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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
c_o b_o the_o departure_n from_o the_o meridian_n in_o the_o course_n between_o both_o place_n then_o make_v that_o one_o leg_n of_o a_o right_a angle_a triangle_n prick_v down_o 17_o 59_o centesme_n the_o difference_n of_o latitude_n between_o those_o place_n out_o of_o the_o same_o equal_a part_n from_o c_o to_o l_o and_o draw_v b_o l_o which_o represent_v the_o course_n and_o distance_n true_o between_o the_o lizard_n &_o bermudas_n and_o the_o extent_n l_o b_o measure_v on_o the_o same_o equal_a part_n show_v the_o distance_n to_o be_v 44_o 31_o centesme_n which_o allow_v twenty_o league_n to_o a_o degree_n be_v 886_o league_n then_o to_o find_v the_o course_n with_o 60_o of_o the_o chord_n set_v one_o foot_n in_o l_o with_o the_o other_o make_v a_o mark_n at_o you_o and_o z_o then_z the_o extent_n z_o y_fw-fr measure_a on_o the_o chord_n show_v the_o rumbe_n to_o be_v 66_o 37′_n from_o the_o meridian_n which_o be_v almost_o 6_o point_n and_o in_o this_o example_n the_o proportion_n do_v not_o err_v any_o thing_n from_o the_o truth_n according_a to_o mercators_n chart_n whereas_o if_o you_o use_v the_o former_a proportion_n by_o the_o middle_a latitude_n the_o rumbe_n will_v have_v be_v 67_o 2′_n from_o the_o meridian_n and_o the_o distance_n 902_o league_n if_o you_o make_v c_o a_o equal_a to_o c_o v_n than_o a_o line_n join_v l_o at_fw-fr shall_v be_v the_o course_n and_o distance_n according_a to_o the_o same_o longitude_n and_o latitude_n lay_v down_o on_o the_o plain_a chart_n and_o thereby_o the_o course_n shall_v be_v 72_o 17′_n from_o the_o meridian_n and_o the_o distance_n 1155_o league_n however_o when_o two_o place_n be_v lay_v down_o true_a at_o first_o in_o their_o rumbe_n distance_n and_o latitude_n on_o the_o plain_a chart_n if_o you_o sail_v home_o in_o or_o near_o the_o same_o rumbe_n the_o plain_a chart_n will_v very_o well_o serve_v to_o keep_v the_o reckon_n upon_o and_o to_o sail_v by_o in_o the_o great_a voyage_n how_o geometrical_o to_o supply_v the_o meridian-line_n of_o mercators_n chart_n general_o that_o the_o find_n of_o the_o true_a rumbe_n between_o two_o place_n may_v not_o be_v a_o proposition_n out_o of_o the_o reach_n of_o geometry_n or_o not_o to_o be_v perform_v by_o scale_n and_o compass_n the_o supply_n thereof_o become_v the_o contemplation_n of_o the_o late_a learned_a mr._n samuel_n forster_n professor_n of_o astronomy_n in_o gresham_n college_n who_o in_o his_o treatise_n of_o a_o ruler_n entitle_v posthuma_n fosteri_n make_v a_o meridian-line_n out_o of_o a_o scale_n of_o secant_v this_o we_o shall_v be_v brief_a in_o and_o show_v how_o near_o it_o come_v to_o the_o truth_n let_v it_o be_v require_v to_o make_v a_o meridian-line_n of_o such_o a_o scantling_n that_o one_o degree_n of_o longitude_n may_v be_v half_a a_o inch_n which_o be_v of_o the_o same_o size_n with_o the_o print_n thereof_o at_o the_o end_n of_o the_o book_n and_o so_o if_o it_o be_v require_v to_o make_v the_o meridian-line_n from_o 43_o to_o 50_o of_o latitude_n the_o former_a extent_n c_o f_o out_o of_o the_o quadrant_a shall_v reach_v in_o this_o line_n f_o l_o from_o 43_o to_o 44_o in_o like_a manner_n the_o secant_fw-la of_o 44_o 30′_n so_o take_v out_o shall_v reach_v from_o 44_o to_o 45._o and_o thus_o may_v the_o whole_a degree_n be_v take_v out_o now_o for_o divide_v they_o into_o centesme_n they_o may_v be_v equal_a division_n and_o yet_o very_a near_o the_o truth_n or_o rather_o first_o divide_v the_o half_a degree_n true_a by_o the_o middle_n secant_fw-la thus_o the_o half_a of_o the_o secant_fw-la of_o 49_o 15′_n shall_v reach_v from_o 49_o to_o 49_o and_o a_o half_a and_o instead_o of_o half_v the_o secant_fw-la it_o may_v be_v take_v out_o to_o half_a the_o radius_fw-la and_o afterward_o the_o part_n of_o each_o half_a degree_n may_v be_v divide_v equal_o and_o so_o if_o you_o will_v divide_v a_o degree_n into_o ten_o part_n each_o part_n will_v be_v 6_o minute_n and_o if_o it_o be_v require_v to_o find_v the_o true_a length_n of_o the_o meridian-line_n from_o 49_o degree_n to_o 49_o degree_n 6_o minute_n the_o ten_o part_n of_o the_o middle_n secant_fw-la to_o wit_n of_o 49_o degree_n 3_o minute_n shall_v be_v the_o length_n require_v and_o so_o on_o successive_o and_o so_o if_o it_o be_v require_v to_o find_v the_o length_n of_o 49_o 7_o ten_o or_o 42′_n i_o say_v that_o 7_o 10_o of_o the_o middle_n secant_fw-la to_o wit_n of_o the_o secant_fw-la of_o 49_o 35_o centesm_n or_o 49_o 21′_n be_v the_o length_n require_v now_o it_o may_v be_v doubt_v that_o the_o make_n of_o the_o meridian-line_n by_o whole_a degree_n be_v not_o near_a enough_o the_o truth_n in_o regard_n the_o table_n at_o first_o be_v make_v by_o the_o add_v of_o the_o secant_v of_o every_o minute_n successive_o together_o and_o the_o learned_a mr._n oughtre_v in_o stead_n of_o add_v the_o secant_v of_o every_o minute_n will_v have_v a_o minute_n divide_v into_o a_o hundred_o a_o thousand_o ten_o thousand_o or_o rather_o a_o million_o of_o part_n and_o the_o secant_v of_o every_o one_o of_o those_o part_n add_v together_o to_o this_o i_o answer_v that_o the_o make_n of_o the_o meridian-line_n by_o whole_a degree_n and_o in_o the_o whole_a do_v not_o breed_v any_o error_n at_o all_o to_o be_v regard_v compare_v with_o the_o make_n of_o it_o up_o successive_o by_o every_o minute_n and_o for_o each_o particular_a degree_n it_o do_v not_o breed_v any_o sensible_a error_n compare_v with_o the_o best_a table_n to_o the_o end_n of_o this_o book_n we_o have_v add_v a_o table_n of_o the_o meridian-line_n to_o every_o second_o centesm_n wherein_o we_o have_v supply_v the_o vacuity_n that_o be_v in_o mr._n gunter_n table_n by_o which_o table_n it_o do_v appear_v that_o the_o meridional_a part_n between_o the_o latitude_n of_o 50_o degree_n and_o 60_o degree_n be_v 17_o ⌊_o 542_o which_o other_o table_n make_v more_o and_o by_o the_o add_v up_o of_o the_o secant_v of_o 51_o degree_n 30_o minute_n 52_o degree_n 30_o minute_n and_o so_o successive_o to_o 59_o degree_n 30_o minute_n they_o will_v amount_v to_o 17_o 547_o the_o difference_n be_v only_o in_o the_o thousand_o part_n of_o a_o degree_n and_o i_o suppose_v there_o be_v no_o mercator_n chart_n make_v wherein_o a_o degree_n of_o longitude_n be_v a_o inch_n the_o big_a i_o have_v see_v be_v but_o half_a a_o inch_n and_o if_o they_o be_v a_o inch_n it_o can_v scarce_o be_v divide_v into_o one_o hundred_o part_n much_o less_o into_o a_o thousand_o and_o so_o in_o every_o part_n of_o the_o meridian-line_n as_o far_o as_o it_o can_v be_v use_v the_o difference_n will_v be_v inconsiderable_a and_o so_o also_o in_o the_o whole_a  _fw-fr deg._n part_v the_o meridional_a part_n for_o 70_o of_o latitude_n by_o our_o table_n and_o mr._n gunter_n be_v 99_o 431._o by_o add_v up_o the_o secant_v of_o 30′_n then_o of_o 1_o 30′_n 2_o 30′_n successive_o to_o 69_o 30′_n they_o be_v 99_o 426._o by_o mr._n wright_n table_n reduce_v to_o degree_n 99_o 444._o calculate_v by_o the_o logarithmical_a tangent_n be_v 99_o 436._o by_o the_o foreign_a table_n of_o maetius_n and_o snellius_n which_o be_v not_o extend_v to_o 80_o 99_o 416._o and_o for_o the_o latter_a part_n of_o the_o objection_n i_o hear_v mr._n oughtred_a be_v make_v a_o new_a table_n of_o they_o according_a to_o his_o own_o mind_n wherein_o it_o be_v probable_a he_o attain_v the_o manner_n of_o add_v up_o those_o secant_v by_o some_o new_a proposition_n in_o regard_n it_o will_v be_v extreme_o tedious_a to_o make_v a_o table_n of_o natural_a secant_v to_o all_o those_o part_n and_o then_o add_v they_o up_o wherein_o he_o long_v since_o desist_v upon_o this_o consideration_n that_o the_o decrease_n will_v happen_v in_o the_o decimal_n part_v remote_a from_o the_o degree_n for_o it_o must_v be_v conceive_v that_o a_o table_n of_o the_o meridian-line_n consist_v of_o the_o sum_n of_o all_o the_o secant_v divide_v by_o radius_fw-la and_o thus_o the_o meridional_a number_n for_o the_o first_o thousand_o centesm_n or_o ten_o degree_n add_v up_o from_o a_o table_n who_o radius_fw-la be_v 1000_o amount_v to_o 1005079_o and_o divide_v by_o the_o radius_fw-la be_v 1005_o 079_o then_o because_o we_o will_v have_v a_o table_n to_o express_v the_o meridian-line_n in_o degree_n allow_v 100_o centesm_n to_o a_o degree_n we_o must_v divide_v the_o former_a number_n by_o 100_o and_o the_o quotient_n be_v 10_o 05079_o which_o be_v the_o very_a number_n in_o our_o table_n save_v that_o our_o table_n be_v not_o continue_v so_o far_o by_o two_o place_n and_o for_o the_o last_o two_o figure_n which_o be_v omit_v we_o add_v in_o a_o unit_fw-la in_o the_o
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
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
meridian_n count_v the_o same_o in_o the_o variation_n column_n upward_o and_o against_o it_o in_o the_o sin_n you_o will_v find_v the_o difference_n of_o latitude_n be_v 923_o league_n and_o 87_o part_n more_o of_o another_o league_n divide_v into_o 100_o part_n but_o if_o the_o difference_n of_o latitude_n be_v require_v for_o any_o other_o distance_n multiply_v the_o sine_fw-la of_o the_o give_a point_n by_o the_o distance_n run_v cut_v off_o three_o figure_n from_o the_o product_n example_n if_o the_o ship_n sail_n 60_o league_n on_o that_o course_n the_o difference_n of_o latitude_n be_v 55_o league_n and_o 38_o centesm_n multiply_v 923_o by_o 60_o and_o the_o product_n cut_v off_o three_o place_n be_v 55_o 380._o to_o find_v the_o departure_n from_o the_o meridian_n the_o proportion_n be_v as_o the_o radius_fw-la be_v to_o the_o distance_n run_v so_o be_v the_o sine_fw-la of_o the_o rumbe_n to_o the_o departure_n ∷_o example_n if_o a_o ship_n sail_v 1000_o league_n south_n south_n west_n count_v the_o point_n sail_v in_o the_o separation_n column_n downward_o be_v the_o three_o column_n and_o on_o the_o right_a hand_n of_o the_o sin_n and_o you_o will_v in_o the_o column_n of_o sines_n find_v the_o departure_n require_v to_o be_v 382_o league_n 68_o centesm_n but_o if_o a_o ship_n sail_v but_o 60_o league_n on_o that_o course_n multiply_v 382_o thereby_o cut_v off_o the_o three_o last_o place_n and_o you_o will_v find_v the_o separation_n or_o departure_n require_v to_o be_v 22_o league_n 92_o centesm_n or_o 22_o 920_o league_n point_v sines_n separ_fw-la tangent_n secant_v angle_n sin_n quintuple_v deg_fw-la part_n ¾_n 49_o 07_o ¼_n 49_o 12_o 1001_o 20_o 2_o 48_o m_o ¾_n 2_o 45_o ½_n 98_o 02_o ½_n 98_o 48_o 1004_o 84_o 5_o 37_o ½_n 4_o 90_o ¼_n 146_o 73_o ¾_n 148_o 32_o 1010_o 94_o 8_o 26_o ¼_n 7_o 33_o 7_o 195_o 09_o 1_o 198_o 91_o 1019_o 59_o 11_o 15_o 9_o 75_o ¾_n 242_o 98_o ¼_n 250_o 48_o 1030_o 89_o 14_o 3_o ¾_n 12_o 14_o ½_n 290_o 28_o ½_n 303_o 36_o 1044_o 99_o 16_o 52_o ½_n 14_o 51_o ¼_n 336_o 89_o ¾_n 357_o 80_o 1062_o 08_o 19_o 41_o ¼_n 16_o 84_o 6_o 382_o 68_o 2_o 414_o 21_o 1082_o 39_o 22_o 30_o 19_o 13_o ¾_n 427_o 55_o ¼_n 472_o 96_o 1106_o 21_o 25_o 18_o ¾_n 21_o 37_o ½_n 471_o 39_o ½_n 534_o 52_o 1133_o 88_o 28_o 7_o ½_n 23_o 56_o ¼_n 514_o 10_o ¾_n 599_o 36_o 1165_o 96_o 30_o 56_o ¼_n 25_o 70_o 5_o 555_o 57_o 3_o 668_o 17_o 1202_o 68_o 33_o 45_o 27_o 77_o ¾_n 595_o 70_o ¼_n 741_o 65_o 1245_o 28_o 36_o 33_o ¾_n 29_o 78_o ½_n 634_o 39_o ½_n 820_o 68_o 1293_o 64_o 39_o 22_o ½_n 31_o 71_o ¼_n 671_o 56_o ¾_n 906_o 34_o 1349_o 61_o 42_o 11_o ¼_n 33_o 57_o 4_o 707_o 10_o 4_o 1000_o 00_o 1414_o 21_o 45_o 35_o 35_o ¾_n 740_o 96_o ¼_n 1103_o 32_o 1489_o 08_o 47_o 48_o ¾_n 37_o 04_o ½_n 773_o 01_o ½_n 1218_o 48_o 1576_o 32_o 50_o 37_o ½_n 38_o 65_o ¼_n 803_o 21_o ¾_n 1348_o 36_o 1678_o 68_o 53_o 26_o ¼_n 40_o 16_o 3_o 831_o 47_o 5_o 1496_o 60_o 1799_o 95_o 56_o 15_o 41_o 57_o ¾_n 857_o 73_o ¼_n 1668_o 37_o 1945_o 14_o 59_o 3_o ¾_n 42_o 88_o ½_n 881_o 91_o ½_n 1870_o 73_o 2121_o 36_o 61_o 52_o ½_n 44_o 09_o ¼_n 903_o 99_o ¾_n 2114_o 20_o 2338_o 88_o 64_o 41_o ¼_n 45_o 19_o 2_o 923_o 87_o 6_o 2414_o 21_o 2613_o 12_o 67_o 30_o 46_o 19_o ¾_n 941_o 53_o ¼_n 2794_o 76_o 2968_o 34_o 70_o 18_o ¾_n 47_o 08_o ½_n 956_o 94_o ½_n 3296_o 50_o 3445_o 69_o 73_o 7_o ½_n 47_o 84_o ¼_n 970_o 03_o ¾_n 3992_o 24_o 4115_o 56_o 75_o 56_o ¼_n 48_o 50_o 1_o 980_o 78_o 7_o 5027_o 33_o 5125_o 83_o 78_o 45_o 49_o 03_o ¾_n 989_o 17_o ¼_n 6741_o 44_o 6808_o 52_o 81_o 33_o ¾_n 49_o 45_o ½_n 995_o 18_o ½_n 10153_o 66_o 10202_o 32_o 84_o 22_o ½_n 49_o 75_o ¼_n 998_o 79_o ¼_n 20355_o 48_o 20380_o 15_o 87_o 11_o ¼_n 49_o 93_o variation_n  _fw-fr  _fw-fr  _fw-fr  _fw-fr  _fw-fr  _fw-fr here_o note_v that_o though_o the_o distance_n sail_v be_v give_v in_o league_n yet_o the_o table_n of_o sines_n may_v be_v so_o alter_v that_o the_o difference_n of_o latitude_n and_o departure_n may_v be_v find_v in_o degree_n and_o centesme_n and_o that_o only_o be_v multiply_v all_o the_o figure_n in_o the_o column_n of_o sines_n by_o the_o number_n 5_o for_o which_o purpose_n we_o add_v the_o last_o column_n of_o quintupuled_a sin_n in_o which_o the_o whole_a degree_n be_v distinguish_v from_o the_o decimal_a part_n example_n if_o a_o ship_n sail_v 1000_o league_n south_n south-west_n against_o the_o second_o point_n count_v upward_o we_o shall_v find_v the_o difference_n of_o latitude_n to_o be_v 46_o 19_o centesme_n but_o count_v downward_o we_o shall_v find_v the_o departure_n from_o the_o meridian_n to_o be_v 19_o degree_n 13_o centesme_n which_o number_n multiply_v by_o 60_o cut_n off_o five_o place_n towards_o the_o right_a hand_n give_v the_o difference_n of_o latitude_n and_o departure_n in_o that_o course_n for_o 60_o league_n to_o be_v 2_o 77140_o part_n diff._n of_o lat._n 1_o 14780_o departure_n of_o which_o result_v we_o need_v take_v no_o more_o place_n than_o 2_o d_o 77._o 1_o 14_o or_o 1_o 15._o by_o these_o two_o proposition_n the_o dead_a reckon_a to_o be_v apply_v to_o the_o plain_a chart_n may_v be_v keep_v after_o the_o form_n prescribe_v by_o mr._n norwood_n and_o according_o the_o difference_n of_o latitude_n and_o departure_n from_o the_o meridian_n be_v here_o express_v in_o degree_n and_o centesme_n for_o the_o three_o traverse_n from_o tenariff_n towards_o nicholas_n island_n which_o we_o make_v our_o former_a example_n course_n distance_n in_o league_n north_n south_n deg._n cent._n east_n west_n d_o cent._n s_o s_o w_n 60_o  _fw-fr 2_o 77_o  _fw-fr 1_o 15_o w_o s_o w_n 80_o  _fw-fr 1_o 53_o  _fw-fr 3_o 69_o s_o b_o e_o ½_n e_o 53_o  _fw-fr 2_o 53_o 0_o 77_o 0_o 00_o  _fw-fr  _fw-fr  _fw-fr 6_o 83_o 0_o 77_o 4_o 84_o when_o some_o course_n increase_v the_o latitude_n and_o other_o decrease_v it_o the_o difference_n between_o the_o north_n and_o south_n column_n find_v by_o substract_v the_o lesser_a from_o the_o great_a give_v the_o difference_n of_o latitude_n so_o likewise_o the_o difference_n of_o the_o east_n and_o west_n column_n give_v the_o departure_n which_o in_o this_o example_n be_v 4_o 07_o this_o difference_n of_o latitude_n and_o departure_n substract_v out_o of_o the_o whole_a difference_n of_o latitude_n and_o of_o longitude_n between_o the_o say_v two_o island_n there_o rest_v 4_o 17_o centesme_n difference_n of_o latitude_n between_o the_o ship_n and_o st._n nicholas_n island_n and_o 3_o 93_o difference_n of_o longitude_n by_o which_o the_o course_n from_o the_o ship_n to_o the_o say_a island_n will_v be_v find_v to_o be_v 43_o 28_o centesm_n to_o the_o westward_n of_o the_o south_n and_o the_o distance_n 5_o 73_o be_v about_o 114_o league_n and_o a_o half_a as_o shall_v afterward_o follow_v the_o difference_n of_o latitude_n and_o the_o rumbe_n give_v to_o find_v the_o departure_n from_o the_o meridian_n and_o distance_n as_o the_o radius_fw-la be_v to_o the_o difference_n of_o latitude_n ∷_o so_n be_v the_o tangent_fw-la of_o the_o rumbe_n to_o the_o departure_n from_o the_o meridian_n ∷_o and_o so_o be_v the_o secant_fw-la of_o the_o rumbe_n to_o the_o distance_n ∷_o example_n for_o the_o departure_n let_v the_o instance_n be_v the_o same_o as_o in_o page_n 24_o where_o we_o suppose_v the_o course_n to_o be_v 3_o point_n from_o the_o meridian_n to_o wit_n s_o w_n b_o s_o and_o the_o correct_a difference_n of_o latitude_n to_o be_v 2_o degree_n 24_o minute_n or_o 2_o degree_n 4_o ten_o or_o 40_o centesm_n by_o which_o multiply_a 668_o the_o tangent_fw-la belong_v to_o 3_o point_n in_o the_o former_a table_n the_o result_n cut_v off_o 5_o figure_n be_v 1_o 60320_o wherefore_o the_o departure_n from_o the_o meridian_n be_v 1_o degree_n 6_o ten_o for_o the_o distance_n in_o like_a manner_n the_o secant_fw-la of_o three_o point_n to_o wit_n 1202_o be_v multiply_v by_o the_o difference_n of_o latitude_n 2_o 40_o cut_n off_o 5_o figure_n be_v 2_o 88480_o wherefore_o the_o distance_n be_v 2_o degree_n 88_o centesme_n at_o 20_o league_n to_o a_o degree_n be_v 50_o league_n and_o three_o quarter_n or_o 50_o 76._o for_o keep_v a_o reckon_n by_o longitude_n which_o be_v applyable_a to_o mercators_n chart_n and_o remove_v the_o error_n of_o the_o plain_a chart_n the_o table_n of_o meridional_a part_n be_v add_v at_o the_o end_n of_o the_o book_n for_o work_v of_o this_o proportion_n as_o the_o radius_fw-la be_v to_o the_o tangent_fw-la of_o the_o rumbe_n ∷_o so_n be_v
the_o meridional_a part_n between_o both_o latitude_n to_o the_o difference_n of_o longitude_n ∷_o which_o proportion_n require_v the_o difference_n of_o latitude_n to_o be_v first_o find_v and_o then_o by_o help_n of_o the_o table_n of_o meridional_a part_n at_o the_o end_n of_o the_o book_n the_o difference_n of_o longitude_n may_v be_v find_v which_o table_n be_v make_v but_o to_o every_o second_o centesm_n have_v half_o the_o difference_n set_v down_o at_o the_o bottom_n of_o the_o page_n which_o add_v to_o the_o meridional_a part_n of_o any_o even_a centesm_n above_o it_o make_v the_o meridional_a part_n for_o each_o odd_a centesm_n thus_o the_o meridional_a part_n for_o 25_o 22_o be_v 26_o 076_o whereto_o add_v 11_o the_o sum_n be_v 26_o 087_o be_v meridional_a part_n for_o 25_o 23_o centesm_n in_o the_o first_o example_n of_o the_o traverse_v the_o difference_n of_o latitude_n be_v 2_o 77_o centesm_n the_o latitude_n of_o tenariff_n be_v 28_o meridional_a part_n 29_o 186_o the_o latitude_n of_o the_o ship_n place_n be_v 25_o 23_o 26_o 087_o the_o meridional_a part_n between_o both_o latitude_n be_v 3_o 099_o which_o be_v multiply_v by_o the_o tangent_fw-la of_o the_o second_o point_n to_o wit_n 414_o cut_v off_o 6_o place_n three_o by_o reason_n the_o radius_fw-la be_v 1000_o and_o three_o more_o for_o the_o decimal_a part_n of_o the_o meridional_a number_n the_o amount_v being_z 1_o 282986_o show_v that_o the_o difference_n of_o longitude_n be_v 1_o degree_n 28_o centesm_n of_o which_o there_o may_v be_v a_o column_n of_o east_n and_o west_n keep_v like_v as_o be_v do_v for_o the_o departure_n and_o this_o cast_v up_o of_o the_o longitude_n may_v be_v ready_o do_v also_o on_o the_o logarithmical_a ruler_n or_o by_o mr._n phillip_n his_o late_a table_n for_o that_o purpose_n without_o calculation_n whereof_o we_o make_v mention_v in_o page_n 37._o we_o have_v before_o say_v that_o a_o table_n of_o sines_n be_v sufficient_a to_o supply_v all_o calculation_n though_o oâ—â—er_v table_n may_v be_v more_o ready_a for_o dispatch_v when_o they_o be_v at_o hand_n as_o we_o have_v show_v concern_v the_o table_n of_o meridional_a part_n in_o page_n 48._o out_o of_o it_o the_o secant_v be_v make_v by_o this_o proportion_n as_o the_o cousin_n of_o any_o arch_n propose_v be_v to_o the_o radius_fw-la ∷_o so_n be_v the_o radius_fw-la to_o the_o secant_fw-la of_o the_o give_a arch_n ∷_o which_o proportion_n hold_v backward_o to_o find_v the_o arch_n in_o the_o table_n of_o sines_n if_o a_o secant_fw-la be_v give_v at_o adventure_n and_o the_o arch_n require_v also_o out_o of_o it_o the_o tangent_n be_v make_v by_o this_o proportion_n as_o the_o cousin_n of_o a_o arch_n be_v to_o the_o sine_fw-la of_o the_o say_a arch_n ∷_o so_n be_v the_o radius_fw-la to_o the_o tangent_fw-la of_o the_o arch_n propose_v ∷_o which_o proportion_n do_v not_o hold_v backward_o to_o find_v the_o arch_n if_o a_o tangent_fw-la be_v give_v in_o which_o case_n the_o secant_fw-la may_v be_v find_v and_o thereby_o the_o arch_n for_o the_o square_n of_o a_o tangent_fw-la more_fw-it the_o square_n of_o the_o radius_fw-la be_v equal_a to_o the_o square_n of_o the_o secant_fw-la in_o this_o case_n we_o have_v also_o the_o proportion_n of_o the_o cousin_n to_o the_o sine_fw-la give_v be_v the_o same_o with_o that_o of_o the_o radius_fw-la to_o the_o tangent_fw-la and_o the_o sum_n oâ—_n their_o square_n give_v be_v equal_a to_o the_o square_n of_o the_o radius_fw-la which_o very_a case_n be_v reduce_v to_o a_o double_a equation_n in_o the_o 33_o question_n of_o mr._n moor_n algebraick_n arithmetic_n and_o by_o which_o the_o radius_fw-la and_o a_o tangent_fw-la be_v give_v either_o the_o sine_fw-la or_o cousin_n may_v be_v find_v without_o find_v the_o secant_fw-la if_o such_o a_o proportion_n as_o this_o be_v propose_v as_o the_o radius_fw-la be_v to_o the_o tangent_fw-la of_o a_o arch_n ∷_o so_n be_v the_o tangent_fw-la of_o another_o arch_n to_o the_o tangent_fw-la of_o a_o four_o arch_n ∷_o it_o may_v be_v resolve_v thus_o without_o make_v a_o table_n of_o tangent_n make_v the_o product_n of_o the_o radius_fw-la and_o of_o the_o cousin_n of_o the_o second_o ark_n the_o divisor_n and_o the_o product_n of_o the_o sin_n of_o the_o second_o and_o three_o ark_n and_o of_o the_o cousin_n of_o the_o three_o ark_n the_o dividend_n and_o the_o quotient_n will_v be_v the_o tangent_fw-la of_o the_o four_o ark_n seek_v in_o like_a manner_n if_o all_o those_o term_n be_v tangent_n the_o product_n of_o the_o radius_fw-la sine_fw-la and_o cousin_n of_o the_o first_o ark_n will_v be_v the_o divisor_n and_o the_o product_n of_o the_o sin_n and_o cosines_n of_o the_o second_o and_o three_o ark_n the_o dividend_n whereby_o may_v be_v find_v the_o quotient_a be_v the_o tangent_fw-la seek_v and_o consequent_o the_o arch_n answer_v thereto_o after_o the_o like_a manner_n with_o due_a regard_n to_o the_o proportion_n for_o make_v the_o secant_v any_o secant_fw-la might_n be_v supply_v and_o a_o proportion_n whole_o in_o secant_v by_o turn_v those_o arke_n into_o their_o compliment_n be_v change_v whole_o into_o sin_n or_o two_o term_n be_v secant_v be_v change_v into_o sin_n by_o alter_v the_o place_n or_o order_n of_o those_o term_n and_o their_o arke_n into_o their_o compliment_n how_o to_o calculate_v a_o table_n of_o natural_a sin_n how_o to_o make_v the_o sine_fw-la of_o any_o ark_n at_o pleasure_n snellius_n in_o his_o cyclometria_n have_v show_v without_o make_v many_o such_o sin_n as_o be_v not_o require_v in_o which_o he_o go_v beyond_o all_o foâ—mer_a writer_n one_o way_n he_o have_v hold_v true_a as_o far_o as_o 30_o of_o the_o quadrant_n out_o of_o which_o sin_n their_o cosines_n may_v be_v make_v and_o so_o the_o quadrant_n fill_v from_o 60_o to_o 90_o and_o then_o by_o find_v the_o sine_fw-la of_o the_o double_a ark_n it_o may_v be_v supply_v from_o 30_o to_o 60_o but_o in_o regard_n the_o performance_n be_v very_o tedious_a and_o therefore_o not_o to_o be_v prosecute_v in_o the_o construction_n of_o a_o whole_a table_n we_o shall_v omit_v to_o mention_v it_o another_o way_n he_o have_v which_o do_v not_o hold_v true_a to_o above_o 1_o 12_o part_n of_o the_o quadrant_n so_o that_o when_o a_o great_a ark_n near_o the_o end_n of_o the_o quadrant_n be_v give_v he_o find_v the_o cousin_n thereof_o first_o and_o thereby_o the_o sine_fw-la or_o if_o the_o ark_n be_v remote_a or_o about_o the_o middle_n of_o the_o quadrant_n he_o find_v the_o sine_fw-la of_o the_o eight_o or_o some_o such_o part_n of_o it_o and_o then_o by_o find_v the_o sine_fw-la of_o the_o double_a ark_n etc._n etc._n backward_o he_o find_v the_o sine_fw-la seek_v but_o how_o to_o calculate_v a_o whole_a table_n by_o proportion_n with_o ease_n be_v attain_v unto_o by_o our_o worthy_a countryman_n mr._n michael_n darie_n who_o discover_v the_o same_o in_o the_o year_n 1651_o and_o communicate_v unto_o i_o the_o proportion_n for_o that_o purpose_n long_o since_o who_o method_n we_o shall_v now_o insist_v upon_o 1._o the_o ready_a way_n be_v to_o assume_v the_o proportion_n of_o the_o diameter_n of_o a_o circle_n to_o its_o circumference_n to_o be_v know_v and_o to_o be_v in_o the_o proportion_n of_o 113_o to_o 355_o of_o which_o see_v maetius_n his_o practical_a geometry_n which_o two_o number_n be_v easy_o remember_v be_v the_o three_o first_o odd_a figure_n of_o the_o digit_n each_o of_o they_o twice_o write_v down_o and_o cut_v asunder_o in_o the_o middle_n 2._o it_o be_v most_o convenient_a that_o the_o radius_fw-la or_o whole_a sine_fw-la shall_v be_v a_o unit_fw-la with_o cipher_n and_o so_o the_o proporrtion_n of_o the_o diameter_n to_o the_o circumference_n find_v by_o the_o two_o number_n above_o be_v such_o as_o 2000000_o to_o 6283185_o so_o far_o true_a and_o to_o find_v it_o true_a to_o as_o many_o place_n of_o figure_n as_o you_o please_v consult_v dr._n wallis_n his_o write_n or_o hungenius_n de_fw-fr magnitudine_fw-la circuli_fw-la 3._o the_o sine_fw-la of_o one_o minute_n or_o centesm_n do_v so_o insensible_o differ_v from_o the_o length_n of_o the_o arch_n to_o which_o it_o belong_v that_o the_o length_n of_o the_o arch_n of_o one_o minute_n or_o centesm_n may_v be_v very_o well_o take_v to_o be_v the_o sine_fw-la thereof_o and_o in_o the_o large_a table_n that_o ever_o be_v publish_v differ_v nothing_o therefrom_o wherefore_o a_o minute_n be_v the_o 21600_o part_n of_o the_o circumference_n of_o a_o whole_a circle_n the_o like_a part_n of_o the_o number_n 6283185_o be_v 290_o be_v the_o sine_fw-la of_o 1_o minute_n 4._o you_o may_v find_v the_o cousin_n of_o one_o minute_n by_o substract_v the_o square_n of_o the_o sine_fw-la of_o one_o minute_n from_o the_o square_n of_o the_o radius_fw