Bibliography

M. Ya. Vygodsky. Foundations of Infinitesimal Calculus.
456 pp., 94 figs., price 4 rub. 70 kop. GTTI, 1923, 2nd ed.

This book presents the elements of analysis in considerably greater detail than is done in ordinary textbooks; 450 pages are devoted to the exposition of the elements of analysis. This, however, gives the author the opportunity to present mathematics in action. The book contains 112 problems, worked out in detail, from the fields of mechanics, physics, chemistry, and technology. The author has succeeded in organically linking the exposition of the methods of mathematical analysis with the practice of solving concrete problems. This makes the theory natural and comprehensible for a reader not yet accustomed to abstract thinking. The author is not afraid to stretch out his exposition; he even allows himself to experiment a little, so that for an approximate estimate of the number \(e\) he devotes 3 pages, to the question of finding the integral of \(\frac{1}{x}\) he devotes 13 pages of theory and 11 pages of exercises. Even the differentiation of the linear function \(ax+b\) occupies 3 pages in the middle of the book; the differential of \(x^2\) is computed over 4 pages, and so on. This circumstance makes the study of the subject from Vygodsky’s book especially attractive. Having worked through all 450 pages, the student will acquire a comparatively small amount of theoretical knowledge, but this knowledge will be firmly mastered, and, most importantly, the reader will learn to apply the knowledge acquired.

A substantial feature of Vygodsky’s book is also his approach to the differential. By means of a large number of examples the author shows how one should set up equations for solving given problems, treating the differential as a very small quantity and neglecting infinitesimals of higher orders. The ability to handle the differential correctly is extremely useful for the engineer and the physicist, and therefore such an approach in the book should be welcomed.

The noted merits of the book explain its well-deserved success. At a conference in the House of Scholars for the evaluation of mathematics textbooks, Vygodsky’s book was put forward for first place and recognized as reflecting definite achievements in the application of Marxist-Leninist methodology to the creation of a Soviet mathematics textbook for higher educational institutions.

The author advances yet another idea in his book: seeking to “explain

Bibliography

from reality the mysterious and incomprehensible operations of the calculus of infinitesimals,” the author systematically carries out what he calls the “classical conception of the differential,” which is, within very broad limits, “relative truth”; simply put: in a good half of the book the author renounces the passage to the limit and considers infinitesimals not as variables, but as constant, though very small, quantities. Such an approach corresponds to the conception of the creators of infinitesimal analysis and in the last century gave rise to many disputes about the true meaning of the calculation of the derivative and of the differential. These discussions seem to us superfluous at the present time, when the concept of the limit simply resolves all contradictions. We are not afraid to show a child the value of a periodic decimal fraction:

\[ 0{,}1111 \ldots = \frac{1}{9}. \]

The concept of the derivative is in essence no more complicated than finding the exact value of a periodic fraction, while the refusal to consider limits in half the book does not make assimilation easier, but, in my view, makes it more difficult. Thus, for example, the author considers in the interval \(a—b\) \(n\) abscissas increasing in a geometric progression:

\[ ah, ah^2, \ldots ah^{n-1}, ah^n = b, \]

and then states: “for infinitely large \(n\) the quantity \(h\) must be set equal to unity.” The reader is bewildered; if \(h = 1\), then \(a = ah = ah^2 = \ldots ah^n = b\), and at the same time \(a < b\). The only salvation is that the reader may not notice this contradiction, but then he has understood nothing of the whole exposition, which gives a method for calculating the integral by summation.

Despite this defect in the exposition (the author regards it as a merit), the book is a valuable aid for beginners in the study of analysis.

A. Shpilrein.