Marx's Mathematical Manuscripts 1881

# Appendix V. The Principles of Differential Calculus according to Boucharlat

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

Of the books of mathematical analysis available to Marx, obviously of the greatest significance for the understanding of his manuscripts is the textbook of Boucharlat, Elementary Treatise on the Differential and Integral Calculus, with which Marx was acquainted in the English version of the third French edition, translated by Blakelock and published in 1828.

This textbook enjoyed a great popularity and was several times reprinted. Its eighth edition with the commentaries of M.H. Laurent, saw the light in Paris in 1881. It was translated into a variety of foreign languages, among them Russian.

Graduate of the Ecole Polytechnique, professor of ‘transcendental’ (higher) mathematics, author of a series of textbooks of mathematics and mechanics, Jean-Louis Boucharlat (1775-1848) was at the same time a poet, and since 1823, professor of literature at the Parisian Atheneum.

No doubt his literary accomplishments and clarity of exposition were responsible in no small part for the popularity of Boucharlat’s textbook. It is clear that Marx did not turn his attention accidentally to the course-book of Boucharlat.

All the same, despite the pretentions of the author to great rigour in his account and to having perfected the ‘algebraic’ method of Lagrange by means of the method of limits (see the introduction to the fifth edition, 1838, p.VIII) the mathematical level of this course was not very elevated. Even in the fifth (of 1838) and not only in the third edition, the English translation of which Marx consulted, the concepts of limit, function, derivative, differential are introduced thus:*

‘1. One variable is said to be a function of another variable, when the first is equal to a certain analytical expression composed of the second; for example, y is a function of x in the following equations:

y = sqrt{a² - x²}, y = x³ - 3bx², y = x²/a , y = b + cx³ .

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‘3. Let us take also the equation

y = x³ (1)

and suppose that when x becomes x + h, y becomes y’, we have then

y’ = (x + h)³

or, by expanding,

y’ = x³ + 3x²h + 3xh² + h³ ;

if from this equation we subtract equation (1) there will remain

y’ - y = 3x²h + 3xh² + h³ ,

and by dividing by h,

(y’ - y)/h = 3x² + 3xh + h² . (2)

‘Let us look at what this result teaches us:

y’ - y represents the increment of the function y when x receives the increment h, because this difference y’ - y is the difference between the new state of the value of the variable y and its original state.

‘On the other hand since the increment of the variable x is h, it follows from this that the expression (y’ - y)/h is the ratio of the increment of the function y to the increment of the variable x. Looking at the second term of equation (2), we see that this ratio decreases together with the decrease of h and that when h becomes zero this ratio is transformed into 3x².

‘Consequently the term 3x² is the limit of the ratio (y’ - y)/h; it approaches this term when we cause h to be decreased.

‘4. Since, in the hypothesis that h = 0 the increment of y also becomes zero, then (y’ - y)/h is transformed into 0/0, and therefore there is obtained from equation (2)

0/0 = 3x² . (3)

‘There is nothing absurd in this equation, since algebra teaches us that 0/0 may represent any value at all. On the other hand it is clear that since division of both parts of a fraction by one and the same number does not change the value of the fraction, we may then conclude that the smallness of the parts of a fraction has no effect at all on its value, and that consequently it may remain the same value, even when its parts attain the last degree of smallness, that is, are transformed to zero.

‘The fraction 0/0 which appears in equation (3) is a symbol which has replaced the ratio of the increment of the function y to the increment of the variable x; since to trace remains in this symbol of the variable, we will represent it by dy/dx; then dy/dx will remind us that the function was y and the variable x. But this dy and dx will not cease to be zero, and we will have

dy/dx = 3x² . (4)

dy/dx, or more precisely its value 3x², is the differential coefficient of the function y.

‘Let us note that since dy/dx is the sign representing the limit 3x² (as equation (4) shows), dx must always be located beneath dy. However, in order to facilitate algebraic operations it is permitted to clear the denominator in equation (4), and we obtain dy = 3x²dx. This expression 3x²dx is called the differential of the function y.’ (pp.1-4)

In §§ 5-8 Boucharlat finds dy in the examples

y = a + 3x², y = ((1 - x³)/(1 - x)) , y = (x² - 2a²) (x² - 3a²) .

In all these these cases the expression for the increased value of y, that is (in Boucharlat’s notation) for y’, is equal to f(x + h) - if y = f(x) - and is represented in the form of a polynomial, expanded in powers of h (with coefficients in x), after which the ratio (y’ - y)/h is easily represented as a polynomial of the same type. Setting h = 0 in this ratio gives dy/dx, and multiplication by dx completes the search for the expression for the differential dy.

‘9. The expression dx is itself the differential of x; let y = x, then y’ = x + h, consequently y’ - y = h, and then (y’ - y)/h = 1. Since the quantity h does not even enter the second term of this equation, it is enough to change (y’ - y)/h to dy/dx which will give dy/dx = 1; consequently, by our hypothesis, dy = dx.

‘10. We find in the same way that the differential of ax is adx; but if we had y = ax + b we also would have obtained adx for the differential, whence it follows that the constant b, unaccompanied by the variable x, provides no term at all upon differentiation or, in other words, has no differential at all.

‘In addition one may note that if y = b, then in the case before us, where a is zero in the equation y = ax + b and where therefore dy/dx = a is now reduced to dy/dx = 1, there is neither limit nor differential.’ (p.6)

We see from the above that according to Boucharlat:

1) There is neither a definition of limit, nor of derivative or differential. All these concepts are explained only in examples, and only such that the ratio (f(x + h) - f(x))/h is represented as a polynomial expanded in powers of h, with coefficients in x. The evaluation of the limit of this ratio as h → 0 is treated as the supposition that h = 0 in the obtained polynomial. Here questions whether there exist other cases, whether in such cases it is possible to ‘differentiate’, and if so, how, do not even arise.

2) The passage from the derivative dy/dx = φ(x) to the differential dy = φ(x)dx is regarded as an unlawful operation, carried out only in order to ‘facilitate’ algebraic calculation.

3) From the fact that for h ≠ 0

>

(f(x + h) - f(x)/h = φ(x, h) , (A)

is drawn the conclusion that for h = 0, that is, when (f(x + h) - f(x))/h loses all meaning, (is transformed into 0/0), equation (A) retains significance, that is, we should obtain

0/0 = φ(x, 0) . (B)

In other words, it is considered φ(x, h) should be defined (and continuous) for h = 0 and that equation (B) follows logically from equation (A) - although the expression 0/0 is without meaning.

4) The limit or differential equalling zero is rationalised as indicating that ‘there is neither limit nor differential’ although at the same time dy and dx are always zeroes (if φ(x)≠0, then the differential, equal to φ(x)⋅0, exists, if φ(x)≡0, then it doesn’t).

It is not surprising that such a treatment of the fundamental concepts of the differential calculus did not satisfy Marx. And in fact the first of his outlines of the opening paragraph of the course-book of Boucharlat (see p.65 of the present edition) contains critical remarks concerning that author. But Marx was displeased in particular with the fact that the fundamental concept of differential calculus - the concept of the differential - appeared without foundation and its introduction justified only because it ‘facilitates algebraic operations’. See the manuscript ‘On the Differential’, p.15).

In §11 of Boucharlat’s book the remark is made, ‘sometimes the increment of the variable is negative; in that case we must put x - h for x, and proceed as before’. In the example y = - ax³ by this means is obtained dy = - 3ax²dx, and the conclusion drawn : ‘We see that this comes to the same thing as supposing dx negative in the differential of y calculated on the hypothesis of a positive increment.’ But for Boucharlat dx is 0. The question of the meaning of ‘negative zero’ never came into his head, however. (In the works of this period there was still no general concept of ‘absolute value’.)

Since the following three paragraphs, §§12-14, are particularly characteristic of Boucharlat’s course-book and since they are related to a variety of passages in the manuscripts of Marx, the text of these paragraphs is reproduced here in full.

‘12. Before proceeding further, we must make one essential remark; viz., that in an equation, of which the second side is a function of x, and which for that reason, we will represent generally by y = f(x), if on changing x into x + h, and arranging the terms according to the powers of h, we find following development:

y’ = A + Bh + Ch² + Dh³ + etc. , (C)

we ought always to have y = A.

‘For if we make h = 0, the second side is reduced to A. In regard to the first side, since we have accented y only, to indicate that y has undergone a certain change on x becoming x + h, it follows necessarily, that when h is 0, we must suppress the accent of y and the equation will be reduced then to

y = A .

‘13. This will give us the means of generalising the process of differentiation. For, if in the equation y = f(x) in which we are supposed to know the expression represented by f(x), we have put x + h in place of x; and after having arranged the terms according to the powers of h, are able to obtain the following development:

y’ = A + Bh + Ch² + Dh³ + etc.

or rather, according to the preceding article,

y’ = y + Bh + Ch² + etc. ,

we shall have

y’ - y = Bh + Ch² + etc. ,

therefore

(y’ - y)/h = B + Ch + etc.

and taking the limit, dy/dx = B; which shows us that the differential coefficient is equal to the coefficient of the term which contains the first power of h, in the development of f(x + h), arranged according to ascending powers of h.

‘14. If instead of one function y, which changes its value in consequences of the increment given to the variable x which it contains, we have two functions, y and z, of that same variable x, and we know how to find separately the differentials of each of these functions, it will be easy, by the following demonstration, to determine the differential of the product zy of these functions. For if we substitute x + h in place of x, in the functions y and z, we shall obtain two developments, which, being arranged according to powers of h, may be represented thus,

y’ = y + Ah + Bh² + etc. , (5)

z’ = z + A’h + B’h² + etc. (6)

Passing to the limit, we shall find

dy/dx = A , dz/dx = A’ ; (7)

multiplying equations (5) and (6) the one by the other, we shall obtained

z’y’ = zy + Azh + Bzh² + etc. +

+ A’yh + AA’h² + etc. +

+ B’yh² + etc. ,

therefore

(z’y’ - zy)/h = Az + A’y + (Bz + AA’ + B’y)h + etc ;

and taking the limit, and indicating, by a point placed before it, the expression to be differentiated, we shall get

d⋅zy/d = Az + A’y ;

and suppressing the common factor dx,

d⋅zy = z⋅dy + y⋅dz .

‘Thus, to find the differential of the product of two variables, we must multiply each by the differential of the other, and add the products.’ (pp.6-8)

In §15 this is correctly used to determine the differential of the product of three variables, in §16 to obtain the differential of the fraction y/z.

In §17 the differential of the power function y = xm for a positive m is obtained from the formula

d⋅xyztu etc./xyztu etc. = dy/x + dy/y + dz/z + dt/t + du/u + etc. (9)

under the supposition that x, y, z, t, u etc. are equal to x and are taken m times.

§18 contains the formula for correctly differentiating a power function.

In §19 by the use of the formula for operation with the differential symbols (having related the problem to previous cases) it is correctly shown in the cases of fractional and negative exponents.

In §20 the differential of a power [function] is obtained immediately by the expansion of (x + h)m according to the binomial theorem of Newton.

In the third edition of Boucharlat’s course-book, The English translation of which Marx used, there is a ‘Note Second’ in the appendices with a title beginning, ‘Considerations which prove the solidity of differentiation ...’ Since this comment attracted Marx’s special attention, its text is introduced here (in part):

‘With the exception of the differentials of circular functions, which, as we have already seen, are readily found by the formulae of trigonometry, all the other monomial differentials, such, for example, as those of xm, ax, log x, etc., have been deduced from the binomial theorem alone. We have, it is true, had recourse to the theorem of MacLaurin, in the determination of the constant A in the exponential formulae, but we might have dispensed with it.’

Later, with the help of formal manipulations of infinite series which are not at all well-founded from the modern point of view, it is shown how this might be done, after which Boucharlat concludes:

‘It follows from this that the principles of differentiation rest all of them on the binomial theorem alone, and since that theorem has been demonstrated, in the elements of algebra, with all the rigour possible, we may conclude that our principles are founded on a firm basis.’ (p.362)

Thus it is clear that Boucharlat adhered to the viewpoint of the ‘algebraic’ differential calculus of Lagrange, which he tried to improve with the help of the concept of limit. His ‘improvement’, however, reduced to the fact that whereas Lagrange wanted to avoid the application of the then not yet well-based concept of limit and simply defined the derivative of f(x) as the coefficient of the first power of h in the expansion

f(x + h) = f(x) + Ah + Bh² + Ch³ + ... , (1)

where A, B, C,... are functions of x, Boucharlat ‘uncovered’ the same derivative (‘differential coefficient’) by means of the passage to the limit, which last, however, consisted simply of taking h = 0 in the expression

(f(x + h) - f(x))/h = A + Bh + Ch² + ... , (2)

which is derived purely formally from equation (1). Boucharlat gave no definition of the concept of ‘limit’ or any sort of commentary on it. He limited himself to hints to the effect that the limit is the last value of the unlimitedly close approach (that is not having a last value) of a variable quantity. No wonder that such a concept of limit could not possibly satisfy Marx.

Marx not only made extracts of this textbook in several of his manuscripts and polemicised with the author regarding the foundations of his methodological essay, but also invested a great deal of effort in the factual examination of the former. Therefore we could hardly do without an acquaintance with the contents of this textbook. Here we produce in detail the contents of the first twenty paragraphs of the course of Boucharlat.