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it_o will_v be_v for_o there_o be_v give_v the_o radius_fw-la c_o d_o and_o the_o tangent_fw-la b_o e_o the_o two_o first_o term_n of_o the_o proportion_n with_o the_o line_n c_o b_o the_o sum_n of_o the_o three_o and_o four_o term_n to_o find_v out_o the_o say_a term_n respective_o and_o it_o will_v hold_v by_o compound_v the_o proportion_n as_o the_o sum_n of_o the_o first_o and_o second_o term_n be_v to_o the_o second_o term_n so_o be_v the_o sum_n of_o the_o three_o and_o four_o term_n to_o the_o four_o term_n that_o be_v as_o c_z d_o +_o b_o e_o be_v to_o b_o e_o so_o be_v c_o f_o +_o f_o b_o =_o c_o b_o to_o f_o b_o see_v 18_o prop._n of_o 5_o of_o euclid_n or_o page_n 18_o of_o the_o english_a clavis_fw-la mathematicae_fw-la of_o the_o famous_a and_o learned_a mr_n oughtre_v after_o the_o same_o manner_n be_v the_o line_n sol_fw-mi or_o proportional_a sin_n make_v that_o be_v also_o such_o a_o line_n that_o any_o ark_n be_v assume_v in_o it_o to_o be_v a_o sine_fw-la the_o 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as_o the_o length_n of_o the_o graduated_a scale_n to_o any_o lesser_a length_n so_o the_o part_n of_o the_o scale_n to_o the_o proportional_a like_o part_n to_o that_o other_o length_n of_o the_o line_n of_o tangent_n on_o the_o left_a edge_n of_o the_o quadrant_n the_o chief_a uses_n of_o this_o scale_n will_v be_v to_o operate_v proportion_n either_o in_o tangent_n alone_a or_o joint_o either_o with_o sines_n or_o equal_a part_n to_o prick_v down_o dial_n and_o to_o proportion_n out_o a_o tangent_fw-la to_o any_o lesser_a radius_fw-la to_o work_v proportion_n in_o tangent_n alone_o 1._o of_o the_o great_a to_o the_o less_o enter_v the_o second_o term_v take_v out_o of_o the_o scale_n upon_o the_o first_o lay_v the_o thread_n to_o the_o other_o foot_n than_o the_o near_a distance_n from_o the_o three_o term_n to_o the_o thread_n be_v take_v out_o and_o measure_v from_o the_o centre_n show_v the_o four_o proportional_a but_o if_o the_o proportion_n be_v of_o the_o less_o to_o the_o 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hence_o it_o may_v be_v observe_v that_o a_o table_n of_o natural_a tangent_n only_o to_o 45_o or_o a_o line_n of_o natural_a tangent_n only_o to_o 45_o may_v serve_v to_o operate_v any_o proportion_n in_o tangent_n whatsoever_o to_o proportion_n on_o out_o a_o tangent_fw-la to_o any_o radius_fw-la enter_v the_o length_n of_o the_o radius_fw-la propose_v upon_o the_o tangent_fw-la of_o 45_o and_o to_o the_o other_o foot_n of_o the_o compass_n lay_v the_o thread_n according_a to_o the_o near_a distance_n then_o if_o the_o respective_a near_a distance_n from_o each_o degree_n of_o the_o tangent_n to_o the_o thread_n be_v take_v out_o they_o shall_v be_v tangent_n to_o the_o assign_v radius_fw-la because_o the_o tangent_n run_v but_o to_o 63_o 26′_n whereas_o there_o may_v be_v occasion_n in_o some_o decline_a dial_n to_o use_v they_o to_o 75_o though_o seldom_o further_o to_o supply_v this_o defect_n they_o may_v be_v suppose_v to_o break_v off_o at_o 60_o and_o be_v supply_v in_o a_o 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altitude_n be_v well_o give_v these_o scale_n in_o their_o use_n presuppose_v the_o hour_n and_o azimuth_n of_o the_o sun_n to_o be_v near_o the_o noon_n meridian_n then_o 60d._o operation_n to_o find_v the_o hour_n take_v the_o distance_n between_o the_o altitude_n and_o the_o declination_n proper_a to_o the_o season_n of_o the_o year_n out_o of_o the_o hour_n scale_n and_o enter_v one_o foot_n of_o this_o extent_n at_o the_o cousin_n of_o the_o declination_n in_o the_o line_n of_o sin_n and_o lay_v the_o thread_n to_o the_o other_o foot_n according_a to_o near_a distance_n it_o show_v the_o hour_n from_o noon_n in_o the_o verse_v sin_n quadruple_v example_n when_o the_o sun_n have_v 23_o 31′_n of_o north_n declination_n and_o 60_o of_o altitude_n the_o hour_n from_o noon_n will_v be_v 13_o 58′_n to_o be_v convert_v into_o time_n when_o the_o hour_n be_v find_v to_o be_v less_o than_o 40_o from_o noon_n the_o former_a extent_n may_v be_v double_v and_o enter_v as_o before_o and_o it_o 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and_o in_o that_o and_o other_o respect_n diversify_v from_o what_o i_o receive_v wherein_o i_o shall_v not_o be_v tedious_a because_o there_o be_v other_o way_n to_o follow_v since_o find_v out_o by_o myself_o the_o chief_a use_n of_o it_o be_v to_o resolve_v the_o two_o case_n of_o the_o four_o axiom_n of_o spherical_a trigonometry_n as_o when_o three_o side_n be_v give_v to_o find_v a_o angle_n or_o two_o side_n with_o the_o angle_n comprehend_v to_o find_v the_o three_o side_n which_o be_v the_o case_n that_o find_v the_o hour_n and_o azimuth_n general_o and_o the_o sun_n altitude_n on_o all_o hour_n for_o the_o hour_n the_o learned_a author_n think_v meet_v to_o add_v a_o zodiaque_fw-la of_o the_o suns-place_n annex_v to_o it_o both_o in_o the_o use_n of_o his_o sector_n as_o also_o
night_n and_o the_o difference_n between_o 90_o and_o either_o of_o those_o ark_n be_v the_o ascensional_a difference_n or_o time_n of_o rise_v and_o set_v from_o 6._o to_o find_v the_o azimuth_n general_o the_o proportion_n for_o this_o purpose_n have_v be_v deliver_v before_o from_o which_o it_o may_v be_v observe_v that_o there_o be_v no_o two_o term_n fix_v and_o therefore_o to_o every_o altitude_n the_o contain_v side_n of_o the_o triangle_n namely_o the_o compliment_n both_o of_o the_o altitude_n and_o latitude_n must_v be_v sum_v and_o difference_v when_o the_o proposition_n be_v to_o be_v perform_v on_o this_o line_n sole_o and_o the_o operation_n will_v be_v after_o the_o same_o manner_n as_o for_o the_o hour_n namely_o with_o a_o parallel_n entrance_n and_o this_o be_v all_o i_o shall_v say_v of_o the_o author_n general_a way_n and_o of_o any_o other_o that_o he_o use_v i_o never_o hear_v of_o those_o way_n that_o follow_v be_v of_o my_o own_o supply_n by_o help_v of_o this_o 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the_o say_v 20d._o this_o line_n sol_n be_v make_v use_n of_o by_o m._n foster_n in_o his_o scale_n for_o dyall_v the_o line_n of_o verse_v sin_n be_v place_v on_o the_o left_a edge_n of_o the_o foreside_n of_o the_o quadrant_n for_o the_o ready_a take_v out_o the_o difference_n of_o the_o verse_v sin_n of_o any_o two_o ark_n and_o to_o measure_v a_o difference_n of_o two_o verse_v sin_n upon_o it_o which_o be_v the_o chief_a use_n i_o shall_v make_v of_o it_o whereas_o to_o operate_v single_o upon_o it_o it_o will_v be_v more_o convenient_a for_o the_o hand_n to_o have_v it_o lie_v on_o the_o right_a edge_n of_o the_o quadrant_n a_o example_n for_o find_v the_o azimuth_n general_o by_o help_n of_o verse_v sin_n in_o the_o limb_n and_o of_o other_o line_n on_o the_o quadrant_n i_o shall_v rehearse_v the_o proportion_n as_o the_o cousin_n of_o the_o latitude_n be_v to_o the_o secant_fw-la of_o the_o altitude_n or_o as_o the_o cousin_n of_o the_o altitude_n be_v to_o the_o secant_fw-la of_o the_o latitude_n so_o be_v the_o difference_n of_o the_o verse_v sin_n of_o the_o sun_n distance_n from_o the_o elevate_v pole_n and_o of_o the_o ark_n of_o difference_n between_o the_o latititude_n and_o altitude_n to_o the_o verse_v sine_fw-la of_o the_o azimuth_n from_o the_o midnight_n meridian_n and_o make_v the_o latter_a clause_n of_o the_o three_o term_v the_o compliment_n of_o the_o sum_n of_o the_o latitude_n and_o altitude_n to_o a_o semicircle_n the_o proportion_n will_v find_v the_o verse_v sine_fw-la of_o the_o azimuth_n from_o the_o noon_n meridian_n example_n altitude_n 51_o 32′_n latitude_n 34_o 32_o compliment_n 55_o 2d_o difference_n 17_o 00_o ☉_o distance_n from_o elevate_a pole_n 66_o 29_o operation_n in_o the_o first_o term_n of_o the_o proportion_n on_o the_o line_n of_o verse_v sin_n take_v the_o distance_n between_o 17_o and_o 66_o 29′_n and_o enter_v it_o twice_o down_o the_o line_n of_o sines_n from_o the_o centre_n take_v the_o near_a distance_n to_o the_o thread_n lay_v over_o the_o secant_fw-la of_o 51_o 32′_n the_o give_v altitude_n and_o enter_v one_o foot_n of_o this_o extent_n at_o the_o sine_fw-la of_o 55_o 28′_n the_o compliment_n of_o the_o latitude_n lay_v the_o thread_n to_o the_o other_o foot_n according_a to_o near_a distance_n and_o in_o the_o line_n of_o verse_v sin_n in_o the_o limb_n it_o will_v lie_v over_o 95_o for_o the_o sun_n azimuth_n from_o the_o midnight_n meridian_n and_o the_o sun_n declination_n suppose_v the_o same_o he_o shall_v have_v the_o like_a azimuth_n from_o the_o north_n in_o our_o latitude_n of_o london_n when_o his_o altitude_n be_v 34_o 32′_n for_o the_o side_n of_o the_o triangle_n be_v the_o same_o another_o example_n to_o find_v it_o in_o the_o verse_v sine_fw-la of_o 90_o latitude_n 47_o 27′_n altitude_n 51_o 32_o sum_n 98_o 59_o compliment_n 81_o 1_o polar_a distance_n 66_o 29_o take_v the_o distance_n in_o the_o line_n of_o sines_n as_o represent_v the_o former_a half_n of_o a_o line_n of_o verse_v sin_n between_o these_o two_o ark_n count_v towards_o the_o centre_n viz._n 66_o 29′_n and_o 81_o 01′_n and_o enter_v this_o extent_n twice_o down_o the_o line_n of_o sines_n from_o the_o centre_n and_o take_v the_o near_a distance_n to_o the_o thread_n lie_v over_o the_o secant_fw-la of_o the_o latitude_n 47_o 27′_n then_o enter_v one_o foot_n of_o this_o extent_n at_o 51_o 32′_n count_v from_o the_o end_n of_o the_o sin_n towards_o the_o centre_n lay_v the_o thread_n to_o the_o other_o foot_n according_a to_o near_a distance_n and_o in_o the_o verse_v sine_fw-la of_o 90_o it_o show_v the_o azimuth_n to_o be_v 65_o from_o the_o south_n in_o this_o our_o northern_a hemispere_n of_o the_o fit_v particular_a scale_n and_o the_o line_n of_o entrance_n thereto_o belong_v this_o scale_n serve_v to_o find_v both_o the_o hour_n and_o azimuth_n in_o the_o latitude_n of_o london_n to_o which_o it_o be_v fit_v in_o the_o equal_a limb_n by_o a_o lateral_a or_o positive_a entrance_n it_o consist_v of_o two_o line_n of_o sines_n the_o great_a be_v 62_o of_o a_o sine_fw-la as_o large_a as_o can_v stand_v upon_o the_o quadrant_n the_o radius_fw-la of_o the_o lesser_a sine_fw-la be_v make_v equal_a to_o 51_o 32′_n of_o this_o great_a be_v fit_v to_o the_o latitude_n the_o scale_n of_o entrance_n stand_v within_o the_o projection_n and_o abut_a on_o the_o line_n of_o sines_n be_v no_o other_o but_o a_o portion_n of_o a_o line_n of_o sines_n who_o radius_fw-la be_v make_v equal_a to_o 38_o 28′_n of_o the_o great_a sine_fw-la of_o the_o fit_a scale_n and_o this_o scale_n of_o entrance_n be_v number_v by_o its_o compliment_n up_o to_o 62_o as_o much_o as_o be_v the_o suns-great_a meridian_n altitude_n in_o this_o latitude_n the_o ground_n of_o this_o scale_n be_v derive_v from_o the_o diagonal_a scale_n the_o length_n whereof_o bear_v such_o proportion_n to_o the_o line_n of_o sines_n whereto_o it_o be_v fit_v as_o the_o secant_fw-la of_o the_o particular_a latitude_n do_v to_o the_o radius_fw-la which_o be_v the_o same_o that_o the_o radius_fw-la bear_v to_o the_o cousin_n of_o the_o latitude_n and_o consequent_o make_v the_o line_n of_o sines_n to_o represent_v the_o fit_a scale_n the_o radius_fw-la of_o that_o sine_fw-la whereto_o it_o be_v fit_v must_v be_v equal_a to_o the_o cousin_n of_o the_o latitude_n and_o so_o we_o need_v no_o particular_a scale_n but_o this_o will_v remove_v the_o particular_a scale_n or_o scale_n of_o entrance_n near_a the_o centre_n and_o will_v not_o have_v be_v so_o ready_a as_o this_o fit_a scale_n however_o hence_o i_o may_v educe_v a_o general_a method_n for_o find_v the_o hour_n and_o azimuth_n in_o the_o limb_n without_o tangent_n or_o secant_v the_o first_o work_n will_v be_v
they_o from_o f_o to_o i_o and_o from_o the_o point_v 1_o ho._n 57_o and_o prick_v they_o from_o f_o to_o k_o and_o from_o the_o point_v 2_o 57_o and_o prick_v they_o from_o f_o to_o e_z and_o from_o the_o point_n thus_o find_v draw_v line_n into_o the_o centre_n then_o for_o the_o other_o side_n of_o the_o square_n make_v c_o f_o the_o radius_fw-la of_o the_o dyall_a tangent_fw-la of_o 3_o hour_n and_o proportion_n out_o tangent_n to_o the_o former_a ark_n namely_o 3′_n and_o prick_v they_o from_o b_o to_o p_o also_o to_o the_o latter_a ark_n 57′_n and_o prick_v they_o from_o a_o to_o n_o 1_o ho._n 3_o and_o prick_v they_o from_o b_o to_z o_o also_o to_o the_o latter_a arks._n 1_o h._n 57_o and_o prick_v they_o from_o a_o to_o m_n 2_o 3_o and_o prick_v they_o from_o b_o to_z l_o also_o to_o the_o latter_a arks._n 2_o 57_o and_o prick_v they_o from_o a_o to_o d_o and_o draw_v line_n from_o these_o termination_n into_o the_o centre_n and_o the_o hour-line_n be_v finish_v after_o the_o same_o manner_n must_v the_o half_n and_o quarter_n be_v finish_v and_o how_o this_o trouble_n in_o proportion_v out_o the_o tangent_n may_v be_v shun_v without_o draw_v any_o line_n on_o the_o plain_a but_o the_o hour-line_n may_v be_v speak_v to_o hereafter_o whereby_o this_o way_n of_o dyall_v and_o those_o that_o follow_v will_v be_v render_v more_o commodious_a last_o the_o style_n may_v be_v prick_v off_o with_o chord_n or_o take_v b_o c_o and_o set_v one_o foot_n in_o f_o with_o that_o extent_n sweep_v the_o touch_n of_o a_o occult_a arch_n and_o from_o c_o draw_v a_o line_n just_o touch_v the_o outward_a extremity_n of_o the_o say_a arch_n and_o it_o shall_v prick_v off_o the_o angle_n of_o the_o style_n height_n above_o the_o substile_a to_o prick_v off_o the_o former_a dyal_n in_o a_o oblique_a parallellogram_n or_o scalenon_fw-mi alias_o unequal_a side_v triangle_n from_o the_o meridian_n first_o in_o a_o oblique_a parallellogram_n draw_v ce_fw-fr the_o meridian_n line_n and_o with_o 60_o of_o a_o line_n of_o chord_n draw_v the_o prick_a arch_n and_o therein_o from_o k_o contrary_a to_o the_o coast_n of_o declination_n prick_v off_o 62_o the_o angle_n of_o 12_o and_o 6_o and_o draw_v the_o line_n c_o d_o for_o the_o say_a hour_n line_v continue_v on_o the_o other_o side_n the_o centre_n and_o out_o of_o a_o line_n of_o sines_n make_v c_o e_o equal_a to_o 65_o the_o compliment_n of_o the_o declination_n then_o take_v out_o the_o sine_fw-la of_o 38_o 28′_n the_o compliment_n of_o the_o latitude_n and_o enter_v it_o in_o the_o line_n d_o c_o so_fw-mi that_o one_o foot_n rest_v at_o d_o the_o other_o turn_v about_o may_v but_o just_o touch_v the_o meridian_n line_n the_o point_n d_o being_n thus_o find_v make_v c_o f_o equal_a to_o c_o d_o and_o with_o the_o side_n c_o f_o and_o c_o e_o make_v the_o parallellogram_n d_o g_o f_o h_o namely_o f_o h_n and_o g_o d_o equal_a to_z c_o e_o and_o e_z g_z and_o e_o h_n equal_a to_o d_o c._n and_o where_o these_o distance_n sweep_v occult_a arch_n therewith_o intersect_n will_v find_v the_o points_z h_z and_o g_z limit_v the_o angle_n of_o the_o parallellogram_n then_z making_z e_z h_z or_o c_z d_o radius_fw-la proportion_n out_o the_o tangent_n of_o 15_o and_o prick_v they_o down_o from_o e_z to_z 1_o and_o 11_o and_o 30_o and_o prick_v they_o down_o from_o e_z to_z 2_o and_o 10_o and_o draw_v line_n into_o the_o centre_n through_o those_o point_n and_o the_o angular_a point_n of_o the_o parallellogram_n at_o h_n and_o g_o and_o there_o will_v be_v 6_o hour_n draw_v beside_o the_o meridian_n line_n or_o hour_n line_n of_o 12._o then_z making_z d_o g_z radius_fw-la proportion_n out_o the_o tangent_fw-la of_o 15_o and_o prick_v it_o down_o from_o d_o upward_o to_o 5_o and_o downward_z to_z 7_o also_o proportion_n out_o the_o tangent_fw-la of_o 30_o and_o prick_v it_o from_o d_o to_z 8_o and_o from_o f_o to_o 4_o and_o draw_v line_n into_o the_o centre_n and_o so_o the_o hour_n line_n be_v 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the_o middle_n of_o the_o projection_n stand_v a_o curve_v or_o bend_a line_n number_v from_o the_o o_o or_o cypher_n both_o way_n one_o way_n to_o 60_o degree_n but_o divide_v to_o 62_o degree_n the_o other_o way_n to_o 20_o degr._fw-la but_o divide_v to_o 23_o deg._n 30′_n the_o invention_n of_o this_o line_n own_v mr._n dary_n for_o the_o author_n thereof_o the_o use_n of_o it_o be_v to_o find_v the_o hour_n or_o azimuth_n in_o that_o particular_a latitude_n whereto_o it_o be_v fit_v by_o the_o extension_n of_o a_o thread_n over_o it_o and_o the_o line_n belong_v to_o it_o the_o line_n belong_v to_o it_o be_v two_o the_o one_o a_o line_n of_o altitude_n and_o declination_n stand_v on_o the_o left_a edge_n of_o the_o quadrant_a be_v no_o other_o but_o a_o line_n of_o sines_n continue_v both_o way_n from_o the_o beginning_n one_o way_n to_o 62_o degree_n the_o other_o way_n to_o 23._o degree_n 30′_n the_o other_o line_n thereto_o belong_v be_v 130_o deg._n of_o a_o line_n of_o verse_v sin_n which_o stand_v next_o without_o the_o projection_n be_v parallel_n to_o the_o left_a edge_n of_o the_o quadrant_a to_o draw_v the_o curve_v draw_v two_o line_n of_o verse_v sin_n it_o matter_n not_o whether_o of_o the_o same_o radius_fw-la or_o no_o nor_o how_o posit_v provide_v they_o be_v parallel_n let_v each_o of_o they_o be_v number_v as_o a_o sine_fw-la both_o way_n from_o the_o middle_n at_o 0_o and_o so_o each_o of_o they_o will_v contain_v two_o line_n of_o sines_n to_o the_o right_a end_n of_o the_o uppermost_a set_v c_o to_o the_o left_a end_n d_o and_o to_o the_o right_a end_n of_o the_o undermost_a set_v a_o and_o to_o the_o left_a end_n b._n first_o note_v that_o there_o be_v a_o certain_a point_n in_o the_o curve_v where_o the_o graduation_n will_v begin_v both_o upward_o and_o downward_o this_o be_v call_v the_o aequinoctial_a point_n to_o find_v it_o lay_v a_o ruler_n from_o a_o to_o the_o compliment_n of_o the_o latitude_n count_v from_o 0_o in_o the_o upper_a scale_n towards_o d_o and_o draw_v a_o line_n from_o a_o to_o it_o then_o count_v it_o the_o other_o way_n towards_o c_o viz._n 38_o degree_n 28′_n for_o the_o co-latitude_n of_o london_n and_o lay_v a_o ruler_n over_o it_o and_o the_o point_n b_o and_o where_o it_o intersect_v the_o line_n before_o draw_v be_v the_o aequinoctial_a point_n to_o be_v graduate_v then_o to_o graduate_v the_o division_n on_o each_o side_n of_o it_o require_v only_o the_o make_n in_o effect_n of_o a_o table_n of_o meridian_n altitude_n to_o every_o degree_n of_o declination_n which_o because_o the_o curve_v will_v also_o serve_v for_o the_o azimuth_n in_o which_o case_n the_o graduation_n of_o the_o curve_v which_o in_o find_v the_o hour_n be_v account_v declination_n must_v be_v account_v altitude_n must_v be_v continue_v to_o 62_o degree_n for_o this_o latitude_n and_o further_a also_o if_o it_o be_v intend_v that_o the_o curve_v shall_v find_v a_o star_n hour_n that_o have_v more_o declination_n to_o 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be_v great_a circle_n first_o know_v that_o the_o reflect_v vertical_a point_n in_o the_o axis_n of_o the_o reflect_v horizon_n will_v always_o be_v find_v in_o the_o reflect_a meridian_n and_o look_v how_o many_o degree_n the_o reflect_a horizon_n differ_v from_o the_o direct_a horizon_n so_o many_o must_v the_o reflect_a axis_n of_o the_o horizon_n differ_v from_o the_o direct_a axis_n of_o the_o horizon_n hence_o the_o reflect_a vertical_a point_n whereby_o the_o reflect_a azimuth-line_n be_v draw_v may_v be_v thus_o find_v take_v that_o thread_n who_o end_n be_v fix_v in_o the_o centre_n of_o the_o glass_n and_o move_v the_o other_o end_n thereof_o to_o &_o fro_o in_o the_o reflect_a meridian_n until_o by_o apply_v one_o side_n of_o a_o quadrant_a thereto_o you_o find_v the_o say_a thread_n depress_v just_o 90_o degree_n or_o perpendicular_a under_o the_o reflect_a horizon_n then_o make_v a_o mark_n or_o point_n where_o the_o other_o end_n of_o the_o say_a thread_n touch_v the_o 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