according_o as_o in_o the_o first_o example_n of_o the_o last_o rule_n at_o last_o i_o discover_v the_o four_o term_v require_v to_o fall_v upon_o the_o transversal_a of_o the_o 24_o fraction_n at_o 4_o digit_n and_o 4_o minim_n of_o the_o say_v 30_o interval_n viz._n at_o the_o point_n k_o so_o likewise_o if_o k_n be_v the_o first_o term_n h_o the_o second_o and_o g_o the_fw-fr three_o work_v towards_o the_o left_a hand_n f_z will_v be_v find_v to_o be_v the_o four_o etc._n etc._n v._o have_v three_o point_n give_v upon_o three_o several_a transversal_n to_o discover_v the_o transversall_a upon_o which_o the_o four_o term_n will_v fall_v and_o also_o the_o point_n of_o that_o transversall_a where_o that_o four_o term_n will_v bear_v like_o distance_n from_o the_o three_o point_n that_o the_o second_o bear_v from_o the_o first_o observe_v this_o direction_n have_v place_v as_o before_o three_o chesseman_n at_o the_o three_o give_v point_n place_z likewise_o three_o other_o plain_a chesseman_n upon_o the_o side_n rank_n of_o alphabet_n at_o the_o right_a end_n of_o the_o fraction_n or_o transversal_n whereupon_o the_o point_n give_v be_v situate_a respective_o this_o do_v by_o the_o second_o rule_n of_o this_o chapter_n find_v upon_o the_o say_a side-rank_n a_o four_o term_n to_o the_o three_o give_v which_o will_v lead_v you_o to_o the_o transversal_a upon_o which_o the_o four_o term_v require_v be_v to_o be_v find_v then_o proceed_v according_a to_o the_o direction_n of_o the_o last_o rule_n you_o will_v discover_v the_o four_o point_n or_o term_v you_o look_v for_o example_n if_o f_z g_o and_z h_z the_o point_n of_o the_o first_o example_n of_o the_o last_o rule_n be_v give_v viz._n upon_o the_o 5_o 19_o and_o 10_o fraction_n as_o before_o in_o this_o case_n i_o place_v a_o plain_a chesseman_n at_o the_o right_a end_n of_o the_o five_o fraction_n another_o at_o the_o same_o end_n of_o the_o ten_o fraction_n and_o a_o three_o at_o the_o like_a end_n of_o the_o nineteenth_o fraction_n and_o in_o work_v downward_o discover_v upon_o that_o side-rank_n of_o alphabet_n by_o the_o second_o rule_n of_o this_o chapter_n a_o four_o term_v correspondent_a to_o the_o other_o three_o give_v term_n which_o four_o term_n lead_v i_o to_o the_o 24_o fraction_n and_o transversal_a upon_o which_o the_o four_o term_n in_o question_n be_v situate_a and_o therefore_o proceed_v thereupon_o as_o in_o the_o first_o example_n of_o the_o last_o rule_n you_o will_v find_v the_o four_o term_v require_v in_o this_o example_n to_o fall_v upon_o the_o transversal_a of_o that_o 24_o fraction_n at_o four_o digit_n and_o four_o minim_n of_o the_o 30_o interval_n viz._n at_o the_o point_n k_o as_o before_o in_o like_a manner_n if_o k_n be_v the_o first_o term_n h_o the_o second_o and_o g_o the_fw-fr three_o in_o mount_v upward_o upon_o the_o side_n rank_n and_o proceed_v upon_o the_o table_n towards_o the_o left_a hand_n as_o i_o do_v before_o towards_o the_o right_a the_o four_o term_n will_v in_o that_o case_n be_v find_v to_o fall_v upon_o the_o transversal_a of_o the_o five_o fraction_n at_o four_o digit_n and_o four_o minim_n of_o the_o three_o interval_n sign_v by_o c_o viz._n at_o the_o point_n f._n so_o if_o g_o be_v the_o first_o term_n h_o the_o second_o and_o k_o the_o three_o the_o fraction_n or_o transversal_a of_o the_o four_o term_n be_v find_v upon_o the_o side_n rank_n and_o i_o guide_v my_o work_n upon_o the_o table_n towards_o the_o right_a hand_n the_o four_o term_n will_v fall_v upon_o the_o transversal_a of_o the_o 15_o fraction_n at_o 4_o digit_n and_o 4_o minim_n of_o the_o 3_o interval_n sign_v by_o c_o viz._n at_o the_o point_n l_o howbeit_o you_o be_v not_o to_o take_v that_o for_o the_o true_a point_n but_o because_o in_o that_o case_n you_o go_v beyond_o the_o table_n towards_o the_o right_a hand_n and_o for_o that_o the_o right_a end_n of_o the_o 15_o fraction_n be_v conceive_v to_o join_v with_o the_o left_a end_n of_o the_o 16_o fraction_n according_a to_o the_o direction_n of_o the_o 9_o 16_o and_o 25_o rule_n of_o the_o first_o chapter_n you_o be_v to_o take_v 4_o digit_n and_o 4_o minim_n of_o the_o transversal_a next_o under_o it_o in_o the_o same_o interval_n and_o so_o the_o true_a point_n require_v will_v be_v in_o that_o case_n find_v to_o reside_v upon_o the_o transversal_a of_o the_o 16_o fraction_n at_o 4_o digit_n and_o 4_o minim_n of_o the_o say_v three_o interval_n viz._n at_o the_o point_n m._n in_o like_a manner_n if_o the_o three_o term_n propound_v be_v h_o g_o and_o f_o and_o a_o four_o term_n be_v require_v answerable_a unto_o they_o in_o that_o case_n the_o proper_a fraction_n or_o transversal_a of_o that_o four_o term_n be_v discover_v upon_o the_o side_n rank_n &_o i_o proceed_v towards_o the_o left_a hand_n the_o four_o term_n will_v fall_v upon_o the_o transversal_a of_o the_o 14_o fraction_n at_o 4_o digit_n and_o 4_o minim_n of_o the_o 30_o interval_n viz._n at_o the_o point_n n._n howbeit_o as_o in_o the_o last_o aforego_v example_n you_o be_v not_o to_o take_v that_o for_o the_o true_a point_n but_o in_o that_o case_n because_o you_o go_v beyond_o the_o table_n towards_o the_o left_a hand_n and_o for_o that_o the_o left_a end_n of_o the_o 14_o fraction_n be_v conceive_v to_o join_v with_o the_o right_a end_n of_o the_o 13_o fraction_n according_a to_o the_o say_v 9_o 16_o and_o 25_o rule_n of_o the_o first_o chapter_n you_o be_v instead_o thereof_o to_o take_v 4_o digit_n and_o 4_o minim_n of_o the_o transversal_a next_o above_o it_o in_o the_o same_o interval_n and_o so_o true_a point_n require_v will_v be_v find_v to_o rest_v upon_o the_o transversal_a of_o the_o 13_o fraction_n at_o 4_o digit_n and_o 4_o minim_n of_o the_o say_v 30_o interval_n viz._n at_o the_o point_n p._n and_o here_o give_v i_o leave_v once_o for_o all_o to_o insert_v this_o direction_n that_o in_o the_o motion_n of_o a_o chesseman_n upon_o the_o table_n when_o you_o be_v constrain_v to_o over-shoot_a the_o table_n either_o on_o the_o right_n or_o left_a hand_n take_v the_o fraction_n next_o to_o it_o either_o above_o or_o below_o it_o viz._n if_o on_o the_o right_a hand_n than_o the_o fraction_n below_o it_o but_o if_o on_o the_o left_a hand_n then_o that_o above_o it_o as_o in_o the_o two_o last_o premise_v example_n you_o find_v it_o practise_v again_o if_o f_z be_v the_o first_o term_n g_o the_o second_n and_o k_o the_o three_o the_o four_o term_n will_v fall_v upon_o the_o three_o fraction_n at_o 4_o digit_n and_o 4_o minim_n of_o the_o three_o interval_n viz._n at_o the_o point_n q_o and_o in_o that_o case_n you_o do_v not_o only_o fall_v off_o at_o the_o low_a end_n of_o the_o side-rank_n take_v it_o again_o at_o the_o top_n but_o likewise_o overshoot_v the_o table_n upon_o the_o right_a hand_n and_o take_v it_o again_o upon_o the_o left_a and_o in_o that_o respect_n take_v not_o the_o fraction_n whereunto_o you_o be_v direct_v by_o the_o four_o term_v find_v in_o the_o side-rank_n but_o take_v the_o next_o under_o it_o on_o the_o other_o side_n if_o k_n be_v the_o first_o term_n h_o the_o second_o and_z f_z the_o three_o the_o four_o term_n will_v reside_v upon_o the_o 27_o fraction_n at_o 4_o digit_n and_o 4_o minim_n of_o the_o 30_o interval_n viz._n at_o the_o point_n r._n and_o in_o that_o case_n also_o you_o do_v not_o only_o mount_v off_o at_o the_o top_n of_o the_o side_n rank_n take_v it_o again_o at_o the_o low_a end_n but_o likewise_o over-shoot_a the_o table_n upon_o the_o left_a hand_n and_o take_v it_o again_o upon_o the_o right_n and_o in_o that_o regard_n also_o take_v not_o the_o fraction_n unto_o which_o you_o be_v direct_v by_o the_o four_o term_v find_v in_o the_o side_n rank_n but_o take_v the_o next_o above_o it_o according_a to_o the_o direction_n of_o the_o aforegoing_a example_n vi_o after_o the_o same_o manner_n may_v you_o also_o discover_v a_o three_o term_n to_o two_o term_n propound_v save_v only_o that_o in_o regard_n the_o second_o term_n do_v in_o a_o sort_n in_o that_o case_n represent_v the_o two_o middle_a term_n you_o be_v to_o double_v the_o digit_n and_o minim_n of_o the_o second_o term_n and_o then_o add_v they_o to_o the_o digit_n and_o minim_n of_o the_o first_o term_n to_o the_o end_n you_o may_v understand_v by_o that_o sum_n how_o far_o to_o advance_v the_o chesseman_n of_o the_o last_o term_n for_o example_n let_v f_o be_v the_o first_o term_n and_o g_o the_fw-fr second_o and_o let_v a_o three_o term_n be_v desire_v here_o chesseman_n be_v place_v at_o the_o term_n give_v and_o likewise_o upon_o the_o side_n rank_n at_o the_o
sell_v for_o 23_o s._n 6_o d._n the_o quarter_n prop._n 3._o one_o number_n be_v give_v to_o be_v multiply_v by_o another_o give_a number_n to_o find_v the_o product_n in_o multiplication_n there_o be_v four_o term_n geometrical_o proportional_a whereof_o the_o first_o be_v always_o a_o unity_n or_o 1_o the_o multiplicator_n and_o multiplicand_v be_v the_o two_o mean_n and_o the_o product_n be_v the_o four_o term_v demand_v for_o as_o 1_o be_v to_o the_o multiplicand_a so_o be_v the_o multiplicator_n to_o the_o product_n or_o as_o 1_o be_v to_o the_o multiplicator_n so_o be_v the_o multiplicand_a to_o the_o product_n now_o a_o unity_n or_o 1_o be_v always_o represent_v at_o the_o begin_n of_o the_o scale_n of_o number_n as_o appear_v by_o the_o eight_o rule_n of_o the_o second_o chapter_n you_o need_v not_o there_o place_v a_o point_a chesseman_n to_o denote_v it_o be_v notorious_a of_o itself_o but_o only_a where_o the_o multiplicand_a or_o multiplicator_n be_v find_v upon_o the_o say_a scale_n when_o therefore_o any_o such_o proposition_n as_o that_o above_o be_v make_v place_v one_o point_v chesseman_n upon_o the_o multiplicator_n and_o another_o upon_o the_o multiplicand_a as_o also_o two_o plain_a chesseman_n upon_o the_o side_n rank_n at_o the_o right_a end_n of_o their_o respective_a fractious_a and_o take_v the_o beginning_n of_o the_o line_n to_o be_v always_o the_o first_o term_n in_o the_o question_n by_o the_o direction_n give_v in_o the_o first_o proposition_n of_o this_o chapter_n find_v out_o a_o four_o term_n to_o those_o three_o term_n propound_v which_o do_v that_o four_o term_n be_v the_o product_v you_o look_v for_o example_n 287_o be_v give_v to_o be_v multiply_v by_o 139_o the_o three_o term_n give_v be_v 1_o 139_o 287_o unto_o which_o if_o a_o four_o be_v seek_v for_o by_o the_o instruction_n deliver_v in_o the_o first_o proposition_n of_o this_o chapter_n it_o will_v be_v find_v upon_o the_o 22_o fraction_n at_o four_o digit_n and_o three_o minim_n of_o the_o 23_o interval_n which_o point_n give_v you_o these_o figure_n 39893_o the_o product_v require_v what_o be_v a_o wedge_n of_o gold_n worth_n that_o weigh_v 4_o ounce_n 6_o peny-weight_n and_o 15_o grain_n at_o 3_o l._n 3_o s._n 2_o d._n the_o ounce_n here_o the_o weight_n of_o the_o wedge_n after_o reduction_n be_v 4.3315_o and_o the_o rate_n of_o a_o ounce_n be_v 3.1582_o and_o therefore_o the_o term_n give_v be_v 1_o 4.3315_o 3.1582_o who_o four_o term_v i_o discover_v to_o fall_v upon_o the_o sixth_o fraction_n at_o two_o digit_n and_o one_o minime_fw-la of_o the_o 33_o interval_n which_o give_v i_o these_o figure_n 1368_o whereof_o i_o take_v the_o two_o first_o for_o pound_n sterling_a and_o the_o other_o two_o for_o the_o decimal_n fraction_n of_o a_o pound_n sterling_n which_o after_o reduction_n amount_v to_o 13_o s._n 7_o d._n and_o somewhat_o more_o for_o common-reason_n dictate_v to_o i_o that_o it_o can_v be_v 136_o l._n nor_o so_o little_a as_o 1_o l._n and_o therefore_o i_o conclude_v the_o product_n to_o be_v 13_o l._n 13_o s._n 7_o d._n as_o before_o be_v the_o value_n of_o the_o 4_o ounc_fw-la 6_o penny_n w._n and_o 15_o grain_n the_o term_n require_v in_o multiplication_n observe_v these_o rule_n in_o perform_v multiplication_n you_o always_o operate_v upon_o the_o table_n towards_o the_o right_a hand_n and_o upon_o the_o side-rank_n of_o alphabet_n always_o downward_o for_o a_o unity_n or_o 1_o be_v always_o the_o first_o term_n you_o always_o begin_v the_o account_n of_o the_o alphabet_n and_o letter_n comprehend_v betwixt_o 1_o and_o the_o term_v place_v next_o to_o it_o from_o the_o left_a side_n of_o the_o table_n which_o will_v always_o tend_v towards_o the_o right_a hand_n and_o then_o by_o consequence_n in_o lay_v down_o the_o like_a distance_n betwixt_o the_o other_o term_n and_o the_o product_n you_o be_v to_o proceed_v the_o same_o way_n viz._n towards_o the_o right_a hand_n for_o the_o like_a reason_n it_o be_v that_o you_o be_v always_o to_o work_v downward_o upon_o the_o side-rank_n because_o there_o also_o you_o be_v to_o begin_v your_o account_n from_o the_o first_o fraction_n be_v that_o whereupon_o 1_o the_o first_o term_n be_v represent_v all_o which_o plain_o appear_v by_o the_o premise_a example_n the_o digit_n and_o minim_n which_o be_v to_o be_v take_v off_o from_o the_o point_n of_o the_o term_n give_v be_v always_o so_o to_o be_v take_v off_o upon_o the_o left_a hand_n and_o never_o upon_o the_o right_n here_o by_o the_o term_n give_v be_v intend_v only_o the_o multiplicand_a and_o multiplicator_n for_o the_o first_o term_n viz._n 1_o have_v no_o digit_n or_o minim_n attend_v it_o be_v represent_v upon_o the_o first_o perpendicular_a at_o the_o begin_n of_o the_o scale_n of_o number_n but_o the_o multiplicand_a and_o multiplicator_n may_v have_v digit_n and_o minim_n attend_v they_o which_o be_v always_o to_o be_v take_v off_o upon_o the_o left_a hand_n according_a to_o the_o direction_n of_o this_o rule_n and_o as_o be_v manifest_a by_o the_o example_n aforego_v prop._n 4._o one_o number_n be_v give_v to_o be_v divide_v by_o another_o give_a number_n to_o find_v the_o quotient_a as_o in_o multiplication_n so_o in_o division_n there_o be_v four_o term_n geomettical_o proportional_a whereof_o the_o divisor_n be_v always_o the_o first_o a_o unity_n or_o 1_o and_o the_o dividend_n the_o two_o mean_a term_n and_o the_o quotient_n be_v the_o four_o term_v require_v for_o as_o the_o divisor_n be_v to_o 1_o so_o be_v the_o dividend_n to_o the_o quotient_a or_o as_o the_o divisor_n be_v to_o the_o dividend_n so_o be_v 1_o to_o the_o quotient_a and_o here_o as_o in_o multiplication_n a_o unity_n or_o 1_o be_v always_o one_o of_o the_o term_n you_o need_v not_o thereat_o place_v a_o point_a chesseman_n to_o denote_v it_o but_o only_o where_o the_o divisor_n and_o dividend_n be_v find_v upon_o the_o scale_n as_o also_o two_o plain_a chesseman_n upon_o the_o side_n rank_n at_o the_o right_a end_n of_o their_o respective_a fraction_n and_o then_o take_v the_o divisor_n to_o be_v always_o the_o first_o term_n in_o the_o question_n by_o the_o direction_n give_v in_o the_o first_o proposition_n of_o this_o chapter_n find_v out_o a_o four_o term_n to_o those_o three_o term_n propound_v which_o do_v that_o four_o term_n be_v the_o quotient_a require_v example_n 39893_o be_v give_v to_o be_v divide_v by_o 287_o the_o three_o term_n give_v be_v 287_o 1_o 39893_o or_o 287_o 39893_o 1_o unto_o which_o if_o a_o four_o term_n be_v find_v out_o by_o the_o instruction_n give_v in_o the_o say_v first_o proposition_n of_o this_o chapter_n it_o will_v be_v find_v at_o the_o second_o hundred_o of_o the_o sixth_o fraction_n which_o give_v this_o number_n 139_o for_o the_o quotient_a require_v so_o likewise_o if_o 3989348_o be_v give_v to_o be_v divide_v by_o 287_o the_o first_o three_o figure_n of_o the_o quotient_n will_v be_v find_v 139_o as_o before_o but_o in_o that_o case_n you_o be_v to_o annex_v unto_o they_o two_o cipher_n to_o make_v the_o quotient_a consist_v of_o five_o place_n for_o that_o in_o this_o question_n the_o divisor_n may_v be_v write_v under_o the_o dividend_n five_o time_n as_o appear_v by_o the_o posture_n of_o the_o number_n hereunto_o annex_v 3_o 9_o 8_o 9_o 3_o 4_o 8_o 2_o 8_o 7_o and_o therefore_o in_o that_o case_n the_o quotient_a require_v will_v be_v 13900_o which_o case_n with_o divers_z other_o as_o they_o happen_v the_o artist_n after_o he_o perfect_o understand_v by_o practice_n the_o nature_n of_o this_o instrument_n will_v be_v well_o able_a by_o discretion_n to_o order_v as_o occasion_n shall_v serve_v if_o a_o pipe_n of_o wine_n contain_v 126_o gallon_n cost_v 25_o l._n 14_o s._n 5_o d._n what_o be_v the_o price_n of_o a_o gallon_n thereof_o according_a to_o the_o same_o rate_n here_o the_o term_n in_o the_o question_n after_o reduction_n be_v 126_o 25.721_o 1_o for_n in_o this_o case_n the_o question_n be_v if_o 126_o gallon_n give_v 25.721_o how_o much_o will_v one_o gallon_n yield_v wherefore_o proceed_v according_a to_o the_o direction_n aforego_v i_o find_v the_o four_o term_n to_o reside_v upon_o the_o 12_o fraction_n at_o three_o digit_n and_o five_o minim_n of_o the_o five_o interval_n where_o i_o find_v these_o figure_n represent_v viz._n 20377_o which_o in_o reason_n i_o conceive_v to_o be_v a_o decimal_n fraction_n of_o a_o pound_n sterling_n and_o after_o reduction_n thereof_o discover_v it_o to_o represent_v 4_o s._n 1_o d._n and_o so_o much_o be_v the_o value_n of_o every_o gallon_n in_o the_o pipe_n and_o the_o quotient_a require_v in_o division_n for_o take_v off_o the_o digit_n and_o minim_n observe_v this_o rule_n when_o the_o digit_n and_o minim_n be_v
for_o the_o performance_n thereof_o observe_v these_o ensue_a direction_n by_o the_o instruction_n deliver_v in_o the_o second_o chapter_n aforego_v find_v the_o number_n give_v upon_o the_o scale_n of_o number_n set_v at_o each_o of_o they_o a_o point_a chesseman_n as_o also_o three_o other_o plain_a chesseman_n upon_o the_o side_n rank_n of_o alphabet_n at_o the_o left_a end_n of_o their_o respective_a fraction_n this_o do_v if_o by_o the_o four_o and_o five_o rule_n of_o the_o last_o chapter_n you_o will_v discover_v a_o four_o term_n to_o the_o three_o term_n propound_v you_o shall_v there_o find_v the_o number_n you_o look_v for_o example_n if_o 12980_o represent_v upon_o the_o five_o fraction_n at_o the_o point_n f_o be_v the_o first_o term_v give_v 32192_o represent_v upon_o the_o nineteenth_o fraction_n at_o the_o point_n g_o the_o second_o and_o 18452_o represent_v upon_o the_o ten_o fraction_n at_o the_o point_n h_z the_o three_o the_o four_o term_n by_o the_o four_o and_o five_o rule_n of_o the_o last_o chapter_n will_v tall_a upon_o the_o four_o and_o twenty_o fraction_n at_o the_o the_o point_n k_o by_o the_o last_o rule_n of_o the_o second_o chapter_n give_v you_o the_o number_n 45907_o the_o four_o proportional_a require_v so_o if_o 45907_o be_v give_v for_o the_o first_o term_n 18452_o for_o the_o second_o and_o 32292_o for_o the_o three_o in_o work_v upward_o upon_o the_o side-rank_n and_o towards_o the_o left_a hand_n upon_o the_o table_n the_o four_o term_n will_v be_v find_v to_o rest_v upon_o the_o five_o fraction_n at_o the_o point_n f_o represent_v 12980_o as_o before_o in_o like_a manner_n if_z g_o viz._n 32292_o be_v the_o first_o term_v give_v h_z viz._n 18452_o the_o second_o and_o k_n viz._n 45907_o the_o three_o the_o four_o term_n will_v fall_v upon_o the_o 15_o fraction_n at_o the_o point_n l_o but_o because_o in_o that_o case_n you_o go_v beyond_o the_o table_n towards_o the_o right_a hand_n you_o be_v to_o take_v instead_o thereof_o according_a to_o the_o direction_n give_v in_o the_o three_o example_n of_o the_o five_o rule_n of_o the_o last_o chapter_n the_o point_n m_o upon_o the_o 16_o fraction_n which_o represent_v 26231_o the_o four_o proportional_a require_v so_o likewise_o if_o h_z represent_v 18452_o be_v the_o first_o term_n g_o represent_v 32292_o the_o second_o and_z f_z represent_v 12980_o the_o three_o the_o four_o term_n will_v reside_v upon_o the_o 14_o fraction_n at_o four_o digit_n and_o four_o minim_n of_o the_o 30_o interval_n viz._n at_o the_o point_n n._n howbeit_o in_o this_o case_n also_o you_o be_v not_o take_v that_o point_n but_o because_o you_o overshoot_v the_o table_n upon_o the_o left_a hand_n you_o be_v instead_o thereof_o to_o take_v the_o digit_n and_o minim_n of_o the_o fraction_n next_o above_o it_o in_o the_o same_o interval_n viz._n the_o point_n p_o upon_o the_o 13_o fraction_n which_o represent_v 22715_o the_o four_o proportional_a require_v according_a to_o the_o four_o example_n of_o the_o say_v five_o rule_n of_o the_o last_o chapter_n again_o if_o f_z viz._n 12980_o be_v the_o first_o term_n g_o viz._n 32292_o the_o second_o and_o k_n viz._n 45707_o the_o three_o in_o this_o case_n the_o four_o term_n will_v according_a to_o the_o five_o example_n of_o the_o five_o rule_n of_o the_o last_o chapter_n at_o last_o reside_v upon_o the_o three_o fraction_n at_o four_o digit_n and_o four_o minim_n of_o the_o three_o interval_n viz._n at_o the_o point_n q_o which_o by_o the_o four_o rule_n of_o the_o second_o chapter_n represent_v 11422_o the_o four_o proportional_a seek_v for_o on_o the_o other_o side_n if_o k_n viz._n 45907_o be_v the_o first_o term_n h_o viz._n 18452_o the_o second_o and_z f_z viz._n 12980_o the_o three_o the_o four_o term_n will_v according_a to_o the_o last_o example_n of_o the_o say_v five_o rule_n of_o the_o last_o chapter_n at_o last_o fall_n upon_o the_o 27_o fraction_n at_o four_o digit_n and_o four_o minim_n of_o the_o 30_o interval_n viz._n at_o the_o point_n r_o which_o represent_v these_o figure_n 52169_o howbeit_o because_o common_a sense_n tell_v i_o that_o the_o four_o term_n to_o the_o other_o three_o last_v give_v term_n can_v be_v so_o great_a nor_o yet_o so_o little_a as_o 521.69_o therefore_o i_o conclude_v the_o term_n require_v to_o be_v in_o this_o case_n 5216.9_o or_o 5217_o ferè_fw-la if_o a_o chest_n of_o sugar_n that_o weigh_v 7_o c._n 2_o qu._n and_o 17_o lb._n cost_v 36_o l._n 14_o s._n 10_o d._n what_o be_v the_o price_n of_o 2_o c_o 1_o q_o and_o 4_o lib._n thereof_o according_a to_o the_o same_o rate_n here_o after_o the_o reduction_n of_o the_o break_a part_n of_o the_o number_n give_v into_o decimal_n the_o first_o term_n be_v 7.6518_o the_o second_o 36.7417_o and_o the_o three_o 2.2857_o with_o which_o three_o term_n work_v upon_o the_o table_n according_a to_o the_o precept_n before_o premise_a i_o find_v the_o four_o term_n to_o be_v fix_v upon_o the_o second_o fraction_n at_o two_o digit_n and_o two_o minim_n of_o the_o 17_o interval_n which_o point_n yield_v i_o these_o figure_n â—0975_n whereof_o i_o take_v the_o two_o fiâ—â—â—_n viz._n 1â—_n for_o 10_o l._n and_o the_o other_o three_o for_o a_o decimal_n fraction_n of_o a_o pound_n sterling_n which_o after_o reduction_n amount_v to_o 19_o s._n 6_o d._n and_o therefore_o i_o conclude_v that_o 2_o c._n 1_o qu._n and_o 4_o lb._n of_o that_o sugar_n be_v worth_a 10_o l._n 19_o s._n 6_o d._n which_o be_v the_o term_n require_v for_o when_o i_o have_v those_o five_o figure_n give_v i_o upon_o the_o table_n for_o the_o four_o term_n common_a reason_n tell_v i_o they_o can_v signify_v 109.75_o for_o that_o be_v too_o great_a nor_o 1.0975_o for_o that_o be_v too_o little_a and_o therefore_o in_o this_o case_n i_o take_v 10.975_o viz._n 10_o l._n 19_o s._n 6_o d._n be_v the_o four_o term_v seek_v for_o now_o from_o this_o example_n and_o the_o rest_n before_o premise_v for_o the_o ready_a work_n of_o the_o digit_n and_o minim_n of_o the_o three_o term_n propound_v this_o general_a rule_n or_o corollary_n may_v be_v infer_v in_o all_o question_n that_o may_v be_v perform_v by_o the_o golden_a rule_n the_o digit_n and_o minim_n to_o be_v take_v off_o from_o the_o first_o term_n be_v always_o so_o take_v off_o from_o that_o side_n of_o the_o first_o term_n which_o incline_v towards_o the_o second_o term_n and_o then_o the_o digit_n and_o minim_n of_o the_o other_o two_o term_n be_v always_o take_v off_o upon_o the_o contrary_a side_n to_o those_o of_o the_o first_o term_n as_o be_v manifest_a by_o all_o the_o example_n aforego_v which_o rule_n be_v always_o due_o observe_v you_o may_v with_o great_a confidence_n proceed_v to_o resolve_v any_o question_n propound_v and_o because_o this_o corollary_n be_v always_o to_o be_v keep_v in_o memory_n i_o have_v express_v it_o in_o this_o distich_n aurata_fw-la in_o regula_n bis_fw-la laeva_fw-la aut_fw-la dextra_fw-la petatur_fw-la dum_fw-la contragreditur_fw-la terminus_n ipsâ—_fw-la prio_fw-la thus_o english_v for_o the_o rule_n of_o three_o each_o hand_n may_v be_v pursue_v two_o time_n while_o that_o the_o foremost_a term_n against_o they_o always_o climb_v prop._n 2._o to_o three_o number_n give_v to_o find_v a_o fourâ—â—_n in_o a_o inversed_a proportion_n this_o rule_n of_o three_o inverse_n be_v thâ—_n same_o with_o that_o of_o the_o rule_n of_o three_o direct_v if_o instead_o of_o the_o first_o term_v you_o take_v the_o three_o term_v give_v to_o be_v the_o first_o in_o the_o question_n by_o transpose_v the_o last_o into_o the_o place_n of_o the_o first_o example_n if_o when_o the_o price_n of_o wheat_n be_v 40_o shilling_n the_o quarter_n a_o penny_n white_a loaf_n weigh_v 8_o ounce_n and_o 9_o peny-weight_n how_o much_o aught_o a_o penny_n white_a loaf_n to_o weigh_v when_o wheat_n be_v at_o 23_o shilling_n six_o penny_n the_o quarter_n here_o the_o term_n give_v be_v viz._n 40_o the_o first_o 8.45_o after_o reduction_n the_o second_o and_o 23.5_o the_o three_o which_o as_o they_o be_v propound_v in_o the_o question_n stand_v in_o this_o form_n 40_o 8.45_o 23.5_o but_o be_v invert_v stand_v thus_o 23.5_o 8.45_o 40_o unto_o which_o three_o have_v by_o the_o direction_n aforego_v make_v search_v upon_o the_o table_n for_o a_o four_o proportional_a you_o shall_v find_v it_o to_o fall_v upon_o the_o sixth_o fraction_n at_o three_o digit_n and_o three_o minim_n of_o the_o 25_o interval_n which_o point_n afford_v you_o these_o figure_n 14.383_o which_o after_o reduction_n amount_v to_o 14_o ounc_fw-la 7_o peny-weight_n 16_o grain_n be_v the_o term_n require_v for_o so_o much_o a_o penny_n white_a loaf_n ought_v to_o weigh_v according_a to_o the_o abovesaid_a rate_n when_o wheat_n be_v
take_v off_o from_o the_o right_a hand_n of_o the_o divisor_n take_v the_o digit_n and_o minim_n place_v on_o the_o left_a hand_n of_o the_o dividend_n and_o when_o on_o the_o left_a hand_n of_o the_o divisor_n take_v they_o from_o the_o right_a hand_n of_o the_o dividend_n for_o the_o ready_a discovery_n and_o take_v off_o the_o digit_n and_o minim_n in_o multiplication_n and_o division_n let_v this_o hexameter_n be_v remember_v multiplicá_fw-fr laeuè_fw-fr sed_fw-la divide_v dextrifinistre_fw-fr multiply_v by_o the_o right_a hand_n with_o both_o the_o hand_n divide_v prop._n 5._o two_o number_n be_v give_v to_o find_v a_o three_o geometrical_o proportional_a unto_o they_o and_o to_o three_o a_o four_o and_o to_o four_o a_o five_o etc._n etc._n this_o proposition_n may_v be_v resolve_v by_o the_o direction_n give_v in_o the_o fix_a rule_n of_o the_o last_o chapter_n for_o have_v two_o term_n give_v and_o place_v chesseman_n upon_o they_o as_o also_o at_o the_o right_a end_n of_o their_o respective_a fraction_n as_o in_o the_o aforegoing_a proposition_n if_o you_o by_o the_o say_v sixth_o rule_n find_v a_o three_o term_n to_o the_o two_o other_o give_v term_n that_o three_o be_v the_o term_n you_o look_v for_o example_n if_o 2_o and_o 4_o be_v the_o two_o term_n give_v a_o three_o proportional_a unto_o they_o by_o the_o fix_a rule_n of_o the_o last_o chapter_n will_v be_v find_v upon_o the_o 33_o fraction_n at_o two_o digit_n and_o two_o minim_n of_o the_o 19_o interval_n which_o point_n represent_v 8_o the_o three_o term_v require_v in_o like_a manner_n you_o may_v proceed_v to_o find_v a_o four_o term_n to_o those_o three_o which_o will_v be_v 16_o and_o a_o five_o to_o those_o 4_o term_n find_v which_o will_v be_v 32_o etc._n etc._n and_o so_o you_o may_v by_o this_o mean_n erect_v a_o rank_n of_o number_n geometrical_o proportional_a which_o in_o arithmetic_n be_v call_v geometrical_a progression_n prop._n 6._o to_o extract_v the_o square_a root_n of_o any_o number_n under_o 10000000000._o first_o prepare_v the_o square-number_n give_v for_o extraction_n as_o in_o vulgar_a arithmetic_n by_o subscribe_v a_o point_n under_o each_o other_o figure_n begin_v with_o the_o last_o first_o so_o these_o number_n follow_v be_v give_v for_o extraction_n and_o prepare_v as_o aforesaid_a will_v stand_v thus_o 3_o 2_o 8_o 1_o 5_o 3_o 2_o 2_o 5_o  _fw-fr  _fw-fr  _fw-fr  _fw-fr 1_o 4_o 5_o 2_o 7_o 5_o 3_o 2_o 2_o 5_o  _fw-fr  _fw-fr  _fw-fr  _fw-fr  _fw-fr and_o so_o many_o point_n as_o be_v in_o that_o manner_n subscribe_v of_o so_o many_o figure_n will_v the_o root_n consist_v viz._n in_o these_o example_n of_o five_o figure_n place_v this_o between_o page_n 64_o and_o 65_o example_n let_v 328153225_o be_v the_o square_a number_n give_v &_o the_o root_n thereof_o require_v this_o number_n admit_v five_o point_n to_o be_v subscribe_v under_o it_o as_o appear_v before_o and_o be_v find_v upon_o the_o 19_o fraction_n at_o five_o digit_n of_o the_o 21_o interval_n also_o the_o half_a distance_n thereof_o by_o the_o seven_o rule_n of_o the_o last_o chapter_n be_v likewise_o discover_v at_o two_o digit_n and_o three_o minim_n of_o the_o 11_o interval_n as_o also_o the_o half_a distance_n in_o the_o side_n rank_n by_o the_o ten_o rule_n of_o the_o say_v last_o chapter_n upon_o the_o ten_o fraction_n wherefore_o if_o i_o place_v another_o point_a chesseman_n upon_o the_o say_v 10_o fraction_n at_o two_o digit_n and_o three_o minim_n of_o the_o 11_o interval_n be_v the_o same_o with_o the_o half_a distance_n upon_o the_o fraction_n of_o the_o number_n give_v that_o point_n will_v discover_v these_o figure_n 18115_o be_v the_o root_n require_v but_o when_o the_o first_o point_n towards_o the_o right_a hand_n happen_v to_o fall_v under_o the_o second_o figure_n of_o the_o number_n give_v and_o there_o be_v also_o a_o odd_a fraction_n upon_o the_o side_n rank_n proceed_v as_o in_o the_o last_o rule_n to_o find_v the_o half_a distance_n upon_o the_o fraction_n of_o the_o number_n give_v and_o also_o upon_o the_o side_n rank_n howbeit_o to_o discover_v the_o true_a fraction_n upon_o which_o the_o root_n in_o such_o case_n be_v to_o be_v find_v account_v three_o alphabet_n downward_o from_o the_o half_a distance_n upon_o the_o side-rank_n in_o regard_n the_o first_o figure_n of_o the_o number_n give_v have_v no_o point_n under_o it_o &_o there_o place_v another_o plain_a chesseman_n also_o in_o regard_n of_o the_o odd_a fraction_n upon_o the_o side_n rank_n account_v in_o in_o like_a manner_n three_o alphabet_n towards_o the_o right_a hand_n from_o the_o half_a distance_n upon_o the_o fraction_n where_o the_o number_n give_v be_v situate_a and_o there_o likewise_o place_v another_o point_a chesseman_n all_o this_o perform_v in_o the_o angle_n of_o position_n where_o the_o last_o place_v chesseman_n meet_v with_o the_o true_a fraction_n of_o the_o root_n before_o find_v upon_o the_o side_n rank_n you_o shall_v discover_v the_o root_n require_v example_n let_v the_o square_a root_n of_o the_o number_n subscribe_v be_v desire_v 1_o 4_o 5_o 2_o 7_o 5_o 3_o 2_o 2_o 5_o  _fw-fr  _fw-fr  _fw-fr  _fw-fr  _fw-fr this_o number_n be_v find_v upon_o the_o sixth_o fraction_n at_o one_o digit_fw-la of_o the_o 31_o interval_n and_o the_o half_a distance_n thereof_o at_o 3_o minim_n of_o the_o 16_o interval_n as_o also_o the_o half_a distance_n in_o the_o side_n rank_n upon_o the_o 3_o fraction_n but_o because_o i_o find_v no_o point_n under_o the_o first_o figure_n i_o account_v upon_o the_o side_n rank_a 3_o alphabet_n downward_o from_o that_o 3_o fraction_n and_o thereupon_o set_v another_o plain_a chesseman_n at_o the_o left_a end_n of_o the_o 21_o fraction_n also_o in_o regard_n in_o this_o case_n the_o five_o fraction_n in_o the_o side_n rank_n be_v a_o odd_a fraction_n i_o likewise_o account_v three_o alphabet_n towards_o the_o right_a hand_n from_o the_o half_a distance_n upon_o the_o fraction_n where_o the_o number_n give_v reside_v and_o thereupon_o place_v another_o point_a chesseman_n at_o three_o minim_n of_o the_o 34_o interval_n all_o this_o thus_o act_v i_o find_v the_o chesseman_n last_o place_v to_o meet_v with_o the_o 21_o fraction_n be_v the_o true_a fraction_n of_o the_o root_n as_o aforesaid_a at_o three_o minim_n of_o the_o say_v 34_o interval_n where_o have_v place_v another_o point_a chesseman_n i_o discover_v these_o figure_n 38115_o be_v the_o root_n seek_v for_o and_o here_o let_v i_o give_v you_o this_o rule_n once_o for_o all_o that_o whensoever_o there_o be_v no_o point_n under_o the_o first_o figure_n of_o the_o number_n give_v you_o be_v to_o account_v upon_o the_o side_n rank_a three_o alphabet_n downward_o from_o the_o half_a distance_n there_o find_v and_o when_o there_o be_v a_o odd_a fraction_n upon_o the_o side_n rank_n you_o be_v likewise_o to_o account_v three_o alphabet_n upon_o the_o fraction_n of_o the_o number_n give_v towards_o the_o right_a hand_n from_o the_o half_a distance_n find_v upon_o that_o fraction_n as_o you_o find_v it_o practise_v in_o the_o last_o example_n note_v also_o that_o when_o you_o have_v more_o figure_n discover_v upon_o the_o table_n for_o the_o root_n than_o the_o number_n give_v require_v those_o that_o exceed_v be_v a_o decimal_n fraction_n belong_v to_o the_o root_n likewise_o when_o a_o mix_a number_n be_v give_v you_o be_v to_o subscribe_v the_o point_n only_o under_o the_o significant_a figure_n thereof_o and_o therefore_o if_o these_o two_o number_n viz._n 4_o 3_o 6_o 2_o 3_o  _fw-fr  _fw-fr and_o 1_o 7_o 6_o 2_o 8_o  _fw-fr  _fw-fr  _fw-fr  _fw-fr be_v give_v the_o square_a root_n of_o the_o first_o will_v be_v 208.86_o and_o of_o the_o other_o 41.985_o all_o which_o observation_n and_o the_o like_a after_o some_o practice_n upon_o the_o table_n common_a reason_n will_v dictate_v unto_o you_o prop._n 7._o to_o extract_v the_o cube-root_n of_o any_o number_n give_v under_o 1000000000000000._o prepare_v the_o cube-number_n give_v to_o be_v extract_v as_o in_o vulgar_a arithmetic_n by_o subscribe_v a_o point_n under_o every_o three_o figure_n and_o beginning_n with_o the_o last_o first_o so_o the_o number_n hereafter_o follow_v prepare_v for_o extraction_n will_v stand_v thus_o 2_o 2_o 1_o 9_o 8_o 9_o 4_o 0_o 6_o 6_o 1_o 2_o 5_o  _fw-fr  _fw-fr  _fw-fr  _fw-fr  _fw-fr  _fw-fr  _fw-fr  _fw-fr and_o so_o many_o point_n as_o be_v in_o this_o manner_n subscribe_v of_o so_o many_o figure_n will_v the_o root_n consist_v according_a to_o the_o aforesaid_a observation_n of_o the_o square-root_n place_v a_o point_a chesseman_n at_o the_o number_n give_v and_o likewise_o another_o upon_o the_o side_n rank_n at_o the_o left_a end_n of_o the_o fraction_n upon_o which_o the_o number_n give_v be_v situate_a this_o do_v by_o the_o eight_o rule_n of_o the_o
last_o chapter_n find_v the_o three_o part_n of_o the_o distance_n betwixt_o the_o point_n of_o the_o number_n give_v and_o the_o left_a end_n of_o the_o fraction_n whereon_o it_o be_v place_v as_o also_o upon_o the_o side_n rank_n by_o the_o eleven_o rule_n of_o the_o same_o chapter_n the_o three_o part_n of_o the_o distance_n betwixt_o that_o fraction_n and_o the_o first_o fraction_n all_o this_o perform_v when_o the_o first_o figure_n of_o the_o number_n give_v towards_o the_o left_a hand_n have_v a_o point_n place_v under_o it_o and_o you_o find_v no_o odd_a fraction_n or_o fraction_n upon_o the_o side_n rank_n than_o the_o point_n where_o the_o three_o part_n of_o the_o distance_n of_o the_o fraction_n of_o the_o number_n give_v meet_v with_o the_o fraction_n of_o the_o three_o part_n of_o the_o distance_n in_o the_o side_n rank_n will_v discover_v unto_o you_o the_o cuâ—â—-â—oot_a require_v example_n let_v 2219894066125_o be_v the_o cube_n number_n to_o be_v extract_v this_o number_n admit_v five_o point_n to_o be_v subscribe_v under_o it_o as_o appear_v above_o and_o be_v find_v upon_o the_o 13_o fraction_n at_o five_o digit_n of_o the_o 17_o interval_n also_o the_o three_o part_n of_o the_o distance_n etc._n etc._n by_o the_o eight_o rule_n of_o the_o last_o chapter_n at_o three_o digit_n and_o four_o minim_n of_o the_o sixth_o interval_n and_o likewise_o the_o three_o part_n of_o the_o distance_n in_o the_o side_n rank_n by_o the_o 11_o rule_n of_o the_o last_o chapter_n upon_o the_o 5_o fraction_n wherefore_o if_o i_o place_v another_o point_a chesseman_n upon_o the_o say_v 5_o fraction_n at_o 3_o digit_n and_o 4_o minim_n of_o the_o 6_o interval_n be_v the_o same_o with_o the_o three_o part_n of_o the_o distance_n upon_o the_o fraction_n of_o the_o number_n give_v that_o point_n represent_v these_o figure_n 13045._o be_v the_o root_n you_o look_v for_o but_o when_o the_o first_o point_n towards_o the_o right_a hand_n happen_v to_o fall_v under_o the_o second_o or_o three_o figure_n of_o the_o number_n give_v and_o there_o be_v also_o one_o or_o two_o odd_a fraction_n upon_o the_o side_n rank_n proceed_v as_o in_o the_o last_o rule_n to_o find_v the_o three_o part_n of_o the_o distance_n etc._n etc._n upon_o the_o fraction_n of_o the_o number_n give_v and_o also_o upon_o the_o side_n rank_n howbeit_o to_o discover_v the_o true_a fraction_n upon_o which_o the_o root_n in_o such_o case_n be_v to_o be_v find_v for_o every_o figure_n which_o the_o say_v first_o point_n have_v towards_o the_o left_a hand_n be_v never_o more_o than_o two_o account_v two_o alphabet_n downward_o from_o the_o three_o part_n of_o the_o distance_n find_v upon_o the_o side_n rank_n and_o there_o place_v another_o plain_a chesseman_n also_o for_o every_o odd_a fraction_n which_o will_v never_o likewise_o exceed_v two_o account_v in_o like_a manner_n two_o alphabet_n towards_o the_o right_a hand_n from_o the_o three_o part_n of_o the_o distance_n upon_o the_o fraction_n where_o the_o number_n give_v be_v situate_a and_o there_o likewise_o place_v another_o point_a chesseman_n all_o this_o perform_v in_o the_o angle_n of_o position_n where_o the_o last_o place_v chesseman_n meet_v with_o the_o true_a fraction_n of_o the_o root_n before_o find_v upon_o the_o side_n rank_n you_o shall_v discover_v the_o root_n require_v example_n let_v the_o cube-root_n of_o the_o number_n under-written_a be_v desire_v 6_o 4_o 1_o 9_o 2_o 1_o 9_o 2_o 0_o 6_o 4_o  _fw-fr  _fw-fr  _fw-fr  _fw-fr  _fw-fr  _fw-fr  _fw-fr this_o number_n be_v find_v upon_o the_o 30_o fraction_n at_o two_o digit_n and_o five_o minim_n of_o the_o three_o interval_n and_o the_o three_o part_n of_o the_o distance_n etc._n etc._n by_o the_o eight_o rule_n of_o the_o last_o chapter_n at_o four_o digit_n four_o minim_n and_o somewhat_o more_o of_o the_o first_o interval_n as_o also_o the_o three_o part_n of_o the_o distance_n in_o the_o side_n rank_n by_o the_o 11_o rule_n of_o the_o last_o chapter_n upon_o the_o 10_o fraction_n but_o because_o i_o find_v the_o first_o point_n of_o the_o number_n give_v to_o have_v a_o figure_n before_o it_o towards_o the_o left_a hand_n i_o account_v two_o alphabet_n downward_o from_o the_o three_o part_n of_o the_o distance_n find_v upon_o the_o side_n rank_n viz._n from_o the_o 10_o fraction_n to_o the_o 22_o fraction_n and_o there_o place_v another_o plain_a chesseman_n which_o 22_o fraction_n be_v the_o fraction_n whereupon_o the_o root_n be_v to_o be_v find_v and_o therefore_o i_o place_v there_o another_o plain_a chesseman_n again_o for_o the_o two_o odd_a fraction_n viz._n the_o 28_o and_o 29_o i_o account_v four_o alphabet_n towards_o the_o right_a hand_n from_o the_o three_o part_n of_o the_o distance_n upon_o the_o fraction_n of_o the_o number_n give_v and_o there_o likewise_o place_v another_o point_a chesseman_n all_o this_o perform_v i_o find_v the_o chesseman_n last_o place_v to_o meet_v with_o the_o 22_o fraction_n be_v the_o true_a fraction_n of_o the_o root_n as_o aforesaid_a at_o four_o digit_n and_o four_o minim_n and_o somewhat_o more_o of_o the_o 25_o interval_n where_o have_v place_v another_o point_a chesseman_n i_o discover_v these_o figure_n 4004_o be_v the_o root_n require_v so_o 172.68_o being_n propound_v to_o be_v extract_v the_o cube-root_n thereof_o will_v be_v find_v upon_o the_o 27_o fraction_n at_o three_o digit_n of_o the_o 31_o interval_n where_o you_o shall_v find_v these_o figure_n represent_v 55686_o whereof_o common_a sense_n tell_v i_o 5_o be_v the_o integer_n and_o the_o rest_n of_o the_o figure_n be_v a_o decimal_n fraction_n of_o the_o root_n so_o as_o in_o that_o example_n the_o true_a root_n seek_v for_o after_o separation_n of_o the_o integral_a part_n from_o the_o break_a part_n thereof_o be_v 5.5686_o here_o i_o may_v proceed_v to_o show_v a_o further_a use_n of_o this_o instrument_n for_o the_o resolve_v of_o divers_a other_o proposition_n in_o arithmetic_n and_o geometry_n as_o betwixt_o two_o number_n give_v to_o discover_v one_o two_o or_o more_o mean_a proportional_n to_o three_o number_n give_v to_o find_v a_o four_o in_o a_o duplicate_v or_o triplicate_v proportion_n to_o work_v rule_n of_o plural_a proportion_n the_o double_a golden_a rule_n direct_v and_o inverse_v the_o rule_n of_o fellowship_n alligation_n false_a position_n etc._n etc._n but_o i_o have_v deem_v these_o at_o present_a sufficient_a to_o satisfy_v the_o curiosity_n of_o the_o practitioner_n who_o in_o obtain_v the_o knowledge_n of_o these_o if_o he_o esteem_v they_o worthy_a his_o pain_n may_v be_v thereby_o so_o perfect_o acquaint_v with_o the_o nature_n of_o the_o table_n that_o he_o may_v afterward_o be_v able_a to_o resolve_v not_o only_o the_o proposition_n abovementioned_a but_o all_o other_o which_o may_v be_v perform_v by_o arithmetic_n either_o vulgar_a or_o artifitial_a and_o perhaps_o upon_o further_a scrutiny_n some_o other_o also_o which_o can_v be_v resolve_v without_o symbolical_a arithmetic_n usual_o call_v algebra_n all_o which_o i_o will_v hereafter_o eudeavour_v also_o to_o explain_v as_o vacancy_n from_o other_o more_o pertinent_a affair_n will_v permit_v together_o with_o the_o fabric_n and_o use_v of_o a_o trigonometricall_a table_n of_o proportion_n for_o the_o resolution_n of_o plain_a and_o spherical_a triangle_n if_o i_o shall_v find_v the_o pain_n herein_o already_o take_v may_v obtain_v grateful_a reception_n this_o tractate_n be_v indeed_o only_o intend_v as_o a_o eschantillon_n or_o glimpse_n of_o that_o which_o may_v be_v perform_v upon_o this_o and_o the_o other_o abovesaid_a table_n applicable_a to_o trigonometry_n praestat_fw-la pauca_fw-la avide_n discere_fw-la quà m_fw-la multa_fw-la eumtaedio_fw-la devorare_fw-la erasm_n in_o coll._n rel_n finis_fw-la
viii_o each_o of_o these_o fraction_n consist_v of_o three_o line_n and_o two_o space_n so_o the_o prick_a line_n which_o you_o find_v place_n under_o each_o fraction_n be_v not_o to_o be_v take_v as_o any_o part_n thereof_o but_o have_v another_o use_n as_o shall_v be_v declare_v in_o the_o proper_a place_n ix_o these_o fraction_n together_o with_o their_o line_n and_o space_n must_v be_v understand_v to_o join_v respective_o one_o to_o another_o in_o such_o sort_n that_o the_o whole_a scale_n of_o number_n may_v be_v conceive_v to_o be_v one_o entire_a and_o continue_a line_n for_o example_n the_o right_a end_n of_o the_o first_o fraction_n mark_v by_o 1_o a._n must_v be_v conceive_v to_o join_v with_o the_o left_a end_n of_o the_o second_o fraction_n sign_v by_o 107_o and_o the_o right_a end_n of_o the_o second_o fraction_n mark_v by_o 2_o b._n must_v be_v understand_v to_o join_v with_o the_o left_a end_n of_o the_o three_o fraction_n note_v by_o 114_o and_o so_o consequent_o of_o the_o rest_n in_o their_o order_n so_o that_o the_o whole_a scale_n of_o number_n begin_v at_o the_o left_a end_n of_o the_o first_o fraction_n sign_v by_o 100_o and_o end_v at_o the_o right_a end_n of_o the_o last_o fraction_n note_v by_o 36_o f._n must_v be_v conceive_v to_o be_v one_o entire_a and_o continue_a line_n as_o aforesaid_a and_o therefore_o by_o far_a consequence_n in_o mount_v up_o ward_n the_o least_a end_n of_o the_o last_o fraction_n sign_v by_o 939_o must_v be_v also_o conceive_v to_o join_v with_o the_o right_a end_n of_o that_o above_o it_o sign_v by_o 35_o e._n and_o so_o of_o the_o rest_n in_o ascend_v upward_o until_o you_o mount_v to_o the_o begin_n of_o the_o scale_n x._o the_o entire_a scale_n of_o number_n be_v first_o divide_v into_o a_o thousand_o unequal_a part_n which_o be_v hereafter_o call_v hundred_o and_o distinguish_v by_o have_v three_o figure_n place_v at_o the_o begin_n of_o each_o of_o they_o so_o 100_o at_o the_o begin_n of_o the_o scale_n be_v the_o figure_n of_o the_o first_o hundred_o 101_o of_o the_o second_o hundred_o 102_o of_o the_o three_o hundred_o 103_o of_o the_o four_o hundred_o etc._n etc._n xi_o each_o of_o these_o hundred_o be_v again_o subdivide_v into_o ten_o other_o unequal_a part_n hereafter_o call_v ten_o and_o each_o ten_o also_o suppose_v to_o be_v again_o divide_v into_o ten_o other_o part_n call_v vnit_v for_o the_o distance_n between_o the_o ten_o be_v small_a they_o will_v not_o admit_v any_o real_a division_n of_o the_o same_o ten_o into_o ten_o other_o part_n and_o therefore_o you_o be_v to_o suppose_v they_o to_o be_v so_o divide_v and_o hereafter_o when_o you_o shall_v have_v occasion_n to_o use_v those_o part_n you_o be_v to_o guess_v at_o they_o as_o to_o direct_v your_o eye_n to_o the_o middle_n of_o they_o when_o you_o be_v to_o take_v five_o of_o these_o unit_n and_o somewhat_o beyond_o the_o middle_n when_o six_o of_o they_o be_v propound_v etc._n etc._n howbeit_o because_o at_o the_o begin_n of_o the_o scale_n of_o number_n the_o distance_n of_o the_o ten_o be_v so_o large_a that_o you_o can_v ready_o in_o manner_n aforesaid_a guess_v at_o the_o unit_v comprehend_v betwixt_o they_o i_o have_v cause_v that_o distance_n upon_o the_o first_o fix_v fraction_n to_o be_v divide_v into_o five_o part_n each_o part_n represent_v two_o unit_n and_o from_o thence_o upon_o the_o six_o fraction_n next_o after_o follow_v into_o two_o part_n each_o part_n represent_v five_o unit_n in_o the_o mean_a time_n distinguish_v the_o ten_o comprehend_v betwixt_o every_o two_o hundred_o by_o sharp_a point_n rise_v from_o the_o middle_a line_n of_o the_o scale_n into_o the_o uppermost_a space_n thereof_o and_o upon_o all_o the_o rest_n of_o the_o scale_n leave_v the_o unit_v to_o be_v guess_v at_o as_o aforesaid_a xii_o to_o describe_v the_o hundred_o and_o ten_o upon_o the_o scale_n of_o number_n have_v first_o prepare_v a_o scale_n of_o 100_o equal_a part_n contain_v in_o length_n the_o hundred_o part_n of_o the_o whole_a intend_a scale_n of_o number_n which_o scale_n of_o equal_a part_n must_v be_v suppose_v to_o be_v divide_v into_o 1000_o equal_a part_n the_o distance_n betwixt_o each_o hundred_o part_n thereof_o be_v suppose_v to_o be_v divide_v into_o ten_o part_n repair_v to_o the_o table_n of_o logarithme_n and_o therein_o observe_v the_o first_o five_o figure_n of_o the_o logarithme_n of_o 1001_o beside_o the_o characteristique_a or_o index_n viz._n 00043_o take_v with_o your_o compass_n the_o distance_n from_o the_o begin_n of_o your_o scale_n of_o equal_a part_n to_o the_o say_v 43_o this_o do_v if_o you_o apply_v that_o extent_n of_o the_o compass_n towards_o the_o right_a hand_n from_o the_o begin_n of_o your_o intend_a scale_n of_o number_n the_o movable_a point_n of_o the_o compass_n will_v fall_v upon_o the_o first_o ten_o of_o that_o scale_n in_o like_a manner_n by_o the_o first_o five_o figure_n of_o the_o logarithme_n of_o 1002_o beside_o the_o index_n viz._n 00086_o you_o may_v mark_v out_o the_o second_o ten_o of_o the_o same_o scale_n and_o so_o consequent_o all_o the_o rest_n in_o their_o due_a order_n example_n if_o it_o be_v propound_v to_o make_v a_o scale_n of_o number_n equal_a to_o this_o whereof_o we_o treat_v this_o scale_n be_v entire_o take_v together_o as_o one_o continue_a scale_n according_a to_o the_o nine_o rule_n aforego_v it_o contain_v in_o length_n 75_o foot_n which_o amount_v to_o 900_o inch_n whereof_o the_o hundred_o part_n be_v nine_o inch_n wherefore_o have_v prepare_v a_o scale_n nine_o inch_n long_o as_o be_v above_o direct_v i_o take_v off_o with_o my_o compass_n the_o part_n 43_o which_o extent_n be_v apply_v from_o the_o begin_n of_o the_o scale_n of_o number_n towards_o the_o right_a hand_n the_o movable_a point_n will_v fall_v upon_o the_o first_o ten_o of_o the_o first_o hundred_o of_o that_o scale_n just_a under_o the_o letter_n z_o so_o likewise_o if_o i_o again_o take_v off_o upon_o the_o scale_n of_o equal_a part_n the_o figure_n 86_o and_o apply_v they_o from_o the_o begin_n of_o the_o scale_n of_o number_n as_o before_o that_o extent_n will_v mark_v out_o the_o second_o ten_o of_o the_o same_o hundred_o just_a under_o the_o letter_n x._o in_o like_a manner_n also_o may_v you_o proceed_v until_o you_o have_v describe_v all_o the_o division_n of_o the_o scale_n of_o number_n as_o you_o see_v here_o draw_v upon_o this_o instrument_n this_o may_v suffice_v to_o have_v speak_v of_o the_o substantial_a part_n or_o body_n of_o the_o table_n itself_o in_o the_o next_o place_n follow_v the_o circumstantial_a part_n or_o appendant_o thereof_o to_o be_v handle_v xiii_o the_o appendant_o of_o the_o table_n be_v either_o external_a and_o place_v without_o it_o or_o internal_a and_o place_v within_o it_o fourteen_o those_o place_v without_o it_o be_v either_o so_o place_v at_o the_o top_n above_o it_o or_o on_o each_o side_n thereof_o viz._n at_o the_o end_n of_o the_o fraction_n xv._o the_o appendant_a place_v at_o the_o top_n above_o it_o be_v the_o whole_a length_n of_o the_o table_n divide_v into_o 36_o equal_a part_n number_v by_o 1_o 2_o 3_o etc._n etc._n to_o 36_o and_o sign_v by_o six_o alphabet_n each_o of_o they_o consist_v of_o six_o letter_n viz._n a_o b_o c_o d_o e_o and_o f._n and_o all_o these_o alphabet_n take_v together_o be_v bereafter_o for_o distinction_n sake_n call_v the_o top-rank_n of_o alphabet_n xvi_o the_o two_o end_n of_o this_o top-rank_n ought_v to_o be_v conceive_v to_o join_v interchangeable_o to_o each_o other_o in_o like_a manner_n as_o if_o the_o alphabet_n and_o letter_n be_v place_v in_o a_o circle_n for_o example_n if_o b_o in_o the_o four_o alphabet_n be_v propound_v and_o i_o be_v to_o account_v from_o that_o letter_n four_o alphabet_n and_o three_o letter_n towards_o the_o right_a hand_n the_o letter_n a_o in_o the_o five_o alphabet_n make_v one_o alphabet_n and_o a_o in_o the_o sixth_o alphabet_n be_v the_o second_o alphabet_n but_o now_o because_o in_o proceed_v to_o account_v another_o alphabet_n i_o shall_v go_v beyond_o the_o right_a end_n of_o the_o line_n for_o the_o three_o alphabet_n i_o take_v a_o in_o the_o first_o alphabet_n and_o for_o the_o four_o i_o take_v a_o in_o the_o second_o alphabet_n and_o so_o have_v i_o all_o the_o four_o alphabet_n demand_v and_o then_o i_o account_v three_o letter_n from_o the_o last_o a_o take_v which_o lead_v i_o to_o the_o letter_n d_o in_o the_o say_v second_o alphabet_n be_v the_o letter_n require_v in_o like_a manner_n if_o i_o be_v to_o proceed_v towards_o the_o left_a hand_n and_o c_o in_o the_o second_o alphabet_n be_v the_o term_n give_v from_o whence_o i_o be_o to_o account_v three_o alphabet_n
and_o five_o letter_n d_o in_o the_o first_o alphabet_n be_v the_o first_o letter_n in_o that_o account_n d_o in_o the_o last_o alphabet_n be_v the_o second_o and_z d_o in_o the_o five_o alphabet_n be_v the_o three_o from_o which_o if_o i_o account_v five_o letter_n the_o same_o way_n viz._n towards_o the_o left_a hand_n at_o last_o i_o shall_v fall_v upon_o e_o in_o the_o four_o alphabet_n which_o be_v the_o letter_n require_v xvii_o the_o appendant_o place_v on_o each_o side_n of_o the_o table_n be_v so_o place_v on_o the_o right_a hand_n or_o on_o the_o left_a xviii_o that_o place_v on_o the_o right_a hand_n be_v another_o like_a rank_n of_o alphabet_n which_o be_v hereafter_o call_v the_o side-rank_n of_o alphabet_n xix_o the_o two_o end_n also_o of_o this_o side-rank_n ought_v to_o be_v conceive_v to_o join_v interchangeable_o to_o each_o other_o as_o those_o of_o the_o top-rank_n for_o example_n if_o d_o in_o the_o three_o alphabet_n be_v propound_v and_o it_o be_v demand_v 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the_o say_v four_o alphabet_n be_v the_o letter_n desire_v xx._n the_o appendant_a place_v on_o the_o left_a hand_n be_v nothing_o else_o but_o a_o rank_n of_o number_n express_v the_o three_o figure_n of_o the_o first_o hundred_o of_o every_o fraction_n respective_o and_o serve_v for_o the_o more_o ready_a find_v out_o of_o number_n upon_o the_o scale_n as_o shall_v be_v more_o clear_o teach_v hereafter_o xxi_o the_o interval_n appendant_o place_v within_o the_o table_n be_v either_o alphabet_n or_o parallel_n the_o alphabet_n be_v nothing_o else_o but_o the_o top-rank_n of_o alphabet_n ten_o time_n repeat_v in_o the_o body_n of_o the_o table_n the_o parallel_n be_v certain_a prick_v line_n which_o cross_a one_o another_o at_o right_a angle_n and_o be_v either_o perpendicular_o or_o transversal_n xxii_o the_o perpendicular_o be_v prick_v line_n draw_v downward_o through_o the_o body_n of_o the_o table_n from_o every_o division_n of_o the_o top-rank_n of_o alphabet_n xxiii_o the_o 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be_v represent_v at_o the_o hundred_o sign_v by_o the_o same_o three_o figure_n so_o 114_o or_o 11400_o or_o 11400000_o or_o 114005321_o be_v all-represented_n at_o the_o first_o hundred_o upon_o the_o three_o fraction_n and_o 322_o or_o 322000_o or_o 32200273_o be_v find_v at_o the_o six_o hundred_o of_o the_o 19_o fraction_n vii_o if_o a_o number_n consist_v of_o two_o place_n or_o beside_o the_o two_o place_n have_v cipher_n in_o the_o three_o four_o or_o five_o place_n thereof_o be_v propound_v it_o be_v represent_v at_o the_o hundred_o which_o have_v those_o two_o figure_n and_o a_o cipher_n annex_v unto_o they_o so_o if_o 13_o or_o 13000_o or_o 130000_o or_o 1300000_o or_o 13000734_o be_v give_v they_o be_v all_o represent_v at_o the_o first_o hundred_o of_o the_o five_o fraction_n viii_o if_o a_o number_n of_o one_o figure_n or_o place_n or_o beside_o that_o one_o place_n have_v cipher_n in_o the_o second_o three_o four_o or_o five_o place_n thereof_o be_v give_v it_o will_v 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the_o 19_o fraction_n the_o number_n or_o figure_n represent_v by_o it_o be_v 32292_o as_o appear_v by_o the_o two_o example_n of_o the_o four_o rule_n aforego_v the_o like_a also_o may_v be_v say_v of_o all_o the_o other_o example_n above_o in_o this_o chapter_n produce_v chap._n iii_o numeration_n upon_o the_o alphabet_n and_o transversal_n i._o numeration_n upon_o the_o alphabet_n and_o transversal_n teach_v how_o to_o discover_v distance_n betwixt_o point_n or_o term_n assign_v thereupon_o ii_o three_o letter_n in_o either_o rank_n of_o alphabet_n be_v propound_v to_o find_v a_o four_o which_o shall_v bear_v like_o distance_n from_o the_o three_o that_o the_o second_o bear_v from_o the_o first_o proceed_v thus_o count_v the_o entire_a alphabet_n and_o letter_n which_o be_v intercept_v betwixt_o the_o letter_n of_o the_o first_o and_o second_o term_n then_o from_o the_o letter_n of_o the_o three_o term_n account_n as_o many_o entire_a alphabet_n and_o letter_n the_o same_o way_n that_o do_v the_o letter_n place_v next_o beyond_o the_o last_o letter_n so_o account_v be_v the_o letter_n require_v example_n in_o the_o top-rank_n of_o alphabet_n let_v e_o in_o the_o first_o alphabet_n a_o in_o the_o three_o and_z b_o in_o the_o four_o be_v give_v in_o this_o case_n i_o place_v three_o point_a chesse-man_n in_o the_o chief_a transversal_a viz._n one_o under_o e_o another_z under_z a_o and_o a_o three_o under_o b_o this_o do_v and_o i_o find_v one_o alphabet_n and_o one_o letter_n betwixt_o the_o two_o first_o term_v e_z and_o a_o and_o account_v the_o like_a from_o b_o in_o the_o four_o alphabet_n towards_o the_o right_a hand_n at_o last_o i_o fall_v upon_o d_o in_o the_o five_o alphabet_n which_o be_v the_o letter_n require_v where_o i_o also_o place_v another_o point_a chesseman_n so_o if_o c_o in_o the_o three_o alphabet_n d_o in_o the_o five_o and_o b_o in_o the_o sixth_o be_v propound_v c_o in_o the_o second_o alphabet_n will_v be_v the_o four_o term_v you_o look_v for_o according_a to_o the_o 16_o rule_n of_o the_o first_o chapter_n again_o if_o f_o in_o the_o five_o alphabet_n c_o in_o the_o three_o and_z d_o in_o the_o first_o be_v give_v in_o this_o case_n work_v towards_o the_o left_a hand_n the_o four_o term_n will_v be_v a_o in_o the_o five_o alphabet_n likewise_o if_o b_o in_o the_o three_o alphabet_n be_v the_o first_o term_n d_o in_o the_o first_o be_v the_o second_o and_o c_o in_o the_o four_o be_v the_o three_o term_n the_o four_o term_n will_v be_v e_o in_o the_o second_o alphabet_n etc._n etc._n after_o the_o same_o manner_n in_o the_o side-rank_n of_o alphabet_n three_o letter_n be_v give_v a_o four_o may_v be_v discover_v by_o this_o rule_n and_o the_o 19_o rule_n