Selected quad for the lemma: end_n

Word A	Word B	Word C	Word D	Occurrence	Frequency Band	MI	MI Band	Prominent
end_n	line_n	perpendicular_a	straight_a	1,765	5	14.0990	5	true

Text snippets containing the quad

ID	Title	Author	Corrected Date of Publication (TCP Date of Publication)	STC	Words	Pages
A34005	The sector on a quadrant, or A treatise containing the description and use of four several quadrants two small ones and two great ones, each rendred many wayes, both general and particular. Each of them accomodated for dyalling; for the resolving of all proportions instrumentally; and for the ready finding the hour and azimuth universally in the equal limbe. Of great use to seamen and practitioners in the mathematicks. Written by John Collins accountant philomath. Also An appendix touching reflected dyalling from a glass placed at any reclination.	Collins, John, 1625-1683.; Lyon, John, professor of mathematics. Appendix touching reflective dialling.; Sutton, Henry, mathematical instrument maker.	1659 (1659)	Wing C5382; ESTC R32501	226,510	415

altitude_n be_v well_o give_v these_o scale_n in_o their_o use_n presuppose_v the_o hour_n and_o azimuth_n of_o the_o sun_n to_o be_v near_o the_o noon_n meridian_n then_o 60d._o operation_n to_o find_v the_o hour_n take_v the_o distance_n between_o the_o altitude_n and_o the_o declination_n proper_a to_o the_o season_n of_o the_o year_n out_o of_o the_o hour_n scale_n and_o enter_v one_o foot_n of_o this_o extent_n at_o the_o cousin_n of_o the_o declination_n in_o the_o line_n of_o sin_n and_o lay_v the_o thread_n to_o the_o other_o foot_n according_a to_o near_a distance_n it_o show_v the_o hour_n from_o noon_n in_o the_o verse_v sin_n quadruple_v example_n when_o the_o sun_n have_v 23_o 31′_n of_o north_n declination_n and_o 60_o of_o altitude_n the_o hour_n from_o noon_n will_v be_v 13_o 58′_n to_o be_v convert_v into_o time_n when_o the_o hour_n be_v find_v to_o be_v less_o than_o 40_o from_o noon_n the_o former_a extent_n may_v be_v double_v and_o enter_v as_o before_o and_o it_o show_v the_o hour_n in_o the_o verse_v sin_n octuple_v and_o when_o the_o hour_n be_v less_o than_o 30_o from_o noon_n the_o former_a extent_n may_v be_v tripled_a and_o enter_v as_o before_o and_o after_o this_o manner_n it_o be_v possible_a to_o make_v the_o whole_a limb_n give_v the_o hour_n next_o noon_n the_o verse_v sine_fw-la duodecuple_v lie_v on_o the_o other_o side_n of_o the_o quadrant_n and_o in_o this_o case_n a_o ark_n must_v first_o be_v find_v in_o the_o limb_n and_o the_o thread_n lay_v over_o the_o say_a ark_n count_v from_o the_o other_o edge_n will_v intersect_v the_o say_v verse_v sine_fw-la at_o the_o ark_n seek_v to_o find_v the_o sun_n azimuth_n take_v the_o distance_n in_o the_o azimuth_n scale_n between_o the_o altitude_n and_o the_o declination_n proper_a to_o the_o season_n of_o the_o year_n and_o enter_v it_o at_o the_o cousin_n of_o the_o altitude_n lay_v the_o thread_n to_o the_o other_o foot_n according_a to_o near_a distance_n it_o will_v show_v the_o azimuth_n in_o the_o verse_v sin_n quadruple_v or_o when_o the_o azimuth_n be_v near_a noon_n according_a to_o the_o former_a restriction_n for_o the_o hour_n the_o extent_n may_v be_v double_v or_o tripled_a and_o the_o answer_v find_v in_o the_o verse_v sin_n octuple_v or_o duodecuple_v as_o be_v do_v for_o the_o hour_n example_n so_o when_o the_o sun_n have_v 23_o 31′_n of_o north_n declination_n his_o altitude_n be_v 60d._o the_o azimuth_n will_v be_v find_v to_o be_v 26_o 21′_n from_o the_o south_n by_o the_o like-reason_n when_o we_o find_v the_o hour_n and_o azimuth_n in_o the_o equal_a limb_n by_o the_o diagonal_a scale_n if_o those_o extent_n have_v be_v double_v the_o hour_n and_o azimuth_n near_o six_o or_o the_o vertical_a may_v have_v be_v find_v in_o a_o line_n of_o sin_n of_o 30_o put_v through_o the_o whole_a limb_n but_o that_o we_o think_v needless_a finis_fw-la the_o descrpition_n and_o use_v of_o a_o great_a universal_a quadrant_n with_o a_o quarter_n of_o stofter_n particular_a projection_n upon_o it_o invert_v contrive_v and_o write_v by_o john_n collins_n accountant_a and_o student_n in_o the_o mathematics_n london_n print_v in_o the_o year_n 1658._o the_o description_n of_o the_o great_a quadrant_n it_o have_v be_v hint_v before_o that_o though_o the_o former_a contrivance_n may_v serve_v for_o a_o small_a quadrant_n yet_o there_o may_v be_v a_o better_a for_o a_o great_a one_o the_o description_n of_o the_o fore-side_n on_o the_o right_a edge_n from_o the_o centre_n be_v place_v a_o line_n of_o sines_n on_o the_o left_a edge_n from_o the_o centre_n a_o line_n of_o verse_v sin_n to_o 180d._o the_o limb_n the_o same_o as_o in_o the_o small_a quadrant_n between_o the_o limb_n and_o the_o centre_n be_v place_v in_o circle_n a_o line_n of_o verse_v sin_n to_o 180_o another_o through_o the_o whole_a limb_n to_o 90d._o the_o line_n of_o lesser_a sin_n and_o secant_v the_o line_n of_o tangent_n the_o quadrat_fw-la and_o shadow_n above_o they_o the_o projection_n with_o the_o declination_n day_n of_o the_o month_n and_o almanac_n on_o the_o left_a edge_n be_v place_v the_o fit_a hour_n and_o azimuth_n scale_n within_o the_o projection_n abut_a against_o the_o sin_n be_v place_v a_o little_a scale_n call_v the_o scale_n of_o entrance_n be_v graduate_v to_o 62_o and_o be_v no_o other_o but_o a_o small_a line_n of_o sines_n number_v by_o the_o compliment_n at_o the_o end_n of_o the_o secant_fw-la be_v put_v on_o the_o verse_v sin_n double_v that_o be_v to_o twice_o the_o radius_fw-la of_o the_o quadrant_n and_o at_o the_o end_n of_o the_o tangent_n tripled_a to_o some_o few_o degree_n to_o give_v the_o hour_n and_o azimuth_n near_o noon_n more_o exact_o the_o description_n of_o the_o backside_n on_o the_o right_a edge_n from_o the_o centre_n be_v place_v a_o line_n of_o equal_a part_n be_v 10_o inch_n precise_a decimal_o subdivide_v on_o the_o outside_n next_o the_o edge_n be_v place_v a_o large_a chord_n to_o 60_o equal_a in_o length_n to_o the_o radius_fw-la of_o the_o line_n of_o sines_n on_o the_o left_a edge_n be_v place_v a_o line_n of_o tangent_n issue_v from_o the_o centre_n continue_v to_o 63_o 26′_n and_o again_o continue_v apart_o from_o 60_o to_o 75_o the_o equal_a limb._n within_o it_o a_o quadrant_n of_o ascension_n divide_v into_o 24_o equal_a hour_n and_o its_o part_n with_o star_n affix_v and_o letter_n grave_v to_o refer_v to_o their_o name_n between_o it_o and_o the_o centre_n be_v place_v a_o circle_n whereof_o there_o be_v but_o three_o quadrant_n graduate_v the_o diameter_n of_o this_o circle_n be_v no_o other_o than_o the_o dyall_a scale_n of_o 6_o hour_n or_o double_a tangent_n divide_v into_o 90d._o two_o quadrant_n or_o the_o half_a of_o this_o circle_n beneath_o the_o diameter_n be_v divide_v into_o 90_o equal_a part_n or_o degree_n the_o upper_a divide_a quadrant_n be_v call_v the_o quadrant_n of_o latitude_n from_o the_o extremity_n of_o the_o say_a quadrant_n and_o perpendicular_a to_o the_o diameter_n be_v graduate_v a_o line_n of_o proportional_a sin_n m_n foster_n call_v it_o the_o line_n sol._n diagonal-wise_a from_o one_o extremity_n of_o the_o quadrant_n of_o latitude_n to_o the_o other_o be_v graduate_v a_o line_n of_o sines_n that_o end_n number_v with_o ●0_n d_o that_o be_v next_o the_o diameter_n be_v of_o the_o same_o radius_fw-la with_o the_o tangent_n opposite_a and_o parallel_a thereto_o from_o 45_o of_o the_o semicircle_n to_o the_o other_o extremity_n of_o the_o diameter_n be_v place_v a_o line_n of_o sines_n equal_a to_o the_o former_a diagonal-wise_a from_o the_o begin_n of_o the_o line_n sol_fw-mi to_o the_o end_n of_o the_o diameter_n be_v graduate_v a_o line_n of_o 60_o chord_n from_o the_o beginning_n of_o the_o diameter_n but_o below_o it_o towards_o 45_o of_o the_o semicircle_n be_v graduate_v the_o projection_n tangent_fw-la alius_fw-la a_o semi-tangent_a to_o 90_o be_v of_o the_o same_o radius_fw-la with_o the_o tangent_n the_o other_o quadrant_n of_o this_o circle_n be_v only_o a_o void_a line_n there_o pass_v through_o it_o from_o the_o centre_n a_o tangent_fw-la of_o 45_o for_o dyall_v divide_v into_o 3_o hour_n with_o its_o quarter_n and_o minute_n below_o the_o diameter_n be_v void_a space_n leave_v to_o graduate_v any_o table_n at_o pleasure_n and_o a_o line_n of_o chord_n may_v be_v there_o place_v most_o of_o these_o line_n and_o the_o projection_n have_v be_v already_o treat_v upon_o in_o the_o use_n of_o the_o small_a quadrant_n those_o that_o be_v add_v shall_v here_o be_v speak_v to_o of_o the_o line_n of_o verse_v sin_n on_o the_o left_a edge_n issue_v from_o the_o centre_n this_o line_n and_o the_o use_n of_o it_o be_v invent_v by_o the_o learned_a mathematician_n m._n samuel_n foster_n of_o gresham_n college_n decease_a from_o who_o i_o receive_v the_o use_n of_o it_o apply_v to_o a_o sector_n i_o shall_v and_o have_v add_v the_o proportion_n to_o be_v wrought_v upon_o it_o and_o in_o that_o and_o other_o respect_n diversify_v from_o what_o i_o receive_v wherein_o i_o shall_v not_o be_v tedious_a because_o there_o be_v other_o way_n to_o follow_v since_o find_v out_o by_o myself_o the_o chief_a use_n of_o it_o be_v to_o resolve_v the_o two_o case_n of_o the_o four_o axiom_n of_o spherical_a trigonometry_n as_o when_o three_o side_n be_v give_v to_o find_v a_o angle_n or_o two_o side_n with_o the_o angle_n comprehend_v to_o find_v the_o three_o side_n which_o be_v the_o case_n that_o find_v the_o hour_n and_o azimuth_n general_o and_o the_o sun_n altitude_n on_o all_o hour_n for_o the_o hour_n the_o learned_a author_n think_v meet_v to_o add_v a_o zodiaque_fw-la of_o the_o suns-place_n annex_v to_o it_o both_o in_o the_o use_n of_o his_o sector_n as_o also_o

six_o we_o may_v educe_v a_o single_a proportion_n applyable_a to_o the_o logarithm_n without_o natural_a table_n for_o calculate_v the_o hour_n of_o the_o day_n to_o all_o altitude_n by_o turn_v the_o three_o term_n be_v a_o difference_n of_o sines_n or_o verse_v sin_n into_o a_o rectangle_n and_o free_v it_o from_o affection_n the_o two_o first_o proportion_n to_o be_v wrought_v be_v fix_v for_o one_o declination_n the_o first_o will_v be_v to_o find_v the_o sun_n altitude_n or_o depression_n at_o six_o the_o second_o will_v be_v to_o find_v half_o the_o difference_n of_o the_o sin_n of_o the_o sun_n meridian_n altitude_n and_o altitude_n seek_v etc._n etc._n as_o before_o define_v the_o proportion_n to_o find_v it_o be_v as_o the_o secant_fw-la of_o 60_o to_o the_o cousin_n of_o the_o declination_n so_o be_v the_o cousin_n of_o the_o latitude_n to_o the_o sine_fw-la of_o a_o four_o arch._n last_o to_o find_v the_o hour_n get_v the_o sum_n and_o difference_n of_o half_a the_o sun_n zenith_n distance_n at_o the_o hour_n of_o six_o and_o of_o half_a his_o zenith_n distance_n to_o any_o other_o propose_v altitude_n or_o depression_n then_o as_o the_o sine_fw-la of_o the_o four_o arch_n be_v to_o the_o sine_fw-la of_o the_o sum_n so_o be_v the_o sine_fw-la of_o the_o difference_n to_o the_o sine_fw-la of_o the_o hour_n from_o six_o towards_o noon_n or_o midnight_n according_a as_o the_o altitude_n or_o depression_n be_v great_a or_o lesser_a than_o the_o altitude_n or_o depression_n at_o six_o observe_v that_o the_o sine_fw-la of_o a_o arch_n great_a than_o a_o quadrant_n be_v the_o sine_fw-la of_o that_o ark_n compliment_n to_o a_o semicircle_n of_o the_o star_n place_v upon_o the_o quadrant_n below_o the_o projection_n all_o the_o star_n place_v upon_o the_o projection_n be_v such_o as_o fall_v between_o the_o tropic_n and_o the_o hour_n may_v be_v find_v by_o they_o with_o the_o projection_n as_o in_o the_o use_n of_o the_o small_a quadrant_n which_o may_v also_o be_v find_v by_o the_o fit_v particular_a scale_n not_o only_o for_o star_n within_o the_o tropic_n but_o for_o all_o other_o without_o when_o their_o altitude_n be_v less_o than_o 62_o and_o likewise_o their_o azimuth_n may_v be_v thereby_o find_v when_o their_o declination_n be_v not_o more_o than_o 62d._o for_o other_o star_n without_o the_o tropic_n they_o may_v be_v put_v on_o below_o the_o projection_n any_o where_o in_o such_o a_o angle_n that_o the_o thread_n lay_v over_o the_o star_n shall_v show_v a_o ark_n in_o the_o limb_n at_o which_o in_o the_o sin_n the_o point_n of_o entrance_n will_v always_o fall_v and_o again_o the_o same_o star_n be_v to_o be_v grave_v at_o its_o altitude_n or_o depression_n at_o six_o in_o the_o sin_n and_o then_o to_o find_v the_o star_n hour_n in_o that_o latitude_n whereto_o they_o be_v fit_v will_v always_o for_o northern_a star_n be_v to_o take_v the_o distance_n in_o the_o line_n of_o sines_n between_o the_o star_n and_o its_o give_a altitude_n and_o to_o enter_v that_o extent_n at_o the_o point_n of_o entrance_n lay_v the_o thread_n to_o the_o other_o foot_n according_a to_o near_a distance_n and_o it_o give_v the_o star_n hour_n in_o the_o equal_a limb_n from_o six_o which_o may_v also_o be_v find_v in_o the_o sin_n by_o a_o parallel_n entrance_n lay_v the_o thread_n over_o the_o star_n example_n let_v the_o altitude_n of_o the_o last_o in_o the_o end_n of_o the_o great_a bear_n tail_n be_v 63_o take_v the_o distance_n between_o it_o and_o the_o star_n which_o be_v grave_v at_o 37_o 30′_n of_o the_o sin_n the_o say_a extent_n enter_v at_o the_o sine_fw-la of_o 23_o the_o ark_n of_o the_o limb_n the_o thread_n intersect_v when_o it_o lie_v over_o the_o say_a star_n and_o by_o lay_v the_o thread_n to_o the_o other_o foot_n you_o will_v find_v that_o star_n hour_n to_o be_v 46_o 11′_n from_o six_o towards_o noon_n meridian_n if_o the_o altitude_n increase_v and_o in_o find_v the_o true_a time_n of_o the_o night_n the_o star_n hour_n must_v be_v always_o reckon_v from_o the_o meridian_n it_o be_v last_o upon_o in_o this_o example_n it_o will_v be_v 5_o minute_n past_o 9_o feré_fw-la of_o the_o quadrant_n of_o ascension_n on_o the_o backside_n this_o quadrant_n be_v divide_v into_o 24_o hour_n with_o their_o quarter_n and_o subdivision_n and_o serve_v to_o give_v the_o right_a ascension_n of_o a_o star_n as_o in_o the_o small_a quadrant_n to_o be_v cast_v up_o by_o the_o pen._n it_o also_o serve_v to_o find_v the_o true_a hour_n of_o the_o night_n with_o compass_n first_o have_v find_v the_o star_n hour_n take_v the_o distance_n on_o the_o quadrant_n of_o ascension_n in_o the_o same_o 12_o hour_n between_o the_o star_n and_o the_o sun_n ascension_n give_v by_o the_o foreside_n of_o the_o quadrant_n the_o say_a extent_n shall_v reach_v from_o the_o star_n hour_n to_o the_o true_a hour_n of_o the_o night_n and_o the_o foot_n of_o the_o compass_n always_o fall_v upon_o the_o quadrant_n which_o extent_n must_v be_v apply_v the_o same_o way_n it_o be_v take_v the_o sun_n foot_n to_o the_o star_n hour_n example_n if_o upon_o the_o 30_o of_o december_n the_o last_o in_o the_o end_n of_o the_o bear_n tail_n be_v find_v to_o be_v 9_o hour_n 05′_n past_o the_o meridian_n it_o be_v last_o upon_o the_o true_a time_n seek_v will_v be_v 16_o minute_n past_o 3_o in_o the_o morning_n another_o example_n for_o the_o bull_n eye_n admit_v the_o altitude_n of_o that_o star_n be_v 39_o d_o that_o star_n hour_n as_o we_o find_v it_o by_o the_o line_n of_o verse_v sin_n be_v 3_o ho_o 3′_n from_o the_o meridian_n if_o the_o altitude_n increase_v than_o that_o star_n hour_n from_o the_o meridian_n it_o be_v last_o upon_o be_v 57_o minute_n past_o 8_o 8_o h_z 57′_n if_o this_o observation_n be_v upon_o the_o 23_o of_o october_n the_o compliment_n of_o the_o sun_n ascension_n will_v be_v 9_o 30_o the_o ascension_n of_o that_o star_n be_v 4_o 16_o the_o true_a time_n of_o the_o night_n will_v be_v forty_o 10_o 43_o three_o minute_n past_o ten_o the_o distance_n between_o the_o star_n and_o the_o sun_n ascension_n be_v apply_v the_o same_o way_n by_o set_v the_o sun_n foot_n at_o the_o star_n hour_n will_v show_v the_o true_a time_n seek_v when_o the_o star_n be_v pass_v the_o meridian_n have_v the_o same_o altitude_n the_o star_n hour_n will_v be_v 3′_n past_o 3_o and_o the_o true_a time_n seek_v will_v be_v 49′_n past_o 4_o in_o the_o next_o morning_n the_o geometrical_a construction_n of_o mr_n foster_n circle_n the_o circle_n on_o the_o back_n side_n of_o the_o quadrant_n whereof_o one_o quarter_n be_v only_o a_o void_a line_n be_v derive_v from_o m._n foster_n treatise_n of_o a_o quadrant_n by_o he_o publish_v in_o an●o_n 1638._o the_o foundation_n and_o use_n whereof_o be_v conceal_v i_o shall_v therefore_o endeavour_v to_o explain_v it_o upon_o the_o centre_n h_n describe_v a_o circle_n and_o draw_v the_o diameter_n a_o c_o pass_v through_o the_o centre_n and_o 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a_o proportion_n then_o here_o be_v do_v but_o not_o conduce_v to_o this_o present_a purpose_n and_o therefore_o not_o mention_v till_o there_o be_v use_n of_o it_o which_o will_v be_v upon_o the_o backside_n of_o a_o great_a quadrant_n of_o a_o different_a contrivance_n from_o this_o upon_o which_o this_o trouble_n with_o the_o tangent_n will_v be_v shun_v a_o example_n for_o this_o case_n as_o the_o tangent_fw-la of_o 65_o to_z tangent_fw-la of_o 24_o so_o the_o tangent_fw-la of_o 20_o to_o what_o tangent_fw-la the_o proportion_n will_v find_v 4_o 10′_n divide_v into_o two_o proportion_n will_v be_v as_o radius_fw-la to_o tangent_fw-la 24_o so_o tangent_fw-la of_o 20_o to_o a_o four_o the_o quantity_n whereof_o need_v not_o be_v measure_v again_o as_o radius_fw-la to_o that_o four_o so_o the_o tangent_fw-la of_o 25_o the_o compliment_n of_o the_o first_o term_n to_o the_o tangent_fw-la of_o 4_o 26′_n the_o four_o tangent_fw-la seek_v operation_n first_o enter_v the_o tangent_fw-la 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hence_o it_o may_v be_v observe_v that_o a_o table_n of_o natural_a tangent_n only_o to_o 45_o or_o a_o line_n of_o natural_a tangent_n only_o to_o 45_o may_v serve_v to_o operate_v any_o proportion_n in_o tangent_n whatsoever_o to_o proportion_n on_o out_o a_o tangent_fw-la to_o any_o radius_fw-la enter_v the_o length_n of_o the_o radius_fw-la propose_v upon_o the_o tangent_fw-la of_o 45_o and_o to_o the_o other_o foot_n of_o the_o compass_n lay_v the_o thread_n according_a to_o the_o near_a distance_n then_o if_o the_o respective_a near_a distance_n from_o each_o degree_n of_o the_o tangent_n to_o the_o thread_n be_v take_v out_o they_o shall_v be_v tangent_n to_o the_o assign_v radius_fw-la because_o the_o tangent_n run_v but_o to_o 63_o 26′_n whereas_o there_o may_v be_v occasion_n in_o some_o decline_a dial_n to_o use_v they_o to_o 75_o though_o seldom_o further_o to_o supply_v this_o defect_n they_o may_v be_v suppose_v to_o break_v off_o at_o 60_o and_o be_v supply_v in_o a_o line_n by_o themselves_o not_o issue_v from_o the_o centre_n or_o only_a prick_n or_o full-point_n make_v at_o each_o quarter_n of_o a_o hour_n for_o the_o 5_o hour_n that_o be_v from_o 60_o to_o 75_o and_o so_o these_o distance_n prick_v again_o from_o the_o centre_n as_o here_o be_v do_v either_o one_o way_n or_o other_o the_o proportion_n will_v hold_v as_o the_o common_a radius_fw-la of_o the_o tangent_n be_v to_o any_o other_o assign_v radius_fw-la so_o be_v the_o difference_n of_o any_o two_o tangent_n to_o the_o common_a radius_fw-la to_o their_o proportional_a difference_n in_o that_o assign_v radius_fw-la and_o so_o have_v proportion_v out_o the_o first_o four_o hour_n the_o 5_o hour_n may_v be_v likewise_o proportion_v out_o and_o prick_v forward_o in_o one_o continue_v straight_o line_n from_o the_o end_n of_o the_o 4_o hour_n to_o work_v proportion_n in_o sines_n and_o tangent_n by_o help_n of_o the_o limb_n and_o line_n of_o tangent_n issue_v from_o the_o centre_n though_o this_o work_n may_v be_v better_o do_v on_o the_o backside_n where_o the_o tangent_n lie_v in_o the_o limb_n and_o the_o sin_n issue_n from_o the_o centre_n and_o where_o also_o there_o be_v a_o secant_fw-la meet_v for_o the_o vary_a of_o some_o proportion_n that_o may_v excur_v yet_o they_o may_v be_v also_o perform_v here_o suppose_v the_o radius_fw-la introduce_v into_o any_o proportion_n wherein_o it_o be_v not_o ingredient_n the_o two_o middle_a term_n not_o be_v of_o the_o same_o kind_n as_o both_o tangent_n or_o both_o sin_n to_o find_v the_o 4_o proportional_a if_o it_o be_v a_o sine_fw-la lay_v the_o thread_n to_o the_o sine_fw-la in_o the_o limb_n be_v one_o of_o the_o middle_a term_n and_o from_o the_o tangent_fw-la be_v the_o other_o middle_n term_v take_v the_o near_a distance_n

sine_fw-la and_o enter_v the_o former_a extent_n between_o the_o scale_n and_o the_o thread_n and_o the_o foot_n of_o the_o compass_n will_v on_o the_o line_n of_o equal_a part_n show_v the_o four_o proportional_a the_o proportion_n for_o find_v the_o altitude_n of_o a_o tower_n at_o one_o station_n by_o the_o measure_a distance_n may_v also_o be_v wrought_v in_o in_o equal_a part_n and_o sin_n for_o as_o the_o cousin_n of_o the_o ark_n at_o first_o station_n to_o the_o measure_a distance_n thereof_o from_o the_o tower_n so_o be_v the_o sine_fw-la of_o the_o say_a ark_n to_o the_o altitude_n of_o the_o tower_n in_o that_o former_a scheme_n the_o measure_a distance_n b_o h_n be_v 85_o and_o the_o angle_n observe_v at_o h_n 48_o d_o 29′_n wherefore_o i_o lie_v the_o thread_n to_o the_o sine_fw-la of_o the_o say_a ark_n in_o the_o limb_n count_v from_o the_o right_a edge_n and_o from_o the_o measure_a distance_n in_o the_o equal_a part_n take_v the_o near_a extent_n to_o the_o thread_n then_o lay_v the_o thread_n to_o the_o cousin_n of_o the_o say_a ark_n in_o the_o limb_n and_o enter_v the_o former_a extent_n between_o the_o thread_n and_o the_o scale_n i_o shall_v find_v the_o foot_n of_o the_o compass_n to_o fall_v upon_o 96_o the_o altitude_n seek_v so_o also_o in_o the_o triangle_n a_o c_o b_o if_o there_o be_v give_v the_o side_n a_o c_o 194_o the_o measure_a distance_n between_o two_o station_n on_o the_o wall_n of_o a_o town_n besiege_v and_o the_o observe_v angle_n at_o a_o 25_o d_o 22′_n at_o c_o 113_o d_o 22′_n if_o b_o be_v a_o battery_n we_o may_v by_o this_o work_n find_v the_o distance_n of_o it_o from_o either_o a_o or_o c_o for_o have_v two_o angle_n give_v all_o the_o three_o be_v give_v it_o therefore_o hold_v as_o the_o sine_fw-la of_o the_o angle_n ot_fw-mi b_o 41d_o 16′_n to_o its_o opposite_a side_n a_o c_o 194_o so_o the_o sine_fw-la of_o the_o angle_n at_o c_o 66d_o 38′_n the_o compliment_n to_o its_o opposite_a side_n b_o a_o 270_o the_o distance_n of_o the_o battery_n from_o a_o such_o proportion_n as_o have_v the_o radius_fw-la in_o they_o will_v be_v more_o easy_o wrought_v we_o shall_v give_v some_o few_o example_n in_o use_n in_o navigation_n 1._o to_o find_v how_o many_o mile_n or_o league_n in_o each_o parralel_n of_o latitude_n answer_v to_o one_o degree_n of_o longitude_n as_o the_o radius_fw-la to_o the_o cousin_n of_o the_o latitude_n so_o the_o number_n of_o miles_n in_o a_o degree_n in_o the_o equinoctial_a to_o the_o number_n of_o miles_n in_o the_o parralel_n so_o in_o 51_o d_o 32′_n of_o latitude_n if_o 60_o mile_n answer_v to_o a_o degree_n in_o the_o equinoctial_a 37_o ‑_o 3_o miles_n shall_v answer_v to_o one_o degree_n in_o this_o parralel_n this_o be_v wrought_v by_o lay_v the_o thread_n to_o 51_o d_o 32′_n in_o the_o limb_n from_o the_o left_a edge_n towards_o the_o right_n then_o take_v the_o near_a distance_n to_o it_o from_o 60_o in_o the_o equal_a part_n which_o measure_v from_o the_o centre_n will_v be_v find_v to_o reach_v to_o 37_o ‑_o 3_o as_o before_o the_o reason_n of_o this_o facile_a operation_n be_v because_o the_o near_a distance_n from_o the_o end_n of_o the_o line_n of_o equal_a part_n to_o the_o thread_n be_v equal_a to_o the_o cousin_n of_o the_o latitude_n the_o scale_n itself_o be_v equal_a to_o the_o radius_fw-la and_o therefore_o need_v not_o be_v take_v out_o of_o a_o scale_n of_o sines_n and_o enter_v upon_o the_o first_o term_v the_o radius_fw-la as_o in_o other_o proportion_n in_o sin_n of_o of_o the_o great_a to_o the_o less_o when_o wrought_v upon_o a_o single_a line_n only_o issue_v from_o the_o centre_n where_o the_o second_o term_n must_v be_v take_v out_o of_o a_o scale_n and_o enter_v upon_o the_o first_o term_n 2._o the_o course_n and_o distance_n give_v to_o find_v the_o difference_n of_o latitude_n in_o league_n or_o mile_n as_o the_o radius_fw-la to_o the_o cousin_n of_o the_o rumb_n from_o the_o meridian_n so_o the_o distance_n sail_v to_o the_o difference_n of_o latitude_n in_o like_a part_n example_n a_o ship_n sail_v s_o w_n by_o w_n that_o be_v on_o a_o rumb_n 56_o d_o 15′_n from_o the_o meridian_n 60_o miles_n the_o difference_n of_o latitude_n in_o miles_n will_v be_v find_v to_o be_v 33_o ‑_o 3_o the_o operation_n be_v all_o one_o with_o the_o former_a lay_v the_o thread_n to_o the_o rumb_n in_o the_o limb_n and_o from_o 60_o take_v the_o near_a distance_n to_o it_o which_o measure_v in_o the_o scale_n of_o equal_a part_n will_v be_v find_v as_o before_o 3._o the_o course_n and_o distance_n give_v to_o find_v the_o departure_n from_o the_o meridian_n alias_o the_o variation_n as_o the_o radius_fw-la to_o the_o sine_fw-la of_o the_o rumb_n from_o the_o meridian_n so_o the_o distance_n sail_v to_o the_o departure_n from_o the_o meridian_n in_o the_o former_a example_n to_o find_v the_o departure_n from_o the_o meridian_n lay_v the_o thread_n to_o the_o rumb_n count_v from_o the_o right_a edge_n towards_o the_o left_a that_o be_v to_o 56_o 15′_n so_o count_v and_o from_o 60_o in_o the_o equal_a part_n be_v the_o miles_n sail_v take_v the_o near_a distance_n to_o it_o this_o extent_n measure_v in_o the_o say_a scale_n will_v be_v find_v to_o be_v 49_o ‑_o 9_o miles_n and_o so_o if_o the_o converse_n of_o this_o be_v to_o be_v wrought_v it_o be_v evident_a that_o the_o mile_n of_o departure_n must_v be_v take_v out_o of_o the_o scale_n of_o equal_a part_n and_o enter_v parralelly_n between_o the_o scale_n and_o the_o thread_n lie_v over_o the_o rumb_n many_o more_o example_n and_o proposition_n may_v be_v illustrate_v but_o these_o be_v sufficient_a those_o that_o will_v use_v a_o quadrant_n for_o this_o purpose_n may_v have_v the_o rumb_n trace_v out_o or_o prick_v upon_o the_o limb_n now_o we_o repair_v to_o the_o backside_n of_o the_o quadrant_n of_o the_o line_n of_o on_o the_o right_a edge_n of_o the_o backside_n the_o use_v of_o this_o line_n be_v manifold_a in_o dyall_v in_o draw_v projection_n in_o work_a proportion_n etc._n etc._n 1._o to_o take_v of_o a_o proportional_a sine_fw-la to_o any_o lesser_a radius_fw-la than_o the_o side_n of_o the_o quadrant_n or_o which_o be_v all_o one_o to_o divide_v any_o line_n short_a in_o length_n then_o the_o whole_a line_n of_o sines_n in_o such_o manner_n as_o the_o same_o be_v divide_v enter_v the_o length_n of_o the_o line_n propose_v at_o 90_o d_o the_o end_n of_o the_o scale_n of_o sines_n and_o to_o the_o other_o foot_n lay_v the_o thread_n according_a to_o near_a distance_n or_o measure_v the_o length_n of_o the_o line_n propose_v on_o the_o line_n of_o sines_n from_o the_o centre_n and_o observe_v to_o what_o sine_fw-la it_o be_v equal_a then_o lay_v the_o thread_n over_o the_o like_a arch_n in_o the_o limb_n and_o the_o near_a distance_n to_o it_o from_o each_o degree_n of_o the_o line_n of_o sines_n shall_v be_v the_o proportional_a part_n seek_v and_o if_o the_o thread_n be_v lay_v over_o 30_o d_o of_o the_o limb_n the_o near_a distance_n to_o it_o will_v be_v sin_n to_o half_a the_o common_a radius_fw-la 2._o from_o a_o line_n of_o sines_n to_o take_v off_o a_o tangent_fw-la the_o proportion_n to_o do_v it_o be_v as_o the_o cousin_n of_o a_o arch_n to_o the_o radius_fw-la of_o the_o line_n propose_v so_o the_o sine_fw-la of_o the_o say_a arch_n to_o the_o tangent_fw-la of_o the_o say_v arch._n enter_v the_o radius_fw-la of_o the_o tangent_fw-la propose_v at_o the_o cousin_n of_o the_o give_v arch_n and_o to_o the_o other_o foot_n lay_v the_o thread_n then_o from_o the_o sine_fw-la of_o that_o arch_n take_v the_o near_a distance_n to_o the_o thread_n this_o extent_n be_v the_o length_n of_o the_o tangent_fw-la seek_v thus_o to_o get_v the_o tangent_fw-la of_o 20_o d_o enter_v the_o radius_fw-la propose_v at_o the_o sine_fw-la of_o 70_o d_o then_o take_v the_o near_a distance_n to_o the_o thread_n from_o the_o sine_fw-la of_o 20_o d_o this_o extent_n be_v the_o tangent_fw-la of_o the_o say_a arch_n in_o reference_n to_o the_o limit_a radius_fw-la otherways_o by_o the_o limb._n lay_v the_o thread_n to_o the_o sine_fw-la of_o that_o arch_n count_v from_o the_o right_a edge_n whereto_o you_o will_v take_v out_o a_o tangent_fw-la and_o enter_v the_o radius_fw-la propose_v down_o the_o line_n of_o sines_n from_o the_o centre_n and_o take_v the_o near_a distance_n to_o the_o thread_n than_o lie_v the_o thread_n to_o the_o like_a arch_n from_o the_o left_a edge_n and_o enter_v the_o extent_n between_o the_o scale_n and_o the_o thread_n the_o distance_n of_o the_o foot_n of_o the_o compass_n from_o the_o centre_n shall_v be_v the_o length_n of_o the_o

sine_fw-la of_o 13_o and_o enter_v one_o foot_n of_o it_o on_o the_o sine_fw-la of_o 38_o 28′_n and_o to_o the_o other_o foot_n lay_v the_o thread_n and_o in_o the_o limb_n it_o show_v the_o amplitude_n seek_v to_o be_v 21_o 12′_n by_o change_v the_o place_n of_o the_o two_o middle_a term_n this_o example_n will_v be_v turn_v into_o a_o parralel_n entrance_n lay_v the_o thread_n to_o the_o compliment_n of_o the_o latitude_n in_o the_o limb_n and_o enter_v the_o sine_fw-la of_o the_o declination_n between_o it_o and_o the_o scale_n and_o you_o will_v find_v the_o same_o ark_n in_o the_o sin_n for_o the_o amplitude_n seek_v as_o be_v before_o find_v in_o the_o limb._n such_o proportion_n of_o the_o great_a to_o the_o less_o wherein_o the_o radius_fw-la be_v not_o ingredient_n that_o have_v two_o fix_a or_o constant_a term_n may_v be_v most_o ready_o perform_v by_o the_o single_a line_n of_o sines_n without_o the_o help_n of_o the_o limb._n a_o example_n for_o find_v the_o sun_n amplitude_n as_o the_o cousin_n of_o the_o latitude_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 sun_n amplitude_n because_o the_o two_o first_o term_n of_o this_o proportion_n be_v fix_v the_o amplitude_n answerable_a to_o every_o degree_n of_o the_o sun_n place_n may_v be_v find_v without_o remove_v the_o thread_n to_o do_v it_o enter_v the_o sine_fw-la of_o the_o sun_n great_a declination_n 23_o 31′_n at_o the_o sine_fw-la of_o the_o latitude_n compliment_n and_o to_o the_o other_o foot_n lay_v the_o thread_n where_o keep_v it_o without_o alteration_n then_o for_o every_o degree_n of_o the_o sun_n place_n count_v in_o the_o sin_n take_v the_o near_a distance_n to_o the_o thread_n and_o measure_v those_o extent_n down_o the_o line_n of_o sines_n from_o the_o centre_n and_o you_o will_v find_v the_o correspondent_a amplitude_n example_n so_o when_o the_o sun_n enter_v ♉_o ♍_o ♏_o ♓_o his_o equinoctial_a distance_n be_v 30_o d_o the_o amplitude_n will_v be_v 18_o d_o 41′_n and_o when_o he_o enter_v ♊_o ♌_o ♐_o ♒_o equinox_n distance_n 60_o d_o the_o amplitude_n will_v be_v 33_o d_o 42′_n and_o when_o he_o enter_v ♋_o ♑_o the_o great_a amplitude_n will_v be_v 39_o 50′_n his_o distance_n from_o the_o near_a equinoctial_a point_n be_v 90_o d._n but_o for_o such_o proportion_n in_o which_o there_o be_v not_o two_o fix_v term_n the_o best_a way_n to_o operate_v they_o will_v be_v by_o the_o joint_a help_n of_o the_o limb_n and_o line_n of_o sines_n a_o example_n for_o find_v the_o time_n of_o the_o day_n the_o sun_n azimuth_n declination_n and_o altitude_n be_v give_v by_o the_o sun_n azimuth_n be_v mean_v the_o angle_n thereof_o from_o the_o midnight_n part_n of_o the_o meridian_n the_o proportion_n be_v 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 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 example_n so_o when_o the_o sun_n have_v 18_o d_o 37′_n north_n declination_n if_o his_o azimuth_n be_v 69_o d_o from_o the_o meridian_n and_o the_o altitude_n 39_o d_o the_o hour_n will_v be_v find_v to_o be_v 49_o d_o 58′_n from_o noon_n so_o if_o there_o be_v give_v the_o hour_n the_o declination_n and_o altitude_n by_o transpose_v the_o order_n of_o the_o former_a proportion_n it_o will_v hold_v 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 the_o cousin_n of_o the_o sun_n declination_n to_o the_o sine_fw-la of_o the_o azimuth_n from_o the_o meridian_n common_o in_o both_o these_o case_n the_o latitude_n be_v also_o know_v and_o the_o affection_n be_v to_o be_v determine_v according_a to_o rule_n former_o give_v a_o proportion_n whole_o in_o secant_v we_o have_v show_v before_o may_v be_v change_v whole_o into_o sines_n but_o the_o like_a mutual_a conversion_n of_o the_o sin_n into_o tangent_n be_v not_o yet_o know_v however_o it_o may_v be_v do_v in_o 〈◊〉_d of_o the_o 16_o case_n wherein_o the_o radius_fw-la be_v ingredient_n for_o instance_n let_v the_o proportion_n be_v to_o find_v the_o time_n of_o sun_n rise_v as_o radius_fw-la to_o tangent_fw-la of_o latitude_n so_o the_o tangent_fw-la of_o the_o declination_n to_o the_o sine_fw-la of_o the_o hour_n from_o 6._o instead_o of_o the_o two_o first_o term_n it_o may_v be_v as_o the_o cousin_n of_o the_o latitude_n to_o the_o sine_fw-la of_o the_o latitude_n than_o instead_o of_o the_o tangent_fw-la of_o the_o declination_n say_v so_o be_v the_o sine_fw-la hereof_o to_o a_o four_o again_o as_o the_o cousin_n of_o the_o declination_n to_o that_o four_o so_o radius_fw-la to_o the_o sine_fw-la of_o the_o hour_n from_o six_o this_o be_v derive_v from_o the_o analemma_n by_o resolve_v a_o triangle_n one_o side_n whereof_o be_v the_o arch_n of_o a_o lesser_a circle_n if_o a_o quadrant_n want_v tangent_n or_o secant_v in_o the_o limb_n but_o may_v admit_v of_o a_o sine_fw-la from_o the_o centre_n the_o tangent_fw-la and_o secant_fw-la of_o the_o latitude_n &c_n &c_n may_v be_v take_v out_o by_o what_o have_v be_v assert_v to_o half_a the_o common_a radius_fw-la and_o mark_v on_o the_o limb_n and_o the_o quadrant_n thereby_o fit_v to_o perform_v most_o of_o the_o proposition_n of_o the_o sphere_n in_o one_o latitude_n and_o how_o to_o supply_v the_o defect_n of_o a_o line_n of_o verse_v sin_n in_o the_o limb_n shall_v afterward_o be_v show_v what_o have_v be_v speak_v concern_v a_o line_n of_o sines_n graduate_v on_o a_o quadrant_n from_o the_o centre_n may_v by_o help_n of_o the_o equal_a limb_n be_v perform_v without_o it_o 1._o a_o proportional_a sine_fw-la may_v be_v take_v off_o to_o any_o diminutive_a radius_fw-la by_o the_o definition_n of_o sines_n the_o right_a sine_fw-la of_o a_o arch_n be_v a_o line_n fall_v from_o the_o end_n of_o that_o arch_n perpendicular_o to_o the_o radius_fw-la draw_v to_o the_o other_o end_n of_o the_o say_a arch_n so_o the_o line_n h_o k_n fall_v perpendicular_o on_o the_o radius_fw-la f_o g_o shall_v be_v the_o sine_fw-la of_o the_o arch_n h_o g_o and_o by_o the_o same_o definition_n the_o line_n g_o i_o fall_v perpendicular_o on_o the_o radius_fw-la f_o h_n shall_v also_o be_v the_o sine_fw-la of_o the_o say_a arch_n and_o whether_o the_o radius_fw-la be_v big_a or_o lesser_a this_o definition_n be_v common_a but_o the_o line_n g_o i_o on_o a_o quadrant_n represent_v the_o near_a distance_n from_o the_o radius_fw-la to_o the_o thread_n therefore_o a_o sine_fw-la may_v be_v take_v off_o from_o the_o limb_n to_o any_o diminutive_a radius_fw-la to_o perform_v which_o enter_v the_o length_n or_o radius_fw-la propose_v down_o the_o straight_a line_n that_o come_v from_o the_o centre_n of_o the_o quadrant_n and_o limit_n the_o limb_n observe_v where_o the_o compass_n rest_v this_o i_o call_v the_o fix_a point_n because_o the_o compass_n must_v be_v set_v down_o at_o it_o at_o every_o take_n off_o then_o to_o take_v off_o the_o sine_fw-la of_o any_o arch_n to_o that_o radius_fw-la lay_v the_o thread_n over_o the_o arch_n count_v in_o the_o limb_n from_o the_o say_a edge_n of_o the_o quadrant_n and_o take_v the_o near_a distance_n to_o it_o for_o the_o length_n of_o the_o sine_fw-la seek_v but_o to_o take_v out_o sines_n to_o the_o radius_fw-la of_o the_o graduated_a limb_n set_v down_o one_o foot_n at_o the_o ark_n in_o the_o limb_n and_o take_v the_o near_a distance_n to_o the_o two_o edge_n line_n of_o the_o limb_n the_o one_o shall_v be_v the_o sine_fw-la the_o other_o cousin_a of_o the_o say_v ark._n 2._o a_o proportion_n in_o sines_n alone_o may_v be_v wrought_v by_o help_n of_o the_o limb_n take_v out_o one_o of_o the_o middle_a term_n by_o the_o former_a prop._n and_o enter_v it_o down_o the_o right_a edge_n from_o the_o centre_n take_v the_o near_a distance_n to_o the_o thread_n lay_v over_o the_o other_o middle_a term_n in_o the_o limb_n count_v from_o right_a edge_n then_o lay_v the_o thread_n to_o the_o first_o term_n in_o the_o limb_n and_o enter_v that_o extent_n between_o the_o right_a edge_n line_n and_o the_o thread_n the_o distance_n of_o the_o foot_n of_o the_o compass_n from_o the_o centre_n be_v the_o length_n of_o the_o sine_fw-la seek_v to_o be_v measure_v in_o the_o limb_n by_o enter_v one_o foot_n of_o that_o extent_n in_o it_o so_o that_o the_o other_o turn_v about_o may_v but_o just_a touch_n one_o of_o the_o edge_n or_o side_n line_n of_o the_o limb_n issue_v from_o the_o centre_n or_o enter_v that_o extent_n at_o the_o concurrence_n of_o the_o limb_n with_o the_o

difference_n of_o the_o say_a sin_n to_o the_o verse_v sine_fw-la of_o the_o azimuth_n as_o before_o from_o noon_n meridian_n example_n the_o 11_o of_o june_n aforesaid_a the_o ☉_o have_v 23_o d_o 31′_n of_o north_n declination_n his_o altitude_n be_v observe_v to_o be_v 18_o d_o 20′_n which_o substract_v from_o 38_o d_o 28′_n the_o remainder_n be_v 20_o d_o 8′_n take_v out_o the_o sine_fw-la thereof_o and_o set_v down_o one_o foot_n at_o the_o sine_fw-la of_o 23_o d_o 31′_n and_o set_v the_o other_o forward_o towards_o 90_o d_o then_o take_v the_o near_a to_o the_o thread_n lay_v over_o the_o secant_fw-la of_o the_o latitude_n 51_o d_o 32′_n enter_v one_o foot_n of_o this_o extent_n at_o the_o compliment_n of_o the_o altitude_n by_o reckon_v the_o altitude_n itself_o from_o 90_o d_o towards_o the_o centre_n and_o the_o thread_n lay_v to_o the_o other_o foot_n cut_v the_o line_n of_o verse_v sin_n at_o 105_o d_o the_o azimuth_n from_o the_o south_n the_o same_o day_n when_o the_o altitude_n be_v more_o than_o the_o colatitude_n suppose_v 60_o d_o 11′_n the_o remainder_n will_v be_v find_v to_o be_v 21_o d_o 43′_n take_v the_o distance_n between_o the_o sine_fw-la thereof_o and_o of_o 23_o d_o 31′_n and_o because_o the_o extent_n be_v but_o small_a enter_v it_o four_o time_n down_o the_o line_n of_o sin_n from_o the_o centre_n and_o take_v the_o near_a distance_n to_o the_o thread_n lay_v over_o the_o secant_fw-la of_o the_o latitude_n which_o enter_v at_o the_o cousin_n of_o the_o altitude_n the_o thread_n lay_v to_o the_o other_o foot_n show_v 25_o d_o in_o the_o quadruple_v verse_v sin_n for_o the_o azimuth_n from_o the_o south_n the_o proportion_n hence_o derive_v for_o the_o amplitude_n as_o the_o cousin_n of_o the_o latitude_n to_o secant_fw-la of_o the_o declination_n etc._n etc._n as_o before_o so_o in_o summer_n be_v the_o sum_n in_o winter_n the_o difference_n of_o the_o sin_n of_o the_o sun_n declination_n and_o of_o the_o compliment_n of_o the_o latitude_n to_o the_o verse_v sine_fw-la of_o the_o amplitude_n from_o noon_n meridian_n the_o proportion_n for_o the_o azimuth_n will_v be_v better_o express_v by_o make_v the_o difference_n to_o be_v a_o difference_n of_o verse_v sin_n how_o the_o verse_v sin_n in_o the_o limb_n may_v be_v spare_v in_o case_n a_o quadrant_n want_v they_o if_o a_o quadrant_n can_v only_o admit_v of_o a_o line_n of_o sin_n from_o the_o centre_n the_o common_a quadrant_n of_o mr_n gunter'_v very_o well_o may_v on_o the_o right_a edge_n above_o the_o margin_n for_o the_o number_n of_o the_o azimuth_n it_o may_v be_v easy_o fit_v for_o any_o or_o many_o latitude_n by_o set_a mark_n or_o prick_v to_o the_o tangent_fw-la and_o secant_fw-la of_o the_o latitude_n in_o the_o limb_n which_o may_v be_v take_v out_o by_o help_n of_o the_o limb_n line_n of_o sines_n or_o by_o protraction_n and_o either_o of_o these_o general_a proportion_n wrought_v upon_o it_o or_o those_o which_o follow_v if_o it_o be_v observe_v that_o whensoever_o the_o thread_n lie_v over_o the_o verse_v sine_fw-la of_o any_o ark_n in_o the_o limb_n it_o also_o at_o the_o same_o time_n lie_v over_o a_o sine_fw-la equal_a to_o half_a that_o verse_v sine_fw-la to_o the_o common_a radius_fw-la now_o because_o the_o sine_fw-la of_o 30_o double_v be_v equal_a to_o the_o radius_fw-la let_v it_o be_v observe_v whether_o the_o sine_fw-la cut_v by_o the_o thread_n be_v great_a or_o less_o than_o 30_o deg_n when_o it_o be_v less_o let_v the_o line_n of_o sines_n represent_v the_o former_a half_n of_o a_o line_n of_o verse_v sin_n and_o take_v the_o sine_fw-la of_o the_o ark_n the_o thread_n lie_v over_o and_o enter_v it_o twice_o forward_o from_o the_o end_n of_o the_o scale_n towards_o the_o centre_n and_o you_o will_v obtain_v the_o verse_v sine_fw-la of_o the_o angle_n seek_v when_o it_o be_v more_o take_v the_o distance_n between_o the_o sine_fw-la of_o 30_o d_o and_o the_o say_v sine_fw-la and_o let_v the_o line_n of_o sin_n represent_v the_o latter_a half_n of_o a_o line_n of_o verse_v sin_n enter_v the_o say_a distance_n twice_o from_o the_o centre_n and_o you_o will_v obtain_v the_o verse_v sine_fw-la of_o the_o arch_n seek_v namely_o the_o sine_fw-la of_o a_o arch_n whereto_o 90_o d_o must_v be_v add_v three_o side_n to_o find_v a_o angle_n a_o general_a proportion_n as_o the_o sine_fw-la of_o one_o of_o the_o side_n include_v a_o angle_n be_v to_o the_o secant_fw-la of_o the_o compliment_n of_o the_o other_o include_v side_n so_o be_v the_o difference_n of_o the_o verse_v sin_n of_o the_o three_o side_n and_o of_o the_o ark_n of_o difference_n between_o the_o two_o include_v side_n to_o the_o verse_v sine_fw-la of_o the_o angle_n seek_v and_o so_o be_v the_o difference_n of_o the_o verse_v sin_n of_o the_o three_o side_n and_o of_o the_o sum_n of_o the_o two_o include_v side_n to_o the_o verse_v sine_fw-la of_o the_o seek_v angle_n compliment_n to_o 180d_o to_o repeat_v the_o converse_n when_o two_o side_n and_o the_o angle_n comprehend_v be_v give_v to_o find_v the_o three_o side_n be_v needless_a if_o one_o of_o the_o contain_v side_n be_v great_a than_o a_o quadrant_n instead_o of_o it_o in_o reference_n to_o the_o two_o first_o term_n of_o the_o proportion_n take_v the_o compliment_n thereof_o to_o 180_o for_o the_o repute_a side_n but_o in_o difference_v or_o sum_v the_o two_o contain_v side_n alter_v it_o not_o and_o further_a note_n that_o the_o same_o verse_v sine_fw-la be_v common_a to_o a_o ark_n less_o than_o a_o semicircle_n and_o to_o its_o compliment_n to_o 360_o d._n the_o operation_n of_o this_o proportion_n will_v be_v whole_o like_o the_o former_a so_o that_o there_o need_v no_o direction_n but_o only_o how_o to_o take_v out_o a_o difference_n of_o two_o verse_v sin_n to_o the_o common_a radius_fw-la see_v this_o quadrant_n of_o so_o small_a a_o radius_fw-la be_v not_o capable_a of_o such_o a_o line_n from_o the_o centre_n and_o here_o note_v that_o the_o 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of_o the_o verse_v sin_n of_o the_o ark_n propose_v to_o measure_v a_o difference_n of_o verse_v sin_n to_o the_o common_a radius_fw-la in_o this_o case_n also_o the_o line_n of_o sin_n must_v sometime_o represent_v the_o former_a sometime_o the_o latter_a half_n of_o a_o line_n of_o verse_v sin_n and_o then_o one_o foot_n of_o the_o difference_n apply_v to_o one_o ark_n the_o other_o will_v fall_v in_o many_o case_n upon_o the_o ark_n seek_v each_o proportion_n various_o express_v so_o that_o possible_o either_o one_o or_o the_o other_o will_v serve_v in_o all_o case_n but_o if_o one_o of_o the_o foot_n of_o the_o compass_n fall_v beyond_o the_o centre_n of_o the_o quadrant_n to_o find_v

how_o much_o it_o fall_v beyond_o it_o bring_v the_o say_a foot_n to_o the_o centre_n and_o let_v the_o other_o fall_v backward_o on_o the_o line_n then_o will_v the_o distance_n between_o the_o say_v other_o foot_n where_o it_o now_o fall_v and_o the_o place_n where_o it_o stand_v before_o be_v equal_a to_o the_o excess_n of_o the_o former_a foot_n beyond_o the_o centre_n which_o according_o thence_o measure_v help_v you_o to_o the_o arch_n seek_v and_o its_o compliment_n both_o at_o once_o with_o due_a regard_n to_o the_o representation_n of_o the_o line_n this_o shall_v be_v well_o observe_v for_o it_o will_v be_v of_o use_n on_o other_o instrument_n a_o difference_n of_o verse_v sin_n thus_o take_v out_o to_o the_o common_a radius_fw-la must_v be_v enter_v but_o once_o down_o from_o the_o centre_n to_o take_v out_o a_o difference_n of_o verse_v sin_n to_o half_a the_o common_a radius_fw-la count_n both_o the_o ark_v propose_v on_o the_o verse_v sin_n in_o the_o limb_n and_o find_v 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the_o sun_n have_v 20_o of_o north_n declination_n if_o his_o altitude_n be_v 40_o the_o hour_n will_v be_v 46_o minute_n past_o 8_o in_o the_o morning_n or_o 14_o minute_n past_o 3_o in_o the_o afternoon_n 4._o the_o converse_n of_o the_o former_a proposition_n will_v be_v to_o find_v the_o altitude_n of_o the_o sun_n at_o any_o hour_n of_o the_o day_n or_o of_o any_o star_n at_o any_o hour_n of_o the_o night_n i_o need_v not_o insist_v on_o this_o have_v show_v the_o manner_n of_o it_o on_o the_o small_a quadrant_a only_o for_o these_o scale_n use_v the_o limb_n instead_o of_o the_o lesser_a sin_n for_o star_n the_o time_n of_o the_o night_n must_v first_o be_v turn_v into_o the_o star_n hour_n and_o then_o the_o work_v the_o same_o as_o for_o the_o sun_n 5._o to_o find_v the_o amplitude_n of_o ehe_fw-ge sun_n or_o star_n take_v out_o the_o declination_n out_o of_o the_o great_a sine_z in_o the_o parralel_n proper_a to_o the_o latitude_n and_o measure_v it_o on_o the_o line_n of_o 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the_o thread_n to_o the_o compliment_n of_o the_o declination_n for_o the_o hour_n or_o the_o compliment_n of_o the_o altitude_n for_o the_o azimuth_n in_o the_o limb_n and_o enter_v the_o respective_a extent_n parralel_o between_o the_o thread_n and_o the_o sin_n and_o find_v the_o answer_n in_o the_o sin_n we_o may_v have_v fit_v one_o scale_n on_o the_o quadrant_a to_o give_v both_o the_o hour_n and_o azimuth_n in_o the_o equal_a limb_n by_o a_o lateral_a entrance_n and_o have_v enlarge_v upon_o many_o more_o proposition_n which_o shall_v be_v handle_v in_o the_o great_a quadrant_n mr_n sutton_n be_v willing_a to_o add_v a_o backside_n to_o this_o scale_n and_o therefore_o have_v put_v on_o particular_a scale_n of_o his_o own_o for_o give_v the_o requisite_n of_o a_o upright_a decliner_n

in_o this_o latitude_n which_o he_o have_v often_o make_v upon_o ruler_n for_o carpenter_n and_o other_o artificer_n and_o diallist_n and_o whereof_o he_o be_v willing_a to_o afford_v they_o a_o print_n whereto_o i_o have_v add_v other_o scale_n for_o give_v the_o hour_n and_o azimuth_n near_o noon_n on_o the_o backside_n be_v draw_v these_o line_n a_o large_a dyall_a scale_n of_o 6_o hour_n or_o double_a tangent_n with_o a_o line_n of_o latitude_n fit_v thereto_o a_o large_a chord_n a_o line_n for_o the_o substiles_a distance_n from_o the_o meridian_n a_o line_n for_o the_o style_n height_n a_o line_n for_o the_o angle_n of_o 12_o and_o 6._o a_o line_n for_o the_o inclination_n of_o meridian_n all_o these_o scale_n relate_v to_o dyall_v a_o azimuth_n scale_v be_v two_o line_n of_o natural_a sin_n of_o the_o same_o radius_fw-la set_v together_o at_o o_o and_o thence_o number_v with_o declination_n this_o scale_n must_v be_v make_v of_o the_o same_o sine_fw-la that_o the_o hour_n scale_v follow_v be_v make_v of_o continue_a from_o o_o one_o way_n to_o 38_o 28′_n and_o the_o other_o way_n to_o 23_o 31′_n or_o further_a at_o pleasure_n but_o number_v from_o the_o beginning_n which_o be_v at_o the_o end_n of_o that_o 38_o 28′_n the_o compliment_n of_o the_o latitude_n with_o 10_o 20′_n etc._n etc._n up_o to_o 60d._o the_o hour_n scale_n be_v no_o other_o than_o a_o line_n of_o sin_n with_o the_o declination_n set_v against_o the_o meridian_n altitude_n in_o the_o latitude_n of_o london_n the_o radius_fw-la of_o which_o sine_z be_v equal_a in_o length_n to_o the_o dyall_a scale_n of_o hour_n of_o the_o use_v of_o these_o scale_n the_o line_n of_o hour_n and_o latitude_n be_v general_a for_o prick_v down_o all_o diall_n with_o centre_n as_o will_v afterward_o be_v show_v in_o the_o use_n of_o the_o great_a quadrant_n and_o by_o help_n of_o the_o scale_n of_o hour_n may_v the_o diameter_n of_o a_o such_o a_o circle_n be_v graduate_v as_o be_v place_v in_o on_o the_o back_n of_o the_o great_a quadrant_n and_o the_o line_n of_o latitude_n will_v serve_v as_o a_o chord_n to_o divide_v the_o upper_a quadrant_n and_o the_o hour_n scale_n or_o line_n of_o sines_n will_v serve_v as_o a_o chord_n to_o divide_v a_o semicircle_n who_o diameter_n be_v equal_a to_o the_o scale_n of_o hour_n into_o 90_o equal_a part_n and_o their_o subdivision_n and_o hereby_o may_v proportion_n in_o sin_n and_o tangent_n or_o tangent_n alone_o be_v wrought_v by_o protraction_n and_o so_o the_o necessary_a ark_n in_o dyall_v find_v general_o as_o be_v do_v by_o mr_n foster_n in_o the_o three_o last_o schem_n of_o his_o posthuma_n this_o will_v easy_o be_v understand_v if_o the_o use_n of_o the_o circle_n on_o this_o quadrant_n be_v well_o apprehend_v the_o particular_a scale_n give_v the_o requisite_a ark_n of_o upright_a decliner_n in_o this_o latitude_n by_o inspection_n for_o count_v the_o plain_n declination_n in_o the_o line_n of_o chord_n and_o a_o square_n lay_v over_o it_o intersect_v all_o those_o ark_n or_o to_o be_v find_v by_o apply_v the_o declination_n take_v out_o of_o the_o chord_n with_o compass_n to_o every_o other_o line_n example_n so_o if_o a_o upright_a plain_n decline_v 35_o from_o the_o meridian_n the_o substiles_a distance_n from_o the_o meridian_n will_v be_v 24_o 30′_n the_o style_n height_n 30_o 38_o the_o inclination_n of_o meridian_n 41_o 49′_n the_o angle_n of_o 12_o and_o 6_o 54_o 10_o these_o particular_a scale_n also_o resolve_v some_o of_o the_o case_n of_o right_a angle_a sphoerical_a triangle_n relate_v to_o the_o motion_n of_o the_o sun_n or_o star_n thus_o of_o the_o line_n of_o the_o style_n height_n account_v the_o declination_n in_o the_o line_n for_o the_o style_n height_n and_o against_o it_o in_o the_o chord_n stand_v the_o amplitude_n of_o the_o sun_n or_o star_n from_o the_o meridian_n example_n for_o amplitude_n so_o when_o the_o sun_n have_v 18_o of_o declination_n his_o amplitude_n will_v be_v 67_o 13′_n from_o the_o meridian_n and_o 29_o 47′_n from_o the_o vertical_a the_o reason_n hereof_o be_v because_o the_o two_o first_o fix_v term_n of_o the_o proportion_n that_o calculate_v the_o style_n height_n be_v the_o radius_fw-la and_o the_o cousin_a of_o the_o latitude_n and_o the_o two_o first_o term_n that_o calculate_v the_o amplitude_n be_v the_o cousin_n of_o the_o latitude_n and_o the_o radius_fw-la and_o therefore_o must_v as_o well_o serve_v in_o this_o case_n as_o in_o that_o on_o this_o style_n line_n may_v be_v find_v the_o sun_n altitude_n on_o all_o hour_n when_o he_o be_v in_o the_o equinoctial_a by_o apply_v the_o hour_n from_o six_o take_v from_o the_o chord_n to_o the_o other_o end_n of_o the_o style_n line_n of_o the_o substile_a line_n hereby_o we_o may_v find_v the_o time_n of_o sun_n rise_v and_o set_v take_v the_o declination_n out_o of_o the_o substilar_a line_n and_o measure_n it_o on_o the_o line_n of_o chord_n example_n so_o when_o the_o sun_n have_v 18_o of_o north_n declination_n the_o ascensional_a difference_n be_v 24_o 9′_n in_o time_n 1_o hour_n 36½_n minute_n and_o so_o much_o the_o sun_n rise_v and_o set_v from_o six_o hereby_o may_v be_v also_o find_v the_o equinoctial_a altitude_n to_o every_o azimuth_n of_o the_o line_n for_o the_o angle_n of_o 12_o and_o 6._o hereby_o we_o may_v find_v the_o time_n when_o the_o sun_n will_v be_v due_a east_n or_o west_n account_v the_o compliment_n of_o the_o declination_n in_o this_o scale_n and_o against_o it_o in_o the_o chord_n stand_v the_o hour_n from_o six_o example_n so_o when_o the_o sun_n have_v 18_o of_o north_n declination_n he_o will_v be_v east_n or_o west_n at_o 7_o in_o the_o morning_n or_o 5_o in_o the_o afternon_n by_o these_o scale_n the_o requisite_n of_o a_o east_n or_o west_n recline_v or_o incline_v dial_n in_o this_o latitude_n may_v be_v find_v 1._o the_o substiles_a distance_n from_o the_o meridian_n account_v the_o compliment_n of_o the_o reclination_n inclination_n in_o the_o chord_n and_o against_o it_o in_o the_o line_n for_o 12_o and_o 6_o stand_v the_o compliment_n of_o the_o angle_n seek_v 2._o for_o the_o style_n height_n apply_v the_o reclination_n in_o the_o lesser_a sin_n on_o the_o diagonal_a scale_n in_o the_o parralel_n proper_a to_o the_o latitude_n to_o the_o great_a sine_fw-la and_o it_o show_v the_o ark_n seek_v 3._o for_o the_o inclination_n of_o meridian_n this_o may_v be_v also_o find_v on_o the_o diagonal_a scale_n when_o the_o substiles_a distance_n be_v not_o more_o than_o the_o latitude_n by_o account_v the_o substiles_a distance_n on_o the_o great_a sine_fw-la and_o apply_v it_o to_o the_o lesser_a 4._o for_o the_o angle_n of_o 12_o and_o six_o account_v the_o compliment_n of_o the_o reclination_n in_o the_o chord_n and_o against_o it_o in_o the_o substilar_a line_n be_v the_o compliment_n of_o the_o angle_n seek_v so_o if_o a_o east_n or_o west_n plain_o recline_v or_o incline_v 35d._o the_o substiles_a distance_n from_o the_o meridian_n will_v be_v 45_o 52′_n the_o style_n height_n 26_o 41_o the_o inclination_n of_o meridian_n 66_o 27_o and_o the_o angle_n of_o 12_o and_o 6_o 56_o 55_o of_o the_o hour_n and_o azimuth_n scales_n this_o scale_n be_v fit_v to_o find_v the_o hour_n from_o noon_n in_o the_o verse_v sine_fw-la augment_v and_o the_o proportion_n to_o be_v wrought_v by_o it_o the_o same_o as_o deliver_v in_o the_o use_n of_o the_o small_a quadrant_n as_o the_o cousin_n of_o the_o declination_n be_v to_o the_o secant_fw-la of_o the_o latitude_n so_o be_v the_o difference_n of_o the_o sin_n of_o the_o sun_n propose_v and_o meridian_n altitude_n to_o the_o verse_v sine_fw-la of_o the_o hour_n from_o noon_n and_o of_o this_o one_o proportion_n we_o make_v two_o by_o introduce_v the_o radius_fw-la as_o the_o radius_fw-la be_v to_o the_o secant_fw-la of_o the_o latitude_n so_o be_v the_o former_a distance_n to_o a_o four_o by_o fit_v the_o radius_fw-la of_o the_o sin_n equal_a in_o length_n to_o the_o secant_fw-la of_o the_o latiude_n this_o first_o proportion_n be_v remove_v for_o the_o say_a difference_n of_o sin_n take_v out_o of_o this_o fit_a scale_n be_v the_o 4_o proportional_a the_o proportion_n that_o remain_v to_o be_v wrought_v upon_o the_o quadrant_n be_v as_o the_o cousin_n of_o the_o declination_n be_v to_o the_o difference_n of_o the_o sin_n take_v out_o of_o this_o fit_a scale_n so_o be_v the_o radius_fw-la to_o the_o verse_v sine_fw-la of_o the_o hour_n from_o noon_n by_o this_o mean_n if_o in_o the_o same_o proportion_n as_o we_o increase_v the_o length_n of_o the_o fit_v scale_n we_o also_o increase_v the_o verse_v sin_n lie_v in_o the_o limb_n we_o may_v find_v the_o hour_n and_o azimuth_n near_o noon_n with_o certainty_n if_o the_o

in_o the_o use_n of_o his_o scale_n publish_v since_o his_o death_n entitle_v posthuma_n fosteri_n that_o the_o sun_n place_n be_v give_v which_o for_o instrumental_a use_n may_v be_v obtain_v by_o know_v on_o what_o day_n of_o each_o month_n the_o sun_n enter_v into_o any_o sign_n and_o allow_v a_o degree_n for_o every_o day_n motion_n come_v by_o it_o prope_fw-la verum_fw-la and_o be_v seek_v in_o the_o annex_v zodiaque_fw-la which_o be_v no_o other_o than_o two_o line_n of_o 90d._o sines_n each_o make_v equal_a to_o the_o sine_fw-la of_o 23_o 31′_n the_o sun_n great_a declination_n just_a against_o it_o stand_v the_o sun_n declination_n if_o account_v in_o the_o verse_v sine_fw-la from_o 90_o each_o way_n but_o this_o for_o want_v of_o room_n and_o because_o the_o declination_n be_v more_o easy_o give_v by_o help_n of_o the_o day_n of_o the_o month_n i_o think_v fit_a to_o omit_v the_o rather_o because_o it_o may_v also_o be_v take_v from_o the_o table_n of_o declination_n but_o from_o hence_o i_o first_o observe_v that_o if_o the_o two_o first_o term_n of_o a_o proportion_n be_v fix_v if_o two_o natural_a line_n proper_a to_o those_o term_n be_v fit_v of_o a_o equal_a length_n and_o posit_v together_o if_o any_o three_o term_n be_v give_v to_o find_v a_o four_o in_o the_o same_o proportion_n it_o will_v be_v give_v by_o inspection_n as_o stand_v against_o the_o three_o but_o if_o the_o line_n stand_v asunder_o or_o a_o difference_n be_v the_o three_o term_n application_n must_v be_v make_v from_o one_o line_n to_o the_o other_o with_o compass_n as_o in_o the_o same_o scale_n there_o be_v also_o fit_v a_o line_n of_o 60_o part_n equal_a in_o length_n to_o the_o radius_fw-la of_o a_o small_a sine_fw-la serve_v to_o give_v the_o mile_n in_o every_o several_a latitude_n answerable_a to_o one_o degree_n of_o longitude_n three_o side_n give_v to_o find_v a_o angle_n the_o proportion_n as_o the_o difference_n of_o the_o verse_v sin_n of_o the_o sum_n and_o difference_n of_o any_o two_o side_n include_v a_o angle_n be_v to_o the_o diameter_n so_o be_v the_o difference_n of_o the_o verse_v sin_n of_o the_o three_o side_n and_o of_o the_o ark_n of_o difference_n between_o the_o two_o include_v side_n to_o the_o verse_v sine_fw-la of_o the_o angle_n seek_v and_o so_o be_v the_o difference_n of_o the_o verse_v sin_n of_o the_o three_o side_n and_o of_o the_o sum_n of_o the_o two_o include_v side_n to_o the_o verse_v sine_fw-la of_o the_o seek_v angle_n compliment_n to_o a_o semicircle_n corollary_n and_o see_v there_o be_v such_o proportion_n between_o the_o latter_a term_n of_o the_o foregoing_a proportion_n as_o between_o the_o former_a omit_v the_o two_o first_o term_n it_o also_o hold_v as_o the_o difference_n of_o the_o verse_v sin_n of_o the_o three_o side_n and_o of_o the_o ark_n of_o difference_n between_o the_o two_o include_v side_n be_v to_o the_o verse_v sine_fw-la of_o the_o angle_n seek_v so_o be_v the_o difference_n of_o the_o verse_v sin_n of_o the_o three_o side_n and_o of_o the_o sum_n of_o the_o two_o include_v side_n to_o the_o verse_v sine_fw-la of_o the_o seek_v angle_n compliment_n to_o 180d._o and_o this_o be_v the_o proportion_n m._n foster_n make_v use_v of_o in_o his_o scale_n page_n 25_o and_o 27._o to_o find_v the_o hour_n and_o azimuth_n by_o protraction_n as_o also_o in_o page_n 68_o in_o dyall_v when_o three_o side_n be_v give_v to_o find_v a_o angle_n by_o constitute_v two_o right_a angle_a equi-angled_n plain_a triangle_n the_o leg_n whereof_o consist_v of_o the_o 4_o term_n of_o this_o proportion_n but_o in_o that_o protraction_n work_n the_o first_o and_o three_o term_n of_o the_o proportion_n be_v give_v together_o with_o the_o sum_n of_o the_o second_o and_o four_o term_n to_o find_v out_o the_o say_a term_n respective_o the_o proportion_n for_o the_o hour_n as_o the_o difference_n of_o the_o verse_v sin_n of_o the_o sum_n and_o difference_n of_o the_o compliment_n of_o the_o latitude_n and_o of_o the_o sun_n or_o star_n distance_n from_o the_o elevate_v pole_n be_v to_o the_o diameter_n or_o verse_v sine_fw-la of_o 180_o so_o be_v the_o difference_n of_o the_o verse_v sin_n of_o the_o compliment_n of_o the_o altitude_n and_o of_o the_o ark_n of_o difference_n between_o the_o compliment_n of_o the_o latitude_n and_o of_o the_o polar_a distance_n to_o the_o verse_v sine_fw-la of_o the_o hour_n from_o noon_n and_o if_o the_o latter_a clause_n of_o the_o three_o term_n be_v the_o sum_n of_o the_o co-latitude_n and_o polar_a distance_n the_o proportion_n will_v find_v the_o verse_v sine_fw-la of_o the_o hour_n from_o midnight_n and_o if_o the_o sum_n of_o any_o two_o ark_n exceed_v a_o semicircle_n take_v its_o compliment_n to_o 360_o for_o the_o same_o verse_v sine_fw-la be_v common_a to_o both_o when_o the_o declination_n be_v towards_o the_o elevate_v pole_n the_o polar_a distance_n be_v the_o compliment_n of_o it_o to_o 90_o and_o when_o towards_o the_o depress_v pole_n the_o polar_a distance_n be_v equal_a to_o the_o sum_n of_o 90_o and_o of_o the_o declination_n add_v together_o example_n let_v the_o sun_n declination_n be_v 15_o 46′_n north_n compliment_n 74_o 14′_n the_o compliment_n of_o the_o latitude_n 38_o 28_o sum_n 112_o 42_o difference_n 35_o 46_o and_o let_v the_o altitude_n be_v 20_o compliment_n 70_o 00_o operation_n take_v the_o distance_n between_o the_o verse_v sin_n of_o 35_o 46′_n and_o of_o 112_o 42′_n and_o enter_v one_o foot_n of_o that_o extent_n at_o the_o end_n of_o the_o verse_v scale_n at_o 180_o lay_v the_o thread_n to_o the_o other_o foot_n according_a to_o near_a distance_n then_o take_v the_o distance_n between_o the_o verse_v sin_n of_o 35_o 46′_n and_o 70_o and_o enter_v that_o extent_n parallelly_n between_o the_o thread_n and_o the_o scale_n and_o the_o other_o foot_n will_v rest_v upon_o the_o verse_v sine_fw-la of_o 77_o 32′_n the_o quantity_n of_o the_o hour_n from_o the_o meridian_n be_v either_o 50′_n past_o 6_o in_o the_o morning_n or_o 10′_n past_o 5_o in_o the_o afternoon_n the_o reader_n may_v observe_v in_o this_o work_n that_o the_o thread_n lie_v over_o a_o star_n by_o enter_v the_o first_o extent_n as_o also_o that_o there_o be_v the_o same_o star_n graduate_v at_o 35_o 46′_n of_o the_o verse_v sine_fw-la and_o this_o no_o other_o than_o the_o bull_n eye_n having_z 15_o 46′_n of_o north_n declination_n for_o which_o star_n in_o this_o latitude_n there_o needs_o be_v no_o sum_v or_o difference_v of_o ark_n in_o regard_n the_o star_n declination_n vary_v not_o so_o to_o find_v that_o star_n hour_n at_o any_o time_n have_v any_o other_o altitude_n only_o lay_v the_o thread_n over_o that_o star_n in_o the_o quadrant_n and_o take_v the_o distance_n between_o the_o star_n in_o the_o scale_n and_o the_o compliment_n of_o its_o altitude_n and_o enter_v that_o extent_n parallelly_n between_o the_o thread_n and_o the_o scale_n and_o it_o find_v the_o star_n hour_n from_o the_o meridian_n thus_o when_o that_o star_n have_v 39_o of_o altitude_n its_o hour_n from_o the_o meridian_n will_v be_v find_v to_o be_v 45_o 54′_n in_o time_n 3_o hour_n 3½′_n which_o to_o get_v the_o true_a time_n of_o the_o night_n must_v be_v turn_v into_o the_o sun_n hour_n by_o help_n of_o the_o nocturnal_a on_o the_o backside_n but_o admit_v the_o sun_n declination_n and_o altitude_n to_o have_v be_v the_o same_o with_o the_o star_n the_o true_a time_n of_o the_o day_n thus_o find_v will_v have_v be_v 56½′_n past_o 9_o in_o the_o morning_n or_o 3½′_n past_o 3_o in_o the_o afternoon_n and_o thus_o the_o reader_n may_v have_v what_o star_n he_o please_v put_v on_o of_o any_o declination_n and_o for_o any_o latitude_n and_o they_o may_v be_v put_v on_o at_o such_o a_o distance_n from_o the_o centre_n that_o the_o distance_n from_o it_o to_o the_o star_n may_v be_v a_o chord_n to_o be_v measure_v in_o the_o limb_n to_o give_v the_o star_n ascensional_a difference_n or_o the_o like_a conclusion_n and_o thus_o the_o thread_n be_v once_o lay_v and_o the_o former_a point_n find_v for_o one_o example_n to_o the_o sun_n declination_n neither_o of_o they_o vary_v that_o day_n which_o be_v a_o ready_a general_a way_n for_o find_v the_o time_n of_o the_o day_n for_o the_o sun_n to_o find_v the_o semidiurnal_a and_o seminocturnal_a arks._n suppose_v the_o sun_n to_o have_v no_o altitude_n and_o the_o compliment_n of_o it_o to_o be_v 90_o and_o then_o work_v by_o the_o former_a precept_n and_o you_o will_v find_v the_o semidiurnal_a ark_n from_o the_o begin_n of_o the_o line_n and_o the_o seminocturnal_a ark_n from_o the_o end_n of_o the_o line_n which_o double_v and_o turn_v into_o time_n show_v the_o length_n of_o the_o day_n and_o

night_n and_o the_o difference_n between_o 90_o and_o either_o of_o those_o ark_n be_v the_o ascensional_a difference_n or_o time_n of_o rise_v and_o set_v from_o 6._o to_o find_v the_o azimuth_n general_o the_o proportion_n for_o this_o purpose_n have_v be_v deliver_v before_o from_o which_o it_o may_v be_v observe_v that_o there_o be_v no_o two_o term_n fix_v and_o therefore_o to_o every_o altitude_n the_o contain_v side_n of_o the_o triangle_n namely_o the_o compliment_n both_o of_o the_o altitude_n and_o latitude_n must_v be_v sum_v and_o difference_v when_o the_o proposition_n be_v to_o be_v perform_v on_o this_o line_n sole_o and_o the_o operation_n will_v be_v after_o the_o same_o manner_n as_o for_o the_o hour_n namely_o with_o a_o parallel_n entrance_n and_o this_o be_v all_o i_o shall_v say_v of_o the_o author_n general_a way_n and_o of_o any_o other_o that_o he_o use_v i_o never_o hear_v of_o those_o way_n that_o follow_v be_v of_o my_o own_o supply_n by_o help_v of_o this_o line_n to_o work_v a_o proportion_n in_o sines_n alone_o wherein_o the_o radius_fw-la lead_v as_o the_o radius_fw-la be_v to_o the_o sine_fw-la of_o any_o ark_n so_o be_v the_o sine_fw-la of_o any_o other_o ark_n to_o the_o sine_fw-la of_o a_o four_o ark._n this_o four_o sine_fw-la as_o i_o have_v say_v before_o be_v demonstrate_v by_o m._n gellibrand_n to_o be_v equal_a to_o half_a the_o difference_n of_o the_o verse_v sin_n of_o the_o sum_n and_o difference_n of_o the_o two_o middle_a term_n of_o the_o proportion_n operation_n let_v the_o proportion_n be_v as_o the_o radius_fw-la be_v to_o the_o sine_fw-la of_o 40_o so_o be_v the_o sine_fw-la of_o 27_o to_o a_o four_o sine_fw-la sum_n 67_o difference_n 13_o take_v the_o distance_n between_o the_o verse_v sin_n of_o the_o say_a sum_n and_o difference_n and_o measure_v it_o on_o the_o line_n of_o sines_n from_o the_o centre_n and_o it_o will_v reach_v to_o 17_o the_o four_o sine_fw-la seek_v by_o help_v of_o this_o line_n may_v the_o division_n of_o the_o line_n sol_fw-mi or_o proportional_a sin_n be_v graduate_v to_o any_o radius_fw-la less_o than_o half_a the_o radius_fw-la of_o the_o quadrant_n the_o canon_n be_v as_o the_o verse_v sine_fw-la of_o any_o ark_n add_v to_o a_o quadrant_n be_v to_o the_o radius_fw-la or_o length_n of_o the_o line_n sol_fw-mi so_o be_v the_o verse_v sine_fw-la of_o that_o ark_n compliment_n to_o 90_o to_o that_o length_n which_o prick_v backward_o from_o the_o end_n of_o the_o radius_fw-la of_o the_o say_a line_n shall_v graduate_v the_o arch_n propose_v example_n suppose_v you_o will_v graduate_v 20_o of_o the_o line_n sol_fw-mi enter_v the_o radius_fw-la of_o the_o say_a line_n upon_o the_o verse_v sine_fw-la of_o 110_o lay_v the_o thread_n to_o the_o other_o foot_n and_o from_o the_o verse_v sine_fw-la of_o 70_o take_v the_o near_a distance_n to_o the_o thread_n which_o prick_v from_o the_o end_n of_o the_o line_n sol_fw-mi towards_o the_o beginning_n and_o it_o shall_v graduate_v the_o say_v 20d._o this_o line_n sol_n be_v make_v use_n of_o by_o m._n foster_n in_o his_o scale_n for_o dyall_v the_o line_n of_o verse_v sin_n be_v place_v on_o the_o left_a edge_n of_o the_o foreside_n of_o the_o quadrant_n for_o the_o ready_a take_v out_o the_o difference_n of_o the_o verse_v sin_n of_o any_o two_o ark_n and_o to_o measure_v a_o difference_n of_o two_o verse_v sin_n upon_o it_o which_o be_v the_o chief_a use_n i_o shall_v make_v of_o it_o whereas_o to_o operate_v single_o upon_o it_o it_o will_v be_v more_o convenient_a for_o the_o hand_n to_o have_v it_o lie_v on_o the_o right_a edge_n of_o the_o quadrant_n a_o example_n for_o find_v the_o azimuth_n general_o by_o help_n of_o verse_v sin_n in_o the_o limb_n and_o of_o other_o line_n on_o the_o quadrant_n i_o shall_v rehearse_v the_o proportion_n as_o the_o cousin_n of_o the_o latitude_n be_v to_o the_o secant_fw-la of_o the_o altitude_n or_o as_o the_o cousin_n of_o the_o altitude_n be_v to_o the_o secant_fw-la of_o the_o latitude_n so_o be_v the_o difference_n of_o the_o verse_v sin_n of_o the_o sun_n distance_n from_o the_o elevate_v pole_n and_o of_o the_o ark_n of_o difference_n between_o the_o latititude_n and_o altitude_n to_o the_o verse_v sine_fw-la of_o the_o azimuth_n from_o the_o midnight_n meridian_n and_o make_v the_o latter_a clause_n of_o the_o three_o term_v the_o compliment_n of_o the_o sum_n of_o the_o latitude_n and_o altitude_n to_o a_o semicircle_n the_o proportion_n will_v find_v the_o verse_v sine_fw-la of_o the_o azimuth_n from_o the_o noon_n meridian_n example_n altitude_n 51_o 32′_n latitude_n 34_o 32_o compliment_n 55_o 2d_o difference_n 17_o 00_o ☉_o distance_n from_o elevate_a pole_n 66_o 29_o operation_n in_o the_o first_o term_n of_o the_o proportion_n on_o the_o line_n of_o verse_v sin_n take_v the_o distance_n between_o 17_o and_o 66_o 29′_n and_o enter_v it_o twice_o down_o the_o line_n of_o sines_n from_o the_o centre_n take_v the_o near_a distance_n to_o the_o thread_n lay_v over_o the_o secant_fw-la of_o 51_o 32′_n the_o give_v altitude_n and_o enter_v one_o foot_n of_o this_o extent_n at_o the_o sine_fw-la of_o 55_o 28′_n the_o compliment_n of_o the_o latitude_n lay_v the_o thread_n to_o the_o other_o foot_n according_a to_o near_a distance_n and_o in_o the_o line_n of_o verse_v sin_n in_o the_o limb_n it_o will_v lie_v over_o 95_o for_o the_o sun_n azimuth_n from_o the_o midnight_n meridian_n and_o the_o sun_n declination_n suppose_v the_o same_o he_o shall_v have_v the_o like_a azimuth_n from_o the_o north_n in_o our_o latitude_n of_o london_n when_o his_o altitude_n be_v 34_o 32′_n for_o the_o side_n of_o the_o triangle_n be_v the_o same_o another_o example_n to_o find_v it_o in_o the_o verse_v sine_fw-la of_o 90_o latitude_n 47_o 27′_n altitude_n 51_o 32_o sum_n 98_o 59_o compliment_n 81_o 1_o polar_a distance_n 66_o 29_o take_v the_o distance_n in_o the_o line_n of_o sines_n as_o represent_v the_o former_a half_n of_o a_o line_n of_o verse_v sin_n between_o these_o two_o ark_n count_v towards_o the_o centre_n viz._n 66_o 29′_n and_o 81_o 01′_n and_o enter_v this_o extent_n twice_o down_o the_o line_n of_o sines_n from_o the_o centre_n and_o take_v the_o near_a distance_n to_o the_o thread_n lie_v over_o the_o secant_fw-la of_o the_o latitude_n 47_o 27′_n then_o enter_v one_o foot_n of_o this_o extent_n at_o 51_o 32′_n count_v from_o the_o end_n of_o the_o sin_n towards_o the_o centre_n lay_v the_o thread_n to_o the_o other_o foot_n according_a to_o near_a distance_n and_o in_o the_o verse_v sine_fw-la of_o 90_o it_o show_v the_o azimuth_n to_o be_v 65_o from_o the_o south_n in_o this_o our_o northern_a hemispere_n of_o the_o fit_v particular_a scale_n and_o the_o line_n of_o entrance_n thereto_o belong_v this_o scale_n serve_v to_o find_v both_o the_o hour_n and_o azimuth_n in_o the_o latitude_n of_o london_n to_o which_o it_o be_v fit_v in_o the_o equal_a limb_n by_o a_o lateral_a or_o positive_a entrance_n it_o consist_v of_o two_o line_n of_o sines_n the_o great_a be_v 62_o of_o a_o sine_fw-la as_o large_a as_o can_v stand_v upon_o the_o quadrant_n the_o radius_fw-la of_o the_o lesser_a sine_fw-la be_v make_v equal_a to_o 51_o 32′_n of_o this_o great_a be_v fit_v to_o the_o latitude_n the_o scale_n of_o entrance_n stand_v within_o the_o projection_n and_o abut_a on_o the_o line_n of_o sines_n be_v no_o other_o but_o a_o portion_n of_o a_o line_n of_o sines_n who_o radius_fw-la be_v make_v equal_a to_o 38_o 28′_n of_o the_o great_a sine_fw-la of_o the_o fit_a scale_n and_o this_o scale_n of_o entrance_n be_v number_v by_o its_o compliment_n up_o to_o 62_o as_o much_o as_o be_v the_o suns-great_a meridian_n altitude_n in_o this_o latitude_n the_o ground_n of_o this_o scale_n be_v derive_v from_o the_o diagonal_a scale_n the_o length_n whereof_o bear_v such_o proportion_n to_o the_o line_n of_o sines_n whereto_o it_o be_v fit_v as_o the_o secant_fw-la of_o the_o particular_a latitude_n do_v to_o the_o radius_fw-la which_o be_v the_o same_o that_o the_o radius_fw-la bear_v to_o the_o cousin_n of_o the_o latitude_n and_o consequent_o make_v the_o line_n of_o sines_n to_o represent_v the_o fit_a scale_n the_o radius_fw-la of_o that_o sine_fw-la whereto_o it_o be_v fit_v must_v be_v equal_a to_o the_o cousin_n of_o the_o latitude_n and_o so_o we_o need_v no_o particular_a scale_n but_o this_o will_v remove_v the_o particular_a scale_n or_o scale_n of_o entrance_n near_a the_o centre_n and_o will_v not_o have_v be_v so_o ready_a as_o this_o fit_a scale_n however_o hence_o i_o may_v educe_v a_o general_a method_n for_o find_v the_o hour_n and_o azimuth_n in_o the_o limb_n without_o tangent_n or_o secant_v the_o first_o work_n will_v be_v

the_o altitude_n at_o 6_o which_o measure_v on_o the_o sin_n will_v be_v find_v to_o be_v 7_o 17′_n for_o the_o winter_n altitude_n to_o that_o hour_n so_o if_o the_o hour_n be_v 60_o from_o 6_o that_o be_v either_o 10_o or_o 2_o the_o summer_n altitude_n will_v be_v find_v to_o be_v 49_o 42′_n and_o the_o winter_n altitude_n 12_o 30′_n and_o this_o may_v be_v find_v in_o the_o verse_v sin_n on_o the_o left_a edge_n account_v as_o a_o sine_fw-la each_o way_n from_o the_o middle_n if_o use_v be_v make_v of_o the_o lesser_a sin_n instead_o of_o the_o limb_n in_o find_v the_o point_n of_o entrance_n as_o also_o in_o lay_v it_o to_o the_o sine_fw-la of_o each_o hour_n from_o 6_o in_o which_o case_n the_o compass_n will_v always_o find_v two_o altitude_n at_o once_o for_o when_o they_o fall_v beyond_o the_o midst_n of_o the_o say_a line_n it_o show_v the_o winter_n altitude_n count_v from_o thence_o towards_o the_o end_n of_o the_o say_v verse_v sines_n have_v find_v the_o four_o sine_fw-la which_o give_v the_o point_n of_o entrance_n as_o before_o the_o altitude_n on_o all_o hour_n may_v be_v find_v by_o the_o verse_v sin_n of_o 90_o in_o the_o limb_n the_o proportion_n will_v be_v as_o the_o radius_fw-la be_v to_o the_o verse_v sine_fw-la of_o the_o hour_n from_o noon_n so_o be_v the_o four_o abovesaid_a to_o the_o difference_n of_o the_o sin_n of_o the_o meridian_n altitude_n and_o of_o the_o altitude_n seek_v but_o for_o hour_n beyond_o 6_o the_o proportion_n will_v be_v as_o the_o radius_fw-la be_v to_o the_o verse_v sine_fw-la of_o the_o hour_n from_o midnight_n so_o be_v the_o four_o abovesaid_a to_o the_o sum_n of_o the_o sin_n of_o the_o sun_n depression_n at_o midnight_n equal_a to_o his_o winter_n meridian_n altitude_n and_o of_o his_o altitude_n seek_v hereby_o also_o we_o may_v find_v two_o altitude_n at_o once_o operation_n lay_v the_o thread_n to_o the_o verse_v sine_fw-la of_o the_o hour_n from_o noon_n and_o from_o the_o point_n of_o entrance_n at_o 34_o 25′_n take_v the_o near_a distance_n to_o it_o the_o say_a extent_n shall_v reach_v from_o the_o summer_n meridian_n altitude_n account_v in_o the_o sin_n to_o the_o altitude_n seek_v also_o from_o the_o winter_n meridian_n altitude_n to_o the_o altitude_n seek_v example_n latitude_n of_o nottingham_n be_v 53_o compliment_n 37_o sun_n declination_n 20_o sum_n be_v the_o summer_n meridian_n altitude_n 57_o difference_n be_v winter_n meridian_n altitude_n 17_o if_o it_o be_v require_v to_o find_v the_o altitude_n for_o the_o hour_n of_o 11_o and_o 1_o the_o extent_n so_o take_v out_o will_v find_v the_o summer_n altitude_n to_o be_v 55_o 00′_n and_o the_o winter_n altitude_n to_o the_o same_o hour_n and_o declination_n 15_o 51_o 10_o and_o 2_o the_o extent_n so_o take_v out_o will_v find_v the_o summer_n altitude_n to_o be_v 49_o 42_o and_o the_o winter_n altitude_n to_o the_o same_o hour_n and_o declination_n 12_o 30_o 9_o and_o 3_o the_o extent_n so_o take_v out_o will_v find_v the_o summer_n altitude_n to_o be_v 42_o 18_o and_o the_o winter_n altitude_n to_o the_o same_o hour_n and_o declination_n 7_o 17_o 8_o and_o 4_o the_o extent_n so_o take_v out_o will_v find_v the_o summer_n altitude_n to_o be_v 33_o 47_o and_o the_o winter_n altitude_n to_o the_o same_o hour_n and_o declination_n 00_o 32_o but_o for_o hour_n more_o remote_a from_o the_o meridian_n than_o 6_o as_o admit_v for_o 5_o in_o the_o morning_n or_o 7_o at_o night_n which_o be_v 75_o from_o the_o north_n meridian_n lay_v the_o thread_n to_o the_o say_a ark_n in_o the_o verse_v sine_fw-la of_o 90_o and_o the_o distance_n from_o the_o point_n of_o entrance_n to_o it_o shall_v reach_v from_o the_o sine_fw-la of_o 57_o the_o meridian_n altitude_n to_o the_o sine_fw-la of_o 24_o 48′_n the_o summer_n altitude_n for_o the_o hour_n 75_o from_o noon_n and_o if_o that_o extent_n be_v prick_v from_o the_o winter_n meridian_n altitude_n it_o will_v reach_v beyond_o the_o centre_n in_o which_o case_n enter_v that_o extent_n upon_o the_o line_n of_o sines_n and_o take_v the_o distance_n between_o the_o point_n of_o limitation_n and_o 17_o which_o will_v be_v measure_v be_v find_v to_o be_v the_o sine_fw-la of_o 7_o 17′_n the_o altitude_n belong_v to_o the_o hour_n 105_o from_o noon_n in_o like_a manner_n the_o altitude_n for_o the_o hour_n 97_o 30_o from_o noon_n that_o be_v 82_o 30′_n from_o midnight_n will_v be_v 11_o 31′_n and_o for_o the_o like_a hour_n from_o noon_n 20_o 17′_n 112_o 30_o from_o noon_n that_o be_v 67_o 30_o from_o midnight_n will_v be_v 3_o 15_o and_o for_o the_o like_a hour_n from_o noon_n 29_o 19_o in_o like_a manner_n it_o may_v have_v be_v find_v in_o the_o verse_v sin_n issue_v from_o the_o centre_n if_o in_o find_v the_o point_n of_o entrance_n and_o in_o lay_v the_o thread_n to_o the_o verse_v sine_fw-la of_o the_o hour_n we_o make_v use_v of_o the_o lesser_a sin_n and_o of_o the_o verse_v sine_fw-la of_o 180_o in_o the_o limb._n for_o the_o azimuth_n two_o of_o the_o former_a proportion_n may_v be_v convenient_o apply_v to_o other_o side_n for_o find_v the_o azimuth_n universal_o as_o the_o radius_fw-la be_v to_o the_o cousin_n of_o the_o latitude_n so_o be_v the_o cousin_n of_o the_o altitude_n to_o a_o four_o sine_fw-la get_v the_o sum_n of_o the_o altitude_n and_o colatitude_n or_o which_o be_v all_o one_o the_o sum_n of_o the_o latitude_n and_o colatitude_n and_o if_o it_o exceed_v a_o quadrant_n take_v its_o compliment_n to_o a_o semicircle_n this_o four_o sine_fw-la be_v equal_a to_o the_o difference_n of_o the_o sin_n of_o this_o compound_v ark_n and_o of_o another_o ark_n to_o be_v thereby_o find_v call_v the_o latter_a ark._n then_o it_o hold_v as_o the_o four_o sine_fw-la be_v to_o the_o radius_fw-la so_o in_o summer_n be_v the_o difference_n but_o in_o winter_n the_o sum_n of_o the_o sin_n of_o this_o latter_a ark_n and_o of_o the_o give_v declination_n to_o the_o sine_fw-la of_o the_o azimuth_n from_o the_o vertical_a when_o the_o latter_a ark_n be_v more_o than_o the_o declination_n the_o azimuth_n will_v be_v find_v from_o the_o vertical_a towards_o the_o noon_n meridian_n otherwise_o towards_o the_o midnight_n meridian_n and_o in_o winter_n always_o towards_o the_o noon_n meridian_n for_o such_o star_n as_o come_v to_o the_o meridian_n between_o the_o zenith_n and_o the_o elevate_a pole_n the_o four_o ark_n will_v never_o exceed_v the_o star_n declination_n and_o their_o azimuth_n will_v be_v always_o find_v from_o the_o vertical_a towards_o the_o meridian_n they_o come_v to_o above_o the_o horizon_n example_n for_o the_o latitude_n of_o nottingham_n compliment_n of_o the_o latitude_n be_v 37_o altitude_n be_v 40_o 40_o sum_n 77_o let_v the_o declination_n be_v 20_o north._n to_o find_v the_o point_n of_o entrance_n take_v the_o near_a distance_n to_o the_o thread_n lay_v over_o 50_o in_o the_o limb_n count_v from_o right_a edge_n from_o the_o sine_fw-la of_o 37_o the_o say_a extent_n measure_v from_o the_o centre_n fall_v upon_o the_o sine_fw-la of_o 27_o 26′_n and_o there_o will_v be_v the_o point_n of_o entrance_n the_o say_a extent_n prick_v from_o 77_o in_o the_o sin_n will_v reach_v to_o the_o sine_fw-la of_o 30_o 51′_n where_o the_o sine_fw-la point_n fall_v last_o the_o distance_n between_o the_o sine_fw-la point_n and_o the_o sine_fw-la of_o 20_o be_v enter_v at_o the_o point_n of_o entrance_n and_o the_o thread_n lay_v to_o the_o other_o foot_n the_o azimuth_n will_v be_v find_v in_o the_o equal_a limb_n to_o be_v 21_o 48′_n from_o the_o east_n or_o west_n southwards_o because_o the_o sine_fw-la point_n fall_v beyond_o the_o declination_n another_o example_n for_o that_o latitude_n the_o declination_n be_v 20_o south_n altitude_n 12_o 30′_n the_o point_n of_o entrance_n will_v fall_v at_o the_o sine_fw-la of_o 36_o the_o sine_fw-la point_n may_v be_v find_v without_o sum_v or_o difference_v of_o ark_n by_o take_v the_o near_a distance_n from_o the_o sine_fw-la of_o the_o latitude_n to_o the_o thread_n lay_v over_o the_o altitude_n count_v in_o the_o limb_n from_o the_o right_a edge_n which_o extent_n be_v add_v to_o the_o sine_fw-la of_o 20_o the_o declination_n the_o whole_a extent_n will_v be_v equal_a to_o the_o sine_fw-la of_o 31_o this_o be_v enter_v on_o the_o point_n of_o entrance_n and_o the_o thread_n lay_v to_o the_o other_o foot_n the_o azimuth_n will_v be_v find_v to_o be_v 61_o 14′_n from_o the_o east_n or_o west_n southwards_o a_o three_o example_n for_o the_o latitude_n of_o london_n 51_o 32′_n let_v it_o be_v require_v to_o find_v the_o azimuth_n of_o the_o middlemost_a star_n in_o the_o great_a bear_n tail_n declination_n be_v 56_o 45′_n let_v the_o altitude_n be_v

44_o 58′_n the_o near_a distance_n from_o the_o sine_fw-la of_o 38_o 28′_n to_o the_o thread_n lay_v over_o the_o altitude_n count_v from_o the_o right_a edge_n will_v find_v the_o point_n of_o entrance_n to_o be_v at_o the_o sine_fw-la of_o 26_o 6′_n the_o near_a distance_n from_o the_o sine_fw-la of_o 51_o 32′_n to_o the_o thread_n lay_v over_o the_o altitude_n count_v from_o the_o right_a edge_n need_v not_o be_v know_v but_o the_o distance_n between_o that_o extent_n and_o the_o sine_fw-la of_o 56_o 45′_n the_o star_n declination_n be_v enter_v on_o the_o point_n of_o entrance_n will_v find_v the_o azimuth_n of_o that_o star_n by_o lay_v the_o thread_n to_o the_o other_o foot_n to_o be_v 40_o from_o the_o east_n or_o west_n northwards_o thus_o we_o find_v it_o the_o general_a way_n and_o so_o it_o will_v also_o be_v find_v by_o the_o fit_v particular_a scale_n for_o the_o hour_n the_o point_n of_o entrance_n and_o sine_fw-la point_n vary_v not_o till_o the_o declination_n change_n but_o for_o the_o azimuth_n they_o vary_v to_o every_o altitude_n to_o find_v the_o azimuth_n in_o the_o verse_v sin_n as_o the_o four_o find_v by_o the_o former_a proportion_n namely_o where_o the_o point_n of_o entrance_n happen_v be_v to_o the_o radius_fw-la so_o be_v the_o difference_n of_o the_o verse_v sin_n of_o the_o polar_a distance_n and_o of_o the_o ark_n of_o difference_n between_o the_o altitude_n and_o the_o latitude_n to_o the_o verse_v sine_fw-la of_o the_o azimuth_n from_o midnight_n meridian_n this_o find_v the_o angle_n itself_o in_o the_o sphere_n and_o so_o be_v the_o difference_n of_o the_o verse_v sin_n of_o the_o polar_a distance_n and_o of_o the_o ark_n of_o residue_n of_o the_o sum_n of_o the_o latitude_n and_o altitude_n take_v from_o a_o semicircle_n to_o the_o verse_v sine_fw-la of_o the_o azimuth_n from_o noon_n meridian_n this_o find_v the_o compliment_n of_o the_o angle_n in_o the_o sphere_n to_o a_o semicircle_n the_o proportion_n to_o find_v it_o from_o midnight_n meridian_n the_o three_o term_n be_v express_v in_o sines_n will_v be_v thus_o get_v the_o sum_n of_o the_o altitude_n and_o colatitude_n and_o when_o it_o exceed_v a_o quadrant_n take_v its_o compliment_n to_o a_o semicircle_n the_o ark_n thus_o find_v be_v call_v the_o compound_v ark._n then_o it_o hold_v as_o the_o four_o find_v before_o be_v to_o the_o radius_fw-la so_o in_o summer_n declination_n be_v the_o difference_n but_o in_o winter_n declination_n the_o sum_n of_o the_o sin_n of_o the_o sun_n or_o star_n declination_n and_o of_o the_o compound_n ark_n to_o the_o verse_v sine_fw-la of_o the_o azimuth_n from_o the_o midnight_n meridian_n of_o the_o place_n use_v this_o proportion_n always_o for_o the_o sun_n or_o star_n when_o they_o come_v to_o the_o meridian_n between_o the_o zenith_n and_o elevate_v pole_n and_o to_o find_v it_o from_o the_o noon_n meridian_n get_v the_o difference_n between_o the_o altitude_n and_o colatitude_n and_o then_o it_o hold_v as_o the_o four_o sine_fw-la find_v before_o be_v to_o the_o radius_fw-la so_o be_v the_o sum_n of_o the_o sin_n of_o the_o say_a ark_n of_o difference_n and_o of_o the_o sun_n declination_n to_o the_o verse_v sine_fw-la of_o the_o azimuth_n from_o the_o noon_n meridian_n in_o summer_n only_o when_o the_o sun_n altitude_n be_v less_o than_o the_o colatitude_n in_o all_o other_o case_n so_o be_v the_o difference_n of_o the_o say_a sin_n to_o the_o verse_v sine_fw-la of_o the_o azimuth_n as_o before_o from_o noon_n meridian_n if_o by_o the_o former_a proportion_n it_o be_v require_v to_o find_v the_o azimuth_n in_o the_o verse_v sine_fw-la of_o 90_o a_o difference_n of_o verse_v sin_n take_v out_o of_o the_o line_n of_o verse_v sin_n on_o the_o left_a edge_n must_v be_v double_v and_o be_v take_v out_o of_o the_o line_n of_o sines_n as_o sometime_o represent_v the_o former_a sometime_o the_o latter_a half_n of_o a_o verse_v sine_fw-la need_v not_o be_v double_v example_n latitude_n of_o nottingham_n 53_o altitude_n of_o the_o sun_n 4_o ark_n of_o difference_n 49_o ☉_o declination_n 20_o north_n the_o polar_a distance_n be_v 70_o the_o point_n of_o entrance_n will_v fall_v at_o the_o sine_fw-la of_o 36_o 54′_n and_o the_o difference_n of_o the_o verse_v sin_n of_o 49_o and_o 70●_n equal_a to_o the_o distance_n between_o the_o sin_n of_o 41_o and_o 20_o be_v enter_v at_o the_o point_n of_o entrance_n and_o the_o tbread_n lay_v to_o the_o other_o foot_n will_v lie_v over_o 61_o 30′_n of_o the_o verse_v sine_fw-la of_o 90_o and_o so_o much_o be_v the_o sun_n azimuth_n from_o the_o north._n another_o example_n for_o find_v it_o from_o the_o south_n when_o the_o altitude_n be_v more_o than_o the_o colatitude_n altitude_n 47_o colatitude_n 37_o of_o nottingham_n difference_n 10_o the_o point_n of_o entrance_n will_v fall_v at_o the_o sine_fw-la of_o 24_o 14′_n find_v by_o take_v the_o near_a extent_n from_o sine_fw-la of_o 37_o to_o the_o thread_n lie_v over_o 43_o of_o the_o limb_n the_o coaltitude_n then_o the_o distance_n between_o the_o sin_n of_o 10_o the_o ark_n of_o difference_n as_o above_o and_o the_o sine_z of_o 20_o the_o sun_n north_n declination_n be_v enter_v at_o the_o point_n of_o entrance_n and_o the_o thread_n lay_v to_o the_o other_o foot_n will_v show_v 53_o 55′_n in_o the_o verse_v sine_fw-la of_o 90_o for_o the_o sun_n azimuth_n from_o the_o south_n a_o three_o example_n when_o the_o altitude_n be_v less_o than_o the_o colatitude_n in_o summer_n compliment_n latitude_n 37_o of_o nottingham_n altitude_n 34_o difference_n 3_o the_o point_n of_o entrance_n will_v fall_v at_o the_o sine_fw-la of_o 29_o 55′_n and_o the_o sum_n of_o the_o sin_n of_o 3_o and_o of_o 20_o the_o sun_n declination_n suppose_v north_n be_v equal_a to_o the_o sine_fw-la of_o 23_o 13′_n which_o extent_n enter_v at_o 29_o 55′_n the_o point_n of_o entrance_n and_o the_o thread_n lay_v to_o the_o other_o foot_n according_a to_o near_a distance_n it_o will_v intersect_v the_o verse_v sine_fw-la of_o 90_o at_o the_o ark_n of_o 77_o 57′_n and_o so_o much_o be_v the_o sun_n azimuth_n from_o the_o south_n and_o if_o there_o be_v no_o verse_v sin_n in_o the_o limb_n find_v a_o ark_n of_o the_o equal_a limb_n and_o enter_v the_o sine_fw-la of_o the_o say_a ark_n down_o the_o line_n of_o sin_n from_o the_o other_o end_n and_o you_o may_v obtain_v the_o verse_v sine_fw-la of_o the_o ark_n seek_v more_o example_n need_v not_o be_v insist_v upon_o have_v find_v the_o point_n of_o entrance_n the_o distance_n between_o the_o verse_v sin_n of_o the_o base_a or_o side_n subtend_v the_o angle_n seek_v and_o of_o the_o ark_n of_o difference_n between_o the_o two_o include_v side_n be_v take_v out_o of_o the_o straight_a line_n of_o verse_v line_n on_o the_o left_a edge_n and_o enter_v at_o the_o point_n of_o entrance_n lay_v the_o thread_n to_o the_o other_o foot_n show_v in_o the_o verse_v sine_fw-la of_o 180_o in_o the_o limb_n the_o angle_n seek_v and_o if_o the_o say_a distance_n or_o extent_n be_v double_v and_o there_o enter_v it_o show_v the_o angle_n seek_v in_o the_o verse_v sine_fw-la of_o 90_o when_o the_o angle_n be_v less_o than_o a_o quadrant_n when_o more_o the_o distance_n between_o the_o verse_v sin_n of_o the_o base_a and_o the_o sum_n of_o the_o leg_n will_v find_v the_o compliment_n of_o the_o angle_n seek_v to_o a_o semicircle_n without_o double_v in_o the_o verse_v sine_fw-la of_o 180_o in_o the_o limb_n with_o double_v in_o the_o verse_v sine_fw-la of_o 90d._o last_o three_o side_n viz._n all_o less_o than_o quadrant_n or_o one_o of_o they_o great_a general_o to_o find_v a_o angle_n in_o the_o equal_a limb_n the_o proportion_n will_v be_v as_o the_o radius_fw-la be_v to_o the_o cousin_n of_o one_o of_o the_o including_z side_n so_o be_v the_o cousin_n of_o the_o other_o includer_n to_o a_o four_o sine_fw-la again_o as_o the_o sine_fw-la of_o one_o of_o the_o includer_n to_o the_o cosecant_v of_o the_o other_o so_o when_o any_o one_o of_o the_o side_n be_v great_a than_o a_o quadrant_n be_v the_o sum_n but_o when_o all_o less_o the_o difference_n of_o the_o four_o sine_fw-la and_o of_o the_o cousin_n of_o the_o three_o side_n to_o the_o cousin_n of_o the_o angle_n seek_v if_o any_o of_o the_o three_o side_n be_v great_a than_o a_o quadrant_n it_o subtend_v a_o obtuse_a angle_n the_o other_o angle_v be_v acute_a but_o when_o they_o be_v all_o less_o than_o quadrant_n if_o the_o 4_o sine_fw-la be_v less_o than_o the_o cousin_n of_o the_o three_o side_n the_o angle_n seek_v be_v acute_a if_o equal_a thereto_o it_o be_v a_o right_a angle_n if_o great_a a_o obtuse_a angle_n from_o the_o proportion_n that_o find_v the_o hour_n from_o

divide_v but_o the_o author_n make_v his_o table_n in_o page_n 5._o without_o double_v to_o be_v graduate_v from_o a_o quadrant_n divide_v into_o 45_o equal_a part_n again_o if_o upon_o the_o centre_n c_o with_o a_o pair_n of_o compass_n each_o degree_n of_o the_o line_n of_o sines_n be_v transfer_v into_o the_o semicircle_n c_o g_o a_o it_o shall_v divide_v it_o into_o 90_o equal_a part_n the_o reason_n whereof_o be_v plain_a because_o the_o sine_fw-la of_o a_o arch_n be_v half_a the_o chord_n of_o twice_o that_o arch_n and_o therefore_o the_o sin_n be_v make_v to_o twice_o the_o radius_fw-la of_o this_o circle_n shall_v be_v transfer_v into_o it_o become_v chord_n of_o the_o like_a arch_n to_o divide_v a_o semicircle_n into_o 90_o equal_a part_n again_o upon_o the_o point_n a_o erect_v a_o line_n of_o tangent_n of_o the_o same_o radius_fw-la with_o the_o former_a sine_fw-la which_o we_o may_v suppose_v to_o be_v infinite_o continue_v here_o we_o use_v a_o portion_n of_o it_o a_o e._n if_o from_o the_o point_n c_o the_o other_o extremity_n of_o the_o diameter_n line_n be_v draw_v cut_v the_o low_a semicircle_n as_o a_o line_n draw_v from_o e_z intersect_v it_o at_o f_o through_o each_o degree_n of_o the_o say_v tangent_fw-la the_o say_v low_a semicircle_n shall_v be_v divide_v into_o 90_o equal_a part_n the_o reason_n be_v evident_a a_o line_n of_o tangent_n from_o the_o centre_n shall_v divide_v a_o quadrant_n into_o 90_o equal_a part_n and_o because_o a_o angle_n in_o the_o circumference_n be_v but_o half_a so_o much_o as_o it_o be_v in_o the_o centre_n be_v transfer_v thither_o a_o whole_a semicircle_n shall_v be_v fill_v with_o no_o more_o part_n the_o chief_a use_n of_o this_o circle_n be_v to_o operate_v proportion_n in_o tangent_n alone_o or_o in_o sines_n and_o tangent_n joint_o build_v upon_o this_o foundation_n that_o equiangle_v plain_a triangle_n have_v their_o side_n proportional_a in_o straight_a line_n it_o will_v be_v evident_a from_o the_o point_n d_o to_o e_z draw_v a_o straight_a line_n intersect_v the_o diameter_n at_o l_o and_o then_o it_o lie_v as_o c_o l_o to_o c_z d_o so_o be_v a_o l_o to_o a_o e_o it_o be_v also_o true_a in_o a_o circle_n provide_v it_o be_v evince_v that_o the_o point_n b_o l_o f_o fall_n in_o a_o straight_a line_n hereof_o i_o have_v a_o geometrical_a demonstration_n which_o will_v require_v more_o scheme_n which_o by_o reason_n of_o its_o length_n and_o difficulty_n i_o think_v fit_a at_o present_a not_o to_o insert_v possible_o a_o easy_a may_v be_v find_v hereafter_o as_o also_o a_o algebraick_n demonstration_n by_o the_o right_n honourable_a the_o lord_n brunkard_n whereby_o after_o many_o algebraick_n inference_n it_o be_v evince_v that_o as_o l_o k_n be_v to_o k_o b_o ∷_o so_o be_v l_o n_o to_o ●_o f_o whence_o it_o will_v follow_v that_o the_o point_n b_o l_o f_o be_v in_o a_o right_a line_n if_o a_o ruler_n be_v lay_v from_o 45_o of_o the_o semicircle_n to_o every_o degree_n of_o the_o quadrant_n of_o latitude_n it_o will_v constitute_v upon_o the_o diameter_n the_o graduation_n of_o the_o line_n sol_fw-mi whereby_o proportion_n in_o sin_n may_v be_v operate_v without_o the_o other_o supply_n from_o the_o same_o scheme_n also_o follow_v the_o construction_n of_o the_o straight_a line_n of_o latitude_n from_o the_o point_n g_o at_o 90_o of_o the_o quadrant_n of_o latitude_n draw_v a_o straight_a line_n to_o c_o and_o transfer_v each_o degree_n of_o the_o quadrant_n of_o latitude_n with_o compass_n one_o foot_n rest_v upon_o c_o into_o the_o say_v straight_a line_n and_o it_o shall_v be_v constitute_v to_o calculate_v it_o the_o line_n of_o latitude_n c_o g_o bear_v such_o proportion_n to_o c_o a_o as_o the_o chord_n of_o 90_o do_v to_o the_o diameter_n which_o be_v the_o same_o that_o the_o sine_fw-la of_o 45_o bear_n to_o the_o radius_fw-la or_o which_o be_v all_o one_o that_o the_o radius_fw-la bear_v to_o the_o secant_fw-la of_o 45_o d_o which_o secant_fw-la be_v equal_a to_o the_o chord_n of_o 90_o d_o from_o the_o diagram_n the_o nature_n of_o the_o line_n of_o latitude_n may_v be_v discover_v any_o two_o line_n be_v draw_v to_o make_v a_o right_a angle_n if_o any_o ark_n of_o the_o line_n of_o latitude_n be_v prick_v off_o in_o one_o of_o those_o line_n retain_v a_o constant_a hipotenusal_a a_o c_o called_z the_o line_n of_o hour_n equal_a to_o the_o diameter_n of_o that_o circle_n from_o whence_o the_o line_n of_o latitude_n be_v constitute_v if_o the_o say_v hipotenusal_a from_o the_o point_n former_o prick_v off_o be_v make_v the_o hipotenusal_a to_o the_o leg_n of_o the_o right_a angle_n former_o prick_v off_o the_o say_a leg_n or_o side_n include_v the_o right_a angle_n shall_v bear_v such_o proportion_n one_o to_o another_o as_o the_o radius_fw-la do_v to_o the_o sine_fw-la of_o the_o ark_n so_o prick_v off_o and_o this_o be_v evident_a from_o the_o schem_n for_o such_o proportion_n as_o a_o c_o bear_v to_o c_o d_o do_v a_o b_o bear_v 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it_o will_v be_v for_o there_o be_v give_v the_o radius_fw-la c_o d_o and_o the_o tangent_fw-la b_o e_o the_o two_o first_o term_n of_o the_o proportion_n with_o the_o line_n c_o b_o the_o sum_n of_o the_o three_o and_o four_o term_n to_o find_v out_o the_o say_a term_n respective_o and_o it_o will_v hold_v by_o compound_v the_o proportion_n as_o the_o sum_n of_o the_o first_o and_o second_o term_n be_v to_o the_o second_o term_n so_o be_v the_o sum_n of_o the_o three_o and_o four_o term_n to_o the_o four_o term_n that_o be_v as_o c_z d_o +_o b_o e_o be_v to_o b_o e_o so_o be_v c_o f_o +_o f_o b_o =_o c_o b_o to_o f_o b_o see_v 18_o prop._n of_o 5_o of_o euclid_n or_o page_n 18_o of_o the_o english_a clavis_fw-la mathematicae_fw-la of_o the_o famous_a and_o learned_a mr_n oughtre_v after_o the_o same_o manner_n be_v the_o line_n sol_fw-mi or_o proportional_a sin_n make_v that_o be_v also_o such_o a_o line_n that_o any_o ark_n be_v assume_v in_o it_o to_o be_v a_o sine_fw-la the_o 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the_o second_o so_o the_o radius_fw-la to_o a_o four_o again_o as_o the_o radius_fw-la be_v to_o that_o four_o so_o be_v the_o three_o term_n give_v to_o the_o four_o proportional_a seek_v in_o illustrate_v the_o matter_n i_o shall_v make_v use_n of_o that_o theorem_n for_o vary_v of_o proportion_n that_o the_o tangent_n of_o arch_n and_o the_o tangent_n of_o their_o compliment_n be_v in_o reciprocal_a proportion_n as_o tangent_fw-la 23_o to_z tangent_fw-la 35_o so_o tangent_fw-la 55_o to_o the_o tangent_fw-la of_o 67d._o in_o work_v of_o this_o proportion_n the_o last_o term_n may_v be_v find_v to_o the_o equal_a semicircle_n or_o on_o the_o diameter_n 1._o in_o the_o semicircle_n extend_v the_o thread_n through_o 23_o on_o the_o diameter_n and_o through_o 3●_n in_o the_o semicircle_n and_o where_o it_o intersect_v the_o circle_n on_o the_o opposite_a side_n there_o hold_v one_o end_n of_o it_o then_o extend_v the_o other_o part_n of_o it_o over_o 55_o in_o the_o diameter_n and_o in_o 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the_o unequal_a quadrant_n and_o where_o it_o intersect_v the_o opposite_a side_n of_o the_o circle_n hold_v it_o then_o extend_v the_o thread_n over_o the_o tangent_fw-la be_v the_o other_o middle_a term_n count_v on_o the_o diameter_n and_o it_o will_v intersect_v the_o graduated_a quadrant_n at_o the_o sine_fw-la seek_v example_n if_o the_o proportion_n be_v as_o the_o tangent_fw-la of_o 14_o to_o the_o sine_fw-la of_o 29_o so_o be_v the_o tangent_fw-la of_o 20_o to_o a_o sine_fw-la the_o four_o proportional_a will_v be_v find_v to_o be_v the_o sine_fw-la of_o 45d._o 2._o if_o a_o tangent_fw-la be_v seek_v the_o middle_a term_n be_v of_o several_a kind_n extend_v the_o thread_n through_o the_o sine_fw-la in_o the_o upper_a quadrant_n be_v the_o first_o term_n and_o through_o the_o tangent_fw-la on_o the_o diameter_n be_v one_o of_o the_o other_o middle_a term_n hold_v it_o at_o the_o intersection_n of_o the_o circle_n on_o the_o opposite_a side_n then_o lay_v the_o thread_n to_o the_o other_o middle_a term_n in_o the_o upper_a quadrant_n and_o on_o the_o diameter_n it_o show_v the_o tangent_fw-la seek_v example_n if_o the_o sun_n amplitude_n and_o vertical_a altitude_n be_v give_v the_o proportion_n from_o the_o analemma_n to_o find_v the_o latitude_n will_v be_v as_o the_o sine_fw-la of_o the_o amplitude_n to_o radius_fw-la so_o be_v the_o sine_fw-la of_o the_o vertical_a altitude_n to_o the_o cotangent_fw-la of_o the_o latitude_n let_v the_o amplitude_n be_v 39_o 54′_n and_o the_o sun_n altitude_n be_v east_n or_o west_n 30_o 39′_n extend_v the_o thread_n through_o 39_o 54′_n the_o amplitude_n count_v in_o the_o upper_a quadrant_n and_o through_o 45_o on_o the_o diameter_n hold_v it_o at_o the_o intersection_n with_o the_o circle_n on_o the_o opposite_a side_n then_o lay_v the_o thread_n over_o 30_o 39′_n the_o vertical_a altitude_n and_o it_o will_v intersect_v the_o diameter_n at_o 38_o 28′_n the_o compliment_n of_o the_o latitude_n seek_v but_o proportion_n derive_v from_o the_o 16_o case_n of_o right_a angle_a spherical_a triangle_n have_v the_o radius_fw-la ingredient_n will_v be_v wrought_v without_o any_o motion_n of_o the_o thread_n a_o example_n for_o find_v the_o sun_n azimuth_n at_o the_o hour_n of_o 6._o as_o the_o radius_fw-la to_o the_o cousin_n of_o the_o latitude_n so_o the_o tangent_fw-la of_o the_o declination_n to_o the_o tangent_fw-la of_o the_o azimuth_n from_o the_o vertical_a towards_o midnight_n meridian_n extend_v the_o thread_n over_o the_o compliment_n of_o the_o latitude_n in_o the_o upper_a quadrant_n and_o over_o the_o declination_n in_o the_o semicircle_n and_o on_o the_o diameter_n it_o show_v the_o azimuth_n seek_v so_o when_o the_o sun_n have_v 15_o of_o declination_n his_o

azimuth_n shall_v be_v 9_o 28′_n from_o the_o vertical_a at_o the_o hour_n of_o 6_o in_o our_o latitude_n of_o london_n another_o example_n 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 extend_v the_o thread_n over_o the_o latitude_n in_o the_o semicircle_n and_o over_o the_o declination_n on_o the_o diameter_n and_o in_o the_o quadrant_n of_o latitude_n it_o show_v the_o ark_n seek_v the_o proportion_n wrought_v be_v as_o the_o radius_fw-la to_o the_o cotangent_fw-la of_o the_o latitude_n so_o be_v the_o tangent_fw-la of_o the_o declination_n to_o the_o sine_fw-la of_o the_o hour_n from_o 6._o example_n so_o when_o the_o sun_n have_v 15_o of_o north_n declination_n in_o our_o latitude_n of_o london_n the_o hour_n will_v be_v find_v 12_o 18′_n from_o 6_o in_o time_n 49⅕′_n past_o 6_o in_o the_o morning_n or_o before_o it_o in_o the_o afternoon_n another_o example_n to_o find_v the_o time_n of_o sun_n rise_v as_o the_o cotangent_fw-la of_o the_o latitude_n to_o radius_fw-la so_o be_v the_o tangent_fw-la of_o the_o declination_n to_o the_o sine_fw-la of_o the_o hour_n from_o 6_o before_o or_o after_o it_o lay_v the_o thread_n to_o the_o compliment_n of_o the_o latitude_n in_o the_o semicircle_n and_o over_o the_o declination_n on_o the_o diameter_n and_o in_o the_o quadrant_n of_o latitude_n it_o show_v the_o time_n seek_v in_o degree_n to_o be_v convert_v into_o common_a time_n by_o allow_v 15_o to_o a_o hour_n and_o 4′_n to_o a_o degree_n so_o in_o the_o latitude_n of_o london_n 51_o 32′_n when_o the_o sun_n have_v 15_o of_o declination_n the_o ascensional_a difference_n or_o time_n of_o rise_v from_o 6_o will_v be_v 19_o 42′_n to_o be_v convert_v into_o common_a time_n as_o before_o by_o what_o have_v be_v say_v it_o appear_v that_o the_o hour_n and_o azimuth_n may_v be_v find_v general_o by_o help_n of_o this_o circle_n and_o diameter_n for_o the_o performance_n whereof_o we_o must_v have_v recourse_n to_o the_o 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now_o by_o the_o second_o case_n of_o right_n angle_v sphoerical_a triangle_n the_o angle_n a_o z_o b_o will_n be_v acute_a wherefore_o the_o angle_n ☉_o z_o b_o be_v 119_o d_o 44′_n the_o sun_n azimuth_n from_o the_o north_n the_o compliment_n be_v 60_o d_o 16′_n be_v the_o angle_n a_o z_o b_o and_o so_o much_o be_v the_o azimuth_n from_o the_o south_n to_o work_v proportion_n in_o sines_n alone_o that_o this_o circle_n may_v be_v capacitate_v to_o try_v any_o case_n of_o sphoerical_a triangle_n there_o be_v add_v line_n to_o it_o namely_o the_o line_n sol_n fall_v perpendicular_o on_o the_o diameter_n from_o the_o end_n of_o the_o

quadrant_n of_o latitude_n whereto_o belong_v the_o two_o parallel_n line_n of_o sines_n in_o the_o opposite_a quadrant_n the_o upermost_a be_v extend_v across_o the_o quadrant_n of_o latitude_n the_o proportion_n not_o have_v the_o radius_fw-la ingredient_n and_o be_v of_o the_o great_a to_o the_o less_o account_v the_o first_o term_n in_o the_o line_n sol_fw-mi and_o the_o second_o in_o the_o upper_a sine_fw-la extend_v the_o thread_n through_o they_o and_o where_o it_o intersect_v the_o opposite_a parallel_n hold_v it_o then_o lay_v the_o thread_n to_o the_o three_o term_n in_o the_o line_n sol_fw-mi and_o it_o will_v intersect_v the_o four_o proportional_a on_o the_o upper_a parallel_n as_o the_o sine_fw-la of_o 30_o to_z the_o sine_z of_o any_o arch_n so_o be_v the_o cousin_n of_o that_o arch_n to_o the_o sine_fw-la of_o the_o double_a arch_n and_o the_o converse_n by_o try_v this_o canon_n the_o use_n of_o these_o line_n will_v be_v sudden_o attain_v example_n as_o the_o 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from_o the_o plain_a by_o some_o term_v the_o line_n of_o deflexion_n and_o be_v the_o ark_n of_o the_o plain_a between_o the_o meridian_n of_o the_o plain_a and_o the_o meridian_n of_o the_o place_n the_o distance_n thereof_o from_o the_o hour-line_n of_o 12_o be_v to_o be_v find_v by_o this_o proportion_n as_o the_o radius_fw-la to_o the_o sine_fw-la of_o the_o plain_n declination_n so_o the_o cotangent_fw-la of_o the_o latitude_n to_o the_o tangent_fw-la of_o the_o substile_a from_o the_o meridian_n 2._o for_o the_o angle_n of_o 12_o and_o 6._o a_o ark_n use_v when_o the_o hour-line_n be_v prick_v down_o from_o the_o meridian_n line_n in_o a_o triangle_n or_o parallellogram_n and_o not_o from_o

the_o substile_a without_o collect_v angle_n at_o the_o pole_n as_o the_o radius_fw-la be_v to_o the_o sine_fw-la of_o the_o plain_n declination_n so_o be_v the_o tangent_fw-la of_o the_o latitude_n to_o the_o tangent_fw-la of_o a_o ark_n the_o compliment_n whereof_o be_v the_o angle_n of_o 12_o and_o 6._o 3._o inclination_n of_o meridian_n be_v a_o ark_n of_o the_o equinoctial_a between_o the_o meridian_n of_o the_o plain_a and_o the_o meridian_n of_o the_o place_n or_o it_o be_v a_o angle_n or_o space_n of_o time_n elapse_v between_o the_o passage_n of_o the_o shadow_n of_o the_o style_n from_o the_o substilar_a line_n into_o the_o meridian_n line_n by_o some_o term_v the_o plain_n difference_n of_o longitude_n and_o not_o improper_o for_o it_o show_v in_o what_o longitude_n from_o the_o meridian_n where_o the_o plain_n be_v the_o say_a plain_n will_v become_v a_o horizontal_n dyal_n and_o the_o style_n height_n show_v the_o latitude_n this_o ark_n be_v use_v in_o calculate_v hour_n distance_n by_o the_o table_n and_o in_o prick_v down_o dial_n by_o the_o line_n of_o latitude_n and_o hour_n from_o the_o substile_a as_o the_o radius_fw-la be_v to_o the_o sine_fw-la of_o the_o latitude_n so_o the_o cotangent_fw-la of_o the_o plain_n declination_n to_o the_o cotangent_fw-la of_o the_o inclination_n of_o meridian_n or_o as_o the_o sine_fw-la of_o the_o latitude_n to_o radius_fw-la so_o be_v the_o tangent_fw-la of_o the_o plain_n declination_n to_o the_o tangent_fw-la of_o inclination_n of_o meridian_n 4._o the_o style_n height_n above_o the_o substile_a as_o the_o radius_fw-la be_v to_o the_o cousin_n of_o the_o latitude_n so_o be_v the_o cousin_n of_o the_o plain_n declination_n to_o the_o sine_fw-la of_o the_o style_n height_n or_o the_o substiles_a distance_n be_v know_v as_o the_o radius_fw-la to_o the_o sine_fw-la of_o the_o substiles_a distance_n from_o the_o meridian_n so_o be_v the_o cotangent_fw-la of_o the_o declination_n to_o the_o tangent_fw-la of_o the_o style_n height_n or_o the_o inclination_n of_o meridian_n be_v know_v as_o the_o radius_fw-la to_o the_o cousin_n of_o the_o inclination_n of_o meridian_n so_o be_v the_o cotangent_fw-la of_o the_o latitude_n to_o the_o tangent_fw-la of_o the_o style_n height_n 5._o last_o for_o the_o distance_n of_o the_o hour-line_n from_o the_o substilar_a line_n as_o the_o radius_fw-la be_v to_o the_o sine_fw-la of_o the_o style_n height_n above_o the_o plain_a so_o be_v the_o tangent_fw-la of_o the_o angle_n at_o the_o pole_n to_o the_o tangent_fw-la of_o the_o hour_n distance_n from_o the_o substilar_a line_n by_o the_o angle_n at_o the_o pole_n be_v mean_v the_o ark_n of_o difference_n between_o the_o ark_n call_v the_o inclination_n of_o meridian_n and_o the_o distance_n of_o any_o hour_n from_o the_o meridian_n for_o all_o hour_n on_o the_o same_o side_n the_o substile_a fall_n and_o the_o sum_n of_o these_o two_o ark_n for_o all_o hour_n on_o the_o other_o side_n the_o substile_a these_o proportion_n be_v sufficient_a for_o all_o plain_n to_o find_v the_o like_a ark_n without_o have_v any_o more_o if_o the_o manner_n of_o refer_v decline_a recline_a incline_v plain_n to_o a_o new_a latitude_n and_o a_o new_a declination_n in_o which_o they_o shall_v stand_v as_o upright_a plain_n be_v but_o well_o explain_v for_o east_n or_o west_n recline_v incline_v plain_n their_o new_a latitude_n be_v the_o compliment_n of_o their_o old_a latitude_n and_o their_o new_a declination_n be_v the_o compliment_n of_o their_o reclination_n inclination_n which_o i_o count_v always_o from_o the_o zenith_n and_o upon_o such_o a_o supposition_n take_v their_o new_a latitude_n and_o declination_n those_o that_o will_v try_v shall_v find_v that_o these_o proportion_n will_v calculate_v all_o the_o ark_n necessary_a to_o such_o dial_n so_o if_o a_o upright_a plain_n decline_v 25_o in_o our_o latitude_n of_o london_n from_o the_o meridian_n the_o substiles_a distance_n from_o the_o meridian_n be_v 18_o 34′_n the_o angle_n of_o 12_o and_o 6_o be_v 62_o 00_o the_o inclination_n of_o meridian_n be_v 30_o 47_o the_o style_n height_n be_v 34_o 19_o to_o delineate_v the_o same_o dial_n from_o the_o substile_a by_o the_o line_n of_o latitude_n and_o scale_n of_o hour_n in_o a_o equicrutal_a triangle_n to_o draw_v a_o upright_o decliner_n an_o upright_o south_n plain_a for_o the_o latitude_n of_o london_n decline_a 25_o eastward_n to_o prick_v down_o this_o dial_n by_o the_o line_n of_o latitude_n and_o scale_n of_o hour_n in_o a_o isoceles_a triangle_n draw_z c_z 12_o the_o meridian_n line_n perpendicular_a to_o the_o horizontal_n line_n of_o the_o plain_a and_o with_o a_o line_n of_o chord_n make_v the_o angle_n f_o c_o 12_o equal_a to_o the_o substiles_a distance_n from_o the_o meridian_n and_o draw_v the_o line_n f_o c_o for_o the_o substile_a draw_v the_o line_n b_o a_o perpendicular_a thereto_o and_o pass_v through_o the_o centre_n at_o c_o and_o out_o of_o the_o line_n of_o latitude_n on_o the_o other_o quadrant_n or_o out_o of_o the_o quadrant_n of_o latitude_n on_o this_o quadrant_n set_v off_o b_o c_z and_o c_o a_o each_o equal_a to_o the_o style_n height_n then_o fit_a in_o the_o scale_n of_o 6_o hour_n proper_a to_o those_o latitude_n so_o that_o one_o extremity_n meet_v at_o a_o the_o other_o may_v meet_v with_o the_o substilar_a line_n at_o f._n then_o get_v the_o difference_n between_o 30_o 47′_n the_o inclination_n of_o meridian_n and_o 30_o the_o next_o hour_n distance_n lesser_a than_o the_o say_a ark_n the_o difference_n be_v 47′_n in_o time_n 3′_n near_o then_o fit_v in_o the_o scale_n of_o hour_n as_o be_v prescribe_v count_n upon_o the_o say_a scale_n hour_n min._n  _fw-fr 0_o 3_o from_o f_o to_o 10_o 1_o 3_o 11_o 2_o 3_o 12_o 3_o 3_o 1_o 4_o 3_o 2_o 5_o 3_o 3_o and_o make_v point_n at_o the_o termination_n with_o a_o pin_n or_o pen_n &_o draw_v line_n from_o those_o point_n into_o the_o centre_n at_o c_o &_o they_o shall_v be_v the_o true_a hour-line_n require_v on_o this_o side_n the_o substile_a again_o fit_v in_o the_o scale_n of_o hour_n from_o b_o to_o f_o count_v from_o that_o end_n at_o b_o the_o former_a ark_n of_o time_n ho_o min_n  _fw-fr 00_o 03_o from_o b_o to_z 4_o 1_o 3_o 5_o 2_o 3_o 6_o 3_o 3_o 7_o 4_o 3_o 8_o 5_o 3_o 9_o and_o make_v point_n at_o the_o termination_n through_o which_o draw_v line_n into_o the_o centre_n and_o they_o shall_v be_v the_o hour_n line_n require_v on_o the_o other_o side_n the_o substile_a the_o like_a must_v be_v do_v for_o the_o half_n and_o quarter_n get_v the_o difference_n between_o the_o half_a hour_n next_z lesser_a in_o this_o example_n 22_o 30′_n under_o the_o ark_n call_v the_o inclination_n of_o meridian_n the_o difference_n be_v 1_o 17′_n in_o time_n 33′_n near_o to_o be_v continual_o augment_v a_o hour_n at_o a_o time_n and_o so_o prick_v off_o as_o before_o be_v do_v for_o the_o whole_a hour_n by_o three_o facile_a proportion_n may_v be_v find_v the_o style_n height_n the_o inclination_n of_o meridian_n and_o the_o substiles_a distance_n from_o the_o plain_n perpendicular_a for_o all_o plain_n decline_a recline_a or_o incline_v which_o be_v sufficient_a to_o prick_v off_o the_o dyal_n after_o the_o manner_n here_o describe_v which_o must_v be_v refer_v to_o another_o place_n if_o the_o scale_n of_o hour_n reach_v above_o the_o plain_a as_o at_o b_o so_o that_o b_o c_o can_v be_v prick_v down_o then_o may_v a_o angle_n be_v prick_v off_o with_o chord_n on_o the_o upper_a side_n the_o substile_a equal_a to_o the_o angle_n f_o c_o a_o on_o the_o under_o side_n and_o thereby_o the_o scale_n of_o hour_n lay_v in_o its_o true_a situation_n have_v first_o find_v the_o point_n f_o on_o the_o under_o side_n to_o prick_v down_o the_o former_a dyal_n in_o a_o rectangular_a ☉_o blong_v or_o long_o square_a figure_n from_o the_o substile_a have_v set_v off_o the_o substilar_a f_o c_o assume_v any_o distance_n in_o it_o as_o at_o f_o to_o be_v the_o radius_fw-la and_o through_o the_o fame_n at_o right_a angle_n draw_v the_o line_n e_o f_o d_o then_o have_v make_v f_o c_o any_o distance_n radius_fw-la take_v out_o the_o sine_fw-la of_o the_o style_n height_n to_o the_o same_o radius_fw-la and_o enter_v it_o at_o the_o end_n of_o the_o scale_n of_o three_o hour_n make_v it_o the_o radius_fw-la of_o a_o tangent_fw-la and_o proportion_n out_o tangent_n to_o 3′_n and_o set_v they_o off_o from_o f_o to_o 10_o 1_o hour_n 3_o and_o set_v they_o off_o from_o f_o to_o g_z 2_o 3_o and_o set_v they_o off_o from_o f_o to_o h_n again_o take_v out_o the_o tangent_n of_o the_o compliment_n of_o the_o first_o ark_n increase_v it_o each_o time_n by_o the_o augmentation_n of_o a_o hour_n namely_o 57′_n and_o prick_v

they_o from_o f_o to_o i_o and_o from_o the_o point_v 1_o ho._n 57_o and_o prick_v they_o from_o f_o to_o k_o and_o from_o the_o point_v 2_o 57_o and_o prick_v they_o from_o f_o to_o e_z and_o from_o the_o point_n thus_o find_v draw_v line_n into_o the_o centre_n then_o for_o the_o other_o side_n of_o the_o square_n make_v c_o f_o the_o radius_fw-la of_o the_o dyall_a tangent_fw-la of_o 3_o hour_n and_o proportion_n out_o tangent_n to_o the_o former_a ark_n namely_o 3′_n and_o prick_v they_o from_o b_o to_o p_o also_o to_o the_o latter_a ark_n 57′_n and_o prick_v they_o from_o a_o to_o n_o 1_o ho._n 3_o and_o prick_v they_o from_o b_o to_z o_o also_o to_o the_o latter_a arks._n 1_o h._n 57_o and_o prick_v they_o from_o a_o to_o m_n 2_o 3_o and_o prick_v they_o from_o b_o to_z l_o also_o to_o the_o latter_a arks._n 2_o 57_o and_o prick_v they_o from_o a_o to_o d_o and_o draw_v line_n from_o these_o termination_n into_o the_o centre_n and_o the_o hour-line_n be_v finish_v after_o the_o same_o manner_n must_v the_o half_n and_o quarter_n be_v finish_v and_o how_o this_o trouble_n in_o proportion_v out_o the_o tangent_n may_v be_v shun_v without_o draw_v any_o line_n on_o the_o plain_a but_o the_o hour-line_n may_v be_v speak_v to_o hereafter_o whereby_o this_o way_n of_o dyall_v and_o those_o that_o follow_v will_v be_v render_v more_o commodious_a last_o the_o style_n may_v be_v prick_v off_o with_o chord_n or_o take_v b_o c_o and_o set_v one_o foot_n in_o f_o with_o that_o extent_n sweep_v the_o touch_n of_o a_o occult_a arch_n and_o from_o c_o draw_v a_o line_n just_o touch_v the_o outward_a extremity_n of_o the_o say_a arch_n and_o it_o shall_v prick_v off_o the_o angle_n of_o the_o style_n height_n above_o the_o substile_a to_o prick_v off_o the_o former_a dyal_n in_o a_o oblique_a parallellogram_n or_o scalenon_fw-mi alias_o unequal_a side_v triangle_n from_o the_o meridian_n first_o in_o a_o oblique_a parallellogram_n draw_v ce_fw-fr the_o meridian_n line_n and_o with_o 60_o of_o a_o line_n of_o chord_n draw_v the_o prick_a arch_n and_o therein_o from_o k_o contrary_a to_o the_o coast_n of_o declination_n prick_v off_o 62_o the_o angle_n of_o 12_o and_o 6_o and_o draw_v the_o line_n c_o d_o for_o the_o say_a hour_n line_v continue_v on_o the_o other_o side_n the_o centre_n and_o out_o of_o a_o line_n of_o sines_n make_v c_o e_o equal_a to_o 65_o the_o compliment_n of_o the_o declination_n then_o take_v out_o the_o sine_fw-la of_o 38_o 28′_n the_o compliment_n of_o the_o latitude_n and_o enter_v it_o in_o the_o line_n d_o c_o so_fw-mi that_o one_o foot_n rest_v at_o d_o the_o other_o turn_v about_o may_v but_o just_o touch_v the_o meridian_n line_n the_o point_n d_o being_n thus_o find_v make_v c_o f_o equal_a to_o c_o d_o and_o with_o the_o side_n c_o f_o and_o c_o e_o make_v the_o parallellogram_n d_o g_o f_o h_o namely_o f_o h_n and_o g_o d_o equal_a to_z c_o e_o and_o e_z g_z and_o e_o h_n equal_a to_o d_o c._n and_o where_o these_o distance_n sweep_v occult_a arch_n therewith_o intersect_n will_v find_v the_o points_z h_z and_o g_z limit_v the_o angle_n of_o the_o parallellogram_n then_z making_z e_z h_z or_o c_z d_o radius_fw-la proportion_n out_o the_o tangent_n of_o 15_o and_o prick_v they_o down_o from_o e_z to_z 1_o and_o 11_o and_o 30_o and_o prick_v they_o down_o from_o e_z to_z 2_o and_o 10_o and_o draw_v line_n into_o the_o centre_n through_o those_o point_n and_o the_o angular_a point_n of_o the_o parallellogram_n at_o h_n and_o g_o and_o there_o will_v be_v 6_o hour_n draw_v beside_o the_o meridian_n line_n or_o hour_n line_n of_o 12._o then_z making_z d_o g_z radius_fw-la proportion_n out_o the_o tangent_fw-la of_o 15_o and_o prick_v it_o down_o from_o d_o upward_o to_o 5_o and_o downward_z to_z 7_o also_o proportion_n out_o the_o tangent_fw-la of_o 30_o and_o prick_v it_o from_o d_o to_z 8_o and_o from_o f_o to_o 4_o and_o draw_v line_n into_o the_o centre_n and_o so_o the_o hour_n line_n be_v finish_v after_o the_o same_o manner_n be_v the_o half_n and_o quarter_n to_o be_v proportion_v out_o and_o prick_v down_o and_o if_o this_o work_n be_v to_o be_v do_v upon_o the_o plain_n itself_o the_o parallel_n f_o h_n will_v excur_v above_o the_o plain_a in_o that_o case_n because_o the_o parallel_n distance_n of_o f_o h_n from_o the_o meridian_n be_v equal_a to_o the_o parallel_a distance_n of_o d_o g_o the_o space_n g._n 8._o may_v be_v set_v from_o h_n to_o 4_o and_o so_o all_o the_o hour_n line_n prick_v down_o to_o prick_v down_o this_o dyal_n in_o a_o scalenon_fw-mi or_o unequal_a side_v triangle_n from_o the_o meridian_n from_o e_z to_z d_o draw_v the_o straight_a line_n d_o e_o and_o from_o the_o same_o point_n draw_v another_o to_o f_o and_o each_o of_o they_o the_o former_a hour_n line_n be_v first_o draw_v shall_v thereby_o be_v divide_v into_o a_o line_n of_o double_a tangent_n or_o scale_v of_o 6_o hour_n such_o a_o one_o as_o be_v in_o the_o diameter_n of_o the_o circle_n on_o this_o quadrant_a or_o on_o the_o right_a edge_n of_o the_o foreside_n and_o therefore_o by_o help_n of_o either_o of_o they_o line_n if_o it_o be_v require_v to_o prick_v down_o the_o dyal_n it_o may_v be_v do_v by_o proportion_v they_o out_o take_v the_o extent_n d_o e_o and_o prick_v it_o from_o one_o extremity_n of_o the_o diameter_n in_o the_o semicircle_n on_o the_o quadrant_a and_o from_o the_o point_n of_o termination_n draw_v a_o line_n with_o black_a lead_n to_o the_o other_o extremity_n which_o will_v easy_o rub_v out_o again_o either_o with_o bread_n or_o leather_n pare_n and_o take_v the_o near_a distance_n from_o 15_o of_o the_o diameter_n to_o the_o say_a line_n and_o the_o say_a extent_n 30_o of_o the_o diameter_n to_o the_o say_a line_n and_o the_o say_a extent_n 45_o of_o the_o diameter_n to_o the_o say_a line_n and_o the_o say_a extent_n shall_v reach_v from_o e_z to_z 11_o and_o from_o d_o to_z 7_o shall_v reach_v from_o e_z to_z 10_o and_o from_o d_o to_z 8_o shall_v reach_v from_o e_z to_z 9_o and_o from_o d_o to_z 9_o and_o the_o like_a must_v be_v do_v for_o the_o line_n e_o f_o enter_v that_o in_o the_o semicirle_n as_o before_o or_o without_o draw_v line_n on_o the_o quadrant_a if_o a_o hole_n be_v drill_v at_o one_o end_n of_o the_o diameter_n and_o a_o thread_n fit_v into_o it_o lay_v the_o thread_n over_o the_o point_n in_o the_o diameter_n and_o take_v the_o near_a distance_n thereto_o last_o from_o a_o line_n of_o chord_n prick_v off_o the_o substilar_a line_n and_o the_o style_n height_n as_o we_o before_o find_v it_o this_o way_n of_o dyall_v in_o a_o parallellogram_n be_v first_o invent_v by_o john_n ferrereus_n a_o spaniard_n long_v since_o and_o afterward_o large_o handle_v by_o clavius_n who_o demonstrate_v it_o and_o show_v how_o to_o fit_v it_o into_o all_o plain_n whatsoever_o albeit_o they_o decline_v recline_v or_o incline_v without_o refer_v they_o to_o a_o new_a latitude_n the_o triangular_a way_n be_v also_o build_v upon_o the_o same_o demonstration_n and_o be_v already_o publish_v by_o mr_n foster_n in_o his_o posthuma_n for_o it_o be_v no_o other_o than_o dyall_v in_o a_o parallellogram_n if_o the_o meridian_n line_n c_o e_o be_v continue_v upward_o and_o c_o e_o set_n off_o upward_o and_o line_n draw_v from_o the_o point_n so_o find_v to_o d_o and_o e_z shall_v constitute_v a_o parallellogram_n a_o advertisement_n about_o observe_v of_o altitude_n imagine_v a_o line_n draw_v from_o the_o begin_n of_o the_o line_n sol_fw-mi to_o the_o end_n of_o the_o diameter_n and_o therein_o suppose_v a_o pair_n of_o sight_n place_v with_o a_o thread_n and_o bullet_n hang_v from_o the_o begin_n of_o the_o say_a line_n as_o from_o a_o centre_n i_o say_v the_o line_n wherein_o the_o sight_n be_v place_v make_v a_o right_a angle_n with_o the_o line_n of_o sin_n on_o the_o other_o side_n of_o sol_n and_o so_o may_v represent_v a_o quadrant_a the_o equal_a limb_n whereof_o be_v either_o represent_v by_o the_o 90_o of_o the_o equal_a semicirle_n or_o by_o the_o 90_o of_o the_o diameter_n and_o thereby_o a_o altitude_n may_v be_v take_v now_o to_o make_v a_o isoceles_a equicrural_a or_o equal_a legged_a triangle_n make_v of_o three_o straight_a ruler_n the_o long_a whereof_o will_n be_v the_o base_a or_o hipotenusal_a line_n thus_o to_o serve_v for_o a_o quadrant_a to_o take_v altitude_n withal_o will_v be_v much_o cheap_a and_o more_o certain_a in_o wood_n than_o the_o great_a arch_a wooden_a frame_v quadrant_n moreover_o the_o say_a diameter_n line_n supply_v all_o the_o use_n of_o the_o

limb_n from_o it_o may_v be_v take_v off_o sin_n tangent_n or_o secant_v as_o be_v do_v from_o the_o limb_n and_o therein_o the_o hour_n and_o azimuth_n find_v general_o by_o help_n of_o the_o line_n of_o sin_n on_o the_o left_a edge_n as_o be_v large_o show_v in_o the_o use_n of_o this_o quadrant_a beside_o its_o use_n in_o dyall_v only_o when_o such_o a_o instrument_n be_v make_v apart_o it_o will_v be_v more_o convenient_a to_o have_v the_o line_n of_o sines_n to_o be_v set_v on_o the_o right_a edge_n and_o the_o diameter_n number_v also_o by_o its_o compliment_n this_o diameter_n or_o double_a tangent_fw-la or_o hipotenusal_a line_n be_v first_o divide_v on_o all_o the_o other_o line_n may_v from_o it_o by_o the_o same_o table_n that_o serve_v to_o graduate_v they_o from_o the_o equal_a limb_n be_v likewise_o inscribe_v and_o here_o let_v i_o put_v a_o period_n to_o the_o use_n of_o this_o quadrant_a gloria_fw-la deo_fw-la the_o description_n of_o a_o universal_a small_a pocket_n quadrant_n this_o quadrant_a have_v only_o one_o face_n on_o the_o right_a edge_n from_o the_o centre_n be_v place_v a_o line_n of_o sin_n divide_v into_o degree_n and_o half_a degree_n up_o to_o 60_o d._n afterward_o into_o whole_a degree_n to_o 80_o d._n on_o the_o left_a edge_n issue_v a_o line_n of_o 10_o equal_a part_n from_o the_o centre_n be_v precise_a 4_o inch_n long_o each_o part_n be_v divide_v into_o 10_o subdivision_n and_o each_o subdivision_n into_o half_n these_o two_o line_n make_v a_o right_a angle_n at_o the_o centre_n and_o between_o they_o include_v a_o projection_n of_o the_o sphere_n for_o the_o latitude_n of_o london_n above_o the_o projection_n be_v put_v on_o in_o quadrant_n of_o circle_n a_o line_n of_o declination_n 4_o quadrant_n for_o the_o day_n of_o the_o month_n above_o they_o the_o name_n of_o 5_o star_n with_o their_o right_a ascension_n grave_v against_o they_o and_o a_o general_a almanac_n beneath_o the_o projection_n be_v put_v on_o in_o quadrant_n of_o circle_n a_o particular_a sine_fw-la and_o secant_fw-la so_o call_v because_o it_o be_v particular_a to_o the_o latitude_n of_o london_n below_o that_o the_o quadrat_fw-la and_o shadow_n below_o that_o a_o line_n of_o tangent_n to_o 45_o d._n last_o of_o all_o the_o equal_a limb_n on_o the_o left_a edge_n be_v place_v the_o dial_a scale_n of_o hour_n 4_o inch_n long_o outwardmost_a on_o the_o right_a edge_n a_o line_n of_o latitude_n fit_v thereto_o within_o the_o line_n of_o sin_n close_o abut_a thereto_o be_v place_v a_o small_a scale_n call_v the_o scale_n of_o entrance_n beginning_n against_o 52_o d._n 35_o min._n of_o the_o sin_n number_v to_o 60_o d._n the_o line_n of_o sin_n that_o issue_v from_o the_o centre_n shall_v for_o a_o particular_a use_n have_v be_v continue_v long_o to_o wit_n to_o a_o secant_fw-la of_o 28_o d._n because_o this_o can_v not_o be_v admit_v the_o say_v secant_fw-la be_v place_v outward_a at_o the_o end_n of_o the_o scale_n of_o entrance_n towards_o the_o limb_n and_o as_o much_o of_o the_o sine_fw-la as_o be_v needful_a place_v at_o its_o due_a distance_n at_o the_o other_o end_n of_o the_o scale_n of_o entrance_n of_o the_o use_n of_o the_o say_v quadrant_a the_o almanac_n have_v be_v large_o speak_v to_o in_o pag._n 12_o and_o 13_o also_o again_o in_o the_o use_n of_o the_o horizontal_n quadrant_a pag._n 11_o 12._o the_o quadrat_fw-la and_o shadow_n from_o pag._n 35_o to_o 44._o the_o line_n of_o latitude_n and_o scale_n of_o hour_n pag._n 250._o again_o from_o page_n to_o 262_o to_o 274_o also_o the_o line_n of_o sin_n equal_a part_n and_o tangent_n in_o other_o part_n of_o the_o book_n the_o use_n of_o the_o projection_n this_o projection_n be_v only_o fit_v for_o find_v the_o hour_n in_o the_o limb_n and_o not_o the_o azimuth_n all_o the_o circular_a line_n on_o it_o be_v parallel_n of_o altitude_n or_o depression_n except_o the_o ecliptic_a and_o horizon_n the_o ecliptic_a be_v know_v by_o the_o character_n of_o the_o sign_n and_o the_o horizon_n lie_v beneath_o it_o be_v number_v with_o 10_o 20_o 30_o 40._o the_o parallel_n of_o altitude_n be_v the_o winter_n parallel_v of_o stofler_n astrolable_a and_o be_v number_v from_o the_o horizon_n upward_o towards_o the_o centre_n the_o parallel_n of_o depression_n which_o supply_v the_o use_n of_o stofler_n summer_n altitude_n be_v number_v downward_o from_o the_o top_n of_o the_o projection_n towards_o the_o limb_n to_o find_v the_o time_n of_o sun_n rise_v and_o his_o amplitude_n lay_v the_o thread_n over_o the_o day_n of_o the_o month_n and_o set_v the_o bead_n to_o the_o ecliptic_a then_o carry_v the_o thread_n and_o bead_n to_o the_o horizon_n and_o the_o thread_n in_o the_o limb_n show_v the_o time_n of_o rise_v and_o the_o bead_n on_o the_o horizon_n the_o quantity_n of_o the_o amplitude_n example_n so_o on_o the_o second_o of_o august_n the_o sun_n declination_n be_v 15_o d_o his_o amplitude_n will_v be_v 24_o d._n 35_o min._n and_o the_o time_n of_o rise_a 41′_n past_o 4_o in_o the_o morning_n to_o find_v the_o hour_n of_o the_o day_n have_v take_v the_o sun_n altitude_n and_o rectify_v the_o bead_n as_o before_o show_v if_o the_o sun_n have_v south_n declination_n bring_v the_o bead_n to_o that_o parallel_n of_o altitude_n on_o which_o the_o sun_n height_n be_v observe_v among_o those_o parallel_n that_o be_v number_v upward_o towards_o the_o centre_n and_o the_o thread_n in_o the_o limb_n show_v the_o time_n of_o the_o day_n eample_n so_o when_o the_o sun_n have_v 15_o d._n of_o south_n declination_n as_o about_o 28_o of_o january_n if_o his_o altitude_n be_v 15_o deg_n the_o time_n of_o the_o day_n will_v be_v 39_o m._n past_o 2_o in_o the_o afternoon_n or_o 21_o m._n past_o 9_o in_o the_o morning_n but_o in_o the_o summer_n half_a year_n bring_v the_o bead_n to_o lie_v on_o those_o parallel_n that_o be_v number_v downward_o to_o the_o limb_n and_o the_o thread_n show_v therein_o the_o time_n of_o the_o day_n seek_v example_n if_o on_o the_o second_o of_o august_n his_o declination_n be_v 15_o d._n his_o altitude_n be_v 40_o d._n the_o true_a time_n of_o the_o day_n will_v be_v 8_o m._n past_o 9_o in_o the_o morning_n or_o 52_o m._n past_o 2_o in_o the_o afternoon_n if_o the_o bead_n will_v not_o meet_v with_o the_o altitude_n give_v among_o those_o parallel_n that_o run_v donwnwards_o towards_o the_o right_a edge_n than_o it_o must_v be_v bring_v to_o those_o parallel_n that_o lie_v below_o the_o horizon_n downward_o towards_o the_o left_a edge_n and_o the_o thread_n in_o the_o limb_n show_v the_o time_n of_o the_o day_n before_o six_o in_o the_o morning_n or_o after_o it_o in_o the_o evening_n in_o summer_n example_n when_o the_o sun_n have_v 15_o d._n of_o north_n declination_n as_o on_o the_o second_o of_o august_n if_o his_o altitude_n be_v 5_o d._n the_o time_n of_o the_o day_n will_v be_v 44_o m._n before_o 6_o in_o the_o morning_n or_o after_o it_o in_o the_o evening_n of_o the_o general_a line_n on_o this_o quadrant_a the_o line_n of_o sin_n on_o the_o right_a edge_n be_v general_a for_o find_v either_o the_o hour_n or_o the_o azimuth_n in_o the_o equal_a limb_n or_o in_o the_o say_a line_n of_o sin_n as_o i_o have_v large_o show_v in_o page_n 231_o for_o find_v the_o hour_n also_o for_o find_v the_o sun_n altitude_n on_o all_o hour_n as_o in_o page_n 234_o for_o find_v the_o azimuth_n from_o page_n 237_o to_o pag._n 2.9_o though_o this_o quadrant_a have_v neither_o secant_v nor_o verse_v sin_n as_o the_o rest_n have_v yet_o both_o may_v be_v easy_o supply_v let_v it_o be_v require_v to_o work_v this_o proportion_n as_o the_o cousin_a of_o the_o declination_n be_v to_o the_o secant_fw-la of_o the_o latitude_n so_o be_v the_o difference_n of_o the_o sin_n of_o the_o sun_n meridian_n and_o give_v altitude_n to_o the_o verse_v sine_fw-la of_o the_o hour_n from_o noon_n before_o or_o after_o six_o the_o hour_n may_v be_v find_v from_o midnight_n by_o the_o proportion_n in_o page_n 230._o let_v the_o radius_fw-la of_o the_o sin_n be_v assume_v to_o represent_v the_o secant_fw-la of_o the_o latitude_n the_o radius_fw-la to_o that_o secant_fw-la will_v be_v the_o cousin_n of_o the_o latitude_n then_o lay_v the_o thread_n to_o the_o compliment_n of_o the_o declination_n in_o the_o limb_n count_v from_o right_a edge_n and_o take_v the_o near_a distance_n to_o it_o i_o say_v that_o extent_n shall_v be_v the_o cousin_n of_o the_o declination_n to_o the_o radius_fw-la of_o the_o secant_fw-la enter_v this_o at_o 90_o d._n of_o the_o line_n of_o sin_n lay_v the_o thread_n to_o the_o other_o foot_n according_a to_o near_a distance_n then_o in_o the_o sin_n take_v the_o

between_o half_a the_o propose_a altitude_n and_o 45_o deg._n thus_o to_o graduate_v 20_o deg_n of_o the_o index_n the_o half_a thereof_o be_v 10_o degree_n which_o take_v from_o 45_o degree_n the_o residue_n be_v 35_o degree_n the_o tangent_fw-la thereof_o prick_v from_o the_o centre_n give_v the_o point_n where_o the_o index_n be_v to_o be_v graduate_v with_o 20_o degree_n of_o the_o hour_n circle_n these_o be_v know_v by_o the_o number_n set_v to_o they_o by_o cross_v the_o parallel_n of_o declination_n and_o by_o issue_v from_o the_o upper_a part_n of_o the_o quadrant_a towards_o the_o limb_n to_o describe_v they_o let_v it_o be_v note_v that_o they_o all_o meet_v in_o a_o point_n in_o the_o meridian_n line_n below_o the_o centre_n of_o the_o quadrant_a the_o distance_n whereof_o from_o the_o centre_n be_v equal_a to_o the_o tangent_fw-la of_o half_a the_o compliment_n of_o the_o latitude_n take_v out_o of_o the_o common_a radius_fw-la which_o at_o london_n will_v be_v the_o tangent_fw-la of_o 19_o deg._n 14′_n the_o former_a point_n which_o may_v be_v call_v the_o pole-point_n limit_n their_o semidiameter_n to_o find_v the_o centre_n prick_v off_o the_o tangent_fw-la of_o the_o latitude_n and_o through_o the_o termination_n raise_v a_o line_n perpendicular_a to_o the_o meridian_n line_n the_o distance_n from_o the_o pole-point_n be_v equal_a to_o the_o secant_fw-la of_o the_o latitude_n must_v be_v make_v radius_fw-la and_o the_o tangent_n of_o 15_o degree_n 30_o degree_n &c._n &c._n prick_v off_o on_o the_o former_a raise_v line_n give_v the_o respective_a centre_n of_o the_o hour_n circle_n the_o distance_n whereof_o from_o the_o pole_n point_n be_v the_o semidiameter_n wherewith_o those_o hour_n circle_n be_v to_o be_v draw_v of_o the_o revert_v tail_n this_o need_v no_o rule_n to_o describe_v it_o be_v make_v by_o the_o continue_n of_o the_o parallel_n of_o declination_n to_o the_o right_a edge_n of_o the_o quadrant_a and_o the_o hour_n circle_n up_o to_o the_o winter_n tropic_n or_o parallel_n of_o declination_n near_o the_o centre_n however_o the_o quantity_n of_o it_o may_v be_v know_v by_o set_v one_o foot_n of_o a_o pair_n of_o compass_n in_o the_o centre_n of_o the_o quadrant_a and_o the_o other_o extend_v to_o 00_o degree_n of_o altitude_n in_o the_o index_n a_o arch_n with_o that_o extent_n sweep_v over_o the_o quadrant_a as_o much_o as_o it_o cut_v off_o will_v be_v the_o revert_v tail_n and_o so_o much_o will_v be_v the_o radius_fw-la of_o a_o quadrant_n make_v of_o this_o projection_n not_o invert_v by_o what_o have_v be_v say_v it_o will_v be_v evident_a to_o the_o judicious_a that_o this_o inversion_n be_v no_o other_o than_o the_o continance_n of_o the_o extent_n of_o one_o quarter_n of_o the_o horizontal_n projection_n which_o otherwise_o can_v not_o with_o convenience_n be_v bring_v upon_o a_o quadrant_a hence_o it_o may_v be_v observe_v that_o have_v assign_v the_o radius_fw-la a_o quadrant_a make_v of_o the_o horizontal_n projection_n without_o inversion_n to_o know_v how_o big_a a_o radius_fw-la it_o will_v require_v when_o invert_v the_o proportion_n will_v hold_v as_o the_o radius_fw-la be_v to_o the_o distance_n of_o the_o intersection_n of_o the_o aequinoctial_a point_n with_o the_o horizon_n from_o the_o centre_n equal_a to_o the_o radius_fw-la of_o the_o say_a projection_n when_o not_o invert_v in_o any_o common_a measure_n so_o be_v the_o tangent_fw-la of_o a_o ark_n compound_v of_o 45_o degree_n and_o of_o half_a the_o sun_n great_a meridian_n altitude_n to_o the_o distance_n between_o the_o centre_n and_o the_o outward_a tropic_n next_o the_o limb_n in_o the_o say_v know_v measure_n when_o invert_v whence_o it_o follow_v that_o between_o the_o tropic_n this_o projection_n can_v be_v invert_v but_o the_o revert_v tail_n will_v be_v but_o small_a and_o may_v be_v draw_v with_o convenience_n without_o inversion_n of_o the_o curve_v line_n and_o scales_n belong_v to_o it_o beyond_o the_o middle_n of_o the_o projection_n stand_v a_o curve_v or_o bend_a line_n number_v from_o the_o o_o or_o cypher_n both_o way_n one_o way_n to_o 60_o degree_n but_o divide_v to_o 62_o degree_n the_o other_o way_n to_o 20_o degr._fw-la but_o divide_v to_o 23_o deg._n 30′_n the_o invention_n of_o this_o line_n own_v mr._n dary_n for_o the_o author_n thereof_o the_o use_n of_o it_o be_v to_o find_v the_o hour_n or_o azimuth_n in_o that_o particular_a latitude_n whereto_o it_o be_v fit_v by_o the_o extension_n of_o a_o thread_n over_o it_o and_o the_o line_n belong_v to_o it_o the_o line_n belong_v to_o it_o be_v two_o the_o one_o a_o line_n of_o altitude_n and_o declination_n stand_v on_o the_o left_a edge_n of_o the_o quadrant_a be_v no_o other_o but_o a_o line_n of_o sines_n continue_v both_o way_n from_o the_o beginning_n one_o way_n to_o 62_o degree_n the_o other_o way_n to_o 23._o degree_n 30′_n the_o other_o line_n thereto_o belong_v be_v 130_o deg._n of_o a_o line_n of_o verse_v sin_n which_o stand_v next_o without_o the_o projection_n be_v parallel_n to_o the_o left_a edge_n of_o the_o quadrant_a to_o draw_v the_o curve_v draw_v two_o line_n of_o verse_v sin_n it_o matter_n not_o whether_o of_o the_o same_o radius_fw-la or_o no_o nor_o how_o posit_v provide_v they_o be_v parallel_n let_v each_o of_o they_o be_v number_v as_o a_o sine_fw-la both_o way_n from_o the_o middle_n at_o 0_o and_o so_o each_o of_o they_o will_v contain_v two_o line_n of_o sines_n to_o the_o right_a end_n of_o the_o uppermost_a set_v c_o to_o the_o left_a end_n d_o and_o to_o the_o right_a end_n of_o the_o undermost_a set_v a_o and_o to_o the_o left_a end_n b._n first_o note_v that_o there_o be_v a_o certain_a point_n in_o the_o curve_v where_o the_o graduation_n will_v begin_v both_o upward_o and_o downward_o this_o be_v call_v the_o aequinoctial_a point_n to_o find_v it_o lay_v a_o ruler_n from_o a_o to_o the_o compliment_n of_o the_o latitude_n count_v from_o 0_o in_o the_o upper_a scale_n towards_o d_o and_o draw_v a_o line_n from_o a_o to_o it_o then_o count_v it_o the_o other_o way_n towards_o c_o viz._n 38_o degree_n 28′_n for_o the_o co-latitude_n of_o london_n and_o lay_v a_o ruler_n over_o it_o and_o the_o point_n b_o and_o where_o it_o intersect_v the_o line_n before_o draw_v be_v the_o aequinoctial_a point_n to_o be_v graduate_v then_o to_o graduate_v the_o division_n on_o each_o side_n of_o it_o require_v only_o the_o make_n in_o effect_n of_o a_o table_n of_o meridian_n altitude_n to_o every_o degree_n of_o declination_n which_o because_o the_o curve_v will_v also_o serve_v for_o the_o azimuth_n in_o which_o case_n the_o graduation_n of_o the_o curve_v which_o in_o find_v the_o hour_n be_v account_v declination_n must_v be_v account_v altitude_n must_v be_v continue_v to_o 62_o degree_n for_o this_o latitude_n and_o further_a also_o if_o it_o be_v intend_v that_o the_o curve_v shall_v find_v a_o star_n hour_n that_o have_v more_o declination_n to_o make_v this_o table_n get_v the_o sum_n and_o difference_n of_o the_o compliment_n of_o the_o latitude_n and_o of_o the_o degree_n intend_v to_o be_v graduate_v and_o if_o the_o sum_n exceed_v 90_o degree_n take_v its_o compliment_n to_o 180_o degree_n instead_o of_o it_o be_v thus_o prepare_v the_o curve_v will_v be_v ready_o make_v to_o graduate_v the_o under_o part_n of_o the_o curve_v account_v the_o sum_n in_o the_o upper_a line_n from_o o_o towards_o d_o and_o from_o the_o point_n a_o in_o the_o under_o line_n draw_v a_o line_n to_o it_o account_v the_o difference_n in_o the_o upper_a lfne_v when_o the_o degree_n propose_v to_o be_v graduate_v be_v less_o than_o the_o compliment_n of_o the_o latitude_n from_o o_o towards_o c_o but_o when_o it_o be_v more_o towards_o d_o and_o from_o the_o point_n b_o lay_v a_o ruler_n over_o it_o and_o where_o the_o ruler_n intersect_v the_o line_n former_o draw_v be_v the_o point_n where_o the_o degree_n propose_v be_v to_o be_v graduate_v example_n let_v it_o be_v require_v to_o find_v the_o point_n where_o 60_o deg_n of_o the_o curve_v be_v to_o be_v graduate_v ark_n propose_v 60_o deg_n co-latitude_n 38_o 28_o  _fw-fr 98_o 28_o sum_n 81_o 32_o difference_n 21_o  _fw-fr 32_o count_n 81_o deg_n 32′_n in_o the_o upper_a line_n from_o o_o towards_o d_o and_o from_o the_o point_n a_o draw_v a_o line_n to_o it_o count_n the_o difference_n 21_o degree_n 32′_n from_o o_o towards_o d_o because_o the_o co-latitude_n be_v less_o than_o the_o ark_n propose_v and_o lay_v a_o ruler_n over_o it_o and_o the_o point_n b_o and_o where_o it_o intersect_v the_o former_a line_n be_v the_o point_n where_o 60_o deg_n of_o the_o curve_v be_v to_o be_v graduate_v on_o the_o low_a side_n another_o example_n let_v it_o be_v propose_v

to_o graduate_v the_o same_o way_n the_o ark_n of_o 30_o degr_n 30_o degr_n co-latitude_n 38_o 28_o sum_n 68_o 28_o difference_n 8_o 28_o count_n 68_o deg_n 28′_n from_o o_o towards_o d_o and_o from_o the_o point_n a_o draw_v a_o line_n to_o it_o again_o in_o the_o say_v upper_a line_n count_v 8_o deg_n 28′_n upward_o from_o o_o towards_o c_o &_o from_o the_o point_n b_o lay_v a_o ruler_n over_o it_o &_o where_o it_o intersect_v the_o line_n last_o draw_v be_v the_o point_n where_o 30_o d._n of_o the_o curve_v be_v to_o be_v graduate_v to_o graduate_v the_o upper_a part_n of_o the_o curve_v require_v no_o other_o direction_n the_o same_o arke_n serve_v if_o the_o account_n be_v but_o make_v the_o other_o way_n and_o in_o account_v the_o sum_n the_o ruler_n lay_v over_o b._n in_o the_o low_a line_n instead_o of_o a_o and_o in_o count_v the_o difference_n over_o a_o instead_o of_o b_o neither_o be_v there_o any_o scheme_n give_v hereof_o the_o practitioner_n need_v only_o let_v the_o upper_a line_n be_v the_o line_n of_o altitude_n on_o the_o left_a edge_n of_o the_o quadrant_a continue_v out_o to_o 90_o deg_n at_o each_o end_n and_o to_o that_o end_n next_o the_o centre_n set_v c_o and_o to_o the_o other_o end_n d._n so_o likewise_o let_v that_o end_n of_o the_o verse_v scale_n next_o the_o right_a edge_n of_o the_o quadrant_a be_v continue_v to_o 180_o deg_n whereto_o set_v a_o and_o at_o the_o other_o end_n b_o and_o then_o if_o these_o direction_n be_v observe_v and_o the_o same_o distance_n and_o position_n of_o the_o line_n retain_v it_o will_v not_o be_v difficult_a to_o constitute_v a_o curve_v in_o all_o respect_n agree_v with_o that_o on_o the_o fore-side_n of_o the_o quadrant_a of_o the_o hour_n and_o azimuth_n scale_v on_o the_o right_a edge_n of_o the_o quadrant_n this_o scale_n stand_v outwardmost_a on_o the_o right_a edge_n of_o the_o quadrant_a and_o consist_v of_o two_o line_n the_o one_o a_o line_n of_o 90_o sin_n make_v equal_a to_o the_o cousin_n of_o the_o latitude_n namely_o to_o the_o sine_fw-la of_o 38_o deg_n 28′_n and_o continue_v the_o other_o way_n to_o 40_o deg_n like_o a_o verse_v sine_fw-la the_o annex_v line_n be_v the_o other_o part_n of_o this_o scale_n be_v a_o line_n of_o natural_a tangent_n begin_v where_o the_o former_a sine_fw-la begin_v the_o tangent_fw-la of_o 38_o deg_n 28′_n be_v make_v equal_a to_o the_o sine_fw-la of_o 90_o deg_n this_o tangent_fw-la be_v continue_v each_o way_n with_o the_o sine_fw-la towards_o the_o limb_n of_o the_o quadrant_a it_o shall_v have_v be_v continue_v to_o 62_o deg_n but_o that_o can_v not_o be_v without_o excursion_n wherefore_o it_o be_v break_v off_o at_o 40_o degree_n and_o the_o residue_n of_o it_o graduate_v below_o and_o next_o under_o the_o verse_v sine_z belonging_z to_z the_o curve_v that_o run_v cross_v the_o quadrant_a be_v continue_v but_o to_o half_a the_o former_a radius_fw-la of_o the_o almanac_n next_o below_o the_o former_a line_n stand_v the_o almanac_n in_o a_o regular_a ob-long_a with_o month_n name_n grave_v on_o each_o side_n of_o it_o below_o the_o almanac_n stand_v the_o quadrate_n and_o shadow_n in_o two_o arke_n of_o circle_n terminate_n against_o 45_o deg_n of_o the_o limb_n below_o they_o a_o line_n of_o 90_o sin_n in_o a_o circle_n equal_a to_o 51_o deg_n 32′_n of_o the_o limb_n break_v off_o below_o the_o straight_a line_n and_o the_o rest_n continue_v above_o it_o below_o these_o be_v put_v on_o in_o circle_n a_o line_n of_o tangent_n to_o 60_o degree_n also_o a_o line_n of_o secant_v to_o 60_o deg_n with_o a_o line_n of_o lesser_a sin_n end_v against_o 30_o deg_n of_o the_o limb_n count_v from_o the_o right_a edge_n where_o the_o graduation_n of_o the_o secant_fw-la begin_v last_o of_o all_o the_o equal_a limb_n prick_v with_o the_o prick_v of_o the_o quadrat_fw-la abut_a upon_o the_o line_n of_o sin_n and_o within_o the_o projection_n stand_v a_o portion_n of_o a_o small_a sine_fw-la number_v with_o its_o compliment_n begin_v against_o 38_o deg_n 28′_n of_o the_o line_n of_o sin_n this_o scale_n be_v call_v the_o scale_n of_o entrance_n upon_o the_o projection_n be_v place_v divers_a star_n how_o they_o be_v inscribe_v shall_v be_v afterward_o show_v the_o description_n of_o the_o backside_n put_v on_o in_o quarter_n or_o quadrant_n of_o circle_n 1_o the_o equal_a limb_n divide_v into_o degree_n as_o also_o into_o hour_n and_o half_n and_o the_o quarter_n prick_v to_o serve_v for_o a_o nocturnal_a 2_o a_o line_n of_o equal_a part_n 3_o a_o line_n of_o superficies_n or_o square_n 4_o a_o line_n of_o solid_n or_o cube_n 5_o a_o tangent_fw-la of_o 45_o degree_n double_v divide_v to_o serve_v for_o a_o dyalling_n tangent_fw-la and_o a_o semitangent_fw-la for_o projection_n 6_o the_o line_n sol_n alias_o a_o line_n of_o proportional_a sin_n 7_o a_o tangent_fw-la of_o 51_o degree_n 32′_n through_o the_o whole_a limb_n 8_o a_o line_n of_o declination_n for_o the_o sun_n to_o 23_o deg_n 31′_n four_o quadrant_n with_o the_o day_n of_o the_o month_n 9_o 10_o 11_o 12_o 13_o the_o sun_n true_a place_n with_o the_o character_n of_o the_o 12_o sign_n 14_o the_o line_n of_o segment_n with_o a_o chord_n before_o they_o begin_v 15_o the_o line_n of_o metal_n and_o equated_a body_n 16_o the_o line_n of_o quadrature_n 17_o the_o line_n of_o inscribe_v body_n 18_o a_o line_n of_o 12_o hour_n of_o ascension_n with_o star_n name_n declination_n and_o ascensional_a difference_n above_o all_o these_o a_o table_n to_o know_v the_o epact_n and_o what_o day_n of_o the_o week_n the_o first_o day_n of_o march_n happen_v upon_o by_o inspection_n continue_v to_o the_o year_n 1700._o all_o these_o between_o the_o limb_n and_o the_o centre_n on_o the_o right_a edge_n a_o line_n of_o equal_a part_n from_o the_o centre_n decimal_o subdivide_v be_v a_o line_n of_o 10_o inch_n also_o a_o dyalling_n tangent_fw-la or_o scale_n of_o 6_o hour_n the_o whole_a length_n of_o the_o quadrant_a not_o issue_v from_o the_o centre_n on_o the_o left_a edge_n a_o tangent_fw-la of_o 63_o deg_n 26′_n from_o the_o centre_n also_o a_o scale_n of_o latitude_n fit_v to_o the_o former_a scale_n of_o hour_n not_o issue_v from_o the_o centre_n and_o below_o it_o a_o small_a chord_n the_o use_v of_o the_o quadrant_n lordsday_n 1657_o 63_o 68_o 74_o ☉_o 85_o 91_o 96_o anno_fw-la 25_o 1_o 26_o 3_o  _fw-fr 4_o 11_o 6_o epact_n monday_n 58_o ☽_o 69_o 75_o 80_o 86_o ☽_o 97_o anno_o 6_o  _fw-fr 7_o 14_o 9_o 15_o  _fw-fr 17_o epact_a tuesday_n 59_o 64_o 70_o ♂_o 81_o 87_o 92_o 98_o anno_fw-la 17_o 12_o 18_o  _fw-fr 20_o 26_o 22_o 28_o epact_n wenesday_n ☿_o 65_o 71_o 76_o 82_o ☿_o 93_o 99_o anno_o  _fw-fr 23_o 29_o 25_o 1_o  _fw-fr 3_o 9_o epact_a thursday_n 60_o 66_o ♃_o 77_o 83_o 88_o 94_o ♃_o anno_o 28_o 4_o  _fw-fr 6_o 12_o 7_o 14_o  _fw-fr epact_n friday_n 61_o 67_o 72_o 78_o ♀_o 89_o 95_o 700_o anno_fw-la 9_o 15_o 11_o 17_o  _fw-fr 18_o 25_o 20_o epact_n saturday_n 62_o ♄_o 73_o 79_o 84_o 90_o ♄_o 701_o anno_o 20_o  _fw-fr 22_o 28_o 23_o 29_o  _fw-fr 1_o epact_a day_n the_o same_o as_o the_o first_o of_o march_n march_n 1_o 8_o 15_o 22_o 29_o november_n august_n 2_o 9_o 16_o 23_o 30_o august_n may_n 3_o 10_o 17_o 24_o 31_o jnuary_n october_n 4_o 11_o 18_o 25_o 0_o october_n april_n 5_o 12_o 19_o 26_o 00_o july_n septem_fw-la 6_o 13_o 20_o 27_o 00_o december_n june_n 7_o 14_o 21_o 28_o 00_o february_n perpetual_a almanac_n of_o the_o use_v of_o the_o projection_n before_o this_o projection_n can_v be_v use_v the_o sun_n declination_n be_v require_v &_o by_o consequence_n the_o day_n of_o the_o month_n for_o the_o ready_a find_v thereof_o there_o be_v repeat_v the_o same_o table_n that_o stand_v on_o the_o backside_n of_o this_o quadrant_a in_o each_o rule_a space_n the_o uppermost_a figure_n signify_v the_o year_n of_o the_o lord_n and_o the_o column_n it_o be_v place_v in_o show_v upon_o what_o day_n of_o the_o week_n the_o first_o day_n of_o march_n happen_v upon_o in_o that_o year_n and_o the_o undermost_a figure_n in_o the_o say_v rule_v space_n show_v what_o be_v the_o epact_n for_o that_o year_n and_o this_o continue_a to_o the_o year_n 1701_o inclusive_a example_n look_v for_o the_o year_n 1660_o i_o find_v the_o figure_n 60_o stand_v in_o thursday_n column_n whence_o i_o may_v conclude_v that_o the_o first_o day_n of_o march_n that_o year_n will_v be_v thursday_n and_o under_o it_o stand_v 28_o for_o the_o epact_n that_o year_n of_o the_o almanac_n have_v as_o before_o find_v what_o day_n of_o the_o week_n the_o first_o day_n of_o march_n happen_v upon_o

one_o day_n variation_n in_o 300_o year_n as_o be_v observe_v by_o mr._n philip_n the_o use_v of_o the_o quadrant_n without_o rectify_v the_o bead_n nothing_o can_v be_v perform_v by_o this_o projection_n except_o find_v the_o sun_n meridian_n altitude_n be_v show_v upon_o the_o index_n by_o the_o intersection_n of_o the_o parallel_n of_o declination_n therewith_o also_o the_o time_n when_o the_o sun_n will_v be_v due_a east_n or_o west_n trace_v the_o parallel_n of_o declination_n to_o the_o right_a edge_n of_o the_o projection_n and_o the_o hour_n it_o there_o intersect_v in_o most_o case_n to_o be_v due_o estimate_v show_v the_o time_n seek_v thus_o when_o the_o sun_n have_v 21_o deg_n of_o north_n declination_n we_o shall_v find_v that_o he_o will_v be_v due_a east_n or_o west_n about_o three_o quarter_n of_o a_o hour_n past_o 4_o in_o the_o afternoon_n or_o a_o quarter_n past_o 7_o in_o the_o morning_n the_o declination_n be_v to_o be_v find_v on_o the_o backside_n of_o the_o quadrant_a by_o lay_v the_o thread_n over_o the_o day_n of_o the_o month_n to_o rectify_v the_o bead._n lay_v the_o thread_n upon_o the_o graduated_a index_n and_o set_v the_o bead_n to_o the_o observe_v or_o give_v altitude_n and_o when_o the_o altitude_n be_v nothing_o or_o when_o the_o sun_n be_v in_o the_o horizon_n set_v the_o bead_n to_o the_o cypher_n on_o the_o graduated_a index_n which_o afterward_o be_v carry_v without_o stretch_v to_o the_o parallel_n of_o declination_n the_o thread_n in_o the_o limb_n show_v the_o amplitude_n or_o azimuth_n and_o the_o bead_n among_o the_o hour_n show_v the_o true_a time_n of_o the_o day_n example_n upon_o the_o 24_o of_o april_n the_o sun_n declination_n will_v be_v find_v to_o 16_o deg_n north._n now_o to_o find_v his_o amplitude_n and_o the_o time_n of_o his_o rise_n lay_v the_o thread_n over_o the_o graduated_a index_n set_v the_o bead_n to_o the_o begin_n of_o the_o graduation_n of_o the_o index_n and_o bring_v it_o without_o stretch_v to_o the_o parallel_n of_o declination_n above_o be_v 16_o d_o and_o the_o thread_n in_o the_o limb_n will_v lie_v over_o 26_o deg_n 18′_n for_o the_o sun_n amplitude_n or_o coast_n of_o rise_v to_o the_o northward_o of_o the_o east_n and_o the_o bead_n among_o the_o hour_n show_v 24_o minute_n past_o 4_o for_o the_o time_n of_o sun_n rise_v which_o double_v give_v the_o length_n of_o the_o night_n 8_o hour_n 49_o min._n in_o like_a manner_n the_o time_n of_o set_v double_v give_v the_o length_n of_o the_o day_n the_o same_o to_o find_v the_o hour_n and_o azimuth_n let_v the_o give_v altitude_n be_v 45_o degree_n have_v rectify_v the_o bead_n to_o the_o say_a altitude_n on_o the_o index_n and_o bring_v it_o to_o the_o intersect_n the_o parallel_n of_o declination_n the_o thread_n lie_v over_o 50_o degree_n 48′_n for_o the_o sun_n azimuth_n from_o the_o south_n and_o the_o bead_n among_o the_o hour_n show_v the_o time_n of_o the_o day_n to_o be_v 41_o minute_n past_o 9_o in_o the_o morning_n or_o 19_o minute_n past_o two_o in_o the_o afternoon_n another_o example_n wherein_o the_o operation_n will_v be_v upon_o the_o revert_v tail_n let_v the_o altitude_n be_v 3_o deg_n 30′_n and_o the_o declination_n 16_o deg_n north_n as_o before_o to_o know_v when_o to_o rectify_v the_o bead_n to_o the_o upper_a or_o nether_a altitude_n will_v be_v no_o matter_n of_o difficulty_n for_o if_o the_o bead_n be_v set_v to_o the_o nether_a altitude_n will_v not_o meet_v with_o the_o parallel_n of_o declination_n than_o set_v it_o to_o the_o upper_a altitude_n and_o it_o will_v meet_v with_o winter_n parallel_n of_o like_a declination_n which_o in_o this_o case_n supply_v the_o turn_n so_o in_o this_o example_n the_o bead_n be_v set_v to_o the_o upper_a altitude_n of_o 3_o deg_n 30′_n and_o carry_v to_o the_o winter_n parallel_n of_o declination_n the_o thread_n in_o the_o limb_n will_v fall_v upon_o 68_o deg_n 28′_n for_o the_o sun_n azimuth_n from_o the_o north_n and_o the_o bead_n among_o the_o hour_n show_v the_o time_n of_o the_o day_n to_o be_v either_o 5_o in_o the_o morning_n or_o 7_o at_o night_n another_o example_n admit_v the_o sun_n have_v 20_o degr_n of_o north_n declination_n as_o about_o the_o 9th_o of_o may_n and_o his_o observe_a altitude_n be_v 56_o deg_n 20′_n have_v rectify_v the_o bead_n thereto_o and_o bring_v it_o to_o intersect_v the_o parallel_n of_o 20_o deg_n among_o the_o hour_n it_o show_v the_o time_n of_o the_o day_n to_o be_v 11_o in_o the_o morning_n or_o 1_o in_o the_o afternoon_n and_o the_o azimuth_n of_o the_o sun_n to_o be_v 26_o deg_n from_o the_o south_n the_o use_v of_o the_o projection_n to_o find_v the_o sun_n altitude_n on_o all_o hour_n or_o azimuth_n will_v be_v but_o the_o converse_n of_o what_o be_v already_o say_v therefore_o one_o example_n shall_v serve_v when_o the_o sun_n have_v 45_o deg_n of_o azimuth_n from_o the_o south_n and_o his_o declination_n 13_o deg_n northwards_o lay_v the_o thread_n over_o 45_o deg_n in_o the_o limb_n and_o where_o the_o thread_n intersect_v the_o parallel_n of_o declination_n thereto_o remove_v the_o bead_n which_o carry_v to_o the_o index_n without_o stretch_v show_v 43_o deg_n 50′_n for_o the_o altitude_n seek_v likewise_o to_o the_o same_o declination_n if_o it_o be_v require_v to_o find_v the_o sun_n altitude_n for_o the_o hour_n of_o 2_o or_o 10._o lay_v the_o thread_n over_o the_o intersection_n of_o the_o hour_n propose_v with_o the_o parallel_n of_o declination_n and_o thereto_o set_v the_o bead_n which_o carry_v to_o the_o index_n show_v the_o altitude_n seek_v namely_o 44_o deg_n 31′_n the_o same_o altitude_n also_o belong_v to_o that_o azimuth_n the_o thread_n in_o the_o former_a position_n lie_v over_o in_o the_o limb_n this_o projection_n be_v of_o worst_a performance_n early_o in_o the_o morning_n or_o late_a in_o the_o evening_n about_o which_o time_n mr._n dary_n curve_v be_v of_o best_a performance_n whereto_o we_o now_o address_v ourselves_o of_o the_o curve_v line_n and_o scale_n thereto_o fit_v this_o as_o we_o have_v say_v before_o be_v the_o ingenious_a invention_n of_o m._n michael_n dary_n derive_v from_o the_o proportionalty_n of_o two_o like_v equiangle_v plain_a triangle_n accommodate_v to_o the_o latitude_n of_o london_n for_o the_o ready_a work_n of_o these_o two_o proportion_n 1._o for_o the_o hour_n as_o the_o cousin_n of_o the_o latitude_n be_v to_o the_o secant_fw-la of_o the_o declination_n so_o be_v the_o difference_n between_o the_o sine_fw-la of_o the_o sun_n propose_v and_o meridian_n altitude_n to_o the_o verse_v sine_fw-la of_o the_o hour_n from_o noon_n and_o the_o converse_n and_o so_o be_v the_o sine_fw-la of_o the_o sun_n meridian_n altitude_n to_o the_o verse_v sine_fw-la of_o the_o semidiurnal_a ark_n 2._o for_o the_o azimuth_n the_o curve_v be_v fit_v to_o find_v it_o from_o the_o south_n and_o not_o from_o the_o north_n and_o the_o proportion_n wrought_v upon_o it_o will_v be_v as_o the_o cousin_n of_o the_o latitude_n be_v to_o the_o secant_fw-la of_o the_o altitude_n so_o be_v the_o difference_n of_o the_o verse_v sin_n of_o the_o sun_n or_o star_n distance_n from_o the_o elevate_a pole_n and_o of_o the_o sum_n of_o the_o compliment_n both_o of_o the_o latitude_n and_o altitude_n to_o the_o verse_v sine_fw-la of_o the_o azimuth_n from_o the_o noon_n meridian_n which_o will_v not_o hold_v backward_o to_o find_v the_o altitude_n on_o all_o azimuth_n because_o the_o altitude_n be_v a_o term_n involve_v both_o in_o the_o second_o and_o three_o term_n of_o the_o former_a proportion_n if_o the_o three_o term_n of_o the_o former_a proportion_n have_v not_o be_v a_o difference_n of_o sines_n or_o verse_v sin_n the_o curve_v line_n will_v have_v be_v a_o straight-line_n and_o the_o three_o term_n always_o count_v from_o one_o point_n which_o though_o in_o the_o use_n it_o may_v seem_v to_o be_v so_o here_o yet_o in_o effect_n the_o three_o term_n for_o the_o hour_n be_v always_o count_v from_o the_o meridian_n altitude_n here_o observe_v that_o the_o thread_n lie_v over_o 12_o or_o the_o end_n of_o the_o verse_v scale_n and_o over_o the_o sun_n meridian_n altitude_n in_o the_o line_n of_o altitude_n it_o will_v also_o upon_o the_o curve_v show_v the_o sun_n declination_n which_o by_o construction_n be_v so_o frame_v that_o if_o the_o distance_n from_o that_o point_n to_o the_o meridian_n altitude_n be_v make_v the_o cousin_n of_o latitude_n the_o distance_n of_o the_o say_a point_n from_o the_o end_n of_o the_o verse_v scale_n number_v with_o 12_o shall_v be_v the_o secant_fw-la of_o the_o declination_n to_o the_o same_o radius_fw-la be_v both_o in_o one_o straight-line_n by_o the_o former_a constitution_n of_o the_o thread_n and_o instead_o

59_o 17_o 04_o 31_o 15_o 03_o 17_o 21_o day_n november_n ☉_o r._n a._n ☉_o decl._n h._n m._n d._n m._n 1_o 15_o 07_o 17_o 38_o 2_o 15_o 11_o 17_o 54_o 3_o 15_o 15_o 18_o 10_o 4_o 15_o 19_o 18_o 26_o 5_o 15_o 23_o 18_o 41_o 6_o 15_o 27_o 18_o 56_o 7_o 15_o 31_o 19_o 11_o 8_o 15_o 36_o 19_o 26_o 9_o 15_o 40_o 19_o 40_o 10_o 15_o 45_o 19_o 53_o 11_o 15_o 49_o 20_o 07_o 12_o 15_o 53_o 20_o 19_o 13_o 15_o 58_o 20_o 32_o 14_o 16_o 02_o 20_o 44_o 15_o 16_o 07_o 20_o 56_o 16_o 16_o 11_o 21_o 08_o 17_o 16_o 15_o 21_o 19_o 18_o 16_o 19_o 21_o 29_o 19_o 16_o 23_o 21_o 39_o 20_o 16_o 28_o 21_o 49_o 21_o 16_o 32_o 21_o 58_o 22_o 16_o 36_o 22_o 08_o 23_o 16_o 40_o 22_o 16_o 24_o 16_o 44_o 22_o 24_o 25_o 16_o 49_o 22_o 32_o 26_o 16_o 53_o 22_o 39_o 27_o 16_o 57_o 22_o 46_o 28_o 17_o 02_o 22_o 52_o 29_o 17_o 06_o 22_o 58_o 30_o 17_o 11_o 23_o 03_o 31_o  _fw-fr  _fw-fr  _fw-fr  _fw-fr day_n december_n ☉_o r._n a._n ☉_o decl._n h._n m._n d._n m._n 1_o 17_o 15_o 23_o 08_o 2_o 17_o 20_o 23_o 13_o 3_o 17_o 25_o 23_o 17_o 4_o 17_o 29_o 23_o 20_o 5_o 17_o 34_o 23_o 23_o 6_o 17_o 38_o 23_o 26_o 7_o 17_o 42_o 23_o 28_o 8_o 17_o 47_o 23_o 29_o 9_o 17_o 51_o 23_o 30_o 10_o 17_o 56_o 23_o 31_o 11_o 18_o 00_o 23_o 31½_n 12_o 18_o 05_o 23_o 31_o 13_o 18_o 09_o 23_o 30_o 14_o 18_o 14_o 23_o 29_o 15_o 18_o 19_o 23_o 27_o 16_o 18_o 24_o 23_o 25_o 17_o 18_o 28_o 23_o 22_o 18_o 18_o 33_o 23_o 19_o 19_o 18_o 37_o 23_o 15_o 20_o 18_o 41_o 23_o 11_o 21_o 18_o 45_o 23_o 07_o 22_o 18_o 49_o 23_o 02_o 23_o 18_o 54_o 22_o 56_o 24_o 18_o 58_o 22_o 50_o 25_o 19_o 03_o 22_o 43_o 26_o 19_o 07_o 22_o 36_o 27_o 19_o 11_o 22_o 29_o 28_o 19_o 16_o 22_o 21_o 29_o 19_o 20_o 22_o 13_o 30_o 19_o 25_o 22_o 04_o 31_o 19_o 30_o 21_o 55_o a_o rectify_v table_n for_o the_o sun_n declination_n  _fw-fr year_n year_n year_n  _fw-fr 1657_o 1661._o 1665_o 1669_o 1673_o 1659._o 1663._o 1667_o 1671_o 1675_o 1660_o 1664_o 1668_o 1672_o 1676_o month_n min._n min._n min._n january_n 3_o s_o 2_o a_o 5_o a_o 4_o s_z 3_o a_o 7_o a_o 5_o s_z 4_o a_o 9_o a_o february_n 5_o s_z 5_o a_o 10_o a_o 5_o s_z 5_o a_o 11_o a_o 6_o s_z 5_o a_o 11_o a_o march_z 6_o s_z 5_o a_o 13_o s_z 5_o a_o 5_o s_z 12_o a_o 5_o a_o 5_o s_z 12_o a_o april_n 5_o a_o 5_o s_z 11_o a_o 5_o a_o 5_o s_z 10_o a_o 4_o a_o 4_o s_z 9_o a_o may_v 4_o a_o 4_o s_z 8_o a_o 3_o a_o 3_o s_z 6_o a_o 2_o a_o 2_o s_z 4_o a_o june_n 1_o a_o 1_o s_o 2_o a_o 0_o s_o 0_o a_o 0_o s_o 1_o s_z 1_o a_o 3_o s_o july_n 2_o s_z 2_o a_o 5_o s_z 3_o s_z 3_o a_o 7_o s_z 4_o s_z 4_o a_o 9_o s_o august_n 5_o s_z 5_o a_o 10_o s_z 5_o s_z 5_o a_o 11_o s_z 6_o s_z 5_o a_o 12_o s_o september_n 6_o s_z 5_o a_o 13_o s_z 6_o a_o 5_o s_z 13_o a_o 6_o a_o 5_o s_z 12_o a_o october_n 6_o a_o 5_o s_z 12_o a_o 5_o a_o 5_o s_z 11_o a_o 4_o a_o 5_o s_z 9_o a_o novem._n 3_o a_o 4_o s_z 7_o a_o 2_o a_o 3_o s_z 5_o a_o 1_o a_o 2_o s_z 3_o a_o decemb._n 0_o a_o 1_o s_z 1_o a_o 1_o s_o 0_o a_o 1_o s_z 2_o s_z 1_o a_o 3_o s_z the_o use_n of_o the_o rectify_v table_n note_v that_o the_o minute_n under_o the_o respective_a year_n be_v to_o be_v add_v or_o substract_v to_o or_o from_o the_o sun_n declination_n in_o the_o former_a table_n as_o be_v note_v with_o the_o letter_n a_o or_o s_o and_o also_o note_v that_o the_o first_o figure_n in_o each_o month_n stand_v for_o the_o first_o 10_o day_n of_o the_o month_n and_o the_o second_o for_o the_o second_o 10_o day_n &_o the_o three_o for_o the_o last_o 10_o day_n except_o in_o march_n or_o september_n which_o in_o march_n will_v be_v the_o first_o 9_o day_n only_o and_o in_o september_n the_o first_o 12_o day_n example_n i_o will_v know_v the_o sun_n declination_n the_o 15_o day_n of_o may_n 1668._o now_o because_o this_o day_n of_o the_o month_n fall_v in_o the_o second_o 10_o day_n i_o look_v in_o the_o table_n under_o the_o year_n 1663._o and_o right_a against_o may_v you_o shall_v find_v that_o in_o the_o second_o place_n of_o the_o month_n stand_v 6_o a_o which_o show_v i_o that_o i_o must_v add_v 6_o minute_n to_o the_o sun_n declination_n in_o the_o former_a table_n 21_o degree_n 5_o min._n that_o stand_v against_o the_o 15_o day_n of_o may_n and_o then_o i_o find_v that_o the_o sun_n will_v have_v 21_o deg_n 11_o min._n of_o north_n declination_n and_o so_o for_o the_o rest_n which_o will_v never_o differ_v above_o two_o minute_n from_o the_o truth_n but_o seldom_o so_o much_o and_o for_o the_o most_o part_n true_a note_v that_o the_o former_a table_n of_o the_o sun_n declination_n be_v fit_v exact_o for_o the_o year_n 1666._o by_o the_o rule_n mr._n wright_n give_v in_o his_o correction_n of_o error_n and_o from_o his_o table_n and_o may_v indifferent_o serve_v for_o the_o year_n 1658_o 1662._o 1670_o 1674_o without_o any_o sensible_a error_n and_o the_o table_n of_o right_a ascension_n will_v not_o vary_v a_o minute_n of_o time_n in_o many_o year_n finis_fw-la error_n in_o the_o horizontal_n quadrant_n page_n 5_o line_n 6_o in_o a_o italian_a letter_n shall_v not_o have_v be_v distinct_a nor_o in_o another_o letter_n from_o the_o former_a line_n page_n 5._o line_n 9_o for_o quarter_n read_v half_a p._n 5._o l._n 13._o r._n of_o a_o quadrant_a p._n 11._o l._n 7._o r._n 63_o d._n 26′_n p._n 19_o l._n 7._o r._n the_o same_o day_n to_o p._n 23._o l._n 17._o r._n and_o end_n at_o 32′_n past_o 9_o p._n 27._o l._n 7._o for_o n_o r_o r._n n_o z._n p._n 28._o l._n 4._o r._n in_o the_o parallel_n p._n 30._o l._n 9_o &_o l._n 10._o r._n 23_o d._n 31′_n p._n 38._o l._n 4._o r._n be_v to_o the_o sine_fw-la p._n 50._o l._n 5._o r._n whereof_o the_o diameter_n a_o appendix_n touch_v reflective_a dial_v by_o john_n lyon_n professor_n of_o this_o or_o any_o other_o part_n of_o the_o mathematics_n near_o somerset_n house_n in_o the_o strand_n london_n print_v anno_fw-la domini_fw-la 1658._o direct_v dial_v by_o a_o hole_n or_o nodus_fw-la to_o draw_v a_o dial_n under_o any_o window_n that_o the_o sun_n shine_v upon_o by_o help_n of_o a_o thread_n fasten_v in_o any_o point_n of_o the_o direct_a axis_n find_v in_o the_o ceiling_n and_o a_o hole_n in_o any_o pane_n of_o glass_n or_o a_o knob_v or_o nodus_fw-la upon_o any_o side_n of_o the_o window_n or_o window-post_n constructio_fw-la first_o draw_v on_o pasteboard_n or_o other_o material_a a_o horizontal_n dial_n for_o the_o latitude_n propose_v then_o by_o help_n of_o the_o sun_n azimuth_n which_o may_v be_v find_v by_o help_n of_o a_o general_a quadrant_n at_o any_o time_n or_o by_o know_v the_o true_a hour_n of_o the_o day_n with_o the_o help_n of_o the_o say_v horizontal_n dial_n and_o draw_v that_o true_a meridian_n from_o the_o hole_n or_o nodus_fw-la propose_v both_o above_o on_o the_o ceiling_n and_o below_o on_o the_o wall_n and_o floor_n of_o the_o room_n so_o that_o if_o a_o right_a line_n be_v extend_v from_o the_o say_a hole_n or_o nodus_fw-la by_o any_o point_n in_o any_o of_o those_o line_n it_o will_v be_v in_o the_o meridian_n circle_n of_o the_o world_n to_o find_v a_o point_n in_o the_o direct_a axis_n of_o the_o world_n which_o will_v ever_o fall_v to_o be_v in_o the_o say_a meridian_n in_o which_o point_n the_o end_n of_o a_o thread_n be_v to_o be_v fasten_v first_o fix_v the_o end_n of_o a_o thread_n or_o small_a silk_n in_o the_o centre_n of_o the_o hole_n or_o nodus_fw-la and_o move_v the_o other_o end_n thereof_o up_o or_o down_o in_o the_o say_a meridian_n former_o draw_v on_o the_o ceiling_n or_o wall_n until_o by_o apply_v the_o side_n of_o a_o quadrant_n to_o that_o thread_n it_o be_v find_v to_o be_v elevate_v equal_a to_o the_o latitude_n of_o the_o place_n so_o be_v that_o thread_n direct_o situate_v parallel_n to_o the_o axis_n of_o the_o world_n and_o the_o point_n where_o the_o end_n of_o that_o thread_n touch_v the_o meridian_n either_o on_o the_o ceiling_n or_o wall_n be_v that_o point_n in_o the_o direct_a axis_n seek_v for_o wherein_o fix_v one_o end_n of_o a_o thread_n which_o thread_n will_v be_v of_o present_a use_n in_o project_v of_o hour-point_n in_o any_o place_n propose_v then_o to_o find_v the_o hour-point_n

either_o under_o the_o window_n or_o any_o other_o convenient_a place_n in_o the_o room_n place_v the_o centre_n of_o the_o say_v horizontal_n dial_n in_o the_o centre_n of_o the_o hole_n or_o nodus_fw-la also_o situate_a the_o say_a dial_n exact_o parallel_v to_o the_o horizon_n and_o the_o meridian_n of_o the_o say_a dial_n in_o the_o meridian_n of_o the_o world_n which_o as_o before_o may_v easy_o be_v do_v if_o at_o that_o instant_n you_o know_v the_o true_a hour_n of_o the_o day_n then_o take_v the_o thread_n who_o end_n be_v fix_v in_o a_o point_n in_o the_o direct_a axis_n and_o move_v it_o to_o and_o fro_o until_o the_o say_a thread_n do_v interpose_v between_o your_o eye_n and_o the_o hour-line_n on_o the_o say_v horizontal_n dial_n which_o you_o intend_v to_o draw_v and_o then_o keep_v your_o eye_n at_o that_o situation_n make_v a_o point_n or_o mark_v in_o any_o place_n where_o you_o please_v or_o under_o the_o window_n so_o that_o the_o say_a thread_n or_o string_n may_v interpose_v between_o that_o point_n or_o mark_v so_o make_v and_o your_o eye_n as_o aforesaid_a which_o say_v point_n so_o sound_a will_v show_v the_o true_a time_n of_o the_o day_n at_o that_o hour_n all_o the_o year_n long_o the_o sun_n shine_v thereon_o so_o will_v that_o point_n together_o with_o the_o say_a thread_n serve_v to_o show_v the_o hour_n instead_o of_o a_o hour-line_n in_o like_a manner_n the_o say_a thread_n fix_v in_o the_o axis_n may_v be_v again_o move_v to_o and_o fro_o until_o the_o say_a thread_n do_v interpose_v between_o the_o eye_n and_o any_o other_o hour-line_n desire_v on_o the_o say_a horizontal_n dial_n and_o then_o as_o before_o make_v another_o point_n or_o mark_v in_o any_o place_n at_o pleasure_n or_o under_o the_o say_a window_n by_o project_v a_o point_n from_o the_o eye_n so_o that_o the_o say_a thread_n also_o interpose_v between_o that_o point_n to_o be_v make_v and_o the_o eye_n so_o will_v that_o point_n so_o find_v show_v the_o true_a time_n 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extend_v a_o thread_n fasten_v in_o the_o centre_n of_o the_o horizontal_n dial_n parallelly_n to_o the_o horizon_n over_o each_o respective_a hour-line_n and_o hold_v it_o steady_a let_v another_o extend_v the_o thread_n fasten_v in_o the_o meridian_n in_o the_o ceiling_n along_o by_o the_o edge_n of_o the_o former_a horizontal_n thread_n and_o so_o this_o latter_a thread_n will_v find_v divers_a point_n on_o the_o ground_n through_o which_o if_o hour-line_n be_v draw_v and_o the_o sun_n shine_v through_o the_o hole_n in_o the_o pane_n of_o glass_n before_o make_v the_o spot_n of_o the_o sun_n on_o the_o ground_n shall_v show_v the_o time_n of_o the_o day_n for_o the_o point_n that_o will_v be_v thus_o find_v on_o the_o beam_n or_o transome_n the_o thread_n fix_v in_o the_o ceiling_n or_o instead_o of_o it_o a_o piece_n of_o tape_n there_o fix_v must_v be_v move_v so_o up_o and_o down_o that_o the_o spot_n of_o the_o sun_n may_v shine_v upon_o it_o and_o be_v extend_v to_o the_o 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