Selected quad for the lemma: hand_n

Word A	Word B	Word C	Word D	Occurrence	Frequency Band	MI	MI Band	Prominent
hand_n	left_a	rank_n	right_a	50,843	5	9.3193	5	true

Text snippets containing the quad

ID	Title	Author	Corrected Date of Publication (TCP Date of Publication)	STC	Words	Pages
A52256	The compleat arithmetician, or, The whole art of arithmetick, vulgar and decimal in a plain and easie method, suitable to the meanest capacity : in which the multiplication and division of numbers of several denominations, and the rule of alligation are more fully explained than in any treatise of this nature, yet extant / by J.N., Philomath.	Newton, John, 1622-1678.	1691 (1691)	Wing N1054; ESTC R4290	77,995	206

imprimis_fw-la ready_a money_n of_o the_o 1000l_n 58.0000_o item_n 808_o ell_n of_o taffety_n worth_a 323_o 2000_o item_n 35_o piece_n of_o dowlace_n worth_n 125.1882_o item_n 55_o piece_n of_o holland_n worth_n 493.6092_o that_o be_v 999_o lib._n 19_o shil_fw-la and_o 11½_n penny_n 999.9974_o the_o account_n for_o the_o 64_o piece_n of_o cloth_n which_o cost_v 30.4_o pound_n sterling_a per_fw-la piece_n that_o be_v in_o all_o 1945.6_o lib._n star_z be_v thus_o first_o then_o for_o the_o worth_n of_o the_o 77_o pistolet_n pistolet_n star_z  _fw-fr pist._n star_z 275._o 1945._o 6_o 77_o 544.768_o second_o for_o the_o worth_n of_o the_o 5546_o florin_n the_o term_n of_o the_o question_n will_v stand_v thus_o 1945.6_o lib._n sterling_a =_o 275_o spanish_a pistolet_n 198_o span._n pisto_n =_o 5821_o florin_n 5546_o florin_n =_o lib._n sterling_a here_o the_o number_n in_o the_o first_o column_n to_o be_v multiply_v together_o be_v 1945._o 6._o 198._o 5546_o and_o the_o product_n be_v 21364789248_o for_o the_o dividend_n and_o the_o number_n in_o the_o second_o column_n to_o be_v multiply_v be_v 275.5821_o who_o product_n be_v 1600775_o for_o the_o divisor_n and_o the_o quotient_a arise_v from_o these_o two_o product_n be_v 1334.6528_o which_o be_v the_o value_n of_o 5546_o florin_n in_o pound_n sterling_a last_o for_o the_o worth_n of_o 114._o 158_o flemish_a the_o term_n of_o the_o question_n will_v stand_v thus_o 1945._o 6_o lib._n sterling_a =_o 274_o spanish_a pistolet_n 198_o spanish_a pisto_n =_o 5821_o florin_n 275_o florin_n =_o 114._o 158_o lib._n flemish_n 114._o 158_o lib._n flem_n =_o pound_n sterling_a where_o neglect_v the_o 275_o pistolet_n in_o one_o column_n and_o 275_o florin_n in_o the_o other_o and_o 114._o 158_o lib._n flemish_n in_o both_o the_o number_n in_o the_o first_o column_n to_o be_v multiply_v be_v 1945._o 6_o and_o 198_o who_o product_n be_v 3852288_o for_o the_o dividend_n and_o 5821_o in_o the_o other_o column_n be_v the_o divisor_n and_o the_o quotient_a arise_v from_o two_o number_n be_v 66._o 1791_o lib._n sterling_a for_o the_o value_n of_o 114._o 158_o lib._n flem._n so_o that_o for_o the_o 64_o piece_n of_o cloth_n the_o factor_n have_v in_o his_o hand_n  _fw-fr sterling_n imprimis_fw-la 77_o pistolet_n worth_a 544._o 7680_o item_n 5546_o florin_n worth_a 1334._o 6528_o item_n 114._o 158_o lib._n flemish_n worth_a 66._o 1791_o and_o the_o balance_n be_v 1945._o 5999_o chap._n xxvii_o the_o rule_n of_o aligation_n 1._o alligation_n be_v a_o art_n by_o which_o we_o resolve_v question_n that_o concern_v the_o mix_n of_o divers_a simple_n together_o 2._o alligation_n be_v twofold_a medial_n and_o alternate_a 3._o alligation_n medial_a be_v when_o the_o several_a quantity_n and_o rate_n of_o divers_a simple_n be_v we_o find_v out_o a_o mean_a rate_n for_o which_o a_o mixture_n make_v of_o these_o simple_n may_v be_v afford_v to_o effect_v this_o the_o sum_n of_o the_o quantity_n be_v give_v with_o the_o value_n of_o all_o the_o simple_n the_o proportion_n be_v as_o the_o sum_n of_o the_o quantity_n be_v to_o the_o value_n of_o all_o the_o simple_n so_o be_v any_o part_n of_o the_o mixture_n propound_v to_o the_o mean_a rate_n that_o be_v require_v example_n let_v 12_o gallon_n of_o canary_n at_o 4_o shil_fw-la the_o gallon_n 36_o gallon_n of_o sherry_n at_o 3_o shilling_n the_o gallon_n and_o 52_o gallon_n of_o white-wine_n at_o 2_o shilling_n 8_o penny_n the_o gallon_n be_v mix_v together_o and_o let_v the_o price_n of_o one_o gallon_n of_o that_o mixture_n be_v require_v the_o sum_n of_o 12_o 36_o and_o 52_o gallon_n be_v 100_o gallon_n and_o the_o value_n of_o 12_o gallon_n of_o canary_n at_o 4_o shilling_n or_o 48_o penny_n by_o the_o gallon_n be_v 576d_o the_o value_n of_o 36_o gallon_n of_o sherry_n at_o 3_o shil_fw-la or_o 36_o penny_n be_v 1296d_o and_o the_o value_n of_o 52_o gallon_n of_o white-wine_n at_o 2_o shilling_n 8_o penny_n the_o gallon_n or_o 32_o penny_n be_v 1664_o penny_n and_o all_o these_o value_v add_v together_o do_v make_v in_o the_o whole_a 3536d_o now_o then_o by_o the_o rule_n of_o three_o 100_o 3536_o 1_o 35_o 36_o 104_o that_o be_v 35_o penny_n and_o 36_o hundred_o of_o a_o penny_n for_o proof_n of_o the_o work_n compare_v the_o total_a value_n of_o the_o several_a simple_n with_o the_o value_n of_o the_o whole_a mixture_n if_o their_o sum_n agree_v the_o work_n be_v true_a as_o in_o this_o example_n  _fw-fr l_o s_o d_o the_o value_n of_o 12_o gallon_n of_o canary_n at_o 4_o shilling_n per_fw-la gallon_n be_v 2_o 08_o 00_o the_o value_n of_o 36_o gallon_n of_o sherry_n at_o 3_o shilling_n the_o gallon_n be_v 5_o 08_o 00_o and_o the_o value_n of_o 52_o gallon_n of_o white_a wine_n at_o 2_o shil_fw-la 8_o penny_n the_o gallon_n be_v 6_o 18_o 08_o all_o which_o amount_v to_o 14_o 14_o 08_o and_o the_o value_n of_o 100_o gallon_n at_o 35d_o and_o 36_o hundred_o of_o a_o penny_n by_o the_o gallon_n be_v 14_o 14_o 08_o as_o before_o 5._o alligation_n alternate_a be_v when_o several_a rate_n of_o divers_a simple_n be_v give_v such_o a_o mixture_n be_v require_v as_o may_v be_v sell_v at_o a_o certain_a mean_a rate_n propound_v 6._o alligation_n alternate_a be_v either_o partial_a or_o total_a 7._o alternation_n partial_a be_v when_o the_o several_a rate_n of_o divers_a simple_n with_o the_o quantity_n of_o one_o of_o they_o be_v give_v and_o the_o several_a quantity_n of_o the_o rest_n be_v require_v in_o such_o sort_n that_o a_o mixture_n be_v make_v according_a to_o the_o give_v quantity_n and_o the_o quantity_n so_o find_v that_o mixture_n may_v bear_v a_o certain_a rate_n propound_v 8._o alternation_n total_a be_v when_o the_o total_a quantity_n of_o all_o the_o simple_n with_o their_o several_a rate_n be_v give_v we_o find_v out_o their_o several_a quantity_n in_o such_o sort_n as_o that_o a_o mixture_n of_o they_o be_v make_v according_a to_o the_o quantity_n so_o find_v that_o mixture_n may_v bear_v a_o certain_a mean_a rate_n propound_v 9_o in_o both_o these_o kind_n of_o alternation_n there_o be_v some_o preparation_n to_o be_v make_v before_o the_o rule_n by_o which_o such_o question_n may_v be_v resolve_v can_v be_v well_o deliver_v 10._o the_o preparation_n to_o be_v make_v do_v partly_o consist_v in_o place_v the_o give_v rate_n of_o the_o thing_n propound_v and_o take_v the_o difference_n between_o those_o rate_n and_o the_o mean_a price_n you_o will_v have_v they_o bear_v 11._o the_o give_v rate_n of_o the_o thing_n propound_v must_v be_v place_v one_o under_o another_o so_o as_o that_o they_o may_v orderly_o increase_v or_o decrease_v and_o the_o mean_a price_n that_o you_o will_v have_v they_o bear_v may_v stand_v on_o the_o right_a hand_n of_o the_o rank_n about_o the_o middle_n for_o example_n let_v the_o give_v rate_n be_v 12._o 24._o 36._o 48._o and_o the_o mean_a price_n 28d_o which_o be_v place_v one_o under_o another_o descend_v i_o 0_o draw_v a_o line_n of_o connexion_n and_o on_o the_o left_a hand_n thereof_o i_o set_v 28_o by_o the_o mean_a price_n by_o itself_o as_o here_o you_o see_v the_o term_n be_v thus_o rank_v in_o their_o due_a order_n link_v they_o together_o by_o certain_a arch_n in_o such_o sort_n as_o that_o one_o that_o be_v great_a than_o the_o mean_a price_n may_v be_v still_o couple_v with_o another_o that_o be_v less_o so_o in_o the_o premise_a example_n 12_o may_v be_v link_v with_o 36_o or_o 48_o and_o 24_o may_v be_v link_v with_o 48_o or_o 36_o and_o the_o work_n will_v stand_v thus_o or_o thus_o the_o term_n be_v thus_o rank_v and_o link_v together_o proceed_v to_o find_v the_o difference_n between_o the_o give_v rate_n and_o the_o mean_a price_n and_o write_v that_o difference_n just_a against_o his_o respective_a yoke-fellow_n thus_o the_o difference_n between_o 12_o and_o 28_o be_v 16_o i_o place_v 16_o against_o 48_o its_o respective_a yoke-fellow_n and_o the_o difference_n between_o 24_o and_o 28_o be_v 4_o i_o place_v 4_o against_o 36_o the_o difference_n between_o 28_o and_o 36_o be_v 8_o i_o place_v 8_o against_o 24_o last_o 20_o be_v the_o difference_n between_o 28_o and_o 48_o i_o place_v 20_o against_o 12_o and_o the_o work_n will_v stand_v according_a to_o the_o first_o way_n of_o link_a the_o rate_n propound_v as_o here_o you_o see_v they_o but_o the_o branch_n be_v link_v after_o the_o other_o manner_n the_o difference_n must_v be_v otherwise_o place_v for_o here_o 24_o have_v 48_o for_o his_o yoke-fellow_n and_o 12_o have_v 36_o for_o he_o 0_o 12._o and_o many_o time_n it_o so_o fall_v out_o that_o one_o of_o the_o give_v rate_n may_v be_v link_v with_o two_o or_o more_o of_o the_o other_o rate_v give_v in_o which_o case_n

be_v no_o reason_n give_v why_o child_n in_o these_o shall_v not_o be_v as_o capable_a of_o this_o and_o other_o mathematical_a learning_n as_o in_o the_o day_n of_o plato_n and_o aristotle_n when_o all_o the_o school_n in_o greece_n do_v teach_v cihldren_n these_o thing_n in_o the_o first_o place_n and_o thus_o you_o have_v at_o once_o not_o only_o the_o reason_n for_o which_o i_o have_v add_v this_o to_o those_o many_o treatise_n of_o this_o kind_n already_o publish_v viz._n that_o the_o memory_n of_o child_n may_v not_o be_v over_o burden_a with_o long_a direction_n and_o tedious_a discourse_n nor_o the_o purse_n of_o parent_n so_o over_o charge_v as_o that_o the_o price_n of_o the_o thing_n shall_v be_v any_o bar_n to_o their_o child_n instruction_n but_o also_o the_o reason_n for_o which_o i_o place_v this_o science_n next_o to_o grammar_n even_o this_o because_o it_o be_v more_o suitable_a to_o child_n capacity_n than_o the_o other_o science_n be_v nay_o will_v be_v soon_o understand_v by_o they_o than_o all_o the_o part_n of_o grammar_n can_v nor_o shall_v the_o art_n of_o geometry_n be_v sever_v from_o arithmetic_n they_o be_v of_o so_o near_o a_o kin_n that_o they_o shall_v indeed_o go_v hand_n in_o hand_n and_o both_o of_o they_o together_o will_v make_v a_o excellent_a introduction_n to_o the_o other_o science_n and_o because_o that_o the_o write_n of_o a_o good_a hand_n be_v as_o necessary_a as_o the_o other_o and_o that_o it_o be_v almost_o as_o hard_o a_o matter_n to_o find_v a_o master_n that_o be_v able_a to_o teach_v youth_n the_o art_n of_o fair_a writing_n i_o mean_v in_o the_o country_n as_o of_o arithmetic_n and_o geometry_n for_o the_o profit_n of_o child_n and_o the_o ease_n of_o schoolmaster_n there_o be_v another_o little_a book_n of_o arithmetic_n call_v the_o country_n schoolmaster_n which_o will_v be_v speedy_o publish_v also_o in_o which_o the_o title_n of_o such_o rule_n of_o arithmetic_n as_o child_n be_v first_o to_o learn_v together_o with_o brief_a direction_n and_o example_n be_v engrave_v in_o a_o fair_a useful_a write_a hand_n as_o will_v with_o the_o master_n diligence_n be_v sufficient_a for_o the_o teach_n of_o child_n as_o well_o to_o write_v fair_o as_o to_o number_v ready_o and_o because_o a_o child_n must_v first_o learn_v to_o write_v his_o letter_n well_o before_o he_o be_v put_v to_o join_v there_o be_v annex_v thereto_o a_o copy_n of_o letter_n both_o great_a and_o small_a suitable_a to_o the_o other_o copy_n of_o which_o thou_o may_v have_v as_o many_o as_o thou_o will_v alone_o and_o upon_o very_o reasonable_a term_n and_o if_o the_o master_n have_v but_o so_o much_o geometry_n as_o to_o teach_v his_o child_n to_o erect_v a_o perpendicular_a and_o draw_v a_o parallel_n line_n he_o shall_v never_o need_v to_o rule_v a_o book_n or_o set_v a_o copy_n which_o will_v be_v a_o wonderful_a ease_n to_o he_o and_o no_o less_o profit_n and_o advantage_n to_o his_o scholar_n for_o which_o i_o pray_v be_v best_a for_o the_o scholar_n to_o rule_v his_o own_o book_n or_o his_o master_n to_o do_v it_o for_o he_o which_o be_v better_o for_o the_o master_n to_o spend_v his_o time_n in_o write_v of_o copy_n or_o direct_v the_o child_n to_o write_v according_a to_o the_o copy_n he_o have_v these_o two_o book_n than_o i_o will_v have_v child_n furnish_v with_o as_o soon_o as_o they_o can_v but_o read_v their_o psalter_n and_o hold_v a_o pen_n in_o their_o hand_n the_o one_o to_o inform_v their_o judgement_n in_o the_o knowledge_n of_o the_o rule_n in_o arithmetic_n and_o the_o other_o to_o teach_v they_o to_o write_v and_o to_o put_v their_o knowledge_n of_o arithmetic_n in_o practice_n and_o as_o they_o shall_v profit_v in_o these_o thing_n so_o to_o be_v set_v forward_o in_o geometry_n and_o the_o other_o science_n for_o which_o god_n willing_a it_o shall_v not_o be_v long_o before_o i_o furnish_v thou_o with_o suitable_a manual_n to_o that_o purpose_n john_n newton_n the_o art_n of_o natural_a arithmetic_n chap._n i._o of_o notation_n 1._o arithmetic_n be_v the_o art_n of_o account_v by_o number_n 2._o there_o be_v three_o kind_n of_o it_o 1._o natural_a 2._o symbolical_a 3._o artificial_a 3._o natural_a arithmetic_n which_o be_v the_o subject_n of_o this_o present_a discourse_n be_v either_o positive_a or_o negative_a 4._o positive_a arithmetic_n be_v either_o single_a or_o comparative_a 5._o single_n be_v that_o which_o be_v wrought_v by_o number_n consider_v alone_o without_o relation_n to_o one_o another_o 6._o the_o part_n of_o single_a arithmetic_n be_v notation_n and_o numeration_n 7._o notation_n be_v the_o write_n down_o of_o any_o sum_n to_o be_v express_v or_o the_o express_n of_o any_o sum_n set_v down_o in_o writing_n 8._o the_o part_n of_o notation_n be_v two_o the_o first_o do_v explain_v the_o general_a value_n of_o the_o note_n by_o which_o number_n be_v to_o be_v express_v 9_o the_o note_n or_o character_n by_o which_o number_n be_v ordinary_o express_v be_v these_o 1._o one_o 2._o two_o 3._o three_o 4._o four_o 5._o five_o 6._o six_o 7._o seven_o 8._o eight_o 9_o nine_o 0._o nothing_o 10._o these_o note_n be_v either_o significant_a figure_n or_o a_o cypher_n 11._o the_o significant_a figure_n be_v the_o first_o nine_o viz._n 1_o 2_o 3_o 4_o 5_o 6_o 7_o 8_o 9_o and_o these_o have_v a_o certain_a signification_n when_o they_o be_v put_v by_o themselves_o but_o any_o of_o these_o be_v join_v with_o itself_o or_o any_o of_o the_o rest_n do_v become_v uncertain_a in_o its_o signification_n for_o every_o of_o these_o figure_n alone_o do_v signify_v itself_o once_o but_o if_o two_o of_o they_o be_v set_v together_o the_o first_o do_v signify_v itself_o once_o the_o second_o do_v signify_v itself_o ten_o time_n as_o in_o 22_o the_o same_o figure_n which_o in_o the_o first_o place_n towards_o the_o right_a hand_n do_v signify_v only_o two_o do_v in_o the_o second_o place_n signify_v ten_o time_n two_o that_o be_v twenty_o 12._o the_o cypher_n do_v indeed_o signify_v nothing_o of_o itself_o but_o be_v set_v before_o or_o after_o the_o rest_n it_o do_v increase_v or_o lessen_v their_o value_n 13._o the_o value_n of_o these_o note_n be_v more_o full_o express_v by_o degree_n and_o period_n 14._o the_o degree_n be_v three_o the_o first_o be_v the_o first_o place_n of_o a_o number_n towards_o the_o right_a hand_n and_o do_v signify_v itself_o once_o the_o second_o degree_n be_v the_o second_o place_n towards_o the_o right_a hand_n and_o do_v signify_v itself_o ten_o time_n the_o three_o be_v the_o three_o place_n towards_o the_o right_a hand_n and_o do_v signify_v one_o hundred_o time_n itself_o so_o these_o figure_n 345_o do_v signify_v three_o hundred_o forty_o five_o 15._o a_o period_n if_o it_o be_v perfect_a do_v consist_v of_o three_o degree_n if_o imperfect_a it_o do_v consist_v of_o few_o and_o it_o be_v either_o simple_a or_o compound_n a_o simple_a period_n if_o it_o be_v perfect_a do_v consist_v of_o three_o degree_n a_o compound_a period_n may_v have_v all_o the_o period_n perfect_a or_o the_o last_o towards_o the_o left_a hand_n may_v be_v imperfect_a thus_o 325_o be_v a_o simple_a perfect_a period_n and_o 325_o 678_o be_v a_o perfect_a compound_n period_n or_o a_o number_n consist_v of_o two_o perfect_a period_n 23_o be_v a_o imperfect_a simple_a period_n and_o 23_o 456_o be_v a_o imperfect_a compound_a period_n 16._o the_o second_o part_n of_o notation_n teach_v how_o to_o read_v a_o number_n that_o be_v express_v in_o writing_n 17._o every_o number_n be_v either_o simple_a or_o mix_v 18._o a_o simple_a number_n be_v that_o who_o part_n be_v of_o one_o and_o the_o same_o kind_n that_o be_v either_o whole_a or_o break_a 19_o a_o whole_a number_n be_v that_o which_o do_v consist_v of_o integer_n or_o entire_a unity_n thus_o 36_o be_v compose_v of_o thirty_o six_o integer_n or_o entire_a unity_n 20._o integer_n of_o one_o simple_a period_n be_v to_o be_v read_v as_o have_v be_v show_v in_o the_o 14_o rule_n and_o integer_n which_o be_v compose_v of_o divers_a period_n be_v to_o be_v read_v much_o after_o the_o same_o manner_n the_o period_n be_v first_o distinguish_v from_o one_o another_o by_o interpose_v a_o short_a stroke_n or_o point_n between_o each_o period_n as_o 15._o 246._o 368_o whence_o 15_o be_v a_o imperfect_a period_n 246_o and_o 368_o be_v perfect_a period_n this_o or_o any_o other_o number_n thus_o distinguish_v into_o period_n may_v easy_o be_v read_v by_o give_v to_o every_o period_n its_o proper_a denomination_n 21._o the_o proper_a denomination_n of_o the_o first_o period_n be_v hundred_o 2._o thousand_o 3._o million_o 4_o thousand_o of_o million_o &_o so_o forward_o increase_v by_o a_o tenfold_a proportion_n as_o far_o

+_o 9_o =_o 23._o and_o 23_o +_o 4_o =_o 27_o subscribe_v 7_o i_o carry_v to_o the_o next_o rank_n say_v 2_o +_o 7_o =_o 9_o 9_o +_o 1_o =_o 10._o 10_o +_o 5_o =_o 15._o 15_o +_o 8_o =_o 23._o 23_o +_o 3_o =_o 26_o subscribe_v 6_o i_o carry_v 2_o to_o the_o next_o rank_n say_v 2_o +_o 4_o =_o 6._o 6_o +_o 5_o =_o 11._o 11_o +_o 8_o =_o 19_o 19_o +_o 7_o =_o 26._o 26_o +_o 6_o =_o 32_o subscribe_v 2_o the_o other_o 3_o i_o set_v in_o the_o next_o rank_n because_o there_o be_v nothing_o more_o to_o be_v add_v and_o so_o the_o total_a sum_n be_v 3267163._o the_o proof_n of_o addition_n be_v thus_o draw_v a_o line_n under_o the_o first_o rank_n of_o number_n then_o add_v together_o all_o the_o remain_a rank_n as_o have_v be_v show_v last_o to_o that_o total_a add_v the_o first_o rank_n of_o number_n if_o that_o be_v the_o same_o with_o the_o first_o total_a find_v the_o addition_n be_v true_a otherwise_o not_o as_o in_o the_o last_o example_n the_o first_o rank_n of_o number_n be_v 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and_o the_o product_n in_o the_o second_o as_o 7_o and_o 8_o be_v give_v right_a against_o those_o number_n be_v 56_o which_o be_v the_o product_n of_o they_o the_o multiplication_n table_n 2.2_o 4_o 2.3_o 6_o 2.4_o 8_o 2.5_o 10_o 2.6_o 12_o 2.7_o 14_o 2.8_o 16_o 2.9_o 18_o 3.3_o 9_o 3.4_o 12_o 3.5_o 15_o 3.6_o 18_o 3.7_o 21_o 3.8_o 24_o 3.9_o 27_o 4.4_o 16_o 4.5_o 20_o 4.6_o 24_o 4.7_o 28_o 4.8_o 32_o 4.9_o 36_o 5.5_o 25_o 5.6_o 30_o 5.7_o 35_o 5.8_o 40_o 5.9_o 45_o 6.6_o

in_o 36_o the_o answer_n be_v 9_o time_n therefore_o i_o place_v 9_o in_o the_o quotient_a and_o 4_o multiply_v by_o 9_o the_o product_n be_v ●●_o which_o b●●ng_v deduct_v from_o ●6_n the_o remainder_n be_v 0_o and_o therefore_o the_o division_n be_v 〈◊〉_d and_o the_o quotient_a require_v be_v 〈◊〉_d 5._o when_o the_o divisor_n be_v also_o a_o composite_a number_n though_o the_o work_n be_v somewhat_o more_o troublesome_a yet_o be_v the_o manner_n of_o work_v the_o same_o with_o the_o former_a &_o therefore_o the_o former_a example_n well_o consider_v will_v make_v way_n for_o that_o which_o follow_v let_v then_o 6639642_o be_v a_o number_n give_v to_o be_v divide_v by_o 978_o and_o be_v set_v down_o as_o be_v before_o direct_v distinguish_v by_o a_o point_n so_o many_o of_o the_o foremost_a place_n of_o the_o dividend_n towards_o the_o left_a hand_n as_o be_v either_o equal_a to_o the_o divisor_n or_o be_v great_a do_v yet_o come_v near_a to_o it_o which_o in_o this_o example_n must_v 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such_o as_o shall_v make_v this_o objection_n let_v they_o make_v the_o three_o part_n of_o a_o pound_n to_o be_v the_o integer_fw-la that_o be_v six_o shilling_n and_o eight_o penny_n so_o may_v they_o have_v the_o decimal_n of_o any_o number_n of_o shilling_n penny_n and_o farthing_n exact_o as_o by_o the_o follow_a operation_n do_v appear_v let_v 6_n 8d_n be_v divide_v into_o 1.000000_o then_o shall_v the_o decimal_n of_o 8d_n be_v 0.100000_o the_o decimal_n of_o 4d_n shall_v be_v 0.050000_o the_o decimal_n of_o 2d_o 0.025000_o the_o decimal_n of_o 1_o penny_n 0.012500_o the_o decimal_n of_o 2_o farthing_n 0.006250_o the_o decimal_n of_o 1_o farthing_n 0.003125_o 6._o the_o like_a may_v be_v do_v for_o weight_n and_o measure_n but_o as_o for_o weight_n there_o will_v be_v no_o great_a advantage_n by_o turn_v they_o into_o decimal_n because_o all_o question_n concern_v they_o may_v be_v more_o convenient_o resolve_v without_o they_o and_o as_o for_o measure_n especial_o long_a 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numeration_n be_v very_o easy_a and_o in_o the_o four_o not_o very_o hard_a to_o effect_v as_o by_o example_n will_v appear_v let_v the_o decimal_n of_o these_o several_a sum_n of_o money_n be_v give_v to_o be_v add_v together_o ●●_o 1●_n 08_o 03_o 10._o 8864583_o 8._o 12_o 09_o 02_o 8._o 6395833_o ●_o ●9_n 11_o 01_o 6._o 4968750_o 1●_n 19_o 10_o 02_o 13._o 9937590_o 40._o ●0_n ●4_n 00_o 40._o 0166666_o draw_v a_o downright_a line_n between_o the_o whole_a number_n and_o the_o decimal_a fraction_n and_o then_o add_v they_o as_o in_o whole_a number_n so_o will_v the_o sum_n be_v 40l_n 00_n 04_o as_o by_o the_o operation_n do_v appear_v 2._o the_o subtraction_n of_o decimal_n be_v as_o easy_a only_o as_o before_o you_o be_v to_o distinguish_v the_o whole_a number_n from_o the_o fraction_n by_o a_o stroke_n or_o point_n and_o then_o proceed_v as_o in_o whole_a number_n have_v be_v direct_v  _fw-fr example_n decimal_n from_o 29._o 11_o 03_o 01_o 29.5635416_o deduct_v 14._o 19_o 11_o 02_o 14.9979166_o remain_v 14._o 11_o 03_o 03_o 14.5656250_o chap._n xvii_o of_o the_o multiplication_n of_o decimal_n 1._o though_o the_o multiplication_n of_o decimal_n do_v not_o at_o all_o differ_v from_o the_o multiplication_n of_o whole_a number_n as_o to_o the_o manner_n of_o the_o work_n and_o so_o i_o may_v be_v as_o brief_a in_o the_o explanation_n thereof_o as_o in_o addition_n and_o subtraction_n yet_o that_o the_o excellent_a use_n of_o decimal_a fraction_n may_v appear_v i_o shall_v illustrate_v 〈◊〉_d rule_n with_o more_o example_n than_o the_o former_a 2._o and_o to_o distinguish_v the_o integer_n from_o the_o decimal_a part_n when_o the_o multiplication_n be_v finish_v consider_v how_o many_o decimal_a part_n there_o be_v in_o both_o number_n give_v and_o cut_v off_o so_o many_o figure_n or_o place_n from_o the_o product_n towards_o the_o right_a hand_n the_o rest_n towards_o the_o left_a be_v integer_n and_o if_o the_o whole_a product_n have_v not_o so_o many_o place_n as_o there_o be_v part_n in_o both_o the_o number_n give_v you_o must_v prefix_v as_o many_o cipher_n before_o the_o product_n towards_o the_o left_a hand_n as_o there_o be_v place_n want_v to_o make_v it_o even_o with_o the_o number_n of_o the_o place_n in_o the_o part_n of_o both_o the_o number_n give_v example_n let_v 2_o pound_n 9_o shilling_n and_o 6_o penny_n be_v give_v to_o be_v multiply_v by_o 2_o pound_n 9_o shilling_n 6_o penny_n the_o decimal_n of_o 9_o shilling_n &_o 6_o penny_n be_v .475_o and_o therefore_o the_o multiplicand_a be_v 2.475_o and_o the_o multiplicator_n also_o be_v 2.475_o the_o product_n be_v 6.125625_o that_o be_v 6_o pound_n 2_o shilling_n 6_o penny_n and_o 625_o part_n 2_o example_n let_v a_o board_n in_o length_n be_v 14_o foot_n 6_o inch_n 3_o quarter_n and_o in_o breadth_n 2_o foot_n 9_o inch_n 3_o quarter_n and_o let_v the_o content_a of_o that_o board_n be_v require_v the_o decimal_n answer_v to_o 6_o inch_n 3_o quarter_n be_v 5625_o and_o therefore_o the_o multiplicand_a be_v .14_o .5625_o and_o the_o decimal_n of_o 9_o inch_n 3_o quarter_n be_v 8125_o and_o therefore_o the_o multiplicator_n be_v 2.8125_o which_o be_v multiply_v together_o as_o whole_a number_n the_o product_v be_v 40.95603125_o that_o be_v 40_o foot_n 11_o inch_n and_o 1_o quarter_n of_o a_o inch_n as_o it_o be_v before_o find_v in_o chap._n 10._o 3._o example_n let_v the_o decimal_n .3625_o be_v give_v to_o be_v multiply_v by_o .2387_o the_o product_n be_v 8652875_o which_o have_v but_o seven_o place_n of_o figure_n whereas_o there_o be_v eight_o place_n of_o part_n in_o both_o the_o number_n give_v i_o set_v a_o cypher_n before_o the_o product_n towards_o the_o left_a hand_n to_o make_v they_o equal_a and_o then_o the_o product_n be_v .08652875_o 3._o by_o these_o few_o example_n the_o easy_a resolution_n of_o some_o question_n by_o decimal_a fraction_n do_v appear_v which_o be_v any_o other_o way_n troublesome_a enough_o and_o as_o to_o the_o manner_n of_o work_v more_o need_v not_o be_v say_v neither_o but_o yet_o there_o be_v some_o contraction_n in_o multiplication_n which_o be_v in_o astronomical_a question_n very_o useful_a and_o also_o in_o make_v of_o table_n of_o interest_n and_o such_o like_a which_o be_v set_v down_o by_o mr._n oughtre_v in_o his_o clavis_fw-la chap._n 4_o it_o be_v to_o find_v any_o part_n of_o the_o product_n towards_o the_o left_a hand_n without_o find_v the_o remainder_n thereof_o towards_o the_o right_a hand_n as_o if_o 583614_o be_v give_v to_o be_v multiply_v by_o 472963_o and_o that_o the_o six_o first_o figure_n of_o the_o product_n be_v require_v towards_o the_o left_a hand_n without_o any_o consideration_n have_v to_o the_o use_n of_o the_o figure_n of_o the_o product_n towards_o the_o right_a hand_n the_o way_n be_v thus_o set_v the_o figure_n of_o your_o multiplicator_n under_o the_o figure_n of_o your_o multiplicand_n but_o the_o quite_o contrary_a way_n as_o here_o you_o see_v then_o beginning_n with_o the_o first_o figure_n f._n towards_o the_o right_a hand_n i_o set_v down_o the_o product_n thereof_o as_o be_v direct_v in_o the_o multiplication_n of_o whole_a number_n which_o do_v i_o multiply_v by_o the_o second_o figure_n but_o begin_v with_o that_o figure_n of_o the_o multiplicand_a which_o stand_v direct_o over_o it_o and_o set_v the_o product_n under_o the_o first_o figure_n of_o the_o first_o product_n and_o so_o the_o rest_n of_o the_o figure_n in_o the_o multiplicator_n have_v still_o regard_v to_o that_o increase_n which_o will_v arise_v out_o of_o the_o figure_n precede_v that_o with_o which_o you_o begin_v your_o multiplication_n so_o shall_v there_o be_v effect_v what_o be_v propound_v as_o by_o the_o operation_n do_v appear_v and_o for_o the_o clear_a manifestation●_n i_o have_v add_v the_o wh●le_a product_n also_o chap._n xviii_o of_o the_o division_n of_o decimal_n 1._o this_o be_v the_o last_o species_n of_o essential_a numeration_n and_o the_o most_o difficult_a but_o not_o much_o more_o difficult_a in_o decimal_a fraction_n than_o in_o whole_a number_n in_o the_o arithmetic_n work_v there_o be_v no_o difference_n between_o they_o at_o all_o that_o difference_n that_o be_v between_o they_o and_o all_o the_o difficulty_n too_o be_v in_o discover_v the_o nature_n of_o the_o quotient_a as_o whether_o it_o be_v only_o a_o fraction_n or_o a_o mix_a number_n or_o which_o be_v integer_n and_o which_o fraction_n or_o if_o fraction_n only_o whether_o the_o first_o place_n be_v ten_o or_o hundred_o or_o of_o some_o other_o denomination_n 2._o for_o the_o resolve_v of_o this_o question_n in_o all_o the_o case_n or_o variety_n that_o can_v possible_o happen_v take_v this_o general_a rule_n the_o first_o figure_n in_o your_o quotient_n will_v be_v always_o of_o the_o same_o degree_n or_o place_n with_o that_o figure_n or_o cypher_n in_o your_o dividend_n which_o stand_v over_o the_o place_n of_o unite_v in_o your_o divisor_n to_o illustrate_v this_o rule_n i_o will_v add_v a_o example_n or_o two_o let_v it_o be_v require_v to_o divide_v a_o mix_a number_n by_o a_o mix_a number_n let_v the_o product_n of_o the_o first_o example_n in_o the_o last_o chapter_n 6.125625_o which_o be_v a_o mix_a number_n be_v give_v to_o be_v divide_v by_o the_o multiplicator_n there_o give_v which_o be_v a_o mix_a number_n also_o to_o wit_n 2.475_o division_n be_v make_v according_a to_o the_o rule_v give_v for_o the_o divide_n of_o whole_a number_n the_o figure_n in_o the_o quotient_n will_v be_v 2475._o now_o it_o remain_v to_o separate_v the_o integer_n in_o this_o quotient_a from_o the_o decimal_a part_n to_o perform_v which_o i_o subscribe_v the_o divisor_n 2.475_o underneath_o the_o first_o dividual_a 6.125_o and_o i_o find_v that_o the_o figure_n 2_o which_o stand_v in_o the_o unite_v place_n in_o the_o divisor_n do_v also_o stand_v under_o the_o place_n of_o unite_v in_o the_o dividend_n and_o therefore_o i_o conclude_v according_a to_o the_o rule_n give_v that_o the_o first_o figure_n in_o the_o quotient_n must_v also_o stand_v in_o the_o first_o place_n of_o integer_n and_o consequent_o that_o the_o next_o place_n on_o the_o right_a hand_n must_v be_v the_o beginning_n of_o the_o fraction_n and_o the_o note_n of_o separation_n must_v be_v put_v between_o 2_o and_o 4_o as_o here_o you_o see_v 2._o example_n let_v the_o mix_a number_n 612.5625_o be_v give_v to_o be_v divide_v by_o .2475_o the_o quotient_n will_v be_v 2475_o as_o before_o but_o place_v the_o divisor_n under_o the_o dividend_n i_o find_v that_o the_o place_n of_o unite_v in_o the_o divisor_n which_o be_v now_o supply_v by_o a_o cipher_n do_v stand_v under_o the_o four_o place_n of_o integer_n in_o the_o dividend_n and_o therefore_o the_o first_o figure_n in_o the_o quotient_a be_v the_o

36_o 6.7_o 42_o 6.8_o 48_o 6.9_o 54_o 7.7_o 49_o 7.8_o 56_o 7.9_o 63_o 8.8_o 64_o 8.9_o 72_o 9.9_o 81_o 8._o the_o multiplication_n of_o single_a figure_n be_v thus_o attain_v the_o multiplication_n of_o compound_a number_n can_v be_v difficult_a 9_o in_o compound_v multiplication_n set_v down_o the_o give_v number_n as_o in_o addition_n and_o subtraction_n so_o 3421_o be_v give_v to_o be_v multiply_v by_o 2_o place_z 2_o the_o multiplicator_n under_o 3421_o the_o multiplicand_a and_o have_v draw_v a_o line_n write_v the_o double_a of_o every_o figure_n in_o the_o multiplicand_a under_o the_o line_n because_o the_o multiplicator_n be_v 2_o in_o the_o same_o degree_n always_o with_o that_o figure_n of_o which_o it_o be_v the_o double_a thus_o the_o double_a of_o one_o be_v 2_o the_o double_a of_o 2_o be_v 4_o the_o double_a of_o 4_o in_o the_o place_n of_o hundred_o be_v 8_o &_o the_o double_a of_o 3_o in_o the_o place_n of_o thousand_o be_v 6_o and_o the_o product_n of_o 3421_o multiply_v by_o 2_o be_v 6842._o 10._o when_o the_o product_n of_o any_o of_o the_o particular_a figure_n exceed_v ten_o place_n the_o excess_n under_o the_o line_n as_o before_o and_o the_o last_o figure_n in_o the_o next_o place_n to_o it_o but_o a_o line_n low_o and_o so_o the_o rest_n until_o the_o multiplication_n be_v finish_v as_o if_o the_o same_o number_n 3421_o be_v to_o be_v multiply_v by_o 8._o every_o figure_n of_o the_o multiplicand_a be_v to_o be_v make_v 8_o time_n as_o much_o as_o it_o be_v and_o subscribe_v in_o their_o proper_a place_n and_o then_o to_o be_v collect_v into_o one_o sum_n thus_o 8_o time_n 1_o be_v 8_o which_o i_o place_v under_o the_o 1._o and_o 8_o time_n 2_o be_v 16_o i_o write_v 6_o in_o the_o place_n of_o ten_o a_o line_n low_o and_o 1_o in_o the_o upper_a line_n in_o the_o place_n of_o hundred_o 8_o time_n 4_o be_v 32_o i_o write_v 2_o in_o the_o low_a line_n in_o the_o place_n of_o hundred_o and_o 3_o in_o the_o upper_a line_n a_o place_n forward_a that_o be_v in_o the_o place_n of_o thousand_o 8_o time_n 3_o be_v 24_o thousand_o i_o write_v 4_o therefore_o in_o the_o place_n of_o thousand_o in_o the_o low_a line_n and_o 2_o a_o place_n forward_a in_o the_o upper_a or_o the_o low_a line_n it_o matter_n not_o which_o because_o it_o be_v the_o last_o figure_n in_o the_o product_n and_o these_o two_o line_n of_o number_n be_v add_v together_o the_o sum_n be_v 27368_o =_o 3421_o ×_o 8._o 11._o but_o this_o may_v be_v more_o compendious_o perform_v if_o that_o every_o figure_n to_o be_v set_v in_o the_o upper_a line_n shall_v be_v commit_v to_o memory_n and_o add_v to_o the_o next_o rank_n so_o 3421_o being_n give_v to_o be_v multiply_v by_o 8_o as_o before_o for_o 8_o time_n 1_o i_o write_v 8_o under_o the_o line_n in_o the_o place_n of_o unite_v for_o 8_o time_n 2_o be_v 16_o i_o write_v 6_o under_o the_o line_n in_o the_o place_n of_o ten_o and_o reserve_v 1_o for_o the_o ten_o it_o exceed_v to_o be_v add_v in_o the_o next_o rank_n the_o place_n of_o hundred_o then_o i_o say_v 8_o time_n 4_o be_v 32_o unto_o which_o if_o i_o add_v 1_o which_o i_o keep_v in_o mind_n the_o whole_a be_v 33_o wherefore_o subscribe_v 3_o in_o the_o next_o rank_n under_o the_o line_n in_o the_o place_n of_o hundred_o and_o carry_v 3_o in_o mind_n for_o the_o three_o ten_o that_o it_o exceed_v i_o proceed_v to_o perform_v the_o rest_n of_o the_o work_n as_o you_o see_v it_o in_o the_o example_n 12._o when_o the_o multiplicand_a and_o the_o multiplicator_n be_v both_o compound_v number_n for_o so_o many_o figure_n as_o be_v in_o the_o multiplicator_n so_o many_o several_a product_n must_v be_v subscribe_v under_o the_o line_n which_o at_o last_o be_v add_v into_o one_o sum_n give_v you_o the_o total_a product_n of_o all_o so_o 3421_o be_v give_v to_o be_v multiply_v by_o 28_o the_o operation_n will_v stand_v thus_o for_o 3421_o be_v multiply_v by_o 8_o the_o product_n be_v 27368_o again_o 342_o i_o be_v multiply_v by_o 2_o the_o product_n be_v 6842_o which_o several_a product_n stand_v in_o their_o due_a order_n and_o add_v together_o do_v make_v 95788_o the_o product_v require_v 13._o when_o in_o the_o multiplicator_n there_o be_v cipher_n between_o the_o significant_a figure_n you_o must_v multiply_v by_o the_o significant_a figure_n only_o neglect_v the_o cipher_n and_o set_v each_o particular_a product_n in_o its_o due_a place_n according_a to_o what_o have_v be_v say_v in_o the_o last_o rule_n and_o as_o be_v do_v in_o these_o example_n follow_v 14._o when_o the_o number_n give_v to_o be_v multiply_v do_v one_o or_o both_o of_o they_o end_n with_o cipher_n neglect_v the_o cyher_n multiply_v by_o the_o significant_a figure_n only_o and_o have_v add_v together_o the_o several_a product_n according_a to_o the_o former_a direction_n annex_v as_o many_o cipher_n to_o the_o sum_n of_o all_o the_o product_n as_o be_v at_o the_o end_n of_o both_o the_o number_n give_v so_o shall_v you_o have_v the_o product_v desire_v as_o in_o the_o example_n follow_v 15._o when_o one_o of_o the_o number_n give_v be_v a_o unite_v with_o nothing_o but_o cipher_n annex_v the_o multiplication_n be_v perform_v by_o annex_v so_o many_o cipher_n to_o the_o multiplicand_a as_o there_o be_v in_o the_o multiplicator_n so_o if_o 4327_o be_v to_o be_v multiply_v by_o 1000_o the_o product_n will_v be_v 4327000._o chap._n v._o of_o division_n 1._o division_n be_v that_o by_o which_o we_o discover_v how_o often_o one_o number_n be_v contain_v in_o another_o and_o thereby_o find_v the_o quotient_a of_o the_o great_a number_n 2._o division_n have_v three_o part_n the_o dividend_n the_o divisor_n and_o the_o quotient_a the_o dividend_n be_v the_o number_n give_v to_o be_v divide_v the_o divisor_n be_v the_o number_n by_o which_o the_o dividend_n be_v divide_v and_o the_o quotient_n be_v the_o number_n which_o the_o division_n do_v produce_v so_o if_o 18_o be_v give_v to_o be_v divide_v by_o 3_o the_o number_n produce_v will_v be_v 6_o for_o 3_o be_v find_v 6_o time_n in_o 18_o and_o here_o 18_o be_v the_o dividend_n 3_o the_o divisor_n and_o 6_o the_o quotient_a 3._o division_n be_v either_o single_a or_o compound_v when_o the_o divisor_n be_v a_o single_a figure_n and_o the_o dividend_n less_o than_o 10_o the_o answer_n may_v be_v give_v by_o memory_n or_o if_o the_o dividend_n be_v the_o product_n of_o two_o single_a figure_n the_o answer_n may_v be_v have_v from_o the_o multiplication_n table_n if_o need_v be_v for_o find_v the_o divisor_n in_o the_o first_o column_n and_o the_o dividend_n in_o the_o second_o the_o other_o figure_n in_o the_o first_o column_n be_v the_o quotient_a seek_v as_o if_o 56_o be_v to_o be_v divide_v by_o 7_o in_o the_o same_o line_n where_o i_o find_v 7_o in_o the_o first_o column_n and_o 56_o in_o the_o second_o i_o find_v 8_o also_o which_o be_v the_o quotient_a 4._o when_o the_o divisor_n be_v a_o single_a figure_n and_o the_o dividend_n a_o number_n of_o many_o place_n make_v a_o crooked_a line_n at_o each_o end_n of_o the_o dividend_n that_o on_o the_o left_a hand_n serve_v for_o the_o place_n of_o the_o divisor_n &_o that_o on_o the_o right_n for_o the_o quotient_a then_o proceed_v gradual_o &_o ask_v how_o often_o the_o divisor_n be_v contain_v in_o the_o dividend_n and_o place_n that_o which_o answer_v the_o question_n in_o the_o quotient_a then_o multiply_v the_o divisor_n by_o that_o figure_n in_o the_o quotient_a &_o subtract_v the_o product_n from_o the_o dividend_n set_v down_o the_o remainder_n for_o example_n let_v 2916_o be_v give_v to_o be_v divide_v by_o 4._o place_v the_o number_n as_o have_v be_v say_v i_o ask_v how_o often_o 4_o be_v contain_v in_o 29_o the_o answer_n be_v 7_o time_n i_o therefore_o put_v 7_o in_o the_o quotient_a than_o i_o multiply_v 4_o by_o 7_o and_o it_o make_v 28_o which_o be_v subtract_v from_o 29_o the_o remainder_n be_v 1._o then_o draw_v a_o stroke_n between_o the_o remainder_n and_o the_o dividend_n i_o draw_v down_o the_o next_o figure_n which_o be_v 1_o also_o and_o so_o the_o new_a dividend_n be_v 11_o than_o i_o ask_v how_o often_o 4_o be_v contain_v in_o 11_o the_o answer_n be_v twice_o therefore_o i_o put_v 2_o in_o the_o quotient_a and_o 4_o multiply_v by_o 2_o makes_z 8_o which_o be_v deduct_v from_o 11_o the_o remainder_n be_v 3_o which_o i_o write_v under_o 11_o and_o draw_v a_o stroke_n between_o they_o as_o before_o and_o draw_v down_o the_o next_o figure_n which_o be_v 6_o and_o so_o the_o new_a divide_v be_v ●●_o then_o i_o ask_v how_o often_o 4_o be_v contain_v

it_o be_v require_v to_o divide_v 102_o pound_n 15_o shilling_n 8_o penny_n into_o 7_o equal_a part_n or_o among_o seven_o man_n you_o must_v place_v the_o number_n as_o have_v be_v show_v before_o then_o ask_v how_o often_o 7_o in_o 10_o the_o answer_n be_v once_o therefore_o i_o write_v 1_o in_o the_o quotient_a and_o deduct_v 7_o out_o of_o 10_o the_o remainder_n be_v 3_o to_o which_o i_o draw_v down_o 2_o the_o next_o figure_n in_o the_o dividend_n and_o it_o make_v 32_o than_o i_o ask_v how_o often_o 7_o be_v contain_v in_o 32_o the_o answer_n be_v 4_o which_o i_o set_v in_o the_o quotient_a than_o i_o multiply_v 7_o by_o 4_o the_o figure_n last_o place_v in_o the_o quotient_a and_o the_o product_n 28_o be_v subtract_v from_o 32_o the_o remainder_n be_v 4_o which_o be_v less_o than_o 7_o i_o multiply_v 4_o by_o 20_o the_o number_n of_o part_n in_o the_o next_o inferior_a denomination_n and_o the_o product_n be_v 80_o to_o which_o i_o add_v the_o 15_o shilling_n in_o the_o question_n and_o then_o the_o sum_n be_v 95_o which_o be_v divide_v by_o seven_o the_o quotient_n be_v 13_o and_o the_o remainder_n 4_o which_o be_v multiply_v by_o 12_o the_o number_n of_o part_n in_o a_o shilling_n the_o product_n be_v 48_o penny_n to_o which_o i_o add_v the_o 8_o penny_n in_o the_o question_n and_o then_o the_o sum_n be_v 56_o which_o be_v divide_v by_o 7_o the_o quotient_n be_v 8_o and_o nothing_o remain_v and_o so_o the_o whole_a quotient_n be_v 14l_o 15_n and_o 8d_n in_o like_a manner_n 8118l_o 11_n 3d._n be_v give_v to_o be_v divide_v by_o 283_o the_o quotient_n will_v be_v 28l_o 13_n 9d_o as_o by_o the_o follow_a operation_n do_v appear_v 3._o when_o a_o number_n of_o diverse_a denomination_n be_v give_v to_o be_v divide_v by_o a_o number_n of_o diverse_a denomination_n also_o it_o will_v be_v convenient_a to_o reduce_v both_o the_o number_n into_o number_n of_o the_o least_o denomination_n give_v before_o you_o begin_v your_o division_n and_o after_o division_n be_v finish_v to_o reduce_v the_o quotient_a into_o the_o several_a denomination_n in_o which_o the_o number_n be_v give_v for_o example_n let_v it_o be_v require_v to_o divide_v 6l_o 2s_n 6d_o 3_o 1_o by_o 2l_o 9_n 6d_o to_o reduce_v this_o dividend_n into_o square_a penny_n the_o ordinary_a way_n of_o reduction_n will_v not_o serve_v for_o this_o dividend_n be_v the_o quotient_a of_o a_o certain_a number_n of_o square_a penny_n divide_v by_o 57600_o the_o number_n of_o square_a penny_n in_o a_o pound_n sterling_a and_o therefore_o to_o make_v this_o sum_n give_v fit_a for_o division_n it_o must_v be_v multiply_v by_o 57600_o and_o to_o do_v this_o it_o will_v be_v proper_a to_o begin_v with_o the_o least_o denomination_n first_o which_o in_o this_o example_n be_v a_o fraction_n and_o a_o fraction_n reduce_v into_o its_o least_o term_n so_o that_o the_o first_o work_n will_v be_v to_o find_v the_o great_a fraction_n to_o which_o the_o fraction_n give_v be_v equivalent_a and_o this_o be_v easy_o do_v for_o 57600_o be_v the_o denominator_fw-la now_o then_o as_o 5_o to_z 3_o so_o 57600_o to_o 34560_o the_o numerator_n seek_v and_o this_o be_v divide_v by_o 4_o the_o number_n of_o farthing_n in_o a_o penny_n give_v i_o the_o last_o remainder_n before_o this_o 34560_o viz._n 8640_o now_o penny_n be_v the_o next_o great_a denomination_n i_o multiply_v 57600_o by_o 6_o the_o number_n of_o penny_n in_o the_o dividend_n the_o product_n be_v 345600_o to_o which_o add_v 8640_o the_o number_n before_o find_v the_o total_a be_v 354240_o which_o be_v divide_v by_o 12_o the_o number_n o●_n penny_n in_o a_o shilling_n the_o quotient_n be_v 2●520_n and_o 〈◊〉_d the_o number_n of_o shilling_n in_o the_o sum_n give_v be_v 2_o i_o multiply_v 57600_o by_o 2_o the_o product_n be_v 115200_o to_o which_o 29520_o be_v add_v the_o total_a be_v 144720_o which_o must_v be_v divide_v by_o 20_o the_o quotient_n will_v be_v 7236_o and_o so_o we_o have_v do_v with_o the_o shilling_n and_o penny_n in_o the_o question_n last_o i_o multiply_v 57600_o by_o 6_o the_o number_n of_o pound_n propound_v and_o the_o product_n be_v 345600_o to_o which_o 7236_o be_v add_v the_o total_a be_v 352836_o and_o so_o at_o length_n the_o number_n give_v be_v fit_v and_o prepare_v for_o division_n the_o divisor_n according_a to_o the_o ordinary_a way_n of_o reduction_n will_v be_v find_v to_o be_v 594_o by_o which_o divide_v 352836_o the_o quotient_n will_v be_v 594_o also_o as_o by_o the_o work_n appear_v and_o this_o quotient_a be_v reduce_v be_v 2l_o ●s_n 6d_o the_o quotient_a seek_v 4._o this_o question_n and_o all_o other_o of_o the_o like_a nature_n may_v be_v also_o resolve_v without_o reduce_v both_o number_n at_o the_o first_o in_o this_o manner_n first_o set_v down_o the_o indices_fw-la of_o the_o several_a denomination_n give_v and_o the_o dividend_n under_o those_o indices_fw-la and_o the_o divisor_n under_o the_o dividend_n then_o ask_v how_o often_o the_o great_a denomination_n in_o the_o divisor_n be_v contain_v in_o the_o great_a denomination_n in_o the_o dividend_n and_o put_v the_o answer_n in_o the_o quotient_a then_o multiply_v all_o the_o denomination_n in_o the_o divisor_n by_o the_o figure_n in_o the_o quotient_a and_o the_o product_n be_v deduct_v from_o the_o dividend_n set_v down_o the_o remainder_n and_o so_o proceed_v to_o the_o next_o and_o work_v in_o like_a manner_n so_o shall_v you_o at_o last_o have_v the_o quotient_a desire_a example_n let_v 6_o pound_n 2_o shilling_n 6_o penny_n ⅗_n be_v give_v to_o be_v divide_v by_o 2_o pound_n 9_o shilling_n 6_o penny_n the_o number_n be_v set_v as_o here_o you_o see_v i_o ask_v how_o often_o 2_o be_v contain_v in_o 6_o and_o the_o answer_n be_v 3_o time_n but_o 2_o pound_n 9_o shilling_n can_v be_v have_v but_o twice_o in_o 6_o pound_n 2_o shilling_n therefore_o i_o put_v 2l_o in_o the_o quotient_a and_o multiply_v the_o divisor_n 2l_o 9_n 6d_o by_o 2._o and_o the_o product_n 4l_o 19_n be_v deduct_v from_o 6l_o 2s_n the_o remainder_n be_v 1l_o 3_n 6d_o and_o so_o have_v i_o wrought_v once_o then_o i_o ask_v how_o often_o 2._o be_v contain_v in_o 1l_o 3_n that_o be_v in_o 23_n the_o answer_n be_v 11_o l._n s._n d._n ●_o ten_o  _fw-fr 6_o 2_o 6_o 00_o 6_o  _fw-fr 2_o 9_o 6_o  _fw-fr  _fw-fr  _fw-fr 4_o 19_o 0_o  _fw-fr  _fw-fr 2_o lib_n 1_o 3_o 6_o 00_o 6_o  _fw-fr 1_o 2_o 3_o 01_o 2_o 9_o shil_fw-la  _fw-fr 1_o 2_o 3_o 4_o  _fw-fr  _fw-fr 1_o 2_o 3_o 4_o 6_o d._n but_o i_o can_v take_v but_o 9_v because_o of_o the_o follow_a figure_n therefore_o i_o put_v 9_o shilling_n in_o the_o quotient_a and_o multiply_v 2l_o 9_n 6d_o by_o 9_o and_o the_o product_n be_v 1l_o 2s_n 3d._n 1_o farthing_n and_o 2_o ten_o which_o be_v deduct_v from_o 1l_o 3_n 6d_o ●_o far_o and_o 6_o ten_o the_o remainder_n be_v 1_o shilling_n 2_o penny_n 3_o farthing_n and_o 4_o ten_o and_o so_o i_o have_v wrought_v twice_o then_o i_o ask_v how_o often_o 2._o in_o 1_n 2d_o and_o the_o answer_n be_v 6_o therefore_o i_o put_v 6_o penny_n in_o the_o quotient_a and_o multiply_v 2_o pound_n 9_o shilling_n and_o 6_o penny_n by_o 6_o and_o the_o product_n be_v 1_o shilling_n 2_o penny_n 3_o farthing_n and_o 4_o ten_o which_o be_v deduct_v from_o 1_o shilling_n 2_o penny_n 3_o farthing_n and_o 4_o ten_o the_o remainder_n be_v nothing_o and_o so_o the_o quotient_n be_v 2l_o 9_o shilling_n and_o 6_o penny_n 5._o in_o astronomical_a fraction_n let_v it_o be_v require_v to_o divide_v 13_o sign_n 19_o degree_n 36_o minute_n 32_o second_n 47_o three_o and_o 20_o fourth_n by_o 17_o degree_n 37_o minute_n 25_o second_o first_o set_v down_o the_o indices_fw-la or_o several_a denomination_n of_o astronomical_a fraction_n give_v and_o under_o they_o write_v the_o number_n of_o the_o part_n belong_v to_o each_o denomination_n as_o they_o be_v propound_v in_o the_o dividend_n and_o set_v your_o divisor_n either_o under_o the_o dividend_n as_o here_o you_o see_v or_o in_o what_o other_o place_n you_o please_v s._n d._n m._n s._n t._n f._n  _fw-fr 13_o 19_o 36_o 32_o 47_o 20_o dividend_n  _fw-fr 17_o 37_o 25_o  _fw-fr  _fw-fr divisor_n 13_o 15_o 20_o 35_o  _fw-fr  _fw-fr 23_o deg_n quotient_a 0_o 04_o 15_o 57_o 47_o  _fw-fr  _fw-fr  _fw-fr 4_o 06_o 43_o 50_o  _fw-fr 14_o min._n  _fw-fr 00_o 09_o 13_o 57_o 20_o  _fw-fr  _fw-fr 00_o 09_o 13_o 57_o 20_o 32_o sec._n then_o beginning_n at_o the_o left_a hand_n ask_v how_o oft_o the_o number_n of_o the_o great_a denomination_n in_o the_o divisor_n can_v be_v have_v in_o number_n of_o the_o great_a denomination_n in_o the_o dividend_n but_o if_o

your_o divisor_n be_v great_a than_o the_o first_o number_n in_o your_o dividend_n place_v your_o divisor_n one_o step_n far_o towards_o your_o right_a hand_n thus_o in_o our_o example_n because_o 17_o be_v not_o contain_v in_o 13_o the_o great_a denomination_n in_o the_o dividend_n therefore_o i_o s●t_v it_o a_o place_n forward_a towards_o the_o right_a hand_n that_o be_v under_o 19_o deg_n and_o then_o reduce_v the_o 13_o sign_n into_o deg_n by_o multiply_v they_o by_o 30_o the_o product_n be_v 390_o to_o which_o the_o 19_o be_v add_v the_o sum_n be_v 409_o in_o which_o 17_o be_v contain_v 23_o time_n therefore_o multiply_v the_o divisor_n 17_o deg_n 37_o min._n 25_o sec._n by_o 23_o the_o product_n be_v 13_o ●ig_n 15_o deg_n 20_o min._n 35_o sec._n which_o be_v deduct_v from_o the_o dividend_n the_o remainder_n be_v 04_o deg_n 15_o min._n 57_o sec._n and_o to_o this_o remainder_n i_o draw_v down_o the_o number_n of_o part_n in_o the_o next_o lesser_a denomination_n and_o then_o i_o ask_v how_o oft_o 17_o be_v contain_v in_o 4_o deg_n 14_o min._n and_o the_o answer_n be_v 14_o time_n which_o i_o place_n in_o the_o quotient_a and_o multiply_v the_o divisor_n 17_o deg_n 37_o min._n 25_o sec._n by_o 14_o the_o product_n be_v 04_o deg_n 06_o min._n 43_o sec._n 50_o three_o which_o be_v deduct_v from_o the_o dividend_n as_o before_o the_o remainder_n will_v be_v 9_o min._n 13_o sec._n 57_o three_o and_o unto_o this_o remainder_n i_o draw_v down_o the_o number_n of_o part_n in_o the_o next_o lesser_a denomination_n and_o then_o proceed_v as_o before_o i_o ask_v how_o oft_o 17_o in_o the_o divisor_n be_v contain_v in_o 9_o min._n 13_o sec._n in_o the_o dividend_n and_o the_o answer_n be_v 32_o which_o i_o set_v in_o the_o quotient_a and_o thereby_o also_o i_o multiply_v the_o divisor_n as_o before_o and_o the_o product_n be_v 9_o min._n 13_o sec._n 57_o three_o 20_o fourth_n there_o remain_v nothing_o and_o so_o at_o last_o i_o find_v the_o quotient_a to_o be_v 23_o deg_n 14_o min._n 32_o sec._n chap._n xii_o of_o the_o reduction_n of_o uulgar_a fraction_n 1._o hitherto_o have_v be_v show_v the_o numeration_n of_o such_o break_a number_n or_o fraction_n as_o be_v many_o time_n express_v like_o integer_n numeration_n of_o such_o break_a number_n or_o fraction_n as_o be_v express_v by_o their_o numerator_n and_o denominator_n now_o follow_v and_o these_o break_a number_n be_v common_o call_v natural_a or_o vulgar_a fraction_n 2._o the_o numeration_n of_o vulgar_a fraction_n be_v either_o accidental_a or_o essential_a 3._o accidental_a numeration_n otherwise_o call_v reduction_n be_v that_o which_o change_v or_o reduce_v the_o term_n of_o the_o part_n give_v without_o change_v or_o alter_v of_o their_o value_n 4._o reduction_n or_o accidental_a numeration_n be_v either_o of_o part_n to_o part_n or_o integer_n to_o part_n 5._o reduction_n of_o part_n to_o part_n be_v that_o which_o find_v other_o part_n that_o be_v proportional_a to_o the_o part_n give_v and_o this_o three_o way_n 1._o by_o reduce_v the_o part_n give_v into_o their_o least_o term_n if_o they_o be_v not_o in_o their_o least_o term_n already_o 2._o by_o reduce_v fraction_n of_o unequal_a denomination_n into_o fraction_n of_o the_o same_o value_n who_o denominator_fw-la shall_v be_v equal_a 3._o by_o reduce_v fraction_n of_o diverse_a denomination_n into_o fraction_n of_o any_o denomination_n that_o shall_v be_v desire_v 6._o reduction_n of_o part_n or_o fraction_n into_o their_o least_o term_n be_v perform_v by_o find_v out_o a_o common_a measure_n to_o both_o part_n of_o the_o fraction_n give_v that_o be_v by_o find_v out_o the_o great_a number_n which_o will_v measure_v or_o divide_v each_o of_o the_o number_n give_v without_o leave_v any_o remainder_n 7._o the_o great_a common_a measure_n of_o two_o number_n give_v may_v be_v find_v by_o divide_v the_o great_a by_o the_o less_o and_o the_o divisor_n by_o the_o remainder_n and_o so_o continual_o until_o nothing_o do_v remain_v the_o last_o divisor_n be_v the_o common_a measure_n seek_v example_n let_v it_o be_v require_v to_o find_v the_o great_a common_a measure_n unto_o 63_o and_o 14_o set_v your_o number_n as_o have_v be_v direct_v in_o division_n of_o whole_a number_n thus_o then_o i_o ask_v how_o often_o 14_o in_o 63_o the_o answer_n be_v 4_o time_n therefore_o i_o put_v 4_o in_o the_o quotiont_n and_o multiply_v 14_o by_o 4_o and_o the_o product_n 56_o i_o set_v under_o the_o dividend_n 63_o and_o substract_v 56_o from_o 63_o the_o remainder_n be_v 7_o than_o i_o divide_v 14_o by_o 7_o the_o quotient_n be_v 2_o and_o the_o remainder_n 0_o therefore_o 7_o the_o last_o divisor_n be_v the_o common_a measure_n seek_v 8._o the_o common_a measure_n of_o any_o two_o number_n which_o do_v constitute_v a_o fraction_n be_v thus_o find_v the_o say_a fraction_n may_v be_v reduce_v into_o its_o least_o term_n by_o divide_v the_o numerator_n and_o denominator_fw-la by_o the_o common_a measure_n for_o the_o quotient_n will_v be_v the_o numerator_n &_o denominator_fw-la of_o the_o fraction_n and_o in_o the_o least_o term_n so_o if_o the_o fraction_n 14_o ●●_o be_v give_v to_o be_v reduce_v into_o the_o least_o term_n the_o common_a measure_n be_v 7_o i_o divide_v 14_o by_o 7_o and_o the_o quotient_n be_v 2_o for_o a_o new_a numerator_n and_o divide_v 63_o by_o 7_o the_o quotient_n be_v 9_o for_o a_o new_a denominator_fw-la and_o so_o the_o fraction_n 14_o 63_o be_v reduce_v into_o its_o least_o term_n that_o be_v into_o the_o fraction_n ●_o ●_o but_o if_o the_o great_a common_a measure_n unto_o the_o numerator_n and_o the_o denominator_fw-la given_z shall_v be_v 1_o such_o fraction_n be_v in_o its_o least_o term_n already_o so_o the_o fraction_n 13_o 34_o can_v be_v reduce_v into_o low_a term_n because_o the_o great_a common_n measure_n be_v 1._o this_o be_v a_o general_a rule_n for_o the_o reduction_n of_o fraction_n into_o their_o least_o term_n but_o yet_o even_a this_o may_v be_v sometime_o shorten_v for_o when_o the_o numerator_n and_o denominator_fw-la be_v even_o number_n they_o may_v be_v measure_v or_o divide_v by_o 2._o as_o if_o 16_o be_v the_o numerator_n and_o 64_o the_o denominator_fw-la draw_v a_o line_n at_o length_n between_o they_o to_o separate_v the_o numerator_n from_o the_o denominator_fw-la and_o cross_v the_o same_o with_o a_o down_o right_a stroke_n near_o the_o fraction_n as_o here_o you_o see_v in_o the_o margin_n then_o take_v the_o half_a of_o 16_o which_o ●●_o be_v 8_o for_o a_o new_a numerator_n and_o the_o half_a of_o 64_o which_o be_v 32_o for_o a_o new_a denominator_fw-la again_o the_o half_a of_o 8_o be_v 4_o and_o the_o half_a of_o 32_o be_v 16_o and_o so_o proceed_v in_o like_a manner_n there_o will_v be_v find_v ¼_n equivalent_a to_o 16_o 64._o the_o like_a abbreviation_n will_v be_v find_v when_o the_o numerator_n and_o denominator_fw-la do_v both_o end_n with_o 5_o or_o one_o of_o they_o end_v with_o 5_o and_o the_o other_o with_o a_o cypher_n for_o then_o the_o fraction_n may_v be_v reduce_v into_o its_o least_o term_n by_o divide_v both_o part_n of_o the_o fraction_n give_v by_o 5._o thus_o if_o it_o be_v require_v to_o reduce_v the_o fraction_n 125_o 345_o divide_v 125_o by_o 5_o the_o quotient_n will_v be_v 5_o and_o divide_v 345_o by_o 5_o the_o quotient_n will_v be_v 19_o and_o so_o the_o fraction_n 135_o 345_o be_v reduce_v to_o the_o fraction_n 5_o 19_o 9_o reduction_n of_o fraction_n to_o the_o same_o denomination_n be_v perform_v by_o multiply_v the_o numerator_n of_o each_o fraction_n into_o all_o the_o denominator_n continual_o except_o its_o own_o for_o so_o the_o several_a product_n which_o arise_v by_o such_o continual_a multiplication_n shall_v be_v the_o new_a numerator_n and_o the_o product_v find_v by_o multiply_v all_o the_o denominator_n continual_o shall_v be_v the_o common_a denominator_fw-la to_o all_o those_o new_a numerator_n as_o in_o the_o follow_a example_n ¾_n and_o 5_o 6_o i_o multiply_v 3_o by_o 6_o and_o they_o make_v 18_o and_o 5_o by_o 4_o and_o they_o make_v 20_o for_o the_o two_o numerator_n than_o i_o multiply_v 4_o by_o 6_o and_o the_o product_n be_v 24_o for_o the_o common_a denominator_fw-la and_o so_o the_o fraction_n ¾_n and_o ⅚_n be_v reduce_v into_o the_o fraction_n 18_o 24_o and_o 20_o 24_o or_o be_v reduce_v into_o their_o least_o term_n 9_o 12_o and_o 10_o 12_o as_o by_o the_o work_n appear_v thus_o if_o the_o fraction_n ⅖_n 3_o 7_o and_o 4_o 9_o ●●_o be_v propound_v here_o i_o multiply_v 2_o by_o 7_o and_o 9_o continual_o and_o they_o make_v 126_o also_o i_o multiply_v 3_o by_o 5_o and_o 9_o and_o the_o product_n be_v 135_o in_o like_a manner_n i_o mulmultiply_v

new_a fraction_n will_v be_v 84_o ●0_n as_o before_o 5_o example_n if_o 4¼_n of_o a_o yard_n of_o cloth_n in_o length_n be_v 1_o yard_n and_o half_a broad_a will_v make_v a_o cloak_n how_o much_o plush_n that_o be_v ¾_n of_o a_o yard_n in_o breadth_n will_v serve_v to_o line_v that_o cloak_n here_o the_o four_o term_v require_v must_v be_v great_a than_o the_o second_o term_n give_v and_o therefore_o the_o lesser_a extreme_a must_v be_v the_o divisor_n and_o the_o less_o extreme_a be_v in_o the_o three_o place_n 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 the_o number_n must_v stand_v as_o here_o you_o see_v breadth_n length_n breadth_n 1_o 1●_n 4¼_n ¾_n or_o 3_o ●_o 17_o 4_o ¾_n and_o ¾_n be_v the_o divisor_n 4_o be_v multiply_v first_o by_o 17_o and_o then_o by_o 3_o the_o last_o product_n be_v 204_o and_o 3_o be_v multiply_v by_o 4_o and_o by_o 2_o the_o last_o product_n be_v 24_o and_o so_o the_o new_a fraction_n or_o quotient_n seek_v be_v 204_o 24_o which_o be_v reduce_v be_v 8½_n yard_n 6._o example_n in_o decimal_a fraction_n if_o 7_o ounce_n 3_o penny_n weight_n and_o 12_o grain_n be_v worth_a 21l_o 11_n 6d_o what_o be_v the_o value_n of_o 1½_n ounce_n the_o decimal_n of_o 3_o penny_n weight_n be_v 0.15_o and_o the_o decimal_n of_o 12_o ounce_n be_v 0.025_o and_o the_o sum_n of_o these_o 2_o decimal_n be_v 0.515_o wherefore_o the_o first_o number_n in_o the_o rule_n of_o three_o be_v 7.175_o again_o the_o decimal_n of_o 10_n 6d_o be_v 0525_o wherefore_o the_o second_o number_n be_v 21.515_o and_o the_o three_o number_n be_v 1.5_o now_o to_o resolve_v this_o question_n the_o 3_o give_v number_n will_v stand_v thus_o 7.175_o −_o 21.525_o −_o 1.5_o last_o multiply_v the_o second_o by_o the_o three_o the_o product_n be_v 32.2875_o which_o be_v divide_v by_o 7.175_o the_o quotient_n be_v 4.5_o that_o be_v 4_o pound_n and_o ten_o shilling_n which_o be_v the_o price_n of_o a_o ounce_n and_o half_a of_o that_o gold_n chap._n xxiii_o of_o the_o rule_n of_o practice_n 1._o when_o the_o rule_n of_o three_o direct_a have_v 1_o or_o a_o integer_fw-la for_o the_o first_o term_n it_o be_v common_o call_v a_o rule_n of_o practice_n not_o only_o for_o the_o speedy_a but_o the_o practical_a resolution_n of_o such_o question_n for_o whereas_o other_o question_n belong_v to_o this_o rule_n do_v for_o the_o most_o part_n require_v multiplication_n and_o division_n these_o in_o the_o ordinary_a way_n be_v resolve_v by_o multiplication_n and_o whereas_o multiplication_n be_v general_o much_o easy_a than_o division_n these_o question_n be_v in_o a_o accustomary_a or_o practical_a way_n for_o the_o most_o part_n perform_v by_o division_n and_o yet_o do_v with_o more_o ease_n than_o they_o can_v be_v by_o multiplication_n nay_o where_o multiplication_n and_o division_n be_v both_o use_v the_o resolution_n be_v yet_o easy_a than_o it_o be_v or_o can_v be_v by_o multiplication_n only_o 2._o the_o question_n to_o be_v resolve_v by_o these_o abbreviation_n of_o the_o rule_n of_o three_o or_o practical_a operation_n do_v for_o the_o most_o part_n come_v under_o one_o or_o other_o of_o these_o five_o case_n the_o price_n of_o 1_o or_o a_o integer_fw-la do_v consist_v either_o 1._o of_o a_o certain_a number_n of_o shilling_n under_o twenty_o 2._o of_o a_o certain_a number_n of_o pound_n and_o shilling_n 3._o of_o a_o certain_a number_n of_o penny_n under_o twelve_o 4._o of_o a_o certain_a number_n of_o shilling_n and_o penny_n 5._o of_o a_o certain_a number_n of_o pound_n shilling_n and_o penny_n with_o the_o part_n of_o a_o penny_n 3._o now_o for_o the_o resolution_n of_o such_o question_n which_o may_v or_o do_v fall_v under_o any_o of_o these_o case_n it_o be_v necessary_a that_o the_o aliquot_fw-la part_n of_o a_o pound_n and_o shilling_n be_v first_o know_v that_o be_v how_o often_o the_o shilling_n of_o penny_n which_o be_v the_o give_v price_n of_o one_o integer_fw-la be_v contain_v in_o a_o pound_n or_o shilling_n without_o leave_v any_o remainder_n thus_o 4_o shilling_n be_v a_o aliquot_fw-la part_n of_o a_o pound_n or_o 20_o shilling_n because_o 4_o may_v be_v take_v 5_o time_n in_o 20_o without_o either_o excess_n or_o defect_n and_o 3_o be_v a_o aliquot_fw-la part_n of_o a_o shilling_n or_o 12_o penny_n for_o 4_o time_n 3_o do_v make_v 12_o but_o 7_o be_v not_o the_o aliquot_fw-la part_n of_o 20_o for_o two_o time_n 7_o be_v less_o than_o 20_o and_o 3_o time_n 7_o be_v more_o in_o like_a manner_n 5_o be_v not_o the_o aliquot_fw-la part_n of_o 12_o for_o 2_o time_n 5_o be_v but_o 10_o and_o 3_o time_n 5_o be_v 15_o the_o one_o want_n of_o 12_o and_o the_o other_o exceed_v it_o but_o yet_o any_o number_n of_o shilling_n under_o 20_o may_v be_v reduce_v to_o the_o aliquot_fw-la part_n of_o a_o pound_n and_o any_o number_n of_o penny_n under_o 12_o may_v be_v reduce_v to_o the_o aliquot_fw-la part_n of_o a_o shilling_n thus_o 7_o shilling_n be_v one_o five_o more_o 3_o twenty_n of_o a_o pound_n and_o 5_o penny_n be_v one_o three_o more_o one_o twelve_v of_o a_o shilling_n or_o one_o four_o more_z one_o six_z such_o aliquot_fw-la part_n of_o a_o pound_n or_o shilling_n as_o be_v most_o useful_a in_o the_o resolution_n of_o question_n of_o this_o nature_n be_v here_o express_v in_o two_o table_n with_o which_o i_o have_v join_v a_o three_o show_v the_o product_n of_o 12_o by_o any_o number_n not_o exceed_v it_o self_n which_o in_o many_o question_n will_v be_v find_v as_o useful_a as_o either_o of_o the_o other_o aliquot_fw-la part_n of_o a_o pound_n aliquot_fw-la part_n of_o a_o shilling_n the_o product_n of_o 12_o by_o any_o number_n less_o  _fw-fr  _fw-fr  _fw-fr pence_n  _fw-fr  _fw-fr 10_o 0_o ½_o 11_o ⅓+⅓+¼_n 2_o ×_o 12._o 24_o 6_o 8_o ⅓_n 10_o ½+⅓_n 3_o ×_o 12._o 36_o 5_o 0_o ¼_n 9_o ½+¼_n 4_o ×_o 12._o 48_o 4_o 0_o ⅕_n 8_o ⅓+●_n ●_o 5_o ×_o 12._o 60_o 3_o 4_o ⅙_n 7_o ¼+⅓_n 6_o ×_o 12._o 72_o 2_o 6_o ⅛_n 6_o ½_o 7_o ×_o 12._o 84_o 2_o 0_o 1_o 10_o 5_o ¼+1_n ●_o 8_o ×_o 12._o 96_o 1_o 8_o 1_o 12_o 4_o ⅓_n 9_o ×_o 12._o 108_o 1_o 4_o 1_o 15_o 3_o ¼_n 10_o ×_o 12._o 120_o 1_o 3_o 1_o ●_o 2_o 1_o ●_o 11_o ×_o 12._o 132_o 1_o 0_o 1_o ●_o 1½_n ●_o ●_o 12_o ×_o 12._o 144_o shil_fw-la pence_n  _fw-fr 1_o 1_o 20_o  _fw-fr these_o thing_n premise_v i_o will_v now_o show_v how_o any_o question_n come_v under_o the_o aforesaid_a five_o case_n may_v be_v resolve_v 1._o case_n where_o the_o price_n of_o a_o integer_fw-la be_v shilling_n only_o 4._o where_o the_o price_n of_o 1_o or_o a_o integer_fw-la be_v two_o shilling_n the_o price_n of_o as_o many_o integer_n as_o you_o will_v may_v be_v discover_v by_o bare_a inspection_n for_o two_o shilling_n be_v the_o ten_o of_o a_o pound_n the_o double_a of_o the_o first_o figure_n towards_o the_o right_a hand_n be_v the_o number_n of_o shilling_n require_v and_o the_o rest_n of_o the_o figure_n be_v so_o many_o pound_n example_n 567_o yard_n at_o 2_o shilling_n the_o yard_n will_v cost_v 56_o pound_n 14_o shilling_n for_o the_o double_a of_o 7_o be_v 14_o which_o i_o write_v down_o by_o itself_o as_o shilling_n then_o take_v the_o rest_n of_o the_o figure_n towards_o the_o left_a hand_n for_o pound_n the_o answer_n be_v 56l_o 14_n 5._o when_o the_o give_v price_n of_o 1_o or_o a_o integer_fw-la be_v a_o even_a number_n of_o shilling_n great_a than_o two_o multiply_v the_o number_n of_o integer_n who_o price_n be_v require_v by_o half_a the_o number_n of_o shilling_n give_v the_o double_a of_o the_o first_o figure_n towards_o the_o right_a hand_n in_o the_o product_n be_v set_v down_o for_o the_o shilling_n apart_o all_o the_o other_o figure_n towards_o the_o left_a hand_n shall_v be_v the_o pound_n require_v example_n let_v the_o price_n of_o 365_o yard_n be_v require_v at_o 14_o shilling_n per_fw-la 1_o yard_n if_o you_o multiply_v 365_o by_o 7_o which_o be_v the_o half_a of_o 14_o the_o number_n of_o shilling_n give_v the_o product_n will_v be_v 2562_o now_o the_o double_a of_o 2_o the_o first_o figure_n 2_o towards_o the_o right_a hand_n be_v 4_o the_o other_o figure_n be_v 256_o pound_n and_o so_o the_o answer_n be_v 256l_o 4s_fw-la here_o note_v that_o 4_o shilling_n be_v the_o 5_o part_n of_o a_o pound_n if_o that_o be_v the_o price_n of_o a_o integer_fw-la it_o will_v be_v all_o one_o to_o multiply_v by_o 2_o or_o divide_v by_o 5_o if_o the_o double_a of_o the_o first_o figure_n in_o the_o product_n towards_o the_o right_a hand_n be_v take_v for_o the_o shilling_n according_a

the_o difference_n ought_v to_o be_v as_o often_o set_v down_o as_o it_o be_v or_o may_v be_v diverse_o link_v thus_o in_o the_o premise_a rate_n give_v if_o you_o change_v the_o mean_a price_n to_o 16_o the_o rate_n must_v be_v link_v and_o the_o difference_n set_v as_o here_o you_o see_v they_o or_o be_v 40_o thus_o 0_o 13._o the_o give_v rate_n be_v thus_o rank_v and_o link_v and_o their_o difference_n take_v all_o question_n in_o alternation_n partial_a may_v be_v resolve_v by_o this_o proportion_n as_o the_o difference_n against_o the_o first_o rate_n be_v to_o the_o several_a difference_n that_o be_v under_o it_o so_o be_v the_o quantity_n propound_v to_o the_o several_a quantity_n that_o be_v require_v example_n suppose_v a_o man_n be_v to_o mix_v 10_o bushel_n of_o wheat_n at_o 48_o penny_n the_o bushel_n with_o rye_n at_o 36d_o the_o bushel_n barley_n at_o 24_o penny_n the_o bushel_n and_o oat_n at_o 12d_o the_o bushel_n so_o that_o the_o whole_a mixture_n may_v be_v sell_v at_o 28d_o the_o bushel_n the_o question_n be_v how_o much_o rye_n how_o much_o barley_n and_o how_o much_o oat_n ought_v to_o be_v add_v to_o the_o 10_o bushel_n of_o wheat_n it_o be_v evident_a by_o the_o difference_n before_o take_v that_o for_o every_o 16_o bushel_n of_o wheat_n i_o ought_v to_o take_v 4_o bushel_n of_o rye_n 8_o bushel_n of_o barley_n and_o 20_o bushel_n of_o oat_n and_o therefore_o i_o say_v 1._o as_o 16._o 4._o 10._o 28_o ●●_o 2._o as_o 16._o 8._o 10._o 5_o 3._o as_o 16._o 20._o 10._o 128_o ●●_o and_o from_o hence_o it_o appear_v that_o 2_o bushel_n and_o a_o half_a of_o rye_n 5_o bushel_n of_o barley_n and_o 12_o bushel_n and_o a_o half_a of_o oat_n be_v put_v to_o 10_o bushel_n of_o wheat_n when_o these_o several_a sort_n of_o grain_n do_v bear_v the_o price_n as_o in_o the_o 11_o rule_n may_v be_v sell_v one_o with_o another_o for_o 28_o penny_n or_o 2_o shilling_n and_o 4_o penny_n the_o bushel_n or_o the_o difference_n be_v take_v according_a to_o the_o second_o way_n of_o link_a they_o express_v the_o same_o rule_n i_o say_v 1._o as_o 4._o 16._o 10._o 40_o bushel_n of_o rye_n 2._o as_o 4._o 20._o 10._o 50_o bushel_n of_o barley_n 3._o as_o 4._o 8._o 10._o 10._o 20_o of_o oat_n and_o a_o mixture_n be_v make_v according_a to_o these_o proportion_n the_o whole_a may_v be_v sell_v at_o 2_o shilling_n 4_o penny_n the_o bushel_n 14._o in_o alternation_n partial_a the_o proof_n be_v likewise_o by_o compare_v the_o total_a value_n of_o the_o several_a simple_n with_o the_o value_n of_o the_o whole_a mixture_n so_o in_o the_o last_o example_n of_o the_o former_a rule_n the_o total_a value_n of_o 10_o bushel_n of_o wheat_n 40_o bushel_n of_o rye_n 50_o bushel_n of_o barley_n and_o 20_o bushel_n of_o oat_n amount_v to_o 14_o pound_n which_o be_v also_o the_o value_n of_o the_o whole_a mixture_n at_o 2_o shilling_n and_o 4_o penny_n the_o bushel_n 15._o in_o alternation_n total_a the_o several_a rate_n and_o mean_a price_n be_v so_o place_v and_o their_o difference_n take_v as_o have_v be_v show_v all_o question_n belong_v thereto_o may_v be_v answer_v by_o this_o proportion_n as_o the_o sum_n of_o all_o the_o difference_n be_v to_o the_o total_a quantity_n of_o all_o the_o simple_n so_o be_v the_o correspondent_a difference_n of_o each_o rate_n to_o the_o respective_a quantity_n of_o the_o same_o rate_n example_n a_o goldsmith_n have_v diverse_a sort_n of_o gold_n viz._n some_o of_o 24_o charact_n fine_a some_o of_o 22_o charact_n some_o of_o 18_o charact_n and_o some_o of_o 16_o charact_n fine_a be_v desirous_a to_o melt_v of_o all_o these_o sort_n so_o much_o together_o as_o may_v make_v a_o mass_n contain_v 60_o ounce_n of_o 21_o charact_n fine_a the_o number_n be_v place_v and_o difference_v as_o have_v be_v show_v and_o be_v here_o express_v i_o say_v 1._o as_o 12._o 60._o 5._o 25_o ounce_n 2._o as_o 12._o 60._o 3._o 15_o ounce_n 3._o as_o 12._o 60._o 1._o 5_o ounce_n 4._o as_o 12._o 60._o 3._o 15_o ounce_n whereupon_o i_o do_v conclude_v that_o 25_o ounce_n of_o 24_o carect_v fine_a 15_o ounce_n of_o 22_o carect_v 5_o ounce_n '_o of_o 18_o carect_v and_o 15_o ounce_n of_o 16_o carect_v fine_a be_v all_o melt_v together_o will_v produce_v a_o mass_n of_o gold_n contain_v 60_o ounce_n of_o 21_o carect_v fine_a as_o be_v require_v in_o the_o question_n propound_v 16._o here_o the_o work_n be_v true_a when_o the_o sum_n of_o the_o quantity_n find_v be_v equal_a to_o the_o total_a quantity_n propound_v thus_o in_o the_o precede_a example_n the_o quantity_n find_v be_v 25._o 15._o 5._o 15._o which_o be_v add_v together_o do_v make_v 60_o and_o the_o quantity_n propound_v be_v 60_o also_o 17._o when_o a_o question_n in_o alligation_n alternate_a do_v consist_v of_o three_o ingredient_n it_o be_v capable_a of_o a_o infinite_a series_n of_o affirmative_a and_o negative_a answer_n and_o when_o the_o question_n do_v consist_v of_o more_o than_o three_o ingredient_n it_o be_v capable_a of_o as_o many_o infinite_a series_n of_o affirmative_a and_o negative_a solution_n as_o you_o please_v to_o impose_v upon_o it_o 18._o in_o alligation_n alternate_a consist_v of_o three_o ingredient_n their_o value_n or_o price_n be_v orderly_o place_v and_o the_o difference_n between_o they_o be_v take_v as_o have_v be_v already_o show_v if_o you_o take_v the_o difference_n of_o every_o two_o value_n and_o place_v they_o against_o the_o three_o you_o shall_v constitute_v two_o continual_a addend_n and_o one_o continual_a subducend_v that_o be_v the_o two_o continual_a addend_n be_v outmost_a and_o the_o continual_a subducend_n be_v inmost_a or_o if_o you_o please_v you_o may_v let_v the_o two_o outmost_a be_v continual_a subducend_v and_o then_o the_o inmost_a will_v be_v a_o continual_a addend_n or_o any_o three_o number_n in_o the_o same_o proportion_n will_v effect_v the_o same_o thing_n but_o it_o be_v best_o to_o take_v the_o three_o lest_o number_n that_o be_v in_o the_o same_o ratio_fw-la for_o example_n let_v the_o value_n or_o price_n of_o the_o three_o ingredient_n be_v 12._o 24._o 36_o and_o let_v the_o mean_a price_n be_v 16_o these_o price_n be_v orderly_o place_v and_o the_o difference_n between_o they_o and_o the_o mean_a price_n will_v be_v as_o here_o you_o see_v they_o now_o then_o deduct_v 12_o from_o 24_o the_o difference_n be_v 12_o which_o i_o place_v against_o 36_o towards_o the_o left_a hand_n and_o the_o difference_n between_o 24_o and_o 36_o be_v 12_o also_o which_o i_o place_n against_o 12_o last_o the_o difference_n between_o 12_o and_o 36_o be_v 24_o which_o i_o place_v against_o 24_o and_o because_o these_o difference_n be_v large_a and_o so_o not_o so_o convenient_a for_o this_o work_n i_o reduce_v they_o to_o the_o least_o number_n in_o the_o same_o proportion_n and_o then_o they_o be_v 1._o 2._o 1._o the_o two_o outmost_a to_o wit_v 1._o and_o 1._o shall_v first_o be_v continual_a addend_n and_o the_o inmost_a or_o middle_a number_n to_o wit_v 2_o let_v be_v a_o continual_a subducend_v and_o then_o some_o of_o the_o many_o answer_n that_o may_v be_v give_v to_o this_o question_n will_v be_v these_o follow_v of_o which_o the_o first_o be_v affirmative_a and_o the_o two_o last_v negative_n but_o make_v the_o two_o outmost_a to_o wit_v 1_o and_o 1_o to_o be_v continual_a subducend_v and_o the_o inmost_a to_o wit_v 2_o a_o continual_a addend_n the_o answer_n to_o this_o question_n will_v be_v these_o follow_v 1_o 16_o 12_o 28_o 27_o 26_o 25_o 24_o 23_o +_o 2_o 24_o 4_o 6_o 8_o 10_o 12_o 14_o 1_o 36_o 4_o 3_o 2_o 1_o 0_o 1_o  _fw-fr  _fw-fr  _fw-fr 36_o 36_o 36_o 36_o 36_o 36_o and_o thus_o you_o may_v proceed_v in_o infinitum_fw-la but_o the_o negative_a answer_n be_v improper_a and_o of_o no_o use_n in_o practice_n it_o will_v be_v sufficient_a to_o go_v as_o far_o as_o they_o be_v affirmative_a and_o proper_a take_v still_o something_o of_o each_o ingredient_n in_o the_o question_n 19_o in_o alligation_n alternate_a consist_v of_o more_o than_o three_o ingredient_n have_v set_v down_o the_o several_a price_n and_o take_v the_o difference_n between_o they_o and_o the_o mean_a price_n you_o may_v constitute_v your_o runner_n from_o any_o three_o that_o be_v together_o and_o keep_v the_o other_o proportion_n for_o stand_a number_n thus_o the_o price_n of_o 4_o ingredient_n be_v 48._o 36._o 24._o 12_o and_o the_o mean_a price_n 16_o the_o price_n and_o difference_n will_v be_v and_o the_o runner_n for_o the_o three_o low_a price_n will_v be_v 1._o 2._o 1._o as_o before_o and_o therefore_o the_o answer_v these_o follow_v +_o 1_o 16_o 48_o 4_o 4_o 4_o 2_o 36_o 4_o 5_o 6_o +_o 1_o