as_o that_o by_o the_o sight_n on_o your_o ruler_n you_o may_v look_v to_o the_o other_o station_n this_o do_v turn_v your_o ruler_n to_o that_o object_n who_o distance_n you_o desire_v to_o know_v and_o observe_v how_o many_o degree_n of_o the_o circle_n be_v cut_v by_o the_o ruler_n as_o suppose_v 36_o degree_n as_o the_o angle_n acd_v in_o fig._n 30._o then_o remove_v your_o instrument_n to_o d_o lay_v the_o ruler_n on_o the_o diameter_n thereof_o and_o then_o turn_v the_o whole_a instrument_n about_o till_o through_o your_o sight_n you_o can_v espy_v the_o mark_n set_v up_o at_o your_o first_o station_n at_o c_o and_o there_o fix_v your_o instrument_n and_o then_o upon_o the_o centre_n of_o your_o circle_n turn_v your_o ruler_n till_o through_o the_o sight_n you_o can_v espy_v the_o object_n who_o distance_n be_v inquire_v suppose_v at_o a_o and_o observe_v the_o degree_n in_o the_o circle_n cut_v by_o the_o ruler_n which_o let_v be_v 112_o which_o be_v the_o angle_n adc_n and_o let_v the_o distance_n between_o your_o two_o station_n be_v dc_o 326_o foot_n so_o have_v you_o two_o angle_n and_o the_o side_n between_o they_o in_o a_o plain_a triangle_n give_v by_o which_o to_o find_v the_o other_o side_n the_o which_o by_o protraction_n may_v be_v do_v as_o have_v be_v show_v in_o the_o five_o proposition_n of_o chapter_n 8._o but_o by_o the_o table_n of_o sines_n and_o tangent_n the_o proportion_n be_v as_o the_o sine_fw-la of_o dac_n be_v to_o dc_o so_o be_v the_o sine_fw-la of_o acd_a to_o the_o side_n ad._n or_o as_o the_o sine_fw-la of_o dac_n be_v to_o the_o give_v side_n dc_o so_o be_v the_o sine_fw-la of_o adc_fw-la to_o the_o side_n ac_fw-la 6._o there_o be_v another_o instrument_n call_v the_o plain_a table_n which_o be_v nothing_o else_o but_o a_o piece_n of_o board_n in_o the_o fashion_n and_o bigness_n of_o a_o ordinary_a sheet_n of_o paper_n with_o a_o little_a frame_n to_o fasten_v a_o sheet_n of_o paper_n upon_o it_o which_o be_v also_o set_v upon_o a_o staff_n you_o may_v by_o help_n of_o your_o ruler_n take_v a_o distance_n therewith_o in_o this_o manner_n have_v measure_v the_o distance_n between_o your_o two_o station_n at_o d_o and_o c_o draw_v upon_o your_o paper_n a_o line_n on_o which_o have_v set_v off_o your_o distance_n place_v your_o instrument_n at_o your_o first_o station_n c_o and_o lay_v your_o ruler_n upon_o the_o line_n so_o draw_v thereon_o turn_v your_o instrument_n till_o through_o the_o sight_n you_o can_v espy_v the_o station_n at_o d_o then_o lay_v your_o ruler_n upon_o the_o point_n c_o turn_v the_o same_o about_o till_o through_o the_o sight_n you_o can_v espy_v the_o object_n at_o a_o and_o there_o draw_v a_o line_n by_o the_o side_n of_o your_o ruler_n and_o remove_v your_o instrument_n to_o d_o and_o lay_v your_o ruler_n upon_o the_o line_n dc_o turn_v the_o instrument_n about_o till_o through_o the_o sight_n you_o can_v espy_v the_o mark_n at_o c_o and_o then_o lay_v your_o ruler_n upon_o the_o 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it_o sufficient_a to_o show_v the_o use_n of_o these_o two_o the_o plain_a table_n and_o theodolite_n 2._o in_o the_o use_n of_o either_o of_o which_o the_o same_o chain_n which_o be_v use_v in_o take_v of_o height_n and_o distance_n be_v not_o so_o proper_a i_o rather_o commend_v that_o which_o be_v know_v by_o the_o name_n of_o gunter_n chain_n which_o be_v four_o pole_n divide_v into_o 100_o link_n be_v as_o i_o conceive_v much_o better_a for_o the_o cast_n up_o the_o content_a of_o a_o piece_n of_o ground_n than_o any_o other_o chain_n that_o i_o have_v yet_o hear_v of_o who_o easy_a use_n shall_v be_v explain_v in_o its_o proper_a place_n 3._o when_o you_o be_v therefore_o enter_v the_o field_n with_o your_o instrument_n whether_o plain_a table_n or_o theodolite_a have_v choose_v out_o your_o station_n let_v visible_a mark_n be_v set_v up_o in_o all_o the_o corner_n thereof_o and_o then_o if_o you_o use_v the_o plain_a table_n make_v a_o mark_n upon_o your_o paper_n represent_v your_o station_n and_o lay_v your_o ruler_n to_o this_o point_n direct_v your_o sight_n to_o the_o several_a corner_n of_o the_o field_n where_o you_o have_v cause_v mark_n to_o be_v set_v up_o and_o draw_v line_n by_o the_o side_n of_o the_o ruler_n upon_o the_o paper_n to_o the_o point_n represent_v your_o station_n then_o measure_v the_o distance_n of_o every_o of_o these_o mark_n from_o your_o instrument_n and_o by_o your_o scale_n set_v those_o distance_n upon_o the_o line_n draw_v upon_o the_o paper_n make_v small_a mark_n at_o the_o end_n of_o every_o such_o distance_n line_n draw_v from_o point_n to_o point_v shall_v give_v you_o upon_o your_o paper_n the_o plot_n of_o the_o field_n by_o which_o plot_n so_o take_v the_o content_a of_o the_o field_n may_v easy_o be_v compute_v example_n let_v fig._n 31._o represent_v a_o field_n who_o plot_n be_v require_v your_o table_n be_v place_v with_o a_o sheet_n of_o paper_n thereupon_o make_v a_o mark_n about_o the_o middle_n of_o your_o table_n as_o at_o a._n apply_v your_o ruler_n from_o this_o mark_n to_o b_o and_o draw_v the_o line_n ab_fw-la then_o with_o your_o chain_n measure_v the_o distance_n thereof_o which_o suppose_v to_o be_v 11_o chain_n 36_o link_n then_o take_v 11_o chain_n 36_o link_n from_o your_o scale_n and_o set_v that_o distance_n from_o a_o to_o b_o and_o at_z b_o make_v a_o mark_n then_o direct_v the_o sight_n to_o c_o draw_v a_o line_n by_o the_o side_n of_o your_o ruler_n as_o before_o and_o measure_v the_o distance_n ac_fw-la which_o suppose_v to_o be_v 7_o chain_n and_o 44_o link_n this_o distance_n must_v be_v take_v from_o your_o scale_n and_o set_v from_o a_o to_o c_o upon_o your_o paper_n and_o in_o this_o manner_n you_o must_v direct_v your_o sight_n from_o mark_n to_o mark_n until_o you_o have_v draw_v the_o line_n and_o set_v down_o the_o distance_n between_o all_o the_o angle_n in_o the_o field_n and_o your_o station_n which_o be_v do_v you_o must_v draw_v the_o line_n from_o one_o point_n to_o another_o till_o you_o conclude_v where_o you_o first_o begin_v so_o will_v those_o line_n bc._n cd_o de._n fg._n and_o gb_n give_v you_o the_o exact_a figure_n of_o the_o field_n 4._o to_o do_v this_o by_o the_o theodolite_a in_o stead_n of_o draw_v line_n upon_o your_o paper_n in_o the_o field_n you_o must_v have_v a_o little_a book_n in_o which_o the_o page_n must_v be_v divide_v into_o five_o column_n in_o the_o first_o column_n whereof_o you_o must_v set_v several_a letter_n to_o signify_v the_o several_a angle_n in_o the_o field_n from_o which_o line_n be_v to_o be_v draw_v to_o your_o place_n of_o stand_v in_o the_o second_o and_o three_o column_n the_o degree_n and_o part_n take_v by_o your_o instrument_n and_o the_o four_o and_o five_o to_o set_v down_o your_o distance_n chain_n and_o link_n this_o be_v in_o readiness_n and_o have_v place_v your_o instrument_n direct_v your_o sight_n to_o the_o first_o mark_n at_o b_o and_o observe_v how_o many_o degree_n be_v comprehend_v between_o the_o diameter_n of_o your_o instrument_n and_o the_o ruler_n and_o set_v they_o in_o the_o second_o and_o three_o column_n of_o your_o book_n against_o the_o letter_n b_o which_o stand_v for_o your_o first_o mark_n then_o measure_v the_o distance_n ab_fw-la as_o before_o and_o set_v that_o down_o in_o the_o four_o and_o five_o column_n and_o so_o proceed_v from_o mark_n to_o mark_n until_o you_o have_v take_v all_o the_o angle_n and_o distance_n in_o the_o field_n which_o suppose_v to_o be_v as_o they_o be_v express_v in_o the_o follow_a table_n  _fw-fr degr._fw-la part_v chain_n link_n b_o 39_o 75_o 11_o 56_o c_o 40_o 75_o 7_o 44_o d_o 96_o
sun_n be_v or_o earth_n motion_n in_o the_o first_o part_n of_o this_o treatise_n we_o have_v speak_v of_o the_o primary_n motion_n of_o the_o planet_n and_o star_n as_o they_o be_v wheel_v about_o in_o their_o diurnal_a motion_n from_o east_n to_o west_n but_o here_o we_o be_v to_o show_v their_o own_o proper_a motion_n in_o their_o several_a orb_n from_o west_n to_o east_n which_o we_o call_v their_o second_o motion_n 1._o and_o these_o orb_n be_v suppose_v to_o be_v elliptical_a as_o the_o ingenious_a repler_n by_o the_o help_n of_o tycho_n accurate_a observation_n have_v demonstrate_v in_o the_o motion_n of_o mars_n and_o mercury_n and_o may_v therefore_o be_v conceive_v to_o be_v the_o figure_n in_o which_o the_o rest_n do_v move_v 2._o here_o than_o we_o be_v to_o consider_v what_o a_o ellipsis_n be_v how_o it_o may_v be_v draw_v and_o by_o what_o method_n the_o motion_n of_o the_o planet_n according_a to_o that_o figure_n may_v be_v compute_v 3._o what_o a_o ellipsis_n be_v apollonius_n pergaeus_n in_o conicis_fw-la claudius_n mydorgius_n and_o other_o have_v well_o define_v and_o explain_v but_o here_o i_o think_v it_o sufficient_a to_o tell_v the_o reader_n that_o it_o be_v a_o long_a circle_n or_o a_o circular_a line_n draw_v within_o or_o without_o a_o long_a square_n or_o a_o circular_a line_n draw_v between_o two_o circle_n of_o different_a diameter_n 4._o the_o usual_a and_o mechanical_a way_n of_o draw_v this_o ellipsis_n be_v thus_o first_o draw_v a_o 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 as_o the_o line_n ap_n in_o figure_n 8_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 xm_o and_o xh_n 5._o then_o take_v a_o piece_n of_o thread_n of_o the_o same_o length_n with_o the_o diameter_n ap_n and_o fasten_v one_o end_n thereof_o in_o the_o point_n m_o and_o the_o other_o in_o the_o point_n h_n and_o with_o your_o pen_n extend_v the_o thread_n thus_o fasten_v to_o the_o point_n a_o and_o from_o thence_o towards_o p_o keep_v the_o thread_n stiff_a upon_o your_o pen_n draw_v a_o line_n from_o a_o by_o b_o to_o p_o the_o line_n so_o draw_v shall_v be_v half_a a_o ellipsis_n and_o in_o like_a manner_n you_o may_v draw_v the_o other_o half_a from_o p_o by_o d_o to_z a._n in_o which_o because_o the_o whole_a thread_n be_v equal_a to_o the_o diameter_n ap._n therefore_o the_o two_o line_n make_v by_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 say_a ellipsis_n be_v also_o equal_a to_o the_o same_o diameter_n ap._n they_o that_o desire_v a_o demonstration_n thereof_o geometrical_o may_v consult_v apollonius_n pergaeus_n claudius_n mydorgius_n or_o other_o in_o their_o treatise_n of_o conical_a section_n this_o be_v sufficient_a for_o our_o present_a purpose_n and_o from_o the_o equality_n of_o these_o two_o line_n with_o the_o diameter_n a_o brief_a method_n of_o calculation_n of_o the_o planet_n place_n in_o a_o ellipsis_n be_v thus_o demonstrate_v by_o dr._n ward_n now_o bishop_n of_o salisbury_n 6._o in_o this_o ellipsis_n h_n denote_v the_o place_n of_o the_o sun_n centre_n to_o which_o the_o true_a motion_n of_o the_o planet_n be_v refer_v m_o the_o other_o focus_n whereunto_o the_o equal_a or_o middle_a motion_n be_v number_v a_o the_o aphelion_n where_o the_o planet_n be_v far_a distant_a from_o the_o sun_n and_o slow_a in_o motion_n p_o the_o perihelion_n where_o the_o planet_n be_v near_a the_o sun_n and_o slow_a in_o motion_n in_o the_o point_v a_o and_o p_o the_o line_n of_o the_o mean_a and_o true_a motion_n do_v convene_v and_o therefore_o in_o either_o of_o these_o place_n the_o planet_n be_v from_o p_o in_o equality_n but_o in_o all_o other_o point_v the_o mean_a and_o true_a motion_n differ_v and_o in_o d_o and_o c_o be_v the_o great_a elliptic_a aequation_n 8._o now_o suppose_v the_o planet_n in_o b_o the_o line_n of_o the_o middle_a motion_n according_a to_o this_o figure_n be_v mb_a the_o line_n of_o the_o true_a motion_n hb_n the_o mean_a anomaly_n amb._n the_o eliptick_a aequation_n or_o prosthaphaeresis_fw-la mbh_n which_o in_o this_o example_n subtract_v from_o ambiguity_n the_o remainder_n ahb_n be_v the_o true_a anomaly_n and_o here_o note_v that_o in_o the_o right_n line_v triangle_n mbh_n the_o side_n mh_o be_v always_o the_o same_o be_v the_o distance_n of_o the_o foci_fw-la the_o other_o two_o side_n mb_a and_o hb_n be_v together_o equal_a to_o ap._n now_o then_o if_o you_o continue_v the_o side_n mb_n till_o be_v be_v equal_a to_o bh_n and_o draw_v the_o line_n he_o in_o the_o right_n line_v triangle_n meh_n we_o have_v give_v i_o =_o ad_fw-la and_o mh_o with_o the_o angle_n emh_o to_o find_v the_o angles_n meh_n and_o mhe_n which_o in_o this_o case_n be_v equal_a because_o ebb_v =_o bh_n by_o contraction_n and_o therefore_o the_o double_a of_o beh_n or_o bhe_n =_o mbh_n which_o be_v the_o angle_n require_v and_o that_o which_o yet_o remain_v to_o be_v do_v be_v the_o find_v the_o place_n of_o the_o aphelion_n the_o true_a excentricity_n or_o distance_n of_o the_o umbilique_a point_n and_o the_o state_v of_o the_o planet_n middle_a motion_n chap._n x._o of_o the_o find_n of_o the_o sun_n apogeon_n quantity_n of_o excentricity_n aend_v middle_a motion_n the_o place_n of_o the_o sun_n apogaeon_fw-mi and_o quantity_n of_o excentricity_n may_v from_o the_o observation_n of_o our_o country_n man_n mr._n edward_n wright_n be_v obtain_v in_o this_o manner_n in_o the_o year_n 1596_o and_o 1497_o the_o sun_n entrance_n into_o ♈_o and_o ♎_o and_o into_o the_o midst_n of_o ♉_o ♌_o â™_o and_o â™’_o be_v as_o in_o the_o table_n follow_v express_v  _fw-fr 1596_o 1597_o  _fw-fr  _fw-fr d._n h._n m._n d._n h._n m._n  _fw-fr january_n 25._o 00.07_o 24._o 05.54_o â™’_o 15_o march_n 9_o 18.43_o 10._o 00.37_o ♈_o 0_o april_n 24._o 21.47_o 25._o 03.54_o ♉_o 15_o july_n 28._o 01.43_o 28._o 09.56_o ♌_o 15_o september_n 12._o 13.48_o 12._o 19.15_o ♎_o 0_o october_n 27._o 15.23_o 27._o 21.50_o â™_o 15_o and_o hence_o the_o sun_n continuance_n in_o the_o northern_a semicircle_n from_o ♈_o to_o ♎_o in_o the_o year_n 1596_o be_v leap_v year_n be_v thus_o find_v  _fw-fr d._n h._n from_o the_o 1._o of_o january_n to_o ☉_o entrance_n ♎_o 256._o 13._o 48._o from_o the_o 1._o of_o jun_n to_o ☉_o entrance_n ♈_o 69._o 18.43_o their_o difference_n 186._o 19.05_o in_o the_o year_n 1597_o from_o the_o 1_o of_o january_n to_o the_o time_n of_o the_o ☉_o entrance_n into_o ♎_o 255._o 19.15_o to_o the_o ☉_o entrance_n into_o ♈_o 69._o 09.37_o their_o difference_n be_v 186._o 18.38_o and_o the_o difference_n of_o the_o sun_n continuance_n in_o these_o ark_n in_o the_o year_n 1596_o and_o 1597_o be_v 27′_n and_o therefore_o the_o mean_a time_n of_o his_o continuance_n in_o those_o ark_n be_v day_n 186._o hour_n 18._o minute_n 51._o second_o 30._o and_o by_o consequence_n his_o continuance_n in_o the_o southern_a semicircle_n that_o be_v from_o ♎_o to_o ♈_o be_v 178_o day_n 11_o hour_n 8_o minute_n and_o 30_o second_n in_o like_a manner_n in_o the_o year_n 1596_o between_o his_o entrance_n into_o ♉_o 15._o and_o â™_o 15_o there_o be_v day_n 185._o 17.36_o and_o in_o the_o year_n 1597_o there_o be_v day_n 185._o 17.56_o and_o to_o find_v the_o middle_a motion_n answer_v to_o day_n 186._o hour_n 18._o minute_n 51._o second_o 30_o i_o say_v as_o 365_o day_n 6_o hour_n the_o length_n of_o the_o julian_n year_n be_v to_o 360_o the_o degree_n in_o a_o circle_n so_o be_v 186_o day_n 18_o hour_n 51′_n 30″_n to_o 184_o degree_n 03′_n 56″_n in_o like_a manner_n the_o mean_a motion_n answer_v to_o 185_o day_n 17_o h._n 46′_n be_v 183_o degree_n 02′_n 09_o apparent_a motion_n from_o ♈_o to_o ♎_o 180._o 00.00_o middle_a motion_n 184._o 03.56_o their_o sum_n 364._o 03.56_o half_a sum_n be_v the_o arch._n sme_o 182._o 01.58_o in_o 1596_o from_o 15_o â™’_o to_o 15_o ♌_o there_n be_v day_n 185_o hour_n 01_o minute_n 36._o in_o 1597._o day_n 135._o hour_n 4._o 02′_n and_o the_o mean_a motion_n answer_v thereunto_o be_v 182_o d._n 30′_n 36″_n apparent_a motion_n from_o 15_o ♉_o to_o 15_o â™_o 180._o middle_a motion_n 185._o 17._o 56._o 181._o 04.53_o half_a sum_n be_v 183._o 32._o 26_o from_o 15_o â™’_o to_o 15_o ♌_o days_n 185._o 04_o h._n 02′_n apparent_a motion_n 180._o middle_a motion_n 182._o 30._o 36_o half_a sum_n 181._o 15._o 18_o now_o then_o in_o fig._n from_o pgc._n 181._o 32._o 26_o deduct_v nkd_v 180_o the_o remainder_n be_v dc+np_n 1._o 32._o 26._o therefore_o dc_o or_o np._n 46._o 13_o who_o sine_fw-la be_v ha._n and_o from_o xpg._n 181._o 15._o 18_o deduct_v
with_o a_o angle_n conterminate_a therewith_o be_v give_v to_o find_v the_o other_o leg._n in_o the_o right_o angle_a spherical_a triangle_n mpr_fw-la the_o give_a leg_n be_v mr_n with_o the_o angle_n m._n the_o leg_n pr._n be_v require_v by_o the_o 3_o rectangle_n cot_n m._n rad_n ∷_o sine_fw-la mr._n tpr_fw-la the_o give_a leg_n rp_n and_o angle_n p._n the_o leg_n mr_n be_v require_v by_o the_o 1._o rectangle_n ctp._n rad_n ∷_o sine_z rp_n tang_n mr._n case_n 5._o a_o leg_n and_o a_o angle_n conterminate_a therewith_o be_v give_v to_o find_v the_o hypotenuse_n in_o the_o right_o angle_a spherical_a triangle_n mpr_fw-la let_v there_o be_v give_v the_o leg_n mr_n and_o the_o angle_n m_o pr._n and_o the_o angle_n p_o to_z find_fw-mi mp_n by_o the_o 5._o rectangle_n t_o mr._n rad_n ∷_o choose_fw-la m._n ct_a mp_n by_o the_o 9_o rectangle_n t_o pr._n rad._n ∷_o choose_fw-la p._n ct_a mp_n case_n 6._o the_o hypotenuse_n and_o a_o leg_n give_v to_o find_v the_o contain_v angle_n in_o the_o right_o angle_a spherical_a triangle_n mpr_fw-la let_v there_o be_v give_v the_o hypotenuse_n mp_n and_o leg_n mr._n pr._n to_o find_v m._n by_o the_o 5._o rectangle_n rad._n ct_a mp_n ∷_o t_o mr._n choose_fw-la m._n by_o the_o 9_o rectangle_n rad._n ct_a mr_n ∷_o t_o pr._n choose_fw-la p._n case_n 7._o the_o hypotenuse_n and_o one_o angle_n give_v to_o find_v the_o other_o angle_n in_o the_o right_o angle_a spherical_a triangle_n mpr_fw-la let_v there_o be_v give_v the_o hypotenuse_n mp_n &_o angle_n m_o p._n to_o find_v the_o angle_n p._n m._n by_o the_o 7._o rectangle_n cot_n m._n rad_n ∷_o choose_fw-la mp_n cot_n p._n by_o the_o 7._o rectangle_n cot_n p._n rad_n ∷_o choose_fw-la mp_n cot_n m._n case_n 8._o the_o oblique_a angel_n give_v to_o find_v the_o hypotenuse_n in_o the_o right_o angle_a spherical_a triangle_n mpr_fw-la let_v there_o be_v give_v the_o angle_n at_o p_o and_o m_o to_o find_v the_o hypotenuse_n pm_n by_o the_o 7._o rectangle_n rad._n ct_v p_o ∷_o cot_n m._n choose_fw-la mp_n case_n 9_o the_o hypotenuse_n and_o a_o angle_n give_v to_o find_v the_o leg_n conterminate_v with_o the_o give_v angle_n in_o the_o right_o angle_a spherical_a triangle_n mpr_fw-la let_v there_o be_v give_v the_o hypotenuse_n pm_n angle_n p._n m._n to_o find_v pr._n mr._n by_o the_o 9_o rectangle_n ct_n pm_n rad_n ∷_o choose_fw-la p._n t_o pr._n by_o the_o 5._o rectangle_n ct_n pm_n rad_n ∷_o choose_fw-la m._n tmr_n case_n 10._o the_o hypotenuse_n and_o a_o angle_n give_v to_o find_v the_o leg_n opposite_a to_o the_o give_v angle_n in_o the_o right_o angle_a spherical_a triangle_n mpr_fw-la let_v there_o be_v give_v the_o hypotenuse_n pm_n and_o the_o angle_n m._n p._n to_o find_v pr._n mr._n by_o the_o 2._o rectangle_n rad._n s_o mp_n ∷_o s_o m._n sine_fw-la pr._n by_o the_o 4._o rectangle_n rad._n s_o mp_n ∷_o s_o p._n sine_fw-la mr_n case_n 11._o a_o leg_n and_o a_o angle_n opposite_a thereunto_o be_v give_v to_o find_v the_o hypotenuse_n in_o the_o right_o angle_a spherical_a triangle_n mpr_fw-la let_v there_o be_v give_v the_o leg_n pr._n mr._n and_o the_o angle_n m_o p_o to_n find_v the_o hypotenuse_n pm_n by_o the_o 2._o rectangle_n s_o m._n rad_n ∷_o s_z pr._n s_o mp_n by_o the_o 4._o rectangle_n s_o p._n rad_n ∷_o s_z mr._n s_o pm_n case_n 12._o the_o hypotenuse_n and_o a_o leg_n give_v to_o find_v the_o angle_n opposite_a to_o the_o give_v leg._n in_o the_o right_o angle_a spherical_a triangle_n pmr_n the_o hypotenuse_n mp_n and_o the_o leg_n mr_n be_v give_v the_o angle_n at_o p_o be_v require_v by_o the_o four_o rectangle_n sine_fw-la mp_n to_o rad_n ∷_o s_z mr._n s_o p._n the_o hypotenuse_n mp_n and_o leg_n pr._n give_v the_o angle_n m_o be_v require_v by_o the_o second_o rectangle_n smp._n rad_n ∷_o s_z pr._n s_o m._n case_n 13._o the_o angle_n and_o leg_n conterminate_v with_o it_o be_v give_v to_o find_v the_o other_o angle_n in_o the_o right_o angle_a spherical_a triangle_n pmr_n let_v there_o be_v give_v the_o angle_n m_o p_o and_o the_o leg_n mr_n pr._n to_o find_v the_o angle_n p._n m._n by_o the_o ten_o rectangle_n rad._n c_n mr_n ∷_o s_o m._n c_n p._n by_o the_o six_o rectangle_n rad._n s_o p_o ∷_o cs_n pr._n cs_n m._n case_n 14._o a_o angle_n and_o a_o leg_n opposite_a thereunto_o be_v give_v to_o find_v the_o other_o angle_n in_o the_o right_o angle_a spherical_a triangle_n mpr_fw-la let_v there_o be_v give_v the_o angle_n p_o m_o and_o the_o leg_n mr_n pr._n to_o find_v the_o angle_n m._n p._n by_o the_o 10._o rectangle_n cs_n mr._n rad_n ∷_o cs_n p._n csm._n by_o the_o 6._o rectangle_n cs_n pr._n rad_n ∷_o cs_n m._n sp._n case_n 15._o the_o oblique_a angle_n give_v to_o find_v a_o leg._n in_o the_o right_o angle_a spherical_a triangle_n mpr_fw-la let_v there_o be_v give_v the_o angle_n at_o m_n and_o p_o to_o find_v the_o leg_n mr_n and_o pr._n by_o the_o 10._o rectangle_n sm._n rad_n ∷_o cs_n p._n c_n mr._n by_o the_o 6._o rectangle_n s_o p._n rad_n ∷_o cs_n m._n cs_n pr._n case_n 16._o the_o hypotenuse_n and_o one_o leg_n give_v to_o find_v the_o other_o leg._n in_o the_o right_o angle_a spherical_a triangle_n mpr_fw-la let_v there_o be_v give_v the_o hypotenuse_n mp_n and_o the_o leg_n pr._n mr_n to_o find_v the_o leg_n mr._n pr._n by_o the_o 8._o rectangle_n cspr._n rad_n ∷_o csmp._n csmr._n csmr._fw-la rad_n ∷_o csmp._n cspr._n thus_o i_o have_v give_v you_o the_o proportion_n by_o which_o the_o 16_o case_n of_o a_o right_o angle_v spherical_a triangle_n may_v be_v resolve_v in_o which_o there_o be_v contain_v 30_o astronomical_a problem_n two_o in_o every_o case_n except_o the_o second_o and_o the_o eight_o in_o both_o which_o case_n there_o be_v but_o two_o problem_n and_o thus_o i_o have_v do_v with_o right_o angle_a spherical_a triangle_n 5._o if_o the_o angle_n at_o the_o base_a be_v both_o acute_a or_o both_o obtuse_a the_o perpendicular_a shall_v fall_v within_o the_o triangle_n but_o if_o one_o of_o the_o angle_n of_o the_o base_a be_v acute_a and_o the_o other_o obtuse_a the_o perpendicular_a shall_v fall_v without_o the_o triangle_n 6._o however_o the_o perpendicular_a fall_v it_o must_v be_v always_o opposite_a to_o a_o know_a angle_n for_o your_o better_a direction_n take_v this_o general_a rule_n from_o the_o end_n of_o a_o side_n give_v be_v adjacent_a to_o a_o angle_n give_v let_v fall_v the_o perpendicular_a as_o in_o the_o triangle_n fp_v in_o fig._n 4._o if_o there_o be_v give_v the_o side_n f_o s_o and_o the_o angle_n at_o saint_n the_o perpendicular_a by_o this_o rule_n must_v fall_v from_o p_o upon_o the_o side_n s_o p_o extend_a if_o need_v require_v but_o if_o there_o be_v give_v the_o side_n p_o saint_n and_o the_o angle_n at_o saint_n the_o perpendicular_a must_v fall_v from_o f_o upon_o the_o side_n f_o s._n 7._o to_o divide_v a_o oblique_a angle_a spherical_a triangle_n into_o two_o right_a by_o let_v fall_n a_o perpendicular_a upon_o the_o globe_n itself_o be_v not_o necessary_a because_o all_o the_o case_n may_v be_v resolve_v without_o it_o but_o in_o projection_n it_o be_v convenient_a to_o inform_v the_o fancy_n and_o see_v the_o reason_n by_o which_o it_o be_v do_v in_o projection_n do_v depend_v upon_o the_o nature_n of_o the_o globe_n i_o will_v here_o show_v it_o both_o way_n first_o upon_o the_o globe_n and_o then_o by_o projection_n a_o oblique_a angle_a spherical_a triangle_n may_v be_v divide_v into_o two_o right_a by_o let_v fall_n a_o perpendicular_a upon_o the_o globe_n itself_o in_o this_o manner_n in_o the_o oblique_a angle_a spherical_a triangle_n fp_v in_o fig._n 4._o let_v it_o be_v require_v to_o let_v fall_v a_o perpendicular_a from_o p_o upon_o the_o side_n f_n suppose_v the_o point_n p_o to_o stand_n in_o the_o zenith_n where_o the_o arch_a f_n shall_v cut_v the_o zodiac_n which_o in_o this_o figure_n be_v at_o k_o make_v a_o mark_n and_o from_o this_o point_n of_o intersection_n of_o the_o circle_n upon_o which_o the_o perpendicular_a be_v to_o fall_v with_o the_o zodiac_n reckon_v 90_o degree_n which_o suppose_v to_o be_v at_o p_o a_o thin_a plate_n of_o brass_n with_o a_o nut_n at_o one_o end_n thereof_o whereby_o to_o fasten_v it_o to_o the_o meridian_n as_o you_o do_v the_o quadrant_n of_o altitude_n be_v graduate_v as_o that_o be_v but_o of_o a_o large_a extent_n for_o that_o a_o quadrant_n in_o this_o case_n will_v not_o suffice_v be_v fasten_v at_o p_o and_o turn_v about_o till_o it_o cut_v the_o point_n l_o in_o the_o zodiac_n will_v describe_v upon_o the_o globe_n the_o arch_n of_o a_o great_a circle_n pel_n intersect_v the_o side_n f_o s_o at_o right_a angle_n in_o the_o point_n e_o