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be_v describe_v out_o of_o the_o angular_a point_n at_o any_o other_o distance_n let_v the_o semidiameter_n be_v turn_v into_o a_o parallel_a chord_n of_o 60_o gr._n then_o take_v the_o chord_n of_o this_o ark_n and_o carry_v it_o parallel_v till_o it_o cross_v in_o like_a chord_n so_o the_o place_n where_o it_o stay_v shall_v give_v the_o quantity_n of_o the_o angle_n as_o in_o the_o former_a example_n if_o i_o make_v the_o semidiameter_n ab_fw-la a_o parallel_a chord_n of_o 60_o gr._n and_o then_o keep_v the_o sector_n at_o that_o angle_n carry_v the_o chord_n bc_n parallel_n till_o it_o stay_v in_o like_a chord_n i_o shall_v find_v it_o to_o stay_v in_o no_o other_o but_o 11_o gr._n 15_o m_n and_o such_o be_v the_o angle_n bac_n 10_o upon_o a_o right_a line_n and_o a_o point_n give_v in_o it_o to_o make_v a_o angle_n equal_a to_o any_o angle_n give_v first_o out_o of_o the_o point_n give_fw-mi describe_v a_o ark_n cut_v the_o same_o line_n then_o by_o the_o 5._o prop_v afore_o 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so_o draw_v shall_v be_v ellipticall_a and_o represent_v the_o houre-circle_n the_o meridian_n pes_z the_o hour_n of_o 12_o at_o noon_n that_o next_o unto_o it_o draw_v through_o 75_o gr_n from_o the_o centre_n the_o hour_n of_o 11_o and_o 1_o that_o which_o be_v draw_v through_o 60_o gr_n from_o the_o centre_n the_o hour_n of_o 10_o and_o 2._o etc._n etc._n then_o have_v respect_n unto_o the_o latitude_n we_o may_v number_v it_o from_o e_z northward_o unto_o z_o and_o there_o place_v the_o zenith_n by_o which_o and_o the_o centre_n the_o line_n draw_fw-mi zan_fw-mi shall_v represent_v
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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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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and_o it_o be_v require_v to_o find_v the_o radius_fw-la let_v bd_o be_v make_v a_o parallel_n sine_fw-la of_o 50_o gr._n by_o apply_v it_o over_o in_o the_o line_n of_o sines_n between_o 50_o and_o 50_o so_o his_o parallel_n radius_fw-la between_o 90_o and_o 90_o shall_v be_v ac_fw-la the_o radius_fw-la require_v 3_o the_o radius_fw-la of_o a_o circle_n or_o the_o right_n sine_fw-la of_o any_o ark_n be_v give_v and_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 let_v the_o radius_fw-la or_o right_a sine_fw-la give_v be_v turn_v into_o his_o parallel_n then_o take_v the_o right_a line_n give_v and_o carry_v it_o parallel_v to_o the_o former_a till_o it_o stay_v in_o like_a sin_n so_o the_o number_n of_o degree_n and_o minute_n where_o it_o stay_v shall_v give_v the_o quantity_n of_o the_o sine_fw-la require_v as_o if_o bd_o be_v the_o give_v sine_fw-la of_o 50_o gr._n and_o bg_n the_o straight_a line_n give_v first_o i_o 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gr_n as_o if_o ac_fw-la be_v the_o radius_fw-la give_v it_o be_v require_v to_o find_v the_o verse_v sine_fw-la of_o 130_o gr_n here_o the_o excess_n above_o 90_o gr_n be_v 40_o gr_n and_o therefore_o the_o verse_v sine_z require_v be_v equal_a to_o the_o radius_fw-la ac_fw-la and_o a_o 40_o both_o be_v set_v together_o 5_o the_o diameter_n or_o radius_fw-la be_v give_v to_o find_v the_o chord_n of_o every_o ark_n the_o sin_n may_v be_v fit_v many_o way_n to_o serve_v for_o chord_n 1_o a_o sine_fw-la be_v the_o half_a of_o the_o chord_n of_o the_o double_a ark_n if_o the_o sine_fw-la be_v double_v it_o give_v the_o chord_n of_o the_o double_a ark_n a_o sine_fw-la of_o 10_o gr_n double_a give_v a_o chord_n of_o 20_o gr_n and_o a_o sine_fw-la of_o 15_o gr_n be_v double_v give_v a_o chord_n of_o 30_o gr_n and_o so_o in_o the_o rest_n as_o here_o bd_o the_o sine_z of_o bc_n a_o ark_n of_o 40_o gr_n be_v double_v give_v be_v the_o chord_n of_o be_v which_o be_v a_o ark_n of_o 80_o gr_n wherefore_o if_o the_o radius_fw-la of_o the_o circle_n give_v be_v equal_a to_o the_o lateral_a radius_fw-la let_v the_o sector_n be_v open_v near_o unto_o his_o length_n so_o that_o both_o the_o line_n of_o sines_n may_v make_v but_o one_o direct_a line_n so_o the_o distance_n on_o the_o sin_n between_o 10_o and_o 10_o shall_v be_v a_o chord_n of_o 20_o the_o distance_n between_o 20_o and_o 20_o shall_v be_v a_o chord_n of_o 40_o and_o the_o distance_n between_o 30_o and_o 30_o shall_v be_v a_o chord_n of_o 60_o and_o so_o in_o the_o rest_n 2_o because_o a_o sine_fw-la be_v the_o half_a of_o the_o chord_n of_o the_o double_a ark_n the_o proportion_n hold_v as_o the_o diameter_n fh_v unto_o the_o radius_fw-la ah_o so_o the_o chord_n be_v unto_o the_o sine_fw-la de_fw-fr or_o the_o chord_n gl_n unto_o the_o sine_fw-la all_o and_o then_o if_o the_o radius_fw-la ah_o be_v put_v for_o the_o diameter_n which_o be_v a_o chord_n of_o 180_o gr_n the_o sine_fw-la de_fw-fr or_o all_o shall_v serve_v for_o a_o chord_n of_o 80_o gr_n and_o the_o semiradius_fw-la which_o be_v the_o sine_fw-la of_o 30_o gr_n shall_v serve_v for_o a_o chord_n of_o 60_o gr_n and_o go_v for_o the_o semidiameter_n of_o a_o circle_n and_o so_o in_o the_o rest_n so_o that_o by_o these_o mean_v we_o shall_v not_o need_v to_o double_v the_o line_n of_o sin_n as_o before_o but_o only_o to_o double_v the_o number_n and_o to_o this_o purpose_n i_o have_v subdivide_v each_o degree_n of_o the_o sin_n into_o two_o that_o so_o they_o may_v show_v how_o far_o the_o half_a degree_n do_v reach_v in_o the_o
the_o vertical_a circle_n pass_v through_o the_o zenith_n and_o nadir_n east_n and_o west_n and_o the_o line_n mah_n cross_v it_o at_o right_a angle_n shall_v represent_v the_o horizon_n these_o two_o be_v divide_v in_o the_o same_o sort_n as_o the_o ecliptic_a and_o the_o equator_fw-la the_o line_n draw_v through_o each_o degree_n of_o the_o semidiameter_n az_n parallel_n to_o the_o horizon_n shall_v be_v the_o circle_n of_o altitude_n and_o the_o division_n in_o the_o horizon_n and_o his_o parallel_n shall_v give_v the_o azimuth_n last_o if_o through_o 18_o gr_n in_o a_o be_v draw_v a_o right_a line_n ik_fw-mi parallel_n to_o the_o horizon_n it_o shall_v show_v the_o time_n when_o the_o day_n break_v and_o the_o end_n of_o the_o twilight_n for_o example_n of_o this_o projection_n let_v the_o place_n of_o the_o sun_n be_v the_o last_o degree_n of_o ♉_o the_o parallel_n pass_v through_o this_o place_n be_v ld_n and_o therefore_o the_o meridian_n altitude_n ml_o and_o the_o depression_n 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e_o 360_o at_o p._n the_o semidiameter_n ap_n aae_n as_o a_o e_o may_v be_v divide_v according_a to_o the_o tangent_n of_o half_a their_o arke_n that_o be_v a_o tangent_fw-la of_o 45_o gr_n which_o be_v always_o equal_a to_o the_o radius_fw-la shall_v give_v the_o semidiameter_n of_o 90_o gr_n a_o tangent_fw-la of_o 40_o gr_n shall_v give_v 80_o gr_n in_o the_o semidiameter_n a_o tangent_fw-la of_o 35_o gr_n shall_v give_v 70._o etc._n etc._n so_o that_o the_o semidiameter_n may_v be_v divide_v in_o such_o sort_n as_o the_o tangent_fw-la on_o the_o side_n of_o the_o sector_n the_o difference_n be_v only_o in_o their_o number_n have_v divide_v the_o circumference_n and_o the_o semidiameter_n we_o may_v easy_o draw_v the_o meridians_fw-la and_o the_o parallel_n by_o the_o help_n of_o the_o sector_n the_o meridians_fw-la be_v to_o be_v draw_v through_o both_o the_o pole_n p_o and_o s_o and_o the_o degree_n before_o graduate_v in_o the_o equator_fw-la the_o distance_n of_o the_o centre_n 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distance_n of_o the_o centre_n of_o each_o parallel_n from_o a_o the_o centre_n of_o the_o plane_n be_v equal_a to_o the_o secant_fw-la of_o the_o same_o parallel_n from_o the_o pole_n and_o the_o semidiameter_n equal_a to_o the_o tangent_fw-la of_o the_o same_o degree_n as_o if_o i_o shall_v draw_v the_o parallel_n of_o 80_o gr_n which_o be_v the_o ten_o from_o the_o pole_n s_o first_o i_o open_v the_o compass_n unto_o ac_fw-la the_o tangent_fw-la of_o 10_o gr_n and_o this_o give_v i_o the_o semidiameter_n of_o this_o parallel_n who_o centre_n be_v a_o little_a from_o s_n in_o such_o distance_n as_o the_o secant_fw-la sc_n be_v long_o than_o the_o radius_fw-la sa_n the_o meridians_fw-la and_o parallel_n be_v draw_v if_o we_o number_v 23_o gr_n 30_o m._n from_o e_z to_z ♋_o northward_o from_o ae_n to_o ♑_o southward_n the_o line_n draw_v from_o ♋_o to_o ♑_o shall_v be_v the_o ecliptic_a which_o be_v divide_v in_o such_o sort_n as_o the_o semidiameter_n ap_n the_o first_o 30_o gr_n from_o a_o to_o ♋_o shall_v stand_v for_o the_o sine_z of_o ♈_o the_o 30_o gr_n next_o follow_v for_o ♉_o the_o rest_n for_o ♊_o ♋_o ♌_o etc._n etc._n in_o their_o order_n if_o far_o we_o have_v respect_n unto_o the_o latitude_n we_o may_v number_v it_o from_o e_z northward_o unto_o z_o and_o there_o place_v the_o zenith_n by_o which_o and_o the_o centre_n the_o line_n draw_fw-mi zan_fw-mi shall_v represent_v the_o vertical_a circle_n and_o the_o line_n mah_n cross_v it_o at_o right_a angle_n shall_v represent_v the_o horizon_n and_o these_o divide_v in_o the_o same_o sort_n as_o ap_n the_o circle_n draw_v through_o each_o degree_n of_o the_o semidiameter_n az_n parallel_n to_o the_o horizon_n shall_v be_v the_o circle_n of_o altitude_n and_o the_o circle_n draw_v through_o the_o horizon_n and_o his_o pole_n shall_v give_v the_o azimuth_n for_o example_n of_o this_o projection_n let_v the_o place_n of_o the_o sun_n be_v in_o the_o begin_n of_o ♒_o the_o parallel_n pass_v through_o this_o place_n be_v ♒_o ☉_o l_o and_o therefore_o the_o meridian_n altitude_n ml_o and_o the_o depression_n below_o the_o horizon_n at_o midnight_n h_n ♒_o the_o semidiurnall_a ark_n l_o ☉_o the_o seminocturnall_a ark_n o_o ♒_o the_o declination_n be_v the_o ascensionall_a difference_n r_o ☉_o the_o amplitude_n of_o ascension_n a_o ☉_o or_o if_o a_o be_v put_v to_o represent_v the_o pole_n of_o the_o world_n then_o shall_v paese_n stand_v for_o the_o equator_fw-la and_o p_o ♋_o s_o ♑_o for_o the_o ecliptic_a and_o the_o rest_n which_o before_o stand_v for_o meridians_fw-la may_v now_o serve_v for_o particular_a horizon_n according_a to_o their_o several_a elevation_n then_o suppose_v the_o place_n of_o the_o sun_n give_v to_o be_v
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 sine_fw-la of_o the_o declination_n measure_v in_o the_o same_o tangent_fw-la line_n shall_v there_o show_v the_o compliment_n of_o the_o angle_n dva_n which_o the_o hour_n line_n of_o 6_o make_v with_o the_o meridian_n then_o have_v the_o point_n d_o take_v out_o the_o semidiameter_n valerio_n and_o prick_v it_o down_o in_o those_o parallel_n from_o d_o unto_o c_o so_o shall_v you_o have_v the_o line_n dc_o and_o ac_fw-la to_o be_v divide_v as_o before_o the_o like_a may_v be_v use_v for_o the_o hour_n line_n upon_o all_o other_o plane_n but_o i_o must_v not_o write_v all_o that_o may_v be_v do_v by_o the_o sector_n it_o may_v suffice_v that_o i_o have_v write_v something_o of_o the_o use_n of_o each_o line_n and_o thereby_o give_v the_o ingenuous_a reader_n occasion_n to_o think_v of_o more_o the_o conclusion_n to_o the_o reader_n it_o be_v well_o know_v to_o many_o of_o you_o that_o this_o sector_n be_v thus_o contrive_v the_o most_o part_n of_o this_o book_n write_v in_o latin_a many_o copy_n transcribe_v and_o disperse_v more_o than_o sixteen_o year_n since_o i_o be_o at_o the_o last_o content_v to_o give_v way_n that_o it_o come_v forth_o in_o english_a not_o that_o i_o think_v it_o worthy_a either_o of_o my_o labour_n or_o the_o public_a view_n but_o partly_o to_o satisfy_v their_o importunity_n who_o not_o understand_v the_o latin_a yet_o be_v at_o the_o charge_n to_o buy_v the_o instrument_n and_o partly_o for_o my_o own_o ease_n for_o as_o it_o be_v painful_a for_o other_o to_o transcribe_v my_o copy_n so_o it_o be_v troublesome_a for_o i_o to_o give_v satisfaction_n herein_o to_o all_o that_o desire_v it_o if_o i_o find_v this_o to_o give_v you_o content_a it_o shall_v encourage_v i_o to_o do_v the_o like_a for_o my_o crosse-staffe_n and_o some_o other_o instrument_n in_o the_o mean_a time_n bear_v with_o the_o printer_n fault_n and_o so_o i_o rest_v gresham_n coll._n 1._o maij._fw-la 1623._o e._n g._n finis_fw-la the_o first_o book_n of_o the_o crosse-staffe_n chap._n i._o of_o the_o description_n of_o the_o staff_n the_o crosse-staffe_n be_v a_o instrument_n well_o know_v to_o our_o seaman_n and_o much_o use_v by_o the_o ancient_a astronomer_n and_o other_o serve_v astronomical_o for_o observation_n of_o altitude_n and_o angle_n of_o distance_n in_o the_o heaven_n geometrical_o for_o perpendicular_a height_n and_o distance_n on_o land_n and_o sea_n the_o description_n and_o several_a use_n of_o it_o be_v extant_a in_o print_n by_o gemma_fw-la frisius_n in_o latin_a in_o english_a by_o dr._n hood_n i_o differ_v something_o from_o they_o both_o in_o the_o projection_n of_o this_o staff_n but_o so_o as_o their_o rule_n may_v be_v apply_v unto_o it_o and_o all_o their_o proposition_n be_v wrought_v by_o it_o and_o therefore_o refer_v the_o reader_n to_o their_o book_n i_o shall_v be_v brief_a in_o the_o explanation_n of_o that_o which_o may_v be_v apply_v from_o they_o unto_o i_o and_o so_o come_v to_o the_o use_n of_o those_o line_n which_o be_v of_o my_o addition_n not_o extant_a heretofore_o the_o necessary_a part_n of_o this_o instrument_n be_v five_o the_o staff_n the_o cross_n and_o the_o three_o sight_n the_o staff_n which_o i_o make_v for_o my_o own_o use_n be_v a_o full_a yard_n in_o length_n that_o so_o it_o may_v serve_v for_o measure_n the_o cross_n belong_v to_o it_o be_v 26_o inch_n â…•_n between_o the_o two_o outward_a sight_n if_o any_o will_v have_v it_o in_o a_o great_a form_n the_o proportion_n between_o the_o staff_n and_o the_o cross_n may_v be_v such_o as_o 360_o unto_o 262._o the_o line_n inscribe_v on_o the_o staff_n be_v of_o four_o sort_n one_o of_o they_o serve_v for_o measure_n and_o protraction_n one_o for_o observation_n of_o angle_n one_o for_o the_o sea-chart_n and_o the_o four_o other_o for_o work_v of_o proportion_n in_o several_a kind_n the_o line_n of_o measure_n be_v a_o inch_n line_n and_o may_v be_v know_v by_o his_o equal_a part_n the_o whole_a yard_n be_v divide_v equal_o into_o 36_o inch_n and_o each_o inch_n subdivide_v first_o into_o ten_o part_n and_o then_o each_o ten_o part_n into_o half_n the_o line_n for_o observation_n of_o angle_n may_v be_v know_v by_o the_o double_a number_n set_v on_o both_o side_n of_o the_o line_n begin_v at_o the_o one_o side_n at_o 20_o and_o end_v at_o 90_o on_o the_o other_o side_n at_o 40_o and_o end_v at_o 180_o and_o this_o be_v divide_v according_a to_o the_o degree_n of_o a_o quadrant_a i_o call_v it_o the_o tangent_fw-la line_n on_o the_o staff_n the_o next_o line_n be_v the_o meridian_n of_o a_o sea-chart_n according_a to_o mercators_n projection_n from_o the_o equinoctial_a to_o 58_o gr_n of_o latitude_n and_o may_v be_v know_v by_o the_o letter_n m_o and_o the_o number_n 1._o 2._o 3._o 4._o unto_o 58._o the_o line_n for_o work_v of_o proportion_n may_v be_v know_v by_o their_o unequal_a division_n and_o the_o number_n at_o the_o end_n of_o each_o line_n 1_o the_o line_n of_o number_n note_v with_o 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
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 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question_n of_o proportion_n between_o line_n and_o superficies_n where_o if_o the_o denomination_n be_v of_o line_n extend_v the_o compass_n from_o the_o first_o to_o the_o second_o number_n of_o the_o same_o denomination_n so_o the_o same_o extent_n be_v double_v shall_v give_v the_o distance_n from_o the_o three_o number_n unto_o the_o four_o the_o diameter_n be_v 14_o the_o content_a of_o the_o circle_n be_v 154_o the_o diameter_n be_v 28_o what_o may_v the_o content_n be_v extend_v the_o compass_n from_o 14_o to_o 28_o the_o same_o extent_n double_v will_v reach_v from_o 154_o to_o 616._o for_o first_o it_o reach_v from_o 154_o unto_o 308_o and_o turn_v the_o compass_n once_o more_o it_o reach_v from_o 308_o unto_o 616_o and_o this_o be_v the_o content_a require_v but_o if_o the_o first_o denomination_n be_v of_o the_o superficial_a content_n extend_v the_o compass_n unto_o the_o half_a of_o the_o distance_n between_o the_o first_o number_n and_o the_o second_o of_o the_o same_o denomination_n so_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 fouth_z the_o content_n of_o a_o circle_n be_v 154_o the_o diameter_n be_v 14_o the_o content_a be_v 616_o what_o may_v the_o diameter_n be_v divide_v the_o distance_n between_o 154_o and_o 616_o into_o two_o equal_a part_n than_o set_v one_o foot_n in_o 14_o the_o other_o will_v reach_v to_o 28_o the_o diameter_n require_v 10_o three_o number_n be_v give_v to_o find_v a_o four_o in_o a_o triplicate_v proportion_n this_o proposition_n concern_v question_n of_o proportion_n between_o line_n and_o solid_n where_o if_o the_o first_o denomination_n be_v of_o line_n extend_v the_o compass_n from_o the_o first_o number_n to_o the_o second_o of_o the_o same_o denomination_n so_o the_o extent_n be_v tripled_a shall_v give_v the_o distance_n from_o the_o three_o number_n unto_o the_o four_o suppose_v the_o diameter_n of_o a_o iron_n bullet_n be_v 4_o inch_n the_o weight_n of_o it_o be_v 9_o l_o the_o diameter_n be_v 8_o inch_n what_o may_v the_o weight_n be_v extend_v the_o compass_n from_o 4_o to_o 8_o the_o same_o extent_n be_v tripled_a will_v reach_v from_o 9_o unto_o 72._o for_o first_o it_o reach_v from_o 9_o unto_o 18_o then_o from_o 18_o to_o 36_o three_o from_o 36_o to_o 72._o and_o this_o be_v the_o weight_n require_v but_o if_o the_o first_o denomination_n shall_v be_v of_o the_o solid_a content_n or_o of_o the_o weight_n extend_v the_o compass_n to_o a_o three_o part_n of_o the_o distance_n between_o the_o first_o number_n and_o the_o second_o of_o the_o same_o denomination_n so_o
observe_v and_o go_v to_o limehouse_n with_o some_o of_o my_o friend_n and_o take_v with_o we_o a_o quadrant_a of_o 3_o foot_n semidiameter_n and_o two_o needle_n the_o one_o above_o 6_o inch_n and_o the_o other_o 10_o inch_n long_o where_o i_o make_v the_o semidiameter_n of_o my_o horizontall_a plane_n az_v 12_o inch_n and_o towards_o night_n the_o 13_o of_o june_n 1622_o i_o make_v observation_n in_o several_a part_n of_o the_o ground_n and_o find_v as_o follow_v alt._n ☉_o azm_n azn_n variat_fw-la gr._n m._n gr._n m._n gr._n m._n gr._n m._n 19_o 0_o 82_o 2_o 75_o 52_o 6_o 10_o 18_o 5_o 80_o 50_o 74_o 44_o 6_o 6_o 17_o 34_o 80_o 0_o 74_o 6_o 5_o 54_o 17_o 0_o 79_o 15_o 73_o 20_o 5_o 55_o 16_o 18_o 78_o 12_o 72_o 32_o 5_o 40_o 16_o 0_o 77_o 50_o 72_o 10_o 5_o 40_o 20_o 10_o 71_o 2_o 64_o 49_o 6_o 13_o 9_o 52_o 70_o 12_o 64_o 25_o 5_o 47_o chap._n vi_o contain_v such_o nautical_a question_n as_o be_v of_o ordinary_a use_n concern_v longitude_n latitude_n rumb_n and_o distance_n 1_o to_o keep_v a_o account_n of_o the_o ship_n way_n the_o way_n that_o the_o ship_n make_v may_v be_v know_v to_o a_o old_a seaman_n by_o experience_n by_o other_o it_o may_v be_v find_v for_o some_o small_a portion_n of_o time_n either_o by_o the_o log_n line_n or_o by_o the_o distance_n of_o two_o know_a mark_n on_o the_o ship_n side_n the_o time_n in_o which_o it_o make_v this_o way_n may_v be_v measure_v by_o a_o watch_n or_o by_o a_o glass_n then_o as_o long_o as_o the_o wind_n continue_v at_o the_o same_o stay_n it_o follow_v by_o proportion_n as_o the_o time_n give_v be_v to_o a_o hour_n so_o the_o way_n make_v to_o a_o hour_n way_n suppose_v the_o time_n to_o be_v 15_o second_o which_o make_v a_o quarter_n of_o a_o minute_n and_o the_o way_n of_o the_o ship_n 88_o foot_n then_o because_o there_o be_v 3600_o second_n in_o a_o hour_n i_o may_v extend_v the_o compass_n in_o the_o line_n of_o number_n from_z 15_o unto_o 3600_o and_o the_o same_o extent_n will_v reach_v from_o 88_o unto_o 21120._o or_o i_o may_v extend_v they_o from_o 15_o unto_o 88_o and_o this_o extent_n will_v reach_v from_o 3600_o unto_o 21120_o which_o show_v that_o a_o hour_n way_n come_v to_o 21120_o foot_n but_o this_o be_v a_o unnecessary_a business_n to_o hearken_v after_o foot_n or_o fathom_n it_o suffice_v our_o seaman_n to_o find_v the_o way_n of_o their_o ship_n in_o league_n or_o mile_n and_o they_o say_v that_o there_o be_v 5_o foot_n in_o a_o pace_n 1000_o pace_n in_o a_o mile_n and_o 60_o mile_n in_o a_o degree_n and_o therefore_o 300000_o foot_n in_o a_o degree_n yet_o compare_v several_a observation_n and_o their_o measure_n with_o our_o foot_n usual_a about_o london_n i_o find_v that_o we_o may_v allow_v 352000_o foot_n to_o a_o degree_n and_o than_o if_o i_o extend_v the_o compass_n in_o the_o line_n of_o number_n from_o 352000_o unto_o 21120_o i_o shall_v find_v the_o same_o extent_n to_o reach_v from_o 20_o league_n the_o measure_n of_o one_o degree_n to_o 1.2_o and_o from_o 60_o mile_n to_o 3.6_o which_o show_v the_o hour_n way_n to_o be_v 1_o league_n and_o 2_o ten_o of_o a_o league_n or_o 3_o mile_n and_o 6_o ten_o of_o a_o mile_n but_o to_o avoid_v these_o fraction_n and_o other_o tedious_a reduction_n i_o suppose_v it_o will_v be_v more_o easy_a to_o keep_v this_o account_n of_o the_o ship_n way_n as_o also_o of_o the_o difference_n of_o latitude_n and_o the_o difference_n of_o longitude_n by_o degree_n and_o part_n of_o degree_n allow_v 100_o part_n to_o each_o degree_n which_o we_o may_v therefore_o call_v by_o the_o name_n of_o centosme_n neither_o will_v this_o be_v hard_a to_o conceive_v for_o if_o 100_o such_o part_n do_v make_v a_o degree_n then_o shall_v 50_o part_n be_v equal_a to_o 30_o minute_n as_o 30_o minute_n be_v equal_a to_o 10_o league_n and_o 5_o part_n shall_v be_v equal_a to_o 3_o minute_n as_o 3_o minute_n be_v equal_a to_o 1_o league_n and_o so_o the_o same_o extent_n as_o before_o will_v reach_v from_o 100_o part_n 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20_o 25_o 30_o mo_z gr._n m_o gr._n m._n gr._n m_o gr._n m._n gr._n m._n gr._n m_o gr._n m._n january_n 16_o 31_o 17_o 24_o 18_o 26_o 19_o 37_o 20_o 57_o 22_o 24_o 23_o 58_o february_n 24_o 17_o 25_o 59_o 27_o 45_o 29_o 35_o 31_o 29_o 33_o 25_o  _fw-fr march_n 34_o 35_o 36_o 33_o 38_o 32_o 40_o 30_o 42_o 27_o 44_o 22_o 46_o 15_o april_n 46_o 37_o 48_o 26_o 50_o 11_o 51_o 50_o 53_o 25_o 54_o 53_o 56_o 15_o may_n 56_o 15_o 57_o 29_o 58_o 35_o 59_o 33_o 60_o 22_o 61_o 2_o 61_o 31_o june_n 61_o 36_o 61_o 54_o 62_o 0_o 61_o 58_o 61_o 45_o 61_o 22_o 60_o 49_o july_n 60_o 49_o 60_o 6_o 59_o 14_o 58_o 13_o 57_o 4_o 55_o 48_o 54_o 24_o august_n 54_o 7_o 52_o 36_o 50_o 59_o 49_o 17_o 47_o 31_o 45_o 41_o 43_o 49_o september_n 43_o 26_o 41_o 30_o 39_o 33_o 37_o 36_o 35_o 38_o 33_o 41_o 31_o 46_o october_n 31_o 46_o 29_o 53_o 28_o 3_o 26_o 16_o 24_o 35_o 22_o 59_o 21_o 29_o november_n 21_o 12_o 19_o 51_o 18_o 39_o 17_o 36_o 16_o 43_o 16_o 0_o 15_o 28_o december_n 15_o 28_o 15_o 7_o 15_o 0_o 15_o 2_o 15_o 17_o 15_o 44_o 16_o 22_o the_o table_n be_v make_v you_o may_v inscribe_v the_o month_n and_o day_n of_o each_o month_n into_o your_o quadrant_a in_o the_o space_n leave_v below_o the_o tropic_n for_o lay_v the_o ruler_n unto_o the_o centre_n a_o and_o 16_o gr_n 31_o m._n in_o the_o quadrant_a bc_n there_o may_v you_o draw_v a_o line_n for_o the_o end_n of_o december_n and_o beginning_n of_o