Selected quad for the lemma: end_n

Word A	Word B	Word C	Word D	Occurrence	Frequency Band	MI	MI Band	Prominent
end_n	draw_v	line_n	perpendicular_a	3,095	5	14.0786	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

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 way_n the_o second_o way_n to_o perform_v this_o be_v demonstrate_v on_o the_o left_a side_n of_o the_o figure_n let_v the_o line_n a_o b_o be_v give_v as_o before_o and_o it_o be_v require_v to_o erect_v the_o line_n a_o d_o perpendicular_o in_o the_o point_n a._n to_o perform_v this_o first_o set_v one_o foot_n of_o your_o compass_n in_o any_o convenient_a point_n at_o pleasure_n as_o at_o l_o and_o open_v the_o other_o foot_n to_o the_o point_n a_o and_o draw_v the_o arch_n n_o a_o m_o then_o lay_v your_o ruler_n to_o the_o centre_n of_o this_o arch_n l_o and_o the_o place_n where_o it_o cross_v the_o line_n a_o b_o which_o be_v at_o n_n and_o draw_v the_o line_n m_o l_o n_o which_o do_v cross_v the_o arch_n n_o a_o m_o in_o the_o point_n m._n last_o lay_v your_o ruler_n to_o this_o cross_n at_o m_o and_o the_o point_n a_o draw_v the_o line_n d_o m_o a_o so_o you_o shall_v have_v your_o desire_n proposition_n 2._o to_o draw_v one_o line_n parallel_n to_o another_o line_n at_o any_o distance_n require_v proposition_n the_o second_o proposition_n line_n what_o be_v mean_v by_o a_o parallel_n line_n a_o line_n be_v say_v to_o be_v parallel_n to_o another_o line_n when_o it_o be_v equal_o distant_a from_o it_o in_o every_o part_n thereof_o thus_o in_o the_o former_a figure_n the_o line_n p_o s_o be_v parallel_n to_o the_o line_n a_o b._n now_o the_o way_n to_o draw_v a_o parallel_n line_n be_v thus_o let_v the_o line_n a_o b_o be_v the_o first_o line_n give_v and_o it_o be_v require_v to_o draw_v the_o line_n p_o s_o parallel_n to_o it_o according_a to_o the_o distance_n p_o a._n first_o open_v your_o compass_n to_o the_o distance_n you_o have_v occasion_n to_o use_v which_o in_o this_o example_n be_v p_o a_o then_o set_v one_o foot_n in_o a_o with_o the_o other_o draw_v the_o arch_n at_o p_o and_o then_o keep_v your_o compass_n at_o the_o same_o distance_n remove_z one_o foot_n to_o b_o and_o with_o the_o other_o draw_v the_o arch_n at_o s_n last_o lay_v your_o ruler_n on_o the_o very_a edge_n of_o these_o two_o arch_n draw_v the_o line_n p_o s_o which_o will_v be_v parallel_n to_o a_o b_o and_o so_o the_o proposition_n be_v perform_v proposition_n 3._o how_o to_o make_v a_o geometrical_a square_n proposition_n the_o three_o proposition_n a_o geometrical_a square_n be_v a_o square_n who_o four_a side_n be_v all_o of_o one_o and_o the_o same_o length_n now_o in_o the_o first_o figure_n let_v the_o line_n a_o b_o be_v the_o side_n of_o such_o a_o square_a and_o it_o be_v require_v to_o make_v a_o square_a of_o that_o length_n and_o breadth_n first_o square_a how_o to_o make_v a_o square_a you_o must_v draw_v the_o line_n a_o b_o according_o to_z the_o length_n give_v then_o erect_v the_o perpendicular_a line_n a_o d_o at_o one_o end_n thereof_o as_o be_v show_v before_o then_o set_v one_o foot_n of_o your_o compass_n in_o the_o corner_n a_o open_z the_o other_z to_z b_o and_o keep_v one_o foot_n still_o in_o a_o with_o the_o other_o across_o the_o perpendicular_a a_o d_o at_z d_o then_o keep_v your_o compass_n at_o the_o same_o distance_n set_v one_o foot_n in_o b_o and_o with_o the_o other_o draw_v a_o short_a arch_n at_o c_o then_o set_v one_o foot_n at_o the_o cross_n at_o d_o and_o with_o the_o other_o across_o the_o arch_n last_o draw_v in_o the_o point_n c_o now_o if_o you_o draw_v line_n through_o these_o mark_n from_o a_o to_o d_o from_z d_o to_z c_z and_z from_z c_z to_z b_o so_o you_o shall_v make_v the_o geometrical_a square_n a_o b_o c_o d_o as_o be_v require_v if_o you_o will_v try_v your_o work_n whether_o you_o have_v make_v it_o true_a or_o no_o then_o set_v one_o foot_n of_o your_o compass_n in_o a_o square_n how_o to_o try_v a_o square_n and_o open_v the_o other_o to_o the_o corner_n at_o c_o then_o with_o that_o distance_n set_v one_o foot_n in_o b_o and_o turn_v the_o other_o to_o the_o corner_n at_o d_o if_o both_o these_o opposite_a corner_n have_v the_o same_o distance_n the_o square_n be_v true_o make_v otherwise_o not_o square_n a_o long_a square_n if_o you_o will_v make_v a_o long_a square_n as_o the_o square_a a_o p_o s_o b_o first_fw-mi you_o may_v draw_v the_o line_n p_o s_o parallel_n to_o a_o b_o and_o according_a to_o the_o length_n of_o the_o side_n of_o your_o square_n then_o erect_v the_o perpendicular_a either_o at_o a_o or_o b_o and_o draw_v the_o opposite_a side_n parallel_n thereunto_o according_a as_o the_o length_n of_o your_o square_n require_v and_o you_o may_v try_v the_o truth_n of_o this_o square_a also_o by_o the_o opposite_a corner_n as_o before_o proposition_n 4._o to_o raise_v a_o perpendicular_a in_o the_o midst_n of_o a_o line_n proposition_n the_o four_o proposition_n in_o this_o second_o figure_n let_v a_o b_o be_v the_o line_n give_v and_o it_o be_v require_v to_o raise_v a_o perpendicular_a in_o the_o point_n c._n figure_n the_o second_o figure_n first_o set_v one_o foot_n of_o your_o compass_n in_o the_o point_n c_o and_o open_v the_o other_o to_o any_o distance_n at_o pleasure_n and_o mark_v the_o give_v line_n therewith_o on_o both_o side_n from_o c_o at_o the_o point_v a_o and_o b_o then_o set_v one_o foot_n of_o your_o compass_n in_o the_o point_n a_o open_v the_o other_o to_o any_o distance_n you_o please_v beyond_o c_o and_o draw_v a_o little_a arch_n above_o the_o line_n at_o f._n then_o with_o the_o same_o distance_n set_v one_o foot_n in_o b_o and_o with_o the_o other_o cross_n the_o arch_n f_o with_o the_o arch_n d._n last_o lay_v your_o ruler_n to_o this_o cross_n and_o the_o point_n c_o and_o draw_v the_o line_n g_o c_o which_o be_v perpendicular_a to_o the_o line_n a_o b_o in_o the_o point_n c_o as_o be_v require_v proposition_n 5._o from_o a_o point_n aloft_o to_o let_v fall_v a_o perpendicular_a upon_o a_o line_n give_v proposition_n the_o five_o proposition_n let_v g_o be_v the_o point_n aloft_o from_o whence_o it_o be_v require_v to_o let_v fall_v a_o perpendicular_a upon_o the_o line_n a_o b_o in_o the_o second_o figure_n first_o set_v one_o foot_n of_o your_o compass_n in_o the_o point_n give_v which_o be_v g_z and_o open_v your_o compass_n so_o wide_o that_o you_o may_v draw_v the_o arch_n a_o h_n b_o which_o may_v cut_v the_o line_n a_o b_o in_o the_o point_v a_o and_o b_o and_o the_o far_o these_o two_o point_n be_v asunder_o so_o much_o the_o better_a then_o keep_v your_o compass_n at_o this_o distance_n set_v one_o foot_n in_o a_o and_o with_o the_o other_o draw_v the_o arch_n e_o then_z remove_z one_o foot_n to_o b_o and_o cross_v the_o last_o arch_n at_o e_o last_o lay_v your_o ruler_n to_o the_o point_n g_o and_o this_o cross_n at_o e_o draw_v the_o line_n g_o c_o e_o so_fw-mi you_o have_v perform_v the_o proposition_n proposition_n 6._o to_o draw_v a_o line_n squire_n wise_a to_o another_o line_n proposition_n the_o six_o proposition_n in_o the_o second_o figure_n let_v a_o b_o be_v the_o line_n give_v and_o it_o be_v require_v to_o draw_v the_o line_n g_o e_o squire_n wise_a 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

which_o will_v cross_v the_o line_n a_o b_o at_o right_a angle_n as_o be_v require_v proposition_n 7._o to_o divide_v a_o line_n give_v into_o two_o equal_a part_n proposition_n the_o seven_o proposition_n in_o the_o second_o figure_n let_v a_o b_o be_v the_o line_n give_v to_o be_v divide_v into_o two_o equal_a part_n first_o set_v one_o foot_n of_o your_o compass_n at_o the_o one_o end_n of_o the_o line_n at_o a_o and_o open_v the_o other_o to_o any_o distance_n above_o half_a the_o line_n &_o therewith_o draw_v two_o little_a arch_n one_o above_o the_o line_n at_o f_o and_o the_o other_o below_o the_o line_n at_o e_o then_o remove_v your_o compass_n to_o b_o the_o other_o end_n of_o the_o line_n and_o cross_v the_o two_o former_a arch_n at_o f_o and_o e_o then_o lay_v your_o ruler_n to_o these_o two_o cross_n f_o and_o e_o and_o draw_v the_o line_n gc_fw-mi e_z which_o will_v divide_v the_o line_n a_o b_o in_o two_o equal_a part_n in_o the_o point_n c_o so_fw-mi that_o a_o c_o be_v the_o one_o half_a and_o c_o b_o the_o other_o proposition_n 8._o to_o raise_v a_o perpendicular_a at_o the_o end_n of_o a_o line_n another_o way_n proposition_n the_o eight_o proposition_n in_o this_o figure_n let_v the_o line_n give_v be_v a_o b_o and_o it_o be_v require_v to_o raise_v a_o perpendicular_a at_o the_o end_n thereof_o at_o b._n here_o you_o may_v note_v that_o if_o the_o three_o side_n of_o a_o triangle_n be_v make_v of_o these_o three_o number_n 3_o 4_o 5_o or_o any_o other_o number_n that_o be_v proportionable_a thereunto_o as_o 6_o 8_o 10_o 9_o 12_o 15_o 12_o 16_o 20_o 30_o 40_o 50_o it_o will_v have_v one_o right_a angle_n which_o will_v be_v opposite_a to_o the_o great_a side_n as_o in_o the_o triangle_n d_o b_o e_o the_o side_n e_o b_o be_v 3_o the_o side_n b_o d_o be_v 4_o and_o the_o side_n d_o e_z be_v 5_o and_o the_o angle_n at_o b_o be_v a_o right_a angle_n proposition_n 9_o to_o make_v one_o angle_n equal_a or_o like_v to_o another_o proposition_n the_o nine_o proposition_n a_o angle_n be_v the_o join_n or_o cross_v of_o two_o line_n angle_n what_o a_o angle_n be_v with_o the_o general_a kind_n of_o angle_n if_o the_o two_o line_n cross_v one_o another_o or_o join_v one_o to_o another_o perpendicular_o than_o they_o be_v say_v to_o make_v a_o right_a angle_n or_o angle_n if_o two_o line_n meet_v or_o cross_a one_o another_o any_o other_o way_n they_o be_v say_v to_o make_v a_o oblique_a angle_n or_o angle_n thus_o in_o the_o three_o figure_n the_o line_n d_o b_o and_o e_o b_o meeting_n in_o the_o point_n b_o make_v a_o right_a angle_n and_o in_o the_o second_o figure_n the_o line_n a_o b_o and_o g_o e_o cross_v one_o another_o in_o the_o point_n c_o make_v four_o right_a angle_n or_o quadrant_n but_o in_o the_o three_o figure_n the_o line_n e_o d_o and_o b_o d_o meet_v in_o the_o point_n d_o be_v say_v to_o make_v a_o oblique_a angle_n now_o these_o oblique_a angle_n if_o they_o be_v less_o than_o a_o right_a angle_n they_o be_v call_v acute_a or_o sharp_a angle_n if_o they_o be_v more_o than_o a_o right_a angle_n they_o be_v call_v obtuse_a or_o blunt_a angle_n now_o for_o example_n of_o the_o proposition_n let_v the_o angle_n e_o d_o b_o be_v the_o appoint_a angle_n and_o it_o be_v require_v to_o make_v the_o angle_n d_o b_o c_o like_v unto_o it_o in_o this_o example_n because_o the_o line_n d_o b_o be_v limit_v and_o be_v common_a to_o both_o the_o angle_n you_o shall_v need_v only_o to_o set_v one_o foot_n of_o your_o compass_n in_o b_o and_o open_v the_o other_o to_o the_o near_a distance_n of_o the_o line_n d_o e_o which_o you_o may_v do_v by_o draw_v the_o little_a arch_n which_o touch_v the_o line_n between_o 3_o and_o 4_o then_o remove_v your_o compass_n to_o d_o and_o draw_v the_o like_a arch_n at_o c_o then_o lay_v your_o ruler_n to_o the_o point_n b_o and_o the_o very_a edge_n of_o this_o arch_n c_o and_o draw_v the_o line_n b_o c_o so_fw-mi shall_v the_o two_o angle_n be_v of_o one_o quantity_n or_o wideness_n as_o be_v desire_v in_o other_o case_n this_o way_n will_v not_o serve_v but_o this_o be_v sufficient_a for_o the_o present_a purpose_n and_o i_o shall_v show_v you_o other_o way_n to_o perform_v that_o in_o the_o next_o chapter_n proposition_n 10._o to_o divide_v a_o line_n into_o any_o number_n of_o equal_a part_n proposition_n the_o ten_o proposition_n in_o the_o three_o figure_n let_v the_o line_n b_o d_o be_v give_v to_o be_v divide_v into_o four_o equal_a part_n first_o from_o the_o end_n d_o draw_v a_o line_n as_o d_o e_z make_v a_o angle_n with_o the_o line_n d_o b_o at_o pleasure_n then_o from_o the_o other_o end_n of_o the_o line_n b_o make_v the_o angle_n d_o b_o c_o equal_a to_o the_o former_a angle_n as_o be_v show_v in_o the_o last_o proposition_n then_o from_o the_o point_n d_o set_v off_o with_o your_o compass_n such_o a_o number_n of_o any_o equal_a part_n as_o lack_v one_o of_o the_o number_n desire_v which_o in_o this_o example_n therefore_o must_v be_v 3_o set_v off_o therefore_o on_o the_o line_n d_o e_o three_o equal_a part_n 1_o 2_o and_z 3_o than_o you_o must_v with_o the_o same_o distance_n of_o your_o compass_n set_v off_o 1_o 2_o and_z 3_o from_o the_o point_n b_o on_o the_o line_n b_o c_o then_z draw_z cross_a line_n from_o the_o last_o number_n in_o the_o one_o line_n to_o the_o first_o in_o the_o other_o that_o be_v from_o 3_o to_o 1_o from_z 2_o to_z 2_o etc._n etc._n and_o these_o line_n will_v divide_v the_o line_n b_o d_o into_o four_o equal_a part_n as_o be_v desire_v proposition_n 11._o to_o bring_v any_o three_o point_n not_o lie_v in_o a_o straight_a line_n into_o a_o circle_n proposition_n the_o eleven_o proposition_n in_o this_o figure_n let_v a_o b_o c_o be_v the_o three_o point_n give_v and_o it_o be_v require_v to_o draw_v a_o circle_n through_o they_o all_o figure_n the_o four_o figure_n set_v one_o foot_n of_o your_o compass_n in_o the_o middle_a point_n at_o b_o and_o open_v your_o compass_n to_o any_o distance_n you_o please_v so_o it_o be_v above_o half_a the_o distance_n between_o b_o and_o either_o of_o the_o other_o mark_n yea_o it_o be_v no_o matter_n if_o need_v be_v though_o it_o reach_v almost_o to_o or_o quite_o beyond_o the_o near_a of_o the_o other_o mark_n and_o draw_v the_o arch_n d_o e_o f_o g_o then_o keep_v your_o compass_n at_o this_o distance_n set_v one_o foot_n in_o a_o and_o with_o the_o other_o draw_v the_o arch_n g_o f_o which_o cross_v the_o former_a arch_n at_o g_o and_o f_o then_o set_v one_o foot_n of_o your_o compass_n in_o the_o three_o point_n c_o and_o with_o the_o other_o draw_v the_o arch_n e_o d_o which_o cross_v the_o first_o arch_n at_o e_o and_o d_o then_o lay_v your_o ruler_n to_o the_o intersection_n of_o these_o arch_n draw_v the_o line_n g_o f_o h_n and_o d_o e_o h_o which_o will_v cross_v one_o another_o in_o the_o point_n h_n this_o cross_n at_o h_n be_v the_o centre_n of_o the_o circle_n therefore_o set_v one_o foot_n of_o your_o compass_n in_o this_o cross_n at_o h_n open_v the_o other_o to_o any_o of_o the_o three_o point_n a_o b_o or_o c_o and_o draw_v the_o circle_n which_o if_o you_o have_v do_v well_o will_v pass_v through_o all_o the_o three_o point_n a_o b_o c_o as_o be_v require_v chap._n ii_o show_v how_o to_o divide_v a_o circle_n several_a way_n which_o will_v be_v needful_a for_o many_o thing_n the_o first_o usual_a division_n of_o a_o circle_n be_v into_o 24_o equal_a part_n according_a to_o the_o 24_o hour_n of_o a_o natural_a day_n which_o be_v thus_o to_o be_v perform_v second_o another_o usual_a and_o necessary_a division_n of_o a_o circle_n be_v to_o divide_v a_o circle_n into_o 360_o equal_a part_n degree_n to_o divide_v a_o circle_n into_o 360_o degree_n for_o in_o all_o question_n of_o astronomy_n and_o in_o the_o calculation_n of_o all_o triangle_n these_o part_n be_v the_o measure_n of_o the_o angle_n so_o that_o every_o arch_n in_o this_o respect_n be_v suppose_v to_o be_v divide_v into_o 360_o equal_a part_n which_o be_v call_v degree_n and_o each_o degree_n be_v suppose_v to_o be_v divide_v into_o 60_o lesser_a part_n call_v minute_n to_o divide_v a_o circle_n after_o this_o manner_n the_o ready_a way_n be_v thus_o first_o draw_v a_o line_n at_o pleasure_n and_o cross_v it_o at_o right_a angle_n with_o another_o line_n and_o draw_v a_o circle_n as_o before_o then_o keep_v your_o compass_n at_o this_o distance_n divide_v the_o circle_n from_o the_o four_o quarter_n into_o 12_o equal_a

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

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 draw_v the_o line_n f_o e_o g_o at_o 5_o deg_fw-la of_o latitude_n according_a to_o your_o observation_n of_o the_o latitude_n mark_v where_o it_o cross_v this_o line_n i_o qui_fw-fr which_o be_v in_o the_o point_n n_n and_o this_o be_v the_o true_a place_n you_o be_v in_o who_o longitude_n be_v 6_o deg_fw-la and_o who_o latitude_n be_v 5_o deg_fw-la north_n in_o like_a manner_n if_o you_o shall_v sail_v upon_o 3_o or_o 4_o several_a rumbe_n before_o you_o can_v make_v a_o observation_n of_o the_o latitude_n your_o best_a way_n will_v be_v to_o draw_v a_o line_n from_o the_o first_o place_n of_o your_o voyage_n to_o that_o present_a place_n according_a to_o your_o dead_a reckon_n or_o at_o least_o from_o the_o last_o place_n where_o you_o make_v a_o fair_a observation_n and_o be_v thereby_o well_o assure_v both_o of_o the_o longitude_n and_o latitude_n thereof_o for_o otherwise_o you_o may_v be_v much_o mistake_v in_o the_o longitude_n of_o your_o place_n as_o for_o instance_n if_o in_o the_o last_o example_n you_o shall_v think_v you_o be_v in_o that_o place_n where_o the_o line_n of_o latitude_n f_o e_o g_o do_v cut_v the_o last_o rumbe_n you_o sail_v upon_o according_a to_o your_o dead_a reckon_n viz._n the_o line_n r_o q_n by_o this_o account_n you_o will_v be_v but_o in_o o_o which_o be_v but_o in_o 5_o deg_fw-la 35_o 100_o of_o longitude_n whereas_o you_o see_v by_o the_o other_o way_n which_o be_v the_o truth_n you_o be_v in_o 6_o deg_fw-la of_o longitude_n so_o that_o the_o difference_n be_v ●_o 100_o which_o be_v very_o considerable_a in_o so_o small_a a_o space_n proposition_n 7._o be_v to_o sail_v from_o one_o place_n to_o another_o but_o by_o reason_n of_o cross_a wind_n or_o the_o coasting_n of_o the_o land_n you_o can_v sail_v thither_o upon_o the_o direct_a point_n of_o the_o compass_n which_o lie_v between_o the_o two_o place_n but_o be_v force_v to_o alter_v your_o course_n several_a time_n yet_o how_o you_o shall_v keep_v your_o account_n of_o 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 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