length_n of_o the_o perpendicular_a c_o a._n the_o analogy_n or_o proportion_n be_v as_o the_o sine_fw-la of_o the_o angle_n at_o c_z be_v to_o the_o log._n of_o a_o b_o so_o be_v the_o sine_fw-la of_o the_o angle_n b_o to_o the_o logar_n of_o c_o a._n or_o as_o the_o radius_fw-la be_v to_o the_o logar_n of_o a_o b_o so_o be_v the_o tangent_fw-la of_o b_o to_o the_o logar_n of_o c_o a._n case_n iu._n the_o hypotenuse_n c_o b_o 225_o the_o angle_n c_o 53_o degr_n 7_o min._n and_o the_o angle_n at_o b_o 36_o degr_n 53_o min._n give_v to_o find_v the_o base_a b_o a_o and_o the_o perpendicular_a c_o a._n draw_v a_o right_a line_n c_o b_o contain_v 225_o of_o your_o line_n of_o equal_a part_n then_o take_v 60_o deg_fw-la out_o of_o your_o line_n of_o chord_n set_v one_o foot_n of_o the_o compass_n in_o b_o and_o with_o the_o other_o describe_v the_o arch_n e_o d_o also_o the_o compass_n continue_v at_o the_o same_o distance_n place_v one_o foot_n in_o c_o and_o with_o the_o other_o 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the_o arch_n from_o d_o to_z e._n then_o from_o b_o through_o the_o point_n e_o draw_v a_o right_a line_n till_o it_o meet_v with_o the_o perpendicular_a before_o draw_v which_o it_o will_v do_v in_o the_o point_n c._n and_o thus_o have_v you_o protract_v your_o triangle_n c_o a_o b._n last_o take_v in_o your_o compass_n the_o length_n of_o the_o hypotenuse_n c_o b_o and_o measure_v it_o upon_o your_o scale_n of_o equal_a part_n and_o you_o shall_v find_v it_o to_o contain_v 225._o the_o analogy_n or_o proportion_n be_v as_o the_o sine_fw-la of_o c_o be_v to_o the_o logarithm_n of_o a_o b_o so_o be_v the_o radius_fw-la to_o the_o logarithm_n of_o c_o b._n case_n vii_o the_o base_a a_o b_o 180_o and_o the_o perpendicular_a c_o at_fw-fr 135_o be_v give_v to_o find_v the_o hypotenuse_n c_o a._n draw_v a_o right_a line_n a_o b_o contain_v 180_o equal_a part_n and_o upon_o the_o end_n a_o erect_v the_o perpendicular_a a_o c_o and_o out_o of_o your_o scale_n of_o equal_a part_n take_v the_o 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the_o end_n of_o the_o line_n wherefore_o take_v 90_o degr_n the_o whole_a line_n and_o set_v that_o distance_n from_o g_o to_z o_fw-mi then_o take_v the_o remainder_n of_o the_o arch_n o_o h_o and_o measure_n that_o upon_o your_o chord_n and_o you_o shall_v find_v it_o to_o contain_v 32_o degr_n which_o add_v to_o 90_o degr_n make_v 122_o degr_n and_o that_o be_v the_o quantity_n of_o the_o angle_n at_o c_o which_o be_v require_v the_o analogy_n or_o proportion_n be_v as_o
and_o k_o h_n both_o parallel_n to_o the_o line_n b_o a._n then_o take_v in_o your_o compass_n the_o distance_n a_o 6_o set_v one_o foot_n of_o they_o in_o c_z and_o with_o the_o other_o describe_v the_o arch_n z_o and_o draw_v the_o line_n a_o x_o so_o that_o it_o only_o touch_v the_o arch_n z._n then_o take_v in_o your_o compass_n the_o distance_n a_o h_n set_v one_o foot_n upon_o the_o line_n a_o c_o move_v it_o along_o till_o the_o other_o foot_n be_v turn_v about_o will_v only_o touch_v the_o line_n a_o z_o and_o where_o the_o point_n of_o the_o compass_n rest_v upon_o the_o line_n a_o c_o which_o it_o will_v do_v at_o e_o through_o that_o point_n draw_v the_o line_n o_o e_o parallel_n to_o b_o a._n so_o shall_v b_o o_o be_v measure_v upon_o your_o line_n of_o chord_n give_v you_o 69_o degr_n 22_o min._n the_o quantity_n of_o the_o inquire_v angle_n at_o b._n case_n viii_o the_o base_a a_o c_o 27_o degr_n 54_o min._n and_o the_o hypotenuse_n a_fw-it b_o 30_o degr_n be_v give_v 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hypotenuse_n a_o b_o 30_o degr_n and_o the_o angle_n at_o the_o base_a a_o 23_o d._n 30_o min._n be_v give_v to_o find_v the_o angle_n at_o the_o perpendicular_a b._n the_o analogy_n or_o proportion_n be_v as_o the_o radius_fw-la be_v to_o the_o cousin_a of_o the_o hypotenuse_n a_fw-it b_o 60_o deg_fw-la so_o be_v the_o tangent_fw-la of_o the_o angle_n at_o the_o base_a a_o 23_o degr_n 30_o min._n to_o the_o co-tangent_a of_o the_o angle_n at_o the_o perpendicular_a b._n first_o draw_v a_o right_a line_n as_o c_o h_n and_o upon_o one_o end_n thereof_o as_o at_o c_o erect_a the_o perpendicular_a c_o a_n and_o with_o the_o distance_n of_o 60_o degr_n of_o your_o line_n of_o chord_n upon_o the_o centre_n c_o describe_v the_o quadrant_n c_o a_o d._n also_o upon_o the_o point_n a_o with_o 60_o degr_n of_o your_o chord_n describe_v the_o quadrant_n a_o b_o c._n be_v thus_o prepare_v first_o take_v 60_o deg_fw-la the_o cousin_a of_o the_o hypotenuse_n a_fw-it b_o and_o set_v they_o from_o a_o to_o o_o also_o take_v 23_o d._n 30_o m._n the_o quantity_n of_o the_o angle_n at_o a_o and_o set_v they_o from_o c_z to_o r_o and_o draw_v the_o line_n a_o m_o f_o and_o the_o line_n o_o g_o parallel_n to_o a_o c._n then_o take_v in_o your_o compass_n the_o distance_n between_o c_o and_z g_z and_o set_v one_o foot_n in_o d_o with_o the_o other_o describe_v the_o arch_a k_o and_o draw_v the_o line_n a_o l_o so_o that_o it_o only_o touch_v the_o arch_n k._n then_o place_v one_o foot_n of_o the_o compass_n in_o f_o take_v the_o least_o distance_n you_o can_v to_o the_o line_n c_o l_o which_o set_v from_o c_o to_o e._n last_o draw_v the_o line_n a_o e_o cut_v the_o quadrant_n c_o b_o in_o n._n so_o the_o distance_n c_o n_n measure_v upon_o your_o chord_n shall_v give_v you_o 20_o degr_n 38_o min._n the_o compliment_n of_o the_o angle_n at_o b_o which_o be_v require_v or_o the_o distance_n b_o n_n will_v give_v you_o 69_o degr_n 22_o min._n the_o angle_n itself_o case_n x._o the_o 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line_n c_o q_n so_o that_o it_o may_v only_o touch_v the_o arch_n p._n then_o set_v one_o foot_n of_o the_o compass_n in_o b_o take_v the_o least_o distance_n between_o b_o and_o the_o line_n c_o q_n which_o distance_n will_v reach_v from_o c_o to_z e._n last_o draw_v the_o line_n a_o e_o cut_v the_o quadrant_n a_o c_o b_o in_o the_o point_n m._n so_o shall_v the_o distance_n c_o m_o be_v take_v in_o the_o compass_n and_o measure_v upon_o the_o line_n of_o chord_n contain_v 27_o degr_n 54_o m._n which_o be_v the_o quantity_n of_o the_o base_a require_v case_n xi_o the_o base_a a_o c_o 27_o degr_n 54_o min._n and_o the_o angle_n at_o the_o base_a a_o 23_o degr_n 30_o min._n be_v give_v to_o find_v the_o perpendicular_a b_o c._o the_o analogy_n or_o proportion_n be_v as_o the_o radius_fw-la be_v to_o the_o sine_fw-la of_o the_o base_a a_o c_o 27_o degr_n 54_o min._n so_o be_v the_o tangent_fw-la of_o the_o angle_n at_o a_o 23_o degr_n 30_o min._n to_o the_o tangent_fw-la of_o the_o 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and_o set_v it_o from_o c_o to_o d_o and_o draw_v the_o line_n d_o b_o which_o will_v contain_v 335_o of_o your_o scale_n of_o equal_a part_n and_o that_o be_v the_o length_n of_o the_o base_a d_o b_o which_o be_v require_v the_o analogy_n or_o proportion_n you_o must_v first_o find_v the_o two_o angle_n at_o d_o and_z b._n then_o make_v choice_n of_o one_o of_o the_o side_n as_o c_o d_o to_o work_v your_o proportion_n by_o then_o as_o the_o sine_fw-la of_o b_o be_v to_o the_o sine_n compliment_n of_o c_o so_o be_v the_o log._n of_o c_o d_o to_o the_o log._n of_o d_o b._n the_o end_n of_o the_o doctrine_n of_o the_o dimension_n of_o right-lined_n triangle_n the_o dimension_n of_o spherical_a triangle_n by_o a_o line_n of_o chord_n the_o three_o exercise_n as_o the_o side_n of_o right-lined_n triangle_n be_v measure_v by_o equal_a part_n and_o their_o angle_n by_o a_o line_n of_o chord_n so_o the_o side_n of_o spherical_a triangle_n be_v arch_n of_o great_a circle_n of_o the_o sphere_n both_o they_o and_o the_o angle_n also_o may_v be_v measure_v by_o chord_n only_o of_o spherical_a triangle_n there_o be_v two_o kind_n as_o there_o be_v of_o right-lined_n namely_o right-angled_n and_o oblique-angled_n of_o both_o which_o there_o be_v 28_o case_n viz._n 16_o of_o right-angled_n and_o 12_o of_o oblique-angled_n and_o to_o observe_v the_o same_o method_n in_o these_o as_o i_o do_v in_o plain_a triangle_n i_o will_v begin_v with_o the_o solution_n i._o of_o right-angled_n spherical_a triangle_n the_o right-angled_n spherical_a triangle_n which_o i_o shall_v make_v use_n of_o for_o my_o example_n shall_v be_v this_o follow_v a_o b_o c_o who_o side_n and_o angle_n contain_v as_o follow_v  _fw-fr  _fw-fr  _fw-fr deg_fw-la m._n the_o side_n a_o b_o contain_v 30_o 00_o b_o c_o 11_o 30_o a_o c_o 27_o 54_o the_o angle_n at_o a_o contain_v 23_o 30_o b_o 69_o 22_o c_o 90_o 00_o case_n i._o the_o base_a a_o c_o 27_o degr_n 54_o min._n and_o the_o perpendicular_a c_o b_o 11_o degr_n 30_o min._n be_v give_v to_o find_v the_o hypotenuse_n a_o b._n the_o analogy_n or_o proportion_n be_v as_o the_o radius_fw-la be_v to_o the_o cousin_a of_o b_o c_z 78_o degr_n 30_o min._n so_o be_v the_o cousin_a of_o a_o c_o 62_o degr_n 6_o min._n to_o the_o cousin_a of_o a_o b._n first_o take_v the_o sum_n and_o the_o difference_n of_o the_o second_o and_o three_o term_n in_o the_o analogy_n or_o proportion_n namely_o the_o sum_n and_o difference_n of_o the_o compliment_n of_o the_o perpendicular_a b_o c_z 11_o degr_n 30_o min._n and_o the_o base_a a_o c_o 27_o degr_n 34_o min._n  _fw-fr degr_fw-fr m._n  _fw-fr  _fw-fr  _fw-fr the_o base_a a_o c_o be_v 27_o 54._o it_o comp._n 62_o 06_o the_o perpendicular_a b_o c_z be_v 11_o 30._o it_o comp._n 78_o 30_o  _fw-fr  _fw-fr  _fw-fr sum_n 140_o 36_o  _fw-fr  _fw-fr  _fw-fr differ_v 16_o 24_o case_n ii_o the_o hypotenuse_n a_o b_o 30_o degr_n and_o the_o angle_n at_o the_o base_a a_o 23_o degr_n 30_o min._n be_v give_v to_o find_v the_o perpendicular_a b_o c._o the_o analogy_n or_o proportion_n be_v as_o the_o radius_fw-la be_v to_o the_o sine_fw-la of_o the_o hypotenuse_n a_fw-it b_o 30_o degr_n so_o be_v the_o sine_fw-la of_o the_o angle_n at_o a_o 23_o degr_n 30_o min._n to_o the_o sine_fw-la of_o the_o perpendicular_a b_o c_o 11._o degr_n 30_o min._n take_v the_o sum_n and_o difference_n of_o the_o angle_n at_o a_o 23_o degr_n 30_o min._n and_o the_o
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
any_o side_n thereof_o a_o ruler_n lay_v upon_o the_o pole_n of_o the_o circle_n which_o be_v to_o be_v measure_v and_o to_o the_o extreme_a end_n of_o the_o side_n of_o the_o triangle_n note_v where_o the_o ruler_n so_o lay_v cut_v the_o meridian_n at_o both_o end_n of_o the_o side_n that_o distance_n take_v in_o your_o compass_n and_o measure_v upon_o the_o line_n of_o chord_n will_v give_v you_o the_o quantity_n of_o the_o side_n of_o the_o triangle_n example_n i._n let_v it_o be_v require_v to_o find_v the_o side_n e_o z_o of_o the_o triangle_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 angular_a point_n e_o it_o will_v cut_v the_o meridian_n in_o m_n and_o a_o ruler_n lay_v to_o z_o will_v cut_v the_o meridian_n in_o z._n so_o the_o distance_n m_o z_o take_v in_o the_o compass_n and_o measure_v upon_o the_o line_n of_o chord_n will_v be_v find_v to_o contain_v 78_o degr_n and_o such_o be_v the_o quantity_n of_o the_o side_n z_o e._n example_n ii_o let_v it_o be_v require_v to_o find_v the_o side_n ☉_o b_o of_o the_o spherical_a triangle_n a_o ☉_o b._n lay_v a_o ruler_n upon_o x_o the_o pole_n of_o the_o circle_n p_o b_o s_o and_o the_o point_n b_o it_o will_v cut_v the_o meridian_n circle_n in_o ae_z also_o lay_v a_o ruler_n from_o x_o to_o ☉_o it_o will_v cut_v the_o meridian_n in_o the_o point_n ♌_o the_o distance_n between_o ae_z and_o ♌_o be_v take_v and_o measure_v on_o the_o line_n of_o chord_n will_v contain_v 20_o d._n and_o such_o be_v the_o quantity_n of_o the_o side_n ☉_o b._n i_o can_v instance_n in_o divers_a other_o example_n concern_v the_o measure_v of_o the_o side_n and_o angle_n of_o triangle_n upon_o the_o projection_n but_o i_o here_o omit_v they_o because_o in_o the_o resolve_v of_o the_o follow_a proposition_n they_o will_v come_v in_o practice_n and_o the_o manner_n of_o the_o performance_n be_v there_o plain_o express_v only_o i_o deem_v it_o convenient_a here_o to_o give_v some_o taste_n thereof_o as_o a_o preparative_a to_o that_o which_o follow_v but_o before_o i_o come_v to_o show_v the_o manner_n of_o resolve_v of_o particular_a question_n in_o astronomy_n geography_n etc._n etc._n i_o will_v declare_v the_o variety_n of_o spherical_a problem_n that_o will_v natural_o arise_v out_o of_o every_o spherical_a triangle_n be_v project_v the_o variety_n of_o spherical_a problem_n natural_o arise_v out_o of_o every_o spherical_a triangle_n both_o right_a and_o oblique-angled_n and_o that_o be_v resolvable_a thereby_o describe_v as_o they_o be_v perspicuous_a to_o the_o eye_n in_o the_o projection_n the_o five_o exercise_n in_o the_o forego_n part_n of_o this_o book_n you_o have_v the_o doctrine_n of_o plain_a and_o spherical_a triangle_n geometrical_o perform_v and_o in_o the_o solution_n of_o right-angled_n spherical_a triangle_n there_o be_v 16_o case_n and_o in_o oblique-angled_n there_o be_v 12_o case_n but_o the_o 16_o case_n of_o right-angled_n triangle_n will_v by_o this_o projective_a way_n be_v reduce_v to_o 5_o case_n and_o the_o 12_o of_o oblique-angled_n will_v be_v reduce_v to_o 6_o so_o that_o in_o both_o there_o will_v be_v but_o 11_o case_n whereas_o before_o there_o be_v 28._o that_o this_o may_v appear_v plain_a to_o the_o reader_n i_o will_v make_v use_n of_o two_o triangle_n in_o the_o projection_n one_o whereof_o shall_v be_v right-angled_n as_o the_o triangle_n p_o o_fw-fr ☉_o right-angled_n at_o o_o and_o the_o other_o shall_v be_v the_o oblique-angled_n triangle_n z_o e_o p._n the_o right-angled_n triangle_n be_v constitute_v by_o the_o intersection_n of_o three_o great_a circle_n of_o the_o sphere_n namely_o of_o p_o o_o a_o arch_n of_o the_o meridian_n ☉_o o_o a_o arch_n of_o the_o horizon_n and_o p_o ☉_o the_o arch_n of_o a_o hour-circle_n the_o oblique-angled_n triangle_n z_o e_o p_o be_v constitute_v also_o of_o three_o arch_n of_o great_a circle_n of_o the_o sphere_n as_o all_o spherical_a triangle_n whatsoever_o be_v namely_o of_o z_o p_o a_o arch_n of_o the_o meridian_n p_o e_o a_o arch_n of_o a_o hour-circle_n and_o z_o e_o a_o arch_n of_o a_o azimuth_n circle_n in_o the_o right-angled_n spherical_a triangle_n p_o ☉_o o_o the_o side_n p_o o_fw-fr be_v the_o latitude_n of_o the_o place_n the_o side_n ☉_o o_fw-fr be_v the_o sun_n amplitude_n from_o the_o north._n the_o side_n p_o ☉_o be_v the_o sun_n distance_n from_o the_o pole_n or_o the_o compliment_n of_o his_o declination_n the_o angle_n ☉_o p_o o_fw-fr be_v the_o hour_n from_o midnight_n or_o from_o the_o north_n part_n of_o the_o meridian_n the_o angle_n p_o ☉_o o_fw-fr be_v the_o angle_n of_o the_o sun_n position_n at_o the_o time_n of_o the_o question_n the_o angle_n p_o o_fw-fr ☉_o be_v the_o right_a angle_n the_o part_n of_o the_o triangle_n be_v declare_v and_o of_o what_o circle_n of_o the_o sphere_n the_o side_n do_v consist_v i_o will_v now_o come_v to_o the_o case_n which_o as_o i_o say_v before_o be_v 5_o in_o every_o right-angled_n triangle_n so_o that_o any_o two_o part_n of_o the_o triangle_n beside_o the_o right_a angle_n be_v give_v i_o will_v show_v in_o every_o of_o the_o 5_o case_n what_o part_n may_v be_v find_v i._o in_o a_o right-angled_n spherical_a triangle_n case_n i._o the_o base_a and_o perpendicular_a be_v give_v to_o find_v the_o other_o part_n of_o the_o triangle_n in_o the_o triangle_n p_o ☉_o o_o here_o be_v give_v p_o o_o the_o latitude_n and_o ☉_o o_o the_o amplitude_n from_o the_o north_n part_n of_o the_o meridian_n by_o which_o you_o may_v find_v 3._o 1._o p_o ☉_o the_o compliment_n of_o the_o sun_n declination_n 2._o ☉_o p_o o_o the_o hour_n from_o midnight_n 3._o p_o ☉_o o_o the_o angle_n of_o the_o sun_n position_n case_n ii_o the_o hypotenuse_n and_o perpendicular_a be_v give_v to_o find_v the_o other_o part_n in_o the_o triangle_n p_o ☉_o o_o here_o be_v give_v p_o o_o the_o latitude_n and_o p_o ☉_o the_o compliment_n of_o the_o sun_n declination_n by_o which_o may_v be_v find_v 6._o 1._o ☉_o o_o the_o amplitude_n from_o the_o north._n 2._o o_o p_o ☉_o the_o hour_n from_o midnight_n 3._o p_o ☉_o o_o the_o angle_n of_o the_o sun_n position_n and_o if_o in_o stead_n of_o the_o perpendicular_a there_o have_v be_v give_v the_o base_a ☉_o o_fw-fr you_o may_v then_o find_v 4._o p_o o_o the_o latitude_n 5._o ☉_o p_o o_o the_o hour_n from_o midnight_n 6._o p_o ☉_o o_o the_o angle_n of_o the_o sun_n position_n case_n iii_o the_o hypotenuse_n and_o a_o angle_n be_v give_v to_o find_v the_o other_o part_n here_o be_v give_v in_o the_o triangle_n p_o ☉_o o_o the_o compliment_n of_o the_o sun_n declination_n p_o ☉_o and_o ☉_o p_o o_o the_o hour_n from_o midnight_n by_o which_o may_v be_v find_v 6._o 1._o ☉_o o_o the_o amplitude_n from_o the_o north._n 2._o p_o o_o the_o latitude_n 3._o c_o ☉_o o_o the_o angle_n of_o the_o sun_n position_n and_o if_o in_o lieu_n of_o the_o angle_n at_o p_o the_o angle_n at_o ☉_o p_o have_v be_v take_v than_o you_o may_v have_v find_v 4._o ☉_o o_o the_o amplitude_n from_o the_o north._n 5._o p_o o_o the_o latitude_n 6._o ☉_o p_o a_o the_o hour_n from_o midnight_n case_n iu._n the_o perpendicular_a or_o base_a and_o either_o of_o the_o angle_n give_v to_o find_v the_o other_o part_n in_o the_o triangle_n p_o ☉_o o_fw-fr let_v there_o be_v give_v the_o amplitude_n ☉_o o_o and_o the_o angle_n of_o the_o sun_n position_n p_o ☉_o o_o by_o which_o you_o may_v find_v 12._o 1._o p_o o_o the_o latitude_n 2._o ☉_o p_o o_o the_o hour_n from_o midnight_n 3._o p_o ☉_o the_o distance_n of_o the_o sun_n from_o the_o pole_n but_o if_o the_o side_n give_v have_v be_v ☉_o o_fw-fr and_o the_o angle_n give_v ☉_o p_o o_o than_o you_o may_v have_v find_v 4._o p_o ☉_o the_o compliment_n of_o the_o sun_n declination_n 5._o p_o o_o the_o latitude_n 6._o p_o ☉_o o_o the_o angle_n of_o the_o sun_n position_n but_o again_o if_o ☉_o o_o and_o ☉_o p_o o_o have_v be_v give_v then_o may_v be_v find_v 7._o ☉_o p_o the_o compliment_n of_o the_o sun_n declination_n 8._o ☉_o o_o the_o amplitude_n from_o the_o north._n 9_o p_o ☉_o o_o the_o angle_n of_o the_o sun_n position_n and_o again_o if_o p_o o_o and_o o_o ☉_o p_o have_v be_v give_v we_o may_v then_o have_v also_o find_v 10._o ☉_o p_o the_o distance_n of_o the_o sun_n from_o the_o pole_n 11._o ☉_o o_o the_o amplitude_n from_o the_o north._n 12._o ☉_o p_o o_o the_o hour_n from_o midnight_n case_n v._o the_o angles_n be_v give_v to_o find_v the_o other_o part_n if_o the_o two_o angel_n p_o ☉_o o_o and_o o_o p_o ☉_o