be_v 13_o deg_n 30_o min._n probl._n 8._o the_o 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 count_v the_o contrary_a way_n as_o in_o the_o last_o problem_n and_o take_v from_o your_o scale_n of_o altitude_n 18_o deg_n for_o twilight_n and_o 13_o deg_n for_o daybreak_n or_o clear_a light_n with_o this_o run_v one_o point_n of_o the_o compass_n along_o the_o line_n of_o hour_n on_o that_o side_n next_o the_o end_n until_o the_o other_o will_v just_o touch_v the_o thread_n and_o then_o the_o former_a point_n give_v the_o respective_a time_n require_v ex._n gr_n at_o 7_o deg_n north_n declination_n day_n break_v 8_o minute_n before_o 4_o but_o twilight_n be_v 3_o hour_n 12_o minute_n in_o the_o morning_n or_o 8_o hour_n 52_o minute_n afternoon_n probl._n 9_o the_o declination_n and_o altitude_n of_o the_o sun_n or_o any_o star_n give_v to_o find_v their_o azimuth_n in_o northern_a declination_n lay_v the_o thread_n to_o the_o altitude_n number_v on_o the_o limb_n of_o the_o movable_a piece_n from_o 60_o 0_o towards_o the_o end_n and_o when_o occasion_n require_v continue_v your_o number_n forward_o upon_o the_o loose_a piece_n and_o take_v the_o declination_n from_o your_o line_n of_o altitude_n with_o this_o distance_n run_v one_o point_n of_o your_o compass_n along_o the_o line_n of_o azimuth_n on_o that_o side_n the_o thread_n next_o the_o head_n until_o the_o other_o just_a touch_n the_o thread_n than_o the_o former_a point_n give_v the_o azimuth_n from_o south_n ex._n gr_n at_o 10_o deg_n declination_n north_n and_o 30_o deg_n altitude_n the_o azimuth_n from_o south_n be_v 64_o deg_n 40_o min._n probl._n 10._o the_o sun_n altitude_n give_v to_o find_v his_o azimuth_n in_o the_o aequator_fw-la lay_v the_o thread_n to_o the_o altitude_n in_o the_o limb_n count_v from_o 60_o 0_o on_o the_o loose_a piece_n towards_o the_o end_n and_o on_o the_o line_n of_o azimuth_n it_o 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altitude_n at_o east_n or_o west_n in_o north_n declination_n and_o by_o consequent_a his_o depression_n in_o south_n declination_n take_v the_o declination_n give_v from_o the_o scale_n of_o altitude_n and_o set_v one_o point_n of_o your_o compass_n in_o 90_o on_o the_o line_n of_o azimuth_n lay_v the_o thread_n to_o the_o other_o point_n on_o that_o side_n 90_o next_o the_o head_n on_o the_o limb_n it_o cut_v the_o altitude_n count_v from_o 60_o 0_o on_o the_o movable_a piece_n ex._n gr_n at_o 10_o deg_n declination_n the_o altitude_n be_v 12_o deg_n 40_o min._n probl._n 13._o the_o declination_n and_o azimuth_n give_v to_o find_v the_o altitude_n of_o the_o sun_n or_o any_o star_n take_v the_o declination_n from_o the_o scale_n of_o altitude_n set_v one_o point_n of_o your_o compass_n in_o the_o azimuth_n give_v then_o in_o north_n declinanation_n turn_v the_o other_o point_n towards_o the_o head_n in_o south_n towards_o the_o end_n and_o thereto_o lay_v the_o thread_n on_o the_o limb_n you_o have_v the_o altitude_n number_v from_o 60_o 0_o on_o the_o movable_a piece_n towards_o the_o end_n ex._n gr_n at_o 7_o deg_n north_n declination_n and_o 48_o deg_n azimuth_n from_o south_n the_o altitude_n be_v 35_o deg_n but_o at_o 7_o deg_n declination_n south_n and_o 50_o deg_n azimuth_n the_o altitude_n be_v only_o 18_o deg_n 30_o min._n probl._n 14._o the_o altitude_n declination_n and_o right_a ascension_n of_o any_o star_n with_o the_o right_a ascension_n of_o the_o sun_n give_v to_o find_v the_o hour_n of_o the_o night_n take_v the_o star_n altitude_n from_o the_o scale_n of_o altitude_n and_o lay_v the_o thread_n to_o his_o declination_n in_o the_o limb_n find_v his_o hour_n from_o the_o last_o meridian_n he_o be_v upon_o as_o you_o do_v for_o the_o sun_n by_o probl._n 5._o if_o the_o star_n be_v past_o the_o south_n this_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 keep_v then_o set_v one_o point_n of_o your_o compass_n in_o the_o sun_n right_a ascension_n number_v upon_o the_o line_n twice_o 12_o or_o 24_o next_o the_o outward_a ledge_n on_o the_o fix_a piece_n extend_v the_o other_o point_n to_o the_o right_a ascension_n of_o the_o star_n number_v upon_o the_o same_o line_n observe_v which_o way_n you_o turn_v the_o point_n of_o your_o compass_n viz._n towards_o the_o head_n or_o end_n with_o this_o distance_n set_v one_o point_n of_o your_o compass_n in_o the_o star_n hour_n before_o find_v count_v on_o the_o same_o line_n and_o turn_v the_o other_o point_n the_o same_o way_n as_o you_o do_v for_o the_o right_a ascension_n it_o give_v the_o true_a hour_n of_o the_o night_n ex._n gr_n the_o 22_o of_o march_n i_o find_v the_o altitude_n of_o the_o lion_n heart_n 45_o deg_n his_o declination_n 13_o d._n 40_o min._n then_o by_o probl._n 5._o i_o find_v his_o hour_n from_o the_o last_o meridian_n 10_o hour_n 5_o min._n the_o right_a ascension_n of_o the_o sun_n be_v 46_o m._n of_o time_n or_o 11_o d._n 30_o m._n of_o the_o circle_n the_o right_a ascension_n of_o the_o lion_n heart_n be_v 9_o hour_n 51_o m._n fere_n of_o time_n or_o 147_o deg_n 43_o m._n of_o a_o circle_n then_o by_o a_o line_n of_o twice_o 12_o you_o may_v find_v the_o true_a hour_n of_o the_o night_n 7_o hour_n 13_o min._n probl._n 15._o the_o right_a ascension_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 time_n of_o night_n give_v to_o find_v the_o altitude_n of_o that_o star_n with_o his_o azimuth_n from_o south_n and_o by_o consequent_a to_o find_v the_o star_n although_o before_o you_o know_v it_o not_o this_o be_v no_o more_o than_o unravel_v the_o last_o problem_n 1_o therefore_o upon_o the_o line_n of_o twice_o 12_o or_o 24_o set_v one_o point_n of_o your_o compass_n in_o the_o right_a ascension_n of_o the_o star_n extend_v the_o other_o to_o the_o right_a ascension_n of_o the_o sun_n upon_o the_o same_o line_n that_o distance_n lay_v the_o same_o way_n upon_o the_o same_o line_n from_o the_o hour_n of_o the_o night_n give_v the_o star_n hour_n from_o the_o last_o meridian_n he_o be_v upon_o this_o find_v by_o probl._n 5._o find_v his_o altitude_n as_o you_o do_v for_o the_o sun_n last_o have_v now_o his_o declination_n and_o altitude_n by_o probl._n 8._o or_o 10._o according_a to_o his_o declination_n you_o will_v soon_o get_v his_o azimuth_n from_o south_n this_o need_v not_o a_o example_n by_o help_v of_o this_o problem_n the_o instrument_n may_v be_v so_o contrive_v as_o to_o be_v one_o of_o the_o best_a tutor_n for_o know_v of_o the_o star_n probl._n 16._o the_o altitude_n and_o azimuth_n of_o any_o star_n give_v to_o find_v his_o declination_n lay_v the_o thread_n to_o the_o altitude_n count_v on_o the_o limb_n from_o 60_o 0_o on_o the_o movable_a piece_n towards_o the_o end_n set_v one_o point_n of_o your_o compass_n in_o the_o azimuth_n take_v the_o near_a distance_n to_o the_o thread_n this_o apply_v to_o the_o scale_n of_o altitude_n give_v the_o declination_n if_o the_o azimuth_n give_v be_v on_o that_o side_n the_o thread_n towards_o the_o end_n the_o declination_n be_v south_n when_o on_o that_o side_n towards_o the_o head_n its_o north._n probl._n 17._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 hour_n of_o night_n give_v to_o find_v the_o star_n right_a ascension_n by_o probl._n 5._o or_o 14._o find_v the_o star_n hour_n from_o the_o meridian_n then_o on_o the_o line_n twice_o 12_o or_o 24_o set_v one_o point_n of_o your_o compass_n in_o the_o star_n hour_n thus_o find_v and_o extend_v the_o other_o to_o the_o hour_n of_o the_o night_n upon_o the_o same_o line_n
style_n height_n above_o the_o substile_a as_o the_o radius_fw-la be_v to_o the_o cousin_n of_o the_o latitude_n so_o be_v the_o cousin_n of_o declination_n to_o the_o sine_fw-la of_o the_o style_n height_n to_o find_v the_o substyles_n distance_n from_o the_o meridian_n as_o the_o radius_fw-la be_v to_o the_o sine_fw-la of_o declination_n so_o be_v the_o co-tangent_a of_o the_o latitude_n to_o the_o tangent_fw-la of_o the_o substyle_n from_o the_o meridian_n to_o find_v the_o angle_n of_o twelve_o and_o six_o as_o the_o co-tangent_a of_o the_o latitude_n be_v to_o the_o radius_fw-la so_o be_v the_o sine_fw-la of_o declination_n to_o the_o co-tangent_a of_o six_o from_o twelve_o to_o find_v the_o inclination_n of_o meridian_n as_o the_o sine_fw-la of_o the_o latitude_n be_v to_o the_o radius_fw-la so_o be_v the_o tangent_fw-la of_o declination_n to_o the_o tangent_fw-la of_o inclination_n of_o meridian_n 6._o all_o north_n decliner_n with_o centre_n have_v the_o angular_a point_n of_o the_o style_n downward_o and_o all_o south_n have_v it_o upward_o 7._o all_o north_n decliner_n without_o centre_n have_v the_o narrow_a end_n of_o the_o style_n downward_o all_o south_n have_v it_o upward_o 8._o in_o all_o decliner_n without_o centre_n take_v so_o much_o of_o the_o style_n as_o you_o think_v convenient_a but_o make_v point_n at_o its_o beginning_n and_o end_n upon_o the_o substyle_n of_o your_o paper_n draught_n and_o transmit_v those_o point_n to_o the_o substyle_n of_o your_o plain_n for_o direction_n in_o place_v your_o style_n thereon_o 9_o in_o all_o north_n decliner_n the_o meridian_n or_o inclination_n of_o meridian_n be_v the_o hour_n line_n of_o twelve_o at_o midnight_n in_o south_n decliner_n at_o noon_n or_o midday_n this_o may_v tell_v you_o the_o true_a name_n of_o the_o hour_n line_n 10._o in_o transmit_v these_o dial_n from_o your_o paper_n draught_n to_o your_o plain_n lay_v the_o horizontal_a of_o your_o paper_n draught_n upon_o the_o horizontal_a line_n of_o the_o plain_a and_o prick_v off_o the_o hour_n and_o substyle_n sect._n 5._o the_o affection_n of_o direct_a recline_a plain_n incline_v plain_n for_o south_n recliner_n and_o north_n incliner_n 1._o the_o difference_n betwixt_o your_o co-latitude_n and_o the_o reclination_n inclination_n be_v the_o elevation_n or_o height_n of_o the_o style_n 2._o when_o the_o reclination_n inclination_n exceed_v your_o co-latitude_n the_o contrary_a pole_n be_v elevate_v so_o much_o as_o the_o excess_n ex._n gr_n a_o north_n recliner_n or_o south_n incliner_n 50._o d._n in_o lat_fw-la 52._o 30._o min._n the_o excess_n of_o the_o reclination_n inclination_n to_o your_o co-latitude_n be_v 12._o d._n 30._o min._n and_o so_o much_o the_o north_n be_v elevate_v on_o the_o recliner_n and_o the_o south_n pole_n on_o the_o incliner_n 3._o when_o the_o reclination_n inclination_n be_v equal_a to_o the_o co-latitude_n it_o be_v a_o polar_n plain_a for_o south_n incliner_n and_o north_n recliner_n 1._o the_o sum_n of_o your_o co-latitude_n and_o the_o reclination_n inclination_n be_v the_o style_n elevation_n 2._o when_o the_o reclination_n inclination_n be_v equal_a to_o your_o latitude_n it_o be_v a_o equinoctial_a plain_n and_o the_o dial_n be_v no_o more_o than_o a_o circle_n divide_v into_o 24._o equal_a part_n have_v a_o wire_n of_o any_o convenient_a length_n place_v in_o the_o centre_n perpendicular_a to_o the_o plain_a for_o the_o style_n 3._o when_o the_o reclination_n inclination_n be_v great_a than_o your_o latitude_n take_v the_o sum_n of_o the_o reclination_n inclination_n of_o your_o co-latitude_n from_o 180._o and_o the_o residue_n or_o remain_v be_v the_o style_n height_n but_o in_o this_o case_n the_o style_n must_v be_v set_v upon_o the_o plain_a as_o if_o the_o contrary_a pole_n be_v elevate_v viz._n these_o north_n recliner_n must_v have_v the_o centre_n of_o the_o style_n upward_o and_o the_o south_n incliner_n have_v it_o downward_o note_n in_o all_o south_n re-in-cliners_a north_n re-in-cliners_a for_o their_o delineation_n the_o style_n height_n be_v to_o be_v call_v the_o co-latitude_n and_o then_o you_o may_v draw_v they_o as_o erect_v direct_a plain_n for_o south_n or_o north_n as_o the_o former_a rule_n shall_v give_v they_o in_o that_o latitude_n which_o be_v the_o compliment_n of_o the_o style_n height_n for_o direct_v east_n and_o west_n recliner_n incliner_n 1._o in_o all_o east_n and_o west_n recliner_n incliner_n you_o may_v refer_v they_o to_o a_o new_a latitude_n and_o new_a declination_n and_o then_o describe_v they_o as_o erect_v decline_a plain_n 2._o their_o new_a latitude_n be_v the_o compliment_n of_o that_o latitude_n where_o the_o plain_n stand_v and_o their_o new_a declination_n be_v the_o compliment_n of_o their_o reclination_n inclination_n but_o to_o know_v which_o way_n you_o be_v to_o account_v this_o new_a declination_n remember_v all_o east_n and_o west_n recliner_n be_v north-east_n and_o north-west_n decliner_n all_o east_n and_o west_n incliner_n be_v southeast_n and_o south-west_n decliner_n 3._o their_o new_a latitude_n and_o declination_n know_v you_o may_v by_o sect_n 4._o par_fw-fr 5._o find_v the_o substyle_n from_o the_o meridian_n height_n of_o the_o style_n angle_v of_o twelve_o and_o six_o and_o inclination_n of_o meridian_n use_v in_o those_o proportion_v the_o new_a latitude_n and_o new_a declination_n instead_o of_o the_o old_a 4._o in_o all_o east_n and_o west_n recliner_n incliner_n with_o centre_n the_o meridian_n lie_v in_o the_o horizontal_a line_n of_o the_o plain_a in_o such_o as_o have_v not_o centre_n its_o parallel_n to_o the_o horizontal_a line_n sect._n 6._o the_o affection_n of_o decline_a recline_a plain_n incline_v plain_n the_o ready_a way_n for_o these_o be_v to_o refer_v they_o to_o a_o new_a latitude_n and_o new_a declination_n by_o the_o subsequent_a proportion_n 1._o to_o find_v the_o new_a latitude_n as_o the_o radius_fw-la be_v to_o the_o cousin_n of_o the_o plain_n declination_n so_o be_v the_o co-tangent_a of_o the_o reclination_n inclination_n to_o the_o tangent_fw-la of_o a_o four_o ark_n in_o south_n recliner_n north_n incliner_n get_v the_o difference_n betwixt_o this_o four_o ark_n and_o the_o latitude_n of_o your_o place_n and_o the_o compliment_n of_o that_o difference_n be_v the_o new_a latitude_n seek_v if_o the_o four_o ark_n be_v less_o than_o your_o old_a latitude_n the_o contrary_a pole_n be_v elevate_v if_o equal_a to_o your_o old_a latitude_n it_o be_v a_o polar_n plain_a in_o south_n incliner_n north_n recliner_n the_o difference_n betwixt_o the_o four_o ark_n and_o the_o compliment_n of_o your_o old_a latitude_n be_v the_o new_a latitude_n if_o the_o four_o ark_n be_v equal_a to_o your_o old_a co-latitude_n they_o be_v equinoctial_a plain_n 2._o to_o find_v the_o new_a declination_n as_o the_o radius_fw-la be_v to_o the_o cousin_n of_o the_o reclination_n inclination_n so_o be_v the_o sine_fw-la of_o the_o old_a declination_n to_o the_o sine_fw-la of_o the_o new_a 3._o to_o find_v the_o angle_n of_o the_o meridian_n with_o the_o horizontal_n line_n of_o the_o plain_a as_o the_o radius_fw-la be_v to_o the_o tangent_fw-la of_o the_o old_a declination_n so_o be_v the_o sine_fw-la of_o reclination_n inclination_n to_o the_o co-tangent_a of_o the_o angle_n of_o the_o meridian_n with_o the_o horizontal_a line_n of_o the_o plain_a this_o give_v the_o angle_n for_o its_o situation_n observe_v in_o north_n incliner_n less_o than_o a_o polar_n the_o meridian_n lie_v above_o that_o end_n of_o the_o horizontall_a line_n contrary_a to_o the_o coast_n of_o declination_n  _fw-fr  _fw-fr below_o  _fw-fr south_n recliner_n more_o than_o a_o polar_n the_o meridian_n lie_v below_o that_o end_n of_o the_o horizontall_a next_o the_o coast_n of_o declination_n  _fw-fr  _fw-fr above_o  _fw-fr north_n recliner_n less_o than_o a_o equinoctial_a the_o meridian_n lie_v above_o that_o end_n of_o the_o horizontalnext_v the_o coast_n of_o declination_n  _fw-fr  _fw-fr below_o in_o north_n recliner_n this_o meridian_n be_v 12._o at_o midnight_n  _fw-fr equal_a to_o a_o equinoctial_a the_o meridian_n descend_v below_o the_o horizontal_n at_o that_o end_n contrary_a to_o the_o coast_n of_o declination_n and_o the_o substyle_n lie_v in_o the_o hour_n line_n of_o six_o  _fw-fr  _fw-fr south_n incliner_n more_o than_o a_o equinoctial_a the_o meridian_n lie_v below_o that_o end_n of_o the_o horizontal_a contrary_n to_o the_o declination_n  _fw-fr  _fw-fr above_o in_o south_n incliner_n this_o meridian_n be_v only_o useful_a for_o draw_v the_o dial_n and_o place_v the_o substyle_n for_o the_o hour_n line_n must_v be_v draw_v through_o the_o centre_n to_o the_o low_a side_n after_o you_o have_v by_o the_o former_a proportion_n and_o rule_v find_v the_o new_a latitude_n new_a declination_n the_o angle_n and_o situation_n of_o the_o meridian_n your_o first_o business_n in_o delineat_v of_o the_o dial_n will_v be_v both_o for_o such_o as_o have_v centre_n and_o
line_n of_o second_o continue_v unto_o 60_o and_o mark_v at_o the_o end_n se._n next_o to_o the_o outward_a edge_n on_o the_o fix_a and_o movable_a piece_n which_o be_v best_a discern_v when_o those_o piece_n be_v open_v to_o the_o full_a length_n be_v a_o line_n of_o meridian_n divide_v to_o 85_o who_o use_n be_v for_o navigation_n in_o describe_v map_n or_o chart_n etc._n etc._n in_o the_o vacant_a space_n you_o may_v have_v a_o line_n of_o chord_n sin_n and_o tangent_n to_o any_o radius_fw-la the_o space_n will_v bear_v and_o what_o other_o any_o one_o think_v best_a of_o as_o a_o line_n of_o latitude_n and_o hour_n etc._n etc._n on_o the_o proportional_a side_n of_o the_o loose_a piece_n be_v line_n for_o measure_v all_o manner_n of_o solid_n as_o timber_n stone_n etc._n etc._n likewise_o for_o gage_v of_o vessel_n either_o in_o wine_n or_o ale_n measure_n on_o the_o outward_a ledge_n of_o the_o movable_a and_o fix_a piece_n both_o which_o in_o use_n must_v be_v stretch_v out_o to_o the_o full_a length_n be_v a_o line_n of_o artificial_a number_n sin_n tangent_n and_o verse_v sin_n the_o first_o mark_v n_o the_o second_o saint_n the_o three_o t_n the_o four_o we_o on_o the_o inward_a ledge_n of_o the_o movable_a piece_n be_v a_o line_n of_o 12_o inch_n divide_v into_o half_n quarter_n half-quarter_n next_o to_o that_o be_v a_o prick_v line_n who_o use_n be_v for_o compute_v of_o weight_n and_o carriage_n last_o a_o line_n of_o foot_n measure_n or_o a_o foot_n divide_v into_o ten_o part_n and_o each_o of_o those_o subdivide_v into_o ten_o or_o twenty_o more_o on_o the_o inward_a ledge_n of_o the_o loose_a piece_n you_o may_v have_v a_o line_n of_o circumference_n diameter_n square_a equal_a and_o square_v inscribe_v there_o will_v still_o be_v requisite_a sight_n a_o thread_n and_o plummet_n and_o if_o any_o go_v to_o the_o price_n of_o a_o slide_a index_n to_o find_v the_o shadow_n from_o the_o plain_n perpendicular_a in_o order_n to_o take_v a_o plain_n declination_n and_o have_v a_o staff_n and_o a_o ballsocket_n the_o instrument_n be_v complete_v with_o its_o furniture_n proceed_v we_o now_o to_o the_o use_n only_o note_n by_o the_o way_n that_o mr._n brown_n have_v for_o conveniency_n of_o carry_v a_o pair_n of_o compass_n pen_n ink_n and_o pencil_n contrive_v the_o fix_a piece_n and_o movable_a both_o to_o be_v hollow_a and_o then_o the_o piece_n that_o cover_v those_o hollow_n do_v one_o supply_n the_o place_n of_o the_o loose_a piece_n for_o take_v altitude_n the_o other_o be_v a_o slide_a rule_n for_o measure_v solid_n and_o gage_v vessel_n without_o compass_n chap._n ii_o some_o use_n of_o the_o quadrantal_a side_n of_o the_o instrument_n probl._n 1._o to_o find_v the_o altitude_n of_o the_o sun_n or_o any_o star_n hang_v the_o thread_n and_o plummet_n upon_o the_o pin_n at_o the_o begin_n of_o the_o line_n of_o sin_n on_o the_o fix_a piece_n and_o have_v two_o sight_n in_o two_o hole_n parallel_v to_o that_o line_n raise_v the_o end_n of_o the_o fix_a piece_n towards_o the_o sun_n until_o the_o ray_n pass_v through_o the_o sight_n but_o when_o the_o sun_n be_v in_o a_o cloud_n or_o you_o take_v the_o altitude_n of_o any_o star_n look_v along_o the_o outward_a ledge_n of_o the_o fix_a piece_n until_o it_o be_v even_o with_o the_o middle_n of_o the_o sun_n or_o star_n then_o on_o the_o limb_n the_o thread_n cut_v the_o degree_n of_o altitude_n if_o you_o reckon_v from_o 0_o 60_o on_o the_o loose_a piece_n towards_o the_o head_n of_o the_o movable_a piece_n probl._n 2._o the_o day_n of_o the_o month_n give_v to_o find_v the_o sun_n place_n declination_n ascensional_a difference_n or_o time_n of_o rise_v and_o set_v with_o his_o right_a ascension_n the_o thread_n lay_v to_o the_o day_n of_o the_o month_n give_v the_o sun_n place_n in_o the_o line_n of_o sign_n reckon_v according_a to_o the_o order_n of_o the_o month_n viz._n forward_o from_o march_n the_o 10_o to_o june_n then_o backward_o to_o december_n and_o forward_o again_o to_o march_v 10._o in_o the_o limb_n you_o have_v the_o sun_n declination_n reckon_v from_o 60_o 0_o on_o the_o movable_a piece_n towards_o the_o head_n for_o north_n towards_o the_o end_n for_o south_n declination_n again_o on_o the_o line_n of_o right_a ascension_n the_o thread_n show_v the_o sun_n right_a ascension_n in_o degree_n or_o hour_n according_a to_o the_o make_n of_o your_o line_n count_v from_o aries_n towards_o the_o head_n and_o so_o back_o again_o according_a to_o the_o course_n of_o the_o sign_n unto_o 24_o hour_n or_o 360_o degree_n last_o on_o the_o line_n of_o hour_n you_o have_v the_o time_n of_o sun_n rise_v and_o set_v which_o turn_v into_o degree_n for_o the_o time_n from_o six_o give_v the_o ascensional_a difference_n ex._n gr_n in_fw-ge lat_fw-ge 52._o deg_n 30_o min._n for_o which_o latitude_n i_o shall_v make_v all_o the_o example_n the_o 22_o day_n of_o march_n i_o lay_v the_o thread_n to_o the_o day_n in_o the_o month_n and_o find_v it_o cut_v in_o the_o sign_n 12_o deg_n 20_o min._n for_o the_o sun_n place_n on_o the_o limb_n 4_o deg_n 43_o min._n for_o the_o sun_n declination_n north._n in_o the_o line_n of_o right_a ascension_n it_o give_v 46_o min._n of_o time_n or_o 11_o deg_n 30_o min._n of_o the_o circle_n last_o on_o the_o line_n of_o hour_n it_o show_v 28_o min._n before_o six_o for_o the_o sun_n rise_v or_o which_o be_v all_o 7_o deg_n for_o his_o ascensional_a difference_n probl._n 3_o the_o declination_n of_o the_o sun_n or_o any_o star_n give_v to_o find_v their_o amplitude_n take_v the_o declination_n from_o the_o scale_n of_o altitude_n with_o this_o distance_n set_v one_o point_n of_o your_o compass_n at_o 90_o on_o the_o line_n of_o azimuth_n apply_v the_o other_o point_n to_o the_o same_o line_n it_o give_v the_o amplitude_n count_v from_o 90_o ex._n gr_n at_o 10_o deg_n declination_n the_o amplitude_n be_v 16_o deg_n 30_o min._n at_o 20_o deg_n declination_n the_o amplitude_n be_v 34_o deg_n probl._n 4._o the_o right_a ascension_n of_o the_o sun_n with_o his_o ascensional_a difference_n give_v to_o find_v the_o oblique_a ascension_n in_o northern_a declination_n the_o difference_n betwixt_o the_o right_a ascension_n and_o ascensional_a difference_n be_v the_o oblique_a ascension_n in_o southern_a declination_n take_v the_o sum_n of_o they_o for_o the_o oblique_a ascension_n ex._n gr_n at_o 11_o deg_n 30_o sec._n right_a ascension_n and_o 6_o deg_n 30_o sec._n ascensional_a difference_n in_o northern_a declination_n the_o oblique_a ascension_n will_v be_v 5_o deg_n in_o southern_a 18_o deg_n probl._n 5._o the_o sun_n altitude_n and_o declination_n or_o the_o day_n of_o the_o month_n give_v to_o find_v the_o hour_n take_v the_o sun_n altitude_n from_o the_o scale_n of_o altitude_n and_o lay_v the_o thread_n to_o the_o declination_n in_o the_o limb_n or_o which_o be_v all_o one_o to_o the_o day_n in_o the_o month_n move_v one_o point_n of_o the_o compass_n along_o the_o line_n of_o hour_n on_o that_o side_n the_o thread_n next_o the_o end_n until_o the_o other_o point_n just_a touch_n the_o thread_n then_o the_o former_a point_n show_v the_o hour_n but_o whether_o it_o be_v before_o or_o after_o noon_n be_v leave_v to_o your_o judgement_n to_o determine_v ex._n gr_n the_o 22_o day_n of_o march_n or_o 4_o deg_n 43_o min._n north_n declination_n and_o 20_o deg_n altitude_n the_o hour_n be_v either_o 47_o minute_n past_o 7_o in_o the_o morning_n or_o 13_o minute_n past_o 4_o afternoon_n probl._n 6._o the_o declination_n of_o the_o sun_n or_o day_n of_o the_o month_n and_o hour_n give_v to_o find_v the_o altitude_n lay_v the_o thread_n to_o the_o day_n or_o declination_n and_o take_v the_o least_o distance_n from_o the_o hour_n to_o the_o thread_n this_o apply_v to_o the_o line_n of_o altitude_n give_v the_o altitude_n require_v ex._n gr_n the_o 5_o day_n of_o april_n or_o 10_o deg_n declination_n north_n at_o 7_o in_o the_o morning_n or_o 5_o afternoon_n the_o altitude_n will_v be_v 17_o deg_n 10_o sec._n and_o better_o probl._n 7._o the_o declination_n and_o hour_n of_o the_o night_n give_v to_o find_v the_o sun_n depression_n under_o the_o horizon_n lay_v the_o thread_n to_o the_o declination_n on_o the_o limb_n but_o count_v the_o contrary_a way_n viz._n from_o 60_o 0_o on_o the_o movable_a piece_n towards_o the_o head_n for_o southern_a and_o towards_o the_o end_n for_o northern_a declination_n this_o do_v take_v the_o near_a distance_n from_o the_o hour_n to_o the_o thread_n and_o apply_v it_o to_o the_o line_n of_o altitude_n you_o have_v the_o degree_n of_o the_o sun_n depression_n ex._n gr_n at_o 5_o deg_n northern_a declination_n &_o 8_o hour_n afternoon_n the_o depression_n
with_o this_o distance_n set_v one_o point_n of_o your_o compass_n in_o the_o right_a ascension_n of_o the_o sun_n and_o turn_v the_o other_o point_n the_o same_o way_n as_o you_o do_v for_o the_o hour_n it_o give_v the_o star_n right_a ascension_n probl._n 18._o the_o meridian_n altitude_n give_v to_o find_v the_o time_n of_o sun-rise_n and_o sunset_n take_v the_o meridian_n altitude_n from_o your_o particular_a scale_n and_o set_v one_o point_n of_o your_o compass_n upon_o the_o point_n 12_o on_o the_o line_n of_o hour_n that_o be_v the_o pin_n at_o the_o end_n lay_v the_o thread_n to_o the_o other_o point_n and_o on_o the_o line_n of_o hour_n the_o thread_n give_v the_o time_n require_v probl._n 19_o to_o find_v any_o latitude_n your_o particular_a scale_n be_v make_v for_o take_v the_o distance_n from_o 90_o on_o the_o line_n of_o azimuth_n unto_o the_o pin_n at_o the_o end_n of_o that_o line_n or_o the_o point_n 12_o this_o apply_v to_o the_o particular_a scale_n give_v the_o compliment_n of_o that_o latitude_n the_o instrument_n be_v make_v for_o probl._n 20._o to_o find_v the_o angle_n of_o the_o substile_a stile_n inclination_n of_o meridian_n and_o six_o and_o twelve_o for_o exact_a decline_a plain_n in_o that_o latitude_n your_o scale_n of_o altitude_n be_v make_v for_o sect._n 1_o to_o find_v the_o distance_n of_o the_o substile_a from_o 12_o or_o the_o plain_n perpendicular_a lay_v the_o thread_n to_o the_o compliment_n of_o declination_n count_v on_o the_o line_n of_o azimuth_n and_o on_o the_o limb_n it_o give_v the_o substile_a count_v from_o 60_o 0_o on_o the_o movable_a piece_n sect._n 2._o to_o find_v the_o angle_n of_o the_o style_n be_v height_n on_o the_o line_n of_o azimuth_n take_v the_o distance_n from_o the_o plain_n declination_n to_o 90._o this_o apply_v to_o the_o scale_n of_o altitude_n give_v the_o angle_n of_o the_o stile_n sect._n 3_o the_o angle_n of_o the_o substile_a give_v to_o find_v the_o inclination_n of_o meridian_n take_v the_o angle_n of_o the_o substile_a from_o the_o scale_n of_o altitude_n and_o apply_v it_o from_o 90_o on_o the_o azimuth_n line_n towards_o the_o end_n the_o figure_n show_v the_o compliment_n of_o inclination_n of_o meridian_n sect._n 4._o to_o find_v the_o angle_n betwixt_o 6_o and_o 12._o take_v the_o declination_n from_o the_o scale_n of_o altitude_n and_o set_v one_o point_n of_o your_o compass_n in_o 90_o on_o the_o line_n of_o azimuth_n lay_v the_o thread_n to_o the_o other_o point_n and_o on_o the_o limb_n it_o give_v the_o compliment_n of_o the_o angle_n seek_v number_v from_o 60_o 0_o on_o the_o movable_a piece_n towards_o the_o end_n this_o last_o rule_n be_v not_o exact_a nor_o be_v it_o here_o worth_a the_o labour_n to_o rectify_v it_o by_o another_o sine_fw-la add_v since_o you_o have_v a_o exact_a proportion_n for_o the_o problem_n in_o the_o treatise_n of_o dial_v chap._n 2_o sect._n 5._o paragr_n 4._o chap._n iii_o some_o use_n of_o the_o line_n of_o natural_a sign_n on_o the_o quadrantal_a side_n of_o the_o fix_a piece_n probl._n 1._o how_o to_o add_v one_o sign_n to_o another_o on_o the_o line_n of_o natural_a sin_n to_o add_v one_o sine_fw-la to_o another_o be_v to_o augment_v the_o line_n of_o one_o sine_fw-la by_o the_o line_n of_o the_o other_o sine_fw-la to_o be_v add_v to_o it_o ex._n gr_n to_o add_v the_o sine_z 15_o to_z the_o sine_z 20_o i_o take_v the_o distance_n from_o the_o begin_n of_o the_o line_n of_o sin_n unto_o 15_o and_o set_v one_o point_n of_o the_o compass_n in_o 20_o upon_o the_o same_o line_n turn_v the_o other_o towards_o 90_o which_o i_o find_v touch_n in_o 37._o so_o that_o in_o this_o case_n for_o we_o regard_v not_o the_o arithmetical_a but_o proportional_a aggregate_v 15_o add_v to_o 20_o upon_o the_o line_n of_o natural_a sin_n be_v the_o sine_fw-la 37_o upon_o that_o line_n and_o from_o the_o begin_n of_o the_o line_n to_o 37_o be_v the_o distance_n i_o be_o to_o take_v for_o the_o sum_n of_o 20_o and_o 15_o sin_n probl._n 2._o how_o to_o subtract_v one_o sine_fw-la from_o another_o upon_o the_o line_n of_o natural_a sin_n the_o substract_n of_o one_o sine_fw-la from_o another_o be_v no_o more_o than_o take_v the_o distance_n from_o the_o lesser_a to_o the_o great_a on_o the_o line_n of_o sin_n and_o that_o distance_n apply_v to_o the_o line_n from_o the_o beginning_n give_v the_o residue_n or_o remainder_n ex._n gr_n to_o subtract_v 20_o from_o 37_o i_o take_v the_o distance_n from_o 20_o to_o 37_o that_o apply_v to_o the_o line_n from_o the_o beginning_n give_v 15_o for_o the_o sine_fw-la remain_v probl._n 3_o to_o work_v proportion_n in_o sin_n alone_o here_o be_v four_o case_n that_o include_v all_o proportion_n in_o sin_n alone_o case_n 1._o when_o the_o first_o term_n be_v radius_fw-la or_o the_o sine_fw-la 90._o lay_v the_o thread_n to_o the_o second_o term_n count_v on_o the_o degree_n upon_o the_o movaeble_a piece_n from_o the_o head_n towards_o the_o end_n then_o number_v the_o three_o on_o the_o line_n of_o sin_n take_v the_o near_a distance_n from_o thence_o to_o the_o thread_n and_o that_o apply_v to_o the_o scale_n from_o the_o beginning_n give_v the_o four_o term_n ex._n gr_n as_o the_o radius_fw-la 90_o be_v to_o the_o sine_fw-la 20_o so_o be_v the_o sine_fw-la 30_o to_o the_o sine_fw-la 10._o case_n 2._o when_o the_o radius_fw-la be_v the_o three_o term_n take_v the_o sine_fw-la of_o the_o second_o term_n in_o your_o compass_n and_o enter_v it_o in_o the_o first_o term_n upon_o the_o line_n of_o sin_n and_o lay_v the_o thread_n to_o the_o near_a distance_n on_o the_o limb_n the_o thread_n give_v the_o four_o term_n ex._n gr_n as_o the_o sine_fw-la 30_o be_v to_o the_o sine_fw-la 20_o so_o be_v the_o radius_fw-la to_o the_o sine_fw-la 43._o 30._o min._n case_n 3._o when_o the_o radius_fw-la be_v the_o second_o term_n provide_v the_o three_o term_n be_v not_o great_a than_o the_o first_o transpose_v the_o term_n the_o method_n of_o transposition_n in_o this_o case_n be_v as_o the_o first_o term_n be_v to_o the_o three_o so_o be_v the_o second_o to_o the_o four_o and_o then_o the_o work_n will_v be_v the_o same_o as_o in_o the_o second_o case_n ex._n gr_n as_o the_o sine_fw-la 30_o be_v to_o the_o radius_fw-la or_o sine_fw-la 90_o so_o be_v the_o sine_fw-la 20_o to_o what_o sine_fw-la which_o transpose_v be_v as_o the_o sine_fw-la 30_o be_v to_o the_o sine_fw-la 20_o so_o be_v the_o radius_fw-la to_o a_o four_o sine_fw-la which_o will_v be_v find_v 43_o 30_o min._n as_o before_o case_n 4._o when_o the_o radius_fw-la be_v none_o of_o the_o three_o term_n give_v in_o this_o case_n when_o both_o the_o middle_a term_n be_v less_o than_o the_o first_o enter_z the_o sine_z of_o the_o second_o term_n in_o the_o first_o and_o lay_v the_o thread_n to_o the_o near_a distance_n take_v the_o near_a extent_n from_o the_o three_o to_o the_o thread_n this_o distance_n apply_v to_o the_o scale_n from_o the_o beginning_n give_v the_o four_o ex._n gr_n as_o the_o sine_fw-la 20_o to_o the_o sine_z 10_o so_o be_v the_o sine_fw-la 30_o to_o the_o sine_fw-la 15._o when_o only_o the_o second_o term_n be_v great_a than_o the_o first_o transpose_v the_o term_n and_o work_n as_o before_o but_o when_o both_o the_o middle_a term_n be_v great_a than_o the_o first_o this_o proportion_n will_v not_o be_v perform_v by_o this_o line_n without_o a_o parallel_n entrance_n or_o double_a radius_fw-la which_o inconveniency_n shall_v be_v remedy_v in_o its_o proper_a place_n when_o we_o show_v how_o to_o work_v proportion_n by_o the_o line_n of_o natural_a sin_n on_o the_o proportional_a or_o sector_n side_n these_o four_o case_n comprise_v the_o method_n of_o work_v all_o proportion_n by_o natural_a sin_n alone_o i_o shall_v propose_v some_o example_n for_o the_o exercise_n of_o young_a practitioner_n and_o therewith_o conclude_v this_o chapter_n probl._n 4._o to_o find_v the_o sun_n amplitude_n in_o any_o latitude_n as_o the_o cousin_n of_o the_o latitude_n be_v to_o the_o sine_fw-la of_o the_o sun_n declination_n so_o be_v the_o radius_fw-la to_o the_o sine_fw-la of_o amplitude_n probl._n 5._o to_o find_v the_o hour_n in_o any_o latitude_n in_o northern_a declination_n proport_n 1._o as_o the_o radius_fw-la to_o the_o sine_fw-la of_o the_o sun_n declination_n so_o be_v the_o sine_fw-la of_o the_o latitude_n to_o the_o sine_fw-la of_o the_o sun_n altitude_n at_o six_o by_o probl._n 2._o subtract_v this_o altitude_n at_o six_o from_o the_o present_a altitude_n and_o take_v the_o difference_n then_o proport_n 2._o as_o the_o cousin_n of_o the_o latitude_n be_v to_o that_o difference_n so_o be_v the_o radius_fw-la to_o a_o four_o sine_fw-la again_o proport_n 3._o as_o the_o
the_o four_o term_v seek_v the_o work_n be_v by_o probl._n 2._o if_o 60_o man_n do_v a_o work_n in_o 5_o day_n how_o long_o will_v 30_o man_n be_v about_o it_o as_o 30_o be_v to_o 60_o so_o be_v 5_o to_o 10._o the_o number_n of_o day_n for_o 30_o man_n in_o the_o work_n again_o when_o the_o four_o term_n be_v to_o be_v less_o than_o the_o single_a denomination_n say_v as_o the_o great_a double_a denomination_n be_v to_o the_o lesser_a double_a denomination_n so_o be_v the_o single_a denomination_n to_o the_o four_o term_v seek_v if_o 30_o man_n do_v a_o work_n in_o 5_o day_n how_o long_o shall_v 60_o be_v do_v of_o it_o as_o 60_o be_v to_o 30_o so_o be_v 5_o to_o 2½_n the_o time_n for_o 60_o man_n in_o the_o work_n probl._n 4._o the_o length_n of_o any_o perpendicular_a with_o the_o length_n of_o the_o shadow_n thereof_o give_v to_o find_v the_o sun_n altitude_n at_o the_o length_n of_o the_o shadow_n upon_o the_o line_n of_o line_n be_v to_o the_o tangent_fw-la 45_o so_o be_v the_o 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20_o 30_o 40_o 50_o 60_o 70_o 80_o 90_o 100_o so_o that_o for_o any_o number_n under_o 100_o the_o figure_n ready_o direct_v you_o ex._n gr_n to_o find_v 79_o on_o the_o line_n of_o number_n count_v 9_o of_o the_o small_a division_n beyond_o 70_o and_o there_o be_v the_o point_n for_o that_o number_n now_o as_o the_o whole_a line_n be_v actual_o divide_v into_o 100_o part_n so_o be_v every_o one_o of_o those_o part_n subdivide_v so_o far_o as_o conveniency_n will_v permit_v actual_o into_o ten_o part_n more_o by_o which_o mean_v you_o have_v the_o whole_a line_n actual_o divide_v into_o 1000_o part_n for_o reckon_v the_o figure_n impress_v 1_o 2_o 3_o 4_o 5_o 6_o 7_o 8_o 9_o to_o be_v 10_o 20_o 30_o 40_o 50_o 60_o 70_o 80_o 90_o and_o the_o other_o figure_n which_o be_v stamp_v 10_o 20_o 30_o 40_o 50_o 60_o 70_o 80_o 90_o 100_o to_o be_v 100_o 200_o 300_o 400_o 500_o 600_o 700_o 800_o 900_o 1000_o you_o may_v enter_v any_o number_n under_o 1000_o upon_o the_o line_n according_a to_o the_o former_a direction_n and_o any_o number_n who_o product_n surmount_v not_o 1000_o may_v be_v wrought_v upon_o this_o line_n but_o where_o the_o product_n exceed_v 1000_o this_o line_n will_v do_v nothing_o accurate_o wherefore_o i_o shall_v willing_o omit_v many_o problem_n mention_v by_o some_o writer_n to_o be_v wrought_v by_o this_o line_n as_o square_n and_o cube_v of_o number_n etc._n etc._n sith_o they_o have_v only_a nicety_n and_o nothing_o of_o exactness_n in_o they_o probl._n 2._o to_o multiply_v two_o number_n give_v by_o the_o line_n of_o number_n the_o proportion_n be_v this_o as_o 1_o on_o the_o line_n be_v to_o the_o multiplicator_n so_o be_v the_o multiplicand_v to_o the_o product_n ex._n gr_n as_o 1_o be_v to_o 4_o so_o be_v 7_o to_z what_o extend_v the_o compass_n from_o the_o first_o term_n viz._n i_o unto_o the_o second_o term_n viz._n 4._o with_o that_o distance_n set_v one_o point_n in_o 7_o the_o three_o term_n turn_v the_o other_o point_n of_o the_o compass_n towards_o the_o same_o end_n of_o the_o rule_n as_o at_o first_o and_o you_o have_v the_o four_o viz._n 28._o there_o be_v only_o one_o difficulty_n remain_v in_o this_o problem_n and_o that_o be_v to_o determine_v the_o number_n of_o place_n or_o figure_n in_o the_o product_n which_o may_v be_v resolve_v by_o this_o general_a rule_n the_o product_n always_o contain_v as_o many_o figure_n as_o be_v in_o the_o multiplicand_a and_o multiplicator_n both_o unless_o the_o two_o first_o figure_n of_o the_o product_n be_v great_a than_o the_o two_o first_o figure_n in_o the_o multiplicator_n and_o then_o the_o product_n must_v have_v one_o figure_n less_o than_o be_v in_o the_o multiplicator_n and_o multiplicand_a both_o ex._n gr_n 47_o multiply_v by_o 25_o be_v 2175_o consist_v of_o four_o figure_n but_o 16_o multiply_v by_o 16_o be_v 240_o consist_v of_o no_o more_o than_o three_o place_n for_o the_o reason_n before_o mention_v i_o here_o for_o distinction_n sake_n call_v the_o 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give_v number_n will_v show_v the_o respective_a mean_a proportional_n probl._n 5._o to_o work_v proportion_n in_o sines_n alone_o by_o the_o artificial_a line_n of_o sines_n extend_v the_o compass_n from_o the_o first_o term_n to_o the_o second_o with_o that_o distance_n set_v one_o point_n in_o the_o three_o term_n and_o the_o other_o point_n give_v the_o four_o only_o observe_v that_o if_o the_o second_o term_n be_v less_o than_o the_o first_o the_o four_o must_v be_v less_o 〈…〉_z or_o if_o the_o second_o term_n exceed_v 〈…〉_z four_o will_v be_v great_a than_o the_o three_o this_o may_v direct_v you_o in_o all_o proportion_n of_o sin_n and_o tangent_n single_o or_o conjunct_o to_o
in_o length_n make_v a_o foot_n as_o the_o square_n in_o foot_n and_o inch_n on_o the_o right_n be_v to_o one_o foot_n on_o the_o left_a so_o be_v the_o point_n sq._n on_o the_o right_n to_o the_o number_n of_o foot_n and_o inch_n on_o the_o left_a for_o a_o foot_n square_n of_o timber_n at_o 18_o inch_n square_a 5_o inch_n ¼_n and_o almost_o half_a a_o quarter_n in_o length_n make_v a_o foot_n when_o your_o timber_n if_o it_o be_v proper_a to_o call_v such_o piece_n by_o that_o name_n be_v under_o 3_o inch_n square_a account_v the_o figure_a division_n on_o the_o right_a line_n from_o the_o letter_n sq._n to_o the_o end_n for_o inch_n and_o each_o sub-division_n twelve_o part_n of_o a_o inch_n so_o that_o every_o three_o of_o they_o make_v a_o quarter_n of_o a_o inch_n then_o the_o proportion_n be_v as_o the_o inch_n and_o quarter_n square_a on_o the_o right_n be_v to_o 100_o on_o the_o left_a so_o be_v the_o point_n 12_o betwixt_o d_o and_o t_o d_o on_o the_o right_n to_o the_o 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need_v no_o compass_n in_o the_o mensuration_n of_o any_o solid_a provide_v the_o line_n for_o solid_a measure_n and_o gauge_v vessel_n be_v double_o impress_v only_o in_o a_o revert_v order_n one_o pair_n of_o line_n proceed_v from_o the_o head_n towards_o the_o end_n and_o the_o other_o pair_n from_o the_o end_n towards_o the_o head_n upon_o the_o slide_a cover_n and_o its_o adjacent_a ledge_n this_o do_v the_o method_n of_o perform_v any_o of_o the_o problem_n mention_v in_o this_o chapter_n be_v easy_a for_o whereas_o you_o be_v before_o direct_v to_o extend_v the_o compass_n from_o the_o first_o term_n to_o the_o second_o and_o with_o that_o distance_n set_v one_o point_n in_o the_o three_o term_n the_o other_o point_n give_v the_o four_o or_o term_v seek_v so_o here_o observe_v the_o line_n as_o before_o slide_v the_o cover_n until_o the_o first_o term_n stand_v direct_o against_o the_o second_o then_o look_v for_o the_o three_o on_o its_o proper_a line_n it_o stand_v exact_o against_o the_o four_o term_n or_o term_v seek_v on_o the_o other_o line_n only_o note_n that_o when_o the_o second_o term_n be_v great_a than_o the_o first_o it_o be_v perform_v by_o that_o pair_n of_o line_n proceed_v from_o the_o head_n towards_o the_o end_n but_o when_o the_o first_o term_n be_v great_a than_o the_o second_o it_o be_v resolve_v by_o that_o pair_n of_o line_n which_o be_v number_v from_o the_o end_n towards_o the_o head_n chap._n x._o the_o gage_n vessel_n either_o for_o wine_n or_o ale_n measure_n probl._n 1_o the_o diameter_n at_o head_n and_o diameter_n at_o boung_n give_v in_o inches_n and_o ten_o part_n of_o a_o inch_n to_o find_v the_o mean_a diameter_n in_o like_a measure_n take_v the_o difference_n in_o inch_n and_o ten_o part_n of_o a_o inch_n between_o the_o two_o diameter_n then_o say_v by_o the_o line_n of_o number_n as_o 1_o be_v to_z 7_o so_o be_v the_o difference_n to_o a_o four_o number_n of_o inch_n and_o ten_o part_n of_o a_o inch_n this_o add_v to_o the_o diameter_n at_o head_n give_v the_o mean_a diameter_n ex._n gr_n at_o 27_o inch_n the_o boung_a and_o 19_o inch_n two_o ten_o at_o the_o head_n the_o difference_n be_v 7_o inch_n 8_o ten_o the_o four_o number_n find_v by_o the_o proportion_n will_v be_v 5_o inch_n 4_o ten_o and_o one_o half_a which_o add_v to_o the_o diameter_n at_o the_o head_n gives_z 24_o inch_n 6_o ten_o and_o one_o half_a ten_o of_o a_o inch_n for_o the_o mean_a diameter_n probl._n 2._o the_o length_n of_o the_o vessel_n and_o the_o mean_a diameter_n give_v in_o inch_n and_o ten_o part_n of_o
which_o end_n of_o the_o rule_n to_o turn_v the_o point_n of_o your_o compass_n for_o find_v the_o four_o term_n ex._n gr_n as_o the_o sine_fw-la 60_o be_v to_o the_o sine_fw-la 40_o so_o be_v the_o sine_fw-la 20_o to_o the_o sine_fw-la 14._o 40._o again_o as_o the_o sine_z 10_o be_v to_o the_o sine_fw-la 20_o so_o be_v the_o sine_fw-la 30_o to_o the_o sine_fw-la 80._o probl._n 6._o to_o work_v proportion_n in_o tangent_n alone_o by_o the_o artificial_a line_n of_o tangent_n for_o this_o purpose_n the_o artificial_a line_n of_o tangent_n must_v be_v imagine_v twice_o the_o length_n of_o the_o rule_n and_o therefore_o for_o the_o great_a conveniency_n it_o be_v double_o number_v viz._n first_o from_o 1_o to_o 45_o which_o be_v the_o radius_fw-la or_o equal_a to_o the_o sine_fw-la 90_o in_o which_o account_n every_o division_n have_v as_o to_o its_o length_n on_o the_o rule_n a_o proportional_a decrease_n second_o it_o be_v number_v back_o again_o from_o 45_o to_o 89_o in_o which_o account_n every_o division_n have_v as_o to_o its_o length_n on_o the_o line_n a_o proportional_a increase_n so_o that_o the_o tangent_fw-la 60_o you_o must_v imagine_v the_o whole_a length_n of_o the_o rule_n and_o so_o much_o more_o as_o the_o distance_n from_o 45_o unto_o 30_o or_o 60_o be_v this_o well_o observe_v all_o proportion_n in_o tangent_n be_v wrought_v after_o the_o same_o manner_n of_o extend_v the_o compass_n from_o the_o first_o term_n to_o the_o second_o and_o that_o distance_n set_v in_o the_o three_o give_v the_o four_o as_o be_v for_o sin_n and_o number_n but_o for_o the_o remedy_n of_o excursion_n since_o the_o line_n be_v no_o more_o than_o half_a the_o length_n we_o must_v imagine_v it_o i_o shall_v lay_v down_o these_o case_n case_n 1._o when_o the_o four_o term_n be_v a_o tangent_fw-la exceed_v 45_o or_o the_o radius_fw-la ex._n gr_n as_o the_o tangent_fw-la 10_o be_v to_o the_o tangent_fw-la 30_o so_o be_v the_o tangent_fw-la 20_o to_o what_o extend_v the_o compass_n from_o 10_o on_o the_o line_n of_o tangent_n to_o 30_o with_o that_o distance_n i_o set_v one_o point_n in_o 20_o and_o find_v the_o other_o point_n reach_v beyond_o 45_o which_o tell_v i_o the_o four_o term_n exceed_v 45_o or_o the_o radius_fw-la wherefore_o with_o the_o former_a extent_n i_o set_v one_o point_n in_o 45_o and_o turn_v the_o other_o towards_o the_o begin_n of_o the_o line_n i_o mark_v where_o it_o touch_v and_o from_o thence_o take_v the_o distance_n to_o the_o three_o term_n i_o have_v the_o excess_n of_o the_o four_o term_n above_o 45_o in_o my_o compass_n wherefore_o with_o this_o last_o distance_n set_v one_o point_n in_o 45_o i_o turn_v the_o other_o upon_o the_o line_n and_o it_o reach_v to_o 50_o the_o tangent_fw-la seek_v case_n 2._o when_o the_o first_o term_n be_v a_o tangent_fw-la exceed_v 45_o or_o the_o radius_fw-la ex._n gr_n as_o the_o tangent_fw-la 50_o be_v to_o the_o tangent_fw-la 20_o so_o be_v the_o tangent_fw-la 30_o to_o what_o because_o the_o second_o term_n be_v less_o than_o the_o first_o i_o know_v the_o four_o must_v be_v less_o than_o the_o three_o all_o the_o difficulty_n be_v to_o get_v the_o true_a extent_n from_o the_o tangent_fw-la 50_o to_o 20._o to_o do_v this_o take_v the_o distance_n from_o 45_o to_o 50_o and_o set_v one_o point_n in_o 20_o the_o second_o term_n turn_v the_o other_o towards_o the_o begin_n of_o the_o line_n mark_v where_o it_o touch_v extend_v the_o compass_n from_o the_o point_n where_o it_o touch_v to_o 45_o and_o you_o will_v have_v the_o same_o distance_n in_o your_o compass_n as_o from_o 50_o to_o 20_o if_o the_o line_n have_v be_v continue_v at_o length_n unto_o 89_o tangent_n with_o this_o distance_n set_v one_o point_n in_o 30_o the_o three_o term_n and_o turn_v the_o other_o towards_o the_o beginning_n because_o you_o know_v the_o four_o must_v be_v less_o and_o it_o give_v 10_o the_o tangent_fw-la seek_v case_n 3._o when_o the_o three_o term_n be_v a_o tangent_fw-la exceed_v 45_o or_o the_o radius_fw-la as_o the_o tangent_fw-la 40_o be_v to_o the_o tangent_fw-la 12_o 40_o min._n so_o be_v the_o tangent_fw-la 65_o to_o what_o extend_v the_o compass_n from_o 40_o to_o 12_o d._n 40_o min._n with_o distance_n set_v one_o point_n in_o 65._o turn_v the_o other_o towards_o 45_o and_o you_o will_v find_v it_o reach_v beyond_o it_o which_o assure_v you_o the_o four_o term_n will_v be_v less_o than_o 45._o therefore_o lay_v the_o extent_n from_o 45_o towards_o the_o beginning_n and_o mark_v where_o it_o touch_v take_v the_o distance_n from_o that_o point_n to_o 65_o and_o lay_v that_o distance_n from_o 45_o towards_o the_o beginning_n it_o give_v 30_o the_o tangent_fw-la seek_v these_o case_n be_v sufficient_a to_o remove_v all_o difficulty_n for_o when_o the_o second_o term_n exceed_v the_o radius_fw-la you_o may_v transpose_v they_o say_v as_o the_o first_o term_n be_v to_o the_o three_o so_o be_v the_o second_o to_o the_o four_o and_o then_o it_o be_v wrought_v by_o the_o three_o case_n i_o suppose_v it_o needless_a to_o add_v any_o thing_n about_o work_a proportion_n by_o sin_n and_o tangent_n conjunct_o since_o enough_o have_v be_v already_o say_v of_o both_o of_o they_o apart_o in_o these_o two_o last_o problem_n and_o the_o work_n be_v the_o same_o when_o they_o be_v intermix_v only_o some_o proportion_n i_o shall_v adjoyn_v and_o leave_v to_o the_o practice_n of_o the_o young_a beginner_n with_o the_o direction_n in_o the_o former_a case_n probl._n 7._o to_o find_v the_o sun_n ascensional_a difference_n in_o any_o latitude_n as_o the_o co-tangent_a of_o the_o latitude_n be_v to_o the_o tangent_fw-la of_o the_o sun_n declination_n so_o be_v the_o radius_fw-la to_o the_o sine_fw-la of_o the_o ascensional_a difference_n probl._n 8._o to_o find_v at_o what_o hour_n the_o sun_n will_v be_v east_n or_o west_n in_o any_o latitude_n as_o the_o tangent_fw-la of_o the_o latitude_n be_v to_o the_o tangent_fw-la of_o the_o sun_n declination_n so_o be_v the_o radius_fw-la to_o the_o cousin_n of_o the_o hour_n from_o noon_n probl._n 9_o the_o latitude_n declination_n of_o the_o sun_n and_o his_o azimuth_n from_o south_n give_v to_o find_v the_o sun_n altitude_n at_o that_o azimuth_n as_o the_o radius_fw-la to_o the_o cousin_n of_o the_o azimuth_n from_o south_n so_o be_v the_o co-tangent_a of_o the_o latitude_n to_o the_o tangent_fw-la of_o the_o sun_n altitude_n in_o the_o equator_fw-la at_o the_o azimuth_n give_v again_o as_o the_o sine_fw-la of_o the_o latitude_n be_v to_o the_o sine_fw-la of_o the_o sun_n declination_n so_o be_v the_o cousin_n of_o the_o sun_n altitude_n in_o the_o equator_fw-la at_o the_o same_o azimuth_n from_o east_n or_o west_n to_o a_o four_o ark_n when_o the_o azimuth_n be_v under_o 90_o and_o the_o latitude_n and_o declination_n be_v under_o the_o same_o pole_n add_v this_o four_o ark_n to_o the_o altitude_n in_o the_o equator_fw-la in_o azimuth_n exceed_v 90_o when_o the_o latitude_n and_o declination_n be_v under_o the_o same_o pole_n take_v the_o equator_fw-la altitude_n out_o of_o the_o four_o ark_n last_o when_o the_o latitude_n and_o declination_n respect_v different_a pole_n take_v the_o four_o ark_n out_o of_o the_o equator_fw-la altitude_n and_o you_o have_v the_o altitude_n seek_v probl._n 10._o the_o azimuth_n altitude_n and_o declination_n of_o the_o sun_n give_v to_o find_v the_o hour_n as_o the_o cousin_n of_o declination_n be_v to_o the_o sine_fw-la of_o the_o sun_n azimuth_n so_o be_v the_o cousin_n of_o the_o altitude_n to_o the_o fine_a of_o the_o hour_n from_o the_o meridian_n proportion_n may_v be_v vary_v eight_o several_a way_n in_o this_o manner_n follow_v 1._o as_o the_o first_o term_n be_v to_o the_o second_o so_o be_v the_o three_o to_o the_o four_o 2._o as_o the_o second_o term_n be_v to_o the_o first_o so_o be_v the_o four_o to_o the_o three_o 3._o as_o the_o three_o term_n be_v to_o the_o first_o so_o be_v the_o four_o to_o the_o second_o 4._o as_o the_o four_o term_n be_v to_o the_o second_o so_o be_v the_o three_o to_o the_o first_o 5._o as_o the_o second_o term_n be_v to_o the_o four_o so_o be_v the_o first_o to_o the_o three_o 6._o as_o the_o first_o term_n be_v to_o the_o three_o so_o be_v the_o second_o to_o the_o four_o 7._o as_o the_o three_o term_n be_v to_o the_o four_o so_o be_v the_o first_o to_o the_o second_o 8._o as_o the_o four_o term_n be_v to_o the_o three_o so_o be_v the_o second_o to_o the_o first_o by_o thesse_n any_o one_o may_v vary_v the_o former_a proportion_n and_o make_v the_o problem_n three_o time_n the_o number_n here_o insert_v ex._n gr_n to_o find_v the_o ascensional_a difference_n in_o problem_n 10_o of_o this_o chapter_n which_o