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
View all documents for the selected quad

Text snippets containing the quad

ID Title Author Corrected Date of Publication (TCP Date of Publication) STC Words Pages
A52487 The seaman's companion being a plain guide to the understanding of arithmetick, geometry, trigonometry, navigation, and astronomy. Applied chiefly to navigation: and furnished with a table of meridional parts, to every third minute: with excellent and easie ways of keeping a reckoning at sea, never in print before. Also, a catalogue of the longitude and latitude of the principal places in the world with other useful things. The third edition corrected and amended. By Matthew Norwood, mariner. Norwood, Matthew. 1678 (1678) Wing N1345A; ESTC R218969 104,181 203

There are 6 snippets containing the selected quad. | View original text

the_o seaman_n companion_n be_v a_o plain_a guide_n to_o the_o understanding_n of_o arithmetic_n geometry_n trigonometry_n navigation_n and_o astronomy_n apply_v chief_o to_o navigation_n and_o furnish_v with_o a_o table_n of_o meridional_a part_n to_o every_o three_o minute_n with_o excellent_a and_o easy_a way_n of_o keep_v a_o reckon_a at_o sea_n never_o in_o print_n before_o also_o a_o catalogue_n of_o the_o longitude_n and_o latitude_n of_o the_o principal_a place_n in_o the_o world_n with_o other_o useful_a thing_n the_o three_o edition_n correct_v and_o amend_v by_o matthew_n norwood_n mariner_n london_n ●●●●●ed_v by_o anne_n godbid_v and_o john_n playford_n for_o william_n fisher_n at_o the_o postern-gate_n near_o tower-hill_n robert_n boulter_n at_o the_o turks-head_n and_o ralph_n smith_n at_o the_o bible_n in_o cornhill_n thomas_n passenger_n at_o the_o three_o bibles_n on_o london-bridge_n and_o richard_n northcot_n next_o st._n peter_n alley_n in_o cornhill_n and_o at_o the_o anchor_n and_o mariner_n on_o fishstreet-hill_n to_o the_o reader_n the_o famous_a and_o ever_o to_o be_v admire_v art_n of_o navigation_n have_v be_v so_o learned_o handle_v and_o write_v of_o not_o only_o in_o all_o other_o language_n but_o also_o in_o our_o mother-tongue_n by_o so_o many_o learned_a and_o able_a man_n both_o of_o former_a and_o of_o our_o present_a age_n that_o it_o may_v seem_v impossible_a to_o write_v any_o thing_n more_o thereof_o that_o have_v not_o already_o be_v do_v by_o other_o yet_o in_o my_o experience_n which_o i_o have_v see_v be_v at_o sea_n in_o several_a vessel_n where_o divers_a young_a mariner_n have_v be_v i_o have_v hear_v this_o general_a complaint_n among_o they_o that_o though_o the_o thing_n that_o be_v chief_o useful_a for_o they_o in_o their_o art_n may_v be_v find_v in_o several_a book_n here_o and_o there_o disperse_v yet_o they_o can_v wish_v that_o there_o be_v such_o a_o book_n contrive_v that_o may_v be_v soley_n useful_a for_o they_o entire_a by_o itself_o which_o then_o will_v be_v more_o convenient_a for_o they_o and_o may_v be_v purchase_v at_o a_o more_o reasonable_a rate_n than_o otherwise_o they_o can_v be_v by_o buy_v of_o so_o many_o sort_n of_o book_n which_o they_o must_v be_v constrain_v to_o do_v if_o ever_o they_o intend_v to_o be_v able_a proficient_n in_o that_o most_o noble_a profession_n of_o navigation_n at_o which_o they_o chief_o aim_v this_o complaint_n of_o they_o be_v one_o chief_a motive_n which_o induce_v i_o to_o collect_v and_o compose_v the_o subsequent_a treatise_n which_o i_o have_v endeavour_v to_o handle_v in_o such_o a_o methodical_a manner_n as_o it_o ought_v to_o be_v read_v and_o practise_v by_o the_o young_a seaman_n for_o first_o there_o be_v a_o treatise_n of_o arithmetic_n contain_v all_o the_o rule_n thereof_o which_o be_v necessary_a for_o the_o seaman_n to_o know_v and_o practice_v all_o or_o most_o of_o the_o question_n thereof_o be_v make_v applicable_a in_o one_o kind_n or_o other_o to_o nautical_a affair_n second_o there_o be_v a_o treatise_n of_o geometry_n contain_v the_o first_o ground_n and_o principle_n thereof_o with_o the_o make_n and_o divide_v of_o the_o mariner_n scale_n and_o compass_n with_o the_o project_v of_o the_o sphere_n in_o plano_n and_o the_o resolve_v of_o many_o question_n in_o astronomy_n which_o be_v useful_a in_o navigation_n thereby_o three_o you_o have_v a_o treatise_n of_o the_o practic_a part_n of_o navigation_n wherein_o be_v show_v after_o a_o new_a experience_a way_n use_v by_o the_o author_n how_o to_o keep_v a_o reckon_a at_o sea_n the_o make_v and_o use_n of_o the_o plain_a sea-chart_n the_o doctrine_n of_o plain_a triangle_n make_v applicable_a to_o such_o question_n in_o navigation_n as_o concern_v course_n distance_n rumb_n difference_n of_o longitude_n and_o departure_n likewise_o a_o table_n of_o meridional_a part_n to_o every_o three_o minute_n and_o the_o application_n and_o use_n thereof_o exemplify_v in_o question_n of_o sail_v by_o mercator_n chart_n with_o table_n of_o longitude_n and_o latitude_n of_o place_n of_o right_a ascension_n and_o season_n of_o certain_a fix_a star_n with_o rule_n to_o keep_v a_o reckon_a and_o to_o find_v the_o latitude_n by_o the_o meridian_n altitude_n of_o the_o sun_n or_o star_n four_o there_o be_v a_o short_a treatise_n of_o astronomy_n wherein_o you_o have_v the_o doctrine_n of_o spherical_a triangle_n apply_v to_o question_n in_o astronomy_n and_o navigation_n this_o be_v the_o brief_a sum_n and_o substance_n of_o the_o follow_a treatise_n which_o i_o commend_v to_o the_o practice_n of_o all_o young_a seaman_n desire_v their_o kind_a acceptance_n of_o these_o my_o first_o labour_n which_o if_o i_o shall_v find_v to_o be_v kind_o entertain_v by_o they_o it_o will_v encourage_v i_o to_o launch_v far_o into_o the_o more_o nice_a and_o critical_a part_n of_o this_o most_o noble_a science_n in_o the_o mean_a time_n i_o commend_v this_o to_o they_o wish_v good_a success_n in_o all_o their_o honest_a and_o laudable_a undertake_n and_o in_o the_o interim_n bid_v they_o farewell_o matthew_n norwood_n the_o content_n of_o arithmetic_n numeration_n 1_o addition_n 2_o subtraction_n 5_o multiplication_n 7_o division_n 10_o reduction_n 16_o the_o golden_a rule_n or_o rule_n of_o three_o 19_o the_o rule_n of_o interest_n and_o interest_n upon_o interest_n 24_o the_o rule_n of_o fellowship_n 25_o of_o geometry_n geometrical_a definition_n 28_o how_n to_o raise_v a_o perpendicular_a 30_o how_o to_o divide_v a_o line_n into_o two_o equal_a part_n 31_o how_n to_o raise_v a_o perpendicular_a on_o the_o end_n of_o a_o line_n ibid._n from_o a_o point_n give_v to_o let_v fall_v a_o perpendicular_a 32_o how_o to_o draw_v parallel_n line_n 33_o how_n to_o make_v a_o geometrical_a square_a 34_o how_o to_o make_v a_o oblong_a or_o square_a who_o length_n and_o breadth_n be_v give_v 34_o how_o to_o make_v a_o diamond-figure_n who_o side_n and_o angle_n shall_v be_v limit_v 35_o to_o make_v a_o rhomboiade_n who_o side_n and_o angle_n be_v give_v 36_o to_o find_v the_o centre_n of_o a_o circle_n 37_o to_o find_v the_o centre_n of_o that_o circle_n which_o shall_v pass_v through_o any_o three_o give_v point_n which_o be_v not_o situate_a in_o a_o straight_a line_n 37_o how_o to_o divide_v a_o circle_n into_o 2_o 4_o 8_o 16_o 32_o equal_a part_n 40_o the_o projection_n of_o the_o mariner_n compass_n 41_o the_o projection_n of_o the_o plain_a scale_n 42_o the_o projection_n of_o the_o sphere_n in_o plano_n 45_o the_o name_n and_o character_n of_o the_o sign_n with_o the_o month_n they_o belong_v to_o 48_o how_o to_o project_n the_o sphere_n 49_o the_o meaning_n of_o certain_a term_n of_o art_n 51_o how_o to_o find_v the_o sun_n meridian_n altitude_n 55_o how_o to_o find_v the_o amplitude_n of_o rise_v or_o set_v ibid._n how_o to_o find_v the_o azimuth_n at_o six_o of_o the_o clock_n 57_o to_o find_v the_o altitude_n at_o six_o 58_o to_o find_v the_o altitude_n the_o sun_n be_v due_a east_n or_o west_n 59_o how_o to_o find_v the_o ascensional_a difference_n ibid._n to_o find_v the_o time_n of_o the_o sun_n rise_v or_o set_v 60_o to_o find_v what_o hour_n it_o be_v the_o sun_n be_v due_a east_n or_o west_n 61_o to_o find_v the_o time_n of_o day_n break_v and_o twilight_n end_v 62_o to_o find_v the_o continuance_n of_o twilight_n ibid._n to_o find_v the_o length_n of_o the_o long_a day_n in_o any_o latitude_n 63_o to_o find_v the_o sun_n place_n and_o right_a ascension_n 68_o to_o find_v the_o sun_n declination_n etc._n etc._n 69_o term_n of_o art_n use_v in_o navigation_n explain_v 70_o proposition_n of_o sail_v by_o the_o plain_a scale_n 72_o question_n in_o navigation_n resolve_v from_o page_n 73_o to_o page_n 92_o of_o a_o travis_n 92_o the_o manner_n of_o keep_v a_o reckon_a at_o sea_n 97_o concern_v the_o variation_n of_o the_o compass_n 102_o the_o use_n of_o a_o plain_a sea-chart_n 108_o of_o oblique_a triangle_n the_o application_n whereof_o be_v in_o ten_o question_n perfect_o explain_v all_o which_o question_n be_v apply_v to_o question_n in_o sail_v and_o be_v wrought_v both_o geometrical_o by_o the_o plain_a scale_n and_o also_o by_o the_o table_n of_o sines_n tangent_n and_o logarithm_n from_o page_n 113_o to_o page_n 146_o a_o table_n of_o meridional_a part_n to_o every_o three_o minute_n 146_o a_o declaration_n of_o the_o table_n 158_o the_o use_n of_o the_o table_n of_o meridional_a part_n exemplify_v in_o five_o question_n appertain_v to_o navigation_n from_o page_n 160_o to_o page_n 166_o of_o the_o longitude_n and_o latitude_n of_o place_n a_o table_n of_o they_o 167_o the_o use_n of_o they_o 171_o how_o to_o keep_v a_o reckon_a of_o the_o longitude_n and_o latitude_n a_o ship_n make_v at_o sea_n 172_o the_o name_n declination_n and_o season_n
fellowship_n this_o rule_n be_v the_o rule_n of_o three_o do_v several_a time_n in_o one_o sum_n or_o question_n for_o here_o be_v several_a stock_n and_o several_a man_n that_o own_o they_o now_o if_o the_o principal_a or_o all_o their_o stock_n gain_v so_o much_o how_o much_o shall_v each_o man_n gain_n according_a to_o the_o stock_n he_o put_v in_o i_o will_v say_v if_o the_o whole_a stock_n gain_v so_o much_o what_o shall_v the_o first_o man_n stock_n gain_v and_o so_o for_o the_o rest_n quest_n 1._o six_o man_n make_v a_o stock_n the_o first_o put_v in_o 30_o l._n the_o second_o 40._o l._n the_o three_o 52_o l._n the_o four_o 58_o l._n the_o five_o 60_o l._n and_o the_o six_o 78_o l._n if_o the_o whole_a stock_n together_o gain_v 200_o l._n what_o shall_v each_o man_n have_v so_o that_o there_o may_v be_v no_o wrong_n first_o find_v the_o whole_a stock_n by_o add_v every_o man_n stock_n together_o which_o be_v 318_o l._n then_o say_v if_o the_o whole_a stock_n 318_o l._n gain_n 200_o l._n what_o shall_v the_o first_o man_n stock_n gain_v  _fw-fr l._n first_o 30_o second_o 40_o three_o 52_o four_o 58_o five_o 60_o sixth_z 78_o  _fw-fr 318_o for_o the_o second_o if_o the_o whole_a stock_n 318_o l._n gain_n 200_o l._n what_o shall_v 40_o l._n gain_n which_o be_v the_o second_o stock_n the_o like_a be_v to_o be_v understand_v of_o the_o rest_n i_o forbear_v to_o work_v they_o as_o be_v out_o of_o my_o intention_n i_o leave_v their_o operation_n to_o your_o genius_n who_o never_o learn_v they_o quest_n 2._o there_o where_o five_o man_n make_v a_o stock_n the_o first_o put_v in_o 200_o l._n the_o second_o 59_o l._n the_o three_o 180_o l._n the_o four_o 78_o l._n the_o five_o 240_o l._n they_o lose_v at_o the_o return_n of_o the_o ship_n 120_o l._n i_o demand_v each_o man_n loss_n proportionable_a to_o his_o venture_n first_o find_v the_o whole_a stock_n by_o add_v every_o man_n venture_n together_o this_o do_v say_v for_o the_o first_o man_n loss_n if_o the_o whole_a stock_n lose_v 120_o l._n what_o shall_v the_o first_o man_n stock_n 200_o l._n lose_v for_o the_o second_o if_o the_o whole_a stock_n lose_v 120_o l._n what_o shall_v the_o second_o man_n lose_v which_o put_v in_o 59_o l_o and_o so_o for_o the_o rest_n the_o way_n to_o prove_v one_o of_o these_o question_n be_v less_o trouble_n than_o the_o rule_n of_o three_o for_o after_o you_o have_v do_v add_v all_o your_o facit_n together_o and_o it_o will_v make_v the_o same_o that_o be_v lose_v in_o the_o whole_a stock_n or_o gain_v in_o the_o whole_a stock_n if_o the_o question_n be_v for_o gain_n and_o this_o stand_v to_o good_a reason_n for_o every_o man_n loss_n or_o gain_n must_v together_o be_v equal_a to_o the_o whole_a or_o else_o they_o do_v not_o contribute_v to_o the_o loss_n or_o enjoy_v the_o gain_n that_o they_o have_v lose_v or_o gain_v there_o be_v more_o intricacy_n in_o this_o rule_n than_o i_o have_v here_o cite_v but_o the_o foundation-work_n lie_v on_o the_o rule_n of_o three_o and_o the_o intricacy_n that_o i_o speak_v of_o will_v be_v understand_v with_o some_o consideration_n i_o have_v thought_n to_o have_v show_v it_o but_o it_o be_v show_v as_o well_o as_o can_v be_v i_o think_v by_o mr._n record_n and_o mr._n hodder_n beside_o it_o will_v take_v up_o much_o paper_n and_o i_o shall_v digress_v from_o my_o intention_n to_o navigation_n what_o i_o have_v show_v since_o i_o treat_v of_o the_o rule_n of_o three_o be_v only_o a_o show_n how_o all_o rule_n depend_v upon_o this_o golden_a rule_n blame_v i_o not_o for_o be_v so_o large_a in_o arithmetic_n but_o both_o that_o and_o all_o fault_n else_o season_n with_o the_o salt_n of_o a_o charitable_a construction_n remember_v that_o navigation_n be_v imperfect_a without_o it_o i_o end_v geometry_n that_o which_o i_o shall_v handle_v in_o geometry_n will_v be_v only_o that_o part_n of_o it_o which_o be_v use_v in_o navigation_n he_o that_o will_v treat_v of_o it_o at_o large_a have_v best_a to_o put_v it_o in_o a_o treatise_n by_o itself_o and_o he_o that_o will_v learn_v to_o be_v a_o good_a geometrician_n let_v he_o apply_v himself_o to_o euclid_n work_n there_o be_v no_o art_n but_o have_v a_o dependency_n on_o it_o and_o navigation_n depend_v much_o on_o it_o which_o enforce_v i_o to_o treat_v upon_o some_o principle_n of_o it_o geometrical_a definition_n a_o point_n or_o prick_n be_v this_o and_o be_v void_a of_o length_n breadth_n or_o thickness_n a_o line_n be_v length_n without_o breadth_n or_o thickness_n and_o be_v proper_o call_v the_o near_a distance_n if_o a_o straight_a line_n between_o two_o place_n but_o if_o not_o straight_o but_o circular_a it_o be_v term_v a_o archippus_n a_o triangle_n be_v when_o three_o line_n meet_v make_v three_o angular_a point_n now_o there_o be_v two_o sort_n right_o line_v and_o spherical_a spherical_a be_v all_o the_o side_n arch_n of_o great_a circle_n there_o be_v three_o sort_n of_o angle_n namely_o obtuse_a acute_a and_o right_a angle_n if_o divers_a circle_n be_v describe_v have_v all_o one_o centre_n they_o be_v call_v concentric_n but_o if_o they_o have_v divers_a centre_n they_o be_v call_v excentric_n the_o circumference_n of_o any_o circle_n consist_v of_o 360_o degree_n every_o degree_n be_v 60_o minute_n the_o diameter_n of_o a_o circle_n be_v a_o line_n draw_v from_o one_o side_n of_o the_o circle_n to_o the_o other_o through_o the_o centre_n as_o a_o c._n the_o semidiameter_n of_o a_o circle_n be_v half_a the_o diameter_n or_o the_o distance_n from_o the_o circumference_n to_o the_o centre_n prop._n i._o to_o raise_v a_o perpendicular_a from_o the_o middle_n of_o a_o line_n give_v the_o line_n give_v be_v a_o b_o divide_v it_o into_o two_o equal_a part_n at_o f_o then_o open_a your_o compass_n to_o any_o convenient_a distance_n above_o half_a the_o length_n of_o the_o give_v line_n and_o fix_v one_o foot_n in_o b_o and_o with_o the_o other_o describe_v the_o arch_n c_o with_o the_o same_o distance_n fix_v one_o foot_n of_o your_o compass_n in_o a_o and_o cross_v the_o other_o arch_n by_o the_o arch_n d_o mind_n where_o they_o intersect_v and_o from_o that_o place_n which_o be_v ⊙_o draw_v a_o line_n to_o the_o middle_n of_o your_o give_a line_n a_o b_o namely_o to_o f_o then_o be_v f_o ⊙_o a_o true_a perpendicular_a from_o the_o middle_n of_o the_o line_n a_o b_o for_o it_o be_v so_o direct_o from_o it_o that_o it_o lean_v no_o way_n prop._n ii_o to_o divide_v a_o line_n into_o two_o equal_a part_n by_o a_o perpendicular_a the_o give_a line_n be_v a_o b_o fix_v one_o foot_n of_o your_o compass_n in_o a_o and_o set_v they_o at_o any_o convenient_a distance_n above_o half_a the_o give_v line_n describe_v the_o arch_n d_o and_o f_o carry_v your_o compass_n to_o b_o with_o the_o same_o distance_n and_o describe_v the_o arch_n c_o and_o e_o so_o as_o they_o may_v cross_v the_o other_o arch_n as_o they_o do_v in_o ⊙_o mind_n it_o and_o by_o those_o intersection_n lay_v your_o scale_n and_o draw_v the_o line_n ⊙_o ⊙_o which_o will_v cut_v the_o give_v line_n in_o the_o midst_n and_o be_v a_o perpendicular_a to_o it_o which_o be_v require_v prop._n iii_o to_o raise_v a_o perpendicular_a to_o the_o end_n of_o a_o line_n give_v here_o your_o give_a line_n be_v a_o b_o fix_v one_o foot_n of_o your_o compass_n in_o that_o end_n of_o your_o give_a line_n which_o you_o will_v raise_v your_o perpendicular_a from_o which_o be_v b_o and_o extend_v your_o compass_n to_o any_o convenient_a distance_n as_o in_o the_o example_n from_o b_o to_z r_z and_o describe_v the_o arch_n r_o c_o d_o continue_v the_o same_o distance_n in_o your_o compass_n and_o fix_v one_o foot_n in_o r_o extend_v the_o other_o upward_o to_o c_o then_o make_v c_o the_o centre_n describe_v the_o arch_n 1_o 2_o 3_o and_o beginning_n at_o r_n set_v off_o that_o distance_n three_o time_n upon_o that_o arch_n as_o 1_o 2_o 3_o from_o the_o place_n where_o your_o compass_n fall_v the_o three_o time_n which_o be_v 3_o draw_v a_o line_n to_o the_o place_n where_o you_o begin_v your_o work_n which_o be_v b_o and_o it_o be_v a_o perpendicular_a to_o your_o give_a line_n another_o way_n after_o you_o have_v describe_v the_o arch_n r_o c_o d_o with_o the_o same_o distance_n which_o describe_v it_o extend_v your_o compass_n from_o r_n to_o c_o from_o c_o with_o the_o same_o distance_n describe_v the_o arch_n u._fw-mi which_o be_v part_n of_o the_o arch_n r_o 1_o 2_o 3_o then_o extend_v the_o same_o distance_n again_o which_o fall_v at_o d_o and_o cross_v the_o arch_n before_o describe_v as_o n_n and_o from_o the_o place_n
of_o their_o intersection_n draw_v your_o line_n to_o the_o end_n of_o your_o give_a line_n and_o it_o shall_v be_v a_o perpendicular_a you_o see_v it_o be_v the_o same_o the_o other_o be_v prop._n iu._n to_o let_v fall_v a_o perpendicular_a from_o a_o point_n to_o a_o give_v line_n let_v the_o give_v line_n be_v d_o a_o the_o point_n from_o whence_o the_o perpendicular_a be_v to_o be_v let_v fall_n be_v at_o c._n from_o the_o point_n c_o draw_v a_o white_a line_n to_o the_o give_v line_n by_o guess_n as_o c_o a_o divide_v it_o into_o two_o equal_a part_n which_o be_v do_v at_o b_o then_o continue_v ½_n the_o line_n c_o a_o which_o be_v a_o b_o or_o c_o b_o in_o your_o compass_n and_o your_o compass_n fix_v one_o foot_n at_o b_o describe_v the_o arch_n c_o d_o and_o where_o it_o cut_v the_o give_v line_n there_o will_v your_o perpendicular_a fall_n from_o the_o give_v point_n for_o c_o d_o be_v perpendicular_a to_o the_o give_v line_n d_o a._n prop._n v._o to_o draw_v a_o line_n parallel_n to_o a_o line_n give_v let_v the_o give_v line_n be_v a_o s_n it_o be_v require_v to_o draw_v a_o line_n so_o that_o the_o two_o line_n may_v run_v at_o both_o end_n one_o by_o the_o other_o and_o never_o meet_v which_o be_v parallel_v one_o to_o the_o other_o open_v your_o compass_n to_o that_o extent_n as_o you_o will_v have_v the_o two_o line_n asunder_o and_o go_v towards_o one_o end_n of_o the_o give_v line_n as_o at_o s_n and_o describe_v the_o arch_n u._fw-mi and_o with_o the_o same_o distance_n come_v towards_o the_o other_o end_n as_o at_o a_o and_o describe_v another_o arch_n which_o be_v n_n and_o by_o the_o top_n of_o these_o two_o arch_n draw_v the_o line_n r_o o_fw-fr which_o be_v parallel_n to_o a_o s._n prop._n vi_o to_o draw_v a_o line_n parallel_n to_o a_o give_v line_n from_o any_o point_n assign_v let_v the_o give_v line_n be_v s_o l_o the_o point_n assign_v be_v a_o take_v the_o distance_n from_o s_n to_o a_o and_o carry_v it_o towards_o the_o other_o end_n of_o the_o give_v line_n as_o at_o l_o describe_v the_o arch_n n_z then_o take_v the_o distance_n from_o l_o to_o s_n and_o fix_v one_o foot_n of_o your_o compass_n in_o the_o give_v point_n a_o across_o the_o arch_n n_z with_o the_o arch_n o_o and_o by_o the_o place_n of_o their_o intersection_n and_o the_o point_n assign_v draw_v a_o line_n which_o shall_v be_v parallel_n to_o s_o l._n prop._n vii_o to_o make_v a_o square_a of_o a_o line_n give_v let_v the_o line_n give_v be_v a_o equal_a to_o which_o draw_v the_o side_n of_o the_o square_a b_o e_z and_o from_o one_o end_n of_o it_o raise_v a_o perpendicular_a and_o by_o it_o set_v off_o the_o length_n of_o a_o as_o here_o from_o e_z the_o perpendicular_a be_v raise_v and_o the_o length_n of_o a_o set_v upon_o it_o which_o be_v the_o side_n of_o the_o square_a e_o d_o continue_v the_o same_o distance_n in_o your_o compass_n and_o go_v to_o d_o and_o describe_v the_o arch_n 8_o carry_v the_o same_o distance_n to_o b_o and_o cross_v the_o arch_n 8_o with_o f_o and_o from_o the_o intersection_n of_o those_o two_o arch_n draw_v the_o side_n c_o d_o and_o c_o b_o which_o make_v the_o square_a bedc_n &_o this_o be_v a_o true_a square_n prop._n viii_o to_o make_v a_o square_a who_o length_n and_o breadth_n be_v give_v these_o sort_n of_o square_n be_v call_v geometrical_a square_n when_o but_o two_o side_n be_v equal_a namely_o the_o two_o long_a side_n or_o the_o two_o short_a side_n the_o angles_n be_v all_o equal_a namely_o right_a angle_n suppose_v the_o length_n of_o the_o square_n be_v a_o the_o breadth_n b_o i_o desire_v to_o make_v it_o first_o draw_v a_o line_n equal_a to_o a_o for_o the_o length_n of_o it_o as_o s_n v_n then_o from_o any_o end_n of_o that_o line_n raise_v a_o perpendicular_a as_o here_o from_o v._o and_o set_v off_o the_o line_n b_o for_o the_o breadth_n of_o it_o upon_o it_o which_o fall_v in_o the_o perpendicular_a line_n at_o l_o then_o take_v the_o length_n of_o a_o and_o describe_v the_o arch_n n_z fix_v your_o compass_n one_o foot_n in_o l_o then_o take_v b_o the_o breadth_n and_o cross_v that_o arch_n by_o another_o fix_v your_o compass_n one_o foot_n in_o s_n and_o draw_v l_o k_n and_o k_o s_o from_o the_o place_n of_o their_o intersection_n as_o you_o do_v in_o the_o other_o thus_o the_o two_o opposite_a side_n in_o this_o square_n be_v equal_a and_o the_o angle_n in_o both_o all_o equal_a for_o they_o be_v right_a angle_n prop._n ix_o to_o make_v a_o diamond_n figure_n of_o a_o line_n and_o a_o angle_n give_v a_o diamond_n figure_n be_v a_o figure_n of_o four_o equal_a side_n but_o the_o angle_n be_v two_o of_o they_o acute_a and_o two_o of_o they_o obtuse_a the_o acute_a angle_n be_v equal_a and_o the_o obtuse_a angle_n be_v equal_a one_o to_o another_o let_v a_o b_o be_v the_o give_v line_n c_o b_o the_o measure_n of_o the_o give_v angle_n a_o be_v the_o angular_a point_n first_o take_v the_o line_n ab_fw-la and_o draw_v a_o line_n of_o its_o length_n for_o one_o side_n of_o the_o figure_n namely_o 8_o 0_o then_o take_v the_o semidiameter_n of_o the_o arch_n c_o b_o which_o be_v c_z a_o and_o fix_v one_o foot_n of_o your_o compass_n in_o 8_o describe_z the_o arch_n s_o 0_o and_o take_v the_o arch_n b_o c_o and_o set_v it_o off_o from_o o_o to_o s_n then_o draw_v the_o line_n s_o 8_o equal_a to_o 8_o o_o this_o do_v keep_v the_o length_n of_o o_o 8_o in_o your_o compass_n and_o from_o s_n and_o o_o describe_v the_o arch_n n_o and_o it_o and_o draw_v the_o side_n o_o r_n and_o s_o r_n as_o you_o do_v in_o the_o other_o figure_n and_o thus_o s_n r_n be_v equal_a to_o 8_o o_o or_o r_o o_o be_v equal_a to_o 8_o s_n and_o the_o opposite_a angle_n also_o equal_a i_o forbear_v to_o show_v the_o reason_n of_o their_o be_v equal_a because_o it_o have_v be_v handle_v by_o other_o and_o indeed_o it_o be_v so_o plain_a that_o with_o a_o little_a consideration_n you_o may_v know_v it_o prop._n x._o to_o make_v a_o rhomboiade_n of_o two_o give_v side_n and_o a_o angle_n include_v a_o rhomboiades_n be_v a_o figure_n who_o opposite_a side_n and_o opposite_a angle_n be_v equal_a as_o a_o geometrical_a square_n be_v but_o in_o this_o they_o differ_v a_o rhomboiades_n have_v never_o a_o right_a angle_n but_o two_o obtuse_a and_o two_o acute_a whereas_o the_o other_o have_v all_o right_a angle_n it_o differ_v from_o the_o diamond_n figure_n also_o for_o in_o one_o the_o side_n be_v all_o equal_a and_o in_o this_o but_o two_o equal_a side_n i_o need_v not_o show_v the_o work_n of_o it_o because_o it_o differ_v not_o from_o a_o diamond_n figure_n only_o in_o take_v the_o two_o side_n apart_o to_o describe_v the_o arch_n at_o l_o ay_o suppose_v you_o may_v conceive_v how_o it_o be_v make_v by_o see_v this_o which_o be_v here_o make_v the_o give_v side_n and_o angle_n be_v s_o o_o n_n i_o have_v not_o set_v down_o the_o arch_n to_o measure_v the_o angle_n at_o o._n i_o suppose_v from_o what_o have_v be_v say_v you_o will_v conceive_v how_o that_o be_v prop._n xi_o to_o find_v the_o centre_n of_o a_o circle_n draw_v a_o line_n from_o side_n to_o side_n of_o the_o circle_n at_o a_o venture_n as_o a_o c_o and_o divide_v that_o line_n into_o two_o equal_a part_n by_o a_o perpendicular_a as_o be_v show_v before_o that_o perpendicular_a line_n draw_v through_o the_o circle_n from_o side_n to_o side_n as_o be_v u_z s_n and_o it_o shall_v be_v the_o diameter_n of_o the_o circle_n the_o half_a of_o which_o be_v the_o semidiameter_n or_o very_a centre_n ⊙_o it_o be_v possible_a to_o find_v the_o centre_n of_o a_o triangle_n after_o the_o same_o manner_n suppose_v the_o triangle_n who_o centre_n you_o will_v find_v be_v a_o c_o n_z divide_v any_o side_n into_o two_o equal_a part_n by_o a_o perpendicular_a and_o it_o will_v go_v through_o the_o centre_n of_o the_o triangle_n as_o the_o side_n a_o c_o be_v divide_v into_o two_o equal_a part_n by_o the_o perpendicular_a b_o s_n then_o i_o say_v b_o s_n go_v through_o the_o centre_n of_o this_o triangle_n but_o to_o find_v whereabouts_o in_o this_o perpendicular_a the_o centre_n of_o the_o triangle_n be_v i_o know_v by_o no_o other_o mean_n but_o by_o remove_v your_o compass_n in_o this_o line_n from_o place_n to_o place_n till_o you_o find_v it_o which_o be_v here_o find_v to_o be_v at_o u._n but_o this_o be_v but_o a_o botch_o way_n and_o with_o a_o little_a more_o labour_v you_o may_v find_v it_o at_o once_o therefore_o mind_v
the_o course_n for_o the_o course_n as_o the_o distance_n run_v a_o b_o 108_o mile_n comp_n arith_n 7,966576_o be_v to_o radius_fw-la  _fw-fr  _fw-fr so_o be_v the_o difference_n of_o latitude_n a_o d_o 90_o mile_n 1,954242_o to_o the_o course_n or_o rumb_n sine_fw-la comp_n b_o 33_o d._n 33_o m._n 9,920818_o for_o the_o departure_n as_o radius_fw-la  _fw-fr to_z the_o distance_n run_v a_o b_o 108_o mile_n 2,0334237_o so_o be_v sine_fw-la the_o course_n a_o 33_o deg_fw-la 33_o min._n 9,9424616_o to_o d_o b_o the_o diff_n of_o longi_fw-la here_o find_v to_o be_v 60_o mile_n 1,7758853_o you_o may_v ask_v the_o reason_n why_o the_o course_n be_v not_o 33_o deg_fw-la 45_o min._n as_o it_o be_v first_o give_v to_o be_v i_o answer_v because_o of_o the_o part_n of_o a_o unite_v that_o the_o difference_n of_o latitude_n be_v set_v down_o more_o than_o it_o shall_v be_v for_o if_o you_o observe_v i_o say_v it_o be_v almost_o 90_o mile_n when_o i_o find_v it_o before_o now_o the_o error_n be_v not_o worth_a mind_v in_o sail_v for_o the_o difference_n that_o the_o fraction_n cause_v be_v but_o 12._o min._n of_o a_o degree_n of_o a_o point_n of_o the_o compass_n which_o be_v no_o more_o than_o the_o 56_o the_o part_n of_o a_o point_n if_o you_o mind_v i_o have_v work_v the_o course_n thus_o find_v and_o it_o do_v not_o alter_v the_o difference_n of_o longitude_n or_o departure_n which_o shall_v be_v 60_o mile_n if_o i_o have_v work_v to_o the_o least_o fraction_n for_o the_o logarithm_n here_o be_v near_a to_o the_o logarithm_n of_o 60_o in_o the_o table_n indeed_o i_o work_v on_o purpose_n thus_o because_o those_o that_o i_o have_v teach_v when_o they_o can_v not_o find_v it_o come_v out_o right_a as_o the_o other_o be_v can_v not_o tell_v where_o the_o fault_n lie_v now_o this_o will_v direct_v they_o to_o know_v that_o some_o part_n of_o a_o unite_v miss_v in_o the_o find_n of_o the_o side_n may_v make_v some_o minute_n difference_n in_o the_o course_n but_o now_o here_o in_o the_o follow_a example_n we_o will_v work_v to_o a_o fraction_n namely_o to_o the_o 10_o the_o or_o 100_o the_o part_n of_o a_o unite_v in_o the_o side_n to_o show_v how_o it_o be_v do_v though_o in_o sail_v if_o you_o work_v to_o half_a a_o unite_v it_o be_v near_o enough_o or_o for_o any_o ordinary_a use_n and_o that_o the_o table_n give_v and_o most_o common_o to_o less_o departure_n from_o the_o meridian_n and_o distance_n run_v give_v to_o find_v the_o course_n and_o difference_n of_o latitude_n question_n v._n admit_v the_o departure_n from_o the_o meridian_n be_v the_o same_o that_o the_o difference_n of_o latitude_n be_v in_o the_o first_o question_n 60_o mile_n the_o distance_n run_v also_o 108_o mile_n i_o demand_v the_o course_n and_o difference_n of_o latitude_n let_v the_o latitude_n you_o set_v from_o be_v 50_o deg_fw-la here_o the_o course_n from_o the_o meridian_n will_v be_v the_o compliment_n of_o the_o course_n there_o and_o the_o difference_n of_o latitude_n will_v be_v the_o difference_n of_o longitude_n or_o departure_n from_o the_o meridian_n there_o but_o instead_o of_o take_v the_o logarithm_n answer_v to_o the_o number_n here_o take_v the_o logarithm_n answer_v to_o ten_o time_n the_o number_n and_o so_o you_o shall_v always_o have_v your_o side_n come_v out_o to_o the_o ten_o part_n of_o a_o unite_v i_o say_v it_o will_v come_v out_o in_o ten_o for_o you_o must_v divide_v by_o 10_o to_o bring_v it_o into_o mile_n after_o it_o be_v do_v now_o instead_o of_o multiply_v your_o give_a side_n by_o 10_o do_v but_o set_v a_o cipher_n more_o to_o the_o number_n and_o it_o be_v do_v as_o now_o for_o 60_o set_v down_o 600_o which_o be_v ten_o time_n 60_o for_o 108_o set_v down_o 1080_o which_o be_v ten_o time_n 108_o and_o thus_o for_o any_o number_n that_o will_v be_v comprehend_v in_o the_o radius_fw-la of_o the_o table_n proportion_n for_o the_o course_n as_o distance_n run_v give_v b_o a_o 1080_o ten_o comp_n arith_n 6,9665762_o be_v to_o radius_fw-la  _fw-fr so_n be_v the_o departure_n from_o the_o meridian_n 600_o ten_o d_o b_o 2,7781512_o to_o the_o sine_fw-la of_o the_o course_n a_o 33_o deg_fw-la 45_o min._n 9,7447274_o for_o the_o difference_n of_o latitude_n as_o radius_fw-la  _fw-fr to_z the_o distance_n run_v 1080_o ten_o b_o at_fw-fr 3,0334327_o so_o be_v the_o sine_fw-la comp_n of_o a_o 33_o deg_fw-la 45_o m._n 9,9198464_o to_o the_o diff._n of_o latitude_n in_o ten_o d_o a_o 898_o ten_o 2,9533708_o divide_v these_o ten_o by_o 10_o and_o it_o give_v the_o mile_n contain_v in_o the_o side_n d_o a_o the_o difference_n of_o latitude_n which_o be_v 89_o mile_n 8_o ten_o of_o a_o mile_n whereas_o before_o it_o be_v 90_o mile_n whis_fw-la be_v 2_o ten_o of_o a_o mile_n more_o now_o when_o you_o have_v the_o side_n thus_o find_v in_o ten_o for_o centism_n work_v for_o the_o angles_n and_o you_o will_v find_v they_o to_o come_v out_o roundly_o alike_o but_o if_o you_o desire_v more_o exactness_n work_v then_o in_o centism_n provide_v that_o the_o number_n in_o centism_n do_v not_o outrun_v the_o table_n a_o centism_n be_v the_o hundred_o part_n of_o a_o unite_v or_o of_o any_o thing_n and_o if_o i_o will_v put_v the_o number_n 60_o into_o centism_n i_o will_v set_v two_o cipher_n behind_o it_o and_o it_o be_v the_o same_o as_o before_o i_o multiply_v 60_o by_o 100_o for_o it_o be_v 6000_o the_o like_a be_v to_o be_v understand_v of_o any_o other_o number_n i_o will_v work_v to_o the_o near_a fraction_n for_o the_o angle_n here_o but_o i_o conceive_v it_o to_o be_v no_o way_n beneficial_a and_o therefore_o i_o will_v refer_v it_o to_o the_o work_n of_o the_o sphere_n where_o it_o be_v of_o ●uch_o more_o use_v this_o find_v the_o angle_n to_o a_o minute_n which_o be_v near_o enough_o it_o be_v like_o that_o some_o will_v be_v so_o curious_a that_o this_o way_n of_o find_v thing_n to_o the_o ten_o or_o hundred_o part_n of_o unit_fw-la will_v not_o suffice_v they_o but_o they_o will_v have_v the_o real_a number_n answer_v to_o the_o logarithm_n to_o the_o 1000_o the_o part_n of_o a_o unite_v because_o i_o find_v it_o of_o no_o real_a consideration_n or_o use_n in_o navigation_n i_o omit_v that_o here_o and_o desire_v such_o to_o look_v in_o book_n 1._o page_n 11._o of_o my_o father_n trigonometry_n and_o there_o it_o be_v plain_o show_v by_o the_o plain_a scale_n this_o question_n differ_v nothing_o from_o the_o other_o in_o its_o operation_n only_o as_o you_o work_v in_o the_o other_o from_o a_o north_n and_o south_n line_n here_o you_o work_v from_o a_o east_n and_o west_n line_n the_o work_n be_v this_o first_o draw_v a_o east_n and_o west_n line_n and_o from_o the_o west_n end_n of_o it_o set_v off_o your_o departure_n from_o the_o meridian_n which_o be_v from_o d_o to_z b_o 60_o mile_n from_o d_o raise_v the_o perpendicular_a d_o a_o which_o be_v a_o north_n and_o south_n line_n and_o draw_v it_o at_o length_n then_o take_v the_o distance_n run_v 108_o mile_n and_o fix_v one_o foot_n of_o your_o compass_n in_o b_o with_o the_o other_o cut_v the_o meridian_n line_n d_o a_o which_o it_o do_v in_o a._n draw_v the_o side_n a_o b_o then_o be_v b_o the_o place_n your_o ship_n be_v at_o after_o her_o sail_v a_o be_v the_o place_n of_o she_o set_v out_o a_o d_o be_v the_o difference_n of_o latitude_n and_o n_n o_o s_o be_v a_o arch_n of_o 60_o deg_fw-la from_o the_o angle_n d_o a_o b_o as_o a_o centre_n the_o measure_n of_o n_n o_o be_v the_o course_n and_o be_v perform_v as_o before_o in_o the_o last_o example_n and_o also_o of_o the_o difference_n of_o latitude_n as_o in_o other_o example_n the_o difference_n of_o latitude_n be_v d_o b_o 90_o mile_n the_o course_n be_v north_n 33_o deg_fw-la 45_o min._n easterly_n d_o a_o b_o or_o n_o o._n the_o plain_a scale_n of_o equal_a part_n will_v not_o resolve_v to_o the_o least_o fraction_n as_o the_o table_n will_v now_o to_o a_o ten_o part_n of_o a_o mile_n i_o confess_v a_o good_a diagonal_a scale_n will_v produce_v thing_n to_o a_o small_a fraction_n if_o you_o will_v know_v what_o latitude_n you_o be_v in_o divide_v 90_o by_o 60_o and_o the_o quotient_n will_v be_v degree_n namely_o 1_o deg_fw-la the_o remainder_n minute_n namely_o 30_o min._n now_o because_o your_o latitude_n be_v northerly_a and_o you_o have_v go_v to_o the_o northwards_o you_o have_v increase_v it_o 1_o deg_fw-la 30_o min._n so_o that_o you_o must_v add_v the_o difference_n of_o latitude_n to_o 50_o deg_fw-la and_o it_o make_v 51_o deg_fw-la 30_o min._n if_o you_o have_v go_v to_o the_o southwards_o you_o must_v
purpose_n to_o show_v the_o truth_n of_o the_o general_a rule_n the_o same_o way_n other_o question_n of_o this_o nature_n be_v wrought_v for_o the_o distance_n that_o the_o southermost_a ship_n sail_v a_o o._n as_o a_o o_o b_o sine_fw-la 67_o deg_fw-la 30_o min._n comp_n arith_n 0,034384_o be_v to_o a_o b_o 58_o mile_n  _fw-fr 2,763428_o so_o be_v sine_fw-la a_o b_o o_fw-fr 45_o deg_fw-la 00_o min._n  _fw-fr 9,849485_o to_z a_o o_fw-fr 44_o 4_o 10_o mile_n  _fw-fr 2,647297_o if_o you_o desire_v to_o find_v the_o difference_n of_o latitude_n between_o the_o two_o place_n b_o and_z a_o you_o have_v the_o distance_n a_o b_o and_o the_o course_n a_o b_o n_z give_v to_o find_v it_o or_o the_o longitude_n if_o you_o desire_v to_o find_v the_o latitude_n the_o ship_n be_v in_o when_o they_o meet_v let_v fall_v a_o perpendicular_a from_o o_o upon_o the_o south_n line_n which_o be_v b_o n_z and_o you_o have_v the_o distance_n o_o b_o given_z 58_o mile_n and_o the_o angle_n o_o b_o r_o 67_o deg_fw-la 30_o min._n to_o find_v it_o or_o the_o departure_n between_o the_o ship_n at_o their_o meeting_n and_o the_o head-land_n o_o r._n the_o like_a be_v to_o be_v understand_v of_o any_o other_o question_n of_o this_o nature_n two_o side_n and_o their_o contain_v angle_n be_v give_v to_o find_v the_o three_o side_n and_o the_o other_o angles_n question_n iii_o two_o ship_n set_v from_o one_o port_n and_o make_v a_o angle_n of_o 58_o deg_fw-la one_o sail_v n_n b_o e_o and_o the_o other_o sail_v n_o w_n 1_o deg_fw-la 45_o min._n westerly_n the_o eastermost_a ship_n sail_v 70_o mile_n and_o come_v to_o her_o port_n the_o westermost_a ship_n sail_v 89_o mile_n and_o come_v to_o her_o port_n i_o demand_v the_o distance_n of_o these_o port_n asunder_o and_o how_o they_o bear_v one_o from_o the_o other_o first_o draw_v the_o north_n and_o south_n line_n a_o s._n let_v a_o b_o represent_v the_o port_n you_o set_v from_o then_o from_o a_o as_o a_o centre_n describe_v the_o arch_n of_o 60_o deg_fw-la f_o s_o d_o from_o s_n to_o the_o eastward_n set_v off_o one_o point_n for_o n_z b_o e_z the_o eastermost_a ship_n course_n draw_v the_o line_n a_o d_o at_o length_n and_o set_v the_o first_o ship_n distance_n 70_o mile_n upon_o it_o which_o be_v a_o b_o then_z from_z a_o draw_z a_o n_z w_o line_n 4_o deg_fw-la 45_o min._n westerly_n and_o set_v off_o the_o westermost_a ship_n distance_n run_v upon_o that_o which_o be_v 89_o mile_n and_o it_o reach_v from_o a_o to_o c_o draw_v the_o line_n c_o b_o from_o the_o end_n of_o the_o other_o two_o side_n which_o be_v the_o distance_n between_o the_o port_n for_o the_o end_n of_o each_o ship_n run_v must_v be_v the_o port_n then_o be_v a_o c_o b_o the_o angle_n of_o the_o bear_n of_o the_o eastermost_a port_n from_o the_o westermost_a or_o a_o b_o c_o the_o bearing_n of_o the_o westermost_a port_n from_o the_o eastermost_a c_z a_o be_v a_o s_o e_o line_n 1_o deg_fw-la 45_o min._n easterly_a from_o c_o see_v how_o many_o point_n be_v contain_v between_o the_o c_o b_o and_o c_o a_o by_o the_o way_n that_o be_v show_v in_o the_o question_n before_o or_o because_o b_o a_o be_v n_o b_o e_o line_n from_z b_o you_o may_v find_v the_o point_n of_o the_o compass_n that_o b_o c_z runs_z upon_z from_z b_o you_o need_v find_v it_o but_o one_o of_o these_o way_n and_o the_o opposite_a point_n must_v be_v the_o other_o as_o i_o have_v show_v before_o then_o take_v the_o length_n of_o the_o side_n c_o b_o which_o be_v the_o distance_n of_o the_o port_n asunder_o i_o find_v their_o distance_n and_o bear_v to_o be_v as_o follow_v the_o distance_n between_o the_o two_o port_n be_v c_z b_o 79_o mile_n the_o bear_v of_o they_o be_v east_n 5_o deg_fw-la 35_o min._n northerly_a or_o west_n 5_o deg_fw-la 35_o min._n southerly_a by_o the_o table_n here_o be_v give_v the_o angle_n at_o a_o 58_o deg_fw-la 00_o min._n the_o side_n a_o b_o 70_o mile_n the_o side_n a_o c_o 89_o mile_n which_o be_v two_o side_n and_o their_o contain_v angle_n and_o here_o be_v require_v a_o c_o b_o or_o a_o b_o c_o and_o the_o side_n b_o c._n first_o for_o the_o angles_n for_o all_o question_n of_o this_o nature_n this_o rule_n be_v beneficial_a and_o will_v work_v your_o question_n note_v that_o in_o all_o plain_a triangle_n the_o sum_n of_o two_o side_n be_v in_o such_o proportion_n to_o their_o difference_n as_o the_o tangent_fw-la of_o the_o half_a sum_n of_o their_o opposite_a angle_n be_v to_o the_o tangent_fw-la of_o a_o angle_n which_o angle_n shall_v be_v the_o difference_n between_o the_o half_a sum_n and_o the_o angles_n so_o that_o if_o it_o be_v add_v to_o the_o half_a sum_n it_o shall_v make_v the_o great_a angle_n if_o it_o be_v subtract_v it_o shall_v make_v the_o lesser_a i_o have_v say_v something_o of_o this_o rule_n before_o which_o make_v i_o demonstrate_v it_o no_o far_o here_o proportion_n as_o the_o sum_n of_o a_o b_o and_o a_o c_o 159_o mile_n comp_n arith_n 6,79860_o be_v to_o their_o difference_n 19_o mile_n  _fw-fr 2,27875_o so_o be_v the_o tan._n of_o the_o half_a sum_n of_o b_o and_o c_o 61_o d._n they_z 10,25624_o to_o tangent_fw-la of_o a_o angle_n 12_o 10_o 9,33359_o this_o add_v to_o the_o half_a sum_n make_v the_o angle_n at_o b_o 73_o d._n 10_o m._n subtract_v it_o make_v the_o angle_n at_o c_o 48_o deg_fw-la 50_o min._n which_o be_v 4_o point_n and_z 3_o deg_fw-la 50_o min._n then_o say_v i_o if_o c_z a_o be_v a_o s_o e_o course_n 1_o deg_fw-la 45_o min._n easterly_a from_o c_o it_o be_v certain_a that_o c_o b_o must_v be_v 4_o point_n and_o 38_o deg_fw-la 50_o min._n from_o it_o which_o be_v east_n 5_o deg_fw-la 35_o min._n northerly_a the_o bear_n of_o the_o eastermost_a port_n from_o the_o westermost_a to_o find_v the_o half_a sum_n of_o the_o angle_n note_v that_o the_o three_o angle_n be_v 180_o d._n therefore_o if_o one_o of_o they_o be_v know_v subtract_v that_o from_o 180_o d._n and_o the_o remainder_n must_v be_v the_o other_o two_o now_o for_o their_o distance_n asunder_o c_o b._n as_o sine_fw-la a_o c_o b_o 48_o deg_fw-la 50_o min._n comp_n arith_n 0,123321_o be_v to_o b_o a_o 70_o mil._n  _fw-fr 2,845098_o so_o be_v sine_fw-la b_o a_o c_o 58_o deg_fw-la co_fw-la min._n  _fw-fr 9,928420_o to_o the_o distance_n of_o the_o port_n asunder_o c_z b_o 78_o 9_o 10_o 2,896839_o the_o three_o side_n of_o a_o triangle_n give_v to_o find_v the_o angles_n question_n iu._n two_o ship_n set_v from_o one_o place_n one_o sail_n n_n b_o e_z 70_o mile_n the_o other_o sail_n between_o the_o north_n and_o the_o west_n 89_o mile_n and_o at_o the_o end_n of_o their_o sail_v they_o be_v distant_a asunder_o 78_o mile_n 9_o 10_o of_o a_o mile_n i_o demand_v what_o course_n the_o westermost_a ship_n sail_v and_o how_o the_o two_o ship_n bear_v one_o from_o the_o other_o first_o draw_v the_o north_n and_o south_n line_n a_o e_o make_v a_o the_o place_n of_o the_o ship_n set_v out_o set_v off_o a_o n_o b_o e_o course_n upon_o the_o arch_n of_o 60_o deg_fw-la i_o e_o which_o be_v e_z d_o because_o the_o eastermost_a ship_n sail_v n_o b_o e_z likewise_z set_v off_o her_o distance_n which_o she_o sail_v upon_o that_o course_n which_o be_v a_o b_o 70_o mile_n then_o take_v the_o other_o ship_n distance_n which_o be_v 89_o mile_n and_o fix_v one_o foot_n of_o your_o compass_n in_o a_o with_o the_o other_o describe_v the_o arch_n at_o c_o last_o take_v the_o distance_n of_o the_o two_o ship_n asunder_o 78_o mile_n 9_o 10_o and_o fix_v one_o foot_n of_o your_o compass_n in_o b_o cross_z that_o arch_a and_o from_o the_o place_n of_o intersection_n draw_v the_o side_n of_o the_o triangle_n c_o a_n and_o c_o b_o then_o measure_v the_o angle_n from_o a_o arch_n of_o 60_o deg_fw-la as_o have_v be_v show_v before_o and_o to_o know_v they_o consider_v that_o a_o b_o be_v a_o n_o b_o e_o line_n so_o that_o a_o f_o will_v be_v upon_o a_o course_n so_o many_o point_n or_o degree_n and_o minute_n from_o a_o n_n b_o e_o as_o there_o be_v point_n or_o degree_n and_o minute_n in_o the_o arch_n d_o f._n second_o for_o the_o bear_n of_o the_o ship_n which_o be_v the_o angle_n at_o b_o consider_v that_o b_o a_o be_v a_o s_o b_o w_n line_n from_o b_o and_o then_o the_o degree_n and_o minute_n or_o point_v contain_v in_o the_o arch_n o_o r_n be_v so_o many_o degree_n and_o minute_n or_o point_n from_o the_o s_n b_o w._n i_o have_v resolve_v the_o demand_n which_o be_v as_o follow_v the_o course_n that_o the_o
which_o be_v a_o c_o and_o cut_v the_o the_o westermost_a head-land_n last_o draw_v out_o a_o south_n line_n from_o a_o which_o be_v a_o d_o and_o cut_v the_o island_n last_o a_o s_o and_o r_o s_o be_v the_o course_n that_o lead_v from_o the_o two_o station_n to_o the_o island_n and_o therefore_o s_o be_v the_o island_n for_o there_o they_o cut_v draw_v a_o line_n from_o every_o one_o of_o they_o as_o from_o s_o to_o o_o and_o from_o o_o to_o i_o from_o i_o to_o s_o and_o take_v the_o distance_n of_o they_o asunder_o then_o for_o their_o bear_v one_o from_o another_o draw_v a_o north_n and_o south_n line_n from_o each_o place_n and_o by_o a_o arch_n of_o 60_o deg_fw-la measure_v it_o as_o have_v be_v show_v by_o the_o table_n first_o i_o consider_v that_o i_o have_v the_o distance_n between_o the_o two_o station_n r_z a_o which_o be_v a_o side_n of_o the_o triangle_n a_o r_o ay_o a_o r_o o_o and_o a_o r_z s_z and_o in_o every_o of_o these_o triangle_n the_o question_n give_v i_o all_o the_o angle_n so_o that_o i_o can_v find_v my_o distance_n from_o my_o station_n to_o the_o headland_n after_o the_o form_n of_o this_o example_n which_o i_o shall_v do_v from_o each_o station_n to_o the_o eastermost_a for_o a_o i._o as_o sine_fw-la r_o i_o a_o 38_o deg_fw-la 45_o min._n comp_n arith_n 0,20347_o to_o a_o r_o 8_o mile_n  _fw-fr 1,90309_o so_o be_v sine_fw-la a_o r_o i_o 28_o deg_fw-la 45_o min._n  _fw-fr 9,68213_o to_o a_o i_o 6_o mile_n 2_o 10_o  _fw-fr 1,78869_o for_n r_o 1._o as_o r_n i_o a_o sine_fw-la 38_o deg_fw-la 45_o min._n comp_n arith_n 0,20347_o to_o a_o r_o 8_o mile_n  _fw-fr 1,90309_o so_o be_v r_o a_o i_o sine_fw-la its_o comp_n to_o 180_o d._n 67_o d._n 30_o m._n 9,96561_o to_o r_n i_o 11_o mile_n 8_o 10_o  _fw-fr 2,07217_o here_z i_o have_v find_v the_o distance_n from_o each_o station_n to_o the_o eastermost_a head-land_n next_o find_v the_o distance_n from_o the_o westermost_a head-land_n to_o the_o first_o station_n a_o o._n for_o a_o o._n as_o sine_fw-la r_o o_o a_o 6_o point_n or_o 67_o d._n 30_o m._n co_fw-la be_v 0,034384_o be_v to_o r_n a_o 8_o mile_n  _fw-fr 1,903090_o so_o be_v o_o r_o a_o 67_o deg_fw-la 30_o min._n sine_fw-la  _fw-fr 9,965615_o to_z a_o o_fw-fr 8_o mile_n  _fw-fr 1,903090_o thus_o you_o may_v find_v r_n o_o r_o s_o and_o a_o s_o for_o you_o have_v the_o same_o thing_n give_v namely_o r_o a_o and_o the_o angles_n here_o we_o have_v find_v the_o side_n a_o i_o and_o a_o o_o and_o because_o a_o i_o be_v upon_o a_o n_n n_o e_o course_n and_o a_o o_o upon_o a_o n_o w_n course_n the_o angles_n contain_v between_o o_o a_o i_o must_v be_v 6_o point_n or_o 67_o deg_fw-la 30_o min._n so_o that_o we_o have_v two_o side_n and_o their_o contain_v angle_n to_o find_v the_o angle_n of_o the_o bear_n of_o these_o headland_n a_o i_o o_o or_o a_o o_o i_o and_o the_o distance_n of_o they_o asunder_o o_o i_o this_o be_v wrought_v as_o the_o three_o question_n be_v thus_o as_o this_o be_v find_v so_o you_o may_v find_v the_o bear_n of_o the_o rest_n of_o the_o place_n one_o from_o the_o other_o if_o there_o be_v twenty_o of_o they_o for_o you_o have_v as_o much_o give_v in_o the_o other_o triangle_n but_o i_o leave_v it_o to_o your_o own_o practice_n also_o to_o make_v such_o experiment_n or_o to_o frame_v such_o question_n two_o side_n and_o a_o contain_v angle_n be_v give_v to_o find_v the_o three_o side_n and_o the_o course_n that_o each_o side_n run_v upon_o provide_v that_o no_o course_n be_v name_v only_o the_o half_a of_o the_o compass_n that_o you_o sail_v in_o or_o the_o quarter_n which_o be_v most_o common_o and_o the_o difference_n of_o latitude_n that_o be_v make_v between_o the_o extent_n of_o the_o two_o side_n question_n vi._n two_o ship_n make_v a_o angle_n of_o 50_o deg_fw-la the_o one_o sail_n between_o the_o south_n and_o the_o west_n 40_o league_n the_o other_o sail_n between_o the_o south_n and_o the_o east_n 50_o league_n and_o at_o the_o end_n of_o their_o sail_v the_o eastermost_a ship_n be_v more_o southerly_a than_o the_o westermost_a so_o that_o they_o differ_v their_o latitude_n 15_o league_n i_o demand_v each_o ship_n course_n and_o their_o distance_n asunder_o the_o place_n the_o ship_n set_v from_o i_o make_v to_o be_v b_o take_v off_o 60_o deg_fw-la from_o your_o scale_n and_o describe_v a_o arch_n make_v b_o your_o centre_n as_o the_o arch_n t_o r_o then_o i_o take_v 50_o deg_fw-la from_o my_o scale_n and_o set_v off_o somewhere_o upon_o that_o arch_n namely_o from_o t_o to_o r_n and_o from_o b_o upon_o the_o course_n that_o lead_v to_o t_o i_o set_v off_o 40_o league_n my_o westermost_a ship_n distance_n and_o so_o conclude_v my_o westermost_a ship_n be_v at_o c_o then_o from_o b_o upon_o the_o course_n that_o lead_v to_o r_n i_o set_v off_o my_o my_o eastermost_a ship_n distance_n run_v which_o be_v 50_o league_n from_o b_o to_o a._n last_o draw_v a_o c_o which_o be_v the_o distance_n of_o the_o ship_n at_o the_o end_n of_o their_o sail_v then_o because_o the_o question_n say_v the_o eastermost_a ship_n be_v 15_o league_n more_o southerly_a in_o latitude_n than_o the_o westermost_a i_o will_v take_v 15_o league_n from_o the_o scale_n of_o equal_a part_n and_o fix_v one_o foot_n of_o my_o compass_n in_o a_o and_o with_o the_o other_o describe_v the_o arch_n i_o s_o n_n and_o draw_v the_o line_n from_o c_o by_o the_o upper_a edge_n of_o the_o circle_n at_o length_n which_o must_v be_v a_o east_n and_o west_n line_n because_o a_o be_v just_a 15_o league_n to_o the_o southwards_o of_o the_o near_a part_n of_o it_o then_o let_v fall_v a_o perpendicular_a from_o b_o upon_z c_z n_z and_o extend_v it_o as_o far_o as_o the_o arch_n of_o 60_o deg_fw-la which_o you_o first_o describe_v t_o r_n and_o it_o cut_v in_o o_o this_o must_v be_v a_o south_n line_n from_o b_o namely_o b_o s_n then_o measure_v upon_o the_o arch_n of_o 60_o deg_fw-la from_o o_o to_o r_n for_o the_o eastermost_a ship_n course_n and_o from_o o_o to_o t_n for_o the_o westermost_a ship_n course_n as_o you_o do_v in_o any_o other_o question_n i_o find_v they_o here_o as_o follow_v the_o eastermost_a ship_n sail_v s_n b_o e_o 4._o d._n 40_o m._n easterly_n the_o westermost_a ship_n sail_v s_o w_n 11_o deg_fw-la southerly_a the_o distance_n asunder_o be_v c_o a_o 39_o league_n by_o proportion_n here_o be_v the_o two_o side_n b_o at_fw-fr 50_o league_n and_o b_o c_o 40_o with_o the_o contain_v angle_n c_o b_o a_o give_a to_o find_v the_o angle_n a_o c_o b_o and_o b_o a_o c_o therefore_o for_o the_o angles_n as_o b_o a_n and_o b_o c_o 90_o league_n comp_n arith_n 7,045757_o be_v to_o their_o difference_n 10_o league_n  _fw-fr 2,000000_o so_o be_v tan._n half_a the_o angle_n b_o at_fw-fr c_o and_o b_o c_o a_o 65_o d._n 0_o m._n 10,331327_o to_o tang._n their_o difference_n 13_o deg_fw-la 24_o min._n 9,377084_o which_o add_v to_o the_o half_a sum_n 65_o deg_fw-la be_v 78_o deg_fw-la 24_o min._n the_o angle_n a_o c_o b_o subtract_v from_o 65_o deg_fw-la it_o make_v 51_o deg_fw-la 36_o min._n the_o angle_n c_o a_o b._n then_o for_o the_o ship_n distance_n asunder_o c_o a._n as_o sine_fw-la a_o c_o b_o 78_o deg_fw-la 24_o min._n comp_n arith_n 0,008962_o be_v to_o b_o a_o 50_o league_n  _fw-fr 2,698970_o so_o be_v sine_fw-la c_o b_o a_o 50_o deg_fw-la 0_o min._n  _fw-fr 9,884254_o to_z c_o a_o 39_o deg_fw-la 1_o 10_o league_n  _fw-fr 2,592186_o here_z we_o have_v find_v the_o require_v angle_n and_o the_o require_a side_n but_o we_o know_v not_o what_o point_n of_o the_o compass_n either_o of_o the_o side_n that_o bound_n the_o triangle_n run_v upon_o but_o this_o which_o we_o have_v find_v make_v way_n to_o it_o for_o now_o we_o know_v the_o side_n a_o c_o to_o be_v 39_o 1_o 10_o league_n and_o if_o you_o let_v fall_v a_o perpendicular_a from_o a_o upon_o the_o east_n and_o west_n line_n c_o n_n it_o cut_v it_o in_o s_n than_o a_o s_n be_v know_v to_o be_v the_o semidiameter_n of_o the_o arch_n i_o n_o s_o you_o 15_o league_n so_o that_o in_o the_o triangle_n a_o s_o c_o right_o angle_v at_o s_n we_o have_v a_o s_n 15_o league_n give_v and_o a_o c_o 39_o 1_o 10_o league_n to_o find_v a_o c_o s_o or_o c_o a_o s_o we_o will_v find_v a_o c_o s_o as_o a_o c_o 39_o 1_o 10_o league_n 7,407823_o be_v to_o radius_fw-la or_o a_o s_o c_o 90_o deg_fw-la  _fw-fr so_n be_v a_o s_n 15_o league_n 2,176091_o to_o sine_fw-la a_o c_o s_o 22_o deg_fw-la