de_fw-fr sectore_fw-la &_o radio_fw-la the_o description_n and_o use_n of_o the_o sector_n in_o three_o book_n the_o description_n and_o use_n of_o the_o crosse-staffe_n in_o other_o three_o book_n for_o such_o as_o be_v studious_a of_o mathematical_a practice_n london_n print_v by_o william_n jones_n and_o be_v to_o be_v sell_v by_o john_n tap_n at_o saint_n magnus_n corner_n 1623._o the_o description_n and_o use_n of_o the_o sector_n for_o such_o as_o be_v studious_a of_o mathematical_a practice_n london_n print_v by_o william_n jones_n 1623._o the_o first_o book_n of_o the_o sector_n chap._n i._n the_o description_n the_o make_n and_o the_o general_a use_n of_o the_o sector_n a_o sector_n in_o geometry_n be_v a_o figure_n comprehend_v of_o two_o right_a line_n contain_v a_o angle_n at_o the_o centre_n and_o of_o the_o circumference_n assume_v by_o they_o this_o geometrical_a instrument_n have_v two_o leg_n contain_v all_o variety_n of_o angle_n &_o the_o distance_n of_o the_o foot_n represent_v the_o subtense_n of_o the_o circumference_n be_v therefore_o call_v by_o the_o same_o name_n it_o contain_v 12_o several_a line_n or_o scale_n of_o which_o 7_o be_v general_n the_o other_o 5_o more_o particular_a the_o first_o be_v the_o scale_n of_o line_n divide_v into_o 100_o equal_a part_n and_o number_v by_o 1.2.3.4.5.6.7.8.9.10_o the_o second_o the_o line_n of_o superficies_n divide_v into_o 100_o unequal_a part_n and_o number_v by_o 1.1.2.3.4.5.6.7.8.9.10_o the_o three_o the_o line_n of_o solid_n divide_v into_o 1000_o unequal_a part_n &_o number_v by_o 1._o 1._o 1._o 2._o 3._o 4._o 5._o 6._o 7._o 8._o 9_o 10._o the_o four_o the_o line_n of_o sines_n and_o chord_n divide_v into_o 90_o degree_n and_o number_v with_o 10._o 20._o 30._o unto_o 90._o these_o four_o line_n of_o line_n of_o superficies_n of_o solid_n and_o of_o sines_n be_v all_o draw_v from_o the_o centre_n of_o the_o sector_n almost_o to_o the_o end_n of_o the_o leg_n they_o be_v draw_v on_o both_o the_o leg_n that_o every_o line_n may_v have_v his_o fellow_n all_o of_o they_o be_v of_o one_o length_n that_o they_o may_v answer_v one_o to_o the_o other_o and_o every_o one_o have_v his_o parallel_n that_o the_o eye_n may_v the_o better_o distinguish_v the_o division_n but_o of_o the_o parallel_v those_o only_o which_o be_v inward_a must_v contain_v the_o true_a division_n there_o be_v three_o other_o general_a line_n which_o because_o they_o be_v infinite_a be_v place_v on_o the_o side_n of_o the_o sector_n the_o first_o a_o line_n of_o tangent_n number_v with_o 10.20.30.40.50.60_o signify_v so_o many_o degree_n from_o the_o begin_n of_o the_o line_n of_o which_o 45_o be_v equal_a to_o the_o whole_a line_n of_o sines_n the_o rest_n follow_v as_o the_o length_n of_o the_o sector_n will_v bear_v the_o second_o a_o line_n of_o secant_v divide_v by_o prick_n into_o 60_o degree_n who_o beginning_n be_v the_o same_o with_o that_o of_o the_o line_n of_o tangent_n to_o which_o it_o be_v join_v the_o three_o be_v the_o meridian_n line_n or_o line_n of_o rumb_n divide_v unequal_o into_o degree_n of_o which_o the_o first_o 70_o be_v almost_o equal_a to_o the_o whole_a line_n of_o sines_n the_o rest_n follow_v unto_o 84_o according_a to_o the_o length_n of_o the_o sector_n of_o the_o particular_a line_n insert_v among_o the_o general_a because_o there_o be_v void_a space_n the_o first_o be_v the_o line_n of_o quadrature_n place_v between_o the_o line_n of_o sines_n and_o note_v with_o 10.9.8.7_o s._n 6.5.90_o q._n the_o second_o the_o line_n of_o segment_n place_v between_o the_o line_n of_o sines_n and_o superficies_n divide_v into_o 50_o part_n and_o number_v with_o 5.6.7.8.9.10_o the_o three_o the_o line_n of_o inscribe_v body_n in_o the_o same_o sphere_n place_v between_o the_o scale_n of_o line_n and_o note_v with_o d._n s._n i._o c._n o._n t._n the_o four_o the_o line_n of_o equated_a body_n place_v between_o the_o line_n of_o line_n and_o solid_n and_o note_v with_o d._n i._o c._n s._n o._n t._n the_o five_o be_v the_o line_n of_o metal_n insert_v with_o the_o line_n of_o equated_a body_n there_o be_v room_n sufficient_a and_o note_v with_o these_o character_n ☉_o ☿_o ♄_o ☽_o ♀_o ♂_o ♃_o there_o remain_v the_o edge_n of_o the_o sector_n and_o on_o the_o one_o i_o have_v set_v a_o line_n of_o inch_n which_o be_v the_o twelth_n part_n of_o a_o foot_n english_a on_o the_o other_o a_o lesser_a line_n of_o tangent_n to_o which_o the_o gnomon_n be_v radius_fw-la 2_o of_o the_o make_n of_o the_o sector_n let_v a_o ruler_n be_v first_o make_v either_o of_o brass_n or_o of_o wood_n like_a unto_o the_o former_a figure_n which_o may_v open_v and_o shut_v upon_o his_o centre_n the_o head_n of_o it_o may_v be_v about_o the_o twelth_n part_n of_o the_o whole_a length_n that_o it_o may_v bear_v the_o movable_a foot_n and_o yet_o the_o most_o part_n of_o the_o division_n may_v fall_v without_o it_o then_o let_v a_o movable_a gnomon_n be_v set_v at_o the_o end_n of_o the_o movable_a foot_n and_o there_o turn_v upon_o a_o axis_n so_o as_o it_o may_v sometime_o stand_v at_o a_o right_a angle_n with_o the_o foot_n and_o sometime_o be_v enclose_v within_o the_o foot_n but_o this_o be_v well_o know_v to_o the_o workman_n for_o draw_v of_o the_o line_n upon_o the_o centre_n of_o the_o sector_n and_o semidiameter_n somewhat_o short_a than_o one_o of_o the_o foot_n draw_v a_o occult_a ark_n of_o a_o circle_n cross_v the_o closure_n of_o the_o inward_a edge_n of_o the_o sector_n about_o the_o letter_n t._n in_o this_o ark_n at_o one_o degree_n on_o either_o side_n from_o the_o edge_n draw_v right_a line_n from_o the_o centre_n fit_v they_o with_o parallel_n and_o divide_v they_o into_o a_o hundred_o equal_a part_n with_o subdivision_n into_o 2.5_o or_o 10._o as_o the_o line_n will_v bear_v but_o let_v the_o number_n set_v to_o they_o be_v only_o 1.2.3.4_o etc._n etc._n unto_o 10._o as_o in_o the_o example_n these_o line_n so_o divide_v i_o call_v the_o line_n or_o scale_n of_o line_n and_o they_o be_v the_o ground_n of_o all_o the_o rest_n in_o this_o ark_n at_o 5_o degree_n on_o either_o side_n from_o the_o edge_n near_o t_o draw_v other_o right_a line_n from_o the_o centre_n and_o fit_v they_o with_o parallel_n these_o shall_v serve_v for_o the_o line_n of_o solid_n then_o on_o the_o other_o side_n of_o the_o sector_n in_o like_a manner_n upon_o the_o centre_n &_o equal_a semidiameter_n draw_v another_o like_a ark_n of_o a_o circle_n &_o here_o again_o at_o one_o near_o degree_n on_o either_o side_n from_o the_o edge_n near_o the_o letter_n q_o draw_v right_a line_n from_o the_o centre_n and_o fit_v they_o with_o parallel_n these_o shall_v serve_v for_o the_o line_n of_o sines_n at_o 5_o degree_n on_o either_o side_n from_o the_o edge_n near_o q_o draw_v other_o right_a line_n from_o the_o centre_n and_o fit_v they_o with_o parallel_n these_o shall_v serve_v for_o the_o line_n of_o superficies_n these_o four_o principal_a line_n be_v draw_v and_o fit_v with_o parallel_n we_o may_v draw_v other_o line_n in_o the_o middle_n between_o the_o edge_n and_o the_o line_n of_o line_n which_o shall_v serve_v for_o the_o line_n of_o inscribe_v body_n and_o other_o between_o the_o edge_n and_o the_o sin_n for_o the_o line_n of_o quadrature_n and_o so_o the_o rest_n as_o in_o the_o example_n 3_o to_o divide_v the_o line_n of_o superficies_n see_v like_o superficies_n do_v hold_v in_o the_o proportion_n of_o their_o homologall_a side_n duplicate_v by_o the_o 29_o pro._n 6_o lib._n euclid_n if_o you_o shall_v find_v mean_a proportional_n between_o the_o whole_a side_n and_o each_o hundred_o part_n of_o the_o like_a side_n by_o the_o 13_o pro._n 6_o lib._n euclid_n all_o of_o they_o cut_v the_o same_o line_n that_o line_n so_o cut_v shall_v contain_v the_o division_n require_v wherefore_o upon_o the_o centre_n a_o and_o semidiamiter_n equal_a to_o the_o line_n of_o line_n describe_v a_o semicircle_n acbd_v with_o ab_fw-la perpendicular_a to_o the_o diameter_n cd_o and_o let_v the_o semidiameter_n ad_fw-la he_o divide_v as_o the_o line_n of_o line_n into_o a_o hundred_o part_n &_o ae_n the_o one_o half_a of_o ac_fw-la divide_v also_o into_o a_o hundred_o part_n so_o shall_v the_o division_n in_o ae_n be_v the_o centre_n from_o whence_o you_o shall_v describe_v the_o semicircle_n c_o 10._o c_o 20._o c_o 30._o &c._n &c._n divide_v the_o lin_v ab_fw-la into_o a_o hundred_o unequal_a part_n and_o this_o line_n ab_fw-la so_o divide_v shall_v be_v the_o line_n of_o superficies_n and_o must_v be_v transfer_v into_o the_o sector_n but_o let_v the_o number_n set_v to_o they_o be_v only_o 1.1.2.3_o unto_o 10_o as_o in_o the_o example_n 4_o to_o
the_o vertical_a circle_n pass_v through_o the_o zenith_n and_o nadir_n east_n and_o west_n and_o the_o line_n mah_n cross_v it_o at_o right_a angle_n shall_v represent_v the_o horizon_n these_o two_o be_v divide_v in_o the_o same_o sort_n as_o the_o ecliptic_a and_o the_o equator_fw-la the_o line_n draw_v through_o each_o degree_n of_o the_o semidiameter_n az_n parallel_n to_o the_o horizon_n shall_v be_v the_o circle_n of_o altitude_n and_o the_o division_n in_o the_o horizon_n and_o his_o parallel_n shall_v give_v the_o azimuth_n last_o if_o through_o 18_o gr_n in_o a_o be_v draw_v a_o right_a line_n ik_fw-mi parallel_n to_o the_o horizon_n it_o shall_v show_v the_o time_n when_o the_o day_n break_v and_o the_o end_n of_o the_o twilight_n for_o example_n of_o this_o projection_n let_v the_o place_n of_o the_o sun_n be_v the_o last_o degree_n of_o ♉_o the_o parallel_n pass_v through_o this_o place_n be_v ld_n and_o therefore_o the_o meridian_n altitude_n ml_o and_o the_o depression_n below_o the_o horizon_n at_o midnight_n hd_v the_o semidiurnall_a ark_n lc_n the_o seminocturnall_a ark_n cd_o the_o declination_n ab_fw-la the_o ascentionall_a difference_n bc_n the_o amplitude_n of_o ascenon_fw-mi ac_fw-la the_o difference_n between_o the_o end_n of_o twilight_n and_o the_o day_n break_v be_v very_o small_a for_o it_o seem_v the_o parallel_n of_o the_o sun_n do_v hardly_o cross_a the_o line_n of_o twilight_n if_o the_o altitude_n of_o the_o sun_n be_v give_v let_v a_o line_n be_v draw_v for_o it_o parallel_v to_o the_o horizon_n so_o it_o shall_v cross_v the_o parallel_n of_o the_o sun_n and_o there_o show_v both_o the_o azimuth_n and_o the_o hour_n of_o the_o day_n as_o if_o the_o place_n of_o the_o sun_n be_v give_v as_o before_o the_o altitude_n in_o the_o morning_n be_v find_v to_o be_v 20_o gr_n the_o line_n fg_v draw_v parallel_n to_o the_o horizon_n through_o 20_o gr_n in_o az_n will_v cross_v the_o parallel_n of_o the_o sun_n in_o ☉_o wherefore_o f_o ☉_o show_v the_o azimuth_n &_o l_o ☉_o the_o quantity_n of_o hour_n from_o the_o meridian_n it_o seem_v to_o be_v about_o half_a a_o hour_n past_o 6_o in_o the_o morning_n and_o yet_o more_o than_o half_a a_o point_n short_a of_o the_o east_n the_o distance_n of_o two_o place_n may_v be_v also_o show_v by_o this_o projection_n their_o latitude_n be_v know_v and_o their_o difference_n of_o longitude_n for_o suppose_v a_o place_n in_o the_o east_n of_o arabia_n have_v 20_o gr_n of_o north_n latitude_n who_o difference_n of_o longitude_n from_o london_n be_v find_v by_o a_o eclipse_n to_o be_v 5_o ho._n ½_n let_v z_o be_v the_o zenith_n of_o london_n the_o parallel_n of_o latitude_n for_o that_o other_o place_n must_v be_v ld_n in_o which_o the_o difference_n of_o longitude_n be_v l_o ☉_o wherefore_o ☉_o represent_v the_o site_n of_o that_o place_n i_o draw_v through_o ☉_o a_o parallel_n to_o the_o horizon_n mh_o cross_v the_o vertical_a az_n near_o about_o 70_o gr_n from_o the_o zenith_n which_o multiply_v by_o 20_o show_v the_o distance_n of_o london_n and_o that_o place_n to_o be_v 1400_o league_n or_o multiply_v by_o 60_o to_o be_v 4200_o mile_n 2_o the_o sphere_n may_v be_v project_v in_o plano_fw-la by_o circular_a line_n as_o in_o the_o general_a astrolabe_n of_o gemma_fw-la frisius_n by_o the_o help_n of_o the_o tangent_fw-la on_o the_o side_n of_o the_o sector_n for_o let_v the_o circle_n give_fw-mi represent_v the_o plane_n of_o the_o general_a meridian_n as_o before_o let_v it_o be_v divide_v into_o four_o part_n and_o cross_v at_o right_a angle_n with_o eae_fw-la the_o equator_fw-la and_o ps_n the_o circle_n of_o the_o hour_n of_o 6_o wherein_o p_o stand_v for_o the_o north_n pole_n and_o s_n for_o the_o south_n pole_n let_v each_o quarter_n of_o the_o meridian_n be_v divide_v into_o 90_o gr_n and_o so_o the_o whole_a into_o 360_o begin_v from_o p_o and_o set_v to_o the_o number_n of_o 10_o 20_o 30._o etc._n etc._n 90_o at_o ae_n 180_o at_o s_n 270_o at_o e_o 360_o at_o p._n the_o semidiameter_n ap_n aae_n as_o a_o e_o may_v be_v divide_v according_a to_o the_o tangent_n of_o half_a their_o arke_n that_o be_v a_o tangent_fw-la of_o 45_o gr_n which_o be_v always_o equal_a to_o the_o radius_fw-la shall_v give_v the_o semidiameter_n of_o 90_o gr_n a_o tangent_fw-la of_o 40_o gr_n shall_v give_v 80_o gr_n in_o the_o semidiameter_n a_o tangent_fw-la of_o 35_o gr_n shall_v give_v 70._o etc._n etc._n so_o that_o the_o semidiameter_n may_v be_v divide_v in_o such_o sort_n as_o the_o tangent_fw-la on_o the_o side_n of_o the_o sector_n the_o difference_n be_v only_o in_o their_o number_n have_v divide_v the_o circumference_n and_o the_o semidiameter_n we_o may_v easy_o draw_v the_o meridians_fw-la and_o the_o parallel_n by_o the_o help_n of_o the_o sector_n the_o meridians_fw-la be_v to_o be_v draw_v through_o both_o the_o pole_n p_o and_o s_o and_o the_o degree_n before_o graduate_v in_o the_o equator_fw-la the_o distance_n of_o the_o centre_n of_o each_o meridian_n from_o a_o the_o centre_n of_o the_o plane_n be_v equal_a to_o the_o tangent_fw-la of_o the_o same_o meridian_n reckon_v from_o the_o general_a meridian_n paese_n and_o the_o semidiameter_n equal_a to_o the_o secant_fw-la of_o the_o same_o degree_n as_o for_o example_n if_o i_o shall_v draw_v the_o meridian_n pb_n which_o be_v the_o ten_o from_o paes_n the_o tangent_fw-la of_o 10_o gr_n give_v i_o ac_fw-la and_o the_o secant_fw-la of_o 10_o gr_n give_v i_o sc_n whereof_o c_z be_v the_o centre_n of_o the_o meridian_n pb_n and_o c_n his_o semidiameter_n so_o of_o a_o tangent_fw-la of_o 20_o gr_n show_v f_o to_o be_v the_o centre_n of_o pd_n the_o twentieth_o meridian_n from_o paes_n and_o agnostus_n a_o tangent_fw-la of_o 23_o gr_n 30_o m._n sheweth_z g_z to_o be_v the_o centre_n of_o p_o 69_o s._n etc._n etc._n the_o parallel_n be_v to_o be_v draw_v through_o the_o degree_n in_o ap_n as_o and_o their_o correspondent_a degree_n in_o the_o general_a meridian_n the_o distance_n of_o the_o centre_n of_o each_o parallel_n from_o a_o the_o centre_n of_o the_o plane_n be_v equal_a to_o the_o secant_fw-la of_o the_o same_o parallel_n from_o the_o pole_n and_o the_o semidiameter_n equal_a to_o the_o tangent_fw-la of_o the_o same_o degree_n as_o if_o i_o shall_v draw_v the_o parallel_n of_o 80_o gr_n which_o be_v the_o ten_o from_o the_o pole_n s_o first_o i_o open_v the_o compass_n unto_o ac_fw-la the_o tangent_fw-la of_o 10_o gr_n and_o this_o give_v i_o the_o semidiameter_n of_o this_o parallel_n who_o centre_n be_v a_o little_a from_o s_n in_o such_o distance_n as_o the_o secant_fw-la sc_n be_v long_o than_o the_o radius_fw-la sa_n the_o meridians_fw-la and_o parallel_n be_v draw_v if_o we_o number_v 23_o gr_n 30_o m._n from_o e_z to_z ♋_o northward_o from_o ae_n to_o ♑_o southward_n the_o line_n draw_v from_o ♋_o to_o ♑_o shall_v be_v the_o ecliptic_a which_o be_v divide_v in_o such_o sort_n as_o the_o semidiameter_n ap_n the_o first_o 30_o gr_n from_o a_o to_o ♋_o shall_v stand_v for_o the_o sine_z of_o ♈_o the_o 30_o gr_n next_o follow_v for_o ♉_o the_o rest_n for_o ♊_o ♋_o ♌_o etc._n etc._n in_o their_o order_n if_o far_o we_o have_v respect_n unto_o the_o latitude_n we_o may_v number_v it_o from_o e_z northward_o unto_o z_o and_o there_o place_v the_o zenith_n by_o which_o and_o the_o centre_n the_o line_n draw_fw-mi zan_fw-mi shall_v represent_v the_o vertical_a circle_n and_o the_o line_n mah_n cross_v it_o at_o right_a angle_n shall_v represent_v the_o horizon_n and_o these_o divide_v in_o the_o same_o sort_n as_o ap_n the_o circle_n draw_v through_o each_o degree_n of_o the_o semidiameter_n az_n parallel_n to_o the_o horizon_n shall_v be_v the_o circle_n of_o altitude_n and_o the_o circle_n draw_v through_o the_o horizon_n and_o his_o pole_n shall_v give_v the_o azimuth_n for_o example_n of_o this_o projection_n let_v the_o place_n of_o the_o sun_n be_v in_o the_o begin_n of_o ♒_o the_o parallel_n pass_v through_o this_o place_n be_v ♒_o ☉_o l_o and_o therefore_o the_o meridian_n altitude_n ml_o and_o the_o depression_n below_o the_o horizon_n at_o midnight_n h_n ♒_o the_o semidiurnall_a ark_n l_o ☉_o the_o seminocturnall_a ark_n o_o ♒_o the_o declination_n be_v the_o ascensionall_a difference_n r_o ☉_o the_o amplitude_n of_o ascension_n a_o ☉_o or_o if_o a_o be_v put_v to_o represent_v the_o pole_n of_o the_o world_n then_o shall_v paese_n stand_v for_o the_o equator_fw-la and_o p_o ♋_o s_o ♑_o for_o the_o ecliptic_a and_o the_o rest_n which_o before_o stand_v for_o meridians_fw-la may_v now_o serve_v for_o particular_a horizon_n according_a to_o their_o several_a elevation_n then_o suppose_v the_o place_n of_o the_o sun_n give_v to_o be_v
of_o may_n if_o you_o observe_v and_o find_v the_o low_a sight_n to_o stay_v at_o 30_o gr_n on_o the_o â—ledgeâ—acke_n of_o the_o bow_n such_o be_v the_o altitude_n for_o the_o declination_n â—ledgeâ—_n 20_o gr_n northward_o the_o altitude_n of_o the_o sun_n between_o the_o â—ledgeâ—wo_a sight_n 40_o gr_n the_o altitude_n of_o the_o equator_fw-la 60_o gr_fw-la and_o thereâ—ledgeâ—re_o the_o latitude_n 30_o gr_n as_o in_o the_o seven_o prop._n 12_o to_o find_v the_o day_n of_o the_o month_n by_o know_v the_o latitude_n of_o the_o place_n and_o observe_v the_o meridian_n altitude_n of_o the_o sun_n place_v your_o three_o sight_n according_a to_o your_o latitude_n the_o hoâ—ledgeâ—zontall_a sight_n to_o the_o centre_n the_o low_a sight_n to_o the_o latitude_n â—ledgeâ—d_v the_o upper_a sight_n among_o the_o month_n then_o when_o the_o â—ledgeâ—nne_n come_v to_o the_o meridian_n observe_v the_o altitude_n look_v â—ledgeâ—_n the_o low_a sight_n through_o the_o horizontall_a and_o keep_v the_o â—ledgeâ—wer_a sight_n still_o at_o the_o latitude_n but_o move_v the_o upper_a sight_n â—ledgeâ—til_o it_o give_v shadow_n upon_o the_o middle_n of_o the_o horizontal_a sight_n â—ledgeâ—_n the_o upper_a sight_n shall_v show_v the_o day_n of_o the_o month_n requiâ—ledgeâ—d_v thus_o in_o our_o latitude_n if_o you_o set_v the_o low_a sight_n to_o 51_o gr_n 30_o â—ledgeâ—_n and_o observe_v find_v the_o altitude_n of_o the_o sun_n between_o â—ledgeâ—at_n and_o the_o upper_a sight_n to_o be_v 28_o gr_n 30_o m._n this_o upper_a sight_n â—ledgeâ—ll_v â—all_n upon_o the_o nine_o of_o october_n and_o the_o twelve_o of_o feâ—ledgeâ—uarâ—e_n and_o if_o yet_o you_o doubt_v which_o oâ—_n they_o two_o be_v the_o day_n â—ledgeâ—u_o may_v expect_v another_o meridian_n altitude_n and_o than_o if_o you_o â—ledgeâ—d_v the_o upper_a sight_n upon_o the_o ten_o of_o october_n and_o the_o eleâ—ledgeâ—nth_n of_o february_n the_o question_n will_v be_v soon_o resolve_v 13_o to_o find_v the_o declination_n of_o any_o unknown_a star_n and_o so_o to_o place_v it_o on_o the_o bow_n by_o know_v the_o latitude_n of_o the_o place_n and_o observe_v the_o meridian_n altitude_n of_o the_o star_n when_o you_o find_v a_o star_n in_o the_o meridian_n that_o be_v fit_a for_o obâ—ledgeâ—uation_n set_a the_o centre_n of_o the_o bow_n to_o your_o eye_n the_o low_a â—ledgeâ—ht_n to_o the_o latitude_n and_o move_v the_o upper_a sight_n up_o or_o down_o â—ledgeâ—till_o you_o see_v the_o horizon_n by_o the_o low_a sight_n and_o the_o star_n by_o the_o upper_a sight_n then_o will_v the_o upper_a sight_n stay_v at_o the_o declination_n and_o place_n of_o the_o star_n thus_o be_v in_o 20_o gr_n of_o north_n latitude_n if_o you_o observe_v anâ—redgeâ—_n find_v the_o meridian_n altitude_n of_o the_o head_n of_o the_o crosier_n to_o bâ—redgeâ—_n 14_o gr_n 50_o m._n the_o upper_a sight_n will_v stay_v at_o 34_o gr_n 50_o m._n and_o therâ—redgeâ—_n may_v you_o place_v this_o star_n for_o by_o this_o observation_n the_o distance_n oâ—redgeâ—_n this_o star_n from_o the_o south_n pole_n shall_v be_v 34_o gr_n 50_o m._n and_o thâ—redgeâ—_n declination_n from_o the_o equator_fw-la 55_o gr_fw-la 10_o m._n and_o so_o for_o the_o resâ—redgeâ—_n the_o star_n which_o i_o mâ—ntioneâ—_n beâ—â—re_o do_v come_v to_o the_o meridian_n in_o this_o order_n after_o the_o first_o point_n of_o aries_n  _fw-fr ho._n mi._n the_o pole_n star_n at_o 0_o 29_o the_o ram_n head_n 1_o 46_o the_o head_n of_o medusa_n 2_o 44_o the_o sidâ—_n of_o perseus_n 2_o 58_o the_o bull_n eye_n 4_o 15_o the_o goat_n 4_o 49_o orion_n left_a shoulder_n 5_o 5_o orion_n girdle_n the_o first_o the_o second_o the_o three_o 5_o 13_o 5_o 17_o 5_o 22_o orion_n right_a shoulder_n 5_o 35_o the_o great_a dog_n 6_o 29_o castor_n 7_o 10_o the_o little_a dog_n 7_o 20_o pollux_n 7_o 22_o the_o hydra_n hart_n 9_o 9_o the_o lion_n hart_n 9_o 48_o the_o great_a bear_n back_o 10_o 40_o first_o in_o gr_n bear_v tail_n 12_o 37_o the_o virgin_n spike_n 13_o 5_o second_o in_o gr_n beaâ—_n tail_n 13_o 9_o third_z in_o gr_n bear_v tail_n 13_o 33_o arcturus_n 13_o 58_o the_o foremost_a gum_n rd_o 14_o 52_o the_o north_n crown_v 15_o 19_o thâ—_n hâ—_n dmost_a guard_n 15_o 25_o scorpion_n hart_n 16_o 7_o the_o harp_n 18_o 24_o praetor_n hart_n 19_o 33_o swan_n tail_n 20_o 29_o fomahant_n 22_o 36_o the_o end_n of_o the_o second_o book_n the_o three_o book_n of_o the_o use_n of_o the_o line_n of_o number_n sin_n and_o tangent_n for_o the_o draw_v of_o houre-line_n on_o all_o sort_n of_o planes_n there_o be_v ten_o several_a sort_n of_o planes_n which_o take_v their_o denomination_n from_o those_o great_a circle_n to_o which_o they_o be_v parallel_n and_o may_v sufficient_o for_o our_o use_n be_v represent_v in_o this_o one_o fundamental_a diagram_n describe_v before_o in_o the_o use_n of_o the_o sector_n and_o be_v know_v by_o their_o horizontall_a and_o perpendicular_a line_n of_o such_o as_o know_v the_o latitude_n of_o the_o place_n and_o the_o circle_n of_o the_o sphere_n 1_o a_o horizontall_a plane_n parallel_v to_o the_o horizon_n here_o represent_v by_o the_o outward_a circle_n esw_v 2_o a_o vertical_a plane_n parallel_v to_o the_o prime_n vertical_a circle_n which_o pass_v through_o the_o zenith_n and_o the_o point_n of_o east_n and_o west_n in_o the_o horizon_n and_o be_v right_n to_o the_o horizon_n and_o the_o meridian_n that_o be_v make_v right_a angle_n with_o they_o both_o this_o be_v represent_v by_o ezw_n 3_o a_o polar_n plane_n parallel_v to_o the_o circle_n of_o the_o hour_n of_o 6_o which_o pass_v through_o the_o pole_n and_o the_o point_n of_o east_n and_o west_n be_v right_a to_o the_o equinoctial_a and_o the_o meridian_n but_o incline_v to_o the_o horizon_n with_o a_o angle_n equal_a to_o the_o latitude_n this_o be_v here_o represent_v by_o epw_n 4_o a_o equinoctial_a plane_n parallel_v to_o the_o equinoctial_a which_o pass_v through_o the_o point_n of_o east_n and_o west_n be_v right_a to_o the_o meridian_n but_o incline_v to_o the_o horizon_n with_o a_o angle_n equal_a to_o the_o compliment_n of_o the_o latitude_n this_o be_v here_o represent_v by_o eaw_n 6_o a_o meridian_n plane_n parallel_v to_o the_o meridian_n the_o circle_n of_o the_o hour_n of_o 12_o which_o pass_v through_o the_o zenith_n the_o pole_n and_o the_o point_n of_o south_n and_o north_n be_v right_a to_o the_o horizon_n and_o the_o prime_n vertical_a this_o be_v here_o represent_v by_o szn_n 7_o a_o meridian_n plane_n incline_v to_o the_o horizon_n parallel_n to_o any_o great_a circle_n which_o pass_v through_o the_o point_n of_o south_n and_o north_n be_v right_a to_o the_o prime_n vertical_a but_o incline_v to_o the_o horizon_n this_o be_v here_o represent_v by_o sgn_n 8_o a_o vertical_a decline_a plane_n parallel_n to_o any_o great_a circle_n which_o pass_v through_o the_o zenith_n be_v right_a to_o the_o horizon_n but_o incline_v to_o the_o meridian_n this_o be_v represent_v by_o bzd._n 9_o a_o polar_n decline_a plane_n parallel_n to_o any_o great_a circle_n which_o pass_v through_o the_o pole_n be_v right_a to_o the_o equinoctial_a but_o incline_v to_o the_o meridian_n this_o be_v here_o represent_v by_o hpq_n 10_o a_o decline_a incline_a plane_n parallel_n to_o any_o great_a circle_n which_o be_v right_a to_o none_o of_o the_o former_a circle_n but_o decline_v from_o the_o prime_n vertical_a and_o incline_v both_o to_o the_o horizon_n and_o the_o meridian_n and_o all_o the_o hour_n circle_n this_o may_v be_v here_o represent_v either_o by_o bmd_a or_o bfd_n or_o bkd_v or_o any_o such_o great_a circle_n which_o pass_v neither_o through_o the_o south_n and_o north_n nor_o east_n and_o west_n point_n nor_o through_o the_o zenith_n nor_o the_o pole_n each_o of_o these_o plane_n except_o the_o horizontall_a have_v two_o face_n whereon_o houre-line_n may_v be_v draw_v and_o so_o there_o be_v 19_o plane_n in_o all_o the_o meridian_n plane_n have_v one_o face_n to_o the_o east_n and_o another_o to_o the_o west_n the_o other_o vertical_a plane_n have_v one_o to_o the_o south_n and_o another_o to_o the_o north_n and_o the_o rest_n one_o to_o the_o zenith_n and_o another_o to_o the_o nadir_n but_o what_o be_v say_v of_o the_o one_o may_v be_v understand_v of_o the_o other_o to_o find_v the_o inclination_n of_o any_o plane_n for_o the_o distinguish_n of_o these_o plane_n we_o may_v find_v whether_o they_o be_v horizontall_a or_o vertical_a or_o incline_v to_o the_o horizon_n and_o how_o much_o they_o incline_v either_o by_o the_o usual_a inclinatorie_n quadrant_a or_o by_o fit_v a_o thread_n and_o plummet_n unto_o the_o sector_n for_o let_v the_o sector_n be_v open_v to_o a_o right_a angle_n the_o line_n of_o sin_n to_o a_o angle_n of_o 92_o gr_n the_o inward_a edge_n of_o the_o sector_n to_o 90_o
20_o 5_o sbw_n 13_o 5_o 10_o 39_o ssw_n 1_o 50_o 1_o 29_o  _fw-fr style_n 0_o 0_o swb_n 9_o 25_o 7_o 38_o sweet_z 20_o 40_o 16_o 58_o swbw_n 31_o 55_o 26_o 45_o wsw_n 43_o 10_o 37_o 11_o wb_n 54_o 25_o 48_o 30_o west_n 65_o 40_o 60_o 48_o wbn_n 76_o 55_o 73_o 58_o wnw_n 88_o 10_o 87_o 44_o these_o angle_n be_v know_v if_o on_o the_o centre_n v_n at_o the_o vertical_a point_n you_o describe_v a_o occult_a circle_n and_o therein_o inscribe_v the_o chord_n of_o these_o angle_n from_o the_o line_n uh_o and_o then_o draw_v right_a line_n through_o the_o vertical_a point_n and_o the_o term_n of_o those_o chord_n the_o line_n so_o draw_v shall_v be_v the_o azimuth_n require_v the_o like_a reason_n hold_v for_o the_o draw_v of_o the_o azimuth_n upon_o all_o other_o incline_v plane_n wheof_o you_o have_v another_o example_n in_o the_o diagram_n belong_v to_o the_o meridian_n incliner_n pag._n 126._o or_o for_o further_a satisfaction_n you_o may_v find_v where_o each_o azimuth_n line_n shall_v cross_v the_o equator_fw-la as_o the_o sine_fw-la of_o 90_o gr_n to_o the_o sine_fw-la of_o the_o latitude_n so_o the_o tangent_fw-la of_o the_o azimuth_n from_o the_o meridian_n to_o the_o tangent_fw-la of_o the_o equator_fw-la from_o the_o meridian_n extend_v the_o compass_n from_o the_o sine_fw-la of_o 90_o gr_n unto_o the_o line_n of_o our_o latitude_n 51_o gr_n 30_o m._n the_o same_o extent_n will_v reach_v in_o the_o line_n of_o tangent_n from_o 10_o gr_n unto_o 7._o gr_n 50._o m._n for_o the_o intersection_n of_o the_o equator_fw-la with_o the_o azimuth_n of_o 10_o gr_n from_o the_o meridian_n again_o the_o same_o extent_n will_v reach_v from_o 20_o gr_n unto_o 15_o gr_n 54_o m._n for_o the_o azimuth_n of_o 20_o gr_n and_o so_o the_o rest_n as_o in_o these_o table_n azim_n equat._n gr._n m._n gr._n m._n 10_o 0_o 7_o 50_o 20_o 0_o 15_o 54_o 30_o 0_o 24_o 20_o 40_o 0_o 43_o 18_o 50_o 0_o 13_o 0_o 60_o 0_o 53_o 35_o 70_o 0_o 65_o 3_o 80_o 0_o 77_o 18_o 90_o 0_o 90_o 0_o azim_n equat._n gr._n m._n gr._n m._n 11_o 15_o 8_o 51_o 22_o 30_o 17_o 58_o 33_o 45_o 27_o 36_o 45_o 0_o 38_o 2_o 56_o 15_o 49_o 30_o 67_o 30_o 62_o 6_o 78_o 45_o 75_o 44_o 90_o 0_o 90_o 0_o by_o which_o you_o may_v see_v that_o the_o azimuth_n 90_o gr_n distant_a from_o the_o meridian_n which_o be_v the_o line_n of_o east_n and_o west_n will_v cross_v the_o equator_fw-la at_o 90_o gr_n from_o the_o meridian_n in_o the_o same_o point_n with_o the_o horizontall_a line_n and_o the_o hour_n of_o 6._o and_o that_o the_o azimuth_n of_o 45_o gr_n will_v cross_v the_o equator_fw-la at_o 38_o gr_n 2_o m._n from_o the_o meridian_n that_o be_v the_o line_n of_o se_n will_v cross_v the_o equator_fw-la at_o the_o hour_n of_o 9_o and_o 28_o m._n in_o the_o morning_n and_o the_o line_n of_o sweet_z at_o 2_o ho._n 32_o min._n in_o the_o afternoon_n and_o so_o for_o the_o rest_n whereby_o you_o may_v examine_v your_o former_a work_n chap._n xx._n to_o describe_v the_o parallel_n of_o the_o horizon_n in_o the_o former_a plane_n the_o parallel_n of_o the_o horizon_n common_o call_v almicanter_n or_o parallel_n of_o altitude_n whereby_o we_o may_v know_v the_o altitude_n of_o the_o sun_n above_o the_o horizon_n have_v such_o respect_n unto_o the_o horizon_n as_o the_o parallel_n of_o declination_n unto_o the_o equator_fw-la and_o so_o may_v be_v describe_v in_o like_a manner_n in_o a_o horizontall_a plane_n these_o parallel_n will_v be_v perfect_a circle_n wherefore_o know_v the_o length_n of_o the_o style_n in_o inch_n and_o part_n and_o the_o distance_n of_o the_o parallel_n from_o the_o horizon_n in_o degree_n and_o minute_n as_o the_o tangent_fw-la of_o 45_o gr_n be_v to_o the_o length_n of_o the_o style_n so_o the_o cotangent_fw-la of_o the_o parallel_n to_o the_o semidiameter_n of_o his_o circle_n thus_o in_o the_o example_n of_o the_o horizontall_a plane_n pag._n 164._o if_o ab_fw-la the_o length_n of_o the_o style_n shall_v be_v 5_o inch_n and_o that_o it_o be_v require_v to_o find_v the_o semidiameter_n of_o the_o parallel_n of_o 62_o gr_n extend_v the_o compass_n from_o the_o tangent_fw-la of_o 45_o gr_n unto_o 5.00_o in_o the_o line_n of_o number_n the_o same_o extent_n will_v reach_v from_o the_o tangent_fw-la of_o 28_o gr_n the_o compliment_n of_o the_o parallel_n unto_o 2.65_o and_o if_o you_o describe_v a_o circle_n on_o the_o centre_n a_o to_o the_o semidiameter_n of_o 2_o inch_n 65_o cent_n it_o shall_v be_v the_o parallel_n require_v in_o all_o upright_a plane_n whether_o they_o be_v direct_v vertical_o or_o decline_v or_o meridian_n plane_n these_o parallel_n will_v be_v conicall_a section_n and_o may_v be_v draw_v through_o their_o point_n of_o intersection_n with_o the_o azimuth_n line_n in_o the_o same_o manner_n as_o the_o parallel_n of_o declination_n through_o their_o point_n of_o intersection_n with_o the_o houre-line_n to_o this_o end_n you_o may_v first_o find_v the_o distance_n between_o the_o top_n of_o the_o style_n and_o the_o azimuth_n and_o then_o the_o distance_n between_o the_o horizon_n and_o the_o parallel_n both_o which_o may_v be_v represent_v in_o this_o manner_n on_o the_o centre_n b_o and_o any_o semidiameter_n bh_n describe_v a_o occult_a ark_n of_o a_o circle_n and_o therein_o inscribe_v the_o chord_n of_o such_o parallel_n of_o altitude_n as_o you_o intend_v to_o draw_v on_o the_o plane_n i_o have_v here_o put_v they_o for_o 15._o 30._o 45_o and_o 60_o gr_n then_o draw_v right_a line_n through_o the_o centre_n and_o the_o term_n of_o those_o chord_n so_o the_o line_n bh_n shall_v be_v the_o horizon_n and_o the_o rest_n the_o line_n of_o altitude_n according_a to_o their_o distance_n from_o the_o horizon_n that_o do_v consider_v your_o plane_n which_o here_o for_o example_n be_v the_o south_n face_n of_o our_o vertical_a plane_n p._n 168_o wherein_o have_v draw_v both_o the_o horizontall_a &_o vertical_a line_n as_o i_o show_v before_o first_o take_v out_o ab_fw-la the_o length_n of_o the_o style_n &_o prick_v that_o down_o in_o this_o horizontall_a line_n from_o b_o unto_o a_o then_o take_v out_o all_o the_o distance_n between_o b_o the_o top_n of_o the_o style_n and_o the_o several_a point_n wherein_o the_o vertical_a line_n do_v cross_v the_o horizontal_a transfer_v they_o into_o this_o horizontal_a line_n bh_n from_o the_o centre_n b_o and_o at_o the_o term_n of_o these_o distance_n erect_v line_n perpendicular_a to_o the_o horizon_n note_v they_o with_o the_o number_n or_o letter_n of_o the_o azimuth_n from_o whence_o they_o be_v take_v so_o these_o perpendicular_o shall_v represent_v those_o azimuth_n and_o the_o several_a distance_n between_o the_o horizon_n and_o the_o line_n of_o altitude_n shall_v give_v the_o like_a distance_n between_o the_o horizontall_a and_o the_o parallel_n of_o altitude_n upon_o the_o azimuth_n in_o your_o plane_n upon_o this_o ground_n it_o follow_v 1_o to_o find_v the_o distance_n between_o the_o top_n of_o the_o style_n and_o the_o several_a point_n wherein_o the_o azimuth_n do_v cross_v the_o horizontall_a line_n have_v draw_v the_o horizontall_a and_o azimuth_n line_n as_o before_o look_v into_o the_o table_n by_o which_o you_o draw_v they_o and_o there_o you_o shall_v have_v the_o angle_n at_o the_o zenith_n then_o as_o the_o cousin_n of_o the_o angle_n at_o the_o zenith_n be_v to_o the_o sine_fw-la of_o 90_o gr_n so_o the_o length_n of_o the_o style_n to_o the_o distance_n require_v azimuth_n angelo_fw-mi see_fw-mi tangent_fw-la secant_fw-la par._n 15._o par._n 30._o gr._n m_o inch_n p._n inch_n p._n inch._n p._n inch._n p._n south_n 0_o 0_o 0_o 0_o 10_o 00_o 2_o 68_o 5_o 77_o sbe_n 11_o 15_o 1_o 99_o 10_o 20_o 2_o 73_o 5_o 90_o sse_n 22_o 30_o 4_o 14_o 10_o 82_o 2_o 90_o 6_o 24_o seb_n 33_o 45_o 6_o 68_o 12_o 03_o 3_o 23_o 6_o 94_o see_fw-la 45_o 0_o 10_o 00_o 14_o 14_o 3_o 80_o 8_o 16_o sebe_fw-la 56_o 15_o 14_o 97_o 18_o 00_o 4_o 82_o 10_o 40_o ease_n 67_o 30_o 24_o 14_o 26_o 13_o 7_o 02_o 15_o 08_o ebb_n 78_o 45_o 50_o 27_o 51_o 26_o 13_o 73_o 29_o 60_o east_n 90_o 0_o infinite_a infinite_a infinite_a infinite_a as_o in_o our_o example_n of_o the_o vertical_a plane_n where_o ab_fw-la the_o length_n of_o the_o style_n be_v suppose_v to_o be_v 10_o inch_n extend_v the_o compass_n from_o the_o sine_fw-la of_o 78_o gr_n 45_o m._n the_o compliment_n of_o 11_o gr_n 15_o m._n the_o angle_n at_o the_o zenith_n belong_v to_o sbe_n and_o sbw_n unto_o the_o sine_fw-la of_o 90_o gr_n the_o same_o extent_n will_v reach_v from_o 10.00_o the_o length_n of_o the_o style_n unto_o 10.20_o for_o the_o distance_n between_o the_o top_n of_o the_o
divide_v the_o line_n of_o solid_n see_v like_o solid_n do_v hold_v in_o the_o proportion_n of_o their_o homologall_a side_n triplicate_v if_o you_o shall_v find_v two_o mean_a proportional_o between_o the_o whole_a side_n &_o each_o thousand_o part_n of_o the_o like_a side_n all_o of_o they_o cut_v the_o same_o two_o right_a line_n the_o former_a of_o those_o line_n so_o cut_v shall_v contain_v the_o division_n require_v wherefore_o upon_o the_o centre_n a_o &_o semidiameter_n equal_a to_o the_o line_n of_o line_n describe_v a_o circle_n and_o divide_v it_o into_o 4_o equal_a part_n cebd_a draw_v the_o cross_a diameter_n cb_n ed._n then_o divide_v the_o semidiameter_n ac_fw-la first_o into_o 10_o equal_a part_n and_o between_o the_o whole_a line_n ad_fw-la &_o of_o the_o ten_o part_n of_o ac_fw-la seek_v out_o two_o mean_v proportional_a line_n a_z and_z ah_o again_o between_o ad_fw-la and_o agnostus_n be_v two_o ten_o part_n of_o ac_fw-la seek_v out_o two_o mean_a proportional_n all_o and_o ak_v and_o so_o forward_o in_o the_o rest_n so_o shall_v the_o line_n ab_fw-la be_v divide_v into_o 10_o unequal_a part_n second_o divide_v each_o ten_o part_n of_o the_o line_n ac_fw-la into_o 10_o more_o and_o between_o the_o whole_a line_n ad_fw-la and_o each_o of_o they_o seek_v out_o two_o mean_a proportional_o as_o before_o so_o shall_v the_o line_n ab_fw-la be_v divide_v now_o into_o a_o hundred_o unequal_a part_n three_o if_o the_o length_n will_v bear_v it_o subdivide_v the_o line_n ac_fw-la once_o again_o each_o part_n into_o ten_o more_o and_o between_o the_o whole_a line_n ad_fw-la and_o each_o subdivision_n seek_v two_o mean_a proportional_o as_o before_o so_o shall_v the_o line_n ab_fw-la be_v now_o divide_v into_o 1000_o part_n but_o the_o ruler_n be_v short_a it_o shall_v suffice_v if_o those_o 10_o which_o be_v near_a the_o centre_n be_v express_v the_o rest_n be_v understand_v to_o be_v so_o divide_v though_o actual_o they_o be_v divide_v into_o no_o more_o than_o 5_o or_o 2._o and_o this_o line_n ab_fw-la so_o divide_v shall_v be_v the_o line_n of_o solid_n and_o must_v be_v transfer_v into_o the_o sector_n but_o let_v the_o number_n set_v to_o they_o be_v only_o 1.1.1.2.3_o etc._n etc._n unto_o 10._o as_o in_o the_o example_n 5_o to_o divide_v the_o line_n of_o sines_n and_o tangent_n on_o the_o side_n of_o the_o sector_n upon_o the_o centre_n a_o and_o semidiameter_n equal_a to_o the_o line_n of_o line_n describe_v a_o semicircle_n abcd_v with_o ab_fw-la perpendicular_a to_o the_o diameter_n cd_o then_o divide_v the_o quadrant_n cb_n bd_o each_o of_o they_o into_o 90._o and_o subdivide_v each_o degree_n into_o 2_o part_n for_o so_o if_o straight_a line_n be_v draw_v parallel_n to_o the_o diameter_n cd_o through_o these_o 90_o and_o their_o subdivision_n they_o shall_v divide_v the_o perpendicular_a ab_fw-la unequal_o into_o 90._o and_o this_o line_n ab_fw-la so_o divide_v shall_v be_v the_o line_n of_o sines_n and_o must_v be_v transfer_v into_o the_o sector_n the_o number_n set_v to_o they_o be_v to_o be_v 10.20.30_o etc._n etc._n unto_o 90_o as_o in_o the_o example_n if_o now_o in_o the_o point_n d_o unto_o the_o diameter_n cd_o we_o shall_v raise_v a_o perpendicular_a de_fw-fr and_o to_o it_o draw_v straight_a line_n from_o the_o centre_n a_o through_o each_o degree_n of_o the_o quadrant_a db._n this_o perpendicular_a so_o divide_v by_o they_o shall_v be_v the_o line_n of_o tangent_n &_o must_v be_v transfer_v unto_o the_o side_n of_o the_o sector_n the_o number_n set_v to_o they_o be_v to_o be_v 10.20.30_o etc._n etc._n as_o in_o the_o example_n if_o between_o a_o and_o d_o another_o straight_a line_n gf_n be_v draw_v parallel_n to_o de_fw-fr it_o will_v be_v divide_v by_o those_o line_n from_o the_o centre_n in_o like_a sort_n as_o de_fw-fr be_v divide_v and_o it_o may_v serve_v for_o a_o lesser_a line_n of_o tangent_n to_o be_v set_v on_o the_o edge_n of_o the_o sector_n these_o line_n of_o sines_n and_o tangent_n may_v yet_o otherwise_o be_v transfer_v into_o the_o sector_n out_o of_o the_o line_n of_o line_n or_o rather_o out_o of_o a_o diagonall_a scale_n equal_a to_o the_o line_n of_o line_n by_o table_n of_o sines_n and_o tangent_n in_o like_a manner_n may_v the_o line_n of_o superficies_n be_v transfer_v by_o table_n of_o square_a root_n and_o the_o line_n of_o solid_n by_o table_n of_o cubique_fw-la root_n which_o i_o leave_v to_o other_o to_o extract_v at_o leisure_n 6_o to_o show_v the_o ground_n of_o the_o sector_n let_v ab_fw-la ac_fw-la represent_v the_o leg_n of_o the_o sector_n then_o sever_n these_o two_o ab_fw-la ac_fw-la be_v equal_a and_o their_o section_n ad_fw-la ae_n also_o equal_a they_o shall_v be_v cut_v proportional_o and_o if_o we_o draw_v the_o line_n bc_n de_fw-fr they_o will_v be_v parallel_n by_o the_o second_o pro._n 6_o lib._n of_o euclid_n and_o so_o the_o triangle_n abc_n ade_n shall_v be_v equiangle_n by_o reason_n of_o the_o common_a angle_n at_o a_o and_o the_o equal_a angle_n at_o the_o base_a and_o therefore_o shall_v have_v the_o side_n proportional_a about_o those_o equal_a angle_n by_o the_o 4_o pro._n 6_o lib._n of_o euclid_n the_o side_n ad_fw-la shall_v be_v to_o the_o side_n ab_fw-la as_o the_o basis_n de_fw-fr unto_o the_o parallel_a basis_n bc_n and_o by_o conversion_n ab_fw-la shall_v be_v unto_o ad_fw-la as_o bc_n unto_o de_fw-fr and_o by_o permutation_n ad_fw-la shall_v be_v unto_o de_fw-fr as_o ab_fw-la to_o bc._n etc._n etc._n so_o that_o if_o ad_fw-la be_v the_o four_o part_n of_o the_o side_n ab_fw-la then_o de_fw-fr shall_v also_o be_v the_o four_o part_n of_o his_o parallel_a basis_n bc_n the_o like_a reason_n hold_v in_o all_o other_o section_n 7_o to_o show_v the_o general_a use_n of_o the_o sector_n there_o may_v some_o conclusion_n be_v wrought_v by_o the_o sector_n even_o then_o when_o it_o be_v shut_v by_o reason_n that_o the_o line_n be_v all_o of_o one_o length_n but_o general_o the_o use_n hereof_o consist_v in_o the_o solution_n of_o the_o golden_a rule_n where_o three_o line_n be_v give_v of_o a_o know_a denominaton_n a_o four_o proportional_a be_v to_o be_v find_v and_o this_o solution_n be_v diverse_a in_o regard_n both_o of_o the_o line_n and_o of_o the_o entrance_n into_o the_o work_n the_o solution_n in_o regard_n of_o the_o line_n be_v sometime_o simple_a as_o when_o the_o work_n be_v begin_v and_o end_v upon_o the_o same_o line_n sometime_o it_o be_v compound_v as_o when_o it_o be_v begin_v on_o one_o kind_n of_o line_n and_o end_v on_o another_o it_o may_v be_v begin_v upon_o the_o line_n of_o line_n &_o finish_v upon_o the_o line_n of_o superficies_n it_o may_v begin_v on_o the_o sin_n and_o end_n on_o the_o tangent_n the_o solution_n in_o regard_n of_o the_o entrance_n into_o the_o work_n may_v be_v either_o with_o a_o parallel_n or_o else_o lateral_a on_o the_o side_n of_o the_o sector_n i_o call_v it_o parallel_v entrance_n or_o enter_v with_o a_o parallel_n when_o the_o two_o line_n of_o the_o first_o denomination_n be_v apply_v in_o the_o parallel_n and_o the_o three_o line_n and_o that_o which_o be_v seek_v for_o be_v on_o the_o side_n of_o the_o sector_n i_o call_v it_o lateral_a entrance_n or_o enter_v on_o the_o side_n of_o the_o sector_n when_o the_o two_o line_n of_o the_o first_o denomination_n be_v one_o the_o side_n of_o the_o sector_n and_o the_o three_o line_n and_o that_o which_o be_v to_o be_v find_v out_o do_v stand_v in_o the_o parallel_n as_o for_o example_n let_v there_o be_v give_v three_o line_n a_o b_o c_o to_o which_o i_o be_o to_o find_v a_o four_o proportional_a let_v a_o measure_v in_o the_o line_n of_o line_n be_v 40_o b_o 50_o and_o c_o 60_o and_o suppose_v the_o question_n be_v this_o if_o 40_o month_n give_v 50_o pound_n what_o shall_v 60_o here_o be_v line_n of_o two_o denomination_n one_o of_o month_n another_o of_o pound_n and_o the_o first_o with_o which_o i_o be_o to_o enter_v must_v be_v that_o of_o 40_o month_n if_o then_o i_o will_v enter_v with_o a_o parallel_n first_o i_o take_v a_o the_o line_n of_o 40_o and_o put_v it_o over_o as_o a_o parallel_n in_o 50_o reckon_v in_o the_o line_n of_o line_n on_o either_o side_n of_o the_o sector_n from_o the_o centre_n so_o as_o it_o may_v be_v the_o base_a of_o a_o isoscheles_a triangle_n bac_n who_o side_n ab_fw-la ac_fw-la be_v equal_a to_z b_o the_o line_n of_o the_o second_o denomination_n then_o the_o sector_n be_v thus_o open_v i_o take_v c_o the_o line_n of_o 60_o between_o the_o foot_n of_o the_o compass_n and_o carry_v they_o parallel_v to_o bc_n i_o find_v they_o to_o cross_v the_o line_n ab_fw-la ac_fw-la on_o the_o side_n of_o the_o
be_v describe_v out_o of_o the_o angular_a point_n at_o any_o other_o distance_n let_v the_o semidiameter_n be_v turn_v into_o a_o parallel_a chord_n of_o 60_o gr._n then_o take_v the_o chord_n of_o this_o ark_n and_o carry_v it_o parallel_v till_o it_o cross_v in_o like_a chord_n so_o the_o place_n where_o it_o stay_v shall_v give_v the_o quantity_n of_o the_o angle_n as_o in_o the_o former_a example_n if_o i_o make_v the_o semidiameter_n ab_fw-la a_o parallel_a chord_n of_o 60_o gr._n and_o then_o keep_v the_o sector_n at_o that_o angle_n carry_v the_o chord_n bc_n parallel_n till_o it_o stay_v in_o like_a chord_n i_o shall_v find_v it_o to_o stay_v in_o no_o other_o but_o 11_o gr._n 15_o m_n and_o such_o be_v the_o angle_n bac_n 10_o upon_o a_o right_a line_n and_o a_o point_n give_v in_o it_o to_o make_v a_o angle_n equal_a to_o any_o angle_n give_v first_o out_o of_o the_o point_n give_fw-mi describe_v a_o ark_n cut_v the_o same_o line_n then_o by_o the_o 5._o prop_v afore_o find_v the_o chord_n of_o the_o angle_n give_v agreeable_a to_o the_o semidiameter_n and_o inscribe_v it_o into_o this_o ark_n so_o a_o right_a line_n draw_v through_o the_o point_n give_v and_o the_o end_n of_o this_o chord_n shall_v be_v the_o side_n that_o make_v up_o the_o angle_n let_v the_o right_a line_n give_v be_v ab_fw-la and_o the_o point_n give_v in_o it_o be_v a_o and_o let_v the_o angle_v give_v be_v 11_o gr_n 15_o m._n here_o i_o open_v the_o compass_n to_o any_o semidiameter_n ab_fw-la but_o as_o oft_o as_o i_o may_v convenient_o to_o the_o lateral_a semiradius_fw-la and_o set_v one_o foot_n in_o a_o i_o describe_v a_o occult_a ark_n bc_n then_o i_o seek_v out_o the_o chord_n of_o 11_o gr_n 15_o m._n and_o take_v it_o with_o the_o compass_n i_o set_v one_o foot_n in_o b_o the_o other_o cross_v the_o ark_n in_o c_o by_o which_o i_o draw_v the_o line_n ac_fw-la and_o it_o make_v up_o the_o angle_n require_v 11_o to_o divide_v the_o circumference_n of_o a_o circle_n into_o any_o part_n require_v if_o 360_o the_o measure_n of_o the_o whole_a circumference_n be_v divide_v by_o the_o number_n of_o part_n require_v the_o quotient_n give_v the_o chord_n which_o be_v find_v will_v divide_v the_o circumference_n so_o a_o chord_n of_o 120_o gr_n will_v divide_v the_o circumference_n into_o 3_o equal_a part_n a_o chord_n of_o 90_o gr_n into_o 4_o part_n a_o chord_n of_o 72_o gr_n into_o 5_o part_n a_o chord_n of_o 60_o gr_n into_o 6_o part_n a_o chord_n of_o 51_o gr_n 26._o into_o 7_o part_n a_o chord_n of_o 45_o gr_n into_o 8_o part_n a_o chord_n of_o 40_o gr_n into_o 9_o part_n a_o chord_n of_o 36_o gr_n into_o 10_o part_n a_o chord_n of_o 32_o gr_n 44_o m._n into_o 11_o part_n a_o chord_n of_o 30_o gr_n into_o 12_o part_n in_o like_a manner_n if_o it_o be_v require_v to_o divide_v the_o circumference_n of_o the_o circle_n who_o semidiameter_n be_v ab_fw-la into_o 32_o first_o i_o take_v the_o semidiameter_n ab_fw-la and_o make_v it_o a_o parallel_a chord_n of_o 60_o gr_n then_o because_o 360_o gr_n be_v divide_v by_o 32_o the_o quotient_n will_v be_v 11_o gr_n 15_o m._n i_o find_v the_o parallel_a chord_n of_o 11_o gr_n 15_o m._n and_o this_o will_v divide_v the_o circumference_n into_o 32._o but_o here_o the_o part_n be_v many_o it_o be_v better_a to_o divide_v it_o first_o into_o few_o and_o after_o to_o come_v over_o it_o again_o as_o first_o to_o divide_v the_o circumference_n into_o 4_o and_o then_o each_o 4_o part_n into_o 8_o or_o otherwise_o as_o the_o part_n may_v be_v divide_v 12_o to_o divide_v a_o right_a line_n by_o extreme_a and_o mean_a proportion_n the_o line_n to_o be_v divide_v by_o extreme_a and_o mean_a proportion_n have_v the_o same_o proportion_n to_o his_o great_a segment_n as_o in_o figure_n inscribe_v in_o the_o same_o circle_n the_o side_n of_o a_o hexagon_n a_o figure_n of_o six_o angle_n have_v to_o a_o side_n of_o a_o decagon_n a_o figure_n of_o ten_o angle_n but_o the_o side_n of_o a_o hexagon_n be_v a_o chord_n of_o 60_o gr_n and_o the_o side_n of_o a_o decagon_n be_v a_o chord_n of_o 36_o gr_n let_v ab_fw-la be_v the_o line_n to_o be_v divide_v if_o i_o make_v ab_fw-la a_o parallel_a chord_n of_o 60_o gr_n and_o to_o this_o semidiameter_n find_v ac_fw-la a_o chord_n of_o 36_o gr_n this_o ac_fw-la shall_v be_v the_o great_a segment_n divide_v the_o whole_a line_n in_o c_o by_o extreme_a and_o mean_a proportion_n so_o that_o as_o ab_fw-la the_o whole_a line_n be_v unto_o ac_fw-la the_o great_a segment_n so_o ac_fw-la the_o great_a segment_n unto_o cb_n the_o lesser_a segment_n or_o let_v ac_fw-la be_v the_o great_a segment_n give_v if_o i_o make_v this_o a_o parallel_a chord_n of_o 36_o gr_n the_o correspondent_a semidiameter_n shall_v be_v the_o whole_a line_n ac_fw-la and_o the_o difference_n cb_n the_o lesser_a segment_n or_o let_v cb_n be_v the_o lesser_a segment_n give_v if_o i_o make_v this_o a_o parallel_a chord_n of_o 36_o gr_n the_o correspondent_a semidiameter_n shall_v be_v great_a segment_n ac_fw-la which_o add_v to_o cb_n give_v the_o whole_a line_n ab_fw-la to_o avoid_v double_v of_o line_n or_o number_n you_o may_v put_v over_o the_o whole_a line_n in_o the_o sin_n of_o 72_o gr_n and_o the_o parallel_n sine_fw-la of_o 36_o gr_n shall_v be_v the_o great_a segment_n or_o if_o you_o put_v over_o the_o whole_a line_n in_o the_o sin_n of_o 54_o gr_n the_o parallel_n sine_fw-la of_o 30_o gr_n shall_v be_v the_o great_a segment_n and_o the_o parallel_n sine_fw-la of_o 18_o gr_n shall_v be_v the_o lesser_a segment_n chap._n iii_o of_o the_o projection_n of_o the_o sphere_n in_o plano_n 1_o the_o sphere_n may_v be_v project_v in_o plano_n in_o straight_a line_n as_o in_o the_o analemma_n if_o the_o semidiameter_n of_o the_o circle_n give_v be_v divide_v in_o such_o sort_n as_o the_o line_n of_o sin_n on_o the_o sector_n as_o if_o the_o radius_fw-la of_o the_o circle_n give_v be_v ae_n the_o circle_n thereon_o describe_v may_v represent_v the_o plane_n of_o the_o general_a meridian_n which_o divide_v into_o four_o equal_a part_n in_o e_o p_o ae_n s_o and_o cross_v at_o right_a angle_n with_o eae_fw-la and_o ps_n the_o diameter_n eae_fw-la shall_v represent_v the_o equator_fw-la and_o ps_n the_o circle_n of_o the_o hour_n of_o 6._o and_o it_o be_v also_o the_o axis_fw-la of_o the_o world_n wherein_o p_o stand_v for_o the_o north_n pole_n and_o s_n for_o the_o south_n pole_n then_o may_v each_o quarter_n of_o the_o meridian_n be_v divide_v into_o 90_o gr_n from_o the_o equator_fw-la towards_o the_o pole_n in_o which_o if_o we_o number_v 23_o gr_n 30_o m._n the_o great_a declination_n of_o the_o sun_n from_o e_o to_o 69_o northwards_o from_o ae_n to_o ♑_o southward_n the_o line_n draw_v from_o 69_o to_o ♑_o shall_v be_v the_o ecliptic_a and_o the_o line_n draw_v parallel_n to_o the_o equator_fw-la through_z ♋_o and_o ♑_o shall_v be_v the_o tropique_n have_v these_o common_a section_n with_o the_o plane_n of_o the_o meridian_n if_o we_o shall_v divide_v each_o diameter_n of_o the_o ecliptic_a into_o 90_o gr_n in_o such_o sort_n as_o the_o sin_n be_v divide_v on_o the_o sector_n the_o first_o 30_o gr_n from_o a_o towards_o 69_o shall_v stand_v for_o the_o sine_z of_o ♈_o the_o 30_o gr_n next_o follow_v for_o ♉_o the_o rest_n for_o ♊_o ♋_o ♌_o etc._n etc._n in_o their_o order_n so_o that_o by_o these_o mean_v we_o have_v the_o place_n of_o the_o sun_n for_o all_o time_n of_o the_o year_n if_o again_o we_o divide_v ap_n as_o in_o the_o like_a sort_n and_o set_v to_o the_o number_n 10._o 20._o 30._o etc._n etc._n unto_o 90_o gr_n the_o line_n draw_v through_o each_o of_o these_o degree_n parallel_v to_o the_o equator_fw-la shall_v show_v the_o declination_n of_o the_o sun_n and_o represent_v the_o parallel_n of_o latitude_n if_o far_o we_o divide_v ae_n aae_n and_o his_o parallel_n in_o the_o like_a sort_n and_o then_o careful_o draw_v a_o line_n through_o each_o 15_o gr_n so_o as_o it_o make_v no_o angle_n the_o line_n so_o draw_v shall_v be_v ellipticall_a and_o represent_v the_o houre-circle_n the_o meridian_n pes_z the_o hour_n of_o 12_o at_o noon_n that_o next_o unto_o it_o draw_v through_o 75_o gr_n from_o the_o centre_n the_o hour_n of_o 11_o and_o 1_o that_o which_o be_v draw_v through_o 60_o gr_n from_o the_o centre_n the_o hour_n of_o 10_o and_o 2._o etc._n etc._n then_o have_v respect_n unto_o the_o latitude_n we_o may_v number_v it_o from_o e_z northward_o unto_o z_o and_o there_o place_v the_o zenith_n by_o which_o and_o the_o centre_n the_o line_n draw_fw-mi zan_fw-mi shall_v represent_v
and_o azimuth_n chap._n i._o of_o the_o description_n of_o the_o quadrant_n have_v describe_v these_o stand_a plane_n i_o will_v now_o show_v the_o most_o of_o these_o conclusion_n by_o a_o small_a quadrant_n this_o may_v be_v do_v general_o for_o all_o latitude_n by_o a_o quarter_n of_o the_o general_a astrolabe_n describe_v before_o in_o the_o use_n of_o the_o sector_n pag._n 58_o and_o particular_o for_o any_o one_o latitude_n by_o a_o quarter_n of_o the_o particular_a astrolabe_n there_o also_o describe_v pag._n 63._o which_o if_o it_o be_v a_o foot_n semidiameter_n may_v show_v the_o azimuth_n unto_o a_o degree_n and_o the_o time_n of_o the_o day_n unto_o a_o minute_n but_o for_o ordinary_a use_v this_o small_a quadrant_n may_v suffice_v which_o may_v be_v make_v portable_a in_o this_o manner_n 1_o upon_o the_o centre_n a_o and_o semidiameter_n ab_fw-la describe_v the_o ark_n bc_n the_o same_o semidiameter_n will_v set_v of_o 60_o gr_n and_o the_o half_a of_o that_o will_v be_v 30_o gr_n which_o be_v add_v to_o the_o former_a 60_o gr_n will_v make_v the_o ark_n bc_n to_o be_v 90_o gr_n the_o four_o part_n of_o the_o whole_a circle_n and_o thence_o come_v the_o name_n of_o a_o quadrant_n 2_o leave_v some_o little_a space_n for_o the_o inscription_n of_o the_o month_n and_o day_n on_o the_o same_o centre_n a_o and_o semidiameter_n at_o describe_v the_o ark_n td_n which_o shall_v serve_v for_o either_o tropic_n 3_o divide_v the_o line_n at_o in_o the_o point_n e_o in_o such_o proportion_n as_o that_o at_o be_v 10000_o ae_n may_v be_v 6556_o and_o there_o draw_v another_o ark_n of_o which_o shall_v serve_v for_o the_o equator_fw-la 5_o this_o part_n of_o the_o ecliptic_a may_v be_v divide_v into_o three_o sign_n and_o each_o sign_n into_o 30_o a_o table_n of_o right_a ascension_n gr._n ♈_o ♉_o ♊_o gr._n m._n gr._n m._n gr._n m._n 0_o 0_o 0_o 27_o 54_o 57_o 48_o 5_o 4_o 35_o 32_o 42_o 63_o 3_o 10_o 9_o 11_o 37_o 35_o 68_o 21_o 15_o 13_o 48_o 42_o 31_o 73_o 43_o 20_o 18_o 27_o 47_o 33_o 79_o 7_o 25_o 23_o 9_o 52_o 38_o 84_o 32_o 30_o 27_o 54_o 57_o 48_o 90_o 0_o gr_n by_o a_o table_n of_o right_a ascension_n make_v as_o before_o pag._n 60._o as_o the_o right_a ascension_n of_o the_o first_o point_n of_o ♉_o being_n 27_o gr_n 54_o m._n you_o may_v lay_v a_o ruler_n to_o the_o centre_n a_o &_o 27_o gr_n 54_o m._n in_o the_o quadrant_n bc_n the_o point_n where_o the_o ruler_n cross_v the_o ecliptic_a shall_v be_v the_o first_o point_n of_o ♉_o in_o like_a manner_n the_o right_a ascension_n of_o the_o first_o point_n of_o ♊_o being_n 57_o gr_n 48_o m._n if_o you_o lay_v a_o ruler_n to_o the_o gr._n part_n 1_o 176_o 2_o 355_o 3_o 537_o 4_o 723_o 5_o 913_o 6_o 1106_o 7_o 1302_o 8_o 1503_o 9_o 1708_o 10_o 1917_o 11_o 2130_o 12_o 2348_o 13_o 2571_o 14_o 2799_o 15_o 3032_o 16_o 3270_o 17_o 3514_o 18_o 3763_o 19_o 4019_o 20_o 4281_o 21_o 4550_o 22_o 4825_o 23_o 5108_o tro_z 5252_o centre_n a_o and_o 57_o gr_n 48_o m._n in_o the_o quadrant_a the_o point_n where_o the_o ruler_n cross_v the_o ecliptic_a shall_v be_v the_o first_o point_n of_o ♊_o and_o so_o for_o the_o rest_n but_o the_o line_n of_o distinction_n between_o sign_n &_o sign_n may_v be_v best_a draw_v from_o the_o centre_n g._n 6_o the_o line_n et_fw-fr between_o the_o equator_fw-la and_o the_o tropic_n which_o i_o call_v the_o line_n of_o declination_n may_v be_v divide_v into_o 23_o gr_n ½_n out_o of_o this_o table_n for_o let_v ae_n the_o semidiameter_n of_o the_o equator_fw-la be_v 10000_o the_o distance_n between_o the_o equator_fw-la and_o 10_o gr_n of_o declination_n may_v be_v 1917_o more_o between_o the_o equator_fw-la and_o 20_o gr_n 4281_o the_o distance_n of_o the_o tropic_n from_o the_o equator_fw-la 5252._o 7_o you_o may_v put_v in_o the_o most_o of_o the_o principal_a star_n between_o the_o equator_fw-la and_o the_o tropic_n of_o ♋_o by_o their_o declination_n from_o the_o equator_fw-la and_o right_a ascension_n from_o the_o next_o equinoctial_a point_n as_o the_o declination_n of_o the_o wing_n of_o pegasus_n be_v 13_o gr_n 7_o m._n the_o right_a ascension_n 358_o gr_n 34_o m._n from_o the_o first_o point_n of_o ♈_o or_o 1_o gr_n 26_o m._n short_a of_o it_o if_o you_o draw_v a_o occult_a parallel_n through_o 13_o gr_n 7_o m._n of_o declination_n and_o then_o lie_v the_o ruler_n to_o the_o centre_n a_o and_o 1_o gr_n 26_o m._n in_o the_o quadrant_a bc_n the_o point_n where_o the_o ruler_n cross_v the_o parallel_n shall_v be_v the_o place_n for_o the_o wing_n of_o pegasus_n to_o which_o you_o may_v set_v the_o name_n and_o the_o time_n when_o he_o come_v to_o the_o south_n in_o this_o manner_n w._n peg._n *_o 23_o ho._n 54_o m._n and_o so_o for_o the_o rest_n of_o these_o five_o or_o any_o other_o star_n  _fw-fr ho._n m._n r._n ascen_n decl._n m_o pegasus_n wing_n *_o 23_o 54_o 1_o 26_o 13_o 7_o arcturus_n *_o 13_o 58_o 29_o 37_o 21_o 10_o lion_n heart_n *_o 9_o 48_o 32_o 58_o 13_o 45_o bull_n eye_v *_o 4_o 15_o 63_o 33_o 15_o 42_o vulture_n heart_n *_o 19_o 33_o 66_o 56_o 7_o 58_o 8_o there_o be_v space_n sufficient_a between_o the_o equator_fw-la and_o the_o centre_n you_o may_v there_o describe_v the_o quadrat_fw-la and_o divide_v each_o of_o the_o two_o side_n far_o from_o the_o centre_n a_o into_o 100_o part_n so_o shall_v the_o quadrant_n be_v prepare_v general_o for_o any_o latitude_n but_o before_o you_o draw_v the_o particular_a line_n you_o be_v to_o fit_v four_o table_n unto_o your_o latitude_n first_o a_o table_n of_o meridian_n altitude_n for_o division_n of_o the_o circle_n of_o day_n and_o month_n which_o may_v be_v thus_o make_v consider_v the_o latitude_n of_o the_o place_n and_o the_o declination_n of_o the_o sun_n for_o each_o day_n of_o the_o year_n if_o the_o latitude_n and_o declination_n be_v alike_o both_o north_n or_o both_o south_n ad_fw-la the_o declination_n to_o the_o compliment_n of_o the_o latitude_n if_o they_o be_v unlike_a one_o north_n and_o the_o other_o south_n subtract_v the_o declination_n from_o the_o compliment_n of_o the_o latitude_n the_o remainder_n will_v be_v the_o meridian_n altitude_n belong_v unto_o the_o day_n thus_o in_o our_o latitude_n of_o 51_o gr_n 30_o m._n northward_o who_o compliment_n be_v 38_o gr_n 30_o m._n the_o declination_n upon_o the_o ten_o day_n of_o june_n will_v be_v 23_o gr_n 30_o m._n northward_o wherefore_o i_o add_v 23_o gr_n 30_o m._n unto_o 38_o gr_n 30_o m._n the_o sum_n of_o both_o be_v 62_o gr_n for_o the_o meridian_n altitude_n at_o the_o ten_o of_o june_n the_o declination_n upon_o the_o ten_o of_o december_n will_v be_v 23_o gr_n 30_o m._n southward_o wherefore_o i_o take_v these_o 23_o gr_n 30_o m._n out_o of_o 38_o gr_n 30_o m._n there_o will_v remain_v 15_o gr_n for_o the_o meridian_n altitude_n at_o the_o ten_o of_o december_n and_o in_o this_o manner_n you_o may_v find_v the_o meridian_n altitude_n for_o each_o day_n of_o the_o year_n and_o set_v they_o down_o in_o a_o table_n dies_fw-la 0_o 5_o 10_o 15_o 20_o 25_o 30_o mo_z gr._n m_o gr._n m._n gr._n m_o gr._n m._n gr._n m._n gr._n m_o gr._n m._n january_n 16_o 31_o 17_o 24_o 18_o 26_o 19_o 37_o 20_o 57_o 22_o 24_o 23_o 58_o february_n 24_o 17_o 25_o 59_o 27_o 45_o 29_o 35_o 31_o 29_o 33_o 25_o  _fw-fr march_n 34_o 35_o 36_o 33_o 38_o 32_o 40_o 30_o 42_o 27_o 44_o 22_o 46_o 15_o april_n 46_o 37_o 48_o 26_o 50_o 11_o 51_o 50_o 53_o 25_o 54_o 53_o 56_o 15_o may_n 56_o 15_o 57_o 29_o 58_o 35_o 59_o 33_o 60_o 22_o 61_o 2_o 61_o 31_o june_n 61_o 36_o 61_o 54_o 62_o 0_o 61_o 58_o 61_o 45_o 61_o 22_o 60_o 49_o july_n 60_o 49_o 60_o 6_o 59_o 14_o 58_o 13_o 57_o 4_o 55_o 48_o 54_o 24_o august_n 54_o 7_o 52_o 36_o 50_o 59_o 49_o 17_o 47_o 31_o 45_o 41_o 43_o 49_o september_n 43_o 26_o 41_o 30_o 39_o 33_o 37_o 36_o 35_o 38_o 33_o 41_o 31_o 46_o october_n 31_o 46_o 29_o 53_o 28_o 3_o 26_o 16_o 24_o 35_o 22_o 59_o 21_o 29_o november_n 21_o 12_o 19_o 51_o 18_o 39_o 17_o 36_o 16_o 43_o 16_o 0_o 15_o 28_o december_n 15_o 28_o 15_o 7_o 15_o 0_o 15_o 2_o 15_o 17_o 15_o 44_o 16_o 22_o the_o table_n be_v make_v you_o may_v inscribe_v the_o month_n and_o day_n of_o each_o month_n into_o your_o quadrant_a in_o the_o space_n leave_v below_o the_o tropic_n for_o lay_v the_o ruler_n unto_o the_o centre_n a_o and_o 16_o gr_n 31_o m._n in_o the_o quadrant_a bc_n there_o may_v you_o draw_v a_o line_n for_o the_o end_n of_o december_n and_o beginning_n of_o
whether_o the_o two_o station_n be_v choose_v at_o the_o one_o end_n of_o the_o breadth_n propose_v or_o without_o it_o or_o within_o it_o if_o the_o line_n between_o the_o station_n be_v perpendicular_a unto_o the_o breadth_n as_o may_v appear_v if_o in_o stead_n of_o the_o station_n at_o a_o and_o h_n we_o make_v choice_n of_o the_o like_a station_n at_o i_o and_o k._n there_o may_v be_v other_o way_n propose_v to_o work_v these_o prop._n by_o hold_v the_o cross_n even_o with_o the_o distance_n and_o the_o staff_n parallel_v with_o the_o height_n but_o these_o will_v prove_v more_o troublesome_a and_o those_o which_o be_v deliver_v be_v sufficient_a and_o the_o same_o with_o those_o which_o other_o have_v set_v down_o under_o the_o name_n of_o the_o jacobs_n staff_n chap._n iii_o the_o use_n of_o the_o tangent_fw-la line_n in_o take_v of_o angles_n 1_o to_o find_v a_o angle_n by_o the_o tangent_fw-la on_o the_o staff_n let_v the_o middle_a sight_n be_v always_o set_v to_o the_o middle_n of_o the_o cross_n note_v with_o 20_o and_o 30_o and_o then_o the_o cross_n draw_v near_o the_o eye_n until_o the_o mark_n may_v be_v see_v close_o within_o the_o sight_n for_o so_o if_o the_o eye_n at_o a_o that_o end_n of_o the_o staff_n which_o be_v note_v with_o 90_o and_o 180_o behold_v the_o mark_n k_o and_o n_n between_o the_o two_o first_o sight_n c_o and_o b_o or_o the_o mark_n k_o and_o p_o between_o the_o two_o outward_a sight_n the_o cross_n be_v draw_v down_o unto_o h_n shall_v stand_v at_o 30_o and_o 60_o in_o the_o tangent_fw-la on_o the_o staff_n it_o show_v that_o the_o angle_n can_n be_v 30_o gr_n the_o angle_n kap_n 60_o gr_n the_o onâ—_n double_a to_o the_o other_o which_o be_v â—he_n reason_n of_o the_o double_a number_n on_o this_o line_n of_o the_o staff_n and_o this_o way_n will_v serve_v for_o any_o angle_n from_o 20_o gr_n towards_o 90_o gr_n or_o from_o 40_o gr_n towards_o 180_o gr_n but_o if_o the_o angle_n be_v less_o than_o 20_o gr_n we_o must_v then_o make_v use_n of_o the_o tangent_fw-la upon_o the_o cross_n 2_o to_o find_v a_o angle_n by_o the_o tangent_fw-la of_o 20_o upon_o the_o cross_n set_a 20_o unto_o 20_o that_o be_v the_o middle_a sight_n to_o the_o midst_n of_o the_o cross_n at_o the_o end_n of_o the_o staff_n note_v with_o 20_o so_o the_o eye_n at_o a_o behold_v the_o mark_n l_o and_o n_o close_o between_o the_o two_o first_o sight_n c_o and_o b_o shall_v seâ—_n they_o in_o a_o angle_n of_o 20_o gr_n if_o the_o mark_n shâ—ll_v be_v near_o together_o as_o be_v m_n and_o n_o then_o draw_v in_o the_o cross_n from_o c_o unto_o e_z if_o they_o be_v far_o asunder_o as_o be_v k_o and_o n_n then_o draw_v out_o the_o cross_n from_o c_o unto_o f_o so_o the_o quantity_n of_o the_o angle_n shall_v be_v still_o find_v in_o the_o cross_n in_o the_o tangent_fw-la of_o 20_o gr_n at_o the_o end_n of_o the_o staff_n and_o this_o will_v serve_v for_o any_o angle_n from_o 0_o gr_n towards_o 35_o gr_n 3_o to_o find_v a_o angle_n by_o the_o tangent_fw-la of_o 30_o upon_o the_o cross_n this_o tangent_fw-la of_o 30_o be_v here_o put_v the_o rather_o that_o the_o end_n of_o the_o staff_n rest_v at_o the_o eye_n the_o hand_n may_v more_o easy_o remove_v the_o cross_n for_o it_o suppose_v the_o radius_fw-la to_o be_v no_o long_o then_o ah_o which_o be_v from_o the_o eye_n at_o the_o end_n of_o the_o staff_n unto_o 30_o gr_n about_o 22_o inch_n and_o 7_o part_n wherefore_o here_o set_v the_o middle_a sight_n unto_o 30_o gr_n on_o the_o staff_n and_o then_o either_o draw_v the_o cross_n in_o or_o out_o until_o the_o mark_n be_v see_v between_o the_o two_o first_o sight_n so_o the_o quantity_n of_o the_o angle_n will_v be_v find_v in_o the_o tangent_fw-la of_o 30_o which_o be_v here_o represent_v by_o the_o line_n gh_o and_o this_o will_v serve_v for_o any_o angle_n from_o 0_o gr_n towards_o 48_o gr_n 4_o to_o observe_v the_o altitude_n of_o the_o sun_n backward_o here_o it_o be_v fit_a to_o have_v a_o horizontall_a sight_n set_v to_o the_o begin_n of_o the_o staff_n and_o then_o may_v you_o turn_v your_o back_n towards_o the_o sun_n and_o your_o cross_n towards_o your_o eye_n if_o the_o altitude_n be_v under_o 45_o gr_n set_v the_o middle_a sight_n to_o 30_o on_o the_o staff_n and_o look_v by_o the_o middle_a sight_n through_o the_o horizontall_a unto_o the_o horizon_n move_v the_o cross_n upward_o or_o downward_o until_o the_o upper_a sight_n do_v shadow_n the_o upper_a half_n of_o the_o horizontall_a sight_n so_o the_o altitude_n will_v be_v find_v in_o the_o tangent_fw-la of_o 30._o if_o the_o altitude_n shall_v be_v more_o than_o 45_o gr_n set_v the_o middle_a sight_n unto_o the_o midst_n of_o the_o cross_n and_o look_v by_o the_o inward_a edge_n of_o the_o low_a sight_n through_o the_o horizontall_a to_o the_o horizon_n move_v the_o middle_a sight_n in_o or_o out_o until_o the_o upper_a sight_n do_v shadow_v the_o upper_a half_n of_o the_o horizontall_a sight_n so_o the_o altitude_n will_v be_v find_v in_o the_o degree_n on_o the_o staff_n between_o 40_o and_o 180._o 5_o to_o set_v the_o staff_n to_o any_o angle_n give_v this_o be_v the_o converse_n of_o the_o former_a prop._n for_o if_o the_o middle_a sight_n be_v set_v to_o his_o place_n and_o degree_n the_o eye_n look_v close_o by_o the_o sight_n as_o before_o can_v but_o see_v his_o object_n in_o the_o angle_n give_v 6_o to_o observe_v the_o altitude_n of_o the_o sun_n another_o way_n set_a the_o middle_a sight_n to_o the_o middle_n of_o the_o cross_n and_o hold_v the_o horizontall_a sight_n downward_o so_o as_o the_o cross_n may_v be_v parallel_n to_o the_o horizon_n then_o be_v the_o staff_n vertical_a and_o if_o the_o outward_a sight_n of_o the_o cross_n do_v shadow_v the_o horizontall_a sight_n the_o compliment_n of_o the_o altitude_n will_v be_v find_v in_o the_o tangent_fw-la on_o the_o staff_n 7_o to_o observe_v a_o altitude_n by_o thread_n and_o plummet_n let_v the_o middle_a sight_n be_v set_v to_o the_o midst_n of_o the_o cross_n and_o to_o that_o end_n of_o the_o staff_n which_o be_v note_v with_o 90_o and_o 180_o then_o have_v a_o thread_n and_o a_o plummet_n at_o the_o begin_n of_o the_o cross_n and_o turn_v the_o cross_n upward_o and_o the_o staff_n towards_o the_o sun_n the_o thread_n will_v fall_v on_o the_o compliment_n of_o the_o altitude_n above_o the_o horizon_n and_o this_o may_v be_v apply_v to_o other_o purpose_n 8_o to_o apply_v the_o line_n of_o inch_n to_o the_o take_n of_o angle_n if_o the_o angle_n be_v observe_v between_o the_o two_o first_o sight_n there_o will_v be_v such_o proportion_n between_o the_o part_n of_o the_o staff_n and_o the_o part_n of_o the_o cross_n as_o between_o the_o radius_fw-la and_o the_o tangent_fw-la of_o the_o angle_n as_o if_o the_o part_n intercept_v on_o the_o staff_n be_v 20_o inch_n the_o part_n on_o the_o cross_n 9_o inch_n then_o by_o proportion_n as_o 20_o unto_o 9_o so_o 100000_o unto_o 45000_o the_o tangent_fw-la of_o 24_o gr_n 14_o m._n but_o if_o the_o angle_n shall_v be_v observe_v between_o the_o two_o outward_a sight_n the_o part_n be_v 20_o and_o 9_o as_o before_o the_o angle_n will_v be_v 48_o gr_n 28_o m._n double_a unto_o the_o former_a in_o all_o these_o there_o be_v a_o regard_n to_o be_v have_v to_o the_o parallax_n of_o the_o eye_n and_o his_o height_n above_o the_o horizon_n in_o observation_n at_o sea_n to_o the_o semidiameter_n of_o the_o sun_n his_o parallax_n and_o refraction_n as_o in_o the_o use_n of_o other_o staff_n and_o so_o this_o will_v be_v as_o much_o or_o more_o than_o that_o which_o have_v be_v heretofore_o perform_v by_o the_o crosse-staffe_n chap._n four_o the_o use_n of_o the_o line_n of_o equal_a part_n join_v with_o the_o line_n of_o chord_n the_o line_n of_o equal_a part_n do_v serve_v also_o for_o protraction_n as_o may_v appear_v by_o the_o former_a diagram_n but_o be_v join_v with_o the_o line_n of_o chord_n which_o i_o place_n upon_o one_o side_n of_o the_o cross_n they_o will_v far_o serve_v for_o the_o protraction_n and_o resolution_n of_o right_a line_n triangle_n whereof_o i_o will_v give_v one_o example_n in_o find_v of_o a_o distance_n at_o two_o station_n otherwise_o then_o in_o the_o second_o cap._n let_v the_o distance_n require_v be_v ab_fw-la at_o a_o the_o first_o station_n i_o make_v choice_n of_o a_o station_n line_n towards_o c_o and_o observe_v the_o angle_n bac_n by_o the_o tangent_fw-la line_n which_o may_v be_v 43_o gr_n 20_o m_o then_o have_v go_v a_o hundred_o pace_n towards_o c_o i_o make_v my_o second_o station_n at_o d_o
january_n then_o lay_v your_o ruler_n to_o the_o centre_n a_o and_o 24_o gr_n 17_o m._n in_o the_o quadrant_a there_o draw_v the_o end_n of_o january_n and_o beginning_n of_o february_n and_o so_o the_o rest_n which_o may_v be_v note_v with_o ay_o f_o m_o a_o m_o i_o etc._n etc._n the_o first_o letter_n of_o each_o month_n and_o will_v here_o fall_v between_o 15_o gr_n and_o 62_o gr_n the_o second_o table_n which_o you_o be_v to_o fit_v may_v serve_v for_o the_o draw_v and_o divide_v of_o the_o horizon_n for_o draw_v of_o the_o horizon_n as_o the_o cotangent_fw-la of_o the_o latitude_n to_o the_o tangent_fw-la of_o the_o great_a declination_n so_o the_o sine_fw-la of_o 90_o gr_n to_o the_o sine_fw-la of_o intersection_n where_o the_o horizon_n shall_v cross_v the_o tropique_n so_o in_o our_o latitude_n of_o 51_o gr_n 30_o m._n we_o shall_v find_v the_o horizon_n to_o cut_v the_o tropic_n in_o 33_o gr_n 9_o m_o wherefore_o if_o you_o lay_v the_o ruler_n to_o the_o centre_n a_o and_o 33_o gr_n 9_o m._n in_o the_o quadrant_a the_o point_n where_o the_o ruler_n cross_v the_o tropic_n shall_v be_v the_o point_n where_o the_o horizon_n cross_v the_o tropic_n and_o if_o you_o find_v a_o point_n at_o h_n in_o the_o line_n ac_fw-la whereon_o set_v the_o compass_n you_o may_v bring_v the_o point_n at_o e_o and_o this_o point_n in_o the_o tropic_n both_o into_o a_o circle_n the_o point_n h_n shall_v be_v the_o centre_n and_o the_o ark_n so_o draw_v shall_v be_v the_o horizon_n then_o for_o the_o division_n of_o this_o horizon_n as_o the_o sine_fw-la of_o 90_o gr_n to_o the_o sine_fw-la of_o the_o latitude_n so_o the_o tangent_fw-la of_o the_o horizon_n to_o the_o tangent_fw-la of_o the_o ark_n in_o the_o quadrant_a which_o shall_v divide_v the_o horizon_n so_o in_o our_o latitude_n of_o 51_o gr_n 30_o m._n we_o shall_v find_v 7_o gr_n 52_o m._n belong_v to_o 10_o gr_n in_o the_o horizon_n and_o 15_o gr_n 54_o m._n belong_v to_o 20_o gr_n and_o so_o the_o rest_n as_o in_o this_o table_n ho_o gr._n m_o ho_o gr._n m_o ho_o gr._n m._n ho_o gr._n m._n ho_o gr._n m._n ho_o gr._n m._n 0_o 0_o 0_o 15_o 11_o 51_o 30_o 24_o 19_o 45_o 38_o 2_o 60_o 53_o 35_o 75_o 71_o 5_o  _fw-fr 0_o 47_o  _fw-fr 12_o 39_o  _fw-fr 25_o 11_o  _fw-fr 39_o 1_o  _fw-fr 54_o 41_o  _fw-fr 72_o 19_o  _fw-fr 1_o 34_o  _fw-fr 13_o 27_o  _fw-fr 26_o 4_o  _fw-fr 40_o 0_o  _fw-fr 55_o 48_o  _fw-fr 73_o 33_o  _fw-fr 2_o 21_o  _fw-fr 14_o 16_o  _fw-fr 26_o 57_o  _fw-fr 41_o 0_o  _fw-fr 56_o 56_o  _fw-fr 74_o 48_o  _fw-fr 3_o 8_o  _fw-fr 15_o 4_o  _fw-fr 27_o 50_o  _fw-fr 42_o 0_o  _fw-fr 58_o 4_o  _fw-fr 76_o 3_o 5_o 3_o 55_o 20_o 15_o 54_o 35_o 28_o 43_o 50_o 43_o 0_o 65_o 59_o 13_o 80_o 77_o 18_o  _fw-fr 4_o 42_o  _fw-fr 16_o 43_o  _fw-fr 29_o 37_o  _fw-fr 44_o 1_o  _fw-fr 60_o 22_o  _fw-fr 78_o 33_o  _fw-fr 5_o 29_o  _fw-fr 17_o 33_o  _fw-fr 30_o 32_o  _fw-fr 45_o 3_o  _fw-fr 61_o 31_o  _fw-fr 79_o 49_o  _fw-fr 6_o 17_o  _fw-fr 18_o 22_o  _fw-fr 31_o 27_o  _fw-fr 46_o 5_o  _fw-fr 62_o 41_o  _fw-fr 81_o 5_o  _fw-fr 7_o 4_o  _fw-fr 19_o 12_o  _fw-fr 32_o 22_o  _fw-fr 47_o 8_o  _fw-fr 62_o 52_o  _fw-fr 82_o 21_o 10_o 7_o 52_o 25_o 20_o 2_o 40_o 33_o 18_o 55_o 48_o 11_o 70_o 65_o 3_o 85_o 83_o 37_o  _fw-fr 8_o 39_o  _fw-fr 20_o 53_o  _fw-fr 34_o 14_o  _fw-fr 49_o 14_o  _fw-fr 66_o 15_o  _fw-fr 84_o 53_o  _fw-fr 9_o 27_o  _fw-fr 21_o 44_o  _fw-fr 35_o 10_o  _fw-fr 50_o 19_o  _fw-fr 67_o 27_o  _fw-fr 86_o 10_o  _fw-fr 10_o 14_o  _fw-fr 22_o 36_o  _fw-fr 36_o 7_o  _fw-fr 51_o 24_o  _fw-fr 68_o 39_o  _fw-fr 87_o 26_o  _fw-fr 11_o 2_o  _fw-fr 23_o 27_o  _fw-fr 37_o 4_o  _fw-fr 52_o 29_o  _fw-fr 69_o 52_o  _fw-fr 88_o 43_o 15_o 11_o 51_o 30_o 24_o 19_o 45_o 38_o 2_o 60_o 53_o 35_o 75_o 71_o 5_o 90_o 90_o 0_o wherefore_o you_o may_v lay_v the_o ruler_n to_o the_o centre_n a_o and_z 7_o gr_n 52_o m._n in_o the_o quadrant_a bc_n the_o point_n where_o the_o ruler_n cross_v the_o horizon_n shall_v be_v 10_o gr_n in_o the_o horizon_n and_o so_o for_o the_o rest_n but_o the_o line_n of_o distinction_n between_o each_o five_o degree_n will_v be_v best_a draw_v from_o the_o centre_n h._n the_o three_o table_n for_o draw_v of_o the_o houre-line_n must_v be_v a_o table_n of_o the_o altitude_n of_o the_o sun_n above_o the_o horizon_n at_o every_o hour_n especial_o when_o he_o come_v to_o the_o equator_fw-la the_o tropique_n and_o some_o other_o intermediate_v declination_n if_o the_o sun_n be_v in_o the_o equator_fw-la and_o so_o have_v no_o declination_n as_o the_o sine_fw-la of_o 90_o gr_n to_o the_o cousin_n of_o the_o latitude_n so_o the_o cousin_n of_o the_o hour_n from_o the_o meridian_n to_o the_o sine_fw-la of_o the_o altitude_n thus_o in_o our_o latitude_n of_o 51_o gr_n 30_o m._n at_o six_o hour_n from_o the_o meridian_n the_o sun_n will_v have_v no_o altitude_n at_o five_o the_o altitude_n will_v be_v 9_o gr_n 17_o m_o at_o four_o 18_o gr_n 8_o m_z at_o three_o 26_o gr_n 7_o m_z at_o two_o 32_o gr_n 37_o m._n at_o one_o 36_o gr_n 58_o m_o at_o noon_n it_o will_v be_v 38_o gr_n 30_o m._n equal_a to_o the_o compliment_n of_o the_o latitude_n if_o the_o sun_n have_v declination_n the_o meridian_n altitude_n will_v be_v find_v as_o before_o for_o the_o table_n of_o day_n and_o month_n if_o the_o hour_n propose_v be_v six_o in_o the_o morning_n or_o six_o at_o night_n as_o the_o sine_fw-la of_o 90_o gr_n to_o the_o sine_fw-la of_o the_o latitude_n so_o the_o sine_fw-la of_o the_o declination_n to_o the_o sine_fw-la of_o the_o altitude_n thus_o in_o our_o latitude_n the_o declination_n of_o the_o sun_n be_v 23_o gr_n 30_o m._n the_o altitude_n will_v be_v find_v to_o be_v 18_o gr_n 11_o m_o the_o declination_n be_v 11_o gr_n 30_o m._n the_o altitude_n will_v be_v 9_o gr_n if_o the_o hour_n propose_v be_v neither_o twelve_o nor_o six_o as_o the_o cousin_n of_o the_o hour_n from_o the_o meridian_n to_o the_o sine_fw-la of_o 90_o gr_n so_o the_o tangent_fw-la of_o the_o latitude_n to_o the_o tangent_fw-la of_o a_o four_o ark_n so_o in_o our_o latitude_n and_o one_o hour_n from_o the_o meridian_n this_o four_o ark_n will_v be_v find_v to_o be_v 52_o gr_n 28_o m._n at_o two_o 55_o gr_n 26_o m._n at_o three_o 60_o gr_n 39_o m._n at_o four_o 68_o gr_n 22_o m._n and_o at_o five_o hour_n from_o the_o meridian_n 78_o gr_n 22_o m._n then_o consider_v the_o declination_n of_o the_o sun_n and_o the_o hour_n propose_v if_o the_o latitude_n and_o declination_n be_v both_o alike_o as_o with_o we_o in_o north_n latitude_n north_n declination_n and_o the_o hour_n fall_v between_o noon_n and_o six_o take_v the_o declination_n out_o of_o the_o four_o ark_n the_o remainder_n shall_v be_v your_o five_o ark_n but_o if_o either_o the_o hour_n fall_v between_o six_o and_o midnight_n or_o the_o latitude_n and_o declination_n shall_v be_v unlike_a add_v the_o declination_n unto_o the_o four_o ark_n and_o the_o sum_n of_o both_o shall_v be_v your_o five_o ark_n or_o if_o the_o sum_n shall_v exceed_v 90_o gr_n you_o may_v take_v the_o compliment_n unto_o 180_o gr_n this_o five_o ark_n be_v know_v as_o the_o sine_fw-la of_o the_o four_o ark_n to_o the_o sine_fw-la of_o the_o latitude_n so_o the_o cousin_n of_o the_o five_o ark_n to_o the_o sine_fw-la of_o the_o altitude_n thus_o in_o our_o latitude_n of_o 51_o gr_n 30_o m._n northward_o the_o sun_n have_v 23_o gr_n 30_o m._n of_o north_n declination_n if_o it_o shall_v be_v require_v to_o find_v the_o altitude_n of_o the_o sun_n for_o seven_o in_o the_o morning_n here_o because_o the_o latitude_n and_o declination_n be_v both_o alike_o to_o the_o northward_o and_o the_o hour_n propose_v fall_v between_o noon_n and_o six_o you_o may_v take_v 23_o gr_n 30_o m._n the_o ark_n of_o the_o declination_n out_o of_o 78_o gr_n 22_o m._n the_o four_o ark_n belong_v to_o the_o five_o hour_n from_o the_o meridian_n so_o there_o will_v remain_v 54_o gr_n 52_o m._n for_o your_o five_o ark_n then_o work_v according_a to_o the_o canon_n you_o shall_v find_v as_o the_o sine_fw-la of_o 78_o gr_n 22_o m._n your_o four_o ark_n to_o the_o sine_fw-la of_o 51_o gr_n 30_o m._n for_o the_o latitude_n so_o the_o sine_fw-la of_o 35_o gr_n 8_o m._n the_o compliment_n of_o your_o five_o ark_n to_o the_o sine_fw-la of_o 27_o gr_n 17_o m._n the_o altitude_n require_v if_o in_o the_o same_o latitude_n and_o declination_n it_o be_v require_v to_o find_v the_o altitude_n for_o five_o in_o the_o morning_n
hour_n past_o 2_o in_o the_o afternoon_n and_o which_o of_o these_o be_v the_o true_a time_n of_o the_o day_n may_v be_v soon_o know_v by_o a_o second_o observation_n for_o if_o the_o sun_n rise_v high_o it_o be_v the_o forenoon_n if_o it_o become_v low_a it_o be_v the_o afternoon_n 5_o the_o hour_n of_o the_o night_n be_v give_v to_o find_v how_o much_o the_o sun_n be_v below_o the_o horizon_n the_o sun_n be_v always_o so_o much_o below_o the_o horizon_n at_o any_o hour_n of_o the_o night_n as_o his_o opposite_a point_n be_v above_o the_o horizon_n at_o the_o like_a hour_n of_o the_o day_n and_o therefore_o the_o bead_n be_v set_v if_o the_o question_n be_v make_v of_o any_o hour_n of_o the_o night_n in_o the_o summer_n then_o move_v it_o to_o the_o like_a hour_n of_o the_o day_n in_o the_o winter_n if_o of_o any_o hour_n of_o the_o night_n in_o winter_n then_o move_v it_o to_o the_o like_a hour_n of_o the_o day_n in_o summer_n so_o the_o degree_n which_o the_o thread_n cut_v in_o the_o quadrant_n shall_v show_v how_o much_o the_o sun_n be_v below_o the_o horizon_n at_o that_o time_n as_o if_o it_o be_v require_v to_o know_v how_o much_o the_o sun_n be_v below_o the_o horizon_n the_o 10_o of_o april_n at_o 4_o of_o the_o clock_n in_o the_o morning_n the_o bead_n be_v set_v to_o his_o place_n according_a to_o the_o time_n in_o the_o summer_n hour_n bring_v it_o to_o 4_o of_o the_o clock_n in_o the_o afternoon_n in_o the_o winter_n hour_n and_o so_o shall_v you_o find_v the_o thread_n to_o cut_v 8_o gr_n and_o about_o 30_o m._n in_o the_o quadrant_a and_o so_o much_o be_v the_o sun_n below_o the_o horizon_n at_o that_o time_n 6_o the_o depression_n of_o the_o sun_n suppose_v to_o give_v the_o hour_n of_o the_o night_n with_o we_o or_o the_o hour_n of_o the_o day_n to_o our_o antipode_n here_o also_o because_o the_o sun_n be_v so_o much_o above_o the_o horizon_n at_o all_o hour_n of_o the_o day_n as_o his_o opposite_a point_n be_v below_o the_o horizon_n at_o the_o like_a hour_n of_o the_o night_n therefore_o first_o set_v the_o bead_n according_a to_o the_o time_n then_o bring_v the_o thread_n to_o the_o degree_n of_o the_o sun_n depression_n below_o the_o horizon_n so_o shall_v the_o bead_n fall_v on_o the_o contrary_a houre-line_n and_o there_o show_v the_o hour_n of_o the_o night_n in_o regard_n of_o we_o which_o be_v the_o like_a hour_n of_o the_o day_n to_o our_o antipode_n as_o if_o the_o 10_o of_o april_n the_o sun_n be_v then_o in_o the_o begin_n of_o ♉_o and_o by_o supposition_n 8_o gr_n 30_o m._n below_o the_o horizon_n in_o the_o east_n it_o be_v require_v to_o know_v what_o time_n of_o the_o night_n it_o be_v first_o set_v the_o bead_n according_a to_o the_o day_n in_o the_o summer_n hour_n then_o bring_v the_o thread_n to_o 8_o gr_n 30_o m._n in_o the_o quadrant_a so_o shall_v the_o bead_n fall_v among_o the_o winter_n hour_n on_o the_o line_n of_o 4_o of_o the_o clock_n in_o the_o afternoon_n wherefore_o to_o our_o antipode_n it_o be_v 4_o of_o the_o clock_n in_o their_o afternoon_n and_o to_o we_o it_o be_v then_o 4_o of_o the_o clock_n in_o the_o morning_n 7_o the_o time_n of_o the_o year_n or_o the_o place_n of_o the_o sun_n be_v give_v to_o find_v the_o begin_n of_o daybreak_a and_o end_n of_o twilight_n this_o proposition_n differ_v little_a from_o the_o former_a for_o the_o day_n be_v say_v to_o begin_v to_o break_v when_o the_o sun_n come_v to_o be_v but_o 18_o gr_n below_o our_o horizon_n in_o the_o east_n and_o twilight_n to_o end_v when_o it_o be_v get_v 18_o gr_n below_o the_o horizon_n in_o the_o west_n wherefore_o let_v the_o bead_n be_v set_v for_o the_o time_n and_o then_o bring_v the_o thread_n to_o 18_o gr_n in_o the_o quadrant_a so_o shall_v the_o bead_n fall_v on_o the_o contrary_a houre-line_n and_o there_o show_v the_o hour_n of_o twilight_n as_o before_o so_o if_o it_o be_v require_v to_o know_v at_o what_o time_n the_o day_n begin_v to_o break_v on_o the_o ten_o of_o april_n the_o sun_n be_v then_o in_o the_o begin_n of_o ♉_o first_o set_v the_o bead_n according_a to_o the_o time_n in_o the_o summer_n hour_n and_o then_o bring_v the_o thread_n to_o 18_o gr_n in_o the_o quadrant_a so_o shall_v the_o bead_n fall_v among_o the_o winter_n hour_n a_o little_a more_o than_o a_o quarter_n before_o 3_o in_o the_o morning_n and_o that_o be_v the_o time_n when_o the_o day_n begin_v to_o break_v upon_o the_o ten_o of_o april_n chap._n vii_o of_o the_o horizon_n the_o horizon_n be_v here_o represent_v by_o the_o ark_n draw_v from_o the_o begin_n of_o declination_n towards_o the_o end_n of_o february_n divide_v unequal_o and_o number_v by_o 10._o 20._o 30._o 40._o 1_o the_o day_n of_o the_o month_n or_o the_o place_n of_o the_o sun_n be_v know_v to_o find_v the_o amplitude_n of_o the_o sun_n rise_v and_o set_v let_v the_o bead_n rectify_v for_o the_o time_n be_v bring_v to_o the_o horizon_n and_o there_o it_o shall_v show_v the_o amplitude_n require_v as_o if_o the_o day_n give_fw-mi be_v the_o 15_o of_o may_n the_o sun_n be_v in_o the_o four_o degree_n of_o ♊_o the_o bead_n rectify_v and_o bring_v to_o the_o horizon_n shall_v there_o fall_v on_o 35_o gr_n 8_o m._n such_o be_v the_o amplitude_n of_o the_o sun_n rise_v from_o the_o east_n and_o of_o his_o set_n from_o the_o west_n which_o amplitude_n be_v always_o northward_o when_o the_o sun_n be_v in_o the_o northern_a sign_n and_o when_o he_o be_v in_o the_o southward_o sign_n always_o southward_o 2_o the_o day_n of_o the_o month_n or_o the_o place_n of_o the_o sun_n be_v give_v to_o find_v the_o ascensionall_a difference_n let_v the_o bead_n rectify_v for_o the_o time_n be_v bring_v to_o the_o horizon_n so_o the_o degree_n cut_v by_o the_o thread_n in_o the_o quadrant_a shall_v show_v the_o difference_n of_o ascension_n as_o if_o the_o day_n give_fw-mi be_v the_o 15_o of_o may_n the_o sun_n be_v in_o the_o four_o degree_n of_o ♊_o let_v the_o bead_n be_v rectify_v and_o bring_v to_o the_o horizon_n so_o shall_v the_o thread_n in_o the_o quadrant_a show_n the_o ascensionall_a difference_n to_o be_v 28_o gr_n and_o about_o 50_o m._n upon_o the_o ascensionall_a difference_n depend_v this_o corollary_n to_o find_v the_o hour_n of_o the_o rise_n and_o set_v of_o the_o sun_n and_o thereby_o the_o length_n of_o the_o day_n and_o night_n the_o time_n of_o the_o sun_n rise_v may_v be_v guess_v at_o by_o the_o 3_o of_o the_o last_o cap._n but_o here_o by_o the_o ascensionall_a difference_n it_o may_v be_v better_o find_v and_o that_o to_o a_o minute_n of_o time_n for_o if_o the_o ascensionall_a difference_n be_v convert_v into_o time_n allow_v a_o hour_n for_o 15_o gr_n and_o 4_o minute_n of_o a_o hour_n for_o each_o degree_n it_o show_v how_o long_o the_o sun_n rise_v before_o six_o of_o the_o clock_n in_o the_o summer_n and_o after_o six_o in_o the_o winter_n as_o if_o the_o day_n give_fw-mi be_v the_o 15_o of_o may_n the_o sun_n be_v in_o the_o four_o of_o ♊_o and_o his_o ascensionall_a difference_n find_v as_o before_o 28_o gr_n 50_o m_o this_o convert_v into_o time_n make_v 1_o ho._n and_o somewhat_o more_o than_o 55_o m._n of_o a_o hour_n wherefore_o the_o sun_n at_o that_o time_n in_o regard_n it_o be_v summer_n rise_v 1_o ho._n and_o full_a 55_o m._n before_o 6_o of_o the_o clock_n and_o so_o have_v the_o quantity_n of_o the_o semidiurnall_a ark_n the_o length_n of_o the_o day_n and_o night_n need_v not_o be_v unknown_a chap._n viii_o of_o the_o five_o star_n i_o may_v have_v put_v in_o more_o star_n but_o these_o may_v suffice_v for_o the_o find_n of_o the_o hour_n of_o the_o night_n at_o all_o time_n of_o the_o year_n and_o first_o i_o make_v choice_n of_o ala_fw-la pegasi_fw-la a_o star_n in_o the_o extremity_n of_o the_o wing_n of_o pegasus_n in_o regard_n it_o want_v but_o 6_o minute_n of_o time_n of_o the_o begin_n of_o ♈_o but_o because_o it_o be_v but_o of_o the_o second_o magnitude_n and_o not_o always_o to_o be_v see_v i_o make_v choice_n of_o four_o more_o one_o for_o each_o quarter_n of_o the_o ecliptic_a as_o of_o oculus_fw-la ♉_o the_fw-fr bull_n eye_n who_o right_a ascension_n convert_v into_o time_n be_v 4_o ho._n 15_o m_o then_o of_o cor_n ♌_o the_fw-fr lion_n heart_n who_o right_a ascension_n be_v 9_o ho._n 48_o m_o next_o of_o arcturus_n who_o right_a ascension_n be_v 13_o h._n 58_o m_o and_o last_o of_o aquila_n or_o the_o vulture_n heart_n who_o right_a ascension_n be_v 19_o h._n 33_o m._n these_o five_o star_n have_v all_o of_o