Selected quad for the lemma: book_n

Word A	Word B	Word C	Word D	Occurrence	Frequency Band	MI	MI Band	Prominent
book_n	difference_n	sum_n	verse_v	36	3	16.3064	5	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

down_o every_o five_o degree_n from_o c_o to_o d_o then_o lay_v the_o beginning_n of_o your_o tangent_n over_o p_o and_o every_o one_o of_o those_o graduation_n you_o may_v by_o the_o view_n see_v what_o latitude_n the_o arch_n pass_v through_o at_o every_o 5_o degree_n difference_n of_o longitude_n from_o the_o perpendicular_a for_o which_o purpose_n the_o tangent_n on_o your_o ruler_n may_v be_v double_o number_v both_o with_o the_o arch_n and_o its_o compliment_n to_o which_o they_o belong_v example_n lay_v the_o ruler_n over_o 25_o degree_n it_o cut_v the_o arch_n at_o e_o and_o the_o extent_n p_o e_o be_v measure_v on_o the_o tangent_n be_v 41_o 51′_n be_v the_o compliment_n of_o the_o latitude_n of_o the_o arch_n at_o that_o difference_n of_o longitude_n wherefore_o the_o latitude_n of_o the_o arch_n at_o 25_o difference_n of_o longitude_n from_o the_o perpendicular_a be_v 48_o 9′_n in_o like_a manner_n the_o latitude_n of_o the_o arch_n for_o 00_o difference_n of_o longitude_n from_o the_o perpendicular_a be_v 50_o 56′_n 05_o 50_o 49_o 10_o 50_o 30_o 15_o 49_o 57_o 20_o 49_o 10_o 25_o 48_o 9_o 30_o 46_o 51_o 35_o 45_o 16_o 40_o 43_o 20_o 45_o 41_o 4_o 50_o 38_o 23_o and_o be_v the_o same_o for_o the_o first_o ten_o degree_n difference_n of_o longitude_n on_o each_o side_n the_o perpendicular_a and_o also_o as_o far_o as_o the_o lesser_a vertical_a angle_n a_o second_o example_n for_o two_o place_n in_o different_a hemisphere_n in_o like_a manner_n if_o we_o suppose_v trinity_n harbour_v to_o be_v in_o as_o much_o south_n latitude_n the_o difference_n of_o longitude_n be_v the_o same_o to_o wit_n 68_o 30′_n place_n p_o l_o from_o p_o to_o i_o and_o draw_v i_o t_o continue_v and_o it_o shall_v represent_v the_o great_a arch_n be_v continue_v from_o i_o upward_o run_v from_o the_o lizard_n towards_o the_o equinoctial_a and_o below_o t_n downward_o run_v also_o from_o the_o suppose_a place_n in_o south_n latitude_n towards_o the_o equinoctial_a where_o in_o this_o projection_n it_o grow_v infinite_a let_v fall_v the_o perpendicular_a p_o qui_fw-fr equal_a to_o the_o radius_fw-la and_o prick_v off_o the_o tangent_n from_o q_o downward_o beyond_o r_n parallel_n to_o h_n t_o and_o if_o you_o set_v p_o i_o from_o p_o to_o k_o so_o as_o to_o meet_v with_o h_n t_o than_o you_o may_v measure_v the_o latitude_n of_o the_o ark_n on_o each_o side_n the_o equator_fw-la the_o the_o same_o way_n by_o lay_v the_o beginning_n of_o the_o tangent_n on_o the_o edge_n of_o the_o ruler_n to_o p_o and_o lay_v the_o ruler_n as_o before_o to_o every_o five_o degree_n of_o longitude_n thus_o the_o latitude_n of_o the_o arch_n at_o 55_o difference_n of_o longitude_n from_o the_o perpendicular_a be_v 44_o 43′_n 60_o 40_o 49_o 65_o 36_o 7_o 70_o 30_o 34_o 75_o 24_o 4_o 80_o 16_o 42_o 85_o 8_o 33_o we_o have_v continue_v the_o tangent_n but_o to_o 65_o at_o r_n for_o want_n of_o room_n however_o without_o any_o great_a excursion_n the_o latitude_n of_o it_o may_v be_v find_v near_o the_o equinoctial_a by_o this_o proportion_n as_o the_o tangent_fw-la of_o the_o perpendicular_a be_v to_o the_o cousin_n of_o the_o angle_n adjacent_a ∷_o so_n be_v the_o radius_fw-la to_o the_o tangent_fw-la of_o the_o arke_n latitude_n example_n let_v it_o be_v require_v to_o find_v the_o latitude_n of_o the_o arch_n at_o 70_o degree_n difference_n of_o longitude_n from_o the_o perpendicular_a the_o compliment_n thereof_o be_v 20_o to_o the_o tangent_fw-la whereof_o draw_v a_o line_n from_o p_o than_o the_o near_a distance_n from_o q_o to_o the_o say_a line_n be_v the_o cousin_n of_o the_o say_a angle_n which_o prick_v from_o h_n to_o s_n a_o ruler_n over_o p_o and_o s_o cut_v this_o tangent-line_n q_o r_n at_o 30_o degree_n 34_o minute_n and_o so_o much_o be_v the_o latitude_n require_v and_o at_o 90_o degree_n from_o the_o perpendicular_a this_o arch_n pass_v over_o the_o equinoctial_a a_o three_o example_n for_o two_o place_n in_o the_o same_o parallel_n to_o wit_n in_o the_o latitude_n of_o 50_o degree_n difference_n of_o longitude_n 92_o degree_n 46_o minute_n thus_o we_o see_v this_o projection_n will_v supply_v all_o the_o case_n that_o can_v be_v put_v and_o be_v in_o effect_n no_o other_o than_o a_o scheme_n for_o the_o carry_v on_o of_o proportion_n and_o the_o sphere_n be_v thus_o reduce_v into_o right_a line_n we_o may_v thence_o raise_v proportion_n for_o calculate_v all_o that_o be_v require_v in_o the_o obliqueangled_n plain_a triangle_n t_o p_o l_o we_o have_v the_o two_o side_n t_o p_o and_o p_o l_o given_z be_v the_o tangent_n of_o the_o compliment_n of_o both_o latitude_n and_o the_o angle_n comprehend_v t_o p_o l_o given_z now_o we_o find_v the_o other_o angle_n at_o t_n and_o l_o by_o this_o proportion_n as_o the_o sum_n of_o the_o two_o side_n be_v to_o their_o difference_n ∷_o so_n be_v the_o tangent_fw-la of_o the_o half_a sum_n of_o the_o opposite_a angle_n to_o the_o tangent_fw-la of_o half_a their_o difference_n ∷_o which_z proportion_n be_v demonstrate_v in_o many_o book_n of_o trigonometrie_n and_o so_o the_o two_o side_n of_o this_o triangle_n be_v the_o tangent_n of_o the_o compliment_n of_o the_o latitude_n but_o as_o the_o sum_n of_o the_o tangent_fw-la of_o any_o two_o arke_n be_v to_o the_o difference_n of_o those_o tangent_n ∷_o so_n be_v the_o sine_fw-la of_o the_o sum_n of_o those_o arke_n to_o the_o sine_fw-la of_o the_o difference_n of_o those_o arke_n see_v this_o demonstrate_v in_o mr._n newtons_n trigonometria_fw-la brittanica_fw-la again_o the_o three_o term_n be_v the_o tangent_fw-la of_o the_o half_a sum_n of_o the_o opposite_a angle_n in_o other_o language_n be_v the_o cotangent_fw-la of_o half_a the_o contain_v angle_n last_o the_o half_a sum_n of_o the_o unknown_a angle_n be_v add_v to_o their_o half_a difference_n the_o sum_n make_v the_o great_a and_o the_o difference_n the_o lesser_a of_o those_o unknown_a angle_n in_o like_a manner_n if_o the_o say_v half_a difference_n be_v add_v to_o half_a the_o contain_v angle_n the_o sum_n be_v the_o great_a of_o those_o angles_n next_o the_o perpendicular_a and_o the_o difference_n be_v the_o lesser_a and_o these_o angle_n we_o call_v the_o vertical_a angle_n the_o proportion_n for_o find_v the_o perpendicular_a have_v the_o hipotenusal_a or_o colatitude_n and_o the_o vertical_a angle_n give_v be_v as_o the_o radius_fw-la be_v to_o the_o cousin_n of_o the_o vertical_a angle_n ∷_o so_n be_v the_o cotangent_fw-la of_o the_o latitude_n to_o the_o targent_n of_o the_o perpendicular_a ∷_o then_z ass_n gn_v the_o difference_n of_o longitude_n from_o the_o perpendicular_a we_o nave_n before_o set_v down_o the_o proportion_n for_o find_v the_o ark_n latitude_n and_o both_o these_o proportion_n may_v be_v bring_v into_o one_o and_o so_o the_o method_n of_o calculation_n arise_v out_o of_o these_o consideration_n sun_n thus_o 1._o for_o a_o parallel_n or_o east_n and_o west_n course_n as_o the_o cousin_n of_o half_a the_o difference_n of_o longitude_n be_v to_o the_o tangent_fw-la of_o the_o common_a latitude_n ∷_o so_n be_v the_o cousin_n of_o the_o ark_n of_o longitude_n from_o the_o perpendicular_a to_o the_o tangent_fw-la of_o the_o ark_n latitude_n answer_v thereto_o ∷_o again_n in_o all_o other_o case_n for_o the_o vertical_a angle_n as_o the_o sine_fw-la of_o the_o sum_n of_o the_o compliment_n of_o both_o latitude_n be_v to_o the_o sine_fw-la of_o their_o difference_n ∷_o so_n be_v the_o tangent_fw-la of_o the_o compliment_n of_o half_a the_o difference_n of_o longitude_n to_o the_o tangent_fw-la of_o half_a the_o difference_n of_o the_o vertical_a angle_n ∷_o which_o ark_n thus_o find_v add_v to_o half_a the_o difference_n of_o longitude_n the_o sum_n be_v the_o great_a vertical_a angle_n which_o if_o it_o exceed_v the_o difference_n of_o longitude_n the_o perpendicular_a fall_n without_o and_o this_o ark_n comprehend_v the_o whole_a angle_n between_o the_o perpendicular_a and_o both_o colatitude_n the_o difference_n between_o the_o four_o proportional_a ark_n and_o half_a the_o difference_n of_o longitude_n be_v the_o angle_n between_o the_o lesser_a colatitude_n and_o the_o perpendicular_a be_v the_o lesser_a vertical_a angle_n when_o one_o place_n be_v in_o south_n latitude_n and_o the_o other_o in_o north_n the_o first_o and_o second_o term_n of_o this_o general_a proportion_n change_v place_n and_o the_o sine_fw-la of_o the_o sum_n become_v the_o sine_fw-la of_o the_o difference_n and_o this_o that_o as_o will_v be_v find_v if_o you_o make_v one_o of_o the_o contain_v side_n great_a than_o a_o quadrant_n the_o polar_a distance_n in_o stead_n of_o the_o colatitude_n and_o then_o for_o the_o sine_fw-la of_o a_o ark_n great_a than_o a_o quadrant_n take_v the_o sine_fw-la of_o that_o ark_n compliment_n to_o a_o

thy_o work_n or_o try_v the_o rumb_n n_o ever_o desist_v but_o let_v we_o have_v more_o of_o thou_o h_n ever_o be_v but_o a_o tangent_fw-la but_o let_v we_o have_v a_o sine_fw-la o_o r_o bosom_n full_a of_o thy_o industrious_a toil_n i_o t_o will_v inform_v the_o weak_a enrich_v our_o soyl._n your_o love_a friend_n sylvanus_n morgan_n the_o content_n of_o the_o first_o book_n in_o the_o proportional_a part._n geometrical_a definition_n page_z 1_o 2_o to_o raise_v perpendicular_o 2_o 3_o 4_o to_o draw_v a_o line_n parallel_n to_o another_o line_n 5_o to_o bring_v three_o point_n into_o a_o circle_n 6_o to_o find_v a_o right_a line_n equal_a to_o the_o arch_n of_o a_o circle_n 9_o 10_o chord_n sin_n tangent_n secant_v verse_v sines_n etc._n etc._n define_v 11_o 12_o 13_o the_o scale_n in_o the_o frontispiece_n describe_v 14_o plain_a triangle_n both_o right_a and_o oblique_v angle_v resolve_v by_o protraction_n from_o p_o 15_o to_o 25_o proportion_n in_o sines_n resolve_v by_o a_o line_n of_o chord_n p_o 21_o to_o 25_o proportion_n in_o tangent_n alone_o so_o resolve_v p._n 25_o 27_o 28_o 29_o 30_o proportion_n in_o sines_n and_o tangent_n resolve_v by_o a_o line_n of_o chord_n p._n 26_o 31_o 32_o 33_o particular_a scheme_n fit_v from_o proportion_n to_o the_o case_n of_o oblique_a angle_a sphoerical_a triangle_n to_o find_v the_o azimuth_n p._n 34_o 35_o as_o also_o the_o amplitude_n p._n 36_o the_o azimuth_n compass_n in_o the_o frontispiece_n describe_v p._n 38_o the_o variation_n find_v by_o the_o azimuth_n compass_n p._n 39_o to_o find_v the_o hour_n of_o the_o day_n p._n 40_o as_o also_o the_o azimuth_n and_o angle_n of_o position_n p._n 41_o to_o find_v the_o sun_n altitude_n on_o all_o hour_n p._n 43_o 46_o also_o the_o distance_n of_o place_n in_o the_o arch_n of_o a_o great_a circle_n p._n 44_o to_o find_v the_o sun_n altitude_n on_o all_o azimuth_n p._n 48_o the_o latitude_n declination_n and_o azimuth_n give_v to_o find_v the_o hour_n p._n 50_o to_o 54_o ☞_o to_o find_v the_o amplitude_n with_o the_o manner_n of_o measure_v a_o sine_fw-la to_o a_o lesser_a radius_fw-la p._n 55_o to_o get_v the_o sun_n altitude_n by_o the_o shadow_n of_o a_o thread_n or_o gnomon_n p._n 56_o the_o content_n of_o the_o treatise_n of_o navigation_n of_o the_o imperfection_n and_o uncertainty_n of_o navigation_n p._n 1_o to_o 5_o to_o measure_v a_o course_n and_o distance_n on_o the_o plain_a chart._n p._n 7_o 8_o 9_o of_o the_o quantity_n of_o a_o degree_n and_o of_o the_o form_n of_o the_o logboard_n p._n 9_o 10_o a_o reckon_a keep_v in_o league_n how_o reduce_v by_o the_o pen_n to_o degree_n and_o centesme_n p._n 11_o 12_o of_o a_o traverse-quadrant_n p._n 13_o a_o traverse_v plait_v on_o the_o plain_a chart_n without_o draw_v line_n thereon_o p._n 14_o to_o 18_o a_o scheme_n with_o direction_n to_o find_v what_o course_n and_o way_n the_o ship_n have_v make_v through_o a_o current_n p._n 18_o to_o 21_o divers_a rule_n for_o correct_v of_o the_o dead_a reckon_a from_o p._n 21_o to_o 33_o of_o the_o error_n of_o the_o plain_a chart._n p._n 33_o and_o how_o such_o chart_n may_v be_v amend_v p._n 34_o to_o find_v the_o rumbe_n between_o two_o place_n p._n 35_o proportion_n have_v one_o term_n the_o middle_a latitude_n how_o far_o to_o be_v trust_v to_o p._n 35_o to_o 38_o to_o find_v the_o rumbe_n between_o two_o place_n by_o a_o line_n of_o chord_n only_o p._n 39_o to_o 42_o the_o meridian-line_n of_o mercators_n chart_n supply_v general_o by_o a_o line_n of_o chord_n p._n 42_o to_o 47_o the_o meridian-line_n divide_v from_o the_o limb_n of_o a_o quadrant_n with_o the_o use_n thereof_o in_o find_v the_o rumbe_n p._n 48_o to_o 51_o the_o error_n commit_v by_o keep_v of_o a_o reckon_a on_o the_o plain_a chart_n remove_v p._n 52_o to_o 54_o of_o the_o nature_n of_o the_o rumbe_n on_o the_o globe_n p._n 55_o to_o 57_o mereators_n chart_n demonstrate_v from_o proportion_n p._n 58_o to_o 60_o objection_n against_o it_o answer_v p._n 60_o to_o 63_o to_o find_v the_o rumbe_n between_o two_o place_n in_o the_o chart._n p._n 64_o distance_n of_o place_n how_o measure_v on_o that_o chart._n p._n 65_o to_o 71_o another_o traverse-quadrant_n fit_v for_o that_o chart_n with_o a_o traverse_v plait_v thereby_o without_o draw_v line_n on_o the_o chart._n p._n 71_o to_o 78_o to_o measure_v a_o course_n and_o distance_n in_o that_o chart_n without_o the_o use_n of_o compass_n p._n 79_o of_o sail_v by_o the_o arch_n of_o a_o great_a circle_n p._n 81_o to_o find_v the_o latitude_n of_o the_o great_a arch_n by_o the_o stereographick_a projection_n p._n 82_o to_o 83_o of_o a_o tangent_fw-la projection_n from_o the_o pole_n for_o find_v the_o latitude_n of_o the_o great_a arch_n p._n 84_o to_o 88_o with_o a_o new_a method_n of_o calculation_n raise_v from_o it_o p._n 89_o 90_o and_o how_o to_o measure_v the_o distance_n in_o the_o arch_n and_o the_o angle_n of_o position_n p._n 91_o another_o tangent_fw-la projection_n from_o the_o equinoctial_a for_o find_v the_o latitude_n of_o the_o arch._n p._n 93_o to_o 100_o and_o how_o to_o find_v the_o vertical_a angle_n and_o arke_n latitude_n geometrical_o p._n 100_o to_o 102_o to_o draw_v a_o curved-line_n in_o mercators_n chart_n resemble_v the_o arch_n with_o a_o example_n for_o find_v the_o course_n and_o distance_n in_o follow_v the_o arch._n p._n 102_o to_o 104_o the_o dead_a reckon_a cast_n up_o by_o arithmetic_n p._n 106_o to_o 108_o a_o brief_a table_n of_o natural_a sin_n tangent_n and_o secant_v for_o each_o point_n of_o the_o compass_n and_o the_o quarter_n p._n 107_o the_o difference_n of_o longitude_n in_o a_o dead_a reckon_a find_v by_o the_o pen._n p._n 109_o that_o a_o table_n of_o natural_a sines_n supply_v the_o want_n of_o all_o other_o table_n p._n 110_o many_o new_a easy_a rule_n and_o proportion_n to_o raise_v a_o table_n of_o natural_a sin_n p._n 111_o to_o 113_o and_o how_o by_o have_v some_o in_o store_n to_o calculate_v any_o other_o sine_fw-la in_o the_o quadrant_n at_o command_n p._n 114_o of_o the_o contrivance_n of_o logarithmical_a table_n of_o number_n sin_n and_o tangent_n and_o how_o the_o want_n of_o natural_a table_n and_o of_o a_o table_n of_o the_o meridian-line_n be_v supply_v from_o they_o p._n 117_o the_o side_n of_o a_o plain_a triangle_n be_v give_v to_o calculate_v the_o angle_n without_o the_o help_n of_o table_n two_o several_a way_n p._n 118_o 119_o a_o instance_n thereof_o in_o calculate_v a_o course_n and_o distance_n p._n 119_o chap._n i._o contain_v geometrical_a definition_n a_o point_n be_v a_o imaginary_a prick_n void_a of_o all_o length_n breadth_n or_o depth_n a_o line_n be_v a_o suppose_a length_n without_o breadth_n or_o depth_n the_o end_n or_o limit_n whereof_o be_v point_n a_o angle_n derive_v from_o the_o word_n angulus_n in_o latin_a which_o signify_v a_o corner_n be_v the_o inclination_n or_o bow_v of_o two_o line_n one_o to_o another_o and_o the_o one_o touch_v the_o other_o and_o not_o be_v direct_o join_v together_o if_o the_o line_n which_o contain_v the_o angle_n be_v right_a line_n then_o be_v it_o call_v a_o right_o line_v angle_n a_o right_a angle_n when_o a_o right_a line_n stand_v upon_o a_o right_a line_n make_v the_o angle_n on_o either_o side_n equal_a each_o of_o these_o angles_n be_v call_v right_a angle_n and_o the_o line_n erect_v be_v call_v a_o perpendicular_a line_n unto_o the_o other_o a_o obtuse_a angle_n be_v that_o which_o be_v great_a than_o a_o right_a angle_n a_o acute_a angle_n be_v that_o which_o be_v less_o than_o a_o right_a angle_n when_o two_o angle_n be_v both_o acute_a or_o obtuse_a they_o be_v of_o the_o same_o kind_n otherwise_o be_v say_v to_o be_v of_o different_a affection_n a_o angle_n be_v common_o denote_v by_o the_o middlemost_a of_o the_o three_o letter_n set_v to_o the_o side_n include_v the_o say_a angle_n the_o quantity_n of_o a_o angle_n be_v measure_v by_o the_o arch_n of_o a_o circle_n describe_v upon_o the_o point_n of_o concurrence_n or_o intersection_n where_o the_o two_o side_n enclose_v the_o say_a angle_n meet_v by_o the_o compliment_n of_o a_o arch_n or_o angle_n be_v mean_v the_o remainder_n of_o that_o arch_n take_v from_o 90_o unless_o it_o be_v express_v the_o compliment_n thereof_o to_o a_o semicircle_n of_o 180d._o a_o circle_n be_v a_o plain_a figure_n contain_v under_o one_o line_n which_o be_v call_v the_o circumference_n thereof_o by_o some_o the_o perimeter_n periphery_n or_o limb_n a_o portion_n or_o part_n thereof_o be_v call_v a_o segment_n the_o centre_n thereof_o be_v a_o point_n in_o the_o very_a midst_n thereof_o from_o which_o point_n all_o right_a line_n draw_v

shall_v show_v and_o another_o true_a chart_n or_o globe_n graduate_v from_o they_o and_o that_o in_o stead_n of_o put_v in_o such_o abundance_n of_o compass_n and_o rumbe-line_n both_o in_o the_o plain_a and_o mercators_n chart_n it_o be_v better_a to_o leave_v they_o out_o and_o to_o put_v into_o some_o spare_a place_n the_o traverse-quadrant_a with_o point_n and_o half_a point_n and_o a_o limb_n divide_v into_o degree_n with_o a_o line_n of_o sines_n or_o else_o the_o sin_n of_o the_o point_n half_n and_o quarter_n only_o 3._o that_o be_v so_o make_v near_o the_o equinoctial_a they_o be_v very_o near_o the_o truth_n and_o may_v there_o very_a well_o serve_v as_o also_o under_o the_o meridian_n and_o for_o short_a distance_n or_o voyage_n and_o how_o to_o take_v away_o the_o error_n of_o the_o chart_n and_o make_v they_o serve_v for_o long_a and_o remote_a voyage_n shall_v be_v handle_v in_o order_n whereto_o the_o first_o proposition_n in_o the_o use_n of_o the_o plain_a chart_n must_v be_v repeat_v propos_fw-fr i._o the_o longitude_n and_o latitude_n of_o any_o two_o place_n be_v give_v to_o find_v the_o true_a course_n or_o rumbe_n between_o those_o place_n and_o the_o distance_n in_o the_o rumbe_n this_o of_o all_o other_o proposition_n in_o navigation_n be_v the_o most_o useful_a and_o withal_o the_o most_o difficult_a and_o have_v as_o learned_a snellius_n well_o observe_v be_v often_o attempt_v by_o the_o learned_a but_o in_o vain_a and_o to_o give_v our_o own_o nation_n its_o due_a repute_n be_v never_o general_o and_o satisfactory_o perform_v by_o any_o man_n till_o our_o late_a famous_a countryman_n mr._n edward_n wright_n invent_v that_o excellent_a chart_n call_v mercators_n chart_n but_o aught_o more_o proper_o to_o be_v call_v wright_n chartley_n the_o meridian-line_n whereof_o require_v a_o table_n to_o be_v make_v by_o the_o perpetual_a addition_n of_o secant_v without_o which_o table_n as_o yet_o there_o be_v no_o proportion_n know_v that_o will_v serve_v to_o calculate_v the_o rumbe_n general_o between_o any_o two_o place_n by_o help_n of_o the_o natural_a table_n of_o sines_n tangent_n and_o secant_v only_o and_o whatsoever_o may_v be_v do_v by_o those_o table_n may_v be_v also_o do_v geometrical_o by_o scheme_n true_a it_o be_v this_o proposition_n may_v be_v perform_v by_o the_o difference_n of_o the_o logarithmical_a tangent_n have_v a_o table_n of_o they_o as_o in_o mr._n norwoods_n epitome_n without_o the_o help_n of_o a_o table_n of_o the_o meridian-line_n but_o as_o yet_o we_o have_v no_o geometrical_a way_n know_v for_o make_v the_o logarithmical-line_n of_o tangent_n nor_o sin_n and_o number_n before_o i_o proceed_v any_o further_a it_o may_v be_v object_v that_o we_o have_v proportion_n in_o our_o english_a book_n deliver_v for_o calculate_v the_o rumbe_n between_o two_o place_n which_o proportion_n may_v be_v perform_v by_o the_o natural_a table_n of_o sines_n and_o tangent_n only_o as_o namely_o in_o mr._n gunter'_v work_n both_o in_o the_o former_a and_o latter_a edition_n in_o the_o three_o edition_n in_o page_n 90._o as_o the_o difference_n of_o latitude_n be_v to_o the_o cousin_n of_o the_o middle_a latitude_n ∷_o so_n be_v the_o difference_n of_o longitude_n to_o the_o tangent_fw-la of_o the_o rumbe_n from_o the_o meridian_n a_o precious_a proportion_n if_o it_o be_v true_a the_o true_a proportion_n be_v as_o the_o meridional_a part_n between_o both_o latitude_n be_v to_o the_o radius_fw-la ∷_o so_n be_v the_o difference_n of_o longitude_n to_o the_o tangent_fw-la of_o the_o rumbe_n ∷_o the_o meridional_a part_n be_v to_o be_v take_v out_o of_o a_o table_n of_o the_o meridian-line_n by_o substract_v the_o meridional_a part_n of_o the_o lesser_a latitude_n from_o the_o meridional_a part_n of_o the_o great_a latitude_n by_o compare_v these_o two_o proportion_n together_o because_o the_o three_o and_o four_o term_n be_v alike_o in_o each_o it_o will_v follow_v that_o we_o may_v calculate_v the_o meridional_a part_n require_v without_o the_o perpetual_a addition_n of_o secant_v by_o this_o proportion_n raise_v out_o of_o the_o two_o former_a term_n of_o each_o proportion_n as_o the_o cousin_n of_o the_o middle_a latitude_n be_v to_o the_o difference_n of_o latitude_n if_o in_o one_o hemisphere_n or_o the_o sum_n of_o both_o latitude_n if_o in_o different_a hemisphere_n ∷_o so_n be_v the_o radius_fw-la to_o the_o meridional_a part_n ∷_o and_o so_o by_o this_o proportion_n the_o meridional_a part_n answer_v to_o the_o latitude_n of_o 50_o shall_v be_v 55_o 157_o but_o be_v in_o truth_n 57_o 909._o 70_o 85_o 459_o by_o the_o table_n 99_o 431._o  _fw-fr  _fw-fr  _fw-fr difference_n 41_o 522._o  _fw-fr  _fw-fr  _fw-fr sum_n 157,340_o which_o sufficient_o refute_v the_o truth_n of_o mr._n gunter'_v proportion_n in_o calculate_v a_o rumbe_n from_o the_o equinoctial_a but_o now_o as_o for_o find_v the_o meridional_a part_n between_o the_o two_o former_a latitude_n in_o one_o or_o both_o hemisphere_n the_o middle_a latitude_n in_o one_o hemisphere_n will_v be_v 60_o and_o the_o meridional_a part_n 40_o 00_o and_o in_o both_o hemisphere_n if_o find_v at_o once_o by_o the_o middle_a latitude_n which_o be_v 10_o be_v 121_o 85_o or_o if_o find_v several_o at_o twice_o 140_o 616_o which_o vary_v from_o the_o truth_n as_o above_o express_v in_o the_o sum_n and_o difference_n we_o may_v conclude_v that_o the_o proportion_n be_v very_o unsound_a and_o intolerable_a for_o any_o great_a difference_n of_o latitude_n but_o mr._n gunter'_v work_n deliver_v no_o caution_n about_o it_o before_o we_o can_v find_v the_o distance_n the_o rumbe_n must_v be_v calculate_v and_o if_o that_o be_v false_a a_o small_a error_n therein_o may_v cause_v a_o considerable_a error_n in_o the_o distance_n where_o the_o difference_n of_o latitude_n be_v not_o above_o five_o degree_n it_o may_v serve_v very_o well_o near_o the_o truth_n from_o the_o equinoctial_a to_o 60_o of_o latitude_n and_o afterward_o to_o 80_o it_o will_v not_o serve_v for_o three_o degree_n difference_n of_o latitude_n and_o in_o all_o case_n the_o cousin_n of_o the_o middle_a latitude_n be_v a_o term_n too_o great_a the_o middle_a latitude_n be_v too_o small_a and_o i_o think_v no_o certain_a rule_n can_v be_v give_v to_o correct_v it_o from_o what_o have_v be_v say_v the_o reader_n may_v take_v due_a caution_n how_o far_o to_o depend_v upon_o such_o proportion_n whereof_o one_o term_n be_v the_o middle_a latitude_n such_o be_v as_o the_o cousin_n of_o the_o middle_a latitude_n to_o the_o radius_fw-la ∷_o or_o as_o the_o radius_fw-la be_v to_o the_o secant_fw-la of_o the_o middle_a latitude_n ∷_o so_n be_v the_o departure_n from_o the_o meridian_n to_o the_o difference_n of_o longitude_n and_o to_o this_o proportion_n mr._n phillip_n his_o late_a table_n of_o secant_v be_v fit_v in_o the_o use_n whereof_o the_o middle_a latitude_n must_v always_o be_v take_v to_o be_v a_o whole_a degree_n unless_o you_o will_v by_o proportion_n find_v the_o difference_n require_v and_o i_o take_v it_o to_o be_v what_o it_o true_o happen_v do_v cast_v up_o the_o course_n and_o distance_n mr._n norwood_n express_v in_o his_o trigonometry_n home_n from_o the_o bermuda_n to_o the_o lizard_n by_o his_o table_n in_o the_o say_a book_n and_o find_v i_o have_v get_v almost_o half_a a_o degree_n of_o longitude_n in_o the_o whole_a too_o much_o by_o reason_n the_o proportion_n be_v not_o sound_a of_o which_o mr._n norwood_n make_v no_o use_n those_o course_n and_o distance_n be_v true_o express_v in_o mr._n phillip_n his_o geometrical_a seaman_n pag._n 31._o whereas_o in_o the_o last_o impression_n of_o mr._n norwoods_n book_n they_o be_v misprinted_n another_o proportion_n of_o this_o kind_n be_v as_o the_o cousin_n of_o the_o middle_a latitude_n be_v to_o the_o sine_fw-la of_o the_o rumbe_n from_o the_o meridian_n ∷_o so_n be_v the_o distance_n sail_v to_o the_o difference_n of_o longitude_n ∷_o so_n also_o there_o may_v be_v two_o proportion_n for_o find_v the_o enlarge_a distance_n as_o to_o the_o radius_fw-la to_o the_o secant_fw-la of_o the_o middle_a latitude_n ∷_o so_n be_v the_o distance_n run_v to_o the_o enlarge_a distance_n on_o mercators_n chart_n ∷_o otherwise_o as_o the_o sine_fw-la of_o the_o rumbe_n from_o the_o meridian_n be_v to_o the_o secant_fw-la of_o the_o middle_a latitude_n ∷_o so_n be_v the_o departure_n from_o the_o meridian_n to_o the_o enlarge_a distance_n ∷_o note_v when_o these_o proportion_n be_v use_v the_o measure_n of_o the_o enlarge_a distance_n must_v be_v take_v from_o the_o degree_n of_o longitude_n in_o mercator_n chart._n another_o proportion_n for_o find_v the_o rumbe_n may_v be_v gather_v from_o mr._n hansons_n addition_n to_o pitiscus_n his_o trigonometry_n and_o be_v as_o the_o difference_n of_o latitude_n be_v to_o the_o half_a sum_n of_o the_o cosines_n of_o

the_o meridional_a part_n between_o both_o latitude_n to_o the_o difference_n of_o longitude_n ∷_o which_o proportion_n require_v the_o difference_n of_o latitude_n to_o be_v first_o find_v and_o then_o by_o help_n of_o the_o table_n of_o meridional_a part_n at_o the_o end_n of_o the_o book_n the_o difference_n of_o longitude_n may_v be_v find_v which_o table_n be_v make_v but_o to_o every_o second_o centesm_n have_v half_o the_o difference_n set_v down_o at_o the_o bottom_n of_o the_o page_n which_o add_v to_o the_o meridional_a part_n of_o any_o even_a centesm_n above_o it_o make_v the_o meridional_a part_n for_o each_o odd_a centesm_n thus_o the_o meridional_a part_n for_o 25_o 22_o be_v 26_o 076_o whereto_o add_v 11_o the_o sum_n be_v 26_o 087_o be_v meridional_a part_n for_o 25_o 23_o centesm_n in_o the_o first_o example_n of_o the_o traverse_v the_o difference_n of_o latitude_n be_v 2_o 77_o centesm_n the_o latitude_n of_o tenariff_n be_v 28_o meridional_a part_n 29_o 186_o the_o latitude_n of_o the_o ship_n place_n be_v 25_o 23_o 26_o 087_o the_o meridional_a part_n between_o both_o latitude_n be_v 3_o 099_o which_o be_v multiply_v by_o the_o tangent_fw-la of_o the_o second_o point_n to_o wit_n 414_o cut_v off_o 6_o place_n three_o by_o reason_n the_o radius_fw-la be_v 1000_o and_o three_o more_o for_o the_o decimal_a part_n of_o the_o meridional_a number_n the_o amount_v being_z 1_o 282986_o show_v that_o the_o difference_n of_o longitude_n be_v 1_o degree_n 28_o centesm_n of_o which_o there_o may_v be_v a_o column_n of_o east_n and_o west_n keep_v like_v as_o be_v do_v for_o the_o departure_n and_o this_o cast_v up_o of_o the_o longitude_n may_v be_v ready_o do_v also_o on_o the_o logarithmical_a ruler_n or_o by_o mr._n phillip_n his_o late_a table_n for_o that_o purpose_n without_o calculation_n whereof_o we_o make_v mention_v in_o page_n 37._o we_o have_v before_o say_v that_o a_o table_n of_o sines_n be_v sufficient_a to_o supply_v all_o calculation_n though_o o●●er_v table_n may_v be_v more_o ready_a for_o dispatch_v when_o they_o be_v at_o hand_n as_o we_o have_v show_v concern_v the_o table_n of_o meridional_a part_n in_o page_n 48._o out_o of_o it_o the_o secant_v be_v make_v by_o this_o proportion_n as_o the_o cousin_n of_o any_o arch_n propose_v be_v to_o the_o radius_fw-la ∷_o so_n be_v the_o radius_fw-la to_o the_o secant_fw-la of_o the_o give_a arch_n ∷_o which_o proportion_n hold_v backward_o to_o find_v the_o arch_n in_o the_o table_n of_o sines_n if_o a_o secant_fw-la be_v give_v at_o adventure_n and_o the_o arch_n require_v also_o out_o of_o it_o the_o tangent_n be_v make_v by_o this_o proportion_n as_o the_o cousin_n of_o a_o arch_n be_v to_o the_o sine_fw-la of_o the_o say_a arch_n ∷_o so_n be_v the_o radius_fw-la to_o the_o tangent_fw-la of_o the_o arch_n propose_v ∷_o which_o proportion_n do_v not_o hold_v backward_o to_o find_v the_o arch_n if_o a_o tangent_fw-la be_v give_v in_o which_o case_n the_o secant_fw-la may_v be_v find_v and_o thereby_o the_o arch_n for_o the_o square_n of_o a_o tangent_fw-la more_fw-it the_o square_n of_o the_o radius_fw-la be_v equal_a to_o the_o square_n of_o the_o secant_fw-la in_o this_o case_n we_o have_v also_o the_o proportion_n of_o the_o cousin_n to_o the_o sine_fw-la give_v be_v the_o same_o with_o that_o of_o the_o radius_fw-la to_o the_o tangent_fw-la and_o the_o sum_n o●_n their_o square_n give_v be_v equal_a to_o the_o square_n of_o the_o radius_fw-la which_o very_a case_n be_v reduce_v to_o a_o double_a equation_n in_o the_o 33_o question_n of_o mr._n moor_n algebraick_n arithmetic_n and_o by_o which_o the_o radius_fw-la and_o a_o tangent_fw-la be_v give_v either_o the_o sine_fw-la or_o cousin_n may_v be_v find_v without_o find_v the_o secant_fw-la if_o such_o a_o proportion_n as_o this_o be_v propose_v as_o the_o radius_fw-la be_v to_o the_o tangent_fw-la of_o a_o arch_n ∷_o so_n be_v the_o tangent_fw-la of_o another_o arch_n to_o the_o tangent_fw-la of_o a_o four_o arch_n ∷_o it_o may_v be_v resolve_v thus_o without_o make_v a_o table_n of_o tangent_n make_v the_o product_n of_o the_o radius_fw-la and_o of_o the_o cousin_n of_o the_o second_o ark_n the_o divisor_n and_o the_o product_n of_o the_o sin_n of_o the_o second_o and_o three_o ark_n and_o of_o the_o cousin_n of_o the_o three_o ark_n the_o dividend_n and_o the_o quotient_n will_v be_v the_o tangent_fw-la of_o the_o four_o ark_n seek_v in_o like_a manner_n if_o all_o those_o term_n be_v tangent_n the_o product_n of_o the_o radius_fw-la sine_fw-la and_o cousin_n of_o the_o first_o ark_n will_v be_v the_o divisor_n and_o the_o product_n of_o the_o sin_n and_o cosines_n of_o the_o second_o and_o three_o ark_n the_o dividend_n whereby_o may_v be_v find_v the_o quotient_a be_v the_o tangent_fw-la seek_v and_o consequent_o the_o arch_n answer_v thereto_o after_o the_o like_a manner_n with_o due_a regard_n to_o the_o proportion_n for_o make_v the_o secant_v any_o secant_fw-la might_n be_v supply_v and_o a_o proportion_n whole_o in_o secant_v by_o turn_v those_o arke_n into_o their_o compliment_n be_v change_v whole_o into_o sin_n or_o two_o term_n be_v secant_v be_v change_v into_o sin_n by_o alter_v the_o place_n or_o order_n of_o those_o term_n and_o their_o arke_n into_o their_o compliment_n how_o to_o calculate_v a_o table_n of_o natural_a sin_n how_o to_o make_v the_o sine_fw-la of_o any_o ark_n at_o pleasure_n snellius_n in_o his_o cyclometria_n have_v show_v without_o make_v many_o such_o sin_n as_o be_v not_o require_v in_o which_o he_o go_v beyond_o all_o fo●mer_a writer_n one_o way_n he_o have_v hold_v true_a as_o far_o as_o 30_o of_o the_o quadrant_n out_o of_o which_o sin_n their_o cosines_n may_v be_v make_v and_o so_o the_o quadrant_n fill_v from_o 60_o to_o 90_o and_o then_o by_o find_v the_o sine_fw-la of_o the_o double_a ark_n it_o may_v be_v supply_v from_o 30_o to_o 60_o but_o in_o regard_n the_o performance_n be_v very_o tedious_a and_o therefore_o not_o to_o be_v prosecute_v in_o the_o construction_n of_o a_o whole_a table_n we_o shall_v omit_v to_o mention_v it_o another_o way_n he_o have_v which_o do_v not_o hold_v true_a to_o above_o 1_o 12_o part_n of_o the_o quadrant_n so_o that_o when_o a_o great_a ark_n near_o the_o end_n of_o the_o quadrant_n be_v give_v he_o find_v the_o cousin_n thereof_o first_o and_o thereby_o the_o sine_fw-la or_o if_o the_o ark_n be_v remote_a or_o about_o the_o middle_n of_o the_o quadrant_n he_o find_v the_o sine_fw-la of_o the_o eight_o or_o some_o such_o part_n of_o it_o and_o then_o by_o find_v the_o sine_fw-la of_o the_o double_a ark_n etc._n etc._n backward_o he_o find_v the_o sine_fw-la seek_v but_o how_o to_o calculate_v a_o whole_a table_n by_o proportion_n with_o ease_n be_v attain_v unto_o by_o our_o worthy_a countryman_n mr._n michael_n darie_n who_o discover_v the_o same_o in_o the_o year_n 1651_o and_o communicate_v unto_o i_o the_o proportion_n for_o that_o purpose_n long_o since_o who_o method_n we_o shall_v now_o insist_v upon_o 1._o the_o ready_a way_n be_v to_o assume_v the_o proportion_n of_o the_o diameter_n of_o a_o circle_n to_o its_o circumference_n to_o be_v know_v and_o to_o be_v in_o the_o proportion_n of_o 113_o to_o 355_o of_o which_o see_v maetius_n his_o practical_a geometry_n which_o two_o number_n be_v easy_o remember_v be_v the_o three_o first_o odd_a figure_n of_o the_o digit_n each_o of_o they_o twice_o write_v down_o and_o cut_v asunder_o in_o the_o middle_n 2._o it_o be_v most_o convenient_a that_o the_o radius_fw-la or_o whole_a sine_fw-la shall_v be_v a_o unit_fw-la with_o cipher_n and_o so_o the_o proporrtion_n of_o the_o diameter_n to_o the_o circumference_n find_v by_o the_o two_o number_n above_o be_v such_o as_o 2000000_o to_o 6283185_o so_o far_o true_a and_o to_o find_v it_o true_a to_o as_o many_o place_n of_o figure_n as_o you_o please_v consult_v dr._n wallis_n his_o write_n or_o hungenius_n de_fw-fr magnitudine_fw-la circuli_fw-la 3._o the_o sine_fw-la of_o one_o minute_n or_o centesm_n do_v so_o insensible_o differ_v from_o the_o length_n of_o the_o arch_n to_o which_o it_o belong_v that_o the_o length_n of_o the_o arch_n of_o one_o minute_n or_o centesm_n may_v be_v very_o well_o take_v to_o be_v the_o sine_fw-la thereof_o and_o in_o the_o large_a table_n that_o ever_o be_v publish_v differ_v nothing_o therefrom_o wherefore_o a_o minute_n be_v the_o 21600_o part_n of_o the_o circumference_n of_o a_o whole_a circle_n the_o like_a part_n of_o the_o number_n 6283185_o be_v 290_o be_v the_o sine_fw-la of_o 1_o minute_n 4._o you_o may_v find_v the_o cousin_n of_o one_o minute_n by_o substract_v the_o square_n of_o the_o sine_fw-la of_o one_o minute_n from_o the_o square_n of_o the_o radius_fw

other_o arch_n be_v equal_a to_o the_o square_n of_o the_o sine_fw-la of_o the_o middle_a arch_n from_o thence_o it_o follow_v that_o the_o sine_fw-la of_o any_o arch_n in_o the_o quadrant_n be_v give_v we_o may_v by_o this_o rule_n and_o extract_v the_o square_a root_n perpetual_o bisect_v or_o find_v the_o sine_fw-la of_o the_o middle_a arch_n between_o it_o and_o 90_o and_o so_o run_v up_o very_o speedy_o towards_o the_o end_n of_o the_o quadrant_n of_o which_o sine_fw-la so_o find_v the_o cousin_n may_v be_v find_v by_o another_o extraction_n and_o then_o you_o have_v the_o sine_fw-la of_o as_o small_a a_o arch_a as_o you_o please_v near_o the_o begin_n of_o the_o quadrant_n from_o which_o by_o proportion_n in_o regard_n the_o length_n of_o the_o sine_fw-la and_o of_o the_o arch_n to_o which_o it_o do_v belong_v have_v no_o sensible_a difference_n you_o may_v find_v the_o sine_fw-la of_o one_o minute_n or_o centesme_fw-fr and_o thence_o by_o the_o former_a proportion_n raise_v a_o whole_a table_n of_o sines_n to_o as_o large_a a_o radius_fw-la as_o you_o please_v albeit_o the_o proportion_n of_o the_o diameter_n to_o the_o circumference_n be_v whole_o unknown_a which_o notwithstanding_o hereby_o may_v be_v find_v for_o have_v make_v a_o sine_fw-la to_o a_o very_a small_a part_n of_o the_o quadrant_n the_o double_a thereof_o be_v the_o side_n of_o a_o polygon_n inscribe_v and_o by_o proportion_n as_o the_o cousin_n so_o find_v be_v to_o its_o sine_fw-la ∷_o so_n be_v the_o radius_fw-la to_o the_o tangent_fw-la of_o the_o arch_n to_o which_o the_o small_a sine_fw-la belong_v ∷_o the_o double_a of_o which_o tangent_fw-la be_v the_o side_n of_o the_o like_a polygon_n circumscribe_v the_o length_n of_o the_o arch_n of_o the_o circle_n contain_v between_o the_o side_n of_o these_o two_o polygon_n be_v great_a than_o the_o side_n of_o the_o inscribe_v polygon_n and_o lesser_a than_o the_o side_n of_o the_o circumscribe_v polygon_n in_o find_v the_o sine_fw-la of_o any_o middle_a arch_n by_o the_o former_a consectary_n to_o shun_v the_o trouble_n of_o multiply_v we_o may_v assume_v the_o radius_fw-la to_o be_v 2_o with_o a_o competent_a number_n of_o cipher_n than_o the_o sine_fw-la of_o 30_o because_o it_o be_v half_a the_o radius_fw-la will_v be_v a_o unit_fw-la with_o as_o many_o cipher_n then_o to_o find_v the_o sine_fw-la of_o 60_o to_o that_o radius_fw-la to_o the_o sine_fw-la of_o 30_o prefix_v to_o the_o left_a hand_n the_o number_n 2_o and_o annex_v the_o cipher_n in_o the_o radius_fw-la to_o the_o right_a hand_n and_o the_o square_a root_n of_o the_o number_n so_o make_v shall_v be_v the_o sine_fw-la require_v and_o the_o like_a for_o find_v the_o sine_fw-la of_o the_o middle_a arch_n between_o the_o radius_fw-la and_o the_o sine_fw-la of_o any_o other_o arch_n give_v and_o retain_v the_o radius_fw-la 2_o with_o cipher_n the_o difference_n between_o the_o sine_fw-la of_o any_o arch_n give_v and_o the_o radius_fw-la have_v the_o figure_n of_o the_o radius_fw-la annex_v thereto_o the_o square_a root_n of_o the_o number_n so_o compose_v shall_v be_v the_o cousin_n of_o the_o middle_a arch_n or_o we_o may_v deliver_v it_o as_o a_o consectary_n from_o former_a proportion_n that_o the_o rectangle_n of_o half_a the_o radius_fw-la and_o of_o the_o difference_n between_o the_o sine_fw-la of_o any_o arch_n give_v and_o the_o radius_fw-la shall_v be_v equal_a to_o the_o square_n of_o the_o cousin_n of_o the_o middle_a arch_n to_o any_o radius_fw-la whatsoever_o we_o forbear_v to_o suit_v arithmetical_a example_n to_o the_o foregoing_a proportion_n and_o consectary_n suppose_v the_o reader_n furnish_v with_o so_o much_o arithmetic_n as_o that_o he_o can_v extract_v the_o square_a root_n and_o work_v the_o golden_a rule_n or_o rule_n of_o three_o see_v page_n 12_o of_o the_o first_o part._n of_o book_n of_o table_n such_o table_n as_o have_v a_o degree_n divide_v into_o 100_o part_n or_o centesm_n be_v to_o be_v prefer_v before_o those_o that_o divide_v the_o degree_n but_o into_o 60_o part_n minute_n or_o sexagesm_n because_o they_o be_v more_o exact_a in_o calculation_n and_o more_o speedy_a in_o find_v the_o part_n proportional_a and_o it_o be_v convenient_a for_o a_o book_n of_o table_n to_o be_v so_o contrive_v that_o the_o natural_a table_n of_o sines_n tangent_n and_o verse_v sin_n to_o 90_o may_v stand_v against_o the_o logarithmical_a table_n of_o sines_n and_o tangent_n in_o a_o portable_a book_n to_o be_v have_v by_o itself_o to_o which_o may_v be_v add_v a_o table_n of_o the_o meridian-line_n to_o each_o or_o every_o second_o centesm_n with_o table_n of_o the_o sun_n declination_n right_a ascension_n and_o of_o the_o longitude_n latitude_n declination_n and_o right_a ascension_n of_o some_o of_o the_o principal_a fix_a star_n and_o of_o the_o longitude_n and_o latitude_n of_o place_n but_o those_o that_o have_v the_o logarithmical_a table_n of_o number_n sin_n and_o tangent_n without_o the_o 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