Selected quad for the lemma: reason_n

Word A	Word B	Word C	Word D	Occurrence	Frequency Band	MI	MI Band	Prominent
reason_n	angle_n	equal_a	triangle_n	2,577	5	14.6378	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

what_o be_v here_o accomplish_v by_o the_o table_n may_v ready_o be_v perform_v by_o a_o meridian-line_n out_o of_o which_o with_o compass_n take_v the_o distance_n between_o both_o latitude_n and_o prick_v it_o from_o l_o towards_o m_n at_o the_o end_n whereof_o raise_v a_o perpendicular_a and_o therein_o prick_v down_o out_o of_o the_o equinoctial_a degree_n or_o equal_a part_n the_o difference_n of_o longitude_n draw_v a_o line_n from_o it_o to_o l_o which_o shall_v pass_v through_o the_o former_a point_n o_o whatsoever_o be_v the_o radius_fw-la whereto_o the_o meridian-line_n be_v fit_v and_o after_o the_o same_o manner_n the_o error_n of_o the_o plain_a chart_n be_v to_o be_v remove_v when_o place_n be_v at_o first_o lay_v down_o in_o it_o according_a to_o their_o longitude_n and_o latitude_n which_o be_v most_o easy_o and_o sudden_o do_v especial_o if_o a_o meridian-line_n on_o a_o rod_n fit_v to_o the_o degree_n of_o longitude_n on_o the_o plain_a chart_n and_o the_o manner_n of_o measure_v a_o westward_o distance_n will_v be_v the_o same_o as_o in_o mercator_n chart._n before_o we_o proceed_v to_o the_o demonstration_n of_o mercators_n chart_n it_o will_v be_v necessary_a to_o search_v into_o the_o nature_n of_o the_o rumbe-line_n on_o the_o globe_n and_o to_o collect_v what_o we_o find_v observable_a in_o foreign_a author_n to_o this_o purpose_n 1._o they_o define_v it_o to_o be_v such_o a_o line_n that_o make_v the_o same_o angle_n with_o every_o meridian_n through_o which_o it_o pass_v and_o therefore_o can_v be_v no_o arch_n of_o a_o great_a circle_n for_o suppose_v such_o a_o circle_n to_o pass_v through_o the_o zenith_n of_o some_o place_n not_o under_o the_o equinoctial_a make_v a_o oblique_a angle_n with_o the_o meridian_n then_o shall_v it_o make_v a_o great_a angle_n with_o all_o other_o meridian_n then_o with_o that_o through_o which_o it_o at_o first_o pass_v therefore_o the_o rumbe_n which_o make_v the_o same_o angle_n with_o every_o meridian_n must_v needs_o be_v a_o line_n curve_v and_o bend_v which_o some_o call_v a_o helix-line_n and_o other_o because_o it_o be_v suppose_v to_o be_v on_o the_o surface_n of_o the_o sphere_n a_o helispherical-line_n 2._o another_o property_n of_o the_o rumbe-line_n be_v that_o it_o lead_v near_a and_o near_o unto_o one_o of_o the_o pole_n but_o never_o fall_v into_o it_o and_o near_o the_o pole_n it_o turn_v often_o round_a the_o same_o in_o many_o spire_n and_o turn_n if_o by_o any_o rumbe_n but_o not_o under_o the_o meridian_n it_o be_v suppose_v possible_a to_o sail_v direct_o under_o the_o pole_n point_n it_o will_v follow_v that_o the_o meridian-line_n of_o mercators_n chart_n shall_v be_v finite_a and_o not_o infinite_a and_o that_o one_o and_o the_o same_o line_n shall_v cut_v infinite_a other_o line_n as_o such_o at_o equal_a angle_n in_o a_o mere_a point_n which_o be_v against_o the_o nature_n and_o definition_n of_o a_o line_n and_o then_o see_v that_o the_o rumbe_n derive_v its_o definition_n from_o the_o angle_n it_o make_v with_o the_o meridian_n the_o nature_n and_o use_n of_o a_o meridian_n under_o the_o pole_n cease_v and_o consequent_o the_o rumbe_n cease_v for_o under_o the_o pole_n no_o star_n or_o other_o immovable_a point_n of_o the_o heaven_n do_v either_o rise_n or_o set_v in_o reference_n whereto_o the_o word_n meridian_n have_v its_o original_n be_v use_v to_o signify_v the_o midday_n time_n or_o high_a noon_n of_o the_o sun_n or_o star_n relate_v to_o their_o proper_a motion_n moreover_o under_o the_o pole_n point_v the_o side_n of_o the_o rumbe-triangle_n in_o the_o sphere_n cease_v wherefore_o the_o angle_n must_v needs_o do_v so_o too_o of_o the_o nature_n of_o the_o triangle_n make_v upon_o the_o globe_n by_o the_o rumbe-line_n and_o in_o the_o whole_a rumbe-triangle_n ae_n t_o m_o the_o whole_a distance_n be_v ae_n m_o the_o whole_a difference_n of_o latitude_n be_v ae_n t_o but_o the_o whole_a departure_n from_o the_o meridian_n be_v not_o give_v in_o any_o one_o triangle_n but_o in_o divers_a triangle_n and_o be_v by_o supposition_n as_o much_o in_o one_o triangle_n as_o another_o to_o wit_n the_o whole_a departure_n from_o the_o meridian_n be_v the_o sum_n of_o a_o b_o c_o d_o e_o f_o g_o h_o i_o k_o l_o m_o each_o of_o which_o portion_n be_v suppose_v to_o be_v equal_a each_o to_o other_o and_o the_o whole_a difference_n of_o longitude_n be_v the_o arch_n ae_n q_n the_o segment_n whereof_o q_n l_o q_o s_o s_o r_o can_v be_v equal_a to_o each_o other_o because_o the_o ark_n or_o departure_n l_o m_o i_o k_o etc._n etc._n be_v suppose_v to_o be_v equal_a each_o to_o other_o not_o in_o the_o same_o but_o in_o different_a parallel_n of_o latitude_n now_o then_o for_o all_o use_n in_o navigation_n we_o suppose_v the_o segment_n of_o the_o rumbe_n m_o k_o k_o h_o and_o the_o rest_n to_o be_v a_o right_a line_n and_o if_o we_o give_v the_o rumbe_n to_o wit_n the_o angle_n l_o k_o m_o and_o the_o side_n k_o l_o we_o may_v by_o the_o doctrine_n of_o plain_a triangle_n find_v the_o side_n k_o m_o the_o distance_n by_o the_o common_a proportion_n as_o the_o cousin_n of_o the_o rumbe_n from_o the_o meridian_n be_v to_o the_o radius_fw-la ∷_o so_n be_v the_o difference_n of_o latitude_n to_o the_o distance_n ∷_o and_n from_o hence_o we_o may_v observe_v another_o property_n in_o the_o rumbe_n as_o namely_o that_o the_o segment_n or_o piece_n thereof_o contain_v between_o two_o parallel_n have_v the_o like_a difference_n of_o latitude_n be_v also_o equal_a to_o each_o other_o so_o that_o the_o like_a distance_n be_v sail_v in_o the_o same_o rumbe_n in_o several_a part_n of_o the_o earth_n shall_v cause_v the_o like_a difference_n or_o alteration_n of_o latitude_n in_o each_o of_o those_o part_n for_o example_n if_o a_o ship_n sail_n from_o the_o latitude_n of_o 10_o north-east_n till_o she_o be_v in_o the_o latitude_n of_o 30_o and_o then_o sail_v from_o thence_o on_o the_o same_o rumbe_n till_o she_o be_v in_o the_o latitude_n of_o 50_o the_o latter_a distance_n shall_v be_v equal_a to_o the_o former_a distance_n and_o the_o like_a shall_v hold_v from_o thence_o to_o 70_o and_o the_o same_o also_o shall_v hold_v from_o 70_o to_o the_o pole_n if_o it_o be_v possible_a to_o sail_v thither_o by_o any_o rumbe_n and_o the_o reason_n be_v because_o in_o the_o triangle_n k_o l_o m_o the_o angle_n at_o k_n and_o l_o with_o the_o side_n l_o k_o be_v equal_a by_o construction_n to_o the_o angle_n at_o h_z and_z i_z and_o the_o side_n i_o h_o in_o the_o triangle_n i_o h_o k_o and_o the_o like_a in_o any_o other_o triangle_n wherefore_o if_o these_o be_v give_v to_o find_v the_o side_n m_o k_n or_o k_o h_o it_o must_v needs_o be_v find_v the_o same_o in_o both_o again_o if_o in_o the_o former_a triangle_n we_o shall_v give_v the_o rumbe_n to_o wit_n the_o angle_n l_o k_o m_o and_o the_o difference_n of_o latitude_n l_o k_o we_o may_v find_v the_o departure_n from_o the_o maridian_n l_o m_z by_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 viz._n tang._n l_o k_o m_o so_o be_v the_o difference_n of_o latitude_n 1_o minute_n to_o wit_n k_o l_o to_o the_o departure_n from_o the_o meridian_n l_o m_o ∷_o now_o such_o proportion_n as_o one_o circle_n have_v to_o another_o such_o proportion_n have_v their_o degree_n semidiameters_a and_o sin_n of_o like_a arch_n one_o unto_o another_o as_o in_o the_o former_a scheme_n a_o c_o be_v the_o radius_fw-la of_o the_o equinoctial_a and_o e_z d_o be_v the_o radius_fw-la of_o a_o parallel_n or_o lesser_a circle_n who_o latitude_n from_o the_o equinoctial_a be_v a_o e_o but_z e_z d_o be_v the_o sine_fw-la of_o the_o ark_n e_o b_o which_o be_v the_o compliment_n of_o the_o latitude_n a_o e._n it_o therefore_o hold_v as_o the_o cousin_n of_o the_o latitude_n be_v to_o the_o radius_fw-la ∷_o or_o rather_o with_o other_o term_n in_o the_o same_o proportion_n as_o the_o radius_fw-la be_v to_o the_o secant_fw-la of_o the_o latitude_n ∷_o so_n be_v a_o mile_n or_o any_o part_n thereof_o or_o number_n of_o miles_n to_o the_o difference_n of_o longitude_n answer_v thereto_o ∷_o and_n because_o the_o two_o first_o term_n of_o the_o proportion_n vary_v not_o it_o will_v hold_v after_o the_o manner_n of_o the_o compound_v rule_n of_o three_o as_o the_o radius_fw-la be_v to_o the_o sum_n of_o the_o secant_v of_o all_o the_o parallel_n or_o latitude_n between_o any_o two_o place_n and_o we_o allow_v a_o parallel_n to_o pass_v through_o every_o minute_n or_o centesme_fw-fr of_o a_o degree_n so_o be_v a_o mile_n or_o any_o number_n or_o part_v thereof_o allot_v to_o every_o parallel_n to_o the_o whole_a difference_n of_o

semicircle_n which_o may_v sometime_o happen_v when_o both_o place_n be_v in_o one_o hemisphere_n if_o the_o sum_n of_o the_o compliment_n exceed_v 90_o degree_n second_o the_o latitude_n of_o the_o great_a ark_n be_v find_v by_o this_o proportion_n which_o be_v in_o a_o manner_n the_o same_o as_o for_o a_o parallel_n course_n as_o the_o cousin_n of_o either_o of_o the_o vertical_a angle_n but_o rather_o the_o lesser_a be_v to_o the_o tangent_fw-la of_o the_o latitude_n of_o that_o place_n to_o which_o it_o be_v adjacent_a ∷_o so_n be_v the_o cousin_n of_o the_o ark_n of_o difference_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 seek_v ∷_o thus_o by_o this_o excellent_a method_n of_o calculation_n we_o dispatch_v that_o at_o two_o operation_n which_o master_n norwood_n and_o other_o do_v not_o attain_v under_o seven_o or_o eight_o which_o render_v the_o sail_a by_o the_o great_a arch_n so_o difficult_a and_o laborious_a that_o none_o care_v to_o practice_v it_o and_o this_o latter_a proportion_n have_v two_o fix_v term_n in_o it_o will_v be_v perform_v on_o a_o serpentine-line_n or_o logarithmical_a ruler_n without_o alter_v the_o index_n or_o compass_n which_o proportion_n be_v vary_v be_v carry_v on_o in_o the_o former_a scheme_n as_o the_o secant_fw-la of_o either_o of_o the_o vertical_a angle_n be_v to_o the_o cotangent_fw-la of_o the_o latitude_n adjacent_a thereto_o ∷_o so_n be_v the_o secant_fw-la of_o the_o difference_n of_o longitude_n from_o the_o perpendicular_a to_o the_o cotangent_fw-la of_o the_o ark_n latitude_n seek_v ∷_o in_o the_o former_a scheme_n make_v the_o perpendicular_a p_o at_fw-fr radius_fw-la p_o t_o become_v the_o secant_fw-la of_o the_o vertical_a angle_n t_o p_o a_o to_o that_o radius_fw-la and_o be_v also_o by_o construction_n the_o tangent_fw-la of_o the_o compliment_n of_o the_o latitude_n next_o that_o angle_n to_z another_z radius_fw-la then_o by_o reason_n of_o the_o proportion_n of_o equality_n which_o we_o have_v former_o handle_v the_o secant_fw-la of_o any_o other_o angle_n from_o the_o perpendicular_a to_o the_o former_a radius_fw-la as_o be_v p_o e_o shall_v be_v also_o the_o cotangent_fw-la of_o the_o ark_n latitude_n to_o the_o latter_a radius_fw-la also_o the_o former_a scheme_n show_v we_o how_o to_o delineate_v proportion_n in_o sines_n and_o tangent_n by_o frame_v of_o rightlined_n oblique_a triangle_n for_o as_o the_o sine_fw-la of_o the_o angle_n at_o l_o be_v to_o its_o opposite_a side_n t_o p_o a_fw-fr tangent_fw-la so_o be_v the_o sine_fw-la of_o the_o angle_n at_o t_o to_o its_o opposite_a side_n p_o l_o another_z tangent_fw-la and_o by_o the_o like_a reason_n proportion_n in_o sines_n alone_o or_o in_o equal_a part_n and_o sin_n etc._n etc._n may_v be_v carry_v on_o in_o the_o angles_n and_o side_n of_o plain_a triangle_n of_o the_o six_o part_n of_o a_o rightangled_n spherical_a triangle_n no_o more_o but_o four_o at_o a_o time_n can_v be_v lay_v down_o in_o right_a line_n and_o angle_n as_o in_o the_o rightangled_n triangle_n p_o a_o t_o to_o wit_n the_o right_a angle_n at_o a_o the_o hipotenusal_a p_o t_n the_o perpendicular_a p_o a_o and_o the_o vertical_a angle_n between_o t_o p_o a_o so_o that_o the_o angle_n at_o t_n be_v the_o compliment_n of_o the_o vertical_a angle_n be_v not_o the_o angle_n of_o position_n in_o the_o sphere_n nor_o be_v the_o side_n a_o t_o the_o measure_n of_o the_o distance_n on_o that_o side_n the_o perpendicular_a however_o both_o these_o ark_n may_v be_v easy_o find_v example_n for_o the_o distance_n prick_v the_o radius_fw-la of_o the_o tangent_n from_o a_o to_o b_o and_z place_z the_o extent_n b_o p_o be_v the_o secant_fw-la of_o the_o perpendicular_a from_o a_o to_o e_o from_z whence_o line_n draw_v to_o l_o and_o t_o shall_v contain_v a_o angle_n equal_a to_o the_o distance_n between_o the_o lizard_n and_o trinity_n harbour_n in_o the_o great_a arch_n to_o measure_v it_o with_o 60_o of_o the_o chord_n set_v one_o foot_n at_o e_o draw_z k_o m_o which_o extent_n measure_v in_o the_o chord_n show_v the_o distance_n to_o be_v 50_o 9′_n at_o 20_o league_n to_o a_o degree_n be_v 1003_o league_n the_o rumbe_n between_o these_o two_o place_n be_v 74_o 17′_n from_o the_o meridian_n and_o the_o distance_n of_o the_o rumbe_n 1034_o league_n and_o after_o the_o same_o manner_n any_o part_n of_o it_o may_v be_v measure_v and_o so_o likewise_o may_v the_o parallel_n distance_n i_o v._o which_o if_o we_o have_v room_n will_v be_v find_v to_o be_v 55_o degree_n and_o a_o half_a also_o the_o distance_n i_o t_n will_v be_v find_v to_o be_v 74_o 7′_n the_o compliment_n whereof_o to_o a_o semicircle_n be_v 105_o 53′_n will_v be_v the_o distance_n between_o the_o lizard_n and_o trinity_n harbour_n as_o we_o suppose_v they_o the_o one_o in_o south_n the_o other_o in_o north_n latitude_n the_o proportion_n carry_v on_o to_o find_v the_o distance_n be_v as_o the_o radius_fw-la be_v to_o the_o sine_fw-la of_o the_o perpendicular_a ∷_o so_n be_v the_o tangent_fw-la of_o the_o vertical_a angle_n to_o the_o tangent_fw-la of_o the_o distance_n ∷_o the_o perpendicular_a a_o p_o be_v a_o tangent_fw-la to_o the_o radius_fw-la a_o b_o and_o if_o we_o make_v the_o extent_n b_o p_o radius_fw-la it_o than_o become_v a_o sine_fw-la as_o be_v evident_a if_o you_o describe_v a_o ark_n from_o p_o therewith_o which_o r●dius_fw-la if_o we_o shall_v place_v from_o p_o outward_a beyond_o c_z and_o suppose_v a_o tangent_fw-la erect_v thereon_o parallel_v to_o a_o t_o then_o do_v the_o extent_n a_o t_o become_v the_o tangent_fw-la of_o the_o four_o proportional_a to_o that_o radius_fw-la which_o that_o it_o may_v be_v measure_v the_o radius_fw-la be_v prick_v from_o a_o to_o e._n to_o find_v the_o angle_n of_o position_n if_o the_o hipotenusal_a and_o one_o legg_n be_v give_v the_o proportion_n to_o find_v the_o angle_n next_o that_o leg_n will_v be_v as_o the_o radius_fw-la be_v to_o the_o tangent_fw-la of_o the_o hipotenusal_a ∷_o so_n be_v the_o tangent_fw-la of_o the_o give_v legg_n to_o the_o cousin_n of_o its_o adjacent_a angle_n ∷_o and_o so_o in_o the_o former_a scheme_n if_o we_o make_v t_o p_o radius_fw-la which_o be_v also_o the_o tangent_fw-la of_o the_o hipotenusal_a then_o do_v p_o a_o be_v the_o tangent_fw-la of_o the_o give_a leg_n become_v the_o sine_fw-la of_o the_o angle_n p_o t_o a_o which_o be_v the_o compliment_n of_o the_o angle_n t_o p_o a_o whence_o we_o may_v observe_v that_o if_o a_o perpendicular_a be_v raise_v at_o the_o end_n of_o the_o tangent_fw-la of_o the_o give_a leg_n the_o tangent_fw-la of_o the_o hipotenusal_a be_v from_o the_o other_o end_n of_o the_o say_a leg_n make_v the_o hipotenusal_a opposite_a to_o the_o right_a angle_n the_o angle_n include_v shall_v be_v the_o angle_n seek_v therefore_o have_v the_o distance_n or_o leg_n a_o t_o give_v with_o its_o radius_fw-la a_o e_o you_o may_v proportion_v out_o the_o hipotenusal_a to_o that_o radius_fw-la and_o with_o it_o set_v one_o foot_n in_o t_o across_o a_o e_o with_o the_o other_o from_o whence_o draw_v a_o line_n to_o t_o the_o angle_n between_o it_o and_o a_o t_o shall_v be_v the_o angle_n of_o position_n require_v otherwise_o more_o ready_o take_v the_o near_a distance_n from_o c_o to_o t_o p_o and_o it_o shall_v be_v the_o cousin_n of_o the_o angle_n of_o position_n if_o we_o make_v e_o at_fw-fr radius_fw-la wherefore_o upon_o a_o describe_v the_o prick_a ark_n n_o a_o ruler_n from_o the_o centre_n e_o just_a touch_v it_o cut_v the_o ark_n x_o n_z at_z n_z which_o ark_n measure_v on_o the_o chord_n be_v 38_o 49′_n the_o compliment_n whereof_o be_v 51_o 11′_n be_v the_o angle_n of_o position_n require_v and_o so_o may_v all_o the_o other_o angle_n of_o position_n be_v find_v if_o there_o be_v any_o need_n of_o they_o the_o proportion_n carry_v on_o be_v as_o the_o radius_fw-la be_v to_o the_o sine_fw-la of_o the_o vertical_a angle_n so_o be_v the_o cousin_n of_o the_o perpendicular_a to_o the_o cousin_n of_o the_o angle_n of_o position_n ∷_o if_o we_o make_v b_o p_o radius_fw-la then_o be_v a_o b_o the_o cousin_n of_o the_o perpendicular_a to_o that_o radius_fw-la and_o the_o near_a distance_n from_o c_o to_o t_o p_o be_v the_o cousin_n of_o the_o angle_n of_o position_n to_o the_o radius_fw-la e_o a._n thus_o we_o have_v show_v the_o use_n of_o this_o pro●ection_n in_o more_o particular_n than_o the_o seaman_n shall_v have_v occasion_n to_o use_v last_o it_o may_v be_v note_v that_o in_o stead_n of_o prick_v down_o a_o new_a line_n of_o tangent_n from_o the_o perpendicular_a in_o every_o question_n that_o shall_v be_v put_v that_o if_o the_o perpendicular_a be_v place_v in_o the_o side_n p_o l_o

give_v i_o have_v show_v in_o a_o treatise_n entitle_v the_o sector_n on_o a_o quadrant_n page_n 139_o 140._o and_o how_o to_o find_v the_o point_n i_o or_o k_o without_o draw_v the_o line_n e_o f_o or_o o_o g_o and_o that_o by_o help_n of_o a_o cross_n or_o intersection_n like_o that_o at_o e_o which_o may_v either_o happen_v within_o or_o without_o the_o outward_a circle_n the_o reader_n may_v attain_v from_o the_o last_o scheme_n for_o find_v the_o amplitude_n the_o converse_n of_o the_o former_a scheme_n for_o find_v the_o hour_n will_v find_v the_o sun_n altitude_n on_o all_o hour_n and_o the_o distance_n of_o place_n in_o the_o arch_n of_o a_o great_a circle_n example_n latitude_n 51_o 32′_n declination_n 23_o 31′_n north._n hour_n 75_o from_o noon_n that_o be_v either_o 7_o in_o the_o morning_n or_o 5_o in_o the_o afternoon_n have_v draw_v the_o semicircle_n its_o diameter_n and_o by_o a_o perpendicular_a from_o the_o centre_n divide_v it_o into_o two_o quadrant_n and_o 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distance_n of_o place_n in_o the_o ark_n of_o a_o great_a circle_n the_o case_n in_o spherical_a triangle_n be_v the_o same_o with_o that_o here_o resolve_v so_o if_o there_o be_v two_o place_n the_o one_o in_o north_n latitude_n 51_o 32′_n the_o other_o in_o north_n latitude_n 23_o 31′_n the_o difference_n of_o longitude_n between_o they_o be_v 75_o their_o distance_n by_o the_o former_a scheme_n will_v be_v find_v to_o be_v 62_o 37′_n but_o if_o the_o latter_a place_n be_v in_o as_o much_o south_n latitude_n than_o their_o distance_n will_v be_v 99_o 30′_n another_o example_n for_o find_v the_o distance_n of_o place_n in_o the_o arch_n of_o a_o great_a circle_n example_n let_v the_o two_o place_n be_v according_a to_o the_o seaman_n calendar_n isle_n of_o lobos_n longitude_n 307_o 41′_n latitude_n 40_o 21′_n south_n lizard_n 18_o 30._o latitude_n 50_o 10_o north._n difference_n of_o longitude_n 289_o 11._o compliment_n 70_o 49′_n have_v describe_v a_o circle_n make_v ae_n m_o 70_o 49′_n m_o i_o the_o latitude_n of_o the_o island_n b_o i_o the_o sine_fw-la thereof_o fall_v perpendicular_o on_o c_o m_n ae_n l_o the_o latitude_n of_o the_o lizard_n l_o a_o the_o sine_fw-la thereof_o make_v a_o e_o equal_a to_o the_o extent_n a_o b_o and_o prick_v b_o i_o from_o l_o upward_o to_o h_n when_o the_o place_n be_v in_o different_a hemisphere_n but_o downward_o to_o k_o when_o in_o the_o same_o hemisphere_n and_o the_o extent_n h_o e_o or_o k_n e_o be_v the_o chord_n of_o the_o ark_n of_o distance_n between_o both_o place_n in_o this_o example_n h_o e_z be_v 109_o 41′_n k_o e_z be_v 48_o 57′_n demonstration_n this_o scheme_n i_o first_o meet_v with_o in_o a_o map_n make_v in_o holland_n the_o foundation_n whereof_o be_v long_o since_o lay_v by_o copernicus_n and_o regiomontanus_n who_o from_o a_o right_n line_v plain_a triangle_n happen_v at_o the_o centre_n of_o the_o sphere_n have_v prescribe_v a_o method_n of_o calculation_n for_o find_v a_o angle_n when_o three_o 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matter_n of_o calculation_n be_v one_o side_n of_o a_o plain_a triangle_n right_o angle_v the_o sum_n or_o difference_n of_o the_o sin_n of_o both_o latitude_n the_o other_o side_n and_o the_o chord_n of_o the_o distance_n be_v the_o hipotenusal_a or_o three_o side_n seek_v thus_o the_o ancient_n by_o calculation_n and_o we_o by_o protraction_n have_v two_o side_n and_o the_o angle_n comprehend_v give_v find_v the_o three_o side_n or_o have_v three_o side_n give_v find_v the_o angle_n opposite_a to_o that_o side_n which_o in_o the_o scheme_n be_v measure_v by_o a_o chord_n as_o by_o result_n from_o the_o three_o side_n there_o will_v be_v get_v the_o two_o extent_n a_o e_o and_o c_o b_o and_o consequent_o the_o intersection_n at_o b_o and_o thence_o the_o angle_n ae_n m_o which_o before_o be_v insist_v on_o in_o find_v the_o azimuth_n and_o hour_n by_o the_o like_a reason_n the_o distance_n of_o star_n may_v be_v find_v from_o their_o longitude_n and_o latitude_n or_o from_o their_o declination_n and_o right_a ascension_n divers_a other_o scheme_n from_o other_o proportion_n may_v be_v add_v for_o find_v the_o hour_n and_o azimuth_n etc._n etc._n which_o which_o i_o be_o loath_a to_o trouble_v the_o reader_n withal_o i_o shall_v add_v another_o scheme_n for_o this_o purpose_n which_o carry_v on_o the_o same_o proportion_n by_o which_o this_o case_n be_v usual_o calculate_v the_o proportion_n be_v express_v in_o a_o treatise_n the_o sector_n on_o a_o quadrant_n page_n 127._o example_n latitude_n 51_o 32′_n declination_n 23_o 31._o hour_n 60_o from_o noon_n have_v draw_v a_o semicircle_n and_o the_o radius_fw-la z_o d_o prick_v the_o latitude_n from_o c_o to_o l_o

which_o snellius_n assert_n be_v that_o the_o tangent_fw-la cut_v off_o to_o wit_n i_o b_o be_v somewhat_o short_a than_o the_o arch_n i_o a_o though_o near_o it_o in_o length_n when_o the_o arch_n be_v not_o above_o 1_o 12_o part_n of_o a_o quadrant_n and_o this_o hugenius_n demonstrate_v in_o his_o book_n de_fw-fr magnitudine_fw-la circuli_fw-la where_o he_o find_v fault_n with_o snellius_n his_o demonstration_n thereof_o now_o the_o proportion_n for_o find_v a_o angle_n raise_v from_o that_o scheme_n lie_v thus_o as_o r_n e_o the_o sum_n of_o the_o double_a of_o the_o hipotenusal_a c_o a_n and_o of_o the_o side_n c_o e_o be_v to_o e_o a_o the_o lesser_a side_n ∷_o so_n be_v r_n i_o the_o triple_a of_o the_o radius_fw-la or_o hipotenusal_a ∷_o to_z i_o b_o the_o length_n of_o the_o arch_n require_v ∷_o propé_fw-fr verum_fw-la this_o proportion_n find_v the_o length_n of_o the_o arch_n make_v the_o hipotenusal_a always_o radius_fw-la whereas_o in_o calculation_n we_o always_o retain_v such_o a_o radius_fw-la whereof_o the_o circumference_n of_o the_o circle_n be_v 360_o now_o the_o diameter_n of_o such_o a_o circle_n will_v be_v find_v by_o the_o number_n in_o page_n 112_o in_o which_o the_o proportion_n of_o the_o circumference_n to_o the_o diameter_n be_v express_v to_o be_v 114_o 5915_o wherefore_o the_o radius_fw-la of_o such_o a_o circle_n be_v 57_o 2957_o and_o the_o triple_a thereof_o be_v 171_o 8871_o then_o retain_v the_o two_o first_o term_n of_o the_o former_a proportion_n we_o may_v make_v this_o number_n the_o thy_o d_o term_n and_o by_o one_o single_a work_n find_v the_o angle_n or_o rather_o take_v the_o half_n of_o all_o four_o term_n the_o proportion_n will_v hold_v as_o the_o sum_n of_o the_o hipotenusal_a and_o of_o half_a the_o great_a leg_n of_o a_o plain_a right_a angle_a triangle_n be_v to_o the_o lesser_a leg_n thereof_o ∷_o so_n be_v 86_o to_o the_o angle_n oppose_v to_o the_o lesser_a leg_n the_o half_a of_o 171_o 8871_o be_v 85_o 9435_o 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square_a root_n of_o the_o sum_n of_o the_o square_n of_o these_o two_o number_n we_o shall_v find_v the_o side_n c_o a_o to_o be_v 5_o 73_o to_o which_o add_v the_o half_a of_o c_o e_o the_o sum_n be_v 782_o the_o divisor_n then_o multiply_v 393_o the_o lesser_a leg_n by_o 86_o the_o product_n be_v 33798_o to_o which_o you_o may_v annex_v cipher_n at_o pleasure_n to_o find_v the_o decimal_a part_n of_o a_o degree_n and_o divide_v by_o 782_o you_o will_v find_v the_o quotient_a to_o be_v 43_o 22_o centesme_n to_o which_o if_o you_o add_v 5_o centesme_n error_n the_o angle_n seek_v be_v 43_o 27_o centesm_n ready_o to_o find_v what_o allowance_n must_v be_v make_v in_o respect_n of_o the_o arch_n find_v you_o may_v repair_v to_o a_o table_n of_o natural_a sin_n and_o take_v the_o two_o leg_n of_o the_o right_a angle_a triangle_n to_o be_v the_o sine_fw-la and_o cousin_n of_o any_o arch_n and_o by_o the_o last_o proportion_n find_v how_o near_o you_o can_v recover_v the_o arch_n again_o 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a_o arch_n of_o 30_o not_o above_o the_o 1_o 21600_o part_n of_o a_o degree_n whereby_o the_o sin_n may_v be_v examine_v and_o a_o angle_n find_v without_o table_n as_o if_o the_o side_n of_o the_o former_a triangle_n be_v give_v by_o take_v c_o e_o out_o of_o c_o a_o there_o remain_v e_z i_o the_o square_a whereof_o add_v to_o the_o square_n of_o e_o a_o the_o sine_fw-la be_v equal_a to_o the_o square_n of_o the_o line_n a_o i_o the_o chord_n whereby_o may_v be_v find_v the_o length_n of_o the_o arch_n i_o a_o to_o the_o give_v radius_fw-la c_o a_o and_o then_o by_o another_o proportion_n the_o measure_n of_o the_o say_a arch_n to_o the_o radius_fw-la of_o such_o a_o circle_n who_o circumference_n be_v 360d._o in_o like_a manner_n if_o the_o side_n of_o a_o oblique_a plain_a triangle_n be_v give_v the_o angle_n thereof_o may_v be_v find_v if_o you_o first_o reduce_v that_o oblique_a triangle_n to_o two_o right_a angle_a plain_a triangle_n which_o be_v perform_v in_o every_o book_n of_o trigonometry_n without_o table_n but_o for_o such_o case_n of_o plain_a triangle_n in_o which_o but_o one_o side_n with_o two_o angle_n be_v give_v to_o find_v the_o other_o side_n in_o regard_n the_o proportion_n for_o such_o case_n require_v sin_n and_o that_o we_o have_v not_o attain_v any_o ready_a way_n to_o make_v the_o sine_fw-la of_o any_o arch_n at_o command_n forbear_v to_o mention_v such_o way_n as_o be_v both_o troublesome_a and_o uncertain_a we_o must_v suppose_v that_o the_o reader_n be_v furnish_v with_o a_o table_n of_o sines_n which_o most_o mariner_n have_v in_o their_o seaman_n calendar_n finis_fw-la errata_fw-la page_n 1_o line_n 31_o for_o will_v read_v shall_v p._n 48._o l._n 30._o for_o 48_o degree_n r._n 84_o degree_n p._n 49_o l._n 3._o for_o angle_n r_o angle_n l._n 4_o for_o sine_fw-la r._n line_n p._n 51._o l._n 25._o for_o follow_a r_o former_a p._n 59_o l._n 14._o for_o or_o r._n of_o p._n 60._o l._n 21._o for_o as_o every_o r._n or_o ever_o p._n 81_o l._n 16._o for_o one_o be_v r._n one_o in_o page_n 88_o l._n 26._o for_o tangent_fw-la r._n tangent_n a_o table_n of_o meridional_a part_n d_o 0_o 1_o 2_o 3_o 4_o 5_o 6_o 7_o 8_o 9_o 0_o 0_o 000_o 1_o 000_o 2_o 000_o 3_o 001_o 4_o 003_o 5_o 006_o 6_o 011_o 7_o 017_o 8_o 02●_n 9_o 037_o 2_o 020_o 020_o 2_o 020_o 0●1_n 023_o 029_o 031_o 037_o 046_o 057_o 4_o 040_o 040_o 04●_n 041_o 043_o 046_o 051_o 057_o 066_o 077_o 6_o 060_o 060_o 060_o 061_o 063_o 066_o 071_o 077_o 086_o 097_o 8_o 080_o 080_o 080_o 081_o 083_o 086_o 091_o 097_o 106_o 117_o 10_o 100_o 1_o 100_o 2_o 100_o 3_o 101_o 4_o 103_o 5_o 106_o 6_o 111_o 7_o 118_o 8_o 127_o 9_o 138_o 12_o 120_o 120_o 120_o 121_o 123_o 126_o 131_o 138_o 147_o 158_o 14_o 140_o 140_o 140_o 141_o 143_o 146_o 151_o 158_o 167_o 178_o 16_o 160_o 160_o 160_o 161_o 163_o 166_o 171_o 178_o 187_o 198_o 18_o 180_o 180_o 180_o 181_o 183_o 186_o 191_o 198_o 207_o 218_o 20_o 200_o 1_o 200_o 2_o 200_o 3_o 201_o 4_o 204_o 5_o 207_o 6_o 212_o 7_o 219_o 8_o 228_o 9_o 239_o 22_o 220_o 220_o 220_o 221_o 224_o 227_o 232_o 239_o 248_o 259_o 24_o 240_o 240_o 240_o 241_o 244_o 247_o 252_o 259_o 268_o 279_o 26_o 260_o 260_o 260_o 261_o 26●_n 267_o 272_o 279_o 288_o 300_o 28_o 280_o 280_o 280_o 281_o 284_o 287_o 292_o 299_o 308_o 320_o 30_o 300_o 1_o 300_o 2_o 300_o 3_o 301_o 4_o 304_o 5_o 307_o 6_o 312_o 7_o 319_o 8_o 329_o 9_o 341_o 32_o 320_o 320_o 320_o 321_o 324_o 327_o 332_o 339_o 349_o 361_o 34_o 340_o 340_o 340_o 341_o 344_o 347_o 352_o