Bibliography

1. G. M. Fikhtengol’ts. Mathematics for Technicians. 568 pp. GIZ, 1926. Price in binding: 6 rubles 25 kopecks.

2. G. Phillips. Differential Calculus. Translated under the editorship and with revision by Prof. V. F. Kagan. 254 pp. GIZ, 1926.

3. G. Phillips. Integral Calculus. Translated with additions by Prof. V. F. Kagan. 441 pp. GIZ, 1927. Price in binding: 4 rubles 25 kopecks.

4. Ya. D. Tamarkin and V. I. Smirnov. A Course in Higher Mathematics for Technicians. Volume I. 2nd ed. 461 pp. GIZ, 1927. Price in binding: 6 rubles 40 kopecks. Volume II. 415 pp. GIZ, 1926. Price: 7 rubles.

The modern development of technology and applied physics requires an ever greater application of mathematics for the conscious solution of problems put forward by practice. Although this need for mathematical knowledge is purely practical, requiring the ability to command the computational apparatus, nevertheless even the higher branches of modern analysis—such as, for example, the theory of integral equations, the calculus of variations, and tensor calculus—find application in our calculations. Thus there arises a need for a special presentation of mathematics, in which chief attention is paid to computational technique, without delving into the philosophical problems of modern mathematics, but, on the other hand, in such a presentation there must also be included those more complex parts of analysis which may be useful in solving problems even when they are still in a state of growth and have not yet received complete clarification. This circumstance apparently explains the large number of mathematical works published by engineers and physicists in the form of journal articles and separate manuals.

This tendency, however, is beginning gradually to penetrate also into textbooks of mathematics intended for non-specialists. The prewar mathematical training of our engineers and physicists is at present clearly insufficient. Among the books more in keeping with the indicated direction, we shall note the books whose titles are listed above.

1. Despite its modest title, Prof. Fikhtengol’ts’s book can serve as an excellent aid and introduction to the study of modern mathematics. The author set himself the goal of “showing mathematics in action,” and he has fully succeeded in this goal. Beginning with an exposition of approximate computations, the theory of the slide rule, and limits, the author proceeds to the study of functions and their graphs, interweaving the exposition of differential calculus, analytic and differential geometry, integral calculus, and differential equations with the solution of practical problems of equal interest to the technician and the physicist. These problems, in most cases, are selected very skillfully and do not obscure the logical connection of the mathematical theory being expounded. The author devotes some space to series, including trigonometric series, the foundations of nomography, and elements of vector algebra. Unfortunately, the author does not make use at all of the method of complex numbers. The book is provided with a subject index.

Bibliography

The book is written in excellent language and is so ample in scope that it is suitable for independent study of the subject.

The only obstacle to the deservedly wide circulation of this useful book is its high price.

2–3. A distinctive feature of Phillips’s book is the brevity of its exposition. This brevity, however, is not achieved at the expense of clarity or precision. Particularly successful, it must be acknowledged, is the selection of examples and problems, to the solution of which more than a quarter of the book is devoted. The book has gained considerably from the inclusion of supplementary chapters, written by V. F. Kagan and devoted to elements of vector calculus and field theory. Unfortunately, this book too lacks an exposition of the method of complex numbers, which at the present time is undoubtedly a gap.

In studying this book, the reader becomes acquainted not only with methods of mathematical computation, but also—what is especially important for us—with the techniques of setting up equations for problems encountered in technology and physics. Without mastering these techniques, acquaintance with higher mathematics is entirely useless for us. One should also note the excellent appearance of this edition, which testifies that GIZ has managed in recent years to improve its publications considerably.

1. The course by Tamarkin and Smirnov, conceived as a textbook for students of technology and physics, considerably exceeds in volume what it is at present possible to require of university students in taking the basic course in mathematics. Life, however, has its own demands, and in the upper years students are compelled to seek textbooks in order to master a whole series of “supplementary chapters” of mathematics, without command of which they cannot properly cope with their principal work. In this respect the book by Ya. D. Tamarkin and V. I. Smirnov will prove extremely useful, since in it the reader will find a whole range of information which until now had to be extracted from specialized and often very difficult works. This circumstance makes the book highly useful also for graduated engineers and physicists who wish to deepen or refresh their mathematical knowledge. Here the reader will find, especially in the small print, highly useful information from differential geometry, the theory of series, vector calculus with its application to analytic and differential geometry, the theory of complex numbers, harmonic analysis, the Fourier integral, and partial differential equations. The authors are preparing the third volume for publication, but already in the two volumes that have appeared they have created a work which will undoubtedly play a major role in deepening the mathematical foundation for the work of our engineers and physicists.

Ya. Shpilrein.