90_o equal_a degree_n and_o subdivide_v each_o degree_n into_o as_o many_o part_n as_o quantity_n will_v give_v leave_n and_o let_v they_o be_v number_v with_o 10_o 20_o 30_o etc._n etc._n unto_o 90_o both_o way_n as_o there_o you_o see_v and_o if_o upon_o the_o centre_n a_o with_o any_o convenient_a distance_n you_o describe_v another_o quadrant_n and_o divide_v it_o into_o six_o equal_a part_n as_o in_o my_o other_o quadrat_fw-la set_v down_o and_o draw_v slope_n line_n to_o each_o 15_o deg_fw-la in_o the_o quadrant_n bc_n and_o number_v they_o backward_o than_o these_o slope_n line_n may_v be_v account_v as_o hour_n in_o the_o art_n of_o dial_v and_o will_v be_v sufficient_a to_o perform_v the_o whole_a art_n of_o dial_v as_o i_o have_v at_o large_a teach_v in_o my_o book_n late_o publish_v call_v the_o complete_a diallist_n this_o do_v by_o lay_v a_o ruler_n upon_o the_o centre_n a_o and_o each_o degree_n in_o the_o quadrant_n bc_n you_o may_v divide_v the_o two_o opposite_a side_n 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kind_n but_o see_v the_o backside_n of_o this_o universal_a instrument_n stand_v yet_o blank_a we_o will_v make_v use_n thereof_o in_o fix_v a_o movable_a planisphere_n of_o brass_n thereon_o on_o which_o planisphere_n we_o may_v project_v the_o sphere_n in_o plano_fw-la with_o straight_a line_n and_o so_o by_o help_n of_o a_o fix_a horizontal_n line_n and_o fix_a almicanter_n we_o shall_v make_v this_o side_n also_o become_v universal_a to_o perform_v most_o spherical_a conclusion_n and_o that_o most_o demonstrative_o and_o general_o for_o all_o latitude_n the_o description_n whereof_o be_v after_o this_o manner_n chap._n ii_o the_o description_n of_o the_o backside_n of_o the_o universrl_n quadrat_fw-la first_o upon_o the_o backside_n draw_v four_o straight_a line_n near_o to_o the_o edge_n and_o parallel_v thereto_o all_o cut_v one_o the_o other_o at_o right_a angle_n as_o here_o in_o the_o figure_n you_o may_v behold_v upon_o which_o line_n you_o shall_v insert_v the_o degree_n 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and_o thereby_o draw_v parallel_a line_n to_o the_o equator_fw-la these_o parallel_n shall_v represent_v the_o declination_n of_o the_o sun_n moon_n and_o star_n and_o the_o latitude_n of_o town_n city_n and_o country_n and_o shall_v divide_v the_o axis_n according_a to_o the_o line_n of_o sines_n if_o again_o we_o divide_v the_o equator_fw-la and_o each_o of_o his_o parallel_n in_o the_o like_a sort_n as_o the_o axe_n of_o the_o world_n be_v divide_v and_o then_o careful_o through_o each_o degree_n draw_v a_o line_n so_o as_o it_o make_v no_o angle_n the_o line_n so_o draw_v shall_v be_v elliptical_a and_o represent_v the_o hour_n circle_n the_o meridian_n circle_n the_o hour_n of_o 12_o at_o noon_n and_o midnight_n and_o that_o 15_o degree_n next_o unto_o it_o draw_v through_o 75_o degree_n from_o the_o centre_n the_o hour_n of_o 11_o &_o 1_o and_o that_o which_o be_v draw_v through_o 60_o degree_n from_o the_o centre_n the_o hour_n of_o 10_o and_o 2_o and_o so_o of_o the_o rest_n let_v these_o hour_n circle_n be_v prick_v to_o know_v they_o from_o the_o other_o meridian_n then_o in_o the_o meridian_n number_n 23_o deg_fw-la 30_o minute_n the_o great_a declination_n of_o the_o sun_n from_o the_o equator_fw-la to_z ♋_o northward_o and_o to_o ♑_o southward_n and_o draw_v the_o line_n ♋_o ♑_o for_o the_o ecliptic_a then_o number_v 23_o degree_n 30_o minute_n from_o the_o north-pole_n northward_o and_o from_o the_o south_n pole_n southward_o draw_v the_o axe_n of_o the_o ecliptic_a which_o let_v be_v divide_v in_o like_a sort_n as_o the_o axis_n of_o the_o world_n be_v divide_v and_o through_o each_o degree_n or_o each_o five_o degree_n draw-line_n parallel_n to_o the_o ecliptic_a these_o line_n so_o draw_v shall_v represent_v the_o latitude_n of_o the_o planet_n and_o fix_v star_n and_o for_o distinction_n sake_n prick_v every_o five_o or_o ten_o as_o before_o now_o if_o we_o divide_v the_o ecliptic_a and_o his_o parallel_n as_o the_o equator_fw-la and_o his_o parallel_n be_v divide_v and_o then_o careful_o draw_v a_o line_n through_o each_o 30_o degree_n so_o as_o it_o make_v no_o angle_n the_o line_n so_o draw_v shall_v represent_v the_o sign_n the_o first_o 30_o degree_n from_o the_o centre_n towards_o ♋_o shall_v stand_v for_o the_o sign_n ♈_o the_o 30_o degree_n next_o follow_v for_o ♉_o and_o so_o of_o the_o rest_n as_o you_o may_v see_v by_o the_o figure_n the_o sphere_n be_v thus_o project_v in_o plain_a let_v we_o resort_v to_o the_o back_n of_o our_o quadrat_fw-la again_o and_o draw_v a_o line_n through_o the_o centre_n cuit_v two_o of_o the_o side_n just_a in_o the_o middle_n so_o shall_v this_o diameter_n make_v right_a angle_n with_o the_o side_n and_o represent_v the_o horizontal_n line_n for_o any_o region_n then_o take_v the_o just_a length_n of_o the_o semidiameter_n of_o your_o plainisphere_n and_o set_v it_o above_o and_o below_o the_o horizon_n at_o both_o end_n thereof_o in_o the_o side_n of_o the_o quadrat_fw-la as_o here_o you_o see_v and_o let_v this_o line_n be_v divide_v in_o like_a sort_n as_o the_o equator_fw-la and_o his_o axis_n be_v divide_v and_o number_v with_o 10_o 20_o 30_o etc._n etc._n unto_o 90_o from_o the_o horizon_n both_o way_n these_o be_v the_o degree_n of_o altitude_n all_o this_o be_v do_v let_v we_o come_v to_o the_o divide_v of_o the_o ecliptic_a which_o incompass_v the_o sphere_n which_o may_v be_v do_v out_o of_o a_o table_n of_o right_a ascension_n for_o place_v your_o sphere_n upon_o the_o centre_n of_o the_o quadration_n the_o back_n side_n put_v some_o pin_n through_o both_o their_o centre_n so_o that_o they_o may_v be_v keep_v both_o at_o a_o stay_n for_o drip_v one_o beside_o the_o other_o then_o turn_v the_o sphere_n about_o until_o the_o beginning_n of_o the_o hour_n come_v to_o the_o
it_o fall_v as_o be_v show_v in_o the_o 28_o chapter_n therefore_o if_o we_o take_v either_o half_a the_o perpendicular_a and_o the_o whole_a base_a or_o the_o whole_a perpendicular_a and_o half_a the_o base_a whereon_o it_o fall_v the_o one_o for_o the_o breadth_n and_o the_o other_o for_o the_o depth_n of_o a_o square_a solid_a we_o may_v measure_v it_o in_o all_o respect_n according_a to_o the_o 44_o chapter_n and_o that_o to_o any_o length_n require_v as_o for_o example_n let_v the_o triangle_n bcd_v be_v the_o base_a of_o a_o prisma_fw-la or_o piece_n of_o timber_n to_o be_v measure_v now_o take_v 10_o the_o whole_a perpendicular_a ab_fw-la for_o the_o breadth_n and_o 8_o the_o half_a base_a cd_o for_o the_o depth_n of_o a_o square_a prisma_fw-la or_o piece_n of_o timber_n we_o may_v find_v the_o solid_a content_n thereof_o by_o the_o 44_o chapter_n and_o that_o according_a to_o any_o length_n require_v chap._n xlvii_o how_o that_o prisma_fw-la or_o piece_n of_o timber_n who_o base_a or_o end_n be_v a_o rhombus_fw-la or_o diamond_n form_n may_v be_v measure_v a_o rhombus_fw-la or_o diamond_n be_v a_o figure_n of_o 4_o equal_a side_n but_o no_o right_a angle_n such_o as_o be_v the_o figure_n abcd._n now_o see_v every_o rhombus_fw-la be_v equal_a in_o their_o superficial_a content_n to_o that_o right_a angle_a parallelogram_n or_o long_o square_a who_o length_n be_v one_o of_o the_o side_n and_o breadth_n equal_a to_o the_o parallel_a distance_n of_o the_o same_o side_n if_o we_o take_v one_o of_o the_o side_n for_o the_o breadth_n and_o the_o length_n of_o the_o perpendicular_a fall_v thereon_o from_o the_o opposite_a angle_n for_o the_o depth_n of_o a_o square_a solid_a we_o may_v find_v the_o solid_a content_n thereof_o by_o the_o 44_o chapter_n as_o for_o example_n let_v the_o rhombus_fw-la abcd_v be_v the_o base_a or_o end_n of_o some_o piece_n of_o timber_n to_o be_v measure_v now_o take_v one_o of_o the_o side_n which_o be_v 10_o for_o the_o breadth_n of_o a_o square_a solid_a and_o the_o perpendicular_a which_o be_v something_o better_a than_o 8â…—_n for_o the_o depth_n of_o the_o same_o solid_a we_o may_v find_v the_o solid_a content_n thereof_o for_o any_o length_n require_v by_o the_o 44_o chapter_n chap._n xlviii_o how_o that_o prisma_fw-la or_o piece_n of_o timber_n who_o base_a or_o end_n be_v a_o rhomboide_n or_o diamond_n like_o may_v be_v measure_v a_o rhomboide_n or_o diamond_n like_o be_v a_o figure_n who_o opposite_a side_n and_o opposite_a angle_n be_v only_o equal_a and_o it_o have_v no_o right_a angle_n such_o as_o be_v the_o figure_n efgh_n now_o see_v that_o the_o rhomboide_n be_v also_o equal_a in_o his_o superficial_a content_n to_o that_o right_a angle_a parallelogram_n who_o length_n be_v one_o of_o the_o side_n and_o breadth_n equal_a to_o the_o parallel_a distance_n of_o the_o same_o side_n if_o we_o take_v one_o of_o the_o side_n for_o the_o breadth_n of_o a_o square_a solid_a and_o the_o length_n of_o the_o perpendicular_a fall_v thereon_o from_o the_o opposite_a angle_n for_o the_o depth_n of_o the_o same_o solid_a we_o may_v find_v the_o solid_a content_n thereof_o by_o the_o 45_o chapter_n let_v this_o rhomboide_n efgh_v be_v the_o base_a of_o a_o piece_n of_o timber_n to_o be_v measure_v now_o if_o you_o take_v 16_o the_o side_n gh_n or_o of_o for_o the_o breadth_n of_o a_o square_a solid_a and_o 10_o the_o length_n of_o the_o perpendicular_a el_n for_o the_o depth_n of_o the_o same_o solid_a you_o may_v find_v the_o solid_a content_n thereof_o by_o the_o 45_o chapter_n for_o any_o length_n require_v chap._n xlix_o how_o that_o prisma_fw-la or_o piece_n of_o timber_n who_o base_a or_o end_n be_v a_o trapeziam_fw-la may_v be_v measure_v a_o trapeziam_n be_v any_o irregular_a foureside_v figure_n of_o what_o fashion_n soever_o as_o this_o figure_n abcd_n be_v a_o trapeziam_fw-la and_o may_v be_v cast_v into_o two_o triangle_n by_o draw_v the_o diagonall_a line_n ac_fw-la and_o so_o each_o triangle_n measure_v as_o be_v before_o show_v which_o be_v do_v add_v the_o content_n of_o they_o both_o together_o &_o you_o shall_v have_v the_o content_a of_o the_o whole_a trapeziam_fw-la or_o when_o you_o have_v draw_v the_o diagonall_a line_n and_o from_o the_o two_o opposite_a angle_n let_v down_o the_o perpendicular_o thereon_o join_v the_o two_o perpendicular_o together_o and_o take_v the_o sum_n thereof_o for_o the_o breadth_n of_o a_o square_a solid_a and_o take_v one_o half_a the_o diagonall_a line_n for_o the_o depth_n of_o the_o same_o solid_a with_o which_o breadth_n and_o depth_n proceed_v by_o the_o 44_o chapter_n according_a to_o your_o length_n give_v and_o you_o shall_v thereby_o find_v the_o solid_a content_n require_v let_v the_o trapeziam_fw-la abcd_v be_v the_o base_a or_o end_n of_o a_o piece_n of_o timber_n now_o if_o you_o take_v the_o two_o perpendicular_o bf_n and_o de_fw-fr and_o join_v they_o together_o they_o will_v make_v 10_o for_o the_o breadth_n of_o a_o square_a solid_a and_o then_o take_v one_o half_a the_o diagonall_a ac_fw-la which_o will_v be_v 8_o for_o the_o depth_n of_o the_o same_o solid_a you_o may_v by_o the_o 44_o chapter_n find_v the_o solid_a content_n thereof_o according_a to_o any_o length_n give_v chap._n l._n how_o that_o prisma_fw-la or_o piece_n of_o timber_n who_o base_a or_o end_n have_v many_o side_n as_o 5_o 6_o 7_o 8_o 9_o 10_o or_o more_o so_o they_o be_v equal_a may_v be_v measure_v many_o side_v figure_n be_v those_o which_o have_v more_o side_n than_o four_o and_o be_v general_o call_v polygon_n as_o this_o figure_n a_o have_v equal_a side_n and_o angle_n be_v call_v a_o polygon_n every_o regular_a polygon_n be_v equal_a in_o the_o superficial_a content_n to_o that_o long_a square_n who_o length_n be_v equal_a to_o one_o half_a the_o perimiter_n and_o breadth_n to_o a_o perpendicular_a draw_v from_o the_o centre_n to_o the_o middle_n of_o any_o side_n thereof_o if_o we_o add_v all_o the_o side_n together_o and_o take_v half_a the_o sum_n for_o the_o breadth_n of_o a_o square_a solid_a and_o then_o take_v the_o perpendicular_a for_o the_o depth_n of_o the_o same_o solid_a we_o may_v by_o the_o 44_o chapter_n find_v the_o solid_a content_n thereof_o let_v the_o figure_n a_o be_v the_o base_a or_o end_n of_o a_o piece_n of_o timber_n to_o be_v measure_v consist_v of_o 5_o equal_a side_n and_o each_o side_n contain_v 12_o inch_n which_o be_v add_v together_o into_o one_o sum_n make_v 60_o the_o half_a whereof_o be_v 30_o for_o the_o breadth_n of_o your_o piece_n then_o take_v the_o length_n of_o the_o perpendicular_a fall_v from_o the_o centre_n a_o to_o the_o midst_n of_o one_o of_o the_o side_n which_o be_v 8_o inch_n for_o the_o depth_n of_o the_o same_o piece_n with_o which_o breadth_n and_o depth_n you_o may_v proceed_v to_o find_v the_o solid_a content_n by_o the_o 44_o chapter_n according_a to_o any_o length_n give_v a_o note_n to_o find_v the_o centre_n of_o those_o equiangle_n figure_n if_o the_o side_n be_v even_o the_o centre_n be_v find_v by_o draw_v line_n from_o one_o angle_n to_o his_o opposite_a angle_n but_o if_o they_o be_v odd_a it_o be_v find_v by_o draw_v line_n from_o the_o middle_n of_o a_o side_n to_o his_o opposite_a angle_n so_o shall_v the_o cut_n or_o intersection_n of_o those_o straight_a line_n show_v the_o place_n of_o the_o centre_n chap._n li._n have_v the_o circumference_n and_o diameter_n of_o a_o cylinder_n give_v in_o inch_n and_o the_o length_n in_o foot_n to_o find_v the_o content_a in_o foot_n as_o 12_o inch_n to_o the_o diameter_n in_o inch_n so_o be_v the_o length_n in_o foot_n to_o a_o four_o number_n then_o as_o 12_o inch_n to_o one_o quarter_n of_o the_o circumference_n so_o be_v this_o four_o number_n to_o the_o solid_a content_n in_o foot_n as_o for_o example_n let_v the_o figure_n a_o be_v a_o cylinder_n or_o some_o round_a piece_n of_o timber_n to_o be_v measure_v who_o circumference_n be_v 44_o inch_n and_o diameter_n 14_o inch_n and_o let_v the_o length_n thereof_o be_v 10_o foot_n now_o for_o to_o find_v the_o solid_a content_n first_o place_z the_o thread_n to_o the_o intersection_n of_o the_o contrary_a parallel_n of_o 12_o with_o the_o right_a parallel_n of_o 14_o and_o it_o will_v cut_v the_o contrary_a parallel_n of_o 10_o at_o the_o right_a parallel_n of_o 11â…”_n now_o where_o this_o right_a parallel_n of_o 11â…”_n cut_v the_o contrary_a parallel_n of_o 12_o there_o place_v the_o thread_n again_o so_o shall_v it_o cut_v the_o contrary_a parallel_n of_o 11_o at_o the_o right_a parallel_n of_o 10_o 7_o 10_o and_o so_o many_o solid_a foot_n be_v contain_v in_o this_o cylinder_n or_o round_a piece_n of_o timber_n chap._n lii_o have_v the_o
place_n where_o you_o will_v have_v the_o beginnng_a of_o ♈_o and_o there_o make_v stay_v of_o it_o and_o mark_v the_o beginning_n of_o ♈_o at_o the_o begin_n of_o the_o hour_n then_o look_v into_o your_o table_n of_o right_a ascension_n &_o see_v what_o hour_n stand_v against_o the_o beginning_n of_o ♉_o and_o you_o shall_v find_v 1_o hour_n 51_o min._n therefore_o number_v 1_o hour_n 51_o min._n among_o the_o hour_n backward_o towards_o the_o left_a hand_n &_o there_o place_v the_o begin_n of_o ♉_o the_o like_a may_v you_o do_v with_o the_o rest_n of_o the_o sign_n and_o every_o degree_n thereof_o remember_v that_o as_o you_o number_v the_o hour_n towards_o the_o right_a hand_n so_o must_v you_o number_v the_o sign_n towards_o the_o left_a then_o last_o of_o all_o prepare_v a_o thin_a narrow_a slip_n of_o brass_n and_o rivet_v it_o on_o with_o the_o sphere_n upon_o the_o quadrat_fw-la so_o that_o the_o sphere_n may_v turn_v about_o at_o all_o time_n when_o need_n shall_v require_v so_o likewise_o let_v it_o be_v rivet_v to_o the_o quadrat_fw-la at_o both_o end_n so_o that_o the_o upper_a edge_n may_v be_v just_a with_o the_o begin_n of_o the_o degree_n of_o altitude_n this_o be_v thus_o fasten_v shall_v serve_v for_o a_o horizon_n general_o for_o all_o latitude_n this_o horizontall_a line_n must_v be_v divide_v as_o the_o equator_fw-la and_o his_o axis_n so_o shall_v it_o natural_a show_v the_o amplitude_n in_o all_o latitude_n chap._n iii_o how_o to_o place_v the_o fix_a star_n in_o the_o sphere_n and_o also_o in_o the_o outward_a eclipticall_a circle_n for_o the_o speedy_a find_n of_o the_o hour_n of_o the_o night_n in_o all_o latitude_n first_o you_o must_v get_v both_o the_o right_a ascension_n and_o declination_n of_o such_o star_n as_o you_o intend_v to_o place_n in_o your_o instrument_n the_o find_v whereof_o shall_v hereafter_o appear_v these_o be_v know_v upon_o the_o star_n parallel_v of_o declination_n seek_v his_o right_a ascension_n among_o the_o meridian_n or_o hour_n circle_n and_o at_o the_o point_n of_o intersection_n betwixt_o the_o star_n declination_n and_o right_a ascension_n there_o place_v the_o star_n as_o for_o example_n suppose_v i_o will_v place_v arcturus_n in_o the_o sphere_n first_o i_o inquire_v his_o right_a ascension_n and_o declination_n either_o by_o the_o proposition_n hereafter_o for_o that_o purpose_n or_o by_o this_o ensue_a table_n and_o so_o find_v his_o right_a ascension_n to_o be_v 13_o hour_n 59_o minute_n and_o his_o declination_n to_o be_v 21_o degree_n 10_o minute_n north_n therefore_o upon_o the_o 21_o parallel_n and_o 10_o minute_n i_o number_v 14_o hour_n from_o ♈_o unto_o ♋_o and_o so_o back_o again_o unto_o 14_o and_o there_o i_o place_v arcturus_n the_o like_a may_v be_v do_v with_o any_o other_o star_n now_o what_o star_n you_o place_n in_o the_o sphere_n the_o same_o place_n in_o the_o outward_a ecliptic_a which_o may_v be_v do_v after_o this_o manner_n enter_v your_o table_n of_o right_a ascension_n with_o the_o right_a ascension_n of_o the_o star_n and_o see_v what_o degree_n of_o the_o ecliptic_a answer_n thereto_o and_o place_v the_o star_n upon_o the_o same_o as_o for_o example_n the_o right_a ascension_n of_o arcturus_n be_v 14_o hour_n i_o enter_v the_o table_n of_o right_a ascension_n therewith_o and_o i_o find_v 2_o degree_n of_o â™_o answer_v thereunto_o therefore_o arcturus_n must_v be_v place_v at_o 2_o degree_n of_o â™_o in_o the_o outward_a ecliptic_a the_o like_a may_v be_v do_v with_o any_o other_o of_o the_o fix_a star_n for_o readiness_n sake_n i_o have_v here_o insert_v a_o table_n of_o right_a ascension_n in_o hour_n and_o minute_n whereby_o the_o outward_a ecliptic_a may_v be_v divide_v and_o the_o star_n therein_o place_v and_o also_o a_o table_n of_o the_o right_a ascension_n and_o declination_n &_o magnitude_n of_o most_o of_o the_o principal_a fix_a star_n rectify_v for_o this_o present_a year_n 1652_o a_o table_n of_o right_a ascension_n for_o every_o degree_n of_o the_o ecliptic_a in_o hour_n and_o minute_n degr._fw-la ♈_o ♉_o ♊_o ♋_o ♌_o â™_o ♎_o â™_o â™_o ♑_o â™’_o ♓_o h._n m._n h._n m._n h._n m._n h._n m._n h._n m._n h._n m._n h._n m._n h._n m._n h._n m._n h._n m._n h._n m._n h._n m._n 1_o 00_o 04_o 01_o 55_o 03_o 55_o 06_o 04_o 08_o 13_o 10_o 12_o 12_o 04_o 13_o 55_o 15_o 55_o 18_o 04_o 20_o 13_o 22_o 12_o 2_o 00_o 07_o 02_o 00_o 04_o 00_o 06_o 09_o 08_o 17_o 10_o 16_o 12_o 07_o 14_o 00_o 16_o 00_o 18_o 09_o 20_o 17_o 22_o 16_o 3_o 00_o 11_o 02_o 03_o 04_o 04_o 06_o 13_o 08_o 21_o 10_o 20_o 12_o 11_o 14_o 03_o 16_o 04_o 18_o 13_o 20_o 21_o 22_o 20_o 4_o 00_o 15_o 02_o 07_o 04_o 08_o 06_o 17_o 08_o 25_o 10_o 24_o 12_o 15_o 14_o 07_o 16_o 08_o 18_o 17_o 20_o 25_o 22_o 24_o 5_o 00_o 18_o 02_o 11_o 04_o 12_o 06_o 22_o 08_o 29_o 10_o 27_o 12_o 18_o 14_o 11_o 16_o 12_o 18_o 22_o 20_o 29_o 22_o 27_o 6_o 00_o 22_o 02_o 15_o 04_o 16_o 06_o 26_o 08_o 34_o 10_o 31_o 12_o 22_o 14_o 15_o 16_o 16_o 18_o 26_o 20_o 34_o 22_o 31_o 7_o 00_o 26_o 02_o 19_o 04_o 20_o 06_o 30_o 08_o 38_o 10_o 35_o 12_o 26_o 14_o 19_o 16_o 20_o 18_o 30_o 20_o 38_o 22_o 35_o 8_o 00_o 29_o 02_o 22_o 04_o 24_o 06_o 35_o 08_o 42_o 10_o 39_o 12_o 29_o 14_o 22_o 16_o 24_o 18_o 35_o 20_o 42_o 22_o 39_o 9_o 00_o 33_o 02_o 26_o 04_o 29_o 06_o 39_o 08_o 46_o 10_o 42_o 12_o 33_o 14_o 26_o 16_o 29_o 18_o 39_o 20_o 46_o 22_o 4â—_n 10_o 00_o 37_o 02_o 30_o 04_o 33_o 06_o 43_o 08_o 50_o 10_o 46_o 12_o 37_o 14_o 30_o 16_o 33_o 18_o 43_o 20_o 50_o 22_o 46_o 11_o 00_o 40_o 02_o 34_o 04_o 38_o 06_o 48_o 08_o 54_o 10_o 50_o 12_o 40_o 14_o 34_o 16_o 38_o 18_o 48_o 20_o 54_o 22_o 50_o 12_o 00_o 44_o 02_o 38_o 04_o 42_o 06_o 52_o 08_o 58_o 10_o 54_o 12_o 44_o 14_o 38_o 16_o 42_o 18_o 52_o 20_o 58_o 22_o 54_o 13_o 00_o 48_o 02_o 42_o 04_o 46_o 06_o 56_o 09_o 02_o 10_o 57_o 12_o 48_o 14_o 42_o 16_o 46_o 18_o 56_o 21_o 02_o 22_o 57_o 14_o 00_o 52_o 02_o 46_o 04_o 50_o 07_o 01_o 09_o 06_o 11_o 01_o 12_o 52_o 14_o 46_o 16_o 50_o 19_o 01_o 21_o 06_o 23_o 01_o 15_o 00_o 55_o 02_o 50_o 04_o 55_o 07_o 05_o 09_o 10_o 11_o 05_o 12_o 55_o 14_o 50_o 16_o 55_o 19_o 05_o 21_o 10_o 23_o 05_o 16_o 00_o 59_o 02_o 54_o 04_o 59_o 07_o 09_o 09_o 14_o 11_o 08_o 12_o 59_o 14_o 54_o 16_o 59_o 19_o 09_o 21_o 14_o 23_o 08_o 17_o 01_o 03_o 02_o 58_o 05_o 04_o 07_o 14_o 09_o 18_o 11_o 12_o 13_o 03_o 14_o 58_o 17_o 04_o 19_o 14_o 21_o 18_o 23_o 12_o 18_o 01_o 06_o 03_o 02_o 05_o 08_o 07_o 18_o 09_o 22_o 11_o 16_o 13_o 06_o 15_o 02_o 17_o 08_o 19_o 18_o 21_o 22_o 23_o 16_o 19_o 01_o 10_o 03_o 06_o 05_o 12_o 07_o 22_o 09_o 26_o 11_o 20_o 13_o 10_o 15_o 06_o 17_o 12_o 19_o 22_o 21_o 26_o 23_o 20_o 20_o 01_o 14_o 03_o 10_o 05_o 16_o 07_o 27_o 09_o 30_o 11_o 23_o 13_o 14_o 15_o 10_o 17_o 16_o 19_o 27_o 21_o 30_o 23_o 23_o 21_o 01_o 18_o 03_o 14_o 05_o 21_o 07_o 31_o 09_o 34_o 11_o 27_o 13_o 18_o 15_o 14_o 17_o 21_o 19_o 31_o 21_o 34_o 23_o 27_o 22_o 01_o 21_o 03_o 18_o 05_o 25_o 07_o 35_o 09_o 37_o 11_o 31_o 13_o 21_o 15_o 18_o 17_o 25_o 19_o 35_o 21_o 37_o 23_o 31_o 23_o 01_o 25_o 03_o 22_o 05_o 30_o 07_o 39_o 09_o 41_o 11_o 34_o 13_o 25_o 15_o 22_o 17_o 30_o 19_o 39_o 21_o 41_o 23_o 34_o 24_o 01_o 29_o 03_o 26_o 05_o 34_o 07_o 44_o 09_o 45_o 11_o 38_o 13_o 29_o 15_o 26_o 17_o 34_o 19_o 44_o 21_o 45_o 23_o 38_o 25_o 01_o 33_o 03_o 31_o 05_o 38_o 07_o 48_o 09_o 49_o 11_o 42_o 13_o 33_o 15_o 31_o 17_o 38_o 19_o 48_o 21_o 49_o 23_o 42_o 26_o 01_o 36_o 03_o 35_o 05_o 42_o 07_o 52_o 09_o 53_o 11_o 49_o 13_o 36_o 15_o 35_o 17_o 42_o 19_o 52_o 21_o 53_o 23_o 45_o 27_o 01_o 40_o 03_o 39_o 05_o 47_o 07_o 56_o 09_o 57_o 11_o 49_o 13_o 40_o 15_o 39_o 17_o 47_o 19_o 56_o 21_o 57_o 23_o 49_o 28_o 01_o 44_o 03_o 43_o 05_o 51_o 08_o 00_o 10_o 01_o 11_o 53_o 13_o 44_o 15_o 43_o 17_o 51_o 20_o 00_o 22_o 01_o 23_o 53_o 29_o 01_o 48_o 03_o 47_o 05_o 56_o 08_o 05_o 10_o 05_o 11_o 56_o 13_o 48_o 15_o 47_o 17_o 56_o 20_o 05_o 22_o 05_o 23_o 56_o 30_o 01_o 52_o 03_o 51_o 06_o 00_o 08_o 09_o 10_o 08_o 12_o 00_o 13_o 52_o 15_o 51_o 18_o 00_o 20_o 09_o 22_o 08_o 24_o 00_o a_o table_n of_o the_o right_a ascension_n
declination_n and_o magnitude_n of_o some_o of_o the_o principal_a fix_a star_n rectify_v for_o this_o present_a year_n 1652._o the_o name_n of_o the_o star_n right_o ascension_n declination_n magnitude_n h._n m._n d._n m._n  _fw-fr the_o pole_n star_n 00_o 32_o 87_o 19_o n_n 2_o andromeda_n girdle_n 00_o 52_o 33_o 48_o n_n 2_o the_o ram_n horn_n 01_o 34_o 17_o 27_o n_n 3_o the_o ram_n head_n 01_o 48_o 21_o 44_o n_n 3_o the_o head_n of_o mednsa_n 02_o 46_o 39_o 32_o n_n 3_o aldebaran_n the_o bull_n eye_n 04_o 16_o 15_o 48_o n_n 1_o hircus_fw-la the_o goat_n 04_o 51_o 45_o 36_o n_n 1_o orion_n left_a foot_n 04_o 58_o 08_o 40_o s_n 1_o orion_n left_a shoulder_n 05_o 06_o 05_o 59_o n_n 2_o the_o first_o in_o orion_n girdle_n 05_o 14_o 00_o 38_o s_n 2_o orion_n right_a shoulder_n 05_o 36_o 07_o 17_o n_n 2_o sirius_n the_o great_a dog_n 06_o 31_o 16_o 13_o s_n 1_o procyon_n the_o lesser_a dog_n 07_o 21_o 06_o 07_o n_n 2_o the_o heart_n of_o hydra_n 09_o 11_o 07_o 09_o s_n 1_o cor_fw-la leonis_fw-la the_o lion_n heart_n 09_o 49_o 13_o 42_o n_n 1_o the_o lion_n neck_n 09_o 50_o 21_o 49_o n_n 2_o the_o lion_n back_o 10_o 54_o 22_o 33_o n_n 2_o the_o foremost_a star_n in_o □_o of_o the_o great_a bear_n 10_o 40_o 58_o 23_o n_n 2_o the_o lion_n tail_n 11_o 30_o 16_o 34_o n_n 1_o the_o root_n of_o the_o great_a bear_n tail_n alio_fw-la tail_n 12_o 38_o 57_o 51_o n_n 2_o the_o virgin_n spike_n 13_o 08_o 09_o 16_o s_n 1_o the_o middlemost_a star_n in_o the_o great_a bear_n 13_o 10_o 56_o 45_o n_n 2_o the_o last_o of_o the_o say_a tail_n 13_o 26_o 51_o 05_o n_n 2_o arcturus_n 13_o 59_o 21_o 10_o n_n 1_o the_o bright_a star_n of_o the_o crown_n 15_o 24_o 27_o 43_o n_n 3_o antare_n the_o scorpion_n heart_n 16_o 08_o 25_o 34_o s_n 1_o the_o bright_a star_n of_o the_o harp_n 18_o 25_o 38_o 30_o n_n 1_o the_o eagle_n heart_n 19_o 33_o 07_o 58_o n_n 2_o the_o bright_a star_n of_o the_o vulture_n 19_o 34_o 08_o 01_o n_n 1_o fomahant_n 22_o 38_o 31_o 25_o s_n 1_o pegasus_n shoulder_n 22_o 48_o 13_o 14_o n_n 2_o pegasus_n leg_n 22_o 52_o 26_o 09_o n_n 2_o chap._n iv_o to_o show_v the_o general_n use_v of_o the_o universal_a quadrat_fw-la in_o the_o use_n of_o this_o quadrat_fw-la let_v we_o for_o distinction_n sake_n call_v those_o parallel_n which_o make_v right_a angle_n with_o the_o line_n whereon_o the_o sight_n stand_v right_a parallel_n and_o the_o other_o which_o run_v parallel_v to_o the_o same_o line_n contrary_a parallel_n the_o one_o signify_v right_o shadow_n the_o other_o contrary_a shadow_n the_o general_n use_v whereof_o consist_v in_o the_o solution_n of_o the_o golden_a rule_n where_o three_o line_n or_o number_n be_v give_v of_o a_o know_a denomination_n a_o four_o proportional_a be_v to_o be_v find_v and_o this_o solution_n be_v diverse_a in_o regard_n of_o the_o signification_n of_o these_o parallel_a line_n and_o also_o of_o the_o entrance_n into_o the_o work_n the_o signification_n of_o these_o parallel_n be_v sometime_o simple_a and_o sometime_o compound_v sometime_o they_o signify_v right_a line_n or_o number_n only_o such_o as_o they_o be_v simple_a in_o themselves_o and_o sometime_o they_o signify_v natural_a sin_n which_o be_v half_a the_o chord_n of_o the_o double_a arch_n and_o sometime_o natural_a tangent_n which_o be_v perpendicular_o at_o the_o end_n of_o the_o diameter_n sometime_o the_o work_n may_v be_v begin_v with_o one_o kind_n of_o signification_n and_o end_v with_o another_o it_o may_v be_v begin_v with_o the_o line_n as_o they_o be_v in_o themselves_o and_o end_v as_o they_o be_v sin_n it_o may_v begin_v as_o they_o be_v sin_n and_o end_n as_o they_o be_v tangent_n the_o solution_n in_o regard_n of_o the_o entrance_n into_o the_o work_n may_v be_v either_o right_a or_o contrary_n i_o call_v that_o right_a entrance_n or_o entrance_n on_o the_o right_a parallel_n when_o the_o two_o line_n or_o number_n of_o the_o first_o denomination_n be_v apply_v or_o find_v in_o the_o right_a parallel_n and_o the_o three_o line_n or_o number_n and_o that_o which_o be_v seek_v for_o on_o the_o contrary_a parallel_n and_o on_o the_o contrary_a i_o call_v that_o contrary_a entrance_n when_o the_o two_o line_n or_o number_n of_o the_o first_o denomination_n be_v apply_v or_o find_v on_o the_o parallel_n of_o contrary_a shadow_n and_o the_o three_o line_n or_o number_n and_o that_o which_o be_v to_o be_v find_v out_o in_o the_o parallel_n of_o right_a shadow_n as_o for_o example_n let_v there_o be_v give_v three_o number_n 50._o 60_o and_o 70._o unto_o which_o i_o be_o to_o find_v a_o four_o proportional_a let_v the_o question_n be_v this_o if_o 50_o pound_n gain_v 60_o shilling_n what_o shall_v 70_o pound_n gain_v here_o be_v number_n of_o two_o denomination_n one_o of_o pound_n the_o other_o of_o shilling_n and_o the_o first_o with_o which_o i_o be_o to_o enter_v must_v be_v that_o of_o 50._o pound_n if_o then_o i_o will_v enter_v my_o question_n upon_o a_o parallel_n of_o right_a shadow_n see_v where_o the_o right_a parallel_n of_o my_o first_o number_n which_o be_v 50_o cut_v the_o contrary_a parallel_n of_o the_o second_o which_o be_v 60_o and_o upon_o the_o point_n of_o intersection_n there_o lie_v the_o thread_n the_o thread_n lie_v in_o this_o position_n look_v along_o upon_o the_o right_a parallel_n of_o the_o three_o number_n which_o be_v 70_o and_o see_v what_o contrary_a parallel_n cut_v it_o direct_o under_o the_o thread_n for_o that_o be_v the_o four_o proportional_a number_n require_v which_o here_o be_v find_v to_o be_v 84_o wherefore_o i_o conclude_v that_o if_o 50_o pound_n gain_n 60_o shilling_n 70_o pound_n will_v gain_v 84_o shilling_n for_o as_o 50_o be_v to_o 60_o so_o be_v 70_o to_o 84_o the_o like_a will_v appear_v if_o you_o enter_v your_o work_n upon_o the_o contrary_a parallel_n for_o it_o matter_v not_o on_o which_o kind_n of_o parallel_n you_o enter_v so_o be_v it_o you_o enter_v your_o first_o and_o three_o number_n on_o the_o same_o kind_n of_o parallel_n so_o shall_v the_o three_o and_o four_o be_v of_o the_o other_o kind_n thus_o much_o for_o the_o general_n use_v of_o this_o fore-side_n of_o the_o universal_a quadrat_fw-la which_o be_v consider_v and_o well_o understand_v there_o be_v nothing_o hard_a in_o that_o which_o follow_v chap._n v._o to_o divide_v a_o line_n give_v into_o any_o number_n of_o equal_a part_n let_v the_o line_n give_v be_v ab_fw-la and_o let_v it_o be_v require_v to_o divide_v the_o same_o into_o 5_o equal_a part_n take_v with_o your_o compass_n the_o give_v line_n ab_fw-la and_o lay_v it_o down_o upon_o any_o parallel_n who_o number_n may_v be_v equal_o divide_v into_o five_o part_n without_o any_o remainder_n as_o lay_v it_o upon_o the_o five_o parallel_n and_o at_o the_o point_n where_o the_o line_n end_v there_o lay_v the_o thread_n the_o thread_n lie_v in_o this_o position_n and_o in_o regard_n that_o 10_o be_v ⅕_n part_n of_o 50_o therefore_o the_o segment_n of_o the_o ten_o parallel_n which_o be_v cut_v by_o the_o thread_n shall_v be_v the_o distance_n which_o shall_v divide_v the_o give_v line_n into_o 5_o equal_a part_n as_o for_o example_n take_v the_o give_v line_n ab_fw-la and_o lay_v it_o upon_o the_o fifty_o contrary_a parallel_n as_o from_o a_o to_o b_o &_o see_v the_o give_v line_n end_v in_o b_o lay_v the_o thread_n upon_o the_o same_o point_n as_o here_o you_o see_v in_o the_o follow_a figure_n then_o in_o regard_n that_o 10_o be_v ⅕_n of_o 50_o therefore_o i_o take_v the_o segment_n cut_v by_o the_o thread_n upon_o the_o ten_o parallel_n viz._n cd_o and_o with_o it_o i_o divide_v the_o give_v line_n ab_fw-la into_o 5_o equal_a part_n as_o here_o you_o see_v the_o fore-side_n of_o the_o quadrat_fw-la chap._n vi_o to_o increase_v a_o line_n in_o a_o give_v proportion_n take_v the_o line_n give_v with_o a_o pair_n of_o compass_n and_o lay_v it_o one_o the_o parallel_n of_o the_o number_n give_v and_o upon_o the_o point_n where_o the_o give_v line_n end_v there_o lay_v the_o thread_n which_o be_v keep_v in_o this_o position_n the_o segment_n of_o the_o parallel_n of_o the_o number_n require_v intercept_v by_o the_o thread_n shall_v give_v the_o line_n require_v let_v a_o be_v a_o line_n give_v to_o be_v increase_v in_o the_o proportion_n of_o 3_o to_o 5._o first_o i_o take_v the_o give_v line_n a_o with_o my_o compass_n and_o in_o regard_n that_o 30_o and_o 50_o be_v the_o same_o with_o 3_o and_o 5_o if_o from_o they_o you_o give_v o_o therefore_o i_o take_v the_o give_v line_n a_o and_o lay_v it_o upon_o the_o thirty_o parallel_n
as_o from_o f_o to_o g_z in_o the_o last_o chapter_n and_o upon_o ay_o the_o point_n where_o the_o line_n end_v there_o i_o lay_v the_o thread_n which_o lie_v in_o this_o position_n cut_v the_o parallel_n of_o 50_o in_o the_o point_n b_o so_o shall_v ab_fw-la be_v the_o length_n of_o the_o line_n b_o which_o be_v require_v chap._n vii_o to_o diminish_v a_o line_n in_o a_o give_v proportion_n to_o diminish_v a_o line_n in_o a_o give_v proportion_n be_v no_o other_o than_o that_o which_o be_v show_v in_o the_o last_o chapter_n but_o only_o whereas_o there_o you_o enter_v with_o the_o lesser_a line_n which_o be_v to_o be_v increase_v upon_o the_o lesser_a parallel_n number_n and_o so_o the_o line_n require_v be_v find_v upon_o the_o great_a parallel_n number_n so_o here_o you_o must_v enter_v with_o the_o great_a line_n which_o be_v to_o be_v diminish_v upon_o the_o great_a parallel_n number_n and_o that_o which_o be_v require_v shall_v be_v find_v upon_o the_o lesser_a parallel_n number_n let_v a_o be_v a_o line_n give_v to_o be_v diminish_v in_o the_o proportion_n of_o 5_o to_o 3_o i_o take_v the_o give_v line_n b_o and_o lay_v it_o upon_o the_o fifty_o parallel_n and_o at_o the_o point_n where_o the_o line_n end_v i_o lay_v the_o thread_n as_o at_o the_o point_n b_o in_o the_o figure_n of_o the_o five_o chapter_n the_o thread_n lie_v in_o this_o position_n i_o see_v it_o cut_v the_o thirty_o parallel_n in_o the_o point_n g_o so_o shall_v the_o distance_n fg_n give_v i_o the_o line_n a_o which_o be_v require_v and_o be_v ⅗_n of_o the_o give_v line_n b._n here_o note_v that_o you_o may_v use_v other_o number_n to_o work_v your_o proportion_n upon_o beside_o these_o for_o you_o may_v multiply_v or_o divide_v the_o proportional_a number_n give_v either_o by_o 2._o 3_o or_o 4_o etc._n etc._n and_o so_o work_v by_o their_o number_n as_o for_o 3_o and_o 5_o we_o may_v work_v by_o 6_o and_o 10_o or_o else_o by_o 9_o and_o 15_o or_o by_o 12_o and_o 20_o or_o 15_o and_o 25_o or_o 18_o and_o 30_o or_o 24_o and_o 40_o or_o 30_o and_o 50_o as_o here_o we_o have_v do_v chap._n viii_o to_o find_v a_o proportion_n betwixt_o two_o or_o more_o right_a line_n take_v the_o great_a line_n betwixt_o your_o compass_n and_o extend_v it_o upon_o that_o parallel_n who_o number_n you_o intend_v to_o make_v the_o denominator_fw-la of_o the_o fraction_n at_o the_o end_n of_o which_o extent_n place_n the_o thread_n the_o thread_n rest_v in_o this_o position_n take_v the_o lesser_a line_n betwixt_o your_o compass_n and_o keep_v one_o foot_n in_o the_o side_n of_o your_o quadrat_fw-la carry_v it_o parallel_v unto_o tthe_n great_a line_n until_o the_o other_o foot_n touch_v the_o thread_n which_o you_o may_v very_o well_o do_v by_o help_n of_o the_o parallel_n so_o shall_v the_o number_n of_o that_o parallel_n whereon_o the_o compass_n shall_v stay_v be_v the_o numerator_n to_o the_o former_a denominator_fw-la let_v the_o line_n give_v be_v ab_fw-la and_o cd_o first_o i_o take_v the_o line_n cd_o and_o place_n it_o upon_o the_o fifty_o parallel_n as_o from_o a_o to_o b_o upon_o which_o point_n b_o i_o lay_v the_o thread_n which_o be_v keep_v in_o this_o position_n i_o take_v the_o lesser_a line_n ab_fw-la betwixt_o my_o compass_n and_o keep_v one_o foot_n in_o the_o side_n of_o the_o quadrat_fw-la eh_fw-it i_o carry_v they_o parallel_v to_o the_o great_a line_n until_o the_o other_o foot_n touch_v the_o thread_n at_o g_o so_o shall_v the_o fix_a foot_n rest_n in_o the_o parallel_n of_o 30_o which_o be_v the_o numerator_n to_o the_o former_a denominator_fw-la wherefore_o i_o conclude_v that_o the_o line_n ab_fw-la be_v 30_o 50_o or_o ⅗_n of_o the_o line_n cd_o or_o the_o line_n ab_fw-la be_v in_o such_o proportion_n to_o the_o line_n cd_o as_o 30_o to_o 50_o which_o be_v require_v but_o if_o the_o line_n cd_o be_v great_a than_o these_o parallel_n can_v contain_v take_v ½_n thereof_o and_o enter_v with_o the_o same_o as_o before_o you_o do_v with_o the_o whole_a line_n so_o likewise_o take_v ½_n the_o lesser_a line_n and_o carry_v it_o parallel_v as_o before_o you_o do_v the_o whole_a line_n and_o you_o shall_v produce_v the_o same_o proportion_n as_o before_o chap._n ix_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 first_o lay_v your_o thread_n by_o chance_n upon_o the_o quadrat_fw-la so_o as_o it_o may_v cut_v the_o parallel_n of_o all_o the_o number_n give_v so_o shall_v the_o thread_n cut_v off_o from_o each_o parallel_n the_o proportional_a line_n require_v which_o may_v be_v take_v with_o your_o compass_n and_o lay_v down_o present_o as_o for_o example_n let_v it_o be_v require_v to_o lay_v down_o four_o straight_a line_n which_o shall_v bear_v such_o proportion_n one_o to_o the_o other_o as_o four_o give_v number_n which_o let_v be_v these_o 60_o 50_o 30_o 10._o then_o open_v the_o thread_n by_o chance_n so_o as_o it_o may_v cut_v the_o parallel_n of_o all_o these_o give_v number_n so_o shall_v the_o thread_n cut_v from_o each_o parallel_n the_o proportional_a line_n require_v which_o you_o may_v take_v from_o your_o quadrat_fw-la and_o lay_v they_o down_o as_o here_o you_o see_v viz._n from_o the_o parallel_n of_o 60_o the_o line_n kl_n from_o the_o parallel_n of_o 50_o the_o line_n ab_fw-la from_o the_o parallel_n of_o 30_o the_o line_n fg_v and_o from_o the_o parallel_n 10_o the_o line_n cd_o chap._n x._o have_v two_o line_n give_v to_o find_v a_o three_o proportional_a line_n to_o they_o first_o place_v both_o the_o line_n give_v on_o the_o contrary_a side_n of_o the_o quadrat_fw-la and_o mark_v how_o many_o parallel_n of_o contrary_a shadow_n they_o do_v extend_v unto_o then_o take_v the_o second_o line_n again_o and_o place_n it_o upon_o the_o right_a side_n of_o the_o quadrat_fw-la and_o note_v the_o extension_n among_o the_o parallel_n of_o right_a shadow_n now_o see_v where_o the_o contrary_a parallel_n of_o the_o first_o line_n cut_v the_o right_a parallel_n of_o the_o second_o line_n and_o there_o lie_v the_o thread_n the_o thread_n lie_v in_o this_o position_n see_v what_o right_a parallel_n be_v cut_v by_o the_o contrary_a parallel_n of_o the_o second_o line_n direct_o under_o the_o thread_n for_o that_o be_v the_o three_o proportional_a line_n require_v let_v the_o two_o line_n give_v be_v eo_fw-la and_o of_o and_o it_o be_v require_v to_o find_v a_o three_o line_n which_o shall_v be_v in_o such_o proportion_n to_o of_o as_o of_o be_v to_o eo_fw-la first_o place_v the_o two_o line_n upon_o the_o contrary_a side_n of_o the_o quadrat_fw-la and_o you_o shall_v find_v the_o line_n eo_fw-la to_o extend_v unto_o 20_o and_o of_o unto_o 30_o now_o extend_v of_o on_o the_o right_a parallel_n also_o which_o will_v be_v likewise_o unto_o 30_o therefore_o where_o the_o contrary_a parallel_n of_o eo_fw-la 20_o cut_v the_o right_a parallel_n of_o of_o 30_o there_o lay_v the_o thread_n as_o at_o p_o the_o thread_n lie_v still_o in_o this_o position_n i_o see_v where_o the_o contrary_a parallel_n of_o of_o 30_o cut_v the_o thread_n which_o be_v at_o g_o upon_o the_o right_a parallel_n of_o 45_o therefore_o fg_v shall_v be_v the_o three_o proportional_a line_n require_v and_o shall_v be_v 45_o of_o such_o division_n as_o eo_fw-la be_v 20_o and_o of_o 30._o to_o perform_v the_o same_o more_o speedy_o with_o the_o number_n for_o as_o 20_o be_v to_o 30_o so_o be_v 30_o to_o a_o three_o number_n therefore_o at_o the_o intersection_n of_o the_o twenty_o contrary_a parallel_n with_o the_o thirty_o right_a parallel_n there_o lay_v the_o thread_n which_o be_v keep_v in_o this_o position_n see_v where_o the_o thirty_o contrary_a parallel_n cut_v the_o thread_n which_o will_v be_v at_o g_o upon_o the_o 45_o parallel_n of_o right_a shadow_n which_o 45._o be_v the_o three_o proportional_a number_n as_o before_o chap._n xi_o have_v three_o line_n give_v to_o find_v a_o four_o proportional_a to_o they_o here_o the_o first_o line_n and_o the_o three_o be_v to_o be_v place_v on_o the_o contrary_a parallel_n and_o the_o second_o on_o the_o right_a parallel_n therefore_o at_o the_o intersection_n of_o the_o contrary_a parallel_n of_o the_o first_o line_n with_o the_o right_a parallel_n of_o the_o second_o there_o lay_v the_o thread_n the_o thread_n lie_v still_o in_o this_o position_n whese_fw-mi the_o contrary_a parallel_n of_o the_o three_o line_n cut_v the_o thread_n there_o be_v the_o right_a parallel_n of_o the_o four_o proportional_a line_n let_v the_o three_o line_n give_v be_v eo_fw-la open_v and_o eke_o and_o let_v it_o be_v require_v to_o find_v a_o four_o line_n which_o shall_v have_v such_o proportion_n to_o eke_o as_o
to_o a_o four_o number_n and_o this_o four_o number_n to_o the_o solid_a content_n in_o foot_n wherefore_o if_o you_o apply_v the_o thread_n to_o the_o intersection_n of_o the_o contrary_a parallel_n of_o 12_o with_o the_o right_a parallel_n of_o 2_o foot_n the_o three_o part_n of_o the_o altitude_n it_o will_v cut_v the_o contrary_a parallel_n of_o 36_o inch_n the_o side_n of_o the_o base_a at_o the_o right_a parallel_n of_o 6_o the_o thread_n lie_v still_o in_o the_o same_o position_n will_v cut_v the_o contrary_a parallel_n of_o this_o 6_o at_o the_o right_a parallel_n of_o 18_o and_o such_o be_v the_o solid_a content_n of_o this_o pyramid_n in_o foot_n which_o be_v require_v now_o if_o you_o will_v measure_v any_o segment_n of_o a_o pyramid_n be_v cut_v off_o observe_v the_o former_a rule_n give_v for_o the_o measure_n of_o the_o segment_n of_o a_o cone_n and_o you_o shall_v soon_o have_v your_o desire_n for_o have_v the_o side_n of_o the_o segment_n of_o the_o pyramid_n with_o the_o semidiameter_n of_o the_o circumscribe_v circle_n at_o each_o base_a you_o may_v by_o the_o 61_o chapter_n find_v the_o altitude_n of_o the_o whole_a great_a pyramid_n as_o if_o it_o have_v not_o be_v cut_v off_o and_o also_o of_o the_o lesser_a pyramid_n which_o be_v suppose_v to_o be_v add_v to_o the_o segment_n to_o make_v it_o a_o perfect_a pyramid_n this_o be_v do_v you_o may_v by_o the_o rule_v deliver_v first_o get_v the_o content_a of_o the_o whole_a great_a pyramid_n and_o then_o of_o the_o lesser_a add_v pyramid_n and_o take_v the_o lesser_a out_o of_o the_o great_a the_o remainder_n will_v be_v the_o content_a of_o the_o give_v segment_n as_o be_v require_v the_o end_n of_o the_o first_o book_n the_o second_o book_n show_v the_o most_o plentiful_a easy_a and_o speedy_a use_n of_o the_o universal_a quadrat_fw-la in_o the_o resolution_n of_o 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 upon_o the_o instrument_n with_o surpass_a facility_n and_o with_o the_o least_o intricacie_n that_o may_v be_v chap._n i._o to_o find_v the_o chord_n of_o any_o arch_n the_o radius_fw-la be_v give_v not_o exceed_v the_o side_n of_o the_o quadrat_fw-la the_o radius_fw-la of_o a_o circle_n be_v the_o semidiameter_n of_o the_o same_o circle_n or_o the_o total_a sine_fw-la or_o sine_fw-la of_o 90_o degree_n so_o ae_n be_v the_o semidiameter_n or_o radius_fw-la upon_o which_o the_o divide_a arch_n eglantine_n be_v dramn_v and_o ab_fw-la be_v the_o radius_fw-la upon_o which_o the_o arch_a bfc_n be_v draw_v and_o a_o chord_n be_v a_o right_a line_n subtend_v a_o arch_n so_o bc_n be_v the_o chord_n of_o the_o arch_a bfc_n for_o the_o find_n of_o which_o chord_n for_o any_o arch_n require_v place_v the_o radius_fw-la give_v upon_o the_o side_n of_o the_o quadrat_fw-la from_o the_o centre_n and_o note_v the_o point_n where_o the_o same_o end_v and_o there_o place_v the_o bead_n then_o open_v the_o thread_n to_o the_o arch_n give_v count_v in_o the_o quadrant_a the_o distance_n between_o the_o point_n note_v in_o the_o side_n of_o the_o quadrat_fw-la and_o the_o head_n shall_v be_v the_o chord_n require_v as_o for_o example_n let_v ab_fw-la be_v the_o radius_fw-la of_o a_o circle_n place_v upon_o the_o side_n of_o the_o quadrat_fw-la and_o let_v it_o be_v require_v to_o find_v a_o chord_n of_o 40_o degree_n first_o i_o place_v the_o bead_n at_o the_o point_n b_o where_o the_o give_v radius_fw-la end_v and_o open_v the_o thread_n to_o 40_o degree_n in_o the_o quadrant_a i_o take_v the_o distance_n betwixt_o the_o point_n b_o and_o the_o bead_n which_o be_v the_o chord_n of_o 40_o degree_n viz._n the_o chord_n of_o the_o arch_a bfc_n draw_v upon_o the_o give_v radius_fw-la ab_fw-la as_o be_v require_v and_o thus_o may_v you_o find_v the_o chord_n of_o any_o arch_n as_o well_o as_o upon_o the_o sector_n chap._n ii_o to_o find_v the_o right_n sine_fw-la of_o any_o arch_n give_v the_o radius_fw-la be_v put_v 1000_o have_v show_v the_o use_n of_o these_o parallel_a line_n as_o they_o stand_v in_o their_o own_o proper_a signification_n it_o rest_v now_o to_o show_v the_o use_n of_o they_o as_o they_o signify_v right_a fine_n and_o tangent_n and_o first_o to_o find_v the_o right_n fine_a of_o any_o arch_n seek_v the_o degree_n give_v upon_o the_o quadrant_a count_v from_o any_o side_n of_o the_o quadrant_a but_o here_o let_v it_o be_v from_o the_o begin_n of_o the_o degree_n at_o e_o so_o shall_v the_o right_a parallel_n run_v through_o the_o degree_n give_v in_o the_o quadrant_a give_v the_o number_n of_o the_o fine_a require_v but_o if_o you_o will_v have_v the_o fine_a in_o a_o right_a line_n take_v the_o length_n of_o the_o contrary_a parallel_n from_o the_o side_n of_o the_o quadrat_fw-la ae_n to_o the_o degree_n give_v in_o the_o quadrat_fw-la and_o that_o shall_v be_v your_o desire_a sine_z whereof_o the_o whole_a side_n be_v radius_fw-la as_o for_o example_n let_v it_o be_v require_v to_o find_v the_o right_n sine_fw-la of_o 53_o degree_n 8_o minute_n i_o look_v 53_o degree_n 8_o minute_n in_o the_o quadrant_a count_v from_o the_o begin_n of_o the_o degree_n at_o e_o unto_z o_o through_o which_o point_n o_o rune_v the_o right_a parallel_n of_o 800_o this_o 800_o be_v the_o right_a sine_fw-la of_o 53_o degree_n 8_o minute_n 100_o be_v radius_fw-la but_o if_o you_o will_v have_v this_o sine_fw-la in_o a_o right_a line_n take_v the_o length_n of_o the_o contrary_a paaallel_n from_o the_o point_n o_o unto_o n_n for_o the_o right_n sine_fw-la of_o 53_o degree_n 8_o minute_n which_o be_v require_v chap._n iii_o to_o find_v the_o arch_n of_o any_o sine_fw-la give_v the_o radius_fw-la be_v put_v 1000_o if_o your_o sine_fw-la be_v give_v in_o number_n seek_v the_o right_a parallel_n of_o the_o number_n give_v for_o where_o that_o parallel_n cut_v the_o quadrant_a there_o be_v the_o arch_n require_v count_v from_o the_o begin_n of_o the_o degree_n at_o e_z as_o before_z but_o if_o your_o sine_fw-la be_v give_v in_o a_o right_a line_n then_o from_o the_o side_n of_o your_o quadrat_fw-la ae_n extend_v your_o give_fw-mi sine_fw-la upon_o any_o of_o the_o contrary_a parallel_n and_o that_o right_a parallel_n that_o shall_v cut_v through_o the_o extreme_a point_n thereof_o shall_v cut_v the_o quadrant_a in_o the_o require_a arch_n as_o for_o example_n let_v 800_o be_v a_o sine_fw-la give_v unto_o which_o it_o be_v require_v to_o find_v the_o arch_n belong_v thereto_o i_o seek_v 800_o in_o the_o right_a parallel_n and_o find_v it_o i_o follow_v it_o to_o the_o quadrant_a and_o there_o i_o see_v it_o cut_v the_o arch_n of_o 53_o degree_n 8_o minute_n count_v from_o e_o which_o 53_o degree_n 8_o minute_n be_v the_o arch_n belong_v to_o the_o sine_fw-la 800_o the_o radius_fw-la being_n 1000_o but_o if_o the_o sine_fw-la be_v give_v in_o a_o right_a line_n take_v it_o betwixt_o your_o compass_n and_o set_v one_o foot_n in_o the_o side_n of_o the_o quadrat_fw-la ae_n extend_v the_o other_o upon_o any_o of_o the_o contrary_a parallel_n as_o here_o i_o set_v one_o foot_n in_o the_o point_n n_n and_o extend_v the_o other_o unto_o o_o through_o which_o point_n the_o right_a parallel_n of_o 800_o run_v and_o cut_v the_o quadrant_a in_o 53_o degree_n 8_o minute_n count_v from_o e_z as_o before_z chap._n iu._n any_o radius_fw-la not_o exceed_v the_o side_n of_o the_o quadrat_fw-la be_v give_v to_o find_v the_o right_n sine_fw-la of_o any_o arch_n or_o angle_v thereunto_o belong_v first_o take_v the_o give_v radius_fw-la and_o place_n it_o upon_o the_o contrary_a parallel_n of_o 100_o which_o be_v the_o total_a sine_fw-la and_o where_o it_o shall_v end_v there_o place_n the_o thread_n which_o be_v so_o place_v see_v where_o it_o cut_v the_o contrary_a parallel_n that_o pass_v through_o the_o give_v arch_n count_v in_o the_o quadrant_a from_o g_z for_o the_o segment_n so_o cut_v which_o lie_v between_o the_o thread_n and_o the_o side_n ae_n be_v the_o fine_a require_v let_v the_o line_n a_o be_v the_o radius_fw-la give_v and_o let_v it_o be_v require_v to_o find_v the_o sine_fw-la of_o 36_o degree_n 53_o minute_n first_o i_o take_v the_o line_n a_o and_o place_n it_o upon_o the_o contrary_a parallel_n of_o 100_o as_o from_o e_z to_z d_o upon_o which_o point_n d_o i_o place_v the_o thread_n this_o do_v i_o count_v the_o give_v arch_n viz._n 36_o degree_n 33_o minute_n from_o the_o end_n of_o the_o quadrant_a at_o g_z unto_o o_o and_o see_v what_o contrary_a parallel_n there_o i_o find_v which_o be_v 60_o intersect_v the_o thread_n at_o m_n so_o shall_v the_o distance_n nm_n be_v the_o line_n b_o the_o sine_fw-la of_o 36_o degree_n 53_o
as_o be_v require_v chap._n lvi_o have_v the_o three_o side_n of_o any_o spherical_a triangle_n to_o find_v a_o angle_n opposite_a to_o any_o of_o they_o let_v the_o triangle_n zp_v be_v give_v whereof_o the_o three_o side_n be_v all_o know_v and_o the_o three_o angle_n all_o unknown_a first_o therefore_o let_v the_o planisphere_n be_v so_o place_v that_o one_o of_o the_o side_n next_o the_o angle_n require_v be_v contain_v between_o the_o pole_n and_o zenith_n or_o place_v the_o sphere_n so_o that_o the_o horizon_n may_v cut_v upon_o the_o limb_n the_o degree_n of_o one_o of_o the_o side_n contain_v the_o angle_n require_v which_o be_v all_o one_o thing_n this_o be_v do_v count_v the_o other_o contain_v side_n from_o the_o pole_n among_o the_o parallel_n and_o the_o side_n subtend_v the_o angle_n require_v always_o from_o the_o zenith_n among_o the_o almicanter_n and_o thereto_o place_v the_o thread_n now_o where_o the_o thread_n cut_v the_o parallel_n of_o the_o second_o contain_v side_n there_o pass_v by_o the_o meridian_n which_o give_v you_o the_o angle_n require_v count_v from_o the_o limb_n thus_o in_o the_o triangle_n z_o ps_n have_v zp_n the_o compliment_n of_o the_o latitude_n and_o ps_n the_o compliment_n of_o the_o sun_n declination_n with_o zs_n the_o compliment_n of_o the_o sun_n altitude_n we_o may_v find_v the_o angle_n of_o the_o hour_n from_o the_o meridian_n zp_v for_o have_v place_v the_o horizon_n to_o the_o compliment_n of_o the_o latitude_n 37_o deg_fw-la 30_o min._n the_o side_n zp_n and_o count_v the_o compliment_n of_o declination_n 78_o deg_fw-la 30_o min._n from_o the_o pole_n among_o the_o parallel_n of_o declination_n or_o the_o declination_n itself_o from_o the_o equator_fw-la 11_o deg_fw-la 30_o min._n and_o also_o 59_o deg_fw-la 15_o min._n among_o the_o almicanter_n from_o the_o zenith_n which_o be_v the_o side_n zs_n or_o the_o compliment_n of_o the_o sun_n altitude_n from_o the_o horizon_n and_o thereto_o apply_v the_o thread_n parallel_v to_o the_o horizon_n it_o shall_v cut_v the_o parallel_n of_o declination_n at_o the_o meridian_n or_o hour_n circle_n of_o 53_o deg_fw-la 40_o min._n from_o the_o limb_n or_o general_a meridian_n and_o such_o be_v the_o angle_n zp_v which_o be_v require_v in_o like_a manner_n may_v we_o find_v the_o angle_n pzs_n which_o be_v the_o azimuth_n from_o the_o north_n for_o let_v the_o sphere_n rest_n as_o before_o with_o the_o side_n zp_n at_o the_o horizon_n if_o we_o count_v the_o side_n zs_n 59_o deg_fw-la 15_o min._n among_o the_o parallel_n count_v from_o the_o pole_n and_o the_o side_n ps_n 78_o deg_fw-la 30_o min._n from_o the_o zenith_n among_o the_o almicanter_n and_o thereto_o apply_v the_o thread_n parallel_v to_o the_o horizon_n it_o shall_v cut_v the_o say_a parallel_n of_o 59_o deg_fw-la 15_o min._n at_o the_o meridian_n of_o 113_o deg_fw-la 17_o min._n for_o in_o this_o manner_n of_o work_n the_o angle_n be_v all_o count_v from_o the_o south_n part_n of_o the_o limb_n this_o 113_o deg_fw-la 17_o min._n be_v the_o angle_n pzs_n which_o be_v require_v then_o for_o to_o find_v the_o three_o angle_n you_o may_v shift_v the_o side_n place_v one_o of_o the_o other_o side_n at_o the_o horizon_n and_o then_o proceed_v as_o before_o thus_o you_o may_v see_v this_o proposition_n which_o be_v most_o useful_a but_o withal_o most_o difficult_a of_o all_o other_o as_o in_o arithmetic_n so_o by_o the_o sector_n and_o crosse-staffe_n and_o also_o by_o the_o fore-side_n of_o this_o universal_a quadrat_fw-la speedy_o perform_v by_o the_o planisphere_n on_o the_o back_n of_o this_o quadrat_fw-la chap._n lvii_o to_o find_v a_o side_n by_o know_v the_o other_o two_o side_n and_o the_o angle_n comprehend_v first_o set_v the_o horizon_n to_o one_o of_o the_o give_v side_n count_v on_o the_o limb_n from_o the_o equator_fw-la and_o count_v the_o other_o side_n from_o the_o pole_n among_o the_o parallel_n then_o count_v the_o give_v angle_n among_o the_o meridians_fw-la from_o the_o limb_n and_o at_o the_o point_n of_o intersection_n between_o that_o meridian_n and_o the_o parallel_n there_o place_v the_o thread_n parallel_v to_o the_o horizon_n which_o you_o may_v do_v by_o help_n of_o the_o almicanter_n and_o note_v what_o almicanter_n the_o thread_n so_o cut_v for_o that_o be_v the_o three_o side_n count_v from_o the_o zenith_n as_o in_o the_o triangle_n zp_v have_v the_o side_n zp_v 37_o deg_fw-la 30_o min._n zs_n 59_o deg_fw-la 15_o min._n with_o the_o angle_n pzs_n 113_o degree_n 17_o min._n we_o may_v find_v the_o side_n ps_n for_o first_o i_o place_v the_o horizon_n at_o 37_o deg_fw-la 30_o min._n and_o count_v the_o other_o side_n zs_n 59_o deg_fw-la 15_o min._n from_o the_o pole_n among_o the_o parallel_n and_o the_o angle_n pzs_n 113_o deg_fw-la 17_o min._n among_o the_o meridians_fw-la from_o the_o south_n part_n of_o the_o limb_n and_o where_o this_o meridian_n of_o 113_o deg_fw-la 17_o min._n cut_v the_o parallel_n of_o 59_o deg_fw-la 15_o min._n there_o i_o place_v the_o thread_n parallel_v to_o the_o horizon_n and_o i_o find_v it_o to_o cut_v the_o almicanter_n at_o 78_o deg_fw-la 30_o min._n from_o the_o zenith_n and_o such_o be_v the_o side_n sparke_n which_o be_v require_v thus_o have_v all_o three_o side_n you_o may_v find_v any_o of_o the_o other_o two_o angle_n by_o the_o last_o chapter_n chap._n lviii_o to_o find_v a_o side_n by_o know_v the_o other_o two_o side_n and_o one_o of_o the_o angle_n next_o the_o inquire_v side_n first_o set_v the_o horizon_n to_o that_o side_n give_v which_o be_v next_o the_o give_v angle_n count_v on_o the_o limb_n from_o the_o equator_fw-la then_o place_v the_o thread_n parallel_n to_o the_o horizon_n upon_o the_o other_o side_n count_v from_o the_o zenith_n among_o the_o almicanter_n now_o where_o the_o thread_n intersect_v the_o angle_n give_v count_v among_o the_o meridians_fw-la from_o the_o limb_n there_o pass_v by_o the_o parallel_n of_o the_o side_n require_v count_v from_o the_o pole_n as_o in_o the_o triangle_n zp_v have_v the_o side_n zp_v the_o compliment_n of_o the_o latitude_n 37_o deg_fw-la 30_o min._n and_o the_o side_n ps_n the_o compliment_n of_o the_o declination_n 78_o deg_fw-la 30_o min._n with_o the_o angle_n pzs_n the_o azimuth_n from_o the_o north_n 113_o deg_fw-la 17_o min._n we_o may_v find_v the_o side_n zs_n the_o compliment_n of_o latitude_n for_o if_o we_o place_v the_o horizon_n to_o the_o side_n zp_v 37_o deg_fw-la 30_o min._n from_o the_o equator_fw-la in_o the_o limb_n and_o then_o place_v the_o thread_n to_o the_o almicanter_n of_o 78_o deg_fw-la 30_o min._n count_v from_o the_o zenith_n it_o shall_v cut_v the_o meridian_n of_o 113_o deg_fw-la 17_o min._n from_o the_o limb_n at_o the_o parallel_n of_o 59_o deg_fw-la 15_o min._n count_v from_o the_o pole_n and_o such_o be_v the_o side_n zs_n which_o be_v require_v chap._n lix_n have_v two_o side_n and_o one_o angle_n to_o find_v any_o of_o the_o other_o two_o angle_n first_o find_v the_o three_o side_n by_o one_o of_o the_o two_o last_o chap._n then_o have_v all_o three_o side_n you_o may_v by_o the_o 56_o chap._n find_v any_o angle_n require_v here_o need_v no_o example_n more_o than_o what_o those_o chapter_n afford_v chap._n lx._n have_v either_o all_o three_o angle_n to_o find_v a_o side_n or_o have_v two_o angle_n and_o one_o side_n to_o find_v any_o of_o the_o other_o two_o side_n to_o find_v a_o side_n by_o know_v the_o three_o angle_n you_o must_v first_o convert_v the_o angle_n into_o side_n by_o the_o 42_o chap._n which_o be_v do_v you_o may_v by_o the_o 56_o chap._n find_v a_o angle_n opposite_a to_o any_o one_o of_o these_o side_n so_o shall_v the_o angle_n which_o be_v thus_o find_v be_v the_o side_n which_o be_v here_o require_v as_o in_o the_o say_v 42_o chap._n so_o likewise_o have_v two_o angle_n &_o one_o side_n you_o may_v by_o the_o say_v 42_o chap._n convert_v these_o angle_n into_o side_n and_o the_o side_n into_o angle_n so_o shall_v you_o have_v two_o side_n and_o one_o angle_n by_o which_o and_o the_o 57_o or_o 58_o chapter_n you_o may_v find_v a_o three_o side_n then_o have_v all_o three_o side_n you_o may_v by_o the_o 56_o chap._n find_v a_o angle_n opposite_a to_o any_o of_o they_o which_o angle_n thus_o find_v shall_v be_v the_o side_n require_v the_o end_n of_o the_o second_o book_n the_o three_o book_n show_v how_o by_o the_o universal_a quadrat_fw-la to_o resolve_v all_o such_o astronomical_a proposition_n as_o be_v of_o ordinary_a use_n as_o well_o in_o the_o art_n of_o navigation_n as_o in_o the_o art_n of_o dial_v with_o the_o resolution_n of_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
rumb_n and_o distance_n chap._n i._o to_o find_v the_o sun_n altitude_n by_o the_o shadow_n of_o a_o gnomon_n set_v perpendicular_a to_o the_o horizon_n to_o perform_v this_o conclusion_n place_v the_o thread_n upon_o the_o intersection_n of_o the_o contrary_a parallel_n of_o the_o shadow_n with_o the_o right_a parallel_n of_o the_o gnomon_n so_o shall_v it_o show_v in_o the_o quadrant_a the_o sun_n altitude_n require_v as_o for_o example_n let_v ab_fw-la in_o the_o follow_a diagram_n be_v the_o gnomon_n or_o substance_n yield_v shadow_n which_o we_o will_v suppose_v to_o be_v 60_o and_o let_v ac_fw-la be_v the_o length_n of_o the_o shadow_n here_o find_v to_o be_v 45_o such_o part_n as_o the_o gnomon_n be_v 60._o now_o place_v the_o thread_n upon_o the_o intersection_n of_o the_o contrary_a parallel_n of_o 45_o with_o the_o right_a parallel_n of_o 60_o it_o will_v cut_v in_o the_o quadrant_n 53_o deg_fw-la 8_o min._n and_o such_o be_v the_o angle_n acb_n the_o altitude_n of_o the_o sun_n require_v or_o if_o the_o shadow_n have_v be_v ad_fw-la which_o here_o be_v find_v to_o be_v 80_o if_o you_o place_v the_o thread_n at_o the_o intersection_n of_o the_o right_a parallel_n of_o 60_o with_o the_o contrary_a parallel_n of_o 80_o it_o will_v cut_v in_o the_o quadrant_n 36_o deg_fw-la 52_o min._n and_o such_o be_v the_o angle_n adb_n the_o altitude_n of_o the_o sun_n require_v chap._n ii_o the_o height_n of_o the_o sun_n be_v give_v to_o find_v the_o length_n of_o the_o right_a shadow_n by_o right_a shadow_n be_v mean_v the_o shadow_n of_o any_o staff_n post_n steeple_n or_o any_o gnomon_n whatsoever_o that_o stand_v at_o right_a angle_n with_o the_o horizon_n the_o one_o end_n thereof_o respect_v the_o zenith_n of_o the_o place_n and_o the_o other_o the_o nadir_n wherefore_o place_n the_o thread_n to_o the_o height_n of_o the_o sun_n give_v in_o the_o quadrant_n and_o it_o shall_v cut_v the_o right_a parallel_n of_o the_o gnomon_n at_o the_o contrary_a parallel_n of_o the_o shadow_n let_v the_o sun_n height_n give_v be_v 53_o deg_fw-la 8_o min._n and_o the_o length_n of_o the_o gnomon_n 60_o wherefore_o if_o you_o place_v the_o thread_n at_o 53_o deg_fw-la 8_o min._n in_o the_o quadrant_n it_o will_v cut_v the_o right_a parallel_n of_o 60_o at_o the_o contrary_a parallel_n of_o 45_o and_o such_o be_v the_o length_n of_o the_o right_a shadow_n ac_fw-la which_o be_v require_v or_o if_o the_o sun_n height_n be_v but_o 36_o deg_fw-la 52_o min._n if_o you_o place_v the_o thread_n to_o 36_o deg_fw-la 52_o min._n in_o the_o quadrant_n it_o will_v cut_v the_o right_a parallel_n of_o 60_o at_o the_o contrary_a parallel_n of_o 80_o and_o such_o be_v the_o length_n of_o the_o right_a shadow_n ad_fw-la which_o be_v require_v chap._n iii_o to_o find_v the_o altitude_n of_o the_o sun_n by_o the_o shadow_n of_o a_o gnomon_n stand_v at_o right_a angle_n with_o any_o perpendicular_a wall_n in_o such_o manner_n that_o it_o may_v lie_v parallel_n to_o the_o horizon_n here_o if_o you_o place_v the_o thread_n to_o the_o intersection_n of_o the_o contrary_a parallel_n of_o the_o gnomon_n with_o the_o right_a parallel_n of_o the_o shadow_n it_o shall_v cut_v the_o quadrant_a at_o the_o sun_n altitude_n require_v as_o for_o example_n let_v ab_fw-la in_o the_o next_o chapter_n be_v some_o perpendicular_a wall_n ac_fw-la a_o gnomon_n make_v right_a angle_n therewith_o who_o length_n be_v 60_o and_o let_v ad_fw-la be_v the_o length_n of_o the_o shadow_n which_o be_v here_o find_v to_o be_v 80_o now_o if_o you_o place_v the_o thread_n to_o the_o intersection_n of_o the_o contrary_a parallel_n of_o 60_o with_o the_o right_a parallel_n of_o 80_o it_o will_v cut_v in_o the_o quadrant_n 5_o deg_fw-la 8_o min._n and_o such_o be_v the_o angle_n acd_v the_o altitude_n of_o the_o sun_n require_v chap._n iu._n the_o height_n of_o the_o sun_n be_v give_v to_o find_v the_o length_n of_o the_o contrary_a shadow_n by_o the_o contrary_a shadow_n be_v understand_v the_o length_n of_o any_o shadow_n that_o be_v make_v by_o a_o staff_n or_o gnomon_n stand_v at_o right_a angle_n against_o any_o perpendicular_a wall_n in_o such_o a_o manner_n that_o it_o may_v lie_v parallel_n to_o the_o horizon_n the_o length_n of_o the_o contrary_a shadow_n do_v increase_v as_o the_o sun_n rise_v in_o height_n whereas_o the_o right_a shadow_n do_v decrease_v in_o length_n as_o the_o sun_n do_v increase_v in_o height_n wherefore_o to_o find_v the_o length_n of_o the_o verse_v shadow_n place_v the_o thread_n upon_o the_o height_n of_o the_o sun_n give_v in_o the_o quadrant_n and_o it_o will_v cut_v the_o contrary_a parallel_n of_o the_o gnomon_n at_o the_o right_a parallel_n of_o the_o shadow_n let_v ab_fw-la be_v a_o perpendicular_a wall_n ac_fw-la a_o gnomon_n make_v right_a angle_n there_o with_o the_o length_n whereof_o be_v 60_o and_o the_o sun_n altitude_n 53_o deg_fw-la 8_o min._n the_o angle_n acd_v now_o if_o you_o place_v the_o thread_n at_o 53_o deg_fw-la 8_o min._n in_o the_o quadrant_a 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 80_o and_o such_o be_v the_o length_n of_o the_o contrary_a shadow_n ad_fw-la which_o be_v require_v chap._n v._o 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 declination_n as_o the_o radius_fw-la be_v in_o proportion_n to_o the_o sine_fw-la of_o the_o sun_n great_a declination_n so_o the_o sine_fw-la of_o the_o sun_n distance_n from_o the_o next_o equinoctial_a point_n to_o the_o sine_fw-la of_o the_o declination_n require_v thus_o in_o the_o right_n angle_v spherical_a triangle_n ♈_o x_o ♉_o in_o the_o follow_a diagram_n have_v the_o base_a ♈_o ♉_o 30_o deg_fw-la which_o be_v the_o sun_n distance_n from_o the_o equinoctial_a point_n ♈_o he_o be_v in_o the_o begin_n of_o ♉_o and_o the_o angle_n x_o ♈_o ♉_o 23_o degree_n 30_o minute_n the_o angle_n of_o the_o sun_n great_a declination_n 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 declination_n require_v 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 degree_n with_o the_o right_n sine_fw-la of_o 23_o degree_n 30_o minute_n it_o will_v cut_v the_o contrary_a sine_fw-la of_o 30_o deg_fw-la at_o the_o right_a sineof_o of_o 11_o deg_fw-la 30_o minute_n or_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 sine_fw-la of_o 30_o deg_fw-la it_o will_v cut_v the_o contrary_a sine_fw-la of_o 23_o deg_fw-la 30_o min._n at_o the_o right_n sine_fw-la of_o 11_o deg_fw-la 30_o min._n &_o so_o much_o be_v the_o side_n ♉_o x_o the_o declination_n of_o the_o sun_n in_o the_o begin_n of_o ♉_o which_o be_v require_v the_o like_a declination_n have_v the_o beginning_n of_o â™_o â™_o &_o ♓_o all_o be_v of_o equal_a distance_n from_o their_o next_o equinoctial_a point_n the_o like_a hold_v of_o all_o the_o other_o point_n of_o the_o ecliptic_a so_o that_o have_v the_o declination_n of_o every_o degree_n in_o one_o quarter_n all_o the_o rest_n be_v know_v chap._n vi_o how_o to_o find_v the_o latitude_n of_o a_o place_n or_o the_o pole_n elevation_n above_o the_o horizon_n the_o declination_n of_o the_o sun_n be_v give_v first_o you_o must_v get_v the_o sun_n meridian_n altitude_n the_o which_o you_o may_v very_o well_o do_v by_o this_o instrument_n after_o this_o manner_n hold_v the_o superficies_n of_o your_o instrument_n perpendicular_a to_o the_o horizon_n and_o the_o centre_n where_o the_o thread_n and_o plummet_n hang_v towards_o the_o sun_n then_o lift_v up_o the_o quadrat_fw-la towards_o the_o sun_n until_o his_o beam_n pass_v through_o both_o his_o sight_n at_o once_o the_o thread_n will_v cut_v in_o the_o quadrant_a the_o degree_n of_o the_o sun_n altitude_n this_o observation_n you_o must_v begin_v a_o little_a before_o noon_n still_o diligent_o observe_v until_o you_o perceive_v the_o sun_n to_o begin_v to_o fall_v again_o then_o mark_v what_o be_v his_o great_a altitude_n will_v serve_v for_o this_o our_o present_a purpose_n have_v get_v the_o meridian_n altitude_n by_o this_o and_o the_o declination_n by_o the_o last_o chapter_n you_o may_v find_v the_o elevation_n of_o the_o pole_n above_o the_o horizon_n as_o follow_v if_o the_o sun_n have_v north_n declination_n then_o subtract_v the_o declination_n out_o of_o the_o meridian_n altitude_n but_o if_o the_o sun_n have_v south_n declination_n then_o add_v the_o declination_n to_o the_o meridian_n altitude_n so_o shall_v the_o sum_n or_o difference_n be_v the_o height_n of_o the_o equinoctial_a which_o be_v take_v out_o of_o the_o quadrant_a or_o 90_o degree_n leave_v the_o elevation_n of_o the_o pole_n
i_o seek_v in_o the_o ecliptic_a and_o follow_v the_o parallel_n that_o pass_v thereby_o to_o the_o horizon_n i_o find_v it_o there_o to_o cut_v 19_o deg_fw-la 7_o min._n count_v from_o the_o centre_n and_o such_o be_v the_o amplitude_n require_v the_o sphere_n be_v set_v to_o the_o latitude_n of_o 52_o deg_fw-la 30_o min._n chap._n xxx_o have_v the_o latitude_n of_o the_o place_n with_o the_o declination_n of_o any_o star_n to_o find_v his_o amplitude_n first_o the_o sphere_n to_o the_o latitude_n be_v set_v if_o the_o star_n be_v therein_o place_v follow_v the_o parallel_n of_o declination_n whereon_o he_o stand_v to_o the_o horizon_n and_o if_o he_o be_v not_o in_o the_o instrument_n get_v his_o declination_n by_o the_o seven_o chap._n or_o otherwise_o by_o some_o table_n for_o that_o purpose_n which_o find_v seek_v it_o in_o the_o axletree-line_n among_o the_o parallel_n of_o declination_n &_o follow_v the_o parallel_n thereof_o to_o the_o horizon_n and_o if_o it_o cut_v the_o horizon_n it_o there_o show_v the_o amplitude_n but_o if_o it_o cut_v it_o not_o then_o do_v not_o that_o star_n set_v or_o go_v down_o under_o our_o horizon_n as_o for_o example_n let_v it_o be_v require_v to_o find_v the_o amplitude_n of_o arcturus_n i_o look_v in_o my_o instrument_n for_o arcturus_n and_o i_o find_v he_o place_v upon_o the_o parallel_n of_o 21_o d._n 10_o min._n of_o north_n declination_n as_o it_o be_v just_a in_o the_o horizon_n at_o 36_o deg_fw-la 23_o min._n from_o the_o centre_n and_o such_o be_v his_o amplitude_n and_o if_o you_o will_v have_v the_o amplitude_n of_o the_o lion_n heart_n look_v in_o your_o instrument_n and_o you_o shall_v find_v he_o place_v upon_o the_o parallel_n of_o 13_o deg_fw-la 42_o min._n of_o north_n declination_n which_o if_o you_o follow_v to_o the_o horizon_n you_o shall_v find_v it_o there_o cut_v 22_o deg_fw-la 54_o min._n and_o such_o be_v the_o amplitude_n of_o the_o lion_n heart_n in_o the_o latitude_n of_o 52_o deg_fw-la 30_o min._n but_o if_o you_o will_v seek_v the_o amplitude_n of_o the_o bright_a star_n call_v the_o harp_n you_o shall_v find_v he_o place_v upon_o the_o parallel_n of_o 38_o deg_fw-la 30_o min._n north_n declination_n but_o this_o parallel_n cut_v not_o the_o horizon_n therefore_o he_o go_v not_o under_o our_o horizon_n at_o all_o in_o the_o aforesaid_a latitude_n of_o 52_o deg_fw-la 30_o min._n but_o when_o he_o be_v in_o his_o low_a direct_o in_o the_o north_n he_o be_v just_a one_o degree_n of_o height_n chap._n xxxii_o have_v the_o declination_n and_o amplitude_n of_o the_o sun_n to_o find_v the_o latitude_n of_o the_o place_n count_n the_o declination_n upon_o the_o axletree-line_n among_o the_o parallel_n of_o declination_n and_o the_o amplitude_n upon_o the_o horizon_n from_o the_o centre_n the_o same_o way_n with_o the_o declination_n then_o move_v the_o sphere_n to_o and_o fro_o until_o the_o amplitude_n and_o the_o parallel_n of_o declination_n meet_v both_o in_o one_o point_n the_o south_n end_n of_o the_o horizon_n shall_v cut_v the_o meridian_n at_o the_o compliment_n of_o the_o latitude_n require_v as_o for_o example_n let_v the_o sun_n declination_n be_v 11_o deg_fw-la 30_o min._n and_o his_o amplitude_n 19_o deg_fw-la 7_o min._n if_o you_o move_v the_o sphere_n until_o the_o 19_o deg_fw-la 7_o min._n one_o the_o horizon_n fall_v direct_o on_o the_o parallel_n of_o 11_o deg_fw-la 30_o min._n the_o south_n end_n of_o the_o horizon_n will_v cut_v the_o limb_n at_o 37_o deg_fw-la 30._o min._n and_o such_o be_v the_o height_n of_o the_o equinoctial_a which_o be_v take_v out_o of_o 90_o deg_fw-la leave_v 52_o deg_fw-la 30_o min._n for_o the_o latitude_n of_o the_o place_n which_o be_v require_v chap._n xxxii_o the_o latitude_n of_o the_o place_n and_o the_o amplitude_n of_o the_o sun_n be_v give_v to_o find_v his_o declination_n first_o set_v the_o sphere_n to_o the_o latitude_n propose_v then_o count_v the_o amplitude_n upon_o the_o horizon_n and_o where_o it_o shall_v end_v shall_v meet_v you_o a_o parallel_n which_o be_v follow_v to_o the_o axletree-line_n will_v there_o show_v you_o the_o declination_n require_v as_o in_o the_o latitude_n of_o 52_o deg_fw-la 30_o min._n the_o amplitude_n be_v 19_o deg_fw-la 7_o min._n if_o you_o first_o place_n the_o sphere_n to_o the_o latitude_n and_o then_o count_v 19_o deg_fw-la 7_o min._n upon_o the_o horizon_n there_o shall_v meet_v you_o a_o parallel_n which_o be_v follow_v to_o the_o axletree_n line_n shall_v there_o give_v you_o 11_o deg_fw-la 30_o min._n the_o declination_n require_v chap._n xxxiii_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 height_n in_o the_o vertical_a circle_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 give_v among_o the_o almicanter_n from_o both_o end_n of_o the_o horizon_n towards_o the_o zenith_n and_o thereto_o place_v the_o thread_n parallel_v to_o the_o horizon_n and_o where_o the_o thread_n so_o place_v cut_v the_o axletree-line_n or_o six_o a_o clock_n hour-line_n there_o be_v the_o altitude_n require_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 if_o you_o count_v this_o 11._o deg_fw-la 30_o min._n from_o both_o end_n of_o the_o horizon_n among_o the_o almicanter_n and_o place_v the_o thread_n thereto_o it_o shall_v lie_v parallel_n to_o the_o horizon_n and_o cut_v the_o axle-tree-line_n at_o 14_o deg_fw-la 33_o min._n and_o such_o be_v the_o sun_n height_n when_o he_o be_v direct_o in_o the_o east_n or_o west_n point_v which_o be_v require_v or_o if_o you_o place_v the_o horizon_n to_o 52_o deg_fw-la 30_o min._n in_o the_o limb_n count_v from_o the_o equator_fw-la and_o count_v the_o declination_n upon_o the_o axletree-line_n viz._n 11_o deg_fw-la 30_o min._n the_o parallel_n thereof_o shall_v cut_v the_o horizon_n at_o 14_o deg_fw-la 33_o min._n for_o the_o height_n of_o the_o sun_n in_o the_o vertical_a circle_n the_o same_o as_o before_o chap._n xxxiv_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 his_o come_n to_o the_o east_n or_o west_n point_n first_o set_v the_o planisphere_n to_o the_o latitude_n give_v then_o place_v the_o thread_n upon_o the_o zenith_n and_o the_o nadir_n so_o shall_v it_o cut_v the_o horizon_n at_o right_a angle_n in_o the_o centre_n and_o the_o parallel_n of_o declination_n at_o the_o time_n from_o six_o count_v among_o the_o meridian_n as_o for_o example_n let_v the_o latitude_n give_v be_v 52_o deg_fw-la 30_o min._n and_o the_o sun_n declination_n 11_o deg_fw-la 30_o min._n first_o place_z the_o horizon_n to_o 37_o deg_fw-la 30_o min._n the_o compliment_n of_o the_o latitude_n below_o the_o south_n end_n of_o the_o equator_fw-la then_o place_v the_o thread_n to_o the_o zenith_n and_o nadir_n that_o be_v to_o 52_o deg_fw-la 30_o min._n above_o the_o south_n end_n of_o the_o equator_fw-la and_o as_o much_o below_o the_o north_n end_v thereof_o the_o thread_n be_v fix_v shall_v cut_v the_o horizon_n at_o right_a angle_n in_o the_o centre_n and_o also_o the_o parallel_n of_o 11_o deg_fw-la 30_o min_fw-mi at_o the_o meridian_n of_o 9_o deg_fw-la or_o in_o time_n at_o 36_o minute_n from_o the_o hour_n of_o six_o and_o such_o be_v the_o distance_n of_o time_n betwixt_o the_o hour_n of_o six_o and_o the_o sun_n come_v to_o the_o east_n or_o west_n point_n which_o be_v require_v chap._n xxxv_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 height_n at_o the_o hour_n of_o six_o first_o set_v the_o horizon_n to_o the_o compliment_n of_o the_o latititude_n below_o the_o south_n end_n of_o the_o equator_fw-la or_o the_o equator_fw-la to_o the_o compliment_n of_o the_o latitude_n above_o the_o south_n end_n of_o the_o horizon_n which_o be_v all_o one_o then_o seek_v the_o declination_n upon_o the_o axletree-line_n and_o place_v the_o thread_n thereunto_o parallel_v to_o the_o horizon_n by_o help_n of_o the_o almicanter_n at_o both_o end_n thereof_o so_o shall_v the_o thread_n show_v among_o those_o almicanter_n the_o height_n of_o the_o sun_n require_v thus_o in_o the_o latitude_n of_o 52_o deg_fw-la 30_o min._n the_o sun_n declination_n be_v 11_o deg_fw-la 30_o min._n i_o first_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 then_o place_v the_o thread_n to_o 11_o deg_fw-la 30_o min._n count_v in_o the_o axletree-line_n and_o move_v the_o end_n thereof_o to_o and_o fro_o until_o the_o thread_n lie_v parallel_n to_o the_o horizon_n which_o will_v be_v when_o both_o end_n of_o the_o thread_n cut_v like_o degree_n from_o the_o horizon_n upon_o
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