Marx's Mathematical Manuscripts 1881

Appendix III. On the Calculus of Zeroes by Leonard Euler

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

In order to understand those places in the manuscripts of Marx at which the ratio dy/dx is regarded as a ratio of zeroes, at times equal to the value of derivative of y with respect to x for all values of x and at the same time something which can be treated as an ordinary fraction - where, for example, the product du/dv ⋅ dv/dx equals the ‘fraction’ du/dx, ‘cancelling’ the dv’s - it is essential to have an acquaintance with Euler’s attempt to construct the differential calculus as a calculus of zeroes. This attempt deserves interpretation as well in view of the fact that Marx specifically refers, in the list of literature appended to his first draft of the history of differential calculus, to chapter III of Euler’s Differential Calculus, and that Marx call Euler’s account of the calculus ‘rational’.

The Differential Calculus by the great mathematician and member of the St Petersburg Academy of Sciences Leonhard Euler was published by the St Petersburg Academy in 1755. The basis for this work lies in the attempt to regard differentials as at the point of equalling zero in quantity, yet at the same time as different from zero: a zero with a ‘history’ of its origin, with various designations (dy, dx and so on) and allowed to be evaluated so that the ratio dy/dx where y = f(x), is distinguished by the fact that it is the derivative f’(x) and can be treated as an ordinary fraction.

Euler undertook this attempt in order to free mathematical analysis from the necessity of treating differentials as actually infinitely small quantities with a clearly contradictory character (appearing to be in some sense zero and non-zero simultaneously). The assertion that ‘pure reason supposedly recognises the possibility that the thousandth part of a cubic foot of substance is devoid of any extent’, Euler considers ‘completely inadequate’ (in the sense of ‘inadmissible’, in context, see the translation [in Russian] of L. Euler, Differential Calculus. Moscow-Leningrad, 1949, p.90).

‘An infinitely small quantity is no different from a vanishingly small one, and thus exactly equal to zero. This includes the definition of infinitely small differentials according to which they are smaller than any given quantity. Actually, if the quantity is to be so small that it is smaller than any possible given quantity, then it could not possibly be not equal to zero; or if it is not equal to zero, then there is a quantity to which it is equal, contrary to the supposition. Thus, if one asks, what is the infinitely small quantity in mathematics, we answer, that it is exactly equal to zero. Consequently, this removes the mystery which is usually attributed to this concept and which for many makes the calculus of infinitely small quantities rather suspicious.’ (p.91)

Since the simple identification of the differential with zero did not yield the differential calculus, Euler introduces ‘various zeroes, establishing for them two types of equality, the ‘arithmetic’ and the ‘geometric’. In the ‘arithmetic’ sense all zeroes equal to a independently of the ‘sort’ of zero which is added to a. In the ‘geometric’ sense of the word, two zeroes are equal only if their ‘ratio’ is equal to unity.

Euler did not clarify what he understands by the ‘ratio’ of two zeroes. It is only clear that he attributes to this ‘ratio’ the usual character of a ratio of non-zero quantities and that in practice by the ratio of two ‘zeroes’ - dy and dx - he intends the same as that which is expressed in modern mathematical analysis by the term lim Δx→0 Δy/Δx, for Euler’s theory of zeroes does not free mathematical analysis from the necessity of the introduction of the concept of limit (and the difficulties attending this concept).

Since for Euler zero becomes various zeroes (and in the ‘geometric’ sense they are not even equal to one another), it is necessary to use a variety of symbols. ‘Two zeroes’, writes Euler, ‘may have any geometric ratio to each other, while from the arithmetic point of view their ratio is the ratio of equality. Therefore, since zeroes may have any ratio between them, in order to express these different ratios different symbols are used, especially when it is necessary to determine the geometric ratio between the two different zeroes. But in the calculus of infinitely small quantities nothing larger is formed than the ratio of various infinitely small quantities. Unless we employ different signs for their designation everything will be an enormous mess and nothing would be distinguishable.’ (p.91)

If, following this interpretation of dx and dy as ‘different’ zeroes, the ratio of which is equal to f’(x), we replace dy/dx = f’(x) with dy = f’(x)dx, then we have an equation the left and right sides of which will be equal both in the ‘arithmetic’ sense and in the ‘geometric’ sense. Actually, the left and right will contain various ‘zeroes’, but all ‘zeroes’, as already noted, are equal in the ‘arithmetic’ sense. Only insofar as the ratio of dy to dx is completely equal to f’(x) - that is, both in the ‘arithmetic’ and ‘geometric’ sense [the ratio dy/dx : f’(x), where y = f(x), is considered unity even if f’(x) = 0] and if the ‘ratio of zeroes is understood correctly as the usual operation of ratio, then we have

dy : f’(x)dx = (dy/dx) : f’(x) = 1 ,

or, in other words, dy and f’(x)dx are also equal in the ‘geometric’ sense.

Obviously, Marx had in min just this ‘complete’ equivalence of the equation (dy/dx) = f’(x) with that of dy = f’(x)dx in the sense not only of the possibility of transition from each of them to the other but also of the treatment of this (and with the strength of this) ‘ratio’ of ‘differential parts’ dy and dx as a usual ratio (as a fraction), whatever the quality of the ‘differential parts’ dy and dx as zeroes (‘various’ zeroes, variously designated), when he transformed the first of these equations into the second ( see ibid, p.147).

For a more detailed account of the Euler zeroes and a history of the ideas related to it the reader may consult the article, A.P. Yushkevich ‘Euler und Lagrange über die Grundlagen der Analysis’, in Samelband zu Ehren des 250 Geburtstages Leonhard Eulers, Berlin, 1959, pp.224-244.

Here we are limiting consideration to two considerations of Euler which are helpful in reading the manuscripts of Marx. The first concerns the concept of the differential as the principal part of the increment of the function. This concept, which plays an essential role in mathematical analysis, particularly in its foundation, Euler introduces in the following way: ‘Let the increment w of the variable x become very small, so that in the expression [for the increment Δy of the function y of x, that is; in] Pw + Qw² + Rw³ + etc.* the terms Qw², Rw³ and all higher orders becomes so small that in an expression not demanding a great degree of precision they may be neglected compared to the first term Pw. Then, knowing the first differential Pdx, we also know, admittedly approximately, the first difference, that will be Pw; this has frequent use in many cases in which analysis is applied to practical tasks’ (p.105, ibid). In other words, having replaced in the differential function y of x (that is, in Pdx, where P is the derivative of y with respect to x) the differential dx, equal to zero according to Euler, with the finite [non-zero] increment w of the variable dx, we obtain the very concept of the differential as the principal part of the increment of the function, the starting point of modern-day course of mathematical analysis.

The analogous concept of the differential as the principal part of the increment of the function is also in the manuscripts of Marx (see the account in manuscript 2768, p. 297 [Yanovskaya, 1968]).

The second consideration concerns the question of the choice of designations specific to differential calculus, that is, of differentials and derivatives. Here interest arises first of all from the fact that Euler interprets the dot designations of Newton as symbolic of the differential, but not the derivative. In fact he writes, ‘the name “fluxions” first used by Newton for the designation of speed of growth, was by analogy carried over to the infinitely small increments which a quantity assumes when it as it were varies’ (p.103). And similarly later, ‘The differentials which they [the English] called “fluxions”, they marked with dots which were placed above the letters, so that y. meant for them the first fluxion of y, y.. the second fluxion, y... the third fluxion and so on.’

This manner of designation, however did not satisfy Euler, and he continues : ‘Although this means of designation depends upon an arbitrary rule, the designation need not be rejected if the number of dots is not large, for they are easily indicated. If, however, it is required to write many dots, this method gives rise to a great deal of confusion and inconvenience. In fact, the tenth differential, or tenth fluxion, is extremely inconvenient to indicate thus y........... where by our means of designation , d10y is given easily. There arise occasions when it is necessary to express differentials of much higher, and even infinite, degree; on those occasions the English method of designation is not at all appropriate.’ (pp. 103-104)

About the analogous identification (in several instances) by Newton and his followers of the ‘fluxions’ x.,y. and so on, with the ‘moments’ (that is, the differentials) τx., τy., and so on (where τ is an ‘infinitely small period of time’) Marx also spoke, when he noted (p.78) ‘τ plays no role in Newton’s analysis of the foundation of functions and therefore may be ignored’, and that Newton himself voluntarily neglected τ (loc.cit.). Marx used the same expression, speaking of the method of Newton, as ‘the differential of y or y, of u or u of z or z’. (see p.79)

We must note in addition that Marx primarily emphasised the Leibnitzian symbology of the differential calculus over the symbology of Newton and his followers (see p.94).



* The Differential Calculus of Euler begins with the calculus of finite differences and the theorem which states that ‘if the variable quantity x assumes an incremental value w, then the consequential value of the increment of any function of x can be expressed as Pw + Qw² + R³ + ... etc., which expression is either finite or continues infinitely.’ (Ibid, p.103, see also p.61) The proof of this theorem is based on the fact that the class of functions considered by Euler consists of power functions: polynomials and elementary transcendental functions expanded into infinite power series which he treats as if they were finite polynomials - Ed.