Bibliography

R. Courant. A Course in Differential and Integral Calculus. Part One. Functions of One Variable. Translated from the German by Yu. Rabinovich and V. Livshits. GIZ, Moscow–Leningrad, 1931, IV + 444 pp.

The course in mathematical analysis by Prof. Courant differs in many essential respects from ordinary courses. Its principal feature is the author’s remarkable ability to approach each question being presented without any constraint of pedagogical tradition, with captivating and deeply serious freshness and impartiality. In the arrangement and treatment of the material, the competent reader at once notices much that is unusual, sometimes striking; but, having considered the matter carefully, he always sees that each “original” device of Courant’s is internally justified, and that, on the contrary, the traditional exposition is insufficiently rational and holds on precisely by virtue of its traditional character.

In concrete terms, the basic distinctive feature of the course is the parallel—not sequential—presentation of differential and integral calculus; moreover, the concept of the definite integral even precedes the concept of the derivative. Such an order of exposition—which, incidentally, has for several years now been adopted among us in advanced higher educational institutions and technical colleges and recommended by the State Academic Council in the resolution of its third session—so far as we know, here for the first time finds a place in a large, serious textbook.

Another noteworthy novelty is the consistent implementation of Klein’s idea of deriving all the properties of the logarithm from its integral definition. A very valuable idea is to include a special large paragraph devoted to the question of the order of growth of functions. And a whole series of other similar interesting and valuable novelties could be noted.

However, in addition to the material and manner of exposition, Courant offers much that is new and valuable. The author refuses on principle to take the logical skeleton of the course as the basis of its structure; for him the subject-matter significance of particular points in his program plays an incomparably greater role. Therefore he often makes use of formal derivation on a par with purely intuitive proofs that have no logical force, and sometimes presents important results without at all being concerned with…

Bibliography

…proofs or postponing it until the “appendices.” For a mathematician who places too high a value on formal rigor, such a manner of exposition may sometimes be somewhat disconcerting; nevertheless everyone must admit that, despite this peculiarity—or thanks to it—Courant’s exposition is on an extraordinarily high scientific level. This fact proves with striking concreteness that, in mathematics, a high scientific level by no means requires logical rigor as an indispensable condition.

Finally, one must note the exceptional attention that Courant devotes to applications—above all to physics and mechanics. This makes his book still more valuable for the natural scientist; however, we believe that the chief attraction of this book for the physicist, mechanic, chemist, and astronomer lies not in this, but in the remarkable concreteness, almost tangibility, with which Courant is able to present the mathematical material. Incidentally, of course, these two aspects are inseparably connected with one another.

The translation, on the whole a good one, suffers from a few annoying lapses; here are examples: “Zeros and infinities of functions” (p. 286); of course, one does not say this in Russian—one should say “infinitely large values,” or, at the very least, “infinite values.” On p. 87: he treats the second derivative as the limit of a ratio of “second-order differences”; this is plainly unsatisfactory, because in such a formulation one is speaking of second-order differences, whereas what is meant is a difference quotient of second order (and this is how it should have been translated).

Finally, one cannot pass over one extremely grievous fact in the external presentation. The word differential, printed with a single “ф” (and so printed throughout the entire book—from the title to the last page), contains an orthographic error, whatever authority the proofreader Giza may have been invoked to cover it. This error will remain an error until people begin printing kommunu with one m, kollektiv with one l, and so on; and one must hope that this will not happen.

A. Khinchin.