namely_o that_o it_o be_v a_o right_a line_n fall_v from_o one_o end_n of_o any_o arch_n perpendicular_o upon_o the_o radius_fw-la draw_v to_o the_o other_o end_n of_o the_o say_a arch_n so_o be_v g_o f_o perpendicular_a to_o c_z a_o be_v the_o sine_fw-la of_o the_o arch_n g_o a_o likewise_o a_o i_o fall_v perpendicular_o on_o c_o g_z therefore_o by_o the_o same_o definition_n be_v also_o the_o sine_fw-la of_o the_o say_v arch._n 3._o the_o verse_v sine_fw-la of_o a_o arch_n be_v that_o part_n of_o the_o diameter_n which_o lie_v between_o the_o right_n sine_fw-la of_o that_o arch_n and_o the_o circumference_n thus_o f_o a_o be_v the_o verse_v sine_fw-la of_o the_o arch_n g_o a_o and_o f_o b_o the_o verse_v sine_fw-la of_o the_o arch_n b_o d_o g._n 4._o if_o unto_o one_o end_n of_o a_o arch_n there_o be_v draw_v a_o radius_fw-la and_o to_o the_o other_o end_n a_o right_a line_n from_o the_o centre_n cut_v the_o circle_n and_o if_o from_o the_o end_n of_o the_o radius_fw-la a_o 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the_o three_o first_o term_n shall_v become_v the_o last_o divers_a proportion_n be_v express_v in_o this_o follow_a book_n which_o if_o the_o reader_n will_v apply_v to_o table_n he_o must_v understand_v that_o when_o a_o side_n or_o a_o angle_n be_v great_a than_o a_o quadrant_n that_o in_o stead_n of_o the_o sine_fw-la tangent_fw-la or_o secant_fw-la of_o such_o a_o ark_n he_o must_v take_v the_o sine_fw-la tangent_fw-la or_o secant_fw-la of_o that_o ark_n compliment_n to_o a_o semicircle_n that_o the_o word_n cousin_n or_o cotangent_fw-la of_o a_o arch_n give_v or_o seek_v signify_v the_o sine_fw-la or_o tangent_fw-la of_o the_o compliment_n of_o the_o arch_n give_v or_o seek_v that_o the_o cousin_n or_o cotangent_fw-la of_o a_o arch_n great_a than_o a_o quadrant_n be_v the_o sine_fw-la or_o tangent_fw-la of_o the_o excess_n of_o that_o arch_n above_o a_o quadrant_n or_o 90_o degree_n 8._o what_o trouble_v the_o ancient_n be_v at_o in_o resolve_v of_o proportion_n by_o 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before_o we_o proceed_v any_o further_a it_o will_v be_v necessary_a to_o describe_v such_o line_n as_o be_v upon_o the_o scale_n intend_v to_o be_v treat_v of_o the_o description_n of_o the_o scale_n in_o the_o frontispiece_n 1._o the_o first_o or_o uppermost_a line_n be_v a_o line_n of_o a_o chord_n number_v to_o 180_o degree_n and_o be_v call_v the_o lesser_a chord_n be_v a_o double_a scale_n the_o undermost_a side_n whereof_o be_v number_v with_o half_a those_o ark_n be_v a_o line_n of_o sines_n and_o be_v call_v the_o great_a sin_n at_o the_o end_n of_o this_o scale_n stand_v another_o little_a scale_n which_o be_v call_v the_o lesser_a sin_n be_v number_v with_o 90_o degree_n and_o both_o these_o scale_n seem_v to_o be_v one_o continue_a scale_n 2._o the_o second_o scale_n be_v another_o line_n of_o chord_n call_v the_o great_a chord_n be_v fit_v to_o the_o same_o radius_fw-la with_o the_o great_a sine_fw-la and_o number_v to_o 90_o degree_n annex_v thereto_o be_v a_o single_a 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whereas_o indeed_o one_o of_o each_o kind_n have_v be_v sufficient_a yea_o the_o sin_n may_v have_v be_v whole_o spare_v for_o throughout_o these_o treatise_n nothing_o be_v more_o require_v of_o necessity_n than_o a_o circle_n divide_v into_o equal_a part_n as_o in_o the_o three_o book_n page_n 4._o and_o a_o right_a line_n divide_v into_o equal_a part_n which_o we_o suppose_v in_o every_o man_n power_n to_o do_v if_o he_o have_v compass_n the_o scheme_n throughout_o these_o book_n be_v fit_v either_o to_o the_o radius_fw-la of_o the_o great_a or_o lesser_a chord_n before_o describe_v that_o be_v to_o say_v they_o be_v draw_v with_o 60_o degree_n of_o the_o one_o or_o the_o other_o of_o those_o line_n of_o chord_n chap._n iii_o show_v how_o all_o the_o common_a case_n of_o plain_a
it_o be_v needful_a it_o be_v not_o necessary_a to_o press_v example_n if_o what_o before_o be_v write_v be_v well_o understand_v especial_o in_o this_o case_n where_o all_o direction_n be_v slippery_a thus_o in_o imitation_n of_o maetius_n a_o hollander_n though_o a_o latin_a author_n we_o have_v prescribe_v several_a rule_n for_o the_o correction_n of_o a_o single_a course_n which_o mr._n phillips_n in_o his_o geometrical_a seaman_n make_v but_o one_o rule_n retain_v always_o the_o same_o course_n and_o correct_v the_o distance_n run_v therein_o by_o draw_v a_o parallel_n through_o the_o observe_v latitude_n and_o so_o for_o many_o course_n they_o be_v first_o bring_v all_o into_o one_o line_n and_o the_o distance_n correct_v by_o the_o same_o rule_n but_o concern_v it_o we_o must_v give_v a_o double_a caution_n first_o that_o no_o three_o place_n can_v be_v lay_v down_o true_a in_o their_o course_n and_o distance_n from_o each_o other_o on_o the_o plain_a chart_n as_o shall_v afterward_o be_v handle_v however_o the_o error_n in_o small_a distance_n will_v be_v inconsiderable_a and_o second_o admit_v they_o can_v the_o say_v general_a direction_n be_v unsound_a but_o the_o near_a the_o truth_n the_o near_o the_o course_n be_v to_o the_o meridian_n and_o when_o all_o the_o course_n do_v either_o increase_v or_o diminish_v the_o latitude_n but_o very_o erroneous_a when_o some_o course_n increase_v and_o other_o lessen_v the_o latitude_n in_o all_o which_o case_n it_o be_v most_o safe_a to_o allot_v to_o the_o variation_n or_o dead_a difference_n of_o latitude_n of_o every_o course_n its_o proportional_a share_n of_o the_o whole_a error_n between_o the_o dead_a and_o observe_v latitude_n and_o then_o to_o correct_v each_o course_n by_o the_o former_a direction_n first_o therefore_o in_o the_o follow_a chart_n let_v we_o suppose_v a_o ship_n to_o sail_v from_o a_o in_o the_o latitude_n of_o 28_o south_n south-west_n almost_o 65_o league_n to_o b_o this_o course_n be_v set_v off_o in_o the_o arch_n e_o f_o and_o by_o the_o dead_a reckon_a she_o shall_v now_o be_v in_o the_o latitude_n of_o 25d._o again_o from_o b_o she_o sail_n south-west_n and_o by_o west_n 72_o league_n to_o c_o which_o course_n be_v three_o point_n from_o the_o west_n be_v set_v off_o in_o the_o arch_n g_o h_n and_o now_o by_o the_o dead_a reckon_a she_o shall_v be_v in_o the_o latitude_n of_o 23_o degree_n whereas_o by_o a_o good_a observation_n she_o be_v find_v to_o be_v in_o the_o latitude_n of_o 23_o 30′_n wherefore_o to_o correct_v this_o reckon_a draw_v the_o line_n c_o a_o which_o be_v the_o compound_n course_n arise_v from_o the_o two_o former_a course_n and_o through_o the_o parallel_n of_o observe_v latitude_n draw_v l_o k_n parallel_n to_o a_o w_n so_o be_v the_o point_n k_n the_o correct_a point_n of_o the_o ship_n place_n according_a to_o mr._n phillip_n and_o agree_v with_o the_o truth_n as_o we_o have_v fit_v the_o example_n but_o now_o as_o to_o the_o other_o way_n of_o correct_v a_o compound_n course_n it_o be_v to_o be_v do_v by_o this_o proportion_n first_o find_v the_o variation_n or_o difference_n of_o latitude_n proper_a to_o each_o course_n than_o it_o hold_v as_o the_o sum_n of_o all_o the_o variation_n or_o difference_n of_o latitude_n be_v to_o the_o whole_a error_n between_o the_o dead_a and_o observe_v latitude_n ∷_o so_n be_v each_o particular_a difference_n of_o latitude_n to_o its_o proportional_a share_n of_o the_o whole_a error_n ∷_o than_o if_o the_o difference_n of_o latitude_n fall_v all_o the_o same_o way_n if_o the_o the_o estimate_v difference_n of_o latitude_n be_v too_o much_o you_o must_v abate_v out_o of_o each_o dead_a latitude_n its_o proportional_a error_n so_o in_o this_o case_n the_o say_a error_n be_v prick_v from_o e_z to_z n._n but_o when_o the_o estimate_v difference_n of_o latitude_n be_v too_o little_a the_o proportional_a error_n must_v be_v add_v to_o each_o difference_n of_o latitude_n then_o prick_v the_o second_o dead_a difference_n of_o latitude_n be_v equal_a to_o e_z s_n from_o n_n to_o o_o and_o place_v the_o say_a extent_n from_o a_o the_o centre_n to_o you_o and_o take_v the_o near_a distance_n to_o m_o a_o as_o before_o and_o prick_v it_o from_o o_o to_o l_o be_v the_o second_o error_n this_o be_v needful_a when_o there_o be_v more_o course_n than_o two_o but_o for_o the_o last_o course_n not_o at_o all_o necessary_a neither_o be_v it_o for_o this_o then_o through_o the_o point_n n_n draw_v the_o line_n n_o f_o parallel_n to_o a_o w_n so_o be_v f_o the_o correct_a point_n of_o the_o ship_n place_n at_o the_o first_o course_n then_o draw_v f_o k_o parallel_n to_o b_o c_o and_o where_o the_o parallel_n of_o latitude_n cut_v it_o as_o at_o k_o be_v the_o correct_a point_n of_o the_o ship_n place_n at_o the_o second_o course_n be_v the_o same_o we_o find_v it_o before_o the_o other_o way_n but_o in_o stead_n of_o the_o second_o course_n and_o distance_n which_o be_v 72_o league_n south-west_n and_o by_o west_n let_v we_o now_o suppose_v the_o ship_n sail_n the_o same_o distance_n from_o the_o point_n b_o northwest_n and_o by_o west_n which_o course_n be_v as_o much_o on_o the_o other_o side_n the_o west_n make_v g_o i_o equal_a to_o g_o h_n and_o draw_v the_o course_n b_o i_o therein_o prick_v off_o the_o former_a distance_n to_o d_o so_o be_v d_o the_o point_n of_o the_o ship_n dead_a reckon_n in_o the_o latitude_n of_o 27_o and_o now_o suppose_v the_o observe_v latitude_n to_o be_v 27_o 30′_n the_o error_n and_o difference_n of_o latitude_n be_v as_o much_o now_o as_o they_o be_v before_o wherefore_o draw_v d_o a_o the_o compound_n rumbe_n draw_v q_o t_o parallel_n to_o a_o w_n &_o where_o it_o cut_v the_o compound_a rumbe_n as_o at_o p_o by_o mâ—_n phillip_n his_o reckon_n be_v the_o correct_a point_n of_o the_o ship_n place_n at_o the_o end_n of_o the_o second_o course_n whereas_o in_o truth_n it_o shall_v happen_v at_o t_o and_o so_o p_o bear_v from_o a_o in_o this_o example_n 76_o 43′_n from_o the_o meridian_n and_o be_v distant_a from_o it_o 43_o league_n and_o a_o half_a whereas_o the_o ship_n true_a course_n from_o a_o to_o t_o be_v 83_o 32′_n from_o the_o meridian_n and_o the_o distance_n almost_o 89_o league_n which_o be_v very_o considerable_a now_o for_o as_o much_o as_o the_o sum_n of_o the_o difference_n of_o latitude_n a_o e_o and_o e_o f_o in_o this_o latter_a example_n be_v equal_a to_o a_o s_n in_o the_o former_a example_n also_o the_o error_n f_o q_n here_o be_v equal_a to_o s_o l_o there_n therefore_o the_o proportional_a part_n of_o each_o error_n will_v be_v the_o same_o as_o before_o then_o if_o some_o course_n decrease_v the_o latitude_n southward_o and_o other_o increase_v it_o north-ward_o if_o the_o dead_a latitude_n be_v too_o little_a as_o in_o this_o example_n consider_v that_o to_o place_v the_o ship_n more_o north-ward_o so_o as_o to_o allot_v to_o each_o difference_n of_o latitude_n its_o proper_a error_n that_o the_o south-ward_o difference_n of_o latitude_n must_v be_v decrease_v or_o lessen_v and_o the_o north-ward_o increase_v wherefore_o the_o proportion_n of_o the_o error_n be_v place_v from_o e_z to_z n_z and_o the_o point_n f_o find_v as_o before_o in_o like_a manner_n if_o the_o dead_a latitude_n be_v too_o much_o to_o bring_v the_o ship_n more_o south-ward_o the_o southern_a difference_n of_o latitude_n must_v be_v increase_v and_o the_o northern_a decrease_v now_o the_o point_n t_n be_v find_v by_o draw_v a_o line_n from_o f_o the_o correct_a point_n of_o the_o first_o course_n parallel_n to_o b_o d_o and_o so_o the_o line_n f_o v_n be_v equal_a to_z b_o d_o be_v the_o ship_n second_o course_n and_o distance_n from_o the_o correct_a point_n f_o then_o in_o regard_n part_n of_o the_o error_n in_o the_o latitude_n be_v suppose_v to_o be_v commit_v as_o well_o in_o the_o latter_a as_o in_o the_o former_a course_n which_o error_n be_v too_o little_a the_o 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the_o distance_n t_o a_o in_o the_o chart_n be_v unknown_a and_o i_o will_v measure_v it_o take_v the_o say_a extent_n and_o prick_v it_o twice_o in_o the_o side_n of_o the_o chart_n from_o h_n and_o it_o will_v reach_v to_o 60_o and_o so_o many_o league_n be_v the_o distance_n require_v and_o so_o any_o other_o extent_n turn_v twice_o over_o show_v the_o distance_n in_o league_n account_v every_o degree_n 10_o league_n and_o be_v turn_v but_o once_o over_o show_v the_o distance_n in_o degree_n and_o decimal_a part_n how_o in_o estimate_v the_o ship_n course_n and_o distance_n to_o allow_v for_o know_a current_n this_o subject_n be_v handle_v by_o mr._n norwood_n in_o his_o seaman_n practice_n at_o the_o end_n and_o by_o mr._n phillip_n in_o his_o advancement_n of_o navigation_n page_n 54_o to_o 64._o as_o also_o how_o to_o how_o find_v they_o out_o by_o compare_v the_o reckon_n outward_o between_o two_o place_n with_o that_o homeward_o wherefore_o i_o shall_v be_v very_o brief_a about_o it_o and_o perform_v that_o with_o scale_n and_o compass_n which_o be_v by_o they_o do_v with_o table_n 1._o if_o you_o stem_v a_o current_n if_o it_o be_v swift_a than_o the_o ship_n way_n you_o fall_v a_o stern_a but_o if_o it_o be_v slow_a you_o get_v on_o head_n so_o much_o as_o be_v the_o difference_n between_o the_o way_n of_o the_o ship_n and_o the_o race_n of_o the_o current_n example_n if_o a_o ship_n sail_v 7_o miles_n north_n in_o a_o hour_n by_o estimation_n against_o a_o current_n that_o set_v south_n
what_o be_v here_o accomplish_v by_o the_o table_n may_v ready_o be_v perform_v by_o a_o meridian-line_n out_o of_o which_o with_o compass_n take_v the_o distance_n between_o both_o latitude_n and_o prick_v it_o from_o l_o towards_o m_n at_o the_o end_n whereof_o raise_v a_o perpendicular_a and_o therein_o prick_v down_o out_o of_o the_o equinoctial_a degree_n or_o equal_a part_n the_o difference_n of_o longitude_n draw_v a_o line_n from_o it_o to_o l_o which_o shall_v pass_v through_o the_o former_a point_n o_o whatsoever_o be_v the_o radius_fw-la whereto_o the_o meridian-line_n be_v fit_v and_o after_o the_o same_o manner_n the_o error_n of_o the_o plain_a chart_n be_v to_o be_v remove_v when_o place_n be_v at_o first_o lay_v down_o in_o it_o according_a to_o their_o longitude_n and_o latitude_n which_o be_v most_o easy_o and_o sudden_o do_v especial_o if_o a_o meridian-line_n on_o a_o rod_n fit_v to_o the_o degree_n of_o longitude_n on_o the_o plain_a chart_n and_o the_o manner_n of_o measure_v a_o westward_o distance_n will_v be_v the_o same_o as_o in_o mercator_n chart._n before_o we_o proceed_v to_o the_o demonstration_n of_o mercators_n chart_n it_o will_v be_v necessary_a to_o search_v into_o the_o nature_n of_o the_o rumbe-line_n on_o the_o globe_n and_o to_o collect_v what_o we_o find_v observable_a in_o foreign_a author_n to_o this_o purpose_n 1._o they_o define_v it_o to_o be_v such_o a_o line_n that_o make_v the_o same_o angle_n with_o every_o meridian_n through_o which_o it_o pass_v and_o therefore_o can_v be_v no_o arch_n of_o a_o great_a circle_n for_o suppose_v such_o a_o circle_n to_o pass_v through_o the_o zenith_n of_o some_o place_n not_o under_o the_o equinoctial_a make_v a_o oblique_a angle_n with_o the_o meridian_n then_o shall_v it_o make_v a_o great_a angle_n with_o all_o other_o meridian_n then_o with_o that_o through_o which_o it_o at_o first_o pass_v therefore_o the_o rumbe_n which_o make_v the_o same_o angle_n with_o every_o meridian_n must_v needs_o be_v a_o line_n curve_v and_o bend_v which_o some_o call_v a_o helix-line_n and_o other_o because_o it_o be_v suppose_v to_o be_v on_o the_o surface_n of_o the_o sphere_n a_o helispherical-line_n 2._o another_o property_n of_o the_o rumbe-line_n be_v that_o it_o lead_v near_a and_o near_o unto_o one_o of_o the_o pole_n but_o never_o fall_v into_o it_o and_o near_o the_o pole_n it_o turn_v often_o round_a the_o same_o in_o many_o spire_n and_o turn_n if_o by_o any_o rumbe_n but_o not_o under_o the_o meridian_n it_o be_v suppose_v possible_a to_o sail_v direct_o under_o the_o pole_n point_n it_o will_v follow_v that_o the_o meridian-line_n of_o mercators_n chart_n shall_v be_v finite_a and_o not_o infinite_a and_o that_o one_o and_o the_o same_o line_n shall_v cut_v infinite_a other_o line_n as_o such_o at_o equal_a angle_n in_o a_o mere_a point_n which_o be_v against_o the_o nature_n and_o definition_n of_o a_o line_n and_o then_o see_v that_o the_o rumbe_n derive_v its_o definition_n from_o the_o angle_n it_o make_v with_o the_o meridian_n the_o nature_n and_o use_n of_o a_o meridian_n under_o the_o pole_n cease_v and_o consequent_o the_o rumbe_n cease_v for_o under_o the_o pole_n no_o star_n or_o other_o immovable_a point_n of_o the_o heaven_n do_v either_o rise_n or_o set_v in_o reference_n whereto_o the_o word_n meridian_n have_v its_o original_n be_v use_v to_o signify_v the_o midday_n time_n or_o high_a noon_n of_o the_o sun_n or_o star_n relate_v to_o their_o proper_a motion_n moreover_o under_o the_o pole_n point_v the_o side_n of_o the_o rumbe-triangle_n in_o the_o sphere_n cease_v wherefore_o the_o angle_n must_v needs_o do_v so_o too_o of_o the_o nature_n of_o the_o triangle_n make_v upon_o the_o globe_n by_o the_o rumbe-line_n and_o in_o the_o whole_a rumbe-triangle_n ae_n t_o m_o the_o whole_a distance_n be_v ae_n m_o the_o whole_a difference_n of_o latitude_n be_v ae_n t_o but_o the_o whole_a departure_n from_o the_o meridian_n be_v not_o give_v in_o any_o one_o triangle_n but_o in_o divers_a triangle_n and_o be_v by_o supposition_n as_o much_o in_o one_o triangle_n as_o another_o to_o wit_n the_o whole_a departure_n from_o the_o meridian_n be_v the_o sum_n of_o a_o b_o c_o d_o e_o f_o g_o h_o i_o k_o l_o m_o each_o of_o which_o portion_n be_v suppose_v to_o be_v equal_a each_o to_o other_o and_o the_o whole_a difference_n of_o longitude_n be_v the_o arch_n ae_n q_n the_o segment_n whereof_o q_n l_o q_o s_o s_o r_o can_v be_v equal_a to_o each_o other_o because_o the_o ark_n or_o departure_n l_o m_o i_o k_o etc._n etc._n be_v suppose_v to_o be_v equal_a each_o to_o other_o not_o in_o the_o same_o but_o in_o different_a parallel_n of_o latitude_n now_o then_o for_o all_o use_n in_o navigation_n we_o suppose_v the_o segment_n of_o the_o rumbe_n m_o k_o k_o h_o and_o the_o rest_n to_o be_v a_o right_a line_n and_o if_o we_o give_v the_o rumbe_n to_o wit_n the_o angle_n l_o k_o m_o and_o the_o side_n k_o l_o we_o may_v by_o the_o doctrine_n of_o plain_a triangle_n find_v the_o side_n k_o m_o the_o distance_n by_o the_o common_a proportion_n as_o the_o cousin_n of_o the_o rumbe_n from_o the_o meridian_n be_v to_o the_o radius_fw-la ∷_o so_n be_v the_o difference_n of_o latitude_n to_o the_o distance_n ∷_o and_n from_o hence_o we_o may_v observe_v another_o property_n in_o the_o rumbe_n as_o namely_o that_o the_o segment_n or_o piece_n thereof_o contain_v between_o two_o parallel_n have_v the_o like_a difference_n of_o latitude_n be_v also_o equal_a to_o each_o other_o so_o that_o the_o like_a distance_n be_v sail_v in_o the_o same_o rumbe_n in_o several_a part_n of_o the_o earth_n shall_v cause_v the_o like_a difference_n or_o alteration_n of_o latitude_n in_o each_o of_o those_o part_n for_o example_n if_o a_o ship_n sail_n from_o the_o latitude_n of_o 10_o north-east_n till_o she_o be_v in_o the_o latitude_n of_o 30_o and_o then_o sail_v from_o thence_o on_o the_o same_o rumbe_n till_o she_o be_v in_o the_o latitude_n of_o 50_o the_o latter_a distance_n shall_v be_v equal_a to_o the_o former_a distance_n and_o the_o like_a shall_v hold_v from_o thence_o to_o 70_o and_o the_o same_o also_o shall_v hold_v from_o 70_o to_o the_o pole_n if_o it_o be_v possible_a to_o sail_v thither_o by_o any_o rumbe_n and_o the_o reason_n be_v because_o in_o the_o triangle_n k_o l_o m_o the_o angle_n at_o k_n and_o l_o with_o the_o side_n l_o k_o be_v equal_a by_o construction_n to_o the_o angle_n at_o h_z and_z i_z and_o the_o side_n i_o h_o in_o the_o triangle_n i_o h_o k_o and_o the_o like_a in_o any_o other_o triangle_n wherefore_o if_o these_o be_v give_v to_o find_v the_o side_n m_o k_n or_o k_o h_o it_o must_v needs_o be_v find_v the_o same_o in_o both_o again_o if_o in_o the_o former_a triangle_n we_o shall_v give_v the_o rumbe_n to_o wit_n the_o angle_n l_o k_o m_o and_o the_o difference_n of_o latitude_n l_o k_o we_o may_v find_v the_o departure_n from_o the_o maridian_n l_o m_z by_o this_o proportion_n as_o the_o radius_fw-la be_v to_o the_o tangent_fw-la of_o the_o rumbe_n viz._n tang._n l_o k_o m_o so_o be_v the_o difference_n of_o latitude_n 1_o minute_n to_o wit_n k_o l_o to_o the_o departure_n from_o the_o meridian_n l_o m_o ∷_o now_o such_o proportion_n as_o one_o circle_n have_v to_o another_o such_o proportion_n have_v their_o degree_n semidiameters_a and_o sin_n of_o like_a arch_n one_o unto_o another_o as_o in_o the_o former_a scheme_n a_o c_o be_v the_o radius_fw-la of_o the_o equinoctial_a and_o e_z d_o be_v the_o radius_fw-la of_o a_o parallel_n or_o lesser_a circle_n who_o latitude_n from_o the_o equinoctial_a be_v a_o e_o but_z e_z d_o be_v the_o sine_fw-la of_o the_o ark_n e_o b_o which_o be_v the_o compliment_n of_o the_o latitude_n a_o e._n it_o therefore_o hold_v as_o the_o cousin_n of_o the_o latitude_n be_v to_o the_o radius_fw-la ∷_o or_o rather_o with_o other_o term_n in_o the_o same_o proportion_n as_o the_o radius_fw-la be_v to_o the_o secant_fw-la of_o the_o latitude_n ∷_o so_n be_v a_o mile_n or_o any_o part_n thereof_o or_o number_n of_o miles_n to_o the_o difference_n of_o longitude_n answer_v thereto_o ∷_o and_n because_o the_o two_o first_o term_n of_o the_o proportion_n vary_v not_o it_o will_v hold_v after_o the_o manner_n of_o the_o compound_v rule_n of_o three_o as_o the_o radius_fw-la be_v to_o the_o sum_n of_o the_o secant_v of_o all_o the_o parallel_n or_o latitude_n between_o any_o two_o place_n and_o we_o allow_v a_o parallel_n to_o pass_v through_o every_o minute_n or_o centesme_fw-fr of_o a_o degree_n so_o be_v a_o mile_n or_o any_o number_n or_o part_v thereof_o allot_v to_o every_o parallel_n to_o the_o whole_a difference_n of_o
proportion_n of_o the_o same_o nature_n and_o after_o the_o same_o manner_n as_o we_o find_v the_o azimuth_n at_o six_o before_o by_o the_o analemma_n be_v the_o say_a proportion_n protract_v yet_o here_o it_o be_v to_o be_v suggest_v that_o in_o the_o analemma_n there_o be_v three_o proportion_n in_o sines_n wrought_v instead_o of_o the_o one_o in_o sines_n and_o tangent_n before_o express_v 1._o as_o namely_o to_o find_v the_o sun_n altitude_n at_o six_o as_o the_o radius_fw-la be_v to_o the_o sine_fw-la of_o the_o latitude_n so_o be_v the_o sine_fw-la of_o the_o declination_n to_o the_o sine_fw-la of_o the_o sun_n height_n at_o six_o 2._o to_o find_v his_o azimuth_n in_o that_o parallel_n of_o altitude_n as_o the_o radius_fw-la be_v to_o the_o cousin_n of_o the_o latitude_n so_o be_v the_o sine_fw-la of_o the_o declination_n to_o the_o sine_fw-la of_o the_o azimuth_n in_o the_o say_a parallel_n 3._o to_o reduce_v it_o to_o the_o common_a radius_fw-la as_o the_o cousin_n of_o the_o altitude_n at_o six_o be_v to_o the_o radius_fw-la so_o be_v the_o sine_fw-la of_o the_o azimuth_n in_o that_o parallel_n to_o the_o sine_fw-la thereof_o in_o the_o common_a radius_fw-la the_o two_o latter_a proportion_n in_o sines_n may_v be_v bring_v into_o one_o as_o i_o have_v show_v in_o a_o treatise_n the_o sector_n on_o a_o quadrant_a pag._n 111_o 114._o and_o that_o will_v be_v as_o the_o cousin_n of_o the_o altitude_n at_o six_o be_v to_o the_o cousin_n of_o the_o latitude_n so_o be_v the_o sine_fw-la of_o the_o declination_n to_o the_o sine_fw-la of_o the_o azimuth_n seek_v and_o thus_o in_o effect_n the_o analemma_n perform_v that_o single_a proportion_n intermingle_v with_o tangent_n after_o a_o more_o laborious_a manner_n in_o sines_n or_o if_o you_o will_v the_o altitude_n at_o six_o be_v find_v it_o hold_v as_o the_o cotangent_fw-la of_o the_o say_a altitude_n be_v to_o the_o radius_fw-la so_o be_v the_o cotangent_fw-la of_o the_o latitude_n to_o the_o sine_fw-la of_o the_o azimuth_n seek_v and_o this_o proportion_n lie_v visible_a in_o the_o analemma_n by_o these_o direction_n derive_v from_o the_o analemma_n together_o with_o the_o proportion_n for_o each_o case_n all_o the_o 16_o case_n of_o right_a angle_a spherical_a triangle_n may_v be_v resolve_v some_o whereof_z seem_v to_o require_v the_o draw_n of_o a_o ellipsis_n as_o namely_o if_o the_o sun_n place_n and_o right_a ascension_n be_v give_v to_o find_v his_o great_a declination_n which_o notwithstanding_o according_a to_o these_o direction_n be_v easy_o shun_v chap._n viii_o show_v how_o to_o come_v by_o the_o sun_n altitude_n or_o height_n without_o instrument_n upon_o any_o flat_a or_o plain_a that_o be_v level_a or_o parallel_v to_o the_o horizon_n erect_v or_o set_v up_o a_o wire_n without_o incline_v or_o lean_v to_o either_o side_n admit_v in_o the_o point_n c_o and_o when_o you_o will_v find_v the_o sun_n altitude_n or_o height_n make_v a_o mark_n at_o that_o instant_n in_o the_o very_a end_n or_o extremity_n of_o its_o shadow_n suppose_v at_o g_o the_o shadow_n be-being_a the_o line_n c_o g._n then_o upon_o the_o same_o flat_a draw_v the_o quadrant_a of_o a_o circle_n c_o a_o f_o with_o 60_o of_o your_o chord_n and_o make_v c_o e_o equal_a to_o the_o height_n of_o the_o wire_n and_o through_o the_o point_n e_o draw_v the_o line_n e_o d_o parallel_n to_o c_o a_o and_o therein_o prick_v down_o the_o length_n of_o the_o shadow_n from_o e_z to_z d_o a_o ruler_n lay_v from_o the_o centre_n to_o d_o cut_v the_o quadrant_a at_o b_o and_o the_o arch_n b_o a_o measure_a on_o the_o chord_n show_v the_o height_n require_v in_o this_o example_n 20_o 25′_n in_o like_a manner_n if_o the_o length_n of_o the_o shadow_n be_v e_o k_o the_o height_n will_v be_v n_o a_o 38_o 16′_n otherwise_o this_o may_v be_v perform_v otherways_o by_o draw_v the_o quadrant_a c_o at_fw-fr f_o upon_o any_o plain_a board_n whatsoever_o then_o stick_v in_o a_o pin_n at_o the_o centre_n c_o and_o hold_v the_o board_n so_o towards_o the_o sun_n that_o the_o shadow_n thereof_o may_v fall_v upon_o the_o line_n c_o a_o then_o imagine_v c_o g_z to_o represent_v or_o supply_v the_o use_n of_o a_o thread_n and_o plummet_n hang_v upon_o the_o pin_n in_o the_o centre_n at_o liberty_n and_o mark_v where_o it_o cut_v the_o arch_n of_o the_o quadrant_a f_o at_fw-fr suppose_v at_o h_n measure_v the_o arch_n f_o h_n on_o the_o line_n of_o chord_n and_o it_o show_v the_o height_n requiâ—ed_v by_o the_o next_o chapter_n we_o shall_v find_v the_o sun_n azimuth_n belong_v to_o the_o altitude_n 20_o 25′_n according_a to_o the_o latitude_n and_o declination_n there_o give_v to_o be_v 31_o 19′_n from_o the_o meridian_n admit_v to_o the_o westward_n of_o the_o south_n then_o do_v the_o shadow_n happen_v so_o much_o to_o the_o eastward_o of_o the_o noâ—th_n wherefore_o if_o 31_o 19′_n be_v set_v off_o in_o the_o quadrant_a from_o the_o line_n of_o shadow_n from_o h_n to_o n_n a_o line_n draw_v into_o the_o centre_n as_o n_n c_o shall_v be_v a_o true_a meridian_n line_n or_o line_n of_o north_n and_o south_n chap._n ix_o show_v the_o resolution_n of_o such_o proposition_n wherein_o the_o sun_n altitude_n iâ—_n either_o give_v or_o seek_v the_o latitude_n of_o the_o place_n declination_n of_o the_o sun_n and_o his_o altitude_n give_v to_o find_v the_o hour_n of_o the_o day_n and_o the_o azimuth_n of_o the_o sun_n example_n declination_n 13_o south_n altitude_n 14_o 40′_n with_o 60_o of_o the_o chord_n draw_v the_o circle_n s_o z_o n_o e_o the_o centre_n whereof_o be_v at_o c_o and_o draw_v the_o diameter_n s_o c_o n_n and_o perpendicular_a thereto_o z_o c_o prick_v off_o the_o latitude_n 51_o 3′_n from_o n_n to_o p_o and_o from_o z_o to_z ae_n and_o draw_v the_o axis_n b_o p_o and_o the_o equator_fw-la ae_n c_o q_n prick_v off_o the_o declination_n from_o ae_n to_z d_o and_o from_o q_o to_o e_o being_z 13_o from_o the_o chord_n and_o draw_v the_o parallel_n of_o declination_n d_o b_o e_o then_o from_o s_n to_o a_o and_z from_z n_z to_z o_o out_o of_o the_o chord_n prick_v off_o 14_o 40′_n the_o altitude_n and_o draw_v the_o parallel_n of_o altitude_n a_o o_o so_o be_v b_o ⊙_o the_o hour_n from_o six_o towards_o noon_n in_o the_o parallel_n of_o declination_n and_o m_o ⊙_o the_o azimuth_n of_o the_o sun_n from_o the_o east_n or_o west_n southwards_o to_o measure_v the_o hour_n set_a the_o extent_n b_o d_o from_z c_z to_o f_o and_o upon_o f_o as_o a_o centre_n with_o the_o extent_n b_o ⊙_o draw_v the_o arch_n g_o a_o ruler_n lay_v from_o c_o just_a touch_v that_o arch_n find_v the_o point_n h_n the_o arch_n n_o h_n measure_v on_o the_o chord_n show_v 45_o and_o so_o much_o be_v the_o hour_n from_o six_o to_o wit_n in_o time_n three_o hour_n either_o nine_o in_o the_o forenoon_n or_o three_o in_o the_o afternoon_n to_o measure_v the_o azimuth_n in_o like_a manner_n set_v m_o a_o from_z c_z to_z r_z and_o upon_o r_n with_o the_o extent_n m_o ⊙_o draw_v the_o arch_n i_o a_o ruler_n lay_v thereto_o from_o c_o cut_n the_o limb_n at_o l_o the_o arch_n l_o s_o measure_v on_o the_o chord_n be_v 44_o 35′_n and_o so_o much_o be_v the_o sun_n to_o the_o southwards_o of_o the_o east_n or_o west_n to_o find_v the_o angle_n of_o position_n place_v the_o compliment_n of_o the_o altitude_n z_o a_o from_o p_o to_o g_o then_z place_z the_o extent_n a_o g_o from_o g_o to_o f_z a_o ruler_n lay_v over_o a_o and_z f_z cut_v the_o parallel_n of_o the_o sun_n declination_n at_o it_o and_z d_o t_z be_v the_o verse_v sine_fw-la of_o the_o angle_n of_o position_n to_o measure_v it_o through_o the_o point_n t_o draw_z k_o t_o y_fw-fr parallel_n to_o the_o axis_n b_o p_o and_o draw_v d_o c_o then_z the_o extent_n c_o y_fw-fr measure_a on_o the_o sin_n show_v the_o compliment_n of_o it_o to_o wit_n 62_o 57′_n therefore_o the_o say_a angle_n be_v 27_o 3′_n otherwise_o with_o the_o extent_n b_o d_o set_v one_o foot_n in_o the_o centre_n at_o c_o with_o the_o other_o across_o the_o former_a parallel_n line_n k_o t_o at_o k_n a_o ruler_n lay_v from_o c_o to_o k_n cut_v the_o limb_n at_o x_o and_o the_o arch_n ae_n x_o be_v 27_o 3′_n the_o measure_n thereof_o as_o before_o the_o compliment_n whereof_o be_v the_o arch_n v_o e_o and_o may_v be_v measure_v from_o the_o south_n pole_n v_n if_o the_o equinoctial_a ae_n q_o be_v not_o draw_v three_o side_n give_v to_o find_v a_o angle_n another_o example_n the_o declination_n be_v north_n latitude_n 51_o 32′_n declination_n 13_o north_n altitude_n 37_o 18′_n in_o the_o follow_a scheme_n upon_o the_o centre_n c_o with_o
be_v draw_v through_o the_o three_o point_n will_v fall_v altogether_o within_o the_o outward_a circle_n and_o may_v very_o well_o be_v describe_v and_o it_o will_v cut_v the_o meridian_n s_o n_n in_o two_o point_n the_o one_o towards_o the_o south_n the_o other_o towards_o the_o north_n a_o ruler_n from_o the_o east_n point_fw-fr o_o over_o each_o of_o the_o former_a point_n will_v give_v the_o sun_n south_n and_o north_n meridian_n altitude_n in_o the_o limb_n and_o if_o the_o ark_n between_o these_o meridian_n altitude_n be_v divide_v into_o half_n the_o point_n in_o the_o middle_n so_o find_v be_v the_o latitude_n point_n l_o the_o ark_n between_o which_o and_z n_z show_v the_o latitude_n require_v after_o the_o same_o manner_n may_v the_o point_n l_o be_v find_v in_o all_o other_o case_n which_o notwithstanding_o to_o avoid_v the_o draw_n of_o a_o whole_a vast_a circle_n we_o shun_v though_o when_o the_o sun_n be_v near_o the_o winter_n tropic_n the_o draw_v of_o part_n thereof_o can_v be_v shun_v without_o the_o help_n of_o a_o steel-bowe_n or_o some_o such_o like_a remedy_n whereby_o to_o describe_v part_n of_o a_o circle_n through_o three_o point_n give_v which_o be_v the_o great_a and_o only_a inconvenience_n of_o the_o stereographick_a projection_n clavius_n handle_v this_o proposition_n after_o another_o manner_n but_o not_o so_o convenient_a and_o make_v no_o further_a use_n of_o it_o but_o here_o we_o shall_v also_o apply_v it_o to_o the_o make_n of_o all_o dial_n that_n have_v centre_n except_o the_o equinoctial_a in_o a_o treatise_n of_o geometrical_a dyall_n from_o page_n 72_o to_o 82_o i_o have_v show_v a_o general_a method_n both_o by_o calculation_n and_o with_o scale_n and_o compass_n how_o from_o three_o shadow_n to_o make_v all_o dial_n with_o centre_n from_o whence_o those_o that_o have_v leisure_n may_v calculate_v the_o ark_n find_v in_o the_o former_a scheme_n whereto_o i_o now_o further_o add_v that_o the_o former_a proposition_n here_o insist_v on_o perform_v the_o same_o the_o three_o shadow_n c_o d_o c_o e_o and_z c_o f_o here_o be_v the_o same_o in_o length_n with_o those_o there_o and_o make_v the_o same_o angle_n at_o the_o centre_n here_o as_o those_o do_v at_o the_o foot_n of_o the_o perpendicular_a style_n there_o and_o suppose_v the_o former_a plain_n to_o be_v a_o decline_a lean_v plain_n in_o some_o other_o latitude_n the_o line_n s_o c_o n_n be_v the_o substilar_a line_n and_o the_o arch_n z_o ae_n be_v the_o style_n height_n the_o same_o as_o we_o find_v it_o in_o that_o treatise_n suppose_v this_o be_v a_o plain_a that_o look_v southward_a not_o much_o lean_v from_o the_o zenith_n if_o the_o former_a shadow_n happen_v in_o the_o summer_n half_a year_n in_o north_n declination_n the_o scheme_n will_v describe_v a_o winter_n parallel_n of_o as_o much_o south_n declination_n also_o if_o the_o sun_n have_v south_n declination_n the_o scheme_n will_v describe_v a_o parallel_n of_o as_o much_o north_n declination_n but_o this_o be_v only_o true_a when_o the_o elevate_a pole_n be_v elevate_v above_o the_o northern_a or_o other_o face_n of_o the_o plain_a to_o illustrate_v this_o in_o the_o scheme_n above_o let_v s_o c_o n_n represent_v the_o horizon_n of_o a_o certain_a place_n the_o latitude_n whereof_o be_v z_o ae_n or_o n_n l_o the_o equator_fw-la be_v represent_v by_o ae_n q_n and_o d_o e_z represent_v a_o parallel_n of_o north_n declination_n let_v z_o c_o represent_v a_o vertical_a or_o upright_a south_n plain_a in_o that_o latitude_n but_o if_o the_o say_v vertical_a plain_n become_v a_o horizontal_n plain_n the_o height_n of_o the_o north_n pole_n will_v be_v z_o l_o and_o the_o height_n of_o the_o south_n pole_n h_o k_n i_o say_v then_o in_o respect_n of_o the_o first_o horizon_n the_o sun_n be_v in_o the_o parallel_n of_o north_n declination_n between_o d_o and_z a_o his_o altitude_n on_o any_o hour_n from_o noon_n be_v the_o angle_n between_o the_o wall_n and_o the_o sun_n above_o the_o vertical_a plain_n z_o a_o in_o the_o first_o horizon_n be_v in_o respect_n of_o the_o second_o horizon_n if_o the_o north_n pole_n be_v elevate_v his_o depression_n under_o the_o same_o upon_o the_o like_a hour_n from_o midnight_n which_o be_v equal_a to_o his_o winter_n altitude_n above_o the_o second_o horizon_n z_o c_o h_o the_o sun_n have_v as_o much_o south_n declination_n be_v between_o b_o and_z g_z in_o the_o parallel_n f_o g_z or_o which_o be_v all_o one_o retain_v the_o sun_n declination_n the_o same_o they_o be_v his_o winter_n altitude_n upon_o the_o like_a hour_n from_o noon_n when_o the_o south_n pole_n k_o be_v elevate_v above_o the_o horizon_n h_o c_o z_o which_o be_v the_o reason_n why_o the_o scheme_n describe_v a_o parallel_n of_o south_n declination_n when_o his_o declination_n be_v as_o much_o north._n to_o place_v the_o meridian_n line_n this_o as_o i_o have_v show_v in_o the_o aforementioned_a treatise_n may_v be_v project_v by_o the_o eye_n on_o the_o plain_a by_o help_n of_o a_o thread_n and_o plummet_n and_o if_o the_o plain_n re_fw-mi clination_n in_o clination_n be_v give_v together_o with_o the_o substiles_a distance_n from_o the_o plain_n perpendicular_a both_o which_o may_v be_v get_v without_o dependence_n on_o the_o sun_n the_o meridian_n place_n may_v be_v calculate_v or_o at_o least_o the_o inclination_n of_o meridian_n as_n be_v there_o suggest_v either_o will_v serve_v but_o for_o the_o place_n of_o the_o same_o and_o for_o find_v the_o latitude_n of_o the_o place_n and_o the_o plain_n declination_n though_o not_o require_v in_o the_o make_n of_o the_o dyal_n the_o converse_n of_o the_o dyall_a scheme_n there_o use_v will_v perform_v it_o we_o shall_v take_v the_o example_n there_o use_v let_v the_o substiles_a distance_n from_o the_o plain_n perpendicular_a be_v 32_o 16′_n the_o style_n height_n 41_o 30._o the_o plain_n inclination_n be_v 15._o upon_o five_o as_o a_o centre_n describe_v the_o circle_n s_o e_o n_n w_n and_o draw_v the_o diameter_n s_o v_n n_n and_o w_n v_o e_o make_v right_a angle_n in_o the_o centre_n then_o assume_v five_o n_n to_o represent_v the_o plain_n perpendicular_a set_v off_o the_o substiles_a distance_n from_o it_o the_o same_o way_n it_o happen_v from_o n_n to_o you_o to_o wit_n 32_o 16′_n and_o from_o you_o prick_v off_o the_o style_v height_n to_o k_o from_o the_o point_n k_o let_v fall_v a_o perpendicular_a on_o the_o substile_a at_o i_o and_o draw_v i_o c_o parallel_n to_o n_n v_n from_o which_o point_n let_v fall_v i_o q_o produce_v perpendicular_a to_o the_o plain_n perpendicular_a v_o n_n and_o make_v q_o t_o equal_a to_o i_o k_o and_o upon_o five_o as_o a_o centre_n with_o the_o extent_n v_o t_o describe_v a_o ark_n in_o the_o other_o quadrant_a of_o the_o same_o semicircle_n as_o at_o b_o then_o from_o the_o same_o end_n of_o the_o horizontal_n line_n prick_v the_o inclination_n from_o e_z to_z r_z upward_o and_o draw_v v_o r_n but_o if_o the_o plain_a recline_n it_o must_v be_v prick_v downward_o towards_o n_n then_o with_o the_o extent_n i_o k_n draw_v a_o line_n parallel_n to_o r_o v_n and_o it_o will_v pass_v through_o the_o former_a ark_n at_o b_o which_o here_o we_o find_v by_o enter_v one_o foot_n of_o that_o extent_n in_o that_o ark_n so_o that_o the_o other_o turn_v about_o will_v but_o just_a touch_n five_o r_n have_v discover_v the_o point_n b_o through_o it_o draw_v b_o o_fw-fr parallel_n to_o e_o c_o then_o take_v the_o near_a distance_n from_o c_o to_o r_z v_z and_o place_n it_o in_o the_o line_n o_o b_o from_o p_o on_o that_o side_n thereof_o which_o be_v far_a from_o r_n v._o here_o we_o place_v it_o from_o p_o to_o m_n then_o from_o the_o centre_n draw_v a_o line_n pass_v through_o the_o point_n m_o as_o do_v five_o x_o and_o it_o shall_v be_v the_o meridian_n line_n require_v through_o the_o point_n o_o draw_v five_o d_o and_o the_o arch_n n_o d_o show_v the_o declination_n of_o the_o plain_a to_o be_v 40_o and_o the_o extent_n v_o o_fw-fr be_v the_o cousin_n of_o the_o latitude_n or_o through_o the_o point_n b_o draw_v g_o b_o l_o parallel_n to_o five_o n_n and_o the_o arch_n n_o l_o be_v the_o measure_n of_o the_o latitude_n of_o the_o place_n to_o wit_n 51_o 32′_n now_o have_v place_v the_o substile_a stile_n and_o meridian_n the_o hour-line_n be_v easy_o inscribe_v either_o in_o a_o circle_n or_o paralellogram_n as_o in_o that_o treatise_n be_v large_o show_v for_o the_o reader_n recreation_n and_o practice_n of_o what_o be_v here_o deliver_v and_o for_o trial_n how_o well_o this_o kind_n of_o dyall_v agree_v with_o other_o kind_n he_o may_v make_v a_o dyal_n true_a to_o any_o know_a latitude_n and_o situation_n and_o
thy_o work_n or_o try_v the_o rumb_n n_o ever_o desist_v but_o let_v we_o have_v more_o of_o thou_o h_n ever_o be_v but_o a_o tangent_fw-la but_o let_v we_o have_v a_o sine_fw-la o_o r_o bosom_n full_a of_o thy_o industrious_a toil_n i_o t_o will_v inform_v the_o weak_a enrich_v our_o soyl._n your_o love_a friend_n sylvanus_n morgan_n the_o content_n of_o the_o first_o book_n in_o the_o proportional_a part._n geometrical_a definition_n page_z 1_o 2_o to_o raise_v perpendicular_o 2_o 3_o 4_o to_o draw_v a_o line_n parallel_n to_o another_o line_n 5_o to_o bring_v three_o point_n into_o a_o circle_n 6_o to_o find_v a_o right_a line_n equal_a to_o the_o arch_n of_o a_o circle_n 9_o 10_o chord_n sin_n tangent_n secant_v verse_v sines_n etc._n etc._n define_v 11_o 12_o 13_o the_o scale_n in_o the_o frontispiece_n describe_v 14_o plain_a triangle_n both_o right_a and_o oblique_v angle_v resolve_v by_o protraction_n from_o p_o 15_o to_o 25_o proportion_n in_o sines_n resolve_v by_o a_o line_n of_o chord_n p_o 21_o to_o 25_o proportion_n in_o tangent_n alone_o so_o resolve_v p._n 25_o 27_o 28_o 29_o 30_o proportion_n in_o sines_n and_o tangent_n resolve_v by_o a_o line_n of_o chord_n p._n 26_o 31_o 32_o 33_o particular_a scheme_n fit_v from_o proportion_n to_o the_o case_n of_o oblique_a angle_a sphoerical_a triangle_n to_o find_v the_o azimuth_n p._n 34_o 35_o as_o also_o the_o amplitude_n p._n 36_o the_o azimuth_n compass_n in_o the_o frontispiece_n describe_v p._n 38_o the_o variation_n find_v by_o the_o azimuth_n compass_n p._n 39_o to_o find_v the_o hour_n of_o the_o day_n p._n 40_o as_o also_o the_o azimuth_n and_o angle_n of_o position_n p._n 41_o to_o find_v the_o sun_n altitude_n on_o all_o hour_n p._n 43_o 46_o also_o the_o distance_n of_o place_n in_o the_o arch_n of_o a_o great_a circle_n p._n 44_o to_o find_v the_o sun_n altitude_n on_o all_o azimuth_n p._n 48_o the_o latitude_n declination_n and_o azimuth_n give_v to_o find_v the_o hour_n p._n 50_o to_o 54_o ☞_o to_o find_v the_o amplitude_n with_o the_o manner_n of_o measure_v a_o sine_fw-la to_o a_o lesser_a radius_fw-la p._n 55_o to_o get_v the_o sun_n altitude_n by_o the_o shadow_n of_o a_o thread_n or_o gnomon_n p._n 56_o the_o content_n of_o the_o treatise_n of_o navigation_n of_o the_o imperfection_n and_o uncertainty_n of_o navigation_n p._n 1_o to_o 5_o to_o measure_v a_o course_n and_o distance_n on_o the_o plain_a chart._n p._n 7_o 8_o 9_o of_o the_o quantity_n of_o a_o degree_n and_o of_o the_o form_n of_o the_o logboard_n p._n 9_o 10_o a_o reckon_a keep_v in_o league_n how_o reduce_v by_o the_o pen_n to_o degree_n and_o centesme_n p._n 11_o 12_o of_o a_o traverse-quadrant_n p._n 13_o a_o traverse_v plait_v on_o the_o plain_a chart_n without_o draw_v line_n thereon_o p._n 14_o to_o 18_o a_o scheme_n with_o direction_n to_o find_v what_o course_n and_o way_n the_o ship_n have_v make_v through_o a_o current_n p._n 18_o to_o 21_o divers_a rule_n for_o correct_v of_o the_o dead_a reckon_a from_o p._n 21_o to_o 33_o of_o the_o error_n of_o the_o plain_a chart._n p._n 33_o and_o how_o such_o chart_n may_v be_v amend_v p._n 34_o to_o find_v the_o rumbe_n between_o two_o place_n p._n 35_o proportion_n have_v one_o term_n the_o middle_a latitude_n how_o far_o to_o be_v trust_v to_o p._n 35_o to_o 38_o to_o find_v the_o rumbe_n between_o two_o place_n by_o a_o line_n of_o chord_n only_o p._n 39_o to_o 42_o the_o meridian-line_n of_o mercators_n chart_n supply_v general_o by_o a_o line_n of_o chord_n p._n 42_o to_o 47_o the_o meridian-line_n divide_v from_o the_o limb_n of_o a_o quadrant_n with_o the_o use_n thereof_o in_o find_v the_o rumbe_n p._n 48_o to_o 51_o the_o error_n commit_v by_o keep_v of_o a_o reckon_a on_o the_o plain_a chart_n remove_v p._n 52_o to_o 54_o of_o the_o nature_n of_o the_o rumbe_n on_o the_o globe_n p._n 55_o to_o 57_o mereators_n chart_n demonstrate_v from_o proportion_n p._n 58_o to_o 60_o objection_n against_o it_o answer_v p._n 60_o to_o 63_o to_o find_v the_o rumbe_n between_o two_o place_n in_o the_o chart._n p._n 64_o distance_n of_o place_n how_o measure_v on_o that_o chart._n p._n 65_o to_o 71_o another_o traverse-quadrant_n fit_v for_o that_o chart_n with_o a_o traverse_v plait_v thereby_o without_o draw_v line_n on_o the_o chart._n p._n 71_o to_o 78_o to_o measure_v a_o course_n and_o distance_n in_o that_o chart_n without_o the_o use_n of_o compass_n p._n 79_o of_o sail_v by_o the_o arch_n of_o a_o great_a circle_n p._n 81_o to_o find_v the_o latitude_n of_o the_o great_a arch_n by_o the_o stereographick_a projection_n p._n 82_o to_o 83_o of_o a_o tangent_fw-la projection_n from_o the_o pole_n for_o find_v the_o latitude_n of_o the_o great_a arch_n p._n 84_o to_o 88_o with_o a_o new_a method_n of_o calculation_n raise_v from_o it_o p._n 89_o 90_o and_o how_o to_o measure_v the_o distance_n in_o the_o arch_n and_o the_o angle_n of_o position_n p._n 91_o another_o tangent_fw-la projection_n from_o the_o 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ship_n be_v by_o the_o dead_a reckon_a or_o estimation_n and_o this_o be_v always_o give_v as_o in_o the_o former_a chart_n between_o tenariff_n and_o st._n nicholas_n island_n by_o the_o point_v 1_o 2_o 3_o in_o the_o south-line_n if_o they_o be_v measure_v in_o the_o other_o side_n of_o the_o chart_n among_o the_o degree_n of_o latitude_n the_o whole_a practice_n of_o the_o art_n of_o navigation_n in_o keep_v a_o due_a reckon_n consist_v chief_o of_o three_o member_n or_o branch_n 1._o a_o experience_a judgement_n in_o estimate_v the_o ship_n way_n in_o her_o course_n upon_o every_o shift_n of_o wind_n allow_v for_o leeward-way_n and_o current_n 2._o in_o due_o estimate_v the_o course_n or_o point_n of_o the_o compass_n on_o which_o the_o ship_n have_v make_v her_o way_n good_a allow_v for_o current_n and_o the_o variation_n of_o the_o compass_n 3._o in_o the_o frequent_a and_o due_a observe_v the_o latitude_n the_o reckon_n arise_v out_o of_o the_o two_o former_a branch_n be_v call_v the_o dead_a reckon_a and_o of_o these_o three_o branch_n there_o ought_v to_o be_v such_o a_o harmony_n and_o consent_n that_o any_o two_o be_v give_v a_o three_o conclusion_n may_v be_v thence_o raise_v with_o truth_n as_o namely_o from_o the_o course_n and_o distance_n to_o find_v the_o latitude_n of_o the_o ship_n place_n or_o by_o the_o course_n and_o difference_n of_o latitude_n to_o find_v the_o distance_n or_o by_o the_o difference_n of_o latitude_n and_o distance_n to_o find_v the_o course_n but_o in_o the_o midst_n of_o so_o many_o uncertainty_n that_o daily_o occur_v in_o the_o practice_n of_o navigation_n a_o joint_a consent_n in_o these_o three_o particular_n be_v hardly_o to_o be_v expect_v and_o when_o a_o error_n arise_v the_o sole_a remedy_n to_o be_v trust_v to_o be_v the_o observation_n of_o the_o latitude_n or_o the_o know_v sounding_n when_o a_o ship_n be_v near_a land_n and_o how_o to_o rectify_v the_o reckon_a by_o the_o observe_v latitude_n we_o shall_v now_o show_v those_o that_o will_v not_o yield_v unto_o truth_n in_o this_o particular_a that_o a_o bout_n 24_o of_o our_o common_a english_a sea_n league_n be_v to_o be_v allow_v to_o vary_v a_o degree_n of_o latitude_n under_o the_o meridian_n do_v put_v themselves_o into_o a_o double_a incapacity_n first_o in_o sail_v direct_o north_n or_o south_n under_o the_o meridian_n where_o there_o be_v no_o current_n find_v their_o reckon_a to_o fall_v short_a of_o the_o observe_v latitude_n they_o take_v it_o to_o be_v a_o error_n in_o their_o judgement_n in_o conclude_v the_o ship_n way_n by_o estimation_n or_o guess_v to_o be_v too_o little_a and_o second_o if_o there_o be_v a_o current_n that_o help_v set_v they_o forward_o that_o there_o be_v a_o near_a agreement_n between_o the_o observe_v and_o the_o dead_a latitude_n they_o conclude_v there_o be_v no_o such_o current_n or_o last_o if_o they_o stem_v the_o current_n they_o conclude_v it_o to_o be_v much_o swift_a than_o in_o truth_n it_o be_v and_o thus_o one_o error_n common_o beget_v another_o but_o suppose_v a_o conformity_n to_o the_o truth_n we_o shall_v prescribe_v four_o precept_n for_o correct_v a_o simple_a or_o single_a course_n prec_n i._n first_o therefore_o if_o a_o ship_n sail_n under_o the_o meridian_n if_o the_o difference_n of_o latitude_n be_v less_o by_o estimation_n than_o it_o be_v by_o observation_n the_o ship_n place_n or_o variation_n only_o be_v to_o be_v correct_v and_o enlarge_v under_o the_o meridian_n and_o the_o error_n iâ—_n to_o be_v impute_v either_o to_o the_o judgement_n in_o guess_v at_o the_o distance_n run_v in_o make_v
c_o b_o the_o departure_n from_o the_o meridian_n in_o the_o course_n between_o both_o place_n then_o make_v that_o one_o leg_n of_o a_o right_a angle_a triangle_n prick_v down_o 17_o 59_o centesme_n the_o difference_n of_o latitude_n between_o those_o place_n out_o of_o the_o same_o equal_a part_n from_o c_o to_o l_o and_o draw_v b_o l_o which_o represent_v the_o course_n and_o distance_n true_o between_o the_o lizard_n &_o bermudas_n and_o the_o extent_n l_o b_o measure_v on_o the_o same_o equal_a part_n show_v the_o distance_n to_o be_v 44_o 31_o centesme_n which_o allow_v twenty_o league_n to_o a_o degree_n be_v 886_o league_n then_o to_o find_v the_o course_n with_o 60_o of_o the_o chord_n set_v one_o foot_n in_o l_o with_o the_o other_o make_v a_o mark_n at_o you_o and_o z_o then_z the_o extent_n z_o y_fw-fr measure_a on_o the_o chord_n show_v the_o rumbe_n to_o be_v 66_o 37′_n from_o the_o meridian_n which_o be_v almost_o 6_o point_n and_o in_o this_o example_n the_o proportion_n do_v not_o err_v any_o thing_n from_o the_o truth_n according_a to_o mercators_n chart_n whereas_o if_o you_o use_v the_o former_a proportion_n by_o the_o middle_a latitude_n the_o rumbe_n will_v have_v be_v 67_o 2′_n from_o the_o meridian_n and_o the_o distance_n 902_o league_n if_o you_o make_v c_o a_o equal_a to_o c_o v_n than_o a_o line_n join_v l_o at_fw-fr shall_v be_v the_o course_n and_o distance_n according_a to_o the_o same_o longitude_n and_o latitude_n lay_v down_o on_o the_o plain_a chart_n and_o thereby_o the_o course_n shall_v be_v 72_o 17′_n from_o the_o meridian_n and_o the_o distance_n 1155_o league_n however_o when_o two_o place_n be_v lay_v down_o true_a at_o first_o in_o their_o rumbe_n distance_n and_o latitude_n on_o the_o plain_a chart_n if_o you_o sail_v home_o in_o or_o near_o the_o same_o rumbe_n the_o plain_a chart_n will_v very_o well_o serve_v to_o keep_v the_o reckon_n upon_o and_o to_o sail_v by_o in_o the_o great_a voyage_n how_o 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