Selected quad for the lemma: book_n

Word A Word B Word C Word D Occurrence Frequency Band MI MI Band Prominent
book_n great_a know_v see_v 2,770 5 3.0783 3 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

and_o the_o remainder_n be_v the_o latitude_n northerly_a if_o the_o sun_n or_o star_n be_v to_o the_o northwards_o and_o have_v south_n declination_n add_v the_o declination_n to_o the_o compliment_n of_o the_o meridian_n altitude_n and_o the_o remainder_n be_v the_o latitude_n southerly_a if_o you_o observe_v the_o sun_n when_o he_o have_v no_o declination_n the_o compliment_n of_o his_o meridian_n altitude_n be_v the_o latitude_n if_o you_o observe_v the_o sun_n or_o star_n in_o the_o zenith_n the_o declination_n be_v the_o latitude_n and_o thus_o much_o for_o the_o find_v the_o latitude_n by_o the_o sun_n meridian_n altitude_n or_o the_o compliment_n thereof_o also_o for_o such_o star_n as_o have_v their_o declination_n from_o the_o aequinoctial_a for_o those_o star_n who_o declination_n be_v their_o distance_n from_o the_o pole_n if_o you_o observe_v they_o under_o the_o pole_n add_v the_o declination_n to_o the_o meridian_n altitude_n if_o above_o the_o pole_n subtract_v their_o declination_n and_o the_o remainder_n be_v the_o latitude_n and_o thus_o much_o for_o find_v the_o latitude_n by_o the_o declination_n and_o meridian_n altitude_n of_o the_o sun_n or_o star_n or_o the_o compliment_n of_o the_o meridian_n altitude_n of_o the_o sun_n or_o such_o star_n who_o declination_n be_v their_o distance_n from_o the_o aequator_fw-la of_o astronomy_n i_o have_v thought_n to_o have_v handle_v the_o doctrine_n of_o spherical_a triangle_n full_o but_o if_o i_o shall_v i_o may_v do_v that_o which_o my_o father_n have_v do_v before_o better_o than_o i_o can_v the_o thing_n which_o induce_v i_o to_o this_o thought_n be_v that_o i_o may_v make_v the_o application_n of_o it_o to_o great_a circle_n sail_v but_o because_o that_o way_n of_o sail_v though_o the_o best_a be_v seldom_o use_v i_o have_v not_o set_v it_o in_o my_o book_n if_o you_o desire_v to_o know_v it_o you_o may_v see_v the_o way_n of_o it_o in_o my_o father_n work_n as_o plain_o as_o i_o think_v it_o can_v possible_o be_v demonstrate_v by_o any_o man_n i_o have_v think_v good_a to_o make_v use_n of_o his_o fundamental_a axiom_n to_o show_v how_o to_o resolve_v those_o question_n in_o the_o sphere_n which_o i_o have_v wrought_v in_o this_o book_n by_o the_o plain_a scale_n though_o i_o confess_v for_o amplitude_n of_o rise_v and_o set_v and_o azimuth_n at_o certain_a give_v hour_n with_o divers_a other_o thing_n that_o be_v chief_o to_o be_v mind_v at_o sea_n for_o the_o variation_n of_o the_o compass_n and_o other_o necessary_a use_n i_o hold_v the_o plain_a scale_n to_o be_v true_a enough_o but_o that_o i_o conceive_v not_o to_o be_v so_o satisfactory_a to_o all_o man_n and_o therefore_o i_o have_v do_v this_o the_o axiom_n be_v this_o the_o sine_fw-la of_o a_o middle_a part_n with_o radius_fw-la be_v equal_a to_o the_o tangent_n of_o the_o extreme_n adjacent_a or_o to_o the_o sine_fw-la compliment_n of_o the_o opposite_a extreme_n now_o for_o brevity_n sake_n it_o be_v common_a to_o use_v these_o character_n for_o this_o character_n +_o signify_v with_o radius_fw-la or_o more_o or_o one_o side_n or_o angle_n with_o a_o another_o for_o equal_a =_o for_o less_o −_o so_o that_o the_o general_a rule_n or_o fundamental_a observation_n which_o we_o purpose_v to_o use_v may_v be_v express_v thus_o the_o sine_fw-la of_o the_o middle_a part_n +_o radius_fw-la be_v =_o to_o the_o tangent_n of_o the_o extreme_n adjacent_a or_o to_o the_o sine_fw-la compliment_n of_o the_o opposite_a extreme_n this_o rule_n be_v right_o understand_v give_v such_o full_a direction_n that_o a_o man_n may_v resolve_v any_o question_n in_o the_o sphere_n by_o it_o which_o make_v i_o judge_v those_o culpable_a of_o a_o fault_n that_o accuse_v he_o of_o prolixity_n that_o make_v it_o in_o so_o brief_a a_o manner_n i_o will_v define_v a_o spherical_a right_a angle_a triangle_n and_o so_o come_v to_o the_o use_n of_o this_o axiom_n if_o you_o have_v two_o thing_n give_v beside_o the_o right_a angle_n to_o find_v a_o three_o as_o you_o always_o have_v the_o first_o thing_n that_o you_o be_v to_o consider_v be_v whether_o it_o be_v a_o adjacent_a or_o a_o opposite_a extreme_a that_o be_v a_o adjacent_a extreme_a when_o the_o give_v part_n and_o the_o require_a part_n join_v for_o example_n in_o adjacent_a extreme_n suppose_v the_o give_v thing_n be_v a_o c_o and_o a_o c_o b_o and_o the_o require_a thing_n be_v c_o b_o here_o you_o see_v that_o a_o c_o be_v join_v to_o c_o and_o so_o be_v c_o b_o therefore_o they_o all_o join_v and_o c_o be_v the_o middle_a part_n for_o it_o be_v between_o b_o c_o and_o c_o a_o the_o extreme_n that_o be_v a_o opposite_a extreme_a that_o have_v two_o thing_n lie_v together_o and_o the_o three_o thing_n lie_v alone_o and_o that_o part_n which_o lie_v alone_o be_v the_o middle_a part_n example_n suppose_v the_o three_o thing_n namely_o the_o two_o give_v thing_n and_o the_o require_a thing_n be_v b_o c_o the_o angle_z at_o a_o and_z c_z a_o then_o i_o conclude_v that_o c_o b_o be_v the_o middle_a part_n and_o that_o it_o be_v a_o opposite_a extreme_a because_o it_o be_v alone_o and_o the_o other_o thing_n namely_o c_o a_n and_o a_o be_v together_o if_o you_o ask_v how_o this_o can_v be_v thus_o a_o c_o b_o be_v between_o c_o b_o and_o c_o a_o and_z a_o b_o be_v between_z c_z b_o and_z the_o angle_z at_o a_o which_o prove_v that_o b_o c_o be_v alone_a and_o a_o c_o and_o the_o angle_n at_o a_o join_v then_o be_v a_o c_o and_o the_o angle_n at_o a_o also_o the_o two_o extreme_n pray_v observe_v this_o for_o a_o rule_n that_o the_o right_a angle_n never_o part_v any_o thing_n for_o if_o c_o b_o the_o side_n b_o a_o and_o the_o angle_n at_o a_o be_v the_o three_o thing_n it_o be_v nevertheless_o a_o adjacent_a extreme_a for_o though_o the_o right_a angle_n be_v between_o the_o side_n b_o a_n and_o b_o c_o yet_o do_v not_o count_v that_o it_o part_v they_o as_o another_o angle_n will_v so_o as_o to_o make_v it_o a_o opposite_a extreme_a or_o if_o the_o give_v and_o require_v thing_n be_v the_o three_o side_n you_o may_v judge_v b_o a_o the_o middle_a part_n because_o it_o be_v alone_o for_o the_o angle_n at_o a_o part_n it_o from_o a_o c_o and_o the_o right_a angle_n at_o b_o be_v between_z b_o c_z and_z it_o now_o never_o count_v that_o the_o right_a angle_n at_o b_o parts_z b_o c_z from_z b_o a_o but_o that_o they_o lie_v together_o and_o a_o c_o lie_v alone_o be_v part_v by_o the_o angle_n c_o from_o c_o b_o and_o by_o the_o angle_n a_o from_o a_o b_o and_o so_o be_v the_o middle_a part_n these_o be_v the_o first_o consideration_n after_o you_o have_v a_o question_v set_v you_o that_o you_o ought_v to_o mind_n without_o regard_v which_o of_o the_o three_o thing_n be_v give_v or_o require_v we_o now_o come_v to_o apply_v this_o general_a axiom_n suppose_v i_o have_v the_o side_n b_o c_o given_z the_o angle_z at_z c_o and_o that_o the_o side_n c_o at_fw-fr be_v require_v from_o what_o have_v be_v say_v i_o know_v c_o to_o be_v the_o middle_a part_n and_o that_o it_o be_v a_o adjacent_a extreme_a i_o will_v apply_v the_o rule_n thus_o it_o say_v the_o sine_fw-la of_o the_o middle_a part_n which_o intimate_v that_o the_o middle_a part_n must_v be_v always_o so_o call_v as_o you_o may_v take_v the_o sine_fw-la of_o it_o now_o because_o it_o be_v note_v to_o be_v a_o compliment_n you_o must_v call_v it_o sine_fw-la compliment_n to_o find_v the_o sine_fw-la of_o it_o this_o do_v i_o will_v go_v to_o one_o of_o the_o extreme_n and_o consider_v what_o the_o axiom_n say_v for_o that_o admit_v it_o be_v the_o extreme_a c_o b_o it_o say_v the_o sine_fw-la of_o the_o middle_a part_n with_o radius_fw-la be_v equal_a to_o the_o tangent_n of_o the_o extreme_n adjacent_a from_o whence_o i_o conclude_v that_o i_o must_v call_v c_o b_o so_o as_o that_o i_o may_v have_v the_o tangent_fw-la of_o it_o now_o c_z b_o be_v not_o note_v to_o be_v a_o compliment_n and_o therefore_o i_o call_v it_o tangent_fw-la c_o b._n last_o for_o the_o other_o extreme_a c_o a_n for_o the_o same_o reason_n i_o must_v call_v it_o so_o as_o that_o i_o may_v take_v the_o tangent_fw-la of_o it_o and_o because_o it_o be_v note_v to_o be_v a_o compliment_n therefore_o i_o call_v it_o tangent_fw-la compliment_n to_o have_v the_o tangent_fw-la of_o it_o now_o as_o i_o go_v along_o thus_o in_o my_o thought_n i_o will_v set_v it_o down_o as_o i_o consider_v it_o and_o it_o frame_v this_o rule_n sine_fw-la come_v c_o +_o radius_fw-la be_v =_o to_o tang._n c_o b_o +_o tang._n comp_n c_o a._n my_o question_n be_v bring_v into_o this_o order_n i_o will_v
the_o course_n a_o 56_o deg_fw-la 15_o min._n 9,824897_o to_o the_o difference_n of_o latitude_n in_o part_n 1968_o part_n 3,293977_o this_o do_v i_o look_v in_o the_o table_n against_o the_o latitude_n that_o be_v give_v i_o 23_o deg_fw-la 9_o min._n and_o the_o meridional_a part_n answer_v to_o it_o be_v 1428_o to_o which_o i_o add_v the_o part_n here_o find_v 1968_o because_o the_o latitude_n be_v increase_v  _fw-fr and_o there_o come_v forth_o 3396_o i_o look_v for_o that_o in_o the_o table_n and_o i_o find_v answer_v to_o it_o the_o latitude_n of_o 49_o deg_fw-la 9_o min._n which_o be_v the_o latitude_n of_o the_o unknown_a place_n b._n for_o the_o distance_n between_o these_o place_n i_o subtract_v the_o latitude_n of_o a_o 23_o deg_fw-la 9_o min._n from_o the_o latitude_n of_o b_o now_o find_v 49_o deg_fw-la 9_o min._n and_o the_o remainder_n be_v the_o difference_n of_o latitude_n between_o they_o 26_o deg_fw-la 0_o min._n which_o convert_v into_o mile_n be_v 1560_o mile_n then_o say_v as_o sine_fw-la comp_n 56_o deg_fw-la 15_o min._n a_o comp_n arith_n 0,255260_o be_v to_o a_o b_o 1560_o mile_n 3,193124_o so_o be_v radius_fw-la  _fw-fr to_z the_o distance_n run_v a_o c_o 2808_o mile_n 3,448384_o i_o have_v thought_n to_o have_v show_v the_o difference_n between_o plano_n and_o mercator_n in_o a_o example_n but_o my_o father_n have_v do_v it_o in_o page_n 170_o of_o his_o book_n if_o you_o be_v desirous_a to_o know_v great_a circle_n sail_v you_o may_v learn_v it_o from_o his_o book_n for_o he_o have_v treat_v of_o it_o large_o and_o indeed_o it_o be_v the_o most_o necessary_a way_n of_o sail_v that_o be_v for_o a_o man_n cut_v his_o way_n much_o short_a in_o many_o run_v than_o the_o usual_a way_n that_o man_n go_v do_v if_o you_o desire_v to_o know_v in_o any_o latitude_n the_o number_n of_o mile_n contain_v in_o any_o number_n of_o degree_n of_o longitude_n convert_v your_o degree_n and_o minute_n of_o longitude_n into_o minute_n and_o say_v as_o radius_fw-la to_o sine_fw-la comp_n that_o latitude_n so_o be_v the_o difference_n of_o longitude_n in_o minute_n to_o the_o difference_n of_o longitude_n in_o mile_n if_o you_o desire_v to_o convert_v mile_n of_o departure_n in_o any_o parallel_n into_o degree_n say_v as_o sine_fw-la comp_n the_o latitude_n or_o the_o parallel_n do_v from_o the_o pole_n be_v to_o radius_fw-la so_o be_v the_o departure_n in_o mile_n to_o the_o difference_n of_o longitude_n in_o minute_n divide_v that_o by_o 60_o and_o the_o quotient_a with_o the_o remainder_n be_v the_o degree_n and_o minute_n contain_v in_o that_o departure_n of_o the_o longitude_n and_o latitude_n of_o place_n before_o i_o come_v to_o show_v how_o to_o keep_v a_o reckon_a by_o mercator_n way_n i_o will_v set_v down_o a_o catalogue_n of_o the_o latitude_n and_o longitude_n of_o some_o place_n in_o degree_n and_o minute_n the_o longitude_n take_v from_o flower_n and_o corve_n place_n name_n latitude_n longitude_n d._n m._n d._n m._n the_o tassel_n north_n 53_o 00_o 35_o 10_o the_o maze_n north_n 52_o 00_o 34_o 29_o the_o willing_a 51_o 30_o 33_o 53_o isle_n of_o wight_n 50_o 24_o 29_o 16_o dover_n 51_o 05_o 32_o 04_o portland_n 50_o 24_o 28_o 16_o the_o start_v 50_o 07_o 27_o 07_o the_o lizard_n 50_o 00_o 25_o 20_o silly_a 50_o 04_o 24_o 00_o vshin_n 48_o 30_o 25_o 30_o cape_n finistre_fw-fr 43_o 08_o 20_o 30_o the_o rock_n of_o lisbon_n 38_o 52_o 20_o 47_o cape_n st._n vincent_n 37_o 00_o 21_o 29_o gilbraltar_n 36_o 01_o 24_o 39_o cape_n blanco_n 20_o 32_o 13_o 09_o cape_n de_fw-fr verde_n 14_o 31_o 13_o 10_o serrat_fw-la lion_n 08_o 00_o 17_o 53_o south_n side_n of_o st._n ann_n 06_o 42_o 17_o 32_o cape_n the_o palmose_n 04_o 00_o 24_o 30_o cape_n terras_fw-la pantas_n 04_o 06_o 30_o 30_o cape_n formosa_fw-la 04_o 04_o 39_o 15_o river_n comrans_n 03_o 19_o 42_o 32_o nethermost_o of_o the_o isle_n farmandosa_n 03_o 26_o 41_o 56_o middle_a of_o the_o isle_n of_o thoma_n 00_o 00_o 40_o 00_o cape_n de_fw-fr lopez_n 10_o 00_o 42_o 01_o cape_n negro_n south_n 16_o 00_o 4_o 521_o cape_n bonesperanza_n south_n 34_o 25_o 52_o 30_o north_n end_n of_o martinass_n 23_o 02_o 17_o 16_o isle_n of_o picos_fw-la 23_o 01_o 17_o 16_o isle_n terrestinam_fw-la 37_o 15_o 23_o 31_o y_o digon_n aluarum_fw-la 38_o 54_o 26_o 45_o ascension_n 08_o 21_o 20_o 16_o st._n ellna_n 16_o 00_o 28_o 21_o cape_n agulhas_n 35_o 00_o 54_o 00_o pomte_n primero_n 32_o 25_o 63_o 06_o cape_n corentas_fw-la 24_o 00_o 69_o 50_o mosambique_n 15_o 10_o 75_o 32_o pemba_fw-la 04_o 41_o 77_o 02_o cape_n derfue_n 10_o 20_o 87_o 40_o cape_n guardafa_n 12_o 01_o 87_o 41_o cape_n de_fw-la raffas_fw-la 22_o 10_o 97_o 29_o serrat_fw-la 21_o 01_o 108_o 35_o goa_n 15_o 40_o 109_o 22_o cape_n cormoram_n 07_o 53_o 112_o 20_o point_n gada_n 06_o 02_o 115_o 00_o northwest_o point_n of_o sumatra_n 05_o 30_o 129_o 02_o the_o flat_a point_n 05_o 51_o 137_o 26_o crequeta_n in_o sinda_n 06_o 02_o 137_o 21_o east_n end_n of_o st._n brandon_n 17_o 15_o 102_o 05_o east_n end_n of_o diego_n roderigo_n 19_o 24_o 99_o 22_o southeast_n harbour_n of_o maurice_n 20_o 11_o 92_o 54_o southeast_n end_n of_o mastorbos_n 20_o 16_o 90_o 22_o cape_n de_fw-fr roman_n 25_o 02_o 83_o 24_o isle_n st._n paulo_n 38_o 30_o 108_o 40_o isle_n st._n maria_n dogusto_n 19_o 02_o 13_o 00_o assemaon_n 20_o 01_o 02_o 42_o east_n end_n of_o the_o foul_a ground_n 18_o 21_o 00_o 00_o isle_n de_fw-la lobas_fw-la at_o rio_n de_fw-fr plato_n 35_o 25_o 343_o 02_o cape_n st._n thoma_n 22_o 21_o 358_o 01_o bay_n todas_n sanctos_fw-la 13_o 15_o 357_o 41_o cape_n st._n augustin_n 08_o 41_o 001_o 12_o isle_n fernando_n 03_o 45_o 005_o 14_o vizia_n 03_o 30_o 003_o 01_o venedo_n st._n paulo_n 01_o 52_o 006_o 25_o the_o west-indies_n barbados_n north_n 13_o 12_o 327_o 42_o ditiatha_fw-mi north_n 16_o 18_o 326_o 32_o st._n christopher_n 17_o 20_o 324_o 52_o tobago_n 11_o 16_o 327_o 41_o east_n side_n of_o trinidado_n 11_o 37_o 326_o 41_o mattinino_n 14_o 48_o 325_o 30_o st._n vincent_n 13_o 10_o 326_o 00_o new_a point_n of_o portorica_n 19_o 02_o 319_o 22_o altanalla_n near_o hispaniola_n 17_o 26_o 313_o 52_o cape_n tibarune_v in_o hispaniola_n 18_o 14_o 310_o 22_o west_n end_n of_o margareta_n 11_o 11_o 321_o 51_o cartagena_n 10_o 49_o 310_o 41_o island_n guanatho_n 16_o 51_o 297_o 11_o west_n end_n of_o cuba_n 22_o 00_o 298_o 44_o cape_n florida_n 25_o 20_o 302_o 35_o bermuda_n island_n the_o body_n of_o it_o 32_o 20_o 325_o 15_o island_n of_o sable_a 43_o 40_o 334_o 02_o cape_n race_n at_o newfoundland_n 46_o 32_o 339_o 32_o cape_n sable_a 43_o 41_o 327_o 21_o cape_n cod_n 41_o 20_o 322_o 01_o cape_n hattarass_n 35_o 40_o 314_o 11_o cape_n henry_n 37_o 00_o 344_o 00_o cape_n charles_n 37_o 20_o 313_o 55_o west_n side_n of_o corve_v 40_o 00_o 000_o 00_o east_n side_n of_o flower_n 37_o 20_o 000_o 00_o the_o road_n of_o fial_n 38_o 50_o 002_o 17_o west_n end_n of_o fial_n 38_o 41_o 002_o 48_o west_n end_n of_o st._n george_n 39_o 00_o 003_o 00_o gratiosa_fw-la 39_o 15_o 003_o 20_o west_n end_n of_o tercera_n north_n 39_o 00_o 03_o 40_o east_n end_n of_o michael_n north_n 38_o 05_o 06_o 10_o east_n end_n of_o maria_n 37_o 00_o 06_o 01_o east_n point_n of_o porto_n santo_n 33_o 05_o 15_o 26_o west_n end_n of_o madera_n 32_o 20_o 13_o 48_o east_n side_n of_o palma_n 28_o 45_o 12_o 48_o north_n point_n of_o gomera_n 28_o 10_o 13_o 20_o north_n side_n of_o ferro_fw-la 27_o 40_o 12_o 30_o north-east_n side_n of_o gam_a 28_o 36_o 14_o 31_o north-east_n point_n of_o grand_fw-fr canaria_n 28_o 20_o 15_o 10_o east_n end_n of_o fortaventure_n 20_o 21_o 17_o 18_o east_n end_n of_o lausareta_n 28_o 53_o 17_o 21_o east_n end_n of_o gratiosa_fw-la 29_o 00_o 17_o 29_o east_n end_n of_o st._n antonio_n 17_o 21_o 04_o 20_o east_n end_n of_o st._n vincent_n 17_o 12_o 05_o 00_o east_n point_n of_o st._n lucia_n 16_o 54_o 05_o 25_o east_n point_n of_o sal_fw-la 16_o 54_o 07_o 30_o east_n side_n of_o bonavista_n 16_o 10_o 08_o 00_o east_n side_n of_o the_o isle_n of_o may_n 15_o 00_o 07_o 30_o east_n side_n of_o st._n jago_n 15_o 10_o 07_o 02_o fogo_n 14_o 36_o 06_o 00_o bramma_n island_n 14_o 24_o 05_o 38_o the_o longitude_n and_o latitudess_n of_o these_o place_n here_o insert_v be_v some_o of_o they_o my_o own_o experience_n and_o the_o rest_n account_n of_o able_a man_n concern_v the_o forego_n table_n he_o that_o be_v bind_v to_o any_o place_n the_o first_o thing_n that_o he_o ought_v to_o consider_v be_v the_o latitude_n of_o each_o place_n and_o the_o
minute_n admit_v i_o have_v these_o several_a number_n of_o hour_n and_o minute_n to_o cast_v up_o in_o one_o sum_n namely_o then_o because_o 60_o minute_n be_v one_o hour_n i_o do_v just_a as_o i_o do_v before_o be_v sure_a in_o the_o place_n of_o minute_n set_v never_o above_o 60_o for_o if_o you_o do_v it_o be_v more_o than_o a_o degree_n ho._n min._n 41_o 07_o 22_o 11_o 12_o 29_o 18_o 32_o 15_o 15_o 109_o 34_o the_o same_o be_v to_o be_v do_v in_o addition_n of_o money_n only_o remember_v in_o the_o place_n of_o penny_n to_o go_v no_o far_o than_o 12_o because_o twelve_o penny_n make_v a_o shilling_n as_o you_o do_v here_o to_o 60_o because_o 60′_n make_v a_o degree_n and_o in_o shilling_n no_o far_o than_o twenty_o because_o twenty_o shilling_n make_v a_o pound_n and_o thus_o in_o any_o thing_n else_o provide_v that_o you_o know_v how_o many_o of_o the_o one_o make_v one_o of_o the_o other_o but_o for_o such_o thing_n they_o be_v not_o so_o necessary_a to_o the_o thing_n we_o purpose_v to_o handle_v and_o therefore_o i_o refer_v you_o to_o mr._n hodder_n book_n it_o be_v of_o small_a price_n and_o much_o matter_n and_o likewise_o very_o plain_a subtraction_n subtraction_n be_v the_o take_v a_o lesser_a number_n from_o a_o great_a and_o to_o see_v what_o remain_v as_o suppose_v a_o man_n be_v bind_v from_o cales_n to_o dover_n which_o be_v 21_o mile_n now_o if_o he_o have_v sail_v 11_o mile_n of_o the_o way_n i_o demand_v what_o he_o have_v yet_o to_o sail_v take_v 11_o from_o 21_o and_o the_o remainder_n be_v 10_o so_o i_o conclude_v he_o have_v 10_o mile_n still_o to_o sail_v before_o he_o come_v to_o dover_n but_o in_o great_a number_n they_o be_v ordinary_o set_v so_o that_o you_o be_v force_v to_o borrow_v so_o that_o they_o be_v something_o hard_a suppose_v the_o distance_n between_o two_o place_n be_v 1799_o mile_n now_o i_o have_v sail_v of_o these_o mile_n 1227_o i_o demand_v what_o number_n of_o mile_n i_o have_v still_o to_o sail_v still_o to_o sail_v 0572_o set_v the_o great_a number_n uppermost_a and_o the_o least_o under_o it_o and_o draw_v a_o line_n between_o they_o and_o a_o line_n under_o they_o both_o and_o say_v 7_o from_z 9_o and_o the_o remainder_n be_v 2_o set_z down_z 2_o then_o 2_o from_o 9_o there_o remain_v 7_o set_v it_o down_o then_o 2_o from_o 7_o and_o the_o remainder_n be_v 5_o whatsoever_o number_n be_v underneath_o so_o much_o be_v still_o to_o sail_v as_o here_o 572_o mile_n but_o now_o i_o will_v give_v you_o a_o example_n wherein_o the_o figure_n in_o the_o lesser_a sum_n be_v some_o of_o they_o big_a than_o in_o the_o great_a from_o which_o they_o must_v be_v take_v now_o in_o such_o a_o case_n borrow_v ten_o and_o add_v to_o the_o upper_a figure_n example_n a_o owe_v to_o b_o 7822_o pound_n of_o tobacco_n now_o he_o have_v pay_v 4998_o l._n of_o it_o i_o demand_v what_o be_v still_o due_a whole_a debt_n 7822_o l_o receive_v 4998_o still_o due_a 2824_o set_v down_o your_o sum_n as_o in_o the_o former_a direction_n and_o say_v 8_o from_o 2_o i_o can_v but_o 8_o from_o 12_o and_o the_o remainder_n be_v 4_o set_v it_o down_o 1_o that_o i_o borrow_v and_o 9_o be_v 10_o from_z 2_o i_o can_v but_o 10_o from_z 12_o and_o the_o remainder_n be_v 2_o set_z down_z 2_o and_z carry_z 1_o then_z 9_o and_o 1_o that_o i_o carry_v be_v 10_o from_z 8_o i_o can_v but_o 10_o from_o 18_o and_o the_o remainder_n be_v 8_o set_v down_o 8_o and_o carry_v 1_o then_o 1_o that_o i_o carry_v and_o 4_o be_v 5_o 5_o from_o 7_o and_o the_o remainder_n be_v 2_o set_v it_o down_o and_o thus_o you_o may_v do_v any_o sum_n of_o one_o denomination_n subtraction_n of_o degree_n and_o minute_n this_o be_v of_o several_a denomination_n observe_v as_o before_o in_o addition_n of_o degree_n and_o minute_n how_o many_o of_o the_o one_o make_v one_o of_o the_o other_o and_o if_o the_o under_o figure_n be_v big_a than_o the_o uppermost_a in_o the_o place_n of_o minute_n borrow_v from_o 60_o the_o denominator_fw-la of_o a_o degree_n and_o subtract_v from_o it_o and_o that_o remainder_n add_v to_o the_o upper_a figure_n and_o set_v it_o down_o but_o in_o the_o place_n of_o degree_n if_o the_o under_o number_n be_v big_a do_v as_o you_o do_v before_o i_o have_v make_v in_o my_o reckon_v difference_n of_o latitude_n north_n 32_o deg_fw-la 48′_n difference_n of_o latitude_n south_n 29_o deg_fw-la 59_o  _fw-fr 02_o deg_fw-la 49_o i_o demand_v what_o way_n i_o have_v differ_v my_o lat._n most_o and_o how_o much_o i_o have_v differ_v my_o latitude_n and_o it_o be_v northerly_a because_o my_o northing_n be_v more_o than_o my_o sothe_v say_v 9_o from_o 8_o i_o can_v but_o 9_o from_o 18_o there_o remain_v 9_o set_v it_o down_o 5_o and_z 1_o that_o i_o borrow_v be_v 6_o 6_o from_z 4_o i_o can_v but_o 6_o from_o 6_o and_o the_o remainder_n be_v 0_o that_o 0_o and_o 4_o be_v 4_o set_v it_o down_o then_z 1_o that_o i_o borrow_v and_o 9_o be_v 10_o from_z 2_o i_o can_v but_o 10_o from_o 12_o and_o there_o remain_v 2_o set_v it_o down_o 1_o that_o i_o carry_v and_o 2_o be_v 3_o 3_o from_o 3_o and_o there_o remain_v 0_o so_o that_o 32_o deg_fw-la 48_o min._n be_v 02_o deg_fw-la 49_o more_o than_o 29_o deg_fw-la 59_o the_o same_o may_v be_v do_v in_o subtraction_n of_o pound_n shilling_n and_o penny_n observe_v but_o how_o many_o of_o the_o one_o go_v to_o make_v one_o of_o the_o other_o be_v sure_a in_o set_v down_o a_o sum_n you_o do_v not_o set_v more_o minute_n in_o the_o place_n of_o minute_n than_o 60_o or_o penny_n in_o the_o place_n of_o penny_n than_o 12_o or_o so_o for_o any_o thing_n else_o set_v no_o more_o than_o go_v to_o make_v one_o in_o the_o next_o denomination_n subtraction_n of_o time_n let_v the_o great_a number_n of_o time_n be_v 272_o hor._n 22_o min._n let_v the_o lesser_a number_n of_o time_n be_v 199_o hor._n 49_o min._n the_o remainder_n of_o time_n be_v 072_o hor._n 33_o min._n you_o may_v try_v this_o over_o it_o be_v do_v as_o the_o former_a and_o needs_o no_o demonstration_n as_o i_o say_v before_o if_o you_o observe_v in_o any_o case_n how_o many_o of_o one_o make_v one_o of_o the_o other_o it_o be_v but_o borrow_v from_o that_o number_n as_o we_o have_v do_v here_o the_o table_n 2_o time_n 2_o be_v 4_o 2_o time_n 3_o be_v 6_o 2_o time_n 4_o be_v 8_o 2_o time_n 5_o be_v 10_o 2_o time_n 6_o be_v 12_o 2_o time_n 7_o be_v 14_o 2_o time_n 8_o be_v 16_o 2_o time_n 9_o be_v 18_o 3_o time_n 3_o be_v 9_o 3_o time_n 4_o be_v 12_o 3_o time_n 5_o be_v 15_o 3_o time_n 6_o be_v 18_o 3_o time_n 7_o be_v 21_o 3_o time_n 8_o be_v 24_o 3_o time_n 9_o be_v 27_o 4_o time_n 4_o be_v 16_o 4_o time_n 5_o be_v 20_o 4_o time_n 6_o be_v 24_o 4_o time_n 7_o be_v 28_o 4_o time_n 8_o be_v 32_o 4_o time_n 9_o be_v 36_o 5_o time_n 5_o be_v 25_o 5_o time_n 6_o be_v 30_o 5_o time_n 7_o be_v 35_o 5_o time_n 8_o be_v 40_o 5_o time_n 9_o be_v 45_o 6_o time_n 6_o be_v 36_o 6_o time_n 7_o be_v 42_o 6_o time_n 8_o be_v 48_o 6_o time_n 9_o be_v 54_o 7_o time_n 7_o be_v 49_o 7_o time_n 8_o be_v 56_o 7_o time_n 9_o be_v 63_o 8_o time_n 8_o be_v 64_o 8_o time_n 9_o be_v 72_o 9_o time_n 9_o be_v 81_o this_o table_n be_v so_o familiar_a to_o any_o one_o that_o i_o need_v to_o say_v but_o little_o concern_v it_o when_o you_o read_v it_o read_v thus_o 2_o time_n 2_o be_v 4_o 2_o time_n 3_o be_v 6_o and_o so_o along_o to_o 2_o time_n 9_o which_o be_v 18_o then_o go_v to_o the_o next_o and_o say_v 3_o time_n 3_o be_v 9_o than_o 3_o time_n 4_o be_v 12_o and_o so_o forward_o now_o i_o wish_v the_o learner_n to_o have_v these_o table_n perfect_a by_o heart_n before_o he_o proceed_v further_o multiplication_n it_o do_v addition_n many_o time_n for_o when_o you_o have_v many_o addition_n it_o make_v one_o work_n of_o it_o and_o also_o of_o two_o give_v number_n it_o increase_v the_o great_a as_n often_o as_o there_o be_v unit_v in_o the_o lesser_a or_o the_o lesser_a as_o often_o as_o there_o be_v unit_v in_o the_o great_a now_o in_o multiply_v there_o be_v three_o thing_n considerable_a first_o the_o multiplicand_a or_o number_n that_o you_o will_v multiply_v second_o the_o multiplier_n or_o number_v you_o multiply_v by_o three_o the_o product_n or_o sum_n produce_v quest_n 1._o i_o demand_v how_o many_o foot_n there_o be_v in_o 320_o yard_n every_o yard_n be_v three_o
westermost_a ship_n steer_v be_v n_z w_n 2_o d._n westerly_n the_o bear_n of_o the_o eastermost_a ship_n from_o the_o westermost_a be_v w_n b_o s_o 5_o deg_fw-la 30_o min._n west_n i_o set_v the_o angle_n by_o the_o plain_a scale_n a_o little_a differ_v because_o the_o scale_n can_v be_v guess_v to_o a_o minute_n or_o 2_o or_o 5_o nor_o the_o mile_n to_o a_o single_a ten_o by_o proportion_n here_o the_o three_o side_n be_v give_v we_o to_o find_v the_o angle_n the_o side_n a_o b_o be_v 70_o mile_n the_o side_n b_o c_o be_v 78_o 9_o 10_o mile_n the_o side_n c_o at_fw-fr be_v 89_o mile_n a_o general_n rule_n in_o oblique_a triangle_n as_o the_o true_a base_a be_v to_o the_o sum_n of_o the_o other_o two_o side_n so_o be_v the_o difference_n of_o the_o side_n to_o the_o alternate_a base_a you_o may_v make_v which_o side_n you_o will_v the_o true_a base_a if_o you_o make_v either_o of_o the_o short_a side_n the_o true_a base_a and_o let_v fall_v a_o perpendicular_a from_o the_o angle_n opposite_a to_o it_o the_o side_n must_v be_v extend_v in_o some_o case_n for_o the_o perpendicular_a will_v fall_v without_o the_o triangle_n and_o the_o extent_n of_o the_o alternate_a base_a will_v fall_v as_o far_o without_o the_o perpendicular_a as_o the_o perpendicular_a be_v without_o the_o length_n of_o the_o true_a base_a but_o if_o you_o make_v the_o long_a side_n the_o true_a base_a the_o perpendicular_a will_v fall_v within_o the_o triangle_n in_o any_o question_n whatsoever_o so_o that_o you_o will_v have_v no_o occasion_n to_o extend_v the_o side_n which_o make_v i_o use_v the_o long_a side_n for_o the_o true_a base_a though_o in_o this_o question_n take_v any_o side_n and_o it_o will_v fall_v within_o here_o the_o side_n c_o at_fw-fr be_v long_a and_o therefore_o we_o will_v make_v that_o the_o true_a base_a take_v the_o length_n of_o the_o short_a side_n which_o be_v a_o b_o and_o fix_v one_o foot_n of_o your_o compass_n in_o b_o the_o angle_n opposite_a to_o the_o true_a base_a cut_v the_o true_a base_a with_o that_o side_n b_o a_o in_o the_o point_n s_o then_o draw_v the_o line_n s_o b_o which_o be_v equal_a to_o a_o b_o the_o perpendicular_a n_n b_o will_v fall_v in_o the_o middle_n between_o s_n and_o a_o which_o be_v in_o the_o point_n n_n then_o shall_v s_n n_n and_z n_z a_o be_v equal_v c_o s_n be_v the_o alternate_a base_a say_v as_o c_z a_o the_o true_a base_a 89_o mile_n comp_n arith_n 7,0506699_o be_v to_o c_o b_o and_o a_o b_o 148_o 9_o 10_o mile_n  _fw-fr 3,1728947_o so_o be_v their_o difference_n 8_o 9_o 10_o mile_n  _fw-fr 1,9493900_o to_o the_o alternate_a base_a s_n c_o 14_o 9_o 10_o mile_n 2,1728946_o subtract_v s_n c_o the_o alternate_a base_a thus_o find_v 14_o 9_o 10_o 89_o 74_o 1_o 10_o 37_o ½_n ten_o from_o c_o a_o the_o true_a base_a 89_o mile_n and_o the_o remainder_n be_v s_n a_o 74_o 1_o 10_o mile_n the_o half_a of_o which_o be_v n_o a_o 37_o mile_n or_o n_o a_o be_v thus_o find_v i_o have_v a_o b_o beside_o it_o in_o the_o rightangled_n triangle_n a_o n_n b_o to_o find_v the_o angle_n at_o a_o say_v then_o as_o a_o b_o 70_o mile_n comp_n arith_n 7,154901_o be_v to_o radius_fw-la  _fw-fr  _fw-fr so_o be_v a_o n_n 37_o mile_n  _fw-fr 2,568201_o to_o sine_fw-la comp_n n_o b_o a_o 58_o deg_fw-la 6_o min._n 9,723102_o which_o be_v sine_fw-la b_o a_o c_o or_o the_o quantity_n of_o the_o arch_n d_o f._n consider_v then_o what_o course_n the_o side_n a_o b_o runs_z upon_o and_o 58_o deg_fw-la 6_o min._n from_o that_o course_n be_v the_o course_n that_o the_o side_n a_o c_o runs_z upon_o example_n a_o b_o run_v upon_o a_o n_n b_o e_o course_n from_o a_o the_o side_n a_o c_o runs_z upon_o a_o course_n 58_o deg_fw-la 6_o min._n from_o it_o to_o the_o northwards_o and_o so_o to_o the_o westward_n 58_o deg_fw-la 6_o min._n be_v 5_o point_n and_o 1_o deg_fw-la 49_o min._n which_o be_v n_n w_n 1_o deg_fw-la 45_o min._n westerly_n what_o this_o be_v above_o a_o northwest_o 1_o deg_fw-la 45_o min._n westerly_n be_v by_o the_o neglect_n of_o take_v the_o absolute_a logarithm_n answer_v to_o 2,1728946_o and_o also_o the_o neglect_n of_o the_o half_a ten_o in_o divide_v you_o see_v it_o be_v but_o 6_o min._n of_o a_o degree_n of_o the_o compass_n from_o the_o other_o in_o the_o question_n before_o this_o which_o be_v no_o sensible_a error_n in_o sail_v and_o indeed_o i_o neglect_v work_v so_o near_o to_o let_v any_o learner_n see_v what_o a_o necessary_a thing_n it_o be_v to_o be_v exact_a the_o angle_n at_o a_o be_v thus_o find_v you_o may_v find_v the_o other_o angle_n at_o pleasure_n and_o also_o know_v what_o point_n the_o side_n b_o c_o runs_z upon_o because_o you_o know_v what_o point_n b_o a_o run_v upon_o from_o b._n for_o the_o angle_n c_o b_o a._n as_o c_z b_o 78_o 9_o 10_o mile_n comp_n arith_n 7,102922_o to_o sine_fw-la b_o a_o c_o 58_o d._n 6_o m._n  _fw-fr 9,928893_o so_o be_v c_o a_o 89_o mile_n  _fw-fr 2,949390_o to_o sine_fw-la c_o b_o a_o 73_o d._n 16_o m._n  _fw-fr 9,981205_o that_o be_v 6_o point_n and_z 5_o deg_fw-la 46_o min._n more_o then_o be_v b_o c_o a_o line_n that_o run_v from_o b_o w_o b_o s_n 5_o deg_fw-la 46_o min._n westerly_n or_o the_o ship_n at_o c_o bear_n from_o the_o ship_n at_o b_o w_o b_o s_n 5_o deg_fw-la 46_o min._n westerly_n then_o must_v b_o bear_v from_o c_o e_z b_o n_z 5_o deg_fw-la 46_o min._n easterly_a and_o thus_o your_o desire_n be_v know_v but_o you_o may_v think_v much_o that_o a_o fraction_n shall_v put_v you_o out_o so_o much_o as_o 6_o min._n and_o if_o you_o know_v no_o better_a you_o may_v be_v afraid_a it_o will_v cause_v some_o difference_n in_o the_o side_n but_o if_o you_o examine_v this_o example_n you_o will_v find_v the_o side_n will_v not_o differ_v for_o this_o a_o full_a ten_o of_o a_o mile_n example_n by_o the_o angle_n to_o find_v the_o side_n to_o agree_v with_o the_o side_n in_o the_o other_o question_n within_o a_o ten_o part_n of_o a_o mile_n as_o sine_fw-la b_o a_o c_o 58_o deg_fw-la 6_o min._n comp_n arith_n 0,07110_o to_o c_o b_o 78_o 9_o 10_o  _fw-fr 2,89707_o so_o be_v sine_fw-la a_o b_o c_o 73_o deg_fw-la 16_o min._n  _fw-fr 9,98120_o to_z c_o a_o 89_o 0_o 10_o mile_n  _fw-fr 2,94938_o here_o it_o be_v evident_a that_o all_o this_o difference_n do_v not_o alter_v the_o side_n the_o ten_o part_n of_o a_o mile_n if_o you_o desire_v to_o see_v far_a proof_n of_o the_o rule_n by_o which_o this_o question_n be_v wrought_v you_o have_v a_o clear_a geometrical_a demonstration_n for_o it_o in_o page_n 19_o and_o 20_o of_o my_o father_n first_o book_n of_o trigonometry_n which_o make_v i_o forbear_v the_o demonstrate_v of_o it_o by_o any_o geometrical_a way_n question_n v._n admit_v i_o be_v sail_v along_o the_o shore_n and_o see_v two_o headland_n the_o eastermost_a bear_v n_n n_n e_o off_o i_o the_o westermost_a bear_v n_o w_n from_o i_o at_o the_o same_o time_n i_o see_v a_o island_n that_o bear_v south_n from_o i_o but_o i_o sail_v west_n 8_o mile_n and_o find_v the_o eastermost_a head-land_n to_o bear_v n_z e_o b_o e_o 5_o deg_fw-la easterly_a from_o i_o the_o westermost_a n_n n_o e_o and_o the_o island_n e_o s_o e_o i_o demand_v how_o these_o place_n bear_v one_o from_o the_o other_o and_o their_o distance_n asunder_o such_o question_n as_o these_o be_v use_v in_o situate_a of_o place_n that_o be_v in_o sight_n one_o of_o the_o other_o or_o as_o a_o man_n sail_n by_o a_o land_n he_o may_v take_v the_o headland_n bear_v one_o from_o the_o other_o and_o also_o the_o imbays_n as_o well_o as_o the_o headland_n if_o he_o be_v not_o too_o far_o out_o this_o question_n be_v wrought_v as_o the_o second_o question_n for_o here_o you_o have_v the_o bear_n of_o the_o place_n you_o note_v in_o both_o station_n and_o also_o the_o course_n that_o you_o steer_v from_o one_o station_n to_o the_o other_o with_o the_o distance_n which_o be_v two_o angle_n and_o a_o side_n to_o find_v the_o other_o two_o side_n which_o be_v your_o distance_n at_o each_o station_n from_o the_o place_n for_o the_o protract_n of_o this_o question_n from_o your_o first_o station_n draw_v a_o line_n upon_o those_o several_a course_n as_o the_o question_n say_v the_o place_n bear_v as_o suppose_v the_o first_o station_n be_v a_o from_o a_o draw_v a_o i_o a_o n_o n_o e_o line_n and_o mind_n that_o line_n must_v be_v cut_v upon_o the_o eastermost_a head-land_n because_o the_o question_n say_v it_o bear_v so_o next_o draw_v a_o n_o w_n line_n from_o a_o
to_o good_a reason_n that_o be_v the_o angle_n in_o this_o triangle_n be_v the_o same_o with_o the_o angle_n in_o the_o question_n therefore_o as_o the_o sum_n of_o these_o side_n be_v in_o proportion_n to_o the_o sum_n of_o the_o side_n there_o so_o be_v the_o sum_n of_o the_o side_n 30_o here_o to_o the_o sum_n of_o the_o side_n that_o be_v correspondent_a to_o it_o there_o or_o run_v upon_o the_o same_o course_n to_o find_v the_o ship_n distance_n that_o sail_v s_o w_n correspondent_a to_o c_z n_z 30_o league_n as_o sum_n of_o the_o side_n of_o the_o triangle_n c_o n_o b_o 95_o le_fw-fr co_fw-la be_v 7,021363_o be_v to_o the_o sum_n of_o the_o side_n in_o the_o quest_n 148_o league_n 3,170261_o so_o be_v the_o side_n c_o n_o 30_o league_n 2,477121_o to_o the_o ship_n distance_n that_o sail_v s_o w_n 46_o league_n 6_o 19_o 2,668745_o this_o side_n be_v thus_o find_v we_o to_o abbreviate_v the_o work_n will_v lay_v it_o down_o in_o the_o triangle_n before_o who_o angle_n be_v equal_a to_o to_o the_o angle_n here_o set_v 46_o 6_o 10_o league_n for_o the_o length_n of_o the_o side_n the_o westermost_a ship_n run_v which_o be_v n_n r._n draw_z r_z s_z parallel_n to_o c_o b_o and_o extend_v n_z b_o till_o it_o intersect_v r_o s_o which_o it_o do_v in_o we_o this_o be_v as_o much_o as_o protract_v of_o it_o anew_o for_o the_o angle_n at_o r_n at_z s_z and_o at_o n_n be_v the_o same_o that_o they_o be_v give_v to_o be_v in_o the_o question_n and_o the_o side_n r_o n_n 47_o league_n as_o it_o be_v find_v you_o may_v measure_v the_o side_n unknown_a by_o the_o plain_a scale_n and_o set_v they_o down_o if_o you_o will_v or_o you_o may_v work_v by_o the_o rule_n of_o three_o use_v this_o proportion_n as_o n_n c_o 30_o league_n be_v to_o n_n r_n 46_o 6_o 10_o leag_fw-mi so_o be_v n_z b_o 41_o 6_o 10_o to_o n_n s_o multiply_v and_o divide_v and_o it_o make_v 64_o 6_o 10_o league_n 14_o 57_o of_o a_o ten_o by_o the_o table_n for_o n_n be_v as_o n_n c_o 30_o league_n comp_n arith_n 7,52287_o to_o n_o r_n 46_o 6_o 10_o league_n  _fw-fr 2,66839_o so_o be_v n_z b_o 41_o 6_o 10_o  _fw-fr 2,61909_o to_z n_o s_z 64_o 6_o 10_o league_n  _fw-fr 2,81035_o we_o may_v have_v do_v this_o by_o the_o table_n put_v the_o angle_n and_o side_n in_o proportion_n as_o we_o have_v do_v all_o along_o in_o the_o other_o triangle_n but_o i_o suppose_v your_o own_o reason_n will_v give_v you_o that_o this_o have_v the_o same_o give_v in_o it_o that_o the_o angle_n c_o n_o b_o have_v and_o be_v wrought_v so_o the_o fraction_n here_o and_o the_o other_o agree_v only_o this_o be_v not_o so_o true_a altogether_o because_o here_o we_o do_v it_o but_o to_o ten_o there_o in_o small_a fraction_n but_o this_o be_v within_o a_o small_a part_n of_o one_o ten_o of_o a_o unite_v for_o the_o side_n r_o s_o as_o sine_fw-la n_o s_o r_n 45_o deg_fw-la 00_o min._n comp_n arith_n 0,15051_o be_v to_o r_o n_n 46_o league_n 6_o 10_o  _fw-fr 2,66839_o so_o be_v sine_fw-la n_n 33_o deg_fw-la 45_o min._n  _fw-fr 9,74473_o to_z r_o s_z 36_o 6_o 10_o league_n  _fw-fr 2,56363_o if_o you_o add_v the_o side_n now_o find_v all_o together_o it_o will_v come_v to_o 2_o 10_o of_o a_o unite_v less_o than_o the_o give_v side_n together_o in_o the_o question_n be_v now_o that_o arise_v by_o neglect_v the_o take_n of_o the_o absolute_a number_n answer_v to_o the_o logarithm_n that_o come_v out_o but_o this_o be_v sufficient_a in_o case_n of_o this_o kind_n if_o you_o have_v set_v it_o in_o pace_n and_o wrought_v to_o the_o ten_o part_n you_o will_v have_v be_v out_o but_o 2_o 10_o of_o a_o pace_n therefore_o use_v your_o mind_n in_o such_o case_n if_o you_o have_v wrought_v to_o centism_n you_o will_v have_v be_v near_a but_o this_o i_o count_v near_o enough_o for_o sail_v and_o i_o be_o sure_a be_v more_o proper_a to_o be_v use_v than_o such_o small_a fraction_n because_o no_o long_a distance_n can_v be_v guess_v to_o a_o mile_n or_o a_o league_n you_o may_v have_v wrought_v your_o last_o proportion_n by_o the_o rule_n of_o three_o as_o you_o do_v the_o other_o three_o side_n of_o a_o triangle_n give_v to_o find_v the_o centre_n question_n ix_o there_o be_v three_o ship_n bind_v to_o one_o place_n the_o eastermost_a be_v distant_a from_o the_o middlemost_a 40_o league_n and_o bear_v s_n e_o b_o e_o the_o middlemost_a be_v distant_a from_o the_o westermost_a 50_o league_n and_o bear_v n_o e_z now_o they_o every_o one_o know_v as_o much_o as_o i_o have_v write_v they_o also_o know_v they_o be_v of_o a_o like_a distance_n from_o this_o port._n i_o desire_v to_o know_v what_o distance_n the_o port_n be_v from_o they_o and_o how_o it_o bear_v from_o each_o ship_n i_o have_v apply_v this_o question_n to_o sail_v here_o but_o before_o in_o this_o book_n i_o have_v use_v it_o in_o a_o thing_n which_o be_v very_o proper_a for_o it_o and_o indeed_o it_o may_v be_v of_o use_v many_o time_n which_o make_v i_o give_v it_o place_n here_o i_o shall_v say_v nothing_o of_o this_o question_n by_o the_o plain_a scale_n for_o it_o be_v do_v as_o before_o you_o see_v but_o we_o will_v do_v it_o by_o the_o table_n of_o tangent_n and_o logarithm_n here_o i_o have_v give_v the_o side_n m_o er_fw-mi 40_o league_n the_o side_n m_o w_n 50_o league_n and_o the_o angle_n at_o m_n namely_o its_o compliment_n to_o 180_o deg_fw-la r_o m_o o_o 78_o deg_fw-la 45_o min._n to_o find_v the_o angle_n m_o ere_fw-we w_n and_o m_o w_n o_z as_o w_n m_o add_v m_o o_o 90_o league_n comp_n arith_n 7,04575_o be_v to_o their_o difference_n 10_o league_n  _fw-fr 2_o so_o be_v the_o tan._n of_o ½_n m_o er_fw-mi w_o and_o m_o w_n oer_o 39_o d._n 22_o m._n 9,91404_o to_o tan._n of_o their_o difference_n 5_o deg_fw-la 12_o min._n 8,95980_o subtract_v from_o 39_o deg_fw-la 22_o min._n and_o the_o remainder_n be_v m_o w_n oer_o 34_o deg_fw-la 50_o min._n add_v to_o it_o make_v m_n oer_o w_n 44_o deg_fw-la 34_o min._n for_o the_o side_n er_fw-mi w._n as_o sine_fw-la m_o o_o w_n 44_o deg_fw-la 34_o min._n comp_n arith_n 0,153824_o to_o w_n m_o 50_o league_n  _fw-fr 2,698970_o so_o be_v sine_fw-la r_o m_o o_o 78_o d._n 45_o m._n its_o comp_n log_n 180_o d._n 9,991573_o to_o w_o o_z 69_o 9_o 10_o league_n  _fw-fr 2,844367_o but_o now_o you_o will_v demand_v of_o i_o how_o we_o shall_v find_v any_o thing_n in_o the_o triangle_n w_n f_o er_fw-mi be_v there_o be_v but_o one_o thing_n know_v in_o it_o namely_o the_o distance_n of_o the_o eastermost_a ship_n from_o the_o westermost_a er_fw-mi w_n i_o will_v tell_v you_o it_o mr._n euclid_n prove_v this_o which_o make_v i_o forbear_v it_o mr._n euclid_n book_n 3_o prop._n 20._o say_v that_o in_o a_o circle_n a_o angle_n at_o the_o centre_n be_v double_a to_o a_o angle_n at_o the_o circumference_n provide_v that_o both_o the_o angle_n have_v to_o their_o base_a one_o and_o the_o same_o part_n of_o the_o circumference_n the_o centre_n be_v f_o here_o the_o triangle_n w_n f_o o_z and_o w_n m_o ere_fw-we have_v w_o o_z to_o both_o their_o base_n then_o must_v b_o f_o a_o be_v double_a to_o i_o m_o o_o or_o c_o b_o a_o double_a to_o d_o m_o i_o that_o be_v to_o the_o arch_n i_o o_o d_o so_o that_o if_o you_o double_a d_o o_fw-fr i_o 101_o deg_fw-la 15_o min._n it_o make_v c_o b_o a_o 202_o deg_fw-la 30_o min._n that_o subtract_v from_o a_o circle_n 360_o deg_fw-la leave_v the_o triangle_n w_n f_o o_z it_o be_v evident_a than_o that_o w_n f_o ere_fw-we be_v double_a to_o r_o m_o i_o because_o r_n m_o i_o be_v what_o i_o o_o d_o want_v of_o a_o semicircle_n as_o w_n f_o ere_fw-we be_v what_o c_o b_o a_o want_v of_o a_o whole_a circle_n which_o be_v double_a to_o it_o example_n  _fw-fr deg_fw-la min._n w_n m_o ere_fw-we be_v 101_o deg_fw-la 30_o min._n double_v it_o be_v c_o b_o a_o 202_o 30_o that_o subtract_v from_o a_o circle_n 360_o 00_o the_o remainder_n be_v w_n f_o oer_o 157_o 30_o and_o this_o subtract_v from_o 180_o 00_o the_o remainder_n be_v f_o w_n o_z and_o f_o oer_o w_n for_o the_o angle_n of_o a_o right_a angle_a triangle_n be_v 180_o d._n 22_o 30_o the_o half_a of_o which_o be_v f_o w_n o_z or_o f_o ere_fw-we w_o 11_o 15_o for_o they_o must_v be_v equal_a because_o the_o side_n opposite_a to_o they_o be_v equal_a namely_o f_o o_z and_o f_o w._n and_o thus_o we_o have_v the_o three_o angle_n of_o the_o triangle_n w_n f_o er_fw-mi and_z the_o side_z w_o o_z to_o
find_v f_o o_z or_o f_o w_n or_o f_o m_o the_o distance_n of_o any_o of_o the_o ship_n from_o the_o port._n for_o the_o distance_n of_o the_o ship_n from_o the_o port._n as_o sine_fw-la w_n f_o oer_o 22_o deg_fw-la 30_o min._n comp_n arith_n 0,417160_o be_v to_o w_o o_z 69_o 9_o 10_o league_n  _fw-fr 2,844477_o so_o be_v sine_fw-la f_o w_n o_z 11_o deg_fw-la 15_o min._n  _fw-fr 9,290235_o to_o any_o of_o the_o ship_n distance_n from_o the_o port_n which_o i_o find_v to_o be_v 35_o 6_o 10_o league_n 2,551872_o i_o need_v not_o count_v the_o triangle_n for_o the_o point_n for_o i_o have_v show_v that_o sufficient_o so_o that_o you_o i_o suppose_v can_v tell_v how_o to_o find_v the_o ship_n bear_v from_o the_o port_n for_o it_o differ_v nothing_o from_o the_o way_n you_o count_v in_o other_o question_n question_n x._o there_o be_v two_o ship_n set_v from_o one_o port_n one_o sail_a n_z w_o b_o n_z 6_o league_n a_o watch_n the_o other_o sail_v n_o e_z 8_o league_n a_o watch_n these_o ship_n arrive_v at_o two_o several_a port_n in_o one_o instant_a of_o time_n the_o port_n be_v 12_o league_n asunder_o i_o demand_v the_o distance_n run_v of_o each_o ship_n and_o the_o port_n bear_v one_o from_o the_o other_o for_o the_o do_n of_o this_o question_n first_o draw_v a_o n_n w_n b_o n_n line_n and_o set_v off_o 6_o league_n upon_o it_o which_o be_v a_o i_o then_o draw_v a_o n_n e_o line_n and_o set_v 8_o league_n off_o upon_o it_o which_o be_v a_o d_o draw_v the_o side_n i_o d_o then_o consider_v that_o as_o a_o i_o be_v to_o a_o d_o so_o be_v the_o westermost_a ship_n run_v to_o the_o eastermost_a so_o that_o the_o side_n a_o d_o and_o a_o i_o extend_v so_o far_o that_o a_o line_n of_o 12_o league_n long_o draw_v parallel_n to_o d_o i_o will_v just_o touch_v the_o extend_a side_n which_o must_v cut_v they_o in_o the_o place_n be_v the_o port_n be_v namely_o a_o b_o in_o b_o and_o a_o i_o in_o c_o measure_v the_o side_n c_o a_n and_o a_o b_o and_o also_o the_o angle_n of_o the_o port_n bear_v a_o b_o c_o or_o a_o c_o b_o and_n count_v they_o as_o have_v be_v show_v before_o i_o find_v the_o side_n a_o b_o the_o eastermost_a ship_n run_v to_o be_v 10_o leag_n 7_o 10._o i_o find_v a_o c_o the_o westermost_a ship_n run_v to_o be_v 8_o league_n the_o westermost_a port_n bear_v from_o the_o eastermost_a west_n 4_o deg_fw-la 14_o min._n southerly_a by_o the_o table_n first_o for_o the_o port_n bear_v one_o from_o the_o other_o a_o d_o i_o equal_a to_o a_o b_o c_o or_o a_o i_o d_o equal_a to_o a_o c_o b_o they_o be_v equal_a because_o i_o d_o and_o c_o b_o be_v parallel_a and_o i_o a_o d_o be_v a_o angle_n opposite_a to_o they_o both_o as_o a_o d_o and_o a_o i_o 14_o league_n comp_n arith_n 6,853871_o be_v to_o their_o difference_n 2_o league_n  _fw-fr 2,301030_o so_o be_v tan._n ½_n a_o i_o d_o and_o a_o d_o i_o 50_o deg_fw-la 37_o min._n 30_o sec_fw-la 10,085698_o to_o tan._n their_o diff._n 9_o deg_fw-la 51_o min._n  _fw-fr 9,240599_o which_o add_v to_o the_o half_a sum_n be_v a_o i_o d_o 60_o deg_fw-la 28_o min._n a_o c_o b_o equal_a to_o a_o d_o i_o or_o a_o b_o c_o the_o bearing_n subtract_v from_o the_o half_a sum_n be_v 40_o deg_fw-la 46_o min._n for_o i_o d._n as_o sine_fw-la d_o 40_o deg_fw-la 46_o min._n 0,185100_o to_o i_o a_o 6_o league_n 0,778151_o so_o be_v sine_fw-la i_o a_o d_o 78_o deg_fw-la 45_o min._n 9,991573_o to_o d_o i_o 9_o league_n 0,954824_o for_o c_o a._n as_o i_o d_o 9_o leag_fw-mi co_fw-la be_v 8,04575_o be_v to_o c_o b_o 12_o league_n 2,07918_o so_o be_v i_o a_o 6_o league_n 1,77815_o to_o a_o c_o 8_o league_n 1,90309_o for_o a_o b._n as_o i_o d_o 9_o leag_fw-mi co_fw-la be_v 8,04575_o be_v to_o c_o b_o 12_o league_n 2,07928_o so_o be_v a_o d_o 8_o league_n 1,90309_o to_o a_o b_o 10_o 7_o 10_o league_n 2,02802_o the_o reason_n that_o the_o proportion_n stand_v thus_o be_v because_o the_o angles_n be_v equal_a in_o each_o triangle_n and_o therefore_o the_o side_n must_v be_v proportional_a thus_o the_o eastermost_a ship_n sail_v 10_o 7_o 10_o league_n the_o westermost_a 8_o league_n and_o the_o eastermost_a port_n bear_v from_o the_o westermost_a east_n 4_o deg_fw-la 14_o min._n northerly_a there_o be_v no_o distance_n that_o a_o man_n sail_n but_o if_o he_o return_v the_o same_o way_n that_o he_o go_v out_o he_o shall_v find_v the_o same_o distance_n backward_o as_o he_o do_v out_o but_o now_o the_o world_n be_v round_o and_o if_o a_o man_n go_v out_o one_o way_n and_z returns_z another_z as_o it_o be_v common_a he_o will_v find_v a_o great_a deal_n of_o difference_n in_o long_a run_v for_o to_o demonstrate_v the_o reason_n of_o this_o i_o shall_v not_o because_o my_o father_n and_o other_o have_v do_v that_o sufficient_o beside_o it_o will_v take_v up_o too_o much_o room_n in_o this_o little_a book_n these_o question_n of_o sail_v may_v be_v understand_v in_o such_o case_n as_o this_o be_v namely_o when_o a_o man_n sail_v the_o same_o way_n home_o as_o he_o do_v out_o but_o because_o the_o wind_n will_v not_o suffer_v a_o man_n to_o do_v so_o it_o be_v necessary_a to_o know_v some_o better_a way_n to_o sail_v by_o let_v a_o man_n return_v never_o so_o contrary_a to_o what_o he_o sail_v outward_o yet_o at_o all_o time_n to_o know_v how_o near_o or_o how_o far_o he_o may_v be_v off_o now_o there_o be_v no_o way_n better_o than_o mercator_n way_n of_o sail_v it_o be_v very_o excellent_a and_o have_v be_v treat_v of_o by_o many_o and_o that_o make_v i_o presume_v to_o think_v that_o i_o may_v treat_v of_o it_o as_o well_o as_o other_o and_o first_o here_o be_v the_o table_n of_o meridional_a part_n by_o which_o you_o be_v to_o work_v a_o table_n of_o meridional_a part_n to_o every_o three_o minute_n latitud_n mer._n part_n d._n m._n 0_o 3_o 3_o  _fw-fr 6_o 6_o  _fw-fr 9_o 9_o  _fw-fr 12_o 12_o  _fw-fr 15_o 15_o  _fw-fr 18_o 18_o  _fw-fr 21_o 21_o  _fw-fr 24_o 24_o  _fw-fr 27_o 27_o  _fw-fr 30_o 30_o  _fw-fr 33_o 33_o  _fw-fr 36_o 36_o  _fw-fr 39_o 39_o  _fw-fr 42_o 42_o  _fw-fr 45_o 45_o  _fw-fr 48_o 48_o  _fw-fr 51_o 51_o  _fw-fr 54_o 54_o  _fw-fr 57_o 57_o 1_o 0_o 60_o  _fw-fr 3_o 63_o  _fw-fr 6_o 66_o  _fw-fr 9_o 69_o  _fw-fr 12_o 72_o  _fw-fr 15_o 75_o  _fw-fr 18_o 78_o  _fw-fr 21_o 81_o  _fw-fr 24_o 84_o  _fw-fr 27_o 87_o  _fw-fr 30_o 90_o  _fw-fr 33_o 93_o  _fw-fr 36_o 96_o  _fw-fr 39_o 99_o  _fw-fr 42_o 102_o  _fw-fr 45_o 105_o  _fw-fr 48_o 108_o  _fw-fr 51_o 111_o  _fw-fr 54_o 114_o  _fw-fr 57_o 117_o 2_o 0_o 120_o  _fw-fr 3_o 123_o  _fw-fr 6_o 126_o  _fw-fr 9_o 129_o  _fw-fr 12_o 132_o  _fw-fr 15_o 135_o  _fw-fr 18_o 138_o  _fw-fr 21_o 141_o  _fw-fr 24_o 144_o  _fw-fr 27_o 147_o  _fw-fr 30_o 150_o  _fw-fr 33_o 153_o  _fw-fr 36_o 156_o  _fw-fr 39_o 159_o  _fw-fr 42_o 162_o  _fw-fr 45_o 165_o  _fw-fr 48_o 168_o  _fw-fr 51_o 171_o  _fw-fr 54_o 174_o  _fw-fr 57_o 177_o 3_o 0_o 180_o  _fw-fr 3_o 183_o  _fw-fr 6_o 186_o  _fw-fr 9_o 189_o  _fw-fr 12_o 192_o  _fw-fr 15_o 195_o  _fw-fr 18_o 198_o  _fw-fr 21_o 201_o  _fw-fr 24_o 204_o  _fw-fr 27_o 207_o  _fw-fr 30_o 210_o  _fw-fr 33_o 213_o  _fw-fr 36_o 216_o  _fw-fr 39_o 219_o  _fw-fr 42_o 222_o  _fw-fr 45_o 225_o  _fw-fr 48_o 228_o  _fw-fr 51_o 231_o  _fw-fr 54_o 234_o  _fw-fr 57_o 237_o 4_o 0_o 240_o  _fw-fr 3_o 243_o  _fw-fr 6_o 246_o  _fw-fr 9_o 249_o  _fw-fr 12_o 252_o  _fw-fr 15_o 255_o  _fw-fr 18_o 258_o  _fw-fr 21_o 262_o  _fw-fr 24_o 264_o  _fw-fr 27_o 267_o  _fw-fr 30_o 270_o  _fw-fr 33_o 273_o  _fw-fr 36_o 276_o  _fw-fr 39_o 279_o  _fw-fr 42_o 282_o  _fw-fr 45_o 285_o  _fw-fr 48_o 288_o  _fw-fr 51_o 291_o  _fw-fr 54_o 294_o  _fw-fr 57_o 297_o 5_o 0_o 300_o  _fw-fr 3_o 303_o  _fw-fr 6_o 306_o  _fw-fr 9_o 309_o  _fw-fr 12_o 312_o  _fw-fr 15_o 315_o  _fw-fr 18_o 318_o  _fw-fr 21_o 321_o  _fw-fr 24_o 324_o  _fw-fr 27_o 327_o  _fw-fr 30_o 330_o  _fw-fr 33_o 333_o  _fw-fr 36_o 337_o  _fw-fr 39_o 340_o  _fw-fr 42_o 343_o  _fw-fr 45_o 346_o  _fw-fr 48_o 349_o  _fw-fr 51_o 353_o  _fw-fr 54_o 355_o  _fw-fr 57_o 358_o 6_o 0_o 361_o  _fw-fr 3_o 364_o  _fw-fr 6_o 367_o  _fw-fr 9_o 370_o  _fw-fr 12_o 373_o  _fw-fr 15_o 376_o  _fw-fr 18_o 379_o  _fw-fr 21_o 382_o  _fw-fr 24_o 385_o  _fw-fr 27_o 388_o  _fw-fr 30_o 391_o  _fw-fr 33_o 394_o  _fw-fr 36_o 397_o  _fw-fr 39_o 400_o  _fw-fr 42_o 403_o  _fw-fr 45_o 406_o  _fw-fr 48_o 409_o  _fw-fr 51_o 412_o  _fw-fr 54_o 415_o  _fw-fr 57_o 418_o