the_o near_a distance_n than_o the_o nd_o from_o the_o right_n sine_fw-la of_o the_o angle_n to_o the_o thread_n shall_v be_v the_o tangent_fw-la require_v example_n i_o make_v ab_fw-la radius_fw-la a_o =_o in_o 50_o the_o cousin_a of_o 40_o than_o the_o =_o sine_fw-la of_o 40_o shall_v be_v be._n again_o if_o i_o make_v ad_fw-la equal_a to_o ab_fw-la the_o =_o cousin_a of_o 30_o viz._n 60_o and_o then_o take_v out_o the_o =_o sine_fw-la of_o 30_o and_o lay_v it_o from_o d_o to_o f_o it_o shall_v be_v a_o angle_n of_o 60_o from_o ab_fw-la or_o 30_o from_o d_o to_o f._n but_o by_o the_o sector_n this_o be_v more_o easy_a use_v x._o to_o take_v out_o ready_o any_o tangent_fw-la above_o 45_o by_o the_o tangent_fw-la to_o 45_o on_o the_o sector-side_n sector_n take_v the_o give_v radius_fw-la make_v it_o a_o =_o in_o the_o co-tangent_a of_o the_o tangent_fw-la require_v then_o the_o =_o tangent_fw-la of_o 45_o shall_v be_v the_o tangent_fw-la require_v example_n i_o will_v have_v a_o tangent_fw-la to_o 80_o degree_n take_v the_o give_v radius_fw-la make_v it_o a_o =_o in_o 10_o the_o compliment_n of_o 80_o then_o the_o =_o tangent_fw-la of_o 45_o shall_v be_v the_o tangent_fw-la of_o 80_o require_v but_o if_o your_o radius_fw-la be_v so_o big_a that_o you_o can_v enter_v it_o then_o take_v the_o half_a or_o a_o quarter_n of_o your_o radius_fw-la and_o then_o =_o 45_o will_v be_v the_o half_a or_o the_o quarter_n of_o the_o tangent_fw-la require_v use_v xi_o how_o to_o work_v proportion_n in_o sines_n alone_o by_o the_o natural_a sin_n there_o be_v 4_o variety_n in_o this_o work_n that_o include_v all_o proportion_n viz._n 1._o when_o the_o sine_fw-la of_o 90_o be_v the_o first_o term_n than_o the_o work_n be_v thus_o lay_v the_o thread_n to_o the_o second_o term_n count_v on_o the_o degree_n from_o the_o head_n quadr_n towards_o the_o loose-piece_n and_o count_v the_o three_o term_n on_o the_o line_n of_o sines_n from_o the_o centre_n downward_o and_o take_v the_o near_a distance_n from_o thence_o to_o the_o thread_n and_o that_o distance_n measure_v from_o the_o centre_n downward_o on_o the_o line_n of_o sines_n give_v the_o four_o term_v require_v example_n as_o sine_fw-la 90_o to_o sine_fw-la 23-30_a so_o be_v 30_o to_o 11-31_a take_v the_o latteral_a second_o term_n sector_n make_v it_o a_o =_o sine_fw-la of_o 90_o then_o take_v out_o the_o =_o three_o term_n and_o measure_v it_o from_o the_o centre_n it_o give_v the_o four_o term_v require_v 2._o when_o the_o sine_fw-la of_o 90_o be_v the_o three_o term_n then_o work_v thus_o take_v the_o sine_fw-la of_o the_o second_o term_n from_o the_o centre_n downward_o quadr._n and_o make_v it_o a_o =_o sine_fw-la in_o the_o first_o term_n lay_v the_o thread_n to_o nd_o then_o on_o the_o degree_n the_o thread_n shall_v give_v the_o four_o term_v require_v example_n as_o the_o sine_fw-la of_o 30_o to_o 23-31_a so_o be_v the_o sine_fw-la of_o 90_o to_o sine_fw-la of_o 52-56_a but_o by_o the_o sector_n sector_n take_v the_o sine_fw-la of_o the_o second_o make_v it_o a_o =_o sine_fw-la of_o the_o first_o term_n then_o take_v out_o =_o 90_o and_o measure_v it_o from_o the_o centre_n and_o it_o shall_v give_v the_o four_o term_v require_v example_n as_o before_o 3._o when_o the_o radius_fw-la or_o sine_fw-la of_o 90_o be_v in_o the_o second_o place_n work_v thus_o quadr._n take_v 90_o from_o a_o lesser_a scale_n as_o the_o uppermost_a sine_fw-la above_o the_o centre_n or_o the_o line_n of_o right-ascentions_a or_o the_o azimuth-scale_n or_o the_o like_a and_o make_v it_o a_o =_o in_o the_o sine_fw-la of_o the_o first_o lay_v the_o thread_n to_o nd_o than_o the_o =_o three_o term_n take_v and_o measure_v on_o the_o same_o scale_n that_o 90_o be_v take_v from_o shall_v give_v the_o four_o term_v require_v example_n as_o 90_o on_o the_o line_n of_o rightascention_n be_v to_o =_o 30_o so_o be_v =_o 20_o to_o 43-12_a measure_v on_o the_o same_o line_n that_o 90_o be_v take_v from_o or_o else_o second_o work_v thus_o as_o 30_o to_o =_o 90_o so_o be_v 20_o to_o =_o 43-12_a by_o carry_v the_o compass_n till_o it_o so_o stay_v as_o that_o the_o foot_n turn_v about_o will_v but_o just_a touch_n the_o thread_n at_o the_o near_a distance_n or_o else_o thus_o three_o by_o transpose_v the_o term_n when_o the_o three_o be_v not_o great_a than_o the_o first_o thus_o as_o the_o first_o to_o the_o three_o so_o be_v the_o second_o term_n to_o the_o four_o where_o the_o radius_fw-la be_v bring_v to_o the_o three_o place_n it_o be_v wrought_v by_o the_o second_o rule_n as_o before_o by_o the_o sector_n take_v a_o small_a sine_fw-la of_o 90_o make_v it_o a_o =_o in_o 30_o then_o the_o =_o sine_fw-la of_o 20_o take_v and_o measure_v on_o the_o small_a sine_fw-la give_v 43-12_a as_o before_o again_o as_o 90_o to_o =_o 30_o so_o be_v =_o 20_o to_o 43-12_a again_o as_o 30_o to_o =_o 90_o so_o be_v =_o 20_o to_o =_o 43-12_a last_o by_o transpose_v as_o 20_o to_o =_o 30_o so_o be_v =_o radius_fw-la to_o 43-12_a as_o before_o 4._o when_o radius_fw-la be_v none_o of_o the_o give_v term_n then_o when_o the_o first_o term_n be_v great_a than_o the_o second_o and_o three_o quadr._n work_v thus_o take_v the_o second_v term_n make_v it_o a_o =_o in_o the_o first_o lay_v the_o thread_n to_o the_o nd_o then_o the_o near_a distance_n from_o the_o three_o term_n to_o the_o thread_n measure_v from_o the_o centre_n downward_o give_v the_o four_o sine_fw-la require_v example_n as_o 20_o to_o 12_o so_o be_v 18_o to_o 10-50_a by_o the_o quadrant_n as_o 12_o to_z =_o 20_o so_o be_v =_o 18_o to_o 10-50_a when_o only_o the_o second_o term_n be_v great_a than_o the_o first_o then_o transpose_v the_o term_n and_o work_v as_o before_o or_o else_o use_v a_o double_a radius_fw-la which_o be_v on_o this_o instrument_n very_o easy_o do_v have_v several_a radiuss_n or_o last_o use_v a_o parallel_n entrance_n or_o answer_v rather_o as_o before_o which_o be_v careful_o wrought_v will_v do_v very_o well_o by_o the_o sector_n the_o same_o manner_n of_o work_n be_v as_o before_o by_o the_o quadrant_n and_o the_o set_v the_o sector_n be_v all_o one_o to_o the_o lay_v the_o thread_n as_o will_v be_v large_o see_v in_o all_o the_o follow_a proposition_n wrought_v both_o by_o the_o artificial_a and_o natural_a line_n of_o number_n sin_n and_o tangent_n as_o follow_v use_v xii_o have_v the_o day_n of_o the_o month_n or_o sun_n place_n give_v to_o find_v his_o declination_n lay_v the_o thread_n on_o the_o day_n of_o the_o month_n in_o the_o calendar_n particular_o and_o in_o the_o line_n of_o degree_n on_o the_o moving-leg_n you_o have_v his_o declination_n either_o northward_o or_o southward_o according_a to_o the_o time_n of_o the_o year_n count_v from_o 600_o towards_o the_o head_n for_o north-declination_n or_o towards_o the_o end_n for_o south-declination_n by_o the_o artificial_a sin_n and_o tangent_n on_o the_o edge_n of_o the_o instrument_n extend_v the_o compass_n from_o the_o sine_fw-la of_o 90_o to_o the_o sine_fw-la of_o 23_o degree_n 31_o minute_n the_o sun_n great_a declination_n the_o same_o extent_n apply_v the_o same_o way_n from_o the_o sine_fw-la of_o the_o sun_n place_n or_o the_o sun_n distance_n from_o the_o next_o equinoctial-point_n shall_v cause_v the_o moving-point_n to_o fall_v on_z the_o sine_z of_o the_o sun_n declination_n this_o be_v the_o general_a way_n of_o work_v example_n the_o extent_n from_o the_o sine_fw-la of_o 90_o to_o the_o sine_fw-la of_o 23-31_a shall_v reach_v from_o the_o sine_fw-la of_o 30_o to_o 11_o deg_n 31_o min._n the_o sun_n declination_n in_o ♉_o taurus_n 30_o degree_n from_o ♈_o aries_n the_o next_o equinoctial-point_n and_o from_o 60_o degree_n the_o sun_n distance_n in_o ♊_o gemini_fw-la 60_o degree_n from_o ♈_o 20_o deg_n 12_o min._n the_o sun_n declination_n then_o this_o be_v the_o manner_n of_o work_v by_o these_o line_n by_o extend_v the_o compass_n from_o the_o first_o to_o the_o second_o term_n i_o shall_v for_o the_o rest_n wave_v this_o large_a repetition_n of_o extend_v the_o compass_n and_o render_v it_o only_o thus_o by_o the_o word_n of_o the_o cannon-general_a in_o all_o book_n as_o sine_fw-la 90_o to_o sine_fw-la 23-31_a so_o be_v the_o sine_fw-la of_o 30_o to_o sine_fw-la 11-31_a lay_v the_o thread_n to_o 23-31_a generally_n on_o the_o degree_n on_o the_o moveable-piece_n count_v from_o the_o head_n towards_o the_o end_n then_o count_v the_o sun_n place_n from_o the_o next_o equinoctial-point_n on_o the_o line_n of_o sines_n from_o the_o centre_n downward_o and_o take_v the_o nd_o from_o thence_o to_o the_o thread_n then_o this_o distance_n be_v measure_v from_o the_o centre_n downward_o
a_o foot_n shut_v be_v draw_v usual_o just_a 5_o degree_n asunder_o or_o rather_o the_o two_o innermost_a line_n on_o each_o leg_n be_v always_o just_a one_o degree_n from_o the_o inside_n so_o that_o if_o you_o put_v a_o center-pin_n in_o the_o line_n of_o tangent_n just_a against_o the_o sine_fw-la of_o 30_o it_o make_v the_o two_o innermost_a line_n that_o come_v from_o the_o centre_n just_a 2_o degree_n asunder_o which_o be_v easy_a to_o remember_v either_o in_o add_v or_o substract_v as_o follow_v two_o way_n 1._o take_v the_o latteral_a sine_fw-la of_o 30_o viz._n the_o measure_n from_o the_o centre_n to_o 30_o the_o compass_n so_o set_v set_v one_o point_n in_o the_o center-pin_n in_o the_o tangent_n just_a against_o 30_o and_o turn_v the_o other_o till_o it_o cut_v the_o common_a line_n in_o the_o line_n of_o sines_n on_o the_o other_o leg_n and_o there_o it_o shall_v show_v what_o angle_n the_o two_o innermost-line_n make_v count_v from_o the_o end_n towards_o the_o head_n and_o two_o degree_n less_o be_v the_o angle_n the_o sector_n stand_v at_o both_o on_o the_o inside_n and_o outside_n the_o leg_n be_v parallel_n which_o number_n must_v near_o agree_v with_o what_o the_o inside_n of_o the_o leg_n cut_v on_o the_o head-semicircle_n or_o there_o be_v a_o mistake_n as_o thus_o for_o example_n suppose_v i_o open_v the_o rule_n at_o all_o adventure_n and_o take_a the_o latteral_a sine_fw-la of_o 30_o from_o the_o sin_n on_o the_o sector-side_n and_o put_v one_o point_n of_o the_o compass_n in_o the_o centre_n on_o the_o tangent_n right_a against_o the_o sine_fw-la of_o 30_o on_o the_o other_o leg_n or_o the_o begin_n of_o the_o secant_v on_o the_o same_o leg_n and_o turn_v the_o other_o point_n to_o the_o line_n of_o sines_n on_o the_o other_o leg_n it_o cut_v the_o sine_fw-la of_o 60_o on_o the_o innermost_a line_n that_o come_v from_o the_o centre_n then_o i_o say_v that_o the_o line_n of_o sines_n and_o tangent_n be_v just_a 30_o degree_n asunder_o and_o the_o inside_n or_o outside_n of_o the_o leg_n but_o 28_o viz._n two_o degree_n less_o as_o a_o glance_n of_o your_o eye_n to_o the_o head_n will_v plain_o show_v 2._o this_o way_n will_v serve_v very_o well_o for_o all_o angle_n above_o 20_o and_o under_o 80_o but_o for_o all_o under_o 20_o and_o above_o 80_o to_o 120_o this_o be_v a_o better_a way_n open_v the_o rule_n to_o any_o angle_n at_o pleasure_n and_o take_v the_o distance_n parallelly_n that_o be_v across_o from_o one_o leg_n to_o the_o other_o between_o the_o center-pin_n at_o 30_o in_o the_o sin_n and_o that_o in_o the_o tangent_n right_a against_o it_o and_o measure_v it_o latteral_o from_o the_o centre_n and_o it_o shall_v show_v the_o sine_fw-la of_o half_a the_o angle_n the_o sin_n and_o tangent_n stand_v at_o and_o one_o degree_n less_o be_v the_o sine_fw-la of_o half_a the_o angle_n the_o sector_n stand_v at_o example_n suppose_v that_o open_v the_o sector_n at_o adventure_n or_o to_o the_o level_n of_o any_o thing_n i_o will_v know_v the_o angle_n it_o stand_v at_o i_o take_v the_o parallel_n distance_n between_o the_o two_o centre_n and_o measure_v it_o latteral_o from_o the_o centre_n i_o find_v it_o give_v i_o the_o sine_fw-la of_o 51_o degree_n viz._n the_o half_a angle_n the_o line_n stand_v at_o or_o 50_o the_o angle_n the_o rule_n stand_v at_o which_o double_v be_v 102_o for_o the_o line_n or_o 100_o for_o the_o leg_n of_o the_o sector_n as_o a_o glance_n of_o the_o eye_n present_o resolve_v by_o the_o inner-edge_n of_o the_o moving-leg_n and_o the_o divide_a semicircle_n 3._o on_o the_o contrary_a will_v you_o set_v the_o leg_n or_o line_n to_o any_o angle_n take_v the_o half_a thereof_o latteral_o or_o one_o degree_n less_o in_o the_o half_a for_o the_o leg_n and_o make_v it_o a_o parallel_n in_o the_o two_o centre_n and_o the_o sector_n be_v so_o set_v according_o example_n i_o will_v set_v the_o leg_n to_o 90_o degree_n or_o a_o just_a square_n take_v out_o the_o latteral_a sine_fw-la of_o 44_o one_o degree_n less_o than_o 45_o the_o half_a of_o 90_o and_o make_v it_o a_o parallel_n in_o the_o two_o centre_n abovesaid_a and_o you_o shall_v find_v the_o leg_n set_v just_a to_o a_o square_a or_o right-angle_n as_o by_o look_v to_o the_o head_n you_o may_v near_o see_v at_o the_o same_o time_n if_o you_o take_v latteral_a 30_o and_o lay_v it_o from_o the_o centre_n according_a to_o the_o first_o rule_n you_o shall_v see_v a_o great_a deficiency_n therein_o as_o above_o be_v hint_v use_v v._o the_o day_n of_o the_o month_n be_v give_v to_o find_v the_o sun_n declination_n true_a place_n in_o the_o zodiac_n right_a ascention_n ascentional_a difference_n or_o rise_v and_o set_v 1._o lay_v the_o thread_n to_o the_o day_n of_o the_o month_n in_o the_o upper_a line_n of_o month_n where_o the_o length_n of_o the_o day_n be_v increase_v or_o in_o the_o lower-line_n when_o the_o day_n be_v decrease_v according_a to_o the_o time_n of_o the_o year_n then_o in_o the_o line_n of_o degree_n you_o have_v his_o declination_n wherein_o note_n that_o if_o the_o thread_n lie_v on_o the_o right_a hand_n of_o 600_o than_o the_o sun_n declination_n be_v northwards_o the_o contrary-way_n be_v southwards_o also_o on_o the_o line_n of_o the_o sun_n be_v right_a ascension_n you_o have_v his_o right_a ascension_n in_o degree_n and_o hour_n count_v one_o hour_n for_o 15_o degree_n as_o the_o month_n proceed_v from_o march_n the_o 10_o or_o equinoctial_a the_o right_a ascention_n be_v then_o 00_o and_o so_o forward_o to_o 24_o hour_n or_o 360_o degree_n as_o the_o month_n and_o day_n proceed_v again_o on_o the_o line_n oâ—_n the_o sun_n be_v true_a place_n you_o have_v the_o sign_n and_o degree_n of_o his_o place_n in_o the_o ecliptic_a aries_n or_o the_o equinoctial-point_n be_v the_o place_n to_o begin_v and_o then_o proceed_v forward_o as_o the_o month_n and_o day_n go_v last_o on_o the_o hour-line_n you_o have_v the_o ascentional-difference_n in_o degree_n and_o minute_n count_v from_o 6_o or_o the_o sun_n rise_v count_v as_o the_o morning_n hour_n proceed_v or_o his_o set_v count_v as_o the_o afternoon_n hour_n proceed_v of_o all_o which_o take_v two_o or_o three_o example_n 1._o for_o march_v the_o 12_o lay_v the_o thread_n to_o the_o day_n and_o extend_v it_o straight_o then_o on_o the_o line_n of_o degree_n it_o show_v near_o 1_o degree_n or_o 54_o minute_n northward_o 2._o the_o sun_n right_a ascension_n be_v in_o time_n 8_o minute_n and_o better_a or_o in_o degree_n 2_o deg_n 5_o minute_n 3._o the_o sun_n place_n be_v 2_o degree_n and_o 16_o minute_n in_o aries_n ♈_o 4._o the_o ascentional_a difference_n be_v 1_o degree_n and_o 10_o minute_n or_o the_o sun_n rise_v 4_o minute_n before_o and_o set_v 4_o minute_n after_o 6._o again_o for_o may_v the_o 10_o the_o thread_n lay_v thereon_o cut_v in_o the_o degree_n 20_o deg_n 9_o min._n for_o northern_a declination_n and_o 57_o deg_n 24_o min._n or_o 3_o hour_n 52_o min._n right_a ascension_n and_o 29_o 37_o in_o ♉_o taurus_n for_o his_o true_a place_n and_o 27_o 12_o for_o difference_n of_o ascension_n or_o rise_v 11_o minute_n after_o 4_o and_o set_v 49_o minute_n after_o 7._o again_o on_o the_o last_o of_o october_n or_o the_o 21_o of_o january_n near_o the_o declination_n be_v 17_o 22_o southwards_o the_o right_a ascension_n for_o october_n 31_o be_v 225_o 53_o for_o january_n 21_o 314_o 21_o the_o true_a place_n for_o october_n 31_o be_v â™_o scorpio_n 18_o deg_n 22_o min_fw-mi but_o for_o january_n 21_o â™’_o aquarius_n 11_o 52_o according_a as_o the_o month_n go_v to_o the_o end_n at_o ♑_o and_o then_o back_o again_o but_o the_o ascentional_a difference_n and_o rise_v and_o set_v be_v very_o near_o the_o same_o at_o both_o time_n viz._n 23_o 10_o and_o rise_v 32_o minute_n and_o more_o after_z 7_o and_o set_v 28_o minute_n less_o after_o 4._o use_v vi_o the_o declination_n of_o the_o sun_n or_o a_o star_n give_v to_o find_v his_o amplitude_n take_v the_o declination_n be_v count_v on_o the_o particular_a scale_n of_o altitude_n between_o your_o compass_n and_o with_o this_o distance_n set_v one_o foot_n in_o 90_o on_o the_o azimuth-line_n the_o other_o point_v apply_v to_o the_o same_o line_n shall_v give_v the_o amplitude_n count_v from_o 90._o example_n the_o declination_n be_v 12_o north_n the_o amplitude_n be_v 19_o deg_n 15_o min._n northwards_o or_o the_o declination_n be_v 20_o south_n the_o amplitude_n be_v 34_o deg_n 10_o min._n southwards_o use_v vii_o the_o right_a ascention_n and_o ascentional-difference_n be_v give_v to_o find_v his_o oblique-ascention_n when_o the_o declination_n be_v north_n than_o the_o difference_n between_o the_o
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the_o figure_n annex_v show_v the_o division_n on_o this_o line_n be_v for_o the_o most_o part_n whole_a degree_n or_o every_o four_o minute_n of_o time_n four_o next_o above_o this_o be_v the_o line_n of_o the_o sun_n place_n in_o the_o zodiac_n note_v with_o ♈_o ♉_o ♊_o ♋_o towards_o the_o head_n then_o back_o again_o with_o ♌_o â™_o ♎_o over_o 600_o in_o the_o degree_n and_o 12_o and_o 24_o in_o the_o line_n of_o the_o sun_n right_a ascension_n then_o towards_o the_o end_n with_o â™_o â™_o ♑_o then_o back_o again_o with_o â™’_o and_o ♓_o be_v the_o character_n of_o the_o 12_o sign_n of_o the_o zodiac_n wherein_o you_o have_v express_v every_o whole_a degree_n as_o the_o number_n of_o they_o do_v show_v
there_o be_v 30_o degree_n in_o one_o sign_n five_o next_o above_o this_o be_v a_o calendar_n of_o month_n and_o day_n every_o single_a day_n be_v express_v and_o three_o or_o more_o letter_n of_o the_o name_n of_o every_o month_n be_v set_v in_o the_o month_n and_o also_o at_o the_o begin_n of_o each_o month_n and_o every_o 10_o day_n note_v with_o a_o prick_n on_o the_o top_n of_o the_o line_n represent_v it_o as_o be_v usual_a in_o such_o work_n six_o next_o over_o the_o month_n be_v the_o line_n to_o find_v the_o hour_n and_o azimuth_n in_o a_o particular_a latitude_n put_v always_o on_o small_a instrument_n and_o very_o rare_o on_o large_a triangular_a quadrant_n for_o sea_n observation_n the_o low_a margin_n whereof_o and_o next_o the_o month_n be_v number_v from_o the_o end_n towards_o the_o head_n with_o 10_o 20_o 30_o 40_o 50_o 60_o 70_o 80_o 90_o 100_o 110_o 120_o 130_o near_o the_o head_n centre_n for_o the_o semi-diurnal_a ark_n of_o the_o sun_n azimuth_n and_o in_o the_o margin_n next_o above_o this_o with_o 4_o 5_o 6_o 7_o 8_o 9_o 10_o 11_o 12_o near_o the_o end_n for_o the_o morning_n hour_n then_o the_o other_o way_n viz._n towards_o the_o head_n on_o the_o other-side_n the_o hour_n line_n with_o 1_o 2_o 3_o 4_o 5_o 6_o 7_o 8_o for_o the_o afternoon_n hour_n seven_o on_o the_o same_o quadrantal-side_n and_o moveable-leg_n on_o the_o spare_a place_n beyond_o the_o month_n towards_o the_o end_n be_v set_v a_o almanac_n and_o the_o name_n of_o 12_o or_o more_o star_n to_o find_v the_o hour_n of_o the_o night_n which_o 12_o star_n be_v note_v with_o 1_o 2_o 3_o 4_o 5_o 6_o 7_o 8_o 9_o 10_o 11_o 12._o among_o the_o degree_n in_o small_a figure_n as_o in_o the_o figure_n eight_o next_o of_o all_o to_o the_o inside_n be_v the_o line_n of_o natural_a verse_v sin_n draw_v to_o the_o centre_n with_o his_o correspondent_a line_n on_o the_o other_o or_o head-leg_n express_v sometime_o in_o a_o prick_a line_n for_o want_v of_o room_n nine_o on_o the_o head-leg_n and_o next_o to_o the_o verse_v sines_n last_o mention_v be_v first_o the_o line_n of_o equal_a part_n or_o line_n of_o line_n and_o on_o the_o same_o common_a line_n wherein_o be_v the_o centre_n be_v the_o line_n of_o natural_a sin_n who_o length_n be_v equal_a to_o the_o measure_n from_o the_o centre_n at_o c_o too_o 600_o on_o the_o moveable-leg_n so_o that_o the_o line_n of_o degree_n be_v a_o tangent_fw-la and_o the_o measure_n from_o c_o to_o any_o tangent_fw-la a_o secant_fw-la to_o the_o same_o radius_fw-la of_o the_o natural_a line_n of_o sines_n and_o line_n also_o beyond_o the_o centre_n c_o on_o the_o same_o common_a middle_a line_n be_v another_o small_a line_n of_o natural_a sin_n who_o length_n be_v equal_a to_o the_o measure_n from_o c_o to_o 60_o on_o the_o loose-piece_n then_o if_o you_o count_v from_o the_o centre_n pin_n at_o 60_o on_o the_o loose-piece_n towards_o the_o end_n of_o the_o movable-leg_n they_o shall_v be_v tangent_n to_o the_o same_o radius_fw-la and_o the_o measure_n from_o the_o centre_n c_o to_o those_o tangent_n shall_v be_v secant_v to_o the_o same_o radius_fw-la which_o may_v be_v well_o to_o be_v order_v to_o a_o three_o or_o four_o part_n of_o the_o former_a from_o the_o centre_n downward_o these_o two_o line_n of_o sines_n be_v best_a figure_v with_o their_o sin_n and_o cosines_n the_o other_o way_n with_o a_o small_a figure_n and_o the_o line_n of_o line_n from_o the_o centre_n downward_o from_o 1_o to_o 10_o where_o 90_o be_v which_o line_n of_o sines_n may_v be_v call_v a_o general_a scale_n for_o all_o latitude_n ten_o next_o to_o this_o towards_o the_o outer-edge_n be_v another_o line_n of_o natural_a sin_n fit_v to_o the_o particular_a line_n of_o hour_n and_o azimuth_n for_o one_o particular_a latitude_n note_v pert._n scale_n of_o altitude_n or_o sines_n eleven_o next_o to_o this_o be_v the_o line_n of_o 29_o ½_o for_o so_o many_o day_n of_o the_o moon_n be_v age_n in_o short_a rule_n of_o the_o whole_a length_n but_o in_o long_a not_o be_v easy_o know_v by_o the_o single_a stroke_n and_o figure_n annex_v to_o those_o stroke_n twelve_o next_o the_o outer-edge_n be_v a_o line_n of_o 24_o hour_n 360_o degree_n or_o 12_o sign_n or_o in_o most_o rule_n inch_n also_o use_v together_o with_o the_o former_a line_n of_o 29_o ½_o and_o as_o a_o theory_n of_o the_o sun_n and_o moon_n and_o ready_a way_n of_o find_v the_o hour_n by_o the_o moon_n or_o fix_a star_n thirteen_o to_o this_o instrument_n also_o belong_v a_o thread_n and_o plummet_n and_o sight_n as_o to_o other_o quadrant_n and_o a_o pair_n of_o compass_n as_o to_o other_o sector_n a_o staff_n and_o ball_n socket_n also_o if_o you_o will_v be_v curious_a and_o accurate_a and_o for_o large_a instrument_n for_o sea_n a_o square_a and_o a_o index_n which_o make_v it_o a_o perfect_a sinical_a quadrant_n and_o two_o slide_a sight_n also_o which_o make_v it_o a_o fore_n and_o back-staff_n and_o bow_n as_o will_v appear_v more_o at_o large_a afterward_o some_o use_v of_o the_o triangule_a quadrant_n for_o land_n and_o sea_n observation_n and_o operation_n chap._n i._o numeration_n on_o the_o line_n graduate_v on_o the_o instrument_n in_o the_o first_o place_n it_o will_v not_o be_v amiss_o to_o hint_n a_o few_o word_n as_o to_o the_o read_v the_o line_n or_o more_o proper_o numeration_n on_o the_o line_n wherein_o take_v notice_n that_o all_o line_n of_o equal_a part_n or_o line_n applicable_a to_o arithmetic_n as_o the_o line_n of_o line_n the_o line_n of_o number_n the_o line_n of_o foot-measure_n and_o the_o like_a wherein_o fraction_n of_o number_n be_v requisite_a they_o be_v most_o common_o account_v in_o a_o decimal_n way_n and_o as_o much_o as_o may_v be_v the_o small_a division_n be_v number_v and_o count_v according_o but_o in_o the_o line_n of_o sines_n tangent_n secant_v and_o chord_n be_v line_n belong_v proper_o to_o a_o circle_n in_o regard_n that_o the_o sexagenary_a fraction_n be_v still_o in_o use_n the_o intermediate_v division_n be_v as_o much_o as_o may_v be_v fit_v to_o that_o way_n of_o account_n viz._n by_o whole_a degree_n where_o they_o come_v close_o together_o or_o the_o line_n of_o no_o great_a use_n and_o if_o more_o room_n be_v to_o half_a degree_n or_o 30_o minute_n and_o sometime_o to_o quarter_n of_o degree_n or_o 15_o minute_n but_o towards_o the_o begin_n of_o the_o line_n of_o natural_a sin_n or_o the_o end_n of_o the_o natural_a tangent_n and_o secant_v where_o the_o degree_n be_v large_a they_o be_v divide_v to_o every_o 10_o minute_n in_o all_o large_a rule_n as_o by_o consider_v and_o account_v you_o may_v plain_o perceive_v take_v two_o or_o three_o example_n of_o each_o kind_n 1._o first_o on_o the_o line_n of_o line_n to_o find_v the_o point_n that_o represent_v 15._o in_o the_o do_n of_o this_o or_o any_o the_o like_a you_o must_v consider_v your_o whole_a scale_n radius_fw-la or_o length_n of_o the_o line_n may_v be_v account_v as_o 1_o as_o 10_o as_o 100_o as_o 1000_o or_o as_o 10000_o and_o no_o further_o can_v be_v applicable_a to_o any_o ordinary_a instrument_n wherein_o observe_v that_o if_o the_o whole_a line_n be_v one_o than_o the_o long_a stroke_n by_o every_o figure_n do_v represent_v one_o ten_o of_o that_o integer_fw-la and_o the_o next_o short_a without_o figure_n be_v hundred_o part_n of_o that_o one_o integer_fw-la and_o a_o 1000th_o part_n be_v estimate_v in_o small_a instrument_n and_o sometime_o express_v in_o large_a but_o the_o hundred_o thousand_o part_n be_v always_o to_o be_v estimate_v by_o the_o eye_n in_o all_o instrument_n whatsoever_o 2._o but_o if_o the_o whole_a line_n of_o line_n shall_v represent_v 10_o as_o it_o usual_o do_v and_o as_o it_o be_v figure_v than_o the_o long_a stroke_n at_o every_o figure_n be_v 1_o and_o the_o next_o long_o be_v ten_o and_o the_o short_a be_v hundred_o part_n and_o the_o thousand_o part_n as_o near_o as_o can_v be_v estimate_v 3._o but_o if_o the_o whole_a line_n represent_v a_o hundred_o as_o here_o in_o our_o present_a example_n than_o the_o long_a stroke_n by_o every_o figure_n represent_v 10_o and_o every_o short_a stroke_n be_v one_o and_o the_o short_a stroke_n be_v ten_o and_o the_o hundred_o part_n as_o much_o as_o can_v be_v estimate_v 4._o but_o if_o the_o whole_a line_n shall_v represent_v a_o 1000_o than_o the_o long_a stroke_n by_o the_o figure_n shall_v represent_v a_o hundred_o and_o every_o short_a 10_o and_o every_o of_o the_o short_a stroke_n be_v one_o integer_fw-la and_o a_o 10_o part_n as_o near_o as_o can_v be_v estimate_v 5._o but_o last_o if_o the_o whole_a line_n represent_v 10000_o than_o
that_o point_n at_o the_o end_n of_o the_o natural-sines_n on_o the_o sector-side_n and_o at_o 90_o and_o 45_o on_o the_o edge_n of_o the_o rule_n for_o the_o artificial_a sin_n and_o tangent_n or_o 10_o on_o the_o line_n of_o number_n and_o 10_o and_o 90_o on_o the_o line_n of_o line_n and_o sines_n on_o the_o quadrantal-side_n of_o the_o instrument_n right-sine_a 2._o a_o right_a sine_fw-la of_o any_o ark_n or_o angle_n be_v the_o measure_n from_o the_o begin_n of_o the_o line_n of_o natural-sines_n to_o that_o point_n on_o that_o line_n of_o sines_n which_o represent_v the_o degree_n and_o minute_n contain_v in_o that_o ark_n or_o angle_n require_v but_o on_o the_o artificial-sines_n we_o respect_v not_o any_o measure_n but_o the_o point_n only_o tangent_fw-la 3._o the_o same_o account_n be_v use_v both_o for_o the_o right-tangent_a and_o secant_fw-la also_o the_o natural-tangent_a take_v from_o the_o begin_n to_o the_o degree_n and_o minute_n require_v the_o artificial_a respect_v the_o point_n only_o chord_n 4._o in_o the_o same_o manner_n count_v for_o the_o secant_v and_o chord_n line_n or_o verse_v sin_n cousin_n 5._o a_o cousin_n or_o sine_fw-la compliment_n of_o any_o ark_n or_o angle_n be_v the_o measure_n from_o the_o point_n represent_v the_o ark_n or_o angle_n count_v from_o 90_o to_o the_o begin_n of_o the_o line_n of_o sines_n be_v in_o effect_n the_o right-sine_a of_o the_o cousin_n of_o the_o ark_n or_o angle_n require_v as_o for_o example_n i_o will_v take_v out_o the_o cousin_n of_o the_o latitude_n of_o london_n which_o be_v 51_o 32_o count_v 51_o 32_o from_o 90_o towards_o the_o beginning_n and_o you_o shall_v find_v your_o account_n to_o end_v at_o the_o right-sine_a of_o 38_o 28_o which_o be_v the_o compliment_n of_o 51_o 32_o for_o both_o put_v together_o make_v 90_o the_o whole_a sine_fw-la or_o radius_fw-la but_o on_o the_o artificial-line_n count_v backward_o to_o the_o point_n require_v without_o mind_v any_o distance_n or_o measure_n till_o you_o come_v to_o proportion_n 6._o a_o lateral_a sine_fw-la tangent_fw-la or_o secant_fw-la lateral-sine_a or_o scale_n of_o equal_a part_n be_v any_o sine_fw-la tangent_fw-la or_o secant_fw-la take_v along_o the_o length_n of_o any_o line_n from_o the_o beginning_n onward_o be_v a_o term_n use_v only_o in_o operation_n with_o a_o sector_n or_o one_o line_n and_o a_o thread_n and_o oppose_v to_o a_o parallel-sine_a tangent_fw-la or_o secant_fw-la the_o thing_n next_o to_o be_v explain_v 7._o a_o parallel_n sine_fw-la tangent_fw-la or_o secant_fw-la parallel_n be_v any_o sine_fw-la tangent_fw-la or_o secant_fw-la take_v across_o from_o one_o leg_n to_o the_o other_o of_o a_o sector_n or_o from_o any_o degree_n and_o minute_n on_o one_o line_n to_o a_o thread_n draw_v straight_o with_o the_o other_o hand_n or_o any_o other_o fix_a line_n whatsoever_o at_o the_o near_a distance_n 8._o the_o near_o distance_n to_o any_o line_n nearest-distance_n be_v thus_o take_v when_o one_o point_n of_o the_o compass_n stand_v in_o any_o one_o point_n and_o the_o line_n be_v lay_v i_o open_v or_o close_o my_o compass_n till_o the_o other_o moveable-foot_n be_v turn_v about_o will_v but_o just_a touch_n or_o cleave_v the_o thread_n but_o if_o you_o be_v to_o lay_v the_o thread_n to_o the_o near_a distance_n than_o one_o point_v of_o the_o compass_n be_v set_v fast_o the_o other_o be_v to_o be_v turn_v about_o and_o the_o thread_n also_o slip_v to_o and_o fro_o till_o the_o compass-point_n shall_v just_o cleave_v the_o thread_n in_o the_o midst_n line_n 9_o to_o add_v one_o sine_fw-la or_o tangent_fw-la to_o a_o sine_fw-la or_o tangent_fw-la be_v to_o take_v the_o right-sine_a or_o tangent_fw-la of_o any_o ark_n or_o angle_n between_o your_o compass_n and_o set_v one_o point_n of_o the_o compass_n in_o the_o point_n of_o the_o other_o number_n and_o then_o to_o see_v how_o far_o the_o other_o point_n will_v extend_v lateral_o example_n to_o add_v the_o sine_fw-la of_o 20_o to_o the_o sine_fw-la of_o 30_o take_v the_o sine_fw-la of_o 20_o between_o your_o compass_n and_o then_o put_v one_o point_n in_o 30_o the_o other_o shall_v reach_v to_o the_o sine_fw-la of_o 51_o 21_o therefore_o the_o distance_n from_o the_o begin_n to_o 51_o 21_o be_v the_o sum_n of_o the_o sine_fw-la of_o 30_o and_o 20_o add_v together_o the_o like_a way_n be_v to_o add_v tangent_n line_n 10._o to_o subtract_v a_o sine_fw-la from_o a_o sine_fw-la or_o a_o tangent_fw-la from_o a_o tangent_fw-la be_v but_o to_o take_v the_o lateral_a lest_o right-sine_a or_o tangent_fw-la between_o your_o compass_n and_o set_v one_o point_n in_o the_o term_n of_o the_o great_a turn_n the_o other_o towards_o the_o beginning_n and_o note_v the_o degree_n and_o minute_n that_o the_o other_o point_n stay_v in_o for_o that_o be_v the_o difference_n or_o remainder_n example_n suppose_v i_o will_v take_v the_o sine_fw-la of_o 10_o degree_n from_o 25_o take_v the_o distance_n 10_o between_o your_o compass_n and_o set_v one_o foot_n in_o 25_o and_o the_o other_o turn_v towards_o the_o beginning_n shall_v reach_v to_o 14_o 23_o the_o residue_n or_o difference_n require_v or_o you_o may_v sometime_o take_v the_o distance_n between_o the_o great_a and_o the_o less_o and_o lay_v this_o from_o the_o beginning_n shall_v give_v the_o remainder_n in_o distance_n on_o the_o sin_n as_o before_o 11._o the_o rectifying-point_n point_n be_v a_o point_n or_o hole_n on_o the_o head_n of_o the_o triangule_a quadrant_n in_o the_o inter-secting_a of_o the_o hour_n and_o azimuth-line_n and_o the_o common_a line_n to_o the_o line_n and_o sin_n on_o the_o head-leg_n in_o which_o point_n you_o be_v when_o the_o rule_n be_v open_a to_o stick_v a_o small_a pin_n to_o look_v to_o the_o object_n who_o altitude_n above_o the_o horizon_n you_o will_v have_v in_o degree_n and_o minute_n of_o term_n use_v in_o dial_v plain_o be_v that_o board_n glass_n or_o flat_a superficies_n you_o intend_v to_o draw_v the_o dial_n upon_o either_o single_a of_o itself_o plain_a or_o join_v to_o some_o other_o pole_n of_o the_o plain_a plain_a be_v a_o imaginary_a point_n in_o the_o horizon_n for_o all_o upright_a dial_n direct_o opposite_a to_o the_o plain_a or_o in_o all_o plain_n a_o point_n every_o way_n 90_o degree_n from_o the_o plain_a declination_n declination_n of_o a_o plain_a be_v only_o the_o number_n of_o degree_n and_o minute_n that_o the_o pole-point_n of_o the_o plain_n be_v distant_a from_o the_o north_n and_o south-point_n of_o the_o horizon_n plain_a the_o perdendiculer-line_n on_o the_o plain_a be_v a_o line_n square_n to_o a_o horizontal-line_n be_v part_n of_o a_o circle_n pass_v through_o the_o zenith_n and_o nadir_n and_o pole-point_n of_o the_o plain_a horizontal-line_n the_o horizontal-line_n be_v a_o line_n draw_v on_o any_o plain_a exact_o parallel_v to_o the_o true_a horizon_n of_o the_o place_n you_o dwell_v in_o inclination_n reclination_n be_v when_o a_o plain_a behold_v the_o zenith-point_n over_o our_o head_n but_o inclination_n be_v when_o a_o plain_a behold_v the_o nadier_n as_o in_o a_o roof_n of_o a_o house_n the_o tiled-part_n recline_v and_o the_o celid-part_n incline_v meridian-line_n the_o meridian-line_n on_o all_o plain_n be_v the_o hour-line_n of_o 12_o but_o the_o meridian_n of_o the_o plain_a be_v the_o great_a circle_n of_o azimuth_n perpendicule_a to_o the_o plain_a be_v the_o same_o with_o the_o perpendiculer-line_n on_o the_o plain_a pass_v through_o the_o point_n of_o declination_n substile_n the_o substile-line_n on_o all_o dial_n be_v that_o line_n wherein_o the_o style_n gnomon_n or_o cock_n of_o the_o dial_n do_v stand_v usual_o count_v from_o 12_o the_o meridian-line_n or_o from_o the_o perpendiculer-line_n which_o in_o all_o erect_a dial_n be_v 12._o the_o style_n of_o a_o dial_n be_v the_o angle_n style_n between_o the_o common_a axis_n of_o the_o world_n and_o the_o plain_a upon_o the_o substile-line_n on_o the_o plain_a on_o all_o dial_n the_o angle_n between_o 12_o and_o 6_o 6._o be_v only_o the_o number_n of_o degree_n and_o minute_n contain_v between_o the_o hour-line_n of_o 12_o and_o the_o hour-line_n of_o 6_o a_o clock_n on_o any_o kind_n of_o plain_a especial_o those_o have_v centre_n the_o inclination_n of_o meridian_n meridian_n be_v the_o number_n of_o degree_n and_o minute_n count_v on_o the_o aequinoctial_a between_o the_o meridian_n or_o hour-line_n of_o 12_o and_o the_o substile_a be_v the_o distance_n between_o the_o meridian_n of_o the_o place_n viz._n 12_o a_o clock_n and_o the_o meridian_n of_o the_o plain_a but_o count_v on_o the_o aequinoctial_a and_o do_v serve_v to_o make_v the_o table_n of_o hour-arks_a at_o the_o pole_n and_o to_o prove_v your_o work_n the_o line_n parallel_v to_o 12_o parallel_n be_v two_o line_n peculiar_a to_o this_o way_n of_o dial_v by_o the_o sector_n and_o be_v only_o
a_o four_o number_n which_o you_o must_v add_v to_o the_o least_o diameter_n to_o make_v a_o mean_a diameter_n have_v thus_o gain_v a_o mean_a diameter_n you_o may_v work_v as_o before_o or_o rather_o thus_o more_o ready_o and_o easy_o by_o the_o line_n of_o number_n thus_o as_o the_o gagepoint_n be_v to_o the_o mean_a diameter_n so_o be_v the_o length_n to_o a_o four_o and_o that_o four_o to_o the_o content_a require_v the_o gagepoint_n for_o wine_n and_o oyl-gallon_n at_o 231_o cube_n inch_n in_o a_o gallon_n be_v 17_o 15_o the_o gagepoint_n for_o ale-gallon_n at_o 282_o ¼_n be_v 18_o 95_o the_o gagepoint_n for_o ale_n or_o beer-gallon_n at_o 288_o be_v 19_o 15_o the_o gagepoint_n for_o a_o corn-gallon_n at_o 272_o ¼_n be_v 18_o 62_o the_o gagepoint_n for_o a_o beer_n barrel_n at_o 10161_o be_v 35_o 96_o the_o gagepoint_n for_o a_o ale_n barrel_n at_o 9032_o be_v 33_o 91_o the_o extent_n of_o the_o compass_n on_o the_o line_n of_o number_n from_o the_o gagepoint_n to_o the_o mean_a diameter_n of_o a_o vessel_n be_v turn_v two_o time_n the_o same_o way_n from_o the_o length_n of_o a_o vessel_n shall_v reach_v to_o the_o content_a of_o the_o vessel_n in_o gallon_n or_o barrel_n according_a to_o the_o nature_n of_o the_o gagepoint_n example_n a_o mean_a diameter_n be_v 30_o and_o the_o length_n 40_o the_o content_a be_v in_o wine-gallon_n 123_o near_o in_o the_o lesser_a ale_n or_o beer-gallon_n 100-½_a in_o the_o great_a ale-gallon_n at_o 288-098_a gallon_n and_o a_o half_a in_o corn-gallon_n at_o 272-¼_a 104_o gallon_n in_o beer_n barrel_n by_o his_o gagepoint_n you_o will_v find_v 2-78_a or_o 2_o three_o quarter_n 2_o 78._o in_o ale_n barrel_n you_o will_v find_v 3_o and_o 11_o of_o a_o hundred_o 3_o 11._o and_o the_o like_a for_o any_o other_o measure_n who_o gagepoint_n be_v know_v problem_n iii_o to_o find_v the_o gagepoint_n of_o any_o measure_n the_o gagepoint_n of_o any_o solid_a measure_n be_v only_o the_o diameter_n of_o a_o circle_n who_o superficial_a content_n be_v equal_a to_o the_o solid_a content_n of_o the_o same_o measure_n as_o thus_o more_o plain_o the_o solid_a content_n of_o a_o wine_n gallon_n be_v 231_o cube_n inch_n now_o if_o you_o have_v a_o circle_n that_o contain_v 231_o superficial_a inch_n the_o diameter_n thereof_o will_v be_v find_v to_o be_v 17_o inch_n and_o 15_o of_o a_o hundred_o as_o by_o the_o seven_o problem_n of_o the_o seven_o chapter_n be_v well_o see_v these_o direction_n may_v serve_v for_o any_o round_a vessel_n either_o close_a or_o open_a yet_o mr._n oughtre_v a_o very_a able_a mathematition_n have_v a_o way_n account_v somewhat_o more_o exact_a and_o consequent_o more_o tedious_a and_o troublesome_a to_o use_v either_o by_o the_o pen_n or_o compass_n which_o be_v this_o you_o must_v measure_v the_o diameter_n at_o head_n and_o bung_n or_o the_o top_n and_o bottom_n in_o inch_n and_o 10th_n the_o length_n also_o by_o the_o same_o measure_n then_o find_v out_o the_o superficial_a content_n of_o the_o circle_n answerable_a to_o those_o two_o diameter_n and_o take_v two_o three_o of_o the_o great_a and_o one_o three_o of_o the_o least_o and_o add_v they_o together_o in_o one_o sum_n which_o sum_n you_o must_v multiply_v by_o the_o length_n in_o inch_n and_o ten_o and_o the_o product_n shall_v be_v the_o content_a in_o cube_n inch_n which_o product_v divide_v by_o 282_o give_v ale_n gallon_n or_o by_o 231_o give_v wine_n gallon_n as_o before_o by_o the_o line_n of_o number_n this_o way_n be_v more_o easy_a and_o ready_a thus_o the_o extent_n from_o 1_o to_o 0-5236_a a_o number_n fit_a for_o 2_o three_o of_o the_o circle_n at_o the_o bung_n so_o be_v the_o square_a of_o the_o diameter_n at_o the_o bung_n to_o a_o four_o then_o again_o as_o 1_o to_o 0-2618_a the_o half_a of_o the_o former_a number_n and_o fit_a for_o one_o three_o of_o the_o circle_n at_o head_n so_o be_v the_o square_a of_o the_o diameter_n at_o head_n to_o a_o four_o these_o two_o 4th_n add_v together_o then_o say_v as_o 231_o for_o wine_n or_o 282_o ¼_n for_o ale-gallon_n be_v to_o the_o sum_n of_o the_o two_o 4th_n add_v together_o so_o be_v the_o length_n to_o the_o content_a in_o wine-gallon_n example_n at_o 18_o inch_n at_o head_n and_o 32_o at_o bung_n that_o old_a example_n the_o square_a of_o 32_o be_v 1024_o the_o square_n of_o 18_o be_v 324_o then_o the_o extent_n of_o the_o compass_n from_o 1_o to_o 0-5236_a shall_v reach_v from_o 1024_o the_o square_a of_o 32_o to_o 536-4_a two_o three_o of_o the_o bung-circle_n again_o the_o extent_n from_o 1_o to_o 0-2618_a shall_v reach_v from_o 324_o the_o square_a of_o 18_o the_o diameter_n at_o head_n to_o 84-9_a the_o sum_n of_o 536-4_a and_o 84-9_a be_v 621-3_a then_o last_o the_o extent_n from_o 231_o the_o 621-3_a shall_v reach_v from_o 40_o the_o length_n to_o 107-58_a or_o 107_o gallon_n and_o a_o half_a and_o better_o the_o content_a in_o wine-gallon_n as_o brief_o as_o can_v be_v do_v this_o way_n but_o if_o you_o take_v the_o diameter_n at_o head_n and_o bung_n with_o a_o line_n call_v oughtred_n gage-line_n and_o set_v the_o measure_n find_v at_o the_o bung_n by_o that_o line_n down_o twice_o and_o the_o measure_n find_v at_o the_o head_n find_v by_o the_o same_o line_n once_o and_o bring_v they_o into_o one_o sum_n then_o multiply_v that_o sum_n by_o the_o length_n of_o the_o vessel_n in_o inch_n and_o 10_o part_n and_o then_o the_o product_n shall_v be_v the_o content_a in_o wine-gallon_n require_v as_o if_o i_o shall_v measure_v a_o cask_n of_o 18_o and_o 32_o as_o before_o right_o against_o 18_o inch_n on_o oughtreds-line_n you_o find_v 0-367_a and_o right_o against_o 32_o you_o shall_v find_v 1-161_a this_o last_o set_v down_o twice_o and_o 0-367_a once_o add_v make_v 2-689_a and_o then_o this_o sum_n multiply_v by_o 40_o make_v 107-56_a be_v very_o near_o to_o the_o former_a operation_n but_o differ_v about_o 2_o gallon_n from_o the_o way_n set_v before_o by_o the_o mean_a diameter_n and_o gagepoint_n by_o reason_n of_o the_o extreme_a swell_n of_o the_o cask_n but_o if_o this_o way_n shall_v prove_v the_o true_a in_o the_o book_n of_o the_o carpenter_n rule_n you_o have_v a_o table_n to_o rectify_v this_o difference_n which_o you_o will_v very_o seldom_o have_v occasion_n to_o use_v note_v also_o that_o this_o line_n call_v oughtred_n gage-line_n be_v very_o excellent_o improve_v to_o find_v the_o content_a of_o great_a vessel_n either_o in_o the_o whole_a or_o inch_n by_o inch_n which_o you_o will_v find_v at_o large_a in_o the_o book_n before_o mention_v also_o the_o use_n of_o the_o line_n call_v diagonal-line_n and_o line_n to_o find_v the_o emptiness_n of_o cask_n and_o to_o measure_v corn-measure_n by_o to_o which_o i_o shall_v for_o the_o present_a refer_v you_o problem_n iu._n the_o diameter_n and_o content_n of_o a_o vessel_n be_v give_v to_o find_v the_o length_n of_o the_o vessel_n extend_v the_o compass_n from_o the_o diameter_n to_o the_o gagepoint_n the_o same_o extent_n twice_o repeat_v from_o the_o content_a shall_v give_v the_o length_n require_v example_n if_o the_o content_a be_v 60_o and_o the_o diameter_n 24_o then_o extend_v the_o compass_n from_o 24_o to_o 17-15_a the_o gagepoint_n for_o wine_n this_o extent_n turn_v twice_o the_o same_o way_n from_o 60_o the_o content_a shall_v reach_v to_o 30_o inch_n and_o 6_o ten_o and_o a_o half_a the_o length_n require_v problem_n v._o the_o length_n and_o content_a of_o a_o vessel_n give_v to_o find_v the_o diameter_n divide_v the_o space_n on_o the_o line_n of_o number_n between_o the_o length_n and_o the_o content_a into_o two_o equal_a part_n the_o compass_n so_o set_v shall_v reach_v the_o same_o way_n from_o the_o gagepoint_n to_o the_o diameter_n of_o the_o vessel_n example_n the_o half_a distance_n between_o 31-65_a the_o length_n and_o 60_o the_o content_a shall_v reach_v the_o same_o way_n from_o 17-15_a the_o gagepoint_n to_o 24_o the_o diameter_n require_v these_o two_o last_o problem_n may_v be_v useful_a for_o cooper_n to_o make_v cask_n of_o any_o length_n diameter_n and_o quantity_n problem_n vi_o to_o find_v what_o be_v want_v in_o any_o close_a cask_n at_o any_o number_n of_o inch_n and_o part_n the_o cask_n lie_v after_o the_o usual_a manner_n with_o the_o bunghole_n uppermost_a from_o the_o bunghole_n to_o the_o superficies_n of_o the_o liquor_n give_v two_o way_n this_o problem_n i_o shall_v resolve_v two_o way_n either_o of_o which_o be_v experiment_a to_o come_v near_o the_o truth_n and_o will_v very_o well_o serve_v till_o a_o better_a come_v to_o light_v the_o one_o by_o a_o line_n of_o segment_n join_v to_o the_o line_n of_o number_n as_o before_o in_o the_o measure_n
extent_n shall_v reach_v in_o the_o same_o line_n from_o the_o true_a hour_n of_o the_o night_n to_o the_o star_n hour_n from_o the_o meridian_n then_o lay_v the_o thread_n to_o the_o star_n declination_n the_o nd_o from_o the_o star_n hour_n in_o the_o line_n of_o hour_n to_o the_o thread_n measure_v on_o the_o particular_a scale_n of_o altitude_n give_v the_o star_n altitude_n then_o by_o his_o declination_n and_o altitude_n you_o may_v soon_o find_v his_o azimuth_n by_o use_v 27._o and_o if_o the_o instrument_n be_v neat_o fix_v to_o a_o foot_n to_o set_v north_n and_o south_n and_o turn_v to_o any_o azimuth_n and_o altitude_n you_o may_v find_v any_o star_n at_o any_o time_n convenient_a and_o visible_a use_v xlvii_o the_o altitude_n and_o azimuth_n of_o any_o star_n be_v give_v to_o find_v his_o declination_n lay_v the_o thread_n to_o the_o altitude_n on_o the_o degree_n count_v from_o 600_o towards_o the_o end_n then_o set_v one_o point_n on_o the_o star_n azimuth_n count_v in_o the_o azimuth_n line_n and_o take_v the_o nd_o from_o thence_o to_o the_o thread_n which_o distance_n measure_v from_o the_o begin_n of_o the_o particular_a scale_n of_o altitude_n shall_v give_v the_o declination_n if_o the_o compass_n stand_v on_o the_o rightside_n of_o the_o thread_n than_o the_o declination_n be_v north_n if_o on_o the_o left_a it_o be_v south_n according_a as_o you_o work_v for_o the_o sun_n azimuth_n in_o a_o particular_a latitude_n use_v xlviii_o the_o altitude_n and_o declination_n of_o any_o star_n with_o the_o right_a ascension_n of_o the_o sun_n and_o the_o true_a hour_n of_o the_o night_n give_v to_o find_v the_o right_a ascension_n of_o that_o star_n first_o by_o the_o 43d_o use_v find_v the_o star_n hour_n viz._n how_o many_o hour_n and_o minute_n it_o want_v of_o come_v to_o or_o be_v past_o the_o meridian_n then_o the_o extent_n of_o the_o compass_n on_o the_o line_n of_o 24_o hour_n on_o the_o head-leg_n from_o the_o star_n hour_n to_o the_o true_a hour_n shall_v reach_v the_o same_o way_n from_o the_o sun_n right_a ascension_n to_o the_o star_n right_a ascension_n on_o the_o line_n of_o twice_o 12_o or_o 24_o hour_n use_v xlix_o to_o find_v when_o any_o fixed-star_n come_v to_o south_n by_o the_o line_n of_o twice_o 12_o or_o 24_o hour_n in_o use_v 42_o section_n 4_o you_o have_v the_o way_n by_o substraction_n with_o its_o caution_n but_o by_o the_o line_n of_o twice_o 12_o or_o 24_o hour_n work_v thus_o count_v the_o sun_n right_a ascension_n on_o that_o line_n and_o take_v the_o distance_n from_o thence_o to_o the_o next_o 12_o backward_o viz._n that_o at_o ♈_o at_o the_o begin_n of_o the_o line_n when_o the_o sun_n right_a ascension_n be_v under_o 12_o hour_n or_o to_o the_o next_o 12_o in_o the_o middle_n of_o the_o rule_n at_o ♎_o when_o the_o sun_n right_a ascension_n be_v above_o 12_o hour_n which_o be_v nothing_o but_o a_o reject_v 12_o for_o more_o conveniency_n then_o the_o same_o extent_n lay_v the_o same_o way_n from_o the_o star_n right_a ascension_n shall_v reach_v to_o the_o star_n come_v to_o south_n or_o the_o extent_n from_o the_o sun_n to_o the_o star_n right_a ascension_n shall_v reach_v the_o same_o way_n from_o 12_o to_o the_o star_n come_v to_o south_n example_n for_o the_o lyons-heart_n august_n 20._o the_o sun_n right_a ascension_n the_o 20_o of_o august_n be_v 10_o hour_n 36_o minute_n the_o right_a ascension_n of_o the_o lions-heart_n be_v 9_o hour_n and_o 50_o min._n therefore_o the_o extent_n from_o 10_o hour_n 35_o min._n to_o the_o beginning_n shall_v reach_v the_o same_o way_n from_o 9_o hour_n 50_o min._n by_o borrow_v 12_o hour_n because_o the_o sun_n right_a ascension_n be_v more_o than_o the_o star_n to_o 11_o hour_n 13_o min._n of_o the_o next_o day_n viz._n at_o a_o quarter_n past_o 11_o or_o at_o 11_o hour_n and_o 13_o min._n the_o same_o day_n where_o you_o may_v observe_v that_o the_o remainder_n be_v above_o 12_o if_o you_o add_v 24_o hour_n the_o time_n of_o sothe_v be_v between_o midnight_n and_o midday_n next_o follow_v use_v l._n to_o find_v what_o two_o day_n in_o the_o year_n be_v of_o equal_a length_n and_o the_o sun_n rise_v and_o set_v lay_v the_o thread_n on_o any_o one_o day_n in_o the_o upper_a line_n of_o month_n and_o day_n and_o at_o the_o same_o time_n the_o thread_n cut_v in_o the_o lower-line_n of_o month_n the_o day_n that_o be_v answerable_a to_o it_o in_o length_n rise_v set_v and_o declination_n and_o other_o requisite_n example_n the_o one_a of_o april_n and_o the_o 21_o of_o august_n be_v day_n of_o equal_a length_n and_o the_o sun_n rise_v and_o set_v be_v the_o same_o on_o both_o those_o day_n only_o in_o the_o upper-line_n the_o day_n be_v increase_v in_o length_n and_o in_o the_o lower-line_n they_o be_v decrease_v use_v li._n to_o find_v how_o many_o degree_n the_o sun_n be_v under_o the_o horizon_n at_o any_o hour_n the_o declination_n and_o hour_n be_v give_v count_n the_o sun_n declination_n on_o the_o degree_n the_o contrary_a way_n viz._n for_o north_n declination_n count_v from_o 600_o towards_o the_o end_n and_o count_v for_o southern_a declination_n towards_o the_o head_n and_o thereunto_o lie_v the_o thread_n then_o take_v the_o near_a distance_n from_o the_o hour_n give_v to_o the_o thread_n this_o distance_n measure_v in_o the_o particular_a scale_n of_o altitude_n shall_v show_v the_o sun_n depression_n under_o the_o horizon_n at_o that_o hour_n example_n january_n the_o 10_o at_o 8_o at_o night_n how_o many_o degree_n be_v the_o sun_n under_o the_o horizon_n on_o that_o day_n and_o hour_n the_o sun_n declination_n be_v about_o 20_o degree_n south_n then_o if_o i_o lay_v the_o thread_n to_o 20_o degree_n of_o declination_n north_n and_o take_v the_o near_a distance_n from_o 8_o to_o the_o thread_n that_o distance_n i_o say_v measure_v in_o the_o particular_a scale_n give_v 34_o degree_n and_o 9_o min._n for_o the_o sun_n depression_n under_o the_o horizon_n of_o 8_o afternoon_n to_o do_v this_o in_o other_o latitude_n you_o be_v to_o find_v the_o sun_n altitude_n at_o 8_o in_o northern_a declination_n by_o use_v 37._o chap._n xvii_o the_o use_n of_o the_o triangule_a quadrant_n in_o find_v of_o height_n and_o distance_n accessable_a or_o inaccessible_a use_v i._o to_o find_v a_o altitude_n at_o one_o station_n first_o the_o triangule_a quadrant_n be_v rectify_v and_o fix_v to_o a_o ball_n and_o socket_n and_o three-legged-staff_n be_v necessary_a in_o these_o operation_n to_o perform_v they_o exact_o especial_o for_o distance_n look_v up_o to_o the_o object_n as_o you_o will_v to_o a_o star_n and_o observe_v what_o degree_n and_o minute_n the_o thread_n cut_v and_o set_v it_o down_o also_o observe_v the_o place_n where_o you_o stand_v at_o the_o time_n of_o observation_n and_o the_o distance_n from_o your_o eye_n to_o the_o ground_n and_o the_o place_n on_o the_o object_n that_o be_v level_a with_o your_o eye_n also_o as_o the_o play_v of_o the_o thread_n and_o plummet_n will_v plain_o show_v also_o you_o must_v have_v the_o measure_n from_o the_o place_n where_o you_o stand_v observe_v to_o the_o point_n exact_o right_a under_o the_o object_n who_o height_n you_o will_v have_v in_o foot_n yard_n perch_n or_o what_o you_o please_v to_o integer_n and_o fraction_n in_o decimal_n if_o it_o may_v be_v also_o note_v that_o in_o all_o right-angle-triangle_n one_o acute_a angle_n be_v always_o the_o compliment_n of_o the_o other_o so_o that_o observe_v or_o find_v one_o by_o observation_n by_o consequence_n you_o have_v the_o other_o by_o take_v that_o from_o 90._o these_o thing_n be_v premise_v the_o operation_n follow_v by_o the_o artificial_a number_n sin_n and_o tangent_n and_o also_o by_o the_o natural_a note_v also_o by_o the_o way_n that_o in_o regard_n the_o compliment_n of_o the_o angle_n observe_v be_v frequent_o use_v if_o you_o count_v the_o degree_n the_o contrary_a way_n that_o be_v to_o say_v from_o the_o head_n you_o shall_v have_v the_o compliment_n require_v as_o have_v be_v oftentimes_o hint_v before_o then_o as_o the_o sine_fw-la of_o the_o angle_n opposite_a to_o the_o measure_a side_n be_v to_o the_o measure_a side_n count_v on_o the_o number_n so_o be_v the_o sine_fw-la of_o the_o angle_n find_v to_o the_o altitude_n or_o height_n require_v on_o number_n example_n at_o one_o station_n stand_v at_o c_o i_o look_v up_o to_o b_o the_o object_n who_o height_n be_v require_v and_o i_o find_v the_o thread_n to_o fall_v on_o 41_o degree_n and_o 45_o minute_n but_o if_o you_o count_v from_o the_o head_n it_o be_v 48-15_a the_o compliment_n thereof_o as_o in_o the_o figure_n you_o see_v also_o the_o measure_n from_o c_o to_o a_o be_v find_v to_o be_v