angle_n namely_o the_o inward_a angle_n general_o be_v equal_a unto_o the_o even_a number_n from_o two_o forward_a but_o the_o outward_a angle_n be_v equal_a but_o to_o 4._o right_a angle_n h._n 5_o a_o rectilineall_a be_v either_o a_o triangle_n or_o a_o triangulate_a as_o before_o of_o a_o line_n be_v make_v a_o lineate_v so_o here_o in_o like_a manner_n of_o a_o triangle_n be_v make_v a_o triangulate_a 6_o a_o triangle_n be_v a_o rectilineall_a figure_n comprehend_v of_o three_o rightlines_n 21._o dj_o therefore_o 7_o a_o triangle_n be_v the_o prime_a figure_n of_o rectilineal_n a_o triangle_n or_o threeside_v figure_n be_v the_o prime_n or_o most_o simple_a figure_n of_o all_o rectilineal_n for_o among_o rectilineall_a figure_n there_o be_v none_o of_o two_o side_n for_o two_o right_a line_n can_v enclose_v a_o figure_n what_o be_v mean_v by_o a_o prime_a figure_n be_v teach_v at_o the_o 7._o e._n iiij_o and_o 8_o if_o a_o infinite_a right_a line_n do_v cut_v the_o angle_n of_o a_o triangle_n it_o do_v also_o cut_v the_o base_a of_o the_o same_o vitell._n 29._o to_o i_o 9_o any_o two_o side_n of_o a_o triangle_n be_v great_a than_o the_o other_o let_v the_o triangle_n be_v a_o e_o i_o i_o say_v the_o side_n a_o i_o be_v short_a than_o the_o two_o side_n a_o e_o and_o e_z i_z because_o by_o the_o 6._o e_fw-la ij_o a_o right_a line_n be_v between_o the_o same_o bound_n the_o short_a therefore_o 10_o if_o of_o three_o right_a line_n give_v any_o two_o of_o they_o be_v great_a than_o the_o other_o and_o periphery_n describe_v upon_o the_o end_n of_o the_o one_o at_o the_o distance_n of_o the_o other_o two_o shall_v meet_v the_o ray_n from_o that_o meeting_n unto_o the_o say_a end_n shall_v make_v a_o triangle_n of_o the_o line_n give_v and_o 11_o if_o two_o equal_a periphery_n from_o the_o end_n of_o a_o right_a line_n give_v and_o at_o his_o distance_n do_v meet_v liâ—es_v draw_v from_o the_o meeting_n unto_o the_o say_a end_n shall_v make_v a_o equilater_n triangle_n upon_o the_o line_n give_v 1_o p.j._n 12_o if_o a_o right_a line_n in_o a_o triangle_n be_v parallel_n to_o the_o base_a it_o do_v cut_v the_o shank_n proportional_o and_o contrariwise_o 2_o p_o five_o i_o as_o here_o in_o the_o triangle_n a_o e_o i_o let_v o_o u._fw-mi be_v parallel_n to_o the_o base_a and_o let_v a_o three_o parallel_n be_v understand_v to_o be_v in_o the_o top_n a_o therefore_o by_o the_o 28._o e.u._n the_o intersegment_n be_v proportional_a the_o converse_n be_v force_v out_o of_o the_o antecedent_n because_o otherwise_o the_o whole_a shall_v be_v less_o than_o the_o part_n for_o if_o o_fw-mi u._fw-mi be_v not_o parallel_v to_o the_o base_a e_o i_o then_z y_z u_z be_v here_o by_o the_o grant_n and_o by_o the_o antecedent_n see_v a_o o_o o_o e_o a_o y_z y_fw-es e_fw-es be_v proportional_a and_o the_o first_o a_o o_o be_v lesser_a than_o a_o y_o the_o three_o o_o e_o the_o second_o must_v be_v lesser_a than_o y_z e_z the_o four_o that_o be_v the_o whole_a than_o the_o part_n 13_o the_o three_o angle_n of_o a_o triangle_n be_v equal_a to_o two_o right_a angle_n 32._o p_o i_o therefore_o 14._o any_o two_o angle_n of_o a_o triangle_n be_v less_o than_o two_o right_a angle_n for_o if_o three_o angle_n be_v equal_a to_o two_o right_a angle_n then_o be_v two_o lesser_a than_o two_o right_a angle_n and_o 15_o the_o one_o side_n of_o any_o triangle_n be_v continue_v or_o draw_v out_o the_o outter_n angle_n shall_v be_v equal_a to_o the_o two_o inner_a opposite_a angle_n therefore_o 16_o the_o say_a outter_n angle_n be_v great_a than_o either_o of_o the_o inner_a opposite_a angle_n 16._o p_o i_o this_o be_v a_o consectary_n follow_v necessary_o upon_o the_o next_o former_a consectary_n 17_o if_o a_o triangle_n be_v equicrural_a the_o angle_n at_o the_o base_a be_v equal_a and_o contrariwise_o 5._o and_o 6._o p.j._n therefore_o 18_o if_o the_o equal_a shank_n of_o a_o triangle_n be_v continue_v or_o draw_v out_o the_o angle_n under_o the_o base_a shall_v be_v equal_a between_o themselves_o and_o 19_o if_o a_o triangle_n be_v a_o equilater_n it_o be_v also_o a_o equiangle_n and_o contrariwise_o and_o 20_o the_o angle_n of_o a_o equilater_n triangle_n do_v countervail_v two_o three_o part_n of_o a_o right_a angle_n regio_fw-la 23._o p_o i_o for_o see_v that_o 3._o angle_n be_v equal_a to_o 2._o 1._o must_v needs_o be_v equal_a to_o â…”_n and_o 21_o six_o equilater_n triangle_n do_v fill_v a_o place_n 22_o the_o great_a side_n of_o a_o triangle_n subtend_v the_o great_a angle_n and_o the_o great_a angle_n be_v subtend_v of_o the_o great_a side_n 19_o and_o 18._o p_o i_o the_o converse_n be_v manifest_a by_o the_o same_o figure_n as_o let_v the_o angle_v a_o e_o i_o be_v great_a than_o the_o angle_n a_o i_o e._n therefore_o by_o the_o same_o 9_o e_z iij._o it_o be_v great_a in_o base_a for_o what_o be_v there_o speak_v of_o angle_n in_o general_a be_v here_o assume_v special_o of_o the_o angle_n in_o a_o triangle_n 23_o if_o a_o right_a line_n in_o a_o triangle_n do_v cut_v the_o angle_n in_o two_o equal_a part_n it_o shall_v cut_v the_o base_a according_a to_o the_o reason_n of_o the_o shank_n and_o contrariwise_o 3._o p_o five_o i_o the_o mingle_a proportion_n of_o the_o side_n and_o angle_n do_v now_o remain_v to_o be_v handle_v in_o the_o last_o place_n the_o converse_n likewise_o be_v demonstrate_v in_o the_o same_o figure_n for_o as_o e_z a_o be_v to_o a_o i_o so_o be_v e_z o_o to_z o_o i_fw-it and_o so_o be_v e_z a_o to_o a_o u._fw-mi by_o the_o 12_o e_fw-la therefore_o a_o i_o and_o a_o u._fw-mi be_v equal_a item_n the_o angle_n e_o a_fw-fr o_o and_o o_o a_o i_o be_v equal_a to_o the_o angle_n at_o you_o and_o i_o by_o the_o 21._o e_o uâ—_n which_o be_v equal_a between_o themselves_o by_o the_o 17._o e._n of_o geometry_n the_o seven_o book_n of_o the_o comparison_n of_o triangle_n 1_o equilater_n triangle_n be_v equiangle_n 8._o p.j._n thus_o forre_v of_o the_o geometry_n or_o affection_n and_o reason_n of_o one_o triangle_n the_o comparison_n of_o two_o triangle_n one_o with_o another_o do_v follow_v and_o first_o of_o their_o rate_n or_o reason_n out_o of_o their_o side_n and_o angle_n whereupon_o triangle_n between_o themselves_o be_v say_v to_o be_v equilater_n and_o equiangle_n first_o out_o of_o the_o equality_n of_o the_o side_n be_v draw_v also_o the_o equality_n of_o the_o angle_n triangle_n therefore_o be_v here_o joint_o call_v equilater_n who_o side_n be_v several_o equal_a the_o first_o to_o the_o first_o the_o second_o to_o the_o second_o the_o three_o to_o the_o three_o although_o every_o several_a triangle_n be_v inequilaterall_a therefore_o the_o equality_n of_o the_o side_n do_v argue_v the_o equality_n of_o the_o angle_n by_o the_o 7._o e_fw-la iij._o as_o here_o 2_o if_o two_o triangle_n be_v equal_a in_o angle_n either_o the_o two_o equicrurals_n or_o two_o of_o equal_a either_o shank_n or_o base_a of_o two_o angle_n they_o be_v equilater_n 4._o and_o 26._o p_o i_o oh_o thus_o if_o two_o triangle_n be_v equal_a in_o their_o angle_n either_o in_o two_o angle_n contain_v under_o equal_a foot_n or_o in_o two_o angle_n who_o side_n or_o base_a of_o both_o be_v equal_a those_o angle_n be_v equilater_n h._n this_o element_n have_v three_o part_n or_o it_o do_v conclude_v two_o triangle_n to_o be_v equilater_n three_o way_n 1._o the_o first_o part_n be_v apparent_a thus_o let_v the_o two_o triangle_n be_v a_o e_o i_o and_o o_o u._fw-mi y_fw-mi because_o the_o equal_a angle_n at_o a_o and_o o_o be_v equicrural_a therefore_o they_o be_v equal_a in_o base_a by_o the_o 7._o e_fw-la iij._o 3_o the_o three_o part_n be_v thus_o force_v in_o the_o triangle_n a_o e_o i_o and_o o_o u._fw-mi y_fw-mi let_v the_o angle_n at_o e_o and_o i_o and_o u_z and_o y_z be_v equal_a as_o afore_o and_o a_o e._n the_o base_a of_o the_o angle_n at_o i_o be_v equal_a to_o o_fw-mi u._fw-mi the_o base_a of_o angle_n at_o y_o i_o say_v that_o the_o two_o triangle_n give_v be_v equilater_n for_o if_o the_o side_n e_o i_o be_v great_a than_o the_o side_n u._fw-mi y_fw-mi let_v e_o s_o be_v cut_v off_o equal_a to_o it_o and_o draw_v the_o right_a line_n a_o s._n therefore_o by_o the_o antecedent_n the_o two_o triangle_n a_o e_o s_o and_o o_o u._fw-mi y_fw-mi equal_a in_o the_o angle_n of_o their_o equal_a shank_n be_v equiangle_n and_o the_o angle_n a_o s_o e_o be_v equal_a to_o the_o angle_n o_o y_fw-fr u._fw-mi which_o be_v equal_a by_o the_o grant_n unto_o the_o angle_n a_o i_o e._n therefore_o a_o s_o e_o be_v equal_a to_o a_o i_o e_o
e_fw-la seven_o and_o so_o forth_o of_o the_o rest_n the_o middle_a triangle_n the_o equal_a angle_n be_v substract_v shall_v have_v their_o other_o angle_n equal_a and_o therefore_o they_o also_o shall_v be_v equiangle_n and_o alike_o by_o the_o same_o secondary_o the_o triangle_n a_o e_o u._fw-mi and_o y_z s_z m_z e_o i_z o_o and_o s_o r_o l_o e_o o_o u._fw-mi and_o s_o l_o m_o to_o wit_n alike_o between_o themselves_o be_v by_o the_o 1_o e_fw-la uj_o in_o a_o double_a reason_n of_o their_o homologall_a side_n e_o u._fw-mi s_z m_z e_z o_o s_o l_o which_o reason_n be_v the_o same_o by_o mean_n of_o the_o common_a side_n therefore_o three_o triangle_n be_v in_o the_o same_o reason_n and_o therefore_o they_o be_v proportional_a and_o by_o the_o three_o composition_n as_o one_o of_o the_o antecedent_n be_v unto_o one_o of_o the_o consequentsâ—_n so_o be_v the_o whole_a quinquangle_n to_o the_o whole_a 5._o a_o triangulate_a be_v a_o quadrangle_n or_o a_o multangle_n the_o part_n of_o this_o partition_n be_v in_o euclid_n and_o yet_o without_o any_o show_n of_o a_o division_n and_o here_o also_o as_o before_o the_o species_n or_o several_a kind_n have_v their_o denomination_n their_o angle_n although_o it_o have_v be_v better_a and_o true_a to_o have_v be_v take_v from_o their_o side_n as_o to_o have_v be_v call_v a_o quadrilater_n or_o a_o multilater_n but_o in_o word_n use_n must_v be_v follow_v as_o a_o master_n 6._o a_o quadrangle_n be_v that_o which_o be_v comprehend_v of_o four_o right_a line_n 22_o d_o i_o 7._o a_o quadrangle_n be_v a_o a_o parallelogramme_n or_o a_o trapezium_fw-la this_o division_n also_o in_o his_o part_n be_v in_o the_o element_n of_o euclid_n but_o without_o any_o form_n or_o show_v of_o a_o division_n but_o the_o difference_n of_o the_o part_n shall_v more_o fit_o be_v distinguish_v thus_o because_o in_o general_a there_o be_v many_o common_a parallel_n 8._o a_o parallelogramme_n be_v a_o quadrangle_n who_o opposite_a side_n be_v parallel_v therefore_o 9_o if_o right_a line_n on_o one_o and_o the_o same_o side_n do_v joint_o bind_v equal_a and_o parallall_a line_n they_o shall_v make_v a_o parallelogramme_n the_o reason_n be_v because_o they_o shall_v be_v equal_a and_o parallel_v between_o themselves_o by_o the_o 26._o e_fw-la v_o and_o 10_o a_o parallelogramme_n be_v equal_a both_o in_o his_o opposite_a side_n and_o angle_n and_o segment_v cut_v by_o the_o diameter_n or_o thus_o the_o opposite_a both_o side_n and_o angle_n and_o segment_v cut_v by_o the_o diameter_n be_v equal_a three_o thing_n be_v here_o conclude_v the_o first_o be_v that_o the_o opposite_a side_n be_v equal_a this_o manifest_a by_o the_o 26_o e_fw-la v_o because_o two_o right_a line_n do_v joint_o bind_v equal_a parallel_n and_o 11_o the_o diameter_n of_o a_o parallelogramme_n be_v cut_v into_o two_o by_o equal_a ray_n as_o in_o the_o three_o figure_n a_o e_o i_o next_o before_o this_o a_o parallelogramme_n have_v common_a with_o a_o circle_n as_o be_v manifest_a at_o the_o 28._o e_fw-la iiij_o and_o 12_o a_o parallelogramme_n be_v the_o double_a of_o a_o triangle_n of_o a_o trinangle_n of_o equal_a base_a and_o height_n 41._o p_o i_o and_o 13_o a_o parallelogramme_n be_v equal_a to_o a_o triangle_n of_o equal_a height_n and_o double_a base_a unto_o it_o è_fw-la 42._o p_o i_o from_o whence_o one_o may_v 14_o to_o a_o triangle_n give_v in_o a_o rectilineall_a angle_n give_v make_v a_o equal_a parallelogramme_n 15_o a_o parallelogramme_n do_v consist_v both_o of_o two_o diagoâ—als_n and_o compliment_n and_o gnomon_n for_o these_o three_o part_n of_o a_o parallelogramme_n be_v much_o use_v in_o geometrical_a work_n and_o business_n and_o therefore_o they_o be_v to_o be_v define_v 16_o the_o diagonall_a be_v a_o particular_a parallelogramme_n have_v both_o a_o angle_n and_o diagonall_a diameter_n common_a with_o the_o whole_a parallelogramme_n 17_o the_o diagonall_a be_v like_a and_o alike_o situate_a to_o the_o whole_a parallelogramme_n è_fw-la 24._o p_o uj._o there_o be_v not_o any_o either_o rate_n or_o proportion_n of_o the_o diagonall_a propound_v only_a similitude_n be_v attribute_v to_o it_o as_o in_o the_o same_o figure_n the_o diagonall_a a_o u._fw-mi y_fw-fr s_o be_v like_a unto_o the_o whole_a parallelogramme_n a_o e_o i_o o._n for_o first_o it_o be_v equianglar_a to_o it_o for_o the_o angle_n at_o a_o be_v common_a to_o they_o both_o and_o that_o be_v equal_a to_o that_o which_o be_v at_o y_o by_o the_o 10._o e_fw-la x_o and_o therefore_o also_o it_o be_v equal_a to_o that_o at_o i_o by_o the_o 10._o e_fw-la x._o then_o the_o angle_v a_o u._fw-mi y_fw-mi and_o a_o s_z y_z be_v equal_a by_o the_o 21._o e_fw-la v_o to_o the_o opposite_a inner_a angle_n at_o e_o and_o o._n therefore_o it_o be_v equiangular_a unto_o it_o again_o it_o be_v proportional_a to_o it_o in_o the_o shank_n of_o the_o equal_a angle_n for_o the_o triangle_n a_o u._fw-mi y_fw-mi and_o a_o e_o i_o be_v alike_o by_o the_o 12_o e_fw-la seven_o because_o u._fw-mi y_fw-mi be_v parallel_v to_o the_o base_a therefore_o as_o a_o u._fw-mi be_v to_o u._fw-mi y_fw-mi so_o be_v a_o i_o to_o e_o i_o then_o as_o u._fw-mi y_z be_v to_z y_z a_o so_o be_v e_z i_z to_z i_z a._n again_o by_o the_o 21_o e_o v_o because_o s_o y_fw-fr be_v parallel_v to_o the_o base_a i_o o_fw-fr as_o a_o y_z be_v to_z y_z s_z so_o be_v a_o i_o to_o i_o o_o therefore_o equiordinate_o as_o u_z y_z be_v to_z y_z s_z so_o be_v e_z i_z to_z i_z o_o item_n as_o s_z y_z be_v to_z y_z a_o so_o be_v i_o o_o to_o i_o a_o and_o as_o y_o a_o be_v to_o a_o s_o so_o be_v i_o a_o to_o a_o o._n therefore_o equiordinate_o as_o y_z s_z be_v to_z s_z a_o so_o be_v i_o o_o to_o o_o a._n last_o as_o s_o a_o be_v unto_o a_o y_fw-es so_o be_v of_o a_o unto_o a_o i_o and_o as_o a_o y_o be_v to_o a_o u._fw-mi so_o be_v a_o i_o unto_o a_o e._n therefore_o equiordinate_o as_o be_v a_o be_v to_o a_o u._fw-mi so_o be_v a_o o_o to_o a_o e._n wherefore_o the_o diagonall_a s_z u._fw-mi be_v proportional_a in_o the_o shank_n of_o equal_a angle_n to_o the_o parallelogramme_n o_o e._n the_o demonstration_n shall_v be_v the_o same_o of_o the_o diagonall_a r_o l._n the_o like_a situation_n be_v manifest_a by_o the_o 21_o e_fw-la iiij_o and_o from_o hence_o also_o be_v manifest_a that_o the_o diagonall_a of_o a_o quadrate_n be_v a_o quadrate_n of_o a_o oblong_a a_o oblong_a of_o a_o rhombe_n a_o rhombe_n of_o a_o rhomboide_n a_o rhomboide_n because_o it_o be_v like_a unto_o the_o whole_a and_o a_o like_a situate_a now_o the_o diagonal_o see_v they_o be_v like_a unto_o the_o whole_a and_o a_o like_a situate_a they_o shall_v also_o be_v like_a between_o themselves_o and_o alike_o situate_a one_o to_o another_o by_o the_o 21_o and_o 22_o e_fw-la iiij_o therefore_o 18._o if_o the_o particular_a parallelogramme_n have_v one_o and_o the_o same_o angle_n with_o the_o whole_a be_v like_a and_o alike_o situate_a unto_o it_o it_o be_v the_o diagonall_a 26_o p_o uj._o as_o for_o example_n let_v the_o particular_a parallelogramme_n a_o u._fw-mi y_fw-fr s_o be_v coangular_a to_o the_o whole_a parallelogramme_n a_o e_o i_o o_o and_o let_v it_o have_v the_o same_o angle_n with_o it_o at_o a_o like_v unto_o the_o whole_a and_o alike_o situate_a unto_o it_o i_o say_v it_o be_v the_o diagonall_a otherwise_o let_v the_o diverse_a diagony_n be_v a_o r_o o_o and_o let_v l_o r_o be_v parallel_a against_o a_o e_o therefore_o a_o l_o r_o s_o shall_v be_v the_o diagonall_a by_o the_o 6_o e_o 15._o now_o therefore_o it_o shall_v be_v by_z 8_o e_o 16_o e_o as_o e_o a_o be_v to_o a_o i_o so_o be_v we_o a_o unto_o a_o l_o again_o by_o the_o grant_n as_o e_z a_o be_v unto_o a_o i_o so_o be_v we_o a_o to_o a_o u._fw-mi therefore_o the_o same_o be_v a_o be_v proportional_a to_o a_o l_o and_o to_o a_o u._fw-mi and_o a_o l_o be_v equal_a to_o a_o u._fw-mi the_o part_n to_o the_o whole_a which_o be_v impossible_a 19_o the_o compliment_n be_v a_o particular_a parallelogramme_n comprehend_v of_o the_o conterminall_a side_n of_o the_o diagonal_o or_o thus_o it_o be_v a_o particular_a parallelogramme_n contain_v under_o the_o next_o adjoin_v side_n of_o the_o diagonal_o 20._o the_o compliment_n be_v equal_a 43_o p_o i_o therefore_o 21._o if_o one_o of_o the_o compliment_n be_v make_v equal_a to_o a_o triangle_n give_v in_o a_o rightlined_n angle_v give_v the_o other_o make_v upon_o a_o right_a line_n give_v shall_v be_v in_o like_a manner_n equal_a to_o the_o same_o triangle_n 44_o p_o i_o as_o if_o thou_o shall_v desire_v to_o have_v a_o parallelogramme_n upon_o a_o
right_a line_n give_v and_o in_o a_o right_n line_v angle_n give_v to_o be_v make_v equal_a to_o a_o triangle_n give_v this_o proposition_n shall_v give_v satisfaction_n and_o 22_o if_o parallelogramme_n be_v continual_o make_v equal_a to_o all_o the_o triangle_n of_o a_o assign_a triangulate_a in_o a_o right_n line_v angle_n give_v the_o whole_a parallelogramme_n shall_v in_o like_a manner_n be_v equal_a to_o the_o whole_a triangulate_a 45_o p_o i_o this_o be_v a_o corollary_n of_o the_o former_a of_o the_o reason_n or_o rate_n of_o a_o parallelogramme_n with_o a_o triangulate_a and_o it_o need_v no_o father_n demonstration_n but_o a_o ready_a and_o steady_a hand_n in_o describe_v and_o work_v of_o it_o here_o thou_o have_v 3_o compliment_n continue_v and_o continue_v the_o parallelogramme_n but_o it_o be_v best_o in_o make_v and_o work_v of_o they_o to_o put_v out_o the_o former_a and_o one_o of_o the_o side_n of_o the_o inferior_a or_o latter_a diagonall_a leaâ—t_v the_o confusion_n of_o line_n do_v hinder_v or_o trouble_v thou_o therefore_o 23._o a_o parallelogramme_n be_v equal_a to_o his_o diagonal_n and_o compliment_n for_o a_o parallelogramme_n do_v consist_v of_o two_o diagonal_n and_o as_o many_o compliment_n wherefore_o a_o parallelogramme_n be_v equal_a to_o his_o part_n and_o again_o the_o part_n be_v equal_a to_o their_o whole_n 24._o the_o gnomon_n be_v any_o one_o of_o the_o diagonall_a with_o the_o two_o compliment_n in_o the_o element_n of_o geometry_n there_o be_v no_o other_o use_n as_o it_o seem_v of_o the_o gnomon_n than_o that_o in_o one_o word_n three_o part_n of_o a_o parallelogramme_n may_v be_v signify_v and_o call_v by_o three_o letter_n a_o e_o i._n otherwise_o gnomon_n be_v a_o perpendicular_a 25._o parallelograme_n of_o equal_a height_n be_v one_o to_o another_o as_o their_o base_n be_v 1_o p_o uj._o therefore_o 26_o parallelogramme_n of_o equal_a height_n upon_o equal_a base_n be_v equal_a 35._o 36_o p_o i_o as_o be_v manifest_a in_o the_o same_o example_n 27_o if_o equiangle_n parallelogramme_n be_v reciprocal_a in_o the_o shank_n of_o the_o equal_a angle_n they_o be_v equal_a and_o contrariwise_o 15_o p_o uj._o therefore_o 28_o if_o four_o right_a line_n be_v proportional_a the_o parallelogramme_n make_v of_o the_o two_o middle_a one_o be_v equal_a to_o the_o equiangle_v parallelogramme_n make_v of_o the_o first_o and_o last_o and_o contrariwise_o e_fw-la 16_o p_o uj._o for_o they_o shall_v be_v equiangle_v parallelogramme_n reciprocal_a in_o the_o shank_n of_o the_o equal_a angle_n and_o 29_o if_o three_o right_a line_n be_v proportional_a the_o parallelogramme_n of_o the_o middle_a one_o be_v equal_a to_o the_o equiangle_v parallelogramme_n of_o the_o extreme_n and_o contrariwise_o it_o be_v a_o consectary_n draw_v out_o of_o the_o former_a of_o geometry_n the_o eleven_o book_n of_o a_o right_a angle_n 1._o a_o parallelogramme_n be_v a_o right_a angle_n or_o a_o obliquangle_n hitherto_o we_o have_v speak_v of_o certain_a common_a and_o general_a matter_n belong_v unto_o parallelogrammesâ—_n special_n do_v follow_v in_o rectangle_v and_o obliquangle_v which_o difference_n as_o be_v aforesaid_a be_v common_a to_o triangle_n and_o triangulate_v but_o at_o this_o time_n we_o find_v no_o fit_a word_n whereby_o to_o distinguish_v the_o general_n 2._o a_o right_a angle_n be_v a_o parallelogramme_n that_o have_v all_o his_o angle_n right_a angle_n as_o in_o a_o e_o i_o o._n and_o here_o hence_o you_o must_v understand_v by_o one_o right_a angle_n that_o all_o be_v right_a angle_n for_o the_o right_a angle_n at_o a_o be_v equal_a to_o the_o opposite_a angle_n at_o i_o by_o the_o 10_o e_fw-la x._o therefore_o 3_o a_o rightangle_n be_v comprehend_v of_o two_o right_a line_n comprehend_v the_o right_a angle_n 1._o d_o ij_o comprehension_n in_o this_o place_n do_v signify_v a_o certain_a kind_n of_o geometrical_a multiplication_n for_o as_o of_o two_o number_n multiply_v between_o themselves_o there_o be_v make_v a_o number_n so_o of_o two_o side_n ductis_fw-la drive_v together_o a_o right_a angle_n be_v make_v and_o yet_o every_o right_a angle_n be_v not_o rational_a as_o before_o be_v manifest_a at_o the_o 12._o e_fw-la iiij_o and_o shall_v after_o appear_v at_o the_o 8_o e._n and_o 4_o four_o right_a angle_n do_v fill_v a_o place_n neither_o be_v it_o any_o matter_n at_o all_o whether_o the_o four_o rectangle_v be_v equal_a or_o unequal_a equilater_n or_o unequilater_n homogeneal_n or_o heterogeneall_n for_o which_o way_n so_o ever_o they_o be_v turn_v the_o angle_n shall_v be_v right_a angle_n and_o therefore_o they_o shall_v fill_v a_o place_n 5_o if_o the_o diameter_n do_v cut_v the_o side_n of_o a_o right_a angle_n into_o two_o aquall_a part_n it_o do_v cut_v it_o perpendicular_o and_o contrariwise_o therefore_o 6_o if_o a_o inscribe_v right_a line_n do_v perpendicular_o cut_v the_o side_n of_o the_o right_a angle_n into_o two_o equal_a part_n it_o be_v the_o diameter_n the_o reason_n be_v because_o it_o do_v cut_v the_o parallelogramme_n into_o two_o equal_a portion_n 7_o a_o right_a angle_n be_v equal_a to_o the_o rightangle_v make_v of_o one_o of_o his_o side_n and_o the_o segment_n of_o the_o other_o as_o here_o the_o four_o particular_a right_a angle_n be_v equal_a to_o the_o whole_a which_o be_v make_v of_o a_o e_fw-it one_o of_o his_o side_n and_o of_o e_o i_o i_fw-it o_o o_fw-mi u._fw-mi u._fw-mi y_fw-mi the_o segment_n of_o the_o other_o last_o every_o arithmetical_a multiplication_n of_o the_o whole_a number_n do_v make_v the_o same_o product_n that_o the_o multiplication_n of_o the_o one_o of_o the_o whole_a number_n give_v by_o the_o part_n of_o the_o other_o shall_v make_v yea_o that_o the_o multiplication_n of_o the_o part_n by_o the_o part_n shall_v make_v this_o proposition_n be_v cite_v by_o ptolomey_n in_o the_o 9_o chapter_n of_o the_o 1_o book_n of_o his_o almage_a 8_o if_o four_o right_a line_n be_v proportional_a the_o rectangle_n of_o the_o two_o middle_a one_o be_v equal_a to_o the_o rectangle_n of_o the_o two_o extreme_n 16._o p_o uj._o 9_o the_o figurate_a of_o a_o rational_a rectangle_n be_v call_v a_o rectinall_n plain_a 16._o d_o seven_o if_o therefore_o the_o base_a of_o a_o rectangle_n be_v 6._o and_o the_o height_n 4._o the_o plot_n or_o content_n shall_v be_v 24._o and_o if_o it_o be_v certain_a that_o the_o rectangle_v content_a be_v 24._o and_o the_o base_a be_v 6._o it_o shall_v also_o be_v certain_a that_o the_o height_n be_v 4._o the_o example_n be_v thus_o this_o manner_n of_o multiplication_n say_v 1_o be_v geometrical_a neither_o be_v there_o here_o of_o line_n make_v line_n as_o there_o of_o unity_n be_v make_v unity_n but_o a_o magnitude_n one_o degree_n high_o to_o wit_n a_o surface_n be_v here_o make_v here_o hence_o be_v the_o geodesy_n or_o manner_n of_o measure_v of_o a_o rectangled_a triangle_n make_v know_v unto_o we_o for_o when_o thou_o shall_v multiply_v the_o shank_n of_o a_o right_a angle_n the_o one_o by_o the_o other_o thou_o do_v make_v the_o whole_a rectangled_a parallelogramme_n who_o half_a be_v a_o triangle_n by_o the_o 12._o e_fw-la x._o of_o geometry_n the_o twelve_o book_n of_o a_o quadrate_n 1_o a_o rectangle_n be_v a_o quadrate_n or_o a_o oblong_a this_o division_n be_v make_v in_o proper_a term_n but_o the_o thing_n itself_o and_o the_o subject_a difference_n be_v common_a out_o of_o the_o angle_n and_o side_n 2_o a_o quadrate_n be_v a_o rectangle_n equilater_n 30._o dj_o plain_n be_v with_o we_o according_a to_o their_o diverse_a nature_n and_o quality_n measure_v with_o divers_a and_o sundry_a kind_n of_o measure_n board_n glass_n and_o paving-stone_n be_v measure_v by_o the_o foot_n cloth_n wainscot_n paint_v pave_v and_o such_o like_a by_o the_o yard_n land_n and_o wood_n by_o the_o perch_n or_o rodde_n of_o measuresâ—_n and_o the_o sundry_a sort_n thereof_o common_o use_v and_o mention_v in_o history_n we_o have_v in_o the_o former_a speak_v at_o large_a yet_o for_o the_o far_a confirmation_n of_o some_o thing_n then_o speak_v and_o here_o again_o now_o upon_o this_o particular_a occasion_n repeat_v it_o shall_v not_o be_v amiss_o to_o hear_v what_o our_o statute_n speak_v of_o these_o three_o sort_n here_o mention_v it_o be_v ordain_v say_v the_o statute_n that_o three_o barley-corne_n dry_a and_o round_a do_v make_v a_o inch_n twelve_o ynche_n do_v make_v a_o foot_n three_o foot_n do_v make_v a_o yard_n five_o yard_n and_o a_o half_a do_v make_v a_o perch_n forty_o perch_n in_o length_n and_o four_o in_o breadth_n do_v make_v a_o acre_n 33._o edwardi_fw-la 1._o de_fw-fr terris_fw-la mensurandis_fw-la item_n de_fw-fr compositione_n vlnarum_fw-la &_o perticarum_fw-la moreover_o observe_v that_o all_o those_o measure_n there_o speak_v of_o be_v only_a length_n these_o here_o now_o last_v repeat_v be_v
bind_v of_o a_o solid_a be_v a_o surface_n 2_o d_o xj_o the_o bind_v of_o a_o line_n be_v a_o point_n and_o yet_o neither_o be_v a_o point_n a_o line_n or_o any_o part_n of_o a_o line_n the_o bind_v of_o a_o surface_n be_v a_o line_n and_o yet_o a_o line_n be_v not_o a_o surface_n or_o any_o part_n of_o a_o surface_n so_o now_o the_o bind_v of_o a_o body_n be_v a_o surface_n and_o yet_o a_o surface_n be_v not_o a_o body_n or_o any_o part_n of_o a_o body_n a_o magnitude_n be_v one_o thing_n a_o bind_v of_o a_o magnitude_n be_v another_o thing_n as_o appear_v at_o the_o 5_o e_fw-la i_o as_o they_o be_v call_v plain_a line_n which_o be_v conceive_v to_o be_v â—â—_o a_o plain_a so_o those_o be_v name_v solid_a both_o line_n and_o surface_n which_o be_v consider_v in_o a_o solid_a and_o their_o perpendicle_n and_o parallelisme_n be_v hither_o to_o be_v recall_v from_o simple_a line_n 3_o if_o a_o right_a line_n be_v unto_o right_a line_n cut_v in_o a_o plain_a underneath_o perpendicular_a in_o the_o common_a intersection_n it_o be_v perependicular_a to_o the_o plain_a beneath_o and_o if_o it_o be_v perpendicular_a it_o be_v unto_o right_a line_n cut_v in_o the_o same_o plain_a perpendicular_a in_o the_o common_a intersection_n è_fw-mi 3_o d_o and_o 4_o pxj._n if_o thou_o shall_v conceive_v the_o right_a line_n a_o e_o i_fw-it o_o u._fw-mi y_fw-mi to_o cut_v one_o another_o in_o the_o plain_a beneath_o in_o the_o common_a intersection_n and_o the_o line_n r_o s_o fall_v from_o above_o to_o be_v to_o every_o one_o of_o they_o perpendicular_a in_o the_o common_a point_n s_o thou_o have_v a_o example_n of_o this_o consectary_n 4_o if_o three_o right_a line_n cut_v one_o another_o be_v unto_o the_o same_o right_a line_n perpendicular_a in_o the_o common_a section_n they_o be_v in_o the_o same_o plain_a 5._o p_o x_o i_o for_o by_o the_o perpendicle_n and_o common_a section_n be_v understand_v a_o equal_a state_n on_o all_o part_n and_o therefore_o the_o same_o plain_a as_o in_o the_o former_a example_n a_o s_o y_z s_z o_o s_o suppose_v they_o to_o be_v to_o s_o r_o the_o same_o lofty_a line_n perpendicular_a they_o shall_v be_v in_o the_o same_o near_a plain_n a_o i_o u_fw-la e_fw-it o_o y._n 5_o if_o two_o right_a line_n be_v perpendicular_a to_o the_o underplaine_n they_o be_v parallel_n and_o if_o the_o one_o of_o two_o parallel_n be_v perpendicular_a to_o the_o under_o plain_a the_o other_o be_v also_o perpendicular_a to_o the_o same_o 6.8_o p_o xj_o 6_o if_o right_a line_n in_o diverse_a plain_n be_v unto_o the_o same_o right_a line_n parallel_v they_o be_v also_o parallel_a between_o themselves_o 9_o p_o xj_o 7_o if_o two_o right_a line_n be_v perpendicular_o the_o first_o from_o a_o point_n above_o unto_o a_o right_a line_n underneath_o the_o second_o from_o the_o common_a section_n in_o the_o plain_a â—nderneath_o a_o three_o from_o the_o say_a point_n perpendicular_a to_o the_o second_o shall_v be_v perpendicular_a to_o the_o plain_a beneath_o è_fw-it 11_o p_o xj_o if_o the_o right_a line_n i_o o_o do_v with_o equal_a angle_n agree_v to_o r_o the_o three_o element_n 8._o if_o a_o right_a line_n from_o a_o point_n assign_v of_o a_o plain_a underneath_o be_v parallel_n to_o a_o right_a line_n perpendicular_a to_o the_o same_o plain_a it_o shall_v also_o be_v perpendicular_a to_o the_o plain_a underneath_o e_o x_o 12_o p_o xj_o 9_o if_o a_o right_a line_n in_o one_o of_o the_o plain_n cut_v perpendicular_a to_o the_o common_a section_n be_v perpendicular_a to_o the_o other_o the_o plain_n be_v perpendicular_a and_o if_o the_o plain_n be_v perpendicular_a a_o right_a line_n in_o the_o one_o perpendicular_a to_o the_o common_a section_n be_v perpendicular_a to_o the_o other_o è_fw-mi 4_o d_o and_o 38_o p_o xj_o 10._o if_o a_o right_a line_n be_v perpendicular_a to_o a_o plain_a all_o plain_n by_o it_o be_v perpendicular_a to_o the_o same_o and_o if_o two_o plain_n be_v unto_o any_o other_o plain_a perpendicular_o the_o common_a section_n be_v perpendicular_a to_o the_o same_o e_o 15_o and_o 19_o p._n xj_o 11._o plain_n be_v parallel_v which_o do_v lean_v no_o way_n 8_o d_o x_o i_o and_o 12._o those_o which_o divide_v by_o a_o common_a perpendicle_n 14_o p_o xj_o it_o be_v also_o out_o of_o the_o definition_n of_o parallel_n at_o the_o 17_o e_o i_o i_o and_o 13._o if_o two_o pair_n of_o right_a in_o they_o be_v joint_o bound_v they_o be_v parallel_v 15_o p_o xj_o the_o same_o will_v fall_v out_o if_o thou_o shall_v imagine_v the_o joint_o bound_v to_o infinite_o draw_v out_o for_o the_o plain_n also_o infinite_o extend_v shall_v be_v parallelly_n 14._o if_o two_o parallel_a plain_n be_v cut_v with_o another_o plain_n the_o common_a section_n be_v parallel_n 16_o p_o xj_o the_o twenty_o second_o book_n of_o p._n ramus_n geometry_n of_o a_o pyramid_n 1._o the_o axis_fw-la of_o a_o solid_a be_v the_o diameter_n about_o which_o it_o be_v turn_v e_fw-la 15,19,22_o d_o x_o i_o 2._o a_o right_a solid_a be_v that_o who_o axis_fw-la be_v perpendicular_a to_o the_o centre_n of_o the_o base_a thus_o serenus_n and_o apollonius_n do_v define_v a_o cone_n and_o a_o cylinder_n and_o these_o only_a euclid_n consider_v yea_o and_o indeed_o stereometry_n entertain_v no_o other_o kind_n of_o solid_a but_o that_o which_o be_v right_a or_o perpendicular_a 3._o if_o solid_n be_v comprehend_v of_o homogeneal_a surface_n equal_a in_o multitude_n and_o magnitude_n they_o be_v equal_a 10_o d_o x_o i_o equality_n of_o line_n and_o surface_n be_v not_o inform_v by_o any_o peculiar_a rule_n far_o than_o out_o of_o reason_n and_o common_a sense_n and_o in_o most_o place_n congruency_n and_o application_n be_v enough_o and_o do_v satisfy_v to_o the_o full_a but_o here_o the_o congruency_n of_o body_n be_v judge_v by_o their_o surface_n two_o cube_n be_v equal_a who_o six_o side_n or_o plain_a surface_n be_v equal_a etc._n etc._n 4._o if_o solid_n be_v comprehend_v of_o surface_n in_o multitude_n equal_a and_o like_a they_o be_v equal_a 9_o d_o x_o i_o this_o be_v a_o consectary_n draw_v out_o of_o the_o general_a definition_n of_o like_a figure_n at_o the_o 19_o e._n iiij_o for_o there_o like_a figure_n be_v define_v to_o be_v equiangle_v and_o proportional_a in_o the_o shank_n of_o the_o equal_a angle_n but_o in_o like_a plain_a solid_n the_o angle_n be_v esteem_v to_o be_v equal_a out_o of_o the_o similitude_n of_o their_o like_a plain_n and_o the_o equal_a shank_n be_v the_o same_o plain_a surface_n and_o therefore_o they_o be_v proportional_a equal_a and_o alike_o 5_o like_a solid_n have_v a_o treble_a reason_n of_o their_o homologall_a side_n and_o two_o mean_a proportional_o 33._o p_o xj_o 8_o p_o xij_o it_o be_v a_o consectary_n draw_v out_o of_o the_o 24_o e._n iiij_o as_o the_o example_n from_o thence_o repeat_v shall_v make_v manifest_a 6_o a_o solid_a be_v plain_a or_o embose_v 7_o a_o plain_a solid_a be_v that_o which_o be_v comprehend_v of_o plain_a surface_n 8_o the_o plain_a angle_n 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40._o p_o xj_o the_o cause_n be_v manifest_a and_o brief_a because_o they_o be_v the_o half_n of_o the_o same_o prisma_fw-la as_o here_o thou_o may_v perccive_v in_o a_o prisma_fw-la cut_v into_o two_o half_n by_o the_o diagoni'_v of_o the_o opposite_a side_n euclid_n do_v demonstrate_v it_o thus_o let_v the_o pentaedra_n a_o e_o i_o o_o u._fw-mi and_o y_z s_o r_o l_o m_o be_v of_o equal_a height_n the_o first_o of_o a_o triangular_a base_a e_o i_o o_o the_o second_o of_o a_o parallelogramme_n base_a s_o l_o double_a unto_o the_o triangular_a now_o let_v both_o of_o they_o be_v double_a and_o make_v up_o so_o that_o first_o be_v nâ—_n the_o second_o y_o s_o r_o l_o v_o f._n now_o again_o by_o the_o grant_n the_o base_a s_o l_o be_v the_o double_a of_o the_o base_a e_o i_o o_o who_o double_a be_v thâ—_n base_a e_o o_o by_o the_o 12_o e_fw-la x._o therefore_o the_o base_n s_o l_o and_o e_o o_o be_v equal_a and_o therefore_o see_v the_o prisma_n by_o the_o grant_n here_o be_v of_o equal_a 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it_o can_v be_v divide_v into_o a_o more_o simple_a solid_a figure_n although_o it_o may_v be_v divide_v into_o a_o infinite_a sort_n of_o other_o figure_n of_o the_o triangle_n all_o plain_n be_v make_v as_o of_o a_o pyramid_n all_o body_n or_o solid_n be_v compoundedâ—_n such_o be_v a_o e_o i._n and_o a_o e_o i_o o._n 12._o a_o rational_a figure_n be_v that_o which_o be_v comprehend_v of_o a_o base_a and_o height_n rational_a between_o themselves_o so_o euclid_n at_o the_o 1._o d._n ij_o say_v that_o a_o rightangled_a parallelogramme_n be_v comprehend_v of_o two_o right_a line_n perpendicular_a one_o to_o another_o videlicet_fw-la one_o multiply_v by_o the_o other_o for_o geometrical_a comprehension_n be_v sometime_o as_o it_o be_v in_o number_n a_o multiplication_n therefore_o if_o you_o shall_v grant_v the_o base_a and_o height_n to_o be_v rational_o between_o themselves_o that_o their_o reason_n i_o mean_v may_v be_v express_v by_o a_o number_n of_o the_o assign_a measure_n than_o the_o number_n of_o their_o â—ides_n be_v multiply_v one_o by_o another_o the_o bigness_n of_o the_o figure_n shall_v be_v express_v therefore_o a_o rational_a figure_n be_v make_v by_o the_o multiply_a of_o two_o rational_a side_n between_o themselves_o therefore_o 13._o the_o number_n of_o a_o rational_a figure_n be_v call_v a_o figurate_a number_n and_o the_o number_n of_o which_o it_o be_v make_v the_o side_n of_o the_o figurate_a as_o if_o a_o right_a angle_a parallelogramme_n be_v comprehend_v of_o the_o base_a four_o and_o the_o height_n three_o the_o rational_a make_v shall_v be_v 12._o which_o we_o here_o call_v the_o figurate_a and_o 4._o and_o 3._o of_o which_o it_o be_v make_v we_o name_v side_n 14._o isoperimetrall_a figure_n be_v figure_n of_o equal_a perimeter_n this_o be_v nothing_o else_o but_o a_o interpretation_n of_o the_o greek_a word_n so_o a_o triangle_n of_o 16._o foot_n about_o be_v a_o isoperimeter_n to_o a_o triangle_n 16._o foot_n about_o to_o a_o quadrate_n 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word_n from_o hence_o have_v three_o line_n give_v be_v the_o invention_n of_o the_o four_o proportional_a and_o out_o of_o that_o have_v two_o line_n give_v arise_v the_o invention_n of_o the_o three_o proportional_a 2_o have_v three_o right_a line_n give_v if_o the_o first_o and_o the_o three_o make_v a_o angle_n and_o knit_v together_o with_o a_o base_a be_v continue_v the_o first_o equal_o to_o the_o second_o the_o three_o infinite_o a_o parallel_n from_o the_o end_n of_o the_o second_o unto_o the_o continuation_n of_o the_o three_o shall_v intercept_v the_o four_o proportional_a 12._o puj._n the_o diagramme_n and_o demonstration_n be_v the_o same_o with_o our_o 31._o e_z or_o 3_o c_o of_o ramus_n 3_o if_o two_o right_a line_n give_v make_v a_o angle_n and_o knit_v together_o with_o a_o base_a be_v continue_v the_o first_o equal_o to_o the_o second_o the_o second_o infinite_o a_o parallel_n to_o the_o base_a from_o the_o end_n of_o the_o first_o continuation_n unto_o the_o second_o 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other_o whether_o you_o do_v cross_v the_o bound_n or_o invert_v they_o here_o for_o demonstration_n sake_n we_o crave_v no_o more_o but_o that_o from_o the_o begin_n of_o a_o antecedent_n make_v a_o parallel_n be_v draw_v to_o the_o second_o consequent_a of_o the_o maker_n unto_o one_o of_o the_o assign_v infinite_o continue_v then_o the_o multiply_a proportion_n shall_v be_v the_o antecedent_n the_o consequent_a the_o antecedent_n the_o consequent_a of_o the_o second_o of_o the_o maker_n every_o way_n the_o reason_n or_o rate_n be_v of_o equality_n the_o antecedent_n the_o consequent_a of_o the_o first_o of_o the_o maker_n the_o parallel_n the_o antecedent_n of_o the_o second_o of_o the_o maker_n by_o the_o 32._o e._n therefore_o by_o multiplication_n of_o proportion_n the_o reason_n of_o the_o parallel_n unto_o the_o consequent_a of_o the_o second_o of_o the_o maker_n that_o be_v by_o the_o fabric_n or_o construction_n and_o the_o 32._o e._n the_o reason_n of_o the_o antecedent_n 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the_o outter_n to_o the_o inner_a contrary_a to_o the_o 15._o e_fw-la v_o i_o therefore_o the_o base_a e_o i_o be_v not_o unequal_a to_o the_o base_a u._fw-mi y_fw-mi but_o equal_a and_o therefore_o as_o above_o be_v say_v the_o two_o triangle_n a_o e_o i_o and_o o_o u._fw-mi y_fw-mi equal_a in_o the_o angle_n of_o their_o equal_a shank_n be_v equilater_n 3._o triangle_n be_v equal_a in_o their_o three_o angle_n and_o yet_o notwithstanding_o it_o be_v not_o therefore_o to_o be_v think_v to_o be_v equiangle_n to_o it_o for_o triangle_n be_v then_o equiangle_n when_o the_o several_a angle_n of_o the_o one_o be_v equal_a to_o the_o several_a angle_n of_o the_o other_o not_o when_o all_o joint_o be_v equal_a to_o all_o therefore_o 4._o if_o two_o angle_n of_o two_o triangle_n give_v be_v equal_a the_o other_o also_o be_v equal_a all_o the_o three_o angle_n be_v equal_a between_o themselves_o by_o the_o 3_o e._n therefore_o if_o from_o equal_a you_o take_v away_o equal_a those_o which_o shall_v remain_v shall_v be_v equal_a 5._o if_o a_o right_a triangle_n equicrural_a to_o a_o triangle_n be_v great_a in_o base_a it_o be_v great_a in_o angle_n and_o contrariwise_o 25._o and_o 24._o pj._n 6._o if_o a_o triangle_n place_v upon_o the_o same_o base_a with_o another_o triangle_n be_v lesser_a in_o the_o inner_a shank_n it_o be_v great_a in_o the_o angle_n of_o the_o shank_n this_o be_v a_o consectary_n draw_v also_o out_o of_o the_o 10_o e_fw-la iij._o as_o here_o in_o the_o triangle_n a_o e_o i_o and_o a_o o_o ay_o within_z it_o and_o upon_o the_o same_o base_a or_o thus_o if_o a_o triangle_n place_v upon_o the_o same_o baâ—e_n with_o another_o triangle_n be_v less_o than_o the_o other_o triangle_n in_o regard_n of_o his_o foot_n those_o foot_n be_v contain_v within_o the_o foot_n of_o the_o other_o triangle_n in_o regard_n of_o the_o angle_n contain_v under_o those_o foot_n it_o be_v great_a h._n 7._o triangle_n of_o equal_a height_n be_v one_o to_o another_o as_o 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triangle_n equiangle_n to_o the_o â—holeâ—_n but_o less_o in_o base_a 14._o if_o two_o trangle_v be_v proportional_a in_o the_o shank_n of_o the_o equal_a angle_n they_o be_v equiangle_n 6_o p_o uj._o 15_o if_o triangle_n proportional_a in_o shank_n and_o alâ—ernly_o parallel_v do_v make_v a_o angle_n between_o they_o their_o base_n be_v but_o one_o right_a line_n continue_v 32_o p._n uj._o or_o thus_o if_o be_v proportional_a in_o their_o foot_n and_o alternate_o parallel_v they_o make_v a_o angle_n in_o the_o midst_n between_o they_o they_o have_v their_o base_n continue_v in_o a_o right_a line_n h._n the_o cause_n be_v out_o of_o the_o 14_o e_fw-la v_o for_o they_o shall_v make_v on_o each_o side_n with_o the_o fall_a line_n a_o i_o two_o angle_n equal_a to_o two_o right_a angle_n 16_o if_o two_o triangle_n have_v one_o angle_n equal_a another_o proportional_a in_o shank_n the_o three_o homogeneal_a they_o be_v equiangle_n 7._o p._n v._n i_o of_o geometry_n the_o eight_o book_n of_o the_o diverse_a kind_n of_o triangle_n 1_o a_o triangle_n be_v either_o right_n angle_v or_o obliquangle_v the_o division_n of_o a_o triangle_n take_v from_o the_o angle_n out_o of_o their_o common_a difference_n i_o mean_v do_v now_o follow_v but_o here_o first_o a_o special_a division_n and_o that_o of_o great_a moment_n as_o hereafter_o shall_v be_v in_o quadrangle_v and_o prisme_n 2_o a_o right_a angle_a triangle_n be_v that_o which_o have_v one_o right_a angle_n a_o obliquangled_a be_v that_o which_o have_v none_o 27._o d_o i_o a_o right_a angle_a triangle_n in_o geometry_n be_v of_o special_a use_n and_o force_n and_o of_o the_o best_a mathematician_n it_o be_v call_v magister_fw-la matheseos_fw-la the_o master_n of_o the_o mathematics_n therefore_o 3_o if_o two_o perpendicular_a line_n be_v knit_v together_o they_o shall_v make_v a_o right_a angle_a triangle_n 4_o if_o the_o angle_n of_o a_o triangle_n at_o the_o base_a be_v a_o right_a angle_n a_o perpendicular_a from_o the_o 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in_o a_o periphery_n and_o do_v differ_v only_o in_o base_a 14_o the_o angle_n in_o opposite_a section_n be_v equal_a to_o two_o right_a angle_n 22._o p_o iij._o the_o reason_n or_o rate_n of_o a_o section_n be_v thus_o the_o similitude_n do_v follow_v 15_o if_o section_n do_v receive_v or_o contain_v equal_v angle_n they_o be_v alike_o e_fw-la 10._o d_o iij._o 16_o if_o like_a section_n be_v upon_o a_o equal_a base_a they_o be_v equal_a and_o contrariwise_o 23,24_o p_o iij._o in_o the_o first_o figure_n let_v the_o base_a be_v the_o same_o and_o if_o they_o shall_v be_v say_v to_o unequal_a section_n and_o one_o of_o they_o great_a than_o another_o the_o angle_n in_o that_o a_o o_o e_o shall_v be_v less_o than_o the_o angle_n a_o i_o e_o in_o the_o lesser_a section_n by_o the_o 16_o e_fw-la uj._o which_o notwithstanding_o by_o the_o grant_n be_v equal_a in_o the_o second_o figure_n if_o one_o section_n be_v put_v upon_o another_o it_o will_v agree_v with_o it_o otherwise_o against_o 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the_o second_o part_n that_o the_o angle_n of_o a_o semicircle_n be_v lesser_a than_o a_o right_a angle_n be_v manifest_a out_o of_o that_o because_o it_o be_v the_o part_n of_o a_o right_a angle_n for_o the_o angle_n of_o the_o semicircle_n a_o i_o e_o be_v a_o part_n of_o the_o rectilineall_a right_a angle_n a_o i_o u._n the_o three_o part_n that_o it_o be_v great_a than_o any_o acute_a angle_n be_v manifest_a out_o of_o the_o 23._o e_fw-la x_o v._n for_o otherwise_o a_o tangent_fw-la be_v not_o on_o the_o same_o part_n one_o only_a and_o no_o more_o the_o four_o part_n be_v thus_o make_v manifest_a the_o angle_n at_o i_o in_o the_o great_a section_n a_o e_fw-it i_fw-it be_v lesser_a than_o a_o right_a angle_n because_o it_o be_v in_o the_o same_o triangle_n a_o e_fw-it i_fw-it which_o at_o a_o be_v right_a angle_n and_o if_o neither_o of_o the_o shank_n be_v by_o the_o centre_n notwithstanding_o a_o angle_n may_v be_v make_v equal_a to_o the_o assign_a in_o the_o same_o section_n 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reason_n of_o the_o angle_n in_o opposite_a section_n do_v follow_v 27_o the_o angle_n in_o opposite_a section_n be_v equal_a in_o the_o alterne_a angle_n make_v of_o the_o secant_fw-la and_o touch_v line_n 32._o p_o iij._o as_o let_v the_o unequal_a section_n be_v e_o i_o o_o and_o e_z a_o o_o the_o tangent_fw-la let_v it_o be_v u._fw-mi e_fw-es y_fw-es and_o the_o angle_n in_o the_o opposite_a section_n e_z a_o o_o and_o e_z i_z o._n i_o say_v they_o be_v equal_a in_o the_o alterne_a angle_n of_o the_o secant_fw-la and_o touch_v line_n o_fw-mi e_fw-es y_fw-es and_o o_o e_o u._n first_o that_o which_o be_v at_o a_o be_v equal_a to_o the_o
quem_fw-la agricola_n &_o alijex_fw-la antiquis_fw-la monumentis_fw-la tradi_fw-la derunt_fw-la now_o by_o any_o one_o of_o these_o know_v and_o compare_v with_o we_o to_o all_o english_a man_n well_o know_v the_o rest_n may_v easy_o be_v proportion_v out_o 2._o the_o thing_n propose_v to_o be_v measure_v be_v a_o magnitude_n magnitudo_fw-la a_o magnitude_n or_o bigness_n be_v the_o subject_n about_o which_o geometry_n be_v busy_v for_o every_o art_n have_v a_o proper_a subject_n about_o which_o it_o do_v employ_v all_o his_o rule_n and_o precept_n and_o by_o this_o especial_o they_o do_v differ_v one_o from_o another_o so_o the_o subject_n of_o grammar_n be_v speech_n of_o logic_n reason_n of_o arithmetic_n number_n and_o so_o now_o of_o geometry_n it_o be_v a_o magnitude_n all_o who_o kind_n difference_n and_o affection_n be_v hereafter_o to_o be_v declare_v 3._o a_o magnitude_n be_v a_o continual_a quantity_n a_o magnitude_n be_v quantitas_fw-la continua_fw-la a_o continue_a or_o 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those_o three_o quadrate_n to_o wit_n of_o a_o i_o his_o own_o quadrate_n and_o of_o e_o i_o two_o the_o first_o i_o o_o the_o second_o o_o e_o by_o the_o 9_o e._n therefore_o the_o excess_n remain_v of_o a_o double_a rectangle_n of_o geometry_n the_o thirteen_o book_n of_o a_o oblong_a 1_o a_o oblong_o be_v a_o rectangle_n of_o inequal_a side_n 31._o d_o i_o this_o second_o kind_n of_o rectangle_n be_v of_o euclid_n in_o his_o element_n proper_o name_v for_o a_o definition_n sake_n only_o the_o rate_n of_o oblong_n be_v very_o copious_a out_o of_o a_o threefold_a section_n of_o a_o right_a line_n give_v sometime_o rational_a and_o expresable_a by_o a_o number_n the_o first_o section_n be_v as_o you_o please_v that_o be_v into_o two_o segment_n equal_a or_o unequal_a from_o whence_o a_o fivefold_a rate_n arise_v 2_o a_o oblong_o make_v of_o a_o whole_a line_n give_v and_o of_o one_o segment_n of_o the_o same_o be_v equal_a to_o a_o rectangle_n make_v of_o both_o the_o segment_n and_o the_o square_a of_o the_o say_a segment_n 3._o p_o ij_o it_o be_v a_o consectary_n out_o of_o the_o 7_o e_fw-la xj_o for_o the_o rectangle_n of_o the_o segment_n and_o the_o quadrate_n be_v make_v of_o one_o side_n and_o of_o the_o segment_n of_o the_o other_o now_o a_o rectangle_n be_v here_o therefore_o propose_v because_o it_o may_v be_v also_o a_o quadrate_n to_o wit_n if_o the_o line_n be_v cut_v into_o two_o equal_a part_n secondary_o 3_o oblong_v make_v of_o the_o whole_a line_n give_v and_o of_o the_o segment_n be_v equal_a to_o the_o quadrate_n of_o the_o whole_a 2_o p_o ij_o this_o be_v also_o a_o consectary_n out_o of_o the_o 4._o e_fw-la xj_o here_o the_o segment_n be_v more_o than_o two_o and_o yet_o notwithstanding_o from_o the_o first_o the_o rest_n may_v be_v take_v for_o one_o see_v that_o the_o particular_a rectangle_n in_o like_a manner_n be_v equal_a to_o they_o this_o proposition_n be_v use_v in_o the_o demonstration_n of_o the_o 9_o e_fw-la xviij_o three_o 4_o two_o oblong_v make_v of_o the_o whole_a line_n give_v and_o of_o the_o one_o segment_n with_o the_o three_o quadrate_n of_o the_o other_o segment_n be_v equal_a to_o the_o quadrate_n of_o the_o whole_a and_o of_o the_o say_a segment_n 7_o p_o ij_o 5_o the_o base_a of_o a_o acute_a triangle_n be_v of_o less_o power_n than_o the_o shank_n be_v by_o a_o double_a oblong_o make_v of_o one_o of_o the_o shank_n and_o the_o one_o segment_n of_o the_o same_o from_o the_o say_a angle_n unto_o the_o perpendicular_a of_o the_o top_n 13._o p.ij._n and_o from_o hence_o be_v have_v the_o segment_n of_o the_o shank_n towards_o the_o angle_n and_o by_o that_o the_o perpendicular_a in_o a_o triangle_n therefore_o 6._o if_o the_o square_a of_o the_o base_a of_o a_o acute_a angle_n be_v take_v out_o of_o the_o square_n of_o the_o shank_n the_o quotient_a of_o the_o half_a of_o the_o remain_n divide_v by_o the_o shank_n shall_v be_v the_o segment_n of_o the_o divide_v shank_n from_o the_o say_a angle_n unto_o the_o perpendicular_a of_o the_o top_n now_o again_o from_o 169_o the_o quadrate_n of_o the_o base_a 13_o take_v 25_o the_o quadrate_n of_o 5_o the_o say_a segment_n and_o the_o remain_n shall_v be_v 144_o for_o the_o quadrate_n of_o the_o perpendicular_a a_o o_o by_o the_o 9_o e_fw-la x_o ij_o here_o the_o perpendicular_a now_o find_v and_o the_o side_n cut_v be_v the_o side_n of_o the_o rectangle_n who_o half_a shall_v be_v the_o content_a of_o the_o triangle_n as_o here_o the_o rectangle_n of_o 21_o and_o 12_o be_v 252_o who_o half_o 126_o be_v the_o content_a of_o the_o triangle_n the_o second_o section_n follow_v from_o whence_o
perpendicular_a and_o 25_o all_o touch-angle_n in_o equal_a periphery_n be_v equal_a but_o in_o unequal_a periphery_n the_o cornicular_a angle_n of_o a_o lesser_a periphery_n be_v great_a than_o the_o cornicular_a of_o a_o great_a 26_o if_o from_o a_o ray_n out_o of_o the_o centre_n of_o a_o periphery_a give_v a_o periphery_n be_v describe_v unto_o a_o point_n assign_v without_o and_o from_o the_o meeting_n of_o the_o assign_a and_o the_o ray_n a_o perpendicular_a fall_n upon_o the_o say_a ray_n unto_o the_o now_o describe_v periphery_a be_v tie_v by_o a_o right_a line_n with_o the_o say_a centre_n a_o right_a line_n draw_v from_o the_o point_n give_v unto_o the_o meeting_n of_o the_o periphery_a give_v and_o the_o knit_a line_n shall_v touch_v the_o assign_a periphery_n 17_o p_o iij._o thus_o much_o of_o the_o secant_v and_o tangent_n several_o it_o follow_v of_o both_o kind_n joint_o together_o 27_o if_o of_o two_o right_a line_n from_o a_o assign_a point_n without_o the_o first_o do_v cut_v a_o periphery_a unto_o the_o concave_n the_o other_o do_v touch_v the_o same_o the_o oblong_a of_o the_o secant_fw-la and_o of_o the_o outter_n segment_n of_o the_o secant_fw-la be_v equal_a to_o the_o quadrate_n of_o the_o tangent_fw-la and_o if_o such_o a_o like_a oblong_o be_v equal_a to_o the_o quadrate_n of_o the_o other_o that_o same_o other_o do_v touch_v the_o periphery_a 36_o and_o 37_o p_o iij._o therefore_o 28._o all_o tangent_n fall_v from_o the_o same_o point_n be_v equal_a or_o touch_v line_n draw_v from_o one_o and_o the_o same_o point_n be_v equal_a h._n because_o their_o quadrate_n be_v equal_a to_o the_o same_o oblong_a and_o 29._o the_o oblong_v make_v of_o any_o secant_fw-la from_o the_o same_o point_n and_o of_o the_o outter_n segment_n of_o the_o secant_fw-la be_v equal_a between_o themselves_o camp_n 36_o p_o iij._o the_o reason_n be_v because_o to_o the_o same_o thing_n and_o 30._o to_o two_o right_a line_n give_v one_o may_v so_o continue_v or_o join_v the_o three_o that_o the_o oblong_a of_o the_o continue_a and_o the_o continuation_n may_v be_v equal_a to_o the_o quadrate_n remain_v vitellio_n 127_o p_o i_o as_o in_o the_o first_o figure_n if_o the_o first_o of_o the_o line_n give_v be_v e_fw-it o_o the_o second_o i_o a_o the_o three_o o_o a._n now_o be_v we_o come_v to_o circular_a geometry_n that_o be_v to_o the_o geometry_n of_o circle_n or_o peripheries_n cut_v and_o touch_v one_o another_o and_o of_o right_a line_n and_o peripheries_n 31._o if_o periphery_n do_v either_o cut_v or_o touch_v one_o another_o they_o be_v eccentrickes_n and_o they_o do_v cut_v one_o another_o in_o two_o point_n only_o and_o these_o by_o the_o touch_n point_n do_v continue_v their_o diameter_n 5._o 6._o 10,11_o 12_o p_o iij._o all_o these_o may_v well_o have_v be_v ask_v but_o they_o have_v also_o their_o demonstration_n ex_fw-la impossibili_fw-la not_o very_o dissicult_a of_o right_a line_n and_o peripheries_n joint_o the_o rate_n be_v but_o one_o 32._o if_o inscript_n be_v equal_a they_o do_v cut_v equal_a periphery_n and_o contrariwise_o 28,29_o p_o iij._o or_o thus_o if_o the_o inscript_n of_o the_o same_o circle_n or_o of_o equal_a circle_n be_v equal_a they_o do_v cut_v equal_a periphery_n and_o contrariwise_o b._n or_o thus_o if_o line_n inscribe_v into_o equal_a circle_n or_o to_o the_o same_o be_v equal_a they_o cut_v equal_a periphery_n and_o contrariwise_o if_o they_o do_v cut_v equal_a periphery_n they_o shall_v themselves_o be_v equal_a schonerâ—_n except_o with_o the_o learned_a rodulphus_fw-la snellius_n you_o do_v understand_v aswell_o two_o equal_a periphery_n to_o be_v give_v as_o two_o equal_a right_a line_n you_o shall_v not_o conclude_v two_o equal_a section_n and_o therefore_o we_o have_v just_o insert_v of_o the_o same_o or_o of_o equal_a circle_n which_o we_o do_v now_o see_v be_v in_o like_a manner_n by_o lazarus_n schonerus_n the_o sixteen_o book_n of_o geometry_n of_o the_o segment_n of_o a_o circle_n 1._o a_o segment_n of_o a_o circle_n be_v that_o which_o be_v comprehend_v outter_o of_o a_o periphery_a sand_n innerly_a of_o a_o râ—ght_a line_n the_o geometry_n of_o segment_n be_v common_a also_o to_o the_o sphere_n but_o now_o this_o same_o general_n be_v hard_o to_o be_v declare_v and_o teach_v and_o the_o segment_n may_v be_v comprehend_v within_o of_o a_o oblique_a line_n either_o single_a or_o manifold_a but_o here_o we_o follow_v those_o thing_n that_o be_v usual_a and_o common_o receive_v first_o therefore_o the_o general_a definition_n be_v set_v foremost_a for_o the_o more_o easy_a distinguish_n of_o the_o species_n and_o several_a kind_n 2._o a_o segment_n of_o a_o circle_n be_v either_o a_o sectour_z or_o a_o sâ—ction_n segmentum_fw-la a_o segment_n and_o sectio_fw-la a_o section_n and_o sector_n a_o sectour_z be_v almost_o the_o same_o in_o common_a acceptation_n but_o they_o shall_v be_v distinguish_v by_o their_o definition_n 3._o a_o sectour_z be_v a_o segment_n inner_o comprehend_v of_o two_o right_a line_n make_v a_o angle_n in_o the_o centre_n which_o be_v call_v a_o angle_n in_o the_o centre_n as_o the_o periphery_n be_v the_o base_a of_o the_o sectour_z 9_o d_o iij._o 4._o a_o angle_n in_o the_o periphery_n be_v a_o angle_n comprehend_v of_o two_o right_a line_n inscribe_v and_o joint_o bound_v or_o meet_v in_o the_o periphery_a 8_o d_o iij._o this_o may_v have_v be_v call_v the_o sectour_z in_o the_o periphery_n to_o wit_n comprehend_v innerly_a of_o two_o right_a line_n joint_o bound_v in_o the_o periphery_a as_o here_o a_o e_o i._n 5._o the_o angle_n in_o the_o centre_n be_v double_a to_o the_o angle_n of_o the_o periphery_a stand_n upon_o the_o same_o base_a 20_o p_o iij._o therefore_o 6._o if_o the_o angle_n in_o the_o periphery_n be_v squall_n to_o the_o angle_n in_o the_o centre_n it_o be_v double_a to_o it_o in_o base_a and_o contrariwise_o this_o follow_v out_o of_o the_o former_a element_n for_o the_o angle_n in_o the_o centre_n be_v double_a to_o the_o angle_n in_o the_o periphery_a stand_n upon_o the_o same_o base_a wherefore_o if_o the_o angle_n in_o the_o periphery_n be_v to_o be_v make_v equal_a to_o the_o angle_n in_o the_o centre_n his_o base_n be_v to_o be_v double_v and_o thence_o shall_v follow_v the_o equality_n of_o they_o both_o s._n 7._o the_o angle_n in_o the_o centre_n or_o periphery_n of_o equal_a circle_n be_v as_o the_o peripheries_n be_v upon_o which_o they_o do_v insist_v and_o contrariwise_o è_fw-it 33_o p_o uj_o and_o 26_o 27_o p_o iij._o here_o be_v a_o double_a proportion_n with_o the_o periphery_n underneath_o of_o the_o angle_n in_o the_o centre_n and_o of_o angle_n in_o the_o periphery_a but_o it_o shall_v suffice_v to_o declare_v it_o in_o the_o angle_n in_o the_o centre_n first_o therefore_o let_v the_o angle_n in_o the_o centre_n a_o e_o i_o and_o o_o u._fw-mi y_fw-mi be_v equal_a the_o base_n a_o i_o and_o o_z y_z shall_v be_v equal_a by_o the_o 11_o e_fw-la seven_o and_o the_o periphery_n a_o i_o and_o o_z y_z by_o the_o 32_o e_fw-la x_o five_o shall_v likewise_o be_v equal_a therefore_o if_o the_o angle_n be_v unequal_a the_o periphery_n likewise_o shall_v be_v equal_a the_o same_o shall_v also_o be_v true_a of_o the_o angle_n in_o the_o periphery_a the_o converse_n in_o like_a manner_n be_v true_a from_o whence_o follow_v this_o consectary_n therefore_o 8._o as_o the_o sectour_z be_v unto_o the_o sectour_z so_o be_v the_o angle_n unto_o the_o angle_n and_o contrariwise_o and_o thus_o much_o of_o the_o sectour_z 9_o a_o section_n be_v a_o segment_n of_o a_o circle_n within_o comprehend_v of_o one_o right_a line_n which_o be_v term_v the_o base_a of_o the_o section_n as_o here_o a_o e_z i_z and_o o_o u._fw-mi y_fw-mi and_o s_o r_o l_o be_v section_n 10._o a_o section_n be_v make_v up_o by_o find_v of_o the_o centre_n 11_o the_o periphery_a of_o a_o section_n be_v divide_v into_o two_o equal_a part_n by_o a_o perpendicular_a divide_v the_o base_a into_o two_o equal_a part_n 20._o p_o iij._o here_o euclid_n do_v by_o congruency_n comprehend_v two_o periphery_n in_o one_o and_o so_o do_v we_o comprehend_v they_o 12_o a_o angle_n in_o a_o section_n be_v a_o angle_n comprehend_v of_o two_o right_a line_n joint_o bound_v in_o the_o base_a and_o in_o the_o periphery_a joint_o bound_v 7_o d_o iij._o or_o thus_o a_o angle_n in_o the_o section_n be_v a_o angle_n comprehend_v under_o two_o right_a line_n have_v the_o same_o term_n with_o the_o base_n and_o the_o term_n with_o the_o circumference_n h._n as_o a_o o_o e_o in_o the_o former_a example_n 13_o the_o angle_n in_o the_o same_o section_n be_v equal_a 21._o p_o iij._o here_o it_o be_v certain_a that_o angle_n in_o a_o section_n be_v indeed_o angle_v
be_v continual_a hitherto_o it_o have_v be_v prove_v that_o the_o quinquangle_v make_v be_v a_o equilater_n and_o plain_a it_o remain_v that_o it_o be_v prove_v to_o be_v equiangle_v let_v therefore_o the_o right_a line_n e_o p_o and_o e_o c_o be_v draw_v i_o say_v that_o the_o angle_n p_o b_o e_o and_o e_o z_o i_o be_v equal_a because_o they_o have_v by_o the_o construction_n the_o base_n of_o equal_a shank_n equal_a be_v to_o wit_n in_o value_n the_o quadruple_a of_o l_o e._n for_o the_o right_a line_n l_o f_o cut_v proportional_o and_o increase_v with_o the_o great_a segment_n d_o f_o that_o be_v f_o c_o be_v cut_v also_o proportional_o by_o the_o 4_o e_fw-la fourteen_o and_o by_o the_o 7_o e_fw-la fourteen_o the_o whole_a line_n proportional_o cut_v and_o the_o lesser_a segment_n that_o be_v c_o p_o be_v of_o treble_a value_n to_o the_o great_a f_o l_o that_o be_v of_o the_o say_v l_o e._n therefore_o e_o l_o and_o l_o c_o that_o be_v e_o c_o and_o c_o p_o that_o be_v e_o p_o be_v of_o quadruple_a power_n to_o e_o l_o and_o therefore_o by_o the_o 14_o e_fw-la xij_o it_o be_v the_o double_a of_o it_o and_o e_fw-it i_fw-it itself_o in_o like_a manner_n by_o the_o fabric_n or_o construction_n be_v the_o double_a of_o the_o same_o therefore_o the_o base_n be_v equal_a and_o after_o the_o same_o manner_n by_o draw_v the_o right_a line_n i_o d_o and_o i_z b_o the_o three_o angle_n b_o p_o i_o shall_v be_v conclude_v to_o be_v equal_a to_o the_o angle_n e_o z_o i._n therefore_o by_o the_o 13_o e_fw-la fourteen_o five_o angle_n be_v equal_a 23._o the_o diagony_n be_v irrational_a unto_o the_o side_n of_o the_o dodecahedrum_fw-la this_o be_v the_o five_o example_n of_o irrationality_n and_o incommensurability_n the_o first_o be_v of_o the_o diagony_n and_o side_n of_o a_o quadrate_n or_o square_n the_o second_o be_v of_o a_o line_n proportional_o cut_v and_o his_o segment_n the_o three_o be_v of_o the_o diameter_n of_o a_o circle_n and_o the_o side_n of_o a_o inscribe_v quinquangle_v the_o four_o be_v of_o the_o diagony_n and_o side_n of_o a_o icosahedrum_fw-la the_o five_o now_o be_v of_o the_o diagony_n and_o side_n of_o a_o dodecahedrum_fw-la 24_o if_o the_o side_n of_o a_o cube_fw-la be_v cut_v proportional_o the_o great_a segment_n shall_v be_v the_o side_n of_o a_o dodecahedrum_fw-la the_o semidiagony_n and_o ray_n of_o the_o circle_n thus_o find_v the_o altitude_n remain_v take_v out_o therefore_o the_o quadrate_n of_o the_o ray_n of_o the_o circle_n 16_o 4_o 225_o out_o of_o the_o quadrate_n of_o the_o semidiagony_n 47._o 12458_o 17161._o the_o side_n of_o the_o remainder_n 3â—_n 2â—14406_n 3861225_o be_v for_o the_o altitude_n or_o height_n who_o â…“_n be_v 5_o 3._o the_o quinquangled_a base_n be_v almost_o 38._o which_o multiply_v by_o 5_o 3_o do_v make_v 63_o â…“_n for_o the_o solidity_n of_o one_o pyramid_n which_o multiply_v by_o 12_o do_v make_v 760._o for_o the_o solidity_n of_o the_o whole_a dodetacedrum_fw-la 25_o there_o be_v but_o five_o ordinate_a solid_a plain_n this_o appear_v plain_o out_o of_o the_o nature_n of_o a_o solid_a angle_n by_o the_o kind_n of_o plain_a figure_n of_o two_o plain_a angle_n a_o solid_a angle_n can_v be_v comprehend_v of_o three_o angle_n of_o a_o ordinate_a triangle_n be_v the_o angle_n of_o a_o tetrahedrum_fw-la comprehend_v of_o four_o a_o octahedrum_fw-la of_o five_o a_o icosahedrum_fw-la of_o six_o none_o can_v be_v comprehend_v for_o six_o such_o like_a plain_a angle_n be_v equal_a to_o 12_o three_o of_o one_o right_a angle_n that_o be_v to_o four_o right_a angle_n but_o plain_a angle_n make_v a_o solid_a angle_n be_v lesser_a than_o four_o right_a angle_n by_o the_o 8_o e_fw-la xxij_o of_o seven_o therefore_o and_o of_o more_o it_o be_v much_o less_o possible_a of_o three_o quadrate_n angle_v the_o angle_n of_o a_o cube_fw-la be_v comprehend_v of_o 4._o such_o angle_v none_o may_v be_v comprehend_v for_o the_o same_o cause_n of_o three_o angle_n of_o a_o ordinate_a 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