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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 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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 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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 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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
such_o as_o admit_v not_o of_o centre_n to_o set_v off_o the_o meridian_n in_o its_o proper_a coast_n and_o quantity_n this_o do_v by_o sect._n 4._o paragr_n 5._o of_o this_o chapter_n find_v the_o substyle_n distance_n from_o the_o meridian_n the_o height_n of_o the_o style_n angle_v of_o twelve_o and_o six_o and_o for_o dial_n without_o centre_n the_o inclination_n of_o meridian_n in_o all_o those_o proportion_n use_v your_o new_a latitude_n and_o new_a declination_n instead_o of_o the_o old_a and_o set_v they_o off_o from_o the_o meridian_n according_a to_o the_o direction_n in_o paragr_n 3._o and_o 4._o you_o may_v draw_v the_o dial_n by_o the_o follow_a rule_n for_o erect_v decline_a plain_n in_o place_v of_o they_o lay_v the_o horizontal_a line_n of_o your_o paper_n draught_n upon_o the_o horizontal_a line_n of_o your_o plain_n and_o prick_v off_o the_o substyle_n and_o hour_n line_n only_o observe_v that_o such_o southeast_n or_o south-west_n recliner_n as_o have_v the_o contrary_a pole_n elevate_v must_v be_v describe_v as_o north-east_n and_o north-west_n decliner_n and_o such_o north-east_n and_o north-west_n incliner_n as_o have_v the_o contrary_a pole_n elevate_v must_v be_v describe_v as_o southeast_n and_o south-west_n decliner_n which_o will_v direct_v you_o which_o way_n to_o set_v off_o the_o substyle_n and_o hour_n line_n of_o six_o from_o the_o meridian_n in_o those_o oblique_a plain_n which_o admit_v of_o centre_n or_o the_o substyle_n from_o the_o meridian_n and_o the_o inclination_n of_o meridian_n from_o the_o substyle_n in_o such_o as_o admit_v not_o of_o centre_n 4._o decline_a polar_n plain_n must_v have_v a_o peculiar_a calculation_n for_o the_o substyle_n and_o inclination_n of_o meridian_n which_o be_v thus_o to_o find_v the_o angle_n of_o the_o substile_a 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 sine_fw-la of_o the_o polar_n plain_n reclination_n so_o be_v the_o tangent_fw-la of_o declination_n to_o the_o co-tangent_a of_o the_o substyle_n distance_n from_o the_o horizontal_a line_n of_o the_o plain_a to_o find_v the_o inclination_n of_o meridian_n as_o the_o radius_fw-la be_v to_o the_o sine_fw-la of_o the_o latitude_n 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 5._o the_o recline_a decline_a polar_n have_v the_o substyle_n lie_v below_o that_o end_n of_o the_o horizontal_a line_n that_o be_v contrary_a to_o the_o coast_n of_o declination_n the_o incline_v decline_a polar_n have_v the_o substyle_n lie_v above_o that_o end_n of_o the_o horizontal_a line_n contrary_a to_o the_o coast_n of_o declination_n chap._n iii_o the_o delineation_n of_o dial_n for_o any_o plain_a superficies_n here_o it_o will_v be_v necessary_a to_o premise_v the_o explication_n of_o some_o few_o term_n and_o symbol_n which_o for_o brevity_n sake_n we_o shall_v hereafter_o make_v use_n of_o ex._n gr_n rad._n denote_v the_o radius_fw-la or_o sine_fw-la 90._o or_o tangent_fw-la 45._o tang._n be_v the_o tangent_fw-la of_o any_o arch_n or_o number_n affix_v to_o it_o cos._n be_v the_o cousin_n of_o any_o arch_n or_o number_n of_o degree_n or_o what_o it_o want_v of_o 90._o ex._n gr_n choose_fw-la 19_o be_v what_o 19_o want_n of_o 90._o that_o be_v 71._o co-tang_n be_v the_o co-tangent_a of_o any_o ark_n or_o number_n affix_v to_o it_o or_o what_o it_o want_v of_o 90._o ex._n gr_n co-tangent_a 30._o be_v what_o 30._o want_n of_o 90._o that_o be_v 60._o =_o this_o be_v a_o note_n of_o equality_n in_o line_n number_n or_o degree_n ex._n gr_n ab_fw-la =_o cd_o that_o be_v the_o line_n ab_fw-la be_v equal_a unto_o or_o of_o the_o same_o length_n as_o the_o line_n cd_o again_o abc_n =_o efg_v that_o be_v the_o angle_n abc_n be_v of_o the_o same_o quantity_n or_o number_n of_o degree_n as_o the_o angle_n efg_v once_o more_o ab_fw-la =_o cd_o =_o fg_v =_o tang_n 15._o that_o be_v ab_fw-la and_o cd_o and_o fg._n be_v all_o of_o the_o same_o length_n and_o that_o length_n be_v the_o tangent_fw-la of_o 15._o d._n â™’_o this_o be_v a_o note_n of_o two_o line_n be_v parallel_n unto_o or_o equidistant_a from_o each_o other_o ex._n gr_n fi._n â™’_o rs._n that_o be_v the_o line_n fl._n be_v parallel_n unto_o or_o equi-distant_a from_o the_o line_n rs._n sect._n 1_o to_o delineate_v a_o horizontal_a dial._n see_v fig._n 9_o first_o draw_v the_o square_n bcde_v of_o what_o quantity_n the_o plain_n will_v permit_v then_o make_v of_o =_o ag_z =_o hd_a =_o he_o =_o sine_fw-la of_o the_o latitude_n and_o ah_o =_o radius_fw-la enter_v hd_n in_o tang_n 45._o and_o keep_v the_o sector_n at_o that_o gage_n set_z off_z he_o =_o hk_v =_o tang_n 15._o and_o ho_o =_o hl_fw-ge =_o tang_n 30._o again_o enter_v fd._n in_o tang_n 45._o and_o set_v off_o fq_n =_o gn_n =_o tang_n 15._o and_o fp_n =_o gm_n =_o tang_n 30._o this_o do_v draw_v aq_n ap._n ad._n ao_o ai._n for_o the_o hour_n line_n of_o 5._o 4._o 3._o 2._o 1._o afternoon_n again_o draw_v ak_v al._n ae_n am._n an._n for_o the_o hour_n line_n of_o 11._o 10._o 9_o 8._o 7._o before_o noon_n the_o line_n fag_n be_v for_o six_o and_o six_o in_o the_o same_o manner_n you_o may_v prick_v the_o quarter_n of_o a_o hour_n reckon_v three_o tangent_n and_o 45._o minute_n for_o a_o quarter_n how_o to_o draw_v the_o hour_n line_n before_o and_o after_o six_o be_v mention_v chap._n 2._o sect._n 1_o sect._n 2._o to_o describe_v a_o erect_v direct_a south_n dial._n see_v fig._n 10._o draw_v abcd._n a_o rect-angle_n parallellogram_n then_o make_v ae_n =_o ebb_v =_o cf_n =_o fd_a =_o choose_fw-la of_o your_o latitude_n and_o of_o =_o ac_fw-la =_o bd_o =_o sine_fw-la of_o your_o latitude_n enter_v cf._n in_o tang_n 45._o and_o lay_v down_o fk_v =_o fl_fw-mi =_o tang_n 15._o and_o fi_n =_o fm_o =_o tang_n 30._o again_o enter_fw-la ac_fw-la in_o tang_n 45._o and_o lay_v down_o agnostus_n =_o boy_n =_o 15._o and_o ah_o =_o bn_n =_o tang_n 30._o with_o a_o ruler_n draw_v the_o line_n eglantine_n eh_n aec_fw-la ef._n eke_o for_o the_o hour_n line_n of_o 7._o 8._o 9_o 10._o 11._o in_o the_o morning_n and_o eo_fw-la en_fw-fr ed._n em_n el._n for_o 5._o 4._o 3._o 2._o 1._o afternoon_n the_o line_n aeb_fw-mi be_v for_o six_o and_o six_o the_o line_n ef._n for_o twelve_o the_o description_n of_o a_o direct_a north-dial_n differ_v nothing_o from_o this_o only_o the_o hour_n line_n from_o sun_n rise_v to_o six_o in_o the_o morning_n and_o from_o six_o in_o the_o evening_n until_o sun_n set_v must_v be_v place_v thereon_o by_o draw_v the_o respective_a morning_n and_o evening_n hour_n beyond_o the_o centre_n as_o in_o the_o horizontal_a see_v fig._n 11._o sect._n 3_o to_o describe_v a_o erect_v direct_a east_n dial._n see_v fig._n 12._o have_v draw_v abcd._n a_o rectangle_n paralellogram_n fix_v upon_o any_o point_n in_o the_o line_n ab_fw-la and_o cd_o for_o the_o line_n of_o six_o provide_v the_o distance_n from_o that_o point_n to_o a._n be_v enter_v radius_fw-la in_o the_o line_n of_o tangent_n the_o distance_n from_o thence_o to_o b._n may_v not_o exceed_v nor_o much_o come_v short_a of_o the_o tangent_fw-la 75._o this_o point_n be_v find_v enter_v the_o distance_n from_o thence_o to_o a._n which_o we_o shall_v call_v 6._o a._n radius_fw-la in_fw-la the_o line_n of_o tangent_n and_o keep_v the_o sector_n at_o that_o gage_n lay_v down_o upon_o the_o line_n ab_fw-la and_o cd_o 6._o 11._o =_o tang_n 75._o 6._o 10_o =_o tang_n 60._o and_o 6._o 9_o =_o 6._o a._n =_o tang_n 45._o and_o 6._o 8._o =_o 6._o =_o 4._o =_o tang_n 30._o last_o 6._o 7_o =_o 6._o 5_o =_o tang_n 15._o draw_v line_n from_o these_o point_n on_o ab_fw-la to_o the_o respective_a point_n on_o cd_o and_o you_o have_v the_o hour_n to_o place_v it_o on_o the_o plain_a draw_v the_o angle_n dce_n =_o co-latitude_n and_o lay_v ed._n on_o the_o horizontal_a line_n of_o the_o plain_a prick_v off_o the_o hour_n the_o same_o rule_v serve_v for_o delineation_n of_o a_o 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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
proportional_a side_n of_o the_o instrument_n the_o general_a rule_n be_v this_o account_v the_o first_o term_n upon_o the_o line_n of_o sines_n from_o the_o centre_n and_o enter_v the_o second_o term_n in_o the_o first_o so_o account_v keep_v the_o sector_n at_o that_o gage_n account_v the_o three_o term_n on_o both_o line_n from_o the_o centre_n and_o take_v the_o distance_n from_o the_o three_o term_n on_o one_o line_n to_o the_o three_o term_n on_o the_o other_o line_n measure_v it_o upon_o the_o line_n of_o sin_n from_o the_o beginning_n and_o you_o have_v the_o four_o term_n ex._n gr_n as_o the_o radius_fw-la be_v to_o the_o sine_fw-la 30_o so_o be_v the_o sine_fw-la 40_o to_o the_o sine_fw-la 18._o 45._o there_o be_v but_o one_o exception_n in_o this_o rule_n and_o that_o be_v when_o the_o second_o term_n be_v great_a than_o the_o first_o yet_o the_o three_o lesser_o than_o the_o first_o and_o in_o this_o case_n transpose_v the_o term_n by_o chap_n 3_o probl._n 3_o case_n 3._o but_o when_o the_o second_o term_n be_v not_o twice_o the_o length_n of_o the_o first_o it_o may_v be_v wrought_v by_o the_o general_a rule_n without_o any_o transposition_n of_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 50_o so_o be_v the_o sine_fw-la 20_o to_o the_o sine_fw-la 31._o 30._o min._n and_o by_o consequent_a when_o the_o three_o term_n be_v great_a than_o the_o first_o provide_v it_o be_v not_o upon_o the_o line_n double_a the_o length_n thereof_o it_o may_v be_v wrought_v by_o transpose_v the_o term_n although_o the_o second_o be_v twice_o the_o length_n of_o the_o first_o ex._n gr_n as_o the_o sine_fw-la 20_o be_v to_o the_o sine_fw-la 60_o so_o be_v the_o sine_fw-la 42_o to_o what_o sine_fw-la which_o transpose_v be_v as_o the_o sine_fw-la 20_o be_v to_o the_o sine_fw-la 42_o so_o be_v the_o sine_fw-la 60_o to_o the_o sine_fw-la 35._o 30._o this_o case_n will_v remove_v the_o inconveniency_n mention_v chap._n 3._o probl._n 3_o case_n 4._o of_o a_o double_a radius_fw-la i_o intend_v there_o to_o have_v adjoin_v the_o method_n of_o work_a proportion_n by_o natural_a tangent_n alone_o and_o by_o natural_a sin_n and_o tangent_n conjunct_o but_o consider_v the_o multiplicity_n of_o proportion_n when_o the_o tangent_n exceed_v 45._o i_o suppose_v it_o too_o troublesome_a for_o beginner_n and_o a_o needless_a variety_n for_o those_o that_o be_v already_o mathematician_n sith_o both_o may_v be_v ease_v by_o the_o artificial_a sin_n and_o tangent_n on_o the_o outward_a ledge_n where_o i_o intend_v to_o treat_v of_o those_o case_n at_o large_a and_o shall_v in_o this_o place_n only_o annex_v some_o proportion_n in_o sines_n alone_o for_o the_o exercise_n of_o young_a beginner_n probl._n 4._o by_o the_o line_n of_o natural_a sin_n to_o lay_v down_o any_o tangent_fw-la or_o secant_fw-la require_v to_o a_o radius_fw-la give_v in_o some_o case_n especial_o for_o dyall_v your_o instrument_n may_v be_v defective_a of_o a_o tangent_fw-la or_o secant_fw-la for_o your_o purpose_n ex._n gr_n when_o the_o tangent_fw-la exceed_v 76_o or_o the_o secant_fw-la be_v more_o than_o 60._o in_o these_o extremity_n use_v the_o follow_a remedy_n first_o for_o a_o tangent_fw-la as_o the_o cousin_n of_o the_o ark_n be_v to_o the_o radius_fw-la give_v so_o be_v the_o sine_fw-la of_o the_o ark_n to_o the_o length_n of_o the_o tangent_fw-la require_v second_o for_o a_o secant_fw-la as_o the_o cousin_n of_o the_o ark_n be_v to_o the_o radius_fw-la give_v so_o be_v the_o sine_fw-la 90_o to_o the_o length_n of_o the_o secant_fw-la require_v probl._n 5._o the_o distance_n from_o the_o next_o equinoctial_a point_n give_v to_o find_v the_o sun_n declination_n as_o the_o radius_fw-la to_o the_o sine_fw-la of_o the_o sun_n great_a declination_n so_o be_v the_o sine_fw-la of_o his_o distance_n from_o the_o next_o equinoctial_a point_n to_o the_o sine_fw-la of_o his_o present_a declination_n probl._n 6._o the_o declination_n give_v to_o find_v the_o sun_n equinoctial_a distance_n as_o the_o sine_fw-la of_o the_o great_a declination_n be_v to_o the_o sine_fw-la of_o the_o present_a declination_n so_o be_v the_o radius_fw-la to_o the_o sine_fw-la of_o his_o equinoctial_a distance_n probl._n 7._o the_o altitude_n declination_n and_o distance_n of_o the_o sun_n from_o the_o meridian_n give_v to_o find_v his_o azimuth_n as_o the_o cousin_n of_o the_o altitude_n to_o the_o cousin_n of_o the_o hour_n from_o the_o meridian_n so_o be_v the_o cousin_n of_o declination_n to_o the_o sine_fw-la of_o the_o azimuth_n chap._n v._n some_o use_n of_o the_o line_n of_o the_o line_n on_o the_o proportional_a side_n of_o the_o instrument_n probl._n 1._o to_o divide_v a_o line_n give_v into_o any_o number_n of_o equal_a part_n see_v fig._n 3._o suppose_v a_o b_o a_o line_n give_v to_o be_v divide_v into_o nine_o equal_a part_n enter_z a_o b_o in_o 9_o and_o 9_o on_o the_o line_n of_o line_n keep_v the_o sector_n at_o that_o gage_n take_v the_o distance_n from_o 8_o on_o one_o side_n to_z 8_o on_o the_o other_o and_o apply_v it_o from_o a_o upon_o the_o line_n a_o b_o which_o reach_v to_o c_o then_o be_v c_o b_o a_fw-fr nine_o part_n of_o the_o line_n a_o b._n by_o this_o mean_v you_o may_v divide_v any_o line_n that_o be_v not_o more_o than_o the_o instrument_n in_o length_n into_o as_o many_o part_n as_o you_o please_v viz._n 10_o 20_o 30_o 40_o 50_o 100_o 500_o etc._n etc._n part_n according_a to_o your_o reckon_n the_o division_n upon_o the_o line_n ex._n gr_n the_o line_n be_v actual_o divide_v into_o 200_o part_n viz._n first_o into_o 10_o mark_v with_o figure_n and_o each_o of_o those_o into_o twenty_o part_n more_o again_o if_o the_o line_n represent_v a_o 1000_o than_o every_o figure_a division_n be_v 100_o the_o second_o or_o short_a division_n be_v 10_o and_o the_o three_o or_o short_a division_n be_v 5._o in_o case_n the_o whole_a line_n be_v 2000_o than_o every_o figure_a division_n be_v 200_o every_o small_a or_o second_v division_n be_v 20_o every_o three_o or_o small_a division_n be_v 10_o etc._n etc._n suppose_v i_o have_v any_o line_n give_v which_o be_v the_o base_a of_o a_o triangle_n who_o content_n be_v 2000_o pole_n and_o i_o demand_v so_o much_o of_o the_o base_a as_o may_v answer_v 1750_o pole_n enter_v the_o whole_a line_n in_o 10_o and_o 10_o at_o the_o end_n of_o the_o line_n of_o line_n and_o keep_v the_o sector_n at_o that_o gage_n now_o the_o whole_a line_n represent_v 2000_o pole_n every_o figure_a division_n be_v 200_o therefore_o 1700_o be_v eight_o and_o a_o half_a of_o the_o figure_a division_n and_o 50_o be_v five_o of_o the_o small_a division_n more_o for_o in_o this_o case_n every_o small_a division_n be_v 10_o as_o be_v before_o express_v wherefore_o set_v one_o point_n of_o the_o compass_n in_o 15_o of_o the_o small_a division_n beyond_o 8_o on_o the_o rule_n i_o extend_v the_o other_o point_n to_o the_o same_o division_n upon_o the_o line_n on_o the_o other_o side_n and_o that_o distance_n be_v 1750_o pole_n in_o the_o base_a of_o the_o triangle_n propose_v how_o ready_a this_o be_v to_o set_v out_o a_o just_a quantity_n in_o any_o plat_n of_o ground_n i_o shall_v show_v in_o a_o scheam_n chap._n 12._o probl._n 2._o to_o work_v proportion_n in_o line_n or_o number_n or_o the_o rule_n of_o three_o direct_v by_o the_o line_n of_o line_n enter_v the_o second_o term_n in_o the_o first_o and_o keep_v the_o sector_n at_o that_o gage_n take_v the_o distance_n betwixt_o the_o three_o on_o one_o line_n to_o the_o three_o on_o the_o other_o line_n that_o distance_n be_v the_o four_o in_o line_n or_o measure_v upon_o the_o line_n from_o the_o centre_n give_v the_o four_o in_o number_n ex._n gr_n as_o 7_o be_v to_z 3_o so_o be_v 21_o to_o 9_o probl._n 3_o to_o work_v the_o rule_n of_o three_o inverse_n or_o the_o back_n rule_n of_o three_o by_o the_o line_n of_o line_n in_o these_o proportion_n there_o be_v always_o three_o term_n give_v to_o find_v a_o four_o and_o of_o the_o three_o give_v term_n two_o be_v of_o one_o denomination_n which_o for_o distinction_n sake_n i_o call_v the_o double_a denomination_n and_o the_o three_o term_n be_v of_o a_o different_a denomination_n from_o those_o two_o which_o i_o therefore_o call_v the_o single_a denomination_n of_o which_o the_o four_o term_v seek_v must_v also_o be_v now_o to_o bring_v these_o into_o a_o direct_a proportion_n the_o rule_n be_v this_o when_o the_o four_o term_v seek_v be_v to_o be_v great_a than_o the_o single_a denomination_n which_o you_o may_v know_v by_o sight_n of_o the_o term_n give_v say_v as_o the_o lesser_a double_a denomination_n be_v to_o the_o great_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 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 length_n of_o the_o perpendicular_a number_v upon_o the_o line_n of_o line_n to_o the_o tangent_fw-la of_o the_o sun_n altitude_n probl._n 5._o to_o find_v the_o altitude_n of_o any_o tree_n steeple_n etc._n etc._n at_o one_o station_n at_o any_o distance_n from_o the_o object_n provide_v the_o ground_n be_v level_a with_o your_o instrument_n look_v to_o the_o top_n of_o the_o object_n along_o the_o outward_a ledge_n of_o the_o fix_a piece_n and_o take_v the_o angle_n of_o its_o altitude_n this_o do_v measure_n by_o foot_n or_o yard_n the_o distance_n from_o your_o stand_n to_o the_o bottom_n of_o the_o object_n then_o say_v as_o the_o cousin_n of_o the_o altitude_n be_v to_o the_o measure_a distance_n number_v upon_o the_o line_n of_o line_n so_o be_v the_o sine_fw-la of_o the_o altitude_n to_o a_o four_o number_n of_o foot_n or_o yard_n according_a to_o the_o measure_n you_o meet_v the_o distance_n to_o this_o four_o add_v the_o height_n of_o your_o eye_n from_o the_o ground_n and_o that_o sum_n give_v the_o number_n of_o foot_n or_o yard_n in_o the_o altitude_n chap._n vi_o how_o to_o work_v proportion_n in_o number_n sin_n or_o tangent_n by_o the_o artificial_a line_n thereof_o on_o the_o outward_a ledge_n the_o general_a rule_n for_o all_o of_o these_o be_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 observe_v whether_o that_o extent_n be_v upward_o or_o downward_o with_o the_o same_o distance_n set_v one_o point_n in_o the_o three_o term_n and_o turn_v the_o other_o point_n the_o same_o way_n as_o at_o first_o it_o give_v the_o four_o but_o in_o tangent_n when_o any_o of_o the_o term_n exceed_v 45_o there_o may_v be_v excursion_n which_o in_o their_o due_a place_n i_o shall_v remove_v probl._n 1._o numeration_n by_o the_o line_n of_o number_n the_o whole_a line_n be_v actual_o divide_v into_o 100_o proportional_a part_n and_o according_o distinguish_v by_o figure_n 1_o 2_o 3_o 4_o 5_o 6_o 7_o 8_o 9_o and_o then_o 10_o 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 multiplicator_n the_o lesser_a of_o the_o two_o number_n although_o it_o may_v be_v either_o of_o they_o at_o pleasure_n probl._n 3_o to_o work_v division_n by_o the_o line_n of_o number_n as_o the_o divisor_n be_v to_o 1_o so_o be_v the_o dividend_n to_o the_o quotient_a suppose_v 800_o to_o be_v divide_v by_o 20_o the_o quotient_n be_v 40._o for_o as_o 20_o be_v to_o 1_o so_o be_v 800_o to_o 40._o to_o know_v how_o many_o figure_n you_o shall_v have_v in_o the_o quotient_a take_v this_o rule_n note_v the_o difference_n of_o the_o number_n of_o place_n or_o figure_n in_o the_o dividend_n and_o divisor_n then_o in_o case_n the_o quantity_n of_o the_o two_o first_o figure_n to_o the_o left_a hand_n in_o your_o divisor_n be_v less_o than_o the_o quantity_n of_o the_o two_o first_o figure_n to_o the_o left_a hand_n in_o your_o dividend_n the_o quotient_n shall_v have_v one_o figure_n more_o than_o the_o number_n of_o difference_n but_o where_o the_o quantity_n of_o the_o two_o first_o figure_n of_o the_o divisor_n be_v great_a than_o the_o quantity_n of_o the_o two_o first_o figure_n of_o the_o dividend_n the_o quotient_n will_v have_v only_o that_o number_n of_o figure_n note_v by_o the_o difference_n ex._n gr_n 245_o divide_v by_o 15._o will_v have_v two_o figure_n in_o the_o quotient_a but_o 16_o divide_a 〈…〉_z â—â—ve_v only_o one_o figure_n in_o the_o quotient_a probl._n 4._o to_o find_v a_o mean_a proportional_a betwixt_o two_o number_n give_v by_o the_o line_n of_o number_n divide_v the_o space_n betwixt_o they_o upon_o the_o line_n of_o number_n into_o two_o equal_a part_n and_o the_o middle_a point_n be_v the_o mean_a proportional_a ex._n gr_n betwixt_o 4_o and_o 16_o the_o mean_a proportional_a be_v 8._o if_o you_o be_v to_o find_v two_o mean_a proportional_n divide_v the_o space_n betwixt_o the_o give_v number_n into_o three_o part_n if_o four_o mean_a proportional_n divide_v it_o into_o five_o part_n and_o the_o several_a point_n betwixt_o the_o two_o 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
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
run_v thus_o 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 ascensional_a difference_n then_o by_o the_o three_o variety_n you_o may_v make_v another_o problem_n viz._n as_o the_o radius_fw-la be_v to_o the_o co-tangent_a of_o the_o latitude_n so_o be_v the_o sine_fw-la of_o the_o sun_n ascensional_a difference_n to_o the_o sine_fw-la of_o his_o declination_n again_o by_o the_o four_o variety_n you_o may_v make_v a_o three_o problem_n thus_o as_o the_o sine_fw-la of_o the_o sun_n ascensional_a difference_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 co-tangent_a of_o the_o latitude_n by_o this_o artifice_n many_o have_v stuff_v their_o book_n with_o bundle_n of_o problem_n chap._n vii_o some_o use_n of_o the_o line_n of_o circumference_n diameter_n square_a equal_a and_o square_n inscribe_v all_o these_o be_v line_n of_o equal_a part_n bear_v such_o proportion_n to_o each_o other_o as_o the_o thing_n signify_v by_o their_o name_n their_o use_n be_v this_o any_o one_o of_o they_o give_v in_o inch_n or_o foot_n etc._n etc._n to_o find_v how_o much_o any_o of_o the_o other_o three_o be_v in_o the_o same_o measure_n suppose_v i_o have_v the_o circumference_n of_o a_o circle_n tree_n or_o cylinder_n give_v in_o inch_n i_o take_v the_o same_o number_n of_o part_n as_o the_o circle_n be_v inch_n from_o the_o line_n of_o circumference_n and_o apply_v that_o distance_n to_o the_o respective_a line_n i_o have_v immediate_o the_o square_a equal_a square_v inscribe_v and_o diameter_n in_o inch_n and_o the_o like_a if_o any_o of_o those_o be_v give_v to_o find_v the_o circumference_n this_o need_v no_o example_n the_o conveniency_n of_o this_o line_n any_o one_o may_v experiment_n in_o stand_v timber_n for_o take_v the_o girth_n or_o circumference_n with_o a_o line_n find_v the_o diameter_n from_o that_o diameter_n abate_v twice_o the_o thickness_n of_o the_o bark_n and_o you_o have_v the_o true_a diameter_n when_o it_o be_v bark_v and_o by_o chap._n 9_o probl._n 5._o you_o will_v guess_v very_o near_o at_o the_o quantity_n of_o timber_n in_o any_o stand_a tree_n chap._n viii_o to_o measure_v any_o kind_n of_o superficies_n as_o board_n glass_n pavement_n walnscot_n hang_n walling_n slate_z or_o tyl_v by_o the_o line_n of_o number_n on_o the_o outward_a ledge_n the_o way_n of_o account_v any_o number_n upon_o or_o work_a proportion_n by_o the_o line_n of_o number_n be_v sufficient_o show_v already_o chap._n 6._o which_o i_o shall_v not_o here_o repeat_v only_o propose_v the_o proportion_n for_o these_o problem_n and_o refer_v you_o to_o those_o direction_n probl._n 1._o the_o breadth_n of_o a_o board_n give_v in_o inch_n to_o find_v how_o many_o inch_n in_o length_n make_v a_o foot_n at_o that_o breadth_n say_v as_o the_o breadth_n in_o inch_n be_v to_o 12_o so_o be_v 12_o to_o the_o 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etc._n etc._n foot_n of_o board_n speedy_o as_o your_o occasion_n require_v probl._n 4._o to_o measure_v wainscot_n hang_n plaster_n etc._n etc._n these_o be_v usual_o compute_v by_o the_o yard_n and_o then_o the_o proportion_n be_v as_o nine_o to_o the_o length_n in_o foot_n and_o inch_n so_o be_v the_o breadth_n or_o depth_n in_o foot_n and_o inch_n to_o the_o content_a in_o yard_n ex._n gr_n at_o 18_o foot_n in_o length_n and_o two_o foot_n in_o breadth_n you_o have_v four_o yard_n probl._n 5._o to_o measure_v mason_n or_o slater_n work_v as_o walling_n tyl_v etc._n etc._n the_o common_a account_n of_o these_o be_v by_o the_o rood_n which_o be_v eighteen_o foot_n square_a that_o be_v 324_o square_a foot_n in_o one_o rood_n and_o then_o the_o proportion_n be_v as_o 324_o to_o the_o length_n in_o foot_n so_o be_v the_o breadth_n in_o foot_n to_o the_o content_a in_o rood_n ex._n gr_n at_o 30_o foot_n in_o length_n and_o 15_o foot_n in_o breadth_n you_o have_v 1_o rood_n 3_o 10_o and_o better_o or_o one_o rood_n 126_o 124_o part_n of_o a_o rood_n chap._n ix_o the_o mensuration_n of_o solid_n as_o timber_n stone_n etc._n etc._n by_o the_o line_n on_o the_o proportional_a side_n of_o the_o loose_a piece_n these_o two_o line_n meet_v upon_o one_o line_n in_o the_o midst_n betwixt_o they_o for_o distinction_n sake_n i_o call_v one_o the_o right_a the_o other_o the_o left_a line_n which_o be_v know_v by_o the_o hand_n they_o stand_v towards_o when_o you_o hold_v up_o the_o piece_n in_o the_o right_a way_n to_o read_v the_o figure_n the_o right_a line_n have_v two_o figure_a partition_n the_o first_o partition_n be_v from_o 3_o at_o the_o begin_n to_o the_o letter_n sq._n every_o figure_a division_n represent_v a_o inch_n and_o each_o subdivision_n quarter_n of_o a_o inch_n the_o next_o partition_n be_v from_o the_o letter_n sq._n unto_o 12_o at_o the_o end_n every_o figure_a division_n signify_v a_o foot_n and_o each_o sub-division_n the_o inch_n in_o a_o foot_n the_o letter_n t_o r_o and_o t_o d_o be_v for_o the_o circumference_n and_o diameter_n in_o the_o next_o measure_v of_o cylinder_n the_o letter_n r_o and_o d._n for_o the_o measure_n of_o timber_n according_a to_o the_o vulgar_a allowance_n when_o the_o four_o part_n of_o the_o girt_n be_v take_v etc._n etc._n the_o letter_n a_o and_o w_n be_v the_o gauger_v point_n for_o ale_n and_o wine_n measure_v last_o the_o figure_n 12_o betwixt_o d_o and_o t_o d._n be_v for_o a_o use_n express_v probl._n 2._o the_o left_a line_n also_o have_v two_o figure_a partition_n proceed_v first_o from_o 1_o at_o the_o begin_n to_o one_o foot_n or_o 12_o inch_n each_o whereof_o be_v subdivide_v into_o quarter_n from_o thence_o again_o to_o 100_o each_o whereof_o to_o 10_o foot_n be_v subdivide_v into_o inch_n etc._n etc._n and_o every_o foot_n be_v figure_v but_o from_o 10_o foot_n to_o 100_o only_o every_o ten_o foot_n be_v figure_v the_o sub-division_n represent_v foot_n the_o method_n of_o work_a proportion_n by_o these_o line_n only_o observe_v the_o side_n be_v the_o same_o as_o by_o the_o line_n of_o number_n viz._n extend_v from_o the_o first_o to_o the_o second_o etc._n etc._n probl._n 1._o to_o reduce_v timber_n of_o unequal_a breadth_n and_o depth_n to_o a_o true_a square_n as_o the_o breadth_n on_o the_o left_a be_v to_o the_o breadth_n on_o the_o right_n so_o be_v the_o depth_n on_o the_o left_a to_o the_o square_n on_o the_o right_a line_n at_o 7_o inch_n breadth_n and_o 18_o inch_n depth_n you_o have_v 11_o inch_n ¼_n and_o better_a for_o the_o true_a square_n probl._n 2._o the_o square_a of_o a_o piece_n of_o timber_n give_v in_o inches_n or_o foot_n and_o inch_n to_o find_v how_o much_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 number_n of_o foot_n in_o length_n on_o the_o left_a to_o make_v a_o foot_n of_o timber_n as_o 2_o inch_n ½_o square_v you_o must_v have_v 23_o foot_n 6_o inch_n and_o somewhat_o better_a for_o the_o length_n of_o a_o foot_n of_o timber_n probl._n 3_o the_o square_a and_o length_n of_o a_o plece_n of_o timber_n give_v to_o find_v the_o content_a as_o the_o point_n sq._n on_o the_o right_n be_v to_o the_o length_n in_o foot_n and_o inch_n on_o the_o left_a so_o be_v the_o square_a in_o foot_n and_o inch_n on_o the_o right_n to_o the_o content_a in_o foot_n on_o the_o left_a at_o 30_o foot_n in_o length_n and_o 15_o inch_n square_a you_o have_v 46_o foot_n ½_o of_o timber_n at_o 20_o foot_n in_o length_n and_o 11_o inch_n square_v you_o have_v 16_o foot_n and_o almost_o ¼_n of_o timber_n when_o you_o have_v a_o great_a piece_n of_o timber_n exceed_v 100_o foot_n which_o you_o may_v easy_o see_v by_o the_o excursion_n upon_o the_o rule_n then_o take_v the_o true_a square_n and_o half_a the_o length_n sinde_v the_o content_a thereof_o by_o the_o former_a proportion_n and_o double_v that_o content_n you_o have_v the_o whole_a content_n probl._n 4._o the_o circumference_n or_o girth_n of_o a_o round_a piece_n of_o timber_n be_v give_v together_o with_o the_o length_n to_o find_v the_o content_a as_o the_o point_n r._n on_o the_o right_n be_v to_o the_o length_n in_o foot_n and_o inch_n on_o the_o left_a so_o be_v the_o circumference_n in_o foot_n and_o inch_n on_o the_o right_n to_o the_o content_a in_o foot_n on_o the_o left_a at_o 20_o foot_n in_o length_n and_o 7_o foot_n in_o girth_n you_o have_v 60_o foot_n of_o timber_n for_o the_o content_a this_o be_v after_o the_o common_a allowance_n for_o the_o waste_n in_o square_n and_o although_o some_o be_v please_v to_o quarrel_n with_o the_o allowancer_n as_o wrong_v the_o seller_n and_o give_v the_o quantity_n less_o than_o in_o truth_n it_o be_v yet_o i_o presume_v when_o they_o buy_v it_o themselves_o they_o scarce_o judge_v those_o chip_n worth_a the_o hew_a and_o have_v as_o low_a thought_n of_o the_o overplus_n as_o other_o have_v of_o that_o their_o admonition_n if_o it_o be_v a_o cylinder_n that_o you_o will_v take_v exact_a content_n of_o then_o say_v as_o the_o point_n t_o r_o on_o the_o right_n be_v to_o the_o length_n on_o the_o left_a so_o be_v the_o gird_v on_o the_o right_n to_o the_o exact_a content_n on_o the_o left_a at_o 15_o foot_n in_o length_n and_o 7_o foot_n in_o girt_n you_o have_v 59_o foot_n of_o solid_a measure_n the_o diameter_n of_o any_o cylinder_n give_v you_o may_v by_o the_o same_o proportion_n find_v the_o content_n place_v the_o point_n d_o instead_o of_o r_o in_o the_o proportion_n for_o the_o usual_a allowance_n and_o the_o point_n t_o d_o for_o the_o exact_a compute_v probl._n 5._o to_o measure_v taper_v timber_n take_v the_o square_a or_o gird_v at_o both_o end_n and_o note_v the_o sum_n and_o difference_n of_o they_o then_o for_o round_a timber_n as_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
divide_v into_o 32_o equal_a part_n call_v rumb_n one_o point_n or_o rumb_n be_v 11_o d._n 15_o min._n of_o a_o circle_n from_o the_o meridian_n two_o point_n or_o rumb_n be_v 22_o d._n 30_o min._n etc._n etc._n of_o the_o rest_n 2._o the_o angle_n which_o the_o needle_n or_o point_n of_o the_o compass_n under_o the_o needle_n make_v with_o the_o meridian_n or_o north_n and_o south_n line_n be_v call_v the_o course_n or_o rumb_n but_o the_o angle_n which_o it_o make_v with_o the_o east_n and_o west_n line_n or_o any_o parallel_n be_v name_v the_o compliment_n of_o the_o course_n or_o rumb_n 3._o the_o departure_n be_v the_o longitude_n of_o that_o port_n from_o which_o you_o set_v sail_v 4._o the_o distance_n run_v be_v the_o number_n of_o mile_n or_o league_n turn_v into_o degree_n that_o you_o have_v sail_v 5._o when_o you_o be_v in_o north_n latitude_n and_o sail_v northward_o add_v the_o difference_n of_o latitude_n to_o the_o latitude_n you_o sail_v from_o and_o when_o you_o be_v in_o north_n latitude_n and_o sail_v southward_o subtract_v the_o difference_n of_o latitude_n from_o the_o latitude_n you_o sail_v and_o you_o have_v the_o latitude_n you_o be_v in_o the_o same_o rule_n be_v to_o be_v observe_v in_o south_n latitude_n 6._o to_o find_v how_o many_o mile_n answer_v to_o one_o degree_n of_o longitude_n in_o any_o latitude_n as_o the_o radius_fw-la be_v to_o the_o number_n 60_o so_o be_v the_o cousin_n of_o the_o latitude_n to_o the_o number_n of_o mile_n for_o one_o degree_n 7._o to_o find_v how_o many_o mile_n answer_v to_o one_o degree_n of_o latitude_n on_o any_o rumb_n as_o the_o cousin_n of_o the_o rumb_n from_o the_o meridian_n be_v to_o the_o number_n 60_o so_o be_v the_o radius_fw-la to_o the_o number_n of_o mile_n the_o most_o material_a question_n in_o navigation_n be_v these_o four_o first_o to_o find_v the_o course_n second_o the_o distance_n run_v three_o the_o difference_n of_o latitude_n four_o the_o difference_n in_o longitude_n and_o any_o two_o of_o these_o be_v give_v the_o other_o two_o be_v ready_o find_v by_o the_o square_a and_o index_n these_o two_o additional_a ruler_n be_v omit_v in_o the_o first_o chapter_n of_o this_o treatise_n since_o they_o be_v only_o for_o navigation_n and_o large_a instrument_n of_o two_o or_o three_o foot_n in_o length_n which_o make_v i_o judge_v their_o description_n most_o proper_a for_o this_o place_n because_o such_o as_o intend_v the_o instrument_n for_o a_o pocket_n companion_n will_v have_v no_o use_n of_o they_o the_o square_n be_v a_o flat_a rule_n have_v a_o piece_n or_o plate_n fasten_v to_o the_o head_n that_o it_o may_v slide_v square_a or_o perpendicular_a to_o the_o outward_a ledge_n of_o the_o fix_a piece_n it_o have_v the_o same_o line_n next_o either_o edge_n on_o the_o upper_a side_n which_o be_v a_o line_n of_o equal_a part_n a_o hundred_o wherein_o be_v equal_a to_o the_o radius_fw-la of_o the_o degree_n on_o the_o outward_a limb_n of_o the_o movable_a piece_n the_o index_n be_v a_o thin_a brass_n rule_v on_o one_o side_n have_v the_o same_o scale_n as_o the_o square_n on_o the_o other_o side_n be_v a_o double_a line_n of_o tangent_n that_o next_o the_o left_a edge_n be_v to_o a_o small_a that_o next_o the_o right_a edge_n to_o a_o large_a radius_fw-la for_o the_o use_n of_o these_o ruler_n you_o must_v have_v a_o line_n of_o equal_a part_n adjoin_v to_o the_o line_n of_o sin_n on_o the_o fix_a piece_n divide_v into_o 10_o part_n stamp_v with_o figure_n each_o of_o those_o divide_v into_o 10_o part_n more_o so_o that_o the_o whole_a line_n be_v divide_v into_o 100_o part_n represent_v degree_n last_o let_v each_o of_o those_o degree_n be_v subdivide_v into_o as_o many_o part_n as_o the_o largeness_n of_o your_o scale_n will_v permit_v for_o compute_v the_o minute_n of_o a_o degree_n the_o index_n be_v to_o move_v upon_o the_o pin_n on_o the_o fix_a piece_n where_o you_o hang_v the_o thread_n for_o take_v altitude_n and_o that_o side_n of_o the_o index_n in_o any_o of_o the_o four_o former_a question_n must_v be_v upward_o which_o have_v the_o scale_n of_o equal_a part_n the_o square_n be_v to_o be_v slide_v along_o the_o outward_a ledge_n of_o the_o fix_a piece_n then_o the_o general_a rule_n be_v these_o the_o difference_n of_o latitude_n be_v account_v on_o the_o line_n of_o equal_a part_n adjoin_v to_o the_o sin_n on_o the_o fix_a piece_n the_o difference_n of_o longitude_n be_v number_v on_o the_o square_n the_o distance_n run_v be_v reckon_v upon_o the_o index_n the_o course_n be_v compute_v upon_o the_o degree_n on_o the_o limb_n from_o the_o head_n towards_o the_o end_n of_o the_o movable_a piece_n but_o when_o any_o will_v work_v these_o problem_n in_o proportion_n let_v they_o note_v the_o distance_n run_v difference_n of_o longitude_n and_o difference_n of_o latitude_n be_v all_o account_v on_o the_o line_n of_o number_n the_o rumb_n or_o course_n be_v either_o a_o sine_fw-la or_o tangent_fw-la this_o premise_v i_o shall_v first_o show_v how_o to_o resolve_v any_o problem_n by_o the_o square_a and_o index_n and_o next_o adjoyn_v the_o proportion_n for_o the_o use_n of_o such_o as_o have_v only_o pocket_v instrument_n probl._n 1._o the_o course_n and_o distance_n run_v give_v to_o find_v the_o difference_n of_o latitude_n and_o difference_n of_o longitude_n apply_v the_o index_n to_o the_o course_n reckon_v on_o the_o limb_n from_o the_o head_n and_o slide_v the_o square_a along_o the_o outward_a ledge_n of_o the_o fix_a piece_n until_o the_o fiducial_a edge_n intersect_v the_o distance_n run_v on_o the_o fiducial_a edge_n of_o the_o index_n then_o at_o the_o point_n of_o intersection_n you_o have_v the_o difference_n of_o longitude_n upon_o the_o square_a and_o on_o the_o line_n of_o equal_a part_n on_o the_o fix_a piece_n the_o square_n show_v the_o difference_n of_o latitude_n the_o proportion_n be_v thus_o as_o the_o radius_fw-la be_v to_o the_o distance_n run_v so_o be_v the_o cousin_n of_o the_o course_n to_o the_o difference_n of_o latitude_n again_o as_o the_o radius_fw-la be_v to_o the_o distance_n run_v so_o be_v the_o sine_fw-la of_o the_o course_n to_o the_o difference_n in_o longitude_n probl._n 2._o the_o course_n and_o difference_n of_o latitude_n give_v to_o find_v the_o distance_n run_v and_o difference_n in_o longitude_n slide_v the_o square_n to_o the_o difference_n of_o latitude_n on_o the_o line_n of_o equal_a part_n upon_o the_o fix_a piece_n and_o set_v the_o index_n to_o the_o course_n on_o the_o limb_n then_o at_o the_o point_n of_o intersection_n of_o the_o square_a and_o index_n on_o the_o square_n be_v the_o difference_n of_o longitude_n on_o the_o index_n the_o distance_n run_v the_o proportion_n be_v thus_o as_o the_o cousin_n of_o the_o course_n be_v to_o the_o difference_n of_o latitude_n so_o be_v the_o radius_fw-la to_o the_o distance_n run_v again_o as_o the_o radius_fw-la be_v to_o the_o sine_fw-la of_o the_o course_n so_o be_v the_o distance_n run_v to_o the_o difference_n of_o longitude_n probl._n 3_o the_o course_n and_o difference_n in_o longitude_n give_v to_o find_v the_o distance_n run_v and_o difference_n of_o latitude_n apply_v the_o index_n to_o the_o course_n on_o the_o limb_n and_o the_o difference_n of_o longitude_n on_o the_o square_n to_o the_o fiducial_a edge_n of_o the_o index_n then_o at_o the_o point_n of_o intersection_n you_o have_v distance_n run_v on_o the_o index_n and_o upon_o the_o line_n of_o equal_a part_n on_o the_o fix_a piece_n the_o square_n show_v the_o difference_n of_o latitude_n the_o proportion_n be_v thus_o as_o the_o sine_fw-la of_o the_o course_n be_v to_o the_o difference_n of_o longitude_n so_o be_v the_o radius_fw-la to_o the_o distance_n run_v again_o as_o the_o radius_fw-la be_v to_o the_o distance_n run_v so_o be_v the_o cousin_n of_o the_o course_n to_o the_o difference_n of_o latitude_n probl._n 4._o the_o distance_n run_v and_o difference_n of_o latitude_n give_v to_o find_v the_o course_n and_o difference_n in_o longitude_n slide_v the_o square_n to_o the_o difference_n of_o latitude_n on_o the_o line_n of_o equal_a part_n on_o the_o fix_a piece_n and_o move_v the_o index_n until_o the_o distance_n run_v number_v thereon_o intersect_v the_o fiducial_a edge_n of_o the_o square_a then_o at_o the_o point_n of_o intersection_n you_o have_v the_o difference_n of_o longitude_n on_o the_o square_n and_o the_o fiducial_a edge_n of_o the_o index_n on_o the_o limb_n show_v the_o course_n as_o the_o distance_n run_v be_v to_o the_o difference_n of_o latitude_n so_o be_v the_o radius_fw-la to_o the_o cousin_n of_o the_o course_n again_o as_o the_o radius_fw-la be_v to_o the_o distance_n run_v so_o be_v the_o sine_fw-la of_o the_o course_n to_o the_o difference_n of_o
set_v half_o the_o co-tangent_a d_o r_o z_o from_o a_o to_o f_o and_o the_o secant_fw-la of_o d_o r_o z_o from_o f_o to_o l_o upon_o the_o centre_n a_o with_o the_o extent_n a_o p._n draw_v the_o ark_n p_o g_o and_o with_o the_o extent_n f_o p_o from_o l_o cross_v the_o ark_n p_o g_z in_o g._n last_o upon_o the_o centre_n g_o with_o the_o extent_n g_o l._n draw_v the_o ark_n r_o d_o f_o l_o and_o your_o triangle_n be_v make_v the_o triangle_n project_v you_o may_v measure_v off_o the_o side_n and_o hypothenuse_n thus_o first_o the_o hypothenuse_n z_o r_o be_v measure_v by_o a_o line_n of_o chord_n second_o a_o ruler_n lay_v to_o l_o d_o cut_v the_o limb_n at_o h_n and_o z_o h_n upon_o a_o line_n of_o chord_n be_v the_o measure_n of_o the_o ark_n z_o d._n three_o draw_v a_o g_o and_o set_v half_a the_o tangent_fw-la d_o r_o z_o from_o a_o to_o five_o apply_v a_o ruler_n to_o five_o d_o it_o cut_v the_o limb_n at_o e_o then_z r_z e_z upon_o a_o line_n of_o chord_n measure_v the_o ark_n r_o d._n note_n the_o radius_fw-la to_o all_o the_o chord_n tangent_n and_o secant_v use_v in_o the_o projection_n and_o measure_v any_o ark_n or_o angle_n be_v the_o semidiameter_n of_o the_o fundamental_a circle_n chap._n xvi_o the_o projection_n and_o solution_n of_o the_o 12_o case_n in_o oblique_a angle_a spherical_a triangle_n in_o six_o case_n see_v fig._n 7._o the_o fundamental_a circle_n n_o h_o z_o m_o be_v always_o suppose_v ready_o draw_v and_o cross_v into_o quadrant_n and_o the_o diameter_n produce_v beyond_o the_o circle_n case_n 1._o the_o three_o side_n z_o p_o p_o z_o and_o z_o s_o give_fw-mi to_o project_v the_o triangle_n by_o a_o line_n of_o chord_n prick_v off_o z_o p_o and_o draw_v the_o diameter_n p_o c_o t_o cross_v it_o at_o right_a angle_n in_o the_o centre_n with_o ae_n c_o e_o set_z half_z the_o co-tangent_a p_o saint_n from_o c_o to_z g_z and_o he_o secant_fw-la p_o s_o from_o c_z to_z r_z upon_o the_o centre_n r_o with_o the_o extent_n r_o g_o draw_z the_o the_o ark_n fgl_n again_o set_v half_o the_o co-tangent_a z_o saint_n from_o c_o to_z d_o and_o the_o tangent_fw-la z_o s_o from_o d_o to_z o_o with_o the_o extent_n o_o d_o upon_o the_o centre_n o_o draw_v the_o ark_n b_o d_o p_o mark_v where_o these_o two_o ark_n intersect_v each_o other_o as_o at_o s._n then_o have_v you_o three_o point_n t_o s_o p_o to_o draw_v that_o ark_n and_o the_o three_o point_n n_o s_o p_o to_o draw_v that_o ark_n which_o make_v up_o your_o triangle_n case_n 2._o give_v two_o side_n z_o saint_n and_o z_o p_o with_o the_o comprehend_v angle_n p_o z_o s_o to_o project_n the_o triangle_n prick_v off_o z_o p_o and_o draw_v pct_n and_o aece_n and_o the_o ark_n b_o d_o p_o by_o case_n 1_o again_o set_v the_o tangent_fw-la of_o half_a the_o excess_n of_o the_o angle_n p_o z_o s_o above_o 90_o from_o c_o to_o w_n and_o co-secant_a of_o that_o excess_n from_o w_n to_o k_o upon_o the_o centre_n k_o with_o the_o extent_n k_n w_n draw_v the_o ark_n n_o w_n z_o which_o cut_v the_o ark_n b_o d_o p_o in_o s._n then_o have_v you_o the_o three_o point_n t_o s_o p_o to_o draw_v that_o ark_n which_o make_v up_o the_o triangle_n case_n 3._o two_o angle_n saint_n z_o p_o and_o z_o p_o s_o with_o the_o comprehend_v side_n z_o p_o give_fw-mi to_o project_v the_o triangle_n prick_v off_o z_o p_o and_o draw_v the_o line_n p_o c_o t_o and_o ae_n c_o e_o by_o case_n 1_o and_o the_o angle_n nwz_n by_o case_n 2._o last_o set_v half_o the_o co-tangent_a zp_n from_o c_o to_o x_o and_o the_o secant_fw-la z_o p_o s_o from_o x_o to_o five_o upon_o the_o centre_n v_n with_o the_o extent_n v_o x_o draw_v the_o ark_n t_o x_o s_o p_o and_o the_o triangle_n be_v make_v case_n 4._o two_o side_n zp_n and_o ps_n with_o the_o angle_n opposite_a to_o one_o of_o they_o szp_n give_v to_o project_v the_o triangle_n prick_v off_o zp_n and_o draw_v pct_n and_o aece_n by_o case_n 1._o and_o the_o angle_n szp_n by_o case_n 2._o last_o by_o case_n 1._o draw_v the_o ark_n fgl_n and_o mark_v where_o it_o intersect_v nwz_n as_o at_o saint_n then_o have_v you_o the_o three_o point_n tsp_v to_o draw_v that_o ark_n and_o make_v up_o the_o triangle_n case_n 5._o two_o angles_n szp_v and_o zp_n with_o the_o side_n opposite_a to_o one_o of_o they_o zs_n give_v to_o project_v the_o triangle_n draw_v the_o ark_n bdp_o by_o case_n 1._o and_o the_o ark_n nwz_n by_o case_n 2._o at_o the_o intersection_n of_o these_o two_o ark_n set_v saint_n with_o the_o tangent_fw-la of_o the_o angle_n zp_v upon_o the_o centre_n c._n sweep_v the_o ark_n vδi_n again_o with_o the_o secant_fw-la of_o the_o ark_n zp_v upon_o the_o centre_n s_o across_o the_o ark_n vδi_n as_o at_o the_o point_v five_o and_o i._o then_o in_o case_n the_o hypothenuse_n be_v less_o than_o a_o quadrant_a as_o here_o the_o point_n v_n be_v the_o centre_n and_o with_o the_o extent_n we_o draw_v the_o ark_n tsp_n which_o make_v up_o the_o triangle_n but_o in_o case_n the_o hypothenuse_n be_v equal_a to_o a_o quadrant_a δ_n be_v the_o centre_n if_o more_o than_o a_o quadrant_a i_o be_v the_o centre_n in_o which_o case_n the_o extent_n from_o δ_n or_o i_o to_o saint_n be_v the_o semidiameter_n of_o the_o ark_n tsp_n case_n 6._o three_o angle_n zps_o and_o pzs_n and_o zsp_v give_v to_o project_v the_o triangle_n see_v fig._n 7._o and_o 8._o the_o angle_n of_o any_o spherical_a triangle_n may_v be_v convert_v into_o their_o opposite_a side_n by_o take_v the_o compliment_n of_o the_o great_a angle_n to_o a_o semicircle_n for_o the_o hypothenuse_n or_o great_a side_n wherefore_o by_o case_n 1._o make_v the_o side_n zp_n in_o fig._n 4._o equal_a to_o the_o angle_n zsp_n in_o fig._n 3_o and_o the_o side_n zs_n in_o fig._n 4._o equal_a to_o the_o angle_n zp_v in_o fig._n 3._o and_o the_o side_n ps_n in_o fig._n 4._o equal_a to_o the_o compliment_n of_o the_o angle_n pzs_n to_o a_o semicircle_n in_o fig._n 3._o then_o be_v your_o triangle_n project_v where_o the_o angle_n zp_v in_o fig._n 4._o be_v the_o side_n zs_n fig._n 3._o again_o the_o angle_n zsp_n fig._n 4._o be_v the_o side_n zp_n in_o fig._n 3._o last_o the_o compliment_n of_o the_o angle_n pzs_n to_o a_o semicircle_n in_o fig._n 4._o be_v the_o measure_n of_o the_o hypothenuse_n or_o side_n p_o s_o in_o fig._n 3._o the_o triangle_n be_v in_o any_o of_o the_o former_a case_n project_v the_o quantity_n of_o any_o side_n or_o angle_n may_v be_v measure_v by_o the_o follow_a rule_n first_o the_o side_n z_o p_o be_v find_v by_o apply_v it_o to_o a_o line_n of_o chord_n second_o cx_o apply_v to_o a_o line_n of_o tangent_n be_v half_a the_o co-tangent_a of_o the_o angle_n zp_v three_o cw_o apply_v to_o a_o line_n of_o tangent_n be_v half_a the_o co-tangent_a of_o the_o excess_n of_o the_o angle_n szp_n above_o 90._o four_o set_v half_o the_o tangent_fw-la of_o the_o angle_n zp_v from_o c_o to_z π_n a_o ruler_n lay_v to_o π_n cut_v the_o limb_n at_o f_o then_o pf_n apply_v to_o a_o line_n of_o chord_n give_v the_o side_n ps_n five_o take_v the_o compliment_n of_o the_o angle_n pzs_n to_o a_o semicircle_n and_o set_v half_a the_o tangent_fw-la of_o that_o compliment_n from_o c_o to_o λ_n a_o ruler_n lay_v to_o λ_n cut_v the_o limb_n at_o b_o and_o zb_n apply_v to_o a_o line_n of_o chord_n give_v the_o side_n zs_n six_o a_o ruler_n lay_v to_o sλ_n cut_v the_o limb_n at_o l._n again_o a_o ruler_n lay_v to_o sπ_n cut_v the_o limb_n at_o φ_n and_o l_o φ_n apply_v to_o a_o line_n of_o chord_n give_v the_o angle_n zsp_n the_o end_n of_o the_o first_o book_n a_o appendix_n to_o the_o first_o book_n the_o sight_n which_o be_v necessary_a for_o take_v any_o altitude_n angle_n or_o distance_n without_o the_o help_n of_o thread_n or_o plummet_n be_v only_o three_o viz._n one_o turn_v sight_n and_o two_o other_o sight_n contrive_v with_o chap_n so_o that_o they_o may_v slide_v by_o the_o inward_a or_o outward_a graduated_a limb_n the_o turn_a sight_n have_v only_o two_o place_n either_o the_o centre_n at_o the_o head_n or_o the_o centre_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 to_o either_o of_o which_o as_o occasion_n require_v it_o be_v fasten_v with_o a_o sorne_n the_o centre_n at_o the_o head_n serve_v for_o the_o graduation_n next_o the_o inward_a limb_n of_o the_o loose_a piece_n and_o the_o centre_n at_o the_o begin_n of_o the_o line_n of_o sin_n serve_v for_o all_o the_o graduation_n
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 σκιογÏαφια_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
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
of_o the_o plain_n declination_n and_o keep_v the_o sector_n at_o that_o gage_n take_v out_o the_o secant_fw-la of_o the_o latitude_n which_o place_n from_o a_o to_o g._n upon_o the_o the_o line_n a._n 12._o and_o again_o from_o h._n which_o be_v the_o intersection_n of_o the_o parallel_n fk_n with_o the_o line_n of_o six_o unto_o f._n this_o do_v lay_v a_o ruler_n to_o the_o point_n f._n and_o g._n and_o draw_v a_o line_n until_o it_o intersect_v ce._n as_o fg._n 3_o last_o enter_v gf_n in_o tang_n 45._o and_o set_v off_o gl_n =_o gn_n =_o tang_n 15._o and_o gm_n =_o go_v =_o tang_n 30._o again_o enter_v hf._n in_o tang_n 45._o and_o set_v off_o hr_n =_o tang_n 15._o and_o hp_n =_o tang_n 30._o a_o ruler_n lay_v to_o these_o point_n and_o the_o centre_n you_o may_v draw_v the_o hour_n line_n from_o six_o in_o the_o morning_n unto_o three_o afternoon_n for_o the_o other_o hour_n line_n do_v thus_o produce_v the_o line_n aec_fw-la and_o likewise_o ha._n beyond_o the_o centre_n until_o they_o intersect_v each_o other_o as_o at_o s._n then_o set_v off_o sy_n sy_n hr_fw-ge and_o s._n 4._o =_o hp_n you_o have_v the_o hour_n point_v after_o three_o in_o the_o afternoon_n until_o six_o although_o none_o be_v proper_a beyond_o the_o hour_n line_n of_o four_o only_o by_o draw_v they_o on_o the_o other_o side_n the_o centre_n they_o help_v you_o to_o the_o hour_n line_n before_o six_o in_o the_o morning_n sect._n 5._o to_o describe_v a_o erect_a decline_a plain_n without_o a_o centre_n see_v fig._n 12._o the_o delineation_n of_o these_o dial_n be_v the_o most_o difficult_a of_o any_o and_o therefore_o i_o shall_v be_v the_o large_a in_o their_o description_n 1._o by_o chap._n 2._o s._n 4._o paragr_n 5._o find_v the_o angle_n of_o the_o style_n substyle_n and_o inclination_n of_o meridian_n 2._o have_v find_v the_o inclination_n of_o meridian_n make_v the_o follow_a table_n for_o the_o distance_n of_o every_o hour_n line_n and_o quarter_n from_o the_o substyle_n where_o i_o take_v for_o example_n a_o north_n plain_a decline_a east_n 72._o d._n 45._o min._n in_o lat_fw-la 52._o 30._o min._n imcl_n mer_fw-fr how_o quart_n 76_o =_o 30._o 12_o 00_o 20_o =_o 15_o 3_o ..._o 16_o =_o 30_o 4_o 00_o 12_o =_o 45_o 4_o 09_o =_o 00_o 4_o 05_o =_o 15_o 4_o ..._o 01_o =_o 30_o 5_o 00_o 02_o =_o 15_o 5_o 06_o =_o 00_o 5_o 09_o =_o 45_o 5_o ..._o 13_o =_o 30_o 6_o 00_o 17_o =_o 15_o 6_o 21_o =_o 00_o 6_o 24_o =_o 45_o 6_o ..._o 28_o =_o 30_o 7_o 00_o 32_o =_o 15_o 7_o 36_o =_o 00_o 7_o 39_o =_o 45_o 7_o ..._o 43_o =_o 30_o 8_o 00_o 47_o =_o 15_o 8_o 51_o =_o 00_o 8_o 54_o =_o 45_o 8_o ..._o 58_o =_o 30_o 9_o 00_o 62_o =_o 15_o 9_o 66_o =_o 00_o 9_o 69_o =_o 45_o 9_o ..._o 73_o =_o 30_o 10_o 00_o the_o manner_n of_o draw_v this_o table_n be_v thus_o the_o inclination_n of_o meridian_n which_o because_o its_o a_o north_n decliner_n be_v twelve_o at_o midnight_n i_o find_v 76._o d._n 30._o min._n now_o consider_v the_o sun_n never_o rise_v till_o more_o than_o half_a a_o hour_n after_o three_o in_o this_o latitude_n i_o know_v that_o one_o quarter_n before_o four_o be_v the_o first_o line_n proper_a for_o this_o plain_a therefore_o reckon_v 15._o d._n for_o a_o hour_n or_o 3._o d._n 45._o m._n for_o a_o quarter_n of_o a_o hour_n i_o find_v three_o hour_n three_o quarter_n the_o distance_n of_o a_o quarter_n before_o four_o from_o midnight_n to_o answer_v 56._o d._n 15._o min._n which_o be_v subtract_v from_o 76._o d._n 30._o min._n the_o inclination_n of_o meridian_n there_n remain_v 20._o d._n 15._o m._n for_o the_o distance_n of_o one_o quarter_n before_o four_o from_o the_o substyle_n again_o from_o 20._o d._n 15._o min._n subtract_v 3._o d._n 45._o min._n the_o quantity_n of_o degree_n for_o one_o quarter_n of_o a_o hour_n and_o there_o remain_v 16._o d._n 30._o min._n for_o the_o distance_n of_o the_o next_o line_n from_o the_o substyle_n which_o be_v the_o hour_n of_o four_o in_o the_o morning_n thus_o for_o every_o quarter_n of_o a_o hour_n continue_v subtract_v 3._o d._n 45._o min._n until_o your_o residue_n or_o remain_v be_v less_o than_o 3._o d._n 45._o min._n and_o then_o first_o subtract_v that_o residue_n out_o of_o 3._o d._n 45._o min._n this_o new_a residue_n give_v the_o quantity_n of_o degree_n for_o that_o line_n on_o the_o other_o side_n the_o substyle_n now_o when_o you_o be_v pass_v to_o the_o other_o side_n of_o the_o substyle_n continue_v add_v 3._o d._n 45._o min._n to_o this_o last_o remain_v for_o every_o quarter_n of_o a_o hour_n and_o so_o make_v up_o the_o table_n for_o what_o hour_n be_v proper_a to_o the_o plain_a 3._o draw_v the_o square_a abcd._n of_o what_o quantity_n the_o plain_n will_v admit_v and_o make_v the_o angle_n cag_fw-mi equal_a to_o the_o angle_n of_o the_o substyle_n with_o twelve_o again_o across_o the_o line_n ag._n in_o any_o two_o convenient_a point_n as_o e._n and_o f._n at_o right_a angle_n by_o the_o line_n kl_n and_o cm_o 4._o take_v the_o distance_n from_o the_o centre_n unto_o 45._o the_o radius_fw-la to_o the_o lesser_a line_n of_o tangent_n which_o be_v continue_v to_o 76._o on_o the_o sector_n side_n enter_v this_o distance_n in_o 45._o on_o the_o large_a line_n of_o tangent_n and_o keep_v the_o sector_n at_o that_o gage_n take_v out_o the_o tang_n 20._o d._n 15._o min._n which_o be_v the_o distance_n of_o the_o first_o line_n from_o the_o substyle_n set_v this_o from_o 73._o d._n 30._o min._n the_o distance_n of_o your_o last_o hour_n line_n from_o the_o substyle_n as_o you_o see_v by_o the_o table_n towards_o the_o end_n upon_o the_o lesser_a line_n of_o tangent_n and_o where_o it_o touch_v as_o here_o at_o 75._o 05._o call_n that_o the_o gage_n tangent_fw-la 5._o enter_v the_o whole_a line_n kl_n in_o the_o gage_n tangent_fw-la which_o in_o this_o example_n be_v 75._o d._n 05._o min._n and_o keep_v the_o sector_n at_o that_o gage_n take_v out_o the_o tangent_fw-la 73._o 30._o min._n which_o be_v your_o last_o hour_n and_o set_v from_o l._n on_o the_o line_n kl_n unto_o v._o again_o take_v out_o the_o tangent_fw-la 20._o d._n 15._o min._n which_o be_v your_o first_o line_n and_o set_v it_o from_o k._n towards_o v._o and_o if_o it_o meet_v in_o v._o it_o prove_v the_o truth_n of_o your_o work_n and_o a_o line_n draw_v through_o v._o parallel_n unto_o ag._n be_v the_o true_a substyle_n line_n then_o keep_v the_o sector_n at_o its_o former_a gage_n set_v off_o the_o tangent_n of_o the_o hour_n and_o quarter_n as_o you_o find_v they_o in_o the_o table_n from_o v._o towards_o k._n and_o from_o v._o towards_o l._n make_v point_n for_o they_o in_o the_o line_n kl_n last_o enter_v the_o radius_fw-la of_o your_o tangent_n to_o these_o hour_n point_v in_o the_o radius_fw-la of_o secant_v and_o set_v off_o the_o secant_fw-la of_o the_o style_n height_n from_o v._o to_o t._n thus_o have_v you_o the_o hour_n point_n and_o style_n on_o one_o line_n of_o contingence_n to_o mark_v they_o out_o upon_o the_o other_o line_n do_v thus_o set_a the_o radius_fw-la to_o the_o hour_n point_v upon_o the_o former_a line_n of_o contingence_n from_o h._n to_o p._n on_o the_o line_n chm._n and_o enter_v hv._n as_o radius_fw-la in_o the_o line_n of_o tangent_n take_v out_o the_o tangent_fw-la of_o the_o style_n height_n and_o set_v from_o p._n to_o r._n again_o enter_v hr._n radius_fw-la in_o your_o line_n of_o tangent_n and_o keep_v the_o sector_n at_o that_o gage_n take_v out_o the_o tangent_n for_o each_o hour_n and_o quarter_n according_a to_o the_o table_n and_o lay_v they_o down_o from_o h._n to_o the_o proper_a side_n of_o the_o substyle_n towards_o c._n or_o m._n and_o apply_v a_o ruler_n to_o the_o respective_a point_n on_o kl_n and_o cm_o draw_v the_o line_n for_o the_o hour_n and_o quarter_n last_o enter_v hr._n radius_fw-la on_o the_o line_n of_o secant_v and_o take_v out_o the_o secant_fw-la of_o the_o style_n height_n set_v it_o from_o h._n to_o s._n and_o draw_v the_o line_n st_n for_o the_o style_n sect._n 6._o to_o describe_v a_o direct_a polar_n dial._n see_v fig._n 15._o draw_v bcde_v a_o rectangle_n paralellogram_n from_o the_o middle_n of_o bc._n to_o the_o middle_n of_o de._n draw_v the_o line_n 12._o or_o substyle_n appoint_v what_o place_n you_o please_v in_o bc._n or_o cd_o for_o the_o hour_n point_n of_o 7._o in_o the_o morning_n and_o 5._o afternoon_n then_o enter_z 12._o 7_o =_o 12._o 5._o in_o the_o tangent_fw-la 75._o and_o set_v off_o 12._o 1._o =_o 12._o 11._o =_o tang_n 15._o and_o 12._o 2._o =_o 12._o 10_o =_o