19. is less than the leffer Number with the Cypher annexed; but the adding this Cypher was multiply167. ing by 10; confequently 10 Times the leffer Number is greater than the greater Number, and therefore cannot be taken from it. 137. Theorem 2. If any Divifor be produced by the continued Multiplication of any Quantities a, b, c, &c. that is, if a, b, c, &c. the Divifor; then, dividing the Dividend by a, and the first Quote by b, the next Quote by c, &c. the laft Quote will be the fame as that found by the Dividend by the whole Divifor. Demonftration. Put = the Dividend, and let m 123. M, =n, b n &c. then n =9, * C = cq, m = bn, x=am: but, by multiplying both Sides of the Equa† 56. tion ncq by b, we have bn + = bcq, ··· | m= •.• [m=b cq; 23. again, multiplying both Sides of this Equation by a we have amabcq, and. abcqx,. dividing by abc we have *q= 2; E. D. 23. 108. a b c . 138. Theorem 3. If, when the Divifor is a compofed Number, the integral Part of the Quote be found by dividing by the Parts of which the Divifor is compofed as fhewn in Art. 120; then we say the Remainder, which would have been found by dividing the whole Dividend by the whole Divifor, may be found by the Rule given in the latter Part of that Article. Demonftration. We fhall demonstrate this Theorem to be true when the Divifor is compofed of three Numbers; and after the fame Manner it may be demonftrated when the Divifor is compofed of four or more Numbers. Let abc the Divifor, x = the Dividend, m the first integral Quote, r the correfponding Remainder, n the fecond Quote, s⇒ the Remainder, q the third Quote, t the Remainder; R the Remainder found by dividing t123. the whole Dividend by the whole Divifor. From the Nature of Divifion abcq+R+x;, taking abcq from † 123. from both Sides of the Equation, we have R*** 36. abcq; and + cq + t = n, bn + s=m, am+r=x; whence, putting am+r for x in the Expreffion for R, we get Ram + r~ abcq; and, taking t from both Sides of the Equation cq+t=n, we get ‡ cq =n—t, which divided by c is q || = ท C t which put for 9, in the laft Expreffion for R; and then - t C =am+rab x n — (because, the laft Term being both multiplied and divided by c, it is plain that may be omitted). =am+rabn+abt. The Equation bn + s=m, being multiplied by a, will give abn + as * = am, and.., writing this Value of am for am in the laft Equation for R, we fhall have Rabn+as+r abn + abt, out of which striking away the contradictory Terins, it will become Ras +r+abt or, which is the fame, tba +sa+r, which is the fame as the Rule in the latter Part of the Article 120. Q. E. D. 9= =3 1 36. * 108. 56. † 123 This Theorem may be demonftrated otherwise: Thus, let the integral Part of the last Quotient, the laft Divifor, the last Remainder; then qc++ the laft Dividend, which in this Cafe is the laft Quotient but one; . if we put b and s for the last Divifor but one, and last Remainder but one, respectively, we fhall have (gc + txb+s =) gcb + bt+s‡=the next preceding Dividend; hence, † 123. if there are but two Divifors c and b, the Remainder is bts; becaufe, dividing qcb + bt + s by cb, the Quotient is q + || C If the young Student should not fee readily the Reafon, that bits bts is the Remainder, that is, why is the fractional Part + of the Quote, or, which is the fame, why cb cannot be contained once in bts, he is defired to obferve, that, being a Remainder belonging to the Divifore, it must at least be one less than in two Divifors. Q. E. D. But, if there are three Divifors, then qcb + bt+s the integral Part of the first Quotient, and if we put a the first Divifor, and rits Remainder, we fhall have 123. (qcbbts x a+r=) qabc + bta +sa+r*➡the firft or given Dividend; and. in three Divifors the Remainder is bta+sa+r, because, qabc+bta+sa+r being divided by abc, the integral Part of the Quotient is q . bta+sa+r and the fractional Part 2. E. D. abc CHAP. VII. Of APPLICATE NUMBERS. 139. WE E hitherto have, in Addition, Subtraction, Multiplication, and Divifion, confined ourfelves to abftract whole Numbers; that is, we have confidered Numbers barely as to the Number of Things, without any Relation to (for we abftracted from, i. e. did not attend to) the particular Kind of Things numbered. Whence abftract Numbers are thofe which are confidered as pure Numbers, without being applied to any particular Subject; but we must now proceed to apply Numbers to particular Things, and the Numbers fo applied are called applicate Numbers, or we are then faid to confider Numbers in the Concrete. Thus, 3, when taken abftractedly, •.', in the Product ç b, the b must at least be taken once more than in the Product bt; and ... e b muft at least be more than bt by bt; and, finces is the Remainder belonging to the Divifor b, it must be less than b, and ..., though it be added to bt, it cannot make up the Defect b; whence, bt+s is always lefs than cb, and.cb cannot be taken once out of that Sum; and confequently bts must be a Remainder. ftractedly,fignifies in general only 3 Things; but, in the Concrete, we fay, 3 Men, 3 Pounds, 3 Yards, &c. 140. When we confider Numbers in the Abstract, whatever is true of thofe Numbers of Things, is true of the fame Numbers in whatever Things they are found; for Inftance, 2 Things is more than i by Thing; fo 2 Yards is 1 more than 1 Yard: But when we confider them in the Concrete, as we must have Regard to the Subjects in which the Numbers are found, as well as to the Numbers themselves, what is true of them taken in the Abftract, may not be fo in the Concrete; for Inftance, 2 Feet is less than 1 Yard, becaufe a Yard is 3 Feet; whereas, in the Abstract, 2 is more than 1; whence it appears, that, in applicate Numbers, we do not barely confider the Number of Things, but have alfo Refpect to the Lengths or Quantities of those Things, in which the Numbers are found. Farther, it must be observed, that, in comparing applicate Numbers, they must be all of one Kind, viz. all Lengths, or Weights, &c. For what Comparison can there be between Yards and Pounds? Or what Relation between Ounces and Bufhels? This it was proper to hint, because on it depends the true Sense, and Poffibility or Impoffibility, of fome Queftions; for Example, if it was required to add 2 Ounces and 2 Bushels together, it is plain the Question would be not only improper, but alfo Nonfenfe and impof fible. 141. As, for the Conveniency of Commerce, it was neceffary to make Ufe of fome Standards, for Weights, Measures, &c. adapted to the different Kinds of Things to be weighed or meafured, it will be neceffary, before we proceed to the Rules relating to applicate Numbers, to give fome Account of those Weights and Meafures, and of the Money made Ufe of in eftimating the Value of the Things weighed or measured. At prefent we fhall confine ourfelves to thofe commonly used in Great Britain. The real Coins now current, and commonly known, are these. 1. Of Copper Money, a Farthing, a Half-penny. 2. Of Silver, Sixpence, a Shilling, Half-crown, a Crown. 3. Of Gold-money, Halfguinea, a Guinea. We have alfo fome foreign Gold Coins current amongst us, viz. a Moidore which paffes for 27 Shillings; another Portugal Piece for 3 Pounds and 12 Shillings, the Half, Quarter, &c. of that Piece. Note, In Scotland, Accounts are kept in Pounds, Shillings, and Pence, Scotch; 12 Pounds Scotch being = 1 Pound English. But (as Mr. Malcolm informs us) they now begin to ufe English Money in their Accounts. Note also, that when £. s. d. q. are written over any Figures (or to the Right-hand of the Figures) they denote Pounds, Shillings, Pence, and Farthings refpectively. Thus, 3 Pounds, 7 Shillings, and 2 Pence 3 Farthings, may be written thus, £. s. d. qs. or 3£. 75. 2 d. 39. Some alfo write the Farthings like Fractions, thus, 3£. 7s. 2 d. 4. 3:7:2:3, The Goldfmiths, &c. exprefs the Fineness of Gold by Caracts, which are not any particular Weight; a Caract only denoting the 24th Part of the |