radiuss_n or_o 400_o instead_o of_o 10_o you_o may_v work_v right-on_a to_o 4_o figure_n but_o then_o the_o radiuss_n must_v be_v very_o small_a or_o the_o instrument_n very_o large_a therefore_o this_o of_o 10_o be_v the_o most_o usual_a i_o shall_v make_v use_n of_o and_o work_v every_o question_n the_o most_o convenient_a way_n that_o by_o a_o frequent_a practice_n the_o young_a beginner_n for_o who_o only_o i_o write_v may_v see_v the_o reason_n and_o nature_n of_o the_o work_n and_o the_o soon_o understand_v it_o for_o a_o conclusion_n of_o this_o rule_n of_o multiplication_n take_v three_o or_o four_o example_n more_o both_o by_o the_o line_n of_o number_n and_o equal-part_n also_o 1._o first_o 15_o foot_n 8_o ten_o multiply_v 9_o foot_n and_o 7_o ten_o by_o the_o line_n of_o number_n the_o extent_n of_o the_o compass_n from_o 1_o to_o 9_o foot_n 7_o ten_o shall_v reach_v the_o same_o way_n from_o 15_o 8_o to_o 153_o 26._o by_o the_o line_n of_o line_n or_o equal_a part_n as_o 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and_o not_o 10_o foot_n and_o a_o half_a and_o better_o which_o be_v as_o to_o the_o right_a number_n of_o figure_n but_o by_o the_o line_n of_o line_n as_o the_o 10_o 75_o from_o any_o scale_n as_o the_o line_n of_o line_n double_v or_o foot-measure_n or_o the_o like_a be_v to_o the_o =_o 10_o so_o be_v the_o =_o 9_o 83_o to_o the_o latteral_a 105_o 6_o ½_o as_o before_o though_o not_o so_o quick_a or_o plain_a as_o by_o the_o line_n of_o number_n 3._o as_o 1_o to_o 1528_o so_o be_v 3522_o to_o 5371616_o the_o true_a answer_n which_o indeed_o be_v to_o more_o place_n than_o possible_a the_o rule_n can_v come_v to_o without_o the_o help_n last_o mention_v but_o if_o the_o question_n have_v be_v thus_o with_o the_o same_o figure_n 15_o foot_n 100_o 28_o part_n by_o 35_o foot_n 100_o 22_o part_n as_o so_o many_o foot_n or_o yard_n and_o hundred_o part_n then_o the_o answer_n will_v be_v as_o before_o 537_o foot_n and_o cut_v off_o four_o figure_n for_o the_o four_o figure_n of_o fraction_n both_o in_o the_o multiplicator_n and_o multiplicand_a viz._n the_o 1616_o which_o make_v near_o 2_o inch_n of_o a_o foot_n more_o or_o 16_o 100_o part_n of_o a_o yard_n more_o which_o in_o ordinary_a measure_n be_v not_o considerable_a by_o the_o line_n of_o line_n 4._o as_o 15_o 28_o to_o =_o 1_o next_o the_o centre_n so_o be_v =_o 35_o 22_o to_o 537_o 1616_o 100000_o to_o multiply_v 3_o pound_n 6_o shilling_n and_o 3_o penny_n by_o itself_o the_o product_n be_v 10._o l._n 19_o s._n 5._o d._n 1._o f._n 7_o 10_o for_o the_o extent_n from_o 1_o to_z 3_o 3125_o the_o decimal_a number_n for_o 3_o â—_o 6._o s._n 3._o d._n shall_v reach_v from_o thence_o to_o 10_o 9726_o which_o reduce_v again_o be_v as_o before_o 10_o 19_o 5_o two_o as_o follow_v note_v that_o in_o this_o way_n of_o multiplication_n by_o the_o pen_n work_v thus_o you_o must_v first_o multiply_v pound_n by_o pound_n one_o over_o the_o other_o as_o 3_o by_o 3_o then_o the_o shilling_n by_o the_o pound_n 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10th_n of_o a_o farthing_n seven_o 6_o time_n 3_o be_v 18_o every_o 5_o whereof_o be_v a_o farthing_n and_o every_o 1_o be_v two_o ten_o of_o a_o farthing_n as_o the_o short_a prick-line_n show_v eight_o 6_o time_n 3_o be_v 18_o or_o 3_o farthing_n and_o 6_o ten_o as_o before_o nine_o and_o last_o to_o the_o right-hand_a 3_o time_n 3_o be_v 9_o where_o note_n that_o there_o go_v 60_o to_o make_v 1_o farthing_n therefore_o 6_o make_v one_o ten_o of_o a_o farthing_n so_o that_o here_o be_v 1_o ten_o and_o ½_o consider_v the_o scheam_n and_o the_o decimal-work_n to_o prove_v it_o exact_o to_o the_o hundred_o of_o million_o of_o a_o pound_n and_o you_o will_v find_v it_o to_o be_v very_o near_o example_n product_v be_v 10._o 19_o 5._o 1._o 7._o ½_o the_o same_o decimal_o which_o sum_n be_v bring_v to_o shilling_n penny_n and_o farthing_n and_o ten_o of_o farthing_n be_v just_a as_o aforesaid_a viz._n or_o else_o find_v the_o square_n of_o the_o least_o denomination_n in_o 20_o s._n and_o divide_v the_o product_n of_o the_o sum_n be_v bring_v to_o that_o least_o denomination_n thereby_o and_o the_o quotient_n shall_v be_v the_o answer_v require_v example_n 960_o the_o farthing_n in_o 20_o s._n square_v be_v 921600._o the_o sum_n of_o 3_o l._n 6_o s._n 3_o d._n in_o farthing_n be_v 3180_o multiply_v by_o itself_o be_v 10112400_o this_o product_v divide_v by_o 921600_o the_o square_a of_o the_o farthing_n in_o 20_o s._n make_v 10_o l._n 896400_o â—216â—â—_n in_o the_o quotient_a which_o reduce_v be_v 10_o l._n 19_o s._n 5_o d._n 1_o 7_o 10._o to_o find_v this_o decimal_a fraction_n be_v very_o easy_a thus_o by_o the_o line_n of_o number_n for_o if_o 20_o shilling_n be_v 1000_o what_o shall_v 6_o shilling_n and_o 3_o penny_n be_v set_a one_o point_n in_o 2_o represent_v 20_o and_o the_o other_o in_o 1_o represent_v 1000_o then_o the_o same_o extent_n lay_v the_o same_o way_n from_o 6_o and_o ¼_n shall_v reach_v to_o 3.125_o the_o decimal_a fraction_n for_o 6_o shilling_n and_o 3_o penny_n or_o by_o inch_n and_o foot-measure_n for_o if_o you_o account_v every_o 8_o of_o a_o inch_n a_o farthing_n than_o every_o inch_n be_v 2_o d._n and_o 6_o inch_n be_v 12_o d._n right_a against_o which_o in_o the_o foot-measure_n be_v the_o decimal_a fraction_n require_v so_o that_o right_a against_o 12_o farthing_n or_o 1_o inch_n and_o ½_o on_o the_o line_n of_o foot-measure_n be_v 125_o the_o decimal_a fraction_n seek_v for_o or_o if_o one_o pound_n be_v the_o integer_fw-la or_o
thus_o the_o first_o and_o three_o term_n be_v bring_v to_o one_o denomination_n also_o you_o see_v that_o the_o demand_v or_o question_n viz._n what_o shall_v 30_o ounce_n cost_v be_v join_v to_o the_o three_o term_n and_o also_o that_o 16_o ounce_n the_o first_o term_n be_v of_o the_o same_o denomination_n therefore_o the_o 3_o s._n must_v needs_o be_v the_o second_o term_n and_o the_o answer_n to_o the_o question_n be_v the_o four_o 6._o have_v thus_o discover_v which_o be_v the_o first_o second_o and_o three_o term_n and_o reduce_v the_o first_o and_o three_o to_o the_o like_a denomination_n then_o the_o work_n by_o the_o line_n of_o number_n be_v always_o thus_o as_o the_o first_o to_o the_o second_o so_o be_v the_o three_o to_o the_o four_o or_o the_o extent_n of_o the_o compass_n upon_o the_o line_n of_o number_n from_o the_o first_o to_o the_o second_o shall_v reach_v the_o same_o way_n from_o the_o three_o to_o the_o four_o require_v as_o 16_o ounce_n be_v to_o 2_o s_o so_o be_v 30_o ounce_n to_o 2_o s_o ¾_n or_o 9_o penny_n or_o as_o 16_o ounce_n be_v to_o 24._o d_o so_o be_v 30_o ounce_n to_o 45_o d_o which_o be_v 3_o s_o 9_o d._n as_o before_o 7._o but_o by_o the_o line_n of_o line_n or_o sector_n if_o you_o will_v work_v on_o one_o scale_n only_o you_o must_v consider_v which_o term_n of_o the_o first_o or_o second_o be_v big_a for_o you_o must_v always_o order_v it_o so_o that_o the_o parallel_n worâ—_n must_v be_v the_o large_a or_o at_o least_o so_o as_o it_o may_v be_v wrought_v and_o as_o much_o as_o may_v be_v that_o the_o four_o term_n may_v be_v a_o latteral_a extent_n as_o the_o first_o always_o be_v for_o than_o it_o be_v wrought_v the_o soon_a and_o also_o the_o exact_a yet_o by_o this_o instrument_n you_o need_v not_o much_o care_n for_o these_o caution_n have_v several_a scale_n of_o equal-part_n to_o begin_v and_o end_v the_o work_n on_o you_o be_v free_v from_o that_o trouble_n as_o thus_o for_o example_n when_o the_o second_o term_n be_v great_a than_o the_o first_o than_o the_o work_n be_v well_o perform_v thus_o two_o way_n as_o the_o latteral_a first_fw-mi 16_o or_o else_o as_o latteral_a third_n 30_o from_o a_o lesser_a scale_n to_o the_o parallel_n second_o 02_o or_o else_o to_o parallel_v first_o 16_o so_o be_v the_o latteral_a third_z 30_o or_o else_o so_o parallel_n second_o 2_o to_o the_o parallel_n four_o 3_o 75_o â—00_o or_o else_o to_o latteral_a four_o 3_o 75_o 100_o by_o the_o same_o scale_n or_o as_o second_o to_z =_o first_o so_o =_o three_o to_z four_o by_o a_o less_o scale_n also_o if_o need_v be_v but_o when_o the_o second_o term_n be_v less_o than_o the_o first_o than_o the_o work_n be_v perform_v thus_o if_o 50_o foot_n of_o timber_n cost_v 40_o s._n what_o shall_v 20_o foot_n cost_n as_o the_o latteral_a second_v 40_o or_o as_o before_o as_o 3d_o 20_o foot_n to_o the_o parallel_n first_o 50_o or_o as_o before_o to_z =_o one_a 50_o foot_n so_o be_v the_o parallel_n three_o 20_o or_o as_o before_o so_o =_o 2d_o 40_o shill_n to_o the_o latteral_a fourth_z 16_o or_o as_o before_o to_o four_o 16_o shill_n 8._o thus_o you_o see_v several_a way_n of_o work_v but_o for_o beginner_n i_o will_v advise_v thus_o brief_o first_o either_o to_o observe_v this_o rule_n of_o change_v the_o term_n from_o the_o first_o to_o the_o second_o viz._n to_o take_v the_o second_o latteral_o and_o make_v it_o a_o parallel_n in_o the_o first_o then_o the_o parallel_n three_o give_v you_o a_o latteral_a answer_n or_o else_o to_o work_v direct_o as_o the_o first_o to_o the_o second_o and_o so_o be_v content_a with_o a_o parallel_n answer_n which_o you_o may_v always_o do_v with_o the_o help_n of_o a_o small_a scale_n when_o need_n require_v it_o note_v the_o figure_n of_o operation_n by_o the_o triangule_a quadrant_n sect._n xii_o the_o rule_n of_o three_o inverse_v 1._o the_o rule_n of_o three_o inverse_v or_o the_o back-rule_n of_o three_o be_v when_o the_o term_n require_v or_o four_o term_n 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10_o cart_n to_o carry_v 60_o load_n for_o note_n as_o in_o the_o former_a rule_n of_o three_o direct_a look_v how_o much_o the_o three_o term_n be_v great_a than_o the_o first_o so_o much_o the_o four_o be_v great_a than_o the_o second_o and_o contrary_o look_v how_o much_o the_o three_o term_n be_v less_o than_o the_o first_o by_o so_o much_o be_v the_o four_o term_n less_o than_o the_o second_o as_o thus_o in_o number_n as_o 2_o be_v to_o 4_o so_o be_v 6_o to_o 12_o for_o as_o 6_o the_o three_o term_n be_v thrice_o as_o much_o as_o 2_o the_o first_o term_n so_o be_v 12_o the_o four_o term_n thrice_o as_o much_o as_o 4_o the_o second_o and_o contrary_o decrease_v as_o 12_o be_v to_z 6_o so_o be_v 4_o to_o 2_o for_o as_o 4_o be_v one_o three_o part_n of_o 12_o so_o be_v 2_o one_o three_o part_n of_o 6._o 2._o but_o now_o in_o this_o rule_n of_o three_o inverse_v or_o the_o back-rule_n of_o three_o it_o be_v contrary_o order_v as_o thus_o look_v how_o much_o the_o three_o term_n be_v great_a or_o lesser_a then_o the_o first_o by_o so_o much_o be_v the_o four_o term_v lesser_a or_o great_a than_o the_o second_o as_o thus_o in_o number_n as_o 5_o be_v to_o 60_o so_o be_v 30_o to_o 2_o 1_o â—_o that_o be_v if_o 5_o s._n be_v 60_o d._n how_o many_o shilling_n be_v 30_o penny_n the_o answer_n be_v 2_o s._n ½_o for_o as_o 30_o be_v great_a than_o 5â—_n so_o be_v 2_o ½_o less_o than_o 60._o again_o as_o 2_o ½_o be_v to_o 30_o so_o be_v 60_o to_o 5_o in_o the_o like_a manner_n pion._n day_n pion._n day_n 18._o 40._o 15._o 48._o 1._o 2._o 3._o 4._o if_o 18_o pioner_n make_v a_o trench_n in_o 40_o day_n how_o many_o pioner_n be_v needful_a to_o perform_v the_o same_o in_o 15_o day_n as_o 40_o to_o 15_o so_o be_v 18_o to_o 48_o here_o as_o the_o three_o be_v lesser_a than_o the_o first_o so_o be_v thâ—_n fourth_z great_a than_o the_o second_o hors._n day_n hors._n day_n 12._o 30._o 24._o 15._o 1._o 2._o 3._o 4._o again_o if_o 12_o horse_n eat_v 20_o bushel_n of_o provender_n in_o 30_o day_n how_o soon_o will_v 24_o horse_n eat_v up_o the_o like_a quantity_n of_o provender_n the_o answer_n be_v in_o 15_o day_n 3._o the_o manner_n of_o work_v this_o rule_n on_o the_o line_n of_o number_n be_v thus_o extend_v the_o compass_n from_o one_o term_n to_o the_o other_o of_o like_a denomination_n the_o same_o extent_n lay_v the_o contrary_a way_n from_o the_o other_o term_n shall_v reach_v to_o the_o answer_v require_v as_o in_o the_o last_o example_n the_o extent_n from_o 12_o to_o 24_o the_o term_n under_o the_o denomination_n of_o horse_n shall_v reach_v the_o contrary_a way_n from_o 30_o to_o 15_o the_o number_n of_o day_n require_v 4._o note_v that_o by_o due_a consideration_n this_o back-rule_n may_v be_v work_v by_o the_o precept_n for_o the_o direct-rule_n thus_o in_o all_o question_n of_o this_o nature_n there_o be_v three_o term_n give_v to_o find_v a_o four_o of_o which_o three_o term_n two_o be_v of_o one_o denomination_n and_o one_o of_o a_o different_a denomination_n of_o which_o the_o four_o must_v always_o be_v which_o in_o the_o first_o rule_n of_o the_o ten_o section_n before_o go_v be_v call_v two_o term_n of_o supposition_n and_o one_o of_o demand_n now_o here_o you_o be_v to_o consider_v that_o 5._o when_o the_o four_o term_v require_v aught_o to_o be_v great_a than_o that_o of_o demand_n which_o by_o reason_n you_o may_v certain_o know_v then_o say_v as_o the_o lesser_a term_n of_o supposition_n be_v to_o the_o great_a so_o be_v the_o
particular_a product_n of_o every_o double_a number_n for_o the_o three_o term_n one_o after_o another_o this_o do_v repeat_v the_o rule_n of_o three_o as_o often_o as_o there_o be_v double_a number_n the_o four_o term_v produce_v from_o those_o operation_n shall_v be_v answer_n to_o the_o question_n require_v viz._n the_o quantity_n of_o each_o man_n gain_n or_o loss_n example_n 25_o &_o 30_o a_n ox_n and_o time_n of_o feed_v multiply_v be_v 750_o 15_o &_o 40_o b_n ox_n and_o time_n of_o feed_v multiply_v be_v 600_o 20_o &_o 40_o c_n ox_n and_o day_n of_o feed_v multiply_v be_v 800_o the_o sum_n 2150_o as_o 215_o to_o 50_o so_o be_v 750_o a_n stock_n to_o 17-9_a a_n rent_n  _fw-fr 600_o b_n 13-19_a b's_n  _fw-fr 800_o c_n 18-12_a c_n 9_o to_o work_v by_o the_o line_n of_o number_n the_o extent_n of_o the_o compass_n from_o 1_o to_o 25_o shall_v reach_v the_o same_o way_n from_o 30_o to_o 750_o the_o first_o product_n of_o a_o a'_v double_a number_n or_o stock_n and_o as_o 1_o to_o 15_o so_o be_v 40_o to_o 600_o the_o product_n of_o b_n double_a number_n and_o stock_n and_o as_o 1_o to_o 20_o so_o be_v 40_o to_o 800_o the_o product_n of_o c_n double_a number_n and_o stock_n which_o three_o product_n add_v make_v 2150_o the_o first_o term_n and_o 50_o be_v the_o second_o term_n and_o 750_o 600_o and_o 800_o the_o three_o product_n several_o the_o three_o term_n then_o the_o extent_n from_o 2150_o to_o 50_o shall_v reach_v the_o same_o way_n from_o 750_o to_o 17-45_a or_o 17_o l._n 9_o s._n and_o from_o 600_o to_o 13-95_a or_o 13_o l._n 19_o s._n and_o from_o 800_o to_o 18-60_a or_o 18_o l._n =_o 12_o s._n the_o several_a answer_n require_v which_o be_v add_v together_o make_v up_o 50_o l._n the_o whole_a rent_n to_o be_v pay_v among_o they_o there_o be_v other_o rule_n of_o arithmetic_n as_o the_o rule_n call_v allegation_n medial_a and_o alternate_a and_o the_o rule_n of_o position_n or_o falsehood_n in_o the_o work_n of_o which_o be_v so_o many_o caution_n in_o order_v the_o number_n before_o you_o come_v to_o the_o proportional_a work_n that_o it_o will_v make_v the_o book_n more_o bulky_a than_o useful_a therefore_o i_o shall_v wave_v it_o and_o refer_v you_o to_o the_o particular_a book_n of_o arithmetic_n as_o that_o of_o mr._n record_n dee_n and_o mellis_n or_o that_o of_o mr._n wingate_n natural_a and_o artificial_a have_v in_o it_o plenty_n of_o example_n and_o other_o also_o as_o johnfâ—us_o jaggers_o or_o moor_n arithmetic_n any_o of_o which_o exceed_v the_o bound_n i_o intend_v for_o this_o whole_a discourse_n i_o shall_v therefore_o pass_v on_o to_o the_o rule_n of_o practice_n in_o several_a kind_n as_o measure_v superfecy_n and_o solid_n and_o rule_n of_o double_a and_o treble_a proportion_n and_o question_n of_o interest_n which_o be_v tedious_a by_o the_o pen_n without_o the_o help_n of_o particular_a table_n and_o very_o easy_a by_o the_o line_n of_o number_n as_o will_v full_o appear_v in_o the_o next_o chapter_n chap._n vii_o the_o use_n of_o the_o line_n of_o number_n in_o measure_v of_o any_o kind_n of_o superficial_a measure_n the_o measure_n that_o be_v common_o use_v in_o this_o work_n be_v a_o foot-rule_n divide_v into_o 100_o part_n or_o else_o into_o 12_o inch_n and_o those_o inch_n into_o half_n and_o quarter_n or_o 8_o part_n or_o inch_n and_o 10_o part_n but_o in_o regard_n that_o the_o number_n do_v most_o fit_o agree_v to_o the_o 100_o part_n of_o a_o foot_n it_o will_v be_v convenient_a here_o to_o show_v how_o to_o reduce_v they_o or_o any_o other_o fraction_n from_o 12_o s._n to_o 10_o s._n or_o any_o other_o whatsoever_o from_o one_o fraction_n to_o the_o other_o which_o by_o the_o line_n of_o number_n be_v quick_o do_v as_o thus_o from_o 12_o to_o 10_o reduction_n extend_v the_o compass_n from_o one_o denominator_fw-la to_o the_o other_o the_o same_o extent_n shall_v reach_v the_o same_o way_n from_o one_o numerator_n to_o the_o other_o example_n as_o 12_o to_o 10_o so_o be_v 6_o half_a of_o 12_o to_z 5_o half_n of_o 10._o again_o as_o 120_o to_o 100_o so_o be_v 30_o a_o four_o of_o 120_o to_o 25_o a_o four_o of_o 100_o which_o two_o line_n of_o inch_n and_o foot-measure_n be_v usual_o set_v together_o on_o rule_n for_o the_o ready_a way_n of_o reduction_n by_o ocular_a inspection_n only_o in_o this_o manner_n as_o in_o the_o figure_n and_o the_o like_a may_v be_v for_o any_o thing_n whatsoever_o as_o mr._n edmond_n windgate_n have_v large_o show_v in_o his_o arithmetic_n which_o line_n be_v next_o to_o the_o line_n of_o number_n on_o your_o rule_n will_v be_v very_o plain_a and_o ready_a in_o the_o use_n of_o the_o line_n of_o number_n for_o foot_n and_o inch_n or_o shilling_n and_o penny_n and_o the_o same_o rule_n of_o reduction_n serve_v for_o all_o manner_n of_o fraction_n for_o as_o the_o denominator_fw-la of_o one_o fraction_n be_v to_o the_o denominator_fw-la of_o the_o other_o which_o in_o the_o decimal_n work_n be_v always_o a_o unite_v with_o one_o two_o or_o more_o cipher_n so_o be_v the_o numerator_n of_o one_o to_o the_o numerator_n of_o the_o other_o and_o note_v that_o the_o operation_n of_o decimal_a number_n and_o their_o fraction_n be_v no_o other_o than_o whole_a number_n except_o only_o the_o cut_n off_o so_o many_o figure_n as_o there_o be_v fraction_n in_o the_o multiplicator_n and_o multiplicand_a after_o any_o multiplication_n as_o in_o the_o follow_a example_n will_v appear_v this_o be_v premise_v i_o come_v next_o to_o the_o work_n problem_n i._o the_o breadth_n of_o a_o oblong_a superficies_n give_v in_o foot_n measure_n to_o find_v how_o much_o in_o length_n make_v one_o foot_n the_o extent_n of_o the_o compass_n from_o the_o breadth_n to_o 1_o shall_v reach_v the_o same_o way_n from_o 1_o to_o the_o length_n require_v example_n at_o 7_o 10_o broad_a as_o 7_o to_o 1_o so_o be_v 1_o to_o 1_o foot_n and_o 43_o part_n the_o breadth_n give_v in_o inch_n to_o find_v how_o much_o make_v a_o foot_n as_o the_o breadth_n in_o inch_n to_o 12_o so_o be_v 12_o to_o the_o length_n of_o a_o foot_n in_o inch_n and_o 10_o part_n example_n at_o 8_o inch_n broad_a you_o must_v have_v 18_o inch_n to_o make_v a_o foot_n for_o the_o extent_n from_o 8_o to_o 12_o shall_v reach_v the_o same_o way_n from_o 12_o to_o 18._o to_o work_v these_o two_o by_o the_o line_n of_o line_n by_o inches_n as_o 1_o to_o =_o 7_o so_o be_v =_o 1_o to_o 1_o 43_o the_o length_n in_o foot-measure_n by_o inch_n as_o 12_o to_o =_o 8_o so_o be_v =_o 12_o to_o 18_o or_o else_o by_o inch_n as_o 8_o to_o =_o 12_o so_o be_v 12_o to_o =_o 18_o the_o length_n in_o inch_n problem_n ii_o have_v the_o breadth_n of_o a_o oblong_a superficies_n give_v in_o foot-measure_n to_o find_v how_o much_o be_v in_o a_o foot_n long_o this_o be_v soon_o wrought_v for_o in_o every_o foot_n long_o there_o be_v just_a as_o much_o as_o the_o breadth_n be_v either_o in_o foot-measure_n or_o inch_n for_o a_o piece_n of_o board_n half_o a_o foot_n broad_a and_o a_o foot_n long_o be_v just_o half_a a_o foot_n problem_n iii_o have_v the_o length_n and_o breadth_n in_o foot-measure_n to_o find_v the_o content_a in_o foot_n the_o extent_n from_o 1_o to_o the_o length_n shall_v reach_v the_o same_o way_n from_o the_o length_n to_o the_o content_a in_o foot_n example_n as_o 1_o to_o 1_o foot_n 50_o the_o breadth_n so_o be_v 11_o foot_n 10_o part_n the_o length_n to_o 16_o foot_n and_o 65_o part_n the_o comment_n require_v the_o breadth_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 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 content_a in_o foot_n require_v example_n at_o 9_o inch_n broad_a and_o 11_o foot_n long_o the_o extent_n from_o 12_o to_o 9_o shall_v reach_v the_o same_o way_n from_o 11_o to_o 8_o foot_n 3_o inch_n or_o ¼_n by_o the_o line_n of_o line_n as_o 11_o to_z =_o 12_o so_o be_v 9_o to_z 8_o ¼_n but_o note_v that_o in_o work_v this_o and_o many_o suchlike_a it_o will_v be_v convenient_a to_o double_v your_o scale_n in_o account_n call_v 10_o at_o the_o end_n 20_o and_o every_o single_a figure_n as_o much_o more_o as_o to_o call_v 12_o and_o 24_o etc._n etc._n so_o that_o in_o this_o operation_n the_o work_n run_v thus_o as_o 11_o take_v from_o the_o line_n of_o line_n count_v 1_o for_o 10_o as_o usual_o to_o =_o 6_o the_o half_a of_o 12_o reckon_v double_a for_o 12_o so_o be_v =_o 4_o ½_o count_v for_o 9_o to_z 8_o ¼_n between_o the_o centre_n and_o 1._o
to_o be_v measure_v thus_o be_v the_o height_n and_o breadth_n take_v etc._n etc._n if_o a_o chimney_n stand_v single_o and_o alone_o not_o lean_v against_o or_o in_o a_o wall_n the_o usual_a way_n be_v to_o gird_v it_o about_o and_o if_o the_o jaume_n be_v but_o a_o brick_n thick_a and_o wrought_v upright_o over_o the_o mantletree_n to_o the_o floor_n then_o i_o say_v gird_v it_o about_o for_o a_o length_n and_o the_o height_n of_o the_o story_n be_v the_o breadth_n at_o a_o brick_n thick_a because_o of_o the_o gather_v together_o to_o make_v room_n for_o the_o next_o hearth_n above_o in_o the_o next_o story_n but_o if_o the_o chimney-back_a be_v a_o party-wall_n the_o wall_n be_v first_o measure_v than_o the_o breast_n and_o the_o depth_n of_o the_o two_o jaume_n be_v one_o side_n and_o the_o height_n of_o the_o story_n another_o side_n to_o be_v multiply_v together_o at_o a_o brick_n and_o a_o half_a thick_a or_o a_o brick_n thick_a according_a as_o the_o jaume_v be_v and_o nothing_o to_o be_v abate_v for_o the_o want_n between_o the_o hearth_n and_o the_o mantletree_n because_o of_o the_o with_o 's_o and_o thicken_n for_o the_o next_o hearth_n for_o measure_v the_o shaft_n of_o the_o chimney_n gird_a with_o a_o line_n round_o about_o the_o least_o place_n of_o they_o for_o one_o side_n and_o the_o height_n for_o the_o other_o side_n at_o a_o brick_n thick_a in_o consideration_n of_o the_o with_o 's_o pargitting_a and_o scaffold_n in_o measure_v of_o ceiling_n a_o foot_n broad_a and_o the_o length_n of_o the_o valley_n be_v always_o to_o be_v allow_v more_o than_o the_o whole_a roof_n also_o the_o length_n of_o the_o rafter_n foot_n above_z or_o beyond_o the_o roof_n when_o rafter_n have_v their_o usual_a pitch_n which_o be_v when_o the_o breadth_n of_o the_o house_n be_v 12_o foot_n the_o rafter_n be_v 9_o foot_n long_o which_o be_v 3_o quarter_n of_o the_o floor_n breadth_n be_v it_o more_o or_o less_o then_o i_o say_v that_o the_o content_a of_o one_o floor_n and_o half_a so_o much_o be_v the_o area_n of_o the_o whole_a roof_n in_o square_n to_o which_o be_v to_o be_v add_v the_o valley_n and_o rafter-feet_n or_o eve_n in_o tile_v and_o also_o a_o deduction_n for_o chimney-room_n and_o gutter_n if_o any_o be_v which_o work_n by_o the_o line_n of_o number_n be_v do_v at_o one_o operation_n thus_o as_o 6666_o be_v to_o the_o length_n of_o the_o house_n so_o be_v the_o breadth_n to_o the_o content_a in_o the_o roof_n example_n a_o house_n 30_o foot_n long_o and_o 20_o foot_n broad_a be_v 900_o foot_n or_o 9_o square_a for_o the_o extent_n from_o 6666_o to_o 20_o shall_v reach_v the_o same_o way_n from_o 30_o to_o 900._o also_o in_o measure_v of_o the_o roof_n as_o to_o carpenter_n work_n by_o the_o square_n there_o be_v to_o be_v allowance_n for_o those_o rafter_n in_o the_o dormer_n and_o gable-end_n on_o which_o no_o tile_n be_v lay_v as_o over-work_n for_o a_o particular_a use_n and_o convenience_n more_o than_o need_v be_v in_o a_o bare_a cover_v or_o roof_n also_o in_o measure_v of_o plasterer_n work_v in_o partition_n and_o wall_n the_o timber_n and_o quarter_n be_v not_o to_o be_v deduct_v out_o of_o the_o render_n for_o work_n only_o except_o when_o the_o workman_n find_v the_o work_n and_o stuff_v also_o then_o subtract_v a_o 6_o part_n for_o the_o quarter_n in_o the_o render_n work_n but_o in_o ceiling_n the_o summer_n which_o be_v see_v be_v always_o abate_v and_o door_n and_o window_n also_o unless_o by_o a_o due_a considerate_a or_o a_o unconsiderate_a bargain_n of_o run_a measure_n thus_o you_o have_v a_o brief_a account_n of_o the_o usual_a order_n use_v among_o workman_n in_o take_v the_o dimension_n of_o a_o house_n viz._n brickwork_n by_o the_o rod_n tile_v and_o carpenters-work_n by_o the_o square_a chimney_n usual_o by_o the_o fire_n and_o plasterer_n and_o painters-work_n by_o the_o yard_n glasier_n by_o the_o foot_n there_o be_v many_o other_o thing_n to_o be_v take_v notice_n of_o in_o the_o carpenter_n bill_n as_o lintel_n mantle-tree_n and_o tassel_n luthern_a light_n and_o other_o light_n both_o architrave_n and_o plain_a light_n sky-light_n or_o cubiloe_n modillean_n cornish_n and_o gutter_a penthouse_n cornish_n timber-front-story_n cellar-door_n and_o door-case_n the_o plank_n and_o curb_n at_o the_o cellar-stair_n dogleg-stair_n and_o open-newel-stair_n canted-stair_n count_v either_o by_o the_o step_n or_o pair_n together_o with_o the_o half_a space_n on_o the_o corner_n of_o the_o open_a newel-stair_n the_o rail_n and_o ballaster_n small_a and_o great_a cornish_a outside-work_n and_o partition_n ceiling_v joyst_n and_o the_o ashler_a board_v partition_n and_o chequer-work_n back-door_n and_o door-case_n window-board_n and_o wall-timber_n plank_n in_o the_o foundation_n palce_v penthouse-floor_n and_o penthouse-roof_n fur_v the_o platform_n centering_n for_o the_o chimney_n trimmer_n girders-end_n end_n of_o brest-summer_n and_o plate_n and_o more_o the_o like_a which_o will_v come_v in_o account_n to_o be_v remember_v and_o set_v down_o according_a as_o the_o building_n be_v also_o with_o due_a allowance_n into_o the_o wall_n that_o way_n the_o end_n of_o the_o joyst_n be_v enter_v or_o lay_v in_o the_o wall_n as_o thus_o if_o it_o be_v frame_v work_n be_v only_o measure_v than_o 9_o inch_n ought_v to_o be_v allow_v into_o each_o wall_n that_o way_n the_o joyst_n end_n be_v lay_v because_o every_o joy_v if_o well_o lay_v shall_v have_v 9_o inch_n at_o least_o hold_v on_o the_o wall_n but_o if_o it_o be_v timber_n and_o board_v both_o to_o be_v measure_v than_o 6_o inch_n only_o be_v a_o competent_a allowance_n because_o the_o timber_n be_v usual_o value_v at_o one_o three_o part_n more_o than_o the_o board_v be_v also_o as_o the_o workman_n do_v think_v on_o this_o the_o workmaster_n may_v not_o forget_v to_o deduct_v for_o stair_n and_o chimney_n also_o where_o work_n and_o stuff_n be_v both_o measure_v though_o for_o work_n only_o it_o may_v be_v very_o well_o allow_v unless_o the_o better_a price_n make_v a_o allowance_n for_o it_o note_v also_o that_o by_o the_o line_n of_o number_n you_o may_v ready_o find_v the_o length_n of_o the_o hip_n and_o rafter_n in_o a_o roof_n of_o any_o largeness_n at_o true_a pitch_n by_o this_o follow_a proportion_n and_o table_n the_o breadth_n of_o the_o house_n be_v 40_o foot_n and_o the_o end_v square_a the_o length_n and_o angle_n be_v as_o in_o the_o table_n at_o the_o usual_a truâ—_n pitch_n  _fw-fr fâ—et_fw-fr 100_o par_fw-fr whole_a breadth_n 40_o 00_o half_a breadth_n 20_o 00_o rafter_n 30_o 00_o hip-rafter_n 36_o 00_o diagonal_a line_n 56_o 57_o half-diagonal_a 28_o 28_o perpendicule_a 22_o 36_o  _fw-fr  _fw-fr deg_n min._n hippolito_n angle_n at_o foot_n 38_o 22_o at_o top_n 51_o 38_o on_o the_o outside_o 116_o 12_o rafter_n angle_n at_o foot_n 48_o 10_o at_o top_n 41_o 50_o for_o any_o other_o house_n by_o the_o number_n thus_o as_o suppose_v 18_o foot_n broad_a the_o extent_n of_o the_o compass_n from_o 40_o the_o breadth_n in_o the_o table_n to_o 18_o the_o breadth_n give_v shall_v reach_v the_o same_o way_n from_o 30_o the_o rafter_n in_o the_o table_n to_o 13-50_a the_o rafter_n require_v and_o from_o 36_o the_o hip_n in_o the_o table_n to_o 16-22_a the_o hip_n require_v and_o from_o 22-36_a the_o perpendicule_a in_o the_o table_n to_o 10-06_a the_o perpendicule_a require_v and_o from_o â—6-57_a the_o diagonal_a in_o the_o table_n to_o â—5-48_a the_o diagonal_a require_v the_o angles_n be_v always_o the_o same_o in_o all_o rooss_n small_a or_o great_a as_o in_o the_o table_n be_v square_a and_o true_a pitch_n if_o you_o will_v have_v direction_n for_o bevel_n or_o taper_n frames_n to_o find_v the_o length_n and_o angle_n of_o raâ—ters_n and_o hip_n you_o may_v have_v it_o at_o large_a in_o a_o appendix_n to_o the_o mirror_n of_o architecture_n or_o vincent_n stamoâ—â—i_n print_v for_o william_n fisher_n at_o the_o postern-gate_n 1669._o by_o which_o direction_n and_o the_o sector_n you_o may_v find_v any_o thing_n that_o be_v there_o set_v down_o as_o also_o by_o the_o triangule_a quadrant_n thread_n and_o compass_n note_v also_o that_o have_v inch_n and_o foot-measure_n together_o you_o may_v present_o by_o inspection_n find_v the_o price_n of_o one_o foot_n have_v the_o price_n of_o the_o square_n and_o the_o contrary_n also_o have_v the_o 12_o inch_n on_o the_o other_o foot_n divide_v into_o 85_o part_n near_o and_o figure_v at_o every_o 8_o with_o 1_o 2_o 3_o 4_o 5_o 6_o 7_o 8_o 9_o 10._o shall_v represent_v penny_n and_o half_a farthing_n then_o at_o any_o price_n the_o rod_n you_o have_v the_o price_n of_o one_o foot_n &_o the_o contrary_n as_o thus_o let_v every_o inch_n represent_v one_o pound_n 〈◊〉_d â—very_n 8_o
the_o fragment_n of_o a_o globe_n but_o the_o other_o be_v by_o a_o way_n find_v out_o by_o mr._n bennit_n a_o cooper_n that_o have_v long_o exercise_v the_o way_n of_o gage_v which_o be_v by_o compare_v a_o cask_n know_v and_o its_o quantity_n of_o emptiness_n to_o a_o cask_n unknown_a and_o its_o inch_n of_o emptiness_n as_o follow_v first_o by_o the_o line_n of_o number_n and_o artificial_a line_n of_o segment_n to_o find_v the_o quantity_n of_o gallon_n that_o any_o vessel_n want_v of_o be_v full_a at_o any_o number_n of_o inch_n from_o the_o inside_n of_o the_o bunghole_n to_o the_o superficies_n of_o the_o liquor_n which_o be_v usual_o call_v inch_n dry_a extend_v the_o compass_n on_o the_o line_n of_o number_n from_o the_o inch_n and_o ten_o diameter_n at_o the_o bung_n to_o 100_o on_o the_o line_n of_o segment_n the_o same_o extent_n apply_v the_o same_o way_n from_o the_o inch_n and_o part_n dry_a shall_v reach_v to_o a_o four_o number_n on_o the_o line_n of_o artificial_a segment_n which_o four_o number_n you_o must_v keep_v or_o if_o you_o will_v you_o may_v use_v the_o inch_n wet_a lay_v the_o same_o extent_n from_o the_o inch_n wet_a and_o that_o also_o will_v on_o the_o segment_n give_v a_o four_o number_n which_o you_o must_v likewise_o keep_v then_o second_o as_o the_o extent_n from_o 1_o to_o the_o whole_a content_a of_o the_o vessel_n in_o wine_n or_o ale-gallon_n so_o be_v the_o four_o number_n keep_v to_o the_o gallon_n of_o emptiness_n or_o fullness_n that_o it_o want_v of_o be_v full_a or_o the_o quantity_n of_o gallon_n in_o the_o vessel_n example_n of_o a_o canary-pipe_n who_o diameter_n at_o bung_n be_v 28_o inch_n and_o 7_o and_o full_a content_n be_v gallon_n 116_o ½_o at_o 12_o inch_n dry_a or_o 16_o inch_n and_o 7_o ten_o wet_v the_o extent_n of_o the_o compass_n from_o 28-7_a to_o 100_o at_o the_o end_n of_o the_o line_n of_o segment_n shall_v reach_v the_o same_o way_n from_o 12_o the_o inch_n dry_a to_o 39_o ½_o on_o the_o line_n of_o segment_n for_o a_o four_o or_o from_o 16-7_a wet_a to_o 60_o 2_o 10_o on_o the_o segment_n for_o his_o four_o also_o which_o two_o 4th_n keep_v then_o second_o the_o extent_n from_o 1_o to_o 116_o ½_o the_o whole_a content_a in_o gallon_n shall_v reaâ—h_v from_o 39_o the_o dry_a four_o on_o the_o line_n of_o number_n to_o 46_o 3_o 10_o for_o the_o gallon_n dry_a or_o want_v or_o the_o same_o extent_n shall_v reach_v the_o same_o way_n on_o the_o line_n of_o number_n from_o 60_o 2_o 10_o the_o four_o number_n for_o wet_a to_o 70_o gallon_n and_o 2_o ten_o in_o the_o vessel_n at_o 16_o inch_n and_o 7_o ten_o wet_a which_o two_o number_n put_v together_o make_v up_o 116_o gallon_n and_o a_o half_a the_o full_a content_a the_o like_a manner_n of_o work_v serve_v for_o any_o cask_n whatsoever_o and_o the_o near_o the_o vessel_n want_v of_o be_v half_o empty_a the_o more_o near_o to_o the_o truth_n will_v your_o work_n be_v and_o the_o most_o error_n in_o very_o round_o and_o swell_a cask_n when_o the_o emptiness_n be_v not_o above_o one_o or_o two_o inch_n but_o in_o vessel_n near_o to_o cillender_n it_o will_v give_v the_o answer_n very_o true_a and_o as_o ready_o as_o any_o way_n whatsoever_o observe_v also_o that_o if_o you_o use_v the_o segment_n in_o take_v the_o want_n you_o must_v abate_v of_o the_o gallon_n find_v till_o you_o come_v to_o the_o 2_o three_o of_o the_o half_a diameter_n that_o be_v to_o say_v the_o rule_n say_v there_o be_v more_o want_v than_o indeed_o there_o be_v and_o that_o somewhat_o considerable_a about_o the_o first_o 6_o inch_n in_o a_o vessel_n of_o 30_o inch_n diameter_n so_o that_o i_o find_v a_o table_n make_v as_o a_o mean_a between_o the_o superficial_a and_o solid_a segment_n will_v do_v the_o work_n the_o true_a and_o best_a of_o any_o other_o or_o else_o use_v the_o mean_a diameter_n and_o mean_a part_n of_o emptiness_n find_v thus_o take_v the_o equade_v diameter_n from_o the_o diameter_n at_o the_o bung_n and_o note_v the_o difference_n then_o half_a this_o difference_n take_v from_o the_o inch_n and_o part_n empty_a give_v the_o mean_a emptiness_n then_o use_v the_o mean_a diameter_n and_o mean_a emptiness_n instead_o of_o the_o other_o and_o the_o work_n be_v more_o exact_a the_o other_o way_n of_o mr._n bennits_n invention_n be_v thus_o first_o you_o be_v to_o fill_v a_o ordinary_a cask_n of_o a_o competent_a magnitude_n as_o 60_o or_o 100_o gallon_n of_o a_o mean_a form_n between_o a_o spheriord_n or_o roundish_n form_n and_o a_o cillenderical_a form_n or_o else_o fill_v two_o cask_n of_o each_o form_n and_o learn_v the_o true_a content_a and_o diameter_n of_o that_o mean_a vessel_n or_o rather_o of_o both_o those_o vessel_n and_o the_o vessel_n be_v full_a draw_v off_o with_o a_o true_a gallon-measure_n and_o on_o the_o draw_v off_o every_o gallon_n take_v the_o exact_a quantity_n of_o inch_n and_o 10_o part_n that_o the_o draw_v off_o of_o every_o gallon_n make_v in_o the_o emptiness_n or_o dryness_n of_o that_o mean_a vessel_n or_o rather_o both_o those_o vessel_n at_o least_o until_o you_o have_v draw_v off_o the_o half_a quantity_n of_o the_o vessel_n which_o number_n of_o gallon_n draw_v off_o and_o the_o inch_n and_o ten_o part_n of_o emptiness_n or_o fullness_n or_o dryness_n or_o wetness_n you_o must_v draw_v into_o a_o table_n or_o insert_v they_o on_o a_o rule_n make_v the_o inch_n as_o equal_a part_n and_o the_o gallon_n and_o his_o proportional_a part_n of_o a_o gallon_n the_o unequal_a part_n then_o with_o the_o line_n of_o number_n and_o this_o mean_a table_n or_o rather_o two_o table_n or_o scale_n which_o you_o may_v put_v on_o a_o rule_n as_o mr._n bennit_n have_v do_v you_o may_v find_v out_o the_o want_n of_o any_o cask_n whatsoever_o either_o spherioid_a or_o cillender-like_a as_o follow_v this_o measure_a cask_n on_o the_o scale_n or_o table_n for_o method_n sake_n and_o avoid_v tautology_n i_o shall_v call_v the_o first_o cask_n and_o the_o vessel_n or_o cask_n who_o want_n you_o will_v know_v i_o shall_v call_v the_o second_o cask_n then_o the_o proportion_n be_v thus_o as_o the_o diameter_n at_o the_o bung_n of_o the_o second_o cask_n be_v to_o the_o bung_a diameter_n of_o the_o first_o cask_n which_o be_v always_o fix_v so_o be_v the_o inch_n dry_a of_o the_o second_o cask_n to_o a_o four_o on_o the_o line_n of_o number_n which_o four_o number_n seek_v on_o the_o inch_n of_o your_o table_n or_o scale_n on_o the_o opposite-part_n of_o your_o scale_n or_o table_n give_v a_o 5_o number_n which_o you_o must_v keep_v then_o as_o the_o whole_a content_a of_o the_o first_o cask_n be_v to_o the_o whole_a content_a of_o the_o second_o cask_n so_o be_v the_o first_o number_n keep_v to_o the_o number_n of_o gallon_n the_o vessel_n want_v of_o be_v full_a at_o so_o many_o inch_n dry_a example_n there_o be_v such_o a_o scale_n make_v on_o purpose_n for_o victualler_n use_v to_o measure_v what_o they_o want_v of_o a_o barrel_n of_o ale_n be_v put_v into_o a_o beer-barrel_n which_o scale_n i_o shall_v here_o use_v to_o try_v this_o former_a example_n by_o suppose_v as_o before_o a_o canary_n pipe_v want_v 12_o inch_n of_o be_v full_a and_o the_o content_a 116_o ½_o gallon_n and_o 28_o inch_n and_o 7_o ten_o diameter_n at_o bung_n the_o extent_n on_o the_o line_n of_o number_n from_o 28-7_a to_o 22-5_a shall_v reach_v from_o 12_o to_o 9-4_a then_o just_a against_o 9_o inch_n and_o 4_o ten_o on_o that_o barrel_n scale_n i_o find_v 14_o gallon_n of_o beer_n which_o be_v 17_o gallon_n and_o a_o half_a of_o wine_n be_v the_o 5_o number_n to_o be_v keep_v then_o the_o extent_n from_o 44_o the_o content_a of_o a_o barrel_n in_o wine-gallon_n to_o 116_o ½_o the_o content_a of_o a_o canary-pipe_n in_o the_o same_o gallon_n shall_v reach_v the_o same_o way_n from_o 17_o ½_o the_o number_n keep_v to_o 46_o and_o near_o a_o half_a the_o gallon_n want_v at_o 12_o inch_n dry_a in_o the_o canary_n pipe_n and_o 46_o gallon_n and_o 3_o quart_n be_v the_o number_n mr._n bennit_n find_v in_o a_o canary-pipe_n by_o measure_v at_o 12_o inch_n dry_a thus_o you_o have_v a_o account_n of_o the_o two_o easy_a mechanic_n way_n to_o discover_v the_o want_v of_o cask_n very_o applicable_a and_o ready_a and_o experiment_v to_o be_v propè_fw-la verum_fw-la the_o gallon_n want_v in_o a_o barrel_n at_o every_o inch_n and_o quarter_n  _fw-fr beer_n gall._n wine_n gall_n  _fw-fr  _fw-fr  _fw-fr gall_n pi_fw-la 100_o gall_n pi_fw-la 100_o gall_n 1000_o  _fw-fr  _fw-fr 0_o 0_o 40_o 0_o 0_o 49_o 0_o 0612_o  _fw-fr  _fw-fr 0_o 1_o 20_o 0_o 1_o 47_o 0_o
the_o near_a distance_n than_o the_o nd_o from_o the_o right_n sine_fw-la of_o the_o angle_n to_o the_o thread_n shall_v be_v the_o tangent_fw-la require_v example_n i_o make_v ab_fw-la radius_fw-la a_o =_o in_o 50_o the_o cousin_a of_o 40_o than_o the_o =_o sine_fw-la of_o 40_o shall_v be_v be._n again_o if_o i_o make_v ad_fw-la equal_a to_o ab_fw-la the_o =_o cousin_a of_o 30_o viz._n 60_o and_o then_o take_v out_o the_o =_o sine_fw-la of_o 30_o and_o lay_v it_o from_o d_o to_o f_o it_o shall_v be_v a_o angle_n of_o 60_o from_o ab_fw-la or_o 30_o from_o d_o to_o f._n but_o by_o the_o sector_n this_o be_v more_o easy_a use_v x._o to_o take_v out_o ready_o any_o tangent_fw-la above_o 45_o by_o the_o tangent_fw-la to_o 45_o on_o the_o sector-side_n sector_n take_v the_o give_v radius_fw-la make_v it_o a_o =_o in_o the_o co-tangent_a of_o the_o tangent_fw-la require_v then_o the_o =_o tangent_fw-la of_o 45_o shall_v be_v the_o tangent_fw-la require_v example_n i_o will_v have_v a_o tangent_fw-la to_o 80_o degree_n take_v the_o give_v radius_fw-la make_v it_o a_o =_o in_o 10_o the_o compliment_n of_o 80_o then_o the_o =_o tangent_fw-la of_o 45_o shall_v be_v the_o tangent_fw-la of_o 80_o require_v but_o if_o your_o radius_fw-la be_v so_o big_a that_o you_o can_v enter_v it_o then_o take_v the_o half_a or_o a_o quarter_n of_o your_o radius_fw-la and_o then_o =_o 45_o will_v be_v the_o half_a or_o the_o quarter_n of_o the_o tangent_fw-la require_v use_v xi_o how_o to_o work_v proportion_n in_o sines_n alone_o by_o the_o natural_a sin_n there_o be_v 4_o variety_n in_o this_o work_n that_o include_v all_o proportion_n viz._n 1._o when_o the_o sine_fw-la of_o 90_o be_v the_o first_o term_n than_o the_o work_n be_v thus_o lay_v the_o thread_n to_o the_o second_o term_n count_v on_o the_o degree_n from_o the_o head_n quadr_n towards_o the_o loose-piece_n and_o count_v the_o three_o term_n on_o the_o line_n of_o sines_n from_o the_o centre_n downward_o and_o take_v the_o near_a distance_n from_o thence_o to_o the_o thread_n and_o that_o distance_n measure_v from_o the_o centre_n downward_o on_o the_o line_n of_o sines_n give_v the_o four_o term_v require_v example_n as_o sine_fw-la 90_o to_o sine_fw-la 23-30_a so_o be_v 30_o to_o 11-31_a take_v the_o latteral_a second_o term_n sector_n make_v it_o a_o =_o sine_fw-la of_o 90_o then_o take_v out_o the_o =_o three_o term_n and_o measure_v it_o from_o the_o centre_n it_o give_v the_o four_o term_v require_v 2._o when_o the_o sine_fw-la of_o 90_o be_v the_o three_o term_n then_o work_v thus_o take_v the_o sine_fw-la of_o the_o second_o term_n from_o the_o centre_n downward_o quadr._n and_o make_v it_o a_o =_o sine_fw-la in_o the_o first_o term_n lay_v the_o thread_n to_o nd_o then_o on_o the_o degree_n the_o thread_n shall_v give_v the_o four_o term_v require_v example_n as_o the_o sine_fw-la of_o 30_o to_o 23-31_a so_o be_v the_o sine_fw-la of_o 90_o to_o sine_fw-la of_o 52-56_a but_o by_o the_o sector_n sector_n take_v the_o sine_fw-la of_o the_o second_o make_v it_o a_o =_o sine_fw-la of_o the_o first_o term_n then_o take_v out_o =_o 90_o and_o measure_v it_o from_o the_o centre_n and_o it_o shall_v give_v the_o four_o term_v require_v example_n as_o before_o 3._o when_o the_o radius_fw-la or_o sine_fw-la of_o 90_o be_v in_o the_o second_o place_n work_v thus_o quadr._n take_v 90_o from_o a_o lesser_a scale_n as_o the_o uppermost_a sine_fw-la above_o the_o centre_n or_o the_o line_n of_o right-ascentions_a or_o the_o azimuth-scale_n or_o the_o like_a and_o make_v it_o a_o =_o in_o the_o sine_fw-la of_o the_o first_o lay_v the_o thread_n to_o nd_o than_o the_o =_o three_o term_n take_v and_o measure_v on_o the_o same_o scale_n that_o 90_o be_v take_v from_o shall_v give_v the_o four_o term_v require_v example_n as_o 90_o on_o the_o line_n of_o rightascention_n be_v to_o =_o 30_o so_o be_v =_o 20_o to_o 43-12_a measure_v on_o the_o same_o line_n that_o 90_o be_v take_v from_o or_o else_o second_o work_v thus_o as_o 30_o to_o =_o 90_o so_o be_v 20_o to_o =_o 43-12_a by_o carry_v the_o compass_n till_o it_o so_o stay_v as_o that_o the_o foot_n turn_v about_o will_v but_o just_a touch_n the_o thread_n at_o the_o near_a distance_n or_o else_o thus_o three_o by_o transpose_v the_o term_n when_o the_o three_o be_v not_o great_a than_o the_o first_o thus_o as_o the_o first_o to_o the_o three_o so_o be_v the_o second_o term_n to_o the_o four_o where_o the_o radius_fw-la be_v bring_v to_o the_o three_o place_n it_o be_v wrought_v by_o the_o second_o rule_n as_o before_o by_o the_o sector_n take_v a_o small_a sine_fw-la of_o 90_o make_v it_o a_o =_o in_o 30_o then_o the_o =_o sine_fw-la of_o 20_o take_v and_o measure_v on_o the_o small_a sine_fw-la give_v 43-12_a as_o before_o again_o as_o 90_o to_o =_o 30_o so_o be_v =_o 20_o to_o 43-12_a again_o as_o 30_o to_o =_o 90_o so_o be_v =_o 20_o to_o =_o 43-12_a last_o by_o transpose_v as_o 20_o to_o =_o 30_o so_o be_v =_o radius_fw-la to_o 43-12_a as_o before_o 4._o when_o radius_fw-la be_v none_o of_o the_o give_v term_n then_o when_o the_o first_o term_n be_v great_a than_o the_o second_o and_o three_o quadr._n work_v thus_o take_v the_o second_v term_n make_v it_o a_o =_o in_o the_o first_o lay_v the_o thread_n to_o the_o nd_o then_o the_o near_a distance_n from_o the_o three_o term_n to_o the_o thread_n measure_v from_o the_o centre_n downward_o give_v the_o four_o sine_fw-la require_v example_n as_o 20_o to_o 12_o so_o be_v 18_o to_o 10-50_a by_o the_o quadrant_n as_o 12_o to_z =_o 20_o so_o be_v =_o 18_o to_o 10-50_a when_o only_o the_o second_o term_n be_v great_a than_o the_o first_o then_o transpose_v the_o term_n and_o work_v as_o before_o or_o else_o use_v a_o double_a radius_fw-la which_o be_v on_o this_o instrument_n very_o easy_o do_v have_v several_a radiuss_n or_o last_o use_v a_o parallel_n entrance_n or_o answer_v rather_o as_o before_o which_o be_v careful_o wrought_v will_v do_v very_o well_o by_o the_o sector_n the_o same_o manner_n of_o work_n be_v as_o before_o by_o the_o quadrant_n and_o the_o set_v the_o sector_n be_v all_o one_o to_o the_o lay_v the_o thread_n as_o will_v be_v large_o see_v in_o all_o the_o follow_a proposition_n wrought_v both_o by_o the_o artificial_a and_o natural_a line_n of_o number_n sin_n and_o tangent_n as_o follow_v use_v xii_o have_v the_o day_n of_o the_o month_n or_o sun_n place_n give_v to_o find_v his_o declination_n lay_v the_o thread_n on_o the_o day_n of_o the_o month_n in_o the_o calendar_n particular_o and_o in_o the_o line_n of_o degree_n on_o the_o moving-leg_n you_o have_v his_o declination_n either_o northward_o or_o southward_o according_a to_o the_o time_n of_o the_o year_n count_v from_o 600_o towards_o the_o head_n for_o north-declination_n or_o towards_o the_o end_n for_o south-declination_n by_o the_o artificial_a sin_n and_o tangent_n on_o the_o edge_n of_o the_o instrument_n extend_v the_o compass_n from_o the_o sine_fw-la of_o 90_o to_o the_o sine_fw-la of_o 23_o degree_n 31_o minute_n the_o sun_n great_a declination_n the_o same_o extent_n apply_v the_o same_o way_n from_o the_o sine_fw-la of_o the_o sun_n place_n or_o the_o sun_n distance_n from_o the_o next_o equinoctial-point_n shall_v cause_v the_o moving-point_n to_o fall_v on_z the_o sine_z of_o the_o sun_n declination_n this_o be_v the_o general_a way_n of_o work_v example_n the_o extent_n from_o the_o sine_fw-la of_o 90_o to_o the_o sine_fw-la of_o 23-31_a shall_v reach_v from_o the_o sine_fw-la of_o 30_o to_o 11_o deg_n 31_o min._n the_o sun_n declination_n in_o ♉_o taurus_n 30_o degree_n from_o ♈_o aries_n the_o next_o equinoctial-point_n and_o from_o 60_o degree_n the_o sun_n distance_n in_o ♊_o gemini_fw-la 60_o degree_n from_o ♈_o 20_o deg_n 12_o min._n the_o sun_n declination_n then_o this_o be_v the_o manner_n of_o work_v by_o these_o line_n by_o extend_v the_o compass_n from_o the_o first_o to_o the_o second_o term_n i_o shall_v for_o the_o rest_n wave_v this_o large_a repetition_n of_o extend_v the_o compass_n and_o render_v it_o only_o thus_o by_o the_o word_n of_o the_o cannon-general_a in_o all_o book_n as_o sine_fw-la 90_o to_o sine_fw-la 23-31_a so_o be_v the_o sine_fw-la of_o 30_o to_o sine_fw-la 11-31_a lay_v the_o thread_n to_o 23-31_a generally_n on_o the_o degree_n on_o the_o moveable-piece_n count_v from_o the_o head_n towards_o the_o end_n then_o count_v the_o sun_n place_n from_o the_o next_o equinoctial-point_n on_o the_o line_n of_o sines_n from_o the_o centre_n downward_o and_o take_v the_o nd_o from_o thence_o to_o the_o thread_n then_o this_o distance_n be_v measure_v from_o the_o centre_n downward_o
of_o the_o latitude_n to_o the_o =_o secant_fw-la of_o the_o sun_n present_a altitude_n lay_v the_o thread_n to_o nd_o so_o be_v the_o sine_fw-la of_o the_o sum_n or_o difference_n of_o the_o sun_n present_a altitude_n &_o vertical_a depression_n in_o winter_n or_o the_o difference_n between_o his_o vertical_a and_o present_a altitude_n in_o summer_n to_o the_o =_o sine_fw-la of_o the_o sun_n azimuth_n at_o that_o altititude_n and_o declination_n yet_o again_o more_o short_a as_o c.t._n lat._n to_o =_o sec._n ☉_o alt._n so_o s._n of_o sum_n or_o difference_n to_o =_o s._n ☉_o azimuth_n but_o note_n that_o in_o latitude_n under_o 45_o when_o the_o compliment_n of_o the_o latitude_n be_v too_o large_a then_o work_v thus_o as_o the_o cousin_a of_o the_o sun_n altitude_n to_o =_o tangent_fw-la of_o the_o latitude_n take_v from_o the_o degree_n on_o the_o moveable-leg_n lay_v the_o thread_n to_o nd_o than_o the_o sine_fw-la of_o the_o sum_n or_o difference_n carry_v parallelly_n shall_v stay_v at_o the_o sun_n azimuth_n require_v if_o the_o tangent_n be_v too_o small_a on_o the_o sector-side_n be_v a_o large_a and_o if_o the_o sin_n be_v too_o great_a on_o the_o head-leg_n there_o be_v a_o less_o find_v the_o vertical_a altitude_n by_o use_n 18_o and_o the_o sum_n or_o difference_n of_o the_o present_a and_o vertical_a altitude_n by_o the_o table_n tangent_n or_o line_n of_o natural_a sin_n as_o before_o show_v then_o the_o canon_n or_o proportion_n run_v thus_o as_o the_o cousin_a of_o the_o sun_n altitude_n to_o the_o tangent_fw-la of_o the_o latitude_n so_o be_v the_o sine_fw-la of_o the_o sum_n or_o difference_n to_o the_o sine_fw-la of_o the_o azimuth_n from_o east_n or_o west_n or_o as_o co-tangent_a latitude_n to_o secant_fw-la of_o the_o sun_n altitude_n so_o be_v the_o sine_fw-la of_o the_o sum_n or_o difference_n to_o the_o sine_fw-la of_o the_o azimuth_n sector_n make_v the_o secant_fw-la of_o the_o sun_n altitude_n a_o =_o co-tangent_a of_o latitude_n then_o the_o sine_fw-la of_o the_o sum_n or_o difference_n shall_v be_v half_a the_o sine_fw-la of_o the_o azimuth_n or_o be_v turn_v twice_o from_o the_o centre_n the_o whole_a sine_z or_o else_o thus_o make_v the_o tangent_fw-la of_o the_o latitude_n a_o =_o cousin_a of_o the_o sun_n altitude_n then_o the_o =_o sine_fw-la of_o the_o sum_n or_o difference_n shall_v be_v the_o sine_fw-la of_o the_o azimuth_n measure_v on_o the_o sine_fw-la equal_a to_o the_o radius_fw-la of_o the_o tangent_n first_o take_v example_n in_o latitude_n 51-32_a declination_n north_n and_o south_n 13-15_a the_o vertical_a altitude_n or_o depression_n be_v 17-01_a and_o the_o present_a altitude_n 20_o the_o azimuth_n for_o south-declination_n will_v be_v find_v to_o be_v 31-45_a from_o south_n the_o depression_n at_o east_n and_o west_n be_v 17-01_a and_o the_o sum_n of_o the_o present_a altitude_n and_o depression_n 39-25_a again_o for_o north-declination_n or_o summertime_n the_o difference_n between_o the_o vertical_a and_o present_a altitude_n be_v 2-54_a and_o the_o azimuth_n from_o south_n will_v be_v find_v to_o be_v 86_o degree_n and_o 15_o minute_n use_v xxviii_o to_o make_v a_o scale_n whereby_o to_o perform_v all_o thâ—se_a proposition_n by_o the_o former_a rule_n agreeable_a to_o the_o triangule_a quadrant_n be_v add_v chief_o as_o a_o demonstration_n of_o the_o instrument_n and_o former_a operation_n then_o make_v de_fw-fr radius_fw-la describe_v the_o circle_n 90_o ei_o and_o divide_v it_o into_o 180_o equal_a degree_n also_o draw_v the_o lesser_a circle_n 90_o f_o to_o the_o radius_fw-la df_n than_o a_o rule_n lay_v to_o the_o centre_n d_o and_o every_o one_o of_o the_o 180_o degree_n shall_v divide_v the_o tangent_fw-la line_n ac_fw-la and_o bc_n into_o 180_o degree_n and_o if_o you_o work_v right_a you_o will_v meet_v with_o all_o the_o former_a point_n f_o g_o 45_o i_o 69-54_a 60-45_a h_n and_o e_z in_o their_o true_a place_n as_o first_o draw_v also_o perpendiculer_o let_v fall_v from_o every_o degree_n in_o the_o circle_n 90_o ei_o to_o the_o line_n db_n shall_v divide_v the_o line_n of_o sines_n d_o 90_o to_o the_o the_o great_a radius_fw-la and_o the_o like_a perpendiculer_o from_o the_o degree_n in_o the_o lesser_a circle_n to_o the_o line_n dam_fw-ge shall_v divide_v the_o lesser_a line_n of_o sines_n also_o the_o extent_n from_o the_o centre_n d_o to_o the_o tangent_fw-la of_o any_o ark_n or_o angle_n in_o the_o line_n ac_fw-la count_v from_o f_o shall_v be_v the_o secant_fw-la to_o that_o ark_n or_o angle_n to_o the_o lesser_a radius_fw-la and_o the_o measure_n from_o the_o centre_n d_o to_o the_o tangent_fw-la of_o any_o ark_n or_o angle_n in_o the_o line_n cb_n but_o count_v from_o e_z shall_v be_v the_o secant_fw-la to_o that_o ark_n or_o angle_n to_o the_o great_a radius_fw-la this_o little_a instrument_n thus_o make_v and_o a_o thread_n fasten_v at_o d_o will_v perform_v any_o proposition_n by_o the_o rule_n here_o insert_v and_o be_v the_o very_a make_n of_o the_o triangule_a quadrant_n or_o you_o may_v put_v these_o line_n on_o a_o rule_n as_o a_o plain_a scale_n and_o use_v they_o thus_o as_o for_o example_n for_o the_o azimuth_n last_o treat_v on_o first_o draw_v a_o straight_a line_n as_o ab_fw-la represent_v the_o line_n ab_fw-la in_o the_o trianguler-quadrant_n then_o appoint_v in_o that_o line_n any_o point_n for_o a_o centre_n as_o c_z then_o for_o this_o proposition_n of_o find_v the_o azimuth_n the_o sin_n and_o tangent_n be_v on_o a_o straight_a scale_n work_v thus_o first_o to_o find_v the_o sun_n altitude_n or_o depression_n in_o the_o vertical-circle_n take_v the_o sine_fw-la of_o the_o latitude_n and_o lay_v it_o from_o c_o to_o 51-30_a then_o take_v out_o the_o sine_fw-la of_o 13-15_a between_o your_o compass_n and_o set_v one_o point_n in_o the_o point_n 51-30_a last_o make_v in_o the_o line_n ab_fw-la and_o strike_v the_o touch_n of_o the_o arch_n at_o d_o and_o draw_v the_o line_n cd_o also_o on_o the_o line_n cb_n lay_v down_o from_o c_o the_o sine_fw-la of_o 90_o out_o of_o the_o scale_n than_o the_o near_a distance_n from_o the_o point_n for_o 90_o in_o cb_n to_o the_o line_n cd_o shall_v be_v the_o sine_fw-la of_o the_o sun_n altitude_n in_o the_o vertical_a in_o summer_n or_o northern_a declination_n or_o his_o depression_n in_o winter_n viz._n 17-01_a then_o as_o before_z on_o the_o line_n of_o sines_n find_v a_o sum_n for_o winter_n or_o a_o difference_n in_o summer_n between_o the_o vertical_a and_o present_a altitude_n now_o suppose_v the_o altitude_n 15_o the_o sum_n be_v 33-30_a or_o the_o difference_n be_v 1-58_a which_o you_o must_v remember_v then_o take_v the_o secant_fw-la of_o 15_o the_o sun_n present_a altitude_n from_o the_o scale_n lay_v it_o from_o c_o to_o e_z then_o take_v out_o the_o co-tangent_a of_o the_o latitude_n between_o your_o compass_n set_v one_o point_n in_o e_o and_o strike_v the_o touch_n of_o a_o ark_n as_o at_o f_o and_o draw_v the_o line_n cf_n then_o take_v the_o sine_fw-la of_o 33-30_a the_o sum_n if_o it_o be_v winter_n or_o 1-58_a if_o it_o be_v summer_n between_o your_o compass_n carry_v one_o point_n in_o the_o line_n cb_n high_a or_o low_o till_o the_o other_o foot_n turn_v about_o will_v but_o just_a touch_n the_o line_n cf_n then_o the_o measure_n from_o thence_o to_o the_o point_n c_o shall_v be_v the_o sine_fw-la of_o the_o azimuth_n require_v viz._n in_o winter_n 43-50_a and_o in_o summer_n 92-30_a from_o the_o south_n because_o the_o present_a altitude_n be_v less_o than_o the_o vertical_a or_o east_n and_o west_n but_o when_o the_o co-tangent_a of_o the_o latitude_n be_v too_o large_a for_o a_o parallel_n entrance_n then_o prick_v off_o first_o the_o tangent_fw-la of_o the_o latitude_n and_o take_v the_o cousin_a of_o the_o sun_n altitude_n to_o work_v in_o a_o parallel_n way_n which_o will_v remedy_v the_o inconvenience_n thus_o you_o see_v that_o by_o draw_v three_o line_n only_o this_o work_n be_v do_v yet_o not_o so_o soon_o by_o far_o as_o by_o the_o instrument_n with_o the_o thread_n which_o represent_v those_o line_n more_o certain_o and_o exact_o after_o the_o same_o way_n of_o operation_n to_o find_v the_o sun_n azimuth_n in_o southern_a declination_n as_o the_o cousin_a of_o the_o latitude_n to_o the_o sine_fw-la of_o the_o sun_n present_a altitude_n t._n so_o be_v the_o sine_fw-la of_o the_o latitude_n to_o a_o four_o sine_fw-la which_o four_o sine_z be_v to_o be_v add_v to_o the_o sun_n amplitude_n for_o that_o time_n on_o a_o line_n of_o natural_a sin_n and_o the_o sum_n observe_v as_o a_o 5_o then_o as_o the_o cousin_a of_o the_o present_a altitude_n be_v to_o the_o sine_fw-la of_o the_o sum_n last_o find_v so_o be_v the_o sine_fw-la of_o 90_o to_o the_o sine_fw-la of_o the_o sun_n azimuth_n
four_o ark_n be_v take_v from_o the_o sun_n distance_n from_o the_o elevate_v pole_n than_o the_o residue_n be_v the_o 5_o ark_n but_o for_o hour_n before_o and_o after_o 6_o add_v the_o four_o ark_n and_o the_o sun_n distance_n from_o the_o pole_n together_o to_o make_v a_o 5_o ark._n then_o say_v as_o the_o cousin_a of_o the_o four_o ark_n to_o the_o sine_fw-la of_o the_o latitude_n so_o be_v the_o cousin_a of_o the_o residue_n or_o sum_n be_v the_o 5_o ark_n to_o the_o sine_fw-la of_o the_o sun_n altitude_n at_o that_o hour_n use_v xxxviii_o the_o latitude_n sun_n azimuth_n and_o declination_n give_v to_o find_v the_o altitude_n or_o height_n thereof_o first_o to_o find_v the_o sun_n altitude_n at_o all_o azimuth_n in_o the_o equinoctial_a t._n as_o sine_fw-la 90_o to_o co-tangent_a latitude_n so_o be_v the_o cousin_a azimuth_n from_o south_n to_o the_o tangent_fw-la of_o the_o sun_n altitude_n in_o aries_n or_o as_o sine_fw-la 90_o to_o the_o cousin_a of_o the_o azimuth_n from_o south_n so_o be_v co-tangent_a lat._n to_o the_o tangent_fw-la of_o the_o sun_n altitude_n at_o that_o azimuth_n in_o the_o equinoctial_a which_o you_o must_v gather_v into_o a_o table_n for_o every_o single_a degree_n then_o as_o the_o sine_fw-la lat._n to_o the_o sine_fw-la of_o the_o sun_n declination_n so_o be_v the_o cousin_a of_o the_o sun_n altitude_n in_o equinoctial_a to_o the_o sine_fw-la of_o a_o four_o ark._n then_o when_o the_o latitude_n and_o declination_n be_v alike_o as_o both_o north_n or_o south_n then_o add_v the_o four_o ark_n and_o the_o altitude_n in_o the_o equator_fw-la together_o and_o the_o sum_n be_v the_o altitude_n require_v but_o in_o wintertime_n when_o the_o latitude_n and_o declination_n be_v unlike_a take_v the_o four_o ark_n out_o of_o the_o reciprocal_a altitude_n in_o the_o equator_fw-la and_o the_o residue_n be_v the_o sun_n altitude_n require_v also_o in_o all_o azimuth_n from_o east_n and_o west_n northwards_o in_o summertime_n also_o you_o must_v use_v substraction_n also_o and_o not_o addition_n as_o the_o rule_n before-going_a suggest_v by_o the_o particular_a quadrant_n work_v thus_o take_v the_o sun_n or_o star_n declination_n from_o the_o particular_a scale_n and_o set_v one_o point_n in_o the_o sun_n azimuth_n on_o the_o azimuth_n line_n and_o with_o the_o other_o lie_v the_o thread_n to_o the_o nd_o the_o right_a way_n and_o on_o the_o degree_n the_o thread_n cut_v the_o altitude_n require_v by_o the_o general_n quadrant_n as_o the_o co-tangent_a latitude_n take_v from_o the_o moving-leg_n or_o loose-piece_n to_o =_o sine_fw-la of_o 90_o lay_v the_o thread_n to_o nd_o so_o be_v the_o =_o cousin_a of_o the_o sun_n azimuth_n from_o south_n to_o the_o tangent_fw-la of_o the_o sun_n altitude_n in_o the_o equator_fw-la at_o that_o reciprocal_a azimuth_n which_o be_v remember_v or_o gather_v into_o a_o table_n together_o then_o say_v as_o the_o cousin_a of_o the_o sun_n altitude_n in_o the_o equator_fw-la to_o the_o =_o sine_fw-la of_o the_o latitude_n lay_v the_o thread_n to_o the_o nd_o so_o be_v the_o =_o sine_fw-la of_o the_o sun_n declination_n to_o the_o sine_fw-la of_o the_o four_o ark._n which_o four_o ark_n be_v to_o be_v add_v or_o substract_v as_o immediate_o before_o be_v direct_v and_o the_o sum_n or_o residue_n shall_v be_v the_o true_a altitude_n require_v example_n at_o 60_o degree_n of_o azimuth_n from_o south_n the_o equinoctial_a altitude_n will_v be_v find_v to_o be_v 21-40_a for_o london_n latitude_n of_o 51-32_a and_o the_o four_o ark_n in_o ♋_o or_o ♑_o be_v 28-16_a then_o 21-40_a the_o sun_n altitude_n at_o 60_o in_o ♈_o and_o 28-16_a the_o reciprocal_a four_o ark_n in_o ♋_o add_v make_v 49-56_a the_o sun_n altitude_n at_o 60_o degree_n from_o the_o south_n in_o ♋_o the_o same_o way_n of_o work_v serve_v for_o the_o sector_n as_o be_v use_v for_o the_o general_n quadrant_n sector_n only_o observe_v to_o set_v the_o sector_n instead_o of_o lay_v the_o thread_n to_o the_o near_a distance_n as_o the_o ingenious_a will_n soon_o perceive_v use_v thirty-nine_o have_v the_o latitude_n declination_n azimuth_n and_o altitude_n to_o find_v the_o hour_n as_o the_o cousin_a of_o the_o sun_n altitude_n sector_n to_o =_o cousin_a of_o the_o sun_n declination_n so_o be_v the_o =_o sine_fw-la of_o the_o sun_n azimuth_n to_o sine_z of_o the_o hour_n or_o else_o thus_o first_o find_v the_o altitude_n at_o that_o azimuth_n and_o then_o at_o that_o altitude_n and_o declination_n quadrant_n the_o hour_n as_o cousin_a of_o declination_n to_o =_o sine_fw-la of_o the_o azimuth_n so_o be_v the_o cousin_a altitude_n to_o =_o sine_fw-la of_o the_o hour_n tangent_n as_o cousin_a declination_n to_o the_o sine_fw-la of_o the_o azimuth_n so_o be_v cousin_a altitude_n to_o sine_fw-la of_o the_o hour_n use_v xl._o have_v the_o latitude_n declination_n hour_n and_o altitude_n to_o find_v the_o azimuth_n general-quadr_a as_o cousin_a of_o declination_n to_o =_o cousin_a of_o the_o sun_n altitude_n so_o be_v =_o sine_fw-la of_o the_o hour_n to_o sine_z of_o the_o azimuth_n quadrant_n first_o find_v the_o altitude_n at_o that_o hour_n and_o then_o the_o azimuth_n at_o that_o altitude_n as_o before_o t._n as_o cousin_a of_o altitude_n to_o sine_fw-la of_o the_o hour_n so_o be_v cousin_a declination_n to_o sine_fw-la of_o the_o azimuth_n from_o south_n or_o north_n as_o the_o hour_n be_v count_v that_o be_v to_o say_v from_o south_n if_o the_o hour_n be_v between_o 6_o at_o morning_n and_o 6_o at_o night_n and_o from_o the_o north_n if_o the_o contrary_a that_o be_v to_o say_v between_z 6_o at_o night_n and_o 6_o next_o morning_n or_o next_o to_o midnight_n use_v xli_o have_v the_o latitude_n and_o the_o sun_n declination_n to_o find_v the_o sun_n azimuth_n at_o 6._o as_o the_o sine_fw-la of_o 90_o t._n to_o the_o cousin_a of_o the_o latitude_n so_o be_v the_o tangent_fw-la of_o the_o sun_n declination_n to_o co-tangent_a of_o the_o sun_n azimuth_n from_o the_o north_n at_o the_o hour_n of_o 6._o first_o find_v the_o sun_n height_n at_o 6_o and_o then_o the_o sun_n azimuth_n at_o that_o altitude_n q._n make_v the_o tangent_fw-la of_o the_o declination_n a_o =_o sine_fw-la of_o 90_o quad_n lay_v the_o thread_n to_o nd_o than_o the_o =_o cousin_a latitude_n shall_v be_v the_o co-tangent_a of_o the_o sun_n azimuth_n from_o the_o north_n at_o 6._o use_v xlii_o to_o find_v the_o amplitude_n azimuth_n rise_v set_v and_o sothe_v of_o the_o fix_a star_n have_v the_o latitude_n altitude_n and_o declination_n or_o time_n of_o the_o year_n give_v first_o for_o the_o amplitude_n quadrant_n take_v the_o star_n declination_n out_o of_o the_o particular_a scale_n of_o altitude_n and_o measure_v it_o from_o 90_o in_o the_o azimuth-line_n and_o count_v the_o same_o way_n and_o the_o other_o point_n shall_v show_v the_o star_n amplitude_n require_v example_n the_o declination_n of_o the_o bull_n eye_n be_v 15-48_a if_o you_o take_v 15-48_a from_o the_o particular_a scale_n and_o lay_v it_o from_o 90_o in_o the_o azimuth-line_n it_o will_v reach_v to_o 26_o degree_n count_v from_o 90_o towards_o either_o end_n the_o same_z as_o for_o the_o sun_n in_o use_n 16._o but_o in_o other_o latitude_n work_v as_o you_o do_v for_o the_o sun_n by_o the_o rule_n in_o the_o 16_o use_v abovesaid_a for_o a_o star_n azimuth_n the_o work_n here_o be_v the_o same_o as_o for_o the_o sun_n thus_o take_v the_o star_n declination_n from_o the_o particular_a scale_n of_o altitude_n or_o sines_n between_o your_o compass_n and_o lay_v the_o thread_n to_o the_o star_n altitude_n count_v from_o 600_o towards_o the_o loose-piece_n then_o carry_v the_o compass_n or_o the_o rightside_n of_o the_o thread_n for_o northern-star_n and_o on_o the_o left-side_n for_o southern-star_n along_o the_o azimuth-line_n till_o the_o other_o foot_n be_v turn_v about_o will_v but_o just_a touch_n the_o thread_n then_o the_o fix_a point_n on_o the_o azimuth-line_n shall_v show_v the_o star_n azimuth_n from_o the_o south_n require_v example_n the_o bull_n eye_n be_v 30_o degree_n high_a shall_v have_v 77_o degree_n and_o 10_o minute_n of_o azimuth_n from_o the_o south_n if_o you_o be_v in_o other_o latitude_n use_v the_o general_a way_n as_o for_o the_o sun_n in_o all_o respect_n have_v the_o same_o declination_n that_o the_o star_n have_v north_n or_o south_n to_o find_v the_o star_n rise_v or_o set_v count_n the_o star_n declination_n on_o the_o degree_n as_o you_o count_v the_o sun_n north_n or_o south_n and_o there_o lie_v the_o thread_n and_o in_o the_o line_n of_o hour_n be_v the_o star_n rise_v or_o set_v when_o the_o star_n right_a ascention_n and_o declination_n be_v equal_a but_o at_o other_o time_n you_o must_v reckon_v thus_o first_o find_v the_o sun_n right_a ascension_n by_o use_v 14_o and_o set_v down_o