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 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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 ♌_o and_o the_o other_o intermediate_v degree_n shall_v be_v the_o centre_n to_o divide_v each_o sign_n into_o 30_o gr_n if_o far_o we_o have_v respect_n unto_o the_o latitude_n we_o may_v the_o meridian_n be_v before_o divide_v number_v it_o from_o p_o northward_o unto_o h_n and_o there_o place_v the_o north_n intersection_n of_o the_o meridian_n and_o horizon_n then_o the_o compliment_n of_o the_o latitude_n be_v number_v from_o p_o southward_o unto_o z_o shall_v there_o give_v the_o zenith_n and_o 90_o gr_n from_o z_o southward_n unto_o f_o shall_v there_o give_v the_o south_n intersection_n of_o the_o meridian_n and_o horizon_n the_o middle_n between_o f_o and_o h_n shall_v be_v g_z the_o centre_n of_o the_o horizon_n ♈_o h_o ♎_o f_o pass_v through_o the_o begin_n of_o ♈_o and_o ♎_o unless_o there_o be_v some_o former_a error_n all_o parallel_n to_o the_o horizon_n may_v be_v find_v in_o like_a sort_n by_o their_o intersection_n with_o the_o meridian_n and_o the_o middle_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
in_o the_o general_a use_n of_o the_o sector_n concern_v lateral_a and_o parallel_a entrance_n it_o may_v suffice_v only_o to_o set_v down_o the_o proportion_n of_o the_o three_o part_n give_v to_o the_o four_o require_v and_o so_o i_o show_v first_o by_o the_o sin_n alone_o in_o a_o rectangle_n triangle_n 1_o to_o find_v a_o side_n by_o know_v the_o base_a and_o the_o angle_n opposite_a to_o the_o require_a side_n as_o the_o radius_fw-la be_v to_o the_o sine_fw-la of_o the_o base_a so_o the_o sine_fw-la of_o the_o opposite_a angle_n to_o the_o sine_fw-la of_o the_o side_n require_v as_o in_o the_o rectangle_n acb_n have_v the_o base_a ab_fw-la the_o place_n of_o the_o sun_n 30_o gr_n from_o the_o equinoctial_a point_n and_o the_o angle_n bac_n of_o 23_o gr_n 30_o m._n the_o great_a declination_n if_o it_o be_v require_v to_o find_v the_o side_n bc_n the_o declination_n of_o the_o sun_n take_v either_o the_o lateral_a sine_fw-la of_o 23_o gr_n 30_o m._n and_o make_v it_o a_o parallel_n 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and_o cab_n the_o second_o the_o side_n bc_n will_v be_v find_v 11_o gr_n 30_o m._n 4_o to_o find_v the_o base_a by_o know_v both_o the_o side_n as_o the_o radius_fw-la to_o the_o sine_fw-la of_o the_o compliment_n of_o the_o one_o side_n so_o the_o sine_fw-la of_o the_o compliment_n of_o the_o other_o side_n to_z the_o sine_z of_o the_o compliment_n of_o the_o base_a require_v so_o in_o the_o rectangle_n acb_n have_v ac_fw-la 27_o gr_fw-la 54_o m._n and_o bc_n 11_o gr_n 30_o m._n the_o base_a ab_fw-la will_v be_v find_v 30_o gr_n 5_o to_o find_v the_o base_a by_o know_v the_o one_o side_n and_o the_o angle_n opposite_a to_o that_o side_n as_o the_o sine_fw-la of_o the_o angle_n give_v to_o the_o sine_fw-la of_o the_o side_n give_v so_o be_v the_o radius_fw-la to_o the_o sine_fw-la of_o the_o base_a require_v so_o in_o the_o rectangle_n bcd_n know_v the_o latitude_n and_o the_o declination_n we_o may_v find_v the_o amplitude_n as_o have_v bc_n the_o side_n of_o the_o declination_n 11_o gr_n 30_o m._n and_o bdc_a the_o angle_n of_o the_o compliment_n of_o the_o latitude_n 38_o gr_n 15_o m._n the_o base_a bd_o which_o be_v the_o amplitude_n will_v be_v find_v to_o be_v 18_o gr_n 47_o m._n 6_o to_o find_v a_o angle_n by_o the_o other_o oblique_a angle_n and_o the_o side_n opposite_a to_o the_o inquire_v angle_n as_o the_o radius_fw-la to_o the_o sine_fw-la of_o the_o compliment_n of_o the_o side_n so_o the_o sine_fw-la of_o the_o angle_n give_v to_o the_o sine_fw-la of_o the_o compliment_n of_o the_o angle_n require_v so_o in_o the_o rectangle_n acb_n have_v the_o angle_n bac_n 23_o gr_n 30_o m._n and_o the_o side_n ac_fw-la 27_o gr_fw-la 54_o m._n the_o angle_n abc_n will_v be_v find_v 69_o gr_n 21_o m._n 7_o to_o find_v a_o angle_n by_o the_o other_o oblique_a angle_n and_o the_o side_n opposite_a to_o the_o angle_n give_v as_o the_o sine_fw-la of_o the_o compliment_n of_o the_o side_n to_o the_o side_n of_o the_o compliment_n of_o the_o angle_n give_v so_o be_v the_o radius_fw-la to_o the_o sine_fw-la of_o the_o angle_n require_v so_o in_o the_o rectangle_n acb_n have_v bac_n 23_o gr_n 30_o m._n and_o bc_n 11_o gr_n 30_o m._n the_o angle_n abc_n will_v be_v find_v 69_o gr_n 21_o m._n 8_o to_o find_v a_o angle_n by_o the_o base_a and_o the_o side_n opposite_a to_o the_o inquire_v angle_n as_o the_o sine_fw-la of_o the_o base_a be_v to_o the_o radius_fw-la so_o the_o sine_fw-la of_o the_o side_n to_o the_o sine_fw-la of_o the_o angle_n require_v so_o in_o the_o rectangle_n bcd_n have_v bd_o 18_o gr_n 47_o m._n and_o bc_n 11_o gr_n 30_o m._n the_o angle_n bdc_n will_v be_v find_v 38_o gr_n 15_o m._n these_o eight_o proposition_n have_v be_v wrought_v by_o the_o sin_n alone_o those_o which_o follow_v require_v joint_a help_n of_o the_o tangent_fw-la and_o forasmuch_o as_o the_o tangent_fw-la can_v not_o well_o be_v extend_v beyond_o 63_o gr_n 30_o m._n i_o shall_v set_v down_o two_o way_n for_o the_o resolution_n of_o each_o proposition_n if_o the_o one_o will_v not_o hold_v the_o other_o may_v 9_o to_o find_v a_o side_n by_o have_v the_o other_o side_n and_o the_o angle_n opposite_a to_o the_o inquire_v sine_fw-la 1_o as_o the_o radius_fw-la to_o the_o sine_fw-la of_o the_o side_n give_v so_o the_o tangent_fw-la of_o the_o angle_n to_o the_o tangent_fw-la of_o the_o side_n require_v 2_o as_o the_o sine_fw-la of_o the_o side_n give_v be_v to_o the_o radius_fw-la so_o the_o tangent_fw-la of_o the_o compliment_n of_o the_o angle_n to_o the_o tangent_fw-la of_o the_o compliment_n of_o the_o side_n require_v so_o in_o the_o rectangle_n acb_n have_v the_o right_a side_n ac_fw-la 27_o gr_fw-la 54_o m_o and_o the_o angle_n bac_n 23_o gr_n 30_o m._n the_o side_n bc_n will_v be_v sound_a to_o be_v 11_o gr_n 30_o m._n 10_o to_o find_v a_o side_n by_o have_v the_o other_o side_n and_o the_o angle_n adjacent_a next_o to_o the_o inquire_v side_n 1_o as_o the_o tangent_fw-la of_o the_o angle_n to_o the_o tangent_fw-la of_o the_o side_n give_v 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20_o 25_o 30_o mo_z gr._n m_o gr._n m._n gr._n m_o gr._n m._n gr._n m._n gr._n m_o gr._n m._n january_n 16_o 31_o 17_o 24_o 18_o 26_o 19_o 37_o 20_o 57_o 22_o 24_o 23_o 58_o february_n 24_o 17_o 25_o 59_o 27_o 45_o 29_o 35_o 31_o 29_o 33_o 25_o  _fw-fr march_n 34_o 35_o 36_o 33_o 38_o 32_o 40_o 30_o 42_o 27_o 44_o 22_o 46_o 15_o april_n 46_o 37_o 48_o 26_o 50_o 11_o 51_o 50_o 53_o 25_o 54_o 53_o 56_o 15_o may_n 56_o 15_o 57_o 29_o 58_o 35_o 59_o 33_o 60_o 22_o 61_o 2_o 61_o 31_o june_n 61_o 36_o 61_o 54_o 62_o 0_o 61_o 58_o 61_o 45_o 61_o 22_o 60_o 49_o july_n 60_o 49_o 60_o 6_o 59_o 14_o 58_o 13_o 57_o 4_o 55_o 48_o 54_o 24_o august_n 54_o 7_o 52_o 36_o 50_o 59_o 49_o 17_o 47_o 31_o 45_o 41_o 43_o 49_o september_n 43_o 26_o 41_o 30_o 39_o 33_o 37_o 36_o 35_o 38_o 33_o 41_o 31_o 46_o october_n 31_o 46_o 29_o 53_o 28_o 3_o 26_o 16_o 24_o 35_o 22_o 59_o 21_o 29_o november_n 21_o 12_o 19_o 51_o 18_o 39_o 17_o 36_o 16_o 43_o 16_o 0_o 15_o 28_o december_n 15_o 28_o 15_o 7_o 15_o 0_o 15_o 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l_o these_o question_n well_o consider_v and_o either_o resolve_v by_o the_o staff_n or_o prick_v down_o on_o the_o chart_n and_o compare_v with_o the_o globe_n and_o the_o common_a sea-chart_n will_v give_v some_o light_n to_o the_o direction_n of_o a_o course_n and_o reduction_n of_o place_n to_o their_o due_a longitude_n which_o be_v now_o foul_o distort_v in_o the_o common_a sea-charts_a a_o appendix_n concern_v the_o description_n and_o use_n of_o a_o instrument_n make_v in_o form_n of_o a_o crossbow_n for_o the_o more_o easy_a find_n of_o the_o latitude_n at_o sea_n the_o former_a prop._n suppose_v the_o latitude_n to_o be_v know_v i_o will_v here_o show_v how_o it_o may_v be_v easy_o observe_v upon_o the_o centre_n a_o and_o semidiameter_n ab_fw-la describe_v a_o ark_n of_o a_o circle_n sbn._n the_o same_o semidiameter_n will_v set_v of_o 60_o gr_n from_z b_o unto_o s_n for_o the_o south_n end_n and_o other_o 60_o gr_n from_z b_o unto_o n_n for_o the_o north_n end_n of_o the_o 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the_o hour_n of_o 4_o after_o noon_n be_v find_v before_o to_o be_v 51_o gr_n 30_o m_o if_o you_o will_v find_v the_o distance_n between_o the_o centre_n and_o the_o equator_fw-la extend_v the_o compass_n from_o the_o sine_fw-la of_o 51_o gr_n 30_o m._n unto_o the_o sine_fw-la of_o 90_o gr_n the_o compliment_n of_o the_o declination_n the_o same_o extent_n will_v reach_v in_o the_o line_n of_o number_n from_z 12_o unto_o 15.33_o and_o such_o be_v the_o distance_n upon_o the_o houre-line_n of_o 4_o between_o the_o centre_n and_o the_o equator_fw-la if_o you_o will_v find_v the_o distance_n upon_o this_o houre-line_n between_o the_o centre_n and_o the_o inner_a tropic_n who_o declination_n be_v know_v to_o be_v 23_o gr_n 30_o m._n add_v the_o declination_n to_o the_o angle_n at_o the_o equator_fw-la so_o the_o angle_n at_o the_o parallel_n will_v be_v 75_o gr_n wherefore_o extend_v the_o compass_n from_o the_o sine_fw-la of_o 75_o gr_n unto_o the_o sine_fw-la of_o 66_o gr_n 30_o m._n 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unto_o ♋_o and_o you_o shall_v have_v the_o same_o intersection_n of_o the_o tropic_n and_o the_o substylar_n as_o before_o &_o the_o like_a reason_n hold_v for_o prick_v down_o of_o the_o rest_n of_o these_o distance_n on_o their_o several_a houre-line_n then_o have_v the_o point_n of_o intersection_n between_o the_o houre-line_n and_o the_o parallel_n you_o may_v join_v they_o all_o in_o a_o crooked_a line_n without_o make_v of_o any_o angle_n the_o line_n so_o draw_v shall_v be_v the_o tropic_n require_v and_o after_o this_o manner_n may_v you_o draw_v any_o other_o parallel_n of_o declination_n whereof_o you_o have_v example_n in_o the_o most_o of_o the_o former_a diagram_n chap._n xiiii_o to_o describe_v the_o parallel_n of_o the_o sign_n in_o any_o of_o the_o former_a plane_n the_o equator_fw-la and_o the_o tropique_n before_o describe_v do_v show_v the_o sun_n entrance_n into_o 4_o of_o the_o sign_n the_o equator_fw-la into_o ♈_o and_o ♎_o the_o one_o tropic_n into_o ♋_o and_o the_o other_o into_o ♑_o the_o rest_n of_o the_o intermediate_v sign_n will_v be_v describe_v in_o the_o same_o manner_n as_o the_o tropique_n if_o first_o we_o know_v their_o declination_n the_o manner_n of_o find_v the_o declination_n not_o only_o of_o the_o begin_n of_o the_o sign_n but_o of_o all_o other_o point_n of_o the_o ecliptic_a be_v before_o set_v down_o in_o 2._o prop._n astronomical_a pag._n 52._o by_o which_o you_o may_v find_v the_o declination_n of_o the_o begin_n of_o ♉_o â™_o â™_o and_o ♓_o to_o be_z 11_o gr_n 30_o m._n and_o of_o ♊_o ♌_o â™_o and_o â™’_o to_o be_v 20_o gr_n 12_o m._n if_o then_o you_o inscribe_v the_o chord_n of_o 11_o gr_n 30_o m._n and_o of_o 20_o gr_n 12_o m._n into_o the_o former_a figure_n bdt_n pag._n 145._o from_o d_o towards_o t_n the_o line_n draw_v from_o b_o through_o the_o term_n of_o those_o chord_n shall_v be_v the_o sign_n require_v and_o with_o these_o declination_n the_o height_n of_o the_o style_n and_o the_o length_n of_o the_o axis_fw-la you_o may_v find_v the_o angle_n at_o the_o parallel_n and_o then_o the_o distance_n between_o the_o centre_n and_o the_o parallel_n which_o be_v prick_v down_o upon_o their_o several_a houre-line_n shall_v give_v you_o the_o point_n of_o intersection_n by_o which_o you_o may_v draw_v the_o parallel_n of_o the_o sign_n as_o in_o the_o figure_n belong_v to_o the_o polar_n plane_n chap._n xv._o to_o describe_v the_o parallel_n of_o the_o length_n of_o the_o day_n in_o any_o of_o the_o former_a plane_n the_o length_n of_o the_o day_n will_v always_o be_v 12_o hour_n long_o when_o the_o sun_n come_v to_o be_v in_o the_o equator_fw-la and_o this_o hold_v in_o all_o latitude_n but_o at_o other_o time_n of_o the_o year_n the_o same_o place_n of_o the_o sun_n will_v not_o give_v the_o same_o length_n of_o the_o day_n in_o another_o latitude_n wherefore_o the_o latitude_n be_v know_v we_o be_v first_o to_o find_v the_o declination_n of_o the_o sun_n agree_v to_o the_o length_n of_o the_o day_n consider_v the_o difference_n between_o the_o length_n of_o a_o equinoctial_a day_n and_o the_o day_n propose_v and_o turn_v the_o time_n into_o degree_n and_o minute_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 half_a the_o difference_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 declination_n if_o then_o you_o inscribe_v the_o chord_n of_o these_o ark_n into_o the_o former_a figure_n bdt_n the_o line_n draw_v from_o b_o through_o the_o term_n of_o these_o ark_n shall_v be_v the_o line_n belong_v to_o the_o diurnal_a arke_n and_o the_o several_a distance_n between_o they_o and_o the_o point_n c_o give_v the_o like_a distance_n between_o the_o centre_n and_o the_o parallel_n of_o the_o length_n of_o the_o day_n upon_o the_o houre-line_n in_o your_o plane_n chap._n xvi_o to_o draw_v the_o old_a unequal_a hour_n in_o the_o former_a plane_n it_o be_v the_o manner_n of_o the_o ancient_n to_o divide_v the_o day_n into_o twelve_o equal_a hour_n and_o the_o night_n into_o twelve_o other_o equal_a hour_n and_o so_o the_o whole_a day_n and_o night_n into_o 24_o hour_n of_o these_o 24_o those_o which_o belong_v unto_o the_o day_n be_v either_o long_o or_o short_a except_v the_o two_o equinoctial_a day_n than_o those_o which_o belong_v unto_o the_o night_n and_o the_o summer_n hour_n always_o long_a than_o the_o hour_n in_o the_o winter_n according_a to_o the_o lengthen_v of_o the_o day_n whereupon_o they_o be_v call_v the_o old_a unequal_a and_o by_o some_o the_o planetary_a hour_n to_o express_v these_o in_o the_o former_a plane_n 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 9_o hour_n the_o other_o for_o 15_o hour_n for_o so_o you_o may_v draw_v a_o straight_a line_n for_o the_o first_o unequal_a hour_n through_o 5_o ho._n 45_o m._n in_o the_o parallel_n of_o 15_o and_o through_o 8_o ho._n 15_o m._n in_o the_o parallel_n of_o 9_o this_o straight_a line_n shall_v pass_v direct_o through_o 7_o ho._n 0_o m._n in_o the_o equator_fw-la and_o so_o cut_v off_o a_o twelve_o part_n of_o the_o ark_n above_o the_o horizon_n both_o from_o these_o two_o parallel_n and_o the_o equator_fw-la and_o be_v continue_v unto_o the_o tropique_n it_o shall_v also_o cut_v off_o about_o a_o twelve_o part_n from_o they_o and_o all_o the_o rest_n of_o the_o parallel_n of_o declination_n without_o any_o sensible_a error_n in_o like_a manner_n may_v you_o draw_v the_o second_o unequal_a hour_n through_o 7_o ho._n in_o the_o parallel_n of_o 15_o through_z 8_o ho._n in_o the_o equator_fw-la