right_a line_n but_o many_o do_v fall_v out_o to_o be_v in_o a_o crooked_a line_n and_o in_o a_o sphere_n a_o cone_n &_o cylinderâ—_n a_o ruler_n may_v be_v apply_v but_o it_o must_v be_v a_o sphearicall_a conicall_a or_o cylindraceall_n but_o by_o the_o example_n of_o a_o right_a line_n do_v vitellio_n 2_o p_o i_o demand_n that_o between_o two_o line_n a_o surface_n may_v be_v extend_v and_o so_o may_v it_o seem_v in_o the_o element_n of_o many_o figure_n both_o plain_a and_o solid_n by_o euclid_n to_o be_v demand_v that_o a_o figure_n may_v be_v describe_v at_o the_o 7._o and_o 8._o e_fw-la ij_o item_n that_o a_o figure_n may_v be_v make_v up_o at_o the_o 8._o 14._o 16._o 23.28_o p._n uj_o which_o be_v of_o plain_n item_n at_o the_o 25._o 31._o 33._o 34._o 36._o 49._o p.xj._n which_o be_v of_o solid_n yet_o notwithstanding_o a_o plain_a surface_n and_o a_o plain_a body_n do_v measure_v their_o rectitude_n by_o a_o right_a line_n so_o that_o jus_o postulandi_fw-la this_o right_a of_o beg_v to_o have_v a_o thing_n grant_v may_v seem_v primary_o to_o be_v in_o a_o right_n plain_a line_n now_o the_o continuation_n of_o a_o right_a line_n be_v nothing_o else_o but_o the_o draw_v out_o far_o of_o a_o line_n now_o draw_v and_o that_o from_o a_o point_n unto_o a_o point_n as_o we_o may_v continue_v the_o right_a line_n a_o e._n unto_o i._o wherefore_o the_o first_o and_o second_o petition_n of_o euâ—lde_n do_v agree_v in_o one_o and_o 7._o to_o set_v at_o a_o point_n assign_v a_o right_a line_n equal_a to_o another_o right_a line_n give_v and_o from_o a_o great_a to_o cut_v off_o a_o part_n equal_a to_o a_o lesser_a 2._o and_o 3._o pj._n therefore_o 8._o one_o right_a line_n or_o two_o cut_v one_o another_o be_v in_o the_o same_o plain_a out_o of_o the_o 1._o and_o 2._o p_o xj_o one_o right_a line_n may_v be_v the_o common_a section_n of_o two_o plain_n yet_o all_o or_o the_o whole_a in_o the_o same_o plain_a be_v one_o and_o all_o the_o whole_a be_v in_o the_o same_o other_o and_o so_o the_o whole_a be_v the_o same_o plain_a two_o right_a line_n cut_v one_o another_o may_v be_v in_o two_o plain_n cut_v one_o of_o another_o but_o then_o a_o plainâ—_n may_v be_v draw_v by_o they_o therefore_o both_o of_o they_o shall_v be_v in_o the_o same_o plain_a and_o this_o plain_n be_v geometrical_o to_o be_v conceive_v because_o the_o same_o plain_a be_v not_o always_o make_v the_o ground_n whereupon_o one_o oblique_a line_n or_o two_o cut_v one_o another_o be_v draw_v when_o a_o periphery_n be_v in_o a_o sphearicall_a neither_o may_v all_o periphery_n cut_v one_o another_o be_v possible_o in_o one_o plain_a and_o 9_o with_o a_o right_a line_n give_v to_o describe_v a_o peripherie_n talus_fw-la the_o nephew_n of_o daedalus_n by_o his_o sister_n be_v say_v in_o the_o viij_o book_n of_o ovid_n metamorphosis_n to_o have_v be_v the_o inventour_n of_o this_o instrument_n for_o there_o he_o thus_o write_v of_o he_o and_o this_o matter_n et_fw-la ex_fw-la uno_fw-la duo_fw-la ferrea_fw-la brachia_fw-la nodo_fw-la junxit_fw-la ut_fw-la aequali_fw-la spatio_fw-la distantibus_fw-la ipsis_fw-la altera_fw-la pars_fw-la staret_fw-la pars_fw-la altera_fw-la duceâ—et_fw-la orbem_fw-la therefore_o 10._o the_o raiâ—s_n of_o the_o same_o or_o of_o a_o equal_a periphery_n be_v equal_a the_o reason_n be_v because_o the_o same_o right_a line_n be_v every_o where_o convert_v or_o turn_v about_o but_o here_o by_o the_o ray_n of_o the_o periphery_a must_v be_v understand_v the_o ray_n the_o figure_n contain_v within_o the_o periphery_n 11._o if_o two_o equal_a periphery_n from_o the_o end_n of_o equal_a shank_n of_o a_o assign_a rectilineall_a angle_n do_v meet_v before_o it_o a_o right_a line_n draw_v from_o the_o meeting_n of_o they_o unto_o the_o top_n or_o point_n of_o the_o angle_n shall_v cut_v it_o into_o two_o equal_a part_n 9_o pj._n hitherto_o we_o have_v speak_v of_o plain_a line_n their_o affection_n follow_v and_o first_o in_o the_o bisection_n or_o divide_v of_o a_o angle_n into_o two_o equal_a part_n 12._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 do_v meet_v on_o each_o side_n of_o the_o same_o a_o right_a line_n draw_v from_o those_o meeting_n shall_v divide_v the_o right_a line_n give_v into_o two_o equal_a part_n 10._o pj._n 13._o if_o a_o right_a line_n do_v stand_v perpendicular_a upon_o another_o right_a line_n it_o make_v on_o each_o side_n right_a angle_n and_o contrary_a wise_a the_o rular_a for_o the_o make_n of_o straight_a line_n on_o a_o plain_a be_v the_o first_o geometrical_a instrument_n the_o compass_n for_o the_o describe_v of_o a_o circle_n be_v the_o second_o the_o norma_n or_o square_n for_o the_o true_a erect_n of_o a_o right_a line_n in_o the_o same_o plain_a upon_o another_o right_a line_n and_o then_o of_o a_o surface_n and_o body_n upon_o a_o surface_n or_o body_n be_v the_o three_o the_o figure_n therefore_o be_v thus_o therefore_o 14._o if_o a_o right_a line_n do_v stand_v upon_o a_o right_a line_n it_o make_v the_o angle_n on_o each_o side_n equal_a to_o two_o right_a angle_n and_o contrariwise_o out_o of_o the_o 13._o and_o 14._o pj._n and_o 15._o if_o two_o right_a line_n do_v cut_v one_o another_o they_o do_v make_v the_o angle_n at_o the_o top_n equal_a and_o all_o equal_a to_o four_o right_a angle_n 15._o pj._n and_o 16._o if_o two_o right_a line_n cut_v with_o one_o right_a line_n do_v make_v the_o inner_a angle_n on_o the_o same_o side_n great_a than_o two_o right_a angle_n those_o on_o the_o other_o side_n against_o they_o shall_v be_v lesser_a than_o two_o right_a angle_n 17._o if_o from_o â—â—oint_n assign_v of_o a_o infinite_a right_a line_n give_v two_o equal_a part_n be_v on_o each_o side_n cut_v off_o and_o then_o from_o the_o point_n of_o those_o section_n two_o equal_a circle_n do_v meet_v a_o right_a line_n draw_v from_o their_o meeting_n unto_o the_o point_n assign_v shall_v be_v perpendicular_a unto_o the_o line_n give_v 11._o pj._n 18._o if_o a_o part_n of_o a_o infinite_a right_a line_n be_v by_o a_o periphery_a from_o a_o point_n give_v without_o cut_v off_o a_o right_a line_n from_o the_o say_a point_n cut_v in_o two_o the_o say_a part_n shall_v be_v perpendicular_a upon_o the_o line_n give_v 12._o pj._n 19_o if_o two_o right_a line_n draw_v at_o length_n in_o the_o same_o plain_a do_v never_o meet_v they_o be_v parallelly_n è_fw-it 35._o dj_o therefore_o 20._o if_o a_o infinite_a right_a line_n do_v cut_v one_o of_o the_o infinite_a right_a parallel_n line_n it_o shall_v also_o cut_v the_o other_o as_o in_o the_o same_o example_n u._fw-mi y._n cut_v a_o e._n it_o shall_v also_o cuâ—_n i_o o._n otherwise_o if_o it_o shall_v not_o cut_v it_o it_o shall_v be_v parallel_n unto_o it_o by_o the_o 18_o e._n and_o that_o against_o the_o grant_n 21._o if_o right_a line_n cut_v with_o a_o right_a line_n be_v pararellell_n they_o do_v make_v the_o inner_a angle_n on_o the_o same_o side_n equal_a to_o two_o right_a angle_n and_o also_o the_o alterne_a angle_n equal_a between_o themselves_o and_o the_o outter_n to_o the_o inner_a opposite_a to_o it_o and_o contrariwise_o 29,28,27_o p_o 1._o the_o cause_n of_o this_o threefold_a propriety_n be_v from_o the_o perpendicular_a or_o plumbline_n which_o fall_v upon_o the_o parallel_n breed_v and_o discover_v all_o this_o variety_n as_o here_o they_o be_v right_a angle_n which_o be_v the_o inner_a on_o the_o same_o part_n or_o side_n item_n the_o alterne_a angle_n item_n the_o inner_a and_o the_o outter_n and_o therefore_o they_o be_v equal_a both_o i_o mean_v the_o two_o inner_a to_o two_o right_a angle_n and_o the_o alterne_a angle_n between_o themselves_o and_o the_o outter_n to_o the_o inner_a opposite_a to_o it_o if_o so_o be_v that_o the_o cut_a line_n be_v oblique_a that_o be_v fall_v not_o upon_o they_o plumbe_v or_o perpendicular_o the_o same_o shall_v on_o the_o contrary_n befall_v the_o parallel_n for_o by_o that_o same_o obliquation_n or_o slant_v the_o right_a line_n remain_v and_o the_o angle_n unaltered_a in_o like_a manner_n both_o one_o of_o the_o inner_a to_o wit_n e_fw-it u._fw-mi y_fw-mi be_v make_v obtuse_a the_o other_o to_o wiâ—_n u._fw-mi y_fw-mi o_o be_v make_v acute_a and_o the_o alterne_a angle_n be_v make_v acute_a and_o obtuse_a as_o also_o the_o outter_n and_o inner_a opposite_a be_v likewise_o make_v acute_a and_o obtuse_a the_o same_o impossibility_n shall_v be_v conclude_v if_o they_o shall_v be_v say_v to_o be_v lesser_a than_o two_o right_a anglesâ—_n the_o second_o and_o three_o part_n may_v be_v conclude_v out_o of_o the_o first_o the_o second_o be_v thus_o twice_o two_o angle_n be_v equal_a to_o two_o right_a
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
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 their_o base_n be_v one_o to_o another_o thus_o far_o of_o the_o reason_n or_o rate_n of_o triangle_n the_o proportion_n of_o triangle_n do_v follow_v and_o first_o of_o a_o right_a line_n with_o the_o base_n it_o be_v a_o consectary_n out_o of_o the_o 16_o e_fw-la iiij_o therefore_o 8._o upon_o a_o equal_a base_a they_o be_v equal_a 9_o if_o a_o right_a line_n draw_v from_o the_o top_n of_o a_o triangle_n do_v cut_v the_o base_a into_o two_o equal_a part_n it_o do_v also_o cut_v the_o triangle_n into_o two_o equal_a part_n and_o it_o be_v the_o diameter_n of_o the_o triangle_n 10._o if_o a_o right_a line_n be_v draw_v from_o the_o top_n of_o a_o triangle_n unto_o a_o point_n give_v in_o the_o base_a so_o it_o be_v not_o in_o the_o midst_n of_o it_o and_o a_o parallel_n be_v draw_v from_o the_o midst_n of_o the_o base_a unto_o the_o side_n a_o right_a line_n draw_v from_o the_o top_n of_o the_o say_a parallel_n unto_o the_o say_a point_n shall_v cut_v the_o triangle_n into_o two_o equal_a part_n 11_o if_o equiangle_v triangle_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._n uj._o or_o thus_o as_o the_o learned_a mâ—_n brigges_n have_v conceive_v it_o if_o two_o triangle_n have_v one_o angle_n be_v reciprocal_a etc._n etc._n the_o converse_n be_v conclude_v by_o the_o same_o sorite_n but_o by_o say_v all_o backward_a for_o you_o a_o unto_o a_o e_fw-la be_v as_o u_z a_o o_o be_v unto_o o_o a_o e_fw-la by_z the_o 7_o e_z and_o as_o e_z a_o i_o by_o the_o grant_n because_o they_o be_v equal_a and_o as_o i_o a_o be_v unto_o a_o o_o by_o the_o same_o wherefore_o u._fw-mi a_o be_v unto_o a_o e_fw-la as_o i_z a_o be_v unto_o a_o o._n 12_o if_o two_o triangle_n be_v equiangle_n they_o be_v proportional_a in_o shankes_n and_o contrariwise_o 4_o and_o 5._o p._n uj._o therefore_o 13._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 off_o from_o it_o a_o 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 top_n shall_v be_v the_o other_o shankeâ—_n and_o contrariwise_o schon_n as_o be_v manifest_a in_o the_o same_o example_n 5_o if_o a_o right_a angle_a triangle_n be_v equicrural_a each_o of_o the_o angle_n at_o the_o base_a be_v the_o halâ—e_n of_o a_o right_a angle_n and_o contrariwise_o therefore_o 6_o if_o one_o angle_n of_o a_o triangle_n be_v equal_a to_o the_o other_o two_o it_o be_v a_o right_a angle_n and_o contrariwise_o schon_n because_o it_o be_v equal_a to_o the_o half_a of_o two_o right_a angle_n by_o the_o 13._o e_o u.j._n and_o 7_o if_o a_o right_a line_n from_o the_o top_n of_o a_o triangle_n cut_v the_o base_a into_o too_o equal_a part_n be_v equal_a to_o the_o bisegment_n or_o half_a of_o the_o base_a the_o angle_n at_o the_o top_n be_v a_o right_a angle_n and_o contrariwise_o schon_n 8_o a_o perpendicular_a in_o a_o triangle_n from_o the_o right_a angle_n to_o the_o base_a do_v cut_v it_o into_o two_o triangle_n like_v unto_o the_o whole_a and_o between_o themselves_o 8._o p_o v_o i_o and_o contrariwise_o schon_n therefore_o 9_o the_o perpendicular_a be_v the_o mean_a proportional_a between_o the_o segment_n or_o portion_n of_o the_o base_a as_o in_o the_o say_a example_n as_o i_o o_o be_v to_o o_o a_o so_o be_v of_o a_o to_z o_o e_o because_o the_o shank_n of_o equal_a angle_n be_v proportional_a by_o the_o 8_o e_o from_o hence_o be_v plato_n mesographus_n invent_v and_o 10_o either_o of_o the_o shank_n be_v proportional_a between_o the_o base_a and_o the_o segment_n of_o the_o base_a next_o adjoin_v for_o as_o e_z i_z be_v unto_o i_o a_o in_o the_o whole_a triangle_n so_o be_v a_o i_o to_o i_o o_o in_o the_o great_a for_o so_o they_o be_v homologall_a side_n which_o do_v subtend_v equal_a angle_n by_o the_o 23_o e_o iiij_o item_n as_o i_o e_o be_v to_z e_z a_o in_o the_o whole_a triangle_n so_o be_v a_o e_o to_z e_z o_o in_o the_o lesser_a triangle_n either_o of_o the_o shank_n be_v proportional_a between_o the_o sum_n and_o the_o difference_n of_o the_o base_a and_o the_o other_o shank_n and_o contrariwise_o if_o one_o side_n be_v proportional_a between_o the_o sum_n and_o the_o difference_n of_o the_o other_o the_o triangle_n give_v be_v a_o rectangle_n m._n h._n brigges_n this_o be_v a_o consectary_n arise_v likewise_o out_o of_o the_o 4_o e._n of_o very_o great_a use_n in_o the_o triangle_n e_o a_fw-fr d_o the_o shank_n a_o d_o 12._o be_v the_o mean_a proportional_a between_n b_o d_o 18._o the_o sum_n of_o the_o base_a a_o e_o 13._o and_o the_o shank_n e_o d_o 5._o and_o 8._o the_o difference_n of_o the_o say_v base_a and_o shank_n for_o if_o thou_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
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
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 the_o first_o part_n like_o section_n upon_o the_o same_o base_a shall_v not_o be_v equal_a but_o congruency_n be_v here_o sufficient_a by_o the_o former_a two_o proposition_n and_o by_o the_o 9_o e_fw-la x_o v._n one_o may_v find_v a_o section_n like_a unto_o another_o assign_v or_o else_o from_o a_o circle_n give_v to_o cut_v off_o one_o like_a unto_o it_o 17_o a_o angle_n of_o a_o section_n be_v that_o which_o be_v comprehend_v of_o the_o bound_n of_o a_o section_n 18_o a_o section_n be_v either_o a_o semicircle_n or_o that_o which_o be_v unequal_a to_o a_o semicircle_n a_o section_n be_v two_o fold_n a_o semicircle_n to_o wit_n when_o it_o be_v cut_v by_o the_o diameter_n or_o unequal_a to_o a_o semicircle_n when_o it_o be_v cut_v by_o a_o line_n lesser_a than_o the_o diameter_n 19_o a_o semicircle_n be_v the_o half_a section_n of_o a_o circle_n or_o it_o be_v that_o which_o be_v make_v the_o diameter_n therefore_o 20_o a_o semicircle_n be_v comprehend_v of_o a_o periphery_a and_o the_o diameter_n 18_o dj_o 21_o the_o angle_n in_o a_o semicircle_n be_v a_o right_a angle_n the_o angle_n of_o a_o semicircle_n be_v lesser_a than_o a_o rectilineall_a right_a angle_n but_o great_a than_o any_o acute_a angle_n the_o angle_n in_o a_o great_a section_n be_v lesser_a than_o a_o right_a angle_n of_o a_o great_a it_o be_v a_o great_a in_o a_o lesser_a it_o be_v great_a of_o a_o lesser_a it_o be_v lesser_a ê_fw-la 31_o and_o 16._o p_o iij._o or_o thus_o the_o angle_n in_o a_o semicircle_n be_v a_o right_a angle_n the_o angle_n of_o a_o semicircle_n be_v less_o than_o a_o right_n rightline_v angle_n but_o great_a than_o any_o acute_a angle_n the_o angle_n in_o the_o great_a section_n be_v less_o than_o a_o right_a angle_n the_o angle_n of_o the_o great_a section_n be_v great_a than_o a_o right_a angle_n the_o angle_n in_o the_o lesser_a section_n be_v great_a than_o a_o right_a angle_n the_o angle_n of_o the_o lesser_a section_n be_v lesser_a than_o a_o right_a angle_n h._n 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 the_o five_o be_v thus_o the_o angle_n of_o the_o great_a section_n e_fw-la a_o i_o be_v great_a than_o a_o right_a angle_n because_o it_o contain_v a_o rightangle_n the_o six_o be_v thus_o the_o angle_n a_o o_o e_o in_o a_o lesser_a section_n be_v great_a than_o a_o right_a angle_n by_o the_o 14_o e_fw-la x_o five_o i_o because_o that_o which_o be_v in_o the_o opposite_a section_n be_v lesser_a than_o a_o right_a angle_n the_o seven_o be_v thus_o the_o angle_n e_o a_o o_o be_v lesser_a than_o a_o rightangle_n because_o it_o be_v part_n of_o a_o right_a angle_n to_o wit_n of_o the_o outter_n angle_n if_o i_o a_o be_v draw_v out_o at_o length_n and_o thus_o much_o of_o the_o angle_n of_o a_o circle_n of_o all_o which_o the_o most_o effectual_a and_o of_o great_a power_n and_o use_n be_v the_o angle_n in_o a_o semicircle_n and_o therefore_o it_o be_v not_o without_o cause_n so_o often_o mention_v of_o aristotle_n this_o geometry_n therefore_o of_o aristotle_n let_v we_o somewhat_o more_o 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angle_n o_o y_fw-mi u._fw-mi and_o e_z u_z y_z by_o the_o former_a part_n item_n a_o u._fw-mi y_fw-mi and_o e_z u_z y_z by_o the_o 14_o e._n therefore_o they_o be_v equal_a between_o themselves_o now_o from_o the_o equal_a take_v away_o e_z u_z y_z the_o common_a angle_n and_o the_o remainder_n the_o alterne_a angle_n at_o u._fw-mi and_o y_z shall_v be_v least_o equal_a the_o three_o be_v thus_o the_o angle_n e_o u._fw-mi y_fw-mi and_o o_z y_z s_z be_v equal_a to_o the_o same_o u._fw-mi y_fw-mi i_fw-it by_o the_o second_o propriety_n and_o by_o the_o 15_o e._n therefore_o they_o be_v equal_a between_o themselves_o if_o they_o be_v oblique_a angle_n as_o here_o the_o line_n one_o slant_v or_o lique_o cross_v one_o another_o the_o angle_n on_o one_o side_n will_v grow_v less_o on_o the_o other_o side_n great_a therefore_o they_o will_v not_o be_v equal_a to_o two_o right_a angle_n against_o the_o grant_n from_o hence_o the_o second_o and_o three_o part_n may_v be_v conclude_v the_o 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o_o and_o let_v a_o e_fw-es u._fw-mi equal_a to_o a_o e_o o_o be_v cut_v off_o from_o the_o great_a a_o e_o i_o and_o let_v e_o u._fw-mi be_v equal_a to_o e_o o._n now_o by_o the_o 2_o e_z seven_o two_o triangle_n a_o e_o u._fw-mi and_o a_o e_o o_o be_v equal_a in_o their_o base_n a_o u._fw-mi and_o a_o o._n item_n a_o o_o and_o e_z i_z be_v great_a than_o a_o i_o and_o a_o o_o and_o a_o o_o be_v equal_a to_o a_o v_o therefore_o o_o ay_o be_v great_a than_o i_o u._fw-mi here_o two_o triangle_n u._fw-mi e_fw-it i_fw-it and_o i_z e_z o_o equal_a in_o two_o shank_n and_o the_o base_a o_o ay_o
a_o inscribe_v quinquangle_v the_o diagony_n of_o a_o icosahedron_n and_o dodecahedron_n be_v irrational_a unto_o the_o side_n 10._o congruall_a or_o agreeable_a magnitude_n be_v those_o who_o part_n be_v apply_v or_o lay_v one_o upon_o another_o do_v fill_v a_o equal_a place_n symmetria_fw-la symmetry_n or_o commensurability_n and_o rate_n be_v from_o number_n the_o next_o affection_n of_o magnitude_n be_v altogether_o geometrical_a congruentia_fw-la congruency_n agreeablenesse_n be_v of_o two_o magnitude_n when_o the_o first_o part_n of_o the_o one_o do_v agree_v to_o the_o first_o part_n of_o the_o other_o the_o mean_a to_o the_o mean_a the_o extreme_n or_o end_n to_o the_o end_n and_o last_o the_o part_n of_o the_o one_o in_o all_o respect_n to_o the_o part_n of_o the_o other_o so_o line_n be_v congruall_a or_o agreeable_a when_o the_o bound_a point_v of_o the_o one_o apply_v to_o the_o bound_a point_n of_o the_o other_o and_o the_o whole_a length_n to_o the_o whole_a lengthe_n 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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 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