Selected quad for the lemma: end_n

Word A	Word B	Word C	Word D	Occurrence	Frequency Band	MI	MI Band	Prominent
end_n	draw_v	line_n	perpendicular_a	3,095	5	14.0786	5	true

Text snippets containing the quad

ID	Title	Author	Corrected Date of Publication (TCP Date of Publication)	STC	Words	Pages
A48262	Mathematicall recreations. Or, A collection of many problemes, extracted out of the ancient and modern philosophers as secrets and experiments in arithmetick, geometry, cosmographie, horologiographie, astronomie, navigation, musick, opticks, architecture, statick, mechanicks, chemistry, water-works, fire-works, &c. Not vulgarly manifest till now. Written first in Greeke and Latin, lately compi'ld in French, by Henry Van Etten, and now in English, with the examinations and augmentations of divers modern mathematicians whereunto is added the description and use of the generall horologicall ring: and the double horizontall diall. Invented and written by William Oughtred.; Récréation mathématique. English.	Oughtred, William, 1575-1660. aut	1653 (1653)	Wing L1790; ESTC R217635	140,825	339

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altitude_n be_v equal_a to_o his_o distance_n from_o any_o of_o the_o equinoctial_a point_n then_o t●e_v sun_n be_v due_a east_n or_o west_n 4_o if_o a_o ship_n be_v between_o the_o equinoctial_a and_o either_o of_o the_o tropic_n the_o sun_n will_v come_v twice_o to_o one_o point_n of_o the_o compass_n in_o the_o forenoon_n that_o be_v in_o one_o and_o the_o same_o position_n 5_o under_o the_o equinoctial_a near_o guinea_n there_o be_v but_o two_o sort_n of_o wind_n all_o the_o year_n 6_o month_n a_o northerly_a wind_n and_o 6_o month_n a_o southerly_a wind_n and_o the_o flux_n of_o the_o sea_n be_v according_o 6_o if_o two_o ship_n under_o the_o equinoctial_a be_v 100_o league_n asunder_o and_o shall_v sail_v northerly_a until_o they_o be_v come_v under_o the_o arctic_a circle_n they_o shall_v then_o be_v but_o 50_o league_n asunder_o 7_o those_o which_o have_v the_o arctic_a circle_n vertical_a when_o the_o sun_n be_v in_o the_o tropic_a of_o cancer_n the_o sun_n set_v not_o but_o 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perpendicular_a line_n i_o a._n augment_v it_o to_o h._n then_o from_o a._n draw_v line_n at_o pleasure_n to_o intersect_v the_o line_n 1._o m._n in_o each_o of_o which_o line_n from_o the_o right_a line_n in_o transfer_v ih_o viz._n kb_n lc.od.pe.qf.mg_n then_o from_o those_o point_n draw_v the_o line_n h.b.c.d.e.f.g._n which_o will_v not_o meet_v with_o the_o line_n in_o and_o yet_o incline_v near_o and_o near_o unto_o it_o problem_n c._n to_o observe_v the_o variation_n of_o the_o compass_n or_o needle_n in_o any_o place_n first_o describe_v a_o circle_n upon_o a_o plain_a so_o that_o the_o sun_n may_v shine_v on_o it_o both_o before_o noon_n and_o afternoon_n in_o the_o centre_n of_o which_o circle_n place_n a_o gn●●on_n or_o wire_n perpendicular_a

&_o a_o line_n strike_v perpendicular_a upon_o it_o apply_v the_o straight_a edge_n unto_o the_o wall_n at_o what_o time_n the_o sun_n shine_v upon_o it_o hold_v the_o board_n parallel_n to_o the_o horizon_n set_v the_o dyal_a thereon_o and_o move_v it_o gentle_o every_o way_n until_o the_o same_o hour_n and_o minute_n be_v show_v in_o both_o dial_n and_o so_o let_v it_o stand_v then_o if_o the_o dyal_a have_v one_o of_o the_o side_n parallel_v to_o the_o meridian_n strike_v a_o line_n along_o that_o side_n upon_o the_o board_n cross_v the_o perpendicular_a or_o else_o with_o a_o bodkin_n make_v a_o point_n upon_o the_o board_n at_o each_o end_n of_o the_o meridian_n and_o take_v away_o the_o instrument_n from_o the_o board_n and_o the_o board_n from_o the_o wall_n lay_v a_o ruler_n to_o those_o two_o point_n and_o draw_v a_o line_n cross_v the_o perpendicular_a for_o the_o angle_n which_o that_o line_n make_v with_o the_o perpendicular_a be_v the_o angle_n of_o the_o decli●nation_n 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a_o circle_n equal_a to_o the_o horizon_n of_o your_o instrument_n in_o which_o circle_n you_o shall_v from_o the_o line_n through_o the_o two_o prick_n measure_v the_o horizontal_n distance_n of_o the_o upright_a shadow_n or_o azumith_n from_o the_o meridian_n that_o way_n towards_o which_o the_o meridian_n be_v draw_v a_o line_n out_o of_o the_o centre_n to_o the_o end_n of_o that_o arch_n measure_v and_o the_o angle_n which_o this_o last_o line_n make_v with_o the_o perpendicular_a shall_v be_v equal_a to_o the_o declination_n of_o the_o wall_n xiii_o use_n how_o to_o place_v the_o dial_n upon_o a_o post_n without_o any_o other_o direction_n but_o itself_o set_a the_o dial_n upon_o the_o post_n with_o the_o stile_n into_o the_o north_n as_o near_o as_o you_o can_v guess_v then_o move_v it_o this_o way_n and_o that_o way_n till_o the_o same_o hour_n and_o minute_n be_v show_v both_o in_o the_o outward_a and_o inward_a dial_n by_o the_o several_a shadow_n as_o have_v be_v already_o teach_v for_o then_o the_o dial_n stand_v in_o its_o true_a situation_n wherefore_o let_v it_o be_v nail_v down_o in_o that_o very_a place_n xiiii_o use_n to_o find_v the_o height_n of_o the_o sun_n at_o high_a noon_n everyday_n seek_v out_o the_o diurnal_a arch_n or_o parallel_n of_o the_o sun_n course_n for_o that_o day_n by_o use_n iii_o and_o with_o a_o pair_n of_o compass_n set_v one_o foot_n in_o the_o centre_n and_o the_o other_o in_o the_o point_n of_o intersection_n of_o that_o parallel_n with_o the_o meridian_n apply_v that_o same_o distance_n unto_o the_o semidiameter_n divide_v for_o that_o measure_n shall_v therein_o show_v the_o degree_n of_o of_o the_o sun_n altitude_n above_o the_o the_o horizon_n that_o day_n at_o high_a noon_n xv_o use_n to_o find_v the_o height_n of_o the_o sun_n at_o any_o hour_n or_o time_n of_o the_o day_n seek_v out_o the_o diurnal_a arch_n or_o parallel_v of_o the_o sun_n course_n for_o that_o day_n and_o mark_v what_o point_n of_o it_o be_v 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the_o centre_n and_o the_o other_o at_o that_o height_n apply_v the_o same_o distance_n unto_o the_o diurnal_a arch_n or_o parallel_v of_o the_o sun_n for_o that_o day_n for_o that_o point_n of_o the_o diurnal_a arch_n upon_o which_o that_o same_o distance_n light_n be_v the_o true_a place_n of_o the_o sun_n upon_o the_o dial_n and_o show_v among_o the_o hour_n line_n the_o true_a time_n of_o the_o day_n xvii_o use._n consideration_n for_o the_o use_n of_o the_o instrument_n in_o the_o night_n in_o such_o question_n as_o concern_v the_o night_n or_o the_o time_n before_o sun_n rise_v and_o after_o sun_n set_v the_o instrument_n represent_v the_o low_a hemisphere_n wherein_o the_o southern_a pole_n be_v elevate_v and_o therefore_o the_o parallel_n which_o be_v above_o the_o aequinoctial_a towards_o the_o centre_n shall_v be_v for_o the_o southern_a or_o winter_n parallel_n and_o those_o beneath_o the_o aequinoctial_a for_o the_o northern_a or_o summer_n parallel_n and_o the_o east_n 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of_o the_o c●epusculum_n or_o twilight_n every_o day_n seek_v the_o declination_n of_o the_o sun_n for_o the_o day_n propose_v by_o use_v ii_o and_o at_o the_o same_o declination_n on_o the_o contrary_a side_n imagine_v a_o parallel_n for_o the_o sun_n that_o night_n and_o with_o a_o pair_n of_o compass_n set_v one_o foot_n in_o the_o centre_n and_o the_o other_o at_o 72_o degree_n upon_o the_o semidiameter_n divide_v apply_v that_o same_o distance_n unto_o the_o sun_n nocturnal_a parallel_n for_o that_o point_n of_o the_o parallel_n upon_o which_o that_o same_o distance_n shall_v light_v sheweth_z among_o the_o hour_n line_n the_o beginning_n of_o the_o twilight_n in_o the_o morning_n or_o the_o end_n of_o the_o twilight_n in_o the_o evening_n xx_o use._n if_o the_o day_n of_o the_o month_n be_v not_o know_v to_o find_v it_o out_o by_o the_o dial_n for_o the_o work_n of_o this_o question_n either_o the_o dial_n must_v be_v fix_v right_o on_o a_o post_n or_o else_o you_o must_v have_v a_o true_a meridian_n line_n draw_v in_o some_o window_n where_o the_o sun_n shine_v wherefore_o suppose_v the_o dial_n to_o be_v just_o set_v either_o upon_o the_o post_n or_o upon_o the_o meridian_n look_v what_o a_o clock_n it_o be_v by_o the_o outward_a dial_n and_o observe_v what_o point_n of_o the_o upright_a shadow_n fall_v upon_o the_o very_a same_o minute_n in_o the_o inner_a dial_n and_o through_o that_o same_o point_n imagine_v a_o parallel_n circle_n for_o the_o sun_n course_n that_o imaginary_a circle_n in_o the_o ecliptic_a shall_v cut_v the_o day_n of_o the_o month_n i_o the_o description_n of_o it_o this_o instrument_n serve_v as_o a_o dial_n to_o find_v the_o hour_n of_o the_o day_n not_o in_o one_o place_n only_o as_o the_o most_o part_n of_o dial_n do_v but_o general_o in_o all_o country_n lie_a north_n of_o the_o aequinoctial_a and_o therefore_o i_o call_v it_o the_o general_a h●rologicall_a ●ing_n it_o consist_v of_o two_o brazen_a circle_n a_o diameter_n and_o a_o little_a ring_n to_o hang_v it_o by_o the_o two_o circle_n be_v so_o make_v that_o though_o they_o be_v to_o be_v set_v at_o right_a angle_n when_o you_o use_v the_o instrument_n yet_o for_o more_o convenient_a carry_v they_o may_v be_v one_o fold_v into_o the_o other_o the_o lesser_a of_o the_o two_o circle_n be_v for_o the_o aequinoctial_a have_v in_o the_o midst_n of_o the_o inner_a side_n or_o thickness_n a_o line_n round_o it_o which_o be_v the_o true_a aequinoctial_a circle_n divide_v into_o twice_o twelve_o hour_n from_o the_o two_o opposite_a point_n in_o which_o it_o be_v fasten_v within_o the_o great_a the_o great_a and_o outer_a of_o the_o two_o circle_n be_v the_o meridian_n one_o quarter_n whereof_o beginning_n at_o one_o of_o the_o point_n in_o which_o the_o equinoctial_a be_v hang_v be_v divide_v into_o ninety_o degree_n the_o diameter_n be_v fasten_v to_o the_o meridian_n in_o two_o opposite_a point_n or_o pole_n o●e_v of_o they_o be_v the_o very_a end_n of_o the_o quadrant_n and_o be_v the_o north_n pole_n wherefore_o it_o be_v perpendicular_a to_o the_o equinoctial_a have_v his_o due_a position_n the_o diameter_n be_v broad_a and_o slit_v in_o the_o middle_n and_o about_o the_o slit_n on_o both_o side_n be_v the_o month_n and_o day_n of_o the_o year_n and_o within_o this_o slit_n be_v a_o little_a slide_a plate_n pierce_v through_o with_o a_o small_a hole_n which_o hole_n in_o the_o motion_n of_o it_o while_o it_o be_v apply_v to_o the_o day_n of_o the_o year_n represent_v the_o axis_n of_o the_o world_n the_o little_a ring_n whereby_o the_o instrument_n hang_v be_v make_v to_o slip_v up_o and_o down_o along_o the_o quadrant_n that_o so_o by_o help_n of_o a_o little_a tooth_n annex_v the_o instrument_n may_v be_v rectify_v to_o any_o elevation_n of_o the_o pole_n ii_o the_o use_n of_o it_o in_o use_v this_o instrument_n first_o the_o tooth_n of_o the_o little_a ring_n must_v careful_o be_v set_v to_o the_o height_n of_o the_o pole_n in_o the_o quadrant_n for_o the_o place_n wherein_o you_o be_v second_o the_o hole_n of_o the_o slide_a plate_n within_o the_o slit_n must_v be_v bring_v exact_o unto_o the_o day_n of_o the_o month_n three_o the_o aeqinoctiall_a be_v to_o be_v draw_v out_o and_o by_o mean_n of_o the_o two_o stud_n in_o the_o meridian_n stay_v it_o it_o be_v to_o be_v set_v perpendicular_a thereto_o four_o guess_v as_o near_o as_o you_o can_v at_o the_o hour_n and_o turn_v the_o hole_n of_o the_o little_a plate_n towards_o it_o last_o hold_v the_o instrument_n up_o by_o the_o little_a ring_n that_o it_o may_v hang_v free_o with_o the_o north_n pole_n thereof_o towards_o the_o north_n and_o move_v it_o gentle_o this_o way_n and_o that_o way_n till_o the_o beam_n of_o the_o sunshining_a through_o that_o hole_n fall_v upon_o that_o middle_a line_n within_o the_o aequinoctial_a for_o there_o shall_v be_v the_o hour_n of_o the_o day_n and_o the_o meridan_n of_o the_o instrument_n shall_v hang_v direct_o north_n and_o south_n these_o instrument_n all_o dial_n be_v make_v in_o brass_n by_o elias_n allen_n dwell_v over_o against_o st._n clement_n church_n without_o temple_n bar_n at_o the_o sign_n of_o the_o horse-shoe_n near_o essex_n gate_n finis_fw-la

full_a so_o in_o c_o there_o be_v now_o but_o 1_o pint_n 5_o in_o b_o and_o 2_o in_o a_o pour_v all_o that_o be_v in_o b_o into_o a_o then_o pour_v the_o wine_n which_o be_v in_o c_o into_o b_o so_o there_o be_v in_o c_o nothing_o in_o b_o only_a 1_o pint_n and_o in_o 7_o a_o 7_o pint_n last_o out_o of_o a_o fill_v the_o pot_n c_o so_fw-mi there_o will_v remain_v in_o a_o 4_o pint_n or_o be_v but_o half_a full_a then_o if_o the_o liquor_n in_o c_o be_v pour_v into_o b_o it_o will_v be_v the_o other_o half_o in_o like_a manner_n may_v be_v take_v the_o half_a of_o a_o vessel_n which_o contain_v 12_o pint_n by_o have_v but_o the_o measure_n 5_o and_o 7_o or_o 5_o and_o 8._o now_o such_o other_o may_v be_v propose_v but_o we_o omit_v many_o in_o one_o and_o the_o same_o nature_n problem_n x._o to_o make_v a_o stick_n stand_v upon_o the_o tip_n of_o one_o finger_n without_o fall_v fasten_v the_o edge_n of_o two_o knife_n or_o such_o like_a of_o equal_a poise_n at_o the_o end_n of_o the_o stick_n lean_v out_o somewhat_o from_o the_o stick_n so_o that_o they_o may_v counterpoise_v one_o another_o the_o stick_n be_v sharp_a at_o the_o end_n and_o hold_v upon_o the_o top_n of_o the_o finger_n will_v there_o rest_v without_o support_v if_o it_o fall_v it_o must_v fall_v together_o and_o that_o perpendicular_a or_o plumb-wise_a or_o it_o must_v fall_v side-wise_a or_o before_o one_o another_o in_o the_o first_o manner_n it_o can_v for_o the_o centre_n of_o gravity_n be_v support_v by_o the_o top_n of_o the_o finger_n and_o see_v that_o each_o part_n by_o the_o knife_n be_v counterpoise_v it_o can_v fall_v sidewise_o therefore_o it_o can_v fall_v no_o wise_a in_o like_a manner_n may_v great_a piece_n of_o timber_n as_o joist_n &c_n &c_n be_v support_v if_o unto_o one_o of_o the_o end_n be_v apply_v convenient_a proportional_a counterpoise_n yea_o a_o lance_n or_o pike_n may_v stand_v perpendicular_a in_o the_o air_n upon_o the_o top_n of_o one_o finger_n or_o place_v in_o the_o midst_n of_o a_o court_n by_o help_n of_o his_o centre_n of_o gravity_n examination_n this_o proposition_n seem_v doubtful_a for_o to_o imagine_v absolute_o that_o a_o pike_n or_o such_o like_a arm_v with_o two_o knife_n or_o other_o thing_n shall_v stand_v upright_o in_o the_o air_n and_o so_o remain_v without_o any_o other_o support_n see_v that_o all_o the_o part_n have_v a_o infinite_a difference_n of_o propensity_n to_o fall_v and_o it_o be_v without_o question_n that_o a_o staff_n so_o accommodate_v upon_o his_o centre_n of_o gravity_n but_o that_o it_o may_v incline_v to_o some_o one_o part_n without_o some_o remedy_n be_v apply_v and_o such_o as_o be_v here_o specify_v in_o the_o problem_n will_v not_o warrant_v the_o thing_n nor_o keep_v it_o from_o fall_v and_o if_o more_o knife_n shall_v be_v place_v about_o it_o it_o shall_v cause_v it_o to_o fall_v more_o swift_o forasmuch_o as_o the_o superior_a part_n by_o reason_n of_o the_o centrical_a motion_n be_v make_v more_o ponderous_a and_o therefore_o less_o in_o rest_n to_o place_v therefore_o this_o prop_n real_o let_v the_o two_o knife_n or_o that_o which_o be_v for_o counterpoise_v be_v long_o always_o then_o the_o staff_n and_o so_o it_o will_v hang_v together_o as_o one_o body_n and_o it_o will_v appear_v admirable_a if_o you_o place_v the_o centre_n of_o gravity_n near_o the_o side_n of_o the_o top_n of_o the_o finger_n or_o point_n for_o it_o will_v then_o hang_v horizontall_a and_o seem_v to_o hang_v only_o by_o a_o touch_n yet_o more_o strange_a if_o you_o turn_v the_o point_n or_o top_n of_o the_o finger_n upside_o down_o problem_n xi_o how_o a_o millstone_n or_o other_o ponderosity_n may_v be_v support_v by_o a_o small_a needle_n without_o break_v o●_n any_o wise_a bow_v the_o same_o let_v a_o needle_n be_v set_v perpendicular_a to_o the_o horizon_n and_o the_o centre_n of_o gravity_n of_o the_o stone_n be_v place_v on_o the_o top_n of_o the_o needle_n it_o be_v evident_a that_o the_o stone_n can_v fall_v forasmuch_o as_o it_o hang_v in_o aequilibra_n or_o be_v counterpoyse_v in_o all_o part_n alike_o and_o moreover_o it_o can_v bow_v the_o needle_n more_o on_o the_o one_o side_n then_o on_o the_o other_o the_o needle_n will_v not_o therefore_o be_v either_o break_a or_o bow_v if_o otherwise_o then_o the_o part_n of_o the_o needle_n must_v penetrate_v and_o sink_v one_o with_o another_o that_o which_o be_v absurd_a and_o impossible_a to_o nature_n therefore_o it_o shall_v be_v support_v the_o experiment_n which_o be_v make_v upon_o trencher_n plate_n or_o such_o like_o lesser_a thing_n do_v make_v it_o most_o credible_a in_o great_a body_n but_o here_o especial_o be_v to_o be_v note_v that_o the_o needle_n ought_v to_o be_v uniform_a in_o matter_n and_o figure_n and_o that_o it_o be_v erect_v perpendicular_a to_o the_o horizon_n and_o last_o that_o the_o centre_n of_o gravity_n be_v exact_o find_v problem_n xii_o to_o make_v three_o knife_n hang_v and_o move_v upon_o the_o point_n of_o a_o needle_n fit_a the_o three_o knife_n in_o form_n of_o a_o balance_n and_o hold_v a_o needle_n in_o your_o hand_n and_o place_v the_o back_n of_o that_o knife_n which_o lie_v crosswise_o to_o the_o other_o two_o upon_o the_o point_n of_o the_o needle_n as_o the_o figure_n here_o show_v you_o for_o then_o in_o blow_v soft_o upon_o they_o they_o will_v easy_o turn_v and_o move_v upon_o the_o point_n of_o the_o needle_n with_o ●ou_n fall_v problem_n xiii_o to_o find_v the_o weight_n of_o smoke_n which_o be_v exhale_v of_o any_o combustible_a body_n whatsoever_o let_v it_o be_v suppose_v that_o a_o great_a heap_n of_o faggot_n or_o a_o load_n of_o straw_n weigh_v 500_o pound_n shall_v be_v fire_v it_o be_v evident_a that_o this_o gross_a substance_n will_v be_v all_o invert_v into_o smoke_n and_o ash_n now_o it_o seem_v that_o the_o smoke_n weigh_v nothing_o see_v it_o be_v of_o a_o thin_a substance_n now_o dilate_v in_o the_o air_n notwithstanding_o if_o it_o be_v gather_v and_o reduce_v into_o the_o thick_a that_o it_o be_v at_o first_o it_o will_v be_v sensible_o weighty_a weigh_v therefore_o the_o ash_n which_o admit_v 50_o pound_n now_o see_v that_o the_o rest_n of_o the_o matter_n be_v not_o lose_v but_o be_v exhale_v into_o smoke_n it_o must_v necessary_o be_v that_o the_o rest_n of_o the_o weight_n to_o wit_n 450_o pound_n must_v be_v the_o weight_n of_o the_o smoke_n require_v examination_n now_o although_o it_o be_v thus_o deliver_v yet_o here_o may_v be_v note_v that_o a_o ponderosity_n in_o his_o own_o medium_n be_v not_o weighty_a for_o thing_n be_v say_v to_o be_v weighty_a when_o they_o be_v out_o of_o their_o place_n or_o medium_n and_o the_o difference_n of_o such_o gravity_n be_v according_a to_o the_o motion_n the_o smoke_n therefore_o certain_o be_v light_n be_v in_o its_o true_a medium_n the_o air_n if_o it_o shall_v change_v his_o medium_n then_o will_v we_o change_v our_o discourse_n problem_n xvi_o many_o thing_n be_v dispose_v circular_a or_o otherwise_o to_o find_v which_o of_o they_o any_o one_o think_v upon_o suppose_v that_o have_v rank_v 10_o thing_n as_o abcdefghik_n circular_a as_o the_o figure_n show_v and_o that_o one_o have_v touch_v or_o think_v upon_o g_o which_o be_v the_o 7_o ask_v the_o party_n at_o what_o letter_n he_o will_v begin_v to_o account_v for_o account_n he_o must_v otherwise_o it_o can_v be_v do_v which_o suppose_v at_o e_z which_o be_v the_o 5_o place_n then_o add_v secret_o to_o this_o 5_o 10_o which_o be_v the_o number_n of_o the_o circle_n and_o it_o make_v 15_o bid_v he_o account_v 15_o backward_a from_o e_z beginning_z his_o account_n with_o that_o number_n he_o think_v upon_o so_o at_o e_z he_o shall_v account_v to_o himself_o 7_o at_z d_o account_n 8_o at_z c_o account_n 9_o etc._n etc._n so_o the_o account_n of_o 15_o will_v exact_o fall_v upon_o g_o the_o thing_n or_o number_n think_v upon_o and_o so_o of_o other_o but_o to_o conceal_v it_o the_o more_o you_o may_v will_v the_o party_n from_o e_z to_o account_v 25_o 35_o etc._n etc._n and_o it_o will_v be_v the_o same_o there_o be_v some_o that_o use_v this_o play_n at_o card_n turn_v upside_o down_o as_o the_o ten_o simple_a card_n with_o the_o king_n and_o queen_n the_o king_n stand_v for_o 12_o and_o the_o queen_n for_o 11_o and_o so_o know_v the_o situation_n of_o the_o card_n and_o think_v a_o certain_a hour_n of_o the_o day_n cause_n the_o party_n to_o account_v from_o what_o card_n he_o please_v with_o this_o proviso_n that_o when_o you_o see_v where_o he_o intend_v to_o account_v set_v 12_o to_o that_o number_n so_o in_o count_v as_o

here_o there_o must_v be_v find_v a_o number_n which_o multiply_v by_o 7_o and_o then_o divide_v by_o 2_o 3_o 4_o 5_o 6_o there_o may_v always_o remain_v a_o number_n less_o by_o 1_o than_o the_o divisor_n now_o the_o first_o number_n which_o arrive_v in_o this_o nature_n be_v 119_o unto_o which_o if_o 420_o be_v add_v make_v 539_o which_o also_o will_v do_v the_o same_o and_o so_o by_o add_v 420_o you_o may_v have_v other_o number_n to_o resolve_v this_o proposition_n problem_n xlviii_o how_o many_o sort_n of_o weight_n in_o the_o least_o manner_n must_v there_o be_v to_o weigh_v all_o sort_n of_o thing_n between_o 1_o pound_n and_o 40_o pound_n and_o so_o unto_o 121_o &_o 364_o pound_n to_o weigh_v thing_n between_v 1_o and_o 40_o take_v number_n in_o triple_a proportion_n so_o that_o their_o sum_n be_v equal_a or_o somewhat_o great_a than_o 40_o as_o be_v the_o number_n 1_o 3.9.27_o i_o say_v that_o with_o ●_o such_o weight_n the_o first_o be_v of_o 1_o pound_n the_o second_o be_v 3_o pound_n the_o three_o be_v 9_o pound_n and_o the_o four_o be_v 27_o any_o weight_n between_o 1_o and_o 40_o pound_n may_v be_v weigh_v as_o admit_v to_o weigh_v 21_o pound_n put_v unto_o the_o thing_n that_o be_v to_o be_v weigh_v the_o 9_o pound_n weight_n then_o in_o the_o other_o balance_n put_v 27_o pound_n and_o 3_o pound_n which_o do_v counterpoise_v 21_o pound_n and_o 9_o pound_n and_o if_o 20_o pound_n be_v to_o be_v weigh_v put_v to_o it_o in_o the_o balance_n 9_o and_o 1_o and_o in_o the_o other_o balance_n put_v 27_o and_o 3_o and_o so_o of_o other_o in_o the_o same_o manner_n take_v those_o 5_o weight_n 1_o 3_o 9_o 27_o 81_o you_o may_v weigh_v with_o they_o between_o 1_o pound_n and_o 121_o pound_n and_o take_v those_o 6_o weight_n as_o 1_o 3_o 9_o 2d_o 81_o 243_o you_o may_v weigh_v even_o from_o 1_o pound_n unto_o 364_o pound_n this_o depend_v upon_o the_o property_n of_o continue_a proportional_n the_o latter_a of_o which_o contain_v twice_o all_o the_o former_a problem_n xlix_o of_o a_o deceitful_a balance_n which_o be_v c●●●ty_n seem_v i●_n be_v just_a because_o it_o hang_v in_o aequilibrio_fw-la notwithstanding_o put_v 12_o pound_n in_o one_o balance_n and_o 11_o in_o the_o other_o it_o will_v remain_v in_o aequilibrio_fw-la aristotle_n make_v mention_n of_o this_o balance_n in_o his_o mechanic_n question_n and_o say_v that_o the_o merchant_n of_o purpose_n in_o his_o time_n use_v they_o to_o deceive_v the_o world_n the_o subtlety_n or_o craft_n of_o which_o be_v thus_o that_o one_o arm_n of_o the_o balance_n be_v long_o than_o another_o by_o the_o same_o proportion_n that_o one_o weight_n be_v heavy_a than_o another_o as_o if_o the_o beam_n be_v 23_o inch_n long_o and_o the_o handle_v place_v so_o that_o 12_o inch_n shall_v be_v on_o one_o side_n of_o it_o and_o 11_o inch_n on_o the_o other_o side_n conditional_o that_o the_o short_a end_n shall_v be_v as_o heavy_a as_o the_o long_o a_o thing_n easy_a to_o be_v do_v then_o afterward_o put_v into_o the_o balance_n two_o unequal_a weight_n in_o such_o proportion_n as_o the_o part_n of_o the_o beam_n have_v one_o unto_o another_o which_o be_v 12_o to_o 11_o but_o so_o that_o the_o great_a be_v place_v in_o the_o balance_n which_o hang_v upon_o the_o short_a part_n of_o the_o beam_n and_o the_o lesser_a weight_n in_o the_o other_o balance_n it_o be_v most_o certain_a that_o the_o balance_n will_v hang_v in_o aequilibrio_fw-la which_o will_v seem_v most_o sincere_a and_o just_a though_o it_o be_v most_o deceitful_a abominable_a and_o false_a the_o reason_n of_o this_o be_v draw_v from_o the_o experiment_n of_o archimedes_n who_o show_v that_o two_o unequal_a weight_n will_v counterpoise_v one_o another_o when_o there_o be_v like_a proportion_n between_o the_o part_n of_o the_o beam_n that_o the_o handle_n separate_v and_o the_o weight_n themselves_o for_o in_o one_o and_o the_o same_o counterpoise_n by_o how_o much_o it_o be_v far_o from_o the_o centre_n of_o the_o handle_n by_o so_o much_o it_o seem_v heavy_a therefore_o if_o there_o be_v a_o diversity_n of_o distance_n that_o the_o balance_n hang_v from_o the_o handle_n there_o must_v necessary_o be_v a_o ineqality_n of_o weight_n in_o these_o balance_n to_o make_v they_o hang_v in_o aequilibrio_fw-la and_o to_o discover_v if_o there_o be_v deceit_n change_v the_o weight_n into_o the_o other_o balance_n for_o as_o soon_o as_o the_o great_a weight_n be_v place_v in_o the_o balance_n that_o hang_v on_o the_o long_a part_n of_o the_o beam_n it_o will_v weigh_v down_o the_o other_o instant_o problem_n l._n to_o heave_v or_o lift_v up_o a_o bottle_n with_o a_o straw_n take_v a_o straw_n that_o be_v not_o bruise_v bow_v it_o that_o it_o make_v a_o angle_n and_o put_v it_o into_o the_o bottle_n so_o that_o the_o great_a end_n be_v in_o the_o neck_n than_o the_o reed_n be_v put_v in_o the_o bow_v part_n will_v cast_v side-wise_a and_o make_v a_o angle_n as_o in_o the_o figure_n may_v be_v see_v then_o may_v you_o take_v the_o end_n which_o be_v out_o of_o the_o bottle_n in_o your_o hand_n and_o heave_v up_o the_o bottle_n and_o it_o be_v so_o much_o sure_a by_o how_o much_o the_o angle_n be_v acuter_n or_o sharp_a and_o the_o end_n which_o be_v bow_v approach_v to_o the_o other_o perpendicular_a part_n which_o come_v out_o of_o the_o bottle_n problem_n li._n how_o in_o the_o middle_n of_o a_o wood_n or_o desert_n without_o the_o sight_n of_o the_o sun_n star_n shadow_n or_o compass_n to_o find_v out_o the_o north_n or_o south_n or_o the_o four_o cardinal_n point_n of_o the_o world_n east_n west_n &_o c_o it_o be_v the_o opinion_n of_o some_o that_o the_o wind_n be_v to_o be_v observe_v in_o this_o if_o it_o be_v hot_a the_o south_n be_v find_v by_o the_o wind_n that_o blow_v that_o way_n but_o this_o observation_n be_v uncertain_a and_o subject_a to_o much_o error_n nature_n will_v help_v you_o in_o some_o measure_n to_o make_v it_o more_o manifest_a than_o any_o of_o the_o former_a from_o a_o tree_n thus_o cut_v a_o small_a tree_n off_o even_o to_o the_o ground_n and_o mark_v the_o many_o circle_n that_o be_v about_o the_o sap_n or_o pith_n of_o the_o tree_n which_o seem_v near_o together_o in_o some_o part_n than_o in_o other_o which_o be_v by_o reason_n of_o the_o sun_n motion_n about_o the_o tree_n for_o that_o the_o humidity_n of_o the_o part_n of_o the_o tree_n towards_o the_o south_n by_o the_o heat_n of_o the_o sun_n be_v rarify_v and_o cause_v to_o extend_v and_o the_o s●n_v not_o give_v such_o heat_n towards_o the_o north-part_n of_o the_o tree_n the_o sap_n be_v lesser_a rarefy_v but_o condense_v by_o which_o the_o circle_n be_v near_o together_o on_o the_o north-part_n than_o on_o the_o south-part_n therefore_o if_o a_o line_n be_v draw_v from_o the_o wide_a to_o the_o narrow_a part_n of_o the_o circle_n it_o shall_v show_v the_o north_n &_o south_n of_o the_o world_n another_o experiment_n may_v be_v thus_o take_v a_o small_a needle_n such_o as_o woman_n work_v with_o place_v it_o gentle_o down_o flatwise_o upon_o still_a water_n and_o it_o will_v not_o sink_v which_o be_v against_o the_o general_a tenet_n that_o iron_n will_v not_o swim_v which_o needle_n will_v by_o little_a and_o little_a turn_n to_o the_o north_n and_o south-point_n but_o if_o the_o needle_n be_v great_a and_o will_v not_o swim_v thrust_v it_o through_o a_o small_a piece_n of_o cork_n or_o some_o such_o like_a thing_n and_o then_o it_o will_v do_v the_o same_o for_o such_o be_v the_o property_n of_o iron_n when_o it_o be_v place_v in_o aequilibrio_fw-la it_o strive_v to_o find_v out_o the_o pole_n of_o the_o world_n or_o point_n of_o north_n and_o south_n in_o a_o manner_n as_o the_o magnes_fw-la do_v examination_n here_o be_v observable_a that_o the_o moisture_n which_o aid_v to_o the_o growth_n of_o the_o tree_n be_v dilate_v and_o rarefy_v by_o the_o meridional_a heat_n and_o contract_v by_o the_o septentrional_a cold_a this_o rarefaction_n work_v upon_o the_o part_n of_o the_o humour_n or_o moisture_n that_o be_v more_o thin_a which_o do_v easy_o dissipate_v and_o evaporate_v which_o evaporation_n carry_v a_o part_n of_o the_o salt_n with_o it_o and_o because_o that_o solidation_n or_o condensation_n so_o that_o there_o be_v leave_v but_o a_o part_n of_o the_o nourishment_n which_o the_o heat_n bake_v up_o and_o consume_v so_o contrary_o on_o the_o other_o side_n the_o condensation_n and_o restrictive_a quality_n of_o the_o moisture_n cause_v less_o evaporation_n and_o perdition_n and_o so_o consequent_o there_o remain_v more_o nourishment_n which_o make_v a_o great_a increase_n on_o that_o side_n than_o on_o the_o other_o

a_o picture_n which_o painter_n know_v well_o enough_o to_o do_v afterward_o put_v a_o candle_n upon_o the_o table_n and_o interpose_v this_o figure_n oblique_o between_o the_o say_a candle_n and_o the_o book_n of_o paper_n where_o you_o desire_v to_o have_v the_o figure_n disguise_v in_o such_o sort_n that_o the_o height_n pass_v athwart_o the_o hole_n of_o the_o picture_n then_o will_v it_o carry_v all_o the_o form_n of_o the_o picture_n upon_o the_o paper_n but_o with_o deformity_n follow_v these_o tract_n and_o mark_v out_o the_o light_n with_o a_o coal_n black_a head_n or_o ink_n and_o you_o have_v your_o desire_n to_o find_v now_o the_o point_n where_o the_o eye_n must_v see_v it_o in_o his_o natural_a form_n it_o be_v accustom_v according_a to_o the_o order_n of_o perspective_n to_o place_v this_o point_n in_o the_o line_n draw_v in_o height_n equal_a to_o the_o largeness_n of_o the_o narrow_a side_n of_o the_o deform_a square_n and_o it_o be_v by_o this_o way_n that_o it_o be_v perform_v problem_n cvii_o how_o a_o cannon_n after_o that_o it_o have_v shoot_v may_v be_v cover_v from_o the_o battery_n of_o the_o enemy_n let_v the_o mouth_n of_o a_o cannon_n be_v i_o the_o cannon_n m._n his_o charge_n no_o the_o wheel_n l_o the_o axletree_n pb_n upon_o which_o the_o cannon_n be_v place_v at_o which_o end_n towards_o b_o be_v place_v a_o pillar_n ae·_a support_v with_o prop_n d_o c_o e_o f_o g_o about_o which_o the_o axletree_n turn_v now_o the_o cannon_n be_v to_o shoot_v it_o retire_v to_o h_n which_o can_v be_v direct_o because_o of_o the_o axletree_n but_o it_o make_v a_o segment_n of_o a_o circle_n and_o hide_v himself_o behind_o the_o wall_n qr_n and_o so_o preserve_v itself_o from_o the_o enemy_n battery_n by_o which_o mean_v one_o may_v avoid_v many_o inconvenience_n which_o may_v arise_v and_o moreover_o one_o man_n may_v more_o easy_o replace_v it_o again_o for_o another_o shot_n by_o help_n of_o pole_n tie_v to_o the_o wall_n or_o other_o help_n which_o may_v multiply_v the_o strength_n problem_n cviii_o how_o to_o make_v a_o lever_n by_o which_o one_o man_n may_v alone_o place_v a_o cannon_n upon_o his_o carriage_n or_o raise_v what_o other_o weight_n he_o will_v first_o place_n two_o thick_a board_n upright_o as_o the_o figure_n show_v pierce_v with_o hole_n alike_o opposite_a one_o unto_o another_o as_o cd_o and_o of_o &_o let_v l_o and_o m_o be_v the_o two_o bar_n of_o iron_n which_o pass_v through_o the_o hole_n gh_o and_o f_o k_n the_o two_o support_n or_o prop_n ab_fw-la the_o cannon_n open_v the_o lever_n r_n the_o two_o notch_n in_o the_o lever_n and_o q_o the_o hook_n where_o the_o burden_n or_o cannon_n be_v tie_v to_o the_o rest_n of_o the_o operation_n be_v ●cill_n that_o the_o young_a scholar_n or_o learner_n can_v fail_v to_o perform_v it_o to_o teach_v minerva_n be_v in_o vain_a and_o it_o be_v to_o mathematician_n injury_n in_o the_o succeed_a age_n problem_n cix_o how_o to_o make_v a_o clock_n with_o one_o only_a wheel_n make_v the_o body_n of_o a_o ordinary_a dial_n and_o divide_v the_o hour_n in_o the_o circle_n into_o 12._o part_n make_v a_o great_a wheel_n in_o height_n above_o the_o axletree_n to_o the_o which_o you_o shall_v place_v the_o cord_n of_o your_o counterpoise_n so_o that_o it_o may_v descend_v that_o in_o 1●_n hour_n of_o time_n your_o index_n or_o needle_n may_v make_v one_o revolution_n which_o may_v be_v know_v by_o a_o watch_n which_o you_o may_v have_v by_o you_o then_o put_v a_o balance_n which_o may_v stop_v the_o course_n of_o the_o wheel_n and_o give_v it_o a_o regular_a motion_n and_o you_o shall_v see_v a_o effect_n as_o just_a from_o this_o as_o from_o a_o clock_n with_o many_o wheel_n problem_n cx_o how_o by_o help_n of_o two_o wheel_n to_o make_v a_o child_n to_o draw_v up_o alone_o a_o hogshead_n of_o water_n at_o a_o time_n and_o be_v draw_v up_o shall_v cast_v out_o itself_o into_o another_o vessel_n as_o one_o will_v have_v it_o let_v r_o be_v the_o pit_n from_o whence_o water_n be_v to_o be_v draw_v p_o the_o hook_n to_o throw_v out_o the_o water_n when_o it_o be_v bring_v up_o this_o hook_n must_v be_v movable_a let_v ab_fw-la be_v the_o axis_n of_o the_o wheel_n sf_n which_o wheel_n have_v divers_a fork_n of_o iron_n make_v at_o g_o equal_o fasten_v at_o the_o wheel_n let_v i_o be_v a_o card_n which_o be_v draw_v by_o k_o to_o make_v the_o wheel_n s_o to_o turn_v which_o wheel_n saint_n bear_v proportion_n to_o the_o wheel_n t_o as_o 8_o to_z ●_o let_v n_o be_v a_o chain_n of_o iron_n to_o which_o be_v tie_v the_o vessel_n o_o and_o the_o other_o which_o be_v in_o the_o pit_n e●_n be_v a_o piece_n of_o wood_n which_o have_v a_o morte_n in_o 1_o and_o ●_o by_o which_o the_o cord_n i_o pass_v tie_v at_o the_o brickwall_n as_o kh_n and_o the_o other_o piece_n of_o timber_n of_o the_o little_a wheel_n as_o m_n mortify_v in_o likewise_o for_o the_o chain_n to_o pass_v through_o draw_v the_o cord_n i_o by_o k_o and_o the_o wheel_n will_v turn_v &_o so_o consequent_o the_o wheel_n t_o which_o will_v cause_v the_o vessel_n o_o to_o raise_v which_o be_v empty_a draw_v the_o cord_n again_o by_o y_o and_o the_o other_o vessel_n which_o be_v in_o the_o pit_n ●ill_v come_v out_o by_o the_o same_o reason_n this_o be_v a_o invention_n which_o will_v save_v labour_n if_o practise_v but_o here_o be_v to_o be_v note_v that_o the_o pit_n must_v be_v large_a enough_o to_o the_o end_n that_o it_o contain_v two_o great_a vessel_n to_o pass_v up_o and_o down_o one_o by_o another_o problem_n cxi_o to_o make_v a_o ladder_n of_o cord_n which_o may_v be_v carry_v in_o one_o pocket_n by_o which_o one_o may_v easy_o mount_v up_o a_o wall_n or_o tree_n alone_o take_v two_o pulley_n a_o &_o d_o unto_o that_o of_o a_o let_v there_o be_v fasten_v a_o cramp_n of_o iron_n as_o b_o and_o at_o d_o let_v there_o be_v fasten_v a_o staff_n of_o a_o foot_n and_o a_o half_a long_o as_o f_o than_o the_o poultry_n a_o place_n a_o hand_n of_o iron_n as_o e_o to_o which_o tie_n a_o cord_n of_o a_o half_a inch_n thick_a which_o may_v be_v of_o silk_n because_o it_o be_v for_o the_o pocket_n then_o strive_v to_o make_v fast_o the_o poultry_n a_o by_o the_o help_n of_o the_o cramp_n of_o iron_n b_o to_o the_o place_n that_o you_o intend_v to_o scale_v and_o the_o staff_n f_o be_v tie_v at_o the_o poultry_n d_o put_v it_o between_v your_o leg_n as_o though_o you_o will_v sit_v upon_o it_o then_o hold_v the_o cord_n c_o in_o your_o hand_n you_o may_v guide_v yourself_o to_o the_o place_n require_v which_o may_v be_v make_v more_o facile_a by_o the_o multiply_a of_o pulley_n this_o secret_a be_v most_o excellent_a in_o war_n and_o for_o lover_n its_o supportablenesse_n avoid_v suspicion_n problem_n cxii_o how_o to_o make_v a_o pump_n who_o strength_n be_v marvellous_a by_o reason_n of_o the_o great_a weight_n of_o water_n that_o it_o be_v able_a to_o bring_v up_o at_o once_o and_o so_o by_o continuance_n let_v 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d be_v the_o height_n of_o the_o case_n about_o two_o or_o three_o foot_n high_a and_o broad_a according_a to_o discretion_n the_o rest_n of_o the_o case_n or_o concavity_n let_v be_v o_o let_v the_o sucker_n of_o the_o pump_n which_o be_v make_v be_v just_a for_o the_o case_n or_o pump_n head_n 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d &_o may_v be_v make_v of_o wood_n or_o brass_n of_o 4_o inch_n thick_a have_v a_o hole_n at_o e_o which_o descend_v raise_v up_o the_o cover_n p_o by_o which_o issue_v forth_o the_o water_n &_o ascend_v or_o raise_v up_o it_o shut_v it_o or_o make_v it_o close_o r_n be_v the_o handle_n of_o the_o sucker_n tie_v to_o the_o handle_n tx_n which_o work_v in_o the_o post_n we_o let_v a_o b_o c_o d_o be_v a_o piece_n of_o brass_n g_o the_o piece_n which_o enter_v into_o the_o hole_n to_o f_o to_o keep_v out_o the_o air_n h_o i_o k_o l_o the_o piece_n tie_v at_o the_o funnel_n or_o pipe_n in_o which_o play_v the_o iron_n rod_n or_o axis_fw-la g_o so_fw-mi that_o it_o pass_v through_o the_o other_o piece_n mn_v which_o be_v tie_v with_o the_o end_n of_o the_o pipe_n of_o brass_n note_v that_o the_o low_a end_n of_o the_o cistern_n ought_v to_o be_v rest_v upon_o a_o gridiron_n or_o iron_n grate_n which_o may_v be_v tie_v in_o the_o pit_n by_o which_o mean_v lift_v up_o and_o put_v down_o the_o handle_n you_o may_v draw_v ten_o time_n more_o water_n than_o otherwise_o you_o can_v problem_n cxiii_o how_o by_o mean_n of_o a_o cistern_n to_o make_v water_n of_o

have_v in_o anoint_v the_o hair_n for_o fear_n of_o touch_v the_o flesh_n for_o this_o composition_n be_v very_o corrosive_a or_o search_a that_o as_o soon_o as_o it_o touch_v the_o flesh_n it_o raise_v blister_n and_o bladder_n very_o painful_a problem_n cxx_o how_o to_o make_v the_o representation_n of_o the_o great_a world_n draw_v salt_n niter_n out_o of_o salt_n earth_n which_o be_v find_v along_o the_o river_n side_n and_o at_o the_o foot_n of_o mountain_n where_o especial_o be_v mineral_n of_o gold_n and_o silver_n mix_v that_o niter_n well_o cleanse_v with_o ♃_o then_o calcinate_v it_o hermetical_o then_o put_v it_o in_o a_o limbeck_n and_o let_v the_o receiver_n be_v of_o glass_n well_o lute_v and_o always_o in_o which_o let_v there_o be_v place_v leaf_n of_o gold_n at_o the_o bottom_n then_o put_v fire_n under_o the_o limbeck_n until_o vapour_n arise_v which_o will_v cleave_v unto_o the_o gold_n augment_v your_o fire_n until_o there_o ascend_v no_o more_o then_o take_v away_o your_o receiver_n and_o close_o it_o hermetical_o and_o make_v a_o lamp_n fire_n under_o it_o until_o you_o may_v see_v present_v in_o it_o that_o which_o nature_n afford_v we_o as_o flower_n tree_n fruit_n fountain_n sun_n moon_n star_n etc._n etc._n behold_v here_o the_o form_n of_o the_o limbeck_n and_o the_o receiver_n a_o represent_v the_o limbeck_n b_o stands_z for_o the_o receiver_n problem_n cxxi_o how_o to_o make_v a_o cone_n or_o a_o pyramidal_a body_n move_v upon_o a_o table_n without_o spring_n or_o other_o artificial_a mean_n so_o that_o it_o shall_v move_v by_o the_o edge_n of_o the_o table_n without_o fall_v this_o proposition_n be_v not_o so_o thorny_a and_o subtle_a as_o it_o seem_v to_o be_v for_o put_v under_o a_o cone_n of_o paper_n a_o beetle_n or_o such_o like_a creature_n you_o shall_v have_v pleasure_n with_o astonishment_n &_o admiration_n to_o those_o which_o be_v ignorant_a in_o the_o cause_n for_o this_o animal_n will_v strive_v always_o to_o free_v 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