Marx's Mathematical Manuscripts 1881

Comparison of D’Alembert's Method to the Algebraic Method

Written: August, 1881;
Source: Marx's Mathematical Manuscripts, New Park Publications, 1983;
First published: in Russian translation, in Pod znamenem marksizma, 1933.

Let us compare d’Alembert’s method to the algebraic one.⁹⁴

I) f(x) or y = x³ ;

a) f(x + h) or y₁ = (x + h)³ = x³ + 3x²h + 3xh² + h³ ;

b) f(x + h) - f(x) or y₁ - y = 3x²h + 3xh² + h³ ;

c) (f(x + h) - f(x))/h or (y₁ - y)/h = 3x² + 3xh + h² ;

if h = 0:

d) 0/0 or dy/dx = 3x² = f’(x) .

II) f(x) or y = x³ ;

a) f(x₁) or y₁ = x₁³ ;

b) f(x₁) - f(x) or y₁ - y = x₁³ - x³ = (x₁ - x)(x₁² + x₁x + x²) ;

c) (f(x₁) - f(x))/(x₁ - x) or (y₁ - y)/(x₁ - x) = x₁² + x₁x + x² .

If x₁ becomes = x, then x₁ - x = 0, hence:

d) 0/0 or dy/dx = (x² + xx + x²) = 3x² .

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 = Δx = h (an undefined, infinitely contractible but always finite difference).⁹⁵

If Δx or h is the increment by which x has increased, then:

a) x₁ = x + Δx, but also in reverse b) x + Δx or x + h = x₁.

The differential calculus begins historically with a); with the fact, that is, that the difference Δx or the increment h (one expresses the same thing as the other: the first negatively as the difference Δx, 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 + Δx [, such that f(x + Δx)] or y₁ = etc.

The mystical differential calculus immediately transforms:

(x + Δx) 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 Δx 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 + Δx), alias (x + h); a development now becomes necessary in which Δx 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)³ now appears in Ia) instead of x³ - 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 x³, namely 3x², as the coefficient in the second term of the series, attached to h. 3x² = 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 + Δx = x + h ,

as x increased by h. The entire problem now consists of uncoupling not the embryonic but the ready-made f’(x) from its factor h and from its other neighbouring terms.

In IIa) in contrast, the increased x₁ enters the algebraic function in exactly the same form as x originally entered it; x³ becomes x₁³. 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 x₁³ - x³ by x₁ - x. Only the existence of the difference x₁³ - x³ made possible its separation into two factors. Since

x₁ - x = h ,

the two factors into which x₁³ - x³ is resolved may also be written h(x₁² + x₁x + x²). This points up a new difference with Ib). h itself as the factor of the preliminary derivative is only derived by means of the expansion of the difference x₁³ - x³ into the product of two factors, while h as the factor of the ‘derivative’, exists just like the latter in Ia), already complete before any difference has been developed at all. That the undefined increase from x to x₁ takes the separated form of the factor h next to x, is assumed from the very beginning in I), but proved (since x₁ - x = h) by means of the derivation in II). Indeed, on the one hand h is undefined in I) while on the other hand it is already fairly well defined, since the undefined increase of x already appears as a separate quantity by which x has increased, and thus as such it enters next to it.

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) = 3x², 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.

⁹⁴ Here Marx wrote, ‘... to the geometric’, a clear slip of the pen.
⁹⁵ As already noted, the source-books employed by Marx did not consider zero a finite quantity. Therefore this passage states that however small the difference, x₁ - x = h becomes, it always remains different from zero.
⁹⁶ Here Marx writes simply x + x. instead of x + Τx..Concerning the origins of such replacement, see pp.78-79 of this edition as well as note 60.