figure_n with_o a_o right_a angle_n accord_v to_o the_o xu_o conclusion_n then_o consider_v the_o likeiamme_n whether_o it_o have_v all_o his_o side_n equal_a or_o not_o for_o if_o they_o be_v all_o equal_a then_o have_v you_o do_v your_o conclusion_n but_o and_o if_o the_o side_n be_v not_o all_o equal_a then_o shall_v you_o make_v one_o right_a line_n just_a as_o long_o as_o two_o of_o those_o unequal_a side_n that_o line_n shall_v you_o divide_v in_o the_o middle_n and_o on_o that_o prick_n draw_v half_a a_o circle_n then_o cut_v from_o that_o diameter_n of_o the_o half_a circle_n a_o certain_a portion_n equal_a to_o the_o one_o side_n of_o the_o likeiamme_n and_o from_o that_o point_n of_o division_n shall_v you_o erect_v a_o perpendicular_a which_o shall_v touch_v the_o edge_n of_o the_o circle_n and_o that_o perpendicular_a shall_v be_v the_o just_a side_n of_o the_o square_a quadrate_n equal_a both_o to_o the_o lykeiamme_n and_o also_o to_o the_o right_n line_v figure_n appoint_v as_o the_o conclusion_n will_v example_n k_o be_v the_o right_n line_v figure_n appoint_v and_o b._n c._n d._n e_o be_v the_o likeiamme_n with_o right_a angle_n equal_a unto_o k_o but_o because_o that_o this_o likeiamme_n be_v not_o a_o square_a quadrate_n i_o must_v turn_v it_o into_o such_o one_o after_o this_o sort_n i_o shall_v make_v one_o right_a line_n as_o long_o as_o two_o unequal_a side_n of_o the_o likeiamme_n that_o line_n here_o be_v f._n g_o which_o be_v equal_a to_o b._n c_o and_o c._n e._n then_o part_n i_o that_o line_n in_o the_o middle_n in_o the_o prick_n m_o and_o on_o that_o prick_n i_o make_v half_o a_o circle_n accord_v to_o the_o length_n of_o the_o diameter_n f._n g._n afterward_o i_o cut_v away_o a_o piece_n from_o f._n g_o equal_a to_o c._n e_o mark_n that_o point_n with_o h._n and_o on_o that_o prick_n i_o erect_v a_o perpendicular_a h._n k_o which_o be_v the_o just_a side_n to_o the_o square_a quadrate_n that_o i_o seek_v for_o therefore_o accord_v to_o the_o doctrine_n of_o the_o ten_o conclusion_n of_o that_o line_n i_o do_v make_v a_o square_a quadrate_n and_o so_o have_v i_o attain_v the_o practice_n of_o this_o conclusion_n the_o xx._n conclusion_n when_o any_o two_o square_a quadrate_n be_v set_v forth_o how_o you_o may_v make_v one_o equal_a to_o they_o both_o first_o draw_v a_o right_a line_n equal_a to_o the_o side_n of_o one_o of_o the_o quadrate_n and_o on_o the_o end_n of_o it_o make_v a_o perpendicular_a equal_a in_o length_n to_o the_o side_n of_o the_o other_o quadrate_n then_o draw_v a_o bias_n line_n between_o those_o two_o other_o line_n make_v thereof_o a_o right_n angel_v triangle_n and_o that_o bias_n line_n will_v make_v a_o square_a quadrate_n equal_a to_o the_o other_o two_o quadrate_n appoint_v example_n a.b._n and_o c._n d_o be_v the_o two_o square_a quadrate_n appoint_v unto_o which_o i_o must_v make_v one_o equal_a square_a quadrate_n first_o therefore_o i_o do_v make_v a_o right_a line_n e._n f_o equal_a to_o one_o of_o the_o side_n of_o the_o square_a quadrate_n a.b._n and_o on_o the_o one_o end_n of_o it_o i_o make_v a_o plumbe_v line_n e._n g_o equal_a to_o the_o side_n of_o the_o other_o quadrate_n d._n c._n then_o draw_v i_o a_o bias_n line_n g._n f_o which_o be_v make_v the_o side_n of_o a_o quadrate_n accord_v to_o the_o ten_o conclusion_n will_v accomplish_v the_o work_n of_o this_o practice_n for_o the_o quadrate_n h._n be_v as_o much_o just_a as_o the_o other_o two_o i_o mean_v a._n b._n and_o d._n c._n the_o xxi_o conclusion_n when_o any_o two_o quadrate_n be_v set_v forth_o how_o to_o make_v a_o squire_n about_o the_o one_o quadrate_n which_o shall_v be_v equal_a to_o the_o other_o quadrate_n determine_v with_o yourself_o about_o which_o quadrate_n you_o will_v make_v the_o squire_n and_o draw_v one_o side_n of_o that_o quadrate_n forth_o in_o lengte_n accord_v to_o the_o measure_n of_o the_o side_n of_o the_o other_o quadrate_n which_o line_n you_o may_v call_v the_o ground_n line_n and_o then_o have_v you_o a_o right_a angle_n make_v on_o this_o line_n by_o a_o other_o side_n of_o the_o same_o quadrate_n therefore_o turn_v that_o into_o a_o right_a corner_a triangle_n accord_v to_o the_o work_n in_o the_o last_o conclusion_n by_o make_v of_o a_o bias_n line_n and_o that_o bias_n line_n will_v perform_v the_o work_n of_o your_o desire_n for_o if_o you_o take_v the_o length_n of_o that_o bias_n line_n with_o your_o compass_n and_o then_o set_v one_o foot_n of_o the_o compass_n in_o the_o far_a angle_n of_o the_o first_o quadrate_n which_o be_v the_o one_o end_n of_o the_o groundline_a and_o extend_v the_o other_o foot_n on_o the_o same_o line_n accord_v to_o the_o measure_n of_o the_o bias_n line_n and_o of_o that_o line_n make_v a_o quadrate_n enclose_n the_o first_o quadrate_n then_o will_v there_o appear_v the_o form_n of_o a_o squire_n about_o the_o first_o quadrate_n which_o squire_n be_v equal_a to_o the_o second_o quadrate_n example_n the_o first_o square_a quadrate_n be_v a._n b._n c._n d_o and_o the_o second_o be_v e._n now_o will_v i_o make_v squire_n about_o the_o quadrate_n a._n b._n c._n d_o which_o shall_v be_v equal_a unto_o the_o quadrate_n e._n therefore_o first_o i_o draw_v the_o line_n a._n d_o more_o in_o length_n accord_v to_o the_o measure_n of_o the_o side_n of_o e_o as_o you_o see_v from_o d._n unto_o f_o and_o so_o the_o hole_n line_n of_o both_o these_o several_a side_n be_v a._n f_o they_o make_v i_o a_o bias_n line_n from_o c_o to_o f_o which_o bias_n line_n be_v the_o measure_n of_o this_o work_n wherefore_o i_o open_v my_o compass_n accord_v to_o the_o length_n of_o that_o bias_n line_n c._n f_o and_o set_v the_o one_o compass_n foot_n in_o a_o and_o extend_v other_o foot_v of_o the_o compass_n towards_o f_o make_v this_o prick_n g_o from_o which_o i_o erect_v a_o plumbe_v line_n g._n h_n and_o so_o make_v out_o the_o square_a quadrate_n a._n g._n h._n k_o who_o side_n be_v equal_a each_o of_o they_o to_o a._n g._n and_o this_o square_n do_v contain_v the_o first_o quadrate_n a._n b._n c._n d_o and_o also_o a_o squire_n g._n h.k_n which_o be_v equal_a to_o the_o second_o quadrate_n e_o for_z as_z the_o last_o conclusion_n declare_v the_o quadrate_n a._n g.h._n k_o be_v equal_a to_o both_o the_o other_o quadrate_v propose_v that_o be_v a._n b._n c.d_n and_o e._n then_o must_v the_o squire_n g._n h.k_n needs_o be_v equal_a to_o e_o consider_v that_o all_o the_o rest_n of_o that_o great_a quadrate_n be_v nothing_o else_o but_o the_o quadrate_n self_n a._n b._n c._n d_o and_o so_o have_v i_o thintent_v of_o this_o conclusion_n the_o xxi_o conclusion_n to_o find_v out_o the_o centre_n of_o any_o circle_n assign_v draw_v a_o cord_n or_o string_n line_n across_o the_o circle_n then_o divide_v into_o two_o equal_a part_n both_o that_o cord_n and_o also_o the_o bow_n line_n or_o arch_n line_n that_o serve_v to_o that_o cord_n and_o from_o the_o prick_v of_o those_o division_n if_o you_o draw_v a_o other_o line_n across_o the_o circle_n it_o must_v needs_o pass_v by_o the_o centre_n therefore_o divide_v that_o line_n in_o the_o middle_n and_o that_o middle_a prick_n be_v the_o centre_n of_o the_o circle_n propose_v example_n let_v the_o circle_n be_v a._n b.c.d_n who_o centre_n i_o shall_v seek_v first_o therefore_o i_o draw_v a_o cord_n across_o the_o circle_n that_o be_v a._n c._n then_o do_v i_o divide_v that_o cord_n in_o the_o middle_n in_o e_o and_o likewaies_o also_o do_v i_o divide_v his_o arch_n line_n a._n b.c_n in_o the_o middle_n in_o the_o point_n b._n afterward_o i_o draw_v a_o line_n from_o b._n to_o e_z and_o so_o cross_v the_o circle_n which_o line_n be_v b._n d_o in_o which_o line_n be_v the_o centre_n that_o i_o seek_v for_o therefore_o if_o i_o part_n that_o line_n b._n d_o in_o the_o middle_n in_o to_o two_o equal_a portion_n that_o middle_a prick_n which_o here_o be_v f_o be_v the_o very_a centre_n of_o the_o say_a circle_n that_o i_o seek_v this_o conclusion_n may_v other_o way_n be_v wrought_v as_o the_o most_o part_n of_o conclusion_n have_v sundry_a form_n of_o practice_n and_o that_o be_v by_o make_v three_o prick_v in_o the_o circumference_n of_o the_o circle_n at_o liberty_n where_o you_o will_v and_o then_o find_v the_o centre_n to_o those_o three_o prick_v which_o work_n because_o it_o serve_v for_o sundry_a use_n i_o think_v meet_v to_o make_v it_o a_o several_a conclusion_n by_o itself_o the_o xxiii_n conclusion_n to_o find_v the_o come_v centre_n belong_v to_o any_o three_o prick_v appoint_v if_o they_o be_v not_o in_o a_o exact_v right_a line_n
the_o angle_n c_o be_v equal_a to_o d._n the_o nine_o theorem_a when_o so_o ever_o in_o any_o triangle_n the_o line_n of_o one_o side_n be_v draw_v forth_o in_o length_n that_o utter_a angle_n be_v great_a than_o any_o of_o the_o two_o inner_a corner_n that_o join_v not_o with_o it_o example_n the_o triangle_n a._n d._n c_o have_v his_o ground_n line_n a._n c._n draw_v forth_o in_o length_n unto_o b_o so_o that_o the_o utter_a corner_n that_o it_o make_v at_o c_o be_v great_a than_o any_o of_o the_o two_o inner_a corner_n that_o lie_v against_o it_o and_o join_v not_o with_o it_o which_o be_v a._n and_o d_o for_o they_o both_o be_v lesser_a than_o a_o right_a angle_n and_o be_v sharp_a angle_n but_o c._n be_v a_o blonte_a angle_n and_o therefore_o great_a than_o a_o right_a angle_n the_o ten_o theorem_a in_o every_o triangle_n any_o two_o corner_n how_o so_o ever_o you_o take_v they_o be_v less_o they_o ij_o right_n corner_n example_n in_o the_o first_o triangle_n e_o which_o be_v a_o threlyke_a and_o therefore_o have_v all_o his_o angle_n sharp_a take_v any_o two_o corner_n that_o you_o will_v and_o you_o shall_v perceive_v that_o they_o be_v lesser_a than_o two_o right_a corner_n for_o in_o every_o triangle_n that_o have_v all_o sharp_a corner_n as_o you_o see_v it_o to_o be_v in_o this_o example_n every_o corner_n be_v less_o than_o a_o right_a corner_n and_o therefore_o also_o every_o two_o corner_n must_v needs_o be_v less_o than_o two_o right_a corner_n furthermore_o in_o that_o other_o triangle_n mark_v with_o m_n which_o have_v two_o sharp_a corner_n and_o one_o right_a any_o two_o of_o they_o also_o be_v less_o than_o two_o right_a angle_n for_o though_o you_o take_v the_o right_a corner_n for_o one_o yet_o the_o other_o which_o be_v a_o sharp_a corner_n be_v less_o than_o a_o right_a corner_n and_o so_o it_o be_v true_a in_o all_o kind_n of_o triangle_n as_o you_o may_v perceive_v more_o plain_o by_o the_o xxij_o theorem_a the_o xi_o theorem_a in_o every_o triangle_n the_o great_a side_n lie_v against_o the_o great_a angle_n example_n as_o in_o this_o triangle_n a._n b._n c_o the_o great_a angle_n be_v c._n and_o a._n b._n which_o be_v the_o side_n that_o lie_v against_o it_o be_v the_o great_a and_o long_a side_n and_o contrary_a way_n as_o a._n c._n be_v the_o short_a side_n so_o b._n which_o be_v the_o angle_n li_v against_o it_o be_v the_o small_a and_o sharp_a angle_n for_o this_o do_v follow_v also_o that_o as_o the_o long_a side_n lie_v against_o the_o great_a angle_n so_o it_o that_o follow_v the_o twelve_o theorem_a in_o every_o triangle_n the_o great_a angle_n lie_v against_o the_o long_a side_n for_o these_o ij_o theorem_n be_v one_o in_o truth_n the_o thirteen_o theorem_a in_o every_o triangle_n any_o ij_o side_n together_o how_o so_o ever_o you_o take_v they_o be_v long_o they_o the_o third_o for_o example_n you_o shall_v take_v this_o triangle_n a.b.c._n which_o have_v a_o veery_a blunt_a corner_n and_o therefore_o one_o of_o his_o side_n great_a a_o good_a deal_n then_o any_o of_o the_o other_o and_o yet_o thr_fw-mi ij_o lesser_a side_n together_o be_v great_a than_o it_o and_o if_o it_o be_v so_o in_o a_o blunt_a angel_v triangle_n it_o must_v needs_o be_v true_a in_o all_o other_o for_o there_o be_v no_o other_o kind_n of_o triangle_n that_o have_v the_o one_o side_n so_o great_a above_o the_o other_o side_n as_o they_o that_o have_v blunt_a corner_n the_o fourtenth_n theorem_a if_o there_o be_v draw_v from_o the_o end_n of_o any_o side_n of_o a_o triangle_n two_o line_n metinge_v within_o the_o triangle_n those_o two_o line_n shall_v be_v less_o than_o the_o other_o two_o side_n of_o the_o triangle_n but_o yet_o the_o corner_n that_o they_o make_v shall_v be_v great_a than_o that_o corner_n of_o the_o triangle_n which_o stand_v over_o it_o example_n a.b.c._n be_v a_o triangle_n on_o who_o ground_n line_n a.b._n there_o be_v draw_v ij_o line_n from_o the_o ij_o end_n of_o it_o i_o say_v from_o a._n and_o b_o and_o they_o meet_v within_o the_o triangle_n in_o the_o point_n d_o wherefore_o i_o say_v that_o as_o those_o two_o line_n a.d._n and_o b._n d_o be_v lesser_a than_o a.c._n and_o b._n c_o so_fw-mi the_o angle_n d._n be_v great_a than_o the_o angle_n c_o which_o be_v the_o angle_n against_o it_o the_o fiftenth_n theorem_a if_o a_o triangle_n have_v two_o side_n equal_a to_o the_o two_o side_n of_o a_o other_o triangle_n but_o yet_o the_o angle_n that_o be_v contain_v between_o those_o side_n great_a than_o the_o like_a angle_n in_o the_o other_o triangle_n then_o be_v his_o ground_n line_v great_a than_o the_o ground_n line_n of_o the_o other_o triangle_n example_n a.b.c._n be_v a_o triangle_n who_o side_n a.c._n and_o b._n c_o be_v equal_a to_o e.d._n and_o d._n f_o the_o two_o side_n of_o the_o triangle_n d._n e._n f_o but_o because_o the_o angle_n in_o d_o be_v great_a than_o the_o angle_n c._n which_o be_v the_o ij_o angle_n contain_v between_o the_o equal_a line_n therefore_o must_v the_o ground_n line_n e._n f._n needs_o be_v great_a thenne_n the_o ground_n line_n a._n b_o as_o you_o see_v plain_o the_o xuj_o theorem_a if_o a_o triangle_n have_v two_o side_n equal_a to_o the_o two_o side_n of_o a_o other_o triangle_n but_o yet_o have_v a_o long_a ground_n line_v they_o that_o other_o triangle_n then_o be_v his_o angle_n that_o lie_v between_o the_o equal_a side_n great_a they_o the_o like_a corner_n in_o the_o other_o triangle_n example_n this_o theorem_a be_v nothing_o else_o but_o the_o sentence_n of_o the_o last_o theorem_a turn_v backward_o and_o therefore_o need_v none_o other_o proof_n neither_o declaration_n than_o the_o other_o example_n the_o sevententh_n theorem_a if_o two_o triangle_n be_v of_o such_o sort_n that_o two_o angle_n of_o the_o one_o be_v equal_a to_o ij_o angle_n of_o the_o other_o and_o that_o one_o side_n of_o the_o one_o be_v equal_a to_o on_o side_n of_o the_o other_o whether_o that_o side_n do_v adjoine_v to_o one_o of_o the_o equal_a corner_n or_o else_o lie_v against_o one_o of_o they_o then_o shall_v the_o other_o two_o side_n of_o those_o triangle_n be_v equal_a together_o and_o the_o third_o corner_n also_o shall_v be_v equal_a in_o those_o two_o triangle_n example_n because_o that_o a._n b._n c_o the_o one_o triangle_n have_v two_o corner_n a._n and_o b_o equal_a to_o d._n e_o that_o be_v two_o corner_n of_o the_o other_o triangle_n d._n e._n f._n and_o that_o they_o have_v one_o side_n in_o theym_n both_o equal_a that_o be_v a._n b_o which_o be_v equaâ—l_o to_o d._n e_o therefore_o shall_v both_o the_o other_o ij_o side_n be_v equal_a one_o to_o a_o other_o as_z a_z c._n and_o b._n c._n equal_a to_o d._n f_o and_o e._n f_o and_o also_o the_o third_o angle_n in_o they_o both_o shall_v be_v equal_v that_o be_v the_o angle_n c._n shall_v be_v equal_a to_o it_o the_o angle_n f._n the_o eightenth_n theorem_a when_o on_o two_o right_a line_n there_o be_v draw_v a_o three_o right_a line_n across_o way_n and_o make_v two_o match_n corner_n of_o the_o one_o line_n equal_a to_o the_o like_o two_o match_n corner_n of_o the_o other_o line_n then_o be_v those_o two_o line_n gemmow_n line_n or_o parallel_n example_n the_o two_o first_o line_n be_v a._n b._n and_o c._n d_o the_o thyrd_o line_n that_o cross_v they_o be_v e._n f._n and_o because_o that_o e._n f._n make_v ij_o match_v angle_n with_o a._n b_o equal_a to_o two_o other_o like_a match_n angle_n on_o c._n d_o that_o be_v to_o say_v e._n g_o equal_a to_o k._n f_o and_o m._n n._n equal_a also_o to_o h_n l._n therefore_o be_v those_o ij_o line_n a._n b._n and_o c._n d._n gemow_n line_n understand_v here_o by_o like_a match_n corner_n those_o that_o go_v one_o way_n as_o do_v e._n g_o and_o k._n f_o lykeways_o n._n m_o and_o h._n l_o for_o as_o e._n g._n and_o h._n l_o other_o n._n m._n and_o k._n f._n go_v not_o one_o way_n so_o be_v not_o they_o like_o match_n corner_n the_o nyntenth_n theorem_a when_o on_o two_o right_a line_n there_o be_v draw_v a_o third_o right_a line_n across_o way_n and_o make_v the_o ij_o over_o corner_n toward_o one_o hand_n equal_a together_o then_o be_v those_o two_o line_n parallel_n and_o in_o like_a manner_n if_o two_o inner_a corner_n towards_o one_o hand_n be_v equal_a to_o ii_o right_a angle_n example_n as_o the_o theorem_a do_v speak_v of_o two_o over_o angle_n so_o must_v you_o understande_v also_o of_o two_o nether_a angle_n for_o the_o judgement_n be_v like_a in_o both_o take_v for_o a_o example_n the_o figure_n of_o the_o last_o theorem_a where_o a._n b_o
and_o c._n d_o be_v call_v parallel_n also_o because_o e._n and_o k_o which_o be_v two_o over_o corner_n be_v equal_a and_o like_o way_n l._n and_o m._n and_o so_o be_v in_o like_a manner_n the_o nether_a corner_n n._n and_o h_n and_o g._n and_o f._n now_o to_o the_o second_o part_n of_o the_o theorem_a those_o two_o line_n a._n b._n and_o c._n d_o shall_v be_v call_v parallel_n because_o the_o ij_o inner_a corner_n as_o for_o example_n those_o two_o that_o be_v towards_o the_o right_a hand_n that_o be_v g._n and_o l._n be_v equal_a by_o the_o first_o part_n of_o this_o nyntenth_n theorem_a therefore_o must_v g._n and_o l._n be_v equal_a to_o two_o right_a angle_n the_o xx_o theorem_a when_o a_o right_a line_n be_v draw_v cross_n over_o two_o right_a gemow_n line_n it_o make_v two_o match_n corner_n of_o the_o one_o line_n equal_a to_o two_o match_n corner_n of_o the_o other_o line_n and_o also_o both_o over_o corner_n of_o one_o hand_n equal_a together_o and_o both_o nether_a corner_n likewaies_o and_o more_o over_o two_o inner_a corner_n and_o two_o utter_a corner_n also_o toward_o one_o hand_n equal_a to_o two_o right_a angle_n example_n because_o a._n b._n and_o c._n d_o in_o the_o last_o figure_n be_v parallel_n therefore_o the_o two_o match_n corner_n of_o the_o one_o line_n as_o e._n g._n be_v equal_a unto_o the_o two_o match_n corner_n of_o the_o other_o line_n that_o be_v k._n f_o and_o lykewaies_o m._n n_n equal_a to_o h._n l._n and_o also_o e._n and_o k._n both_o over_o corner_n of_o the_o left_a hand_n equal_a together_o and_o so_o be_v m._n and_o l_o the_o two_o over_o corner_n on_o the_o right_n hand_n in_o like_a manner_n n._n and_o h_n the_o two_o nether_a corner_n on_o the_o left_a hand_n equal_a each_o to_o other_o and_o g._n and_o f._n the_o two_o nether_a angle_n on_o the_o right_a hand_n equal_v together_o ¶_o farthermore_o yet_o g._n and_o l._n the_o two_o inner_a angle_n on_o the_o right_a hand_n be_v equal_a to_o two_o right_a angle_n and_o so_o be_v m._n and_o f._n the_o two_o utter_a angle_n on_o the_o same_o hand_n in_o like_a manner_n shall_v you_o say_v of_o n._n and_o k._n the_o two_o inner_a corner_n on_o the_o left_a hand_n and_o of_o e._n and_o h._n the_o two_o utter_a corner_n on_o the_o same_o hand_n and_o thus_o you_o see_v the_o agreeable_a sentence_n of_o these_o three_o theorem_n to_o tend_v to_o this_o purpose_n to_o declare_v by_o the_o angle_n how_o to_o judge_v parallel_n and_o contrary_a way_n how_o you_o may_v by_o parallel_n judge_v the_o proportion_n of_o the_o angle_n the_o xxi_o theorem_a what_o so_o ever_o line_n be_v parallel_n to_o any_o other_o line_n those_o same_o be_v parallel_n together_o example_n a._n b._n be_v a_o gemow_n line_n or_o a_o parellele_n unto_o c._n d._n and_o e._n f_o like_o way_n be_v a_o parallel_n unto_o c._n d._n wherefore_o it_o follow_v that_o a._n b._n must_v needs_o be_v a_o parallel_n unto_o e._n f._n the_o xxij_o theorem_a in_o every_o triangle_n when_o any_o side_n be_v draw_v forth_o in_o length_n the_o utter_a angle_n be_v equal_a to_o the_o ij_o inner_a angle_n that_o lie_v against_o it_o and_o all_o iij._o inner_a angle_n of_o any_o triangle_n be_v equal_a to_o ij_o right_n angle_n example_n the_o triangle_n be_v a._n d._n e._n and_o the_o side_n a._n e._n draw_v forth_o unto_o b_z there_o be_v make_v a_o utter_a corner_n which_o be_v c_o and_o this_o utter_a corner_n c_o be_v equal_a to_o both_o the_o inner_a corner_n that_o lie_v against_o it_o which_o be_v a._n and_o d._n and_o all_o three_o inner_a corner_n that_o be_v to_o say_v a._n d._n and_o e_o be_v equal_a to_o two_o right_a corner_n whereof_o it_o follow_v that_o all_o the_o three_o corner_n of_o any_o one_o triangle_n be_v equal_a to_o all_o the_o three_o corner_n of_o everye_o other_o triangle_n for_o what_o so_o ever_o thing_n be_v equal_a to_o anny_n one_o three_o thing_n those_o same_o be_v equal_a togitther_o by_o the_o first_o common_a sentence_n so_o that_o because_o all_o the_o three_o angle_n of_o every_o triangle_n be_v equal_a to_o two_o right_n angle_n and_o all_o right_n angle_v be_v equal_a together_o by_o the_o four_o request_n therefore_o must_v it_o needs_o follow_v that_o all_o the_o three_o corner_n of_o every_o triangle_n account_v they_o together_o be_v equal_a to_o iij._o corner_n of_o any_o other_o triangle_n take_v all_o together_o the_o xxiii_o theorem_a when_o any_o ij_o right_n line_n do_v touch_v and_o couple_v two_o other_o right_a line_n which_o be_v equal_a in_o length_n and_o parallel_n and_o if_o those_o two_o line_n be_v draw_v toward_o one_o hand_n then_o be_v they_o also_o equal_a together_o and_o parallel_n example_n a._n b._n and_o c._n d._n be_v ij_o right_a line_n and_o parallel_n and_o equal_a in_o length_n and_o they_o be_v touch_v and_o join_v together_o by_o ij_o other_o line_n a._n c._n and_o b._n d_o this_o be_v so_o and_o a._n c._n and_o b._n d._n be_v draw_v toward_o one_o side_n that_o be_v to_o say_v both_o toward_o the_o left_a hand_n therefore_o be_v a_o c._n and_o b._n d._n both_o equal_a and_o also_o parallel_v the_o xxiiij_o theorem_a in_o any_o likeiamme_n the_o two_o contrary_a side_n be_v equal_a together_o and_o so_o be_v each_o two_o contrary_a angle_n and_o the_o bias_n line_n that_o be_v draw_v in_o it_o do_v divide_v it_o into_o two_o equal_a portion_n example_n here_o be_v two_o likeiamme_n join_v together_o the_o one_o be_v a_o long_o square_a a._n b._n e_o and_o the_o other_o be_v a_o losengelike_a d._n c._n e._n f._n which_o ij_o likeiamme_n be_v prove_v equal_a together_o because_o they_o have_v one_o ground_n line_n that_o be_v f._n e_o and_o be_v make_v between_o one_o pair_n of_o gemow_n line_n i_o mean_v a._n d._n and_o e._n h._n by_o this_o theorem_a may_v you_o know_v the_o art_n of_o the_o right_a measuringe_v of_o likeiamme_n as_o in_o my_o book_n of_o measure_v i_o will_v more_o plain_o declare_v the_o xxvi_o theorem_a all_o likeiamme_n that_o have_v equal_a ground_n line_n and_o be_v draw_v between_o one_o pair_n of_o parallel_n be_v equal_a together_o example_n first_o you_o must_v mark_v the_o difference_n between_o this_o theorem_a and_o the_o last_o for_o the_o last_o theorem_a presuppose_a to_o the_o diverse_a likeiamme_n one_o ground_n line_n common_a to_o they_o but_o this_o theorem_a do_v presuppose_v a_o diverse_a ground_n line_n for_o every_o like_a iamme_n only_o mean_v they_o to_o be_v equal_a in_o length_n though_o they_o be_v diverse_a in_o numbbe_n as_o for_o example_n in_o the_o last_o figure_n there_o be_v two_o parallel_n a._n d._n and_o e._n h_n and_o between_o they_o be_v draw_v three_o likeiamme_n the_o first_o be_v a._n b._n e._n f_o the_o second_o be_v e._n c._n d._n f_o and_o the_o third_o be_v c._n g._n h._n d._n the_o first_o and_o the_o second_o have_v one_o ground_n line_n that_o be_v e._n f._n and_o therefore_o in_o so_o much_o as_o they_o be_v between_o one_o pair_n of_o parallel_n they_o be_v equal_a accord_v to_o the_o five_o and_o twenty_o theorem_a but_o the_o third_o likeiamme_n that_o be_v c._n g._n h._n d._n have_v his_o ground_n line_n g._n h_n several_a from_o the_o other_o but_o yet_o equal_a unto_o it_o wherefore_o the_o three_o likeiam_fw-la be_v equal_a to_o the_o other_o two_o first_o likeiamme_n and_o for_o a_o proof_n that_o g.h._n be_v the_o ground_n line_n of_o the_o three_o likeiamme_n be_v equal_a to_o e._n f_o which_o be_v the_o ground_n line_n to_o both_o the_o other_o likeiam_n that_o may_v be_v thus_o declare_v g.h._n be_v equal_a to_o c._n d_o sing_v they_o be_v the_o contrary_a side_n of_o one_o likeiamme_n by_o the_o four_o and_o twenty_o theorem_a and_o so_o be_v c.d._n and_o e._n f._n by_o the_o same_o theorem_a therefore_o sing_v both_o those_o ground_n line_n e.f._n and_o g._n h_n be_v equal_a to_o one_o third_o line_n that_o be_v c.d._n they_o must_v needs_o be_v equal_a together_o by_o the_o first_o common_a sentence_n the_o xxvii_o theorem_a all_o triangle_n have_v one_o ground_n line_n a_o stand_n between_o one_o pair_n of_o parallel_n be_v equal_a together_o example_n a.b._n and_o c._n f._n be_v two_o gemowe_a line_n between_o which_o there_o be_v make_v two_o triangle_n a._n d._n e._n and_o d._n e._n b_o so_fw-mi that_o d._n e_o be_v the_o common_a ground_n line_n to_o they_o both_o wherefore_o it_o do_v follow_v that_o those_o two_o triangle_n a.d.e._n and_o d.e.b._n be_v equal_a each_o to_o other_o the_o xxviij_o theorem_a all_o triangle_n that_o have_v like_o long_a ground_n line_n and_o be_v make_v between_o one_o pair_n of_o gemow_n line_n be_v
for_o it_o be_v of_o like_a distance_n as_o be_v the_o line_n m.n._n now_o say_v i_o that_o a._n d_o be_v the_o diameter_n be_v the_o long_a of_o all_o those_o line_n and_o also_o of_o any_o other_o that_o may_v be_v draw_v within_o that_o circle_n and_o the_o other_o line_n m._n n_n be_v long_a than_o f._n g_o because_o it_o be_v near_a to_o the_o centre_n of_o the_o circle_n then_o f._n g._n also_o the_o line_n f._n g_o be_v short_a than_o the_o line_n b._n c._n for_o because_o it_o be_v far_o from_o the_o centre_n than_o be_v the_o line_n b._n c._n and_o thus_o may_v you_o judge_v of_o all_o line_n draw_v in_o any_o circle_n how_o to_o know_v the_o proportion_n of_o their_o length_n by_o the_o proportion_n of_o their_o distance_n and_o contrary_a way_n how_o to_o discern_v the_o proportion_n of_o their_o distance_n by_o their_o lengthe_n if_o you_o know_v the_o proportion_n of_o their_o length_n and_o to_o speak_v of_o it_o by_o the_o way_n it_o be_v a_o maruaylouse_a thing_n to_o consider_v that_o a_o man_n may_v know_v a_o exact_v proportion_n between_o two_o thing_n and_o yet_o can_v not_o name_v nor_o attain_v the_o precise_a quantitee_n of_o those_o two_o thing_n as_o for_o exaunple_n if_o two_o square_n be_v set_v forth_o whereof_o the_o one_o contain_v in_o it_o five_o square_a foot_n and_o the_o other_o contain_v five_o and_o forty_o foot_n of_o like_a square_a foot_n i_o be_o not_o able_a to_o tell_v no_o nor_o yet_o any_o man_n live_v what_o be_v the_o precyse_n measure_n of_o the_o side_n of_o any_o of_o those_o two_o square_n and_o yet_o i_o can_v prove_v by_o unfallible_a reason_n that_o their_o side_n be_v in_o a_o triple_a proportion_n that_o be_v to_o say_v that_o the_o side_n of_o the_o great_a square_n which_o contain_v xlu_o foot_n be_v three_o time_n so_o long_o just_a as_o the_o side_n of_o the_o lesser_a square_n that_o include_v but_o five_o foot_n but_o this_o seem_v to_o be_v speak_v out_o of_o ceason_n in_o this_o place_n therefore_o i_o will_v omit_v it_o now_o reserve_v the_o exacter_n declaration_n thereof_o to_o a_o more_o convenient_a place_n and_o time_n and_o will_v proceed_v with_o the_o residew_n of_o the_o theorem_n appoint_v for_o this_o book_n the_o lxi_o theorem_a if_o a_o right_a line_n be_v draw_v at_o any_o end_n of_o a_o diameter_n in_o perpendicular_a form_n and_o do_v make_v a_o right_a angle_n with_o the_o diameter_n that_o right_a line_n shall_v light_v without_o the_o circle_n and_o yet_o so_o joint_o knit_v to_o it_o that_o it_o be_v not_o possible_a to_o draw_v any_o other_o right_a line_n between_o that_o say_a line_n and_o the_o circumference_n of_o the_o circle_n and_o the_o angle_n that_o be_v make_v in_o the_o semicircle_n be_v great_a than_o any_o sharp_a angle_n that_o may_v be_v make_v of_o right_a line_n but_o the_o other_o angle_n without_o be_v lesser_a than_o any_o that_o can_v be_v make_v of_o right_a line_n example_n in_o this_o circle_n a._n b.c_n the_o diameter_n be_v a._n c_o the_o perpendicular_a line_n which_o make_v a_o right_a angle_n with_o the_o diameter_n be_v e._n a_o which_o line_n fall_v without_o the_o circle_n and_o yet_o join_v so_o exact_o unto_o it_o that_o it_o be_v not_o possible_a to_o draw_v a_o other_o right_a line_n between_o the_o circumference_n of_o the_o circle_n and_o it_o which_o thing_n be_v so_o plain_o see_v of_o the_o eye_n that_o it_o need_v no_o far_a declaration_n for_o every_o man_n will_v easy_o consent_v that_o between_o the_o crooked_a line_n a._n f_o which_o be_v a_o part_n of_o the_o circumference_n of_o the_o circle_n and_o a._n e_o which_o be_v the_o say_v perpendicular_a line_n there_o can_v none_o other_o line_n be_v draw_v in_o that_o place_n where_o they_o make_v the_o angle_n now_o for_o the_o residue_n of_o the_o theorem_a the_o angle_n d._n a._n b_o which_o be_v make_v in_o the_o semicircle_n be_v great_a than_o any_o sharp_a angle_n that_o may_v be_v make_v of_o right_n line_n and_o yet_o be_v it_o a_o sharp_a angle_n also_o in_o as_o much_o as_o it_o be_v lesser_a than_o a_o right_a angle_n which_o be_v the_o angle_n e._n a.d_n and_o the_o residue_n of_o that_o right_a angle_n which_o lie_v without_o the_o circle_n that_o be_v to_o say_v e._n a.b_n be_v lesser_a than_o any_o sharp_a angle_n that_o can_v be_v make_v of_o right_a line_n also_o for_o as_o it_o be_v before_o rehearse_v there_o can_v no_o right_a line_n be_v draw_v to_o the_o angle_n between_o the_o circumference_n and_o the_o right_a line_n e.a._n then_o must_v it_o needs_o follow_v that_o there_o can_v be_v make_v no_o lesser_a angle_n of_o right_a line_n and_o again_o if_o there_o can_v be_v no_o lesser_a than_o the_o one_o then_o do_v it_o soon_o appear_v that_o there_o can_v be_v no_o great_a than_o the_o other_o for_o they_o two_o do_v make_v the_o whole_a right_a angle_n so_o that_o if_o any_o corner_n can_v be_v make_v great_a than_o the_o one_o part_n then_o shall_v the_o residue_n be_v lesser_a than_o the_o other_o part_n so_o that_o other_o both_o part_n must_v be_v false_a or_o else_o both_o grant_v to_o be_v true_a the_o lxij_o theorem_a if_o a_o right_a line_n do_v touch_v a_o circle_n and_o a_o other_o right_a line_n draw_v from_o the_o centre_n of_o tge_a circle_n to_o the_o point_n where_o they_o touch_v that_o line_n which_o be_v drawenne_n from_o the_o centre_n shall_v be_v a_o perpendicular_a line_n to_o the_o touch_n line_n example_n the_o circle_n be_v a._n b._n c_o and_o his_o centre_n be_v f._n the_o touch_n line_n be_v d._n e_o and_o the_o point_n where_o they_o touch_v be_v c._n now_o by_o reason_n that_o a_o right_a line_n be_v draw_v from_o the_o centre_n f._n unto_o c_o which_o be_v the_o point_n of_o the_o touch_n therefore_o say_v the_o theorem_a that_o the_o say_a line_n f._n c_o must_v needs_o be_v a_o perpendicular_a line_n unto_o the_o touch_n line_n d.e._n the_o lxiij_o theorem_a if_o a_o right_a line_n do_v touch_v a_o circle_n and_o a_o other_o right_a line_n be_v draw_v from_o the_o point_n of_o their_o touchinge_a so_o that_o it_o do_v make_v right_a corner_n with_o the_o touch_n line_n then_o shall_v the_o centre_n of_o the_o circle_n be_v in_o that_o same_o line_n so_o draw_v example_n the_o circle_n be_v a._n b._n c_o and_o the_o centre_n of_o it_o be_v g._n the_o touch_n line_n be_v d._n c.e_n and_o the_o point_n where_o it_o touch_v be_v c._n now_o it_o appear_v manifest_a that_o if_o a_o right_a line_n be_v draw_v from_o the_o point_n where_o the_o touch_n line_n do_v join_v with_o the_o circle_n and_o that_o the_o say_a line_n do_v make_v right_a corner_n with_o the_o touch_n line_n then_o must_v it_o needs_o go_v by_o the_o centre_n of_o the_o circle_n and_o then_o consequent_o it_o must_v have_v the_o say_a centre_n in_o he_o for_o if_o the_o say_a line_n shall_v go_v beside_o the_o centre_n as_o f._n c._n do_v then_o do_v it_o not_o make_v right_a angle_n with_o the_o touch_n line_n which_o in_o the_o theorem_a be_v suppose_v the_o lxiiij_o theorem_a if_o a_o angle_n be_v make_v on_o the_o centre_n of_o a_o circle_n and_o a_o other_o angle_n make_v on_o the_o circumference_n of_o the_o same_o circle_n and_o their_o ground_n line_n be_v one_o common_a portion_n of_o the_o circumference_n then_o be_v the_o angle_n on_o the_o centre_n twice_o so_o great_a as_o the_o other_o angle_n on_o the_o circumference_n example_n the_o cirle_n be_v a._n b._n c._n d_o and_o his_o centre_n be_v e_o the_o angle_n on_o the_o centre_n be_v c._n e.d_n and_o the_o angle_n on_o the_o circumference_n be_v c._n a._n d_o t_z their_z come_v ground_n line_n be_v c._n f.d_n now_o say_v i_o that_o the_o angle_n c._n e._n d_o which_o be_v one_o the_o centre_n be_v twice_o so_o great_a as_o the_o angle_n c._n a.d_n which_o be_v on_o the_o circumference_n the_o lxv_o theorem_a those_o angle_n which_o be_v make_v in_o one_o cantle_n of_o a_o circle_n must_v needs_o be_v equal_a together_o example_n before_o i_o declare_v this_o theorem_a by_o a_o example_n it_o shall_v be_v needful_a to_o declare_v what_o be_v to_o be_v understande_v by_o the_o word_n in_o this_o theorem_a for_o the_o sentence_n can_v not_o be_v know_v unless_o the_o very_a meaning_n of_o the_o word_n be_v first_o understand_v therefore_o when_o it_o speak_v of_o angel_n make_v in_o one_o cantle_n of_o a_o circle_n it_o be_v this_o to_o be_v understand_v that_o the_o angle_n must_v touch_v the_o circumference_n and_o the_o line_n that_o do_v enclose_v that_o angle_n must_v be_v draw_v to_o the_o extremity_n of_o that_o line_n which_o make_v the_o cantle_n of_o the_o
with_o my_o compass_n i_o take_v the_o length_n of_o a._n b._n and_o set_v the_o one_o foot_n of_o my_o compass_n in_o c_o and_o draw_v a_o arch_a line_n with_o the_o other_o foot_n likewaies_n i_o take_v the_o length_n of_o e._n f_o and_o set_v one_o foot_n in_o d_o and_o with_o the_o other_o foot_n i_o make_v a_o arch_a line_n cross_v the_o other_o arch_n and_o the_o prick_n of_o their_o meeting_n which_o be_v g._n shall_v be_v the_o third_o corner_n of_o the_o triangle_n for_o in_o all_o such_o kynde_n of_o work_v to_o make_v a_o tryangle_n if_o you_o have_v one_o line_n draw_v there_o remain_v nothing_o else_o but_o to_o find_v where_o the_o pitch_n of_o the_o third_o corner_n shall_v be_v for_o two_o of_o they_o must_v needs_o be_v at_o the_o two_o cande_n of_o the_o line_n that_o be_v draw_v the_o xiii_o conclusion_n if_o you_o have_v a_o line_n appoint_v and_o a_o point_n in_o it_o limit_v how_o you_o may_v make_v on_o it_o a_o right_a line_v angle_n equal_a to_o a_o other_o right_o line_v angle_n all_o ready_a assign_v first_o draw_v a_o line_n against_o the_o corner_n assign_v and_o so_o be_v it_o a_o triangle_n then_o take_v heed_n to_o the_o line_n and_o the_o point_n in_o it_o assign_v and_o consider_v if_o that_o line_n from_o the_o prick_n to_o this_o end_n be_v as_o long_o as_o any_o of_o the_o side_n that_o make_v the_o triangle_n assign_v and_o if_o it_o be_v long_o enough_o then_o prick_v out_o there_o the_o length_n of_o one_o of_o the_o line_n and_o then_o work_v with_o the_o other_o two_o line_n accord_v to_o the_o last_o conclusion_n make_v a_o triangle_n of_o three_o like_a line_n to_o that_o assign_a triangle_n if_o it_o be_v not_o long_o enough_o thenne_v lengthen_v it_o first_o and_o afterward_o do_v as_o i_o have_v say_v before_o example_n let_v the_o angle_n appoint_v be_v a._n b._n c_o and_o the_o corner_n assign_v b._n farthermore_o let_v the_o limit_v line_n be_v d._n g_o and_o the_o prick_v assign_v d._n first_o therefore_o by_o draw_v the_o line_n a._n c_o i_o make_v the_o triangle_n a.b.c._n then_o consider_v that_o d._n g_o be_v long_a thanne_n a._n b_o you_o shall_v cut_v out_o a_o line_n from_o d._n towards_o g_o equal_a to_o a._n b_o as_z for_o example_n d_o f._n then_o measure_v out_o the_o other_o ij_o line_n and_o work_n with_o they_o according_a as_o the_o conclusion_n with_o the_o first_o also_o and_o the_o second_o teach_v you_o and_o then_o have_v you_o do_v the_o xiiii_o conclusion_n to_o make_v a_o square_a quadrate_n of_o any_o right_a line_n appoint_v first_o make_v a_o plumbe_v line_n unto_o your_o line_n appoint_v which_o shall_v light_v at_o one_o of_o the_o end_n of_o it_o accord_v to_o the_o five_o conclusion_n and_o let_v it_o be_v of_o like_a length_n as_o your_o first_o line_n be_v then_o open_a your_o compass_n to_o the_o just_a length_n of_o one_o of_o they_o and_o set_v one_o foot_n of_o the_o compass_n in_o the_o end_n of_o the_o one_o line_n and_o with_o the_o other_o foot_n draw_v a_o arch_n line_n there_o as_o you_o think_v that_o the_o fowerth_o corner_n shall_v be_v after_o that_o set_v the_o one_o foot_n of_o the_o same_o compass_n unstur_v in_o the_o cande_v of_o the_o other_o line_n and_o draw_v a_o other_o arch_n line_n across_o the_o first_o arch_n line_n and_o the_o point_n that_o they_o do_v cross_v in_o be_v the_o prick_n of_o the_o four_o corner_n of_o the_o square_a quadrate_n which_o you_o seek_v for_o therefore_o draw_v a_o line_n from_o that_o prick_n to_o the_o eande_n of_o each_o line_n and_o you_o shall_v thereby_o have_v make_v a_o square_a quadrate_n example_n a._n b._n be_v the_o line_n propose_v of_o which_o i_o shall_v make_v a_o square_a quadrate_n therefore_o first_o i_o make_v a_o plumbe_v line_n unto_o it_o which_o shall_v light_o in_o a_o and_o that_o plumb_n line_n be_v a._n b_o then_o open_v i_o my_o compass_n as_o wide_o as_o the_o length_n of_o a._n b_o or_o b._n c_o for_o they_o must_v be_v both_o equal_a and_o i_o set_v the_o one_o foot_n of_o the_o end_n in_o c_o and_o with_o the_o other_o i_o make_v a_o arch_n line_n nigh_o unto_o d_z afterward_o i_o set_v the_o compass_n again_o with_o one_o foot_n in_o b_o a_z and_z with_o the_o other_o foot_n i_o make_v a_o arch_n line_n across_o the_o first_o arch_n line_n in_o d_o and_o from_o the_o prick_n of_o their_o cross_n i_o draw_v two_o line_n one_o to_o b_o a_o and_o a_o other_o to_o c_o and_o so_o have_v i_o make_v the_o square_a quadrate_n that_o i_o intend_v the_o xv._o conclusion_n to_o make_v a_o likeiamme_n equal_a to_o a_o triangle_n appoint_v and_o that_o in_o a_o right_n line_v angle_n limit_v first_o from_o one_o of_o the_o angle_n of_o the_o triangle_n you_o shall_v draw_v a_o gemowe_a line_n which_o shall_v be_v a_o parallel_n to_o that_o side_n of_o the_o triangle_n on_o which_o you_o will_v make_v that_o likeiamme_n then_o on_o one_o end_n of_o the_o side_n of_o the_o triangle_n which_o lie_v against_o the_o gemowe_a line_n you_o shall_v draw_v forth_o a_o line_n unto_o the_o gemow_n line_n so_o that_o one_o angle_n that_o come_v of_o those_o two_o line_n be_v like_a to_o the_o angle_n which_o be_v limit_v unto_o you_o then_o shall_v you_o divide_v into_o ij_o equal_a part_n that_o side_n of_o the_o triangle_n which_o bear_v that_o line_n and_o from_o the_o prick_n of_o that_o division_n you_o shall_v raise_v a_o other_o line_n parallel_n to_o that_o former_a line_n and_o continewe_v it_o unto_o the_o first_o gemowe_n line_n and_o they_o of_o those_o two_o last_o gemowe_n line_n and_o the_o first_o gemowe_n line_n with_o the_o half_a side_n of_o the_o triangle_n be_v make_v a_o lykeiamme_n equal_a to_o the_o triangle_n appoint_v and_o have_v a_o angle_n like_o to_o a_o angle_n limit_v accord_v to_o the_o conclusion_n example_n b._n c._n g_o be_v the_o triangle_n appoint_v unto_o which_o i_o must_v make_v a_o equal_a likeiamme_n and_o d_o be_v the_o angle_n that_o the_o likeiamme_n must_v have_v therefore_o first_o intend_v to_o erect_v the_o likeiamme_n on_o the_o one_o side_n that_o the_o ground_n line_n of_o the_o triangle_n which_o be_v b._n g._n i_o do_v draw_v a_o gemow_n line_n by_o c_o and_o make_v it_o parallel_v to_o the_o ground_n line_n b._n g_o and_o that_o new_a gemow_n line_n be_v a._n h._n then_o do_v i_o raise_v a_o line_n from_o b._n unto_o the_o gemowe_a line_n which_o line_n be_v a._n b_o and_o make_v a_o angle_n equal_a to_o d_o that_o be_v the_o appoint_a angle_n accord_v as_o the_o eight_o conclusion_n teach_v and_o that_o angle_n be_v b._n a._n e._n then_o to_o proceed_v i_o do_v part_n in_o the_o middle_n the_o say_a ground_n line_n b._n g_o in_o the_o prick_n f_o from_o which_o prick_n i_o draw_v to_o the_o first_o gemowe_n line_n a._n h._n a_o other_o line_n that_o be_v parallale_n to_o a._n b_o and_o that_o line_n be_v e._n f._n now_o say_v i_o that_o the_o likeiamme_n b._n a._n e._n f_o be_v equal_a to_o the_o triangle_n b._n c._n g._n and_o also_o that_o it_o have_v one_o angle_n that_o be_v b._n a._n e._n like_v to_o d._n the_o angle_n that_o be_v limit_v and_o so_o have_v i_o my_o intent_n the_o prose_n of_o the_o equalnes_n of_o those_o two_o figure_n do_v depend_v of_o the_o xli_o proposition_n of_o euclides_n first_o book_n and_o be_v the_o xxxi_o proposition_n of_o this_o second_o book_n of_o theoremis_n which_o say_v that_o when_o a_o tryangle_n and_o a_o likeiamme_n be_v make_v between_o two_o self_n same_o gemow_n line_n and_o have_v their_o ground_n line_n of_o one_o length_n then_o be_v the_o likeiamme_n double_a to_o the_o triangle_n whereof_o it_o follow_v that_o if_o two_o such_o figure_n so_o draw_v differ_v in_o their_o ground_n line_n only_o so_o that_o the_o ground_n line_n of_o the_o likeiamme_n be_v but_o half_a the_o ground_n line_n of_o the_o triangle_n then_o be_v those_o two_o figure_n equal_a as_o you_o shall_v more_o at_o large_a perceive_v by_o the_o book_n of_o theoremis_n in_o the_o xxxi_o theorem_a the_o xvi_n conclusion_n to_o make_v a_o likeiamme_n equal_a to_o a_o triangle_n appoint_v accord_v to_o a_o angle_n limit_v and_o on_o a_o line_n also_o assign_v in_o the_o last_o conclusion_n the_o side_n of_o your_o likeiamme_n be_v leave_v to_o your_o liberty_n though_o you_o have_v a_o angle_n appoint_v now_o in_o this_o conclusion_n you_o be_v somewhat_o more_o restrain_v of_o liberty_n since_o the_o line_n be_v limit_v which_o must_v be_v the_o side_n of_o the_o likeiamme_n therefore_o thus_o shall_v you_o proceed_v first_o accord_v to_o the_o last_o conclusion_n make_v a_o likeiamme_n
right_a angle_n then_o draw_v four_o line_n that_o may_v join_v the_o four_o end_n of_o those_o diameter_n one_o to_o a_o other_o and_o then_o have_v you_o make_v a_o square_a quadrate_n in_o the_o circle_n appoint_v example_n a._n b._n c._n d._n be_v the_o circle_n assign_v and_o a._n c._n and_o b._n d._n be_v the_o two_o diameter_n which_o cross_n in_o the_o centre_n e_o and_o make_v four_o right_a corner_n then_o do_v i_o make_v four_o other_o line_n that_o be_v a._n b_o b._n c_o c._n d_o and_o d._n a_o which_o do_v join_v together_o the_o four_o end_n of_o the_o ij_o diameter_n and_o so_o be_v the_o square_a quadrate_n make_v in_o the_o circle_n assign_v as_o the_o conclusion_n will_v the_o xxxiiii_o conclusion_n to_o make_v a_o square_a quadrate_n about_o annye_n circle_v assign_v draw_v two_o diameter_n in_o cross_a way_n so_o that_o they_o make_v four_o right_a angle_n in_o the_o centre_n then_o with_o your_o compass_n take_v the_o length_n of_o the_o half_a diameter_n and_o set_v one_o foot_n of_o the_o compass_n in_o each_o end_n of_o those_o diameter_n draw_v two_o arch_n line_n at_o every_o pitchinge_v of_o the_o compass_n so_o shall_v you_o have_v viij_o arch_n line_n then_o if_o you_o mark_v the_o prick_v wherein_o those_o arch_a line_n do_v cross_a and_o draw_v between_o those_o iiij_o prick_v iiij_o right_a line_n then_o have_v you_o make_v the_o square_a quadrate_n accord_v to_o the_o request_n of_o the_o conclusion_n example_n a.b.c._n be_v the_o circle_n assign_v in_o which_o first_o i_o draw_v two_o diameter_n in_o cross_a way_n make_v iiij_o right_a angle_n and_o those_o ij_o diameter_n be_v a.c._n and_o b.d._n then_o set_v i_o my_o compass_n which_o be_v open_v according_a to_o the_o semidiameter_n of_o the_o say_a circle_n fix_v one_o foot_n in_o the_o end_n of_o every_o semidiameter_n and_o draw_v with_o the_o other_o foot_n two_o arch_n line_n one_o on_o every_o side_n as_o first_o when_o i_o set_v the_o one_o foot_n in_o a_o then_o with_o the_o other_o foot_n i_o do_v make_v two_o arch_n line_n one_o in_o e_o and_o a_o other_o in_o f._n then_o set_v i_o the_o one_o foot_n of_o the_o compass_n in_o b_o and_o draw_v two_o arch_n line_n f._n and_o g._n like_o wise_a set_v the_o compass_n foot_n in_o c_o i_o draw_v two_o other_o arch_n line_n g._n and_o h_n and_o on_o d._n i_o make_v two_o other_o h._n and_o e._n then_o from_o the_o crossinge_n of_o those_o eighte_o arch_n line_n i_o draw_v iiij_o straight_a line_n that_o be_v to_o say_v e._n f_o and_o f._n g._n also_o g._n h_n and_o h._n e_o which_o iiij_o straight_o line_n do_v make_v the_o square_a quadrate_n that_o i_o shall_v draw_v about_o the_o circle_n assign_v the_o xxxv_o conclusion_n to_o draw_v a_o circle_n in_o any_o square_a quadrate_n appoint_v first_o divide_v every_o side_n of_o the_o quadrate_n into_o two_o equal_a part_n and_o so_o draw_v two_o line_n between_o each_o two_o contrary_a point_n and_o where_o those_o two_o line_n do_v cross_a there_o be_v the_o centre_n of_o the_o circle_n then_o set_v the_o one_o foot_n of_o the_o compass_n in_o that_o point_n and_o stretch_v forth_o the_o other_o foot_n according_a to_o the_o length_n to_o half_a one_o of_o those_o line_n and_o so_o make_v a_o compass_n in_o the_o square_a quadrate_n assign_v example_n a_o b.c.d._n be_v the_o quadrate_n appoint_v in_o which_o i_o must_v make_v a_o circle_n therefore_o first_o i_o do_v divide_v every_o side_n in_o ij_o equal_a part_n and_o draw_v ij_o line_n across_o between_o each_o ij_o contrary_n prick_v as_o you_o see_v e._n g_o and_o f._n h_n which_o meet_v in_o k_o and_o therefore_o shall_v k_o be_v the_o centre_n of_o the_o circle_n then_o do_v i_o set_v one_o foot_n of_o the_o compass_n in_o k._n and_o open_v the_o other_o as_o wide_a as_o k._n e_o and_o so_o draw_v a_o circle_n which_o be_v make_v ancordinge_v to_o the_o conclusion_n the_o xxxvi_o conclusion_n to_o draw_v a_o circle_n about_o a_o square_a quadrate_n draw_v ij_o line_n between_o the_o iiij_o corner_n of_o the_o quadrate_n and_o where_o they_o meet_v in_o cross_n there_o be_v the_o centre_n of_o the_o circle_n that_o you_o seek_v for_o then_o set_v one_o foot_n of_o the_o compass_n in_o that_o centre_n and_o extend_v the_o other_o foot_n unto_o one_o corner_n of_o the_o quadrate_n and_o so_o may_v you_o draw_v a_o circle_n which_o shall_v just_o enclose_v the_o quadrate_n propose_v example_n a._n b._n c._n d._n be_v the_o square_a quadrate_n propose_v about_o which_o i_o must_v make_v a_o circle_n therefore_o do_v i_o draw_v ij_o line_n across_o the_o square_a quadrate_n from_o angle_n to_o angle_v as_o you_o see_v a._n c._n &_o b._n d._n and_o where_o they_o ij_o do_v cross_n that_o be_v to_o say_v in_o e._n there_o set_v i_o the_o one_o foot_n of_o the_o compass_n as_o in_o the_o centre_n and_o the_o other_o foot_n i_o do_v extend_v unto_o one_o angle_n of_o the_o quadrate_n as_o for_o exampse_n to_o a_o and_o so_o make_v a_o compass_n which_o do_v just_o enclose_v the_o quadrate_n according_a to_o the_o mind_n of_o the_o conclusion_n the_o xxxvii_o conclusion_n to_o make_v a_o twileke_a triangle_n which_o shall_v have_v every_o of_o the_o ij_o angle_n that_o lie_v about_o the_o ground_n line_n double_a to_o the_o other_o corner_n first_o make_v a_o circle_n and_o divide_v the_o circumference_n of_o it_o into_o fyve_o equal_a part_n and_o thenne_n draw_v from_o one_o prick_n which_o you_o will_v two_o line_n to_o ij_o other_o prick_v that_o be_v to_o say_v to_o the_o iij._o and_o iiij_o prick_v count_v that_o for_o the_o first_o wherhence_o you_o draw_v both_o those_o line_n then_o draw_v the_o three_o line_n to_o make_v a_o triangle_n with_o those_o other_o two_o and_o you_o have_v do_v according_a to_o the_o conclusion_n and_o have_v make_v a_o twelike_a triangle_n who_o ij_o corner_n about_o the_o ground_n line_n be_v each_o of_o theym_a double_a to_o the_o other_o corner_n example_n a_o b._n c._n be_v the_o circle_n which_o i_o have_v divide_v into_o five_o equal_a portion_n and_o from_o one_o of_o the_o prick_v which_o be_v a_o i_o have_v draw_v ij_o line_n a._n b._n and_o a._n c_o which_o be_v draw_v to_o the_o three_o and_o iiij_o prick_v then_o draw_v i_o the_o three_o line_n c._n b._n which_o be_v the_o ground_n line_n and_o make_v the_o triangle_n that_o i_o will_v have_v for_o the_o angle_n c._n be_v double_a to_o the_o angle_n a_o and_o so_o be_v the_o angle_n b._n also_o the_o xxxvii_o conclusion_n to_o make_v a_o cinkangle_n of_o equal_a side_n and_o equal_a corner_n in_o any_o circle_n appoint_v divide_v the_o circle_n appoint_v into_o five_o equal_a part_n as_o you_o do_v in_o the_o last_o conclusion_n and_o draw_v ij_o line_n from_o every_o prick_n to_o the_o other_o ij_o that_o be_v next_o unto_o it_o and_o so_o shall_v you_o make_v a_o cinkangle_n after_o the_o meaning_a of_o the_o conclusion_n example_n you_o see_v here_o this_o circle_n a._n b._n c._n d._n e._n divide_v into_o five_o equal_a portion_n and_o from_o each_o prick_n ij_o line_n draw_v to_o the_o other_o ij_o next_o prick_v so_o from_o a._n be_v draw_v ij_o line_n one_o to_o b_o and_z the_z other_z to_z e_z and_o so_o from_o c._n one_o to_o b._n and_o a_o other_o to_o d_o and_o likewise_o of_o the_o rest_n so_o that_o you_o have_v not_o only_o learn_v hereby_o how_o to_o make_v a_o sinkangle_n in_o any_o circle_n but_o also_o how_o you_o shall_v make_v a_o like_a figure_n speedy_o whanne_n and_o where_o you_o will_v only_o draw_v the_o circle_n for_o the_o intent_n readylye_o to_o make_v the_o other_o figure_n i_o mean_v the_o cinkangle_n thereby_o the_o thirty-nine_o conclusion_n how_o to_o make_v a_o cinkangle_n of_o equal_a side_n and_o equal_a angle_n about_o any_o circle_n appoint_v divide_v first_o the_o circle_n as_o you_o do_v in_o the_o last_o conclusion_n into_o five_o equal_a portion_n and_o draw_v five_o semidiameter_n in_o the_o circle_n then_o make_v five_o touch_v line_n in_o such_o sort_n that_o every_o touch_v line_n make_v two_o right_a angle_n with_o one_o of_o the_o semidiameter_n and_o those_o five_o touch_n line_n will_v make_v a_o cinkangle_n of_o equal_a side_n and_o equal_a angle_n example_n a._n b._n c._n d._n e._n be_v the_o circle_n appoint_v which_o be_v divide_v into_o five_o equal_a part_n and_o unto_o every_o prycke_n be_v draw_v a_o semidiameter_n as_o you_o see_v then_o do_v i_o make_v a_o touch_n line_n in_o the_o prick_n b_o which_o be_v f._n g_o make_v ij_o right_n angle_n with_o the_o semidiameter_n b_o and_o like_o way_n on_o c._n be_v make_v g._n h_n on_o d._n stand_v h._n k_o
prick_v that_o be_v nigh_a together_o and_o there_o will_v appear_v right_o draw_v the_o figure_n of_o fiftene_o side_n and_o angle_n equal_a and_o so_o do_v with_o any_o other_o figure_n of_o what_o number_a of_o side_n so_o ever_o it_o be_v finis_fw-la the_o second_o book_n of_o the_o principle_n of_o geometry_n contain_v certain_a theorem_n which_o may_v be_v call_v approve_v truth_n and_o be_v as_o it_o be_v the_o most_o certain_a ground_n whereon_o the_o practic_a conclusion_n of_o geometry_n be_v found_v whereunto_o be_v annex_v certain_a declaration_n by_o example_n for_o the_o right_a understanding_n of_o the_o same_o to_o the_o end_n that_o the_o simple_a reader_n may_v not_o just_o complain_v of_o hardness_n or_o obscuritee_n and_o for_o the_o same_o cause_n be_v the_o demonstration_n and_o just_a profess_v omit_v until_o a_o more_o convient_a time_n 1551._o if_o truth_n may_v try_v itself_o by_o reason_n prudent_a skill_n if_o reason_n may_v prevail_v by_o right_a and_o rule_v the_o rage_n of_o will_n i_o dare_v the_o trial_n bide_v for_o truth_n that_o i_o pretend_v and_o though_o some_o list_v at_o i_o repine_v just_a truth_n shall_v i_o defend_v the_o preface_n unto_o the_o theorem_n i_o doubt_v not_o gentle_a reader_n but_o as_o my_o argument_n be_v strange_a and_o unacquainted_a with_o the_o vulgar_a tougue_n so_o shall_v i_o of_o many_o man_n be_v strange_o talk_v of_o and_o as_o straunglye_o judge_v some_o man_n will_v say_v peradventure_o i_o may_v have_v better_o employ_v my_o time_n in_o some_o pleasant_a history_n comprisinge_v matter_n of_o chivasrye_n some_o other_o will_v more_o have_v preise_v my_o travail_n if_o i_o have_v spend_v the_o like_a time_n in_o some_o moral_a matter_n other_o in_o decise_v some_o controversy_n of_o religion_n and_o yet_o some_o man_n as_o i_o judge_n will_v not_o mislike_v this_o kind_n of_o matter_n but_o then_o will_v they_o wish_v that_o i_o have_v use_v a_o more_o certain_a order_n in_o place_n both_o the_o proposition_n and_o theorem_n and_o also_o a_o more_o exacter_n proof_n of_o each_o of_o they_o both_o by_o demonstration_n mathematical_a some_o also_o will_v mislike_v my_o shortenes_n and_o simple_a plainness_n as_o other_o of_o other_o affection_n diverse_o shall_v espy_v somewhat_o that_o they_o shall_v think_v blame_v worthy_a and_o shall_v miss_v somewhat_o that_o they_o will_v wish_v to_o have_v be_v here_o use_v so_o that_o every_o man_n shall_v give_v his_o verdict_n of_o i_o according_a to_o his_o fantasy_n unto_o who_o joinct_o i_o make_v this_o my_o first_o answer_n that_o as_o they_o be_v many_o and_o in_o opinion_n very_o diverse_a so_o be_v it_o scarce_o possible_a to_o please_v they_o all_o with_o any_o one_o argument_n of_o what_o kind_n so_o ever_o it_o be_v and_o for_o my_o second_o answer_n i_o say_v thus_o that_o if_o annye_n one_o argument_n may_v please_v they_o all_o then_o shall_v they_o be_v thankful_a unto_o i_o for_o this_o kind_n of_o matter_n for_o neither_o be_v there_o any_o matter_n more_o strange_a in_o the_o english_a tungue_n than_o this_o whereof_o never_o book_n be_v write_v before_o now_o in_o that_o tungue_n and_o therefore_o ought_v to_o delight_n all_o they_o that_o desire_v to_o understand_v strange_a matter_n as_o most_o man_n common_o do_v and_o again_o the_o practice_n be_v so_o pleasant_a in_o usinge_v and_o so_o profitable_a in_o appliing_n that_o who_o so_o ever_o do_v the_o delight_n in_o any_o of_o both_o ought_v not_o of_o right_a to_o mislike_v this_o art_n and_o if_o any_o man_n shall_v like_v the_o art_n welle_v for_o itself_o but_o shall_v mislike_v the_o form_n that_o i_o have_v use_v in_o teaching_n of_o it_o to_o he_o i_o shall_v say_v first_o that_o i_o do_v wish_v with_o he_o that_o some_o other_o man_n which_o can_v better_o have_v do_v it_o have_v show_v his_o good_a will_n and_o use_v his_o diligence_n in_o such_o sort_n that_o i_o may_v have_v be_v thereby_o occasion_v just_o to_o have_v leave_v of_o my_o labour_n or_o after_o my_o travail_n to_o have_v suppress_v my_o book_n but_o sith_o no_o man_n have_v yet_o attempt_v the_o like_a as_o far_o as_o i_o can_v learn_v i_o trust_v all_o such_o as_o be_v not_o exercise_v in_o the_o study_n of_o geometrye_n shall_v find_v great_a ease_n and_o furtherance_n by_o this_o simple_a plain_a and_o easy_a form_n of_o write_n and_o shall_v perceive_v the_o exact_v work_v of_o theon_n and_o other_o that_o write_v on_o euclid_n a_o great_a deal_n the_o sumner_n by_o this_o blunt_a delineation_n afore_o hand_n to_o they_o teach_v for_o i_o dare_v presuppose_v of_o they_o that_o thing_n which_o i_o have_v set_v in_o myself_o and_o have_v mark_v in_o other_o that_o be_v to_o say_v that_o it_o be_v not_o easy_a for_o a_o man_n that_o shall_v travail_v in_o a_o strange_a art_n to_o understand_v at_o the_o begin_v both_o the_o thing_n that_o be_v teach_v and_o also_o the_o just_a reason_n why_o it_o be_v so_o and_o by_o experience_n of_o teachinge_a i_o have_v try_v it_o to_o be_v true_a for_o whenne_n i_o have_v teach_v the_o proposition_n as_o it_o import_v in_o meaning_n and_o annex_v the_o demonstration_n with_o all_o i_o do_v perceive_v that_o it_o be_v a_o great_a trouble_n and_o a_o painful_a vexation_n of_o mind_n to_o the_o learner_n to_o comprehend_v both_o those_o thing_n at_o one_o and_o therefore_o do_v i_o prove_v first_o to_o make_v they_o to_o understande_v the_o sense_n of_o the_o proposition_n and_o then_o afterward_o do_v they_o conceive_v the_o demonstration_n much_o soon_o when_o they_o have_v the_o sentence_n of_o the_o proposition_n first_o ingraft_v in_o their_o mind_n this_o thing_n cause_v i_o in_o both_o these_o book_n to_o omit_v the_o demonstration_n and_o to_o use_v only_o a_o plain_a form_n of_o declaration_n which_o may_v best_o serve_v for_o the_o first_o introduction_n which_o example_n hat_n be_v use_v by_o other_o learned_a man_n before_o now_o for_o not_o only_a georgius_n joachimus_fw-la rheticus_fw-la but_o also_o boetius_fw-la that_o wittye_v clerk_n do_v set_v forth_o some_o whole_a book_n of_o euclid_n without_o any_o demonstration_n or_o any_o other_o declaration_n at_o al._n but_o &_o if_o i_o shall_v hereafter_o perceive_v that_o it_o may_v be_v a_o thankful_a travail_n to_o set_v forth_o the_o proposition_n of_o geometry_n with_o demonstration_n i_o will_v not_o refuse_v to_o do_v it_o and_o that_o with_o sundry_a varietee_n of_o demonstration_n both_o pleasant_a and_o profitable_a also_o and_o then_o will_v i_o in_o like_a manner_n prepare_v to_o set_v forth_o the_o other_o book_n which_o now_o be_v leave_v unprint_v by_o occasion_n not_o so_o much_o of_o the_o charge_n in_o cut_v of_o the_o figure_n as_o for_o other_o just_a hynderance_n which_o i_o trust_v hereafter_o shall_v be_v remedy_v in_o the_o mean_a season_n if_o any_o man_n muse_v why_o i_o have_v set_v the_o conclusion_n before_o the_o teoreme_n sing_v many_o of_o the_o theorem_n seem_v to_o include_v the_o cause_n of_o some_o of_o the_o conclusion_n and_o therefore_o ought_v to_o have_v go_v before_o they_o as_o the_o cause_n go_v before_o the_o effect_n hereunto_o i_o say_v that_o although_o the_o cause_n do_v go_v before_o the_o effect_n in_o order_n of_o nature_n yet_o in_o order_n of_o teach_v the_o effect_n must_v be_v first_o declare_v and_o than_o the_o cause_n thereof_o show_v for_o so_o shall_v man_n best_o understand_v thing_n first_o to_o learn_v that_o such_o thing_n be_v to_o be_v wrought_v and_o secondary_o what_o they_o be_v and_o what_o they_o do_v import_v and_o than_o three_o what_o be_v the_o cause_n thereof_o an_o other_o cause_n why_o that_o the_o theorem_n be_v put_v after_o the_o conclusion_n be_v this_o when_o i_o write_v these_o first_o cunclusion_n which_o be_v iiiij_fw-la year_n pass_v i_o think_v not_o then_o to_o have_v add_v any_o theorem_n but_o next_o unto_o the_o conclusion_n to_o have_v teach_v the_o order_n how_o to_o have_v apply_v they_o to_o work_v for_o draw_v of_o plot_n &_o such_o like_a use_n but_o afterward_o consider_v the_o great_a commodity_n that_o they_o serve_v for_o and_o the_o light_n that_o they_o do_v geve_v to_o all_o sort_n of_o practice_n geometrical_a beside_o other_o more_o notable_a benefit_n which_o shall_v be_v declare_v more_o special_o in_o a_o place_n convenient_a i_o think_v best_a to_o geve_v you_o some_o taste_n of_o theym_n and_o the_o pleasant_a contemplation_n of_o such_o geometrical_a proposition_n which_o may_v serve_v diverselye_o in_o other_o book_n for_o the_o demonstration_n and_o proof_n of_o all_o geometrical_a work_v and_o in_o they_o as_o well_o as_o in_o the_o proposition_n i_o have_v draw_v in_o the_o linearie_a example_n many_o time_n more_o line_n than_o be_v speak_v of_o in_o the_o explication_n of_o
and_o on_o e_o be_v set_v k._n l_o so_fw-mi that_o of_o those_o u_o touch_n line_n be_v make_v the_o u_o side_n of_o a_o cinkeangle_n accord_v to_o the_o conclusion_n an_o other_o way_n an_o other_o way_n also_o may_v you_o draw_v a_o cinkeangle_n about_o a_o circle_n draw_v first_o a_o cinkeangle_n in_o the_o circle_n which_o be_v a_o easy_a thing_n to_o do_v by_o the_o doctrine_n of_o the_o xxxvij_o conclusion_n and_o then_o draw_v u_o touch_v line_n which_o shall_v be_v just_a parallel_n to_o the_o u_o side_n of_o the_o cinkeangle_n in_o the_o circle_n forsee_a that_o one_o of_o they_o do_v not_o cross_v overthwarte_o a_o other_o and_o then_o have_v you_o do_v the_o exaumple_v of_o this_o because_o it_o be_v easy_a i_o leave_v to_o your_o own_o exercise_n the_o xl._n conclusion_n to_o make_v a_o circle_n in_o any_o appoint_a cinkeangle_n of_o equal_a side_n and_o equal_a corner_n draw_v a_o plumbe_v line_n from_o any_o one_o corner_n of_o the_o cinkeangle_n unto_o the_o middle_n of_o the_o side_n that_o lie_v just_a against_o that_o angle_n and_o do_v likewaies_o in_o draw_v a_o other_o line_n from_o some_o other_o corner_n to_o the_o middle_n of_o the_o side_n that_o lie_v against_o that_o corner_n also_o and_o those_o two_o line_n will_v meet_v in_o cross_n in_o the_o prick_n of_o their_o cross_n shall_v you_o judge_v the_o centre_n of_o the_o circle_n to_o be_v therefore_o set_v one_o foot_n of_o the_o compass_n in_o that_o prick_n and_o extend_v the_o other_o to_o the_o end_n of_o the_o line_n that_o touch_v the_o middle_a of_o one_o side_n which_o you_o list_v and_o so_o draw_v a_o circle_n and_o it_o shall_v be_v just_o make_v in_o the_o cinkeangle_n accord_v to_o the_o conclusion_n example_n the_o cinkeangle_v assign_v be_v a._n b._n c._n d._n e_o in_o which_o i_o must_v make_v a_o circle_n wherefore_o i_o draw_v a_o right_a line_n from_o the_o one_o angle_n as_o from_o b_o to_o the_o middle_n of_o the_o contrary_a side_n which_o be_v e._n d_o and_o that_o middle_a prick_n be_v f._n then_o lykewaies_o from_o a_o other_o corner_n as_z from_z e_z i_o draw_v a_o right_a line_n to_o the_o middle_n of_o the_o side_n that_o lie_v against_o it_o which_o be_v b._n c._n and_o that_o prick_n be_v g._n now_o because_o that_o these_o two_o line_n do_v cross_a in_o h_n i_o say_v that_o h._n be_v the_o centre_n of_o the_o circle_n which_o i_o will_v make_v therefore_o i_o set_v one_o foot_n of_o the_o compass_n in_o h_n and_o extend_v the_o other_o foot_n unto_o g_o or_o f._n which_o be_v the_o end_n of_o the_o line_n that_o light_o in_o the_o middle_n of_o the_o side_n of_o that_o cinkeangle_n and_o so_o make_v i_o a_o circle_n in_o the_o cinkangle_n right_a as_o the_o conclusion_n mean_v the_o xli_o conclusion_n to_o make_v a_o circle_n about_o any_o assign_a cinkeangle_n of_o equal_a side_n and_o equal_a corner_n draw_v two_o line_n within_o the_o cinkeangle_n from_o two_o corner_n to_o the_o middle_n on_o the_o two_o contrary_a side_n as_o the_o last_o conclusion_n teach_v and_o the_o point_n of_o their_o cross_n shall_v be_v the_o centre_n of_o the_o circle_n that_o i_o seek_v for_o then_o set_v i_o one_o foot_n of_o the_o compass_n in_o that_o centre_n and_o the_o other_o foot_n i_o extend_v to_o one_o of_o the_o angle_n of_o the_o cinkangle_n and_o so_o draw_v i_o a_o circle_n about_o the_o cinkangle_v assign_v example_n an_o other_o way_n also_o an_o other_o way_n may_v i_o do_v it_o thus_o presuppose_v any_o three_o corner_n of_o the_o cinkangle_n to_o be_v three_o prick_v appoint_v unto_o which_o i_o shall_v find_v the_o centre_n and_o then_o draw_v a_o circle_n touching_n they_o all_o three_o accord_v to_o the_o doctrine_n of_o the_o seventene_n one_o and_o twenty_o and_o two_o and_o twenty_o conclusion_n and_o when_o i_o have_v find_v the_o centre_n then_o do_v i_o draw_v the_o circle_n as_o the_o samc_a conclusion_n do_v teach_v and_o this_o forty_o conclusion_n also_o the_o xlii_o conclusion_n to_o make_v a_o siseangle_n of_o equal_a side_n and_o equal_a angle_n in_o any_o circle_n assign_v if_o the_o centre_n of_o the_o circle_n be_v not_o know_v then_o seek_v out_o the_o centre_n according_a to_o the_o doctrine_n of_o the_o sixetenth_fw-mi conclusion_n and_o with_o your_o compass_n take_v the_o quantitee_n of_o the_o semidiameter_n just_o and_o then_o set_v one_o foot_n in_o one_o prick_n of_o the_o circumference_n of_o the_o circle_n and_o with_o the_o other_o make_v a_o mark_n in_o the_o circumference_n also_o toward_o both_o side_n then_o set_v one_o foot_n of_o the_o compass_n stedy_o in_o each_o of_o those_o new_a prick_v and_o point_n out_o two_o other_o prick_v and_o if_o you_o have_v do_v well_o you_o shall_v peceave_v that_o there_o will_v be_v but_o even_o six_o such_o division_n in_o the_o circumference_n whereby_o it_o do_v well_o appear_v that_o the_o side_n of_o any_o siseangle_v make_v in_o a_o circle_n be_v equal_a to_o the_o semidiameter_n of_o the_o same_o circle_n example_n the_o circle_n be_v b._n c._n d._n e._n f._n g_o who_o centre_n i_o find_v to_o be_v a._n therefore_o i_o set_v one_o foot_n of_o the_o compass_n in_o a_o and_o do_v extend_v the_o other_o foot_n to_o b_o thereby_o take_v the_o semidiameter_n then_o set_v i_o one_o foot_n of_o the_o compass_n unremoved_a in_o b_z and_o mark_v with_o the_o other_o foot_n on_o each_o side_n c._n and_o g._n then_o from_o c._n i_o mark_v d_o and_z from_z d_z e_o from_o e._n mark_v i_o f._n and_o then_o have_v i_o but_o one_o space_n just_a unto_o g._n and_o so_o have_v i_o make_v a_o just_a siseangle_n of_o equal_a side_n and_o equal_a angle_n in_o a_o circle_n appoint_v the_o xliii_o conclusion_n to_o make_v a_o siseangle_n of_o equal_a side_n and_o equal_a angle_n about_o any_o circle_n assign_v the_o xliiii_o conclusion_n to_o make_v a_o circle_n in_o any_o siseangle_v appoint_v of_o equal_a side_n and_o equal_a angle_n the_o xlv_o conclusion_n to_o make_v a_o circle_n about_o any_o siseangle_v limit_v of_o equal_a side_n and_o equal_a angle_n because_o you_o may_v easy_o conjecture_v the_o make_n of_o these_o figure_n by_o that_o that_o be_v say_v before_o of_o cinkangle_n only_o consider_v that_o there_o be_v a_o difference_n in_o the_o number_a of_o the_o side_n i_o think_v best_a to_o leave_v these_o unto_o your_o own_o device_n that_o you_o shall_v study_v in_o some_o thing_n to_o exercise_v your_o wit_n withal_o and_o that_o you_o may_v have_v the_o better_a occasion_n to_o perceive_v what_o difference_n there_o be_v between_o each_o two_o of_o those_o conclusion_n for_o though_o it_o seem_v one_o thing_n to_o make_v a_o siseangle_n in_o a_o circ_n e_o and_o to_o make_v a_o circle_n about_o a_o siseangle_n yet_o shall_v you_o perceive_v that_o it_o be_v not_o one_o thing_n neither_o be_v those_o two_o conclusion_n wrought_v one_o way_n like_a waise_v shall_v you_o think_v of_o those_o other_o two_o conclusion_n to_o make_v a_o siseangle_n about_o a_o circle_n and_o to_o make_v a_o circle_n in_o a_o siseangle_n though_o the_o figure_n be_v one_o in_o fashion_n when_o they_o be_v make_v yet_o be_v they_o not_o one_o in_o work_v as_o you_o may_v well_o perceive_v by_o the_o thirty_o seven_o thirty_o viij_o thirty-nine_o and_o xl_o conclusion_n in_o which_o the_o same_o work_n be_v teach_v touch_v a_o circle_n and_o a_o cinkangle_n yet_o this_o much_o will_v i_o say_v for_o your_o help_n in_o work_n that_o when_o you_o shall_v seek_v the_o centre_n in_o a_o size_n angle_n whether_o it_o be_v to_o make_v a_o circle_v in_o it_o other_o about_o it_o you_o shall_v draw_v the_o two_o cross_a line_n from_o one_o angle_n to_o the_o other_o angle_n that_o lie_v against_o it_o and_o not_o to_o the_o middle_n of_o any_o side_n as_o you_o do_v in_o the_o cinkangle_n the_o xlvi_o conclusion_n to_o make_v a_o figure_n of_o fifteen_o equal_a side_n and_o angle_n in_o any_o circle_n appoint_v this_o rule_n be_v general_a that_o how_o many_o side_n the_o figure_n shall_v have_v that_o shall_v be_v draw_v in_o any_o circle_n into_o so_o many_o part_n just_o must_v the_o circle_n be_v divide_v and_o therefore_o it_o be_v the_o more_o easy_a work_n common_o to_o draw_v a_o figure_n in_o a_o circle_n then_o to_o make_v a_o circle_n in_o a_o other_o figure_n now_o therefore_o to_o end_v this_o conclusion_n divide_v the_o circle_n first_o into_o five_o part_n and_o then_o each_o of_o they_o into_o three_o part_n again_o or_o else_o first_o divide_v it_o into_o three_o part_n and_o then_o each_o of_o they_o into_o five_o other_o part_n as_o you_o listen_v and_o can_v most_o readilye_o then_o draw_v line_n between_o every_o two_o
sensible_a principle_n and_o this_o be_v the_o cause_n why_o all_o learned_a man_n do_v approve_v the_o certainty_n of_o geometry_n and_o consequent_o of_o the_o other_o art_n mathematical_a which_o have_v the_o ground_n as_o arithmetic_n music_n and_o astronomy_n above_o all_o other_o art_n and_o science_n that_o be_v use_v amongst_o man_n thus_o much_o have_v i_o say_v of_o the_o first_o principle_n and_o now_o will_v i_o go_v on_o with_o the_o theorem_n which_o i_o do_v only_o by_o example_n declae_fw-la mind_v to_o reserve_v the_o proof_n to_o a_o peculiar_a book_n which_o i_o will_v then_o set_v forth_o when_o i_o perceive_v this_o to_o be_v thankful_o take_v of_o the_o reader_n of_o it_o the_o theorem_n of_o geometry_n brieflye_v declare_v by_o short_a example_n the_o first_o theorem_a when_o two_o triangle_n be_v so_o draw_v that_o the_o one_o of_o they_o have_v ij_o side_n equal_a to_o ij_o side_n of_o the_o other_o triangle_n and_o that_o the_o angle_n enclose_v with_o those_o side_n be_v equal_a also_o in_o both_o triangle_n then_o be_v the_o third_o side_n likewise_o equal_a in_o they_o and_o the_o whole_a triangle_n be_v of_o one_o greatness_n and_o every_o angle_n in_o the_o one_o equal_a to_o his_o match_n angle_n in_o the_o other_o i_o mean_v those_o angle_n that_o be_v enclose_v with_o like_a side_n example_n this_o triangle_n a.b.c._n have_v ij_o side_n that_o be_v to_o say_v c.a._n and_o c._n b_o equal_a to_o ij_o side_n of_o the_o other_o triangle_n f._n g.h_o for_o a._n c._n be_v equal_a to_o f._n g_o and_o b.c._n be_v equal_a to_o g.h._n and_o also_o the_o angle_n c._n contain_v between_o f._n g_o and_o g._n h_n for_o both_o of_o they_o answer_v to_o the_o eight_o part_n of_o a_o circle_n therefore_o do_v it_o remain_v that_o a._n b._n which_o be_v the_o third_o line_n in_o the_o first_o triangle_n do_v agree_v in_o length_n with_o f._n h_n which_o be_v the_o three_o line_n in_o that_o second_o triangle_n &_o the_o hole_n triangle_n a.b.c._n must_v needs_o be_v equal_a to_o the_o hole_n triangle_n f.g.h._n and_o every_o corner_n equal_a to_o his_o match_n that_o be_v to_o say_v a._n equal_a to_o f_o b._n to_o h_n and_o c._n to_o g_o for_o those_o be_v call_v match_n corner_n which_o be_v enclose_v with_o like_a side_n other_o else_o do_v lie_v against_o like_a side_n the_o second_o theorem_a in_o twileke_a triangle_n the_o ij_o corner_n that_o be_v about_o the_o ground_n line_n be_v equal_a together_o and_o if_o the_o side_n that_o be_v equal_a be_v draw_v out_o in_o length_n they_o will_v the_o corner_n that_o be_v under_o the_o ground_n line_n be_v equal_a also_o together_o example_n a.b.c._n be_v a_o twileke_a triangle_n for_o the_o one_o side_n a._n c_o be_v equal_a to_o the_o other_o side_n b.c._n and_o therefore_o i_o say_v that_o the_o inner_a corner_n a._n and_o b_o which_o be_v about_o the_o ground_n line_n that_o be_v a.b._n be_v equal_a to_o gither_o and_o far_o if_o c._n a._n and_o c._n b._n be_v draw_v forth_o unto_o d_z and_z e._n as_o you_o see_v that_o i_o have_v draw_v they_o then_o say_v i_o that_o the_o two_o utter_a angle_n under_o a._n and_o b_o be_v equal_a also_o together_o as_o the_o theorem_fw-la say_fw-la the_o proof_n whereof_o as_o of_o all_o the_o rest_n shall_v appear_v in_o euclid_n who_o i_o intend_v to_o set_v forth_o in_o english_a with_o sundry_a new_a addition_n if_o i_o may_v perceive_v that_o it_o will_v be_v thankful_o take_v the_o third_o theorem_a if_o in_o annye_n triangle_n there_o be_v two_o angle_n equal_a together_o then_o shall_v the_o side_n that_o lie_v against_o those_o angle_n be_v equal_a also_o example_n this_o triangle_n a.b.c._n have_v two_o corner_n equal_v each_o to_o other_o that_o be_v a._n and_o b_o as_o i_o do_v by_o supposition_n limit_v wherefore_o it_o follow_v that_o the_o side_n a._n c_o be_v equal_a to_o that_o other_o side_n b._n c_o for_o the_o side_n a._n c_o lie_v against_o the_o angle_n b_o and_o the_o side_n b._n c_o lie_v against_o the_o angle_n a._n the_o four_o theorem_a when_o two_o line_n be_v draw_v from_o the_o end_n of_o any_o one_o line_n and_o meet_v in_o any_o point_n it_o be_v not_o possible_a to_o draw_v two_o other_o line_n of_o like_a length_n each_o to_o his_o match_n that_o shall_v begin_v at_o the_o same_o point_n and_o end_n in_o any_o other_o point_n than_o the_o two_o first_o do_v example_n the_o first_o line_n be_v a._n b_o on_o which_o i_o have_v erect_v two_o other_o line_n a._n c_o and_o b._n c_o that_o meet_a in_o the_o prick_n c_o wherefore_o i_o say_v it_o be_v not_o possible_a to_o draw_v ij_o other_o line_n from_o a._n and_o b._n which_o shall_v mete_v in_o one_o point_n as_o you_o see_v a._n d._n and_o b.d._n mete_v in_o d._n but_o that_o the_o match_n line_n shall_v be_v unequal_a i_o mean_v by_o match_n line_n the_o two_o line_n on_o one_o side_n that_o be_v the_o ij_o on_o the_o right_a hand_n or_o the_o ij_o on_o the_o left_a hand_n for_o as_o youse_a in_o this_o example_n a._n d._n be_v long_o they_o a._n c_o and_o b.c._n be_v long_o than_o b.d._n and_o it_o be_v not_o possible_a that_o a._n c._n and_o a._n d._n shall_v be_v of_o one_o length_n if_o b._n d._n and_o b.c._n be_v like_o long_o for_o if_o one_o couple_n of_o match_n line_n be_v equal_a as_o the_o same_o example_n a.e._n be_v equal_a to_o a.c._n in_o length_n then_o must_v b._n e_o need_v be_v unequal_a to_o b.c._n as_o you_o see_v it_o be_v here_o short_a the_o fifte_n theorem_a if_o two_o triangle_n have_v there_o ij_o side_n equal_a one_o to_o another_o and_o their_o ground_n line_n equal_a also_o then_o shall_v their_o corner_n which_o be_v contain_v between_o like_a side_n be_v equal_a one_o to_o the_o other_o example_n because_o these_o two_o triangle_n a.b._n c_o and_o d.e.f._n have_v two_o side_n equal_v one_o to_o a_o other_o for_o a._n c._n be_v equal_a to_o d._n f_o and_o b.c._n be_v equal_a to_o e._n f_o and_o again_o their_o ground_n line_n a.b._n and_o d.e._n be_v like_a in_o length_n therefore_o be_v each_o angle_n of_o the_o one_o triangle_n equal_a to_o each_o angle_n of_o the_o other_o compare_v together_o those_o angle_n that_o be_v contain_v within_o like_o side_n so_o be_v a._n equal_a to_o d_o b._n to_o e_z and_o c._n to_o f_o for_o they_o be_v contain_v within_o like_a side_n as_o before_o be_v say_v the_o sixth_o theorem_a when_o any_o right_a line_n stand_v on_o a_o other_o the_o ij_o angle_n that_o they_o make_v other_o be_v both_o right_a angle_n or_o else_o equal_a to_o two_o right_a angle_n example_n a.b._n be_v a_o right_a line_n and_o on_o it_o there_o do_v light_v another_o right_a line_n draw_v from_o c._n perpendicular_o on_o it_o therefore_o say_v i_o that_o the_o two_o angle_n that_o they_o do_v make_v be_v two_o right_a angle_n as_o may_v be_v judge_v by_o the_o definition_n of_o a_o right_a angle_n but_o in_o the_o second_o part_n of_o the_o example_n where_o a.b._n be_v still_o the_o right_a line_n on_o which_o d._n stand_v inslope_o way_n the_o two_o angle_n that_o be_v make_v of_o they_o be_v not_o right_a angle_n but_o yet_o they_o be_v equal_a to_o two_o right_a angle_n for_o so_o much_o as_o the_o one_o be_v to_o great_a more_o than_o a_o right_a angle_n so_o much_o just_a be_v the_o other_o to_o little_a so_o that_o both_o together_o be_v equal_a to_o two_o right_a angle_n as_o you_o may_v perceive_v the_o seven_o theorem_a if_o two_o line_n be_v draw_v to_o any_o one_o prick_n in_o a_o other_o line_n and_o those_o two_o line_n do_v make_v with_o the_o first_o line_n two_o right_a angle_n other_o such_o as_o be_v equal_a to_o two_o right_a angle_n and_o that_o toward_o one_o hand_n than_o those_o two_o line_n do_v make_v one_o straight_o line_n example_n a.b._n be_v a_o straight_o line_n on_o which_o there_o do_v light_v two_o other_o line_n one_o from_o d_o and_o the_o other_o from_o c_o but_o consider_v that_o they_o meet_v in_o one_o prick_n e_o and_o that_o the_o angle_n on_o one_o hand_n be_v equal_a to_o two_o right_a corner_n as_o the_o last_o theorem_a do_v declare_v therefore_o may_v d.e._n and_o e.c._n be_v count_v for_o one_o right_a line_n the_o eight_o theorem_a when_o two_o line_n do_v cut_v one_o a_o other_o cross_a way_n they_o do_v make_v their_o match_n angle_v equal_a example_n what_o match_n angle_n be_v i_o have_v tell_v you_o in_o the_o definition_n of_o the_o term_n and_o here_o a_o and_o b._n be_v match_n corner_n in_o this_o example_n as_o be_v also_o c._n and_o d_o so_o that_o the_o corner_n a_o be_v equal_a to_o b_o and_o