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
right_a ascension_n and_o the_o ascentional-difference_n be_v the_o oblique-ascention_n but_o in_o southern_a declination_n the_o sum_n of_o the_o right_a ascension_n and_o difference_n of_o asscention_n be_v the_o oblique_a ascension_n example_n on_o or_o between_o the_o 25_o and_o 26_o of_o july_n the_o oblique-ascention_n be_v by_o substraction_n 112_o 15_o on_o the_o 30_o of_o october_n the_o oblique-ascention_n be_v 337_o 45_o by_o addition_n use_v viii_o the_o day_n of_o the_o month_n or_o sun_n declination_n and_o altitude_n be_v give_v to_o find_v the_o hour_n of_o the_o day_n take_v the_o sun_n altitude_n from_o the_o particular_a scale_n of_o altitude_n set_v one_o point_n of_o the_o compass_n in_o the_o centre_n at_o the_o begin_n of_o that_o line_n and_o open_v the_o other_o to_o the_o degree_n and_o minute_n of_o the_o sun_n altitude_n count_v on_o that_o line_n then_o lay_v the_o thread_n on_o the_o day_n of_o the_o month_n or_o declination_n and_o there_o keep_v it_o then_o carry_v the_o compass_n set_v at_o the_o former_a distance_n along_o the_o line_n of_o hour_n perpendicule_a to_o the_o thread_n till_o the_o other_o point_n be_v turn_v about_o will_v but_o just_a touch_n the_o thread_n the_o compass_n stand_v between_o the_o thread_n and_o the_o hour_n 12_o than_o the_o fix_a point_n in_o the_o hour-line_n shall_v show_v the_o hour_n and_o minute_n require_v but_o whether_o it_o be_v the_o fore_n or_o afternoon_n your_o judgement_n or_o a_o second_o observation_n must_v determine_v example_n on_o the_o first_o of_o august_n in_o the_o morning_n at_o 20_o degree_n of_o altitude_n you_o shall_v find_v it_o to_o be_v just_a 52_o minute_n past_o 6_o but_o at_o the_o same_o altitude_n in_o the_o afternoon_n it_o be_v 7_o minute_n past_o 5_o at_o night_n in_o the_o latitude_n of_o 51_o 32_o for_o london_n use_v ix_o the_o sun_n declination_n and_o altitude_n give_v to_o find_v the_o sun_n azimuth_n from_o the_o south-part_n of_o the_o horizon_n first_o by_o the_o four_o use_n find_v the_o sun_n declination_n count_v the_o same_o on_o the_o particular_a scale_n and_o take_v the_o distance_n between_o your_o compass_n then_o lay_v the_o thread_n to_o the_o sun_n altitude_n count_v the_o same_o way_n as_o the_o southern-declination_n be_v from_o 600_o towards_o the_o loose-piece_n and_o when_o need_n require_v on_o the_o loose-piece_n then_o carry_v the_o compass_n along_o the_o azimuth-line_n on_o the_o rightside_n of_o the_o thread_n that_o be_v between_o the_o thread_n and_o the_o head_n when_o the_o declination_n be_v northward_o and_o on_o the_o left-side_n of_o the_o thread_n that_o be_v between_o the_o thread_n and_o the_o end_n when_o the_o declination_n be_v southward_o so_o as_o the_o compass_n set_v to_o the_o declination_n as_o before_o and_o one_o point_v stay_v on_o the_o azimuth-line_n and_o the_o other_o turn_v about_o shall_v but_o just_a touch_n the_o thread_n at_o the_o near_a distance_n then_o i_o say_v the_o fixed-point_n shall_v in_o the_o azimuth-line_n show_v the_o suns-azimuth_n require_v example_n 1._o the_o sun_n be_v in_o the_o equinoctial_a and_o have_v no_o declination_n you_o have_v nothing_o to_o take_v with_o your_o compass_n but_o only_o lay_v the_o thread_n to_o the_o altitude_n count_v from_o 600_o towards_o the_o loose-piece_n and_o in_o the_o azimuth-line_n it_o cut_v the_o azimuth_n require_v example_n at_o 25_o degree_n high_a you_o shall_v find_v the_o sun_n azimuth_n to_o be_v 54_o 10_o at_o 32_o degree_n high_a you_o shall_v find_v 38_o 20_o the_o azimuth_n again_o at_o 20_o degree_n of_o declination_n take_v 20_o from_o the_o particular_a scale_n and_o at_o 10_o degree_n of_o altitude_n lay_v the_o thread_n to_o 10_o count_v as_o before_o then_o if_o you_o carry_v the_o compass_n on_o the_o rightside_n for_o north-declination_n you_o shall_v find_v 109_o 30_o from_o south_n but_o if_o you_o carry_v they_o on_o the_o left-side_n for_o south-declination_n you_o shall_v find_v 38_o 30_o from_o south_n the_o rest_n of_o the_o use_v you_o shall_v have_v more_o ample_o afterward_o chap._n vi_o the_o use_n of_o the_o line_n of_o number_n on_o the_o edge_n and_o the_o line_n of_o line_n on_o the_o quadrantal-side_n or_o on_o the_o sector-side_n be_v all_o as_o one_o have_v show_v the_o way_n of_o numeration_n on_o the_o line_n as_o in_o chapter_n the_o first_o also_o to_o add_v or_o subtract_v one_o line_n or_o number_n to_o or_o from_o one_o another_o as_o in_o chapter_n four_o explanation_n the_o 9th_o i_o come_v now_o to_o work_v the_o rule_n of_o multiplication_n and_o division_n and_o the_o rule_n of_o three_o direct_a and_o reverse_v both_o by_o the_o artificial_a and_o natural-line_n and_o first_o by_o the_o artificial_a be_v the_o most_o easy_a and_o then_o by_o the_o natural-line_n both_o on_o the_o sector_n and_o triangule_a quadrant_n be_v alike_o and_o i_o work_v they_o together_o first_o because_o i_o will_v avoid_v tautology_n second_o because_o thereby_o be_v better_o see_v the_o harmony_n between_o they_o and_o which_o be_v best_a and_o speedy_a three_o because_o it_o be_v a_o way_n not_o yet_o as_o i_o know_v of_o go_v by_o any_o other_o and_o last_o of_o all_o because_o one_o may_v explain_v the_o other_o the_o geometrical_a figure_n be_v the_o same_o with_o the_o instrumental-work_n by_o the_o natural_a way_n sect._n i._o to_o multiply_v one_o number_n by_o another_o 1._o by_o the_o line_n of_o number_n on_o the_o edge_n artificial_o thus_o extend_v the_o compass_n from_o 1_o to_o the_o multiplicator_n the_o same_o extent_n apply_v the_o same_o way_n from_o the_o multiplicand_a will_v cause_v the_o other_o point_n to_o fall_v on_o the_o product_n require_v example_n let_v 8_o be_v give_v to_o be_v multiply_v by_o 6_o if_o you_o set_v one_o point_n of_o the_o compass_n in_o 1_o either_o at_o the_o beginning_n or_o at_o the_o middle_n or_o at_o the_o end_n it_o matter_n not_o which_o yet_o the_o middle_n 1_o on_o the_o head-leg_n be_v for_o the_o most_o part_n the_o most_o convenient_a and_o open_v the_o other_o to_o 6_o or_o 8_o it_o matter_n not_o which_o for_o 6_o time_n 8_o and_o 8_o time_n 6_o be_v alike_o but_o yet_o you_o may_v mind_v the_o precept_n if_o you_o will_v the_o same_o extent_n lay_v the_o same_o way_n from_o 8_o shall_v reach_v to_o 48_o the_o product_v require_v which_o without_o these_o parenthesis_n be_v thus_o the_o extent_n from_o 1_o to_o 6_o shall_v reach_v the_o same_o way_n from_o 8_o to_o 48._o or_o the_o extent_n from_o 1_o to_o 8_o shall_v reach_v the_o same_o way_n from_o 6_o to_o 48._o the_o product_v require_v by_o the_o natural-line_n on_o the_o sector-side_n or_o triangule_a quadrant_n with_o a_o thread_n and_o compass_n the_o work_n be_v thus_o 1._o for_o the_o most_o part_n it_o be_v wrought_v by_o change_v the_o term_n from_o the_o artificial_a way_n as_o thus_o the_o former_a way_n be_v as_o 1_o to_o 6_o so_o be_v 8_o to_o 48_o or_o as_o 1_o to_o 8_o so_o be_v 6_o to_o 48_o but_o by_o the_o sector_n it_o be_v thus_o as_o the_o latteral_a 6_o take_v from_o the_o centre_n towards_o the_o end_n be_v to_o the_o parallel_n 10_o &_o 10_o set_v over_o from_o 10_o to_o 10_o at_o the_o end_n count_v as_o 1_o so_o be_v the_o parallel-distance_n between_o 8_o &_o 8_o on_o the_o line_n of_o line_n take_v across_o from_o one_o leg_n to_o the_o other_o to_o the_o latteral-distance_n from_o the_o centre_n to_o 48_o the_o product_v require_v or_o short_a thus_o as_o the_o latteral_a 8_o to_o the_o parallel_n 10_o so_o be_v the_o parallel_n 6_o to_o the_o latteral_a 48._o see_v figure_n i._n 2._o another_o way_n may_v you_o work_v without_o alter_v the_o term_n from_o the_o artificial_a way_n as_o thus_o by_o a_o double_a radius_fw-la take_v the_o latteral-extent_n from_o the_o centre_n to_o 1_o or_o from_o 10_o to_o 9_o if_o the_o beginning_n be_v defective_a make_v this_o a_o parallel_n in_o 6_o &_o 6_o than_o the_o latteral-extent_n from_o the_o centre_n to_o 8_o of_o the_o 10_o part_n between_o figure_n and_o figure_n shall_v reach_v across_o from_o 48_o to_o 48_o as_o before_o see_v fig._n ii_o the_o same_o work_n as_o be_v do_v by_o the_o sector_n be_v do_v by_o the_o line_n of_o line_n and_o thread_n on_o the_o quadrant-side_n that_o if_o your_o sector_n be_v put_v together_o as_o a_o triangule_a quadrant_n you_o may_v work_v any_o thing_n by_o it_o as_o well_o as_o by_o the_o sector_n in_o this_o manner_n or_o by_o the_o scale_n and_o compass_n as_o in_o the_o figure_n i._o and_o first_o as_o above_z sector-wise_a take_v the_o extent_n from_o the_o centre_n to_o 6_o latteral_o between_o your_o compass_n set_v one_o point_n in_o 10_o and_o with_o the_o other_o lie_v the_o thread_n in_o the_o near_a distance_n
or_o from_o 12_o as_o it_o be_v figure_v example_n on_o april_n 20_o at_o 30_o deg_n 20_o min._n of_o altitude_n latitude_n 51-32_a the_o hour_n will_v be_v find_v to_o be_v just_a 2_o hour_n from_o 6_o or_o just_a 8._o again_o on_o the_o 10_o of_o november_n at_o 8_o deg_n 25_o min._n high_a it_o be_v just_a 3_o hour_n from_o 6_o or_o 9_o a_o clock_n in_o the_o forenoon_n or_o 3_o afternoon_n or_o somewhat_o differ_v thus_o take_v the_o sine_fw-la of_o the_o sum_n or_o difference_n of_o the_o sun_n present_a altitude_n and_o altitude_n at_o 6_o and_o make_v it_o a_o =_o in_o the_o cousin_a of_o the_o latitude_n and_o lay_v the_o thread_n to_o the_o near_a distance_n then_o take_v out_o the_o =_o secant_fw-la of_o the_o declination_n beyond_o 90_o and_o make_v it_o a_o =_o sine_fw-la of_o 90_o and_o lay_v the_o thread_n to_o the_o near_a distance_n on_o the_o degree_n it_o shall_v show_v the_o hour_n from_o 6_o require_v tang._n first_o by_z use_v 25_o find_v the_o sun_n height_n at_o 6_o or_o depression_n in_o winter_n then_o by_o the_o former_a 2d_o find_v the_o sum_n or_o difference_n between_o the_o altitude_n at_o 6_o and_o the_o sun_n present_a altitude_n but_o if_o you_o have_v table_n of_o natural_a sin_n and_o tangent_n then_o in_o winter_n add_v the_o natural_a sin_n of_o the_o two_o altitude_n together_o and_o in_o summer_n subtract_v the_o lesser_a out_o of_o the_o great_a and_o find_v the_o ark_n of_o difference_n more_o exact_o then_o as_o the_o cousin_a of_o the_o latitude_n to_o the_o secant_fw-la of_o the_o declination_n count_v beyond_o 90_o as_o much_o forward_a as_o from_o 90_o to_o the_o cousin_a of_o the_o sun_n declination_n so_o be_v the_o sine_fw-la of_o the_o sum_n or_o difference_n to_o the_o hour_n from_o 6_o require_v or_o else_o â—hus_o as_o the_o cousin_a of_o the_o latitude_n to_o the_o sine_fw-la of_o the_o sum_n or_o difference_n so_o be_v sine_fw-la of_o 90_o to_o a_o four_o then_o as_o the_o cousin_a of_o the_o sun_n declination_n to_o that_o four_o so_o be_v sine_fw-la 90_o to_o the_o hour_n from_o 6._o by_o the_o sector_n take_v the_o secant_fw-la of_o the_o sun_n declination_n make_v it_o a_o =_o in_o the_o cousin_a of_o the_o latitude_n then_o take_v out_o the_o =_o sine_fw-la of_o the_o sum_n or_o difference_n and_o turn_v it_o twice_o from_o the_o centre_n latteraâ—ly_o and_o it_o shall_v be_v the_o sine_fw-la of_o the_o hour_n from_o 6_o require_v example_n april_n 20_o the_o sun_n declination_n be_v 15_o degree_n and_o the_o sun_n height_n at_o 6_o then_o be_v 11_o deg_n 42_o min._n now_o the_o natural_a sine_fw-la of_o 11-42_a 20278_o take_v from_o the_o natural_a sine_fw-la of_o 30_o deg_n 20_o min._n 50502_o the_o sun_n present_a altitude_n the_o residue_n be_v 30224_o the_o sine_fw-la of_o 17_o deg_n 35_o min._n and_o a_o half_a then_o the_o secant_fw-la of_o 15_o make_v a_o =_o sine_fw-la of_o 38-28_a and_o the_o sector_n so_o set_v the_o =_o sine_fw-la of_o 17-35_a ½_o turn_v latteral_o twice_o from_o the_o centre_n shall_v reach_v to_o 30_o the_o sine_fw-la of_o 2_o hour_n from_o 6_o the_o hour_n require_v use_v xxvii_o have_v the_o latitude_n the_o sun_n declination_n and_o altitude_n to_o find_v the_o sun_n azimuth_n quadrant_n take_v the_o declination_n from_o the_o particular_a scale_n of_o sines_n for_o the_o particular_a latitude_n the_o instrument_n be_v make_v for_o then_o count_v the_o give_v altitude_n on_o the_o degree_n from_o 600_o towards_o the_o loose-piece_n and_o sometime_o on_o the_o loose-piece_n also_o and_o thereunto_o lie_v the_o thread_n then_o carry_v the_o compass_n so_o set_v along_o the_o azimuth-line_n on_o the_o rightside_n of_o the_o thread_n in_o northern-declination_n and_o on_o the_o left-side_n in_o southern-declination_n till_o the_o other_o foot_n turn_v about_o will_v but_o just_a touch_n the_o thread_n then_o the_o fixed-point_n shall_v stay_v at_o the_o sun_n true_a azimuth_n require_v take_v two_o or_o three_o example_n 1._o first_o when_o the_o sun_n be_v in_o the_o equinoctial_a and_o have_v no_o declination_n than_o there_o be_v nothing_o to_o take_v between_o your_o compass_n but_o just_a to_o lay_v the_o thread_n to_o the_o sun_n altitude_n count_v from_o 600_o on_o the_o loose-piece_n towards_o the_o end_n then_o 〈◊〉_d the_o azimuth-line_n it_o cut_v the_o azimuth_n from_o the_o south_n require_v example_n at_o 00_o degree_n high_a the_o azimuth_n be_v 90_o from_o south_n and_o at_o 10_o degree_n high_a it_o be_v 77-5_a at_o 20_o high_a the_o azimuth_n be_v 62-45_a at_o 30_o degree_n high_a it_o be_v 43-30_a at_o 34_o degree_n high_a it_o be_v 32_o degree_n of_o azimuth_n from_o south_n and_o at_o 38-28_a degree_n high_a it_o be_v just_a south_n 2._o second_o at_o 10_o degree_n of_o declination_n northward_o and_o 20_o degree_n of_o altitude_n take_v 10_o degree_n from_o the_o particular_a scale_n and_o lay_v the_o thread_n to_o the_o sun_n present_a altitude_n as_o before_o and_o carry_v the_o compass_n on_o the_o rightside_n of_o the_o thread_n on_o the_o azimuth-line_n till_o the_o other_o foot_n be_v turn_v about_o will_v but_o just_a touch_n it_o then_o shall_v the_o point_n rest_n at_o 80_o degree_n 42_o min._n of_o azimuth_n from_o the_o south_n 3._o but_o if_o the_o declimation_n be_v the_o same_o to_o the_o southwards_o and_o the_o altitude_n also_o the_o same_o then_o carry_v the_o compass_n on_o the_o left-side_n of_o the_o thread_n on_o the_o azimuth-line_n till_o the_o other_o foot_n turn_v about_o will_v but_o just_a touch_n it_o and_o you_o shall_v find_v the_o point_n to_o stay_v at_o 41_o deg_n 10_o min._n the_o true_a azimuth_n from_o the_o south_n require_v note_v that_o any_o thing_n as_o thick_a as_o the_o rule_n lay_v by_o the_o rule_n and_o the_o thread_n draw_v over_o it_o will_v keep_v the_o thread_n steady_a till_o you_o get_v the_o near_a distance_n more_o true_o first_o by_o the_o 18_o use_n find_v the_o sun_n altitude_n in_o the_o vertical_a circle_n or_o circle_n of_o east_n and_o west_n thus_o general-quadr_a take_v the_o sine_fw-la of_o the_o sun_n declination_n and_o set_v one_o foot_n in_o the_o sine_fw-la of_o latitude_n lay_v the_o thread_n to_o nd_o and_o in_o the_o degree_n you_o shall_v have_v the_o altitude_n at_o east_n and_o west_n require_v which_o vertical_a altitude_n in_o summer_n or_o northern_a declination_n you_o must_v subtract_v out_o of_o the_o sun_n present_a altitude_n or_o take_v the_o lesser_a from_o the_o great_a to_o find_v a_o difference_n but_o in_o winter_n you_o must_v add_v this_o depression_n in_o the_o vertical_a circle_n to_o the_o sun_n present_a altitude_n to_o get_v a_o sum_n which_o must_v be_v do_v on_o a_o line_n of_o natural_a sin_n or_o by_o the_o table_n of_o natural_a sin_n as_o before_o in_o the_o hour_n by_o lay_v it_o over_o or_o under_o the_o centre_n and_o take_v from_o that_o note_a point_n to_o the_o sun_n present_a altitude_n all_o that_o day_n then_o take_v the_o distance_n from_o the_o centre_n to_o the_o tangent_fw-la of_o the_o sun_n present_a altitude_n on_o the_o loose-piece_n which_o be_v the_o secant_fw-la of_o the_o sun_n present_a altitude_n and_o lay_v it_o from_o the_o centre_n on_o the_o line_n of_o sines_n and_o note_v the_o place_n then_o take_v the_o distance_n from_o 60_o on_o the_o loose-piece_n to_o the_o co-tangent_a of_o the_o latitude_n by_o count_v 10_o 20_o 30_o etc._n etc._n from_o 60_o towards_o the_o moveable-leg_n between_o your_o compass_n then_o set_v one_o point_n on_o the_o secant_fw-la of_o the_o sun_n altitude_n last_o find_v and_o note_v on_o the_o line_n of_o sines_n and_o with_o the_o other_o lay_v the_o thread_n to_o the_o near_a distance_n and_o there_o keep_v it_o by_o note_v what_o degree_n day_n of_o the_o month_n or_o hour_n &_o minute_n or_o azimuth_n it_o cut_v then_o take_v the_o distance_n on_o the_o sin_n from_o the_o sine_fw-la of_o the_o sun_n vertical_a altitude_n to_o his_o present_a altitude_n for_o a_o difference_n in_o summer_n or_o the_o distance_n from_o a_o point_n make_v beyond_o the_o centre_n equal_a to_o the_o sine_fw-la of_o the_o sun_n vertical_a depression_n to_o the_o sun_n present_a altitude_n for_o a_o sum_n in_o winter_n then_o have_v this_o distance_n of_o sum_n or_o difference_n for_o winter_n or_o summer_n between_o your_o compass_n carry_v one_o point_v parallelly_n on_o the_o line_n of_o sines_n till_o the_o other_o be_v turn_v about_o shall_v just_o touch_v the_o thread_n at_o the_o nd_o the_o place_n where_o the_o point_n stay_v shall_v be_v the_o azimuth_n from_o east_n or_o west_n as_o it_o be_v figure_v from_o the_o centre_n or_o from_o north_n or_o south_n count_v from_o 90._o which_o work_n in_o brief_a may_v be_v sufficient_o word_v thus_o as_o co-tangent_a
the_o compliment_n thereof_o to_o 12_o hour_n and_o the_o star_n rightascention_n and_o the_o hour_n of_o rise_v the_o thread_n cut_v and_o add_v they_o into_o one_o sum_n and_o the_o sum_n if_o under_o 12_o be_v the_o time_n of_o his_o rise_n in_o common_a hour_n or_o if_o you_o add_v the_o hour_n of_o set_v that_o the_o thread_n show_v it_o shall_v give_v his_o set_n example_n if_o you_o lay_v the_o thread_n to_o 15-48_a the_o declination_n of_o the_o bull_n eye_n in_o the_o hour-line_n it_o cut_v 4_o hour_n 36_o min._n for_o rise_v or_o 7-24_a for_o his_o set_v then_o if_o you_o work_v for_o april_n the_o 23d_o the_o sun_n right_a ascension_n then_o be_v 2-44_a and_o the_o compliment_n thereof_o to_o 12_o be_v 9-16_a and_o the_o star_n rightascention_n be_v 4_o hour_n and_o 16_o minute_n and_o the_o hour_n cut_v be_v 4-36_a for_o rise_v and_o the_o three_o number_n viz._n 9-16_a the_o compliment_n of_o the_o sun_n right_a ascension_n and_o 4-16_a the_o star_n right_a ascension_n and_o 4-36_a the_o hour_n of_o rise_v the_o thread_n cut_v be_v add_v make_v 18-8_a from_o which_o take_v 12_o rest_z 6-8_a the_o time_n that_o the_o bull_n eye_n rise_v on_o april_n 23_o and_o if_o you_o add_v 7-24_a the_o time_n of_o set_v that_o the_o thread_n cut_v there_o come_v forth_o 8-56_a viz._n one_o hour_n and_o 32_o min._n after_o the_o sun_n to_o find_v the_o time_n of_o a_o star_n come_v to_o south_n subtract_v the_o right_a ascension_n of_o the_o sun_n from_o the_o right_a ascension_n of_o the_o star_n increase_v by_o 24_o when_o you_o can_v do_v without_o and_o the_o remainder_n if_o less_o than_o 12_o be_v the_o time_n between_o 12_o at_o noon_n and_o 12_o at_o night_n but_o if_o the_o remainder_n be_v more_o than_o 12_o it_o be_v the_o time_n between_o midnight_n and_o midday_n follow_v example_n the_o lyons-heart_n who_o right_a ascension_n be_v 9-50_a will_v come_v to_o the_o south_n on_o march_n 10_o at_o 9-48_a the_o sun_n right_a ascension_n be_v then_o only_o 2_o minute_n by_o the_o line_n of_o 24_o hour_n say_v or_o work_v thus_o extend_v the_o compass_n from_o the_o sun_n right_a ascension_n to_o the_o star_n right_a ascension_n that_o distance_n lay_v the_o same_o way_n from_o 12_o at_o the_o middle_n or_o at_o the_o beginning_n shall_v reach_v to_o the_o time_n of_o the_o star_n come_v to_o south_n to_o find_v the_o time_n of_o the_o star_n continuance_n above_o the_o horizon_n first_o find_v what_o the_o sun_n semi-diurnal-ark_n be_v have_v the_o same_o declination_n and_o that_o double_a be_v the_o whole_a time_n of_o continuance_n or_o if_o you_o shall_v add_v and_o subtract_v it_o to_o or_o from_o the_o time_n of_o the_o star_n come_v to_o south_n you_o shall_v find_v the_o time_n of_o set_v or_o rise_v or_o else_o by_o lay_v the_o thread_n to_o the_o star_n declination_n it_o show_v the_o ascentional_a difference_n in_o this_o latitude_n which_o add_v in_o those_o star_n that_o have_v north_n declination_n or_o substract_v in_o southern_a to_o 6_o hour_n give_v the_o semi-diurnal_a ark_n of_o the_o star_n above_o the_o horizon_n example_n the_o eye_n be_v ascentional-difference_n be_v one_o hour_n and_o 24_o minute_n which_o add_v to_o 6_o hour_n because_o of_o northern_a declination_n make_v 7-24_a for_o the_o semi-diurnal-ark_n or_o 14_o 48′_n for_o the_o whole_a time_n of_o be_v above_o the_o horizon_n note_v that_o to_o work_v this_o for_o other_o latitude_n the_o sun_n ascentional-difference_n be_v to_o be_v find_v for_o that_o latitude_n you_o be_v in_o and_o the_o operation_n be_v general_a for_o all_o place_n to_o find_v a_o meridian_n line_n by_o the_o sun_n on_o any_o flat_a horizontal-plain_a set_v up_o a_o straight_a wire_n in_o the_o centre_n of_o a_o circle_n or_o hold_v up_o a_o thread_n or_o plummet_n till_o the_o shadow_n of_o the_o thread_n cut_v the_o centre_n and_o any_o where_o in_o the_o circumference_n which_o two_o point_n you_o must_v note_v then_o immediate_o take_v the_o sun_n altitude_n and_o find_v the_o sun_n azimuth_n and_o count_v so_o many_o degree_n in_o the_o circle_n the_o right_a way_n as_o the_o sun_n azimuth_n come_v to_o from_o the_o point_n of_o the_o shadow_n mark_v in_o the_o circumference_n and_o draw_v that_o line_n for_o a_o true_a meridian-line_n this_o work_n be_v best_a do_v before_o 10_o in_o the_o morning_n and_o after_o two_o afternoon_n or_o in_o the_o night_n by_o two_o plumb-line_n set_v in_o a_o rightline_n with_o the_o north-star_n at_o a_o right_a situation_n use_v xliii_o to_o find_v the_o hour_n of_o the_o night_n by_o the_o fix_a star_n first_o find_v the_o star_n altitude_n by_o look_v along_o the_o fix_a or_o moveable-leg_n to_o the_o middle_n of_o the_o star_n let_v the_o thread_n with_o a_o weighty_a plummet_n play_v even_o by_o the_o degree_n between_o your_o thumb_n and_o forefinger_n to_o the_o end_n you_o may_v command_v the_o thread_n and_o know_v whether_o it_o play_v well_o or_o no_o by_o feeling_n then_o take_v the_o altitude_n find_v from_o the_o particular_a scale_n of_o sines_n and_o lay_v the_o thread_n over_o the_o star_n declination_n which_o for_o readiness_n sake_n be_v mark_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 according_a to_o the_o figure_n set_v to_o the_o 12_o name_n of_o the_o 12_o star_n on_o the_o rule_n and_o then_o carry_v the_o compass_n as_o you_o do_v in_o find_v the_o hour_n by_o the_o sun_n you_o shall_v find_v how_o much_o the_o star_n want_v or_o be_v past_o the_o meridian_n which_o be_v call_v the_o stars-hour_n and_o note_v that_o if_o the_o star_n be_v past_o the_o south_n it_o be_v a_o afternoon_n hour_n if_o not_o come_v to_o the_o south_n a_o morning_n hour_n which_o you_o must_v remember_v also_o know_v the_o sun_n right_a ascension_n set_v one_o point_n of_o the_o compass_n in_o the_o sun_n right_a ascension_n count_v in_o the_o line_n of_o twice_o 12_o or_o 24_o hour_n on_o the_o outward-leg_n of_o the_o fixed-piece_n next_o to_o the_o particular_a scale_n of_o sines_n and_o open_v the_o other_o to_o the_o star_n right_a ascension_n note_v which_o way_n you_o turn_v the_o compass_n for_o the_o same_o extent_n apply_v the_o same_o way_n from_o the_o star_n hour_n last_o find_v shall_v show_v the_o true_a hour_n of_o the_o night_n require_v example_n suppose_v on_o the_o 10_o of_o january_n i_o shall_v observe_v the_o altitude_n of_o the_o bull_n eye_n to_o be_v 20_o degree_n if_o you_o take_v 20_o degree_n the_o altitude_n from_o the_o particular_a scale_n and_o lay_v the_o thread_n on_o 15-48_a the_o star_n declination_n northward_o and_o measure_n from_o the_o hour-scale_n the_o near_a distance_n to_o the_o thread_n you_o shall_v find_v the_o compass-point_n to_o stay_v at_o 6-49_a on_o the_o east-side_n of_o the_o meridian_n suppose_v also_o the_o sun_n right_a ascension_n the_o same_o day_n be_v 8_o hour_n and_o 12_o minute_n then_o the_o extent_n from_o 8_o hour_n 12_o minute_n on_o the_o line_n of_o twice_o 12_o hour_n the_o sun_n right_a ascension_n to_o 4-16_a the_o star_n right_a ascension_n shall_v reach_v the_o same_o way_n from_o 6-49_a the_o star_n hour_n to_o 2-53_a the_o true_a hour_n use_v xliv_o to_o find_v the_o hour_n of_o the_o night_n by_o the_o moon_n first_o by_o a_o almanac_n or_o ephemerides_n find_v the_o moon_n age_n and_o true_a place_n for_o the_o present_a time_n then_o by_o lay_v the_o thread_n on_o the_o moon_n place_n you_o may_v have_v her_o right_a ascension_n and_o also_o the_o sun_n right_a ascension_n and_o by_o the_o moon_n altitude_n take_v from_o the_o particular_a scale_n and_o the_o thread_n lay_v over_o the_o moon_n place_n you_o find_v what_o the_o moon_n want_v or_o be_v past_o come_v to_o south_n which_o be_v call_v the_o moon_n hour_n then_o by_o the_o line_n of_o 24_o hour_n say_v as_o the_o sun_n right_a ascension_n be_v to_o the_o moon_n right_a ascension_n so_o be_v the_o moon_n hour_n last_o find_v to_o the_o true_a hour_n example_n suppose_v that_o on_o the_o 8_o of_o january_n about_o 40_o min._n after_o 3_o there_o be_v a_o new_a moon_n then_o note_v that_o the_o sun_n true_a place_n be_v the_o moon_n true_a place_n and_o consequent_o their_o right_a ascension_n and_o the_o moon_n hour_n and_o altitude_n be_v the_o same_o with_o the_o sun_n therefore_o as_o 8_o hour_n 04_o min._n the_o sun_n right_a ascension_n be_v to_o 8-04_a the_o moon_n right_a ascension_n so_o be_v the_o moon_n hour_n at_o any_o altitude_n to_o the_o sun_n true_a hour_n again_o suppose_v that_o on_o the_o one_a quarter-day_n the_o moon_n be_v go_v 90_o degree_n from_o the_o sun_n to_o find_v her_o place_n then_o do_v thus_o set_v one_o point_n in_o the_o moon_n
place_n the_o change-day_n and_o open_v the_o other_o to_o the_o beginning_n or_o the_o end_n of_o the_o line_n of_o 24_o hour_n then_o the_o same_o extent_n apply_v the_o contrary_a way_n from_o 6_o hour_n or_o 7_o day_n and_o a_o half_a the_o moon_n age_n shall_v give_v 28_o deg_n 58_o min._n ♈_o to_o which_o you_o must_v add_v 7_o degree_n and_o 30_o minute_n the_o sun_n place_n between_o and_o the_o sum_n shall_v be_v the_o moon_n true_a place_n require_v viz._n 6-28_a degree_n in_o ♉_o example_n if_o the_o moon_n change_v on_o the_o 8_o day_n the_o first_o quarter_n be_v 7_o day_n and_o a_o half_a after_o will_v be_v on_o the_o 15_o day_n late_a at_o night_n then_o the_o difference_n between_o the_o sun_n and_o moon_n right_a ascension_n will_v be_v find_v to_o be_v near_o 6_o hour_n for_o the_o sun_n right_a ascension_n january_n 15_o be_v 8-32_a and_o the_o moon_n right_a ascension_n the_o same_o day_n be_v about_o 8_o degree_n and_o a_o half_a in_o ♉_o be_v 2_o hour_n and_o 28_o minute_n if_o you_o take_v the_o distance_n between_o they_o on_o the_o 24_o hour_n it_o be_v near_o 6_o hour_n which_o be_v the_o difference_n of_o time_n between_o the_o moon_n and_o the_o sun_n hour_n again_o for_o the_o full_a moon_n on_o the_o 22_o day_n near_o 4_o hour_n after_o noon_n the_o moon_n age_n be_v 14_o day_n ¾_n if_o you_o add_v 12_o hour_n or_o 6_o sign_n to_o the_o moon_n place_n aâ—_n the_o change_n you_o shall_v find_v ♋_o 29-0_a to_o which_o if_o you_o add_v 14-45_a the_o day_n between_o the_o new_a and_o full_a you_o shall_v find_v ♌_o 13_o deg_n 45_o min._n for_o the_o moon_n place_n the_o sun_n right_a ascension_n the_o 22_o day_n be_v 9_o hour_n and_o the_o moon_n the_o same_o day_n at_o 1_o afternoon_n be_v 9_o hour_n also_o or_o rather_o 12_o difference_n so_o that_o the_o sun_n hour_n and_o the_o moon_n be_v equal_a only_o one_o be_v north_n and_o the_o other_o south_n again_o for_o the_o last_o quarter_n 22_o ¼_n day_n or_o 18_o hour_n add_v and_o 22_o degree_n also_o together_o makes_z â™_o 22_o deg_n 11_o min._n for_o the_o moon_n place_n by_o help_n of_o which_o to_o find_v the_o moon_n hour_n by_o her_o altitude_n above_o the_o horizon_n find_v by_o observation_n or_o without_o regard_v the_o sun_n or_o moon_n right_a ascension_n have_v her_o true_a age_n and_o hour_n say_v thus_o as_o 12_o on_o the_o line_n of_o 24_o hour_n be_v to_o the_o moon_n age_n in_o the_o line_n of_o her_o age_n so_o be_v the_o moon_n hour_n to_o the_o true_a hour_n for_o the_o extent_n from_o 12_o in_o the_o middle_n to_o the_o moon_n age_n under_o or_o over_o the_o middle_n shall_v reach_v the_o same_o way_n on_o the_o same_o line_n from_o the_o moon_n hour_n to_o the_o true_a hour_n the_o like_a work_n serve_v to_o find_v the_o hour_n of_o the_o night_n by_o any_o planet_n as_o saturn_n mars_n or_o jupiter_n which_o be_v see_v to_o shine_v very_o brave_a and_o bright_a in_o winter_n evening_n and_o have_v learn_v their_o place_n by_o their_o distance_n from_o the_o fix_a star_n or_o by_o the_o ephemerides_n than_o their_o altitude_n and_o place_n will_v find_v their_o hour_n from_o the_o meridian_n and_o the_o compare_v their_o right_a ascension_n with_o the_o sun_n give_v the_o true_a hour_n as_o before_o in_o the_o fixed-star_n use_v xlv_o to_o find_v the_o moon_n place_n and_o declination_n without_o the_o ephemerides_n somewhat_o near_o first_o observe_v when_o the_o moon_n be_v in_o the_o meridian_n and_o then_o find_v her_o altitude_n and_o take_v the_o same_o from_o the_o particular_a scale_n between_o your_o compass_n then_o set_v one_o point_n in_o the_o hour_n 12_o and_o lay_v the_o thread_n to_o nd_o and_o on_o the_o degree_n it_o shall_v show_v the_o moon_n declination_n and_o in_o the_o line_n of_o the_o sun_n place_n the_o moon_n present_a place_n count_v her_o progress_n orderly_o from_o the_o last_o change-day_n or_o new_a moon_n when_o she_o be_v with_o the_o sun_n otherwise_o thus_o observe_v what_o hour_n the_o moon_n show_v on_o any_o sundial_n at_o the_o same_o instance_n by_o the_o fix_a star_n or_o other_o way_n find_v the_o true_a hour_n then_o the_o extent_n from_o the_o moon_n hour_n to_o the_o the_o true_a hour_n shall_v reach_v the_o same_o way_n from_o 12_o to_o the_o moon_n age_n right_a against_o which_o be_v her_o come_n to_o south_n at_o which_o time_n you_o may_v find_v her_o true_a altitude_n and_o so_o come_v by_o her_o declination_n yet_o again_o for_o her_o age_n and_o place_n according_a to_o mr._n street_n and_o mr._n blundevil_n add_v the_o epact_n the_o month_n and_o day_n of_o month_n in_o one_o sum_n count_v the_o month_n from_o march_n by_z call_z march_z the_o first_o month_n april_n the_o second_o etc._n etc._n then_o that_o sum_n if_o under_o 30_o be_v the_o moon_n age_n but_o if_o the_o sum_n be_v above_o 30_o then_o subtract_v 30_o and_o the_o remainder_n be_v the_o moon_n age_n when_o the_o month_n have_v 31_o day_n but_o if_o the_o month_n have_v but_o 30_o or_o less_o than_o 30_o day_n then_o subtract_v but_o 29_o and_o the_o remainder_n be_v the_o moon_n age._n or_o thus_o add_v to_o the_o epact_n for_o the_o present_a year_n and_o in_o january_n 0_o in_o february_n 2_o in_o march_z 1_o in_o april_n 2_o in_o may_v 3_o in_o june_n 4_o in_o july_n 5_o in_o august_n 6_o in_o september_n 8_o in_o october_n 8_o in_o november_n 10_o in_o december_n 10_o and_o the_o sum_n if_o under_o 30_o or_o the_o excess_n above_o 30_o add_v to_o the_o day_n of_o the_o month_n abate_v 30_o if_o need_v be_v give_v the_o moon_n age_n that_o day_n but_o substract_v from_o 30_o leave_v the_o day_n of_o her_o change_n in_o that_o month_n or_o from_o the_o 〈…〉_z month_n example_n july_n 10._o 1668._o the_o epact_n that_o year_n be_v 26_o and_o the_o number_n for_o july_n be_v 5_o the_o excess_n above_o 30_o be_v 1_o which_o add_v to_o any_o day_n of_o the_o month_n as_o to_o 10_o gives_z 11_o for_o the_o moon_n age_n july_n 10._o 1668._o then_o for_o the_o moon_n place_n multiply_v the_o moon_n age_n by_o 4_o and_o the_o product_v divide_v by_o 10_o the_o quotient_n give_v the_o sign_n and_o the_o remainder_n multiply_v by_o 3_o gives_z the_o degree_n which_o you_o must_v add_v to_o the_o sun_n place_n that_o day_n to_o find_v out_o the_o moon_n place_n for_o that_o day_n of_o her_o age._n example_n on_o july_n 10._o 1668_o the_o moon_n age_n be_v 11_o which_o multiply_v by_o 4_o make_v 44_o and_o 44_o divide_v by_o 10_o gives_z 4_o sign_n in_o the_o quotient_a and_o 4_o the_o remainder_n multiply_v by_o 3_o make_v 12_o degree_n more_o which_o add_v to_o cancer_n 29_o degree_n the_o sun_n place_n on_o the_o 10_o day_n of_o july_n make_v 11_o degree_n in_o sagittarius_fw-la the_o moon_n place_n the_o same_o day_n propè_fw-la verum_fw-la or_o rather_o by_o the_o rule_n thus_o on_o the_o line_n of_o 24_o hour_n by_o particular_a scale_n have_v the_o moon_n place_n to_o find_v her_o age_n by_o the_o line_n of_o 24_o hour_n the_o extent_n from_o the_o sun_n true_a place_n to_o the_o moon_n true_a place_n shall_v reach_v the_o same_o way_n from_o 0_o day_n to_o the_o day_n of_o her_o age._n or_o contrary_o have_v the_o moon_n true_a age_n to_o find_v her_o true_a place_n the_o extent_n from_o 0_o day_n old_a to_o the_o moon_n true_a age_n shall_v reach_v the_o same_o way_n from_o the_o sun_n true_a place_n to_o the_o moon_n or_o have_v the_o moon_n true_a place_n at_o the_o new_a moon_n to_o find_v her_o place_n any_o day_n of_o her_o age_n after_o the_o extent_n from_o ♈_o to_o the_o moon_n true_a place_n at_o the_o change_n shall_v reach_v the_o same_o way_n from_o the_o day_n of_o her_o true_a age_n to_o her_o true_a place_n add_v as_o many_o degree_n to_o the_o number_n find_v as_o the_o moon_n be_v day_n old_a then_o have_v her_o place_n and_o age_n it_o be_v easy_a to_o find_v the_o moon_n hour_n and_o then_o her_o true_a hour_n but_o i_o fear_v i_o spend_v herein_o too_o much_o time_n on_o a_o uncertain_a subject_n use_v xlvi_o the_o right_a ascention_n and_o declination_n of_o any_o star_n with_o the_o sun_n right_a ascension_n and_o the_o hour_n of_o the_o night_n give_v to_o find_v the_o altitude_n and_o azimuth_n of_o that_o star_n and_o thereby_o to_o know_v the_o star_n if_o you_o know_v it_o not_o before_o set_a one_o point_n of_o the_o compass_n in_o the_o star_n right_a ascension_n find_v in_o the_o line_n of_o twice_o 12_o hour_n and_o open_v the_o other_o to_o the_o sun_n right_a ascension_n find_v in_o the_o same_o line_n then_o this_o
the_o description_n and_o use_v of_o the_o trianguler-quadrant_a be_v a_o particular_a and_o general_a instrument_n useful_a at_o land_n or_o sea_n both_o for_o observation_n and_o operation_n more_o universal_o useful_a portable_a and_o convenient_a than_o any_o other_o yet_o discover_v with_o its_o uses_n in_o arithmetic_n geometry_n superficial_a and_o solid_a astronomy_n dyall_v three_o way_n gage_v navigation_n in_o a_o method_n not_o before_o use_v by_o john_n brown_n philomath_n london_n print_v by_o john_n derby_n for_o john_n wingfield_n and_o be_v to_o be_v sell_v at_o his_o house_n in_o crutched-fryer_n and_o by_o john_n brown_a at_o the_o sphere_n and_o sundial_n in_o the_o minory_n and_o by_o john_n seller_n at_o the_o hermitage-stair_n in_o wapping_n 1671._o to_o the_o reader_n friendly_a reader_n thou_o have_v once_o more_o present_v to_o thy_o view_n a_o further_a improvement_n and_o use_n of_o the_o sector_n under_o the_o name_n of_o the_o triangule_a quadrant_n so_o call_v from_o the_o shape_n thereof_o in_o the_o year_n 1660_o it_o be_v my_o lot_n first_o to_o apply_v and_o improve_v this_o former_a contrivance_n of_o mr._n samuel_n foster_n on_o a_o quadrant_n to_o a_o joint_a rule_n or_o sector_n and_o do_v in_o 1661._o publish_v my_o present_a thought_n thereof_o in_o a_o small_a discourse_n under_o the_o name_n of_o the_o joint_n rule_n since_o then_o through_o my_o persuasion_n and_o assistance_n another_o piece_n be_v publish_v 1667_o by_o i._o t._n under_o the_o name_n of_o the_o semicircle_n on_o a_o sector_n but_o neither_o of_o these_o that_o be_v to_o say_v neither_o my_o own_o nor_o he_o speak_v what_o i_o will_v have_v it_o speak_v neither_o have_v i_o hope_v ever_o to_o produce_v a_o discourse_n either_o for_o method_n or_o matter_n worthy_a or_o become_v so_o excellent_a universal_a and_o useful_a a_o instrument_n for_o the_o most_o mathematical_a occasion_n be_v for_o acurateness_n conveniency_n cheapness_n and_o universality_n before_o all_o other_o for_o 1._o if_o it_o be_v make_v of_o wood_n if_o the_o wood_n keep_v but_o straight_o it_o be_v as_o true_a to_o be_v make_v use_n of_o as_o of_o metal_n 2._o it_o may_v be_v make_v of_o any_o radius_fw-la or_o bigness_n and_o yet_o in_o little_a room_n in_o comparison_n of_o other_o quadrant_n 3._o more_o convenient_a to_o use_v wheâ—_n large_a than_o other_o quadrant_n 4._o as_o to_o the_o projection_n for_o 〈◊〉_d and_o azimuth_n particular_o use_v only_o two_o line_n of_o natural_a sin_n thâ—_n thread_n and_o compass_n for_o those_o twâ—_n difficult_a and_o many_o more_o easy_a proâ—positions_n 5._o the_o neat_a conveniency_n of_o great_a and_o a_o less_o radius_fw-la doublâ—_n treble_a or_o quadruple_a one_o to_o another_o 6._o the_o convenient_a contrivance_n that_o happen_v to_o it_o of_o three_o instrument_n in_o one_o viz._n a_o sector_n quadrant_n and_o gunter_n rule_n all_o three_o convenient_o in_o one_o the_o consideration_n of_o these_o thing_n and_o the_o love_n and_o willingness_n i_o always_o have_v to_o the_o communicate_v of_o they_o to_o other_o have_v put_v i_o on_o this_o hard_a task_n of_o write_v this_o collection_n of_o the_o use_n thereof_o wherein_o i_o do_v most_o hearty_o beg_v thy_o pardon_n and_o acceptance_n to_o accept_v in_o good_a part_n the_o willing_a endeavour_n of_o my_o poor_a ability_n which_o i_o doubt_v not_o but_o to_o have_v from_o most_o that_o know_v i_o for_o first_o my_o insufficiency_n â—n_o the_o tongue_n art_n and_o science_n second_o my_o meanness_n and_o poverty_n in_o the_o world_n for_o these_o employment_n which_o take_v up_o so_o much_o of_o a_o man_n time_n and_o ability_n to_o perform_v they_o to_o purpose_n my_o plead_v my_o excuse_n for_o first_o here_o be_v the_o product_n of_o more_o than_o two_o year_n improvement_n of_o more_o than_o vacant_a hour_n with_o the_o great_a disadvantage_n of_o take_v three_o week_n at_o time_n to_o do_v that_o which_o three_o day_n together_o may_v have_v as_o well_o if_o not_o much_o better_a perform_v and_o at_o last_o to_o calâ—_n the_o assistance_n of_o two_o other_o to_o unâ—dertake_v the_o charge_n thereof_o to_o midwife_n it_o into_o the_o world_n thus_o as_o widow_n mite_n be_v accept_v which_o be_v offer_v in_o sinceriâ—ty_n so_o i_o hope_v will_v i_o though_o atâ—tended_v with_o much_o disorder_n as_o 〈◊〉_d method_n more_o uncouthness_n as_o 〈◊〉_d style_n and_o matter_n what_o it_o be_v it_o 〈◊〉_d as_o at_o first_o compose_v for_o i_o can_v nerver_o get_v time_n nor_o liberty_n from_o 〈◊〉_d daily_a trade_n and_o call_v to_o tranâ—scribe_v it_o twice_o yet_o be_v it_o not_o do_v at_o any_o timâ—_n careless_o but_o with_o good_a will_n and_o free_a intent_n of_o plainness_n and_o usefulâ—ness_n for_o the_o public_a good_a of_o other_o as_o well_o as_o my_o own_o recreation_n 〈◊〉_d delight_n the_o gunter_n rule_n the_o quadrant_n and_o sector_n i_o need_v not_o commend_v they_o be_v so_o well_o know_v already_o but_o 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deg_n 23_o min._n in_o ♉_o his_o true_a place_n require_v as_o sine_fw-la of_o 23-31_a the_o sun_n great_a declination_n t._n to_o sine_fw-la of_o 90_o so_o sine_fw-la of_o 12-00_a the_o sun_n present_a declination_n to_o sine_fw-la of_o sun_n distance_n from_o ♈_o or_o ♎_o 31-23_a which_o by_o consider_v the_o time_n of_o the_o year_n give_v his_o true_a place_n by_o look_v on_o the_o month_n and_o line_n of_o sun_n place_n on_o the_o quadrantal-side_n take_v the_o sine_fw-la of_o the_o present_a declination_n generally_n make_v it_o a_o =_o sine_fw-la in_o the_o great_a declination_n lay_v the_o thread_n to_o nd_o and_o on_o the_o degree_n the_o thread_n shall_v give_v the_o sun_n distance_n from_o ♈_o or_o ♎_o require_v example_n as_o before_o make_v sine_fw-la of_o the_o give_v sun_n declination_n sector_n a_o =_o sine_fw-la in_o the_o sun_n great_a declination_n then_o =_o sine_fw-la of_o 90_o measure_v from_o the_o centre_n be_v the_o =_o sine_fw-la of_o the_o sun_n distance_n from_o ♈_o or_o ♎_o require_v or_o count_v 30_o deg_n for_o one_o sign_n and_o the_o centre_n for_o the_o next_o equinoctial-point_n and_o 90_o for_o the_o two_o tropic_n of_o cancer_n and_o capricorn_n ♋_o ♑_o use_v fourteen_o the_o sun_n place_n or_o day_n of_o the_o month_n and_o great_a declination_n give_v to_o find_v his_o right_a ascension_n from_o the_o same_o equinoctial_a lay_v the_o thread_n to_o the_o day_n of_o the_o month_n quadr._n or_o place_n give_v and_o in_o the_o line_n of_o the_o sun_n right_a ascension_n you_o have_v his_o right_a ascension_n in_o degree_n or_o hour_n and_o minute_n count_v 4_o minute_n for_o every_o degree_n example_n on_o the_o 9th_o of_o april_n near_o night_n the_o sun_n be_v then_o enter_v ♉_o the_o sun_n right_a ascension_n will_v be_v 1_o hour_n 52_o min._n or_o 28_o degree_n of_o right_a ascension_n distant_a from_o ♈_o as_o the_o sine_fw-la of_o 90_o t._n to_o the_o sine_fw-la compliment_n of_o the_o sun_n great_a declination_n or_o c.s._n of_o 23-31_a count_v backward_o from_o 90_o which_o will_v be_v at_o the_o sine_fw-la of_o 66-29′_a so_o be_v the_o tangent_fw-la of_o the_o sun_n distance_n from_o the_o next_o equinoctial-point_n to_o the_o tangent_fw-la of_o the_o sun_n right_a ascension_n from_o the_o same_o equinoctial-point_n generally_n take_v the_o cousin_a of_o the_o great_a declination_n from_o the_o centre_n downward_o be_v the_o sine_fw-la of_o 66-29′_a make_v it_o a_o =_o sine_fw-la of_o 90_o lay_v the_o thread_n to_o nd_o and_o note_v what_o degree_n and_o minuit_fw-la it_o cut_v for_o this_o be_v fix_v to_o this_o proportion_n then_o take_v the_o tangent_fw-la of_o the_o sun_n distance_n from_o the_o next_o equinoctial-point_n from_o the_o centre_n at_o 600_o on_o the_o degree_n towards_o the_o end_n and_o lay_v it_o on_o the_o sin_n from_o the_o centre_n downward_o and_o note_v the_o point_n where_o it_o stay_v for_o the_o nd_o from_o thence_o to_o the_o thread_n shall_v be_v the_o tangent_fw-la of_o the_o sun_n right_a ascension_n require_v note_v that_o if_o the_o sun_n distance_n from_o ♈_o or_o ♎_o be_v above_o 45_o degree_n than_o the_o tangent_n on_o the_o loose-piece_n be_v to_o be_v use_v instead_o of_o the_o tangent_n on_o the_o moveable-leg_n or_o by_o sines_n only_o thus_o or_o take_v sine_fw-la of_o the_o present_a sun_n declination_n make_v it_o a_o =_o in_o the_o sine_fw-la of_o the_o sun_n great_a declination_n and_o lay_v the_o thread_n to_o nd_o then_o take_v =_o cousin_a of_o the_o sun_n great_a declination_n and_o make_v it_o a_o =_o in_o cousin_a of_o the_o sun_n present_a declination_n and_o lay_v the_o thread_n to_o nd_o and_o in_o the_o degree_n it_o cut_v the_o sun_n right_a ascension_n require_v make_v cousin_a of_o 23-31_a sector_n viz._n the_o right_a sine_fw-la of_o 66-29_a a_o =_o sine_fw-la of_o 90_o than_o the_o =_o tangent_fw-la of_o the_o sun_n distance_n from_o ♈_o or_o ♎_o be_v the_o =_o tangent_fw-la of_o the_o sun_n right_a ascension_n from_o the_o same_o point_n of_o ♈_o or_o ♎_o as_o at_o 30_o from_o ♈_o it_o be_v 28_o degree_n or_o 1_o hour_n and_o 52_o minute_n from_o ♈_o near_o use_v xv._n have_v the_o sun_n right_a ascension_n and_o great_a declination_n to_o find_v the_o angle_n of_o the_o ecliptic_a and_o meridian_n as_o sine_fw-la 90_o to_o sine_fw-la 23-31_a sine_fw-la so_o be_v the_o cousin_a of_o the_o sun_n right_a ascension_n to_o the_o cousin_a of_o the_o angle_n of_o the_o ecliptic_a and_o meridian_n lay_v the_o thread_n to_o 23-31_a quadr._n count_v on_o the_o degree_n from_o the_o head_n then_o count_v the_o cousin_a of_o the_o right_a ascension_n from_o the_o centre_n downward_o or_o the_o sine_fw-la from_o 90_o upward_o and_o take_v the_o nd_o from_o thence_o to_o the_o thread_n and_o measure_v it_o from_o the_o centre_n and_o it_o shall_v reach_v to_o the_o cousin_a of_o the_o angle_n require_v example_n the_o right_a ascention_n be_v 30_o degree_n or_o 2_o hour_n the_o angle_n shall_v be_v 69-50_a sector·_n make_v the_o right_a sine_fw-la of_o 23-31_a a_o =_o sine_fw-la of_o 90_o then_o the_o =_o cousin_a of_o 30_o viz._n =_o 60_o shall_v make_v the_o sine_fw-la of_o 69-50_a the_o angle_n of_o the_o ecliptic_a and_o meridian_n use_v xvi_o have_v the_o latitude_n and_o declination_n of_o the_o sun_n or_o star_n to_o find_v the_o sun_n or_o star_n amplitude_n at_o rise_v or_o set_v quadr._n take_v the_o sun_n declination_n from_o the_o particular_a scale_n of_o sines_n and_o lay_v it_o from_o 6_o in_o the_o hour_n or_o azimuth-line_n and_o it_o shall_v give_v the_o amplitude_n from_o south_n as_o it_o be_v figure_v or_o from_o east_n or_o west_n count_v from_o 90_o observe_v to_o turn_v the_o compass_n the_o same_o way_n from_o 90_o or_o 6_o as_o the_o declination_n be_v northward_o or_o southwards_o example_n the_o sun_n declination_n be_v 10_o degree_n northward_o the_o sun_n amplitude_n or_o line_n be_v 106-12_a from_o the_o south_n or_o 16-12_a from_o the_o east-point_n sines_n as_o cousin_a of_o the_o latitude_n to_o s._n 90_o so_o be_v s._n of_o the_o sun_n declination_n to_o s._n of_o the_o amplitude_n general_o take_v the_o sine_fw-la of_o the_o sun_n declination_n make_v it_o a_o =_o in_o the_o cousin_a of_o the_o latitude_n and_o lay_v the_o thread_n to_o the_o near_a distance_n sector_n and_o on_o the_o degree_n the_o thread_n shall_v show_v the_o true_a amplitude_n require_v make_v the_o right_a sine_fw-la of_o the_o sun_n declination_n a_o =_o in_o cousin_a latitude_n then_o =_o 90_o take_v and_o measure_v from_o the_o centre_n give_v the_o amplitude_n or_o line_n use_v xvii_o have_v the_o same_o amplitude_n and_o declination_n to_o find_v the_o latitude_n as_o s._n of_o the_o sun_n amplitude_n sine_fw-la to_o s._n the_o sun_n declination_n so_o be_v s_o 90_o to_o cousin_a latitude_n take_v the_o sine_fw-la of_o the_o sun_n declination_n quadr._n set_v one_o point_n in_o the_o sine_fw-la of_o the_o sun_n amplitude_n lay_v the_o thread_n to_o nd_o and_o on_o the_o degree_n it_o show_v the_o compliment_n of_o the_o latitude_n require_v example_n the_o declination_n be_v 20_o degree_n and_o the_o amplitude_n 33-15_a the_o compliment_n of_o the_o latitude_n will_v be_v 38-28_a count_v from_o the_o head_n toward_z the_o end_n make_v the_o right_n sine_fw-la of_o the_o sun_n declination_n sector_n a_o =_o sine_fw-la in_o the_o sun_n amplitude_n then_o the_o =_o sine_fw-la of_o 90_o shall_v be_v the_o cousin_a of_o the_o latitude_n require_v use_v xviii_o have_v the_o latitude_n and_o sun_n declination_n to_o find_v his_o altitude_n at_o east_n or_o west_n common_o call_v the_o vertical-circle_n or_o azimuth_n of_o east_n or_o west_n take_v the_o sun_n declination_n from_o the_o particular_a line_n of_o sines_n q._n set_v one_o point_n in_o 90_o on_o the_o
azimuth-line_n and_o lay_v the_o thread_n to_o the_o nd_o and_o on_o the_o degree_n it_o show_v the_o altitude_n require_v count_v from_o 600_o towards_o the_o end_n as_o s._n latitude_n s._n of_o 90_o s._n so_o s._n of_o sun_n declination_n to_o s._n sun_n height_n at_o east_n or_o west_n quad._n take_v the_o sine_fw-la of_o the_o sun_n declination_n make_v it_o a_o =_o in_o the_o sine_fw-la of_o the_o latitude_n and_o lay_v the_o thread_n to_o nd_o and_o on_o the_o degree_n it_o shall_v show_v the_o sun_n altitude_n at_o east_n and_o west_n require_v example_n declination_n 10._o latitude_n 51-32_a the_o altitude_n be_v 12_o degree_n and_o 50_o minute_n as_o s._n of_o the_o sun_n declination_n to_o =_o s._n of_o latitude_n sector_n so_o be_v the_o =_o s._n of_o 90_o to_o s._n of_o vertical_a altitude_n use_v xix_o have_v the_o latitude_n and_o sun_n declination_n to_o find_v the_o time_n when_o the_o sun_n will_v be_v due_a east_n or_o west_n have_v get_v the_o altitude_n by_o the_o last_o rule_n take_v it_o from_o the_o particular_a sine_fw-la q._n then_o lay_v the_o thread_n to_o the_o sun_n declination_n count_v on_o the_o degree_n then_o set_v one_o point_n in_o the_o hour-line_n so_o as_o the_o other_o turn_v about_o shall_v but_o just_a touch_n the_o thread_n and_o the_o compass-point_n shall_v stay_v at_o the_o hour_n and_o minuit_fw-la of_o time_n require_v as_o tangent_fw-la latitude_n to_o sine_fw-la 90_o t._n so_o be_v the_o tangent_fw-la of_o the_o sun_n declination_n to_o cousin_a of_o the_o hour_n or_o as_o sine_fw-la 90_o to_o tangent_fw-la sun_n declination_n so_o be_v co-tangent-latitude_n to_o cousin_a of_o the_o hour_n from_o noon_n example_n latitude_n 51-32_a declination_n 10_o the_o sun_n will_v be_v due_a east_n at_o 6-32_a and_o west_n at_o 5-28_a take_v the_o tangent_fw-la of_o the_o latitude_n on_o the_o loose-piece_n quad._n count_v from_o 60_o towards_o the_o moveable-leg_n or_o else_o from_o 600_o on_o the_o moving-leg_n or_o degree_n according_a as_o the_o latitude_n be_v above_o or_o under_o 45_o degree_n and_o lay_v it_o from_o the_o centre_n downward_o and_o note_v the_o point_n where_o it_o end_v then_o take_v from_o the_o same_o tangent_fw-la the_o tangent_fw-la of_o the_o sun_n declination_n and_o set_v one_o foot_n in_o the_o point_n last_o note_v lie_v the_o thread_n to_o nd_o than_o the_o =_o sine_fw-la of_o 90_o shall_v be_v the_o sine_fw-la of_o the_o hour_n from_o 6._o or_o by_o the_o sin_n only_o work_v thus_o take_v the_o sine_fw-la of_o the_o sun_n declination_n make_v it_o a_o =_o in_o sine_fw-la of_o the_o latitude_n lay_v the_o thread_n to_o nd_o then_o take_v nd_o from_o the_o cousin_a latitude_n to_o the_o thread_n then_o set_v one_o foot_n in_o the_o cousin_a of_o the_o sun_n declination_n lay_v the_o thread_n to_o nd_o and_o on_o the_o degree_n it_o give_v the_o hour_n from_o noon_n as_o it_o be_v figure_v or_o the_o hour_n from_o 6_o count_v from_o the_o head_n count_v 4_o minute_n for_o every_o degree_n make_v the_o small_a tangent_fw-la of_o the_o latitude_n if_o above_o 45_o sector_n take_v from_o the_o centre_n a_o =_o sine_fw-la of_o 90_o then_o the_o tangent_fw-la of_o the_o sun_n declination_n take_v from_o the_o same_o small_a tangent_fw-la and_o carry_v paralle_o till_o it_o stay_v in_o like_a sin_n shall_v be_v the_o sine_fw-la of_o the_o hour_n from_o 6._o or_o as_o before_o by_o sines_n only_o make_v sine_fw-la declination_n a_o =_o sine_fw-la latitude_n then_o take_v =_o cousin_a latitude_n and_o make_v it_o a_o =_o cousin_a of_o the_o sun_n declination_n then_o take_v =_o 90_o and_o lay_v it_o from_o the_o centre_n it_o give_v the_o sine_fw-la of_o the_o hour_n from_o 6._o use_v xx._n have_v the_o latitude_n and_o sun_n declination_n to_o find_v the_o ascentional_a difference_n or_o the_o sun_n rise_v and_o set_v and_o oblique_a ascension_n lay_v the_o thread_n to_o the_o day_n of_o the_o month_n or_o to_o the_o sun_n declination_n or_o true_a place_n q._n or_o to_o his_o right_a ascension_n for_o the_o thread_n be_v lay_v to_o any_o one_o of_o they_o be_v then_o also_o lay_v to_o all_o the_o rest_n then_o in_o the_o azimuth-line_n it_o cut_v the_o ascentional_a difference_n if_o it_o you_o count_v from_o 90_o or_o the_o sun_n rise_v as_o you_o count_v the_o morning_n hour_n or_o his_o set_v count_v the_o afternoon_n hour_n the_o oblique_a ascension_n be_v find_v out_o for_o the_o six_o northern_a sign_n oblique-ascention_n or_o summer_n half-year_n by_o substract_v the_o sun_n difference_n of_o ascension_n out_o of_o the_o sun_n right_a ascension_n but_o for_o the_o other_o winter-half_a year_n or_o six_o southern_a sign_n it_o be_v find_v by_o add_v the_o sun_n difference_n of_o ascension_n to_o his_o right_a ascension_n this_o sum_n in_o winter_n and_o the_o remainder_n as_o abovesaid_a in_o summer_n shall_v be_v the_o sun_n oblique_a ascension_n require_v tan._n as_o co-tangent_a lat._n to_o sine_fw-la 90_o to_o be_v the_o tangent_fw-la of_o the_o sun_n declination_n to_o the_o sine_fw-la of_o the_o sun_n ascentional_a difference_n quad._n take_v the_o co-tangent_a latitude_n from_o the_o loose_a or_o moveable-piece_n as_o it_o be_v above_o or_o under_o 45_o degree_n make_v it_o a_o =_o in_o sine_fw-la 90_o lay_v the_o thread_n to_o nd_o then_o take_v the_o tangent_fw-la of_o the_o sun_n declination_n from_o the_o same_o tangent_n and_o carry_v it_o =_o till_o it_o stay_v in_o the_o part_n that_o the_o other_o foot_n turn_v about_o will_v but_o just_a touch_n the_o thread_n which_o part_n shall_v be_v the_o sine_fw-la of_o the_o sun_n ascentional_a difference_n require_v or_o â—hus_o by_o sines_n only_o make_v the_o sine_fw-la of_o declination_n a_o =_o cousin_a of_o the_o latitude_n lay_v the_o thread_n to_o nd_o then_o take_v the_o =_o sine_fw-la of_o latitude_n make_v it_o a_o =_o in_o cousin_a of_o the_o declination_n and_o lay_v the_o thread_n to_o nd_o and_o on_o the_o degree_n it_o shall_v cut_v the_o sun_n ascentional-difference_n require_v which_o be_v turn_v into_o time_n by_o count_v 4_o minute_n for_o every_o degree_n and_o add_v to_o or_o take_v from_o 6_o gives_z the_o sun_n rise_v in_o summer_n or_o winter_n make_v the_o co-tangent_a latitude_n sector_n a_o =_o sine_fw-la of_o 90_o then_o take_v tangent_fw-la of_o the_o sun_n declination_n and_o carry_v it_o =_o till_o it_o stay_v in_o like_a part_n viz._n the_o sine_fw-la of_o the_o sun_n ascentional_a difference_n require_v example_n otherwise_o as_o sine_fw-la 90_o to_o =_o tangent_fw-la 38-28_a so_o be_v =_o tangent_fw-la of_o 23-31_a the_o sun_n great_a declination_n to_o the_o sine_fw-la of_o the_o sun_n great_a ascentional_a difference_n 33_o deg_n and_o 12_o min._n use_v xxi_o the_o latitude_n and_o sun_n declination_n give_v to_o find_v the_o sun_n meridian_n altitude_n when_o the_o latitude_n and_o declination_n be_v both_o alike_o viz._n both_o north_n or_o both_o south_n then_o subtract_v the_o declination_n out_o of_o the_o latitude_n or_o the_o less_o from_o the_o great_a and_o the_o remainder_n shall_v be_v the_o compliment_n of_o the_o sun_n meridian_n altitude_n but_o when_o they_o be_v unlike_a then_o add_v they_o together_o and_o the_o sum_n shall_v be_v the_o compliment_n of_o the_o meridian_n altitude_n the_o contrary_a work_n serve_v when_o the_o compliment_n of_o the_o latitude_n and_o declination_n be_v give_v to_o find_v the_o meridian_n altitude_n lay_v the_o thread_n to_o the_o declination_n count_v on_o the_o degree_n from_o 600_o the_o right_a way_n towards_o the_o head_n for_o north_n and_o towards_o the_o end_n for_o south_n declination_n then_o take_v the_o near_a distance_n from_o the_o center-prick_a at_o 12_o in_o the_o hour-line_n to_o the_o thread_n this_o distance_n measure_v on_o the_o particular-line_n of_o sines_n shall_v show_v the_o sun_n meridian-altitude_n require_v use_v xxi_o the_o latitude_n and_o hour_n from_o the_o midnight_n meridian_n give_v to_o find_v the_o angle_n of_o the_o sun_n position_n viz._n the_o angle_n between_o the_o hour_n and_o azimuth-line_n in_o the_o centre_n of_o the_o sun_n t._n as_o sine_fw-la 90_o to_o cousin_a of_o the_o latitude_n so_o be_v the_o sine_fw-la of_o the_o hour_n from_o midnight_n to_o the_o sine_fw-la of_o the_o angle_n of_o position_n example_n as_o sine_fw-la 90_o to_o cousin_a latitude_n 38-28_a so_o be_v the_o cousin_a of_o the_o hour_n from_o midnight_n 120_o for_o which_o you_o must_v use_v 60_o to_o 32-34_a the_o angle_n of_o position_n q._n take_v the_o distance_n from_o the_o hour_n to_o the_o 90_o azimuth_n on_o the_o hour-line_n and_o measure_v it_o in_o the_o particular_a sin_n and_o it_o shall_v show_v the_o angle_n of_o position_n require_v this_o hold_v in_o the_o equinoctial_a take_v cousin_a latitude_n quad_n make_v it_o a_o =_o in_o sine_fw-la 90_o then_o take_v
out_o the_o =_o cousin_a of_o the_o hour_n from_o the_o meridian_n and_o it_o shall_v be_v the_o sine_fw-la of_o the_o sun_n position_n make_v cousin_a latitude_n a_o =_o sine_fw-la 90_o sector_n then_o =_o cousin_a of_o the_o hour_n shall_v be_v sine_z of_o the_o sun_n position_n note_v the_o angle_n of_o the_o sun_n position_n may_v be_v vary_v and_o it_o be_v general_o the_o angle_n make_v in_o the_o centre_n of_o the_o sun_n by_o his_o meridian_n or_o hour-circle_n be_v a_o circle_n pass_v through_o the_o pole_n of_o the_o world_n and_o the_o centre_n of_o the_o sun_n and_o any_o other_o principal_a circle_n as_o the_o meridian_n the_o horizon_n or_o any_o azimuth_n the_o anguler-point_n be_v always_o the_o centre_n of_o the_o sun_n use_v xxii_o the_o sun_n declination_n give_v to_o find_v the_o beginning_n and_o end_n of_o twilight_n or_o daybreak_n lay_v the_o thread_n to_o the_o declination_n on_o the_o degree_n but_o count_v the_o contrary_a way_n q._n viz._n south-declination_n towards_o the_o 〈…〉_z north-declination_n towards_o the_o 〈◊〉_d then_o take_v 18_o degree_n from_o the_o particular_a 〈◊〉_d sin_n for_o twilight_n or_o 13_o degree_n for_o daybreak_n or_o clear_a light_n then_o carry_v this_o distance_n of_o 18_o for_o twilight_n or_o 13_o for_o daybreak_n along_o the_o line_n of_o hour_n on_o that_o side_n of_o the_o thread_n next_o the_o end_n till_o the_o other_o foot_n turn_v about_o will_v but_o just_a touch_n the_o thread_n then_o shall_v the_o point_n show_v the_o time_n of_o twilight_n or_o daybreak_n require_v example_n the_o sun_n declination_n be_v 12_o degree_n north_n the_o twilight_n continue_v till_o 9_o hour_n 24_o minute_n or_o it_o begin_v in_o the_o morning_n at_o 38_o minute_n after_o 2_o but_o the_o daybreak_n be_v not_o till_o 22_o minute_n after_o 3_o in_o the_o morning_n or_o 38_o minute_n after_o 8_o at_o night_n and_o last_v no_o long_o to_o work_v this_o for_o any_o other_o place_n where_o the_o latitude_n do_v vary_v do_v thus_o find_v the_o hour_n that_o answer_v to_o 18_o degree_n of_o altitude_n in_o as_o much_o declination_n the_o contrary_a way_n and_o that_o shall_v be_v the_o time_n of_o twilight_n or_o at_o 13_o degree_n for_o daybreak_n according_a to_o the_o rule_n in_o the_o 26_o use_n where_o the_o way_n how_o be_v large_o handle_v to_o the_o 33_o use_n both_o way_n general_o use_v xxiii_o to_o find_v for_o what_o latitude_n your_o instrument_n be_v particular_o make_v for_o take_v the_o near_a distance_n from_o the_o centre_n on_o the_o head-leg_n quadr._n to_o the_o azimuth-line_n on_o the_o moveable-leg_n this_o distance_n measure_v on_o the_o particular_a scale_n of_o sines_n shall_v show_v the_o latitude_n require_v or_o the_o extent_n from_o 0_o to_o 90_o on_o the_o azimuth-line_n shall_v show_v the_o compliment_n of_o the_o latitude_n be_v measure_v as_o before_o use_v xxiv_o have_v the_o meridian_n altitude_n give_v to_o find_v the_o time_n of_o sun_n rising_n or_o set_v true_a place_n or_o declination_n take_v the_o sun_n meridian_n altitude_n from_o the_o particular_a scale_n quadrant_n and_o set_v on_o point_n in_o ☉_o on_o the_o azimuth-line_n lay_v the_o thread_n to_o the_o nd_o and_o on_o the_o hour-line_n it_o show_v the_o time_n of_o rise_v or_o set_v and_o on_o the_o degree_n the_o declination_n and_o the_o rest_n in_o their_o respective_a line_n example_n the_o meridian_n altitude_n be_v 50_o the_o sun_n rise_v at_o 5_o and_o set_v at_o 7._o use_v xxv_o the_o latitude_n and_o declination_n give_v to_o find_v the_o sun_n height_n at_o 6._o lay_v the_o thread_n to_o the_o day_n of_o the_o month_n or_o declination_n quadrant_n then_o take_v the_o nd_o from_o the_o hour-point_n of_o 06_o and_o 6_o to_o the_o thread_n and_o that_o distance_n measure_v on_o the_o particular_a scale_n of_o sines_n shall_v be_v the_o sun_n altitude_n at_o 6_o in_o summer_n time_n or_o his_o depression_n under_o the_o horizon_n in_o the_o winter_n time_n t._n as_o sine_fw-la of_o 90_o to_o sine_fw-la of_o the_o sun_n declination_n so_o be_v sine_fw-la latitude_n to_o sine_fw-la of_o the_o sun_n altitude_n at_o 6._o quad._n count_n the_o sun_n declination_n on_o the_o degree_n from_o 90_o towards_o the_o end_n and_o there_o lie_v the_o thread_n then_o the_o least_o distance_n from_o the_o sine_fw-la of_o the_o latitude_n to_o the_o thread_n measure_v from_o the_o centre_n downward_o shall_v be_v the_o sine_fw-la of_o the_o sun_n altitude_n at_o 6._o make_v the_o sine_fw-la of_o the_o declination_n a_o =_o sine_fw-la of_o 90_o sector_n then_o the_o =_o sine_fw-la of_o the_o latitude_n shall_v be_v the_o sine_fw-la of_o the_o sun_n height_n at_o 6._o example_n latitude_n 51-32_a declination_n 23-31_a the_o height_n at_o 6_o be_v 18_o deg_n 13_o min._n use_v xxvi_o have_v the_o latitude_n the_o sun_n declination_n and_o altitude_n to_o find_v the_o hour_n of_o the_o day_n take_v the_o sun_n altitude_n quadrant_n from_o the_o particular_a scale_n of_o sines_n between_o the_o compass_n then_o lay_v the_o thread_n to_o the_o day_n of_o the_o month_n or_o declination_n then_o carry_v the_o compass_n along_o the_o line_n of_o hour_n between_o the_o thread_n and_o the_o end_n till_o the_o other_o point_n be_v turn_v about_o will_v but_o just_a touch_n the_o thread_n and_o then_o the_o fix_a point_n shall_v show_v the_o true_a hour_n and_o min._n require_v in_o the_o fore_n or_o afternoon_n if_o you_o be_v in_o doubt_n which_o it_o be_v than_o another_o observation_n present_o after_o will_v determine_v it_o example_n may_n 10_o at_o 30_o degree_n of_o altitude_n the_o hour_n will_v be_v 32_o minute_n after_o 7_o in_o the_o morning_n or_o 28_o minute_n after_o 4_o in_o the_o afternoon_n this_o work_n be_v somewhat_o more_o difficult_a than_o the_o former_a i_o shall_v part_v it_o thus_o 1._o first_o to_o find_v the_o hour_n the_o sun_n be_v in_o the_o equinoctial_a take_v the_o sine_fw-la of_o the_o sun_n altitude_n quad._n make_v it_o a_o =_o cousin_a of_o the_o latitude_n lay_v the_o thread_n to_o nd_o and_o on_o the_o degree_n it_o shall_v give_v the_o hour_n from_o 12_o as_o it_o be_v figure_v count_v 15_o degree_n for_o a_o hour_n or_o from_o 6_o count_v from_o the_o head_n at_o 90._o example_n latitude_n 51-30_a altitude_n 20_o the_o hour_n be_v 8_o &_o 12′_n in_o the_o forenoon_n or_o 3-48′_a in_o the_o afternoon_n the_o same_o by_o artificial_a sin_n &_o tangent_n as_o cousin_a latitude_n to_o sine_fw-la 90_o so_o be_v the_o sine_fw-la of_o the_o sun_n altitude_n to_o sine_fw-la of_o the_o hour_n from_o 6._o sector_n make_v s._n ☉_o altitude_n a_o =_o s._n in_o ☉_o latitude_n then_o take_v out_o =_o s._n 90_o and_o it_o shall_v be_v the_o sine_fw-la of_o the_o hour_n from_o 6._o 2._o the_o latitude_n declination_n and_o altitude_n give_v to_o find_v the_o hour_n at_o any_o time_n quad_n first_o by_o the_o 25_o use_n find_v the_o sun_n altitude_n or_o depression_n at_o 6_o then_o in_o summertime_n lay_v this_o distance_n from_o the_o centre_n downward_o and_o in_o wintertime_n lay_v it_o upward_o from_o the_o centre_n towards_o the_o end_n of_o the_o head-leg_n and_o note_v that_o point_n for_o that_o day_n or_o degree_n of_o declination_n for_o by_o take_v the_o distance_n from_o thence_o to_o the_o sun_n altitude_n on_o the_o general_n scale_n you_o have_v add_v or_o substract_v the_o altitude_n at_o 6_o to_z or_o from_o the_o present_a altitude_n for_o by_o take_v the_o distance_n from_o that_o note_a point_n over_o or_o under_o the_o centre_n to_o the_o sun_n present_a altitude_n you_o have_v in_o summer_n the_o difference_n between_o the_o sun_n present_a altitude_n and_o his_o altitude_n at_o 6._o and_o in_o winter_n you_o have_v the_o sum_n of_o the_o present_a altitude_n and_o the_o altitude_n at_o 6._o this_o operation_n be_v plain_o hint_v at_o in_o the_o four_o chapter_n and_o 9th_o and_o 10_o section_n which_o be_v understand_v take_v the_o whole_a operation_n in_o short_a term_n thus_o count_v the_o sun_n declination_n from_o 90_o towards_o the_o end_n and_o thereunto_o lie_v the_o thread_n the_o near_a distance_n from_o the_o sine_fw-la of_o the_o latitude_n to_o the_o thread_n be_v the_o sun_n height_n or_o depression_n at_o 6_o in_o winter_n use_v the_o sum_n of_o in_o summer_n the_o difference_n between_o the_o sun_n altitude_n at_o 6_o and_o his_o present_a altitude_n with_o this_o distance_n between_o your_o compass_n set_v one_o point_n in_o the_o cousin_a of_o the_o latitude_n lay_v the_o thread_n to_o nd_o then_o take_v the_o nd_o from_o 90_o to_o the_o thread_n then_o set_v one_o foot_n in_o the_o cousin_a of_o the_o sun_n declination_n and_o lay_v the_o thread_n to_o nd_o and_o on_o the_o degree_n it_o give_v the_o hour_n require_v from_o 6_o count_v from_o 90_o
which_o shall_v be_v the_o verse_v sine_fw-la of_o the_o hour_n from_o the_o north_n meridian_n or_o midnight_n or_o if_o you_o take_v the_o distance_n from_o the_o difference_n to_o the_o co-altitude_n and_o carry_v that_o =_o till_o it_o stay_v in_o like_a sin_n it_o shall_v be_v the_o hour_n from_o noon_n count_v the_o centre_n 12_o at_o noon_n the_o middle_a at_o 90_o the_o two_o six_o and_o 180_o at_o the_o end_n for_o 12_o at_o night_n use_v xxxv_o have_v the_o latitude_n the_o sun_n altitude_n and_o distance_n from_o the_o elevate_v pole_n to_o find_v his_o true_a azimuth_n from_o south_n or_o north_n by_o natural_a verse_v sines_n first_o of_o the_o co-altitude_n and_o co-latitude_n find_v the_o sum_n and_o difference_n by_o addition_n and_o substraction_n second_o count_v the_o sum_n and_o difference_n from_o the_o centre_n and_o take_v the_o distance_n between_o they_o with_o compass_n on_o the_o verse_v sin_n three_o make_v it_o a_o =_o verse_v sine_fw-la of_o 180_o and_o so_o keep_v the_o sector_n four_o take_v the_o distance_n between_o the_o sum_n and_o the_o sun_n distance_n from_o the_o pole_n count_v the_o centre_n the_o elevate_v pole_n and_o 90_o the_o equinoctial_a and_o carry_v it_o =_o till_o it_o stay_v in_o like_a part_n which_o shall_v be_v the_o azimuth_n from_o south_n or_o if_o you_o take_v the_o distance_n from_o the_o difference_n to_o the_o sun_n distance_n from_o the_o pole_n and_o carry_v it_o as_o before_o it_o shall_v stay_v at_o the_o verse_v sine_fw-la of_o the_o azimuth_n from_o the_o north_n part_n of_o the_o horizon_n these_o five_o general_a way_n of_o find_v the_o hour_n and_o azimuth_n be_v not_o all_o needful_a to_o be_v learn_v by_o every_o one_o but_o to_o delight_v the_o ingenious_a and_o to_o hold_v forth_o the_o usefulness_n of_o the_o instrument_n and_o to_o supply_v defect_n that_o at_o some_o time_n may_v happen_v by_o excursion_n and_o as_o a_o fourfold_a testimony_n to_o show_v the_o harmony_n in_o several_a way_n of_o operation_n the_o first_o particular_a way_n and_o this_o last_o by_o verse_v sin_n be_v most_o easy_a and_o comprehensive_a of_o any_o other_o use_v xxxvi_o to_o work_v the_o last_o without_o the_o line_n of_o verse_v sin_n note_v that_o if_o for_o want_n of_o room_n the_o verse_v sin_n be_v set_v but_o on_o one_o leg_n than_o it_o be_v to_o be_v lay_v at_o the_o near_a distance_n instead_o of_o like_a part_n after_o the_o manner_n of_o use_v the_o thread_n on_o the_o general_n quadrant_n also_o if_o you_o have_v it_o not_o at_o all_o than_o the_o azimuth-line_n for_o the_o particular_a latitude_n and_o if_o that_o be_v too_o large_a the_o little_a line_n of_o sines_n beyond_o the_o centre_n will_v supply_v this_o defect_n very_o well_o thus_o first_o turn_v the_o radius_fw-la or_o whole_a length_n of_o that_o line_n of_o sines_n two_o time_n from_o the_o centre_n downward_o which_o in_o sea-instrument_n will_v most_o convenient_o stay_v at_o 30_o on_o the_o large_a line_n of_o sines_n or_o general_a scale_n as_o be_v hint_v in_o the_o 28_o use_n be_v just_a 4_o time_n as_o much_o one_o as_o the_o other_o for_o a_o point_n represent_v 180_o of_o verse_v sin_n to_o set_v the_o compass_n in_o when_o you_o lay_v the_o thread_n to_o nd_o and_o to_o take_v any_o verse_v sine_fw-la above_o 90_o degree_n this_o be_v premise_v the_o operation_n be_v thus_o example_n lat._n 51-32_a ☉_o dist._n from_o pole_n 80_o ☉_o alt._n 25_o to_o find_v the_o hour_n quad._n the_o sum_n of_o colat_a 38-28_a and_o 80_o be_v 118-28_a and_o the_o difference_n be_v 41-32_a now_o in_o regard_n the_o sum_n be_v above_o 90_o count_v the_o centre_n 90_o 10_o on_o the_o small_a sin_n 100_o and_o 20_o on_o the_o same_o sin_n 110_o and_o 28_o deg_n 28_o min_fw-mi 118_o deg_n 28_o min._n turn_v this_o distance_n the_o other_o way_n from_o the_o centre_n downward_o and_o note_v that_o place_n for_o the_o point_n represent_v the_o sum_n on_o the_o verse_v sin_n then_o the_o extent_n between_o this_o sum_n and_o the_o difference_n 41-32_a as_o the_o small_a figure_n reckon_v it_o be_v take_v between_o your_o compass_n set_v one_o point_n in_o 180_o the_o point_n first_o find_v and_o lay_v the_o thread_n to_o nd_o and_o there_o keep_v it_o or_o observe_v where_o it_o cut_v then_o take_v the_o distance_n between_o the_o verse_v sine_fw-la of_o the_o difference_n count_v as_o the_o small_a figure_n be_v reckon_v and_o the_o sine_fw-la of_o the_o sun_n altitude_n 25_o as_o the_o great_a figure_n be_v reckon_v from_o the_o centre_n towards_o the_o end_n and_o carry_v this_o extent_n parallel_o along_o the_o great_a line_n of_o sin_n till_o the_o other_o point_n will_v but_o just_a touch_n the_o thread_n at_o nd_o then_o i_o say_v the_o measure_n from_o that_o point_n to_o the_o centre_n measure_v on_o the_o small_a sin_n as_o verse_v sin_n shall_v be_v the_o verse_v sine_fw-la of_o the_o hour_n require_v viz._n 62_o from_o south_n or_o 7_o hour_n 52_o minute_n from_o midnight_n this_o rule_n or_o use_v be_v long_a far_o in_o word_v than_o the_o operation_n need_v be_v in_o work_v for_o if_o you_o shall_v approve_v of_o this_o way_n the_o add_v of_o two_o brass_n center-pin_n will_v show_v you_o the_o two_o point_n most_o use_v very_o ready_o and_o the_o thread_n be_v soon_o lay_v than_o the_o leg_n can_v be_v open_v or_o shut_v and_o the_o instrument_n keep_v its_o triangule_a form_n as_o it_o be_v in_o during_o the_o time_n of_o observation_n use_v xxxvii_o have_v the_o latitude_n sun_n declination_n and_o hour_n to_o find_v his_o altitude_n this_o problem_n be_v not_o of_o such_o use_n as_o the_o contrary_a viz._n have_v the_o altitude_n to_o 〈◊〉_d the_o hour_n it_o shall_v suffice_v to_o hint_n only_o two_o â—ayes_v the_o most_o convenient_a and_o first_o by_o the_o particular_a quadrant_n lay_v the_o thread_n to_o the_o day_n or_o declination_n than_o the_o nd_o from_o the_o hour_n to_o the_o thread_n measure_v in_o the_o particular_a scale_n of_o altitude_n shall_v show_v the_o sun_n altitude_n require_v second_o by_o the_o verse_v sin_n 1._o first_o of_o the_o co-latitude_n and_o sun_n distance_n from_o the_o pole_n find_v the_o sum_n and_o difference_n 2._o take_v the_o distance_n between_o they_o and_o make_v it_o a_o =_o verse_v sine_fw-la of_o 180_o by_o set_v the_o sector_n or_o lay_v the_o thread_n to_o nd_o 3._o then_o take_v the_o =_o verse_v sine_fw-la of_o the_o hour_n and_o lay_v it_o latteral_o from_o the_o sum_n and_o it_o shall_v give_v the_o compliment_n of_o the_o altitude_n require_v this_o work_n be_v the_o same_o both_o by_o the_o sector_n or_o general_a quadrant_n as_o be_v show_v in_o use_n â—he_v 36th_o and_o be_v nothing_o else_o but_o a_o backward_a work_n but_o the_o altitude_n at_o any_o azimuth_n be_v not_o so_o to_o be_v do_v to_o do_v the_o same_o by_o the_o natural-sines_n first_o have_v the_o latitude_n and_o the_o sun_n declination_n find_v the_o sun_n altitude_n or_o depression_n at_o 6_o and_o note_v the_o point_n either_o below_o or_o above_o or_o in_o the_o centre_n as_o be_v large_o show_v in_o use_v the_o 26_o where_o the_o altitude_n be_v give_v to_o find_v the_o hour_n in_o any_o latitude_n then_o lay_v the_o thread_n to_o the_o hour_n count_v in_o the_o degree_n either_o from_o 12_o or_o 6_o then_o take_v the_o nd_o from_o the_o cousin_a of_o the_o sun_n declination_n and_o make_v it_o a_o =_o in_o the_o sine_fw-la of_o 90_o lay_v the_o thread_n to_o the_o nd_o then_o the_o nd_o from_o the_o sine_fw-la compliment_n of_o the_o latitude_n to_o the_o thread_n shall_v reach_v from_o the_o note_a point_n for_o the_o sun_n altitude_n or_o depression_n at_o 6_o to_o the_o sun_n altitude_n require_v example_n latitude_n 51-32_a declination_n 23-71_a a_o 8_o or_o 4_o viz._n 2_o hour_n from_o 6_o southwards_o the_o altitude_n will_v be_v find_v to_o be_v 36-42′_a 1._o for_o the_o altitude_n at_o 6_o at_o any_o time_n of_o the_o year_n say_v as_o the_o sine_fw-la of_o 90_o tang._n to_o sine_fw-la of_o the_o latitude_n so_o be_v the_o sine_fw-la of_o the_o sun_n declination_n to_o the_o sine_fw-la of_o the_o sun_n altitude_n at_o 6._o 2._o for_o the_o sun_n altitude_n at_o any_o hour_n or_o quarter_n in_o aries_n or_o libra_n the_o equinoctial_a as_o sine_fw-la 90_o to_o cousin_a latitude_n so_o be_v the_o sine_fw-la of_o the_o sun_n distance_n from_o 6_o in_o degree_n to_o sine_fw-la of_o the_o sun_n altitude_n 3._o for_o the_o sun_n altitude_n at_o all_o other_o hour_n or_o time_n of_o the_o year_n as_o sine_fw-la 90_o to_o co-tangent_a latitude_n so_o be_v sine_fw-la of_o the_o sun_n distance_n from_o 6_o to_o the_o tangent_fw-la of_o a_o four_o ark_n in_o the_o tangent_n which_o
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
what_o ibid._n circle_n of_o position_n what_o ibid._n of_o term_n in_o astronomy_n what_o a_o sphere_n be_v page_z 50_o of_o ten_o point_n and_o ten_o circle_n of_o the_o sphere_n page_n 51_o the_o 2_o pole_n of_o the_o world_n or_o equinoctial_a ibid._n the_o 2_o pole_n of_o the_o zodiac_n page_n 52_o the_o 2_o equinoctial-point_n ibid._n the_o 2_o solstitial-point_n page_n 53_o the_o zenith_n and_o nadir_n page_n 54_o the_o horizon_n the_o meridian_n the_o equinoctial_a the_o zodiac_n the_o 2_o colour_n the_o 2_o tropic_n and_o 2_o polar_a circle_n page_n 55_o 56_o 58_o hour_n azimuth_n almicanter_n declination_n latitude_n longitude_n right_a ascention_n page_n 59_o 60_o oblique_a ascension_n difference_n of_o ascension_n amplitude_n circle_n and_o angle_n of_o position_n what_o they_o be_v page_z 61_o 62_o to_o rectify_v the_o triangule_a quadrant_n page_n 63_o to_o observe_v or_o find_v the_o sun_n altitude_n page_n 64_o to_o try_v if_o any_o thing_n be_v level_a or_o upright_o page_n 66_o to_o find_v what_o angle_n the_o sector_n stand_v at_o at_o any_o open_n or_o to_o set_v the_o sector_n to_o any_o angle_n require_v page_n 67_o 68_o the_o day_n of_o the_o month_n give_v to_o find_v the_o sun_n declination_n true_a place_n right_a ascention_n or_o rise_v and_o set_v by_o inspection_n only_o page_n 71_o to_o find_v the_o sun_n amplitude_n and_o difference_n of_o ascension_n and_o oblique_a ascension_n page_n 73_o to_o find_v the_o hour_n of_o the_o day_n page_n 74_o to_o find_v the_o sun_n azimuth_n page_n 75_o the_o use_n of_o the_o line_n of_o number_n and_o the_o use_n of_o the_o line_n of_o line_n both_o on_o the_o triangule_a quadrant_n and_o sector_n one_o after_o another_o in_o most_o example_n to_o multiply_v one_o number_n by_o another_o page_z 78_o a_o help_n to_o multiply_v true_o page_n 85_o a_o crabbed_a question_n of_o multiplication_n page_n 90_o precept_n of_o reduction_n page_n 94_o to_o divide_v one_o number_n by_o another_o page_z 95_o a_o caution_n in_o division_n page_n 97_o to_o 2_o line_n or_o number_n give_v to_o find_v a_o 3d_o in_o geometrical_a proportion_n page_n 98_o any_o one_o side_n of_o a_o figure_n be_v give_v to_o find_v all_o the_o rest_n or_o to_o find_v a_o proportion_n between_o two_o or_o more_o line_n or_o number_n page_n 99_o to_o lay_v down_o any_o number_n of_o part_n on_o a_o line_n to_o any_o radius_fw-la page_n 100_o to_o divide_v a_o line_n into_o any_o number_n of_o part_n page_z 102_o to_o find_v a_o geometrical_a mean_a proportion_n between_o two_o line_n or_o number_n three_o way_n page_n 104_o to_o make_v a_o square_a equal_a to_o a_o oblong_a page_n 107_o or_o to_o a_o triangle_n ibid._n to_o find_v a_o proportion_n between_o unlike_a superficies_n page_n 108_o to_o make_v one_o superficies_n like_o another_o superficies_n and_o equal_a to_o a_o three_o page_z 109_o the_o diameter_n and_o content_a of_o a_o circle_n be_v give_v to_o find_v the_o content_a of_o another_o circle_n by_o have_v his_o diameter_n page_n 111_o to_o find_v the_o square-root_n of_o a_o number_n ibid._n to_o find_v the_o cube-root_n of_o a_o number_n page_n 113_o to_o find_v two_o mean_v proportional_n between_o two_o line_n or_o number_n give_v page_n 116_o the_o diameter_n and_o content_a of_o a_o globe_n be_v give_v to_o find_v the_o content_a of_o another_o globe_n who_o diameter_n also_o be_v give_v page_n 118_o the_o proportion_n between_o the_o weight_n and_o magnitude_n of_o metal_n page_n 119_o the_o weight_n and_o magnitude_n of_o a_o body_n of_o one_o kind_n of_o metal_n be_v give_v to_o find_v the_o magnitude_n of_o a_o body_n of_o another_o metal_n of_o equal_a weight_n page_n 121_o the_o magnitude_n of_o two_o body_n of_o several_a metal_n have_v the_o weight_n of_o one_o give_v to_o find_v the_o weight_n of_o the_o other_o page_z 122_o the_o weight_n and_o magnitude_n of_o one_o body_n of_o any_o metal_n be_v give_v and_o another_o body_n like_o unto_o the_o former_a be_v to_o be_v make_v of_o any_o other_o metal_n to_o find_v the_o diameter_n or_o magnitude_n of_o it_o page_z 123_o to_o divide_v a_o line_n or_o number_n by_o extreme_a and_o mean_a proportion_n page_n 124_o three_o line_n or_o number_n give_v to_o find_v a_o four_o in_o geometrical_a proportion_n page_n 128_o the_o nature_n &_o reason_n of_o the_o golden_a rule_n page_n 129_o the_o rule_n of_o three_o inverse_v with_o several_a caution_n and_o example_n page_n 132_o the_o double_a and_o compound_v rule_n of_o three_o direct_v and_o reverse_v with_o example_n page_n 139_o the_o rule_n of_o fellowship_n with_o example_n page_n 148_o the_o use_n of_o the_o line_n of_o number_n in_o superficial_a measure_n and_o the_o part_n on_o the_o rule_n page_n 154_o the_o breadth_n give_v in_o foot-measure_n to_o find_v the_o length_n of_o one_o foot_n page_n 156_o the_o breadth_n give_v in_o inch_n to_o find_v how_o much_o in_o length_n make_v one_o foot_n ibid._n the_o breadth_n give_v to_o find_v how_o much_o be_v in_o a_o foot-long_a page_n 157_o have_v the_o length_n and_o breadth_n give_v in_o foot-measure_n to_o find_v the_o content_a in_o foot_n ibid._n have_v the_o breadth_n give_v in_o inch_n and_o length_n in_o foot_n to_o find_v the_o content_a in_o foot_n page_n 158_o have_v the_o length_n &_o breadth_n give_v in_o inches_n to_o find_v the_o content_n in_o superficial_a inch_n page_n 160_o have_v the_o length_n &_o breadth_n give_v in_o inches_n to_o find_v the_o content_a in_o foot_n superficial_a page_n 161_o the_o length_n and_o breadth_n of_o a_o oblong_o give_v to_o find_v the_o side_n of_o a_o square_n equal_a to_o it_o page_z 163_o the_o diameter_n of_o a_o circle_n give_v to_o find_v the_o circumference_n square_a equal_a square_n inscribe_v and_o content_a page_n 164_o the_o content_a of_o a_o circle_n give_v to_o find_v the_o diameter_n or_o circumference_n page_n 166_o 167_o certain_a rule_n to_o measure_v several_a figure_n page_n 108_o a_o segment_n of_o a_o circle_n give_v to_o find_v the_o true_a diameter_n and_o area_n thereof_o page_n 169_o a_o table_n to_o divide_v the_o line_n of_o segment_n page_n 170_o the_o use_n of_o it_o in_o part_n page_z 171_o the_o measure_n of_o triangle_n tapeziaes_n romboide_n poligon_n and_o oval_n page_n 172_o 173_o a_o table_n of_o the_o proportion_n between_o the_o side_n and_o area_n of_o regular_a poligon_n and_o the_o use_n thereof_o for_o any_o other_o page_z 174_o 175_o to_o make_v a_o oval_a equal_a to_o a_o circle_n and_o the_o contrary_a two_o way_n page_n 175_o 176_o the_o length_n and_o breadth_n of_o any_o oblong_a superficies_n give_v in_o foot_n to_o find_v the_o content_a in_o yard_n page_n 177_o the_o length_n and_o breadth_n give_v in_o foot_n and_o part_n to_o find_v the_o content_a in_o rod_n page_z 179_o the_o near_a way_n to_o measure_v a_o party_n wall_n page_z 180_o to_o multiply_v and_o reduce_v any_o length_n breadth_n or_o thickness_n of_o a_o wall_n to_o one_o brick_n and_o a_o half_a at_o one_o operation_n page_n 183_o example_n at_o six_o several_a thickness_n page_n 184_o to_o find_v the_o gage-point_n for_o this_o reduce_n page_n 185_o at_o one_o open_v of_o the_o compass_n to_o find_v how_o many_o rod_n quarter_n and_o foot_n in_o any_o sum_n under_o 10_o rod_n page_n 186_o the_o usual_a and_o ready_a equal_a way_n to_o measure_v tile_v and_o chimney_n page_n 187_o of_o plaisterers-work_n or_o painters-work_n page_n 188_o of_o particular_n of_o work_n usual_o mention_v in_o a_o carpenters-bill_n with_o caution_n page_n 189_o 190_o at_o any_o breadth_n of_o a_o house_n to_o find_v the_o rafter_n and_o hip-rafter_n length_n and_o angle_n by_o the_o line_n of_o number_n ready_o page_n 191_o the_o price_n of_o one_o foot_n be_v give_v to_o find_v the_o price_n of_o a_o rod_n or_o a_o square_a of_o brickwork_n or_o floore_v by_o inspection_n page_n 193_o at_o any_o length_n of_o a_o land_n give_v to_o find_v how_o much_o in_o breadth_n make_v one_o acre_n page_n 194_o a_o useful_a table_n in_o measure_v land_n and_o the_o use_n thereof_o in_o several_a example_n page_n 196_o 197_o the_o length_n and_o breadth_n give_v in_o perch_n to_o find_v the_o content_a in_o square_n perch_n pole_n or_o rod_n page_n 200_o the_o length_n and_o breadth_n in_o perch_n to_o find_v the_o content_a in_o acre_n ibid._n the_o length_n and_o breadth_n give_v in_o chain_n to_o find_v the_o content_n in_o square_a acre_n quarter_n and_o link_n page_n 201_o to_o measure_v a_o triangle_n at_o once_o without_o half_v the_o base_a or_o area_n ibid._n to_o reduce_v statute-measure_n or_o acre_n to_o customary_a and_o the_o contrary_a ibid._n a_o table_n to_o make_v scale_n to_o do_v it_o by_o measure_v or_o inspection_n with_o example_n page_n