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53_o 56_o 1_o 356_o 76_o 11_o 0_o 01_o 58_o 2_o 359_o 52_o 22_o 0_o 18_o 17_o 3_o 359_o 28_o 30_o 0_o 04_o 74_o b_o 4_o 000_o 03_o 00_o 0_o 06_o 30_o 5_o 359_o 79_o 11_o 0_o 07_o 89_o 6_o 359_o 55_o 19_o 0_o 09_o 47_o 7_o 359_o 31_o 30_o 0_o 11_o 05_o b_o 8_o 000_o 05_o 97_o 0_o 12_o 64_o 9_o 359_o 82_o 08_o 0_o 14_o 22_o 10_o 359_o 58_o 19_o 0_o 15_o 78_o 11_o 359_o 34_o 30_o 0_o 17_o 36_o b_o 12_o 000_o 08_o 97_o 0_o 18_o 94_o 13_o 359_o 85_o 08_o 0_o 20_o 52_o 14_o 359_o 00_o 19_o 0_o 22_o 11_o 15_o 359_o 37_o 30_o 0_o 23_o 69_o b_o 16_o 000_o 11_o 97_o 0_o 25_o 25_o 17_o 359_o 88_o 08_o 0_o 26_o 83_o 18_o 359_o 64_o 19_o 0_o 28_o 41_o 19_o 359_o 40_o 28_o 0_o 30_o 00_o b_o 20_o 000_o 14_o 97_o 0_o 31_o
25_o degr_n pc_n be_v give_v by_o substract_v 25_o degr_n from_o pz_n 38_o degr_n 47_o min._n the_o compliment_n of_o the_o pole_n height_n the_o angle_n cp_n 1_o be_v 15_o degree_n one_o hour_n distance_n and_o the_o angle_n at_o c_o right_v we_o may_v find_v c_o 1_o by_o the_o first_o case_n of_o right_n angle_v spherical_a triangle_n for_o as_o the_o radius_fw-la 90_o 10.000000_o be_v to_o the_o sine_fw-la of_o pc_n 13.47_o 9.367237_o so_o be_v the_o tangent_fw-la of_o cp_n 1._o 15._o 9.428052_o to_o the_o tangent_fw-la of_o c_o 1_o 3.57_o 8.795289_o and_o this_o be_v all_o the_o variety_n save_v only_o increase_v the_o angle_n at_o p_o i_o need_v not_o reiterate_v the_o work_n 3._o of_o south_n recline_v more_o than_o the_o pole_n this_o plane_n in_o the_o fundamental_a scheme_n be_v represent_v by_o the_o prick_a circle_n eaw_n of_o which_o in_o the_o same_o latitude_n let_v the_o reclination_n be_v 55_o degree_n from_o which_o if_o you_o deduct_v pz_fw-mi 38_o deg_n 47_o min._n 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10._o to_o draw_v the_o hour-line_n upon_o any_o direct_a north_n recline_v or_o incline_v plane_n the_o direct_a north_n recline_v plane_n have_v the_o same_o variety_n that_o the_o south_n have_v for_o either_o the_o plane_n may_v recline_v from_o the_o zenith_n just_a to_o the_o equinoctial_a and_o then_o it_o be_v a_o polar_a plane_n as_o i_o call_v it_o before_o because_o the_o pole_n of_o the_o plane_n lie_v in_o the_o pole_n of_o the_o world_n or_o else_o the_o plane_n may_v recline_v more_o or_o less_o than_o the_o equinoctial_a and_o consequent_o their_o pole_n do_v fall_v above_o or_o under_o the_o pole_n of_o the_o world_n and_o the_o hour_n line_n do_v likewise_o differ_v from_o the_o former_a of_o the_o polar_a plain_n this_o place_n be_v well_o know_v to_o be_v a_o circle_n divide_v into_o 24_o equal_a part_n which_o may_v be_v do_v by_o draw_v a_o circle_n with_o the_o line_n of_o chord_n and_o then_o take_v the_o distance_n of_o 15_o degree_n from_o the_o same_o chord_n draw_v straight_o line_n from_o the_o centre_n through_o those_o equal_a division_n you_o have_v the_o houre-line_n desire_v the_o houre-line_n be_v draw_v erect_v a_o straight_a pin_n of_o wire_n upon_o the_o centre_n of_o whâ—â—_n length_n you_o please_v and_o the_o dial_n be_v finish_v yet_o see_v our_o latitude_n be_v capable_a of_o no_o more_o than_o 16_o hour_n and_o a_o half_a the_o six_o hour_n next_o the_o south_n part_n of_o the_o meridian_n 11_o 10_o 9_o 1_o 2_o and_o 3_o may_v be_v leave_v out_o as_o useless_a nor_o can_v the_o recline_a face_n serve_v any_o long_a then_o during_o the_o sun_n abode_n in_o the_o north_n part_n of_o the_o zodiac_n and_o the_o incline_a face_n the_o rest_n of_o the_o year_n because_o this_o plain_n be_v parallel_n to_o the_o equinoctial_a which_o the_o sun_n cross_v twice_o in_o a_o year_n these_o thing_n perform_v to_o your_o like_n let_v the_o hour_n of_o 12_o be_v place_v upon_o the_o meridian_n and_o the_o whole_a plain_n raise_v to_o a_o angle_n equal_a to_o the_o compliment_n of_o your_o latitude_n the_o which_o in_o this_o example_n be_v 38_o deg_n 47_o min._n so_o be_v this_o polar_a plain_n and_o dial_n rectify_v to_o show_v the_o true_a hour_n of_o the_o day_n 2._o of_o north_n recline_v less_o than_o the_o equator_fw-la the_o next_o sort_n be_v of_o such_o recline_a plain_n as_o fall_v between_o the_o zenith_n and_o the_o equator_fw-la and_o in_o the_o scheme_n be_v represent_v by_o the_o prick_a circle_n efw_o suppose_v to_o recline_v 25_o degree_n from_o the_o zenith_n which_o be_v add_v to_o pz_n 38_o deg_n 47_o min._n the_o compliment_n of_o the_o pole_n elevation_n the_o aggregate_v be_v pf_n 63_o deg_n 47_o min._n the_o height_n of_o the_o pole_n or_o stile_n above_o the_o plane_n and_o therefore_o in_o the_o triangle_n pf1_n we_o have_v give_v pf_n and_o the_o angle_n at_o p_o to_o find_v f1_n the_o first_o hour_n distance_n from_o the_o meridian_n upon_o the_o plain_a for_o which_o the_o proportion_n be_v as_o the_o radius_fw-la 90_o 10.000000_o be_v to_o the_o sine_fw-la of_o pf_n 63.47_o 9.951677_o so_o be_v the_o tangent_fw-la of_o fp1_n 15_o 9.428052_o to_o the_o tangent_fw-la of_o f1_n 13.48_o 9.379729_o in_o compute_v the_o other_o hour_n distance_n there_o be_v no_o other_o variety_n but_o increase_v the_o angle_n at_o p_o as_o before_o we_o show_v 3._o of_o north_n recline_v more_o than_o the_o equator_fw-la the_o last_o sort_n be_v of_o such_o recline_a plain_n as_o fall_v between_o the_o horizon_n and_o equator_fw-la represent_v in_o the_o fundamental_a scheme_n by_o the_o prick_a circle_n ebw_o suppose_v to_o recline_v 70_o deg_n and_o because_o the_o equator_fw-la cut_v the_o axis_n of_o the_o world_n at_o right_a angle_n all_o plane_n that_o be_v parallel_v thereunto_o have_v the_o height_n of_o their_o style_n full_a 90_o deg_n above_o the_o plane_n and_o by_o how_o much_o any_o plane_n recline_v from_o the_o zenith_n more_o than_o the_o equator_fw-la by_o so_o much_o less_o than_o 90_o be_v the_o height_n of_o the_o stile_n proper_a to_o it_o and_o therefore_o if_o you_o add_v pz_n 38_o deg_n 47_o min._n the_o height_n of_o the_o equator_fw-la unto_o zb_n 70_o deg_n the_o reclination_n of_o the_o plain_a the_o total_a be_v pb_v 108_o deg_n 47_o mi._n who_o complemenc_n to_o 180_o be_v the_o arch_a b_n 71_o deg_n 53_o min._n the_o height_n of_o the_o pole_n above_o the_o plain_a to_o calculate_v the_o hour_n line_n thereof_o we_o must_v suppose_v the_o meridian_n pfb_n and_o the_o hour_n circle_v p1_n p2_n p3_n etc._n etc._n to_o be_v continue_v till_o they_o meet_v in_o the_o south_n pole_n then_o will_v the_o proportion_n be_v the_o same_o as_o before_o as_o the_o radius_fw-la 90_o 10.000000_o to_o the_o sine_fw-la of_o pb_n 71.53_o 9.977033_o so_o be_v the_o tangent_fw-la of_o 1pb_n 15_o 9.428052_o to_o the_o tangent_fw-la of_o b1_n 14.27_o 9.405085_o and_o so_o be_v the_o other_o hour_n distance_n to_o be_v compute_v as_o in_o all_o the_o other_o plane_n the_o geometrical_a projection_n the_o projection_n of_o these_o plane_n be_v but_o little_o differ_v from_o those_o in_o the_o last_o probl._n for_o the_o place_v the_o hour_n and_o erect_v the_o stile_n they_o be_v the_o same_o and_o must_v be_v elevate_v to_o a_o angle_n above_o the_o horizon_n equal_a to_o the_o compliment_n of_o their_o reclination_n which_o in_o the_o north_n recline_v less_o than_o the_o equator_fw-la be_v in_o our_o example_n 65_o degree_n and_o in_o this_o plane_n the_o hour_n about_o the_o meridian_n that_o be_v from_o 10_o in_o the_o morning_n till_o 2_o in_o the_o afternoon_n can_v never_o receive_v any_o shadow_n by_o reason_n of_o the_o plane_n small_a reclination_n from_o the_o
zenith_n and_o therefore_o needless_a to_o put_v they_o on_o in_o the_o north_n recline_v more_o than_o the_o equator_fw-la the_o plane_n in_o our_o example_n must_v be_v elevate_v 120_o degr_n above_o the_o horizon_n and_o the_o style_n of_o both_o must_v point_v to_o the_o north_n pole_n last_o as_o all_o other_o plane_n have_v two_o face_n respect_v the_o contrary_a part_n of_o the_o heaven_n so_o these_o recliner_n have_v opposite_a side_n look_v downward_o the_o nadir_n as_o those_o do_v towards_o the_o zenith_n and_o may_v be_v therefore_o make_v by_o the_o same_o rule_n or_o if_o you_o will_v spare_v that_o labour_n and_o make_v the_o same_o dial_n serve_v for_o the_o opposite_a side_n turn_v the_o centre_n of_o the_o incliner_n downward_o which_o be_v upward_o in_o the_o recliner_n and_o those_o upward_o in_o the_o incliner_n which_o be_v downward_o in_o the_o recliner_n and_o after_o this_o conversion_n let_v the_o hour_n on_o the_o right_a hand_n of_o the_o meridian_n in_o the_o recliner_n become_v on_o the_o left_a hand_n in_o the_o incliner_n and_o contrary_o so_o have_v you_o do_v what_o you_o desire_v and_o this_o be_v a_o general_a rule_n for_o the_o opposite_a side_n of_o all_o plane_n probl._n 11._o to_o draw_v the_o hour-line_n upon_o a_o decline_a recline_a or_o decline_v incline_a plane_n decline_a recline_a plane_n have_v the_o same_o variety_n that_o be_v in_o the_o former_a recline_a north_n and_o south_n for_o either_o the_o declination_n may_v be_v such_o that_o the_o recline_a plane_n will_v fall_v just_a upon_o the_o pole_n and_o then_o it_o be_v call_v a_o decline_a equinoctial_a or_o it_o may_v fall_v above_o or_o under_o the_o pole_n and_o then_o it_o be_v call_v a_o south_n decline_v cast_v and_o west_n recliner_n on_o the_o other_o side_n the_o declination_n may_v be_v such_o that_o the_o recline_a plane_n shall_v fall_v just_a upon_o the_o intersection_n of_o the_o meridian_n and_o equator_fw-la and_o than_o it_o be_v call_v a_o decline_a polar_n or_o it_o may_v fall_v above_o or_o under_o the_o say_a intersection_n and_o then_o it_o be_v call_v a_o north_n decline_v east_n and_o west_n recliner_n the_o three_o variety_n of_o south_n recliner_n be_v represent_v by_o the_o three_o circle_n ahb_n fall_v between_o the_o pole_n of_o the_o world_n and_o the_o zenith_n agb_v just_a upon_o the_o pole_n and_o aeb_n between_o the_o pole_n and_o the_o horizon_n and_o the_o particular_a pole_n of_o each_o plane_n be_v so_o much_o elevate_v above_o the_o horizon_n upon_o the_o azimuth_n dzc_a cross_v the_o base_a at_o right_a angle_n as_o the_o plane_n itself_o recline_v from_o the_o zenith_n note_v in_o the_o scheme_n with_o i_o k_o and_o l._n 1._o of_o the_o equinoctial_a decline_a and_o recline_a plane_n this_o plane_n represent_v by_o the_o circle_n agb_n have_v his_o base_a azb_n decline_v 30_o degree_n from_o the_o east_n and_o west_n line_n ezw_n equal_a to_o the_o declination_n of_o the_o south_n pole_n thereof_o 30_o degree_n from_o saint_n the_o south_n part_n of_o the_o meridian_n easterly_n unto_o d_o recline_v from_o the_o zenith_n upon_o the_o azimuth_n czd_v the_o quantity_n zg_v 34_o degree_n 53_o min._n and_o pass_v through_o the_o pole_n at_o p._n set_v off_o the_o reclination_n zg_v from_o d_o to_o k_o and_o k_o shall_v represent_v the_o pole_n of_o the_o recline_a plane_n so_o much_o elevate_v above_o the_o horizon_n at_o d_o as_o the_o circle_n agb_n represent_v the_o plane_n decline_v from_o the_o zenith_n z_o from_o p_o the_o pole_n of_o the_o world_n to_o k_o the_o pole_n of_o the_o plane_n draw_v a_o arch_n of_o a_o great_a circle_n pk_n thereby_o the_o better_a to_o inform_v the_o fancy_n in_o the_o rest_n of_o the_o work_n and_o if_o any_o be_v desirous_a to_o any_o declination_n give_v to_o fit_v a_o plane_n recline_v just_a to_o the_o pole_n or_o any_o reclination_n be_v give_v to_o find_v the_o declination_n proper_a to_o it_o this_o diagram_n will_v satisfy_v they_o therein_o for_o in_o the_o triangle_n zgp_n we_o have_v limit_v first_o the_o hypothenusal_a pz_n 38_o degree_n 47_o min._n second_o the_o angle_n at_o the_o base_a pzg_n the_o plane_n declination_n 30_o degree_n hence_o to_o find_v the_o base_a gz_n by_o the_o seven_o case_n of_o right_n angle_v spherical_a triangle_n the_o proportion_n be_v as_o the_o radius_fw-la 90_o 10.000000_o to_o the_o cousin_a of_o gzp_n 30_o 9.937531_o so_o the_o tangent_fw-la of_o pz_n 38.47_o 9.900138_o to_o the_o tangent_fw-la of_o gz_n 34.53_o 9.837669_o the_o reclination_n require_v if_o the_o declination_n be_v require_v to_o a_o reclination_n give_v then_o by_o the_o 13_o case_n of_o right_n angle_v spherical_a triangle_n the_o proportion_n be_v as_o the_o radius_fw-la 90_o 10.000000_o to_o the_o tangent_fw-la of_o zg_v 34.53_o 9.837669_o so_o the_o co-tangent_a of_o pz_n 38.47_o 10.099861_o o_o the_o cousin_a of_o gzp_n 39_o 9.937530_o and_o now_o to_o calculate_v the_o hour-line_n of_o this_o dial_n you_o be_v to_o find_v two_o thing_n first_o the_o arch_n of_o the_o plane_n or_o distance_n of_o the_o meridian_n and_o substile_a from_o the_o horizontal_a line_n which_o in_o this_o scheme_n be_v pb_n the_o intersection_n of_o the_o recline_a plane_n with_o the_o horizon_n be_v at_o b._n and_o second_o the_o distance_n of_o the_o meridian_n of_o the_o place_n szpn_n from_o the_o meridian_n of_o the_o plane_n pk_n which_o be_v have_v the_o dial_n be_v easy_o make_v wherefore_o in_o the_o triangle_n zgp_n right_o angle_v at_o g_o you_o have_v the_o angle_n gzp_v give_v 30_o degree_n the_o declination_n and_o zp_v 38_o degr_n 47_o min._n the_o compliment_n of_o the_o pole_n to_o find_v gp_n and_o therefore_o by_o the_o eight_o case_n of_o right_n angle_v spherical_a triangle_n the_o proportion_n be_v as_o the_o radius_fw-la 90_o 10.000000_o to_o the_o sine_fw-la of_o zp_n 38.47_o 9.793863_o so_o be_v the_o sine_fw-la of_o gzp_n 30_o 9.698970_o to_o the_o sine_fw-la of_o gp_n 18.12_o 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the_o stile_n also_o which_o be_v erect_v over_o the_o substile_a and_o the_o dial_n place_v according_a to_o the_o declination_n 30_o deg_n easterly_a and_o the_o whole_a plain_n raise_v to_o a_o angle_n of_o 55_o deg_n 47_o min._n the_o compliment_n of_o the_o reclination_n the_o shadow_n of_o the_o stile_n shall_v give_v the_o hour_n of_o the_o day_n desire_v 2._o to_o draw_v the_o hour_n line_n upon_o a_o south_n recline_v plain_a decline_a east_n or_o west_n which_o pass_v between_o the_o zenith_n and_o the_o pole_n in_o these_o kind_n of_o decline_a recline_a plain_n the_o south_n pole_n be_v elevate_v above_o the_o plane_n as_o be_v clear_a by_o the_o circle_n ahb_n represent_v the_o same_o which_o fall_v between_o the_o zenith_n and_o the_o north_n pole_n and_o therefore_o hide_v