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the_o vertical_a circle_n pass_v through_o the_o zenith_n and_o nadir_n east_n and_o west_n and_o the_o line_n mah_n cross_v it_o at_o right_a angle_n shall_v represent_v the_o horizon_n these_o two_o be_v divide_v in_o the_o same_o sort_n as_o the_o ecliptic_a and_o the_o equator_fw-la the_o line_n draw_v through_o each_o degree_n of_o the_o semidiameter_n az_n parallel_n to_o the_o horizon_n shall_v be_v the_o circle_n of_o altitude_n and_o the_o division_n in_o the_o horizon_n and_o his_o parallel_n shall_v give_v the_o azimuth_n last_o if_o through_o 18_o gr_n in_o a_o be_v draw_v a_o right_a line_n ik_fw-mi parallel_n to_o the_o horizon_n it_o shall_v show_v the_o time_n when_o the_o day_n break_v and_o the_o end_n of_o the_o twilight_n for_o example_n of_o this_o projection_n let_v the_o place_n of_o the_o sun_n be_v the_o last_o degree_n of_o ♉_o the_o parallel_n pass_v through_o this_o place_n be_v ld_n and_o therefore_o the_o meridian_n altitude_n ml_o and_o the_o depression_n 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several_a metal_n and_o equal_a weight_n have_v the_o magnitude_n of_o the_o one_o to_o find_v the_o magnitude_n of_o the_o rest_n take_v the_o magnitude_n give_v out_o of_o the_o line_n of_o solid_n and_o to_o it_o open_v the_o sector_n in_o the_o point_n belong_v to_o the_o mettle_n give_v so_o the_o parallel_n take_v from_o between_o the_o point_n of_o the_o other_o metal_n and_o measure_v in_o the_o line_n of_o solid_n shall_v give_v the_o magnitude_n of_o their_o body_n thus_o have_v cube_n or_o sphere_n of_o equal_a weight_n but_o several_a metal_n we_o shall_v find_v that_o if_o those_o of_o tin_n contain_v 10000_o d_o â—he_v other_o of_o iron_n will_v contain_v 9250_o those_o of_o copper_n 8222_o those_o of_o silver_n 7161_o those_o of_o lead_v 6435_o those_o full_a of_o quicksilver_n 5453_o and_o those_o of_o gold_n 3895._o 2_o in_o like_a body_n of_o several_a metal_n and_o equal_a magnitude_n have_v the_o weight_n of_o one_o to_o find_v the_o weight_n of_o the_o rest_n 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distance_n from_o the_o end_n of_o the_o sector_n as_o in_o the_o sector_n of_o fourteen_o inch_n long_o where_o we_o find_v 1_o and_o 13_o it_o show_v that_o division_n to_o be_v 1_o inch_n from_o the_o near_a end_n and_o 13_o inch_n from_o the_o far_a end_n of_o the_o sector_n the_o other_o half_o contain_v a_o line_n of_o lesser_a tangent_n to_o which_o the_o gnomon_n be_v radius_fw-la they_o be_v here_o continue_v to_o 75_o gr_n and_o if_o there_o be_v need_n to_o produce_v they_o far_o take_v 45_o out_o of_o the_o number_n of_o degree_n require_v and_o double_a the_o remainder_n so_o the_o tangent_fw-la and_o secant_fw-la of_o this_o double_a remainder_n be_v add_v shall_v make_v up_o the_o tangent_fw-la of_o the_o degree_n require_v as_o if_o ab_fw-la be_v the_o radius_fw-la and_o bc_n the_o tangent_fw-la line_n it_o be_v require_v to_o find_v the_o tangent_fw-la of_o 75_o gr_n if_o we_o take_v 45_o gr_n out_o of_o 75_o gr_n the_o remainder_n be_v 30_o gr_n and_o the_o double_a 60_o gr_n who_o tangent_fw-la be_v bd_o and_o the_o secant_fw-la be_v ad_fw-la if_o then_o we_o add_v ad_fw-la to_o bd_o it_o make_v bc_n the_o tangent_fw-la of_o 75_o gr_n which_o be_v require_v in_o like_a sort_n the_o secant_fw-la of_o 61_o gr_n add_v to_o the_o tangent_fw-la of_o 61_o gr_n give_v the_o tangent_fw-la of_o 75_o gr_n 30_o m._n and_o the_o secant_fw-la of_o 62_o gr_n add_v to_o the_o tangent_fw-la of_o 62_o gr_n give_v the_o tangent_fw-la of_o 76_o gr_n and_o so_o in_o the_o rest_n the_o use_n of_o this_o line_n may_v be_v to_o observe_v the_o altitude_n of_o the_o sun_n hold_v the_o sector_n so_o as_o the_o tangent_fw-la bc_n may_v be_v vertical_a and_o the_o gnomon_n basilius_n parallel_v to_o the_o horizon_n then_o turn_v the_o gnomon_n towards_o the_o sun_n so_o that_o it_o may_v cast_v a_o shadow_n upon_o the_o tangent_fw-la and_o the_o end_n of_o the_o shadow_n shall_v show_v the_o altitude_n of_o the_o sun_n so_o if_o the_o end_n of_o the_o gnomon_n at_o a_o do_v give_v a_o shadow_n unto_o h_n it_o show_v that_o the_o altitude_n be_v 38_o gr_n ½_n if_o unto_o d_o then_o 60_o gr_n and_o so_o in_o the_o rest_n there_o be_v another_o use_n of_o this_o tangent_fw-la line_n for_o the_o draw_v of_o the_o hour_n line_n upon_o any_o ordinary_a plane_n whereof_o i_o will_v set_v down_o these_o proposition_n 1_o to_o draw_v the_o hour_n line_n upon_o a_o horizontall_a plane_n 2_o to_o draw_v the_o hour_n line_n upon_o a_o direct_a vertical_a plane_n first_o draw_v a_o right_a line_n ac_fw-la for_o the_o horizon_n and_o the_o equator_fw-la and_o cross_v it_o at_o the_o point_n a_o about_o the_o middle_n of_o the_o line_n with_o ab_fw-la another_o right_a line_n which_o may_v serve_v for_o the_o meridian_n and_o the_o hour_n of_o 12_o then_o take_v out_o 15_o gr_n out_o of_o the_o tangent_n and_o prick_v they_o down_o in_o the_o equator_fw-la on_o both_o side_n from_o 12_o so_o the_o one_o point_n shall_v serve_v for_o the_o hour_n of_o 11_o and_o the_o other_o for_o the_o hour_n of_o 1._o again_o take_v out_o the_o tangent_fw-la of_o 30_o gr_n and_o prick_v it_o down_o in_o the_o equator_fw-la on_o both_o side_n from_o 12_o so_o the_o one_o of_o these_o point_n shall_v serve_v for_o the_o hour_n of_o 10_o and_o the_o other_o for_o the_o hour_n of_o 2._o in_o like_a manner_n may_v you_o prick_v down_o the_o tangent_fw-la of_o 45_o gr_n for_o the_o hour_n of_o 9_o and_o 3_o and_o the_o tangent_fw-la of_o 60_o gr_n for_o the_o hour_n of_o 8_o and_o 4_o and_o the_o tangent_fw-la of_o 75_o gr_n for_o the_o hour_n of_o 7_o and_o 5._o
of_o may_n if_o you_o observe_v and_o find_v the_o low_a sight_n to_o stay_v at_o 30_o gr_n on_o the_o â—ledgeâ—acke_n of_o the_o bow_n such_o be_v the_o altitude_n for_o the_o declination_n â—ledgeâ—_n 20_o gr_n northward_o the_o altitude_n of_o the_o sun_n between_o the_o â—ledgeâ—wo_a sight_n 40_o gr_n the_o altitude_n of_o the_o equator_fw-la 60_o gr_fw-la and_o thereâ—ledgeâ—re_o the_o latitude_n 30_o gr_n as_o in_o the_o seven_o prop._n 12_o to_o find_v the_o day_n of_o the_o month_n by_o know_v the_o latitude_n of_o the_o place_n and_o observe_v the_o meridian_n altitude_n of_o the_o sun_n place_v your_o three_o sight_n according_a to_o your_o latitude_n the_o hoâ—ledgeâ—zontall_a sight_n to_o the_o centre_n the_o low_a sight_n to_o the_o latitude_n â—ledgeâ—d_v the_o upper_a sight_n among_o the_o month_n then_o when_o the_o â—ledgeâ—nne_n come_v to_o the_o meridian_n observe_v the_o altitude_n look_v â—ledgeâ—_n the_o low_a sight_n through_o the_o horizontall_a and_o keep_v the_o â—ledgeâ—wer_a sight_n still_o at_o the_o latitude_n but_o move_v the_o upper_a sight_n â—ledgeâ—til_o it_o give_v shadow_n upon_o the_o middle_n of_o the_o horizontal_a sight_n â—ledgeâ—_n the_o upper_a sight_n shall_v show_v the_o day_n of_o the_o month_n requiâ—ledgeâ—d_v thus_o in_o our_o latitude_n if_o you_o set_v the_o low_a sight_n to_o 51_o gr_n 30_o â—ledgeâ—_n and_o observe_v find_v the_o altitude_n of_o the_o sun_n between_o â—ledgeâ—at_n and_o the_o upper_a sight_n to_o be_v 28_o gr_n 30_o m._n this_o upper_a sight_n â—ledgeâ—ll_v â—all_n upon_o the_o nine_o of_o october_n and_o the_o twelve_o of_o feâ—ledgeâ—uarâ—e_n and_o if_o yet_o you_o doubt_v which_o oâ—_n they_o two_o be_v the_o day_n â—ledgeâ—u_o may_v expect_v another_o meridian_n altitude_n and_o than_o if_o you_o â—ledgeâ—d_v the_o upper_a sight_n upon_o the_o ten_o of_o october_n and_o the_o eleâ—ledgeâ—nth_n of_o february_n the_o question_n will_v be_v soon_o resolve_v 13_o to_o find_v the_o declination_n of_o any_o unknown_a star_n and_o so_o to_o place_v it_o on_o the_o bow_n by_o know_v the_o latitude_n of_o the_o place_n and_o observe_v the_o meridian_n altitude_n of_o the_o star_n when_o you_o find_v a_o star_n in_o the_o meridian_n that_o be_v fit_a for_o obâ—ledgeâ—uation_n set_a the_o centre_n of_o the_o bow_n to_o your_o eye_n the_o low_a â—ledgeâ—ht_n to_o the_o latitude_n and_o move_v the_o upper_a sight_n up_o or_o down_o â—ledgeâ—till_o you_o see_v the_o horizon_n by_o the_o low_a sight_n and_o the_o star_n by_o the_o upper_a sight_n then_o will_v the_o upper_a sight_n stay_v at_o the_o declination_n and_o place_n of_o the_o star_n thus_o be_v in_o 20_o gr_n of_o north_n latitude_n if_o you_o observe_v anâ—redgeâ—_n find_v the_o meridian_n altitude_n of_o the_o head_n of_o the_o crosier_n to_o bâ—redgeâ—_n 14_o gr_n 50_o m._n the_o upper_a sight_n will_v stay_v at_o 34_o gr_n 50_o m._n and_o therâ—redgeâ—_n may_v you_o place_v this_o star_n for_o by_o this_o observation_n the_o distance_n oâ—redgeâ—_n this_o star_n from_o the_o south_n pole_n shall_v be_v 34_o gr_n 50_o m._n and_o thâ—redgeâ—_n declination_n from_o the_o equator_fw-la 55_o gr_fw-la 10_o m._n and_o so_o for_o the_o resâ—redgeâ—_n the_o star_n which_o i_o mâ—ntioneâ—_n beâ—â—re_o do_v come_v to_o the_o meridian_n in_o this_o order_n after_o the_o first_o point_n of_o aries_n  _fw-fr ho._n mi._n the_o pole_n star_n at_o 0_o 29_o the_o ram_n head_n 1_o 46_o the_o head_n of_o medusa_n 2_o 44_o the_o sidâ—_n of_o perseus_n 2_o 58_o the_o bull_n eye_n 4_o 15_o the_o goat_n 4_o 49_o orion_n left_a shoulder_n 5_o 5_o orion_n girdle_n the_o first_o the_o second_o the_o three_o 5_o 13_o 5_o 17_o 5_o 22_o orion_n right_a shoulder_n 5_o 35_o the_o great_a dog_n 6_o 29_o castor_n 7_o 10_o the_o little_a dog_n 7_o 20_o pollux_n 7_o 22_o the_o hydra_n hart_n 9_o 9_o the_o lion_n hart_n 9_o 48_o the_o great_a bear_n back_o 10_o 40_o first_o in_o gr_n bear_v tail_n 12_o 37_o the_o virgin_n spike_n 13_o 5_o second_o in_o gr_n beaâ—_n tail_n 13_o 9_o third_z in_o gr_n bear_v tail_n 13_o 33_o arcturus_n 13_o 58_o the_o foremost_a gum_n rd_o 14_o 52_o the_o north_n crown_v 15_o 19_o thâ—_n hâ—_n dmost_a guard_n 15_o 25_o scorpion_n hart_n 16_o 7_o the_o harp_n 18_o 24_o praetor_n hart_n 19_o 33_o swan_n tail_n 20_o 29_o fomahant_n 22_o 36_o the_o end_n of_o the_o second_o book_n the_o three_o book_n of_o the_o use_n of_o the_o line_n of_o number_n sin_n and_o tangent_n for_o the_o draw_v of_o houre-line_n on_o all_o sort_n of_o planes_n there_o be_v ten_o several_a sort_n of_o planes_n which_o take_v their_o denomination_n from_o those_o great_a circle_n to_o which_o they_o be_v parallel_n and_o may_v sufficient_o for_o our_o use_n be_v represent_v in_o this_o one_o fundamental_a diagram_n describe_v before_o in_o the_o use_n of_o the_o sector_n and_o be_v know_v by_o their_o horizontall_a and_o perpendicular_a line_n of_o such_o as_o know_v the_o latitude_n of_o the_o place_n and_o the_o circle_n of_o the_o sphere_n 1_o a_o horizontall_a plane_n parallel_v to_o the_o horizon_n here_o represent_v by_o the_o outward_a circle_n esw_v 2_o a_o vertical_a plane_n parallel_v to_o the_o prime_n vertical_a circle_n which_o pass_v through_o the_o zenith_n and_o the_o point_n of_o east_n and_o west_n in_o the_o horizon_n and_o be_v right_n to_o the_o horizon_n and_o the_o meridian_n that_o be_v make_v right_a angle_n with_o they_o both_o this_o be_v represent_v by_o ezw_n 3_o a_o polar_n plane_n parallel_v to_o the_o circle_n of_o the_o hour_n of_o 6_o which_o pass_v through_o the_o pole_n and_o the_o point_n of_o east_n and_o west_n be_v right_a to_o the_o equinoctial_a and_o the_o meridian_n but_o incline_v to_o the_o horizon_n with_o a_o angle_n equal_a to_o the_o latitude_n this_o be_v here_o represent_v by_o epw_n 4_o a_o equinoctial_a plane_n parallel_v to_o the_o equinoctial_a which_o pass_v through_o the_o point_n of_o east_n and_o west_n be_v right_a to_o the_o meridian_n but_o incline_v to_o the_o horizon_n with_o a_o angle_n equal_a to_o the_o compliment_n of_o the_o latitude_n this_o be_v here_o represent_v by_o eaw_n 6_o a_o meridian_n plane_n parallel_v to_o the_o meridian_n the_o circle_n of_o the_o hour_n of_o 12_o which_o pass_v through_o the_o zenith_n the_o pole_n and_o the_o point_n of_o south_n and_o north_n be_v right_a to_o the_o horizon_n and_o the_o prime_n vertical_a this_o be_v here_o represent_v by_o szn_n 7_o a_o meridian_n plane_n incline_v to_o the_o horizon_n parallel_n to_o any_o great_a circle_n which_o pass_v through_o the_o point_n of_o south_n and_o north_n be_v right_a to_o the_o prime_n vertical_a but_o incline_v to_o the_o horizon_n this_o be_v here_o represent_v by_o sgn_n 8_o a_o vertical_a decline_a plane_n parallel_n to_o any_o great_a circle_n which_o pass_v through_o the_o zenith_n be_v right_a to_o the_o horizon_n but_o incline_v to_o the_o meridian_n this_o be_v represent_v by_o bzd._n 9_o a_o polar_n decline_a plane_n parallel_n to_o any_o great_a circle_n which_o pass_v through_o the_o pole_n be_v right_a to_o the_o equinoctial_a but_o incline_v to_o the_o meridian_n this_o be_v here_o represent_v by_o hpq_n 10_o a_o decline_a incline_a plane_n parallel_n to_o any_o great_a circle_n which_o be_v right_a to_o none_o of_o the_o former_a circle_n but_o decline_v from_o the_o prime_n vertical_a and_o incline_v both_o to_o the_o horizon_n and_o the_o meridian_n and_o all_o the_o hour_n circle_n this_o may_v be_v here_o represent_v either_o by_o bmd_a or_o bfd_n or_o bkd_v or_o any_o such_o great_a circle_n which_o pass_v neither_o through_o the_o south_n and_o north_n nor_o east_n and_o west_n point_n nor_o through_o the_o zenith_n nor_o the_o pole_n each_o of_o these_o plane_n except_o the_o horizontall_a have_v two_o face_n whereon_o houre-line_n may_v be_v draw_v and_o so_o there_o be_v 19_o plane_n in_o all_o the_o meridian_n plane_n have_v one_o face_n to_o the_o east_n and_o another_o to_o the_o west_n the_o other_o vertical_a plane_n have_v one_o to_o the_o south_n and_o another_o to_o the_o north_n and_o the_o rest_n one_o to_o the_o zenith_n and_o another_o to_o the_o nadir_n but_o what_o be_v say_v of_o the_o one_o may_v be_v understand_v of_o the_o other_o to_o find_v the_o inclination_n of_o any_o plane_n for_o the_o distinguish_n of_o these_o plane_n we_o may_v find_v whether_o they_o be_v horizontall_a or_o vertical_a or_o incline_v to_o the_o horizon_n and_o how_o much_o they_o incline_v either_o by_o the_o usual_a inclinatorie_n quadrant_a or_o by_o fit_v a_o thread_n and_o plummet_n unto_o the_o sector_n for_o let_v the_o sector_n be_v open_v to_o a_o right_a angle_n the_o line_n of_o sin_n to_o a_o angle_n of_o 92_o gr_n the_o inward_a edge_n of_o the_o sector_n to_o 90_o
divide_v the_o line_n of_o solid_n see_v like_o solid_n do_v hold_v in_o the_o proportion_n of_o their_o homologall_a side_n triplicate_v if_o you_o shall_v find_v two_o mean_a proportional_o between_o the_o whole_a side_n &_o each_o thousand_o part_n of_o the_o like_a side_n all_o of_o they_o cut_v the_o same_o two_o right_a line_n the_o former_a of_o those_o line_n so_o cut_v shall_v contain_v the_o division_n require_v wherefore_o upon_o the_o centre_n a_o &_o semidiameter_n equal_a to_o the_o line_n of_o line_n describe_v a_o circle_n and_o divide_v it_o into_o 4_o equal_a part_n cebd_a draw_v the_o cross_a diameter_n cb_n ed._n then_o divide_v the_o semidiameter_n ac_fw-la first_o into_o 10_o equal_a part_n and_o between_o the_o whole_a line_n ad_fw-la &_o of_o the_o ten_o part_n of_o ac_fw-la seek_v out_o two_o mean_v proportional_a line_n a_z and_z ah_o again_o between_o ad_fw-la and_o agnostus_n be_v two_o ten_o part_n of_o ac_fw-la seek_v out_o two_o mean_a proportional_n all_o and_o ak_v and_o so_o forward_o in_o the_o rest_n so_o shall_v the_o line_n ab_fw-la be_v divide_v into_o 10_o unequal_a part_n second_o divide_v each_o ten_o part_n of_o the_o line_n ac_fw-la into_o 10_o more_o and_o between_o the_o whole_a line_n ad_fw-la and_o each_o of_o they_o seek_v out_o two_o mean_a proportional_o as_o before_o so_o shall_v the_o line_n ab_fw-la be_v divide_v now_o into_o a_o hundred_o unequal_a part_n three_o if_o the_o length_n will_v bear_v it_o subdivide_v the_o line_n ac_fw-la once_o again_o each_o part_n into_o ten_o more_o and_o between_o the_o whole_a line_n ad_fw-la and_o each_o subdivision_n seek_v two_o mean_a proportional_o as_o before_o so_o shall_v the_o line_n ab_fw-la be_v now_o divide_v into_o 1000_o part_n but_o the_o ruler_n be_v short_a it_o shall_v suffice_v if_o those_o 10_o which_o be_v near_a the_o centre_n be_v express_v the_o rest_n be_v understand_v to_o be_v so_o divide_v though_o actual_o they_o be_v divide_v into_o no_o more_o than_o 5_o or_o 2._o and_o this_o line_n ab_fw-la so_o divide_v shall_v be_v the_o line_n of_o solid_n and_o must_v be_v transfer_v into_o the_o sector_n but_o let_v the_o number_n set_v to_o they_o be_v only_o 1.1.1.2.3_o etc._n etc._n unto_o 10._o as_o in_o the_o example_n 5_o to_o divide_v the_o line_n of_o sines_n and_o tangent_n on_o the_o side_n of_o the_o sector_n upon_o the_o centre_n a_o and_o semidiameter_n equal_a to_o the_o line_n of_o line_n describe_v a_o semicircle_n abcd_v with_o ab_fw-la perpendicular_a to_o the_o diameter_n cd_o then_o divide_v the_o quadrant_n cb_n bd_o each_o of_o they_o into_o 90._o and_o subdivide_v each_o degree_n into_o 2_o part_n for_o so_o if_o straight_a line_n be_v draw_v parallel_n to_o the_o diameter_n cd_o through_o these_o 90_o and_o their_o subdivision_n they_o shall_v divide_v the_o perpendicular_a ab_fw-la unequal_o into_o 90._o and_o this_o line_n ab_fw-la so_o divide_v shall_v be_v the_o line_n of_o sines_n and_o must_v be_v transfer_v into_o the_o sector_n the_o number_n set_v to_o they_o be_v to_o be_v 10.20.30_o etc._n etc._n unto_o 90_o as_o in_o the_o example_n if_o now_o in_o the_o point_n d_o unto_o the_o diameter_n cd_o we_o shall_v raise_v a_o perpendicular_a de_fw-fr and_o to_o it_o draw_v straight_a line_n from_o the_o centre_n a_o through_o each_o degree_n of_o the_o quadrant_a db._n this_o perpendicular_a so_o divide_v by_o they_o shall_v be_v the_o line_n of_o tangent_n &_o must_v be_v transfer_v unto_o the_o side_n of_o the_o sector_n the_o number_n set_v to_o they_o be_v to_o be_v 10.20.30_o etc._n etc._n as_o in_o the_o example_n if_o between_o a_o and_o d_o another_o straight_a line_n gf_n be_v draw_v parallel_n to_o de_fw-fr it_o will_v be_v divide_v by_o those_o line_n from_o the_o centre_n in_o like_a sort_n as_o de_fw-fr be_v divide_v and_o it_o may_v serve_v for_o a_o lesser_a line_n of_o tangent_n to_o be_v set_v on_o the_o edge_n of_o the_o sector_n these_o line_n of_o sines_n and_o tangent_n may_v yet_o otherwise_o be_v transfer_v into_o the_o sector_n out_o of_o the_o line_n of_o line_n or_o rather_o out_o of_o a_o diagonall_a scale_n equal_a to_o the_o line_n of_o line_n by_o table_n of_o sines_n and_o tangent_n in_o like_a manner_n may_v the_o line_n of_o superficies_n be_v transfer_v by_o table_n of_o square_a root_n and_o the_o line_n of_o solid_n by_o table_n of_o cubique_fw-la root_n which_o i_o leave_v to_o other_o to_o extract_v at_o leisure_n 6_o to_o show_v the_o ground_n of_o the_o sector_n let_v ab_fw-la ac_fw-la represent_v the_o leg_n of_o the_o sector_n then_o sever_n these_o two_o ab_fw-la ac_fw-la be_v equal_a and_o their_o section_n ad_fw-la ae_n also_o equal_a they_o shall_v be_v cut_v proportional_o and_o if_o we_o draw_v the_o line_n bc_n de_fw-fr they_o will_v be_v parallel_n by_o the_o second_o pro._n 6_o lib._n of_o euclid_n and_o so_o the_o triangle_n abc_n ade_n shall_v be_v equiangle_n by_o reason_n of_o the_o common_a angle_n 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sector_n in_o d_o and_o e_o number_v with_o 75_o wherefore_o i_o conclude_v the_o line_n ad_fw-la or_o ae_n be_v the_o four_o proportional_a and_o the_o correspondent_a number_n 75_o which_o be_v require_v but_o if_o i_o will_v enter_v on_o the_o side_n of_o the_o sector_n then_o will_v i_o dispose_v the_o line_n of_o the_o first_o denomination_n a_o and_o c_o in_o the_o line_n of_o line_n on_o both_o side_n of_o the_o sector_n in_o ab_fw-la ac_fw-la &_o in_o ad_fw-la ae_n so_o as_o they_o shall_v all_o meet_v in_o the_o centre_n a_o and_z then_z taking_z b_o the_o line_n of_o the_o second_o denomination_n put_v it_o over_o as_o a_o parallel_n in_o bc_n that_o it_o may_v be_v the_o basis_n of_o the_o isocheles_a triangle_n bac_n who_o side_n ab_fw-la ac_fw-la be_v equal_a to_o a_o the_o first_o line_n of_o the_o first_o denomination_n for_o so_o the_o sector_n be_v thus_o open_v the_o other_o parallel_n from_o d_o to_z e_z shall_v be_v the_o four_o proportional_a which_o be_v require_v 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between_o the_o line_n give_v as_o they_o be_v line_n by_o the_o five_o prop._n of_o line_n then_o open_a the_o sector_n in_o the_o line_n of_o superficies_n according_a to_o his_o number_n to_o the_o quantity_n of_o the_o one_o and_o a_o parallel_n take_v between_o the_o point_n of_o the_o number_n belong_v to_o the_o other_o line_n shall_v be_v the_o mean_a proportional_a let_v the_o line_n give_v be_v a_o and_o c._n the_o proportion_n between_o they_o as_o they_o be_v line_n will_v be_v find_v by_o the_o five_o prop._n of_o line_n to_o be_v as_o 4_o to_o 9_o wherefore_o i_o take_v the_o line_n c_o and_o put_v it_o over_o in_o the_o line_n of_o superficies_n between_o 9_o and_o 9_o and_o keep_v the_o sector_n at_o this_o angle_n his_o parallel_n between_o 4_o and_o 4_o do_v give_v i_o b_o for_o the_o mean_a proportional_a then_o for_o proof_n of_o the_o operation_n i_o may_v take_v this_o line_n b_o and_o put_v it_o over_o between_o 9_o and_o 9_o so_o his_o parallel_n between_o 4_o 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5_o 3_o  _fw-fr 37_o 34_o 32_o 29_o 26_o 24_o 21_o 19_o 16_o 13_o 11_o 8_o 5_o 3_o  _fw-fr 36_o 33_o 31_o 28_o 26_o 23_o 21_o 18_o 15_o 13_o 10_o 8_o 5_o 3_o  _fw-fr 35_o 32_o 30_o 27_o 25_o 22_o 20_o 17_o 15_o 12_o 10_o 7_o 5_o 2_o 3_o 34_o 30_o 29_o 26_o 24_o 22_o 19_o 17_o 14_o 12_o 10_o 7_o 5_o 2_o  _fw-fr 33_o 30_o 28_o 26_o 23_o 21_o 19_o 16_o 14_o 12_o 9_o 7_o 5_o 2_o  _fw-fr 31_o 29_o 27_o 24_o 22_o 20_o 18_o 16_o 13_o 11_o 9_o 7_o 4_o 2_o  _fw-fr 30_o 28_o 25_o 23_o 21_o 19_o 17_o 15_o 13_o 11_o 8_o 6_o 4_o 2_o 4_o 28_o 26_o 24_o 22_o 20_o 18_o 16_o 14_o 12_o 10_o 8_o 6_o 4_o 2_o  _fw-fr 27_o 25_o 23_o 21_o 19_o 17_o 15_o 13_o 11_o 10_o 8_o 6_o 4_o 2_o  _fw-fr 25_o 23_o 21_o 20_o 18_o 16_o 14_o 13_o 11_o 9_o 7_o 5_o 4_o 2_o  _fw-fr 23_o 22_o 20_o 18_o 17_o 15_o 13_o 12_o 10_o 8_o 7_o 5_o 3_o 2_o 5_o 22_o 20_o 18_o 17_o 15_o 14_o 12_o 11_o 9_o 8_o 6_o 5_o 3_o 2_o  _fw-fr 20_o 18_o 17_o 16_o 14_o 13_o 11_o 10_o 8_o 7_o 6_o 4_o 3_o 1_o  _fw-fr 18_o 17_o 15_o 14_o 13_o 12_o 10_o 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9_o or_o one_o of_o they_o in_o the_o point_n c_o then_o take_v out_o the_o semidiameter_n hence_v and_o prick_v it_o down_o in_o those_o parallel_n from_o c_o unto_o d_o and_o draw_v right_a line_n from_o a_o unto_o c_o and_o from_o five_o unto_o d_o the_o line_n vd_n shall_v be_v the_o hour_n of_o 6_o and_o if_o you_o divide_v these_o line_n ac_fw-la and_o dc_o in_o such_o sort_n as_o you_o divide_v the_o like_a line_n dc_o in_o the_o horizontall_a plane_n you_o shall_v have_v all_o the_o hour_n point_v require_v or_o you_o may_v find_v the_o point_n d_o in_o the_o hour_n of_o 6_o without_o knowledge_n either_o of_o h_n or_o c._n for_o have_v prick_v down_o hence_v in_o the_o meridian_n line_n and_o ae_n in_o the_o horizontall_a line_n and_o draw_v parallel_n to_o the_o meridian_n through_o the_o point_n at_o e_o you_o may_v take_v the_o tangent_fw-la of_o the_o latitude_n out_o of_o the_o sector_n and_o fit_v it_o over_o in_o the_o sin_n of_o 90_o and_o 90_o so_o the_o parallel_n 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the_o letter_n n_n divide_v unequal_o into_o 1000_o part_n and_o number_v with_o 1._o 2._o 3._o 4._o unto_o 10._o 2_o the_o line_n of_o artificial_a tangent_n be_v note_v with_o the_o letter_n t_o divide_v unequal_o into_o 45_o degree_n and_o number_v both_o way_n for_o the_o tangent_fw-la and_o the_o compliment_n 3_o the_o line_n of_o artificial_a sin_n note_v with_o the_o letter_n s_o divide_v unequal_o into_o 90_o degree_n and_o number_v with_o 1._o 2._o 3._o 4._o unto_o 90._o 4_o the_o line_n of_o verse_v sin_n for_o more_o easy_a find_v the_o hour_n and_o azimuth_n note_v with_o v_n divide_v unequal_o into_o about_o 164_o gr_n 50_o m._n number_v backward_o with_o 10._o 20._o 30._o unto_o 164._o thus_o there_o be_v seven_o line_n inscribe_v on_o the_o staff_n there_o be_v five_o line_n more_o inscribe_v on_o the_o cross_n 1_o a_o tangent_fw-la line_n of_o 36_o gr_n 3_o m._n number_v by_o 5._o 10._o 15._o â—ledgeâ—nto_fw-it 35_o the_o midst_n whereof_o be_v at_o 20_o gr_n and_o therefore_o i_o call_v it_o â—ledgeâ—he_v tangent_fw-la of_o 20_o and_o this_o have_v respect_n unto_o 20_o gr_n in_o the_o tangent_fw-la on_o the_o staff_n 2_o a_o tangent_fw-la line_n of_o 49_o gr_n 6_o m._n number_v by_o 5._o 10._o 15._o unâ—ledgeâ—o_fw-la 45_o the_o midst_n whereof_o be_v at_o 30_o gr_n and_o have_v respect_n unto_o â—ledgeâ—0_n gr_n in_o the_o tangent_fw-la on_o the_o staff_n whereupon_o i_o call_v it_o the_o â—ledgeâ—angent_fw-la of_o 30._o 3_o a_o line_n of_o inch_n number_v with_o 1._o 2._o 3._o unto_o 26_o each_o inch_n â—ledgeâ—qually_o subdivide_v into_o ten_o part_n answerable_a to_o the_o inch_n line_n â—ledgeâ—pon_o the_o staff_n 4_o a_o line_n of_o several_a chord_n one_o answerable_a to_o a_o circle_n of_o â—ledgeâ—welue_fw-mi inch_n semidiameter_n number_v with_o 10._o 20._o 30._o unto_o â—ledgeâ—0_n another_o to_o a_o semidiameter_n of_o a_o circle_n of_o six_o inch_n and_o â—ledgeâ—he_v three_o to_o a_o semidiameter_n of_o a_o circle_n of_o three_o inch_n both_o â—ledgeâ—umbred_a with_o 10._o 20._o 30._o unto_o 90._o 5_o a_o continuation_n of_o the_o meridian_n line_n from_o 57_o gr_n of_o laâ—ledgeâ—tude_n unto_o 76_o gr_n and_o from_o 76_o gr_n to_o 84_o gr_n for_o the_o inscription_n of_o these_o line_n the_o first_o for_o measure_n be_v â—ledgeâ—qually_o divide_v into_o inch_n and_o ten_o part_n of_o inch_n the_o tangent_fw-la on_o the_o staff_n for_o observation_n of_o angle_n with_o â—ledgeâ—e_fw-la tangent_fw-la of_o 20_o and_o the_o tangent_fw-la of_o 30_o on_o the_o cross_n may_v â—ledgeâ—l_v three_o be_v inscribe_v out_o of_o the_o ordinary_a table_n of_o tangent_n the_o â—ledgeâ—affe_n be_v 36_o inch_n in_o length_n the_o radius_fw-la for_o the_o tangent_fw-la â—ledgeâ—_n the_o staff_n will_v be_v 13_o
whether_o the_o two_o station_n be_v choose_v at_o the_o one_o end_n of_o the_o breadth_n propose_v or_o without_o it_o or_o within_o it_o if_o the_o line_n between_o the_o station_n be_v perpendicular_a unto_o the_o breadth_n as_o may_v appear_v if_o in_o stead_n of_o the_o station_n at_o a_o and_o h_n we_o make_v choice_n of_o the_o like_a station_n at_o i_o and_o k._n there_o may_v be_v other_o way_n propose_v to_o work_v these_o prop._n by_o hold_v the_o cross_n even_o with_o the_o distance_n and_o the_o staff_n parallel_v with_o the_o height_n but_o these_o will_v prove_v more_o troublesome_a and_o those_o which_o be_v deliver_v be_v sufficient_a and_o the_o same_o with_o those_o which_o other_o have_v set_v down_o under_o the_o name_n of_o the_o jacobs_n staff_n chap._n iii_o the_o use_n of_o the_o tangent_fw-la line_n in_o take_v of_o angles_n 1_o to_o find_v a_o angle_n by_o the_o tangent_fw-la on_o the_o staff_n let_v the_o middle_a sight_n be_v always_o set_v to_o the_o middle_n of_o the_o cross_n note_v with_o 20_o and_o 30_o and_o then_o the_o cross_n draw_v near_o the_o eye_n until_o the_o mark_n may_v be_v see_v close_o within_o the_o sight_n for_o so_o if_o the_o eye_n at_o a_o that_o end_n of_o the_o staff_n which_o be_v note_v with_o 90_o and_o 180_o behold_v the_o mark_n k_o and_o n_n between_o the_o two_o first_o sight_n c_o and_o b_o or_o the_o mark_n k_o and_o p_o between_o the_o two_o outward_a sight_n the_o cross_n be_v draw_v down_o unto_o h_n shall_v stand_v at_o 30_o and_o 60_o in_o the_o tangent_fw-la on_o the_o staff_n it_o show_v that_o the_o angle_n can_n be_v 30_o gr_n the_o angle_n kap_n 60_o gr_n the_o onâ—_n double_a to_o the_o other_o which_o be_v â—he_n reason_n of_o the_o double_a number_n on_o this_o line_n of_o the_o staff_n and_o this_o way_n will_v serve_v for_o any_o angle_n from_o 20_o gr_n towards_o 90_o gr_n or_o from_o 40_o gr_n towards_o 180_o gr_n but_o if_o the_o angle_n be_v less_o than_o 20_o gr_n we_o must_v then_o make_v use_n of_o the_o tangent_fw-la upon_o the_o cross_n 2_o to_o find_v a_o angle_n by_o the_o tangent_fw-la of_o 20_o upon_o the_o cross_n set_a 20_o unto_o 20_o that_o be_v the_o middle_a sight_n to_o the_o midst_n of_o the_o cross_n at_o the_o end_n of_o the_o staff_n note_v with_o 20_o so_o the_o eye_n at_o a_o behold_v the_o mark_n l_o and_o n_o close_o between_o the_o two_o first_o sight_n c_o and_o b_o shall_v seâ—_n they_o in_o a_o angle_n of_o 20_o gr_n if_o the_o mark_n shâ—ll_v be_v near_o together_o as_o be_v m_n and_o n_o then_o draw_v in_o the_o cross_n from_o c_o unto_o e_z if_o they_o be_v far_o asunder_o as_o be_v k_o and_o n_n then_o draw_v out_o the_o cross_n from_o c_o unto_o f_o so_o the_o quantity_n of_o the_o angle_n shall_v be_v still_o find_v in_o the_o cross_n in_o the_o tangent_fw-la of_o 20_o gr_n at_o the_o end_n of_o the_o staff_n and_o this_o will_v serve_v for_o any_o angle_n from_o 0_o gr_n towards_o 35_o gr_n 3_o to_o find_v a_o angle_n by_o the_o tangent_fw-la of_o 30_o upon_o the_o cross_n this_o tangent_fw-la of_o 30_o be_v here_o put_v the_o rather_o that_o the_o end_n of_o the_o staff_n rest_v at_o the_o eye_n the_o hand_n may_v more_o easy_o remove_v the_o cross_n for_o it_o suppose_v the_o radius_fw-la to_o be_v no_o long_o then_o ah_o which_o be_v from_o the_o eye_n at_o the_o end_n of_o the_o staff_n unto_o 30_o gr_n about_o 22_o inch_n and_o 7_o part_n wherefore_o here_o set_v the_o middle_a sight_n unto_o 30_o gr_n on_o the_o staff_n and_o then_o either_o draw_v the_o cross_n in_o or_o out_o until_o the_o mark_n be_v see_v between_o the_o two_o first_o sight_n so_o the_o quantity_n of_o the_o angle_n will_v be_v find_v in_o the_o tangent_fw-la of_o 30_o which_o be_v here_o represent_v by_o the_o line_n gh_o and_o this_o will_v serve_v for_o any_o angle_n from_o 0_o gr_n towards_o 48_o gr_n 4_o to_o observe_v the_o altitude_n of_o the_o sun_n backward_o here_o it_o be_v fit_a to_o have_v a_o horizontall_a sight_n set_v to_o the_o begin_n of_o the_o staff_n and_o then_o may_v you_o turn_v your_o back_n towards_o the_o sun_n and_o your_o cross_n towards_o your_o eye_n if_o the_o altitude_n be_v under_o 45_o gr_n set_v the_o middle_a sight_n to_o 30_o on_o the_o staff_n and_o look_v by_o the_o middle_a sight_n through_o the_o horizontall_a unto_o the_o horizon_n move_v the_o cross_n upward_o or_o downward_o until_o the_o upper_a sight_n do_v shadow_n the_o upper_a half_n of_o the_o horizontall_a sight_n so_o the_o altitude_n will_v be_v find_v in_o the_o tangent_fw-la of_o 30._o if_o the_o altitude_n shall_v be_v more_o than_o 45_o gr_n set_v the_o middle_a sight_n unto_o the_o midst_n of_o the_o cross_n and_o look_v by_o the_o inward_a edge_n of_o the_o low_a sight_n through_o the_o horizontall_a to_o the_o horizon_n move_v the_o middle_a sight_n in_o or_o out_o until_o the_o upper_a sight_n do_v shadow_v the_o upper_a half_n of_o the_o horizontall_a sight_n so_o the_o altitude_n will_v be_v find_v in_o the_o degree_n on_o the_o staff_n between_o 40_o and_o 180._o 5_o to_o set_v the_o staff_n to_o any_o angle_n give_v this_o be_v the_o converse_n of_o the_o former_a prop._n for_o if_o the_o middle_a sight_n be_v set_v to_o his_o place_n and_o degree_n the_o eye_n look_v close_o by_o the_o sight_n as_o before_o can_v but_o see_v his_o object_n in_o the_o angle_n give_v 6_o to_o observe_v the_o altitude_n of_o the_o sun_n another_o way_n set_a the_o middle_a sight_n to_o the_o middle_n of_o the_o cross_n and_o hold_v the_o horizontall_a sight_n downward_o so_o as_o the_o cross_n may_v be_v parallel_n to_o the_o horizon_n then_o be_v the_o staff_n vertical_a and_o if_o the_o outward_a sight_n of_o the_o cross_n do_v shadow_v the_o horizontall_a sight_n the_o compliment_n of_o the_o altitude_n will_v be_v find_v in_o the_o tangent_fw-la on_o the_o staff_n 7_o to_o observe_v a_o altitude_n by_o thread_n and_o plummet_n let_v the_o middle_a sight_n be_v set_v to_o the_o midst_n of_o the_o cross_n and_o to_o that_o end_n of_o the_o staff_n which_o be_v note_v with_o 90_o and_o 180_o then_o have_v a_o thread_n and_o a_o plummet_n at_o the_o begin_n of_o the_o cross_n and_o turn_v the_o cross_n upward_o and_o the_o staff_n towards_o the_o sun_n the_o thread_n will_v fall_v on_o the_o compliment_n of_o the_o altitude_n above_o the_o horizon_n and_o this_o may_v be_v apply_v to_o other_o purpose_n 8_o to_o apply_v the_o line_n of_o inch_n to_o the_o take_n of_o angle_n if_o the_o angle_n be_v observe_v between_o the_o two_o first_o sight_n there_o will_v be_v such_o proportion_n between_o the_o part_n of_o the_o staff_n and_o the_o part_n of_o the_o cross_n as_o between_o the_o radius_fw-la and_o the_o tangent_fw-la of_o the_o angle_n as_o if_o the_o part_n intercept_v on_o the_o staff_n be_v 20_o inch_n the_o part_n on_o the_o cross_n 9_o inch_n then_o by_o proportion_n as_o 20_o unto_o 9_o so_o 100000_o unto_o 45000_o the_o tangent_fw-la of_o 24_o gr_n 14_o m._n but_o if_o the_o angle_n shall_v be_v observe_v between_o the_o two_o outward_a sight_n the_o part_n be_v 20_o and_o 9_o as_o before_o the_o angle_n will_v be_v 48_o gr_n 28_o m._n double_a unto_o the_o former_a in_o all_o these_o there_o be_v a_o regard_n to_o be_v have_v to_o the_o parallax_n of_o the_o eye_n and_o his_o height_n above_o the_o horizon_n in_o observation_n at_o sea_n to_o the_o semidiameter_n of_o the_o sun_n his_o parallax_n and_o refraction_n as_o in_o the_o use_n of_o other_o staff_n and_o so_o this_o will_v be_v as_o much_o or_o more_o than_o that_o which_o have_v be_v heretofore_o perform_v by_o the_o crosse-staffe_n chap._n four_o the_o use_n of_o the_o line_n of_o equal_a part_n join_v with_o the_o line_n of_o chord_n the_o line_n of_o equal_a part_n do_v serve_v also_o for_o protraction_n as_o may_v appear_v by_o the_o former_a diagram_n but_o be_v join_v with_o the_o line_n of_o chord_n which_o i_o place_n upon_o one_o side_n of_o the_o cross_n they_o will_v far_o serve_v for_o the_o protraction_n and_o resolution_n of_o right_a line_n triangle_n whereof_o i_o will_v give_v one_o example_n in_o find_v of_o a_o distance_n at_o two_o station_n otherwise_o then_o in_o the_o second_o cap._n let_v the_o distance_n require_v be_v ab_fw-la at_o a_o the_o first_o station_n i_o make_v choice_n of_o a_o station_n line_n towards_o c_o and_o observe_v the_o angle_n bac_n by_o the_o tangent_fw-la line_n which_o may_v be_v 43_o gr_n 20_o m_o then_o have_v go_v a_o hundred_o pace_n towards_o c_o i_o make_v my_o second_o station_n at_o d_o
point_n i_o will_v fall_v in_o the_o latitude_n of_o 51_o gr_n 0_o m._n and_o the_o point_n k_o in_o the_o latitude_n of_o 51_o gr_n 30_o m._n but_o the_o longitude_n of_o i_o will_v be_v only_o 1_o gr_n 30_o m._n and_o the_o longitude_n of_o k_o only_o 3_o gr_n 57_o m._n which_o be_v 33_o m_o westward_o from_o the_o meridian_n of_o the_o place_n to_o which_o the_o ship_n be_v bind_v such_o be_v the_o difference_n between_o both_o these_o charts_n chap._n vi_o the_o use_n of_o the_o line_n of_o number_n 1_o have_v two_o number_n give_v to_o find_v a_o three_o in_o continual_a proportion_n a_o four_o a_o five_o and_o so_o forward_o extend_v the_o compass_n from_o the_o first_o number_n unto_o the_o second_o then_o may_v you_o turn_v they_o from_o the_o second_o to_o the_o three_o and_o from_o the_o three_o to_o the_o four_o and_o so_o forward_o let_v the_o two_o number_n give_v be_v 2_o and_o 4._o extend_v the_o compass_n from_o 2_o to_o 4_o then_o may_v you_o turn_v they_o from_o 4_o to_o 8_o and_o from_o 8_o to_o 16_o and_o from_o 16_o to_o 32_o and_o from_o 32_o to_o 64_o and_o from_o 64_o to_o 128._o or_o if_o the_o one_o foot_n of_o the_o compass_n be_v set_v to_o 64_o the_o other_o fall_v out_o of_o the_o line_n you_o may_v set_v it_o to_o another_o 64_o near_a the_o begin_n of_o the_o line_n and_o there_o the_o other_o foot_n will_v reach_v to_o 128_o and_o from_o 128_o you_o may_v turn_v they_o to_o 256_o and_o so_o forward_o or_o if_o the_o two_o first_o number_n give_v be_v 10_o and_o 9_o extend_v the_o compass_n from_o 10_o at_o the_o end_n of_o the_o line_n back_o unto_o 9_o then_o may_v you_o turn_v they_o from_o 9_o unto_o 8.1_o and_o from_o 8.1_o unto_o 7.29_o and_o so_o if_o the_o two_o first_o number_n give_v be_v 1_o and_o 9_o the_o three_o will_v be_v find_v to_o be_v 81_o the_o four_o 729_o with_o the_o same_o extent_n of_o the_o compass_n in_o the_o same_o manner_n if_o the_o two_o first_o number_n be_v 10_o and_o 12_o you_o may_v find_v the_o three_o proportional_a to_o be_v 14.4_o the_o four_o 17.28_o and_o with_o the_o same_o extent_n of_o the_o compass_n if_o the_o two_o first_o number_n be_v 1_o and_o 12_o the_o three_o will_v be_v find_v to_o be_v 144_o and_o the_o four_o to_o be_v 1728._o 2_o have_v two_o extreme_a number_n give_v to_o find_v a_o mean_a proportional_a between_o they_o divide_v the_o space_n between_o the_o extreme_a number_n into_o two_o equal_a part_n and_o the_o foot_n of_o the_o compass_n will_v stay_v at_o the_o mean_a proportional_a so_o the_o extreme_a number_n give_v be_v 8_o and_o 32_o the_o mean_a between_o they_o will_v be_v find_v to_o be_v 16_o which_o may_v be_v prove_v by_o the_o former_a prop._n where_o it_o be_v show_v that_o as_o 8_o to_o 16_o so_o be_v 16_o to_o 32._o 3_o to_o find_v the_o square_a root_n of_o any_o number_n give_v the_o square_a root_n be_v always_o the_o mean_a proportional_a between_o 1_o and_o the_o number_n give_v and_o therefore_o to_o be_v find_v by_o divide_v the_o space_n between_o they_o into_o two_o equal_a part_n so_o the_o root_n of_o 9_o be_v 3_o and_o the_o root_n of_o 81_o be_v 9_o and_o the_o root_n of_o 144_o be_v 12._o 4_o have_v two_o extreme_a number_n give_v to_o find_v two_o mean_a proportional_n between_o they_o divide_v the_o space_n between_o the_o two_o extreme_a number_n give_v into_o three_o equal_a part_n as_o if_o the_o extreme_a number_n give_v be_v 8_o and_o 27_o divide_v the_o space_n between_o they_o into_o three_o equal_a part_n the_o foot_n of_o the_o compass_n will_v stand_v in_o 12_o and_o 18._o 5_o to_o find_v the_o cubique_fw-la root_n of_o a_o number_n give_v the_o cubique_fw-la root_n be_v always_o the_o first_o of_o two_o mean_a proportional_n between_o 1_o and_o the_o number_n give_v and_o therefore_o to_o be_v find_v by_o divide_v the_o space_n between_o they_o into_o three_o equal_a part_n so_o the_o root_n of_o 1728_o will_v be_v find_v to_o be_v 12._o the_o root_n of_o 17280_o be_v almost_o 26_o and_o the_o root_n of_o 172800_o be_v almost_o 56._o 6_o to_o multiply_v one_o number_n by_o another_o extend_v the_o compass_n from_o 1_o to_o the_o multiplicator_n the_o same_o extent_n apply_v the_o same_o way_n shall_v reach_v from_o the_o multiplicand_a to_o the_o product_n as_o if_o the_o number_n to_o be_v multiply_v be_v 25_o and_o 30_o either_o extend_v the_o compass_n from_o 1_o to_o 25_o and_o the_o same_o extent_n will_v give_v the_o distance_n from_o 30_o to_o 750_o or_o extend_v they_o from_o 1_o to_o 30_o and_o the_o same_o extent_n shall_v reach_v from_o 25_o to_o 750._o 7_o to_o divide_v one_o number_n by_o another_o extend_v the_o compass_n from_o the_o divisor_n to_o 1_o the_o same_o extent_n shall_v reach_v from_o the_o dividend_n to_o the_o quotient_a so_o if_o 750_o be_v to_o be_v divide_v by_o 25_o the_o quotient_n will_v be_v find_v to_o be_v 30._o 8_o three_o number_n be_v give_v to_o find_v a_o four_o proportional_a this_o golden_a rule_n the_o most_o useful_a of_o all_o other_o be_v perform_v with_o like_a ease_n for_o extend_v the_o compass_n from_o the_o first_o number_n to_o the_o second_o the_o same_o extent_n shall_v give_v the_o distance_n from_o the_o three_o to_o the_o four_o as_o for_o example_n the_o proportion_n between_o the_o diameter_n and_o the_o circumference_n be_v say_v to_o be_v such_o as_o 7_o to_o 22_o if_o the_o diameter_n be_v 14_o how_o much_o be_v the_o circumference_n extend_v the_o compass_n from_o 7_o to_o 22_o the_o same_o extent_n shall_v give_v the_o distance_n from_o 14_o to_o 44_o or_o extend_v they_o from_o 7_o to_o 14_o and_o the_o same_o extent_n shall_v reach_v from_o 22_o to_o 44._o either_o of_o these_o way_n may_v be_v try_v on_o several_a place_n of_o this_o line_n but_o that_o place_n be_v best_a where_o the_o seete_fw-mi of_o the_o compass_n may_v stand_v near_o together_o 9_o three_o number_n be_v give_v to_o find_v a_o four_o in_o a_o duplicate_v proportion_n this_o proposition_n concern_v 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