indefinite_o that_o be_v to_o know_v as_o much_o as_o they_o can_v without_o propound_v to_o themselves_o any_o limit_a question_n or_o they_o inquire_v into_o the_o cause_n of_o some_o determine_a appearance_n or_o endeavour_v to_o find_v out_o the_o certainty_n of_o something_o in_o question_n as_o what_o be_v the_o cause_n of_o light_n of_o heat_n of_o gravity_n of_o a_o figure_n propound_v and_o the_o like_a or_o in_o what_o subject_n any_o propound_a accident_n be_v inhaerent_a or_o what_o may_v conduce_v most_o to_o the_o generation_n of_o some_o propound_a effect_n from_o many_o accident_n or_o in_o what_o manner_n particular_a cause_n ought_v to_o be_v compound_v for_o the_o production_n of_o some_o certain_a effect_n now_o according_a to_o this_o variety_n of_o thing_n in_o question_n sometime_o the_o analyticall_a method_n be_v to_o be_v use_v and_o sometime_o the_o syntheticall_a 4_o but_o to_o those_o that_o search_v after_o science_n indefinite_o which_o consist_v in_o 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a_o square_a this_o square_n be_v to_o be_v resolve_v into_o a_o plain_a terminate_v with_o a_o certain_a number_n of_o equal_a and_o straight_a line_n and_o right_a angle_n for_o by_o this_o resolution_n we_o have_v these_o thing_n universal_a or_o agreeable_a to_o all_o matter_n namely_o line_n plain_n which_o contain_v superficies_n terminate_v angle_n straightness_n rectitude_n and_o equality_n and_o if_o we_o can_v find_v out_o the_o cause_n of_o these_o we_o may_v compound_v they_o all_o together_o into_o the_o cause_n of_o a_o square_n again_o if_o any_o man_n propound_v to_o himself_o the_o conception_n of_o gold_n he_o may_v by_o resolve_v come_v to_o the_o idea_n of_o solid_a visible_a heavy_a that_o be_v tend_v to_o the_o centre_n of_o the_o earth_n or_o downward_o and_o many_o other_o more_o universal_a than_o gold_n itself_o and_o these_o he_o may_v resolve_v again_o till_o he_o come_v to_o such_o thing_n as_o be_v most_o universal_a and_o in_o this_o manner_n by_o 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savour_n etc._n etc._n any_o other_o cause_n then_o motion_n reside_v partly_o in_o the_o object_n that_o work_v upon_o our_o sense_n and_o partly_o in_o ourselves_o in_o such_o manner_n as_o that_o it_o be_v manifest_o some_o kind_n of_o motion_n though_o we_o can_v without_o ratiocination_n come_v to_o know_v what_o kind_n for_o though_o many_o can_v understand_v till_o it_o be_v in_o some_o sort_n demonstrate_v to_o they_o that_o all_o mutation_n consist_v in_o motion_n yet_o this_o happen_v not_o from_o any_o obscurity_n in_o the_o thing_n itself_o for_o it_o be_v not_o intelligible_a that_o any_o thing_n can_v depart_v either_o from_o rest_n or_o from_o the_o motion_n it_o have_v except_o by_o motion_n but_o either_o by_o have_v their_o natural_a discourse_n corrupt_v with_o former_a opinion_n receive_v from_o their_o master_n or_o else_o for_o this_o that_o they_o do_v not_o at_o all_o bend_v their_o mind_n to_o the_o inquire_v out_o of_o truth_n 6_o by_o the_o knowledge_n therefore_o of_o universall_n and_o of_o their_o cause_n which_o be_v the_o first_o principle_n by_o which_o we_o know_v the_o 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d of_o thing_n we_o have_v in_o the_o first_o place_n their_o definition_n which_o be_v nothing_o but_o the_o explication_n of_o our_o simple_a conception_n for_o example_n he_o that_o have_v a_o true_a conception_n of_o place_n can_v be_v ignorant_a of_o this_o definition_n place_n be_v that_o space_n which_o be_v possess_v or_o fill_v adaequate_o by_o some_o body_n and_o so_o he_o that_o conceive_v motion_n aright_o can_v but_o know_v that_o motion_n be_v the_o privation_n of_o one_o place_n and_o the_o acquisition_n of_o another_o in_o the_o next_o place_n we_o have_v their_o generation_n or_o description_n as_o for_o example_n that_o a_o line_n be_v make_v by_o the_o motion_n of_o a_o point_n superficies_n by_o the_o motion_n of_o a_o line_n and_o one_o motion_n by_o another_o motion_n etc._n etc._n it_o remain_v that_o we_o inquire_v what_o motion_n beget_v such_o and_o such_o effect_n as_o what_o motion_n make_v a_o straight_a line_n and_o what_o a_o circular_a what_o motion_n thrust_v what_o draw_v and_o by_o what_o way_n what_o make_v a_o thing_n which_o be_v see_v or_o hear_v to_o be_v see_v or_o hear_v sometime_o in_o one_o manner_n sometime_o in_o another_o now_o the_o method_n of_o this_o kind_n of_o enquiry_n be_v compositive_a for_o first_o we_o be_v to_o observe_v what_o effect_v a_o body_n move_v produce_v when_o we_o consider_v nothing_o in_o it_o beside_o its_o motion_n and_o we_o see_v present_o that_o this_o make_v a_o line_n or_o length_n next_o what_o the_o motion_n of_o a_o long_a body_n produce_v which_o we_o find_v to_o be_v superficies_n and_o so_o forward_o till_o we_o see_v what_o the_o effect_n of_o simple_a motion_n be_v and_o then_o in_o like_a manner_n we_o be_v to_o observe_v what_o proceed_v from_o the_o addition_n multiplication_n substraction_n and_o division_n of_o these_o motion_n and_o what_o effect_n what_o figure_n and_o what_o property_n they_o produce_v from_o which_o kind_n of_o contemplation_n spring_v that_o part_n of_o philosophy_n which_o be_v call_v geometry_n from_o this_o consideration_n of_o what_o be_v produce_v by_o simple_a motion_n we_o be_v to_o pass_v to_o the_o consideration_n of_o what_o effect_n one_o body_n move_v work_v upon_o another_o and_o because_o there_o may_v be_v motion_n in_o all_o the_o several_a part_n of_o a_o body_n yet_o so_o as_o that_o the_o whole_a body_n remain_v still_o in_o the_o same_o place_n we_o must_v inquire_v first_o what_o motion_n cause_v such_o and_o such_o motion_n in_o the_o whole_a that_o be_v when_o one_o body_n invade_v another_o body_n which_o be_v either_o at_o rest_n or_o in_o motion_n what_o way_n and_o with_o what_o swiftness_n the_o invade_v body_n shall_v move_v and_o again_o what_o motion_n this_o second_o body_n will_v generate_v in_o a_o three_o and_o so_o forward_o from_o which_o contemplation_n shall_v be_v draw_v that_o part_n of_o philosophy_n which_o treat_v of_o motion_n in_o the_o three_o place_n we_o must_v proceed_v to_o the_o enquiry_n of_o such_o effect_n as_o be_v make_v by_o the_o motion_n of_o the_o part_n of_o any_o body_n as_o how_o it_o come_v to_o pass_v that_o thing_n when_o they_o be_v the_o same_o yet_o seem_v not_o to_o be_v the_o same_o but_o change_v and_o here_o the_o thing_n we_o search_v after_o be_v sensible_a quality_n such_o as_o light_v colour_n transparency_n opacity_n sound_v odour_n savour_n heat_n cold_a and_o the_o like_a which_o because_o they_o can_v be_v know_v till_o we_o know_v the_o cause_n of_o sense_n itself_o therefore_o the_o consideration_n of_o the_o cause_n of_o see_v hear_v smell_a taste_v and_o touch_v belong_v to_o this_o three_o place_n and_o all_o those_o quality_n and_o change_n above_o mention_v be_v to_o be_v refer_v to_o
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 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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 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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
uniform_o the_o straight_a line_n p_o c._n seeing_n therefore_o the_o two_o straight_a line_n a_o p_o and_o p_o c_o be_v describe_v in_o the_o time_n a_o e_o with_o the_o same_o increase_n of_o impetus_fw-la wherewith_o the_o crooked_a line_n a_o b_o c_o be_v describe_v in_o the_o same_o time_n a_o e_o that_o be_v see_v the_o line_n a_o p_o c_o and_o the_o line_n a_o b_o c_o be_v transmit_v by_o the_o same_o body_n in_o the_o same_o time_n &_o with_o equal_a velocity_n the_o line_n themselves_o be_v equal_a which_o be_v to_o be_v demonstrate_v by_o the_o same_o method_n if_o any_o of_o the_o semiparabolaster_n in_o the_o table_n of_o the_o 3d_o article_n of_o the_o precedent_a chapter_n be_v exhibit_v may_v be_v find_v a_o straight_a line_n equal_a to_o the_o crooked_a line_n thereof_o namely_o by_o divide_v the_o diameter_n into_o two_o equal_a part_n and_o proceed_v as_o before_o yet_o no_o man_n hitherto_o have_v compare_v any_o crooked_a with_o any_o straight_a line_n though_o 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between_o the_o straight_a line_n which_o be_v reflect_v and_o the_o line_n reflect_v 1_o if_o two_o straight_a line_n which_o fall_v upon_o another_o straight_a line_n be_v be_v parallel_n their_o reflect_a line_n shall_v be_v also_o parallel_a let_v the_o two_o straight_a line_n ab_fw-la and_o cd_o in_o the_o 1_o figure_n which_o fall_n upon_o the_o straight_a line_n of_o at_o the_o point_n b_o and_o d_o be_v parallel_n and_o let_v the_o line_n reflect_v from_o they_o be_v bg_n and_o dh_n i_o say_v bg_n and_o dh_n be_v also_o parallel_n for_o the_o angle_v abe_n and_o cde_v be_v equal_a by_o reason_n of_o the_o parallellelisme_n of_o ab_fw-la and_o cd_o and_o the_o angle_n gbf_n and_o hdf_n be_v equal_a to_o they_o by_o supposition_n for_o the_o line_n bg_n and_o dh_n be_v reflect_v from_o the_o line_n ab_fw-la and_o cd_o wherefore_o bg_n and_o dh_n be_v parallel_n 2_o if_o two_o straight_a line_n draw_v from_o the_o same_o point_n fall_v upon_o another_o straight_a line_n the_o line_n reflect_v from_o they_o if_o they_o be_v draw_v out_o the_o other_o way_n will_v meet_v in_o a_o angle_n equal_a to_o the_o angle_n of_o the_o incident_a line_n from_o the_o point_n ac_fw-la in_o the_o 2d_o figure_v let_v the_o two_o straight_a line_n ab_fw-la and_o ad_fw-la be_v draw_v and_o let_v they_o fall_v upon_o the_o straight_a line_n eke_o at_o the_o point_n b_o and_o d_o and_o let_v the_o line_n by_o and_o dg_n be_v reflect_v from_o they_o i_o say_v ib_n and_o gd_v do_v converge_n and_o that_o if_o they_o be_v produce_v on_o the_o other_o side_n of_o the_o line_n eke_o they_o shall_v meet_v as_o in_o f_o and_o that_o the_o angle_n bfd_n shall_v be_v equal_a to_o the_o angle_n bad_a for_o the_o angle_n of_o reflection_n ibk_n be_v equal_a to_o the_o angle_n
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
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
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
and_o the_o property_n of_o straight_a parallel_n 13_o the_o circumference_n of_o circle_n be_v to_o one_o another_o as_o their_o diameter_n be_v 14_o in_o triangle_n straight_a line_n parallel_v to_o the_o base_n be_v to_o one_o another_o as_o the_o part_n of_o the_o side_n which_o they_o cut_v off_o from_o the_o vertex_fw-la 15_o by_o what_o fraction_n of_o a_o straight_a line_n the_o circumference_n of_o a_o circle_n be_v make_v 16_o that_o a_o angle_n of_o contingence_n be_v quantity_n but_o of_o a_o different_a kind_n from_o that_o of_o a_o angle_n simple_o so_o call_v and_o that_o it_o can_v neither_o add_v nor_o take_v away_o any_o thing_n from_o the_o same_o 17_o that_o the_o inclination_n of_o plain_n be_v angle_v simple_o so_o call_v 18_o a_o solid_a angle_n what_o it_o be_v 19_o what_o be_v the_o nature_n of_o asymptotes_n 20_o situation_n by_o what_o it_o be_v determine_v 21_o what_o be_v like_o situation_n what_o be_v figure_n and_o what_o be_v like_o figure_n 1_o between_o two_o point_n give_v the_o short_a line_n be_v that_o who_o extreme_a point_n can_v be_v draw_v further_o asunder_o without_o alter_v the_o quantity_n that_o be_v without_o alter_v the_o proportion_n of_o that_o line_n to_o any_o other_o line_n give_v for_o the_o magnitude_n of_o a_o line_n be_v compute_v by_o the_o great_a distance_n which_o may_v be_v between_o its_o extreme_a point_n so_o that_o any_o one_o line_n whether_o it_o be_v extend_v or_o bow_v have_v always_o one_o and_o the_o same_o length_n because_o it_o can_v have_v but_o one_o great_a distance_n between_o its_o extreme_a point_n and_o see_v the_o action_n by_o which_o a_o straight_a line_n be_v make_v crooked_a or_o contrary_o a_o crooked_a line_n be_v make_v straight_o be_v nothing_o but_o the_o bring_n of_o its_o extreme_a point_n near_o to_o one_o another_o or_o the_o set_n of_o they_o further_o asunder_o a_o crooked_a line_n may_v right_o be_v define_v to_o be_v that_o who_o extreme_a point_n may_v be_v 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b_o h_o d_o in_o i_o if_o x_o c_o be_v draw_v shall_v cut_v the_o same_o arch_n in_o the_o same_o point_n i_o it_o will_v be_v as_o true_a that_o the_o arch_n b_o i_o be_v equal_a to_o the_o radius_fw-la b_o c_o as_o it_o be_v true_a that_o the_o arch_n b_o f_o be_v equal_a to_o the_o straight_a line_n b_o t_o and_o draw_v x_o k_o it_o will_v cut_v the_o arch_n b_o i_o in_o the_o midst_n in_o i_o also_o draw_v a_o i_o and_o produce_v it_o to_o the_o tangent_fw-la b_o c_o in_o k_o the_o straight_a line_n b_o k_o will_v be_v the_o tangent_fw-la of_o the_o arch_n b_o i_o which_o arch_n be_v equal_a to_o half_a the_o radius_fw-la and_o the_o same_o straight_a line_n b_o k_o will_v be_v equal_a to_o the_o straight_a line_n k_o i._n i_o say_v all_o this_o be_v true_a if_o the_o precede_a demonstration_n be_v true_a and_o consequent_o the_o proportional_a section_n of_o the_o arch_n and_o its_o tangent_fw-la proceed_v hitherto_o but_o it_o be_v manifest_a by_o the_o golden_a rule_n that_o take_v b_o h_o double_a to_o b_o t_o the_o line_n x_o h_o shall_v not_o cut_v off_o the_o arch_n b_o e_o which_o be_v double_a to_o the_o arch_n b_o f_o but_o a_o much_o great_a for_o the_o magnitude_n of_o the_o straight_a line_n x_o m_n x_o b_o and_o m_o e_o be_v know_v in_o number_n the_o magnitude_n of_o the_o straight_a line_n cut_v off_o in_o the_o tangent_fw-la by_o the_o straight_a line_n x_o e_z produce_v to_o the_o tangent_fw-la may_v also_o be_v know_v and_o it_o will_v be_v find_v to_o be_v less_o than_o b_o h_z wherefore_o the_o straight_a line_n xh_v be_v draw_v will_v cut_v off_o a_o part_n of_o the_o arch_n of_o the_o quadrant_n great_a than_o the_o arch_n b_o e._n but_o i_o shall_v speak_v more_o full_o in_o the_o next_o article_n concern_v the_o magnitude_n of_o the_o arch_n b_o 1._o and_o let_v this_o be_v 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 the_o section_n of_o the_o arch_n b_o f._n 3_o i_o shall_v now_o attempt_v the_o same_o by_o argument_n draw_v from_o the_o nature_n of_o the_o crookedness_n of_o the_o circle_n itself_o but_o i_o shall_v first_o set_v down_o some_o premise_n necessary_a for_o this_o speculation_n and_o first_o if_o a_o straight_a line_n be_v bow_v into_o a_o arch_n of_o a_o circle_n equal_a to_o it_o as_o when_o a_o stretch_a thread_n which_o touch_v a_o right_o cylinder_n be_v so_o bow_v in_o every_o point_n that_o it_o be_v every_o where_o coincident_a with_o the_o perimeter_n of_o the_o base_a of_o the_o cylinder_n the_o flexion_n of_o that_o line_n will_v be_v equal_a in_o all_o its_o point_n and_o consequent_o the_o crookedness_n of_o the_o arch_n of_o a_o circle_n be_v every_o where_o uniform_a which_o need_v no_o other_o demonstration_n than_o this_o that_o the_o perimeter_n of_o a_o circle_n be_v a_o uniform_a line_n second_o and_o consequent_o if_o two_o unequal_a arch_n of_o the_o same_o circle_n be_v make_v by_o the_o bow_n of_o two_o straight_a line_n equal_a to_o they_o the_o flexion_n of_o the_o long_a line_n while_o it_o be_v bow_v into_o the_o great_a arch_n be_v great_a than_o the_o flexion_n of_o the_o short_a line_n while_o it_o be_v bow_v into_o the_o lesser_a arch_n according_a to_o the_o proportion_n of_o the_o arch_n themselves_o and_o consequent_o the_o crookedness_n of_o the_o great_a arch_n be_v to_o the_o crookedness_n of_o the_o lesser_a arch_n as_o the_o great_a arch_n be_v to_o the_o lesser_a arch._n three_o if_o two_o unequal_a circle_n and_o a_o straight_a line_n touch_v one_o another_o in_o the_o same_o point_n the_o crookedness_n of_o any_o arch_n take_v in_o the_o lesser_a circle_n will_v be_v great_a than_o the_o crookedness_n of_o a_o arch_n equal_a to_o it_o take_v in_o the_o great_a circle_n in_o reciprocal_a proportion_n to_o that_o of_o the_o radii_fw-la with_o which_o the_o circle_n be_v describe_v or_o which_o be_v all_o one_o any_o straight_a line_n be_v draw_v from_o the_o point_n of_o contact_n till_o it_o cut_v both_o the_o circumference_n as_o the_o part_n of_o that_o straight_a line_n cut_v off_o by_o the_o circumference_n of_o the_o great_a circle_n to_o that_o part_n which_o be_v cut_v off_o by_o the_o circumference_n of_o the_o lesser_a circle_n for_o let_v a_o b_o and_o a_o c_z in_o the_o second_o figure_n be_v two_o circle_n touch_v one_o another_o and_o the_o straight_a line_n a_o d_o in_o the_o point_n a_o and_o let_v their_o centre_n be_v e_o and_o f_o and_o let_v it_o be_v suppose_v that_o as_o a_o e_o be_v to_o a_o f_o so_o be_v the_o arch_n a_o b_o to_o the_o arch_n a_o h._n i_o say_v the_o crookedness_n of_o the_o arch_n a_o c_o be_v too_o the_o crookedness_n of_o the_o arch_n a_o h_n as_o a_o e_o be_v to_o a_o f._n for_o let_v the_o straight_a line_n a_o d_o be_v suppose_v to_o be_v equal_a to_o the_o arch_n a_o b_o and_o the_o straight_a line_n a_o g_o to_o the_o arch_n a_o c_o and_o let_v a_o d_o for_o example_n be_v double_a to_o a_o g._n therefore_o by_o reason_n of_o the_o likeness_n of_o the_o arch_n a_o b_o and_o a_o c_o the_o straight_a line_n a_o b_o will_v be_v double_a to_o the_o straight_a line_n a_o c_o and_o the_o radius_fw-la a_o e_o double_v to_o the_o radius_fw-la a_o f_o and_o the_o arch_n a_o b_o double_a to_o the_o arch_n a_o h._n and_o because_o the_o straight_a line_n a_o d_o be_v so_o bow_v to_o be_v coincident_a with_o the_o arch_n a_o b_o equal_v to_o it_o as_o the_o straight_a line_n a_o g_o be_v bow_v to_o be_v coincident_a with_o the_o arch_n a_o c_o equal_a
equal_a that_o no_o inequality_n can_v be_v discover_v between_o they_o i_o will_v therefore_o leave_v this_o to_o be_v further_o search_v into_o for_o though_o it_o be_v almost_o out_o of_o doubt_n that_o the_o straight_a line_n bp_o and_o the_o arch_n bmd_v be_v equal_a yet_o that_o may_v not_o be_v receive_v without_o demonstration_n and_o mean_n of_o demonstration_n the_o circular_a line_n admit_v none_o that_o be_v not_o ground_v upon_o the_o nature_n of_o flexion_n or_o of_o angle_n but_o by_o that_o way_n i_o have_v already_o exhibit_v a_o straight_a line_n equal_a to_o the_o arch_n of_o a_o quadrant_n in_o the_o first_o and_o second_o aggression_n it_o remain_v that_o i_o prove_v dt_n to_o be_v equal_a to_o the_o sine_fw-la of_o 45_o degree_n in_o basilius_n produce_v let_v hence_v he_o take_v equal_a to_o the_o sine_fw-la of_o 45_o degree_n and_o draw_v and_o produce_v uh_o it_o will_v cut_v the_o arch_n of_o the_o quadrant_n cna_fw-mi in_o the_o midst_n in_o n_o and_o the_o same_o arch_n again_o in_o o_o and_o the_o straight_a line_n dc_o in_o t_o so_o that_o dt_n will_v be_v equal_a to_o the_o sine_fw-la of_o 45_o degree_n or_o to_o the_o straight_a line_n hence_v also_o the_o straight_a line_n uh_o will_v be_v equal_a to_o the_o straight_a line_n he_o or_o the_o sine_fw-la of_o 60_o degree_n for_o the_o square_n of_o ave_fw-la be_v equal_a to_o two_o square_n of_o the_o semiradius_fw-la and_o consequent_o the_o square_a of_o uh_o be_v equal_a to_o three_o square_n of_o the_o semiradius_fw-la but_o he_o be_v a_o mean_a proportional_a between_o the_o semiradius_fw-la and_o three_o semiradii_fw-la and_o therefore_o the_o square_a of_o he_o be_v equal_a to_o three_o square_n of_o the_o semiradius_fw-la wherefore_o he_o be_v eqval_n to_o hv_v but_o because_o ad_fw-la be_v cut_v in_o the_o midst_n in_o h_n therefore_o uh_o and_o ht_n be_v equal_a and_o therefore_o also_o dt_n be_v equal_a to_o the_o sine_fw-la of_o 45_o degree_n in_o the_o radius_fw-la basilius_n let_v bx_n be_v take_v equal_a to_o the_o sine_fw-la of_o 45_o degree_n for_o so_o vx_n will_v be_v equal_a to_o the_o radius_fw-la and_o it_o will_v be_v as_o valerio_n to_o ah_o the_o semiradius_fw-la so_o vx_n the_o radius_fw-la to_o xn_n the_o sine_fw-la of_o 45_o degree_n wherefore_o uh_o produce_v pass_n through_o n._n last_o upon_o the_o centre_n five_o with_o the_o radius_fw-la va_o let_v the_o arch_n of_o a_o circle_n be_v draw_v cut_v uh_o in_o y_o which_o be_v do_v vy_z will_v be_v equal_a to_o ho_o for_o ho_o be_v by_o construction_n equal_a to_o the_o sine_fw-la of_o 45_o degree_n and_o yh_n will_v be_v equal_a to_o ot_fw-mi &_o therefore_o vt_fw-la pass_v through_o o._n all_o which_o be_v to_o be_v demonstrate_v i_o will_v here_o add_v certain_a problem_n of_o which_o if_o any_o analyst_n can_v make_v the_o construction_n he_o will_v thereby_o be_v able_a to_o judge_v clear_o of_o what_o i_o have_v now_o say_v concern_v the_o dimension_n of_o a_o circle_n now_o these_o problem_n be_v nothing_o else_o at_o least_o to_o sense_n but_o certain_a symptom_n accompany_v the_o construction_n of_o the_o first_o and_o three_o figure_n of_o this_o chapter_n describe_v therefore_o again_o the_o square_a abcd_n in_o the_o five_o figure_n and_o the_o three_o quadrant_n abdella_n bca_n and_o dac_n let_v the_o diagonal_n ac_fw-la &_o bd_o be_v draw_v cut_v the_o arch_n bhd_v &_o cia_n in_o the_o middle_n in_o h_n and_o i_o &_o the_o straight_a line_n of_o and_o gl_n divide_v the_o square_n abcd_v into_o four_o equal_a square_n and_o trisect_v the_o arch_n bhd_v and_o cia_n namely_o bhd_v in_o k_o and_o m_n and_o cia_n in_o m_n and_o o._n then_o divide_v the_o arch_n bk_n in_o the_o midst_n in_o p_o let_v qp_v the_o sine_fw-la of_o the_o arch_n bp_o be_v draw_v and_o produce_v to_o r_o so_o that_o qr_n be_v double_a to_o qp_v and_o connect_v kr_n let_v it_o be_v produce_v one_o way_n to_o bc_n in_o saint_n and_o the_o other_o way_n to_o basilius_n produce_v in_o t._n also_o let_v bv_o be_v make_v triple_a to_o b_n and_o consequent_o by_o the_o second_o article_n of_o this_o chapter_n equal_a to_o the_o arch_n bd._n this_o construction_n be_v the_o same_o with_o that_o of_o the_o first_o figure_n which_o i_o think_v fit_a to_o renew_v discharge_v of_o all_o line_n but_o such_o as_o be_v necessary_a for_o my_o present_a purpose_n in_o the_o first_o place_n therefore_o if_o hence_v be_v draw_v cut_v the_o arch_n bhd_v in_o x_o and_o the_o side_n dc_o in_o z_o i_o desire_v some_o analyst_n will_v if_o he_o can_v give_v a_o reason_n why_o the_o straight_a line_n te_fw-la and_o tc_n shall_v cut_v the_o arch_n bd_o the_o one_o in_o y_o the_o other_o in_o x_o so_o as_o to_o make_v the_o arch_n by_o equal_a to_o the_o arch_n yx_n or_o if_o they_o be_v not_o equal_a that_o he_o will_v determine_v their_o difference_n second_o if_o in_o the_o side_n dam_fw-ge the_o straight_a line_n dam_fw-ge be_v take_v equal_a to_o dz_n and_o valerio_n be_v draw_v why_o va_o and_o ub_z shall_v be_v equal_a or_o if_o they_o be_v not_o equal_a what_o be_v the_o difference_n three_o draw_v zb_a parallel_n and_o equal_a to_o the_o side_n cb_n cut_v the_o arch_n bhd_v in_o c_o and_o draw_v the_o straight_a line_n ac_fw-la and_o produce_v it_o to_o bv_v in_o d_o why_o admetus_n shall_v be_v equal_a and_o parallel_v to_o the_o straight_a line_n hence_v and_o consequent_o equal_a also_o to_o the_o arch_n bd._n four_o draw_v eke_o the_o sine_fw-la of_o the_o arch_n bk_n &_o take_v in_o ea_z produce_v of_o equal_a to_o the_o diagonal_a ac_fw-la and_o connect_v fc_n why_o fc_n shall_v pass_v through_o a_o which_o point_n be_v give_v the_o length_n of_o the_o arch_n bhd_v be_v also_o give_v and_o c_o and_o why_o fe_o and_o fc_n shall_v be_v equal_a or_o if_o not_o why_o unequal_a five_o draw_v fz_n i_o desire_v he_o will_v show_v why_o it_o be_v equal_a to_o bv_v or_o to_o the_o arch_n bd_o or_o if_o they_o be_v not_o equal_a what_o be_v their_o difference_n six_o grant_v fz_n to_o be_v equal_a to_o the_o arch_n bd_o i_o desire_v he_o will_v determine_v whether_o it_o fall_v all_o without_o the_o arch_n bca_n or_o cut_v the_o same_o or_o touch_v it_o and_o in_o what_o point_n seven_o the_o semicircle_n bdg_n be_v complete_v why_o give_v be_v draw_v and_o produce_v shall_v pass_v through_o x_o by_o which_o point_n x_o the_o length_n of_o the_o arch_n bd_o be_v determine_v and_o the_o same_o give_v be_v yet_o further_o produce_v to_o dc_o in_o h_z why_o add_v which_o be_v equal_a to_o the_o arch_n bd_o shall_v pass_v through_o that_o point_n h._n eight_o upon_o the_o centre_n of_o the_o square_n abcd_v which_o let_v be_v k_o the_o arch_n of_o the_o quadrant_a eil_n be_v draw_v cut_v eke_o produce_v in_o i_o why_o the_o draw_v straight_a line_n ix_o shall_v be_v parallel_n to_o the_o side_n cd_o nine_o in_o the_o side_n basilius_n and_o bc_n take_v bl_a and_o bm_n several_o equal_a to_o half_a bv_n or_o to_o the_o arch_n bh_n and_o draw_v mn_a parallel_n and_o equal_a to_o the_o side_n basilius_n cut_v the_o arch_n bd_o in_o o_o why_o the_o straight_a line_n which_o connect_v vl_a shall_v pass_v through_o the_o point_n o_o ten_o i_o will_v know_v of_o he_o why_o the_o straight_a line_n which_o connect_n ah_o shall_v be_v equal_a to_o bm_v or_o if_o not_o how_o much_o it_o differ_v from_o it_o the_o analyst_n that_o can_v solve_v these_o problem_n without_o know_v first_o the_o length_n of_o the_o arch_n bd_o or_o use_v any_o other_o know_v method_n then_o that_o which_o proceed_v by_o perpetual_a bisection_n of_o a_o angle_n or_o be_v draw_v from_o the_o consideration_n of_o the_o nature_n of_o flexion_n shall_v do_v more_o than_o ordinary_a geometry_n be_v able_a to_o perform_v but_o if_o the_o dimension_n of_o a_o circle_n can_v be_v find_v by_o any_o other_o method_n then_o i_o have_v either_o find_v it_o or_o it_o be_v not_o at_o all_o to_o be_v find_v from_o the_o know_a length_n of_o the_o arch_n of_o a_o quadrant_n and_o from_o the_o proportional_a division_n of_o the_o arch_n and_o of_o the_o tangent_fw-la bc_n may_v be_v deduce_v the_o section_n of_o a_o angle_n into_o any_o give_a proportion_n as_o also_o the_o square_n of_o the_o circle_n the_o square_n of_o a_o give_v sector_n and_o many_o the_o like_a proposition_n which_o it_o be_v not_o necessary_a here_o to_o demonstrate_v i_o will_v therefore_o only_o exhibit_v a_o straight_a line_n equal_a to_o the_o spiral_n of_o archimedes_n and_o so_o dismiss_v this_o speculation_n 5_o the_o length_n of_o the_o perimeter_n of_o a_o circle_n be_v find_v that_o straight_a line_n be_v also_o find_v which_o
weaken_v by_o little_a and_o little_a but_o this_o can_v be_v do_v but_o by_o the_o long_a continuance_n of_o action_n or_o by_o action_n often_o repeat_v and_o therefore_o custom_n beget_v that_o facicility_n which_o be_v common_o and_o right_o call_v habit_n and_o it_o may_v be_v define_v thus_o habit_n be_v motion_n make_v more_o easy_a and_o ready_a by_o custom_n that_o be_v to_o say_v by_o perpetual_a endeavour_n or_o by_o iterate_a endeavour_n in_o a_o way_n differ_v from_o that_o in_o which_o the_o motion_n proceed_v from_o the_o beginning_n and_o oppose_v such_o endeavour_n as_o resist_v and_o to_o make_v this_o more_o perspicuous_a by_o example_n we_o may_v observe_v that_o when_o one_o that_o have_v no_o skill_n in_o music_n first_o put_v his_o hand_n to_o a_o instrument_n he_o can_v after_o the_o first_o stroke_n carry_v to_o his_o hand_n to_o the_o place_n where_o he_o will_v make_v the_o second_o stroke_n without_o take_v it_o back_o by_o a_o new_a endeavour_n and_o as_o it_o be_v beginning_n again_o pass_v from_o the_o first_o to_o the_o second_o nor_o will_v he_o be_v able_a to_o go_v on_o to_o the_o three_o place_n without_o another_o new_a endeavour_n but_o he_o will_v be_v force_v to_o draw_v back_o his_o hand_n again_o and_o so_o successive_o by_o renew_v his_o endeavour_n at_o every_o stroke_n till_o at_o the_o last_o by_o do_v this_o often_o and_o by_o compound_v many_o interrupt_a motion_n or_o endeavour_n into_o one_o equal_a endeavour_n he_o be_v able_a to_o make_v his_o hand_n go_v ready_o on_o from_o stroke_n to_o stroke_n in_o that_o order_n and_o way_n which_o be_v at_o the_o first_o design_v nor_o be_v habit_n to_o be_v observe_v in_o live_a creature_n only_o but_o also_o in_o body_n inanimate_a for_o we_o find_v that_o when_o the_o lath_n of_o a_o crossbow_n be_v strong_o bend_v and_o will_v if_o the_o impediment_n be_v remove_v return_n again_o with_o great_a force_n if_o it_o remain_v a_o long_a time_n bend_v it_o will_v get_v such_o a_o habit_n that_o when_o it_o be_v loose_v and_o leave_v to_o its_o own_o freedom_n it_o will_v not_o only_o not_o restore_v itself_o but_o will_v require_v as_o much_o force_n for_o the_o bring_n of_o it_o back_o to_o its_o first_o posture_n as_o it_o do_v for_o the_o bend_n of_o it_o at_o the_o first_o chap._n xxiii_o of_o the_o centre_n of_o equiponderation_n of_o body_n press_v dottard_n in_o straight_o parallel_v line_n 1_o definition_n and_o supposition_n 2_o two_o plain_n of_o equiponderation_n be_v nâ—â—_n parallel_v 3_o the_o centre_n of_o equiponderation_n be_v in_o every_o plain_n of_o equiponderation_n 4_o the_o moment_n of_o equal_a ponderant_n be_v to_o one_o another_o as_o their_o distance_n from_o the_o centre_n of_o the_o scale_n 5,6_o the_o moment_n of_o unequal_a ponderant_n have_v their_o proportion_n to_o one_o another_o compound_v of_o the_o proportion_n of_o their_o weight_n and_o distance_n from_o the_o centre_n of_o the_o scale_n reciprocal_o take_v 7._o if_o two_o ponderant_n have_v their_o moment_n and_o distance_n from_o the_o centre_n of_o the_o scale_n in_o reciprocal_a proportion_n they_o be_v equal_o poise_v and_o contrary_o 8_o if_o the_o part_n of_o any_o ponderant_fw-la press_v the_o beam_n of_o the_o scale_n every_o where_o equal_o all_o the_o part_n cut_v out_o off_o reckon_v from_o the_o centre_n of_o the_o scale_n will_v have_v their_o moment_n in_o the_o same_o proportion_n with_o that_o of_o the_o part_n of_o a_o triangle_n cut_v off_o from_o the_o vertex_fw-la by_o straight_a line_n parallel_v to_o the_o base_a 9_o the_o diameter_n of_o equiponderation_n of_o figure_n which_o be_v deficient_a according_a to_o commensurable_a proportion_n of_o their_o altitude_n and_o base_n divide_v the_o axis_n so_o that_o the_o part_n take_v next_o the_o vertex_fw-la be_v to_o the_o other_o part_n as_o the_o complete_a figure_n to_o the_o deficient_a figure_n 10_o the_o diameter_n of_o equiponderation_n of_o the_o compliment_n of_o the_o half_a of_o any_o of_o the_o say_v deficient_a figure_n divide_v that_o line_n which_o be_v draw_v through_o the_o vertex_fw-la parallel_n to_o the_o base_a so_o that_o the_o part_n next_o the_o vertex_fw-la be_v to_o the_o other_o part_n as_o the_o complete_a figure_n to_o the_o compliment_n 11_o the_o centre_n of_o equiponderation_n of_o the_o half_a of_o any_o of_o the_o desicient_a figure_n in_o the_o first_o row_n of_o the_o table_n of_o the_o 3d._a article_n of_o the_o 17_o chapter_n may_v be_v find_v out_o by_o the_o number_n of_o the_o second_o row_n 12_o the_o centre_n of_o equiponderation_n of_o the_o half_a of_o any_o of_o the_o figure_n in_o the_o second_o row_n of_o the_o same_o table_n may_v be_v find_v out_o by_o the_o number_n of_o the_o four_o row_n 13_o the_o centre_n of_o equiponderation_n of_o the_o half_a of_o any_o of_o the_o figure_n in_o the_o same_o table_n be_v know_v the_o centre_n of_o the_o excess_n of_o the_o same_o figure_n above_o a_o triangle_n of_o the_o same_o altitude_n and_o base_a be_v also_o know_v 14_o the_o centre_n of_o equiponderation_n of_o a_o solid_a sector_n be_v in_o the_o axis_n so_o divide_v that_o the_o part_n next_o the_o vertex_fw-la be_v to_o the_o whole_a axis_n want_v half_a the_o axis_n of_o the_o portion_n of_o the_o sphere_n as_o 3_o to_o 4._o 1_o definition_n 1_o a_o scale_n be_v a_o straight_a line_n who_o middle_a point_n be_v immovable_a all_o the_o rest_n of_o its_o point_n be_v at_o liberty_n and_o that_o part_n of_o the_o scale_n which_o reach_v from_o the_o centre_n to_o either_o of_o the_o weight_n be_v call_v the_o beam_n 2_o equiponderation_n be_v when_o the_o endeavour_n of_o one_o body_n which_o press_v one_o of_o the_o beam_n resist_v the_o endeavour_n of_o another_o body_n press_v the_o other_o beam_n so_o that_o neither_o of_o they_o be_v move_v and_o the_o body_n when_o neither_o of_o they_o be_v move_v be_v say_v to_o be_v equal_o poise_v 3_o weight_n be_v the_o aggregate_v of_o all_o the_o endeavour_n by_o which_o all_o the_o point_n of_o that_o body_n which_o press_v the_o beam_n tend_v downward_o in_o line_n parallel_v to_o one_o another_o and_o the_o body_n which_o press_v be_v call_v the_o ponderant_fw-la 4_o moment_n be_v the_o power_n which_o the_o ponderant_fw-la have_v to_o move_v the_o beam_n by_o reason_n of_o a_o determine_a situation_n 5_o the_o plain_a of_o equiponderation_n be_v that_o by_o which_o the_o ponderant_fw-la be_v so_o divide_v that_o the_o moment_n on_o both_o side_n remain_v equal_a 6_o the_o diameter_n of_o equiponderation_n be_v the_o common_a section_n of_o the_o two_o plain_n of_o equiponderation_n and_o be_v in_o the_o straight_a line_n by_o which_o the_o weight_n be_v hang_v 7_o the_o centre_n of_o equiponderation_n be_v the_o common_a point_n of_o the_o two_o diameter_n of_o equiponderation_n supposition_n 1_o when_o two_o body_n be_v equal_o poise_v if_o weight_n be_v add_v to_o one_o of_o they_o and_o not_o to_o the_o other_o their_o equiponderation_n cease_v 2_o when_o two_o ponderant_n of_o equal_a magnitude_n and_o of_o the_o same_o species_n or_o matter_n press_v the_o beam_n on_o both_o side_n at_o equal_a distance_n from_o the_o centre_n of_o the_o scale_n their_o moment_n be_v equal_a also_o when_o two_o body_n endeavour_v at_o equal_a distance_n from_o the_o centre_n of_o the_o scale_n if_o they_o be_v of_o equal_a magnitude_n and_o of_o the_o same_o species_n their_o moment_n be_v equal_a 2_o no_o two_o plain_n of_o equiponderation_n be_v parallel_n let_v a_o b_o c_o d_o in_o the_o first_o figure_n be_v any_o ponderant_fw-la whatsoever_o and_o in_o it_o let_v e_o f_o be_v a_o plain_a of_o equiponderation_n parallel_n to_o which_o let_v any_o other_o plain_o be_v draw_v as_o g_z h._n i_o say_v g_o h_n be_v not_o a_o plain_a of_o equiponderation_n for_o see_v the_o part_n a_o e_o f_o d_o and_o e_o b_o c_o f_o of_o the_o ponderant_fw-la a_o b_o c_o d_o be_v equal_o poise_v and_o the_o weight_n e_o g_o h_o f_o be_v add_v to_o the_o part_n a_o e_o f_o d_o and_o nothing_o be_v add_v to_o the_o part_n e_o b_o c_o f_o but_o the_o weight_n e_o g_o h_o f_o be_v take_v from_o it_o therefore_o by_o the_o first_o supposition_n the_o part_n a_o g_o h_o d_o and_o g_o b_o c_o h_o will_v not_o be_v equal_o poise_v and_o consequent_o g_z h_n be_v not_o a_o plain_a of_o equiponderation_n wherefore_o no_o two_o plain_n of_o equiponderation_n etc._n etc._n which_o be_v to_o be_v prove_v 3_o the_o centre_n of_o equiponderation_n be_v in_o every_o plain_n of_o equiponderation_n for_o if_o another_o plain_n of_o equiponderation_n be_v take_v it_o will_v not_o by_o the_o last_o article_n be_v parallel_n to_o the_o former_a plain_n and_o therefore_o both_o those_o plain_n will_v
time_n of_o their_o descent_n can_v be_v easy_o measure_v with_o sufficient_a exactness_n and_o also_o because_o the_o place_n near_o the_o pole_n be_v inaccessible_a nevertheless_o this_o we_o know_v that_o by_o how_o much_o the_o near_o we_o come_v to_o the_o pole_n by_o so_o much_o the_o great_a be_v the_o flake_n of_o the_o snow_n that_o fall_v and_o by_o how_o much_o the_o more_o swift_o such_o body_n descend_v as_o be_v fluid_a and_o dissipable_a by_o so_o much_o the_o small_a be_v the_o particle_n into_o which_o they_o be_v dissipate_v 5_o suppose_v therefore_o this_o to_o be_v the_o cause_n of_o the_o descent_n of_o heavy_a body_n it_o will_v follow_v that_o their_o motion_n will_v be_v accelerate_a in_o such_o manner_n as_o that_o the_o space_n which_o be_v transmit_v by_o they_o in_o the_o several_a time_n will_v have_v to_o one_o another_o the_o same_o proportion_n which_o the_o odd_a number_n have_v in_o succession_n from_o unity_n for_o if_o the_o straight_a line_n ea_fw-la be_v divide_v into_o any_o number_n of_o equal_a part_n the_o heavy_a body_n descend_v will_v by_o reason_n of_o the_o perpetual_a action_n of_o the_o diurnal_a motion_n receive_v from_o the_o air_n in_o every_o one_o of_o those_o time_n in_o every_o several_a point_n of_o the_o straight_a line_n ea_fw-la a_o several_a new_a and_o equal_a impulsion_n and_o therefore_o also_o in_o every_o one_o of_o those_o time_n it_o will_v acquire_v a_o several_a and_o equal_a degree_n of_o celerity_n and_o from_o hence_o it_o follow_v by_o that_o which_o galilaeus_fw-la have_v in_o his_o dialogue_n of_o motion_n demonstrate_v that_o heavy_a body_n descend_v in_o the_o several_a time_n with_o such_o difference_n of_o transmit_v space_n as_o be_v equal_a to_o the_o difference_n of_o the_o square_a number_n that_o succeed_v one_o another_o from_o unity_n which_o square_a number_n be_v 1_o 4_o 9_o 16_o etc._n etc._n their_o difference_n be_v 3_o 5_o 7_o that_o be_v to_o say_v the_o odd_a number_n which_o succeed_v another_o from_o unity_n against_o this_o cause_n of_o gravity_n which_o i_o have_v give_v it_o will_v perhaps_o be_v object_v that_o if_o a_o heavy_a body_n be_v place_v in_o the_o bottom_n of_o some_o hollow_a cylinder_n of_o iron_n or_o adamant_n and_o the_o bottom_n be_v turn_v upward_o the_o body_n will_v descend_v though_o the_o air_n above_o can_v depress_v it_o much_o less_o accelerate_v its_o motion_n but_o it_o be_v to_o be_v consider_v that_o there_o can_v be_v no_o cylinder_n or_o cavern_n but_o such_o as_o be_v support_v by_o the_o earth_n and_o be_v so_o support_v be_v together_o with_o the_o earth_n carry_v about_o by_o its_o diurnal_a motion_n for_o by_o this_o mean_v the_o bottom_n of_o the_o cylinder_n will_v be_v as_o the_o superficies_n of_o the_o earth_n and_o by_o thrust_v off_o the_o next_o and_o low_a air_n will_v make_v the_o uppermost_a air_n depress_v the_o heavy_a body_n which_o be_v at_o the_o top_n of_o the_o cylinder_n in_o such_o manner_n as_o be_v above_o explicate_v 6_o the_o gravity_n of_o water_n be_v so_o great_a as_o by_o experience_n we_o find_v it_o be_v the_o reason_n be_v demand_v by_o many_o why_o those_o that_o dive_v how_o deep_a soever_o they_o go_v under_o water_n do_v not_o at_o all_o feel_v the_o weight_n of_o the_o water_n which_o lie_v upon_o they_o and_o the_o cause_n seem_v to_o be_v this_o that_o all_o body_n by_o how_o much_o the_o heavy_a they_o be_v by_o so_o much_o the_o great_a be_v the_o endeavour_n by_o which_o they_o tend_v downward_o but_o the_o body_n of_o a_o man_n be_v heavy_a than_o so_o much_o water_n as_o be_v equal_a to_o it_o in_o magnitude_n and_o therefore_o the_o endeavour_n downward_o of_o a_o man_n body_n be_v great_a than_o that_o of_o water_n and_o see_v all_o endeavour_n be_v motion_n the_o body_n also_o of_o a_o man_n will_v be_v carry_v towards_o the_o bottom_n with_o great_a velocity_n than_o so_o much_o water_n wherefore_o there_o be_v great_a reaction_n from_o the_o bottom_n and_o the_o endeavour_v upward_o be_v equal_a to_o the_o endeavour_n downward_o whether_o the_o water_n be_v press_v by_o water_n or_o by_o another_o body_n which_o be_v heavy_a than_o water_n and_o therefore_o by_o these_o two_o opposite_a equal_a endeavour_n the_o endeavour_n both_o way_n in_o the_o water_n be_v take_v away_o and_o consequent_o those_o that_o dive_v be_v not_o at_o all_o press_v by_o it_o coral_n from_o hence_o also_o it_o be_v manifest_a that_o water_n in_o water_n have_v no_o weight_n at_o all_o because_o all_o the_o part_n of_o water_n both_o the_o part_n above_o and_o the_o part_n that_o be_v direct_o under_o tend_v towards_o the_o bottom_n with_o equal_a endeavour_n and_o in_o the_o same_o straight_a line_n 7_o if_o a_o body_n float_v upon_o the_o water_n the_o weight_n of_o that_o body_n be_v equal_a to_o the_o weight_n of_o so_o much_o water_n as_o will_v fill_v the_o place_n which_o the_o immersed_a part_n of_o the_o body_n take_v up_o within_o the_o water_n let_v of_o in_o the_o 3d_o figure_n be_v a_o body_n float_v in_o the_o water_n abcd_v and_o let_v the_o part_n e_o be_v above_o and_o the_o other_o part_n f_o under_o the_o water_n i_o say_v the_o weight_n of_o the_o whole_a body_n of_o be_v equal_a to_o the_o weight_n of_o so_o much_o water_n as_o the_o space_n f_o will_v receive_v for_o see_v the_o weight_n of_o the_o body_n of_o force_v the_o water_n out_o of_o the_o space_n f_o and_o place_v it_o upon_o the_o superficies_n ab_fw-la where_o it_o press_v downward_o it_o follow_v that_o from_o the_o resistance_n of_o the_o bottom_n there_o will_v also_o be_v a_o endeavour_n upward_o and_o see_v again_o that_o by_o this_o endeavour_n of_o the_o water_n upward_o the_o body_n of_o be_v lift_v up_o it_o follow_v that_o if_o the_o endeavour_n of_o the_o body_n downward_o be_v not_o equal_a to_o the_o endeavour_n of_o the_o water_n upward_o either_o the_o whole_a body_n of_o will_n by_o reason_n of_o that_o inequality_n of_o their_o endeavour_n or_o moment_n be_v raise_v out_o of_o the_o water_n or_o else_o it_o will_v descend_v to_o the_o bottom_n but_o it_o be_v suppose_v to_o stand_v so_o as_o neither_o to_o ascend_v nor_o descend_v wherefore_o there_o be_v a_o aequilibrium_fw-la between_o the_o two_o endeavour_n that_o be_v to_o say_v the_o weight_n of_o the_o body_n of_o be_v equal_a to_o the_o weight_n of_o so_o much_o water_n as_o the_o space_n f_o will_v receive_v which_o be_v to_o be_v prove_v 8_o from_o hence_o it_o follow_v that_o any_o body_n of_o how_o great_a magnitude_n soever_o provide_v it_o consist_v of_o matter_n less_o heavy_a than_o water_n may_v nevertheless_o float_v upon_o any_o quantity_n of_o water_n how_o little_a soever_o let_v abcd_n in_o the_o four_o figure_n be_v a_o vessel_n and_o in_o it_o let_v efgh_o be_v a_o body_n consist_v of_o matter_n which_o be_v less_o heavy_a than_o water_n and_o let_v the_o space_n agcf_n be_v fill_v with_o water_n i_o say_v the_o body_n efgh_o will_v not_o sink_v to_o the_o bottom_n dc_o for_o see_v the_o matter_n of_o the_o body_n efgh_n be_v less_o heavy_a than_o water_n if_o the_o whole_a space_n without_o abcd_n be_v full_a of_o water_n yet_o some_o part_n of_o the_o body_n efgh_o as_o efik_n will_v be_v above_o the_o water_n and_o the_o weight_n of_o so_o much_o water_n as_o will_v fill_v the_o space_n ighk_n will_v be_v equal_a to_o the_o weight_n of_o the_o whole_a body_n efgh_o and_o consequent_o gh_a will_v not_o touch_v the_o bottom_n dc_o as_o for_o the_o side_n of_o the_o vessel_n it_o be_v no_o matter_n whether_o they_o be_v hard_a or_o fluid_a for_o they_o serve_v only_o to_o terminate_v the_o water_n which_o may_v be_v do_v as_o well_o by_o water_n as_o by_o any_o other_o matter_n how_o hard_o soever_o and_o the_o water_n without_o the_o vessel_n be_v terminate_v somewhere_o so_o as_o that_o it_o can_v spread_v no_o further_o the_o part_n therefore_o efig_n will_v be_v extant_a above_o the_o water_n agcf_n which_o be_v contain_v in_o the_o vessel_n wherefore_o the_o body_n efgh_o will_v also_o float_v upon_o the_o water_n agcf_n how_o little_a soever_o that_o water_n be_v which_o be_v to_o be_v demonstrate_v 9_o in_o the_o four_o article_n of_o the_o 26_o chapter_n there_o be_v bring_v for_o the_o prove_v of_o vacuum_n the_o experiment_n of_o water_n enclose_v in_o a_o vessel_n which_o water_n the_o orifice_n above_o be_v open_v be_v eject_v upward_o by_o the_o impulsion_n of_o the_o air_n it_o be_v therefore_o demand_v see_v water_n be_v heavy_a than_o air_n how_o that_o can_v be_v do_v let_v the_o second_o figure_n of_o the_o same_o 26_o chap._n be_v consider_v where_o the_o water_n be_v with_o great_a force_n inject_v by_o
that_o the_o proportion_n of_o the_o first_o antecedent_n to_o the_o first_o consequent_a be_v the_o same_o with_o that_o of_o the_o second_o antecedent_n to_o the_o second_o consequent_a and_o when_o four_o magnitude_n be_v thus_o to_o one_o another_o in_o geometrical_a proportion_n they_o be_v call_v proportional_n and_o by_o some_o more_o brief_o analogisme_n and_o great_a proportion_n be_v the_o proportion_n of_o a_o great_a antecedent_n to_o the_o same_o consequent_a or_o of_o the_o same_o antecedent_n to_o a_o less_o consequent_a and_o when_o the_o proportion_n of_o the_o first_o antecedent_n to_o the_o first_o consequent_a be_v great_a than_o that_o of_o the_o second_o antecedent_n to_o the_o second_o consequent_a the_o four_o magnitude_n which_o be_v so_o to_o one_o another_o may_v be_v call_v hyperlogisme_n less_o proportion_n be_v the_o proportion_n of_o a_o less_o antecedent_n to_o the_o same_o consequent_a or_o of_o the_o same_o antecedent_n to_o a_o great_a consequent_a and_o when_o the_o proportion_n of_o the_o first_o antecedent_n to_o the_o first_o consequent_a be_v less_o than_o that_o of_o the_o second_o to_o the_o second_o the_o four_o magnitude_n may_v be_v call_v hypologisme_n 5_o one_o arithmetical_a proportion_n be_v the_o same_o with_o another_o arithmetical_a proportion_n when_o one_o of_o the_o antecedent_n exceed_v its_o consequent_a or_o be_v exceed_v by_o it_o as_o much_o as_o the_o other_o antecedent_n exceed_v its_o consequent_a or_o be_v exceed_v by_o it_o and_o therefore_o in_o four_o magnitude_n arithmetical_o proportional_a the_o sum_n of_o the_o extreme_n be_v equal_a to_o the_o sum_n of_o the_o mean_n for_o if_o a._n b_o c._n d_o be_v arithmetical_o proportional_a and_o the_o difference_n on_o both_o side_n be_v the_o same_o excess_n or_o the_o same_o defect_n e_o then_o b+c_n if_o a_o be_v great_a than_o b_o will_v be_v equal_a to_o a_o −_o e+c_n and_o a+d_n will_v be_v equal_a to_o a+c_n −_o e_o but_o a_o −_o e+c_n and_o a+c_fw-la −_o e_o be_v equal_a or_o if_o a_o be_v less_o than_o b_o than_o b+c_n will_v be_v equal_a to_o a+e+c_n and_o a+d_n will_v be_v equal_a to_o a+c+e_v but_o a+e+c_n and_o a+c+e_o be_v equal_a also_o if_o there_o be_v never_o so_o many_o magnitude_n arithmetical_o proportional_a the_o sum_n of_o they_o all_o will_v be_v equal_a to_o the_o product_n of_o half_a the_o number_n of_o the_o term_n multiply_v by_o the_o sum_n of_o the_o extreme_n for_o if_o a._n b_o c._n d_o e._n f_o be_v arithmetical_o proportional_a the_o couple_n a+f_n b+e_n c+d_n will_v be_v equal_a to_o one_o another_o and_o their_o sum_n will_v be_v equal_a to_o a+f_o multiply_v by_o the_o number_n of_o their_o combination_n that_o be_v by_o half_a the_o number_n of_o the_o term_n if_o of_o four_o unequal_a magnitude_n any_o two_o together_o take_v be_v equal_a to_o the_o other_o two_o together_o take_v than_o the_o great_a and_o the_o least_o of_o they_o will_v be_v in_o the_o same_o combination_n let_v the_o unequal_a magnitude_n be_v a_o b_o c_o d_o and_o let_v a+b_n be_v equal_a to_o c+d_v &_o let_v a_o be_v the_o great_a of_o they_o all_o i_o say_v b_o will_v be_v the_o least_o for_o if_o it_o may_v be_v let_v any_o of_o the_o rest_n as_o d_o be_v the_o least_o see_v therefore_o a_o be_v great_a than_o c_z and_o b_o then_z d_o a+b_n will_v be_v great_a than_o c+d_n which_o be_v contrary_a to_o what_o be_v suppose_v if_o there_o be_v any_o four_o magnitude_n the_o sum_n of_o the_o great_a and_o least_o the_o sum_n of_o the_o mean_n the_o difference_n of_o the_o two_o great_a and_o the_o difference_n of_o the_o two_o lest_o will_v be_v arithmetical_o proportional_a for_o let_v there_o be_v four_o magnitude_n whereof_o a_o be_v the_o great_a d_o the_o least_o and_o b_o and_o c_o the_o mean_n i_o say_v a+d_v b+c_n a_o −_o b._n c_o −_o d_o be_v arithmetical_o proportional_a for_o the_o difference_n between_o the_o first_o antecedent_n and_o its_o consequent_a be_v this_o a+d_v −_o b_o −_o c_o and_o the_o difference_n between_o the_o second_o antecedent_n and_o its_o consequent_a this_o a_o −_o b_o −_o c+d_n but_o these_o two_o difference_n be_v equal_a and_o therefore_o by_o this_o 5_o article_n a+d_v b+c_n a_o −_o b._n c_o −_o d_o be_v arithmetical_o proportional_a if_o of_o four_o magnitude_n two_o be_v equal_a to_o the_o other_o two_o they_o will_v be_v in_o reciprocal_a arithmetical_a proportion_n for_o let_v a+b_n be_v equal_a to_o c+d_v i_o say_v a._n c_o d._n b_o be_v arithmetical_o proportional_a for_o if_o they_o be_v not_o let_v a._n c_o d._n e_o suppose_n e_o to_o be_v great_a or_o less_o than_o b_o be_v arithmetical_o proportional_a and_o then_o a+e_n will_v be_v equal_a to_o c+d_v wherefore_o a+b_n and_o c+d_v be_v not_o equal_a which_o be_v contrary_a to_o what_o be_v suppose_v 6_o one_o geometrical_a proportion_n be_v the_o same_o with_o another_o geometrical_a proportion_n when_o the_o same_o cause_n produce_v equal_a effect_n in_o equal_a time_n determine_v both_o the_o proportion_n if_o a_o point_n uniformly_n move_v describe_v two_o line_n either_o with_o the_o same_o or_o different_a velocity_n all_o the_o part_n of_o they_o which_o be_v contemporary_a that_o be_v which_o be_v describe_v in_o the_o same_o time_n will_v be_v two_o to_o two_o in_o geometrical_a proportion_n whether_o the_o antecedent_n be_v take_v in_o the_o same_o line_n or_o not_o for_o from_o the_o point_n a_o in_o the_o 10_o figure_n at_o the_o end_n of_o the_o 14_o chapter_n let_v the_o two_o line_n a_o d_o a_o g_o be_v describe_v with_o uniform_a motion_n and_o let_v there_o be_v take_v in_o they_o two_o part_n ab_fw-la ae_n and_o again_o two_o other_o part_n ac_fw-la of_o in_o such_o manner_n that_o ab_fw-la ae_n be_v contemporary_a and_o likewise_o ac_fw-la of_o contemporary_a i_o say_v first_o take_z the_o antecedent_n ab_fw-la ac_fw-la in_o the_o line_n ad_fw-la and_o the_o consequent_n ae_n of_o in_o the_o line_n agnostus_n that_o ab_fw-la ac_fw-la ae_n of_o be_v proportional_n for_o see_v by_o the_o 8_o chapter_n and_o the_o 15_o article_n velocity_n be_v motion_n consider_v as_o determine_v by_o a_o certain_a length_n or_o line_n in_o a_o certain_a time_n transmit_v by_o it_o the_o quantity_n of_o the_o line_n ab_fw-la will_v be_v determine_v by_o the_o velocity_n and_o time_n by_o which_o the_o same_o ab_fw-la be_v describe_v and_o for_o the_o same_o reason_n the_o quantity_n of_o the_o line_n ac_fw-la will_v be_v determine_v by_o the_o velocity_n and_o time_n by_o which_o the_o same_o ac_fw-la be_v describe_v and_o therefore_o the_o proportion_n of_o ab_fw-la to_o ac_fw-la whether_o it_o be_v proportion_n of_o equality_n or_o of_o excess_n or_o defect_n be_v determine_v by_o the_o velocity_n and_o time_n by_o which_o ab_fw-la ac_fw-la be_v describe_v but_o see_v the_o motion_n of_o the_o point_n a_o upon_o ab_fw-la and_o ac_fw-la be_v uniform_a they_o be_v both_o desribe_v with_o equal_a velocity_n and_o therefore_o whether_o one_o of_o they_o have_v to_o the_o other_o the_o proportion_n of_o majority_n or_o of_o minority_n the_o sole_a cause_n of_o that_o proportion_n be_v the_o difference_n of_o their_o time_n and_o by_o the_o same_o reason_n it_o be_v evident_a that_o the_o proportion_n of_o ae_n to_o of_o be_v determine_v by_o the_o difference_n of_o their_o time_n only_o see_v therefore_o ab_fw-la ae_n as_o also_o ac_fw-la of_o be_v contemporary_a the_o difference_n of_o the_o time_n in_o which_o ab_fw-la and_o ac_fw-la be_v describe_v be_v the_o same_o with_o that_o in_o which_o ae_n and_o of_o be_v describe_v wherefore_o the_o proportion_n of_o ab_fw-la to_o ac_fw-la and_o the_o proportion_n of_o ae_n to_o of_o be_v both_o determine_v by_o the_o same_o cause_n but_o the_o cause_n which_o so_o determine_v the_o proportion_n of_o both_o work_v equal_o in_o equal_a time_n for_o it_o be_v uniform_a motion_n and_o therefore_o by_o the_o last_o precedent_n definition_n the_o proportion_n of_o ab_fw-la to_o ac_fw-la be_v the_o same_o with_o that_o of_o ae_n to_o of_o and_o consequent_o ab_fw-la ac_fw-la af._n of_o be_v proportional_n which_o be_v the_o first_o second_o take_v the_o antecedent_n in_o different_a line_n i_o say_v ab_fw-la ae_n ac_fw-la of_o be_v proportional_n for_o see_v ab_fw-la ae_n be_v describe_v in_o the_o same_o time_n the_o difference_n of_o the_o velocity_n in_o which_o they_o be_v describe_v be_v the_o sole_a cause_n of_o the_o proportion_n they_o have_v to_o one_o another_o and_o the_o same_o may_v be_v say_v of_o the_o proportion_n of_o ac_fw-la to_o af._n but_o see_v both_o the_o line_n ad_fw-la and_o agnostus_n be_v pass_v over_o by_o uniform_a motion_n the_o difference_n of_o the_o velocity_n in_o which_o ab_fw-la ae_n be_v describe_v will_v be_v the_o same_o with_o the_o
fhg_n and_o the_o straight_a line_n da_fw-la db_fw-la and_o dc_o proportional_a to_o the_o straight_a line_n he_o hf_n and_o hg_n i_o say_v the_o three_o point_n a_o b_o and_o c_o have_v like_a situation_n with_o the_o three_o point_n e_o f_o &_o g_o or_o be_v place_v alike_o for_o if_o he_o be_v understand_v to_o be_v lay_v upon_o da_fw-la so_o that_o the_o point_n h_n be_v in_o d_o the_o point_n f_o will_v be_v in_o the_o straight_a line_n db_n by_o reason_n of_o the_o equality_n of_o the_o angel_n adb_n and_o ehf_n and_o the_o point_n g_o will_v be_v in_o the_o straight_a line_n dc_o by_o reason_n of_o the_o equality_n of_o the_o angel_n bdc_n and_o fhg_a and_o the_o strright_a line_n ab_fw-la and_o of_o as_o also_o bc_n and_o fg_n will_v be_v parallel_n because_o ad._n ed_z bh_n fh_o cd_o gh_o be_v proportional_n by_o construction_n and_o therefore_o the_o distance_n between_o the_o point_v a_o and_o b_o and_o the_o points_z b_o and_o c_o will_v be_v proportional_a to_o the_o distance_n between_o the_o point_n e_o and_o f_o and_o the_o point_v f_o and_o g._n wherefore_o in_o the_o situation_n of_o the_o point_v a_o b_o and_o c_o and_o the_o situation_n of_o the_o point_n e_o f_o and_o g_o the_o angle_n in_o the_o same_o order_n be_v equal_a so_o that_o their_o situation_n differ_v in_o nothing_o but_o the_o inequality_n of_o their_o distance_n from_o one_o another_o and_o of_o their_o distance_n from_o the_o point_n d_o and_o h._n now_o in_o both_o the_o order_n of_o point_n those_o inequality_n be_v equal_a for_o ab_fw-la bc_n ef._n fg_v which_o be_v their_o distance_n from_o one_o another_o as_o also_o da._n db._n dc_o he._n hf._n hg_v which_o be_v their_o distance_n from_o the_o assume_v points_z d_o and_o h_z be_v proportional_n their_o difference_n therefore_o consist_v sole_o in_o the_o magnitude_n of_o their_o distance_n but_o by_o the_o definition_n of_o like_a chap._n 11._o art_n 2_o those_o thing_n which_o differ_v only_o in_o magnitude_n be_v like_a wherefore_o the_o point_v a_o b_o and_o c_o have_v to_o one_o another_o like_a situation_n with_o the_o point_n e_o f_o and_o g_o or_o be_v place_v alike_o which_o be_v to_o be_v prove_v figure_n be_v quantity_n determine_v by_o the_o situation_n or_o place_v of_o all_o its_o extreme_a point_n now_o i_o call_v those_o point_n extreme_a which_o be_v contiguous_a to_o the_o place_n which_o be_v without_o the_o figure_n in_o line_n therefore_o and_o superficies_n all_o point_n may_v be_v call_v extreme_a but_o in_o solid_n only_o those_o which_o be_v in_o the_o superficies_n that_o include_v they_o like_a figure_n be_v those_o who_o extreme_a point_n in_o one_o of_o they_o be_v all_o place_v like_o all_o the_o extreme_a point_n in_o the_o other_o for_o such_o figure_n differ_v in_o nothing_o but_o magnitude_n and_o like_o figure_n be_v alike_o place_v when_o in_o both_o of_o they_o the_o homologal_a straight_a line_n that_o be_v the_o straight_a line_n which_o connect_v the_o point_n which_o answer_v one_o another_o be_v parallel_n and_o have_v their_o proportional_a side_n incline_v the_o same_o way_n and_o see_v every_o straight_a line_n be_v like_o every_o other_o straight_a line_n and_o every_o plain_a like_o every_o other_o plain_n when_o nothing_o but_o plainness_n be_v consider_v if_o the_o line_n which_o include_v plain_n or_o the_o superficies_n which_o include_v solid_n have_v their_o proportion_n know_v it_o will_v not_o be_v hard_a to_o know_v whether_o any_o figure_n be_v like_a or_o unlike_a to_o another_o propound_a figure_n and_o thus_o much_o concern_v the_o first_o ground_n of_o philosophy_n the_o next_o place_n belong_v to_o geometry_n in_o which_o the_o quantity_n of_o figure_n be_v seek_v out_o from_o the_o proportion_n of_o line_n and_o angle_n wherefore_o it_o be_v necessary_a for_o he_o that_o will_v study_v geometry_n to_o know_v first_o what_o be_v the_o nature_n of_o quantity_n proportion_n angle_n and_o figure_n have_v therefore_o explain_v these_o in_o the_o three_o last_o chapter_n i_o think_v â—it_a to_o add_v they_o to_o this_o part_n and_o so_o pass_v to_o the_o next_o of_o the_o proportion_n of_o motion_n and_o magnitude_n chap._n xv._n of_o the_o nature_n property_n and_o diverse_a consideration_n of_o motion_n and_o endeavour_n 1_o repetition_n of_o some_o principle_n of_o the_o doctrine_n of_o motion_n former_o set_v down_o 2_o other_o principle_n add_v to_o they_o 3_o certain_a theorem_n concern_v the_o nature_n of_o motion_n 4_o diverse_a consideration_n of_o motion_n 5_o the_o way_n by_o which_o the_o first_o endeavour_n of_o body_n move_v â—endoth_a 6_o in_o motion_n which_o be_v make_v by_o concourse_n one_o of_o the_o movent_n cease_v the_o endeavour_n be_v make_v by_o the_o way_n by_o which_o the_o rest_n tend_v 7_o the_o endeavour_n of_o any_o move_v body_n which_o have_v its_o motion_n in_o the_o circumference_n of_o a_o circle_n part_n from_o the_o same_o proceed_v afterward_o in_o a_o straight_a line_n which_o touch_v the_o circle_n 8_o how_o much_o great_a the_o velocity_n or_o magnitude_n be_v of_o a_o movent_fw-la so_o much_o great_a be_v the_o efficacy_n thereof_o upon_o any_o other_o body_n in_o its_o way_n 1_o the_o next_o thing_n in_o order_n to_o be_v treat_v of_o be_v motion_n and_o magnitude_n which_o be_v the_o most_o common_a accident_n of_o all_o body_n this_o place_n therefore_o most_o proper_o belong_v to_o the_o element_n of_o geometry_n but_o because_o this_o part_n of_o philosophy_n have_v be_v improve_v by_o the_o best_a wit_n of_o all_o age_n have_v afford_v great_a plenty_n of_o matter_n than_o can_v well_o be_v thrust_v together_o within_o the_o narrow_a limit_n of_o this_o discourse_n i_o think_v fit_a to_o admonish_v the_o reader_n that_o before_o he_o proceed_v further_o he_o take_v into_o his_o hand_n the_o work_v of_o euclid_n archimedes_n apollonius_n and_o other_o as_o well_o ancient_a as_o modern_a writer_n for_o to_o what_o end_n be_v it_o to_o do_v over_o again_o that_o which_o be_v already_o do_v the_o little_a therefore_o that_o i_o shall_v say_v concern_v geometry_n in_o some_o of_o the_o follow_a chapter_n shall_v be_v such_o only_a as_o be_v new_a and_o conduce_v to_o natural_a philosophy_n i_o have_v already_o deliver_v some_o of_o the_o principle_n of_o this_o doctrine_n in_o the_o 8_o &_o 9_o chapter_n which_o i_o shall_v brief_o put_v together_o here_o that_o the_o reader_n in_o go_v on_o may_v have_v their_o light_n near_o at_o hand_n first_o therefore_o in_o the_o 8_o chap._n and_o 10_o article_n motion_n be_v define_v to_o be_v the_o continual_a privation_n of_o one_o place_n and_o acquisition_n of_o another_o second_o it_o be_v there_o show_v that_o whatsoever_o be_v move_v be_v move_v in_o time_n three_o in_o the_o same_o chap._n 11._o article_n i_o have_v define_v rest_n to_o be_v when_o a_o body_n remain_v for_o some_o time_n in_o one_o place_n four_o it_o be_v there_o show_v that_o whatsoever_o be_v move_v be_v not_o in_o any_o determine_a place_n as_o also_o that_o the_o same_o have_v be_v move_v be_v still_o move_v and_o will_v yet_o be_v move_v so_o that_o in_o every_o part_n of_o that_o space_n in_o which_o motion_n be_v make_v we_o may_v consider_v three_o time_n namely_o the_o past_a the_o present_a and_o the_o future_a time_n five_o in_o the_o 15_o article_n of_o the_o same_o chapter_n i_o have_v define_v velocity_n or_o swiftness_n to_o be_v motion_n consider_v as_o power_n namely_o that_o power_n by_o which_o a_o body_n moved_n may_v in_o a_o certain_a time_n transmit_v a_o certain_a length_n which_o also_o may_v more_o brief_o be_v enunciate_v thus_o velocity_n be_v the_o quantity_n of_o motion_n determine_v by_o time_n and_o line_n six_o in_o the_o same_o chap._n 16._o article_n i_o have_v show_v that_o motion_n be_v the_o measure_n of_o time_n seven_o in_o the_o same_o chap._n 17_o art_n i_o have_v define_v motion_n to_o be_v equal_o swift_a when_o in_o equal_a time_n equal_a length_n be_v transmit_v by_o they_o eight_o in_o the_o 18_o article_n of_o the_o same_o chapter_n motion_n be_v define_v to_o be_v equal_a when_o the_o swiftness_n of_o one_o move_v body_n compute_v in_o every_o part_n of_o its_o magnitude_n be_v equal_a to_o the_o swiftness_n of_o another_o compute_v also_o in_o every_o part_n of_o its_o magnitude_n from_o whence_o it_o be_v to_o be_v note_v that_o motion_n equal_a to_o one_o another_o and_o motion_n equal_o swift_a do_v not_o signify_v the_o same_o thing_n for_o when_o two_o horse_n draw_v abr_a the_o motion_n of_o both_o be_v great_a than_o the_o motion_n of_o either_o of_o they_o single_o but_o the_o swiftness_n of_o both_o together_o be_v but_o equal_a to_o that_o of_o either_o nine_o in_o the_o 19_o article_n of_o the_o
same_o chapter_n i_o have_v show_v that_o whatsoever_o be_v at_o rest_n will_v always_o be_v at_o rest_n unless_o there_o be_v some_o other_o body_n beside_o it_o which_o by_o get_v into_o its_o place_n suffer_v it_o no_o long_o to_o remain_v at_o rest._n and_o that_o whatsoever_o be_v move_v will_v always_o be_v move_v unless_o there_o be_v some_o other_o body_n beside_o it_o which_o hinder_v its_o motion_n ten_o in_o the_o 9_o chapter_n and_o 7_o article_n i_o have_v demonstrate_v that_o when_o any_o body_n be_v move_v which_o be_v former_o at_o rest_n the_o immediate_a efficient_a cause_n of_o that_o motion_n be_v in_o some_o other_o move_v and_o contiguous_a body_n eleven_o i_o have_v show_v in_o the_o same_o place_n that_o whatsoever_o be_v move_v will_v always_o be_v move_v in_o the_o same_o way_n and_o with_o the_o same_o swiftness_n if_o it_o be_v not_o hinder_v by_o some_o other_o move_v and_o contiguous_a body_n 2_o to_o which_o principle_n i_o shall_v here_o add_v these_o that_o follow_v first_o i_o define_v endeavour_n to_o be_v motion_n make_v in_o less_o space_n and_o time_n then_o can_v be_v give_v that_o be_v less_o than_o can_v be_v determine_v or_o assign_v by_o exposition_n or_o number_n that_o be_v motion_n make_v through_o the_o length_n of_o a_o point_n and_o in_o a_o instant_a or_o point_n of_o time_n for_o the_o explayn_v of_o which_o definition_n it_o must_v be_v remember_v that_o by_o a_o point_n be_v not_o to_o be_v understand_v that_o which_o have_v no_o quantity_n or_o which_o can_v by_o any_o mean_n be_v divide_v for_o there_o be_v no_o such_o thing_n in_o nature_n but_o that_o who_o quantity_n be_v not_o at_o all_o consider_v that_o be_v whereof_o neither_o quantity_n nor_o any_o part_n be_v compute_v in_o demonstration_n so_o that_o a_o point_n be_v not_o to_o be_v take_v for_o a_o indivisible_a but_o for_o a_o undivided_a thing_n as_o also_o a_o instant_a be_v to_o be_v take_v for_o a_o undivided_a and_o not_o for_o a_o indivisible_a time_n in_o like_a manner_n endeavour_n be_v to_o be_v conceive_v as_o motion_n but_o so_o as_o that_o neither_o the_o quantity_n of_o the_o time_n in_o which_o nor_o of_o the_o line_n in_o which_o it_o be_v make_v may_v in_o demonstration_n be_v at_o all_o bring_v into_o comparison_n with_o the_o quantity_n of_o that_o time_n or_o of_o that_o line_n of_o which_o it_o be_v a_o part_n and_o yet_o as_o a_o point_n may_v be_v compare_v with_o a_o point_n so_o one_o endeavour_n may_v be_v compare_v with_o another_o endeavour_n and_o one_o may_v be_v find_v to_o be_v great_a or_o less_o than_o another_o for_o if_o the_o vertical_a point_n of_o two_o angle_n be_v compare_v they_o will_v be_v equal_a or_o unequal_a in_o the_o same_o proportion_n which_o the_o angle_n themselves_o have_v to_o one_o another_o or_o if_o a_o straight_a line_n cut_v many_o circumference_n of_o concentric_a circle_n the_o inequality_n of_o the_o point_n of_o intersection_n will_v be_v in_o the_o same_o proportion_n which_o the_o perimeter_n have_v to_o one_o another_o and_o in_o the_o same_o manner_n if_o two_o motion_n begin_v and_o end_n both_o together_o their_o endeavour_n will_v be_v equal_a or_o unequal_a according_a to_o the_o proportion_n of_o their_o velocity_n as_o we_o see_v a_o bullet_n of_o lead_n descend_v with_o great_a endeavour_n than_o a_o ball_n of_o wool_n second_o i_o define_v impetus_fw-la or_o quickness_n of_o motion_n to_o be_v the_o swiftness_n or_o velocity_n of_o the_o body_n move_v but_o consider_v in_o the_o several_a point_n of_o that_o time_n in_o which_o it_o be_v move_v in_o which_o sense_v impetus_fw-la be_v nothing_o else_o but_o the_o quantity_n or_o velocity_n of_o endeavour_n but_o consider_v with_o the_o whole_a time_n it_o be_v the_o whole_a velocity_n of_o the_o body_n move_v take_v together_o throughout_o all_o the_o time_n and_o equal_a to_o the_o product_n of_o a_o line_n represent_v the_o time_n multiply_v into_o a_o line_n represent_v the_o arithmetical_o mean_v impetus_fw-la or_o quickness_n which_o arithmetical_a mean_n what_o it_o be_v be_v define_v in_o the_o 29_o article_n of_o the_o 13_o chapter_n and_o because_o in_o equal_a time_n the_o way_n that_o be_v pass_v be_v as_o the_o velocity_n and_o the_o impetus_fw-la be_v the_o velocity_n they_o go_v withal_o reckon_v in_o all_o the_o several_a point_n of_o the_o time_n it_o follow_v that_o during_o any_o time_n whatsoever_o howsoever_o the_o impetus_fw-la be_v increase_v or_o decrease_v the_o length_n of_o the_o way_n pass_v over_o shall_v be_v increase_v or_o decrease_v in_o the_o same_o proportion_n and_o the_o same_o line_n shall_v represent_v both_o the_o way_n of_o the_o body_n move_v and_o the_o several_a impetus_fw-la or_o degree_n of_o swiftness_n wherewith_o the_o way_n be_v pass_v over_o and_o if_o the_o body_n move_v be_v not_o a_o point_n but_o a_o straight_a line_n move_v so_o as_o that_o every_o point_n thereof_o make_v a_o several_a straight_a line_n the_o plain_a describe_v by_o its_o motion_n whether_o uniform_a accelerate_a or_o retard_v shall_v be_v great_a or_o less_o the_o time_n be_v the_o same_o in_o the_o same_o proportion_n with_o that_o of_o the_o impetus_fw-la reckon_v in_o one_o motion_n to_o the_o impetus_fw-la reckon_v in_o the_o other_o for_o the_o reason_n be_v the_o same_o in_o parallellogram_n and_o their_o side_n for_o the_o same_o cause_n also_o if_o the_o body_n move_v be_v a_o plain_a the_o solid_a describe_v shall_v be_v still_o great_a or_o less_o in_o the_o proportion_n of_o the_o several_a impetus_fw-la or_o quickness_n reckon_v through_o one_o line_n to_o the_o several_a impetus_fw-la reckon_v through_o another_o this_o understand_v let_v abcd_v in_o the_o first_o figure_n of_o the_o 17_o chapter_n be_v a_o parallelogram_n in_o which_o suppose_v the_o side_n ab_fw-la to_o be_v move_v parallel_o to_o the_o opposite_a side_n cd_o decrease_v all_o the_o way_n till_o it_o vanish_v in_o the_o point_n c_o and_o so_o describe_v the_o figure_n abefc_n the_o point_n b_o as_o ab_fw-la decrease_v will_v therefore_o describe_v the_o line_n befc_n and_o suppose_v the_o time_n of_o this_o motion_n design_v by_o the_o line_n cd_o and_o in_o the_o same_o time_n cd_o suppose_v the_o side_n ac_fw-la to_o be_v move_v parallelly_n and_o uniform_o to_o bd._n from_o the_o point_n o_o take_v at_o adventure_n in_o the_o line_n cd_o draw_v or_o parallel_v to_o bd_o cut_v the_o line_n befc_n in_o e_o and_o the_o side_n ab_fw-la in_o r._n and_o again_o from_o the_o point_n qui_fw-fr take_v also_o at_o adventure_n in_o the_o line_n cd_o draw_v q_n parallel_v to_o bd_o cut_v the_o line_n befc_n in_o f_o and_o the_o side_n ab_fw-la in_o saint_n and_o draw_v eglantine_n and_o fh_n parallel_n to_o cd_o cut_v ac_fw-la in_o g_z and_o h._n last_o suppose_v the_o same_o construction_n do_v in_o all_o the_o point_n possible_a of_o the_o line_n befc_n i_o saâ—_n that_o as_o the_o proportion_n of_o the_o swiftness_n wherewith_o qf_n oe_o db_n and_o all_o the_o rest_n supopse_v to_o be_v draw_v parallel_n to_o db_n and_o terminate_v in_o the_o line_n befc_n be_v to_o the_o proportion_n of_o their_o several_a time_n design_v by_o the_o several_a parallel_n hf_n ge_n ab_fw-la and_o all_o the_o rest_n suppose_v to_o be_v draw_v parallel_n to_o the_o line_n of_o time_n cd_o and_o terminate_v in_o the_o line_n befc_n the_o aggregate_v to_o the_o aggregate_v so_o be_v the_o area_n or_o plain_a dbefc_n to_o the_o area_n or_o plain_a acfeb_n for_o as_o ab_fw-la decrease_v continual_o by_o the_o line_n befc_n vanish_v in_o the_o time_n cd_o into_o the_o point_n c_o so_o in_o the_o same_o time_n the_o line_n dc_o continual_o decrease_v vanish_v by_o the_o same_o line_n cfeb_n into_o the_o point_n b_o and_o the_o point_n d_o describe_v in_o that_o decrease_a motion_n the_o line_n db_n equal_a to_o the_o line_n ac_fw-la describe_v by_o the_o point_n a_o in_o the_o decrease_a motion_n of_o a_o &_o b_o &_o their_o swiftness_n be_v therefore_o equal_a again_o because_o in_o the_o time_n ge_n the_o point_n o_o describe_v the_o line_n oe_v and_o in_o the_o same_o time_n the_o point_n r_o describe_v the_o line_n re_fw-mi the_o line_n oe_n shall_v be_v to_o the_o line_n re_fw-mi as_o the_o swiftness_n wherewith_o oe_n be_v describe_v to_o the_o swiftness_n wherewith_o re_n be_v describe_v in_o like_a manner_n because_o in_o the_o same_o time_n hf_n the_o point_n qui_fw-fr describe_v the_o line_n qf_n and_o the_o point_n s_o the_o line_n sf_n it_o shall_v be_v as_o the_o swiftness_n by_o which_o qf_n be_v describe_v to_o the_o swiftness_n by_o which_o sf_n be_v describe_v so_o the_o line_n itself_o qf_o to_o the_o line_n itself_o sf_n and_o so_o in_o all_o the_o line_n that_o can_v possible_o be_v
point_n f_o of_o the_o time_n a_o f_o the_o impetus_fw-la acquire_v by_o f_o k_o and_o let_v d_o e_o be_v the_o length_n pass_v through_o in_o the_o time_n a_o b_o with_o impetus_fw-la uniformly_n increase_v i_o say_v the_o length_n d_o e_o be_v to_o the_o length_n transmit_v in_o the_o time_n a_o f_o as_o the_o time_n a_o b_o multiply_v into_o the_o mean_a of_o the_o impetus_fw-la increase_a through_o the_o time_n a_o b_o be_v to_o the_o time_n a_o f_o multiply_v into_o the_o mean_a of_o the_o impetus_fw-la increase_a through_o the_o time_n a_o f._n for_o see_v the_o triangle_n a_o b_o i_o be_v the_o whole_a velocity_n of_o the_o body_n move_v in_o the_o time_n a_o b_o till_o the_o impetus_fw-la acquire_v by_o b_o i_o and_o the_o triangle_n a_o f_o k_n the_o whole_a velocity_n of_o the_o body_n move_v in_o the_o time_n a_o f_o with_o impetus_fw-la increase_n till_o there_o be_v acquire_v the_o impetus_fw-la f_o k_n the_o length_n d_o e_z to_o the_o length_n acquire_v in_o the_o time_n a_o f_o with_o impetus_fw-la increase_n from_o rest_n in_o a_o till_o there_o be_v acquire_v the_o impetus_fw-la f_o k_n will_v be_v as_o the_o triangle_n a_o b_o i_o to_o the_o triangle_n a_o f_o k_n that_o be_v if_o the_o triangle_n a_o b_o i_o and_o a_o f_o k_n be_v like_o in_o duplicate_v proportion_n of_o the_o time_n a_o b_o to_o the_o time_n a_o f_o but_o if_o unlike_a in_o the_o proportion_n compound_v of_o the_o proportion_n of_o a_o b_o to_z b_o ay_o &_o of_o a_o k_o to_o a_o f._n wherefore_o as_o a_o b_o i_o be_v to_o a_o f_o k_n so_o let_v d_o e_o be_v to_o d_o p_o for_o so_o the_o length_n transmit_v in_o the_o time_n a_o b_o with_o impetus_fw-la increase_n to_z b_o i_o will_v be_v to_o the_o length_n transmit_v in_o the_o time_n a_o f_o with_o impetus_fw-la increase_n to_z f_o k_n as_o the_o triangle_n a_o b_o i_o be_v to_o the_o triangle_n a_o f_o k_n but_o the_o triangle_n a_o b_o i_o be_v make_v by_o the_o multiplication_n of_o the_o time_n a_o b_o into_o the_o mean_a of_o the_o impetus_fw-la increase_a to_z b_o i_o and_o the_o triangle_n a_o f_o k_n be_v make_v by_o the_o multiplication_n of_o the_o time_n a_o f_o into_o the_o mean_a of_o the_o impetus_fw-la increase_a to_z f_o k_n and_o therefore_o the_o length_n d_o e_o which_o be_v transmit_v in_o the_o time_n a_o b_o with_o impetus_fw-la increase_n to_z b_o i_o to_o the_o length_n d_o p_o which_o be_v transmit_v in_o the_o time_n a_o f_o with_o impetus_fw-la increase_n to_z f_o k_n be_v as_o the_o product_n which_o be_v make_v of_o the_o time_n a_o b_o multiply_v into_o its_o mean_a impetus_fw-la to_o the_o product_n of_o the_o time_n a_o f_o multiply_v also_o into_o its_o mean_a impetus_fw-la which_o be_v to_o be_v prove_v corol._n 1_o in_o motion_n uniformly_n accelerate_a the_o proportion_n of_o the_o length_n transmit_v to_o that_o of_o their_o time_n be_v compound_v of_o the_o proportion_n of_o their_o time_n to_o their_o time_n and_o impetus_fw-la to_o impetus_fw-la corol._n 2_o in_o motion_n uniformly_n accelerate_a the_o length_n transmit_v in_o equal_a time_n take_v in_o continual_a succession_n from_o the_o begin_n of_o motion_n be_v as_o the_o difference_n of_o square_a number_n begin_v from_o unity_n namely_o as_o 3_o 5_o 7_o etc._n etc._n for_o if_o in_o the_o first_o time_n the_o length_n transmit_v be_v as_o 1_o in_o the_o first_o and_o second_o time_n the_o length_n transmit_v will_v be_v as_o 4_o which_o be_v the_o square_a of_o 2_o and_o in_o the_o three_o first_o time_n it_o will_v be_v as_o 9_o which_o be_v the_o square_a of_o 3_o and_o in_o the_o four_o first_o time_n as_o 16_o and_o so_o on_o now_o the_o difference_n of_o these_o square_n be_v 3_o 5_o 7_o etc._n etc._n corol._n 3_o in_o motion_n uniformly_n accelerate_a from_o rest_n the_o length_n transmit_v be_v to_o another_o length_n transmit_v uniform_o in_o the_o same_o time_n but_o with_o such_o impetus_fw-la as_o be_v acquire_v by_o the_o accelerate_a motion_n in_o the_o last_o point_n of_o that_o time_n as_o a_o triangle_n to_o a_o parallelogram_n which_o have_v their_o altitude_n and_o base_a common_a for_o see_v the_o length_n d_o e_o in_o the_o same_o 1_o figure_n be_v pass_v through_o with_o velocity_n as_o the_o triangle_n a_o b_o i_o it_o be_v necessary_a that_o for_o the_o pass_n through_o of_o a_o length_n which_o be_v double_a to_o d_o e_o the_o velocity_n be_v as_o the_o parallelogram_n a_o i_o for_o the_o parallelogram_n a_o i_o be_v double_a to_o the_o triangle_n a_o b_o 1._o 4_o in_o motion_n which_o beginning_n from_o rest_n be_v so_o accelerate_a that_o the_o impetus_fw-la thereof_o increase_v continual_o in_o proportion_n duplicate_v to_o the_o proportion_n of_o the_o time_n in_o which_o it_o be_v make_v a_o length_n transmit_v in_o one_o time_n will_v be_v to_o a_o length_n transmit_v in_o another_o time_n as_o the_o product_v make_v by_o the_o mean_a impetus_fw-la multiply_v into_o the_o time_n of_o one_o of_o those_o motion_n to_o the_o product_n of_o the_o mean_a impetus_fw-la multiply_v into_o the_o time_n of_o the_o other_o motion_n for_o let_v a_o b_o in_o the_o 2d_o figure_v represent_v a_o time_n in_o who_o first_o instant_n a_o let_v the_o impetus_fw-la be_v as_o the_o point_n a_o but_o as_o the_o time_n proceed_v so_o let_v the_o impetus_fw-la increase_n continual_o in_o duplicate_v proportion_n to_o that_o of_o the_o time_n till_o in_o the_o last_o point_n of_o time_n b_o the_o impetus_fw-la acquire_v by_o b_o i_o then_o take_v the_o point_n f_o any_o where_o in_o the_o time_n a_o b_o let_v the_o impetus_fw-la f_o k_n acquire_v in_o the_o time_n a_o f_o be_v ordinate_o apply_v to_o that_o point_n f._n see_v therefore_o the_o proportion_n of_o f_o k_o to_o b_o i_o be_v suppose_v to_o be_v duplicate_v to_o that_o of_o a_o f_o to_o a_o b_o the_o proportion_n of_o a_o f_o to_o a_o b_o will_v be_v subduplicate_v to_o that_o of_o f_o k_o to_o b_o i_o and_o that_o of_o a_o b_o to_o a_o f_o will_v be_v by_o chap._n 13._o article_n 16_o duplicate_v to_o that_o of_o b_o i_o to_o f_o k_n and_o consequent_o the_o point_n k_o will_v be_v in_o a_o parabolical_a line_n who_o diameter_n be_v a_o b_o and_o base_a b_o i_o and_o for_o the_o same_o reason_n to_o what_o point_n soever_o of_o the_o time_n a_o b_o the_fw-fr impetus_fw-la acquire_v in_o that_o time_n be_v ordinate_o apply_v the_o straight_a line_n design_v that_o impetus_fw-la will_v be_v in_o the_o same_o parabolical_a line_n a_o k_o i._n wherefore_o the_o mean_a impetus_fw-la multiply_v into_o the_o whole_a time_n a_o b_o will_v be_v the_o parabola_fw-la a_o k_o i_o b_o equal_a to_o the_o parallelogram_n a_o m_o which_o parallelogram_n have_v for_o one_o side_n the_o line_n of_o time_n a_o b_o and_o for_o the_o other_o the_o line_n of_o the_o impetus_fw-la a_o l_o which_o be_v two_o three_o of_o the_o impetus_fw-la b_o i_o for_o every_o parabola_fw-la be_v equal_a to_o two_o three_o of_o that_o parallelogram_n with_o which_o it_o have_v its_o altitude_n and_o base_a common_a wherefore_o the_o whole_a velocity_n in_o a_o b_o will_v be_v the_o parallelogram_n a_o m_o as_o be_v make_v by_o the_o multiplication_n of_o the_o impetus_fw-la a_o l_o into_o the_o time_n a_o b._n and_o in_o like_a manner_n if_o f_o n_o be_v take_v which_o be_v two_o three_o of_o the_o impetus_fw-la f_o k_n and_o the_o parallelogram_n f_o o_fw-fr be_v complete_v f_o o_o will_v be_v the_o whole_a velocity_n in_o the_o time_n a_o f_o as_o be_v make_v by_o the_o uniform_a impetus_fw-la a_fw-it o_o or_o f_o n_o multiply_v into_o the_o time_n a_o f._n let_v now_o the_o length_n transmit_v in_o the_o time_n a_o b_o and_o with_o the_o velocity_n a_o m_o be_v the_o straight_a line_n d_o e_z and_o last_o let_v the_o length_n transmit_v in_o the_o time_n a_o f_o with_o the_o velocity_n a_o n_o be_v d_o p_o ay_o say_v that_o as_o a_o m_n be_v to_o a_o n_o or_o as_o the_o parabola_fw-la a_o k_o i_o b_o to_o the_o parabola_fw-la a_o f_o k_n so_o be_v d_o e_z to_z d_o p._n for_o as_o a_o m_n be_v to_o f_o l_o that_o be_v as_o a_o b_o be_v to_o a_o f_o so_o let_v d_o e_o be_v to_o d_o g._n now_o the_o proportion_n of_o a_o m_n to_o a_o n_o be_v compound_v of_o the_o proportion_n of_o a_o m_n to_o f_o l_o and_o of_o f_o l_o to_o a_o n._n but_o as_o a_o m_n to_o f_o l_o so_o by_o construction_n be_v d_o e_z to_z d_o g_z and_o as_o f_o l_o be_v to_o a_o n_o see_v the_o time_n in_o both_o be_v the_o same_o namely_o a_o f_o so_o be_v the_o
be_v show_v at_o the_o end_n of_o the_o 3d_o article_n of_o the_o 15_o chapter_n the_o cause_n therefore_o of_o their_o restitution_n be_v some_o motion_n either_o of_o the_o part_n of_o the_o ambient_fw-la or_o of_o the_o part_n of_o the_o body_n compress_v or_o extend_v but_o the_o part_n of_o the_o ambient_fw-la have_v no_o endeavour_v which_o contribute_v to_o their_o compression_n or_o extension_n nor_o to_o the_o set_n of_o they_o at_o liberty_n or_o restitution_n it_o remain_v therefore_o that_o from_o the_o time_n of_o their_o compression_n or_o extension_n there_o be_v leave_v some_o endeavour_n or_o motion_n by_o which_o the_o impediment_n be_v remove_v every_o part_n resume_v its_o former_a place_n that_o be_v to_o say_v the_o whole_a restore_v itself_o 14_o in_o the_o carriage_n of_o body_n if_o that_o body_n which_o carry_v another_o hit_v upon_o any_o obstacle_n or_o be_v by_o any_o mean_n sudden_o stop_v and_o that_o which_o be_v carry_v be_v not_o stop_v it_o will_v go_v on_o till_o its_o motion_n be_v by_o some_o external_a impediment_n take_v away_o for_o i_o have_v demonstrate_v in_o the_o 8_o chapter_n at_o the_o 19_o article_n that_o motion_n unless_o it_o be_v hinder_v by_o some_o external_a resistance_n will_v be_v continue_v eternal_o with_o the_o same_o celerity_n and_o in_o the_o seven_o article_n of_o the_o 9th_o chap._n that_o the_o action_n of_o a_o external_a agent_n be_v of_o no_o effect_n without_o contact_n when_o therefore_o that_o which_o carry_v another_o thing_n be_v stop_v that_o stop_n do_v not_o present_o take_v away_o the_o motion_n of_o that_o which_o be_v carry_v it_o will_v therefore_o proceed_v till_o its_o motion_n be_v by_o little_a and_o little_o extinguish_v by_o some_o external_a resistance_n which_o be_v to_o be_v prove_v though_o experience_n alone_o have_v be_v sufficient_a to_o prove_v this_o in_o like_a manner_n if_o that_o body_n which_o carry_v another_o be_v put_v from_o rest_n into_o sudden_a motion_n that_o which_o be_v carry_v will_v not_o be_v move_v forward_o together_o with_o it_o but_o will_v be_v leave_v behind_o for_o the_o contiguous_a part_n of_o the_o body_n carry_v have_v almost_o the_o same_o motion_n with_o the_o body_n which_o carry_v it_o and_o the_o remote_a part_n will_v receive_v different_a velocity_n according_a to_o their_o different_a distance_n from_o the_o body_n that_o carry_v they_o namely_o the_o more_o remote_a the_o part_n be_v the_o less_o will_v be_v their_o degree_n of_o velocity_n it_o be_v necessary_a therefore_o that_o the_o body_n which_o be_v carry_v be_v leave_v according_o more_a or_o less_o behind_o and_o this_o also_o be_v manifest_a by_o experience_n when_o at_o the_o start_n forward_o of_o the_o horse_n the_o rider_n fall_v backward_o 15_o in_o percussion_n therefore_o when_o one_o hard_a body_n be_v in_o some_o small_a â—art_n of_o it_o strike_v by_o another_o with_o great_a force_n it_o be_v not_o necessary_a that_o the_o whole_a body_n shall_v yield_v to_o the_o stroke_n with_o the_o same_o celerity_n with_o which_o the_o strike_a part_n yield_v for_o the_o rest_n of_o the_o part_n receive_v their_o motion_n from_o the_o motion_n of_o the_o part_n strike_v and_o yield_a which_o motion_n be_v less_o propagate_v every_o way_n towards_o the_o side_n than_o it_o be_v direct_o forward_o and_o hence_o it_o be_v that_o sometime_o very_o hard_a body_n which_o be_v erect_v can_v hardly_o be_v make_v to_o stand_v be_v more_o easy_o break_v then_o throw_v down_o by_o a_o violent_a stroke_n when_o nevertheless_o if_o all_o their_o part_n together_o be_v by_o any_o weak_a motion_n thrust_v forward_o they_o will_v easy_o be_v cast_v down_o 16_o though_o the_o difference_n between_o trusion_n and_o percussion_n consist_v only_o in_o this_o that_o in_o trusion_n the_o motion_n both_o of_o the_o movent_fw-la and_o moved_n body_n begin_v both_o together_o in_o their_o very_a contact_n and_o in_o percussion_n the_o strike_a body_n be_v first_o move_v and_o afterward_o the_o body_n strike_v yet_o their_o effect_n be_v so_o different_a that_o it_o seem_v scarce_o possible_a to_o compare_v their_o force_n with_o one_o another_o i_o say_v any_o effect_n of_o percussion_n be_v propound_v as_o for_o example_n the_o stroke_n of_o a_o beetle_n of_o any_o weight_n assign_v by_o which_o a_o pile_n of_o any_o give_a length_n be_v to_o be_v drive_v into_o earth_n of_o any_o tenacity_n give_v it_o seem_v to_o i_o very_o hard_o if_o not_o impossible_a to_o define_v with_o what_o weight_n or_o with_o what_o stroke_n and_o in_o what_o time_n the_o same_o pile_n may_v be_v drive_v 〈◊〉_d a_o depth_n assign_v into_o the_o same_o earth_n the_o cause_n of_o which_o difficulty_n be_v this_o that_o the_o velocity_n of_o the_o percutient_fw-la be_v to_o be_v compare_v with_o the_o magnitude_n of_o the_o ponderant_fw-la now_o velocity_n see_v it_o be_v compute_v by_o the_o length_n of_o space_n transmit_v be_v to_o be_v account_v but_o as_o one_o dimension_n but_o weight_n be_v as_o a_o solid_a thing_n be_v measure_v by_o the_o dimension_n of_o the_o whole_a body_n and_o there_o be_v no_o comparison_n to_o be_v make_v of_o a_o solid_a body_n with_o a_o length_n that_o be_v with_o a_o line_n 17_o if_o the_o internal_a part_n of_o a_o body_n be_v at_o rest_n or_o retain_v the_o same_o situation_n with_o one_o another_o for_o any_o time_n how_o little_a soever_o there_o can_v in_o those_o part_n be_v generate_v any_o new_a motion_n or_o endeavour_v whereof_o the_o efficient_a cause_n be_v not_o without_o the_o body_n of_o which_o they_o be_v part_n for_o if_o any_o small_a part_n which_o be_v comprehend_v within_o the_o superficies_n of_o the_o whole_a body_n be_v suppose_v to_o be_v now_o at_o rest_n and_o by_o and_o by_o to_o be_v move_v that_o part_n must_v of_o necessity_n receive_v its_o motion_n from_o some_o move_v and_o contiguous_a body_n but_o by_o supposition_n there_o be_v no_o such_o move_v and_o contiguous_a part_n within_o the_o body_n wherefore_o if_o there_o be_v any_o endeavour_n or_o motion_n or_o change_v of_o situation_n in_o the_o internal_a part_n of_o that_o body_n it_o must_v needs_o arise_v from_o some_o efficient_a cause_n that_o be_v without_o the_o body_n which_o contain_v they_o which_o be_v to_o be_v prove_v 18_o in_o hard_a body_n therefore_o which_o be_v compress_v or_o extend_v if_o that_o which_o compress_v or_o extend_v they_o be_v take_v away_o they_o restore_v themselves_o to_o their_o former_a place_n or_o situation_n it_o must_v needs_o be_v that_o that_o endeavour_v or_o motion_n of_o their_o internal_a part_n by_o which_o they_o be_v able_a to_o recover_v their_o former_a place_n or_o situation_n be_v not_o extinguish_v when_o the_o force_n by_o which_o they_o be_v compress_v or_o extend_v be_v take_v away_o therefore_o when_o the_o lath_n of_o a_o cross-bow_n bend_v do_v as_o soon_o as_o it_o be_v at_o liberty_n restore_v itself_o though_o to_o he_o that_o judge_n by_o sense_n both_o it_o and_o all_o its_o part_n seem_v to_o be_v at_o rest_n yet_o he_o that_o judge_v by_o reason_n do_v not_o account_v the_o take_v away_o of_o impediment_n for_o a_o efficient_a cause_n nor_o conceive_v that_o without_o a_o efficient_a cause_n any_o thing_n can_v pass_v from_o rest_n to_o motion_n will_v conclude_v that_o the_o part_n be_v already_o in_o motion_n before_o they_o begin_v to_o restore_v themselves_o 19_o action_n and_o reaction_n proceed_v in_o the_o same_o line_n but_o from_o opposite_a term_n for_o see_v reaction_n be_v nothing_o but_o endeavour_n in_o the_o patient_n to_o restore_v itself_o to_o that_o situation_n from_o which_o it_o be_v force_v by_o the_o agent_n the_o endeavour_n or_o motion_n both_o of_o the_o agent_n and_o patient_n or_o reagent_n will_v be_v propagate_v between_o the_o same_o term_n yet_o so_o as_o that_o in_o action_n the_o term_n from_o which_o be_v in_o reaction_n the_o term_n to_o which_o and_o see_v all_o action_n proceed_v in_o this_o manner_n not_o only_o between_o the_o opposite_a term_n of_o the_o whole_a line_n in_o which_o it_o be_v propagate_v but_o also_o in_o all_o the_o part_n of_o that_o line_n the_o term_n from_o which_o and_o to_o which_o both_o of_o the_o action_n and_o reaction_n will_v be_v in_o the_o same_o line_n wherefore_o action_n and_o reaction_n proceed_v in_o the_o same_o line_n etc._n etc._n 20_o to_o what_o have_v be_v say_v of_o motion_n i_o will_v add_v what_o i_o have_v to_o say_v concern_v habit._n habit_n therefore_o be_v a_o generation_n of_o motion_n not_o of_o motion_n simple_o but_o a_o easy_a conduct_v of_o the_o move_v body_n in_o a_o certain_a and_o design_a way_n and_o see_v it_o be_v attain_v by_o the_o weaken_n of_o such_o endeavour_n as_o divert_v its_o motion_n therefore_o such_o endeavour_n be_v to_o be_v
that_o which_o the_o line_n of_o incidence_n make_v with_o that_o line_n which_o from_o the_o point_n of_o refraction_n be_v draw_v perpendicular_a to_o the_o separate_a superficies_n 8_o the_o angle_n of_o incidence_n be_v the_o compliment_n to_o a_o right_a angle_n of_o the_o angle_n of_o inclination_n and_o so_o in_o the_o first_o figure_n the_o refraction_n be_v make_v in_o a_o b_o f._n the_o refracted_a line_n be_v b_o f._n the_o line_n of_o incidence_n be_v a_o b._n the_o point_n of_o incidence_n and_o of_o refraction_n be_v b._n the_o refract_n or_o separate_a superficies_n be_v d_o b_o e._n the_o line_n of_o incidence_n produce_v direct_o be_v a_o b_o c_o the_o perpendicular_a to_o the_o separate_a superficies_n be_v b_o h._n the_o angle_n of_o refraction_n be_v c_o b_o f._n the_o angle_n refract_v be_v h_n b_o f._n the_o angle_n of_o inclination_n be_v a_o b_o g_z or_o h_o b_o c._n the_o angle_n of_o incidence_n be_v a_o b_o d._n 9_o moreover_o the_o thin_a medium_fw-la be_v understand_v to_o be_v that_o in_o which_o there_o be_v less_o resistance_n to_o motion_n or_o to_o the_o generation_n of_o motion_n &_o the_o thick_a that_o wherein_o there_o be_v great_a resistance_n 10_o and_o that_o medium_fw-la in_o which_o there_o be_v equal_a resistance_n every_o where_o be_v a_o homogeneous_a medium_fw-la all_o other_o medium_n be_v heterogeneous_a 2_o if_o a_o body_n pass_v or_o there_o be_v generation_n of_o motion_n from_o one_o medium_fw-la to_o another_o of_o different_a density_n in_o a_o line_n perpendicular_a to_o the_o separate_a superficies_n there_o will_v be_v no_o refraction_n for_o see_v on_o every_o side_n of_o the_o perdendicular_a all_o thing_n in_o the_o medium_n be_v suppose_v to_o be_v like_a and_o equal_a if_o the_o motion_n itself_o be_v suppose_v to_o be_v perpendicular_a the_o inclination_n also_o will_v be_v equal_a or_o rather_o none_o at_o all_o and_o therefore_o there_o can_v be_v no_o cause_n from_o which_o refraction_n may_v be_v infer_v to_o be_v on_o one_o side_n of_o the_o perpendicular_a which_o will_v not_o conclude_v the_o same_o refraction_n to_o be_v on_o the_o other_o side_n which_o be_v so_o refraction_n on_o one_o side_n will_v destroy_v refraction_n on_o the_o other_o side_n and_o consequent_o either_o the_o refracted_a line_n will_v be_v every_o where_o which_o be_v absurd_a or_o there_o will_v be_v no_o refracted_a line_n at_o all_o which_o be_v to_o be_v demonstrate_v corol._n it_o be_v manifest_a from_o hence_o that_o the_o cause_n of_o refraction_n consist_v only_o in_o the_o obliquity_n of_o the_o line_n of_o incidence_n whether_o the_o incident_a body_n penetrate_v both_o the_o medium_n or_o without_o penetrate_v propagate_v motion_n by_o pressure_n only_o 3_o if_o a_o body_n without_o any_o change_n of_o situation_n of_o its_o internal_a part_n as_o a_o stone_n be_v move_v oblique_o out_o of_o the_o thin_a medium_fw-la and_o proceed_v penetrate_v the_o thick_a medium_fw-la and_o the_o thick_a medium_fw-la be_v such_o as_o that_o its_o internal_a part_n be_v move_v restore_v themselves_o to_o their_o former_a situation_n the_o angle_n refract_v will_v be_v great_a than_o the_o angle_n of_o inclination_n for_o let_v d_o b_o e_o in_o the_o same_o first_o figure_n be_v the_o separate_a superficies_n of_o two_o medium_n and_o let_v a_o body_n as_o a_o stone_n throw_v be_v understand_v to_o be_v move_v as_o be_v suppose_v in_o the_o straight_a line_n a_o b_o c_o and_o let_v a_o b_o be_v in_o the_o thin_a medium_fw-la as_o in_o the_o air_n and_o b_o c_o in_o the_o thick_a as_o in_o the_o water_n i_o say_v the_o stone_n w_z ch_z be_v throw_v be_v move_v in_o the_o line_n a_o b_o will_n not_o proceed_v in_o the_o line_n b_o c_o but_o in_o some_o other_o line_n namely_o that_o with_o which_o the_o perpendicular_a b_o h_n make_v the_o refracted_a angle_n h_o b_o f_o great_a than_o the_o angle_n of_o inclination_n h_n b_o c._n for_o see_v the_o stone_n come_v from_o a_o and_o fall_v upon_o b_o makes_z that_o which_o be_v at_o b_o proceed_v towards_o h_n and_o that_o the_o like_a be_v do_v in_o all_o the_o straight_a line_n which_o be_v parallel_v to_o b_o h_n and_o see_v the_o part_n move_v restore_v themselves_o by_o contrary_a motion_n in_o the_o same_o line_n there_o will_v be_v contrary_a motion_n generate_v in_o h_n b_o and_o in_o all_o the_o straight_a line_n which_o be_v parallel_v to_o it_o wherefore_o the_o motion_n of_o the_o stone_n will_v be_v make_v by_o the_o concourse_n of_o the_o motion_n in_o a_o g_o that_o be_v in_o d_o b_o and_o in_o g_z b_o that_o be_v in_o b_o h_n and_o last_o in_o h_n b_o that_o be_v by_o the_o concourse_n of_o three_o motion_n but_o by_o the_o concourse_n of_o the_o motion_n in_o a_o g_o and_o b_o h_n the_o stone_n will_v be_v carry_v to_o c_o and_o therefore_o by_o add_v the_o motion_n in_o h_n b_o it_o will_v be_v carry_v high_a in_o some_o other_o line_n as_o in_o b_o f_o and_o make_v the_o angle_n h_o b_o f_o great_a than_o the_o angle_n h_o b_o c._n and_o from_o hence_o may_v be_v derive_v the_o cause_n why_o body_n which_o be_v throw_v in_o a_o very_a oblique_a line_n if_o either_o they_o be_v any_o thing_n flat_a or_o be_v throw_v with_o great_a force_n will_v when_o they_o fall_v upon_o the_o water_n be_v cast_v up_o again_o from_o the_o water_n into_o the_o air_n for_o let_v a_o b_o in_o the_o 2d_o figure_n be_v the_o superficies_n of_o the_o water_n into_o which_o from_o the_o point_n c_o let_v a_o stone_n be_v throw_v in_o the_o straight_a line_n c_o a_o make_v with_o the_o line_n b_o a_o produce_v a_o very_a little_a angle_n c_o a_o d_o and_o produce_v b_o a_o indefinite_o to_o d_o let_v c_o d_o be_v draw_v perpendicular_a to_o it_o and_o a_o e_z parallel_n to_o c_o d._n the_o stone_n therefore_o will_v be_v move_v in_o c_o a_o by_o the_o concourse_n of_o two_o motion_n in_o c_o d_o and_o d_o a_o who_o velocity_n be_v as_o the_o line_n themselves_o c_o d_o and_o d_o a._n and_o from_o the_o motion_n in_o c_o d_o and_o all_o its_o parallel_n downward_o as_o soon_o as_o the_o stone_n fall_v upon_o a_o there_o will_v be_v reaction_n upward_o because_o the_o water_n restore_v itself_o to_o its_o former_a situation_n if_o now_o the_o stone_n be_v throw_v with_o sufficient_a obliquity_n that_o be_v if_o the_o straight_a line_n c_o d_o be_v short_a enough_o that_o be_v if_o the_o endeavour_n of_o the_o stone_n downward_o be_v less_o than_o the_o reaction_n of_o the_o water_n upward_o that_o be_v less_o than_o the_o endeavour_v it_o have_v from_o its_o own_o gravity_n for_o that_o may_v be_v the_o stone_n will_v by_o reason_n of_o the_o excess_n of_o the_o endeavour_n which_o the_o water_n have_v to_o restore_v itself_o above_o that_o which_o the_o stone_n have_v downward_o be_v raise_v again_o above_o the_o superficies_n a_o b_o and_o be_v carry_v high_a be_v reflect_v in_o a_o line_n which_o go_v high_o as_o the_o line_n a_o g._n 4_o if_o from_o a_o point_n whatsoever_o the_o medium_fw-la be_v endeavour_v be_v propagate_v every_o way_n into_o all_o the_o part_n of_o that_o medium_fw-la and_o to_o the_o same_o endeavour_n there_o be_v oblique_o oppose_v another_o medium_fw-la of_o a_o different_a nature_n that_o be_v either_o thin_a or_o thick_a that_o endeavour_n will_v be_v so_o refract_v that_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 will_v be_v as_o the_o density_n of_o the_o first_o medium_fw-la to_o the_o density_n of_o the_o second_o medium_fw-la reciprocal_o take_v first_o let_v a_o body_n be_v in_o the_o thin_a medium_fw-la in_o a_o figure_n 3d._n and_o let_v it_o be_v understand_v to_o have_v endeavour_v every_o way_n and_o consequent_o that_o its_o endeavour_n proceed_v in_o the_o line_n a_o b_o and_o a_o b_o to_o which_o let_v b_o b_o the_o superficies_n of_o the_o thick_a medium_fw-la be_v oblique_o oppose_v in_o b_o and_o b_o so_o that_o a_o b_o and_o a_o b_o be_v equal_a and_o let_v the_o straight_a line_n b_o b_o be_v produce_v both_o way_n from_o the_o point_n b_o and_o b_o let_v the_o perpendicular_n b_o c_o and_o b_o c_o be_v draw_v and_o upon_o the_o centre_n b_o and_o b_o and_o at_o the_o equal_a distance_n b_o a_n and_o b_o a_o let_v the_o circle_n a_o c_o and_o a_o c_o be_v describe_v cut_v b_o c_o and_o b_o c_o in_o c_o and_o c_o and_o the_o same_o c_o b_o and_o c_o b_o produce_v in_o d_o and_o d_o as_o also_o a_o b_o and_o a_o b_o produce_v in_o e_z and_o e._n then_o from_o the_o point_n a_o to_o the_o straight_a line_n b_o c_o and_o b_o c_o let_v the_o perpendicular_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
will_v remain_v no_o cause_n at_o all_o why_o the_o water_n shall_v be_v force_v out_o wherefore_o the_o assertion_n of_o vacuum_n be_v repugnant_a to_o the_o very_a experiment_n which_o be_v here_o bring_v to_o establish_v it_o many_o other_o phaenomena_n be_v usual_o bring_v for_o vacuum_n as_o those_o of_o weather-glass_n aeolipile_n wind-gun_n etc._n etc._n which_o will_v all_o be_v very_o hard_a to_o be_v salve_v unless_o water_n be_v penetrable_a by_o air_n without_o the_o intermixture_n of_o empty_a space_n but_o now_o see_v air_n may_v with_o no_o great_a endeavour_n pass_v through_o not_o only_a water_n but_o any_o other_o fluid_a body_n though_o never_o so_o stubborn_a as_o quicksilver_n these_o phaenomena_n prove_v nothing_o nevertheless_o it_o may_v in_o reason_n be_v expect_v that_o he_o that_o will_v take_v away_o vacuum_n shall_v without_o vacuum_n show_v we_o such_o cause_n of_o these_o phaenomena_n as_o shall_v be_v at_o least_o of_o equal_a if_o not_o great_a probability_n this_o therefore_o shall_v be_v do_v in_o the_o follow_a discourse_n when_o i_o come_v to_o speak_v of_o these_o phaenomena_n in_o their_o proper_a place_n but_o first_o the_o most_o general_a hypothesis_n of_o natural_a philosophy_n be_v to_o be_v premise_v and_o see_v that_o supposition_n be_v put_v for_o the_o true_a cause_n of_o apparent_a effect_n every_o supposition_n except_o such_o as_o be_v absurd_a must_v of_o necessity_n consist_v of_o some_o suppose_a possible_a motion_n for_o rest_n can_v never_o be_v the_o essicient_a cause_n of_o any_o thing_n &_o motion_n suppose_v body_n movable_a of_o which_o there_o be_v three_o kind_n fluid_a consistent_a and_o mix_v of_o both_o fluid_fw-la be_v those_o who_o part_n may_v by_o very_o weak_a endeavonr_n be_v separate_v from_o one_o another_o and_o consistent_a those_o for_o the_o separation_n of_o who_o part_n great_a force_n be_v to_o be_v apply_v there_o be_v therefore_o degree_n of_o consistency_n which_o degree_n by_o comparison_n with_o more_o or_o less_o consistent_a have_v the_o name_n of_o hardness_n or_o softness_n wherefore_o a_o fluid_a body_n be_v always_o divisible_a into_o body_n equal_o fluid_a as_o quantity_n into_o quantity_n and_o soft_a body_n of_o whatsoever_o degree_n of_o softness_n into_o soft_a body_n of_o the_o same_o degree_n and_o though_o many_o man_n seem_v to_o conceive_v no_o other_o difference_n of_o fluidity_n but_o such_o as_o arise_v from_o the_o different_a magnitude_n of_o the_o part_n in_o which_o sense_n dust_n though_o of_o diamond_n may_v be_v call_v fluid_a yet_o i_o understand_v by_o fluidity_n that_o which_o be_v make_v such_o by_o nature_n equal_o in_o every_o part_n of_o the_o fluid_a body_n not_o as_o dust_n be_v fluid_a for_o so_o a_o house_n which_o be_v fall_v in_o piece_n may_v be_v call_v fluid_fw-la but_o in_o such_o manner_n as_o water_n seem_v fluid_a and_o to_o divide_v itself_o into_o part_n perpetual_o fluid_a and_o this_o be_v well_o understand_v i_o come_v to_o my_o supposition_n 5_o first_o therefore_o i_o suppose_v that_o the_o immense_a space_n which_o we_o call_v the_o world_n be_v the_o aggregate_v of_o all_o body_n which_o be_v either_o consistent_a &_o visible_a as_o the_o earth_n and_o the_o star_n or_o invisible_a as_o the_o small_a atom_n which_o be_v disseminated_a through_o the_o whole_a space_n between_o the_o earth_n and_o the_o star_n and_o last_o that_o most_o fluid_a aether_n which_o so_o fill_v all_o the_o rest_n of_o the_o universe_n as_o that_o it_o leave_v in_o it_o no_o empty_a place_n at_o all_o second_o i_o suppose_v with_o copernicus_n that_o the_o great_a body_n of_o the_o world_n which_o be_v both_o consistent_a and_o permanent_a have_v such_o order_n among_o themselves_o as_o that_o the_o sun_n have_v the_o first_o place_n mercury_n the_o second_o venus_n the_o three_o the_o earth_n with_o the_o moon_n go_v about_o it_o the_o four_o mars_n the_o five_o jupiter_n with_o his_o attendant_n the_o six_o saturn_n the_o seven_o and_o after_o these_o the_o fix_a star_n have_v their_o several_a distance_n from_o the_o sun_n three_o i_o suppose_v that_o in_o the_o sun_n &_o the_o rest_n of_o the_o planet_n there_o be_v and_o always_o have_v be_v a_o simple_a circular_a motion_n four_o i_o suppose_v that_o in_o the_o body_n of_o the_o air_n there_o be_v certain_a other_o body_n intermingle_v which_o be_v not_o fluid_a but_o withal_o that_o they_o be_v so_o small_a that_o they_o be_v not_o preceptible_a by_o sense_n and_o that_o these_o also_o have_v their_o proper_a simple_a motion_n and_o be_v some_o of_o they_o more_o some_o less_o hard_a or_o consistent_a five_o i_o suppose_v with_o kepler_n that_o as_o the_o distance_n between_o the_o sun_n and_o the_o earth_n be_v to_o the_o distance_n between_o the_o moon_n and_o the_o earth_n so_o the_o distance_n between_o the_o moon_n and_o the_o earth_n be_v to_o the_o semidiameter_n of_o the_o earth_n as_o for_o the_o magnitude_n of_o the_o circle_n and_o the_o time_n in_o which_o they_o be_v describe_v by_o the_o body_n which_o be_v in_o they_o i_o will_v suppose_v they_o to_o be_v such_o as_o shall_v seem_v most_o agreeable_a to_o the_o phaenomena_n in_o question_n 6_o the_o cause_n of_o the_o different_a season_n of_o the_o year_n and_o of_o the_o several_a variation_n of_o day_n and_o night_n in_o all_o the_o part_n of_o the_o superficies_n of_o the_o earth_n have_v be_v demonstrate_v first_o by_o copernicus_n and_o since_o by_o kepler_n galilaeus_fw-la and_o other_o from_o the_o supposition_n of_o the_o earth_n diurnal_a revolution_n about_o its_o own_o axis_n together_o with_o its_o annual_a motion_n about_o the_o sun_n in_o the_o ecliptic_a according_a to_o the_o order_n of_o the_o sign_n and_o three_o by_o the_o annual_a revolution_n of_o the_o same_o earth_n about_o its_o own_o centre_n contrary_a to_o the_o order_n of_o the_o sign_n i_o suppose_v with_o copernicus_n that_o the_o diurnal_a revolution_n be_v from_o the_o motion_n of_o the_o earth_n by_o which_o the_o aequinoctial_a circle_n be_v describe_v about_o it_o and_o as_o for_o the_o other_o two_o annual_a motion_n they_o be_v the_o efficient_a cause_n of_o the_o earth_n be_v carry_v about_o in_o the_o ecliptic_a in_o such_o manner_n as_o that_o its_o axis_n be_v always_o keep_v parallel_n to_o itself_o which_o parallelisme_n be_v for_o this_o reason_n introduce_v lest_o by_o the_o earth_n annual_a revolution_n its_o pole_n shall_v seem_v to_o be_v necessary_o carry_v about_o the_o sun_n contrary_a to_o experience_n i_o have_v in_o the_o 10_o artic._n of_o the_o â—â—th_v chap._n demonstrate_v from_o the_o supposition_n of_o simple_a circular_a motion_n in_o the_o sun_n that_o the_o earth_n be_v so_o carry_v about_o the_o sun_n as_o that_o its_o axis_n be_v thereby_o keep_v always_o parallel_v to_o itself_o wherefore_o from_o these_o two_o suppose_a motion_n in_o the_o sun_n the_o one_o simple_a circular_a motion_n the_o other_o circular_a motion_n about_o it_o own_v centre_n it_o may_v be_v demonstrate_v that_o the_o year_n have_v both_o the_o same_o variation_n of_o day_n and_o night_n as_o have_v be_v demonstrate_v by_o copernicus_n for_o if_o the_o circle_n abcd_v in_o the_o 3d_o figure_n be_v the_o ecliptic_a who_o centre_n be_v e_o and_o diameter_n aec_fw-la and_o the_o earth_n be_v place_v in_o a_o &_o the_o sun_n be_v move_v in_o the_o little_a circle_n fghi_fw-la namely_o according_a to_o the_o order_n f_o g_o h_o &_o i_o it_o have_v be_v demonstrate_v that_o a_o body_n place_v in_o a_o will_v be_v move_v in_o the_o same_o order_n through_o the_o point_n of_o the_o ecliptic_a a_o b_o c_o &_o d_o and_o will_v always_o keep_v its_o axis_n parallel_n to_o its_o self_n but_o if_o as_o i_o have_v suppose_v the_o earth_n also_o be_v move_v with_o simple_a circular_a motion_n in_o a_o plain_a that_o pass_v through_o a_o cut_v the_o plain_a of_o the_o ecliptic_a so_o as_o that_o the_o common_a section_n of_o both_o the_o plain_n be_v in_o ac_fw-la thus_o also_o the_o axis_n of_o the_o earth_n will_v be_v keep_v always_o parallel_v to_o itself_o for_o let_v the_o centre_n of_o the_o earth_n be_v move_v about_o in_o the_o circumference_n of_o the_o epicycle_n who_o diameter_n be_v lak_a which_o be_v a_o part_n of_o the_o straight_a line_n lac_fw-la therefore_o lak_a the_o diameter_n of_o the_o epicycle_n pass_v through_o the_o centre_n of_o the_o earth_n will_v be_v in_o the_o plain_a of_o the_o ecliptic_a wherefore_o see_v that_o by_o reason_n of_o the_o earth_n simple_a motion_n both_o in_o the_o ecliptic_a and_o in_o its_o epicycle_n the_o straight_a line_n lak_a be_v keep_v always_o parallel_v to_o itself_o every_o other_o straight_o line_n also_o take_v in_o the_o body_n of_o the_o earth_n and_o consequent_o its_o axis_n will_v in_o like_a manner_n be_v keep_v always_o parallel_v
reach_v the_o zodiac_n of_o the_o fix_a star_n will_v fall_v still_o upon_o the_o same_o fix_a star_n because_o the_o whole_a orb_n a_o b_o c_o d_o be_v suppose_v to_o have_v no_o magnitude_n at_o all_o in_o respect_n of_o the_o great_a distance_n of_o the_o fix_a star_n suppose_v now_o the_o sun_n to_o be_v in_o c_o it_o remain_v that_o i_o show_v the_o cause_n why_o the_o earth_n be_v near_a to_o the_o sun_n when_o in_o its_o annual_a motion_n it_o be_v find_v to_o be_v in_o d_o then_o when_o it_o be_v in_o b._n and_o i_o take_v the_o cause_n to_o be_v this_o when_o the_o earth_n be_v in_o the_o begin_n of_o capricorn_n at_o b_o the_o sun_n appear_v in_o the_o begin_n of_o cancer_n at_o d_o &_o then_o be_v the_o midst_n of_o summer_n but_o in_o the_o midst_n of_o summer_n the_o northern_a part_n of_o the_o earth_n be_v towards_o the_o sun_n which_o be_v almost_o all_o dry_a land_n contain_v all_o europe_n and_o much_o the_o great_a part_n of_o asia_n and_o america_n but_o when_o the_o earth_n be_v in_o the_o begin_n of_o cancer_n at_o d_o it_o be_v the_o midst_n of_o winter_n and_o that_o part_n of_o the_o earth_n be_v towards_o the_o sun_n which_o contain_v those_o great_a sea_n call_v the_o south_n sea_n and_o the_o indian_a sea_n which_o be_v of_o far_o great_a extent_n than_o all_o the_o dry_a land_n in_o that_o hemisphere_n wherefore_o by_o the_o last_o article_n of_o the_o 21_o chapter_n when_o the_o earth_n be_v in_o d_o it_o will_v come_v near_o to_o its_o first_o movent_fw-la that_o be_v to_o the_o sun_n which_o be_v in_o it_o that_o be_v to_o say_v the_o earth_n be_v near_a to_o the_o sun_n in_o the_o midst_n of_o winter_n when_o it_o be_v in_o d_o then_o in_o the_o midst_n of_o summer_n when_o it_o be_v b_o and_o therefore_o during_o the_o winter_n the_o sun_n be_v in_o its_o perigaeum_n and_o in_o its_o apogaeum_n during_o the_o summer_n and_o thus_o i_o have_v show_v a_o possible_a cause_n of_o the_o excentricity_n of_o the_o earth_n which_o be_v to_o be_v do_v i_o be_o therefore_o of_o keplers_n opinion_n in_o this_o that_o he_o attribute_n the_o excentricity_n of_o the_o earth_n to_o the_o difference_n of_o the_o part_n thereof_o and_o suppose_v one_o part_n to_o be_v affect_v and_o another_o disaffect_v to_o the_o sun_n and_o i_o dissent_v from_o he_o in_o this_o that_o he_o think_v it_o to_o be_v by_o magnetic_a virtue_n and_o that_o this_o magnetic_a virtue_n or_o attraction_n and_o thrust_v back_o of_o the_o earth_n be_v wrought_v by_o immateriate_a species_n which_o can_v be_v because_o nothing_o can_v give_v motion_n but_o a_o body_n move_v and_o contiguous_a for_o if_o those_o body_n be_v not_o move_v which_o be_v contiguous_a to_o a_o body_n unmoved_a how_o this_o body_n shall_v begin_v to_o be_v move_v be_v not_o imaginable_a as_o have_v be_v demonstrate_v in_o the_o seven_o article_n of_o the_o 9th_o chapter_n and_o often_o inculcate_v in_o other_o place_n to_o the_o end_n that_o philosopher_n may_v at_o last_o abstain_v from_o the_o use_n of_o such_o unconceivable_a connexion_n of_o word_n i_o dissent_v also_o from_o he_o in_o this_o that_o he_o say_v the_o similitude_n of_o body_n be_v the_o cause_n of_o their_o mutual_a attraction_n for_o if_o it_o be_v so_o i_o see_v no_o reason_n why_o one_o egg_n shall_v not_o be_v attract_v by_o another_o if_o therefore_o one_o part_n of_o the_o earth_n be_v more_o affect_v by_o the_o sun_n than_o another_o part_n it_o proceed_v from_o this_o that_o one_o part_n have_v more_o water_n the_o other_o more_o dry_a land_n and_o from_o hence_o it_o be_v as_o i_o show_v above_o that_o the_o earth_n come_v near_a to_o the_o sun_n when_o it_o shine_v upon_o that_o part_n where_o there_o be_v more_o water_n then_o when_o it_o shine_v upon_o that_o where_o there_o be_v more_o dry_a land_n 9_o this_o excentricity_n of_o the_o earth_n be_v the_o cause_n why_o the_o way_n of_o its_o annual_a motion_n be_v not_o a_o perfect_a circle_n but_o either_o a_o elliptical_a or_o almost_o a_o elliptical_a line_n as_o also_o why_o the_o axis_n of_o the_o earth_n be_v not_o keep_v exact_o parallel_v to_o itself_o in_o all_o place_n but_o only_o in_o the_o equinoctial_a point_n now_o see_v i_o have_v say_v that_o the_o moon_n be_v carry_v about_o by_o the_o earth_n in_o the_o same_o manner_n that_o the_o earth_n be_v by_o the_o sun_n and_o that_o the_o earth_n go_v about_o the_o sun_n in_o such_o manner_n as_o that_o it_o show_v sometime_o one_o hemisphere_n sometime_o the_o other_o to_o the_o sun_n it_o remain_v to_o be_v inquire_v why_o the_o moon_n have_v always_o one_o and_o the_o same_o face_n turn_v towards_o the_o earth_n suppose_v therefore_o the_o sun_n to_o be_v move_v with_o simple_a motion_n in_o the_o little_a circle_n f_o g_o h_o i_o in_o the_o four_o figure_n who_o centre_n be_v it_o and_o let_v ♈_o ♋_o ♎_o ♑_o be_v the_o annual_a circle_n of_o the_o earth_n and_o a_o the_o beginning_n of_o libra_n about_o the_o point_n a_o let_v the_o little_a circle_n l_o k_o be_v describe_v and_o in_o it_o let_v the_o centre_n of_o the_o earth_n be_v understand_v to_o be_v move_v with_o simple_a motion_n and_o both_o the_o sun_n &_o the_o earth_n to_o be_v move_v according_a to_o the_o order_n of_o the_o sign_n upon_o the_o centre_n a_o let_v the_o way_n of_o the_o moon_n m_o n_o o_o p_o be_v describe_v and_o let_v q_o r_o be_v the_o diameter_n of_o a_o circle_n cut_v the_o globe_n of_o the_o moon_n into_o two_o hemisphere_n whereof_o one_o be_v see_v by_o we_o when_o the_o moon_n be_v at_o the_o full_a and_o the_o other_o be_v turn_v from_o we_o the_o diameter_n therefore_o of_o the_o moon_n q_o o_o r_o will_n be_v perpendicular_a to_o the_o straight_a line_n t_o a._n wherefore_o the_o moon_n be_v carry_v by_o reason_n of_o the_o motion_n of_o the_o earth_n from_o o_o towards_o p._n but_o by_o reason_n of_o the_o motion_n of_o the_o sun_n if_o it_o be_v in_o p_o it_o will_v at_o the_o same_o time_n be_v carry_v from_o p_o towards_o o_o and_o by_o these_o two_o contrary_a movent_n the_o straight_a line_n q_o r_o will_v be_v turn_v about_o and_o in_o a_o quadrant_n of_o the_o circle_n m_o n_o o_o p_o it_o will_v be_v turn_v so_o much_o as_o make_v the_o four_o part_n of_o its_o whole_a conversion_n wherefore_o when_o the_o moon_n be_v in_o p_o q_o r_o will_v be_v parallel_n to_o the_o straight_a line_n m_o o._n second_o when_o the_o moon_n be_v in_o m_o the_o straight_a line_n q_o r_o will_v by_o reason_n of_o the_o motion_n of_o the_o earth_n be_v in_o m_o o._n but_o by_o the_o work_n of_o the_o sun_n motion_n upon_o it_o in_o the_o quadrant_a p_o m_z toâ—_n same_o q_o r_o will_v be_v turn_v so_o much_o as_o make_v another_o quarter_n of_o its_o whole_a conversion_n when_o therefore_o the_o moon_n be_v in_o m_o q_o r_o will_v be_v perpendicular_a to_o the_o straight_a line_n o_o m._n by_o the_o same_o reason_n when_o the_o moon_n be_v in_o n_o q_o r_o will_v be_v parallel_n to_o the_o straight_a line_n m_o o_fw-mi and_o the_o moon_n return_v to_o o_o the_o same_o q_o r_o will_v return_v to_o its_o first_o place_n and_o the_o body_n of_o the_o moon_n will_v in_o one_o entire_a period_n make_v also_o one_o entire_a conversion_n upon_o her_o own_o axis_n in_o the_o make_n of_o which_o it_o be_v manifest_a that_o one_o and_o the_o same_o face_n of_o the_o moon_n be_v always_o turn_v towards_o the_o earth_n and_o if_o any_o diameter_n be_v take_v in_o that_o little_a circle_n in_o which_o the_o moon_n be_v suppose_v to_o be_v carry_v about_o with_o simple_a motion_n the_o same_o effect_n will_v follow_v for_o if_o there_o be_v no_o action_n from_o the_o sun_n every_o diameter_n of_o the_o moon_n will_v be_v carry_v about_o always_o parallel_v to_o itself_o wherefore_o i_o have_v give_v a_o possible_a cause_n why_o one_o and_o the_o same_o face_n of_o the_o moon_n be_v always_o turn_v towards_o the_o earth_n but_o it_o be_v to_o be_v note_v that_o when_o the_o moon_n be_v without_o the_o ecliptic_a we_o do_v not_o always_o see_v the_o same_o face_n precise_o for_o we_o see_v only_o that_o part_n which_o be_v illuminate_v but_o when_o the_o moon_n be_v without_o the_o ecliptic_a that_o part_n which_o be_v towards_o we_o be_v not_o exact_o the_o same_o with_o that_o which_o be_v illuminate_v 10_o to_o these_o three_o simple_a motion_n one_o of_o the_o sun_n another_o of_o the_o moon_n and_o the_o three_o of_o the_o earth_n in_o their_o own_o little_a circle_n f_o g_o h_o i_o l_o k_o &_o q_o r_o together_o with_o the_o diurnal_a
of_o this_o evocation_n and_o swell_v and_o such_o as_o agree_v with_o the_o rest_n of_o the_o phaenomena_n of_o heat_n may_v be_v think_v to_o have_v give_v the_o cause_n of_o the_o heat_n of_o the_o sun_n it_o have_v be_v show_v in_o the_o 5_o article_n of_o the_o 21_o chapter_n that_o the_o fluid_fw-la medium_fw-la which_o we_o call_v the_o air_n be_v so_o move_v by_o the_o simple_a circular_a motion_n of_o the_o sun_n as_o that_o all_o its_o part_n even_o the_o least_o do_v perpetual_o change_v place_n with_o one_o another_o which_o change_n of_o place_n be_v that_o which_o there_o i_o call_v fermentation_n from_o this_o fermentation_n of_o the_o air_n i_o have_v in_o the_o 8_o article_n of_o the_o last_o chapter_n demonstrate_v that_o the_o water_n may_v be_v draw_v up_o into_o the_o cloud_n and_o i_o shall_v now_o show_v that_o the_o fluid_a part_n may_v in_o like_a manner_n by_o the_o same_o fermentation_n be_v draw_v out_o from_o the_o internal_a to_o the_o external_a part_n of_o our_o body_n for_o see_v that_o wheresoever_o the_o fluid_fw-la medium_fw-la be_v contiguous_a to_o the_o body_n of_o any_o live_a creature_n there_o the_o part_n of_o that_o medium_fw-la be_v by_o perpetual_a change_n of_o place_n separate_v from_o one_o another_o the_o contiguous_a part_n of_o the_o live_a creature_n must_v of_o necessity_n endeavour_v to_o enter_v into_o the_o space_n of_o the_o separate_a part_n for_o otherwise_o those_o part_n suppose_v there_o be_v no_o vacuum_n will_v have_v no_o place_n to_o go_v into_o and_o therefore_o that_o which_o be_v most_o fluid_a and_o separable_a in_o the_o part_n of_o the_o live_a creature_n which_o be_v contiguous_a to_o the_o medium_fw-la will_v go_v first_o out_o and_o into_o the_o place_n thereof_o will_v succeed_v such_o other_o part_n as_o can_v most_o easy_o transpire_v through_o the_o poâ—es_n of_o the_o skin_n and_o from_o hence_o it_o be_v necessary_a that_o the_o rest_n of_o the_o part_n which_o be_v not_o separate_v 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fire_n be_v generate_v now_o by_o fire_n i_o do_v not_o understand_v a_o body_n distinct_a from_o matter_n combustible_a or_o glow_a as_o wood_n or_o iron_n but_o the_o matter_n itself_o not_o simple_o and_o always_o but_o then_o only_o when_o it_o shine_v and_o heat_v he_o therefore_o that_o render_v a_o cause_n possible_a and_o agreeable_a to_o the_o rest_n of_o the_o phaenomena_n namely_o whence_o and_o from_o what_o action_n both_o the_o shine_v and_o heat_n proceed_v may_v be_v think_v to_o have_v give_v a_o possible_a cause_n of_o the_o generation_n of_o fire_n let_v therefore_o abc_n in_o the_o first_o figure_n be_v a_o sphere_n or_o the_o portion_n of_o a_o sphere_n who_o centre_n be_v d_o and_o let_v it_o be_v transparent_a and_o homogeneous_a as_o crystal_n glass_n or_o water_n and_o object_v to_o the_o sun_n wherefore_o the_o foremost_a part_n abc_n will_v by_o the_o simple_a motion_n of_o the_o sun_n by_o which_o it_o thrust_v forward_o the_o medium_fw-la be_v wrought_v 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see_v therefore_o the_o medium_fw-la within_o the_o sphere_n be_v thick_a than_o that_o without_o it_o the_o refracted_a line_n ak_v will_v recede_v further_o from_o the_o perpendicular_a km_n then_o quoth_fw-mi will_v recede_v from_o the_o same_o wherefore_o quoth_fw-mi will_n fall_n between_o the_o refracted_a line_n and_o the_o perpendicular_a let_v therefore_o the_o refracted_a line_n be_v kp_n cut_a foe_n in_o p_o and_o for_o the_o same_o reason_n the_o straight_a line_n lp_n will_v be_v the_o refracted_a line_n of_o the_o straight_a line_n cl._n wherefore_o see_v the_o beam_n be_v nothing_o else_o but_o the_o way_n in_o which_o the_o motion_n be_v propagate_v the_o motion_n about_o p_o will_v be_v so_o much_o more_o vehement_a than_o the_o motion_n about_o abc_n by_o how_o much_o the_o base_a of_o the_o portion_n abc_n be_v great_a than_o the_o base_a of_o a_o like_a portion_n in_o the_o sphere_n who_o centre_n be_v p_o and_o who_o magnitude_n be_v equal_a to_o that_o of_o the_o 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heat_n that_o be_v to_o say_v fire_n be_v generate_v 6_o the_o shine_a of_o glow-worm_n some_o kind_n of_o rot_a wood_n and_o of_o a_o kind_n of_o stone_n make_v at_o bolognia_n may_v have_v one_o common_a cause_n namely_o the_o expose_v of_o they_o to_o the_o hot_a sun_n we_o find_v by_o experience_n that_o the_o bolonian_a stone_n shine_v not_o unless_o it_o be_v so_o expose_v and_o after_o it_o have_v be_v expose_v it_o shine_v but_o for_o a_o little_a time_n namely_o as_o long_o as_o it_o retain_v a_o certain_a degree_n of_o heat_n and_o the_o cause_n may_v be_v that_o the_o part_n of_o which_o it_o be_v make_v may_v together_o with_o heat_n have_v simple_a motion_n imprint_v in_o they_o by_o the_o sun_n which_o if_o it_o be_v so_o it_o be_v necessary_a that_o it_o shine_v in_o the_o dark_a as_o
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