a_o b_o c_o d_o b_o k._n 2_o a_o n_o i_o b_o be_v proportional_o and_o by_o take_v the_o half_n of_o the_o three_o &_o the_o four_o a_o b_o c_o d+b_o e._n 2_o a_o b_o c_o d_o a_o b_o 1._o a_o n_o i_o b_o be_v also_o proportional_n which_o be_v to_o be_v prove_v 10_o from_o what_o have_v be_v say_v of_o deficient_a figure_n describe_v in_o a_o parallelogram_n may_v be_v find_v out_o what_o proportion_n space_n transmit_v with_o accelerate_a motion_n in_o determine_a time_n have_v to_o the_o time_n themselves_o according_a as_o the_o move_a body_n be_v accelerate_a in_o the_o several_a time_n with_o one_o or_o more_o degree_n of_o velocity_n for_o let_v the_o parallelogram_n a_o b_o c_o d_o in_o the_o 6_o figure_n and_o in_o it_o the_o threesided_n figure_n d_o e_o b_o c_o be_v describe_v and_o let_v f_o g_z be_v draw_v any_o where_o parallel_n to_o the_o base_a cut_v the_o diagonal_a b_o d_o in_o h_n and_o the_o crooked_a line_n b_o e_o d_o in_o e_z &_o let_v the_o proportion_n of_o b_o c_o to_o b_o f_o be_v for_o example_n triplicate_v to_o that_o of_o f_o g_o to_o f_o e_z whereupon_o the_o figure_n d_o e_o b_o c_o will_n be_v triple_a to_o its_o compliment_n b_o e_o d_o a_o and_o in_o like_a manner_n i_o f_o be_v draw_v parallel_n to_o b_o c_o the_o threesided_n figure_n e_o k_n b_o f_o will_v be_v triple_a to_o its_o compliment_n b_o k_n e_o i._n wherefore_o the_o part_n of_o the_o deficient_a figure_n cut_v off_o from_o the_o vertex_fw-la by_o straight_a line_n parallel_v to_o the_o base_a namely_o d_o e_o b_o c_o and_o e_o k_n b_o f_o will_v be_v to_o one_o another_o as_o the_o parallellogram_n a_o c_o and_o i_o f_o that_o be_v in_o proportion_n compound_v of_o the_o proportion_n of_o the_o altitude_n and_o base_n see_v therefore_o the_o proportion_n of_o the_o altitude_n b_o c_o to_o the_o altitude_n b_o f_o be_v triplicate_v to_o the_o proportion_n of_o the_o base_a d_o c_o to_o the_o base_a f_o e_o the_o figure_n d_o e_o b_o c_o to_o the_o figure_n e_o k_n b_o f_o will_n be_v quadruplicate_v to_o the_o proportion_n of_o the_o same_o d_o c_o to_o f_o e._n and_o by_o the_o same_o method_n may_v be_v find_v out_o what_o proportion_n any_o of_o the_o say_a threesided_n figure_n have_v to_o any_o part_n of_o the_o same_o cut_v off_o from_o the_o vertex_fw-la by_o a_o straight_a line_n parallel_n to_o the_o base_a now_o as_o the_o say_a figure_n be_v understand_v to_o be_v describe_v by_o the_o continual_a decrease_a of_o the_o base_a as_o of_o c_o d_o for_o example_n till_o it_o end_v in_o a_o point_n as_o in_o b_o so_o also_o they_o may_v be_v understand_v to_o be_v describe_v by_o the_o continual_a increase_n of_o a_o point_n as_o of_o b_o till_o it_o acquire_v any_o magnitude_n as_o that_o of_o c_o d._n suppose_v now_o the_o figure_n b_o e_o d_o c_o to_n be_v describe_v by_o the_o increase_n of_o the_o point_n b_o to_o the_o magnitude_n c_o d._n seeing_n therefore_o the_o propor_n ion_n of_o b_o c_o to_o b_o f_o be_v triplicate_v to_o that_o of_o c_o d_o to_o f_o e_o the_o proportion_n of_o f_o e_o to_o c_o d_o will_n by_o conversion_n as_o i_o shall_v present_o demonstrate_v be_v triplicate_v to_o that_o b_o f_o to_o b_o c._n wherefore_o if_o the_o straight_a line_n b_o c_o be_v take_v for_o the_o measure_n of_o the_o time_n in_o which_o the_o point_n b_o be_v move_v the_o figure_n e_o k_n b_o f_o will_v represent_v the_o sum_n of_o all_o the_o increase_a velocity_n in_o the_o time_n b_o f_o and_o the_o figure_n d_o e_o b_o c_o will_n in_o like_a manner_n represent_v the_o sum_n of_o all_o the_o increase_a velocity_n in_o the_o time_n b_o c._n seeing_n therefore_o the_o proportion_n of_o the_o figure_n e_o k_n b_o f_o to_o the_o figure_n d_o e_o b_o c_o be_v compound_v of_o the_o proportion_n of_o altitude_n to_o altitude_n and_o base_a to_o base_a and_o see_v the_o proportion_n of_o f_o e_o to_o c_z d_o be_v triplicate_v to_o that_o of_o b_o f_o to_o b_o c_z the_o proportion_n of_o the_o figure_n e_o k_n b_o f_o to_o the_o figure_n d_o e_o b_o c_o will_n be_v quadruplicate_v to_o that_o of_o b_o f_o to_o b_o c_z that_o be_v the_o proportion_n of_o the_o sum_n of_o the_o velocity_n in_o the_o time_n b_o f_o to_o the_o sum_n of_o the_o velocity_n in_o the_o time_n b_o c_o will_n be_v quadruplicate_v to_o the_o proportion_n of_o b_o f_o to_o b_o c._n wherefore_o if_o a_o body_n be_v move_v from_o b_o with_o velocity_n so_o increase_a that_o the_o velocity_n acquire_v in_o the_o time_n b_o f_o be_v to_o the_o velocity_n acquire_v in_o the_o time_n b_o c_o in_o triplicate_v proportion_n to_o that_o of_o the_o time_n themselves_o b_o f_o to_n b_o c_o and_o the_o body_n be_v carry_v to_o f_o in_o the_o time_n b_o f_o the_o same_o body_n in_o the_o time_n b_o c_o will_n be_v carry_v through_o a_o line_n equal_a to_o the_o five_o proportional_a from_o b_o f_o in_o the_o continual_a proportion_n of_o b_o f_o to_o b_o c._n and_o by_o the_o same_o manner_n of_o work_v we_o may_v determine_v what_o space_n be_v transmit_v by_o velocity_n increase_n according_o to_o any_o other_o proportion_n it_o remain_v that_o i_o demonstrate_v the_o proportion_n of_o f_o e_o to_o c_o d_o to_o be_v triplicate_v to_o that_o of_o b_o f_o to_o b_o c._n seeing_n therefore_o the_o proportion_n of_o c_o d_o that_o be_v of_o f_o g_o to_o f_o e_z be_v subtriplicate_v to_o that_o of_o b_o c_o to_o b_o f_o the_o proportion_n of_o f_o g_o to_o f_o e_o will_v also_o be_v subtriplicate_v to_o that_o of_o f_o g_o to_o f_o h._n wherefore_o the_o proportion_n of_o f_o g_o to_o f_o h_n be_v triplicate_v to_o that_o of_o f_o g_o that_o be_v of_o c_o d_o to_o f_o e._n but_o in_o four_o continual_a proportional_n of_o which_o the_o least_o be_v the_o first_o the_o proportion_n of_o the_o first_o to_o the_o four_o by_o the_o 16_o art_n of_o the_o 13_o chap._n be_v subtriplicate_v to_o the_o proportion_n of_o the_o three_o to_o the_o same_o fourh_fw-mi wherefore_o the_o proportion_n of_o f_o h_n to_o g_o f_o be_v subtriplicate_v to_o that_o of_o f_o e_o to_o c_z d_o and_o therefore_o the_o proportion_n of_o f_o e_o to_o c_z d_o be_v triplicate_v to_o that_o of_o f_o h_n to_o f_o g_o that_o be_v of_o b_o f_o to_o b_o c_o which_o be_v to_o be_v prove_v it_o may_v from_o hence_o be_v collect_v that_o when_o the_o velocity_n of_o a_o body_n be_v increase_v in_o the_o same_o proportion_n with_o that_o of_o the_o time_n the_o degree_n of_o velocity_n above_o one_o another_o proceed_v as_o number_n do_v in_o immediate_a succession_n from_o unity_n namely_o as_o 1_o 2_o 3_o 4_o etc._n etc._n and_o when_o the_o velocity_n be_v increase_v in_o proportion_n duplicate_v to_o that_o of_o the_o time_n the_o degree_n proceed_v as_o number_n from_o unity_n skip_v one_o as_o 1_o 3_o 5_o 7_o etc._n etc._n last_o when_o the_o proportion_n of_o the_o velocity_n be_v triplicate_v to_o those_o of_o the_o time_n the_o progression_n of_o the_o degree_n be_v as_o that_o of_o number_n from_o unity_n skip_v two_o in_o every_o place_n as_o 1_o 4_o 7_o 10_o etc._n etc._n and_o so_o of_o other_o proportion_n for_o geometrical_a proportional_n when_o they_o be_v take_v in_o every_o point_n be_v the_o same_o with_o arithmetical_a proportional_n 11_o moreover_o it_o be_v to_o be_v note_v that_o as_o in_o quantity_n which_o be_v make_v by_o any_o magnitude_n decrease_v the_o proportion_n of_o the_o figure_n to_o one_o another_o be_v as_o the_o proportion_n of_o the_o altitude_n to_o those_o of_o the_o base_n so_o also_o it_o be_v in_o those_o which_o be_v make_v with_o motion_n decrease_v which_o motion_n be_v nothing_o else_o but_o that_o power_n by_o which_o the_o describe_v figure_n be_v great_a or_o less_o and_o therefore_o in_o the_o description_n of_o archimedes_n his_o spiral_n which_o be_v do_v by_o the_o continual_a diminution_n of_o the_o semidiameter_n of_o a_o circle_n in_o the_o same_o proportion_n in_o which_o the_o circumference_n be_v diminish_v the_o space_n which_o be_v contain_v within_o the_o semidiameter_n and_o the_o spiral_n line_n be_v a_o three_o part_n of_o the_o whole_a circle_n for_o the_o semidiameter_n of_o circle_n in_o as_o much_o as_o circle_n be_v understand_v to_o be_v make_v up_o of_o the_o aggregate_v of_o they_o be_v so_o many_o sector_n and_o therefore_o in_o the_o description_n of_o a_o spiral_n the_o sector_n which_o describe_v it_o be_v diminish_v in_o duplicate_v proportion_n to_o the_o diminution_n of_o the_o circumference_n of_o the_o circle_n in_o
angle_n nbk_n which_o be_v do_v boy_n will_v be_v the_o line_n of_o reflection_n from_o the_o line_n of_o incidence_n nb._n last_o from_o the_o incident_a line_n lc_n let_v the_o reflect_a line_n co_n be_v draw_v cut_v boy_n at_o o_o and_o make_v the_o angle_n cob_n i_o say_v the_o angle_n cob_n be_v equal_a to_o the_o angle_n z._n let_v nb_n be_v produce_v till_o it_o meet_v with_o the_o straight_a line_n lc_n produce_v in_o p._n see_v therefore_o the_o angle_n lcm_n be_v by_o construction_n equal_a to_o twice_o the_o angle_n bac_n together_o with_o the_o angle_n z_o the_o angle_n npl_n which_o be_v equal_a to_o lcm_n by_o reason_n of_o the_o parallel_n np_n and_o mc_n will_v also_o be_v equal_a to_o twice_o the_o same_o angle_n bac_n together_o with_o the_o angle_n z._n and_o see_v the_o two_o straight_a line_n oc_n and_o ob_fw-la fall_n from_o the_o point_n o_o upon_o the_o point_n c_o and_o b_o and_o their_o reflect_a line_n lc_a and_o nb_n meet_v in_o the_o point_n p_o the_o angle_n npl_n will_v be_v equal_a to_o twice_o the_o angle_n bac_n together_o with_o the_o angle_n cop_z but_o i_o have_v already_o prove_v the_o angle_n npl_n to_o be_v equal_a to_o twice_o the_o angle_n bac_n together_o with_o the_o angle_n z._n therefore_o the_o angle_n cop_z be_v equal_a to_o the_o angle_n z_o wherefore_o two_o point_n in_o the_o circumference_n of_o a_o circle_n be_v give_v i_o have_v draw_v etc._n etc._n which_o be_v to_o be_v do_v but_o if_o it_o be_v require_v to_o draw_v the_o incident_a line_n from_o a_o point_n within_o the_o circle_n so_o that_o the_o line_n reflect_v from_o they_o may_v contain_v a_o angle_n equal_a to_o the_o angle_n z_o the_o same_o method_n be_v to_o be_v use_v save_v that_o in_o this_o case_n the_o angle_n z_o be_v not_o to_o be_v add_v to_o twice_o the_o angle_n bac_n but_o to_o be_v take_v from_o it_o 9_o if_o a_o straight_a line_n fall_v upon_o the_o circumference_n of_o a_o circle_n be_v produce_v till_o it_o reach_v the_o semidiameter_n and_o that_o part_n of_o it_o which_o be_v intercept_v between_o the_o circumference_n and_o the_o semidiameter_n be_v equal_a to_o that_o part_n of_o the_o semidiameter_n which_o be_v between_o the_o point_n of_o concourse_n and_o the_o centre_n the_o reflect_a line_n will_v be_v parallel_n to_o the_o semidiameter_n let_v any_o line_n ab_fw-la in_o the_o 9th_o figure_n be_v the_o semidiameter_n of_o the_o circle_n who_o centre_n be_v a_o and_o upon_o the_o circumference_n bd_o let_v the_o straight_a line_n cd_o fall_n and_o be_v produce_v till_o it_o cut_v ab_fw-la in_o e_z so_o that_z ed_z and_o ea_z may_v be_v equal_a &_o from_o the_o incident_a line_n cd_o let_v the_o line_n df_n be_v reflect_v i_o say_v ab_fw-la and_o df_n will_v be_v parallel_n let_v agnostus_n be_v draw_v through_o the_o point_n d._n see_v therefore_o ed_z and_o ea_z be_v equal_a the_o angle_v eda_n and_o ead_n will_v also_o be_v equal_a but_o the_o angle_n fdg_n and_o eda_n be_v equal_a for_o each_o of_o they_o be_v half_a the_o angle_n edh_n or_o fdc_n wherefore_o the_o angle_n fdg_n and_o ead_n be_v equal_a and_o consequent_o df_n and_o ab_fw-la be_v parallel_n which_o be_v to_o be_v prove_v corollahy_fw-fr if_o ea_fw-la be_v great_a than_o ed_z than_o df_n and_o ab_fw-la be_v produce_v will_v concur_v but_o if_o ea_z be_v less_o than_o ed_z then_z ba_z and_o dh_n be_v produce_v will_v concur_v 10_o if_o from_o a_o point_n within_o a_o circle_n two_o straight_a line_n be_v draw_v to_o the_o circumference_n and_o their_o reflect_a line_n meet_v in_o the_o circumference_n of_o the_o same_o circle_n the_o angle_n make_v by_o the_o line_n of_o reflection_n will_v be_v a_o three_o part_n of_o the_o angle_n make_v by_o the_o line_n of_o incidence_n from_o the_o point_n b_o in_o the_o 10_o figure_n take_v within_o the_o circle_n who_o centre_n be_v a_o let_v the_o two_o straight_a line_n bc_n and_o bd_o be_v draw_v to_o the_o circumference_n and_o let_v their_o reflect_a lines_n ce_fw-fr and_o de_fw-fr meet_v in_o the_o circumference_n of_o the_o same_o circle_n at_o the_o point_n e._n i_o say_v the_o angle_n ce_v will_v be_v a_o three_o part_n of_o the_o angle_n cbd_v let_v ac_fw-la and_o ad_fw-la be_v draw_v see_v therefore_o the_o angle_n ce_v and_o cbd_v together_o take_v be_v equal_a to_o twice_o the_o angle_n god_n as_o have_v be_v demonstrate_v in_o the_o 5_o article_n and_o the_o angle_n god_n twice_o take_v be_v quadruple_a to_o the_o angle_n ce_v the_o angle_n ce_v and_o cbd_v together_o take_v will_v also_o be_v equal_a to_o the_o angle_n ce_v four_o time_n take_v and_o therefore_o if_o the_o angle_n ce_v be_v take_v away_o on_o both_o side_n there_o will_v remain_v the_o angle_n cbd_v on_o one_o side_n equal_a to_o the_o angle_n ce_v thrice_o take_v on_o the_o other_o side_n which_o be_v to_o be_v demonstrate_v coral_n therefore_o a_o point_n be_v give_v within_o a_o circle_n there_o may_v be_v draw_v two_o line_n from_o it_o to_o the_o circumference_n so_o as_o their_o reflect_a line_n may_v meet_v in_o the_o circumference_n for_o it_o be_v but_o trisect_v the_o angle_n cbd_v which_o how_o it_o may_v be_v do_v shall_v be_v show_v in_o the_o follow_a chapter_n chap._n xx._n of_o the_o dimension_n of_o a_o circle_n and_o the_o division_n of_o angle_n or_o arch_n 1_o the_o dimension_n of_o a_o circle_n near_o determine_v in_o number_n by_o archimedes_n and_o other_o 2_o the_o first_o attempt_n for_o the_o find_v out_o of_o the_o dimension_n of_o a_o circle_n by_o line_n 3_o the_o second_o attempt_n for_o the_o find_v out_o of_o the_o dimension_n of_o a_o circle_n from_o the_o consideration_n of_o the_o nature_n of_o crookedness_n 4_o the_o three_o attempt_n and_o some_o thing_n propound_v to_o be_v further_o search_v into_o 5_o the_o equation_n of_o the_o spiral_n of_o archimedes_n with_o a_o straight_a line_n 6_o of_o the_o analysis_n of_o geometrician_n by_o the_o power_n of_o line_n 1_o in_o the_o compare_v of_o a_o arch_n of_o a_o circle_n with_o a_o straight_a line_n many_o and_o great_a geometrician_n even_o from_o the_o most_o ancient_a time_n have_v exercise_v their_o wit_n and_o more_o have_v do_v the_o same_o if_o they_o have_v not_o see_v their_o pain_n though_o undertake_v for_o the_o common_a good_a if_o not_o bring_v to_o perfection_n vilify_v by_o those_o that_o envy_v the_o praise_n of_o other_o man_n among_o those_o ancient_a writer_n who_o work_n be_v come_v to_o our_o hand_n archimedes_n be_v the_o first_o that_o bring_v the_o length_n of_o the_o perimeter_n of_o a_o circle_n within_o the_o limit_n of_o number_n very_o little_o differ_v from_o the_o truth_n demonstrate_v the_o same_o to_o be_v less_o than_o three_o diameter_n and_o a_o seven_o part_n but_o great_a than_o three_o diameter_n and_o ten_o seventy_o one_o part_n of_o the_o diameter_n so_o that_o suppose_v the_o radius_fw-la to_o consist_v of_o 10000000_o equal_a part_n the_o arch_n of_o a_o quadrant_n will_v be_v between_o 15714285_o and_o 15_o 04225_o of_o the_o same_o part_n in_o our_o time_n ludovicus_n van_n cullen_n &_o willebrordus_fw-la snellius_n with_o joint_a endeavour_n have_v come_v yet_o near_o to_o the_o truth_n and_o pronounce_v from_o true_a principle_n that_o the_o arch_n of_o a_o quadrant_n put_v as_o before_o 10000000_o for_o radius_fw-la differ_v not_o one_o whole_a unity_n from_o the_o number_n 15707963_o which_o if_o they_o have_v exhibit_v their_o arithmetical_a operation_n and_o no_o man_n have_v discover_v any_o error_n in_o that_o long_a work_n of_o they_o have_v be_v demonstrate_v by_o they_o this_o be_v the_o further_a progress_n that_o have_v be_v make_v by_o the_o way_n of_o number_n and_o they_o that_o have_v proceed_v thus_o far_o deserve_v the_o praise_n of_o industry_n nevertheless_o if_o we_o consider_v the_o benefit_n which_o be_v the_o scope_n at_o which_o all_o speculation_n shall_v aim_v the_o improvement_n they_o have_v make_v have_v be_v little_a or_o none_o for_o any_o ordinary_a man_n may_v much_o soon_o &_o more_o accurate_o find_v a_o straight_a line_n equal_a to_o the_o perimeter_n of_o a_o circle_n and_o consequent_o square_v the_o circle_n by_o wind_v a_o small_a thread_n about_o a_o give_v cylinder_n than_o any_o geometrician_n shall_v do_v the_o same_o by_o divide_v the_o radius_fw-la into_o 10000000_o equal_a part_n but_o though_o the_o length_n of_o the_o circumference_n be_v exact_o set_v out_o either_o by_o number_n or_o mechanical_o or_o only_o by_o chance_n yet_o this_o will_v contribute_v no_o help_n at_o all_o towards_o the_o section_n of_o angles_n unless_o happy_o these_o two_o problem_n to_o divide_v a_o give_v angle_n according_a to_o any_o proportion_n assign_v and_o to_o find_v a_o
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article_n of_o the_o 9_o chapter_n it_o will_v always_o proceed_v in_o the_o same_o way_n and_o with_o the_o same_o swiftness_n and_o if_o its_o endeavour_n be_v in_o space_n which_o be_v fill_v yet_o see_v endeavour_n be_v motion_n that_o which_o stand_v next_o in_o its_o way_n shall_v be_v remove_v and_o endeavour_v further_o and_o again_o remove_v that_o which_o stand_v next_o &_o so_o infinite_o wherefore_o the_o propagation_n of_o endeavour_n from_o one_o part_n of_o full_a space_n to_o another_o proceed_v infinite_o beside_o it_o reach_v in_o any_o instant_n to_o any_o distance_n how_o great_a soever_o for_o in_o the_o same_o instant_n in_o which_o the_o first_o part_n of_o the_o full_a medium_fw-la remove_v that_o which_o be_v next_o it_o the_o second_o also_o remove_v that_o part_n which_o be_v next_o to_o it_o and_o therefore_o all_o endeavour_n whether_o it_o be_v in_o empty_a or_o in_o full_a space_n proceed_v not_o only_o to_o any_o distance_n how_o great_a soever_o but_o also_o in_o any_o time_n how_o little_a soever_o that_o be_v in_o a_o instant_n nor_o make_v it_o any_o matter_n that_o endeavour_v by_o proceed_n grow_v weak_a and_o weak_a till_o at_o last_o it_o can_v no_o long_o be_v perceive_v by_o sense_n for_o motion_n may_v be_v insensible_a and_o i_o do_v not_o here_o examine_v thing_n by_o sense_n and_o experience_n but_o by_o reason_n 8_o when_o two_o movent_n be_v of_o equal_a magnitude_n the_o swift_a of_o they_o work_v with_o great_a force_n than_o the_o slow_a upon_o a_o body_n that_o resist_v their_o motion_n also_o if_o two_o movent_n have_v equal_a velocity_n the_o great_a of_o they_o work_v with_o more_o force_n than_o the_o less_o for_o where_o the_o magnitude_n be_v equal_a the_o movent_fw-la of_o great_a velocity_n make_v the_o great_a impression_n upon_o that_o body_n upon_o which_o it_o fall_v and_o where_o the_o velocity_n be_v equal_a the_o movent_fw-la of_o great_a magnitude_n fall_v upon_o the_o same_o point_n or_o a_o equal_a part_n of_o another_o body_n lose_v less_o of_o its_o velocity_n because_o the_o resist_a body_n work_v only_o upon_o that_o part_n of_o the_o movent_fw-la which_o it_o touch_v and_o therefore_o abate_v the_o impetus_fw-la of_o that_o part_n only_o whereas_o in_o the_o mean_a time_n the_o part_n which_o be_v not_o touch_v proceed_v and_o retain_v their_o whole_a force_n till_o they_o also_o come_v to_o be_v touch_v and_o their_o force_n have_v some_o effect_n wherefore_o for_o example_n in_o battery_n a_o long_a than_o a_o short_a piece_n of_o timber_n of_o the_o same_o thickness_n and_o velocity_n and_o a_o thick_a than_o a_o slendere_a piece_n of_o the_o same_o length_n and_o velocity_n work_v a_o great_a effect_n upon_o the_o wall_n chap._n xvi_o of_o motion_n accelerate_a and_o vniform_a and_o of_o motion_n by_o concourse_n 1_o the_o velocity_n of_o any_o body_n in_o what_o time_n
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
which_o it_o be_v inscribe_v so_o that_o the_o compliment_n of_o the_o spiral_n that_o be_v that_o space_n in_o the_o circle_n which_o be_v without_o the_o spiral_n line_n be_v double_a to_o the_o space_n within_o the_o spiral_n line_n in_o the_o same_o manner_n if_o there_o be_v take_v a_o mean_a proportional_a every_o where_o between_o the_o semidiameter_n of_o the_o circle_n which_o contain_v the_o spiral_n and_o that_o part_n of_o the_o semidiameter_n which_o be_v within_o the_o same_o there_o will_v be_v make_v another_o figure_n which_o will_v be_v half_a the_o circle_n and_o to_o conclude_v this_o rule_n serve_v for_o all_o such_o space_n as_o may_v be_v describe_v by_o a_o line_n or_o superficies_n decrease_v either_o in_o magnitude_n or_o power_n so_o that_o if_o the_o proportion_n in_o which_o they_o decrease_v be_v commensurable_a to_o the_o proportion_n of_o the_o time_n in_o which_o they_o decrease_v the_o magnitude_n of_o the_o figure_n they_o describe_v will_v be_v know_v 12_o the_o truth_n of_o that_o proposition_n which_o i_o demonstrate_v in_o the_o second_o article_n which_o be_v the_o foundation_n of_o all_o that_o have_v be_v say_v concern_v deficient_a figure_n may_v be_v derive_v from_o the_o element_n of_o philosophy_n as_o have_v iâ—â—_n original_n in_o this_o that_o all_o equality_n and_o inequality_n between_o two_o effect_n that_o be_v all_o proportion_n proceed_v from_o and_o be_v determine_v by_o the_o equal_a and_o unequal_a cause_n of_o those_o effect_n or_o from_o the_o proportion_n which_o the_o cause_n concur_v to_o one_o effect_n have_v to_o the_o cause_n which_o concur_v to_o the_o produce_v of_o the_o other_o effect_n and_o that_o therefore_o the_o proportion_n of_o quantity_n be_v the_o same_o with_o the_o proportion_n of_o their_o cause_n see_v therefore_o two_o deficient_a figure_n of_o which_o one_o be_v the_o compliment_n of_o the_o other_o be_v make_v one_o by_o motion_n decrease_v in_o a_o certain_a time_n and_o proportion_n the_o other_o by_o the_o loss_n of_o motion_n in_o the_o same_o time_n the_o cause_n which_o make_v and_o determine_v the_o quantity_n of_o both_o the_o figure_n so_o that_o they_o can_v be_v no_o other_o than_o they_o be_v differ_v only_o in_o this_o that_o the_o proportion_n by_o which_o the_o quantity_n which_o generate_v the_o figure_n proceed_v in_o describe_v of_o the_o same_o that_o be_v the_o proportion_n of_o the_o remainder_n of_o all_o the_o time_n and_o altitude_n may_v be_v other_o proportion_n then_o those_o by_o which_o the_o same_o generate_a quantity_n decrease_v in_o make_v the_o compliment_n of_o that_o figure_n that_o be_v the_o proportion_n of_o the_o quantity_n which_o generate_v the_o figure_n continual_o diminish_v wherefore_o as_o the_o proportion_n of_o the_o quantity_n in_o which_o motion_n be_v lose_v be_v to_o that_o of_o the_o decrease_a quantity_n by_o which_o the_o deficient_a figure_n be_v generate_v so_o will_v the_o defect_n or_o compliment_n be_v to_o the_o figure_n itself_o which_o be_v generate_v 13_o there_o be_v also_o other_o quantity_n which_o be_v determinable_a from_o the_o knowledge_n of_o their_o cause_n namely_o from_o the_o comparison_n of_o the_o motion_n by_o which_o they_o be_v make_v and_o that_o more_o easy_o then_o from_o the_o common_a element_n of_o geometry_n for_o example_n that_o the_o superficies_n of_o any_o portion_n of_o a_o sphere_n be_v equal_a to_o that_o circle_n who_o radius_fw-la be_v a_o straight_a line_n draw_v from_o the_o pole_n of_o the_o portion_n to_o the_o circumference_n of_o its_o base_a i_o may_v demonstrate_v in_o this_o manner_n let_v b_o a_o c_o in_o the_o 7_o figure_n be_v a_o portion_n of_o a_o sphere_n who_o axis_n be_v a_o e_o &_o who_o base_a be_v b_o c_o &_o let_v a_o b_o be_v the_o straight_a line_n draw_v from_o the_o pole_n a_o to_o the_o base_a in_o b_o and_o let_v a_o d_o equal_a to_o a_o b_o touch_v the_o great_a circle_n b_o a_o c_o in_o the_o pole_n a._n it_o be_v to_o be_v prove_v that_o the_o circle_n make_v by_o the_o radius_fw-la a_o d_o be_v equal_a to_o the_o superficies_n of_o the_o portion_n b_o a_o c._n let_v the_o plain_n a_o e_o b_o d_o be_v understand_v to_o make_v a_o revolution_n about_o the_o axis_n a_o e_o &_o it_o be_v manifest_a that_o by_o the_o straight_a line_n a_o d_o a_o circle_n will_v be_v describe_v and_o by_o the_o arch_n a_o b_o the_o superficies_n of_o a_o portion_n of_o a_o sphere_n and_o last_o by_o the_o subtense_n a_o b_o the_o superficies_n of_o a_o right_a cone_n now_o see_v both_o the_o straight_a line_n a_o b_o and_o the_o arch_n a_o b_o make_v one_o and_o the_o same_o revolution_n and_o both_o of_o they_o have_v the_o same_o extreme_a point_n a_o and_o b_o the_o cause_n why_o the_o the_o spherical_a superficies_n which_o be_v make_v by_o the_o arch_n be_v great_a than_o the_o conical_a superficies_n which_o be_v make_v by_o the_o subtense_n be_v that_o a_o b_o the_o arch_n be_v great_a than_o a_o b_o the_o subtense_n and_o the_o cause_n why_o it_o be_v great_a consist_v in_o this_o that_o although_o they_o be_v both_o draw_v from_o a_o to_o b_o yet_o the_o subtense_n be_v draw_v straight_o but_o the_o arch_n angular_o namely_o according_a to_o that_o angle_n which_o the_o arch_n make_v with_o the_o subtense_n which_o angle_n be_v equal_a to_o the_o angle_n d_o a_o b_o for_o a_o angle_n of_o contingence_n add_v nothing_o to_o a_o angle_n of_o a_o segment_n as_o have_v be_v show_v in_o the_o 14_o chapter_n at_o the_o 16_o article_n wherefore_o the_o magnitude_n of_o the_o angle_n d_o a_o b_o be_v the_o cause_n why_o the_o superficies_n of_o the_o portion_n describe_v by_o the_o arch_n a_o b_o be_v great_a than_o the_o superficies_n of_o the_o right_a cone_n describe_v by_o the_o subtense_n a_o b._n again_o the_o cause_n why_o the_o circle_n describe_v by_o the_o tangent_fw-la a_o d_o be_v great_a than_o the_o superficies_n of_o the_o right_a cone_n describe_v by_o the_o subtense_n a_o b_o notwitstand_v that_o the_o tangent_fw-la and_o the_o subtense_n be_v equal_a and_o both_o move_v round_o in_o the_o same_o time_n be_v this_o that_o a_o d_o stand_v at_o right_a angle_n to_o the_o axis_n but_o a_o b_o oblique_o which_o obliquity_n consist_v in_o the_o same_o angle_n d_o a_o b._n see_v therefore_o the_o quantity_n of_o the_o angle_n d_o a_o b_o be_v that_o which_o make_v the_o excess_n both_o of_o the_o superficies_n of_o the_o portion_n and_o of_o the_o circle_n make_v by_o the_o radius_fw-la a_o d_o above_o the_o superficies_n of_o the_o right_a cone_n describe_v by_o the_o subtense_n a_o b_o it_o follow_v that_o both_o the_o superficies_n of_o the_o portion_n and_o that_o of_o the_o circle_n do_v equal_o exceed_v the_o superficies_n of_o the_o cone_n wherefore_o the_o circle_n make_v by_o a_o d_o or_o a_o b_o and_o the_o spherical_a superficies_n make_v by_o the_o arch_n a_o b_o be_v equal_a to_o one_o another_o which_o be_v to_o be_v prove_v â—4_n if_o these_o deficient_a figure_n which_o i_o have_v describe_v in_o a_o 〈◊〉_d be_v capable_a of_o exact_a description_n than_o any_o number_n of_o mean_a proportional_n may_v be_v find_v out_o between_o two_o straight_a line_n give_v for_o example_n in_o the_o parallelogram_n a_o b_o c_o d_o in_o the_o 8_o figure_n let_v the_o threesided_n figure_n of_o two_o mean_n be_v describe_v which_o many_o call_v a_o cubical_a parabola_fw-la and_o let_v r_o and_o s_o be_v two_o give_v straight_o line_n between_o which_o if_o it_o be_v require_v to_o find_v two_o mean_a proportional_n it_o may_v be_v do_v thus_o let_v it_o be_v as_o r_o to_o saint_n so_o b_o c_z to_z b_o f_o and_o let_v f_o e_z be_v draw_v parallel_n to_o b_o a_o and_o cut_v the_o crooked_a line_n in_o e_z then_z through_z e_z let_v g_o h_n be_v draw_v parallel_n and_o equal_a to_o the_o straight_a line_n a_o d_o and_o cut_v the_o diagonal_a b_o d_o in_o i_o for_o thus_o we_o have_v g_o i_o the_o great_a of_o two_o mean_n between_o g_o h_n and_o g_o e_o as_o appear_v by_o the_o description_n of_o the_o figure_n in_o the_o four_o article_n wherefore_o if_o it_o be_v as_o g_z h_z to_z g_z ay_o so_o r_o to_o another_o line_n t_o that_o t_n will_v be_v the_o great_a of_o two_o mean_n between_o r_o and_o s._n and_o therefore_o if_o it_o be_v again_o as_o r_o to_o t_n so_o t_n to_o another_o line_n x_o that_o will_v be_v do_v which_o be_v require_v in_o the_o same_o manner_n four_o mean_a proportional_n may_v be_v find_v out_o by_o the_o description_n of_o a_o threesided_n figure_n of_o four_o mean_n and_o so_o any_o other_o number_n of_o mean_n etc._n etc._n chap._n xviii_o of_o the_o equation_n of_o straight_a line_n with_o the_o
draw_v from_o the_o proposition_n which_o prove_v the_o same_o but_o the_o cause_n of_o his_o construction_n be_v in_o the_o thing_n themselves_o and_o consist_v in_o motion_n or_o in_o the_o concourse_n of_o motion_n wherefore_o those_o proposition_n in_o which_o analysis_n end_v be_v definition_n but_o such_o as_o signify_v in_o what_o manner_n the_o construction_n or_o generation_n of_o the_o thing_n proceed_v for_o otherwise_o when_o he_o go_v back_o by_o synthesis_n to_o the_o proof_n of_o his_o problem_n he_o will_v come_v to_o no_o demonstration_n at_o all_o there_o be_v no_o true_a demonstration_n but_o such_o as_o be_v scientifical_a and_o no_o demonstration_n be_v scientifical_a but_o that_o which_o proceed_v from_o the_o knowledge_n of_o the_o cause_n from_o which_o the_o construction_n of_o the_o problem_n be_v draw_v to_o collect_v therefore_o what_o have_v be_v say_v into_o few_o word_n analysis_n be_v ratiocination_n from_o the_o suppose_a construction_n or_o generation_n of_o a_o thing_n to_o the_o efficient_a cause_n or_o coefficient_a cause_n of_o that_o which_o be_v construct_v or_o generate_v and_o synthesis_n be_v ratiocination_n from_o the_o first_o cause_n of_o the_o construction_n continue_v through_o all_o the_o middle_a cause_n till_o we_o come_v to_o the_o thing_n itself_o which_o be_v construct_v or_o generate_v but_o because_o there_o be_v many_o mean_n by_o which_o the_o same_o thing_n may_v be_v generate_v or_o the_o same_o problem_n be_v construct_v therefore_o neither_o do_v all_o geometrician_n nor_o do_v the_o same_o geometrician_n always_o use_v one_o and_o the_o same_o method_n for_o if_o to_o a_o certain_a quantity_n give_v it_o be_v require_v to_o construct_v another_o quantity_n equal_a there_o may_v be_v some_o that_o will_v inquire_v whether_o this_o may_v not_o be_v do_v by_o mean_n of_o some_o motion_n for_o there_o be_v quantity_n who_o equality_n and_o inequality_n may_v be_v argue_v from_o motion_n and_o time_n as_o well_o as_o from_o congruence_n and_o there_o be_v motion_n by_o which_o two_o quantity_n whether_o line_n or_o superficies_n though_o one_o of_o they_o be_v crooked_a the_o other_o straight_o may_v be_v make_v congruous_a or_o coincident_a and_o this_o method_n archimedes_n make_v use_v of_o in_o his_o book_n de_fw-fr spiralibus_fw-la also_o the_o equality_n or_o inequality_n of_o two_o quantity_n may_v be_v find_v out_o and_o demonstrate_v from_o the_o consideration_n of_o weight_n as_o the_o same_o archimedes_n do_v in_o his_o quadrature_n of_o the_o parabola_fw-la beside_o equality_n and_o equality_n be_v find_v out_o often_o by_o the_o division_n of_o the_o two_o quantity_n into_o part_n which_o be_v consider_v as_o undivisible_a as_o cavallerius_n bonaventura_n have_v do_v in_o our_o time_n and_o archimedes_n often_o last_o the_o same_o be_v perform_v by_o the_o consideration_n of_o the_o power_n of_o line_n or_o the_o root_n of_o those_o power_n and_o by_o the_o multiplication_n division_n addition_n and_o substraction_n as_o also_o by_o the_o extraction_n of_o the_o root_n of_o those_o power_n or_o by_o find_v where_o straight_o line_n of_o the_o same_o proportion_n terminate_v for_o example_n when_o any_o number_n of_o straight_a line_n how_o many_o soever_o be_v draw_v from_o a_o straight_a line_n and_o pass_v all_o through_o the_o same_o point_n look_v what_o proportion_n they_o have_v and_o if_o their_o part_n continue_v from_o the_o point_n retain_v every_o where_o the_o same_o proportion_n they_o shall_v all_o terminate_v in_o a_o straight_a line_n and_o the_o same_o happen_v if_o the_o point_n be_v take_v between_o two_o circle_n so_o that_o the_o place_n of_o all_o their_o point_n of_o termination_n make_v either_o straight_a line_n or_o circumference_n of_o circle_n and_o be_v call_v plain_a place_n so_o also_o when_o straight_o parallel_a line_n be_v apply_v to_o one_o straight_a line_n if_o the_o part_n of_o the_o straight_a line_n to_o which_o they_o be_v apply_v be_v to_o one_o another_o in_o proportion_n duplicate_v to_o that_o of_o the_o contiguous_a apply_v line_n they_o will_v all_o terminate_v in_o a_o conical_a section_n which_o section_n be_v the_o place_n of_o their_o termination_n be_v call_v a_o solid_a place_n because_o it_o serve_v for_o the_o find_v out_o of_o the_o quantity_n of_o any_o equation_n which_o consist_v of_o three_o dimension_n there_o be_v therefore_o three_o way_n of_o find_v out_o the_o cause_n of_o equality_n or_o inequality_n between_o two_o give_v quantity_n namely_o first_o by_o the_o computation_n of_o motion_n for_o by_o equal_a motion_n &_o equal_a time_n equal_a space_n be_v describe_v and_o ponderation_n be_v motion_n second_o by_o indivisibles_n because_o all_o the_o part_n together_o take_v be_v equal_a to_o the_o whole_a and_o three_o by_o the_o power_n for_o when_o they_o be_v equal_a their_o root_n also_o be_v equal_a and_o contrary_o the_o power_n be_v equal_a when_o their_o root_n be_v equal_a but_o if_o the_o question_n be_v much_o complicate_v there_o can_v by_o any_o of_o these_o way_n be_v constitute_v a_o certain_a rule_n from_o the_o supposition_n of_o which_o of_o the_o unknown_a quantity_n the_o analysis_n may_v best_o begin_v nor_o out_o of_o the_o variety_n of_o equation_n that_o at_o first_o appear_v which_o we_o be_v best_a to_o choose_v but_o the_o success_n will_v depend_v upon_o dexterity_n upon_o former_o acquire_v science_n and_o many_o time_n upon_o fortune_n for_o no_o man_n can_v ever_o be_v a_o good_a analyst_n without_o be_v first_o a_o good_a geometrician_n nor_o do_v the_o rule_n of_o analysis_n make_v a_o geometrician_n as_o synthesis_n do_v which_o begin_v at_o the_o very_a element_n and_o proceed_v by_o a_o logical_a use_n of_o the_o same_o for_o the_o true_a teach_n of_o geometry_n be_v by_o synthesis_n according_a to_o euclides_n method_n and_o he_o that_o have_v euclid_n for_o his_o master_n may_v be_v a_o geometrician_n without_o vieta_n though_o vieta_n be_v a_o most_o admirable_a geometrician_n but_o he_o that_o have_v vieta_n for_o his_o master_n not_o so_o without_o euclid_n and_o as_o for_o that_o part_n of_o analysis_n which_o work_v by_o the_o power_n though_o it_o be_v esteem_v by_o some_o geometrician_n not_o the_o chief_a to_o be_v the_o best_a way_n of_o solve_v all_o problem_n yet_o it_o be_v a_o thing_n of_o no_o great_a extent_n it_o be_v all_o contain_v in_o the_o doctrine_n of_o rectangle_v and_o rectangle_v solid_n so_o that_o although_o they_o come_v to_o a_o equation_n which_o determine_v the_o quantity_n seek_v yet_o they_o can_v sometime_o by_o art_n exhibit_v that_o quantity_n in_o a_o plain_a but_o in_o some_o conique_a section_n that_o be_v as_o geometrician_n say_v not_o geometrical_o but_o mechanical_o now_o such_o problem_n as_o these_o they_o call_v solid_a and_o when_o they_o can_v exhibit_v the_o quantity_n seek_v for_o with_o the_o help_n of_o a_o conique_n section_n they_o call_v it_o a_o lineary_a problem_n and_o therefore_o in_o the_o quantity_n of_o angle_n and_o of_o the_o arch_n of_o circle_n there_o be_v no_o use_n at_o all_o of_o the_o analytic_n which_o proceed_v by_o the_o power_n so_o that_o the_o ancient_n pronounce_v it_o impossible_a to_o exhibit_v in_o a_o plain_a the_o division_n of_o angles_n except_o bisection_n and_o the_o bisection_n of_o the_o bisected_a part_n otherwise_o then_o mechanical_o for_o pappus_n before_o the_o 31_o proposition_n of_o his_o four_o book_n distinguish_v and_o define_v the_o several_a kind_n of_o problem_n say_v that_o some_o be_v plain_a other_o solid_a and_o other_o lineary_a those_o therefore_o which_o may_v be_v solve_v by_o straight_a line_n and_o the_o circumference_n of_o circle_n that_o be_v which_o may_v be_v describe_v with_o the_o rule_n and_o compass_n without_o any_o other_o instrument_n be_v fit_o call_v plain_a for_o the_o line_n by_o which_o such_o problem_n be_v find_v out_o have_v their_o generation_n in_o a_o plain_n but_o those_o which_o be_v solve_v by_o the_o use_n of_o some_o one_o or_o more_o conique_a section_n in_o their_o construction_n be_v call_v solid_a because_o their_o construction_n can_v be_v make_v without_o use_v the_o superficies_n of_o solid_a figure_n namely_o of_o cones_n there_o remain_v the_o three_o kind_n which_o be_v call_v lineary_a because_o other_o line_n beside_o those_o already_o mention_v be_v make_v use_n of_o in_o their_o construction_n etc._n etc._n and_o a_o little_a after_o he_o say_v of_o this_o kind_n be_v the_o spiral_n line_n the_o quadratrice_n the_o conchoeides_n and_o the_o cissoeides_n and_o geometrician_n think_v it_o no_o small_a fault_n when_o for_o the_o find_v out_o of_o a_o plain_a problem_n any_o man_n make_v use_v of_o conique_n or_o new_a line_n now_o he_o rank_n the_o trisection_n of_o a_o angle_n among_o solid_a problem_n and_o the_o quinquesection_n among_o lineary_a but_o what_o be_v the_o ancient_a geometrician_n
to_o itself_o so_o that_o in_o what_o part_n soever_o of_o the_o ecliptic_a the_o centre_n of_o the_o epicycle_n be_v find_v and_o in_o what_o part_n soever_o of_o the_o epicycle_n the_o centre_n of_o the_o earth_n be_v find_v at_o the_o same_o time_n the_o axis_n of_o the_o earth_n will_v be_v parallel_n to_o the_o place_n where_o the_o same_o axis_n will_v have_v be_v if_o the_o centre_n of_o the_o earth_n have_v never_o go_v out_o of_o the_o ecliptic_a now_o as_o i_o have_v demonstrate_v the_o simple_a annual_a motion_n of_o the_o earth_n from_o the_o supposition_n of_o simple_a motion_n in_o the_o sun_n so_o from_o the_o supposition_n of_o simple_a motion_n in_o the_o earth_n may_v be_v demonstrate_v the_o monthly_a simple_a motion_n of_o the_o moon_n for_o if_o the_o name_n be_v but_o change_v the_o demonstration_n will_v be_v the_o same_o and_o therefore_o need_v not_o be_v repeat_v 7_o that_o which_o make_v this_o supposition_n of_o the_o sun_n simple_a motion_n in_o the_o epicycle_n fghi_fw-mi probable_a be_v first_o that_o the_o period_n of_o all_o the_o planet_n be_v not_o only_o describe_v about_o the_o sun_n but_o so_o describe_v as_o that_o they_o be_v all_o contain_v within_o the_o zodiac_n that_o be_v to_o say_v within_o the_o latitude_n of_o about_o 16_o degree_n for_o the_o cause_n of_o this_o seem_v to_o depend_v upon_o some_o power_n in_o the_o sun_n especial_o in_o that_o part_n of_o the_o sun_n which_o respect_v the_o zodiac_n second_o that_o in_o the_o whole_a coâ—passe_n of_o the_o heaven_n there_o appear_v no_o other_o body_n from_o which_o the_o cause_n of_o this_o phaenomenon_n can_v in_o probability_n be_v derive_v beside_o i_o can_v not_o imagine_v that_o so_o many_o and_o such_o various_a motion_n of_o the_o planet_n shall_v have_v no_o dependence_n at_o all_o upon_o one_o another_o but_o by_o suppose_v motive_n power_n in_o the_o sun_n we_o suppose_v motion_n also_o for_o power_n to_o move_v without_o motion_n be_v no_o power_n at_o all_o i_o have_v therefore_o suppose_v that_o there_o be_v in_o the_o sun_n for_o the_o govern_n of_o the_o primary_n planet_n and_o in_o the_o earth_n for_o the_o govern_n of_o the_o moon_n such_o motion_n as_o be_v receive_v by_o the_o primary_n planet_n and_o by_o the_o moon_n make_v they_o necessary_o appear_v to_o we_o in_o such_o manner_n as_o we_o see_v they_o whereas_o that_o circular_a motion_n which_o be_v common_o attribute_v to_o they_o about_o a_o fix_a axis_n which_o be_v call_v conversion_n be_v a_o motion_n of_o their_o part_n only_o and_o not_o of_o their_o whole_a body_n be_v insufficient_a to_o salve_v their_o appearance_n for_o see_v whatsoever_o be_v so_o move_v have_v no_o endeavour_n at_o all_o towards_o those_o part_n which_o be_v without_o the_o circle_n they_o have_v no_o power_n to_o propagate_v any_o endeavour_n to_o such_o body_n as_o be_v place_v without_o it_o and_o as_o for_o they_o that_o suppose_v this_o may_v be_v do_v by_o magnetical_a virtue_n or_o by_o incorporeal_a and_o immaterial_a species_n they_o suppose_v no_o natural_a cause_n nay_o no_o cause_n at_o all_o for_o there_o be_v no_o such_o thing_n as_o a_o incorporeal_a movent_fw-la and_o magnetical_a virtue_n be_v a_o thing_n altogether_o unknown_a and_o whensoever_o it_o shall_v be_v know_v it_o will_v be_v find_v to_o be_v a_o motion_n of_o body_n it_o remain_v therefore_o that_o if_o the_o primary_n planet_n be_v carry_v about_o by_o the_o sun_n and_o the_o moon_n by_o the_o earth_n they_o have_v the_o simple_a circular_a motion_n of_o the_o sun_n and_o the_o earth_n for_o the_o cause_n of_o their_o circulation_n otherwise_o if_o they_o be_v not_o carry_v about_o by_o the_o sun_n and_o the_o earth_n but_o that_o every_o planet_n have_v be_v move_v as_o it_o be_v now_o move_v ever_o since_o it_o be_v make_v there_o will_v be_v of_o their_o motion_n no_o cause_n natural_a for_o either_o these_o motion_n be_v concreated_a with_o their_o body_n and_o their_o cause_n be_v supernatural_a or_o they_o be_v coeternal_a with_o they_o and_o so_o they_o have_v no_o cause_n at_o all_o for_o whatsoever_o be_v eternal_a be_v never_o generate_v i_o may_v add_v beside_o to_o confirm_v the_o probability_n of_o this_o simple_a motion_n that_o almost_o all_o learned_a man_n be_v now_o of_o the_o same_o opinion_n with_o copernicus_n concern_v the_o parallelisme_n of_o the_o axis_n of_o the_o earth_n it_o seem_v to_o i_o to_o be_v more_o agreeable_a to_o truth_n or_o at_o least_o more_o handsome_a that_o it_o shall_v be_v cause_v by_o simple_a circular_a motion_n alone_o than_o by_o two_o motion_n one_o in_o the_o ecliptic_a and_o the_o other_o about_o the_o earth_n own_o axis_n the_o contrary_a way_n neither_o of_o they_o simple_a nor_o either_o of_o they_o such_o as_o may_v be_v produce_v by_o any_o motion_n of_o the_o sun_n i_o think_v best_a therefore_o to_o retain_v this_o hypothesis_n of_o simple_a motion_n and_o from_o it_o to_o derive_v the_o cause_n of_o as_o many_o of_o the_o phaenomena_n as_o i_o can_v and_o to_o let_v such_o alone_a as_o i_o can_v not_o deduce_v frm_n thence_o it_o will_v perhaps_o be_v object_v that_o although_o by_o this_o supposition_n the_o reason_n may_v be_v give_v of_o the_o parallelisme_n of_o the_o axis_n of_o the_o earth_n and_o of_o many_o other_o appearance_n nevertheless_o see_v it_o be_v do_v by_o place_v the_o body_n of_o the_o sun_n in_o the_o centre_n of_o that_o orb_n which_o the_o earth_n describe_v with_o its_o annual_a motion_n the_o supposition_n itself_o be_v false_a because_o this_o annual_a orb_n be_v excentrique_fw-la to_o the_o sun_n in_o the_o first_o place_n therefore_o let_v we_o examine_v what_o that_o excentricity_n be_v and_o whence_o it_o proceed_v 8_o let_v the_o annual_a circle_n of_o the_o earth_n abcd_v in_o the_o same_o 3d_o figure_n be_v divide_v into_o four_o equal_a part_n by_o the_o straight_a line_n ac_fw-la &_o bd_v cut_v one_o another_o in_o the_o centre_n e_o and_o let_v a_o be_v the_o beginning_n of_o libra_n b_o of_o capricorn_n c_o of_o aries_n and_o d_o of_o cancer_n and_o let_v the_o whole_a orb_n abcd_v be_v understand_v according_a to_o copernicus_n to_o have_v every_o way_n so_o great_a distance_n from_o the_o zodiac_n of_o the_o fix_a star_n that_o it_o be_v in_o comparison_n with_o it_o but_o as_o a_o point_n let_v the_o earth_n be_v now_o suppose_v to_o be_v in_o the_o begin_n of_o libra_n at_o a._n the_o sun_n therefore_o will_v appear_v in_o the_o begin_n of_o aries_n at_o c._n wherefore_o if_o the_o earth_n be_v move_v from_o a_o to_o b_o the_o apparent_a motion_n of_o the_o sun_n will_v be_v from_o c_o to_o the_o begin_n of_o cancer_n in_o d_o and_o the_o earth_n be_v move_v forward_o from_o b_o to_z c_o the_o sun_n also_o will_v appear_v to_o be_v move_v forward_o to_o the_o begin_n of_o libra_n in_o a_o wherefore_o cda_n will_v be_v the_o summer_n arch_n and_o the_o winter_n arch_n will_v be_v abc_n now_o in_o the_o time_n of_o the_o sun_n apparent_a motion_n in_o the_o summer_n arch_n there_o be_v number_v 186¾_n day_n and_o consequent_o the_o earth_n make_v in_o the_o same_o time_n the_o same_o number_n of_o diurnal_a conversion_n in_o the_o arch_a abc_n and_o therefore_o the_o earth_n in_o its_o motion_n through_o the_o arch_n cda_fw-mi will_v make_v only_a 178½_n diurnal_a conversion_n wherefore_o the_o arch_a a_o b_o c_o aught_o to_o be_v great_a than_o the_o arch_n c_o d_o a_o by_o 8¼_n day_n that_o be_v to_o say_v by_o almost_o so_o many_o degree_n let_v the_o arch_a a_o r_o as_o also_o c_o s_o be_v each_o of_o they_o a_o arch_n of_o two_o degree_n and_o 1_o 16._o wherefore_o the_o arch_n r_o b_o s_o will_v be_v great_a than_o the_o semicircle_n a_o b_o c_o by_o 4_o degree_n and_o â…›_n and_o great_a than_o the_o arch_n s_o d_o r_o by_o 8_o degree_n and_o ¼_n the_o equinox_n therefore_o will_v be_v in_o the_o point_n r_o &_o s_o and_o therefore_o also_o when_o the_o earth_n be_v in_o r_o the_o sun_n will_v appear_v in_o s._n wherefore_o the_o true_a place_n of_o the_o sun_n will_v be_v in_o it_o that_o be_v to_o say_v without_o the_o centre_n of_o the_o earth_n annual_a motion_n by_o the_o quantity_n of_o the_o sine_fw-la of_o the_o arch_a a_o r_o or_o the_o sine_fw-la of_o two_o degree_n and_o 16_o minute_n now_o this_o sine_fw-la put_v 100000_o for_o the_o radius_fw-la will_v be_v near_o 3580_o part_n thereof_o and_o so_o munh_o be_v the_o excentricity_n of_o the_o earth_n annual_a motion_n provide_v that_o that_o motion_n be_v in_o a_o perfect_a circle_n and_o s_z &_o r_o be_v the_o equinoctial_a point_n and_o the_o straight_a line_n his_o r_o &_o c_o asleep_o produce_v both_o way_n till_o they_o
of_o the_o part_n of_o those_o plant_n make_v a_o odorous_a liquor_n so_o also_o of_o air_n pass_v through_o the_o same_o plant_n while_o they_o be_v grow_v be_v make_v odorous_a air_n and_o thus_o also_o it_o be_v with_o the_o juice_n and_o spirit_n which_o be_v breed_v in_o live_v creature_n 16_o that_o odorous_a body_n may_v be_v make_v more_o odorous_a by_o contrition_n proceed_v from_o this_o that_o be_v break_v into_o many_o part_n which_o be_v all_o odorous_a the_o air_n which_o by_o respiration_n be_v draw_v from_o the_o object_n towards_o the_o organ_n do_v in_o its_o passage_n touch_v upon_o all_o those_o part_n and_o receive_v their_o motion_n now_o the_o air_n touch_v the_o superficies_n only_o and_o a_o body_n have_v less_o superficies_n while_o it_o be_v whole_a than_o all_o its_o part_n together_o have_v after_o it_o be_v reduce_v to_o powder_n it_o follow_v that_o the_o same_o odorous_a body_n yield_v less_o smell_n while_o it_o be_v whole_a than_o it_o will_v do_v after_o it_o be_v break_v into_o small_a part_n and_o thus_o much_o of_o smell_v 17_o the_o taste_n follow_v who_o generation_n have_v this_o difference_n from_o that_o of_o the_o sight_n hear_v and_o smell_a that_o by_o these_o we_o have_v sense_n of_o remote_a object_n whereas_o we_o taste_v nothing_o but_o what_o be_v contiguous_a and_o do_v immediate_o touch_v either_o the_o tongue_n or_o palate_n or_o both_o from_o whence_o it_o be_v evident_a that_o the_o cuticle_n of_o the_o tongue_n and_o palate_n and_o the_o nerve_n insert_v into_o they_o be_v the_o first_o organ_n of_o taste_n and_o because_o from_o the_o concussion_n of_o the_o part_n of_o these_o there_o follow_v necessary_o a_o concussion_n of_o the_o pia_fw-la mater_fw-la that_o the_o action_n communicate_v to_o these_o be_v propagate_v to_o the_o brain_n and_o from_o thence_o to_o the_o far_a organ_n namely_o the_o heart_n in_o who_o reaction_n consist_v the_o nature_n of_o sense_n now_o that_o savour_v as_o well_o as_o odour_n do_v not_o only_o move_v the_o brain_n but_o the_o stomach_n also_o as_o be_v manifest_a by_o the_o loathe_n that_o be_v cause_v by_o they_o both_o they_o that_o consider_v the_o organ_n of_o both_o these_o sense_n will_v not_o wonder_v at_o all_o see_v the_o tongue_n the_o palate_n &_o the_o nostril_n have_v one_o and_o the_o same_o continue_a cuticle_n derive_v from_o the_o dura_n mater_n and_o that_o effluvium_n have_v nothing_o to_o do_v in_o the_o sense_n of_o taste_v be_v manifest_a from_o this_o that_o there_o be_v no_o taste_n where_o the_o organ_n and_o the_o object_n be_v not_o contiguous_a by_o what_o variety_n of_o motion_n the_o different_a kind_n of_o taste_n which_o be_v innumerable_a may_v be_v distinguish_v i_o know_v not_o i_o may_v with_o other_o derive_v they_o from_o the_o divers_a figure_n of_o those_o atom_n of_o which_o whatsoever_o may_v be_v taste_v consist_v or_o from_o the_o diverse_a motion_n which_o i_o may_v by_o way_n of_o supposition_n attribute_v to_o those_o atom_n conjecture_v not_o without_o some_o likelihood_n of_o truth_n that_o such_o thing_n as_o taste_v sweet_a have_v their_o particle_n move_v with_o slow_a circular_a motion_n and_o their_o figure_n spherical_a which_o make_v they_o smooth_a and_o please_a to_o the_o organ_n that_o bitter_a thing_n have_v circular_a motion_n but_o vehement_a and_o their_o figure_n full_a of_o angle_n by_o which_o they_o trouble_v the_o organ_n and_o that_o sour_a thing_n have_v straight_o and_o reciprocal_a motion_n and_o their_o figure_n long_o and_o small_a so_o that_o they_o cut_v and_o wound_v the_o organ_n and_o in_o like_a manner_n i_o may_v assign_v for_o the_o cause_n of_o other_o taste_v such_o several_a motion_n and_o figure_n of_o atom_n as_o might_n in_o probability_n seem_v to_o be_v the_o true_a cause_n but_o this_o will_v be_v to_o revolt_v from_o philosophy_n to_o divination_n 18_o by_o the_o touch_n we_o feel_v what_o body_n be_v cold_a or_o hot_a though_o they_o be_v distant_a from_o we_o other_o as_o hard_a soft_a rough_a and_o smooth_a we_o can_v feel_v unless_o they_o be_v contiguous_a the_o organ_n of_o touch_n be_v every_o one_o of_o those_o membrane_n which_o be_v continue_v from_o the_o pia_fw-la mater_fw-la be_v so_o diffuse_v throughout_o the_o whole_a body_n as_o that_o no_o part_n of_o it_o can_v be_v press_v but_o the_o pia_n mater_n be_v press_v together_o with_o it_o whatsoever_o therefore_o press_v it_o be_v feel_v as_o hard_a or_o soft_a that_o be_v to_o say_v as_o more_o or_o less_o hard._n and_o as_o for_o the_o sense_n of_o rough_a it_o be_v nothing_o else_o but_o innumerable_a perception_n of_o hard_a and_o hard_o succeed_v one_o another_o by_o short_a interval_n both_o of_o time_n and_o place_n for_o we_o take_v notice_n of_o rough_a and_o smooth_a as_o also_o of_o magnitude_n and_o figure_n not_o only_o by_o the_o touch_n but_o also_o by_o memory_n for_o though_o some_o thing_n be_v touch_v in_o one_o point_n yet_o rough_a and_o smooth_a like_a quantity_n and_o figure_n be_v not_o perceive_v but_o by_o the_o flux_n of_o a_o point_n that_o be_v to_o say_v we_o have_v no_o sense_n of_o they_o without_o time_n and_o we_o can_v have_v no_o sense_n of_o time_n without_o memory_n chap._n xxx_o of_o gravity_n 1_o a_o thick_a body_n do_v not_o contain_v more_o matter_n unless_o also_o more_o place_n than_o a_o thin_a 2_o that_o the_o descent_n of_o heavy_a body_n proceed_v not_o from_o their_o own_o appetite_n but_o from_o some_o power_n of_o the_o earth_n 3_o the_o difference_n of_o gravity_n proceed_v from_o the_o difference_n of_o the_o impetus_fw-la with_o which_o the_o element_n whereof_o heavy_a body_n be_v make_v do_v fall_v upon_o the_o earth_n 4_o the_o cause_n of_o the_o descent_n of_o heavy_a body_n 5_o in_o what_o proportion_n the_o descent_n of_o heavy_a body_n be_v accelerate_a 6_o why_o those_o that_o dive_v do_v not_o when_o they_o be_v under_o water_n feel_v the_o weight_n of_o the_o water_n above_o they_o 7_o the_o weight_n of_o a_o body_n that_o float_v be_v equal_a to_o the_o weight_n of_o so_o much_o water_n as_o will_v fill_v the_o space_n which_o the_o immersed_a part_n of_o the_o body_n take_v up_o within_o the_o water_n 8_o if_o a_o body_n be_v light_a than_o water_n then_o how_o big_a soever_o that_o body_n be_v it_o will_v float_v upon_o any_o quantity_n of_o water_n how_o little_a soever_o 9_o how_o water_n may_v be_v lift_v up_o and_o force_v out_o of_o a_o vessel_n by_o air._n 10_o why_o a_o bladder_n be_v heavy_a when_o blow_v full_a of_o air_n then_o when_o it_o be_v empty_a 11_o the_o cause_n of_o the_o ejection_n upward_o of_o heavy_a body_n from_o a_o windgun_n 12_o the_o cause_n of_o the_o ascent_n of_o water_n in_o a_o weatherglass_n 13_o the_o cause_n of_o motion_n upward_o in_o live_v creature_n 14_o that_o there_o be_v in_o nature_n a_o kind_n of_o body_n heavy_a than_o air_n which_o nevertheless_o be_v not_o by_o sense_n distinguishable_a from_o it_o 15_o of_o the_o cause_n of_o magnetical_a virtue_n 1_o in_o the_o 21_o chapter_n i_o have_v define_v thick_a and_o thin_a as_o that_o place_n require_v so_o as_o that_o by_o thick_a be_v signify_v a_o more_o resist_a body_n and_o by_o thin_a a_o body_n less_o resist_v follow_v the_o custom_n of_o those_o that_o have_v before_o i_o discourse_v of_o refraction_n now_o if_o we_o consider_v the_o true_a and_o vulgar_a signification_n of_o those_o word_n we_o shall_v find_v they_o to_o be_v name_n collective_a that_o be_v to_o say_v name_n of_o multitude_n as_o thick_a to_o be_v that_o which_o take_v up_o more_o part_n of_o a_o space_n give_v &_o thin_a that_o which_o contain_v few_o part_n of_o the_o same_o magnitude_n in_o the_o same_o space_n or_o in_o a_o space_n equal_a to_o it_o thick_o therefore_o be_v the_o same_o with_o frequent_a as_o a_o thick_a troop_n and_o thin_a the_o same_o with_o unfrequent_a as_o a_o thin_a rank_n thin_a of_o house_n not_o that_o there_o be_v more_o matter_n in_o one_o place_n then_o in_o another_o equal_a place_n but_o a_o great_a quantity_n of_o some_o name_v body_n for_o there_o be_v not_o less_o matter_n or_o body_n indefinite_o take_v in_o a_o desert_n then_o there_o be_v in_o a_o city_n but_o few_o house_n or_o few_o men._n nor_o be_v there_o in_o a_o thick_a rank_n a_o great_a quantity_n of_o body_n but_o a_o great_a number_n of_o soldier_n then_o in_o a_o thin_a wherefore_o the_o multitude_n &_o paucity_n of_o the_o part_n contain_v within_o the_o same_o space_n do_v constitute_v density_n and_o rarity_n whether_o those_o part_n be_v separate_v by_o vacuum_n or_o by_o air_n but_o the_o consideration_n of_o this_o be_v not_o of_o any_o great_a moment_n in_o philosophy_n and_o therefore_o i_o let_v
in_o like_a manner_n be_v follow_v by_o the_o noxious_a matter_n contain_v in_o cb_n &_o by_o this_o mean_v the_o pit_n be_v for_o that_o time_n make_v healthful_a out_o of_o this_o history_n which_o i_o write_v only_o to_o such_o as_o have_v have_v experience_n of_o the_o truth_n of_o it_o without_o any_o design_n to_o support_v my_o philosophy_n with_o story_n of_o doubtful_a credit_n may_v be_v collect_v the_o follow_v possible_a cause_n of_o this_o phaenomenon_n namely_o that_o there_o be_v a_o certain_a matter_n fluid_fw-la &_o most_o transparent_a and_o not_o much_o light_a than_o water_n which_o break_v out_o of_o the_o earth_n fill_v the_o pit_n to_o c_o and_o that_o in_o this_o matter_n as_o in_o water_n both_o fire_n and_o live_a creature_n be_v extinguish_v 15_o about_o the_o nature_n of_o heavy_a body_n the_o great_a difficulty_n arise_v from_o the_o contemplation_n of_o those_o thing_n which_o make_v other_o heavy_a body_n ascend_v to_o they_o such_o be_v jet_n amber_n and_o the_o loadstone_n but_o that_o which_o trouble_v man_n most_o be_v the_o loadstone_n which_o be_v also_o call_v lapis_fw-la herculeus_fw-la a_o stone_n though_o otherwise_o despicable_a yet_o of_o so_o great_a power_n that_o it_o take_v up_o iron_n from_o the_o earth_n and_o hold_v it_o suspend_v in_o the_o air_n as_o hercules_n do_v antaeus_n nevertheless_o we_o wonder_v at_o it_o somewhat_o the_o less_o because_o we_o see_v jet_n draw_v up_o straw_n which_o be_v heavy_a body_n though_o not_o so_o heavy_a as_o iron_n but_o as_o for_o jet_n it_o must_v first_o be_v excite_v by_o rub_v that_o be_v to_o say_v by_o motion_n to_o and_o fro_o whereas_o the_o loadstone_n have_v sufficient_a excitation_n from_o its_o own_o nature_n that_o be_v to_o say_v from_o some_o internal_a principle_n of_o motion_n peculiar_a to_o itself_o now_o whatsoever_o be_v move_v be_v move_v by_o some_o contiguous_a and_o move_v body_n as_o have_v be_v former_o demonstrate_v and_o from_o hence_o it_o follow_v evident_o that_o the_o first_o endeavour_n which_o iron_n have_v towards_o the_o loadstone_n be_v cause_v by_o the_o motion_n of_o that_o air_n which_o be_v contiguous_a to_o the_o iron_n also_o that_o this_o motion_n be_v generate_v by_o the_o motion_n of_o the_o next_o air_n and_o so_o on_o successive_o till_o by_o this_o succession_n we_o find_v that_o the_o motion_n of_o all_o the_o intermediate_v air_n take_v its_o beginning_n from_o some_o motion_n which_o be_v in_o the_o loadstone_n itself_o which_o motion_n because_o the_o loadstone_n seem_v to_o be_v at_o rest_n be_v invisible_a it_o be_v therefore_o certain_a that_o the_o attractive_a power_n of_o the_o loadstone_n be_v nothing_o else_o but_o some_o motion_n of_o the_o small_a particle_n thereof_o suppose_v therefore_o that_o those_o small_a body_n of_o which_o the_o loadstone_n be_v in_o the_o bowel_n of_o the_o earth_n compose_v have_v by_o nature_n such_o motion_n or_o endeavour_v as_o be_v above_o attribute_v to_o jet_v namely_o a_o reciprocal_a motion_n in_o a_o line_n too_o short_a to_o be_v see_v both_o those_o stone_n will_v have_v one_o &_o the_o same_o cause_n of_o attraction_n now_o in_o what_o manner_n and_o in_o what_o order_n of_o work_v this_o cause_n produce_v the_o effect_n of_o attraction_n be_v the_o thing_n to_o be_v inquire_v and_o first_o we_o know_v that_o when_o the_o string_n of_o a_o lute_n or_o viol._n be_v strike_v the_o vibration_n that_o be_v the_o reciprocal_a motion_n of_o that_o string_n in_o the_o same_o straight_a line_n cause_v like_o vibration_n in_o another_o string_n which_o have_v like_o tension_n we_o know_v also_o that_o the_o dregs_o or_o small_a sand_n which_o sink_v to_o the_o bottom_n of_o a_o vessel_n will_v be_v raise_v up_o from_o the_o bottom_n by_o any_o strong_a and_o reciprocal_a agitation_n of_o the_o water_n stir_v with_o the_o hand_n or_o with_o a_o staff_n why_o therefore_o shall_v not_o reciprocal_a motion_n of_o the_o part_n of_o the_o loadstone_n contribute_v as_o much_o towards_o the_o move_n of_o iron_n for_o if_o in_o the_o loadstone_n there_o be_v suppose_v such_o reciprocal_a motion_n or_o motion_n of_o the_o part_n forward_o and_o backward_o it_o will_v follow_v that_o the_o like_a motion_n will_v be_v propagate_v by_o the_o air_n to_o the_o iron_n and_o consequent_o that_o there_o will_v be_v in_o all_o the_o part_n of_o the_o iron_n the_o same_o reciprocation_n or_o motion_n forward_o and_o backward_o and_o from_o hence_o also_o it_o will_v follow_v that_o the_o intermediate_v air_n between_o the_o stone_n and_o the_o iron_n will_v by_o little_a and_o little_o be_v thrust_v away_o and_o the_o air_n be_v thrust_v away_o the_o body_n of_o the_o loadstone_n and_o the_o iron_n will_v necessary_o come_v together_o the_o possible_a cause_n therefore_o why_o the_o loadstone_n and_o jet_n draw_v to_o they_o the_o one_o iron_n the_o other_o straw_n may_v be_v this_o that_o those_o attract_v body_n have_v reciprocal_a motion_n either_o in_o a_o straight_a line_n or_o in_o a_o elliptical_a line_n when_o there_o be_v nothing_o in_o the_o nature_n of_o the_o attract_a body_n which_o be_v repugnant_a to_o such_o a_o motion_n but_o why_o the_o loadstone_n if_o with_o the_o help_n of_o cork_n it_o float_v at_o liberty_n upon_o the_o top_n of_o the_o water_n shall_v from_o any_o position_n whatsoever_o so_o place_n itself_o in_o the_o plain_a of_o the_o meridian_n as_o that_o the_o same_o point_n which_o at_o one_o time_n of_o its_o be_v at_o rest_n respect_v the_o pole_n of_o the_o earth_n shall_v at_o all_o other_o time_n respect_v the_o same_o pole_n the_o cause_n may_v be_v this_o that_o the_o reciprocal_a motion_n which_o i_o suppose_v to_o be_v in_o the_o part_n of_o the_o stone_n be_v make_v in_o a_o line_n parallel_n to_o the_o axis_n of_o the_o earth_n and_o have_v be_v in_o those_o part_n ever_o since_o the_o stone_n be_v generate_v see_v therefore_o the_o stone_n while_o it_o remain_v in_o the_o mine_n and_o be_v carry_v about_o together_o with_o the_o earth_n by_o its_o diurnal_a motion_n do_v by_o length_n of_o time_n get_v a_o habit_n of_o be_v move_v in_o a_o line_n which_o be_v perpendicular_a to_o the_o line_n of_o its_o reciprocal_a motion_n it_o will_v afterward_o though_o its_o axis_fw-la be_v remove_v from_o the_o parallel_n situation_n it_o have_v with_o the_o axis_fw-la of_o the_o earth_n retain_v its_o endeavour_n of_o return_v to_o that_o situation_n again_o and_o all_o endeavour_n be_v the_o beginning_n of_o motion_n and_o nothing_o intervening_a that_n may_v hinder_v the_o same_o the_o loadstone_n will_v therefore_o return_v to_o its_o former_a situation_n for_o any_o piece_n of_o iron_n that_o have_v for_o a_o long_a time_n rest_v in_o the_o plain_a of_o the_o meridian_n whensoever_o it_o be_v force_v from_o that_o situation_n and_o afterward_o leave_v to_o its_o own_o liberty_n again_o will_v of_o itself_o return_v to_o lie_v in_o the_o meridian_n again_o which_o return_n be_v cause_v by_o the_o endeavour_n it_o acquire_v from_o the_o diurnal_a motion_n of_o the_o earth_n in_o the_o parallel_a circle_n which_o be_v perpendicular_a to_o the_o meridian_n if_o iron_n be_v rub_v by_o the_o loadstone_n draw_v from_o one_o pole_n to_o the_o other_o two_o thing_n will_v happen_v one_o that_o the_o iron_n will_v acquire_v the_o same_o direction_n with_o the_o loadstone_n that_o be_v to_o say_v that_o it_o will_v lie_v in_o the_o meridian_n and_o have_v its_o axis_n and_o pole_n in_o the_o same_o position_n with_o those_o of_o the_o stone_n the_o other_o that_o the_o like_a pole_n of_o the_o stone_n and_o of_o the_o iron_n will_v avoid_v one_o another_o and_o the_o unlike_a pole_n approach_v one_o another_o and_o the_o cause_n of_o the_o former_a may_v be_v this_o that_o iron_n be_v touch_v by_o motion_n which_o be_v not_o reciprocal_a but_o draw_v the_o same_o way_n from_o pole_n to_o pole_n there_o will_v be_v imprint_v in_o the_o iron_n also_o a_o endeavour_n from_o the_o same_o pole_n to_o the_o same_o pole_n for_o see_v the_o loadstone_n differ_v from_o iron_n no_o otherwise_o then_o as_o ore_n from_o metal_n there_o will_v be_v no_o repugnance_n at_o all_o in_o the_o iron_n to_o receive_v the_o same_o motion_n which_o be_v in_o the_o stone_n from_o whence_o it_o follow_v that_o see_v they_o be_v both_o affect_v alike_o by_o the_o diurnal_a motion_n of_o the_o earth_n they_o will_v both_o equal_o return_v to_o their_o situation_n in_o the_o meridian_n whensoever_o they_o be_v put_v from_o the_o same_o also_o of_o the_o late_a this_o may_v be_v the_o cause_n that_o as_o the_o loadstone_n in_o touch_v the_o iron_n do_v by_o its_o action_n imprint_v in_o the_o iron_n a_o endeavour_n towards_o one_o of_o the_o pole_n suppose_v towards_o the_o north_n pole_n so_o reciprocal_o the_o