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the_o 6_o a_o clock_n line_n 4._o take_v the_o lesser_a parallelogram_n and_o put_v it_o into_o its_o place_n as_o close_o to_o the_o end_n a_o b_o as_o it_o will_v go_v and_o when_o it_o be_v there_o lay_v observe_v where_o the_o thread_n lie_v upon_o the_o low_a area_n and_o draw_v the_o line_n upon_o which_o it_o lie_v which_o must_v be_v the_o line_n 90_o 90_o upon_o which_o the_o scale_n that_o be_v inscribe_v must_v take_v their_o beginning_n and_o likewise_o you_o be_v to_o take_v the_o long_a radius_fw-la o_fw-la m_o and_o as_o the_o plate_n there_o lie_v set_v one_o foot_n of_o that_o radius_fw-la upon_o o_fw-la and_o let_v the_o other_o reach_n as_o far_o as_o it_o will_v which_o must_v be_v upon_o the_o rise_a limb_n beyond_o the_o inner_a edge_n of_o it_o so_o much_o as_o m_n n_o come_v to_o in_o that_o point_n you_o be_v to_o make_v one_o centre_n so_o again_o turn_v your_o lesser_a plate_n about_o you_o must_v lay_v it_o in_o the_o contrary_a way_n close_o to_o a_o b_o 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ellipsis_n the_o same_o thread_n will_v show_v the_o azimuth_n in_o the_o graduation_n upon_o the_o limb_n 3._o to_o find_v the_o ascensionall_a difference_n the_o semidiurnall_a and_o seminocturnall_a arke_n with_o the_o time_n of_o their_o rise_n and_o setting_n in_o general_a note_n this_o if_o the_o sun_n or_o star_n decline_v northward_o than_o the_o semidiurnall_a ark_n be_v great_a than_o six_o hour_n if_o southward_o then_o less_o the_o upper_a plate_n therefore_o be_v rectify_v either_o to_o the_o sun_n place_n or_o to_o the_o star_n declination_n the_o thread_n will_v show_v upon_o the_o middle_a scale_n the_o ascensionall_a difference_n if_o you_o count_v the_o hour_n
another_o slide_a plate_n of_o the_o same_o breadth_n that_o the_o former_a be_v name_o m_n o_o here_o equal_a to_o m_o o_fw-fr in_o the_o former_a and_o m_o n_o must_v in_o both_o be_v just_o equal_a that_o it_o may_v both_o slip_v in_o the_o former_a cavity_n so_o as_o just_a to_o fill_v it_o and_o that_o the_o same_o limb_n centre_n and_o thread_n may_v in_o both_o agree_v a_o lesser_a ellipsis_n i_o say_v must_v be_v describe_v which_o though_o it_o be_v less_o yet_o it_o must_v be_v like_a and_o proportional_a to_o the_o former_a therefore_o assume_v any_o length_n as_o m_n p_o for_o the_o long_a radius_fw-la you_o must_v to_o that_o radius_fw-la make_v m_o r_o and_o m_o s_o equal_o to_o the_o sine_fw-la of_o your_o latitude_n and_o so_o describe_v and_o divide_v it_o as_o former_o be_v order_v then_o again_o to_o this_o new_a ellipsis_n there_o must_v a_o new_a scale_n of_o tangent_n be_v make_v who_o length_n must_v be_v limit_v as_o the_o former_a be_v in_o this_o manner_n to_o m_n p_o as_o radius_fw-la find_v the_o cousin_a of_o your_o latitude_n and_o make_v that_o cousin_a a_o radius_fw-la or_o tangent_fw-la of_o 45_o gr_n and_o according_a thereunto_o continue_v the_o scale_n of_o tangent_n as_o far_o as_o it_o will_v go_v it_o must_v begin_v upon_o the_o same_o line_n that_o the_o other_o scale_n of_o tangent_n begin_v and_o must_v go_v the_o same_o way_n with_o it_o and_o in_o this_o new_a way_n of_o order_v the_o scale_n and_o the_o ellipse_n it_o must_v be_v consider_v what_o the_o other_o part_n of_o the_o instrument_n do_v signify_v the_o end_n of_o the_o low_a plate_n upon_o which_o the_o scale_n begin_v must_v be_v account_v for_o the_o north_n the_o other_o end_n c_o d_o be_v to_o be_v take_v for_o south_n the_o side_n line_n be_v the_o meridian_n the_o degree_n of_o the_o limb_n be_v to_o be_v use_v as_o the_o degree_n of_o the_o equinoctial_a the_o thread_n apply_v to_o those_o degree_n be_v the_o meridian_n come_v from_o the_o centre_n which_o be_v the_o north_n pole_n the_o tangent_n be_v the_o degree_n of_o altitude_n the_o ellipsis_n note_v out_o that_o almicantar_n which_o the_o thread_n index_n stand_v unto_o upon_o the_o tangent_fw-la scale_n and_o further_o it_o be_v to_o be_v here_o note_v that_o if_o the_o meridian_n or_o thread_n come_v from_o the_o centre_n be_v right_o divide_v those_o division_n or_o part_n of_o it_o shall_v signify_v the_o degree_n of_o declination_n from_o the_o equinoctial_a but_o the_o inconvenience_n of_o it_o be_v that_o because_o one_o ellipsis_n be_v to_o represent_v every_o almicantar_n therefore_o the_o radius_fw-la can_v possible_o be_v at_o all_o time_n the_o same_o in_o length_n but_o must_v vary_v according_a to_o the_o remove_n or_o several_a position_n of_o the_o say_a ellipsis_n that_o be_v when_o the_o ellipsis_n signify_v the_o horizon_n or_o stand_v at_o the_o begin_n of_o the_o scale_n without_o any_o altitude_n at_o all_o than_o the_o radius_fw-la m_o o_fw-fr in_o the_o first_o move_v plate_n or_o m_o p_o in_o the_o second_o move_v plate_n be_v the_o scale_n of_o declination_n but_o at_o other_o time_n when_o the_o ellipsis_n signify_v any_o almicantar_n then_o must_v the_o secant_fw-la of_o that_o almicantar_n of_o altitude_n be_v take_v as_o the_o radius_fw-la for_o that_o particular_a work_n but_o always_o the_o radius_fw-la of_o what_o length_n soever_o it_o be_v must_v be_v divide_v as_o a_o whole_a line_n of_o sines_n the_o great_a part_n beginning_n towards_o the_o centre_n the_o least_o part_n end_v at_o the_o remote_a end_n from_o the_o centre_n and_o yet_o then_o again_o the_o numeration_n of_o the_o part_n or_o declination_n of_o it_o must_v begin_v at_o the_o remote_a end_n and_o must_v end_v in_o the_o centre_n at_o the_o pole_n it_o may_v be_v apply_v to_o use_v in_o the_o motion_n of_o the_o star_n but_o that_o will_v be_v troublesome_a therefore_o it_o shall_v suffice_v to_o make_v it_o useful_a only_o for_o the_o sun_n the_o best_a way_n that_o for_o the_o present_a i_o know_v whereby_o to_o divide_v this_o radius_fw-la thus_o of_o several_a length_n be_v this_o divide_v the_o length_n from_o the_o centre_n t_o to_o c_o into_o 90_o sin_n the_o great_a part_n beginning_n at_o the_o centre_n but_o number_v the_o contrary_a way_n this_o line_n may_v be_v draw_v from_o the_o centre_n t_o into_o some_o part_n of_o the_o limb_n where_o it_o may_v have_v room_n to_o receive_v division_n then_o with_o your_o compass_n take_v the_o radius_fw-la t_o v_o equal_a to_o m_n o_o and_o from_o the_o end_n of_o the_o former_a scale_n open_v the_o thread_n to_o the_o least_o distance_n and_o where_o the_o thread_n stay_v there_o make_v a_o mark_n upon_o the_o limb_n v_o d_o for_o there_o be_v a_o new_a scale_n to_o begin_v for_o the_o secant_v that_o be_v mention_v before_o then_o lay_v the_o thread_n from_o t_o to_o 80_o in_o the_o opposite_a limb_n and_o move_v the_o ellipsis_n till_o the_o point_n o_o fall_v just_a under_o the_o thread_n and_o take_v t_o o_fw-fr in_o your_o compass_n and_o put_v in_o that_o distance_n as_o before_o so_o shall_v you_o put_v in_o 10_o gr_n into_o this_o new_a scale_n in_o like_a manner_n lay_v the_o thread_n to_o 70_o 60_o etc._n etc._n and_o bring_v the_o ellipsis_n till_o the_o point_n o_o lie_v under_o it_o so_o shall_v t_n o_o in_o these_o several_a position_n give_v the_o length_n to_o be_v inscribe_v in_o this_o new_a scale_n for_o 20_o gr_n and_o 30_o gr_n etc._n etc._n all_o the_o rest_n must_v be_v do_v in_o like_a manner_n till_o you_o have_v put_v in_o as_o many_o as_o will_v come_v within_o the_o reach_n of_o the_o limb_n which_o will_v be_v upon_o 60_o gr_n or_o thereabouts_o the_o like_a shall_v also_o be_v do_v for_o the_o lesser_a ellipsis_n work_v in_o the_o same_o manner_n by_o the_o thread_n and_o the_o point_n p._n these_o be_v call_v the_o new_a inscribe_v scale_n the_o former_a line_n of_o sines_n serve_v to_o they_o both_o indifferent_o now_o when_o thing_n be_v thus_o fit_v the_o use_n of_o they_o will_v be_v such_o as_o in_o particular_a astrolabe_fw-la be_v vulgar_o know_v some_o of_o the_o uses_n be_v here_o mention_v 1._o have_v the_o sun_n declination_n and_o altitude_n to_o find_v the_o hour_n and_o azimuth_n first_o for_o north_n declination_n set_a the_o ellipsis_n to_o the_o altitude_n count_v in_o the_o tangent_fw-la scale_n then_o count_v the_o same_o altitude_n in_o the_o new_a inscribe_v scale_n of_o secant_v and_o thereto_o lie_v the_o thread_n afterward_o take_v the_o least_o distance_n from_o the_o sun_n declination_n count_v in_o the_o forementioned_a inscribe_v scale_n of_o sines_n to_o the_o thread_n set_a one_o foot_n of_o that_o distance_n in_o the_o centre_n and_o extend_v the_o other_o till_o it_o across_o the_o ellipsis_n and_o where_o it_o cross_v thither_o apply_v the_o thread_n which_o in_o the_o limb_n will_v show_v the_o hour_n and_o scruple_n in_o degree_n require_v the_o same_o point_n of_o the_o compass_n do_v also_o immediate_o show_v upon_o the_o ellipsis_n the_o azimuth_n seek_v for_o if_o the_o compass_n do_v cross_v the_o ellipsis_n twice_o as_o sometime_o it_o will_v take_v that_o cross_v that_o be_v further_a from_o the_o meridian_n second_o for_o south_n declination_n the_o work_n be_v in_o a_o manner_n the_o same_o only_o you_o must_v note_v that_o the_o thread_n will_v cross_v the_o ellipsis_n twice_o and_o you_o be_v in_o this_o case_n to_o take_v that_o cross_v which_o be_v near_a to_o the_o meridian_n by_o suppose_v the_o altitude_n to_o be_v 00_o that_o be_v by_o lay_v the_o ellipsis_n upon_o the_o line_n t_o v_n and_o use_v the_o declination_n as_o before_o you_o may_v find_v both_o the_o amplitude_n and_o ascensionall_a difference_n as_o before_o and_o what_o else_o do_v thereon_o depend_v but_o because_o these_o thing_n be_v do_v near_o the_o horizon_n will_v not_o prove_v good_a especial_o when_o the_o declination_n be_v little_a from_o the_o equinoctial_a and_o because_o the_o work_n at_o the_o best_a be_v but_o troublesome_a i_o shall_v here_o break_v off_o suppose_v that_o i_o have_v already_o write_v too_o much_o a_o demonstration_n of_o the_o ellipticall_a dial_n upon_o a_o horizontall_a plain_n show_v the_o reason_n why_o the_o same_o dial_n by_o a_o upright_a index_n shall_v show_v the_o true_a hour_n the_o reason_n be_v principal_o deduce_v from_o the_o sphere_n itself_o and_o second_o from_o the_o orthographical_a projection_n of_o the_o sphere_n upon_o the_o plain_a of_o the_o horizon_n which_o as_o it_o do_v represent_v the_o sphere_n itself_o so_o it_o do_v perform_v the_o same_o conclusion_n with_o the_o like_a certainty_n that_o the_o sphere_n do_v and_o that_o upon_o this_o general_a ground_n that_o
and_o lay_v horizontal_o with_o the_o meridian_n line_n of_o it_o just_a north_n and_o south_n and_o if_o further_o there_o be_v a_o upright_a index_n set_v in_o the_o zenith_n point_n at_o z_o then_o shall_v the_o shadow_n of_o that_o erect_a index_n represent_v the_o azimuth_n in_o which_o the_o sun_n be_v and_o if_o you_o note_v where_o it_o cut_v the_o parallel_n wherein_o the_o sun_n for_o that_o day_n be_v the_o same_o shadow_n will_v show_v among_o the_o hour_n of_o that_o parallel_n the_o time_n of_o the_o day_n this_o conclusion_n be_v true_a upon_o any_o horizontal_n projection_n the_o stereographicall_a as_o well_o as_o the_o orthographical_a for_o the_o straight_a line_n or_o azimuth_n whether_o shadow_n or_o index_n come_v from_o the_o zenith_n and_o cut_v through_o any_o hour_n in_o any_o parallel_n do_v show_v the_o azimuth_n in_o which_o the_o sun_n must_v be_v at_o that_o hour_n for_o the_o day_n of_o that_o parallel_n and_o further_o because_o all_o the_o projection_n be_v make_v upon_o the_o plain_a of_o the_o horizon_n which_o therefore_o must_v be_v the_o fundamental_a circle_n of_o the_o projection_n and_o which_o for_o that_o cause_n be_v and_o must_v be_v as_o all_o fundamental_a circle_n of_o any_o projection_n be_v equal_o divide_v therefore_o a_o upright_a index_n stand_v in_o z_o the_o centre_n of_o it_o and_o so_o answer_v to_o the_o zenith_n line_n of_o the_o sphere_n and_o the_o degree_n of_o the_o horizontall_a circle_n answer_v to_o the_o degree_n of_o the_o horizon_n in_o the_o sphere_n i_o say_v the_o shadow_n of_o that_o index_n do_v real_o show_v the_o true_a azimuth_n of_o the_o sun_n or_o the_o true_a angle_n of_o position_n that_o the_o sun_n at_o any_o time_n make_v with_o the_o meridian_n line_n and_o further_o the_o same_o shadow_n or_o azimuth_n where_o it_o cross_v the_o parallel_n of_o the_o day_n which_o parallel_n be_v divide_v into_o its_o proper_a part_n like_v to_o the_o part_n of_o the_o equinoctial_a by_o the_o meridian_n circle_n issue_v out_o of_o the_o pole_n it_o note_v out_o also_o the_o hour_n of_o the_o day_n so_o that_o the_o projection_n make_v in_o this_o manner_n with_o a_o upright_a index_n and_o divide_a parallel_n will_v by_o the_o shadow_n of_o that_o index_n show_v the_o true_a hour_n of_o the_o day_n upon_o that_o parallel_n that_o be_v proper_a for_o the_o day_n if_o the_o meridian_n of_o the_o projection_n lie_v in_o the_o true_a meridian_n of_o the_o horizon_n the_o next_o thing_n to_o be_v show_v be_v how_o the_o ellipticall_a equinoctial_a may_v supply_v the_o use_n of_o all_o the_o parallel_n that_o be_v to_o say_v how_o the_o equinoctial_a be_v make_v movaable_a and_o the_o index_n stand_v always_o still_o may_v be_v fit_v to_o represent_v any_o of_o the_o sun_n parallel_n and_o because_o as_o be_v say_v before_o it_o be_v every_o way_n like_o to_o each_o of_o the_o parallel_n it_o will_v therefore_o be_v only_o require_v to_o give_v some_o rule_n how_o the_o say_v equinoctial_a may_v be_v at_o any_o time_n place_v in_o a_o like_a position_n to_o the_o zenith_n point_n at_o z_o that_o any_o parallel_n have_v to_o the_o same_o point_n so_o as_o that_o any_o right_a line_n or_o azimuth_n be_v draw_v from_o thence_o may_v cut_v a_o like_a part_n or_o point_n in_o the_o equinoctial_a that_o it_o do_v in_o the_o parallel_n let_v the_o parallel_n be_v h_n e_o i_o will_v remove_v the_o equinoctial_a ae_n a_o so_o as_o that_o it_o may_v have_v a_o like_a position_n to_o the_o point_n z_o that_o the_o parallel_n h_o e_o hath_z by_o the_o corollary_n of_o the_o first_o lemma_n it_o may_v represent_v it_o by_o the_o corollary_n of_o the_o second_o lemma_n the_o way_n of_o it_o will_v be_v easy_a for_o first_o the_o two_o ellipse_n h_n e_o and_o ae_n a_o have_v their_o centre_n upon_o the_o meridian_n line_n ae_n h_o z_o p_o as_o upon_o one_o of_o their_o common_a diameter_n second_o the_o parallel_n h_n e_o cut_n the_o six_o a_o clock_n circle_n or_o meridian_n p_o e_o a_o in_o the_o point_n e_o make_v e_o a_o equal_a in_o radiall_a degree_n to_o h_n ae_n for_o the_o parallel_n be_v every_o where_o equidistant_a to_o the_o equator_fw-la and_o the_o meridional_a ark_n therefore_o intercept_v between_o they_o such_o as_o be_v e_z a_o and_o h_o ae_n must_v be_v equal_a so_o that_o e_z a_o be_v the_o declination_n of_o that_o parallel_n from_o the_o equinoctial_a if_o therefore_o the_o line_n or_o azimuth_n z_o e_o be_v draw_v infinite_o through_o the_o point_n of_o 6_o in_o the_o parallel_n or_o through_o the_o declination_n of_o that_o parallel_n count_v in_o the_o circle_n of_o six_o and_o then_o the_o equinoctial_a still_o keep_v its_o centre_n upon_o the_o common_a diameter_n ae_n p_o be_v slip_v up_o till_o the_o point_n a_o thereof_o do_v concur_v with_o the_o infinite_a line_n draw_v before_o at_o g_o in_o thus_o do_v i_o say_v that_o the_o equinoctial_a be_v in_o a_o like_a situation_n to_o the_o point_n z_o that_o the_o parallel_n h_o e_z be_v in_o to_o the_o same_o point_n and_o consequent_o that_o any_o right_a line_n or_o azimuth_n from_o z_o will_v cut_v the_o same_o rather_o like_a part_n in_o one_o that_o it_o do_v in_o the_o other_o as_o by_o the_o corollary_n of_o the_o second_o lemma_n will_v appear_v the_o next_o thing_n to_o be_v inquire_v be_v what_o the_o line_n a_o g_o be_v or_o how_o it_o must_v be_v find_v and_o estimate_v for_o this_o purpose_n consider_v that_o z_o p_o e_o and_o z_o c_o o_fw-fr be_v two_o triangle_n rectangle_v at_o p_o and_o c_o and_o have_v a_o common_a angle_n at_o z_o therefore_o as_o z_o p_o cousin_a of_o the_o latitude_n to_o z_o c_o the_fw-fr radius_fw-la so_o p_o e_o the_o co-tangent_a of_o the_o declination_n to_o the_o tangent_fw-la of_o c_o o._n or_o as_o the_o radius_fw-la z_o c_o to_o the_o cousin_a of_o the_o latitude_n z_o p_o so_o be_v the_o tangent_fw-la of_o e_z a_o the_o parallel_n declination_n to_o the_o tangent_fw-la of_o a_o o_o that_o be_v to_o the_o line_n a_o g_o which_o be_v the_o tangent_fw-la of_o a_o o_o because_o the_o ark_n a_o o_fw-fr be_v express_v in_o its_o full_a quantity_n without_o any_o enlarge_n or_o fore-shortning_a and_o the_o right_a line_n a_o g_o stand_v to_o it_o as_o a_o tangent_fw-la thereof_o stand_v at_o the_o end_n of_o the_o radius_fw-la z_o a._n now_o consider_v that_o if_o z_o c_o be_v consider_v as_o radius_fw-la then_o must_v the_o tangent_fw-la of_o e_z a_o be_v the_o tangent_fw-la of_o the_o parellel_n declination_n estimate_v to_o the_o same_o radius_fw-la and_o because_o z_o p_o be_v in_o the_o self_n same_o proportion_n to_o a_o c_o therefore_o if_o z_o p_o the_o cousin_a of_o the_o latitude_n be_v estimate_v for_o a_o radius_fw-la then_o must_v a_o g_o be_v esteem_v as_o the_o tangent_fw-la of_o this_o parallel_n declination_n to_o the_o say_v radius_fw-la z_o p._n and_o hence_o it_o will_v follow_v that_o if_o z_o p_o the_o cousin_a of_o the_o latitude_n be_v take_v as_o that_o cousin_a to_o the_o radius_fw-la z_o c_o or_o z_o a_o which_o be_v the_o long_a radius_fw-la of_o the_o ellipticall_a equinoctial_a be_v count_v as_o a_o radius_fw-la or_o tangent_fw-la of_o 45_o gr_n and_o so_o be_v divide_v and_o continue_v as_o occasion_n shall_v be_v it_o be_v to_o be_v note_v i_o say_v that_o this_o scale_n thus_o limit_v will_v be_v a_o right_a scale_n for_o the_o requisite_a motion_n of_o the_o ellipticall_a equinoctial_a to_o such_o a_o position_n in_o respect_n of_o the_o zenith_n point_n as_o that_o it_o may_v represent_v any_o parallel_n who_o declination_n be_v know_v if_o it_o be_v remove_v to_o the_o same_o declination_n count_v upon_o that_o scale_n and_o be_v set_v in_o that_o like_a posture_n any_o azimuth_n or_o shadow_n of_o the_o upright_a index_n that_o will_v pass_v through_o any_o hour_n point_n of_o the_o parallel_n will_v also_o pass_v through_o the_o same_o point_n of_o the_o equinoctial_a and_o so_o this_o one_o may_v serve_v for_o they_o all_o this_o give_v the_o reason_n of_o make_v the_o zodiac_n or_o annual_a scale_n mention_v in_o the_o former_a treatise_n i_o say_v it_o give_v the_o reason_n of_o the_o second_o way_n mention_v for_o lesser_a latitude_n for_o if_o either_o the_o sign_n of_o the_o zodiac_n or_o the_o month_n of_o the_o year_n be_v put_v in_o according_a to_o their_o declination_n take_v in_o this_o scale_n it_o will_v be_v all_o one_o to_o set_v your_o ellipsis_n to_o the_o sign_n or_o month_n that_o it_o be_v to_o set_v it_o to_o the_o declination_n since_o they_o be_v make_v one_o to_o answer_v to_o the_o other_o then_o for_o the_o first_o way_n for_o great_a latitude_n because_o as_o the_o cousin_a of_o
of_o shall_v be_v to_o take_v away_o that_o inconvenience_n in_o use_n which_o be_v in_o the_o former_a and_o therefore_o suppose_v the._n style_n or_o index_n to_o be_v wried_a from_o the_o proper_a meridian_n of_o the_o plain_a to_o the_o meridian_n of_o the_o place_n or_o to_o the_o point_n of_o 12._o it_o may_v as_o well_o be_v put_v to_o any_o other_o just_a hour_n as_o to_o 12_o but_o be_v it_o so_o in_o this_o example_n and_o let_v the_o point_n be_v l_o select_v from_o all_o the_o rest_n from_o the_o select_a point_n l_o draw_v the_o diameter_n l_o a_o m_o and_o from_o the_o select_a point_n l_o divide_v the_o circle_n into_o 24_o equal_a part_n or_o hour_n then_o from_o each_o two_o hour_n point_v correspondent_o distant_a on_o both_o side_n the_o diameter_n l_o m_o draw_n right_n line_n as_o g_z h_z and_o n_o o_o and_o g_o o_o which_o will_v be_v all_o parallel_n to_o the_o diameter_n l_o m._n draw_v these_o line_n i_o say_v and_o continue_v they_o until_o they_o meet_v with_o the_o horologicall_a line_n r_o a_o s_o sufficient_o prolong_v both_o way_n by_o this_o draught_n thus_o make_v the_o horologicall_a line_n r_o a_o s_o shall_v be_v both_o prolong_v to_o h_z o_o and_o also_o divide_v just_a as_o a_o line_n of_o sines_n so_o the_o part_n of_o it_o be_v in_o the_o same_o proportion_n that_o the_o part_n of_o the_o diameter_n g_o a_o g_o be_v which_o be_v right_a sines_n both_o way_n from_o the_o centre_n or_o verse_v sin_n from_o g_o through_o a_o to_o g._n ¶_o note_v that_o here_o as_o former_o you_o may_v assume_v the_o circle_n of_o what_o extent_n you_o list_v and_o shall_v find_v most_o fit_v your_o purpose_n it_o be_v plain_a then_o by_o this_o example_n first_o that_o the_o radius_fw-la of_o the_o circle_n a_o r_o be_v to_o be_v enlarge_v according_a to_o the_o secant_fw-la of_o the_o equinoctial_a distance_n between_o the_o proper_a meridian_n of_o the_o plain_a and_o that_o just_a hour_n whatever_o it_o be_v inro_fw-la which_o the_o index_n be_v make_v to_o point_n as_o you_o see_v that_o ark_n to_o be_v here_o l_o c_o or_o g_o r_o or_o g_o s_o who_o secant_fw-la be_v a_o h_z or_o a_o o_o in_o respect_n of_o the_o assume_v radius_fw-la a_o r._n and_o second_o that_o h_o o_o thus_o enlarge_v be_v divide_v like_o a_o double_a line_n of_o sines_n as_o it_o ought_v to_o be_v and_o for_o number_v the_o hour_n if_o a_o be_v 12_o and_o the_o index_n look_v towards_o the_o south_n then_o must_v o_o be_v 2_o and_o the_o other_o o_o must_v be_v 6_o and_o the_o first_o o_o 10_o at_o night_n and_o h_o must_v be_v 9_o in_o the_o morning_n the_o other_o h_z 6_o in_o the_o morning_n and_o the_o first_o h_o 3_o in_o the_o morning_n but_o if_o the_o index_n have_v not_o point_v up_o into_o 12_o at_o l_o but_o into_o some_o other_o hour_n as_o p_o then_o must_v a_o diameter_n have_v be_v draw_v at_o first_o from_o p_o to_o the_o centre_n a_o quite_o through_o and_o all_o the_o hour_n parallel_v to_o p_o a_o and_o then_o the_o horologicall_a line_n will_v have_v be_v thus_o number_v with_o 7_o at_o the_o two_o extreme_n and_o 1_o in_o the_o midst_n at_o a._n and_o here_o be_v one_o special_a thing_n to_o be_v note_v as_o former_o pag._n 125_o but_o this_o here_o be_v always_o so_o and_o in_o all_o oblique_a orison_n that_o in_o our_o climate_n where_o the_o day_n be_v in_o summer_n long_o than_o 12_o hour_n the_o shadow_n of_o the_o index_n do_v real_o go_v forward_o and_o backward_o as_o to_o make_v it_o plain_a if_o the_o dial_n be_v describe_v upon_o a_o horizontall_a plain_n for_o instance_n and_o it_o be_v our_o long_a day_n than_o the_o shadow_n of_o the_o index_n will_v begin_v at_o 4_o a_o clock_n figure_v below_o the_o line_n and_o will_v go_v thence_o towards_o the_o left_a hand_n to_o 5_o 6_o and_o 7._o at_z 7_o it_o will_v return_v back_o towards_o the_o right_a hand_n through_o 8_o 9_o 10_o 11_o 12_o 1_o 2_o 3_o 4_o 5_o 6_o until_o 7_o and_o then_o be_v arrive_v at_o 7_o the_o utmost_a point_n on_o the_o right_a hand_n it_o will_v come_v back_o again_o towards_o 8_o and_o 9_o to_o the_o left_a hand_n this_o for_o the_o place_n limit_v and_o divide_v of_o the_o horologicall_a line_n ¶_o three_o concern_v the_o zodiac_n how_o it_o must_v be_v place_v and_o limit_v consider_v of_o these_o thing_n the_o zodiac_n in_o this_o case_n of_o wry_v the_o index_n from_o the_o plain_n proper_a meridian_n must_v be_v augment_v in_o length_n and_o change_v in_o situation_n from_o what_o it_o be_v in_o the_o former_a case_n where_o the_o index_n look_v up_o into_o the_o plain_n proper_a meridian_n yet_o i_o will_v here_o first_o begin_v with_o that_o way_n so_o far_o as_o it_o go_v and_o then_o proceed_v to_o what_o be_v further_a needful_a let_v the_o figure_n pag._n 137_o be_v here_o repeat_v to_o the_o radius_fw-la a_o r_o the_o tangent_fw-la of_o 23_o and_o a_o half_a be_v a_o f_o and_o the_o same_o be_v there_o augment_v thus_o as_o the_o radius_fw-la to_o the_o secant_fw-la of_o the_o pole_n elevation_n above_o the_o plain_a so_o let_v of_o be_v suppose_v to_o be_v to_o ab_fw-la so_o that_o if_o the_o index_n be_v point_v direct_o to_o the_o meridian_n point_n in_o the_o equinoctial_a which_o be_v proper_a to_o the_o plain_a than_o the_o business_n be_v at_o a_o end_n for_o a_o b_o must_v be_v the_o length_n of_o the_o zodiac_n and_o stand_v square_a to_o r_o a_o s._n but_o now_o we_o here_o suppose_v the_o index_n to_o be_v wried_a from_o that_o point_n as_o much_o as_o the_o ark_n c_o l_o come_v too_o which_o turn_v aside_o will_v both_o lengthen_v the_o zodiac_n again_o and_o turn_v it_o out_o of_o its_o perpendicular_a stand_v to_o r_o a_o s._n which_o thing_n be_v thus_o to_o be_v effect_v to_o a_o f_o as_o a_o radius_fw-la find_v the_o tangent_fw-la of_o the_o pole_n elevation_n above_o the_o plain_a suppose_v that_o tangent_fw-la be_v g_z h._n then_o say_v as_o the_o radius_fw-la to_o the_o tangent_fw-la of_o c_o l_o the_o quantity_n of_o wry_v so_o g_z h_z to_o x_o z._n make_v b_o k_o equal_a to_o x_o z_o and_o parallel_n to_o a_o r_o and_o draw_v a_o k_o so_o shall_v a_o k_o be_v the_o length_n and_o shall_v also_o give_v in_o this_o position_n the_o situation_n of_o the_o zodiac_n the_o manner_n how_o to_o divide_v it_o need_v not_o be_v repeat_v because_o it_o have_v be_v often_o set_v down_o heretofore_o you_o must_v only_o remember_v to_o place_v this_o line_n upon_o that_o coast_n of_o a_o b_o which_o be_v contrary_a to_o the_o coast_n of_o the_o index_n deviation_n or_o wry_v from_o the_o proper_a meridian_n point_n of_o the_o plain_a this_o work_n may_v either_o be_v protract_v or_o calculate_v or_o wrought_v instrumental_o as_o shall_v best_o be_v like_v of_o another_o way_n i_o will_v here_o add_v to_o do_v the_o same_o work_n which_o may_v perhaps_o be_v not_o inferior_a to_o the_o former_a let_v all_o thing_n preparatorie_n in_o the_o former_a figure_n be_v here_o again_o repeat_v r_o a_o the_o radius_fw-la and_o a_o f_o the_o tangent_fw-la of_o 23_o degree_n and_o a_o half_a to_o that_o radius_fw-la then_o to_o a_o f_o as_o radius_fw-la find_v the_o sine_fw-la and_o cousin_a of_o the_o pole_n elevation_n above_o the_o plain_a let_v f_o p_o be_v the_o sine_fw-la and_o a_o p_o the_o cousin_a make_v a_o m_o equal_a to_o a_o p_o and_o from_o m_o draw_v the_o line_n m_o n_o make_v the_o angle_n n_o m_o c_o equal_a to_o the_o angle_n b_o a_o c_o pag._n 142_o and_o stand_v on_o the_o contrary_a coast_n which_o be_v the_o angle_n of_o the_o declination_n of_o the_o index_n from_o a_o c_o in_o this_o figure_n the_o proper_a meridian_n of_o the_o plain_a then_o say_v as_o the_o radius_fw-la be_v to_o the_o tangent_fw-la of_o the_o reclination_n of_o the_o index_n which_o be_v the_o compliment_n of_o d_o a_o c_o pag._n 142_o the_o elevation_n of_o the_o index_n above_o the_o plain_a so_o be_v the_o sine_fw-la f_o p_o to_o a_o four_o quantity_n suppose_v the_o same_o to_o be_v d_o g_o make_v m_z n_o equal_a to_o d_o g_o and_o draw_v the_o straight_a line_n a_o â—_o so_o shall_v a_o n_o in_o this_o figure_n be_v equal_a to_o a_o k_o in_o the_o precedent_n and_o the_o angle_n n_o a_o c_o be_v here_o equal_a to_o the_o angle_n k_o a_o b_o in_o the_o former_a so_o that_o a_o n_o must_v be_v the_o length_n of_o the_o half_a zodiac_n and_o will_v lie_v in_o its_o true_a position_n by_o this_o work_n this_o may_v be_v do_v either_o by_o calculation_n protraction_n or_o instrument_n as_o all_o other_o work_n may_v