Selected quad for the lemma: end_n

Word A	Word B	Word C	Word D	Occurrence	Frequency Band	MI	MI Band	Prominent
end_n	angle_n	draw_v	line_n	2,677	5	9.7995	5	true

Text snippets containing the quad

ID	Title	Author	Corrected Date of Publication (TCP Date of Publication)	STC	Words	Pages
A44322	Lectures de potentia restitutiva, or, Of spring explaining the power of springing bodies : to which are added some collections viz. a description of Dr. Pappins wind-fountain and force-pump, Mr. Young's observation concerning natural fountains, some other considerations concerning that subject, Captain Sturmy's remarks of a subterraneous cave and cistern, Mr. G.T. observations made on the Pike of Teneriff, 1674, some reflections and conjectures occasioned thereupon, a relation of a late eruption in the Isle of Palma / by Robert Hooke ...	Hooke, Robert, 1635-1703.; Papin, Denis, 1647-1714.; Young, James.; Sturmy, Samuel, 1633-1669.; G. T.	1678 (1678)	Wing H2619; ESTC R38967	35,527	58

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_o +_o 17._o at_o the_o end_n of_o the_o three_o as_o the_o the_o root_n of_o 51._o that_o be_v of_o 19_o +_o 17_o +_o 15._o at_o the_o end_n of_o the_o four_o as_o the_o root_n of_o 64._o that_o be_v of_o 19_o +_o 17_o +_o 15_o +_o 13._o 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 √_o 19_o +_o 17_o +_o 15_o +_o 13_o +_o 11_o +_o 9_o +_o 7_o +_o 5_o +_o 3_o +_o 1_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 abc_n 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 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d d_o d_o and_o let_v cd_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 dam_fw-ge and_o from_o any_o point_n of_o the_o line_n ac_fw-la as_o bb._n draw_v line_n parallel_v to_o cd_o cut_v the_o line_n da_fw-la in_o e_o e_o the_o line_n be_v be_v 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 be_v be_v cd_o will_v 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 ab_fw-la ab_fw-la ac_fw-la 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 ac_fw-la a_o particular_a power_n therefore_o imagine_v infinite_a line_n draw_v from_o every_o point_n of_o ac_fw-la parallel_n to_o cd_o till_o they_o touch_v the_o line_n ad_fw-la they_o will_v all_o of_o they_o fill_v and_o compose_v the_o triangle_n acd_v the_o triangle_n therefore_o acd_v will_v 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 abe_n abe_n will_v represent_v the_o aggregate_v of_o all_o the_o power_n 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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 ac_fw-la 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 cggf_n and_o ordinates_n draw_v from_o any_o point_n of_o ca_n 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 bg_n bg_n these_o line_n bg_n bg_n 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 cdeb_fw-mi cdeb_fw-mi for_o put_v ac_fw-la =_o to_o a_o and_o ab_fw-la =_o b_o bg_n will_v always_o be_v equal_a to_o √_o aa_o bb_n 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 bg_n bg_n of_o to_o find_v the_o time_n correspond_v on_o the_o diameter_n ac_fw-la draw_v a_o parabola_fw-la chf_n 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 bh_n bh_n of_o be_v in_o the_o same_o proportion_n with_o the_o root_n of_o the_o space_n cb_n cb_n ca_n then_o make_v gb_n to_o hb_v as_o hb_v to_o ib_n and_o through_o the_o point_n ciif_v draw_v the_o curve_v ciiif_fw-mi 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 cb_n cb_n ca._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 cdea_n than_o the_o velocity_n will_v be_v the_o ordinates_n in_o a_o ellipse_n as_o c_o 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d n_o 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 of_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 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d ee+a_fw-mi represent_v the_o power_n and_o c_o 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d o_o the_o ellipsis_n of_o velocity_n who_o ordinates_n b_o 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d b_o 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d ao_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 b2_n and_o from_o thence_o let_v go_v b2ea_n be_v the_o triangle_n of_o its_o power_n the_o ordinates_n of_o the_o circle_n bgl_n 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 bif_n be_v the_o line_n 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 line_n of_o the_o motion_n of_o the_o body_n so_o shoot_v shall_v be_v move_v in_o a_o parabola_fw-la or_o a_o line_n very_o near_o it_o i_o say_v very_o near_o it_o by_o reason_n that_o the_o impediment_n of_o the_o medium_n do_v hinder_v the_o exactness_n of_o it_o if_o it_o be_v deliver_v with_o treble_a quadruple_a quintuple_a sextuple_a etc._n etc._n the_o first_o velocity_n it_o shall_v touch_v the_o floor_n at_o almost_o treble_a quadruple_a quintuple_a sextuple_a etc._n etc._n the_o first_o distance_n i_o shall_v not_o need_v to_o show_v the_o reason_n why_o it_o be_v move_v in_o a_o parabola_fw-la it_o have_v be_v sufficient_o demonstrate_v long_o since_o by_o many_o other_o if_o the_o body_n be_v deliver_v by_o the_o spring_n at_o the_o floor_n but_o shoot_v by_o some_o angle_n upward_o know_v withwhat_o velocity_n the_o same_o be_v move_v when_o deliver_v and_o with_o what_o inclination_n to_o the_o perpendicular_a the_o same_o be_v direct_v and_o the_o true_a velocity_n of_o a_o fall_a body_n you_o may_v easy_o know_v the_o length_n of_o the_o jactus_fw-la or_o shot_n and_o the_o time_n it_o will_v spend_v in_o pass_v that_o length_n this_o be_v find_v by_o compare_v the_o time_n of_o its_o ascent_n with_o the_o time_n of_o the_o descent_n of_o heavy_a body_n the_o ascent_n of_o any_o body_n be_v easy_o know_v by_o compare_v its_o velocity_n with_o the_o angle_n of_o inclination_n let_v ab_fw-la then_o in_o the_o five_o figure_n represent_v 16_o foot_n or_o the_o space_n descend_v by_o a_o heavy_a body_n in_o a_o second_o minute_n of_o time_n if_o a_o body_n be_v shoot_v from_o b_o in_o the_o line_n bf_n with_o a_o velocity_n as_o much_o swift_a than_o that_o equal_a motion_n of_o 16_o foot_n in_o a_o second_o as_o this_o line_n bf_o be_v long_o than_o ab_fw-la the_o body_n shall_v fall_v at_o e_o for_o in_o the_o same_o space_n of_o time_n that_o the_o oblique_a equal_a motion_n will_v make_v it_o ascend_v from_o bd_v to_o ac_fw-la will_v the_o accelerate_a direct_a motion_n downward_o move_v it_o from_o ac_fw-la to_o bd_v and_o therefore_o at_o the_o end_n of_o the_o space_n of_o one_o second_o when_o the_o motion_n do_v equal_a and_o balance_v each_o other_o the_o body_n must_v be_v in_o the_o same_o horizontal_n line_n in_o which_o it_o be_v at_o first_o but_o remove_v asunder_o by_o the_o space_n be_v and_o for_o the_o point_v it_o pass_v through_o in_o all_o the_o intermediate_v space_n this_o method_n will_v determine_v it_o let_v the_o parallelogram_n abpq_fw-fr then_o represent_v the_o whole_a velocity_n of_o the_o ascent_n of_o a_o body_n by_o a_o equal_a motion_n of_o 16_o foot_n in_o a_o second_o and_o the_o triangle_n pqr_v represent_v the_o whole_a velocity_n of_o the_o accelerate_a descend_v motion_n pb_n be_v then_o the_o velocity_n with_o which_o the_o body_n be_v shoot_v and_o p_o be_v the_o point_n of_o rest_n where_o the_o power_n of_o gravity_n begin_v to_o work_v on_o the_o body_n and_o make_v it_o descend_v now_o draw_v line_n parallel_v to_o aqr_n as_o s_z tu_fw-la sy_n give_v the_o velocity_n of_o the_o point_n to_o ascend_v and_o tu_o the_o velocity_n of_o the_o same_o point_n to_o descend_v again_o pb_v signify_v the_o space_n ascend_v and_o ptu_fw-la the_o space_n descend_v so_o that_o subtract_v the_o descent_n from_o the_o ascent_n you_o have_v the_o height_n above_o the_o line_n bd_v the_o consideration_n of_o this_o and_o the_o equal_a progress_n forward_o will_v give_v the_o intermediate_v velocity_n and_o determine_v the_o point_n of_o the_o parabola_fw-la now_o have_v the_o jactus_fw-la give_v by_o this_o scheme_n or_o scale_n appropriate_v to_o the_o particular_a velocity_n wherewith_o any_o body_n be_v move_v in_o this_o or_o that_o line_n of_o inclination_n it_o will_v be_v easy_a to_o find_v what_o velocity_n in_o any_o inclination_n will_v throw_v it_o to_o any_o length_n for_o in_o any_o inclination_n as_o the_o square_n of_o the_o velocity_n thus_o find_v in_o this_o scale_n for_o any_o inclination_n be_v to_o the_o square_n of_o any_o other_o velocity_n so_o be_v the_o distance_n find_v by_o this_o scale_n to_o the_o distance_n answer_v to_o the_o second_o velocity_n i_o have_v not_o now_o time_n to_o enlarge_v upon_o this_o speculation_n which_o will_v afford_v matter_n enough_o to_o fill_v a_o volume_n by_o which_o all_o the_o difficulty_n about_o impress_v and_o receive_v motion_n and_o the_o velocity_n and_o effect_n result_v will_v be_v easy_o resolve_v nor_o have_v i_o now_o time_n to_o mention_v the_o great_a number_n of_o use_n that_o be_v and_o may_v be_v make_v of_o spring_n in_o mechanic_n contrivance_n but_o shall_v only_o add_v that_o of_o all_o springy_a body_n there_o be_v none_o comparable_a to_o the_o air_n for_o the_o vastness_n of_o its_o power_n of_o extension_n and_o contraction_n upon_o this_o principle_n i_o remember_v to_o have_v see_v long_o since_o in_o wadham_n college_n in_o the_o garden_n of_o the_o learned_a dr._n wilkins_n late_a bishop_n of_o chester_n a_o fountain_n so_o contrive_v as_o by_o the_o spring_n of_o the_o include_v air_n to_o throw_v up_o to_o a_o great_a height_n a_o large_a and_o last_a stream_n of_o water_n which_o water_n be_v first_o force_v into_o the_o leaden_a cistern_n thereof_o by_o two_o force_n pump_n which_o do_v alternate_o work_v and_o so_o condense_a the_o air_n include_v into_o a_o small_a room_n the_o contrivance_n of_o which_o engine_n be_v not_o unknown_a to_o the_o ancient_n as_o hero_n in_o his_o spiritalia_fw-la do_v sufficient_o manifest_a nor_o be_v they_o want_v in_o apply_v it_o to_o very_o good_a use_n namely_o for_o engine_n for_o quench_a fire_n as_o vitruvius_n by_o the_o help_n of_o the_o ingenious_a mounseur_fw-fr claude_n perrault_n interpretation_n have_v acquaint_v we_o in_o the_o twelve_o chapter_n of_o his_o ten_o book_n where_o he_o endeavour_v to_o describe_v ctesibius_n his_o engine_n for_o quench_v fire_n not_o long_o since_o a_o german_a here_o in_o england_n have_v add_v a_o further_a improvement_n thereof_o by_o convey_v the_o constant_a stream_n of_o water_n through_o pipe_n make_v of_o well_o tan_a and_o liquor_v leather_n join_v together_o to_o any_o convenient_a length_n by_o the_o help_n of_o brazen_a screw_n by_o which_o the_o stream_n of_o water_n may_v be_v convey_v to_o any_o convenient_a place_n through_o narrow_a and_o otherwise_o inaccessible_a passage_n the_o ingenious_a dr._n denys_n pappin_n have_v add_v a_o further_a improvement_n that_o 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