CALCULUS Differentialis — 5th ed. (1815)

CALCULUS Differentialis, is a method of differencing quantities, or of finding an infinitely small quantity, which being taken infinite times, shall be equal to a given quantity; or, it is the arithmetic of the infinitely small differences of variable quantities.

The foundation of this calculus is an infinitely small quantity, or an infinitesimal, which is a portion of a quantity incomparable to that quantity, or that is less than any assignable one, and therefore accounted as nothing; the error accruing by omitting it being less than any assignable one. Hence two quantities, only differing by an infinitesimal, are reputed equal. Thus, in astronomy, the diameter of the earth is an infinitesimal, in respect of the distance of the fixed stars; and the same holds in abstract quantities. The term, infinitesimal, therefore, is merely relative, and involves a relation to another quantity; and does not denote any real ens or being. Now infinitesimals are called differentials, or differential quantities, when they are considered as the differences of two quantities. Sir Isaac Newton calls them moments; considering them as the momentary increments of quantities, v. g. of a line generated by the flux of a point, or of a surface by the flux of a line. The differential calculus, therefore, and the doctrine of fluxions, are the same thing under different names; the former given by M. Leibnitz, and the latter by Sir Isaac Newton: each of whom lays claim to the discovery. There is, indeed, a difference in the manner of expressing the quantities resulting from the different views wherein the two authors consider the infinitesimals: the one as moments, the other as differences. Leibnitz, and most foreigners, express the differentials of quantities by the same letters as variable ones, only prefixing the letter d: thus the differential of x is called dx; and that of y, dy: now dx is a positive quantity, if x continually increases; negative, if it decrease. The English, with Sir Isaac Newton,

Newton, instead of $dx$ write $\dot{x}$ (with a dot over it), for $dy$ , $y$ , &c. which foreigners object against, on account of that confusion of points, which they imagine arises when differentials are again differenced; besides, that the printers are more apt to overlook a point than a letter. Stable quantities being always expressed by the first letters of the alphabet $a = \dot{a}$ , $b = \dot{b}$ , $c = \dot{c}$ ; wherefore $d(x+y-a) = dx+dy$ , and $d(x-y+a) = dx-dy$ . So that the differencing of quantities is easily performed by the addition or subtraction of their compounds.

To difference quantities that multiply each other; the rule is, first, multiply the differential of one factor into the other factor, the sum of the two factors is the differential sought: thus, the quantities being $x, y$ , the differential will be $x dy + y dx$ , i. e. $d(xy) = x dy + y dx$ . Secondly, If there be three quantities mutually multiplying each other, the factum of the two must then be multiplied into the differential of the third; thus suppose $vxy$ , let $v = t$ , then $vxy = ty$ ; consequently $d(vxy) = t dy + y dt$ : but $dt = v dx + x dv$ . These values, therefore, being substituted in the antecedent differential, $t dy + y dt$ , the result is, $d(vxy) = v x dy + v y dx + x y dv$ . Hence it is easy to apprehend how to proceed, where the quantities are more than three. If one variable quantity increase, while the other $y$ decreases, it is evident $y dx - x dy$ will be the differential of $xy$ .

To difference quantities that mutually divide each other; the rule is, first, multiply the differential of the divisor into the dividend; and on the contrary, the differential of the dividend into the divisor; subtract the last product from the first, and divide the remainder by the square of the divisor, the quotient is the differential of the quantities mutually dividing each other. See FLUXIONS.