denominator_n than_o one_o which_o kind_n of_o break_a number_n be_v common_o set_v down_o with_o the_o particle_n of_o between_o they_o as_o five_o twelfth_n of_o one_o third_n be_v to_o be_v write_v thus_o 5_o 12_o of_o 1_o â—â—_o three_o fourth_n of_o seven_o eighth_n thus_o ¾_n of_o 7_o â—_o and_o so_o of_o any_o other_o 11._o the_o thing_n to_o be_v express_v by_o break_a number_n be_v chief_o the_o part_n or_o fraction_n of_o money_n weight_n measure_n time_n motion_n and_o thing_n account_v by_o the_o dozen_o of_o the_o three_o first_o of_o these_o there_o be_v infinite_a kind_n and_o variety_n according_a to_o the_o law_n and_o custom_n of_o particular_a country_n those_o use_v in_o this_o kingdom_n be_v most_o proper_a for_o we_o to_o know_v and_o the_o knowledge_n of_o they_o will_v be_v sufficient_a to_o direct_v we_o in_o the_o use_n of_o those_o in_o foreign_a nation_n if_o at_o any_o time_n there_o be_v a_o occasion_n for_o they_o 12._o the_o several_a piece_n of_o coin_n or_o denomination_n of_o money_n use_v in_o england_n in_o reference_n to_o account_v be_v pound_n shilling_n penny_n and_o farthing_n who_o particular_a value_n be_v as_o follow_v 4._o farthing_n make_v 1._o penny_n 12._o penny_n make_v 1._o shilling_n 20._o shilling_n make_v 1._o pound_n sterling_n and_o according_a to_o these_o value_n a_o pound_n sterling_n be_v esteem_v a_o integer_fw-la and_o may_v be_v divide_v into_o 20_o part_n call_v shilling_n and_o therefore_o one_o shilling_n be_v a_o break_a number_n of_o a_o pound_n sterling_n &_o by_o the_o former_a direction_n be_v to_o be_v write_v thus_o 1_o â—â—_o l._n that_o be_v one_o twenty_o of_o a_o pound_n again_o a_o shilling_n may_v be_v divide_v into_o twelve_o equal_a part_n call_v penny_n and_o so_o one_o penny_n be_v a_o fraction_n of_o a_o shilling_n and_o be_v to_o be_v write_v thus_o â—_o 12_o s._n that_o be_v one_o twelve_o of_o a_o shilling_n last_o a_o penny_n may_v be_v divide_v into_o four_o equal_a part_n call_v farthing_n and_o so_o one_o farthing_n be_v a_o fraction_n of_o a_o penny_n and_o write_v thus_o ¼_n d._n that_o be_v one_o four_o of_o a_o penny_n or_o thus_o ¼_n of_o 1_o 12_o s._n that_o be_v one_o four_o of_o one_o twelve_o of_o a_o shilling_n or_o thus_o ¼_n of_o 1_o 12_o of_o 1_o 20_o l._n that_o be_v one_o four_o of_o one_o twelve_o of_o one_o twenty_o of_o a_o pound_n sterling_n 13._o now_o than_o though_o the_o true_a and_o natural_a way_n of_o express_v break_a number_n be_v by_o their_o numerator_n and_o denominator_n as_o have_v be_v show_v yet_o the_o break_a number_n or_o know_v part_n of_o money_n weight_n measure_n and_o such_o like_a be_v for_o more_o convenient_a operation_n common_o express_v like_o integer_n so_o that_o if_o 12_o shilling_n seven_o penny_n half_a penny_n farthing_n be_v to_o be_v express_v in_o figure_n the_o ordinary_a &_o most_o usual_a way_n be_v thus_o 12_n −_o 07d_o 03_o f._n but_o the_o say_v twelve_o shilling_n seven_o penny_n half_a penny_n farthing_n be_v distinct_o consider_v as_o fraction_n of_o a_o pound_n sterling_n the_o way_n to_o write_v they_o proper_o be_v thus_o 12_o shilling_n be_v twelve_o twentieth_n of_o a_o pound_n sterling_n and_o write_v thus_o 12_o 2d_o l._n 7_o penny_n be_v seven_o twelfth_n of_o one_o twenty_o of_o a_o pound_n sterling_n and_o write_v thus_o 7_o 12_o of_o 1_o 20_o l._n 3_o farthing_n be_v three_o fourth_n of_o one_o twelve_o of_o one_o twenty_o of_o a_o pound_n and_o write_v thus_o ¾_n of_o 1_o 12_o of_o 1_o 20_o l._n 14._o the_o weight_v use_v in_o england_n be_v of_o two_o sort_n troy_n weight_n and_o averdupois_n 15._o the_o several_a piece_n or_o denomination_n of_o troy_n weight_n be_v pound_n ounce_n penny_n weight_n and_o grain_n who_o particular_a value_n be_v as_o follow_v 24_o grain_n make_v 1_o peny_n weight_n 20_o penny_n weight_n make_v 1_o ounce_n 12_o ounce_n make_v 1_o pound_n troy_n 16._o the_o weight_v use_v by_o apothecary_n be_v derive_v from_o a_o pound_n troy_n the_o which_o be_v subdivide_v as_o in_o the_o follow_a table_n lb_v a_o pound_n troy_n be_v equal_a to_o 12_o ounce_n â„¥_o a_o ounce_n be_v equal_a to_o 8_o dram_fw-la Ê’_n a_o dram_n be_v equal_a to_o 3_o scruple_n ℈_o a_o scruple_n be_v equal_a to_o 20_o grain_n 17._o but_o beside_o troy_n weight_n there_o be_v another_o kind_n of_o weight_n use_v in_o england_n call_v averdupois_n weight_n a_o pound_n whereof_o be_v equal_a unto_o 14_o ounce_n and_o twelve-peny_n weight_n troy_n 18._o this_o averdupois_n weight_n be_v either_o great_a or_o small_a 19_o the_o great_a averdupois_n weight_n be_v when_o a_o hundred_o consist_v of_o 112_o pound_n averdupois_n be_v the_o integer_fw-la and_o subdivide_v into_o half_n and_o quarter_n each_o quarter_n contemn_v 28_o pound_n 20._o the_o small_a averdupois_n weight_n be_v when_o a_o pound_n be_v the_o integer_fw-la each_o pound_n be_v subdivide_v into_o 16_o ounce_n each_o ounce_n into_o 16_o dram_n and_o each_o dram_n into_o 4_o quarter_n and_o because_o many_o person_n have_v occasion_n to_o use_v both_o and_o be_v perhaps_o furnish_v but_o with_o one_o i_o have_v here_o exhibit_v a_o table_n for_o the_o speedy_a convert_n of_o the_o part_n of_o a_o pound_n troy_n into_o the_o part_n of_o a_o pound_n averdupois_n and_o the_o contrary_a dram_fw-la averdupois_n  _fw-fr peny-weight_n troy_n grain_n decimal_n of_o a_o grain_n  _fw-fr ounce_n averdupois_n ounce_n troy_n peny-weight_n grain_n 1_o  _fw-fr 01_o 03_o 375_o  _fw-fr 1_o 00_o 18_o 06_o 2_o  _fw-fr 2_o â—6_n â—50_n  _fw-fr 2_o 01_o 16_o 12_o 3_o  _fw-fr 3_o 10_o 125_o  _fw-fr 3_o 2_o 14_o 18_o 4_o dram_n averdupois_n 4_o 13_o 500_o  _fw-fr 4_o 3_o 13_o  _fw-fr 5_o  _fw-fr 5_o 16_o 875_o  _fw-fr 5_o 4_o 11_o 06_o 6_o  _fw-fr 6_o 20_o 250_o  _fw-fr 6_o 5_o 9_o 12_o 7_o  _fw-fr 7_o 23_o 625_o  _fw-fr 7_o 6_o 7_o 18_o 8_o weight_n be_v equal_a to_o 9_o 03_o 000_o  _fw-fr 8_o 7_o 06_o  _fw-fr 9_o  _fw-fr 10_o 06_o 375_o  _fw-fr 9_o 8_o 04_o 06_o 10_o  _fw-fr 11_o 19_o 750_o  _fw-fr 10_o 9_o 02_o 12_o 11_o  _fw-fr 12_o 03_o 125_o  _fw-fr 11_o 10_o 00_o 18_o 12_o  _fw-fr 13_o 16_o 500_o  _fw-fr 12_o 10_o 19_o 00_o 13_o  _fw-fr 14_o 19_o 875_o  _fw-fr 13_o 11_o 17_o 06_o 14_o  _fw-fr 15_o 23_o 250_o  _fw-fr 14_o 12_o 15_o 12_o 15_o  _fw-fr 17_o 02_o 625_o  _fw-fr 15_o 13_o 13_o 18_o 16_o  _fw-fr 18_o 06_o 000_o  _fw-fr 16_o 14_o 12_o 00_o 21._o the_o measure_v use_v in_o england_n be_v of_o two_o sort_n capacity_n or_o length_n 22._o the_o measure_n of_o capacity_n be_v produce_v from_o weight_n and_o be_v also_o of_o two_o sort_n liquid_a or_o dry_a 23._o the_o liquid_a measure_n be_v those_o in_o which_o all_o kind_n of_o liquid_a substance_n be_v measure_v and_o be_v express_v in_o the_o table_n follow_v 1_o pound_n of_o wheat_n troy_n weight_n make_v 1_o pint._n 2_o pint_n make_v 1_o quart_n 2_o quart_n make_v 1_o pottle_n 2_o pottle_n make_v 1_o gallon_n 8_o gallon_n make_v 1_o firkin_n of_o ale_n soap_n herring_n 9_o gallon_n make_v 1_o firkin_n of_o beer_n 10½_n gallon_n make_v 1_o firk_v of_o salmon_n or_o eel_n 2_o firkins_n make_v 1_o kilderkin_n 2_o kilderkin_n make_v 1_o barrel_n 42_o gallon_n make_v 1_o tierce_n of_o wine_n 63_o gallon_n make_v 1_o hogshead_n 2_o hogshead_n make_v 1_o pipe_n or_o but._n 2_o pipe_n or_o butt_n make_v 1_o tau_n of_o wine_n 24._o dry_a measure_n be_v those_o in_o which_o all_o kind_n of_o dry_a substance_n be_v measure_v as_o grain_n sea-coal_n salt_n and_o such_o like_a and_o be_v express_v in_o the_o table_n follow_v 1_o pint_n make_v 1_o pint._n 2_o pint_n make_v 1_o quart_n 2_o quart_n make_v 1_o pottle_n 2_o pottle_n make_v 1_o gallon_n 2_o gallon_n make_v 1_o peck_n 4_o peck_v make_v 1_o bushel_n land_n measure_n 5_o peck_v make_v 1_o bushel_n water_n measure_n 8_o bushel_n make_v 1_o quarter_n 4_o quarter_n make_v 1_o chalder_n 5_o quarter_n make_v 1_o wey_n 25._o long_o measure_n be_v as_o follow_v 3_o barley_n corn_n in_o length_n make_v 1_o inch._n 12_o inch_n make_v 1_o foot_n 3_o foot_n make_v 1_o yard_n 3_o foot_n 9_o inch_n make_v 1_o ell._n 6_o foot_n make_v 1_o fathom_n 5_o yard_n and_o ½_o make_v 1_o pole_n or_o perch_n 40_o pole_n make_v 1_o furlong_n 8_o furlong_n make_v 1_o english_a mile_n 26._o in_o superficial_a or_o square_a measure_n 40_o square_a pole_n or_o perch_n make_v 1_o rood_n or_o quarter_n of_o a_o acre_n and_o 4_o rood_n a_o acre_n 27._o a_o table_n of_o time_n be_v this_o that_o follow_v 1_o minute_n make_v 1_o minute_n 60_o minute_n make_v 1_o hour_n 24_o hour_n make_v 1_o day_n 7_o day_n
make_v 1_o week_n 4_o week_n make_v 1_o month_n of_o 28_o day_n 13_o month_n 1_o day_n and_o 6_o hour_n make_v 1_o year_n not_o exact_o but_o very_o near_o 28._o a_o year_n be_v that_o space_n of_o time_n in_o which_o the_o sun_n do_v finish_v course_n through_o the_o circle_n in_o the_o heaven_n call_v the_o zodiac_n which_o be_v in_o 365_o day_n 5_o hour_n 4_o min._n 29._o the_o zodiac_n by_o be_v astronomer_n divide_v or_o suppose_v to_o be_v into_o twelve_o equal_a part_n call_v sign_n who_o name_n and_o character_n be_v these_o aries_n ♈_o taurus_n ♉_o gemini_fw-la ♊_o cancer_n ♋_o leo_n ♌_o virgo_fw-la â™_o libra_n ♎_o scorpio_n â™_o sagittarius_fw-la â™_o capricornus_fw-la ♑_o aquarius_n â™’_o pisces_fw-la ♓_o and_o each_o of_o these_o sign_n into_o 30_o part_n call_v degree_n so_o that_o this_o and_o all_o other_o circle_n be_v suppose_v to_o be_v divide_v into_o 12_o time_n 30_o part_n or_o degree_n that_o be_v into_o 360_o each_o degree_n into_o 60_o minute_n each_o minute_n into_o 60_o second_n each_o second_o into_o 60_o three_o etc._n etc._n 30._o of_o thing_n account_v by_o the_o dozen_o a_o gross_a be_v the_o integer_fw-la consist_v of_o 12_o dozen_o and_o each_o dozen_o of_o twelve_o particular_n 31._o a_o improper_a fraction_n or_o break_a number_n be_v that_o who_o numeration_n be_v great_a than_o the_o denominator_fw-la as_o 54_o 12_o foot_n that_o be_v fifty_o and_o four_o twelfth_n of_o a_o foot_n and_o this_o may_v be_v well_o call_v a_o improper_a fraction_n see_v it_o will_v not_o admit_v of_o the_o definition_n of_o a_o true_a break_a number_n because_o it_o be_v great_a than_o that_o whole_a whereas_o a_o fraction_n be_v proper_o but_o a_o part_n of_o the_o whole_a 32._o a_o mix_a number_n be_v that_o which_o beside_o the_o integer_n or_o entire_a unity_n of_o which_o it_o consist_v have_v also_o a_o break_a number_n annex_v as_o in_o this_o improper_a fraction_n 54_o 12_o if_o you_o divide_v the_o numerator_n 54_o by_o the_o denominator_fw-la 12_o it_o will_v be_v reduce_v into_o the_o mix_a number_n 4_o 6_o 12_o of_o which_o 4_o be_v the_o whole_a part_n and_o 6_o 12_o the_o break_a number_n or_o fraction_n 33._o and_o this_o i_o hope_n be_v sufficient_a to_o show_v what_o be_v mean_v by_o a_o fraction_n and_o how_o all_o fraction_n whether_o proper_a or_o improper_a be_v to_o be_v express_v and_o read_v which_o be_v the_o notation_n of_o they_o the_o next_o thing_n propound_v concern_v fraction_n be_v their_o numeration_n whether_o such_o fraction_n be_v express_v by_o their_o true_a and_o natural_a way_n that_o be_v by_o their_o numerator_n and_o denominator_n or_o whether_o they_o be_v express_v like_o integer_n as_o the_o know_a part_n of_o money_n be_v express_v by_o pound_n shilling_n penny_n and_o farthing_n and_o the_o know_a part_n of_o troy_n weight_n by_o pound_n ounce_n pennyweight_n and_o grain_n of_o these_o and_o the_o like_a break_a number_n which_o be_v express_v like_o integer_n we_o speak_v of_o first_o chap._n vii_o of_o the_o numeration_n of_o such_o break_a number_n as_o be_v express_v like_o integer_n 1._o numeration_n of_o such_o as_o be_v express_v like_o integer_n be_v twofold_a 2._o accidental_a and_o essential_a 3._o accidental_a numeration_n be_v otherwise_o call_v reduction_n 4._o reduction_n be_v either_o descend_v or_o ascend_v 5._o reduction_n descend_v be_v when_o a_o number_n of_o a_o great_a denomination_n be_v give_v it_o be_v require_v to_o find_v how_o many_o of_o a_o lesser_a denomination_n be_v equal_a in_o value_n to_o that_o give_v number_n of_o the_o great_a as_o when_o it_o be_v require_v to_o find_v how_o many_o shilling_n be_v contain_v in_o 34_o pound_n or_o how_o many_o penny_n in_o 325_o shilling_n or_o how_o many_o hour_n in_o 365_o day_n 6._o reduction_n descend_v be_v perform_v by_o multiplication_n for_o if_o the_o give_v number_n of_o integer_n of_o a_o great_a denomination_n be_v multiply_v by_o the_o number_n of_o integer_n contain_v in_o the_o next_o inferior_a denomination_n the_o product_n shall_v show_v how_o many_o of_o that_o inferior_a denomination_n be_v contain_v in_o the_o integer_n of_o the_o great_a denomination_n give_v for_o example_n let_v 34_o pound_n be_v the_o great_a denomination_n give_v and_o let_v it_o be_v require_v to_o show_v how_o many_o shilling_n be_v in_o 34_o pound_n shilling_n be_v the_o next_o inferior_a denomination_n unto_o pound_n and_o that_o every_o pound_n do_v contain_v 20_o shilling_n as_o have_v be_v show_v if_o you_o multiply_v 34_o by_o 20_o the_o product_n be_v 680_o the_o number_n of_o shilling_n require_v 8_o in_o like_a manner_n 680_o shilling_n will_v be_v reduce_v into_o 8160_o penny_n if_o you_o multiply_v 680_o by_o 12_o the_o number_n of_o penny_n in_o a_o shilling_n and_o 8160_o penny_n will_v be_v reduce_v into_o farthing_n 32640_o if_o you_o multiply_v 8160_o by_o 4_o the_o number_n of_o farthing_n in_o a_o penny_n the_o like_a method_n be_v to_o be_v observe_v in_o weight_n and_o measure_n or_o any_o thing_n else_o that_o be_v or_o may_v be_v subdivide_v into_o inferior_a denomination_n thus_o 26_o pound_n troy_n will_v 8_o be_v reduce_v into_o 312_o ounce_n and_o 312_o ounounce_v into_o 6240_o penny_n weight_n and_o 6240_o penny_n weight_n into_o 149760_o grain_n as_o by_o the_o operation_n in_o the_o margin_n it_o do_v appear_v and_o in_o this_o manner_n may_v any_o number_n of_o a_o great_a denomination_n give_v be_v reduce_v into_o the_o least_o denomination_n into_o which_o the_o great_a be_v suppose_v to_o be_v subdivide_v reduction_n ascend_v be_v when_o a_o number_n of_o a_o lesser_a denomination_n be_v give_v it_o be_v require_v to_o find_v how_o many_o of_o a_o great_a denomination_n be_v equal_a in_o value_n to_o that_o give_v number_n of_o the_o lesser_a as_o when_o it_o be_v require_v to_o find_v how_o many_o penny_n be_v contain_v 32640_o farthing_n or_o how_o many_o shilling_n in_o 8160_o penny_n or_o how_o many_o day_n in_o 8760_o hour_n and_o this_o kind_n of_o reduction_n call_v reduction_n ascend_v be_v perform_v by_o division_n for_o if_o the_o number_n of_o integer_n give_v be_v divide_v by_o the_o number_n of_o integer_n in_o the_o next_o superior_a or_o great_a denomination_n the_o quotient_n shall_v be_v the_o number_n of_o integer_n seek_v so_o 32640_o farthing_n be_v divide_v by_o 4_o the_o number_n of_o farthing_n in_o a_o penny_n the_o quotient_n be_v 8160_o the_o number_n of_o penny_n contain_v in_o 32640_o farthing_n in_o like_a manner_n if_o 8160_o penny_n be_v divide_v by_o 12_o the_o number_n of_o penny_n in_o a_o shilling_n the_o quotient_n will_v be_v 680_o the_o number_n of_o shilling_n in_o 8160_o penny_n and_o last_o if_o 680_o shilling_n be_v divide_v by_o 20_o the_o number_n of_o shilling_n in_o a_o pound_n the_o quotient_n will_v be_v 34_o the_o number_n of_o pound_n in_o 680_o shilling_n the_o like_a may_v be_v do_v by_o any_o other_o integer_n of_o any_o know_a denomination_n give_v chap._n viii_o of_o the_o addition_n of_o number_n that_o be_v of_o diverse_a denomination_n 1._o accidental_a numeration_n of_o such_o break_a number_n as_o be_v express_v like_o integer_n have_v be_v show_v that_o which_o be_v essential_a now_o follow_v 2._o essential_a numeration_n do_v consist_v of_o addition_n subduction_n multiplication_n and_o division_n 3._o when_o the_o number_n propound_v to_o be_v add_v be_v of_o diverse_a denomination_n you_o must_v begin_v with_o the_o least_o denomination_n first_o and_o set_v down_o their_o sum_n under_o that_o inferior_a denomination_n if_o their_o sum_n be_v few_o than_o the_o number_n of_o part_n in_o the_o next_o great_a denomination_n but_o if_o their_o sum_n be_v more_o than_o the_o number_n of_o part_n in_o the_o next_o great_a denomination_n set_v down_o the_o excess_n if_o equal_a set_v down_o a_o cypher_n and_o the_o rest_n must_v be_v add_v to_o the_o next_o superior_a denomination_n and_o for_o memory_n sake_n it_o may_v be_v set_v down_o under_o that_o denomination_n to_o which_o it_o be_v to_o be_v add_v as_o in_o the_o example_n follow_v 1_o in_o which_o i_o begin_v with_o the_o farthing_n first_o and_o say_v 3_o +_o 2_o =_o 5_o that_o be_v one_o penny_n and_o a_o farthing_n wherefore_o set_v 1_v down_o under_o the_o denomination_n of_o farthing_n i_o carry_v a_o penny_n to_o the_o denomination_n of_o penny_n and_o say_v 1_o +_o 7_o =_o 8_o and_o 8_o +_o 4_o =_o 12_o and_o 12_o +_o 5_o =_o 17_o penny_n that_o be_v 1_o shilling_n and_o 5_o penny_n wherefore_o i_o set_v down_o 5_o in_o the_o low_a rank_n under_o the_o denomination_n of_o penny_n and_o 1_o in_o the_o line_n above_o it_o under_o the_o denomination_n of_o shilling_n and_o say_v 1_o +_o 8_o =_o 9_o and_o 9_o +_o 19_o =_o
7_o shilling_n the_o value_n of_o one_o escus_fw-la d'or_fw-la who_o product_n be_v 896_o shilling_n and_o then_o the_o question_n must_v be_v thus_o state_v if_o 14_o shilling_n be_v worth_a 1_o pistolet_n how_o many_o pistolet_n be_v 896_o worth_n facit_fw-la 64_o pistolet_n 7._o sometime_o a_o question_n belong_v to_o this_o rule_n must_v be_v prepare_v for_o solution_n by_o multiplication_n and_o addition_n sometime_o by_o multiplication_n and_o subtraction_n as_o in_o these_o example_n follow_v the_o first_o of_o which_o require_v multiplication_n and_o addition_n a_o butcher_n send_v his_o man_n with_o 216_o pound_n sterling_a to_o buy_v cattle_n ox_n at_o 11_o pound_n apiece_o cow_n at_o 2_o pound_n a_o piece_n colt_n at_o one_o pound_n 5_o shilling_n hog_n at_o 1_o pound_n 15_o shilling_n a_o piece_n and_o of_o each_o a_o like_a number_n the_o question_n be_v how_o many_o of_o each_o he_o may_v buy_v for_o that_o money_n to_o resolve_v this_o question_n i_o must_v first_o reduce_v the_o pound_n sterling_a into_o shilling_n by_o multiplication_n and_o 216_o pound_n be_v multiply_v by_o 20_o the_o product_n be_v 4320_o shilling_n and_o the_o several_a prize_n of_o the_o cattle_n must_v be_v reduce_v into_o shilling_n also_o and_o thus_o 11_o pound_n for_o a_o ox_n be_v 220_o shilling_n the_o price_n of_o a_o cow_n be_v 40_o shilling_n the_o price_n of_o a_o colt_n 25_o shilling_n and_o the_o price_n of_o a_o hog_n be_v 35_o shilling_n the_o sum_n of_o these_o several_a prize_n be_v 320_o shilling_n this_o do_v the_o question_n may_v be_v thus_o state_v if_o 320_o shilling_n buy_v 1_o of_o each_o sort_n how_o many_o shall_v 4160_o buy_v facit_fw-la 13._o 2._o example_n in_o the_o which_o the_o question_n be_v to_o be_v prepare_v by_o multiplication_n and_o subtraction_n if_o 128_o gallon_n of_o water_n run_v into_o a_o cistern_n in_o a_o hour_n time_n from_o one_o cock_n and_o 174_o gallon_n of_o water_n run_v into_o another_o cistern_n in_o a_o hour_n time_n also_o but_o not_o begin_v to_o run_v till_o 4_o hour_n after_o the_o first_o in_o what_o time_n will_v the_o quantity_n of_o water_n run_v out_o of_o both_o be_v equal_a to_o resolve_v this_o question_n one_o of_o the_o term_n give_v must_v be_v prepare_v by_o multiplication_n and_o another_o by_o subtraction_n first_o there_o must_v be_v compute_v how_o much_o water_n be_v run_v out_o of_o the_o first_o cock_n before_o the_o second_o begin_v and_o that_o be_v 4_o time_n 128_o gallon_n that_o be_v 512_o second_o it_o must_v be_v consider_v how_o many_o gallon_n more_o do_v run_v out_o of_o the_o last_o cock_n than_o do_v out_o of_o the_o first_o in_o a_o hour_n time_n which_o by_o subtract_v 128_o from_o 174_o i_o find_v to_o be_v 46_o and_o hence_o this_o question_n be_v to_o be_v thus_o state_v if_o i_o gain_v 46_o gallon_n in_o one_o hour_n in_o how_o many_o hour_n shall_v i_o gain_v 512_o gallon_n facit_fw-la 11._o 8._o sometime_o a_o question_n belong_v to_o this_o rule_n must_v be_v prepare_v for_o solution_n by_o arithmetical_a or_o geometrical_a proportion_n as_o in_o these_o question_n follow_v a_o man_n go_v a_o journey_n spend_v one_o shilling_n the_o first_o day_n 3_o the_o second_o 5_o the_o three_o 7_o the_o four_o still_o increase_a his_o expense_n by_o this_o proportion_n till_o he_o have_v spend_v 1500_o shilling_n i_o demand_v in_o how_o many_o day_n that_o sum_n be_v spend_v to_o resolve_v this_o question_n the_o sum_n of_o such_o a_o arithmetical_a progression_n for_o some_o certain_a time_n must_v be_v first_o compute_v as_o suppose_v for_o thirty_o day_n which_o will_v be_v find_v to_o be_v 900_o shilling_n and_o hence_o it_o may_v be_v thus_o conclude_v if_o 900_o shilling_n last_v i_o 30_o day_n how_o long_o shall_v 1500_o shilling_n last_v i_o facit_fw-la 50._o another_o question_n may_v be_v this_o if_o a_o piece_n of_o cloth_n contain_v 40_o yard_n do_v cost_v i_o 30_o pound_n at_o what_o rate_n must_v that_o cloth_n be_v sell_v to_o gain_v 12_o pound_n in_o the_o hundred_o before_o this_o question_n can_v be_v resolve_v i_o must_v find_v the_o price_n of_o one_o yard_n by_o a_o proportion_n geometrical_a by_o way_n of_o preparation_n thus_o if_o 4â—_n yard_n cost_v 50_o pound_n what_o shall_v one_o yard_n cost_n facit_fw-la 1¼_n then_o i_o say_v if_o 100_o pound_n be_v increase_v to_o 112_o to_o what_o shall_v 1¼_n be_v increase_v answer_v 1â…–_n chap._n xxii_o the_o rule_n of_o three_o in_o fraction_n 1._o fraction_n as_o i_o have_v already_o show_v be_v of_o 3_o sort_n viz._n such_o as_o be_v many_o time_n express_v like_o integer_n such_o as_o be_v express_v by_o their_o numerator_n and_o denominator_n and_o such_o as_o have_v a_o unite_v with_o cipher_n annex_v for_o their_o denominator_fw-la and_o now_o know_v by_o the_o name_n of_o decimal_a fraction_n in_o each_o of_o these_o i_o will_v exemplify_v this_o rule_n by_o some_o few_o example_n and_o first_o in_o such_o fraction_n as_o be_v express_v like_o integer_n 2._o your_o question_n be_v state_v bring_v your_o number_n propound_v into_o the_o least_o name_n mention_v or_o as_o low_o as_o you_o desire_v the_o question_n to_o be_v answer_v in_o then_o multiply_v the_o second_o and_o three_o and_o divide_v the_o product_n by_o the_o first_o if_o your_o question_n belong_v to_o the_o rule_n of_o three_o direct_v or_o multiply_v the_o first_o and_o second_o and_o divide_v by_o the_o three_o if_o it_o belong_v to_o the_o rule_n of_o three_o inverse_n the_o quotient_a be_v reduce_v shall_v be_v the_o answer_n require_v example_n if_o 7_o lib._n 8_o ounce_n of_o currant_n cost_v 2s_n 7d_o what_o shall_v 100½_n 13_o lib._n cost_n in_o this_o question_n the_o first_o and_o three_o term_n must_v be_v reduce_v into_o ounce_n and_o the_o second_o term_n into_o penny_n and_o then_o the_o question_n will_v stand_v thus_o if_o 112_o ounce_n cost_v 31_o penny_n what_o shall_v 1267_o ounce_n cost_n facit_fw-la 359d_o 1_o 64_o 112_o farthing_n that_o be_v 1l_o 9_n 11d_o 1f_o â…˜_n of_o a_o farthing_n a_o second_o example_n may_v be_v this_o a_o man_n do_v borrow_v of_o his_o friend_n 246_o lib._n 7_o shilling_n and_o 6_o penny_n for_o 27_o week_n for_o how_o long_a time_n must_v he_o lend_v his_o friend_n 329_o lib._n 13_o shilling_n 4d_o to_o require_v his_o kindness_n here_o it_o be_v plain_a that_o the_o four_o term_v require_v must_v be_v less_o than_o the_o second_o term_n give_v and_o therefore_o the_o great_a extreme_a must_v be_v the_o divisor_n and_o the_o number_n be_v reduce_v and_o place_v as_o have_v be_v direct_v they_o will_v stand_v thus_o 59130_o 187_o 79120_o in_o which_o the_o great_a extreme_a being_n in_o the_o three_o place_n do_v plain_o show_v that_o this_o question_n do_v belong_v to_o the_o rule_n of_o three_o inverse_n and_o therefore_o as_o 3_o example_n in_o sexagenary_a number_n if_o 48_o fir_v 28_o second_n give_v 60_o minute_n what_o shall_v 25_o minute_n 12_o second_o give_v according_a to_o the_o former_a rule_n give_v in_o multiplication_n 25_o fir_v 12_o second_o be_v multiply_v by_o 60_o minute_n the_o product_n be_v 25_o degree_n 12_o fir_v 00_o second_n which_o be_v divide_v by_o 48_o fir_v 28_o second_n the_o quotient_n will_v be_v 31_o fir_v 08_o second_n 4._o example_n in_o vulgar_a fraction_n if_o â…”_n of_o a_o yard_n of_o velvet_n be_v sell_v for_o â…˜_n of_o a_o pound_n sterling_a what_o shall_v â…ž_n of_o a_o yard_n cost_n if_o respect_n be_v have_v to_o the_o rule_n of_o the_o multiplication_n and_o division_n of_o fraction_n already_o deliver_v in_o the_o 14_o chapter_n the_o work_n of_o this_o rule_n in_o fraction_n will_v be_v the_o same_o as_o in_o whole_a number_n thus_o â…˜_n and_o â…ž_n be_v multiply_v together_o the_o product_n will_v be_v 28_o 40_o which_o be_v divide_v by_o â…”_n 7_o â…”_n the_o quotient_n be_v 84_o â—0_n or_o 1_o lib._n 4_o â—0_n or_o 1_o 20_o and_o so_o the_o answer_n be_v 1_o pound_n and_o one_o shilling_n otherwise_o thus_o multiply_v the_o denominator_fw-la of_o the_o first_o number_n by_o the_o numerator_n of_o the_o second_o and_o three_o continual_o so_o shall_v the_o last_o product_n be_v a_o new_a numerator_n then_o multiply_v the_o numerator_n of_o the_o first_o number_n by_o the_o denominator_n of_o the_o second_o and_o three_o number_n continual_o so_o shall_v the_o last_o product_n be_v a_o new_a denominator_fw-la and_o this_o fraction_n shall_v be_v the_o quotient_a desire_a thus_o the_o denominator_fw-la 3_o being_n multiply_v first_o by_o 4_o and_o then_o by_o 7_o the_o last_o product_n be_v 84_o and_o the_o numerator_n 2_o be_v multiply_v first_o by_o 5_o and_o then_o by_o 8_o the_o last_o product_n will_v be_v 80_o for_o a_o new_a denominator_fw-la and_o so_o the_o