the_o angle_n at_o a_o be_v the_o compliment_n of_o the_o latitude_n and_o the_o angle_n at_o c_o the_o vertical_a angle_n make_v between_o the_o equator_fw-la and_o the_o vertical_a circle_n pass_v by_o the_o sun_n at_o c_o perpendicular_o to_o the_o horizon_n ab_fw-la wherefore_o know_v the_o base_a ac_fw-la to_o be_v 65_o degree_n 23_o minute_n and_o the_o angle_n at_o a_o to_o be_v 37_o degree_n 30_o minute_n i_o may_v find_v the_o side_n bc_n for_o if_o i_o place_v the_o thread_n to_o the_o intersection_n of_o the_o contrary_a sine_fw-la of_o 90_o degree_n with_o the_o right_n sine_fw-la of_o 65_o degree_n 22_o minute_n it_o will_v cut_v the_o contrary_a sine_fw-la of_o 37_o degree_n 30_o minute_n at_o the_o right_n sine_fw-la of_o 33_o degree_n 36_o minute_n and_o such_o be_v the_o side_n bc_n which_o be_v require_v chap._n xxiv_o to_o find_v a_o side_n by_o know_v the_o base_a and_o the_o other_o side_n as_o the_o sine_fw-la of_o the_o compliment_n of_o the_o side_n give_v to_z the_o sine_z of_o the_o compliment_n of_o the_o base_a so_o be_v the_o radius_fw-la to_o the_o sine_fw-la of_o the_o compliment_n of_o the_o side_n require_v thus_o in_o the_o rectangle_n abc_n have_v ac_fw-la 65_o degree_n 23_o minute_n and_o ab_fw-la 60_o degree_n let_v it_o be_v require_v to_o find_v the_o side_n bc_n place_v the_o thread_n at_o the_o intersection_n of_o the_o contrary_a sine_fw-la of_o 30_o degree_n the_o compliment_n of_o ab_fw-la with_o the_o right_n sine_fw-la of_o 24_o degree_n 37_o minute_n the_o compliment_n of_o ac_fw-la and_o it_o shall_v cut_v the_o contrary_a sine_fw-la of_o 90_o degree_n at_o the_o right_n sine_fw-la of_o 56_o degree_n 24_o minute_n the_o compliment_n whereof_o be_v 33_o degree_n 36_o minute_n the_o side_n bc_n which_o be_v require_v chap._n xxv_o to_o find_v a_o side_n by_o know_v the_o two_o oblique_a angle_n as_o the_o sine_fw-la of_o either_o angle_n to_o the_o cousin_n of_o the_o other_o angle_n so_o be_v the_o sine_fw-la of_o 90_o degree_n to_o the_o cousin_n of_o the_o side_n opposite_a to_o the_o second_o angle_n so_o in_o the_o rectangle_n abc_n have_v acb_n for_o the_o first_o angle_n 72_o degree_n 16_o minute_n and_o bac_n for_o the_o second_o 37_o degree_n 30_o minute_n it_o be_v require_v to_o find_v the_o side_n bc_n wherefore_o i_o place_v the_o the_o thread_n at_o the_o intersection_n of_o the_o contrary_a sine_fw-la of_o 72_o degree_n 16_o minute_n with_o the_o right_n sine_fw-la of_o 52_o degree_n 30_o minute_n the_o compliment_n of_o the_o angle_n at_o a_o and_o i_o find_v it_o cut_v the_o contrary_a sine_fw-la of_o 90_o degree_n at_o the_o right_n sine_fw-la of_o 56_o degree_n 24_o minute_n who_o compliment_n be_v 33_o degree_n 36_o minute_n the_o side_n bc_n as_o be_v require_v chap._n xxvi_o to_o find_v the_o base_a by_o know_v both_o the_o side_n as_o the_o radius_fw-la to_o the_o cousin_n of_o the_o one_o side_n so_o the_o cousin_n of_o the_o other_o side_n to_o the_o cousin_n of_o the_o base_a thus_o in_o the_o rectangle_n abc_n know_v the_o latitude_n of_o the_o place_n with_o the_o altitude_n of_o the_o sun_n have_v no_o declination_n we_o may_v find_v the_o hour_n of_o the_o day_n as_o have_v bc_n the_o altitude_n of_o the_o sun_n 33_o degree_n 36_o minute_n and_o the_o angle_n bac_n the_o compliment_n of_o the_o latitude_n 37_o degree_n 30_o minute_n the_o base_a ac_fw-la which_o be_v the_o hour_n of_o the_o day_n count_v from_o sun_n rise_v or_o set_v will_v be_v find_v by_o the_o quadrat_fw-la for_o if_o you_o apply_v the_o thread_n to_o the_o intersection_n of_o the_o contrary_a sine_fw-la of_o 37_o degree_n 30_o minute_n with_o the_o right_n sine_fw-la of_o 33_o degree_n 36_o minute_n it_o will_v cut_v the_o contrary_a sine_fw-la of_o 90_o degree_n at_o the_o right_n sine_fw-la of_o 65_o degree_n 23_o minute_n and_o such_o be_v the_o base_a ac_fw-la which_o be_v reduce_v into_o time_n by_o allow_v 15_o degree_n to_o a_o hour_n and_o for_o every_o 4_o minute_n a_o degree_n give_v 4_o hour_n 22_o minute_n for_o the_o time_n after_o sun_n rise_v or_o before_o sun_n set_v so_o that_o if_o it_o be_v in_o the_o forenoon_n it_o be_v 10_o a_o clock_n and_o 22_o minute_n the_o sun_n then_o rise_v at_o 6_o and_o if_o it_o be_v in_o the_o afternoon_n it_o be_v 38_o minute_n past_o one_o chap._n xxviii_o to_o find_v a_o angle_n by_o know_v the_o other_o oblique_a angle_n and_o the_o side_n opposite_a to_o the_o inquire_v angle_n a_o the_o radius_fw-la to_o the_o sine_fw-la of_o the_o angle_n give_v so_o the_o cousin_n of_o the_o side_n to_o the_o cousin_n of_o the_o angle_n require_v thus_o in_o the_o rectangle_n abc_n have_v the_o angle_n bac_n 37_o degree_n 30_o minute_n with_o the_o side_n ab_fw-la 60_o degree_n to_o find_v the_o angle_n acb_n i_o place_v the_o thread_n at_o the_o intersection_n of_o the_o contrary_a sine_fw-la of_o 90_o degree_n with_o the_o right_n sine_fw-la of_o 37_o degree_n 30_o minute_n so_o will_v it_o cut_v the_o contrary_a sine_fw-la of_o 30_o degree_n the_o compliment_n of_o the_o side_n ab_fw-la at_o the_o right_n sine_fw-la of_o 17_o degree_n 44_o minute_n the_o compliment_n whereof_o be_v 72_o degree_n 16_o minute_n the_o angle_n acb_n which_o be_v require_v chap._n xxix_o to_o find_v a_o angle_n by_o know_v the_o other_o oblique_a angle_n and_o the_o side_n opposite_a to_o the_o angle_n give_v as_o the_o cousin_n of_o the_o side_n to_o the_o cousin_n of_o the_o angle_n give_v so_o be_v the_o radius_fw-la to_o the_o sine_fw-la of_o the_o angle_n require_v so_o in_o the_o rectangle_n abc_n have_v the_o angle_n bac_n 37_o degree_n 30_o minute_n and_o the_o side_n bc_n 33_o degree_n 36_o minute_n to_o find_v the_o angle_n acb_n place_n the_o thread_n at_o the_o intersection_n of_o the_o contrary_a sine_fw-la of_o 56_o degree_n 24_o minute_n the_o compliment_n of_o bc_n with_o the_o right_n sine_fw-la of_o 52_o degree_n 30_o minute_n the_o compliment_n of_o the_o angle_n bac_n and_o it_o will_v cut_v the_o contrary_a sine_fw-la of_o 90_o degree_n at_o the_o right_n sine_fw-la of_o 72_o degree_n 16_o minute_n the_o angle_n acb_n which_o be_v require_v chap._n xxx_o to_o find_v a_o angle_n hy_z know_v the_o base_a and_o the_o side_n opposite_a to_o the_o inquire_v angle_n as_o the_o sine_fw-la of_o the_o base_a to_o the_o sine_fw-la of_o 90_o degree_n so_o the_o sine_fw-la of_o the_o side_n give_v to_z the_o sine_z of_o the_o angle_n require_v thus_o in_o the_o rectangle_n abc_n have_v the_o base_a ac_fw-la 65_o degree_n 23_o minute_n and_o the_o side_n bc_n 33_o degree_n 36_o minute_n to_o find_v the_o angle_n bac_n i_o place_v the_o thread_n to_o the_o intersection_n of_o the_o contrary_a sine_fw-la of_o 65_o degree_n 23_o minute_n with_o the_o right_n sine_fw-la of_o 90_o degree_n and_o i_o find_v it_o cut_v the_o contrary_a sine_fw-la of_o 33_o degree_n 36_o minute_n at_o the_o right_n sine_fw-la of_o 37_o degree_n 30_o minute_n the_o angle_n bac_n which_o be_v require_v thus_o far_o by_o sin_n alone_o but_o now_o we_o must_v require_v joint_a help_n of_o tangent_n and_o forasmuch_o as_o the_o tangent_n extend_v but_o to_o 45_o degree_n which_o be_v equal_a to_o the_o side_n of_o the_o quadrat_fw-la i_o shall_v set_v down_o the_o two_o way_n for_o the_o resolution_n of_o each_o proposition_n deliver_v by_o master_n gunter_n and_o if_o one_o will_v not_o hold_v the_o other_o will_v observe_v the_o rule_v give_v to_o this_o purpose_n in_o the_o 8_o chapter_n chap._n xxxi_o to_o find_v a_o side_n by_o know_v the_o other_o side_n and_o the_o angle_n opposite_a to_o the_o inquire_v side_n as_o the_o radius_fw-la to_o the_o sine_fw-la of_o the_o side_n give_v so_o be_v the_o tangent_fw-la of_o the_o angle_n to_o the_o tangent_fw-la of_o the_o side_n require_v or_o as_o the_o sine_fw-la of_o the_o side_n give_v be_v to_o the_o radius_fw-la so_o the_o cotangent_fw-la of_o the_o angle_n to_o the_o cotangent_fw-la of_o the_o side_n require_v thus_o in_o the_o rectangle_n abc_n have_v the_o side_n ab_fw-la 60_o degree_n and_o the_o angle_n bac_n 37_o degree_n 30_o minute_n to_o find_v the_o side_n bc_n i_o place_v the_o thread_n at_o the_o intersection_n of_o the_o contrary_a sine_fw-la of_o 90_o degree_n with_o the_o right_n sine_fw-la of_o 60_o degree_n and_o it_o do_v cut_v the_o contrary_a tangent_fw-la of_o 37_o degree_n 30_o minute_n at_o the_o right_n tangent_fw-la of_o 33_o degree_n 36_o minute_n the_o side_n bc_n which_o be_v require_v chap._n xxxii_o to_o find_v a_o side_n by_o know_v the_o other_o side_n and_o the_o angle_n next_o to_o the_o inquire_v side_n as_o the_o tangent_fw-la of_o the_o angle_n give_v to_o the_o tangent_fw-la of_o the_o side_n
of_o a_o rectangle_n spherical_a triangle_n now_o follow_v the_o resolution_n of_o all_o manner_n of_o spherical_a triangle_n whatsoever_o in_o any_o spherical_a triangle_n whatsoever_o chap._n xli_o to_o find_v a_o side_n opposite_a to_o a_o angle_n give_v by_o know_v one_o side_n and_o two_o angle_n the_o one_o opposite_a to_o the_o give_v side_n the_o other_o to_o the_o side_n require_v as_o the_o sine_fw-la of_o the_o opposite_a angle_n to_o the_o side_n give_v be_v to_o the_o sine_fw-la of_o that_o side_n give_v so_o be_v the_o sine_fw-la of_o the_o angle_n opposite_a to_o the_o side_n require_v to_o the_o sine_fw-la of_o the_o side_n require_v thus_o in_o the_o follow_a triangle_n pzs_n have_v the_o side_n pz_fw-mi 37_o degree_n 30_o minute_n and_o the_o angle_n zp_v 53_o degree_n 40_o minute_n with_o the_o angle_n zsp_v 34_o degree_n 47_o minute_n we_o may_v find_v the_o side_n zs_n for_o if_o you_o apply_v the_o thread_n to_o the_o intersection_n of_o the_o contrary_a sine_fw-la of_o 34_o degree_n 47_o minute_n with_o the_o right_n sine_fw-la of_o 37_o degree_n 30_o minute_n it_o shall_v cut_v the_o contrary_a sine_fw-la of_o 53_o degree_n 40_o minute_n at_o the_o right_n sine_fw-la of_o 59_o degree_n 15_o minute_n and_o such_o be_v the_o side_n zs_n which_o be_v require_v chap._n xl._n to_o find_v a_o angle_n opposite_a to_o a_o side_n give_v by_o know_v one_o angle_n and_o two_o side_n the_o one_o opposite_a to_o the_o give_v angle_n the_o other_o to_o the_o angle_n require_v as_o the_o sine_fw-la of_o the_o side_n opposite_a to_o the_o angle_n give_v be_v to_o the_o sine_fw-la of_o that_o angle_n give_v so_o be_v the_o sine_fw-la of_o the_o side_n opposite_a to_o the_o angle_n require_v to_o the_o sine_fw-la of_o the_o angle_n require_v thus_o in_o the_o triangle_n pzs_n have_v the_o angle_n pzs_n 113_o degree_n 17_o minute_n the_o azimuth_n from_o the_o north_n and_o the_o side_n ps_n 78_o degree_n 30_o minute_n the_o compliment_n of_o the_o sun_n declination_n with_o the_o side_n zs_n 59_o degree_n 15_o minute_n the_o compliment_n of_o the_o sun_n altitude_n we_o may_v find_v the_o angle_n zp_v which_o be_v the_o hour_n from_o the_o meridian_n for_o if_o we_o place_v the_o thread_n at_o the_o intersection_n of_o the_o contrary_a sine_fw-la of_o 78_o degree_n 30_o minute_n with_o the_o right_n sine_fw-la of_o 66_o degree_n 43_o minute_n the_o outward_a angle_n at_o z_o because_o the_o inward_a angle_n exceed_v a_o quadrant_a it_o will_v cut_v the_o contrary_a sine_fw-la of_o 59_o degree_n 15_o minute_n at_o the_o right_n sine_fw-la of_o 53_o degree_n 40_o minute_n which_o be_v the_o hour_n from_o the_o meridian_n as_o be_v require_v chap._n xli_o to_o find_v a_o angle_n by_o know_v the_o three_o side_n in_o respect_n of_o the_o three_o side_n of_o a_o not_o right_o angle_v spherical_a triangle_n let_v that_o which_o subtend_v the_o angle_n require_v be_v call_v the_o base_a and_o the_o other_o two_o the_o contain_v side_n then_o first_o take_v the_o sum_n and_o difference_n of_o the_o lesser_a side_n and_o the_o compliment_n or_o excess_n of_o the_o great_a side_n and_o add_v the_o sin_n of_o the_o say_a sum_n and_o difference_n together_o the_o half_a whereof_o shall_v be_v the_o first_o find_v sine_fw-la then_o out_o of_o the_o sine_fw-la of_o the_o sum_n take_v this_o first_o find_v sine_fw-la and_o note_v the_o remainder_n now_o if_o every_o of_o the_o three_o side_n be_v less_o than_o a_o quadrant_a take_v the_o difference_n between_o the_o remainder_n note_v and_o the_o cousin_n of_o the_o base_a but_o if_o any_o of_o the_o three_o side_n be_v great_a than_o a_o quadrant_a add_v the_o remainder_n note_v to_o the_o cousin_n or_o excess_n of_o the_o base_a so_o shall_v the_o sum_n or_o difference_n be_v the_o second_o find_v sine_fw-la then_o say_v as_o the_o first_o find_v sine_fw-la to_o the_o second_o sound_n sine_fw-la so_o be_v the_o radius_fw-la to_o a_o four_o fine_a now_o when_o every_o side_n of_o the_o triangle_n be_v less_o than_o a_o quadrant_a if_o the_o remainder_n note_v be_v less_o than_o the_o cousin_n of_o the_o base_a the_o arch_n of_o this_o four_o sine_fw-la must_v be_v take_v out_o of_o 90_o degree_n but_o if_o the_o remainder_n note_v be_v great_a than_o the_o cousin_n of_o the_o base_a then_o add_v the_o arch_n of_o this_o four_o sine_fw-la to_o 90_o degree_n so_o shall_v the_o sum_n or_o difference_n be_v the_o angle_n require_v but_o when_o one_o side_n be_v great_a than_o a_o quadrant_a and_o that_o side_n be_v make_v the_o base_a then_o always_o add_v this_o four_o arch_n to_o 90_o degree_n to_o make_v up_o your_o angle_n require_v and_o if_o the_o side_n exceed_v a_o quadrant_a be_v one_o of_o the_o contain_v side_n than_o always_o take_v this_o four_o arch_n out_o of_o 90_o degeee_n and_o the_o remainder_n shall_v be_v the_o angle_n require_v as_o in_o the_o triangle_n zp_v have_v the_o side_n zp_n the_o compliment_n of_o the_o latitude_n 37_o degree_n 30_o minute_n and_o the_o side_n ps_n the_o compliment_n of_o the_o declination_n 78_o degree_n 30_o minute_n with_o the_o base_a zs_n the_o compliment_n of_o the_o sun_n altitude_n 59_o degree_n 15_o minute_n we_o may_v find_v the_o angle_n zp_v the_o hour_n of_o the_o day_n for_o the_o side_n zp_n the_o compliment_n of_o the_o latitude_n be_v 37_o degree_n 30_o minute_n and_o the_o compliment_n of_o the_o side_n ps_n the_o declination_n of_o the_o sun_n be_v 11_o degree_n 30_o minute_n their_o sum_n be_v 49_o degree_n 00_o minute_n and_o their_o difference_n 26_o degree_n 00_o minute_n and_o if_o we_o put_v the_o side_n of_o the_o quadrat_fw-la 1000_o the_o sine_fw-la of_o the_o sum_n 49_o degree_n 00_o minute_n be_v 754_o and_o the_o sine_fw-la of_o the_o difference_n 26_o degree_n 00_o minute_n be_v 438_o these_o two_o sin_n join_v together_o be_v 1192_o the_o half_a whereof_o be_v 596_o for_o the_o first_o find_v sine_fw-la which_o take_v out_o of_o the_o sine_fw-la of_o the_o sum_n leave_v 158_o which_o remainder_n i_o note_v &_o because_o either_o of_o these_o side_n be_v less_o than_o a_o quadrant_a i_o take_v the_o difference_n between_o this_o remainder_n note_v and_o the_o sine_fw-la of_o the_o compliment_n of_o the_o base_a zs_n be_v 30_o degree_n 45_o minute_n the_o sine_fw-la whereof_o be_v 511_o out_o of_o which_o i_o take_v the_o remainder_n note_v and_o the_o remainder_n be_v 353_o for_o the_o second_o find_v sine_fw-la then_o place_v the_o thread_n at_o the_o intersection_n of_o the_o contrary_a parallel_n of_o 596_o with_o the_o right_a parallel_n of_o 353_o it_o will_v cut_v the_o contrary_a sine_fw-la of_o 90_o degree_n at_o the_o right_n sine_fw-la of_o 36_o degree_n 20_o minute_n and_o in_o regard_n the_o side_n be_v all_o less_o than_o quadrant_n and_o the_o remainder_n note_v less_o than_o the_o cousin_n of_o the_o base_a i_o take_v this_o 36_o degree_n 20_o minute_n out_o of_o 90_o degree_n and_o there_o rest_v 53_o degree_n 40_o minute_n for_o the_o hour_n from_o the_o meridian_n which_o be_v require_v the_o like_a may_v you_o do_v with_o any_o of_o the_o other_o angle_n as_o to_o find_v the_o azimuth_n or_o the_o angle_n of_o the_o sun_n position_n in_o regard_n of_o pole_n and_o zenith_n but_o you_o shall_v have_v a_o more_o speedy_a way_n to_o perform_v this_o problem_n by_o the_o planisphere_n on_o the_o back_n of_o the_o quadrat_fw-la for_o by_o know_v the_o three_o side_n of_o any_o spherical_a triangle_n you_o shall_v find_v a_o angle_n as_o speedy_o as_o if_o you_o have_v the_o other_o two_o angle_n also_o chap._n xlii_o to_o find_v a_o side_n by_o know_v the_o three_o angle_n in_o any_o spherical_a triangle_n the_o side_n may_v be_v change_v into_o angle_n and_o the_o angle_n into_o side_n by_o take_v first_o for_o any_o one_o angle_n and_o his_o subtend_a side_n the_o remainder_n to_o a_o semicircle_n then_o shall_v the_o operation_n be_v the_o same_o as_o in_o the_o former_a chapter_n thus_o in_o the_o triangle_n zp_v have_v the_o angle_n zp_v 53_o degree_n 40_o minute_n and_o zsp_v 34_o degree_n 47_o minute_n with_o s_n zp_v 113_o degree_n 17_o minute_n if_o you_o take_v the_o great_a angle_n of_o 113_o degree_n 17_o minute_n out_o of_o 180_o degree_n there_o will_v remain_v 66_o degree_n 43_o minute_n then_o as_o if_o you_o have_v a_o triangle_n of_o three_o know_a side_n one_o of_o 53_o degree_n 40_o minute_n another_o of_o 34_o degree_n 47_o minute_n and_o a_o three_o of_o 66_o degree_n 43_o minute_n you_o may_v find_v a_o angle_n opposite_a to_o any_o one_o of_o these_o side_n by_o the_o last_o chapter_n so_o shall_v the_o angle_v thus_o find_v be_v the_o side_n require_v if_o it_o be_v not_o the_o angle_n
the_o ascensional_a difference_n 1_o hour_n 1_o minute_n add_v to_o 6_o hour_n make_v 7_o hour_n 1_o minute_n the_o semidiurnal_a arch_n or_o the_o time_n of_o sun_n set_v which_o be_v double_v give_v 14_o hour_n 2_o minute_n for_o the_o diurnal_a arch_n this_o be_v take_v out_o of_o 24_o hour_n leave_v 9_o hour_n 58_o minute_n for_o the_o nocturnal_a arch_n the_o half_a of_o which_o 9_o hour_n 58_o minute_n be_v 4_o hour_n 59_o minute_n the_o time_n of_o sun_n rise_v or_o 7_o hour_n 1_o minute_n take_v out_o of_o 12_o hour_n leave_v 4_o hour_n 59_o minute_n the_o time_n of_o sun_n rise_v but_o if_o the_o declination_n be_v southwards_o then_o take_v this_o 1_o hour_n 1_o minute_n the_o ascensional_a difference_n out_o of_o 6_o hour_n and_o there_o will_v rest_v 4_o hour_n 59_o minute_n for_o the_o semidiurnal_a arch_n which_o be_v double_v give_v 9_o hour_n 58_o minute_n for_o the_o length_n of_o the_o day_n the_o compliment_n unto_o 24_o hour_n be_v 14_o hour_n 2_o minute_n the_o length_n of_o the_o night_n the_o half_a of_o which_o be_v 7_o hour_n 1_o min._n the_o time_n of_o sunrising_n chap._n xix_o have_v the_o latitude_n of_o the_o place_n &_o the_o declination_n of_o the_o sun_n to_o find_v the_o time_n of_o the_o beginning_n or_o end_v of_o twilight_n the_o beginning_n of_o twilight_n be_v when_o the_o sun_n approach_v within_o 18_o degree_n of_o the_o horizon_n in_o the_o east_n and_o the_o end_n of_o the_o same_o at_o 18_o deg_fw-la depression_n in_o the_o west_n wherefore_o to_o find_v the_o same_o work_n thus_o first_o by_o the_o 16_o chap._n get_v the_o height_n of_o the_o sun_n at_o the_o hour_n of_o six_o if_o the_o declination_n be_v north_n but_o if_o it_o be_v south_n take_v his_o depression_n at_o six_o for_o all_o be_v one_o thing_n if_o his_o quantity_n of_o declination_n be_v alike_o the_o height_n or_o depression_n at_o six_o be_v know_v if_o the_o declination_n be_v north_n add_v the_o sine_fw-la of_o this_o height_n to_o the_o sine_fw-la of_o 18_o deg_fw-la but_o if_o the_o declination_n be_v south_n take_v the_o difference_n betwixt_o the_o sine_fw-la of_o his_o depression_n and_o the_o sine_fw-la of_o 18_o deg_fw-la so_o shall_v the_o sum_n of_o difference_n be_v the_o second_o proportional_a number_n then_o say_v as_o the_o cousin_n of_o the_o latitude_n to_o this_o second_o proportional_a so_o be_v the_o radius_fw-la to_o a_o four_o proportional_a sine_fw-la then_o as_o the_o cousin_n of_o the_o declination_n to_o this_o four_o sine_fw-la so_o be_v the_o radius_fw-la to_o the_o sine_fw-la of_o the_o quantity_n of_o time_n between_o the_o beginning_n or_o end_v of_o twilight_n and_o the_o hour_n of_o six_o thus_o in_o the_o diagram_n of_o the_o 6_o chap._n have_v the_o declination_n of_o the_o sun_n ♈_o o_fw-fr 11_o deg_fw-la 30_o min._n north_n and_o thereby_o the_o sun_n height_n at_o six_o ro_n 9_o deg_fw-la 6_o min._n we_o have_v give_v we_o in_o the_o triangle_n byo_n first_o the_o angle_n ybo_n the_o compliment_n of_o the_o latitude_n 37_o deg_fw-la 30_o min._n and_o then_o the_o side_n o_fw-fr y_fw-fr 467_o the_o radius_fw-la being_n 1000_o this_o side_n be_v make_v up_o of_o the_o sine_fw-la of_o the_o sun_n height_n at_o six_o 158_o and_o the_o sine_fw-la of_o 18_o deg_fw-la 309_o or_o by_o add_v of_o or_o 158_o to_o r_o y_fw-fr 309_o we_o have_v o_o y_fw-fr 467_o for_o the_o second_o proportional_a these_o be_v know_v with_o the_o right_a angle_n at_o you_o we_o may_v find_v the_o base_a bo_z by_o the_o 10_o chapter_n or_o 27_o chap._n of_o the_o second_o book_n for_o if_o we_o place_v the_o thread_n upon_o the_o intersection_n of_o the_o contrary_a sine_fw-la of_o 37_o deg_fw-la 30_o min._n with_o the_o right_a parallel_n of_o 467_o it_o will_v cut_v the_o contrary_a sine_fw-la of_o 90_o deg_fw-la at_o the_o right_n sine_fw-la of_o 50_o deg_fw-la 7_o min._n which_o be_v the_o base_a bo_o but_o see_v the_o radius_fw-la of_o this_o sine_z bo_z be_v but_o the_o cousin_n of_o the_o declination_n oh_o 78_o deg_fw-la 30_o min._n therefore_o i_o place_v the_o thread_n again_o at_o the_o intersection_n of_o the_o contrary_a sine_fw-la of_o 78_o deg_fw-la 30_o min._n with_o the_o right_n sine_fw-la of_o this_o 50_o degr_n 7_o min._n and_o it_o will_v cut_v the_o contrary_a sine_fw-la of_o 90_o deg_fw-la at_o the_o right_n sine_fw-la of_o 51_o deg_fw-la 33_o min._n and_o such_o be_v the_o distance_n of_o time_n between_o the_o hour_n of_o six_o and_o the_o beginning_n or_o end_v of_o twilight_n be_v 3_o hour_n 26_o min._n now_o at_o all_o time_n if_o you_o take_v this_o distance_n out_o of_o six_o hour_n your_o remainder_n shall_v be_v the_o beginning_n of_o twilight_n and_o if_o you_o add_v this_o distance_n find_v unto_o 6_o hour_n the_o sum_n will_v be_v the_o end_n of_o the_o same_o except_o it_o be_v in_o those_o latitude_n where_o the_o depression_n of_o 6_o be_v great_a than_o 18_o deg_fw-la and_o then_o you_o must_v in_o that_o latitude_n and_o south_n declination_n who_o depression_n at_o six_o be_v more_o than_o 18_o deg_fw-la work_n by_o the_o contrary_a rule_n and_o when_o the_o sun_n have_v north_n declination_n if_o you_o take_v the_o ascensional_a difference_n out_o of_o the_o distance_n of_o time_n between_o the_o hour_n of_o six_o and_o the_o beginning_n or_o end_v of_o twilight_n you_o shall_v have_v the_o distance_n of_o time_n between_o snn_n rise_v or_o set_v and_o the_o beginning_n or_o end_v of_o twilight_n and_o when_o the_o declination_n be_v south_n add_v the_o ascensional_a difference_n to_o the_o distance_n of_o time_n find_v and_o you_o shall_v have_v the_o quantity_n of_o twilight_n as_o before_o as_o for_o example_n let_v the_o distance_n of_o time_n between_o the_o beginning_n or_o end_v of_o twilight_n and_o the_o hour_n of_o six_o be_v find_v to_o be_v 3_o hour_n 26_o min._n if_o i_o take_v this_o 3_o hour_n 26_o min._n out_o of_o 6_o hour_n there_o will_v remain_v 2_o hour_n 34_o min_fw-mi for_o the_o beginning_n of_o twilight_n and_o if_o i_o add_v this_o 3_o hour_n 26_o min._n to_o six_o hour_n it_o will_v make_v 9_o hour_n 26_o min._n for_o the_o end_n of_o twilight_n and_o now_o be_v the_o sun_n have_v north_n declination_n if_o i_o take_v the_o difference_n of_o ascension_n 1_o hour_n 1_o min._n as_o it_o be_v find_v by_o the_o last_o chapter_n out_o of_o 3_o hour_n 26_o min._n there_o will_v remain_v 2_o hour_n 25_o min._n for_o the_o quantity_n of_o twilight_n before_o or_o after_o sun_n rise_v or_o set_v chap._n xx._n have_v the_o distance_n of_o the_o sun_n from_o the_o next_o equinoctial_a point_n to_o find_v his_o right_a ascension_n as_o the_o radius_fw-la to_o the_o cousin_n of_o the_o great_a declination_n so_o be_v the_o tangent_fw-la of_o the_o distance_n to_o the_o tangent_fw-la of_o the_o right_a ascension_n thus_o in_o the_o right_a angle_a triangle_n ♈_o x_o ♉_o in_o the_o diagram_n of_o the_o 6_o chap._n have_v the_o angle_n x_o ♈_o ♉_o the_o sun_n great_a declination_n 23_o deg_fw-la 30_o min._n and_o the_o base_a ♈_o ♉_o the_o distance_n of_o the_o sun_n from_o the_o equinoctial_a point_n ♈_o 30_o deg_fw-la with_o the_o right_a angle_n at_o x_o we_o may_v find_v the_o side_n ♈_o x_o the_o sun_n right_a ascension_n by_o the_o 33_o chap._n of_o the_o second_o book_n for_o if_o we_o place_v the_o thread_n at_o the_o intersection_n of_o the_o contrary_a sine_fw-la of_o 90_o deg_fw-la with_o the_o right_n tangent_fw-la of_o 30_o deg_fw-la it_o will_v cut_v the_o contrary_a sine_fw-la of_o 66_o degree_n 30_o minute_n at_o the_o right_n tangent_fw-la of_o 27_o degree_n 54_o minute_n and_o so_o much_o be_v the_o side_n ♈_o x_o the_o right_a ascension_n require_v chap._n xxi_o have_v the_o right_a ascension_n of_o the_o sun_n or_o star_n together_o with_o the_o difference_n of_o their_o ascension_n to_o find_v the_o oblique_a ascension_n and_o descension_n if_o the_o declination_n of_o the_o sun_n or_o star_n be_v north_n subtract_v the_o ascensional_a difference_n from_o the_o right_a ascension_n and_o the_o residue_n will_v be_v the_o oblique_a ascension_n but_o if_o you_o add_v they_o together_o the_o sum_n will_v be_v the_o oblique_a descension_n but_o if_o the_o declination_n be_v south_n add_v the_o ascensional_a difference_n and_o right_a ascension_n together_o so_o will_v the_o sum_n thereof_o be_v the_o oblique_a ascension_n but_o if_o you_o take_v the_o ascensional_a difference_n out_o of_o the_o right_a ascension_n the_o remainder_n will_v be_v the_o oblique_a descension_n as_o for_o example_n the_o right_a ascension_n of_o the_o sun_n in_o the_o begin_n of_o ♉_o being_z find_v by_o the_o last_o chapter_n to_o be_v 27_o degree_n 54_o minute_n and_o the_o ascensional_a
difference_n of_o the_o same_o point_n by_o the_o 18_o chap._n to_o be_v 15_o deg_fw-la 22_o min._n if_o i_o take_v 15_o deg_fw-la 22_o min._n out_o of_o 27_o deg_fw-la 54_o min._n because_o the_o declination_n be_v north_n there_o will_v remain_v 12_o deg_fw-la 32_o min._n for_o the_o oblique_a ascension_n likewise_o if_o i_o add_v 15_o deg_fw-la 22_o min._n to_o 27_o degree_n 54_o minute_n the_o sum_n will_v be_v 43_o degree_n 16_o minute_n for_o the_o oblique_a descension_n chap._n xxii_o have_v the_o azimuth_n the_o sun_n altitude_n and_o the_o declination_n to_o find_v the_o hour_n of_o the_o day_n as_o the_o cousin_n of_o the_o declination_n to_o the_o sine_fw-la of_o the_o azimuth_n so_o be_v the_o cousin_n of_o the_o altitude_n to_o the_o sine_fw-la of_o the_o hour_n thus_o in_o the_o triangle_n zp_n ♉_o in_o the_o diagram_n of_o the_o 6_o chapter_n have_v the_o side_n p_o ♉_o the_o compliment_n of_o declination_n 78_o degree_n 30_o min._n and_o the_o side_n z_o ♉_o the_o compliment_n of_o the_o sun_n altitude_n 64_o degree_n 4_o minute_n with_o the_o outward_a angle_n ♉_o zp_n the_o azimuth_n 74_o degree_n 22_o min._n we_o may_v find_v the_o angle_n zp_n ♉_o the_o hour_n from_o the_o meridian_n by_o the_o 40_o chapter_n of_o the_o second_o book_n for_o if_o we_o place_v the_o thread_n to_o the_o intersection_n of_o the_o contrary_a sine_fw-la of_o 78_o degree_n 30_o minute_n with_o the_o right_n sine_fw-la of_o 74_o degree_n 22_o minute_n it_o will_v cut_v the_o contrary_a sine_fw-la of_o 64_o degree_n 4_o minute_n at_o the_o right_n sine_fw-la of_o 62_o degree_n 6_o minute_n and_o such_o be_v the_o angle_n zp_n ♉_o the_o hour_n from_o the_o meridian_n or_o in_o time_n 4_o hour_n 8_o minute_n which_o be_v require_v chap._n xxiii_o have_v the_o hour_n of_o the_o day_n the_o sun_n altitude_n and_o the_o declination_n to_o find_v the_o azimuth_n as_o the_o cousin_n of_o the_o sun_n altitude_n to_o the_o sine_fw-la of_o the_o hour_n from_o the_o meridian_n so_o be_v the_o cousin_n of_o the_o declination_n to_o the_o sine_fw-la of_o the_o azimuth_n thus_o in_o the_o triangle_n zp_n ♉_o of_o the_o diagram_n in_o the_o 6_o chapter_n have_v the_o side_n z_o ♉_o the_o compliment_n of_o the_o sun_n altitude_n 64_o degree_n 4_o minute_n and_o the_o side_n p_o ♉_o the_o compliment_n of_o declination_n 78_o degree_n 30_o minute_n with_o the_o angle_n zp_n ♉_o the_o hour_n from_o the_o meridian_n 62_o degree_n 6_o minute_n we_o may_v find_v the_o angle_n pz_fw-fr ♉_o the_o azimuth_n by_o the_o 40_o chapter_n of_o the_o second_o book_n for_o if_o we_o place_v the_o thread_n upon_o the_o intersection_n of_o the_o contrary_a sine_fw-la of_o 64_o degree_n 4_o minute_n with_o the_o right_n sine_fw-la of_o 62_o degree_n 6_o minute_n it_o will_v cut_v the_o contrary_a sine_fw-la of_o 78_o degree_n 30_o minute_n at_o the_o right_n sine_fw-la of_o 74_o degree_n 22_o minute_n and_o such_o be_v the_o outward_a angle_n pz_n ♉_o the_o sun_n azimuth_n from_o the_o south_n which_o be_v require_v chap._n xxiv_o the_o latitude_n of_o the_o place_n the_o altitude_n and_o declination_n of_o the_o sun_n be_v give_v to_o find_v the_o hour_n of_o the_o day_n thus_o in_o the_o oblique_a spherical_a triangle_n zp_n ♉_o in_o the_o diagram_n of_o the_o 6_o chapter_n have_v the_o three_o side_n viz._n zp_v the_o compliment_n of_o the_o latitude_n 37_o degree_n 30_o minute_n and_o z_o ♉_o the_o compliment_n of_o the_o sun_n altitude_n 64_o degree_n 4_o minute_n with_o p_o ♉_o the_o compliment_n of_o declination_n 78_o degree_n 30_o minute_n we_o may_v find_v the_o angle_n zp_n ♉_o the_o hour_n from_o the_o meridian_n by_o the_o 41_o chapter_n of_o the_o second_o book_n but_o by_o this_o instrument_n more_o ready_o thus_o first_o by_o the_o 16_o chapter_n get_v the_o sun_n height_n or_o depression_n at_o the_o hour_n of_o 6_o and_o if_o the_o declination_n be_v north_n take_v the_o distance_n betwixt_o this_o sine_fw-la and_o the_o sine_fw-la of_o the_o altitude_n give_v either_o by_o substraction_n or_o with_o your_o compass_n this_o distance_n either_o count_v or_o place_n upon_o the_o contrary_a sine_fw-la of_o the_o compliment_n of_o the_o latitude_n and_o thereto_o place_v the_o thread_n the_o thread_n be_v thus_o place_v will_v cut_v the_o contrary_a sine_fw-la of_o 90_o degree_n at_o a_o right_a sine_fw-la or_o parallel_n which_o follow_v to_o the_o contrary_a sine_fw-la of_o the_o compliment_n of_o declination_n and_o thereto_o place_v the_o thread_n so_o shall_v it_o cut_v the_o contrary_a sine_fw-la of_o 90_o degree_n at_o the_o right_n sine_fw-la of_o the_o hour_n from_o six_o for_o as_o the_o cousin_n of_o the_o latitude_n to_o the_o distance_n betwixt_o the_o sine_fw-la of_o the_o height_n at_o six_o and_o the_o sine_fw-la of_o the_o altitude_n give_v so_o be_v the_o radius_fw-la to_o a_o four_o sine_fw-la then_o as_o the_o cousin_n of_o declination_n to_o this_o four_o sine_fw-la so_o be_v the_o radius_fw-la to_o the_o sine_fw-la of_o the_o hour_n from_o six_o thus_o in_o the_o diagram_n of_o the_o 6_o chapter_n have_v the_o sun_n declination_n eglantine_n or_o ♈_o o_fw-fr 11_o degree_n 30_o minute_n and_o the_o angle_n p_o ♈_o b_o the_o latitude_n of_o the_o place_n 52_o degree_n 30_o minute_n we_o have_v by_o the_o 16_o chapter_n the_o sun_n height_n at_o 6_o ro_n 9_o deg_fw-la 6_o min._n the_o sine_fw-la whereof_o be_v 158_o the_o radius_fw-la being_n 1000_o this_o sine_fw-la be_v compare_v with_o the_o sine_fw-la of_o 25_o deg_fw-la 56_o min._n 437_o the_o sine_fw-la of_o the_o sun_n altitude_n r_o a_o and_o take_v ro_n 158_o out_o of_o r_o a_o 437_o there_o will_v remain_v o_o a_o 279_o by_o which_o and_o the_o angle_n a_o ♉_o o_o the_o compliment_n of_o the_o latitude_n with_o the_o right_a angle_n at_o a_o we_o may_v find_v ♉_o o_fw-fr the_o hour_n from_o six_o for_o if_o we_o apply_v the_o thread_n to_o the_o intersection_n of_o the_o contrary_a sine_fw-la of_o 37_o deg_fw-la 30_o min._n with_o the_o right_a parallel_n of_o 279_o it_o will_v cut_v the_o contrary_a sine_fw-la of_o 90_o degree_n at_o the_o right_n sine_fw-la of_o about_o 27¼_n degree_n but_o see_v this_o 27¼_n degree_n which_o be_v o_fw-fr ♉_o be_v but_o upon_o the_o parallel_n of_o declination_n go_v therefore_o i_o place_v the_o thread_n again_o at_o the_o intersection_n of_o this_o 27¼_n with_o the_o contrary_a sine_fw-la of_o 78_o deg_fw-la 30_o min._n the_o compliment_n of_o declination_n go_v and_o it_o will_v cut_v the_o contrary_a sine_fw-la of_o 90_o deg_fw-la at_o the_o right_n sine_fw-la of_o 27_o deg_fw-la 54_o min._n and_o so_o much_o be_v ♈_o x_o or_o the_o angle_n ♈_o px_n the_o hour_n from_o six_o which_o be_v require_v and_o here_o note_v that_o if_o the_o altitude_n give_v be_v great_a than_o the_o altitude_n at_o six_o then_o be_v the_o time_n find_v to_o the_o southward_a of_o six_o but_o if_o it_o be_v lesser_a then_o be_v it_o to_o the_o northward_a but_o if_o the_o declination_n be_v southward_o then_o add_v the_o sine_fw-la of_o his_o depression_n at_o six_o to_o the_o sine_fw-la of_o the_o altitude_n give_v and_o with_o the_o same_o proceed_v in_o all_o respect_n as_o before_o and_o you_o shall_v have_v your_o desire_n all_o these_o kind_n of_o proposition_n upon_o this_o instrument_n when_o you_o be_v well_o acquaint_v therewith_o be_v soon_o wrought_v then_o speak_v chap._n xxv_o the_o latitude_n of_o the_o place_n the_o declination_n and_o altitude_n of_o the_o sun_n be_v give_v to_o find_v the_o azimuth_n thus_o in_o the_o obliquespherical_a triang_n zp_n ♉_o in_o the_o diagram_n of_o the_o 6_o chapter_n have_v the_o three_o side_n viz._n zp_n the_o compliment_n of_o latitude_n 37_o degree_n 30_o minute_n and_o z_o ♉_o the_o compliment_n of_o the_o sun_n altitude_n 64_o degree_n 4_o minute_n with_o the_o side_n p_o ♉_o the_o compliment_n of_o the_o declination_n 78_o degree_n 30_o minute_n we_o may_v find_v the_o angle_n pz_fw-fr ♉_o by_o the_o 41_o chapter_n of_o the_o second_o book_n but_o we_o will_v proceed_v as_o follow_v little_a differ_v from_o the_o former_a chapter_n first_o therefore_o if_o the_o sun_n have_v north_n declination_n get_v his_o height_n in_o the_o vertical_a circle_n by_o the_o 13_o or_o 14_o chapter_n and_o compare_v the_o sine_fw-la thereof_o with_o the_o sine_fw-la of_o the_o altitude_n give_v take_v the_o lesser_a out_o of_o the_o great_a and_o note_v the_o remainder_n then_o say_v as_o the_o cousin_n of_o the_o latitude_n to_o this_o remainder_n so_o be_v the_o sine_fw-la of_o the_o latitude_n to_o a_o four_o sine_fw-la then_o as_o the_o cousin_n of_o the_o sun_n altitude_n to_o this_o four_o sine_fw-la so_o be_v the_o radius_fw-la to_o the_o sine_fw-la of_o the_o azimuth_n
the_o almicanter_n the_o thread_n be_v thus_o place_v upon_o the_o 11_o deg_fw-la 30_o min._n in_o the_o axletree-line_n and_o parallel_v to_o the_o horizon_n will_v cut_v at_o each_o end_n thereof_o 9_o deg_fw-la 6_o min._n among_o the_o almicanter_n and_o such_o be_v the_o height_n of_o the_o sun_n at_o the_o hour_n of_o six_o or_o if_o you_o place_v the_o equator_fw-la 52_o deg_fw-la 30_o min._n above_o the_o south_n end_n of_o the_o horizon_n and_o then_o count_v 11_o deg_fw-la 30_o min._n upon_o the_o horizon_n where_o you_o shall_v find_v meet_v you_o a_o parallel_n which_o be_v follow_v to_o the_o axletree-line_n shall_v there_o give_v you_o 9_o deg_fw-la 6_o min._n for_o the_o altitude_n of_o the_o sun_n at_o six_o as_o before_o chap._n xxxvi_o have_v the_o latitude_n of_o the_o place_n and_o the_o declination_n of_o the_o sun_n to_o find_v his_o azimuth_n at_o the_o hour_n of_o six_o first_o set_v the_o sphere_n according_a to_o the_o latitude_n give_v and_o count_v the_o declination_n upon_o the_o horizon_n and_o look_v what_o parallel_n there_o meet_v you_o the_o same_o follow_v to_o the_o axtree-line_n where_o it_o will_v show_v you_o the_o azimuth_n desire_v as_o in_o the_o latitude_n of_o 52_o deg_fw-la 30_o min._n the_o declination_n of_o the_o sun_n be_v 11_o deg_fw-la 30_o min._n first_o i_o place_v the_o equator_fw-la at_o 37_o deg_fw-la 30_o min._n above_o the_o south_n end_n of_o the_o horizon_n and_o count_v the_o declination_n 11_o deg_fw-la 30_o min_fw-mi upon_o the_o horizon_n where_o i_o find_v to_o meet_v i_o the_o parallel_n of_o 7_o deg_fw-la 4_o min._n and_o such_o be_v the_o azimuth_n from_o the_o east_n or_o west_n point_v northwards_o chap._n xxxvii_o have_v the_o latitude_n of_o the_o place_n and_o the_o declination_n of_o the_o sun_n to_o find_v the_o ascensional_a difference_n and_o consequent_o the_o time_n of_o sunrising_n and_o set_v with_o the_o diurnal_a and_o nocturnal_a arch_n first_o set_v the_o planisphere_n to_o the_o latitude_n give_v and_o reckon_v the_o declination_n upon_o the_o axletree-line_n follow_v low_o the_o parallel_n thereof_o to_o the_o horizon_n where_o you_o shall_v find_v meet_v you_o the_o meridian_n of_o the_o ascensional_a difference_n count_v from_o the_o hour_n of_o six_o and_o this_o same_o meridian_n give_v you_o the_o time_n of_o sun_n set_v count_v from_o the_o south_n part_n of_o the_o limb_n or_o general_a meridian_n and_o likewise_o the_o time_n of_o his_o rise_n count_v from_o the_o north_n part_n of_o the_o limb_n and_o now_o if_o you_o still_o note_v this_o intersection_n of_o the_o horizon_n and_o sun_n parallel_n and_o count_v the_o hour_n thereupon_o between_o the_o south_n part_n of_o the_o limb_n and_o the_o horizon_n you_o shall_v have_v the_o sun_n semidiurnal_a arch_n and_o from_o the_o horizon_n to_o the_o north_n part_n of_o the_o limb_n his_o seminocturnal_a arch_n both_o which_o be_v double_v give_v you_o both_o the_o diurnal_a and_o nocturnal_a arch_n as_o for_o example_n let_v the_o latitude_n give_v be_v 52_o degree_n 30_o minute_n and_o the_o declination_n 11_o degree_n 30_o minute_n first_o i_o set_v the_o equator_fw-la 37_o degree_n 30_o minute_n above_o the_o horizon_n then_o count_v 11_o degree_n 30_o minute_n upon_o the_o axtree-line_n i_o follow_v the_o parallel_n thereof_o to_o the_o horizon_n where_o i_o find_v meet_v i_o the_o meridian_n of_o 15_o deg_fw-la 22_o min._n and_o such_o be_v the_o difference_n of_o ascensionals_n require_v now_o upon_o the_o same_o parallel_n of_o declination_n i_o seek_v how_o many_o hour_n i_o can_v find_v betweeene_v the_o south_n part_n of_o the_o limb_n and_o the_o horizon_n which_o be_v 7_o hour_n 1_o minute_n and_o so_o much_o be_v the_o semidiurnal_a arch_n and_o the_o time_n of_o sun_n set_v which_o be_v double_v make_v 14_o hour_n 2_o minute_n the_o diurnal_a arch_n so_o likewise_o count_v the_o hour_n between_o the_o north_n part_n of_o the_o limb_n and_o the_o horizon_n i_o find_v they_o 4_o hour_n 59_o minute_n and_o such_o be_v the_o seminocturnal_a arch_n and_o the_o time_n of_o sunrising_n which_o be_v double_v give_v 9_o hour_n 58_o min._n for_o the_o nocturnal_a arch_n the_o semidiurual_a and_o seminocturnal_a arch_n of_o any_o star_n who_o declination_n be_v know_v may_v be_v find_v in_o the_o self_n same_o manner_n chap._n xxxviii_o have_v the_o latitude_n of_o the_o place_n and_o the_o declination_n of_o the_o sun_n to_o find_v the_o time_n of_o the_o beginning_n and_o end_v of_o twilight_n first_o set_v the_o sphere_n to_o the_o latitude_n give_v then_o count_v the_o declination_n upon_o the_o axtree-line_n either_o northward_o or_o southward_o according_a as_o it_o be_v and_o then_o place_v the_o thread_n parallel_n to_o the_o horizon_n to_o the_o 18_o deg_fw-la of_o depression_n below_o the_o same_o and_o note_v where_o it_o cut_v the_o parallel_n of_o declination_n for_o those_o hour_n to_o the_o southward_n of_o the_o thread_n give_v the_o time_n of_o twilight_n end_v and_o the_o other_o to_o the_o northward_o the_o time_n of_o its_o beginning_n as_o for_o example_n let_v the_o latitude_n give_v be_v 52_o degree_n 30_o minute_n and_o the_o declination_n of_o the_o sun_n 11_o deg_fw-la 30_o min._n first_o i_o set_v the_o equator_fw-la 37_o deg_fw-la 30_o min._n above_o the_o horizon_n and_o count_v 11_o deg_fw-la 30_o min._n upon_o the_o axletree-line_n northward_o because_o his_o declination_n be_v towards_o that_o pole_n i_o place_v the_o thread_n parallel_v to_o the_o horizon_n 18_o deg_fw-la below_o the_o same_o and_o i_o find_v it_o cut_v the_o say_a parallel_n of_o 11_o deg_fw-la 30_o min._n at_o 2_o hour_n 34_o minute_n from_o the_o north_n part_n of_o the_o limb_n and_o such_o be_v the_o time_n of_o the_o begin_n of_o twilight_n so_o do_v it_o likewise_o cut_v the_o same_o parallel_n at_o 9_o hour_n 26_o minute_n from_o the_o south_n part_n of_o the_o meridian_n and_o such_o be_v the_o time_n of_o the_o end_n of_o twilight_n which_o be_v require_v chap._n thirty-nine_o the_o latitude_n of_o the_o place_n the_o altitude_n and_o declination_n of_o the_o sun_n be_v give_v to_o find_v the_o hour_n of_o the_o day_n first_o place_v the_o planisphere_n to_o the_o latitude_n give_v and_o seek_v his_o declination_n on_o the_o axletree-line_n among_o the_o parallel_n of_o declination_n then_o place_v the_o thread_n parallel_n to_o the_o horizon_n to_o the_o degree_n of_o the_o sun_n altitude_n count_v from_o the_o horizon_n among_o the_o almicanter_n and_o where_o the_o thread_n so_o place_v shall_v cut_v the_o parallel_n of_o declination_n there_o pass_v by_o the_o meridian_n which_o give_v you_o the_o hour_n of_o the_o day_n as_o for_o example_n in_o the_o latitude_n of_o 52_o deg_fw-la 30_o min._n the_o declination_n of_o the_o sun_n be_v 11_o deg_fw-la 30_o min._n and_o his_o altitude_n 25_o deg_fw-la 56_o min._n i_o set_v the_o equator_fw-la 37_o deg_fw-la 30_o min._n above_o the_o horizon_n and_o count_v 11_o deg_fw-la 30_o min._n upon_o the_o axletree-line_n then_o place_v the_o thread_n to_o 25_o deg_fw-la 56_o min._n among_o the_o almicanter_n i_o find_v it_o cut_v the_o parallel_n of_o 11_o deg_fw-la 30_o min._n at_o four_o hour_n 8_o minute_n from_o the_o meridian_n and_o so_o much_o be_v the_o hour_n of_o the_o day_n if_o the_o observation_n be_v make_v in_o the_o afternoon_n but_o if_o the_o observation_n be_v make_v before_o noon_n than_o be_v it_o 7_o a_o clock_n and_o 52_o min._n in_o the_o forenoon_n again_o let_v the_o latitude_n and_o declination_n be_v the_o same_o as_o before_z and_o the_o altitude_n give_v 42_o deg_fw-la 26_o min._n now_o if_o i_o place_v the_o thread_n to_o 42_o deg_fw-la 26_o min._n among_o the_o almicanter_n it_o will_v cut_v the_o parallel_n of_o 11_o deg_fw-la 30_o min._n at_o just_a two_o hour_n from_o the_o meridian_n so_o shall_v it_o be_v either_o ten_o a_o clock_n in_o the_o morning_n or_o two_o afternoon_n according_a to_o the_o observation_n chap._n xl._n the_o latitude_n of_o the_o place_n the_o sun_n altitude_n and_o declination_n be_v give_v to_o find_v his_o azimuth_n first_o set_v the_o sphere_n to_o the_o latitude_n give_v and_o then_o count_v your_o altitude_n among_o the_o parallel_n of_o declination_n and_o your_o declination_n among_o the_o almicanter_n so_o shall_v your_o meridian_n become_v azimuth_n and_o the_o work_n will_v be_v the_o same_o as_o in_o the_o former_a chapter_n only_o here_o by_o the_o way_n i_o will_v have_v you_o note_v that_o as_o your_o parallel_n and_o almicanter_n change_v office_n so_o must_v your_o north_n and_o south_n part_n of_o the_o general_a meridian_n change_v name_n so_o that_o when_o your_o thread_n cut_v any_o parallel_n at_o any_o azimuth_n if_o you_o count_v that_o azimuth_n from_o the_o south_n part_n of_o the_o limb_n it_o shall_v
be_v the_o azimuth_n from_o the_o north_n and_o if_o you_o count_v it_o from_o the_o north_n part_n of_o the_o limb_n it_o shall_v be_v the_o azimuth_n from_o the_o south_n and_o here_o also_o note_n that_o your_o altitude_n give_v must_v be_v always_o count_v among_o the_o parallel_n of_o north_n declination_n and_o when_o the_o declination_n be_v south_n the_o thread_n must_v lie_v below_o the_o horizon_n and_o when_o the_o declination_n be_v north_n it_o must_v lie_v above_o the_o same_o as_o for_o example_n in_o the_o latitude_n of_o 52_o deg_fw-la 30_o min._n let_v the_o declination_n of_o the_o sun_n be_v 11_o deg_fw-la 30_o min._n and_o his_o altitude_n 25_o deg_fw-la 56_o min._n first_o i_o set_v the_o equator_fw-la 37_o deg_fw-la 30_o min._n above_o the_o south_n end_n of_o the_o horizon_n and_o count_v 25_o deg_fw-la 56_o min._n among_o the_o parallel_n of_o declination_n than_o i_o place_v the_o thread_n to_o 11_o deg_fw-la 30_o min._n above_o the_o horizon_n because_o the_o declination_n be_v north_n and_o i_o find_v it_o cut_v the_o parallel_n of_o 25_o deg_fw-la 56_o min._n at_o the_o azimuth_n of_o 74_o deg_fw-la 22_o minute_n count_v from_o the_o north_n part_n of_o the_o limb_n and_o such_o be_v the_o azimuth_n from_o the_o south_n and_o if_o you_o count_v the_o azimuth_n from_o the_o south_n part_n of_o the_o limb_n you_o shall_v find_v the_o thread_n cut_v the_o azimuth_n of_o 105_o deg_fw-la 38_o min._n which_o be_v the_o azimuth_n from_o the_o north._n again_o let_v the_o latitude_n and_o the_o declination_n be_v the_o same_o as_o before_o and_o the_o altitude_n of_o the_o sun_n 30_o deg_fw-la 45_o min._n now_o the_o thread_n be_v place_v to_o 11_o deg_fw-la 30_o min._n count_v among_o the_o almicanter_n above_o the_o horizon_n will_v cut_v the_o parallel_n of_o 30_o deg_fw-la 45_o min._n at_o the_o azimuth_n of_o 66_o deg_fw-la 30_o min._n count_v from_o the_o north_n part_n of_o the_o limb_n and_o such_o be_v the_o azimuth_n from_o the_o south_n a_o three_o example_n let_v the_o latitude_n be_v the_o same_o and_o the_o sun_n declination_n 11_o deg_fw-la 30_o min._n south_n and_o his_o altitude_n 13_o deg_fw-la 20_o min._n now_o if_o you_o place_v the_o thread_n 11_o deg_fw-la 30_o min._n below_o the_o horizon_n it_o will_v cut_v the_o parallel_n of_o 13_o deg_fw-la 20_o min._n at_o the_o azimuth_n of_o 49_o deg_fw-la 49_o min._n count_v from_o the_o north_n part_n of_o the_o limb_n and_o such_o be_v the_o azimuth_n from_o the_o south_n as_o be_v require_v chap._n xli_o the_o longitude_n and_o latitude_n of_o any_o planet_n or_o fix_a star_n be_v give_v to_o sinde_v his_o declination_n and_o right_a ascension_n first_o seek_v the_o star_n latitude_n among_o the_o parallel_n of_o latitude_n and_o upon_o the_o parallel_n of_o latitude_n count_v his_o longitude_n and_o where_o this_o count_n end_v there_o be_v the_o true_a place_n of_o the_o star_n both_o in_o respect_n of_o longitude_n and_o latitude_n this_o point_n be_v find_v look_v what_o parallel_n of_o declination_n pass_v thereby_o for_o that_o be_v the_o declination_n of_o the_o star_n from_o the_o equinoctial_a and_o also_o look_v what_o meridian_n meet_v you_o there_o for_o that_o be_v the_o right_a ascension_n of_o the_o star_n count_v from_o the_o equinoctial_a point_n aries_n as_o for_o example_n let_v the_o declination_n and_o right_a ascension_n of_o the_o bright_a star_n arcturus_n be_v require_v who_o latitude_n from_o the_o ecliptic_a be_v 31_o degree_n north_n and_o longitude_n in_o the_o 19_o degree_n 22_o minute_n ♎_o now_o look_v the_o 19_o degree_n 22_o minute_n ♎_o upon_o the_o parallel_n of_o 31_o degree_n north_n latitude_n i_o find_v there_o to_o meet_v i_o the_o parallel_n of_o 21_o deg_fw-la 10_o min._n north_n declination_n and_o also_o the_o meridian_n of_o just_a 14_o hour_n of_o time_n from_o the_o equinoctial_a point_n aries_n and_o such_o be_v the_o declination_n and_o right_a ascension_n of_o arcturus_n which_o be_v require_v chap._n xlii_o have_v the_o declination_n and_o right_a ascension_n of_o any_o fix_a star_n to_o find_v his_o longitude_n and_o latitude_n first_o seek_v the_o star_n declination_n among_o the_o parallel_n of_o declination_n upon_o which_o count_n his_o right_a ascension_n and_o where_o this_o count_n end_v shall_v meet_v you_o both_o the_o circle_n of_o his_o longitude_n and_o the_o parallel_n of_o his_o latitude_n as_o for_o example_n let_v the_o longitude_n and_o latitude_n of_o acturus_fw-la be_v require_v who_o declination_n be_v 21_o degree_n 10_o north_n and_o right_a ascension_n 14_o hour_n now_o upon_o the_o parallel_n of_o 21_o degree_n 10_o minute_n north_n declination_n i_o count_v 14_o hour_n from_o the_o equinoctial_a point_n aries_n and_o there_o i_o find_v to_o meet_v i_o the_o parallel_n of_o 31_o degree_n north_n latitude_n and_o also_o the_o circle_n of_o longitude_n for_o 19_o degree_n 22_o minute_n ♎_o and_o such_o be_v the_o longitude_n and_o latitude_n of_o arcturus_n which_o be_v require_v chap._n xliii_o to_o find_v the_o culmination_n or_o sothe_v of_o any_o of_o the_o fix_a star_n as_o also_o of_o the_o moon_n and_o other_o planet_n first_o for_o the_o fix_a star_n if_o it_o be_v a_o star_n which_o be_v place_v in_o your_o instrument_n his_o culmination_n be_v speedy_o to_o be_v find_v after_o this_o manner_n at_o the_o estimate_n time_n of_o the_o star_n culmination_n seck_fw-mi the_o sun_n place_n in_o some_o ephemeris_fw-la then_o seek_v the_o star_n who_o culmination_n you_o desire_v in_o the_o outward_a ecliptic_a which_o encompass_v the_o planisphere_n and_o bring_v the_o south_n point_n of_o the_o hour_n circle_n which_o be_v the_o point_n where_o the_o hour_n both_o begin_v and_o end_v at_o 24_o for_o i_o count_v the_o hour_n from_o the_o noon_n of_o the_o one_o day_n to_o 24_o at_o the_o noon_n of_o the_o day_n follow_v direct_o to_o the_o star_n so_o shall_v the_o place_n of_o the_o sun_n in_o the_o same_o ecliptic_a point_v out_o the_o time_n of_o culmination_n upon_o the_o hour_n circle_n as_o for_o example_n upon_o the_o 20_o of_o september_n 1652_o it_o be_v require_v to_o find_v the_o time_n of_o the_o 7_o star_n come_v to_o the_o south_n which_o i_o guess_v will_v be_v about_o two_o a_o clock_n in_o the_o morning_n at_o which_o time_n the_o sun_n be_v about_o 7_o deg_fw-la 20_o min._n ♎_o now_o find_v the_o 7_o star_n in_o the_o instrument_n i_o move_v about_o the_o hour_n circle_n until_o the_o south_n point_v thereof_o or_o 24_o hour_n stand_v direct_o against_o the_o 7_o star_n and_o then_o look_v for_o 7_o deg_fw-la 20_o min._n ♎_o i_o find_v it_o stand_v direct_o against_o 15_o hour_n in_o the_o hour_n circle_n wherefore_o i_o conclude_v that_o the_o 7_o star_n come_v to_o the_o south_n that_o morning_n at_o three_o of_o the_o clock_n again_o upon_o the_o 20_o of_o december_n i_o will_v know_v the_o time_n of_o the_o 7_o star_n culmination_n which_o i_o suppose_v may_v be_v about_o 8_o at_o night_n at_o which_o time_n i_o find_v the_o sun_n in_o 10_o deg_fw-la ♑_o now_o the_o south_n point_n of_o the_o hour-circle_n be_v set_v against_o the_o 7_o star_n as_o before_o i_o find_v 10_o deg_fw-la of_o ♑_o to_o stand_n against_o 8_o hour_n 45_o min._n in_o the_o hour-circle_n whereby_o it_o do_v appear_v that_o the_o 7_o star_n come_v to_o the_o south_n that_o afternoon_n at_o 8_o of_o the_o clock_n and_o 45_o minute_n but_o if_o you_o will_v have_v the_o culmination_n of_o any_o of_o the_o planet_n or_o of_o any_o other_o star_n not_o place_v in_o your_o instrument_n you_o must_v first_o get_v their_o right_a ascension_n by_o the_o 41_o chap._n which_o be_v find_v follow_v the_o circle_n thereof_o to_o the_o ecliptic_a line_n and_o it_o will_v there_o show_v you_o the_o sine_fw-la and_o degree_n whereunto_o you_o must_v set_v the_o south_n point_n of_o the_o hour_n circle_n which_o be_v do_v the_o place_n of_o the_o sun_n will_v point_v out_o the_o hour_n require_v as_o for_o example_n let_v it_o be_v require_v to_o find_v the_o time_n of_o the_o moon_n come_v to_o the_o south_n upon_o the_o 30_o of_o december_n 1652_o which_o i_o suppose_v may_v be_v about_o 8_o of_o the_o clock_n at_o night_n at_o which_o time_n i_o find_v the_o moon_n in_o 1_o deg_fw-la 47_o min._n ♊_o with_o latitude_n 4_o deg_fw-la 36_o min._n north_n and_o so_o by_o the_o 41_o chapter_n her_o right_a ascension_n be_v 58_o deg_fw-la 40_o min._n which_o circle_n of_o 58_o deg_fw-la 40_o min._n right_a ascension_n i_o follow_v to_o the_o ecliptic_a line_n and_o there_o i_o find_v it_o cut_v 00_o deg_fw-la 50_o min._n ♊_o therefore_o i_o set_v the_o south_n point_n of_o the_o hour_n circle_n to_o 00_o deg_fw-la 50_o min._n ♊_o and_o then_o look_v
the_o description_n and_o use_n of_o the_o universal_a quadrat_fw-la by_o which_o be_v perform_v with_o great_a expedition_n the_o whole_a doctrine_n of_o triangle_n both_o plain_a and_o spherical_a two_o several_a way_n with_o ease_n and_o exactness_n also_o the_o resolution_n of_o such_o proposition_n as_o be_v most_o useful_a in_o astronomy_n navigation_n and_o dial_v by_o which_o be_v also_o perform_v the_o proportion_v of_o line_n and_o superficies_n the_o measure_n of_o all_o manner_n of_o land_n board_n glass_n timber_n stone_n etc._n etc._n by_o thomas_n stirrup_n philomathemat_n london_n print_v by_o r._n &_o w._n leybourn_n for_o tho._n pierrepont_n at_o the_o sun_n in_o paul_n churchyard_n 1655._o honi_fw-fr soit_fw-fr qvi_fw-fr mal_fw-fr y_fw-fr pense_fw-fr to_o the_o reader_n courteous_a reader_n i_o have_v here_o present_v to_o thy_o view_n the_o description_n and_o use_n of_o a_o instrument_n which_o for_o its_o excellent_a and_o general_a use_n be_v call_v the_o universal_a quadrat_fw-la for_o by_o it_o be_v of_o a_o portable_a size_n be_v perform_v with_o incredible_a expedition_n plainness_n facility_n and_o pleasure_n the_o whole_a doctrine_n trigonometrical_a as_o well_o plain_a as_o spherical_a and_o that_o two_o several_a way_n from_o whence_o a_o innumerable_a sort_n of_o conclusion_n mathematical_a may_v be_v resolve_v more_o than_o i_o have_v here_o write_v for_o hereby_o be_v resolve_v as_o many_o proposition_n as_o by_o any_o quadrant_n circle_n cylinder_n dial_n horoscope_n astrolabe_n globe_n sector_n crossstaff_n or_o any_o such_o like_a instrument_n heretofore_o devise_v and_o that_o general_o for_o all_o latitude_n yea_o and_o if_o you_o fit_a sight_n according_o to_o the_o planisphere_n on_o the_o backside_n as_o i_o be_v intend_v to_o have_v show_v have_v not_o time_n call_v i_o so_o fast_o away_o so_o as_o they_o move_v about_o with_o the_o planisphere_n the_o horizon_n will_v show_v upon_o the_o limb_n the_o quantity_n of_o any_o angle_n take_v in_o the_o field_n as_o well_o as_o the_o theodolite_a plain_a table_n circumferentor_n or_o peractor_n or_o any_o other_o such_o like_a instrument_n so_o that_o hereby_o it_o do_v appear_v to_o be_v a_o excellent_a companion_n for_o most_o sort_n of_o man_n but_o especial_o for_o navigator_n or_o traveller_n for_o hereby_o they_o may_v find_v altitude_n and_o declination_n of_o sun_n or_o star_n and_o thereby_o the_o latitude_n of_o the_o place_n where_o they_o be_v where_o also_o they_o may_v find_v the_o amplitude_n and_o azimuth_n and_o such_o like_a conclusion_n whereby_o they_o may_v find_v the_o variation_n of_o their_o compass_n so_o likewise_o may_v they_o ready_o find_v the_o time_n of_o sunrising_n and_o set_v with_o the_o beginning_n and_o end_v of_o twilight_n the_o hour_n of_o the_o day_n by_o the_o sun_n and_o the_o hour_n of_o the_o night_n by_o the_o fix_a star_n and_o planet_n with_o the_o time_n of_o their_o rise_n sothe_v and_o set_v and_o that_o very_o easy_o and_o speedy_o with_o the_o resolution_n of_o such_o nautical_a proposition_n as_o be_v of_o most_o frequent_a use_n among_o they_o the_o whole_a treatise_n consist_v of_o three_o book_n the_o first_o whereof_o show_v the_o construction_n and_o general_a use_n of_o the_o instrument_n and_o how_o thereby_o to_o resolve_v all_o such_o proposition_n as_o may_v be_v apply_v to_o right_a line_n only_o as_o the_o proportion_v of_o line_n and_o superficies_n the_o measure_n of_o all_o manner_n of_o land_n board_n glass_n timbet_n stone_n and_o such_o like_a the_o second_o sheweth_z the_o most_o plentiful_a easy_a and_o speedy_a resolution_n of_o the_o whole_a doctrine_n trigonometrical_a both_o plain_a and_o spherical_a and_o that_o two_o several_a way_n with_o surpass_a facility_n the_o three_o sheweth_z how_o to_o resolve_v all_o such_o proposition_n astronomical_a as_o be_v of_o most_o frequent_a use_n in_o the_o art_n of_o navigation_n and_o dialling_n with_o the_o resolution_n of_o all_o such_o nautical_a proposition_n as_o be_v of_o ordinary_a use_n among_o seaman_n concern_v longitude_n latitude_n rumbe_n and_o distance_n all_o which_o i_o very_o well_o know_v may_v more_o exact_o be_v perform_v by_o the_o canon_n of_o sines_n tangent_n and_o logarithme_n yet_o nothing_o so_o speedy_o as_o on_o the_o instrument_n but_o for_o observation_n a_o instrument_n must_v be_v have_v wherewith_o if_o thou_o like_v it_o thou_o be_v here_o fit_v both_o for_o observation_n and_o operation_n thus_o desire_v thou_o to_o accept_v of_o this_o instrument_n from_o i_o as_o i_o here_o free_o present_v it_o to_o thou_o which_o i_o wish_v even_o so_o to_o further_o thou_o as_o i_o know_v it_o sufficient_a for_o the_o performance_n of_o much_o more_o than_o i_o have_v speak_v of_o it_o farewell_o the_o content_n of_o the_o first_o book_n the_o description_n of_o the_o fore-side_n of_o the_o instrument_n page_z 1_o the_o description_n of_o the_o back_n side_n page_z 4_o how_o to_o place_v the_o fix_a star_n in_o the_o instrument_n page_z 7_o the_o general_n use_v of_o the_o instrument_n page_z 12_o to_o divide_v a_o line_n into_o any_o number_n of_o equal_a part_n page_z 14_o to_o increase_v a_o line_n in_o a_o give_v proportion_n page_n 16_o to_o diminish_v a_o line_n in_o a_o give_v proportion_n page_n 17_o to_o find_v a_o proportion_n betwixt_o two_o or_o more_o right_a line_n page_n 18_o to_o lay_v down_o sudden_o 2_o 3_o or_o more_o right_a line_n in_o any_o proportion_n require_v page_n 19_o to_o two_o line_n give_v to_o find_v a_o three_o proportional_a page_n 20_o to_o three_o line_n give_v to_o find_v a_o four_o proportional_a page_n 22_o to_o find_v a_o mean_a proportional_a line_n between_o two_o right_a line_n give_v page_n 24_o to_o find_v the_o square_a root_n of_o a_o number_n page_n 26_o to_o divide_v a_o line_n by_o extreme_a and_o mean_a proportion_n page_n 27_o have_v either_o segment_n of_o a_o line_n divide_v by_o extreme_a and_o mean_a proportion_n to_o find_v the_o other_o segment_n &_o so_o the_o whole_a line_n page_n 28_o to_o divide_v a_o line_n in_o power_n page_z 29_o to_o augment_v a_o superficies_n in_o any_o proportion_n page_n 30_o to_o diminish_v a_o superficies_n in_o any_o proporton_n page_n 31_o to_o find_v the_o perpendicular_a of_o a_o equilateral_a triangle_n page_n 32_o to_o find_v the_o perpendicular_a of_o any_o right_a angle_a triangle_n page_n 34_o to_o find_v the_o perpendicular_a of_o any_o right_n line_v triangle_n page_n 36_o to_o find_v the_o proportion_n between_o two_o or_o more_o like_a superficies_n page_n 37_o to_o add_v two_o geometrical_a square_n or_o circle_n together_o into_o one_o square_a or_o circle_n page_n 38_o to_o subtract_v one_o square_a or_o circle_n from_o another_o and_o to_o produce_v the_o remainder_n in_o a_o three_o square_a or_o circle_n page_z 40_o to_o make_v a_o square_a equal_a to_o any_o superficies_n page_n 41_o the_o length_n and_o breadth_n of_o a_o parallellogram_n give_v in_o perch_n to_o to_o find_v the_o content_a in_o acre_n page_n 42_o the_o length_n and_o breadth_n of_o a_o parallellogram_n give_v in_o chain_n to_o find_v the_o content_a in_o acre_n page_n 43_o the_o perpendicular_a and_o base_a of_o a_o triangle_n give_v in_o perch_n to_o find_v the_o content_a in_o acre_n page_n 44_o the_o perpendicular_a and_o base_a of_o a_o triangle_n give_v in_o chain_n to_o find_v the_o content_a in_o acre_n page_n 45_o of_o reduce_v of_o one_o kind_n of_o measure_n to_o another_o page_z 46_o to_o find_v the_o scale_n by_o which_o any_o plot_n be_v plot_v page_n 47_o to_o find_v the_o breadh_fw-mi of_o the_o acre_n at_o any_o length_n give_v page_n 48_o to_o find_v the_o content_n of_o any_o board_n page_n 49_o have_v the_o breadth_n of_o a_o board_n to_o find_v the_o length_n of_o a_o foot_n there_o of_o page_z 50_o to_o measure_n taper_a board_n page_n 50_o have_v the_o circumference_n of_o a_o circle_n to_o find_v the_o diameter_n page_n 51_o have_v the_o diameter_n of_o a_o circle_n to_o find_v the_o circumference_n page_n 52_o to_o find_v the_o content_n of_o any_o circle_n page_z 54_o to_o find_v the_o content_a of_o any_o part_n or_o segment_n of_o a_o circle_n page_z 55_o to_o find_v the_o side_n of_o a_o square_a equal_a to_o any_o circle_n page_z 56_o to_o find_v the_o content_a of_o any_o square_a solid_a page_n 58_o to_o measure_v a_o triangular_a prisma_fw-la page_n 60_o to_o measure_n timber_n who_o base_a be_v a_o rhombus_fw-la page_n 61_o to_o measure_n timber_n who_o base_a be_v a_o rhomboide_n page_n 62_o ta_o measure_n timber_n who_o base_a be_v a_o trapezia_n page_n 63_o to_o measure_n timber_n who_o side_n be_v many_o page_n 65_o to_o measure_v round_a timber_n page_n 66_o to_o find_v how_o much_o in_o length_n make_v a_o foot_n solid_a of_o any_o round_a piece_n of_o timber_n page_n 68_o another_o
way_n to_o find_v the_o content_a of_o a_o cylinder_n page_n 69_o how_o otherwise_o to_o find_v the_o length_n of_o a_o foot_n of_o any_o cylinder_n page_n 70_o to_o find_v the_o solid_a content_n of_o any_o cone_n page_n 73_o to_o find_v the_o altitude_n of_o a_o cone_n page_n 77_o how_o to_o measure_n any_o segment_n of_o a_o cone_n or_o any_o round_a taper_a piece_n of_o timber_n page_n 78_o how_o to_o find_v the_o solid_a content_n of_o any_o pyramid_n or_o segment_n thereof_o be_v cut_v off_o page_z 80_o the_o content_n of_o the_o second_o book_n to_o find_v the_o chord_n of_o any_o arch_a page_n 83_o to_o find_v the_o right_n sine_fw-la of_o any_o arch_a page_n 85_o to_o find_v the_o arch_n of_o any_o sine_fw-la page_z 86_o the_o radius_fw-la be_v give_v with_o a_o straight_a line_n resemble_v a_o sine_fw-la to_o find_v the_o quantity_n of_o that_o unknown_a sine_fw-la page_z 89_o the_o use_n of_o the_o parallel_n as_o they_o signify_v natural_a sin_n and_o tangent_n page_n 90_o of_o the_o resolution_n of_o right_n line_v triangle_n page_n 95_o another_o way_n to_o resolve_v right_o line_v triangle_n page_n 102_o of_o the_o resolution_n of_o right_n angle_v spherical_a triangle_n by_o the_o foreside_n of_o the_o quadrat_fw-la page_n 116_o to_o resolve_v any_o spherical_a triangle_n whatsoever_o by_o the_o foreside_n page_n 131_o of_o the_o resolution_n of_o right_n angle_v spherical_a triangle_n by_o the_o backside_n page_n 141_o how_o to_o resolve_v any_o spherical_a triangle_n whatsoever_o by_o the_o backside_n of_o the_o instrument_n page_n 146_o the_o content_n of_o the_o three_o book_n to_o find_v the_o sun_n altitude_n by_o the_o shadow_n of_o a_o perpendicular_a gnomon_n page_n 151_o to_o find_v the_o length_n of_o the_o right_a shadow_n page_n 152_o to_o find_v the_o height_n of_o the_o sun_n by_o the_o shadow_n of_o a_o gnomon_n lie_v parallel_n to_o the_o horizon_n page_n 153_o to_o find_v the_o length_n of_o the_o contrary_a shadow_n page_n 154_o to_o find_v the_o sun_n declination_n page_n 155_o to_o find_v the_o latitude_n of_o the_o place_n page_z 156_o to_o find_v the_o declination_n of_o the_o sun_n or_o star_n by_o observation_n page_n 158_o to_o find_v the_o amplitude_n of_o the_o sun_n or_o star_n page_n 160_o to_o find_v the_o pole_n elevation_n page_n 161_o to_o find_v the_o sun_n declination_n page_n 162_o to_o find_v the_o sun_n place_n page_z 159_o to_o find_v the_o sun_n height_n in_o the_o vertical_a circle_n page_n 163_o to_o find_v the_o time_n when_o the_o sun_n will_v come_v to_o the_o due_a east_n or_o west_n page_n 164_o to_o find_v the_o sun_n altitude_n at_o the_o hour_n of_o six_o page_n 165_o to_o find_v the_o sun_n azimuch_o at_o the_o hour_n of_o six_o page_n 165_o to_o find_v the_o ascensional_a difference_n page_n 166_o to_o find_v the_o beginning_n and_o end_v of_o twilight_n page_n 167_o to_o find_v the_o sun_n right_a ascension_n page_n 170_o to_o find_v the_o oblique_a ascension_n page_n 171_o to_o find_v the_o hour_n of_o the_o day_n page_n 171_o to_o find_v the_o azimuth_n page_n 172_o to_o find_v the_o hour_n of_o the_o day_n page_n 173_o to_o find_v the_o azimuth_n page_n 175_o to_o find_v the_o sun_n place_n and_o declination_n page_n 178_o to_o find_v the_o sun_n right_a ascension_n page_n 179_o to_o find_v the_o sun_n amplitude_n page_n 179_o to_o find_v the_o amplitude_n of_o the_o star_n page_n 180_o to_o find_v the_o latitude_n of_o the_o place_n page_z 181_o to_o find_v the_o sun_n declination_n page_n 182_o to_o find_v the_o sun_n height_n in_o the_o vertical_a circle_n page_n 182_o to_o find_v the_o time_n when_o the_o sun_n will_v be_v due_a east_n or_o west_n page_n 183_o to_o find_v the_o sun_n height_n at_o the_o hour_n of_o six_o page_n 184_o to_o find_v the_o sun_n azimuth_n at_o the_o hour_n of_o six_o page_n 185_o to_o find_v the_o ascensional_a difference_n of_o the_o sun_n or_o star_n page_n 185_o to_o find_v the_o beginning_n and_o end_v of_o twilight_n page_n 187_o to_o find_v the_o hour_n of_o the_o day_n page_n 187_o to_o find_v the_o azimuth_n page_n 188_o to_o find_v the_o right_a ascension_n and_o declination_n of_o the_o planet_n or_o fix_a star_n page_n 190_o to_o find_v the_o longitude_n and_o latitude_n of_o the_o star_n page_n 191_o to_o find_v the_o culmination_n of_o any_o of_o the_o fix_a star_n moon_n or_o other_o planet_n page_n 191_o to_o find_v the_o time_n of_o the_o rise_n and_o set_v of_o any_o of_o the_o fix_a star_n moon_n or_o other_o planet_n page_n 193_o to_o find_v the_o exact_a hour_n of_o the_o night_n by_o the_o star_n page_n 194_o to_o find_v how_o many_o mile_n answer_v to_o one_o deg_fw-la of_o longitude_n in_o any_o latitude_n page_n 198_o to_o find_v what_o difference_n of_o longitude_n belong_v to_o one_o degree_n of_o distance_n in_o any_o latitude_n page_n 199_o to_o find_v how_o many_o league_n do_v answer_v to_o one_o degree_n of_o latitude_n upon_o any_o rumb_n give_v page_n 200_o to_o find_v the_o difference_n of_o latitude_n page_n 201_o to_o find_v the_o distance_n upon_o the_o rumb_n page_n 202_o to_o find_v the_o rumb_n page_n 202_o to_o find_v the_o difference_n of_o longitude_n page_n 203_o to_o find_v the_o difference_n of_o longitude_n page_n 204_o to_o find_v the_o rumb_n page_n 205_o to_o find_v the_o difference_n of_o lat_fw-la &_o long_o page_n 206_o to_o find_v the_o distance_n upon_o the_o rumb_n page_n 208_o to_o find_v the_o distance_n between_o the_o ship_n and_o the_o land_n page_n 250_o to_o find_v the_o distance_n of_o any_o ship_n from_o the_o land_n page_n 207_o to_o find_v the_o distance_n of_o two_o ship_n at_o road_n page_n 029_o the_o fore-side_n of_o the_o universal_a quadrat_fw-la the_o backside_n of_o the_o universal_a quadrat_fw-la delineavit_fw-la antonius_n thompson_n the_o first_o book_n show_v the_o description_n and_o general_a use_n of_o the_o universal_a quadrat_fw-la and_o how_o thereby_o to_o resolve_v all_o such_o proposition_n as_o may_v be_v apply_v unto_o right_a line_n as_o the_o proportion_v of_o line_n and_o superficies_n the_o measure_n of_o all_o manner_n of_o land_n board_n glass_n timber_n stone_n and_o such_o like_a chap._n i._n the_o description_n and_o make_n of_o the_o universal_a quadrat_fw-la a_o quadrat_fw-la or_o square_a geometrical_a be_v a_o figure_n of_o four_o equal_a side_n who_o angle_n be_v all_o right_a angle_n within_o the_o limit_n of_o which_o figure_n we_o intend_v to_o describe_v a_o universal_a instrument_n for_o the_o speedy_a performance_n of_o most_o mathematical_a concusion_n and_o that_o general_o for_o all_o latitude_n first_o therefore_o prepare_v a_o piece_n of_o box_n or_o brass_n and_o after_o you_o have_v make_v it_o very_o smooth_a and_o plain_a cut_v it_o direct_o four_o square_a with_o four_o equal_a side_n and_o angle_v all_o right_a your_o wood_n or_o brass_n be_v thus_o prepare_v draw_v thereupon_o four_o right_a line_n equal_o distant_a or_o parallel_a to_o the_o edge_n all_o of_o they_o cut_v one_o the_o other_o at_o right_a angle_n let_v the_o parallel_n distance_n of_o these_o line_n from_o the_o edge_n be_v so_o much_o as_o that_o you_o may_v place_v the_o figure_n to_o the_o division_n limb_n like_a as_o you_o may_v perceive_v by_o the_o figure_n this_o be_v do_v divide_v each_o side_n of_o the_o square_n into_o 100_o equal_a part_n or_o rather_o 1000_o if_o quantity_n will_v give_v leave_n for_o the_o large_a the_o instrument_n be_v the_o more_o exact_o will_v the_o conclusion_n wrought_v thereby_o be_v perform_v but_o the_o instrument_n not_o so_o portable_a then_o from_o each_o division_n of_o the_o one_o side_n draw_v parallel_a line_n to_o the_o like_a division_n on_o the_o opposite_a side_n the_o like_a do_v by_o the_o other_o two_o side_n so_o shall_v the_o whole_a face_n of_o the_o instrument_n be_v fill_v with_o parallel_a line_n both_o way_n and_o cross_v each_o other_o at_o right_a angle_n according_a to_o the_o common_a cynical_a quadrant_n and_o for_o the_o better_a distinction_n of_o these_o parallel_n you_o may_v prick_v every_o five_o so_o may_v you_o guide_v your_o eye_n along_o upon_o they_o the_o better_a this_o be_v do_v make_v choice_n of_o one_o of_o the_o angle_n for_o a_o centre_n as_o here_o we_o have_v do_v at_o a_o from_o which_o you_o shall_v number_v the_o two_o next_o side_n by_o 10_o 20_o 30_o etc._n etc._n unto_o 100_o and_o place_n figure_n to_o they_o in_o the_o margin_n as_o you_o see_v in_o the_o figure_n then_o set_v one_o foot_n of_o your_o compass_n in_o the_o centre_n a_o with_o the_o other_o open_v to_o the_o full_a length_n of_o one_o of_o the_o side_n draw_v the_o quadrant_n bc_n over_o all_o the_o parallel_n this_o quadrant_n divide_v into_o
for_o the_o sun_n place_n which_o at_o the_o estimate_n time_n be_v in_o 20_o deg_fw-la 12_o min._n ♑_o i_o find_v it_o to_o stand_v against_o 8_o hour_n 30_o min._n wherefore_o i_o conclude_v that_o the_o moon_n come_v to_o the_o south_n that_o night_n at_o 8_o of_o the_o clock_n and_o 30_o min._n chap._n xliv_o to_o find_v the_o time_n of_o the_o rise_n or_o set_v of_o any_o of_o the_o fix_a star_n and_o also_o of_o the_o moon_n or_o other_o planet_n when_o you_o will_v find_v the_o time_n of_o any_o star_n rise_v first_o get_v his_o seminocturnal_a arch_n according_a to_o the_o 37_o chap._n which_o put_v unto_o 12_o hour_n and_o if_o the_o star_n be_v place_v in_o the_o instrument_n bring_v that_o number_n of_o hour_n to_o the_o star_n but_o if_o the_o star_n be_v not_o in_o the_o instrument_n find_v his_o place_n by_o the_o last_o chap._n and_o bring_v this_o sum_n of_o 12_o hour_n and_o the_o star_n seminocturnal_a arch_n thereto_o so_o shall_v the_o place_n of_o the_o sun_n point_v out_o the_o the_o time_n of_o his_o rise_n as_o for_o example_n the_o 30_o of_o december_n 1652_o let_v the_o rise_n of_o arcturus_n be_v require_v which_o i_o suppose_v may_v be_v about_o 10_o of_o the_o clock_n in_o the_o night_n at_o which_o time_n the_o sun_n be_v in_o 20_o deg_fw-la 22_o min._n of_o ♑_o first_fw-mi by_o the_o 37_o chap._n i_o find_v his_o seminocturnal_a arch_n to_o be_v 4_o hour_n 3_o min._n which_o put_v unto_o 12_o hour_n make_v 16_o hour_n 3_o min._n and_o in_o regard_n arcturus_n be_v place_v in_o the_o instrument_n i_o bring_v 16_o hour_n 3_o minunto_n he_o so_o shall_v the_o sun_n in_o 20_o deg_fw-la 22_o min._n ♑_o point-out_a 10½_n hour_n for_o the_o time_n of_o arcturus_n rise_v the_o say_v 30_o of_o december_n in_o the_o night_n and_o if_o you_o will_v have_v the_o time_n of_o his_o set_n bring_v unto_o he_o his_o semidiurnal_a arch_n which_o you_o may_v find_v to_o be_v 7_o hour_n 59_o min._n so_o shall_v the_o sun_n in_o 19_o deg_fw-la 56_o min._n ♑_o which_o be_v his_o place_n of_o the_o estimate_n time_n of_o the_o star_n set_v point_n to_o 2_o hour_n 32_o min._n in_o the_o hour_n circle_n for_o the_o time_n of_o arcturus_n set_v afternoon_n in_o the_o day_n and_o year_n aforesaid_a and_o thus_o may_v you_o find_v the_o time_n of_o any_o other_o star_n or_o planet_n rise_v or_o set_v if_o you_o first_o get_v his_o declination_n by_o the_o 41_o chap._n and_o then_o his_o semidiurnal_a and_o seminocturnal_a arch_n by_o the_o 37_o chap._n and_o then_o by_o the_o last_o chap._n find_v the_o place_n whereunto_o you_o must_v bring_v his_o semidiurnal_a or_o seminocturnal_a arch_n as_o you_o there_o do_v the_o south_n point_n for_o his_o culmination_n so_o shall_v the_o sun_n place_n at_o the_o estimate_n time_n point_v out_o the_o time_n require_v chap._n xlv_o to_o find_v the_o exact_a hour_n of_o the_o night_n speedy_o by_o the_o star_n first_o with_o your_o instrument_n observe_v the_o star_n horary_a distance_n from_o the_o meridian_n that_o be_v first_o take_v his_o height_n and_o then_o find_v his_o hour_n from_o the_o meridian_n as_o you_o do_v for_o the_o sun_n in_o the_o 39_o chap._n in_o all_o respect_n but_o here_o i_o will_v have_v you_o note_v that_o in_o take_v the_o height_n of_o a_o star_n by_o the_o common_a sight_n the_o hole_n be_v so_o small_a that_o you_o can_v perceive_v a_o star_n through_o they_o and_o therefore_o for_o this_o purpose_n you_o may_v have_v two_o great_a sight_n hole_n for_o the_o star_n and_o two_o lesser_a one_o for_o the_o sun_n or_o rather_o as_o i_o use_v to_o make_v they_o for_o myself_o and_o my_o friend_n in_o all_o my_o night_n dial_n which_o be_v thus_o in_o the_o sight_n next_o my_o eye_n i_o make_v a_o great_a hole_n and_o for_o the_o sight_n by_o the_o centre_n i_o set_v up_o a_o flat_a pin_n of_o brass_n perpendicular_a to_o the_o plate_n the_o breadth_n of_o this_o pin_n will_v be_v such_o that_o the_o shadow_n thereof_o may_v not_o full_o fill_v the_o hole_n of_o the_o other_o sight_n so_o that_o when_o the_o sun_n shine_v and_o the_o shadow_n of_o the_o pin_n fall_v upon_o the_o hole_n of_o the_o other_o sight_n there_o may_v some_o small_a light_n of_o the_o sun_n pass_v through_o the_o hole_n on_o both_o side_n of_o the_o shadow_n by_o which_o mean_v you_o may_v take_v the_o height_n of_o the_o sun_n as_o exact_o as_o by_o fine_a hole_n and_o you_o may_v take_v the_o height_n of_o a_o star_n better_o with_o this_o pin_n then_o with_o great_a sight_n hole_n but_o now_o to_o the_o matter_n in_o hand_n the_o horary_a distance_n of_o the_o star_n from_o the_o meridian_n be_v find_v seek_v it_o in_o the_o hour_n circle_n if_o the_o star_n be_v to_o the_o westward_n of_o the_o meridian_n but_o if_o the_o star_n be_v to_o the_o eastward_o take_v the_o compliment_n of_o his_o horary_a distance_n to_o 24_o hour_n and_o seek_v that_o in_o the_o hour_n circle_n which_o be_v find_v bring_v it_o to_o the_o star_n or_o to_o his_o place_n find_v as_o in_o 43_o chap._n so_o shall_v the_o place_n of_o the_o sun_n point_v out_o the_o hour_n require_v as_o for_o example_n upon_o the_o 16_o of_o november_n 1652_o it_o be_v serene_a air_n i_o desire_v to_o see_v what_o time_n of_o the_o morning_n it_o be_v for_o which_o purpose_n i_o make_v use_n of_o arcturus_n and_o by_o observation_n i_o sound_v his_o height_n to_o be_v 27_o deg_fw-la 12_o min._n towards_o the_o east_n now_o his_o declination_n be_v 21_o deg_fw-la 10_o min._n north_n i_o find_v by_o the_o 39_o chap._n that_o he_o be_v 4_o hour_n 50_o min._n from_o the_o south_n and_o in_o regard_n he_o be_v on_o the_o east_n side_n of_o the_o meridian_n i_o take_v the_o compliment_n of_o 4_o hour_n 50_o min._n to_o 24_o hour_n which_o be_v 19_o hour_n 10_o min._n this_o 19_o hour_n 10_o min._n i_o seek_v in_o the_o hour_n circle_n and_o bring_v it_o to_o arcturus_n now_o the_o sun_n at_o the_o estimate_n timebeing_n in_o 4_o deg_fw-la 40_o min._n â™_o point_v to_o 17_o hour_n in_o the_o hour_n circle_n which_o be_v direct_o 5_o of_o the_o clock_n in_o the_o morning_n the_o like_a may_v be_v do_v by_o any_o of_o the_o planet_n as_o i_o will_v make_v appear_v by_o these_o two_o example_n follow_v which_o i_o make_v in_o the_o day_n time_n by_o the_o glitter_a planet_n venus_n the_o 16_o of_o november_n 1652_o i_o make_v these_o two_o follow_a observation_n by_o the_o planet_n venus_n in_o the_o day_n time_n as_o to_o the_o find_n of_o the_o hour_n by_o the_o star_n she_o be_v about_o her_o great_a distance_n from_o the_o sun_n the_o say_v 16_o day_n be_v serene_a and_o i_o be_v abroad_o and_o in_o a_o convenient_a place_n where_o i_o may_v see_v a_o clear_a horizon_n and_o see_v venus_n shine_v so_o bright_a direct_o at_o the_o sun_n rise_v i_o can_v do_v no_o less_o then_o make_v observation_n by_o she_o to_o see_v how_o her_o longitude_n latitude_n right_a ascension_n declination_n and_o likewise_o the_o hour_n find_v by_o she_o will_v agree_v with_o the_o time_n of_o sun_n rise_v wherefore_o first_o i_o observe_v her_o altitude_n by_o my_o instrument_n which_o be_v but_o small_a i_o can_v not_o make_v so_o exact_a observation_n as_o otherwise_o imight_o have_v do_v but_o as_o it_o be_v i_o find_v her_o altitude_n just_a 30_o deg_fw-la now_o the_o estimate_n time_n be_v sun_n rise_v which_o be_v that_o morning_n at_o 8_o a_o clock_n and_o 1_o min._n in_o our_o latitude_n of_o 52_o deg_fw-la 30_o min._n i_o look_v in_o my_o almanac_n for_o the_o longitude_n and_o latitude_n of_o ♀_o which_o i_o find_v according_a to_o mr._n vincent_n wing_n ephemeris_fw-la to_o be_v in_o 18_o deg_fw-la ♎_o with_o 2_o deg_fw-la north_n latitude_n then_o by_o the_o 41_o chap._n i_o find_v her_o declination_n to_o be_v 5_o deg_fw-la 14_o min._n south_n and_o so_o by_o the_o 39_o chap._n her_o hour_n from_o the_o meridian_n to_o be_v no_o more_o but_o 57_o min._n so_o likewise_o by_o the_o 41_o chap._n i_o find_v she_o right_a ascension_n to_o be_v 13_o hour_n 9_o min._n which_o circle_n i_o follow_v to_o the_o ecliptic_a line_n where_o it_o give_v i_o 18_o deg_fw-la 45_o min._n ♎_o for_o the_o place_n of_o ♀_o in_o the_o outward_a ecliptic_a unto_o which_o 18_o deg_fw-la 45_o min._n ♎_o i_o bring_v the_o compliment_n of_o 0_o hour_n 57_o min._n which_o be_v 23_o hour_n 3_o min._n so_o do_v the_o sun_n in_o 4_o deg_fw-la 47_o min._n â™_o being_n his_o place_n at_o the_o time_n of_o his_o rise_n point_v out_o 20_o hour_n 1_o min._n for_o the_o time_n require_v agree_v exact_o with_o the_o
time_n of_o sun_n rise_v be_v also_o a_o good_a proof_n that_o ♀_o was_z than_o in_o the_o aforesaid_a place_n both_o in_o respect_n of_o longitude_n and_o latitude_n a_o second_o example_n the_o same_o day_n a_o while_n after_o noon_n i_o observe_v the_o altitude_n of_o ♀_o again_o which_o i_o find_v to_o be_v 16_o deg_fw-la 56_o min._n the_o sun_n be_v then_o in_o 5_o deg_fw-la â™_o and_o ♀_o in_o 18_o deg_fw-la 10_o min._n ♎_o which_z 2_o deg_fw-la north_n latitude_n wherefore_o by_o the_o 41_o chap._n i_o find_v her_o declination_n to_o be_v 5_o deg_fw-la 18_o min._n south_n and_o so_o by_o the_o 39_o chap._n her_o distance_n from_o the_o meridian_n 3_o hour_n 32_o min._n so_o likewise_o by_o the_o say_v 41_o chap._n her_o right_a ascension_n be_v 13_o hour_n 10_o min._n which_o circle_n of_o right_a ascension_n i_o follow_v to_o the_o ecliptic_a line_n where_o it_o give_v i_o the_o 19_o deg_fw-la of_o ♎_o for_o the_o place_n of_o ♀_o in_o the_o outward_a ecliptic_a unto_o which_o 19_o deg_fw-la ♎_o i_o bring_v 3_o hour_n 32_o min._n of_o the_o hour_n circle_n so_o do_v the_o sun_n in_o 5_o deg_fw-la â™_o point_n to_o 0_o hour_n 30_o min._n in_o the_o hour_n circle_n whereby_o it_o do_v appear_v that_o it_o be_v then_o half_a a_o hour_n past_o 12_o of_o the_o clock_n in_o the_o day_n time_n and_o in_o deed_n so_o it_o be_v by_o my_o fix_a sun_n dial_n be_v a_o exact_a dial_n of_o a_o yard_n square_n perhaps_o some_o will_v think_v it_o strange_a that_o the_o hour_n of_o the_o day_n shall_v be_v find_v by_o this_o star_n or_o that_o this_o star_n shall_v appear_v so_o bright_a at_o the_o noon_n time_n of_o the_o day_n that_o the_o hour_n shall_v thereby_o be_v find_v there_o be_v no_o eclipse_n of_o the_o sun_n at_o that_o time_n but_o indeed_o it_o be_v not_o so_o strange_a as_o its_o true_a as_o some_o of_o my_o friend_n can_v testify_v thus_o have_v show_v on_o both_o side_n of_o the_o universal_a quadrat_fw-la how_o to_o resolve_v such_o proposition_n astronomical_a as_o be_v most_o useful_a for_o seaman_n and_o dialler_n and_o such_o like_a artist_n i_o will_v now_o show_v likewise_o how_o to_o resolve_v such_o nautical_a proposition_n as_o be_v of_o ordinary_a use_n concern_v longitude_n latitude_n rumb_n and_o distance_n chap._n xlvi_o to_o find_v how_o many_o minute_n or_o mile_n answer_v to_o one_o degree_n of_o longitude_n in_o any_o latitude_n require_v in_o sail_v by_o the_o compass_n the_o course_n hold_v sometime_o upon_o a_o great_a circle_n sometime_o upon_o a_o parallel_n to_o the_o equator_fw-la but_o most_o common_o upon_o crooked_a line_n wind_v towards_o one_o of_o the_o pole_n which_o line_n be_v well_o know_v by_o the_o name_n of_o rombs_n if_o the_o course_n hold_v upon_o a_o great_a circle_n it_o be_v either_o north_n or_o south_n under_o some_o one_o of_o the_o meridians_fw-la or_o else_o east_n or_o west_n under_o the_o equator_fw-la and_o in_o these_o case_n every_o degree_n require_v a_o allowance_n of_o 20_o league_n or_o 60_o mile_n every_o 20_o league_n make_v a_o degree_n difference_n in_o sail_v upon_o those_o circle_n so_o that_o here_o need_v no_o further_a precept_n than_o this_o divide_v your_o league_n sail_v by_o 20_o and_o you_o shall_v have_v the_o degree_n of_o distance_n and_o contrary_o multiply_v the_o degree_n of_o distance_n by_o 20_o so_o shall_v you_o have_v the_o number_n of_o league_n sail_v but_o if_o the_o course_n hold_v east_n or_o west_n upon_o any_o of_o the_o parallel_n to_o the_o equator_fw-la to_o find_v how_o many_o mile_n do_v answer_v to_o one_o degree_n of_o the_o equinoctial_a take_v this_o proportion_n as_o the_o radius_fw-la to_o the_o cousin_n of_o the_o latitude_n so_o be_v 60_o mile_n the_o measure_n of_o one_o degree_n at_o the_o equator_fw-la to_o the_o number_n of_o mile_n answer_v to_o one_o degree_n of_o longitude_n in_o the_o give_v latitude_n thus_o in_o the_o latitude_n of_o 50_o deg_fw-la if_o i_o place_v the_o thread_n upon_o the_o intersection_n of_o the_o contrary_a sine_fw-la of_o 90_o deg_fw-la with_o the_o right_n sine_fw-la of_o 40_o deg_fw-la which_o be_v the_o compliment_n to_o 50_o deg_fw-la it_o will_v cut_v the_o contrary_a parallel_n of_o 60_o at_o the_o right_a parallel_n of_o 38_o 34_o 60_o and_o so_o many_o mile_n do_v answer_v to_o one_o degree_n of_o longitude_n in_o the_o latitude_n of_o 50_o deg_fw-la again_o in_o the_o latitude_n of_o 52_o deg_fw-la 30_o min._n i_o place_v the_o thread_n to_o the_o intersection_n of_o the_o contrary_a sine_fw-la of_o 90_o deg_fw-la with_o the_o right_n sine_fw-la of_o 37_o deg_fw-la 30_o min._n and_o it_o cut_v the_o contrary_a parallel_n of_o 60_o at_o the_o right_a parallel_n of_o 36_o 31_o 60_o and_o so_o many_o mile_n do_v answer_v to_o one_o degree_n of_o longitude_n in_o the_o latitude_n of_o 52_o deg_fw-la 30_o min._n as_o be_v require_v or_o if_o you_o place_v the_o bead_n to_o 60_o on_o the_o side_n of_o the_o quadrat_fw-la and_o then_o open_a the_o thread_n to_o the_o angle_n of_o the_o latitude_n as_o here_o to_o 50_o deg_fw-la the_o bead_n will_v cut_v the_o contrary_a parallel_n of_o 38_o 34_o 60_o as_o in_o the_o first_o example_n and_o if_o you_o open_v the_o thread_n to_o the_o angle_n of_o 52_o deg_fw-la 30_o min._n the_o bead_n will_v cut_v the_o contrary_a parallel_n of_o 36_o 31_o 60_o as_o in_o the_o latter_a example_n chap._n xlvii_o to_o find_v what_o difference_n of_o longitude_n be_v long_v to_o one_o degree_n or_o 20_o league_n of_o distance_n upon_o any_o parallel_n of_o latitude_n as_o the_o cousin_n of_o the_o latitude_n to_o the_o sine_fw-la of_o 90_o degree_n so_o be_v the_o distance_n of_o 20_o league_n on_o the_o parallel_n to_o the_o difference_n of_o longitude_n require_v thus_o in_o the_o latitude_n of_o 50_o deg_fw-la if_o you_o place_v the_o thread_n to_o the_o intersection_n of_o the_o contrary_a sine_fw-la of_o 40_o deg_fw-la with_o the_o right_n sine_fw-la of_o 90_o deg_fw-la it_o will_v cut_v the_o contrary_a parallel_n of_o 20_o league_n at_o the_o right_a parallel_n of_o 31_o league_n and_o something_o better_o or_o 1_o deg_fw-la 33_o min._n and_o such_o be_v the_o difference_n of_o longitude_n at_o the_o equator_fw-la answer_v to_o 20_o league_n of_o distance_n on_o the_o parallel_n of_o 50_o deg_fw-la of_o latitude_n again_o in_o the_o latitude_n of_o 52_o deg_fw-la 30_o min._n if_o you_o place_v the_o thread_n to_o the_o intersection_n of_o the_o contrary_a sine_fw-la of_o the_o compliment_n of_o the_o latitude_n 37_o deg_fw-la 30_o min._n with_o the_o right_n sine_fw-la of_o 90_o deg_fw-la it_o will_v cut_v the_o contrary_a parallel_n of_o 20_o league_n at_o the_o right_a parallel_n of_o almost_o 33_o league_n or_o 1_o deg_fw-la 39_o min._n and_o such_o be_v the_o difference_n of_o longitude_n answer_v to_o 20_o league_n of_o distance_n in_o the_o parallel_n of_o 52_o degree_n 30_o min._n chap._n xlviii_o to_o find_v how_o many_o league_n do_v answer_v to_o one_o degree_n of_o latitude_n upon_o any_o romb_n require_v as_o the_o cousin_n of_o the_o romb_n from_o the_o meridian_n to_o 20_o league_n or_o one_o degree_n upon_o the_o meridian_n so_o be_v the_o radius_fw-la to_o the_o league_n answer_v to_o one_o degree_n upon_o the_o romb_n sail_v from_o the_o north_n parallel_n of_o 40_o degree_n upon_o a_o course_n northeast_n and_o by_o east_n which_o be_v the_o five_o romb_n from_o the_o meridian_n or_o 56_o deg_fw-la 15_o min._n the_o compliment_n whereof_o be_v 33_o deg_fw-la 45_o min._n it_o be_v require_v how_o many_o league_n the_o ship_n shall_v run_v before_o it_o shall_v raise_v the_o pole_n one_o degree_n wherefore_o i_o place_v the_o thread_n at_o the_o intersection_n of_o the_o contrary_a sine_fw-la of_o 33_o deg_fw-la 45_o min_fw-mi with_o the_o right_n sine_fw-la of_o 90_o deg_fw-la and_o it_o will_v cut_v the_o contrary_a parallel_n of_o 20_o league_n at_o at_o the_o right_a parallel_n of_o 36_o league_n and_o so_o much_o way_n must_v the_o ship_n make_v upon_o the_o five_o rumb_n from_o the_o meridian_n before_o she_o can_v raise_v or_o lay_v the_o pole_n one_o degree_n chap._n xlix_o to_o find_v how_o much_o any_o ship_n have_v raise_v or_o depress_v the_o pole_n know_v the_o course_n she_o have_v make_v and_o the_o league_n she_o have_v sail_v as_o the_o sine_fw-la of_o 90_o degree_n to_o the_o cousin_n of_o the_o rumb_n from_o the_o meridian_n so_o be_v the_o distance_n upon_o the_o rumb_n to_o the_o difference_n between_o both_o latitude_n thus_o know_v the_o course_n to_o be_v on_o the_o five_o rumb_n or_o 56_o degr_n 15_o min._n from_o the_o meridian_n the_o compliment_n whereof_o be_v 33_o deg_fw-la 45_o min._n and_o the_o league_n sail_v 120_o we_o may_v find_v