G. Wieleitner. How Modern Mathematics Was Born. Translated from the German under the editorship of Prof. A. Ya. Khinchin. GIZ, 1927. Price 1 ruble 10 kopecks.

The title of G. Wieleitner’s book does not fully correspond to its contents: by “modern” mathematics the author understands exclusively analytic geometry and infinitesimal analysis, and does not touch upon other mathematical disciplines, such as higher algebra, number theory, probability theory, etc., not to mention still more modern ones such as the theory of functions, set theory, mathematical statistics, non-Euclidean geometry, and so on. On the other hand, the book presents not only the history of the emergence but also the very essence of the two named sciences, since the author assumes in his readers “as little mathematical preparation as possible” (p. 71), as a result of which he finds it necessary to set forth such elementary matters as the area of a triangle and a parallelogram (p. 63), the evaluation of the fraction $\dfrac{x^2 - 1}{x - 1}$ for $x = 1$ (p. 93), the sum of the numbers of the natural series (p. 83), and the like. Such an expansion of the main task, given the small size of the book, could lead to extreme compression and superficiality in the exposition of the foundations of higher mathematics, and, despite the author’s skill, they are hardly likely to be intelligible to persons already unfamiliar with analytic geometry and analysis; all the more so because later on the author himself seems to presuppose such familiarity, using terms and concepts that have not previously been explained: for example, oblique coordinates (p. 25), diameters of an ellipse (p. 41), the formula for the distance from a point to a straight line (p. 51), the natural logarithm (p. 70), and so forth. The solving of examples such as those proposed (on p. 16) will also be difficult for unprepared persons: to construct curves for the functions $y = \dfrac{1}{x^2}$, $y = \dfrac{1 + x}{1 - x}$, and so on, from particular values of $x$, since the reader may put $x = 0$ in the first example and $x = 1$ in the second, while nothing has been said about infinite values or continuity.

More successful is the historical part of the book, which is distinguished by a lively exposition and by the fullness and novelty of the material. Especially valuable is the account of Fermat’s work on analytic geometry, which is usually insufficiently illuminated in courses on the history of mathematics. But even here the author’s desire to embrace almost all epochs in the development of the science and the activity of many scholars leads in places to an overly superficial treatment of the subject; for example, only a few words are said about the projective method on p. 35; about many major scholars there is only one or two lines; on p. 70 the merit of founding analysis is ascribed exclusively to Weierstrass, without mention of Cauchy and others.

There are very few editorial oversights and misprints in the book: let us note only, on p. 79, last line, the omission of the word “circle”; on p. 87—9th line from the top—“equal” instead of “equilibria”; on p. 49, at the bottom, misprints that distort the meaning in the equalities in the solution of Pappus’s problem.

The remarks made do not diminish the important significance of the appearance of Wieleitner’s book, which, given the extreme poverty of our literature on the history of mathematics, will be very useful for those persons who, possessing some preparation in higher mathematics, wish to deepen their knowledge by becoming acquainted with the evolution of its basic concepts. Let us add that the translation has been carried out in exemplary fashion; the book is excellently produced and inexpensive.

I. Chistyakov.