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prick_v down_o 30_o degree_n 58_o minute_n for_o the_o angle_n give_v and_o draw_v the_o line_n a_o f_o c_o then_o from_o the_o end_n of_o the_o side_n prick_v off_o prick_v the_o other_o side_n from_o b_o to_z c_z or_o e_z and_o so_o the_o angle_n b_o c_o a_o or_o b_o e_o a_o be_v the_o angle_n seek_v but_o which_o of_o the_o two_o can_v possible_o be_v determine_v unless_o the_o affection_n be_v also_o give_v to_o wit_n whether_o it_o be_v obtuse_a or_o acute_a though_o some_o of_o our_o writer_n affirm_v it_o may_v be_v determine_v by_o draw_v the_o triangle_n as_o true_a as_o you_o can_v then_o upon_o the_o angular_a point_n c_o describe_v a_o arch_n with_o 60_o of_o the_o chord_n and_o measure_v the_o say_a arch_n in_o the_o chord_n continue_v the_o side_n if_o need_v be_v and_o it_o show_v the_o quantity_n of_o the_o say_a angle_n to_o be_v 53_o 6′_n and_o the_o compliment_n thereof_o to_o 180_o be_v 126_o 54′_n be_v the_o measure_n of_o the_o angle_n b_o e_o a_o because_o the_o angle_n b_o e_o c_o be_v equal_a to_o the_o angle_n b_o c_o e._n how_o to_o frame_v such_o triangle_n who_o side_n shall_v be_v all_o whole_a number_n be_v show_v in_o our_o english_a ramus_n page_n 155_o 156._o 2._o from_o what_o be_v give_v to_o find_v the_o three_o side_n and_o the_o other_o angle_n in_o this_o case_n also_o unless_o the_o quality_n of_o the_o angle_n opposite_a to_o the_o great_a side_n be_v determine_v the_o three_o side_n will_v be_v doubtful_a to_o wit_n it_o may_v be_v either_o a_o e_o or_o a_o c_o which_o extent_n measure_v on_o the_o equal_a part_n show_v the_o side_n according_o and_o the_o three_o angle_n to_o be_v measure_v as_o before_o this_o mr._n norwood_n do_v not_o make_v a_o case_n because_o a_o angle_n must_v be_v first_o find_v and_o determine_v before_o the_o three_o side_n can_v be_v find_v and_o then_o it_o will_v be_v resolve_v by_o the_o follow_a case_n 3._o two_o angle_n with_o a_o side_n opposite_a to_o one_o of_o they_o give_v to_o find_v 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limb_n as_o before_o here_o observe_v that_o every_o proportion_n without_o the_o radius_fw-la may_v be_v make_v into_o two_o single_a proportion_n with_o the_o radius_fw-la in_o each_o thus_o as_o the_o radius_fw-la be_v to_o one_o of_o the_o middle_a term_n so_o be_v the_o other_o middle_a term_n to_o a_o four_o proportional_a again_o as_o the_o first_o term_n be_v to_o the_o radius_fw-la so_o be_v the_o four_o proportional_a before_o find_v to_o the_o true_a proportional_a seek_v from_o which_o consideration_n the_o former_a scheme_n be_v contrive_v two_o other_o general_a way_n for_o work_a proportion_n in_o sines_n let_v the_o proportion_n be_v as_o the_o sine_fw-la of_o 70_o to_o the_o sine_fw-la of_o 50_o so_o be_v the_o sine_fw-la of_o 35d._o to_o what_o sine_fw-la answer_n 27_o 50′_n have_v draw_v the_o quadrant_n d_o e_o with_o 60_o of_o the_o chord_n and_o its_o two_o radii_fw-la d_o c_o c_o e_z at_o right_a angle_n in_o the_o centre_n prick_v down_o from_o the_o chord_n one_o of_o the_o middle_a term_n to_o wit_n 35_o from_o d_o to_o h_n and_o draw_v a_o line_n into_o the_o centre_n and_o upon_o the_o say_a line_n from_o the_o centre_n to_o a_o prick_v down_o out_o of_o the_o sin_n the_o other_o middle_a term_n to_o wit_n 50_o and_o through_o the_o point_n a_o draw_v the_o line_n a_o b_o parallel_n to_o d_o c_o then_o count_v the_o first_o term_n from_o d_o to_z g_z 70_o and_o draw_v a_o line_n from_o the_o centre_n which_o pass_v through_o a_o b_o at_o f_o and_o the_o extent_n c_o f_o measure_v on_o the_o sin_n be_v 27_o 50′_n the_o four_o proportional_a and_o thus_o the_o first_o term_n may_v be_v vary_v as_o much_o as_o you_o please_v otherwise_o place_v the_o sine_fw-la of_o the_o first_o term_n to_o wit_n of_o 70_o which_o in_o this_o example_n be_v the_o near_a distance_n from_o g_z to_z d_o c_z so_o from_o the_o centre_n that_o it_o may_v cross_v a_o b_o produce_v when_o need_n require_v in_o this_o example_n it_o cross_v it_o at_o ay_o a_o ruler_n over_o the_o centre_n and_o the_o point_n i_o cut_v the_o limb_n at_o k_o and_o the_o arch_n d_o k_n be_v 27_o 50′_n be_v the_o measure_n of_o the_o four_o proportional_a as_o before_o when_o it_o can_v be_v there_o place_v to_o wit_n as_o when_o the_o sine_fw-la of_o the_o first_o term_n be_v short_a than_o c_z b_o the_o four_o proportional_a be_v more_o than_o the_o radius_fw-la and_o the_o arch_n of_o the_o first_o term_n be_v count_v from_o d_o towards_z e_z a_o line_n
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
other_o end_n of_o the_o line_n at_o b_o and_o through_o these_o two_o point_n draw_v a_o right_a line_n and_o it_o shall_v be_v the_o parallel_n require_v this_o way_n though_o it_o be_v not_o so_o geometrical_a as_o the_o former_a yet_o in_o other_o respect_n may_v be_v much_o more_o convenient_a and_o certain_a enough_o prop._n 4._o to_o draw_v the_o arch_n of_o a_o circle_n through_o any_o three_o point_n not_o lie_v in_o a_o straight_a line_n in_o the_o figure_n adjoin_v let_v a_o b_o c_o be_v the_o three_o point_n give_v and_o let_v it_o be_v require_v to_o draw_v a_o circle_n that_o may_v pass_v through_o they_o all_o set_a one_o foot_n in_o the_o middle_a point_n at_o b_o and_o open_v the_o compass_n to_o above_o one_o half_a of_o the_o distance_n of_o the_o further_a point_n therefrom_o or_o to_o any_o other_o competent_a extent_n and_o therewith_o draw_v the_o obscure_a arch_n d_o e_o f_o h_o with_o the_o same_o extent_n set_v one_o foot_n in_o the_o point_n c_o draw_v the_o arch_n f_o h_n again_o with_o the_o same_o extent_n set_v one_o foot_n in_o the_o point_n a_o draw_v the_o arch_n d_o e_o then_o lay_v a_o ruler_n to_o the_o intersection_n of_o these_o arch_n draw_v the_o line_n d_o g_o and_o g_o h_n which_o will_v cross_v each_o other_o at_o the_o point_n g_o and_o there_o be_v the_o centre_n of_o the_o circle_n seek_v where_o set_v one_o foot_n of_o the_o compass_n and_o extend_v the_o other_o to_o any_o of_o the_o three_o point_n describe_v the_o arch_n of_o a_o circle_n which_o shall_v pass_v through_o the_o three_o point_n require_v prop._n 5._o two_o line_n be_v give_v incline_v each_o to_o other_o so_o as_o they_o seem_v to_o make_v a_o acute_a angle_n if_o they_o be_v produce_v to_o find_v their_o angular_a point_n of_o concurrence_n or_o meeting_n without_o produce_v or_o continue_v the_o say_a line_n and_o if_o they_o be_v multiply_v the_o other_o way_n from_z d_o to_o h_n and_z from_z c_z to_z e_z than_o the_o line_n e_o h_n and_o f_o g_o be_v produce_v find_v the_o same_o point_n i_o without_o continue_v either_o of_o the_o line_n first_o give_v and_o with_o much_o more_o certainty_n a_o oblong_a a_o rectangle_n a_o right_a angle_a parralellogram_n or_o a_o long_a square_n be_v all_o word_n of_o one_o and_o the_o same_o signification_n and_o signify_v a_o flat_a figure_n have_v only_a length_n and_o breadth_n the_o four_o angle_n whereof_o be_v right_a angle_n the_o opposite_a side_n whereof_o be_v equal_a in_o proportion_n the_o product_n of_o two_o term_n or_o number_n be_v call_v their_o rectangle_n or_o oblong_v because_o if_o the_o side_n of_o a_o flat_a be_v divide_v into_o as_o many_o part_n as_o there_o be_v unit_v in_o each_o multiplyer_n line_n rule_v over_o those_o part_n will_v make_v as_o many_o small_a square_n as_o there_o be_v unit_v in_o the_o product_n and_o the_o whole_a figure_n itself_o will_v have_v the_o shape_n of_o a_o long_a square_n a_o rhombus_fw-la or_o diamond_n be_v a_o figure_n with_o four_o equal_a side_n but_o no_o right_a angle_n but_o a_o rhomboide_n or_o diamond-like_a figure_n be_v such_o a_o figure_n who_o opposite_a side_n and_o opposite_a angle_n only_o be_v equal_a either_o of_o these_o figure_n be_v common_o call_v oblique_a angle_a parralellogram_n thus_o either_o of_o the_o figure_n a_o b_o f_o e_o or_o b_o c_o e_z d_o be_v rhomboide_v or_o oblique_a angle_a parralellogram_n this_o forego_v figure_n be_v much_o use_v in_o dyall_v thereby_o to_o set_v off_o the_o hour-line_n admit_v the_o side_n a_o b_o and_o b_o e_o be_v give_v and_o it_o be_v require_v on_o both_o side_n of_o b_o e_o to_o make_v two_o oblique_a parralellogram_n who_o opposite_a side_n shall_v be_v equal_a to_o the_o line_n give_v this_o may_v be_v do_v either_o by_o draw_v a_o line_n through_o the_o point_n e_o parallel_n to_o a_o b_o c_o and_o then_o make_v f_o e_o e_z d_o and_o b_o c_o each_o equal_a to_o a_o b_o and_o through_o those_o point_n draw_v the_o side_n of_o the_o parralellogram_n or_o continue_v a_o b_o and_o make_v b_o c_o equal_a thereto_o and_o with_o the_o extent_n b_o e_z upon_o a_o and_o c_o draw_v the_o cross_n of_o a_o ark_n at_o f_o and_o d_o again_o upon_o e_o with_o the_o extent_n a_o b_o draw_v oâ—her_o ark_v cross_v the_o former_a at_o f_o and_o those_o cross_n or_o intersection_n limit_v the_o extremity_n of_o the_o side_n of_o the_o parralellogram_n a_o line_n draw_v within_o a_o four-sided_n figure_n from_o one_o corner_n to_o another_o be_v call_v a_o diagonal-line_n a_o parralellipipedon_n be_v a_o solid_a figure_n contain_v under_o six_o four-sided_n figure_n whereof_o those_o which_o be_v opposite_a be_v parallel_n and_o be_v well_o represent_v by_o two_o or_o many_o dice_n set_v one_o upon_o another_o or_o by_o the_o case_n of_o a_o clock-weight_n to_o find_v a_o right_a line_n equal_a to_o the_o circumference_n of_o a_o circle_n give_v let_v the_o give_v circle_n be_v b_o d_o c_o divide_v the_o upper_a semicircle_n b_o d_o c_o into_o half_n at_o d_o and_o the_o low_a semicircle_n into_o three_o equal_a part_n and_o draw_v the_o line_n d_o e_o d_o f_o which_o cut_v the_o diameter_n at_o g_o and_o h_n and_o make_v g_o i_o equal_a to_o g_o h_n then_o be_v the_o length_n d_o i_o a_o little_a more_o than_o the_o length_n of_o the_o quadrant_a b_o d_o neither_o do_v the_o excess_n amount_v unto_o one_o part_n of_o the_o diameter_n b_o c_o if_o it_o be_v divide_v into_o five_o thousand_o and_o four_o time_n the_o extent_n d_o i_o will_v be_v a_o little_a more_o than_o the_o whole_a circumference_n of_o the_o circle_n to_o find_v a_o right_a line_n equal_a to_o any_o arch_n of_o a_o give_v circle_n let_v c_o d_o be_v a_o arch_n of_o a_o give_v circle_n less_o than_o a_o quadrant_n whereto_o it_o be_v require_v to_o find_v a_o right_a line_n equal_a divide_v the_o arch_n c_o d_o into_o half_n at_o e_o and_o make_v the_o right_a line_n f_o g_o equal_a to_o the_o chord_n c_o d_o and_o make_v f_o h_n equal_a to_o twice_o c_o e_o and_z place_z one_o three_o part_n of_o the_o distance_n between_o g_z and_o h_n from_o h_n to_o i_o and_o the_o whole_a line_n f_o i_o will_v be_v nigh_o equal_a in_o length_n to_o the_o arch_n c_o d_o but_o so_o near_o the_o truth_n that_o if_o the_o line_n f_o i_o be_v divide_v into_o 1200_o equal_a part_n one_o of_o those_o part_n add_v thereto_o will_v make_v it_o too_o great_a albeit_o the_o ark_n c_o d_o be_v equal_a to_o a_o quadrant_n but_o in_o lesser_a arch_n the_o difference_n will_v be_v less_o and_o if_o the_o give_v arch_n be_v less_o than_o 60_o degree_n or_o one_o six_o part_n of_o the_o whole_a circle_n the_o line_n find_v will_v not_o want_v one_o six_o thousand_o part_n of_o its_o true_a length_n but_o when_o the_o give_v arch_n be_v great_a than_o a_o quadrant_a it_o may_v be_v find_v at_o twice_o thrice_o or_o four_o time_n by_o former_a direction_n these_o two_o proposition_n be_v take_v out_o of_o hugenius_n de_fw-fr magnitudine_fw-la circuli_fw-la page_n 20_o 21._o in_o dyall_v to_o shun_v draw_v of_o line_n on_o a_o plain_a it_o may_v be_v of_o frequent_a use_n to_o prick_v off_o a_o angle_n by_o sines_n or_o tangent_n in_o stead_n of_o chord_n it_o will_v therefore_o be_v necessary_a to_o define_v these_o kind_n of_o line_n 2._o the_o right_a sine_fw-la of_o a_o arch_n be_v half_a the_o chord_n of_o twice_o that_o arch_n thus_o g_o f_o be_v the_o half_a of_o g_o l_o be_v the_o sine_fw-la of_o the_o arch_n g_o a_o half_a of_o the_o arch_n g_o a_o l_o whence_o it_o follow_v that_o the_o right_a sine_fw-la of_o a_o arch_n less_o than_o a_o quadrant_a be_v also_o the_o right_a sine_fw-la of_o that_o ark_n residue_n from_o a_o semicircle_n because_o as_o be_v show_v above_o the_o chord_n of_o a_o ark_n be_v the_o same_o both_o to_o a_o ark_n lesser_a than_o a_o quadrant_a and_o to_o its_o compliment_n to_o a_o semicircle_n what_o a_o arch_n want_v of_o a_o quadrant_a be_v call_v the_o compliment_n thereof_o thus_o the_o arch_n d_o g_o be_v the_o compliment_n of_o the_o arch_n a_o g_o and_z h_z g_z be_v the_o sine_fw-la of_o the_o arch_n d_o g_o or_o which_o be_v all_o one_o it_o be_v the_o cousin_n of_o the_o arch_n a_o g_o and_o the_o line_n h_o g_o being_n equal_a to_o c_o f_o it_o follow_v that_o the_o right_a sine_fw-la of_o the_o compliment_n of_o a_o arch_n be_v equal_a to_o that_o part_n of_o the_o diameter_n which_o lie_v between_o that_o arch_n and_o the_o centre_n from_o the_o former_a scheme_n also_o follow_v another_o definition_n of_o a_o right_a sine_fw-la as_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 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