Let us compare d Alembert s method to the algebraic one.94 I) f(x) or y = x ; a) f(x + h) or y1 = (x + h) = x + 3x h + 3xh + h ; b) f(x + h) - f(x) or y1 - y = 3x h + 3xh + h ; c) (f(x + h) - f(x))/h or (y1 - y)/h = 3x + 3xh + h ; d) 0/0 or dy/dx = 3x = f (x) . II) f(x) or y = x ; a) f(x1) or y1 = x1 ; b) f(x1) - f(x) or y1 - y = x1 - x = (x1 - x)(x1 + x1x + x ) ; c) (f(x1) - f(x))/(x1 - x) or (y1 - y)/(x1 - x) = x1 + x1x + x . d) 0/0 or dy/dx = (x + xx + x ) = 3x . It is the same in both so far: if the independent variable x increases, so does the dependent [variable] y. Everything depends on how the increase of x is expressed. If x becomes x1, then x1 - x = x = h (an undefined, infinitely contractible but always finite difference).95 If x or h is the increment by which x has increased, then: a) x1 = x + x, but also in reverse b) x + x or x + h = x1. The differential calculus begins historically with a); with the fact, that is, that the difference x or the increment h (one expresses the same thing as the other: the first negatively as the difference x, the second positively as the increment h) exists independently next to the quantity x, whose increment it is and thus which it expresses as increased, but increased by h. It thereby achieves the advantage from the very beginning, that the original function of the variable corresponding to this general expression, as soon as it increases, is expressed in a binomial of a defined degree, and therefore from the very beginning the binomial theorem is applicable to it. Already, in fact, we have a binomial on the general, the left-hand, side, namely x + x [, such that f(x + x)] or y1 = etc. The mystical differential calculus immediately transforms: (x + x) into (x + dx) or according to Newton, x + x..96 Thereby we have also immediately obtained on the right-hand, the algebraic, side a binomial in x + dx or x + x. which may be treated as an ordinary binomial. The transformation from x to dx or x. is assumed a prori rather than rejected on mathematical grounds, so that later the mystical suppression of terms of the developed binomial becomes possible. D Alembert begins with (x + dx) but corrects the expression to (x + x), alias (x + h); a development now becomes necessary in which x or h is transformed into dx, but all of that development really proceeds (das ist auch alle Entwicklung, die wirklich vorgeht). Whether it begin falsely with (x + dx) or correctly with (x + h), this undefined binomial placed in the given algebraic function of x transforms into a binomial of a defined degree - such as (x + h) now appears in Ia) instead of x - and even into a binomial in which in the first case dx, in the other case h appears as its last term, and also in the expansion as well as merely a factor to which the functions derived from the binomial are externally attached (behaftet). Therefore we find right in Ia) the complete first derivative of x , namely 3x , as the coefficient in the second term of the series, attached to h. 3x = f (x) remains unchanged from now on. It is itself derived by means of no sort of process of differentiation at all but rather provided from the very beginning by means of the binomial theorem, indeed because from the very beginning we have represented the increased x as a binomial, x + x = x + h , In IIa) in contrast, the increased x1 enters the algebraic function in exactly the same form as x originally entered it; x becomes x1 . The derivative f (x) can only be obtained at the end by means of two successive differentiations, and those of quite distinct character indeed. In equation Ib) the difference f(x + h) - f (x) or y1 - y now prepares the arrival of the symbolic differential coefficient; in real terms, however, all that changes is that it moves out of second rank into the first rank of the series and therefore makes possible its liberation from h. In IIb) we obtain the expression of differences on both sides; it has been so developed on the algebraic side that (x1 - x) appears as a factor beside a derived function in x and x1 which was obtained by means of the division of x1 - x by x1 - x. Only the existence of the difference x1 - x made possible its separation into two factors. Since x1 - x = h , In Ic), f (x) is now freed of its factor h; we thus obtain on the left-hand side (y1 - y)/h or (f(x + h) - f(x))/h, thus a still finite expression of the differential coefficient. On the other side, however, we have reached the point where, when we set h = 0 in (f(x + h) - f(x))/h, and this transforms into 0/0 = dy/dx , we obtain on one side in Id) the symbolic differential coefficient and on the other f (x), which appeared complete already in Ia) but now has been freed of its neighbouring terms and stands alone on the right-hand side. Positive development only proceeds on the left-hand side, since here the symbolic differential coefficient is produced. On the right-hand side the development consists only of freeing f(x) = 3x , already found in Ia) by means of the binomial, from its original impediment. The transformation of h into 0 or x1 - x = 0 has only this negative meaning on the right-hand side. In IIc), by contrast, a preliminary derivative is only obtained by dividing both sides by x1 - x ( = x). Finally, in IId) the definitive derivative is obtained by the positive setting of x1 = x. This x1 = x means, however, setting at the same time x1 - x = 0, and therefore transforms the finite ratio (y1 - y)/(x1 - x) on the left-hand side to 0/0 or dy/dx . In I) the derivative is no more found by setting x1 - x = 0 or h = 0 than it is in the mystical differential method. In both cases the neighbouring terms of the f (x) which appeared complete from the very beginning have been tossed aside, now in a mathematically correct manner, there by means of a coup d etat.
Marx's Mathematical Manuscripts 1881