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.

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*.

*y _{1} = u_{1}z_{1},*

*y _{1} - y = u_{1}z_{1} - uz = z_{1}(u_{1} - u) + u(z_{1} - z),*

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

Now on the right-hand side let *x _{1} = x*, so that

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 free^{15} of all symbolic expressions, for example, *mx ^{m-1}* when 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

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

become *0/0*, *0/0*, because *x _{1}* has become

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

*(u _{1} - u)/(x_{1} - x) = Δu/Δx ;*

one does not first obtain the numerator = 0 because one has begun with it and set u_{1} - u = 0, but rather the numerator only becomes 0 or u_{1} - u = 0 because the denominator, the difference of the independent variable quantities x, that is x_{1} - 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 begin^{18} 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

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), u _{1} - u = f(x_{1}) - f(x) ,*

and

*z = φ(x); z _{1} - z = φ(x_{1}) - (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)* :

*(u _{1} - u)/(x_{1} - x) = (f(x_{1}) - f(x))/(x_{1} - x) , (z_{1} - z)/(x_{1} - x) = (φ(x_{1}) - φ(x))/(x_{1} - x)*

is generated immediately by the process of taking the derivative. The process has now reached the point where *x _{1}* is set

*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

*(y _{1} - y)/(x_{1} - x) = (f(x_{1}) - f(x))/(x_{1} - x)*.

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 = x ^{4} ; 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 *x ^{4}* and

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 = x ^{4}, 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 + x ^{4}(3x² +2ax)dx/dx* ; and then

*dy/dx = (x³ + ax²)4x³ + x ^{4}(3x² + 2ax) ;*

therefore

*dy = {(x³ + ax²)4x³ + x ^{4}(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.

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

*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 ratio^{24} reduced to its absolute minimum:

*(y _{1} - y)/(x_{1} - x) = (f(x_{1}) - f(x))/(x_{1} - 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 *x _{1} - x = 0* achieves in

*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(x _{1}) = y_{1} ,*

[then]

*f(x _{1}) - f(x) = y_{1} - y* or

*f¹(x) (x _{1} - x) = y_{1} - y* or

The preliminary derivative must^{†} contain expressions in *x _{1}* and

*f¹(x) = (y _{1} - y)/(x_{1} - x)* or

We now substitute into *f¹(x)*

*x _{1} = x* so that

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 differential^{26} 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,*

*y _{1} = x_{1} ;*

*y = uz .*

*y _{1} - y* or

I) *(y _{1} - y)/(x_{1} - x)* or

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 *x _{1} - 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 , 3u _{1}² = y_{1}* ,

then

2) *x³ + ax² = u ; x _{1}³ + ax_{1}² = u_{1} *.

We deal with equation 1) for the present:

*3u _{1}² - 3u² = y_{1} - y ,*

*3(u _{1}² - u²) = y_{1} - y ,*

*3(u _{1} - u) (u_{1} + u) = y_{1} - y ,*

*3(u _{1} + u) = (y_{1} - y)/(u_{1} - u) or Δy/Δu .*

On the left-hand side u_{1} is now set = u, so that u_{1} - 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):

*x _{1}³ + ax_{1}² - x³ - ax² = u_{1} - u ,*

*(x _{1}³ - x³) + a(x_{1}² - x²) = u_{1} - u ,*

*(x _{1} - x) (x_{1}² + x_{1}x + x²) + a(x_{1} - x) (x_{1} + x) = u_{1} - u ,*

*(x _{1}² + x_{1}x + x²) + a(x_{1} + x) = (u_{1} - u)/(x_{1} - x)* or

We set x_{1} = x on the left-hand side, so that x_{1} - 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), y _{1} = f(u_{1}), y_{1} - y = f(u_{1}) - f(u),*

so that therefore

2) *u = φ(x), u _{1} = φ(x_{1}), u_{1} - u = φ(x_{1}) - φ(x) .*

From the difference under 1) comes:

*(y _{1} - y)/(u_{1} - u) = (f(u_{1}) - f(u))/(u_{1} - 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:

*(u _{1} - u)/(x_{1} - x) = (φ(x_{1}) - φ(x))/(x_{1} - 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/dx ^{31} = 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 *(y _{1} - y)/(x_{1} - x)* corresponding to