touch_v a_o spiral_n at_o the_o end_n of_o its_o first_o conversion_n for_o upon_o the_o centre_n a_o in_o the_o six_o figure_n let_v the_o circle_n bcde_v be_v describe_v and_o in_o it_o let_v archimedes_n his_o spiral_n afghb_n be_v draw_v beginning_z at_z a_o and_o end_n at_o b._n through_o the_o centre_n a_o let_v the_o straight_a line_n ce_fw-fr be_v draw_v cut_v the_o diameter_n bd_o at_o right_n angle_n and_o let_v it_o be_v produce_v to_o i_o so_o that_o ai_fw-fr be_v equal_a to_o the_o perimeter_n bcdeb_fw-mi therefore_o ib_n be_v draw_v will_v touch_v the_o spiral_n afghb_n in_o b_o which_o be_v demonstrate_v by_o archimedes_n in_o his_o book_n de_fw-fr spiralibus_fw-la and_o for_o a_o straight_a line_n equal_a to_o the_o give_v spiral_n afghb_n it_o may_v be_v find_v thus_o let_v the_o straight_a line_n a_z which_o be_v equal_a to_o the_o perimeter_n bcde_v be_v bisect_v in_o k_o and_o take_v kl_n equal_a to_o the_o radius_fw-la ab_fw-la let_v the_o rectangle_n il_fw-fr be_v complete_v let_v ml_o be_v understand_v to_o be_v the_o axis_fw-la and_o kl_n the_o base_a of_o a_o parabola_fw-la and_o let_v mk_n be_v the_o crooked_a line_n thereof_o now_o if_o the_o point_n m_o be_v conceive_v to_o be_v so_o move_v by_o the_o concourse_n of_o two_o movent_n the_o one_o from_o in_o to_o kl_n with_o velocity_n increase_a continual_o in_o the_o same_o proportion_n with_o the_o time_n the_o other_o from_o ml_o to_o ik_fw-mi uniform_o that_o both_o those_o motion_n begin_v together_o in_o m_n and_o end_n in_o k_o galilaeus_fw-la have_v demonstrate_v that_o by_o such_o motion_n of_o the_o point_n m_o the_o crooked_a line_n of_o a_o parabola_fw-la will_v be_v describe_v again_o if_o the_o point_n a_o be_v conceive_v to_o be_v move_v uniform_o in_o the_o straight_a line_n ab_fw-la and_o in_o the_o same_o time_n to_o be_v carry_v round_o upon_o the_o centre_n a_o by_o the_o circular_a motion_n of_o all_o the_o point_n between_o a_o and_o b_o archimedes_n have_v demonstrate_v that_o by_o such_o motion_n will_v be_v describe_v a_o spiral_n line_n and_o see_v the_o circle_n of_o all_o these_o motion_n be_v concentric_a in_o a_o and_o the_o interior_a circle_n be_v always_o less_o than_o the_o exterior_a in_o the_o proportion_n of_o the_o time_n in_o which_o ab_fw-la be_v pass_v over_o with_o uniform_a motion_n the_o velocity_n also_o of_o the_o circular_a motion_n of_o the_o point_n a_o will_v continual_o increase_v proportional_o to_o the_o time_n and_o thus_o far_o the_o generation_n of_o the_o parabolical_a line_n mk_n and_o of_o the_o spiral_n line_n afghb_n be_v like_a but_o the_o uniform_a motion_n in_o ab_fw-la concur_v with_o circular_a motion_n in_o the_o perimeter_n of_o all_o the_o concentric_a circle_n describe_v that_o circle_n who_o centre_n be_v a_o and_o perimeter_n bcde_v and_o therefore_o that_o circle_n be_v by_o the_o coral_n of_o the_o first_o article_n of_o the_o 16_o chapter_n the_o aggregate_v of_o all_o the_o velocity_n together_o take_v of_o the_o point_n a_o whilst_o it_o describe_v the_o spiral_n afghb_n also_o the_o rectangle_n iklm_n be_v the_o aggregate_v of_o all_o the_o velocity_n together_o take_v of_o the_o point_n m_o while_o it_o describe_v the_o crooked_a line_n mk_n and_o therefore_o the_o whole_a velocity_n by_o which_o the_o parabolicall_a line_n mk_n be_v describe_v be_v to_o the_o whole_a velocity_n with_o which_o the_o spiral_n line_n afghb_n be_v describe_v in_o the_o same_o time_n as_o the_o rectangle_n iklm_n be_v to_o the_o circle_n bcde_v that_o be_v to_o the_o triangle_n aib_n but_o because_o a_z be_v bisect_v in_o k_o &_o the_o straight_a line_n in_o &_o ab_fw-la be_v equal_a therefore_o the_o rectangle_n iklm_n and_o the_o triangle_n aib_n be_v also_o equal_a wherefore_o the_o spiral_n line_n afghb_n and_o the_o parabolical_a line_n mk_n be_v describe_v with_o equal_a velocity_n and_o in_o equal_a time_n be_v equal_a to_o one_o another_o now_o in_o the_o first_o article_n of_o the_o 18_o chapter_n a_o straight_a line_n be_v find_v out_o equal_a to_o any_o parabolical_a line_n wherefore_o also_o a_o straight_a line_n be_v find_v out_o equal_a to_o a_o give_v spiral_n line_n of_o the_o first_o revolution_n describe_v by_o archimedes_n which_o be_v to_o be_v do_v 6_o in_o the_o six_o chapter_n which_o be_v of_o method_n that_o which_o i_o shall_v there_o have_v speak_v of_o the_o analytic_n of_o geometrician_n i_o think_v fit_a to_o defer_v because_o i_o can_v not_o there_o have_v be_v understand_v as_o not_o have_v then_o so_o much_o as_o name_v line_n superficies_n solid_n equal_a and_o unequal_a etc._n etc._n wherefore_o i_o will_v in_o this_o place_n set_v down_o my_o thought_n concern_v it_o analysis_n be_v continual_a reason_v from_o the_o definition_n of_o the_o term_n of_o a_o proposition_n we_o suppose_v true_a and_o again_o from_o the_o definition_n of_o the_o term_n of_o those_o definition_n and_o so_o on_o till_o we_o come_v to_o some_o thing_n know_v the_o composition_n whereof_o be_v the_o demonstration_n of_o the_o truth_n or_o falsity_n of_o the_o first_o supposition_n and_o this_o composition_n or_o demonstration_n be_v that_o we_o call_v synthesis_n analytica_n therefore_o be_v that_o art_n by_o which_o our_o reason_n proceed_v from_o something_o suppose_v to_o principle_n that_o be_v to_o prime_a proposition_n or_o to_o such_o as_o be_v know_v by_o these_o till_o we_o have_v so_o many_o know_a proposition_n as_o be_v sufficient_a for_o the_o demonstration_n of_o the_o truth_n or_o falsity_n of_o the_o thing_n suppose_v synthetica_fw-la be_v the_o art_n itself_o of_o demonstration_n synthesis_n therefore_o and_o analysis_n differ_v in_o nothing_o but_o in_o proceed_v forward_o or_o backward_o and_o logistica_fw-la comprehend_v both_o so_o that_o in_o the_o analysis_n or_o synthesis_n of_o any_o question_n that_o be_v to_o say_v of_o any_o problem_n the_o term_n of_o all_o the_o proposition_n ought_v to_o be_v convertible_a or_o if_o they_o be_v enunciate_v hypothetical_o the_o truth_n of_o the_o consequent_a ought_v not_o only_o to_o follow_v out_o of_o the_o truth_n of_o its_o antecedent_n but_o contrary_o also_o the_o truth_n of_o the_o antecedent_n must_v necessary_o be_v infer_v from_o the_o truth_n of_o the_o consequent_a for_o otherwise_o when_o by_o resolution_n we_o be_v arrive_v at_o principle_n we_o can_v by_o composition_n return_v direct_o back_o to_o the_o thing_n seek_v for_o for_o those_o term_n which_o be_v the_o first_o in_o analysis_n will_v be_v the_o last_o in_o synthesis_n as_o for_o example_n when_o in_o resolve_v we_o say_v these_o two_o rectangle_v be_v equal_a and_o therefore_o their_o side_n be_v reciprocal_o proportional_a we_o must_v necessary_o in_o compound_v say_v the_o side_n of_o these_o rectangle_v be_v reciprocal_o proportional_a and_o therefore_o the_o rectangle_v themselves_o be_v equal_a which_o we_o can_v not_o say_v â—…ss_n rectangle_v have_v their_o side_n reciprocal_o proportional_a and_o rectangle_v be_v equal_a be_v term_n convertible_a now_o in_o every_o analysis_n that_o which_o be_v seek_v be_v the_o proportion_n of_o two_o quantity_n by_o which_o proportion_n a_o figure_n be_v describe_v the_o quantity_n seek_v for_o may_v be_v expose_v to_o sense_n and_o this_o exposition_n be_v the_o end_n and_o solution_n of_o the_o question_n or_o the_o construction_n of_o the_o problem_n and_o see_v analysis_n be_v reason_v from_o something_o suppose_v till_o we_o come_v to_o principle_n that_o be_v to_o definition_n or_o to_o theorem_n former_o know_v and_o see_v the_o same_o reason_n tend_v in_o the_o last_o place_n to_o some_o equation_n we_o can_v therefore_o make_v no_o end_n of_o resolve_v till_o we_o come_v at_o last_o to_o the_o cause_n themselves_o of_o equality_n and_o inequality_n or_o to_o theorem_n former_o demonstrate_v from_o those_o cause_n and_o so_o have_v a_o sufficient_a number_n of_o those_o theorem_n for_o the_o demonstration_n of_o the_o thing_n seek_v for_o and_o see_v also_o that_o the_o end_n of_o the_o analytic_n be_v either_o the_o construction_n of_o such_o a_o problem_n as_o be_v possible_a or_o the_o detection_n of_o the_o impossibility_n thereof_o whensoever_o the_o problem_n may_v be_v solve_v the_o analyst_n must_v not_o stay_v till_o he_o come_v to_o those_o thing_n which_o contain_v the_o efficient_a cause_n of_o that_o whereof_o he_o be_v to_o make_v construction_n but_o he_o must_v of_o necessity_n stay_v when_o he_o come_v to_o prime_a proposition_n and_o these_o be_v definition_n these_o definition_n therefore_o must_v contain_v the_o efficient_a cause_n of_o his_o construction_n i_o say_v of_o his_o construction_n not_o of_o the_o conclusion_n which_o he_o demonstrate_v for_o the_o cause_n of_o the_o conclusion_n be_v contain_v in_o the_o premise_a proposition_n that_o be_v to_o say_v the_o truth_n of_o the_o proposition_n he_o prove_v be_v
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 understand_v to_o be_v draw_v further_o asunder_o and_o a_o straight_a line_n to_o be_v that_o who_o extreme_a point_n can_v be_v draw_v further_o asunder_o and_o comparative_o a_o more_o crooked_a to_o be_v that_o line_n who_o extreme_a point_n be_v near_o to_o one_o another_o then_o those_o of_o the_o other_o suppose_v both_o the_o line_n to_o be_v of_o equal_a length_n now_o howsoever_o a_o line_n be_v bow_v it_o make_v always_o a_o sinus_n or_o cavity_n sometime_o on_o one_o side_n sometime_o on_o another_o so_o that_o the_o same_o crooked_a line_n may_v either_o have_v its_o whole_a cavity_n on_o one_o side_n only_o or_o it_o may_v have_v it_o part_n on_o one_o side_n and_o part_n on_o other_o side_n which_o be_v well_o understand_v it_o will_v be_v easy_a to_o understand_v the_o follow_a comparison_n of_o straight_a and_o crooked_a line_n first_o if_o a_o straight_a &_o a_o crooked_a line_n have_v their_o extreme_a point_n common_a the_o crooked_a line_n be_v long_a than_o the_o 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and_o huge_a volume_n of_o ethic_n be_v think_v unsufficient_a if_o what_o they_o teach_v have_v be_v certain_a and_o well_o demonstrate_v what_o then_o can_v be_v imagine_v to_o be_v the_o cause_n that_o the_o write_n of_o those_o man_n have_v increase_v science_n and_o the_o write_n of_o these_o have_v increase_v nothing_o but_o word_n save_v that_o the_o former_a be_v write_v by_o man_n that_o know_v and_o the_o late_a by_o such_o as_o know_v not_o the_o doctrine_n they_o teach_v only_o for_o ostentation_n of_o their_o wit_n and_o eloquence_n nevertheless_o i_o deny_v not_o but_o the_o read_n of_o some_o such_o book_n be_v very_o delightful_a for_o they_o be_v most_o eloquent_o write_v and_o contain_v many_o clear_a wholesome_a and_o choice_a sentence_n which_o yet_o be_v not_o universal_o true_a though_o by_o they_o universal_o pronounce_v from_o whence_o it_o come_v to_o pass_v that_o the_o circumstance_n of_o time_n place_n and_o person_n be_v change_v they_o be_v no_o less_o frequent_o make_v use_n of_o to_o confirm_v wicked_a man_n in_o their_o purpose_n then_o to_o make_v they_o understand_v the_o precept_n of_o civil_a duty_n now_o that_o which_o be_v chief_o want_v in_o they_o be_v a_o true_a and_o certain_a rule_n of_o our_o action_n by_o which_o we_o may_v know_v whether_o that_o we_o undertake_v be_v just_a or_o unjust_a for_o it_o be_v to_o no_o purpose_n to_o be_v bid_v in_o every_o thing_n to_o do_v right_o before_o there_o be_v a_o certain_a rule_n and_o measure_n of_o right_o establish_v which_o no_o man_n hitherto_o have_v establish_v see_v therefore_o from_o the_o not_o know_v of_o civil_a duty_n that_o be_v from_o the_o want_n of_o moral_a science_n proceed_v civil_a war_n and_o the_o great_a calamity_n of_o mankind_n we_o may_v very_o well_o attribute_n to_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 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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 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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
another_o other_o are_z 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d or_o of_o several_a form_n moreover_o of_o such_o as_o be_v crooked_a some_o be_v continual_o crooked_a other_o have_v part_n which_o be_v not_o crooked_a 4_o if_o a_o straight_a line_n be_v move_v in_o a_o plain_a in_o such_o manner_n that_o while_o one_o end_n of_o it_o stand_v still_o the_o whole_a line_n be_v carry_v round_o about_o till_o it_o come_v again_o into_o the_o same_o place_n from_o whence_o it_o be_v first_o move_v it_o will_v describe_v a_o plain_a superficies_n which_o will_v be_v terminate_v every_o way_n by_o that_o crooked_a line_n which_o be_v make_v by_o that_o end_n of_o the_o straight_a line_n which_o be_v carry_v round_o now_o this_o superficies_n be_v call_v a_o circle_n and_o of_o this_o circle_n the_o unmoved_a point_n be_v the_o the_o centre_n the_o crooked_a line_n which_o terminate_v it_o the_o perimeter_n and_o every_o part_n of_o that_o crooked_a line_n a_o circumference_n or_o arch_n the_o 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that_o plain_n which_o contain_v both_o its_o extreme_a point_n for_o see_v both_o its_o extreme_a point_n be_v in_o the_o plain_a that_o straight_a line_n which_o describe_v the_o plain_n will_v pass_v through_o they_o both_o and_o if_o one_o of_o they_o be_v make_v a_o centre_n and_o at_o the_o distance_n between_o both_o a_o circumference_n be_v describe_v who_o radius_fw-la be_v the_o straight_a line_n which_o describe_v the_o plain_a that_o circumference_n will_v pass_v through_o the_o other_o point_n wherefore_o between_o the_o two_o propound_a point_n there_o be_v one_o straight_a line_n by_o the_o definition_n of_o a_o circle_n contain_v whole_o in_o the_o propound_a plain_n and_o therefore_o if_o another_o straight_a line_n may_v be_v draw_v between_o the_o same_o point_n and_o yet_o not_o be_v contain_v in_o the_o same_o plain_a it_o will_v follow_v that_o between_o two_o point_n two_o straight_a line_n may_v be_v draw_v which_o have_v be_v demonstrate_v to_o be_v 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connect_v those_o point_n will_v also_o be_v in_o the_o same_o plain_a 6_o two_o line_n be_v say_v to_o touch_v one_o another_o which_o be_v both_o draw_v to_o one_o and_o the_o same_o point_n will_v not_o cut_v one_o another_o though_o they_o be_v produce_v produce_v i_o say_v in_o the_o same_o manner_n in_o which_o they_o be_v generate_v and_o therefore_o if_o two_o straight_a line_n touch_v one_o another_o in_o any_o one_o point_n they_o will_v be_v contiguous_a through_o their_o whole_a length_n also_o two_o line_n continual_o crooked_a will_v do_v the_o same_o if_o they_o be_v congruous_a and_o be_v apply_v to_o one_o another_o according_a to_o their_o congruity_n otherwise_o if_o they_o be_v incongruous_o apply_v they_o will_v as_o all_o other_o crooked_a line_n touch_v one_o another_o where_o they_o touch_v but_o in_o one_o point_n only_o which_o be_v manifest_a from_o this_o that_o there_o can_v be_v no_o congruity_n between_o a_o straight_a line_n and_o a_o line_n that_o be_v continual_o crooked_a for_o otherwise_o the_o same_o line_n may_v be_v both_o straight_a and_o crooked_a beside_o when_o a_o straight_a line_n touch_v a_o crooked_a line_n if_o the_o straight_a line_n be_v never_o so_o little_o move_v about_o upon_o the_o point_n of_o contact_n it_o will_v cut_v the_o crooked_a line_n for_o see_v it_o touch_v it_o but_o in_o one_o point_n if_o it_o incline_v any_o way_n it_o will_v do_v more_o than_o touch_v it_o that_o be_v it_o will_v either_o be_v congruous_a to_o it_o or_o it_o will_v cut_v it_o but_o it_o can_v be_v congruous_a to_o it_o and_o therefore_o it_o will_v cut_v it_o 7_o a_o angle_n according_a to_o the_o most_o general_a acception_n of_o the_o word_n may_v be_v thus_o define_v when_o two_o line_n or_o many_o superficies_n concur_v in_o one_o sole_a point_n and_o diverge_n every_o where_o else_o the_o quantity_n of_o that_o divergence_n be_v a_o angle_n and_o a_o angle_n be_v of_o two_o â—orts_n for_o first_o it_o may_v be_v make_v by_o the_o concurrence_n of_o line_n and_o then_o it_o be_v a_o superficial_a angle_n or_o by_o the_o concurrence_n of_o superficies_n and_o then_o it_o be_v call_v a_o solid_a angle_n again_o from_o the_o two_o way_n by_o which_o two_o line_n may_v diverge_n from_o one_o another_o superficial_a angle_n be_v divide_v into_o two_o kind_n for_o two_o straight_a line_n which_o be_v apply_v to_o one_o another_o and_o be_v contiguous_a in_o their_o whole_a length_n may_v be_v separate_v or_o puly_v open_a in_o such_o manner_n that_o their_o concurrence_n in_o one_o point_n will_v still_o remain_v and_o this_o separation_n or_o open_v may_v be_v either_o by_o circular_a motion_n the_o centre_n whereof_o be_v their_o point_n of_o concurrence_n and_o the_o line_n will_v still_o retain_v their_o straightness_n the_o quantity_n of_o which_o separation_n or_o divergence_n be_v a_o angle_n
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
of_o those_o two_o movent_n the_o body_n will_v be_v carry_v through_o the_o semipabolical_a crooked_a line_n a_o g_o d._n for_o let_v the_o parallelelogram_n a_o b_o d_o c_o be_v complete_v &_o from_o the_o point_n e_o taken_z any_o where_o in_o the_o straight_a line_n a_o b_o let_v e_o f_o be_v draw_v parallel_n to_o a_o c_o and_o cut_v the_o crooked_a line_n in_o g_o and_o last_o through_o the_o point_n g_o let_v a_o i_o be_v draw_v parallel_n to_o the_o straight_a line_n a_o b_o and_o c_o d._n seeing_n therefore_o the_o proportion_n of_o a_o b_o to_o a_o e_o be_v by_o supposition_n duplicate_v to_o the_o proportion_n of_o e_o f_o to_o e_z g_z that_o be_v of_o the_o time_n a_o c_o to_o the_o time_n a_o h_n at_o the_o same_o time_n when_o a_o c_o be_v in_o e_o f_o a_o b_o will_v be_v in_o h_n i_o and_o therefore_o the_o move_a body_n will_v be_v in_o the_o common_a point_n g._n and_o so_o it_o will_v always_o be_v in_o what_o part_n soever_o of_o a_o b_o the_o point_n e_o be_v take_v wherefore_o the_o move_a body_n will_v always_o be_v find_v in_o the_o parabolical_a line_n a_o g_o d_o which_o be_v to_o be_v demonstrate_v 10_o if_o a_o body_n be_v carry_v by_o two_o movent_n together_o which_o meet_v in_o any_o give_a angle_n and_o be_v move_v the_o one_o uniformly_n the_o other_o with_o impetus_fw-la increase_n from_o rest_n till_o it_o be_v equal_a to_o that_o of_o the_o uniform_a motion_n and_o with_o such_o acceleration_n that_o the_o proportion_n of_o the_o length_n transmit_v be_v every_o where_o triplicate_v to_o that_o of_o the_o time_n in_o which_o they_o be_v transmit_v the_o line_n in_o which_o that_o body_n be_v move_v will_v be_v the_o crooked_a line_n of_o the_o first_o semiparabolaster_n of_o two_o mean_n who_o baâ—e_n be_v the_o impetus_fw-la last_v acquire_v let_v the_o straight_a line_n a_o b_o in_o the_o 6_o figure_n be_v move_v uniformly_n to_o c_z d_o and_o let_v another_o movent_fw-la a_o c_o be_v move_v at_o the_o same_o time_n to_o b_o d_o with_o motion_n so_o accelerate_a that_o the_o proportion_n of_o the_o length_n transmit_v by_o every_o where_o triplicate_v to_o the_o proportion_n of_o their_o time_n and_o let_v the_o impetus_fw-la acquire_v in_o the_o end_n of_o that_o motion_n be_v b_o d_o equal_a to_o the_o straight_a line_n a_o c_o &_o last_o let_v a_o d_o be_v the_o crooked_a line_n of_o the_o first_o semiparabolaster_n of_o two_o mean_n i_o say_v that_o by_o the_o concourse_n of_o the_o two_o movent_n together_o the_o body_n will_v be_v always_o in_o that_o crooked_a line_n a_o d._n for_o let_v the_o parallelogram_n a_o b_o d_o c_o be_v complete_v and_o from_o the_o point_n e_o taken_z any_o where_o in_o the_o straight_a line_n a_o b_o let_v e_o f_o be_v draw_v parallel_n to_o a_o c_o and_o cut_v the_o crooked_a line_n in_o g_z and_o through_o the_o point_n g_o let_v h_n i_o be_v draw_v parallel_n to_o the_o straight_a line_n a_o b_o and_o c_o d._n seeing_n therefore_o the_o proportion_n of_o a_o b_o to_z a_o e_z be_v by_o supposition_n triplicate_v to_o the_o proportion_n of_o e_o f_o to_o e_z g_z that_o be_v of_o the_o time_n a_o c_o to_o the_o time_n a_o h_n at_o the_o same_o time_n when_o a_o c_o be_v in_o e_o f_o a_o b_o will_v be_v in_o h_n i_o and_o therefore_o the_o move_a body_n will_v be_v in_o the_o common_a point_n g._n and_o so_o it_o will_v always_o be_v in_o what_o part_n soever_o of_o a_o b_o the_o point_n e_o be_v take_v and_o by_o consequent_a the_o body_n will_v always_o be_v in_o the_o crooked_a line_n a_o g_o d_o which_o be_v to_o be_v demonstrate_v 11_o by_o the_o same_o method_n it_o may_v be_v show_v what_o line_n it_o be_v that_o it_o make_v by_o the_o motion_n of_o a_o body_n carry_v by_o the_o concourse_n of_o any_o two_o movent_n which_o be_v move_v one_o of_o they_o uniformly_n the_o other_o with_o acceleration_n but_o in_o such_o proportion_n of_o space_n and_o time_n as_o be_v explicable_a by_o number_n as_o duplicate_v triplicate_v etc._n etc._n or_o such_o as_o may_v be_v design_v by_o any_o break_a number_n whatsoever_o for_o which_o this_o be_v the_o rule_n let_v the_o two_o number_n of_o the_o length_n &_o time_n be_v add_v together_o &_o let_v their_o sum_n be_v the_o denominator_fw-la of_o a_o fraction_n who_o numerator_n must_v be_v the_o number_n of_o the_o length_n seek_v this_o fraction_n in_o the_o table_n of_o the_o three_o article_n of_o the_o 17_o chapter_n and_o the_o line_n seek_v will_v be_v that_o which_o denominate_v the_o threesided_n figure_n note_v on_o the_o left_a hand_n and_o the_o kind_n of_o it_o will_v be_v that_o which_o be_v number_v above_o over_o the_o fraction_n for_o example_n let_v there_o be_v a_o concourse_n of_o two_o movement_n whereof_o one_o be_v move_v uniformly_n the_o other_o with_o motion_n so_o accelerate_a that_o the_o space_n be_v to_o the_o time_n as_o 5_o to_o 3._o let_v a_o fraction_n be_v make_v who_o denominator_fw-la be_v the_o sum_n of_o 5_o and_o 3_o and_o the_o numerator_n 5_o namely_o the_o fraction_n â…_n seek_v in_o the_o table_n and_o you_o will_v find_v â…_n to_o be_v the_o three_o in_o that_o row_n which_o belong_v to_o the_o threesided_n figure_n of_o four_o mean_n wherefore_o the_o line_n of_o motion_n make_v by_o the_o concourse_n of_o two_o such_o movent_n as_o be_v last_o of_o all_o describe_v will_v be_v the_o crooked_a line_n of_o the_o three_o parabolaster_n of_o four_o mean_n 12_o if_o motion_n be_v make_v by_o the_o concourse_n of_o two_o movent_n whereof_o one_o be_v move_v uniformly_n the_o other_o begin_n from_o rest_n in_o the_o angle_n of_o concourse_n with_o any_o acceleration_n whatsoever_o the_o movent_fw-la which_o be_v move_v uniformly_n shall_v put_v forward_o the_o move_a body_n in_o the_o several_a parallel_n space_n less_o then_o if_o both_o the_o movent_n have_v uniform_a motion_n and_o still_o less_o and_o less_o as_o the_o motion_n of_o the_o other_o movent_fw-la be_v more_o and_o more_o accelerate_a let_v the_o body_n be_v place_v in_o a_o in_o the_o seven_o figure_n and_o be_v move_v by_o two_o movent_n by_o one_o with_o uniform_a motion_n from_o the_o straight_a line_n a_o b_o to_o the_o straight_a line_n c_o d_o parallel_n to_o it_o and_o by_o the_o other_o with_o any_o acceleration_n from_o the_o straight_a line_n a_o c_o to_o the_o straight_a line_n b_o d_o parallel_n to_o it_o and_o in_o the_o parallelogram_n a_o b_o d_o c_o let_v a_o space_n be_v take_v between_o any_o two_o parallel_n e_o f_o and_o g_o h._n i_o say_v that_o while_o the_o movent_fw-la a_o c_o passes_z through_o the_o latitude_n which_o be_v between_o e_o f_o and_o g_o h_n the_o body_n be_v less_o move_v forward_o from_o a_o b_o towards_z c_z d_o than_o it_o will_v have_v be_v if_o the_o motion_n from_o a_o c_o to_z b_o d_o have_v be_v uniform_a for_o suppose_v that_o while_o the_o body_n be_v make_v to_o descend_v to_o the_o parallel_n e_o f_o by_o the_o power_n of_o the_o movent_fw-la from_o a_o c_o towards_z b_o d_o the_o same_o body_n in_o the_o same_o time_n be_v move_v forward_o to_o any_o point_n f_o in_o the_o line_n e_o f_o by_o the_o power_n of_o the_o movent_fw-la from_o a_o b_o towards_z c_z d_o and_o let_v the_o straight_a line_n a_o f_o be_v draw_v and_o produce_v indeterminate_o cut_v g_z h_o in_o h._n see_v therefore_o it_o be_v as_o a_o e_z to_z a_o g_z so_z e_o f_o to_o g_z h_z if_o a_o c_z shall_v descend_v towards_o b_o d_o with_o uniform_a motion_n the_o body_n in_o the_o time_n g_o h_n for_o i_o make_v a_o c_o and_o its_o parallel_n the_o measure_n of_o time_n will_v be_v find_v in_o the_o point_n h._n but_o because_o a_o c_o be_v suppose_v to_o be_v move_v towards_o b_o d_o which_o motion_n continual_o accelerate_a that_o be_v in_o great_a proportion_n of_o space_n to_o space_n then_o of_o time_n to_o time_n in_o the_o time_n g_o h_n the_o body_n will_v be_v in_o some_o parallel_n beyond_o it_o as_o between_o g_o h_n and_o b_o d._n suppose_v now_o that_o in_o the_o end_n of_o the_o time_n g_o h_n it_o be_v in_o the_o parallel_n i_o k_n &_o in_o i_o k_o let_v i_o l_o be_v take_v equal_a to_o g_o h._n when_o therefore_o the_o body_n be_v in_o the_o parallel_n i_o k_n it_o will_v be_v in_o the_o point_n l._n wherefore_o when_o it_o be_v in_o the_o parallel_n g_o h_n it_o be_v in_o some_o point_n between_o g_z and_o h_n as_o in_o the_o point_n m_o but_o if_o both_o the_o motion_n have_v be_v uniform_a it_o have_v be_v in_o the_o point_n h_n and_o therefore_o while_o the_o movent_fw-la
crooked_a line_n of_o parabolas_fw-la and_o other_o figure_n make_v in_o imitation_n of_o parabolas_fw-la 1_o to_o find_v a_o straight_a line_n equal_a to_o the_o crooked_a line_n of_o a_o semiparabola_fw-la 2_o to_o find_v a_o straight_a line_n equal_a to_o the_o crooked_a line_n of_o the_o first_o semiparabolaster_n or_o to_o the_o crooked_a line_n of_o any_o other_o of_o the_o deficient_a figure_n of_o the_o table_n of_o the_o 3d._a article_n of_o the_o prâ—â—edent_a chapter_n 1_o aparabola_fw-la be_v give_v to_o find_v a_o straight_a line_n equal_a to_o the_o crooked_a line_n of_o the_o semiparabola_fw-la let_v the_o parabolical_a line_n give_v be_v abc_n in_o the_o first_o figure_n and_o the_o diameter_n find_v be_v ad_fw-la and_o the_o base_a draw_v dc_o and_o the_o parallelogram_n adce_n be_v complete_v draw_v the_o straight_a line_n ac_fw-la then_o divide_v ad_fw-la into_o two_o equal_a part_n in_o f_o draw_v fh_o equal_a and_o parallel_v to_o dc_o cut_v ac_fw-la in_o k_o and_o the_o parabolical_a line_n in_o o_o and_o between_o fh_n and_o foyes_n take_v a_o mean_a proportional_a fp_n and_o draw_v ao_o ap_n and_o pc_n i_o say_v that_o the_o two_o line_n ap_n and_o pc_n take_v together_o as_o one_o line_n be_v equal_a to_o the_o parabolical_a line_n aboc_n for_o the_o line_n aboc_fw-fr be_v a_o parabolical_a line_n be_v generate_v by_o the_o concourse_n of_o two_o motion_n one_o uniform_a from_o a_o to_o e_o the_o other_z in_o the_o same_o time_n uniform_o accelerate_a from_o rest_n in_o a_o to_o d._n and_o because_o the_o motion_n from_o a_o to_o e_z be_v uniform_a ae_n may_v represent_v the_o time_n of_o both_o those_o motion_n from_o the_o begin_n to_o the_o end_n let_v therefore_o ae_n be_v the_o time_n and_o consequent_o the_o line_n ordinate_o apply_v in_o the_o semiparabola_fw-la will_v design_n the_o part_n of_o time_n wherein_o the_o body_n that_o describe_v the_o line_n aboc_n be_v in_o every_o point_n of_o the_o same_o so_o that_o as_o at_o the_o end_n of_o the_o time_n ae_n or_o dc_o it_o be_v in_o c_o so_o at_o the_o end_n of_o the_o time_n foyes_n it_o will_v be_v in_o o._n and_o because_o the_o velocity_n in_o ad_fw-la be_v increase_v uniform_o that_o be_v in_o the_o same_o proportion_n with_o the_o time_n the_o same_o line_n ordinate_o apply_v in_o the_o semiparabola_fw-la will_n design_n also_o the_o continual_a augmentation_n of_o the_o impetus_fw-la till_o it_o be_v at_o the_o great_a design_v by_o the_o base_a dc_o therefore_o suppose_v uniform_a motion_n in_o the_o line_n of_o in_o the_o time_n fk_v the_o body_n in_o a_o by_o the_o concourse_n of_o the_o two_o uniform_a motion_n in_o of_o and_o fk_n will_v be_v move_v uniform_o in_o the_o line_n ak_v and_o quoth_fw-mi will_v be_v the_o increase_n of_o the_o impetus_fw-la or_o swiftness_n gain_v in_o the_o time_n fk_n and_o the_o line_n ao_o will_v be_v uniform_o describe_v by_o the_o concourse_n of_o the_o two_o uniform_a motion_n in_o of_o and_o foe_n in_o the_o time_n foyes_n from_o o_o draw_v ol_n parallel_v to_o aec_fw-la cut_v ac_fw-la in_o l_o &_o draw_v ln_o parallel_n to_o dc_o cut_v aec_fw-la in_o n_o and_o the_o parabolical_a line_n in_o m_n and_o produce_v it_o on_o the_o other_o side_n to_o ad_fw-la in_o i_o and_o in_o in_o and_o ill_n will_v be_v by_o the_o construction_n of_o a_o parabola_fw-la in_o continual_a proportion_n &_o equal_a to_o the_o three_o line_n fh_o fp_n and_o foyes_n and_o a_o straight_a line_n parallel_n to_o aec_fw-la pass_v through_o m_n will_v fall_v on_o p_o and_o therefore_o open_a will_v be_v the_o increase_n of_o impetus_fw-la gain_v in_o the_o time_n foyes_n or_o il._n last_o produce_v pm_v to_o cd_o in_o q_o and_o qc_n or_o mn_v or_o ph_n will_v be_v the_o increase_n of_o impetus_fw-la proportional_a to_o the_o time_n fp_n or_o in_o or_o dq_n suppose_v now_o uniform_a motion_n from_o h_n to_o c_o in_o the_o time_n ph._n see_v therefore_o in_o the_o time_n fp_n with_o uniform_a motion_n and_o the_o impetus_fw-la increase_v in_o proportion_n to_o the_o time_n be_v describe_v the_o straight_a line_n ap_n and_o in_o the_o rest_n of_o the_o time_n and_o impetus_fw-la namely_o ph_n be_v describe_v the_o line_n cp_n uniform_o it_o follow_v that_o the_o whole_a line_n apc_n be_v describe_v with_o the_o whole_a impetus_fw-la and_o in_o the_o same_o time_n wherewith_o be_v describe_v the_o parabolicall_a line_n abc_n and_o therefore_o the_o line_n apc_n make_v of_o the_o two_o straight_a line_n ap_n and_o pc_n be_v equal_a to_o the_o parabolical_a line_n abc_n which_o be_v to_o be_v prove_v 2_o to_o find_v a_o straight_a line_n equal_a to_o the_o crooked_a line_n of_o the_o first_o semiparabolaster_n let_v abc_n be_v the_o crooked_a line_n of_o the_o first_o semiparabolaster_n ad_fw-la the_o diameter_n dc_o the_o base_a and_o let_v the_o parallelogram_n complete_v be_v adce_n who_o diagonal_a be_v ac_fw-la divide_v the_o diameter_n into_o two_o equal_a part_n in_o f_o and_o draw_v fh_o equal_a and_o parallel_v to_o dc_o â—utting_v ac_fw-la in_o k_o the_o crooked_a line_n in_o o_o and_o aec_fw-la in_o h._n then_o draw_v ol_n parallel_v to_o aec_fw-la cut_v ac_fw-la in_o l_o and_o draw_v ln_o parallel_n to_o the_o base_a dc_o cut_v the_o crooked_a line_n in_o m_n and_o the_o straight_a line_n aec_fw-la in_o n_o and_o produce_v it_o on_o the_o other_o side_n to_o ad_fw-la in_o i._n last_o through_o the_o point_n m_o draw_v pmq_fw-fr parallel_n and_o equal_a to_o hc_n cut_v fh_n in_o p_o and_o join_v cp_n ap_n and_o ao_o i_o say_v the_o two_o straight_a line_n ap_n and_o pc_n be_v equal_a to_o the_o crooked_a line_n aboc_n for_o the_o line_n aboc_fw-fr be_v the_o crooked_a line_n of_o the_o first_o semiparabolaster_n be_v generate_v by_o the_o concourse_n of_o two_o motion_n one_o uniform_a from_o a_o to_o e_o the_o other_z in_o the_o same_o time_n accelerate_a from_o rest_n in_o a_o to_o d_o so_o as_o that_o the_o impetus_fw-la increase_v in_o proportion_n perpetual_o triplicate_v to_o that_o of_o the_o increase_n of_o the_o time_n or_o which_o be_v all_o one_o the_o length_n transmit_v be_v in_o proportion_n triplicate_v to_o that_o of_o the_o time_n of_o their_o transmission_n for_o as_o the_o impetus_fw-la or_o quickness_n increase_v so_o the_o length_n transmit_v increase_n also_o and_o because_o the_o motion_n from_o a_o to_o e_z be_v uniform_a the_o line_n ae_n may_v serve_v to_o represent_v the_o time_n and_o consequent_o the_o line_n ordinate_o draw_v in_o the_o semiparabolaster_n will_v design_n the_o part_n of_o time_n wherein_o the_o body_n beginning_n from_o rest_n in_o a_o describe_v by_o its_o motion_n the_o crooked_a line_n aboc_n and_o because_o dc_o which_o represent_v the_o great_a acquire_v impetus_fw-la be_v equal_a to_o ae_n the_o same_o ordinate_a line_n will_v represent_v the_o several_a augmentation_n of_o the_o impetus_fw-la increase_a from_o rest_n in_o a._n therefore_o suppose_v uniform_a motion_n from_o a_o to_o f_o in_o the_o time_n fk_n there_o will_v be_v describe_v by_o the_o concourse_n of_o the_o two_o uniform_a motion_n of_o and_o fk_v the_o line_n ak_v uniform_o and_o quoth_fw-mi will_v be_v the_o increase_n of_o impetus_fw-la in_o the_o time_n fk_n and_o by_o the_o concourse_n of_o the_o two_o uniform_a motion_n in_o of_o and_o foe_n will_v be_v describe_v the_o line_n ao_o uniform_o through_o the_o point_n l_o draw_v the_o straight_a line_n lmn_n parallel_v to_o dc_o cut_v the_o straight_a line_n ad_fw-la in_o i_o the_o crooked_a line_n abc_n in_o m_n and_o the_o straight_a line_n aec_fw-la in_o n_o and_o through_o the_o point_n m_o the_o straight_a line_n pmq_n parallel_n and_o equal_a to_o hc_n cut_v dc_o in_o q_o and_o fh_n in_o p._n by_o the_o concourse_n therefore_o of_o the_o two_o uniform_a motion_n in_o of_o and_o fp_n in_o the_o time_n fp_n will_v be_v uniform_o describe_v the_o straight_a line_n ap_n and_o lm_o or_o open_v will_v be_v the_o increase_n of_o impetus_fw-la to_o be_v add_v for_o the_o time_n foyes_n and_o because_o the_o proportion_n if_o in_o to_o i_o l_o be_v triplicate_v to_o the_o proportion_n of_o i_o n_o to_o i_o m_n the_o proportion_n of_o f_o h_n to_o f_o o_o will_v also_o be_v triplicate_v to_o the_o proportion_n of_o f_o h_n to_o f_o p_o and_o the_o proportional_a impetus_fw-la gain_v in_o the_o time_n f_o p_o be_v p_o h._n so_o that_o f_o h_n be_v equal_a to_z p_o c_o which_o design_v the_o whole_a impetus_fw-la acquire_v by_o the_o acceleration_n there_o be_v no_o more_o increase_n of_o impetus_fw-la to_o be_v compute_v now_o in_o the_o time_n p_o h_n suppose_v a_o uniform_a motion_n from_o h_n to_o c_o and_o by_o the_o two_o uniform_a motion_n in_o c_o h_n and_o h_n p_o will_v be_v describe_v
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
to_o be_v blame_v who_o make_v use_v of_o the_o quadratrix_n for_o the_o find_v out_o of_o a_o straight_a line_n equal_a to_o the_o arch_n of_o a_o circle_n and_o pappus_n himself_o be_v he_o faulty_a when_o he_o find_v out_o the_o trisection_n of_o a_o angle_n by_o the_o help_n of_o a_o hyperbole_n or_o be_o i_o in_o the_o wrong_n who_o think_v i_o have_v find_v out_o the_o construction_n of_o both_o these_o problem_n by_o the_o rule_n and_o compass_n only_o neither_o they_o nor_o i._o for_o the_o ancient_n make_v use_v of_o this_o analysis_n which_o proceed_v by_o the_o power_n and_o with_o they_o it_o be_v a_o fault_n to_o do_v that_o by_o a_o more_o remote_a power_n which_o may_v be_v do_v by_o a_o near_o as_o be_v a_o argument_n that_o they_o do_v not_o sufficient_o understand_v the_o nature_n of_o the_o thing_n the_o virtue_n of_o this_o kind_n of_o analysis_n consist_v in_o the_o change_n and_o turn_v and_o toss_v of_o rectangle_v and_o analogisme_n and_o the_o skill_n of_o analyst_n be_v mere_a logic_n by_o which_o they_o be_v able_a methodical_o to_o find_v out_o whatsoever_o lie_v hide_v either_o in_o the_o subject_a or_o predicate_v of_o the_o conclusion_n seek_v for_o but_o this_o do_v not_o proper_o belong_v to_o algebra_n or_o the_o analytic_n specious_a symbolical_a or_o cossick_n which_o be_v as_o i_o may_v say_v the_o brachygraphy_n of_o the_o analytic_n and_o a_o art_n neither_o of_o teach_v nor_o learn_v geometry_n but_o of_o register_n with_o brevity_n and_o celerity_n the_o invention_n of_o geometrician_n for_o though_o it_o be_v easy_a to_o discourse_v by_o symbol_n of_o very_a remote_a proposition_n yet_o whether_o such_o discourse_n deserve_v to_o be_v think_v very_o profitable_a when_o it_o be_v make_v without_o any_o idea_n of_o the_o thing_n themselves_o i_o know_v not_o chap._n xxi_o of_o circular_a motion_n 1_o in_o simple_a motion_n every_o straight_a line_n take_v in_o the_o body_n move_v be_v so_o carry_v that_o it_o be_v always_o parallel_a to_o the_o place_n in_o which_o it_o former_o be_v 2_o if_o circular_a motion_n be_v make_v about_o a_o rest_a centre_n and_o in_o that_o circle_n there_o be_v a_o epicyle_n who_o revolution_n be_v make_v the_o contrary_a way_n in_o such_o manner_n that_o in_o equal_a time_n it_o make_v equal_a angle_n every_o straight_a line_n take_v in_o that_o epicycle_n will_v be_v so_o carry_v that_o it_o will_v always_o be_v parallel_n to_o the_o place_n in_o which_o it_o former_o be_v 3_o the_o property_n of_o simple_a motion_n 4_o if_o a_o fluid_a body_n be_v move_v with_o simple_a circular_a motion_n all_o the_o point_v take_v in_o it_o will_v describe_v their_o circle_n in_o time_n proportional_a to_o the_o distance_n from_o the_o centre_n 5_o simple_z motion_n dissipate_v heterogeneous_a and_o congregate_n homogeneous_a body_n 6_o if_o a_o circle_n make_v by_o a_o movent_fw-la move_v with_o simple_a motion_n be_v commensurable_a to_o another_o circle_n make_v by_o a_o point_n which_o be_v carry_v about_o by_o the_o same_o movent_fw-la all_o the_o point_n of_o both_o the_o circle_n will_v at_o some_o time_n return_v to_o the_o same_o situation_n 7_o if_o a_o sphere_n have_v simple_a motion_n its_o motion_n will_v more_o dissipate_v heterogeneous_a body_n by_o how_o much_o it_o be_v more_o remote_a from_o the_o pole_n 8_o if_o the_o simple_a circular_a motion_n of_o a_o fluid_a body_n be_v hinder_v by_o a_o body_n which_o be_v not_o fluid_a the_o fluid_a body_n will_v spread_v itself_o upon_o the_o superficies_n of_o that_o body_n 9_o circular_a motion_n about_o a_o fix_a centre_n cast_v off_o by_o the_o tangent_fw-la such_o thing_n as_o lie_v upon_o the_o circumference_n and_o stick_v not_o to_o it_o 10_o such_o thing_n as_o be_v move_v with_o simple_a circular_a motition_n beget_v simple_a circular_a motion_n 11_o if_o that_o which_o be_v so_o move_v have_v one_o side_n hard_o and_o the_o other_o side_n fluid_a its_o motion_n will_v not_o be_v perfect_o circular_a 1_o i_o have_v already_o define_v simple_a motion_n to_o be_v that_o in_o which_o the_o several_a point_n take_v in_o a_o move_a body_n do_v in_o several_a equal_a time_n describe_v several_a equal_a arch_n and_o therefore_o in_o simple_a circular_a motion_n it_o be_v necessary_a that_o every_o straight_a line_n take_v in_o the_o move_v body_n be_v always_o carry_v parallel_n to_o itself_o which_o i_o thus_o demonstrate_v first_o let_v a_o b_o in_o the_o first_o figure_n be_v any_o straight_a line_n take_v in_o any_o solid_a body_n and_o let_v ad_fw-la be_v any_o arch_n draw_v upon_o any_o centre_n c_o and_o radius_fw-la ca._n let_v the_o point_n b_o be_v understand_v to_o describe_v towards_o the_o same_o part_n the_o arch_n be_v like_a and_o equal_a to_o the_o arch_n ad._n now_o in_o the_o same_o time_n in_o which_o the_o point_n a_o transmits_n the_o arch_n ad_fw-la the_o point_n b_o which_o by_o reason_n of_o its_o simple_a motion_n be_v suppose_v to_o be_v carry_v with_o velocity_n equal_a to_o that_o of_o a_o will_v transmit_v the_o arch_n be_v and_o at_o the_o end_n of_o the_o same_o time_n the_o whole_a ab_fw-la will_n be_v in_o de_fw-fr and_o therefore_o ab_fw-la and_o de_fw-fr be_v equal_v and_o see_v the_o arch_n ad_fw-la and_o be_v be_v like_a and_o equal_a their_o subtend_v straight_o line_n ad_fw-la and_o be_v will_v also_o be_v equal_a and_o therefore_o the_o four_o side_v figure_n abde_n will_v be_v a_o parallelogram_n wherefore_o ab_fw-la be_v carry_v parallel_n to_o itself_o and_o the_o same_o may_v be_v prove_v by_o the_o same_o method_n if_o any_o other_o straight_o line_n be_v take_v in_o the_o same_o move_v body_n in_o which_o the_o straight_a line_n ab_fw-la be_v take_v so_o that_o all_o straight_a line_n take_v in_o a_o body_n move_v with_o simple_a circular_a motion_n will_v be_v carry_v parallel_n to_o themselves_o coral_n 1_o it_o be_v manifest_a that_o the_o same_o will_v also_o happen_v in_o any_o body_n which_o have_v simple_a motion_n though_o not_o circular_a for_o all_o the_o point_n of_o any_o straight_a line_n whatsoever_o will_v describe_v line_n though_o not_o circular_a yet_o equal_a so_o that_o though_o the_o crooked_a line_n ad_fw-la and_o be_v be_v not_o arch_n of_o circle_n but_o of_o parabolas_fw-la ellipse_n or_o of_o any_o other_o figure_n yet_o both_o they_o and_o their_o subtense_n and_o the_o straight_a line_n which_o join_v they_o will_v be_v equal_a and_o parallel_v coral_n 2_o it_o be_v also_o manifest_a that_o the_o radii_fw-la of_o the_o equal_a circle_n ad_fw-la and_o be_v or_o the_o axis_n of_o a_o sphere_n will_v be_v so_o carry_v as_o to_o be_v always_o parallel_a to_o the_o place_n in_o which_o they_o former_o be_v for_o the_o straight_a line_n bf_n draw_v to_o the_o centre_n of_o the_o arch_n be_v be_v equal_a to_o the_o radius_fw-la ac_fw-la will_v also_o be_v equal_a to_o the_o straight_a line_n fe_o or_o cd_o and_o the_o angle_n bfe_n will_v be_v equal_a to_o the_o angle_n acd_v now_o the_o intersection_n of_o the_o straight_a line_n ca_n and_o be_v be_v at_o g_o the_o angle_n cge_n see_v be_v and_o ad_fw-la be_v parallel_n will_v be_v equal_a to_o the_o angle_n dac_n but_o the_o angle_n ebf_n be_v equal_a to_o the_o same_o angle_n dac_n and_o therefore_o the_o angle_n cge_n and_o ebf_n be_v also_o equal_a wherefore_o ac_fw-la and_o bf_n be_v parallel_v which_o be_v to_o be_v demonstrate_v 2_o let_v there_o be_v a_o circle_n give_v in_o the_o second_o figure_n who_o centre_n be_v a_o and_o radius_fw-la ab_fw-la and_o upon_o the_o centre_n b_o and_o any_o radius_fw-la bc_n let_v the_o epicycle_n cde_v be_v describe_v let_v the_o centre_n b_o be_v understand_v to_o be_v carry_v about_o the_o centre_n a_o and_o the_o whole_a epicycle_n with_o it_o till_o it_o be_v coincident_a with_o the_o circle_n fgh_o who_o centre_n be_v i_o and_o let_v bajazet_n be_v any_o angle_n give_v but_o in_o the_o time_n that_o the_o centre_n b_o be_v move_v to_o i_o let_v the_o epicycle_n cde_v have_v a_o contrary_a revolution_n upon_o its_o own_o centre_n namely_o from_o e_z by_z d_o to_z c_z according_a to_o the_o same_o proportion_n that_o be_v in_o such_o manner_n that_o in_o both_o the_o circle_n equal_a angle_n be_v make_v in_o equal_a time_n i_o say_v aec_fw-la the_o axis_n of_o the_o epicycle_n will_v be_v always_o carry_v parallel_n to_o itself_o let_v the_o angle_n fig_n be_v make_v equal_a to_o the_o angle_n bajazet_n if_o and_o ab_fw-la will_v therefore_o be_v parallel_n and_o how_o much_o the_o axis_n agnostus_n have_v depart_v from_o its_o former_a place_n ac_fw-la the_o measure_n of_o which_o progression_n be_v the_o angle_n cag_fw-mi or_o cbd_v which_o i_o suppose_v equal_a to_o it_o so_o much_o in_o the_o same_o time_n have_v the_o
alike_o and_o together_o with_o they_o and_o afterward_o meet_v with_o more_o body_n like_o themselves_o they_o will_v unite_v and_o become_v great_a body_n wherefore_o homogeneous_a body_n be_v congregat_v and_o heterogenous_a dissipate_v by_o simple_a motion_n in_o a_o medium_fw-la where_o they_o natural_o float_v again_o such_o as_o be_v in_o a_o fluid_fw-la medium_fw-la do_v not_o float_v but_o sink_v if_o the_o motion_n of_o the_o fluid_fw-la medium_fw-la be_v strong_a enough_o will_v be_v stir_v up_o and_o carry_v away_o by_o that_o motion_n and_o consequent_o they_o will_v be_v hinder_v from_o return_v to_o that_o place_n to_o which_o they_o sink_v natural_o and_o in_o which_o only_o they_o will_v unite_v and_o out_o of_o which_o they_o be_v promiscuous_o carry_v that_o be_v they_o be_v disorderly_o mingle_v now_o this_o motion_n by_o which_o homogeneous_a body_n be_v congregat_v and_o heterogeneous_a be_v scatter_v be_v that_o which_o be_v common_o call_v fermentation_n from_o the_o latin_a fervere_fw-la as_o the_o greek_n have_v their_o 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d which_o signify_v the_o same_o from_o 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d ferveo_fw-la for_o seething_n make_v all_o the_o part_n of_o the_o water_n change_v their_o place_n and_o the_o part_n of_o any_o thing_n that_o be_v throw_v into_o it_o will_v go_v several_a way_n according_a to_o their_o several_a nature_n and_o yet_o all_o fervour_n or_o seething_n be_v not_o cause_v by_o fire_n for_o new_a wine_n and_o many_o other_o thing_n have_v also_o their_o fermentation_n and_o fervour_n to_o which_o fire_n contribute_v little_a and_o some_o time_n nothing_o but_o when_o in_o fermentation_n we_o find_v heat_n it_o be_v make_v by_o the_o fermentation_n 6_o four_o in_o what_o time_n soever_o the_o movent_fw-la who_o centre_n be_v a_o in_o the_o 2d_o figure_n move_v in_o kln_n shall_v by_o any_o number_n of_o revolution_n that_o be_v when_o the_o perimeter_n by_o and_o kln_v be_v commensurable_a have_v describe_v a_o line_n equal_a to_o the_o circle_n which_o pass_v through_o the_o point_n b_o an_o i_o in_o the_o same_o time_n all_o the_o point_n of_o the_o float_a body_n who_o centre_n be_v b_o shall_v return_v to_o have_v the_o same_o situation_n in_o respect_n of_o the_o movent_fw-la from_o which_o they_o depart_v for_o see_v it_o be_v as_o the_o distance_n basilius_n that_o be_v as_o the_o radius_fw-la of_o the_o circle_n which_o pass_v through_o by_o be_v to_o the_o perimeter_n itself_o by_o so_o the_o radius_fw-la of_o the_o circle_n kln_n be_v to_o the_o perimeter_n kln_n and_o see_v the_o velocity_n of_o the_o point_n b_o and_o k_o be_v equal_a the_o time_n also_o of_o the_o revolution_n in_o ib_n to_o the_o time_n of_o one_o revolution_n in_o kln_n will_v be_v as_o the_o perimeter_n by_o to_o the_o perimeter_n kln_n and_o therefore_o so_o many_o revolution_n in_o kln_n as_o together_o take_v be_v equal_a to_o the_o perimeter_n by_o will_v be_v finish_v in_o the_o same_o time_n in_o which_o the_o whole_a perimeter_n by_o be_v finish_v &_o therefore_o also_o the_o point_n l_o n_o f_o &_o h_n or_o any_o of_o the_o rest_n will_v in_o the_o same_o time_n return_v to_o the_o same_o situation_n from_o which_o they_o depart_v and_o this_o may_v be_v demonstrate_v whatsoever_o be_v the_o point_v consider_v wherefore_o all_o the_o point_n shall_v in_o that_o time_n return_v to_o the_o same_o situation_n which_o be_v to_o be_v prove_v from_o hence_o it_o follow_v that_o if_o the_o perimeter_n by_o and_o lkn_o be_v not_o commensurable_a than_o all_o the_o point_n will_v never_o return_v to_o have_v the_o same_o situation_n or_o configuration_n in_o respect_n of_o one_o another_o 7_o in_o simple_a motion_n if_o the_o body_n move_v be_v of_o a_o spherical_a figure_n it_o have_v less_o force_n towards_o its_o pole_n then_o towards_o its_o middle_n to_o dissipate_v heterogeneous_a or_o to_o congregate_v homogeneous_a body_n let_v there_o be_v a_o sphere_n as_o in_o the_o three_o figure_n who_o centre_n be_v a_o and_o diameter_n bc_n &_o let_v it_o be_v conceive_v to_o be_v move_v with_o simple_a circular_a motion_n of_o which_o motion_n let_v the_o axis_n be_v the_o straight_a line_n de_fw-fr cut_v the_o diameter_n bc_n at_o right_a angle_n in_o a._n let_v now_o the_o circle_n which_o be_v describe_v by_o any_o point_n b_o of_o the_o sphere_n have_v bf_n for_o its_o diameter_n and_o take_v fg_v equal_a to_o bc_n and_o divide_v it_o in_o the_o middle_n in_o h_n the_o centre_n of_o the_o sphere_n a_o will_v when_o half_o a_o revolution_n be_v finish_v lie_v in_o h._n and_o see_v hf_n and_o ab_fw-la be_v equal_a a_o circle_n describe_v upon_o the_o centre_n h_n with_o the_o radius_fw-la hf_n or_o hg_n will_v be_v equal_a to_o the_o circle_n who_o centre_n be_v a_o and_o radius_fw-la ab_fw-la and_o if_o the_o same_o motion_n be_v continue_v the_o point_n b_o will_n at_o the_o end_n of_o another_o half_a revolution_n return_v to_o the_o place_n from_o whence_o it_o begin_v to_o be_v move_v and_o therefore_o at_o the_o end_n of_o half_a a_o revolution_n the_o point_n b_o will_v be_v carry_v to_o f_o and_o the_o whole_a hemisphere_n dbe_n into_o that_o hemisphere_n in_o which_o be_v the_o point_n l_o k_o and_o f._n wherefore_o that_o part_n of_o the_o fluid_fw-la medium_fw-la which_o be_v contiguous_a to_o the_o point_n f_o will_v in_o the_o same_o time_n go_v back_o the_o length_n of_o the_o straight_a line_n bf_n and_o in_o the_o return_n of_o the_o point_n f_o to_o b_o that_o be_v of_o g_z to_z c_z the_o fluid_fw-la medium_fw-la will_v go_v back_o as_o much_o in_o a_o straight_a line_n from_o the_o point_n c._n and_o this_o be_v the_o effect_n of_o simple_a motion_n in_o the_o middle_n of_o the_o sphere_n where_o the_o distance_n from_o the_o pole_n be_v great_a let_v now_o the_o point_n i_o be_v take_v in_o the_o same_o sphere_n near_o to_o the_o pole_n e_o and_o through_o it_o let_v the_o straight_a line_n ik_fw-mi be_v draw_v parallel_n to_o the_o straight_a line_n bf_n cut_v the_o arch_n fl_fw-mi in_o k_o &_o the_o axis_n hl_n in_o m_n then_o connect_v hk_n upon_o hf_n let_v the_o perpendicular_a kn_n be_v draw_v in_o the_o same_o time_n therefore_o that_o b_o comes_z to_z f_o the_o point_n i_o will_v come_v to_o k_o bf_n and_o ik_fw-mi being_n equal_a and_o describe_v with_o the_o same_o velocity_n now_o the_o motion_n in_o ik_fw-mi to_o the_o fluid_fw-la medium_fw-la upon_o which_o it_o work_v namely_o to_o that_o part_n of_o the_o medium_fw-la which_o be_v contiguous_a to_o the_o point_n k_o be_v oblique_a whereas_o if_o it_o proceed_v in_o the_o straight_a line_n hk_n it_o will_v be_v perpendicular_a and_o therefore_o the_o motion_n which_o proceed_v in_o ik_fw-mi have_v less_o power_n then_o that_o which_o proceed_v in_o hk_n with_o the_o same_o velocity_n but_o the_o motion_n in_o hk_n and_o hf_n do_v equal_o thrust_v back_o the_o medium_fw-la and_o therefore_o the_o part_n of_o the_o sphere_n at_o k_o move_v the_o medium_fw-la less_o than_o the_o part_n at_o f_o namely_o so_o much_o less_o as_o kn_n be_v less_o than_o hf._n wherefore_o also_o the_o same_o motion_n have_v less_o power_n to_o disperse_v heterogeneous_a and_o to_o congregate_v homogeneous_a body_n when_o it_o be_v near_o than_o when_o it_o be_v more_o remote_a from_o the_o pole_n which_o be_v to_o be_v prove_v corollary_n it_o be_v also_o necessary_a that_o in_o plain_n which_o be_v perpendicular_a to_o the_o axis_n and_o more_o remote_a than_o the_o pole_n itself_o from_o the_o middle_n of_o the_o sphere_n this_o simple_a motion_n have_v no_o effect_n for_o the_o axis_n de_fw-fr with_o simple_a motion_n describe_v the_o superficies_n of_o a_o cylinder_n and_o towards_o the_o base_n of_o the_o cylinder_n there_o be_v in_o this_o motion_n no_o endeavour_n at_o all_o 8_o if_o in_o a_o fluid_fw-la medium_fw-la move_v about_o as_o have_v be_v say_v with_o simple_a motion_n there_o be_v conceive_v to_o float_v some_o other_o spherical_a body_n which_o be_v not_o fluid_a the_o part_n of_o the_o medium_fw-la which_o be_v stop_v by_o that_o body_n will_v endeavour_v to_o spread_v themselves_o every_o way_n upon_o the_o superficies_n of_o it_o and_o this_o be_v manifest_a enough_o by_o experience_n namely_o by_o the_o spread_a of_o water_n pour_v out_o upon_o a_o pavement_n but_o the_o reason_n of_o it_o may_v be_v this_o see_v the_o sphere_n a_o in_o the_o 3d_o figure_n be_v move_v towards_o b_o the_o medium_fw-la also_o in_o which_o it_o be_v move_v will_v have_v the_o same_o motion_n but_o because_o in_o this_o motion_n it_o fall_v upon_o a_o body_n not_o liquid_a as_o g_o so_o that_o it_o can_v go_v on_o and_o see_v the_o small_a part_n of_o the_o medium_fw-la can_v not_o go_v forward_o
in_o a_o straight_a line_n perpendicular_a to_o its_o superficies_n in_o that_o point_n in_o which_o it_o be_v press_v let_v abcd_n in_o the_o first_o figure_n be_v a_o hard_a body_n and_o let_v another_o body_n fall_v upon_o it_o in_o the_o straight_a line_n ea_fw-la with_o any_o inclination_n or_o without_o inclination_n press_v it_o in_o the_o point_n a._n i_o say_v the_o body_n so_o press_v &_o not_o penetrate_v it_o will_v give_v to_o the_o part_n a_o a_o endeavour_n to_o yield_v or_o recede_v in_o a_o straight_a line_n perpendicular_a to_o the_o line_n ad._n for_o let_v ab_fw-la be_v perpendicular_a to_o ad_fw-la and_o let_v basilius_n be_v produce_v to_o f._n if_o therefore_o of_o be_v coincident_a with_o ae_n it_o be_v of_o itself_o manifest_v that_o the_o motion_n in_o ea_fw-la will_v make_v a_o to_o endeavour_v in_o the_o line_n ab_fw-la let_v now_o ea_z be_v oblique_a to_o ad_fw-la and_o from_o the_o point_n e_o let_v the_o straight_a line_n aec_fw-la be_v draw_v cut_v ad_fw-la at_o right_a angle_n in_o d_o and_o let_v the_o rectangle_v abcd_v and_o adef_n be_v complete_v i_o have_v show_v in_o the_o 8_o article_n of_o the_o 16_o chapter_n that_o the_o body_n will_v be_v carry_v from_o e_z to_o a_o by_o the_o concourse_n of_o two_o uniform_a motion_n the_o one_o in_o of_o and_o its_o parallel_n the_o other_o in_o ed_z and_o its_o parallel_n but_o the_o motion_n in_o of_o and_o its_o parallel_n whereof_o da_fw-la be_v one_o contribute_v nothing_o to_o the_o body_n in_o a_o to_o make_v it_o endeavour_n or_o press_v towards_o b_o and_o therefore_o the_o whole_a endeavour_n which_o the_o body_n have_v in_o the_o incline_v line_n ea_fw-la to_o pass_v or_o press_v the_o straight_a line_n ad_fw-la it_o have_v it_o all_o from_o the_o perpendicular_a motion_n or_o endeavour_n in_o fa._n wherefore_o the_o body_n e_o after_o it_o be_v in_o a_o will_v have_v only_o that_o perpendicular_a endeavour_n which_o proceed_v from_o the_o motion_n in_o favorina_n that_o be_v in_o ab_fw-la which_o be_v to_o be_v prove_v 7_o if_o a_o hard_a body_n fall_v upon_o or_o press_v another_o body_n penetrate_v the_o same_o its_o endeavour_n after_o its_o first_o penetration_n will_v be_v neither_o in_o the_o incline_v line_n produce_v nor_o in_o the_o perpendicular_a but_o sometime_o betwixt_o both_o sometime_o without_o they_o let_v eag_n in_o the_o same_o â—_o figure_n be_v the_o incline_v line_n produce_v and_o first_o let_v the_o passage_n through_o the_o medium_fw-la in_o which_o ea_z be_v be_v easy_a than_o the_o passage_n through_o the_o medium_fw-la in_o which_o agnostus_n be_v as_o soon_o therefore_o as_o the_o body_n be_v within_o the_o medium_fw-la in_o which_o be_v agnostus_n it_o will_v find_v great_a resistance_n to_o its_o motion_n in_o da_fw-la and_o its_o parallel_n than_o it_o do_v while_o it_o be_v above_o ad_fw-la and_o therefore_o below_o ad_fw-la it_o will_v proceed_v with_o slow_a motion_n in_o the_o parallel_n of_o da_fw-la then_o above_o it_o wherefore_o the_o motion_n which_o be_v compound_v of_o the_o two_o motion_n in_o of_o and_o ed_z will_v be_v slow_a below_n ad_fw-la then_o above_o it_o and_o therefore_o also_o the_o body_n will_v not_o proceed_v from_o a_o in_o ea_fw-la produce_v but_o below_o it_o see_v therefore_o the_o endeavour_n in_o ab_fw-la be_v generate_v by_o the_o endeavour_n in_o favorina_n if_o to_o the_o endeavour_n in_o favorina_n there_o be_v add_v the_o endeavour_n in_o da_fw-la which_o be_v not_o all_o take_v away_o by_o the_o immersion_n of_o the_o point_n a_o into_o the_o low_a medium_fw-la the_o body_n will_v not_o proceed_v from_o a_o in_o the_o perpendicular_a ab_fw-la but_o beyond_o it_o namely_o in_o some_o straight_a line_n between_o ab_fw-la and_o agnostus_n as_o in_o the_o line_n ah_o second_o let_v the_o passage_n through_o the_o medium_fw-la ea_fw-la be_v less_o easy_a than_o that_o through_o ag._n the_o motion_n therefore_o which_o be_v make_v by_o the_o concourse_n of_o the_o motion_n in_o of_o and_o fb_v be_v slow_a above_n ad_fw-la then_o below_o it_o and_o consequent_o the_o endeavour_n will_v not_o proceed_v from_o a_o in_o ea_fw-la produce_v but_o beyond_o it_o as_o in_o ai._n wherefore_o if_o a_o hard_a body_n fall_v which_o be_v to_o be_v prove_v this_o divergency_n of_o the_o straight_a line_n ah_o from_o the_o straight_a line_n agnostus_n be_v that_o which_o the_o writer_n of_o optic_n common_o call_v refraction_n which_o when_o the_o passage_n be_v easy_a in_o the_o first_o then_o in_o the_o second_o medium_fw-la be_v make_v by_o diverge_v from_o the_o line_n of_o inclination_n towards_o the_o perpendicular_a and_o contrary_o when_o the_o passage_n be_v not_o so_o easy_a in_o the_o first_o medium_fw-la by_o depart_v far_o from_o the_o perpendicular_a 8_o by_o the_o 6_o theorem_a it_o be_v manifest_a that_o the_o force_n of_o the_o movent_fw-la may_v be_v so_o place_v as_o that_o the_o body_n move_v by_o it_o may_v proceed_v in_o a_o way_n almost_o direct_o contrary_a to_o that_o of_o the_o movent_fw-la as_o we_o see_v in_o the_o motion_n of_o ship_n for_o let_v ab_fw-la in_o the_o 2d_o figure_n represent_v a_o ship_n who_o length_n from_o the_o prow_n to_o the_o poop_n be_v ab_fw-la and_o let_v the_o wind_n lie_v upon_o it_o in_o the_o straight_a parallel_n line_n cb_n de_fw-fr and_o fg_a and_o let_v de_fw-fr and_o fg_n be_v cut_v in_o e_z and_o g_z by_o a_o straight_a line_n draw_v from_o b_o perpendicular_a to_o ab_fw-la also_o let_v be_v and_o eglantine_n be_v equal_a and_o the_o angle_n abc_n any_o angle_n how_o small_a soever_o then_o between_o bc_n and_o basilius_n let_v the_o straight_a line_n by_o be_v draw_v and_o let_v the_o sail_n be_v conceive_v to_o be_v spread_v in_o the_o same_o line_n by_o and_o the_o wind_n to_o fall_v upon_o it_o in_o the_o point_n l_o m_o and_o b_o from_o which_o point_n perpendicular_a to_o by_o let_v bk_n mq_n and_o lp_n be_v draw_v last_o let_v en_fw-fr and_o go_v be_v draw_v perpendicular_a to_o bg_n and_o cut_v bk_n in_o h_n and_o k_n and_o let_v hn_n and_o quoth_fw-mi be_v make_v equal_a to_o one_o another_o and_o several_o equal_a to_o basilius_n i_o say_v the_o ship_n basilius_n by_o the_o wind_n fall_v upon_o it_o in_o cb_n de_fw-fr fg_n and_o other_o line_n parallel_v to_o they_o will_v be_v carry_v forward_o almost_o opposite_a to_o the_o wind_n that_o be_v to_o say_v in_o a_o way_n almost_o contrary_a to_o the_o way_n of_o the_o movent_fw-la for_o the_o wind_n that_o blow_v in_o the_o line_n cb_n will_v as_o have_v be_v show_v in_o the_o 6_o article_n give_v to_o the_o point_n b_o a_o endeavour_n to_o proceed_v in_o a_o straight_a line_n perpendicular_a to_o the_o straight_a line_n by_o that_o be_v in_o the_o straight_a line_n bk_n and_o to_o the_o point_n m_o and_o l_o a_o endeavour_n to_o proceed_v in_o the_o straight_a line_n mq_n and_o lp_n which_o be_v parallel_v to_o bk_n let_v now_o the_o measure_n of_o the_o time_n be_v bg_n which_o be_v divide_v in_o the_o middle_n in_o e_z &_o let_v the_o point_n b_o be_v carry_v to_o h_n in_o the_o time_n be._n in_o the_o same_o time_n therefore_o by_o the_o wind_n blow_v in_o dm_o &_o fl_fw-mi and_o as_o many_o other_o line_n as_o may_v be_v draw_v parallel_n to_o they_o the_o whole_a ship_n will_v be_v apply_v to_o the_o straight_a line_n hn._n also_o at_o the_o end_n of_o the_o second_o time_n eglantine_n it_o will_v be_v apply_v to_o the_o straight_a line_n quoth_fw-mi wherefore_o the_o ship_n will_v always_o go_v forward_o and_o the_o angle_n it_o make_v with_o the_o wind_n will_v be_v equal_a to_o the_o angle_n abc_n how_o small_a soever_o that_o angle_n be_v and_o the_o way_n it_o make_v will_n in_o every_o time_n be_v equal_a to_o the_o straight_a line_n eh_n i_o say_v thus_o it_o will_v be_v if_o the_o ship_n may_v be_v move_v with_o as_o great_a celerity_n sidewayes_o from_o basilius_n towards_o quoth_fw-mi as_o it_o may_v be_v move_v forward_o in_o the_o line_n basilius_n but_o this_o be_v impossible_a by_o reason_n of_o the_o resistance_n make_v by_o the_o great_a quantity_n of_o water_n which_o press_v the_o side_n much_o exceed_v the_o resistance_n make_v by_o the_o much_o small_a quantity_n which_o press_v the_o prow_n of_o the_o ship_n so_o that_o the_o way_n the_o ship_n make_v sidewayes_o be_v scarce_o sensible_a and_o therefore_o the_o point_n b_o will_v proceed_v almost_o in_o the_o very_a line_n basilius_n make_v with_o the_o wind_n the_o angle_n abc_n how_o acute_a soever_o that_o be_v to_o say_v it_o will_v proceed_v almost_o in_o the_o straight_a line_n bc_n that_o be_v in_o a_o way_n almost_o contrary_a to_o the_o way_n of_o the_o movent_fw-la which_o be_v to_o be_v demonstrate_v but_o the_o sail_n in_o by_o must_v
hearer_n the_o sound_n will_v come_v strong_a than_o it_o will_v do_v through_o the_o open_a air_n and_o the_o cause_n not_o only_o the_o possible_a but_o the_o certain_a and_o manifest_a cause_n be_v this_o that_o the_o air_n which_o be_v move_v by_o the_o first_o breath_n and_o carry_v forward_o in_o the_o trunk_n be_v not_o diffuse_v as_o it_o will_v be_v in_o the_o open_a air_n and_o be_v consequent_o bring_v to_o the_o ear_n almost_o with_o the_o same_o velocity_n with_o which_o it_o be_v first_o breathe_v out_o whereas_o in_o the_o open_a air_n the_o first_o motion_n diffuse_v itself_o every_o way_n into_o circle_n such_o as_o be_v make_v by_o the_o throw_n of_o a_o stone_n into_o a_o stand_a water_n where_o the_o velocity_n grow_v less_o and_o less_o as_o the_o undulation_n proceed_v further_a and_o further_o from_o the_o begin_n of_o its_o motion_n the_o second_o be_v this_o that_o if_o the_o trunk_n be_v short_a and_o the_o end_n which_o be_v apply_v to_o the_o mouth_n be_v wide_a than_o that_o which_o be_v apply_v to_o the_o ear_n thus_o also_o the_o sound_n will_v be_v strong_a than_o if_o it_o be_v make_v in_o the_o open_a air_n and_o the_o cause_n be_v the_o same_o namely_o that_o by_o how_o much_o the_o wide_a end_n of_o the_o trunk_n be_v less_o distant_a from_o the_o begin_n of_o the_o sound_n by_o so_o much_o the_o less_o be_v the_o diffusion_n the_o three_o that_o it_o be_v easy_a for_o one_o that_o be_v within_o a_o chamber_n to_o hear_v what_o be_v speak_v without_o then_o for_o he_o that_o stand_v without_o to_o hear_v what_o be_v speak_v within_o for_o the_o window_n and_o other_o inlet_n of_o the_o move_v air_n be_v as_o the_o wide_a end_n of_o the_o trunk_n and_o for_o this_o reason_n some_o creature_n seem_v to_o hear_v the_o better_a because_o nature_n have_v bestow_v upon_o they_o wide_o and_o capacious_a ear_n the_o four_o be_v this_o that_o though_o he_o which_o stand_v upon_o the_o sea_n shore_n can_v hear_v the_o collision_n of_o the_o two_o near_a wave_n yet_o neverthess_n he_o hear_v the_o roar_n of_o the_o whole_a sea_n and_o the_o cause_n seem_v to_o be_v this_o that_o though_o the_o several_a collision_n move_v the_o organ_n yet_o they_o be_v not_o several_o great_a enough_o to_o cause_v sense_n whereas_o nothing_o hinder_v but_o that_o all_o of_o they_o together_o may_v make_v sound_a 3_o that_o body_n when_o they_o be_v strike_v do_v yield_v some_o a_o more_o grave_a other_o a_o more_o acute_a sound_n the_o cause_n may_v consist_v in_o the_o difference_n of_o the_o time_n in_o which_o the_o part_n strike_v and_o force_v out_o of_o their_o place_n return_v to_o the_o same_o place_n again_o for_o in_o some_o body_n the_o restitution_n of_o the_o move_v part_n be_v quick_a in_o other_o slow_a and_o this_o also_o may_v be_v the_o cause_n why_o the_o part_n of_o the_o organ_n which_o be_v move_v by_o the_o medium_fw-la return_v to_o their_o rest_n again_o sometime_o soon_o sometime_o late_a now_o by_o how_o 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the_o force_n of_o some_o strong_a wind_n rake_v rather_o then_o strike_v such_o hard_a body_n as_o it_o fall_v upon_o on_o the_o contrary_a when_o i_o use_v the_o word_n clear_a i_o do_v not_o understand_v such_o a_o sound_n as_o may_v be_v easy_o and_o distinct_o hear_v for_o so_o whisper_v will_v be_v clear_a but_o such_o as_o be_v make_v by_o somewhat_o that_o be_v break_v and_o such_o as_o be_v clamour_v tinkle_a the_o sound_n of_o a_o trumpet_n etc._n etc._n and_o to_o express_v it_o significant_o in_o one_o word_n noise_z and_o see_v no_o sound_n be_v make_v but_o by_o the_o concourse_n of_o two_o body_n at_o the_o least_o by_o which_o concourse_n it_o be_v necessary_a that_o there_o be_v as_o well_o reaction_n as_o action_n that_o be_v to_o say_v one_o motion_n opposite_a to_o another_o it_o follow_v that_o according_a as_o the_o proportion_n between_o those_o two_o opposite_a motion_n be_v diversify_v so_o the_o sound_n which_o be_v make_v will_v be_v different_a from_o one_o another_o and_o whensoever_o the_o proportion_n between_o they_o be_v so_o great_a as_o that_o the_o motion_n of_o one_o of_o the_o body_n be_v insensible_a if_o compare_v with_o the_o motion_n of_o the_o other_o than_o the_o sound_n will_v not_o be_v of_o the_o same_o kind_n as_o when_o the_o wind_n fall_v very_o oblique_o upon_o a_o hard_a body_n or_o when_o a_o hard_a body_n be_v carry_v swift_o through_o the_o air_n for_o then_o there_o be_v make_v that_o sound_n which_o i_o call_v a_o hoarse_a sound_n in_o greek_a 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d therefore_o the_o breath_n blow_v with_o violence_n from_o the_o mouth_n make_v a_o hiss_v because_o in_o go_v out_o it_o rake_n the_o superficies_n of_o the_o lip_n who_o reaction_n against_o the_o force_n of_o the_o breath_n be_v not_o sensible_a and_o this_o be_v the_o cause_n why_o the_o wind_n have_v that_o hoarse_a sound_n also_o if_o two_o body_n how_o hard_o soever_o be_v rub_v together_o with_o no_o great_a pressure_n they_o make_v a_o hoarse_a sound_n and_o this_o hoarse_a sound_n when_o it_o be_v make_v as_o i_o have_v say_v by_o the_o air_n rake_v the_o superficies_n of_o a_o hard_a body_n seem_v to_o be_v nothing_o but_o the_o divide_n of_o the_o air_n into_o innumerable_a and_o very_a small_a file_n for_o the_o asperity_n of_o the_o superficies_n do_v by_o the_o eminency_n of_o its_o innumerable_a part_n divide_v or_o cut_v in_o piece_n the_o air_n that_o slide_v upon_o it_o 4_o noise_n or_o that_o which_o i_o call_v clear_a sound_n be_v make_v two_o way_n one_o by_o two_o hoarse_a sound_n make_v by_o opposite_a motion_n the_o other_o by_o collision_n or_o by_o the_o sudden_a pull_v asunder_o of_o two_o body_n whereby_o their_o small_a particle_n be_v put_v into_o commotion_n or_o be_v already_o in_o commotion_n sudden_o restore_v themselves_o again_o which_o motion_n make_v impression_n upon_o the_o medium_fw-la be_v propagate_v to_o the_o organ_n of_o hear_v and_o see_v there_o be_v in_o this_o collision_n or_o divulsion_n a_o endeavour_n in_o the_o particle_n of_o one_o body_n opposite_a to_o the_o endeavour_n of_o the_o particle_n of_o the_o other_o body_n there_o will_v also_o be_v make_v in_o the_o organ_n of_o hear_v a_o like_a opposition_n of_o endeavour_n that_o be_v to_o say_v of_o motion_n and_o consequent_o the_o sound_n arise_v from_o thence_o will_v be_v make_v by_o two_o opposite_a motion_n that_o be_v to_o say_v by_o two_o opposite_a hoarse_a sound_n in_o one_o and_o the_o same_o part_n of_o the_o organ_n for_o as_o i_o have_v already_o say_v a_o hoarse_a sound_n suppose_v the_o sensible_a motion_n of_o but_o one_o of_o the_o body_n and_o this_o opposition_n of_o motion_n in_o the_o organ_n be_v the_o cause_n why_o two_o body_n make_v a_o noise_n when_o they_o be_v either_o sudden_o strike_v against_o one_o another_o or_o sudden_o break_v asunder_o 5_o this_o be_v grant_v and_o see_v withal_o that_o thunder_n be_v make_v by_o the_o vehement_a eruption_n of_o the_o air_n out_o of_o the_o cavity_n of_o congeal_v cloud_n the_o cause_n of_o the_o great_a noise_n or_o clap_n may_v be_v the_o sudden_a break_n asunder_o of_o the_o ice_n for_o in_o this_o action_n it_o be_v necessary_a that_o there_o be_v not_o only_o a_o great_a concussion_n of_o the_o small_a particle_n of_o the_o break_a part_n but_o also_o that_o this_o concussion_n by_o be_v communicate_v to_o the_o air_n be_v carry_v to_o the_o organ_n of_o hear_v &_o make_v impression_n upon_o it_o and_o then_o from_o the_o first_o reaction_n of_o the_o organ_n proceed_v that_o first_o and_o great_a sound_n which_o be_v make_v by_o the_o collision_n of_o the_o part_n whilst_o they_o restore_v themselves_o and_o see_v there_o be_v
that_o be_v strike_v be_v not_o only_o sufficient_o hard_a but_o have_v also_o the_o particle_n of_o which_o it_o consist_v like_a to_o one_o another_o both_o in_o hardness_n and_o figure_n such_o as_o be_v the_o particle_n of_o glass_n and_o metal_n which_o be_v first_o melt_v do_v afterward_o settle_v and_o harden_v the_o sound_v it_o yield_v will_v because_o the_o motion_n of_o its_o part_n and_o their_o reciprocation_n be_v like_a and_o uniform_a be_v uniform_a and_o pleasant_a and_o be_v more_o or_o less_o last_a according_a as_o the_o body_n strike_v have_v gteat_a or_o less_o magnitude_n the_o possible_a cause_n therefore_o of_o sound_n uniform_a and_o harsh_a and_o of_o their_o long_a or_o short_a duration_n may_v be_v one_o and_o the_o same_o likeness_n and_o unlikeness_n of_o the_o internal_a part_n of_o the_o sound_a body_n in_o respect_n both_o of_o their_o figure_n and_o hardness_n beside_o if_o two_o plain_a body_n of_o the_o same_o matter_n and_o of_o equal_a thickness_n 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nor_o favourable_a wind_n increase_v the_o light_n unless_o it_o be_v so_o strong_a as_o to_o remove_v the_o object_n further_o off_o or_o bring_v it_o near_o to_o the_o eye_n for_o the_o wind_n that_o be_v to_o say_v the_o air_n move_v do_v not_o by_o its_o interposition_n between_o the_o object_n and_o the_o eye_n work_v otherwise_o then_o it_o will_v do_v if_o it_o be_v still_o and_o calm_a for_o where_o the_o pressure_n be_v perpetual_a one_o part_n of_o the_o air_n be_v no_o soon_o carry_v away_o but_o another_o by_o succeed_v it_o receive_v the_o same_o impression_n which_o the_o part_n carry_v away_o have_v receive_v before_o but_o in_o the_o generation_n of_o sound_n the_o first_o collision_n or_o break_v asunder_o beat_v away_o &_o drive_v out_o of_o its_o place_n the_o near_a part_n of_o the_o air_n and_o that_o to_o a_o considerable_a distance_n and_o with_o considerable_a velocity_n and_o as_o the_o circle_n grow_v by_o their_o remoteness_n wide_o and_o wide_o so_o the_o air_n be_v more_o &_o more_o dissipate_v have_v also_o its_o motion_n more_o &_o more_o weaken_a whensoever_o therefore_o the_o air_n be_v so_o strike_v as_o to_o cause_n sound_v if_o the_o wind_n fall_v upon_o it_o it_o will_v move_v it_o all_o near_o to_o the_o ear_n if_o it_o blow_v that_o way_n and_o further_o from_o it_o if_o it_o blow_v the_o contrary_a way_n so_o that_o according_a as_o it_o blow_v from_o or_o towards_o the_o object_n so_o the_o sound_n which_o be_v hear_v will_v seem_v to_o come_v from_o a_o near_o or_o remote_a place_n and_o the_o action_n by_o reason_n of_o the_o unequal_a distance_n be_v strengthen_v or_o debilitate_a from_o hence_o may_v be_v understand_v the_o reason_n why_o the_o voice_n of_o such_o as_o be_v say_v to_o speak_v in_o their_o belly_n though_o it_o be_v utter_v near_o hand_n be_v nevertheless_o hear_v by_o those_o that_o suspect_v nothing_o as_o if_o it_o be_v a_o great_a way_n off_o for_o have_v no_o former_a thought_n of_o any_o 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