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:* y - y = 3x h + 3xh + h , (y - y)/h = 3x + 3xh + h . (2) In 5-8 Boucharlat finds dy in the examples y = a + 3x , y = ((1 - x )/(1 - x)) , y = (x - 2a ) (x - 3a ) . 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 = (x) to the differential dy = (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 = (x, h) , (A) 0/0 = (x, 0) . (B) 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 (x) 0, then the differential, equal to (x) 0, exists, if (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 = - ax by this means is obtained dy = - 3ax dx, 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. 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) 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): 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: 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 + Bh + Ch + ... , (1) (f(x + h) - f(x))/h = A + Bh + Ch + ... , (2)
Marx's Mathematical Manuscripts 1881