declination_n and_o magnitude_n of_o some_o of_o the_o principal_a fix_a star_n rectify_v for_o this_o present_a year_n 1652._o the_o name_n of_o the_o star_n right_o ascension_n declination_n magnitude_n h._n m._n d._n m._n  _fw-fr the_o pole_n star_n 00_o 32_o 87_o 19_o n_n 2_o andromeda_n girdle_n 00_o 52_o 33_o 48_o n_n 2_o the_o ram_n horn_n 01_o 34_o 17_o 27_o n_n 3_o the_o ram_n head_n 01_o 48_o 21_o 44_o n_n 3_o the_o head_n of_o mednsa_n 02_o 46_o 39_o 32_o n_n 3_o aldebaran_n the_o bull_n eye_n 04_o 16_o 15_o 48_o n_n 1_o hircus_fw-la the_o goat_n 04_o 51_o 45_o 36_o n_n 1_o orion_n left_a foot_n 04_o 58_o 08_o 40_o s_n 1_o orion_n left_a shoulder_n 05_o 06_o 05_o 59_o n_n 2_o the_o first_o in_o orion_n girdle_n 05_o 14_o 00_o 38_o s_n 2_o orion_n right_a shoulder_n 05_o 36_o 07_o 17_o n_n 2_o sirius_n the_o great_a dog_n 06_o 31_o 16_o 13_o s_n 1_o procyon_n the_o lesser_a dog_n 07_o 21_o 06_o 07_o n_n 2_o the_o heart_n of_o hydra_n 09_o 11_o 07_o 09_o s_n 1_o cor_fw-la leonis_fw-la the_o lion_n heart_n 09_o 49_o 13_o 42_o n_n 1_o the_o lion_n neck_n 09_o 50_o 21_o 49_o n_n 2_o the_o lion_n back_o 10_o 54_o 22_o 33_o n_n 2_o the_o foremost_a star_n in_o □_o of_o the_o great_a bear_n 10_o 40_o 58_o 23_o n_n 2_o the_o lion_n tail_n 11_o 30_o 16_o 34_o n_n 1_o the_o root_n of_o the_o great_a bear_n tail_n alio_fw-la tail_n 12_o 38_o 57_o 51_o n_n 2_o the_o virgin_n spike_n 13_o 08_o 09_o 16_o s_n 1_o the_o middlemost_a star_n in_o the_o great_a bear_n 13_o 10_o 56_o 45_o n_n 2_o the_o last_o of_o the_o say_a tail_n 13_o 26_o 51_o 05_o n_n 2_o arcturus_n 13_o 59_o 21_o 10_o n_n 1_o the_o bright_a star_n of_o the_o crown_n 15_o 24_o 27_o 43_o n_n 3_o antare_n the_o scorpion_n heart_n 16_o 08_o 25_o 34_o s_n 1_o the_o bright_a star_n of_o the_o harp_n 18_o 25_o 38_o 30_o n_n 1_o the_o eagle_n heart_n 19_o 33_o 07_o 58_o n_n 2_o the_o bright_a star_n of_o the_o vulture_n 19_o 34_o 08_o 01_o n_n 1_o fomahant_n 22_o 38_o 31_o 25_o s_n 1_o pegasus_n shoulder_n 22_o 48_o 13_o 14_o n_n 2_o pegasus_n leg_n 22_o 52_o 26_o 09_o n_n 2_o chap._n iv_o to_o show_v the_o general_n use_v of_o the_o universal_a quadrat_fw-la in_o the_o use_n of_o this_o quadrat_fw-la let_v we_o for_o distinction_n sake_n call_v those_o parallel_n which_o make_v right_a angle_n with_o the_o line_n whereon_o the_o sight_n stand_v right_a parallel_n and_o the_o other_o which_o run_v parallel_v to_o the_o same_o line_n contrary_a parallel_n the_o one_o signify_v right_o shadow_n the_o other_o contrary_a shadow_n the_o general_n use_v whereof_o consist_v in_o the_o solution_n of_o the_o golden_a rule_n where_o three_o line_n or_o number_n be_v give_v of_o a_o know_a denomination_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 of_o the_o signification_n of_o these_o parallel_a line_n and_o also_o of_o the_o entrance_n into_o the_o work_n the_o signification_n of_o these_o parallel_n be_v sometime_o simple_a and_o sometime_o compound_v sometime_o they_o signify_v right_a line_n or_o number_n only_o such_o as_o they_o be_v simple_a in_o themselves_o and_o sometime_o they_o signify_v natural_a sin_n which_o be_v half_a the_o chord_n of_o the_o double_a arch_n and_o sometime_o natural_a tangent_n which_o be_v perpendicular_o at_o the_o end_n of_o the_o diameter_n sometime_o the_o work_n may_v be_v begin_v with_o one_o kind_n of_o signification_n and_o end_v with_o another_o it_o may_v be_v begin_v with_o the_o line_n as_o they_o be_v in_o themselves_o and_o end_v as_o they_o be_v sin_n it_o may_v begin_v as_o they_o be_v sin_n and_o end_n as_o they_o be_v 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 right_a or_o contrary_n i_o call_v that_o right_a entrance_n or_o entrance_n on_o the_o right_a parallel_n when_o the_o two_o line_n or_o number_n of_o the_o first_o denomination_n be_v apply_v or_o find_v in_o the_o right_a parallel_n and_o the_o three_o line_n or_o number_n and_o that_o which_o be_v seek_v for_o on_o the_o contrary_a parallel_n and_o on_o the_o contrary_a i_o call_v that_o contrary_a entrance_n when_o the_o two_o line_n or_o number_n of_o the_o first_o denomination_n be_v apply_v or_o find_v on_o the_o parallel_n of_o contrary_a shadow_n and_o the_o three_o line_n or_o number_n and_o that_o which_o be_v to_o be_v find_v out_o in_o the_o parallel_n of_o right_a shadow_n as_o for_o example_n let_v there_o be_v give_v three_o number_n 50._o 60_o and_o 70._o unto_o which_o i_o be_o to_o find_v a_o four_o proportional_a let_v the_o question_n be_v this_o if_o 50_o pound_n gain_v 60_o shilling_n what_o shall_v 70_o pound_n gain_v here_o be_v number_n of_o two_o denomination_n one_o of_o pound_n the_o other_o of_o shilling_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 50._o pound_n if_o then_o i_o will_v enter_v my_o question_n upon_o a_o parallel_n of_o right_a shadow_n see_v where_o the_o right_a parallel_n of_o my_o first_o number_n which_o be_v 50_o cut_v the_o contrary_a parallel_n of_o the_o second_o which_o be_v 60_o and_o upon_o the_o point_n of_o intersection_n there_o lie_v the_o thread_n the_o thread_n lie_v in_o this_o position_n look_v along_o upon_o the_o right_a parallel_n of_o the_o three_o number_n which_o be_v 70_o and_o see_v what_o contrary_a parallel_n cut_v it_o direct_o under_o the_o thread_n for_o that_o be_v the_o four_o proportional_a number_n require_v which_o here_o be_v find_v to_o be_v 84_o wherefore_o i_o conclude_v that_o if_o 50_o pound_n gain_n 60_o shilling_n 70_o pound_n will_v gain_v 84_o shilling_n for_o as_o 50_o be_v to_o 60_o so_o be_v 70_o to_o 84_o the_o like_a will_v appear_v if_o you_o enter_v your_o work_n upon_o the_o contrary_a parallel_n for_o it_o matter_v not_o on_o which_o kind_n of_o parallel_n you_o enter_v so_o be_v it_o you_o enter_v your_o first_o and_o three_o number_n on_o the_o same_o kind_n of_o parallel_n so_o shall_v the_o three_o and_o four_o be_v of_o the_o other_o kind_n thus_o much_o for_o the_o general_n use_v of_o this_o fore-side_n of_o the_o universal_a quadrat_fw-la which_o be_v consider_v and_o well_o understand_v there_o be_v nothing_o hard_a in_o that_o which_o follow_v chap._n v._o to_o divide_v a_o line_n give_v into_o any_o number_n of_o equal_a part_n let_v the_o line_n give_v be_v ab_fw-la and_o let_v it_o be_v require_v to_o divide_v the_o same_o into_o 5_o equal_a part_n take_v with_o your_o compass_n the_o give_v line_n ab_fw-la and_o lay_v it_o down_o upon_o any_o parallel_n who_o number_n may_v be_v equal_o divide_v into_o five_o part_n without_o any_o remainder_n as_o lay_v it_o upon_o the_o five_o parallel_n and_o at_o the_o point_n where_o the_o line_n end_v there_o lay_v the_o thread_n the_o thread_n lie_v in_o this_o position_n and_o in_o regard_n that_o 10_o be_v ⅕_n part_n of_o 50_o therefore_o the_o segment_n of_o the_o ten_o parallel_n which_o be_v cut_v by_o the_o thread_n shall_v be_v the_o distance_n which_o shall_v divide_v the_o give_v line_n into_o 5_o equal_a part_n as_o for_o example_n take_v the_o give_v line_n ab_fw-la and_o lay_v it_o upon_o the_o fifty_o contrary_a parallel_n as_o from_o a_o to_o b_o &_o see_v the_o give_v line_n end_v in_o b_o lay_v the_o thread_n upon_o the_o same_o point_n as_o here_o you_o see_v in_o the_o follow_a figure_n then_o in_o regard_n that_o 10_o be_v ⅕_n of_o 50_o therefore_o i_o take_v the_o segment_n cut_v by_o the_o thread_n upon_o the_o ten_o parallel_n viz._n cd_o and_o with_o it_o i_o divide_v the_o give_v line_n ab_fw-la into_o 5_o equal_a part_n as_o here_o you_o see_v the_o fore-side_n of_o the_o quadrat_fw-la chap._n vi_o to_o increase_v a_o line_n in_o a_o give_v proportion_n take_v the_o line_n give_v with_o a_o pair_n of_o compass_n and_o lay_v it_o one_o the_o parallel_n of_o the_o number_n give_v and_o upon_o the_o point_n where_o the_o give_v line_n end_v there_o lay_v the_o thread_n which_o be_v keep_v in_o this_o position_n the_o segment_n of_o the_o parallel_n of_o the_o number_n require_v intercept_v by_o the_o thread_n shall_v give_v the_o line_n require_v let_v a_o be_v a_o line_n give_v to_o be_v increase_v in_o the_o proportion_n of_o 3_o to_o 5._o first_o i_o take_v the_o give_v line_n a_o with_o my_o compass_n and_o in_o regard_n that_o 30_o and_o 50_o be_v the_o same_o with_o 3_o and_o 5_o if_o from_o they_o you_o give_v o_o therefore_o i_o take_v the_o give_v line_n a_o and_o lay_v it_o upon_o the_o thirty_o parallel_n
to_o a_o four_o number_n and_o this_o four_o number_n to_o the_o solid_a content_n in_o foot_n wherefore_o if_o you_o apply_v the_o thread_n to_o the_o intersection_n of_o the_o contrary_a parallel_n of_o 12_o with_o the_o right_a parallel_n of_o 2_o foot_n the_o three_o part_n of_o the_o altitude_n it_o will_v cut_v the_o contrary_a parallel_n of_o 36_o inch_n the_o side_n of_o the_o base_a at_o the_o right_a parallel_n of_o 6_o the_o thread_n lie_v still_o in_o the_o same_o position_n will_v cut_v the_o contrary_a parallel_n of_o this_o 6_o at_o the_o right_a parallel_n of_o 18_o and_o such_o be_v the_o solid_a content_n of_o this_o pyramid_n in_o foot_n which_o be_v require_v now_o if_o you_o will_v measure_v any_o segment_n of_o a_o pyramid_n be_v cut_v off_o observe_v the_o former_a rule_n give_v for_o the_o measure_n of_o the_o segment_n of_o a_o cone_n and_o you_o shall_v soon_o have_v your_o desire_n for_o have_v the_o side_n of_o the_o segment_n of_o the_o pyramid_n with_o the_o semidiameter_n of_o the_o circumscribe_v circle_n at_o each_o base_a you_o may_v by_o the_o 61_o chapter_n find_v the_o altitude_n of_o the_o whole_a great_a pyramid_n as_o if_o it_o have_v not_o be_v cut_v off_o and_o also_o of_o the_o lesser_a pyramid_n which_o be_v suppose_v to_o be_v add_v to_o the_o segment_n to_o make_v it_o a_o perfect_a pyramid_n this_o be_v do_v you_o may_v by_o the_o rule_v deliver_v first_o get_v the_o content_a of_o the_o whole_a great_a pyramid_n and_o then_o of_o the_o lesser_a add_v pyramid_n and_o take_v the_o lesser_a out_o of_o the_o great_a the_o remainder_n will_v be_v the_o content_a of_o the_o give_v segment_n as_o be_v require_v the_o end_n of_o the_o first_o book_n the_o second_o book_n show_v the_o most_o plentiful_a easy_a and_o speedy_a use_n of_o the_o universal_a quadrat_fw-la in_o the_o resolution_n of_o the_o whole_a doctrine_n trigonometrical_a as_o well_o plain_a as_o spherical_a and_o that_o two_o several_a way_n upon_o the_o instrument_n with_o surpass_a facility_n and_o with_o the_o least_o intricacie_n that_o may_v be_v chap._n i._o to_o find_v the_o chord_n of_o any_o arch_n the_o radius_fw-la be_v give_v not_o exceed_v the_o side_n of_o the_o quadrat_fw-la the_o radius_fw-la of_o a_o circle_n be_v the_o semidiameter_n of_o the_o same_o circle_n or_o the_o total_a sine_fw-la or_o sine_fw-la of_o 90_o degree_n so_o ae_n be_v the_o semidiameter_n or_o radius_fw-la upon_o which_o the_o divide_a arch_n eglantine_n be_v dramn_v and_o ab_fw-la be_v the_o radius_fw-la upon_o which_o the_o arch_a bfc_n be_v draw_v and_o a_o chord_n be_v a_o right_a line_n subtend_v a_o arch_n so_o bc_n be_v the_o chord_n of_o the_o arch_a bfc_n for_o the_o find_n of_o which_o chord_n for_o any_o arch_n require_v place_v the_o radius_fw-la give_v upon_o the_o side_n of_o the_o quadrat_fw-la from_o the_o centre_n and_o note_v the_o point_n where_o the_o same_o end_v and_o there_o place_v the_o bead_n then_o open_v the_o thread_n to_o the_o arch_n give_v count_v in_o the_o quadrant_a the_o distance_n between_o the_o point_n note_v in_o the_o side_n of_o the_o quadrat_fw-la and_o the_o head_n shall_v be_v the_o chord_n require_v as_o for_o example_n let_v ab_fw-la be_v the_o radius_fw-la of_o a_o circle_n place_v upon_o the_o side_n of_o the_o quadrat_fw-la and_o let_v it_o be_v require_v to_o find_v a_o chord_n of_o 40_o degree_n first_o i_o place_v the_o bead_n at_o the_o point_n b_o where_o the_o give_v radius_fw-la end_v and_o open_v the_o thread_n to_o 40_o degree_n in_o the_o quadrant_a i_o take_v the_o distance_n betwixt_o the_o point_n b_o and_o the_o bead_n which_o be_v the_o chord_n of_o 40_o degree_n viz._n the_o chord_n of_o the_o arch_a bfc_n draw_v upon_o the_o give_v radius_fw-la ab_fw-la as_o be_v require_v and_o thus_o may_v you_o find_v the_o chord_n of_o any_o arch_n as_o well_o as_o upon_o the_o sector_n chap._n ii_o to_o find_v the_o right_n sine_fw-la of_o any_o arch_n give_v the_o radius_fw-la be_v put_v 1000_o have_v show_v the_o use_n of_o these_o parallel_a line_n as_o they_o stand_v in_o their_o own_o proper_a signification_n it_o rest_v now_o to_o show_v the_o use_n of_o they_o as_o they_o signify_v right_a fine_n and_o tangent_n and_o first_o to_o find_v the_o right_n fine_a of_o any_o arch_n seek_v the_o degree_n give_v upon_o the_o quadrant_a count_v from_o any_o side_n of_o the_o quadrant_a but_o here_o let_v it_o be_v from_o the_o begin_n of_o the_o degree_n at_o e_o so_o shall_v the_o right_a parallel_n run_v through_o the_o degree_n give_v in_o the_o quadrant_a give_v the_o number_n of_o the_o fine_a require_v but_o if_o you_o will_v have_v the_o fine_a in_o a_o right_a line_n take_v the_o length_n of_o the_o contrary_a parallel_n from_o the_o side_n of_o the_o quadrat_fw-la ae_n to_o the_o degree_n give_v in_o the_o quadrat_fw-la and_o that_o shall_v be_v your_o desire_a sine_z whereof_o the_o whole_a side_n be_v radius_fw-la as_o for_o example_n let_v it_o be_v require_v to_o find_v the_o right_n sine_fw-la of_o 53_o degree_n 8_o minute_n i_o look_v 53_o degree_n 8_o minute_n in_o the_o quadrant_a count_v from_o the_o begin_n of_o the_o degree_n at_o e_o unto_z o_o through_o which_o point_n o_o rune_v the_o right_a parallel_n of_o 800_o this_o 800_o be_v the_o right_a sine_fw-la of_o 53_o degree_n 8_o minute_n 100_o be_v radius_fw-la but_o if_o you_o will_v have_v this_o sine_fw-la in_o a_o right_a line_n take_v the_o length_n of_o the_o contrary_a paaallel_n from_o the_o point_n o_o unto_o n_n for_o the_o right_n sine_fw-la of_o 53_o degree_n 8_o minute_n which_o be_v require_v chap._n iii_o to_o find_v the_o arch_n of_o any_o sine_fw-la give_v the_o radius_fw-la be_v put_v 1000_o if_o your_o sine_fw-la be_v give_v in_o number_n seek_v the_o right_a parallel_n of_o the_o number_n give_v for_o where_o that_o parallel_n cut_v the_o quadrant_a there_o be_v the_o arch_n require_v count_v from_o the_o begin_n of_o the_o degree_n at_o e_z as_o before_z but_o if_o your_o sine_fw-la be_v give_v in_o a_o right_a line_n then_o from_o the_o side_n of_o your_o quadrat_fw-la ae_n extend_v your_o give_fw-mi sine_fw-la upon_o any_o of_o the_o contrary_a parallel_n and_o that_o right_a parallel_n that_o shall_v cut_v through_o the_o extreme_a point_n thereof_o shall_v cut_v the_o quadrant_a in_o the_o require_a arch_n as_o for_o example_n let_v 800_o be_v a_o sine_fw-la give_v unto_o which_o it_o be_v require_v to_o find_v the_o arch_n belong_v thereto_o i_o seek_v 800_o in_o the_o right_a parallel_n and_o find_v it_o i_o follow_v it_o to_o the_o quadrant_a and_o there_o i_o see_v it_o cut_v the_o arch_n of_o 53_o degree_n 8_o minute_n count_v from_o e_o which_o 53_o degree_n 8_o minute_n be_v the_o arch_n belong_v to_o the_o sine_fw-la 800_o the_o radius_fw-la being_n 1000_o but_o if_o the_o sine_fw-la be_v give_v in_o a_o right_a line_n take_v it_o betwixt_o your_o compass_n and_o set_v one_o foot_n in_o the_o side_n of_o the_o quadrat_fw-la ae_n extend_v the_o other_o upon_o any_o of_o the_o contrary_a parallel_n as_o here_o i_o set_v one_o foot_n in_o the_o point_n n_n and_o extend_v the_o other_o unto_o o_o through_o which_o point_n the_o right_a parallel_n of_o 800_o run_v and_o cut_v the_o quadrant_a in_o 53_o degree_n 8_o minute_n count_v from_o e_z as_o before_z chap._n iu._n any_o radius_fw-la not_o exceed_v the_o side_n of_o the_o quadrat_fw-la be_v give_v to_o find_v the_o right_n sine_fw-la of_o any_o arch_n or_o angle_v thereunto_o belong_v first_o take_v the_o give_v radius_fw-la and_o place_n it_o upon_o the_o contrary_a parallel_n of_o 100_o which_o be_v the_o total_a sine_fw-la and_o where_o it_o shall_v end_v there_o place_n the_o thread_n which_o be_v so_o place_v see_v where_o it_o cut_v the_o contrary_a parallel_n that_o pass_v through_o the_o give_v arch_n count_v in_o the_o quadrant_a from_o g_z for_o the_o segment_n so_o cut_v which_o lie_v between_o the_o thread_n and_o the_o side_n ae_n be_v the_o fine_a require_v let_v the_o line_n a_o be_v the_o radius_fw-la give_v and_o let_v it_o be_v require_v to_o find_v the_o sine_fw-la of_o 36_o degree_n 53_o minute_n first_o i_o take_v the_o line_n a_o and_o place_n it_o upon_o the_o contrary_a parallel_n of_o 100_o as_o from_o e_z to_z d_o upon_o which_o point_n d_o i_o place_v the_o thread_n this_o do_v i_o count_v the_o give_v arch_n viz._n 36_o degree_n 33_o minute_n from_o the_o end_n of_o the_o quadrant_a at_o g_z unto_o o_o and_o see_v what_o contrary_a parallel_n there_o i_o find_v which_o be_v 60_o intersect_v the_o thread_n at_o m_n so_o shall_v the_o distance_n nm_n be_v the_o line_n b_o the_o sine_fw-la of_o 36_o degree_n 53_o
of_o 16½_n foot_n contain_v 8_o acre_n and_o it_o be_v demand_v how_o many_o acre_n it_o will_v contain_v if_o it_o be_v measure_v with_o a_o chain_n of_o 18_o foot_n to_o the_o perch_n these_o kind_n of_o proposition_n be_v wrought_v by_o the_o backward_a rule_n of_o three_o after_o a_o duplicate_v proportion_n but_o by_o exchange_v the_o place_n of_o the_o first_o and_o three_o number_n by_o put_v each_o in_o other_o place_n it_o will_v become_v the_o forward_a rule_n of_o three_o and_o may_v be_v resolve_v after_o this_o manner_n place_v the_o thread_n upon_o the_o intersection_n of_o the_o contrary_a parallel_n of_o 18_o with_o the_o right_a parallel_n of_o 8_o and_o it_o will_v cut_v the_o contrary_a parallel_n of_o 16⅓_n at_o the_o right_a parallel_n of_o 7_o 33_o 100_o now_o where_o this_o right_a parallel_n of_o 7_o 33_o 100_o cross_v the_o former_a contrary_a parallel_n of_o 18_o there_o lay_v the_o thread_n again_o so_o shall_v it_o cut_v the_o contrary_a parallel_n of_o 16½_n at_o the_o right_a parallel_n of_o 6_o 72_o 100_o which_o show_v the_o content_a to_o be_v 6_o acre_n 72_o part_n such_o as_o 100_o make_v a_o acre_n measure_v by_o a_o 18_o foot_n pole_n chap._n xxxi_o have_v the_o plot_n of_o a_o plain_a with_o the_o content_n in_o acre_n to_o find_v the_o scale_n by_o which_o it_o be_v plot_v suppose_v the_o plane_n abcd_v in_o the_o twenty_o sixth_o chapter_n contain_v just_a 8_o acre_n if_o i_o shall_v measure_v it_o with_o a_o scale_n of_o 12_o in_o a_o inch_n the_o length_n ab_fw-la will_v be_v 60_o pole_n and_o the_o breadth_n ad_fw-la 30_o 18_o 25_o pole_n and_o the_o content_n will_v be_v find_v by_o the_o 16_o chapter_n to_o be_v about_o 11½_n acre_n where_o it_o shall_v be_v but_o 8_o acre_n therefore_o by_o the_o twelve_o chapter_n find_v a_o mean_a proportional_a number_n between_o 8_o and_o 11½_n which_o will_v be_v about_o 9½_n and_o something_o better_o then_o will_v the_o proportion_n hold_v as_o this_o mean_a proportional_a to_o the_o scale_n you_o measure_v by_o so_o the_o content_a of_o the_o plot_n to_o the_o scale_n by_o which_o it_o be_v plot_v wherefore_o i_o place_v the_o thread_n upon_o the_o intersection_n of_o the_o contrary_a parallel_n of_o 9½_n with_o the_o right_a parallel_n of_o 12_o so_o shall_v i_o see_v it_o cut_v the_o contrary_a parallel_n of_o 8_o at_o the_o right_a parallel_n of_o 10_o which_o 10_o be_v the_o scale_n by_o which_o it_o be_v plot_v chap._n xxxii_o have_v the_o length_n of_o the_o furlong_n give_v in_o perch_n to_o find_v the_o breadth_n of_o the_o acre_n in_o perch_n as_o the_o length_n in_o perch_n unto_o 40_o so_o be_v 4_o to_o the_o breadth_n in_o perch_n so_o the_o length_n of_o the_o furlong_n be_v 20_o perch_n i_o lay_v the_o thread_n at_o the_o intersection_n of_o the_o contrary_a parallel_n of_o 20_o with_o the_o right_a parallel_n of_o 40_o so_o will_v it_o cut_v the_o contrary_a parallel_n of_o 4_o at_o the_o right_a parallel_n of_o 8_o which_o be_v the_o breadth_n of_o the_o acre_n give_v if_o the_o length_n have_v be_v 30_o the_o breadth_n will_v have_v be_v 5⅓_n and_o so_o of_o the_o rest_n chap._n xxxiv_o have_v the_o length_n of_o the_o furlong_n give_v in_o chain_n to_o find_v the_o breadth_n of_o the_o acre_n in_o chain_n as_o the_o length_n in_o chain_n unto_o 5_o so_o be_v 2_o to_o the_o breadth_n in_o chain_n measure_n so_o the_o length_n of_o the_o furlong_n be_v 4_o chain_n lay_v the_o thread_n upon_o the_o intersection_n of_o the_o contrary_a parallel_n of_o 4_o with_o the_o right_a parallel_n of_o 5_o so_o shall_v it_o cut_v the_o contrary_a parallel_n of_o 2_o at_o the_o right_a parallel_n of_o 2_o chain_n 50_o link_n and_o such_o be_v the_o breadth_n require_v chap._n xxxiv_o have_v the_o breadth_n of_o any_o board_n give_v in_o inch_n and_o the_o length_n in_o foot_n to_o find_v the_o content_a in_o foot_n as_o 12_o inch_n to_o the_o breadth_n in_o inch_n so_o the_o length_n in_o foot_n to_o the_o content_a in_o foot_n thus_o in_o the_o figure_n a_o the_o length_n be_v 10_o foot_n and_o the_o breadth_n 16_o inch_n if_o i_o place_v the_o thread_n at_o the_o intersection_n of_o the_o contrary_a parallel_n of_o 12_o with_o the_o right_a parallel_n of_o 16_o it_o will_v cut_v the_o contrary_a parallel_n of_o 10_o at_o the_o right_a parallel_n of_o 13½_n and_o so_o many_o square_a foot_n be_v in_o that_o board_n note_v with_o the_o let_n a._n chap._n xxxv_o have_v the_o breadth_n of_o any_o board_n give_v in_o inch_n to_o find_v the_o length_n of_o a_o foot_n superficial_a in_o inch_n measure_n as_o the_o breadth_n of_o the_o board_n give_v in_o inch_n be_v to_o 12_o inch_n so_o be_v 12_o inch_n to_o the_o length_n of_o a_o foot_n in_o inch_n measure_n thus_o in_o the_o figure_n a_o of_o the_o last_o chapter_n be_v a_o board_n who_o breadth_n be_v 16_o inch_n if_o i_o place_v the_o thread_n at_o the_o intersection_n of_o the_o contrary_a parallel_n of_o 16_o with_o the_o right_a parallel_n of_o 12_o it_o will_v cut_v the_o contrary_a parallel_n of_o 12_o at_o the_o right_a parallel_n of_o 9_o and_o such_o be_v the_o length_n of_o a_o foot_n of_o that_o board_n who_o breadth_n be_v 16_o inch_n chap._n xxxvi_o how_o to_o measure_v board_n which_o be_v taper_n grow_v or_o wide_a at_o one_o end_n then_o at_o the_o other_o first_o add_v the_o breadth_n at_o both_o end_n together_o and_o take_v the_o one_o half_a thereof_o for_o a_o mean_a breadth_n then_o say_v as_o in_o the_o 35_o chapter_n as_o 12_o inch_n to_o this_o mean_a breadth_n in_o inch_n so_o the_o length_n in_o foot_n to_o the_o content_a in_o foot_n as_o for_o example_n let_v this_o figure_n b_o be_v a_o taper_a board_n to_o be_v measure_v who_o length_n be_v 21_o foot_n and_o let_v the_o breadth_n at_o one_o end_n be_v 18_o inch_n and_o at_o the_o other_o 14_o now_o add_v 14_o to_o 18_o the_o sum_n will_v be_v 32_o the_o half_a whereof_o be_v 16_o inch_n for_o a_o mean_a breadth_n then_o if_o you_o apply_v the_o thread_n to_o the_o contrary_a parallel_n of_o 12_o with_o the_o right_a parallel_n of_o 16_o it_o will_v cut_v the_o contrary_a parallel_n of_o 21_o at_o the_o right_a parallel_n of_o 28_o and_o so_o many_o square_a foot_n of_o flat_a measure_n be_v contain_v in_o this_o taper_a board_n chap._n xxxvii_o have_v the_o circumference_n of_o a_o circle_n to_o find_v the_o diameter_n as_o 22_o be_v to_o 7_o so_o be_v the_o circumference_n to_o the_o diameter_n thus_o in_o the_o follow_a circle_n a_o the_o circumference_n be_v 81_o 5_o 7_o if_o i_o place_v the_o thread_n at_o the_o intersection_n of_o the_o contrary_a parallel_n of_o 22_o with_o the_o right_a parallel_n of_o 81_o 5_o 7_o it_o will_v cut_v the_o contrary_a parallel_n of_o 7_o at_o the_o right_a parallel_n of_o 26_o and_o such_o be_v the_o diameter_n bc_n as_o be_v require_v chap._n xxxviii_o have_v the_o diameter_n of_o a_o circle_n to_o find_v the_o circumference_n as_o 7_o be_v to_o 22_o so_o be_v the_o diameter_n to_o the_o circumference_n thus_o in_o this_o circle_n a_o the_o diameter_n be_v 26_o if_o i_o place_v the_o thread_n to_o the_o intersection_n of_o the_o contrary_a parallel_n of_o 7_o with_o the_o right_a parallel_n of_o 26_o it_o will_v cut_v the_o contrary_a parallel_n of_o 22_o at_o the_o right_a parallel_n of_o 81_o 5_o 7_o and_o such_o be_v the_o circumference_n require_v chap._n thirty-nine_o have_v the_o chord_n of_o any_o arch_n with_o the_o perpendicular_a fall_n from_o the_o middle_n of_o the_o say_a arch_n to_o the_o chord_n to_o find_v the_o diameter_n as_o the_o perpendicular_a to_o half_a the_o chord_n line_n so_o be_v half_a the_o say_a chord_n to_o a_o four_o line_n or_o number_n which_o be_v add_v to_o the_o aforesaid_a perpendicular_a give_v the_o diameter_n require_v thus_o in_o the_o segment_n fegh_n in_o the_o former_a circle_n have_v the_o perpendicular_a or_o verse_v sine_fw-la eh_n 8_o and_o the_o chord_n fg_v 24_o the_o half_a whereof_o be_v 12_o the_o line_n gh_o we_o may_v find_v the_o diameter_n for_o if_o we_o place_v the_o thread_n upon_o the_o intersection_n of_o the_o contrary_a parallel_n of_o 8_o with_o the_o right_a parallel_n of_o 12_o it_o will_v cut_v the_o contrary_a parallel_n of_o 12_o at_o the_o right_a parallel_n of_o 18_o unto_o which_o if_o you_o add_v 8_o you_o shall_v have_v 26_o for_o the_o whole_a diameter_n de_fw-fr as_o be_v require_v chap._n xl._n have_v the_o circumference_n and_o diameter_n of_o a_o circle_n give_v in_o inch_n to_o find_v the_o superficial_a content_n in_o foot_n every_o circle_n be_v near_o equal_a to_o that_o right_a angle_a triangle_n of_o who_o
contain_v side_n the_o one_o be_v equal_a to_o the_o semidiameter_n the_o other_o to_o the_o circumference_n or_o to_o that_o right_a angle_a parallelogram_n who_o breadth_n be_v equal_a to_o the_o semidiameter_n and_o length_n to_o the_o semicircumference_n or_o else_o to_o that_o right_a angle_a parallelogram_n who_o breadth_n be_v equal_a to_o the_o diameter_n and_o length_n to_o the_o one_o quarter_n of_o the_o circumference_n or_o breadth_n equal_a to_o the_o one_o quarter_n of_o the_o circumference_n and_o length_n to_o the_o diameter_n which_o be_v all_o one_o wherefore_o as_o 144_o inch_n to_o the_o diameter_n in_o inch_n so_o one_o quarter_n of_o the_o circumference_n in_o inch_n to_o the_o content_a in_o foot_n thus_o in_o the_o former_a circle_n becd_v have_v the_o diameter_n bc_n 26_o inch_n and_o one_o quarter_n of_o the_o circumference_n bd_o or_o cd_o 20_o 3_o 7_o inch_n we_o may_v find_v the_o content_a in_o foot_n for_o if_o we_o place_v the_o thread_n at_o the_o intersection_n of_o the_o contrary_a parallel_n of_o 144_o with_o the_o right_a parallel_n of_o 26_o it_o will_v cut_v the_o contrary_a parallel_n of_o 20_o â—_o at_o the_o right_a parallel_n of_o 3_o 5_o 8_o and_o something_o more_o and_o such_o be_v the_o content_a superficial_a in_o foot_n measure_n as_o be_v require_v chap._n xli_o to_o find_v the_o superficial_a content_n of_o any_o part_n or_o segment_n of_o a_o circle_n every_o part_n of_o a_o circle_n whether_o it_o be_v semicircle_n quadrant_n or_o any_o other_o sector_n of_o a_o circle_n be_v near_o equal_a to_o that_o right_a angle_a parallelogram_n or_o long_a square_n who_o length_n and_o breadth_n be_v the_o one_o equal_a to_o the_o semidiameter_n the_o other_o to_o the_o one_o half_a of_o the_o arch_a line_n or_o to_o the_o whole_a diameter_n and_o one_o quarter_n of_o the_o arch_a line_n and_o therefore_o may_v be_v measure_v by_o the_o 26_o or_o 27_o or_o by_o this_o last_o chapter_n for_o if_o you_o will_v measure_v the_o semicircle_n abdc_n in_o the_o 39_o chapter_n you_o may_v take_v the_o semidiameter_n ab_fw-la or_o ac_fw-la for_o your_o breadth_n and_o the_o one_o half_a of_o his_o arch_a line_n bdc_n which_o be_v bd_o or_o cd_o 20_o 3_o 7_o for_o your_o length_n or_o if_o you_o will_v measure_v the_o quadrant_n abdella_n you_o may_v take_v the_o semidiameter_n 13_o for_o your_o length_n and_o half_o the_o arch_n bd_o 10_o 3_o 14_o for_o your_o breadth_n the_o like_a hold_v for_o any_o sector_n of_o a_o circle_n whatsoever_o but_o if_o you_o will_v measure_v the_o segment_n fdgh_o you_o must_v first_o measure_v the_o plain_a of_o afdga_n by_o the_o former_a rule_n and_o then_o the_o triangle_n agf_n by_o the_o 28_o or_o 29_o chapter_n or_o by_o the_o 41_o chapter_n take_v the_o perpendicular_a for_o the_o breadth_n and_o half_o the_o base_a for_o the_o length_n of_o a_o right_a angle_a parallelogram_n the_o content_a of_o which_o triangle_n be_v add_v to_o the_o content_a of_o afdga_n give_v the_o content_a of_o the_o segment_n fdgh_o and_o if_o you_o will_v measure_v the_o segment_n fge_n you_o must_v first_o measure_v the_o sector_n egaf_n and_o from_o the_o content_n take_v the_o content_a of_o the_o triangle_n afg_v so_o shall_v the_o remainder_n be_v the_o content_a of_o the_o segment_n fge_n as_o be_v require_v chap._n xlii_o have_v the_o circumference_n of_o a_o circle_n to_o find_v the_o side_n of_o a_o square_a equal_a to_o the_o same_o circle_n as_o 1000_o to_o the_o circumference_n so_o be_v 282_o to_o the_o side_n of_o the_o square_n thus_o in_o the_o circle_n of_o the_o 38_o chapter_n have_v the_o circumference_n 81_o 5_o 7_o we_o may_v find_v the_o side_n of_o a_o square_a equal_a thereunto_o for_o if_o we_o apply_v the_o thread_n to_o the_o intersection_n of_o the_o contrary_a parallel_n of_o 1000_o with_o the_o right_a parallel_n of_o 282_o it_o will_v cut_v the_o contrary_a parallel_n of_o 81_o 5_o 7_o at_o the_o right_a parallel_n of_o 23_o 43_o 1000_o and_o such_o be_v the_o side_n of_o that_o square_n which_o be_v equal_a to_o the_o same_o circle_n chap._n lxiii_o have_v the_o diameter_n of_o a_o circle_n to_o find_v the_o side_n of_o a_o square_a equal_a to_o the_o same_o circle_n as_o 1000_o to_o the_o diameter_n so_o be_v 886_o to_o the_o side_n of_o the_o square_a thus_o in_o the_o circle_n of_o the_o 38_o chapter_n have_v the_o diameter_n 26_o inch_n we_o may_v find_v the_o side_n of_o a_o square_a equal_a thereunto_o for_o if_o we_o place_v the_o thread_n upon_o the_o intersection_n of_o the_o contrary_a parallel_n of_o 1000_o with_o the_o right_a parallel_n of_o 886_o it_o will_v cut_v the_o contrary_a parallel_n of_o 26_o at_o the_o right_a parallel_n of_o 23_o 43_o 100_o and_o such_o be_v the_o side_n of_o the_o square_n require_v chap._n xliv_o have_v the_o breadth_n and_o depth_n of_o a_o square_a solid_a give_v in_o inch_n and_o the_o length_n in_o foot_n to_o find_v the_o content_a in_o foot_n as_o 12_o inch_n be_v to_o the_o breadth_n in_o inch_n so_o be_v the_o length_n in_o foot_n to_o the_o superficial_a content_n of_o that_o same_o side_n in_o foot_n then_o again_o as_o 12_o inch_n be_v to_o the_o depth_n in_o inch_n so_o be_v this_o superficial_a content_n to_o the_o solid_a content_n in_o foot_n as_o for_o example_n let_v the_o figure_n a_o be_v a_o square_a piece_n of_o timber_n to_o be_v measure_v who_o breadth_n be_v 18_o inch_n and_o depth_n 14_o and_o let_v the_o length_n thereof_o be_v 10_o foot_n then_o for_o to_o find_v the_o solid_a content_n first_o place_n the_o thread_n to_o the_o intersection_n of_o the_o contrary_a parallel_n of_o 12_o with_o the_o right_a parallel_n of_o 18_o and_o it_o shall_v cut_v the_o contrary_a parallel_n of_o 10_o at_o the_o right_a parallel_n of_o 15_o and_o so_o many_o foot_n of_o flat_a measure_n be_v contain_v in_o the_o broad_a side_n then_o look_v where_o this_o right_a parallel_n of_o 15_o intersect_v the_o contrary_a parallel_n of_o 12_o and_o there_o place_v the_o thread_n again_o so_o shall_v it_o cut_v the_o contrary_a parallel_n of_o 14_o at_o the_o right_a parallel_n of_o 17½_n foot_n and_o so_o many_o cubicall_a foot_n be_v contain_v in_o this_o piece_n of_o square_a timber_n note_v with_o the_o letter_n a._n chap._n xlv_o have_v the_o breadth_n and_o depth_n of_o a_o square_a solid_a give_v in_o inch_n to_o find_v the_o length_n of_o a_o foot_n solid_a in_o inch_n measure_n as_o the_o breadth_n in_o inch_n be_v to_o 12_o inch_n so_o be_v 12_o inch_n to_o a_o four_o number_n which_o be_v the_o length_n of_o a_o foot_n superficial_a at_o the_o breadth_n give_v then_o as_o the_o depth_n in_o inch_n be_v to_o 12_o inch_n so_o be_v this_o four_o number_n to_o the_o length_n of_o a_o foot_n solid_a thus_o in_o the_o square_a piece_n of_o timber_n in_o the_o last_o chapter_n have_v the_o breadth_n 18_o inch_n and_o the_o depth_n 14_o inch_n we_o may_v find_v how_o many_o inch_n in_o length_n will_v make_v a_o foot_n solid_a of_o that_o piece_n for_o if_o we_o place_v the_o thread_n to_o the_o intersection_n of_o the_o contrary_a parallel_n of_o 18_o with_o the_o right_a parallel_n of_o 12_o it_o will_v cut_v the_o contrary_a parallel_n of_o 12_o at_o the_o right_a parallel_n of_o 8_o and_o such_o be_v the_o length_n of_o a_o foot_n superficial_a the_o breadth_n be_v 18_o inch_n then_o again_o place_v the_o thread_n to_o the_o intersection_n of_o the_o contrary_a parallel_n of_o 14_o with_o the_o right_a parallel_n of_o 12_o and_o it_o will_v cut_v the_o contrary_a parallel_n of_o this_o 8_o at_o the_o right_a parallel_n of_o 6_o â—_o and_o so_o many_o inch_n in_o length_n will_v make_v a_o foot_n solid_a of_o that_o piece_n who_o breadth_n be_v 18_o inch_n and_o depth_n 14._o have_v in_o this_o chapter_n show_v how_o you_o may_v by_o the_o universal_a quadrat_fw-la measure_v any_o square_a solid_a or_o prisma_fw-la who_o base_n or_o end_n be_v equal_a like_a and_o parallel_v i_o will_v now_o in_o some_o chapter_n follow_v show_v how_o to_o reduce_v any_o prisma_fw-la of_o what_o fashion_n soever_o the_o base_a or_o end_n thereof_o be_v into_o a_o square_a solid_a and_o so_o by_o this_o chapter_n to_o measure_v all_o manner_n of_o prisma_n or_o all_o manner_n of_o timber_n and_o stone_n who_o base_n or_o end_n be_v equal_a like_a and_o parallel_v chap._n xlvi_o how_o that_o prisma_fw-la or_o piece_n of_o timber_n who_o base_a or_o end_n be_v a_o triangle_n may_v be_v measure_v all_o triangle_n be_v equal_a in_o their_o superficial_a content_n to_o half_a that_o right_a angle_a parallelogram_n who_o length_n and_o breadth_n be_v the_o one_o equal_a to_o the_o perpendicular_a and_o the_o other_o to_o the_o side_n whereon_o
it_o fall_v as_o be_v show_v in_o the_o 28_o chapter_n therefore_o if_o we_o take_v either_o half_a the_o perpendicular_a and_o the_o whole_a base_a or_o the_o whole_a perpendicular_a and_o half_a the_o base_a whereon_o it_o fall_v the_o one_o for_o the_o breadth_n and_o the_o other_o for_o the_o depth_n of_o a_o square_a solid_a we_o may_v measure_v it_o in_o all_o respect_n according_a to_o the_o 44_o chapter_n and_o that_o to_o any_o length_n require_v as_o for_o example_n let_v the_o triangle_n bcd_v be_v the_o base_a of_o a_o prisma_fw-la or_o piece_n of_o timber_n to_o be_v measure_v now_o take_v 10_o the_o whole_a perpendicular_a ab_fw-la for_o the_o breadth_n and_o 8_o the_o half_a base_a cd_o for_o the_o depth_n of_o a_o square_a prisma_fw-la or_o piece_n of_o timber_n we_o may_v find_v the_o solid_a content_n thereof_o by_o the_o 44_o chapter_n and_o that_o according_a to_o any_o length_n require_v chap._n xlvii_o how_o that_o prisma_fw-la or_o piece_n of_o timber_n who_o base_a or_o end_n be_v a_o rhombus_fw-la or_o diamond_n form_n may_v be_v measure_v a_o rhombus_fw-la or_o diamond_n be_v a_o figure_n of_o 4_o equal_a side_n but_o no_o right_a angle_n such_o as_o be_v the_o figure_n abcd._n now_o see_v every_o rhombus_fw-la be_v equal_a in_o their_o superficial_a content_n to_o that_o right_a angle_a parallelogram_n or_o long_o square_a who_o length_n be_v one_o of_o the_o side_n and_o breadth_n equal_a to_o the_o parallel_a distance_n of_o the_o same_o side_n if_o we_o take_v one_o of_o the_o side_n for_o the_o breadth_n and_o the_o length_n of_o the_o perpendicular_a fall_v thereon_o from_o the_o opposite_a angle_n for_o the_o depth_n of_o a_o square_a solid_a we_o may_v find_v the_o solid_a content_n thereof_o by_o the_o 44_o chapter_n as_o for_o example_n let_v the_o rhombus_fw-la abcd_v be_v the_o base_a or_o end_n of_o some_o piece_n of_o timber_n to_o be_v measure_v now_o take_v one_o of_o the_o side_n which_o be_v 10_o for_o the_o breadth_n of_o a_o square_a solid_a and_o the_o perpendicular_a which_o be_v something_o better_a than_o 8â…—_n for_o the_o depth_n of_o the_o same_o solid_a we_o may_v find_v the_o solid_a content_n thereof_o for_o any_o length_n require_v by_o the_o 44_o chapter_n chap._n xlviii_o how_o that_o prisma_fw-la or_o piece_n of_o timber_n who_o base_a or_o end_n be_v a_o rhomboide_n or_o diamond_n like_o may_v be_v measure_v a_o rhomboide_n or_o diamond_n like_o be_v a_o figure_n who_o opposite_a side_n and_o opposite_a angle_n be_v only_o equal_a and_o it_o have_v no_o right_a angle_n such_o as_o be_v the_o figure_n efgh_n now_o see_v that_o the_o rhomboide_n be_v also_o equal_a in_o his_o superficial_a content_n to_o that_o right_a angle_a parallelogram_n who_o length_n be_v one_o of_o the_o side_n and_o breadth_n equal_a to_o the_o parallel_a distance_n of_o the_o same_o side_n if_o we_o take_v one_o of_o the_o side_n for_o the_o breadth_n of_o a_o square_a solid_a and_o the_o length_n of_o the_o perpendicular_a fall_v thereon_o from_o the_o opposite_a angle_n for_o the_o depth_n of_o the_o same_o solid_a we_o may_v find_v the_o solid_a content_n thereof_o by_o the_o 45_o chapter_n let_v this_o rhomboide_n efgh_v be_v the_o base_a of_o a_o piece_n of_o timber_n to_o be_v measure_v now_o if_o you_o take_v 16_o the_o side_n gh_n or_o of_o for_o the_o breadth_n of_o a_o square_a solid_a and_o 10_o the_o length_n of_o the_o perpendicular_a el_n for_o the_o depth_n of_o the_o same_o solid_a you_o may_v find_v the_o solid_a content_n thereof_o by_o the_o 45_o chapter_n for_o any_o length_n require_v chap._n xlix_o how_o that_o prisma_fw-la or_o piece_n of_o timber_n who_o base_a or_o end_n be_v a_o trapeziam_fw-la may_v be_v measure_v a_o trapeziam_n be_v any_o irregular_a foureside_v figure_n of_o what_o fashion_n soever_o as_o this_o figure_n abcd_n be_v a_o trapeziam_fw-la and_o may_v be_v cast_v into_o two_o triangle_n by_o draw_v the_o diagonall_a line_n ac_fw-la and_o so_o each_o triangle_n measure_v as_o be_v before_o show_v which_o be_v do_v add_v the_o content_n of_o they_o both_o together_o &_o you_o shall_v have_v the_o content_a of_o the_o whole_a trapeziam_fw-la or_o when_o you_o have_v draw_v the_o diagonall_a line_n and_o from_o the_o two_o opposite_a angle_n let_v down_o the_o perpendicular_o thereon_o join_v the_o two_o perpendicular_o together_o and_o take_v the_o sum_n thereof_o for_o the_o breadth_n of_o a_o square_a solid_a and_o take_v one_o half_a the_o diagonall_a line_n for_o the_o depth_n of_o the_o same_o solid_a with_o which_o breadth_n and_o depth_n proceed_v by_o the_o 44_o chapter_n according_a to_o your_o length_n give_v and_o you_o shall_v thereby_o find_v the_o solid_a content_n require_v let_v the_o trapeziam_fw-la abcd_v be_v the_o base_a or_o end_n of_o a_o piece_n of_o timber_n now_o if_o you_o take_v the_o two_o perpendicular_o bf_n and_o de_fw-fr and_o join_v they_o together_o they_o will_v make_v 10_o for_o the_o breadth_n of_o a_o square_a solid_a and_o then_o take_v one_o half_a the_o diagonall_a ac_fw-la which_o will_v be_v 8_o for_o the_o depth_n of_o the_o same_o solid_a you_o may_v by_o the_o 44_o chapter_n find_v the_o solid_a content_n thereof_o according_a to_o any_o length_n give_v chap._n l._n how_o that_o prisma_fw-la or_o piece_n of_o timber_n who_o base_a or_o end_n have_v many_o side_n as_o 5_o 6_o 7_o 8_o 9_o 10_o or_o more_o so_o they_o be_v equal_a may_v be_v measure_v many_o side_v figure_n be_v those_o which_o have_v more_o side_n than_o four_o and_o be_v general_o call_v polygon_n as_o this_o figure_n a_o have_v equal_a side_n and_o angle_n be_v call_v a_o polygon_n every_o regular_a polygon_n be_v equal_a in_o the_o superficial_a content_n to_o that_o long_a square_n who_o length_n be_v equal_a to_o one_o half_a the_o perimiter_n and_o breadth_n to_o a_o perpendicular_a draw_v from_o the_o centre_n to_o the_o middle_n of_o any_o side_n thereof_o if_o we_o add_v all_o the_o side_n together_o and_o take_v half_a the_o sum_n for_o the_o breadth_n of_o a_o square_a solid_a and_o then_o take_v the_o perpendicular_a for_o the_o depth_n of_o the_o same_o solid_a we_o may_v by_o the_o 44_o chapter_n find_v the_o solid_a content_n thereof_o let_v the_o figure_n a_o be_v the_o base_a or_o end_n of_o a_o piece_n of_o timber_n to_o be_v measure_v consist_v of_o 5_o equal_a side_n and_o each_o side_n contain_v 12_o inch_n which_o be_v add_v together_o into_o one_o sum_n make_v 60_o the_o half_a whereof_o be_v 30_o for_o the_o breadth_n of_o your_o piece_n then_o take_v the_o length_n of_o the_o perpendicular_a fall_v from_o the_o centre_n a_o to_o the_o midst_n of_o one_o of_o the_o side_n which_o be_v 8_o inch_n for_o the_o depth_n of_o the_o same_o piece_n with_o which_o breadth_n and_o depth_n you_o may_v proceed_v to_o find_v the_o solid_a content_n by_o the_o 44_o chapter_n according_a to_o any_o length_n give_v a_o note_n to_o find_v the_o centre_n of_o those_o equiangle_n figure_n if_o the_o side_n be_v even_o the_o centre_n be_v find_v by_o draw_v line_n from_o one_o angle_n to_o his_o opposite_a angle_n but_o if_o they_o be_v odd_a it_o be_v find_v by_o draw_v line_n from_o the_o middle_n of_o a_o side_n to_o his_o opposite_a angle_n so_o shall_v the_o cut_n or_o intersection_n of_o those_o straight_a line_n show_v the_o place_n of_o the_o centre_n chap._n li._n have_v the_o circumference_n and_o diameter_n of_o a_o cylinder_n give_v in_o inch_n and_o the_o length_n in_o foot_n to_o find_v the_o content_a in_o foot_n as_o 12_o inch_n to_o the_o diameter_n in_o inch_n so_o be_v the_o length_n in_o foot_n to_o a_o four_o number_n then_o as_o 12_o inch_n to_o one_o quarter_n of_o the_o circumference_n so_o be_v this_o four_o number_n to_o the_o solid_a content_n in_o foot_n as_o for_o example_n let_v the_o figure_n a_o be_v a_o cylinder_n or_o some_o round_a piece_n of_o timber_n to_o be_v measure_v who_o circumference_n be_v 44_o inch_n and_o diameter_n 14_o inch_n and_o let_v the_o length_n thereof_o be_v 10_o foot_n now_o for_o to_o find_v the_o solid_a content_n first_o place_z the_o thread_n to_o the_o intersection_n of_o the_o contrary_a parallel_n of_o 12_o with_o the_o right_a parallel_n of_o 14_o and_o it_o will_v cut_v the_o contrary_a parallel_n of_o 10_o at_o the_o right_a parallel_n of_o 11â…”_n now_o where_o this_o right_a parallel_n of_o 11â…”_n cut_v the_o contrary_a parallel_n of_o 12_o there_o place_v the_o thread_n again_o so_o shall_v it_o cut_v the_o contrary_a parallel_n of_o 11_o at_o the_o right_a parallel_n of_o 10_o 7_o 10_o and_o so_o many_o solid_a foot_n be_v contain_v in_o this_o cylinder_n or_o round_a piece_n of_o timber_n chap._n lii_o have_v the_o
minute_n as_o be_v require_v chap._n v._o the_o right_a sine_fw-la of_o any_o arch_n be_v give_v to_o find_v the_o radius_fw-la first_o take_v the_o give_v sine_fw-la and_o place_n it_o from_o the_o side_n ae_n of_o your_o quadrat_fw-la along_o upon_o that_o contrary_a parallel_n which_o cut_v the_o degree_n of_o the_o sine_fw-la give_v count_v in_o the_o quadrant_a from_o g_z and_o where_o the_o sine_fw-la end_v there_o place_n the_o thread_n so_o shall_v it_o show_v upon_o the_o contrary_a parallel_n of_o 100_o the_o radius_fw-la require_v let_v the_o line_n b_o in_o the_o forego_n chapter_n be_v a_o give_fw-mi sine_fw-la of_o 36_o degree_n 53_o minute_n first_o place_v it_o from_o n_n to_o m_n upon_o that_o parallel_n which_o run_v through_o the_o degree_n of_o 36_o and_o 53_o minute_n as_o at_o o_o then_o upon_o the_o point_n m_o i_o place_v the_o thread_n so_o do_v it_o cut_v the_o contrary_a parallel_n of_o 100_o in_o the_o point_n d_o make_o de_fw-fr the_o radius_fw-la which_o be_v the_o line_n a_o in_o the_o forego_n chapter_n as_o be_v require_v chap._n vi_o the_o radius_fw-la be_v give_v with_o a_o straight_a line_n resemble_v a_o sine_fw-la to_o find_v the_o quantity_n of_o that_o unknown_a sine_fw-la first_o place_v the_o give_v radius_fw-la upon_o the_o contrary_a parallel_n of_o 100_o and_o thereto_o apply_v the_o thread_n then_o take_v the_o right_a sine_fw-la give_v and_o set_v one_o foot_n of_o your_o compass_n in_o the_o side_n of_o the_o quadrat_fw-la carry_v it_o parallel_v to_o the_o former_a till_o the_o other_o foot_n cut_v the_o thread_n and_o there_o stay_v for_o the_o parallel_n where_o the_o compass_n so_o rest_v shall_v cut_v the_o quadrant_a in_o the_o degree_n answer_v to_o that_o unknown_a sine_fw-la give_v as_o if_o the_o line_n a_o in_o the_o four_o chapter_n be_v the_o radius_fw-la give_v and_z b_o the_o straight_a line_n resemble_v a_o sine_fw-la first_o i_o place_v the_o line_n a_o from_z e_z to_z d_o upon_o the_o contrary_a parallel_n of_o 100_o and_o apply_v the_o thread_n thereto_o the_o thread_n lie_v in_o this_o position_n i_o take_v the_o line_n b_o and_o fix_v one_o foot_n in_o the_o side_n of_o the_o quadrat_fw-la ae_n i_o carry_v it_o parallel_v to_o the_o former_a until_o the_o other_o foot_n touch_v the_o thread_n so_o will_v one_o foot_n rest_n in_o the_o point_n n_n and_o the_o other_o in_o m_n upon_o the_o parallel_n of_o 60_o which_o parallel_n cut_v the_o quadrat_fw-la in_o the_o point_n o_o at_o 36_o degree_n 53_o minute_n count_v from_o g_o this_o 36_o degree_n 53_o minute_n be_v the_o arch_n of_o which_o the_o give_a line_n b_o be_v the_o sine_fw-la the_o line_n a_o being_n radius_fw-la chap._n vii_o the_o use_n of_o these_o parallel_n as_o they_o signifienaturall_a sin_n i_o tell_v you_o in_o the_o four_o chapter_n of_o the_o forego_n book_n that_o these_o line_n do_v sometime_o signify_v themselves_o alone_o sometime_o sin_n and_o sometime_o tangent_n and_o now_o for_o distinction_n sake_n when_o i_o call_v they_o parallel_n i_o will_v have_v you_o to_o understand_v they_o as_o they_o be_v in_o their_o own_o proper_a signification_n and_o when_o i_o call_v they_o sines_n i_o will_v have_v they_o understand_v as_o the_o sin_n of_o those_o arch_n through_o which_o they_o run_v and_o when_o i_o call_v they_o tangent_n i_o will_v have_v they_o understand_v as_o the_o tangent_fw-la of_o those_o degree_n against_o which_o they_o butt_v in_o the_o side_n of_o the_o quadrant_a and_o further_o for_o distincton_n whereas_o in_o their_o own_o signification_n i_o call_v they_o right_a and_o contrary_a parallel_n so_o now_o i_o call_v they_o right_a and_o contrary_a sin_n and_o right_a and_o contrary_a tangent_n these_o line_n as_o they_o signify_v sin_n have_v like_o use_n in_o find_v a_o four_o proportional_a sine_fw-la as_o the_o ordinary_a canon_n of_o natural_a sin_n and_o the_o manner_n of_o find_v it_o be_v always_o such_o as_o in_o this_o example_n as_o the_o sine_fw-la of_o 90_o degree_n unto_o the_o sine_fw-la of_o 30_o degree_n so_o be_v the_o sine_fw-la of_o 23_o degree_n 30_o minute_n unto_o a_o four_o sine_fw-la wherefore_o place_n the_o thread_n at_o the_o intersection_n of_o the_o contrary_a sine_fw-la of_o 90_o degree_n with_o the_o right_n sine_fw-la of_o 30_o degree_n and_o it_o will_v cut_v the_o contrary_a sine_fw-la of_o 23_o degree_n 30_o minute_n at_o the_o right_n sine_fw-la of_o 11_o degree_n 30_o minute_n and_o such_o be_v the_o four_o proportional_a sine_fw-la require_v or_o you_o may_v place_v the_o thread_n at_o the_o intersection_n of_o the_o contrary_a sine_fw-la of_o 90_o degree_n with_o the_o right_n sine_fw-la of_o 23_o degree_n 30_o minute_n and_o it_o shall_v cut_v the_o contrary_a sine_fw-la of_o 30_o degree_n at_o the_o right_n sine_fw-la of_o 11_o degree_n 30_o minute_n as_o before_o and_o thus_o may_v all_o the_o rest_n of_o the_o sinical_a proposition_n be_v wrought_v both_o way_n chap._n viii_o the_o use_n of_o these_o parallel_n as_o they_o signify_v both_o sin_n and_o tangent_n here_o the_o work_n be_v in_o a_o manner_n the_o same_o as_o before_o as_o will_v appear_v by_o this_o example_n as_o the_o sine_fw-la of_o 90_o degree_n to_o the_o sine_fw-la of_o 51_o degr_n 30_o min._n so_o be_v the_o tangent_fw-la of_o 30_o degree_n to_o a_o four_o tangent_fw-la thus_o place_v the_o thread_n upon_o the_o intersection_n of_o the_o contrary_a sine_fw-la of_o 90_o degree_n with_o the_o right_n sine_fw-la of_o 51_o degree_n 30_o minute_n it_o will_v cut_v the_o contrary_a tangent_fw-la of_o 30_o degree_n at_o the_o right_n tangent_fw-la of_o 24_o degree_n 20_o minute_n or_o if_o you_o place_v the_o thread_n at_o the_o intersection_n of_o the_o contrary_a sine_fw-la of_o 90_o degree_n with_o the_o right_n tangent_fw-la of_o 30_o degree_n it_o will_v cut_v the_o contrary_a sine_fw-la of_o 51_o degree_n 30_o minute_n at_o the_o right_n tangent_fw-la of_o 24_o degree_n 20_o minute_n which_o be_v the_o four_o proportional_a tangent_fw-la require_v and_o this_o manner_n of_o work_n will_v hold_v until_o the_o tangent_n be_v great_a than_o 45_o degree_n the_o side_n of_o the_o quadrat_fw-la but_o when_o the_o give_v tangent_fw-la or_o tangent_n exceed_v 45_o degree_n the_o side_n of_o the_o quadrat_fw-la if_o the_o two_o first_o give_v number_n be_v sines_n and_o the_o three_o a_o tangent_fw-la set_v the_o second_o sine_fw-la in_o the_o place_n of_o the_o first_o and_o the_o first_o sine_fw-la in_o the_o place_n of_o the_o second_o and_o in_o steed_n of_o the_o give_v tangent_fw-la take_v his_o compliment_n unto_o a_o quadrant_a so_o shall_v the_o four_o proportional_a be_v the_o compliment_n of_o the_o tangent_fw-la require_v as_o for_o example_n in_o this_o proportion_n as_o the_o sine_fw-la of_o 90_o degree_n to_o the_o sine_fw-la of_o 20_o degree_n so_o the_o tangent_fw-la of_o 70_o degree_n to_o a_o four_o tangent_fw-la and_o because_o the_o tangent_fw-la of_o 70_o degree_n be_v great_a than_o 45_o degree_n the_o side_n of_o the_o quadrat_fw-la therefore_o i_o turn_v the_o proportion_n thus_o as_o the_o sine_fw-la of_o 20_o degree_n to_o the_o sine_fw-la of_o 90_o degree_n so_o the_o co-tangent_a of_o 70_o degree_n to_o the_o co-tangent_a of_o the_o sum_n require_v thus_o place_v the_o thread_n upon_o the_o intersection_n of_o the_o contrary_a sine_fw-la of_o 20_o degree_n with_o the_o right_n tangent_fw-la of_o 20_o degree_n the_o compliment_n of_o 70_o degree_n it_o shall_v cut_v the_o contrary_a sine_fw-la of_o 90_o degree_n at_o the_o right_n tangent_fw-la of_o the_o compliment_n of_o the_o arch_n require_v but_o the_o thread_n intersect_v not_o the_o say_v contrary_a sine_fw-la of_o 90_o degree_n upon_o the_o quadrat_fw-la but_o beside_o it_o which_o never_o happen_v but_o when_o the_o one_o tangent_fw-la be_v great_a than_o 45_o degree_n and_o the_o other_o less_o wherefore_o you_o be_v to_o note_v that_o at_o any_o such_o time_n though_o the_o thread_n cut_v not_o the_o contrary_a sine_fw-la of_o 90_o degree_n upon_o the_o quadrat_fw-la yet_o be_v the_o thread_n by_o the_o former_a position_n place_v at_o the_o true_a angle_n of_o the_o tangent_fw-la require_v which_o be_v 46_o degree_n 47_o minute_n the_o compliment_n whereof_o 43_o degree_n 13_o minute_n be_v the_o arch_n require_v and_o if_o the_o first_o give_v number_n be_v tangent_n and_o the_o three_o a_o sine_fw-la take_v the_o compliment_n of_o the_o two_o tangent_n and_o exchange_v their_o place_n as_o in_o the_o example_n follow_v but_o if_o one_o be_v more_o and_o the_o other_o less_o as_o in_o this_o proportion_n as_o the_o tangent_fw-la of_o 70_o degree_n to_o the_o tangent_fw-la of_o 43_o degree_n 13_o minute_n so_o the_o sine_fw-la of_o 90_o degree_n to_o a_o four_o sine_fw-la herein_o regard_v the_o tangent_fw-la of_o 70_o degree_n be_v more_o
rumb_n and_o distance_n chap._n i._o to_o find_v the_o sun_n altitude_n by_o the_o shadow_n of_o a_o gnomon_n set_v perpendicular_a to_o the_o horizon_n to_o perform_v this_o conclusion_n place_v the_o thread_n upon_o the_o intersection_n of_o the_o contrary_a parallel_n of_o the_o shadow_n with_o the_o right_a parallel_n of_o the_o gnomon_n so_o shall_v it_o show_v in_o the_o quadrant_a the_o sun_n altitude_n require_v as_o for_o example_n let_v ab_fw-la in_o the_o follow_a diagram_n be_v the_o gnomon_n or_o substance_n yield_v shadow_n which_o we_o will_v suppose_v to_o be_v 60_o and_o let_v ac_fw-la be_v the_o length_n of_o the_o shadow_n here_o find_v to_o be_v 45_o such_o part_n as_o the_o gnomon_n be_v 60._o now_o place_v the_o thread_n upon_o the_o intersection_n of_o the_o contrary_a parallel_n of_o 45_o with_o the_o right_a parallel_n of_o 60_o it_o will_v cut_v in_o the_o quadrant_n 53_o deg_fw-la 8_o min._n and_o such_o be_v the_o angle_n acb_n the_o altitude_n of_o the_o sun_n require_v or_o if_o the_o shadow_n have_v be_v ad_fw-la which_o here_o be_v find_v to_o be_v 80_o if_o you_o place_v the_o thread_n at_o the_o intersection_n of_o the_o right_a parallel_n of_o 60_o with_o the_o contrary_a parallel_n of_o 80_o it_o will_v cut_v in_o the_o quadrant_n 36_o deg_fw-la 52_o min._n and_o such_o be_v the_o angle_n adb_n the_o altitude_n of_o the_o sun_n require_v chap._n ii_o the_o height_n of_o the_o sun_n be_v give_v to_o find_v the_o length_n of_o the_o right_a shadow_n by_o right_a shadow_n be_v mean_v the_o shadow_n of_o any_o staff_n post_n steeple_n or_o any_o gnomon_n whatsoever_o that_o stand_v at_o right_a angle_n with_o the_o horizon_n the_o one_o end_n thereof_o respect_v the_o zenith_n of_o the_o place_n and_o the_o other_o the_o nadir_n wherefore_o place_n the_o thread_n to_o the_o height_n of_o the_o sun_n give_v in_o the_o quadrant_n and_o it_o shall_v cut_v the_o right_a parallel_n of_o the_o gnomon_n at_o the_o contrary_a parallel_n of_o the_o shadow_n let_v the_o sun_n height_n give_v be_v 53_o deg_fw-la 8_o min._n and_o the_o length_n of_o the_o gnomon_n 60_o wherefore_o if_o you_o place_v the_o thread_n at_o 53_o deg_fw-la 8_o min._n in_o the_o quadrant_n it_o will_v cut_v the_o right_a parallel_n of_o 60_o at_o the_o contrary_a parallel_n of_o 45_o and_o such_o be_v the_o length_n of_o the_o right_a shadow_n ac_fw-la which_o be_v require_v or_o if_o the_o sun_n height_n be_v but_o 36_o deg_fw-la 52_o min._n if_o you_o place_v the_o thread_n to_o 36_o deg_fw-la 52_o min._n in_o the_o quadrant_n it_o will_v cut_v the_o right_a parallel_n of_o 60_o at_o the_o contrary_a parallel_n of_o 80_o and_o such_o be_v the_o length_n of_o the_o right_a shadow_n ad_fw-la which_o be_v require_v chap._n iii_o to_o find_v the_o altitude_n of_o the_o sun_n by_o the_o shadow_n of_o a_o gnomon_n stand_v at_o right_a angle_n with_o any_o perpendicular_a wall_n in_o such_o manner_n that_o it_o may_v lie_v parallel_n to_o the_o horizon_n here_o if_o you_o place_v the_o thread_n to_o the_o intersection_n of_o the_o contrary_a parallel_n of_o the_o gnomon_n with_o the_o right_a parallel_n of_o the_o shadow_n it_o shall_v cut_v the_o quadrant_a at_o the_o sun_n altitude_n require_v as_o for_o example_n let_v ab_fw-la in_o the_o next_o chapter_n be_v some_o perpendicular_a wall_n ac_fw-la a_o gnomon_n make_v right_a angle_n therewith_o who_o length_n be_v 60_o and_o let_v ad_fw-la be_v the_o length_n of_o the_o shadow_n which_o be_v here_o find_v to_o be_v 80_o now_o if_o you_o place_v the_o thread_n to_o the_o intersection_n of_o the_o contrary_a parallel_n of_o 60_o with_o the_o right_a parallel_n of_o 80_o it_o will_v cut_v in_o the_o quadrant_n 5_o deg_fw-la 8_o min._n and_o such_o be_v the_o angle_n acd_v the_o altitude_n of_o the_o sun_n require_v chap._n iu._n the_o height_n of_o the_o sun_n be_v give_v to_o find_v the_o length_n of_o the_o contrary_a shadow_n by_o the_o contrary_a shadow_n be_v understand_v the_o length_n of_o any_o shadow_n that_o be_v make_v by_o a_o staff_n or_o gnomon_n stand_v at_o right_a angle_n against_o any_o perpendicular_a wall_n in_o such_o a_o manner_n that_o it_o may_v lie_v parallel_n to_o the_o horizon_n the_o length_n of_o the_o contrary_a shadow_n do_v increase_v as_o the_o sun_n rise_v in_o height_n whereas_o the_o right_a shadow_n do_v decrease_v in_o length_n as_o the_o sun_n do_v increase_v in_o height_n wherefore_o to_o find_v the_o length_n of_o the_o verse_v shadow_n place_v the_o thread_n upon_o the_o height_n of_o the_o sun_n give_v in_o the_o quadrant_n and_o it_o will_v cut_v the_o contrary_a parallel_n of_o the_o gnomon_n at_o the_o right_a parallel_n of_o the_o shadow_n let_v ab_fw-la be_v a_o perpendicular_a wall_n ac_fw-la a_o gnomon_n make_v right_a angle_n there_o with_o the_o length_n whereof_o be_v 60_o and_o the_o sun_n altitude_n 53_o deg_fw-la 8_o min._n the_o angle_n acd_v now_o if_o you_o place_v the_o thread_n at_o 53_o deg_fw-la 8_o min._n in_o the_o quadrant_a it_o will_v cut_v the_o contrary_a parallel_n of_o 60_o at_o the_o right_a parallel_n of_o 80_o and_o such_o be_v the_o length_n of_o the_o contrary_a shadow_n ad_fw-la which_o be_v require_v chap._n v._o have_v the_o distance_n of_o the_o sun_n from_o the_o next_o equinoctial_a point_n to_o find_v his_o declination_n as_o the_o radius_fw-la be_v in_o proportion_n to_o the_o sine_fw-la of_o the_o sun_n great_a declination_n so_o the_o sine_fw-la of_o the_o sun_n distance_n from_o the_o next_o equinoctial_a point_n to_o the_o sine_fw-la of_o the_o declination_n require_v thus_o in_o the_o right_n angle_v spherical_a triangle_n ♈_o x_o ♉_o in_o the_o follow_a diagram_n have_v the_o base_a ♈_o ♉_o 30_o deg_fw-la which_o be_v the_o sun_n distance_n from_o the_o equinoctial_a point_n ♈_o he_o be_v in_o the_o begin_n of_o ♉_o and_o the_o angle_n x_o ♈_o ♉_o 23_o degree_n 30_o minute_n the_o angle_n of_o the_o sun_n great_a declination_n with_o the_o right_a angle_n at_o x_o we_o may_v find_v the_o side_n ♉_o x_o the_o declination_n require_v for_o if_o we_o place_v the_o thread_n at_o the_o intersection_n of_o the_o contrary_a sine_fw-la of_o 90_o degree_n with_o the_o right_n sine_fw-la of_o 23_o degree_n 30_o minute_n it_o will_v cut_v the_o contrary_a sine_fw-la of_o 30_o deg_fw-la at_o the_o right_a sineof_o of_o 11_o deg_fw-la 30_o minute_n or_o if_o we_o place_v the_o thread_n at_o the_o intersection_n of_o the_o contrary_a sine_fw-la of_o 90_o deg_fw-la with_o the_o right_n sine_fw-la of_o 30_o deg_fw-la it_o will_v cut_v the_o contrary_a sine_fw-la of_o 23_o deg_fw-la 30_o min._n at_o the_o right_n sine_fw-la of_o 11_o deg_fw-la 30_o min._n &_o so_o much_o be_v the_o side_n ♉_o x_o the_o declination_n of_o the_o sun_n in_o the_o begin_n of_o ♉_o which_o be_v require_v the_o like_a declination_n have_v the_o beginning_n of_o â™_o â™_o &_o ♓_o all_o be_v of_o equal_a distance_n from_o their_o next_o equinoctial_a point_n the_o like_a hold_v of_o all_o the_o other_o point_n of_o the_o ecliptic_a so_o that_o have_v the_o declination_n of_o every_o degree_n in_o one_o quarter_n all_o the_o rest_n be_v know_v chap._n vi_o how_o to_o find_v the_o latitude_n of_o a_o place_n or_o the_o pole_n elevation_n above_o the_o horizon_n the_o declination_n of_o the_o sun_n be_v give_v first_o you_o must_v get_v the_o sun_n meridian_n altitude_n the_o which_o you_o may_v very_o well_o do_v by_o this_o instrument_n after_o this_o manner_n hold_v the_o superficies_n of_o your_o instrument_n perpendicular_a to_o the_o horizon_n and_o the_o centre_n where_o the_o thread_n and_o plummet_n hang_v towards_o the_o sun_n then_o lift_v up_o the_o quadrat_fw-la towards_o the_o sun_n until_o his_o beam_n pass_v through_o both_o his_o sight_n at_o once_o the_o thread_n will_v cut_v in_o the_o quadrant_a the_o degree_n of_o the_o sun_n altitude_n this_o observation_n you_o must_v begin_v a_o little_a before_o noon_n still_o diligent_o observe_v until_o you_o perceive_v the_o sun_n to_o begin_v to_o fall_v again_o then_o mark_v what_o be_v his_o great_a altitude_n will_v serve_v for_o this_o our_o present_a purpose_n have_v get_v the_o meridian_n altitude_n by_o this_o and_o the_o declination_n by_o the_o last_o chapter_n you_o may_v find_v the_o elevation_n of_o the_o pole_n above_o the_o horizon_n as_o follow_v if_o the_o sun_n have_v north_n declination_n then_o subtract_v the_o declination_n out_o of_o the_o meridian_n altitude_n but_o if_o the_o sun_n have_v south_n declination_n then_o add_v the_o declination_n to_o the_o meridian_n altitude_n so_o shall_v the_o sum_n or_o difference_n be_v the_o height_n of_o the_o equinoctial_a which_o be_v take_v out_o of_o the_o quadrant_a or_o 90_o degree_n leave_v the_o elevation_n of_o the_o pole_n
i_o seek_v in_o the_o ecliptic_a and_o follow_v the_o parallel_n that_o pass_v thereby_o to_o the_o horizon_n i_o find_v it_o there_o to_o cut_v 19_o deg_fw-la 7_o min._n count_v from_o the_o centre_n and_o such_o be_v the_o amplitude_n require_v the_o sphere_n be_v set_v to_o the_o latitude_n of_o 52_o deg_fw-la 30_o min._n chap._n xxx_o have_v the_o latitude_n of_o the_o place_n with_o the_o declination_n of_o any_o star_n to_o find_v his_o amplitude_n first_o the_o sphere_n to_o the_o latitude_n be_v set_v if_o the_o star_n be_v therein_o place_v follow_v the_o parallel_n of_o declination_n whereon_o he_o stand_v to_o the_o horizon_n and_o if_o he_o be_v not_o in_o the_o instrument_n get_v his_o declination_n by_o the_o seven_o chap._n or_o otherwise_o by_o some_o table_n for_o that_o purpose_n which_o find_v seek_v it_o in_o the_o axletree-line_n among_o the_o parallel_n of_o declination_n &_o follow_v the_o parallel_n thereof_o to_o the_o horizon_n and_o if_o it_o cut_v the_o horizon_n it_o there_o show_v the_o amplitude_n but_o if_o it_o cut_v it_o not_o then_o do_v not_o that_o star_n set_v or_o go_v down_o under_o our_o horizon_n as_o for_o example_n let_v it_o be_v require_v to_o find_v the_o amplitude_n of_o arcturus_n i_o look_v in_o my_o instrument_n for_o arcturus_n and_o i_o find_v he_o place_v upon_o the_o parallel_n of_o 21_o d._n 10_o min._n of_o north_n declination_n as_o it_o be_v just_a in_o the_o horizon_n at_o 36_o deg_fw-la 23_o min._n from_o the_o centre_n and_o such_o be_v his_o amplitude_n and_o if_o you_o will_v have_v the_o amplitude_n of_o the_o lion_n heart_n look_v in_o your_o instrument_n and_o you_o shall_v find_v he_o place_v upon_o the_o parallel_n of_o 13_o deg_fw-la 42_o min._n of_o north_n declination_n which_o if_o you_o follow_v to_o the_o horizon_n you_o shall_v find_v it_o there_o cut_v 22_o deg_fw-la 54_o min._n and_o such_o be_v the_o amplitude_n of_o the_o lion_n heart_n in_o the_o latitude_n of_o 52_o deg_fw-la 30_o min._n but_o if_o you_o will_v seek_v the_o amplitude_n of_o the_o bright_a star_n call_v the_o harp_n you_o shall_v find_v he_o place_v upon_o the_o parallel_n of_o 38_o deg_fw-la 30_o min._n north_n declination_n but_o this_o parallel_n cut_v not_o the_o horizon_n therefore_o he_o go_v not_o under_o our_o horizon_n at_o all_o in_o the_o aforesaid_a latitude_n of_o 52_o deg_fw-la 30_o min._n but_o when_o he_o be_v in_o his_o low_a direct_o in_o the_o north_n he_o be_v just_a one_o degree_n of_o height_n chap._n xxxii_o have_v the_o declination_n and_o amplitude_n of_o the_o sun_n to_o find_v the_o latitude_n of_o the_o place_n count_n the_o declination_n upon_o the_o axletree-line_n among_o the_o parallel_n of_o declination_n and_o the_o amplitude_n upon_o the_o horizon_n from_o the_o centre_n the_o same_o way_n with_o the_o declination_n then_o move_v the_o sphere_n to_o and_o fro_o until_o the_o amplitude_n and_o the_o parallel_n of_o declination_n meet_v both_o in_o one_o point_n the_o south_n end_n of_o the_o horizon_n shall_v cut_v the_o meridian_n at_o the_o compliment_n of_o the_o latitude_n require_v as_o for_o example_n let_v the_o sun_n declination_n be_v 11_o deg_fw-la 30_o min._n and_o his_o amplitude_n 19_o deg_fw-la 7_o min._n if_o you_o move_v the_o sphere_n until_o the_o 19_o deg_fw-la 7_o min._n one_o the_o horizon_n fall_v direct_o on_o the_o parallel_n of_o 11_o deg_fw-la 30_o min._n the_o south_n end_n of_o the_o horizon_n will_v cut_v the_o limb_n at_o 37_o deg_fw-la 30._o min._n and_o such_o be_v the_o height_n of_o the_o equinoctial_a which_o be_v take_v out_o of_o 90_o deg_fw-la leave_v 52_o deg_fw-la 30_o min._n for_o the_o latitude_n of_o the_o place_n which_o be_v require_v chap._n xxxii_o the_o latitude_n of_o the_o place_n and_o the_o amplitude_n of_o the_o sun_n be_v give_v to_o find_v his_o declination_n first_o set_v the_o sphere_n to_o the_o latitude_n propose_v then_o count_v the_o amplitude_n upon_o the_o horizon_n and_o where_o it_o shall_v end_v shall_v meet_v you_o a_o parallel_n which_o be_v follow_v to_o the_o axletree-line_n will_v there_o show_v you_o the_o declination_n require_v as_o in_o the_o latitude_n of_o 52_o deg_fw-la 30_o min._n the_o amplitude_n be_v 19_o deg_fw-la 7_o min._n if_o you_o first_o place_n the_o sphere_n to_o the_o latitude_n and_o then_o count_v 19_o deg_fw-la 7_o min._n upon_o the_o horizon_n there_o shall_v meet_v you_o a_o parallel_n which_o be_v follow_v to_o the_o axletree_n line_n shall_v there_o give_v you_o 11_o deg_fw-la 30_o min._n the_o declination_n require_v chap._n xxxiii_o have_v the_o latitude_n of_o the_o place_n and_o the_o declination_n of_o the_o sun_n to_o find_v his_o height_n in_o the_o vertical_a circle_n first_o set_v the_o sphere_n to_o the_o latitude_n give_v then_o count_v the_o declination_n give_v among_o the_o almicanter_n from_o both_o end_n of_o the_o horizon_n towards_o the_o zenith_n and_o thereto_o place_v the_o thread_n parallel_v to_o the_o horizon_n and_o where_o the_o thread_n so_o place_v cut_v the_o axletree-line_n or_o six_o a_o clock_n hour-line_n there_o be_v the_o altitude_n require_v as_o in_o the_o latitude_n of_o 52_o deg_fw-la 30_o min._n the_o declination_n of_o the_o sun_n be_v 11_o deg_fw-la 30_o min._n if_o you_o count_v this_o 11._o deg_fw-la 30_o min._n from_o both_o end_n of_o the_o horizon_n among_o the_o almicanter_n and_o place_v the_o thread_n thereto_o it_o shall_v lie_v parallel_n to_o the_o horizon_n and_o cut_v the_o axle-tree-line_n at_o 14_o deg_fw-la 33_o min._n and_o such_o be_v the_o sun_n height_n when_o he_o be_v direct_o in_o the_o east_n or_o west_n point_v which_o be_v require_v or_o if_o you_o place_v the_o horizon_n to_o 52_o deg_fw-la 30_o min._n in_o the_o limb_n count_v from_o the_o equator_fw-la and_o count_v the_o declination_n upon_o the_o axletree-line_n viz._n 11_o deg_fw-la 30_o min._n the_o parallel_n thereof_o shall_v cut_v the_o horizon_n at_o 14_o deg_fw-la 33_o min._n for_o the_o height_n of_o the_o sun_n in_o the_o vertical_a circle_n the_o same_o as_o before_o chap._n xxxiv_o have_v the_o latitude_n of_o the_o place_n and_o the_o declination_n of_o the_o sun_n to_o find_v the_o time_n of_o his_o come_n to_o the_o east_n or_o west_n point_n first_o set_v the_o planisphere_n to_o the_o latitude_n give_v then_o place_v the_o thread_n upon_o the_o zenith_n and_o the_o nadir_n so_o shall_v it_o cut_v the_o horizon_n at_o right_a angle_n in_o the_o centre_n and_o the_o parallel_n of_o declination_n at_o the_o time_n from_o six_o count_v among_o the_o meridian_n as_o for_o example_n let_v the_o latitude_n give_v be_v 52_o deg_fw-la 30_o min._n and_o the_o sun_n declination_n 11_o deg_fw-la 30_o min._n first_o place_z the_o horizon_n to_o 37_o deg_fw-la 30_o min._n the_o compliment_n of_o the_o latitude_n below_o the_o south_n end_n of_o the_o equator_fw-la then_o place_v the_o thread_n to_o the_o zenith_n and_o nadir_n that_o be_v to_o 52_o deg_fw-la 30_o min._n above_o the_o south_n end_n of_o the_o equator_fw-la and_o as_o much_o below_o the_o north_n end_v thereof_o the_o thread_n be_v fix_v shall_v cut_v the_o horizon_n at_o right_a angle_n in_o the_o centre_n and_o also_o the_o parallel_n of_o 11_o deg_fw-la 30_o min_fw-mi at_o the_o meridian_n of_o 9_o deg_fw-la or_o in_o time_n at_o 36_o minute_n from_o the_o hour_n of_o six_o and_o such_o be_v the_o distance_n of_o time_n betwixt_o the_o hour_n of_o six_o and_o the_o sun_n come_v to_o the_o east_n or_o west_n point_n which_o be_v require_v chap._n xxxv_o have_v the_o latitude_n of_o the_o place_n and_o the_o declination_n of_o the_o sun_n to_o find_v his_o height_n at_o the_o hour_n of_o six_o first_o set_v the_o horizon_n to_o the_o compliment_n of_o the_o latititude_n below_o the_o south_n end_n of_o the_o equator_fw-la or_o the_o equator_fw-la to_o the_o compliment_n of_o the_o latitude_n above_o the_o south_n end_n of_o the_o horizon_n which_o be_v all_o one_o then_o seek_v the_o declination_n upon_o the_o axletree-line_n and_o place_v the_o thread_n thereunto_o parallel_v to_o the_o horizon_n by_o help_n of_o the_o almicanter_n at_o both_o end_n thereof_o so_o shall_v the_o thread_n show_v among_o those_o almicanter_n the_o height_n of_o the_o sun_n require_v thus_o in_o the_o latitude_n of_o 52_o deg_fw-la 30_o min._n the_o sun_n declination_n be_v 11_o deg_fw-la 30_o min._n i_o first_o set_v the_o equator_fw-la 37_o deg_fw-la 30_o min._n above_o the_o horizon_n and_o then_o place_v the_o thread_n to_o 11_o deg_fw-la 30_o min._n count_v in_o the_o axletree-line_n and_o move_v the_o end_n thereof_o to_o and_o fro_o until_o the_o thread_n lie_v parallel_n to_o the_o horizon_n which_o will_v be_v when_o both_o end_n of_o the_o thread_n cut_v like_o degree_n from_o the_o horizon_n upon_o