Selected quad for the lemma: end_n

Word A	Word B	Word C	Word D	Occurrence	Frequency Band	MI	MI Band	Prominent
end_n	draw_v	line_n	require_v	1,461	5	9.1279	5	false

Text snippets containing the quad

ID	Title	Author	Corrected Date of Publication (TCP Date of Publication)	STC	Words	Pages
A96647	Willsfords arithmetick, naturall, and artificiall: or, decimalls. Containing the science of numbers, digested in three books. Made compendious and facile for all ingenious capacities, viz: merchants, citizens, sea-men, accomptants, &c. Together with the theorie and practice united in a sympathetical proportion betwixt lines and numbers, in their quantitites and qualities, as in respect of form, figure, magnitude and affection: demonstrated by geometrie, illustrated by calculations, and confirmed with variety of examples in every species. / By Thomas Willsford, Gent.	Willsford, Thomas.; Vaughan, Robert, engraver.	1656 (1656)	Wing W2874; Thomason E1584_1; ESTC R204128	162,697	364

difference_n between_o 8_o &_o 10_o and_o also_o 10_o &_o 16_o be_v 2_o &_o 6_o that_o be_v as_o 1_o to_z 3_o or_o ⅓_n and_o so_o will_v these_o difference_n prove_v if_o increase_v or_o diminish_v by_o any_o one_o common_a number_n as_o in_o the_o example_n do_v appear_v by_o these_o former_a ground_n be_v all_o question_n solve_v in_o this_o kind_n and_o not_o otherwise_o unless_o some_o geometrical_a progression_n be_v employ_v in_o the_o operation_n as_o square_n cube_n square_a square_n etc._n etc._n which_o divers_a author_n have_v introduce_v and_o amplify_v this_o rule_n of_o false_a position_n very_o much_o and_o to_o effect_v that_o which_o before_o be_v not_o think_v circumscriptable_a or_o within_o any_o bound_n but_o cossick_n number_n yet_o all_o their_o rule_n involve_v with_o such_o obscurity_n that_o those_o way_n be_v more_o difficult_a to_o find_v and_o less_o certain_a to_o continue_v in_o then_o be_v the_o illustrious_a algebra_n itself_o which_o i_o have_v reserve_v for_o my_o next_o work_n if_o these_o my_o labour_n and_o direction_n be_v acceptable_a and_o so_o i_o will_v proceed_v the_o common_a way_n and_o direction_n in_o the_o double_a rule_n of_o false_a position_n it_o be_v a_o common_a adagy_n that_o there_o be_v more_o way_n to_o the_o wood_n than_o one_o the_o plain_a road_n or_o ready_a tract_n in_o most_o thing_n ought_v to_o be_v follow_v which_o in_o this_o i_o shall_v endeavour_v evident_o to_o show_v &_o this_o way_n first_o draw_v two_o line_n like_o a_o st._n andrews_n cross_n as_o in_o this_o figure_n then_o make_v two_o supposition_n for_o the_o thing_n require_v but_o both_o of_o one_o denomination_n as_o in_o respect_n of_o quantity_n or_o quality_n the_o first_o false_a position_n place_n at_o the_o upper_a end_n of_o the_o cross_n upon_o your_o left_a hand_n the_o other_o on_o the_o right_n as_o in_o this_o 8_o &_o 12_o underneath_o they_o direct_o place_v the_o error_n respective_o from_o which_o false_a position_n it_o do_v arise_v as_o under_o 8_o set_v 6_o and_z beneath_z 12_o insert_v 14_o note_v they_o with_o the_o sign_n as_o whether_o more_o or_o less_o in_o this_o both_o be_v note_v with_o ✚_o than_o multiply_v the_o error_n by_o their_o contrary_a suppose_a number_n that_o be_v crossewise_o the_o first_o false_a position_n by_o the_o second_o error_n and_o the_o second_o position_n by_o the_o first_o error_n as_o 6_o by_o 12_o be_v 72_o and_o 14_o by_o 8_o be_v 112._o thus_o far_o it_o be_v general_a next_o observe_v whether_o the_o sign_n be_v alike_o or_o no_o if_o both_o in_o excess_n or_o both_o defective_a subtract_v the_o lesser_a from_o the_o great_a product_n and_o the_o remainder_n shall_v be_v the_o dividend_n and_o the_o difference_n of_o error_n the_o divisor_n in_o this_o example_n 40_o be_v to_o be_v divide_v by_o 8_o the_o quotient_n will_v be_v 5_o the_o number_n require_v which_o place_n upon_o the_o right_a hand_n of_o the_o cross_n betwixt_o the_o second_o supposition_n and_o its_o error_n and_o if_o the_o whole_a number_n or_o quantity_n know_v be_v 20_o the_o other_o must_v be_v 15_o which_o place_n against_o it_o upon_o the_o left_a hand_n of_o the_o cross_n as_o in_o the_o figure_n but_o if_o the_o error_n be_v of_o different_a kind_n as_o the_o one_o ✚_o in_o excess_n and_o the_o other_o be_v defective_a the_o sum_n of_o all_o such_o product_n must_v be_v the_o dividend_n and_o the_o sum_n of_o the_o error_n must_v be_v divisor_n the_o quotient_n will_v be_v the_o number_n seek_v and_o by_o either_o way_n it_o will_v be_v always_o in_o the_o same_o denomination_n with_o the_o false_a suppose_a number_n as_o shall_v be_v illustrate_v by_o example_n question_v 1_o there_o be_v a_o excise_n upon_o all_o good_n sell_v by_o wholesale_n man_n and_o by_o a_o edict_n if_o any_o man_n render_v not_o a_o true_a account_n the_o penalty_n if_o discover_v be_v double_a the_o commodity_n yet_o all_o tradesman_n allow_v to_o be_v as_o obscure_a as_o they_o can_v the_o truth_n be_v affirm_v it_o be_v demand_v of_o one_o man_n what_o he_o have_v sell_v who_o reply_v 2_o piece_n of_o canvas_n 2_o of_o fustian_n and_o 7_o of_o holland_n every_o piece_n a_o crown_n more_o than_o other_o and_o one_o piece_n of_o the_o best_a be_v 3_o time_n the_o price_n of_o the_o worst_a what_o be_v the_o value_n of_o the_o 11_o piece_n of_o cloth_n in_o this_o proposition_n i_o take_v the_o last_o example_n in_o which_o i_o choose_v the_o mean_a price_n and_o make_v two_o supposition_n thereof_o and_o suppose_v the_o price_n of_o that_o piece_n to_o be_v 8_o crown_n and_o be_v there_o be_v 10_o p_o ece_n more_o every_o one_o dear_a by_o 5_o shilling_n the_o best_a than_o must_v consequent_o be_v 18_o crown_n which_o according_a to_o the_o supposition_n shall_v contain_v the_o worst_a three_o time_n whereas_o here_o 3_o time_n 8_o be_v 24_o from_o whence_o take_v 18_o the_o remainder_n be_v 6_o for_o the_o first_o error_n and_o too_o much_o for_o the_o second_o supposition_n you_o may_v take_v a_o less_o but_o here_o i_o will_v again_o suppose_v a_o great_a as_o 12_o crown_n and_o be_v it_o be_v of_o the_o same_o denomination_n the_o best_a piece_n shall_v be_v worth_a 22_o crown_n and_o that_o equal_a unto_o three_o time_n the_o supposition_n but_o 3_o time_n 12_o be_v 36_o from_o whence_o take_v 22_o the_o remainder_n will_v be_v 14_o for_o the_o second_o error_n which_o place_n under_o 12_o the_o supposition_n with_o the_o sign_n of_o more_o this_o second_o error_n multiply_v by_o 8_o the_o first_o supposition_n produce_v 112_o and_o 6_o the_o first_o error_n multiply_v by_o 12_o the_o second_o supposition_n the_o product_n will_v be_v 72_o the_o difference_n betwixt_o these_o product_n be_v 40_o for_o the_o dividend_n the_o difference_n of_o error_n 8_o the_o divisor_n the_o quotient_a 5_o for_o the_o true_a number_n require_v and_o then_o the_o best_a piece_n must_v be_v 15_o crown_n be_v 3_o time_n the_o value_n of_o the_o other_o the_o price_n of_o all_o the_o rest_n will_v be_v easy_o discover_v as_o thus_o two_o piece_n of_o canvas_n the_o first_o 1_o l._n 5_o s._n the_o other_o 1_o l._n 10_o s._n then_o two_o piece_n of_o fustian_n the_o first_o 5_o s._n more_o than_o the_o last_o piece_n of_o canvas_n viz_o 1_o l._n 15_o s._n the_o other_o 2_o l._n the_o piece_n then_o for_o the_o 7_o piece_n of_o holland_n the_o first_o 2_o l._n 5_o s._n the_o second_o 2_o l._n 10_o s._n the_o three_o 2_o l._n 15_o s._n the_o four_o 3_o l._n the_o five_o 3_o l._n 5_o s._n the_o sixth_o 3_o l._n 10._o the_o last_o 3_o l._n 15._o that_o be_v 15_o crown_n and_o three_o time_n the_o first_o piece_n in_o value_n according_a unto_o the_o state_n of_o the_o question_n the_o total_a sum_n be_v 27_o l._n 10_o s._n the_o thing_n require_v this_o be_v a_o sufficient_a trial_n and_o yet_o for_o to_o please_v all_o according_a unto_o the_o former_a rule_n as_o 8_o be_v to_z 6_o so_o will_v 4_o be_v to_o 3_o which_o four_o 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which_o show_v the_o second_o error_n for_o to_o be_v 5_o defective_a these_o error_n multiply_v crossewise_o into_o their_o suppose_a number_n will_v produce_v 40_o &_o 130_o their_o difference_n 90_o for_o the_o dividend_n the_o difference_n of_o error_n 8_o the_o divisor_n the_o quotient_a 11_o ¼_n for_o the_o lesser_a number_n require_v three_o time_n that_o be_v 33_o ¾_n their_o total_a 45_o the_o question_n solve_v by_o the_o first_o rule_n the_o proportion_n be_v as_o 8_o to_o 13_o so_o 2_o unto_o 3_o 2_o 8_o or_o ¼_n which_o 3_o ¼_n according_a to_o the_o first_o rule_n must_v be_v add_v unto_o 8_o the_o first_o supposition_n the_o sum_n will_v be_v 11_o ¼_n

but_o if_o the_o progression_n begin_v with_o any_o other_o figure_n that_o be_v the_o progression_n term_v then_o multiply_v that_o term_n by_o the_o number_n of_o progression_n the_o product_n will_v be_v the_o extreme_a require_v and_o the_o contrary_a if_o the_o great_a extreme_a be_v give_v divide_v it_o by_o the_o number_n of_o progression_n the_o quotient_n will_v give_v the_o lesser_a extreme_a as_o if_o 2_o be_v the_o lesser_a extreme_a and_o the_o progression_n what_o will_v the_o 6_o place_n be_v the_o answer_n to_o this_o be_v 12_o &_o thus_o they_o will_v stand_v 2_o 4_o 6_o 8_o 10_o 12_o if_o by_o 3_o what_o the_o 7_o place_n the_o answer_n will_v be_v 21_o as_o it_o will_v thus_o appear_v 3_o 6_o 9_o 12_o 15_o 18_o 21_o &_o if_o the_o great_a extreme_a be_v give_v to_o find_v the_o less_o as_o 32_o who_o progression_n be_v to_o 8_o place_n therefore_o divide_v 32_o by_o 8_o the_o quotient_n be_v 4_o the_o less_o extreme_a and_o the_o order_n of_o progression_n as_o 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one_o less_o than_o the_o number_n of_o the_o say_a progression_n subtract_v the_o product_n from_o the_o great_a extreme_a give_v and_o the_o remainder_n will_v be_v the_o first_o number_n of_o that_o progression_n continue_v as_o for_o example_n admit_v 44_o be_v the_o extreme_a propound_v which_o have_v proceed_v by_o 10_o in_o five_o progression_n 10_o multiply_v by_o 4_o produce_v 40_o which_o take_v from_o 44_o there_o will_v remain_v 4_o for_o the_o first_o number_n and_o the_o lesser_a extreme_a as_o by_o these_o number_n will_v appear_v viz_o 4_o 14_o 24_o 34_o 44._o again_o admit_v 29_o to_o be_v the_o great_a extreme_a proceed_v by_o 2_o in_o 13_o progression_n i_o say_v 12_o multiply_v by_o 2_o produce_v 24_o which_o subtract_v from_o 29_o leave_v 5_o for_o the_o first_o number_n as_o 5_o 7_o 9_o 11_o 13_o 15_o 17_o 19_o 21_o 23_o 25_o 27_o 29_o and_o so_o the_o like_a of_o any_o other_o continue_a progression_n with_o a_o mean_a proportional_a and_o one_o extreme_a be_v give_v to_o find_v the_o other_o extreme_a and_o the_o number_n of_o progression_n example_n 4._o any_o number_n in_o arithmetical_a proportion_n the_o sum_n of_o the_o extreme_n be_v double_a unto_o the_o mean_a by_o the_o former_a theorem_n and_o therefore_o it_o be_v evident_a that_o either_o of_o the_o extreme_n take_v from_o the_o mean_a double_a will_v give_v the_o other_o extreme_a as_o 4_o 6_o 8_o and_z 6_o double_v be_v 12_o from_z whence_o take_v 4_o and_o there_o will_v remain_v 8_o or_o subtract_v 8_o and_o there_o will_v remain_v 4_o and_o consequent_o the_o two_o extreme_n be_v give_v the_o mean_a be_v also_o include_v be_v half_o the_o sum_n of_o two_o extreme_n and_o to_o find_v the_o number_n of_o progression_n subtract_v the_o lesser_a from_o the_o great_a and_o divide_v the_o remainder_n by_o the_o difference_n of_o progression_n the_o quotient_n will_v be_v always_o one_o less_o than_o the_o number_n as_o if_o 5_o &_o 29_o be_v two_o extreme_n who_o difference_n be_v 2_o the_o difference_n between_o they_o be_v 24_o which_o divide_v by_o 2_o the_o difference_n of_o progression_n the_o quotient_n will_v be_v 12_o unto_o which_o add_v one_o the_o number_n be_v 13_o the_o whole_a series_n or_o order_n of_o the_o progression_n viz_o 5_o 7_o 9_o 11_o 13_o 15_o 17_o 19_o 21_o 23_o 25_o 27_o 29._o for_o the_o clear_a explanation_n of_o this_o suppose_v two_o man_n be_v to_o go_v from_o london_n to_o edinburgh_n by_o computation_n 292_o mile_n they_o take_v horse_n together_o one_o resolve_v for_o to_o travel_v 30_o mile_n every_o day_n the_o other_o 6_o mile_n the_o first_o day_n the_o second_o day_n 12_o and_o so_o proceed_v in_o arithmetical_a progression_n of_o 6_o until_o he_o shall_v arrive_v at_o his_o journey_n end_n and_o here_o it_o be_v require_v to_o know_v which_o of_o they_o come_v first_o to_o edinburgh_n or_o in_o how_o many_o day_n one_o will_v overtake_v the_o other_o first_o observe_v that_o 30_o be_v here_o the_o mean_a proportional_a and_o 6_o the_o lesser_a extreme_a and_o the_o great_a 54_o the_o mean_v double_v be_v equal_a unto_o the_o sum_n of_o the_o extreme_n that_o be_v 60_o as_o by_o the_o former_a rule_n where_o the_o number_n of_o progression_n will_v be_v find_v 9_o and_o in_o so_o many_o day_n these_o traveller_n will_v overtake_v one_o the_o other_o have_v go_v 270_o mile_n which_o be_v 9_o time_n 30_o equal_a to_o the_o other_o as_o by_o these_o number_n 6_o 12_o 18_o 24_o 30_o 36_o 42_o 48_o 54_o in_o this_o series_n you_o may_v observe_v 30_o the_o mean_a exceed_v the_o other_o first_o day_n journey_n as_o it_o be_v exceed_v by_o the_o last_o and_o consequent_o first_fw-mi at_o his_o journey_n end_n for_o the_o next_o day_n will_v be_v 60_o mile_n wherein_o he_o ride_v as_o fast_v again_o as_o the_o other_o and_o have_v but_o 22_o mile_n to_o ride_v i_o will_v leave_v they_o and_o end_v this_o paragraph_n and_o proceed_v no_o far_o in_o natural_a progression_n paragraph_n vi_o treat_v of_o geometrical_a progression_n and_o proportion_n continue_v and_o interrupt_v with_o the_o addition_n of_o those_o number_n geometrical_a progression_n do_v consist_v of_o more_o than_o two_o number_n not_o of_o equal_a difference_n in_o number_n but_o of_o like_a proportion_n in_o quality_n and_o quantity_n viz_o 1_o 2_o 4_o 8_o 16_o 32_o 64_o etc._n etc._n this_o be_v call_v a_o double_a proportion_n every_o one_o be_v but_o half_a the_o succeed_a number_n or_o these_o 1_o 3_o 9_o 27_o or_o 2_o 6_o 18_o 54_o denominate_v a_o triple_a proportion_n every_o number_n contain_v the_o precedent_n 3_o time_n so_o 2_o 8_o 32_o 128_o or_o 3_o 12_o 48_o a_o quadruple_a proportion_n every_o succeed_a number_n be_v 4_o time_n the_o precedent_n and_o so_o of_o all_o other_o number_n of_o what_o quality_n or_o quantity_n soever_o proceed_v in_o this_o kind_n and_o be_v nominate_v geometrical_a proportion_n continue_v proportion_n interrupt_v or_o break_v off_o be_v when_o the_o progression_n be_v discontinue_v in_o the_o proportion_n of_o the_o number_n and_o consist_v of_o four_o place_n or_o progression_n at_o the_o least_o viz_o 3_o 6_o 7_o 14_o wherein_o the_o second_o and_o the_o three_o number_n that_o be_v 6_o &_o 7_o differ_v in_o the_o proportion_n not_o as_o 3_o to_o 6_o or_o 7_o to_z 14_o which_o be_v a_o double_a proportion_n and_o so_o 1_o 3_o 27_o 81_o be_v a_o geometrical_a progression_n interrupt_v for_o 1_o &_o 3_o &_o 27_o &_o 81_o be_v in_o a_o triple_a proportion_n but_o not_o the_o second_o and_o the_o three_o 27_o contain_v 3_o nine_o time_n theorem_a 1._o geometrical_a progression_n be_v either_o ascend_a or_o descend_v in_o the_o former_a proportion_n and_o if_o continue_a and_o proceed_v from_o a_o unite_v ●he_v first_o from_o 1_o be_v call_v the_o root_n or_o first_o quantity_n the_o second_o be_v the_o square_a or_o the_o second_o quantity_n the_o three_o the_o cube_n or_o three_o quantity_n the_o four_o be_v biquadrat_n the_o square_a square_a or_o four_o quantity_n the_o five_o be_v call_v the_o surde_fw-la solid_a or_o five_o quantity_n the_o sixth_o be_v denominate_v the_o square_a cube_n or_o the_o sixth_o quantity_n etc._n etc._n all_o which_o be_v explain_v in_o these_o number_n viz_o 1_o be_v neither_o number_n nor_o quantity_n 2_o the_o root_n 4_o the_o square_a 8_o the_o cube_n 16_o the_o biquadrat_n make_v by_o the_o square_n of_o 4_o next_o 32_o be_v the_o surde_fw-la solid_a make_v by_o the_o multiplication_n of_o the_o biquadrat_n and_o the_o root_n or_o the_o product_n of_o the_o square_a and_o the_o cube_n 64_o be_v the_o square_a cube_n that_o be_v the_o square_n make_v of_o the_o cube_n multiply_v in_o itself_o all_o which_o