Selected quad for the lemma: end_n

Word A	Word B	Word C	Word D	Occurrence	Frequency Band	MI	MI Band	Prominent
end_n	compass_n	foot_n	line_n	1,942	5	9.5106	5	true

Text snippets containing the quad

ID	Title	Author	Corrected Date of Publication (TCP Date of Publication)	STC	Words	Pages
A74866	The geometrical sea-man: or, the art of navigation performed by geometry. Shewing how all the three kinds of sayling, viz. by the plain chart, by Mercators chart, by a great circle. may be easily and exactly performed by a plain ruler and a pair of compasses, without arithmeticall calculation. / By Henry Phillippes.	Phillippes, Henry, d. 1677?	1652 (1652)	Thomason E652_10; ESTC R205892	65,784	93

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part_n as_o before_o then_o close_v your_o compass_n divide_v each_o of_o these_o 12_o part_n into_o 3_o so_o you_o shall_v have_v in_o all_o 36_o part_n than_o you_o may_v easy_o with_o your_o pen_n divide_v each_o of_o these_o part_n into_o 10_o little_a part_n each_o of_o which_o stand_v for_o a_o degree_n and_o so_o you_o may_v number_v they_o as_o in_o the_o middle_a circle_n of_o the_o figure_n a_o three_o usual_a division_n of_o a_o circle_n be_v into_o 32_o equal_a part_n part_n to_o divide_v a_o circle_n into_o 32_o part_n according_a to_o the_o number_n of_o the_o point_n of_o the_o compass_n which_o may_v be_v thus_o perform_v first_o draw_v the_o line_n of_o east_n and_o west_n and_o cross_v it_o at_o right_a angle_n with_o the_o line_n of_o north_n and_o south_n and_o draw_v the_o circle_n as_o before_o then_o keep_v your_o compass_n at_o that_o distance_n set_v one_o foot_n where_o the_o line_n of_o east_n do_v cross_v the_o circle_n and_o with_o the_o other_o draw_v 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chart._n first_o make_v the_o square_a a_o b_o c_o d_o of_o what_o length_n and_o breadth_n you_o please_v and_o divide_v each_o side_n into_o as_o many_o equal_a part_n as_o your_o occasion_n require_v and_o then_o draw_v straight_a line_n through_o these_o part_n cross_v one_o another_o at_o right_a angle_n and_o so_o make_v many_o little_a geometrical_a square_n each_o of_o which_o you_o may_v suppose_v to_o contain_v one_o degree_n in_o longitude_n and_o latitude_n chapter_n latitude_n according_a to_o account_v 20_o league_n be_v in_o one_o degree_n &_o so_o each_o 10_o part_n will_v be_v 2_o leagus_n but_o it_o be_v somewhat_o more_o as_o you_o may_v see_v in_o the_o three_o proposition_n of_o this_o

1,00_o col._n last_o line_n 3._o for_o 0_o 8_o read_z 0_o 58._o the_o geometrical_a seaman_n or_o the_o art_n of_o navigation_n perform_v by_o geometry_n chap._n i._o contain_v some_o geometrical_a proposition_n which_o will_v be_v of_o frequent_a use_n proposition_n 1._o how_o to_o erect_v a_o perpendicular_a line_n at_o the_o end_n of_o a_o line_n proposition_n the_o first_o proposition_n be_v what_o a_o perpendicular_a line_n be_v a_o perpendicular_a line_n be_v a_o line_n that_o stand_v direct_o upright_a from_o another_o line_n as_o in_o the_o figure_n the_o line_n b_o c_o be_v a_o perpendicular_a line_n to_o the_o line_n a_o b._n now_o in_o the_o figure_n there_o be_v two_o way_n of_o raise_v it_o the_o one_o on_o the_o right_a side_n and_o the_o other_o on_o the_o left_a first_o let_v the_o line_n a_o b_o be_v give_v and_o it_o be_v require_v to_o erect_v the_o line_n b_o c_o perpendicular_o to_o it_o at_o the_o end_n of_o the_o line_n in_o the_o point_n b._n way_n another_o 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to_o it_o so_o that_o it_o may_v cross_v it_o at_o right_a angle_n first_o open_v your_o compass_n at_o pleasure_n and_o set_v one_o foot_n in_o the_o line_n at_o b_o with_o the_o other_o make_v two_o short_a arch_n one_o above_o the_o line_n at_o d_o and_o the_o other_o below_o the_o line_n at_o e._n then_o with_o the_o same_o distance_n set_v one_o foot_n in_o a_o and_o with_o the_o other_o across_o the_o two_o former_a arch_n at_o d_o and_z e._n last_o lay_v your_o ruler_n by_o these_o two_o crosses_z d_o and_z e_z draw_v the_o line_n g_o e_o

you_o must_v do_v thus_o first_o from_o the_o point_n i_o set_v the_o first_o 40_o double_a league_n upon_o the_o rumbe_n n_o w_n by_o n_n which_o will_v end_v at_o r._n then_o from_o the_o point_n r_o draw_v the_o rumbe_n n_o e_o by_z e_z which_o be_v the_o line_n r_o q_n and_o set_v thereon_o the_o 40_o double_a league_n from_o r_n to_o q_n thus_o you_o will_v find_v q_o to_o be_v the_o place_n you_o shall_v be_v in_o according_a to_o your_o dead_a reckon_n which_o be_v in_o 5_o d._n 5_o 10_o and_o somewhat_o 55._o somewhat_o 5d._o 55._o more_o of_o north_n latitude_n whereas_o by_o your_o observation_n you_o find_v that_o you_o be_v but_o in_o 5_o deg_fw-la of_o north_n latitude_n now_o to_o know_v the_o true_a place_n where_o you_o be_v in_o respect_n of_o the_o longitude_n because_o you_o have_v sail_v upon_o two_o rumbe_n draw_v the_o line_n i_o qui_fw-fr from_o i_o the_o first_o place_n you_o set_v sail_n from_o to_o q_o the_o place_n of_o your_o dead_a reckon_n and_o then_o 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your_o way_n so_o that_o you_o may_v know_v at_o any_o time_n what_o longitude_n and_o latitude_n you_o be_v in_o and_o how_o the_o place_n you_o be_v bind_v to_o bear_n from_o you_o and_o how_o far_o you_o want_v to_o it_o chart._n 7._o the_o manner_n of_o keep_v your_o reckon_n upon_o the_o chart._n former_a this_o proposition_n contain_v the_o use_n and_o practice_v of_o all_o the_o former_a for_o example_n suppose_v you_o be_v to_o sail_v from_o the_o place_n i_o in_o the_o former_a chart_n which_o be_v under_o the_o equinoctial_a and_o in_o 5_o degree_n of_o longitude_n unto_o the_o place_n h_n which_o have_v 5_o deg_fw-la of_o longitude_n and_o 10_o degree_n of_o north_n latitude_n here_o the_o direct_a way_n from_o i_o to_o h_n lie_v full_a north_n but_o suppose_v that_o you_o can_v sail_v upon_o this_o point_n but_o be_v force_v first_o to_o run_v n_z w_o by_z n_z 36_o double_a league_n and_o then_o n_z e_z by_z e_z 36_o double_a league_n more_o the_o question_n be_v what_o be_v the_o longitude_n and_o latitude_n of_o this_o place_n and_o how_o far_o it_o be_v distant_a from_o the_o place_n h_n and_o upon_o what_o point_n of_o the_o compass_n it_o lie_v from_o it_o first_o from_o the_o point_n i_o draw_v the_o rumbe_n n_o w_n by_o n_n and_o set_v off_o thereon_o 36_o double_a league_n from_o i_o to_o m._n then_o from_o this_o point_n m_o draw_v the_o rumbe_n n_o e_o by_z e_z and_o set_v off_o thereon_o the_o 36_o double_a league_n which_o you_o have_v sail_v upon_o it_o from_o m_n to_o n_n thus_o you_o shall_v find_v that_o n_n be_v the_o place_n wherein_o you_o be_v who_o longitude_n be_v fix_v degree_n and_o who_o latitude_n be_v five_o degree_n now_o if_o you_o lay_v a_o ruler_n from_o this_o point_n n_n to_o the_o place_n you_o be_v bind_v to_o which_o be_v h_n and_o draw_v the_o line_n h_o n_n this_o line_n be_v the_o direct_a way_n to_o the_o place_n you_o be_v bind_v and_o by_o the_o help_n your_o circle_n or_o scale_v of_o rumbe_n you_o shall_v find_v that_o it_o lie_v north_n by_o west_n or_o the_o first_o rumbe_n from_o the_o meridian_n westward_o last_o if_o you_o set_v one_o end_n of_o your_o compass_n in_o n_n and_o open_v the_o other_o to_o h_n and_o measure_v that_o distance_n in_o the_o side_n of_o the_o chart_n you_o will_v find_v it_o to_o be_v about_o 5_o degree_n 1_o ●●_o or_o 51_o double_a league_n and_o so_o much_o you_o want_v to_o the_o end_n of_o your_o voyage_n proposition_n 8._o how_o to_o know_v the_o distance_n of_o any_o cape_n headland_n or_o island_n from_o you_o which_o you_o can_v see_v at_o two_o distant_a place_n you_o 8._o to_o know_v the_o distance_n of_o any_o cape_n from_o you_o suppose_v that_o sail_v on_o the_o sea_n you_o espy_v a_o island_n or_o cape_n lie_v at_o the_o first_o sight_n just_a north-east_n from_o you_o and_o then_o sail_v forward_o upon_o your_o way_n which_o lie_v full_a north_n to_o the_o distance_n of_o 5_o league_n you_o then_o observe_v that_o the_o island_n lie_v full_a east_n from_o you_o the_o question_n be_v to_o know_v the_o distance_n of_o this_o isle_n from_o either_o of_o these_o two_o place_n in_o such_o question_n as_o this_o you_o may_v suppose_v each_o degree_n in_o the_o former_a chart_n to_o stand_v now_o but_o for_o a_o league_n chart_n these_o two_o follow_a proposition_n rather_o belong_v to_o the_o plain_a table_n then_o the_o chart_n and_o let_v the_o first_o place_n where_o you_o espy_v the_o island_n be_v at_o a_o now_o because_o the_o island_n lie_v north-east_n from_o this_o place_n draw_v the_o line_n a_o b_o which_o be_v n_n e_o from_o a._n then_o count_v the_o 5_o league_n which_o you_o have_v sail_v upon_o your_o course_n which_o be_v full_a north_n in_o the_o meridian_n line_n from_o a_o to_o f_o and_o because_o from_o this_o place_n the_o island_n do_v lie_v full_a east_n therefore_o from_o this_o point_n f_o draw_v the_o east_n line_n f_o e_o g_o and_o mark_v where_o this_o line_n do_v cross_v the_o former_a line_n a_o e_o of_o n_o e_o from_o a_o which_o be_v in_o the_o point_n e._n this_o therefore_o must_v needs_o be_v the_o place_n of_o the_o island_n who_o distance_n if_o you_o take_v with_o your_o compass_n and_o measure_n in_o the_o side_n of_o the_o chart_n you_o shall_v find_v that_o the_o place_n e_o be_v distant_a from_o a_o 7_o league_n and_o almost_o 1_o 15_o part_n of_o a_o league_n and_o from_o f_o just_a 5_o league_n proposition_n league_n a_o double_a use_n of_o this_o proposition_n and_o by_o this_o mean_n if_o you_o know_v the_o longitude_n and_o latitude_n of_o this_o isle_n or_o cape_n you_o may_v the_o more_o certain_o know_v the_o truth_n of_o your_o account_n and_o if_o need_v be_v correct_v it_o or_o if_o you_o know_v not_o the_o place_n before_o you_o may_v set_v it_o down_o in_o your_o chart_n by_o its_o longitude_n and_o latitude_n which_o you_o find_v it_o to_o be_v in_o according_a to_o the_o best_a account_n you_o can_v make_v by_o your_o observation_n proposition_n 9_o by_o observe_v upon_o what_o rumbe_n many_o place_n lie_v from_o you_o at_o two_o several_a station_n to_o find_v the_o distance_n of_o those_o place_n and_o their_o true_a posture_n and_o bear_v one_o from_o another_o proposition_n 9_o to_o find_v out_o the_o true_a distance_n and_o bear_v of_o many_o place_n the_o use_n of_o this_o proposition_n as_o in_o the_o former_a proposition_n you_o do_v for_o one_o place_n so_o in_o this_o you_o may_v do_v for_o many_o and_o this_o will_v be_v of_o good_a use_n for_o hereby_o sail_v in_o sight_n of_o any_o coast_n you_o may_v find_v out_o how_o the_o

because_o when_o you_o be_v to_o go_v any_o long_a voyage_n it_o will_v be_v needful_a for_o you_o first_o to_o make_v a_o general_a map_n of_o your_o whole_a voyage_n by_o the_o lesser_a line_n whereby_o you_o may_v know_v the_o course_n and_o distance_n thereof_o in_o general_a and_o then_o to_o make_v three_o or_o four_o other_o charts_n by_o the_o great_a line_n upon_o which_o with_o your_o ruler_n and_o compass_n you_o may_v set_v down_o your_o daily_a course_n and_o distance_n more_o exact_o also_o i_o have_v make_v these_o two_o line_n in_o such_o proportion_n that_o the_o one_o be_v the_o ten_o part_n of_o the_o other_o that_o so_o that_o they_o may_v both_o agree_v with_o the_o scale_n upon_o the_o quadrant_n now_o the_o way_n to_o make_v one_o of_o these_o charts_n be_v very_o easy_a line_n to_o make_v a_o sea_n chartley_n by_o these_o meridian_n line_n and_o much_o after_o the_o manner_n of_o the_o plain_a chart_n for_o first_o you_o may_v draw_v the_o line_n of_o east_n and_o west_n a_o b_o of_o what_o length_n you_o please_v and_o divide_v it_o into_o equal_a part_n or_o degree_n than_o you_o may_v erect_v a_o perpendicular_a line_n either_o at_o one_o of_o the_o end_n of_o the_o line_n or_o in_o any_o of_o the_o division_n towards_o the_o midst_n of_o the_o line_n and_o then_o draw_v the_o other_o parallel_n of_o longitude_n parallel_v thereto_o so_o far_o it_o be_v all_o one_o with_o the_o plain_a chart_n but_o when_o you_o come_v to_o draw_v the_o parallel_n of_o latitude_n you_o must_v not_o make_v they_o all_o equal_a though_o they_o must_v be_v all_o parallel_n each_o to_o other_o but_o you_o must_v either_o with_o your_o compass_n take_v they_o out_o of_o the_o quadrant_n or_o which_o be_v more_o easy_a lay_v a_o scroll_n of_o paper_n to_o the_o meridian_n line_n which_o be_v ready_o draw_v to_o your_o hand_n and_o so_o mark_v out_o the_o degree_n of_o latitude_n upon_o the_o scroll_n of_o paper_n and_o then_o lay_v that_o scroll_n to_o the_o side_n of_o your_o chart_n you_o may_v transfer_v the_o degree_n of_o latitude_n into_o the_o side_n of_o your_o chart_n and_o through_o they_o draw_v the_o parallel_n and_o set_v fit_a number_n to_o they_o as_o in_o the_o figure_n the_o figure_n of_o a_o general_a sea-chart_n contain_v almost_o a_o eight_o part_n of_o the_o globe_n north_n now_o though_o this_o be_v not_o a_o general_a chart_n of_o the_o whole_a globe_n yet_o it_o may_v be_v call_v a_o general_a chart_n in_o respect_n of_o other_o which_o will_v serve_v only_o for_o a_o lesser_a portion_n of_o the_o globe_n for_o this_o chart_n contain_v almost_o a_o eight_o part_n of_o the_o globe_n and_o may_v be_v fit_v to_o set_v forth_o any_o part_n thereof_o for_o if_o you_o change_v the_o number_n of_o the_o longitude_n if_o the_o latitude_n be_v northward_o it_o will_v serve_v as_o it_o now_o stand_v but_o if_o the_o latitude_n be_v southward_o you_o must_v turn_v the_o bottom_n upward_o if_o you_o have_v occasion_n in_o one_o 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 before_o with_o another_o little_a line_n and_o so_o the_o cross_n of_o these_o two_o line_n will_v show_v you_o the_o point_n where_o the_o place_n must_v be_v suppose_v to_o stand_v example_n example_n thus_o suppose_v the_o longitude_n of_o the_o summer_n land_n to_o be_v 300_o degree_n and_o the_o latitude_n thereof_o 32_o degree_n 25_o minute_n you_o will_v find_v that_o it_o must_v be_v set_v at_o s_n upon_o the_o chart_n proposition_n 2._o the_o longitude_n and_o latitude_n of_o two_o place_n be_v know_v to_o find_v the_o rumbe_n which_o you_o must_v sail_v upon_o to_o go_v direct_o from_o the_o one_o place_n to_o the_o other_o example_n 2._o by_o the_o longitude_n and_o latitude_n of_o two_o place_n to_o find_v the_o rumbe_n example_n suppose_v the_o one_o place_n to_o be_v the_o summer_n land_n who_o longitude_n and_o latitude_n we_o will_v suppose_v to_o be_v as_o be_v before_o set_v down_o let_v the_o other_o place_n be_v the_o lizard_n who_o latitude_n be_v about_o 50_o 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

place_n but_o it_o be_v only_o their_o distance_n in_o the_o rumbe_n so_o that_o if_o the_o two_o place_n be_v not_o both_o under_o the_o equinoctial_a or_o both_o in_o one_o meridian_n than_o there_o be_v somewhat_o a_o near_a cut_n between_o the_o two_o place_n than_o the_o rumbe_n point_v out_o which_o sometime_o especial_o near_o the_o pole_n be_v very_o considerable_a but_o this_o be_v not_o all_o the_o benefit_n which_o come_v by_o this_o way_n of_o sail_v way_n second_o it_o be_v the_o most_o convenient_a way_n but_o many_o time_n when_o your_o course_n lie_v near_o the_o east_n and_o west_n this_o way_n be_v far_o more_o convenient_a for_o if_o you_o shall_v sail_v full_a east_n or_o west_n you_o must_v altogether_o depend_v upon_o your_o dead_a reckon_n have_v no_o way_n to_o help_v yourself_o by_o the_o observation_n of_o the_o latitude_n but_o now_o if_o you_o sail_v by_o the_o arch_n of_o a_o great_a circle_n between_v two_o such_o place_n you_o not_o only_o go_v the_o 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 ●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 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their_o three_o point_n be_v their_o correspondent_a degree_n in_o the_o diameter_n a_o b._n by_o these_o three_o point_n you_o may_v find_v the_o centre_n and_o so_o draw_v the_o arch_n as_o be_v show_v in_o the_o first_o chapter_n but_o to_o save_v that_o labour_n you_o must_v know_v that_o the_o centre_n all_o lie_n in_o the_o diameter_n line_n which_o must_v be_v extend_v beyond_o the_o circle_n and_o then_o the_o centre_n be_v thus_o find_v out_o the_o diameter_n c_o d_o be_v divide_v into_o half_a tangent_n as_o before_o if_o for_o every_o degree_n you_o account_v two_o begin_v from_o the_o centre_n e_o so_fw-mi you_o shall_v have_v the_o centre_n of_o the_o meridians_fw-la then_o if_o you_o set_v one_o foot_n of_o your_o compass_n in_o that_o centre_n and_o open_v the_o other_o to_o the_o pole_n a_o or_o b_o it_o will_v pass_v through_o the_o correspondent_a degree_n or_o three_o point_n in_o the_o diameter_n c_o d_o on_o the_o other_o side_n of_o the_o centre_n so_o the_o 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circle_n that_o your_o compass_n will_v not_o reach_v they_o you_o may_v then_o bridle_v a_o thin_a ruler_n that_o will_v bend_v with_o a_o double_a string_n like_o a_o cross_a bow_n and_o then_o by_o twist_v the_o string_n together_o you_o may_v by_o little_a and_o little_a set_v it_o to_o what_o bend_v you_o please_v till_o it_o shall_v cut_v the_o three_o point_n of_o your_o arch_n you_o will_v draw_v and_o then_o with_o your_o pen_n you_o may_v draw_v your_o arch_n which_o if_o the_o ruler_n be_v all_o of_o one_o thickness_n and_o so_o bend_v in_o all_o place_n alike_o it_o will_v be_v very_o true_a your_o compass_n will_v reach_v the_o centre_n very_o well_o till_o you_o come_v to_o 60_o degree_n but_o afterward_o you_o must_v be_v force_v to_o use_v this_o or_o some_o such_o like_a way_n to_o help_v yourself_o the_o large_a you_o make_v your_o draught_n and_o the_o more_o meridians_fw-la and_o parallel_v you_o draw_v in_o it_o so_o much_o the_o better_a it_o be_v therefore_o if_o you_o can_v make_v it_o so_o large_a that_o you_o may_v draw_v meridians_fw-la and_o parallel_n through_o every_o degree_n which_o you_o may_v do_v very_o well_o in_o a_o sheet_n of_o large_a paper_n in_o a_o lesser_a draught_n you_o may_v draw_v every_o second_o degree_n which_o be_v the_o least_o i_o will_v wish_v you_o to_o do_v last_o to_o save_v time_n and_o labour_n in_o draw_v of_o these_o blank_n for_o every_o question_n when_o you_o have_v make_v a_o little_a trial_n and_o know_v how_o to_o draw_v they_o then_o draw_v two_o good_a large_a one_o of_o one_o and_o the_o same_o size_n which_o you_o may_v do_v very_o well_o by_o draw_v the_o same_o line_n in_o both_o before_o you_o stir_v your_o compass_n from_o their_o distance_n then_o six_o the_o one_o of_o these_o to_o the_o other_o by_o their_o centre_n so_o that_o they_o may_v be_v turn_v round_o and_o the_o uppermost_a of_o these_o be_v draw_v in_o fine_a thin_a paper_n and_o a_o little_a oil_v you_o may_v easy_o see_v through_o it_o all_o the_o line_n of_o the_o other_o and_o thus_o you_o shall_v have_v a_o jewel_n a_o this_o will_v be_v somewhat_o like_a mr._n blagraves_n mathematical_a jewel_n instrument_n whereby_o this_o and_o most_o other_o question_n of_o spherical_a triangle_n may_v be_v resolve_v have_v thus_o show_v the_o draw_v of_o this_o projection_n i_o shall_v now_o come_v to_o show_v you_o the_o use_n of_o it_o in_o several_a example_n the_o first_o example_n shall_v be_v the_o forementioned_a voyage_n from_o the_o summer-island_n to_o the_o lizard_n the_o latitude_n of_o the_o summer-island_n be_v 32_o degree_n 25_o minute_n north_n circle_n the_o use_n of_o this_o projection_n in_o find_v out_o the_o great_a circle_n and_o the_o latitude_n of_o the_o lizard_n be_v 50_o degree_n north_n and_o their_o difference_n of_o longitude_n be_v 70_o degree_n and_o it_o be_v require_v to_o know_v first_o the_o latitude_n and_o longitude_n by_o which_o 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 second_o the_o angle_n of_o position_n from_o the_o first_o place_n to_o the_o second_o three_o the_o near_a distance_n between_o the_o two_o place_n to_o perform_v this_o first_o you_o must_v set_v down_o upon_o your_o draught_n example_n first_o example_n the_o first_o place_n which_o be_v the_o summer-island_n according_a to_o the_o latitude_n thereof_o which_o be_v 32_o degree_n 25_o minute_n in_o the_o outmost_a circle_n at_o s._n second_o s._n note_v well_o which_o way_n the_o first_o place_n bear_v from_o the_o second_o and_o herein_o you_o must_v regard_v how_o the_o second_o place_n do_v bear_v from_o the_o first_o if_o the_o second_o place_n lie_v west_n from_o the_o first_o than_o you_o must_v set_v down_o the_o first_o place_n on_o the_o east_n or_o right_a side_n of_o the_o circle_n but_o if_o the_o second_o place_n lie_v eastward_o from_o the_o first_o as_o it_o do_v in_o this_o example_n than_o you_o must_v set_v down_o the_o first_o place_n on_o the_o westside_n of_o the_o circle_n as_o it_o be_v here_o at_o s._n then_o from_o the_o point_n s_o through_o the_o centre_n e_o draw_v the_o diameter_n line_n s_o e_o k_n and_o cross_v it_o at_o right_a angle_n with_o the_o line_n m_o e_o n._n then_o account_v 70_o degree_n which_o be_v the_o difference_n of_o longitude_n of_o the_o two_o place_n in_o the_o diameter_n c_o e_z from_z c_z to_o 70_o mark_v that_o meridian_n arch_n &_o thereupon_o mark_v out_o the_o latitude_n of_o the_o other_o place_n which_o be_v 50_o degree_n at_o l._n thus_o the_o two_o place_n be_v set_v down_o according_a to_o their_o latitude_n and_o the_o difference_n of_o their_o longitude_n at_o s_n and_o l._n now_o to_o help_v you_o to_o draw_v the_o arch_n of_o a_o great_a circle_n between_o these_o two_o place_n s_n &_o l_o you_o have_v these_o three_o point_n s_n l_o &_o k_o by_o which_o you_o may_v find_v the_o centre_n of_o the_o arch_n which_o be_v at_o m_o in_o the_o line_n n_o m_o therefore_o set_v one_o foot_n of_o your_o compass_n in_o m_n and_o open_v the_o other_o to_o any_o of_o the_o three_o point_n draw_v the_o arch_n s_o l_o k._n this_o arch_n be_v the_o great_a circle_n that_o pass_v through_o these_o two_o place_n by_o which_o you_o shall_v find_v all_o the_o thing_n desire_v arch_n the_o longitude_n and_o latitude_n of_o the_o arch_n as_o first_o if_o you_o will_v know_v by_o what_o longitude_n and_o latitude_n this_o arch_n do_v pass_v which_o be_v the_o thing_n most_o needful_a to_o be_v know_v if_o you_o trace_v the_o way_n of_o this_o arch_n through_o the_o meridians_fw-la and_o parallel_n of_o the_o draught_n you_o will_v find_v they_o to_o agree_v with_o the_o former_a

table_n hereof_o for_o every_o five_o or_o ten_o degree_n for_o at_o 10_o degree_n of_o longitude_n from_o s_n the_o arch_n pass_v through_o 39_o degree_n of_o latitude_n at_o 20_o through_o 43_o ½_n and_o so_o of_o the_o rest_n position_n the_o angle_n of_o position_n second_o if_o you_o will_v know_v the_o angle_n of_o position_n from_o s_n to_o l_o then_z observe_z in_o what_o point_n the_o arch_n s_o l_o k_n do_v cross_v the_o line_n n_o m_o which_o be_v at_o t_o then_o take_v the_o distance_n n_o t_o and_o measure_v it_o in_o the_o semidiameter_n c_o e_z from_z c_z toward_z e_z and_o it_o will_v reach_v almost_o to_o 49_o degree_n which_o show_v the_o angle_n of_o position_n to_o be_v north-easter_o almost_o 49_o degree_n place_n the_o distance_n of_o the_o place_n three_o if_o you_o will_v know_v the_o distance_n of_o the_o two_o place_n you_o must_v with_o your_o compass_n take_v the_o distance_n of_o the_o two_o place_n s_n and_o l_o and_o measure_v it_o in_o that_o meridian_n which_o agree_v with_o the_o angle_n of_o position_n viz._n 49_o degree_n you_o shall_v find_v it_o will_v reach_v from_o a_o to_o five_o now_o if_o you_o reckon_v the_o degree_n of_o the_o parallel_n of_o latitude_n from_o a_o to_o this_o point_n v_n you_o shall_v have_v the_o distance_n which_o be_v 53_o degree_n and_o a_o half_a likewise_o you_o may_v measure_v any_o part_n of_o this_o arch_n s_o l_o in_o this_o meridian_n a_o v._o if_o you_o always_o set_v one_o foot_n in_o s_o and_o open_v the_o other_o to_o the_o point_n require_v and_o then_o set_v one_o foot_n in_o a_o and_o the_o other_o will_v show_v the_o distance_n of_o that_o place_n thus_o the_o distance_n will_v be_v find_v out_o as_o exact_o as_o by_o any_o other_o geometrical_a way_n but_o in_o regard_n of_o the_o smallness_n of_o the_o projection_n you_o may_v mistake_v some_o few_o mile_n or_o league_n but_o if_o you_o be_v to_o sail_v from_o the_o lizard_n to_o the_o summer-island_n arch_n the_o difference_n in_o sail_v forward_o &_o backward_o by_o the_o arch_n than_o you_o must_v first_o set_v down_o the_o latitude_n of_o the_o lizard_n on_o the_o other_o side_n of_o the_o circle_n as_o i_o note_v before_o &_o so_o the_o work_n will_v fall_v out_o much_o as_o it_o do_v before_o for_o the_o longitude_n and_o latitude_n of_o the_o arch_n will_v be_v the_o same_o only_o account_v they_o backward_o the_o distance_n will_v be_v the_o same_o viz._n 53_o degree_n and_o a_o half_a only_o it_o must_v be_v measure_v in_o another_o meridian_n according_a to_o the_o angle_n of_o position_n from_o the_o lizard_n which_o will_v be_v about_o 81_o degree_n so_o that_o in_o effect_n all_o be_v the_o same_o only_o the_o angle_n of_o position_n which_o be_v of_o little_a use_n but_o to_o find_v out_o the_o scale_n of_o the_o distance_n so_o that_o if_o you_o regard_v it_o well_o one_o labour_n will_v serve_v to_o find_v your_o way_n outward_a and_o homeward_n i_o may_v here_o show_v you_o how_o to_o perform_v the_o parallel_n question_n but_o because_o such_o question_n may_v with_o more_o ease_n and_o certainty_n be_v perform_v by_o the_o former_a way_n i_o shall_v not_o spend_v time_n about_o it_o i_o shall_v only_o instance_n in_o two_o sort_n of_o voyage_n which_o can_v be_v perform_v by_o the_o other_o projection_n and_o in_o such_o case_n as_o these_o there_o will_v be_v some_o need_n of_o this_o way_n and_o not_o else_o first_o when_o one_o of_o the_o place_n be_v under_o the_o equinoctial_a and_o the_o other_o towards_o one_o of_o the_o pole_n the_o other_o be_v when_o the_o one_o place_n have_v north_n latitude_n and_o the_o other_o south_n suppose_v you_o be_v to_o sail_v from_o the_o island_n of_o st._n thomas_n situation_n example_n of_o 2_o place_n in_o another_o manner_n of_o situation_n which_o lie_v under_o the_o equinoctial_a and_o have_v about_o 35_o degree_n of_o longitude_n to_o the_o strait_n of_o magellan_n which_o have_v about_o 53_o degree_n of_o south_n latitude_n and_o differ_v in_o longitude_n from_o the_o former_a place_n 9●_n degree_n to_o the_o westward_o now_o it_o be_v require_v to_o find_v out_o the_o arch_n of_o the_o great_a circle_n between_o these_o two_o place_n and_o the_o longitude_n and_o latitude_n of_o this_o arch_n with_o the_o angle_n of_o position_n and_o the_o distance_n of_o the_o two_o place_n to_o perform_v this_o first_o set_v down_o the_o isle_n of_o st._n thomas_n which_o be_v under_o the_o equinoctial_a at_o the_o one_o end_n of_o the_o equinoctial_a line_n at_o d_o then_o account_v 90_o degree_n of_o longitude_n from_o d_o to_z e_z there_o be_v the_o meridian_n of_o the_o strait_n of_o magellan_n whereupon_o you_o must_v mark_v out_o the_o latitude_n thereof_o which_o be_v 53_o degree_n at_o w_n so_o you_o shall_v have_v these_o three_o point_n d_o w_n c_o by_o which_o you_o may_v find_v the_o centre_n and_o draw_v the_o arch_n d_o w_n c_o now_o this_o part_n of_o the_o circle_n from_o d_o to_z w_o be_v the_o arch_n of_o the_o great_a circle_n which_o lie_v between_o these_o two_o place_n by_o which_o you_o may_v find_v all_o the_o other_o thing_n require_v as_o first_o for_o the_o longitude_n and_o latitude_n of_o this_o arch_n they_o be_v find_v out_o by_o note_v where_o it_o cross_v the_o circle_n of_o longitude_n and_o latitude_n in_o the_o draught_n which_o you_o shall_v find_v for_o every_o ten_o degree_n to_o be_v as_o in_o this_o table_n long_o latitude_n deg_n d._n m._n 10_o 12_o 58_o 20_o 24_o 25_o 30_o 22_o 34_o 40_o 40_o 28_o 50_o 45_o 31_o 60_o 48_o 58_o 70_o 51_o 16_o 80_o 52_o 35_o 90_o 53_o 0_o second_o for_o the_o angle_n of_o rule_n of_o the_o difference_n of_o longitude_n be_v just_a 90_o deg_fw-la the_o angle_n of_o position_n be_v ready_a measure_v in_o the_o semidiameter_n e_o b_o be_v the_o distance_n b_o w_n which_o be_v 37_o degree_n but_o at_o other_o time_n must_v follow_v the_o rule_n position_n between_o the_o two_o place_n this_o be_v show_v by_o the_o arch_n dw_n cross_v the_o semidiameter_n e_o b_o so_fw-mi that_o if_o you_o take_v the_o distance_n b_o w_n and_o measure_v it_o in_o the_o diameter_n c_o d_o it_o will_v reach_v from_o c_o to_o 37_o degree_n which_o be_v the_o angle_n require_v &_o the_o situation_n show_v it_o to_o be_v south_n westerly_a last_o for_o the_o distance_n of_o the_o two_o place_n if_o you_o take_v the_o distance_n d_o w_n and_o measure_n it_o in_o the_o 37_o meridian_n line_n according_a to_o the_o angle_n of_o of_o position_n it_o will_v reach_v from_o a_o to_o the_o equinoctial_a line_n degree_n likewise_o the_o distance_n d_o w_n need_v no_o other_o measure_v but_o must_v needs_o be_v 90_o degree_n which_o show_v the_o distance_n to_o be_v 90_o degree_n the_o last_o example_n shall_v be_v of_o two_o place_n the_o one_o be_v on_o the_o one_o side_n and_o the_o other_o on_o the_o other_o side_n of_o the_o equinoctial_a as_o suppose_v you_o be_v to_o sail_v from_o the_o summer-isle_n to_o the_o cape_n of_o good_a hope_n situation_n example_n of_o place_n in_o another_o situation_n the_o latitude_n of_o the_o summer-isle_n be_v 32_o degree_n 25_o north_n and_o the_o latitude_n of_o the_o cape_n of_o good_a hope_n be_v 35_o degree_n south_n and_o suppose_v the_o difference_n of_o longitude_n between_o these_o two_o place_n to_o be_v 90_o degree_n and_o it_o be_v require_v to_o find_v out_o the_o arch_n of_o the_o great_a circle_n between_o these_o two_o place_n according_a to_o the_o longitude_n and_o latitude_n thereof_o the_o angle_n of_o position_n and_o the_o distance_n of_o the_o two_o place_n to_o perform_v this_o first_o you_o must_v set_v down_o the_o first_o place_n according_a to_o the_o latitude_n thereof_o in_o the_o outmost_a circle_n at_o s_n and_o draw_v the_o diameter_n s_o k_n to_o which_o you_o must_v draw_v the_o line_n n_o m_o squirewise_o at_o right_a angle_n then_o count_v the_o difference_n of_o longitude_n which_o be_v 90_o degree_n from_o c_o the_o meridian_n of_o the_o cape_n of_o good_a hope_n will_v fall_v in_o the_o line_n a_o b_o which_o you_o must_v mark_v out_o according_a to_o its_o latitude_n 35_o degree_n south_n at_o x_o then_o by_o these_o three_o point_n s_n x_o k_o find_v the_o centre_n which_o will_v be_v in_o the_o line_n m_o n_n extend_v and_o so_o draw_v the_o arch_n s_o x_o k._n now_o first_o for_o the_o longitude_n and_o latitude_n of_o this_o arch_n you_o may_v find_v they_o by_o see_v how_o this_o arch_n do_v cross_v the_o circle_n of_o longitude_n and_o latitude_n in_o the_o draught_n which_o for_o every_o ten_o degree_n of_o longitude_n be_v as_o follow_v long._n latitude_n  _fw-fr deg._n deg._n m._n