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〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d that_o be_v alexander_n unto_o aristotle_n send_v greeting_n you_o have_v not_o do_v well_o to_o put_v forth_o those_o book_n of_o secret_a philosophy_n entitle_v 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d for_o wherein_o shall_v we_o excel_v other_o if_o that_o knowledge_n that_o we_o have_v study_v shall_v be_v make_v come_v to_o all_o other_o man_n namely_o sith_o our_o desire_n be_v to_o excelle_n other_o man_n in_o experience_n and_o knowledge_n rather_o than_o in_o power_n and_o strength_n farewell_n by_o which_o letter_n it_o appear_v that_o he_o esteem_v learning_n and_o knowledge_n above_o power_n of_o man_n and_o the_o like_a judgement_n do_v he_o utter_v when_o he_o behold_v the_o state_n of_o diogenes_n cinicus_n adiudginge_v it_o the_o best_a state_n next_o to_o his_o own_o so_o that_o he_o say_v if_o i_o be_v not_o alexander_n i_o will_v wish_v to_o be_v diogenes_n whereby_o appear_v how_o he_o esteem_v learning_n and_o what_o felicity_n he_o 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length_n of_o the_o short_a line_n that_o enclose_v the_o angle_n then_o set_v one_o foot_n of_o the_o compass_n in_o the_o very_a point_n of_o the_o angle_n and_o with_o the_o other_o foot_n draw_v a_o compass_a arch_n from_o the_o one_o line_n of_o the_o angle_n to_o the_o other_o that_o arch_n shall_v you_o divide_v in_o half_a and_o they_o draw_v a_o line_n from_o the_o angle_n to_o the_o middle_n of_o that_o arch_n and_o so_o the_o angle_n be_v divide_v into_o ij_o equal_a part_n example_n let_v the_o triangle_n be_v a._n b.c_n they_o set_v i_o one_o foot_n of_o
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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 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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
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 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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
they_o which_o be_v do_v to_o this_o intent_n that_o if_o any_o man_n list_v to_o learn_v the_o demonstration_n by_o heart_n as_o some_o learned_a man_n have_v judge_v best_a to_o do_v those_o same_o man_n shall_v find_v the_o linearye_a exaumple_n to_o serve_v for_o this_o purpose_n and_o to_o want_v no_o thing_n needful_a to_o the_o just_a proof_n whereby_o this_o book_n may_v be_v well_o approve_v to_o be_v more_o complete_a than_o many_o man_n will_v suppose_v it_o and_o thus_o for_o this_o time_n i_o will_v make_v a_o end_n without_o any_o large_a declaration_n of_o the_o commoditee_n of_o this_o art_n or_o any_o far_o answer_v to_o that_o may_v be_v object_v against_o my_o handle_v of_o it_o willing_a they_o that_o myslike_v it_o not_o to_o meddle_v with_o it_o and_o unto_o those_o that_o will_v not_o disdain_v the_o study_n of_o it_o i_o promise_v all_o such_o aid_n as_o i_o shall_v be_v able_a to_o show_v for_o their_o far_a proceding_n both_o in_o the_o same_o and_o in_o all_o 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make_v and_o use_n of_o a_o instrument_n whereby_o you_o may_v not_o only_o measure_v the_o distance_n at_o one_o of_o all_o place_n that_o you_o can_v see_v together_o how_o much_o each_o one_o be_v from_o you_o and_o every_o one_o from_o other_o but_o also_o thereby_o to_o draw_v the_o plot_n of_o any_o countreie_n that_o you_o shall_v come_v in_o as_o just_o as_o may_v be_v by_o man_n diligence_n and_o labour_n the_o use_n both_o of_o the_o globe_n and_o the_o sphere_n and_o therein_o also_o of_o the_o art_n of_o navigation_n and_o what_o instrument_n serve_v best_a thereunto_o and_o of_o the_o true_a latitude_n and_o longitude_n of_o region_n and_o town_n euclides_n work_v in_o four_o part_n with_o diverse_a demonstration_n arithmetical_a and_o geometrical_a or_o linearie_a the_o first_o part_n of_o platte_a form_n the_o second_o of_o numbre_n and_o quantitee_n surde_fw-la or_o irrational_a the_o three_o of_o body_n and_o solid_a form_n the_o fourthe_n of_o perspective_n and_o other_o thing_n thereto_o annex_v beside_o these_o i_o have_v other_o sundry_a work_v partly_o end_v and_o partly_o to_o be_v end_v of_o the_o peregrination_n of_o man_n and_o the_o original_n of_o all_o nation_n the_o state_n of_o time_n and_o mutation_n of_o reasme_n the_o image_n of_o a_o perfect_a common_a wealth_n with_o diverse_a other_o work_v in_o natural_a science_n of_o the_o wonderful_a work_n and_o effect_n in_o beast_n plant_n and_o mineral_n of_o which_o at_o this_o time_n i_o will_v omit_v the_o argument_n because_o they_o do_v appertain_v little_a to_o this_o art_n and_o handle_v other_o matter_n in_o a_o other_o sort_n to_o have_v or_o leave_v now_o may_v you_o choose_v no_o pain_n to_o please_v will_v i_o refuse_v the_o theorem_n of_o geometry_n before_o which_o be_v set_v forth_o certain_a grauntable_a request_n which_o serve_v for_o demonstration_n mathematical_a that_o from_o any_o prick_n to_o one_o other_o there_o may_v be_v draw_v a_o right_a line_n as_o for_o example_n a_o be_v the_o one_o prick_n and_o b._n the_o other_o you_o may_v draw_v between_o they_o from_o the_o one_o to_o the_o other_o that_o be_v to_o say_v from_o a._n unto_fw-la b_o and_o from_o b._n to_o a._n that_o any_o right_a line_n of_o measurable_a length_n may_v be_v draw_v forth_o long_o and_o straight_o example_n of_o a._n b_o which_o as_o it_o be_v a_o line_n of_o measurable_a length_n so_o may_v it_o be_v draw_v forth_o far_o as_o for_o exaumple_v unto_o c_o and_o that_o in_o true_a streightenes_n without_o crokinge_v that_o upon_o any_o centre_n there_o may_v be_v make_v a_o circle_n of_o any_o quamtitee_n that_o a_o man_n will_v let_v the_o centre_n be_v set_v to_o be_v a_o what_o shall_v hinder_v a_o man_n to_o draw_v a_o circle_n about_o it_o of_o what_o quantitee_n that_o he_o lu_v as_o you_o see_v the_o form_n here_o other_o big_a or_o less_o as_o it_o shall_v like_o he_o to_o do_v that_o all_o right_a angle_n be_v equal_a each_o to_o other_o set_a for_o a_o example_n a._n and_o b_o of_z which_z two_z though_o a._n seem_v the_o great_a angle_n to_o some_o man_n of_o small_a experience_n it_o happen_v only_o because_o that_o the_o line_n about_o a_o be_v long_o they_o the_o line_n about_o b_o as_o you_o may_v prove_v by_o draw_v they_o long_o for_o so_o shall_v b._n seem_v the_o great_a angle_n â–ª_o if_o you_o make_v his_o line_n long_o than_o the_o line_n that_o make_v the_o angle_n a._n and_o to_o prove_v it_o by_o demonstration_n i_o say_v thus_o if_o any_o ij_o right_n corner_n be_v not_o equal_a than_o one_o right_a corner_n be_v great_a than_o a_o other_o but_o that_o corner_n which_o be_v great_a than_o a_o right_a angle_n be_v a_o blunt_a corner_n by_o his_o definition_n so_o must_v one_o corner_n be_v both_o a_o right_a corner_n and_o a_o blunt_a corner_n also_o which_o be_v not_o possible_a and_o again_o the_o lesser_a right_a corner_n must_v be_v a_o sharp_a corner_n by_o his_o definition_n because_o it_o be_v less_o than_o a_o right_a angle_n which_o thing_n be_v impossible_a therefore_o i_o conclude_v that_o all_o right_a angle_n be_v equal_a if_o one_o right_a line_n do_v cross_v two_o other_o right_a line_n and_o make_v ij_o inner_a corner_n of_o one_o side_n lesser_o they_o ij_o right_a corner_n it_o be_v certain_a that_o if_o those_o two_o line_n be_v draw_v forth_o right_o on_o that_o side_n that_o the_o sharp_a inner_a corner_n be_v they_o will_v at_o length_n mete_v together_o and_o cross_v on_o a_o other_o the_o ij_o line_n be_v as_o a._n b._n and_o c._n d_o and_o the_o three_o line_n cross_v they_o as_o do_v here_o e._n f_o make_v ij_o inner_a corn_n as_o be_v g.h._n lesser_o than_o two_o right_a corner_n fithech_v of_o they_o be_v less_o than_o a_o right_a corner_n as_o your_o eye_n may_v judge_v then_o say_v i_o if_o those_o ij_o line_n a.b._n and_o c.d._n be_v draw_v in_o length_n on_o that_o side_n that_o g._n and_o h._n be_v the_o will_n at_o length_n meet_v and_o cross_a one_o a_o other_o two_o right_a line_n make_v no_o platte_a form_n a_o platte_a form_n as_o you_o hear_v before_o have_v both_o length_n and_o hredthe_n and_o be_v enclose_v with_o line_n as_o with_o his_o bound_n but_o ij_o right_n line_n can_v enclose_v all_o the_o bond_n of_o any_o platte_a form_n take_v for_o a_o example_n first_o these_o two_o right_a line_n a_o b._n and_o a._n c_o which_o meet_v together_o in_o a_o but_o yet_o can_v be_v call_v a_o platte_a form_n because_o there_o be_v no_o bond_n from_o b._n to_o c_o but_o if_o you_o will_v draw_v a_o line_n between_o they_o two_o that_o be_v from_o b._n to_n c_o then_o will_v it_o be_v a_o platte_a form_n that_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
if_o a_o right_a line_n be_v divide_v as_o chance_n happen_v the_o iiij_o long_a square_n that_o may_v be_v make_v of_o that_o whole_a line_n and_o one_o of_o his_o part_n with_o the_o square_n of_o the_o other_o part_n shall_v be_v equal_a to_o the_o square_a that_o be_v make_v of_o the_o whole_a line_n and_o the_o say_v first_o portion_n join_v to_o he_o in_o length_n as_o one_o whole_a line_n example_n the_o first_o line_n be_v a._n b_o and_o be_v divide_v by_o c._n into_o two_o unequal_a part_n as_o happen_v the_o longsquare_n of_o it_o and_o his_o lesser_a portion_n a._n c_o be_v four_o time_n draw_v the_o first_o be_v e._n g.m._n k_o the_o second_o be_v k._n m._n q.o_o the_o three_o be_v h._n k._n r._n s_o and_o the_o fourthe_n be_v k.l.s._n t._n and_o where_o as_o it_o appear_v that_o one_o of_o the_o little_a square_n i_o mean_v k._n l.p_n o_o be_v reckon_v twice_o once_o as_o par_fw-fr cell_n of_o the_o second_o longsquare_n and_o again_o as_o part_n of_o the_o third_o longsquare_n to_o avoid_v ambiguite_o you_o may_v place_v one_o instead_o of_o it_o a_o other_o square_n of_o equalitee_n with_o it_o that_o be_v to_o say_v d._n e.k.h_o which_o be_v at_o no_o time_n accompethy_o as_o parcel_n of_o any_o one_o of_o they_o and_o then_o have_v you_o iiij_o long_a square_n distinct_o make_v of_o the_o whole_a line_n a._n b_o and_o his_o lesser_a portion_n a.c._n and_o within_o they_o be_v there_o a_o great_a full_a square_a p.q._n t._n v._n which_o be_v the_o just_a square_n of_o b._n c_o be_v the_o great_a portion_n of_o the_o line_n a.b._n and_o that_o those_o five_o square_n do_v make_v just_a as_o much_o as_o the_o whole_a square_n of_o that_o long_a line_n d._n g_o which_o be_v as_o long_o as_o a._n b_o and_o a.c._n join_v together_o it_o may_v be_v judge_v easy_o by_o the_o eye_n since_o that_o one_o great_a square_n do_v comprehend_v in_o it_o all_o the_o other_o five_o square_n that_o be_v to_o say_v four_o longsquares_n as_o be_v before_o mention_v and_o one_o full_a square_n which_o be_v the_o intent_n of_o the_o theorem_a the_o xliij_o theorem_a if_o a_o right_a line_n be_v divide_v into_o ij_o equal_a part_n first_o and_o one_o of_o those_o part_n again_o into_o other_o ij_o part_n as_o chance_n hapen_v the_o square_a that_o be_v make_v of_o the_o last_o part_n of_o the_o line_n so_o divide_v and_o the_o square_a of_o the_o residue_n of_o that_o whole_a line_n be_v double_a to_o the_o square_n of_o half_a that_o line_n and_o to_o the_o square_n of_o the_o middle_a portion_n of_o the_o same_o line_n example_n the_o line_n to_o be_v divide_v be_v a._n b_o and_o be_v part_v in_o c._n into_o two_o equal_a part_n and_o then_o c._n b_o be_v divide_v again_o into_o two_o part_n in_o d_o so_o that_o the_o meaning_n of_o the_o theorem_a be_v that_o the_o square_a of_o d._n b._n which_o be_v the_o latter_a part_n of_o the_o line_n and_o the_o square_a of_o a._n d_o which_o be_v the_o residue_n of_o the_o whole_a line_n those_o two_o square_n i_o say_v be_v double_a to_o the_o square_n of_o the_o one_o half_a of_o the_o line_n and_o to_o the_o square_n of_o c._n d_o which_o be_v the_o middle_a portion_n of_o those_o three_o division_n which_o thing_n that_o you_o may_v more_o easy_o perceive_v i_o have_v draw_v four_o square_n whereof_o the_o great_a be_v mark_v with_o f._n be_v the_o square_a of_o a.d._n the_o next_o which_o be_v mark_v with_o g_o be_v the_o square_a of_o half_a the_o line_n that_o be_v of_o a._n c_o and_o the_o other_o two_o little_a square_n mark_v with_o f._n and_o h_n be_v both_o of_o one_o bigness_n by_o reason_n that_o i_o do_v divide_v c.b._n into_o two_o equal_a part_n so_o that_o you_o may_v take_v the_o square_a f_o for_o the_o square_n of_o d._n b_o and_o the_o square_a h_n for_o the_o square_n of_o c.d._n now_o i_o think_v you_o doubt_v not_o but_o that_o the_o square_a e._n and_o the_o square_a f_o be_v double_a so_o much_o as_o the_o square_a g._n and_o the_o square_a h_n which_o thing_n the_o eayser_n be_v to_o be_v understande_v because_o that_o the_o great_a square_n have_v in_o his_o side_n iij._o quarter_n of_o the_o first_o line_n which_o multiply_v by_o itself_o make_v nine_o quarter_n and_o the_o square_a f._n contain_v but_o one_o quarter_n so_o that_o both_o do_v make_v ten_o quarter_n then_o g._n contain_v iiij_o quarter_n sing_v his_o side_n contain_v two_o and_o h._n contain_v but_o one_o quarter_n which_o both_o make_v but_o five_o quarter_n and_o that_o be_v but_o half_a of_o ten_o whereby_o you_o may_v easylye_o conjecture_v that_o the_o meaning_a of_o the_o theorem_a be_v verify_v in_o the_o figure_n of_o this_o example_n the_o xliiij_o theorem_a if_o a_o right_a line_n be_v divide_v into_o ij_o part_n equal_o and_o a_o other_o portion_n of_o a_o right_a line_n annex_v to_o that_o first_o line_n the_o square_a of_o this_o whole_a line_n so_o compound_v and_o the_o square_a of_o the_o portion_n that_o be_v annex_v be_v double_a as_o much_o as_o the_o square_n of_o the_o half_a of_o the_o first_o line_n and_o the_o square_a of_o the_o other_o half_o join_v in_o one_o with_o the_o annex_v portion_n as_o one_o whole_a line_n example_n the_o line_n be_v a._n b_o and_o be_v divide_v first_o tnto_n two_o equal_a part_n in_o c_o and_o they_o be_v there_o annex_v to_o it_o a_o other_o portion_n which_o be_v b.d._n now_o say_v the_o theorem_a that_o the_o square_n of_o a._n d_o and_o the_o square_a of_o b._n d_o be_v double_a to_o the_o square_n of_o a._n c_o and_o to_o the_o square_n of_o c.d._n the_o line_n a.b._n contain_v four_o part_n then_o must_v needs_o his_o half_n contain_v ij_o part_n of_o such_o part_n i_o suppose_v b.d._n which_o be_v the_o amnex_v line_n to_o contain_v three_o so_o shall_v the_o hole_n line_n comprehend_v seven_o part_n and_o his_o square_a xlix_o part_n whereunto_o if_o you_o ad_fw-la the_o square_v of_o the_o annex_v line_n which_o make_v nine_o than_o those_o both_o do_v yield_v lviij_o which_o must_v be_v double_a to_o the_o square_n of_o the_o half_a line_n with_o the_o annex_v portion_n the_o half_a line_n by_o itself_o contain_v but_o two_o part_n and_o therefore_o his_o square_n do_v make_v four_o the_o half_a line_n with_o the_o annex_v portion_n contain_v five_o and_o the_o square_a of_o it_o be_v twenty-five_o now_o put_v four_o to_o twenty-five_o and_o it_o make_v just_a xxix_o the_o even_o half_a of_o fifty_o and_o eight_o whereby_o appear_v the_o truth_n of_o the_o theorem_a the_o xlu_o theorem_a in_o all_o triangle_n that_o have_v a_o blunt_a angle_n the_o square_a of_o the_o side_n that_o lie_v against_o the_o blunt_a angle_n be_v great_a than_o the_o two_o square_n of_o the_o other_o two_o side_n by_o twice_o as_o much_o as_o be_v comprehend_v of_o the_o one_o of_o those_o two_o side_n enclose_n the_o blunt_a corner_n and_o that_o portion_n of_o the_o same_o line_n be_v draw_v forth_o in_o length_n which_o lie_v between_o the_o say_a blunt_a corner_n and_o a_o perpendicular_a line_n light_a on_o it_o and_o draw_v from_o one_o of_o the_o sharp_a angle_n of_o the_o foresay_a triangle_n example_n for_o the_o declaration_n of_o this_o theorem_a and_o the_o next_o also_o who_o use_n be_v wonderful_a in_o the_o practice_n of_o geometry_n and_o in_o measuring_a especial_o it_o shall_v be_v needful_a to_o declare_v that_o every_o triangle_n that_o have_v no_o right_a angle_n as_o those_o be_v which_o be_v call_v as_o in_o the_o book_n of_o practice_n be_v declare_v sharp_a corner_a triangle_n and_o blunt_a cover_a triangle_n yet_o may_v they_o be_v bring_v to_o have_v a_o right_a angle_n either_o by_o part_v they_o into_o two_o lesser_a triangle_n or_o else_o by_o add_v a_o other_o triangle_n unto_o they_o which_o may_v be_v a_o great_a help_n for_o the_o aid_n of_o measuring_a as_o more_o large_o shall_v be_v set_v forth_o in_o the_o book_n of_o measuring_a but_o for_o this_o present_a place_n this_o form_n will_v i_o use_v which_o theon_n also_o use_v to_o add_v one_o triangle_n unto_o a_o other_o to_o bring_v the_o blunt_a corner_a triangle_n into_o a_o right_a angle_a triangle_n whereby_o the_o proportion_n of_o the_o square_n of_o the_o side_n in_o such_o a_o blunt_a corner_a triangle_n may_v the_o better_o be_v know_v the_o square_a of_o the_o line_n a._n b_o be_v the_o great_a square_n mark_v with_o e._n the_o square_a of_o a._n c_o be_v the_o mean_a square_n mark_v with_o f._n the_o square_a of_o b._n c_o be_v the_o least_o square_n mark_v with_o g._n and_o the_o long_a square_n mark_v with_o k_o be_v set_v in_o steed_n of_o
circle_n they_o must_v it_o need_v be_v other_o a_o blunt_a angle_n or_o else_o a_o sharp_a angle_n and_o in_o no_o wise_a a_o right_a angle_n for_o if_o the_o cantle_n wherein_o the_o angle_n be_v make_v be_v great_a than_o the_o half_a circle_n then_o be_v that_o angle_n a_o sharp_a angle_n and_o general_o the_o great_a the_o cantle_n be_v the_o lesser_a be_v the_o angle_n comprise_v in_o that_o cantle_n and_o contrary_a way_n the_o lesser_a any_o cantle_n be_v the_o great_a be_v the_o angle_n that_o be_v make_v in_o it_o wherefore_o it_o must_v needs_o follow_v that_o the_o angle_n make_v in_o a_o cantle_n less_o than_o a_o semicircle_n must_v needs_o be_v great_a than_o a_o right_a angle_n so_o the_o angle_n by_o b_o be_v make_v of_o the_o right_a line_n a._n b_o and_o the_o right_a line_n b._n d_o be_v a_o just_a right_a angle_n because_o it_o be_v make_v in_o a_o semicircle_n but_o the_o angle_n make_v by_o a_o which_o be_v make_v of_o the_o right_a line_n a._n b_o and_o of_o the_o right_a line_n a._n d_o be_v lesser_a than_o a_o right_a angle_n and_o be_v name_v a_o sharp_a angle_n for_o as_o much_o as_o it_o be_v make_v in_o a_o cantle_n of_o a_o circle_n great_a than_o a_o semicircle_n and_o contrary_a way_n the_o angle_n by_o c_o be_v make_v of_o the_o right_a line_n b._n c_o and_o of_o the_o right_a line_n c._n d_o be_v great_a than_o a_o right_a angle_n and_o be_v name_v a_o blunt_a angle_n because_o it_o be_v make_v in_o a_o cantle_n of_o a_o circle_n lesser_a than_o a_o semicircle_n but_o now_o touch_v the_o other_o angle_n of_o the_o cantle_n i_o say_v accord_v to_o the_o theorem_a that_o the_o two_o angle_n of_o the_o great_a cantle_n which_o be_v by_o b._n and_o d_o as_z be_v before_o declare_v be_v great_a each_o of_o they_o then_o a_o right_a angle_n and_o the_o angle_n of_o the_o lesser_a cantle_n which_o be_v by_o the_o same_o letter_n b_o and_z d_z but_o be_v on_o the_o other_o side_n of_o the_o cord_n be_v lesser_a each_o of_o they_o then_o a_o right_a angle_n and_o be_v therefore_o sharp_a corner_n the_o lxxiiij_o theorem_a if_o a_o right_a line_n do_v touch_v a_o circle_n and_o from_o the_o point_n where_o they_o touch_v a_o right_a line_n be_v draw_v across_o the_o circle_n and_o divide_v it_o the_o angle_n that_o the_o say_v line_n do_v make_v with_o the_o touch_n line_n be_v equal_a to_o the_o angle_n which_o be_v make_v in_o the_o cantle_n of_o the_o same_o circle_n on_o the_o contrary_a side_n of_o the_o line_n aforesaid_a example_n the_o circle_n be_v a._n b.c.d_n and_o the_o touch_n line_n be_v e._n f._n the_o point_n of_o the_o touch_v be_v d_o from_o which_o point_n i_o suppose_v the_o line_n d._n b_o to_o be_v draw_v across_o the_o circle_n and_o to_o divide_v it_o into_o two_o cantle_n whereof_o the_o great_a be_v b._n a.d_n and_o the_o lesser_a be_v b._n c.d_v and_o in_o each_o of_o they_o a_o angle_n draw_v for_o in_o the_o great_a cantle_n the_o angle_n be_v by_o a_o and_o be_v make_v of_o the_o right_a 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line_n b.d._n the_o proof_n of_o all_o these_o i_o do_v reserve_v as_o i_o have_v often_o say_v to_o a_o convenient_a book_n wherein_o they_o shall_v be_v all_o set_v at_o large_a the_o lxxu_o theorem_a in_o any_o circle_n when_o two_o right_a line_n do_v cross_a one_o a_o other_o the_o likeiamme_n that_o be_v make_v of_o the_o portion_n of_o the_o one_o line_n shall_v be_v equal_a to_o the_o lykeiamme_n make_v of_o the_o part_n of_o the_o other_o line_n because_o this_o theorem_a do_v serve_v to_o many_o use_n and_o will_v be_v well_o understande_v i_o have_v set_v forth_o two_o example_n of_o it_o in_o the_o first_o the_o line_n by_o their_o cross_n do_v make_v their_o portion_n somewhat_o towards_o a_o equality_n in_o the_o second_o the_o portion_n of_o the_o line_n be_v very_o far_o from_o a_o equality_n and_o yet_o in_o both_o these_o and_o in_o all_o other_o the_o theorem_a be_v true_a in_o the_o first_o example_n the_o circle_n be_v a._n b.c.d_n in_o which_o the_o one_o line_n 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that_o you_o may_v somewhat_o whet_v your_o own_o wit_n in_o practise_v this_o theorem_a accord_v to_o the_o doctrine_n of_o the_o nineteenth_o conclusion_n the_o lxxvi_o theorem_a if_o a_o point_n be_v mark_v without_o a_o circle_n and_o from_o that_o point_n two_o right_a line_n draw_v to_o the_o circle_n so_o that_o the_o one_o of_o they_o do_v run_v cross_v the_o circle_n and_o the_o other_o do_v touch_v the_o circle_n only_o the_o long_a square_n that_o be_v make_v of_o that_o whole_a line_n which_o cross_v the_o circle_n and_o the_o portion_n of_o it_o that_o lie_v between_o the_o utter_a circumference_n of_o the_o circle_n and_o the_o point_n shall_v be_v equal_a to_o the_o full_a square_n of_o the_o other_o line_n that_o only_o touch_v the_o circle_n example_n the_o circle_n be_v d._n b.c_n and_o the_o point_n without_o the_o circle_n be_v a_o from_o which_o point_n there_o be_v draw_v one_o line_n across_o the_o circle_n and_o that_o be_v a._n d.c_n and_o a_o other_o line_n be_v draw_v from_o the_o say_a prick_n to_o the_o marge_n or_o edge_n of_o the_o circumference_n of_o the_o circle_n and_o do_v only_o touch_v it_o that_o be_v the_o line_n a.b._n and_o of_o that_o first_o line_n a._n d.c_n you_o may_v perceive_v one_o part_n of_o it_o which_o be_v a._n d_o to_o lie_v without_o the_o circle_n between_o the_o utter_a circumference_n of_o it_o and_o the_o point_n assign_v which_o be_v a._n no_o we_o concern_v the_o meaning_n of_o the_o theorem_a if_o you_o make_v a_o longsquare_n of_o the_o whole_a line_n a._n c_o and_o of_o that_o part_n of_o it_o that_o lie_v between_o the_o circumference_n and_o the_o point_n which_o be_v a._n d_o that_o long_o square_n shall_v be_v equal_a to_o the_o full_a square_n of_o the_o touch_n line_n a._n b_o accord_v not_o only_a as_o this_o figure_n she_o w_v but_o also_o the_o say_v nyneteenth_fw-mi conclusion_n do_v prove_v if_o you_o lyste_v to_o examyne_n the_o one_o by_o the_o other_o the_o lxxvij_o theorem_a if_o a_o point_n be_v assign_v without_o a_o circle_n and_o from_o that_o point_n two_o right_a line_n be_v draw_v to_o the_o circle_n so_o that_o the_o one_o do_v cross_v the_o circle_n and_o the_o other_o do_v end_n at_o the_o circumference_n and_o that_o the_o longsquare_n of_o the_o line_n which_o cross_v the_o circle_n make_v with_o the_o portion_n of_o the_o same_o line_n be_v without_o the_o circle_n between_o the_o utter_a circumference_n and_o the_o point_n assign_v do_v equal_o agree_v with_o the_o just_a square_n of_o that_o line_n that_o end_v at_o the_o circumference_n then_o be_v that_o line_n so_o end_v on_o the_o circumference_n a_o touch_n line_n unto_o that_o circle_n example_n in_o as_o much_o as_o this_o theorem_a be_v nothing_o else_o but_o the_o sentence_n of_o the_o last_o theorem_a before_o convert_v therefore_o it_o shall_v not_o be_v needful_a to_o use_v any_o other_o example_n than_o the_o same_o for_o as_o in_o that_o other_o theorem_a because_o the_o one_o line_n be_v a_o touch_n line_n therefore_o it_o make_v a_o square_a just_a equal_a with_o the_o longsquare_n make_v of_o that_o whole_a line_n which_o cross_v the_o circle_n and_o his_o portion_n li_v without_o the_o same_o circle_n so_o say_v this_o theorem_a that_o if_o the_o just_a square_n of_o the_o line_n that_o end_v on_o the_o circumference_n be_v equal_a to_o that_o longsquare_n which_o be_v make_v as_o for_o his_o long_a side_n of_o the_o whole_a line_n which_o come_v from_o the_o point_n assign_v and_o cross_v the_o circle_n and_o for_o his_o other_o short_a side_n be_v make_v of_o the_o portion_n of_o the_o same_o line_n li_v between_o the_o circumference_n of_o the_o circle_n and_o the_o point_n assign_v then_o be_v that_o line_n which_o end_v on_o the_o circumference_n a_o right_a touch_n line_n that_o be_v to_o say_v if_o the_o full_a square_n of_o the_o right_a line_n a._n b_o be_v equal_a to_o the_o longsquare_n make_v of_o the_o whole_a line_n a._n c_o as_o one_o of_o his_o line_n and_o of_o his_o portion_n a._n d_o as_o his_o other_o line_n then_o must_v it_o needs_o be_v that_o the_o line_n a._n b_o be_v a_o right_a touch_n line_n unto_o the_o circle_n d._n b.c._n and_o thus_o for_o this_o time_n i_o make_v a_o end_n of_o the_o theorem_n finis_fw-la imprint_v at_o london_n in_o paul_n churcheyarde_n at_o the_o sign_n of_o the_o brazen_a serpent_n by_o reynold_n wolf_n cum_fw-la privilegio_fw-la ad_fw-la imprimendum_fw-la solum_fw-la anno_fw-la domini_fw-la m.d.li_o