Marx's Mathematical Manuscripts 1881

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 *y _{1} = (x + h)³ = x³ + 3x²h + 3xh² + h³* ;

b) *f(x + h) - f(x)* or *y _{1} - y = 3x²h + 3xh² + h³* ;

c) *(f(x + h) - f(x))/h* or *(y _{1} - 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(x _{1})* or

b) *f(x _{1}) - f(x)* or

c) *(f(x _{1}) - f(x))/(x_{1} - x)* or

If *x _{1}* becomes

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 *x _{1}*, then

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

a) *x _{1} = x + Δx*, but also in reverse b)

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 *y _{1} =* 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 *x _{1}* enters the algebraic function in exactly the same form as

In equation Ib) the difference f(x + h) - f’(x) or *y _{1} - 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

In IIb) we obtain the expression of differences on both sides; it has been so developed on the algebraic side that *(x _{1} - x)* appears as a factor beside a derived function in

*x _{1} - x = h ,*

the two factors into which *x _{1}³ - x³* is resolved may also be written

In Ic), *f’(x)* is now freed of its factor h; we thus obtain on the left-hand side *(y _{1} - y)/h* or

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 *x _{1} - 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 *x _{1} - x ( = x)*.

Finally, in IId) the *definitive derivative* is obtained by the positive setting of *x _{1} = x*. This

In I) the ‘derivative’ is no more found by setting *x _{1} - x = 0* or

^{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, *x _{1} - x = h* becomes, it always remains different from zero.