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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
more_o to_o wit_n 180_o 54._o but_o suppose_v two_o place_n to_o be_v in_o the_o same_o latitude_n and_o to_o have_v but_o 58_o 33_o centesm_n difference_n of_o longitude_n the_o example_n will_v be_v the_o same_o with_o one_o of_o those_o before_o put_v between_o the_o berbadoe_n and_o st._n helen_n and_o the_o distance_n find_v by_o the_o middle_a arch_n or_o latitude_n be_v 62_o 00_o and_o the_o same_o by_o the_o middle_a space_n 61_o 38_o centesm_n but_o shall_v be_v in_o truth_n 64_o 7._o also_o in_o the_o former_a example_n between_o the_o berbadoe_n and_o the_o lizard_n the_o true_a distance_n be_v 58_o 54_o and_o by_o the_o middle_a latitude_n or_o arch_n be_v 58_o 7_o but_o by_o the_o middle_a space_n it_o be_v 57_o 66._o also_o in_o the_o example_n between_o the_o bermuda_n and_o the_o lizard_n the_o true_a distance_n be_v find_v to_o be_v 44_o 31_o centesm_n by_o the_o middle_a arch_n or_o latitude_n it_o be_v 45_o 13_o centesm_n and_o by_o the_o middle_a space_n it_o be_v 44_o 97_o centesm_n where_o place_n be_v in_o the_o same_o latitude_n or_o parallel_n the_o compass_n must_v be_v set_v down_o in_o the_o meridian-line_n at_o the_o latitude_n give_v and_o the_o half_a extent_n apply_v both_o upward_o and_o downward_o as_o before_o and_o if_o the_o distance_n be_v large_a the_o measure_n thereof_o will_v be_v much_o more_o erroneous_a than_o when_o the_o rumbe_n lie_v near_o the_o meridian_n some_o example_n of_o a_o parallel_n distance_n let_v there_o be_v two_o place_n in_o the_o latitude_n of_o 35d._o  _fw-fr  _fw-fr true_a distance_n distance_n by_o the_o meridian-line_n difference_n of_o longitude_n 180_o 147_o 45_o 123_o 93._o 18_o 14_o 74_o 13_o 98._o another_o example_n of_o two_o place_n in_o the_o latitude_n of_o 50d._o  _fw-fr  _fw-fr true_a distance_n distance_n by_o the_o meridian-line_n difference_n of_o longitude_n 180_o 115_o 8_o 111_o 87._o 18_o 11_o 58_o 11_o 59_o a_o three_o example_n of_o two_o place_n in_o the_o latitude_n of_o 70d._o  _fw-fr  _fw-fr true_a distance_n distance_n by_o the_o meridian-line_n difference_n of_o longitude_n 180_o 61_o 57_o 76_o 41._o 18_o 6_o 15_o 6_o 18._o from_o which_o example_n we_o may_v observe_v that_o a_o large_a distance_n can_v be_v so_o certain_o measure_v in_o the_o meridian-line_n as_o a_o small_a one_o whereof_o mr._n wrâ—ght_v be_v very_o sensible_a and_o therefore_o prescribe_v rule_n for_o the_o measure_n of_o a_o small_a part_n of_o the_o distance_n at_o a_o time_n and_o argue_v for_o the_o truth_n thereof_o but_o where_o the_o whole_a extent_n between_o two_o place_n be_v not_o above_o ten_o of_o the_o degree_n of_o longitude_n i_o see_v nothing_o to_o the_o contrary_a but_o that_o it_o may_v well_o enough_o be_v measure_v in_o the_o meridian-line_n and_o so_o for_o a_o great_a distance_n we_o may_v measure_v a_o ten_o or_o a_o twenty_o part_n of_o the_o whole_a and_o by_o multiply_v the_o know_a part_n find_v the_o whole_a for_o the_o ready_a perform_n whereof_o another_o scale_n of_o equal_a part_n whereof_o the_o degree_n be_v twice_o as_o large_a as_o those_o in_o the_o scale_n of_o longitude_n will_v be_v of_o much_o conveniency_n and_o ease_n example_n suppose_v it_o be_v require_v to_o measure_v the_o distance_n between_o the_o point_n f_o and_o d_o in_o the_o former_a chart_n take_v the_o same_o distance_n and_o measure_v it_o in_o the_o inch_n find_v how_o much_o it_o be_v to_o wit_n 1_o inch_n 61_o centesmâ—_n then_o take_v the_o same_o number_n out_o of_o the_o scale_n of_o longitude_n and_o set_v one_o foot_n at_o the_o middle_a latitude_n to_o wit_n 19_o 18_o centesm_n the_o other_o will_v reach_v northwards_o to_o 20_o 69_o centesm_n and_o southwards_o to_o 17_o 64_o centesm_n the_o difference_n of_o which_o two_o ark_n be_v 3_o 05_o the_o distance_n seek_v which_o allow_v 20_o league_n to_o a_o degree_n be_v 61_o league_n as_o before_o and_o this_o be_v more_o easy_o do_v then_o to_o take_v the_o half_a of_o any_o extent_n and_o by_o the_o same_o reason_n you_o may_v find_v the_o middle_a space_n between_o both_o latitude_n so_o also_o when_o you_o will_v measure_v the_o ten_o part_n of_o a_o great_a distance_n measure_v the_o whole_a extent_n in_o the_o scale_n of_o longitude_n and_o take_v the_o twenty_o part_n find_v by_o the_o pen_n or_o memory_n out_o of_o the_o say_a scale_n and_o set_v it_o at_o the_o middle_a latitude_n or_o middle_a space_n turn_v the_o other_o foot_n in_o the_o meridian-line_n both_o upward_o and_o downward_o and_o the_o degree_n so_o intercept_v will_v be_v the_o ten_o part_n of_o the_o whole_a distance_n now_o the_o take_n of_o the_o twenty_o or_o forty_o part_n of_o a_o extent_n be_v easy_o do_v by_o help_n of_o these_o two_o scale_n of_o equal_a part_n suppose_v i_o will_v find_v the_o twenty_o part_n of_o 3_o inch_n or_o degree_n in_o the_o great_a scale_n i_o say_v 3_o of_o the_o small_a part_n in_o the_o lesser_a scale_n be_v the_o length_n require_v and_o so_o the_o twenty_o part_n of_o 3_o inch_n 5_o ten_o be_v three_o and_o a_o half_a of_o the_o small_a part_n in_o the_o lesser_a scale_n and_o the_o half_a of_o that_o be_v the_o forty_o part_n of_o the_o whole_a i_o need_v not_o insist_v further_o upon_o these_o way_n of_o measure_v see_v i_o have_v before_o deliver_v other_o which_o as_o they_o be_v more_o ready_a in_o the_o practice_n so_o also_o they_o be_v build_v upon_o better_a foundation_n to_o keep_v a_o reckon_a on_o the_o true_a chart._n here_o i_o shall_v insist_v upon_o a_o new_a method_n never_o before_o publish_v which_o will_v render_v this_o chart_n very_o easy_a and_o acceptable_a to_o seaman_n and_o have_v make_v our_o example_n that_o before_o lay_v down_o be_v the_o same_o with_o that_o in_o the_o plain_a chart_n we_o shall_v here_o also_o retain_v the_o same_o traverse_n the_o first_o operation_n be_v to_o find_v the_o latitude_n the_o first_o course_n the_o ship_n sail_n be_v south_n south-west_n 60_o league_n from_o tenariff_n to_o protract_v this_o traverse_v i_o shall_v make_v use_n of_o another_o traverse-quadrant_a big_a than_o that_o which_o be_v use_v before_o which_o may_v be_v make_v upon_o state_n take_v 60_o league_n out_o of_o the_o scale_n of_o longitude_n t_o w_n and_o enter_v it_o in_o the_o traverse-quadrant_a on_o the_o second_o point_n from_o c_o to_o a_o the_o near_a distance_n from_o a_o to_o c_o w_n prick_v in_o the_o scale_n of_o inch_n from_o w_n to_o a_o +_o and_o it_o show_v i_o now_o that_o the_o ship_n be_v in_o the_o latitude_n of_o 25_o 23_o centesm_n in_o the_o meridian-line_n t_o s_n set_v the_o figure_n 1_o to_o this_o latitude_n second_o to_o find_v the_o difference_n of_o longitude_n take_v the_o extent_n t_o 1_o in_o the_o meridian-line_n and_o enter_v it_o so_o in_o the_o traverse-quadrant_a on_o the_o second_o rumbe_n that_o one_o foot_n rest_v thereon_o as_o at_o a_o the_o other_o turn_v about_o may_v but_o just_a touch_n c_o w_n then_o be_v the_o near_a distance_n from_o a_o to_o c_o s_o the_o difference_n of_o longitude_n require_v to_o wit_n 1_o 24_o centesm_n which_o prick_v in_o the_o scale_n of_o longitude_n from_o t_o towards_o w_n and_o set_v the_o figure_n 1_o at_o it_o three_o to_o plot_v the_o traverse_n point_n set_a one_o foot_n of_o the_o say_a extent_n at_o 1_o in_o the_o south_n line_n of_o the_o chart_n and_o with_o the_o other_o draw_v a_o small_a ark_n at_o a_o then_o take_v the_o extent_n t_o 1_o out_o of_o the_o south_n line_n and_o set_v one_o foot_n at_o 1_o in_o the_o west_n line_n with_o the_o other_o across_o the_o former_a ark_n at_o a_o and_o there_o be_v the_o point_n where_o the_o ship_n be_v at_o the_o end_n of_o this_o first_o traverse_n demonstration_n the_o proportion_n for_o find_v the_o difference_n of_o latitude_n we_o have_v before_o handle_v the_o proportion_n for_o find_v the_o difference_n of_o longitude_n be_v as_o the_o radius_fw-la be_v to_o the_o meridional_a part_n between_o any_o two_o latitude_n ∷_o so_n be_v the_o tangent_fw-la of_o the_o rumbe_n to_o the_o difference_n of_o longitude_n ∷_o the_o extent_n t_o 1_n be_v take_v out_o of_o the_o south_n line_n be_v the_o part_n of_o the_o meridian-line_n between_o the_o latitude_n of_o 28_o and_o the_o latitude_n of_o the_o ship_n place_n namely_o 25_o 23_o centesm_n which_o be_v enter_v as_o before_o in_o the_o traverse-quadrant_a at_o a_o become_v the_o radius_fw-la to_o the_o tangent_fw-la of_o that_o rumbe_n and_o so_o the_o tangent_fw-la of_o the_o say_a rumbe_n to_o that_o radius_fw-la be_v the_o near_a distance_n from_o a_o to_o c_o s_o become_v the_o difference_n of_o longitude_n require_v the_o proportion_n for_o find_v it_o be_v due_o observe_v
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 equinoctial_a for_o find_v the_o latitude_n of_o the_o arch._n p._n 93_o to_o 100_o and_o how_o to_o find_v the_o vertical_a angle_n and_o arke_n latitude_n geometrical_o p._n 100_o to_o 102_o to_o draw_v a_o curved-line_n in_o mercators_n chart_n resemble_v the_o arch_n with_o a_o example_n for_o find_v the_o course_n and_o distance_n in_o follow_v the_o arch._n p._n 102_o to_o 104_o the_o dead_a reckon_a cast_n up_o by_o arithmetic_n p._n 106_o to_o 108_o a_o brief_a table_n of_o natural_a sin_n tangent_n and_o secant_v for_o each_o point_n of_o the_o compass_n and_o the_o quarter_n p._n 107_o the_o difference_n of_o longitude_n in_o a_o dead_a reckon_a find_v by_o the_o pen._n p._n 109_o that_o a_o table_n of_o natural_a sines_n supply_v the_o want_n of_o all_o other_o table_n p._n 110_o many_o new_a easy_a rule_n and_o proportion_n to_o raise_v a_o table_n of_o natural_a sin_n p._n 111_o to_o 113_o and_o how_o by_o have_v some_o in_o store_n to_o calculate_v any_o other_o sine_fw-la in_o the_o quadrant_n at_o command_n p._n 114_o of_o the_o contrivance_n of_o logarithmical_a table_n of_o number_n sin_n and_o tangent_n and_o how_o the_o want_n of_o natural_a table_n and_o of_o a_o table_n of_o the_o meridian-line_n be_v supply_v from_o they_o p._n 117_o the_o side_n of_o a_o plain_a triangle_n be_v give_v to_o calculate_v the_o angle_n without_o the_o help_n of_o table_n two_o several_a way_n p._n 118_o 119_o a_o instance_n thereof_o in_o calculate_v a_o course_n and_o distance_n p._n 119_o chap._n i._o contain_v geometrical_a definition_n a_o point_n be_v a_o imaginary_a prick_n void_a of_o all_o length_n breadth_n or_o depth_n a_o line_n be_v a_o suppose_a length_n without_o breadth_n or_o depth_n the_o end_n or_o limit_n whereof_o be_v point_n a_o angle_n derive_v from_o the_o word_n angulus_n in_o latin_a which_o signify_v a_o corner_n be_v the_o inclination_n or_o bow_v of_o two_o line_n one_o to_o another_o and_o the_o one_o touch_v the_o other_o and_o not_o be_v direct_o join_v together_o if_o the_o line_n which_o contain_v the_o angle_n be_v right_a line_n then_o be_v it_o call_v a_o right_o line_v angle_n a_o right_a angle_n when_o a_o right_a line_n stand_v upon_o a_o right_a line_n make_v the_o angle_n on_o either_o side_n equal_a each_o of_o these_o angles_n be_v call_v right_a angle_n and_o the_o line_n erect_v be_v call_v a_o perpendicular_a line_n unto_o the_o other_o a_o obtuse_a angle_n be_v that_o which_o be_v great_a than_o a_o right_a angle_n a_o acute_a angle_n be_v that_o which_o be_v less_o than_o a_o right_a angle_n when_o two_o angle_n be_v both_o acute_a or_o obtuse_a they_o be_v of_o the_o same_o kind_n otherwise_o be_v say_v to_o be_v of_o different_a affection_n a_o angle_n be_v common_o denote_v by_o the_o middlemost_a of_o the_o three_o letter_n set_v to_o the_o side_n include_v the_o say_a angle_n the_o quantity_n of_o a_o angle_n be_v measure_v by_o the_o arch_n of_o a_o circle_n describe_v upon_o the_o point_n of_o concurrence_n or_o intersection_n where_o the_o two_o side_n enclose_v the_o say_a angle_n meet_v by_o the_o compliment_n of_o a_o arch_n or_o angle_n be_v mean_v the_o remainder_n of_o that_o arch_n take_v from_o 90_o unless_o it_o be_v express_v the_o compliment_n thereof_o to_o a_o semicircle_n of_o 180d._o a_o circle_n be_v a_o plain_a figure_n contain_v under_o one_o line_n which_o be_v call_v the_o circumference_n thereof_o by_o some_o the_o perimeter_n periphery_n or_o limb_n a_o portion_n or_o part_n thereof_o be_v call_v a_o segment_n the_o centre_n thereof_o be_v a_o point_n in_o the_o very_a midst_n thereof_o from_o which_o point_n all_o right_a line_n draw_v
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a_o and_o b_o with_o any_o sufficient_a extent_n you_o may_v make_v a_o intersection_n underneath_o and_o lay_v a_o ruler_n to_o that_o and_o the_o upper_a intersection_n and_o thereby_o find_v the_o point_n c._n prop._n 3._o to_o draw_v one_o line_n parallel_n to_o another_o line_n at_o any_o distance_n require_v in_o this_o figure_n let_v the_o line_n a_o b_o be_v give_v and_o let_v it_o be_v require_v to_o draw_v the_o line_n c_o d_o parallel_n thereto_o according_a to_o the_o distance_n of_o a_o c._n first_o open_v the_o compass_n to_o the_o distance_n require_v then_o set_v one_o foot_n in_o a_o or_o further_o in_o with_o the_o other_o draw_v the_o touch_n of_o a_o arch_n at_o c_o then_o retain_v the_o same_o extent_n of_o the_o compass_n set_v down_o the_o compass_n at_o b_o and_o with_o the_o former_a extent_n draw_v the_o touch_n of_o a_o arch_n at_o d_o then_o lay_v a_o ruler_n to_o the_o outwardmost_a edge_n of_o these_o two_o ark_n draw_v the_o right_a line_n c_o d_o which_o will_v be_v the_o parallel_n require_v this_o proposition_n will_v be_v of_o frequent_a use_n in_o dyall_v now_o the_o draw_v of_o such_o piece_n of_o ark_n as_o may_v deface_v the_o plain_a may_v be_v shun_v for_o have_v open_v the_o compass_n to_o the_o assign_a parallel_n distance_n set_v down_o one_o foot_n opposite_a to_o one_o end_n of_o the_o line_n propose_v a_o b_o so_o as_o the_o other_o but_o just_a turn_v about_o may_v touch_v the_o say_a line_n and_o it_o will_v find_v one_o point_n again_o find_v another_o point_n in_o like_a manner_n opposite_a to_o the_o
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 distance_n f_o v_n be_v to_o be_v enlarge_v and_o where_o the_o parallel_n of_o observe_v latitude_n cut_v it_o as_o at_o t_o 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 and_o though_o what_o we_o have_v here_o perform_v be_v do_v by_o the_o draw_n of_o many_o line_n yet_o by_o help_n of_o the_o traverse-quadrant_a it_o may_v may_v be_v insert_v into_o the_o chart_n without_o draw_v any_o line_n therein_o at_o all_o for_o in_o each_o the_o course_n and_o correct_v difference_n of_o latitude_n be_v give_v and_o that_o two_o thing_n be_v sufficient_a to_o dispatch_v the_o work_n we_o have_v show_v before_o those_o that_o be_v prompt_a in_o plain_a triangle_n may_v
scheme_n the_o vertical_a angle_n be_v the_o angle_n at_o the_o pole_n on_o each_o side_n the_o perpendicular_a be_v measure_v from_o the_o point_n b_o and_o the_o chord_n of_o those_o angle_n from_o the_o point_n a_o be_v the_o cosines_n of_o those_o angles_n and_o so_o the_o extent_n a_o l_o be_v the_o cousin_n of_o the_o lesser_a vertical_a angle_n which_o prick_v from_o a_o in_o the_o line_n a_o s_o the_o point_n find_v we_o may_v call_v the_o vertical_a point_n then_o prick_v down_o the_o sin_n of_o every_o five_o degree_n from_o 90_o as_o of_o 85_o 80_o &c_n &c_n in_o the_o line_n i_o b_o from_o i_o towards_o b_o and_o lay_v a_o ruler_n over_o the_o vertical_a point_n and_o all_o those_o sine_fw-la point_v divide_v the_o diameter_n into_o as_o many_o point_n then_o prick_v down_o the_o sine_fw-la of_o the_o latitude_n to_o wit_n 50_o in_o the_o semicircle_n a_o b_o from_o a_o upward_o towards_o b_o and_o a_o ruler_n over_o the_o say_a latitude_n point_n and_o all_o the_o former_a point_n find_v in_o the_o diameter_n will_v cut_v the_o under_o semicircle_n b_o a_o in_o as_o many_o point_n or_o arke_n more_o which_o be_v count_v or_o measure_v from_o b_o downward_o towards_o a_o be_v the_o respective_a latitude_n of_o the_o great_a arch_n seek_v nota_fw-la if_o a_o line_n of_o sin_n equal_a to_o the_o diameter_n be_v graduate_v on_o the_o floap_n edge_n of_o a_o ruler_n the_o sin_n in_o the_o line_n b_o i_o may_v be_v easy_o prick_v down_o with_o the_o pen_n by_o the_o edge_n thereof_o without_o compass_n and_o the_o arke_n in_o the_o under_o semicircle_n b_o a_o may_v be_v ready_o measure_v by_o view_n by_o lay_v the_o beginning_n of_o that_o line_n of_o sin_n to_o the_o point_n b_o and_o move_v the_o edge_n of_o the_o ruler_n to_o each_o respective_a ark_n before_o find_v in_o the_o say_a semicircle_n here_o note_n that_o a_o semicircle_n be_v divide_v into_o 90_o equal_a part_n the_o distance_n of_o each_o degree_n from_o one_o end_n of_o the_o diameter_n be_v sin_n of_o those_o arke_n the_o whole_a diameter_n be_v their_o radius_fw-la wherefore_o the_o use_n of_o sines_n be_v as_o geometrical_a as_o the_o use_n of_o chord_n in_o the_o same_o scheme_n the_o reader_n may_v first_o find_v the_o perpendicular_a and_o then_o the_o distance_n on_o each_o side_n of_o it_o by_o proportion_n before_o set_v down_o now_o for_o the_o latitude_n of_o the_o great_a ark._n have_v some_o of_o these_o way_n find_v the_o latitude_n of_o the_o arch_n make_v a_o mercator_n chart_n for_o your_o voyage_n lay_v down_o the_o two_o place_n suppose_v the_o lizard_n at_o l_o in_o the_o follow_a chart_n in_o the_o latitude_n of_o 50_o degree_n and_o trinity_n harbour_n at_o t_o in_o the_o latitude_n of_o 36_o degree_n the_o difference_n of_o longitude_n to_o wit_n l_o m_o be_v 68_o degree_n and_o a_o half_a then_o because_o the_o perpendicular_a fall_n between_o both_o place_n count_v off_o the_o lesser_a vertical_a angle_n 14_o degree_n 63_o centesm_n from_o l_o to_o p_o and_o there_o raise_v the_o perpendicular_a p_o r._n this_o follow_a chart_n we_o have_v make_v as_o large_a as_o the_o page_n will_v admit_v and_o have_v a_o meridian-line_n fit_v to_o the_o degree_n of_o longitude_n in_o that_o chart_n by_o the_o edge_n thereof_o on_o the_o perpendicular_a p_o r_n you_o may_v prick_v down_o the_o latitude_n of_o the_o arch_n before_o find_v and_o 5_o degree_n difference_n of_o longitude_n on_o each_o side_n of_o the_o perpendicular_a at_o a_o time_n and_o thereby_o graduate_v the_o respective_a point_n or_o cross_n through_o which_o the_o curve_v prick_v line_n be_v draw_v example_n now_o to_o the_o apprehension_n the_o right_a line_n l_o t_o will_v seem_v near_a than_o to_o sail_v along_o by_o the_o prick_a arch_n also_o it_o may_v seem_v improbable_a that_o a_o ship_n bind_v from_o the_o lizard_n to_o trinity_n harbour_n in_o virginia_n be_v a_o place_n near_o the_o equinoctial_a by_o 14_o degree_n of_o latitude_n shall_v yet_o run_v into_o a_o more_o northwardly_a latitude_n than_o the_o lizard_n to_o which_o i_o answer_v that_o mercators_n chart_n be_v no_o projection_n of_o the_o sphere_n do_v not_o represent_v thing_n to_o the_o fancy_n as_o they_o be_v in_o the_o sphere_n but_o as_o they_o be_v accommodate_v to_o that_o chart_n by_o which_o trinity_n harbour_n in_o that_o chart_n bear_v from_o the_o lizard_n 74_o 17′_n from_o the_o meridian_n which_o be_v above_o 6½_n point_v to_o the_o westward_n of_o the_o south_n to_o wit_n west_n and_o by_o south_n ¼_n of_o a_o point_n and_o one_o degree_n 39_o minute_n more_o southwardly_a and_o the_o distance_n in_o the_o rumbe_n be_v 1034_o league_n whereas_o the_o distance_n in_o the_o arch_n if_o it_o can_v be_v precise_o follow_v be_v 1003_o league_n and_o follow_v it_o from_o point_n to_o point_n through_o every_o 5_o degree_n difference_n of_o longitude_n from_o the_o perpendicular_a it_o be_v 1020_o league_n almost_o be_v less_o than_o the_o distance_n in_o the_o rumbe_n by_o 14_o league_n as_o appear_v by_o calculation_n or_o the_o chart_n itself_o the_o respective_a course_n and_o distance_n to_o be_v sail_v  _fw-fr  _fw-fr course_n  _fw-fr distance_n  _fw-fr  _fw-fr deg._n min._n  _fw-fr league_n part_n from_o l_o to_o 1_o 80_o 25_o to_o the_o westward_n of_o the_o north_n 60_o 06_o 1_o 2_o 84_o 14½_n 63_o 86_o 2_o 3_o 88_o 00½_n 61_o 02_o 3_o 4_o 88_o 00½_n to_o the_o westward_n of_o the_o south_n 61_o 02_o 4_o 5_o 84_o 14½_n 63_o 86_o 5_o 6_o 80_o 14½_n 64_o 88_o 6_o 7_o 76_o 28_o 66_o 66_o 7_o 8_o 72_o 51_o 69_o 18_o 8_o 9_o 68_o 56_o 72_o 32_o 9_o 10_o 65_o 30_o 69_o 20_o 10_o 11_o 61_o 32_o 81_o 40_o 11_o 12_o 59_o 00_o 87_o 76_o 12_o 13_o 55_o 02_o 93_o 86_o 13_o t_n 52_o 20_o 77_o 88_o  _fw-fr  _fw-fr  _fw-fr whole_a distance_n 1019_o 96._o this_o example_n serve_v full_o to_o explain_v the_o sail_v by_o the_o great_a arch_n though_o it_o may_v not_o be_v safe_a to_o follow_v it_o by_o reason_n of_o hale_v too_o near_o the_o coast_n of_o ireland_n of_o arithmetical_a navigation_n all_o performance_n by_o the_o pen_n require_v the_o help_n of_o some_o table_n by_o which_o the_o question_v propose_v may_v be_v speedy_o resolve_v if_o such_o table_n be_v at_o hand_n and_o if_o no_o other_o table_n be_v at_o hand_n but_o that_o of_o natural_a sin_n it_o will_v serve_v to_o do_v the_o whole_a work_n and_o how_o such_o a_o table_n may_v be_v make_v or_o the_o sine_fw-la of_o any_o ark_n at_o command_n we_o shall_v handle_v for_o conveniency_n and_o dispatch_n there_o follow_v a_o brief_a table_n of_o natural_a sin_n tangent_n and_o secant_v to_o every_o one_o of_o the_o eight_o point_n of_o the_o compass_n and_o their_o quarter_n from_o the_o meridian_n and_o such_o a_o kind_n of_o table_n but_o false_a print_v be_v in_o the_o work_v of_o maetius_n the_o use_n whereof_o he_o explain_v in_o some_o of_o the_o follow_a proposition_n give_v seek_v  _fw-fr 1_o the_o longitude_n and_o latitude_n of_o two_o place_n rumbe_n which_o be_v require_v at_o the_o begin_n of_o a_o voyage_n distance_n 2._o the_o rumb_n and_o distance_n give_v difference_n of_o latitude_n whereby_o dead_a reckon_a keep_v  _fw-fr departure_n from_o the_o meridian_n 3._o difference_n of_o latude_n and_o rumbe_n departure_n from_o the_o meridian_n whereby_o the_o dead_a reckon_a be_v correct_v  _fw-fr distance_n 4._o difference_n of_o latude_n and_o distance_n rumbe_n departure_n  _fw-fr to_o give_v the_o difference_n of_o latitude_n and_o departure_n whereby_o in_o some_o case_n to_o correct_v the_o dead_a reckon_a be_v the_o same_o with_o the_o first_o proposition_n and_o that_o we_o shall_v handle_v last_o 5._o rumbe_n difference_n of_o latitude_n departure_n distance_n 6._o departure_n rumbe_n distance_n difference_n of_o latitude_n these_o two_o last_o proposition_n be_v of_o small_a use_n some_o example_n of_o the_o use_n of_o the_o say_a table_n 1._o in_o keep_v the_o dead_a reckon_a as_o the_o radius_fw-la be_v to_o the_o distance_n run_v ∷_o so_n be_v the_o cousin_n of_o the_o rumbe_n to_o the_o difference_n of_o latitude_n the_o table_n of_o sines_n be_v number_v with_o the_o point_n of_o the_o compass_n on_o the_o left_a side_n in_o the_o variation_n column_n from_o the_o bottom_n upward_o and_o show_v the_o difference_n of_o latitude_n to_o 1000_o league_n sail_v upon_o any_o rumbe_n take_v the_o figure_n that_o stand_v against_o the_o point_n of_o the_o compass_n as_o far_o as_o the_o comma_n and_o the_o other_o two_o figure_n beyond_o it_o may_v be_v take_v in_o if_o more_o preciseness_n be_v require_v so_o if_o a_o ship_n sail_v 1000_o league_n south_n south_n west_n be_v the_o second_o point_n from_o the_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
former_a direction_n have_v find_v the_o angle_n c_o a_o b_o it_o be_v equal_a to_o the_o side_n z_o p_o in_o the_o first_o triangle_n which_o be_v the_o compliment_n of_o the_o latitude_n require_v and_o the_o compliment_n of_o the_o angle_n a_o c_o b_o in_o the_o second_o triangle_n be_v take_v to_o 180_o be_v equal_a to_o the_o side_n ⊙_o p_o in_o the_o first_o triangle_n the_o sun_n distance_n from_o the_o elevate_a pole_n require_v example_n in_o this_o scheme_n the_o latitude_n s_o t_o 30_o who_o compliment_n be_v z_o t_o 60_o equal_a to_o the_o side_n of_o the_o triangle_n a_o b._n z_o you_o be_v the_o polar_a distance_n 69_o 46′_n the_o compliment_n whereof_o be_v n_n y._n l_o g_o be_v the_o sine_fw-la of_o 6_o 36′_n in_o that_o parallel_n the_o excess_n of_o the_o azimuth_n above_o a_o quadrant_a from_o the_o north_n c_o f_o be_v equal_a to_o c_o g_o the_o extent_n g_o z_o reach_n from_o f_o to_o a_o and_o the_o arch_n z_o at_fw-fr be_v 35_o 59′_n be_v equal_a to_o c_o b_o in_o the_o second_o triangle_n or_o to_o the_o angle_n z_o ⊙_o p_o in_o the_o first_o triangle_n be_v there_o the_o angle_n of_o position_n to_o find_v the_o hour_n make_v n_n p_o equal_a to_o s_o t_o drawing_z c_z p_o and_o set_v off_o 69_o 46′_n from_o p_o to_o d_o and_z e_z and_o draw_v the_o parallel_n d_o e._n then_o draw_v the_o parallel_n of_o altitude_n h_n a_o and_o b_o ⊙_o be_v the_o sine_fw-la of_o the_o hour_n from_o six_o c_o r_n be_v equal_a to_o b_o d_o upon_z r_z with_o b_o ⊙_o describe_v k_o a_o ruler_n from_o the_o centre_n just_o touch_v it_o find_v m_n and_o the_o arch_n m_o o_fw-fr be_v 38_o 28′_n be_v the_o angle_n c_o a_o p_o in_o the_o second_o triangle_n equal_a to_o the_o side_n z_o p_o in_o the_o first_o triangle_n which_o be_v the_o compliment_n of_o the_o latitude_n there_o require_v to_o find_v the_o angle_n of_o position_n place_n z_o h_n from_o p_o to_o f_o and_o make_v f_o g_o equal_a to_z f_o h_n a_o ruler_n from_o h_n to_o g_o cut_n d_o e_z at_z i_o upon_o r_z with_o b_o i_o draw_v q_n a_o ruler_n from_o the_o centre_n touch_v it_o find_v five_o and_o the_o ark_n o_o v_n be_v 66_o 29′_n be_v the_o measure_n of_o the_o angle_n a_o c_o b_o in_o the_o second_o triangle_n which_o be_v equal_a to_o the_o side_n ⊙_o p_o in_o the_o first_o triangle_n the_o sun_n distance_n from_o the_o pole_n require_v and_o thus_o this_o question_n have_v exercise_v most_o of_o the_o rule_n before_o deliver_v and_o thus_o sole_o upon_o the_o consideration_n of_o the_o projection_n of_o the_o sphere_n i_o have_v show_v how_o all_o the_o twenty_o eight_o common_a case_n may_v be_v resolve_v orthographical_o chap._n x._o show_v how_o to_o project_v all_o the_o case_n of_o right_a angle_a spherical_a triangle_n and_o because_o projection_n of_o the_o sphere_n do_v not_o depend_v upon_o proportion_n for_o the_o resolve_v of_o any_o case_n propound_v but_o on_o the_o contrary_a proportion_n be_v derive_v from_o projection_n i_o shall_v therefore_o further_o enlarge_v in_o show_v how_o the_o right_a angle_a case_n be_v resolve_v without_o the_o knowledge_n of_o the_o proportion_n in_o project_v the_o right_a angle_a case_n orthographical_o the_o triangle_n v_o a_o ⊙_o right_a angle_v at_o a_o shall_v be_v the_o example_n use_v in_o which_o the_o arch_n s_o ae_n be_v equal_a to_o the_o compliment_n of_o the_o latitude_n at_o london_n 38_o 28′_n equal_a to_o the_o angle_n a_o v_o ⊙_o the_o hipotenusal_a five_o ⊙_o be_v the_o hour_n from_o six_o the_o leg_n a_o ⊙_o the_o altitude_n or_o sun_n height_n at_o the_o time_n propose_v the_o leg_n v_o a_o the_o sun_n azimuth_n from_o the_o vertical_a the_o angle_n a_o ⊙_o v_o the_o angle_n between_o the_o sun_n azimuth_n circle_n and_o the_o equinoctial_a case_n 1_o 2_o 3._o give_v a_o leg_n and_o its_o adjacent_a angle_n to_o find_v the_o rest_n data_fw-la a_o five_o 31d._o angle_z a_o v_o ⊙_o 38_o 28′_n have_v draw_v the_o quadrant_n s_o z_o and_o s_o v_n and_o z_o v_n at_o right_a angle_n in_o the_o centre_n place_n the_o give_v angle_n 38_o 28′_n from_o s_n to_o ae_n and_o draw_v ae_n v_n draw_v ae_n l_o parallel_n to_o z_o v_n then_o prick_v the_o sine_fw-la of_o the_o give_a leg_n from_o five_o to_o a_o and_o f_o the_o near_a distance_n from_o a_o to_o ae_n v_n prick_v from_o l_o to_z b._n 1._o to_o find_v the_o other_o side_n or_o leg._n a_o ruler_n over_o five_o and_o b_o cut_v the_o limb_n at_o d_o and_o the_o arch_n s_o d_o be_v 22_o 16′_n be_v the_o measure_n of_o the_o side_n a_o ⊙_o 2._o to_o find_v the_o hipotenusal_a through_o d_o draw_v d_o ⊙_o g_o parallel_n to_o s_o v_n and_o the_o extent_n v_o ⊙_o measure_v on_o the_o sin_n be_v 37_o 30′_n the_o hipotenusal_a seek_v 3._o to_o find_v the_o angle_n v_o ⊙_o a._n through_o f_o draw_v f_o e_o parallel_n to_o five_o s_n and_o make_v v_o c_o equal_a to_o five_o ⊙_o a_o ruler_n from_o the_o centre_n over_o c_o cuâ—s_v the_o limb_n at_o l_o and_o the_o arch_n s_o l_o being_n 57_o 55′_n be_v the_o measure_n of_o the_o angle_n seek_v otherwise_o make_v v_n i_o equal_a to_o five_o c_o then_z take_z the_o distance_z between_z c_z and_o z_o and_o the_o say_a extent_n shall_v reach_v from_o i_o to_o l_o in_o the_o limb_n as_o before_o after_o the_o same_o manner_n the_o point_n d_o might_n have_v be_v find_v also_o the_o extent_n v_o f_o may_v be_v double_v or_o tripled_a upward_o and_o at_o the_o end_n thereof_o a_o perpendicular_a raise_v according_o multiply_v or_o increase_v the_o extent_n v_o ⊙_o and_o one_o foot_n rest_v in_o the_o centre_n across_o the_o say_v perpendicular_a and_o a_o ruler_n lay_v over_o the_o point_n of_o intersection_n or_o cross_v and_o the_o centre_n v_o shall_v pass_v through_o the_o limb_n at_o l_o as_o before_z otherwise_o place_v v_n ⊙_o from_o v_z to_o q_n and_o upon_o q_o with_o five_o a_o describe_v the_o ark_n h_n a_o ruler_n from_o the_o centre_n just_o touch_v the_o outward_a extremity_n of_o it_o cut_v the_o limb_n at_o l_o as_o before_o the_o arch_n s_o l_o be_v the_o angle_n seek_v or_o to_o keep_v the_o say_a arch_n more_o remote_a from_o the_o centre_n prick_v v_o ⊙_o twice_z from_o five_o to_o p_o then_o double_a the_o extent_n v_o a_o and_o therewith_o upon_o p_o describe_v the_o ark_n k_o a_o rule_n from_o the_o centre_n just_o touch_v the_o outward_a extremity_n of_o that_o ark_n cut_v the_o limb_n at_o l_o as_o before_z these_o case_n be_v the_o most_o difficult_a as_o require_v a_o ellipsis_n to_o be_v draw_v which_o we_o have_v shun_v and_o may_v be_v otherwise_o avoid_v according_a to_o the_o example_n in_o page_n 51_o i_o think_v fit_a first_o to_o handle_v the_o triangle_n resolve_v be_v not_o represent_v unless_o there_o be_v a_o elliptical_a a_o k_n draw_v from_o ⊙_o to_z a._n the_o find_n of_o the_o point_n d_o be_v carry_v on_o from_o a_o proportion_n of_o this_o kind_n as_o the_o radius_fw-la ae_n l_o to_o the_o tangent_fw-la of_o the_o ark_n s_o ae_n be_v the_o measure_n of_o the_o give_v angle_n so_o be_v the_o sine_fw-la of_o the_o give_a leg_n l_o b_o proportion_a out_o to_o that_o radius_fw-la to_o the_o tangent_fw-la of_o the_o ark_n s_o d_o be_v the_o measure_n of_o the_o leg_n seek_v case_n 4_o 5_o 6._o give_v a_o side_n and_o its_o opposite_a angle_n to_o find_v the_o rest_n data_fw-la the_o side_n a_o five_o 31d._o the_o angle_n v_o ⊙_o a_o 57_o 55′_n place_v the_o give_v angle_n at_o the_o centre_n of_o the_o quadrant_a wherefore_o prick_v it_o to_o wit_n 57_o 55′_n from_o s_n to_o l_o and_o draw_v l_o v._n then_o place_v the_o give_v leg_n from_o s_n to_o d_o to_z wâ—t_v 31_o and_o draw_v d_o ⊙_o g_o parallel_n to_o s_o v._n to_o measure_v the_o ark_n seek_v 1_o the_o hipotenusal_a the_o extent_n v_o ⊙_o apply_v to_o the_o sin_n be_v 37_o 30′_n be_v the_o measure_n thereof_o 2._o the_o other_o leg._n place_n d_o g_o from_o five_o to_o r_n and_o upon_o r_n with_o the_o extent_n g_o ⊙_o draw_v the_o ark_n h_n a_o ruler_n from_o the_o centre_n touch_v the_o extremity_n of_o it_o cut_v the_o limb_n at_o e_o and_o the_o arch_n s_o e_z be_v 22_o 16′_n be_v the_o measure_n of_o the_o other_o leg._n 3._o the_o other_o angle_n draw_v e_o f_o parallel_n to_o s_o v_n and_o make_v v_o c_o equal_a to_o five_o ⊙_o a_o ruler_n from_o the_o centre_n over_o c_o cut_v the_o limb_n at_o ae_n and_o the_o arch_n s_o ae_n be_v 38_o 28′_n be_v the_o measure_n of_o the_o other_o oblique_a angle_n last_o the_o triangle_n
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 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