the_o same_o quality_n as_o every_o man_n be_v a_o live_a creature_n some_o man_n be_v a_o live_a creature_n or_o no_o man_n be_v wise_a some_o man_n be_v not_o wise._n of_o these_o iâ—_n the_o universal_a be_v true_a the_o particular_a will_v be_v true_a also_o contrary_a be_v universal_a proposition_n of_o different_a quality_n as_o every_o man_n be_v happy_a no_o man_n be_v happy_a and_o of_o these_o if_o one_o be_v true_a the_o other_o be_v false_a also_o they_o may_v both_o be_v false_a as_o in_o the_o example_n give_v subcontrary_n be_v particular_a proposition_n of_o different_a quality_n as_o some_o man_n be_v learned_a some_o man_n be_v not_o learned_a which_o can_v be_v both_o false_a but_o they_o may_v be_v both_o true_a contradictory_n be_v those_o that_o differ_v both_o in_o quantity_n and_o quality_n as_o every_o man_n be_v a_o live_a creature_n some_o man_n be_v not_o a_o live_a creature_n which_o can_v neither_o be_v both_o true_a nor_o both_o false_a 18_o a_o proposition_n be_v say_v to_o follow_v from_o two_o other_o proposition_n when_o these_o be_v grant_v to_o be_v true_a it_o can_v be_v deny_v but_o the_o other_o be_v true_a also_o for_o example_n let_v these_o two_o proposition_n every_o man_n be_v a_o live_a creature_n and_o every_o live_a creature_n be_v a_o body_n be_v suppose_v true_a that_o be_v that_o body_n be_v the_o name_n of_o every_o live_a creature_n and_o live_v creature_n the_o name_n of_o every_o man._n see_v therefore_o if_o these_o be_v understand_v to_o be_v true_a it_o can_v be_v understand_v that_o body_n be_v not_o the_o name_n of_o every_o man_n that_o be_v that_o every_o man_n be_v a_o body_n be_v false_a this_o proposition_n will_v be_v say_v to_o follow_v from_o those_o two_o or_o to_o be_v necessary_o infer_v from_o they_o 19_o that_o a_o true_a proposition_n may_v follow_v from_o false_a proposition_n may_v happen_v sometime_o but_o false_a from_o true_a never_o for_o if_o these_o every_o man_n be_v a_o stone_n and_o every_o stone_n be_v a_o live_a creature_n which_o be_v both_o false_a be_v grant_v to_o be_v true_a it_o be_v grant_v also_o that_o live_a creature_n be_v the_o name_n of_o every_o stone_n and_o stone_n of_o every_o man_n that_o be_v that_o live_v creature_n be_v the_o name_n of_o every_o man_n that_o be_v to_o say_v this_o proposition_n every_o man_n be_v a_o live_a creature_n be_v true_a as_o it_o be_v indeed_o true_a wherefore_o a_o true_a proposition_n may_v sometime_o follow_v from_o false_a but_o if_o any_o two_o proposition_n be_v true_a a_o false_a one_o can_v never_o follow_v from_o they_o for_o if_o true_a follow_v from_o false_a for_o this_o reason_n only_o that_o the_o false_a be_v grant_v to_o be_v true_a than_o truth_n from_o two_o truth_n grant_v will_v follow_v in_o the_o same_o manner_n 20_o now_o see_v none_o but_o a_o true_a proposition_n will_v follow_v from_o true_a and_o that_o the_o understanding_n of_o two_o proposition_n to_o be_v true_a be_v the_o cause_n of_o understanding_n that_o also_o to_o be_v true_a which_o be_v deduce_v from_o they_o the_o two_o antecedent_n proposition_n be_v common_o call_v the_o cause_n of_o the_o infer_v proposition_n or_o conclusion_n and_o from_o hence_o it_o be_v that_o logician_n say_v the_o premise_n be_v cause_n of_o the_o conclusion_n which_o may_v pass_v though_o it_o be_v not_o proper_o speak_v for_o though_o understanding_n be_v the_o cause_n of_o understanding_n yet_o speech_n be_v not_o the_o cause_n of_o speech_n but_o when_o they_o say_v the_o cause_n of_o the_o property_n of_o any_o thing_n be_v the_o thing_n itself_o they_o speak_v absurd_o eor_fw-la example_n if_o a_o figure_n be_v propound_v which_o be_v triangular_a see_v every_o triangle_n have_v all_o its_o angle_n together_o equal_a to_o two_o right_a angle_n from_o whence_o it_o follow_v that_o all_o the_o angle_n of_o that_o figure_n be_v equal_a to_o two_o right_a angle_n they_o say_v for_o this_o reason_n that_o that_o figure_n be_v the_o cause_n of_o that_o equality_n but_o see_v the_o figure_n do_v not_o itself_o make_v its_o angle_n and_o therefore_o can_v be_v say_v to_o be_v the_o efficient-cause_n they_o call_v it_o the_o formall-cause_n whereas_o in_o deed_n it_o be_v no_o cause_n at_o all_o nor_o do_v the_o property_n of_o any_o figure_n follow_v the_o figure_n but_o have_v its_o be_v at_o the_o same_o time_n with_o it_o only_o the_o knowledge_n of_o the_o figure_n go_v before_o the_o knowledge_n of_o the_o property_n and_o one_o knowledge_n be_v true_o the_o cause_n of_o another_o knowledge_n namely_o the_o efficient-cause_n and_o thus_o much_o concern_v proposition_n which_o in_o the_o progress_n of_o philosophy_n be_v the_o first_o step_v like_o the_o move_n towards_o of_o one_o foot_n by_o the_o due_a addition_n of_o another_o step_v i_o shall_v proceed_v to_o syllogism_n and_o make_v a_o complete_a pace_n of_o which_o in_o the_o next_o chapter_n chap._n iu._n of_o syllogism_n 1_o the_o definition_n of_o syllogism_n 2_o in_o a_o syllogism_n there_o be_v but_o three_o term_n 3_o major_n minor_n and_o middle_a term_n also_o major_a and_o minor_a proposition_n what_o they_o be_v 4_o the_o middle_a term_n in_o every_o syllogism_n ought_v to_o be_v determine_v in_o both_o the_o proposition_n to_o one_o and_o the_o same_o thing_n 5_o from_o two_o particular_a proposition_n nothing_o can_v be_v conclude_v 6_o a_o syllogism_n be_v the_o collection_n of_o two_o proposition_n into_o one_o sum_n 7_o the_o figure_n of_o a_o syllogism_n what_o it_o be_v 8_o what_o be_v in_o the_o mind_n answer_v to_o a_o syllogism_n 9_o the_o first_o indirect_a figure_n how_o it_o be_v make_v 10_o the_o second_o indirect_a figure_n how_o make_v 11_o how_o the_o three_o indirect_a figure_n be_v make_v 12_o there_o be_v many_o mood_n in_o every_o figure_n but_o most_o of_o they_o useless_a in_o philosophy_n 13_o a_o hypotheticall_a syllogism_n when_o equipollent_a to_o a_o categoricall_a 1._o a_o speech_n consist_v of_o three_o proposition_n from_o two_o of_o which_o the_o three_o follow_v be_v call_v a_o syllogism_n and_o that_o which_o follow_v be_v call_v the_o conclusion_n the_o other_o two_o premise_n for_o example_n this_o speech_n every_o man_n be_v a_o live_a creature_n every_o live_a creature_n be_v a_o body_n therefore_o every_o man_n be_v a_o body_n be_v a_o syllogism_n because_o the_o three_o proposition_n follow_v from_o the_o two_o first_o that_o be_v if_o those_o be_v grant_v to_o be_v true_a this_o must_v also_o be_v grant_v to_o be_v true_a 2_o from_o two_o proposition_n which_o have_v not_o one_o term_n common_a no_o conclusion_n can_v follow_v and_o therefore_o no_o syllogism_n can_v be_v make_v of_o they_o for_o let_v any_o two_o premise_n a_o man_n be_v a_o live_a creature_n a_o tree_n be_v a_o plant_n be_v both_o of_o they_o true_a yet_o because_o it_o can_v be_v collect_v from_o they_o that_o plant_n be_v the_o name_n of_o a_o man_n or_o man_n the_o name_n of_o a_o plant_n it_o be_v not_o necessary_a that_o this_o conclusion_n a_o man_n be_v a_o plant_n shall_v be_v true_a corollary_n therefore_o in_o the_o premise_n of_o a_o syllogism_n there_o can_v be_v but_o three_o term_n beside_o there_o can_v be_v no_o term_n in_o the_o conclusion_n which_o be_v not_o in_o the_o premise_n for_o let_v any_o two_o premise_n be_v a_o man_n be_v a_o live_a creature_n a_o live_a creature_n be_v a_o body_n yet_o if_o any_o other_o term_n be_v put_v in_o the_o conclusion_n as_o man_n be_v two_o footed_a though_o it_o be_v true_a it_o can_v follow_v from_o the_o premise_n because_o from_o they_o it_o can_v be_v collect_v that_o the_o name_n two_o footed_a belong_v to_o a_o man_n and_o therefore_o again_o in_o every_o syllogism_n there_o can_v be_v but_o three_o term_n 3_o of_o these_o term_n that_o which_o be_v the_o predicate_a in_o the_o conclusion_n be_v common_o call_v the_o major_a that_o which_o be_v the_o subject_a in_o the_o conclusion_n the_o minor_a and_o the_o other_o be_v the_o middle_a term_n as_o in_o this_o syllogism_n a_o man_n be_v a_o live_a creature_n a_o live_a creature_n be_v a_o body_n therefore_o a_o man_n be_v a_o body_n body_n be_v the_o major_n man_n the_o minor_a and_o live_a creature_n the_o middle_a term._n also_o of_o the_o premise_n that_o in_o which_o the_o major_a term_n be_v find_v be_v call_v the_o major_a proposition_n and_o that_o which_o have_v the_o minor_a term_n the_o minor_a proposition_n 4_o if_o the_o middle_a term_n be_v not_o in_o both_o the_o premise_n determine_v to_o one_o and_o the_o same_o singular_a thing_n no_o conclusion_n will_v follow_v nor_o syllogism_n be_v make_v for_o let_v the_o minor_a term_n be_v man_n the_o middle_a term_n live_v creature_n and_o the_o major_a term_n
distance_n and_o the_o magnitude_n and_o splendour_n be_v see_v in_o more_o line_n and_o distance_n than_o one_o as_o it_o be_v in_o reflection_n or_o refraction_n than_o neither_o that_o splendour_n nor_o apparent_a magnitude_n be_v in_o the_o sun_n itself_o and_o therefore_o the_o body_n of_o the_o sun_n can_v be_v the_o subject_a of_o that_o splendour_n and_o magnitude_n and_o for_o the_o same_o reason_n the_o air_n and_o other_o part_n will_v be_v reject_v till_o at_o last_o nothing_o remain_v which_o can_v be_v the_o subject_a of_o that_o splendour_n and_o magnitude_n but_o the_o sentient_n itself_o and_o this_o method_n in_o regard_n the_o subject_n be_v divide_v into_o part_n be_v analitycall_a and_o in_o regard_n the_o property_n both_o of_o the_o subject_a and_o accident_n be_v compare_v with_o the_o accident_n concern_v who_o subject_n the_o enquiry_n be_v make_v it_o be_v syntheticall_a 10_o but_o when_o we_o seek_v after_o the_o cause_n of_o any_o propound_a effect_n we_o must_v in_o the_o first_o place_n get_v into_o our_o mind_n a_o exact_a notion_n or_o idea_n of_o that_o which_o we_o call_v cause_n namely_o that_o a_o cause_n be_v the_o sum_n or_o aggregate_v of_o all_o such_o accident_n both_o in_o the_o agent_n and_o the_o patient_n as_o concur_v to_o the_o produce_v of_o the_o effect_v propound_v all_o which_o exist_v together_o it_o can_v be_v understand_v but_o that_o the_o effect_n exi_v with_o they_o or_o that_o it_o can_v possible_o exist_v if_o any_o one_o of_o they_o be_v absent_a this_o be_v know_v in_o the_o next_o place_n we_o must_v examine_v single_o every_o accident_n that_o accompany_v or_o praecede_v the_o effect_n as_o far_o forth_o as_o it_o seem_v to_o conduce_v in_o any_o manner_n to_o the_o production_n of_o the_o same_o and_o see_v whether_o the_o propound_a effect_n may_v be_v conceive_v to_o exist_v without_o the_o existence_n of_o any_o of_o those_o accident_n and_o by_o this_o mean_n separate_v such_o accident_n as_o do_v not_o concur_v from_o such_o as_o concur_v to_o produce_v the_o say_a effect_n which_o be_v do_v we_o be_v to_o put_v together_o the_o concur_v accident_n and_o consider_v whether_o we_o can_v possible_o conceive_v that_o when_o these_o be_v all_o present_a the_o effect_v propound_v will_v not_o follow_v and_o if_o it_o be_v evident_a that_o the_o effect_n will_v follow_v then_o that_o aggregate_v of_o accident_n be_v the_o entire_a cause_n otherwise_o not_o but_o we_o must_v still_o search_v out_o and_o put_v together_o other_o accident_n for_o example_n if_o the_o cause_n of_o light_n be_v propound_v to_o be_v seek_v out_o first_o we_o examine_v thing_n without_o we_o and_o find_v that_o whensoever_o light_n appear_v there_o be_v some_o principal_a object_n as_o it_o be_v the_o fountain_n of_o light_n without_o which_o we_o can_v have_v any_o perception_n of_o light_n and_o therefore_o the_o concurrence_n of_o that_o object_n be_v necessary_a to_o the_o generation_n of_o light_n next_o we_o consider_v the_o medium_fw-la and_o find_v that_o unless_o it_o be_v dispose_v in_o a_o certain_a manner_n namely_o that_o it_o be_v transparent_a though_o the_o object_n remain_v the_o same_o yet_o the_o effect_n will_v not_o follow_v and_o therefore_o the_o concurrence_n of_o transparency_n be_v also_o necessary_a to_o the_o generation_n of_o light_n three_o we_o observe_v our_o own_o body_n and_o find_v that_o by_o the_o indisposition_n of_o the_o eye_n the_o brain_n the_o nerve_n and_o the_o heart_n that_o be_v by_o obstruction_n stupidity_n and_o debility_n we_o be_v deprive_v of_o light_n so_o that_o a_o fit_a disposition_n of_o the_o organ_n to_o receive_v impression_n from_o without_o be_v likewise_o a_o necessary_a part_n of_o the_o cause_n of_o light_n again_o of_o all_o the_o accident_n inhaerent_a in_o the_o object_n there_o be_v none_o that_o can_v conduce_v to_o the_o effect_n of_o light_n but_o only_a action_n or_o a_o certain_a motion_n which_o can_v be_v conceive_v to_o be_v want_v whensoever_o the_o effect_n be_v present_a for_o that_o any_o thing_n may_v shine_v it_o be_v not_o requisite_a that_o it_o be_v of_o such_o or_o such_o magnitude_n or_o figure_n or_o that_o the_o whole_a body_n of_o it_o be_v move_v out_o of_o the_o place_n it_o be_v in_o unless_o it_o may_v perhaps_o be_v say_v that_o in_o the_o sun_n or_o other_o body_n that_o which_o cause_v light_n be_v the_o light_n it_o have_v in_o itself_o which_o yet_o be_v but_o a_o trifle_a exception_n see_v nothing_o be_v mean_v thereby_o but_o the_o cause_n of_o light_n as_o if_o any_o man_n shall_v say_v that_o the_o cause_n of_o light_n be_v that_o in_o the_o sun_n which_o produce_v it_o it_o remain_v therefore_o that_o the_o action_n by_o which_o light_n be_v generate_v be_v motion_n only_o in_o the_o part_n of_o the_o object_n which_o be_v understand_v we_o may_v easy_o conceive_v what_o it_o be_v the_o medium_fw-la contribute_v namely_o the_o continuation_n of_o that_o motion_n to_o the_o eye_n and_o last_o what_o the_o eye_n and_o the_o rest_n of_o the_o organ_n of_o the_o sentient_n contribute_v namely_o the_o continuation_n of_o the_o same_o motion_n to_o the_o last_o organ_n of_o sense_n the_o heart_n and_o in_o this_o manner_n the_o cause_n of_o light_n may_v be_v make_v up_o of_o motion_n continue_v from_o the_o original_a of_o the_o same_o motion_n to_o the_o original_a of_o vital_a motion_n light_n be_v nothing_o but_o the_o alteration_n of_o vital_a motion_n make_v by_o the_o impression_n upon_o it_o of_o motion_n continue_v from_o the_o object_n but_o i_o give_v this_o only_a for_o a_o example_n for_o i_o shall_v speak_v more_o at_o large_a of_o light_n and_o the_o generation_n of_o it_o in_o its_o proper_a place_n in_o the_o mean_a time_n it_o be_v manifest_a that_o in_o the_o search_n out_o of_o cause_n there_o be_v need_v partly_o of_o the_o analyticall_a and_o partly_o of_o the_o syntheticall_a method_n of_o the_o analyticall_a to_o conceive_v how_o circumstance_n conduce_v several_o to_o the_o production_n of_o effect_n and_o of_o the_o syntheticall_a for_o the_o add_v together_o and_o compound_v of_o what_o they_o can_v effect_v single_o by_o themselves_o and_o thus_o much_o may_v serve_v for_o the_o method_n of_o invention_n it_o remain_v that_o i_o speak_v of_o the_o method_n of_o teach_v that_o be_v of_o demonstration_n and_o of_o the_o mean_n by_o which_o we_o demonstrate_v 11_o in_o the_o method_n of_o invention_n the_o use_n of_o word_n consist_v in_o this_o that_o they_o may_v serve_v for_o mark_n by_o which_o whatsoever_o we_o have_v find_v out_o may_v be_v recall_v to_o memory_n for_o without_o this_o all_o our_o invention_n perish_v nor_o will_v it_o be_v possible_a for_o we_o to_o go_v on_o from_o principle_n beyond_o a_o syllogism_n or_o two_o by_o reason_n of_o the_o weakness_n of_o memory_n for_o example_n if_o any_o man_n by_o consider_v a_o triangle_n set_v before_o he_o shall_v find_v that_o all_o its_o angle_n together_o take_v be_v equal_a to_o two_o right_a angle_n and_o that_o by_o think_v of_o the_o same_o tacit_o without_o any_o use_n of_o word_n either_o understand_v or_o express_v and_o it_o shall_v happen_v afterward_o that_o another_o triangle_n unlike_o the_o former_a or_o the_o same_o in_o different_a situation_n shall_v be_v offer_v to_o his_o consideration_n he_o will_v not_o know_v ready_o whether_o the_o same_o property_n be_v in_o this_o last_o or_o no_o but_o will_v be_v force_v as_o often_o as_o a_o different_a triangle_n be_v bring_v before_o he_o and_o the_o difference_n of_o triangle_n be_v infinite_a to_o begin_v his_o contemplation_n anew_o which_o he_o will_v have_v no_o need_n to_o do_v if_o he_o have_v the_o use_n of_o name_n for_o every_o universal_a name_n denote_v the_o conception_n we_o have_v of_o infinite_a singular_a thing_n nevertheless_o as_o i_o say_v above_o they_o serve_v as_o mark_v for_o the_o help_n of_o our_o memory_n whereby_o we_o register_v to_o ourselves_o our_o own_o invention_n but_o not_o as_o sign_n by_o which_o we_o declare_v the_o same_o to_o other_o so_o that_o a_o man_n may_v be_v a_o philosopher_n alone_o by_o himself_o without_o any_o master_n adam_n have_v this_o capacity_n but_o to_o teach_v that_o be_v to_o demonstrate_v suppose_v two_o at_o the_o least_o and_o syllogistical_a speech_n 12_o and_o see_v teach_v be_v nothing_o but_o lead_v the_o mind_n of_o he_o we_o teach_v to_o the_o knowledge_n of_o our_o invention_n in_o that_o track_n by_o which_o we_o attain_v the_o same_o with_o our_o own_o mind_n therefore_o the_o same_o method_n that_o serve_v for_o our_o invention_n will_v serve_v also_o for_o demonstration_n to_o other_o save_v that_o we_o omit_v the_o first_o part_n of_o method_n which_o proceed_v from_o the_o sense_n of_o thing_n to_o universal_a principle_n which_o
of_o incidence_n abe_n and_o to_o the_o angle_n ibk_n its_o vertical_a angle_n ebf_n be_v equal_a and_o therefore_o the_o angle_n abe_n be_v equal_a to_o the_o angle_n ebf_n again_o the_o angle_n ade_n be_v equal_a to_o the_o angle_n of_o reflection_n gdk_n that_o be_v to_o its_o vertical_a angle_n edf_n and_o therefore_o the_o two_o angle_n abdella_n and_o adb_n of_o the_o triangle_n abdella_n be_v one_o by_o one_o equal_a to_o the_o two_o angle_n fbd_v and_o fdb_n of_o the_o triangle_n fbd_v wherefore_o also_o the_o three_o angle_n bad_a be_v equal_a to_o the_o three_o angle_n bfd_n which_o be_v to_o be_v prove_v corollary_n 1._o if_o the_o straight_a line_n of_o be_v draw_v it_o will_v be_v perpendicular_a to_o the_o straight_a line_n eke_o for_o both_o the_o angle_n at_o e_o will_v be_v equal_a by_o reason_n of_o the_o equality_n of_o the_o two_o angle_n abe_n and_o fbe_n and_o of_o the_o two_o side_n ab_fw-la and_o fb_fw-la corollary_n 2._o if_o upon_o any_o point_n between_o b_o and_o d_o there_o fall_v a_o straight_a line_n as_o ac_fw-la who_o reflect_a line_n be_v ch_z this_o also_o produce_v beyond_o c_o will_n fall_n upon_o f_o which_o be_v evident_a by_o the_o demonstration_n above_o 3_o if_o from_o two_o point_n take_v without_o a_o circle_n two_o straight_a parallel_n line_n draw_v not_o opposite_o but_o from_o the_o same_o part_n fall_v upon_o the_o circumference_n the_o line_n reflect_v from_o they_o if_o produce_v they_o meet_v within_o the_o circle_n will_v make_v a_o angle_n double_a to_o that_o which_o be_v make_v by_o two_o straight_a line_n draw_v from_o the_o centre_n to_o the_o point_n of_o incidence_n let_v the_o two_o straight_a parallel_n ab_fw-la and_o dc_o in_o the_o 3d_o figure_n fall_v upon_o the_o circumference_n bc_n at_o the_o point_n b_o and_o c_o and_o let_v the_o centre_n of_o the_o circle_n be_v e_o and_o let_v ab_fw-la reflect_v be_v bf_n and_o dc_o reflect_v be_v cg_n and_o let_v the_o line_n fb_n and_o gc_n produce_v meet_v within_o the_o circle_n in_o h_n and_o let_v ebb_n and_o aec_fw-la be_v connect_v i_o say_v the_o angle_n fhg_n be_v double_a to_o the_o angle_n bec_n for_o see_v ab_fw-la and_o dc_o be_v parallel_n and_o ebb_n cut_v ab_fw-la in_o b_o the_o same_o ebb_n produce_v will_v cut_v dc_o somewhere_o let_v it_o cut_v it_o in_o d_o &_o let_v dc_o be_v produce_v howsoever_o to_o i_o and_o let_v the_o intersection_n of_o dc_o &_o bf_o be_v at_o k._n the_o angle_n therefore_o ich_n be_v external_a to_o the_o triangle_n ckh_n will_v be_v equal_a to_o the_o two_o opposite_a angle_n ckh_n and_o chk_n again_o ice_n be_v external_a to_o the_o triangle_n cde_v be_v equal_a to_o the_o two_o angle_n at_o d_o and_o e._n wherefore_o the_o angle_n ich_n be_v double_a to_o the_o angle_n ice_n be_v equal_a to_o the_o angle_n at_o d_o and_o e_o twice_o take_v and_o therefore_o the_o two_o angle_n ckh_n and_o chk_n be_v equal_a to_o the_o two_o angle_n at_o d_o and_o e_o twice_o take_v but_o the_o angle_n ckh_n be_v equal_a to_o the_o angle_n d_o and_o abdella_n that_o be_v d_o twice_o take_v for_o ab_fw-la and_o dc_o be_v parallel_n the_o altern_a angle_n d_o and_o abdella_n be_v equal_a wherefore_o chk_n that_o be_v the_o angle_n fhg_n be_v also_o equal_a to_o the_o angle_n at_o e_o twice_o take_v which_o be_v to_o be_v prove_v corollary_n if_o from_o two_o point_n take_v within_o a_o circle_n two_o straight_a parallel_n fall_v upon_o the_o circumference_n the_o line_n reflect_v from_o they_o shall_v meet_v in_o a_o angle_n double_a to_o that_o which_o be_v make_v by_o two_o straight_a line_n draw_v from_o the_o centre_n to_o the_o point_n of_o incidence_n for_o the_o parallel_v lb_n and_o i_fw-mi falling_z upon_o the_o point_n b_o and_o c_o be_v reflect_v in_o the_o line_n bh_n and_o ch_z and_o make_v the_o angle_n at_o h_n double_a to_o the_o angle_n at_o e_o as_o be_v but_o now_o demonstrate_v 4_o if_o two_o straight_a line_n draw_v from_o the_o same_o point_n without_o a_o circle_n fall_v upon_o the_o circumference_n and_o the_o line_n reflect_v from_o they_o be_v produce_v meet_v within_o the_o circle_n they_o will_v make_v a_o angle_n equal_a to_o twice_o that_o angle_n which_o be_v make_v by_o two_o straight_a line_n draw_v from_o the_o centre_n to_o the_o point_n of_o incidence_n together_o with_o the_o angle_n which_o the_o incident_a line_n themselves_o make_v let_v the_o two_o straight_a line_n ab_fw-la and_o ac_fw-la in_o the_o four_o figure_n be_v draw_v from_o the_o point_n a_o to_o the_o circumference_n of_o the_o circle_n who_o centre_n be_v d_o and_o let_v the_o line_n reflect_v from_o they_o be_v be_v and_o cg_n and_o be_v produce_v make_v within_o the_o circle_n the_o angle_n h_n also_o let_v the_o two_o straight_a line_n db_a and_o dc_o be_v draw_v from_o the_o centre_n d_o to_o the_o point_n of_o incidence_n b_o and_o c._n i_o say_v the_o angle_n h_n be_v equal_a to_o twice_o the_o angle_n at_o d_o together_o with_o the_o angle_n at_o a._n for_o let_v ac_fw-la be_v produce_v howsoever_o to_o i._o therefore_o the_o angle_n ch_z which_o be_v external_a to_o the_o triangle_n ckh_n will_v be_v equal_a to_o the_o two_o angle_n gkh_n and_o chk_n again_o the_o angle_n icd_n which_o be_v external_a to_o the_o triangle_n cld_v will_v be_v equal_a to_o the_o two_o angle_n cld_a and_o cdl_o but_o the_o angle_n ich_n be_v double_a to_o the_o angle_n icd_n and_o be_v therefore_o equal_a to_o the_o angle_n cld_a and_o cdl_o twice_o take_v wherefore_o the_o angle_v ckh_n and_o chk_n be_v equal_a to_o the_o angle_n cld_a and_o cdl_o twice_o take_v but_o the_o angle_n cld_v be_v external_a to_o the_o triangle_n alb_n be_v equal_a to_o the_o two_o angle_n labernele_n &_o lba_n &_o consequent_o cld_v twice_o take_v be_v equal_a to_o labernele_n &_o labernele_n twice_o take_v wherefore_o ckh_n &_o chk_n be_v equal_a to_o the_o angle_n cdl_o together_o with_o labernele_n and_o lba_n twice_o take_v also_o the_o angle_n ckh_n be_v equal_a to_o the_o angle_n labernele_n once_o and_o abk_n that_o be_v lba_n twice_o take_v wherefore_o the_o angle_n chk_n be_v equal_a to_o the_o remain_a angle_n cdl_o that_o be_v to_o the_o angle_n at_o d_o twice_o take_v and_o the_o angle_n labernele_n that_o be_v the_o angle_n at_o a_o once_o take_v which_o be_v to_o be_v prove_v corollary_n if_o two_o straight_a converge_a line_n as_o i_fw-mi and_o mb_a fall_n upon_o the_o concave_a circumference_n of_o a_o circle_n their_o reflect_a line_n as_o ch_z and_o bh_n will_v meet_v in_o the_o angle_n h_n equal_a to_o twice_o the_o angle_n d_o together_z with_o the_o angle_n at_o a_o make_v by_o the_o incident_a line_n produce_v or_o if_o the_o incident_a line_n be_v hb_a and_o i_fw-mi who_o reflect_v line_n ch_n and_o bm_n meet_v in_o the_o point_n n_o the_o angle_n cnb_n will_v be_v equal_a to_o twice_o the_o angle_n d_o together_z with_o the_o angle_n ckh_n make_v by_o the_o line_n of_o incidence_n for_o the_o angle_n cnb_n be_v equal_a to_o the_o angle_n h_n that_o be_v to_o twice_o the_o angle_n d_o together_o with_o the_o two_o angle_n a_o and_o nbh_n that_o be_v kba_n but_o the_o angle_v kba_n and_o a_o be_v equal_a to_o the_o angle_n ckh_n wherefore_o the_o angle_n cnb_n be_v equal_a to_o twice_o the_o angle_n d_o together_z with_o the_o angle_n ckh_n make_v by_o the_o line_n of_o incidence_n i_fw-mi and_o hb_v produce_v to_o k._n 5_o if_o two_o straight_a line_n draw_v from_o one_o point_n fall_v upon_o the_o concave_a circumference_n of_o a_o circle_n and_o the_o angle_n they_o make_v be_v less_o than_o twice_o the_o angle_n at_o the_o centre_n the_o line_n reflect_v from_o they_o and_o meet_v within_o the_o circle_n will_v make_v a_o angle_n which_o be_v add_v to_o the_o angle_n of_o the_o incident_a line_n will_v be_v equal_a to_o twice_o the_o angle_n at_o the_o centre_n let_v the_o two_o line_n ab_fw-la and_o ac_fw-la in_o the_o 5_o figure_n draw_v from_o the_o point_n a_o fall_v upon_o the_o concave_a circumference_n of_o the_o circle_n who_o centre_n be_v d_o &_o let_v their_o reflect_a line_n be_v and_o ce_fw-fr meet_v in_o the_o point_n e_o also_o let_v the_o angle_v a_o be_v less_o than_o twice_o the_o angle_n d._n i_o say_v the_o angle_v a_o and_o e_o together_o take_v be_v equal_a to_o twice_o the_o angle_n d._n for_o let_v the_o straight_a line_n ab_fw-la and_o aec_fw-la cut_v the_o straight_a line_n dc_o and_o db_n in_o the_o point_n g_o and_o h_o and_o the_o angle_n bhc_n will_v be_v equal_a to_o the_o two_o angle_n ebh_n and_o e_o also_o the_o same_o angle_n bhc_n will_v be_v equal_a to_o the_o two_o angle_n d_o and_o dch_n
which_o make_v equal_a angle_n on_o either_o side_n of_o the_o perpendicular_a be_v produce_v to_o the_o tangent_fw-la they_o will_v be_v equal_a 12_o there_o be_v in_o euclid_n a_o definition_n of_o straight-lined_n parallel_n but_o i_o do_v not_o find_v that_o parallel_n in_o general_a be_v any_o where_o define_v and_o therefore_o for_o a_o universal_a definition_n of_o they_o i_o say_v that_o any_o two_o line_n whatsoever_o straight_o or_o crooked_a as_o also_o any_o two_o superficies_n be_v parallel_n when_o two_o equal_a straight_a line_n wheresoever_o they_o fall_v upon_o they_o make_v always_o equal_a angle_n with_o each_o of_o they_o from_o which_o definition_n it_o follow_v first_o that_o any_o two_o straight_a line_n not_o incline_v opposite_a way_n fall_v upon_o two_o other_o straight_a line_n which_o be_v parallel_n and_o intercept_n equal_a part_n in_o both_o of_o they_o be_v themselves_o also_o equal_a and_o parallel_v as_o if_o ab_fw-la and_o cd_o in_o the_o three_o figure_n incline_v both_o the_o same_o way_n fall_n upon_o the_o parallel_n ac_fw-la and_o bd_o and_o ac_fw-la and_o bd_o be_v equal_a ab_fw-la and_o cd_o will_n also_o be_v equal_a and_o parallel_v for_o the_o perpendicular_o be_v and_o df_n be_v draw_v the_o right_a angle_n ebb_v and_o fdh_n will_v be_v equal_a wherefore_o see_v of_o and_o bd_o be_v parallel_n the_o angle_v eba_n and_o fdc_n will_v be_v equal_a now_o if_o dc_o be_v not_o equal_a to_o basilius_n let_v any_o other_o straight_o line_n equal_a to_o basilius_n be_v draw_v from_o the_o point_n d_o which_o see_v it_o can_v fall_v upon_o the_o point_n c_o let_v it_o fall_v upon_o g._n whereore_n agnostus_n will_v be_v either_o great_a or_o less_o than_o bd_o and_o therefore_o the_o angle_n eba_n and_o fdg_n be_v not_o equal_a as_o be_v suppose_v wherefore_o ab_fw-la and_o cd_o be_v equal_a which_o be_v the_o first_o again_o because_o they_o make_v equal_a angle_n with_o the_o perpendicular_o be_v and_o df_n therefore_o the_o angle_n cdh_n will_v be_v equal_a to_o the_o angle_n abdella_n and_o by_o the_o definition_n of_o parallel_n ab_fw-la and_o cd_o will_v be_v parallel_n which_o be_v the_o second_o that_o plain_n which_o be_v include_v both_o way_n within_o parallel_a line_n be_v call_v a_o parallelogram_n 1_o corollary_n from_o this_o last_o it_o follow_v that_o the_o angles_n abdella_n and_o cdh_n be_v equal_a that_o be_v that_o a_o straight_a line_n as_o bh_n fall_v upon_o two_o parallel_n as_o ab_fw-la and_o cd_o make_v the_o internal_a angle_n abdella_n equal_a to_o the_o external_a and_o opposite_a angle_n cdh_n 2_o coral_n and_o from_o hence_o again_o it_o follow_v that_o a_o straight_a line_n fall_v upon_o two_o parallel_n make_v the_o alternate_a angle_n equal_a that_o be_v the_o angle_n agf_n in_o the_o four_o figure_n equal_a to_o the_o angle_n gfd_n for_o see_v gfd_n be_v equal_a to_o the_o external_a opposite_a angle_n egb_n it_o will_v be_v also_o equal_a to_o its_o vertical_a angle_n agf_n which_o be_v alternate_a to_o gfd_n 3_o coral_n that_o the_o internal_a angle_n on_o the_o same_o side_n of_o the_o line_n fg_v be_v equal_a to_o two_o right_a angle_n for_o the_o angle_n at_o f_o namely_o gfc_a and_o gfd_n be_v equal_a to_o two_o right_a angle_n but_o gfd_n be_v equal_a to_o its_o alternate_a angle_n agf_n wherefore_o both_o the_o angel_n gfc_n and_o agf_n which_o be_v internal_a on_o the_o same_o side_n of_o the_o line_n fg_v be_v equal_a to_o two_o right_a angle_n 4_o coral_n that_o the_o three_o angle_n of_o a_o straight-lined_n plain_a triangle_n be_v equal_a to_o two_o right_a angle_n and_o any_o side_n be_v produce_v the_o external_a angle_n will_v be_v equal_a to_o the_o two_o opposite_a internal_a angle_n for_o if_o there_o be_v draw_v by_o the_o vertex_fw-la of_o the_o plain_a triangle_n abc_n figure_n 5._o a_o parallel_n to_o any_o of_o the_o side_n as_o to_o ab_fw-la the_o angle_n a_o and_o b_o will_v be_v equal_a to_o their_o alternate_a angle_n e_z &_o f_o &_o the_o angle_n c_o be_v common_a but_o by_o the_o 10_o article_n the_o three_o angle_n e_o c_o and_o f_o be_v equal_a to_o two_o right_a angle_n and_o therefore_o the_o three_o angle_n of_o the_o triangle_n be_v equal_a to_o the_o same_o which_o be_v the_o first_o again_o the_o two_o angle_n b_o and_o d_o be_v equal_a to_o two_o right_a angle_n by_o the_o 10_o article_n wherefore_o take_v away_o b_o there_o will_v remain_v the_o angle_n a_o and_o c_o equal_v to_o the_o angle_n d_o which_o be_v the_o second_o 5_o coral_n if_o the_o angle_n a_o and_o b_o be_v equal_a the_o side_n ac_fw-la and_o cb_n will_v also_o be_v equal_a because_o ab_fw-la and_o of_o be_v parallel_n and_o on_o the_o contrary_a if_o the_o side_n ac_fw-la and_o cb_n be_v equal_a the_o angle_n a_o and_o b_o will_v also_o be_v equal_a for_o if_o they_o be_v not_o equal_a let_v the_o angle_n b_o and_o g_o be_v equal_a wherefore_o see_v gb_n and_o of_o be_v parallel_n and_o the_o angle_n g_o and_o b_o equal_v the_o side_n gc_a and_o cb_n will_v also_o be_v equal_a and_o because_o cb_n and_o ac_fw-la be_v equal_a by_o supposition_n cg_n and_o ca_n will_v also_o be_v equal_a which_o can_v be_v by_o the_o 11_o article_n 6_o coral_n from_o hence_o it_o be_v manifest_a that_o if_o two_o radii_fw-la of_o a_o circle_n be_v connect_v by_o a_o straight_a line_n the_o angle_n they_o make_v with_o that_o connect_a line_n will_v be_v equal_a to_o one_o another_o and_o if_o there_o be_v add_v that_o segment_n of_o the_o circle_n which_o be_v subtend_v by_o the_o same_o line_n which_o connect_v the_o radii_fw-la than_o the_o angle_n which_o those_o radii_fw-la make_v with_o the_o circumference_n will_v also_o be_v equal_a to_o one_o another_o for_o a_o straight_a line_n which_o subtend_v any_o arch_n make_v equal_a angle_n with_o the_o same_o because_o if_o the_o arch_n and_o the_o subtense_n be_v divide_v in_o the_o middle_n the_o two_o half_n of_o the_o segment_n will_v be_v congruous_a to_o one_o another_o by_o reason_n of_o the_o uniformity_n both_o of_o the_o circumference_n of_o the_o circle_n and_o of_o the_o straight_a line_n 13_o perimeter_n of_o circle_n be_v to_o one_o another_o as_o their_o semidiameter_n be_v for_o let_v there_o be_v any_o two_o circle_n as_o in_o the_o first_o figure_n bcd_v the_o great_a and_o efg_v the_o lesser_a have_v their_o common_a centre_n at_o a_o and_o let_v their_o semidiameter_n be_v ac_fw-la and_o ae_n i_o say_v ac_fw-la have_v the_o same_o proportion_n to_o ae_n which_o the_o perimeter_n bcd_n have_v to_o the_o perimeter_n efg_v for_o the_o magnitude_n of_o the_o semidiameter_n ac_fw-la and_o ae_n be_v determine_v by_o the_o distance_n of_o the_o point_n c_o and_o e_o from_o the_o centre_n a_o and_o the_o same_o distance_n be_v acquire_v by_o the_o uniform_a motion_n of_o a_o point_n from_o a_o to_o c_o in_o such_o manner_n that_o in_o equal_a time_n the_o distance_n acquire_v be_v equal_a but_o the_o perimeter_n bcd_v and_o efg_n be_v also_o determine_v by_o the_o same_o distance_n of_o the_o point_n c_o and_o e_o from_o the_o centre_n a_o and_o therefore_o the_o perimeter_n bcd_v and_o efg_a as_o well_o as_o the_o semidiameter_n ac_fw-la and_o ae_n have_v their_o magnitude_n determine_v by_o the_o same_o cause_n which_o cause_n make_v in_o equal_a time_n equal_a space_n wherefore_o by_o the_o 13_o chapter_n and_o 6_o article_n the_o perimeter_n of_o circle_n and_o their_o semidiameter_n be_v proportional_n which_o be_v to_o be_v prove_v 14_o if_o two_o straight_a line_n w_z ch_n constitute_v a_o angle_n be_v cut_v by_o straight-lined_n parallel_n the_o intercept_a parallel_n will_v be_v to_o one_o another_o as_o the_o parts_z w_z ch_z they_o cut_v off_o from_o the_o vertex_fw-la let_v the_o straight_a line_n ab_fw-la and_o ac_fw-la in_o the_o 6_o figure_n make_v a_o angle_n at_o a_o &_o be_v cut_v by_o the_o two_o straight-lined_n parallel_n bc_n and_o de_fw-fr so_o that_o the_o part_n cut_v off_o from_o the_o vertex_fw-la in_o either_o of_o those_o line_n as_o in_o ab_fw-la may_v be_v ab_fw-la and_o ad._n i_o say_v the_o parallel_n bc_n and_o de_fw-fr be_v to_o one_o another_o as_o the_o part_n ab_fw-la and_o ad._n for_o let_v ab_fw-la be_v divide_v into_o any_o number_n of_o equal_a part_n as_o into_z of_o fd_n db_n and_o by_o the_o point_v f_o and_o d_o let_v fg_n and_o de_fw-fr be_v draw_v parallel_n to_o the_o base_a bc_n and_o cut_v ac_fw-la in_o g_z and_o e_z and_o again_o by_o the_o point_n g_o and_o e_o let_v other_o straight_a line_n be_v draw_v parallel_n to_o ab_fw-la and_o cut_v bc_n in_o h_n and_o i._n if_o now_o the_o point_n a_o be_v understand_v to_o be_v move_v uniform_o over_o ab_fw-la and_o in_o the_o
same_o time_n b_o be_v move_v to_o c_o and_o all_o the_o point_n f_o d_o and_o b_o be_v move_v uniform_o and_o with_o equal_a swiftness_n over_o fg_n de_fw-fr and_o bc_n then_o shall_v b_o pass_v over_o bh_n equal_a to_o fg_v in_o the_o same_o time_n that_o a_o pass_v over_o of_o and_o of_o and_o fg_n will_v be_v to_o one_o another_o as_o their_o velocity_n be_v and_o when_o a_o be_v in_o f_o d_o will_v be_v in_o k_o when_o a_o be_v in_o d_o d_o will_v be_v in_o e_z and_o in_o what_o manner_n the_o point_n a_o pass_n by_o the_o point_n f_o d_o and_o b_o in_o the_o same_o manner_n the_o point_n b_o will_v pass_v by_o the_o point_v h_n i_o and_o c_o &_o the_o straight_a line_n fg_v dk_n ke_n bh_n he_o &_o i_fw-mi be_v equal_a by_o reason_n of_o their_o parallelisme_n and_o therefore_o as_o the_o velocity_n in_o ab_fw-la be_v to_o the_o velocity_n 〈◊〉_d bc_n so_o be_v ad_fw-la to_o de_fw-fr but_o as_o the_o velocity_n in_o ab_fw-la be_v to_o the_o velocity_n in_o bc_n so_o be_v ab_fw-la to_o bc_n that_o be_v to_o say_v all_o the_o parallel_n will_v be_v several_o to_o all_o the_o part_n cut_v off_o from_o the_o vertex_fw-la as_o of_o be_v to_o fg._n wherefore_o af._n gf_n ad._n de_fw-fr ab_fw-la bc_n be_v proportional_n the_o subtense_n of_o equal_a angle_n in_o different_a circle_n as_o the_o straight_a line_n bc_n and_o fe_o in_o the_o 1_o figure_n be_v to_o one_o another_o as_o the_o arch_n which_o they_o subtend_v for_o by_o the_o 8_o article_n the_o arch_n of_o equal_a angle_n be_v to_o one_o another_o as_o their_o perimeter_n be_v and_o by_o the_o 13_o art_n the_o perimeter_n as_o their_o semidiameter_n but_o the_o the_o subtense_n bc_n and_o fe_o be_v parallel_n to_o one_o another_o by_o reason_n of_o the_o equality_n of_o the_o angle_n which_o they_o make_v with_o the_o semidiameter_n and_o therefore_o the_o same_o subtense_n by_o the_o last_o precedent_n article_n will_v be_v proportional_a to_o the_o semidiameter_n that_o be_v to_o the_o perimeter_n that_o be_v to_o the_o arch_n which_o they_o subtend_v 15_o if_o in_o a_o circle_n any_o number_n of_o equal_a subtense_n be_v place_v immediate_o after_o one_o another_o and_o straight_a line_n be_v draw_v from_o the_o extreme_a point_n of_o the_o first_o subtense_n to_o the_o extreme_a point_n of_o all_o the_o rest_n the_o first_o subtense_n be_v produce_v will_v make_v with_o the_o second_o subtense_n a_o external_a angle_n double_a to_o that_o which_o be_v make_v by_o the_o same_o first_o subtense_n and_o a_o tangent_fw-la to_o the_o circle_n touch_v it_o in_o the_o extreme_a point_n thereof_o and_o if_o a_o straight_a line_n which_o subtend_v two_o of_o those_o arch_n be_v produce_v it_o will_v make_v a_o external_a angle_n with_o the_o three_o subtense_n triple_a to_o the_o angle_n which_o be_v make_v by_o the_o tangent_fw-la with_o the_o first_o subtense_n and_o so_o continual_o for_o with_o the_o radius_fw-la ab_fw-la in_o the_o seven_o figure_n let_v a_o circle_n be_v describe_v &_o in_o it_o let_v any_o number_n of_o equal_a subtense_n bc_n cd_o &_o de_fw-fr be_fw-mi place_v also_o let_v bd_o &_o be_v be_v draw_v &_o by_o produce_v bc_n bd_o &_o be_v to_o any_o distance_n in_o g_z h_o and_o i_o let_v they_o make_v angle_n with_o the_o subtense_n which_o succeed_v one_o another_o namely_o the_o external_a angle_n gcd_v and_o hde_v last_o let_v the_o tangent_fw-la kb_fw-la be_v draw_v make_v with_o the_o first_o subtense_n the_o angle_n kbc_n i_o say_v the_o angle_n gcd_n be_v double_a to_o the_o angle_n kbc_n and_o the_o angle_n hde_v triple_a to_o the_o same_o angle_n kbc_n for_o if_o ac_fw-la be_v draw_v cut_v bd_o in_o m_o and_o from_o the_o point_n c_o there_o be_v draw_v lc_n perpendicular_a to_o the_o same_o ac_fw-la then_o cl_n and_o md_o will_v be_v parallel_n by_o reason_n of_o the_o right_a angle_n at_o c_o and_o m_o and_o therefore_o the_o alterne_a angle_n lcd_v and_o bdc_n will_v be_v equal_a as_o also_o the_o angel_n bdc_n and_o cbd_n will_v be_v equal_a because_o of_o the_o equality_n of_o the_o straight_a line_n bc_n and_o cd_o wherefore_o the_o angle_n gcd_n be_v double_a to_o either_o of_o the_o angel_n cbd_v or_o cdb_n and_o therefore_o also_o the_o angle_n gcd_n be_v double_a to_o the_o angle_n lcd_v that_o be_v to_o the_o angle_n kbc_n again_o cd_o be_v parallel_n to_o be_v by_o reason_n of_o the_o equality_n of_o the_o angel_n cbe_n and_o deb_n and_o of_o the_o straight_a line_n cb_n and_o de_fw-fr and_o therefore_o the_o angle_n gcd_v and_o gbe_n be_v equal_a and_o consequent_o gbe_n as_o also_o deb_n be_v double_a to_o the_o angle_n kbc_n but_o the_o external_a angle_n hde_v be_v equal_a to_o the_o two_o internal_a deb_n and_o dbe_n and_o therefore_o the_o angle_n hde_v be_v triple_a to_o the_o angle_n kbc_n etc._n etc._n which_o be_v to_o be_v prove_v 1_o corollary_n from_o hence_o it_o be_v manifest_a that_o the_o angel_n kbc_n and_o cbd_v as_o also_o that_o all_o the_o angle_n that_o be_v comprehend_v by_o two_o straight_a line_n meet_v in_o the_o circumference_n of_o a_o circle_n and_o insist_v upon_o equal_a arch_n be_v equal_a to_o one_o another_o 2_o coral_n if_o the_o tangent_fw-la bk_n be_v move_v in_o the_o circumference_n with_o uniform_a motion_n about_o the_o centre_n b_o it_o will_v in_o equal_a time_n cut_v off_o equal_a arch_n and_o will_v pass_v over_o the_o whole_a perimeter_n in_o the_o same_o time_n in_o which_o itself_o describe_v a_o semiperimeter_n about_o the_o centre_n b._n 3_o coral_n from_o hence_o also_o we_o may_v understand_v what_o it_o be_v that_o determine_v the_o bend_n or_o curvation_n of_o a_o straight_a line_n into_o the_o circumference_n of_o a_o circle_n namely_o that_o it_o be_v fraction_n continual_o increase_a in_o the_o same_o manner_n as_o number_n from_o one_o upward_o increase_v by_o the_o continual_a addition_n of_o unity_n for_o the_o indefinite_a straight_a line_n kb_n be_v break_v in_o b_o according_a to_o any_o angle_n as_o that_o of_o kbc_n &_o again_o in_o c_z according_a to_o a_o double_a angle_n and_o in_o d_o according_a to_o a_o angle_n which_o be_v triple_a and_o in_o e_z according_a to_o a_o angle_n which_o be_v quadruple_a â—o_o the_o first_o angle_n and_o so_o continual_o there_o will_v be_v describe_v a_o figure_n which_o will_v indeed_o be_v rectilineal_a if_o the_o break_a part_n be_v consider_v as_o haâ—ing_v magnitude_n but_o if_o they_o be_v understand_v to_o be_v the_o least_o tâ—aâ—_n can_v 〈◊〉_d â—â—at_o be_v as_o so_o many_o point_n then_o the_o figure_n describe_v will_v â—ot_n be_v rectilineal_a but_o a_o circle_n who_o circumference_n wâ—…_n break_a line_n 4_o 〈…〉_z be_v say_v in_o this_o present_a article_n it_o may_v 〈…〉_z angle_z in_o the_o centre_n be_v double_a to_o a_o angle_n in_o the_o circumference_n of_o the_o same_o circle_n if_o the_o intercept_a arch_n be_v equal_a for_o see_v that_o straight_a line_n by_o who_o motion_n a_o angle_n be_v determine_v pass_v over_o equal_a arch_n in_o equal_a time_n as_o well_o from_o the_o centre_n as_o from_o the_o circumference_n and_o while_o that_o which_o be_v from_o the_o circumference_n be_v pass_v over_o half_a its_o own_o perimeter_n it_o pass_v in_o the_o same_o time_n over_o the_o whole_a perimeter_n of_o that_o which_o be_v from_o the_o centre_n the_o arch_n w_z ch_n it_o cut_v off_o in_o the_o perimeter_n who_o centre_n be_v a_o will_v be_v double_a to_o those_o which_o it_o make_v in_o its_o own_o semiperimeter_n who_o centre_n be_v b._n but_o in_o equal_a circle_n as_o arch_n be_v to_o one_o another_o so_o also_o be_v angle_n it_o may_v also_o be_v demonstrate_v that_o the_o external_a angle_n make_v by_o a_o subtense_n produce_v and_o the_o next_o equal_a subtense_n be_v equal_a to_o a_o angle_n from_o the_o centre_n insist_v upon_o the_o same_o arch_n as_o in_o the_o last_o diagram_n the_o angle_n gcd_n be_v equal_a to_o the_o angle_n god_n for_o the_o external_a angle_n gcd_v be_v double_a to_o the_o angle_n cbd_v and_o the_o angle_n god_n insist_v upon_o the_o same_o arch_a cd_o be_v also_o double_a to_o the_o same_o angle_n cbd_v or_o kbc_n 16_o a_o angle_n of_o contingence_n if_o it_o be_v compare_v with_o a_o angle_n simple_o so_o call_v how_o little_a soever_o have_v such_o proportion_n to_o it_o as_o a_o point_n have_v to_o a_o line_n that_o be_v no_o proportion_n at_o all_o nor_o any_o quantity_n for_o first_o a_o angle_n of_o contingence_n be_v make_v by_o continual_a flexion_n so_o that_o in_o the_o generation_n of_o it_o there_o be_v no_o circular_a motion_n at_o all_o in_o which_o consist_v the_o nature_n of_o a_o angle_n simple_o so_o call_v and_o therefore_o it_o can_v be_v
and_o in_o like_a manner_n the_o angle_n bgc_n will_v be_v equal_a to_o the_o two_o angle_n acd_v &_o a_o &_o the_o same_o angle_n bgc_n will_v be_v also_o equal_a to_o the_o two_o angle_n dbg_n and_o d._n wherefore_o the_o four_o angle_n ebh_n e_o acd_a and_o a_o be_v equal_a to_o the_o four_o angle_n d_o dch_n dbg_n and_o d._n if_o therefore_o equal_v be_v take_v away_o on_o both_o side_n namely_o on_o one_o side_n acd_v and_o ebh_n and_o on_o the_o other_o side_n dch_n and_o dbg_n for_o the_o angle_n ebh_n be_v equal_a to_o the_o angle_n dbg_n and_o the_o angle_n acd_v equal_a to_o the_o angle_n dch_o the_o remainder_n on_o both_o side_n will_v be_v equal_a namely_o on_o one_o side_n the_o angle_v a_o and_o e_o and_o on_o the_o other_o the_o angle_n d_o twice_z take_v wherefore_o the_o angle_n a_o and_o e_o be_v equal_a to_o twice_o the_o angle_n d._n corollary_n if_o the_o angle_n a_o be_v great_a than_o twice_o the_o angle_n d_o their_o reflect_a â—â—ines_n will_v diverge_n for_o by_o the_o corollary_n of_o the_o three_o proposition_n if_o the_o angle_n a_o be_v equal_a to_o twice_o the_o angle_n d_o the_o reflect_a line_n be_v and_o ce_fw-fr will_v be_v parallel_n and_o if_o it_o be_v less_o they_o will_v concur_v as_o have_v now_o be_v demonstrate_v and_o therefore_o if_o it_o be_v great_a the_o reflect_a line_n be_v and_o ce_fw-fr will_n diverge_n and_o consequent_o if_o they_o be_v produce_v the_o other_o way_n they_o will_v concur_v and_o make_v a_o angle_n equal_a to_o the_o excess_n of_o the_o angle_n a_o above_o twice_o the_o angle_n d_o as_o be_v evident_a by_o the_o four_o article_n 6_o if_o through_o any_o one_o point_n two_o unequal_a chord_n be_v draw_v cut_v one_o another_o either_o within_o the_o circle_n or_o if_o they_o be_v produce_v without_o it_o and_o the_o centre_n of_o the_o circle_n be_v not_o place_v between_o they_o and_o the_o line_n reflect_v from_o they_o concur_v wheresoever_o there_o can_v through_o the_o point_n through_o which_o the_o former_a line_n be_v draw_v be_v draw_v another_o straight_a line_n who_o reflect_a line_n shall_v pass_v through_o the_o point_n where_o the_o two_o former_a reflect_a line_n concur_v let_v any_o two_o unequal_a chord_n as_o bk_n and_o ch_n in_o the_o 6_o figure_n be_v draw_v through_o the_o point_n a_o in_o the_o circle_n bc_n and_o let_v their_o reflect_a line_n bd_o and_o ce_fw-fr meet_v in_o f_o and_o let_v the_o centre_n not_o be_v between_o ab_fw-la and_o ac_fw-la and_o from_o the_o point_n a_o let_v any_o other_o straight_o line_n as_o agnostus_n be_v draw_v to_o the_o circumference_n between_o b_o and_o c._n i_o say_v gn_n which_o pass_v through_o the_o point_n f_o where_o the_o reflect_v line_n bd_o and_o ce_fw-fr meet_a will_v not_o be_v the_o reflect_a line_n of_o ag._n for_o let_v the_o arch_n bl_n be_v take_v equal_a to_o the_o arch_n bg_n and_o the_o straight_a line_n bm_n equal_a to_o the_o straight_a line_n basilius_n and_o lm_o be_v draw_v let_v it_o be_v produce_v to_o the_o circummference_n in_o o._n see_v therefore_o basilius_n and_o bm_n be_v equal_a and_o the_o arch_n bl_n equal_a to_o the_o arch_n bg_n and_o the_o angle_n mbl_n equal_a to_o the_o angle_n abg_n agnostus_n and_o ml_o will_n also_o be_v equal_a and_o produce_v give_v to_o the_o circumference_n in_o i_o the_o whole_a line_n lo_o and_o give_v will_n in_o like_a manner_n be_v equal_a but_o lo_o be_v great_a than_o gfn_n as_o shall_v present_o be_v demonstrate_v and_o therefore_o also_o give_v be_v great_a than_o gn_n wherefore_o the_o angle_n ngc_a and_o igb_n be_v not_o equal_a wherefore_o the_o line_n gfn_n be_v not_o reflect_v from_o the_o line_n of_o incidence_n agnostus_n and_o consequent_o no_o other_o straight_a line_n beside_o ab_fw-la and_o ac_fw-la which_o be_v draw_v through_o the_o point_n a_o and_o faâ—ls_n upon_o the_o circumference_n bc_n can_v be_v reflect_v to_o the_o point_n f_o which_o be_v to_o be_v demonstrate_v it_o remain_v that_o i_o prove_v lo_o to_o be_v great_a than_o gn_n which_o i_o shall_v do_v in_o this_o manner_n lo_o and_o gn_v cut_v one_o another_o in_o p_o and_o pl_z be_v great_a than_o pg._n see_v now_o lp_v pg_n pn_n po_n be_v proportional_n therefore_o the_o two_o extreme_n lp_v and_o po_n together_o take_v that_o be_v lo_o be_v great_a than_o pg_n and_o pn_n together_o take_v that_o be_v gn_n which_o remain_v to_o be_v prove_v 7_o but_o if_o two_o equal_a chord_n be_v draw_v through_o one_o point_n within_o a_o circle_n and_o the_o line_n reflect_v from_o they_o meet_v in_o another_o point_n than_o another_o straight_a line_n may_v be_v draw_v between_o they_o through_o the_o former_a point_n who_o reflect_a line_n shall_v pass_v through_o the_o late_a point_n let_v the_o two_o equal_a chord_n bc_n and_o ed_z in_o the_o seven_o figure_n cut_v one_o another_o in_o the_o point_n a_o within_o the_o circle_n bcd_v and_o let_v their_o reflect_a line_n ch_n and_o diego_n meet_v in_o the_o point_n f._n then_o divide_v the_o arch_n cd_o equal_o in_o g_z let_v the_o two_o chord_n gk_n and_o gl_n be_v draw_v through_o the_o point_v a_o and_o f._n i_o say_v gl_n will_v be_v the_o line_n reflect_v from_o the_o chord_n kg_v for_o the_o four_o chord_n bc_n ch_z ed_z and_o di_z be_v by_o supposition_n all_o equal_a to_o one_o another_o and_o therefore_o the_o arch_a bch_n be_v equal_a to_o the_o arch_n edi_n as_o also_o the_o angle_n bch_o to_o the_o angle_n edi_n &_o the_o angle_n amc_n to_o its_o vertical_a angle_n fmd_v and_o the_o straight_a line_n dm_o to_o the_o straight_a line_n cm_o and_o in_o like_a manner_n the_o straight_a line_n ac_fw-la to_o the_o straight_a line_n fd_n and_o the_o chord_n cg_n and_o gd_n be_v draw_v will_v also_o be_v equal_a as_o also_o the_o angle_n fdg_n and_o acg_n in_o the_o equal_a segment_n gdi_n and_o gcb_n wherefore_o the_o straight_a line_n fg_v and_o agnostus_n be_v equal_a and_o therefore_o the_o angle_n fgd_v be_v equal_a to_o the_o angle_n agc_n that_o be_v the_o angle_n of_o incidence_n equal_a to_o the_o angle_n of_o reflection_n wherefore_o the_o line_n gl_n be_v reflect_v from_o the_o incident_a line_n kg_v which_o be_v to_o be_v prove_v corollary_n by_o the_o very_a sight_n of_o the_o figure_n it_o be_v manifest_a that_o if_o g_z be_v not_o the_o middle_a point_n between_o c_o and_o d_o the_o reflect_a line_n gl_n will_v not_o pass_v through_o the_o point_n f._n 8_o two_o point_n in_o the_o circumference_n of_o a_o circle_n be_v give_v to_o draw_v two_o straight_a line_n to_o they_o so_o as_o that_o their_o reflect_a line_n may_v be_v parallel_n or_o contain_v any_o angle_n give_v in_o the_o circumference_n of_o the_o circle_n who_o centre_n be_v a_o in_o the_o 8_o figure_v let_v the_o two_o point_n b_o and_o c_o be_v give_v and_o let_v it_o be_v require_v to_o draw_v to_o they_o from_o two_o point_n take_v without_o the_o circle_n two_o incident_a line_n so_o that_o their_o reflect_a line_n may_v first_o be_v parallel_n let_v ab_fw-la and_o ac_fw-la be_v draw_v as_o also_o any_o incident_a line_n dc_o with_o its_o reflect_a line_n cf_n and_o let_v the_o angle_n ecd_v be_v make_v double_a to_o the_o angle_n a_o and_o let_v hb_n be_v draw_v parallel_n to_o aec_fw-la and_o produce_v till_o it_o meet_v with_o dc_o produce_v in_o i._n last_o produce_v ab_fw-la indefinite_o to_o k_o let_v gb_n be_v draw_v so_o that_o the_o angle_n gbk_n may_v be_v equal_a to_o the_o angle_n hbk_n and_o then_o gb_n will_v be_v the_o reflect_a line_n of_o the_o incident_a line_n hb_n i_o say_v dc_o and_o hb_n be_v two_o incident_a line_n who_o reflect_a line_n cf_n and_o bg_n be_v parallel_n for_o see_v the_o angle_n ecd_n be_v double_a to_o the_o angle_n bac_n the_o angle_n hic_fw-la be_v also_o by_o reason_n of_o the_o parallel_n aec_fw-la and_o he_o double_a to_o the_o same_o bac_n therefore_o also_o fc_a and_o gb_n namely_o the_o line_n reflect_v from_o the_o incident_a line_n dc_o and_o hb_n be_v parallel_n wherefore_o the_o first_o thing_n require_v be_v do_v second_o let_v it_o be_v require_v to_o draw_v to_o the_o point_n b_o &_o c_o two_o straight_a line_n of_o incidence_n so_o that_o the_o line_n reflect_v from_o they_o may_v contain_v the_o give_v angle_n z._n to_o the_o angle_n ecd_v make_v at_o the_o point_n c_o let_v there_o be_v add_v on_o one_o side_n the_o angle_n dcl_o equal_a to_o half_a z_o and_o on_o the_o other_o side_n the_o angle_n ecm_n equal_a to_o the_o angle_n dcl_o and_o let_v the_o straight_a line_n bn_n be_v draw_v parallel_n to_o the_o straight_a line_n cm_o and_o let_v the_o angle_n kbo_n be_v make_v equal_a to_o the_o
angle_n nbk_n which_o be_v do_v boy_n will_v be_v the_o line_n of_o reflection_n from_o the_o line_n of_o incidence_n nb._n last_o from_o the_o incident_a line_n lc_n let_v the_o reflect_a line_n co_n be_v draw_v cut_v boy_n at_o o_o and_o make_v the_o angle_n cob_n i_o say_v the_o angle_n cob_n be_v equal_a to_o the_o angle_n z._n let_v nb_n be_v produce_v till_o it_o meet_v with_o the_o straight_a line_n lc_n produce_v in_o p._n see_v therefore_o the_o angle_n lcm_n be_v by_o construction_n equal_a to_o twice_o the_o angle_n bac_n together_o with_o the_o angle_n z_o the_o angle_n npl_n which_o be_v equal_a to_o lcm_n by_o reason_n of_o the_o parallel_n np_n and_o mc_n will_v also_o be_v equal_a to_o twice_o the_o same_o angle_n bac_n together_o with_o the_o angle_n z._n and_o see_v the_o two_o straight_a line_n oc_n and_o ob_fw-la fall_n from_o the_o point_n o_o upon_o the_o point_n c_o and_o b_o and_o their_o reflect_a line_n lc_a and_o nb_n meet_v in_o the_o point_n p_o the_o angle_n npl_n will_v be_v equal_a to_o twice_o the_o angle_n bac_n together_o with_o the_o angle_n cop_z but_o i_o have_v already_o prove_v the_o angle_n npl_n to_o be_v equal_a to_o twice_o the_o angle_n bac_n together_o with_o the_o angle_n z._n therefore_o the_o angle_n cop_z be_v equal_a to_o the_o angle_n z_o wherefore_o two_o point_n in_o the_o circumference_n of_o a_o circle_n be_v give_v i_o have_v draw_v etc._n etc._n which_o be_v to_o be_v do_v but_o if_o it_o be_v require_v to_o draw_v the_o incident_a line_n from_o a_o point_n within_o the_o circle_n so_o that_o the_o line_n reflect_v from_o they_o may_v contain_v a_o angle_n equal_a to_o the_o angle_n z_o the_o same_o method_n be_v to_o be_v use_v save_v that_o in_o this_o case_n the_o angle_n z_o be_v not_o to_o be_v add_v to_o twice_o the_o angle_n bac_n but_o to_o be_v take_v from_o it_o 9_o if_o a_o straight_a line_n fall_v upon_o the_o circumference_n of_o a_o circle_n be_v produce_v till_o it_o reach_v the_o semidiameter_n and_o that_o part_n of_o it_o which_o be_v intercept_v between_o the_o circumference_n and_o the_o semidiameter_n be_v equal_a to_o that_o part_n of_o the_o semidiameter_n which_o be_v between_o the_o point_n of_o concourse_n and_o the_o centre_n the_o reflect_a line_n will_v be_v parallel_n to_o the_o semidiameter_n let_v any_o line_n ab_fw-la in_o the_o 9th_o figure_n be_v the_o semidiameter_n of_o the_o circle_n who_o centre_n be_v a_o and_o upon_o the_o circumference_n bd_o let_v the_o straight_a line_n cd_o fall_n and_o be_v produce_v till_o it_o cut_v ab_fw-la in_o e_z so_o that_z ed_z and_o ea_z may_v be_v equal_a &_o from_o the_o incident_a line_n cd_o let_v the_o line_n df_n be_v reflect_v i_o say_v ab_fw-la and_o df_n will_v be_v parallel_n let_v agnostus_n be_v draw_v through_o the_o point_n d._n see_v therefore_o ed_z and_o ea_z be_v equal_a the_o angle_v eda_n and_o ead_n will_v also_o be_v equal_a but_o the_o angle_n fdg_n and_o eda_n be_v equal_a for_o each_o of_o they_o be_v half_a the_o angle_n edh_n or_o fdc_n wherefore_o the_o angle_n fdg_n and_o ead_n be_v equal_a and_o consequent_o df_n and_o ab_fw-la be_v parallel_n which_o be_v to_o be_v prove_v corollahy_fw-fr if_o ea_fw-la be_v great_a than_o ed_z than_o df_n and_o ab_fw-la be_v produce_v will_v concur_v but_o if_o ea_z be_v less_o than_o ed_z then_z ba_z and_o dh_n be_v produce_v will_v concur_v 10_o if_o from_o a_o point_n within_o a_o circle_n two_o straight_a line_n be_v draw_v to_o the_o circumference_n and_o their_o reflect_a line_n meet_v in_o the_o circumference_n of_o the_o same_o circle_n the_o angle_n make_v by_o the_o line_n of_o reflection_n will_v be_v a_o three_o part_n of_o the_o angle_n make_v by_o the_o line_n of_o incidence_n from_o the_o point_n b_o in_o the_o 10_o figure_n take_v within_o the_o circle_n who_o centre_n be_v a_o let_v the_o two_o straight_a line_n bc_n and_o bd_o be_v draw_v to_o the_o circumference_n and_o let_v their_o reflect_a lines_n ce_fw-fr and_o de_fw-fr meet_v in_o the_o circumference_n of_o the_o same_o circle_n at_o the_o point_n e._n i_o say_v the_o angle_n ce_v will_v be_v a_o three_o part_n of_o the_o angle_n cbd_v let_v ac_fw-la and_o ad_fw-la be_v draw_v see_v therefore_o the_o angle_n ce_v and_o cbd_v together_o take_v be_v equal_a to_o twice_o the_o angle_n god_n as_o have_v be_v demonstrate_v in_o the_o 5_o article_n and_o the_o angle_n god_n twice_o take_v be_v quadruple_a to_o the_o angle_n ce_v the_o angle_n ce_v and_o cbd_v together_o take_v will_v also_o be_v equal_a to_o the_o angle_n ce_v four_o time_n take_v and_o therefore_o if_o the_o angle_n ce_v be_v take_v away_o on_o both_o side_n there_o will_v remain_v the_o angle_n cbd_v on_o one_o side_n equal_a to_o the_o angle_n ce_v thrice_o take_v on_o the_o other_o side_n which_o be_v to_o be_v demonstrate_v coral_n therefore_o a_o point_n be_v give_v within_o a_o circle_n there_o may_v be_v draw_v two_o line_n from_o it_o to_o the_o circumference_n so_o as_o their_o reflect_a line_n may_v meet_v in_o the_o circumference_n for_o it_o be_v but_o trisect_v the_o angle_n cbd_v which_o how_o it_o may_v be_v do_v shall_v be_v show_v in_o the_o follow_a chapter_n chap._n xx._n of_o the_o dimension_n of_o a_o circle_n and_o the_o division_n of_o angle_n or_o arch_n 1_o the_o dimension_n of_o a_o circle_n near_o determine_v in_o number_n by_o archimedes_n and_o other_o 2_o the_o first_o attempt_n for_o the_o find_v out_o of_o the_o dimension_n of_o a_o circle_n by_o line_n 3_o the_o second_o attempt_n for_o the_o find_v out_o of_o the_o dimension_n of_o a_o circle_n from_o the_o consideration_n of_o the_o nature_n of_o crookedness_n 4_o the_o three_o attempt_n and_o some_o thing_n propound_v to_o be_v further_o search_v into_o 5_o the_o equation_n of_o the_o spiral_n of_o archimedes_n with_o a_o straight_a line_n 6_o of_o the_o analysis_n of_o geometrician_n by_o the_o power_n of_o line_n 1_o in_o the_o compare_v of_o a_o arch_n of_o a_o circle_n with_o a_o straight_a line_n many_o and_o great_a geometrician_n even_o from_o the_o most_o ancient_a time_n have_v exercise_v their_o wit_n and_o more_o have_v do_v the_o same_o if_o they_o have_v not_o see_v their_o pain_n though_o undertake_v for_o the_o common_a good_a if_o not_o bring_v to_o perfection_n vilify_v by_o those_o that_o envy_v the_o praise_n of_o other_o man_n among_o those_o ancient_a writer_n who_o work_n be_v come_v to_o our_o hand_n archimedes_n be_v the_o first_o that_o bring_v the_o length_n of_o the_o perimeter_n of_o a_o circle_n within_o the_o limit_n of_o number_n very_o little_o differ_v from_o the_o truth_n demonstrate_v the_o same_o to_o be_v less_o than_o three_o diameter_n and_o a_o seven_o part_n but_o great_a than_o three_o diameter_n and_o ten_o seventy_o one_o part_n of_o the_o diameter_n so_o that_o suppose_v the_o radius_fw-la to_o consist_v of_o 10000000_o equal_a part_n the_o arch_n of_o a_o quadrant_n will_v be_v between_o 15714285_o and_o 15_o 04225_o of_o the_o same_o part_n in_o our_o time_n ludovicus_n van_n cullen_n &_o willebrordus_fw-la snellius_n with_o joint_a endeavour_n have_v come_v yet_o near_o to_o the_o truth_n and_o pronounce_v from_o true_a principle_n that_o the_o arch_n of_o a_o quadrant_n put_v as_o before_o 10000000_o for_o radius_fw-la differ_v not_o one_o whole_a unity_n from_o the_o number_n 15707963_o which_o if_o they_o have_v exhibit_v their_o arithmetical_a operation_n and_o no_o man_n have_v discover_v any_o error_n in_o that_o long_a work_n of_o they_o have_v be_v demonstrate_v by_o they_o this_o be_v the_o further_a progress_n that_o have_v be_v make_v by_o the_o way_n of_o number_n and_o they_o that_o have_v proceed_v thus_o far_o deserve_v the_o praise_n of_o industry_n nevertheless_o if_o we_o consider_v the_o benefit_n which_o be_v the_o scope_n at_o which_o all_o speculation_n shall_v aim_v the_o improvement_n they_o have_v make_v have_v be_v little_a or_o none_o for_o any_o ordinary_a man_n may_v much_o soon_o &_o more_o accurate_o find_v a_o straight_a line_n equal_a to_o the_o perimeter_n of_o a_o circle_n and_o consequent_o square_v the_o circle_n by_o wind_v a_o small_a thread_n about_o a_o give_v cylinder_n than_o any_o geometrician_n shall_v do_v the_o same_o by_o divide_v the_o radius_fw-la into_o 10000000_o equal_a part_n but_o though_o the_o length_n of_o the_o circumference_n be_v exact_o set_v out_o either_o by_o number_n or_o mechanical_o or_o only_o by_o chance_n yet_o this_o will_v contribute_v no_o help_n at_o all_o towards_o the_o section_n of_o angles_n unless_o happy_o these_o two_o problem_n to_o divide_v a_o give_v angle_n according_a to_o any_o proportion_n assign_v and_o to_o find_v a_o
together_o take_v so_o that_o the_o part_n next_o the_o vertex_fw-la be_v triple_a to_o the_o other_o part_n or_o to_o the_o whole_a straight_a line_n as_o 3_o to_o 4._o for_o let_v a_o b_o c_o in_o the_o 9th_o fig._n be_v the_o sector_n of_o a_o sphere_n who_o vertex_fw-la be_v the_o centre_n of_o the_o sphere_n a_o who_o axis_n be_v a_o d_o and_o the_o circle_n upon_o b_o c_z be_v the_o common_a base_a of_o the_o portion_n of_o the_o sphere_n and_o of_o the_o cone_n who_o vertex_fw-la be_v a_o the_o axis_n of_o which_o portion_n be_v e_o d_o and_o the_o half_a thereof_o f_o d_o and_o the_o axis_n of_o the_o cone_n a_o e._n last_o let_v a_o g_o be_v ¾_n of_o the_o straight_a line_n a_o f._n i_o say_v g_o be_v the_o centre_n of_o equiponderation_n of_o the_o sector_n a_o b_o c._n let_v the_o straight_a line_n f_o h_n be_v draw_v of_o any_o length_n make_v right_a angle_n with_o a_o f_o at_o f_o and_o draw_v the_o straight_a line_n a_o h_n let_v the_o triangle_n a_o f_o h_n be_v make_v then_o upon_o the_o same_o centre_n a_o let_v any_o arch_n i_o k_o be_v draw_v cut_v a_o d_o in_o l_o and_o its_o chord_n cut_v a_o d_o in_o m_n and_o divide_v m_n l_o equal_o in_o n_o let_v n_o o_o be_v draw_v parallel_n to_o the_o straight_a line_n f_o h_n and_o meet_v with_o the_o straight_a line_n a_o h_n in_o o._n see_v now_o b_o d_o c_z be_v the_o spherical_a superficies_n of_o the_o portion_n cut_v off_o with_o a_o plain_a pass_v through_o b_o c_z and_o cut_v the_o axis_n at_o right_a angle_n and_o see_v f_o h_n divide_v e_o d_o the_o axis_n of_o the_o portion_n into_o two_o equal_a part_n in_o f_o the_o centre_n of_o equiponderation_n of_o the_o superficies_n b_o d_o c_o will_n be_v in_o f_o by_o the_o 8_o article_n and_o for_o the_o same_o reason_n the_o centre_n of_o equiponderation_n of_o the_o superficies_n i_o l_o k_n k_o be_v in_o the_o straight_a line_n a_o c_o will_v be_v in_o n._n and_o in_o like_a manner_n if_o there_o be_v draw_v between_o the_o centre_n of_o the_o sphere_n a_o and_o the_o outermost_a spherical_a superficies_n of_o the_o sector_n arch_n infinite_a in_o number_n the_o centre_n of_o equiponderation_n of_o the_o spherical_a superficies_n in_o which_o those_o arch_n be_v will_v be_v find_v to_o be_v in_o that_o part_n of_o the_o axis_n which_o be_v intercept_v between_o the_o superficies_n itself_o and_o a_o plain_a pass_v along_o by_o the_o chord_n of_o the_o arch_n and_o cut_v the_o axis_n in_o the_o middle_n at_o right_a angle_n let_v it_o now_o be_v suppose_v that_o the_o moment_n of_o the_o outermost_a spherical_a superficies_n b_o d_o c_o be_v f_o h._n see_v therefore_o the_o superficies_n b_o d_o c_o be_v to_o the_o superficies_n i_o l_o k_o in_o proportion_n duplicate_v to_o that_o of_o the_o arch_n b_o d_o c_o to_o the_o arch_n i_o l_o k_n that_o be_v of_o b_o e_z to_z ay_o m_o that_o be_v of_o f_o h_n to_o n_o o_o let_v it_o be_v as_o f_o h_n to_o n_o o_o so_o n_o o_o to_o another_o n_o p_o and_o again_o as_o n_o o_o to_o n_o p_o so_o n_o p_o to_o another_o n_o q_o and_o let_v this_o be_v do_v in_o all_o the_o straight_a line_n parallel_v to_o the_o base_a f_o h_n that_o can_v possible_o be_v draw_v between_o the_o base_a and_o the_o vertex_fw-la of_o the_o triangle_n a_o f_o h._n if_o then_o through_o all_o the_o point_n q_o there_o be_v draw_v the_o crooked_a line_n a_o q_o h_n the_o figure_n a_o f_o h_o q_o a_o will_v be_v the_o compliment_n of_o the_o first_o three-sided_n figure_n of_o two_o mean_n and_o the_o same_o will_v also_o be_v the_o moment_n of_o all_o the_o spherical_a superficies_n of_o which_o the_o solid_a sector_n a_o b_o c_o d_o be_v compound_v and_o by_o consequent_a the_o moment_n of_o the_o sector_n itself_o let_v now_o f_o h_n be_v understand_v to_o be_v the_o semidiameter_n of_o the_o base_a of_o a_o right_a cone_n who_o side_n be_v a_o h_n and_o axis_n a_o f._n wherefore_o see_v the_o base_n of_o the_o cones_n which_o pass_v through_o f_o and_o n_o and_o the_o rest_n of_o the_o point_n of_o the_o axis_n be_v in_o proportion_n duplicate_v to_o that_o of_o the_o straight_a line_n f_o h_n and_o n_o o_o etc._n etc._n the_o moment_n of_o all_o the_o base_n together_o that_o be_v of_o the_o whole_a cone_n will_v be_v the_o figure_n itself_o a_o f_o h_o q_o a_o and_o therefore_o the_o centre_n of_o equiponderation_n of_o the_o cone_n a_o f_o h_n be_v the_o same_o with_o that_o of_o the_o solid_a sector_n wherefore_o see_v a_o g_z be_v ¾_n of_o the_o axis_n a_o f_o the_o centre_n of_o equiponderation_n of_o the_o cone_n a_o f_o h_n be_v in_o g_z and_o therefore_o the_o centre_n of_o the_o solid_a sector_n be_v in_o g_o also_o and_o divide_v the_o part_n a_o f_o of_o the_o axis_n so_o that_o a_o g_o be_v triple_a to_o g_o f_o that_o be_v a_o g_z be_v to_o a_o f_o as_o 3_o to_o 4_o which_o be_v to_o be_v demonstrate_v note_v that_o when_o the_o sector_n be_v a_o hemisphere_n the_o axis_n of_o the_o cone_n vanish_v into_o that_o point_n which_o be_v the_o centre_n of_o the_o sphere_n and_o therefore_o it_o add_v nothing_o to_o half_a the_o axis_n of_o the_o portion_n wherefore_o if_o in_o the_o axis_n of_o the_o hemisphere_n there_o be_v take_v from_o the_o centre_n ¾_n of_o half_a the_o axis_n that_o be_v 3_o â—_o of_o the_o semidiameter_n of_o the_o sphere_n there_o will_v be_v the_o centre_n of_o equiponderation_n of_o the_o hemisphere_n chap._n xxiv_o of_o refraction_n and_o reflection_n 1_o definition_n 2_o in_o perpendicular_a motion_n there_o be_v no_o refraction_n 3_o thing_n throw_v out_o of_o a_o thin_a into_o a_o thick_a medium_fw-la be_v so_o refract_v that_o the_o angle_n refract_v be_v great_a than_o the_o angle_n of_o inclination_n 4_o endeavour_v which_o from_o one_o point_n tend_v every_o way_n will_v be_v so_o refract_v at_o that_o the_o sine_fw-la of_o the_o angle_n refract_v will_v be_v to_o the_o sine_fw-la of_o the_o angle_n of_o inclination_n as_o the_o density_n of_o the_o first_o medium_fw-la be_v to_o the_o density_n of_o the_o second_o medium_fw-la reciprocal_o take_v 5_o the_o sine_z of_o the_o refracted_a angle_n in_o one_o inclination_n be_v to_o the_o sine_fw-la of_o the_o refracted_a angle_n in_o another_o inclination_n as_o the_o sine_fw-la of_o the_o angle_n of_o that_o inclination_n be_v to_o the_o sine_fw-la of_o the_o angle_n of_o this_o inclination_n 6_o if_o two_o line_n of_o incidence_n have_v equal_a inclination_n be_v the_o one_o in_o a_o thin_a the_o other_o in_o a_o thick_a medium_fw-la the_o sine_fw-la of_o the_o angle_n of_o inclination_n will_v be_v a_o mean_a proportional_a between_o the_o two_o sin_n of_o the_o refracted_a angle_n 7_o if_o the_o angle_n of_o inclination_n be_v semirect_v and_o the_o line_n of_o inclination_n be_v in_o the_o thick_a medium_fw-la and_o the_o proportion_n of_o their_o density_n be_v the_o same_o with_o that_o of_o the_o diagonal_a to_o the_o side_n of_o a_o square_a and_o the_o separate_a superficies_n be_v plain_a the_o refracted_a line_n will_v be_v in_o the_o separate_a superficies_n 8_o if_o a_o body_n be_v carry_v in_o a_o straight_a line_n upon_o another_o body_n and_o do_v not_o penetrate_v the_o same_o but_o be_v reflect_v from_o it_o the_o angle_n of_o reflection_n will_v be_v equal_a to_o the_o angle_n of_o incidence_n 9_o the_o same_o happen_v in_o the_o generation_n of_o motion_n in_o the_o line_n of_o incidence_n 1_o definition_n 1_o refraction_n be_v the_o break_n of_o that_o straight_a line_n in_o which_o a_o body_n be_v move_v or_o its_o action_n will_v proceed_v in_o one_o and_o the_o same_o medium_fw-la into_o two_o straight_a line_n by_o reason_n of_o the_o different_a nature_n of_o the_o two_o medium_n 2_o the_o former_a of_o these_o be_v call_v the_o line_n of_o incidence_n the_o late_a the_o refract_v line_n 3_o the_o point_n of_o refraction_n be_v the_o common_a point_n of_o the_o line_n of_o incidence_n and_o of_o the_o refracted_a line_n 4_o the_o refract_a superficies_n which_o also_o be_v the_o separate_a superficies_n of_o the_o two_o medium_n be_v that_o in_o which_o be_v the_o point_n of_o refraction_n 5_o the_o angle_n refract_v be_v that_o which_o the_o refracted_a line_n make_v in_o the_o point_n of_o refraction_n with_o that_o line_n which_o from_o the_o same_o point_n be_v draw_v perpendicular_a to_o the_o separate_a superficies_n in_o a_o different_a medium_fw-la 6_o the_o angle_n of_o refraction_n be_v that_o which_o the_o refracted_a line_n make_v with_o the_o line_n of_o incidence_n produce_v 7_o the_o angle_n of_o inclination_n be_v
in_o the_o point_n b_o will_v be_v compound_v of_o the_o proportion_n of_o a_o b_o to_o a_o n_o and_o of_o a_o n_o to_o a_o b._n wherefore_o set_v in_o order_n a_o b_o a_o n_o a_o b_o the_o moment_n of_o o_o to_o the_o moment_n of_o c_o will_v be_v as_o the_o first_o to_o the_o last_o that_o be_v as_o a_o b_o to_o a_o b._n their_o moment_n therefore_o be_v equal_a and_o consequent_o the_o plain_n which_o pass_v through_o a_o will_v by_o the_o five_o definition_n be_v a_o plain_a of_o equiponderation_n wherefore_o they_o will_v be_v equal_o poise_v as_o be_v to_o be_v prove_v now_o the_o converse_n of_o this_o be_v manifest_a for_o if_o there_o be_v equiponderation_n and_o the_o proportion_n of_o the_o weight_n and_o distance_n be_v not_o reciprocal_a then_o both_o the_o weight_n will_v always_o have_v the_o same_o moment_n although_o one_o of_o they_o have_v more_o weight_n add_v to_o it_o or_o its_o distance_n change_v corollary_n when_o ponderant_n be_v of_o the_o same_o species_n and_o their_o moment_n be_v equal_a their_o magnitude_n and_o distance_n from_o the_o centre_n of_o the_o scale_n will_v be_v reciprocal_o proportional_a for_o in_o homogeneous_a body_n it_o be_v as_o weight_n to_o waight_n so_o magnitude_n to_o magaltude_n 8_o if_o to_o the_o whole_a length_n of_o the_o beam_n there_o be_v apply_v a_o parallelogram_n or_o a_o parallelopipedum_n or_o a_o prisma_fw-la or_o a_o cylinder_n or_o the_o superficies_n of_o a_o cylinder_n ot_fw-mi of_o a_o sphere_n or_o of_o any_o portion_n of_o a_o sphere_n or_o prisma_fw-la the_o part_n of_o any_o of_o they_o cut_v off_o with_o plain_n parallel_v to_o the_o base_a will_v have_v their_o moment_n in_o the_o same_o proportion_n with_o the_o part_n of_o a_o triangle_n which_o have_v its_o vertex_fw-la in_o the_o centre_n of_o the_o scale_n and_o for_o one_o of_o its_o side_n the_o beam_n itself_o which_o part_n be_v cut_v off_o by_o plain_n parallel_v to_o the_o base_a first_o let_v the_o rectangled_a parallelogram_n a_o b_o c_o d_o in_o the_o four_o figure_n be_v apply_v to_o the_o whole_a length_n of_o the_o beam_n a_o b_o and_o produce_v c_o b_o howsoever_o to_o e_z let_v the_o triangle_n a_o b_o e_o be_v describe_v let_v now_o any_o part_n of_o the_o parallelogram_n as_o a_o f_o be_v cut_v off_o by_o the_o plain_n f_o g_o parallel_n to_o the_o base_a c_o b_o and_o let_v f_o g_z be_v produce_v to_o a_o e_o in_o the_o point_n h._n i_o say_v the_o moment_n of_o the_o whole_a a_o b_o c_o d_o to_o the_o moment_n of_o its_o part_n a_o f_o be_v as_o the_o triangle_n a_o b_o e_o to_o the_o triangle_n a_o g_o h_n that_o be_v in_o proportion_n duplicate_v to_o that_o of_o the_o distance_n from_o the_o centre_n of_o the_o scale_n for_o the_o parallelogram_n a_o b_o c_o d_o being_n divide_v into_o equal_a part_n infinite_a in_o number_n by_o straight_a line_n draw_v parallel_n to_o the_o base_a and_o suppose_v the_o moment_n of_o the_o straight_a line_n c_o b_o to_o be_z b_o e_z the_o moment_n of_o the_o straight_a line_n f_o g_o will_v by_o the_o seven_o article_n be_v g_z h_z and_o the_o moment_n of_o all_o the_o straight_a line_n of_o that_o parallelogram_n will_v be_v so_o many_o straight_a line_n in_o the_o triangle_n a_o b_o e_o drawn_z parallel_n to_o the_o base_a b_o e_z all_o which_o parallel_v together_o take_v be_v the_o moment_n of_o the_o whole_a parallelogram_n a_o b_o c_o d_o and_o the_o same_o parallel_n do_v also_o constitute_v the_o superficies_n of_o the_o triangle_n a_o b_o e._n wherefore_o the_o moment_n of_o the_o parallelogram_n a_o b_o c_o d_o be_v the_o triangle_n a_o b_o e_o and_o for_o the_o same_o reason_n the_o moment_n of_o the_o parallelogram_n a_o f_o be_v the_o triangle_n a_o g_o h_n and_o therefore_o the_o moment_n of_o the_o whole_a parallelogram_n to_o the_o moment_n of_o a_o parallelogram_n which_o be_v part_n of_o the_o same_o be_v as_o the_o triangle_n a_o b_o e_o to_o the_o triangle_n a_o g_o h_n or_o in_o proportion_n duplicate_v to_o that_o of_o the_o beam_n to_o which_o they_o be_v apply_v and_o what_o be_v here_o demonstrate_v in_o the_o case_n of_o a_o parallelogram_n may_v be_v understand_v to_o serve_v for_o that_o of_o a_o cylinder_n and_o of_o a_o prisma_fw-la and_o their_o superficies_n as_o also_o for_o the_o superficies_n of_o a_o sphere_n of_o a_o hemisphere_n or_o any_o portion_n of_o a_o sphere_n for_o the_o part_n of_o the_o superficies_n of_o a_o sphere_n have_v the_o same_o proportion_n with_o that_o of_o the_o part_n of_o the_o axis_n cut_v off_o by_o the_o same_o parallel_n by_o which_o the_o part_n of_o the_o superficies_n be_v cut_v off_o as_o archimedes_n have_v demonstrate_v and_o therefore_o when_o the_o part_n of_o any_o of_o these_o figure_n be_v equal_a and_o at_o equal_a distance_n from_o the_o centre_n of_o the_o scale_n their_o moment_n also_o be_v equal_a in_o the_o same_o manner_n as_o they_o be_v in_o parallelograms_n second_o let_v the_o parallelogram_n a_o k_o i_o b_o not_o be_v rectangle_v the_o straight_a line_n i_o b_o will_v nevertheless_o press_v the_o point_n b_o perpendicular_o in_o the_o straight_a line_n b_o e_z &_o the_o straight_a line_n l_o g_o will_n press_v the_o point_n g_z perpendicular_o in_o the_o straight_a line_n g_o h_n and_o all_o the_o rest_n of_o the_o straight_a line_n which_o be_v parallel_v to_o i_o b_o will_v do_v the_o like_a whatsoever_o therefore_o the_o moment_n be_v which_o be_v assign_v to_o the_o straight_a line_n i_o b_o as_o here_o for_o example_n it_o be_v suppose_v to_o be_v b_o e_o if_o a_o e_o be_v draw_v the_o moment_n of_o the_o whole_a parallelogram_n a_o i_o will_v be_v the_o triangle_n a_o b_o e_o and_o the_o moment_n of_o the_o part_n a_o l_o will_v be_v the_o triangle_n a_o g_o h._n wherefore_o the_o moment_n of_o any_o ponderant_fw-la which_o have_v its_o side_n equal_o apply_v to_o the_o beam_n whether_o they_o be_v apply_v perpendicular_o or_o oblique_o will_v be_v always_o to_o the_o moment_n of_o a_o part_n of_o the_o same_o in_o such_o proportion_n as_o the_o whole_a triangle_n have_v to_o a_o part_n of_o the_o same_o cut_v off_o by_o a_o plain_a which_o be_v parallel_n to_o the_o base_a 9_o the_o centre_n of_o equiponderation_n of_o any_o figure_n which_o be_v deficient_a according_a to_o commensurable_a proportion_n of_o the_o altitude_n and_o base_a diminish_v and_o who_o complete_a figure_n be_v either_o a_o parallelogram_n or_o a_o cylinder_n or_o a_o parallelopipedum_n divide_v the_o axis_n so_o that_o the_o part_n next_o the_o vertex_fw-la to_o the_o other_o part_n be_v as_o the_o complete_a figure_n to_o the_o deficient_a figure_n for_o let_v c_o i_o a_o p_o e_o in_o the_o 5_o figure_n be_v a_o deficient_a figure_n who_o axis_n be_v a_o b_o and_o who_o complete_a figure_n be_v c_o d_o f_o e_o and_o let_v the_o axis_n a_o b_o be_v so_o divide_v in_o z_o that_o a_o z_o be_v to_o z_o b_o as_o c_o d_o f_o e_o be_v to_z c_o i_o a_o p_o e._n i_o say_v the_o centre_n of_o equiponderation_n of_o the_o figure_n c_o i_o a_o p_o e_o will_n be_v in_o the_o point_n z._n first_o that_o the_o centre_n of_o equiponderation_n of_o the_o figure_n c_o i_o ap_n e_z be_v somewhere_o in_o the_o axis_n a_o b_o be_v manifest_a of_o itself_o and_o therefore_o a_o b_o be_v a_o diameter_n of_o equiponderation_n let_v a_o e_o be_v draw_v and_o let_v b_o e_o be_v put_v for_o the_o moment_n of_o the_o straight_a line_n c_o e_o the_o triangle_n a_o b_o e_o will_n therefore_o by_o the_o 3d_o article_n be_v the_o moment_n of_o the_o complete_a figure_n c_o d_o f_o e._n let_v the_o axis_n a_o b_o be_v equal_o divide_v in_o l_o and_o let_v g_o l_o h_o be_v draw_v parallel_n and_o equal_a to_o the_o straight_a line_n c_o e_o cut_v the_o crooked_a line_n c_o i_o a_o p_o e_o in_o i_o and_o p_o and_o the_o straight_a line_n a_o c_o and_o a_o e_o in_o k_o and_o m._n moreover_o let_v z_o o_fw-fr be_v draw_v parallel_n to_o the_o same_o c_o e_z and_o let_v it_o be_v as_o l_o g_z to_o l_o i_o so_o l_o m_o to_o another_o l_o n_o and_o let_v the_o same_o be_v do_v in_o all_o the_o rest_n of_o the_o straight_a line_n possible_a parallel_n to_o the_o base_a and_o through_o all_o the_o point_n n_o let_v the_o line_n a_o n_o e_o be_v draw_v the_o threesided_n figure_n a_o n_o e_o b_o will_n therefore_o be_v the_o moment_n of_o the_o figure_n c_o i_o a_o p_o e._n now_o the_o triangle_n a_o b_o e_o be_v by_o the_o 9th_o article_n of_o the_o 17_o chapter_n to_o the_o threesided_n figure_n a_o n_o e_o b_o as_o a_o b_o c_o d_o +_o
a_o f_o and_o a_o f_o be_v draw_v a_o f_o therefore_o will_v be_v the_o sine_fw-la of_o the_o angle_n of_o inclination_n of_o the_o straight_a line_n a_o b_o and_o a_o f_z the_o sine_z of_o the_o angle_n of_o inclination_n of_o the_o straight_a line_n a_o h_z which_o two_o inclination_n be_v by_o construction_n make_v equal_a i_o say_v as_o the_o density_n of_o the_o medium_fw-la in_o which_o be_v b_o c_z and_o b_o c_o be_v to_o the_o density_n of_o the_o medium_fw-la in_o which_o be_v b_o d_o and_o b_o d_o so_o be_v the_o sine_fw-la of_o the_o angle_n refract_v to_o the_o sine_fw-la of_o the_o angle_n of_o inclination_n let_v the_o straight_a line_n f_o g_z be_v draw_v parallel_n to_o the_o straight_a line_n a_o b_o meet_v with_o the_o straight_a line_n b_o b_o produce_v in_o g._n see_v therefore_o a_o f_o and_o b_o g_o be_v also_o parallel_n they_o will_v be_v equal_a and_o consequent_o the_o endeavour_n in_o a_o f_o be_v propagate_v in_o the_o same_o time_n in_o which_o the_o endeavour_n in_o b_o g_o will_v be_v propagate_v if_o the_o medium_fw-la be_v of_o the_o same_o density_n but_o because_o b_o g_z be_v in_o a_o thick_a medium_fw-la that_o be_v in_o a_o medium_fw-la which_o resist_v the_o endeavour_n more_o than_o the_o medium_fw-la in_o which_o a_o f_o be_v the_o endeavour_n will_v be_v propagate_v less_o in_o b_o g_z than_o in_o a_o f_o according_a to_o the_o proportion_n which_o the_o density_n of_o the_o medium_fw-la in_o which_o a_o f_o be_v have_v to_o the_o density_n of_o the_o medium_fw-la in_o which_o b_o g_z be_v let_v therefore_o the_o density_n of_o the_o medium_fw-la in_o which_o b_o g_z be_v be_v to_o the_o density_n of_o the_o medium_fw-la in_o which_o a_o f_o be_v as_o b_o g_z be_v to_z b_o h_n and_o let_v the_o measure_n of_o the_o time_n be_v the_o radius_fw-la of_o the_o circle_n let_v h_n i_o be_v draw_v parallel_n to_o b_o d_o meet_v with_o the_o circumference_n in_o i_o and_o from_o the_o point_n i_o let_v i_o k_o be_v draw_v perpendicular_a to_o b_o d_o which_o be_v do_v b_o h_n and_o i_o k_o will_v be_v equal_a and_o i_o k_o will_v be_v to_o a_o f_o as_o the_o density_n of_o the_o medium_fw-la in_o which_o be_v a_o f_o be_v to_o the_o density_n of_o the_o medium_fw-la in_o which_o be_v i_o k._n see_v therefore_o in_o the_o time_n a_o b_o which_o be_v the_o radius_fw-la of_o the_o circle_n the_o endeavour_n be_v propagate_v in_o a_o f_o in_o the_o thin_a medium_fw-la it_o will_v be_v propagate_v in_o the_o same_o time_n that_o be_v in_o the_o time_n b_o i_o in_o the_o thick_a medium_fw-la from_o k_o to_o i._o therefore_o b_o i_o be_v the_o refracted_a line_n of_o the_o line_n of_o incidence_n a_o b_o and_o i_o k_o be_v the_o sine_fw-la of_o the_o angle_n refract_v and_o a_o f_o the_o sine_fw-la of_o the_o angle_n of_o inclination_n wherefore_o see_v i_o k_o be_v to_o a_o f_o as_o the_o density_n of_o the_o medium_fw-la in_o which_o be_v a_o f_o to_o the_o density_n of_o the_o medium_fw-la in_o which_o be_v i_o k_o it_o will_v be_v as_o the_o density_n of_o the_o medium_fw-la in_o which_o be_v a_o f_o or_o b_o c_z to_o the_o density_n of_o the_o medium_fw-la in_o which_o be_v i_o k_o or_o b_o d_o so_o the_o sine_fw-la of_o the_o angle_n refract_v to_o the_o sine_fw-la of_o the_o angle_n of_o inclination_n and_o by_o the_o same_o reason_n it_o may_v be_v show_v that_o as_o the_o density_n of_o the_o thin_a medium_fw-la be_v to_o the_o density_n of_o the_o thick_a medium_fw-la so_o will_v k_o i_o the_o sine_fw-la of_o the_o angle_n refract_v be_v to_o a_o f_o the_o sine_fw-la of_o the_o angle_n of_o inclination_n second_o let_v the_o body_n which_o endeavour_v every_o way_n be_v in_o the_o thick_a medium_fw-la at_o i._n if_o therefore_o both_o the_o medium_n be_v of_o the_o same_o density_n the_o endeavour_n of_o the_o body_n in_o i_o b_o will_v tend_v direct_o to_o l_o and_o the_o sine_fw-la of_o the_o angle_n of_o inclination_n l_o m_o will_v be_v equal_a to_o i_o k_o or_o b_o h._n but_o because_o the_o density_n of_o the_o medium_fw-la in_o which_o be_v ik_fw-mi to_o the_o density_n of_o the_o medium_fw-la in_o which_o be_v l_o m_o be_v as_o bh_n to_o b_o g_z that_o be_v to_o a_o f_o the_o endeavour_n will_v be_v propagate_v further_o in_o the_o medium_fw-la in_o which_o l_o m_o be_v then_o in_o the_o medium_fw-la in_o which_o i_o k_o be_v in_o the_o proportion_n of_o density_n to_o density_n that_o be_v of_o m_o l_o to_o a_o f._n wherefore_o b_o a_o be_v draw_v the_o angle_n refract_v will_v be_v c_o b_o a_o and_o its_z sine_z a_o f._n but_o l_o m_o be_v the_o sine_fw-la of_o the_o angle_n of_o inclination_n and_o therefore_o again_o as_o the_o density_n of_o one_o medium_fw-la be_v to_o the_o density_n of_o the_o different_a medium_fw-la so_o reciprocal_o be_v the_o sine_fw-la of_o the_o angle_n refract_v to_o the_o sine_fw-la of_o the_o angle_n of_o inclination_n which_o be_v to_o be_v demonstrate_v in_o this_o demonstration_n i_o have_v make_v the_o separate_a superficies_n b_o b_o plain_a by_o construction_n but_o though_o it_o be_v concave_a or_o convex_a the_o theorem_a will_v nevertheless_o be_v true_a for_o the_o refraction_n be_v make_v in_o the_o point_n b_o of_o the_o plain_a separate_a superficies_n if_o a_o crooked_a line_n as_o p_o qui_fw-fr be_v draw_v touch_v the_o separate_a line_n in_o the_o point_n b_o neither_o the_o refracted_a line_n b_o i_o nor_o the_o perpendicular_a b_o d_o will_v be_v alter_v and_o the_o refracted_a angle_n k_o b_o i_o as_o also_o its_o sine_fw-la k_o i_o will_v be_v still_o the_o same_o they_o be_v 5_o the_o sine_z of_o the_o angle_n refract_v in_o one_o inclination_n be_v to_o the_o sine_fw-la of_o the_o angle_n refract_v in_o another_o inclination_n as_o the_o sine_fw-la of_o the_o angle_n of_o that_o inclination_n to_o the_o sine_fw-la of_o the_o angle_n of_o this_o inclination_n for_o see_v the_o sine_fw-la of_o the_o refracted_a angle_n be_v to_o the_o sine_fw-la of_o the_o angle_n of_o inclination_n whatsoever_o that_o inclination_n be_v as_o the_o density_n of_o one_o medium_fw-la to_o the_o density_n of_o the_o other_o medium_fw-la the_o proportion_n of_o the_o sine_fw-la of_o the_o refracted_a angle_n to_o the_o sine_fw-la of_o the_o angle_n of_o inclination_n will_v be_v compound_v of_o the_o proportion_n of_o density_n to_o density_n and_o of_o the_o sine_fw-la of_o the_o angle_n of_o one_o inclination_n to_o the_o sine_fw-la of_o the_o angle_n of_o the_o other_o inclination_n but_o the_o proportion_n of_o the_o density_n in_o the_o same_o homogeneous_a body_n be_v suppose_v to_o be_v the_o same_o wherefore_o refracted_a angle_n in_o different_a inclination_n be_v as_o the_o sin_n of_o the_o angle_n of_o those_o inclination_n which_o be_v to_o be_v demonstrate_v 6_o if_o two_o line_n of_o incidence_n have_v equal_a inclination_n be_v the_o one_o in_o a_o thin_a the_o other_o in_o a_o thick_a medium_fw-la the_o sine_fw-la of_o the_o angle_n of_o their_o inclination_n will_v be_v a_o mean_a proportional_a between_o the_o two_o sin_n of_o their_o angle_n refract_v for_o let_v the_o straight_a line_n ab_fw-la in_o the_o same_o 3d_o figure_n have_v its_o inclination_n in_o the_o thin_a medium_fw-la and_o be_v refract_v in_o the_o thick_a medium_fw-la in_o b_o i_o and_o let_v e_o b_o have_v as_o much_o inclination_n in_o the_o thick_a medium_fw-la and_o be_v refract_v in_o the_o thin_a medium_fw-la in_o b_o saint_n and_o let_v r_o s_o the_o sine_fw-la of_o the_o angle_n refract_v be_v draw_v i_o say_v the_o straight_a line_n r_o saint_n a_o f_o and_o i_o k_o be_v in_o continual_a proportion_n for_o it_o be_v as_o the_o density_n of_o the_o thick_a medium_fw-la to_o the_o density_n of_o the_o thin_a medium_fw-la so_o r_o s_o to_o a_o f._n but_o it_o be_v also_o as_o the_o density_n of_o the_o same_o thick_a medium_fw-la to_o that_o of_o the_o same_o thin_a medium_fw-la so_o of_o to_o ik_fw-mi wherefore_o r_o s._n a_o f_o a_o f._n i_o k_o be_v propoortional_n that_o be_v r_o s_o a_o f_o and_o i_o k_o be_v in_o continual_a proportion_n and_o a_o f_o be_v the_o mean_a proportional_a which_o be_v to_o be_v prove_v 7_o if_o the_o angle_n of_o inclination_n be_v semirect_v and_o the_o line_n of_o inclination_n be_v in_o the_o thick_a medium_fw-la and_o the_o proportion_n of_o the_o density_n be_v as_o that_o of_o a_o diagonal_a to_o the_o side_n of_o its_o square_n and_o the_o separate_a superficies_n be_v plain_a the_o refracted_a line_n will_v be_v in_o that_o separate_a superficies_n for_o in_o the_o circle_n a_o c_o in_o the_o four_o figure_n let_v the_o angle_n
of_o the_o niter_n be_v generate_v that_o vehement_a motion_n and_o inflammation_n and_o last_o when_o there_o be_v no_o more_o action_n from_o the_o niter_fw-la that_o be_v to_o say_v when_o the_o volatile_a part_n of_o the_o niter_n be_v fly_v out_o there_o be_v find_v about_o the_o side_n a_o certain_a white_a substance_n which_o be_v throw_v again_o into_o the_o fire_n will_v grow_v red_a hot_a again_o but_o will_v not_o be_v dissipate_v at_o least_o unless_o the_o fire_n be_v augment_v if_o now_o a_o possible_a cause_n of_o this_o be_v find_v out_o the_o same_o will_v also_o be_v a_o possible_a cause_n why_o a_o grain_n of_o gunpowder_n set_v on_o fire_n do_v expand_v itself_o with_o such_o vehement_a motion_n and_o shine_v and_o it_o may_v be_v cause_v in_o this_o manner_n let_v the_o particle_n of_o which_o niter_n consist_v be_v suppose_v to_o be_v some_o of_o they_o hard_o other_o watery_a and_o the_o rest_n aethereal_a also_o let_v the_o hard_a particle_n be_v suppose_v to_o be_v spherical_o hollow_a like_o small_a bubble_n so_o that_o many_o of_o they_o grow_v together_o may_v constitute_v a_o body_n who_o little_a cavern_n be_v fill_v with_o a_o substance_n which_o be_v either_o watery_a or_o aethereal_a or_o both_o as_o soon_o therefore_o as_o the_o hard_a particle_n be_v dissipate_v the_o watery_a and_o aethereal_a particle_n will_v necessary_o fly_v out_o and_o as_o they_o fly_v of_o necessity_n blow_v strong_o the_o burn_a coal_n and_o brimstone_n which_o be_v mingle_v together_o whereupon_o there_o will_v follow_v a_o great_a expansion_n of_o light_n with_o vehement_a flame_n and_o a_o violent_a dissipation_n of_o the_o particle_n of_o the_o niter_n the_o brimstone_n and_o the_o coal_n wherefore_o i_o have_v give_v a_o possible_a cause_n of_o the_o force_n of_o fire_a gunpowder_n it_o be_v manifest_a from_o hence_o that_o for_o the_o render_n of_o the_o cause_n why_o a_o bullet_n of_o lead_n or_o iron_n shoot_v from_o a_o piece_n of_o ordnance_n fly_v with_o so_o great_a velocity_n there_o be_v no_o necessity_n to_o introduce_v such_o rarefaction_n as_o by_o the_o common_a definition_n of_o it_o make_v the_o same_o matter_n to_o have_v sometime_o more_o sometime_o less_o quantity_n which_o be_v unconceivable_a for_o every_o thing_n be_v say_v to_o be_v great_a or_o less_o as_o it_o have_v more_o or_o less_o quantity_n the_o violence_n with_o which_o a_o bullet_n be_v thrust_v out_o of_o a_o gun_n proceed_v from_o the_o swiftness_n of_o the_o small_a particle_n of_o the_o fire_a powder_n at_o least_o it_o may_v proceed_v from_o that_o cause_n without_o the_o supposition_n of_o any_o empty_a space_n 11_o beside_o by_o the_o attrition_n or_o rub_v of_o one_o body_n against_o another_o as_o of_o wood_n against_o wood_n we_o find_v that_o not_o only_o a_o certain_a degree_n of_o heat_n but_o fire_v itself_o be_v sometime_o generate_v for_o such_o motion_n be_v the_o reciprocation_n of_o pressure_n sometime_o one_o way_n sometime_o the_o other_o and_o by_o this_o reciprocation_n whatsoever_o be_v fluid_a in_o both_o the_o piece_n of_o wood_n be_v force_v hither_o and_o thither_o and_o consequent_o to_o a_o endeavour_n of_o get_v out_o and_o at_o last_o by_o break_v out_o make_v fire_n 12_o now_o light_n be_v distinguish_v into_o first_o second_o third_z and_o so_o on_o infinite_o and_o we_o call_v that_o first_o light_n which_o be_v in_o the_o first_o lucid_a body_n as_o the_o sun_n fire_n etc._n etc._n second_o that_o which_o be_v in_o such_o body_n as_o be_v not_o transparent_a be_v illuminate_v by_o the_o sun_n as_o the_o moon_n a_o wall_n etc._n etc._n and_o three_o that_o which_o be_v in_o body_n not_o transparent_a but_o illuminate_v by_o second_o light_n etc._n etc._n 13_o colour_n be_v light_a but_o trouble_a light_n namely_o such_o as_o be_v generate_v by_o perturb_a motion_n as_o shall_v be_v make_v manifest_a by_o the_o red_a yellow_a blue_a and_o purple_a which_o be_v generate_v by_o the_o interposition_n of_o a_o diaphanous_a prisma_fw-la who_o opposite_a base_n be_v triangular_a between_o the_o light_n and_o that_o which_o be_v enlighten_v for_o let_v there_o be_v a_o prisma_fw-la of_o glass_n or_o of_o any_o other_o transparent_a matter_n which_o be_v of_o great_a density_n than_o air_n and_o let_v the_o triangle_n abc_n be_v the_o base_a of_o this_o prisma_fw-la also_o let_v the_o straight_a line_n de_fw-fr be_v the_o diameter_n of_o the_o sun_n body_n have_v oblique_a position_n to_o the_o straight_a line_n ab_fw-la and_o let_v the_o sunbeam_n pass_v in_o the_o line_n da_fw-la and_o ebc_n and_o last_o let_v the_o straight_a line_n da_fw-la and_o aec_fw-la be_v produce_v indefinite_o to_o f_o and_o g._n see_v therefore_o the_o straight_a line_n da_fw-la by_o reason_n of_o the_o density_n of_o the_o glass_n be_v refract_v towards_o the_o perpendicular_a let_v the_o line_n refract_v at_o the_o point_n a_o be_v the_o straight_a line_n ah_o and_o again_o see_v the_o medium_fw-la below_o ac_fw-la be_v thin_a than_o that_o above_o it_o the_o other_o refraction_n which_o will_v be_v make_v there_o will_v diverge_n from_o the_o perpendicular_a let_v therefore_o this_o second_o refracted_a line_n be_v ai._n also_o let_v the_o same_o be_v do_v at_o the_o point_n c_o by_o make_v the_o first_o refract_v line_n to_o be_v ck_n and_o the_o second_o cl._n see_v therefore_o the_o cause_n of_o the_o refraction_n in_o the_o point_n a_o of_o the_o straight_a line_n of_o ab_fw-la be_v the_o excess_n of_o the_o resistance_n of_o the_o medium_fw-la in_o ab_fw-la above_o the_o resistance_n of_o the_o air_n there_o must_v of_o necessity_n be_v reaction_n from_o the_o point_n a_o towards_o the_o point_n b_o and_o consequent_o the_o medium_fw-la at_o a_o within_o the_o triangle_n abc_n will_v have_v its_o motion_n trouble_v that_o be_v to_o say_v the_o straight_a motion_n in_o of_o and_o ah_o will_v be_v mix_v with_o the_o transverse_a motion_n between_o the_o same_o of_o and_o ah_o represent_v by_o the_o short_a transverse_a line_n in_o the_o triangle_n afh_v again_o see_v at_o the_o point_n a_o of_o the_o straight_a line_n ac_fw-la there_o be_v a_o second_o refraction_n from_o ah_o in_o ai_fw-fr the_o motion_n of_o the_o medium_fw-la will_v again_o be_v perturb_v by_o reason_n of_o the_o transverse_a reaction_n from_o a_o towards_o c_o represent_v likewise_o by_o the_o short_a transverse_a line_n in_o the_o triangle_n ahi_n and_o in_o the_o same_o manner_n there_o be_v a_o double_a perturbation_n represent_v by_o the_o transverse_a line_n in_o the_o triangle_n cgk_n and_o ckl_n but_o as_o for_o the_o light_n between_o a_z and_o cg_n it_o will_v not_o be_v perturb_v because_o if_o there_o be_v in_o all_o the_o point_n of_o the_o straight_a line_n ab_fw-la and_o ac_fw-la the_o same_o action_n which_o be_v in_o the_o point_v a_o and_o c_o then_z the_o plain_a of_o the_o triangle_n cgk_n will_v be_v every_o where_o coincident_a with_o the_o plain_a of_o the_o triangle_n afh_v by_o which_o mean_v all_o will_v appear_v alike_o between_o a_o and_o c._n beside_o it_o be_v to_o be_v observe_v that_o all_o the_o reaction_n at_o a_o tend_v towards_o the_o illuminate_v part_n which_o be_v between_o a_o and_o c_o and_o consequent_o perturb_v the_o first_o light_n and_o on_o the_o contrary_a that_o all_o the_o reaction_n at_o c_o tend_v towards_o the_o part_n without_o the_o triangle_n or_o without_o the_o prisma_fw-la abc_n where_o there_o be_v none_o but_o second_o light_n and_o that_o the_o triangle_n afh_n show_v that_o perturbation_n of_o light_n which_o be_v make_v in_o the_o glass_n itself_o as_o the_o triangle_n ahi_n show_v that_o perturbation_n of_o light_n which_o be_v make_v below_o the_o glass_n in_o like_a manner_n that_o cgk_n show_v the_o perturbation_n of_o light_n within_o the_o glass_n and_o ckl_n that_o which_o be_v below_o the_o glass_n from_o whence_o there_o be_v four_o divers_a motion_n or_o four_o different_a illumination_n or_o colour_n who_o difference_n appear_v most_o manifest_o to_o the_o sense_n in_o a_o prisma_fw-la who_o base_a be_v a_o equilateral_a triangle_n when_o the_o sunbeam_n that_o pass_v through_o it_o be_v receive_v upon_o a_o white_a paper_n for_o the_o triangle_n afh_n appear_v red_a to_o the_o sense_n the_o triangle_n ahi_n yellow_a the_o triangle_n cgk_n green_n and_o approach_v to_o blue_n and_o last_o the_o triangle_n ckl_n appear_v purple_a it_o be_v therefore_o evident_a that_o when_o weak_a but_o first_o light_n pass_v through_o a_o more_o resist_a diaphanous_a body_n as_o glass_n the_o beam_n which_o fall_v upon_o it_o tranvers_o make_v redness_n and_o when_o the_o same_o first_o light_n be_v strong_a as_o it_o be_v in_o the_o thin_a medium_fw-la below_o the_o straight_a line_n ac_fw-la the_o transverse_a beam_n make_v yellowness_n also_o when_o second_o light_n be_v strong_a