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). 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 x 0 y/ x, 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 , 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 ber 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 y of the function y of x, that is; in] Pw + Qw + Rw + etc.* the terms Qw , Rw 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) x., y., and so on (where is an infinitely small period of time ) Marx also spoke, when he noted (p.78) plays no role in Newton s analysis of the foundation of functions and therefore may be ignored , and that Newton himself voluntarily neglected (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). ____________
Marx's Mathematical Manuscripts 1881