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equal_a cantle_n therefore_o can_v not_o they_o be_v like_o cantle_n for_o necessary_o it_o go_v together_o that_o all_o cantle_n of_o circle_n make_v upon_o equal_a right_a line_n if_o they_o be_v like_a they_o must_v be_v equal_a also_o the_o lxix_o theorem_a in_o equal_a circle_n such_o angle_n as_o be_v equal_a be_v make_v upon_o equal_a arch_a line_n of_o the_o circumference_n whether_o the_o angle_n light_n on_o the_o circumference_n or_o on_o the_o centre_n example_n first_o i_o have_v set_v for_o a_o exaumple_v two_o equal_a circle_n that_o be_v a._n b._n c._n d_o who_o centre_n be_v k_o and_o the_o second_o circle_n e._n f._n g._n h_n and_o his_o centre_n l_o and_o in_o each_o of_o they_o be_v there_o make_v two_o angle_n one_o on_o the_o circumference_n and_o the_o other_o on_o the_o centre_n of_o each_o circle_n and_o they_o be_v all_o make_v on_o two_o equal_a arch_n line_n that_o be_v b.c.d._n the_o one_o and_o f.g.h._n the_o other_o now_o say_v the_o theorem_a that_o if_o 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equal_a together_o example_n example_n of_o this_o theorem_a you_o may_v see_v in_o the_o last_o figure_n where_o as_o six_o triangle_n make_v between_o those_o two_o gemowe_a line_n a._n b._n and_o c._n f_o the_o first_o triangle_n be_v a._n c._n d_o the_o second_o be_v a._n d.e_n the_o third_o be_v a._n d.b_n the_o four_o be_v a._n b._n e_o the_o fifte_n be_v d._n e.b_n and_o the_o sixth_o be_v b._n e.f_n of_o which_o fix_v triangle_n a._n d.e._n and_o d._n e._n b._n be_v equal_a because_o they_o have_v one_o common_a ground_n line_n and_o so_o likewise_o a.b.e._n and_o a.b._n d_o who_o come_v ground_n line_n be_v a._n b_o but_o a.c.d._n be_v equal_a to_o b._n e.f_n be_v both_o between_o one_o couple_n of_o parallel_n not_o because_o they_o have_v one_o ground_n line_n but_o because_o they_o have_v their_o ground_n line_n equal_a for_o c.d._n be_v equal_a to_o e._n f_o as_o you_o may_v declare_v thus_o c._n d_o be_v equal_a to_o a.b._n by_o the_o four_o and_o twenty_o theorem_a for_o theiare_v two_o contrary_a side_n of_o one_o lykeiamme_n a._n c.d.b_n and_o e._n f_o by_o the_o same_o theorem_a be_v equal_a to_o a._n b_o for_o they_o be_v the_o two_o the_o contrary_a side_n of_o the_o likeiamme_n a._n e.f.b_n wherefore_o c.d._n must_v needs_o be_v equal_a to_o e.f._n like_o wise_a the_o triangle_n a._n c.d_n be_v equal_a to_o a._n b.e_n because_o they_o be_v make_v between_o one_o pair_n of_o parallel_n and_o have_v their_o groundlines_n like_a i_o mean_v c._n d._n and_o a._n b._n again_o a._n d.e_n be_v equal_a to_o each_o of_o they_o both_o for_o his_o ground_n line_n d._n e_o be_v equal_a to_o a._n b_o in_z so_o much_o as_o they_o be_v the_o contrary_a side_n of_o one_o likeiamme_n that_o be_v the_o long_a square_a a._n b._n d._n e._n and_o thus_o may_v you_o prove_v the_o equalnes_n of_o all_o the_o rest_n the_o xxix_o theorem_a alequal_a triangle_n that_o be_v make_v on_o one_o ground_n line_n and_o rise_v one_o way_n must_v needs_o be_v between_o one_o pair_n 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this_o theorem_a do_v presuppose_v the_o ground_n line_n to_o be_v diverse_a but_o yet_o of_o one_o length_n as_o a._n c._n d_o and_o b._n e.f_n as_o they_o be_v ij_o equal_a triangle_n approve_v by_o the_o eighte_o and_o twenty_o theorem_fw-la so_o in_o the_o same_o theorem_fw-la it_o be_v declare_v that_o their_o ground_n line_n be_v equal_a together_o that_o be_v c._n d_o and_o e._n f_o now_o this_o be_v true_a and_o consider_v that_o they_o be_v make_v toward_o one_o side_n it_o follow_v that_o they_o be_v make_v between_o one_o pair_n of_o parallel_n when_o i_o say_v draw_v toward_o one_o side_n i_o mean_v that_o the_o triangle_n must_v be_v draw_v other_o both_o upward_o from_o one_o parallel_n other_o else_o both_o downward_o for_o if_o the_o one_o be_v draw_v upward_o and_o the_o other_o downward_o then_o be_v they_o draw_v between_o two_o pair_n of_o parallel_n presupposinge_v one_o to_o be_v draw_v by_o their_o ground_n line_n and_o then_o do_v they_o rise_v towards_o 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that_o theibe_v make_v between_o one_o pair_n of_o parallel_n and_o hereof_o may_v you_o perceive_v the_o reason_n why_o in_o measuring_a the_o platte_a of_o a_o triangle_n you_o must_v multiply_v the_o perpendicular_a line_n by_o half_a the_o ground_n line_n or_o else_o the_o hole_n ground_n line_n by_o half_a the_o perpendicular_a for_o by_o any_o of_o these_o both_o way_n be_v there_o make_v a_o lykeiamme_n equal_a to_o half_a such_o a_o one_o as_o shall_v be_v make_v on_o the_o same_o hole_n ground_n line_n with_o the_o triangle_n and_o between_o one_o pair_n of_o parallel_n therefore_o as_o that_o lykeiamme_n be_v double_a to_o the_o triangle_n so_o the_o half_a of_o it_o must_v needs_o be_v equal_a to_o the_o triangle_n compare_v the_o xu_o conclusion_n with_o this_o theorem_a the_o xxxij_o theorem_a in_o all_o likeiamme_n where_o there_o be_v more_o than_o one_o make_v about_o one_o bias_n line_n the_o fill_n square_n of_o every_o of_o they_o must_v needs_o be_v equal_a 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h._n c._n e._n n._n be_v one_o square_a likeiamme_n and_o l._n m._n g._n c._n be_v a_o other_o which_o both_o be_v make_v about_o one_o bias_n line_n that_o be_v n._n m_o than_o k._n l._n h._n c._n and_o c._n e._n f._n g._n be_v two_o syll_a square_n for_o they_o do_v syll_v up_o the_o side_n of_o the_o two_o first_o square_v sykeiamme_n in_o such_o sort_n that_o of_o all_o they_o four_o be_v make_v one_o great_a general_a square_a k._n m.f.n._n now_o to_o the_o sentence_n of_o the_o theorem_a i_o say_v that_o the_o two_o fill_v squre_n h._n k._n l._n c._n and_o c._n e._n f._n g._n be_v both_o equal_a together_o as_o it_o shall_v be_v declare_v in_o the_o book_n of_o proof_n because_o they_o be_v the_o fill_n squre_v of_o two_o likeiamme_n make_v about_o one_o bias_n line_n as_o the_o exaumple_v sheweth_z confer_v the_o twelfthe_o conclusion_n with_o this_o the_o oreme_a the_o xxxiij_o theorem_a in_o all_o right_n angule_v triangle_n the_o square_a of_o that_o side_n which_o lie_v against_o the_o right_a angle_n be_v equal_a to_o the_o two_o square_n of_o both_o the_o other_o side_n example_n a.b.c._n be_v a_o triangle_n have_v a_o right_a angle_n in_o b._n wherefore_o it_o solow_v that_o the_o square_n of_o a._n c_o which_o be_v the_o side_n that_o lie_v against_o the_o right_a angle_n shall_v be_v as_o much_o as_o the_o two_o square_n of_o a._n b._n and_o b._n c._n which_o be_v the_o other_o two_o side_n ¶_o by_o the_o square_n of_o any_o sign_n you_o must_v understande_v a_o figure_n make_v just_a square_n have_v all_o his_o iiij_o side_n equal_a to_o that_o line_n whereof_o it_o be_v the_o square_a so_o be_v a._n c.f_n the_o square_a of_o a.c._n lykewais_o a._n b._n d._n
be_v equal_a together_o and_o contrary_a way_n if_o they_o be_v equal_a together_o they_o be_v also_o equal_o distant_a from_o that_o least_o line_n for_o the_o declaration_n of_o this_o proposition_n it_o shall_v not_o need_v to_o use_v any_o other_o example_n then_o that_o which_o be_v bring_v for_o the_o explication_n of_o this_o last_o theorem_a by_o which_o you_o may_v without_o any_o teachinge_a casy_o perceive_v both_o the_o meaning_n and_o also_o the_o truch_n of_o this_o proposition_n the_o l_o iiij_o theorem_a if_o a_o point_n be_v set_v forth_o in_o a_o circle_n and_o from_o that_o point_n unto_o the_o circumference_n many_o line_n draw_v of_o which_o more_o than_o two_o be_v equal_a together_o then_o be_v that_o point_n the_o centre_n of_o that_o circle_n example_n the_o circle_n be_v a._n b.c_n and_o with_o init_fw-la i_o have_v set_v four_o for_o a_o example_n three_o prick_v which_o be_v d.e._n and_o f_o and_o from_o every_o one_o of_o they_o i_o have_v draw_v at_o the_o jest_n iiij_o line_n unto_o the_o circumference_n of_o the_o circle_n but_o from_o d_o i_o have_v draw_v more_o yet_o may_v it_o appear_v ready_o unto_o your_o eye_n that_o of_o all_o the_o line_n which_o be_v draw_v from_o e._n and_o f_o unto_o the_o circumference_n there_o be_v but_o two_o equal_a and_o more_o can_v not_o be_v for_o g.e._n nor_o e.h._n have_v none_o other_o equal_a to_o they_o nor_o can_v not_o have_v any_o be_v draw_v from_o the_o fame_n point_fw-fr e._n no_o more_o can_v l._n f_o or_o f._n k_o have_v any_o line_n equal_a to_o either_o of_o they_o be_v draw_v from_o the_o same_o point_n f._n and_o yet_o from_o either_o of_o those_o two_o point_n be_v there_o draw_v two_o line_n equal_a together_o as_o a._n e_o be_v equal_a to_o e._n b_o and_o b._n f_o be_v equal_a to_o f._n c_o but_o there_o can_v no_o three_o line_n be_v draw_v equal_a to_o either_o of_o these_o two_o couple_n and_o that_o be_v by_o reason_n that_o they_o be_v draw_v from_o a_o point_n distaunte_v from_o the_o centre_n of_o the_o circle_n but_o from_o d_z although_o there_o be_v seven_o line_n draw_v to_o the_o circumference_n yet_o all_o be_v equal_a because_o it_o be_v the_o centre_n of_o the_o circle_n and_o therefore_o if_o you_o draw_v never_o so_o mannye_n more_o from_o it_o unto_o the_o circumference_n all_o shall_v be_v equal_a so_o that_o this_o be_v the_o privilege_n as_o it_o be_v of_o the_o centre_n and_o therefore_o no_o other_o point_n can_v have_v above_o two_o equal_a line_n draw_v from_o it_o unto_o the_o circumference_n and_o from_o all_o poitte_v you_o may_v draw_v ij_o equal_a line_n to_o the_o circumference_n of_o the_o cirle_n whether_o that_o point_n be_v within_o the_o circle_n or_o without_o it_o the_o l_o v._n theorem_a no_o circle_n can_v cut_v a_o other_o circle_n in_o more_o point_n than_o two_o example_n the_o first_o circle_n be_v a._n b.f.e_n the_o second_o circle_n be_v b._n c._n d_o f_o and_o they_o cross_v one_o a_o other_o in_o b._n and_o in_o e_o and_o in_o no_o more_o point_n neither_o be_v it_o possible_a that_o they_o shall_v but_o other_o figure_n there_o be_v which_o may_v cut_v a_o circle_n in_o four_o part_n as_o you_o see_v in_o this_o example_n where_o i_o have_v set_v forth_o one_o ton_n form_n and_o one_o eye_n form_n and_o each_o of_o they_o cut_v every_o of_o their_o two_o circle_n into_o four_o part_n but_o as_o they_o be_v irregulare_fw-la form_n that_o be_v to_o say_v such_o form_n as_o have_v no_o precise_a measure_n neither_o proportion_n in_o their_o draughte_n so_o can_v there_o scarce_o be_v make_v any_o certain_a theorem_fw-la of_o they_o but_o circle_n be_v regular_fw-la form_n that_o be_v to_o say_v such_o form_n as_o have_v in_o their_o protracture_n a_o just_a and_o certain_a proportion_n so_o that_o certain_a and_o determinate_a truth_n may_v be_v affirm_v of_o they_o since_o they_o be_v uniform_a and_o unchaungable_a the_o lvi_o theorem_a if_o two_o circle_n be_v so_o draw_v that_o the_o one_o be_v within_o the_o other_o and_o that_o they_o touch_v one_o a_o other_o if_o a_o line_n be_v draw_v by_o both_o their_o centre_n and_o so_o forth_o in_o length_n that_o line_n shall_v run_v to_o that_o point_n where_o the_o circle_n do_v touch_v example_n the_o one_o circle_n which_o be_v the_o great_a and_o uttermost_a be_v a._n b._n c_o the_o other_o circle_n that_o be_v the_o lesser_a and_o be_v draw_v within_o the_o first_o be_v a._n d._n e._n the_o centre_n of_o the_o great_a circle_n be_v f_o and_o the_o centre_n of_o the_o lesser_a circle_n be_v g_o the_o point_n where_o they_o touch_v be_v a._n and_o now_o you_o may_v see_v the_o truth_n of_o the_o theorem_a so_o plain_o that_o it_o need_v no_o far_a declaration_n for_o you_o may_v see_v that_o draw_v a_o line_n from_o f._n to_o g_z and_o so_o forth_o in_o length_n until_o it_o come_v to_o the_o circumference_n it_o will_v light_o in_o the_o very_a point_n a_o where_o the_o circle_n touch_v one_o a_o other_o the_o lvij_o theorem_a if_o two_o circle_n be_v draw_v so_o one_o without_o and_o other_o that_o their_o edge_n do_v touch_v and_o a_o right_a line_n be_v drawenne_n from_o the_o centre_n of_o the_o oneto_o the_o centre_n of_o the_o other_o that_o line_n shall_v pass_v by_o the_o place_n of_o their_o touch_v example_n the_o first_o circle_n be_v a._n b.e_n and_o his_o centre_n be_v k_o the_o second_o circle_n be_v d_o b._n c_o and_o his_o centre_n be_v h_n the_o point_n where_o they_o do_v touch_n be_v b._n now_o do_v you_o see_v that_o the_o line_n k._n h_n which_o be_v draw_v from_o k_o that_o be_v centre_n of_o the_o first_o circle_n unto_o h_n be_v centre_n of_o the_o second_o circle_n do_v pass_v as_o it_o must_v needs_o by_o the_o point_n b_o which_o be_v the_o very_a point_n where_o they_o do_v to_o touch_v together_o the_o lviij_o theorem_a one_o circle_n can_v not_o touch_v a_o other_o in_o more_o point_n than_o one_o whether_o they_o touch_v within_o or_o without_o example_n for_o the_o declaration_n of_o this_o theorem_a i_o have_v draw_v iiij_o circle_n the_o first_o be_v a._n b._n c_o and_o his_o centre_n h._n the_o second_o be_v a._n d._n g_o and_o his_o centre_n f._n the_o three_o be_v l._n m_o and_o his_o centre_n k._n the_o four_o be_v d._n g.l.m_n and_o his_o centre_n e._n now_o as_o you_o perceive_v the_o second_o circle_n a._n d.g_n touch_v the_o first_o in_o the_o inner_a side_n inso_v much_o as_o it_o be_v draw_v within_o the_o other_o and_o yet_o it_o touch_v he_o but_o in_o one_o point_n that_o be_v to_o say_v in_o a_o so_o like_o way_n the_o three_o he_o as_o you_o may_v see_v but_o in_o one_o place_n and_o now_o as_o for_o the_o four_o circle_n it_o be_v draw_v to_o declare_v the_o diversity_n between_o touch_v and_o cut_v or_o cross_v for_o one_o circle_n may_v cross_v and_o cut_v a_o great_a many_o other_o circle_n yet_o can_v be_v not_o cut_v any_o one_o in_o more_o place_n than_o two_o as_o the_o five_o and_o fifty_o theorem_a affirm_v the_o lix_o theorem_a in_o every_o circle_n those_o line_n be_v to_o be_v count_v equal_a which_o be_v in_o like_a distance_n from_o the_o centre_n and_o contrary_a way_n they_o be_v in_o like_a distance_n from_o the_o centre_n which_o be_v equal_a example_n in_o this_o figure_n you_o see_v first_o the_o circle_n draw_v which_o be_v a._n b.c.d_n and_o his_o centre_n be_v e._n in_o this_o circle_n also_o there_o be_v draw_v two_o line_n equal_o distant_a from_o the_o centre_n for_o the_o line_n a._n b_o and_o the_o line_n d._n c_o be_v just_a of_o one_o distance_n from_o the_o centre_n which_o be_v e_z and_o therefore_o be_v they_o of_o one_o length_n again_o they_o be_v of_o one_o length_n as_o shall_v be_v prove_v in_o the_o book_n of_o profess_v and_o therefore_o then_o distance_n from_o the_o centre_n be_v all_o one_o the_o lx_o theorem_a in_o every_o circle_n the_o long_a line_n be_v the_o diameter_n and_o of_o all_o the_o other_o line_n they_o be_v still_o long_a that_o be_v next_o unto_o the_o centre_n and_o they_o be_v the_o short_a that_o be_v far_a distant_a from_o it_o example_n in_o this_o circle_n a._n b.c.d_n i_o have_v draw_v first_o the_o diameter_n which_o be_v a._n d_o which_o pass_v as_o it_o must_v by_o the_o centre_n e_o then_o have_v i_o draw_v ij_o other_o line_n as_o m._n n_n which_o be_v near_o the_o centre_n and_o f._n g_o that_o be_v far_a from_o the_o centre_n the_o four_o line_n also_o on_o the_o other_o side_n of_o the_o diameter_n that_o be_v b._n c_o be_v near_a to_o the_o centre_n than_o the_o line_n f._n g_o
the_o compass_n in_o b_o and_o with_o the_o other_o i_o draw_v the_o arch_n d._n e_o which_o i_o part_v into_o ij_o equal_a part_n in_o f_o and_o they_o draw_v a_o line_n from_o b_o to_z f_o &_o so_o i_o have_v my_o intent_n the_o four_o concl._n to_o divide_v any_o measurable_a line_n into_o ij_o equal_a part_n open_v your_o compass_n to_o the_o just_a length_n of_o the_o line_n and_o they_o set_v one_o foot_n stedde_o at_o the_o one_o end_n of_o the_o line_n &_o with_o the_o other_o foot_n draw_v a_o arch_n of_o a_o circle_n against_o the_o middle_n of_o the_o line_n both_o over_o it_o and_o also_o under_o it_o then_o do_v like_o waise_v at_o the_o other_o end_n of_o the_o line_n and_o mark_v where_o those_o arch_n line_n do_v meet_v crossewaies_o and_o between_o those_o ij_o prick_n draw_v a_o line_n and_o it_o shall_v cut_v the_o first_o line_n in_o two_o equal_a portion_n example_n the_o line_n be_v a._n b._n accord_v to_o which_o i_o open_v the_o compass_n and_o make_v four_o arch_n line_n which_o meet_v in_o c._n and_o d_o then_o draw_v i_o a_o line_n from_o c_o so_fw-mi have_v i_o my_o purpose_n this_o conclusion_n serve_v for_o make_v of_o quadrate_n and_o squire_n beside_o many_o other_o commodity_n howbeit_o it_o may_v be_v do_v more_o readylye_o by_o this_o conclusion_n that_o follow_v next_o the_o five_o conclusion_n to_o make_v a_o plum_n line_n or_o any_o prick_n that_o you_o will_v in_o any_o right_a line_n appoint_v example_n the_o line_n be_v a.b._n the_o prick_n on_o which_o i_o shall_v make_v the_o plum_n line_n be_v c._n then_o open_v i_o the_o compass_n as_o wide_o as_o a_o c_o and_o set_v one_o foot_n in_o c._n and_o with_o the_o other_o do_v i_o mark_v out_o c.a._n and_o c._n b_o then_o open_v i_o the_o compass_n as_o wide_o as_o a._n b_o and_o make_v ij_o arch_a line_n which_o do_v cross_v in_o d_o and_o so_o have_v i_o do_v how_o be_v it_o it_o happen_v so_o sommetymes_o that_o the_o prick_n on_o which_o you_o will_v make_v the_o perpendicular_a or_o plum_n line_n be_v so_o never_o the_o eand_n of_o your_o line_n that_o you_o can_v not_o extend_v any_o notable_a length_n from_o it_o to_o the_o one_o end_n of_o the_o line_n and_o if_o so_o be_v it_o then_o that_o you_o may_v not_o draw_v your_o line_n long_o from_o that_o end_n then_o do_v this_o conclusion_n require_v a_o new_a aid_n for_o the_o last_o devise_n will_v not_o serve_v in_o such_o case_n therefore_o shall_v you_o do_v thus_o if_o your_o line_n be_v of_o any_o notable_a length_n divide_v it_o into_o five_o part_n and_o if_o it_o be_v not_o so_o long_o that_o it_o may_v yield_v five_o notable_a part_n then_o make_v a_o other_o line_n at_o will_n and_o part_n it_o into_o five_o equal_a portion_n so_o that_o three_o of_o those_o part_n may_v be_v find_v in_o your_o line_n then_o open_v your_o compass_n as_o wide_o as_o three_o of_o these_o five_o measure_n be_v and_o set_v the_o one_o foot_n of_o the_o compass_n in_o the_o prick_n where_o you_o will_v have_v the_o plum_n line_n to_o light_n which_o i_o call_v the_o first_o prick_n and_o with_o the_o other_o foot_n draw_v a_o arch_n line_n right_a over_o the_o prick_n as_o you_o can_v aim_v it_o then_o open_v your_o compass_n as_o wide_o as_o all_o five_o measure_n be_v and_o set_v the_o one_o foot_n in_o the_o four_o prick_n and_o with_o the_o other_o foot_n draw_v a_o other_o arch_a line_n cross_v the_o first_o and_o where_o they_o two_o do_v cross_n thence_o draw_v a_o line_n to_o the_o point_n where_o you_o will_v have_v the_o perpendicular_a line_n to_o light_n and_o you_o have_v do_v example_n the_o line_n be_v a._n b._n and_o a._n be_v the_o prick_n on_o which_o the_o perpendicular_a line_n must_v light_v therefore_o i_o divide_v a._n b._n into_o five_o part_n equal_a then_o do_v i_o open_v the_o compass_n to_o the_o wideness_n of_o three_o part_n that_o be_v a._n d._n and_o let_v one_o foot_n stay_v in_o a._n and_o with_o the_o other_o i_o make_v a_o arch_n line_n in_o c._n afterward_o i_o open_v the_o compass_n as_o wide_o as_o a.b._n that_o be_v as_o wide_a as_o all_o five_o part_n and_o set_v one_o foot_n in_o the_o four_o prick_n which_o be_v e_z drawing_z a_o arch_a line_n with_o the_o other_o foot_n in_o c._n also_o then_o do_v i_o draw_v thence_o a_o line_n unto_o a_o and_o so_o have_v i_o do_v but_o and_o if_o the_o line_n be_v to_o short_a to_o be_v part_v into_o five_o part_n i_o shall_v divide_v it_o into_o iij._o part_n only_o as_o you_o see_v the_o line_n f._n g_o and_o then_o make_v d._n a_o other_o line_n as_o be_v k._n l._n which_o i_o divide_v into_o u_o such_o division_n as_o f._n g._n contain_v iij_o then_o open_v i_o the_o compaa_v as_o wide_a as_o four_o part_n which_o be_v k._n m._n and_o so_o set_v i_o one_o foot_n of_o the_o compass_n in_o f_o and_o with_o the_o other_o i_o draw_v a_o arch_a line_n towards_o h_n then_o open_v i_o the_o compass_n as_o wide_o as_o k._n l._n that_o be_v all_o u_o part_n and_o set_v one_o foot_n in_o g_o that_o be_v the_o iij._o prick_n and_o with_o the_o other_o i_o draw_v a_o arch_a line_n towards_o h._n also_o and_o where_o those_o two_o arch_a line_n do_v cross_a which_o be_v by_o h._n thence_o draw_v i_o a_o line_n unto_o f_o and_o that_o make_v a_o very_o plumbe_v line_n to_o f._n g_o as_z my_o desire_n be_v the_o manner_n of_o work_n of_o this_o conclusion_n be_v like_a to_o the_o second_o conclusion_n but_o the_o reason_n of_o it_o do_v depend_v of_o the_o xlvi_o proposition_n of_o the_o first_o book_n of_o euclid_n an_o other_o way_n yet_o set_v one_o foot_n of_o the_o compass_n in_o the_o prick_n on_o which_o you_o will_v have_v the_o plumbe_v line_n to_o light_n and_o stretch_v forth_o other_o foot_n towards_o the_o long_a end_n of_o the_o line_n as_o wide_o as_o you_o can_v for_o the_o length_n of_o the_o line_n and_o so_o draw_v a_o quarter_n of_o a_o compass_n or_o more_o then_o without_o stir_v of_o the_o compass_n set_v one_o foot_n of_o it_o in_o the_o same_o line_n where_o as_o the_o circularline_n do_v begin_v and_o extend_v other_o in_o the_o circular_a line_n set_v a_o mark_n where_o it_o do_v light_a then_o take_v half_a that_o quantity_n more_o thereunto_o and_o by_o that_o prick_n that_o end_v the_o last_o part_n draw_v a_o line_n to_o the_o prick_n assign_v and_o it_o shall_v be_v a_o perpendicular_a example_n a._n b._n be_v the_o line_n appoint_v to_o which_o i_o must_v make_v a_o perpendicular_a line_n to_o light_v in_o the_o prick_n assign_v which_o be_v a._n therefore_o do_v i_o set_v one_o foot_n of_o the_o compass_n in_o a_o and_o extend_v the_o other_o unto_o d._n make_v a_o part_n of_o a_o circle_n more_o than_o a_o quarter_n that_o be_v d._n e._n then_o do_v i_o set_v one_o foot_n of_o the_o compass_n unalter_a in_o d_o and_o stretch_v the_o other_o in_o the_o circular_a line_n and_o it_o do_v light_a in_o f_o this_o space_n between_o d._n and_o f._n i_o divide_v into_o half_n in_o the_o prick_n g_o which_o half_o i_o take_v with_o the_o compass_n and_o set_v it_o beyond_o f._n unto_o h_n and_o therefore_o be_v h._n the_o point_n by_o which_o the_o perpendicular_a line_n must_v be_v draw_v so_o say_v i_o that_o the_o line_n h._n a_o be_v a_o plumbe_v line_n to_o a._n b_o as_z the_o conclusion_n will_v the_o vi_o conclusion_n to_o draw_v a_o straight_a line_n from_o any_o prick_n that_o be_v not_o in_o a_o line_n and_o to_o make_v it_o perpendicular_a to_o a_o other_o line_n open_v your_o compass_n so_o wide_a that_o it_o may_v extend_v somewhat_o far_o they_o from_o the_o prick_n to_o the_o line_n than_o set_v the_o one_o foot_n of_o the_o compass_n in_o the_o prick_n and_o with_o the_o other_o shall_v you_o draw_v a_o compass_a line_n that_o shall_v cross_v that_o other_o first_o line_n in_o two_o place_n now_o if_o you_o divide_v that_o arch_a line_n into_o two_o equal_a part_n and_o from_o the_o middle_a prick_n thereof_o unto_o the_o prick_n without_o the_o line_n you_o draw_v a_o straight_a line_n it_o shall_v be_v a_o plumbe_v line_n to_o that_o first_o line_n accord_v to_o the_o conclusion_n example_n c._n be_v the_o appoint_a prick_n from_o which_o unto_o the_o line_n a._n b._n i_o must_v draw_v a_o perpendicular_a therefore_o i_o open_v the_o compass_n so_o wide_a that_o it_o may_v have_v one_o foot_n in_o c_o and_o other_o to_o reach_v over_o the_o line_n and_o with_o that_o foot_n i_o draw_v a_o arch_a line_n as_o you_o see_v between_o a._n and_o b_o which_o arch_a
it_o be_v to_o be_v note_v that_o though_o every_o small_a arch_n of_o a_o great_a circle_n do_v seem_v to_o be_v a_o right_a line_n yet_o in_o very_a deed_n it_o be_v not_o so_o for_o every_o part_n of_o the_o circumference_n of_o all_o circle_n be_v compass_v though_o in_o little_a arch_n of_o great_a circle_n the_o eye_n can_v discern_v the_o crokednes_n yet_o reason_n do_v always_o declare_v it_o therefore_o iij._n prick_v in_o a_o exact_a right_a line_n can_v not_o be_v bring_v into_o the_o circumference_n of_o a_o circle_n but_o and_o if_o they_o be_v not_o in_o a_o right_a line_n how_o so_o ever_o they_o stand_v thus_o shall_v you_o find_v their_o common_a centre_n open_v your_o compass_n so_o wide_a that_o it_o be_v some_o what_o more_o than_o the_o half_a distance_n of_o two_o of_o those_o prick_v then_o set_v the_o one_o foot_n of_o the_o compass_n in_o the_o one_o prick_n and_o with_o the_o other_o foot_n draw_v a_o arch_n sign_n towards_o the_o other_o prick_n then_o again_o put_v the_o foot_n of_o your_o compass_n in_o the_o second_o prick_n and_o with_o the_o other_o foot_n make_v a_o arch_n line_n that_o may_v cross_v the_o first_o arch_a line_n in_o ij_o place_n now_o as_o you_o have_v do_v with_o those_o two_o prick_v so_o do_v with_o the_o middle_a prick_n and_o the_o third_o that_o remain_v then_o draw_v ij_o line_n by_o the_o point_n where_o those_o arch_n line_n do_v cross_a and_o where_o those_o two_o line_n do_v meet_v there_o be_v the_o centre_n that_o you_o seek_v for_o example_n the_o xxiiii_o conclusion_n to_o draw_v a_o touch_n line_n unto_o a_o circle_n from_o any_o point_n assign_v here_o must_v you_o understand_v that_o the_o prick_n must_v be_v without_o the_o circle_n else_o the_o conclusion_n be_v not_o possible_a but_o the_o prick_n or_o point_n be_v without_o the_o circle_n thus_o shall_v you_o proceed_v open_v your_o compass_n so_o that_o the_o one_o foot_n of_o it_o may_v be_v set_v in_o the_o centre_n of_o the_o circle_n and_o the_o other_o foot_n on_o the_o prick_v appoint_v and_o so_o draw_v a_o other_o circle_n of_o that_o largeness_n about_o the_o same_o centre_n and_o it_o shall_v govern_v you_o certain_o in_o make_v the_o say_a touch_n line_n for_o if_o you_o draw_v a_o line_n from_o the_o prick_n appoint_v unto_o the_o centre_n of_o the_o circle_n and_o mark_v the_o place_n where_o it_o do_v cross_v the_o lesser_a circle_n and_o from_o that_o point_n erect_v a_o plumbe_v line_n that_o shall_v touch_v the_o edge_n of_o the_o utter_a circle_n and_o mark_v also_o the_o place_n where_o that_o plumbe_v line_n cross_v that_o utter_a circle_n and_o from_o that_o place_n draw_v a_o other_o line_n to_o the_o centre_n take_v heed_n where_o it_o cross_v the_o lesser_a circle_n if_o you_o draw_v a_o plumbe_v line_n from_o that_o prick_n unto_o the_o edge_n of_o the_o great_a circle_n that_o line_n i_o say_v be_v a_o tooth_n line_n draw_v from_o the_o point_n assign_v according_a to_o the_o meaning_n of_o this_o conclusion_n example_n let_v the_o circle_n be_v call_v b.c._n d_o and_o his_o centre_n e_z and_z the_o prick_v assign_v a_o open_v your_o compass_n now_o of_o such_o wideness_n that_o the_o one_o foot_n may_v be_v set_v in_o e_o which_o is_z the_o centre_n of_o the_o circle_n &_o the_o other_o in_o a_o which_o is_z the_o point_n assign_v &_o so_o make_v a_o other_o great_a circle_n as_o here_o be_v a._n f_o g_o they_o draw_v a_o line_n from_o a._n unto_fw-la e_o and_o where_o that_o line_n do_v cross_a the_o inner_a circle_n which_o here_o be_v in_o the_o prick_n b._n there_o erect_v a_o plumb_n line_n unto_o the_o line_n a.e._n and_o let_v that_o plumb_n line_n touch_v the_o utter_a circle_n as_o it_o do_v here_o in_o the_o point_n f_o so_o shall_v b.f._n be_v that_o plumbe_v line_n then_o from_o f._n unto_o e._n draw_v a_o other_o line_n which_o shall_v be_v f._n e_o and_o it_o will_v cut_v the_o inner_a circle_n as_o it_o do_v here_o in_o the_o point_n c_o from_o which_o point_n c._n if_o you_o erect_v a_o plumb_n line_n unto_o a_o then_o be_v that_o line_n a._n c_o the_o touch_n line_n which_o you_o shall_v find_v notwithstanding_o that_o this_o be_v a_o certain_a way_n to_o find_v any_o touch_n line_n and_o a_o demonstrable_a form_n yet_o more_o easy_o by_o many_o fold_n may_v you_o find_v and_o make_v any_o such_o line_n with_o a_o true_a ruler_n layinge_v the_o edge_n of_o the_o ruler_n to_o the_o edge_n of_o the_o circle_n and_o to_o the_o prick_n and_o so_o draw_v a_o right_a line_n as_o this_o example_n show_v where_o the_o circle_n be_v e_o the_z prick_v assign_v be_v a._n and_o the_o ruler_n c._n d._n by_o which_o the_o touch_n line_n be_v draw_v and_o that_o be_v a_o b_o and_z as_z this_o way_n be_v light_a to_o do_v so_o be_v it_o certain_a enough_o for_o any_o kind_n b_o of_o work_v the_o xxv_n conclusion_n when_o you_o have_v any_o piece_n of_o the_o circumference_n of_o a_o circle_n assign_v how_o you_o may_v make_v out_o the_o whole_a circle_n agre_v thereunto_o first_o seek_v out_o the_o centre_n of_o that_o arch_n according_a to_o the_o do_v trine_n of_o the_o sevententh_fw-mi conclusion_n and_o then_o set_v one_o foot_n of_o your_o compass_n in_o the_o centre_n and_o extend_v the_o other_o foot_n un_fw-fr to_o the_o edge_n of_o the_o arch_n or_o piece_n of_o the_o circumference_n it_o be_v easy_a to_o draw_v the_o whole_a circle_n example_n a_o piece_n of_o a_o old_a pillar_n be_v find_v like_o inform_v to_o this_o figure_n a.d.b._n now_o to_o know_v how_o much_o the_o compass_n of_o the_o hole_n pillar_n be_v see_v by_o this_o part_n it_o appear_v that_o it_o be_v round_o thus_o shall_v you_o do_v make_v in_o a_o table_n the_o like_a draught_n of_o that_o circumference_n by_o the_o self_n patron_n use_v it_o as_o it_o be_v a_o crooked_a ruler_n then_o make_v three_o prick_v in_o that_o arch_n line_n as_o i_o have_v make_v c._n d._n and_o e._n and_o then_o find_v out_o the_o common_a centre_n to_o they_o all_o as_o the_o xvij_o conclusion_n teach_v and_o that_o centre_n be_v here_o f_o now_o set_v one_o foot_n of_o your_o compass_n in_o f_o and_o the_o other_o in_o c._n d_o other_z in_z e_z and_o so_o make_v a_o compass_n you_o have_v your_o whole_a intent_n the_o xxvi_o conclusion_n to_o find_v the_o centre_n to_o any_o arch_n of_o a_o circle_n if_o so_o be_v it_o that_o you_o desire_v to_o find_v the_o centre_n by_o any_o other_o way_n then_o by_o those_o three_o prick_v consider_v that_o sometime_o you_o can_v not_o have_v so_o much_o space_n in_o the_o thing_n where_o the_o arch_n be_v draw_v as_o shall_v serve_v to_o make_v those_o four_o bow_n line_n then_o shall_v you_o do_v thus_o parte_fw-la that_o arch_n line_n into_o two_o part_n equal_a other_o unequal_a it_o make_v no_o force_n and_o unto_o each_o portion_n draw_v a_o cord_n other_o a_o string_n line_n and_o then_o accord_v as_o you_o do_v in_o one_o arch_n in_o the_o xvi_o conclusion_n so_o do_v in_o both_o those_o arch_n here_o that_o be_v to_o say_v divide_v the_o arch_n in_o the_o middle_n and_o also_o the_o cord_n and_o draw_v then_o a_o line_n by_o those_o two_o devision_n so_o then_o be_v you_o sure_a that_o that_o line_n go_v by_o the_o centre_n afterward_o do_v lykewaies_o with_o the_o other_o arch_n and_o his_o cord_n and_o where_o those_o two_o line_n do_v cross_a there_o be_v the_o centre_n that_o you_o seek_v for_o example_n the_o arch_n of_o the_o circle_n be_v a._n b._n c_o unto_o which_o i_o must_v seek_v a_o centre_n therefore_o first_o i_o do_v divide_v it_o into_o two_o part_n the_o one_o of_o they_o be_v a._n b_o and_o the_o other_o be_v b._n c._n then_o do_v i_o cut_v every_o arch_n in_o the_o middle_n so_o be_v e._n the_o middle_a of_o a._n b_o and_o g._n be_v the_o middle_n of_o b.c._n likewaies_n i_o take_v the_o middle_n of_o their_o cord_n which_o i_o mark_v with_o f._n and_o h_n set_v f._n by_z e_z and_o h._n by_o g._n then_o draw_v i_o a_o line_n from_o e._n to_o f_o and_o from_o g._n to_o h_n and_o they_o do_v cross_v in_o d_o wherefore_o say_v i_o that_o d._n be_v the_o centre_n that_o i_o seek_v for_o the_o xxvii_n conclusion_n to_o draw_v a_o circle_n within_o a_o triangle_n appoint_v for_o this_o conclusion_n and_o all_o other_o like_a you_o must_v understande_v that_o when_o one_o figure_n be_v name_v to_o be_v within_o a_o other_o that_o be_v not_o other_o way_n to_o be_v understande_v but_o that_o either_o every_o side_n of_o the_o inner_a figure_n do_v touch_v every_o corner_n of_o the_o other_o other_z else_o every_o
in_o that_o one_o point_n f_o and_o those_o iij._o angle_n be_v equal_a to_o the_o iij._o angle_n of_o the_o triangle_n assign_v which_o thing_n do_v plain_o appear_v in_o so_o much_o as_o they_o be_v equal_a to_o ij_o right_n angle_n as_o you_o may_v guess_v by_o the_o sixth_o theorem_a and_o the_o three_o angle_n of_o everye_o triangle_n be_v equal_a also_o to_o ij_o right_a angle_n as_o the_o two_o and_o twenty_o theorem_a do_v show_v so_o that_o because_o they_o be_v equal_a to_o one_o third_o thing_n they_o must_v needs_o be_v equal_a together_o as_o the_o common_a sentence_n say_v then_o do_v i_o draw_v a_o line_n from_o g._n to_o h_n and_o that_o line_n make_v a_o triangle_n f.g.h._n who_o angle_n be_v equal_a to_o the_o angle_n of_o the_o triangle_n appoint_v and_o this_o triangle_n be_v draw_v in_o a_o circle_n as_o the_o conclusion_n do_v will_n the_o proof_n of_o this_o conclusion_n do_v appear_v in_o the_o seventy_o and_o iiij_o theorem_a the_o xxx_n conclusion_n to_o make_v a_o triangle_n about_o a_o circle_n assign_v which_o shall_v have_v corner_n equal_a to_o the_o corner_n of_o any_o triangle_n appoint_v first_o draw_v forth_o in_o length_n the_o one_o side_n of_o the_o triangle_n assign_v so_o that_o thereby_o you_o may_v have_v ij_o utter_a angle_n unto_o which_o two_o utter_a angle_n you_o shall_v make_v ij_o other_o equal_a on_o the_o centre_n of_o the_o circle_n propose_v draw_v three_o half_a diameter_n from_o the_o circumference_n which_o shall_v enclose_v those_o ij_o angle_n they_o draw_v iij._o touch_v line_n which_o shall_v make_v ij_o right_n angle_n each_o of_o they_o with_o one_o of_o those_o semidiameter_n those_o iij._o line_n will_v make_v a_o triangle_n equal_o corner_a to_o the_o triangle_n assign_v and_o that_o triangle_n be_v draw_v about_o a_o circle_n appoint_v as_o the_o conclusion_n do_v wil._n example_n a._n b.c_n be_v the_o triangle_n assign_v and_o g._n h.k_n be_v the_o circle_n appoint_v about_o which_o i_o must_v make_v a_o triangle_n have_v equal_a angle_n to_o the_o angle_n of_o that_o triangle_n a.b.c._n first_o therefore_o i_o draw_v a.c._n which_o be_v one_o of_o the_o side_n of_o the_o triangle_n in_o length_n that_o there_o may_v appear_v two_o utter_a angle_n in_o that_o triangle_n as_o you_o see_v b._n a._n d_o and_o b._n c_o e._n then_o draw_v i_o in_o the_o circle_n appoint_v a_o semidiameter_n which_o be_v here_o h._n f_o for_o f._n be_v the_o centre_n of_o the_o circle_n g._n h.k._n then_o make_v i_o on_o that_o centre_n a_o angle_n equal_a to_o the_o utter_a angle_n b._n a._n d_o and_o that_o angle_n be_v h.f._n k._n likewaies_n on_o the_o same_o centre_n by_o draw_v a_o other_o semidiameter_n i_o make_v a_o other_o angle_n h._n f._n g_o equal_a to_o the_o second_o utter_a angle_n of_o the_o triangle_n which_o be_v b._n c._n e._n and_o thus_o have_v i_o make_v three_o semidiameter_n in_o the_o circle_n appoint_v then_o at_o the_o end_n of_o each_o semidiameter_n i_o draw_v a_o touch_n line_n which_o shall_v make_v right_a angle_n with_o the_o semidiameter_n and_o those_o three_o touch_n line_n mete_v as_o you_o see_v and_o make_v the_o triangle_n l._n m._n n_n which_o be_v the_o triangle_n that_o i_o shall_v make_v for_o it_o be_v draw_v about_o a_o circle_n assign_v and_o have_v corner_n equal_a to_o the_o corner_n of_o the_o triangle_n appoint_v for_o the_o corner_n m._n be_v equal_a to_o c._n likewaies_n l._n to_o a_o and_o n._n to_o b_o which_o thing_n you_o shall_v better_o perceive_v by_o the_o vi_o theorem_a as_o i_o will_v declare_v in_o the_o book_n of_o proof_n the_o xxxi_n conclusion_n to_o make_v a_o portion_n of_o a_o circle_n on_o any_o right_a line_n assign_v which_o shall_v contain_v a_o angle_n equal_a to_o a_o right_n line_v angle_n appoint_v the_o angle_n appoint_v may_v be_v a_o sharp_a angle_n a_o right_a angle_n other_o a_o blunt_a angle_n so_o that_o the_o work_n must_v be_v diverse_o handle_v according_a to_o the_o diversity_n of_o the_o angle_n but_o consider_v the_o hardenes_n of_o those_o several_a work_v i_o will_v omit_v they_o for_o a_o more_o meet_a time_n and_o at_o this_o time_n will_v she_o we_o you_o one_o light_a way_n which_o serve_v for_o all_o kind_n of_o angle_n and_o that_o be_v this_o when_o the_o line_n be_v propose_v and_o the_o angle_n assign_v you_o shall_v join_v that_o line_n propose_v so_o to_o the_o other_o two_o line_n contain_v the_o angle_n assign_v that_o you_o shall_v make_v a_o triangle_n of_o theym_n for_o the_o easy_a dooinge_a whereof_z you_o may_v enlarge_v or_o shorten_v as_o you_o see_v cause_n nigh_o of_o the_o two_o line_n contain_v the_o angle_n appoint_v and_o when_o you_o have_v make_v a_o triangle_n of_o those_o iij._o line_n then_o accord_v to_o the_o doctrine_n of_o the_o seven_o and_o twety_a coclusion_n make_v acircle_n about_o that_o triangle_n and_o so_o have_v you_o wrought_v the_o request_n of_o this_o conclusion_n which_o yet_o you_o may_v work_v by_o the_o twenty_o and_o eight_o conclusion_n also_o so_o that_o of_o your_o line_n appoint_v you_o make_v one_o side_n of_o the_o triangle_n be_v equal_a to_o the_o angleassign_v as_o yourself_o mai_fw-gr easy_o guess_v example_n first_o for_o example_n of_o a_o sharp_a angle_n let_v a._n stand_v &_o b.c._n shall_v be_v that_o line_n assign_v then_o do_v i_o make_v a_o triangle_n by_o add_v b._n c_o as_o a_o third_o side_n to_o those_o other_o ij_o which_o do_v include_v the_o angle_n assign_v and_o that_o triangle_n be_v d_o e._n f_o so_o that_o e._n f._n be_v the_o line_n appoint_v and_o d._n be_v the_o angle_n assign_v then_o do_v i_o draw_v a_o portion_n of_o a_o circle_n about_o that_o triangle_n from_o the_o one_o end_n of_o that_o line_n assign_v unto_o the_o other_o that_o be_v to_o say_v from_o e._n a_o long_a by_o d._n unto_o f_o which_o portion_n be_v evermore_o great_a than_o the_o half_a of_o the_o circle_n by_o reason_n that_o the_o angle_n be_v a_o sharp_a angle_n but_o if_o 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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