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what_o day_n of_o the_o week_n the_o first_o of_o march_n will_v happen_v in_o the_o year_n of_o our_o lord_n 1687._o operation_n divisor_n 7_o 2_o the_o even_o four_o of_o the_o year_n the_o remainder_n neglect_v 301_o quotient_n 1687_o 421_o 2100_o 21_o 10_o 7_o 3_o remains_o because_o three_o remains_o the_o first_o of_o march_n in_o that_o year_n happen_v on_o a_o tuesday_n in_o the_o year_n 1679_o nothing_o remain_v therefore_o it_o happen_v on_o a_o saturday_n proposition_n 2._o the_o day_n of_o the_o week_n on_o which_o the_o first_o of_o march_n happen_v on_o any_o year_n be_v know_v and_o remember_v to_o find_v on_o what_o day_n of_o the_o week_n any_o day_n of_o any_o month_n in_o the_o say_a year_n happen_v to_o perform_v this_o proposition_n the_o follow_a verse_n be_v in_o effect_n a_o perpetual_a almanac_n be_v to_o be_v record_v &_o lay_v up_o in_o memory_n all_o evil_a chance_n grievous_a evil_n bring_v fierce_a death_n attend_v foul_a chance_n govern_v in_o this_o 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thy_o work_n or_o try_v the_o rumb_n n_o ever_o desist_v but_o let_v we_o have_v more_o of_o thou_o h_n ever_o be_v but_o a_o tangent_fw-la but_o let_v we_o have_v a_o sine_fw-la o_o r_o bosom_n full_a of_o thy_o industrious_a toil_n i_o t_o will_v inform_v the_o weak_a enrich_v our_o soyl._n your_o love_a friend_n sylvanus_n morgan_n the_o content_n of_o the_o first_o book_n in_o the_o proportional_a part._n geometrical_a definition_n page_z 1_o 2_o to_o raise_v perpendicular_o 2_o 3_o 4_o to_o draw_v a_o line_n parallel_n to_o another_o line_n 5_o to_o bring_v three_o point_n into_o a_o circle_n 6_o to_o find_v a_o right_a line_n equal_a to_o the_o arch_n of_o a_o circle_n 9_o 10_o chord_n sin_n tangent_n secant_v verse_v sines_n etc._n etc._n define_v 11_o 12_o 13_o the_o scale_n in_o the_o frontispiece_n describe_v 14_o plain_a triangle_n both_o right_a and_o oblique_v angle_v resolve_v by_o protraction_n from_o p_o 15_o to_o 25_o proportion_n in_o sines_n resolve_v by_o a_o line_n of_o chord_n p_o 21_o to_o 25_o proportion_n in_o tangent_n alone_o so_o resolve_v p._n 25_o 27_o 28_o 29_o 30_o proportion_n in_o sines_n and_o tangent_n resolve_v by_o a_o line_n of_o chord_n p._n 26_o 31_o 32_o 33_o particular_a scheme_n fit_v from_o proportion_n to_o the_o case_n of_o oblique_a angle_a sphoerical_a triangle_n to_o find_v the_o azimuth_n p._n 34_o 35_o as_o also_o the_o amplitude_n p._n 36_o the_o azimuth_n compass_n in_o the_o frontispiece_n describe_v p._n 38_o the_o variation_n find_v by_o the_o azimuth_n compass_n p._n 39_o to_o find_v the_o hour_n of_o the_o day_n p._n 40_o as_o also_o the_o azimuth_n and_o angle_n of_o position_n p._n 41_o to_o find_v the_o sun_n altitude_n on_o all_o hour_n p._n 43_o 46_o also_o the_o distance_n of_o place_n in_o the_o arch_n of_o a_o great_a circle_n p._n 44_o to_o find_v the_o sun_n altitude_n on_o all_o azimuth_n p._n 48_o the_o latitude_n declination_n and_o azimuth_n give_v to_o find_v the_o hour_n p._n 50_o to_o 54_o ☞_o to_o find_v the_o amplitude_n with_o the_o manner_n of_o measure_v a_o sine_fw-la to_o a_o lesser_a radius_fw-la p._n 55_o to_o get_v the_o sun_n altitude_n by_o the_o shadow_n of_o a_o thread_n or_o gnomon_n p._n 56_o the_o content_n of_o the_o treatise_n of_o navigation_n of_o the_o imperfection_n and_o uncertainty_n of_o navigation_n p._n 1_o to_o 5_o to_o measure_v a_o course_n and_o distance_n on_o the_o plain_a chart._n p._n 7_o 8_o 9_o of_o the_o quantity_n of_o a_o degree_n and_o of_o the_o form_n of_o the_o logboard_n p._n 9_o 10_o a_o reckon_a keep_v in_o league_n how_o reduce_v by_o the_o pen_n to_o degree_n and_o centesme_n p._n 11_o 12_o of_o a_o traverse-quadrant_n p._n 13_o a_o traverse_v plait_v on_o the_o plain_a chart_n without_o draw_v line_n thereon_o p._n 14_o to_o 18_o a_o scheme_n with_o direction_n to_o find_v what_o course_n and_o way_n the_o ship_n have_v make_v through_o a_o current_n p._n 18_o to_o 21_o divers_a rule_n for_o correct_v of_o the_o dead_a reckon_a from_o p._n 21_o to_o 33_o of_o the_o error_n of_o the_o plain_a chart._n p._n 33_o and_o how_o such_o chart_n may_v be_v amend_v p._n 34_o to_o find_v the_o rumbe_n between_o two_o place_n p._n 35_o proportion_n have_v one_o term_n the_o middle_a latitude_n how_o far_o to_o be_v trust_v to_o p._n 35_o to_o 38_o to_o find_v the_o rumbe_n between_o two_o place_n by_o a_o line_n of_o chord_n only_o p._n 39_o to_o 42_o the_o meridian-line_n of_o mercators_n chart_n supply_v general_o by_o a_o line_n of_o chord_n p._n 42_o to_o 47_o the_o meridian-line_n divide_v from_o the_o limb_n of_o a_o quadrant_n with_o the_o use_n thereof_o in_o find_v the_o rumbe_n p._n 48_o to_o 51_o the_o error_n commit_v by_o keep_v of_o a_o reckon_a on_o the_o plain_a chart_n remove_v p._n 52_o to_o 54_o of_o the_o nature_n of_o the_o rumbe_n on_o the_o globe_n p._n 55_o to_o 57_o mereators_n chart_n demonstrate_v from_o proportion_n p._n 58_o to_o 60_o objection_n against_o it_o answer_v p._n 60_o to_o 63_o to_o find_v the_o rumbe_n between_o two_o place_n in_o the_o chart._n p._n 64_o distance_n of_o place_n how_o measure_v on_o that_o chart._n p._n 65_o to_o 71_o another_o traverse-quadrant_n fit_v for_o that_o chart_n with_o a_o traverse_v plait_v thereby_o without_o draw_v line_n on_o the_o chart._n p._n 71_o to_o 78_o to_o measure_v a_o course_n and_o distance_n in_o that_o chart_n without_o the_o use_n of_o compass_n p._n 79_o of_o sail_v by_o the_o arch_n of_o a_o great_a circle_n p._n 81_o to_o find_v the_o latitude_n of_o the_o great_a arch_n by_o the_o stereographick_a projection_n p._n 82_o to_o 83_o of_o a_o tangent_fw-la projection_n from_o the_o pole_n for_o find_v the_o latitude_n of_o the_o great_a arch_n p._n 84_o to_o 88_o with_o a_o new_a method_n of_o calculation_n raise_v from_o it_o p._n 89_o 90_o and_o how_o to_o measure_v the_o distance_n in_o the_o arch_n and_o the_o angle_n of_o position_n p._n 91_o another_o tangent_fw-la projection_n from_o the_o equinoctial_a for_o find_v the_o latitude_n of_o the_o arch._n p._n 93_o to_o 100_o and_o how_o to_o find_v the_o vertical_a angle_n and_o arke_n latitude_n geometrical_o p._n 100_o to_o 102_o to_o draw_v a_o curved-line_n in_o mercators_n chart_n resemble_v the_o arch_n with_o a_o example_n for_o find_v the_o course_n and_o distance_n in_o follow_v the_o arch._n p._n 102_o to_o 104_o the_o dead_a reckon_a cast_n up_o by_o arithmetic_n p._n 106_o to_o 108_o a_o brief_a table_n of_o natural_a sin_n tangent_n and_o secant_v for_o each_o point_n of_o the_o compass_n and_o the_o quarter_n p._n 107_o the_o difference_n of_o longitude_n in_o a_o dead_a reckon_a find_v by_o the_o pen._n p._n 109_o that_o a_o table_n of_o natural_a sines_n supply_v the_o want_n of_o all_o other_o table_n p._n 110_o many_o new_a easy_a rule_n and_o proportion_n to_o raise_v a_o table_n of_o natural_a sin_n p._n 111_o to_o 113_o and_o how_o by_o have_v some_o in_o store_n to_o calculate_v any_o other_o 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into_o two_o equal_a part_n the_o point_n of_o intersection_n be_v call_v aries_n and_o libra_n the_o one_o half_a of_o it_o do_v decline_v into_o the_o north_n the_o other_o half_o into_o the_o south_n as_o much_o as_o the_o pole_n thereof_o be_v distant_a from_o the_o pole_n of_o the_o world_n namely_o 23_o 31′_n and_o likewise_o pass_v through_o the_o two_o solstitial_a point_n it_o be_v ordinary_a breadth_n or_o latitude_n be_v 12_o degree_n but_o late_a writer_n make_v it_o 14_o or_o 16_o by_o reason_n of_o the_o wander_n of_o mars_n and_o venus_n a_o line_n divide_v the_o breadth_n thereof_o into_o two_o half_n be_v call_v the_o ecliptic_a line_n because_o the_o eclipse_n of_o the_o sun_n and_o moon_n be_v always_o under_o that_o line_n it_o be_v circumference_n be_v divide_v into_o 12_o part_n call_v the_o 12_o sign_n whereof_o each_o contain_v 30d._o the_o name_n and_o character_n of_o the_o 12_o sign_n be_v aries_n ♈_o taurus_n ♉_o gemini_fw-la ♊_o cancer_n ♋_o leo_n ♌_o virgo_fw-la â™_o libra_n ♎_o scorpius_n â™_o sagittarius_fw-la â™_o capricornus_fw-la ♑_o aquarius_n â™’_o pisces_fw-la ♓_o the_o six_o former_a be_v the_o northern_a and_o the_o six_o latter_a the_o southern_a sign_n of_o the_o colour_n these_o be_v two_o great_a circle_n and_o be_v no_o other_o than_o two_o meridian_n pass_v through_o both_o the_o pole_n of_o the_o world_n cross_v one_o another_o therein_o at_o right_a angle_n and_o divide_v the_o equinoctial_a and_o the_o zodiac_n into_o four_o equal_a part_n make_v thereby_o the_o four_o season_n of_o the_o year_n the_o colour_n of_o the_o equinox_n be_v so_o call_v because_o it_o pass_v through_o the_o equinoctial_a point_n of_o aries_n and_o libra_n show_v thereby_o the_o beginning_n of_o the_o spring_n and_o autumn_n when_o the_o day_n and_o night_n be_v equal_a the_o other_o colour_n pass_v through_o the_o two_o solstitial_a or_o tropical_a point_n of_o cancer_n and_o capricorn_n show_v the_o beginning_n of_o the_o summer_n and_o winter_n at_o which_o two_o time_n the_o day_n be_v long_a and_o short_a the_o very_a beginning_n of_o cancer_n where_o the_o colour_n cross_v the_o ecliptic_a line_n be_v call_v the_o point_n of_o the_o summer_n solstice_n to_o which_o place_n when_o the_o sun_n come_v he_o can_v approach_v no_o near_o the_o zenith_n but_o return_v towards_o the_o equinoctial_a again_o the_o arch_n of_o the_o meridian_n or_o colour_n contain_v betwixt_o the_o summer_n solstice_n and_o the_o equator_fw-la be_v call_v the_o great_a declination_n of_o the_o sun_n of_o the_o four_o lesser_a circle_n the_o tropic_n be_v two_o lesser_a circle_n parallel_n to_o the_o equinoctial_a limit_v the_o sun_n great_a declination_n towards_o both_o the_o pole_n that_o towards_o the_o north_n pole_n be_v call_v the_o tropic_a of_o cancer_n because_o the_o
to_o the_o say_a word_n shall_v be_v the_o same_o day_n of_o the_o week_n the_o first_o of_o march_n happen_v upon_o example_n the_o word_n fierce_a belong_v to_o the_o month_n september_n and_o the_o first_o letter_n of_o the_o say_a word_n be_v f_z be_v the_o six_o letter_n of_o the_o alphabet_n a_o b_o c_o d_o e_o f_o wherefore_o the_o six_o day_n of_o september_n be_v the_o same_o day_n of_o the_o week_n that_o the_o first_o of_o march_n happen_v upon_o which_o in_o the_o year_n 1687_o will_v be_v on_o tuesday_n and_o then_o by_o add_v perpetual_o seven_o we_o may_v find_v on_o what_o day_n of_o the_o month_n all_o the_o tuesday_n in_o that_o month_n will_v happen_v thus_o the_o 6_o 13_o 20_o 27_o day_n of_o september_n in_o the_o year_n 1687_o will_v be_v all_o of_o they_o tuesdays_n then_o if_o it_o be_v require_v to_o know_v on_o what_o day_n of_o the_o week_n the_o 29_o of_o september_n or_o quarterday_n will_v happen_v we_o may_v conclude_v it_o to_o be_v on_o a_o thursday_n because_o the_o 27_o day_n happen_v on_o a_o tuesday_n and_o the_o like_a for_o the_o month_n of_o december_n the_o word_n proper_a whereto_o be_v foul_a and_o then_o in_o that_o year_n because_o the_o 27_o day_n be_v tuesday_n we_o may_v conclude_v the_o 25_o day_n be_v christmas_n day_n to_o happen_v on_o the_o lord_n day_n the_o day_n of_o the_o week_n be_v give_v to_o find_v what_o day_n of_o what_o month_n it_o be_v the_o month_n must_v be_v give_v and_o also_o the_o number_n of_o the_o week_n in_o the_o say_a month_n as_o whether_o it_o be_v the_o first_o second_o three_o or_o four_o week_n in_o the_o month_n otherwise_o no_o almanac_n whatsoever_o can_v resolve_v this_o proposition_n wherefore_o let_v it_o be_v require_v to_o know_v what_o day_n of_o the_o month_n friday_n in_o the_o three_o week_n of_o january_n happen_v in_o the_o year_n 1687_o the_o word_n proper_a to_o january_n be_v chance_n begin_v with_o the_o three_o letter_n of_o the_o alphabet_n wherefore_o the_o 3_o 10_o 17_o day_n of_o that_o month_n be_v tuesdays_n wherefore_o friday_n in_o the_o three_o week_n of_o that_o month_n happen_v on_o the_o 20_o day_n of_o that_o month_n because_o the_o 17_o be_v tuesday_n the_o foundation_n of_o which_o almanac_n be_v this_o that_o the_o 1_o march_v 5_o april_n 3_o may_n 7_o june_n 5_o july_n 2_o august_n 6_o september_n 4_o october_n 1_o november_n 6_o december_n 3_o january_n 7_o february_n do_v for_o ever_o happen_v in_o the_o same_o year_n which_o always_o begin_v the_o first_o of_o march_n and_o end_n on_o the_o last_o of_o february_n on_o the_o same_o day_n of_o the_o week_n which_o every_o year_n vari_v as_o do_v the_o first_o of_o march_n for_o the_o more_o ready_a find_n whereof_o this_o proposition_n may_v be_v add_v the_o day_n of_o the_o week_n on_o which_o any_o day_n of_o any_o month_n happen_v be_v know_v to_o find_v on_o what_o day_n of_o the_o week_n the_o first_o day_n of_o march_n happen_v that_o year_n example_n in_o the_o year_n 1660_o admit_v it_o be_v know_v or_o shall_v be_v remember_v that_o the_o 21_o of_o may_n happen_v on_o monday_n the_o three_o word_n of_o the_o verse_n be_v the_o word_n proper_a to_o may_v be_v chance_n the_o first_o letter_n whereof_o be_v c_o the_o three_o letter_n of_o the_o alphabet_n wherefore_o the_o 3_o 10_o 17_o 24_o day_n of_o the_o same_o month_n happen_v on_o the_o same_o day_n of_o the_o week_n that_o the_o first_o of_o march_n happen_v upon_o now_o the_o 24_o day_n of_o that_o month_n will_v be_v thursday_n because_o the_o 21_o day_n be_v monday_n wherefore_o the_o first_o of_o march_n happen_v that_o year_n on_o a_o thursday_n chap._n iu._n show_v the_o nature_n of_o sines_n a_o arch_n be_v give_v to_o take_v out_o the_o sine_fw-la or_o cousin_n thereof_o to_o the_o common_a radius_fw-la whereby_o the_o defect_n of_o a_o line_n of_o sines_n be_v supply_v if_o one_o be_v not_o at_o hand_n draw_v the_o two_o radii_fw-la ae_n c_o and_o c_o p_o make_v right_a angle_n at_o c_o the_o centre_n then_o with_o 60_o of_o the_o chord_n upon_o the_o say_a centre_n describe_v the_o quadrant_n ae_n p_o and_o thereon_o prick_v off_o the_o ark_n propound_v admit_v 50_o from_o ae_n to_o f_o the_o near_a distance_n from_o f_o to_o ae_n c_o be_v the_o sine_fw-la of_o 50_o and_o the_o near_a distance_n from_o p_o c_o be_v the_o cousin_n to_o wit_n the_o sine_fw-la of_o 40_o degree_n to_o prick_v off_o a_o arch_n or_o angle_n by_o sines_n in_o stead_n of_o chord_n or_o to_o find_v what_o arch_n belong_v to_o any_o sine_fw-la propose_v admit_v i_o will_v prick_v off_o a_o arch_n of_o 50_o degree_n by_o a_o line_n of_o sines_n first_o with_o the_o sine_fw-la of_o 90_o degree_n the_o radius_fw-la draw_v the_o quadrant_n ae_n p_o c_o then_z with_o the_o sine_fw-la of_o the_o arch_n propose_v upon_o ae_n as_o a_o centre_n describe_v the_o arch_n a_o b_o d_o a_o line_n draw_v from_o c_o the_o centre_n into_o the_o limb_n so_o as_o to_o touch_v the_o extremity_n of_o that_o ark_n at_o b_o shall_v make_v a_o angle_n of_o 50_o degree_n with_o the_o line_n ae_n c_o the_o arch_n ae_n f_o be_v a_o arch_n of_o 50_o and_o will_v be_v find_v so_o much_o if_o measure_v on_o the_o great_a chord_n but_o when_o a_o arch_a to_o be_v prick_v off_o by_o sines_n be_v great_a it_o be_v best_a to_o prick_v off_o the_o compliment_n thereof_o from_o the_o other_o radius_fw-la thus_o upon_o p_o as_o a_o centre_n with_o the_o sine_fw-la of_o 40_o describe_v the_o arch_a k_o a_o line_n from_o the_o centre_n touch_v that_o arch_n find_v the_o point_n f_o in_o the_o limb_n as_o before_o otherwise_o prick_v the_o sine_fw-la of_o 50_o from_o c_z to_z e_z and_o with_o the_o say_a extent_n upon_o ae_n as_o a_o centre_n describe_v a_o arch_n at_o a_o a_o line_n draw_v from_o e_o touch_v the_o extremity_n of_o the_o arch_n at_o a_o will_v pass_v through_o the_o point_n f_o in_o the_o limb_n as_o before_o and_o the_o same_o point_n may_v have_v be_v find_v by_o prick_v the_o sine_fw-la of_o 40_o on_o the_o other_o radius_fw-la c_o ae_n after_o the_o same_o manner_n three_o the_o extent_n c_o e_o so_o enter_v in_o the_o quadrant_a that_o one_o foot_n rest_v thereon_o may_v but_o just_a touch_n ae_n c_o find_v the_o point_n f_o as_o before_o to_o proportion_v out_o a_o sine_fw-la to_o a_o lesser_a radius_fw-la admit_v i_o will_v make_v f_o e_o a_o line_n of_o 90_o sines_n and_o to_o that_o radius_fw-la will_v prick_v down_o the_o sine_fw-la of_o 20_o first_o prick_v the_o radius_fw-la e_o f_o from_o c_z to_z r_z then_o prick_v off_o 20_o from_o the_o chord_n from_o ae_n to_o h_n and_o draw_v the_o line_n h_o c_o into_o the_o centre_n the_o near_a distance_n from_o r_n to_o the_o say_a line_n be_v the_o sine_fw-la of_o 20_o to_o the_o radius_fw-la f_o e_o and_o be_v equal_a to_o e_o l._n otherwise_o without_o draw_v liâ—es_n from_o each_o respective_a arch_n into_o the_o centre_n draw_v the_o line_n f_o c_o into_o the_o centre_n and_o from_o f_o set_v off_o a_o quadrant_a each_o way_n to_o m_n and_o n_n and_o from_o those_o point_n prick_v off_o the_o arch_n propose_v to_o s_n and_o t_o a_o ruler_n lay_v over_o those_o point_n cut_v the_o line_n f_o c_o at_o g_z and_o the_o near_a distance_n from_o g_z to_z c_z e_z be_v equal_a to_o e_o l_o as_o before_z but_o when_o a_o line_n of_o sines_n be_v at_o hand_n these_o quadrant_n and_o arch_n need_v not_o be_v prick_v off_o only_o set_v off_o the_o sine_fw-la of_o 20_o to_o the_o common_a radius_fw-la from_o c_z to_z g_z and_o the_o near_a distance_n from_o g_z to_z c_z e_z be_v the_o sine_fw-la require_v equal_a to_o e_o l_o as_o before_z a_o sine_fw-la be_v give_v in_o a_o parallel_n or_o lesser_a radius_fw-la to_o reduce_v it_o to_o the_o common_a radius_fw-la and_o thereby_o find_v to_o what_o arch_n it_o belong_v let_v e_o l_o be_v the_o sine_fw-la of_o a_o arch_n in_o the_o parallel_n f_o e_o prick_v f_o e_o from_z c_z to_z r_z and_o upon_o r_n as_o a_o centre_n with_o the_o extent_n l_o e_o describe_z the_o arch_n v_n a_o ruler_n lay_v from_o the_o centre_n touch_v the_o extremity_n of_o that_o arch_n find_v the_o point_n h_n in_o the_o limb_n and_o the_o arch_n ae_n h_n measure_v on_o the_o chord_n be_v 20_o be_v the_o arch_a proper_a to_o l_o e_o and_o this_o be_v the_o best_a way_n when_o a_o arch_n be_v not_o very_o great_a and_o be_v the_o same_o use_v in_o find_v the_o sun_n right_a ascension_n otherwise_o by_o sines_n prick_v e_o l_o from_o c_z to_o i_o a_o
proportion_n of_o the_o same_o nature_n and_o after_o the_o same_o manner_n as_o we_o find_v the_o azimuth_n at_o six_o before_o by_o the_o analemma_n be_v the_o say_a proportion_n protract_v yet_o here_o it_o be_v to_o be_v suggest_v that_o in_o the_o analemma_n there_o be_v three_o proportion_n in_o sines_n wrought_v instead_o of_o the_o one_o in_o sines_n and_o tangent_n before_o express_v 1._o as_o namely_o to_o find_v the_o sun_n altitude_n at_o six_o as_o the_o radius_fw-la be_v to_o the_o sine_fw-la of_o the_o latitude_n so_o be_v the_o sine_fw-la of_o the_o declination_n to_o the_o sine_fw-la of_o the_o sun_n height_n at_o six_o 2._o to_o find_v his_o azimuth_n in_o that_o parallel_n of_o altitude_n 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 sine_fw-la of_o the_o declination_n to_o the_o sine_fw-la of_o the_o azimuth_n in_o the_o say_a parallel_n 3._o to_o reduce_v it_o to_o the_o common_a radius_fw-la as_o the_o cousin_n of_o the_o altitude_n at_o six_o be_v to_o the_o radius_fw-la so_o be_v the_o sine_fw-la of_o the_o azimuth_n in_o that_o parallel_n to_o the_o sine_fw-la thereof_o in_o the_o common_a radius_fw-la the_o two_o latter_a proportion_n in_o sines_n may_v be_v bring_v into_o one_o as_o i_o have_v show_v in_o a_o treatise_n the_o sector_n on_o a_o quadrant_a pag._n 111_o 114._o and_o that_o will_v be_v as_o the_o cousin_n of_o the_o altitude_n at_o six_o be_v to_o the_o cousin_n of_o the_o latitude_n so_o be_v the_o sine_fw-la of_o the_o declination_n to_o the_o sine_fw-la of_o the_o azimuth_n seek_v and_o thus_o in_o effect_n the_o analemma_n perform_v that_o single_a proportion_n intermingle_v with_o tangent_n after_o a_o more_o laborious_a manner_n in_o sines_n or_o if_o you_o will_v the_o altitude_n at_o six_o be_v find_v it_o hold_v as_o the_o cotangent_fw-la of_o the_o say_a altitude_n be_v to_o the_o radius_fw-la so_o be_v the_o cotangent_fw-la of_o the_o latitude_n to_o the_o sine_fw-la of_o the_o azimuth_n seek_v and_o this_o proportion_n lie_v visible_a in_o the_o analemma_n by_o these_o direction_n derive_v from_o the_o analemma_n together_o with_o the_o proportion_n for_o each_o case_n all_o the_o 16_o case_n of_o right_a angle_a spherical_a triangle_n may_v be_v resolve_v some_o whereof_z seem_v to_o require_v the_o draw_n of_o a_o ellipsis_n as_o namely_o if_o the_o sun_n place_n and_o right_a ascension_n be_v give_v to_o find_v his_o great_a declination_n which_o notwithstanding_o according_a to_o these_o direction_n be_v easy_o shun_v chap._n viii_o show_v how_o to_o come_v by_o the_o sun_n altitude_n or_o height_n without_o instrument_n upon_o any_o flat_a or_o plain_a that_o be_v level_a or_o parallel_v to_o the_o horizon_n erect_v or_o set_v up_o a_o wire_n without_o incline_v or_o lean_v to_o either_o side_n admit_v in_o the_o point_n c_o and_o when_o you_o will_v find_v the_o sun_n altitude_n or_o height_n make_v a_o mark_n at_o that_o instant_n in_o the_o very_a end_n or_o extremity_n of_o its_o shadow_n suppose_v at_o g_o the_o shadow_n be-being_a the_o line_n c_o g._n then_o upon_o the_o same_o flat_a draw_v the_o quadrant_a of_o a_o circle_n c_o a_o f_o with_o 60_o of_o your_o chord_n and_o make_v c_o e_o equal_a to_o the_o height_n of_o the_o wire_n and_o through_o the_o point_n e_o draw_v the_o line_n e_o d_o parallel_n to_o c_o a_o and_o therein_o prick_v down_o the_o length_n of_o the_o shadow_n from_o e_z to_z d_o a_o ruler_n lay_v from_o the_o centre_n to_o d_o cut_v the_o quadrant_a at_o b_o and_o the_o arch_n b_o a_o measure_a on_o the_o chord_n show_v the_o height_n require_v in_o this_o example_n 20_o 25′_n in_o like_a manner_n if_o the_o length_n of_o the_o shadow_n be_v e_o k_o the_o height_n will_v be_v n_o a_o 38_o 16′_n otherwise_o this_o may_v be_v perform_v otherways_o by_o draw_v the_o quadrant_a c_o at_fw-fr f_o upon_o any_o plain_a board_n whatsoever_o then_o stick_v in_o a_o pin_n at_o the_o centre_n c_o and_o hold_v the_o board_n so_o towards_o the_o sun_n that_o the_o shadow_n thereof_o may_v fall_v upon_o the_o line_n c_o a_o then_o imagine_v c_o g_z to_o represent_v or_o supply_v the_o use_n of_o a_o thread_n and_o plummet_n hang_v upon_o the_o pin_n in_o the_o centre_n at_o liberty_n and_o mark_v where_o it_o cut_v the_o arch_n of_o the_o quadrant_a f_o at_fw-fr suppose_v at_o h_n measure_v the_o arch_n f_o h_n on_o the_o line_n of_o chord_n and_o it_o show_v the_o height_n requiâ—ed_v by_o the_o next_o chapter_n we_o shall_v find_v the_o sun_n azimuth_n belong_v to_o the_o altitude_n 20_o 25′_n according_a to_o the_o latitude_n and_o declination_n there_o give_v to_o be_v 31_o 19′_n from_o the_o meridian_n admit_v to_o the_o westward_n of_o the_o south_n then_o do_v the_o shadow_n happen_v so_o much_o to_o the_o eastward_o of_o the_o noâ—th_n wherefore_o if_o 31_o 19′_n be_v set_v off_o in_o the_o quadrant_a from_o the_o line_n of_o shadow_n from_o h_n to_o n_n a_o line_n draw_v into_o the_o centre_n as_o n_n c_o shall_v be_v a_o true_a meridian_n line_n or_o line_n of_o north_n and_o south_n chap._n ix_o show_v the_o resolution_n of_o such_o proposition_n wherein_o the_o sun_n altitude_n iâ—_n either_o give_v or_o seek_v the_o latitude_n of_o the_o place_n declination_n of_o the_o sun_n and_o his_o altitude_n give_v to_o find_v the_o hour_n of_o the_o day_n and_o the_o azimuth_n of_o the_o sun_n example_n declination_n 13_o south_n altitude_n 14_o 40′_n with_o 60_o of_o the_o chord_n draw_v the_o circle_n s_o z_o n_o e_o the_o centre_n whereof_o be_v at_o c_o and_o draw_v the_o diameter_n s_o c_o n_n and_o perpendicular_a thereto_o z_o c_o prick_v off_o the_o latitude_n 51_o 3′_n from_o n_n to_o p_o and_o from_o z_o to_z ae_n and_o draw_v the_o axis_n b_o p_o and_o the_o equator_fw-la ae_n c_o q_n prick_v off_o the_o declination_n from_o ae_n to_z d_o and_o from_o q_o to_o e_o being_z 13_o from_o the_o chord_n and_o draw_v the_o parallel_n of_o declination_n d_o b_o e_o then_o from_o s_n to_o a_o and_z from_z n_z to_z o_o out_o of_o the_o chord_n prick_v off_o 14_o 40′_n the_o altitude_n and_o draw_v the_o parallel_n of_o altitude_n a_o o_o so_o be_v b_o ⊙_o the_o hour_n from_o six_o towards_o noon_n in_o the_o parallel_n of_o declination_n and_o m_o ⊙_o the_o azimuth_n of_o the_o sun_n from_o the_o east_n or_o west_n southwards_o to_o measure_v the_o hour_n set_a the_o extent_n b_o d_o from_z c_z to_o f_o and_o upon_o f_o as_o a_o centre_n with_o the_o extent_n b_o ⊙_o draw_v the_o arch_n g_o a_o ruler_n lay_v from_o c_o just_a touch_v that_o arch_n find_v the_o point_n h_n the_o arch_n n_o h_n measure_v on_o the_o chord_n show_v 45_o and_o so_o much_o be_v the_o hour_n from_o six_o to_o wit_n in_o time_n three_o hour_n either_o nine_o in_o the_o forenoon_n or_o three_o in_o the_o afternoon_n to_o measure_v the_o azimuth_n in_o like_a manner_n set_v m_o a_o from_z c_z to_z r_z and_o upon_o r_n with_o the_o extent_n m_o ⊙_o draw_v the_o arch_n i_o a_o ruler_n lay_v thereto_o from_o c_o cut_n the_o limb_n at_o l_o the_o arch_n l_o s_o measure_v on_o the_o chord_n be_v 44_o 35′_n and_o so_o much_o be_v the_o sun_n to_o the_o southwards_o of_o the_o east_n or_o west_n to_o find_v the_o angle_n of_o position_n place_v the_o compliment_n of_o the_o altitude_n z_o a_o from_o p_o to_o g_o then_z place_z the_o extent_n a_o g_o from_o g_o to_o f_z a_o ruler_n lay_v over_o a_o and_z f_z cut_v the_o parallel_n of_o the_o sun_n declination_n at_o it_o and_z d_o t_z be_v the_o verse_v sine_fw-la of_o the_o angle_n of_o position_n to_o measure_v it_o through_o the_o point_n t_o draw_z k_o t_o y_fw-fr parallel_n to_o the_o axis_n b_o p_o and_o draw_v d_o c_o then_z the_o extent_n c_o y_fw-fr measure_a on_o the_o sin_n show_v the_o compliment_n of_o it_o to_o wit_n 62_o 57′_n therefore_o the_o say_a angle_n be_v 27_o 3′_n otherwise_o with_o the_o extent_n b_o d_o set_v one_o foot_n in_o the_o centre_n at_o c_o with_o the_o other_o across_o the_o former_a parallel_n line_n k_o t_o at_o k_n a_o ruler_n lay_v from_o c_o to_o k_n cut_v the_o limb_n at_o x_o and_o the_o arch_n ae_n x_o be_v 27_o 3′_n the_o measure_n thereof_o as_o before_o the_o compliment_n whereof_o be_v the_o arch_n v_o e_o and_o may_v be_v measure_v from_o the_o south_n pole_n v_n if_o the_o equinoctial_a ae_n q_o be_v not_o draw_v three_o side_n give_v to_o find_v a_o angle_n another_o example_n the_o declination_n be_v north_n latitude_n 51_o 32′_n declination_n 13_o north_n altitude_n 37_o 18′_n in_o the_o follow_a scheme_n upon_o the_o centre_n c_o with_o
60_o of_o the_o chord_n describe_v a_o circle_n prick_v one_o of_o the_o side_n namely_o the_o colatitude_n 38_o 28′_n from_o z_o to_o p_o in_o the_o limb_n and_o from_o p_o prick_v the_o sun_n polar_n distance_n 77_o to_o d_o and_z e_z and_o draw_v the_o parallel_n of_o declination_n d_o e_o from_z z_o prick_v off_o the_o compliment_n of_o the_o altitude_n to_o a_o and_o o_o to_o wit_n 52_o 42′_n and_o draw_v the_o parallel_n of_o altitude_n a_o o_o so_o be_v the_o hour_n and_o azimuth_n find_v in_o these_o two_o parallel_n without_o draw_v any_o more_o line_n the_o draw_v of_o the_o axis_n c_o p_o be_v only_o to_o divide_v the_o parallel_n d_o e_z into_o half_n at_o b_o likewise_o the_o draw_v of_o c_o z_o divide_v the_o parallel_n a_o o_o into_o half_n at_o m_n which_o may_v be_v find_v without_o draw_v line_n by_o lay_v a_o ruler_n from_o c_o to_o z_o and_o p_o or_o if_o you_o will_v draw_v the_o horizontal_n line_n s_o n_n pass_v through_o the_o centre_n and_o above_o it_o the_o altitude_n may_v be_v set_v from_o s_n to_o a_o and_z from_z n_z to_z o_o in_o like_a manner_n if_o the_o equinoctial_a be_v draw_v the_o declination_n be_v north_n be_v to_o be_v set_v above_o it_o the_o manner_n of_o measure_v be_v the_o same_o as_o before_o to_o find_v the_o hour_n c_o f_o be_v make_v equal_a to_o b_o d_o then_o upon_o f_o with_o the_o extent_n b_o ⊙_o draw_v the_o arch_n g_o a_o ruler_n from_o the_o cenâ—er_n touch_v g_z find_v h_n and_o the_o arch_n h_o n_n be_v 45_o be_v the_o hour_n from_o noon_n for_o the_o azimuth_n c_o r_n be_v make_v equal_a to_o m_o a_o then_o upon_o r_n with_o m_o ⊙_o draw_z i_o a_o ruler_n from_o the_o centre_n touch_v it_o find_n l_o and_o the_o arch_n l_o s_o measure_v on_o the_o chord_n be_v 30_o and_o so_o much_o be_v the_o azimuth_n to_o the_o southwards_o of_o the_o east_n or_o west_n to_o find_v the_o angle_n of_o position_n here_o also_o z_o a_n be_v place_v from_o p_o to_o g_o and_o then_o g_o f_o be_v make_v equal_a to_o g_o a_o a_o ruler_n from_o a_o to_o f_o cut_v the_o parallel_n of_o declination_n at_o it_o and_z d_o t_z be_v the_o verse_v sine_fw-la of_o the_o angle_n of_o position_n which_o be_v measure_v as_o in_o the_o former_a example_n for_o find_v it_o will_v be_v find_v to_o be_v 33_o 34′_n another_o example_n for_o south_n declination_n 13_o retain_v the_o former_a latitude_n a_o new_a scheme_n need_v not_o be_v draw_v the_o declination_n be_v as_o much_o south_n let_v the_o altitude_n be_v 20_o 25′_n prick_v the_o altitude_n from_o s_n to_o q_o and_o from_o n_n to_o x_o and_o draw_v the_o parallel_n q_o x_o where_o it_o crosseâ—h_v the_o parallel_n of_o declination_n set_v v._o join_v c_o x_o and_o draw_v v_o r_o parallel_n to_o w_n c._n to_o find_v the_o hour_n upon_o f_o as_o a_o centre_n with_o the_o extent_n b_o v_n describe_v the_o arch_a k_o a_o ruler_n from_o the_o centre_n touch_v it_o find_v t_n the_o arch_n n_o t_n be_v 60_o be_v the_o hour_n from_o six_o to_o wit_n either_o ten_o in_o the_o morning_n or_o two_o in_o the_o afternoon_n to_o find_v the_o azimuth_n the_o extent_n c_o r_o measure_v on_o the_o sin_n be_v 58_o 41′_n and_o so_o much_o be_v the_o sun_n azimuth_n to_o the_o southward_n of_o the_o east_n or_o west_n in_o this_o our_o northern_a hemisphere_n a_o three_o example_n for_o find_v the_o azimuth_n in_o the_o former_a example_n it_o may_v be_v observe_v that_o the_o azimuth_n be_v always_o find_v in_o the_o parallel_n of_o altitude_n which_o towards_o the_o zenith_n grow_v very_o small_a and_o consequent_o this_o way_n of_o find_v the_o azimuth_n in_o latitude_n between_o or_o near_o the_o tropic_n and_o sometime_o in_o our_o own_o latitude_n when_o the_o sun_n have_v much_o altitude_n will_v be_v very_o inconvenient_a for_o remedy_v whereof_o let_v it_o be_v note_v that_o the_o azimuth_n may_v be_v always_o find_v in_o the_o parallel_n of_o latitude_n example_n for_o the_o latitude_n of_o the_o barbados_n 13_o ⊙_o declination_n 20_o north_n altitude_n 52_o 27′_n in_o the_o first_o scheme_n have_v draw_v the_o horizon_n s_o n_n its_o axis_n c_o z_o the_o parallel_n of_o altitude_n b_o a_o 52_o 27′_n set_v off_o the_o latitude_n from_o n_n to_o p_o 13_o and_o draw_v the_o axis_n c_o p_o prick_v off_o the_o compliment_n of_o the_o declination_n 70_o from_o p_o to_o d_o and_z e_z and_o draw_v the_o parallel_n of_o declination_n d_o e_o then_o be_v f_o ⊙_o the_fw-fr sine_fw-la of_o the_o azimuth_n from_o east_n or_o west_n northwards_o f_o a_o being_n radius_fw-la which_o be_v place_v from_o c_o to_o g_z and_o the_o arch_n i_o describe_v with_o f_o ⊙_o a_o ruler_n from_o the_o centre_n touch_v it_o find_v s_n l_o 16_o the_o measure_n of_o the_o azimuth_n in_o the_o limb_n but_o thus_o to_o transfer_v from_o a_o less_o to_o a_o great_a will_v breed_v much_o incertainty_n and_o the_o parallel_n of_o altitude_n near_o the_o zenith_n decrease_v very_o much_o for_o remedy_v this_o inconvenience_n change_v the_o name_n latitude_n and_o altitude_n and_o fit_v the_o analemma_n thereto_o for_o the_o bare_a change_n of_o the_o name_n of_o the_o two_o contain_v side_n of_o a_o triangle_n do_v not_o alter_v the_o quantity_n of_o the_o angle_n comprehend_v then_o will_v the_o new_a latitude_n be_v 52_o 27′_n and_o the_o new_a altitude_n 13_o 00._o i_o say_v the_o azimuth_n be_v the_o same_o as_o it_o be_v in_o the_o latitude_n of_o 13_o when_o the_o altitude_n be_v 52_o 27′_n also_o the_o hour_n from_o noon_n in_o the_o new_a latitude_n be_v equal_a to_o the_o angle_n of_o position_n in_o the_o old_a latitude_n and_o the_o angle_n of_o position_n in_o the_o new_a latitude_n be_v equal_a to_o the_o hour_n from_o noon_n in_o the_o old_a latitude_n the_o old_a or_o first_o scheme_n may_v very_o well_o serve_v if_o you_o set_v off_o 70_o the_o compliment_n of_o the_o declination_n each_o way_n from_o a_o the_o end_n of_o the_o parallel_n of_o altitude_n and_o then_o through_o the_o point_n p_o draw_v the_o parallel_n of_o altitude_n parallel_v to_o the_o horizon_n but_o here_o we_o have_v draw_v a_o second_o scheme_n where_o n_o p_o be_v 52_o 27′_n p_o d_o and_o p_o e_o 70_o through_o which_o point_n be_v draw_v the_o parallel_n d_o e_o also_o s_n b_o and_z n_z a_o be_v 13_o through_o which_o be_v draw_v the_o parallel_n b_o a_o then_o be_v f_o ⊙_o the_fw-fr sine_fw-la of_o the_o sun_n azimuth_n to_o the_o northwards_o of_o the_o east_n to_o the_o radius_fw-la f_o a_o which_o be_v place_v from_o c_o to_o g_z upon_o which_o with_o f_o ⊙_o be_v the_o arch_n i_o describe_v and_o the_o azimuth_n s_o l_o sound_v to_o be_v 16_o as_o before_o nota_fw-la also_o in_o the_o first_o scheme_n or_o in_o both_o that_o the_o radius_fw-la c_o g_o might_n have_v be_v double_v and_o prick_v upon_o n_n s_o produce_v then_o also_o must_v the_o extent_n f_o ⊙_o haué_fw-fr be_v double_v and_o the_o arch_n i_o therewith_o describe_v also_o the_o hour_n of_o the_o day_n may_v after_o the_o same_o manner_n be_v find_v in_o the_o parallel_n of_o latitude_n which_o may_v be_v convenient_a for_o star_n that_o have_v much_o declination_n in_o that_o case_n the_o declination_n and_o latitude_n must_v change_v name_n and_o thus_o when_o three_o side_n be_v give_v to_o find_v a_o angle_n we_o may_v find_v it_o in_o the_o analemma_n by_o call_v the_o compliment_n of_o those_o side_n the_o declination_n the_o latitude_n and_o the_o altitude_n or_o depression_n at_o pleasure_n also_o when_o three_o angle_n be_v give_v to_o find_v a_o side_n those_o angles_n must_v be_v change_v into_o side_n by_o take_v the_o compliment_n of_o the_o great_a angle_n to_o a_o semicircle_n and_o write_v it_o and_o the_o other_o angle_n down_o to_o their_o opposite_a side_n in_o another_o triangle_n as_o in_o the_o scheme_n follow_v wherein_o as_o the_o angle_n be_v change_v into_o side_n so_o be_v the_o side_n change_v into_o angle_n and_o then_o the_o case_n will_v be_v the_o same_o as_o before_o two_o side_n with_o the_o angle_n comprehend_v to_o find_v the_o three_o side_n and_o both_o the_o other_o angle_n by_o this_o case_n may_v be_v find_v the_o sun_n altitude_n on_o all_o hour_n and_o the_o distance_n of_o place_n in_o the_o arch_n of_o a_o great_a circle_n first_o the_o sun_n altitude_n on_o all_o hour_n thereby_o be_v mean_v that_o if_o the_o hour_n of_o the_o day_n the_o declination_n and_o latitude_n be_v give_v the_o sun_n altitude_n proper_a to_o that_o hour_n or_o his_o depression_n may_v be_v find_v upon_o the_o centre_n c_o with_o 60_o of_o the_o
the_fw-fr sine_fw-la of_o 30_o the_o azimuth_n thereto_o again_o make_v n_n g_o the_o lesser_a altitude_n 14_o 15′_n and_o draw_v the_o parallel_n thereof_o h_n g_z which_o be_v radius_fw-la prick_v down_o the_o sine_fw-la of_o 20_o the_o give_v azimuth_n thereto_o from_o h_n to_o i_o because_o it_o be_v to_o the_o northwards_o of_o the_o east_n through_o the_o point_n b_o i_o draw_v the_o line_n d_o e_o and_o it_o shall_v be_v the_o parallel_n of_o declination_n divide_v the_o ark_n d_o e_z into_o half_n at_o p_o and_o the_o arch_n p_o n_n be_v the_o latitude_n 51_o 32′_n place_n z_o p_o from_o s_n to_o ae_n and_o the_o arch_n ae_n d_o be_v the_o declination_n 23_o 31′_n moreover_o the_o amplitude_n hour_n etc._n etc._n be_v give_v by_o the_o scheme_n though_o not_o require_v to_o calculate_v the_o latitude_n from_o the_o say_a scheme_n from_o the_o point_n i_o draw_v i_o l_o parallel_n to_o h_n f_o and_o produce_v b_o f_o to_o l_o then_o be_v i_o l_o equal_v to_o h_n f_o the_o difference_n of_o the_o sin_n of_o both_o the_o give_v altitude_n and_o b_o l_o be_v equal_a to_o the_o sum_n of_o b_o f_o and_o h_n i_o and_o the_o angle_n b_o i_o l_o be_v equal_a to_o the_o latitude_n of_o the_o place_n the_o first_o work_n be_v to_o find_v b_o f_o and_o h_n i_o in_o the_o parallel_n of_o altitude_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 the_o give_v altitude_n be_v the_o radius_fw-la of_o the_o parallel_n so_o be_v the_o sine_fw-la of_o the_o azimuth_n from_o east_n or_o west_n belong_v to_o the_o say_a altitude_n to_o the_o sine_fw-la of_o the_o say_a azimuth_n in_o its_o parallel_n of_o altitude_n whereby_o at_o two_o operation_n may_v be_v find_v b_o f_o and_o h_n 1._o when_o the_o azimuth_n be_v on_o different_a side_n the_o vertical_a take_v the_o sum_n when_o on_o the_o same_o side_n the_o difference_n between_o the_o four_o term_n find_v by_o each_o operation_n than_o it_o hold_v as_o the_o difference_n of_o the_o sin_n of_o both_o the_o give_v altitude_n be_v to_o the_o aforesaid_a sum_n or_o difference_n so_o be_v the_o radius_fw-la to_o the_o tangent_fw-la of_o the_o latitude_n the_o two_o first_o term_n of_o this_o latter_a proportion_n in_o this_o scheme_n be_v as_o ay_o l_o to_o l_o b._n or_o it_o may_v be_v express_v in_o two_o proportion_n thus_o as_o the_o difference_n of_o the_o sin_n of_o both_o the_o give_v altitude_n be_v to_o the_o cousin_n of_o the_o great_a altitude_n so_o be_v the_o sine_fw-la of_o the_o azimuth_n from_o the_o east_n thereto_o belong_v to_o a_o four_o number_n again_o as_o the_o difference_n of_o the_o sin_n of_o both_o the_o give_v altitude_n be_v to_o the_o cousin_n of_o the_o lesser_a altitude_n so_o be_v the_o sine_fw-la of_o the_o azimuth_n thereto_o belong_v to_o a_o seven_o number_n the_o sum_n of_o the_o four_o and_o seven_o number_n when_o the_o azimuth_n fall_v on_o different_a side_n the_o vertical_a but_o their_o difference_n when_o they_o happen_v on_o the_o same_o side_n be_v equal_a to_o the_o tangent_fw-la of_o the_o latitude_n or_o it_o may_v be_v thus_o deliver_v multiply_v the_o cousin_n of_o each_o altitude_n by_o the_o sine_fw-la of_o the_o azimuth_n thereto_o belong_v and_o there_o will_v arise_v two_o product_n the_o sum_n of_o these_o product_n when_o the_o azimuth_n happen_v on_o different_a side_n but_o the_o difference_n of_o they_o when_o they_o happen_v on_o the_o same_o side_n the_o vertical_a divide_v by_o the_o difference_n of_o the_o sin_n of_o both_o the_o give_v altitude_n and_o the_o quotient_n will_v be_v the_o tangent_fw-la of_o the_o latitude_n seek_v this_o suppose_v the_o operation_n to_o be_v perform_v by_o the_o natural_a table_n of_o sines_n and_o tangent_n now_o have_v the_o latitude_n either_o of_o the_o altitude_n with_o the_o azimuth_n theteto_o belong_v we_o have_v two_o side_n and_o the_o angle_n comprehend_v give_v to_o find_v the_o three_o side_n whereby_o be_v find_v the_o declination_n require_v how_o from_o three_o shadow_n happen_v the_o same_o day_n on_o a_o horizontal_n plain_n to_o calculate_v the_o latitude_n etc._n etc._n as_o also_o to_o perform_v it_o by_o scale_n and_o compass_n with_o another_o way_n for_o find_v the_o azimuth_n i_o have_v show_v in_o a_o treatise_n of_o geometrical_a dyall_n whereto_o the_o reader_n be_v refer_v finis_fw-la errata_fw-la page_n 47._o line_n 28._o for_o 38_o 28′_n read_v 51_o 32′_n page_n 49._o line_n 23._o for_o latitude_n read_v altitude_n the_o mariners_z plain_a scale_n new_a plain_v the_o three_o part_n or_o book_n show_v the_o uses_n of_o a_o line_n of_o chord_n only_o in_o resolve_v or_o project_v all_o the_o 28_o case_n of_o spherical_a triangle_n on_o the_o stereographick_n or_o circular_a projection_n also_o how_o from_o three_o shadow_n of_o a_o wire_n on_o a_o horizontal_n plain_n take_v the_o same_o day_n to_o find_v the_o latitude_n of_o the_o place_n the_o sun_n declination_n etc._n etc._n whence_o follow_v a_o general_a method_n of_o dyall_v from_o three_o shadow_n or_o by_o help_n of_o two_o shadow_n etc._n etc._n with_o other_o proposition_n of_o excellent_a use_n to_o seaman_n and_o practitioner_n in_o the_o mathematics_n be_v contrive_v to_o be_v have_v either_o alone_a or_o with_o the_o other_o part_n write_a by_o john_n collins_n of_o london_n penman_n accountant_a philomathet_fw-la london_n print_v by_o tho._n johnson_n for_o francis_n cossinet_n and_o be_v to_o be_v sell_v at_o the_o anchor_n and_o mariner_n in_o tower-street_n and_o by_o hen._n sutton_n mathematical_a instrument-maker_n in_o thread-needle-street_n behind_o the_o royal_a exchange_n 1659._o courteous_a reader_n this_o part_n of_o the_o treatise_n be_v also_o contrive_v to_o go_v alone_o wherein_o we_o use_v no_o other_o scale_n but_o only_o a_o line_n of_o chord_n or_o rather_o the_o equal_a division_n of_o a_o quadrant_n of_o a_o circle_n and_o according_o have_v divide_v the_o same_o in_o the_o second_o scheme_n of_o this_o book_n have_v shun_v the_o use_n of_o tangent_n semitangent_n and_o secant_v which_o other_o use_v upon_o scale_n or_o sector_n for_o this_o purpose_n the_o demonstration_n of_o this_o projection_n the_o reader_n will_v find_v sufficient_o handle_v in_o the_o six_o book_n of_o aquilonius_fw-la his_o optic_n in_o guidoubaldus_n his_o theoric_n of_o planisphaere_n see_v also_o clavius_n his_o astrolabe_n and_o the_o two_o former_a author_n have_v also_o handle_v the_o orthographick_a projection_n the_o spherical_a definition_n necessary_a to_o be_v understand_v by_o the_o reader_n be_v handle_v in_o the_o second_o part._n not_o will_v further_o to_o detain_v thou_o i_o rest_v a_o well-willer_n to_o the_o advancement_n of_o knowledge_n john_n collins_n the_o content_n certain_a definition_n and_o notion_n about_o spherical_a triangle_n page_z 1_o the_o 16_o right_a angle_a case_n project_v from_o page_n 2_o to_o 12_o the_o scheme_n for_o chord_n use_v in_o this_o book_n 4_o how_o quadrantal_a triangle_n arise_v from_o the_o right_a angleâ—_n 5_o how_o otherwise_o in_o less_o room_n than_o a_o semicircle_n to_o project_v the_o right_a angle_a case_n 12_o the_o 12_o oblique_a case_n project_v from_o page_n 13_o to_o 22_o how_o it_o arise_v from_o projection_n that_o the_o angle_n of_o one_o oblique_a triangle_n be_v equal_a to_o the_o side_n of_o another_o 16_o how_o it_o arise_v from_o projection_n that_o in_o some_o case_n the_o sun_n and_o star_n have_v two_o altitude_n upon_o one_o and_o the_o same_o azimuth_n 24_o how_o from_o three_o shadow_n on_o a_o horizontal_n plain_n to_o find_v the_o latitude_n of_o the_o place_n the_o sun_n declination_n amplitude_n and_o the_o azimuth_n of_o those_o shadow_n from_o page_n 25_o to_o 28_o a_o general_a method_n thence_o derive_v for_o inscribe_v the_o style_n and_o substile_n into_o such_o dial_n as_o have_v centre_n 29_o also_o a_o scheme_n for_o place_v the_o meridian_n line_n in_o such_o dial_n 31_o the_o sun_n declination_n two_o altitude_n with_o the_o difference_n of_o azimuth_n be_v give_v how_o by_o projection_n to_o find_v the_o latitude_n of_o the_o place_n and_o the_o azimuth_n of_o these_o shadow_n etc._n etc._n 33_o 34_o a_o method_n of_o dyall_v thence_o issue_v 34_o how_o from_o the_o former_a data_fw-la only_o alter_v the_o difference_n of_o azimuth_n into_o the_o difference_n of_o time_n to_o find_v the_o latitude_n etc._n etc._n 35_o after_o the_o same_o manner_n the_o declination_n and_o right_a ascension_n of_o two_o fix_a star_n be_v give_v as_o also_o the_o altitude_n of_o they_o both_o be_v at_o any_o period_n of_o time_n observe_v to_o find_v the_o latitude_n etc._n etc._n 36_o spherical_a triangle_n solve_v by_o projection_n a_o spherical_a triangle_n be_v suppose_v to_o be_v describe_v on_o the_o surface_n or_o convexity_n of_o the_o
instrument_n to_o be_v cut_v in_o brass_n with_o paper_n print_v fit_v up_o for_o sale_n which_o will_v be_v of_o excellent_a use_n to_o seaman_n surveyor_n and_o all_o that_o be_v mathematical_o studious_a i_o remain_v thy_o friend_n desirous_a of_o the_o advancement_n of_o knowledge_n john_n collins_n the_o content_n a_o double_a scale_n of_o chord_n use_v in_o this_o book_n and_o describe_v page_z 1_o 2._o spherical_a definition_n from_o page_n 2_o to_o 9_o all_o the_o point_n ark_n and_o circle_n define_v represent_v to_o the_o view_n in_o a_o scheme_n of_o the_o analemma_n p._n 10_o to_o 14_o a_o general_a almanac_n in_o two_o verse_n p._n 15_o the_o manner_n of_o measure_v and_o proportion_v out_o sin_n by_o a_o line_n of_o chord_n p._n 18_o 19_o a_o general_a rule_n in_o two_o verse_n for_o find_v the_o sun_n place_n 21_o to_o find_v the_o sun_n declination_n and_o right_a ascension_n 23_o 24_o to_o find_v the_o sun_n amplitude_n height_n at_o six_o vertical_a height_n time_n of_o rise_v etc._n etc._n 25_o 26_o the_o sixteen_o right_a angle_a case_n resolve_v by_o proportion_n of_o four_o several_a kind_n p._n 27_o to_o 32_o to_o find_v the_o sun_n height_n without_o instrument_n 32_o to_o find_v the_o hour_n azimuth_n and_o angle_n of_o position_n p._n 33_o to_o 39_o to_o find_v the_o sun_n altitude_n on_o all_o hour_n 40_o 41_o to_o find_v the_o distance_n of_o place_n etc._n etc._n 42_o 43_o all_o the_o oblique_a case_n solve_v p._n 46._o p._n 53_o to_o find_v the_o altitude_n on_o all_o azimuth_n p._n 46_o to_o 52_o all_o the_o sixteen_o case_n of_o right_a angle_a spherical_a triangle_n project_v and_o otherwise_o resolve_v p._n 57_o to_o 63_o the_o longitude_n and_o latitude_n of_o a_o star_n give_v to_o find_v its_o declination_n and_o right_a ascension_n p._n 64_o two_o azimuth_n and_o two_o altitude_n give_v to_o find_v the_o latitude_n and_o declination_n by_o projection_n p._n 65_o with_o proportion_n to_o find_v the_o same_o by_o calculation_n 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