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be_v 51_o degr_n 30_o min._n there_o the_o aequinoctial_a be_v elevate_v 38_o degr_n 30_o min._n which_o be_v so_o much_o as_o 51_o degr_n 30_o min._n want_v of_o 90_o degr_n and_o it_o cut_v the_o horizon_n in_o the_o point_n of_o east_n and_o west_n unto_o this_o circle_n when_o the_o sun_n come_v which_o be_v twice_o every_o year_n namely_o about_o the_o 10._o of_o march_n and_o the_o 12._o of_o september_n it_o cause_v the_o day_n and_o night_n to_o be_v of_o equal_a length_n all_o the_o world_n over_o this_o circle_n be_v note_v in_o the_o scheme_n with_o ae_n a_o ae_z and_o cut_v the_o horizon_n in_o the_o point_n a_o which_o represent_v both_o the_o east_n and_o west_n point_v thereof_o iv_o of_o the_o ecliptic_a this_o also_o be_v a_o great_a circle_n of_o the_o sphere_n and_o in_o the_o northern_a hemisphere_n where_o the_o north_n pole_n be_v visible_a above_o the_o horizon_n and_o the_o south_n pole_n not_o visible_a be_v elevate_v above_o the_o aequinoctial_a circle_n so_o 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circle_n thus_o note_v z_o o_o n_o z_o f_o n_o z_o a_o n_o and_z z_o g_o n._n viii_o of_o the_o tropic_n the_o tropic_n be_v lesser_a circle_n of_o the_o sphere_n divide_v it_o unequal_o and_o be_v draw_v parallel_n to_o the_o aequinoctial_a at_o 23_o degr_n 30_o min._n distance_n therefrom_o equal_a to_o the_o sun_n be_v great_a declination_n on_o either_o side_n that_o tropic_n which_o be_v on_o the_o north-side_n be_v call_v the_o tropic_a of_o cancer_n to_o which_o when_o the_o sun_n come_v which_o be_v but_o once_o in_o a_o year_n about_o the_o 10._o of_o june_n it_o make_v the_o long_a day_n to_o all_o the_o northern_a inhabitant_n of_o the_o world_n and_o the_o short_a night_n the_o other_o tropic_a which_o be_v on_o the_o southside_n of_o the_o aequinoctial_a be_v call_v the_o tropic_a of_o capricorn_n to_o which_o when_o the_o sun_n come_v which_o be_v about_o the_o 11._o of_o december_n it_o make_v the_o short_a day_n and_o the_o long_a night_n to_o all_o northern_a 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the_o sphere_n both_o great_a and_o small_a have_v his_o proper_a pole_n which_o pole_n of_o all_o the_o great_a circle_n be_v 90_o degree_n or_o a_o quadrant_n of_o a_o circle_n distant_a from_o the_o circle_n itself_o the_o pole_n of_o the_o circle_n in_o this_o projection_n be_v as_o follow_v z_o and_z n_n be_v the_o pole_n of_o h_n a_o o_o the_o horizon_n p_o and_z s_n be_v the_o pole_n of_o ae_n a_o ae_z the_o aequinoctial_a o_o and_o h_n be_v the_o pole_n of_o z_o a_o n_o the_o prime_n vertical_a q_n and_o r_n be_v the_o pole_n of_o the_o ecliptic_a ae_n and_o ae_z be_v the_o pole_n of_o p_o a_o s_o the_o axis_n of_o the_o world_n the_o pole_n of_o these_o five_o circle_n be_v all_o in_o the_o meridian_n and_o so_o there_o need_v no_o far_a precept_n for_o the_o find_n of_o they_o and_o the_o pole_n of_o the_o meridian_n be_v the_o centre_n thereof_o but_o for_o the_o three_o azimuth_n circle_n they_o fall_v in_o several_a point_n of_o the_o horizon_n and_o the_o three_o hour-circle_n in_o certain_a point_n in_o the_o aequinoctial_a how_o to_o find_v which_o point_n shall_v be_v show_v afterward_o in_o due_a place_n a_o be_v the_o pole_n of_o the_o meridian_n z_o h_o n_o o._n t_o be_v the_o pole_n of_o the_o azimuth_n circle_n z_o f_o n._n g_o be_v the_o pole_n of_o the_o azimuth_n circle_n z_o ☉_o n._n ☉_o be_v the_o pole_n of_o the_o azimuth_n circle_n z_o g_o n._n x_o be_v the_o pole_n of_o the_o hour-circle_n p_o b_o s_o you_o be_v the_o pole_n of_o the_o hour-circle_n p_o d_o s_o v_o be_v the_o pole_n of_o the_o hour-circle_n p_o c_o s_o the_o pole_n of_o the_o world_n p_o and_o s_o be_v also_o the_o pole_n of_o the_o
tropic_n and_o of_o all_o the_o parallel_n of_o declination_n and_o the_o zenith_n and_o nadir_n z_o and_o n_o be_v the_o pole_n of_o all_o the_o parallel_n of_o altitude_n have_v sufficient_o acquaint_v the_o reader_n with_o the_o several_a circle_n line_n point_n and_o pole_n belong_v to_o every_o circle_n i_o will_v now_o proceed_v to_o my_o intend_a purpose_n namely_o to_o project_v or_o lay_v down_o in_o plano_n all_o these_o circle_n line_n point_n and_o pole_n in_o their_o true_a position_n how_o to_o project_v the_o sphere_n upon_o the_o plain_a of_o the_o meridian_n first_o take_v 60_o degr_n of_o your_o line_n of_o chord_n and_o with_o that_o distance_n upon_o the_o point_n a._n as_o a_o centre_n describe_v the_o circle_n z_o h_o n_o o_o represent_v the_o meridian_n within_o which_o circle_n all_o the_o rest_n be_v to_o be_v project_v and_o cross_v it_o with_o the_o two_o diameter_n h_n a_o o_o the_o horizon_n and_o z_o a_o n_o the_o prime_n vertical_a second_o because_o the_o latitude_n of_o the_o place_n for_o which_o you_o draw_v your_o projection_n viz._n london_n be_v 51_o degr_n 30_o min._n take_v 51_o degr_n 30_o min._n from_o your_o line_n of_o chord_n and_o set_v they_o upon_o the_o meridian_n from_o z_o to_z ae_n and_o from_o n_n to_o ae_z and_o draw_v the_o line_n ae_n a_o ae_z for_o the_o aequinoctial_a also_o set_v 51_o degr_n 30_o m._n from_o o_o to_o p_o and_o from_o h_n to_o s_n and_o draw_v the_o line_n p_o a_o s_o represent_v the_o axis_n of_o the_o world_n and_o the_o hour-circle_n of_o 6_o a_o clock_n three_o take_v 23_o degr_n 30_o min._n the_o quantity_n of_o the_o sun_n be_v great_a declination_n and_o also_o of_o the_o distance_n of_o the_o two_o tropic_n from_o the_o aequinoctial_a and_o set_v they_o upon_o the_o meridian_n from_o ae_n to_o ♋_o above_o the_o aequinoctial_a and_o also_o from_o ae_n to_o ♑_o below_o the_o aequinoctial_a in_o like_a manner_n set_v the_o same_o distance_n of_o 23_o degr_n 30_o min._n from_o ae_z to_o ♋_o above_o the_o aequinoctial_a and_o from_o ae_z to_o ♑_o below_o it_o this_o do_v lay_v a_o ruler_n upon_o the_o point_n ae_n and_o ♋_o and_o it_o will_v cut_v the_o axis_n of_o the_o world_n p_o a_o s_o in_o the_o point_n i._o so_o a_o circle_n draw_v which_o shall_v pass_v through_o these_o three_o point_n ♋_o i_o ♋_o shall_v be_v the_o tropic_a of_o cancer_n again_o lay_v a_o ruler_n to_o ae_n and_o ♑_o and_o it_o will_v cut_v the_o axis_n in_o the_o point_n k._n so_o a_o circle_n draw_v through_o ♑_o k_n ♑_o shall_v be_v the_o tropic_a of_o capricorn_n but_o to_o show_v how_o you_o may_v find_v the_o centre_n upon_o which_o these_o tropical_a circle_n be_v to_o be_v describe_v i_o must_v make_v this_o diversion_n but_o for_o those_o parallel_n of_o altitude_n which_o fall_n near_o the_o horizon_n those_o circle_n or_o parallel_n of_o declination_n which_o fall_v near_o to_o the_o aequinoctial_a those_o hour-circle_n which_o fall_v near_o to_o the_o axis_n of_o the_o world_n or_o hour_n of_o six_o and_o those_o azimuth_n circle_n which_o be_v near_o to_o the_o prime_n vertical_a or_o azimuth_n of_o east_n and_o west_n those_o that_o make_v mathematical_a instrument_n have_v a_o instrument_n call_v a_o bow_n which_o by_o the_o help_n of_o one_o or_o more_o screw_n according_a to_o the_o length_n of_o the_o bow_n may_v be_v extend_v to_o touch_v any_o three_o point_n which_o lie_v near_o in_o a_o straight_a line_n by_o the_o edge_n of_o which_o bow_n you_o may_v draw_v your_o hour-circle_n azimuth_n parallel_n of_o declination_n and_o altitude_n as_o easy_o as_o you_o may_v draw_v a_o right_a line_n by_o the_o edge_n of_o a_o ruler_n but_o to_o return_v again_o to_o our_o projection_n four_o draw_v a_o right_a line_n ♋_o a_o ♑_o between_o the_o two_o tropic_n touch_v the_o tropic_a of_o cancer_n above_o the_o horizon_n at_o ♋_o and_o the_o tropic_a of_o capricorn_n below_o the_o horizon_n at_o the_o point_n ♑_o this_o circle_n have_v upon_o it_o the_o character_n of_o the_o 12_o sign_n of_o the_o zodiac_n which_o be_v to_o be_v put_v on_o in_o this_o manner_n take_v 23_o d._n 30_o min._n out_o of_o your_o line_n of_o chord_n and_o set_v they_o from_o p_o to_o q_o and_o from_o s_n to_o r_n which_o point_n q_n and_o r_n be_v the_o two_o pole_n of_o the_o ecliptic_a then_o take_v 60_o degr_n from_o your_o line_n of_o chord_n and_o set_v they_o from_o q_o to_o 1_o and_o from_o q_o to_o 3._o also_o set_v the_o same_o distance_n from_o ♋_o to_o 2_o and_o from_o ♑_o to_o 4._o this_o do_v lay_v a_o ruler_n to_o the_o pole_n r_o and_o the_o figure_n 1_o it_o will_v cut_v the_o ecliptic_a in_o the_o point_n ♊_o and_o ♌_o the_o ruler_n lay_v to_o r_n and_o 2_o will_v cut_v it_o in_o the_o point_n ♉_o and_o â™_o and_o lay_v to_o r_n and_o 4_o in_o â™_o and_o ♓_o and_o lay_v to_o r_n and_o 3_o in_o â™_o and_o â™’_o so_o have_v you_o the_o true_a point_n for_o the_o sun_n entrance_n into_o every_o sign_n and_o if_o you_o will_v have_v every_o ten_o degree_n of_o each_o sign_n divide_v every_o of_o the_o space_n ♋_o 1_n 12_o 2_o q_o q_o 4_o 4_o 3_o and_o 3_o ♑_o into_o three_o equal_a part_n so_o will_v each_o part_n contain_v 10_o d._n and_o a_o ruler_n lay_v to_o each_o of_o they_o and_o the_o point_n r_o shall_v give_v you_o the_o point_n upon_o the_o ecliptic_a answer_v to_o the_o 10._o degr_n of_o every_o sign_n and_o in_o the_o same_o manner_n may_v you_o if_o your_o projection_n be_v large_a put_v on_o every_o degree_n five_o for_o the_o put_n on_o of_o the_o hour-circle_n consider_v how_o far_o the_o circle_n you_o be_v to_o put_v on_o be_v distant_a from_o the_o meridian_n and_o set_v so_o many_o degree_n upon_o the_o meridian_n from_o the_o aequinoctial_a a_o ruler_n lay_v from_o z_o to_o those_o degree_n will_v cross_v the_o aequinoctial_a and_o through_o that_o point_n in_o the_o aequinoctial_a where_o the_o ruler_n so_o cross_v the_o hour-circle_n will_v pass_v example_n the_o hour-circle_n p_o b_o s_o in_o this_o projection_n be_v distant_a from_o the_o meridian_n 62_o d._n 46_o m._n wherefore_o take_v 62_o d._n 46_o m._n from_o your_o chord_n and_o set_v they_o from_o a_o to_o b_o then_o lay_v a_o ruler_n from_o z_o to_o b_o it_o will_v cut_v the_o aequinoctial_a in_o b_o through_o which_o point_n the_o hour-circle_n of_o 62_o d._n 46_o m._n must_v pass_v to_o find_v the_o centre_n of_o this_o hour-circle_n and_o so_o of_o any_o other_o repair_v to_o the_o former_a scheme_n for_o find_v of_o the_o centre_n of_o the_o parallel_n of_o altitude_n and_o declination_n and_o because_o this_o hour-circle_n be_v distant_a from_o the_o meridian_n 62_o degr_n 46_o min._n take_v 62_o degr_n 46_o min._n from_o your_o line_n of_o chord_n and_o set_v they_o upon_o the_o quadrant_n a_o b_o c_o from_o c_o to_o l_o and_o draw_v the_o line_n a_o l_o m._n so_o shall_v the_o line_n a_o l_o m_o be_v the_o semidiameter_n of_o the_o hour-circle_n p_o b_o s_o which_o be_v take_v in_o your_o compass_n and_o set_v upon_o your_o projection_n from_o b_o upon_o the_o line_n b_o ae_n be_v extend_v shall_v there_o give_v you_o the_o centre_n of_o that_o hour-circle_n and_o in_o the_o same_o manner_n may_v the_o centre_n of_o all_o the_o rest_n be_v find_v six_o the_o azimuth_n circle_n be_v to_o be_v draw_v upon_o the_o projection_n and_o the_o centre_n of_o they_o find_v in_o all_o respect_n as_o the_o hour-circle_n be_v so_o the_o azimuth_n circle_n z_o ☉_o n_o be_v 56_o degr_n 40_o min._n from_o the_o meridian_n take_v 56_o degr_n 41_o min._n out_o of_o your_o line_n of_o chord_n and_o set_v they_o upon_o the_o meridian_n of_o your_o projection_n from_o o_o to_o d_o then_o lay_v a_o ruler_n unto_o z_o and_z d_o it_o will_v cut_v the_o horizon_n in_o the_o point_n ☉_o through_o which_o the_o azimuth_n of_o 56_o degr_n 41_o min._n z_o ☉_o n_o must_v pass_v then_o to_o find_v its_o centre_n repair_v to_o the_o former_a scheme_n for_o find_v of_o centre_n and_o upon_o the_o quadrant_n a_o b_o c_o set_n 56_o degr_n 41_o min._n of_o your_o chord_n from_o c_o to_z n_z and_o draw_v the_o line_n a_o u._fw-mi o_o so_o shall_v the_o line_n a_o u._fw-mi o_o be_v the_o semidiameter_n of_o the_o azimuth_n circle_n z_o ☉_o n_o which_o be_v take_v in_o your_o compass_n and_o set_v upon_o your_o projection_n from_o ☉_o upon_o the_o line_n ☉_o h_n be_v extend_v shall_v there_o give_v you_o the_o centre_n of_o the_o azimuth_n circle_n z_o ☉_o n._n and_o in_o this_o manner_n may_v the_o centre_n of_o any_o other_o azimuth_n circle_n be_v find_v
and_o here_o note_v i._o that_o the_o centre_n of_o all_o azimuth_n circle_n fall_v in_o the_o horizon_n h_o a_o d_o be_v extend_v where_o need_n be_v the_o centre_n of_o all_o the_o hour-circle_n fall_v in_o the_o aequinoctial_a line_n ae_n a_o ae_z be_v extend_v the_o centre_n of_o the_o tropic_n and_o parallel_n of_o declination_n fall_v in_o the_o axis_n of_o the_o world_n p_o a_o s_o extend_v and_o the_o centre_n of_o the_o circle_n of_o altitude_n fall_v in_o the_o prime_n vertical_a circle_n z_o a_o n._n note_v ii_o that_o if_o the_o middle_a point_n of_o any_o hour-circle_n do_v not_o fall_v just_a in_o the_o aequinoctial_a or_o any_o azimuth_n circle_v just_a in_o the_o horizon_n but_o on_o either_o side_n of_o they_o than_o you_o may_v find_v the_o centre_n by_o the_o geometrical_a proposition_n at_o the_o begin_n of_o this_o book_n though_o there_o be_v other_o way_n to_o find_v the_o centre_n upon_o the_o projection_n itself_o which_o i_o omit_v for_o that_o i_o will_v not_o cumber_v the_o scheme_n with_o unnecessary_a line_n seven_o every_o circle_n in_o the_o projection_n have_v its_o proper_a pole_n as_o be_v before_o intimate_v now_o for_o the_o find_n of_o they_o you_o be_v to_o note_v that_o the_o pole_n of_o every_o great_a circle_n be_v 90_o degr_n or_o a_o quadrant_n of_o a_o circle_n distant_a from_o the_o circle_n itself_o upon_o that_o line_n which_o cut_v the_o circle_n at_o right_a angle_n thus_o the_o pole_n of_o all_o the_o hour-circle_n be_v upon_o the_o aequinoctial_a and_o the_o pole_n of_o all_o the_o azimuth_n upon_o the_o horizon_n now_o if_o you_o will_v find_v the_o pole_n of_o the_o hour-circle_n p_o d_o s_o lay_v a_o ruler_n upon_o p_o and_o d_o and_o it_o will_v cut_v the_o meridian_n circle_n in_o e_o then_o take_v 90_o degr_n of_o your_o line_n of_o chord_n and_o set_v they_o from_o e_z to_z f_o a_o ruler_n lay_v from_o p_o to_o f_o will_v cut_v the_o aequinoctial_a in_o you_o so_o be_v y_o the_o pole_n of_o the_o hour-circle_n p_o d_o s_o last_o the_o find_n of_o the_o pole_n of_o the_o azimuth_n circle_n be_v the_o same_o with_o the_o hour-circle_n so_o if_o you_o will_v find_v the_o pole_n of_o the_o azimuth_n circle_n z_o g_o n_o lay_v a_o ruler_n upon_o z_o and_z g_z it_o will_v cut_v the_o meridian_n circle_n in_o g_o then_o set_v 90_o degr_n of_o your_o chord_n from_o g_o to_o d_o so_o a_o ruler_n lay_v from_o z_o to_o d_o will_v cut_v the_o horizon_n h_o a_o o_o in_o the_o point_n ☉_o which_o point_v ☉_o be_v the_o pole_n of_o the_o azimuth_n circle_n z_o g_o n._n and_o thus_o have_v you_o find_v the_o pole_n of_o one_o of_o the_o hour_n and_o one_o of_o the_o azimuth_n circle_n and_o by_o the_o same_o manner_n of_o work_n you_o may_v find_v the_o pole_n of_o all_o the_o rest_n as_o the_o pole_n of_o the_o hour-circle_n p_o d_o s_o will_n be_v find_v at_o y_o the_o pole_n of_o the_o hour-circle_n p_o c_o s_o will_n be_v find_v at_o five_o the_o pole_n of_o the_o hour-circle_n p_o a_o s_o will_v be_v find_v at_o ae_n or_o ae_z the_o pole_n of_o the_o hour-circle_n p_o b_o s_o will_n be_v find_v at_o x_o the_o pole_n of_o the_o azimuth_n circle_n z_o g_o n_o will_v be_v find_v at_o ☉_o the_o pole_n of_o the_o azimuth_n circle_n z_o a_o n_o will_v be_v find_v at_o h_n or_o o_o the_o pole_n of_o the_o azimuth_n circle_n z_o f_o n_o will_v be_v find_v at_o t_o the_o pole_n of_o the_o azimuth_n circle_n z_o ☉_o n_o will_v be_v find_v at_o g_o the_o pole_n of_o the_o horizon_n h_o a_o o_o be_v z_o and_o n_o the_o zenith_n and_o nadir_n the_o pole_n of_o the_o aequinoctial_a ae_n a_o ae_z be_v p_o and_o s_o the_o pole_n of_o the_o world_n the_o pole_n of_o the_o ecliptic_a ♋_o at_fw-fr ♑_o are_z q_n and_o r._n thus_o have_v i_o give_v you_o at_o large_a a_o plain_a and_o easy_a method_n how_o to_o project_v the_o sphere_n upon_o the_o plain_a of_o the_o meridian_n circle_n by_o help_n of_o the_o line_n of_o chord_n only_o upon_o which_o projection_n by_o the_o intersection_n or_o cross_v of_o the_o several_a circle_n thereof_o be_v constitute_v divers_a spherical_a triangle_n some_o right-angled_n and_o other_o oblique-angled_n by_o the_o resolve_v of_o which_o triangle_n variety_n of_o question_n appertain_v to_o astronomy_n geography_n and_o navigation_n may_v with_o speed_n and_o exactness_n be_v resolve_v but_o before_o i_o come_v to_o show_v the_o manner_n of_o work_v particular_a question_n of_o any_o kind_n it_o will_v be_v expedient_a that_o i_o show_v you_o 1._o how_o to_o measure_v or_o find_v the_o quantity_n of_o the_o side_n and_o angle_n of_o a_o spherical_a triangle_n as_o they_o be_v here_o project_v and_o 2._o how_o to_o project_n or_o lay_v down_o a_o angle_n or_o side_n of_o any_o quantity_n that_o shall_v be_v require_v i._o a_o spherical_a triangle_n be_v project_v how_o to_o find_v the_o quantity_n of_o any_o angle_n thereof_o lay_v a_o ruler_n to_o the_o angular_a point_n and_o the_o extremity_n of_o the_o side_n contain_v the_o angle_n they_o be_v continue_v to_o quadrant_n and_o note_v where_o the_o ruler_n cut_v the_o meridian_n or_o outward_a circle_n at_o both_o which_o place_n make_v mark_n upon_o the_o meridian_n the_o distance_n between_o those_o two_o mark_n be_v measure_v upon_o your_o line_n of_o chord_n shall_v give_v you_o the_o quantity_n of_o the_o angle_n require_v example_n i._n in_o the_o triangle_n p_o ☉_o o_o in_o the_o projection_n let_v it_o be_v require_v to_o find_v the_o quantity_n of_o the_o angle_n ☉_o p_o o._n first_n lay_v a_o ruler_n upon_o the_o angular_a point_n p_o and_o to_o the_o extreme_a end_n of_o the_o side_n p_o ☉_o and_o p_o o_o they_o be_v extend_v to_o quadrant_n which_o be_v to_o that_o circle_n which_o measure_v that_o angle_n as_o the_o aequinoctial_a measure_n all_o the_o angle_n at_o p_o the_o pole_n of_o the_o world_n the_o horizon_n all_o the_o angle_n at_o z_o the_o zenith_n etc._n etc._n so_o the_o ruler_n lay_v from_o p_o to_o ae_z will_v cut_v the_o meridian_n in_o ae_z and_o be_v lay_v from_o p_o to_o b_o it_o will_v cut_v the_o meridian_n in_o the_o point_n b._n the_o distance_n b_o ae_z be_v take_v in_o your_o compass_n and_o measure_v upon_o your_o line_n of_o chord_n will_v be_v find_v to_o contain_v 62_o degr_n 46_o min._n which_o be_v the_o quantity_n of_o the_o angle_n ☉_o p_o o._n but_o if_o upon_o the_o point_n p_o you_o be_v to_o project_v a_o angle_n to_o contain_v 62_o degr_n 46_o min._n then_o take_v 90_o degr_n of_o your_o chord_n and_o set_v they_o from_o p_o to_o ae_z and_o through_o the_o centre_n a_o draw_v the_o line_n ae_n a_o ae_z then_o take_v 62_o degr_n 46_o m._n out_o of_o your_o line_n of_o chord_n and_o set_v they_o from_o ae_z to_o b_o and_o lay_v a_o ruler_n from_o p_o to_o b_o it_o will_v cut_v ae_n a_o ae_z in_o the_o point_n b_o the_o circle_n p_o b_o s_o being_n draw_v the_o angle_z at_z p_o will_v contain_v 62_o degr_n 46_o min._n example_n ii_o let_v it_o be_v require_v to_o find_v the_o quantity_n of_o the_o angle_n z_o e_o p._n lay_v a_o ruler_n to_o ☉_o the_o pole_n of_o the_o circle_n z_o e_o n_o and_o the_o point_n e_o it_o will_v cut_v the_o meridian_n circle_n in_o m_n from_o m_n set_v 90_o degr_n to_o z_o a_o ruler_n lay_v from_o ☉_o to_o z_o will_v cut_v the_o circle_n z_o e_o n_o it_o be_v extend_v beyond_o the_o zenith_n z_o at_o the_o point_n δ._n again_o lay_v a_o ruler_n upon_o you_o the_o pole_n of_o the_o circle_n p_o e_o s_o and_o it_o will_v cut_v the_o meridian_n circle_n in_o five_o set_v 90_o degr_n from_o five_o to_o x_o upon_o the_o meridian_n a_o ruler_n lay_v from_o young_a to_o x_o will_v cut_v the_o circle_n p_o e_o s_o in_o y._n this_o do_v lay_v a_o ruler_n from_o e_o to_o δ_n and_o it_o will_v cut_v the_o meridian_n in_o θ_o also_o lay_v the_o ruler_n from_o e_z to_z y_z it_o will_v cut_v the_o meridian_n in_o λ_n the_o distance_n θ_o λ_n be_v take_v in_o your_o compass_n and_o apply_v to_o your_o line_n of_o chord_n will_v be_v find_v to_o contain_v 21_o degr_n 45_o min._n and_o such_o be_v the_o quantity_n of_o the_o angle_n z_o e_o p._n these_o two_o sort_n of_o angle_n be_v the_o most_o troublesome_a to_o find_v their_o quantity_n and_o therefore_o i_o have_v instance_a in_o they_o there_o be_v other_o angle_n in_o the_o projection_n which_o render_v their_o measure_n to_o the_o eye_n without_o far_a instruction_n for_o find_v their_o quantity_n ii_o a_o spherical_a triangle_n be_v project_v to_o find_v the_o quantity_n of_o
upper_a wing_n of_o pegasus_n 341_o 53_o 13_o 21_o n_n 2_o in_o the_o tip_n of_o pegasus_n wing_n 358_o 52_o 13_o 15_o n_n 2_o prop._n vii_o the_o latitude_n of_o the_o place_n and_o the_o sun_n declination_n be_v give_v to_o find_v at_o what_o hour_n the_o sun_n will_v be_v upon_o the_o true_a east_n or_o west_n point_n upon_o the_o projection_n there_o be_v two_o right-angled_n spherical_a triangle_n by_o either_o of_o which_o this_o proposition_n may_v be_v solve_v the_o one_o be_v the_o triangle_n z_o p_o o_o make_v by_o the_o intersection_n of_o z_o o_o a_o arch_n of_o the_o prime_n vertical_a p_o o_o a_o arch_n of_o a_o hour-circle_n and_o z_o p_o a_o arch_n of_o the_o meridian_n in_o which_o triangle_n there_o be_v give_v z_o p_o the_o perpendicular_a the_o compliment_n of_o the_o latitude_n of_o the_o place_n 38_o degr_n 30_o min._n and_o the_o hypotenuse_n p_o o_fw-fr the_o compliment_n of_o the_o sun_n declination_n 70_o degr_n to_o find_v the_o angle_n at_o the_o perpendicular_a z_o p_o o_o which_o you_o may_v do_v by_o the_o 14._o case_n of_o right-angled_n spherical_a triangle_n the_o other_o triangle_n be_v o_o c_o a_o right-angled_n at_o c_o and_o be_v constitute_v of_o o_o c_o a_o arch_n of_o a_o hour-circle_n c_o a_o a_o arch_n of_o the_o aequinoctial_a and_o o_o a_o a_o arch_n of_o the_o prime_n vertical_a in_o which_o triangle_n you_o have_v give_v 1._o the_o perpendicular_a o_o c_o the_o sun_n declination_n 2._o the_o angle_n at_o the_o base_a c_o a_o o_o the_o latitude_n 51_o degr_n 30_o min._n to_o find_v the_o base_a c_o a._n thus_o have_v the_o perpendicular_a and_o the_o angle_n at_o the_o base_a you_o may_v find_v the_o base_a c_o at_fw-fr as_o follow_v this_o be_v the_o analogy_n or_o proportion_n as_o the_o tangent_fw-la of_o the_o latitude_n 51_o degr_n 30_o min._n be_v to_o the_o tangent_fw-la of_o the_o sun_n declination_n 20_o degr_n so_o be_v the_o radius_fw-la 90_o degr_fw-la to_o the_o cousin_a of_o the_o hour_n from_o noon_n to_o resolve_v the_o proposition_n by_o the_o projection_n lay_v a_o ruler_n upon_o p_o the_o pole_n of_o the_o world_n and_o the_o angle_n c_o of_o your_o triangle_n the_o ruler_n will_v cut_v the_o meridian_n circle_n in_o the_o point_n g_o so_o g_o ae_n be_v take_v in_o your_o compass_n and_o measure_v upon_o your_o line_n of_o chord_n will_v be_v find_v to_o contain_v 73_o degr_n 10_o min._n which_o convert_v into_o hour_n and_o minute_n will_v be_v 4_o hour_n and_o about_o 53_o min._n so_o that_o the_o sun_n when_o he_o have_v 20_o degr_n of_o declination_n will_v come_v to_o the_o east_n point_v at_o 7_o min._n past_o 7_o in_o the_o morning_n and_o will_v be_v due_a west_n 53_o min._n after_o 4_o in_o the_o afternoon_n prop._n viii_o have_v the_o latitude_n of_o the_o place_n and_o the_o sun_n declination_n give_v to_o find_v what_o altitude_n the_o sun_n shall_v have_v when_o he_o be_v upon_o the_o true_a east_n or_o west_n point_n this_o proposition_n may_v be_v resolve_v by_o either_o or_o both_o of_o the_o triangle_n mention_v in_o the_o last_o proposition_n for_o in_o the_o triangle_n z_o p_o o_o you_o have_v give_v p_o z_o the_o perpendicular_a and_o p_o o_o the_o hypotenuse_n to_o find_v z_o o_o the_o base_a by_o the_o 8._o case_n of_o right-angled_n spherical_a triangle_n but_o in_o the_o triangle_n o_o c_o a_o you_o have_v give_v 1._o the_o perpendicular_a o_fw-la c_o the_o sun_n declination_n 20_o degr_n 2._o the_o angle_n at_o the_o base_a o_o a_o c_o the_o latitude_n of_o the_o place_n 51_o degr_n 30_o m._n to_o find_v the_o hypotenuse_n o_o a_o for_o which_o this_o be_v the_o analogy_n or_o proportion_n as_o the_o sine_fw-la of_o the_o latitude_n 51_o degr_n 30_o min._n be_v to_o the_o sine_fw-la of_o the_o declination_n 20_o degr_n so_o be_v the_o radius_fw-la 90_o degr_fw-la to_o the_o sine_fw-la of_o the_o sun_n altitude_n be_v due_a east_n or_o west_n 25_o degr_n 55_o min._n to_o resolve_v the_o proposition_n by_o the_o projection_n lay_v a_o ruler_n upon_o o_o one_o of_o the_o pole_n of_o the_o prime_n vertical_a and_o to_o the_o angle_n o_o of_o the_o triangle_n a_o ruler_n thus_o lay_v will_v cut_v the_o meridian_n circle_n in_o the_o point_n p_o so_o the_o distance_n h_o p_o be_v take_v and_o measure_v upon_o the_o line_n of_o chord_n will_v be_v 25_o degr_n 55_o min._n and_o such_o height_n will_v the_o sun_n have_v when_o he_o be_v either_o east_n or_o west_n prop._n ix_o the_o latitude_n of_o the_o place_n and_o the_o declination_n of_o the_o sun_n be_v give_v to_o find_v what_o altitude_n the_o sun_n shall_v have_v at_o six_o of_o the_o clock_n for_o find_v of_o the_o triangle_n upon_o the_o projection_n which_o will_v resolve_v this_o and_o the_o follow_a proposition_n you_o must_v suppose_v another_o azimuth_n circle_n to_o be_v draw_v in_o the_o projection_n from_o z_o to_o n_n and_o through_o that_o point_n where_o the_o parallel_n of_o declination_n ♊_o ☉_o ♌_o and_o the_o axis_n of_o the_o world_n or_o hour-circle_n of_o six_o p_o a_o s_o do_v across_o each_o other_o the_o draw_v of_o which_o azimuth_n circle_n i_o purposely_o omit_v chief_o because_o the_o scheme_n in_o that_o place_n be_v more_o cumber_v with_o line_n and_o letter_n than_o any_o other_o part_n thereof_o but_o you_o may_v well_o enough_o for_o the_o solving_a of_o these_o two_o proposition_n imagine_v it_o to_o be_v draw_v the_o pole_n whereof_o be_v at_o *_o this_o azimuth_n circle_n be_v suppose_v to_o be_v draw_v you_o have_v upon_o the_o projection_n two_o triangle_n like-angled_n which_o will_v perform_v the_o work_n of_o resolve_v this_o proposition_n in_o one_o of_o which_o you_o have_v give_v the_o base_a which_o be_v the_o compliment_n of_o the_o declination_n and_o the_o perpendicular_a which_o be_v the_o compliment_n of_o the_o latitude_n to_o find_v the_o hypotenuse_n which_o be_v the_o compliment_n of_o the_o sun_n altitude_n require_v this_o triangle_n may_v be_v resolve_v by_o the_o first_o case_n aforego_v in_o the_o other_o triangle_n there_o will_v be_v give_v the_o hypotenuse_n which_o be_v the_o sun_n declination_n and_o the_o angle_n at_o the_o base_a which_o be_v the_o latitude_n to_o find_v the_o perpendicular_a which_o be_v the_o sun_n altitude_n at_o six_o a_o clock_n to_o find_v which_o this_o be_v the_o analogy_n or_o proportion_n as_o the_o radius_fw-la 90_o degr_fw-la be_v to_o the_o sine_fw-la of_o the_o sun_n declination_n 20_o degr_n so_o be_v the_o sine_fw-la of_o the_o latitude_n 51_o degr_n 30_o min._n to_o the_o sine_fw-la of_o the_o sun_n altitude_n at_o six_o 15_o degr_n 30_o min._n to_o resolve_v the_o proposition_n by_o the_o projection_n lay_v a_o ruler_n upon_o the_o point_n *_o and_o that_o point_v where_o the_o parallel_n of_o declination_n ♊_o ☉_o ♌_o cross_v the_o axis_n or_o hour_n of_o six_o the_o ruler_n thus_o lay_v will_v cut_v the_o meridian_n circle_n in_o the_o point_n g._n so_o o_o g_o be_v measure_v upon_o the_o chord_n will_v give_v you_o 15_o degr_n 30_o min._n and_o such_o altitude_n will_v the_o sun_n have_v at_o the_o hour_n of_o six_o in_o the_o latitude_n of_o 51_o degr_n 30_o min._n when_o he_o have_v 20_o degr_n of_o declination_n prop._n x._o the_o latitude_n of_o the_o place_n and_o the_o declination_n of_o the_o sun_n be_v give_v to_o find_v what_o azimuth_n the_o sun_n shall_v have_v at_o six_o a_o clock_n the_o two_o triangle_n that_o be_v suppose_v in_o the_o last_o proposition_n to_o be_v draw_v upon_o the_o projection_n will_v resolve_v this_o proposition_n also_o but_o see_v the_o triangle_n be_v not_o draw_v but_o suppose_v i_o will_v only_o give_v you_o the_o analogy_n and_o then_o the_o way_n of_o work_v it_o upon_o the_o projection_n the_o analogy_n or_o proportion_n as_o the_o cousin_a of_o the_o latitude_n 38_o degr_n 30_o min._n be_v to_o the_o radius_fw-la 90_o degr_fw-la so_o be_v the_o co-tangent_a of_o the_o sun_n declination_n 70_o degr_n to_o the_o tangent_fw-la of_o the_o sun_n azimuth_n count_v from_o the_o north_n part_n of_o the_o meridian_n 77_o degr_n 14_o min._n to_o resolve_v the_o proposition_n upon_o the_o projection_n lay_v a_o ruler_n to_o the_o zenith-point_n z_o and_o upon_o the_o point_n where_o the_o parallel_n of_o declination_n cut_v the_o hour-circle_n of_o six_o the_o ruler_n thus_o lay_v will_v cut_v the_o meridian_n circle_n in_o r_o so_o the_o distance_n r_o o_fw-fr be_v measure_v upon_o the_o line_n of_o chord_n will_v contain_v 77_o degr_n 14_o min._n which_o be_v the_o azimuth_n from_o the_o north_n part_n of_o the_o meridian_n the_o distance_n n_o r_o measure_v upon_o the_o chord_n will_v give_v you_o 12_o degr_n 46_o min._n which_o be_v the_o azimuth_n