Selected quad for the lemma: book_n

Word A	Word B	Word C	Word D	Occurrence	Frequency Band	MI	MI Band	Prominent
book_n	call_v	name_n	part_n	1,946	5	4.1439	3	false

Text snippets containing the quad

ID	Title	Author	Corrected Date of Publication (TCP Date of Publication)	STC	Words	Pages
A80170	Navigation by the mariners plain scale new plain'd: or, A treatise of geometrical and arithmetical navigation; wherein sayling is performed in all the three kindes by a right line, and a circle divided into equal parts. Containing 1. New ways of keeping of a reckoning, or platting of a traverse, both upon the plain and mercators chart ... 2. New rules for estimating the ships way through currents, and for correcting the dead reckoning. 3. The refutation of divers errors, and of the plain chart, and how to remove the error committed thereby ... as also a table thereof made to every other centesm. 4. A new easie method of calculation for great circle-sayling, with new projections, schemes and charts ... 5. Arithmetical navigation, or navigation performed by the pen, if tables were wanting ... By John Collins of London, Pen-man, accomptant, philomathet.	Collins, John, 1625-1683.	1659 (1659)	Wing C5378A; Thomason E1002_1; ESTC R207824	177,819	300

instrument_n to_o be_v cut_v in_o brass_n with_o paper_n print_v fit_v up_o for_o sale_n which_o will_v be_v of_o excellent_a use_n to_o seaman_n surveyor_n and_o all_o that_o be_v mathematical_o studious_a i_o remain_v thy_o friend_n desirous_a of_o the_o advancement_n of_o knowledge_n john_n collins_n the_o content_n a_o double_a scale_n of_o chord_n use_v in_o this_o book_n and_o describe_v page_z 1_o 2._o spherical_a definition_n from_o page_n 2_o to_o 9_o all_o the_o point_n ark_n and_o circle_n define_v represent_v to_o the_o view_n in_o a_o scheme_n of_o the_o analemma_n p._n 10_o to_o 14_o a_o general_a almanac_n in_o two_o verse_n p._n 15_o the_o manner_n of_o measure_v and_o proportion_v out_o sin_n by_o a_o line_n of_o chord_n p._n 18_o 19_o a_o general_a rule_n in_o two_o verse_n for_o find_v the_o sun_n place_n 21_o to_o find_v the_o sun_n declination_n and_o right_a ascension_n 23_o 24_o to_o find_v the_o sun_n amplitude_n height_n at_o six_o vertical_a height_n time_n of_o rise_v etc._n etc._n 25_o 26_o the_o sixteen_o right_a angle_a case_n resolve_v by_o proportion_n of_o four_o several_a kind_n p._n 27_o to_o 32_o to_o find_v the_o sun_n height_n without_o instrument_n 32_o to_o find_v the_o hour_n azimuth_n and_o angle_n of_o position_n p._n 33_o to_o 39_o to_o find_v the_o sun_n altitude_n on_o all_o hour_n 40_o 41_o to_o find_v the_o distance_n of_o place_n etc._n etc._n 42_o 43_o all_o the_o oblique_a case_n solve_v p._n 46._o p._n 53_o to_o find_v the_o altitude_n on_o all_o azimuth_n p._n 46_o to_o 52_o all_o the_o sixteen_o case_n of_o right_a angle_a spherical_a triangle_n project_v and_o otherwise_o resolve_v p._n 57_o to_o 63_o the_o longitude_n and_o latitude_n of_o a_o star_n give_v to_o find_v its_o declination_n and_o right_a ascension_n p._n 64_o two_o azimuth_n and_o two_o altitude_n give_v to_o find_v the_o latitude_n and_o declination_n by_o projection_n p._n 65_o with_o proportion_n to_o find_v the_o same_o by_o calculation_n p._n 66_o 67_o page_n 2._o line_n 17._o obliterate_v superficies_n of_o the_o scale_n use_v in_o this_o book_n though_o this_o treatise_n bear_v the_o name_n of_o the_o mariner_n plain_a scale_n new_o plain_v yet_o the_o scale_n intend_v thereby_o be_v cut_v in_o the_o frontispiece_n of_o the_o first_o book_n nothing_o more_o be_v necessary_o require_v in_o the_o performance_n of_o this_o book_n then_o from_o the_o common_o know_a division_n of_o a_o circle_n into_o 360_o equal_a part_n call_v degree_n to_o prick_v down_o any_o number_n of_o degree_n less_o than_o 180_o and_o a_o quadrant_a divide_v into_o nine_o equal_a part_n and_o one_o of_o those_o part_n for_o convenience_n below_o the_o diamater_n into_o 10_o sub-division_n call_v degree_n may_v very_o well_o serve_v the_o turn_n which_o the_o reader_n ingenuity_n will_v furnish_v himself_o withal_o in_o any_o place_n if_o he_o have_v but_o compass_n yet_o for_o expedition_n a_o line_n of_o sines_n be_v often_o make_v use_n of_o in_o the_o follow_a diagram_n the_o equal_a division_n of_o the_o semicircle_n be_v transfer_v into_o the_o line_n of_o chord_n in_o the_o diameter_n by_o set_v one_o foot_n of_o the_o compass_n in_o a_o which_o be_v call_v the_o lesser_a chord_n those_o chord_n be_v number_v by_o the_o half_a arch_n to_o which_o they_o belong_v become_v a_o line_n of_o sin_n of_o the_o same_o radius_fw-la with_o the_o diameter_n of_o the_o semicircle_n and_o be_v call_v the_o great_a sin_n to_o that_o radius_fw-la there_o be_v fit_v a_o chord_n of_o 60_o call_v the_o great_a chord_n that_o the_o reader_n may_v be_v supply_v with_o both_o for_o the_o lesser_a chord_n will_v not_o serve_v to_o prick_v off_o a_o arch_n in_o a_o circle_n of_o twice_o the_o radius_fw-la whereto_o it_o be_v fit_v unless_o the_o say_a chord_n be_v double_v in_o a_o right_a line_n before_o it_o be_v pirck_v into_o the_o circumference_n the_o sin_n to_o the_o chord_n in_o the_o diameter_n be_v graduate_v on_o the_o radius_fw-la or_o line_n c_o b_o by_o draw_v line_n through_o each_o degree_n of_o the_o quadrant_n ab_fw-la db_fw-la and_o be_v call_v the_o lesser_a sin_n there_o be_v also_o add_v a_o scale_n of_o equal_a part_n and_o rumbe_n for_o other_o protraction_n but_o we_o use_v neither_o of_o they_o in_o this_o book_n the_o scheme_n in_o the_o book_n be_v fit_v either_o to_o the_o lesser_a or_o great_a chord_n here_o describe_v every_o degree_n of_o a_o line_n of_o sines_n or_o chord_n we_o suppose_v to_o be_v divide_v into_o 60_o part_n which_o we_o call_v minute_n which_o in_o the_o follow_a operation_n be_v guess_v at_o for_o a_o small_a instrument_n will_v not_o admit_v of_o so_o many_o sub-division_n chap._n i._n spherical_a definition_n before_o we_o proceed_v to_o the_o resolution_n of_o any_o particular_a question_n it_o will_v be_v necessary_a to_o premise_v the_o common_a spherical_a definition_n and_o to_o show_v how_o the_o analemma_n repesent_v they_o the_o word_n sphere_n though_o as_o herigonius_n show_v it_o be_v take_v in_o a_o fourfold_a sense_n yet_o i_o think_v it_o not_o necessary_a to_o define_v above_o two_o of_o they_o 1._o therefore_o a_o sphere_n may_v according_a to_o theodosius_n be_v understand_v to_o be_v a_o solid_a superficies_n or_o round_a body_n contain_v under_o one_o surface_n in_o the_o middle_n whereof_o there_o be_v a_o point_n whence_o all_o right_a line_n draw_v unto_o the_o circumference_n be_v equal_a and_o be_v make_v by_o the_o turn_n round_o of_o half_a a_o circle_n till_o it_o end_v where_o it_o begin_v 2._o it_o be_v take_v for_o a_o certain_a round_a instrument_n consist_v of_o divers_a circle_n whereby_o the_o motion_n of_o the_o heaven_n and_o the_o situation_n of_o the_o whole_a world_n be_v most_o convenient_o represent_v for_o the_o better_a explanation_n whereof_o astronomer_n do_v imagine_v 10_o principal_a point_n and_o 10_o circle_n to_o be_v in_o the_o hollow_a inside_n of_o the_o first_o movable_a sphere_n which_o be_v common_o draw_v upon_o any_o globe_n or_o sphere_n beside_o divers_a other_o circle_n which_o be_v not_o delineate_v but_o only_o apprehend_v in_o the_o fancy_n the_o point_n be_v the_o two_o pole_n of_o the_o world_n the_o two_o pole_n of_o the_o zodiac_n the_o two_o equinoctial_a point_n the_o two_o solstitial_a point_n and_o the_o zenith_n and_o nadir_n the_o pole_n of_o the_o world_n be_v two_o point_n which_o be_v diametrical_o or_o direct_o opposite_a to_o one_o another_o about_o which_o the_o whole_a frame_n of_o heaven_n move_v from_o the_o east_n into_o the_o west_n whereof_o one_o be_v perpetual_o see_v by_o we_o and_o be_v call_v the_o arctic_a or_o north_n pole_n the_o other_o be_v hide_v from_o we_o and_o direct_o opposite_a to_o the_o former_a be_v call_v the_o antarctick_a or_o south_n pole_n a_o right_a line_n imagine_v to_o be_v draw_v from_o the_o one_o pole_n to_o the_o other_o be_v call_v the_o axis_n or_o axletree_n of_o the_o world_n the_o axis_n differ_v from_o the_o diameter_n because_o that_o every_o right_a line_n draw_v through_o the_o centre_n of_o the_o sphere_n and_o limit_v on_o each_o side_n of_o the_o surface_n of_o the_o sphere_n be_v a_o diameter_n but_o not_o a_o axis_n unless_o the_o sphere_n move_v round_o about_o it_o the_o pole_n of_o the_o zodiac_n be_v two_o point_n diametrical_o opposite_a upon_o which_o the_o heaven_n move_v from_o the_o west_n into_o the_o east_n one_o of_o they_o be_v towards_o the_o north_n distant_a from_o the_o arctic_a or_o north_n pole_n 23_o degree_n 31_o minute_n the_o other_o be_v towards_o the_o south_n and_o be_v as_o much_o distant_a from_o the_o south_n pole_n a_o degree_n be_v the_o 360_o part_n of_o any_o whole_a circle_n and_o a_o minute_n be_v the_o 60_o part_n of_o a_o degree_n but_o of_o late_a some_o divide_v a_o degree_n into_o 100_o part_n which_o be_v call_v centesme_n or_o centesimal_a minute_n in_o define_v some_o of_o the_o point_n we_o must_v refer_v to_o circle_n afterward_o to_o be_v define_v the_o equinoctial_a point_n be_v in_o the_o beginning_n of_o aries_n and_o libra_n to_o which_o when_o the_o sun_n come_v he_o make_v the_o day_n and_o night_n of_o a_o equal_a length_n throughout_o the_o whole_a world_n to_o wit_n in_o the_o begin_n of_o aries_n about_o the_o 11_o of_o march_n which_o be_v account_v the_o beginning_n of_o the_o spring_n and_o in_o the_o begin_n of_o libra_n about_o the_o 13_o of_o september_n which_o be_v the_o begin_n of_o autumn_n the_o point_n of_o the_o summer_n solstice_n be_v in_o the_o begin_n of_o cancer_n to_o which_o when_o the_o sun_n come_v as_o about_o the_o 12_o of_o june_n be_v the_o begin_n of_o summer_n and_o the_o long_a day_n in_o the_o year_n the_o point_n of_o the_o winter_n solstice_n be_v

namely_o that_o it_o be_v a_o right_a line_n fall_v from_o one_o end_n of_o any_o arch_n perpendicular_o upon_o the_o radius_fw-la draw_v to_o the_o other_o end_n of_o the_o say_a arch_n so_o be_v g_o f_o perpendicular_a to_o c_z a_o be_v the_o sine_fw-la of_o the_o arch_n g_o a_o likewise_o a_o i_o fall_v perpendicular_o on_o c_o g_z therefore_o by_o the_o same_o definition_n be_v also_o the_o sine_fw-la of_o the_o say_v arch._n 3._o the_o verse_v sine_fw-la of_o a_o arch_n be_v that_o part_n of_o the_o diameter_n which_o lie_v between_o the_o right_n sine_fw-la of_o that_o arch_n and_o the_o circumference_n thus_o f_o a_o be_v the_o verse_v sine_fw-la of_o the_o arch_n g_o a_o and_o f_o b_o the_o verse_v sine_fw-la of_o the_o arch_n b_o d_o g._n 4._o if_o unto_o one_o end_n of_o a_o arch_n there_o be_v draw_v a_o radius_fw-la and_o to_o the_o other_o end_n a_o right_a line_n from_o the_o centre_n cut_v the_o circle_n and_o if_o from_o the_o end_n of_o the_o radius_fw-la a_o 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sin_n tangent_n and_o secant_v of_o which_o at_o large_a see_v mr._n newtons_n trigonometria_fw-la brittanica_fw-la for_o abbreviate_v way_n and_o something_o we_o shall_v add_v about_o it_o in_o the_o arithmetical_a part_n of_o navigation_n 7._o the_o table_n be_v make_v their_o chief_a use_n be_v to_o work_v the_o rule_n of_o three_o or_o golden_a rule_n arithmetical_o by_o multiply_v the_o second_o and_o three_o term_n of_o any_o proportion_n and_o divide_v by_o the_o first_o and_o thereby_o to_o resolve_v all_o proposition_n relate_v either_o to_o plain_a or_o spherical_a triangle_n which_o in_o line_n be_v perform_v by_o draw_v a_o line_n parallel_n to_o the_o side_n of_o a_o triangle_n and_o where_o four_o term_n either_o in_o sines_n tangent_n secant_v verse_v sines_n be_v express_v as_o the_o first_o to_o the_o second_o so_o be_v the_o three_o to_o the_o four_o it_o imply_v a_o proportion_n and_o that_o the_o second_o and_o three_o term_n be_v to_o be_v multiply_v together_o and_o the_o product_v divide_v by_o the_o first_o the_o proportion_n or_o reason_n of_o two_o number_n or_o reference_n of_o one_o to_o the_o other_o be_v measure_v by_o the_o quotient_a the_o one_o be_v divide_v by_o the_o other_o a_o proportion_n be_v then_o say_v to_o be_v direct_a when_o the_o three_o term_n bear_v such_o proportion_n to_o the_o four_o as_o the_o first_o to_o the_o second_o four_o number_n be_v say_v to_o be_v proportional_a when_o as_o often_o as_o the_o first_o and_o second_o be_v the_o one_o contain_v in_o the_o other_o so_o often_o be_v the_o three_o and_o four_o the_o one_o contain_v in_o the_o other_o reciprocal_a proportion_n be_v when_o the_o four_o term_n bear_v such_o proportion_n to_o the_o three_o as_o the_o first_o do_v to_o the_o second_o a_o proportion_n be_v say_v to_o hold_v alternate_o when_o the_o second_o and_o three_o term_n thereof_o change_v place_n and_o invers_o when_o the_o order_n of_o the_o term_n be_v so_o alter_v that_o one_o of_o the_o three_o first_o term_n shall_v become_v the_o last_o divers_a proportion_n be_v express_v in_o this_o follow_a book_n which_o if_o the_o reader_n will_v apply_v to_o table_n he_o must_v understand_v that_o when_o a_o side_n or_o a_o angle_n be_v great_a than_o a_o quadrant_n that_o in_o stead_n of_o the_o sine_fw-la tangent_fw-la or_o secant_fw-la of_o such_o a_o ark_n he_o must_v take_v the_o sine_fw-la tangent_fw-la or_o secant_fw-la of_o that_o ark_n compliment_n to_o a_o semicircle_n that_o the_o word_n cousin_n or_o cotangent_fw-la of_o a_o arch_n give_v or_o seek_v signify_v the_o sine_fw-la or_o tangent_fw-la of_o the_o compliment_n of_o the_o arch_n give_v or_o seek_v that_o the_o cousin_n or_o cotangent_fw-la of_o a_o arch_n great_a than_o a_o quadrant_n be_v the_o sine_fw-la or_o tangent_fw-la of_o the_o excess_n of_o that_o arch_n above_o a_o quadrant_n or_o 90_o degree_n 8._o what_o trouble_v the_o ancient_n be_v at_o in_o resolve_v of_o proportion_n by_o multiplication_n and_o division_n be_v wonderful_o abbreviate_v by_o a_o admirable_a invention_n call_v logarithme_n where_o by_o frame_v and_o substitute_v other_o number_n in_o stead_n of_o the_o former_a multiplication_n be_v change_v into_o addition_n and_o division_n into_o substraction_n of_o which_o also_o see_v the_o former_a book_n trigonometria_fw-la brittanica_fw-la 9_o what_o may_v be_v perform_v by_o either_o of_o the_o former_a kind_n of_o table_n may_v also_o with_o a_o line_n of_o chord_n and_o equal_a part_n be_v perform_v but_o not_o so_o near_o the_o truth_n without_o they_o and_o that_o either_o by_o project_v or_o represent_v the_o sphere_n on_o a_o flat_a or_o plain_a as_o we_o have_v handle_v in_o the_o second_o and_o three_o part_n or_o by_o protract_v and_o delineat_v of_o such_o proportion_n as_o may_v be_v wrought_v by_o the_o table_n and_o this_o in_o some_o measure_n be_v the_o intend_a subject_n of_o our_o subsequent_a discourse_n 10._o therefore_o 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natural_a may_v supply_v the_o want_n of_o they_o example_n 1._o let_v the_o natural_a sine_fw-la of_o 38_o be_v require_v the_o logarithmical_a sine_fw-la of_o that_o ark_n be_v 9_o 789342_o because_o we_o may_v make_v the_o radius_fw-la of_o the_o natural_a sin_n to_o be_v a_o unit_fw-la and_o all_o the_o rest_n to_o be_v decimal_a part_n thereof_o reject_v the_o first_o figure_n of_o the_o logarithm_n or_o characterisk_n be_v 9_o and_o seek_v the_o remain_a figure_n of_o the_o logarithmical_a sine_fw-la to_o wit_n 789342_o among_o the_o logarithm_n of_o absolute_a number_n and_o you_o will_v find_v the_o absolute_a number_n answer_v thereto_o to_o be_v 61566_o near_o and_o that_o be_v the_o natural_a sine_fw-la of_o 38_o degree_n require_v 2._o the_o table_n of_o the_o meridian-line_n may_v be_v supply_v by_o this_o proportion_n raise_v out_o of_o mr._n bond_n addition_n to_o mr._n gunter'_v work_n as_o 75795_o be_v to_o 60_o ∷_o 100_o ∷_o so_n be_v the_o difference_n of_o the_o logarithmical_a tangent_n of_o 45_o and_o of_o a_o ark_n compound_v or_o make_v of_o 45_o and_o of_o half_a the_o give_v ark_n to_o the_o meridional_a part_n belong_v to_o the_o give_v ark_n ∷_o if_o you_o use_v the_o number_n 60_o you_o will_v make_v such_o a_o sexagesimal_n table_n as_o mr._n wright_n or_o mr._n norwoods_n but_o if_o you_o use_v the_o number_n 100_o than_o you_o will_v produce_v a_o centesimal_n table_n like_o mr._n gunter_n or_o that_o in_o mr._n roe_n table_n in_o which_o the_o ark_n to_o which_o the_o meridional_a table_n be_v fit_v be_v degree_n and_o every_o other_o minute_n but_o the_o tabular_a number_n be_v degree_n and_o decimal_n be_v 〈◊〉_d very_o good_a table_n of_o that_o kind_n and_o much_o full_a than_o either_o mr._n gunter_n or_o mr._n norwoods_n the_o table_n at_o the_o end_n of_o this_o book_n be_v of_o the_o same_o kind_n but_o fit_v to_o every_o second_o centesm_n of_o a_o degree_n in_o stead_n of_o every_o second_o minute_n the_o ground_n of_o the_o former_a proportion_n be_v that_o the_o logarithmical_a tangent_n above_o 45_o account_v every_o half_a degree_n for_o a_o whole_a one_o be_v in_o the_o same_o ratio_fw-la or_o proportion_n with_o a_o table_n of_o the_o meridian-line_n whence_o also_o it_o follow_v for_o instrumental_a use_n that_o a_o line_n of_o logarithmical_a tangent_n will_v supply_v the_o defect_n of_o the_o meridian-line_n the_o number_n above_o use_v to_o wit_n 75795_o be_v not_o the_o difference_n between_o the_o logarithmical_a tangent_n of_o 45_o and_o 45½_n d_o though_o near_o it_o be_v the_o difference_n of_o the_o logarithmical_a tangent_n of_o 45_o and_o of_o 45_o and_o one_o centesm_n more_o multiply_v by_o 50_o and_o for_o to_o avoid_v the_o trouble_n of_o division_n in_o work_v this_o proportion_n it_o be_v convenient_a either_o to_o have_v a_o table_n of_o the_o say_a number_n multiply_v by_o all_o the_o nine_o digit_n or_o rather_o to_o alter_v the_o proportion_n so_o as_o a_o unit_fw-la may_v be_v the_o first_o term_n and_o then_o make_v such_o a_o table_n for_o the_o second_o term_n the_o say_a proportion_n in_o a_o manner_n will_v be_v wrought_v whole_o by_o addition_n and_o substraction_n what_o former_a age_n perform_v by_o table_n this_o latter_a age_n have_v endeavour_v in_o some_o respect_n to_o perform_v without_o they_o snellius_n in_o his_o cyclometria_n show_v we_o how_o the_o side_n of_o a_o plain_a right_a angle_a triangle_n be_v give_v we_o may_v without_o table_n find_v the_o angle_n of_o that_o triangle_n in_o the_o right_n angle_v plain_a triangle_n c_o e_o a_o with_o the_o radius_fw-la c_o a_o describe_v a_o semicircle_n produce_v the_o diameter_n and_o therein_o make_v d_o r_z ●qual_a to_o the_o radius_fw-la and_o draw_v r_o b_o passing_n through_o a_o till_o it_o meet_v with_o the_o tangent_fw-la i_o b_o at_z b_o now_o that_o

three_o side_n first_o which_o be_v the_o thing_n require_v in_o the_o three_o case_n yet_o it_o may_v be_v resolve_v from_o proportion_n immediate_o without_o the_o knowledge_n of_o the_o three_o side_n as_o i_o have_v explain_v in_o a_o treatise_n of_o geometrical_a dyall_n as_o also_o in_o the_o first_o book_n of_o the_o mariner_n plain_a scale_n new_o plain_v there_o remain_v yet_o six_o other_o case_n to_o wit_n 7._o two_o side_n with_o a_o angle_n opposite_a to_o one_o of_o they_o to_o find_v the_o ●hird_n side_n 8._o from_o the_o same_o data_fw-la or_o part_n of_o the_o triangle_n give_v to_o find_v ●●e_z angle_v include_v that_o be_v the_o angle_n between_o the_o two_o give_v ●●des_n 9_o to_o find_v the_o angle_n opposite_a to_o the_o other_o side_n the_o eight_o and_o nine_o case_n also_o require_v the_o three_o side_n to_o be_v find_v first_o before_o the_o angle_n include_v or_o the_o angle_n opposite_a to_o the_o other_o side_n can_v be_v find_v from_o this_o projection_n as_o to_o my_o knowledge_n yet_o be_v easy_o resolve_v from_o proportion_n without_o find_v the_o say_v three_o side_n at_o all_o as_o in_o the_o say_v first_o part._n 10._o two_o angle_n with_o a_o side_n opposite_a to_o one_o of_o they_o to_o find_v the_o three_o angle_n 11._o to_o find_v the_o interjacent_a side_n or_o side_n between_o they_o 12._o to_o find_v the_o side_n opposite_a to_o the_o other_o angle_n and_o these_o three_o case_n be_v the_o same_o with_o the_o seven_o eight_o and_o nine_o when_o the_o angle_n be_v change_v into_o side_n and_o the_o side_n into_o a_o angle_n these_o six_o latter_a case_n i_o call_v the_o doubtful_a case_n because_o sometime_o a_o double_a answer_n must_v be_v give_v and_o both_o true_a and_o when_o these_o case_n will_v happen_v and_o when_o not_o i_o have_v show_v at_o large_a in_o a_o treatise_n call_v the_o sector_n on_o a_o quadrant_a 3._o two_o side_n with_o a_o angle_n opposite_a to_o they_o to_o find_v the_o three_o side_n by_o this_o case_n if_o the_o sun_n true_a azimuth_n and_o altitude_n thereto_o be_v observe_v at_o any_o time_n off_o the_o meridian_n and_o his_o declination_n give_v the_o latitude_n may_v be_v find_v example_n give_v sun_n declination_n 23_o 31′_n north._n azimuth_n 20_o to_o southwards_o of_o the_o vertical_a altitude_n 44_o 39′_n demand_v the_o latitude_n z_o ae_n 51_o 32′_n in_o every_o scheme_n of_o the_o analemma_n observe_v that_o the_o near_a distance_n of_o the_o point_n of_o the_o sun_n place_n in_o his_o parallel_n of_o altitude_n from_o the_o equinoctial_a line_n be_v equal_a to_o the_o sine_fw-la of_o his_o declination_n from_o whence_o the_o resolution_n of_o this_o proposition_n easy_o follow_v have_v with_o 60_o degree_n of_o the_o chord_n upon_o the_o centre_n c_o draw_fw-mi the_o semicircle_n s_o z_o n_o the_o diameter_n s_o c_o n_o the_o axis_n of_o the_o horizon_n z_o c_o out_z of_o the_o chord_n prick_v off_o the_o give_v altitude_n from_o s_n to_o a_o and_z from_z n_z to_z b_o and_o draw_v the_o parallel_n thereof_o a_o f_o d_o b_o then_z ptoportion_n out_o the_o sine_fw-la of_o 20_o degree_n to_o the_o radius_fw-la a_o d_o draw_v the_o line_n a_o c_o and_o therein_o from_o the_o sin_n prick_v off_o 20_o degree_n from_o c_o to_o 20_o degree_n and_o take_v the_o near_a distance_n to_o c_o d_o and_o set_v that_o extent_n from_o d_o to_o the_o point_n f_o then_o take_v 23_o degree_n 31_o minute_n from_o the_o sin_n and_o set_v one_o foot_n in_o f_o with_o the_o other_o draw_v the_o arch_n l_o a_o right_a line_n draw_v from_o c_o just_o touch_v the_o outward_a extremity_n of_o that_o arch_n as_o do_v ae_n c_o shall_v be_v the_o equinoctial_a and_o so_o the_o arch_n s_o ae_n be_v the_o compliment_n of_o the_o latitude_n and_o z_o ae_n the_o latitude_n seek_v in_o this_o case_n 51_o 32′_n by_o this_o case_n of_o triangle_n the_o declination_n and_o latitude_n be_v give_v the_o sun_n altitude_n or_o depression_n to_o each_o assign_a azimuth_n may_v be_v find_v this_o case_n may_v be_v do_v without_o the_o draw_n of_o any_o ellipsis_n and_o that_o divers_a way_n though_o it_o have_v not_o hitherto_o be_v so_o repute_v in_o order_n whereto_o observe_v that_o any_o two_o contain_v side_n of_o a_o triangle_n may_v change_v name_n as_o in_o the_o former_a example_n if_o the_o latitude_n be_v 44_o 39′_n the_o altitude_n to_o that_o azimuth_n will_v be_v 38_o 28′_n consequent_o by_o the_o same_o scheme_n and_o manner_n of_o work_n if_o the_o azimuth_n be_v prick_v off_o in_o the_o parallel_n of_o latitude_n the_o respective_a altitude_n thereto_o may_v be_v find_v but_o this_o scheme_n i_o have_v further_o insist_v upon_o in_o the_o first_o part._n another_o manner_n of_o operation_n let_v the_o latitude_n be_v 51_o 32′_n north._n the_o sun_n declination_n 23_o 31′_n north._n and_o let_v it_o be_v require_v to_o find_v the_o sun_n respective_a altitude_n to_o the_o azimuth_n of_o 20_o and_o 60_o on_o each_o side_n from_o the_o vertical_a circle_n upon_o the_o centre_n c_o describe_v a_o arch_n big_a than_o a_o semicircle_n and_o draw_v the_o horizontal_n line_n s_o c_o n_o and_o its_o axis_n z_o c_o at_o right_a angle_n thereto_o and_o set_v off_o the_o latitude_n from_o s_n to_o a_o and_o draw_v the_o parallel_n thereof_o l_o a_o set_v off_o the_o declination_n likewise_o from_o n_z to_z e_z and_o draw_v the_o parallel_n thereof_o d_o e_z which_o be_v easy_o do_v by_o prick_v off_o the_o like_a arch_n above_o s._n now_o to_o proportion_n out_o the_o sin_n to_o the_o radius_fw-la a_o l_o draw_v the_o line_n a_o c_o and_o there_o prick_v down_o the_o sin_n of_o 20_o and_o 60_o and_o take_v the_o near_a distance_n therefrom_o to_o the_o line_n c_o z_o and_o transfer_v or_o place_v those_o respective_a extent_n into_o the_o parallel_n of_o latitude_n l_o a_o prick_v they_o down_o from_o l_o then_z one_o foot_n of_o the_o compass_n rest_v in_o c_o take_v the_o distance_n to_o 20_o and_o 60_o in_o the_o parallel_n of_o latitude_n l_o a_o and_o transfer_v or_o place_v they_o into_o the_o parallel_n of_o declination_n d_o e_o and_o there_o number_v they_o as_o before_z then_o take_v the_o distance_n between_o the_o zenith_n at_o z_o and_o 20_o in_o the_o parallel_v of_o latitude_n l_o a_o and_o the_o say_a extent_n shall_v reach_v from_o 20_o in_o the_o parallel_n of_o declination_n d_o e_o to_z f_o upward_o and_o to_o h_n downward_o the_o arch_n n_o f_o be_v 44_o 39′_n be_v the_o sun_n height_n for_o 20_o of_o azimuth_n to_o the_o southwards_o of_o the_o east_n or_o west_n and_o the_o arch_n n_o h_n 14_o 15′_n be_v the_o sun_n height_n for_o the_o like_a azimuth_n to_o the_o northwards_o of_o the_o east_n or_o west_n also_o the_o distance_n between_o 60_o in_o the_o parallel_n of_o latitude_n and_o the_o zenith_n z_o reach_v from_o 60_o in_o the_o parallel_n of_o declination_n to_o q_o upward_o and_o to_o r_n downward_o the_o ark_n n_o qui_fw-fr be_v 59_o 21′_n be_v the_o sun_n height_n for_o 60_o of_o azimuth_n to_o the_o southwards_o of_o the_o east_n or_o west_n and_o the_o ark_n n_o r_n be_v 9_o 43′_n be_v the_o sun_n depression_n in_o summer_n for_o 60_o of_o azimuth_n to_o the_o northwards_o of_o the_o east_n or_o west_n it_o be_v also_o his_o winter_n altitude_n for_o 60_o of_o azimuth_n from_o east_n or_o west_n southwards_o when_o his_o declination_n be_v 23_o 31′_n south_n which_o be_v as_o much_o as_o we_o suppose_v it_o to_o be_v north_n the_o ground_n of_o the_o former_a scheme_n be_v that_o the_o sun_n in_o his_o parallel_n of_o altitude_n have_v the_o like_a distance_n from_o the_o zenith_n as_o in_o his_o parallel_n of_o declination_n also_o his_o distance_n from_o the_o centre_n in_o his_o parallel_n of_o altitude_n be_v the_o same_o with_o his_o distance_n from_o the_o centre_n in_o his_o parallel_n of_o declination_n whence_o follow_v another_o delineation_n of_o the_o analemma_n suitable_a to_o this_o case_n for_o we_o may_v place_v the_o parallel_n of_o declination_n parallel_v to_o the_o horizon_n which_o now_o must_v represent_v the_o equinoctial_a then_o if_o the_o sine_fw-la of_o the_o azimuth_n be_v prick_v down_o in_o any_o give_a parallel_n of_o altitude_n the_o distance_n of_o that_o point_n from_o the_o centre_n and_o zenith_n be_v both_o give_v then_o make_v the_o sun_n to_o have_v the_o like_a distance_n in_o his_o parallel_n of_o declination_n from_o the_o centre_n as_o he_o have_v in_o his_o parallel_n of_o latitude_n and_o moreover_o if_o from_o the_o say_v point_v his_o distance_n from_o the_o zenith_n in_o his_o parallel_n of_o altitude_n be_v apply_v upward_o it_o will_v give_v the_o latitude_n or_o zenith_n seek_v from_o the_o

meridian_n count_v the_o same_o in_o the_o variation_n column_n upward_o and_o against_o it_o in_o the_o sin_n you_o will_v find_v the_o difference_n of_o latitude_n be_v 923_o league_n and_o 87_o part_n more_o of_o another_o league_n divide_v into_o 100_o part_n but_o if_o the_o difference_n of_o latitude_n be_v require_v for_o any_o other_o distance_n multiply_v the_o sine_fw-la of_o the_o give_a point_n by_o the_o distance_n run_v cut_v off_o three_o figure_n from_o the_o product_n example_n if_o the_o ship_n sail_n 60_o league_n on_o that_o course_n the_o difference_n of_o latitude_n be_v 55_o league_n and_o 38_o centesm_n multiply_v 923_o by_o 60_o and_o the_o product_n cut_v off_o three_o place_n be_v 55_o 380._o to_o find_v the_o departure_n from_o the_o meridian_n the_o proportion_n be_v as_o the_o radius_fw-la be_v to_o the_o distance_n run_v so_o be_v the_o sine_fw-la of_o the_o rumbe_n to_o the_o departure_n ∷_o example_n if_o a_o ship_n sail_v 1000_o league_n south_n south_n west_n count_v the_o point_n sail_v in_o the_o separation_n column_n downward_o be_v the_o three_o column_n and_o on_o the_o right_a hand_n of_o the_o sin_n and_o you_o will_v in_o the_o column_n of_o sines_n find_v the_o departure_n require_v to_o be_v 382_o league_n 68_o centesm_n but_o if_o a_o ship_n sail_v but_o 60_o league_n on_o that_o course_n multiply_v 382_o thereby_o cut_v off_o the_o three_o last_o place_n and_o you_o will_v find_v the_o separation_n or_o departure_n require_v to_o be_v 22_o league_n 92_o centesm_n or_o 22_o 920_o league_n point_v sines_n separ_fw-la tangent_n secant_v angle_n sin_n quintuple_v deg_fw-la part_n ¾_n 49_o 07_o ¼_n 49_o 12_o 1001_o 20_o 2_o 48_o m_o ¾_n 2_o 45_o ½_n 98_o 02_o ½_n 98_o 48_o 1004_o 84_o 5_o 37_o ½_n 4_o 90_o ¼_n 146_o 73_o ¾_n 148_o 32_o 1010_o 94_o 8_o 26_o ¼_n 7_o 33_o 7_o 195_o 09_o 1_o 198_o 91_o 1019_o 59_o 11_o 15_o 9_o 75_o ¾_n 242_o 98_o ¼_n 250_o 48_o 1030_o 89_o 14_o 3_o ¾_n 12_o 14_o ½_n 290_o 28_o ½_n 303_o 36_o 1044_o 99_o 16_o 52_o ½_n 14_o 51_o ¼_n 336_o 89_o ¾_n 357_o 80_o 1062_o 08_o 19_o 41_o ¼_n 16_o 84_o 6_o 382_o 68_o 2_o 414_o 21_o 1082_o 39_o 22_o 30_o 19_o 13_o ¾_n 427_o 55_o ¼_n 472_o 96_o 1106_o 21_o 25_o 18_o ¾_n 21_o 37_o ½_n 471_o 39_o ½_n 534_o 52_o 1133_o 88_o 28_o 7_o ½_n 23_o 56_o ¼_n 514_o 10_o ¾_n 599_o 36_o 1165_o 96_o 30_o 56_o ¼_n 25_o 70_o 5_o 555_o 57_o 3_o 668_o 17_o 1202_o 68_o 33_o 45_o 27_o 77_o ¾_n 595_o 70_o ¼_n 741_o 65_o 1245_o 28_o 36_o 33_o ¾_n 29_o 78_o ½_n 634_o 39_o ½_n 820_o 68_o 1293_o 64_o 39_o 22_o ½_n 31_o 71_o ¼_n 671_o 56_o ¾_n 906_o 34_o 1349_o 61_o 42_o 11_o ¼_n 33_o 57_o 4_o 707_o 10_o 4_o 1000_o 00_o 1414_o 21_o 45_o 35_o 35_o ¾_n 740_o 96_o ¼_n 1103_o 32_o 1489_o 08_o 47_o 48_o ¾_n 37_o 04_o ½_n 773_o 01_o ½_n 1218_o 48_o 1576_o 32_o 50_o 37_o ½_n 38_o 65_o ¼_n 803_o 21_o ¾_n 1348_o 36_o 1678_o 68_o 53_o 26_o ¼_n 40_o 16_o 3_o 831_o 47_o 5_o 1496_o 60_o 1799_o 95_o 56_o 15_o 41_o 57_o ¾_n 857_o 73_o ¼_n 1668_o 37_o 1945_o 14_o 59_o 3_o ¾_n 42_o 88_o ½_n 881_o 91_o ½_n 1870_o 73_o 2121_o 36_o 61_o 52_o ½_n 44_o 09_o ¼_n 903_o 99_o ¾_n 2114_o 20_o 2338_o 88_o 64_o 41_o ¼_n 45_o 19_o 2_o 923_o 87_o 6_o 2414_o 21_o 2613_o 12_o 67_o 30_o 46_o 19_o ¾_n 941_o 53_o ¼_n 2794_o 76_o 2968_o 34_o 70_o 18_o ¾_n 47_o 08_o ½_n 956_o 94_o ½_n 3296_o 50_o 3445_o 69_o 73_o 7_o ½_n 47_o 84_o ¼_n 970_o 03_o ¾_n 3992_o 24_o 4115_o 56_o 75_o 56_o ¼_n 48_o 50_o 1_o 980_o 78_o 7_o 5027_o 33_o 5125_o 83_o 78_o 45_o 49_o 03_o ¾_n 989_o 17_o ¼_n 6741_o 44_o 6808_o 52_o 81_o 33_o ¾_n 49_o 45_o ½_n 995_o 18_o ½_n 10153_o 66_o 10202_o 32_o 84_o 22_o ½_n 49_o 75_o ¼_n 998_o 79_o ¼_n 20355_o 48_o 20380_o 15_o 87_o 11_o ¼_n 49_o 93_o variation_n  _fw-fr  _fw-fr  _fw-fr  _fw-fr  _fw-fr  _fw-fr here_o note_v that_o though_o the_o distance_n sail_v be_v give_v in_o league_n yet_o the_o table_n of_o sines_n may_v be_v so_o alter_v that_o the_o difference_n of_o latitude_n and_o departure_n may_v be_v find_v in_o degree_n and_o centesme_n and_o that_o only_o be_v multiply_v all_o the_o figure_n in_o the_o column_n of_o sines_n by_o the_o number_n 5_o for_o which_o purpose_n we_o add_v the_o last_o column_n of_o quintupuled_a sin_n in_o which_o the_o whole_a degree_n be_v distinguish_v from_o the_o decimal_a part_n example_n if_o a_o ship_n sail_v 1000_o league_n south_n south-west_n against_o the_o second_o point_n count_v upward_o we_o shall_v find_v the_o difference_n of_o latitude_n to_o be_v 46_o 19_o centesme_n but_o count_v downward_o we_o shall_v find_v the_o departure_n from_o the_o meridian_n to_o be_v 19_o degree_n 13_o centesme_n which_o number_n multiply_v by_o 60_o cut_n off_o five_o place_n towards_o the_o right_a hand_n give_v the_o difference_n of_o latitude_n and_o departure_n in_o that_o course_n for_o 60_o league_n to_o be_v 2_o 77140_o part_n diff._n of_o lat._n 1_o 14780_o departure_n of_o which_o result_v we_o need_v take_v no_o more_o place_n than_o 2_o d_o 77._o 1_o 14_o or_o 1_o 15._o by_o these_o two_o proposition_n the_o dead_a reckon_a to_o be_v apply_v to_o the_o plain_a chart_n may_v be_v keep_v after_o the_o form_n prescribe_v by_o mr._n norwood_n and_o according_o the_o difference_n of_o latitude_n and_o departure_n from_o the_o meridian_n be_v here_o express_v in_o degree_n and_o centesme_n for_o the_o three_o traverse_n from_o tenariff_n towards_o nicholas_n island_n which_o we_o make_v our_o former_a example_n course_n distance_n in_o league_n north_n south_n deg._n cent._n east_n west_n d_o cent._n s_o s_o w_n 60_o  _fw-fr 2_o 77_o  _fw-fr 1_o 15_o w_o s_o w_n 80_o  _fw-fr 1_o 53_o  _fw-fr 3_o 69_o s_o b_o e_o ½_n e_o 53_o  _fw-fr 2_o 53_o 0_o 77_o 0_o 00_o  _fw-fr  _fw-fr  _fw-fr 6_o 83_o 0_o 77_o 4_o 84_o when_o some_o course_n increase_v the_o latitude_n and_o other_o decrease_v it_o the_o difference_n between_o the_o north_n and_o south_n column_n find_v by_o substract_v the_o lesser_a from_o the_o great_a give_v the_o difference_n of_o latitude_n so_o likewise_o the_o difference_n of_o the_o east_n and_o west_n column_n give_v the_o departure_n which_o in_o this_o example_n be_v 4_o 07_o this_o difference_n of_o latitude_n and_o departure_n substract_v out_o of_o the_o whole_a difference_n of_o latitude_n and_o of_o longitude_n between_o the_o say_v two_o island_n there_o rest_v 4_o 17_o centesme_n difference_n of_o latitude_n between_o the_o ship_n and_o st._n nicholas_n island_n and_o 3_o 93_o difference_n of_o longitude_n by_o which_o the_o course_n from_o the_o ship_n to_o the_o say_a island_n will_v be_v find_v to_o be_v 43_o 28_o centesm_n to_o the_o westward_n of_o the_o south_n and_o the_o distance_n 5_o 73_o be_v about_o 114_o league_n and_o a_o half_a as_o shall_v afterward_o follow_v the_o difference_n of_o latitude_n and_o the_o rumbe_n give_v to_o find_v the_o departure_n from_o the_o meridian_n and_o distance_n as_o the_o radius_fw-la be_v to_o the_o difference_n of_o latitude_n ∷_o so_n be_v the_o tangent_fw-la of_o the_o rumbe_n to_o the_o departure_n from_o the_o meridian_n ∷_o and_o so_o be_v the_o secant_fw-la of_o the_o rumbe_n to_o the_o distance_n ∷_o example_n for_o the_o departure_n let_v the_o instance_n be_v the_o same_o as_o in_o page_n 24_o where_o we_o suppose_v the_o course_n to_o be_v 3_o point_n from_o the_o meridian_n to_o wit_n s_o w_n b_o s_o and_o the_o correct_a difference_n of_o latitude_n to_o be_v 2_o degree_n 24_o minute_n or_o 2_o degree_n 4_o ten_o or_o 40_o centesm_n by_o which_o multiply_a 668_o the_o tangent_fw-la belong_v to_o 3_o point_n in_o the_o former_a table_n the_o result_n cut_v off_o 5_o figure_n be_v 1_o 60320_o wherefore_o the_o departure_n from_o the_o meridian_n be_v 1_o degree_n 6_o ten_o for_o the_o distance_n in_o like_a manner_n the_o secant_fw-la of_o three_o point_n to_o wit_n 1202_o be_v multiply_v by_o the_o difference_n of_o latitude_n 2_o 40_o cut_n off_o 5_o figure_n be_v 2_o 88480_o wherefore_o the_o distance_n be_v 2_o degree_n 88_o centesme_n at_o 20_o league_n to_o a_o degree_n be_v 50_o league_n and_o three_o quarter_n or_o 50_o 76._o for_o keep_v a_o reckon_n by_o longitude_n which_o be_v applyable_a to_o mercators_n chart_n and_o remove_v the_o error_n of_o the_o plain_a chart_n the_o table_n of_o meridional_a part_n be_v add_v at_o the_o end_n of_o the_o book_n for_o work_v of_o this_o proportion_n as_o the_o radius_fw-la be_v to_o the_o tangent_fw-la of_o the_o rumbe_n ∷_o so_n be_v