90_o begin_v again_o with_o 10._o 20._o 30_o etc._n etc._n be_v for_o that_o the_o use_n of_o this_o meridian_n in_o reference_n to_o its_o division_n in_o degree_n require_v no_o more_o than_o that_o number_n 12._o the_o horizon_n be_v a_o great_a circle_n without_o the_o globe_n which_o divide_v the_o upper_a part_n of_o heaven_n from_o the_o low_a so_o that_o the_o one_o half_a be_v always_o above_o that_o circle_n and_o the_o other_o under_o it_o 13._o the_o pole_n of_o this_o circle_n be_v two_o the_o one_o direct_o over_o our_o head_n and_o be_v call_v the_o zenith_n the_o other_o be_v under_o foot_n and_o be_v call_v the_o nadir_n 14._o the_o horizon_n be_v either_o rational_a or_o sensible_a 15._o the_o rational_a horizon_n be_v that_o which_o divide_v the_o heaven_n and_o the_o earth_n into_o two_o equal_a part_n which_o though_o it_o can_v be_v perceive_v and_o distinguish_v by_o the_o eye_n yet_o may_v be_v conceive_v iâ—_n our_o mind_n in_o which_o respect_n all_o the_o star_n may_v be_v conceive_v to_o rise_v and_o set_v as_o in_o our_o view_n 16._o the_o visible_a horizon_n be_v that_o circle_n which_o the_o eye_n do_v make_v at_o its_o far_a extent_n of_o sight_n when_o the_o body_n in_o any_o particular_a place_n do_v turn_v itself_o round_o of_o these_o two_o circle_n there_o need_v no_o more_o to_o be_v say_v at_o present_a only_o we_o may_v observe_v that_o it_o be_v ingenious_o devise_v by_o those_o who_o first_o think_v upon_o it_o to_o set_v one_o meridian_n and_o one_o horizon_n without_o the_o globe_n to_o avoid_v the_o confusion_n if_o not_o the_o impossibility_n of_o draw_v a_o several_a meridian_n and_o a_o several_a horizon_n for_o every_o place_n which_o must_v have_v be_v do_v if_o this_o or_o the_o like_a device_n have_v not_o be_v think_v upon_o 17._o beside_o these_o two_o great_a circle_n without_o the_o globe_n there_o be_v 4_o other_o great_a circle_n draw_v upon_o the_o globe_n itself_o beside_o the_o meridian_n 1._o the_o aequator_fw-la or_o equinoctial_a circle_n 2._o the_o zodiac_n 3._o the_o aequinoctial_a colour_n 4._o solstitial_a colour_n and_o these_o four_o circle_n be_v immutable_a that_o be_v in_o whatsoever_o part_n of_o the_o world_n you_o be_v these_o circle_n have_v no_o variation_n as_o the_o other_o two_o have_v 18._o the_o aequator_fw-la be_v a_o great_a circle_n draw_v upon_o the_o globe_n in_o the_o middle_n between_o the_o two_o pole_n and_o plain_o divide_v the_o globe_n into_o two_o equal_a part_n 19_o the_o aequator_fw-la be_v the_o measure_n of_o the_o motion_n of_o the_o primum_fw-la mobile_n for_o 15_o degree_n of_o this_o circle_n do_v always_o arise_v in_o a_o hour_n time_n the_o which_o do_v clear_o show_v that_o the_o whole_a heaven_n be_v turn_v round_o by_o equal_a interval_n in_o the_o space_n of_o one_o day_n or_o 24_o hour_n 20._o in_o this_o circle_n the_o declination_n of_o the_o star_n be_v compute_v from_o the_o mid-heaven_n towards_o the_o north_n or_o south_n 21._o this_o circle_n give_v denomination_n to_o the_o aequinox_n for_o the_o sun_n do_v twice_o in_o a_o year_n and_o no_o more_o cross_v this_o circle_n to_o wit_n when_o he_o enter_v the_o first_o point_n of_o aries_n and_o libra_n and_o then_o he_o make_v the_o day_n and_o the_o night_n equal_a his_o entrance_n into_o aries_n be_v in_o march_n and_o be_v call_v the_o vernal_a equinox_n and_o his_o entrance_n into_o libra_n be_v in_o september_n and_o be_v call_v the_o autumnal_a equinox_n 22._o and_o from_o one_o certain_a point_n in_o this_o circle_n the_o longitude_n of_o place_n upon_o the_o earth_n be_v reckon_v and_o the_o latitude_n of_o place_n be_v reckon_v from_o this_o circle_n towards_o the_o north_n or_o the_o south_n poles_n 23._o the_o zodiac_n be_v a_o great_a circle_n draw_v upon_o the_o globe_n cut_v the_o aequinoctial_a point_n at_o oblique_a angle_n for_o although_o it_o divide_v the_o whole_a world_n into_o two_o equal_a part_n in_o reference_n to_o its_o own_o pole_n yet_o in_o reference_n to_o the_o pole_n of_o the_o world_n it_o have_v a_o oblique_a motion_n 24._o the_o pole_n of_o this_o circle_n be_v as_o far_o distant_a from_o the_o pole_n of_o the_o world_n as_o the_o great_a obliquity_n thereof_o be_v from_o the_o equinoctial_a that_o be_v 23_o degree_n and_o 31_o minute_n or_o thereabouts_o 25._o this_o circle_n do_v differ_v from_o all_o other_o circle_n upon_o the_o globe_n in_o this_o other_o circle_n to_o speak_v proper_o have_v longitude_n assign_v they_o but_o no_o latitude_n but_o this_o have_v both_o whereas_o other_o circle_n be_v in_o reference_n to_o their_o longitude_n or_o rotundity_n only_o divide_v into_o 360_o degree_n this_o circle_n in_o respect_n of_o its_o latitude_n be_v suppose_v to_o be_v divide_v into_o 16_o degree_n in_o latitude_n 26._o the_o zodiac_n then_o in_o respect_n of_o longitude_n be_v common_o divide_v into_o 360_o degree_n as_o other_o circle_n be_v but_o more_o peculiarly_a in_o respect_n of_o its_o self_n it_o be_v divide_v into_o 12_o part_n call_v sign_n and_o each_o sign_n into_o 30_o degree_n and_o 12_o time_n 30_o do_v make_v 360._o 27._o the_o 12_o sign_n into_o which_o the_o zodiac_n be_v divide_v have_v these_o name_n and_o character_n aries_n ♈_o taurus_n ♉_o gemini_fw-la ♊_o cancer_n ♋_o leo_n ♌_o virgo_fw-la â™_o libra_n ♎_o scorpio_n â™_o sagittarius_fw-la â™_o capricornus_fw-la ♑_o aquarius_n â™’_o and_o pisces_fw-la ♓_o 28._o these_o two_o circle_n of_o the_o equator_fw-la and_o zodiack_n be_v cross_v by_o two_o other_o great_a circle_n which_o be_v call_v colour_n they_o be_v draw_v through_o the_o pole_n of_o the_o world_n and_o cut_v one_o another_o as_o well_o as_o the_o equator_fw-la at_o right_a angle_n one_o of_o they_o pass_v through_o the_o intersection_n of_o the_o equinoctial_a point_n and_o be_v call_v the_o equinoctial_a colour_n the_o other_o pass_v through_o the_o point_n of_o the_o great_a distance_n of_o the_o zodiac_n from_o the_o equator_fw-la and_o be_v call_v the_o solstitial_a colour_n 29._o the_o other_o great_a circle_n describe_v upon_o the_o globe_n be_v the_o meridian_n where_o we_o must_v not_o think_v much_o to_o hear_v of_o the_o meridian_n again_o that_o of_o brass_n without_o the_o globe_n be_v to_o serve_v all_o turn_n and_o the_o globe_n be_v frame_v to_o apply_v itself_o thereto_o the_o meridian_n upon_o the_o globe_n will_v easy_o be_v perceive_v to_o be_v of_o a_o new_a and_o another_o use_n 30._o the_o meridian_n upon_o the_o globe_n be_v either_o the_o great_a or_o the_o less_o not_o that_o the_o great_a be_v any_o great_a than_o the_o less_o for_o they_o have_v all_o one_o and_o the_o same_o centre_n and_o equal_o pass_v through_o the_o pole_n of_o the_o earth_n but_o those_o which_o be_v call_v less_o be_v of_o less_o use_n than_o that_o which_o be_v call_v the_o great_a 31._o the_o great_a be_v otherwise_o call_v the_o fix_v and_o first_o meridian_n to_o which_o the_o less_o be_v second_o and_o respective_o movable_a the_o great_a meridian_n be_v as_o it_o be_v the_o landmark_n of_o the_o whole_a sphere_n from_o whence_o the_o longitude_n of_o the_o earth_n or_o any_o part_n thereof_o be_v account_v and_o it_o be_v the_o only_a circle_n which_o pass_v through_o the_o pole_n be_v graduate_v or_o divide_v into_o degree_n not_o the_o whole_a circle_n but_o the_o half_a because_o the_o longitude_n be_v to_o be_v reckon_v round_o about_o the_o earth_n 32._o the_o lesser_a meridian_n be_v those_o black_a line_n which_o you_o see_v to_o pass_v through_o the_o pole_n and_o succeed_a the_o great_a at_o 10_o and_o 10_o degree_n as_o in_o most_o globe_n or_o at_o 15_o and_o 15_o degree_n difference_n as_o in_o some_o every_o place_n never_o so_o little_o more_o east_n or_o west_n than_o another_o have_v proper_o a_o several_a meridian_n yet_o because_o of_o the_o huge_a distance_n of_o the_o earth_n from_o the_o heaven_n there_o be_v no_o sensible_a difference_n between_o the_o meridian_n of_o place_n that_o be_v less_o than_o one_o degree_n of_o longitude_n asunder_o and_o therefore_o the_o geographer_n as_o well_o as_o the_o astronomer_n allow_v a_o new_a meridian_n to_o every_o degree_n of_o the_o equator_fw-la which_o will_v be_v 180_o in_o all_o but_o except_o the_o globe_n be_v make_v of_o a_o extreme_a and_o a_o unusual_a diameter_n so_o many_o will_v stand_v too_o thick_a for_o the_o description_n therefore_o most_o common_o they_o put_v down_o but_o 18_o that_o be_v at_o 10_o degree_n distance_n from_o one_o another_o the_o special_a use_n of_o the_o lesser_a meridian_n being_n to_o make_v a_o quick_a dispatch_n in_o the_o account_n of_o the_o longitude_n other_o set_v down_o but_o 12_o at_o 15_o degree_n difference_n aim_v at_o this_o that_o the_o meridian_n might_n be_v distant_a
from_o one_o another_o a_o full_a part_n of_o time_n or_o a_o hour_n for_o see_v that_o the_o sun_n be_v carry_v 15_o degree_n of_o the_o equinoctial_a every_o hour_n the_o meridian_n set_v at_o that_o distance_n must_v make_v a_o hour_n difference_n in_o the_o rise_n or_o set_v of_o the_o sun_n in_o those_o place_n which_o differ_v 15_o degree_n in_o longitude_n and_o to_o this_o purpose_n also_o upon_o the_o north_n end_n of_o the_o globe_n without_o the_o brass_n meridian_n there_o be_v a_o small_a circle_n of_o brass_n set_v and_o divide_v into_o two_o equal_a part_n and_o each_o of_o they_o into_o twelve_o that_o be_v twenty_o four_o all_o to_o show_v the_o hour_n of_o the_o day_n and_o night_n in_o any_o place_n where_o the_o day_n and_o night_n exceed_v not_o 24_o hour_n for_o which_o purpose_n it_o have_v a_o little_a brass_n pin_n turn_v about_o upon_o the_o pole_n and_o point_v to_o the_o several_a hour_n which_o be_v therefore_o the_o index_n horarius_fw-la or_o hour_n index_n 33._o have_v describe_v the_o great_a circle_n frame_v without_o and_o draw_v upon_o the_o globe_n we_o will_v now_o describe_v the_o lesser_a circle_n also_o and_o these_o lesser_a circle_n be_v call_v parallel_n that_o be_v such_o as_o be_v in_o all_o place_n equal_o distant_a from_o the_o equator_fw-la and_o these_o circle_n how_o little_a soever_o be_v suppose_v to_o be_v divide_v into_o 360_o degree_n but_o these_o degree_n be_v not_o so_o large_a as_o in_o the_o great_a circle_n but_o do_v proportionable_o decrease_v according_a to_o the_o radius_fw-la by_o which_o they_o be_v draw_v 34._o these_o lesser_a circle_n be_v either_o the_o tropic_n or_o the_o polar_a circle_n 35._o the_o tropic_n be_v two_o small_a circle_n draw_v upon_o the_o globe_n one_o beyond_o the_o equator_fw-la towards_o the_o north_n pole_n and_o the_o other_o towards_o the_o south_n show_v the_o way_n which_o the_o sun_n make_v in_o his_o diurnal_a motion_n when_o he_o be_v at_o his_o great_a distance_n from_o the_o equator_fw-la either_o north_n or_o south_n these_o circle_n be_v call_v tropic_n 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d that_o be_v from_o the_o sun_n return_v for_o the_o sun_n come_v to_o these_o circle_n he_o be_v at_o his_o great_a distance_n from_o the_o equator_fw-la and_o in_o the_o same_o moment_n of_o time_n slope_v as_o it_o be_v his_o course_n he_o return_v near_a and_o near_o to_o the_o equator_fw-la again_o 36._o these_o tropical_a circle_n do_v show_v the_o point_n of_o heaven_n in_o which_o the_o sun_n do_v make_v either_o the_o long_a day_n or_o the_o short_a day_n in_o the_o year_n according_a as_o he_o be_v in_o the_o northern_a or_o the_o southern_a tropic_n and_o be_v draw_v at_o 23_o degree_n and_o a_o half_a distant_a from_o the_o equator_fw-la 37._o the_o polar_a circle_n be_v two_o lesser_a circle_n draw_v upon_o the_o globe_n at_o the_o radius_fw-la of_o 23_o degree_n and_o a_o half_a distant_a from_o the_o pole_n of_o the_o world_n show_v thereby_o the_o pole_n of_o the_o zodiac_n which_o be_v so_o many_o degree_n distant_a from_o the_o equator_fw-la on_o both_o side_n thereof_o 38._o these_o polar_a circle_n be_v 66_o degree_n and_o a_o half_a distant_a from_o the_o equator_fw-la and_o 43_o degree_n distant_a from_o his_o near_a tropic_n they_o be_v call_v the_o arctic_a and_o antarctick_a circle_n 39_o the_o arctic_a circle_n be_v that_o which_o be_v describe_v about_o the_o arctic_a pole_n and_o pass_v almost_o through_o the_o middle_n of_o the_o head_n of_o the_o great_a bear_n it_o be_v call_v the_o arctic_a circle_n 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d from_o the_o two_o conspicuous_a star_n towards_o the_o north_n call_v the_o great_a and_o the_o lesser_a bear_n 40._o the_o antarctick_a circle_n be_v that_o which_o be_v describe_v about_o the_o antarctick_a or_o south_n pole_n it_o be_v so_o call_v 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d 〈◊〉_d that_o be_v from_o be_v opposite_a to_o the_o great_a and_o lesser_a bear_n have_v thus_o describe_v the_o globe_n or_o astronomical_a instrument_n by_o which_o the_o frame_n of_o the_o world_n be_v represent_v to_o our_o view_n i_o will_v proceed_v to_o show_v the_o use_n for_o which_o it_o be_v intend_v chap._n ii_o of_o the_o distinction_n and_o affection_n of_o spherical_a line_n or_o arch_n the_o use_n of_o the_o globe_n as_o to_o practise_v be_v either_o such_o as_o concern_v the_o heaven_n or_o the_o earth_n in_o either_o of_o which_o if_o we_o shall_v descend_v unto_o particular_n the_o use_n will_v be_v more_o in_o number_n than_o a_o short_a treatise_n will_v contain_v see_v therefore_o that_o all_o problem_n which_o concern_v the_o globe_n may_v be_v best_a and_o most_o accurate_o resolve_v by_o the_o doctrine_n of_o spherical_a triangle_n we_o will_v contract_v these_o use_n of_o the_o globe_n which_o otherwise_o may_v prove_v infinite_a to_o such_o problem_n as_o come_v within_o the_o compass_n of_o the_o 28_o case_n of_o right_n and_o oblique_a angle_a spherical_a triangle_n 2._o and_o that_o the_o nature_n of_o spherical_a triangle_n may_v be_v the_o better_o understand_v and_o by_o which_o of_o the_o 28_o case_n the_o particular_a problem_n may_v be_v best_a resolve_v i_o will_v set_v down_o some_o general_n definition_n and_o affection_n which_o do_v belong_v to_o such_o line_n or_o arch_n of_o which_o the_o triangle_n must_v be_v frame_v with_o the_o part_n and_o affection_n of_o those_o triangle_n and_o how_o the_o thing_n give_v and_o require_v in_o they_o may_v be_v represent_v and_o resolve_v upon_o and_o by_o the_o globe_n as_o also_o how_o they_o may_v be_v represent_v and_o resolve_v by_o the_o projection_n of_o the_o sphere_n and_o by_o the_o canon_n of_o triangle_n 3._o a_o spherical_a triangle_n than_o be_v a_o figure_n consist_v of_o three_o arch_n of_o the_o great_a circle_n upon_o the_o superficies_n of_o a_o sphere_n or_o globe_n every_o one_o being_n less_o than_o a_o semicircle_n 4._o a_o great_a circle_n be_v that_o which_o divide_v the_o sphere_n or_o globe_n into_o two_o equal_a part_n and_o thus_o the_o horizon_n equator_fw-la zodiac_n and_o meridian_n before_n describe_v be_v all_o of_o they_o great_a circle_n and_o of_o these_o circle_n or_o any_o other_o there_o must_v be_v three_o arch_n to_o make_v a_o triangle_n and_o every_o one_o of_o these_o arch_n several_o must_v be_v less_o than_o a_o semicircle_n to_o make_v this_o plain_a in_o fig._n 1._o the_o straight_a line_n har_n do_v represent_v the_o horizon_n pr._n the_o height_n of_o the_o pole_n above_o the_o horizon_n pm_v a_o meridian_n and_o these_o three_o arch_n by_o their_o intersect_v one_o another_o do_v visible_o constitute_v four_o spherical_a triangle_n 1._o pmr_n 2._o pmh_o 3._o shm._n 4._o smr_n and_o every_o arch_n be_v less_o than_o a_o semicircle_n as_o in_o the_o triangle_n pmr_n the_o arch_n pr._n be_v less_o than_o the_o semicircle_n pr_n the_o arch_a mr_n be_v less_o than_o the_o semicircle_n amr_n and_o the_o arch_n pm_n be_v less_o than_o the_o semicircle_n pm_v the_o like_a may_v be_v show_v in_o the_o other_o triangle_n 5._o spherical_a or_o circular_a line_n be_v parallel_n or_o angular_a 6._o parallel_v arch_n or_o circle_n be_v such_o as_o be_v draw_v upon_o the_o same_o centre_n within_o without_z or_o equal_a to_o another_o arch_n or_o circle_n thus_o in_o fig._n 1._o the_o arch_n ♋_o m_o ♋_o and_o ♑_o o_o ♑_o are_z though_o lesser_a circle_n parallel_n to_o the_o equinoctial_a ae_n a_o q_o and_o do_v in_o that_o scheme_n represent_v the_o tropic_n of_o cancer_n and_o capricorn_n the_o manner_n of_o describe_v they_o or_o any_o other_o parallel_n circle_n be_v thus_o set_v off_o their_o distance_n from_o the_o great_a circle_n to_o which_o you_o be_v to_o draw_v a_o parallel_n with_o your_o compass_n by_o help_n of_o your_o line_n of_o chord_n which_o in_o this_o example_n be_v 23_o degree_n and_o a_o half_a from_o ae_n to_o ♋_o then_o draw_v the_o line_n a_o ♋_o and_o upon_o the_o 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489050_o 489177_o 489304_o 489432_o 489559_o 489687_o  _fw-fr 492_o 489814_o 489931_o 490049_o 490166_o 490283_o 490401_o 490538_o 490675_o 490812_o 490950_o  _fw-fr 493_o 491087_o 491214_o 491342_o 491469_o 491596_o 491724_o 491815_o 491978_o 492105_o 492233_o  _fw-fr 494_o 492360_o 492487_o 492615_o 492742_o 492869_o 492997_o 493124_o 493251_o 493378_o 493505_o  _fw-fr 495_o 493633_o 493760_o 493888_o 494015_o 494143_o 494270_o 494397_o 494525_o 494652_o 494779_o  _fw-fr 496_o 494007_o 495034_o 495161_o 495288_o 495416_o 495543_o 495670_o 495798_o 495925_o 496052_o  _fw-fr 497_o 496180_o 496307_o 496434_o 496561_o 496689_o 496816_o 496943_o 497071_o 497198_o 497327_o  _fw-fr 498_o 497453_o 497580_o 497708_o 497835_o 497962_o 498090_o 498217_o 498344_o 498471_o 498599_o  _fw-fr 499_o 499726_o 498853_o 498981_o 499108_o 499235_o 499363_o 499490_o 499616_o 499773_o 499871_o  _fw-fr finis_fw-la cosmographia_fw-la the_o second_o part._n or_o the_o doctrine_n of_o the_o primum_fw-la mobile_n a_o introduction_n to_o astronomy_n the_o first_o part._n of_o the_o primum_fw-la mobile_n chap._n i._n of_o the_o general_n subject_a of_o astronomy_n astronomy_n be_v a_o science_n concern_v the_o measure_n and_o motion_n of_o the_o sphere_n and_o star_n 2._o astronomy_n have_v two_o part_n the_o first_o be_v absolute_a and_o the_o other_o comparative_a 3._o the_o absolute_a part_n of_o astronomy_n be_v that_o which_o treat_v of_o the_o measure_n and_o motion_n of_o the_o orb_n and_o star_n absolute_o without_o respect_n to_o any_o distinction_n of_o time_n 4._o the_o comparative_a part_n of_o astronomy_n be_v that_o which_o treat_v 〈◊〉_d the_o motion_n of_o the_o star_n in_o reference_n to_o some_o certain_a distinction_n of_o time_n 5._o the_o absolute_a part_n of_o astronomy_n treat_v of_o the_o primum_fw-la mobile_n or_o diurnal_a motion_n of_o all_o the_o celestial_a orb_n or_o sphere_n 6._o the_o primum_fw-la mobile_n or_o diurnal_a motion_n of_o the_o heaven_n be_v that_o motion_n by_o which_o the_o several_a sphere_n be_v move_v round_o the_o world_n in_o a_o day_n 〈…〉_z from_o east_n towards_o west_n and_o â—o_o forward_o from_o west_n towards_o east_n and_o so_o continual_o return_v to_o the_o same_o point_n from_o whence_o they_o begin_v their_o motion_n 7._o this_o first_o and_o common_a motion_n of_o the_o heaven_n will_v be_v best_a understand_v by_o help_n of_o a_o instrument_n call_v a_o globe_n which_o be_v a_o artificial_a representation_n of_o the_o heaven_n or_o the_o earth_n and_o water_n under_o that_o form_n and_o figure_n of_o roundness_n which_o they_o be_v suppose_v to_o have_v 8._o this_o representation_n or_o description_n of_o the_o visible_a world_n be_v by_o circle_n great_a and_o small_a some_o of_o which_o be_v express_v upon_o and_o other_o be_v frame_v without_o the_o globe_n 9_o the_o circle_n without_o the_o globe_n be_v chief_o two_o the_o meridian_n and_o the_o horizon_n the_o one_o of_o brass_n and_o the_o other_o of_o wood_n and_o these_o two_o circle_n be_v variable_a or_o mutable_a for_o although_o there_o be_v but_o one_o horizon_n and_o one_o meridian_n in_o respect_n of_o the_o whole_a world_n or_o in_o respect_n of_o the_o whole_a heaven_n and_o earth_n yet_o in_o respect_n of_o the_o particular_a part_n of_o heaven_n or_o rather_o in_o respect_n of_o the_o diverse_a province_n country_n and_o city_n on_o the_o earth_n there_o be_v diverse_a both_o orison_n and_o meridian_n 10._o the_o meridian_n than_o be_v a_o great_a circle_n without_o the_o globe_n divide_v the_o globe_n and_o consequent_o the_o day_n and_o night_n into_o two_o equal_a part_n from_o the_o north_n and_o south_n end_v whereof_o a_o strong_a wire_n of_o brass_n or_o iron_n be_v draw_v or_o suppose_v to_o be_v draw_v through_o the_o centre_n of_o the_o globe_n represent_v the_o axis_n of_o the_o earth_n by_o mean_n whereof_o the_o whole_a globe_n turn_v round_o within_o the_o say_a circle_n so_o that_o any_o part_n may_v be_v bring_v direct_o under_o this_o brass_n meridian_n at_o pleasure_n 11._o this_o brass_n meridian_n be_v divide_v into_o 4_o equal_a part_n or_o quadrant_n and_o each_o of_o they_o be_v subdivide_v into_o 90_o degree_n that_o be_v 360_o for_o the_o whole_a circle_n the_o reason_n why_o this_o circle_n be_v not_o divide_v in_o 360_o degree_n throughout_o but_o still_o stop_v at_o
the_o one_o be_v right_a or_o perpendicular_a to_o the_o other_o thus_o the_o brass_n meridian_n cut_v the_o horizon_n at_o right_a angle_n and_o thus_o the_o meridian_n draw_v upon_o the_o globe_n as_o well_o as_o the_o brass_n meridian_n do_v all_o of_o they_o cut_v the_o equator_fw-la at_o right_a angle_n 10._o a_o oblique_a spherical_a angle_n be_v that_o which_o be_v contain_v by_o two_o arch_n of_o the_o great_a circle_n in_o the_o superficies_n of_o the_o sphere_n not_o be_v right_a or_o perpendicular_a to_o one_o another_o 11._o a_o oblique_a spherical_a angle_n be_v obtuse_a or_o acute_a 12._o a_o obtuse_a spherical_a angle_n be_v that_o which_o be_v great_a than_o a_o right_a angle_n a_o acute_a be_v that_o which_o be_v less_o than_o a_o right_a angle_n 13._o if_o two_o of_o the_o great_a circle_n of_o the_o sphere_n shall_v pass_v through_o one_o another_o pole_n those_o two_o great_a circle_n shall_v cut_v one_o another_o at_o right_a angle_n thus_o the_o brazen_a meridian_n do_v intersect_v the_o equinoctial_a and_o horizon_n 14._o if_o two_o of_o the_o great_a circle_n of_o the_o sphere_n shall_v intersect_v one_o another_o and_o pass_v through_o each_o other_o pole_n they_o shall_v intersect_v one_o another_o at_o unequal_a or_o oblique_a angle_n the_o angle_n upon_o the_o one_o side_n of_o the_o intersection_n be_v obtuse_a or_o more_o than_o a_o right_n and_o the_o angle_n upon_o the_o other_o side_n of_o the_o intersection_n be_v acute_a or_o less_o than_o a_o right_n thus_o in_o fig._n 1._o the_o arch_n pm_n do_v intersect_v the_o meridian_n and_o horizon_n but_o not_o in_o the_o pole_n of_o either_o therefore_o the_o angle_n hpm_n upon_o one_o side_n of_o the_o intersection_n of_o that_o arch_n with_o the_o meridian_n be_v more_o than_o a_o right_a angle_n and_o the_o angle_n mpr_fw-la upon_o the_o other_o side_n of_o the_o intersection_n be_v less_o and_o so_o likewise_o the_o angle_n pmh_o upon_o the_o one_o side_n of_o the_o intersection_n of_o the_o arch_n pm_n with_o the_o horizon_n hr_n be_v great_a than_o a_o right_a angle_n and_o the_o angle_n rmp_n upon_o the_o other_o side_n of_o the_o intersection_n be_v less_o than_o a_o right_n 15._o a_o spherical_a angle_n be_v measure_v by_o the_o arch_n of_o a_o great_a circle_n describe_v from_o the_o angular_a point_n between_o the_o side_n of_o the_o angle_n those_o side_n be_v continue_v unto_o quadrant_n thus_o the_o arch_n of_o the_o equator_fw-la tq_fw-la in_o fig._n 1._o be_v the_o measure_n of_o the_o angle_n mpr_fw-la or_o tpq_n the_o side_n pt_v and_o pq_n be_v quadrant_n and_o the_o measure_n thereof_o in_o the_o projection_n may_v thus_o be_v find_v lay_v a_o ruler_n from_o p_o to_o t_o and_o it_o will_v cut_v the_o primitive_a circle_n in_o five_o and_o the_o arch_n vq_n be_v take_v in_o your_o compass_n and_o apply_v to_o your_o line_n of_o chord_n will_v give_v the_o quantity_n of_o the_o angle_n propound_v 16._o the_o compliment_n of_o a_o spherical_a arch_n or_o angle_n be_v so_o much_o as_o it_o want_v of_o a_o quadrant_n if_o the_o arch_n or_o angle_n give_v be_v less_o than_o a_o quadrant_n or_o so_o much_o as_o it_o want_v of_o a_o semicircle_n if_o it_o be_v more_o than_o a_o quadrant_n 17._o a_o arch_n of_o a_o great_a circle_n cut_v the_o arch_n of_o another_o great_a circle_n shall_v intersect_v one_o another_o at_o right_a angle_n or_o make_v two_o angle_n which_o be_v take_v together_o shall_v be_v equal_a unto_o two_o right_a thus_o in_o fig._n 1._o the_o axis_n pas_n or_o equinoctial_a colour_n do_v cut_v the_o equator_fw-la ae_n a_o q_o at_o right_a angle_n but_o the_o meridian_n pms_n do_v cut_v the_o horizon_n hmr_n at_o oblique_a angle_n make_v the_o angle_n pmr_n less_o than_o a_o right_n and_o the_o angle_n smr_n more_o than_o a_o right_n and_o both_o together_o equal_a to_o a_o semicircle_n 18._o from_o these_o general_a definition_n proper_a to_o spherical_a line_n or_o arch_n the_o general_a affection_n of_o these_o arch_n may_v easy_o be_v discern_v i_o mean_v the_o various_a position_n of_o the_o globe_n of_o the_o earth_n in_o respect_n of_o all_o and_o singular_a the_o inhabitant_n thereof_o 19_o and_o the_o whole_a body_n of_o the_o sphere_n or_o globe_n in_o respect_n of_o the_o horizon_n be_v look_v upon_o by_o the_o earth_n inhabitant_n either_o in_o a_o parallel_n a_o right_a or_o a_o oblique_a sphere_n 20._o a_o parallel_n sphere_n be_v when_o one_o of_o the_o pole_n of_o the_o world_n be_v elevate_v above_o the_o horizon_n to_o the_o zenith_n the_o other_o depress_v as_o low_a as_o the_o nadir_n and_o the_o equinoctial_a line_n join_v with_o the_o horizon_n they_o which_o there_o inhabit_v if_o any_o such_o be_v see_v not_o the_o sun_n or_o other_o star_n rise_v or_o set_v or_o high_a or_o low_a in_o their_o diurnal_a revolution_n and_o see_v that_o the_o sun_n traverse_v the_o whole_a zodiac_n in_o a_o year_n and_o that_o half_a the_o zodiac_n be_v above_o the_o horizon_n and_o half_a under_o it_o it_o come_v to_o pass_v that_o the_o sun_n set_v not_o with_o they_o for_o the_o space_n of_o six_o month_n nor_o give_v they_o any_o light_n for_o the_o space_n of_o other_o six_o month_n and_o so_o make_v but_o one_o day_n and_o night_n of_o the_o whole_a year_n 21._o a_o right_a sphere_n be_v when_o both_o the_o pole_n of_o the_o world_n do_v lie_v in_o the_o horizon_n and_o the_o equinoctial_a circle_n be_v at_o his_o great_a distance_n from_o it_o pass_v through_o the_o zenith_n of_o the_o place_n and_o in_o this_o position_n of_o the_o sphere_n all_o the_o celestial_a body_n sun_n moon_n and_o other_o planet_n and_o fix_v star_n by_o the_o daily_o turn_v about_o of_o the_o heaven_n do_v direct_o ascend_v above_o and_o also_o direct_o descend_v below_o the_o horizon_n because_o the_o motion_n which_o they_o make_v in_o their_o daily_a motion_n do_v cut_v the_o horizon_n perpendicular_o and_o as_o it_o be_v at_o right_a angle_n in_o this_o position_n of_o the_o sphere_n all_o the_o star_n may_v be_v observe_v to_o rise_v and_o set_v in_o a_o equal_a space_n of_o time_n and_o to_o continue_v as_o long_o above_o the_o horizon_n as_o they_o do_v under_o it_o the_o day_n and_o night_n to_o those_o inhabitant_n be_v always_o of_o a_o equal_a length_n 22._o a_o oblique_a sphere_n be_v when_o the_o axis_n of_o the_o world_n be_v neither_o direct_a nor_o parallel_v to_o the_o horizon_n be_v incline_v oblique_o towards_o both_o side_n of_o the_o horizon_n as_o in_o fig._n 1._o whence_o it_o come_v to_o pass_v that_o so_o much_o as_o one_o of_o the_o pole_n be_v elevate_v above_o the_o horizon_n upon_o the_o one_o side_n so_o much_o be_v the_o other_o depress_v under_o the_o horizon_n upon_o the_o other_o side_n and_o in_o this_o position_n of_o sphere_n the_o day_n be_v sometime_o long_o than_o the_o night_n sometime_o short_a and_o sometime_o of_o equal_a length_n when_o the_o sun_n be_v in_o either_o of_o the_o equinoctial_a point_n the_o day_n and_o night_n be_v equal_a but_o when_o he_o decline_v from_o the_o equator_fw-la towards_o the_o elevate_a pole_n the_o day_n be_v observe_v to_o increase_v and_o when_o he_o decline_v from_o the_o equator_fw-la towards_o the_o opposite_a pole_n or_o the_o pole_n depress_v the_o day_n do_v decrease_v as_o be_v manifest_a in_o fig._n 1._o for_o when_o the_o sun_n rise_v at_o m_n the_o line_n m_o ♋_o above_o the_o horizon_n be_v the_o semidiurnal_a arch_n of_o the_o long_a day_n when_o he_o rise_v at_o c_o the_o arch_n c_o ♑_o above_o the_o horizon_n be_v the_o semidiurnal_a arch_n of_o the_o short_a day_n and_o when_o he_o rise_v at_o a_o the_o day_n and_o night_n be_v of_o equal_a length_n the_o semidiurnal_a arch_n aae_n be_v equal_a to_o the_o seminocturnal_a arch_n aq_n chap._n iii_o of_o the_o kind_n and_o part_n of_o spherical_a triangle_n and_o how_o to_o project_v the_o same_o upon_o the_o plane_n of_o the_o meridian_n have_v show_v what_o a_o spherical_a triangle_n be_v and_o of_o what_o circle_n it_o be_v compose_v with_o the_o general_a affection_n of_o such_o line_n i_o will_v now_o show_v how_o many_o several_a sort_n of_o triangle_n there_o be_v of_o what_o circular_a part_n they_o do_v consist_v and_o such_o affection_n proper_a to_o they_o as_o will_v render_v the_o sopition_n of_o they_o more_o clear_a and_o certain_a 2._o spherical_a triangle_n be_v either_o right_a or_o oblique_a 3._o a_o right_o angle_v spherical_a triangle_n be_v that_o which_o have_v one_o or_o more_o right_a angle_n 4._o a_o spherical_a triangle_n which_o have_v three_o right_a angle_n have_v always_o his_o three_o side_n quadrant_n as_o in_o fig._n 1._o the_o spherical_a triangle_n azr_n the_o angles_n zra_n raz_n and_o azr_n be_v
how_o a_o right_a or_o oblique_a angle_a spherical_a triangle_n may_v be_v project_v upon_o the_o plane_n of_o the_o meridian_n as_o well_o as_o delineate_v upon_o the_o globe_n we_o will_v now_o consider_v the_o several_a triangle_n usual_o represent_v upon_o the_o globe_n with_o the_o several_a astronomical_a and_o geographical_a problem_n contain_v in_o they_o and_o resolve_v by_o they_o 28._o the_o spherical_a triangle_n usual_o represent_v upon_o the_o globe_n be_v eight_o whereof_o there_o be_v five_o right_a angle_a triangle_n have_v their_o denomination_n from_o their_o hypotenusas_n the_o first_o be_v call_v the_o elliptical_a triangle_n who_o hypotenusa_fw-la be_v a_o arch_n of_o the_o ecliptic_a the_o leg_n thereof_o be_v arch_n of_o the_o aequator_fw-la and_o meridian_n this_o be_v represent_v upon_o the_o globe_n by_o the_o triangle_n adf_n in_o fig._n 1._o in_o which_o the_o five_o circular_a part_n beside_o the_o right_a angle_n be_v 1._o the_o hypotenuse_n or_o arch_n of_o the_o ecliptic_a af._n 2._o the_o leg_n or_o arch_n of_o the_o aequator_fw-la ad._n 3._o the_o leg_n or_o arch_n of_o the_o meridian_n df._n 4._o the_o oblique_a angle_n of_o the_o equator_fw-la with_o the_o ecliptic_a and_o the_o sun_n great_a declination_n daf_a 5._o the_o oblique_a angle_n of_o the_o ecliptic_a and_o meridian_n or_o the_o angle_n of_o the_o sun_n position_n afd_v the_o two_o next_o i_o call_v meridional_a because_o the_o hypotenusas_n in_o they_o both_o be_v arch_n of_o a_o meridian_n one_o of_o these_o be_v note_v with_o the_o letter_n mpr_fw-la in_o fig._n 1._o in_o which_o the_o five_o circular_a part_n be_v 1._o the_o hypotenusa_fw-la or_o arch_n of_o a_o meridian_n pm_n 2._o the_o leg_n or_o arch_n of_o the_o horizon_n mr_n the_o sun_n azimuth_n north._n 3._o the_o leg_n or_o arch_n of_o the_o brass_n meridian_n represent_v the_o height_n of_o the_o pole_n pr._n 4._o the_o oblique_a angle_n of_o the_o meridian_n upon_o the_o globe_n with_o the_o brass_n meridian_n or_o angle_n of_o the_o hour_n from_o midnight_n p._n 5._o the_o oblique_a angle_n of_o the_o sun_n meridian_n with_o the_o horizon_n or_o the_o compliment_n of_o the_o sun_n angle_n of_o position_n pmr_n the_o other_o right_o angle_v meridional_a triangle_n be_v note_v with_o the_o letter_n aeg_fw-la in_o fig._n 1._o in_o which_o the_o 5_o circular_a part_n be_v 1._o the_o hypotenusa_fw-la or_o present_a declination_n of_o the_o sun_n ae_n 2._o the_o leg_n or_o sun_n amplitude_n at_o the_o hour_n of_o six_o ag._n 3._o the_o other_o leg_n or_o sun_n height_n at_o the_o same_o time_n eglantine_n 4._o the_o angle_n of_o the_o meridian_n with_o the_o horizon_n or_o angle_n of_o the_o pole_n elevation_n eag_n 5._o the_o angle_n of_o the_o meridian_n with_o the_o azimuth_n or_o the_o angle_n of_o the_o sun_n position_n aeg_n the_o four_o right_o angle_v spherical_a triangle_n i_o call_v a_o azimuth_n triangle_n because_o the_o hypotenusa_fw-la do_v cut_v the_o horizon_n in_o the_o east_n and_o west_n azimuth_n as_o be_v represent_v by_o the_o triangle_n adu._n in_o fig._n 1._o in_o which_o the_o 5_o circular_a part_n be_v 1._o the_o hypotenusa_fw-la or_o arch_n of_o the_o sun_n or_o star_n altitude_n au._n 2._o the_o leg_n or_o declination_n of_o the_o sun_n or_o star_n dv_o 3._o the_o other_o leg_n or_o right_a ascension_n of_o the_o sun_n or_o star_n ad._n 4._o the_o oblique_a angle_n or_o angle_n of_o the_o pole_n elevation_n dav_n 5._o the_o other_o oblique_a angle_n or_o angle_n of_o the_o sun_n or_o star_n position_n dva_n the_o five_o and_o last_o right_o angle_v spherical_a triangle_n that_o i_o shall_v mention_v i_o call_v a_o horizontal_n triangle_n because_o the_o hypotenusa_fw-la thereof_o be_v a_o arch_n of_o the_o horizon_n and_o be_v represent_v by_o the_o triangle_n amt_v in_o fig._n 1._o in_o which_o the_o 5_o circular_a part_n be_v 1._o the_o hypotenusa_fw-la and_o arch_n of_o the_o horizon_n or_o amplitude_n of_o the_o sun_n at_o his_o rise_n or_o set_v am._n 2._o the_o leg_n contain_v the_o sun_n or_o star_n declination_n tm_n 3._o the_o other_o leg_n or_o ascensional_a difference_n at_o that_o be_v the_o difference_n between_o dt_n the_o right_a ascension_n and_o da_fw-la the_o oblique_a angle_n 4._o the_o oblique_a angle_n of_o the_o horizon_n and_o equator_fw-la or_o height_n of_o the_o equator_fw-la tam_fw-la 5._o the_o other_o oblique_a angle_n or_o angle_n of_o the_o horizon_n and_o meridian_n amt_v the_o oblique_a angle_a spherical_a triangle_n usual_o represent_v upon_o the_o globe_n be_v three_o the_o first_o i_o call_v the_o complimental_a triangle_n because_o the_o side_n thereof_o be_v all_o compliment_n and_o this_o be_v represent_v by_o the_o triangle_n fzp_n in_o fig._n 1._o who_o circular_a part_n be_v 1._o the_o compliment_n of_o the_o pole_n elevation_n zp_n 2._o the_o compliment_n of_o the_o sun_n declination_n fp_n 3._o the_o compliment_n of_o the_o sun_n altitude_n or_o almicantar_n fz_n 4._o the_o sun_n azimuth_n or_o distance_n from_o the_o north_n fzp_n 5._o the_o hour_n of_o the_o day_n or_o distance_n of_o the_o sun_n from_o noon_n zpf_n 6._o the_o angle_n of_o the_o sun_n position_n zfp_n the_o second_o oblique_a angle_a spherical_a triangle_n i_o call_v a_o geographical_a or_o nautical_a triangle_n because_o it_o serve_v to_o resolve_v those_o problem_n which_o concern_v geographie_n and_o navigation_n and_o this_o be_v also_o represent_v by_o the_o triangle_n fzp_n in_o fig._n 1._o who_o part_n be_v 1._o the_o compliment_n of_o latitude_n as_o before_o zp_v 2._o the_o distance_n between_o the_o two_o place_n at_o z_o and_o f_o or_o side_n fz_n 3._o the_o compliment_n of_o the_o latitude_n of_o the_o place_n at_o f_o or_o side_n fp_n 4._o the_o difference_n of_o longitude_n between_o the_o two_o place_n at_o z_o and_o f_o or_o the_o angle_n fbz._n 5._o the_o point_n of_o the_o compass_n lead_v from_o z_o to_o f_o or_o angle_n fzp_n 6._o the_o point_n of_o the_o compass_n lead_v from_o f_o to_o z_o or_o angle_n zfp_n the_o three_o oblique_a angle_a spherical_a triangle_n be_v call_v a_o polar_a triangle_n because_o one_o side_n thereof_o be_v the_o distance_n between_o the_o pole_n of_o the_o world_n and_o the_o pole_n of_o the_o zodiac_n this_o triangle_n be_v represent_v upon_o the_o celestial_a globe_n by_o the_o triangle_n fsp_n in_o fig._n 4._o in_o which_o the_o circular_a part_n be_v 1._o the_o distance_n between_o the_o pole_n of_o the_o world_n and_o the_o pole_n of_o the_o ecliptic_a or_o the_o arch_n sp._n 2._o the_o compliment_n of_o the_o star_n declination_n fp_n 3._o the_o compliment_n of_o the_o star_n north_n latitude_n from_o the_o ecliptic_a or_o the_o arch_a f_n 4._o the_o angle_n of_o the_o star_n right_a ascension_n fp_v 5._o the_o compliment_n of_o the_o star_n longitude_n fsp_n 6._o the_o angle_n of_o the_o star_n position_n sfp_n 29._o and_o thus_o at_o length_n i_o have_v perform_v what_o be_v propose_v in_o the_o 15_o of_o this_o chapter_n that_o be_v i_o have_v show_v how_o the_o several_a circle_n of_o the_o globe_n may_v be_v project_v upon_o the_o plane_n of_o the_o meridian_n the_o several_a useful_a triangle_n that_o be_v describe_v by_o such_o projection_n with_o such_o astronomical_a proposition_n as_o be_v contain_v and_o resolveable_a by_o those_o triangle_n and_o although_o the_o most_o accurate_a way_n of_o resolution_n be_v by_o the_o doctrine_n of_o trigonometry_n and_o the_o canon_n of_o line_n and_o tangent_n yet_o it_o be_v not_o impertinent_a to_o do_v the_o same_o upon_o the_o globe_n itself_o which_o as_o to_o the_o side_n be_v easy_a but_o to_o measure_v or_o lay_v down_o the_o angles_n be_v sometime_o a_o little_a laborious_a in_o the_o right_o angle_a spherical_a triangle_n aebz_n in_o fig._n 1._o the_o measure_n of_o the_o angle_n aezb_n be_v reckon_v in_o the_o horizon_n from_o h_n to_o c_z but_o to_o lay_v down_o or_o measure_v the_o angle_n aebz_n the_o ready_a way_n be_v to_o describe_v the_o triangle_n again_o make_v aez_n =_o aeb_fw-mi and_o aeb_fw-mi =_o aez_n so_o will_v the_o angle_n aebz_n stand_v where_o the_o angle_n aezb_n be_v and_o may_v be_v measure_v in_o the_o horizon_n as_o the_o other_o be_v and_o so_o in_o the_o oblique_a angle_a spherical_a triangle_n fzp_v in_o fig._n 1._o the_o angle_n at_o z_o and_o p_o be_v easy_o measure_v or_o lay_v down_o upon_o the_o globe_n but_o to_o perform_v the_o same_o with_o the_o angle_n zfp_n you_o may_v represent_v it_o at_o the_o pole_n or_o zenith_n and_o find_v the_o measure_n in_o the_o equator_fw-la or_o horizon_n 30._o and_o now_o have_v as_o i_o hope_v sufficient_o prepare_v the_o young_a student_n for_o the_o first_o part_n of_o astronomy_n the_o doctrine_n of_o the_o primum_fw-la mobile_n by_o show_v how_o the_o heaven_n and_o the_o earth_n
from_o the_o meridian_n and_o declination_n to_o find_v his_o azimuth_n the_o azimuth_n of_o the_o sun_n be_v a_o arch_n of_o the_o horizon_n intercept_v between_o the_o meridian_n and_o the_o vertical_a line_n pass_v by_o the_o sun_n be_v understand_v by_o the_o angle_n hzc_n in_o fig._n 1._o or_o arch_a hc_n and_o in_o all_o the_o variety_n of_o the_o last_o problem_n may_v be_v find_v by_o the_o first_o case_n of_o oblique_a angle_a spherical_a triangle_n thus_o in_o the_o triangle_n zbp._n as_o the_o sine_fw-la of_o bz_n 71._o 88_o comp._n ar._n 0_o 022090â—_n be_v to_o the_o sine_fw-la of_o bpz_n 60._o 9.9375306_o so_o be_v the_o sine_fw-la of_o bp_o 90._o 10.0000000_o to_o the_o sine_fw-la of_o bzp._n 65._o 67._o 9.9596209_o in_o the_o triangle_n zfp_n i_o say_v s._n zf_n s._n zpf_n s._n fp_n s._n fzp_n in_o the_o triangle_n zbp._n i_o say_v sine_fw-la zb._n sine_fw-la zpb_fw-la sine_fw-la bp._n sine_fw-la bzp._n problem_n 12._o the_o pole_n elevation_n with_o the_o sun_n altitude_n and_o declination_n give_v to_o find_v his_o azimuth_n in_o the_o oblique_a angle_a spherical_a triangle_n fzp_v in_o fig._n 1._o let_v there_o be_v give_v 1._o fp_n 67._o 97_o the_o compliment_n of_o the_o ☉_o declination_n 2._o zp_n 38._o 47_o the_o compliment_n of_o the_o pole_n elevation_n 3._o fz_n 45._o 46_o the_o compliment_n of_o the_o ☉_o altitude_n and_o let_v the_o angle_n fzp_v the_o ☉_o azimuth_n be_v require_v by_o the_o 11_o case_n of_o oblique_a angle_a spherical_a triangle_n as_o the_o sine_fw-la zp_n x_o sine_fw-la fz_n be_v to_o the_o square_n of_o radius_fw-la so_o be_v the_o sine_fw-la 1_o 2_o z_o of_o the_o side_n zp_v x_o 1_o 2_o z_o cr_n zf_n to_o the_o square_n of_o the_o sine_fw-la of_o half_a the_o angle_n fzp_n the_o sum_n of_o the_o three_o side_n be_v 151.89_o the_o half_a sum_n be_v 75.945_o from_o which_o deduct_v pz_n 38._o 47._o the_o difference_n be_v 37.475_o and_o the_o difference_n between_o 75.945_o and_o fz_n be_v 30._o 495._o sine_fw-la of_o pz_n 38._o 47._o comp._n ar._n 0.2061365_o sine_fw-la of_o fz_n 45._o 45._o comp._n ar._n 0.1471308_o s._n 1_o 2_o z_o cr_n pz_n 37._o 475._o 9.7842000_o s._n 1_o 2_o z_o cr_n fz_n 30._o 495._o 9.7054045_o square_n of_o the_o sine_fw-la of_o 1_o 2_o fzp_a 19.8428618_o sine_fw-la of_o 57_o 94._o 9.9214309_o the_o double_a whereof_o be_v 115.88_o the_o ☉_o azimuth_n from_o the_o north._n and_o the_o compliment_n 64.12_o be_v the_o ☉_o azimuth_n from_o the_o south_n problem_n 13._o to_o find_v the_o point_n of_o the_o ecliptic_a culminate_v and_o its_o altitude_n before_o we_o can_v know_v what_o sign_n and_o degree_n of_o the_o ecliptic_a be_v in_o the_o medium_fw-la coeli_fw-la we_o must_v find_v the_o right_a ascension_n thereof_o to_o do_v which_o we_o must_v add_v the_o sun_n right_a ascension_n to_o the_o time_n afternoon_n be_v reduce_v into_o degree_n and_o minute_n of_o the_o aequator_fw-la the_o sum_n be_v the_o right_a ascension_n of_o the_o medium_fw-la coeli_fw-la example_n let_v the_o time_n give_v be_v march_v the_o 20._o 1674._o at_o one_o of_o the_o clock_n in_o the_o afternoon_n at_o which_o time_n the_o sun_n place_n be_v in_o ♈_o 10_o deg_n 23_o centesm_n to_o find_v the_o right_a ascension_n thereof_o in_o the_o right_o angle_a spherical_a triangle_n adf_n in_o fig._n 1._o we_o have_v give_v the_o angle_n of_o the_o sun_n be_v great_a declination_n daf_a 23._o 525_o and_o the_o sun_n distance_n from_o the_o next_o equinoctial_a point_n of_o 10._o 23._o therefore_o by_o the_o nine_o case_n of_o right_o angle_v spherical_a triangle_n as_o the_o ct_n af._n 10._o 23._o 10.7435974_o be_v to_o radius_fw-la 10.0000000_o so_o be_v c_n daf_a 22._o 525._o 19.9623154_o to_o it_o ad_fw-la 9_o 39_o 9.2187180_o to_o which_o add_v the_o equinoctial_a degree_n answer_v to_o one_o hour_n viz._n 15._o the_o sum_n be_v 24.39_o the_o right_a ascension_n of_o the_o mid_a heaven_n hence_o to_o find_v the_o point_n culminate_v in_o the_o right_o angle_a spherical_a triangle_n adf_n in_o fig._n 1._o we_o have_v give_v ad_fw-la 24._o 39_o and_o daf_a 23._o 525_o to_o find_v af._n therefore_o by_o the_o five_o case_n of_o right_o angle_v spherical_a triangle_n as_o it_o ad_fw-la 24._o 39_o 10.6564908_o be_v to_o radius_fw-la 10.0000000_o so_o be_v c_n daf_a 23._o 525._o 9.9623154_o to_o ct_n of_o 26._o 31._o 10.3058246_o therefore_o the_o point_n culminate_v be_v ♈_o 26._o 31._o to_o find_v the_o altitude_n thereof_o above_o the_o horizon_n we_o have_v give_v in_o the_o same_o triangle_n daf_n 23._o 525._o and_o of_o 26._o 31._o to_o find_v df._n therefore_o by_o the_o ten_o case_n of_o right_o angle_v spherical_a triangle_n as_o radius_fw-la 10.0000000_o be_v to_o we_o of_o 26_o 31._o 9.6466268_o so_o be_v s_o daf_n 23._o 525._o 9.6011352_o to_o the_o s_o df_n 10._o 19_o 9.2477628_o which_o be_v the_o north_n declination_n of_o the_o point_n of_o the_o ecliptic_a culminate_v and_o be_v add_v to_o the_o height_n of_o the_o aequator_fw-la at_o london_n 38._o 47_o the_o sum_n be_v 48.66_o the_o altitude_n of_o the_o mid_a heaven_n as_o be_v require_v problem_n 14._o have_v the_o great_a obliquity_n of_o the_o ecliptic_a together_o with_o the_o distance_n of_o the_o point_n give_v from_o the_o equinoctial_a to_o find_v the_o meridian_n angle_n or_o intersection_n of_o the_o meridian_n with_o the_o ecliptic_a have_v draw_v the_o primitive_a circle_n hzrn_v in_o fig._n 5._o represent_v the_o meridian_n and_o the_o two_o diameter_n har_o and_o zan_fw-mi set_v off_o the_o height_n of_o the_o pole_n from_o r_o to_o p._n 51._o 53_o and_o from_o n_o to_o saint_n and_o draw_v the_o diameter_n pas_n for_o the_o axis_n of_o the_o world_n and_o ae_n aq_n for_o the_o aequator_fw-la this_o do_v the_o right_a ascension_n of_o the_o mid_a heaven_n be_v give_v as_o in_o the_o last_o problem_n 24.39_o with_o the_o point_n culminate_v ♈_o 26.31_o and_o the_o declination_n thereof_o 10.19_o if_o you_o set_v 10_o deg_n 19_o centesme_n from_o ae_n to_o f_o and_o e_o to_o x_o you_o may_v draw_v the_o diameter_n fax_fw-la and_o cad_z at_o right_a angle_n thereunto_o and_o because_o the_o imum_fw-la coeli_fw-la be_v direct_o opposite_a to_o the_o point_n culminate_v that_o be_v in_o ♎_o 26.31_o if_o you_o set_v 26.31_o from_o x_o to_o b_o a_o ruler_n lay_v from_o c_o to_o b_o will_v cut_v the_o diameter_n fx_n in_o g_z and_o then_o make_v xh_n z_o xb_v you_o have_v the_o three_o point_n b_o g_z h_z by_o which_o to_o draw_v that_o circle_n which_o will_v cut_v the_o aequator_fw-la ae_n aq_n in_fw-it ♎_o and_o so_o you_o have_v the_o three_o point_n x_o ♎_o f_o by_o which_o to_o describe_v the_o arch_n of_o the_o ecliptic_a ♈_o f_o ♎_o x._o and_o in_o the_o right_o angle_a spherical_a triangle_n ♈_o aef_n we_o have_v give_v the_o angle_n ae_n ♈_o f._n 23.525_o the_o sun_n be_v great_a declination_n and_o ♈_o f._n 26._o 31._o the_o point_n culminate_v to_o find_v the_o angle_n ♈_o f_o ae_n therefore_o by_o the_o seven_o case_n of_o right_o angle_v spherical_a triangle_n as_o the_o ct_n ae_n ♈_o f._n 23._o 525._o 10.3611802_o be_v to_o the_o radius_fw-la 10.0000000_o so_o be_v the_o c_n ♈_o f._n 26._o 31._o 9.9525062_o to_o the_o cot_n ♈_o fae_z 68_o 60._o 9.5913260_o which_o be_v the_o angle_n of_o the_o ecliptic_a with_o the_o meridian_n problem_n 15._o to_o find_v the_o angle_n orient_a or_o altitude_n of_o the_o nonagesime_fw-la degree_n of_o the_o ecliptic_a in_o fig._n 5._o the_o pole_n of_o the_o ecliptic_a ♈_o f_o ♎_o x_o be_v at_o m_o and_o so_o you_o have_v the_o three_o point_n z_o m_z n_o to_o draw_v the_o vertical_a circle_n z_o k_o n_o cut_v the_o ecliptic_a at_o right_a angle_n in_o the_o point_n a_o and_o then_o in_o the_o right_o angle_a spherical_a triangle_n f_o a_fw-fr z_o we_o have_v give_v fz_o 41._o 34_o the_o compliment_n of_o fh_n the_o altitude_n of_o the_o mid_a heaven_n and_o the_o angle_n a_fw-fr fz_fw-fr 68_o 68_o the_o angle_n of_o the_o ecliptic_a with_o the_o meridian_n to_o find_v z_o a._n therefore_o by_o the_o ten_o case_n of_o right_o angle_v spherical_a triangle_n as_o the_o radius_fw-la to_o the_o sine_fw-la of_o fz_n 41._o 34._o 9_o 819889â—_n so_o be_v the_o sine_fw-la of_o z_o f_o a._n 68_o 68_o 9.9691128_o to_o the_o sine_fw-la of_o z_o a._n 37._o 97._o 9.7891027_o who_o be_v compliment_n be_v ak_v the_o measure_n of_o the_o angle_n agk_v 52._o 03_o the_o angle_n of_o the_o ecliptic_a with_o the_o horizon_n or_o altitude_n of_o the_o nonagesime_fw-la degree_n problem_n 16._o to_o find_v the_o place_n of_o nonagosime_fw-la degree_n of_o the_o ecliptic_a in_o fig._n 5._o f_o represent_v the_o point_n of_o the_o ecliptic_a in_o the_o mid_a heaven_n which_o according_a to_o problem_n
be_v go_v about_o this_o revolution_n be_v call_v the_o cycle_n of_o the_o sun_n take_v its_o name_n from_o the_o sunday_n letter_n of_o which_o it_o show_v all_o the_o change_n that_o it_o can_v have_v by_o reason_n of_o the_o bissextile_a or_o leap-year_n to_o find_v which_o of_o the_o 28_o the_o present_a be_v add_v nine_o to_o the_o year_n of_o our_o lord_n because_o this_o circle_n be_v so_o far_o go_v about_o at_o the_o time_n of_o christ_n birth_n and_o divide_v the_o whole_a by_o 28_o what_o remain_v be_v the_o present_a year_n if_o nothing_o remain_v the_o cycle_n be_v out_o and_o that_o you_o must_v call_v the_o last_o year_n of_o the_o cycle_n or_o 28._o 9_o this_o intercalation_n of_o one_o day_n in_o four_o year_n do_v occasion_n the_o letter_n f_o to_o be_v twice_o repeat_v in_o february_n in_o which_o month_n the_o day_n be_v add_v that_o be_v the_o letter_n f_o be_v set_v to_o the_o 24_o and_o 25_o day_n of_o that_o month_n and_o in_o such_o a_o year_n s._n mathias_n 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of_o the_o situation_n and_o distance_n of_o the_o part_n but_o can_v be_v fit_v to_o represent_v the_o figure_n of_o the_o whole_a 4._o the_o description_n therefore_o of_o the_o whole_a by_o planisphere_n be_v much_o better_o because_o it_o represent_v the_o face_n of_o the_o earth_n upon_o a_o plain_a in_o its_o own_o proper_a spherical_a figure_n as_o upon_o the_o globe_n itself_o this_o description_n can_v well_o be_v contrive_v upon_o so_o few_o as_o one_o circle_n or_o more_o than_o two_o suppose_v then_o the_o globe_n to_o be_v divide_v into_o two_o equal_a part_n or_o hemisphere_n which_o can_v be_v do_v but_o by_o a_o great_a circle_n and_o therefore_o it_o must_v be_v do_v by_o the_o aequator_fw-la or_o meridian_n for_o the_o colour_n be_v all_o one_o with_o the_o meridian_n the_o horizon_n can_v fix_v and_o the_o zodiac_n have_v nothing_o to_o do_v here_o 5._o suppose_v then_o the_o globe_n to_o be_v flat_v upon_o the_o plain_a of_o the_o aequator_fw-la and_o you_o have_v the_o first_o way_n of_o projection_n divide_v the_o globe_n into_o the_o north_n and_o south_n hemisphere_n in_o this_o projection_n the_o pole_n be_v the_o centre_n the_o aequator_fw-la be_v the_o circumference_n divide_v into_o 360_o degree_n of_o longitude_n the_o paralle_n be_v whole_a circle_n the_o meridian_n be_v straight_o line_n the_o parallel_n be_v parallel_n indeed_o and_o the_o meridian_n equidistant_o concur_v and_o therefore_o all_o the_o degree_n be_v equal_a after_o this_o way_n of_o projection_n ptolemy_n describe_v that_o part_n of_o the_o habitable_a world_n which_o be_v discover_v to_o his_o time_n 6._o suppose_v the_o globe_n to_o be_v flat_v upon_o the_o plain_a of_o the_o meridian_n and_o you_o have_v the_o other_o way_n of_o projection_n the_o aequator_fw-la here_o be_v a_o straight_a line_n the_o great_a meridian_n a_o whole_a circle_n in_o this_o section_n the_o meridian_n do_v not_o equidistant_o concur_v the_o parallel_n be_v not_o parallel_n indeed_o and_o therefore_o the_o degree_n be_v all_o unequal_a however_o this_o latter_a way_n be_v that_o which_o be_v now_o most_o and_o indeed_o altogether_o in_o use_n 7._o particular_a map_n be_v but_o limb_n of_o the_o globe_n and_o therefore_o though_o they_o be_v draw_v asunder_o yet_o be_v they_o still_o to_o be_v do_v with_o that_o proportion_n as_o a_o remember_a eye_n may_v sudden_o acknowledge_v and_o join_v they_o to_o the_o whole_a body_n the_o projection_n be_v most_o common_o upon_o a_o parallelogram_n in_o which_o the_o latitude_n be_v to_o be_v express_v by_o paralle_n from_o north_n to_o south_n and_o the_o longtitude_n by_o meridian_n from_o west_n to_o east_n at_o 10_o or_o 15_o degree_n distance_n as_o you_o please_v and_o may_v be_v draw_v either_o by_o circle_n or_o right_a line_n but_o if_o they_o be_v right_a line_n the_o meridian_n be_v not_o to_o be_v draw_v parallel_n but_o incline_v and_o concur_v to_o show_v the_o nature_n of_o the_o whole_a whereof_o they_o be_v such_o part_n for_o the_o graduation_n the_o degree_n of_o longitude_n be_v most_o common_o divide_v upon_o the_o north_n and_o south_n side_n of_o the_o parallelogram_n the_o degree_n of_o latitude_n upon_o the_o east_n and_o west_n side_n or_o otherwise_o upon_o the_o most_o eastern_a or_o western_a meridian_n of_o the_o map_n within_o the_o square_n but_o it_o have_v seem_v good_a to_o some_o in_o these_o particular_a description_n to_o make_v no_o graduation_n or_o projection_n at_o all_o but_o to_o put_v the_o matter_n off_o to_o a_o scale_n of_o miles_n and_o leave_v the_o rest_n to_o be_v believe_v the_o difference_n of_o miles_n in_o several_a country_n be_v great_a but_o it_o will_v be_v enough_o to_o know_v that_o the_o italian_a and_o english_a be_v reckon_v for_o all_o one_o and_o four_o of_o these_o do_v make_v a_o german_a mile_n two_o a_o french_a league_n the_o swedish_n or_o danish_a mile_n consist_v of_o 5_o miles_n english_a and_o somewhat_o more_o sixty_o common_a english_a and_o italian_a mile_n answer_v to_o a_o degree_n of_o a_o great_a circle_n now_o as_o the_o mile_n of_o several_a country_n do_v very_o much_o differ_v so_o those_o of_o the_o same_o do_v not_o very_o much_o agree_v and_o therefore_o the_o scale_n be_v common_o write_v upon_o with_o magna_n mediocria_fw-la and_o parva_n to_o show_v the_o difference_n in_o some_o map_n you_o shall_v find_v the_o mile_n thus_o hidden_o set_v down_o and_o the_o meaning_n be_v that_o you_o shall_v measure_v the_o milliaria_fw-la magna_fw-la upon_o the_o lowermost_a line_n the_o parva_n upon_o the_o uppermost_a and_o the_o mediocria_fw-la upon_o the_o middlemost_a scala_fw-la milliarium_fw-la the_o use_n of_o the_o scale_n be_v for_o the_o measure_v the_o distance_n of_o place_n in_o the_o map_n by_o set_v one_o foot_n of_o your_o compass_n in_o the_o little_a circle_n represent_v one_o place_n and_o the_o other_o foot_n in_o the_o like_o little_a circle_n represent_v another_o the_o compass_n keep_v at_o that_o distance_n be_v apply_v to_o the_o scale_n will_v show_v the_o number_n of_o great_a or_o middle_a mile_n according_a as_o the_o inhabitant_n of_o those_o place_n be_v know_v to_o reckon_v soli_fw-la deo_fw-la gloria_fw-la a_o view_n of_o the_o more_o notable_a epochae_fw-la epochae_fw-la year_n of_o the_o julian_n period_n month_n the_o julian_n period_n 1_o jan._n 1_o creation_n of_o the_o world_n 765_o jan._n 1_o aera_fw-la of_o the_o olympiades_n 3938_o july_n 8_o the_o building_n of_o rome_n 4961_o ap._n 21_o epochae_n of_o nabonasser_n 3667_o feb._n 26_o the_o beginning_n of_o metons_n circle_n 4281_o june_n 26_o the_o beginning_n of_o the_o period_n of_o calippus_n 4384_o june_n 28_o the_o death_n of_o alexander_n the_o great_a 4390_o no._n 12_o aera_fw-la of_o the_o caldee_n 4403_o oct._n 15_o the_o aera_fw-la of_o dionysrus_fw-la 4429_o mar._n 25_o the_o beginning_n of_o the_o christian_a aera_fw-la fall_v in_o the_o 4713_o year_n of_o the_o julian_n 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2555_o 0_o 0_o 0_o 8_o 0_o 0_o 0_o  _fw-fr 2922_o 0_o 0_o 0_o  _fw-fr 8_o 0_o 0_o 0_o  _fw-fr 2920_o 0_o 0_o 0_o 9_o 0_o 0_o 0_o  _fw-fr 3287_o 2_o 5_o 0_o  _fw-fr 9_o 0_o 0_o 0_o  _fw-fr 3285_o 0_o 0_o 0_o 10_o 0_o 0_o 0_o  _fw-fr 3652_o 5_o 0_o 0_o  _fw-fr 10_o 0_o 0_o 0_o  _fw-fr 3650_o 0_o 0_o 0_o day_n in_o julian_n month_n day_n in_o egyptian_a month_n day_n in_o persian_a month_n common_a bissex_n thoth_v 30_o pharvadin_n 30_o january_n 31_o 30_o paophi_fw-la 60_o aripehast_a 60_o february_n 59_o 60_o athyr_n 90_o chortat_fw-la 90_o march_v 90_o 91_o chaeae_fw-la 120_o tirma_fw-la 120_o april_n 120_o 121_o tybi_n 150_o mertat_fw-la 150_o may_n 151_o 152_o michir_n 180_o sachriur_fw-fr 180_o june_n 181_o 182_o phamenoth_n 210_o