Selected quad for the lemma: end_n

Word A	Word B	Word C	Word D	Occurrence	Frequency Band	MI	MI Band	Prominent
end_n	draw_v	line_n	require_v	1,461	5	9.1279	5	false

Text snippets containing the quad

ID	Title	Author	Corrected Date of Publication (TCP Date of Publication)	STC	Words	Pages
A64223	The semicircle on a sector in two books. Containing the description of a general and portable instrument; whereby most problems (reducible to instrumental practice) in astronomy, trigonometry, arithmetick, geometry, geography, topography, navigation, dyalling, &c. are speedily and exactly resolved. By J. T.	Taylor, John, 1666 or 7-1687.	1667 (1667)	Wing T533B; ESTC R221720	52,688	153

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 cut_v the_o azimuth_n from_o south_n ex._n gr_n at_o 25_o deg_n altitude_n the_o azimuth_n be_v 53_o deg_n at_o 30_o deg_n altitude_n the_o azimuth_n be_v 41_o deg_n 30_o min._n fere_n probl._n 11._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 the_o azimuth_n in_o southern_a declination_n lay_v the_o thread_n to_o the_o altitude_n number_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 and_o take_v the_o declination_n from_o the_o scale_n of_o altitude_n then_o carry_v one_o point_n of_o your_o compass_n on_o the_o line_n of_o azimuth_n on_o that_o side_n the_o thread_n next_o the_o end_n until_o the_o other_o just_a touch_n the_o thread_n which_o do_v the_o former_a point_n give_v the_o azimuth_n from_o south_n ex._n gr_n at_o 15_o deg_n altitude_n and_o 6_o deg_n south_n declination_n the_o azimuth_n be_v 58_o deg_n 30_o min._n probl._n 12._o the_o declination_n give_v to_o find_v the_o sun_n 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

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

cousin_n of_o the_o declination_n to_o that_o four_o sine_fw-la so_o be_v the_o radius_fw-la to_o the_o sine_fw-la of_o the_o hour_n from_o six_o probl._n 6._o to_o find_v the_o hour_n in_o any_o latitude_n when_o the_o sun_n be_v in_o the_o equinoctial_a as_o the_o cousin_n of_o the_o latitude_n be_v to_o the_o sine_fw-la of_o altitude_n so_o be_v the_o radius_fw-la to_o the_o sine_fw-la of_o the_o hour_n from_o six_o probl._n 7._o to_o find_v the_o hour_n in_o any_o latitude_n in_o southern_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 depression_n at_o six_o add_v the_o sine_fw-la of_o depression_n to_o the_o present_a altitude_n by_o probl._n 1._o then_o proport_n 2._o as_o the_o cousin_n of_o the_o latitude_n be_v to_o that_o sum_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 cousin_n of_o declination_n be_v to_o the_o four_o sine_fw-la so_o be_v the_o radius_fw-la to_o the_o sine_fw-la of_o the_o hour_n from_o six_o probl._n 8._o to_o find_v the_o sun_n azimuth_n in_o any_o latitude_n in_o northern_a declination_n proport_n 1._o as_o the_o sine_fw-la of_o the_o latitude_n to_o the_o sine_fw-la of_o declination_n so_o be_v the_o radius_fw-la to_o the_o sine_fw-la of_o altitude_n at_o east_n or_o west_n by_o probl._n 2._o subtract_v this_o from_o the_o present_a altitude_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 residue_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 cousin_n of_o the_o altitude_n be_v to_o that_o four_o sine_fw-la so_o be_v the_o radius_fw-la to_o the_o sine_fw-la of_o the_o azimuth_n from_o east_n or_o west_n probl._n 9_o to_o find_v the_o azimuth_n for_o any_o latitude_n when_o the_o sun_n be_v in_o the_o equator_fw-la proport_n 1._o as_o the_o cousin_n of_o the_o latitude_n to_o the_o sine_fw-la of_o altitude_n so_o be_v the_o sine_fw-la of_o the_o latitude_n to_o a_o four_o sine_fw-la proport_n 2._o as_o the_o cousin_n of_o altitude_n to_o that_o four_o sine_fw-la so_o be_v the_o radius_fw-la to_o the_o sine_fw-la of_o the_o azimuth_n from_o east_n or_o west_n probl._n 10._o to_o find_v the_o azimuth_n for_o any_o latitude_n in_o southern_a declination_n proport_n 1._o as_o the_o cousin_n of_o the_o latitude_n to_o the_o sine_fw-la of_o altitude_n so_o be_v the_o sine_fw-la of_o the_o latitude_n to_o a_o four_o have_v by_o probl._n 4._o find_v the_o sun_n amplitude_n add_v it_o to_o this_o four_o sine_fw-la by_o probl._n 1._o and_o say_v as_o the_o cousin_n of_o the_o altitude_n be_v to_o the_o sum_n so_o be_v the_o radius_fw-la to_o the_o sine_fw-la of_o the_o azimuth_n from_o east_n or_o west_n the_o term_n mention_v in_o the_o 5_o seven_o 8_o 10_o problem_n be_v appropriate_v unto_o we_o that_o live_v on_o the_o north_n side_n the_o equator_fw-la in_o case_n they_o be_v apply_v to_o such_o latitude_n as_o lie_v on_o the_o south_n side_n the_o equator_fw-la then_o what_o be_v now_o call_v northern_a declination_n name_n southern_a and_o what_o be_v here_o style_v southern_a declination_n term_v northern_a and_o all_o the_o proportion_n with_o the_o operation_n be_v the_o same_o these_o proportion_n to_o find_v the_o hour_n and_o azimuth_n may_v be_v all_o ready_o wrought_v by_o the_o line_n of_o artificial_a sin_n only_o the_o addition_n and_o substraction_n must_v always_o be_v wrought_v upon_o the_o line_n of_o natural_a sin_n chap._n iu._n some_o use_n of_o the_o line_n on_o the_o proportional_a side_n of_o the_o instrument_n viz._n the_o line_n of_o natural_a sin_n tangent_n and_o secant_v probl._n 1._o to_o lay_v down_o any_o sine_fw-la tangent_fw-la or_o secant_fw-la to_o a_o radius_fw-la give_v see_v fig._n 1._o if_o you_o be_v to_o lay_v down_o a_o sine_fw-la enter_v the_o radius_fw-la give_v in_o 90_o and_o 90_o upon_o the_o line_n of_o sines_n keep_v the_o sector_n at_o that_o gage_n set_v one_o point_n of_o your_o compass_n in_o the_o sine_fw-la require_v upon_o one_o line_n and_o extend_v the_o other_o point_n to_o the_o same_o sine_fw-la upon_o the_o other_o line_n this_o distance_n be_v the_o length_n of_o the_o sine_fw-la require_v to_o the_o give_v radius_fw-la ex._n gr_n suppose_v a._n b._n the_o radius_fw-la give_v and_o i_o require_v the_o sine_fw-la 40._o proportional_a to_o that_o radius_fw-la enter_v a._n b._n in_o 90_o and_o 90_o keep_v the_o sector_n at_o that_o gage_n i_o take_v the_o distance_n betwixt_o 40_o on_o one_o side_n to_o 40_o on_o the_o other_o that_o be_v c._n d._n the_o sine_fw-la require_v the_o work_n be_v the_o same_o to_o lay_v down_o a_o tangent_fw-la to_o any_o radius_fw-la give_v provide_v you_o enter_v the_o give_v radius_fw-la in_o 45_o and_o 45_o on_o the_o line_n of_o tangent_n only_o observe_v if_o the_o tangent_fw-la require_v be_v less_o than_o 45._o you_o must_v enter_v the_o radius_fw-la in_o 45._o and_o 45_o next_o the_o end_n of_o the_o rule_n but_o when_o the_o tangent_fw-la require_v exceed_v 45._o enter_n the_o radius_fw-la give_v in_o 45_o and_o 45_o betwixt_o the_o centre_n and_o end_n and_o keep_v the_o sector_n at_o that_o gage_n take_v out_o the_o tangent_fw-la require_v this_o be_v so_o plain_a there_o need_v no_o example_n to_o lay_v down_o a_o secant_fw-la to_o any_o radius_fw-la give_v be_v no_o more_o than_o to_o enter_v the_o radius_fw-la in_o the_o two_o pin_n at_o the_o begin_n of_o the_o line_n of_o secant_v and_o keep_v the_o sector_n at_o that_o gage_n take_v the_o distance_n from_o the_o number_n of_o the_o secant_fw-la require_v on_o one_o side_n to_o the_o same_o number_n on_o the_o other_o side_n and_o that_o be_v the_o secant_fw-la seek_v at_o the_o radius_fw-la give_v the_o use_n of_o this_o problem_n will_v be_v sufficient_o see_v in_o delineat_v dial_n and_o project_v the_o sphere_n probl._n 2._o to_o lay_v down_o any_o angle_n require_v by_o the_o line_n of_o sines_n tangent_n and_o secant_v see_v fig._n 2._o there_o be_v two_o way_n of_o protract_v a_o angle_n by_o the_o line_n of_o sines_n first_o if_o you_o use_v the_o sin_n in_o manner_n of_o chord_n then_o have_v draw_v the_o line_n a_o b_o at_o any_o distance_n of_o your_o compass_n set_v one_o point_n in_o b_o and_o draw_v a_o mark_n to_o intersect_v the_o line_n b_o a_o as_o e_z f._n enter_v this_o distance_n b_o f_o in_o 30_o and_o 30_o upon_o the_o line_n of_o sines_n and_o keep_v the_o sector_n at_o that_o gage_n take_v out_o the_o sine_fw-la of_o half_a the_o angle_n require_v and_o set_v one_o point_n where_o f_o intersect_v b_o a_n turn_z the_o other_z toward_z e_z and_o make_v the_o mark_n e_o with_o a_o ruler_n draw_v b_o e_z and_o the_o angle_n e_o b_o f_o be_v the_o angle_n require_v which_o here_o be_v 40._o d._n a_o second_o method_n by_o the_o line_n of_o sines_n be_v thus_o enter_v b_o at_fw-fr radius_fw-la in_o the_o line_n of_o sines_n and_o keep_v the_o sector_n at_o that_o gage_n take_v out_o the_o sine_fw-la of_o your_o angle_n require_v with_o that_o distance_n set_v one_o point_n of_o your_o compass_n in_o a_o sweep_v the_o ark_n d_o a_o line_n draw_v from_o b_o by_o the_o connexity_n of_o the_o ark_n d_o make_v the_o angle_n a_o b_o c_o 40_o d._n as_o before_o to_o protract_v a_o angle_n by_o the_o line_n of_o tangent_n be_v easy_o do_v draw_v b_o a_o the_o radius_fw-la upon_o a_o erect_v a_o perpendicular_a a_o c_o enter_z b_o a_o in_o 45_o and_o 45_o on_o the_o line_n of_o tangent_n and_o take_v out_o the_o tangent_fw-la require_v as_o here_o 40_o set_v it_o from_o a_o to_o c._n last_o draw_v b_o c_o and_o the_o angle_n c_o b_o at_fw-fr be_v 40_o d._n as_o before_o in_o case_n you_o will_v protract_v a_o angle_n by_o the_o line_n of_o secant_v draw_z b_o a_o and_o upon_o a_o erect_a the_o perpendicular_a a_o c_o enter_z a_o b_o in_o the_o begin_n of_o the_o line_n of_o secant_v and_o take_v out_o the_o secant_fw-la of_o the_o angle_n with_o that_o distance_n set_v one_o point_n of_o your_o compass_n in_o b_o with_o the_o other_o across_o the_o perpendicular_a a_o c_o as_o in_o c._n this_o do_v lay_v a_o ruler_n to_o b_o and_o the_o point_n of_o intersection_n and_o draw_v the_o line_n b_o c._n so_o have_v you_o again_o the_o angle_n c_o b_o a._n 40._o d._n by_o another_o projection_n these_o variety_n be_v here_o insert_v only_o to_o satisfy_v a_o friend_n and_o recreate_v the_o young_a practitioner_n in_o try_v the_o truth_n of_o his_o projection_n probl._n 3_o to_o work_v proportion_n in_o sines_n alone_o by_o the_o line_n of_o natural_a sin_n on_o the_o

next_o the_o outward_a limb_n of_o the_o movable_a and_o loose_a piece_n both_o yet_o because_o it_o be_v requisite_a to_o have_v pin_n to_o keep_v the_o loose_a piece_n close_o in_o its_o place_n you_o may_v have_v two_o sight_n more_o to_o supply_v their_o place_n which_o sometime_o you_o may_v make_v use_n of_o and_o so_o the_o number_n of_o sight_n may_v be_v five_o viz._n two_o slide_a sight_n one_o turn_v sight_n and_o two_o pin_n sight_n to_o put_v into_o the_o hole_n at_o the_o end_n of_o the_o fix_a and_o movable_a piece_n to_o hold_v the_o tenon_n of_o the_o loose_a piece_n close_o joint_v every_o one_o of_o these_o sight_n have_v a_o fiducial_a or_o perpendicular_a line_n draw_v down_o the_o middle_n of_o they_o from_o the_o top_n to_o the_o bottom_n where_o this_o line_n touch_v the_o graduation_n on_o the_o limb_n be_v the_o point_n of_o observation_n the_o place_n of_o these_o sight_n have_v a_o oval_a proportion_n about_o the_o middle_n of_o they_o only_o leave_v a_o small_a bar_n of_o brass_n to_o conduct_v the_o fiducial_a line_n down_o the_o oval_a cavity_n and_o support_v a_o little_a brass_n knot_n with_o a_o sight_n hole_n in_o it_o in_o the_o middle_n of_o that_o bar_n which_o be_v ever_o the_o point_n to_o be_v look_v at_o there_o be_v two_o way_n of_o observe_v a_o altitude_n with_o help_n of_o these_o sight_n the_o one_o when_o we_o turn_v our_o face_n towards_o the_o object_n this_o be_v call_v a_o forward_a observation_n in_o which_o you_o must_v always_o set_v the_o turn_a sight_n next_o your_o eye_n this_o way_n of_o observation_n will_v not_o exact_o give_v a_o altitude_n above_o 45_o degree_n the_o other_o way_n of_o observe_v a_o altitude_n be_v peculiar_a to_o the_o sun_n in_o a_o bright_a day_n when_o we_o turn_v our_o back_n towards_o the_o sun_n this_o be_v term_v a_o backward_a observation_n wherein_o you_o must_v have_v one_o of_o the_o slide_a sight_n next_o your_o eye_n and_o the_o turn_a sight_n towards_o the_o horizon_n this_o serve_v to_o take_v the_o sun_n altitude_n without_o thread_n or_o plummet_n when_o it_o be_v near_o the_o zenith_n probl._n 1._o to_o find_v the_o sun_n altitude_n by_o a_o forward_a observation_n serve_v the_o turn_a sight_n to_o the_o centre_n of_o those_o graduation_n you_o please_v to_o make_v use_n of_o whether_o on_o the_o inward_a or_o outward_a limb_n and_o place_v the_o two_o slide_a sight_n upon_o the_o respective_a limb_n to_o that_o centre_n this_o do_v look_v by_o the_o knot_n of_o the_o turn_a sight_n move_v the_o instrument_n upward_o or_o downward_o until_o you_o see_v the_o knot_n of_o one_o of_o the_o slide_a sight_n direct_o against_o the_o sun_n then_o move_v the_o other_o slide_a sight_n until_o the_o knot_n of_o the_o turn_a sight_n and_o the_o knot_n of_o this_o other_o slide_a sight_n be_v against_o the_o horizon_n then_o the_o degree_n intercept_v betwixt_o the_o fiducial_a line_n of_o the_o slide_a sight_n on_o the_o limb_n show_v the_o altitude_n require_v probl._n 2._o to_o find_v the_o distance_n of_o any_o two_o star_n etc._n etc._n by_o a_o forward_a observation_n serve_v the_o turn_a sight_n to_o either_o centre_n and_o apply_v the_o two_o slide_a sight_n to_o the_o respective_a limb_n hold_v the_o instrument_n with_o the_o proportional_a side_n downward_o and_o apply_v the_o turn_a sight_n to_o your_o eye_n so_o move_v the_o two_o slide_a sight_n either_o near_o together_o or_o further_a asunder_o that_o you_o may_v by_o the_o knot_n of_o the_o turn_a sight_n see_v both_o object_n even_o with_o the_o knot_n of_o their_o respective_a slide_a sight_n then_o will_v the_o degree_n intercept_v betwixt_o the_o fiducial_a line_n of_o the_o object_n sight_n on_o the_o limb_n show_v the_o true_a distance_n by_o this_o mean_v you_o may_v take_v any_o angle_n for_o survey_v etc._n etc._n probl._n 3_o to_o find_v the_o sun_n altitude_n by_o a_o backward_a observation_n serve_v the_o turn_a sight_n to_o the_o centre_n at_o the_o begin_n of_o the_o line_n of_o sin_n and_o apply_v one_o of_o the_o slide_a sight_n to_o the_o outward_a limb_n of_o the_o loose_a piece_n and_o the_o other_o to_o the_o outward_a limb_n of_o the_o movable_a piece_n and_o turn_v your_o back_n towards_o the_o sun_n set_v the_o slide_a sight_n upon_o the_o movable_a piece_n next_o your_o eye_n and_o slide_v it_o upward_o or_o downward_o towards_o the_o end_n or_o head_n until_o you_o see_v the_o shadow_n of_o the_o little_a bar_n or_o edge_n of_o the_o sight_n on_o the_o loose_a piece_n fall_v direct_o on_o the_o little_a bar_n on_o the_o turn_a sight_n and_o at_o the_o same_o time_n the_o bar_n of_o the_o sight_n next_o your_o eye_n and_o the_o bar_n of_o the_o turn_a sight_n to_o be_v in_o a_o direct_a line_n with_o the_o horizon_n then_o will_v the_o degree_n on_o the_o limb_n intercept_v betwixt_o the_o fiducial_a line_n of_o the_o slide_a sight_n if_o you_o take_v the_o shadow_n of_o the_o bar_n or_o betwixt_o the_o fiducial_a line_n of_o the_o slide_a sight_n next_o your_o eye_n and_o the_o edge_n of_o the_o other_o slide_a sight_n when_o you_o take_v the_o shadow_n of_o the_o edge_n be_v the_o true_a altitude_n require_v σκιογραφια_n or_o the_o art_n of_o dyall_v for_o any_o plain_a superficies_n lib_fw-la ii_o chap._n i._n the_o distinction_n of_o plain_n with_o rule_n for_o know_v of_o they_o all_o plain_a superficies_n be_v either_o horizontal_a or_o such_o as_o make_v angle_n with_o the_o horizon_n horizontal_n plain_n be_v those_o that_o lie_v upon_o a_o exact_a level_n or_o flat_a plain_n that_o make_v angle_n with_o the_o horizon_n be_v of_o three_o sort_n 1._o such_o as_o make_v right_a angle_n with_o the_o horizon_n general_o know_v by_o the_o name_n of_o erect_v or_o upright_a plain_n 2._o such_o as_o make_v acute_a angle_n with_o the_o horizon_n or_o have_v their_o upper_a edge_n lean_v towards_o you_o usual_o term_v incline_v plain_n 3._o such_o as_o make_v obtuse_a angle_n with_o the_o horizon_n or_o have_v their_o upper_a edge_n fall_v from_o you_o common_o call_v recline_v plain_n all_o these_o three_o sort_n be_v either_o direct_a viz._n east_n west_n north_n south_n or_o else_o decline_a from_o south_n towards_o east_n or_o west_n from_o north_n towards_o east_n or_o west_n all_o plain_a superficies_n whatsoever_o be_v comprise_v under_o one_o of_o these_o term_n but_o before_o we_o treat_v of_o the_o affection_n or_o delineation_n of_o dial_n for_o they_o it_o will_v be_v requisire_n to_o acquaint_v you_o with_o the_o nature_n of_o any_o plain_a which_o may_v be_v find_v by_o the_o follow_a problem_n probl._n 1._o to_o find_v the_o reclination_n of_o any_o plain_a apply_v the_o outward_a ledge_n of_o the_o movable_a piece_n to_o the_o plain_a with_o the_o head_n upward_o and_o reckon_v what_o number_n of_o degree_n the_o thread_n cut_v on_o the_o limb_n begin_v your_o account_n at_o 30._o on_o the_o loose_a piece_n and_o continue_v it_o towards_o 60_o 0_o on_o the_o movable_a piece_n you_o have_v the_o angle_n of_o reclination_n if_o the_o thread_n fall_v direct_o on_o 60_o 0_o upon_o the_o movable_a piece_n it_o be_v a_o horizontal_a if_o on_o 30._o on_o the_o loose_a piece_n it_o be_v a_o erect_v plain_a probl._n 2._o to_o find_v the_o inclination_n of_o any_o plain_a apply_v the_o outward_a ledge_n of_o the_o fix_a piece_n to_o the_o plain_a with_o the_o head_n upward_o and_o what_o number_n of_o degree_n the_o thread_n cut_v upon_o the_o limb_n of_o the_o loose_a piece_n be_v the_o compliment_n of_o the_o plain_n inclination_n probl._n 3_o to_o draw_v a_o horizontal_n line_n upon_o any_o plain_a apply_v the_o proportional_a side_n of_o the_o instrument_n to_o the_o plain_a and_o move_v the_o end_n of_o the_o fix_a piece_n upward_o or_o downward_o until_o the_o thread_n fall_v direct_o on_o 60_o 0._o upon_o the_o loose_a piece_n then_o draw_v a_o line_n by_o the_o outward_a ledge_n of_o the_o fix_a piece_n its_o horizontal_a or_o parallel_v to_o the_o horizon_n probl._n 4._o to_o draw_v a_o perpendicular_a line_n upon_o any_o plain_a when_o the_o sun_n shine_v hold_v up_o a_o thread_n with_o a_o plummet_n against_o the_o plain_a and_o make_v two_o point_n at_o any_o distance_n in_o the_o shadow_n of_o the_o thread_n upon_o the_o plain_a lay_v a_o ruler_n to_o these_o point_n and_o the_o line_n you_o draw_v be_v a_o perpendicular_a probl._n 5._o to_o find_v the_o declination_n of_o any_o plain_a apply_v the_o outward_a ledge_n of_o the_o fix_a piece_n to_o the_o horizontal_a line_n of_o your_o plain_n hold_v your_o instrument_n parallel_v to_o the_o horizon_n this_o do_v lift_v up_o the_o thread_n and_o plumber_n until_o the_o shadow_n of_o the_o thread_n fall_v direct_o upon_o the_o pin_n hole_n on_o the_o fix_a