for_o the_o most_o part_n be_v of_o such_o sort_n that_o compare_v together_o they_o have_v such_o proportion_n the_o one_o to_o the_o other_o booâ—e_n which_o can_v not_o be_v express_v by_o any_o number_n certain_a and_o therefore_o be_v call_v irrational_a line_n he_o in_o the_o ten_o book_n have_v write_v &_o teach_v which_o lineâ—_z be_v commensurable_a or_o incommensurable_a the_o one_o to_o the_o other_o and_o of_o the_o diversity_n of_o kind_n of_o irrational_a line_n with_o all_o the_o condition_n &_o propriety_n of_o they_o and_o thus_o have_v euclid_n in_o these_o ten_o foresay_a book_n full_o &_o most_o plenteous_o in_o a_o marvelous_a order_n teach_v whatsoever_o seem_v necessary_a and_o requisite_a to_o the_o knowledge_n of_o all_o superficial_a figure_n of_o what_o sort_n &_o form_n so_o ever_o they_o be_v now_o in_o these_o book_n follow_v he_o entreat_v of_o figure_n of_o a_o other_o kind_n namely_o of_o bodily_a figure_n follow_v as_o of_o cube_n pyramid_n cones_n column_n cilinder_n parallelipipedon_n sphere_n and_o such_o othersâ—_n and_o show_v the_o diversity_n of_o they_o the_o generation_n and_o production_n of_o they_o and_o demonstrate_v with_o great_a and_o wonderful_a art_n their_o propriety_n and_o passion_n with_o all_o their_o nature_n and_o condition_n he_o also_o compare_v one_o oâ—_n they_o to_o a_o other_o whereby_o to_o know_v the_o reason_n and_o proportion_n of_o the_o one_o to_o the_o other_o chief_o of_o the_o five_o body_n which_o be_v call_v regular_a body_n element_n and_o these_o be_v the_o thing_n of_o all_o other_o entreat_v of_o in_o geometry_n most_o worthy_a and_o of_o great_a dignity_n and_o as_o it_o be_v the_o end_n and_o final_a intent_n of_o the_o whole_a be_v of_o geometry_n and_o for_o who_o cause_n have_v be_v write_v and_o speak_v whatsoever_o have_v hitherto_o in_o the_o former_a book_n be_v say_v or_o write_v as_o the_o first_o book_n be_v a_o ground_n and_o a_o necessary_a entry_n to_o all_o the_o râ—st_o â—ollowing_v so_o be_v this_o eleven_o book_n a_o necessary_a entry_n and_o ground_n to_o the_o rest_n which_o follow_v 〈◊〉_d and_o as_o that_o contain_v the_o declaration_n of_o word_n and_o definition_n of_o thinger_n requisite_a to_o the_o knowledge_n of_o superficial_a figure_n and_o entreat_v of_o line_n and_o of_o their_o division_n and_o section_n which_o be_v the_o term_n and_o limit_n of_o superficial_a figure_n so_o in_o this_o book_n be_v set_v forth_o the_o declaration_n of_o word_n and_o definition_n of_o thing_n pertain_v to_o solid_a and_o corporal_a figure_n and_o also_o of_o superficiece_n which_o be_v the_o term_n &_o limit_n of_o solid_n moreover_o of_o the_o division_n and_o intersection_n of_o they_o and_o diverse_a other_o thing_n without_o which_o the_o knowledge_n of_o bodily_a and_o solid_a form_n can_v not_o be_v attain_a unto_o and_o first_o be_v set_v the_o definition_n as_o follow_v definition_n a_o solid_a or_o body_n be_v that_o which_o have_v length_n breadth_n and_o thickness_n definition_n and_o the_o term_n or_o limit_v of_o a_o solid_a be_v a_o superficies_n there_o be_v three_o kind_n of_o continual_a quantity_n a_o line_n a_o superficies_n and_o a_o solid_a or_o body_n the_o beginning_n of_o all_o which_o as_o before_o have_v be_v say_v be_v a_o point_n which_o be_v indivisible_a two_o of_o these_o quantity_n namely_o a_o line_n and_o a_o superficies_n be_v define_v of_o euclid_n before_o in_o his_o first_o book_n but_o the_o three_o kind_n namely_o a_o solid_a or_o body_n he_o there_o define_v not_o as_o a_o thing_n which_o pertain_v not_o then_o to_o his_o purpose_n but_o here_o in_o this_o place_n he_o set_v the_o definition_n thereof_o as_o that_o which_o chief_o now_o pertain_v to_o his_o purpose_n and_o without_o which_o nothing_o in_o these_o thing_n can_v profitable_o be_v teach_v a_o solid_a say_v he_o be_v that_o which_o have_v length_n breadth_n and_o thickness_n or_o depth_n there_o be_v as_o before_o have_v be_v teach_v three_o reason_n or_o mean_n of_o measure_v which_o be_v call_v common_o dimension_n namely_o length_n breadth_n and_o thickness_n these_o dimension_n be_v ascribe_v unto_o quantity_n only_o by_o these_o be_v all_o kind_n of_o quantity_n define_v â—â—_o be_v count_v perfect_a or_o imperfect_a according_a as_o they_o be_v partaker_n of_o few_o or_o more_o of_o they_o as_o euclid_n define_v a_o line_n ascribe_v unto_o it_o only_o one_o of_o these_o dimension_n namely_o length_n wherefore_o a_o line_n be_v the_o imperfect_a kind_n of_o quantity_n in_o define_v of_o a_o superficies_n he_o ascribe_v unto_o it_o two_o dimension_n namely_o length_n and_o breadth_n whereby_o a_o superficies_n be_v a_o quantity_n of_o
but_o not_o every_o pyramid_n a_o tetrahedron_n and_o in_o deed_n psellus_n in_o number_v of_o these_o five_o solid_n or_o body_n call_v a_o tetrahedron_n a_o pyramid_n in_o manifest_a word_n pyramid_n this_o i_o say_v may_v make_v flussas_n &_o other_o as_o i_o think_v it_o do_v to_o omit_v the_o definition_n of_o a_o tetrahedron_n in_o this_o place_n as_o sufficient_o comprehend_v within_o the_o definition_n of_o a_o pyramid_n give_v before_o but_o why_o then_o do_v he_o not_o count_v that_o definition_n of_o a_o pyramid_n faulty_a for_o that_o it_o extend_v itself_o to_o large_a and_o comprehend_v under_o it_o a_o tetrahedron_n which_o differ_v from_o a_o pyramid_n by_o that_o it_o be_v contain_v of_o equal_a triangle_n as_o he_o not_o so_o advise_o do_v before_o the_o definition_n of_o a_o prism_n definition_n 23_o a_o octohedron_n be_v a_o solid_a or_o bodily_a figure_n contain_v under_o eight_o equal_a and_o equilater_n triangle_n as_o a_o cube_n be_v a_o solid_a figure_n contain_v under_o six_o superficial_a figure_n of_o four_o side_n or_o square_n which_o be_v equilater_n equiangle_n and_o equal_v the_o one_o to_o the_o other_o so_o be_v a_o octohedron_n a_o solid_a figure_n contain_v under_o eight_o triangle_n which_o be_v equilater_n and_o equal_a the_o one_o to_o the_o other_o as_o you_o may_v in_o these_o two_o figure_n here_o set_v behold_v whereof_o the_o first_o be_v draw_v according_a as_o this_o solid_a be_v common_o describe_v upon_o a_o plain_a superficies_n the_o second_o be_v draw_v as_o it_o be_v describe_v by_o art_n upon_o a_o plain_a to_o show_v bodilike_o and_o in_o deed_n although_o the_o second_o appear_v to_o the_o eye_n more_o bodilike_o yet_o as_o i_o before_o note_v in_o a_o cube_n for_o the_o understanding_n of_o diverse_a proposition_n in_o these_o five_o book_n follow_v be_v the_o first_o description_n of_o more_o use_n yea_o &_o of_o necessity_n for_o without_o it_o you_o can_v not_o conceive_v the_o draught_n of_o line_n and_o section_n in_o any_o one_o of_o the_o eight_o side_n which_o be_v sometime_o in_o the_o description_n of_o some_o of_o those_o proposition_n require_v wherefore_o to_o the_o consideration_n of_o this_o first_o description_n imagine_v first_o that_o upon_o the_o upper_a face_n of_o the_o superficies_n of_o the_o parallelogram_n abcd_o be_v describe_v a_o pyramid_n have_v his_o four_o triangle_n afb_o afc_n cfd_n and_o dfb_n equilater_n and_o equiangle_n and_o concur_v in_o the_o point_n f._n then_o conceive_v that_o on_o the_o low_a face_n of_o the_o superficies_n of_o the_o former_a parallelogram_n be_v describe_v a_o other_o pyramid_n have_v his_o four_o triangle_n aeb_n aec_o ce_z &_o deb_n equilater_n and_o equiangle_n and_o concur_v in_o the_o point_n e._n for_o so_o although_o somewhat_o gross_o by_o reason_n the_o triangle_n can_v not_o be_v describe_v equilater_n you_o may_v in_o a_o plain_a perceive_v the_o form_n of_o this_o solid_a and_o by_o that_o mean_n conceive_v any_o line_n or_o section_n require_v to_o be_v draw_v in_o any_o of_o the_o say_v eight_o triangle_n which_o be_v the_o side_n of_o that_o body_n 24_o a_o dodecahedron_n be_v a_o solid_a or_o bodily_a figure_n contain_v under_o twelve_o equal_a equilater_n definition_n and_o equiangle_n pentagons_n as_o a_o cube_n a_o tetrahedron_n and_o a_o octohedron_n be_v contain_v under_o equal_a plain_a figure_n a_o cube_n under_o square_n the_o other_o two_o under_o triangle_n so_o be_v this_o solid_a figure_n contain_v under_o twelve_o equilater_n equiangle_n and_o equal_a pentagons_n or_o figure_n of_o five_o side_n as_o in_o these_o two_o figure_n here_o set_v you_o may_v perceive_v of_o which_o the_o first_o which_o thing_n also_o be_v before_o note_v of_o a_o cube_n a_o tetrahedron_n and_o a_o octohedron_n be_v the_o common_a description_n of_o it_o in_o a_o plain_a the_o other_o be_v the_o description_n of_o it_o by_o art_n upon_o a_o plain_a to_o make_v it_o to_o appear_v somewhat_o bodilike_o the_o first_o description_n in_o deed_n be_v very_o obscure_a to_o conceive_v but_o yet_o of_o necessity_n it_o must_v so_o neither_o can_v it_o otherwise_o be_v in_o a_o plain_a describe_v to_o understand_v those_o proposition_n of_o euclid_n in_o these_o five_o book_n a_o follow_v which_o concern_v the_o same_o for_o in_o it_o although_o rude_o may_v you_o see_v all_o the_o twelve_o pentagons_n which_o shall_v in_o deed_n be_v all_o equal_a equilater_n and_o equiangle_n and_o now_o how_o you_o may_v somewhat_o conceive_v the_o first_o figure_n describe_v in_o the_o plain_a to_o be_v a_o body_n imagine_v first_o the_o pentagon_n abcde_v â—o_o be_v upon_o a_o ground_n plain_a superficies_n then_o imagine_v the_o pentagon_n fghkl_o to_o be_v on_o high_a opposite_a unto_o the_o pentagon_n abcde_o and_o between_o those_o two_o pentagons_n there_o will_v be_v ten_o pentagons_n pull_v up_o five_o from_o the_o five_o side_n of_o the_o ground_n pentagon_n namely_o from_o the_o side_n ab_fw-la the_o pentagon_n abonm_n from_o the_o side_n bc_o the_o pentagon_n bcqpo_n from_o the_o side_n cd_o the_o pentagon_n cdsrq_n from_o the_o side_n de_fw-fr the_o pentagon_n deut_v from_o the_o side_n ea_fw-la the_o pentagon_n eamxv_n the_o other_o five_o pentagons_n have_v each_o one_o of_o their_o side_n common_a with_o one_o of_o the_o side_n of_o the_o pentagon_n fghkl_o which_o be_v opposite_a unto_o the_o pentagon_n in_o the_o ground_n superficies_n namely_o these_o be_v the_o other_o five_o pentagons_n fgnmx_n ghpon_n hkrqp_n klrst_n lfxut_n so_o here_o you_o may_v behold_v twelve_o pentagons_n which_o if_o you_o imagine_v to_o be_v equal_a equilater_n &_o equiangle_n and_o to_o be_v lift_v up_o you_o shall_v although_o somewhat_o rude_o conceive_v the_o bodily_a form_n of_o a_o pentagon_n and_o some_o light_n it_o will_v geve_v to_o the_o understanding_n of_o certain_a proposition_n of_o the_o five_o book_n follow_v concern_v the_o same_o definition_n 25_o a_o icosahedron_n be_v a_o solid_a or_o bodily_a figure_n contain_v under_o twenty_o equal_a and_o equilater_n triangle_n these_o â—ive_a solid_n now_o last_v define_v namely_o a_o cube_n a_o tetrahedron_n a_o octohedron_n a_o dodecahedron_n and_o a_o icosahedron_n be_v call_v regular_a body_n body_n as_o in_o plain_a superficiece_n those_o be_v call_v regular_a figure_n who_o side_n and_o angle_n be_v equal_a as_o be_v equilater_n triangle_n equilater_n pentagon_n hexagon_n &_o such_o like_a so_o in_o solid_n such_o only_o be_v count_v and_o call_v regular_a which_o be_v comprehend_v under_o equal_a plain_a superficiece_n which_o have_v equal_a side_n and_o equal_a angle_n as_o all_o these_o five_o foresay_a have_v as_o manifest_o appear_v by_o their_o definition_n which_o be_v all_o give_v by_o this_o propriety_n of_o equality_n of_o their_o superficiece_n which_o have_v also_o their_o side_n and_o angle_n equal_a and_o in_o all_o the_o course_n of_o nature_n there_o be_v no_o other_o body_n of_o this_o condition_n and_o perfection_n but_o only_o these_o five_o wherefore_o they_o have_v ever_o of_o the_o ancient_a philosopher_n be_v have_v in_o great_a estimation_n and_o admiration_n and_o have_v be_v think_v worthy_a of_o much_o contemplation_n about_o which_o they_o have_v bestow_v most_o diligent_a study_n and_o endeavour_n to_o search_v out_o the_o nature_n &_o property_n of_o they_o they_o be_v as_o it_o be_v the_o end_n and_o perfection_n of_o all_o geometry_n for_o who_o sake_n be_v write_v whatsoever_o be_v write_v in_o geometry_n they_o be_v as_o man_n say_v first_o invent_v by_o the_o most_o witty_a pythagoras_n then_o afterward_o set_v forth_o by_o the_o divine_a plato_n and_o last_o of_o all_o marvellous_o teach_v and_o declare_v by_o the_o most_o excellent_a philosopher_n euclid_n in_o these_o book_n follow_v and_o ever_o since_o wonderful_o embrace_v of_o all_o learned_a philosopher_n body_n the_o knowledge_n of_o they_o contain_v infinite_a secret_n of_o nature_n pithagâ—ras_n timeus_n and_o plato_n by_o they_o search_v out_o the_o composition_n of_o the_o world_n with_o the_o harmony_n and_o preservation_n thereof_o and_o apply_v these_o â—ive_a solid_n to_o the_o simple_a part_n thereof_o the_o pyramid_n or_o tetrahedron_n they_o ascribe_v to_o the_o â—ire_n fire_n for_o that_o it_o ascend_v upward_o according_a to_o the_o figure_n of_o the_o pyramid_n to_o the_o air_n they_o ascribe_v the_o octohedron_n air_n for_o that_o through_o the_o subtle_a moisture_n which_o it_o have_v it_o extend_v itself_o every_o way_n to_o the_o one_o side_n and_o to_o the_o other_o accord_v as_o that_o figure_n do_v unto_o the_o water_n they_o assign_v the_o ikosahedron_n for_o that_o it_o be_v continual_o flow_v and_o move_v water_n and_o as_o it_o be_v make_v angleâ—_n 〈…〉_o â—ide_v according_a to_o that_o figure_n and_o to_o the_o earth_n they_o attribute_v a_o cube_n earth_n as_o to_o a_o thing_n stableâ—_n 〈◊〉_d and_o sure_a as_o the_o figure_n
signify_v last_o of_o all_o a_o dodecahedron_n heaven_n for_o that_o it_o be_v make_v of_o pâ—ntagoâ—_n who_o angle_n be_v more_o ample_a and_o large_a than_o the_o angle_n of_o the_o other_o body_n and_o by_o that_o â—eaâ—â—â—_n draw_v more_o â—â—_o roundness_n 〈◊〉_d &_o to_o the_o form_n and_o nature_n of_o a_o sphere_n they_o assign_v to_o a_o sphere_n namely_o 〈…〉_o who_o so_o will_v 〈…〉_o in_o his_o tineus_n shall_v â—ead_a of_o these_o figure_n and_o of_o their_o mutual_a proportionâ—â—â—raunge_a maâ—terâ—_n which_o hâ—re_o be_v not_o to_o be_v entreat_v of_o this_o which_o be_v say_v shall_v be_v sufficient_a for_o the_o 〈◊〉_d of_o they_o and_o for_o thâ—_z declaration_n of_o their_o definition_n after_o all_o these_o definition_n here_o set_v of_o euclid_n flussas_n have_v add_v a_o other_o definition_n which_o 〈◊〉_d of_o a_o parallelipipedon_n which_o because_o it_o have_v not_o hitherto_o of_o euclid_n in_o any_o place_n be_v define_v and_o because_o it_o be_v very_o good_a and_o necessary_a to_o be_v have_v i_o think_v good_a not_o to_o omit_v it_o thus_o it_o be_v a_o parallelipipedon_n be_v a_o solid_a figure_n comprehend_v under_o four_o plain_a quadrangle_n figure_n parallelipipedon_n of_o which_o those_o which_o be_v opposite_a be_v parallel_n because_o these_o five_o regular_a body_n here_o define_v be_v not_o by_o these_o figure_n here_o set_v so_o full_o and_o lively_o express_v that_o the_o studious_a beholder_n can_v thorough_o according_a to_o their_o definition_n conceive_v they_o i_o have_v here_o give_v of_o they_o other_o description_n draw_v in_o a_o plain_n by_o which_o you_o may_v easy_o attain_v to_o the_o knowledge_n of_o they_o for_o if_o you_o draw_v the_o like_a form_n in_o matter_n that_o will_v bow_v and_o geve_v place_n as_z most_o apt_o you_o may_v do_v in_o fine_a past_v paper_n such_o as_o pastwive_n make_v woman_n paste_n of_o &_o they_o with_o a_o knife_n cut_v every_o line_n fine_o not_o through_o but_o half_a way_n only_o if_o they_o you_o bow_v and_o bend_v they_o according_o you_o shall_v most_o plain_o and_o manifest_o see_v the_o form_n and_o shape_n of_o these_o body_n even_o as_o their_o definition_n show_v and_o it_o shall_v be_v very_o necessary_a for_o you_o to_o hadâ—â—tore_v of_o that_o past_v paper_n by_o you_o for_o so_o shall_v yoâ—_n upon_o it_o 〈…〉_o the_o form_n of_o other_o body_n as_o prism_n and_o parallelipopedon_n 〈…〉_o set_v forth_o in_o these_o five_o book_n follow_v and_o see_v the_o very_a 〈◊〉_d of_o thâ—se_a body_n there_o mention_v which_o will_v make_v these_o book_n concern_v body_n as_o easy_a unto_o you_o as_o be_v the_o other_o book_n who_o figure_n you_o may_v plain_o see_v upon_o a_o plain_a superficies_n describe_v thiâ—_z figurâ—_n which_o consi_v of_o twâ—luâ—â—quilâ—â—â—â—_n and_o equiangle_n pâ—ntâ—â—â—â—â—_n upoâ—_n the_o foresay_a matter_n and_o fine_o cut_v as_o before_o be_v â—â—ught_v tâ—â—â—lâ—uâ—n_v line_n contain_v within_o thâ—_z figurâ—_n and_o bow_n and_o fold_n the_o penâ—â—gonâ—_n according_o and_o they_o will_v so_o close_o together_o thaâ—_z thâ—y_n will_v â—â—kâ—_n thâ—_z very_a form_n of_o a_o dodecahedron_n dâ—dâ—â—â—â—edron_n icosaâ—edron_n if_o you_o describe_v this_o figure_n which_o consist_v of_o twenty_o equilater_n and_o equiangle_n triangle_n upon_o the_o foresay_a matter_n and_o fine_o cut_v as_o before_o be_v show_v the_o ninâ—tâ—ne_a line_n which_o be_v contain_v within_o the_o figure_n and_o then_o bow_n and_o fold_n they_o according_o they_o will_v in_o such_o sort_n close_o together_o that_o therâ—_n will_v be_v make_v a_o perfect_v form_n of_o a_o icosahedron_n because_o in_o these_o five_o book_n there_o be_v sometime_o require_v other_o body_n beside_o the_o foresay_a five_o regular_a body_n as_o pyramid_n of_o diverse_a form_n prism_n and_o other_o i_o have_v here_o set_v forth_o three_o figure_n of_o three_o sundry_a pyramid_n one_o have_v to_o his_o base_a a_o triangle_n a_o other_o a_o quadrangle_n figure_n the_o other_o â—_o pentagonâ—_n which_o if_o you_o describe_v upon_o the_o foresay_a matter_n &_o fine_o cut_v as_o it_o be_v before_o teach_v the_o line_n contain_v within_o each_o figure_n namely_o in_o the_o first_o three_o line_n in_o the_o second_o four_o line_n and_o in_o the_o three_o five_o line_n and_o so_o bend_v and_o fold_v they_o according_o they_o will_v so_o close_o together_o at_o the_o top_n that_o they_o will_v â—ake_v pyramid_n of_o that_o form_n that_o their_o base_n be_v of_o and_o if_o you_o conceive_v well_o the_o describe_v of_o these_o you_o may_v most_o easy_o describe_v the_o body_n of_o a_o pyramid_n of_o what_o form_n so_o ever_o you_o will._n because_o these_o five_o book_n follow_v be_v somewhat_o hard_o for_o young_a beginner_n by_o reason_n they_o must_v in_o the_o figure_n describe_v in_o a_o plain_a imagine_v line_n and_o superficiece_n to_o be_v elevate_v and_o erect_v the_o one_o to_o the_o other_o and_o also_o conceive_v solid_n or_o body_n which_o for_o that_o they_o have_v not_o hitherto_o be_v acquaint_v with_o will_v at_o the_o first_o sight_n be_v somewhat_o strange_a unto_o they_o i_o have_v for_o their_o more_o â—ase_a in_o this_o eleven_o book_n at_o the_o end_n of_o the_o demonstration_n of_o every_o proposition_n either_o set_v new_a figure_n if_o they_o concern_v the_o elevate_n or_o erect_v of_o line_n or_o superficiece_n or_o else_o if_o they_o concern_v body_n i_o have_v show_v how_o they_o shall_v describe_v body_n to_o be_v compare_v with_o the_o construction_n and_o demonstration_n of_o the_o proposition_n to_o they_o belong_v and_o if_o they_o diligent_o weigh_v the_o manner_n observe_v in_o this_o eleven_o book_n touch_v the_o description_n of_o new_a figure_n agree_v with_o the_o figure_n describe_v in_o the_o plain_a it_o shall_v not_o be_v hard_o for_o they_o of_o themselves_o to_o do_v the_o like_a in_o the_o other_o book_n follow_v when_o they_o come_v to_o a_o proposition_n which_o concern_v either_o the_o elevate_n or_o erect_v of_o line_n and_o superficiece_n or_o any_o kind_n of_o body_n to_o be_v imagine_v ¶_o the_o 1._o theorem_a the_o 1._o proposition_n that_o part_n of_o a_o right_a line_n shall_v be_v in_o a_o ground_n plain_a superficies_n &_o part_v elevate_v upward_o be_v impossible_a for_o if_o it_o be_v possible_a let_v part_n of_o the_o right_a line_n abc_n namely_o the_o part_n ab_fw-la be_v in_o a_o ground_n plain_a superficies_n and_o the_o other_o part_n thereof_o namely_o bc_o be_v elevate_v upward_o and_o produce_v direct_o upon_o the_o ground_n plain_a superficies_n the_o right_a line_n ab_fw-la beyond_o the_o point_n b_o unto_o the_o point_n d._n wherefore_o unto_o two_o right_a line_n give_v abc_n and_o abdella_n the_o line_n ab_fw-la be_v a_o common_a section_n or_o part_n which_o be_v impossible_a impossibility_n for_o a_o right_a line_n can_v not_o touch_v a_o right_a line_n in_o 〈◊〉_d point_n then_o one_o uâ—lesse_o those_o right_n be_v exact_o agree_v and_o lay_v the_o one_o upon_o the_o other_o wherefore_o that_o part_n of_o a_o right_a line_n shall_v be_v in_o a_o ground_n plain_a superficies_n and_o part_v elevate_v upward_o be_v impossible_a which_o be_v require_v to_o be_v prove_v this_o figure_n more_o plain_o set_v forth_o the_o foresay_a demonstration_n if_o you_o elevate_v the_o superficies_n wherein_o the_o line_n bc._n an_o other_o demonstration_n after_o flâ—sâ—â—s_n if_o it_o be_v possible_a let_v there_o be_v a_o right_a line_n abg_o who_o part_n ab_fw-la let_v be_v in_o the_o ground_n plain_a superficies_n aed_n flussas_n and_o let_v the_o rest_n thereof_o bg_o be_v elevate_v on_o high_a that_o be_v without_o the_o plain_n aed_n then_o i_o say_v that_o abg_o be_v not_o one_o right_a line_n for_o forasmuch_o as_o aed_n be_v a_o plain_a superficies_n produce_v direct_o &_o equal_o upon_o the_o say_a plain_n aed_n the_o right_a line_n ab_fw-la towards_o d_z which_o by_o the_o 4._o definition_n of_o the_o first_o shall_v be_v a_o right_a line_n and_o from_o some_o one_o point_n of_o the_o right_a line_n abdella_n namely_o from_o c_o draâ—_n unto_o the_o point_n g_o a_o right_a line_n cg_o wherefore_o in_o the_o triangle_n 〈…〉_o the_o outward_a angâ—â—_n abâ—_n be_v equal_a to_o the_o two_o inward_a and_o opposite_a angle_n by_o the_o 32._o of_o the_o first_o and_o therefore_o it_o be_v less_o than_o two_o right_a angle_n by_o the_o 17._o of_o the_o same_o wherefore_o the_o line_n abg_o forasmuch_o as_o it_o make_v a_o angle_n be_v not_o â—_o right_n line_n wherefore_o that_o part_n of_o a_o right_a line_n shall_v be_v in_o a_o ground_n plain_a superficies_n and_o part_v elevate_v upward_o be_v impossible_a if_o you_o mark_v well_o the_o figure_n before_o add_v for_o the_o plain_a declaration_n of_o euclides_n demonstration_n iâ—_z will_v not_o be_v hard_o for_o you_o to_o coâ—â—â—â—e_v this_o figure_n which_o ulyss_n put_v for_o his_o demonstration_n â—_o wherein_o
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
side_n wherefore_o the_o whole_a parallelipipedon_n ae_n be_v equal_a and_o like_a to_o the_o whole_a parallelipipedon_n yu._n extend_v by_o the_o second_o petition_n the_o line_n doctor_n and_o wv_n until_o they_o concur_v and_o let_v they_o concur_v in_o the_o point_n q._n and_o by_o the_o 31._o of_o the_o first_o by_o the_o point_n t_o draw_v unto_o the_o line_n rq_n a_o parallel_a line_n t_o 4_o and_o extend_v due_o the_o line_n ta_o and_o do_v until_o they_o concur_v and_o let_v they_o concur_v in_o the_o point_n ✚_o and_o make_v perfect_v the_o solid_n qy_n and_o ri._n now_o the_o solid_a qy_n who_o plain_n base_a be_v the_o parallelogram_n ry_o and_o the_o opposite_a side_n unto_o the_o base_a the_o parallelogram_n qb_n be_v equal_a to_o the_o solid_a yv_n who_o base_a be_v the_o parallelogram_n ry_o and_o the_o opposite_a side_n unto_o the_o base_a the_o parallelogram_n we_o for_o they_o consist_v upon_o one_o and_o the_o self_n fame_n base_a namely_o ry_o and_o be_v under_o one_o and_o the_o self_n same_o altitude_n and_o their_o stand_a line_n namely_o rq_n rv_n ta_o tw_n sn_a sd_v yb_n and_o yz_n be_v in_o the_o self_n same_o right_a line_n namely_o qw_o and_o nz_n but_o the_o solid_a yv_n be_v prove_v equal_a to_o the_o solid_a ae_n wherefore_o also_o the_o solid_a yq_n be_v equal_a to_o the_o solid_a ae_n now_o forasmuch_o as_o the_o parallelogram_n ruwt_fw-mi be_v equal_a to_o the_o parallelogram_n qt_n by_o the_o 35._o of_o the_o first_o and_o the_o parallelogram_n ab_fw-la be_v equal_a to_o the_o parallelogram_n rw_n therefore_o the_o parallelogram_n qt_o be_v equal_a to_o the_o parallelogram_n ab_fw-la and_o the_o parallelogram_n cd_o be_v equal_a to_o the_o parallelogram_n ab_fw-la by_o supposition_n wherefore_o the_o parallelogram_n cd_o be_v equal_a to_o the_o parallelogram_n qt_n and_o there_o be_v a_o certain_a other_o superficies_n namely_o dt_n wherefore_o by_o the_o 7._o of_o the_o five_o as_o the_o base_a cd_o be_v to_o the_o base_a dt_n so_o be_v the_o base_a qt_n to_o the_o base_a dt_n and_o forasmuch_o as_o the_o whole_a parallelipipedon_n ci_o be_v cut_v by_o the_o plain_a superficies_n rf_n which_o be_v a_o parallel_n to_o either_o of_o the_o opposite_a plain_a superficiece_n therefore_o as_o the_o base_a cd_o be_v to_o the_o base_a dt_n so_o be_v the_o solid_a cf_o to_o the_o solid_a ri_n by_o the_o 25._o of_o the_o eleven_o and_o by_o the_o same_o reason_n also_o forasmuch_o as_o the_o whole_a parallelipipedon_n qi_n be_v cut_v by_o the_o plain_a superficies_n ry_o which_o be_v a_o parallel_n to_o either_o of_o the_o opposite_a plain_a superficiece_n therefore_o as_o the_o base_a qt_n be_v to_o the_o base_a dt_n so_o be_v the_o solid_a qy_n to_o the_o solid_a ri._n but_o as_o the_o base_a cd_o be_v to_o the_o base_a dt_n so_o be_v the_o base_a qt_n to_o the_o base_a td_n wherefore_o by_o the_o 11._o of_o the_o five_o as_o the_o solid_a cf_o be_v to_o the_o solid_a ri_n so_o be_v the_o solid_a qy_n to_o the_o solid_a ri._n wherefore_o either_o of_o these_o solid_n cf_o and_o qy_n have_v to_o the_o solid_a ri_n one_o and_o the_o same_o proportion_n wherefore_o the_o solid_a cf_o be_v equal_a to_o the_o solid_a qy_n but_o it_o be_v prove_v that_o the_o solid_a qy_n be_v equal_a to_o the_o solid_a ae_n wherefore_o also_o the_o solid_a cf_o be_v equal_a to_o the_o solid_a ae_n but_o now_o suppose_v that_o the_o stand_a line_n namely_o agnostus_n hk_o case_n be_v lm_o cx_o open_v df_o and_o r_n be_v not_o erect_v perpendicular_o to_o the_o base_n ab_fw-la and_o cd_o then_o also_o i_o say_v that_o the_o solid_a ae_n be_v equal_a to_o the_o solid_a cf._n draw_v by_o the_o 11._o of_o the_o eleven_o unto_o the_o ground_n plain_a superficiece_n ab_fw-la and_o cd_o from_o these_o point_n king_n e_o c_o m_o p_o f_o x_o saint_n these_o perpendicular_a line_n kn_n et_fw-la gv_n mz_n pw_n fie_o xq_n and_o si_fw-it and_o draw_v these_o right_a line_n nt_v nv_n zv_n zt_n why_o wq_n iq_fw-la and_o jy_n now_o by_o that_o which_o have_v before_o be_v prove_v in_o this_o 31._o proposition_n the_o solid_a kz_o be_v equal_a to_o the_o solid_a piperollo_n for_o they_o consist_v upon_o equal_a you_o base_n namely_o km_o and_o ps_n and_o be_v under_o one_o and_o the_o self_n same_o altitude_n who_o stand_a line_n also_o be_v erect_v perpendicular_o to_o the_o base_n but_o the_o solid_a kz_o be_v equal_a to_o the_o solid_a ae_n by_o the_o 29._o of_o the_o eleven_o and_o the_o solid_a piperollo_n be_v by_o the_o same_o equal_a to_o the_o solid_a cf_o for_o they_o consist_v upon_o one_o and_o the_o self_n same_o base_a and_o be_v under_o one_o &_o the_o self_n same_o altitude_n who_o stand_a line_n be_v upon_o the_o self_n same_o right_a line_n wherefore_o also_o the_o solid_a ae_n be_v equal_a to_o the_o solid_a ce._n wherefore_o parallelipipedon_n consist_v upon_o equal_a base_n and_o be_v under_o one_o and_o the_o self_n same_o altitude_n be_v equal_a the_o one_o to_o the_o other_o which_o be_v require_v to_o be_v demonstrate_v the_o demonstration_n of_o the_o first_o case_n of_o this_o proposition_n be_v very_o hard_a to_o conceive_v by_o the_o figure_n describe_v for_o it_o in_o a_o plain_n and_o yet_o before_o m._n do_v you_o invent_v that_o figure_n which_o we_o have_v there_o place_v for_o it_o it_o be_v much_o hard_o for_o both_o in_o the_o greek_a and_o latin_a euclid_n it_o be_v very_o ill_o make_v and_o it_o be_v in_o a_o manner_n impossible_a to_o conceive_v by_o it_o the_o construction_n and_o demonstration_n thereto_o appertain_v wherefore_o i_o have_v here_o describe_v other_o figure_n which_o first_o describe_v upon_o past_v paper_n or_o such_o like_a matter_n and_o then_o order_v they_o in_o manner_n follow_v as_o touch_v the_o solid_a ae_n in_o the_o first_o case_n i_o need_v not_o to_o make_v any_o new_a description_n for_o it_o be_v plain_a enough_o to_o conceive_v as_o it_o be_v there_o draw_v although_o you_o may_v for_o your_o more_o ease_n of_o imagination_n describe_v of_o past_v paper_n a_o parallelipipedoâ—_n have_v his_o side_n equal_a with_o the_o side_n of_o the_o parallelipipedon_n ae_n before_o describe_v and_o have_v also_o the_o six_o parallelogram_n thereof_o contain_v under_o those_o side_n equiangle_n with_o the_o six_o parallelogram_n of_o that_o figure_n each_o side_n and_o each_o angle_n equal_a to_o his_o correspondent_a side_n and_o to_o his_o correspondent_a angle_n but_o concern_v the_o other_o solid_a when_o you_o have_v describe_v these_o three_o figure_n upon_o past_v paper_n where_o note_n for_o the_o proportion_n of_o each_o line_n to_o make_v your_o figure_n of_o past_v paper_n equal_a with_o the_o figure_n before_o describe_v upon_o the_o plain_n let_v your_o line_n open_v cx_o r_n df_o etc_n etc_n namely_o the_o rest_n of_o the_o stand_a line_n of_o these_o figure_n be_v equal_a to_o the_o stand_a line_n open_v cx_o r_n df_o etc_n etc_n of_o that_o figure_n likewise_o let_v the_o line_n oc_fw-fr cr_n rd_o do_v etc_n etc_n namely_o the_o side_n which_o contain_v the_o base_n of_o these_o figure_n be_v equal_a to_o the_o line_n oc_fw-fr cr_n rd_o do_v etc_n etc_n namely_o to_o the_o side_n which_o contain_v the_o base_n of_o that_o figure_n moreover_o let_v the_o line_n px_n xâ—_n sf_n fp_n etc_n etc_n namely_o the_o rest_n of_o the_o line_n which_o contain_v the_o upper_a superficiece_n of_o these_o figure_n be_v equal_a to_o the_o line_n px_n xs_n sf_n fâ—_n etc_n etc_n namely_o to_o the_o rest_n of_o the_o line_n which_o contain_v the_o upper_a superficiece_n of_o that_o figure_n to_o have_v describe_v all_o those_o foresay_a line_n of_o these_o figure_n equal_a to_o all_o the_o line_n of_o that_o figure_n will_v have_v require_v much_o more_o space_n than_o here_o can_v be_v spare_v i_o have_v make_v they_o equal_v only_o to_o the_o half_n of_o those_o line_n but_o by_o the_o example_n of_o these_o you_o may_v if_o you_o will_v describe_v the_o like_a figure_n have_v their_o line_n equal_a to_o the_o whole_a line_n of_o the_o figure_n in_o the_o plain_n each_o line_n to_o his_o correspondent_a line_n when_o i_o say_v you_o have_v as_o before_o be_v teach_v describe_v these_o three_o figure_n cut_v fine_o the_o line_n xc_o sir_n fd_v of_o the_o first_o figure_n and_o the_o line_n sir_n it_o and_o i_o ✚_o of_o the_o second_o figure_n likewise_o the_o line_n â—r_o nq_n zw_n and_o it_o of_o the_o three_o figure_n and_o fold_v these_o figure_n according_o which_o you_o can_v not_o choose_v but_o do_v if_o you_o mark_v well_o the_o letter_n of_o every_o line_n as_o touch_v the_o second_o case_n you_o need_v no_o new_a figure_n for_o it_o be_v plain_a to_o see_v by_o the_o figure_n now_o
be_v manifest_a by_o the_o last_o of_o the_o eleven_o and_o now_o that_o both_o the_o prism_n take_v together_o be_v great_a than_o the_o half_a of_o the_o whole_a pyramid_n hereby_o it_o be_v manifest_a for_o that_o either_o of_o they_o may_v be_v divide_v into_o two_o pyramid_n of_o which_o the_o one_o be_v a_o triangular_a pyramid_n equal_a to_o one_o of_o the_o two_o pyramid_n into_o which_o together_o with_o the_o two_o prism_n be_v divide_v the_o whole_a pyramid_n and_o the_o other_o be_v a_o quadrangled_a pyramid_n double_a to_o the_o other_o pyramid_n wherefore_o it_o be_v plain_a that_o the_o two_o prism_n take_v together_o be_v three_o quarter_n of_o the_o whole_a pyramid_n divide_v but_o if_o you_o be_v desirous_a to_o know_v the_o proportion_n between_o they_o read_v the_o â—_o of_o this_o book_n but_o now_o here_o to_o this_o purpose_n it_o shall_v be_v sufficient_a to_o know_v that_o the_o two_o prism_n take_v together_o do_v exceed_v in_o quantity_n the_o two_o partial_a pyramid_n take_v together_o into_o which_o together_o with_o the_o two_o prism_n the_o whole_a pyramid_n be_v divide_v ¶_o the_o 4._o theorem_a the_o 4._o proposition_n if_o there_o be_v two_o pyramid_n under_o equal_a altitude_n have_v triangle_n to_o their_o base_n and_o either_o of_o those_o pyramid_n be_v divide_v into_o two_o pyramid_n equal_v the_o one_o to_o the_o other_o and_o like_v unto_o the_o whole_a and_o into_o two_o squall_n prism_n and_o again_o if_o in_o either_o of_o the_o pyramid_n make_v of_o the_o two_o first_o pyramid_n be_v still_o observe_v the_o same_o order_n and_o manner_n then_o as_o the_o base_a of_o the_o one_o pyramid_n be_v to_o the_o base_a of_o the_o other_o pyramid_n so_o be_v all_o the_o prism_n which_o be_v in_o the_o one_o pyramid_n to_o all_o the_o prism_n which_o be_v in_o the_o other_o be_v equal_a in_o multitude_n with_o they_o svppose_v that_o there_o be_v two_o pyramid_n under_o equal_a altitude_n have_v triangle_n to_o their_o base_n namely_o abc_n and_o def_n and_o have_v to_o their_o top_n the_o point_n g_o and_o h._n and_o let_v either_o of_o these_o pyramid_n be_v divide_v into_o two_o pyramid_n equal_v the_o one_o to_o the_o other_o and_o like_v unto_o the_o whole_a and_o into_o two_o equal_a prism_n according_a to_o the_o method_n of_o the_o former_a proposition_n and_o again_o let_v either_o of_o those_o pyramid_n soâ—_n make_v of_o the_o two_o first_o pyramid_n be_v imagine_v to_o be_v after_o the_o same_o order_n divide_v and_o so_o do_v continual_o then_o i_o say_v that_o as_o the_o base_a abc_n be_v to_o the_o base_a def_n so_o be_v all_o the_o prism_n which_o be_v in_o the_o pyramid_n abcg_o to_o all_o the_o prism_n which_o be_v in_o the_o pyramid_n defh_o be_v equal_a in_o multitude_n with_o they_o for_o forasmuch_o as_o the_o line_n bx_n be_v equal_a to_o the_o line_n xc_o and_o the_o line_n all_o to_o the_o line_n lc_n for_o as_o we_o see_v in_o the_o construction_n pertain_v to_o the_o former_a proposition_n all_o the_o six_o side_n of_o the_o whole_a pyramid_n be_v each_o divide_v into_o two_o equal_a part_n the_o like_a of_o which_o construction_n be_v in_o this_o proposition_n also_o suppose_v therefore_o the_o line_n xl_o be_v a_o parallel_n to_o the_o line_n ab_fw-la &_o the_o triangle_n abc_n be_v like_a to_o the_o triangle_n lxc_n by_o the_o corollary_n of_o the_o second_o of_o the_o six_o and_o by_o the_o same_o reason_n the_o triangle_n def_n be_v like_a to_o the_o triangle_n rwf_n and_o forasmuch_o as_o the_o line_n bc_o be_v double_a to_o the_o line_n cx_o and_o the_o line_n fe_o to_o the_o line_n fw_o therefore_o as_o the_o line_n bc_o be_v to_o the_o line_n cx_o so_o be_v the_o line_n of_o to_o the_o line_n fw_o and_o upon_o the_o line_n bc_o and_o cx_o be_v describe_v rectiline_a figure_n like_a and_o in_o like_a sort_n set_v namely_o the_o triangle_n abc_n and_o lxc_n and_o upon_o the_o line_n of_o and_o fw_o be_v also_o describe_v rectiline_a figure_n like_a and_o in_o like_a sort_n set_v namely_o the_o triangle_n def_n &_o rwfâ—_n but_o if_o there_o be_v four_o right_a line_n proportional_a the_o rectiline_a figure_n describe_v of_o they_o be_v like_a and_o in_o like_a sort_n set_v shall_v also_o be_v proportional_a by_o the_o 22._o of_o the_o six_o wherefore_o as_o the_o triangle_n abc_n be_v to_o the_o triangle_n lxc_n so_o be_v the_o triangle_n def_n to_o the_o triangle_n rwf_n wherefore_o alternate_o by_o the_o 16._o of_o the_o five_o as_o the_o triangle_n abc_n be_v to_o the_o triangle_n def_n so_o be_v the_o triangle_n lxc_n to_o the_o triangle_n rwf_n assumpt_n but_o as_o the_o triangle_n lxc_n be_v to_o the_o triangle_n rwf_n so_o be_v the_o prism_n who_o base_a be_v the_o triangle_n lxc_n and_o the_o opposite_a side_n unto_o it_o the_o triangle_n omnes_n to_o the_o prism_n who_o base_a be_v the_o triangle_n rwf_n and_o the_o opposite_a side_n unto_o it_o the_o triangle_n stv_n by_o the_o corollary_n of_o the_o 40._o of_o the_o eleven_o for_o these_o prism_n be_v under_o one_o &_o the_o self_n same_o altitude_n namely_o under_o the_o half_a of_o the_o altitude_n of_o the_o whole_a pyramid_n which_o pyramid_n be_v suppose_v to_o be_v under_o one_o and_o the_o self_n same_o altitude_n this_o be_v also_o prove_v in_o the_o assumpt_n follow_v wherefore_o by_o the_o 11._o of_o the_o five_o as_o the_o triangle_n abc_n be_v to_o the_o triangle_n def_n so_o be_v the_o prism_n who_o base_a be_v the_o triangle_n lxc_n and_o the_o opposite_a side_n unto_o it_o the_o triangle_n omnes_n to_o the_o prism_n who_o base_a be_v the_o triangle_n rwf_n and_o the_o opposite_a side_n unto_o it_o the_o triangle_n stv_n and_o forasmuch_o as_o there_o be_v two_o prism_n in_o the_o pyramid_n abcg_o equal_a the_o one_o to_o the_o other_o &_o two_o prism_n also_o in_o the_o pyramid_n defh_o equal_a the_o one_o to_o the_o other_o therefore_o as_o the_o prism_n who_o base_a be_v the_o parallelogram_n bklx_fw-fr and_o the_o opposite_a side_n unto_o it_o the_o line_n more_n be_v to_o the_o prism_n who_o base_a be_v the_o triangle_n lxc_n and_o the_o opposite_a side_n unto_o it_o the_o triangle_n omnes_n so_o be_v the_o prism_n who_o base_a be_v the_o parallelogram_n perw_n and_o the_o opposite_a unto_o it_o the_o line_n n-ab_n to_o the_o prism_n who_o base_a be_v the_o triangle_n rwf_n and_o the_o opposite_a side_n unto_o it_o the_o triangle_n stv_n wherefore_o by_o composition_n by_o the_o 18._o of_o the_o five_o as_o the_o prism_n kbxlmo_n &_o lxcmno_n be_v to_o the_o prism_n lxcmno_n so_o be_v the_o prism_n pewrst_a and_o rwfstv_n to_o the_o prism_n rwfstv_n wherefore_o alternate_o by_o the_o 16._o of_o the_o five_o as_o the_o two_o prism_n kbxlmo_n and_o lxcmno_n be_v to_o the_o two_o prism_n pewrst_a and_o rwfstv_n so_o be_v the_o prism_n lxcmno_n to_o the_o prism_n rwfstv_n but_o as_o the_o prism_n lxcmno_n be_v to_o the_o prism_n rwfstv_n so_o have_v we_o prove_v that_o the_o base_a lxc_n be_v to_o the_o base_a rwf_n and_o the_o base_a abc_n to_o the_o base_a def_n wherefore_o by_o the_o 16._o of_o the_o five_o as_o the_o triangle_n abc_n be_v to_o the_o triangle_n def_n so_o be_v both_o the_o prism_n which_o be_v in_o the_o pyramid_n abcg_o to_o both_o the_o prism_n which_o be_v in_o the_o pyramid_n defh_o and_o in_o like_a sort_n if_o we_o divide_v the_o other_o pyramid_n after_o the_o self_n same_o manner_n namely_o the_o pyramid_n omng_v and_o the_o pyramid_n stuh_o as_o the_o base_a omnes_n be_v to_o the_o base_a stv_n so_o shall_v both_o the_o prism_n that_o be_v in_o the_o pyramid_n omng_v be_v to_o both_o the_o prism_n which_o be_v in_o the_o pyramid_n stuh_o but_o as_o the_o base_a omnes_n be_v to_o the_o base_a stv_n so_o be_v the_o base_a abc_n to_o the_o base_a def_n wherefore_o by_o the_o 11._o of_o the_o five_o as_o the_o base_a abc_n be_v to_o the_o base_a def_n so_o be_v the_o two_o prism_n that_o be_v in_o the_o pyramid_n abcg_o to_o the_o two_o prism_n that_o be_v in_o the_o pyramid_n defh_o and_o the_o two_o prism_n that_o be_v in_o the_o pyramid_n omng_v to_o the_o two_o prism_n that_o be_v in_o the_o pyramid_n stuh_o and_o the_o four_o prism_n to_o the_o four_o prism_n and_o so_o also_o shall_v it_o follow_v in_o the_o prism_n make_v by_o divide_v the_o two_o pyramid_n aklo_n and_o dpr_n and_o of_o all_o the_o other_o pyramid_n in_o general_a be_v equal_a in_o multitude_n andrea_n that_o as_o the_o triangle_n lxc_n be_v to_o the_o triangle_n rwf_n so_o be_v the_o prism_n who_o base_a be_v the_o triangle_n lxc_n and_o the_o opposite_a side_n omnes_n assumpt_n to_o the_o prism_n who_o base_a be_v the_o triangle_n rwf_n and_o the_o
semicircle_n to_o be_v the_o plain_a superficies_n which_o pass_v by_o it_o shall_v make_v in_o the_o superficies_n of_o the_o sphere_n a_o circle_n and_o it_o be_v manifest_a that_o be_v also_o a_o great_a circle_n for_o the_o diameter_n of_o the_o sphere_n which_o be_v also_o the_o diameter_n of_o the_o semicircle_n and_o therefore_o also_o of_o the_o circle_n be_v by_o the_o 15._o of_o the_o three_o great_a then_o all_o the_o right_a line_n draw_v in_o the_o circle_n or_o sphere_n which_o circle_n shall_v have_v both_o one_o centre_n be_v both_o also_o in_o that_o one_o plain_a superficies_n be_v by_o which_o the_o sphere_n be_v cut_v suppose_v that_o that_o section_n or_o circle_n in_o the_o great_a sphere_n be_v bcde_v and_o in_o the_o less_o be_v the_o circle_n fgh_n draw_v the_o diameter_n of_o those_o two_o circle_n in_o such_o sort_n that_o they_o make_v right_a angle_n and_o let_v those_o diameter_n be_v bd_o and_o ce._n construction_n and_o let_v the_o line_n agnostus_n be_v part_n of_o the_o line_n ab_fw-la be_v the_o semidiameter_n of_o the_o less_o sphere_n and_o circle_n as_z ab_fw-la be_v the_o semidiameter_n of_o the_o great_a sphere_n and_o great_a circle_n both_o the_o sphere_n and_o circle_n have_v one_o and_o the_o same_o centre_n now_o two_o circle_n that_o be_v bcde_v and_o fgh_a consist_v both_o about_o one_o and_o the_o self_n same_o centre_n be_v give_v let_v there_o be_v describe_v by_o the_o proposition_n next_o go_v before_o in_o the_o great_a circle_n bcde_v a_o polygonon_n figure_n consist_v of_o equal_a and_o even_a side_n not_o touch_v the_o less_o circle_n fgh_a and_o let_v the_o side_n of_o that_o figure_n in_o the_o four_o part_n of_o the_o circle_n namely_o in_o be_v be_v bk_o kl_o lm_o and_o me._n and_o draw_v a_o right_a line_n from_o the_o point_n king_n to_o the_o point_n a_o and_o extend_v it_o to_o the_o point_n n._n and_o by_o the_o 12._o of_o the_o eleven_o from_o the_o a_o raise_v up_o to_o the_o superficies_n of_o the_o circle_n bcde_v a_o perpendicular_a line_n axe_n and_o let_v it_o light_v upon_o the_o superficies_n of_o the_o great_a sphere_n in_o the_o point_n x._o noteâ—_n and_o by_o the_o line_n axe_n and_o by_o either_o of_o these_o line_n bd_o and_o kn_n extend_v plain_a superficiece_n now_o by_o that_o which_o be_v before_o speak_v those_o plain_a superficiece_n shall_v in_o the_o perceive_v superficies_n of_o the_o sphere_n make_v two_o great_a circle_n let_v their_o semicircle_n consist_v upon_o the_o diameter_n bd_o and_o kn_n be_v bxd_v and_o kxn_o and_o forasmuch_o satisfy_v as_o the_o line_n xa_n be_v erect_v perpendicular_o to_o the_o plain_a superficies_n of_o the_o circle_n bcde_v therefore_o all_o the_o plain_a superficiece_n which_o be_v draw_v by_o the_o line_n xa_n be_v erect_v perpendicular_o to_o the_o superficies_n of_o the_o circle_n bcde_v by_o the_o 18._o of_o the_o eleven_o wherefore_o the_o semicircle_n bxd_v and_o kxn_o be_v erect_v perpendicular_o to_o the_o plain_a superficies_n of_o the_o circle_n bcde_v and_o forasmuch_o as_o the_o semicircle_n bed_n bxd_v and_o kxn_o be_v equal_a for_o they_o consist_v upon_o equal_a diameter_n â—d_a and_o kn_n therefore_o also_o the_o four_o part_n or_o quarter_n of_o those_o circle_n namely_o be_v bx_n and_o kx_n be_v equal_a the_o one_o to_o the_o other_o wherefore_o how_o many_o side_n of_o a_o polygonon_n figure_n there_o be_v in_o the_o four_o part_n or_o quarter_n be_v so_o many_o also_o be_v there_o in_o the_o other_o four_o part_n or_o quarter_n bx_n and_o kx_n equal_a to_o the_o right_a line_n bk_o kl_o lm_o and_o me._n let_v those_o side_n be_v describe_v and_o let_v they_o be_v boy_n open_v pr._n rx_n k_n st_z tu_fw-la and_o vx_n follow_v and_o draw_v these_o right_a line_n so_o tâ—_n and_o vâ—_n and_o from_o the_o point_n oh_o and_o â—_o draw_v to_o the_o plain_a superficies_n of_o the_o circle_n bcde_v perpendicular_a line_n which_o perpendicular_a line_n will_v fall_v upon_o the_o common_a section_n of_o the_o plain_a superficiece_n namely_o upon_o the_o line_n bd_o &_o kn_n by_o the_o 38._o of_o the_o eleven_o for_o that_o the_o plain_a superficiece_n of_o the_o semicircle_n bxd_v kxn_o be_v erect_v perpendicular_o to_o the_o plain_a superficies_n of_o the_o circle_n bcde_v let_v those_o perpendicular_a line_n be_v gz_n demonstration_n and_o sweet_a and_o draw_v a_o right_a line_n from_o the_o point_n z_o to_o the_o point_n w._n and_o forasmuch_o as_o in_o the_o equal_a semicircle_n bxd_v and_o kxn_o the_o right_a line_n boy_n and_o k_n be_v equal_a from_o the_o end_n whereof_o be_v draw_v perpendicular_a line_n oz_n and_o sweet_a therefore_o by_o the_o corollary_n of_o the_o 35._o of_o the_o eleven_o the_o line_n oz_n be_v equal_a to_o the_o line_n sweet_a &_o the_o line_n bz_o be_v equal_a to_o the_o line_n kw_n flussas_n prove_v this_o a_o other_o way_n thus_o forasmuch_o as_o in_o the_o triangle_n swk_n and_o ozb_n the_o two_o angle_n swk_n and_o ozb_n be_v equal_a for_o that_o by_o construction_n they_o be_v right_a angle_n and_o by_o the_o 27._o of_o three_o the_o angle_n wk_n and_o zbo_n be_v equal_a for_o they_o subtend_v equal_a circumference_n sxn_v and_o oxd_v and_o the_o side_n sk_n be_v equal_a to_o the_o side_n ob_fw-la as_o it_o have_v before_o be_v prove_v wherefore_o by_o the_o 26._o of_o the_o first_o the_o other_o side_n &_o angle_n be_v equal_a namely_o the_o line_n oz_n to_o the_o line_n sweet_a and_o the_o line_n bz_o to_o the_o line_n kw_n but_o the_o whole_a line_n basilius_n be_v equal_a to_o the_o whole_a line_n ka_o by_o the_o definition_n of_o a_o circle_n wherefore_o the_o residue_n za_n be_v equal_a to_o the_o residue_n wa_n wherefore_o the_o line_n zw_n be_v a_o parallel_n to_o the_o line_n bk_o by_o the_o 2._o of_o the_o six_o and_o forasmuch_o as_o either_o of_o these_o line_n oz_n &_o sweet_a be_v erect_v perpendicular_o to_o the_o plain_a superficies_n of_o the_o circle_n bcde_v therefore_o the_o line_n oz_n be_v a_o parallel_n to_o the_o line_n sweet_a by_o the_o 6._o of_o the_o eleven_o and_o it_o be_v prove_v that_o it_o be_v also_o equal_a unto_o it_o wherefore_o the_o line_n wz_n and_o so_o be_v also_o equal_a and_o parallel_n by_o the_o 7_o of_o the_o eleven_o and_o the_o 33._o of_o the_o first_o and_o by_o the_o 3._o of_o the_o first_o and_o forasmuch_o as_o wz_n be_v a_o parallel_n to_o so_o but_o zw_n be_v a_o parallel_n be_v to_o kb_v wherefore_o so_o be_v also_o a_o parallel_n to_o kb_v by_o the_o 9_o of_o the_o eleven_o and_o the_o line_n boy_n and_o k_n do_v knit_v they_o together_o wherefore_o the_o four_o side_v figure_n book_n be_v in_o one_o and_o the_o self_n same_o plain_a superficies_n for_o by_o the_o 7._o of_o the_o eleven_o if_o there_o be_v any_o two_o parallel_n right_a line_n and_o if_o in_o either_o of_o they_o be_v take_v a_o point_n at_o allaventure_n a_o right_a line_n draw_v by_o these_o point_n be_v in_o one_o &_o the_o self_n same_o plain_a superficies_n with_o the_o parallel_n and_o by_o the_o same_o reason_n also_o every_o one_o of_o the_o four_o side_v figure_n sopt_v and_o tpkv_n be_v in_o one_o and_o thâ—_z self_n same_o plain_a superficies_n and_o the_o triangle_n vrx_n be_v also_o in_o one_o and_o the_o self_n same_o plain_a superficies_n by_o the_o â—_o of_o the_o eleven_o now_o if_o we_o imagine_v right_a line_n draw_v from_o the_o point_n oh_o s_o p_o t_o r_o five_o to_o the_o point_n 〈◊〉_d there_o shall_v be_v describe_v a_o polyhedron_n or_o a_o solid_a figure_n of_o many_o side_n between_o the_o circumference_n bx_n and_o kx_n compose_v of_o pyramid_n who_o base_n be_v the_o four_o side_v figure_n bkos_n sopt_v tprv_n and_o the_o triangle_n vrx_n and_o top_n the_o point_n a._n and_o if_o in_o every_o one_o of_o the_o side_n kâ—l_o lm_o and_o i_o we_o use_v the_o self_n same_o construction_n that_o we_o do_v in_o bk_o and_o moreover_o in_o the_o other_o three_o quadrant_n or_o quarter_n and_o also_o in_o the_o other_o half_n of_o the_o sphere_n there_o shall_v then_o be_v make_v a_o polyhedron_n or_o solid_a figure_n consist_v of_o many_o side_n describe_v in_o the_o sphere_n which_o polyhedron_n be_v make_v of_o the_o pyramid_n who_o base_n be_v the_o foresay_a four_o side_v figure_n and_o the_o triangle_n vrk_n and_o other_o which_o be_v in_o the_o self_n same_o order_n with_o they_o and_o common_a top_n to_o they_o all_o in_o the_o point_n a._n now_o i_o say_v that_o the_o foresaid_a polihedron_n solid_a of_o many_o side_n touch_v not_o the_o superficies_n of_o the_o less_o sphere_n in_o which_o be_v the_o circle_n fgh_o construction_n draw_v by_o the_o 11._o of_o the_o eleven_o from_o the_o point_n a_o to_o the_o
place_n from_o whence_o first_o it_o begin_v to_o be_v move_v it_o shall_v pass_v by_o the_o point_n f_o g_o h_o and_o the_o octohedron_n shall_v be_v comprehend_v in_o a_o sphere_n i_o say_v also_o that_o it_o be_v comprehend_v in_o the_o sphere_n give_v for_o forasmuch_o as_o the_o line_n lk_n be_v equal_a to_o the_o line_n km_o by_o position_n and_o the_o line_n ke_o be_v common_a to_o they_o both_o and_o they_o contain_v right_a angle_n by_o the_o 3._o definition_n of_o the_o eleven_o therefore_o by_o the_o 4._o of_o the_o first_o the_o base_a le_fw-fr be_v equal_a to_o the_o base_a em_n and_o forasmuch_o as_o the_o angle_n lem_n be_v a_o right_a angle_n by_o the_o 31._o of_o the_o three_o for_o it_o be_v in_o a_o semicircle_n as_o have_v be_v prove_v therefore_o the_o square_a of_o the_o line_n lm_o be_v double_a to_o the_o square_n of_o the_o line_n le_fw-fr by_o the_o 47._o of_o the_o first_o again_o forasmuch_o as_o the_o line_n ac_fw-la be_v equal_a to_o the_o line_n bc_o therefore_o the_o line_n ab_fw-la be_v double_a to_o the_o line_n bc_o by_o the_o definition_n of_o a_o circle_n but_o as_o the_o line_n ab_fw-la be_v to_o the_o line_n bc_o so_o be_v the_o square_a of_o the_o line_n ab_fw-la to_o the_o square_n of_o the_o line_n bd_o by_o the_o corollary_n of_o the_o 8._o and_o _o of_o the_o six_o wherefore_o the_o square_a of_o the_o line_n ab_fw-la be_v double_a to_o the_o square_n of_o the_o line_n bd._n and_o it_o be_v prove_v that_o the_o square_a of_o the_o line_n lm_o be_v double_a to_o the_o square_n of_o the_o line_n le._n wherefore_o the_o square_a of_o the_o line_n bd_o be_v equal_a to_o the_o square_n of_o the_o line_n le._n for_o the_o line_n eh_o which_o be_v equal_a to_o the_o line_n lf_n be_v put_v to_o be_v equal_a to_o the_o line_n db._n wherefore_o the_o square_a of_o the_o line_n ab_fw-la be_v equal_a to_o the_o square_n of_o the_o line_n lm_o wherefore_o the_o line_n ab_fw-la be_v equal_a to_o the_o line_n lm_o and_o the_o line_n ab_fw-la be_v the_o diameter_n of_o the_o sphere_n give_v wherefore_o the_o line_n lm_o be_v equal_a to_o the_o diameter_n of_o the_o sphere_n give_v wherefore_o the_o octoedron_n be_v contain_v in_o the_o sphere_n give_v le._n and_o it_o be_v also_o prove_v that_o the_o diameter_n of_o the_o sphere_n be_v in_o power_n double_a to_o the_o side_n of_o the_o octohedron_n wherefore_o there_o be_v make_v a_o octohedron_n and_o it_o be_v comprehend_v in_o the_o sphere_n give_v wherein_o be_v comprehend_v the_o pyramid_n and_o it_o be_v prove_v that_o the_o diameter_n of_o the_o sphere_n be_v in_o power_n double_a to_o the_o side_n of_o the_o octohedrn_n which_o be_v require_v to_o be_v do_v and_o to_o be_v prove_v certain_a corollary_n add_v by_o flussas_n first_o corollary_n the_o side_n of_o a_o pyramid_n be_v in_o power_n sesquitertia_fw-la to_o the_o side_n of_o a_o octahedron_n inscribe_v in_o the_o same_o sphere_n for_o forasmch_v as_o the_o diameter_n be_v in_o power_n double_a to_o the_o side_n of_o the_o octohedron_n therefore_o of_o what_o part_n the_o diameter_n contain_v in_o power_n 6._o of_o the_o same_o the_o side_n of_o the_o octohedron_n contain_v in_o power_n 3._o but_o of_o what_o part_n the_o diameter_n contain_v 6._o of_o the_o same_o the_o side_n of_o the_o pyramid_n contain_v 4._o by_o the_o 13._o of_o this_o book_n wherefore_o of_o what_o part_n the_o side_n of_o the_o pyramid_n contain_v 4._o of_o the_o same_o the_o side_n of_o the_o octohedron_n contain_v 3._o second_o corollary_n a_o octohedron_n be_v divide_v into_o two_o equal_a and_o like_a pyramid_n the_o common_a base_n of_o these_o pyramid_n be_v set_v upon_o every_o square_n contain_v of_o the_o side_n of_o the_o octohedron_n upon_o which_o square_a be_v set_v the_o â—â—_o triangle_n of_o the_o octohedron_n campane_n which_o pyramid_n be_v by_o the_o â—_o definition_n of_o the_o eleven_o equal_a and_o like_a and_o the_o foresay_a square_n common_a to_o those_o pyramid_n be_v the_o half_a of_o the_o square_n of_o the_o diameter_n of_o the_o sphere_n for_o it_o be_v the_o square_a of_o the_o side_n of_o the_o octohedron_n three_o corollary_n the_o three_o diameter_n of_o the_o octohedron_n do_v cut_v the_o one_o the_o other_o perpendicular_o into_o two_o equal_a part_n in_o the_o centre_n of_o the_o sphere_n which_o contain_v the_o say_a octohedron_n as_o it_o be_v manifest_a by_o the_o three_o diameter_n eglantine_n fh_o and_o lm_o which_o cut_v the_o one_o the_o other_o in_o the_o centre_n king_n equal_o and_o perpendicular_o ¶_o the_o 3._o problem_n the_o 15._o proposition_n to_o make_v a_o solid_a call_v a_o cube_fw-la and_o to_o comprehend_v it_o in_o the_o sphere_n give_v namely_o that_o sphere_n wherein_o the_o former_a two_o solid_n be_v comprehendâ—dâ—_n and_o to_o prove_v that_o the_o diameter_n of_o the_o sphere_n be_v in_o power_n treble_a to_o the_o side_n of_o the_o cube_fw-la take_v the_o diameter_n of_o the_o sphere_n give_v namely_o ab_fw-la and_o divide_v it_o in_o the_o point_n câ—_n so_o that_o let_v the_o line_n ac_fw-la be_v double_a to_o the_o line_n bc_o by_o the_o 9_o of_o the_o six_o and_o upon_o the_o line_n ab_fw-la describe_v a_o semicircle_n adb_o and_o by_o the_o 11._o of_o the_o first_o from_o the_o pâ—ynt_n c_o râ—yse_v up_o unto_o the_o line_n ab_fw-la a_o perpendicular_a line_n cd_o and_o draw_v a_o right_a linâ—_n db._n and_o describe_v a_o square_a efgh_o have_v every_o one_o of_o his_o side_n equal_a to_o the_o line_n db_o and_o from_o the_o point_n e_o f_o g_o h_o raise_v up_o by_o the_o 12._o of_o the_o eleven_o unto_o the_o plain_a superficies_n of_o the_o square_a efgh_o perpendicular_a line_n eke_o fl_fw-mi gm_n and_o hn_o and_o let_v every_o one_o of_o the_o line_n eke_o fl_fw-mi gm_n and_o hn_o be_v put_v equal_a to_o one_o of_o the_o line_n of_o fg_o gh_o or_o he_o which_o be_v the_o side_n of_o the_o square_n and_o draw_v these_o right_a line_n kl_o lm_o mn_a and_o nk_v demonstration_n wherefore_o there_o be_v make_v a_o cube_fw-la namely_o fn_o which_o be_v contain_v under_o six_o equal_a square_n construction_n now_o it_o be_v require_v to_o comprehend_v the_o same_o cube_fw-la in_o the_o sphere_n give_v and_o to_o prove_v that_o the_o diameter_n of_o the_o sphere_n be_v in_o power_n ble_o to_o the_o side_n of_o the_o cube_fw-la demonstration_n draw_v these_o right_a line_n kg_v and_o eglantine_n and_o forasmuch_o as_o the_o angle_n keg_n be_v a_o right_a angle_n for_o that_o the_o line_n ke_o be_v erect_v perpendicular_o to_o the_o plain_a superficies_n eâ—_n and_o therefore_o also_o to_o the_o right_a line_n eglantine_n by_o the_o 2._o definition_n of_o the_o eleven_o wherefore_o a_o semicircle_n describe_v upon_o the_o line_n kg_v shall_v book_n pass_v by_o the_o point_n e._n again_o forasmuch_o as_o the_o line_n fg_o be_v erect_v perpendicular_o to_o either_o of_o these_o line_n fl_fw-mi and_o fe_o by_o the_o definition_n of_o a_o square_a &_o by_o the_o 2._o definition_n of_o the_o eleven_o therefore_o the_o line_n fg_o be_v erect_v perpendicular_o to_o the_o plain_a superficies_n fk_o by_o the_o 4._o of_o the_o eleven_o wherefore_o if_o we_o draw_v a_o right_a line_n from_o the_o point_n f_o to_o the_o point_n king_n the_o line_n gf_o shall_v be_v erect_v perpendicular_o to_o the_o line_n kf_o by_o the_o 2._o definition_n of_o the_o eleven_o and_o by_o the_o same_o reason_n again_o a_o semicircle_n describe_v upon_o the_o line_n gk_o shall_v pass_v also_o by_o the_o point_n f._n and_o likewise_o shall_v it_o pass_v by_o the_o rest_n of_o the_o point_n of_o the_o angle_n of_o that_o cube_fw-la if_o now_o the_o diameter_n kg_v abide_v fix_v the_o semicircle_n be_v turn_v round_o about_o until_o it_o return_v into_o the_o self_n same_o place_n from_o whence_o it_o begin_v first_o to_o be_v move_v the_o cube_fw-la shall_v be_v comprehend_v in_o a_o sphere_n i_o say_v also_o that_o it_o be_v comprehend_v in_o the_o sphere_n give_v demonstration_n for_o forasmuch_o as_o the_o line_n gf_o be_v equal_a to_o the_o linâ—â—e_n and_o the_o angle_n fletcher_n be_v a_o right_a angle_n therefore_o the_o square_a of_o the_o line_n eglantine_n be_v by_o the_o 47._o of_o the_o first_o double_a to_o the_o square_n of_o the_o line_n â—f_n but_o the_o line_n of_o be_v equal_a to_o the_o line_n eke_o wherefore_o the_o square_a of_o the_o line_n eglantine_n be_v double_a to_o the_o square_n of_o the_o line_n eke_o wherefore_o the_o square_n of_o eglantine_n and_o eke_o that_o be_v the_o square_a of_o the_o line_n gk_o by_o the_o 47._o of_o the_o first_o be_v treble_a to_o the_o square_n of_o the_o line_n eke_o and_o forasmuch_o as_o the_o line_n ab_fw-la be_v treble_a to_o the_o line_n bc_o but_o
f._n and_o draw_v these_o right_a line_n favorina_n fb_o fc_o fd_o fe_o wherefore_o those_o line_n do_v divide_v the_o angle_n of_o the_o pentagon_n into_o two_o equal_a part_n in_o the_o point_n a_o b_o c_o d_o e_o by_o the_o 4._o of_o the_o first_o and_o forasmuch_o as_o the_o five_o angle_n that_o be_v at_o the_o point_n f_o aâ—e_fw-fr equal_a to_o four_o right_a angle_n by_o the_o corollary_n of_o the_o 15._o of_o the_o first_o and_o they_o be_v equal_v the_o one_o to_o the_o other_o by_o the_o 8._o of_o the_o first_o therefore_o one_o of_o those_o angle_n as_o â—or_a example_n sake_n the_o angle_n afb_o be_v a_o fiâ—th_v part_n less_o than_o a_o right_a angle_n wherefore_o the_o angle_n remain_v namely_o fab._n &_o abf_n be_v one_o right_a angle_n and_o a_o five_o part_n over_o but_o the_o angle_n fab._n be_v equal_a to_o the_o angle_n fbc_n wherefore_o the_o whole_a angle_n abc_n be_v one_o of_o the_o angle_n of_o the_o pentagon_n be_v a_o right_a angle_n and_o a_o five_o part_v more_o than_o a_o right_a angle_n which_o be_v require_v to_o be_v prove_v ¶_o a_o corollary_n add_v by_o flussas_n now_o let_v we_o teach_v how_o those_o five_o solid_n have_v each_o like_a inclination_n of_o their_o base_n â—iâ—st_v let_v we_o take_v a_o pyramid_n and_o divide_v one_o of_o the_o side_n thereof_o into_o two_o equal_a part_n and_o from_o the_o two_o angle_n opposite_a unto_o that_o side_n dâ—aw_v perpendiculars_n which_o shall_v fall_v upon_o the_o section_n by_o the_o corollary_n of_o the_o 12._o of_o the_o thirteen_o rational_a and_o at_o the_o say_a point_n of_o division_n as_o may_v easy_o be_v prove_v wherefore_o they_o shall_v contain_v the_o angâ—e_n of_o the_o inclination_n of_o the_o plain_n by_o the_o 4._o definition_n of_o the_o eleven_o which_o angle_n be_v subtend_v of_o the_o opposite_a side_n of_o the_o pyramid_n now_o forasmuch_o as_o the_o rest_n of_o the_o angle_n of_o the_o inclination_n of_o the_o plain_n of_o the_o pyramid_n be_v contain_v under_o two_o perpendicular_a line_n of_o the_o triangle_n and_o be_v subtend_v of_o the_o side_n of_o the_o pyramid_n it_o follow_v by_o the_o 8._o of_o the_o firâ—t_o that_o those_o angle_n be_v equal_a wherâ—foâ—e_o by_o the_o 5._o definition_n of_o the_o eleven_o the_o superficiece_n be_v in_o like_a sort_n incline_v the_o one_o to_o the_o other_o 〈…〉_o one_o of_o the_o side_n of_o a_o cube_n be_v divide_v into_o two_o equal_a part_n if_o from_o the_o say_a section_n be_v draw_v in_o two_o of_o the_o base_n thereof_o two_o perpendicular_a line_n they_o shall_v be_v parallel_n and_o equal_a to_o the_o side_n of_o the_o square_n which_o contain_v a_o right_a angle_n and_o forasmuch_o as_o all_o the_o angle_n of_o the_o base_n of_o the_o cube_n be_v right_a angle_n therefore_o those_o perpendicular_n fall_v upon_o the_o section_n of_o the_o side_n common_a to_o the_o two_o base_n shall_v contâ—yne_v a_o right_a angle_n by_o the_o 10._o of_o the_o eleven_o which_o self_n angle_n be_v the_o angle_n of_o inclination_n by_o the_o 4._o definition_n of_o the_o eleven_o and_o be_v subtend_v of_o the_o diameter_n of_o the_o base_a of_o the_o cube_n and_o by_o the_o same_o reason_n may_v we_o prove_v that_o the_o rest_n of_o the_o angle_n of_o the_o inclination_n of_o the_o base_n of_o the_o cube_fw-la be_v right_a angle_n wherefore_o the_o inclination_n of_o the_o superficiece_n of_o the_o cube_fw-la the_o one_o to_o the_o other_o be_v equal_a by_o the_o 5._o definition_n of_o the_o eleven_o in_o a_o octohedron_n take_v the_o diameter_n which_o couple_v the_o two_o opposite_a angle_n incline_v and_o from_o those_o opposite_a angle_n draw_v to_o one_o and_o the_o selâ—e_a same_o side_n of_o the_o octohedron_n in_o two_o base_n thereof_o two_o perpendicular_a line_n which_o shall_v divide_v that_o side_n into_o two_o equal_a part_n and_o perpendicular_o by_o the_o corollary_n of_o the_o 12._o of_o the_o thirteen_o wherefore_o those_o perpendicular_n shall_v contain_v the_o angle_n of_o the_o inclination_n of_o the_o base_n by_o the_o 4._o definition_n of_o the_o eleven_o and_o the_o same_o angle_n be_v subtend_v of_o the_o diameter_n of_o the_o octohedron_n wherefore_o the_o rest_n of_o the_o angle_n after_o the_o same_o manner_n describe_v in_o the_o rest_n of_o the_o base_n be_v comprehend_v and_o subtend_v of_o equal_a side_n shall_v by_o the_o 8._o of_o the_o first_o be_v equal_a the_o one_o to_o the_o other_o and_o therefore_o the_o inclination_n of_o the_o plain_n in_o the_o octohedron_n shall_v by_o the_o 5._o definition_n of_o the_o eleven_o be_v equal_a in_o a_o icosahedron_n let_v there_o be_v draw_v from_o the_o angle_n of_o two_o of_o the_o base_n incline_v to_o one_o side_n common_a to_o both_o the_o say_v base_n perpendicular_n which_o shall_v contain_v the_o angle_n of_o the_o inclination_n of_o the_o base_n by_o the_o 4._o definition_n of_o the_o eleven_o which_o angle_n be_v subtend_v of_o the_o right_a line_n which_o subtend_v the_o angle_n of_o the_o pentagon_n which_o contain_v five_o side_n of_o the_o icosahedron_n by_o the_o 16._o of_o this_o book_n for_o it_o couple_v the_o two_o opposite_a angle_n of_o the_o triangle_n which_o be_v join_v together_o wherefore_o the_o rest_n of_o the_o angle_n of_o the_o inclination_n of_o the_o base_n be_v after_o the_o same_o manner_n find_v out_o they_o shall_v be_v contain_v under_o equal_a side_n and_o subtend_v of_o equal_a base_n and_o therefore_o by_o the_o 8._o of_o the_o fiâ—st_n those_o angle_n shall_v be_v equal_a wherefore_o also_o all_o the_o inclination_n of_o the_o base_n of_o the_o icosahedron_n the_o one_o to_o the_o other_o shalbâ—_n equal_a by_o the_o 5._o definition_n of_o the_o eleven_o in_o a_o dodecahedron_n incline_v from_o the_o two_o opposite_a angle_n of_o two_o next_o pentagon_n draw_v to_o their_o common_a side_n perpendicular_a line_n pass_v by_o the_o centre_n of_o the_o say_a pentagon_n which_o shall_v where_o they_o fall_v divide_v the_o side_n into_o two_o equal_a part_n by_o the_o 3._o of_o the_o three_o for_o the_o base_n of_o a_o dodecahedron_n be_v contain_v in_o a_o circle_n and_o the_o angle_n contaynâ—d_v under_o those_o perpendicular_a line_n be_v the_o inclination_n of_o those_o base_n by_o the_o 4._o definition_n of_o the_o eleven_o and_o the_o foresay_a opposite_a angle_n be_v couple_v by_o a_o right_a line_n equal_a to_o the_o right_a line_n which_o couple_v the_o opposite_a section_n into_o two_o equal_a part_n of_o the_o side_n of_o the_o dodecahedron_n by_o the_o 33._o of_o the_o first_o for_o they_o couple_v together_o the_o half_a side_n of_o the_o dodecahedron_n which_o half_n be_v parallel_n and_o equal_a by_o the_o 3._o corollary_n of_o the_o 17._o of_o this_o book_n which_o couple_v line_n also_o be_v equal_a by_o the_o same_o corollary_n wherefore_o the_o angle_n be_v contain_v of_o equal_a perpendicular_a line_n and_o subtend_v of_o equal_a couple_a line_n shall_v by_o the_o 8._o of_o the_o first_o be_v equal_a and_o they_o be_v the_o angle_n of_o the_o inclination_n wherefore_o the_o base_n of_o the_o dodecahedron_n be_v in_o like_a sort_n incline_v the_o one_o to_o the_o other_o by_o the_o 5._o definition_n of_o the_o eleven_o flussas_n after_o this_o teach_v how_o to_o know_v the_o rationality_n or_o irrationality_n of_o the_o side_n of_o the_o triangle_n which_o contain_v the_o angle_n of_o the_o inclination_n of_o the_o superficiece_n of_o the_o foresay_a body_n in_o a_o pyramid_n the_o angle_n of_o the_o inclination_n be_v contain_v under_o two_o perpâ—dicular_a line_n of_o the_o triangle_n rational_a and_o be_v subtend_v of_o the_o side_n of_o the_o pyramid_n now_o the_o side_n of_o the_o pyramid_n be_v in_o power_n sesquitertia_fw-la to_o the_o perpendicular_a line_n by_o the_o corollary_n of_o the_o 12._o of_o this_o book_n and_o therefore_o the_o triangle_n contain_v of_o those_o perpendicular_a line_n and_o the_o side_n of_o pyramid_n have_v his_o side_n rational_a &_o commensurable_a in_o power_n the_o one_o to_o the_o other_o forasmuch_o as_o the_o two_o side_n of_o a_o cube_n or_o right_a line_n equal_a to_o they_o subtend_v under_o the_o diameter_n of_o one_o of_o the_o base_n rational_a do_v make_v the_o angle_n of_o the_o inclination_n and_o the_o diameter_n of_o the_o cube_fw-la be_v in_o power_n sesquialter_fw-la to_o the_o diameter_n of_o the_o base_a which_o diameter_n of_o the_o base_a be_v in_o power_n double_a to_o the_o side_n by_o the_o 47._o of_o the_o first_o therefore_o those_o line_n be_v rational_a and_o commensurable_a in_o power_n in_o a_o octohedron_n rational_a who_o two_o perpendicular_n of_o the_o base_n contain_v the_o angle_n of_o the_o inclination_n of_o the_o octohedron_n which_o angle_n also_o be_v subtend_v of_o the_o diameter_n of_o the_o octohedron_n the_o diameter_n be_v in_o power_n
overthrow_v and_o overwhelm_v the_o whole_a world_n he_o be_v utter_o rude_a and_o ignorant_a in_o the_o greek_a tongue_n so_o that_o certain_o he_o never_o read_v euclid_n in_o the_o greek_a nor_o of_o like_a translate_v out_o of_o the_o greek_a but_o have_v it_o translate_v out_o of_o the_o arabike_a tongue_n the_o arabian_n be_v man_n of_o great_a study_n and_o industry_n and_o common_o great_a philosopher_n notable_a physician_n and_o in_o mathematical_a art_n most_o expert_a so_o that_o all_o kind_n of_o good_a learning_n flourish_v and_o reign_v amongst_o they_o in_o a_o manner_n only_o these_o man_n turn_v whatsoever_o good_a author_n be_v in_o the_o greek_a tongue_n of_o what_o art_n and_o knowledge_n so_o ever_o it_o be_v into_o the_o arabike_a tongue_n and_o from_o thence_o be_v many_o of_o they_o turn_v into_o the_o latin_a and_o by_o that_o mean_v many_o greek_a author_n come_v to_o the_o hand_n of_o the_o latin_n and_o not_o from_o the_o first_o fountain_n the_o greek_a tongue_n wherein_o they_o be_v first_o write_v as_o appear_v by_o many_o word_n of_o the_o arabike_a tongue_n yet_o remain_v in_o such_o book_n as_o be_v zenith_n nadir_n helmuayn_fw-mi helmuariphe_n and_o infinite_a such_o other_o which_o arabian_n also_o in_o translate_n such_o greek_a work_n be_v accustom_v to_o add_v as_o they_o think_v good_a &_o for_o the_o full_a understanding_n of_o the_o author_n many_o thing_n as_o be_v to_o be_v see_v in_o diverse_a author_n as_o namely_o in_o theodosius_n de_fw-la sphera_fw-la where_o you_o see_v in_o the_o old_a translation_n which_o be_v undoubteldy_a out_o of_o the_o arabike_a many_o proposition_n almost_o every_o three_o or_o four_o leaf_n some_o such_o copy_n of_o euclid_n most_o likely_a do_v campanus_n follow_v wherein_o he_o find_v those_o proposition_n which_o he_o have_v more_o &_o above_o those_o which_o be_v find_v in_o the_o greek_a set_v out_o by_o hypsicles_n and_o that_o not_o only_o in_o this_o 15._o book_n but_o also_o in_o the_o 14._o book_n wherein_o also_o you_o find_v many_o proposition_n more_o they_o be_v find_v in_o the_o greek_a set_v out_o also_o by_o hypsicles_n likewise_o in_o the_o book_n before_o you_o shall_v find_v many_o proposition_n add_v and_o many_o invert_v and_o set_v out_o of_o order_n far_o otherwise_o than_o they_o be_v place_v in_o the_o greek_a examplar_n flussas_n also_o a_o diligent_a restorer_n of_o euclid_n a_o man_n also_o which_o have_v well_o deserve_v of_o the_o whole_a art_n of_o geometry_n have_v add_v moreover_o in_o this_o book_n as_o also_o in_o the_o former_a 14._o book_n he_o add_v 8._o proposition_n 9_o proposition_n of_o his_o own_o touch_v the_o inscription_n and_o circumscription_n 〈…〉_o body_n very_o singular_a â—ndoubtedly_o and_o witty_a all_o which_o for_o that_o nothing_o shall_v want_v to_o the_o desirous_a lover_n of_o knowledge_n i_o have_v faithful_o with_o no_o small_a pain_n turn_v and_o whereas_o flâ—ssâ—â—_n in_o the_o begin_n of_o the_o eleven_o book_n namely_o in_o the_o end_n of_o the_o definition_n there_o â—eâ—_n put_v two_o definition_n of_o the_o inscription_n and_o circumscription_n of_o solid_n or_o corporal_a figure_n within_o or_o about_o the_o one_o the_o other_o which_o certain_o be_v not_o to_o be_v reject_v yet_o for_o that_o until_o this_o present_a 15._o book_n there_o be_v no_o mention_n make_v of_o the_o inscription_n or_o circumscription_n of_o these_o body_n i_o think_v it_o not_o so_o convenient_a thâ—râ—_n to_o place_v they_o but_o to_o refer_v they_o to_o the_o begin_n of_o this_o 15._o book_n where_o they_o be_v in_o manner_n of_o necessity_n require_v to_o the_o elucidation_n of_o the_o proposition_n and_o dâ—monstrationâ—_n of_o the_o same_o the_o definition_n be_v these_o definition_n 1._o a_o solid_a figure_n be_v then_o â—aid_v to_o be_v inscribe_v in_o a_o solid_a figure_n when_o the_o angle_n of_o the_o figure_n inscribe_v touch_n together_o at_o one_o time_n either_o the_o angle_n of_o the_o figure_n circumscribe_v or_o the_o superficiece_n or_o the_o side_n definition_n 2._o a_o solid_a figure_n be_v then_o say_v to_o be_v circumscribe_v about_o a_o solid_a figure_n when_o together_o at_o one_o time_n either_o the_o angle_n or_o the_o superficiece_n or_o the_o side_n of_o the_o figure_n circumscribe_v â—ouch_v the_o angle_n of_o the_o figure_n inscribe_v in_o the_o fourâ—â—_n book_n in_o the_o definition_n of_o the_o inscription_n or_o circumscription_n of_o plain_a rectiline_n figure_v one_o with_o in_o or_o about_o a_o other_o be_v require_v that_o all_o the_o angle_n of_o the_o figuâ—â—_n inscribe_v shall_v at_o one_o time_n touch_v all_o the_o side_n of_o the_o figure_n circumscribe_v but_o in_o the_o five_o regular_a solid_n â—o_o who_o chief_o these_o two_o definition_n pertain_v for_o that_o the_o number_n of_o their_o angle_n superficiece_n &_o side_n be_v not_o equal_a one_o compare_v to_o a_o other_o it_o be_v not_o of_o necessity_n that_o all_o the_o angle_n of_o the_o solid_a inscribe_v shall_v together_o at_o one_o time_n touch_v either_o all_o the_o angle_n or_o all_o the_o superficiece_n or_o all_o the_o side_n of_o the_o solid_a circumscribe_v but_o it_o be_v sufficient_a that_o those_o angle_n of_o the_o inscribe_v solid_a which_o touch_n do_v at_o one_o time_n together_o each_o touch_n some_o one_o angle_n of_o the_o figure_n circumscribe_v or_o some_o one_o base_a or_o some_o one_o side_n so_o that_o if_o the_o angle_n of_o the_o inscribe_v figure_n do_v at_o one_o time_n touch_v the_o angle_n of_o the_o figure_n circumscribe_v none_o of_o they_o may_v at_o the_o same_o time_n touch_v either_o the_o base_n or_o the_o side_n of_o the_o same_o circumscribe_v figure_n and_o so_o if_o they_o touch_v the_o base_n they_o may_v touch_v neither_o angle_n nor_o side_n and_o likewise_o if_o they_o touch_v the_o side_n they_o may_v touch_v neither_o angle_n nor_o base_n and_o although_o sometime_o all_o the_o angle_n of_o the_o figure_n inscribe_v can_v not_o touch_v either_o the_o angle_n or_o the_o base_n or_o the_o side_n of_o the_o figure_n circumscribe_v by_o reason_n the_o number_n of_o the_o angle_n base_n or_o side_n of_o the_o say_a figure_n circumscribe_v want_v of_o the_o number_n of_o the_o angle_n of_o the_o â—igure_n inscribe_v yet_o shall_v those_o angle_n of_o the_o inscribe_v figure_n which_o touch_n so_o touch_v that_o the_o void_a place_n leave_v between_o the_o inscribe_v and_o circumscribe_v figure_n shall_v on_o every_o side_n be_v equal_a and_o like_a as_o you_o may_v afterward_o in_o this_o fifteen_o book_n most_o plain_o perceive_v ¶_o the_o 1._o proposition_n the_o 1._o problem_n in_o a_o cube_n give_v to_o describe_v admonish_v a_o trilater_n equilater_n pyramid_n svppose_v that_o the_o cube_fw-la give_v be_v abcdefgh_o in_o the_o same_o cube_fw-la it_o be_v require_v to_o inscribe_v a_o tetrahedron_n draw_v these_o right_a line_n ac_fw-la construction_n ce_fw-fr ae_n ah_o eh_o hc_n demonstration_n now_o it_o be_v manifest_a that_o the_o triangle_n aec_o ahe_o ahc_n and_o i_o be_v equilater_n for_o their_o side_n be_v the_o diameter_n of_o equal_a square_n wherefore_o aech_n be_v a_o trilater_n equilater_n pyramid_n or_o tetrahedron_n &_o it_o be_v inscribe_v in_o the_o cube_fw-la give_v by_o the_o first_o definition_n of_o this_o book_n which_o be_v require_v to_o be_v do_v ¶_o the_o 2._o proposition_n the_o 2._o problem_n in_o a_o trilater_n equilater_n pyramid_n give_v to_o describe_v a_o octohedron_n svppose_v that_o the_o trilater_n equilater_n pyramid_n give_v be_fw-mi abcd_o who_o side_n let_v be_v divide_v into_o two_o equal_a part_n in_o the_o point_n e_o z_o i_o king_n l_o t._n construction_n and_o draw_v these_o 12._o right_a line_n ez_o zi_n je_n kl_o lt_n tk_n eke_o kz_o zl_n li_n it_o and_o te_fw-la which_o 12._o right_a line_n be_v demonstration_n by_o the_o 4._o of_o the_o first_o equal_a for_o they_o subtend_v equal_a plain_a angle_n of_o the_o base_n of_o the_o pyramid_n and_o those_o equal_a angle_n be_v contain_v under_o equal_a side_n namely_o under_o the_o half_n of_o the_o side_n of_o the_o pyramid_n wherefore_o the_o triangle_n tkl_n tli_n tie_v tek_v zkl_n zli_n zib_n zek_n be_v equilater_n and_o they_o limitate_v and_o contain_v the_o solid_a tklezi_n wherefore_o the_o solid_a tklezi_n be_v a_o octohedron_n by_o the_o 23._o definition_n of_o the_o eleven_o and_o the_o angle_n of_o the_o same_o octohedron_n do_v touch_v the_o side_n of_o the_o pyramid_n abcd_o in_o the_o point_n e_o z_o i_o t_o edward_n l._n wherefore_o the_o octohedron_n be_v inscribe_v in_o the_o pyramid_n by_o the_o 1._o definition_n of_o this_o book_n wherefore_o in_o the_o trilater_n equilater_n pyramid_n give_v be_v inscribe_v a_o octohedron_n which_o be_v require_v to_o be_v do_v a_o corollary_n add_v by_o flussas_n hereby_o it_o be_v manifest_a that_o a_o pyramid_n be_v cut_v into_o two_o
triple_a to_o a_o extreme_a &_o mean_a propartion_n forasmuch_o as_o in_o the_o â—â—_o corollary_n of_o the_o 13._o of_o the_o fiuâ—â—enth_n it_o be_v prove_v that_o the_o side_n of_o a_o dodecahedron_n inscribe_v in_o a_o cube_fw-la be_v the_o less_o segment_n of_o the_o side_n of_o that_o cube_fw-la divide_v by_o a_o extreme_a and_o mean_a proportion_n and_o the_o side_n of_o the_o dodecahedron_n circumscribe_v about_o the_o same_o cube_fw-la be_v the_o great_a segment_n of_o the_o side_n of_o the_o same_o cube_fw-la which_o thing_n also_o be_v teach_v in_o the_o 13._o of_o the_o fifteen_o the_o side_n of_o the_o dodecahedron_n circumscribe_v shall_v be_v to_o the_o side_n of_o the_o dodecahedron_n inscribe_v as_o the_o great_a segment_n of_o a_o right_a line_n divide_v by_o a_o extreme_a and_o mean_a proportion_n be_v to_o the_o less_o segment_n of_o the_o same_o which_o proportion_n be_v call_v a_o extreme_a and_o mean_a proportion_n by_o the_o definition_n and_o by_o the_o 30._o of_o six_o but_o the_o proportion_n of_o like_a solid_a prolihedron_n be_v triple_a to_o the_o proportion_n of_o the_o sideâ—_n of_o like_a proportion_n by_o the_o corollary_n of_o the_o 17._o of_o the_o twelve_o wherefore_o the_o proportion_n of_o the_o dodecahedron_n circumscribe_v about_o the_o cube_fw-la be_v to_o the_o dodecahedron_n inscribe_v in_o the_o same_o cube_fw-la in_o triple_a proportion_n of_o the_o side_n join_v together_o by_o a_o extreme_a and_o mean_a proportion_n the_o proportion_n therefore_o of_o â—_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 triple_a to_o a_o extreme_a and_o mean_a proportion_n the_o 3._o proposition_n in_o every_o equiangle_n and_o equilater_n pentagon_n a_o perpendicular_a draw_v from_o one_o of_o the_o angle_n to_o the_o base_a be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n by_o a_o right_a line_n subtend_v the_o same_o angle_n svppose_v that_o abcdf_n be_v aâ—_z equiangle_n and_o equilater_n pentagon_n construction_n and_o from_o one_o of_o the_o angle_n namely_o from_o a_o let_v there_o be_v draw_v to_o the_o base_a cd_o a_o perpendicular_a agnostus_n and_o let_v the_o line_n bf_o subtend_v the_o angle_n bap_n demonstration_n which_o line_n bf_o let_v the_o line_n ad_fw-la cut_v in_o the_o point_n i_o then_o i_o say_v that_o the_o line_n bf_o cut_v the_o line_n agnostus_n by_o a_o extreme_a and_o mean_a proportion_n for_o forasmuch_o as_o the_o angle_n give_v and_o gab_n be_v equal_a by_o the_o 27._o of_o the_o three_o and_o the_o angle_n aâ—f_n and_o afb_o be_v equal_a by_o the_o 5._o of_o the_o firstâ—_o therefore_o the_o â—â—gles_n remain_v at_o the_o point_n e_o of_o the_o triangle_n aeb_fw-mi and_o aef_n be_v equal_a for_o that_o they_o be_v the_o residue_v of_o two_o right_a angle_n by_o the_o corollary_n of_o the_o 32._o of_o the_o first_o but_o the_o angle_n egc_n be_v by_o construction_n a_o right_a angleâ—_n wherefore_o the_o line_n bf_o &_o cd_o be_v parallel_n by_o the_o 28._o of_o the_o first_o wherefore_o as_o the_o line_n diego_n be_v to_o the_o line_n ja_z so_o iâ—_z the_o line_n ge_z to_o the_o line_n ea_fw-la by_o the_o 2._o of_o the_o six_o but_o the_o line_n dam_n be_v in_o the_o poinâ—_n i_o divide_v by_o a_o extreme_a and_o mean_a proportion_n by_o the_o 8._o of_o the_o thirteen_o wherefore_o the_o line_n ga_o be_v in_o the_o point_n e_o divide_v by_o a_o extreme_a and_o mean_a proportion_n by_o the_o â—_o of_o the_o fourteen_o wherefore_o in_o every_o equiangle_n and_o equilater_n pentagon_n a_o perpendicular_a draw_v from_o one_o of_o the_o angle_n to_o the_o base_a be_v divide_v by_o a_o exâ—reme_a and_o mean_a proportion_n by_o a_o right_a line_n subtend_v the_o same_o angleâ—_n ¶_o a_o corollary_n the_o line_n which_o subtend_v the_o angle_n of_o a_o pentagon_n be_v a_o parallel_n to_o the_o side_n opposite_a unto_o the_o angle_n as_o it_o be_v manifest_a in_o the_o line_n â—f_a and_o cd_o the_o 4._o proposition_n if_o from_o the_o angle_n of_o the_o base_a of_o a_o book_n pyramid_n be_v draw_v to_o the_o opposite_a side_n right_a line_n cut_v the_o say_a side_n by_o a_o extreme_a and_o mean_a proportion_n they_o shall_v contain_v the_o bise_n of_o the_o icosahedron_n inscribe_v in_o the_o pyramid_n which_o base_n shall_v be_v inscribe_v in_o a_o equilater_n triangle_n who_o angle_n cut_v the_o side_n of_o the_o base_a of_o the_o pyramid_n by_o a_o extreme_a and_o mean_a proportion_n construction_n svppose_v that_o abg_o be_v the_o base_a of_o a_o pyramid_n in_o which_o let_v be_v inscribe_v a_o equilater_n triangle_n fkh_n which_o be_v do_v by_o devide_v the_o side_n into_o two_o equal_a part_n and_o in_o â—his_n triangle_n let_v there_o be_v inscribe_v the_o base_a of_o the_o icosahedron_n inscribe_v in_o the_o pyramid_n which_o be_v describe_v by_o devide_v the_o side_n fk_o kh_o hf_o by_o a_o extreme_a &_o mean_a proportion_n in_o the_o point_n c_o d_o e_o by_o the_o 19_o of_o the_o fivetenth_n again_o let_v the_o side_n of_o the_o pyramid_n namely_o ab_fw-la bg_o and_o ga_o be_v divide_v by_o a_o extreme_a and_o meâ—ne_a proportion_n in_o the_o point_n i_o m_o l_o by_o the_o 30._o of_o the_o six_o and_o draw_v these_o right_a line_n be_o bl_o give_v demonstration_n then_o i_o say_v that_o those_o line_n describe_v the_o triangle_n cde_o of_o the_o icosahedron_n for_o forasmuch_o as_o the_o line_n bg_o and_o fh_o be_v parallel_n by_o the_o 2._o of_o the_o six_o by_o the_o point_n d_o let_v the_o line_n odn_n be_v draw_v parallel_n to_o either_o of_o the_o line_n bg_o &_o fh_o wherefore_o the_o triangle_n hdn_n shall_v be_v like_v to_o the_o triangle_n hkg_v by_o the_o corollary_n of_o the_o 2._o of_o the_o six_o wherefore_o either_o of_o these_o line_n dn_o and_o nh_v shall_v be_v equal_a to_o the_o line_n dh_o the_o great_a segment_n of_o the_o line_n kh_o or_o fh_o and_o forasmuch_o as_o the_o line_n foyes_n be_v a_o parallel_n to_o the_o line_n hk_o and_o the_o line_n odd_a to_o the_o line_n fhâ—_n the_o line_n odd_a shall_v be_v equal_a to_o the_o whole_a line_n fh_o in_o the_o parallelogram_n fodh_n by_o the_o 34._o of_o the_o â—irst_n wherefore_o as_o the_o whole_a line_n fh_o be_v to_o the_o greaâ—er_a segment_n fe_o so_o shall_v the_o line_n equal_a to_o they_o be_v namely_o the_o line_n odd_a to_o the_o line_n dn_o by_o the_o 7._o of_o the_o five_o wherefore_o the_o line_n on_o be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n in_o the_o point_n d_o by_o the_o 2._o of_o the_o fourteen_o but_o the_o triangle_n aod_n afe_n and_o abm_n be_v like_o the_o one_o to_o the_o other_o and_o so_o also_o be_v the_o triangle_n adn_n aeh_n and_o amg_n by_o the_o corollary_n of_o the_o second_o of_o the_o sixths_n wherefore_o as_z fe_o be_v to_o eh_o so_o be_v odd_a to_o dn_o and_o bm_n to_o mg_o wherefore_o the_o line_n be_o cut_v the_o line_n fh_o and_o on_o like_v unto_o the_o line_n bg_o in_o the_o point_n e_o d_o m_o describe_v ed_z the_o side_n of_o the_o triangle_n of_o the_o icosahedron_n ecd_v which_o be_v describe_v in_o the_o section_n e_o c_o d_o by_o supposition_n and_o by_o the_o same_o reason_n the_o line_n bl_o and_o give_v shall_v describe_v the_o other_o side_n ec_o and_o cd_o of_o the_o same_o triangle_n by_o the_o point_n e_o let_v there_o be_v draw_v to_o give_v a_o parallel_a line_n peq_n now_o forasmuch_o as_o the_o line_n bm_n and_o fe_o be_v parallel_n the_o line_n be_o be_v in_o the_o point_n e_o cut_v like_a to_o the_o line_n ab_fw-la in_o the_o point_n fletcher_n by_o the_o 2._o of_o the_o six_o wherefore_o the_o line_n ae_n be_v equal_a to_o the_o line_n em_n and_o unto_o the_o line_n em_n also_o be_v equal_a either_o of_o the_o line_n gd_o and_o diego_n which_o â—re_n cut_v lâ—ke_v unto_o the_o foresaid_a line_n again_o forasmuch_o as_o in_o the_o triangle_n adi_n the_o line_n diego_n and_o ep_n be_v parallel_n as_o the_o line_n diego_n be_v to_o the_o line_n ep_n so_o be_v the_o line_n ad_fw-la to_o the_o line_n ae_n but_o as_o the_o line_n ad_fw-la be_v to_o the_o line_n ae_n so_o be_v the_o line_n dg_o to_o the_o line_n eq_n by_o the_o 2._o of_o the_o six_o wherefore_o as_o the_o line_n diego_n be_v to_o the_o line_n ep_n so_o be_v the_o line_n dg_o to_o the_o line_n eq_n and_o alternate_o as_o the_o line_n diego_n be_v to_o the_o line_n dg_o so_o be_v the_o line_n ep_n to_o the_o line_n eq_n but_o the_o line_n diego_n and_o ig_n be_v equal_a wherefore_o also_o the_o line_n ep_n and_o eq_n be_v equal_a and_o forasmuch_o as_o the_o line_n ah_o be_v equal_a to_o the_o line_n fh_o who_o great_a segment_n be_v the_o line_n hnâ—_n therefore_o the_o whole_a
line_n a_fw-la be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n in_o the_o point_n h_o by_o the_o â—_o of_o the_o thirteen_o but_o as_o the_o line_n a_o be_v to_o the_o line_n ah_o so_o be_v the_o line_n ad_fw-la to_o the_o line_n ae_n by_o the_o 2._o of_o six_o for_o the_o lineâ—_z fh_o and_o on_o be_v parallel_n and_o again_o as_o the_o line_n ad_fw-la be_v to_o the_o line_n ae_n so_o by_o the_o same_o be_v the_o line_n agnostus_n to_o the_o line_n aq_n and_o the_o line_n ai_fw-fr to_o the_o line_n ap_n for_o the_o line_n pq_n and_o give_v be_v parallel_n wherefore_o the_o line_n agnostus_n and_o a_z be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n in_o the_o point_v q_n &_o p_o &_o the_o line_n aq_n shall_v be_v the_o great_a segment_n of_o the_o line_n agnostus_n or_o ab_fw-la and_o forasmuch_o as_o the_o wholâ—_n line_n agnostus_n be_v to_o the_o great_a segment_n aq_n as_o the_o great_a segment_n a_z be_v to_o the_o residue_n ap_n the_o line_n aâ—_z shall_v be_v the_o less_o segment_n of_o the_o whole_a line_n aâ—_z or_o ag._n wherefore_o the_o liâ—â—_n peq_n which_o by_o the_o point_n e_o pass_v parallelwise_o to_o the_o line_n give_v cut_v the_o line_n agnostus_n and_o basilius_n by_o a_o extreme_a and_o mean_a proportion_n in_o the_o point_n queen_n and_o p._n and_o by_o the_o same_o reason_n the_o line_n â—r_n which_o by_o the_o point_n c_o pass_v parallelwise_o to_o the_o line_n be_o shall_v fall_v upon_o the_o section_n p_o and_o r_o so_o also_o shall_v the_o line_n rq_n which_o by_o the_o point_n d_z pass_v parallelwise_o to_o the_o line_n bl_o fall_v upoâ—_n the_o section_n rq_n wherefore_o either_o of_o the_o line_n pe_n and_o eq_n shall_v be_v equal_a to_o the_o line_n cd_o in_o the_o parallelogram_n pd_v and_o qc_n by_o the_o 34._o of_o the_o first_o and_o forasmuch_o as_o the_o line_n pe_n and_o eq_n be_v equal_a the_o line_n pc_n cr_n rd_o and_o dq_n shall_v be_v likewise_o equal_a wherefore_o the_o triangle_n prq_n iâ—â—quilater_n and_o cut_v the_o side_n of_o the_o base_a of_o the_o pyramiâ—_n in_o the_o point_n p_o q_o r_o by_o a_o extreme_a and_o mean_a proportion_n and_o in_o it_o be_v inscribe_v the_o base_a ecd_v of_o the_o icosahedron_n contain_v in_o the_o foresaid_a pyramid_n if_o therefore_o from_o the_o angle_n of_o the_o base_a of_o a_o pyramid_n be_v draw_v to_o the_o opposite_a side_n right_a line_n cut_v the_o say_a side_n by_o a_o extreme_a and_o mean_a proportion_n they_o shall_v contain_v the_o base_a of_o the_o icosahedron_n inscribe_v in_o the_o pyramid_n which_o base_a shall_v be_v inscribe_v in_o a_o equilater_n triangle_n who_o angle_n cut_v the_o side_n of_o the_o base_a of_o the_o pyramid_n by_o a_o extreme_a &_o mean_a proportion_n ¶_o a_o corollary_n the_o side_n of_o a_o icosahedron_n inscribe_v in_o a_o octohedron_n be_v the_o great_a segment_n of_o the_o line_n which_o be_v draw_v from_o the_o angle_n of_o the_o base_a of_o the_o octohedron_n cut_v the_o opposite_a side_n by_o a_o extreme_a and_o mean_a proportion_n for_o by_o the_o 16._o of_o the_o fifteen_o fkh_n be_v the_o base_a of_o the_o octohedron_n which_o contain_v the_o base_a of_o the_o icosahedron_n cde_v unto_o which_o triangle_n fkh_n the_o triangle_n hkg_n be_v equal_a as_o have_v be_v prove_v by_o the_o point_n h_o draw_v unto_o the_o line_n menander_n a_o parallel_a line_n ht_o cut_v the_o line_n dn_o in_o the_o point_n s._n wherefore_o es_fw-ge dt_fw-ge and_o et_fw-la be_v parallelogram_n and_o therefore_o the_o line_n eh_o and_o mt_n be_v equal_a and_o the_o line_n em_n and_o ht_o be_v like_o cut_n in_o the_o point_n d_o and_o s_o by_o the_o 34._o of_o the_o first_o wherefore_o the_o great_a segment_n of_o the_o line_n ht_o be_v the_o line_n his_o which_o be_v equal_a to_o ed_z the_o side_n of_o the_o icosahedron_n but_o by_o the_o 2._o of_o the_o six_o the_o line_n tk_n be_v cut_v like_o to_o the_o line_n hk_o by_o the_o parallel_n dm_o and_o therefore_o by_o the_o 2._o of_o the_o fourteen_o it_o be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n but_o the_o line_n tm_n be_v equal_a to_o the_o line_n eh_o wherefore_o also_o the_o line_n tk_n be_v equal_a to_o the_o line_n of_o or_o dh_o wherefore_o the_o residue_v eh_o and_o tg_n be_v equal_a for_o the_o whole_a line_n fh_a and_o kg_o be_v equal_a wherefore_o kg_o the_o side_n of_o the_o triangle_n hkg_n be_v in_o the_o point_n t_o divide_v by_o a_o extreme_a and_o mean_a proportion_n in_o the_o point_n t_o by_o the_o right_a line_n ht_o and_o the_o great_a segment_n thereof_o be_v the_o line_n ed_z the_o side_n of_o the_o icosahedron_n inscribe_v in_o the_o octohedron_n who_o base_a be_v the_o triangle_n hkg_v or_o the_o triangle_n fkh_n which_o be_v equal_a to_o the_o triangle_n hkg_v by_o the_o 16._o of_o the_o fifteen_o ¶_o the_o 5._o proposition_n the_o side_n of_o a_o pyramid_n divide_v by_o a_o extreme_a and_o mean_a proportion_n make_v the_o less_o segment_n in_o power_n double_a to_o the_o side_n of_o the_o icosahedron_n inscribe_v in_o it_o svppose_v that_o abg_o be_v the_o base_a of_o a_o pyramid_n construction_n and_o let_v the_o base_a of_o the_o icosahedron_n inscribe_v in_o it_o be_v cde_o describe_v of_o three_o right_a line_n which_o be_v draw_v from_o the_o angle_n of_o the_o base_a abg_o cut_v the_o opposite_a side_n by_o a_o extreme_a and_o mean_a proportion_n by_o the_o former_a proposition_n namely_o of_o these_o three_o line_n be_o by_o and_o give_v then_o i_o say_v that_z a_z the_o less_o segment_n of_o the_o side_n aâ—_z be_v in_o power_n duple_n to_o ce_fw-fr the_o side_n of_o the_o icosahedron_n for_o forasmuch_o as_o by_o the_o former_a proposition_n it_o be_v prove_v that_o the_o triangle_n cde_o be_v inscribe_v in_o a_o equilater_n triangle_n demonstration_n who_o angle_n cut_v the_o side_n of_o abg_o the_o base_a of_o the_o pyramid_n by_o a_o extreme_a and_o mean_a proportion_n let_v that_o triangle_n be_v fhk_v cut_v the_o line_n ab_fw-la in_o the_o point_n f._n wherefore_o the_o less_o segment_n favorina_n be_v equal_a to_o the_o segment_n ai_fw-fr by_o the_o 2._o of_o the_o fourteen_o for_o the_o line_n ab_fw-la and_o ag_z be_v cut_v like_o moreover_o the_o side_n fh_o of_o the_o triangle_n fhk_n be_v in_o the_o point_n d_o cut_n into_o two_o equal_a part_n as_o in_o the_o former_a proposition_n it_o be_v prove_v and_o fce_v also_o by_o the_o same_o be_v a_o parallelogram_n wherefore_o the_o line_n ce_fw-fr and_o fd_o be_v equal_a by_o the_o 33_o of_o the_o first_o and_o forasmuch_o as_o the_o line_n fh_o subtend_v the_o angle_n bag_n of_o a_o equilater_n triangle_n which_o angle_n be_v contain_v under_o the_o great_a segment_n ah_o and_o the_o less_o segment_n afâ—_n therefore_o the_o line_n fh_o be_v in_o power_n double_a to_o the_o line_n of_o or_o to_o the_o line_n ai_fw-fr the_o less_o segment_n by_o the_o corollary_n of_o the_o 16._o of_o the_o fifteen_o but_o the_o same_o line_n fh_o be_v in_o power_n quadruple_a to_o the_o line_n ce_fw-fr by_o the_o 4._o of_o the_o second_o for_o the_o line_n fh_o be_v double_a to_o the_o line_n ce_fw-fr wherefore_o the_o line_n ai_fw-fr be_v the_o half_a of_o the_o square_n of_o the_o line_n fh_o be_v in_o power_n duple_n to_o the_o line_n ce_fw-fr to_o which_o the_o line_n fh_o be_v in_o power_n quadruple_a wherefore_o the_o side_n agnostus_n of_o the_o pyramid_n be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n make_v thâ—_z less_o segment_n ai_fw-fr in_o power_n duple_n to_o the_o side_n ce_fw-fr of_o the_o icosahedron_n inscribe_v in_o it_o ¶_o a_o corollary_n the_o side_n of_o a_o icosahedron_n inscribe_v in_o a_o pyramid_n be_v a_o residual_a line_n for_o the_o diameter_n of_o the_o sphere_n which_o contain_v the_o five_o regular_a body_n be_v rational_a be_v in_o power_n sesquialtera_fw-la to_o the_o side_n of_o the_o pyramid_n by_o the_o 13._o of_o the_o thirteen_o and_o therefore_o the_o side_n of_o the_o pyramid_n be_v rational_a by_o the_o definition_n which_o side_n be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n make_v the_o less_o segment_n a_o residual_a line_n by_o the_o 6._o of_o the_o thirteen_o wherefore_o the_o side_n of_o the_o icosahedron_n be_v commensurable_a to_o the_o same_o less_o segment_n for_o the_o square_n of_o the_o side_n of_o the_o icosahedron_n be_v the_o half_a of_o the_o square_n of_o the_o say_v less_o segment_n be_v a_o residual_a line_n by_o that_o which_o be_v add_v after_o the_o 103._o of_o the_o ten_o book_n ¶_o the_o 6._o proposition_n the_o side_n of_o a_o cube_n contain_v in_o power_n half_o the_o side_n of_o a_o equilater_n triangular_a pyramid_n inscribe_v in_o the_o say_a
of_o the_o icosidodecahedron_n now_o let_v we_o prove_v that_o it_o be_v contain_v in_o the_o sphere_n give_v who_o diameter_n be_v nl_o give_v forasmuch_o as_o perpendicular_n draw_v from_o the_o centre_n of_o the_o dodecahedron_n to_o the_o middle_a section_n of_o his_o side_n be_v the_o half_n of_o the_o line_n which_o couple_n the_o opposite_a middle_a section_n of_o the_o side_n of_o the_o dodecahedron_n by_o the_o 3._o corollary_n of_o the_o 17._o of_o the_o thirteen_o which_o line_n also_o by_o the_o same_o corollary_n do_v in_o the_o centre_n divide_v the_o one_o the_o other_o into_o two_o equal_a part_n therefore_o right_a line_n draw_v from_o that_o point_n to_o the_o angle_n of_o the_o icosidodecahedron_n which_o be_v set_v in_o those_o middle_a section_n be_v equal_a which_o line_n be_v 30._o in_o number_n according_a to_o the_o number_n of_o the_o side_n of_o the_o dodecahedron_n for_o every_o one_o of_o the_o angle_n of_o the_o icosidodecahedron_n be_v set_v in_o the_o middle_a section_n of_o every_o one_o of_o the_o side_n of_o the_o dodecahedron_n wherefore_o make_v the_o centre_n the_o centre_n of_o the_o dodecahedron_n and_o the_o space_n any_o one_o of_o the_o line_n draw_v from_o the_o centre_n to_o the_o middle_a section_n describe_v a_o sphere_n and_o it_o shall_v pass_v by_o all_o the_o angle_n of_o the_o icosidodecahedron_n and_o shall_v contain_v it_o and_o forasmuch_o as_o the_o diameter_n of_o this_o solid_a be_v that_o right_a line_n who_o great_a segment_n be_v the_o side_n of_o the_o cube_fw-la inscribe_v in_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 which_o side_n be_v ni_fw-fr by_o supposition_n wherefore_o that_o solid_a be_v contain_v in_o the_o sphere_n give_v who_o diameter_n be_v put_v to_o be_v the_o line_n nl_o ¶_o a_o advertisement_n of_o flussas_n to_o the_o understanding_n of_o the_o nature_n of_o this_o icosidodecahedron_n you_o must_v well_o conceive_v the_o passion_n and_o propriety_n of_o both_o those_o solid_n of_o who_o base_n iâ—_z consist_v namely_o of_o the_o icosahedron_n and_o of_o the_o dodecahedron_n and_o although_o in_o it_o the_o base_n be_v place_v opposite_o yet_o hâ—uâ—_n they_o one_o to_o the_o other_o one_o &_o the_o sâ—me_a inclination_n by_o reason_n whereof_o there_o he_o hide_v in_o it_o the_o action_n and_o pâ—â—â—â—ons_n of_o the_o other_o regular_a solid_n and_o i_o will_v have_v think_v iâ—_z not_o impertinent_a to_o the_o purpose_n to_o have_v set_v forth_o the_o inscription_n and_o circumscription_n of_o this_o solid_a if_o wâ—nt_n of_o tâ—hâ—_n have_v not_o hinder_v but_o to_o the_o end_n the_o reader_n may_v the_o better_a aâ—â—aine_n to_o the_o understanding_n thereof_o i_o have_v here_o follow_v brief_o set_v forth_o how_o it_o may_v in_o oâ—_n about_o every_o one_o of_o the_o five_o regular_a solid_n be_v inscribe_v or_o circumscribe_v by_o the_o help_n whereof_o â—e_n may_v with_o small_a travail_n or_o rather_o none_o at_o all_o so_o that_o he_o have_v well_o pâ—ysed_v and_o consider_v the_o dâ—monstrations_n pertain_v to_o the_o foresay_a fiâ—e_n regular_a solid_n demonstrate_v both_o the_o inscription_n of_o the_o say_a solid_n in_o it_o and_o the_o inscription_n of_o it_o in_o the_o say_a solid_n ¶_o of_o the_o inscription_n and_o circumscription_n of_o a_o icosidodecâ—hedron_n a_o icosidodecahedron_n may_v contain_v the_o other_o five_o regular_a body_n for_o it_o will_v receive_v the_o angle_n of_o a_o dodecahedron_n in_o the_o centre_n of_o the_o triangle_n which_o subtend_v the_o solid_a angle_n of_o the_o dodecahedron_n which_o solid_a angle_n be_v 20._o in_o number_n and_o be_v place_v in_o the_o same_o order_n in_o which_o the_o solid_a angle_n of_o the_o dodecahedron_n take_v away_o or_o subtend_v by_o they_o be_v and_o by_o that_o reason_n it_o shall_v receive_v a_o cube_n and_o a_o pyramid_n contain_v in_o the_o dodecahedron_n when_o as_o the_o angle_n of_o the_o one_o be_v set_v in_o the_o angle_n of_o the_o other_o a_o icosidodecahedron_n receive_v a_o octohedron_n in_o the_o angle_n cut_v the_o six_o opposite_a section_n of_o the_o dodecahedron_n even_o as_o if_o it_o be_v a_o simple_a dodecahedron_n and_o it_o contain_v a_o icosahedron_n place_v the_o 12._o angle_n of_o the_o icosahedron_n in_o the_o self_n same_o centre_n of_o the_o 12._o pentagons_n it_o may_v also_o by_o the_o same_o reason_n be_v inscribe_v in_o every_o one_o of_o the_o five_o regular_a body_n namely_o in_o a_o pyramid_n if_o you_o place_v 4._o triangular_a base_n concentrical_a with_o 4._o base_n of_o the_o pyramid_n after_o the_o same_o manner_n that_o you_o inscribe_v a_o icosahedron_n in_o a_o pyramiâ—_n so_o likewise_o may_v it_o be_v inscribe_v in_o a_o octohedron_n if_o you_o make_v 8._o base_n thereof_o concentrical_a with_o the_o 8._o base_n of_o the_o octohedron_n it_o shall_v also_o be_v inscribe_v in_o a_o cube_n if_o you_o place_v the_o angle_n which_o receive_v the_o octohedron_n inscribe_v in_o it_o in_o the_o centre_n of_o the_o base_n of_o the_o cube_n moreover_o you_o shall_v inscribe_v it_o in_o a_o icosahedron_n when_o the_o triangle_n compass_v in_o of_o the_o pentagon_n base_n be_v concentrical_a with_o the_o triangle_n which_o make_v a_o solid_a angle_n of_o the_o icosahedron_n final_o it_o shall_v be_v inscribe_v in_o a_o dodecahedron_n if_o you_o place_v every_o one_o of_o the_o angle_n thereof_o in_o the_o middle_a section_n of_o the_o side_n of_o the_o dodecahedron_n according_a to_o the_o order_n of_o the_o construction_n thereof_o the_o opposite_a plain_a superficiece_n also_o of_o this_o solid_a be_v parallel_n for_o the_o opposite_a solid_a angle_n be_v subtend_v of_o parallel_n plain_a superficiece_n as_o well_o in_o the_o angle_n of_o the_o dodecahedron_n subtend_v by_o â—riâ—ngleâ—_n aâ—_z in_o the_o angle_n of_o the_o icosahedron_n subtend_v of_o pentagons_n which_o thing_n may_v easy_o be_v demonstrate_v moreover_o in_o thiâ—_z solid_a be_v infinitâ—_n propertiâ—â—_n &_o passion_n spring_v of_o the_o solideâ—_n whereof_o â—t_a be_v compose_v whereforâ—_n it_o be_v manifest_a that_o a_o dodecahedron_n &_o a_o icosahedron_n mix_v be_v transform_v into_o one_o &_o the_o self_n same_o solid_a of_o a_o icosidodecahedron_n a_o cube_fw-la also_o and_o a_o octohedron_n be_v mix_v and_o alter_v into_o a_o other_o solid_a namely_o into_o one_o and_o the_o same_o exoctohedron_n but_o a_o pyramid_n be_v transform_v into_o a_o simple_a and_o perfect_a solid_a namely_o into_o a_o octohedron_n if_o we_o will_v frame_v these_o two_o solid_n join_v together_o into_o one_o solid_a this_o only_a must_v we_o observe_v in_o the_o pentagon_n of_o a_o dodecahedron_n inscribe_v a_o like_a pentagon_n so_o that_o let_v the_o angle_n of_o the_o pentagon_n inscribe_v be_v set_v in_o the_o middle_a section_n of_o the_o side_n of_o the_o pentagon_n circumscribe_v and_o then_o upon_o the_o say_a pentagon_n inscribe_v let_v there_o be_v set_v a_o solid_a angle_n of_o a_o icosahedron_n and_o so_o observe_v the_o self_n same_o order_n in_o every_o one_o of_o the_o base_n of_o the_o dodecahedron_n and_o the_o solid_a angle_n of_o the_o icosahedron_n set_v upon_o these_o pentagon_n shall_v produce_v a_o solid_a consist_v of_o the_o whole_a dodecahedron_n and_o of_o the_o whole_a icosahedron_n in_o like_a sort_n if_o in_o every_o base_a of_o the_o icosahedron_n the_o side_n be_v divide_v into_o two_o equal_a part_n be_v inscribe_v a_o equilater_n triangle_n and_o upon_o every_o one_o of_o those_o equilater_n triangle_n be_v set_v a_o solid_a angle_n of_o a_o dodecahedron_n there_o shall_v be_v produce_v the_o self_n same_o solid_a consist_v of_o the_o whole_a icosahedron_n &_o of_o the_o whole_a dodecahedron_n and_o after_o the_o same_o order_n if_o in_o the_o base_n of_o a_o cube_fw-la be_v inscribe_v square_n subtend_v the_o solid_a angle_n of_o a_o octohedron_n or_o in_o the_o base_n of_o a_o octohedron_n be_v inscribe_v equilater_n triangle_n subtend_v the_o solid_a angle_n of_o a_o cube_fw-la there_o shall_v be_v produce_v a_o solid_a consist_v of_o either_o of_o the_o whole_a solid_n namely_o of_o the_o whole_a cube_fw-la and_o of_o the_o whole_a octohedron_n but_o equilater_n triangle_n inscribe_v in_o the_o base_n of_o a_o pyramid_n have_v their_o angle_n set_v in_o the_o middle_a section_n of_o the_o side_n of_o the_o pyramid_n and_o the_o solid_a angle_n of_o a_o pyramid_n set_v upon_o the_o say_a equilater_n triangle_n there_o shall_v be_v produce_v a_o solid_a consist_v of_o two_o equal_a and_o like_a pyramid_n and_o now_o if_o in_o these_o solid_n thus_o compose_v you_o take_v away_o the_o solid_a angle_n there_o shall_v be_v restore_v again_o the_o first_o compose_v solid_n namely_o the_o solid_a angle_n take_v away_o from_o a_o dodecahedron_n and_o a_o icosahedron_n compose_v into_o one_o there_o shall_v be_v leave_v a_o icosidodecahedâ—on_n the_o solid_a angle_n take_v away_o
subtend_v angle_n of_o triangle_n like_v unto_o the_o triangle_n who_o angle_n the_o line_n ac_fw-la bf_o and_o pl_z subtend_n be_v cut_v into_o two_o equal_a part_n in_o the_o point_n z_o an_o i_o by_o the_o 4._o of_o the_o six_o so_o also_o be_v the_o other_o line_n nv_n xm_o d_n qt_fw-la which_o be_v equal_a unto_o the_o line_n hk_o &_o ge_z cut_v in_o like_a sort_n and_o they_o shall_v cut_v the_o line_n ac_fw-la bf_o and_o pl_z like_z wherefore_o the_o line_n ko_o which_o be_v equal_a to_o rz_n shall_v make_v the_o great_a segment_n the_o line_n ro_n which_o be_v equal_a to_o the_o line_n zk_n for_o the_o great_a segment_n of_o the_o rz_n be_v the_o line_n zk_n and_o therefore_o the_o line_n oi_o shall_v be_v the_o less_o segment_n when_o as_o the_o whole_a line_n ri_n be_v equal_a to_o the_o whole_a line_n rz_n wherefore_o the_o square_n of_o the_o whole_a line_n ko_o and_o of_o the_o less_o segment_n oi_o be_v triple_a to_o the_o square_n of_o the_o great_a segment_n ro_n by_o the_o 4._o of_o the_o thirteen_o wherefore_o the_o line_n ki_v which_o contain_v in_o power_n the_o two_o line_n ko_o and_o oi_o be_v in_o power_n triple_a to_o the_o line_n ro_n by_o the_o 47._o of_o the_o first_o for_o the_o angle_n koi_n be_v a_o right_a angle_n and_o forasmuch_o as_o the_o line_n fe_o and_o fg_o which_o be_v the_o less_o segment_n of_o the_o side_n of_o the_o octohedron_n be_v equal_a and_o the_o line_n fk_o be_v common_a to_o they_o both_o and_o the_o angle_n kfg_n and_o kfe_n of_o the_o triangle_n of_o the_o octohedron_n be_v equal_a the_o base_n kg_v and_o ke_o shall_v by_o the_o 4_o of_o the_o first_o be_v equal_a and_o therefore_o the_o angle_n kie_fw-mi and_o kig_n which_o they_o subtend_v be_v equal_a by_o the_o 8._o of_o the_o first_o wherefore_o they_o be_v right_a angle_n by_o the_o 13._o of_o the_o first_o wherefore_o the_o right_a line_n ke_o which_o contain_v in_o power_n the_o two_o line_n ki_v and_o â—e_n by_o the_o 47._o of_o the_o first_o be_v in_o power_n quadruple_a to_o the_o line_n ro_n or_o je_n for_o the_o line_n ri_n be_v prove_v to_o be_v in_o power_n triple_a to_o the_o same_o line_n ro_n but_o the_o line_n ge_z be_v double_a to_o the_o line_n je_n wherefore_o the_o line_n ge_z be_v also_o in_o power_n 〈…〉_o pf_n and_o by_o the_o same_o reason_n may_v be_v prove_v that_o the_o â—est_n of_o the_o eleven_o solid_a angle_n of_o the_o 〈◊〉_d be_v 〈…〉_o the_o section_n of_o every_o one_o of_o the_o side_n of_o the_o octohedron_n namely_o in_o the_o point_n e_o n_o five_o h_o â—_o m_o â—_o d_o s_o q_o t._n wherefore_o there_o be_v 12._o angle_n of_o the_o icosahedron_n moreover_o forasmuch_o as_o every_o one_o of_o the_o base_n of_o the_o octohedron_n do_v each_o contain_v triangle_n of_o the_o icosahedron_n 〈…〉_o pyramid_n abcâ—fp_n which_o be_v the_o half_a of_o the_o octohedron_n the_o triangle_n fcp_n receive_v in_o thâ—_z section_n of_o his_o side_n the_o â—_o triangle_n gm_n and_o the_o triangle_n cpb_n contain_v the_o triangle_n nxs_n and_o thâ—_z triangle_n â—ap_n contain_v the_o triangle_n hnd_n and_o moreover_o the_o triangle_n apf_n contain_v the_o triangle_n edge_n and_o the_o same_o may_v be_v prove_v in_o the_o opposite_a pyramid_n abcfl_fw-mi wherefore_o there_o shall_v be_v eight_o triangleâ—_n and_o forasmuch_o as_o beside_o these_o triangle_n to_o every_o one_o of_o the_o solid_a angle_n of_o the_o octohedron_n 〈◊〉_d subtend_v two_o triangle_n as_o the_o triangle_n keg_n and_o meg_n to_o the_o angle_n f_o and_o the_o triangle_n hnv_n and_o xnv_n to_o the_o angle_n b_o also_o the_o triangle_n nd_n and_o â—ds_n to_o the_o angle_n p_o likewise_o the_o triangleâ—_n dhk_n and_o qhk_v to_o the_o angle_n a_o moreover_o the_o triangle_n eqt_a and_o vqt_a to_o the_o angle_n l_o and_o final_o the_o triangle_n sxm_n and_o txm_n to_o the_o angle_n c_o these_o 12._o triangle_n be_v add_v to_o thâ—â—_n for_o 〈◊〉_d triangle_n shall_v produce_v _o triangle_n equal_a and_o equilater_v couple_v together_o which_o shall_v male_a a_o icosahedron_n by_o the_o 25._o definition_n of_o the_o eleven_o and_o it_o shall_v be_v inscribe_v in_o the_o octohedron_n give_v abcâ—â—l_o by_o the_o first_o definition_n of_o this_o book_n for_o the_o 1â—_o angle_n thereof_o be_v set_v in_o 1â—_o like_a section_n of_o the_o side_n of_o the_o octohedron_n wherefore_o in_o a_o octohedron_n give_v be_v inscribe_v a_o icosahedron_n ¶_o first_n corollary_n the_o side_n of_o a_o equilater_n triangle_n be_v divide_v by_o a_o extreme_a and_o mean_a proportion_n a_o right_a line_n subtend_v within_o the_o triangle_n the_o angle_n which_o be_v contain_v under_o the_o great_a segment_n and_o the_o less_o be_v in_o power_n duple_n to_o the_o less_o segment_n of_o the_o same_o side_n for_o the_o line_n ke_o which_o subtend_v the_o angle_n kfe_n of_o the_o triangle_n afl_fw-mi which_o angle_n kfe_n be_v contain_v under_o the_o two_o segment_n kf_a &_o fe_o be_v prove_v equal_a 〈◊〉_d the_o line_n hk_o which_o contain_v in_o power_n the_o two_o less_o segment_n ha_o and_o ak_o by_o the_o 47._o of_o the_o â—â—rst_a foâ—_n 〈◊〉_d angle_n hak_n be_v 〈…〉_o second_o corollary_n the_o base_n of_o the_o icosahedron_n be_v concentrical_a that_o be_v have_v one_o and_o the_o self_n same_o centre_n with_o the_o base_n of_o the_o octohedron_n which_o contain_v it_o for_o suppose_v that_o 〈…〉_o octohedron_n 〈◊〉_d ecd_v the_o base_a of_o a_o icosahedron_n and_o let_v the_o centre_n of_o the_o base_a abg_o be_v the_o point_n f._n and_o draw_v these_o right_a line_n favorina_n fb_o fc_o and_o fe_o now_o than_o the_o 〈…〉_o to_o the_o two_o line_n fb_o and_o bc_o for_o they_o be_v line_n draw_v from_o the_o centre_n and_o be_v also_o less_o segment_n and_o they_o contain_v the_o 〈…〉_o ¶_o the_o 17._o problem_n the_o 17._o proposition_n in_o a_o octohedron_n give_v to_o inscribe_v a_o dodecahedron_n construction_n svppose_v that_o the_o octohedron_n give_v be_v abgdec_n who_o 12._o â—ides_n let_v be_v cut_v by_o a_o extreme_a and_o mean_a proportion_n as_o in_o the_o former_a proposition_n it_o be_v manifest_a that_o of_o the_o right_a line_n which_o couple_v thâ—se_a section_n be_v make_v 20._o triangle_n of_o which_o 8._o be_v concentrical_a with_o the_o base_n of_o the_o octohedron_n by_o the_o second_o corollary_n of_o the_o former_a proposition_n if_o therefore_o in_o every_o one_o of_o the_o centre_n of_o the_o 20._o triangle_n be_v inscribe_v by_o the_o 1._o of_o this_o book_n every_o one_o of_o the_o â—â—_o â—â—gles_n of_o the_o dodecahedron_n demonstration_n we_o shall_v find_v that_o â—_o angle_n of_o the_o dodecahedron_n be_v set_v in_o the_o 8._o centre_n of_o the_o base_n of_o the_o octohedron_n namely_o these_o angle_n i_o u._fw-mi ct_n oh_o m_o a_o p_o and_o x_o and_o of_o the_o other_o 12._o solid_a angle_n there_o be_v two_o in_o the_o centre_n of_o the_o two_o triangle_n which_o have_v one_o side_n common_a under_o every_o one_o of_o the_o solid_a angle_n of_o the_o octohedron_n namely_o under_o the_o solid_a angle_n a_o the_o two_o solid_a angle_n king_n z_o under_o the_o solid_a angle_n b_o the_o two_o solid_a angle_n h_o t_o under_o the_o solid_a angle_n g_o the_o two_o solid_a angle_n in_o v_z under_o the_o solid_a angle_n d_o the_o two_o solid_a angle_n f_o l_o under_o the_o solid_a angle_n e_o the_o two_o solid_a angle_n s_o n_o under_o the_o solid_a angle_n c_o the_o two_o solid_a angle_n queen_n r_o and_o forasmuch_o as_o in_o the_o octohedron_n be_v six_o solid_a angle_n under_o they_o shall_v be_v subtend_v 12._o solid_a angle_n of_o the_o dodâ—cahedron_n and_o so_o be_v mâ—de_v 20._o solid_a angle_n compose_v of_o 12._o equal_a and_o equilater_v superficial_a pentagon_n as_o it_o be_v 〈◊〉_d by_o the_o 5._o of_o this_o book_n which_o therefore_o contain_v a_o dodecahedron_n by_o the_o 24._o definition_n of_o the_o eleven_o and_o it_o be_v inscribe_v in_o the_o octohedron_n by_o the_o 1._o definition_n of_o this_o book_n for_o that_o every_o one_o of_o the_o base_n of_o the_o octohedron_n do_v receive_v angle_n thereof_o wherefore_o in_o a_o octohedron_n give_v be_v inscribe_v a_o dodecahedron_n ¶_o the_o 18._o problem_n the_o 18._o proposition_n in_o a_o trilater_n and_o equilater_n pyramid_n to_o inscribe_v a_o cube_n construction_n svppose_v that_o there_o be_v a_o trilater_n equilater_n pyramid_n who_o base_a let_v be_v abc_n and_o â—oppe_v the_o point_n d._n and_o let_v it_o be_v comprehend_v in_o a_o sphereâ—_n by_o the_o 13._o of_o the_o 〈◊〉_d and_o lâ—â—_n the_o centre_n of_o that_o sphere_n be_v the_o point_n e._n and_o from_o the_o solid_a angle_n a_o b_o c_o d_o draw_v right_a line_n pass_v by_o the_o centre_n e_o unto_o the_o opposite_a base_n of_o