Selected quad for the lemma: power_n

Word A	Word B	Word C	Word D	Occurrence	Frequency Band	MI	MI Band	Prominent
power_n	length_n	line_n	square_n	12,261	5	15.3418	5	true

Text snippets containing the quad

ID	Title	Author	Corrected Date of Publication (TCP Date of Publication)	STC	Words	Pages
A44320	Lectiones Cutlerianæ, or, A collection of lectures, physical, mechanical, geographical, & astronomical made before the Royal Society on several occasions at Gresham Colledge : to which are added divers miscellaneous discourses / by Robert Hooke ...	Hooke, Robert, 1635-1703.	1679 (1679)	Wing H2617; ESTC R4280	276,083	420

four_o as_o the_o root_n of_o 64._o that_o be_v of_o 19+17+15+13_n at_o the_o end_n of_o the_o ten_o or_o whole_a as_o the_o root_n of_o 100_o that_o be_v as_o equal_a to_o 100_o now_o since_o the_o velocity_n be_v in_o the_o same_o proportion_n to_o the_o root_n of_o the_o space_n as_o the_o root_n of_o the_o space_n be_v to_o the_o time_n it_o be_v easy_a to_o determine_v the_o particular_a time_n in_o which_o every_o one_o of_o these_o space_n be_v pass_v for_o divide_v the_o space_n by_o the_o velocity_n correspond_v the_o quotient_n give_v the_o particular_a time_n to_o explain_v this_o more_o intelligible_o let_v a_o in_o the_o four_o figure_n represent_v the_o end_n of_o a_o spring_n not_o bend_v or_o at_o least_o counterpoise_v in_o that_o posture_n by_o a_o power_n fix_v to_o it_o and_o movable_a with_o it_o draw_v the_o line_n a_o b_o c_o and_o let_v it_o represent_v the_o way_n in_o which_o the_o end_n of_o the_o spring_n by_o additional_a power_n be_v to_o be_v move_v draw_v to_o the_o end_n of_o it_o c_o at_o right_a angle_n the_o line_n c_o δ_n d_o d_o and_o let_v c_o d_o represent_v the_o power_n that_o be_v sufficient_a to_o bend_v or_o move_v the_o end_n of_o the_o spring_n a_o to_o c_o then_o draw_v the_o line_n d_o a_o and_o from_o any_o point_n of_o the_o line_n a_o c_o as_o b_o b._n draw_v line_n parallel_v to_o c_o d_o cut_v the_o line_n d_o a_o in_o e_o e_o the_o line_n b_o e_o b_o e_o will_v represent_v the_o respective_a power_n requisite_a to_o bend_v the_o end_n of_o the_o spring_n a_o to_o b_o which_o line_n b_o e_z b_o e_o c_o d_o will_n be_v in_o the_o same_o proportion_n with_o the_o length_n of_o the_o bent_n of_o the_o spring_n a_o b_o a_o b_o a_o c._n and_o because_o the_o spring_n have_v in_o every_o point_n of_o the_o line_n of_o bend_v a_o c_o a_o particular_a power_n therefore_o imagine_v infinite_a line_n draw_v from_o every_o point_n of_o a_o c_o parallel_n to_o c_o d_o till_o they_o touch_v the_o line_n a_o d_o they_o will_v all_o of_o they_o fill_v and_o compose_v the_o triangle_n a_o c_o d._n the_o triangle_n therefore_o a_o c_o d_o will_n represent_v the_o aggregate_v of_o the_o power_n of_o the_o spring_n bend_v from_o a_o to_o c_o and_o the_o lesser_a triangle_n a_o b_o e_o a_o b_o e_o will_n represent_v the_o aggregate_v of_o all_o the_o power_n of_o the_o spring_n bend_v from_o a_o to_o b_o b_o and_o the_o spring_n bend_v to_o any_o point_n of_o the_o line_n a_o c_o and_o let_v go_v from_o thence_o will_v exert_v in_o its_o return_n to_o a_o all_o those_o power_n which_o be_v equal_a to_o the_o respective_a ordinates_n b_o e_o b_o e_o in_o the_o triangle_n the_o sum_n of_o all_o which_o make_v up_o the_o triangle_n a_o b_o e_o a_o b_o e._n and_o the_o aggregate_v of_o the_o power_n with_o which_o it_o return_v from_o any_o point_n as_o from_o c_o to_o any_o point_n of_o the_o space_n c_o a_o as_o to_o b_o b_o be_v equal_a to_o the_o trapezium_fw-la c_o d_o e_o b_o c_o d_o e_o b_o or_o the_o excess_n of_o the_o great_a triangle_n above_o the_o less_o have_v therefore_o show_v a_o image_n to_o represent_v the_o flexure_n and_o the_o power_n so_o as_o plain_o to_o solve_v and_o answer_v all_o question_n and_o problem_n concern_v they_o in_o the_o next_o place_n i_o come_v to_o represent_v the_o velocity_n appropriate_v to_o the_o several_a power_n the_o velocity_n then_o be_v always_o in_o a_o subduplicate_v proportion_n of_o the_o power_n that_o be_v as_o the_o root_n of_o the_o power_n impress_v and_o the_o power_n impress_v be_v as_o the_o trapezium_fw-la or_o the_o excess_n of_o the_o triangle_n or_o square_n of_o the_o whole_a space_n to_o be_v pass_v above_o the_o square_n of_o the_o space_n yet_o unpassed_a if_o upon_o the_o centre_n a_o and_o space_n a_o c_o c_z being_z the_o point_n from_o which_o the_o spring_n be_v suppose_v let_v go_v a_o circle_n be_v describe_v as_o c_o g_o g_o f_o and_o ordinates_n draw_v from_o any_o point_n of_o c_o a_o the_o space_n to_o be_v past_a as_o from_o b_o b_o to_o the_o say_a circle_n as_o b_o g_z b_o g_o these_o line_n b_o g_z b_o g_o will_v represent_v the_o velocity_n of_o the_o spring_n return_v from_o c_o to_z b_o b_o etc._n etc._n the_o say_v ordinates_n being_n always_o in_o the_o same_o proportion_n with_o the_o root_n of_o the_o trapezium_n c_o d_o e_o b_o c_o d_o e_o b_o for_o putting_z a_o c_o =_o to_o a_o and_o a_o b_o =_o b_o b_o g_o will_n always_o be_v equal_a to_o the_o square_a of_o the_o ordinate_a be_v always_o equal_a to_o the_o rectangle_n of_o the_o intercept_a part_n of_o the_o diameter_n have_v thus_o find_v the_o velocity_n to_o wit_n b_o g_o b_o g_o a_o f_o to_o find_v the_o time_n correspond_v on_o the_o diameter_n a_o c_o draw_v a_o parabola_fw-la c_o h_o f_o who_o vertex_fw-la be_v c_o and_o which_o pass_v through_o the_o point_n f._n the_o ordinates_n of_o this_o parabola_fw-la b_o h_n b_o h_n a_o f_o be_v in_o the_o same_o proportion_n with_o the_o root_n of_o the_o space_n c_o b_o c_o b_o c_o a_o then_z making_z g_z b_o to_z h_z b_o as_o h_z b_o to_z ay_o b_o and_o through_o the_o point_n c_o i_o i_o f_o draw_v the_o curve_v c_z i_o i_o i_o f_o the_o respective_a ordinates_n of_o this_o curve_v shall_v represent_v the_o proportionate_a time_n that_o the_o spring_n spend_v in_o return_v the_o space_n c_o b_o c_o b_o c_o a._n if_o the_o power_n or_o stiffness_n of_o the_o spring_n be_v great_a than_o what_o i_o before_o suppose_v and_o therefore_o must_v be_v express_v by_o the_o triangle_n c_o de_fw-fr a._n then_o the_o velocity_n will_v be_v the_o ordinates_n in_o a_o ellipse_n as_o c_o γ_n γ_n n_n great_a than_o the_o circle_n as_o it_o will_v also_o if_o the_o power_n be_v the_o same_o and_o the_o bulk_n move_v by_o the_o spring_n be_v less_o then_o will_v the_o s-like_a line_n of_o time_n meet_v with_o the_o line_n a_o f_o at_o a_o point_n as_o x_o within_o the_o point_n f._n but_o if_o the_o power_n of_o the_o spring_n be_v weak_a than_o i_o suppose_v then_o will_v c_o δ_n e_o e_o a_o represent_v the_o power_n and_o c_o γ_n γ_n o_o the_o ellipsis_n of_o velocity_n who_o ordinates_n b_o γ_n b_o γ_n a_o o_o will_v give_v the_o particular_a velocity_n and_o the_o s-like_a line_n of_o time_n will_v extend_v beyond_o n._n the_o same_o will_v happen_v suppose_v the_o body_n move_v by_o the_o spring_n to_o be_v proportionate_o heavy_a and_o the_o power_n of_o the_o spring_n the_o same_o with_o the_o first_o and_o suppose_v the_o power_n of_o the_o spring_n the_o same_o as_o at_o first_o bend_v only_o to_o b_o 2_o and_o from_o thence_o let_v go_v b_o 2_o e_z a_o be_v the_o triangle_n of_o its_o power_n the_o ordinates_n of_o the_o circle_n b_o g_o l_o be_v the_o line_n of_o its_o velocity_n and_o the_o ordinates_n of_o the_o s-like_a line_n b_o i_o f_o be_v the_o live_v of_o time_n have_v thus_o show_v you_o how_o the_o velocity_n of_o a_o spring_n may_v be_v compute_v it_o will_v be_v easy_a to_o calculate_v to_o what_o distance_n it_o will_v be_v able_a to_o shoot_v or_o throw_v any_o body_n that_o be_v move_v by_o it_o and_o this_o must_v be_v do_v by_o compare_v the_o velocity_n of_o the_o ascent_n of_o a_o body_n throw_v with_o the_o velocity_n of_o the_o descent_n of_o gravity_n allowance_n be_v also_o make_v for_o the_o resistance_n and_o impediment_n of_o the_o medium_n through_o which_o it_o pass_v for_o instance_n suppose_v a_o bow_n or_o spring_n fix_v at_o 16_o foot_n above_o a_o horizontal_n floor_n which_o be_v near_o the_o space_n that_o a_o heavy_a body_n from_o rest_n will_v descend_v perpendicular_o in_o a_o second_o of_o time_n if_o a_o spring_n deliver_v the_o body_n in_o the_o horizontal_n line_n with_o a_o velocity_n that_o move_v it_o 16_o foot_n in_o a_o second_o of_o time_n then_o shall_v it_o fall_v at_o 16_o foot_n from_o the_o perpendicular_a point_n on_o the_o floor_n over_o which_o it_o be_v deliver_v with_o such_o velocity_n and_o by_o its_o motion_n shall_v describe_v in_o the_o air_n or_o space_n through_o which_o it_o pass_v a_o parabola_fw-la if_o the_o spring_n be_v bend_v to_o twice_o the_o former_a tension_n so_o as_o to_o deliver_v the_o body_n with_o double_a the_o velocity_n in_o a_o horizontal_n line_n that_o be_v with_o a_o velocity_n that_o move_v 32_o foot_n in_o a_o second_o then_o shall_v the_o body_n touch_v the_o floor_n in_o a_o point_n very_o near_o at_o 32_o foot_n from_o the_o aforesaid_a perpendicular_a point_n and_o the_o

minute_n of_o time_n their_o motion_n be_v of_o such_o a_o velocity_n impress_v from_o the_o ambient_fw-la on_o the_o two_o extreme_a particle_n 1_o and_o 8._o first_o if_o by_o any_o external_a power_n on_o the_o two_o extreme_n 1_o and_o 8_o they_o be_v remove_v further_o asunder_o as_o to_o cd_o then_o shall_v all_o the_o vibrative_a particle_n be_v proportionable_o extend_v and_o the_o number_n of_o vibration_n and_o consequent_o of_o occursion_n be_v reciprocal_o diminish_v and_o consequent_o their_o endeavour_n of_o recede_v from_o each_o other_o be_v reciprocal_o diminish_v also_o for_o suppose_v this_o second_o dimension_n of_o length_n be_v to_o the_o first_o as_o 3_o to_o 2_o the_o length_n of_o the_o vibration_n and_o consequent_o of_o occursion_n be_v reciprocal_o diminish_v for_o whereas_o i_o suppose_v 1000000_o in_o a_o second_o of_o the_o former_a here_o can_v be_v but_o 666666_o in_o this_o and_o consequent_o the_o spring_n inward_a must_v be_v in_o proportion_n to_o the_o extension_n beyond_o its_o natural_a length_n second_o if_o by_o any_o external_a force_n the_o extreme_a particle_n be_v remove_v a_o three_o part_n near_o together_o than_o the_o external_a natural_a force_n be_v always_o the_o same_o both_o in_o this_o and_o the_o former_a instance_n which_o be_v the_o balance_n to_o it_o in_o its_o natural_a state_n the_o length_n of_o the_o vibration_n shall_v be_v proportionable_o diminish_v and_o the_o number_n of_o they_o and_o consequent_o of_o the_o occursion_n be_v reciprocal_o augment_v and_o instead_o of_o 1000000_o there_o shall_v be_v 1500000._o in_o the_o next_o place_n for_o fluid_fw-la body_n among_o which_o the_o great_a instance_n we_o have_v be_v air_n though_o the_o same_o be_v in_o some_o proportion_n in_o all_o other_o fluid_a body_n the_o air_n than_o be_v a_o body_n consist_v of_o particle_n so_o small_a as_o to_o be_v almost_o equal_a to_o the_o particle_n of_o the_o heterogeneous_a fluid_a medium_n encompass_v the_o earth_n it_o be_v bound_v but_o on_o one_o side_n namely_o towards_o the_o earth_n and_o be_v indefinite_o extend_v upward_o be_v only_o hinder_v from_o fly_v away_o that_o way_n by_o its_o own_o gravity_n the_o cause_n of_o which_o i_o shall_v some_o other_o time_n explain_v it_o consist_v of_o the_o same_o particle_n single_a and_o separate_v of_o which_o water_n and_o other_o fluid_n do_v conjoin_v and_o compound_v and_o be_v make_v of_o particle_n exceed_v small_a its_o motion_n to_o make_v its_o balance_n with_o the_o rest_n of_o the_o earthy_a body_n be_v exceed_v swift_a and_o its_o vibrative_a space_n exceed_v large_a comparative_a to_o the_o vibrative_a space_n of_o other_o terrestrial_a body_n i_o suppose_v that_o of_o the_o air_n next_o the_o earth_n in_o its_o natural_a state_n may_v be_v 8000_o time_n great_a than_o that_o of_o steel_n and_o above_o a_o thousand_o time_n great_a than_o that_o of_o common_a water_n and_o proportionable_o i_o suppose_v that_o its_o motion_n must_v be_v eight_o thousand_o time_n swift_a than_o the_o former_a and_o above_o a_o thousand_o time_n swift_a than_o the_o late_a if_o therefore_o a_o quantity_n of_o this_o body_n be_v enclose_v by_o a_o solid_a body_n and_o that_o be_v so_o contrive_v as_o to_o compress_v it_o into_o less_o room_n the_o motion_n thereof_o suppose_v the_o heat_n the_o same_o will_v continue_v the_o same_o and_o consequent_o the_o vibration_n and_o occursion_n will_v be_v increase_v in_o reciprocal_a proportion_n that_o be_v if_o it_o be_v condense_v into_o half_a the_o space_n the_o vibration_n and_o occursion_n will_v be_v double_a in_o number_n if_o into_o a_o quarter_n the_o vibration_n and_o occursion_n will_v be_v quadruple_a etc._n etc._n again_o if_o the_o contain_v vessel_n be_v so_o contrive_v as_o to_o leave_v it_o more_o space_n the_o length_n of_o the_o vibration_n will_v be_v proportionable_o enlarge_v and_o the_o number_n of_o vibration_n and_o occursion_n will_v be_v reciprocal_o diminish_v that_o be_v if_o it_o be_v suffer_v to_o extend_v to_o twice_o its_o former_a dimension_n its_o vibration_n will_v be_v twice_o as_o long_o and_o the_o number_n of_o its_o vibration_n and_o occursion_n will_v be_v few_o by_o half_a and_o consequent_o its_o endeavour_n outward_a will_v be_v also_o weak_a by_o half_a these_o explanation_n will_v serve_v mutatis_fw-la mutandis_fw-la for_o explain_v the_o spring_n of_o any_o other_o body_n whatsoever_o it_o now_o remain_v that_o i_o show_v how_o the_o constitution_n of_o springy_a body_n be_v such_o the_o vibration_n of_o a_o spring_n or_o a_o body_n move_v by_o a_o spring_n equal_o and_o uniform_o shall_v be_v of_o equal_a duration_n whether_o they_o be_v great_a or_o less_o i_o have_v here_o already_o show_v then_o that_o the_o power_n of_o all_o spring_n be_v proportionate_a to_o the_o degree_n of_o flexure_n viz._n one_o degree_n of_o flexure_n or_o one_o space_n bend_v have_v one_o power_n two_o have_v two_o and_o three_o have_v three_o and_o so_o forward_o and_o every_o point_n of_o the_o space_n of_o flexure_n have_v a_o peculiar_a power_n and_o consequent_o there_o be_v infinite_a point_n of_o the_o space_n there_o must_v be_v infinite_a degree_n of_o power_n and_o consequent_o all_o those_o power_n begin_v from_o nought_o and_o end_v at_o the_o last_o degree_n of_o tension_n or_o bend_n add_v together_o into_o one_o sum_n or_o aggregate_v will_v be_v in_o duplicate_v proportion_n to_o the_o space_n bend_a or_o degree_n of_o flexure_n that_o be_v the_o aggregate_v of_o the_o power_n of_o the_o spring_n tend_v from_o its_o quiescent_n posture_n by_o all_o the_o intermediate_v point_n to_o one_o space_n be_v it_o what_o length_n you_o please_v be_v equal_a or_o in_o the_o same_o proportion_n to_o the_o square_n of_o one_o suppose_v the_o say_a space_n infinite_o divisible_a into_o the_o fraction_n of_o one_o to_o two_o be_v equal_a or_o in_o the_o same_o proportion_n to_o the_o square_n of_o two_o that_o be_v four_o to_o three_o be_v equal_a or_o in_o the_o same_o proportion_n to_o the_o square_n of_o three_o that_o be_v nine_o and_o so_o forward_o and_o consequent_o the_o aggregate_v of_o the_o first_o space_n will_v be_v one_o of_o the_o second_o space_n will_v be_v three_o of_o the_o three_o space_n will_v be_v five_o of_o the_o four_o will_v be_v seven_o and_o so_o onward_o in_o a_o arithmetical_a proportion_n be_v the_o degree_n or_o excess_n by_o which_o these_o aggregate_n exceed_v one_o another_o the_o spring_n therefore_o in_o return_v from_o any_o degree_n of_o flexure_n to_o which_o it_o have_v be_v bend_v by_o any_o power_n receive_v at_o every_o point_n of_o the_o space_n return_v a_o impulse_n equal_a to_o the_o power_n of_o the_o spring_n in_o that_o point_n of_o tension_n and_o in_o return_v the_o whole_a it_o receive_v the_o whole_a aggregate_v of_o all_o the_o force_n belong_v to_o the_o great_a degree_n of_o that_o tension_n from_o which_o it_o return_v so_o a_o spring_n bend_v two_o space_n in_o its_o return_n receive_v four_o degree_n of_o impulse_n that_o be_v three_o in_o the_o first_o space_n return_v and_o one_o in_o the_o second_o so_o bend_v three_o space_n it_o receive_v in_o its_o whole_a return_n nine_o degree_n of_o impulse_n that_o be_v five_o in_o the_o first_o space_n return_v three_o in_o the_o second_o and_o one_o in_o the_o three_o so_o bend_v ten_o space_n it_o receive_v in_o its_o whole_a return_n one_o hundred_o degree_n of_o impulse_n to_o wit_n nineteen_o in_o the_o first_o seventeen_o in_o the_o second_o fifteen_o in_o the_o three_o thirteen_o in_o the_o four_o eleven_o in_o the_o five_o nine_o in_o the_o six_o seven_o in_o the_o seven_o five_o in_o the_o eight_o three_o in_o the_o nine_o and_o one_o in_o the_o ten_o now_o the_o comparative_a velocity_n of_o any_o body_n move_v be_v in_o subduplicate_v proportion_n to_o the_o aggregate_n or_o sum_n of_o the_o power_n by_o which_o it_o be_v move_v therefore_o the_o velocity_n of_o the_o whole_a space_n return_v be_v always_o in_o the_o same_o proportion_n with_o those_o space_n they_o be_v both_o subduplicate_v to_o the_o power_n and_o consequent_o all_o the_o time_n shall_v be_v equal_a next_o for_o the_o velocity_n of_o the_o part_n of_o the_o space_n return_v they_o will_v be_v always_o proportionate_a to_o the_o root_n of_o the_o aggregate_n of_o the_o power_n impress_v in_o every_o of_o these_o space_n for_o in_o the_o last_o instance_n where_o the_o spring_n be_v suppose_v bend_v ten_o space_n the_o velocity_n at_o the_o end_n of_o the_o first_o space_n return_v shall_v be_v as_o the_o root_n of_o 19_o at_o the_o end_n of_o the_o second_o as_o the_o root_n of_o 36._o that_o be_v of_o 19+17_n at_o the_o end_n of_o the_o three_o as_o the_o root_n of_o 51._o that_o be_v of_o 19+17_n +15._o at_o the_o end_n of_o the_o