to_o be_v multiply_v 2._o the_o multiplier_n which_o be_v the_o number_n by_o which_o the_o multiplicand_a be_v multiply_v 3._o the_o product_n which_o be_v the_o sum_n produce_v by_o the_o multiply_a of_o the_o two_o former_a together_o and_o here_o note_v that_o the_o product_n do_v contain_v the_o multiplicand_a so_o many_o time_n as_o there_o be_v unite_v in_o the_o multiplier_n thus_o for_o the_o definition_n of_o multiplication_n now_o for_o the_o work_n thereof_o by_o the_o rod_n for_o which_o this_o be_v the_o rule_n first_o set_v down_o upon_o your_o paper_n the_o multiplicand_a and_o orderly_o under_o it_o the_o multiplier_n it_o matter_n not_o great_o which_o of_o the_o two_o give_v number_n be_v make_v multiplicand_v or_o multiplier_n but_o it_o be_v usual_a and_o best_a to_o make_v the_o great_a number_n multiplicand_v and_o the_o lesser_a multiplier_n then_o draw_v a_o line_n with_o your_o pen_n under_o they_o and_o have_v tabulate_v your_o multiplicand_n or_o great_a number_n look_v what_o number_n in_o your_o rod_n stand_v against_o the_o first_o figure_n towards_o your_o right_a hand_n and_o that_o number_n which_o you_o shall_v find_v upon_o your_o rod_n stand_v against_o that_o first_o figure_n find_v in_o your_o tabulat_fw-la set_v down_o under_o your_o line_n which_o you_o former_o draw_v under_o your_o multiplicand_a and_o multiplier_n and_o have_v so_o do_v with_o the_o first_o figure_n of_o your_o multiplier_n do_v so_o with_o the_o rest_n set_v they_o down_o one_o under_o another_o remove_v every_o figure_n one_o place_n more_o towards_o the_o left_a hand_n then_o that_o which_o go_v before_o it_o as_o be_v do_v in_o common_a multiplication_n and_o as_o you_o see_v in_o the_o follow_a example_n example_n 1._o let_v it_o be_v require_v to_o multiply_v 3496_o by_o 489._o as_o if_o it_o be_v require_v to_o know_v how_o much_o 489_o time_n 3496_o will_v amount_v unto_o first_o set_v down_o your_o give_a number_n 3496_o and_o 489_o one_o under_o another_o and_o draw_v your_o line_n under_o they_o as_o here_o you_o see_v do_v second_o 3496_o your_o multiplicand_a be_v tabulate_v and_o 9_o be_v the_o first_o figure_n to_o the_o right_a hand_n in_o your_o multiplier_n look_v upon_o your_o rod_n what_o sum_n there_o stand_v against_o 9_o in_o the_o side_n of_o your_o tabulat_fw-la and_o you_o shall_v find_v as_o by_o the_o rule_n in_o the_o second_o prop._n of_o the_o five_o chap._n you_o be_v direct_v 31464_o which_o be_v the_o product_n of_o 3496_o multiply_v by_o 9_o wherefore_o set_v down_o this_o number_n 31464_o under_o your_o line_n as_o you_o see_v in_o the_o example_n three_o look_v what_o sum_n upon_o the_o rod_n stand_v against_o 8_o which_o be_v the_o second_o figure_n of_o your_o multiplier_n and_o you_o shall_v find_v 27968_o set_v this_o number_n under_o the_o former_a move_v it_o one_o place_n forward_o towards_o the_o left_a hand_n four_o look_v what_o sum_n upon_o the_o rod_n stand_v against_o 4_o which_o be_v the_o three_o figure_n in_o your_o multiplier_n and_o you_o shall_v find_v 13984_o which_o set_v down_o under_o the_o other_o one_o place_n more_o to_o the_o left_a hand_n last_o under_o these_o three_o sum_n draw_v a_o line_n and_o add_v the_o three_o sum_n together_o and_o they_o make_v 1709544_o which_o be_v the_o product_n of_o 3496_o multiply_v by_o 489_o and_o this_o 1709544_o the_o product_n contain_v 3496_o the_o multiplicand_a 489_o time_n practise_v well_o this_o first_o example_n and_o compare_v it_o with_o the_o rod_n as_o they_o be_v tabulate_v in_o figure_n 4_o at_o the_o begin_n of_o the_o book_n as_o also_o with_o the_o rule_n in_o the_o five_o chapter_n and_o you_o may_v perform_v any_o multiplication_n however_o i_o will_v give_v you_o one_o or_o two_o more_o example_n and_o some_o other_o way_n of_o multiplication_n example_n 2._o let_v it_o be_v require_v to_o multiply_v the_o same_o sum_n 3496_o by_o 261._o set_a the_o number_n down_o as_o here_o be_v do_v then_o look_v upon_o the_o rod_n for_o the_o product_n of_o 3496_o by_o 1_o and_o you_o shall_v find_v it_o to_o be_v the_o same_o wherefore_o set_v down_o 3496_o under_o the_o line_n then_o look_v upon_o the_o rod_n for_o the_o product_n of_o 3496_o by_o 6_o and_o you_o shall_v find_v it_o to_o be_v 20976_o which_o set_v down_o under_o the_o other_o number_n one_o place_n more_o towards_o the_o left_a hand_n again_o look_v in_o the_o rod_n for_o the_o product_n of_o 3496_o multiply_v by_o 2_o and_o you_o shall_v find_v it_o to_o be_v 6992_o which_o set_v down_o under_o the_o other_o two_o last_o draw_v a_o line_n under_o they_o and_o add_v the_o three_o number_n together_o in_o order_n as_o they_o stand_v and_o the_o sum_n of_o they_o will_v be_v 912456_o which_o be_v the_o product_n of_o 3496_o multiply_v by_o 261._o example_n 3._o let_v it_o be_v require_v to_o multiply_v the_o same_o number_n 3496_o by_o 520._o set_a down_o your_o number_n as_o here_o you_o see_v do_v then_o because_o the_o first_o figure_n of_o your_o multiplier_n towards_o your_o right_a hand_n be_v a_o cypher_n whole_o omit_v it_o and_o multiply_v 3496_o by_o 52_o only_a so_o shall_v you_o find_v the_o product_n of_o 3496_o by_o 2_o to_o be_v 6992_o which_o set_v down_o also_o the_o product_n by_o 5_o will_v be_v 17480_o which_o set_v down_o under_o the_o other_o one_o place_n further_o then_o draw_v a_o line_n and_o add_v these_o two_o sum_n together_o and_o they_o make_v 181792_o to_o the_o which_o if_o you_o add_v a_o cypher_n for_o the_o cypher_n which_o you_o omit_v in_o your_o multiplier_n the_o sum_n will_v be_v 1817920_o which_o be_v the_o product_n of_o 3496_o by_o 520._o example_n 4._o let_v it_o be_v require_v to_o multiply_v the_o same_o 3496_o by_o 7003_o set_n down_o your_o number_n as_o before_o and_o as_o you_o see_v here_o do_v then_o have_v tabulate_v 3496_o see_v what_o the_o product_n thereof_o be_v upon_o the_o rod_n be_v multiply_v by_o 3_o the_o first_o figure_n in_o your_o multiplier_n and_o you_o shall_v find_v it_o to_o be_v 10488_o which_o set_v down_o under_o the_o line_n then_o the_o two_o next_o place_n of_o your_o multiplier_n be_v cipher_n make_v two_o prick_n under_o the_o former_a number_n one_o under_o 8_o the_o other_z under_z 4_o as_o you_o see_v in_o the_o example_n or_o instead_o of_o 2_o prick_v you_o may_v make_v two_o cipher_n then_o look_v in_o the_o rod_n for_o the_o product_n of_o 3496_o by_o 7_o and_o you_o shall_v find_v it_o to_o be_v 24472_o which_o set_v down_o under_o the_o other_o sum_n begin_v your_o number_n at_o the_o four_o place_n or_o beyond_o the_o two_o prick_n or_o cipher_n last_o draw_v a_o line_n and_o add_v these_o two_o sum_n together_o and_o their_o sum_n be_v 24482488_o which_o be_v the_o product_n of_o 3496_o multiply_v by_o 7003._o thus_o have_v you_o four_o example_n in_o multiplication_n in_o which_o be_v include_v all_o the_o variety_n that_o may_v at_o any_o time_n happen_v in_o that_o rule_n viz._n two_o where_o the_o multiplier_n consist_v all_o of_o figure_n as_o in_o the_o first_o and_o second_o example_n they_o do_v another_z where_o the_o latter_a place_n of_o the_o multiplier_n consist_v of_o a_o cypher_n and_o this_o last_o example_n where_o cipher_n be_v intermix_v among_o the_o figure_n and_o thus_o much_o for_o this_o kind_n of_o multiplication_n but_o before_o i_o leave_v i_o will_v show_v you_o another_o form_n of_o multiplication_n whereas_o in_o the_o forego_n form_n of_o multiplication_n which_o be_v the_o best_a and_o most_o usual_a only_o i_o insert_v this_o follow_n for_o variety_n you_o begin_v your_o rod_n be_v tabulate_v with_o that_o figure_n of_o your_o multiplier_n which_o stand_v next_o your_o right_a hand_n but_o there_o be_v no_o necessity_n for_o that_o for_o you_o may_v begin_v with_o that_o figure_n which_o stand_v next_o to_o your_o left_a hand_n and_o by_o so_o do_v and_o place_v your_o several_a product_n one_o place_n more_o to_o the_o right_a hand_n as_o you_o do_v before_o place_v they_o to_o the_o left_a hand_n those_o product_n add_v together_o in_o the_o form_n they_o then_o stand_v shall_v produce_v a_o product_n equal_a to_o the_o former_a example_n for_o our_o example_n we_o will_v take_v the_o first_o example_n before-going_a at_o the_o begin_n of_o this_o chapter_n where_o it_o be_v require_v to_o multiply_v 3496_o by_o 489._o set_a the_o number_n down_o as_o before_o in_o that_o first_o example_n and_o as_o you_o see_v here_o do_v then_o 3496_o be_v tabulate_v look_v upon_o your_o rod_n for_o the_o product_n thereof_o multiply_v by_o 4_o which_o be_v the_o first_o figure_n of_o your_o multiplier_n towards_o your_o left_a hand_n and_o you_o shall_v find_v the_o product_n thereof_o to_o be_v 13984_o which_o
multiple_n no_o not_o the_o least_o of_o all_o find_v in_o the_o left_a rod_n and_o the_o plate_n may_v be_v substract_v from_o the_o forego_n remain_v than_o a_o cypher_n o_fw-la must_v be_v put_v under_o that_o point_n for_o the_o quotume_n the_o remain_v be_v untouched_a and_o abide_v as_o before_o ii_o caution_n and_o if_o the_o aforesaid_a sum_n to_o be_v take_v away_o can_v be_v take_v from_o the_o figure_n go_v before_o its_o point_n the_o small_a multiple_n must_v be_v add_v which_o the_o next_o upper_a quotume_n in_o the_o cubick_a rod_n do_v show_v in_o the_o rod_n who_o sum_n may_v be_v tuken_v away_o therefrom_o example_n of_o the_o cubick_a extraction_n let_v 22022635627_o be_v a_o number_n give_v who_o cube_n root_n you_o desire_v set_v down_o your_o number_n and_o point_v it_o begin_v at_o 7_o the_o last_o figure_n towards_o the_o right_a hand_n and_o so_o under_o every_o three_o figure_n and_o draw_v two_o parallel_n line_n under_o it_o and_o it_o will_v stand_v in_o this_o manner_n look_v in_o your_o rod_n for_o the_o extract_a the_o cube_n root_n for_o the_o near_a cube_n root_n of_o the_o figure_n of_o your_o give_a number_n stand_v before_o the_o first_o point_n towards_o your_o left_a hand_n namely_o for_o the_o near_a cube_n root_n of_o the_o number_n less_o than_o 22_o which_o you_o shall_v find_v to_o be_v 2_o which_o set_v between_o the_o two_o line_n just_a under_o the_o first_o point_n and_o its_o cube_n which_o be_v 8_o set_v under_o the_o line_n and_o subtract_v it_o from_o the_o figure_n above_o the_o line_n namely_o from_o 22_o and_o there_o will_v remain_v 14_o which_o place_n orderly_o above_o then_o will_v your_o work_n stand_v thus_o and_o the_o work_n of_o your_o first_o point_n finish_v second_o for_o the_o find_n of_o the_o root_n belong_v to_o the_o second_o prick_n triple_a the_o quotume_n or_o figure_n which_o be_v under_o the_o first_o prick_n namely_o 2_o and_o it_o be_v 6_o find_v therefore_o a_o rod_n which_o have_v 6_o at_o the_o head_n thereof_o and_o lay_v that_o rod_n by_o the_o side_n of_o your_o cubick_a rod_n towards_o the_o right_a hand_n then_o triple_a the_o square_n of_o 2_o which_o be_v 4_o and_o it_o make_v 12_o which_o find_v among_o the_o rod_n place_n by_o the_o side_n of_o the_o cubick_a rod_n towards_o the_o left_a hand_n then_o from_o the_o rod_n which_o lie_v on_o the_o left_a hand_n of_o the_o cubick_a rod_n and_o the_o cubick_a rod_n itself_o find_v the_o near_o lesser_a number_n than_o the_o figure_n stand_v before_o the_o second_o prick_n namely_o less_o than_o 14022_o and_o in_o the_o nine_o place_n you_o shall_v find_v 11529_o which_o write_v by_o itself_o as_o in_o the_o margin_n and_o over_o 9_o the_o last_o figure_n towards_o the_o right_a hand_n draw_v first_o a_o line_n between_o set_v its_o quotume_n and_o by_o it_o its_o square_n 81_o in_o the_o same_o order_n as_o you_o find_v they_o stand_v in_o your_o cubick_a rod._n then_o write_v under_o 1_o its_o multiple_n which_o be_v show_v right-ward_n against_o 1_o in_o the_o cubick_a rod_n and_o be_v the_o single_a figure_n 6_o and_o under_o 8_o write_v the_o multiple_n that_o it_o show_v right-ward_n against_o 8_o in_o the_o cubick_a rod_n which_o be_v 48_o and_o these_o three_o multiple_n so_o write_v crosswise_o below_o the_o line_n and_o add_v together_o as_o in_o the_o margin_n do_v produce_v 16389_o which_o because_o they_o can_v be_v take_v from_o the_o upper_a figure_n stand_v before_o the_o second_o point_n namely_o from_o 14022_o the_o number_n 9_o before_o take_v be_v to_o be_v reject_v as_o be_v too_o great_a and_o instead_o of_o 819_o by_o the_o second_o caution_n the_o next_o high_o note_n in_o the_o plate_n be_v to_o be_v take_v which_o be_v 648_o and_o the_o multiple_n that_o these_o do_v show_v namely_o the_o octuple_a among_o the_o left_a rod_n which_o be_v 10112_o and_o the_o quadruple_a among_o the_o right_a rod_n which_o be_v 24_o and_o the_o sextuple_a among_o the_o right_a rod_n which_o be_v 36_o be_v add_v crosswise_o as_o in_o the_o margin_n do_v produce_v 13952_o which_o substract_v from_o 14022_o the_o figure_n stand_v before_o the_o second_o prick_n there_o remain_v 70_o for_o the_o remain_n of_o the_o second_o prick_n and_o let_v there_o be_v take_v for_o the_o quotume_n of_o the_o second_o prick_n the_o right-most_a of_o the_o choose_a figure_n 648_o which_o be_v 8_o which_o place_n under_o the_o second_o point_n between_o the_o line_n so_o be_v the_o second_o figure_n of_o your_o root_n find_v and_o your_o work_n will_v stand_v thus_o three_o put_v the_o triple_a of_o the_o precedent_a quotume_n viz._n 28_o between_o the_o line_n which_o be_v 84_o be_v take_v out_o of_o the_o rod_n and_o put_v they_o on_o the_o rightside_n of_o the_o cubick_a rod_n and_o get_v the_o triple_a of_o the_o squate_a of_o the_o same_o 28_o which_o may_v be_v find_v to_o be_v 2352_o which_o take_v out_o of_o the_o rod_n and_o place_n on_o the_o left-side_n of_o the_o cubick_a rod_n and_o of_o the_o multiple_n on_o the_o left-hand_a rod_n and_o the_o simple_a single_a figure_n upon_o the_o cubick_a rod_n the_o least_o of_o which_o being_n 235201_o there_o be_v none_o so_o little_a that_o may_v be_v substract_v from_o the_o figure_n belong_v to_o the_o three_o point_n namely_o from_o 70635_o therefore_o by_o the_o first_o caution_n the_o remain_v abide_v or_o continue_v as_o they_o be_v you_o must_v put_v a_o cypher_n under_o the_o three_o point_n for_o the_o three_o quotume_n belong_v to_o the_o three_o point_n and_o thus_o the_o operation_n of_o the_o three_o point_n be_v accomplish_v and_o the_o work_n will_v stand_v as_o follow_v four_o set_v the_o triple_a of_o the_o forego_n quotume_n viz._n 280_o which_o be_v 840_o on_o the_o right-hand_a and_o the_o triple_a of_o the_o square_n of_o the_o same_o 280_o which_o be_v 225200_o on_o the_o left-hand_a and_o the_o cubick_a rod_n between_o they_o then_o out_o of_o the_o left-most_a multiple_n choose_v that_o which_o be_v next_o less_o than_o the_o figure_n belong_v to_o the_o four_o point_n namely_o 70635627_o which_o be_v this_o 70560027_o which_o stand_v against_o 3_o on_o the_o tabulat_v wherefore_o write_v this_o number_n 70560027_o upon_o paper_n as_o in_o the_o margin_n with_o a_o line_n over_o it_o and_o set_v over_o the_o line_n the_o quotient_a 3_o over_o its_o right-most_a figure_n and_o the_o square_a of_o the_o say_v quotume_n 3_o which_o be_v 9_o left-ward_n thereof_o and_o the_o noncuple_n find_v in_o the_o right-hand_a rod_n which_o be_v 7560_o write_v under_o 9_o let_v these_o two_o multiple_n be_v add_v as_o in_o the_o margin_n and_o the_o sum_n will_v be_v 70625627_o which_o substract_v from_o the_o figure_n forego_v the_o four_o prick_n and_o there_o will_v nothing_o remain_v therefore_o let_v the_o right-most_a of_o the_o figure_n of_o 93_o viz._n 3_o be_v place_v under_o the_o four_o and_o last_o point_n for_o the_o four_o and_o last_o quotume_n of_o the_o root_n and_o so_o the_o whole_a and_o perfect_a cubick_a root_n of_o the_o give_v number_n 22022635627_o be_v 2803_o and_o be_v nothing_o remain_v it_o be_v a_o perfect_a cubick_a number_n the_o like_a be_v to_o be_v do_v in_o other_o number_n but_o i_o shall_v forbear_v to_o give_v you_o any_o more_o example_n there_o fall_v out_o in_o this_o all_o the_o variety_n that_o at_o any_o time_n may_v happen_v for_o the_o general_n rule_n and_o the_o two_o caution_n before_o premise_v be_v here_o make_v applicable_a to_o practice_n wherefore_o to_o this_o treatise_n for_o the_o present_a i_o will_v put_v a_o end_n
other_o side_n another_o for_o the_o extract_a of_o the_o cube_n root_n the_o figure_n whereof_o be_v at_o the_o begin_n of_o the_o book_n note_v with_o figure_n but_o i_o shall_v forbear_v to_o say_v any_o thing_n of_o they_o till_o i_o come_v to_o show_v you_o how_o to_o extract_v the_o square_a and_o cube_n root_n by_o the_o help_n of_o they_o and_o the_o rods._n of_o a_o board_n with_o a_o frame_n upon_o which_o to_o lay_v your_o rod_n when_o any_o operation_n be_v to_o be_v wrought_v by_o they_o know_v by_o the_o name_n of_o a_o tabulat_fw-la in_o the_o use_n of_o these_o rod_n care_n be_v to_o be_v have_v first_o of_o the_o orderly_a lay_n of_o they_o and_o then_o second_o for_o the_o keep_n of_o they_o in_o that_o position_n till_o your_o work_n be_v end_v for_o the_o effect_n whereof_o both_o neat_o and_o certain_o there_o be_v a_o little_a table_n or_o frame_n contrive_v contain_v in_o breadth_n 1_o 20_o of_o a_o inch_n more_o than_o the_o length_n of_o the_o rod_n and_o in_o length_n at_o pleasure_n but_o it_o may_v well_o be_v about_o once_o and_o a_o half_a the_o length_n of_o the_o breadth_n it_o ought_v to_o be_v make_v of_o a_o thin_a piece_n of_o pear_n or_o walnut-tree_n or_o of_o such_o matter_n as_o your_o box_n or_o case_n be_v make_v of_o and_o it_o may_v very_o commodious_o be_v contrive_v to_o be_v put_v into_o the_o box_n as_o i_o ever_o have_v they_o make_v to_o do_v for_o that_o i_o find_v it_o inconvenient_a to_o carry_v loose_a upon_o the_o superficies_n of_o this_o board_n close_o to_o one_o of_o the_o edge_n thereof_o must_v be_v glue_v or_o otherwise_o fasten_v with_o pin_n a_o small_a piece_n of_o the_o same_o matter_n and_o also_o of_o the_o same_o length_n breadth_n and_o thickness_n of_o one_o of_o your_o rod_n which_o must_v be_v divide_v into_o 9_o equal_a part_n and_o line_n draw_v across_o the_o piece_n so_o will_v there_o be_v 9_o square_n in_o which_o you_o must_v grave_a or_o stamp_v the_o nine_o digit_n begin_v with_o 1_o at_o the_o top_n and_o so_o descend_v by_o 2_o 3_o 4_o to_o 9_o at_o the_o bottom_n thereof_o and_o it_o be_v necessary_a that_o these_o figure_n as_o also_o those_o which_o be_v at_o the_o head_n of_o every_o of_o your_o rod_n be_v grave_v or_o stamp_v of_o something_o a_o big_a figure_n than_o the_o other_o figure_n of_o your_o rod_n be_v under_o the_o end_n of_o this_o ledge_n begin_v at_o the_o figure_n and_o so_o continue_v the_o whole_a length_n of_o the_o board_n must_v another_o ledge_n of_o the_o same_o matter_n and_o thickness_n as_o the_o other_o be_v glue_v or_o pin_v and_o then_o be_v your_o tabulat_fw-la finish_v a_o figure_n whereof_o you_o have_v at_o the_o begin_n of_o the_o book_n note_v with_o figure_n 4_o it_o be_v call_v a_o tabulat_fw-la for_o that_o when_o the_o rod_n be_v lay_v thereon_o for_o any_o operation_n to_o be_v wrought_v by_o they_o we_o usual_o say_v the_o rod_n be_v tabulate_v be_v thus_o prepare_v with_o rod_n and_o tabulat_v you_o be_v ready_a for_o the_o work_n intend_v by_o they_o and_o for_o which_o chief_o they_o be_v invent_v thus_o much_o for_o the_o fabric_n inscription_n and_o number_v of_o these_o rod_n let_v we_o now_o come_v to_o show_v the_o use_v of_o they_o chap._n iu._n to_o what_o use_v these_o rod_n general_o serve_v the_o chief_a uses_n to_o which_o these_o small_a rod_n serve_v unto_o i_o in_o part_n intimate_v at_o the_o beginning_n to_o which_o effect_n i_o shall_v repeat_v it_o again_o for_o by_o they_o all_o manner_n of_o multiplication_n and_o division_n as_o also_o of_o the_o extraction_n of_o both_o the_o root_n either_o square_a or_o cube_n be_v so_o facilly_a and_o expeditious_o perform_v and_o that_o by_o the_o help_n of_o addition_n and_o substraction_n only_o that_o it_o be_v as_o i_o may_v well_o say_v inconceivable_a for_o here_o be_v no_o charge_n at_o all_o require_v of_o the_o memory_n and_o you_o shall_v assure_o take_v your_o quotient_a figure_n in_o division_n always_o certain_a neither_o too_o great_a nor_o to_o little_a a_o inconvenience_n so_o prejudicial_a that_o i_o leave_v it_o to_o the_o censure_n of_o such_o as_o have_v find_v it_o to_o their_o great_a loss_n of_o time_n and_o other_o vexation_n which_o it_o have_v put_v they_o to_o but_o cease_v to_o say_v more_o of_o their_o property_n i_o will_v now_o come_v to_o show_v their_o use_n chap._n v._o how_o to_o apply_v or_o lay_v down_o any_o number_n by_o the_o rods._n prop._n i._n any_o number_n be_v give_v how_o to_o tabulate_v or_o lay_v down_o the_o same_o by_o the_o rods._n let_v it_o be_v require_v to_o tabulate_v or_o lay_v down_o this_o number_n 3_o 4_o 9_o 6._o first_o from_o among_o your_o set_v of_o rod_n or_o out_o of_o your_o case_n take_v four_o of_o they_o of_o which_o let_v one_o of_o they_o have_v the_o figure_n 3_o at_o the_o top_n thereof_o and_o lay_v it_o upon_o your_o tabulat_fw-la close_o to_o the_o edge_n thereof_o then_o second_o take_v another_o rod_n from_o your_o case_n which_o have_v the_o figure_n 4_o at_o the_o top_n of_o it_o and_o lay_v that_o also_o upon_o your_o tabulat_fw-la close_o by_o the_o side_n of_o the_o other_o three_o take_v another_o rod_n which_o have_v the_o figure_n 9_o at_o the_o top_n of_o it_o and_o lay_v that_o upon_o your_o tabulat_fw-la close_o by_o the_o other_o two_o and_o last_o take_v a_o four_o rod_n have_v the_o figure_n 6_o at_o the_o head_n thereof_o and_o lay_v that_o also_o upon_o your_o tabulat_fw-la close_o by_o the_o rest_n these_o four_o rod_n thus_o take_v and_o lay_v upon_o the_o tabulat_fw-la you_o shall_v see_v in_o the_o uppermost_a row_n which_o stand_v against_o the_o figure_n 1_o on_o the_o side_n of_o your_o tabulat_fw-la these_o four_o figure_n 3_o 4_o 9_o 6_o that_o be_v 3496_o equal_a to_o your_o give_a number_n in_o the_o second_o row_n against_o the_o figure_n 2_o of_o your_o tabulat_fw-la you_o shall_v find_v the_o double_a thereof_o in_o the_o three_o against_o the_o figure_n 3_o you_o shall_v find_v the_o triple_a thereof_o in_o the_o four_o the_o quadruple_a thereof_o in_o the_o five_o the_o quintuple_a and_o so_o on_o the_o nine_o and_o last_o in_o which_o you_o shall_v find_v the_o noncuple_n of_o the_o number_n give_v prop._n ii_o how_o these_o rod_n will_v appear_v when_o tabulate_v and_o be_v tabulate_v how_o to_o read_v the_o multiplication_n of_o that_o number_n so_o tabulate_v by_o any_o of_o the_o nine_o digit_n the_o four_o rod_n be_v tabulate_v according_a to_o the_o precept_n deliver_v in_o the_o precede_a proposition_n they_o will_v appear_v exact_o as_o they_o be_v represent_v in_o figure_n 4_o at_o the_o begin_n of_o the_o book_n which_o figure_n lively_o represent_v the_o four_o rod_n lie_v upon_o the_o tabulat_fw-la which_o mind_n well_o for_o upon_o the_o true_a tabulate_v and_o right_o read_v of_o the_o rod_n so_o tabulate_v depend_v the_o whole_a work_n the_o rod_n thus_o tabulate_v and_o as_o you_o see_v they_o in_o the_o figure_n 4_o do_v to_o the_o eye_n appear_v in_o the_o form_n of_o a_o glass-window_n every_o pane_n thereof_o represent_v a_o rhomboyade_n or_o diamond_n form_n in_o the_o read_n of_o the_o figure_n which_o be_v in_o these_o several_a rhomboyade_n or_o diamond_n form_n observe_v these_o few_o direction_n follow_v which_o will_v full_o illustrate_v the_o whole_a business_n intend_v and_o therefore_o especial_o to_o be_v mind_v note_v i._o that_o the_o figure_n upon_o the_o rod_n be_v to_o be_v read_v beginning_n at_o the_o right_a hand_n and_o read_n towards_o the_o left_a which_o be_v contrary_a to_o our_o common_a course_n of_o read_v and_o write_v which_o be_v from_o the_o left_a hand_n towards_o the_o right_n ii_o that_o in_o every_o rhomboyade_n or_o diamond_n there_o be_v either_o one_o figure_n or_o two_o figure_n but_o never_o more_o than_o two_n iii_o if_o there_o be_v but_o one_o figure_n in_o a_o rhombus_fw-la than_o that_o figure_n be_v the_o figure_n to_o be_v set_v down_o alone_o be_v it_o either_o a_o figure_n or_o a_o cypher_n but_o if_o there_o be_v two_o figure_n in_o a_o rhomboyade_n as_o for_o the_o most_o part_n there_o be_v then_o add_v they_o two_o figure_n together_o and_o set_v down_o their_o sum_n in_o one_o figure_n iu._n but_o if_o the_o sum_n of_o the_o two_o figure_n in_o one_o rhomboyade_n or_o diamond_n do_v exceed_v ten_o than_o you_o must_v set_v down_o the_o overplus_n above_o ten_o and_o keep_v one_o in_o mind_n which_o one_o you_o must_v carry_v to_o the_o next_o rhomboyades_n v._o note_v that_o the_o first_o towards_o your_o right_a hand_n and_o the_o last_o towards_o your_o left_a hand_n be_v but_o half_a rhomboyade_n or_o diamond_n and_o never_o
have_v in_o they_o more_o than_o one_o figure_n only_o but_o all_o between_o they_o be_v whole_a one_o and_o for_o the_o most_o part_n have_v two_o figure_n in_o they_o vi_o if_o in_o either_o rhomboyade_n or_o half_a rhomboyade_n you_o find_v no_o figure_n but_o cipher_n you_o must_v not_o neglect_v but_o set_v they_o down_o as_o if_o they_o be_v figure_n ¶_o these_o rule_n be_v right_o understand_v all_o that_o follow_v will_v be_v familiar_a and_o easy_a and_o these_o i_o shall_v explain_v by_o example_n follow_v example_n for_o the_o illustration_n of_o the_o precede_a rule_n we_o will_v make_v use_n of_o those_o rod_n which_o be_v before_o tabulate_v therefore_o have_v recourse_n to_o figure_n 4_o at_o the_o begin_n of_o the_o book_n where_o this_o number_n 3496_o be_v tabulate_v the_o figure_n at_o the_o top_n of_o the_o four_o rod_n be_v these_o 3_o 4_o 9_o 6._o which_o signify_v the_o former_a give_v number_n 3496_o and_o this_o number_n stand_v against_o the_o figure_n 1_o on_o the_o side_n of_o the_o tabulat_fw-la then_o i_o say_v that_o the_o figure_n in_o the_o next_o row_n stand_v against_o the_o figure_n â_o of_o the_o tabulat_fw-la and_o double_v thereunto_o which_o i_o thus_o prove_v repair_v to_o the_o rod_n as_o they_o lie_v upon_o the_o tabulat_fw-la and_o in_o that_o row_n which_o lie_v against_o the_o figure_n 2_o you_o shall_v find_v in_o the_o first_o half_a rhomboyade_n towards_o your_o right_a hand_n where_o by_o rule_n 1_o you_o must_v begin_v the_o figure_n 2_o wherefore_o set_v down_o with_o your_o pen_n upon_o paper_n the_o figure_n 2._o in_o the_o next_o rhomboyade_n in_o the_o same_o row_v you_o shall_v find_v 8_o and_o 1_o which_o add_v make_v 9_o set_z down_z 9_o on_o the_o left_a hand_n of_o 2_o in_o the_o next_o rhombus_fw-la you_o shall_v find_v 8_o and_o 1_o again_o which_o be_v 9_o also_o set_v down_o 9_o on_o the_o left_a hand_n of_o the_o other_o and_o in_o the_o last_o rhomboyade_n you_o shall_v find_v only_o 6_o wherefore_o set_v down_o 6_o on_o the_o left_a hand_n of_o 9_o so_o have_v you_o in_o all_o 6992_o which_o be_v double_a to_o 3496._o again_o the_o figure_n in_o the_o row_n which_o stand_v against_o the_o figure_n 3_o in_o the_o tabulat_fw-la be_v triple_a to_o 3496_o for_o in_o the_o first_o half_a rhomboyade_n towards_o your_o right_a hand_n you_o have_v 8_o set_v down_o 8._o in_o the_o next_o rhom_n you_o have_v 7_o and_o 1_o which_o be_v 8_o set_z down_z 8_o again_z in_o the_o next_o you_o have_v 2_o and_o 2_o which_o be_v 4_o set_v down_o 4._o in_o the_o next_o rhom_n you_o have_v 9_o and_o 1_o which_o make_v 10_o set_v down_o 0_o and_o carry_v 1_o but_o it_o be_v the_o last_o rhom_n and_o because_o there_o be_v never_o another_o to_o carry_v the_o 1_o unto_o you_o must_v therefore_o set_v it_o down_o so_o have_v you_o this_o number_n 10488_o which_o be_v triple_a to_o 3496._o again_o the_o figure_n stand_v against_o 4_o in_o the_o tabulat_fw-la be_v quadruple_a to_o 3496_o for_o in_o the_o half_a rhom_n you_o have_v 4_o set_v it_o down_o in_o the_o next_o 6_o and_o 2_o which_o be_v 8_o set_z that_o down_o in_o the_o next_o 6_o and_o 3_o which_o be_v 9_o set_z that_o down_o in_o the_o next_o 2_o and_o 1_o which_o be_v 3_o set_z that_o down_o and_o in_o the_o last_o half_a rhom_n you_o have_v 1_o which_o also_o set_v down_o so_o have_v you_o 13984_o which_o be_v quadruple_a to_o 3496._o also_o the_o figure_n against_o 5_o in_o the_o tabulat_fw-la the_o first_o be_v a_o cypher_n therefore_o put_v down_o 0_o the_o next_o be_v 5_o and_o 3_o which_o be_v 8_o set_z down_z 8_o the_o next_o be_v 0_o and_o 4_o set_z down_z 4_o the_o next_o be_v 5_o and_o 2_o that_o be_v 7_o set_z down_z 7_o and_o the_o last_o be_v 1_o therefore_o set_v down_o 1_o so_o have_v you_o in_o all_o 17480_o which_o be_v quintuple_a to_o 3496._o against_o 6_o in_o the_o tabulat_fw-la you_o have_v in_o the_o first_o place_n 6_o set_v it_o down_o then_o in_o the_o next_o 4_o and_o 3_o that_o be_v 7_o set_z that_o down_o in_o the_o next_o 4_o and_o 5_o that_o be_v 9_o set_z 9_o down_o in_o the_o next_o you_o have_v 8_o and_o 2_o that_o be_v 10_o set_v down_o 0_o and_o carry_v 1_o to_o the_o next_o rhom_n where_o you_o find_v only_o 1_o to_o which_o add_v the_o 1_o which_o you_o carry_v from_o the_o rhom_n before_o and_o it_o make_v 2_o set_z down_z 2_o so_o have_v you_o 20976_o which_o be_v six_o time_n 3496._o against_o 7_o in_o the_o tabulat_fw-la you_o have_v first_o 2_o set_v it_o down_o then_o 3_o and_o 4_o which_o be_v 7_o set_z 7_o down_o in_o the_o next_o 8_o and_o 6_o which_o be_v 14_o which_o be_v above_o 10_o set_z down_z 4_o and_o carry_v 1_o to_o the_o next_o rhom_n where_o you_o have_v 2_o and_o 1_o which_o be_v 3_o and_z 1_o which_o you_o carry_v make_v 4_o set_z down_z 4_o then_o in_o the_o last_o place_n you_o have_v only_o 2_o which_o set_v down_o so_o have_v you_o in_o all_o 24472_o which_o be_v septuple_a to_o 3496_o or_o seven_o time_n as_o much_o against_o 8_o in_o the_o tabulat_fw-la you_o have_v first_o 8_o which_o set_v down_o then_o 2_o and_o 4_o which_o be_v 6_o set_z 6_o down_o then_o 2_o and_o 7_o which_o be_v 9_o set_z 9_o down_o then_o 4_o and_o 3_o which_o be_v 7_o set_z 7_o down_o and_o last_o 2_o set_z that_o down_o so_o have_v you_o 27968_o which_o be_v octuple_a to_o 3496_o or_o eight_o time_n as_o much_o last_o against_o 9_o in_o the_o tabulat_fw-la you_o have_v in_o the_o first_o place_n 4_o set_z that_o down_o in_o the_o next_o you_o have_v 1_o and_o 5_o which_o be_v 6_o set_z 6_o down_o in_o the_o next_o place_n you_o have_v 6_o and_o 8_o which_o be_v 14_o set_z down_z 4_o and_o carry_v 1_o to_o the_o next_o rhom_n where_o you_o find_v 7_o and_o 3_o that_o be_v 10_o which_o with_o 1_o which_o you_o carry_v make_v 11_o set_z down_z 1_o and_o carry_v 1_o to_o the_o next_o rhom_n where_o you_o find_v only_o 2_o and_o the_o 1_o carried_z makes_z 3_o therefore_o set_v down_o 3_o and_o so_o you_o have_v 31464_o which_o be_v noncuple_n to_o 3496_o or_o nine_o time_n as_o much_o as_o the_o tabulated_a number_n thus_o have_v i_o give_v you_o example_n in_o show_v you_o how_o the_o number_n upon_o the_o rod_n be_v to_o be_v read_v and_o write_v down_o and_o in_o the_o delivery_n of_o this_o example_n i_o have_v make_v the_o whole_a work_n which_o be_v to_o follow_v so_o plain_a and_o easy_a that_o the_o mean_a capacity_n i_o think_v if_o he_o can_v but_o tell_v his_o figure_n and_o add_v any_o two_o figure_n together_o he_o may_v by_o this_o here_o deliver_v read_v or_o write_v down_o any_o number_n that_o can_v be_v tabulate_v and_o that_o you_o may_v thorough_o understand_v this_o chapter_n before_o you_o proceed_v further_o i_o will_v give_v you_o the_o product_n of_o 7009078_o multiply_v by_o all_o the_o nine_o digit_n which_o i_o will_v have_v yourself_o to_o tabulate_v and_o see_v if_o you_o find_v your_o work_n by_o your_o rod_n to_o agree_v with_o those_o which_o be_v here_o write_v which_o number_n if_o they_o do_v you_o need_v not_o scruple_n at_o the_o most_o difficult_a that_o can_v be_v propose_v to_o you_o therefore_o study_v it_o and_o try_v it_o 7009078_o be_v multiply_v by_o 2_o 3_o 4_o 5_o 6_o 7_o 8_o 9_o produce_v 7009078_o 14018156_o 21027234_o 28036312_o 35045390_o 42054468_o 49063546_o 56072624_o 63081702_o thus_o have_v i_o sufficient_o describe_v these_o rod_n and_o the_o manner_n of_o number_v upon_o they_o and_o now_o i_o think_v it_o time_n to_o apply_v they_o to_o that_o use_n for_o which_o they_o be_v intend_v namely_o the_o more_o difficult_a part_n of_o arithmetic_n as_o multiplication_n division_n and_o extraction_n of_o root_n but_o first_o let_v i_o give_v you_o a_o admonition_n concern_v addition_n and_o substraction_n whereas_o it_o be_v the_o difficult_a operaration_n of_o arithmetic_n which_o by_o the_o benefit_n of_o these_o rod_n the_o inventor_n chief_o aim_v at_o of_o which_o kind_n he_o esteem_v multiplication_n division_n and_o extraction_n of_o the_o square_a and_o cube_n root_n he_o omit_v to_o say_v any_o thing_n concern_v addition_n and_o substraction_n in_o thing_n obvious_a to_o every_o tyro_n he_o therefore_o omit_v they_o begin_v to_o show_v the_o use_n of_o his_o rod_n in_o multiplication_n who_o method_n i_o shall_v here_o follow_v chap._n vi_o multiplication_n by_o the_o rods._n in_o multiply_v by_o the_o rod_n you_o be_v to_o consider_v as_o in_o vulgar_a arithmetic_n three_o term_n thing_n or_o number_n viz._n 1._o the_o multiplicand_a which_o be_v the_o number_n
set_v down_o second_o look_v the_o product_n of_o 3496_o by_o 8_o your_o second_a figure_n and_o you_o shall_v find_v it_o to_o be_v 27968_o which_o must_v not_o be_v set_v down_o as_o in_o the_o other_o first_o example_n but_o as_o you_o see_v it_o in_o this_o 8_o the_o first_o figure_n thereof_o must_v be_v set_v one_o place_n forward_o towards_o the_o right_a hand_n as_o in_o the_o other_o it_o be_v set_v a_o place_n backward_o towards_o the_o left_a last_o seek_v in_o your_o rod_n for_o the_o product_n of_o 3496_o by_o 9_o your_o last_o figure_n and_o you_o shall_v find_v it_o to_o be_v 13984_o which_o set_v under_o the_o other_o two_o number_n yet_o one_o place_n more_o to_o the_o right_a hand_n so_o a_o line_n be_v draw_v under_o and_o these_o three_o number_n add_v together_o produce_v 1709544_o equal_a to_o that_o in_o the_o first_o example_n and_o that_o you_o may_v the_o better_o see_v the_o difference_n of_o the_o work_n i_o have_v set_v they_o one_o by_o the_o other_o as_o in_o the_o first_o example_n as_o in_o this_o example_n one_o example_n more_o in_o multiplication_n which_o shall_v be_v for_o advertisement_n and_o direction_n i_o will_v give_v and_o so_o conclude_v multiplication_n i_o say_v in_o the_o general_a rule_n for_o work_v of_o multiplication_n at_o the_o begin_n of_o this_o chapter_n that_o it_o matter_a not_o which_o of_o your_o number_n be_v make_v the_o multiplicand_a or_o which_o the_o multiplier_n of_o which_o i_o will_v here_o give_v you_o a_o precedent_n where_o the_o lesser_a number_n shall_v be_v tabulate_v and_o the_o great_a number_n only_o set_v down_o and_o i_o will_v work_v it_o here_o according_a to_o this_o last_o way_n of_o multiplication_n and_o the_o example_n shall_v be_v as_o follow_v example_n let_v it_o be_v require_v to_o multiply_v 868437_o by_o 3496_o and_o let_v 3496_o the_o lesser_a number_n be_v tabulate_v let_v the_o number_n be_v set_v as_o you_o here_o see_v than_o 3496_o be_v tabulate_v begin_v with_o the_o first_o figure_n towards_o the_o leave_v hand_n of_o your_o multiplier_n which_o here_o be_v 8_o and_o upon_o your_o rod_n find_v the_o product_n of_o 3496_o multiply_v by_o 8_o which_o be_v 27968_o set_v that_o down_o under_o the_o line_n then_o find_v the_o product_n of_o 3496_o by_o 6_o the_o second_o figure_n of_o your_o multiplier_n and_o you_o shall_v find_v that_o to_o be_v 20976_o set_v this_o number_n under_o the_o former_a one_o place_n more_o towards_o the_o right_a hand_n again_o the_o three_o figure_n of_o your_o product_n be_v 8_o who_o product_n be_v 27968_o as_o before_o set_z that_o under_o the_o other_o still_o one_o place_n more_o to_o the_o right_a hand_n in_o this_o manner_n do_v with_o the_o other_o figure_n of_o the_o multiplier_n as_o 4_o the_o next_o figure_n who_o product_n be_v 13984_o which_o also_o set_v down_o a_o place_n forward_o so_o also_o the_o product_n of_o 3_o which_o be_v 10488_o which_o set_v down_o and_o last_o of_o 7_o which_o be_v 24472._o all_z these_o product_n be_v set_v down_o in_o the_o order_n as_o you_o see_v they_o in_o the_o margin_n if_o you_o add_v they_o together_o the_o sum_n of_o they_o will_v be_v 3036055752_o which_o be_v the_o product_n of_o 3496_o multiply_v by_o 868437_o the_o lesser_a number_n be_v tabulate_v other_o way_n of_o multiplication_n i_o can_v have_v add_v but_o these_o i_o esteem_v sufficient_a chap._n vii_o division_n by_o the_o rods._n as_o in_o multiplication_n so_o in_o division_n there_o be_v three_o number_n term_n or_o thing_n require_v viz._n 1._o the_o dividend_n or_o number_n to_o be_v divide_v 2._o the_o divisor_n or_o number_n by_o which_o the_o dividend_n be_v divide_v and_o 3._o the_o quotient_a which_o be_v the_o number_n issue_v from_o the_o dividend_n be_v divide_v by_o the_o divisor_n and_o this_o quotient_n do_v always_o consist_v of_o so_o many_o unite_v as_o the_o divisor_n be_v time_n contain_v in_o the_o dividend_n thus_o much_o for_o the_o definition_n of_o division_n now_o let_v we_o come_v to_o the_o practice_n of_o it_o by_o the_o rod_n to_o perform_v which_o this_o be_v the_o rule_n tabulate_v the_o divisor_n which_o be_v always_o the_o lesser_a number_n of_o the_o two_o give_v and_o set_v down_o the_o dividend_n and_o set_v the_o divisor_n on_o the_o left_a hand_n and_o draw_v a_o crooked_a line_n on_o the_o right_a hand_n for_o your_o quotient_n as_o in_o common_a arithmetic_n then_o look_v upon_o your_o tabulated_a rod_n always_o for_o the_o number_n less_o than_o the_o number_n in_o the_o first_o figure_n of_o your_o dividend_n and_o what_o figure_n stand_v against_o that_o number_n on_o the_o edge_n of_o your_o tabulat_fw-la must_v be_v the_o figure_n you_o must_v put_v in_o your_o quotient_n and_o that_o number_v you_o must_v always_o subtract_v from_o the_o figure_n of_o your_o dividend_n and_o to_o the_o remainder_n add_v another_o figure_n so_o proceed_v from_o figure_n to_o figure_n till_o your_o division_n be_v whole_o end_v example_n let_v it_o be_v require_v to_o divide_v 1709544_o by_o 3496._o have_v tabulate_v 3496_o set_v down_o your_o dividend_n your_o divisor_n on_o the_o left_a hand_n thereof_o and_o a_o crooked_a line_n for_o the_o quotient_a on_o the_o right_a hand_n thereof_o as_o by_o the_o rule_n precede_v you_o be_v direct_v and_o as_o you_o see_v do_v in_o the_o example_n adjoin_v and_o because_o at_o your_o first_o set_v down_o of_o your_o divisor_n 3496_o it_o will_v reach_v if_o it_o be_v set_v under_o your_o dividend_n 1709544_o as_o far_o as_o the_o figure_n 5_o therefore_o under_o the_o figure_n 5_o make_v a_o prick_n to_o intimate_v how_o far_o you_o be_v go_v on_o in_o your_o work_n and_o under_o this_o prick_n draw_v a_o line_n quite_o under_o your_o dividend_n then_o be_v your_o sum_n set_v down_o ready_a for_o work_v and_o will_v appear_v as_o here_o you_o see_v your_o sum_n thus_o prepare_v ask_v how_o often_o can_v you_o have_v 3496_o in_o 17095_o look_v in_o your_o tabulated_a rod_n for_o 17095_o which_o you_o can_v there_o find_v but_o the_o near_a number_n thereunto_o among_o the_o rod_n which_o be_v less_o than_o 17095_o for_o you_o must_v always_o take_v a_o less_o number_n be_v 13984_o which_o number_n stand_v against_o the_o figure_n 4_o in_o the_o tabulat_fw-la wherefore_o set_v 4_o in_o your_o quotient_n and_o 13984_o under_o the_o line_n and_o subtract_v 13984_o from_o 17095_o and_o there_o will_v remain_v 3111_o so_o be_v the_o first_o part_n of_o your_o division_n end_v and_o your_o work_n will_v stand_v thus_o then_o make_v another_o prick_n under_o 4_o the_o next_o figure_n of_o your_o dividend_n so_o will_v the_o remain_a number_n be_v 31114_o then_o look_v among_o your_o rod_n for_o the_o number_n 31114_o or_o the_o near_o less_o than_o it_o and_o the_o near_a less_o you_o shall_v find_v to_o be_v 27968_o which_o stand_v against_o 8_o in_o your_o tabulat_fw-la put_v 8_o in_o your_o quotient_n and_o set_v 27968_o under_o 31114_o and_o subtract_v 27968_o from_o 31114_o so_o will_v there_o remain_v 3146_o which_o set_v over_o head_n so_o be_v the_o second_o part_n of_o your_o division_n end_v and_o your_o work_n will_v appear_v thus_o last_o make_v another_o prick_n under_o the_o next_o figure_n of_o your_o dividend_n which_o be_v 4_o also_o make_v the_o remain_a number_n to_o be_v 31464_o seek_v among_o your_o tabulated_a rod_n for_o this_o number_n or_o the_o near_a less_o but_o look_v you_o shall_v find_v the_o very_a number_n against_o which_o stand_v on_o your_o tabulat_fw-la the_o figure_n 9_o set_z 9_o in_o the_o quotient_a and_o the_o number_n 31464_o under_o the_o line_n and_o subtract_v it_o from_o 31464_o the_o remainder_n which_o stand_v above_o the_o line_n and_o nothing_o remain_v and_o be_v there_o be_v never_o another_o figure_n in_o your_o dividend_n your_o division_n be_v end_v and_o your_o work_n will_v stand_v thus_o and_o 3496_o be_v contain_v in_o 1709544_o 489_o time_n another_o example_n and_o by_o another_o way_n of_o division_n let_v it_o be_v require_v to_o divide_v 912456_o by_o 3496_o set_v down_o your_o dividend_n and_o divisor_n draw_v a_o crooked_a line_n for_o your_o quotient_n and_o also_o make_v a_o prick_n under_o the_o four_o figure_n of_o your_o dividend_n and_o draw_v a_o line_n under_o your_o dividend_n so_o be_v your_o sum_n prepare_v to_o be_v divide_v and_o will_v stand_v thus_o then_o your_o divisor_n 3496_o be_v tabulate_v look_v among_o your_o rod_n for_o the_o near_a number_n to_o 9124_o which_o be_v less_o and_o you_o shall_v find_v it_o to_o be_v 6992_o against_o which_o stand_v on_o your_o tabulat_fw-la the_o figure_n 2_o set_z 2_o in_o the_o quotient_a and_o this_o number_n under_o the_o line_n and_o subtract_v it_o from_o 9124_o and_o there_o
serve_v for_o the_o square_a root_n have_v the_o word_n square_n write_v over_o head_n that_o for_o the_o cube_n root_n have_v cube_n write_v over_o head_n thus_o have_v give_v you_o the_o fabric_n and_o inscription_n of_o these_o rod_n i_o will_v now_o show_v you_o their_o use_n and_o first_o concern_v the_o extract_a of_o the_o square-root_n in_o extract_v of_o the_o square-root_n you_o must_v as_o in_o common_a arithmetic_n when_o you_o have_v set_v down_o your_o number_n make_v a_o prick_n under_o the_o first_o figure_n towards_o your_o right_a hand_n and_o so_o successive_o under_o every_o second_o figure_n then_o under_o those_o prick_n draw_v two_o line_n parallel_v whereinto_o set_v the_o figure_n of_o your_o root_n as_o you_o find_v they_o your_o number_n be_v thus_o place_v and_o prick_v as_o before_o be_v direct_v and_o as_o in_o the_o follow_a example_n you_o see_v do_v you_o may_v proceed_v to_o extract_n the_o root_n thereof_o as_o follow_v example_n 1._o let_v it_o be_v require_v to_o find_v the_o square-root_n of_o this_o number_n 12418576_o first_o make_v a_o prick_n under_o 6_o another_z under_z 5_o another_z under_z 1_o and_z another_z under_z 2_o under_o which_o point_n draw_v two_o line_n in_o which_o you_o must_v place_v your_o root_n and_o then_o will_v your_o number_n stand_v thus_o take_v the_o rod_n for_o extract_v of_o the_o square-root_n and_o look_v in_o the_o first_o row_n or_o column_n thereof_o for_o the_o near_a number_n you_o can_v there_o find_v less_o than_o 12_o which_o be_v as_o far_o as_o the_o first_o prick_n in_o your_o number_n reach_v and_o you_o shall_v find_v 9_o against_o which_o in_o the_o three_o column_n you_o shall_v find_v 3_o set_z 3_o under_o the_o first_o point_n between_o the_o line_n and_o 9_o under_o the_o line_n and_o substract_v 9_o from_o 12_o there_o will_v remain_v 3_o which_o set_v over_o 12_o so_o will_v your_o number_n stand_v thus_o then_o in_o the_o middle_a column_n of_o your_o rod_n between_o 9_o and_o 3_o there_o stand_v 6_o take_v therefore_o one_o of_o your_o rod_n which_o have_v 6_o at_o the_o top_n thereof_o and_o lay_v it_o upon_o your_o tabulat_fw-la by_o the_o left_a side_n of_o your_o square_a rod_n then_o be_v there_o be_v 341_o to_o the_o next_o prick_n seek_v the_o near_a number_n less_o upon_o your_o two_o rod_n and_o you_o shall_v find_v the_o next_o less_o to_o be_v 325_o against_o which_o in_o the_o last_o column_n of_o your_o square_a rod_n stand_v 5_o therefore_o place_v 5_o under_o your_o second_o prick_n and_o set_v 325_o under_o 341_o and_o substract_v it_o from_o 341_o there_o will_v remain_v 16_o which_o set_v over_o head_n then_o will_v the_o sum_n appear_v thus_o and_o in_o the_o middle_a column_n of_o your_o square_a rod_n against_o this_o 5_o there_o stand_v 10_o for_o this_o 10_o you_o shall_v take_v a_o rod_n that_o have_v 10_o at_o the_o top_n but_o be_v there_o be_v no_o such_o take_v therefore_o one_o that_o have_v a_o cypher_n and_o place_n that_o between_o your_o square_a rod_n and_o your_o rod_n of_o 6_o and_o change_v your_o rod_n 6_o for_o one_o of_o 7_o then_o shall_v you_o have_v upon_o your_o tabulat_v one_o rod_n of_o 7_o another_z of_o 0_o and_o your_o square_a rod._n thus_o must_v you_o always_o do_v when_o the_o number_n in_o the_o middle_a column_n exceed_v 10._o then_o look_v upon_o your_o sum_n you_o shall_v find_v 1685_o to_o your_o three_o prick_n look_v therefore_o upon_o your_o rod_n for_o the_o near_o less_o number_n which_o you_o shall_v find_v to_o be_v 1404_o against_o which_o stand_v 2_o in_o the_o last_o column_n set_v 2_o between_o the_o line_n under_o the_o three_o prick_n and_o 1404_o under_o 1685_o and_o substract_v it_o from_o 1685_o and_o there_o will_v remain_v 281_o which_o place_n above_o so_o will_v your_o sum_n stand_v thus_o and_o because_o the_o number_n stand_v against_o in_o the_o middle_a column_n of_o your_o square_a rod_n between_o 1404_o and_o 2_o be_v 4_o set_v 4_o under_o your_o last_o prick_n and_o take_v a_o rod_n of_o 4_o and_o put_v it_o between_o your_o square_a rod_n and_o your_o rod_n of_o 0_o and_o because_o 28176_o remain_v upon_o your_o sum_n to_o the_o last_o prick_n look_v upon_o your_o rod_n for_o the_o near_a number_n thereunto_o and_o you_o shall_v find_v the_o very_a number_n itself_o to_o stand_v against_o the_o figure_n 4_o set_v therefore_o 28176_o below_o and_o subtract_v it_o from_o that_o above_o and_o there_o will_v remain_v nothing_o which_o denote_v the_o number_n 12418576_o to_o be_v a_o square_a number_n and_o the_o root_n thereof_o to_o be_v 3524_o and_o the_o work_n finish_v will_v stand_v thus_o this_o sum_n have_v it_o be_v wrought_v by_o that_o second_o way_n of_o division_n which_o i_o show_v in_o chapter_n 7_o it_o will_v stand_v as_o follow_v square_a root_n caution_n if_o at_o any_o time_n you_o look_v for_o the_o remainder_n upon_o your_o rod_n and_o you_o can_v find_v it_o there_o you_o must_v then_o place_v a_o cypher_n between_o the_o line_n and_o proceed_v to_o the_o next_o figure_n as_o by_o try_v this_o other_o example_n which_o i_o have_v insert_v for_o practice_n will_v appear_v another_o example_n add_v for_o practice_n chap._n x._o concern_v the_o extraction_n of_o the_o cube_n root_n there_o be_v somewhat_o more_o difficulty_n in_o extract_v of_o the_o cube_n then_o of_o the_o square_a root_n wherefore_o before_o i_o come_v to_o example_n i_o will_v deliver_v the_o manner_n of_o the_o operation_n together_o with_o such_o caution_n as_o be_v to_o be_v observe_v in_o the_o performance_n thereof_o all_o which_o immediate_o follow_v in_o this_o general_z rule_n write_v down_o the_o number_n who_o cube_n root_n you_o be_v to_o extract_n and_o under_o the_o first_o figure_n towards_o the_o right_a hand_n make_v a_o prick_n or_o point_n and_o so_o under_o every_o three_o figure_n towards_o the_o left_a hand_n till_o you_o come_v to_o the_o end_n of_o your_o number_n under_o these_o prick_n draw_v two_o parallel_n line_n as_o you_o do_v in_o extract_v the_o square_a root_n between_o which_o line_n you_o be_v to_o place_v the_o figure_n of_o your_o root_n as_o you_o find_v they_o then_o beginning_n at_o the_o figure_n or_o figure_n of_o the_o left_a hand_n prick_v and_o go_v forward_o towards_o the_o right_a hand_n extract_v by_o help_n of_o the_o rod_n for_o extract_v tract_v the_o cube_n root_n their_o root_n or_o if_o the_o true_a number_n be_v not_o on_o the_o plate_n than_o the_o near_a less_o and_o place_v this_o root_n which_o never_o exceed_v one_o figure_n between_o the_o line_n and_o under_o its_o point_n and_o take_v its_o cube_n from_o the_o uppermost_a figure_n which_o stand_v before_o or_o leftwards_o of_o the_o first_o point_n and_o note_v the_o remainder_n above_o second_o keep_v the_o triple_a of_o this_o root_n sound_n in_o the_o head_n or_o top_n of_o the_o rod_n and_o triple_a the_o square_n of_o the_o same_o root_n and_o set_v this_o triple_a on_o the_o head_n of_o the_o rod_n and_o apply_v it_o leftwards_o of_o the_o cubick_a rod_n and_o the_o reserve_a rod_n or_o rod_n right-ward_n the_o cubick_a rod_n be_v in_o the_o midst_n between_o they_o and_o out_o of_o the_o left_a hand_n rod_n and_o the_o cubick_a rod_n together_o pick_z or_o find_v out_o the_o multiple_n or_o next_z less_o number_n than_o the_o figure_n precede_v the_o second_o point_n which_o write_v apart_o in_o a_o pater_fw-la and_o note_v its_o quotume_n over_o its_o utmost_a right-hand_a figure_n and_o write_v the_o square_n of_o that_o quotume_n left-ward_n from_o the_o quotume_n itself_o even_o in_o that_o order_n as_o you_o find_v they_o in_o your_o cubick_a rod_n and_o under_o every_o several_a figure_n of_o this_o square_a write_v their_o multiple_n find_v right-ward_n even_o such_o as_o the_o figure_n themselves_o do_v show_v so_o that_o every_o multiple_n may_v end_v under_o its_o figure_n or_o quotume_n then_o add_v together_o these_o multiple_n crosswise_o and_o take_v their_o sum_n from_o the_o figure_n forego_v the_o second_o point_n and_o write_v the_o remainder_n over_o they_o but_o write_v the_o right-hand_a quotume_n before_o note_v under_o the_o second_o point_n between_o the_o line_n for_o the_o second_o figure_n or_o quotume_n of_o the_o root_n and_o so_o be_v perform_v the_o operation_n of_o the_o second_o point_n which_o you_o must_v repeat_v through_o the_o several_a point_n even_o to_o the_o last_o but_o in_o the_o practice_n by_o this_o rule_n you_o may_v sometime_o be_v at_o a_o stand_n wherefore_o to_o this_o general_z rule_v that_o there_o may_v be_v no_o obstacle_n i_o will_v add_v these_o two_o calltion_n 1._o caution_n but_o in_o all_o operation_n and_o point_n it_o must_v be_v observe_v that_o if_o no_o
will_v remain_v 2132_o to_o which_o number_n add_v the_o next_o figure_n of_o your_o dividend_n namely_o 5_o and_o it_o make_v 21325_o under_o which_o number_n draw_v a_o line_n then_o will_v your_o sum_n stand_v thus_o then_o among_o your_o rod_n seek_v the_o near_a number_n to_o 21325_o and_o you_o shall_v find_v 20976_o to_o be_v the_o near_a number_n less_o against_o which_o in_o your_o tabulat_fw-la stand_v 6_o set_z 6_o in_o the_o quotient_a and_o 20976_o under_o the_o line_n substract_v it_o from_o 21325_o which_o when_o you_o have_v do_v there_o will_v remain_v 349_o to_o 349_o add_v the_o next_o figure_n in_o your_o dividend_n which_o be_v 6_o your_o last_o figure_n and_o it_o make_v 3496_o under_o which_o draw_v a_o line_n and_o your_o work_n will_v stand_v as_o here_o you_o see_v this_o do_v look_v among_o your_o rod_n for_o the_o near_a number_n to_o 3496_o and_o you_o shall_v find_v the_o exact_a number_n at_o the_o top_n of_o the_o rod_n against_o which_o stand_v the_o figure_n 1_o on_o the_o tabulat_fw-la set_v 1_o in_o the_o quotient_a and_o subtract_v 3496_o from_o 3496_o the_o remainder_n be_v nothing_o and_o so_o be_v your_o division_n end_v the_o work_n stand_v thus_o and_o 3496_o the_o divisor_n be_v contain_v in_o 912456_o the_o dividend_n 261_o time_n a_o three_o example_n ready_a wrought_v by_o the_o last_o and_o best_a way_n of_o division_n i_o will_v only_o set_v it_o down_o ready_o wrought_v leave_v the_o practice_n of_o it_o to_o yourself_o let_v it_o be_v require_v to_o divide_v 73020506_o by_o 3496._o this_o sum_n thus_o divide_v produce_v in_o the_o quotient_a 20886_o and_o 3050_o remain_v so_o that_o the_o quotient_a with_o fraction_n and_o all_o be_v which_o show_v that_o 3496_o the_o divisor_n be_v contain_v in_o 73020506_o the_o dividend_n 20886_o time_n and_o 3050_o remain_v this_o example_n well_o practise_v together_o with_o they_o before-going_a be_v sufficient_a instruction_n for_o any_o student_n whatever_o and_o he_o that_o can_v perform_v these_o need_n not_o despair_v of_o the_o most_o difficult_a that_o can_v be_v propose_v and_o so_o i_o conclude_v with_o division_n chap._n viii_o concern_v the_o rule_n of_o three_o or_o golden_a rule_n both_o direct_a and_o reverse_v or_o reciprocal_a to_o discourse_n of_o this_o rule_n at_o large_a be_v to_o run_v into_o a_o labyrinth_n for_o it_o be_v the_o performance_n of_o work_a multiplication_n and_o division_n by_o the_o rod_n that_o be_v here_o aim_v at_o and_o he_o that_o can_v multiply_v and_o divide_v may_v command_v this_o golden_a rule_n wherefore_o i_o will_v show_v you_o the_o nature_n or_o order_n of_o place_v the_o number_n and_o also_o the_o manner_n of_o work_v a_o example_n in_o either_o of_o they_o the_o rule_n of_o three_o be_v that_o rule_n which_o teach_v by_o have_v three_o number_n in_o proportion_n one_o to_o another_o give_v to_o find_v a_o four_o which_o shall_v be_v in_o proportion_n to_o they_o also_o in_o this_o rule_n direct_v the_o four_o number_n which_o be_v seek_v be_v to_o have_v the_o same_o proportion_n to_o the_o three_o as_o the_o second_o number_n have_v to_o the_o first_o as_o if_o the_o three_o number_n give_v be_v 2-4_a and_o 8_o say_v as_o 2_o be_v to_z 4_o so_o be_v 8_o to_o what_o multiply_v 4_o by_o 8_o that_o be_v the_o second_o number_n by_o the_o three_o and_o the_o product_n will_v be_v 32_o which_o divide_v by_o 2_o the_o first_o number_n the_o quotient_n will_v be_v 16_o which_o be_v the_o four_o number_n in_o proportion_n to_o the_o three_o as_o the_o second_o be_v to_o the_o first_o for_z as_o 4_o the_o second_o number_n contain_v 2_o the_o first_o number_n twice_o so_o 16_o the_o four_o number_n contain_v 8_o the_o three_o number_n twice_o also_o but_o in_o the_o reciprocal_a rule_n of_o three_o there_o the_o proportion_n be_v not_o as_o the_o first_o to_o the_o second_o so_o the_o three_o to_o the_o four_o but_o as_o the_o first_o be_v to_o the_o three_o so_o be_v the_o second_o to_o the_o four_o as_o if_o the_o number_n be_v 3_o 4_o and_z 6_o say_v as_o 3_o the_o first_o number_n be_v to_o 6_o the_o three_o number_n so_o be_v 4_o the_o second_o number_n to_o what_o multiply_v 4_o the_o second_o number_n by_o 3_o the_o first_o number_n the_o product_n be_v 12_o which_o divideby_o 6_o the_o three_o number_n and_o the_o quotient_n will_v be_v 2_o for_o as_o 6_o the_o three_o number_n contain_v 3_o the_o first_o number_n twice_o so_z 4_o the_o second_o number_n contain_v 2_o the_o four_o number_n twice_o also_o and_o in_o this_o consist_v the_o difference_n between_o the_o direct_a and_o reciprocal_a rule_n of_o three_o a_o question_n in_o each_o rule_n 1._o in_o the_o direct_a rule_n if_o four_o man_n eat_v two_o peck_v of_o corn_n in_o one_o week_n how_o many_o peck_v will_v serve_v a_o hundred_o man_n the_o same_o time_n man_n peck_v men._n 4_o 2_o 100_o multiply_v 2_o the_o second_o number_n by_o 100_o the_o three_o number_n the_o product_n will_v be_v 200_o which_o divide_v by_o 4_o the_o first_o number_n and_o the_o quotient_n will_v be_v 50_o and_o so_o many_o peck_v will_v suffice_v 100_o man_n the_o same_o time_n 2._o in_o the_o reciprocal_a if_o twelve_o man_n do_v any_o piece_n of_o work_n in_o 8_o day_n how_o many_o man_n must_v be_v employ_v to_o do_v the_o same_o piece_n of_o work_n in_o 2_o day_n day_n man_n day_n 8_o 12_o 2._o multiply_v 8_o the_o first_o number_n by_o 12_o the_o second_o their_o product_n be_v 96_o which_o divide_v by_o 2_o the_o three_o number_n the_o quotient_n will_v be_v 48_o and_o so_o many_o man_n will_v do_v the_o same_o work_n in_o 2_o day_n for_o as_o 8_o day_n be_v to_o 2_o day_n so_o be_v 12_o man_n to_o 48_o man_n etc._n etc._n chap._n ix_o of_o the_o extraction_n of_o root_n the_o extraction_n of_o root_n which_o be_v the_o difficult_a part_n of_o multiplication_n and_o division_n be_v expeditious_o and_o certain_o perform_v by_o the_o rod_n for_o the_o easy_a and_o expedite_a performance_n of_o which_o there_o be_v two_o rod_n on_o purpose_n one_o for_o the_o square_n the_o other_o for_o the_o cube_n root_n of_o which_o i_o will_v speak_v first_o of_o their_o fabric_n second_o of_o their_o use_n of_o the_o fabric_n of_o the_o rod_n for_o extract_v of_o root_n of_o the_o same_o matter_n and_o of_o the_o same_o length_n and_o thickness_n of_o your_o other_o rod_n let_v there_o be_v make_v another_o rod_n but_o three_o time_n the_o breadth_n of_o the_o former_a the_o inscription_n on_o one_o side_n serve_v to_o extract_v the_o square_n and_o that_o on_o the_o other_o side_n for_o the_o cube_n root_n each_o of_o which_o be_v divide_v into_o three_o row_v or_o colume_n that_o which_o serve_v for_o the_o square_a root_n have_v in_o the_o top_n or_o uppermost_a square_n between_o the_o diagonal_a thereof_o these_o figure_n 0-1_a in_o the_o second_o 0-4_a in_o the_o three_o 0-9_a in_o the_o four_o 1-6_a in_o the_o five_o 2-5_a in_o the_o six_o 3-6_a in_o the_o seven_o 4-9_a in_o the_o eight_o 6-4_a and_o in_o the_o nine_o or_o lowermost_a 8â_a which_o be_v the_o square_a number_n belong_v to_o the_o nine_o digit_n in_o the_o second_o column_n of_o the_o same_o rod_n in_o the_o first_o square_n be_v inscribe_v 2_o in_o the_o second_o 4_o in_o the_o three_o 6_o in_o the_o four_o 8_o in_o the_o five_o 10_o in_o the_o six_o 12_o in_o the_o seven_o 14_o in_o the_o eight_o 16_o and_o in_o the_o nine_o 18._o in_o the_o last_o or_o three_o column_n there_o be_v the_o nine_o digit_n orderly_o descend_v namely_o 1_o 2_o 3_o 4_o 5_o 6_o 7_o 8_o 9_o this_o rod_n thus_o make_v be_v fit_v for_o the_o square_a root_n that_o which_o serve_v for_o the_o cube_n root_n have_v in_o the_o top_n or_o uppermost_a square_n of_o the_o first_o column_n towards_o the_o left_a hand_n between_o the_o diagonal_a thereof_o these_o figure_n 0-01_a in_o the_o second_o 0-08_a in_o the_o three_o 0-27_a in_o the_o four_o 0-64_a in_o the_o five_o 1-25_a in_o the_o six_o 2-16_a in_o the_o seven_o 3-43_a in_o the_o eight_o 5-12_a and_o in_o the_o nine_o 7-29_a which_o be_v cube_n number_n orderly_o descend_v the_o second_o column_n of_o this_o rod_n contain_v these_o square_a number_n 1_o 4_o 9_o 16_o 25_o 36_o 49_o 64_o 81_o orderly_o descend_v the_o three_o and_o last_o column_n of_o this_o rod_n have_v in_o it_o the_o nine_o digit_n 1_o 2_o 3_o 4_o 5_o 6_o 7_o 8_o 9_o orderly_o descend_v also_o this_o rod_n thus_o prepare_v and_o inscribe_v be_v fit_a for_o extract_v of_o the_o square_a and_o cube_n root_n a_o figure_n of_o either_o side_n whereof_o you_o have_v at_o the_o begin_n of_o the_o book_n that_o which_o