you_o may_v first_o find_v what_o altitude_n the_o sun_n will_v have_v when_o he_o come_v to_o be_v due_a east_n or_o west_n and_o then_o expect_v until_o he_o come_v to_o that_o altitude_n so_o the_o declination_n of_o the_o plane_n shall_v be_v such_o as_o the_o angle_n contain_v between_o the_o line_n ho_o and_o the_o line_n of_o the_o shadow_n have_v distinguish_v the_o plane_n the_o next_o care_n will_v be_v for_o the_o place_n of_o the_o style_n and_o the_o draw_v of_o the_o houre-line_n the_o style_n will_v be_v as_o the_o axis_fw-la of_o the_o world_n sometime_o parallel_v to_o the_o plane_n sometime_o perpendicular_a sometime_o cut_v the_o plane_n with_o oblique_a angle_n the_o houre-line_n will_v be_v either_o parallel_a one_o to_o the_o other_o or_o meet_v in_o a_o centre_n with_o equal_a angle_n or_o meet_v with_o unequal_a angle_n if_o the_o style_n be_v perpendicular_a to_o the_o plane_n the_o angle_n at_o the_o centre_n will_v be_v equal_a and_o this_o fall_v out_o only_o in_o the_o south_n and_o north_n face_n of_o a_o equinoctial_a plane_n if_o the_o style_n be_v parallel_v to_o the_o plane_n the_o houre-line_n will_v be_v also_o parallel_v one_o to_o another_o and_o this_o fall_v out_o in_o all_o polar_n plane_n as_o in_o the_o east_n and_o west_n meridian_n plane_n parallel_v to_o the_o circle_n of_o the_o hour_n of_o 12_o in_o the_o upper_a and_o low_o direct_v polar_n parallel_v to_o the_o circle_n of_o the_o hour_n of_o 6_o and_o in_o the_o upper_a and_o low_a decline_a polar_n which_o be_v parallel_v to_o any_o of_o the_o other_o hour_n circle_n but_o in_o the_o horizontall_a and_o all_o other_o plane_n the_o style_n will_v cut_v the_o plane_n with_o a_o acute_a angle_n and_o the_o hour_n line_n will_v meet_v at_o the_o root_n of_o the_o style_n and_o there_o make_v unequal_a angle_n chap._n i._o to_o draw_v the_o houre-line_n in_o a_o equinoctial_a plane_n a_o equinoctial_a plane_n be_v that_o which_o be_v parallel_v to_o the_o equinoctial_a circle_n here_o represent_v by_o eaw_n wherein_o the_o space_n between_o the_o hour_n circle_v be_v equal_a there_o be_v no_o need_n of_o further_a precept_n but_o only_o to_o draw_v a_o circle_n and_o to_o divide_v it_o into_o 24_o equal_a part_n for_o the_o 24_o hour_n and_o subdivide_v each_o hour_n into_o half_n and_o quarter_n and_o then_o to_o set_v up_o the_o style_n perpendicular_a to_o the_o plane_n in_o the_o centre_n of_o the_o circle_n the_o help_n which_o these_o line_n of_o proportion_n do_v here_o afford_v we_o be_v only_o in_o the_o division_n of_o the_o circle_n which_o may_v be_v do_v ready_o by_o that_o which_o i_o show_v before_o pag._n 29._o for_o example_n suppose_v the_o semidiameter_n of_o the_o equinoctial_a circle_n to_o be_v six_o inch_n and_o that_o it_o be_v require_v to_o know_v the_o distance_n of_o the_o houre-point_n each_o from_o other_o here_o each_o hour_n be_v 15_o gr_n distant_a from_o other_o i_o extend_v the_o compass_n from_o the_o sine_fw-la of_o 30_o gr_n unto_o the_o sine_fw-la of_o 7_o gr_n 30_o m._n the_o half_a of_o 15_o gr_n and_o i_o find_v the_o same_o extent_n to_o reach_v in_o the_o line_n of_o number_n from_o 6.00_o unto_o 1.56_o or_o in_o cross_a work_n i_o extend_v they_o from_o the_o sine_fw-la of_o 30_o gr_n unto_o 6.00_o in_o the_o line_n of_o number_n the_o same_o extent_n will_v reach_v from_o the_o sine_fw-la of_o 7_o gr_n 30_o m._n unto_o 1.56_o in_o the_o line_n of_o number_n which_o show_v that_o in_o a_o circle_n of_o six_o inch_n semidiameter_n the_o distance_n of_o the_o houre-point_n each_o from_o other_o will_v be_v about_o 1_o inch_n and_o 56_o centesme_n or_o part_n of_o 100_o the_o like_a reason_n hold_v for_o the_o inscribe_v of_o all_o other_o chord_n in_o the_o prop._n follow_v chap._n ii_o to_o draw_v the_o houre-line_n in_o a_o direct_a polar_n plane_n a_o direct_a polar_n plane_n be_v that_o which_o be_v parallel_v to_o the_o hour_n of_o 6_o here_o represent_v by_o epw_n wherein_o the_o style_n will_v be_v parallel_n to_o the_o plane_n and_o the_o houre-line_n parallel_v one_o to_o the_o other_o and_o therefore_o may_v be_v best_a draw_v by_o that_o which_o i_o have_v show_v in_o the_o use_n of_o the_o sector_n they_o may_v be_v also_o draw_v by_o the_o help_n of_o these_o line_n of_o proportion_n in_o this_o manner_n first_o draw_v a_o right_a line_n we_o for_o the_o horizon_n and_o the_o equator_fw-la and_o cross_v it_o at_o the_o point_n c_o about_o the_o middle_n of_o the_o line_n with_o cb_n another_o right_a line_n which_o may_v serve_v for_o the_o meridian_n and_o the_o hour_n of_o 12_o and_o must_v also_o be_v the_o substylar_a line_n wherein_o the_o style_n shall_v stand_v then_o to_o proportion_v the_o style_n unto_o the_o plane_n consider_v the_o length_n of_o the_o horizontall_a line_n and_o what_o houre-line_n you_o will_v have_v to_o fall_v on_o your_o plane_n for_o the_o distance_n of_o any_o one_o houre-line_n from_o the_o meridian_n be_v know_v we_o may_v find_v both_o the_o length_n of_o the_o style_n and_o the_o distance_n of_o the_o rest_n because_o as_o the_o tangent_fw-la of_o the_o hour_n give_v be_v to_o the_o distance_n from_o the_o meridian_n so_o the_o tangent_fw-la of_o 45_o gr_n to_o the_o height_n of_o the_o style_n suppose_v the_o length_n of_o the_o horizontall_a line_n to_o be_v 12_o inch_n and_o that_o it_o be_v require_v to_o put_v on_o all_o the_o houre-line_n from_o 7_o in_o the_o morning_n unto_o 5_o in_o the_o evening_n here_o we_o have_v 5_o hour_n and_o 6_o inch_n on_o either_o side_n the_o meridian_n wherefore_o i_o allow_v 15_o gr_n for_o a_o hour_n and_o extend_v the_o compass_n the_o compass_n from_o the_o tangent_fw-la of_o 75_o gr_n the_o measure_n of_o 5_o hour_n unto_o the_o tangent_fw-la of_o 45_o gr_n i_o find_v the_o same_o extent_n to_o reach_v in_o the_o line_n of_o number_n from_o 6.00_o to_o about_o 1.61_o this_o show_v both_o the_o height_n of_o the_o style_n and_o the_o distance_n of_o the_o houre-point_n of_o 9_o &_o 3_o from_o the_o meridian_n to_o be_v 1_o inch_n 61_o part_n to_o find_v the_o length_n of_o the_o tangent_fw-la between_o the_o substylar_a and_o the_o houre-point_n as_o the_o tangent_fw-la of_o 45_o gr_n to_o the_o tangent_fw-la of_o the_o hour_n so_o the_o height_n of_o the_o style_n to_o the_o length_n of_o the_o tangent_fw-la line_n between_o the_o substylar_n and_o the_o houre-point_n thus_o have_v find_v the_o length_n of_o the_o style_n in_o our_o example_n to_o be_v 1.61_o if_o i_o exend_v the_o compass_n from_o the_o tangent_fw-la of_o 45_o gr_n unto_o the_o tangent_fw-la of_o 15_o gr_n the_o measure_n of_o the_o first_o hour_n from_o the_o substylar_n i_o shall_v find_v the_o same_o extent_n to_o reach_v in_o the_o line_n of_o number_n from_o 1.61_o unto_o 0.43_o for_o the_o length_n of_o the_o tangent_fw-la between_o the_o substylar_a and_o the_o houre-point_n of_o 11_o and_o 1._o if_o i_o extend_v they_o from_o the_o tangent_fw-la of_o 45_o gr_n unto_o the_o tangent_fw-la of_o 75_o gr_n the_o measure_n of_o the_o five_o hour_n i_o shall_v find_v they_o to_o reach_v in_o the_o line_n of_o number_n from_o 1.61_o unto_o 6.00_o for_o the_o length_n of_o the_o tangent_fw-la from_o the_o substylar_n to_o the_o houre-point_n of_o 7_o and_o 5._o for_o howsoever_o it_o be_v the_o same_o distance_n in_o the_o line_n of_o tangent_n from_o 45_o unto_o 75_o as_o from_o 45_o unto_o 15_o yet_o because_o 75_o be_v more_o and_o 15_o less_o than_o 45_o the_o tangent_fw-la line_n that_o answer_v to_o they_o will_v be_v according_o more_a or_o less_o than_o the_o length_n of_o the_o style_n ho._n an._n po_n tang._n gr._n m._n in._n par_n 12_o 0_o 0_o 0_o 0_o 11.1_o 15_o 0_o 0_o 43_o 10.2_o 30_o 0_o 0_o 93_o 9.3_o 45_o 0_o 1_o 61_o 8.4_o 60_o 0_o 2_o 79_o 7.5_o 75_o 0_o 6_o 0_o 6.6_o 90_o 0_o infin_fw-fr again_o if_o i_o extend_v they_o from_o 45_o gr_n in_o the_o tangent_n unto_o 30_o gr_n the_o measure_n of_o the_o second_o hour_n i_o shall_v find_v they_o to_o reach_v in_o the_o line_n of_o number_n from_o 1.61_o unto_o 0.93_o for_o the_o hour_n of_o 10_o and_o 2_o if_o i_o extend_v they_o from_o the_o tangent_fw-la of_o 45_o gr_n unto_o the_o tangent_fw-la of_o 60_o gr_n for_o the_o four_o hour_n i_o shall_v find_v they_o to_o reach_v in_o the_o line_n of_o number_n from_o 1.61_o unto_o 2_o 79_o and_o such_o be_v the_o length_n of_o the_o tangent_fw-la line_n from_o the_o substylar_n unto_o the_o hour_n of_o 8_o and_o 4._o and_o the_o like_a reason_n hold_v for_o the_o inscribe_v of_o all_o other_o tangent_fw-la line_n in_o the_o proposition_n follow_v but_o for_o such_o tangent_n
like_a manner_n a_o chord_n of_o 24_o gr_n 20_o m._n for_o the_o hour_n of_o 10_o and_o 2_o and_o a_o chord_n of_o 38_o gr_n 3_o m._n for_o the_o hour_n of_o 9_o and_o 3_o and_o so_o for_o the_o rest_n of_o the_o hour_n their_o half_n and_o quarter_n 4_o i_o draw_v right_a line_n through_o the_o centre_n and_o the_o term_n of_o these_o chord_n and_o these_o line_n so_o draw_v be_v the_o houre-line_n require_v last_o i_o set_v up_o the_o style_n over_o the_o meridian_n so_o as_o it_o may_v cut_v the_o plane_n in_o the_o centre_n and_o there_o make_v a_o angle_n with_o the_o meridian_n equal_a to_o the_o latitude_n of_o the_o place_n so_o it_o shall_v represent_v the_o axis_fw-la of_o the_o world_n and_o be_v true_o place_v for_o cast_v of_o the_o shadow_n upon_o the_o houre-line_n in_o a_o horizontall_a plane_n chap._n v._o to_o draw_v the_o houre-line_n in_o a_o vertical_a plane_n a_o vertical_a plane_n be_v that_o which_o be_v parallel_n to_o the_o prime_n vertical_a circle_n here_o represent_v by_o ezw_n it_o have_v two_o face_n one_o to_o the_o north_n the_o other_o to_o the_o south_n in_o each_o of_o they_o the_o substylar_n will_v be_v the_o same_o with_o the_o meridian_n line_n and_o the_o angle_n of_o the_o style_n above_o the_o plane_n will_v be_v equal_a to_o zp_v the_o compliment_n of_o the_o latitude_n the_o triangle_n here_o consider_v be_v make_v by_o the_o vertical_a the_o meridian_n and_o the_o houre-circle_n in_o which_o we_o know_v the_o side_n zp_n the_o angle_n at_o the_o pole_n and_o the_o right_a angle_n at_o the_o zenith_n and_o therefore_o may_v find_v the_o ark_n of_o the_o vertical_a between_o the_o meridian_n and_o the_o houre-circle_n after_o this_o manner_n as_o the_o sine_fw-la of_o 90_o gr_n be_v to_o the_o cousin_n of_o the_o latitude_n so_o the_o tangent_fw-la of_o the_o hour_n to_o the_o tangent_fw-la of_o the_o houre-line_n from_o the_o meridian_n extend_v the_o compass_n from_o the_o sine_fw-la of_o 90_o gr_n to_o the_o sine_fw-la of_o the_o compliment_n of_o the_o latitude_n so_o the_o same_o extent_n shall_v reach_v from_o the_o tangent_fw-la of_o the_o hour_n to_o the_o tangent_fw-la of_o the_o houre-line_n from_o the_o meridian_n thus_o in_o the_o latitude_n of_o 51_o gr_n 30_o m._n i_o extend_v the_o compass_n from_o the_o sine_fw-la of_o 90_o gr_n to_o the_o sine_fw-la of_o 38_o gr_n 30_o m._n and_o find_v the_o same_o extent_n to_o reach_v from_o the_o tangent_fw-la of_o 15_o gr_n to_o the_o tangent_fw-la of_o 9_o gr_n 28_o m_o for_o the_o distance_n of_o the_o first_o hour_n from_o the_o meridian_n and_o from_o the_o tangent_fw-la of_o 75_o gr_n unto_o the_o tangent_fw-la of_o 66_o gr_n 42_o m._n for_o the_o five_o hour_n and_o so_o in_o the_o rest_n as_o in_o the_o table_n follow_v these_o ark_n be_v know_v i_o may_v come_v to_o the_o plane_n and_o there_o by_o help_n of_o a_o thread_n and_o plummet_n draw_v a_o vertical_a line_n serve_v both_o for_o the_o meridian_n and_o the_o hour_n of_o 12_o and_o the_o substylar_n then_o may_v i_o draw_v a_o occult_a vertical_a circle_n and_o therein_o inscribe_v the_o chord_n of_o those_o former_a ark_n and_o draw_v the_o houre-line_n and_o set_v up_o the_o style_n as_o before_o in_o the_o horizontall_a plane_n ho._n ang._n po_n arc._n pla_z gr._n m._n gr._n m._n 12_o 0_o 0_o 0_o 0_o  _fw-fr 3_o 45_o 2_o 20_o  _fw-fr 7_o 30_o 4_o 41_o  _fw-fr 11_o 15_o 7_o 3_o 1_o 15_o 0_o 9_o 28_o  _fw-fr 18_o 45_o 11_o 56_o  _fw-fr 22_o 30_o 14_o 27_o  _fw-fr 26_o 15_o 17_o 4_o 2_o 30_o 0_o 19_o 46_o  _fw-fr 33_o 45_o 22_o 35_o  _fw-fr 37_o 30_o 25_o 32_o  _fw-fr 41_o 15_o 28_o 38_o 3_o 45_o 0_o 31_o 54_o  _fw-fr 48_o 45_o 35_o 22_o  _fw-fr 52_o 30_o 39_o 3_o  _fw-fr 56_o 15_o 42_o 58_o 4_o 60_o 0_o 47_o 9_o  _fw-fr 63_o 45_o 51_o 36_o  _fw-fr 67_o 30_o 56_o 20_o  _fw-fr 71_o 15_o 61_o 23_o 5_o 75_o 0_o 66_o 42_o  _fw-fr 78_o 45_o 72_o 17_o  _fw-fr 82_o 30_o 78_o 3_o  _fw-fr 86_o 15_o 84_o 0_o 6_o 90_o 0_o 90_o 0_o if_o it_o be_v the_o south_n face_n of_o the_o plane_n the_o centre_n will_v be_v upward_o and_o the_o style_n must_v point_v downward_o if_o the_o north_n face_n the_o centre_n must_v be_v in_o the_o low_a part_n of_o the_o meridian_n line_n and_o the_o style-point_n upward_o in_o all_o such_o place_n as_o be_v to_o the_o northward_o of_o the_o equinoctial_a line_n as_o it_o may_v appear_v by_o consider_v how_o the_o line_n do_v fall_v in_o the_o fundamental_a diagram_n chap._n vi_o to_o draw_v the_o houre-line_n in_o a_o vertical_a incline_a plane_n all_o those_o plane_n that_o have_v their_o horizontall_a line_n lie_v east_n and_o west_n be_v in_o that_o respect_n say_v to_o be_v vertical_a if_o they_o be_v also_o upright_o and_o pass_v through_o the_o zenith_n they_o be_v direct_a vertical_o if_o they_o incline_v to_o the_o pole-they_a be_v direct_a polar_n if_o to_o the_o equinoctial_a they_o be_v proper_o call_v equinoctial_a plane_n and_o be_v describe_v before_o if_o to_o none_o of_o these_o three_o point_n they_o be_v then_o call_v by_o the_o general_a name_n of_o incline_v vertical_n these_o may_v incline_v either_o to_o the_o north_n part_n of_o the_o horizon_n or_o to_o the_o south_n and_o each_o of_o they_o have_v two_o face_n one_o to_o the_o zenith_n the_o other_o to_o the_o nadir_n in_o which_o we_o be_v first_o to_o consider_v the_o height_n of_o the_o pole_n above_o the_o plane_n by_o compare_v the_o inclination_n of_o the_o plane_n to_o the_o horizon_n with_o the_o latitude_n of_o the_o place_n as_o in_o our_o latitude_n of_o 51_o gr_n 30_o m._n if_o the_o inclination_n of_o the_o plane_n eiw_n shall_v be_v 13_o gr_n northward_o that_o be_v if_o in_o the_o ark_n of_o the_o meridian_n between_o the_o plane_n and_o the_o north_n part_n of_o the_o horizon_n shall_v be_v 13_o gr_n we_o may_v take_v these_o 13_o gr_n out_o of_o pn_n 51_o gr_n 30_o m._n the_o elevation_n of_o the_o pole_n above_o the_o horizon_n and_o there_o will_v remain_v pi_n 38_o gr_n 30_o m._n for_o the_o elevation_n of_o the_o north_n pole_n above_o the_o upper_a face_n of_o the_o plane_n and_o therefore_o 38_o gr_n 30_o m._n for_o the_o height_n of_o the_o south_n pole_n above_o the_o low_a face_n of_o the_o plane_n or_o if_o the_o inclination_n of_o the_o plane_n shall_v be_v find_v to_o be_v 62_o gr_n to_o the_o southward_n we_o may_v number_v they_o in_o the_o meridian_n from_o s_n the_o southpart_n of_o the_o horizon_n unto_o l_o and_o there_o draw_v the_o ark_n elw_n represent_v this_o plane_n so_o the_o ark_n of_o the_o meridian_n pl_z shall_v give_v the_o height_n of_o the_o north_n pole_n above_o the_o upper_a face_n of_o this_o plane_n to_o be_v 66_o gr_n 30_o m._n and_o therefore_o the_o height_n of_o the_o south_n pole_n above_o the_o low_a face_n of_o the_o plane_n be_v also_o 66_o gr_n 30_o m._n in_o like_a manner_n if_o the_o inclination_n of_o the_o plane_n eyw_n shall_v be_v 15_o gr_n southward_o that_o be_v if_o sy_n the_o ark_n of_o the_o meridian_n between_o the_o south_n part_n of_o the_o horizon_n and_o the_o plane_n shall_v be_v 15_o gr_n the_o height_n of_o the_o north_n pole_n above_o the_o upper_a face_n of_o the_o plane_n and_o the_o height_n of_o the_o south_n pole_n above_o the_o low_a face_n of_o the_o plane_n will_v be_v also_o find_v to_o be_v 66_o gr_n 30_o m._n but_o if_o the_o plane_n shall_v fall_v between_o the_o zenith_n and_o the_o north_n pole_n then_o will_v the_o north_n pole_n be_v elevate_v above_o the_o low_a face_n and_o the_o south_n pole_n above_o the_o upward_a face_n of_o the_o plane_n as_o may_v appear_v by_o the_o projection_n of_o the_o sphere_n in_o the_o fundamental_a diagram_n then_o in_o the_o triangle_n make_v by_o the_o plane_n the_o meridian_n and_o the_o houre-circle_n we_o have_v the_o side_n which_o be_v the_o height_n of_o the_o pole_n above_o the_o plane_n together_o with_o the_o angle_n at_o the_o pole_n and_o the_o right_a angle_n at_o the_o intersection_n of_o the_o meridian_n with_o the_o plane_n by_o which_o we_o may_v find_v the_o ark_n of_o the_o plane_n between_o the_o meridian_n and_o the_o houre-circle_n after_o this_o manner_n as_o the_o sine_fw-la of_o 90_o gr_n be_v to_o the_o sine_fw-la of_o the_o pole_n above_o the_o plane_n so_o the_o tangent_fw-la of_o the_o hour_n to_o the_o tangent_fw-la of_o the_o houre-line_n from_o the_o meridian_n thus_o in_o the_o former_a example_n where_o pi_n the_o height_n of_o the_o pole_n above_o the_o plane_n be_v find_v to_o be_v 38_o gr_n 30_o m._n if_o you_o shall_v extend_v the_o compass_n from_o the_o sine_fw-la of_o
open_v the_o compass_n to_o the_o near_a extent_n between_o the_o pole_n and_o the_o plane_n and_o there_o i_o place_v the_o letter_n r_n to_o make_v this_o rectangle_n triangle_n prz_n 2_o i_o extend_v the_o compass_n from_o the_o sine_fw-la of_o 90_o gr_n unto_o the_o sine_fw-la of_o 38_o gr_n 30_o m._n the_o compliment_n of_o the_o latitude_n and_o the_o same_o extent_n will_v reach_v from_o the_o sine_fw-la of_o 65_o gr_n 40_o m._n the_o compliment_n of_o the_o declination_n unto_o the_o sine_fw-la of_o 34_o gr_n 33_o m._n or_o i_o may_v extend_v they_o from_o the_o sine_fw-la of_o 90_o gr_n unto_o the_o sine_fw-la of_o 60_o gr_n the_o compliment_n of_o the_o inclination_n of_o the_o meridians_fw-la and_o the_o same_o extent_n will_v reach_v from_o the_o tangent_fw-la of_o 38_o gr_n 30_o m._n the_o compliment_n of_o the_o latitude_n unto_o the_o tangent_fw-la of_o 34_o gr_n 33_o m._n and_o such_o be_v the_o ark_n pr._n the_o height_n of_o the_o pole_n above_o the_o plane_n 3_o i_o may_v extend_v the_o compass_n from_o the_o sine_fw-la of_o 90_o gr_n unto_o the_o sine_fw-la of_o 24_o gr_n 20_o m._n the_o declination_n give_v and_o the_o same_o extent_n will_v reach_v from_o the_o tangent_fw-la of_o 38_o gr_n 30_o m._n the_o compliment_n of_o the_o latitude_n unto_o the_o tangent_fw-la of_o 38_o gr_n 8_o m._n and_o such_o be_v the_o ark_n zr_n the_o distance_n of_o the_o substylar_n from_o the_o meridian_n 4_o that_o i_o may_v find_v the_o distance_n of_o each_o houre-line_n from_o the_o substylar_n i_o consider_v the_o angle_n of_o inclination_n of_o the_o meridians_fw-la rpz_fw-fr and_o there_o see_v how_o that_o pz_v the_o meridian_n of_o the_o place_n which_o be_v the_o hour_n of_o 12_o be_v 30_o gr_n distance_n from_o pr._n the_o meridian_n of_o the_o plane_n and_o that_o one_o face_n of_o the_o plane_n be_v open_a to_o the_o south-west_n and_o the_o other_o to_o the_o north-east_n this_o meridian_n of_o the_o plane_n fall_v to_o be_v the_o same_o with_o the_o hour_n of_o 2_o otherwise_o with_o the_o hour_n of_o 10_o therefore_o allow_v 15_o gr_n for_o a_o hour_n the_o hour_n of_o 1_o rpo_n will_v be_v 15_o gr_n and_o rfx_v the_o hour_n of_o 11_o 45_o gr_n distant_a from_o pr._n the_o proper_a meridian_n of_o the_o plane_n and_o so_o i_o gather_v the_o inclination_n of_o the_o rest_n of_o the_o houre-circle_n towards_o this_o meridian_n according_a to_o their_o angle_n at_o the_o pole_n as_o in_o this_o table_n then_o take_v my_o compass_n in_o my_o hand_n i_o extend_v they_o from_o the_o sine_fw-la of_o 90_o gr_n unto_o the_o sine_fw-la of_o 34_o gr_n 33_o m._n the_o height_n of_o the_o pole_n above_o the_o plane_n and_o find_v they_o to_o reach_v in_o the_o line_n of_o tangent_n from_o 15_o gr_n the_o inclination_n of_o the_o hour_n of_o 1_o to_z 8_o gr_n 38_o m._n for_o the_o ark_n of_o 1_o from_o the_o substylar_n and_o from_o 30_o gr_n unto_o 18_o gr_n 8_o m._n for_o the_o hour_n of_o 12_o agreeable_a to_o the_o three_o prop._n and_o from_o 45_o gr_n unto_o 29_o gr_n 33_o m._n for_o the_o hour_n of_o 11_o and_o so_o the_o rest_n which_o i_o also_o set_v down_o in_o a_o table_n hor._n ang._n po_n ar._n pla._n gr._n m._n gr._n m._n 8_o 90_o 0_o 90_o 0_o 9_o 75_o 0_o 64_o 42_o 10_o 60_o 0_o 44_o 30_o 11_o 45_o 0_o 29_o 33_o 12_o 30_o 0_o 18_o 8_o 1_o 15_o 0_o 8_o 38_o 2_o merid_n substyl_o 3_o 15_o 0_o 8_o 38_o 4_o 30_o 0_o 18_o 8_o 5_o 45_o 0_o 29_o 33_o 6_o 60_o 0_o 44_o 30_o 7_o 75_o 0_o 64_o 42_o 8_o 90_o 0_o 90_o 0_o these_o ark_n be_v thus_o find_v will_v serve_v for_o the_o draw_v of_o the_o houre-line_n both_o on_o the_o south-west_n face_n and_z the_o northwest_o face_n of_o the_o plane_n for_o come_v to_o the_o plane_n 1_o by_o the_o help_n of_o a_o thread_n and_o plummet_n i_o draw_v a_o vertical_a line_n serve_v both_o for_o the_o meridian_n of_o the_o place_n and_o the_o hour_n of_o 12._o 2_o in_o this_o meridian_n line_n i_o make_v choice_n of_o a_o centre_n at_o c_o in_o the_o upper_a part_n of_o the_o line_n if_o it_o be_v the_o south_n face_n as_o here_o we_o suppose_v it_o that_o the_o style_n may_v have_v room_n to_o point_v downward_o but_o in_o the_o low_a part_n of_o the_o line_n if_o it_o be_v the_o north_n face_n of_o the_o plane_n for_o there_o the_o style_n must_v point_v upward_o and_o upon_o this_o centre_n i_o describe_v a_o occult_a circle_n represent_v the_o decline_a vertical_a belong_v to_o the_o plane_n 4_o according_a to_o the_o table_n of_o the_o arke_n of_o the_o plane_n from_o the_o substylar_n i_o find_v a_o chord_n of_o 8_o gr_n 38_o m._n and_o inscribe_v it_o into_o this_o circle_n from_o the_o substylar_n towards_o the_o meridian_n for_o the_o hour_n of_o 1._o in_o like_a manner_n a_o chord_n of_o 29_o gr_n 23_o m._n for_o the_o hour_n of_o 11_o and_o a_o chord_n of_o 44_o gr_n 30_o m._n for_o the_o hour_n of_o 10_o and_o so_o for_o the_o rest_n of_o the_o hour_n their_o half_n and_o quarter_n 5_o i_o draw_v right_a line_n through_o the_o centre_n and_o the_o term_n of_o these_o chord_n and_o these_o line_n so_o draw_v be_v the_o houre-line_n require_v last_o i_o set_v up_o the_o style_n over_o the_o substylar_n so_o as_o it_o may_v cut_v the_o plane_n in_o the_o centre_n and_o there_o make_v a_o angle_n with_o the_o substylar_n of_o 34_o gr_n 33_o m._n according_a to_o the_o height_n of_o the_o pole_n above_o the_o plane_n so_o it_o shall_v represent_v the_o axis_fw-la of_o the_o world_n and_o be_v true_o place_v for_o cast_v of_o the_o shadow_n upon_o the_o houre-line_n in_o this_o decline_a plane_n a_o second_o example_n after_o the_o like_a manner_n if_o in_o our_o latitude_n a_o upright_a plane_n shall_v decline_v 85_o gr_n from_o the_o prime_n vertical_a the_o one_o face_n of_o it_o be_v open_a to_o the_o northwest_o and_o the_o other_o to_o the_o southeast_n we_o may_v in_o some_o sort_n represent_v it_o by_o the_o vertical_a qzh_n and_o then_o work_v as_o before_o 1_o the_o angle_n zpt_n the_o inclination_n of_o the_o two_o meridians_fw-la will_v be_v find_v to_o be_v 86_o gr_n 5_o m._n so_o that_o pt_v the_o meridian_n of_o this_o plane_n will_v here_o fall_v between_o the_o houre-circle_n of_o 6_o and_o 7_o from_o the_o meridian_n 2_o the_o ark_n pt_v the_o height_n of_o the_o pole_n above_o the_o plane_n will_v be_v only_o 3_o gr_n 6_o m._n 3_o the_o ark_n zt_n the_o distance_n of_o the_o substylar_n from_o the_o meridian_n 38_o gr_n 23_o m._n 4_o the_o table_n of_o the_o angle_n at_o the_o pole_n will_v be_v also_o gather_v by_o compare_v the_o meridian_n of_o the_o plane_n with_o the_o rest_n of_o the_o houre-circle_n for_o the_o angle_n tpz_n between_o pt_v the_o meridian_n of_o the_o plane_n pz_n the_o meridian_n of_o the_o place_n and_o the_o hour_n of_o 12_o be_v 86_o gr_n 5_o m._n allow_v 15_o gr_n for_o a_o hour_n the_o hour_n of_o 11_o ½_n will_v be_v 78_o gr_n 35_o m._n and_o the_o hour_n of_o 11_o 71_o gr_n 5_o m._n distant_a from_o the_o meridian_n of_o the_o plane_n &_o so_o the_o rest_n of_o the_o hour_n as_o in_o the_o second_o column_n of_o this_o table_n hor._n ang._n po_n ar._n pla._n c_o f_o c_o g_o gr._n m._n gr._n m._n in._n par._n in._n par._n 12_o 86_o 5_o 38_o 23_o 91_o 08_o 79_o 21_o  _fw-fr 78_o 35_o 15_o 3_o 30_o 92_o 26_o 89_o 11_o 71_o 5_o 9_o 6_o 18_o 42_o 16_o 02_o  _fw-fr 63_o 35_o 6_o 13_o 12_o 52_o 10_o 89_o 10_o 56_o 5_o 4_o 36_o 9_o 25_o 8_o 05_o 9_o 41_o 5_o 2_o 42_o 5_o 43_o 4_o 72_o 8_o 26_o 5_o 1_o 31_o 3_o 05_o 2_o 65_o 7_o 11_o 5_o 0_o 36_o 1_o 20_o 1_o 05_o  _fw-fr merid._n substyl_o 0_o 0_o 0_o 0_o 6_o 3_o 55_o 0_o 13_o 0_o 44_o 0_o 38_o 5_o 18_o 55_o 1_o 4_o 2_o 15_o 1_o 86_o 4_o 33_o 55_o 2_o 5_o 4_o 18_o 3_o 64_o 3_o 48_o 55_o 3_o 33_o 7_o 13_o 6_o 20_o 2_o 63_o 55_o 6_o 20_o 12_o 77_o 11_o 10_o  _fw-fr 71_o 25_o 9_o 10_o 18_o 56_o 16_o 14_o 1_o 78_o 55_o 15_o 28_o 31_o 82_o 27_o 67_o  _fw-fr 86_o 25_o 40_o 55_o 99_o 67_o 86_o 68_o then_o have_v the_o height_n of_o the_o pole_n above_o the_o plane_n and_o these_o angle_n at_o the_o pole_n the_o arke_n of_o the_o plane_n between_o the_o substylar_n and_o the_o houre-circle_n will_v be_v find_v as_o in_o the_o three_o column_n these_o ark_n be_v find_v will_v serve_v for_o the_o draw_v of_o the_o houre-line_n on_o either_o face_n of_o this_o plane_n 1_o by_o the_o help_n of_o a_o thread_n and_o plummet_n i_o draw_v zc_n
 _fw-fr 18_o 45_o 3_o 39_o  _fw-fr 22_o 30_o 4_o 14_o  _fw-fr 26_o 15_o 4_o 93_o 10_o 30_o 0_o 5_o 77_o  _fw-fr 33_o 45_o 6_o 68_o  _fw-fr 37_o 30_o 7_o 67_o  _fw-fr 41_o 15_o 8_o 77_o 11_o 45_o 0_o 10_o 00_o  _fw-fr 48_o 45_o 11_o 40_o  _fw-fr 52_o 30_o 13_o 03_o  _fw-fr 56_o 15_o 14_o 97_o 12_o 60_o 0_o 17_o 32_o  _fw-fr 63_o 45_o 20_o 28_o  _fw-fr 67_o 30_o 24_o 14_o  _fw-fr 71_o 15_o 29_o 46_o 1_o 75_o 0_o 37_o 32_o  _fw-fr 78_o 45_o 50_o 27_o  _fw-fr 82_o 30_o 75_o 96_o  _fw-fr 86_o 15_o 152_o 57_o 2_o 90_o 0_o infinite_a 5_o these_o length_n be_v thus_o find_v and_o set_v down_o in_o the_o table_n i_o take_v out_o 17_o inch_n 32_o cent_n and_o prick_v they_o in_o the_o equator_fw-la from_z c_o unto_o a_o for_o the_o hour_n of_o 12_o and_o 37_o inch_n 32_o cent_n and_o prick_v they_o down_o for_o the_o hour_n of_o 1._o and_o so_o the_o rest_n of_o the_o houre-point_n 6_o this_o do_v if_o i_o draw_v right_a line_n through_o each_o of_o these_o point_n cross_v the_o equator_fw-la at_o right_a angle_n they_o shall_v be_v the_o houre-line_n require_v and_o if_o i_o set_v the_o style_n over_o the_o substylar_n so_o as_o the_o edge_n of_o it_o may_v be_v parallel_n to_o the_o plane_n and_o the_o height_n of_o it_o be_v ten_o inch_n equal_a to_o the_o former_a radius_fw-la bc_n it_o shall_v represent_v the_o axis_fw-la of_o the_o world_n and_o be_v true_o place_v for_o cast_v of_o the_o shadow_n upon_o the_o houre-line_n in_o this_o decline_a polar_n plane_n chap._n x._o to_o draw_v the_o houre-line_n in_o a_o decline_a incline_a plane_n if_o a_o plane_n shall_v decline_v from_o the_o prime_n vertical_a and_o incline_v to_o the_o horizon_n and_o yet_o not_o lie_v even_o with_o the_o pole_n of_o the_o world_n it_o be_v then_o call_v a_o decline_a incline_a plane_n of_o these_o there_o be_v several_a sort_n for_o the_o inclination_n be_v northward_o the_o plane_n may_v fall_v between_o the_o horizon_n and_o the_o pole_n as_o the_o circle_n bmd_v in_o the_o fundamental_a diagram_n or_o between_o the_o zenith_n and_o the_o pole_n as_o bfd_n or_o the_o inclination_n may_v be_v southward_o and_o so_o be_v represent_v by_o bkd_a it_o may_v also_o fall_v either_o below_o the_o intersection_n of_o the_o meridian_n and_o the_o equator_fw-la or_o above_o it_o and_o each_o of_o these_o have_v two_o face_n the_o upper_a towards_o the_o zenith_n and_o the_o low_a towards_o the_o nadir_n wherein_o have_v the_o latitude_n of_o the_o place_n with_o the_o declination_n and_o inclination_n of_o the_o plane_n we_o be_v far_a to_o consider_v 1_o the_o ark_n of_o the_o meridian_n between_o the_o pole_n and_o the_o plane_n 2_o the_o inclination_n of_o the_o plane_n to_o the_o meridian_n 3_o the_o ark_n of_o the_o plane_n between_o the_o horizon_n and_o the_o meridian_n 4_o the_o angle_n of_o inclination_n between_o both_o meridians_fw-la 5_o the_o height_n of_o the_o pole_n above_o the_o plane_n 6_o the_o distance_n of_o the_o substylar_n from_o the_o meridian_n 7_o the_o distance_n of_o each_o houre-line_n from_o the_o substylar_n and_o all_o these_o seven_o may_v be_v represent_v in_o the_o fundamental_a diagram_n as_o in_o this_o example_n in_o our_o latitude_n of_o 51_o gr_n 30_o m._n a_o plane_n be_v propose_v decline_v from_o the_o vertical_a 24_o gr_n 20_o m._n and_o incline_v northward_o 36_o gr_n the_o upper_a face_n lie_v open_a to_o the_o south-west_n the_o low_a to_o the_o north-east_n if_o i_o number_v these_o 24_o gr_n 20_o m._n in_o the_o horizon_n from_o e_z to_z b_o and_o there_o draw_v the_o line_n bcd_n it_o shall_v represent_v the_o horizontall_a line_n of_o the_o plane_n then_o cross_v it_o at_o right_a angle_n with_o the_o plane_n hzq_n draw_v through_o the_o zenith_n i_o number_v 36_o gr_n for_o the_o inclination_n from_o q_o unto_o m_n and_o there_o draw_v the_o circle_n bmd_v cross_v the_o meridian_n in_o the_o point_n a_o this_o circle_n so_o draw_v shall_v represent_v the_o plane_n propose_v and_o because_o it_o do_v not_o pass_v through_o the_o pole_n be_v therefore_o no_o polar_n but_o a_o ordinary_a decline_a incline_a plane_n 1_o the_o ark_n of_o the_o meridian_n of_o the_o place_n between_o the_o pole_n and_o the_o plane_n be_v here_o represent_v by_o pa_n and_o may_v be_v find_v by_o resolve_v the_o triangle_n dn_n a_o wherein_o the_o angle_n at_o n_n be_v know_v to_o be_v a_o right_a angle_n the_o angle_n at_o d_o be_v the_o angle_n of_o inclination_n the_o side_n d_o n_n the_o compliment_n of_o the_o declination_n which_o be_v know_v as_o the_o sine_fw-la of_o 90_o gr_n to_o the_o cousin_n of_o declination_n so_o the_o tangent_fw-la of_o inclination_n to_o the_o horizon_n to_o the_o tangent_fw-la of_o the_o meridian_n between_o the_o horizon_n and_o the_o plane_n extend_v the_o compass_n from_o the_o sine_fw-la of_o 90_o gr_n unto_o the_o sine_fw-la of_o 65_o gr_n 40_o m._n the_o compliment_n of_o the_o declination_n the_o same_o extent_n will_v reach_v from_o the_o tangent_fw-la of_o 36_o gr_n the_o inclination_n propose_v unto_o the_o tangent_fw-la of_o 33_o gr_n 30_o m._n and_o such_o be_v the_o ark_n of_o the_o meridian_n na_fw-mi between_o the_o horizon_n and_o the_o plane_n this_o ark_n na_n be_v compare_v with_o the_o ark_n np_n which_o be_v the_o elevation_n of_o the_o pole_n above_o the_o horizon_n and_o be_v here_o suppose_a to_o be_v 51_o gr_n 30_o m._n the_o difference_n n_o a_o come_v to_o 18_o gr_n and_o such_o be_v the_o ark_n of_o the_o meridian_n require_v between_o the_o pole_n and_o the_o plane_n 2_o the_o inclination_n of_o the_o plane_n to_o the_o meridian_n be_v here_o represent_v by_o the_o angle_n n_n add_v and_o may_v be_v find_v by_o that_o which_o we_o have_v give_v in_o the_o former_a triangle_n dn_n a._n for_o as_o the_o sine_fw-la of_o 90_o gr_n to_o the_o sine_fw-la of_o the_o declination_n from_o the_o vertical_a so_o the_o sine_fw-la of_o inclination_n to_o the_o horizon_n to_o the_o cousin_n of_o the_o inclination_n to_o the_o meridian_n extend_v the_o compass_n from_o the_o sine_fw-la of_o 90_o gr_n unto_o the_o sine_fw-la of_o 24_o gr_n 20_o m._n the_o declination_n of_o the_o plane_n the_o same_o extent_n will_v reach_v from_o the_o sine_fw-la of_o 36_o gr_n the_o inclination_n give_v unto_o the_o cousin_n of_o 76_o gr_n and_o such_o be_v n_n add_v the_o angle_n of_o inclination_n between_o the_o plane_n d_o a_o and_z n_z a_o the_o meridian_n of_o the_o place_n or_o as_o the_o sine_fw-la of_o the_o ark_n of_o the_o meridian_n between_o the_o horizon_n and_o the_o plane_n be_v to_o the_o sine_fw-la of_o 90_o gr_n so_o the_o cotangent_fw-la of_o the_o declination_n to_o the_o tangent_fw-la of_o the_o inclination_n to_o the_o meridian_n extend_v the_o compass_n from_o the_o sine_fw-la of_o 33_o gr_n 30_o m._n the_o ark_n of_o the_o meridian_n between_o the_o horizon_n and_o the_o plane_n unto_o the_o sine_fw-la of_o 90_o gr_n the_o same_o extent_n will_v reach_v from_o the_o 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tangent_fw-la of_o 69_o gr_n 54_o m._n and_o such_o be_v d_o a_o the_o ark_n of_o the_o plane_n between_o the_o horizon_n and_o the_o meridian_n of_o the_o place_n 4_o the_o inclination_n of_o meridians_fw-la be_v here_o represent_v by_o the_o angle_n apb_n for_o if_o i_o let_v down_o a_o perpendicular_a pb_n from_o the_o pole_n unto_o the_o plane_n this_o perpendicular_a shall_v be_v the_o meridian_n of_o the_o plane_n and_o we_o shall_v have_v another_o triangle_n abp_n wherein_o the_o angle_n at_o b_o be_v a_o right_a angle_n because_o of_o the_o perpendicular_a the_o angle_n at_o a_o be_v the_o inclination_n of_o the_o plane_n to_o the_o meridian_n of_o the_o place_n and_o the_o side_n p_o a_fw-fr be_v the_o
be_v describe_v out_o of_o the_o angular_a point_n at_o any_o other_o distance_n let_v the_o semidiameter_n be_v turn_v into_o a_o parallel_a chord_n of_o 60_o gr._n then_o take_v the_o chord_n of_o this_o ark_n and_o carry_v it_o parallel_v till_o it_o cross_v in_o like_a chord_n so_o the_o place_n where_o it_o stay_v shall_v give_v the_o quantity_n of_o the_o angle_n as_o in_o the_o former_a example_n if_o i_o make_v the_o semidiameter_n ab_fw-la a_o parallel_a chord_n of_o 60_o gr._n and_o then_o keep_v the_o sector_n at_o that_o angle_n carry_v the_o chord_n bc_n parallel_n till_o it_o stay_v in_o like_a chord_n i_o shall_v find_v it_o to_o stay_v in_o no_o other_o but_o 11_o gr._n 15_o m_n and_o such_o be_v the_o angle_n bac_n 10_o upon_o a_o right_a line_n and_o a_o point_n give_v in_o it_o to_o make_v a_o angle_n equal_a to_o any_o angle_n give_v first_o out_o of_o the_o point_n give_fw-mi describe_v a_o ark_n cut_v the_o same_o line_n then_o by_o the_o 5._o prop_v afore_o find_v the_o chord_n of_o the_o angle_n give_v agreeable_a to_o the_o semidiameter_n and_o inscribe_v it_o into_o this_o ark_n so_o a_o right_a line_n draw_v through_o the_o point_n give_v and_o the_o end_n of_o this_o chord_n shall_v be_v the_o side_n that_o make_v up_o the_o angle_n let_v the_o right_a line_n give_v be_v ab_fw-la and_o the_o point_n give_v in_o it_o be_v a_o and_o let_v the_o angle_v give_v be_v 11_o gr_n 15_o m._n here_o i_o open_v the_o compass_n to_o any_o semidiameter_n ab_fw-la but_o as_o oft_o as_o i_o may_v convenient_o to_o the_o lateral_a semiradius_fw-la and_o set_v one_o foot_n in_o a_o i_o describe_v a_o occult_a ark_n bc_n then_o i_o seek_v out_o the_o chord_n of_o 11_o gr_n 15_o m._n and_o take_v it_o with_o the_o compass_n i_o set_v one_o foot_n in_o b_o the_o other_o cross_v the_o ark_n in_o c_o by_o which_o i_o draw_v the_o line_n ac_fw-la and_o it_o make_v up_o the_o angle_n require_v 11_o to_o divide_v the_o circumference_n of_o a_o circle_n into_o any_o part_n require_v if_o 360_o the_o measure_n of_o the_o whole_a circumference_n be_v divide_v by_o the_o number_n of_o part_n require_v the_o quotient_n give_v the_o chord_n which_o be_v find_v will_v divide_v the_o circumference_n so_o a_o chord_n of_o 120_o gr_n will_v divide_v the_o circumference_n into_o 3_o equal_a part_n a_o chord_n of_o 90_o gr_n into_o 4_o part_n a_o chord_n of_o 72_o gr_n into_o 5_o part_n a_o chord_n of_o 60_o gr_n into_o 6_o part_n a_o chord_n of_o 51_o gr_n 26._o into_o 7_o part_n a_o chord_n of_o 45_o gr_n into_o 8_o part_n a_o chord_n of_o 40_o gr_n into_o 9_o part_n a_o chord_n of_o 36_o gr_n into_o 10_o part_n a_o chord_n of_o 32_o gr_n 44_o m._n into_o 11_o part_n a_o chord_n of_o 30_o gr_n into_o 12_o part_n in_o like_a manner_n if_o it_o be_v require_v to_o divide_v the_o circumference_n of_o the_o circle_n who_o semidiameter_n be_v ab_fw-la into_o 32_o first_o i_o take_v the_o semidiameter_n ab_fw-la and_o make_v it_o a_o parallel_a chord_n of_o 60_o gr_n then_o because_o 360_o gr_n be_v divide_v by_o 32_o the_o quotient_n will_v be_v 11_o gr_n 15_o m._n i_o find_v the_o parallel_a chord_n of_o 11_o gr_n 15_o m._n and_o this_o will_v divide_v the_o circumference_n into_o 32._o but_o here_o the_o part_n be_v many_o it_o be_v better_a to_o divide_v it_o first_o into_o few_o and_o after_o to_o come_v over_o it_o again_o as_o first_o to_o divide_v the_o circumference_n into_o 4_o and_o then_o each_o 4_o part_n into_o 8_o or_o otherwise_o as_o the_o part_n may_v be_v divide_v 12_o to_o divide_v a_o right_a line_n by_o extreme_a and_o mean_a proportion_n the_o line_n to_o be_v divide_v by_o extreme_a and_o mean_a proportion_n have_v the_o same_o proportion_n to_o his_o great_a segment_n as_o in_o figure_n inscribe_v in_o the_o same_o circle_n the_o side_n of_o a_o hexagon_n a_o figure_n of_o six_o angle_n have_v to_o a_o side_n of_o a_o decagon_n a_o figure_n of_o ten_o angle_n but_o the_o side_n of_o a_o hexagon_n be_v a_o chord_n of_o 60_o gr_n and_o the_o side_n of_o a_o decagon_n be_v a_o chord_n of_o 36_o gr_n let_v ab_fw-la be_v the_o line_n to_o be_v divide_v if_o i_o make_v ab_fw-la a_o parallel_a chord_n of_o 60_o gr_n and_o to_o this_o semidiameter_n find_v ac_fw-la a_o chord_n of_o 36_o gr_n this_o ac_fw-la shall_v be_v the_o great_a segment_n divide_v the_o whole_a line_n in_o c_o by_o extreme_a and_o mean_a proportion_n so_o that_o as_o ab_fw-la the_o whole_a line_n be_v unto_o ac_fw-la the_o great_a segment_n so_o ac_fw-la the_o great_a segment_n unto_o cb_n the_o lesser_a segment_n or_o let_v ac_fw-la be_v the_o great_a segment_n give_v if_o i_o make_v this_o a_o parallel_a chord_n of_o 36_o gr_n the_o correspondent_a semidiameter_n shall_v be_v the_o whole_a line_n ac_fw-la and_o the_o difference_n cb_n the_o lesser_a segment_n or_o let_v cb_n be_v the_o lesser_a segment_n give_v if_o i_o make_v this_o a_o parallel_a chord_n of_o 36_o gr_n the_o correspondent_a semidiameter_n shall_v be_v great_a segment_n ac_fw-la which_o add_v to_o cb_n give_v the_o whole_a line_n ab_fw-la to_o avoid_v double_v of_o line_n or_o number_n you_o may_v put_v over_o the_o whole_a line_n in_o the_o sin_n of_o 72_o gr_n and_o the_o parallel_n sine_fw-la of_o 36_o gr_n shall_v be_v the_o great_a segment_n or_o if_o you_o put_v over_o the_o whole_a line_n in_o the_o sin_n of_o 54_o gr_n the_o parallel_n sine_fw-la of_o 30_o gr_n shall_v be_v the_o great_a segment_n and_o the_o parallel_n sine_fw-la of_o 18_o gr_n shall_v be_v the_o lesser_a segment_n chap._n iii_o of_o the_o projection_n of_o the_o sphere_n in_o plano_n 1_o the_o sphere_n may_v be_v project_v in_o plano_n in_o straight_a line_n as_o in_o the_o analemma_n if_o the_o semidiameter_n of_o the_o circle_n give_v be_v divide_v in_o such_o sort_n as_o the_o line_n of_o sin_n on_o the_o sector_n as_o if_o the_o radius_fw-la of_o the_o circle_n give_v be_v ae_n the_o circle_n thereon_o describe_v may_v represent_v the_o plane_n of_o the_o general_a meridian_n which_o divide_v into_o four_o equal_a part_n in_o e_o p_o ae_n s_o and_o cross_v at_o right_a angle_n with_o eae_fw-la and_o ps_n the_o diameter_n eae_fw-la shall_v represent_v the_o equator_fw-la and_o ps_n the_o circle_n of_o the_o hour_n of_o 6._o and_o it_o be_v also_o the_o axis_fw-la of_o the_o world_n wherein_o p_o stand_v for_o the_o north_n pole_n and_o s_n for_o the_o south_n pole_n then_o may_v each_o quarter_n of_o the_o meridian_n be_v divide_v into_o 90_o gr_n from_o the_o equator_fw-la towards_o the_o pole_n in_o which_o if_o we_o number_v 23_o gr_n 30_o m._n the_o great_a declination_n of_o the_o sun_n from_o e_o to_o 69_o northwards_o from_o ae_n to_o ♑_o southward_n the_o line_n draw_v from_o 69_o to_o ♑_o shall_v be_v the_o ecliptic_a and_o the_o line_n draw_v parallel_n to_o the_o equator_fw-la through_z ♋_o and_o ♑_o shall_v be_v the_o tropique_n have_v these_o common_a section_n with_o the_o plane_n of_o the_o meridian_n if_o we_o shall_v divide_v each_o diameter_n of_o the_o ecliptic_a into_o 90_o gr_n in_o such_o sort_n as_o the_o sin_n be_v divide_v on_o the_o sector_n the_o first_o 30_o gr_n from_o a_o towards_o 69_o shall_v stand_v for_o the_o sine_z of_o ♈_o the_o 30_o gr_n next_o follow_v for_o ♉_o the_o rest_n for_o ♊_o ♋_o ♌_o etc._n etc._n in_o their_o order_n so_o that_o by_o these_o mean_v we_o have_v the_o place_n of_o the_o sun_n for_o all_o time_n of_o the_o year_n if_o again_o we_o divide_v ap_n as_o in_o the_o like_a sort_n and_o set_v to_o the_o number_n 10._o 20._o 30._o etc._n etc._n unto_o 90_o gr_n the_o line_n draw_v through_o each_o of_o these_o degree_n parallel_v to_o the_o equator_fw-la shall_v show_v the_o declination_n of_o the_o sun_n and_o represent_v the_o parallel_n of_o latitude_n if_o far_o we_o divide_v ae_n aae_n and_o his_o parallel_n in_o the_o like_a sort_n and_o then_o careful_o draw_v a_o line_n through_o each_o 15_o gr_n so_o as_o it_o make_v no_o angle_n the_o line_n so_o draw_v shall_v be_v ellipticall_a and_o represent_v the_o houre-circle_n the_o meridian_n pes_z the_o hour_n of_o 12_o at_o noon_n that_o next_o unto_o it_o draw_v through_o 75_o gr_n from_o the_o centre_n the_o hour_n of_o 11_o and_o 1_o that_o which_o be_v draw_v through_o 60_o gr_n from_o the_o centre_n the_o hour_n of_o 10_o and_o 2._o etc._n etc._n then_o have_v respect_n unto_o the_o latitude_n we_o may_v number_v it_o from_o e_z northward_o unto_o z_o and_o there_o place_v the_o zenith_n by_o which_o and_o the_o centre_n the_o line_n draw_fw-mi zan_fw-mi shall_v represent_v
24_o gr_n of_o ♉_o his_o longitude_n shall_v be_v pi_n his_o right_a ascension_n ph_n his_o declination_n he_o and_o if_o the_o place_n give_fw-mi be_v 19_o gr_n of_o ♌_o his_o longitude_n shall_v be_v pk_n his_o right_a ascension_n pn_n his_o declination_n nk_v again_o the_o declination_n bring_v to_o the_o horizon_n of_o the_o place_n shall_v there_o show_v the_o ascentionall_a difference_n amplitude_n of_o ascension_n and_o the_o like_a conclusion_n of_o the_o globe_n but_o i_o intend_v not_o here_o to_o show_v the_o use_n of_o the_o astrolabe_n but_o the_o use_n of_o the_o sector_n in_o projection_n and_o after_o this_o manner_n may_v a_o nocturnal_a be_v project_v to_o show_v the_o hour_n of_o the_o night_n whereof_o i_o will_v set_v down_o a_o type_n for_o the_o use_n of_o seaman_n it_o consist_v as_o you_o see_v of_o two_o part_n the_o one_o be_v a_o plane_n divide_v equal_o according_a to_o the_o 24_o hour_n of_o the_o day_n and_o each_o hour_n into_o quarter_n or_o minute_n as_o the_o plane_n will_v bear_v the_o line_n from_o the_o centre_n to_o xii_o stand_v for_o the_o meridian_n and_o xii_o stand_v for_o the_o hour_n of_o 12_o at_o midnight_n the_o other_o part_n be_v a_o rundle_n for_o such_o star_n as_o be_v near_o the_o north_n pole_n together_o with_o the_o twelve_o month_n and_o the_o day_n of_o each_o month_n fit_v to_o the_o right_a ascension_n of_o the_o star_n those_o that_o have_v occasion_n to_o see_v the_o south_n pole_n may_v do_v the_o like_a for_o the_o southern_a constellation_n and_o put_v they_o in_o a_o rundle_n on_o the_o back_n of_o this_o plane_n and_o so_o it_o may_v serve_v for_o all_o the_o world_n the_o use_n of_o this_o nocturnal_a be_v easy_a and_o ready_a for_o look_v up_o to_o the_o pole_n and_o see_v what_o star_n be_v near_o the_o meridian_n then_o place_v the_o rundle_n to_o the_o like_a situation_n so_o the_o day_n of_o the_o month_n will_v show_v the_o hour_n of_o the_o night_n 3_o the_o sphere_n may_v be_v project_v in_o plano_fw-la by_o circular_a line_n as_o in_o the_o particular_a astrolabe_n of_o joh._n stophlerin_n by_o help_n of_o the_o tangent_fw-la as_o before_o for_o let_v the_o circle_n give_fw-mi represent_v the_o tropic_n of_o ♑_o let_v it_o be_v divide_v into_o four_o part_n and_o cross_v at_o right_a angle_n with_o ac_fw-la the_o equinoctial_a colour_n and_o mb_v the_o solstitial_a colour_n and_o general_a meridian_n the_o centre_n p_o represent_v the_o pole_n of_o the_o world_n let_v each_o quarter_n be_v divide_v into_o 90_o gr_n and_o so_o the_o whole_a into_o 360_o begin_v from_o a_o towards_o b._n the_o meridian_n pm_n or_o pb_n may_v be_v divide_v according_a to_o the_o tangent_fw-la of_o half_a his_o ark_n so_o as_o the_o ark_n from_o the_o north_n pole_n to_o the_o tropic_n of_o ♑_o be_v 90_o gr_n and_o 23_o gr_n 30_o m._n that_o be_v 113_o gr_n 80_o m._n and_o the_o half_a ark_n 56_o gr_n 45_o m._n the_o meridian_n shall_v be_v divide_v into_o 90_o gr_n and_o 23_o gr_n 30_o m._n in_o such_o sort_n as_o the_o tangent_fw-la of_o 56_o gr_n 45_o m._n on_o the_o side_n of_o the_o sector_n be_v divide_v into_o degree_n and_o half_a degree_n of_o which_o pae_n the_o ark_n of_o the_o equator_fw-la 90_o gr_fw-la from_o the_o pole_n shall_v be_v give_v by_o the_o tangent_fw-la of_o 45_o gr_n and_o p_o 69_o the_o ark_n of_o the_o summer_n tropic_n 66_o gr_n 30_o m._n from_o the_o pole_n shall_v be_v give_v by_o the_o tangent_fw-la of_o 33_o gr_n 15_o m._n and_o the_o circle_n draw_v upon_o the_o centre_n p_o through_z ae_n and_o ♋_o shall_v be_v the_o equator_fw-la and_o the_o summer_n tropic_n have_v the_o equator_fw-la and_o both_o the_o tropique_n the_o ecliptic_a ♈_o ♋_o ♎_o ♑_o shall_v be_v draw_v from_o the_o one_o tropic_n to_o the_o other_o through_o the_o intersection_n of_o the_o equator_fw-la and_o the_o equinoctial_a colour_n and_o it_o may_v be_v divide_v first_o into_o the_o twelve_o sign_n after_o this_o manner_n pe_n the_o ark_n of_o the_o pole_n of_o the_o ecliptic_a 23_o gr_n 30_o m._n from_o the_o pole_n of_o the_o world_n shall_v be_v give_v by_o the_o tangent_fw-la of_o 11_o gr_n 45_o m._n the_o centre_n of_o the_o circle_n of_o longitude_n pass_v through_o this_o pole_n e_o ♈_o and_o ♎_o shall_v be_v find_v at_o d_o somewhat_o below_o b_o by_o the_o tangent_fw-la of_o 66_o gr_n 30_o m._n then_o through_o d_o draw_v a_o occult_a line_n parallel_n to_o ac_fw-la and_o divide_v it_o on_o each_o side_n from_o d_o in_o such_o sort_n as_o the_o tangent_fw-la be_v divide_v on_o the_o side_n of_o the_o sector_n allow_v 45_o gr_n to_o be_v equal_a to_o de._n so_o the_o thirtith_n degree_n from_o d_o towards_o the_o right_a hand_n shall_v be_v the_o centre_n of_o the_o circle_n of_o longitude_n pass_v through_o e_o ♉_o and_o â™_o the_o sixtith_n degree_n the_o centre_n of_o ♊_o e_o â™_o the_o thirtith_n degree_n from_o d_o towards_o the_o left_a hand_n the_o centre_n of_o ♓_o e_o â™_o the_o sixtith_n the_o centre_n of_o â™’_o e_o 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between_o those_o intersection_n be_v always_o the_o centre_n for_o example_n of_o this_o projection_n let_v ☉_o the_o place_n of_o the_o sun_n give_fw-mi be_v 10_o gr_n of_o ♉_o a_o right_a line_n draw_v from_o p_o through_o this_o place_n unto_o the_o equator_fw-la shall_v there_o show_v his_o right_a ascension_n ♈_o k_n and_o his_o declination_n k_o ☉_o then_o may_v we_o on_o the_o centre_n p_o and_z semidiameter_n ☉_o p_o draw_v a_o occult_a parallel_n of_o declination_n cross_v the_o horizon_n in_o l_o and_o m_o the_o meridian_n in_o g_z and_z n._n so_o the_o right_a line_n pl_z and_o pm_n produce_v shall_v show_v the_o time_n of_o the_o sun_n rise_v and_o set_v ♈_o q_o the_o difference_n of_o ascension_n ♎_o r_o the_o difference_n of_o descension_n ♈_o l_o the_o amplitude_n of_o his_o rise_n and_o ♎_o m_o the_o amplitude_n of_o his_o set_n lgm_n show_v the_o length_n of_o the_o day_n ln_o m_n the_o length_n of_o the_o night_n zg_v show_v his_o distance_n from_o the_o zenith_n 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centre_n z_o represent_v the_o zenith_n let_v each_o quarter_n of_o the_o horizon_n be_v divide_v into_o 90_o gr_n and_o so_o the_o whole_a into_o 360_o gr_n begin_n from_o n_n and_o set_v to_o the_o number_n of_o 10._o 20._o 30._o etc._n etc._n 90_o at_o e_o 180_o at_o s_n 270_o at_o five_o 360_o at_o n._n the_o semidiameter_n zn_o zs_n may_v be_v divide_v according_a to_o the_o tangent_fw-la of_o half_a their_o arke_n so_o as_o the_o ark_n from_o the_o zenith_n to_o the_o horizon_n be_v 90_o gr_n and_o the_o half_a ark_n 45_o gr_n the_o semidiameter_n be_v to_o be_v divide_v in_o such_o sort_n as_o the_o tangent_fw-la of_o 45_o gr_n as_o
gr_n and_o let_v a_o thread_n and_o plummet_n be_v hang_v upon_o a_o line_n parallel_n to_o the_o edge_n of_o one_o of_o the_o leg_n so_o that_o leg_n shall_v be_v vertical_a and_o the_o other_o leg_n parallel_v to_o the_o horizon_n if_o the_o plane_n seem_v to_o be_v level_a with_o the_o horizon_n you_o may_v try_v it_o by_o set_v the_o horizontall_a leg_n of_o the_o sector_n to_o the_o plane_n and_o hold_v the_o other_o leg_n upright_o for_o than_o if_o the_o thread_n shall_v fall_v on_o his_o plummet_n line_n which_o way_n soever_o you_o turn_v the_o sector_n it_o be_v a_o horizontall_a plane_n if_o the_o one_o end_n of_o the_o plane_n be_v high_a than_o the_o other_o and_o yet_o not_o vertical_a it_o be_v a_o incline_a plane_n and_o you_o may_v find_v the_o inclination_n in_o this_o manner_n first_o hold_v the_o vertical_a leg_n of_o the_o sector_n upright_o and_o turn_v the_o horizontall_a leâ—_n about_o until_o it_o lie_v close_o with_o the_o plane_n and_o the_o thread_n fall_v 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declination_n the_o axis_fw-la to_o give_v denomination_n unto_o the_o plane_n for_o example_n in_o a_o vertical_a plane_n here_o represent_v by_o ezw_n the_o horizontall_a line_n be_v ecw_o the_o same_o with_o the_o line_n of_o east_n and_o west_n and_o therefore_o no_o declination_n the_o perpendicular_a cross_v it_o cz_n the_o same_o with_o the_o vertical_a line_n draw_v from_o the_o centre_n to_o the_o zenith_n right_a unto_o the_o horizon_n and_o therefore_o no_o inclination_n the_o axis_fw-la of_o the_o plane_n be_v scn_n the_o same_o with_o the_o meridian_n line_n draw_v from_o the_o south_n to_o the_o north_n and_o according_o give_v the_o denomination_n to_o the_o plane_n for_o the_o plane_n have_v two_o face_n and_o the_o axis_fw-la two_o pole_n s_n and_o n_o the_o pole_n s_n fall_v direct_o into_o the_o south_n do_v cause_n that_o face_n to_o which_o it_o be_v next_o to_o be_v call_v the_o south_n face_n and_o the_o other_o pole_n at_o n_n point_v into_o the_o north_n do_v give_v the_o denomination_n to_o the_o other_o face_n and_o make_v it_o to_o be_v call_v the_o north_n face_n of_o this_o plane_n in_o like_a manner_n in_o the_o decline_a incline_v plane_n here_o represent_v by_o bfd_n the_o horizontall_a line_n be_v bcd_v which_o cross_v the_o prime_n vertical_a line_n ecw_o and_o therefore_o it_o be_v call_v a_o decline_a plane_n according_a to_o the_o angle_n of_o declination_n ecb_n or_o wcd_n the_o perpendicular_a to_o this_o horizontall_a line_n be_v cf_n where_o the_o point_n f_o fall_v in_o the_o plane_n qzh_n perpendicular_a to_o the_o plane_n propose_v between_o the_o zenith_n and_o the_o north_n part_n of_o the_o horizon_n and_o therefore_o it_o be_v call_v a_o plane_n incline_v to_o the_o northward_o according_a to_o the_o ark_n fq_n or_o the_o angle_n fcq_n the_o axis_fw-la of_o the_o plane_n be_v here_o represent_v by_o the_o line_n ck_n where_o the_o pole_n k_o be_v 90_o gr_n distant_a from_o the_o plane_n and_o so_o be_v as_o much_o above_o the_o horizon_n at_o h_n and_o the_o other_o pole_n as_o much_o below_o the_o horizon_n at_o q_o as_o the_o plane_n at_o f_o be_v distant_a from_o the_o zenith_n and_o this_o pole_n k_o here_o fall_v between_o the_o meridian_n and_o the_o prime_n vertical_a circle_n into_o the_o south-west_n part_n of_o the_o world_n this_o upper_a face_n of_o the_o plane_n be_v therefore_o call_v the_o south-west_n face_n and_o the_o low_a the_o north-east_n face_n of_o the_o plane_n the_o declination_n from_o the_o prime_n vertical_a may_v be_v find_v by_o the_o needle_n in_o the_o usual_a inclinatorie_a quadrant_n or_o rather_o by_o compare_v the_o horizontall_a line_n draw_v upon_o the_o plane_n with_o the_o azimuth_n of_o the_o sun_n and_o the_o meridian_n line_n in_o such_o sort_n as_o before_o we_o find_v the_o variation_n of_o the_o magnetical_a needle_n for_o take_v any_o board_n that_o have_v one_o side_n straight_o and_o draw_v the_o line_n ho_o parallel_n to_o that_o side_n and_o the_o line_n zm_n perpendicular_a unto_o it_o and_o on_o the_o centre_n z_o make_v a_o semicircle_n hmo_n this_o do_v hold_v the_o board_n to_o the_o plane_n so_o as_o ho_o may_v be_v parallel_n to_o bd_o the_o horizontall_a line_n on_o the_o plane_n and_o the_o board_n parallel_v to_o the_o horizon_n then_o the_o sun_n shine_v upon_o it_o hold_v out_o a_o thread_n and_o plummet_n so_o as_o the_o thread_n be_v vertical_a the_o shadow_n of_o the_o sun_n may_v fall_v on_o the_o centre_n z_o and_o draw_v the_o line_n of_o shadow_n az_n represent_v the_o common_a section_n which_o the_o azimuth_n of_o the_o sun_n make_v with_o the_o plane_n of_o the_o horizon_n and_o let_v another_o take_v the_o altitude_n of_o the_o sun_n at_o the_o same_o instant_n so_o by_o resolve_v a_o triangle_n as_o i_o show_v before_o pag._n 65._o you_o may_v find_v what_o azimuth_n the_o sun_n be_v in_o when_o he_o give_v shadow_n upon_o az_n suppose_v the_o azimuth_n to_o be_v as_o before_o pag._n 64._o 72_o gr_n 52_o m._n from_o the_o north_n to_o the_o westward_n and_o therefore_o 17_o gr_n 8_o m._n from_o the_o west_n we_o may_v allow_v these_o 17_o gr_n 8_o m._n from_o a_o unto_o five_o and_o draw_v the_o line_n zv_n and_o so_o we_o have_v the_o true_a west_n point_n of_o the_o prime_n vertical_a line_n then_o allow_v 90_o gr_n from_o five_o unto_o s_o we_o have_v the_o south_n point_n of_o the_o meridian_n line_n zs_n and_o the_o angle_n hzv_n shall_v give_v the_o declination_n of_o the_o plane_n from_o the_o vertical_a and_o the_o angle_n ozs_n the_o declination_n of_o the_o plane_n from_o the_o meridian_n or_o we_o may_v take_v out_o only_o the_o angle_n azh_n which_o the_o line_n of_o shadow_n make_v with_o the_o horizontall_a line_n of_o the_o plane_n and_o compare_v it_o with_o the_o angle_n azv_n which_o the_o line_n of_o shadow_n make_v with_o the_o prime_n vertical_a and_o so_o here_o if_o azv_v the_o sun_n azimuth_n shall_v be_v 17_o gr_n 8_o m._n past_o the_o west_n and_o yet_o the_o line_n of_o shadow_n az_v 7_o gr_n 12_o m._n short_a of_o the_o plane_n the_o declination_n of_o the_o plane_n shall_v be_v 24_o gr_n 20_o m._n as_o may_v appear_v by_o the_o site_n of_o the_o plane_n and_o the_o circle_n if_o the_o altitude_n of_o the_o sun_n be_v take_v at_o such_o time_n as_o the_o shadow_n of_o the_o thread_n fall_v on_o bd_o or_o ho_o and_o then_o a_o triangle_n resolve_v the_o declination_n of_o the_o plane_n will_v be_v such_o as_o the_o azimuth_n of_o the_o sun_n from_o the_o prime_n vertical_a if_o at_o such_o a_o time_n as_o the_o shadow_n fall_v on_o mz_n the_o declination_n will_v be_v such_o as_o the_o azimuth_n of_o the_o sun_n from_o the_o meridian_n if_o it_o be_v a_o fair_a summer_n day_n
as_o fall_v under_o 45_o gr_n i_o may_v better_o use_v across_o work_n and_o extend_v the_o compass_n from_o the_o tangent_fw-la of_o 45_o gr_n unto_o 1.61_o in_o the_o line_n of_o number_n so_o shall_v i_o find_v the_o same_o extent_n to_o reach_v from_o 30_o gr_n in_o the_o tangent_n to_o 93_o part_n in_o the_o line_n of_o number_n for_o the_o distance_n of_o the_o second_o hour_n and_o from_o 15_o gr_n in_o the_o tangent_n to_o 43_o part_n for_o the_o distance_n of_o the_o first_o hour_n from_o the_o meridian_n or_o if_o this_o extent_n from_o 45_o gr_n backward_o to_o 1.61_o be_v too_o large_a for_o the_o compass_n i_o may_v extend_v forward_o from_o the_o tangent_fw-la of_o 5_o gr_n 43_o m._n to_o 1.61_o part_n in_o the_o line_n of_o number_n and_o the_o same_o extent_n shall_v reach_v from_o 15_o gr_n in_o the_o tangent_n to_o 43_o part_n in_o the_o line_n of_o number_n for_o the_o distance_n of_o the_o first_o hour_n and_o from_o 30_o gr_n to_o 93_o part_n for_o the_o distance_n of_o the_o second_o hour_n as_o before_o have_v find_v the_o length_n of_o the_o tangent_fw-la line_n in_o inch_n and_o part_n of_o inch_n and_o prick_v they_o in_o the_o equator_fw-la on_o both_o side_n of_o the_o meridian_n from_o the_o centre_n c_o if_o we_o draw_v right_a line_n through_o each_o of_o those_o point_n cross_v the_o equator_fw-la at_o right_a angle_n they_o shall_v be_v the_o houre-line_n require_v and_o if_o we_o set_v a_o style_n over_o the_o meridian_n so_o as_o the_o edge_n of_o it_o be_v parallel_v to_o the_o plane_n and_o the_o height_n of_o it_o be_v as_o much_o above_o the_o meridian_n as_o the_o distance_n between_o the_o meridian_n and_o the_o houre-point_n of_o 3_o or_o 9_o it_o shall_v represent_v the_o axis_fw-la of_o the_o world_n and_o be_v true_o place_v for_o the_o cast_n of_o the_o shadow_n upon_o the_o houre-line_n in_o a_o polar_n plane_n horae_n ang._n po_n tangle_n gr._n m._n in._n pa_n 6_o 0_o 0_o  _fw-fr 3_o 45_o 165_o  _fw-fr 7_o 30_o 1_o 32_o  _fw-fr 11_o 15_o 1_o 99_o 7_o 15_o 0_o 2_o 68_o  _fw-fr 18_o 45_o 3_o 39_o  _fw-fr 22_o 30_o 4_o 14_o  _fw-fr 26_o 15_o 4_o 93_o 8_o 30_o 0_o 5_o 77_o  _fw-fr 33_o 45_o 6_o 68_o  _fw-fr 37_o 30_o 7_o 67_o  _fw-fr 41_o 15_o 8_o 77_o 9_o 45_o 0_o 10_o 00_o  _fw-fr 48_o 45_o 11_o 40_o  _fw-fr 52_o 30_o 13_o 03_o  _fw-fr 56_o 15_o 14_o 97_o 10_o 60_o 0_o 17_o 32_o  _fw-fr 63_o 45_o 20_o 28_o  _fw-fr 67_o 30_o 24_o 14_o  _fw-fr 71_o 15_o 29_o 46_o 11_o 75_o 0_o 37_o 32_o  _fw-fr 78_o 45_o 50_o 27_o  _fw-fr 82_o 30_o 75_o 96_o  _fw-fr 86_o 15_o 152_o 57_o 12_o 90_o 0_o infin_fw-fr chap._n iii_o to_o draw_v the_o houre-line_n in_o a_o meridian_n plane_n a_o meridian_n plane_n be_v that_o which_o be_v parallel_v to_o the_o meridian_n circle_v here_o represent_v by_o szn_n it_o have_v two_o face_n one_o to_o the_o east_n and_o the_o other_o to_o the_o west_n in_o each_o of_o they_o the_o style_n will_v be_v parallel_n to_o the_o plane_n and_o the_o houre-line_n parallel_v one_o to_o the_o other_o as_o in_o a_o polar_n plane_n the_o difference_n be_v only_o in_o the_o place_n of_o the_o equator_fw-la &_o in_o number_v of_o the_o hour_n for_o suppose_v the_o length_n of_o the_o style_n cb_n to_o be_v ten_o inch_n the_o length_n of_o the_o tangent_fw-la line_n belong_v to_o the_o first_o hour_n will_v be_v 2_o in_o 68_o p._n the_o length_n of_o the_o second_o 5_o in_o 77_o p._n as_o in_o the_o table_n then_o the_o tangent_fw-la of_o 15_o gr_n be_v prick_v down_o in_o the_o equator_fw-la on_o both_o side_n from_o 6_o shall_v serve_v for_o the_o hour_n of_o 5_o and_o 7_o and_o the_o tangent_fw-la of_o 30_o gr_n for_o the_o hour_n of_o 4_o and_o 8_o and_o so_o in_o the_o rest_n this_o do_v if_o we_o draw_v right_a line_n through_o each_o of_o these_o point_n cross_v the_o equator_fw-la at_o right_a angle_n they_o shall_v be_v the_o houre-line_n require_v and_o if_o we_o set_v a_o style_n over_o the_o hour_n of_o 6_o so_o as_o the_o edge_n of_o it_o may_v be_v parallel_n to_o the_o plane_n and_o the_o height_n of_o it_o may_v be_v equal_a to_o the_o distance_n between_o the_o hour_n of_o 6_o and_o 9_o in_o the_o equator_fw-la it_o shall_v represent_v the_o axis_fw-la of_o the_o world_n and_o be_v true_o place_v for_o the_o cast_n of_o the_o shadow_n upon_o the_o houre-line_n in_o a_o meridian_n plane_n chap._n four_o to_o draw_v the_o houre-line_n in_o a_o horizontall_a plane_n a_o horizontall_a plane_n be_v that_o which_o be_v parallel_v to_o the_o horizon_n here_o represent_v by_o the_o outward_a circle_n esw_v in_o which_o the_o diameter_n sn_a draw_v from_o the_o south_n to_o the_o north_n may_v go_v both_o for_o the_o meridian_n line_n and_o the_o meridian_n circle_n z_o for_o the_o zenith_n p_o for_o the_o pole_n of_o the_o world_n and_o the_o circle_n draw_v through_o p_o for_o the_o houre-circle_n of_o 1.2.3.4_o etc._n etc._n as_o they_o be_v number_v from_o the_o meridian_n these_o houre-circle_n consider_v with_o the_o meridian_n &_o the_o horizon_n do_v make_v diverse_a triangle_n pn_n 1_o pn_n 2_o pn_n 3_o in_o which_o we_o have_v know_v first_o the_o right_a angle_n at_o n_n the_o north_n intersection_n of_o the_o meridian_n and_o the_o horizon_n second_o the_o side_n pn_n the_o ark_n of_o the_o meridian_n between_o the_o pole_n and_o the_o horizon_n which_o be_v always_o equal_a to_o the_o latitude_n of_o the_o place_n three_o the_o angle_n at_o the_o pole_n make_v by_o the_o meridian_n and_o the_o houre-circle_n the_o angle_n np_fw-mi 1_o be_v 15_o gr_n np_n 2_o 30_o gr_n each_o hour_n 15_o gr_n more_o than_o other_o each_o half_a hour_n 7_o gr_n 30_o m._n each_o quarter_n 3._o gr_n 45._o m._n and_o these_o three_o be_v know_v we_o may_v find_v the_o ark_n of_o the_o horizon_n between_o the_o meridian_n and_o the_o houre-circle_n n_o 1_o n_z 2_o n_z 3_o etc._n etc._n for_o as_o the_o sine_fw-la of_o 90_o gr_n be_v to_o the_o sine_fw-la of_o the_o latitude_n so_o the_o tangent_fw-la of_o the_o hour_n to_o the_o tangent_fw-la of_o the_o houre-line_n from_o the_o meridian_n ho._n ang._n po_n arc._n pla_z gr._n m._n gr._n m._n 12_o 0_o 0_o 0_o 0_o  _fw-fr 3_o 45_o 2_o 56_o  _fw-fr 7_o 30_o 5_o 52_o  _fw-fr 11_o 15_o 8_o 51_o 1_o 15_o 0_o 11_o 50_o  _fw-fr 18_o 45_o 14_o 52_o  _fw-fr 22_o 30_o 17_o 57_o  _fw-fr 26_o 15_o 21_o 6_o 2_o 30_o 0_o 24_o 20_o  _fw-fr 33_o 45_o 27_o 36_o  _fw-fr 37_o 30_o 31_o 0_o  _fw-fr 41_o 15_o 34_o 28_o 3_o 45_o 0_o 38_o 3_o  _fw-fr 48_o 45_o 41_o 45_o  _fw-fr 52_o 30_o 45_o 34_o  _fw-fr 56_o 15_o 49_o 30_o 4_o 60_o 0_o 53_o 35_o  _fw-fr 63_o 45_o 57_o 47_o  _fw-fr 67_o 30_o 62_o 6_o  _fw-fr 71_o 15_o 66_o 33_o 5_o 75_o 0_o 71_o 6_o  _fw-fr 78_o 45_o 75_o 45_o  _fw-fr 82_o 30_o 80_o 25_o  _fw-fr 86_o 15_o 85_o 13_o 6_o 90_o 0_o 90_o 0_o extend_v the_o compass_n from_o the_o sine_fw-la of_o 90_o gr_n to_o the_o sine_fw-la of_o the_o latitude_n so_o the_o same_o extent_n shall_v reach_v from_o the_o tangent_fw-la of_o the_o hour_n to_o the_o tangent_fw-la of_o the_o houre-line_n from_o the_o meridian_n thus_o the_o latitude_n be_v 51_o gr_n 30_o m._n i_o extend_v the_o compass_n from_o the_o sine_fw-la of_o 90_o gr_n to_o the_o sine_fw-la of_o 51_o gr_n 30_o m._n &_o find_v the_o same_o extent_n to_o reach_v from_o the_o tangent_fw-la of_o 3_o gr_n 45_o m._n unto_o the_o tangent_fw-la of_o 2_o gr_n 56_o m._n for_o the_o distance_n of_o the_o first_o quarter_n from_o the_o meridian_n and_o from_o the_o tangent_fw-la of_o 7_o gr_n 30_o m._n unto_o the_o tangent_fw-la of_o 5_o gr_n 52_o m._n for_o the_o half_a hour_n and_o from_o the_o tangent_fw-la of_o 11_o gr_n 15_o m._n to_o the_o tangent_fw-la of_o 8_o gr_n 51_o m._n for_o the_o three_o quarter_n and_o from_o the_o tangent_fw-la of_o 15_o gr_n 0_o m._n unto_o 11_o gr_n 50_o m._n for_o the_o first_o hour_n and_o so_o in_o the_o rest_n this_o do_v i_o come_v to_o the_o plane_n and_o there_o according_a as_o the_o line_n do_v fall_v in_o the_o fundamental_a diagram_n 1_o i_o draw_v a_o right_a line_n sn_a serve_v for_o the_o meridian_n the_o hour_n of_o 12_o and_o the_o substylar_n 2_o in_o this_o meridian_n i_o make_v choice_n of_o a_o centre_n at_o c_o and_o there_o describe_v a_o occult_a circle_n represent_v the_o horizon_n 3_o i_o find_v a_o chord_n of_o 11_o gr_n 50_o m._n and_o inscribe_v it_o into_o this_o circle_n on_o either_o side_n of_o the_o meridian_n for_o the_o hour_n of_o 11_o &_o 1_o in_o
90_o gr_n to_o the_o sine_fw-la of_o 38_o gr_n 30_o m._n the_o same_o extent_n will_v reach_v from_o the_o tangent_fw-la of_o 15_o gr_n unto_o the_o tangent_fw-la of_o 9_o gr_n 28_o m._n for_o the_o distance_n of_o the_o first_o hour_n from_o the_o meridian_n and_o from_o 30_o gr_n unto_o 19_o gr_n 46_o m._n for_o the_o second_o hour_n and_o so_o forward_o as_o in_o the_o direct_a vertical_a and_o for_o the_o two_o last_o example_n you_o may_v extend_v the_o compass_n from_o the_o sine_fw-la of_o 90_o gr_n unto_o the_o sine_fw-la of_o 66_o gr_n 30_o m_o so_o the_o same_o extent_n shall_v reach_v in_o the_o line_n of_o tangent_n from_o 15_o gr_n unto_o 13_o gr_n 48_o m._n for_o the_o first_o hour_n from_o 75_o gr_n unto_o 73_o gr_n 43_o m._n for_o the_o five_o hour_n from_o 30_o gr_n unto_o 27_o gr_n 54_o m._n for_o the_o second_o hour_n from_o 60_o gr_n unto_o 57_o gr_n 48_o m._n for_o the_o four_o hour_n and_o from_o 45_o gr_n unto_o 42_o gr_n 31_o m._n for_o the_o three_o hour_n from_o the_o meridian_n these_o ark_n be_v know_v you_o may_v first_o draw_v the_o horizontall_a line_n and_o cross_v it_o in_o the_o middle_n with_o a_o perpendicular_a that_o may_v serve_v both_o for_o the_o meridian_n and_o the_o hour_n of_o 12_o and_o the_o substylar_n then_o know_v which_o pole_n be_v elevate_v above_o the_o plane_n you_o may_v according_o make_v choice_n of_o a_o fit_a point_n in_o the_o meridian_n for_o the_o centre_n of_o your_o houre-line_n and_o thence_o describe_v a_o occult_a ark_n of_o a_o circle_n inscribe_v the_o chord_n of_o those_o former_a ark_n and_o draw_v the_o hour_n line_n and_o set_v up_o the_o style_n as_o i_o show_v before_o in_o the_o horizontall_a plane_n chap._n vii_o to_o draw_v the_o houre-line_n in_o a_o vertical_a decline_a plane_n all_o upright_a plane_n whereon_o a_o man_n may_v draw_v a_o vertical_a line_n be_v in_o this_o respect_n say_v to_o be_v vertical_a if_o they_o shall_v also_o stand_v direct_o east_n and_o west_n they_o be_v direct_a vertical_o if_o direct_o north_n and_o south_n they_o be_v proper_o call_v meridian_n plane_n and_o be_v describe_v before_o if_o they_o behold_v none_o of_o these_o four_o principal_a part_n of_o the_o world_n but_o shall_v stand_v between_o the_o prime_n vertical_a and_o the_o meridian_n they_o be_v then_o call_v by_o the_o general_a name_n of_o decline_a vertical_n these_o have_v two_o face_n one_o to_o the_o south_n the_o other_o to_o the_o northward_o which_o may_v be_v distinguish_v in_o these_o northern_a part_n of_o the_o world_n after_o this_o manner_n if_o the_o sun_n come_v to_o the_o meridian_n shall_v shine_v upon_o the_o plane_n it_o be_v the_o south_n face_n if_o not_o it_o be_v the_o north_n face_n of_o that_o plane_n again_o if_o the_o sun_n shall_v shine_v upon_o the_o plane_n at_o high_a noon_n and_o yet_o long_o in_o the_o forenoon_n then_o in_o the_o afternoon_n it_o be_v the_o southeast_n face_n if_o long_a in_o the_o afternoon_n then_o in_o the_o forenoon_n it_o be_v the_o south-west_n face_n of_o the_o plane_n but_o how_o much_o the_o declination_n come_v to_o be_v best_a find_v as_o before_o when_o the_o declination_n be_v find_v there_o be_v four_o thing_n more_o to_o be_v consider_v before_o we_o can_v come_v to_o the_o draw_v of_o the_o houre-line_n and_o all_o four_o represent_v in_o the_o fundamental_a diagram_n 1_o the_o meridian_n of_o the_o plane_n and_o his_o inclination_n to_o the_o meridian_n of_o the_o place_n let_v the_o ark_n ezw_n represent_v the_o prime_n vertical_a and_o bzd_v a_o decline_a vertical_a according_a to_o the_o angle_n of_o declination_n ezb_n the_o meridian_n of_o the_o place_n be_v represent_v by_o pzs_n cross_v the_o vertical_a ezw_n at_o right_a angle_n at_o the_o zenith_n in_o the_o point_n z_o but_o the_o proper_a meridian_n of_o the_o plane_n will_v be_v pr._n which_o be_v a_o perpendicular_a let_v down_o from_o the_o pole_n unto_o the_o decline_a vertical_a and_o cross_v it_o with_o right_a angle_n in_o the_o point_n r_o so_o the_o angle_n rpz_n shall_v show_v the_o inclination_n of_o the_o two_o meridians_fw-la and_o may_v thus_o be_v find_v in_o the_o triangle_n prz_n we_o know_v the_o angle_n at_o r_n to_o be_v a_o right_a angle_n and_o the_o angle_n at_o z_o for_o it_o be_v the_o compliment_n of_o the_o declination_n and_o the_o base_a pz_n for_o it_o be_v the_o compliment_n of_o the_o latitude_n and_o therefore_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 90_o gr_n so_o the_o tangent_fw-la of_o the_o declination_n to_o the_o tangent_fw-la of_o the_o inclination_n require_v 2_o the_o height_n of_o the_o style_n above_o the_o plane_n this_o be_v here_o represent_v by_o the_o perpendicular_a ark_n pr._n and_o may_v be_v find_v by_o that_o which_o we_o have_v know_v in_o the_o former_a triangle_n prz_n for_o as_o the_o sine_fw-la of_o 90_o gr_n to_o the_o cousin_n of_o the_o latitude_n so_o the_o cousin_n of_o the_o declination_n to_o the_o sine_fw-la of_o the_o height_n of_o the_o style_n or_o if_o you_o please_v to_o make_v use_n of_o the_o angle_n of_o the_o inclination_n of_o the_o two_o meridians_fw-la the_o proportion_n will_v hold_v as_o the_o sine_fw-la of_o 90_o gr_n to_o the_o cousin_n of_o the_o inclination_n of_o meridians_fw-la so_o the_o cotangent_fw-la of_o the_o latitude_n to_o the_o tangent_fw-la of_o the_o height_n of_o the_o style_n 3_o the_o distance_n of_o the_o substylar_n from_o the_o meridian_n this_o be_v here_o represent_v by_o the_o ark_n zr_n and_o may_v be_v find_v by_o that_o which_o we_o have_v know_v in_o the_o former_a triangle_n prz_n as_o the_o sine_fw-la of_o 90_o gr_n to_o the_o sine_fw-la of_o the_o declination_n so_o the_o cotangent_fw-la of_o the_o latitude_n to_o the_o tangent_fw-la of_o the_o substylar_n from_o the_o meridian_n 4_o the_o distance_n of_o each_o houre-line_n from_o the_o substylar_n the_o distance_n of_o the_o houre-line_n from_o the_o substylar_n be_v here_o represent_v by_o those_o ark_n of_o the_o decline_a vertical_a belong_v to_o the_o plane_n which_o be_v intercept_v between_o the_o proper_a meridian_n of_o the_o plane_n and_o the_o houre-circle_n to_o this_o purpose_n we_o have_v diverse_a triangle_n make_v by_o the_o decline_a plane_n together_o with_o his_o proper_a meridian_n and_o the_o houre-circle_n in_o these_o we_o have_v know_v first_o the_o right_a angle_n at_o the_o intersection_n of_o the_o proper_a meridian_n with_o the_o plane_n then_o the_o side_n which_o be_v the_o height_n of_o the_o pole_n above_o the_o plane_n and_o three_o the_o angle_n at_o the_o pole_n for_o know_v the_o angle_n of_o inclination_n between_o the_o meridian_n of_o the_o plane_n and_o the_o meridian_n of_o the_o place_n which_o be_v always_o the_o hour_n of_o 12_o we_o may_v find_v the_o angle_n between_o the_o meridian_n of_o the_o plane_n and_o the_o hour_n of_o 1_o by_o allow_v in_o 15_o gr_n and_o the_o angle_n between_o the_o meridian_n of_o the_o plane_n and_o the_o hour_n of_o 2_o by_o allow_v in_o 30_o gr_n and_o so_o for_o the_o rest_n which_o be_v know_v we_o may_v find_v the_o ark_n of_o the_o plane_n from_o the_o substylar_n to_o the_o houre-circle_n in_o this_o manner_n as_o the_o sine_fw-la of_o 90_o gr_n to_o the_o sine_fw-la of_o the_o height_n of_o the_o pole_n above_o the_o plane_n so_o the_o tangent_fw-la of_o the_o hour_n from_o the_o proper_a meridian_n to_o the_o tangent_fw-la of_o the_o houre-line_n from_o the_o substylar_n thus_o in_o our_o latitude_n of_o 51_o gr_n 30_o m._n if_o the_o declination_n of_o a_o upright_a plane_n shall_v be_v find_v to_o be_v 24_o gr_n 20_o m._n from_o the_o prime_n vertical_a the_o one_o face_n open_a to_o the_o south-west_n the_o other_o to_o the_o northwest_o i_o may_v number_v these_o 24_o gr_n 20_o m._n in_o the_o horizon_n of_o the_o fundamental_a diagram_n from_o e_z unto_o b_o according_a to_o the_o situation_n of_o the_o plane_n and_o there_o draw_v the_o vertical_a bzd_fw-mi which_o shall_v represent_v the_o plane_n propose_v then_o take_v the_o compass_n into_o my_o hand_n 1_o i_o may_v extend_v they_o from_o the_o sine_fw-la of_o the_o latitude_n 51_o gr_n 30_o m._n unto_o the_o sine_fw-la of_o 90_o gr_n the_o same_o extent_n will_v reach_v in_o the_o line_n of_o tangent_n from_o 24_o gr_n 20_o m._n the_o declination_n give_v to_o about_o 30_o gr_n and_o such_o be_v zpr_fw-la the_o angle_n of_o inclination_n between_o the_o meridian_n of_o the_o place_n &_o the_o meridian_n of_o the_o plane_n and_o therefore_o the_o meridian_n of_o the_o plane_n will_v here_o fall_v upon_o the_o circle_n of_o the_o second_o hour_n from_o the_o meridian_n of_o the_o place_n as_o it_o may_v also_o appear_v by_o
a_o vertical_a plane_n serve_v both_o for_o the_o meridian_n of_o the_o place_n and_o the_o hour_n of_o 12._o 2_o in_o this_o meridian_n line_n i_o make_v choice_n of_o a_o centre_n in_o the_o upper_a part_n of_o the_o line_n if_o it_o have_v be_v the_o southern_a face_n of_o the_o plane_n but_o here_o in_o c_o the_o low_a part_n of_o the_o line_n because_o we_o suppose_v it_o to_o be_v the_o northwest_o face_n of_o the_o plane_n and_o the_o style_n must_v point_v upward_o and_o upon_o this_o centre_n i_o describe_v a_o occult_a circle_n represent_v the_o decline_a vertical_a belong_v to_o this_o plane_n 4_o the_o substylar_n be_v draw_v i_o may_v inscribe_v the_o chord_n of_o the_o ark_n of_o the_o plane_n from_o the_o substylar_n and_o draw_v the_o houre-line_n and_o set_v up_o the_o style_n as_o in_o the_o former_a plane_n or_o the_o ark_n of_o the_o plane_n from_o the_o substylar_n be_v find_v as_o before_o we_o may_v draw_v the_o houre-line_n upon_o the_o plane_n otherwise_o then_o by_o chord_n for_o have_v draw_v the_o houre-line_n as_o in_o the_o last_o figure_n upon_o paper_n or_o paist-boord_n we_o shall_v find_v the_o most_o part_n of_o they_o in_o this_o and_o such_o like_a plane_n that_o have_v great_a declination_n to_o fall_v so_o close_o together_o that_o they_o can_v hardly_o be_v discern_v wherefore_o to_o draw_v they_o at_o large_a to_o the_o best_a advantage_n of_o the_o plane_n i_o leave_v out_o the_o centre_n and_o draw_v they_o by_o tangent_n as_o in_o the_o polar_n plane_n 1_o i_o consider_v the_o length_n and_o breadth_n of_o the_o plane_n whereon_o i_o be_o to_o draw_v the_o houre-line_n which_o i_o suppose_v to_o be_v a_o square_a who_o side_n be_v 36_o inch_n and_o find_v that_o the_o little_a square_a abde_n will_v contain_v both_o the_o substylar_n and_o all_o those_o houre-line_n which_o be_v require_v in_o the_o great_a square_a azcq_n 2_o i_o draw_v two_o parallel_a line_n fn_n gm_n cross_v the_o substylar_n at_o right_a angle_n in_o the_o point_v f_o &_o g_o so_o as_o they_o may_v best_o cross_v all_o the_o houre-line_n and_o yet_o the_o one_o be_v distant_a from_o the_o other_o as_o far_o as_o the_o plane_n will_v give_v i_o leave_v and_o i_o find_v by_o the_o sight_n of_o the_o figure_n that_o if_o ab_fw-la the_o side_n of_o the_o lesser_a square_n shall_v be_v 36_o inch_n the_o line_n cf_n will_v be_v about_o 115_o inch_n and_o the_o line_n cg_n about_o 100_o inch_n and_o therefore_o fg_v 15_o inch_n again_o that_o the_o point_n f_o will_v fall_v about_o 6_o inch_n below_o the_o upper_a horizontall_a side_n ab_fw-la and_o about_o 12_o inch_n from_o the_o next_o vertical_a side_n bd_o for_o i_o need_v not_o here_o stand_v upon_o part_n 3_o because_o these_o two_o parallel_a line_n be_v tangent_fw-la line_n in_o respect_n of_o circle_n draw_v upon_o the_o semidiameter_n cf_n cg_n and_o such_o tangent_n as_o belong_v to_o the_o arke_n of_o the_o plane_n between_o the_o substylar_n and_o the_o houre-line_n the_o proportion_n will_v hold_v as_o the_o tangent_fw-la of_o 45_o gr_n to_o the_o tangent_fw-la of_o the_o ark_n of_o the_o plane_n so_o the_o length_n of_o the_o semidiameter_n to_o the_o length_n of_o the_o tangent_fw-la line_n as_o for_o example_n the_o ark_n of_o the_o plane_n between_o the_o substylar_n and_o the_o hour_n of_o 1_o be_v 15_o gr_n 28_o m._n in_o the_o former_a table_n the_o semidiameter_n cf_n 115_o inch_n and_o the_o semidiameter_n cg_n 100_o inch_n wherefore_o i_o extend_v the_o compass_n from_o the_o tangent_fw-la of_o 45_o gr_n unto_o the_o tangent_fw-la of_o 15_o gr_n 28_o m._n the_o same_o extent_n will_v reach_v from_o 115_o in_o the_o line_n of_o number_n unto_o 31_o 82_o which_o show_v the_o length_n of_o the_o tangent_fw-la line_n between_o f_o in_o the_o substylar_n and_o the_o houre-line_n of_o 1_o to_o be_v 31_o inch_n 82_o cent_n or_o part_n of_o 100_o again_o the_o same_o extent_n will_v reach_v from_o 100_o unto_o 27_o 67_o and_o such_o be_v the_o length_n of_o the_o lesser_a tangent_fw-la from_fw-la g_o to_o the_o hour_n of_o 1._o the_o like_a reason_n hold_v for_o the_o length_n of_o the_o other_o tangent_n from_o the_o substylar_n to_o the_o rest_n of_o the_o hour_n as_o in_o the_o table_n as_o also_o for_o the_o height_n of_o the_o style_n above_o these_o tangent_fw-la line_n and_o so_o the_o angle_n of_o the_o style_n above_o the_o plane_n be_v 3_o gr_n 6_o m._n the_o height_n fk_n will_v be_v find_v to_o be_v 6_o inch_n 23_o cent_n and_o the_o height_n gl_n 5_o inch_n 42_o cent_n where_o the_o reader_n may_v observe_v that_o if_o the_o extent_n from_o the_o tangent_fw-la of_o 45_o gr_n to_o the_o tangent_fw-la of_o 3_o gr_n 6_o m._n or_o to_o 115_o in_o the_o line_n of_o number_n be_v too_o large_a for_o his_o compass_n he_o may_v use_v the_o tangent_fw-la of_o 5_o gr_n 43_o m._n in_o stead_n of_o the_o tangent_fw-la of_o 45_o gr_n as_o i_o note_v before_o pag._n 100_o 4_o have_v find_v these_o length_n and_o height_n and_o set_v they_o down_o in_o a_o table_n i_o come_v to_o the_o plane_n here_o resemble_v by_o the_o lesser_a square_a abde_n where_o i_o begin_v with_o a_o occult_a vertical_a fh_n about_o 12_o inch_n from_o the_o side_n bd_o and_o upon_o the_o centre_n f_o about_o 6_o inch_n below_o the_o side_n ab_fw-la describe_v a_o occult_a ark_n of_o a_o circle_n 5_o into_o this_o ark_n i_o first_o inscribe_v a_o chord_n of_o 38_o gr_n 23_o m._n the_o distance_n of_o the_o substylar_n from_o the_o meridian_n to_o make_v the_o angle_n hfg_v equal_a to_o the_o zct_n so_o the_o line_n fg_v shall_v be_v the_o substylar_n and_o then_o another_o chord_n of_o 51_o gr_n 37_o m._n the_o compliment_n of_o this_o distance_n to_o make_v up_o the_o right_a angle_n gfn_n so_o the_o line_n fn_n shall_v be_v the_o great_a of_o the_o two_o tangent_fw-la line_n before_o mention_v 6_o i_o set_v off_o 15_o inch_n from_o f_o unto_o g_z towards_o the_o centre_n and_o through_o g_o draw_v the_o lesser_a tangent_fw-la line_n gm_n parallel_v to_o the_o former_a 7_o these_o two_o occult_a tangent_fw-la line_n be_v thus_o draw_v i_o look_v unto_o the_o former_a table_n for_o the_o hour_n of_o 1_o and_o there_o find_v the_o ark_n of_o the_o plane_n between_o the_o substylar_n and_o the_o hour_n of_o 1_o to_o be_v 15_o gr_n 28_o m._n and_o the_o length_n belong_v to_o it_o in_o the_o great_a tangent_fw-la line_n to_o be_v 31_o inch_n 82_o cent_n in_o the_o lesser_a tangent_fw-la line_n 27_o inch_n 67_o cent_n wherefore_o i_o take_v out_o 31_o inch_n 82_o part_n and_o prick_v they_o down_o in_o the_o great_a tangent_fw-la from_o f_o to_o n_n and_o then_o 27_o inch_n 67_o part_n and_o prick_v they_o down_o in_o the_o lesser_a tangent_fw-la from_fw-la g_o to_o m_n and_o draw_v the_o mn_a for_o the_o hour_n of_o 1_o which_o if_o it_o be_v produce_v will_v cross_v the_o substylar_a fg_n in_o the_o centre_n c_o and_o there_o make_v the_o angle_n fcn_n 15_o gr_n 28_o m._n the_o like_a reason_n hold_v for_o the_o draw_v of_o all_o the_o rest_n of_o the_o houre-line_n last_o i_o set_v up_o the_o style_n right_o over_o the_o substylar_n so_o as_o the_o height_n fk_n may_v be_v 6_o inch_n 23_o cent_n and_o the_o height_n gl_n 5_o inch_n 42_o cent_n then_o shall_v kl_n represent_v the_o axis_fw-la of_o the_o world_n and_o if_o it_o be_v produce_v will_v cross_v the_o substylar_a fg_n in_o the_o centre_n c_o and_o there_o make_v the_o angle_n fck_n to_o be_v 3_o gr_n 6_o m._n and_o so_o be_v true_o place_v for_o cast_v of_o the_o shadow_n upon_o the_o houre-line_n in_o this_o decline_a plane_n chap._n viii_o to_o draw_v the_o houre-line_n in_o a_o meridian_n incline_v plane_n all_o those_o plane_n wherein_o the_o horizontall_a line_n be_v the_o same_o with_o the_o meridian_n line_n be_v therefore_o call_v meridian_n plane_n if_o they_o be_v right_a to_o the_o horizon_n they_o be_v call_v by_o the_o general_a name_n of_o meridian_n plane_n without_o far_a addition_n and_o be_v describe_v before_o if_o they_o lean_a to_o the_o horizon_n they_o be_v then_o call_v meridian_n incliner_n these_o may_v incline_v either_o to_o the_o east_n part_n of_o the_o horizon_n or_o to_o the_o west_n and_o each_o of_o they_o have_v two_o face_n the_o upper_a towards_o the_o zenith_n the_o low_a towards_o the_o nadir_n wherein_o know_v the_o latitude_n of_o the_o place_n and_o the_o inclination_n of_o the_o plane_n to_o the_o horizon_n we_o be_v to_o consider_v 1_o the_o inclination_n of_o the_o meridian_n of_o the_o plane_n to_o the_o meridian_n of_o the_o place_n 2_o the_o height_n of_o the_o pole_n above_o the_o plane_n 3_o the_o
distance_n of_o the_o substylar_n from_o the_o meridian_n 4_o the_o distance_n of_o each_o houre-line_n from_o the_o substylar_n and_o all_o these_o four_o be_v represent_v in_o the_o fundamental_a diagram_n as_o in_o this_o example_n in_o our_o latitude_n of_o 51_o gr_n 30_o m._n a_o meridian_n plane_n incline_v eastward_o 50_o gr_n these_o 50_o gr_n i_o number_n in_o the_o vertical_a circle_n from_o e_z unto_o g_z according_a to_o the_o inclination_n of_o the_o plane_n and_o there_o draw_v the_o ark_n sgn_v represent_v the_o plane_n propose_v then_o i_o let_v down_o a_o perpendicular_a ark_n pv_o from_o the_o pole_n to_o the_o plane_n serve_v for_o the_o meridian_n of_o the_o plane_n so_o have_v i_o a_o rectangle_n triangle_n pun_v wherein_o the_o base_a pn_n be_v the_o height_n of_o the_o pole_n above_o the_o north_n part_n of_o the_o horizon_n and_o the_o angle_n pnv_n the_o compliment_n of_o the_o inclination_n to_o the_o horizon_n and_o these_o be_v know_v 1_o i_o may_v find_v the_o angle_n npv_n of_o inclination_n of_o the_o two_o meridians_fw-la for_o as_o the_o cousin_n of_o the_o latitude_n be_v to_o the_o sine_fw-la of_o 90_o gr_n so_o the_o tangent_fw-la of_o inclination_n to_o the_o horizon_n to_o the_o tangent_fw-la of_o inclination_n of_o meridians_fw-la extend_v the_o compass_n from_o the_o sine_fw-la of_o 38_o gr_n 30_o m._n the_o compliment_n of_o the_o latitude_n unto_o the_o sine_fw-la of_o 90_o gr_n the_o same_o extent_n will_v reach_v from_o the_o tangent_fw-la of_o 50_o gr_n 0_o m._n the_o inclination_n of_o the_o plane_n to_o the_o horizon_n unto_o the_o tangent_fw-la of_o 62_o gr_n 25_o m._n and_o such_o be_v the_o inclination_n of_o the_o meridian_n of_o the_o plane_n to_o the_o meridian_n of_o the_o place_n which_o be_v resolve_v into_o time_n do_v give_v about_o 4_o hour_n and_o 10_o m._n from_o the_o meridian_n for_o the_o place_n of_o the_o substylar_n among_o the_o houre-line_n 2_o the_o height_n of_o the_o pole_n above_o the_o plane_n be_v here_o represent_v by_o the_o quantity_n of_o the_o ark_n of_o the_o proper_a meridian_n pv_n between_o the_o pole_n and_o the_o plane_n and_o may_v be_v know_v by_o that_o which_o we_o have_v give_v in_o the_o former_a triangle_n pun_v for_o as_o the_o sine_fw-la of_o 90_o gr_n to_o the_o sine_fw-la of_o the_o latitude_n so_o the_o cousin_n of_o the_o inclination_n to_o the_o horizon_n to_o the_o sine_fw-la of_o the_o height_n of_o the_o pole_n above_o the_o plane_n extend_v the_o compass_n from_o the_o sine_fw-la of_o 90_o gr_n unto_o 51_o gr_n 30_o m._n the_o sine_fw-la of_o the_o latitude_n the_o same_o extent_n will_v reach_v from_o the_o sine_fw-la of_o 40_o gr_n the_o compliment_n of_o the_o inclination_n of_o the_o plane_n to_o the_o horizon_n unto_o the_o sine_fw-la of_o 30_o gr_n 12_o m._n or_o as_o the_o sine_fw-la of_o 90_o gr_n to_o the_o cousin_n of_o inclination_n of_o meridians_fw-la so_o the_o tangent_fw-la of_o the_o latitude_n to_o the_o tangent_fw-la of_o the_o height_n of_o the_o pole_n above_o the_o plane_n extend_v the_o compass_n from_o the_o sine_fw-la of_o 90_o gr_n unto_o the_o tangent_fw-la of_o 51_o gr_n 30_o m._n the_o latitude_n of_o the_o place_n the_o same_o extent_n will_v reach_v from_o the_o sine_fw-la of_o 27_o gr_n 35_o m._n the_o compliment_n of_o the_o inclination_n of_o the_o two_o meridians_fw-la unto_o the_o tangent_fw-la of_o 30_o gr_n 12_o m._n and_o such_o be_v pv_o the_o height_n of_o the_o pole_n above_o the_o plane_n and_o such_o must_v be_v the_o height_n of_o the_o style_n above_o the_o substylar_n 3_o the_o distance_n of_o the_o substylar_n from_o the_o meridian_n be_v here_o represent_v by_o nv_n the_o ark_n of_o the_o plane_n between_o the_o two_o meridians_fw-la and_o may_v be_v find_v by_o that_o which_o we_o have_v give_v at_o the_o first_o in_o the_o former_a triangle_n pun_v for_o as_o the_o sine_fw-la of_o 90_o gr_n to_o the_o sine_fw-la of_o the_o inclination_n to_o the_o horizon_n so_o the_o tangent_fw-la of_o the_o latitude_n to_o the_o tangent_fw-la of_o the_o substylar_n from_o the_o meridian_n extend_v the_o compass_n from_o the_o sine_fw-la of_o 90_o gr_n unto_o the_o tangent_fw-la of_o 51_o gr_n 30_o m._n the_o latitude_n of_o the_o place_n the_o same_o extent_n will_v reach_v from_o the_o sine_fw-la of_o 50_o gr_n the_o inclination_n of_o the_o plane_n to_o the_o horizon_n unto_o 43_o gr_n 55_o m._n and_o such_o be_v the_o ark_n nv_n the_o distance_n of_o the_o substylar_n from_o the_o meridian_n 4_o the_o distance_n of_o the_o houre-line_n from_o the_o substylar_n be_v here_o also_o represent_v by_o those_o arke_n of_o the_o plane_n which_o be_v here_o intercept_v between_o the_o proper_a meridian_n and_o the_o houre-circle_n and_o may_v be_v find_v by_o that_o which_o we_o have_v give_v in_o the_o triangle_n make_v by_o the_o plane_n with_o his_o proper_a meridian_n and_o the_o houre-circle_n for_o the_o angle_n at_o five_o between_o the_o plane_n and_o the_o proper_a meridian_n be_v well_o know_v to_o be_v a_o right_a angle_n and_o the_o side_n pv_n be_v the_o height_n of_o the_o pole_n above_o the_o plane_n and_o the_o angle_n at_o the_o pole_n between_o the_o proper_a meridian_n and_o the_o houre-circle_n be_v easy_o gather_v into_o a_o table_n the_o angle_n vpn_n between_o up_o the_o proper_a meridian_n of_o the_o plane_n and_o pn_n the_o general_a meridian_n of_o the_o place_n be_v 62_o gr_n 25_o m._n the_o angle_n between_o the_o proper_a meridian_n and_o the_o circle_n of_o the_o hour_n of_o 11_o will_v be_v 77_o gr_n 25_o m._n and_o the_o angle_n belong_v to_o the_o hour_n of_o 1_o 47_o gr_n 25_o m_o and_o so_o the_o rest_n of_o the_o angle_n at_o the_o pole_n then_o as_o the_o sine_fw-la of_o 90_o gr_n to_o the_o sine_fw-la of_o the_o pole_n above_o the_o plane_n so_o the_o tangent_fw-la of_o the_o angle_n at_o the_o pole_n to_o the_o tangent_fw-la of_o the_o houre-line_n from_o the_o substylar_n hor._n ang._n po._n arc._n pla._n gr._n m._n gr._n m._n 11_o 77_o 25_o 66_o 4_o 12_o 62_o 25_o 43_o 55_o 1_o 47_o 25_o 28_o 41_o 2_o 32_o 25_o 17_o 43_o 3_o 17_o 25_o 8_o 58_o 4_o 2_o 25_o 1_o 13_o  _fw-fr merid_n substyl_o 5_o 12_o 35_o 6_o 26_o 6_o 27_o 35_o 14_o 44_o 7_o 42_o 35_o 24_o 48_o 8_o 57_o 35_o 38_o 23_o 9_o 72_o 35_o 58_o 3_o 10_o 87_o 35_o 85_o 12_o wherefore_o i_o extend_v the_o compass_n from_o the_o sine_fw-la of_o 90_o gr_n unto_o the_o sine_fw-la of_o 30_o gr_n 12_o m._n the_o height_n of_o the_o pole_n above_o the_o plane_n and_o i_o find_v the_o same_o extent_n to_o reach_v in_o the_o line_n of_o tangent_n from_o 77_o gr_fw-la 25_o m._n unto_o 66_o gr_n 4_o m._n for_o the_o distance_n belong_v to_o the_o hour_n of_o 11_o and_o from_o the_o tangent_fw-la of_o 62_o gr_n 25_o m._n to_o 43_o gr_n 55_o m._n for_o the_o hour_n of_o 12._o as_o when_o i_o find_v the_o distance_n of_o the_o substylar_n from_o the_o meridian_n and_o so_o for_o the_o rest_n of_o the_o ark_n of_o plane_n between_o 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an._n po_n arc._n pla._n an._n equ_n c_o ♈_o c_o ♋_o c_o ♑_o gr._n m._n gr._n m._n gr._n m._n in._n p._n in._n p._n in._n p._n substy_n 0_o 0_o 0_o 0_o 55_o 27_o 14_o 57_o 20_o 80_o 11_o 21_o 1_o 3_o 15_o 0_o 8_o 38_o 54_o 30_o 14_o 74_o 21_o 36_o 11_o 25_o 12_o 4_o 30_o 0_o 18_o 8_o 51_o 30_o 15_o 33_o 23_o 44_o 11_o 40_o 11_o 5_o 45_o 0_o 29_o 33_o 45_o 45_o 16_o 75_o 29_o 06_o 11_o 76_o 10_o 6_o 60_o 0_o 44_o 30_o 36_o 0_o 20_o 00_o 50_o 84_o 12_o 77_o 9_o 7_o 75_o 0_o 64_o 42_o 20_o 36_o 34_o 10_o infin_fw-fr 15_o 82_o 8_o 8_o 90_o 0_o 90_o 0_o 0_o 0_o infinite_a  _fw-fr 27_o 60_o if_o then_o it_o be_v require_v to_o find_v the_o angle_n which_o the_o houre-line_n of_o 4_o after_o noon_n do_v make_v with_o the_o plane_n of_o the_o equator_fw-la that_o be_v the_o angle_n c_o 4_o b_o contain_v between_o the_o houre-line_n c_o 4_o and_z the_o line_n b_o 4_o draw_v from_o the_o top_n of_o the_o style_n unto_o the_o intersection_n of_o the_o houre-line_n of_o 4_o with_o the_o 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and_o through_o 9_o ho._n in_o the_o parallel_n of_o 9_o and_o so_o in_o the_o rest_n as_o in_o this_o table_n horae_n 15_o aeq_n 9_o ho._n m._n ho_o ho._n m._n 0_o 4_o 30_o 6_o 7_o 30_o 1_o 5_o 45_o 7_o 8_o 15_o 2_o 7_o 0_o 8_o 9_o 0_o 3_o 8_o 15_o 9_o 9_o 45_o 4_o 9_o 30_o 10_o 10_o 30_o 5_o 10_o 45_o 11_o 11_o 15_o 6_o 12_o 0_o 12_o 12_o 0_o 7_o 1_o 15_o 1_o 0_o 45_o 8_o 2_o 30_o 2_o 1_o 30_o 9_o 3_o 45_o 3_o 2_o 15_o 10_o 5_o 0_o 4_o 3_o 0_o 11_o 6_o 15_o 5_o 3_o 45_o 12_o 7_o 30_o 6_o 4_o 30_o and_o of_o these_o unequal_a hour_n you_o have_v a_o far_a example_n in_o the_o diagram_n belong_v to_o the_o polar_n decline_a plane_n pag._n 130._o chap._n xvii_o to_o draw_v the_o hour_n from_o sun_n rise_v and_o sun_n set_v in_o the_o former_a plane_n to_o know_v how_o many_o hour_n be_v pass_v since_o the_o sun_n rise_v or_o how_o many_o remain_n to_o the_o sun_n set_v first_o draw_v the_o common_a houre-line_n the_o equator_fw-la and_o the_o tropique_n as_o before_o then_o describe_v two_o occult_a parallel_n of_o the_o length_n of_o the_o day_n one_o for_o 8_o hour_n and_o the_o other_o for_o 16_o hour_n for_o so_o you_o may_v draw_v the_o first_o hour_n from_o the_o sun_n rise_v through_o the_o common_a hour_n of_o 5_o in_o the_o parallel_n of_o 16_o of_o 7_o in_o the_o equator_fw-la and_o of_o 9_o in_o the_o parallel_n of_o 8._o in_o like_a manner_n the_o second_o hour_n from_o sun_n rise_v through_o the_o common_a hour_n of_o 6_o in_o the_o parallel_n of_o 16_o of_o 8_o in_o the_o equator_fw-la and_o of_o 10_o in_o the_o parallel_n of_o 8._o and_o so_o the_o rest_n in_o their_o order_n the_o first_o hour_n before_o sun_n set_v or_o the_o 23_o hour_n from_o the_o last_o sun_n set_v may_v be_v draw_v in_o like_a sort_n through_o the_o common_a hour_n of_o 3_o after_o noon_n in_o the_o parallel_n of_o 8_o of_o 5_o in_o the_o equator_fw-la and_o of_o 7_o in_o the_o parallel_n of_o 16._o the_o second_o hour_n before_o sun_n set_v or_o the_o 22_o hour_n after_o the_o 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