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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
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
which_o be_v do_v the_o excentricity_n of_o the_o earth_n will_v be_v cf._n see_v therefore_o the_o annual_a motion_n of_o the_o earth_n be_v in_o the_o circumference_n of_o a_o ellipsis_n of_o which_o ♑_o ♋_o be_v the_o great_a axis_n ab_fw-la can_v be_v the_o lesser_a axis_n for_o ab_fw-la and_o ♑_o ♋_o be_v equal_a wherefore_o the_o earth_n pass_v through_o a_o &_o b_o will_v either_o pass_v above_o ♑_o as_o through_o g_o or_o pass_v through_o ♑_o will_n fall_n between_o c_o and_o a_o it_o be_v no_o matter_n which_o let_v it_o pass_v therefore_o through_o g_o and_o let_v gl_o be_v take_v equal_a to_o the_o straight_a line_n ♑_o ♋_o and_o divide_v gl_o equal_o in_o i_o gi_z will_v be_v equal_a to_o ♑_o ♋_o &_o ill_a equal_a to_o f_o ♋_o and_o consequent_o the_o point_n i_o will_v cut_v the_o excentricity_n cf_n into_o two_o equal_a part_n and_o take_v ih_v equal_a to_o if_o hi_o will_v be_v the_o whole_a excentricity_n if_o now_o a_o straight_a line_n namely_o the_o line_n ♎_o i_o ♈_o be_v draw_v through_o i_o parallel_n to_o the_o straight_a line_n ab_fw-la and_o ed_z the_o way_n of_o the_o sun_n in_o summer_n namely_o the_o arch_n ♎_o g_o ♈_o will_v be_v great_a than_o his_o way_n in_o winter_n by_o 8_o degree_n and_o ¼_n wherefore_o the_o true_a aequinox_n will_v be_v in_o the_o straight_a line_n ♎_o i_o ♈_o and_o therefore_o the_o ellipsis_n of_o the_o earth_n annual_a motion_n will_v not_o pass_v through_o a_o g_o b_o &_o l_o but_o through_o ♎_o g_o ♈_o &_o l._n wherefore_o the_o annual_a motion_n of_o the_o earth_n be_v in_o the_o ellipsis_n ♎_o g_o ♈_o l_o and_o can_v be_v the_o excentricity_n be_v salve_v in_o any_o other_o line_n and_o this_o perhaps_o be_v the_o reason_n why_o kepler_n against_o the_o opinion_n of_o all_o the_o astronomer_n of_o former_a time_n think_v fit_a to_o bisect_v the_o excentricity_n of_o the_o earth_n or_o according_a to_o the_o ancient_n of_o the_o sun_n not_o by_o diminish_v the_o quantity_n of_o the_o same_o excentricity_n because_o the_o true_a measure_n of_o that_o quantity_n be_v the_o difference_n by_o which_o the_o summer_n arch_n exceed_v the_o winter_n arch_n but_o by_o take_v for_o the_o centre_n of_o the_o ecliptic_a of_o the_o great_a orb_n the_o point_n c_o near_o to_o f_o &_o so_o place_v the_o whole_a great_a orb_n as_o much_o near_a to_o the_o ecliptic_a of_o the_o fix_a star_n towards_o ♋_o as_o be_v the_o distance_n between_o c_o &_o i._o for_o see_v the_o whole_a great_a orb_n be_v but_o as_o a_o point_n in_o respect_n of_o the_o immense_a distance_n of_o the_o fix_a star_n the_o two_o straight_a line_n ♎_o ♈_o and_o ab_fw-la be_v produce_v both_o way_n to_o the_o beginning_n of_o aries_n and_o libra_n will_v fall_v upon_o the_o same_o point_n of_o the_o sphere_n of_o the_o fix_a star_n let_v therefore_o the_o diameter_n of_o the_o earth_n mn_v be_v in_o the_o plain_a of_o the_o earth_n annual_a motion_n if_o now_o the_o earth_n be_v move_v by_o the_o sun_n simple_a motion_n in_o the_o circumference_n of_o the_o ecliptic_a about_o the_o centre_n i_o this_o diameter_n will_v be_v keep_v always_o parallel_v to_o itself_o and_o to_o the_o straight_a line_n gl_o but_o see_v the_o earth_n be_v move_v in_o the_o circumference_n of_o a_o ellipsis_n without_o the_o ecliptic_a the_o point_n n_o whilst_o it_o pass_v through_o ♎_o ♑_o ♈_o will_n go_v in_o a_o lesser_a circumference_n than_o the_o point_n m_o and_o consequent_o as_o soon_o as_o ever_o it_o begin_v to_o be_v move_v it_o will_v lose_v its_o parallelisme_n with_o the_o straight_a line_n ♑_o ♋_o so_o that_o mn_v produce_v will_n at_o last_o cut_v the_o straight_a line_n gl_o produce_v and_o contrary_o as_o soon_o as_o mn_n be_v pass_v ♈_o the_o earth_n make_v its_o way_n in_o the_o internal_a ellipticall_a line_n ♈_o l_o ♎_o the_o same_o mn_v produce_v towards_o m_o will_v cut_v lg_v produce_v and_o when_o the_o earth_n have_v almost_o finish_v its_o whole_a circumference_n the_o same_o mn_v shall_v again_o make_v a_o right_a angle_n with_o a_o line_n draw_v from_o the_o centre_n i_o a_o little_a short_a of_o the_o point_n from_o which_o the_o earth_n begin_v its_o motion_n and_o there_o the_o next_o year_n shall_v be_v one_o of_o the_o aequinoctial_a point_n namely_o near_o the_o end_n of_o â™_o the_o other_o shall_v be_v opposite_a to_o it_o near_o the_o end_n of_o ♓_o and_o thus_o the_o point_n in_o which_o the_o day_n and_o night_n be_v make_v equal_a do_v every_o year_n fall_v back_o but_o with_o so_o slow_a a_o motion_n that_o in_o a_o whole_a year_n it_o make_v but_o 51_o first_o minute_n and_o this_o relapse_n be_v contrary_a to_o the_o order_n of_o the_o sign_n be_v common_o call_v the_o praecession_n of_o the_o aequinox_n of_o which_o i_o have_v from_o my_o former_a supposition_n deduce_v a_o possible_a cause_n which_o be_v to_o be_v do_v according_a to_o what_o i_o have_v say_v concern_v the_o cause_n of_o the_o excentricity_n of_o the_o earth_n and_o according_a to_o kepler_n who_o for_o the_o cause_n thereof_o suppose_v one_o part_n of_o the_o earth_n to_o be_v affect_v to_o the_o sun_n the_o other_o part_n to_o be_v disaffect_v the_o apogaeum_n &_o perigaeum_n of_o the_o sun_n shall_v be_v move_v every_o year_n in_o the_o same_o order_n and_o with_o the_o same_o velocity_n with_o which_o the_o aequinoctial_a point_n be_v move_v and_o their_o distance_n from_o they_o shall_v always_o be_v the_o quadrant_a of_o a_o circle_n which_o seem_v to_o be_v otherwise_o for_o astronomer_n say_v that_o the_o aequinox_n be_v now_o the_o one_o about_o 28_o degree_n go_v back_o from_o the_o first_o star_n of_o aries_n the_o other_o as_o much_o from_o the_o begin_n of_o libra_n so_o that_o the_o apogaeum_n of_o the_o sun_n or_o the_o aphelium_n of_o the_o earth_n ought_v to_o be_v about_o the_o 28_o degree_n of_o cancer_n but_o it_o be_v reckon_v to_o be_v in_o the_o seven_o degree_n see_v therefore_o we_o have_v not_o sufficient_a evidence_n of_o the_o 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d that_o so_o it_o be_v it_o be_v in_o vain_a to_o seek_v for_o the_o 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d why_o it_o be_v so_o wherefore_o as_o long_o as_o the_o motion_n of_o the_o apogaeum_n be_v not_o observable_a by_o reason_n of_o the_o slowness_n thereof_o and_o as_o long_o as_o it_o remaive_v doubtful_a whether_o their_o distance_n from_o the_o aequinoctial_a point_n be_v more_o or_o less_o than_o a_o quadrant_a precise_o so_o long_v it_o may_v be_v lawful_a for_o i_o to_o think_v they_o proceed_v both_o of_o they_o with_o equal_a velocity_n also_o i_o do_v not_o at_o all_o meddle_v with_o the_o cause_n of_o the_o excentricity_n of_o saturn_n jupiter_n mars_n and_o mercury_n nevertheless_o see_v the_o excentricity_n of_o the_o earth_n may_v as_o i_o have_v show_v be_v cause_v by_o the_o unlike_a constitution_n of_o the_o several_a part_n of_o the_o earth_n which_o be_v alternate_o turn_v towards_o the_o sun_n it_o be_v credible_a also_o that_o like_a effect_n may_v be_v produce_v in_o these_o other_o planet_n from_o their_o have_v their_o superficies_n of_o unlike_a part_n and_o this_o be_v all_o i_o shall_v say_v concern_v sidereal_a philosophy_n and_o though_o the_o cause_n i_o have_v here_o suppose_v be_v not_o the_o true_a cause_n of_o these_o phaenomena_n yet_o i_o have_v demonstrate_v that_o they_o be_v sufficient_a to_o produce_v they_o according_a to_o what_o i_o at_o first_o propound_v chap._n xxvii_o of_o light_n heatt_z and_o of_o colour_n 1_o of_o the_o immense_a magnitude_n of_o some_o body_n and_o the_o unspeakable_a littleness_n of_o other_o 2_o of_o the_o cause_n of_o the_o light_n of_o the_o sun_n 3_o how_o light_n heat_v 4_o the_o generation_n of_o fire_n from_o the_o sun_n 5_o the_o generation_n of_o fire_n from_o collision_n 6_o the_o cause_n of_o light_n in_o glow-worm_n rot_a wood_n and_o the_o bolonian_a stone_n 7_o the_o cause_n of_o light_n in_o the_o concussion_n of_o sea-water_n 8_o the_o cause_n of_o flame_n spark_n and_o colliquation_n 9_o the_o cause_n why_o wet_a hay_n sometime_o burn_v of_o its_o own_o accord_n also_o the_o cause_n of_o lightning_n 10_o the_o cause_n of_o the_o force_n of_o gunpowder_n and_o what_o be_v to_o be_v ascribe_v to_o the_o coal_n what_o to_o the_o brimstone_n and_o what_o to_o the_o nitre_n 11_o how_o heat_n be_v cause_v by_o attrition_n 12_o the_o distinction_n of_o light_n into_o first_o second_o etc._n etc._n 13_o the_o cause_n of_o the_o colour_n we_o see_v in_o look_v through_o a_o prisma_fw-la of_o glass_n namely_o of_o red_a yellow_a blue_a and_o violet_a colour_n 14_o why_o the_o moon_n and_o the_o star_n appear_v red_a in_o the_o horizon_n then_o in_o
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
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
to_o be_v settle_v any_o where_o as_o at_o h._n if_o now_o the_o heat_n of_o the_o air_n be_v augment_v the_o water_n will_v descend_v below_o h_n and_o if_o the_o heat_n be_v diminish_v it_o will_v ascend_v above_o it_o which_o though_o it_o be_v certain_o know_v to_o be_v true_a by_o experience_n the_o cause_n nevertheless_o have_v not_o as_o yet_o be_v discover_v in_o the_o 6_o and_o 7_o article_n of_o the_o 27_o chapter_n where_o i_o consider_v the_o cause_n of_o cold_a i_o have_v show_v that_o fluid_a body_n be_v make_v cold_a by_o the_o pressure_n of_o the_o air_n that_o be_v to_o say_v by_o a_o constant_a wind_n that_o press_v they_o for_o the_o same_o cause_n it_o be_v that_o the_o superficies_n of_o the_o water_n be_v press_v at_o f_o and_o have_v no_o place_n to_o which_o it_o may_v retire_v from_o this_o pressure_n beside_o the_o cavity_n of_o the_o cylinder_n between_o h_n and_o e_o it_o be_v therefore_o necessary_o force_v thither_o by_o the_o cold_a and_o consequent_o it_o ascend_v more_o or_o less_o according_a as_o the_o cold_a be_v more_o or_o less_o increase_v and_o again_o as_o the_o heat_n be_v more_o intense_a or_o the_o cold_a more_o remiss_a the_o same_o water_n will_v be_v depress_v more_o or_o less_o by_o its_o own_o gravity_n that_o be_v to_o say_v by_o the_o cause_n of_o gravity_n above_o explicate_v 13_o also_o live_v creature_n though_o they_o be_v heavy_a can_v by_o leap_v swim_v &_o fly_v raise_v themselves_o to_o a_o certain_a degree_n of_o height_n but_o they_o can_v do_v this_o except_o they_o be_v support_v by_o some_o resist_a body_n as_o the_o earth_n the_o water_n and_o the_o air_n for_o these_o motion_n have_v their_o beginning_n from_o the_o contraction_n by_o the_o help_n of_o the_o muscle_n of_o the_o body_n animate_v for_o to_o this_o contraction_n there_o succee_v a_o distension_n of_o their_o whole_a body_n by_o which_o distension_n the_o earth_n the_o water_n or_o the_o air_n which_o support_v they_o be_v press_v and_o from_o hence_o by_o the_o reaction_n of_o those_o press_a body_n live_v creature_n acquire_v a_o endeavour_n upward_o but_o such_o as_o by_o reason_n of_o the_o gravity_n of_o their_o body_n be_v present_o lose_v again_o by_o this_o endeavour_n therefore_o it_o be_v that_o live_v creature_n raise_v themselves_o up_o a_o little_a way_n by_o leap_v but_o to_o no_o great_a purpose_n but_o by_o swim_v &_o fly_v they_o raise_v themselves_o to_o a_o great_a height_n because_o before_o the_o effect_n of_o their_o endeavour_n be_v quite_o extinguish_v by_o the_o gravity_n of_o their_o body_n they_o can_v renew_v the_o same_o endeavour_n again_o that_o by_o the_o power_n of_o the_o soul_n without_o any_o antecedent_n contraction_n of_o the_o muscle_n or_o the_o help_n of_o something_o to_o support_v he_o any_o man_n can_v be_v able_a to_o raise_v his_o body_n upward_o be_v a_o childish_a conceit_n for_o if_o it_o be_v true_a a_o man_n may_v raise_v himself_o to_o what_o height_n he_o please_v 14_o the_o diaphanous_a medium_fw-la which_o surround_v the_o eye_n on_o all_o fides_fw-la be_v invisible_a nor_o be_v air_n to_o be_v see_v in_o air_n nor_o water_n in_o water_n nor_o any_o thing_n but_o that_o which_o be_v more_o opacous_a but_o in_o the_o confine_n of_o two_o diaphanous_a body_n one_o of_o they_o may_v be_v distinguish_v from_o the_o other_o it_o be_v not_o therefore_o a_o thing_n so_o very_o ridiculous_a for_o ordinary_a people_n to_o think_v all_o that_o space_n empty_a in_o which_o we_o say_v be_v air_n it_o be_v the_o work_n of_o reason_n to_o make_v we_o conceive_v that_o the_o air_n be_v any_o thing_n for_o by_o which_o of_o our_o sense_n be_v it_o that_o we_o take_v notice_n of_o the_o air_n see_v we_o neither_o see_v nor_o hear_v nor_o taste_n nor_o smell_v nor_o feel_v it_o to_o be_v any_o thing_n when_o we_o feel_v heat_n we_o do_v not_o impute_v it_o to_o the_o air_n but_o to_o the_o fire_n nor_o do_v we_o say_v the_o air_n be_v cold_a but_o we_o ourselves_o be_v cold_a and_o when_o we_o feel_v the_o wind_n we_o rather_o think_v something_o be_v come_v then_o that_o any_o thing_n be_v already_o come_v also_o we_o do_v not_o at_o all_o feel_v the_o weight_n of_o water_n in_o water_n much_o less_o of_o air_n in_o air_n that_o we_o come_v to_o know_v that_o to_o be_v a_o body_n which_o we_o call_v air_n it_o be_v by_o reason_v but_o it_o be_v from_o one_o reason_n only_o namely_o because_o it_o be_v impossible_a for_o remote_a body_n to_o work_v upon_o our_o organ_n of_o sense_n but_o by_o the_o help_n of_o body_n intermediate_v without_o which_o we_o can_v have_v no_o sense_n of_o they_o till_o they_o come_v to_o be_v contiguous_a wherefore_o from_o the_o sense_n alone_o without_o reason_v from_o effect_n we_o can_v have_v sufficient_a evidence_n of_o the_o nature_n of_o body_n for_o there_o be_v underground_o in_o some_o mine_n of_o coal_n a_o certain_a matter_n of_o a_o middle_a nature_n between_o water_n and_o aire_n which_o nevertheless_o can_v by_o sense_n be_v distinguish_v from_o air_n for_o it_o be_v as_o diaphanous_a as_o the_o pure_a air_n and_o as_o far_o as_o sense_n can_v judge_v equal_o penetrable_a but_o if_o we_o look_v upon_o the_o effect_n it_o be_v like_a that_o of_o water_n for_o when_o that_o matter_n break_v out_o of_o the_o earth_n into_o one_o of_o those_o pit_n it_o fill_v the_o same_o either_o total_o or_o to_o some_o degree_n and_o if_o a_o man_n or_o fire_n be_v then_o let_v down_o into_o it_o it_o extinguish_v they_o in_o almost_o as_o little_a time_n as_o water_n will_v do_v but_o for_o the_o better_a understanding_n of_o this_o phaenomenon_n i_o shall_v describe_v the_o 6_o figure_n in_o which_o let_v a_o b_o represent_v the_o pit_n of_o the_o mine_n and_o let_v part_n thereof_o namely_o c_o b_o be_v suppose_v to_o be_v fill_v with_o that_o matter_n if_o now_o a_o light_a candle_n be_v let_v down_o into_o it_o below_o c_o it_o will_v as_o sudden_o be_v extinguish_v as_o if_o it_o be_v thrust_v into_o water_n also_o if_o a_o grate_n fill_v with_o coal_n thorough_o kindle_v and_o burn_v never_o so_o bright_o be_v let_v down_o as_o soon_o as_o ever_o it_o be_v below_o c_o the_o fire_n will_v begin_v to_o grow_v pale_a and_o short_o after_o lose_v its_o light_n be_v extinguish_v no_o otherwise_o then_o if_o it_o be_v quench_v in_o water_n but_o if_o the_o grate_n be_v draw_v up_o again_o present_o while_o the_o coal_n be_v still_o very_o hot_a the_o fire_n will_v by_o little_a and_o little_o be_v kindle_v again_o and_o shine_v as_o before_o there_o be_v indeed_o between_o this_o matter_n &_o water_v this_o considerable_a difference_n that_o it_o neither_o wet_v nor_o stick_v to_o such_o thing_n as_o be_v put_v down_o into_o it_o as_o water_n do_v which_o by_o the_o moisture_n it_o leave_v hinder_v the_o kindle_v again_o of_o the_o matter_n once_o extinguish_v in_o like_a manner_n if_o a_o man_n be_v let_v down_o below_o c_o he_o will_v present_o fall_v into_o a_o great_a difficulty_n of_o breathe_v and_o immediate_o after_o into_o a_o swoon_n and_o die_v unless_o he_o be_v sudden_o draw_v up_o again_o they_o therefore_o that_o go_v down_o into_o these_o pit_n have_v this_o custom_n that_o as_o soon_o as_o ever_o they_o feel_v themselves_o sick_a they_o shake_v the_o rope_n by_o which_o they_o be_v let_v down_o to_o signify_v they_o be_v not_o well_o and_o to_o the_o end_n that_o they_o may_v speedy_o be_v pull_v up_o again_o for_o if_o a_o man_n be_v draw_v out_o too_o late_a void_a of_o sense_n and_o motion_n they_o dig_v up_o a_o turf_n and_o put_v his_o face_n and_o mouth_n into_o the_o fresh_a earth_n by_o which_o mean_v unless_o he_o be_v quite_o dead_a he_o come_v to_o himself_o again_o by_o little_a and_o little_a and_o recover_v life_n by_o the_o breathe_n out_o as_o it_o be_v of_o that_o suffocate_a matter_n which_o he_o have_v suck_v in_o while_o he_o be_v in_o the_o pit_n almost_o in_o the_o same_o manner_n as_o they_o that_o be_v drown_v come_v to_o themselves_o again_o by_o vomit_v up_o the_o water_n but_o this_o do_v not_o happen_v in_o all_o mine_n but_o in_o some_o only_o and_o in_o those_o not_o always_o but_o often_o in_o such_o pit_n as_o be_v subject_a to_o it_o they_o use_v this_o remedy_n they_o dig_v another_o pit_n as_o de_fw-fr close_o by_o it_o of_o equal_a depth_n and_o join_v they_o both_o together_o with_o one_o common_a channel_n of_o they_o make_v a_o fire_n in_o the_o bottom_n e_o which_o carry_v out_o at_o d_o the_o air_n contain_v in_o the_o pit_n de_fw-fr and_o this_o draw_v with_o it_o the_o air_n contain_v in_o the_o channel_n of_o which_o
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