reduce_v their_o epoche_n to_o the_o meridian_n of_o london_n the_o most_o famous_a city_n of_o this_o our_o island_n of_o great_a britain_n for_o which_o reason_n and_o for_o the_o easiness_n of_o the_o calculation_n which_o our_o own_o countryman_n have_v invent_v and_o we_o here_o use_v the_o treatise_n itself_o do_v carry_v the_o style_n of_o astronomia_fw-la britannica_fw-la and_o whether_o this_o method_n of_o find_v the_o first_o inequality_n will_v agree_v with_o observation_n can_v be_v know_v until_o a_o geometrical_a way_n of_o find_v the_o excentricity_n &_o aphelion_n of_o the_o planet_n in_o a_o ellipsis_n be_v also_o propound_v to_o we_o and_o if_o at_o last_o when_o all_o thing_n in_o this_o elliptical_a astronomy_n shall_v geometrical_o be_v demonstrate_v the_o place_n of_o the_o planet_n thus_o find_v shall_v not_o agree_v with_o observation_n it_o will_v be_v hard_a to_o say_v in_o what_o particular_a the_o fault_n do_v lie_v and_o therefore_o we_o must_v not_o then_o impute_v it_o to_o the_o first_o inequality_n much_o less_o now_o when_o the_o geometry_n of_o that_o be_v demonstrate_v &_o other_o part_n be_v defective_a but_o a_o new_a hypothesis_n must_v be_v think_v of_o and_o until_o this_o shall_v be_v effect_v as_o there_o be_v no_o table_n extant_a by_o which_o the_o place_n of_o the_o planet_n can_v be_v compute_v with_o so_o much_o ease_n so_o none_o can_v be_v expect_v by_o which_o their_o place_n may_v be_v compute_v with_o more_o probability_n of_o truth_n then_o by_o these_o which_o we_o must_v now_o submit_v to_o thy_o censure_n and_o do_v commend_v to_o thy_o perusal_n john_n newton_n the_o content_n of_o the_o several_a book_n or_o part_n of_o astronomia_fw-la britannica_fw-la the_o first_o book_n the_o preface_n page_n 1._o the_o sun_n great_a declination_n be_v give_v to_o find_v his_o declination_n in_o any_o point_n of_o the_o ecliptic_a 9_o how_o to_o find_v the_o declination_n of_o a_o planet_n or_o fix_a star_n with_o latitude_n 10_o how_o to_o find_v the_o right_a ascension_n of_o any_o point_n of_o the_o ecliptic_a 13_o how_o to_o find_v the_o right_a ascension_n of_o a_o planet_n or_o other_o star_n with_o latitude_n 15_o the_o elevat_fw-la of_o the_o pole_n &_o declin_n of_o the_o sun_n give_v to_o find_v his_o amplit_fw-mi 16_o the_o meridian_n altitude_n and_o declination_n of_o the_o sun_n with_o the_o pole_n elevation_n give_v to_o find_v his_o true_a place_n in_o the_o zodiac_n 17_o have_v the_o meridian_n altitude_n of_o a_o unknown_a star_n and_o the_o distance_n thereof_o from_o a_o know_a star_n to_o find_v the_o right_a ascension_n of_o the_o unknown_a star_n ibid._n have_v the_o declin_n and_o right_a ascension_n of_o a_o star_n give_v to_o find_v the_o longitude_n and_o latitude_n thereof_o 19_o how_o to_o find_v the_o ascensionall_a difference_n 20_o how_o to_o find_v the_o oblique_a ascension_n or_o descension_n of_o any_o point_n of_o the_o ecliptic_a 21_o the_o pole_n elevation_n and_o the_o sun_n declination_n be_v give_v to_o find_v his_o altitude_n at_o any_o time_n assign_v 22_o have_v the_o sun_n great_a declination_n with_o his_o distance_n from_o the_o next_o equinoctional_a point_n to_o find_v the_o meridian_n angle_n or_o intersection_n of_o the_o meridian_n with_o the_o ecliptic_a 26_o to_o find_v the_o angle_n of_o the_o meridian_n with_o the_o horizon_n ibid._n the_o pole_n elevation_n with_o the_o sun_n altitude_n and_o declination_n give_v to_o find_v his_o azimuth_n ibid._n how_o to_o erect_v a_o figure_n of_o heaven_n 27_o to_o find_v the_o angle_n of_o the_o ecliptic_a with_o the_o horizon_n or_o the_o altitude_n of_o the_o nonagesime_fw-la deg_n together_o with_o its_o distance_n from_o the_o mid-heaven_n 36_o to_o find_v the_o parallactical_a angle_n or_o angle_v of_o the_o ecliptic_a with_o the_o vertical_a circle_n â—7_n the_o elevation_n of_o the_o pole_n and_o declination_n of_o the_o sun_n give_v to_o find_v the_o time_n when_o he_o will_v be_v due_a east_n and_o west_n 39_o the_o elevation_n of_o the_o pole_n with_o the_o sun_n declination_n and_o altitude_n give_v to_o find_v his_o distance_n from_o the_o meridian_n 40_o to_o find_v the_o time_n of_o the_o sun_n rise_v and_o set_v with_o the_o length_n of_o the_o day_n and_o night_n ibid._n to_o find_v the_o distance_n of_o a_o star_n from_o the_o meridian_n 41_o to_o find_v the_o elevation_n of_o the_o pole_n above_o any_o circle_n of_o position_n ibid._n of_o the_o ark_n of_o direction_n what_o it_o be_v and_o how_o to_o find_v it_o 45_o how_o to_o direct_a the_o mid-heaven_n and_o the_o imum_fw-la coeli_fw-la 46_o how_o to_o direct_a the_o ascendent_n or_o significator_n posit_v in_o the_o sign_n ascend_v ibid._n how_o to_o direct_v a_o significator_n posit_v in_o the_o sign_n descend_v 47_o however_o to_o find_v the_o arch_n of_o the_o equator_fw-la whereby_o be_v make_v the_o general_a table_n of_o position_n 48_o how_o by_o the_o general_a table_n of_o position_n to_o make_v a_o particular_a table_n for_o any_o latitude_n there_o express_v 49_o of_o the_o doctrine_n of_o the_o sphere_n in_o table_n 51_o the_o second_o book_n of_o the_o year_n civil_a and_o astronomical_a 57_o of_o the_o figure_n which_o the_o planet_n describe_v in_o their_o motion_n 66_o of_o the_o line_n and_o method_n to_o be_v use_v for_o the_o find_n of_o a_o planet_n true_a longitude_n from_o the_o aphelion_n in_o this_o figure_n 72_o of_o the_o proportion_n by_o which_o the_o motion_n of_o the_o planet_n do_v increase_v from_o the_o aphelion_n to_o the_o perihelion_n 74_o of_o the_o inequality_n of_o the_o earth_n annual_a motion_n and_o of_o the_o diameter_n in_o which_o the_o aphelion_n and_o perihelion_n be_v place_v 81_o of_o state_v the_o earth_n middle_a motion_n by_o sundry_a observation_n 90_o to_o calculate_v the_o sun_n true_a place_n and_o distance_n from_o the_o earth_n 94_o of_o the_o aequation_n of_o civil_a day_n 97_o of_o the_o theory_n and_o motion_n of_o the_o moon_n 98_o to_o calculate_v the_o true_a motion_n of_o the_o moon_n by_o table_n 109_o to_o find_v the_o moon_n true_a latitude_n and_o place_n in_o the_o ecliptic_a 113_o of_o the_o motion_n of_o the_o fix_a star_n 115_o of_o the_o motion_n of_o saturn_n 116_o of_o the_o motion_n of_o jupiter_n 121_o of_o the_o motion_n of_o mars_n 124_o of_o the_o motion_n of_o venus_n 128_o of_o the_o motion_n of_o mercury_n 133_o of_o the_o semidiameter_n of_o the_o sun_n moon_n and_o shadow_n of_o the_o earth_n 136_o of_o the_o proportion_n and_o magnitude_n of_o the_o three_o great_a body_n the_o sun_n moon_n and_o the_o earth_n 142_o of_o the_o proportion_n between_o the_o orb_n of_o the_o superior_a and_o inferior_a planet_n and_o the_o orb_n of_o the_o earth_n 144_o to_o find_v the_o mean_a conjunction_n and_o opposition_n of_o the_o sun_n &_o moon_n 151_o to_o find_v the_o true_a opposition_n and_o conjunction_n of_o the_o sun_n and_o moon_n 152_o to_o find_v whether_o there_o will_v be_v a_o eclipse_n or_o not_o 154_o to_o find_v the_o quantity_n of_o a_o lunar_a eclipse_n ibid._n to_o find_v the_o duration_n of_o a_o lunar_a eclipse_n or_o the_o continuance_n of_o the_o total_a darkness_n where_o the_o eclipse_n be_v total_a 155_o to_o find_v the_o moon_n latitude_n at_o the_o beginning_n &_o end_n of_o the_o eclipse_n 156_o to_o find_v the_o middle_n of_o the_o eclipse_n or_o great_a darkness_n 157_o of_o the_o calculation_n of_o the_o sun_n eclipse_n 159_o to_o find_v the_o parallax_n of_o longitude_n and_o latitude_n 160_o to_o find_v the_o visible_a motion_n of_o the_o moon_n from_o the_o sun_n for_o any_o time_n assign_v 162_o to_o find_v the_o time_n of_o the_o visible_a conjunction_n of_o the_o sun_n and_o moon_n ibid._n to_o find_v the_o visible_a latitude_n of_o the_o moon_n at_o the_o time_n of_o the_o visible_a conjunction_n 163_o to_o find_v the_o quantity_n of_o a_o solar_a eclipse_n 164_o to_o find_v the_o beginning_n and_o end_v of_o the_o sun_n eclipse_n ibid._n to_o find_v the_o visible_a latitude_n of_o the_o moon_n at_o the_o beginning_n and_o end_n of_o the_o sun_n eclipse_n 165_o to_o delineat_a the_o eclipse_n of_o the_o sun_n and_o moon_n 166_o the_o use_n of_o the_o table_n of_o refraction_n 168_o the_o index_n of_o the_o table_n a_o view_n of_o the_o more_o notable_a epochae_n 2_o a_o table_n show_v the_o dominical_a letter_n in_o both_o account_n 4_o a_o table_n show_v the_o golden_a number_n &_o epact_n in_o both_o account_n ibid._n the_o anticipation_n of_o the_o gregorian_a calendar_n ibid._n a_o table_n of_o movable_a feast_n in_o both_o account_n 5_o a_o table_n of_o fix_a feast_n 6_o a_o catalogue_n of_o place_n their_o latitude_n and_o distance_n in_o longitude_n from_o the_o meridian_n of_o london_n 8_o a_o table_n to_o convert_v sexagenary_a
move_v and_o how_o mechanical_o to_o draw_v this_o ellipticall_a figure_n of_o their_o motion_n upon_o a_o plane_n as_o to_o the_o sphere_n 1_o we_o suppose_v that_o the_o sun_n be_v place_v in_o the_o middle_n of_o the_o world_n in_o or_o about_o the_o centre_n of_o the_o sphere_n of_o the_o fix_a star_n and_o have_v no_o circular_a motion_n but_o centrall_a only_o 2_o that_o the_o earth_n be_v one_o of_o the_o planet_n and_o with_o her_o annual_a motion_n about_o the_o sun_n describe_v her_o orb_n between_o the_o orb_n of_o mars_n and_o venus_n 3_o that_o the_o moon_n be_v move_v about_o the_o earth_n as_o her_o centre_n and_o so_o in_o her_o annual_a motion_n have_v respect_n both_o to_o the_o earth_n and_o to_o the_o centre_n of_o the_o earth_n orb_n the_o sun_n 4_o that_o the_o orb_n of_o venus_n be_v next_o under_o the_o orb_n of_o the_o earth_n and_o the_o orb_n of_o mercury_n between_o the_o sun_n &_o the_o orb_n of_o venus_n next_o above_o the_o orb_n of_o the_o earth_n we_o suppose_v the_o orb_n of_o mars_n the_o orb_n of_o jupiter_n next_o above_o mars_n and_o the_o orb_n of_o saturn_n next_o to_o the_o orb_n of_o the_o fix_a star_n according_a to_o these_o suppose_a principle_n we_o will_v have_v immediate_o show_v the_o method_n of_o calculation_n but_o that_o the_o mechanical_a way_n of_o draw_v a_o ellpsis_n do_v if_o not_o demostrate_a yet_o at_o least_o illustrate_v that_o method_n a_o ellipsis_n by_o the_o help_n of_o a_o thread_n may_v be_v mechanical_o make_v thus_o first_o draw_v a_o right_a line_n to_o that_o length_n which_o you_o will_v have_v the_o great_a diameter_n to_o be_v which_o let_v be_v a_o p_o and_o from_o the_o middle_n of_o this_o line_n at_o x_o set_v off_o with_o your_o compass_n the_o equal_a distance_n x_o m_n and_o x_o h._n then_o take_v a_o piece_n of_o three_o of_o the_o same_o length_n with_o the_o diameter_n ap_n &_o fasten_v one_o end_n of_o the_o three_o in_o the_o point_n m_o and_o the_o other_o at_o h_n &_o with_o your_o pen_n extend_v the_o thread_n thus_o fasten_v to_o a_o &_o from_o thence_o towards_o p_o keep_v the_o three_o stiff_o upon_o your_o pen_n draw_v a_o line_n from_o p_o by_o b_o to_o a_o the_o line_n so_o draw_v shall_v be_v a_o ellipsis_n in_o which_o because_o the_o whole_a thread_n be_v equal_a to_o the_o diameter_n a_o p_o therefore_o the_o two_o line_n make_v by_o the_o thread_n in_o draw_v of_o the_o ellipsis_n must_v in_o every_o point_n of_o the_o ellipsis_n be_v also_o equal_a to_o the_o fame_n diameter_n a_o p_o they_o that_o desire_v a_o demonstration_n thereof_o geometrical_o may_v consult_v with_o apollonius_n pergaeus_n claudius_n mydorgius_n oâ—_n other_o in_o their_o treatise_n of_o conicall_a section_n for_o our_o present_a purpose_n this_o be_v sufficient_a and_o from_o the_o equality_n of_o those_o two_o line_n with_o the_o diameter_n a_o brief_a method_n of_o calculation_n be_v thus_o demonstrate_v by_o dr._n warde_n let_v the_o line_n m_o e_o be_v equal_a to_o a_o p_o and_o draw_v the_o line_n h_o b_o and_o h_o e_o then_o in_o the_o plain_a triangle_n m_o h_o e_o have_v the_o side_n m_n e_o equal_v to_o the_o diameter_n and_o m_o h_n the_o distance_n of_o the_o umbilique_a point_n with_o the_o angle_n h_o m_o e_o the_o angle_n m_o e_o h_o and_o m_o h_o e_o shall_v be_v give_v also_o but_o the_o angle_n b_o e_o h_o and_o b_o h_o e_o be_v equal_a because_o the_o side_n b_o h_n and_o b_o e_o be_v equal_a by_o construction_n and_o therefore_o if_o you_o subtract_v the_o angle_n b_o e_o h_o from_o the_o angle_n m_o h_o e_o the_fw-mi will_v remain_v the_o angle_n at_o the_o sun_n m_o h_o b_o which_o be_v a_o planet_n true_a longitude_n from_o the_o aphelion_n or_o the_o equated_a anomaly_n and_o of_o these_o three_o thing_n propound_v to_o be_v give_v the_o side_n m_o e_o be_v by_o construction_n make_v equal_a to_o the_o diameter_n a_o p_o how_o the_o angle_n h_o m_o e_o and_o the_o side_n m_o h_n must_v be_v have_v shall_v plain_o appear_v by_o that_o which_o follow_v chap._n 4._o of_o the_o proportion_n by_o which_o the_o motion_n of_o the_o planet_n do_v increase_v from_o the_o aphelion_n to_o the_o perihelion_n though_o the_o equal_a motion_n of_o the_o planet_n be_v to_o be_v reckon_v as_o have_v be_v say_v in_o diverse_a parallel_n circle_n about_o the_o axis_n of_o the_o cone_n who_o diameter_n must_v still_o increase_v from_o the_o top_n of_o the_o cone_n at_o a_o to_o the_o base_a b_o c_o that_o the_o motion_n of_o the_o planet_n in_o the_o ellipsis_n may_v increase_v also_o yet_o in_o the_o calculation_n we_o can_v convenient_o reckon_v the_o middle_a motion_n in_o any_o more_o circle_n than_o one_o and_o therefore_o it_o must_v be_v prove_v that_o the_o angle_n comprehend_v between_o the_o line_n draw_v from_o e_z to_o that_o point_n in_o the_o diameter_n of_o the_o ellipsis_n which_o be_v make_v the_o common_a centre_n of_o the_o circle_n of_o middle_a motion_n and_o frâ—â—_n that_o centre_n to_o the_o planet_n in_o the_o ellipsis_n be_v always_o equal_a to_o the_o aâ—gle_a of_o middle_a motion_n comprehend_v between_o the_o semidiameter_n of_o the_o planet_n proper_a circle_n of_o middle_a motion_n and_o the_o line_n dâ—â—wn_v from_o the_o centre_n of_o that_o circle_n to_o the_o planet_n in_o the_o ellipsis_n this_o bullâ—aldus_n take_v for_o grant_v and_o dr._n ward_n do_v thus_o demonstrate_v f_o m_o g_o be_v a_o circle_n of_o middle_a motion_n who_o centre_n be_v in_o the_o axis_n of_o the_o cone_n and_o in_o the_o umbilique_a point_n of_o the_o ellipsis_n m_o but_o a_o planet_n be_v in_o the_o ellipsis_n at_o r_o the_o proper_a circle_n of_o its_o middle_a motion_n be_v l_o β_n q_n and_o the_o angle_n comprehend_v by_o the_o radius_fw-la of_o that_o circle_n lβ_n and_o the_o line_n draw_v from_o β_n the_o centre_n thereof_o to_o r_o the_o place_n of_o the_o planet_n be_v equal_a to_o the_o angle_n comprehend_v between_o the_o line_n e_o m_o and_o the_o line_n draw_v from_o m_n to_o r_o in_o the_o ellipsis_n for_o thus_o there_o be_v make_v two_o right_a angle_a triangle_n m_n h_o r_o and_o β_n h_n r_o in_o which_o the_o side_n h_o r_o be_v common_a to_o both_o for_o it_o be_v set_v at_o right_a angle_n in_o the_o ellipsis_n and_o in_o the_o circle_n equant_fw-la and_o the_o side_n m_o h_n and_o β_n h_n be_v equal_a by_o what_o have_v be_v already_o say_v in_o the_o 2_o chapter_n therefore_o the_o angle_n r_o m_o h_n be_v equal_a to_o the_o angle_n r_o β_n h_n and_o by_o consequence_n r_o β_n l_o and_o r_o m_o e_o be_v equal_a also_o but_o the_o angle_n r_o β_n l_o be_v the_o angle_n of_o the_o middle_a motion_n of_o a_o planet_n from_o the_o aphelion_n or_o the_o angle_n of_o the_o simple_a excentricke_a anomaly_n and_o therefore_o the_o angle_n r_o m_o e_o be_v the_o mean_a anomaly_n also_o who_o compliment_n to_o a_o semicircle_n be_v the_o angle_n m_o e_o h_o in_o the_o diagram_n of_o the_o precede_a chapter_n when_o a_o planet_n therefore_o descend_v from_o e_z to_o k_o the_o angle_n at_o the_o axis_fw-la of_o the_o cone_n or_o at_o m_o the_o umbilique_a point_n of_o the_o ellipsis_n do_v always_o increase_v and_o therefore_o the_o mean_a anomaly_n be_v increase_v for_o the_o angle_n of_o its_o circle_n equant_fw-la do_v answer_n to_o more_o degree_n at_o the_o axis_fw-la and_o those_o angle_n also_o be_v always_o degree_n or_o part_n of_o great_a circle_n and_o therefore_o the_o planet_n increase_v in_o the_o swiftness_n of_o their_o motion_n in_o such_o proportion_n as_o the_o circle_n diameter_n or_o radii_fw-la of_o those_o circle_n equant_fw-la do_fw-mi increase_n for_o demonstration_n whereof_o let_v the_o line_n e_o ψ_n be_v make_v parallel_n to_o the_o axis_fw-la of_o the_o cone_n then_o shall_v the_o part_n of_o the_o semidiameter_n of_o the_o circle_n equant_fw-la comprehend_v between_o those_o parallel_n e_o ψ_n and_o a_o i_o be_v equal_a to_o the_o semidiameter_n of_o the_o least_o circle_n e_o o_fw-fr and_o the_o part_n intercept_v between_o that_o parallel_n e_o ψ_n and_o the_o side_n of_o the_o triangle_n a_o b_o shall_v be_v the_o proportional_a excess_n above_o the_o least_o motion_n in_o e_z o._n and_o the_o mean_a acceleration_n be_v in_o that_o circle_n equant_fw-la who_o plain_a parallel_n to_o the_o base_a of_o the_o cone_n do_v pass_v by_o the_o centre_n of_o the_o ellipsis_n that_o be_v in_o the_o circle_n v_o d_o t_o for_o see_v that_o m_n x_o &_o z_o x_o be_v equal_a as_o also_o five_o d_o &_o z_o x_o five_o d_o must_v needs_o be_v the_o difference_n between_o the_o semidiameter_n of_o the_o least_o circle_n e_o l_o &_o
f_o the_o compliment_n of_o the_o pole_n elevation_n h_o g_o the_o compliment_n of_o the_o sun_n altitude_n to_o find_v g_o f_o h_o the_o angle_n of_o the_o sun_n distance_n from_o the_o meridian_n 1_o the_o side_n h_o g_z 44._o 54111_o  _fw-fr 2_o the_o side_n f_o h_n 67._o 97090_o  _fw-fr 3_o the_o side_n g_o f_o 38._o 46667_o  _fw-fr sum_n 150._o 97868_o  _fw-fr â—alf_a sum_n 75._o 48934_o  _fw-fr â—â—ne_n of_o f_o h_n 67._o 97090_o co_fw-la be_v 0._o 0329234_o â—â—ne_n of_o g_o f_o 38._o 46667_o co_fw-la be_v 0._o 2061682_o differ_v g_o f_o ½_o sum_n 37._o 02267_o 9_o 7796909_o differ_v f_o h_n ½_o sum_n 07._o 51844_o 9_o 1167578_o quadrat_fw-la of_o the_o sine_fw-la of_o half_a the_o angle_n  _fw-fr 19_o 1355403_o which_o bisect_v be_v sine_fw-la of_o 21._o 69295_o 9_o 5677701_o and_o the_o double_a thereof_o be_v 43._o 38590._o the_o sun_n distance_n from_o the_o meridian_n and_o convert_v into_o time_n give_v two_o hour_n 89259_o part_n chap._n 21._o to_o find_v the_o time_n of_o the_o sun_n rise_v and_o set_v with_o the_o length_n of_o the_o day_n and_o night_n the_o ascensional_a difference_n of_o the_o sun_n be_v add_v to_o the_o semidiurnal_a arch_n in_o a_o right_a sphere_n that_o be_v to_o 90_o degree_n in_o the_o northern_a sign_n or_o substract_v from_o it_o in_o the_o southern_a there_o sum_n or_o difference_n will_v be_v the_o semidiurnal_a arch_n which_o double_v be_v the_o day_n arch_n and_o the_o compliment_n to_o 360_o be_v the_o night_n arch_n which_o bisect_v be_v the_o time_n of_o the_o sun_n rise_v and_o the_o day_n arch_n bisect_v be_v the_o time_n of_o his_o set_n as_o when_o the_o sun_n be_v in_o ten_o degree_n of_o gemini_fw-la his_o ascensional_a difference_n be_v find_v to_o be_v 30._o 61613_o the_o quadrant_n add_v 90._o the_o semidiurnal_a arch_n 120._o 61613_o the_o diurnal_a arch_n 241._o 23226_o who_o be_v compliment_n 118._o 76774_o convert_v into_o time_n give_v 7_o hour_n 91078_o part_n which_o bisect_v give_v the_o time_n of_o the_o sun_n rise_v 3_o hour_n 95539_o part_n or_o a_o little_a before_o 4_o of_o the_o clock_n chap._n 22._o to_o find_v the_o distance_n of_o a_o star_n from_o the_o meridian_n if_o a_o star_n be_v between_o the_o mid-heaven_n and_o the_o horoscope_n deduct_v the_o right_a ascension_n of_o the_o mid-heaven_n from_o the_o right_a ascension_n of_o the_o star_n what_o remain_v be_v the_o distance_n from_o the_o meridian_n if_o a_o star_n be_v between_o the_o mid-heaven_n and_o the_o 7_o house_n deduct_v the_o right_a ascension_n of_o the_o star_n from_o the_o right_a ascension_n of_o the_o mid-heaven_n and_o what_o remain_v be_v the_o distance_n as_o before_o if_o a_o star_n be_v between_o the_o 7_o house_n and_o the_o imum_fw-la coeli_fw-la or_o four_o house_n deduct_v the_o right_a ascension_n of_o the_o imum_fw-la coeli_fw-la from_o the_o right_a ascension_n of_o the_o star_n and_o what_o remain_v be_v the_o distance_n from_o the_o meridan_n if_o a_o star_n be_v between_o the_o ascendant_n and_o the_o imum_fw-la coeli_fw-la deduct_v the_o right_a ascension_n of_o the_o star_n from_o the_o right_a ascension_n of_o the_o imum_fw-la coeli_fw-la and_o what_o remain_v be_v the_o distance_n from_o the_o meridian_n as_o before_o for_o example_n in_o the_o precede_a figure_n the_o right_a ascension_n of_o the_o mid-heaven_n be_v 072_o deg_n 82_o part_n the_o sun_n be_v in_o the_o 12_o house_n and_o his_o right_a ascension_n 155._o 97_o from_o which_o deduct_v the_o right_a ascension_n of_o the_o m._n c._n 72._o 82_o the_o distance_n of_o the_o sun_n from_o the_o meridian_n be_v 83._o 15_o chap._n 23._o to_o find_v the_o elevation_n of_o the_o pole_n above_o any_o circle_n of_o position_n a_o circle_n of_o position_n be_v as_o it_o be_v a_o certain_a horizon_n upon_o which_o the_o point_n or_o star_n propose_v do_v arise_v pass_v by_o the_o two_o intersection_n of_o the_o horizon_n with_o the_o meridian_n and_o may_v be_v either_o above_o or_o under_o the_o earth_n in_o respect_n of_o the_o place_n for_o which_o the_o figure_n be_v erect_v a_o star_n posit_v in_o the_o 1_o 2_o 3_o 4_o 5_o 6_o house_n be_v under_o the_o earth_n 7_o 8_o 9_o 10_o 11_o 12_o house_n be_v above_o the_o earth_n thus_o in_o the_o annex_v diagram_n a_o h_n c_o be_v a_o circle_n of_o position_n pass_v by_o the_o horizontal_n point_n of_o the_o significator_n at_o h_n and_o the_o two_o intersection_n of_o the_o horizon_n of_o the_o place_n at_o a_o and_o c_o and_o l_o m_o be_v the_o elevation_n of_o the_o pole_n above_o this_o horizon_n of_o the_o star_n or_o circle_n of_o position_n to_o find_v which_o there_o must_v be_v know_v 1._o the_o latitude_n of_o the_o place_n 2._o the_o declination_n of_o the_o star_n or_o point_n propose_v 3._o the_o distance_n thereof_o from_o the_o meridian_n hence_o to_o find_v the_o angle_n of_o inclination_n of_o the_o circle_n of_o position_n with_o the_o meridian_n the_o proportion_n be_v as_o follow_v 1._o as_o the_o radius_fw-la to_o the_o tangent_fw-la of_o the_o compliment_n of_o the_o star_n declination_n so_o be_v the_o cousin_n of_o the_o star_n distance_n from_o the_o meridian_n to_o the_o tangent_fw-la of_o the_o first-arch_n to_o which_o the_o pole_n of_o the_o place_n be_v add_v or_o subtractod_a from_o it_o according_a to_o the_o follow_a direction_n their_o sum_n or_o difference_n be_v the_o second_o arch_n if_o the_o distance_n of_o the_o star_n from_o the_o meridian_n be_v more_o than_o 90_o and_o the_o declination_n south_n under_o the_o earth_n or_o north_n above_o it_o subtract_v the_o first_o arch_n from_o the_o pole_n elevation_n and_o what_o remain_v be_v the_o second_o arch_n if_o the_o distance_n of_o a_o star_n from_o the_o meridian_n be_v less_o than_o 90_o and_o the_o declination_n south_n under_o the_o earth_n or_o north_n above_o it_o add_v the_o pole_n elevation_n to_o the_o first_o arch_n and_o their_o aggâ—gate_n if_o less_o than_o 90_o be_v the_o second_o arch_n if_o more_o than_o â—0_n the_o compliment_n thereof_o if_o the_o distance_n of_o a_o star_n from_o the_o meridian_n be_v either_o more_o or_o less_o than_o 90_o and_o the_o declination_n north_n under_o the_o earth_n or_o south_n above_o it_o subtract_v the_o elevation_n of_o the_o pole_n from_o the_o first_o arch_n and_o what_o remaineâ—â—â—s_v the_o second_o arch_n if_o the_o distance_n of_o a_o star_n from_o the_o meridian_n be_v a_o just_a quadrant_a the_o angle_n of_o inclination_n may_v be_v find_v at_o one_o operation_n as_o in_o the_o four_o example_n 2_o as_o the_o sine_fw-la of_o the_o first_o arch_n find_v be_v to_o the_o cotongent_a of_o the_o star_n distance_n from_o the_o meridian_n so_o be_v the_o sine_fw-la of_o the_o second_o arch_n find_v to_o the_o cotangent_fw-la of_o the_o angle_n of_o inclination_n then_o to_o find_v the_o elevation_n of_o the_o pole_n above_o the_o circle_n of_o position_n the_o analogy_n be_v 3_o as_o the_o radius_fw-la to_o the_o sine_fw-la of_o the_o pole_n of_o the_o place_n so_o be_v the_o sine_fw-la of_o the_o angle_n of_o inclination_n to_o the_o sine_fw-la of_o the_o pole_n of_o the_o circle_n 1_o example_n let_v the_o distance_n of_o a_o star_n from_o the_o meridian_n be_v more_o than_o 90_o viz._n 97_o deg_n and_o the_o declination_n of_o the_o star_n 31_o deg_n north_n above_o the_o earth_n the_o pole_n of_o the_o place_n 45._o then_o in_o the_o oblique_a spherical_a triangle_n h_n m_o c_o we_o have_v limit_v 1_o the_o side_n m_o c_o the_o pole_n elevation_n 45_o degree_n 2_o the_o side_n h_o m_o the_o compliment_n of_o the_o star_n declination_n 59_o degree_n 3_o the_o angle_n h_o m_o c_o the_o star_n distance_n from_o the_o meridian_n 97_o or_o instead_o thereof_o the_o acue_n angle_n i_o m_o h_o 83_o the_o compliment_n of_o the_o other_o to_o a_o semicircle_n hence_o to_o find_v i_o m_o the_o proportion_n be_v as_o the_o radius_fw-la 90_o 10._o 000000â—_n to_o the_o tangent_fw-la of_o h_n m_o 59_o 10._o 2212262_o so_o be_v the_o cousin_n of_o i_o m_o h_o 83_o 9_o 085894â—_n to_o the_o tangent_fw-la of_o i_o m_o 11._o 47_o 9_o 3071206_o which_o be_v substract_v from_o 45_o the_o pole_n elevation_n there_o rest_v 33._o 53_o the_o second_o arch_n 2_o as_o the_o sine_fw-la of_o i_o m_o 11._o 47_o 9_o 2985361_o to_o the_o cotangent_fw-la of_o i_o m_o h_o 83_o 9_o 089143â—_n so_o be_v the_o sine_fw-la of_o i_o c_o 33._o 53_o 9_o 742232â—_n to_o the_o cotangent_fw-la of_o h_n c_o m_o 71._o 17_o 9_o 5328404_o the_o angle_n of_o inclination_n 3_o as_o the_o radius_fw-la 90_o 10._o 0000000_o to_o the_o sine_z of_o c_o m_o 45_o 9_o 8494850_o so_o be_v the_o sine_fw-la of_o h_n c_o m._n 71._o 17_o 9_o 9761116_o to_o the_o sine_z of_o l_o m._n
know_v upon_o what_o day_n of_o the_o week_n any_o feast_n or_o day_n of_o the_o month_n shall_v be_v the_o catalogue_n of_o place_n do_v serve_v to_o show_v the_o height_n of_o the_o pole_n in_o those_o place_n and_o the_o difference_n of_o the_o meridian_n of_o any_o place_n in_o the_o catalogue_n from_o that_o of_o london_n the_o letter_n s_o note_n that_o the_o distance_n be_v westward_o a_o that_o it_o be_v eastward_o the_o figure_n under_o the_o title_n of_o time_n be_v hour_n and_o decimal_a part_n of_o a_o hour_n the_o earth_n or_o any_o star_n come_v soon_o or_o late_a to_o the_o meridian_n of_o that_o place_n than_o that_o of_o london_n if_o the_o time_n of_o a_o lunar_a eclipse_n then_o or_o other_o appearance_n be_v give_v at_o london_n afternoon_n 8_o hour_n 23_o part_n and_o the_o time_n when_o this_o happen_v at_o uraniburge_n be_v inquire_v there_o be_v find_v in_o the_o catalogue_n for_o uraniburge_n 0_o hour_n 83_o part_n a_o if_o therefore_o according_a to_o the_o letter_n a_o 83_o part_n be_v add_v to_o the_o time_n give_v it_o make_v 9_o hour_n 06_o part_n for_o the_o time_n at_o uraniburge_n but_o if_o the_o time_n of_o another_o place_n be_v to_o be_v reduce_v to_o the_o time_n at_o london_n the_o difference_n be_v to_o be_v apply_v with_o the_o contrary_a title_n and_o that_o these_o reduction_n whether_o in_o time_n or_o motion_n may_v be_v the_o better_o compare_v with_o those_o book_n that_o be_v write_v in_o the_o old_a sexagenary_a form_n we_o have_v add_v table_n for_o the_o ready_a convert_n of_o sexagenary_a part_n into_o decimall_n and_o the_o contrary_a the_o first_o of_o these_o table_n be_v for_o the_o convert_n of_o the_o minute_n and_o second_o etc._n etc._n of_o a_o degree_n in_o motion_n and_o the_o other_o of_o the_o part_n of_o a_o day_n in_o time_n a_o example_n in_o each_o will_v be_v a_o sufficient_a explanation_n let_v it_o be_v require_v to_o find_v the_o decimall_n answer_v to_o 37′_n 25″_n 16‴_n 5_o '_o '_o '_o '_o 29_o '_o '_o '_o '_o '_o in_o motion_n in_o the_o first_o page_n of_o the_o table_n i_o find_v 37′_n 12″_n which_o be_v the_o near_a less_o and_o 62_o answer_v thereunto_o and_o in_o the_o three_o column_n of_o the_o second_o page_n in_o the_o top_n of_o the_o page_n i_o find_v 12″_n in_o which_o column_n i_o find_v 25_o second_n and_o in_o the_o six_o and_o last_o column_n of_o that_o page_n right_a against_o 25″_n i_o find_v this_o number_n 36111111_o which_o be_v annex_v to_o 62._o the_o decimall_n of_o 37_o minute_n 25_o second_n be_v 6236111111_o and_o the_o decimall_n of_o 16_o thud_n 0000740741_o the_o decimall_n of_o 5_o fourth_n 0000003858_o the_o decimall_n of_o 29_o fifth_n 0000000373_o their_o sum_n 6236856083_o be_v the_o decimall_n seek_v 2._o example_n again_o if_o it_o be_v require_v to_o find_v the_o decimall_n of_o 8_o hour_n 17_o minute_n 8_o second_o 5_o three_o 12_o fourth_n 9_o fift_n in_o the_o first_o column_n of_o the_o table_n entitle_v a_o table_n to_o convert_v the_o hour_n and_o minute_n of_o a_o day_n into_o decimall_n i_o find_v 7_o hour_n 12_o minute_n and_o in_o the_o second_o column_n the_o figure_n 3_o then_o look_v the_o 12_o minute_n in_o the_o top_n of_o the_o page_n i_o cast_v my_o eye_n downward_o in_o that_o column_n till_o i_o come_v to_o 8_o hour_n 17_o minute_n and_o in_o the_o last_o column_n of_o the_o page_n against_o 8_o hour_n 17_o minute_n i_o find_v this_o number_n 451388889_o and_o therefore_o the_o decimall_n of_o 8_o hour_n 17_o minute_n be_v 3451388889_o the_o decimall_n of_o 8_o second_o 925926_o the_o decimall_n of_o 5_o three_o 009645_o the_o decimall_n of_o 12_o fourth_n 0387_o the_o decimal_n of_o 9_o fifth_n 0005_o their_a aggregate_v 3452324852_o be_v the_o decimall_n seek_v to_o find_v the_o part_n of_o a_o degree_n in_o motion_n or_o of_o a_o day_n in_o time_n answer_v to_o any_o decimal_n give_v be_v but_o the_o contrary_a work_n to_o the_o former_a example_n as_o if_o it_o be_v require_v to_o find_v the_o part_n of_o a_o degree_n answer_v to_o 6236856083_o the_o 2_o first_o figure_n of_o this_o decimall_n be_v 62_o which_o be_v seek_v in_o the_o first_o page_n of_o the_o table_n give_v i_o 37′_n 12_o and_o 62_o being_n subtract_v from_o 6236856083_o the_o remainder_n will_v be_v 36856083_o which_o be_v seek_v in_o the_o last_o column_n my_o near_a number_n be_v 36111111_o and_o right_a against_o that_o number_n under_o 12_o in_o the_o top_n of_o the_o page_n i_o find_v 25_o therefore_o 37′_n 25″_n be_v the_o part_n of_o a_o degree_n answer_v to_o the_o decimall_n give_v but_o if_o you_o will_v find_v the_o three_o fourth_n and_o fifth_n from_o 36856083_o subtract_v 36111111_o the_o remainder_n be_v 749972_o which_o be_v seek_v among_o the_o decimal_n of_o the_o three_o give_v i_o 16_o three_o and_o this_o number_n to_o be_v subtract_v from_o it_o 740741_o and_o the_o remainder_n 004231_o be_v seek_v among_o the_o decimal_n of_o the_o fourth_n give_n i_o 5_o fourth_n and_o this_o number_n to_o be_v subtract_v from_o it_o 3858_o and_o the_o remainder_n 373_o seek_v among_o the_o decimal_n of_o the_o fifth_n give_v i_o 29_o fifth_n and_o so_o the_o part_n of_o a_o degree_n answer_v to_o the_o decimall_n give_v be_v 37_o minute_n 25_o second_n 16_o three_o 11_o fourth_n and_o 29_o fifth_n thus_o may_v you_o also_o find_v the_o part_n of_o a_o day_n in_o time_n answer_v to_o any_o decimall_n give_v the_o next_o thing_n to_o be_v do_v towards_o the_o find_n of_o the_o annual_a revolution_n of_o the_o planet_n be_v to_o find_v their_o entrance_n into_o any_o point_n of_o the_o zodiac_n desire_v and_o that_o may_v be_v do_v thus_o have_v the_o place_n of_o the_o planet_n take_v by_o observation_n before_o and_o after_o its_o entrance_n into_o the_o point_n desire_v subtract_v the_o observe_v place_n next_o before_o from_o the_o observe_v place_n next_o after_o and_o the_o remainder_n shall_v show_v you_o the_o apparent_a motion_n answerable_a to_o the_o time_n between_o those_o observation_n subtract_v also_o the_o former_a place_n from_o the_o place_n in_o the_o point_n desire_v and_o note_v their_o difference_n for_o as_o the_o former_a remainder_n that_o be_v the_o apparent_a motion_n between_o the_o observation_n be_v to_o the_o time_n between_o those_o observation_n so_o be_v this_o difference_n to_o the_o time_n between_o the_o first_o observation_n and_o the_o planet_n entrance_n into_o the_o point_n desire_v thus_o we_o be_v to_o deal_v with_o those_o observation_n that_o we_o ourselves_o shall_v make_v but_o one_o man_n age_n not_o be_v distance_n enough_o between_o the_o observation_n from_o whence_o the_o middle_a motion_n may_v be_v right_o state_v we_o must_v take_v some_o observation_n upon_o trust_n and_o find_v the_o middle_a motion_n by_o compare_v the_o observation_n make_v in_o former_a age_n with_o those_o of_o our_o own_o of_o the_o sun_n or_o earth_n take_v this_o example_n follow_v the_o vernal_a equinox_n observe_v 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13333_o s._n now_o then_o to_o find_v the_o distance_n of_o the_o moon_n from_o the_o earth_n in_o this_o eclipse_n the_o earth_n semidiameter_n be_v one_o degree_n let_v fec_n represent_v the_o true_a horizon_n bde_v the_o vertical_a at_o paris_n e_o the_o centre_n of_o the_o earth_n d_o the_o city_n of_o paris_n the_o moon_n true_a altitude_n aef_n deg_n 5766778_o the_o observe_v altitude_n adg_n deg_n 5711916._o the_o parallacticall_a angle_n dae_n deg_n 0._o 34861._o therefore_o in_o the_o triangle_n ade_n we_o have_v give_v all_o the_o angle_n and_o the_o find_v de_fw-fr one_o semidiameter_n of_o the_o earth_n to_o find_v ae_n for_o which_o the_o anolagy_a be_v as_o the_o sine_fw-la of_o dae_n 0_o d._n 54861_o co_fw-la be_v 2._o 0188745_o to_o the_o side_n de_fw-fr 1_o  _fw-fr 0._o 0000000_o so_o be_v the_o sine_fw-la of_o adb_fw-la 32_o d._n 88084_o 9_o 7347147_o to_o the_o side_n ae_n 56._o 70_o 1._o 7535892_o this_o foundation_n be_v lay_v he_o proceed_v to_o the_o rest_n and_o to_o show_v how_o we_o may_v possible_o fall_v into_o some_o absurdity_n he_o suppose_v the_o moon_n distance_n from_o the_o earth_n in_o this_o eclipse_n to_o be_v but_o 55_o semidiameter_n or_o the_o side_n bc_n in_o the_o follow_a figure_n the_o apparent_a angle_n of_o the_o semidiameter_n of_o the_o earth_n shadow_n chi_fw-mi 0._o 73583_o aef_n represent_v the_o sun_n his_o semidiameter_n ae_n the_o angle_n of_o his_o apparent_a semidiameter_n when_o he_o be_v perig._n age_n 16._o 45_o or_o in_o decimall_n 27916_o bhg_n represent_v the_o earth_n bg_n the_o semidiameter_n thereof_o hence_o to_o find_v he_o in_o the_o triangle_n hic_fw-la the_o proportion_n be_v as_o the_o sine_fw-la of_o hic_fw-la 89._o 26417_o co_fw-la be_v 0._o 0000358_o to_o the_o side_n hc_fw-mi 54_o 1._o 7323937_o so_o be_v the_o radius_fw-la hci_fw-la 90_o 10._o 0000000_o to_o the_o hypothenusal_a he_o 54._o 004_o 1._o 7324295_o 2._o in_o the_o triangle_n hbi_n we_o have_v give_v the_o side_n bh_n 1._o and_o 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the_o angle_n k_o i_o g_o 0._o 31857._o for_o â—â—_o i_o k_o 55_o co_fw-la be_v 8._o 2596373_o to_o radius_fw-la  _fw-fr 10._o 0000000_o so_o be_v k_o g_o 0._o 3065_o 1._o 4864305_o to_o the_o tangent_fw-la of_o k_n i_o g_o 0._o 31857_o 7._o 7450678_o and_o the_o angle_n b_o d_o g_o be_v equal_a thereunto_o but_o so_o the_o angle_n of_o the_o sun_n apparent_a semidiameter_n a_o g_o e_o 27916_o by_o observation_n be_v less_o than_o the_o angle_n a_o d_o e_o which_o be_v absurd_a and_o therefore_o some_o part_n assume_v be_v false_a the_o semidiameter_n of_o the_o sun_n and_o moon_n must_v not_o be_v change_v constant_a experience_n agree_v with_o these_o observation_n in_o this_o eclipse_n therefore_o bullialdus_n do_v take_v