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chart_n to_o set_v down_o both_o north_n and_o south_n latitude_n than_o you_o must_v draw_v the_o like_a parallel_n of_o latitude_n below_o the_o equinoctial_a as_o these_o be_v above_o it_o now_o i_o will_v show_v you_o how_o the_o several_a proposition_n which_o be_v perform_v by_o the_o plain_a chart_n may_v be_v perform_v by_o this_o and_o wherein_o they_o agree_v and_o wherein_o they_o differ_v proposition_n 1._o know_v the_o longitude_n and_o latitude_n of_o any_o place_n to_o set_v it_o upon_o the_o chart._n chart._n 1._o by_o the_o longitude_n and_o latitede_n to_o find_v the_o point_n of_o any_o place_n in_o the_o chart._n this_o must_v be_v do_v as_o in_o the_o plain_a chart_n for_o first_o lay_v your_o ruler_n by_o the_o longitude_n of_o the_o place_n you_o must_v draw_v a_o little_a occult_a line_n as_o near_o the_o latitude_n of_o the_o place_n as_o you_o can_v guess_v then_o lay_v your_o ruler_n to_o the_o latitude_n of_o the_o place_n across_o that_o line_n you_o draw_v 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deg_fw-la and_o let_v the_o longitude_n thereof_o be_v suppose_v to_o be_v 10_o degr_n so_o the_o difference_n of_o the_o longitude_n of_o the_o two_o place_n will_v be_v 70_o deg_fw-la as_o mr._n norwood_n both_o in_o his_o book_n of_o the_o doctrine_n of_o triangle_n and_o his_o seaman_n practice_n suppose_v they_o to_o be_v though_o as_o he_o say_v in_o one_o place_n he_o do_v not_o think_v they_o to_o be_v so_o far_o distant_a and_o it_o be_v require_v to_o find_v the_o rumbe_n this_o proposition_n must_v also_o be_v perform_v as_o in_o the_o plain_a chart_n for_o first_o the_o two_o place_n must_v be_v set_v upon_o the_o chart_n according_a to_o their_o longitude_n and_o latitude_n which_o will_v be_v at_o s_n and_o l_o then_o draw_v a_o strait_a line_n from_o s_n to_o l_o this_o represent_v the_o direct_a way_n between_o the_o two_o place_n now_o to_o know_v what_o rumbe_n this_o be_v open_v your_o compass_n to_o the_o radius_fw-la of_o your_o scale_n of_o rumbe_n and_o set_v one_o foot_n of_o your_o compass_n in_o s_n with_o the_o other_o draw_v the_o arch_n r_o m_o then_o set_v one_o foot_n of_o your_o compass_n in_o r_n open_v the_o other_o to_o the_o cross_v of_o the_o line_n and_o the_o arch_n at_o m_n and_o measure_v that_o distance_n on_o your_o scale_n of_o chord_n or_o rumbe_n so_o shall_v you_o find_v it_o to_o be_v 71_o deg_fw-la 21_o min._n or_o the_o sixth_o rumbe_n and_o somewhat_o above_o a_o quarter_n of_o a_o rumbe_n from_o the_o meridian_n proposition_n 3._o know_v the_o longitude_n and_o latitude_n of_o two_o place_n to_o know_v how_o far_o they_o be_v distant_a one_o from_o another_o place_n 3._o to_o measure_v the_o distance_n of_o place_n let_v the_o two_o place_n be_v as_o be_v before_o say_a s_n and_o l_o it_o be_v require_v to_o find_v their_o distance_n in_o the_o work_a this_o proposition_n there_o be_v some_o difference_n from_o the_o plain_a chart_n for_o whereas_o there_o you_o measure_v the_o distance_n of_o place_n by_o one_o and_o the_o same_o scale_n of_o equal_a part_n here_o you_o will_v have_v use_n of_o many_o scale_n according_a to_o the_o latitude_n of_o the_o place_n way_n mr._n gunter_n way_n now_o the_o ordinary_a way_n prescribe_v by_o mr._n gunter_n to_o perform_v this_o be_v thus_o open_v your_o compass_n to_o the_o distance_n of_o the_o two_o place_n and_o then_o set_v your_o compass_n in_o the_o meridian_n line_n so_o that_o the_o one_o point_n of_o the_o compass_n may_v stand_v just_a so_o much_o above_o the_o great_a latitude_n as_o the_o other_o do_v below_o the_o lesser_a latitude_n and_o so_o the_o degree_n between_o they_o be_v the_o distance_n this_o way_n may_v serve_v for_o small_a distance_n as_o master_n gunter_n use_v it_o but_o in_o great_a distance_n it_o will_v not_o always_o hold_v true_a and_o beside_o it_o be_v somewhat_o troublesome_a to_o set_v the_o compass_n just_a as_o much_o above_o the_o one_o latitude_n as_o below_o the_o other_o as_o in_o this_o example_n if_o you_o take_v the_o distance_n s_o l_o in_o your_o compass_n and_o measure_v it_o so_o in_o the_o meridian_n line_n it_o will_v reach_v from_o about_o 16_o degree_n to_o about_o 66_o degree_n and_o a_o half_a that_o be_v 16_o degree_n and_o a_o half_a above_o 50_o degree_n the_o great_a latitude_n and_o 16_o degree_n and_o a_o half_a below_o 32_o degree_n 25_o minute_n the_o lesser_a latitude_n and_o so_o the_o degree_n intercept_v between_o the_o point_n of_o the_o compass_n be_v about_o 50_o degree_n and_o a_o half_a whereas_o the_o distance_n of_o the_o two_o place_n be_v almost_o 55_o degree_n but_o you_o shall_v find_v the_o distance_n more_o exact_o latitude_n the_o way_n to_o measure_v the_o distance_n of_o
quadrant_n a_o b_o this_o line_n r_o t_n be_v the_o tangent_fw-la line_n which_o you_o must_v divide_v into_o degree_n as_o you_o see_v in_o the_o figure_n by_o draw_v straight_o line_n from_o the_o centre_n a_o to_o the_o limb_n of_o the_o quadrant_n then_o transfer_v this_o line_n to_o the_o side_n of_o the_o quadrant_n a_o b_o and_o a_o d_o and_o then_o set_v one_o foot_n of_o your_o compass_n in_o the_o centre_n a_o open_v the_o other_o to_o the_o several_a degree_n in_o the_o line_n a_o b_o or_o a_o d_o and_o draw_v the_o arch_n now_o you_o must_v know_v that_o these_o arch_n be_v the_o parallel_n of_o latitude_n and_o the_o straight_a line_n draw_v from_o the_o centre_n be_v meridian_n line_n or_o the_o line_n of_o longitude_n the_o arch_n of_o latitude_n you_o must_v number_v as_o in_o the_o figure_n but_o the_o line_n of_o longitude_n you_o may_v number_v as_o your_o occasion_n require_v this_o be_v a_o projection_n of_o a_o part_n of_o the_o globe_n in_o plano_fw-la by_o natural_a tangent_n you_o may_v if_o you_o please_v when_o occasion_n require_v divide_v a_o circle_n into_o four_o quadrant_n and_o draw_v the_o line_n of_o longitude_n from_o the_o centre_n and_o number_v they_o to_o 360_o and_o likewise_o describe_v the_o circle_n of_o latitude_n round_o about_o the_o centre_n and_o you_o may_v make_v this_o projection_n as_o large_a or_o as_o little_a as_o you_o will_v by_o the_o table_n of_o natural_a tangent_n if_o you_o lengthen_v or_o shorten_v your_o radius_fw-la a_o table_n of_o natural_a tangent_n the_o radius_fw-la being_n 1000_o part_n d._n tan_n d._n tangle_n d._n tang._n d._n tangent_fw-la 1_o 017_o 24_o 445_o 46_o 1,036_o 69_o 02,605_o 2_o 035_o 25_o 466_o 47_o 1,072_o 70_o 02,747_o 3_o 052_o 26_o 488_o 48_o 1,112_o 71_o 02_o 904_o 4_o 070_o 27_o 510_o 49_o 1_o 150_o 72_o 03_o 078_o 5_o 087_o 28_o 532_o 50_o 1_o 192_o 73_o 03_o 271_o 6_o 105_o 29_o 554_o 51_o 1,235_o 74_o 03,487_o 7_o 123_o 30_o 577_o 52_o 1_o 280_o 75_o 03,732_o 8_o 141_o 31_o 601_o 53_o 1_o 327_o 76_o 04_o 011_o 9_o 158_o 32_o 624_o 54_o 1,376_o 77_o 04,331_o 10_o 176_o 33_o 649_o 55_o 1,428_o 78_o 04,705_o 11_o 194_o 34_o 675_o 56_o 1,483_o 79_o 05,144_o 12_o 213_o 35_o 700_o 57_o 1_o 540_o 80_o 05_o 671_o 13_o 231_o 36_o 727_o 58_o 1,600_o 81_o 06_o 313_o 14_o 249_o 37_o 754_o 59_o 1,664_o 82_o 07,115_o 15_o 268_o 38_o 781_o 60_o 1_o 732_o 83_o 08_o 144_o 16_o 287_o 39_o 810_o 61_o 1,804_o 84_o 09_o 514_o 17_o 306_o 40_o 839_o 62_o 1_o 881_o 85_o 11,430_o 18_o 325_o 41_o 869_o 63_o 1,963_o 86_o 14,300_o 19_o 344_o 42_o 900_o 64_o 2_o 050_o 87_o 19,081_o 20_o 364_o 43_o 933_o 65_o 2_o 144_o 88_o 28_o 636_o 21_o 384_o 44_o 966_o 66_o 2_o 246_o 89_o 57,290_o 22_o 404_o 45_o 1000_o 67_o 2,356_o 90_o infinite_a 23_o 424_o  _fw-fr rad._n 68_o 2_o 475_o  _fw-fr  _fw-fr let_v your_o radius_fw-la be_v of_o what_o length_n you_o please_v first_o divide_v it_o into_o 10_o equal_a part_n and_o then_o subdivide_v each_o of_o those_o part_n into_o 10_o so_o you_o shall_v have_v 100_o part_n in_o your_o line_n than_o you_o may_v if_o you_o can_v divide_v each_o of_o these_o 100_o part_n into_o 10_o so_o you_o shall_v have_v 1000_o but_o this_o last_o division_n will_v be_v needless_a for_o you_o may_v by_o your_o eye_n guess_v at_o the_o proportion_n ill_a part_n have_v thus_o fit_v your_o scale_n of_o equal_a part_n you_o may_v prick_v down_o the_o line_n of_o tangent_n out_o of_o this_o table_n note_v after_o you_o be_v pass_v 45_o degree_n in_o the_o table_n the_o figure_n before_o the_o comma_n show_v the_o whole_a radius_fw-la or_o how_o many_o time_n the_o whole_a radius_fw-la be_v contain_v therein_o and_o the_o three_o follow_a figure_n the_o part_n to_o be_v reckon_v upon_o the_o scale_n as_o before_o you_o will_v find_v this_o table_n necessary_a either_o when_o you_o will_v make_v a_o large_a tangent_fw-la line_n to_o serve_v for_o place_n only_o near_o the_o pole_n or_o when_o you_o will_v make_v a_o very_a little_a tangent_fw-la line_n that_o so_o you_o may_v bring_v in_o the_o degree_n near_o the_o equinoctial_a into_o your_o quadrant_n easy_a the_o flank_n be_v make_v will_v serve_v for_o many_o example_n so_o that_o the_o work_n will_v be_v very_o easy_a have_v thus_o draw_v this_o blank_a quadrant_n you_o must_v set_v down_o therein_o the_o two_o place_n you_o be_v to_o sail_v between_o according_a to_o their_o latitude_n and_o longitude_n and_o then_o only_o by_o your_o ruler_n draw_v a_o straight_a line_n from_o the_o one_o place_n to_o the_o other_o and_o this_o straight_a line_n will_v represent_v the_o great_a circle_n which_o pass_v between_o the_o two_o place_n and_o will_v exact_o cross_v those_o degree_n of_o longitude_n and_o latitude_n which_o you_o must_v sail_v by_o for_o the_o example_n example_n example_n and_o proof_n hereof_o i_o shall_v take_v mr._n norwoods_n example_n of_o a_o voyage_n from_o the_o summer_n land_n to_o the_o lizard_n the_o latitude_n of_o the_o summer-iland_n be_v 32_o degree_n 25_o minute_n let_v the_o longitude_n thereof_o be_v suppose_v to_o be_v â—00_o degree_n the_o latitude_n of_o the_o lizard_n be_v near_o 50_o degree_n the_o difference_n of_o longitude_n between_v the_o two_o place_n be_v suppose_v to_o be_v 70_o degree_n so_o that_o the_o longitude_n of_o the_o lizard_n will_v be_v 10_o degree_n and_o it_o be_v require_v to_o know_v by_o what_o longitude_n and_o latitude_n the_o arch_n of_o a_o great_a circle_n draw_v between_o these_o two_o place_n do_v pass_v example_n the_o work_n of_o the_o example_n first_o let_v the_o line_n a_o b_o represent_v the_o meridian_n of_o the_o summer_n land_n upon_o which_o you_o must_v mark_v out_o their_o latitude_n 32_o degree_n 25_o minute_n at_o b_o and_o because_o the_o longitude_n thereof_o be_v 300_o set_v down_o 100l_n at_o the_o end_n of_o the_o line_n a_o b_o so_fw-mi the_o summer-iland_n shall_v be_v set_v down_o according_a to_o their_o longitude_n and_o latitude_n then_o count_v still_o forward_o the_o degree_n of_o the_o difference_n of_o longitude_n till_o you_o come_v to_o 70_o degree_n in_o the_o limb_n of_o the_o quadrant_a and_o there_o draw_v the_o line_n a_o c_o 70_o this_o line_n will_v represent_v the_o meridian_n of_o the_o lizard_n and_o upon_o this_o line_n you_o must_v mark_v out_o the_o latitude_n of_o the_o lizard_n which_o be_v 50_o degree_n at_o c_o then_o lay_v your_o ruler_n to_o these_o two_o mark_n at_o b_o and_o c_o and_o draw_v the_o straight_a line_n b_o c._n this_o line_n b_o c_o will_n represent_v the_o arch_n of_o the_o great_a circle_n between_o these_o two_o place_n and_o if_o you_o guide_v your_o eye_n along_o in_o this_o line_n you_o may_v ready_o and_o true_o perceive_v by_o what_o longitude_n and_o latitude_n you_o shall_v sail_v for_o mark_v well_o where_o this_o line_n cross_v the_o arch_n of_o latitude_n and_o the_o line_n of_o longitude_n and_o that_o show_v the_o true_a longitude_n and_o latitude_n of_o the_o arch_n of_o the_o great_a circle_n according_a to_o your_o desire_n proof_n the_o proof_n now_o the_o truth_n hereof_o will_n more_o evident_o appear_v if_o you_o compare_v the_o latitude_n and_o longitude_n which_o this_o line_n intersect_v with_o this_o table_n thereof_o calculate_v by_o mr._n circle_n mr._n in_o the_o ten_o problem_n of_o sail_v by_o the_o arch_n of_o a_o great_a circle_n norwood_n for_o every_o five_o degree_n of_o longitude_n longitude_n latitude_n de._n or_o difference_n of_o longitude_n d._n deg._n m._n 310_o 00_o 32_o 25_o 305_o 05_o 35_o 52_o 300_o 10_o 38_o 51_o 315_o 15_o 41_o 24_o 320_o 20_o 43_o 34_o 325_o 25_o 45_o 24_o 330_o 30_o 46_o 54_o 335_o 35_o 48_o 07_o 340_o 40_o 49_o 04_o 345_o 45_o 49_o 47_o 350_o 50_o 50_o 15_o 355_o 55_o 50_o 31_o 360_o 60_o 50_o 33_o 005_o 65_o 50_o 23_o 010_o 70_o 50_o 00_o now_o you_o may_v hereby_o see_v that_o the_o line_n b_o c_o in_o the_o point_n g_z do_v cross_v the_o 305_o or_o the_o 5_o degree_n of_o longitude_n from_o b_o almost_o at_o the_o arch_n of_o 36_o degree_n of_o latitude_n just_a as_o the_o table_n show_v it_o shall_v at_o 35_o degree_n 52_o minute_n of_o latitude_n again_o the_o line_n b_o c_o do_v cross_a the_o 310_o or_o the_o 10_o degree_n of_o longitude_n from_o b_o in_o the_o point_n h_o almost_o at_o the_o arch_n of_o 39_o degree_n of_o latitude_n agree_v with_o the_o table_n which_o show_v it_o to_o be_v in_o 38_o degree_n 51_o minute_n and_o so_o in_o all_o the_o rest_n it_o so_o near_o agree_v that_o if_o you_o take_v any_o care_n in_o make_v of_o this_o blank_a map_n to_o draw_v the_o arch_n
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near_a way_n but_o also_o may_v alter_v your_o latitude_n many_o degree_n whereby_o your_o account_n may_v be_v often_o rectify_v less_o rectify_v so_o in_o the_o example_n of_o the_o summer_n land_n the_o distance_n by_o the_o rumbe_n be_v 3299_o mile_n the_o distance_n by_o the_o arch_n be_v 3204_o mile_n that_o be_v 95_o mile_n less_o as_o for_o example_n suppose_v you_o be_v to_o sail_v from_o spain_n to_o virginia_n both_o which_o lie_v near_o the_o parallel_n of_o 40_o degree_n and_o suppose_v the_o difference_n of_o longitude_n between_v two_o such_o place_n in_o the_o parallel_n of_o 40_o to_o be_v 70_o degree_n the_o distance_n of_o these_o two_o place_n measure_v in_o the_o parallel_n of_o 40_o which_o be_v the_o rumbe_n that_o lead_v between_v the_o two_o place_n be_v east_n and_o west_n be_v 53_o degree_n 62_o 100_o but_o their_o distance_n in_o the_o arch_n of_o a_o great_a circle_n be_v but_o 52_o degree_n 08_o 100_o that_o be_v 1_o degree_n 54_o 100_o less_o but_o this_o as_o say_v be_v but_o the_o least_o part_n of_o the_o benefit_n that_o come_v by_o this_o way_n of_o sail_v the_o chief_a be_v this_o that_o in_o sail_v between_o two_o such_o place_n by_o the_o arch_n of_o a_o great_a circle_n you_o will_v first_o in_o the_o one_o half_a of_o the_o way_n raise_v the_o pole_n 5_o degree_n 69_o 100_o and_o then_o in_o the_o other_o half_o depress_v the_o pole_n as_o much_o so_o that_o in_o your_o whole_a voyage_n you_o will_v alter_v the_o latitude_n 11_o degree_n 38_o â—0_n &_o so_o by_o the_o observation_n of_o the_o latitude_n you_o may_v rectify_v your_o dead_a reckon_n very_o well_o which_o you_o can_v do_v sail_v in_o the_o parallel_n thus_o you_o see_v this_o way_n of_o sail_v be_v not_o only_o the_o near_a but_o the_o best_a way_n now_o concern_v this_o way_n of_o sail_v there_o have_v be_v but_o little_o write_v by_o any_o subject_n few_o have_v write_v of_o this_o subject_n and_o therefore_o i_o shall_v be_v the_o more_o large_a in_o this_o captain_n saltonstall_n in_o his_o book_n call_v the_o navigator_n have_v say_v somewhat_o how_o to_o direct_v a_o parallel_n course_n but_o for_o any_o other_o course_n he_o have_v say_v nothing_o and_o what_o he_o show_v be_v to_o be_v perform_v by_o arithmetic_n master_n norwood_n in_o his_o book_n of_o trigonometry_n have_v add_v as_o a_o appendix_n many_o problem_n of_o sail_v by_o the_o arch_n of_o a_o great_a circle_n whereby_o those_o who_o both_o can_v and_o will_v take_v the_o pain_n may_v by_o calculation_n find_v out_o all_o thing_n necessary_a in_o this_o way_n of_o sail_v but_o those_o way_n of_o calculation_n as_o they_o be_v very_o difficult_a to_o the_o unlearned_a so_o they_o be_v tedious_a to_o those_o that_o have_v the_o best_a skill_n and_o therefore_o i_o hope_v it_o will_v be_v well_o accept_v if_o i_o here_o show_v you_o how_o the_o same_o may_v be_v perform_v by_o geometry_n both_o plain_o and_o speedy_o and_o yet_o with_o as_o much_o exactness_n as_o need_v be_v require_v know_v the_o chief_a thing_n to_o be_v know_v and_o in_o the_o pursuance_n hereof_o i_o shall_v keep_v as_o close_o to_o master_n norwood_n as_o i_o can_v both_o in_o his_o proposition_n and_o example_n that_o thereby_o you_o may_v see_v how_o near_o my_o plain_a line_n will_v approach_v to_o the_o exactness_n of_o his_o calculation_n now_o if_o you_o observe_v he_o there_o be_v these_o three_o thing_n which_o must_v be_v find_v out_o in_o every_o example_n first_o the_o distance_n of_o the_o two_o place_n in_o the_o arch_n of_o a_o great_a circle_n second_o the_o angle_n of_o position_n from_o the_o one_o place_n to_o the_o other_o three_o to_o find_v out_o what_o longitude_n and_o latitude_n the_o arch_n of_o the_o great_a circle_n do_v pass_v through_o between_o the_o two_o place_n place_n to_o find_v the_o distance_n of_o two_o place_n for_o the_o first_o of_o these_o know_v the_o longitude_n and_o latitude_n of_o two_o place_n to_o find_v their_o distance_n in_o the_o arch_n of_o a_o great_a circle_n which_o be_v always_o the_o near_a distance_n i_o may_v show_v you_o how_o to_o perform_v this_o in_o the_o first_o place_n but_o i_o here_o pass_v it_o by_o for_o these_o reason_n first_o because_o master_n wright_n master_n blundevile_n and_o captain_n saltonstall_n have_v all_o of_o they_o demonstrate_v it_o in_o their_o book_n already_o and_o second_o because_o the_o chief_a benefit_n in_o this_o way_n of_o sail_v do_v not_o so_o much_o consist_v in_o save_v of_o a_o little_a way_n as_o in_o sail_v the_o most_o convenient_a way_n that_o be_v so_o as_o you_o may_v alter_v your_o latitude_n most_o and_o so_o your_o reckon_n may_v be_v the_o more_o certain_a for_o though_o near_o the_o pole_n the_o difference_n of_o the_o distance_n of_o two_o place_n in_o the_o arch_n of_o a_o great_a circle_n and_o in_o their_o rumbe_n may_v be_v considerable_a yet_o in_o most_o voyage_n it_o be_v not_o as_o in_o the_o forename_a example_n of_o two_o place_n in_o the_o parallel_n of_o 40_o degree_n the_o difference_n by_o calculation_n be_v find_v to_o be_v but_o one_o degree_n 54_o 100_o which_o be_v scarce_o considerable_a in_o the_o whole_a voyage_n be_v 52_o degree_n three_o it_o will_v be_v somewhat_o difficult_a &_o it_o require_v great_a curiosity_n in_o draw_v of_o those_o line_n prescribe_v by_o they_o so_o exact_o that_o you_o may_v come_v to_o the_o knowledge_n of_o the_o distance_n any_o thing_n near_o last_o all_o that_o trouble_n be_v needless_a for_o though_o in_o calculation_n this_o distance_n must_v be_v find_v out_o first_o that_o so_o you_o may_v find_v out_o the_o rest_n of_o the_o proposition_n follow_v yet_o in_o this_o way_n i_o be_o about_o to_o show_v that_o which_o follow_v no_o way_n depend_v upon_o the_o true_a knowledge_n of_o this_o distance_n it_o shall_v be_v sufficient_a therefore_o for_o the_o present_a to_o tell_v you_o that_o this_o way_n be_v always_o somewhat_o the_o near_a way_n for_o the_o second_o of_o these_o proposition_n which_o be_v to_o know_v the_o angle_n of_o position_n from_o the_o one_o place_n to_o the_o other_o operation_n the_o angle_n of_o position_n be_v needless_a in_o this_o operation_n though_o this_o must_v be_v find_v out_o in_o calculation_n before_o you_o can_v proceed_v any_o further_a yet_o in_o this_o work_n it_o be_v more_o needless_a than_o the_o former_a proposition_n and_o therefore_o may_v be_v very_o well_o omit_v but_o now_o for_o the_o three_o proposition_n pass_v to_o find_v out_o the_o longitude_n and_o latitude_n by_o which_o the_o great_a circle_n do_v pass_v which_o be_v the_o find_v out_o by_o what_o longitude_n and_o latitude_n the_o great_a circle_n must_v pass_v between_o the_o two_o place_n this_o be_v the_o very_a end_n aim_v at_o in_o all_o the_o work_n may_v be_v thus_o attain_v first_o draw_v the_o follow_a quadrant_n a_o d_o b_o and_o divide_v it_o into_o degree_n then_o consider_v of_o what_o length_n your_o tangent_fw-la line_n 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