arch_n of_o fifteen_o degree_n beneath_o the_o horizontal_n line_n must_v be_v take_v in_o the_o describe_v periphery_n for_o the_o hour_n before_o six_o in_o the_o morning_n and_o six_o in_o the_o evening_n and_o the_o line_n of_o the_o shadow_n must_v be_v draw_v the_o perpendicular_a style_n must_v also_o be_v erect_v from_o the_o centre_n furthermore_o in_o the_o horizontal_n plain_a if_o that_o the_o plain_a of_o the_o scioterick_n be_v not_o yet_o erect_v the_o meridian_n line_n must_v be_v find_v and_o the_o line_n of_o the_o aequinoctial_a rise_n and_o set_v and_o so_o it_o must_v be_v place_v on_o or_o above_o this_o plain_a of_o the_o scioterick_n that_o the_o horizontal_n line_n of_o the_o scioterick_n may_v be_v parallel_n to_o this_o line_n of_o the_o rise_n and_o set_v so_o the_o shadow_n of_o the_o style_n shall_v show_v the_o beginning_n of_o the_o hour_n at_o every_o day_n of_o the_o year_n but_o because_o the_o sun_n only_o illustrate_v this_o one_o superficies_n of_o this_o plain_a half_a a_o year_n and_o the_o other_o another_o half_a year_n therefore_o in_o both_o the_o superficies_n a_o scioterick_n must_v be_v make_v after_o the_o appoint_a mode_n lay_v down_o before_o that_o on_o one_o side_n of_o it_o in_o the_o time_n of_o summer_n and_o spring_n in_o the_o other_o in_o the_o time_n of_o autumn_n the_o hour_n may_v be_v know_v by_o the_o benefit_n of_o the_o shadow_n the_o line_n of_o the_o circle_n which_o show_v the_o place_n of_o the_o sun_n in_o the_o ecliptic_a or_o the_o entrance_n of_o the_o sun_n into_o the_o twelve_o sign_n of_o the_o zodiac_n and_o which_o do_v represent_v the_o parallel_n which_o the_o sun_n describe_v in_o the_o heaven_n by_o his_o circumvolution_n may_v easy_o be_v draw_v on_o this_o aequinoctial_a scioterick_n for_o let_v a_o certain_a magnitude_n of_o the_o style_n be_v take_v and_o let_v it_o be_v accurate_o divide_v into_o ten_o part_n and_o one_o of_o thsee_v ten_o into_o ten_o other_o part_n that_o the_o whole_a line_n may_v be_v conceive_v to_o be_v cut_v into_o a_o hundred_o particle_n then_o from_o a_o table_n of_o declination_n let_v the_o declination_n of_o the_o sun_n be_v except_v the_o five_o the_o ten_o the_o fifteen_o the_o twenty_o the_o twenty_o five_o the_o thirty_o degree_n of_o aries_n or_o the_o first_o the_o fifteen_o degree_n of_o taurus_n the_o first_o the_o fifteen_o degree_n of_o taurus_n the_o first_o the_o fifteen_o degree_n of_o gemini_fw-la the_o first_o degree_n of_o cancer_n and_o let_v the_o tangent_n be_v take_v from_o the_o mathematical_a canon_n moreover_o from_o the_o centre_n of_o the_o horologe_n in_o the_o interval_n of_o the_o tangent_fw-la of_o compliment_n of_o the_o five_o degree_n of_o aries_n let_v the_o periphery_a of_o the_o circle_n be_v describe_v this_o will_v note_v the_o entrance_n of_o the_o sun_n into_o the_o five_o degree_n of_o aries_n and_o the_o twenty_o five_o of_o virgo_fw-la and_o the_o parallel_n of_o the_o sun_n for_o that_o day_n viz._n when_o the_o diurnal_a extremity_n of_o the_o shadow_n by_o its_o circumvolution_n shall_v fall_v on_o this_o describe_v periphery_n it_o shall_v be_v a_o sign_n that_o the_o sun_n be_v in_o the_o five_o degree_n of_o aries_n or_o the_o twenty_o five_o of_o virgo_fw-la after_o the_o same_o mode_n let_v the_o periphery_n be_v describe_v in_o the_o interval_n of_o the_o compliment_n of_o the_o ten_o and_o the_o twenty_o degree_n of_o aries_n the_o first_o and_o the_o fifteen_o of_o taurus_n the_o first_o and_o the_o fifteen_o of_o gemini_fw-la and_o the_o first_o degree_n of_o cancer_n those_o will_v show_v the_o parallel_n of_o the_o sun_n in_o those_o point_n and_o also_o in_o the_o point_n of_o the_o 20_o degree_n of_o virgo_fw-la the_o 10_o and_o the_o first_o of_o virgo_fw-la the_o 15_o of_o leo_n and_o the_o first_o of_o leo_n and_o the_o 15_o degree_n of_o cancer_n after_o the_o same_o mode_n on_o the_o other_o side_n of_o the_o scioterick_n let_v the_o periphery_n be_v describe_v for_o the_o parallel_n of_o the_o sun_n in_o the_o first_o degree_n of_o libra_n and_o the_o 25_o of_o pisces_fw-la in_o the_o 10_o of_o libra_n and_o the_o 20_o of_o pisces_fw-la in_o the_o 15_o of_o libra_n and_o the_o 15_o of_o pisces_fw-la in_o the_o first_o of_o scorpio_n and_o the_o first_o of_o pisces_fw-la in_o the_o 15_o of_o scorpio_n and_o the_o 15_o of_o aquarius_n and_o in_o the_o first_o degree_n of_o sagittarius_fw-la and_o the_o first_o of_o aquarius_n unto_o every_o one_o of_o these_o periphery_n the_o character_n of_o the_o sign_n of_o the_o zodiac_n must_v be_v ascribe_v proposition_n xxii_o to_o describe_v a_o horizontal_n scioterick_n or_o a_o horizontal_n plain_n describe_v a_o horizontal_n scioterick_n or_o horizontal_n plain_a describe_v by_o the_o globe_n let_v the_o pole_n and_o meridian_n be_v elevate_v for_o the_o latitude_n of_o the_o place_n which_o meridian_n be_v more_o conspicuous_a than_o the_o other_o line_n in_o the_o superficies_n both_o for_o colour_n and_o magnitude_n let_v it_o be_v bring_v under_o the_o brazen_a meridian_n let_v the_o index_n be_v place_v at_o the_o hour_n of_o twelve_o let_v the_o globe_n be_v turn_v round_o until_o the_o index_n show_v the_o hour_n one_o or_o eleven_o or_o until_o 15_o degree_n of_o the_o aequator_fw-la do_v pass_v the_o brazen_a meridian_n in_o this_o situation_n of_o the_o globe_n let_v the_o degree_n intercept_v between_o the_o brazen_a meridian_n and_o the_o meridian_n of_o the_o globe_n be_v number_v on_o the_o wooden_a horizon_n and_o let_v this_o hour_n be_v note_v for_o the_o hour_n of_o one_o after_o noon_n and_o eleven_o before_o noon_n then_o let_v the_o globe_n be_v turn_v again_o until_o the_o index_n show_v the_o hour_n 11_o or_o 10_o and_o let_v the_o degree_n intercept_v between_o those_o two_o meridian_n the_o brazen_a one_o and_o that_o assume_v be_v note_v for_o the_o 10_o or_o 11_o hour_n after_o the_o same_o manner_n let_v it_o be_v do_v for_o the_o hour_n 9_o and_o 3_o for_o 8_o and_o 4_o for_o 7_o and_o 5_o for_o 6_o and_o 6_o but_o we_o shall_v not_o want_v this_o hour_n for_o 5_o and_o 7_o for_o 4_o and_o 8_o for_o 3_o and_o 9_o these_o degree_n be_v thus_o note_v for_o every_o ascribe_v hour_n let_v the_o meridian_n line_n be_v find_v on_o the_o horizontal_n plain_a and_o for_o any_o point_n of_o this_o line_n let_v the_o periphery_a of_o the_o circle_n be_v describe_v as_o from_o a_o centre_n and_o let_v it_o be_v draw_v perpendicular_o from_o the_o centre_n to_o the_o same_o on_o either_o side_n this_o shall_v be_v the_o line_n of_o the_o shadow_n at_o the_o hour_n 6_o before_o noon_n and_o 6_o after_o noon_n the_o meridian_n line_n be_v the_o line_n of_o the_o shadow_n of_o the_o hour_n 12._o in_o the_o describe_v periphery_a let_v the_o arch_n before_o note_a be_v cut_v of_o begin_v from_o the_o meridian_n line_n towards_o the_o line_n of_o the_o hour_n 6_o before_z and_z after_o noon_n first_o the_o arch_n note_v for_o 11_o and_o 1_o then_o for_o the_o hour_n 10_o and_o 2_o for_o 9_o and_o 3_o for_o 8_o and_o 4_o etc._n etc._n the_o arch_n thus_o cut_v off_o let_v the_o line_n be_v draw_v from_o the_o centre_n to_o those_o bound_n these_o shall_v be_v the_o line_n of_o the_o shadow_n in_o the_o beginning_n and_o end_n of_o the_o other_o hour_n but_o the_o style_n must_v be_v so_o elevate_v from_o the_o centre_n of_o the_o horologe_n above_o the_o meridian_n line_n that_o the_o angle_n which_o it_o make_v with_o it_o may_v be_v equal_a to_o the_o latitude_n of_o the_o place_n or_o elevation_n of_o the_o pole_n but_o it_o be_v more_o commodious_a to_o make_v some_o triangle_n who_o angle_n at_o the_o basis_n be_v equal_a to_o the_o latitude_n of_o the_o place_n if_o the_o declination_n be_v make_v on_o paper_n let_v the_o line_n be_v draw_v from_o the_o centre_n which_o from_o the_o periphery_n may_v take_v a_o arch_n equal_a to_o the_o latitude_n of_o the_o place_n the_o numeration_n be_v from_o the_o meridian_n line_n and_o let_v the_o triangle_n be_v cut_v out_o to_o be_v place_v above_o the_o meridian_n line_n so_o the_o shadow_n will_v show_v the_o hour_n the_o make_n of_o this_o scioterick_n be_v easy_a without_o a_o globe_n proposition_n xxiii_o to_o describe_v a_o scioterick_n on_o a_o vertical_a plain_n which_o may_v direct_o regard_v the_o east_n and_o west_n aequinoctial_a what_o a_o scioterick_n what_o the_o make_n of_o this_o be_v perfect_v after_o the_o same_o mode_n which_o we_o use_v in_o the_o horizontal_n if_o that_o the_o pole_n be_v not_o elevate_v according_a to_o the_o latitude_n of_o the_o place_n but_o according_a to_o the_o compliment_n of_o it_o and_o then_o the_o style_n also_o be_v elevate_v above_o the_o meridian_n
in_o e_o b_o which_o show_v the_o 52_o deg_fw-la of_o the_o arch_n b_o c_o shall_v be_v the_o projecture_n of_o the_o arctic_a pole_n let_v the_o point_n in_o e_o d_o be_v note_v with_o the_o letter_n p_o which_o represent_v the_o 52_o deg_fw-la of_o the_o arch_n d_o c_o by_o account_v from_o c_o to_z d_o shall_v be_v the_o projecture_n of_o the_o intersection_n of_o the_o aequator_fw-la and_o the_o meridian_n of_o london_n let_v the_o letter_n qui_fw-fr be_v note_v and_o from_o that_o towards_o the_o letter_n p_o let_v the_o number_n of_o the_o degree_n 1_o 2_o 3_o etc._n etc._n be_v ascribe_v also_o from_o q_o towards_o d_o and_z from_z b_o towards_z p_o viz._n 52_o 53_o 54_o 55_o etc._n etc._n then_o the_o point_n be_v take_v from_o p_o of_o the_o equal_a degree_n viz._n 99_o and_o 99_o also_o 88_o and_o 88_o let_v these_o be_v describe_v about_o these_o part_n as_o the_o diameter_n of_o the_o periphery_n of_o the_o circle_n which_o shall_v represent_v the_o parallel_n or_o circle_n of_o latitude_n and_o the_o tropic_n and_o polary_a circle_n with_o the_o aequator_fw-la meridian_n for_o the_o describe_v the_o meridian_n to_o describe_v the_o meridian_n first_o let_v a_o periphery_n be_v describe_v through_o the_o point_v a_o p_o c_o that_o shall_v show_v the_o meridian_n which_o be_v 90_o degree_n absent_a from_o london_n his_o centre_n shall_v be_v m_n in_o b_o d_o protract_v into_o the_o point_n n_n which_o show_v the_o antarctick_a pole_n let_v p_o n_n the_o diameter_n be_v draw_v through_o m_n parallel_n to_o a_o c_o which_o be_v f_o h_n protract_v from_o both_o part_n in_o k_o l._n moreover_o let_v the_o circle_n p_o h_o n_o f_o be_v divide_v into_o 360_o deg_fw-la and_o right_a line_n from_o the_o point_n p_o to_o every_o deg_fw-la or_o only_o by_o application_n of_o the_o rule_n which_o shall_v cut_v the_o line_n k_o f_o h_o l._n the_o circle_n must_v be_v describe_v through_o every_o point_n of_o the_o section_n and_o both_o the_o pole_n p_o n_o as_o through_o three_o give_v point_n which_o shall_v represent_v all_o the_o meridian_n the_o centre_n of_o the_o arch_n to_o be_v describe_v be_v seat_v in_o the_o same_o k_o l_o viz._n those_o which_o be_v find_v by_o the_o former_a section_n but_o to_o be_v take_v with_o this_o condition_n that_o the_o most_o remote_a centre_n at_o l_o be_v choose_v for_o the_o near_a meridian_n from_o b_o d_o n_o towards_o a_o and_o for_o the_o second_o the_o second_o from_o this_o the_o circle_n of_o the_o latitude_n and_o the_o meridian_n be_v thus_o describe_v it_o be_v easy_a to_o inscribe_v the_o place_n of_o the_o earth_n on_o a_o map_n and_o the_o situation_n of_o they_o all_o to_o london_n will_v be_v conspicuous_a moreover_o to_o affix_v the_o rule_n to_o the_o place_n of_o london_n the_o same_o part_n shall_v be_v bring_v in_o into_z which_z e_z b_o be_v divide_v and_o the_o number_n of_o degree_n must_v be_v ascribe_v so_o the_o rule_n be_v bring_v round_o unto_o every_o place_n we_o shall_v present_o know_v both_o how_o great_a a_o interval_n they_o lie_v from_o amsterdam_n and_o in_o what_o quater_fw-la they_o lie_v in_o respect_n of_o it_o now_o how_o by_o the_o benefit_n of_o the_o globe_n such_o a_o map_n shall_v be_v make_v we_o shall_v show_v in_o the_o four_o mode_n of_o particular_a map_n the_o first_o mode_n of_o geographical_a particular_a map_n we_o have_v speak_v of_o the_o make_n of_o general_n or_o universal_a map_n now_o it_o be_v require_v that_o we_o shall_v teach_v the_o composition_n of_o particular_a or_o special_a map_n america_n of_o particular_a or_o special_a map_n as_o of_o asia_n africa_n europe_n america_n the_o part_n therefore_o of_o the_o earth_n which_o we_o will_v represent_v on_o the_o map_n be_v either_o great_a or_o small_a it_o great_a as_o asia_n africa_n europe_n america_n it_o will_v be_v necessary_a to_o institute_v a_o declination_n according_a to_o the_o modes_n explain_v for_o general_a map_n but_o in_o divers_a part_n sundey_a way_n be_v more_o commodious_a africa_n and_o america_n because_o the_o aequator_fw-la pass_v through_o they_o be_v not_o commodious_o exhibit_v by_o the_o first_o mode_n but_o most_o apt_o by_o the_o second_o the_o eye_n be_v place_v in_o the_o plain_a of_o the_o aequator_fw-la above_o the_o middle_a meridian_n between_o the_o extreme_n which_o shut_v up_o africa_n or_o america_n therefore_o in_o these_o map_n the_o aequator_fw-la be_v a_o right_a line_n but_o the_o parallel_n and_o the_o meridian_n be_v the_o arch_n of_o the_o circle_n but_o to_o represent_v asia_n and_o europe_n the_o first_o and_o six_o mode_n be_v more_o commodious_a but_o for_o the_o polary_a land_n or_o frigid_a zone_n we_o have_v say_v that_o the_o first_o mode_n be_v most_o apt_a in_o the_o explication_n of_o the_o same_o first_o therefore_o a_o straight_a line_n must_v be_v draw_v upon_o the_o plain_a for_o the_o meridian_n of_o the_o place_n unto_o which_o we_o will_v have_v the_o eye_n hang_v over_o and_o that_o must_v be_v divide_v into_o degree_n according_a to_o the_o method_n explain_v in_o the_o precede_a modes_n and_o which_o shall_v be_v degree_n of_o latitude_n the_o number_n of_o which_o must_v be_v ascribe_v then_o from_o the_o table_n must_v be_v extract_v the_o latitude_n of_o both_o parallel_n viz._n that_o which_o terminate_v the_o region_n from_o both_o side_n which_o represent_v the_o pole_n the_o degree_n of_o the_o latitude_n of_o these_o must_v be_v note_v in_o the_o right_a line_n or_o the_o meridian_n of_o the_o eye_n and_o through_o those_o point_n straight_o perpendicular_a line_n must_v be_v draw_v which_o enclose_v the_o map_n towards_o the_o northern_a and_o southern_a quarter_n then_o parallel_n and_o meridian_n must_v be_v draw_v at_o every_o degree_n and_o the_o place_n inscribe_v until_o the_o map_n be_v perfect_v the_o second_o mode_n of_o describe_v particular_a map_n map_n the_o second_o mode_n of_o particular_a map_n artificer_n be_v wont_a to_o use_v another_o method_n in_o region_n not_o so_o large_a but_o only_o moderate_a or_o small_a first_o a_o tranverse_n line_n be_v draw_v in_o the_o extremity_n of_o the_o table_n for_o the_o circle_n of_o latitude_n in_o which_o the_o end_n of_o the_o region_n respect_v the_o aequator_fw-la be_v to_o be_v draw_v in_o that_o so_o many_o part_n be_v take_v equal_o through_o how_o many_o deg_fw-la of_o longitude_n that_o region_n be_v extend_v from_o that_o part_n then_o from_o the_o middle_n of_o this_o line_n a_o perpendicular_a be_v draw_v which_o have_v so_o many_o part_n as_o there_o be_v deg_fw-la of_o longitude_n between_o the_o bound_n of_o that_o region_n towards_o the_o aequator_fw-la and_o the_o pole_n but_o how_o great_a these_o part_n shall_v be_v be_v know_v from_o the_o proportion_n of_o the_o deg_fw-la of_o the_o first_o circle_n which_o be_v great_a to_o the_o deg_fw-la of_o parallel_n which_o be_v represent_v from_o the_o low_a transverse_a line_n through_o the_o term_n of_o this_o perpendicular_a another_o perpendicular_a or_o parallel_v to_o the_o inferior_a line_n be_v draw_v in_o which_o so_o many_o deg_fw-la of_o longitude_n must_v be_v take_v as_o be_v in_o the_o low_a line_n and_o equal_a to_o they_o of_o the_o low_a line_n if_o these_o latitude_n be_v not_o much_o distant_a from_o the_o aequator_fw-la or_o mutual_a from_o themselves_o but_o if_o the_o distance_n from_o the_o aequator_fw-la be_v great_a or_o if_o the_o excess_n of_o the_o ultimate_a latitude_n of_o the_o region_n be_v great_a above_o that_o which_o be_v more_o near_o the_o aequator_fw-la the_o part_n to_o be_v take_v in_o the_o transverse_a line_n shall_v not_o be_v equal_a to_o the_o part_n of_o the_o inferior_a line_n but_o they_o ought_v to_o be_v lesser_a according_a to_o proportion_n which_o the_o degree_n of_o this_o more_o remote_a parallel_n have_v to_o the_o degree_n of_o the_o inferior_a line_n which_o proportion_n be_v know_v from_o the_o table_n we_o have_v place_v in_o the_o four_o chapter_n iu._n see_v chap._n iu._n after_o the_o part_n be_v thus_o take_v for_o the_o deg_fw-la of_o longitude_n in_o the_o superior_a and_o inferior_a line_n the_o right_a line_n be_v to_o be_v draw_v through_o the_o beginning_n and_o end_n of_o the_o part_n of_o the_o same_o number_n which_o right_a line_n shall_v represent_v the_o meridian_n line_n then_o through_o every_o deg_fw-la of_o its_o perpendicular_a which_o we_o have_v order_v to_o be_v erect_v from_o the_o middle_a point_n of_o the_o inferior_a line_n line_n parallel_n to_o that_o low_a line_n must_v be_v draw_v through_o the_o beginning_n of_o every_o degree_n which_o shall_v show_v the_o parallel_n of_o latitude_n in_o the_o last_o place_n place_n must_v be_v inscribe_v at_o the_o point_n in_o which_o the_o parallel_n of_o every_o
definition_n spring_n summer_n autumn_n and_o winter_n may_v be_v attribute_v to_o all_o place_n of_o the_o earth_n neither_o be_v it_o easy_a to_o find_v out_o any_o other_o mode_n of_o define_v they_o so_o that_o they_o may_v agree_v with_o all_o place_n now_o these_o definition_n be_v lay_v down_o let_v we_o come_v to_o the_o matter_n itself_o proposition_n iii_o the_o celestial_a summer_n of_o the_o place_n of_o the_o earth_n which_o lie_v between_o the_o tropic_a of_o cancer_n and_o the_o pole_n arctic_a begin_v with_o the_o entrance_n of_o the_o sun_n into_o the_o first_o degree_n of_o cancer_n viz._n the_o 21_o of_o june_n and_o end_n with_o the_o entrance_n of_o the_o sun_n into_o the_o first_o degree_n of_o libra_n viz._n the_o 21_o of_o september_n and_o that_o together_o at_o once_o in_o all_o those_o place_n so_o that_o autumn_n be_v in_o those_o place_n the_o sun_n go_v from_o the_o first_o of_o libra_n unto_o the_o first_o of_o capricorn_n the_o winter_n whilst_o the_o sun_n move_v from_o the_o first_o of_o capricorn_n to_o the_o first_o of_o aries_n the_o spring_n whilst_o the_o sun_n move_v from_o the_o first_o degree_n of_o aries_n unto_o the_o first_o of_o cancer_n year_n further_o concern_v the_o season_n of_o the_o year_n the_o truth_n of_o this_o proposition_n be_v easy_o show_v by_o the_o antecedent_n definition_n and_o may_v be_v demonstrate_v on_o the_o globe_n and_o in_o universal_a map_n for_o the_o sun_n come_v to_o the_o first_o degree_n of_o cancer_n have_v the_o least_o distance_n in_o the_o meridies_n from_o the_o vertex_n of_o every_o one_o of_o the_o place_n of_o the_o northern_a temperate_a and_o frigid_a zone_n after_o the_o same_o mode_n the_o sun_n in_o the_o first_o degree_n of_o libra_n have_v a_o moderate_a distance_n from_o those_o vertex_n in_o the_o first_o of_o capricorn_n a_o great_a in_o the_o first_o of_o aries_n a_o moderate_a and_o he_o ascend_v to_o a_o more_o great_a which_o be_v apparent_a both_o from_o the_o declination_n of_o the_o sun_n and_o from_o the_o globe_n and_o map_n therefore_o it_o be_v infer_v by_o the_o definition_n lay_v down_o before_o that_o the_o summer_n the_o winter_n and_o the_o spring_n of_o those_o place_n begin_z and_o end_v in_o those_o day_n we_o have_v speak_v of_o the_o summer_n of_o those_o place_n of_o the_o earth_n which_o lie_v between_o the_o tropic_a of_o capricorn_n and_o the_o antarctic_a pole_n or_o those_o of_o the_o southern_a zone_n temperate_a and_o frigid_a begin_v with_o the_o entrance_n of_o the_o sun_n into_o the_o first_o of_o capricorn_n viz._n 21_o of_o december_n and_o end_n with_o the_o entrance_n of_o the_o sun_n into_o the_o first_o of_o aries_n viz._n the_o 21_o of_o march._n the_o autumn_n of_o those_o place_n begin_v with_o the_o entrance_n of_o the_o sun_n into_o the_o first_o of_o aries_n and_o end_n with_o the_o entrance_n of_o the_o sun_n into_o the_o first_o of_o cancer_n viz._n the_o 21_o of_o june_n with_o this_o the_o winter_n of_o those_o place_n begin_v which_o end_v with_o the_o entrance_n of_o the_o sun_n into_o the_o first_o of_o libra_n viz._n 21_o of_o september_n and_o with_o this_o their_o spring_n begin_v and_o end_v with_o the_o entrance_n of_o the_o sun_n into_o the_o first_o of_o capricorn_n viz._n 21_o of_o december_n where_o the_o summer_n begin_v again_o these_o be_v show_v after_o the_o same_o mode_n by_o the_o definition_n deliver_v and_o by_o the_o globe_n or_o map_n by_o which_o we_o show_v the_o former_a because_o in_o the_o first_o degree_n of_o capricorn_n the_o sun_n have_v the_o least_o distance_n from_o the_o vertex_n of_o those_o place_n in_o the_o first_o of_o aries_n a_o moderate_a and_o descend_v to_o the_o less_o in_o the_o first_o of_o cancer_n the_o great_a in_o the_o first_o of_o libra_n a_o moderate_a and_o ascend_v to_o a_o great_a but_o the_o celestial_a summer_n spring_n autumn_n and_o winter_n of_o the_o place_n of_o the_o earth_n which_o lie_v in_o the_o torrid_a zone_n between_o the_o tropic_a of_o cancer_n and_o capricorn_n do_v not_o begin_v on_o one_o and_o the_o same_o day_n of_o the_o year_n but_o on_o divers_a day_n in_o every_o place_n of_o diverse_a parallel_n or_o of_o a_o diverse_a latitude_n of_o this_o zone_n now_o the_o place_n of_o the_o torrid_a zone_n be_v threefold_a viz._n the_o place_n of_o the_o aequator_fw-la the_o northern_a place_n of_o the_o torrid_a zone_n and_o the_o southern_a place_n of_o the_o torrid_a zone_n 1._o the_o place_n lie_v in_o the_o aequator_fw-la have_v this_o peculiar_a to_o they_o that_o they_o enjoy_v two_o summer_n two_o winter_n two_o spring_n season_n and_o two_o autumn_n and_o that_o in_o every_o year_n so_o that_o in_o half_a a_o year_n they_o have_v or_o aught_o to_o have_v those_o four_o season_n according_a to_o our_o definition_n and_o the_o celestial_a law_n they_o have_v again_o the_o same_o four_o season_n from_o the_o 21_o of_o september_n to_o the_o 21_o of_o march_n september_n half_n a_o year_n be_v from_o the_o 21_o of_o march_n to_o the_o 21_o of_o september_n viz._n one_o summer_n whilst_o the_o sun_n move_v from_o the_o first_o degree_n of_o aries_n to_o the_o second_o of_o taurus_n from_o the_o 21_o of_o march_n to_o the_o 22_o of_o april_n autumn_n whilst_o the_o sun_n move_v from_o the_o second_o degree_n of_o taurus_n to_o the_o first_o of_o cancer_n from_o the_o 22_o of_o april_n to_o the_o 21_o of_o june_n zodiac_n the_o motion_n of_o â—he_n sun_n in_o the_o 12_o sign_n of_o the_o zodiac_n winter_n whilst_o the_o sun_n move_v from_o the_o first_o degree_n of_o cancer_n to_o the_o second_o of_o leo_n from_o the_o 21_o of_o june_n to_o the_o 19_o of_o august_n the_o spring_n whilst_o the_o sun_n move_v from_o the_o 28_o degree_n of_o leo_n to_o the_o first_o of_o libra_n from_o the_o 19_o of_o august_n to_o the_o 21_o of_o september_n the_o other_o summer_n whilst_o the_o sun_n move_v from_o the_o first_o degree_n of_o libra_n to_o the_o second_o of_o scorpio_n from_o the_o 21_o of_o september_n to_o the_o 22_o of_o october_n the_o other_o autumn_n whilst_o the_o sun_n move_v from_o the_o second_o degree_n of_o scorpio_n to_o the_o first_o of_o capricorn_n from_o the_o 22_o of_o october_n to_o the_o 21_o of_o december_n the_o other_o winter_n whilst_o the_o sun_n move_v from_o the_o first_o degree_n of_o capricorn_n to_o the_o 28_o of_o aquarius_n from_o the_o 21_o of_o december_n to_o the_o 19_o of_o february_n the_o other_o spring_n be_v whilst_o the_o sun_n do_v move_v from_o the_o 28_o degree_n of_o aquarius_n to_o the_o first_o of_o aries_n from_o the_o 19_o of_o february_n to_o the_o 21_o of_o march._n all_o these_o be_v easy_o demonstrate_v from_o the_o definition_n lay_v down_o because_o that_o the_o sun_n in_o the_o first_o degree_n of_o aries_n and_o in_o the_o first_o of_o libra_n have_v the_o less_o distance_n in_o the_o meridies_n from_o the_o vertices_fw-la of_o the_o place_n lie_v under_o the_o aequator_fw-la for_o it_o have_v none_o because_o it_o be_v vertical_a unto_o they_o therefore_o then_o do_v the_o summer_n begin_v then_o in_o the_o second_o degree_n of_o taurus_n and_o the_o second_o of_o scorpio_n where_o the_o declination_n of_o the_o sun_n be_v 11_o degree_n 45_o minute_n it_o acquire_v a_o mean_a distance_n depart_v to_o a_o great_a then_o therefore_o the_o autumn_n do_v begin_v moreover_o when_o he_o be_v in_o the_o first_o degree_n of_o cancer_n and_o the_o first_o of_o capricorn_n he_o have_v a_o great_a distance_n from_o the_o place_n of_o the_o aequator_fw-la therefore_o then_o do_v the_o winter_n begin_v final_o on_o the_o 28_o degree_n of_o leo_n and_o the_o 28_o of_o aquarius_n he_o receive_v a_o moderate_a distance_n from_o the_o place_n of_o the_o aequator_fw-la 10_o degree_n 45_o minute_n ascend_v towards_o the_o least_o and_o therefore_o then_o do_v the_o spring_n season_v begin_v these_o be_v understand_v more_o perspicuous_o from_o the_o globe_n therefore_o here_o these_o season_n may_v be_v distinguish_v thus_o according_a to_o the_o celestial_a law_n notwithstanding_o the_o terrestrial_a season_n be_v in_o many_o place_n of_o the_o aequator_fw-la otherwise_o observe_v as_o we_o shall_v show_v in_o the_o follow_a proposition_n 2._o all_o the_o place_n of_o the_o earth_n lie_v under_o the_o torrid_a northern_a zone_n have_v the_o end_n of_o the_o autumn_n and_o the_o beginning_n of_o the_o winter_n together_o both_o at_o one_o time_n viz._n the_o 21_o of_o december_n but_o they_o have_v not_o together_o the_o beginning_n and_o end_n of_o the_o summer_n and_o spring_n as_o also_o the_o autumn_n but_o different_a place_n have_v they_o in_o several_a day_n season_n other_o observation_n about_o the_o begin_n of_o the_o season_n for_o the_o end_n of_o the_o
perioeci_n and_o what_o hour_n they_o do_v not_o see_v it_o together_o let_v the_o place_n of_o the_o stay_n of_o the_o sun_n above_o the_o horizon_n of_o the_o place_n give_v be_v find_v at_o the_o day_n give_v and_o let_v the_o time_n of_o his_o stay_n beneath_o the_o horizon_n that_o be_v the_o quantity_n of_o the_o day_n and_o the_o night_n be_v find_v half_o the_o difference_n between_o the_o quantity_n of_o the_o day_n and_o the_o night_n will_v show_v the_o hour_n or_o part_n of_o the_o hour_n in_o which_o the_o sun_n first_o rise_v to_o one_o place_n before_o he_o set_v to_o another_o and_o set_v late_a also_o to_o that_o place_n than_o he_o arise_v to_o this_o chap._n xxix_o of_o the_o computation_n of_o time_n in_o the_o divers_a place_n of_o the_o earth_n proposition_n i._o the_o hour_n of_o one_o place_n be_v give_v in_o the_o globe_n to_o find_v the_o hour_n of_o another_o place_n give_v out_o by_o the_o globe_n the_o hour_n of_o the_o place_n be_v find_v out_o let_v the_o place_n who_o hour_n be_v give_v be_v bring_v to_o the_o brazen_a meridian_n the_o index_n to_o that_o hour_n of_o the_o horary_a cycle_n such_o as_o be_v give_v let_v the_o globe_n be_v turn_v round_o until_o the_o other_o give_v place_n come_v under_o the_o meridian_n the_o index_n in_o that_o situation_n of_o the_o globe_n will_v show_v the_o hour_n demand_v of_o this_o other_o place_n proposition_n ii_o the_o hour_n of_o our_o place_n be_v give_v or_o of_o some_o other_o place_n in_o the_o globe_n to_o exhibit_v on_o the_o globe_n all_o those_o place_n in_o which_o at_o that_o hour_n the_o meridies_n be_v also_o those_o in_o which_o it_o be_v midnight_n also_o those_o in_o which_o be_v what_o hour_n we_o please_v the_o problem_n shall_v be_v propound_v concern_v the_o earth_n if_o we_o will_v act_v scientifical_o for_o it_o be_v a_o affection_n of_o the_o earth_n understand_v the_o same_o concern_v many_o other_o follow_v problem_n let_v the_o place_n give_v be_v bring_v to_o the_o meridian_n the_o index_n to_o the_o give_v hour_n of_o the_o horary_a cycle_n let_v the_o globe_n be_v turn_v round_o until_o the_o index_n show_v the_o 12_o hour_n of_o the_o meridies_n so_o the_o place_n which_o be_v discover_v to_o be_v subject_a to_o the_o superior_a semicircle_n of_o the_o meridian_n from_o the_o elevate_a pole_n to_o the_o pole_n depress_v be_v those_o which_o have_v the_o meridies_n at_o the_o time_n give_v but_o if_o the_o globe_n be_v turn_v round_o that_o the_o index_n may_v show_v the_o 12_o inferior_a hour_n the_o place_n which_o be_v discover_v to_o be_v subject_a to_o the_o same_o semicircle_n of_o the_o meridian_n be_v those_o in_o which_o the_o midnight_n than_o shall_v be_v if_o we_o desire_v place_n in_o which_o be_v any_o hour_n let_v the_o globe_n be_v turn_v until_o the_o index_n show_v that_o hour_n if_o the_o place_n subject_n to_o the_o semicircle_n of_o the_o meridian_n be_v those_o that_o be_v seek_v proposition_n iii_o the_o altitude_n of_o the_o sun_n be_v give_v the_o day_n of_o the_o year_n and_o the_o latitude_n of_o the_o place_n to_o find_v the_o hour_n at_o the_o time_n of_o that_o altitude_n day_n rule_n for_o the_o find_v the_o hour_n of_o the_o day_n let_v the_o pole_n be_v elevate_v for_o the_o give_v latitude_n of_o the_o place_n from_o the_o give_v day_n let_v the_o place_n of_o the_o sun_n be_v find_v in_o the_o ecliptic_a and_o let_v that_o be_v note_v in_o the_o ecliptic_a of_o the_o globe_n and_o bring_v to_o the_o meridian_n then_o let_v the_o quadrant_n be_v apply_v to_o the_o vertex_fw-la and_o let_v the_o degree_n of_o the_o give_v altitude_n be_v note_v in_o it_o and_o let_v the_o index_n be_v place_v at_o the_o 12_o hour_n of_o the_o horary_a cycle_n then_o let_v the_o globe_n and_o the_o quadrant_n be_v move_v until_o the_o note_a place_n of_o the_o sun_n agree_v with_o the_o note_a point_n of_o the_o quadrant_n in_o that_o situation_n the_o sun_n will_v show_v the_o hour_n demand_v proposition_n iu._n a_o quarter_n be_v give_v in_o which_o the_o sun_n be_v behold_v sometime_o of_o the_o day_n give_v and_o the_o latitude_n of_o a_o place_n be_v give_v to_o find_v the_o hour_n of_o the_o day_n compass_n mariner_n observe_v the_o quarter_n of_o the_o sun_n on_o the_o compass_n let_v all_o be_v do_v as_o in_o the_o precede_a proposition_n that_o the_o quadrant_n may_v be_v apply_v to_o the_o vertex_fw-la let_v his_o end_n or_o extremity_n be_v bring_v to_o that_o quarter_n of_o the_o horizon_n which_o be_v observe_v and_o let_v the_o globe_n be_v turn_v round_o until_o that_o point_n of_o the_o sun_n come_v to_o the_o quadrant_n in_o this_o situation_n the_o index_n will_v show_v the_o hour_n of_o the_o day_n proposition_n v._o the_o sun_n shine_v by_o the_o benefit_n of_o the_o globe_n to_o know_v the_o hour_n of_o the_o place_n give_v or_o the_o latitude_n thereof_o which_o be_v give_v let_v the_o pole_n be_v elevate_v for_o the_o give_v latitude_n of_o the_o place_n and_o let_v the_o globe_n be_v place_v at_o the_o four_o quarter_n of_o the_o world_n then_o let_v a_o needle_n be_v fix_v perpendicular_o at_o the_o place_n of_o the_o sun_n in_o the_o ecliptic_a or_o which_o be_v better_o let_v the_o spherical_a gnomon_n be_v apply_v to_o the_o ecliptic_a so_o that_o the_o apex_n of_o the_o gnomon_n fix_v on_o the_o place_n of_o the_o sun_n and_o so_o let_v it_o be_v bring_v to_o the_o meridian_n and_o the_o index_n to_o the_o 12_o hour_n let_v the_o globe_n be_v turn_v until_o the_o needle_n make_v no_o shadow_n on_o the_o globe_n in_o this_o situation_n the_o index_n will_v show_v the_o demand_v hour_n proposition_n vi_o a_o hour_n of_o our_o numeration_n be_v give_v to_o find_v what_o hour_n it_o be_v from_o the_o rise_n of_o the_o sun_n that_o be_v the_o babylonish_n or_o norimbergian_a hour_n in_o time_n past_o the_o babylonian_n and_o now_o the_o inhabitant_n of_o norimberg_n and_o some_o other_o people_n reckon_v 24_o hour_n from_o one_o rise_v of_o the_o sun_n to_o the_o rise_n of_o the_o sun_n the_o next_o day_n let_v the_o pole_n be_v elevate_v from_o the_o latitude_n of_o the_o place_n give_v and_o the_o place_n of_o the_o sun_n be_v find_v from_o the_o day_n give_v let_v it_o be_v bring_v to_o the_o meridian_n the_o index_n to_o the_o 12_o hour_n of_o the_o horary_a cycle_n let_v the_o globe_n be_v turn_v until_o the_o index_n show_v the_o hour_n give_v then_o the_o globe_n remain_v immovable_a let_v the_o index_n be_v reduce_v to_o 12_o which_o be_v do_v let_v the_o globe_n be_v turn_v from_o the_o set_n to_o the_o rise_n until_o the_o place_n of_o the_o sun_n appear_v in_o the_o oriental_a horizon_n and_o in_o the_o horary_a cycle_n let_v the_o hour_n be_v reckon_v from_o 12_o towards_o the_o east_n or_o rise_n even_o to_o the_o index_n for_o these_o be_v the_o babylonish_n or_o norimberg_n hour_n seek_v for_o proposition_n vii_o on_o the_o contrary_a the_o hour_n be_v give_v from_o the_o babylonish_n rise_v to_o find_v out_o the_o hour_n of_o our_o numeration_n which_o be_v from_o midnight_n or_o midnoon_n let_v the_o pole_n be_v elevate_v for_o the_o latitude_n of_o the_o place_n give_v let_v the_o place_n of_o the_o sun_n be_v note_v in_o the_o ecliptic_a and_o bring_v to_o the_o oriental_a horizon_n the_o index_n to_o the_o 12_o hour_n let_v the_o globe_n be_v turn_v towards_o the_o west_n until_o the_o index_n show_v the_o hour_n give_v on_o the_o cycle_n from_o the_o east_n which_o be_v do_v let_v the_o index_n be_v reduce_v to_o the_o 12_o hour_n and_o then_o let_v the_o globe_n again_o be_v move_v until_o the_o place_n of_o the_o sun_n be_v bring_v back_o to_o the_o semicircle_n of_o the_o meridian_n which_o be_v next_o pass_v through_o and_o let_v the_o hour_n be_v number_v from_o 12_o to_o the_o index_n towards_o that_o quarter_n unto_o which_o the_o motion_n of_o the_o globe_n be_v make_v so_o shall_v be_v find_v the_o hour_n of_o our_o number_n from_o the_o meridies_n or_o midnight_n proposition_n viii_o a_o hour_n of_o our_o reckon_a be_v give_v to_o find_v what_o hour_n it_o be_v from_o the_o precede_a set_v of_o the_o sun_n that_o be_v the_o italian_a hour_n hour_n of_o italian_a hour_n at_o this_o day_n in_o many_o place_n of_o italy_n and_o in_o time_n pass_v in_o greece_n they_o number_v 24_o hour_n from_o one_o set_v of_o the_o sun_n to_o the_o follow_v or_o next_o set_v to_o find_v out_o which_o we_o must_v thus_o do_v from_o the_o hour_n of_o our_o numeration_n let_v the_o pole_n be_v elevate_v for_o the_o latitude_n of_o the_o place_n give_v let_v the_o place_n of_o the_o sun_n in_o the_o
or_o heaven_n which_o the_o meridian_n line_n be_v find_v be_v easy_a to_o do_v by_o the_o mariner_n compass_n or_o the_o magnetic_a needle_n the_o globe_n be_v thus_o place_v at_o every_o moment_n of_o the_o day_n when_o the_o sun_n shine_v on_o the_o globe_n may_v be_v see_v the_o part_n of_o the_o earth_n illuminate_v and_o the_o part_n not_o illuminate_v those_o place_n which_o lie_v in_o the_o middle_a semicircle_n of_o the_o part_n illuminate_v be_v those_o which_o will_v have_v the_o meridies_n at_o that_o moment_n of_o time_n to_o those_o which_o be_v seat_v in_o the_o oriental_a semicircle_n divide_v the_o illuminate_v part_n from_o the_o part_n not_o illuminate_v the_o sun_n set_v but_o to_o those_o which_o be_v in_o the_o occidental_a semicircle_n separate_v the_o illuminate_v part_n from_o the_o part_n not_o illuminate_v the_o sun_n rise_v to_o find_v out_o the_o place_n of_o the_o sun_n in_o the_o ecliptic_a let_v the_o needle_n or_o spherical_a gromon_n be_v move_v hither_o and_o thither_o perpendicular_o about_o the_o middle_n of_o the_o part_n illuminate_v until_o it_o make_v no_o shadow_n and_o let_v the_o point_n in_o the_o globe_n be_v note_v 〈◊〉_d for_o this_o be_v bring_v to_o the_o meridian_n here_o will_v show_v the_o declination_n of_o the_o ecliptic_a point_n in_o which_o the_o sun_n be_v at_o the_o time_n of_o the_o observation_n whence_o according_a to_o the_o condition_n of_o the_o time_n to_o wit_n spring_n summer_n autumn_n or_o winter_n the_o place_n of_o the_o sun_n shall_v be_v know_v and_o thence_o the_o day_n of_o the_o year_n also_o the_o place_n in_o the_o globe_n unto_o which_o the_o needle_n be_v affix_v give_v no_o shadow_n be_v that_o to_o which_o the_o sun_n be_v vertical_a at_o that_o moment_n of_o time_n and_o the_o parallel_n pass_v through_o this_o place_n will_v exhibit_v all_o the_o place_n in_o which_o the_o sun_n will_v be_v vertical_a on_o that_o day_n moreover_o to_o find_v the_o hour_n of_o the_o place_n in_o which_o the_o globe_n be_v so_o place_v or_o hang_v let_v that_o place_n be_v bring_v to_o the_o meridian_n to_o which_o the_o sun_n be_v vertical_a the_o index_n to_o the_o 12_o hour_n of_o the_o horary_a circle_n and_o let_v the_o globe_n be_v turn_v round_o until_o our_o place_n or_o that_o in_o which_o the_o globe_n be_v seat_v do_v come_v to_o the_o meridian_n the_o index_n will_v show_v the_o hour_n but_o because_o the_o globe_n can_v be_v turn_v round_o when_o it_o be_v affix_v by_o the_o iron_n style_n to_o the_o horizontal_n plain_a therefore_o it_o will_v be_v convenient_a that_o the_o quadrant_n be_v tie_v to_o the_o pole_n or_o part_n of_o the_o circle_n of_o the_o periphery_n 113_o â—_o â—_o for_o here_o the_o arch_n be_v bring_v to_o the_o place_n of_o the_o needle_n will_v show_v the_o declination_n of_o the_o sun_n from_o the_o aequator_fw-la whence_o the_o place_n of_o the_o sun_n and_o the_o day_n of_o the_o year_n shall_v be_v find_v the_o same_o arch_n will_v show_v the_o degree_n in_o the_o aequator_fw-la from_o whence_o if_o that_o the_o degree_n be_v number_v to_o the_o brazen_a meridian_n and_o these_o degree_n be_v change_v into_o hour_n or_o part_n of_o hour_n hour_n fifteen_o degree_n make_v a_o hour_n you_o shall_v have_v the_o hour_n of_o the_o place_n if_o so_o be_v that_o the_o sun_n be_v between_o the_o occident_n and_o the_o brazen_a meridian_n that_o be_v of_o our_o place_n but_o if_o that_o it_o be_v between_o the_o east_n and_o our_o meridian_n the_o hour_n find_v out_o must_v be_v subtract_v from_o 12_o and_o the_o remain_a number_n will_v show_v the_o hour_n from_o midnight_n if_o that_o such_o a_o brazen_a arch_n be_v adjoin_v to_o the_o pole_n of_o the_o globe_n as_o i_o have_v describe_v 113_o ½_n degree_n it_o may_v be_v bore_v through_o from_o the_o end_n even_o to_o 47_o degree_n that_o be_v from_o the_o departure_n of_o the_o sun_n from_o the_o aequator_fw-la and_o a_o turn_a plate_n be_v insert_v in_o it_o which_o may_v bear_v the_o perpendicular_a style_n and_o so_o there_o will_v neither_o be_v need_n of_o a_o needle_n or_o of_o a_o spherical_a gnomon_n and_o the_o operation_n will_v be_v less_o obnoxious_a to_o error_n proposition_n ii_o the_o terrestrial_a globe_n be_v â—o_o place_v as_o in_o the_o former_a proposition_n be_v declare_v it_o will_v also_o show_v when_o the_o moon_n shine_v to_o what_o people_n at_o any_o moment_n of_o time_n in_o which_o it_o be_v above_o our_o horizon_n it_o be_v conspicuous_a to_o who_o it_o arise_v to_o who_o it_o set_v and_o to_o who_o it_o be_v vertical_a these_o be_v all_o manifest_a from_o the_o precede_a proposition_n 1._o see_v proposit_n 1._o proposition_n iii_o by_o how_o much_o the_o place_n of_o the_o earth_n be_v remote_a from_o the_o parallel_n of_o the_o sun_n on_o any_o day_n by_o so_o much_o the_o sun_n be_v elevate_v to_o a_o lesser_a altitude_n in_o the_o same_o hour_n above_o their_o horizon_n let_v the_o place_n in_o the_o same_o meridian_n be_v take_v in_o the_o globe_n for_o these_o do_v reckon_v all_o the_o same_o hour_n and_o that_o at_o once_o then_o let_v a_o parallel_n be_v describe_v for_o any_o assume_v day_n and_o it_o will_v be_v manifest_a that_o any_o point_n of_o this_o parallel_n be_v far_a distant_a from_o the_o more_o remote_a place_n than_o from_o the_o place_n more_o near_o the_o sun_n therefore_o be_v above_o the_o point_n of_o this_o parallel_n will_v be_v far_a distant_a from_o the_o vertex_fw-la of_o the_o remote_a place_n than_o from_o the_o vertex_fw-la of_o those_o that_o be_v near_o and_o therefore_o he_o shall_v be_v less_o elevate_v over_o the_o horizon_n of_o those_o place_n than_o of_o these_o proposition_n iv_o by_o how_o much_o the_o place_n of_o the_o earth_n be_v more_o remote_a from_o the_o aequator_fw-la or_o more_o near_o the_o pole_n by_o so_o much_o the_o more_o the_o part_n of_o the_o horizon_n be_v distant_a in_o which_o the_o sun_n rise_v on_o the_o day_n of_o the_o solstice_n and_o the_o day_n of_o the_o winter_n as_o also_o those_o in_o which_o he_o set_v the_o same_o be_v true_a concern_v the_o moon_n and_o all_o the_o planet_n take_v what_o place_n you_o please_v of_o a_o diverse_a distance_n from_o the_o aequator_fw-la and_o let_v the_o pole_n be_v elevate_v for_o the_o latitude_n of_o every_o one_o of_o they_o and_o let_v the_o point_n be_v note_v in_o the_o horizon_n in_o which_o the_o tropic_n of_o capricorn_n and_o cancer_n cut_v it_o a_o comparison_n be_v make_v the_o truth_n of_o the_o proposition_n will_v appear_v this_o be_v also_o show_v the_o same_o way_n by_o how_o much_o the_o place_n be_v more_o remote_a from_o the_o aequator_fw-la by_o so_o much_o the_o more_o the_o sun_n in_o his_o aequinoctial_a rise_n be_v distant_a in_o the_o east_n on_o every_o day_n of_o the_o year_n the_o astronomer_n term_v it_o the_o rise_a amplitude_n proposition_n v._o star_n place_v between_o the_o parallel_n of_o any_o place_n lie_v without_o the_o aequator_fw-la and_o the_o pole_n be_v less_o elevate_v above_o the_o horizon_n of_o the_o place_n between_o this_o parallel_n and_o the_o other_o pole_n of_o those_o situate_v there_o than_o above_o the_o horizon_n of_o the_o place_n situate_v between_o this_o parallel_n and_o the_o near_a pole_n etc._n of_o the_o elevation_n of_o star_n etc._n etc._n the_o parallel_n of_o any_o star_n may_v be_v design_v on_o the_o terrestrial_a globe_n or_o a_o point_n only_o note_v for_o a_o star_n and_o any_o place_n more_o remote_a from_o the_o pole_n be_v assume_v design_v the_o parallel_n of_o the_o place_n then_o take_v another_o place_n situate_v towards_o the_o other_o pole_n the_o stay_n of_o the_o star_n above_o the_o horizon_n of_o both_o place_n may_v be_v find_v and_o the_o truth_n of_o the_o proposition_n will_v be_v manifest_a proposition_n vi_o in_o place_n situate_a in_o and_o near_o the_o aequator_fw-la the_o sun_n and_o star_n direct_o ascend_v above_o the_o horizon_n even_o to_o the_o meridian_n and_o so_o descend_v again_o but_o in_o place_n situate_v above_o the_o aequator_fw-la they_o oblique_o ascend_v and_o descend_v and_o so_o much_o the_o more_o oblique_o by_o how_o much_o the_o place_n be_v more_o remote_a from_o the_o aequator_fw-la star_n of_o the_o ascension_n and_o descension_n of_o the_o sun_n and_o star_n let_v any_o parallel_n of_o the_o sun_n be_v describe_v on_o the_o globe_n such_o as_o some_o already_o be_v delineate_v on_o the_o globe_n viz._n the_o aequator_fw-la the_o tropic_n and_o some_o intermedial_a one_o then_o let_v the_o pole_n be_v place_v in_o the_o very_a horizon_n that_o it_o may_v be_v the_o horizon_n of_o the_o place_n of_o the_o aequator_fw-la and_o it_o will_v be_v evident_a that_o the_o point_n of_o the_o
degree_n and_o b_o n_o l_o be_v equal_a to_o the_o angle_n a_o s_o l_o or_o b_o a_o r._n therefore_o the_o arch_n of_o the_o angle_n n_o b_o l_o be_v equal_a to_o the_o arch_n b_o m_o by_o which_o p_o b_o be_v great_a than_o 90_o degree_n or_o than_o p_o m_o so_o also_o the_o arch_n p_o c._n if_o we_o take_v the_o semidiameter_n of_o the_o sun_n according_a unto_o ptolemy_n of_o 5_o 2_o semidiameter_n of_o the_o earth_n but_o the_o distance_n a_o s_o 1168_o semidiameter_n these_o i_o say_v be_v lay_v down_o the_o arch_n m_o b_o will_v be_v find_v 13_o minute_n in_o which_o the_o sun_n illustrate_v the_o earth_n more_o than_o half_a m_o p_o q._n corollary_n when_o therefore_o the_o centre_n of_o the_o sun_n rise_v to_o some_o place_n than_o his_o limbus_n or_o edge_n rise_v to_o the_o people_n which_o inhabit_v in_o the_o parallel_n of_o the_o horizon_n situate_v 13_o minute_n beneath_o the_o horizon_n also_o after_o the_o same_o mode_n to_o those_o to_o who_o he_o set_v and_o when_o his_o centre_n set_v than_o his_o limbus_n yet_o remain_v conspicuous_a until_o the_o centre_n set_v to_o the_o people_n which_o be_v remote_a 13_o minute_n from_o our_o horizon_n proposition_n xiii_o the_o height_n of_o a_o mountain_n be_v give_v to_o find_v how_o much_o soon_o the_o sun_n seem_v to_o rise_v in_o the_o vertex_fw-la of_o the_o same_o than_o at_o the_o foot_n or_o root_n of_o the_o mountain_n and_o how_o much_o late_a it_o set_v from_o the_o give_a altitude_n 5._o see_v chap._n 9_o proposit_n 5._o by_o the_o five_o proposition_n in_o the_o nine_o chapter_n let_v the_o interval_n or_o arch_n from_o which_o the_o vertex_fw-la of_o the_o mountain_n may_v be_v discover_v or_o in_o the_o bind_v of_o which_o a_o line_n so_o draw_v from_o the_o vertex_fw-la of_o the_o mountain_n that_o it_o may_v be_v the_o tangent_fw-la of_o the_o earth_n refract_v the_o same_o for_o this_o line_n show_v the_o first_o ray_n which_o may_v come_v from_o a_o direct_a passage_n from_o the_o sun_n to_o the_o vertex_fw-la of_o the_o mountain_n moreover_o the_o point_n of_o the_o earth_n in_o which_o this_o be_v touch_v by_o the_o line_n be_v the_o place_n to_o which_o the_o sun_n arise_v when_o he_o begin_v to_o be_v see_v on_o the_o vertex_fw-la of_o the_o mountain_n and_o the_o arch_n interrupt_v between_o that_o point_n and_o the_o foot_n of_o the_o mountain_n be_v equal_a to_o that_o in_o which_o the_o sun_n be_v depress_v as_o yet_o beneath_o the_o horizon_n of_o the_o foot_n of_o the_o mountain_n when_o he_o be_v apparent_a in_o the_o vertex_fw-la therefore_o the_o problem_n be_v reduce_v hither_o the_o depression_n of_o the_o sun_n beneath_o the_o horizon_n be_v give_v to_o find_v the_o time_n which_o be_v spend_v whilst_o the_o sun_n move_v from_o the_o depression_n to_o the_o horizon_n whence_o also_o it_o will_v be_v manifest_a that_o this_o time_n be_v also_o diverse_a in_o the_o divers_a day_n of_o the_o year_n therefore_o let_v the_o place_n of_o the_o root_n of_o the_o mountain_n be_v note_v on_o the_o globe_n and_o let_v the_o pole_n be_v clevate_v for_o the_o latitude_n of_o the_o same_o let_v the_o quadrant_n be_v affix_v to_o the_o vertex_fw-la the_o place_n of_o the_o sun_n be_v find_v in_o the_o ecliptic_a from_o any_o day_n take_v let_v it_o be_v note_v also_o the_o point_n of_o the_o ecliptic_a opposite_a to_o the_o place_n of_o the_o sun_n then_o let_v this_o opposite_a point_n be_v bring_v to_o the_o occidental_a horizon_n and_o let_v the_o index_n be_v place_v at_o the_o hour_n 12._o this_o be_v do_v let_v the_o degree_n of_o depression_n before_o find_v be_v note_v in_o the_o quadrant_n and_o the_o opposite_a point_n be_v turn_v above_o the_o horizon_n until_o it_o have_v a_o altitude_n equal_a to_o the_o arch_n of_o the_o depression_n which_o will_v be_v discover_v from_o the_o application_n of_o the_o quadrant_n so_o the_o place_n of_o the_o sun_n beneath_o the_o oriental_a horizon_n will_v have_v that_o depression_n and_o the_o index_n in_o the_o horaty_a circle_n will_v show_v the_o time_n intercopt_v between_o that_o depression_n of_o he_o and_o his_o emersion_n above_o the_o horizon_n but_o because_o in_o this_o case_n we_o do_v almost_o work_v only_o by_o minute_n therefore_o it_o be_v better_a to_o calculate_v it_o than_o to_o search_v after_o it_o on_o the_o globe_n now_o you_o shall_v find_v it_o if_o that_o the_o altitude_n of_o the_o mountain_n be_v place_v 3_o stadias_fw-la or_o ¾_n of_o a_o german_a mile_n because_o the_o arch_n of_o the_o depression_n be_v about_o three_o degree_n night_n the_o mountain_n of_o caucasus_n and_o casius_n according_a to_o aristotle_n and_o pliny_n be_v illustrate_v with_o the_o sun_n ray_n to_o the_o three_o part_n of_o the_o night_n and_o if_o the_o latitude_n of_o the_o foot_n of_o the_o mountain_n be_v 38_o degree_n and_o the_o place_n of_o the_o sun_n about_o the_o middle_n of_o leo_n the_o time_n in_o which_o the_o sun_n be_v behold_v be_v soon_o in_o the_o vertex_fw-la than_o at_o the_o foot_n of_o the_o mountain_n by_o 13_o minute_n hence_o it_o be_v manifest_a that_o that_o be_v not_o so_o probable_a which_o aristotle_n relate_v of_o the_o high_a part_n of_o caucasus_n and_o pliny_n of_o the_o top_n of_o mount_n casius_n that_o they_o before_o the_o rise_n and_o after_o the_o set_n of_o the_o sun_n be_v illustrate_v with_o the_o sun_n beam_n even_o to_o the_o three_o part_n of_o the_o night_n now_o how_o great_a a_o altitude_n be_v require_v for_o this_o shall_v be_v show_v in_o the_o follow_a proposition_n proposition_n fourteen_o the_o time_n be_v give_v in_o which_o the_o sun_n be_v soon_o discern_v on_o the_o vertex_fw-la of_o the_o mountain_n than_o at_o the_o foot_n of_o the_o same_o to_o find_v the_o altitude_n of_o the_o mountain_n let_v the_o pole_n be_v elevate_v on_o the_o globe_n for_o the_o latitude_n of_o the_o root_n of_o the_o mountain_n and_o the_o point_n be_v note_v which_o be_v oppose_v to_o the_o place_n of_o the_o sun_n in_o the_o ecliptic_a let_v the_o arch_n of_o the_o depression_n of_o the_o sun_n beneath_o the_o horizon_n for_o the_o give_v time_n be_v find_v then_o from_o this_o arch_n as_o from_o a_o interval_n from_o whence_o the_o vertex_fw-la of_o the_o mountain_n be_v discover_v the_o altitude_n of_o the_o mountain_n must_v be_v search_v after_o by_o the_o four_o proposition_n of_o the_o nine_o chapter_n proposition_n xv._o the_o place_n of_o the_o moon_n be_v give_v in_o the_o zodiac_n together_o with_o its_o latitude_n to_o find_v out_o or_o show_v all_o those_o place_n on_o the_o globe_n to_o which_o the_o moon_n be_v vertical_a in_o the_o circumrotation_n of_o that_o day_n let_v the_o place_n of_o the_o moon_n take_v from_o the_o ephemerides_n be_v note_v in_o the_o ecliptic_a then_o let_v one_o end_n of_o the_o quadrant_n be_v apply_v to_o the_o pole_n of_o the_o ecliptic_a the_o other_o to_o the_o point_v note_v in_o the_o ecliptic_a or_o to_o the_o place_n of_o the_o moon_n and_o let_v the_o degree_n of_o the_o latitude_n of_o the_o moon_n be_v account_v on_o the_o quadrant_n and_o let_v a_o mark_n be_v make_v at_o the_o term_n of_o the_o numeration_n on_o the_o globe_n then_o this_o be_v bring_v to_o the_o meridian_n and_o a_o chalk_n apply_v let_v a_o parallel_n be_v describe_v which_o the_o moon_n that_o day_n do_v describe_v by_o her_o circumvolution_n and_o all_o the_o place_n situate_v in_o this_o parallel_n be_v those_o demand_v after_o the_o same_o mode_n we_o act_n with_o the_o other_o planer_n if_o their_o longitude_n and_o latitude_n be_v give_v proposition_n xvi_o the_o place_n of_o the_o moon_n be_v give_v in_o the_o zodiac_n and_o its_o latitude_n and_o the_o day_n of_o the_o year_n to_o find_v the_o hour_n in_o which_o she_o arise_v in_o any_o place_n give_v and_o in_o which_o she_o set_v also_o in_o which_o she_o make_v midnight_n let_v the_o pole_n be_v elevate_v for_o the_o latitude_n of_o the_o place_n of_o the_o earth_n give_v let_v 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the_o latitude_n of_o the_o place_n know_v and_o the_o distance_n of_o the_o unknown_a place_n be_v turn_v into_o degree_n and_o the_o angle_n comprehend_v from_o the_o plaga_fw-la give_v from_o these_o three_o give_v the_o opposite_a angle_n to_o the_o distance_n must_v be_v seek_v for_o for_o this_o will_v exhibit_v the_o longitude_n of_o the_o other_o place_n from_o the_o know_a place_n but_o on_o the_o globe_n and_o mariner_n chart_n the_o place_n be_v thus_o find_v let_v the_o pole_n be_v elevate_v for_o the_o latitude_n of_o the_o place_n give_v let_v the_o quadrant_n be_v apply_v to_o the_o vertex_fw-la and_o let_v the_o other_o extremity_n be_v apply_v to_o the_o give_v plaga_fw-la of_o the_o horizon_n then_o the_o distance_n give_v be_v turn_v into_o degree_n let_v it_o be_v reckon_v on_o the_o quadrant_n from_o the_o vertex_fw-la the_o term_n of_o the_o numeration_n shall_v be_v the_o place_n seek_v for_o on_o the_o globe_n but_o if_o that_o the_o longitude_n be_v only_o 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vertex_fw-la and_o the_o other_o extremity_n to_o the_o give_v plaga_fw-la of_o the_o horizon_n to_o wit_n in_o which_o the_o three_o unknown_a place_n be_v place_v to_o lie_v at_o this_o same_o know_a place_n the_o point_n in_o which_o the_o margin_n of_o the_o quadrant_n cut_v the_o periphery_n before_o draw_v with_o chalk_n be_v the_o three_o place_n demand_v on_o map_n it_o be_v thus_o do_v let_v a_o line_n be_v draw_v from_o one_o give_v place_n for_o the_o give_v quarter_n of_o the_o three_o place_n after_o the_o same_o mode_n let_v the_o line_n of_o the_o quarter_n be_v draw_v from_o the_o other_o give_v place_n the_o point_n in_o which_o these_o two_o line_n mutual_o cut_v one_o another_o be_v the_o place_n demand_v after_o the_o same_o mode_n we_o shall_v do_v on_o the_o earth_n if_o that_o we_o will_v act_n scientifical_o neither_o in_o science_n do_v we_o value_v hindrance_n and_o impediment_n so_o that_o we_o may_v comprehend_v the_o mode_n in_o our_o mind_n the_o calculation_n in_o which_o our_o unknown_a longitude_n of_o a_o place_n be_v find_v from_o these_o give_v we_o leave_v to_o the_o instruction_n of_o a_o tutor_n if_o that_o he_o have_v apt_a and_o capable_a scholar_n but_o more_o than_o enough_o have_v be_v say_v concern_v the_o invention_n of_o longitude_n the_o ample_a use_n of_o which_o we_o have_v explain_v in_o the_o 2d_o proposition_n here_o shall_v be_v add_v a_o table_n of_o the_o longitude_n and_o latitude_n of_o the_o chief_a place_n of_o the_o earth_n which_o the_o author_n have_v collect_v and_o do_v here_o insert_v but_o be_v but_o short_a and_o have_v map_n of_o the_o several_a kingdom_n of_o the_o world_n in_o the_o other_o part_n or_o volumn_n to_o which_o the_o latitude_n and_o longitude_n be_v add_v they_o be_v think_v convenient_a to_o be_v omit_v here_o and_o refer_v the_o reader_n to_o the_o particular_a map_n by_o which_o you_o may_v easy_o find_v the_o latitude_n and_o longitude_n of_o any_o place_n desire_v navigation_n the_o fix_a star_n as_o to_o their_o declination_n and_o ascension_n of_o great_a use_n in_o geography_n and_o navigation_n moreover_o see_v that_o there_o be_v great_a use_n of_o declination_n and_o ascension_n of_o the_o fix_a star_n both_o in_o geography_n and_o navigation_n i_o shall_v here_o add_v a_o catalogue_n of_o the_o star_n of_o the_o first_o magnitude_n with_o their_o declination_n and_o direct_a ascension_n at_o the_o year_n 1650._o for_o it_o be_v know_v from_o astronomy_n that_o in_o progress_n of_o time_n a_o change_n be_v make_v in_o these_o by_o reason_n of_o the_o proper_a motion_n of_o the_o star_n above_o the_o pole_n of_o the_o ecliptic_a but_o in_o the_o use_n it_o be_v convenient_a to_o have_v such_o a_o table_n of_o all_o the_o star_n because_o we_o have_v not_o always_o a_o conveniency_n of_o use_v the_o same_o star_n but_o we_o only_o lie_v down_o these_o for_o exercise_n and_o for_o the_o try_v the_o propose_a problem_n in_o these_o this_o business_n belong_v to_o astronomy_n but_o the_o use_n be_v notable_a both_o in_o other_o science_n and_o also_o in_o geography_n astronomy_n show_v how_o a_o declination_n and_o direct_a ascension_n may_v be_v find_v at_o every_o year_n a_o table_n of_o the_o declination_n and_o right_a ascension_n of_o the_o star_n for_o the_o year_n 1650._o the_o letter_n s_o show_v the_o northern_a declination_n and_o the_o letter_n a_o the_o southern_a the_o name_n of_o the_o star_n declination_n right_o ascension_n of_o the_o first_o maginitude_n deg_fw-la min._n deg_fw-la min._n oculus_fw-la tauri_n 15_o 46_o s_n 64_o 0_o regulus_n or_o cor_fw-la leonis_fw-la 13_o 39_o s_n 147_o 27_o cauda_fw-la leonis_fw-la 16_o 32_o s_n 172_o 59_o spica_fw-la virgin_n 9_o 17_o a_o 196_o 44_o cor_fw-la scorpii_n 25_o
pay_v which_o be_v not_o of_o facility_n and_o lester_n expense_n for_o the_o make_n of_o one_o globe_n but_o most_o apt_a and_o prompt_a for_o the_o make_n of_o innumerable_a of_o the_o same_o magnitude_n and_o less_o expense_n of_o which_o i_o shall_v speak_v in_o the_o three_o place_n yet_o the_o foundation_n of_o the_o constouction_n of_o it_o depend_v on_o this_o description_n conceive_v on_o the_o globe_n moreover_o where_o peculiar_a terrestrial_a globe_n be_v to_o be_v make_v in_o brass_n of_o a_o notable_a magnitude_n and_o the_o place_n of_o the_o earth_n be_v to_o be_v design_v on_o its_o superficies_n as_o brince_n that_o favour_v the_o mathematics_n be_v wont_a to_o have_v they_o holstein_n a_o great_a globe_n a_o prepare_v for_o frederick_n duke_n of_o holstein_n as_o for_o example_n frederick_n duke_n of_o holstein_n have_v order_v such_o a_o globe_n to_o be_v make_v who_o oavity_n be_v to_o be_v so_o great_a that_o one_o may_v commodious_o set_v in_o it_o and_o in_o 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which_o the_o table_n of_o longitude_n and_o latitude_n which_o be_v to_o be_v use_v in_o the_o designation_n of_o place_n do_v acknowledge_v for_o the_o first_o or_o from_o whence_o they_o number_v the_o longitude_n and_o latitude_n of_o the_o other_o place_n and_o therefore_o if_o the_o place_n first_o give_v be_v not_o that_o which_o the_o table_n acknowledge_v let_v the_o longitude_n of_o the_o first_o give_v place_n be_v take_v from_o the_o table_n and_o let_v so_o many_o degree_n be_v number_v from_o that_o point_n in_o the_o aequator_fw-la where_o the_o meridian_n of_o the_o first_o place_n cut_v it_o the_o term_n of_o the_o numeration_n shall_v be_v the_o point_n for_o the_o first_o meridian_n of_o the_o table_n now_o unto_o what_o quarter_n the_o numeration_n must_v be_v make_v be_v know_v viz._n towards_o the_o west_n but_o what_o part_n of_o the_o globe_n be_v to_o be_v take_v for_o the_o occidental_a quarter_n and_o what_o for_o the_o oriental_a quarter_n from_o the_o meridian_n of_o the_o first_o place_n you_o shall_v thus_o know_v so_o place_v the_o globe_n that_o the_o semicircle_n of_o the_o meridian_n conta_v the_o first_o point_n may_v be_v uppermost_a the_o other_o beneath_o and_o regard_v the_o globe_n now_o the_o pole_n arctic_a shall_v be_v hear_v the_o antarctick_a more_o remote_a if_o that_o the_o place_n give_v near_o the_o arctic_a pole_n but_o if_o near_o the_o antarctick_a than_o the_o antarctick_a must_v be_v place_v near_o we_o so_o that_o hemisphere_n which_o be_v in_o our_o right_a hand_n must_v be_v take_v for_o the_o occidental_a place_n and_o the_o other_o for_o the_o oriental_a but_o in_o the_o section_n of_o the_o first_o meridian_n note_v in_o the_o aequator_fw-la let_v those_o number_n be_v ascribe_v to_o the_o degree_n of_o the_o aequator_fw-la beginning_n from_o that_o section_n viz._n 10_o 20_o 30_o and_o so_o on_o and_o then_o so_o must_v it_o be_v do_v in_o represent_v for_o any_o other_o place_n let_v the_o 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other_o give_v than_o to_o be_v use_v for_o the_o make_n of_o a_o entire_a globe_n for_o it_o use_v the_o distance_n of_o place_n let_v the_o great_a periphery_n or_o the_o arch_n of_o the_o great_a periphery_n be_v draw_v through_o the_o globe_n and_o in_o this_o from_o the_o give_v point_n let_v the_o arch_n be_v take_v as_o much_o as_o the_o distance_n of_o the_o other_o place_n be_v from_o the_o place_n first_o give_v the_o term_n of_o the_o arch_n shall_v be_v other_o place_n then_o if_o you_o will_v design_n any_o three_o place_n take_v by_o the_o interval_n of_o the_o compass_n the_o distance_n of_o that_o three_o place_n from_o the_o other_o two_o even_o now_o design_v and_o from_o these_o as_o from_o centre_n let_v the_o arch_n be_v describe_v by_o these_o interval_n of_o the_o compass_n the_o point_n in_o which_o these_o arch_n mutual_o cut_v one_o another_o be_v the_o point_n of_o the_o three_o place_n but_o as_o i_o have_v say_v that_o this_o mode_n be_v not_o commodious_a 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globe_n the_o three_o mode_n of_o make_v of_o globe_n the_o three_o mode_n of_o exhibit_v and_o represent_v the_o superficies_n and_o place_n of_o the_o earth_n in_o the_o give_v globe_n be_v that_o which_o artificer_n use_v in_o the_o make_n of_o all_o globe_n both_o celestial_a and_o terrestrial_a except_o those_o great_a one_o of_o which_o i_o have_v now_o speak_v which_o have_v nothing_o of_o compendiousness_n or_o commendation_n from_o the_o facility_n if_o that_o the_o place_n of_o the_o earth_n be_v but_o only_o to_o be_v represent_v from_o one_o superficies_n of_o the_o globe_n but_o it_o be_v to_o be_v do_v on_o the_o superficies_n of_o the_o globe_n of_o the_o same_o magnitude_n this_o practice_n have_v great_a prerogative_n before_o the_o other_o for_o the_o mode_n be_v thus_o the_o superficies_n of_o the_o globe_n and_o the_o earth_n be_v conceive_v to_o be_v divide_v into_o twelve_o part_n or_o more_o if_o the_o globe_n be_v to_o be_v make_v of_o a_o large_a form_n through_o the_o meridian_n draw_v from_o pole_n to_o pole_n so_o that_o in_o any_o two_o meridian_n the_o 12_o part_n of_o a_o superficies_n be_v include_v from_o pole_n to_o pole_n then_o on_o a_o plain_n let_v the_o like_a figure_n be_v include_v in_o such_o a_o part_n of_o the_o 12_o in_o two_o arch_n which_o then_o in_o the_o globe_n make_v the_o half_a periphery_n of_o the_o meridian_n and_o in_o many_o meridian_n draw_v through_o every_o degree_n of_o the_o aequator_fw-la and_o divide_v into_o portion_n and_o segment_n of_o the_o parallel_n afford_v a_o kind_n of_o lettice_a work_n the_o portion_n of_o the_o aequator_fw-la be_v in_o the_o midst_n all_o the_o meridian_n end_n in_o the_o pole_n than_o c._n e_o meridian_n be_v take_v for_o the_o first_o which_o the_o table_n of_o longitude_n acknowledge_v let_v the_o degree_n be_v note_v from_o it_o in_o the_o aequator_fw-la the_o number_n be_v ascribe_v so_o that_o the_o degree_n of_o longitude_n of_o every_o place_n may_v â—e_v account_v then_o in_o every_o one_o of_o these_o place_n represent_v the_o 12_o part_n of_o the_o superficies_n of_o the_o globe_n let_v the_o place_n be_v note_v for_o the_o place_n of_o the_o earth_n every_o one_o at_o his_o degree_n of_o longitude_n and_o latitude_n which_o be_v extract_v from_o the_o table_n and_o the_o name_n be_v ascribe_v to_o the_o table_n and_o the_o tract_n of_o the_o river_n and_o bay_n draw_v as_o also_o of_o the_o land_n these_o be_v thus_o describe_v on_o paper_n or_o wood_n then_o make_v a_o incision_n and_o engrave_v according_a to_o that_o exemplar_n in_o plate_n of_o brass_n which_o then_o be_v fit_a for_o the_o print_n press_n which_o be_v afterward_o apply_v and_o join_v to_o the_o superficies_n of_o the_o globe_n so_o that_o its_o end_n may_v touch_v the_o axis_n or_o pole_n of_o the_o globe_n yet_o in_o many_o the_o paper_n do_v not_o touch_v the_o pole_n but_o be_v so_o make_v only_o to_o touch_v the_o arctic_a or_o antarctick_a circle_n and_o peculiar_a paper_n be_v take_v for_o the_o polary_a space_n for_o so_o they_o be_v more_o easy_o apply_v especial_o in_o great_a one_o so_o in_o the_o superficies_n of_o this_o globe_n all_o the_o place_n of_o the_o earth_n be_v exhibit_v to_o which_o be_v then_o add_v a_o brass_n meridian_n and_o horizon_n with_o a_o foot_n horary_a circle_n and_o a_o index_n mode_n the_o thing_n worthy_a of_o note_n in_o this_o mode_n there_o be_v two_o thing_n in_o this_o description_n which_o require_v a_o more_o full_a explication_n all_o the_o rest_n i_o suppose_v to_o be_v plain_a and_o intelligible_a first_o after_o what_o mode_n these_o 12_o or_o 24_o part_n be_v to_o be_v describe_v according_a to_o the_o example_n of_o which_o the_o engrave_v in_o brass_n must_v be_v make_v second_o how_o plain_a paper_n can_v be_v apply_v to_o the_o superficies_n of_o the_o globe_n the_o first_o be_v thus_o donâ—â—ommodiously_o enough_o for_o example_n let_v the_o 12_o portion_n of_o the_o hemisphere_n from_o the_o pole_n to_o the_o aequator_fw-la be_v apply_v to_o the_o globe_n first_o from_o the_o know_a diameter_n of_o the_o globe_n let_v the_o quantity_n of_o the_o great_a periphery_n be_v find_v out_o according_a to_o the_o proportion_n of_o archimedes_n or_o the_o other_o proportion_n of_o the_o periphery_a to_o the_o diameter_n for_o example_n let_v the_o diameter_n of_o the_o globe_n be_v two_o foot_n and_o let_v the_o longitude_n of_o the_o foot_n in_o the_o note_a paper_n be_v divide_v into_o 10_o digit_n and_o the_o 10_o digit_n into_o 10_o grain_n that_o there_o may_v be_v 100_o part_n in_o a_o foot_n let_v it_o be_v do_v so_o that_o as_o 7_o be_v to_o 22_o so_o 200_o be_v to_o 628_o 4_o 7_o part_n or_o 6_o 28_o 200_o foot_n for_o the_o periphery_a the_o four_o part_n of_o this_o that_o be_v the_o quadrant_n of_o the_o periphery_n shall_v be_v of_o 157_o 1_o 7_o hundred_o or_o 1_o 57_o 10071_o foot_n and_o the_o 12_o part_n of_o 52_o 19_o 21_o hundred_o or_o ½_n a_o foot_n and_o 2_o hundred_o and_o 19_o 21_o of_o a_o hundred_o these_o be_v find_v let_v a_o long_a line_n of_o 52_o 19_o 21_o hundred_o be_v draw_v on_o the_o paper_n from_o the_o ascribe_v scale_n from_o the_o middle_n of_o this_o line_n let_v a_o long_a perpendicular_a of_o 157_o 19_o 21_o hundred_o be_v erect_v which_o shall_v be_v the_o quadrant_n its_o extremity_n shall_v be_v the_o pole_n and_o may_v be_v divide_v into_o degree_n you_o have_v the_o longitude_n of_o one_o degree_n if_o you_o divide_v 628_o â—_o 7_o by_o 360_o then_o let_v a_o periphery_n be_v describe_v from_o the_o pole_n through_o the_o begin_n of_o every_o degree_n or_o of_o every_o ten_o they_o shall_v be_v parallel_n in_o these_o periphery_n from_o both_o part_n of_o the_o draw_v perpendicular_a let_v that_o part_n be_v cut_v off_o by_o the_o compass_n as_o much_o as_o be_v the_o 1_o 24_o of_o the_o periphery_a now_o how_o great_a it_o be_v in_o the_o opposite_a scale_n be_v know_v from_o the_o proportion_n of_o the_o parallel_n to_o the_o aequator_fw-la which_o we_o have_v deliver_v in_o the_o end_n of_o the_o iv_o chapter_n 4._o see_v chap._n 4._o so_o the_o point_n be_v sign_v in_o every_o periphery_a and_o arch_a you_o please_v a_o line_n must_v be_v draw_v through_o they_o and_o part_n of_o the_o paper_n perminate_v by_o these_o line_n must_v be_v cut_v off_o for_o this_o be_v apply_v to_o the_o globe_n will_v possess_v 1_o 12_o of_o the_o hemisphere_n now_o the_o application_n be_v easy_o perform_v viz._n if_o that_o the_o portion_n be_v small_a for_o in_o these_o the_o distance_n between_o straight_a and_o crooked_a be_v little_o discover_v especial_o of_o the_o earth_n when_o the_o paper_n have_v first_o
place_n and_o the_o meridian_n of_o latitude_n do_v meet_v so_o a_o geographical_a map_n for_o a_o give_v region_n shall_v be_v complete_v the_o three_o mode_n of_o describe_v particular_a map_n etc._n the_o three_o mode_n of_o particular_a map_n as_o province_n etc._n etc._n in_o represent_v the_o province_n of_o a_o small_a tract_n we_o use_v another_o method_n which_o we_o have_v explain_v before_o viz._n that_o the_o map_n may_v more_o accurate_o exhibit_v the_o situation_n of_o one_o place_n to_o another_o and_o the_o distance_n of_o place_n the_o method_n consist_v in_o this_o that_o we_o may_v find_v the_o angle_n of_o the_o position_n of_o one_o place_n to_o another_o by_o mathematical_a instrument_n and_o then_o apt_o transfer_v into_o the_o chart._n for_o example_n sake_n let_v there_o be_v five_o place_n of_o any_o region_n to_o be_v dispose_v in_o the_o table_n according_a to_o its_o situation_n and_o distance_n we_o shall_v call_v those_o five_o place_n a_o b_o c_o d_o e._n first_o we_o shall_v choose_v from_o these_o that_o a_o from_o which_o the_o rest_n or_o most_o of_o they_o may_v convenient_o be_v behold_v and_o a_o instrument_n be_v apply_v we_o shall_v observe_v the_o angle_n of_o position_n at_o every_o place_n viz._n the_o angle_n between_o the_o meridian_n line_n of_o the_o place_n in_o which_o we_o observe_v and_o between_o the_o quarter_n of_o the_o other_o place_n b_o c_o d_o e._n moreover_o in_o the_o chart_n in_o which_o we_o will_v represent_v those_o place_n we_o may_v take_v the_o letter_n a_o and_o cut_v the_o periphery_a describe_v from_o thence_o into_o degree_n which_o be_v not_o necessary_a if_o we_o have_v a_o semicircle_n divide_v or_o some_o other_o geometrical_a instrument_n fit_a for_o that_o purpose_n we_o shall_v assume_v one_o diameter_n for_o the_o meridian_n line_n of_o the_o place_n a_o viz._n f_o a_o g_o the_o other_o perpendicular_a to_o this_o h_n a_o k_o will_v show_v the_o aequinoctial_a rise_n in_o the_o extremity_n h_n the_o setting_z k._n f_o be_v the_o northern_a cardo_n c_o the_o southern_a let_v the_o angle_n of_o the_o position_n of_o the_o place_n b_o to_o a_o be_v observe_v of_o 30_o deg_fw-la from_o the_o south_n towards_o the_o east_n we_o shall_v number_v in_o the_o quadrant_n g_o h_n so_o many_o degree_n and_o shall_v draw_v the_o line_n from_o a_o through_o that_o degree_n this_o shall_v represent_v the_o place_n b_o from_o the_o place_n a._n after_o the_o same_o mode_n the_o quarter_n of_o the_o other_o place_n d_o e_o must_v be_v note_v on_o the_o chart_n if_o they_o be_v observe_v then_o another_o place_n be_v to_o be_v choose_v from_o b_o c_o d_o e_o who_o distance_n from_o a_o be_v know_v or_o find_v out_o for_o example_n the_o place_n b_o and_o in_o that_o the_o instrument_n be_v apply_v the_o quarter_n observe_v of_o the_o three_o other_o place_n c_o d_o e._n this_o be_v do_v we_o put_v in_o our_o table_n the_o scale_n of_o mile_n or_o league_n which_o we_o take_v 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help_n of_o the_o terrestrial_a globe_n exhibit_v on_o a_o plain_a the_o situation_n and_o distance_n of_o place_n remote_a from_o one_o to_o another_o and_o of_o divers_a kingdom_n yea_o the_o whole_a superficies_n of_o the_o earth_n so_o that_o any_o place_n give_v may_v seem_v to_o occupy_v the_o middle_n as_o we_o have_v show_v in_o the_o six_o mode_n so_o that_o this_o mode_n may_v be_v reckon_v to_o the_o mode_n for_o general_a map_n but_o it_o be_v better_a not_o to_o extend_v the_o mode_n beyond_o the_o hemisphere_n for_o distance_n i_o determine_v to_o set_v down_o before_o your_o eye_n on_o a_o chart_n the_o situation_n of_o all_o the_o place_n to_o london_n and_o their_o distance_n from_o this_o place_n first_o in_o the_o chart_n let_v the_o middle_a point_n be_v take_v for_o london_n let_v the_o letter_n a_o be_v note_v from_o that_o let_v the_o periphery_n f_o h_o g_o k_n be_v describe_v let_v f_o g_o be_v the_o meridian_n line_n or_o the_o line_n of_o the_o north_n and_o south_n let_v h_n k_o be_v the_o line_n of_o the_o east_n and_o west_n f_o may_v show_v the_o north_n g_o the_o south_n h_n the_o east_n and_o k_o the_o west_n let_v every_o one_o of_o the_o quadrant_n be_v divide_v into_o 90_o degree_n then_o in_o the_o globe_n let_v london_n be_v bring_v to_o the_o brazen_a meridian_n and_o let_v the_o pole_n be_v elevate_v according_a to_o the_o latitude_n of_o london_n let_v the_o quadrant_n be_v affix_v to_o the_o vertex_fw-la and_o apply_v to_o every_o place_n who_o situation_n we_o will_v represent_v to_o london_n on_o the_o chart._n for_o example_n the_o beginning_n middle_a and_o end_n of_o france_n so_o to_o the_o bound_n of_o italy_n spain_n hungary_n sweden_n and_o the_o like_a and_o let_v the_o angle_n be_v note_v which_o the_o quadrant_n make_v with_o the_o meridian_n in_o every_o application_n that_o be_v the_o angle_n of_o the_o position_n of_o those_o place_n to_o london_n moreover_o the_o degree_n of_o the_o quadrant_n between_o london_n and_o every_o place_n that_o be_v the_o distance_n of_o every_o one_o of_o they_o this_o do_v lay_v aside_o the_o globe_n and_o on_o the_o chart_n let_v the_o line_n be_v draw_v from_o a_o for_o the_o quarter_n of_o every_o one_o of_o the_o place_n viz._n which_o may_v make_v such_o angle_n with_o the_o meridian_n line_n as_o be_v note_v before_o and_o that_o between_o the_o cardinal_n note_v point_n how_o we_o may_v supercede_v this_o labour_n of_o draw_v of_o line_n we_o shall_v show_v anon_o in_o these_o line_n of_o the_o quarter_n the_o point_n must_v be_v find_v out_o for_o the_o place_n by_o take_v the_o distance_n from_o london_n which_o we_o may_v do_v by_o a_o double_a method_n for_o the_o place_n be_v either_o remove_v a_o little_a interval_n which_o we_o will_v note_v or_o by_o a_o large_a interval_n but_o so_o it_o be_v best_o to_o make_v a_o small_a map_n or_o the_o place_n be_v remove_v a_o great_a interval_n and_o so_o you_o must_v form_v a_o map_n of_o a_o great_a bulk_n in_o the_o two_o case_n it_o be_v sufficient_a to_o make_v a_o scale_n of_o degree_n by_o divide_v some_o line_n into_o equal_a part_n every_o one_o of_o which_o may_v represent_v a_o degree_n from_o this_o scale_n we_o take_v the_o distance_n of_o every_o place_n before_o note_a with_o the_o interval_n of_o the_o compass_n and_o bring_v they_o into_o the_o line_n of_o the_o quarter_n of_o every_o place_n the_o term_n shall_v be_v note_v with_o the_o appellation_n of_o the_o place_n region_n a_o scale_n of_o equal_a part_n for_o degree_n not_o sufficient_a for_o large_a map_n with_o remote_a region_n and_o so_o we_o shall_v note_v all_o the_o place_n in_o the_o table_n about_o london_n but_o if_o you_o must_v make_v a_o map_n of_o a_o large_a form_n and_o the_o remote_a region_n must_v be_v note_v it_o be_v not_o sufficient_a to_o take_v a_o scale_n of_o equal_a part_n for_o degree_n but_o the_o line_n must_v be_v divide_v by_o another_o mode_n viz._n according_a to_o the_o rule_n of_o perspective_n because_o in_o this_o mode_n we_o place_v the_o eye_n beneath_o the_o horizon_n of_o london_n in_o the_o place_n of_o the_o antipode_n and_o take_v the_o horizon_n for_o the_o glass_n if_o we_o be_v mind_v to_o represent_v a_o hemisphere_n or_o a_o part_n great_a lesser_a than_o the_o hemisphere_n than_o we_o take_v a_o plain_a parallel_n to_o the_o horizon_n which_o may_v be_v distant_a from_o it_o by_o so_o great_a a_o arch_n as_o be_v the_o part_n to_o be_v represent_v by_o the_o hemisphere_n therefore_o let_v the_o periphery_a of_o the_o circle_n be_v describe_v in_o another_o chart_n m_o the_o centre_n n_n o_o one_o diameter_n p_o q_o another_o perpendicular_a
the_o distance_n of_o place_n in_o degree_n and_o part_n of_o degree_n which_o degree_n must_v be_v turn_v into_o mile_n or_o other_o measure_n which_o we_o will_v have_v but_o if_o the_o interval_n be_v great_a than_o can_v be_v convenient_o take_v with_o compass_n we_o must_v do_v as_o in_o the_o vii_o proposition_n 7._o see_v proposition_n 7._o but_o because_o the_o planisphere_n be_v more_o apt_a for_o use_v especial_o by_o seaman_n and_o many_o love_n to_o solve_v problem_n by_o it_o and_o the_o use_n of_o this_o problem_n be_v frequent_a i_o shall_v also_o propound_v this_o method_n by_o the_o planisphere_n there_o be_v two_o case_n of_o this_o problem_n for_o either_o the_o give_v longitude_n of_o the_o place_n be_v one_o and_o the_o same_o or_o the_o difference_n of_o 180_o to_o wit_n if_o they_o lie_v in_o the_o same_o meridian_n or_o the_o longitude_n be_v diverse_a if_o it_o be_v the_o same_o there_o be_v no_o need_n of_o another_o method_n but_o that_o difference_n of_o latitude_n may_v be_v turn_v into_o mile_n viz._n that_o every_o latitude_n be_v the_o distance_n of_o place_n in_o degree_n but_o if_o the_o latitude_n be_v of_o divers_a species_n to_o wit_n one_o north_n the_o other_o south_n the_o degree_n of_o latitude_n shall_v be_v add_v if_o the_o difference_n be_v of_o 180_o degree_n viz._n if_o in_o divers_a semicircle_n of_o the_o same_o circle_n of_o longitude_n we_o must_v do_v after_o the_o same_o mode_n which_o be_v easy_a for_o any_o one_o to_o collect_v but_o it_o be_v otherwise_o if_o that_o the_o longitude_n of_o the_o place_n be_v unequal_a that_o be_v if_o the_o place_n shall_v be_v situate_v in_o divers_a meridian_n and_o without_o the_o aequator_fw-la problem_n case_n which_o vary_v the_o solution_n of_o this_o problem_n but_o it_o will_v be_v useful_a for_o the_o distinct_a understanding_n of_o the_o problem_n to_o reckon_v the_o case_n which_o vary_v the_o solution_n and_o most_o of_o they_o have_v a_o most_o easy_a solution_n as_o will_v be_v manifest_a by_o example_n which_o the_o studious_a aught_o to_o examine_v 1._o if_o the_o longitude_n of_o the_o place_n be_v the_o same_o and_o they_o be_v the_o same_o cognominated_a latitude_n in_o this_o case_n the_o difference_n of_o latitude_n be_v the_o very_a distance_n in_o degree_n which_o may_v be_v change_v into_o mile_n or_o other_o measure_n 2._o if_o the_o longitude_n of_o place_n be_v the_o same_o but_o the_o latitude_n be_v of_o a_o divers_a name_n one_o northern_a the_o other_o southern_a in_o this_o case_n the_o latitude_n shall_v be_v add_v in_o one_o sum_n this_o shall_v show_v the_o distance_n in_o degree_n 3._o if_o the_o difference_n of_o longitude_n be_v of_o 180_o degree_n and_o be_v of_o a_o like_a cognominated_a latitude_n the_o compliment_n of_o the_o latitude_n shall_v be_v take_v at_o 90_o degree_n or_o the_o distance_n of_o the_o place_n from_o the_o pole_n and_o they_o shall_v be_v add_v the_o same_o will_v show_v the_o distance_n in_o degree_n 4._o if_o the_o difference_n of_o longitude_n be_v of_o 180_o degree_n and_o the_o latitude_n be_v of_o a_o diverse_a name_n let_v the_o difference_n of_o latitude_n be_v take_v and_o substract_v from_o 180_o degree_n or_o the_o semicircle_n the_o remain_a number_n shall_v exhibit_v the_o distance_n in_o degree_n 5._o it_o both_o place_n shall_v be_v in_o the_o aequator_fw-la the_o difference_n of_o longitude_n be_v the_o very_a distance_n 6._o if_o the_o latitude_n of_o place_n shall_v be_v one_o and_o the_o same_o and_o not_o great_a than_o 20_o degree_n and_o the_o difference_n of_o longitude_n small_a we_o must_v enter_v with_o that_o latitude_n the_o table_n of_o magnitude_n lay_v down_o in_o the_o iv_o chapter_n and_o we_o must_v except_v the_o quantity_n of_o one_o degree_n then_o we_o shall_v take_v the_o difference_n of_o longitude_n and_o turn_v these_o deg_fw-la into_o the_o except_a mile_n or_o measure_n 7._o but_o if_o the_o longitude_n and_o latitude_n be_v divers_a or_o if_o the_o latitude_n be_v the_o same_o but_o yet_o great_a than_o 20_o degree_n and_o the_o difference_n of_o longitude_n be_v some_o what_o great_a which_o be_v usual_a in_o many_o example_n in_o this_o case_n we_o must_v not_o use_v the_o same_o compendium_n but_o the_o solution_n be_v more_o difficult_a and_o in_o this_o case_n the_o problem_n be_v chief_o propound_v we_o have_v show_v the_o solution_n by_o the_o globe_n the_o method_n by_o the_o planisphere_n be_v this_o let_v the_o rule_n of_o the_o planisphere_n be_v bring_v to_o the_o latitude_n of_o one_o place_n or_o to_o the_o degree_n of_o the_o elevation_n of_o the_o pole_n then_o let_v the_o difference_n of_o longitude_n be_v number_v in_o the_o meridian_n begin_v from_o the_o other_o part_n and_o wherein_o the_o point_n may_v be_v observe_v where_o this_o meridian_n terminate_a the_o numeration_n cut_v the_o parallel_n of_o the_o other_o place_n of_o latitude_n let_v the_o end_n of_o the_o index_n be_v place_v above_o this_o point_n this_o do_v let_v the_o rule_n be_v apply_v to_o the_o line_n of_o the_o aequator_fw-la the_o number_n of_o the_o parallel_n intercept_v between_o the_o pole_n and_o the_o index_n be_v the_o seek_v for_o distance_n in_o the_o degree_n problem_n the_o solution_n of_o the_o problem_n thus_o the_o problem_n be_v solve_v by_o the_o planisphere_n there_o be_v another_o method_n find_v out_o by_o maurolicus_n which_o by_o the_o stroke_n of_o the_o line_n on_o the_o circle_n teach_v by_o a_o pleasant_a operation_n to_o exhibit_v the_o distance_n from_o which_o lineary_a description_n also_o be_v deduce_v a_o mode_n in_o which_o the_o problem_n be_v solve_v by_o calculation_n let_v a_o certain_a periphery_n of_o the_o circle_n be_v describe_v in_o the_o centre_n e_o one_o semidiamiter_n b_o e_o let_v the_o arch_n b_o a_o be_v take_v equal_a to_o the_o difference_n of_o the_o longitude_n of_o the_o place_n if_o the_o difference_n take_v be_v great_a than_o 180_o deg_fw-la the_o compliment_n of_o this_o difference_n be_v at_o 360_o degree_n and_o let_v the_o semidiameter_n a_o e_o be_v draw_v then_o let_v the_o arch_n a_o f_o towards_z b_o be_v take_v equal_a to_o the_o latitude_n of_o the_o place_n a_o and_o from_o b_o the_o arch_n b_o c_o equal_a to_o the_o latitude_n of_o the_o place_n b_o let_v g_o i_o be_v let_v down_o perpendicular_a from_o g_z on_z b_o e_z and_z f_o h_n from_o f_o on_o a_o e._n let_v i_o h_n be_v draw_v and_o above_o this_o the_o point_n i_o and_o h_n must_v be_v erect_v perpendicular_a i_o l_o equal_a to_o i_o g_z and_z h_z k_o equal_a to_o h_n f_o on_o the_o same_o quarter_n if_o the_o latitude_n of_o the_o place_n shall_v be_v cognominal_a but_o if_o they_o be_v of_o a_o divers_a name_n than_o i_o l_o shall_v be_v draw_v from_o one_o quarter_n to_o the_o right_a line_n i_o h_n and_o h_n k_o from_o the_o other_o this_o do_v the_o right_a line_n l_o k_n shall_v be_v stretch_v to_o the_o demand_v distance_n or_o the_o arch_n of_o it_o shall_v be_v subtend_v which_o shall_v show_v the_o distance_n of_o degree_n therefore_o by_o the_o interval_n of_o the_o compass_n k_o l_o let_v the_o arch_n b_o x_o be_v take_v this_o shall_v represent_v the_o distance_n in_o degree_n this_o mode_n of_o maurolicus_n be_v take_v from_o the_o solution_n of_o spherical_a triangle_n neither_o will_v this_o lineary_a method_n exhibit_v a_o accurate_a distance_n although_o the_o practice_n be_v pleasant_a and_o easy_a but_o only_o the_o method_n by_o number_n or_o the_o trigonometry_n of_o spherical_a triangle_n exhibit_v a_o accurate_a distance_n for_o let_v there_o be_v have_v a_o spherical_a triangle_n in_o which_o two_o side_n be_v give_v viz._n the_o distance_n of_o the_o place_n from_o the_o pole_n the_o compliment_n of_o latitude_n and_o the_o angle_n contain_v who_o measure_n be_v the_o difference_n of_o longitude_n the_o three_o side_n be_v demand_v for_o the_o find_n of_o which_o although_o there_o be_v many_o method_n yet_o the_o most_o general_a be_v this_o first_o if_o that_o the_o latitude_n of_o place_n be_v cognominal_a let_v it_o be_v bring_v to_o pass_v that_o as_o the_o quadrant_n of_o the_o whole_a sinus_n be_v to_o the_o right_a angle_n contain_v under_o the_o sinus_n of_o the_o distance_n of_o the_o place_n from_o the_o pole_n so_o be_v it_o towards_o the_o sinus_n of_o the_o difference_n of_o the_o longitude_n if_o it_o be_v great_a than_o 180_o degree_n let_v his_o compliment_n be_v take_v at_o 360_o degree_n to_o a_o certain_a four_o number_n then_o let_v the_o difference_n of_o latitude_n be_v take_v and_o the_o sinus_n of_o this_o compliment_n moreover_o let_v the_o four_o number_n find_v out_o before_o be_v compare_v with_o this_o sinus_n if_o it_o become_v equal_a the_o distance_n
of_o the_o place_n shall_v be_v 90_o degree_n if_o it_o be_v lesser_a let_v it_o be_v substract_v and_o the_o residue_n shall_v be_v the_o sinus_n of_o the_o arch_n who_o compliment_n be_v the_o distance_n of_o the_o place_n if_o the_o four_o be_v find_v great_a than_o the_o say_v sinus_n let_v this_o be_v subtract_v from_o that_o and_o the_o residue_n shall_v be_v the_o sinus_n of_o the_o arch_n which_o be_v add_v to_o 90_o degree_n will_v exhibit_v the_o distance_n seek_v in_o degree_n which_o must_v be_v convert_v into_o a_o itinerary_a distance_n 2._o if_o the_o latitude_n be_v of_o a_o divers_a name_n viz._n one_o northern_a the_o other_o southern_a let_v the_o place_n of_o the_o antipode_n be_v take_v for_o either_o place_n of_o it_o and_o the_o distance_n of_o it_o may_v be_v find_v from_o the_o other_o place_n according_a to_o the_o say_a method_n for_o the_o latitude_n of_o this_o shall_v be_v the_o same_o with_o that_o of_o that_o place_n but_o of_o the_o same_o name_n with_o the_o other_o place_n therefore_o in_o a_o spherical_a triangle_n there_o shall_v be_v two_o give_v side_n and_o the_o angle_n be_v the_o compliment_n of_o the_o difference_n of_o the_o longitude_n of_o the_o place_n at_o 180_o degree_n or_o a_o excess_n above_o 180_o if_o this_o difference_n shall_v be_v great_a than_o 180_o therefore_o the_o distance_n between_o one_o place_n and_o the_o antipode_n of_o the_o other_o place_n be_v find_v you_o have_v also_o the_o distance_n of_o those_o place_n for_o this_o be_v the_o compliment_n of_o the_o former_a to_o 180_o degree_n as_o have_v be_v say_v in_o the_o former_a proposition_n in_o place_n near_o and_o not_o much_o distant_a from_o the_o aequator_fw-la viz._n not_o beyond_o 18_o degree_n we_o use_v a_o more_o easy_a though_o not_o a_o apodictical_a method_n which_o shall_v exhibit_v a_o distance_n not_o much_o diverse_a from_o the_o true_a viz._n we_o take_v the_o quadrant_n of_o the_o difference_n of_o longitude_n and_o also_o of_o latitude_n we_o add_v the_o quadrant_n and_o from_o the_o aggregate_v extract_v the_o quadrate_n root_n this_o will_v show_v the_o difference_n not_o much_o different_a from_o the_o true_a or_o thus_o act_n in_o a_o more_o certain_a method_n which_o may_v also_o be_v apply_v to_o place_n beyond_o the_o 20_o degree_n of_o latitude_n from_o the_o table_n of_o the_o quantity_n of_o the_o parallel_n except_o the_o proportion_n of_o the_o great_a parallel_n of_o latitude_n to_o the_o aequator_fw-la and_o as_o the_o quantity_n of_o the_o aequator_fw-la be_v to_o the_o quantity_n of_o the_o parallel_n so_o be_v the_o difference_n of_o longitude_n to_o the_o other_o or_o to_o the_o difference_n of_o longitude_n take_v in_o the_o parallel_n of_o a_o great_a latitude_n let_v this_o quantity_n be_v assume_v for_o the_o difference_n of_o latitude_n and_o do_v as_o before_o problem_n table_n of_o logarithm_n useful_a for_o the_o solution_n of_o this_o problem_n the_o solution_n of_o this_o problem_n be_v easy_a if_o we_o apply_v table_n of_o logarithm_n and_o resolve_v a_o triangle_n oblique_a angle_n into_o two_o right_a angle_n so_o there_o will_v be_v need_n of_o no_o multiplication_n or_o division_n proposition_n x._o the_o latitude_n of_o two_o place_n be_v give_v and_o the_o quarter_n in_o which_o one_o be_v situate_v from_o the_o other_o to_o find_v the_o distance_n this_o problem_n be_v the_o same_o with_o the_o trigonometrical_a abstract_n two_o side_n be_v give_v in_o a_o spherical_a triangle_n and_o a_o angle_n which_o be_v opposite_a to_o one_o give_v side_n to_o find_v a_o three_o side_n for_o the_o two_o give_v side_n be_v in_o this_o geographical_a problem_n the_o distance_n of_o these_o two_o place_n from_o the_o pole_n and_o the_o angle_n opposite_a to_o either_o side_n be_v the_o angle_n of_o position_n or_o the_o angle_n of_o one_o quarter_n of_o the_o place_n to_o the_o other_o or_o the_o compliment_n of_o this_o angle_n at_o 180_o degree_n the_o solution_n of_o this_o problem_n be_v thus_o perform_v by_o the_o globe_n let_v the_o first_o meridian_n be_v take_v for_o the_o meridian_n of_o the_o place_n who_o quarter_n be_v not_o give_v at_o the_o other_o and_o in_o this_o meridian_n let_v the_o point_n of_o latitude_n be_v note_v for_o this_o place_n then_o let_v the_o pole_n be_v elevate_v for_o the_o latitude_n of_o the_o other_o place_n and_o the_o quadrant_n affix_v to_o the_o vertex_fw-la but_o let_v the_o other_o end_n be_v apply_v to_o the_o quarter_n or_o degree_n of_o the_o horizon_n for_o the_o give_v quarter_n then_o let_v the_o globe_n be_v turn_v round_o until_o the_o point_n note_v in_o the_o first_o meridian_n come_v to_o the_o quadrant_n so_o the_o arch_n of_o the_o quadrant_n intercept_v between_o the_o vertex_fw-la and_o that_o point_n be_v the_o demand_v distance_n of_o the_o two_o place_n you_o shall_v also_o have_v the_o difference_n of_o longitude_n in_o the_o aequator_fw-la viz._n the_o arch_n of_o the_o aequator_fw-la intercept_v between_o the_o brazen_a and_o first_o meridian_n proposition_n xi_o the_o longitude_n of_o two_o place_n be_v give_v the_o latitude_n of_o one_o place_n and_o the_o quarter_n in_o which_o this_o other_o place_n lie_v at_o this_o to_o find_v out_o the_o distance_n here_o we_o have_v again_o a_o spherical_a triangle_n who_o side_n be_v the_o distance_n of_o the_o place_n from_o the_o pole_n and_o the_o mutual_a distance_n of_o the_o place_n themselves_o in_o which_o one_o side_n be_v give_v viz_o the_o distance_n of_o one_o place_n from_o the_o pole_n and_o two_o angle_n be_v give_v one_o who_o measure_n be_v the_o difference_n of_o longitude_n the_o other_o be_v know_v from_o the_o give_v quarter_n of_o the_o other_o place_n from_o these_o three_o give_v the_o side_n be_v demand_v which_o be_v opposite_a to_o the_o angle_n of_o the_o difference_n of_o longitude_n the_o solution_n may_v easy_o be_v perform_v by_o the_o globe_n and_o by_o the_o planisphere_n and_o very_o exact_o by_o a_o logarithmical_a calculation_n as_o also_o by_o the_o common_a computation_n we_o will_v only_o show_v the_o method_n which_o the_o globe_n afford_v although_o it_o be_v more_o easy_a by_o the_o planisphere_n but_o that_o which_o be_v do_v by_o the_o globe_n represent_v the_o triangle_n herein_o the_o method_n which_o the_o globe_n afford_v herein_o let_v the_o first_o meridian_n be_v take_v for_o the_o meridian_n of_o the_o place_n who_o latitude_n be_v not_o give_v and_o let_v the_o degree_n of_o the_o difference_n of_o the_o longitude_n of_o the_o place_n be_v account_v in_o the_o aequator_fw-la let_v the_o term_n be_v note_v with_o chalk_n and_o bring_v to_o the_o brazen_a meridian_n so_o this_o shall_v represent_v the_o meridian_n of_o the_o other_o place_n let_v the_o degree_n of_o the_o give_v latitude_n be_v reckon_v on_o it_o and_o the_o globe_n remain_v fix_v let_v the_o pole_n be_v elevate_v for_o that_o latitude_n let_v the_o quadrant_n be_v affix_v to_o the_o vertex_fw-la and_o the_o other_o end_n to_o the_o give_v quarter_n of_o the_o horizon_n in_o this_o situation_n of_o the_o globe_n the_o point_n in_o which_o the_o quadrant_n cut_v the_o first_o meridian_n shall_v represent_v that_o other_o place_n and_o the_o arch_n of_o the_o quadrant_n which_o be_v intercept_v between_o the_o vertex_fw-la and_o the_o point_n be_v the_o distance_n demand_v also_o by_o the_o same_o method_n the_o latitude_n of_o this_o other_o place_n be_v have_v proposition_n xii_o the_o distance_n of_o two_o place_n situate_v in_o the_o same_o meridian_n or_o of_o the_o same_o longitude_n be_v give_v in_o the_o quarter_n in_o which_o that_o three_o place_n lie_v from_o those_o two_o to_o find_v the_o distance_n of_o this_o three_o place_n from_o both_o of_o they_o here_o again_o we_o have_v a_o spherical_a triangle_n who_o three_o side_n be_v the_o distance_n between_o those_o three_o place_n and_o one_o place_n be_v give_v viz._n the_o distance_n of_o two_o place_n which_o must_v be_v turn_v into_o degree_n except_o it_o be_v so_o give_v and_o the_o two_o adjacent_a angle_n be_v give_v the_o two_o other_o side_n be_v seek_v leave_v the_o method_n which_o perform_v it_o by_o calculation_n and_o the_o planisphere_n although_o they_o be_v more_o accurate_a we_o shall_v only_o deliver_v that_o which_o solve_v it_o by_o the_o globe_n and_o place_v it_o more_o before_o the_o eye_n let_v the_o degree_n of_o two_o place_n distant_a be_v take_v on_o the_o brazen_a meridian_n where_o you_o please_v and_o let_v the_o term_n be_v note_v so_o that_o these_o may_v represent_v the_o place_n who_o distance_n be_v give_v then_o let_v the_o pole_n be_v elevate_v for_o the_o latitude_n of_o one_o of_o these_o term_n let_v the_o quadrant_n be_v affix_v to_o the_o vertex_fw-la and_o apply_v to_o the_o give_v quarter_n in_o
now_o this_o problem_n be_v the_o same_o with_o that_o to_o find_v out_o the_o meridian_n line_n and_o the_o north_n and_o south_n quarter_n for_o these_o be_v know_v it_o be_v easy_a to_o know_v the_o rest_n first_o by_o the_o star_n viz._n in_o the_o night_n the_o bear_n or_o the_o helice_n and_o polary_a star_n so_o call_v in_o the_o extremity_n of_o the_o tail_n of_o the_o vrsa_fw-la minor_n of_o great_a same_o among_o the_o ancient_n which_o show_v the_o north_n quarter_n whence_o all_o the_o rest_n be_v find_v for_o the_o face_n be_v turn_v to_o the_o north_n the_o east_n be_v at_o the_o right_a hand_n and_o the_o west_n on_o the_o left_a the_o line_n of_o which_o quarter_n at_o right_a angle_n cut_v the_o line_n of_o the_o north_n and_o south_n and_o these_o cardinal_n quarter_n be_v find_v it_o be_v easy_a to_o find_v the_o intermedial_a quarter_n unto_o which_o purpose_n that_o there_o may_v be_v no_o need_n of_o a_o description_n they_o have_v a_o circle_n make_v with_o the_o quarter_n who_o northern_a line_n be_v place_v above_o the_o northern_a line_n of_o any_o place_n the_o other_o quarter_n at_o one_o sight_n be_v discover_v but_o in_o the_o day_n they_o seek_v out_o the_o quarter_n by_o the_o rise_n or_o set_v of_o the_o sun_n as_o we_o have_v show_v in_o the_o xxviii_o chapter_n 28._o see_v chap._n 28._o 2._o the_o other_o method_n of_o the_o ancient_n for_o the_o know_v of_o quarter_n be_v the_o knowledge_n of_o the_o situation_n or_o extension_n of_o the_o shore_n and_o one_o promontory_n to_o the_o other_o for_o see_v the_o quarter_n of_o this_o extension_n be_v know_v to_o they_o either_o from_o the_o map_n or_o from_o observation_n and_o experience_n they_o may_v in_o navigation_n by_o see_v they_o know_v the_o other_o quarter_n for_o one_o quarter_n be_v know_v all_o the_o rest_n be_v know_v therefore_o the_o ancient_n do_v not_o far_o depart_v from_o the_o coast_n viz._n that_o they_o may_v know_v the_o quarter_n by_o the_o benefit_n of_o the_o know_a quarter_n of_o the_o extension_n of_o shore_n for_o they_o can_v not_o always_o use_v the_o method_n of_o the_o star_n and_o the_o rise_n and_o set_v of_o the_o sun_n 3._o the_o three_o method_n of_o the_o ancient_n of_o the_o knowledge_n of_o the_o quarter_n be_v the_o observe_v course_n of_o the_o ship_n for_o go_v from_o any_o place_n and_o guide_v the_o ship_n to_o the_o know_a quarter_n they_o be_v able_a from_o the_o mutation_n of_o the_o course_n of_o the_o ship_n to_o know_v the_o quarter_n 4._o hence_o it_o be_v manifest_a that_o the_o chief_a cause_n of_o the_o dangerous_a and_o imperfect_a navigation_n of_o the_o ancient_n be_v the_o ignorance_n of_o a_o method_n by_o which_o every_o where_n in_o the_o middle_n of_o the_o vast_a ocean_n they_o may_v know_v the_o quarter_n and_o so_o that_o quarter_n unto_o which_o the_o ship_n be_v to_o be_v steer_v for_o as_o i_o have_v say_v the_o method_n by_o the_o star_n and_o the_o rise_n and_o set_v sun_n can_v be_v apply_v on_o all_o day_n and_o on_o the_o hour_n of_o every_o day_n for_o the_o mark_n from_o the_o situation_n of_o the_o shore_n fail_v in_o the_o mid_a sea_n in_o the_o night_n neither_o be_v it_o safe_a enough_o in_o the_o day_n time_n the_o three_o method_n from_o the_o observe_v course_n of_o the_o ship_n have_v not_o place_n when_o the_o ship_n be_v toss_v by_o boisterous_a wind_n and_o tempest_n from_o one_o quarter_n to_o another_o and_o in_o this_o casual_o lie_v the_o chief_a difficulty_n this_o i_o think_v fit_a to_o admonish_v concern_v the_o modes_n of_o the_o ancient_n for_o the_o find_v out_o the_o meridian_n line_n and_o the_o north_n and_o south_n by_o reason_n that_o the_o imperfection_n of_o these_o be_v the_o cause_n of_o the_o dangerous_a and_o small_a navigation_n of_o the_o ancient_n see_v that_o they_o be_v never_o able_a to_o commit_v themselves_o to_o the_o vast_a ocean_n and_o therefore_o never_o know_v those_o region_n between_o which_o the_o ocean_n be_v interpose_v of_o which_o the_o chief_a be_v all_o america_n never_o yet_o full_o know_v but_o at_o this_o day_n the_o method_n of_o know_v the_o quarter_n in_o all_o place_n and_o of_o find_v out_o the_o line_n of_o the_o north_n and_o south_n be_v facile_a by_o the_o benefit_n of_o the_o admirable_a propriety_n which_o the_o loadstone_n and_o all_o iron_n touch_v by_o it_o have_v be_v find_v to_o have_v viz._n that_o all_o magnetick_n not_o hinder_v by_o other_o in_o any_o place_n direct_v their_o point_n almost_o to_o the_o same_o quarter_n for_o there_o be_v two_o opposite_a point_n in_o the_o loadstone_n whereof_o one_o always_o and_o in_o all_o place_n turn_v itself_o to_o the_o north_n or_o the_o adjacent_a quarter_n the_o other_o to_o the_o south_n and_o so_o also_o the_o other_o point_n of_o the_o magnes_n respect_v the_o other_o quarter_n viz._n every_o point_n its_o particular_a quarter_n but_o all_o of_o they_o be_v not_o consider_v but_o only_o those_o two_o point_n which_o as_o i_o have_v say_v do_v convert_v themselves_o to_o the_o north_n and_o south_n which_o be_v term_v the_o pole_n of_o the_o magnes_n one_o northern_a the_o other_o southern_a and_o the_o same_o virtue_n much_o to_o admiration_n be_v communicate_v to_o the_o needle_n but_o by_o a_o invert_v and_o contrary_a operation_n of_o nature_n for_o the_o end_n of_o the_o lamine_n or_o needle_n which_o be_v touch_v at_o the_o north_n pole_n of_o the_o magnes_n do_v not_o convert_v itself_o to_o the_o north_n but_o to_o the_o south_n and_o that_o end_n which_o be_v rub_v at_o the_o south_n pole_n of_o the_o loadstone_n turn_v not_o to_o the_o south_n but_o to_o the_o north._n these_o point_n of_o the_o needle_n be_v also_o term_v the_o pole_n loadstone_n the_o virtue_n of_o the_o loadstone_n although_o therefore_o the_o loadstone_n and_o the_o iron_n touch_v by_o it_o have_v very_o many_o notable_a property_n yet_o all_o may_v be_v refer_v to_o two_o species_n or_o head_n one_o be_v that_o 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it_o will_v show_v the_o quarter_n demand_v but_o the_o magnetic_a needle_n be_v more_o easy_a for_o use_v who_o end_n be_v touch_v at_o the_o south_n pole_n of_o the_o magnes_n for_o if_o that_o this_o needle_n be_v place_v in_o the_o middle_n upon_o a_o sharp_a perpendicular_a pin_n so_o that_o it_o can_v free_o turn_v round_o the_o needle_n rest_v will_v show_v by_o one_o of_o its_o end_n the_o north_n quarter_n and_o by_o the_o other_o the_o south_n from_o what_o have_v be_v say_v it_o be_v easy_a to_o make_v a_o nautical_a instrument_n proposition_n ii_o to_o make_v a_o mariner_n compass_n compass_n of_o the_o make_n the_o mariner_n compass_n let_v the_o describe_v circle_n on_o any_o paper_n be_v divide_v into_o 32_o quarter_n or_o degree_n and_o let_v one_o of_o these_o deg_fw-la be_v take_v for_o the_o north_n quarter_n be_v ascribe_v with_o these_o appellation_n viz._n with_o a_o peculiar_a sign_n the_o flower_n de_fw-fr luce_n and_o the_o find_v out_o point_v for_o the_o other_o quarter_n viz._n south_n east_n west_n north-east_n north-west_n as_o we_o have_v propound_v they_o in_o the_o diagram_n in_o the_o xx_o chapter_n mariner_n term_v this_o chart_n the_o rose_n then_o let_v the_o magnetic_a needle_n be_v so_o affix_v beneath_o the_o chart_n that_o the_o middle_n of_o the_o needle_n may_v be_v beneath_o its_o centre_n and_o the_o north_n pole_n of_o the_o needle_n may_v be_v subject_v to_o the_o line_n of_o the_o paper_n unto_o which_o we_o ascribe_v the_o northern_a quarter_n moreover_o the_o paper_n be_v so_o make_v with_o the_o needle_n lie_v under_o let_v it_o be_v put_v upon_o the_o point_a pin_n that_o it_o may_v have_v a_o free_a circumrotation_n so_o the_o index_n of_o the_o