Max Planck. Introduction to General Mechanics. Translated under the editorship of Prof. N. P. Kasterin. State Publishing House, 1927. Pp. 249. Price 2 rubles 40 kopecks.

“Difficulties that a beginner has to struggle with in taking the first steps in the field of theoretical physics often have to do not so much with the mathematical form as with the physical content of the train of thought being presented. It is not the handling of equations, but their formulation and interpretation—that is what is most difficult for the beginner; to help him in this is the chief aim of the proposed manual.” (From the preface to the book under review.)

It is hard to formulate with greater clarity the task facing those who teach theoretical physics, whether orally or in writing. If broad circles of representatives of the exact sciences, in their pedagogical activity, were guided by the idea expressed in the words quoted from M. Planck, this would undoubtedly lead to an increase in the effectiveness of instruction in higher education.

The choice of material and the mode of exposition in the book under consideration correspond to a considerable degree to the aim set by the author (we may note, for example, the “dialogue” on the possibility of transferring a pair of forces, § 82). A significant number of problems and examples analyzed in detail in the text helps to clarify the basic propositions of mechanics. The author returns to some of these basic propositions several times, striving to reveal their physical content and to approach them from different points of view. Only in isolated cases is this plan of exposition, perhaps, disturbed to the detriment of the subject. Thus, for example, it would seem appropriate, alongside the complete theory of the heavy symmetrical gyroscope, to give an approximate theory of it, proceeding directly from the law of moments (and not by introducing simplifications into the exact formula for precession), which would help to clarify the physical side of the question. Perhaps the physical content of the concepts of a rigid body and rigid constraints has not been sufficiently brought out; apparently it is nowhere explicitly indicated in precisely which systems of reference (in particular, one at rest relative to the stars) the law of inertia is valid, although sufficient attention is devoted to questions concerning the transformation of systems relative to frames of reference.

With great satisfaction one may note that the principle of relativity in its classical form (invariance with respect to the Galilean transformation) is one of the guiding lines of the exposition. Only by giving this principle due attention can one prepare the reader for the perception of methods of reasoning so characteristic of modern physics. Let us note in particular the establishment of a connection between the law of conservation of energy and the principle of equality of action and reaction on the basis of the principle of relativity (§ 129), as well as the consideration of the question of the character of the transformation of forces under a transformation of the reference system (§ 57).

Quite appropriate, too, is the treatment of the differential equations of Hamilton–Jacobi, usually not presented in elementary textbooks, which in recent years have acquired such great significance for the theory of the atom.

Unfortunately, the author chiefly uses the coordinate method of analytic geometry, which lacks geometric visuality and, moreover, inevitably leads to encumbering the book with cumbersome algebraic calculations. True, he also introduces some notations, but for the most part only for the purpose of a more concise recording of results obtained by direct operation with the separate components of vectors. Thus, for example, in § 18 the term component of velocity is introduced for the derivative of a Cartesian coordinate with respect to time; after this it is proved that velocity is a vector. In the following paragraph the same arguments are repeated as applied to the components of acceleration, etc.; nevertheless, later on the author operates not with the vectors of velocity or acceleration, but with their components. Meanwhile, the application of vector calculus from the very first steps of acquaintance with physics, and in particular with mechanics, greatly helps to clarify the visually geometric meaning of the relations between physical quantities, not to mention the substantial simplification of mathematical calculations. Moreover, this calculus has in recent times ceased to be the possession of a narrow circle of specialists; it is therefore time to recognize full civil rights for vector calculus, and a textbook of mechanics that does not make use of all the advantages of this method cannot, in our opinion, be regarded as a perfect textbook. Unfortunately, such a textbook, adapted for teaching, does not yet exist in the educational literature—not only in ours, but also in foreign literature.

The language of the book under review is compressed; some remarks are set forth laconically, so that full understanding of them (of these remarks) is accessible only to a comparatively well-prepared reader. Familiarity with analytic geometry and differential and integral calculus is, of course, assumed.

I. Timm.