Marx's Mathematical Manuscripts 1881

On the Differential¹³

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

I

1) Let f(x) or y = uz be a function to be differentiated; u and z are both functions dependent on the independent variable x. They are independent variables with respect to the function y, which depends on them, and thus on x.

y₁ = u₁z₁,

y₁ - y = u₁z₁ - uz = z₁(u₁ - u) + u(z₁ - z),

(y₁ - y)/(x₁ - x) or Δy/Δx = z₁⋅(u₁ - u)/(x₁ - x) + u⋅(z₁ - z)/(x₁ - x) = (z₁⋅Δu)/(Δx) + (u⋅Δz)/(Δx) .^*

Now on the right-hand side let x₁ = x, so that x₁ - x = 0, likewise u₁ - u = 0, z₁ - z = 0; so that the factor z₁ in z₁⋅(u₁ - u)/(x₁ - x) also goes to z; finally on the left-hand side y₁ - y = 0. Therefore:

A) dy/dx = z⋅du/dx + u⋅dz/dx .

Which equation, when all its terms are multiplied by the common denominator dx, becomes

B) dy or d(uz) = z⋅du + u⋅dz. ¹⁴

2) Consider for the time being the first equation A):

dy/dx = z⋅du/dx + u⋅dz/dx .

In equations with only one variable dependent on x, the final result has always been

dy/dx = f’(x) ,

and f’(x), the first derived function^*2 of f(x), has been free¹⁵ of all symbolic expressions, for example, mx^m-1 when x^m is the original function of the independent variable x. As a direct result of the process of differentiation which f(x) had to pass through in order to be transformed into f’(x), its shadow image (Doppelgänger) 0/0 or dy/dx appeared as the symbolic equivalent on the left-hand side opposite f’(x), the real differential coefficient.¹⁶ Alternately 0/0 or dy/dx found its real equivalent in f’(x).

In equation A) by contrast, f’(x), the first derivative of uz, itself includes symbolic differential coefficients, which are therefore present on both sides while on neither is there a real value. Since, however, uz has been handled in the same manner as the earlier functions of x with only one independent variable, this contrast is obviously a result of the peculiar character of the beginning function itself, namely uz. A more complete treatment of this is found under 3).

For the moment, it remains to be seen whether there are any twists in the derivation of equation A).

On the right-hand side

(u₁ - u)/(x₁ - x) or Δu/Δx and (z₁ - z)/(x₁ - x) or Δz/Δx

become 0/0, 0/0, because x₁ has become = x, so that x₁ - x = 0. In place of 0/0, 0/0 we put du/dx, dz/dx without further ado. Was that permissible, since these 0/0 figure here as the multipliers of the variables u and z respectively, while in case with one independent variable the single symbolic differential coeficient - 0/0 or dy/dx - has no multiplier other tahan the constant, 1?

If we place the primitive problematic form of du/dx, dz/dx on the right-hand side it becomes: z⋅0/0 + u⋅0/0. If we then multiply z and u by the numerators of the 0/0 accompanying them, we obtain: 0/0 + 0/0; and since the variables z and u themselves become = 0,¹⁷ as are their derivatives as well, so that [we obtain] finally:

0/0 = 0 and not z⋅du/dx + u⋅dz/dx .

This procedure, however, is mathematically false.

Let us take, for example

(u₁ - u)/(x₁ - x) = Δu/Δx ;

one does not first obtain the numerator = 0 because one has begun with it and set u₁ - u = 0, but rather the numerator only becomes 0 or u₁ - u = 0 because the denominator, the difference of the independent variable quantities x, that is x₁ - x, has become = 0.

Therefore what arises opposite the variables u and z is not 0 but (0/0), whose numerator in this form remains inseparable from its denominator. Consequently as a multiplier 0/0 then could nullify its coefficients only when and so far as

0/0 = 0.

Even in the usual algebra it would be false, in the case where a product P⋅m/n takes the form P⋅0/0 , to conclude immediately that it must be = 0 , although it may be set always = 0 here, since we can begin¹⁸ the nullification arbitrarily with numerator or denominator.

For example, P⋅(x² - a²)/(x - a). Let [because x = a] x² be set = a² , so that x² - a² = 0; we then obtain: P⋅0/0 = 0/0, and the last [term] may be set = 0, since 0/0 can just as readily be 0 as any other number.

By contrast, let us reduce x² - a² to its factors, so that we obtain P.((x - a)(x + a))/(x - a) = P(x + a), and since x = a,¹⁹ = 2Pa .

Successive differentiation - for example, of x³, where 0/0 first becomes = 0 only in the fourth derivative, since in the third the variable x has run out and is replaced by a constant - proves that 0/0 becomes = 0 only under completely defined conditions.

In our case, however, where the origin of 0/0, 0/0 is known to be the differential expression of Δz/Δx, Δu/Δx respectively, the two deserve, as above, the ‘uniform’ (die Uniform) dz/dx , du/dx .

3) In the equations, such as y = x^m, y = a^x etc., which have been treated previously, an original function of x stands opposite a y ‘dependent’ on it.

In y = uz, both sides contain ‘dependent [variables]’. While here y depends directly on u and z, so in turn u and z [depend] as well on x. This specific character of the original function uz necessarily stamps on its ‘derivatives’ as well.

That u is a function of x, and z another function of x is represented by:

u = f(x), u₁ - u = f(x₁) - f(x) ,

and

z = φ(x); z₁ - z = φ(x₁) - (x) .

But neither the beginning equation for f(x) nor for φ(x) leads to an original function of x, that is, a definite value^*3 in x. Consequently u and z figure as mere names, as symbols of functions of x; therefore as well only the general forms of this ratio of dependence (Abhängigkeitsverhältnis) :

(u₁ - u)/(x₁ - x) = (f(x₁) - f(x))/(x₁ - x) , (z₁ - z)/(x₁ - x) = (φ(x₁) - φ(x))/(x₁ - x)

is generated immediately by the process of taking the derivative. The process has now reached the point where x₁ is set = x, so that x₁ - x = 0, and those general forms are transformed to

du/dx = df(x)/dx , dz/dx = dφ(x)/dx ,

and the symbolic differential coefficients du/dx , dz/dx become as such incorporated into the ‘derivatives’.

In equations with only one dependent variable, dy/dx has no other content at all than du/dx , dz/dx have here. It is also merely the symbolic differential expression of

(y₁ - y)/(x₁ - x) = (f(x₁) - f(x))/(x₁ - x).²⁰

Although the nature of du/dx , dz/dx - that is, of symbolic coefficients in general - is in no way altered when they appear within the derivative itself, and so on the right-hand side of the differential equation as well, nonetheless their role and the character of the equation are thereby altered.

Let us represent the original function of uz, in combination, by f(x), and their first ‘derivative’ by f’(x),

dy/dx = z⋅du/dx + u⋅dz/dx

then becomes:

dy/dx = f’(x) .

We have obtained this very general form for equations with only one dependent variable. In both cases the beginning forms of dy/dx arose from the process of taking the derivative (Ableitungsprozesse), which transforms f(x) into f’(x). So soon, therefore, as f(x) becomes f’(x), dy/dx stands opposite the latter as its own symbolic expression, as its shadow image (Dopelgänger) or symbolic equivalent.

In both cases, therefore, dy/dx plays the same role.

It is otherwise with du/dx , dz/dx. Together with the other elements of f’(x), into which they are incorporated, in dy/dx they meet with their symbolic expression or their symbolic equivalent, but they themselves do not stand opposite the f’(x), φ’(x) whose symbolic shadow images they would be in turn. They are brought into the world unilaterally, shadow figures lacking the body which cast them, symbolic differential coefficients without the real differential coefficients, that is, without the corresponding equivalent ‘derivative’. Thus the symbolic differential coefficient becomes the autonomous starting point whose real equivalent is first to be found. The initiative is thus shifted from the right-hand pole, the algebraic, to the left-hand one, the symbolic. Thereby, however, the differential calculus also appears as a specific type of calculation which already operates independently on its own ground (Boden). For its starting points du/dx , dz/dx belong only to it and are mathematical quantities characteristic of it. And this inversion of the method arose as a result of the algebraic differentiation of uz. The algebraic method therefore inverts itself into its exact opposite, the differential method.^*4

Now, what are the corresponding ‘derivatives’ of the symbolic differential coefficients du/dx , dz/dx ? The beginning equation y = uz provides no data for the resolution of this question. This last [question] may still be answered if one substitutes arbitrary original functions of x for u and z. For example,

u = x⁴ ; z = x³ + ax² .

Thereby, however, the symbolic differential coefficients du/dx , dz/dx are suddenly transformed into operational symbols (Operationssymbole), into symbols of the process which must be carried out with x⁴ and x³ + ax² in order to find their ‘derivatives’. Originally having arisen as the symbolic expression of the ‘derivative’ and thus already finished, the symbolic differential coefficient now plays the rolw of the symbol of the operation of differentiation which is yet to be completed.

At the same time the equation

dy/dx = z⋅du/dx + u⋅dz/dx ,

from the beginning purely symbolic, because lacking a side free of symbols has been transformed into a general symbolic operational equation.

I remark further that^*5 from the early part of the 18th century right down to the present day, the general task of the differential calculus has usually been formulated as follows: to find the real equivalent of the symbolic differential coefficient.

4)

A) dy/dx = z⋅du/dx + u⋅dz/dx .

This is obviously not the simplest expression of equation A), since all its terms have the denominator dx in common. Let this be struck out, and then:

B) d(uz) or dy = z⋅du + u⋅dz .

Any trace in B) of its origin in A) has disappeared. It is therefore equally as valid when u and z depend on x as when they depend only reciprocally on one another, without any relation to x at all.²¹ From the beginning it has been a symbolic equation and from the beginning could have served as a symbolic operational equation. In the present case it means, that when

that is = a product of any arbitrary number of variables multiplied together, then dy = a sum of products, in each one of which one of the factors is treated as a variable while the other factors are treated as constants, etc.

For our purpose, namely the further investigation of the differential of y in general, form B) nonetheless will not do. We therefore set:

u = x⁴, z = x³ + ax² .

so that

du = 4x³dx, dz = (3x² + 2ax)dx ,

as was proved earlier for equations with only one dependent variable. These values of du, dz are brought into equation A), so that

A) dy/dx = (x³ + ax²)(4x³dx)/dx + x⁴(3x² +2ax)dx/dx ; and then

dy/dx = (x³ + ax²)4x³ + x⁴(3x² + 2ax) ;

therefore

dy = {(x³ + ax²)4x³ + x⁴(3x² + 2ax)}dx .

The expression in brackets is the first derivative of uz; since, however, uz = f(x), its derivative is = f’(x); we now substitute the latter in place of the algebraic function, and so:

dy = f’(x)dx .

We have already obtained the same result from an arbitrary equation with only one variable. For example:

y = x^m,

dy/dx = mx^m-1 = f’(x) ,

dy = f’(x)dx .

In general we have if y = f(x), whether this function of x is now an original function in x or contains a dependent variable, then always dy = df(x) and df(x) = f’(x)dx, and so:

B) dy = f’(x)dx is the most generally valid form of the differential of y. This would be demonstrable immediately also if given f(x) were f(x, z), that is a function of two mutually independent variables. For our purposes, however, this would be superfluous.

II

1) The differential

dy = f’(x)dx

appears right away to be more suspicious than the differential coefficient

dy/dx = f’(x)

from which it is derived.

In dy/dx = 0/0 the numerator and denominator are inseparably bound; in dy = f’(x)dx they are apparently separated, so that one is forced to the conclusion that it is only a disguised expression for

0 = f’(x)⋅0 or 0 = 0 ,

whereupon ‘nothing’s to be done’ (‘nix zu wolle’).

A French mathematician of the first third of the 19th century, who is clear in a completely different manner than the well-known [to you] ‘elegant’ Frenchman,²² has drawn a connection between the differential method and Lagrange’s algebraic method: - Boucharlat says:

If for example dy/dx = 3x², then ‘dy/dx alias 0/0, or rather its value 3x², is the differential coefficient of the function y. Since dy/dx is thus the symbol which represents the value 3x², dx must always stay (stehn)^*6 under dy, but in order to facilitate algebraic operation we treat dy/dx as an ordinary fraction and dy/dx = 3x² as an ordinary equation. By removing the denominator from the equation one obtains the result

dy = 3x²dx,

which expression is called the differential of y’.²³

Thus in order ‘to facilitate algebraic operation’, one introduces a demonstrably false formula which one baptises the ‘differential’.

In fact the situation is now so nasty.

In 0^*7/0 the numerator is inseparable from the denominator, but why? Because both only express a ratio if they are not separated, something like (dans l’espèce) the ratio²⁴ reduced to its absolute minimum:

(y₁ - y)/(x₁ - x) = (f(x₁) - f(x))/(x₁ - x) ,

where the numerator goes to 0 because the denominator has done so. Separated, both are 0; they lose their symbolic meaning, their reason.

As soon, however, as x₁ - x = 0 achieves in dx a form which is manifested without modification as the vanished difference in the independent variable x, so that dy as well is a vanished difference in the function of x or in the dependent [variable] y, then the separation of the denominator from the numerator becomes a completely permissible operation. Wherever dx stands now, such a change of position leaves the ratio of dy to dx undisturbed, dy = f’(x)dx thus appears to us to be an alternative form of

dy/dx = f’(x)

and may always be substituted for the latter.²⁵

2) The differential dy = f’(x)dx arose from A) by means of a direct algebraic derivation (see I,4), while the algebraic derivation of equation A) had already shown that the differential symbol, somewhat like (dans l’espéce) the symbolic differential coefficient which originally emerged as a purely symbolic expression of the algebraically performed process of differentiation, necessarily inverts into an independent starting point, into a symbol of an operation yet to be performed, into an operational symbol, and thus the symbolic equations which have emerged along the algebraic route also invert into symbolic operational equations (Operationsgleichungen).

We are thus doubly correct in treating the differential y = f’(x)dx as a symbolic operational equation. So we now know a priori, that if

y = f(x) [then] dy = df(x) ,

that if the operation of differentiation indicated by df(x) is performed on f(x), the result is dy = f’(x)dx, and that from this result finally

dy/dx = f’(x) .

As well, however, from the first moment that the differential functions as the starting point of the calculus, the inversion of the algebraic method of differentiation is complete, and the differential calculus itself therefore appears, a unique, specific method of calculating with variable quantities.

In order to make this more graphic I will combine at once all the algebraic methods which I have used, while setting simply f(x) in place of a fixed algebraic expression in x, and the ‘preliminary derivative’ (see the first manuscript^*8) will be designated as f¹(x) to distinguish from the definitive ‘derivative’, f’(x). Then, if

f(x) = y, f(x₁) = y₁ ,

[then]

f(x₁) - f(x) = y₁ - y or Δy ,

f¹(x) (x₁ - x) = y₁ - y or Δy.

The preliminary derivative must^† contain expressions in x₁ and x exactly like the factor (x₁ - x) with the single exception when f(x) is an original function to the first power:

f¹(x) = (y₁ - y)/(x₁ - x) or Δy/Δx .

We now substitute into f¹(x)

x₁ = x so that x₁ - x = 0 ,

and thus obtain:

f’(x) = 0/0 or dy/dx

and finally

f’(x)dx = dy or dy = f’(x)dx

The differential of y is therefore the conclusion of an algebraic development; it becomes the starting point for differential calculus operating on its own ground. dy, the differential²⁶ of y - considered in isolation, that is, without its [real] equivalent - here immediately plays the same role as Δy in the algebraic method; and the differential of x, dx, the same role as Δx does there.

If we had, in

Δy/Δx = f¹(x)

cleared the denominator, then

I) Δy = f¹(x) Δx .

On the other hand, beginning with the differential calculus as a separate, complete type of calculating - and this point of departure has been itself derived algebraically - we start immediately with the differential expression of I), namely:

II) dy = f’(x)dx .

3) Since the symbolic differential equation (Gleichung des Differentials) arises simply by the algebraic handling of the most elementary functions with only one independent variable, it appears that the inversion of the method (Umschlag in der Methode) could have been developed in a much more simple manner than happened with the example

y = uz .

The most elementary functions are those of the first degree; they are:

a) y = x, which leads to the differential coefficient dy/dx = 1, so that the differential is dy = dx.

b) y = x ± ab; it leads to the differential coefficient dy/dx = 1, so that again the differential is dy = dx.

c) y = ax; it leads to the differential coefficient dy/dx = a, so that the differential is dy = adx.

Let us take the simplest case of all (under a)). Then:

y = x,

y₁ = x₁ ;

y = uz .

y₁ - y or Δy = x₁ - x or Δx .

I) (y₁ - y)/(x₁ - x) or Δy/Δx = 1. Thus also Δy = Δx. In Δy/Δx x₁ is now set = x, or x₁ - x = 0, and thus:

II) 0/0 or dy/dx = 1; so that dy = dx.

Right at the start, as soon, as we obtain I) Δy/Δx = 1, we are forced to operate further on the left-hand side, since on the right-hand side is the constant, 1. And therein the inversion of the method, which throws the initiative from the right-hand side to the left-hand side, once and for all from the group up proves to be in fact the first word of the algebraic method itself.

Let’s look at the matter more closely.

The real result was:

I) Δy/Δx = 1.

II) 0/0 or dy/dx = 1.

Since both I) and II) lead to the same result we may choose between them. The setting of x₁ - x = 0 appears in any case to be a superfluous and therefore an arbitrary operation. Further: we operate from here on in II) on the left-hand side, since on the right-hand side ‘ain’t no way’, so that we obtain:

0/0 or d²y/dx² = 0 .

The final conclusion would be that 0/0 = 0, so that the method is erroneous with which 0/0 was obtained. At the first use^*9 it leads to nothing new, and at the second to exactly nothing.²⁷

Finally: we know from algebra that if the second sides of two equations are identical, so also must the first sides be. It therefore follows that:

dy/dx = Δy/Δx .

Since, however, both x and y, the variable dependent on x, are variable quantities, Δx while remaining a finite difference may be infinitely shortened; in other words it can approach 0 as closely as one wants, so that it becomes infinitely small; therefore the Δy dependent on it does so as well. Further, since dy/dx = Δy/Δx it follows therefrom that dy/dx really signifies, not the extravagant 0/0, but rather the Sunday dress (Sonntagsuniform) of Δy/Δx, as soon as the latter functions as a ratio of infinitely small differences, hence differently from the usual difference calculation.

For its part the differential dy = dx has no meaning, or more correctly only as much meaning as we have discovered for both differentials in the analysis of dy/dx. Were we to accept the interpretation just given,²⁸ we could then perform miraculous operations with the differential, such as for example showing the role of adx in the determination of the subtangent of the parabola, which by no means requires that the nature of dx and dy really be understood.

4) Before I proceed to section III, which sketches the historical path of development of the differential calculus on an extremely condensed scale, here is one more example of the algebraic method applied previously. In order graphically to distinguish it I will place the given function on the left-hand side, which will always be the side of the initiative, since we always write from left to right, so that the general equation is:

x^m + Px^m-1 + etc. + Tx + U = 0 ,

and not

0 = x^m + Px^m-1 + etc. + Tx + U .

If the function y and the independent variable x are divided into two equations, of which the first expresses y as a function of the variable u, while on the other hand the second expresses u as a function of x, then both symbolic differential coefficients in combination are to be found.²⁹ Assuming:

1) 3u² = y , 3u₁² = y₁ ,

then

2) x³ + ax² = u ; x₁³ + ax₁² = u₁ .

We deal with equation 1) for the present:

3u₁² - 3u² = y₁ - y ,

3(u₁² - u²) = y₁ - y ,

3(u₁ - u) (u₁ + u) = y₁ - y ,

3(u₁ + u) = (y₁ - y)/(u₁ - u) or Δy/Δu .

On the left-hand side u₁ is now set = u, so that u₁ - u = 0, then:

3(u + u) = dy/du ,

3(2u) = dy/du ,

6u = dy/du .

We now substitute for u its value x³ + ax², so that:

3) 6(x³ + ax²) = dy/du .

Now applying ourselves to equation 2):

x₁³ + ax₁² - x³ - ax² = u₁ - u ,

(x₁³ - x³) + a(x₁² - x²) = u₁ - u ,

(x₁ - x) (x₁² + x₁x + x²) + a(x₁ - x) (x₁ + x) = u₁ - u ,

(x₁² + x₁x + x²) + a(x₁ + x) = (u₁ - u)/(x₁ - x) or Δu/Δx .

We set x₁ = x on the left-hand side, so that x₁ - x = 0 .

(x² + xx + x²) + a(x + x) = du/dx .

4) 3x² + 2ax = du/dx .

We now multiply equations 3) and 4) together, so that:

5) 6(x³ + ax²) (3x² + 2ax) = dy/du ⋅ du/dx = dy/dx .³⁰

Thus, by algebraic means the operational formula:

dy/dx = dy/du ⋅ du/dx ,

has been found, which is also occasionally applicable to equations with two independent variables.

____________

The above example shows that it is not witchcraft to transform a development demonstrated from given functions into a completely general form. Assume:

1) y = f(u), y₁ = f(u₁), y₁ - y = f(u₁) - f(u),

so that therefore

2) u = φ(x), u₁ = φ(x₁), u₁ - u = φ(x₁) - φ(x) .

From the difference under 1) comes:

(y₁ - y)/(u₁ - u) = (f(u₁) - f(u))/(u₁ - u) ; dy/du = df(u)/du ,

however, since du(u) = f’(u)du,

dy/du = f’(u)du/du ;

consequently

3) dy/du = f’(u) .

From the difference under 2) follows:

(u₁ - u)/(x₁ - x) = (φ(x₁) - φ(x))/(x₁ - x) , du/dx = dφ(x)/dx ,

and since dφ(x) = φ’(x)dx ,

du/dx = φ’(x)dx/dx ,

so that:

4) du/dx = φ’(x) .

We multiply equation 3) by 4), so that:

5) dy/du ⋅ du/dx or dy/dx³¹ = f’(u)⋅φ’(x) Q.E.D.

N. III. The conclusion of this second instalment will follow, as soon as I consult John Landen at the Museum.³²

¹³ The manuscript dates from 1881. On the envelope attached to the manuscript is written ‘II For Fred’ (II Für Fred). Marx calls this manuscript the ‘second instalment’ (see p.33), since, in it he continues to set forth the views at which he arrived in the process of studying mathematics. Engels showed the manuscript to S[amuel] Moore and conveyed the latter’s comments to Marx in his letter of November 21, 1882 (see p.xxix). The manuscript ‘On the Differential’ was first published (not in full) in Russian translation in the 1933 collection Marxism and Science (Marksizm I estestvoznanie), pp.16-25; and in the journal Under the Banner of Marxism (Pod znamenem marksizma), 1933, No.1.
^* The last part of the equation was apparently added by Engels - Ed.
¹⁴ Marx thus assumes here that the functions u and z, which, as subsequently becomes clear, are defined by means of the equations u = f(x), z = φ(x) (where f(x) and φ(x) are expressions ‘in the variable x’), are differentiable functions of x. The fact that no further information on the functions f(x) and φ(x) is required to prove the theorem on the differential of the product of two functions, is reflected in Marx’s graphic comments regarding du/dx, dz/dx : ‘shadow figures lacking the body which cast them, symbolic differential coefficients without the real differential coefficients, that is without the corresponding equivalent “derivative” ’(see p.20). Marx also discusses this specifically in his rough draft essays on the differential. Here and hereafter we shall write d(uz) instead of the contraction d⋅uz which Marx used in his manuscripts.
^*2 Synonymous with ‘derivative’ - Trans.
¹⁵ The symbols for derivative and differential which are specific to differential calculus are intended here.
¹⁶ In the literature of the 18th-19th centuries the derivative was often called the ‘differential coefficient’, which is obviously related to the definition of the derivative as the coefficient of the first power of the increment h of the independent variable x in the expansion of the expression f(x + h) into a series of powers of h. The adjective ‘real’ refers to the fact that the expression for f’(x) contains no symbols which are restricted to differential calculus.
¹⁷ This way of speaking, in which as a result of multiplication by zero ‘the variables u and z themselves become equal to zero,’ is explained by the fact that in Marx’s time there still existed widespread conceptions of mathematical operations on numbers as changing the numbers themselves: the addition of the positive number b to a ‘increases the number a’, the multiplication of a by 0 ‘changes the number a to zero’, and so on. These conceptions were put on a scientific basis only in the 20th century.
¹⁸ The words ‘since we can begin the nullification arbitrarily with numerator or denominator’ obviously mean that the predefinition of an expression of the form f(x)/g(x), which at x = a becomes 0/0 and therefore loses any meaning, may be established for x = a in a number of different ways. If we wish to preserve in the predefinition that property of the ordinary fraction which makes it equal to zero when the numerator is equal to zero, then the value of f(a)/g(a) must be zero. ‘To begin the nullification with the numerator’ in this case simply means to set f(a)/g(a) equal to zero. Since, however, a fraction with a denominator of 0 does not exist, ‘to begin the nullification with the denominator’ makes it impossible to retain in the predefinition anything of the properties of an ordinary fraction with a zero denominator. But if for all x ≠ a f(x)/g(x) = φ(x) = φ(a)), then it is natural to set f(a)/g(a) equal to φ(a), retaining in this manner the equation f(x)/g(x) = φ(x) even for x = a. If the numerator is also transformed to zero because the denominator is set at zero, then the words ‘begin the nullification with the denominator’ may be explained naturally as denoting: predefine in the above-mentioned manner, that is, ‘using continuity’. In the books which Marx used, even including the large Traité of Lacroix, the preservation of the equation f(a)/g(a) = φ(a) in the case of f(a) = g(a) = 0 was considered independent, in general, of whatever may have been ‘derived’; it was a necessary consequence of the metaphysical law of the continuity of ‘all real numbers’.
¹⁹ There is a slip of the pen here in the text: instead of x = a there appears x² = a². Instead of correcting it, someone, apparently Moore, made insertion marks in the text in pencil, after which he observed, ‘und da x² = a² ∴ x = ± a = = 2Pa oder 0,’ that is ‘and since x² = a², then x = ± a, [whence P(x + a)] = 2Pa or [=] 0’. Such an interpretation, however, clearly does not agree with the overall context.
^*3 ‘Definite expression’ is meant - Ed.
²⁰ Marx here calls the expression dy/dx, which was obtained by the transition from a ratio of finite differences to the derivative, the symbolic differential expression for (y₁ - y)/(x₁ - x) corresponding to (f(x₁) - f(x))/(x₁ - x) .
^*4 The draft of the work ‘On the Differential’ (4148, Pl.16-17) contains this paragraph: ‘du/dx , dz/dx thrown over. Born within the derivative, they, together with the remaining elements of the same, meet in dy/dx their own symbolic expression, therefore their symbolic equivalent. But they themselves exist without equivalent, real differential coefficients, that is without the derivative f’(x), φ’(x) whose symbolic expression they in turn had been. They are the completed differential symbols whose real values figure as shadows whose bodies are to be sought first. The problem has thus been turned around before one’s eyes. The symbolic differential coefficients have become autonomous starting points, for whom the equivalent, the real differential coefficient or the corresponding derived function, is first to be sought. Thereby the initiative has been shifted from the right-hand pole to the left. Since this inversion of the method originated from the algebraic manipulation of the function uz, it has itself been demonstrated algebraically.’ - Ed.
^*5 The following is in the draft: ‘save for a few exceptions.’ - Ed.
²¹ Apparently this concerns the case where the choice of independent variable is not necessarily fixed, where either u or z may be used as the independent variable. In general, if u and z may be considered to be interchangeable functions of one and the same independent variable, then assigning a value to either one of u and z determines the value of the independent variable and, of course, the value of the other function as well. In other words, what is intended here is the (?)invariance of the symbolic operational equation with respect to the choice of independent variable.
²² Apparently the word ‘dir’ (to you) in the phrase ‘der(?) dir bekannte’ (which is known to you) was omitted during recopying, although it is preserved in the notebooks. It is to be understood that this concerns the French mathematician L.B. Francoeur, about whom Engels wrote to Marx in the letter of May 30, 1864. The word in quotation marks, ‘elegant’, refers to Engels’s comment, ‘Einzelnes ist sehr elegant’ (‘Someone is very elegant’), and contains, obviously, a hint of an ironic relationship of Engels to the author under discussion. Francoeur, like Boucharlat and some others, tried to combine the ‘algebraic’ method of Lagrange (see pp.24) with the differential calculus of Leibnitz, all the while operating with the symbols of differentials. Marx’s note of irony about the ‘clarity’ with which this was done, concerns both Boucharlat and Francoeur. The first, in order to ‘facilitate algebraic operation’, introduced an absurd formula; the second, suggested that the differential ‘appears synonymous to the derivative and differs from it only ambiguously’, consequently, he also wrote, ‘the derivative of x is x´ = 1 or dx = 1’.
^*6 The draft has: ‘remain’ (stehn bleiben) - Ed.
²³ The extract in quotation marks is a text translated from the French of the books of J.-L. Boucharlat. See, for example Elémens de calcul differential et de calcul intégral fifth edition, 1838, p.4.
^*7 The draft has : ‘In the form 0/0’ - Ed.
²⁴ The reduction to its ‘absolute minimum’ here obviously implies the stated predefinition of the ratio by continuity at x₁ = x; that is, in essence, the transition to the limit where x₁→x.
²⁵ See Appendix III, ‘On the Calculus of Zeroes of Leonhard Euler’, p.160
^*8 See ‘On the Concept of the Derived Function’, p.3 above - Ed.
^† The draft has: ‘must as a rule’ - Ed.
²⁶ Marx here makes a distinction between the differential particles (die Differentiellen) dx and dy, which represent the ‘removed’ differences Δx and Δy, and the differential (das Differential) dy, which is defined by the equations dy = f’(x)dx . (1) This last equation can be treated as an operational formula which makes it possible to find the derivative f’(x) by means of the already determined differentials dy and dx, transforming equation (1) to its equivalent (see note [24]) dy/dx = f’(x) (2)
^*9 Original: ‘coup’, French for ‘strike’, ‘blow’ - Trans.
²⁷ Marx’s argument against applying the method of treatment which already took place in the ‘algebraic’ differentiation of the simplest functions of first order consists of the following: 1) the step which consists of assuming x₁ = x is superfluous, since the preliminary derivative here already agrees with the final one; that is, that which is specific to the ‘algebraic’ method of differentiation does not come to light; 2) thee extension to the general case of attributes of differential functions of the first order may lead to the completely erroneous conclusion that all derivatives of higher order, beginning with the second, must be equal to zero.
²⁸ That is, consider dy/dx a ratio of infinitely small quantities, as Leibnitz and Newton had done already.
²⁹ That is, to find the derivative of y with respect to x, considering y as a function of x, given by the two equations: 1) y = 3u², 2) u = x³ + ax² .
³⁰ Marx assumes here that it has already been established that it is correct to operate with differentials as if they were ordinary fractions (see p.24 and Appendix V, p.173).
³¹ At this point in the manuscript Moore made the following note in pencil ‘On p.12(5) this is proved for the concrete case there investigated. Should it not be proved instead of assumed for the general case also?’ [English is garbled in text; recovered from Russian translation - Trans.] This note, however, is based on a misunderstanding. The ‘development demonstrated from given functions’ consisted of the symbolic expressions dy/du and du/dx which had been obtained as a result of differentiation. Since, as Marx has already assumed, it is correct to operate with such expressions as if they were conventional fractions, the conclusion was natural that (dy/du)⋅(du/dx)=dy/dx .
³² Marx did not write section III apparently because he did not succeed in carrying out his intention of studying John Landen’s book in the British Museum (see Appendix IV).

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