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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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two_o square_n make_v of_o b._n c_o and_o c._n d._n for_o as_o the_o short_a side_n be_v the_o just_a length_n of_o c._n d_o so_o the_o other_o long_a side_n be_v just_a twice_o so_o long_o as_o b._n c_o wherefore_o i_o say_v now_o accord_v to_o the_o theorem_a that_o the_o greatte_fw-mi square_n e_z be_v more_o than_o the_o other_o two_o square_n f._n and_o g_o by_o the_o quantitee_n of_o the_o long_o square_a k_o whereof_o i_o reserve_v the_o proof_n to_o a_o more_o convenient_a place_n where_o i_o will_v also_o teach_v the_o reason_n how_o to_o find_v the_o length_n of_o all_o such_o perpendicular_a line_n and_o also_o of_o the_o line_n that_o be_v draw_v between_o the_o blunt_a angle_n and_o the_o perpendicular_a line_n with_o sundry_a other_o very_o pleasant_a conclusion_n the_o xlvi_o theorem_a in_o sharp_a corner_a triangle_n the_o square_a of_o any_o side_n that_o lie_v against_o a_o sharp_a corner_n be_v lesser_a than_o the_o two_o square_n of_o the_o other_o two_o side_n by_o 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be_v by_o a._n for_o declaration_n of_o the_o figure_n the_o square_n mark_v with_o e._n be_v the_o square_a of_o b._n c_o which_o be_v the_o side_n that_o lie_v against_o the_o sharp_a angle_n the_o square_n mark_v with_o c._n be_v the_o square_a of_o a._n b_o and_o the_o square_n mark_v with_o f._n be_v the_o square_a of_o a._n c_o and_o the_o two_o long_a square_n mark_v with_o h._n k_o be_v make_v of_o the_o hole_n line_n a._n b_o and_o one_o of_o his_o portion_n a._n d._n and_o truth_n it_o be_v that_o the_o square_a e._n be_v lesser_a than_o the_o other_o two_o square_n c._n and_o f._n by_o the_o quantitee_n of_o those_o two_o long_a square_n h._n and_o k._n whereby_o you_o may_v consider_v again_o a_o other_o proportion_n of_o equalitee_n that_o be_v to_o say_v that_o the_o square_a e._n with_o the_o two_o long_a square_n h._n k_o be_v just_a equal_a to_o the_o other_o two_o square_n c._n and_o f._n and_o so_o may_v you_o make_v as_o it_o be_v a_o other_o theorem_a that_o in_o all_o sharp_a corner_a triangle_n where_o a_o perpendicular_a line_n be_v draw_v from_o one_o angle_n to_o the_o side_n that_o lie_v against_o it_o the_o square_a of_o any_o one_o side_n with_o the_o ij_o longesquares_n make_v of_o that_o hole_n line_n whereon_o the_o perpendicular_a line_n do_v light_o and_o of_o that_o portion_n of_o it_o which_o join_v to_o that_o side_n who_o square_a be_v all_o ready_a take_v those_o three_o figure_n i_o say_v be_v equal_a to_o the_o ij_o square_n of_o the_o other_o ij_o side_n of_o the_o triangle_n in_o which_o you_o must_v understand_v that_o the_o side_n on_o which_o the_o perpendiculare_a false_v be_v thrice_o use_v yet_o be_v his_o square_n but_o one_o be_v mention_v for_o twice_o he_o be_v take_v for_o one_o side_n of_o the_o two_o long_a square_n and_o as_o i_o have_v thus_o make_v as_o it_o be_v a_o other_o theorem_a out_o of_o this_o forty_o and_o six_o theorem_a so_o may_v i_o out_o of_o it_o and_o the_o other_o that_o go_v next_o before_o make_v as_o manny_n as_o will_v suffice_v for_o a_o whole_a book_n so_o that_o when_o they_o shall_v be_v apply_v to_o practice_v and_o consequent_o to_o express_v their_o benefit_n no_o man_n that_o have_v not_o well_o weigh_v their_o wonderful_a commoditee_n will_v credit_n the_o posibilitie_n of_o their_o wonderful_a use_n and_o large_a aid_n in_o knowledge_n but_o all_o this_o will_v i_o remit_v to_o a_o place_n convenient_a the_o xlvij_o theorem_a if_o ij_o point_n be_v mark_v in_o the_o circumference_n of_o a_o circle_n and_o a_o right_a line_n draw_v from_o the_o one_o to_o the_o other_o that_o line_n must_v needs_o fall_v with_o in_o the_o circle_n example_n the_o circle_n be_v a._n b.c.d_n the_o ij_o point_n be_v a._n b_o the_o right_a line_n that_o be_v drawenne_n from_o the_o one_o to_o the_o other_o be_v the_o line_n a._n b_o which_o as_o you_o see_v must_v needs_o light_v within_o the_o circle_n so_o if_o you_o put_v the_o point_n to_o be_v a._n d_o or_o d._n c_o or_o a._n c_o other_o b._n c_o or_o b._n d_o inany_n of_o these_o case_n you_o see_v that_o the_o line_n that_o be_v draw_v from_o the_o one_o prick_v to_o the_o other_o do_v evermore_o run_v within_o the_o edge_n of_o the_o circle_n else_o can_v it_o be_v no_o right_a line_n howbeit_o that_o a_o crooked_a line_n especial_o be_v more_o crooked_a than_o the_o portion_n of_o the_o circumference_n may_v be_v draw_v from_o point_n to_o point_n without_o the_o circle_n but_o the_o theorem_a speak_v only_o of_o right_a line_n and_o not_o of_o crooked_a line_n the_o xlviij_o theorem_a if_o a_o right_a line_n pass_v by_o the_o centre_n of_o a_o circle_n do_v cross_v a_o other_o right_a line_n within_o the_o same_o circle_n pass_v beside_o the_o centre_n if_o be_v divide_v the_o say_a line_n into_o two_o equal_a part_n then_o do_v they_o make_v all_o their_o angle_n right_a and_o contrary_a way_n if_o they_o make_v all_o their_o angle_n right_a then_o do_v the_o long_a line_n cut_v the_o short_a in_o two_o part_n example_n the_o circle_n be_v a._n b._n c._n d_o the_o line_n that_o pass_v by_o the_o centre_n be_v a._n e._n c_o the_o line_n that_o go_v beside_o the_o centre_n be_v d._n b._n now_o say_v i_o that_o the_o line_n a._n e._n c_o do_v cut_v that_o other_o line_n d._n b._n into_o two_o just_a part_n and_o therefore_o all_o their_o four_o angle_n be_v right_a angle_n and_o contrary_a way_n because_o all_o their_o angle_n be_v right_a angle_n therefore_o it_o must_v be_v true_a that_o the_o great_a cut_v the_o lesser_a into_o two_o equal_a part_n acordinge_v as_o the_o theorem_a will_v the_o xlix_o theorem_a if_o two_o right_a line_n draw_v in_o a_o circle_n do_v cross_v one_o a_o other_o and_o do_v not_o pass_v by_o the_o centre_n every_o of_o they_o do_v not_o divide_v the_o other_o into_o two_o equal_a partion_n example_n the_o circle_n be_v a._n b._n c._n d_o and_o the_o centre_n be_v e_o the_o one_o line_n a._n c_o and_o the_o other_o be_v b._n d_o which_o two_o line_n cross_v one_o a_o other_o but_o yet_o they_o go_v not_o by_o the_o centre_n wherefore_o accord_v to_o the_o word_n of_o the_o theorem_a each_o of_o they_o do_v cut_v the_o other_o into_o equal_a portion_n for_o as_o you_o may_v easy_o judge_v a_o c._n have_v one_o portion_n long_o and_o a_o other_o short_a and_o so_o like_o wise_a b._n d._n howbeit_o it_o be_v not_o so_o to_o be_v understand_v but_o one_o of_o they_o may_v be_v divide_v into_o ij_o even_o part_n but_o both_o to_o be_v cut_v equal_o in_o the_o middle_n be_v not_o possible_a unless_o both_o pass_n through_o the_o centre_n therefore_o much_o rather_o when_o both_o go_v beside_o the_o centre_n it_o can_v not_o be_v that_o each_o of_o theym_n shall_v be_v just_o part_v into_o ij_o even_o part_n the_o l._n theorem_a if_o two_o circle_n cross_a and_o cut_v one_o a_o other_o then_o have_v not_o they_o both_o one_o centre_n example_n this_o theorem_a seem_v of_o itself_o so_o manifest_a that_o it_o nead_v neither_o demonstration_n neither_o declaration_n yet_o for_o the_o plain_a understanding_n of_o it_o i_o have_v set_v forth_o a_o figure_n here_o where_o ij_o circle_n be_v draw_v so_o that_o one_o of_o they_o do_v cross_v the_o other_o as_o you_o see_v in_o the_o point_n b._n and_o g_o and_o their_o centre_n appear_v at_o the_o first_o sight_n to_o be_v diverse_a for_o the_o centre_n of_o the_o one_o be_v f_o and_o the_o centre_n of_o the_o other_o be_v e_o which_o diffre_v as_o far_o a_o sondre_fw-fr as_o the_o edge_n of_o the_o circle_n where_o they_o
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