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nothing_o will_v remain_v double_v then_o the_o root_n and_o place_v the_o first_o figure_n of_o it_o betwixt_o the_o next_o point_n as_o 2_o time_n 2_o be_v 4_o which_o set_v under_o 3_o now_z 4_o can_v be_v have_v in_o 3_o therefore_o put_v a_o cypher_n in_o the_o quotient_a and_o likewise_o under_o the_o next_o point_n beneath_o 2_o to_o supply_v the_o place_n only_o and_o have_v cancel_v they_o both_o double_v the_o root_n say_v 2_o time_n nothing_o be_v nothing_o for_o which_o put_v a_o cypher_n betwixt_o the_o next_o two_o point_n then_o double_a the_o 2_o and_o place_v that_o 4_o one_o degree_n to_o the_o left_a hand_n of_o it_o as_o under_z 2_o then_o see_v how_o many_o time_n 4_o will_v be_v find_v in_o 32_o which_o will_v now_o be_v 8_o time_n place_v it_o in_o the_o quotient_a and_o also_o under_o the_o last_o point_n then_o multiply_v this_o by_o 8_o that_o be_v in_o itself_o the_o product_n be_v 64_o which_o take_v out_o of_o 64_o nothing_o remain_v than_o 8_o time_n 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to_o proceed_v or_o have_v read_v they_o over_o think_v the_o best_a to_o be_v at_o first_o and_o nothing_o in_o the_o rest_n my_o pain_n of_o write_v it_o will_v again_o renew_v that_o i_o have_v lose_v my_o time_n and_o trouble_v you_o if_o so_o then_o let_v the_o press_n be_v quite_o exile_v lest_o that_o more_o virgin_n paper_n be_v defile_v and_o where_o i_o have_v the_o unhappy_a author_n be_v may_v these_o be_v wind_v sheet_n to_o bury'm_n in_o and_o bind_v to_o rest_n from_o sect_n of_o silly_a man_n as_o from_o assault_n of_o a_o malicious_a pen_n may_v dark_a oblivion_n undertake_v the_o trust_n to_o shrine_n they_o up_o with_o other_o deed_n in_o dust_n and_o never_o suffer_v more_o to_o come_v to_o light_n to_o stand_v the_o censure_n of_o a_o proselyte_n yet_o when_o these_o sect_n of_o zoilist_n shall_v be_v dead_a may_v this_o revive_v and_o be_v by_o artist_n read_v who_o will_v my_o fault_n with_o mercy_n overlook_v and_o find_v it_o be_v no_o r_n write_v the_o book_n 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by_o demonstration_n you_o the_o arch_n sustain_v envy_n and_o folly_n shall_v as_o cipher_n stand_v and_o while_o they_o give_v your_o figure_n the_o left_a hand_n augment_v the_o it_o value_n the_o success_n of_o those_o who_o by_o invective_n do_v advance_v their_o foe_n i_o must_v suspend_v my_o sense_n since_o every_o line_n command_v our_o gratitude_n and_o speak_v a_o mine_n of_o public_a good_a yet_o i_o have_v hope_n i_o may_v discharge_v one_o think_v at_o least_o and_o free_o say_v while_o man_n the_o advantage_n of_o your_o work_n divide_v our_o profit_n and_o your_o praise_n be_v multiply_v edward_n boteler_n a_o index_n or_o content_n of_o the_o principal_n of_o the_o first_o book_n in_o each_o section_n and_o paragraph_n of_o natural_a arithmetic_n sect._n 1_o parag._n 1_o the_o definition_n element_n and_o species_n of_o arithmetic_n page_n 1_o addition_n the_o first_o species_n of_o arithmetic_n parag_v 2._o pag._n 6_o subtraction_n the_o second_o species_n parag_v 3._o p._n 15_o multiplication_n the_o three_o par_fw-fr 4._o p._n 26_o division_n the_o four_o par_fw-fr 5._o p._n 40_o the_o second_o section_n show_v the_o definition_n term_n and_o value_v of_o fraction_n with_o their_o reduction_n from_o one_o denomination_n to_o another_o parag_v 1._o pag._n 63_o addition_n of_o fraction_n proper_a improper_a or_o compound_v parag_v 2._o p._n 86_o subtraction_n of_o fraction_n either_o proper_a improper_a or_o mix_v par_fw-fr 3._o p._n 93._o multiplication_n of_o fraction_n proper_a improper_a or_o compound_v par_fw-fr 4._o p._n 103_o division_n of_o fraction_n or_o break_a number_n either_o proper_a improper_a or_o mix_v par_fw-fr 5._o p._n 109_o a_o index_n show_v the_o principal_a content_n of_o each_o paragraph_n in_o the_o second_o book_n definition_n of_o quadrat_fw-la root_n page_n 121_o the_o extract_a of_o the_o square_a root_n in_o whole_a number_n and_o fraction_n parag_v 1._o pag._n 122_o to_o extract_v the_o quadrat_fw-la root_n from_o a_o irrational_a number_n in_o any_o proportion_n assign_v pag._n 128_o the_o definition_n of_o a_o cube_n p._n 136_o the_o extraction_n of_o the_o cube_n root_n both_o in_o whole_a and_o break_a number_n parag_v 2._o pag._n 137_o general_n rule_n for_o extract_v of_o cubique_a root_n pag._n 142_o the_o extract_a of_o cubique_a root_n from_o irrational_a number_n in_o any_o give_a proportion_n pag._n 150_o the_o reason_n and_o cause_n of_o the_o operation_n in_o extract_v of_o cubique_a root_n pag._n 159_o how_o to_o extract_v all_o other_o root_n in_o whole_a number_n or_o fraction_n as_o biquadrat_n square_a cube_n etc._n etc._n parag_v 3._o pag._n 162_o how_o to_o extract_v all_o root_n in_o any_o proportion_n that_o shall_v be_v assign_v pag._n 165_o a_o reference_n of_o number_n as_o in_o relation_n to_o their_o quantity_n and_o quality_n parag_v 4._o p._n 166_o natural_a progression_n and_o arithmetical_a proportion_n with_o their_o addition_n par_fw-fr 5._o p._n 172_o geometrical_a progression_n and_o proportion_n with_o the_o addition_n of_o their_o number_n par_fw-fr 6._o p._n 179_o universal_a axiom_n in_o arithmetic_n accommodate_v to_o the_o rule_n of_o practice_n and_o proportion_n parag_v 7._o p._n 190_o canon_n in_o arithmetic_n with_o definition_n and_o divers_a rule_n of_o proportion_n single_a and_o double_a direct_a and_o reverse_v par_fw-fr 8._o p._n 201_o the_o rule_n of_o practice_n direct_v and_o reverse_v both_o in_o whole_a number_n &_o fraction_n par_fw-fr 9_o p._n 212_o the_o double_a rule_n of_o three_o direct_a and_o reverse_v perform_v at_o two_o operation_n or_o at_o one_o in_o whole_a number_n or_o fraction_n par_fw-fr 10._o p._n 220_o the_o rule_n of_o society_n or_o company_n both_o single_a and_o double_a by_o whole_a or_o break_a number_n either_o in_o gain_n or_o loss_n par_fw-fr 11._o p._n 226_o the_o rule_n of_o alligation_n or_o mixture_n of_o divers_a simples_n according_a to_o a_o common_a price_n or_o in_o proportion_n to_o any_o quantity_n par_fw-fr 12._o p._n 233_o the_o rule_n of_o alligation_n in_o compound_v of_o physical_a simples_n according_a to_o their_o quality_n in_o any_o of_o their_o degree_n par_fw-fr 13._o p._n 250_o the_o rule_n of_o single_a false_a position_n or_o by_o false_a suppose_a number_n to_o discover_v the_o truth_n par_fw-fr 14._o pag._n 262_o the_o double_a rule_n of_o false_a wherein_o by_o false_a position_n and_o error_n the_o truth_n be_v discover_v par_fw-fr 15._o pag._n 269_o a_o index_n or_o content_n of_o all_o the_o principal_n in_o each_o section_n or_o chapter_n of_o this_o the_o three_o book_n of_o artificial_a arithmetic_n sect._n 1_o chap._n 1._o the_o definition_n of_o decimall_n and_o the_o reduction_n of_o vulgar_a fraction_n to_o artificial_a number_n pag._n 287_o annotation_n of_o decimall_n p._n 291_o numeration_n of_o decimall_n p._n 293_o addition_n of_o decimall_n with_o whole_a number_n and_o fraction_n commix_v together_o ch_z 2._o 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too_o far_o let_v not_o your_o inquisition_n go_v examine_v often_o whether_o right_a or_o no_o with_o too_o much_o haste_n you_o may_v over_o run_v the_o mark_n which_o oft_o do_v lie_n conceal_v within_o the_o dark_a and_o false_a conception_n of_o the_o sight_n there_o be_v for_o all_o that_o glister_v be_v not_o gold_n we_o see_v this_o error_n will_v unmask_v in_o truth_n disguise_n with_o all_o her_o paint_a train_n of_o fallacy_n which_o various_a be_v so_o leave_v no_o rule_n behind_o for_o you_o may_v be_v in_o darkness_n till_o you_o find_v or_o soon_o to_o light_v you_o will_v this_o dilemma_n bring_v from_o art_n and_o nature_n that_o there_o be_v no_o such_o thing_n aequation_n will_v the_o thing_n that_o be_v seek_v produce_v or_o show_v the_o quaerie_n false_a or_o not_o of_o use_n rule_v i_o will_v give_v as_o many_o as_o you_o will_v need_v farewell_o and_o in_o the_o name_n of_o god_n proceed_v festina_fw-la lent_fw-la serius_n absolvit_fw-la qui_fw-la nimium_fw-la 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4_o four_o 6_o six_o 9_o nine_o 0_o nothing_o &c._n &c._n the_o next_o degree_n or_o second_o figure_n towards_o the_o left_a hand_n be_v ten_o time_n its_o own_o quantity_n the_o three_o place_n a_o hundred_o time_n the_o four_o degree_n a_o thousand_o time_n the_o five_o place_n ten_o thousand_o time_n the_o sixth_o figure_n in_o order_n be_v a_o hundred_o thousand_o time_n its_o own_o number_n the_o seven_o place_n a_o million_o the_o eight_o ten_o million_o the_o nine_o place_n a_o hundred_o million_o and_o so_o proceed_v without_o end_n increase_a every_o degree_n in_o this_o manner_n follow_v reiterate_v unite_v ten_o and_o hundred_o a_o table_n of_o numeration_n continue_v degree_n 1_o 1_o one_o 1_o 1_o degree_n 2_o 10_o tennes_n 21_o 2_o 3_o 100_o hundred_o 321_o 3_o 4_o 1000_o thousand_o 4321_o 4_o 5_o 10000_o ten_o thousand_o 54321_o 5_o 6_o 100000_o hundred_o thousand_o 654321_o 6_o 7_o 1000000_o million_o 7654321_o 7_o 8_o 10000000_o ten_o million_o 87654321_o 8_o 9_o 100000000_o hundred_o million_o 987654321_o 9_o by_o this_o table_n of_o numeration_n you_o may_v observe_v how_o the_o figure_n by_o degree_n do_v increase_v by_o ten_o proceed_v from_o the_o right_a hand_n towards_o the_o left_a in_o value_n according_a to_o their_o place_n as_o for_o example_n the_o sinister_a table_n begin_v with_o a_o unite_v unto_o which_o annex_v a_o cypher_n and_o then_o the_o second_o degree_n will_v be_v 10_o that_o be_v in_o value_n ten_o time_n the_o first_o unto_o that_o annex_v another_o cypher_n and_o then_o in_o the_o three_o place_n it_o be_v 100_o a_o hundred_o that_o be_v ten_o time_n great_a than_o the_o second_o degree_n the_o four_o place_n 1000_o a_o thousand_o that_o be_v ten_o time_n 100_o etc._n etc._n the_o second_o table_n i_o also_o here_o begin_v with_o a_o unite_v before_o which_o towards_o the_o left_a hand_n 1_o place_n 2_o and_o then_o in_o the_o second_o degree_n it_o make_v twenty_o so_o both_o the_o figure_n 21_o and_o 2_o in_o that_o place_n 10_o time_n its_o own_o number_n then_o put_v 3_o before_o they_o and_o it_o will_v be_v the_o hundred_o place_n and_o the_o character_n be_v 3_o denote_v three_o hundred_o the_o three_o figure_n make_v in_o all_o 321_o three_o hundred_o twenty_o and_o one_o the_o four_o degree_n obtain_v the_o thousand_o place_n be_v ten_o time_n great_a than_o the_o three_o degree_n and_o have_v the_o first_o figure_n before_o it_o 4_o the_o value_n of_o the_o four_o figure_n will_v be_v 4321_o that_o be_v four_o thousand_o three_o hundred_o twenty_o and_o one_o the_o next_o degree_n be_v ten_o thousand_o and_o the_o character_n be_v 5_o it_o must_v be_v fifty_o thousand_o and_o the_o whole_a value_n of_o those_o five_o figure_n 54321_o and_o so_o all_o other_o proceed_v in_o order_n by_o ten_o yet_o the_o better_a for_o to_o effect_v the_o computation_n of_o great_a number_n when_o their_o degree_n can_v easy_o be_v contain_v in_o memory_n than_o it_o be_v necessary_a for_o to_o put_v period_n unto_o every_o century_n or_o three_o figure_n and_o so_o number_v they_o in_o unite_v ten_o hundred_o and_o then_o in_o express_v of_o their_o value_n observe_v their_o place_n and_o the_o quantity_n of_o each_o figure_n as_o it_o be_v simple_o of_o itself_o and_o note_n that_o after_o the_o first_o period_n their_o place_n be_v thousand_o and_o after_o the_o second_o period_n million_o and_o after_o the_o next_o period_n million_o of_o million_o and_o so_o proceed_v with_o reiteration_n of_o unite_v ten_o and_o hundred_o until_o their_o period_n be_v put_v unto_o they_o by_o this_o mean_n avoid_v error_n arise_v in_o number_v of_o great_a quantity_n and_o mistake_v their_o degree_n as_o in_o this_o example_n shall_v be_v illustrate_v and_o suppose_v this_o number_n give_v who_o quantity_n be_v require_v viz_o 1_o 234_o 567_o 890_o which_o have_v point_v read_v thus_o one_o million_o of_o million_o two_o hundred_o thirty_o four_o million_o five_o hundred_o sixty_o seven_o thousand_o eight_o hundred_o and_o ninety_o if_o this_o have_v be_v the_o number_n give_v 1_o 0_o 0_o 0_o 0_o 0_o 0_o 0_o 0_o 0_o 0_o it_o will_v have_v be_v thus_o express_v ten_o million_o of_o million_o or_o thus_o ten_o thâ—usand_o million_o observe_v this_o order_n well_o and_o then_o the_o computation_n of_o any_o number_n will_v prove_v as_o easy_a as_o they_o be_v necessary_a for_o to_o be_v know_v or_o express_v a_o according_v to_o their_o place_n in_o degree_n number_n be_v also_o for_o to_o be_v consider_v whether_o it_o be_v of_o one_o or_o several_a denâ—minations_n those_o be_v say_v of_o one_o denomination_n when_o they_o do_v consist_v of_o one_o number_n weight_n or_o measure_n whether_o they_o be_v all_o of_o one_o entire_a number_n or_o in_o the_o same_o part_n all_o number_n of_o several_a
in_o their_o own_o figure_n and_o from_o thence_o be_v call_v circular_a number_n 7_o multiply_v by_o 7_o end_n in_o 9_o 3_o 1_o and_o fourthly_a in_o its_o self_n 8_o by_o 8_o end_n in_o 4_o 2_o 6_o and_o fourthly_a in_o its_o self_n again_o 9_o multiply_v by_o 9_o will_v end_v in_o 1_o and_o in_o every_o second_o multiplication_n be_v terminate_v in_o 9_o it_o be_v own_o root_n or_o number_n again_o produce_v multiplication_n be_v a_o quadrature_n and_o have_v this_o analogy_n or_o proportion_n with_o a_o superficial_a square_n call_v in_o geometry_n the_o second_o quantity_n a_o figure_n compose_v of_o line_n who_o side_n be_v divide_v into_o part_n and_o intersect_v with_o paralle's_n or_o equidistant_a line_n as_o be_v the_o last_o figure_n a_o b_o c_o d_o make_v thus_o both_o square_n and_o right_o angle_a parallellogram_n equal_a to_o the_o number_n multiply_v in_o themselves_o together_o as_o for_o example_n the_o table_n or_o square_v a_o c_o have_v every_o side_n divide_v into_o 9_o equal_a part_n as_o admit_v pole_n yard_n foot_n inch_n or_o what_o you_o please_v suppose_v these_o side_n as_o a_o b_o and_z a_o d_o in_o foot_n and_o intersect_v with_o line_n the_o whole_a square_n will_v contain_v 81_o square_a foot_n for_o the_o true_a superficial_a content_n of_o it_o so_o if_o 9_o be_v multiply_v by_o 9_o the_o product_n will_v be_v the_o same_o or_o if_o the_o long_a square_n a_o f_o e_o d_o be_v give_v the_o superficial_a content_n of_o this_o geometrical_a figure_n call_v a_o parallelogram_n will_v be_v 45_o square_a foot_n and_o so_o be_v the_o product_n of_o 9_o multiply_v by_o 5_o if_o it_o be_v require_v to_o know_v how_o many_o foot_n there_o be_v in_o a_o yard_n square_a 3_o foot_n make_v a_o yard_n in_o length_n therefore_o if_o every_o side_n of_o such_o a_o square_n be_v divide_v into_o 3_o equal_a part_n and_o intersect_v with_o right_a line_n there_o will_v be_v find_v 9_o square_a foot_n as_o in_o the_o table_n will_v appear_v and_o so_o 3_o multiply_v by_o 3_o will_v produce_v 9_o and_o in_o these_o the_o proportion_n be_v continue_v as_o a_o unite_v be_v unto_o the_o number_n or_o side_n of_o the_o square_n give_v so_o will_v the_o side_n be_v to_o the_o whole_a square_n or_o as_o 1_o to_o the_o multiplier_n so_o will_v the_o multiplicand_a be_v unto_o the_o product_n as_o in_o the_o former_a example_n as_o 1_o to_o a_o b_o 9_o so_o will_v a_o d_o 9_o be_v to_o 81_o the_o whole_a square_n or_o product_n or_o as_o 1_o to_o a_o f_o 5_o so_o a_o d._n 9_o in_o proportion_n to_o 45_o the_o long_a square_n a_o f_o e_o d_o and_o so_o the_o like_a of_o any_o other_o in_o this_o kind_n whether_o great_a or_o less_o and_o so_o much_o for_o the_o sympathy_n betwixt_o multiplication_n and_o geometrical_a square_n of_o the_o second_o quantity_n the_o way_n and_o form_n of_o multiplication_n when_o there_o be_v more_o than_o one_o significant_a figure_n in_o the_o multiplicand_a multiplier_n or_o in_o both_o of_o they_o when_o there_o be_v several_a figure_n to_o be_v multiply_v both_o in_o the_o multiplier_n and_o multiplicand_n set_v down_o first_o the_o great_a number_n and_o under_o that_o the_o lesser_a according_a to_o their_o degree_n or_o place_n as_o unite_v under_o unite_v ten_o under_o ten_o hundred_o underneath_o hundred_o etc._n etc._n as_o have_v be_v direct_v in_o divers_a other_o example_n already_o this_o do_v draw_v a_o line_n underneath_o they_o both_o whereby_o to_o separate_v they_o from_o the_o number_n increase_a by_o their_o mutual_a multiplication_n proceed_v from_o the_o right_a hand_n in_o order_n towards_o the_o left_a every_o figure_n of_o the_o multiplier_n must_v be_v increase_v or_o multiply_v into_o every_o particular_a figure_n of_o the_o multiplicand_a from_o whence_o there_o will_v arise_v so_o many_o product_n as_o there_o be_v figure_n in_o the_o multiplier_n the_o first_o figure_n in_o every_o product_n ought_v to_o be_v place_v exact_o under_o that_o figure_n which_o multiply_v and_o so_o in_o order_n with_o a_o convenient_a distance_n towards_o the_o left_a hand_n for_o all_o the_o several_a product_n must_v be_v add_v together_o whereby_o to_o find_v the_o result_n or_o total_a of_o they_o and_o in_o every_o figure_n as_o you_o multiply_v set_v down_o the_o product_n if_o under_o 10_o but_o if_o a_o decimall_n or_o decimall_n subscribe_v a_o cypher_n if_o a_o compound_a number_n write_v down_o the_o significant_a figure_n and_o keep_v the_o decimall_n or_o decimall_n in_o your_o mind_n and_o as_o you_o multiply_v the_o next_o figure_n add_v they_o in_o as_o unite_v and_o so_o proceed_v until_o every_o figure_n of_o the_o multiplier_n be_v increase_v by_o all_o the_o figure_n of_o the_o multiplicand_v in_o order_n as_o be_v say_v already_o and_o shall_v be_v now_o illustrate_v with_o several_a paradigma_n follow_v to_o find_v how_o many_o day_n there_o be_v contain_v in_o a_o common_a year_n consist_v of_o 52_o week_n and_o one_o day_n example_n 1._o have_v set_v down_o the_o 52_o multiplicand_n 52_o the_o multiplier_n 7_o the_o product_n 364_o the_o odd_a day_n 1_o the_o total_a of_o day_n in_o a_o year_n 365_o week_n subscribe_v underneath_o it_o the_o day_n in_o one_o week_n which_o be_v 7_o and_o be_v a_o simple_a number_n i_o set_v it_o under_o 2_o in_o the_o unite_z place_n and_o since_o there_o be_v so_o many_o day_n in_o one_o week_n there_o must_v be_v 52_o time_n so_o many_o in_o a_o year_n beside_o the_o odd_a day_n then_o say_v 7_o time_n 2_o make_v 14_o for_o which_o subscribe_v 4_o below_o the_o line_n and_o right_a under_o 7_o and_o keep_v the_o decimall_n in_o your_o mind_n they_o say_v 7_o time_n 5_o as_o by_o the_o table_n of_o multiplication_n will_v make_v 35_o and_o the_o 1_o decimall_n in_o mind_n make_v 36_o therefore_o subscribe_v 6_o right_a underneath_o the_o 5_o and_o keep_v the_o 3_o decimall_n in_o mind_n but_o since_o there_o be_v no_o more_o write_v it_o down_o and_o the_o product_n be_v 364_o the_o number_n of_o day_n in_o 52_o week_n and_o be_v there_o be_v one_o day_n more_o in_o a_o year_n add_v that_o in_o and_o then_o the_o total_a will_v be_v 365_o the_o true_a number_n of_o day_n in_o the_o sun_n annual_a revolution_n the_o thing_n require_v the_o day_n and_o part_n of_o a_o year_n be_v know_v to_o find_v the_o number_n of_o hour_n contain_v in_o it_o example_n 2._o first_o set_v down_o the_o great_a multiplicand_v 365_o multiplier_n 24_o the_o product_n 1460_o  _fw-fr 730_o the_o total_a 8760_o number_n and_o then_o the_o multiplier_n according_a to_o the_o place_n of_o the_o figure_n in_o this_o 24_o be_v the_o hour_n contain_v in_o a_o natural_a day_n and_o then_o say_v 4_o time_n 5_o be_v 20_o for_o which_o subscribe_v a_o cypher_n and_o keep_v 2_o decimall_n in_o mind_n then_o 4_o time_n 6_o be_v 24_o &_o 2_o be_v 26_o that_o be_v 6_o and_o go_v 2_o then_o say_v 4_o time_n 3_o be_v 12_o and_z 2_o in_o mind_n be_v 14_o and_o be_v there_o be_v no_o more_o figure_n in_o the_o multiplicand_a write_v they_o down_o and_o then_o the_o first_o product_n will_v be_v 1460_o then_o go_v to_o the_o second_o figure_n of_o the_o multiplier_n which_o here_o be_v 2_o and_o say_v 2_o time_n 5_o be_v 10_o subscribe_v a_o cypher_n under_o 2_o the_o multiplier_n and_o go_v 1_o decimall_n than_o 2_o time_n 6_o be_v 12_o and_o 1_o make_v 13_o that_o be_v 3_o and_o go_v 1_o than_o 2_o time_n 3_o be_v 6_o and_z 1_o in_o mind_n make_v 7_o which_o write_v down_o under_o the_o 1_o and_o then_o add_v the_o product_n together_o the_o total_a will_v be_v 8760_o the_o number_n of_o hour_n in_o 365_o day_n and_o since_o the_o julian_n account_n make_v the_o magnitude_n of_o the_o year_n for_o to_o consist_v of_o 6_o hour_n more_o add_v 6_o unto_o the_o product_n and_o the_o total_a will_v be_v 8766_o hour_n contain_v in_o a_o common_a year_n to_o find_v how_o many_o minute_n be_v contain_v in_o a_o year_n example_n 3._o here_o you_o be_v to_o know_v the_o multiplicand_a 8766_o multiplier_n 60_o  _fw-fr 0000_o  _fw-fr 52596_o product_n 525960_o part_n or_o minute_n of_o a_o hour_n which_o be_v 60_o those_o must_v be_v the_o multiplier_n and_o 8766_o the_o hour_n contain_v in_o a_o year_n the_o multiplicand_a then_o say_v no_o time_n 6_o or_o 6_o time_n nothing_o be_v nothing_o therefore_o subscribe_v so_o many_o cypher_n as_o there_o be_v figure_n in_o the_o multiplicand_a and_o then_o begin_v with_o 6_o and_o say_v 6_o time_n 6_o be_v 36_o there_o set_v down_o 6_o beneath_o the_o line_n right_o underneath_o the_o multiplier_n 6_o then_o say_v 6_o time_n 6_o make_v 36_o and_o 3_o decimall_n in_o mind_n will_v
be_v 39_o there_o subscribe_v 9_o and_o go_v 3_o then_z 6_o time_n 7_o will_v be_v 42_o and_o 3_o in_o mind_n be_v 45_o that_o be_v 5_o and_z ge_z 4_o then_z 6_o time_n 8_o be_v 48_o and_o 4_o decimall_n in_o mind_n will_v make_v 52_o which_o subscribe_v there_o be_v no_o more_o figure_n to_o be_v multiply_v the_o product_n of_o these_o be_v 525960_o and_o so_o many_o minute_n there_o be_v in_o a_o vulgar_a year_n but_o this_o and_o all_o such_o question_n be_v perform_v a_o ready_a way_n as_o you_o shall_v see_v by_o the_o example_n follow_v breviate_n or_o compendious_a abridgement_n in_o multiplication_n exemplify_v example_n 4._o if_o any_o number_n be_v give_v multiplicand_v 1000_o multiplier_n 25_o the_o product_n 25000_o for_o to_o be_v multiply_v which_o have_v only_a cypher_n with_o a_o unite_v upon_o the_o left_a hand_n be_v 1_o be_v no_o number_n and_o consequent_o multiply_v not_o annex_v all_o the_o cypher_n to_o the_o significant_a figure_n towards_o the_o right_a hand_n and_o the_o work_n be_v end_v as_o suppose_v 1000_o soldier_n and_o every_o one_o receive_v for_o his_o pay_n 25_o crown_n the_o product_n will_v be_v 25000_o crown_n for_o the_o whole_a sum_n pay_v and_o so_o for_o any_o other_o as_o 10_o 100_o 1000_o 10000_o etc._n etc._n but_o if_o there_o be_v any_o other_o number_n give_v to_o be_v multiply_v with_o cypher_n follow_v any_o significant_a figure_n or_o figure_n in_o all_o such_o case_n neglect_v the_o cypher_n and_o make_v the_o lesser_a number_n than_o multiplier_n and_o proceed_v as_o before_o and_o to_o the_o product_n annex_v all_o the_o cypher_n that_o be_v towards_o the_o right_a hand_n either_o in_o the_o multiplier_n multiplicand_n or_o in_o both_o of_o they_o as_o in_o the_o former_a example_n where_o 60_o be_v the_o multiplier_n if_o 8766_o have_v be_v multiply_v by_o 6_o the_o product_n will_v have_v be_v 52596_o to_o which_o if_o a_o cypher_n have_v be_v annex_v the_o sum_n will_v have_v be_v 525960_o as_o before_o when_o cypher_n in_o any_o multiplier_n be_v include_v between_o significant_a figure_n the_o multiplication_n may_v be_v easy_o abreviate_v thus_o example_n 5._o admit_v 9080_o be_v a_o number_n multiplicand_v 908_o multiplier_n 309_o  _fw-fr 8172_o  _fw-fr 2724_o product_n 280572_o give_v for_o to_o be_v multiply_v by_o 3090_o here_o according_a unto_o the_o last_o paradigma_n i_o do_v neglect_v the_o cypher_n in_o both_o number_n and_o so_o set_v down_o 908_o and_o under_o that_o 309_o then_o draw_v a_o line_n underneath_o they_o as_o in_o the_o table_n and_o beginning_n with_o 9_o the_o first_o multiplier_n say_v 9_o time_n 8_o be_v 72_o both_o which_o figure_n subscribe_v because_o a_o cypher_n follow_v in_o the_o multiplicand_a then_o say_v 9_o time_n 9_o be_v 81_o which_o also_o set_v down_o there_o be_v no_o more_o to_o come_v the_o cypher_n be_v next_o which_o make_v nothing_o but_o only_o assume_v a_o place_n therefore_o go_v unto_o the_o next_o significant_a figure_n which_o be_v 3_o and_o by_o reason_n of_o the_o cypher_n in_o the_o multiplicand_a you_o may_v say_v 3_o time_n 8_o be_v 24_o and_o 3_o time_n 9_o be_v 27_o which_o subscribe_v in_o order_n observe_v punctual_o to_o place_v the_o first_o figure_n of_o that_o product_n under_o the_o figure_n of_o the_o multiplier_n as_o in_o this_o 4_o under_o 3_o and_o so_o in_o order_n proceed_v towards_o the_o left_a hand_n by_o this_o mean_v the_o place_n of_o the_o cypher_n be_v gain_v and_o be_v between_o the_o first_o and_o last_o product_n can_v be_v nothing_o nor_o have_v a_o place_n in_o add_v they_o together_o the_o product_n of_o these_o figure_n be_v 280572_o and_o be_v the_o multiplier_n and_o multiplicand_n have_v either_o of_o they_o a_o cypher_n in_o the_o first_o place_n on_o the_o right_a hand_n annex_v the_o two_o cypher_n to_o this_o sum_n and_o then_o the_o total_a will_v be_v 28,057,200_o the_o solution_n require_v when_o any_o number_n be_v give_v for_o to_o be_v multiply_v by_o one_o figure_n or_o by_o two_o if_o one_o be_v a_o unite_v it_o may_v be_v perform_v compendious_o thus_o example_n 6._o it_o be_v require_v to_o know_v  _fw-fr 112_o multiplicand_n a_o 896_o the_o product_n b_o 5376_o the_o total_a a_o b_o 14336_o how_o many_o ounce_n there_o be_v in_o 800_o li._n of_o the_o merchant_n weight_n first_o set_v down_o their_o hundred_o which_o be_v 112_o li._n then_o draw_v a_o line_n underneath_o it_o and_z 8_o be_v the_o multiplier_n you_o may_v keep_v it_o in_o mind_n and_o say_v 8_o time_n 2_o be_v 16_o that_o be_v 6_o and_o go_v 1_o then_o 8_o time_n 1_o be_v 8_o &_o 1_o will_v be_v 9_o which_o subscribe_v then_o say_v again_o 8_o time_n 1_o be_v but_o 8_o which_o have_v set_v down_o the_o product_n of_o 8_o hundred_o weight_n be_v 896_o lib._n as_o in_o the_o table_n which_o be_v to_o be_v reduce_v into_o ounce_n 16_o of_o they_o make_v 1_o li._n a_o unite_v be_v here_o in_o the_o second_o place_n towards_o the_o left_a hand_n multiply_v not_o but_o will_v produce_v the_o multiplicand_a and_o remove_v it_o one_o degree_n far_o according_a to_o my_o former_a prescribe_a rule_n of_o multiplication_n therefore_o in_o all_o such_o case_n as_o these_o you_o may_v neglect_v the_o unite_v and_o keep_v the_o other_o figure_n in_o mind_n for_o your_o multiplier_n and_o place_v your_o product_n a_o degree_n backward_o as_o in_o this_o example_n a_o be_v the_o multiplicand_a for_o to_o be_v increase_v by_o 16_o or_o 6_o keep_v in_o mind_n then_o say_v 6_o time_n 6_o be_v 36_o that_o be_v 6_o and_o go_v 3_o decimall_n than_o 6_o time_n 9_o be_v 54_o and_o 3_o will_v be_v 57_o which_o be_v 7_o and_o reserve_v 5_o then_o say_v 6_o time_n 8_o will_v make_v 48_o and_o 5_o be_v 53_o which_o figure_n subscribe_v there_o be_v no_o more_o to_o be_v multiply_v so_o this_o product_n be_v at_o b_o 5376_o which_o be_v place_v one_o degree_n towards_o the_o right_a hand_n whereby_o the_o unite_v place_n in_o a_o the_o multiplicand_a may_v stand_v in_o the_o decimall_n place_n of_o the_o product_n and_o so_o it_o will_v have_v do_v according_a to_o the_o common_a way_n of_o multiplication_n but_o then_o beneath_o it_o as_o it_o be_v now_o above_o it_o these_o two_o number_n must_v be_v add_v together_o who_o total_a be_v a_o b_o 14,336_o the_o true_a number_n of_o ounce_n in_o 8_o c_o lib._n gross_a weight_n the_o line_n draw_v between_o a_o &_o b_o be_v here_o for_o direction_n only_o and_o in_o your_o work_n be_v better_a omit_v how_o to_o multiply_v the_o short_a way_n with_o two_o figure_n when_o the_o first_o figure_n on_o the_o right_a hand_n be_v a_o unite_v example_n 7._o it_o be_v require_v to_o find_v multiplicand_v 6336_o product_n 25344_o the_o total_a 259776_o how_o many_o inch_n there_o be_v in_o 41_o miles_n one_o contain_v 320_o perch_n every_o perch_n in_o length_n 16_o foot_n and_o a_o half_a and_o 12_o inch_n the_o length_n of_o one_o foot_n now_o in_o a_o pole_n there_o be_v 33_o half_a foot_n which_o multiply_v by_o 6_o inch_n the_o product_n will_v be_v 198_o the_o number_n of_o inch_n in_o a_o perch_n or_o pole_n that_o multiply_v by_o 320_o the_o perch_n in_o a_o mile_n or_o by_o 32_o the_o product_n will_v be_v 6336_o to_o which_o add_v a_o cypher_n and_o the_o product_n be_v then_o 63,360_o the_o number_n of_o inch_n in_o a_o mile_n and_o 41_o be_v the_o distance_n give_v now_o omit_v the_o cypher_n set_v down_o 6336_o which_o be_v the_o multiplicand_a and_o 41_o be_v to_o be_v the_o multiplier_n a_o unite_v can_v only_o produce_v 6336_o and_o be_v the_o 1_o be_v to_o the_o right_a hand_n set_v the_o product_n of_o the_o other_o figure_n one_o degree_n towards_o the_o left_a hand_n keep_v that_o multiplier_n in_o your_o mind_n and_o say_v 4_o time_n 6_o be_v 24_o that_o be_v 4_o and_o go_v 2_o then_z 4_o time_n 3_o be_v 12_o and_z 2_o makes_z 14_o that_o be_v 4_o and_o go_v 1_o then_z 4_o time_n 3_o be_v 12_o and_o 1_o decimall_n in_o mind_n be_v 13_o that_o be_v 3_o and_o go_v 1_o then_z 4_o time_n 6_o be_v 24_o and_o 1_o be_v 25_o the_o total_a be_v 259776_o to_o which_o annex_v the_o cypher_n omit_v on_o the_o right_a hand_n the_o total_a be_v 2,597,760_o the_o number_n of_o inch_n contain_v in_o 41_o miles_n the_o thing_n seek_v example_n 8._o if_o you_o be_v to_o multiply_v by_o 11_o 28800_o 28800_o 28800_o 28800_o 316800_o 316800_o it_o be_v all_o one_o whether_o you_o place_v the_o number_n give_v as_o the_o product_n a_o degree_n forward_o or_o backward_o for_o it_o comprehend_v both_o the_o last_o example_n in_o one_o as_o in_o this_o paradigma_n will_v appear_v there_o be_v 11_o half_a
etc._n example_n 4._o admit_v the_o number_n give_v who_o be_v quadrat_fw-la root_n to_o be_v extract_v be_v 5676_o which_o number_n be_v irrational_a the_o root_n be_v require_v in_o such_o proportion_n as_o that_o the_o denominator_fw-la of_o the_o fraction_n shall_v be_v 3_o to_o perform_v this_o or_o the_o like_a square_n the_o number_n give_v in_o who_o part_n the_o root_n be_v require_v then_o multiply_v that_o square_n by_o the_o number_n propound_v from_o whence_o extract_v the_o quadrat_fw-la root_n and_o then_o divide_v the_o root_n find_v by_o the_o denominator_fw-la of_o the_o fraction_n give_v then_o note_n what_o point_v the_o number_n propound_v will_v have_v have_v and_o the_o other_o number_n or_o number_n shall_v be_v the_o numerator_n of_o a_o fraction_n unto_o the_o denominator_fw-la give_v as_o in_o this_o example_n 5676_o be_v the_o number_n propound_v 3_o the_o proportion_n give_v which_o multiply_v in_o itself_o be_v 9_o and_o that_o by_o 5676_o produce_v 51084_o as_o in_o the_o margin_n which_o number_n will_v bear_v 3_o point_n as_o under_o 4_o the_o cypher_n and_o beneath_o the_o figure_n of_o 5_o who_o near_a root_n be_v 2_o which_o be_v the_o first_o root_n you_o shall_v not_o need_v for_o to_o subscribe_v under_o 5_o but_o set_v it_o in_o the_o quotient_a which_o square_v be_v 4_o and_o subtract_v from_o 5_o the_o remainder_n be_v 1_o which_o place_n over_o it_o and_o cancel_v the_o 5_o that_o do_v double_v the_o root_n as_o 2_o time_n 2_o be_v 4_o which_o place_n betwixt_o the_o next_o two_o point_n then_o see_v how_o many_o time_n 4_o will_v be_v contain_v in_o 11_o which_o will_v be_v twice_o then_o set_v 2_o in_o the_o quotient_a and_o under_o the_o next_o point_n beneath_o the_o cypher_n then_o say_v 2_o time_n 2_o be_v 4_o which_o place_n beneath_o it_o and_o also_o multiply_v the_o root_n 2_o by_o 4_o who_o product_n be_v 8_o which_o 84_o subtract_v from_o 110_o over_o it_o the_o remainder_n will_v be_v 26_o which_o place_n overhead_o as_o in_o division_n then_o double_a the_o two_o root_n find_v and_o always_o place_v the_o first_o betwixt_o the_o next_o two_o point_n and_o so_o proceed_v in_o order_n towards_o the_o left_a hand_n then_o look_v how_o many_o time_n 4_o be_v in_o 26_o over_o it_o and_o you_o will_v find_v 6_o which_o put_v in_o the_o quotient_a and_o also_o under_o the_o next_o point_n towards_o the_o right_a hand_n then_o take_v the_o square_a of_o 6_o that_o be_v multiply_v 6_o by_o 6_o the_o product_n be_v 36_o that_o be_v 6_o and_o go_v 3_o decimal_n then_o say_v 6_o time_n 4_o be_v 24_o and_o 3_o in_o mind_n be_v 27_o that_o be_v 7_o and_o go_v 2_o then_o multiply_v the_o last_o root_n by_o the_o next_o 4_o which_o produce_v 24_o and_o 2_o in_o mind_n make_v 26_o which_o write_v down_o be_v there_o be_v no_o more_o and_o subtract_v they_o from_o the_o figure_n above_o they_o uncancel_v as_o 6_o out_o of_o 14_o there_o will_v remain_v 8_o than_o 1_o borrow_a and_o 7_o makes_z 8_o which_o take_v from_o 8_o nothing_o remain_v and_o so_o likewise_o 6_o from_o 6_o and_o 2_o out_o of_o 2_o now_o the_o root_n be_v 226._o which_v be_v extract_v in_o thirdâ—_o divide_v it_o by_o 3_o the_o quotient_n will_v be_v 75â…“_n for_o the_o quadrat_fw-la root_n of_o 5676_o and_o extract_v according_a unto_o the_o demonstration_n the_o root_n will_v have_v be_v 75_o 51_o 1519_o but_o neither_o of_o they_o exact_o true_a for_o their_o square_n will_v not_o produce_v the_o number_n give_v to_o extract_v the_o quadrat_fw-la root_n from_o any_o irrational_a number_n with_o a_o fraction_n who_o denominator_fw-la shall_v be_v a_o decimall_n as_o 1_o 10_o 1_o 100_o 1_o 1000_o 1_o 10000_o 1_o 100000_o etc._n etc._n example_n 5._o in_o extract_v of_o a_o quadrat_fw-la root_n from_o a_o irrational_a number_n where_o no_o denominator_fw-la be_v assign_v but_o the_o near_a unto_o truth_n be_v require_v this_o way_n be_v better_a than_o the_o first_o more_o compendious_a than_o the_o second_o and_o much_o more_o exact_a than_o either_o as_o for_o example_n let_v 168_o be_v a_o number_n give_v who_o square_a root_n be_v require_v in_o decimal_n all_o that_o be_v now_o to_o be_v do_v be_v only_o to_o take_v any_o even_a number_n of_o cypher_n as_o you_o please_v annex_v those_o to_o the_o number_n give_v on_o the_o right_a hand_n as_o in_o this_o paradigma_n where_o 168_o by_o annex_v of_o cypher_n be_v make_v 1680000_o the_o root_n 12_o 96_o 100_o observe_v what_o point_v there_o will_v be_v under_o the_o cypher_n so_o many_o figure_n or_o place_n must_v be_v sever_v from_o the_o root_n extract_v cut_v off_o from_o the_o right_a hand_n to_o make_v the_o numerator_n of_o a_o fraction_n who_o denominator_fw-la shall_v be_v a_o unite_v with_o as_o many_o cypher_n annex_v unto_o it_o as_o the_o numerator_n have_v figure_n or_o place_n the_o reason_n of_o this_o be_v first_o wherefore_o the_o cypher_n be_v in_o couple_n be_v evident_a they_o be_v square_a number_n as_o 10_o time_n 10_o be_v 100_o which_o have_v 2_o cypher_n the_o square_a of_o 100_o have_v 4_o cypher_n 1000_o will_v have_v 6_o place_n 10000_o must_v possess_v 8_o place_n 100000_o will_v have_v 10_o place_n etc._n etc._n now_o these_o square_n multiply_v into_o any_o number_n propound_v can_v increase_v it_o but_o so_o many_o cypher_n as_o by_o the_o four_o example_n in_o multiplication_n of_o whole_a number_n neither_o shall_v you_o need_v for_o to_o divide_v the_o root_n extract_v by_o the_o proportion_n give_v as_o by_z 10_o by_o 100_o 1000_o etc._n etc._n as_o in_o my_o breviate_n of_o division_n be_v explain_v but_o sever_v so_o many_o figure_n in_o the_o quotient_a as_o there_o be_v point_n under_o the_o cypher_n of_o the_o square_n annex_v as_o be_v abovesaid_a so_o according_a to_o the_o demonstration_n &_o the_o first_o example_n the_o square_a root_n of_o 168_o will_v be_v 12_o and_o the_o remainder_n 24_o for_o the_o numerator_n of_o a_o fraction_n who_o denominator_fw-la be_v double_a the_o root_n with_o a_o unite_v add_v to_o it_o that_o be_v 25_o so_o it_o will_v stand_v thus_o 12_o 24_o 25_o to_o try_v if_o the_o root_n be_v true_o extract_v the_o square_a of_o 12_o be_v 144_o to_o which_o add_v the_o remainder_n 24_o the_o sum_n will_v be_v 168_o as_o before_o if_o the_o root_n have_v be_v extract_v in_o ten_o it_o will_v have_v be_v 12_o 9_o 10_o which_o be_v less_o than_o the_o former_a but_o here_o it_o be_v extract_v in_o 1_o 100_o by_o annex_v 4_o cypher_n to_o it_o &_o then_o it_o be_v find_v as_o in_o the_o scheme_n 1296_o that_o be_v 12_o 96_o 100_o or_o reduce_v 12_o 24_o 25_o as_o before_o if_o 6_o cypher_n have_v be_v annex_v to_o it_o than_o it_o have_v be_v 168000000_o and_o the_o quadrat_fw-la root_n 12961_o that_o be_v 12_o 961_o 1000_o which_o come_v near_o the_o truth_n err_v not_o a_o unite_v in_o 1000_o and_o so_o the_o great_a number_n of_o cypher_n you_o take_v in_o this_o kind_n the_o near_a you_o will_v approach_v unto_o the_o 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as_o they_o can_v the_o truth_n be_v affirm_v it_o be_v demand_v of_o one_o man_n what_o he_o have_v sell_v who_o reply_v 2_o piece_n of_o canvas_n 2_o of_o fustian_n and_o 7_o of_o holland_n every_o piece_n a_o crown_n more_o than_o other_o and_o one_o piece_n of_o the_o best_a be_v 3_o time_n the_o price_n of_o the_o worst_a what_o be_v the_o value_n of_o the_o 11_o piece_n of_o cloth_n in_o this_o proposition_n i_o take_v the_o last_o example_n in_o which_o i_o choose_v the_o mean_a price_n and_o make_v two_o supposition_n thereof_o and_o suppose_v the_o price_n of_o that_o piece_n to_o be_v 8_o crown_n and_o be_v there_o be_v 10_o p_o ece_n more_o every_o one_o dear_a by_o 5_o shilling_n the_o best_a than_o must_v consequent_o be_v 18_o crown_n which_o according_a to_o the_o supposition_n shall_v contain_v the_o worst_a three_o time_n whereas_o here_o 3_o time_n 8_o be_v 24_o from_o whence_o take_v 18_o the_o remainder_n be_v 6_o for_o the_o first_o error_n and_o too_o much_o for_o 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the_o other_o the_o price_n of_o all_o the_o rest_n will_v be_v easy_o discover_v as_o thus_o two_o piece_n of_o canvas_n the_o first_o 1_o l._n 5_o s._n the_o other_o 1_o l._n 10_o s._n then_o two_o piece_n of_o fustian_n the_o first_o 5_o s._n more_o than_o the_o last_o piece_n of_o canvas_n viz_o 1_o l._n 15_o s._n the_o other_o 2_o l._n the_o piece_n then_o for_o the_o 7_o piece_n of_o holland_n the_o first_o 2_o l._n 5_o s._n the_o second_o 2_o l._n 10_o s._n the_o three_o 2_o l._n 15_o s._n the_o four_o 3_o l._n the_o five_o 3_o l._n 5_o s._n the_o sixth_o 3_o l._n 10._o the_o last_o 3_o l._n 15._o that_o be_v 15_o crown_n and_o three_o time_n the_o first_o piece_n in_o value_n according_a unto_o the_o state_n of_o the_o question_n the_o total_a sum_n be_v 27_o l._n 10_o s._n the_o thing_n require_v this_o be_v a_o sufficient_a trial_n and_o yet_o for_o to_o please_v all_o according_a unto_o the_o former_a rule_n as_o 8_o be_v to_z 6_o so_o will_v 4_o be_v to_o 3_o which_o four_o proportional_a number_n take_v from_o 8_o the_o first_o supposition_n the_o remainder_n will_v be_v 5._o the_o number_n require_v the_o error_n be_v both_o in_o excess_n question_n 2._o to_o divide_v any_o number_n propound_v into_o any_o part_n that_o shall_v be_v require_v and_o those_o part_n for_o to_o be_v in_o any_o proportion_n one_o unto_o another_o that_o shall_v be_v assign_v the_o give_a number_n here_o be_v 45_o for_o to_o be_v divide_v into_o 2_o part_n and_o those_o to_o be_v in_o a_o triple_a proportion_n one_o unto_o the_o other_o the_o cross_n make_v as_o before_o let_v 8_o be_v the_o first_o supposition_n and_o the_o least_o part_n then_o 24_o shall_v have_v be_v the_o great_a the_o sum_n of_o they_o be_v 32_o but_o the_o number_n give_v be_v 45_o therefore_o this_o supposition_n be_v 13_o defective_a as_o by_o the_o error_n do_v appear_v then_o suppose_v 10_o for_o the_o number_n require_v the_o triple_a of_o it_o be_v 30_o the_o sum_n of_o they_o both_o be_v 40_o 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in_o proportion_n unto_o the_o decimall_n numerator_n require_v which_o be_v 5_o 10_o equal_a to_o ½_n thus_o the_o second_o example_n be_v â…—_n which_o be_v to_o be_v convert_v unto_o a_o decimall_n and_o thus_o as_o 5_o to_z 3_o so_o 10_o will_v be_v to_o 6_o so_o the_o fraction_n be_v 6_o 10._o the_o three_o example_n be_v of_o ¼_n which_o be_v make_v 25_o 100_o equal_a to_o ¼_n the_o four_o example_n be_v â…_n reduce_v to_o a_o decimall_n of_o 625_o 1000_o equal_a to_o â…_n be_v in_o the_o same_o proportion_n with_o the_o vulgar_a fraction_n from_o whence_o it_o be_v derive_v in_o compose_v of_o these_o decimall_n fraction_n there_o need_v no_o multiplication_n as_o lib_n 1._o sect_n 1._o parag_v 4_o example_n 4._o but_o annex_v cypher_n unto_o the_o numerator_n of_o the_o vulgar_a fraction_n give_v as_o occasion_n require_v but_o my_o second_o way_n of_o division_n be_v best_a as_o lib_n 1._o sect_n 1._o parag_v 5._o example_n 6._o by_o which_o mean_v you_o may_v continue_v 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find_v these_o in_o this_o as_o â…“_n who_o numerator_n convert_v to_o a_o decimall_n will_v be_v 3333_o that_o be_v 3333_o 10000_o which_o be_v very_o near_o equal_a to_o â…“_n as_o by_o the_o first_o proposition_n of_o this_o example_n in_o the_o second_o 2_o 7_o be_v propound_v who_o numerator_n 2_o multiply_v by_o 100000_o or_o which_o be_v all_o one_o annex_v 5_o cypher_n to_o it_o and_o then_o it_o will_v be_v 200000_o which_o divide_v by_o 7_o the_o vulgar_a fraction_n denominator_fw-la the_o quotient_n will_v be_v 28570_o for_o the_o new_a numerator_n and_o will_v stand_v thus_o 28570_o 100000_o or_o thus_o 2857_o 10000_o equal_a to_o the_o former_a as_o lib_n 1._o sect_n 2._o parag_v 1._o parad_n 4._o of_o this_o last_o decimall_n fraction_n if_o the_o numerator_n have_v be_v require_v but_o of_o 3_o place_n viz_o 285_o 1000_o add_v 1_o unto_o the_o unite_v place_n and_o so_o make_v it_o 286_o 1000_o and_o the_o reason_n be_v evident_a â—86_n 1000_o be_v 3_o 10_o too_o much_o whereas_o 285_o 1000_o be_v 7_o 10_o defective_a and_o of_o two_o error_n the_o least_o be_v to_o be_v choose_v and_o so_o for_o any_o other_o fraction_n in_o decimall_n rule_n 2._o the_o annotation_n of_o decimall_n fraction_n and_o how_o denominate_v decimall_n fraction_n be_v a_o table_n of_o annotation_n unite_v 3_o 3_o l_o 0_o prime_n 1_v 1_o 9_o second_o 2_o 5_o 3_o s_n 8_o three_o 3_o 6_o 7_o fourths_o 4_o 5_o 1_o d_o 6_o fift_n 5_o 1_o 5_o sixt_n 6_o 0_o 4_o seventh_n 7_o 4_o 2_o q_o 3_o eight_o 8_o 1_o 2_o ninth_n 9_o 6_o 1_o ten_o 10_o 7_o ¼_n i_o degree_n two_o iii_o iv_o thus_o denominate_v descend_v from_o a_o unite_v the_o first_o place_n below_o a_o integer_fw-la be_v call_v a_o prime_n the_o next_o unto_o that_o be_v second_o the_o place_n beneath_o they_o be_v three_o etc._n etc._n denote_v according_a to_o the_o degree_n series_n or_o order_n of_o the_o arithmetical_a figure_n descend_v by_o ten_o to_o what_o number_n of_o place_n you_o please_v or_o the_o state_n of_o the_o question_n shall_v require_v this_o table_n be_v make_v unto_o ten_o place_n although_o 5_o or_o 6_o will_v be_v abundant_o sufficient_a for_o most_o question_n in_o the_o uppermost_a row_n of_o this_o table_n note_v i_o stand_v o_o prime_n 9_o second_o 8_o three_o 7_o fourths_o etc._n etc._n and_o so_o any_o other_o number_n reckon_v according_a to_o their_o degree_n descend_v from_o a_o unite_v as_o in_o the_o second_o row_n two_o who_o first_o place_n must_v be_v always_o note_v with_o this_o mark_n or_o a_o point_n only_o prefix_v before_o they_o as_o thus_o 25_o denote_v 2_o prime_n &_o 5_o second_o the_o three_o row_n show_v a_o number_n compound_v of_o integer_n and_o fraction_n together_o continue_v unto_o ten_o place_n and_o according_a to_o the_o quality_n of_o this_o artificial_a arithmetic_n comprehend_v as_o in_o the_o four_o row_n 3_o l._n 3_o s._n 1_o d._n 2_o q_n &_o ¼_n all_o in_o one_o sum_n as_o shall_v be_v make_v evident_a hereafter_o according_a to_o this_o one_o unite_n make_v 10_o prime_n prime_a second_o second_v three_o third_z have_v four_o fift_n five_o sixt_n sixth_o sevenths_n seven_o eight_o eight_o ninth_n nine_o ten_o last_n prescribe_v rule_n in_o all_o fraction_n of_o artificial_a or_o decimall_n arithmetic_n the_o degree_n proceed_v by_o 10_o so_o that_o one_o unite_v contain_v 10_o prime_n one_o prime_n 10_o second_o one_o second_o 10_o three_o one_o three_o make_n 10_o fourths_o &c._n &c._n so_o a_o unite_v of_o any_o one_o denomination_n make_v 10_o in_o the_o succeed_a degree_n descend_v and_o 5_o be_v the_o half_a of_o any_o precedent_a decimall_n as_o 5_o second_o be_v half_a a_o prime_n and_o 5_o prime_n half_n a_o integer_fw-la or_o unite_v whether_o it_o be_v of_o number_n weight_n or_o measure_n that_o be_v of_o any_o divisible_a thing_n rule_v 3_o the_o numeration_n of_o decimall_n fraction_n without_o denominator_n numerator_n of_o all_o decimall_n fraction_n be_v number_v from_o a_o unite_v descend_v towards_o the_o right_a hand_n contrary_a to_o all_o number_n in_o natural_a arithmetic_n as_o by_o the_o front_n of_o this_o pyramid_n will_v appear_v which_o figure_n be_v divide_v into_o two_o part_n contain_v upon_o the_o right_a hand_n the_o numerator_n of_o fraction_n descend_v in_o 8_o degree_n of_o decimall_n as_o by_o the_o numeral_a letter_n on_o the_o right_a side_n of_o the_o pyramid_n appear_v from_o 1_o 10_o of_o a_o unite_v 1_o 10000000_o upon_o the_o left_a hand_n in_o this_o figure_n do_v stand_v all_o the_o respective_a denominator_n contain_v between_o they_o at_o the_o basis_n of_o these_o stand_v letter_n of_o the_o alphabet_n over_o the_o capital_a letter_n the_o 9_o significant_a figure_n be_v inscribe_v ascend_v by_o integer_n according_a unto_o natural_a arithmetic_n viz_o over_o a_o the_o unite_v place_n over_z b_o the_o place_n of_o ten_o above_o c_o the_o degree_n of_o hundred_o etc._n etc._n ascend_v ten_o every_o degree_n or_o place_n towards_o the_o left_a hand_n against_o these_o upon_o the_o right_a hand_n stand_v as_o many_o small_a roman_a letter_n with_o arithmetical_a figure_n over_o they_o to_o show_v the_o order_n of_o decimall_n from_o a_o integer_fw-la or_o unite_v descend_v towards_o the_o right_a hand_n viz_o a_o denote_v the_o unite_z place_n over_o which_o stand_v 1_o and_o that_o under_o the_o point_n represent_v the_o place_n of_o integer_n b_o note_v the_o prime_n be_v ten_o time_n less_o than_o the_o former_a c_o stands_z under_o second_o the_o three_o degree_n or_o place_n of_o hundred_o descend_v be_v ten_o time_n inferior_a to_o the_o last_o and_o thus_o these_o decrease_n from_o a_o unite_v by_o ten_o as_o integer_n increase_v and_o as_o from_o a_o unite_v one_o ascend_v the_o other_z descends_z both_o in_o the_o same_o order_n and_o without_o end_n or_o limitation_n as_o in_o numeration_n continue_v the_o denominator_n in_o this_o kind_n of_o artificial_a arithmetic_n be_v total_o to_o be_v omit_v observe_v to_o make_v good_a the_o place_n of_o the_o numerator_n unto_o prime_n inclusive_a by_o prefix_v cypher_n as_o thus_o 1_o for_o 1_o 10_o and_z −_o 01_o for_o 1_o 100_o and_z −_o 001_o for_o 1_o 1000_o see_v lib_n 1._o pag_n 3._o by_o which_o mean_v the_o denominator_fw-la of_o any_o decimall_n be_v always_o know_v be_v a_o unite_v with_o so_o many_o cypher_n as_o the_o numerator_n have_v place_n and_o for_o find_v all_o decimall_n fraction_n the_o follow_a rule_n declare_v and_o illustrate_v with_o example_n rule_n 4._o a_o general_a method_n for_o reduction_n of_o vulgar_a fraction_n unto_o decimall_n without_o any_o sensible_a error_n as_o thus_o this_o differ_v nothing_o real_o from_o the_o 1_o rule_n &_o yet_o depend_v upon_o the_o two_o last_o for_o here_o you_o be_v
fraction_n you_o may_v find_v their_o decimall_n thus_o for_o 1_o ou_fw-fr take_v 1_o 12_o the_o decimall_n −_o 083333_o for_o 16_o p._n the_o fraction_n be_v reduce_v to_o â…˜_n of_o 1_o 12_o that_o be_v 1_o 1â—_n the_o decimall_n −_o 066667._o next_o for_o 5_o g._n the_o vulgar_a fraction_n 1_o 1152_o the_o decimall_n be_v −_o 000168_o the_o ½_n g._n be_v ½_n of_o 1_o 24_o of_o 1_o 20_o of_o 1_o 12._o that_o be_v 1_o â—000087_n the_o total_a be_v 1_o lb_n −_o 150955_o in_o the_o same_o manner_n you_o may_v take_v the_o particular_n as_o in_o the_o second_o table_n to_o 5_o place_n who_o sum_n will_v be_v 1_o −_o 15096_o not_o differ_v a_o unite_v from_o the_o former_a in_o the_o five_o place_n herein_o you_o be_v to_o note_v that_o although_o these_o be_v fraction_n give_v yet_o their_o totall_n do_v make_v one_o integer_fw-la part_v from_o the_o place_n of_o prime_n with_o a_o point_n the_o three_o table_n be_v of_o liquid_a measure_n viz_o gallon_n and_o their_o inferior_a part_n to_o be_v add_v together_o as_o 13_o g._n 2_o q._n 1_o p_o ¾_n to_o be_v add_v unto_o 26_o g._n 1_o q._n 0_o p_o ¼_n the_o sum_n of_o these_o be_v 40_o gallâ—ns_n but_o to_o add_v they_o according_a to_o their_o particular_n begin_v first_o with_o 2_o q_o that_o be_v ½_n who_o decimal_n be_v always_o 5._o next_o 1_o q._n be_v ¼_n the_o decimall_n −_o 25_o than_o 1_o p._n be_v ½_n of_o ¼_n that_o be_v â…›_n the_o decimall_n 125._o then_o ¾_n of_o ½_n of_o ¼_n be_v 3_o 32_o the_o decimall_n be_v −_o 09375._o last_o ¼_n of_o ½_n of_o ¼_n be_v 1_o 32_o and_o for_o the_o decimall_n −_o 03125_o these_o number_n be_v all_o rational_a and_o so_o no_o fraction_n remain_v here_o observe_v that_o all_o ten_o in_o the_o place_n of_o prime_n be_v integer_n as_o be_v say_v before_o and_o must_v be_v add_v as_o unite_v if_o there_o be_v any_o whole_a number_n or_o inscribe_v beyond_o the_o point_n or_o prime_a line_n as_o in_o the_o second_o and_o three_o example_n addition_n of_o these_o artificial_a number_n differ_v not_o from_o the_o common_a or_o vulgar_a way_n but_o in_o this_o and_o place_v they_o according_a to_o their_o degree_n and_o so_o much_o for_o this_o chapter_n which_o be_v prove_v by_o subtraction_n as_o in_o natural_a arithmetic_n sect._n i._n chap._n iii_o subtraction_n of_o decimall_n with_o integer_n and_o fraction_n commix_v together_o if_o several_a  _fw-fr  _fw-fr  _fw-fr  _fw-fr l._n  _fw-fr 1_o −_o 9260417_o 2_o 20_o 275_o  _fw-fr −_o 3989583_o  _fw-fr 8_o 2875_o re._n −_o 5270834_o re._n 11_o 9875_o  _fw-fr l._n  _fw-fr  _fw-fr l._n  _fw-fr 3_o 40_o 1666667_o 4_o 100_o  _fw-fr −_o 325_o  _fw-fr −_o 008333_o re._n 39_o 8416667_o re._n 99_o 991667_o sum_n be_v give_v to_o be_v subtract_v reduce_v they_o unto_o totall_n and_o place_v they_o according_a to_o their_o degree_n as_o in_o the_o last_o chapter_n and_o then_o subtract_v the_o lesser_a number_n from_o the_o great_a and_o if_o the_o decimall_n be_v unequal_a that_o defect_n may_v be_v supply_v with_o point_n or_o suppose_v cypher_n to_o be_v annex_v unto_o they_o in_o all_o thing_n else_o except_o their_o degree_n it_o differ_v nothing_o from_o the_o subtraction_n of_o whole_a number_n as_o by_o these_o 4_o example_n shall_v be_v make_v evident_a in_o the_o first_o be_v propound_v 18_o s._n 6_o d._n 1_o q._n from_o whence_o 7_o s._n 11_o d._n 3_o q._n be_v to_o be_v take_v in_o decimall_n which_o to_o find_v i_o have_v show_v already_o yet_o for_o your_o ease_n one_o more_o shall_v be_v inscribe_v and_o first_o for_o 18_o s._n the_o vulgar_a fraction_n be_v 9_o 10_o the_o decimall_n −_o 9_o then_z 6_o d._n be_v ½_n of_o 1_o 20_o that_o be_v 1_o 40_o the_o artificial_a number_n −_o 025_o than_o 1_o q._n be_v ¼_n of_o 1_o 12_o of_o 1_o 20_o that_o be_v 1_o 960_o the_o decimall_n −_o 0010417._o the_o total_a −_o 9260417_o and_o in_o this_o manner_n you_o will_v find_v the_o second_o number_n to_o be_v −_o 3989583_o which_o take_v from_o the_o former_a there_o will_v remain_v −_o 5270834._o which_o be_v the_o decimall_n of_o 10_o s_o 6_o d._n ½_n the_o true_a remainder_n of_o the_o vulgar_a number_n if_o subtract_v in_o the_o second_o table_n 8_o l._n 5_o s._n 9_o do_v be_v give_v to_o be_v subtract_v from_o 20_o l_o 5_o s._n 6_o d._n the_o decimall_n of_o 5_o s._n be_v −_o 25_o of_o 6_o d._n −_o 025_o the_o sum_n be_v 20_o l_o 275._o the_o total_a of_o 5_o s._n 9_o d._n be_v −_o 2875_o so_o the_o lesser_a sum_n be_v 8_o l._n −_o 2875_o the_o difference_n be_v 11_o l._n −_o 9875_o according_a to_o the_o table_n which_o be_v 19_o s._n 9_o d._n the_o three_o table_n be_v 6_o s._n 6_o d._n to_o be_v subtract_v from_o 40_o l_o 3_o s._n 4_o d._n the_o sum_n of_o the_o decimall_n answer_v 3_o s._n 4_o d._n be_v −_o 1666667_o −_o the_o decimall_n belong_v to_o 6_o s._n 6_o d._n be_v −_o 325_o to_o which_o annex_v cypher_n or_o make_v point_n as_o in_o the_o table_n and_o subtract_v they_o according_a to_o vulgar_a or_o natural_a arithmetic_n the_o difference_n be_v 39_o l._n and_o this_o decimall_n −_o 8416667_o equal_a to_o 16_o s._n 10_o d._n the_o four_o and_o last_o example_n be_v of_o 100_o l._n from_o whence_o 2_o d._n be_v to_o be_v deduct_v who_o decimall_n be_v −_o 008333_o suppose_v cypher_n or_o point_n represent_v their_o place_n and_o subtract_v the_o decimall_n as_o in_o the_o table_n where_o you_o will_v find_v the_o remainder_n 99_o l._n −_o 991667_o the_o decimall_n of_o 19_o s._n 10_o d._n answer_v the_o truth_n and_o your_o expectation_n too_o this_o chap_n be_v try_v by_o the_o last_o and_o that_o prove_v by_o this_o as_o by_o subtraction_n in_o natural_a arithmetic_n to_o which_o i_o refer_v you_o for_o the_o form_n and_o reason_n and_o for_o the_o practic_a part_n to_o this_o chapter_n in_o 4_o example_n sect._n i._n chap._n iv_o multiplication_n of_o decimall_n commix_v with_o whole_a and_o break_a number_n the_o increase_n of_o decimal_n 1_o 2_o l_o 456_o −_o 91875_o −_o 15_o −_o 25_o 2280_o 459375_o 456_o 183750_o 6840_o −_o 2296875_o by_o integer_n or_o mix_v number_n differ_v nothing_o from_o multiplication_n in_o natural_a arithmetic_n yet_o 4_o thing_n in_o this_o be_v particular_o to_o be_v observe_v first_o for_o conveniency_n make_v that_o number_n which_o consist_v of_o most_o place_n the_o multiplicand_a although_o it_o be_v oftentimes_o the_o least_o in_o quantity_n second_o you_o must_v distinguish_v the_o decimall_n from_o the_o integer_n if_o there_o be_v any_o with_o a_o point_n or_o prime_a line_n three_o cut_v off_o with_o a_o prime_a line_n so_o many_o place_n of_o the_o product_n number_v from_o the_o right_a hand_n towards_o the_o left_a as_o there_o be_v fraction_n both_o in_o the_o multiplier_n and_o multiplicand_n all_o the_o figure_n from_o thence_o to_o the_o left_a hand_n be_v integer_n and_o those_o to_o the_o right_n be_v decimall_n fraction_n or_o this_o may_v be_v do_v with_o a_o point_n to_o note_v the_o place_n of_o prime_n four_o if_o the_o product_n shall_v have_v few_o place_n than_o the_o term_n give_v have_v decimall_n fraction_n those_o place_n must_v be_v make_v good_a by_o prefix_v cypher_n on_o the_o left_a hand_n this_o will_v often_o happen_v when_o the_o place_n of_o prime_n second_o or_o three_o shall_v be_v cypher_n all_o which_o by_o example_n will_v be_v illustrate_v in_o the_o first_o of_o these_o two_o table_n there_o be_v propound_v 456_o l._n to_o be_v multiply_v by_o 3_o s._n who_o decimall_n be_v 15_o and_o according_a unto_o multiplication_n in_o whole_a number_n the_o product_n will_v be_v that_o be_v 68_o l._n 4_o 10_o or_o 8_o s._n and_o so_o will_v 456_o 1_o multiply_v by_o 3_o 20_o which_o be_v the_o vulgar_a fraction_n of_o 3_o s._n the_o multiplier_n propound_v and_o in_o the_o second_o example_n 18_o s._n 4_o d._n ½_n be_v to_o be_v multiply_v by_o 5_o s._n the_o decimall_n fraction_n of_o the_o multiplicand_n be_v −_o 91875._o and_o 5_o s._n be_v −_o 25_o as_o a_o decimall_n these_o multiply_v will_v produce_v −_o 2296875_o and_o be_v there_o be_v 7_o place_n in_o both_o term_n the_o whole_a product_n be_v but_o a_o decimall_n fraction_n and_o must_v have_v a_o point_n prefix_v as_o in_o the_o example_n the_o fraction_n of_o several_a denomination_n may_v be_v reduce_v unto_o a_o single_a fraction_n as_o 1323_o 1440_o or_o 147_o 160_o and_o this_o unto_o a_o decimall_n as_o before_o be_v show_v in_o reduction_n of_o artificial_a number_n example_n 2._o in_o the_o first_o of_o these_o 1_o 2_o table_n there_o be_v give_v a_o whole_a number_n with_o a_o fraction_n to_o