Read This!

The MAA Online book review column

Briefly Noted

January 2001

"What differential calculus is, and, in general, analysis of the infinite might be, can hardly be explained to those innocent of any knowledge of it. Nor can we here offer a definition at the beginning of this dissertation as is sometimes done in other disciplines. It is not that there is no clear definition of this calculus; rather, the fact is that in order to understand the definition there are concepts that must first be understood."

This is quintessential Euler: all the cards are on the table. He won't try to explain what the calculus is in his introduction, because you can only understand that by actually learning it. But keep reading, he's going to try anyway. He explains, in the next few paragraphs, that the calculus deals with changing quantities called variables, which he illustrates by considering "a shot fired from a cannon with a charge of gunpowder." This situation involves many quantities, he says, some of which are to be considered constant and others are variables. He goes on to define functions and to talk about their "vanishing increments," i.e., their differentials.

And so it goes on. The introduction has quite a bit more to say about the calculus in general, and includes a discussion of what in fact differentials really are. Euler argues that they are really just equal to zero. Thus, a derivative (which he thinks of as a ratio between two differentials) turns out to be a ratio between two quantities, both of which are zero, but which nevertheless makes sense as a ratio!

But the introduction is just one of the fascinating things about this book. Consider the table of contents:

1. On Finite Differences
2. On the Use of Differences in the Theory of Series
3. On the Infinite and Infinitely Small
4. On the Nature of Differentials of Each Order
5. On the Differentiation of Algebraic Functions of One Variable
6. On the Differentiation of Transcendental Functions
7. On the Differentiation of Functions of Two or More Variables
8. On the Higher Differentiation of Differential Formulas
9. On Differential Equations

This volume on Kolmogorov is the twentieth in the History of Mathematics series published jointly by the American Mathematical Society and the London Mathematical Society. It is a collection of essays, translated from the Russian, dealing with Kolmogorov and his mathematics. The essays originally appeared in two different books. The first was an obituary volume called "Kolmogorov in Remembrance," the second a collection of essays on the history of mathematics from which two essays by Kolmogorov have been taken. The result is an interesting package. The first article, by A. N. Shiryaev, is a comprehensive biography that focuses mostly on Kolmogorov's mathematics. There follow several essays of a more personal nature by students and colleagues. Finally, the two essays by Kolmogorov himself, one on P. S. Alexandrov and one on Newton's influence on modern mathematical thought, allow us to get to know him in a different way. The volume is rounded off by a detailed bibliography of Kolmogorov's work. [Fernando Q. Gouvêa]

Most MAA members don't need to be introduced to Ross Honsberger. His many books (nine so far in the "Dolciani Mathematical Expositions") have made him well known to all who enjoy reading about mathematics, and especially about mathematical problems. Mathematical Chestnuts from Around the World is a collection of "elementary gems," mostly dealing with Euclidean geometry, combinatorics, and combinatorial geometry, with hints here and there of number theory and algebra. The problems come, as advertised, from many different countries (Poland, Ireland, Bulgaria, the Philippines...) and many different sources (Pi Mu Epsilon Journal, The Mathematical Gazette, Quantum...). The author says that these essays "are intended as mathematical entertainment," and those who have read his other books know that he can be trusted to deliver. [Fernando Q. Gouvêa]

Publication Data

Foundations of the Differential Calculus, by Leonhard Euler, translated by John D. Blanton. Springer-Verlag, 2000. Hardcover, 194pp, $ ISBN 0-387-98534-4.

Kolmogorov in Perspective (History of Mathematics, volume 20). American Mathematical Society, 2000. Hardcover, 230 pp., $49.00 ($39.00 to AMS members). ISBN 0-8218-0872-9.

Fernando Q. Gouvêa (fqgouvea@colby.edu) is the editor of FOCUS and MAA Online. He teaches both "History of Mathematics" and "Number Theory", among others, at Colby College. He is a number theorist whose main research focus is on p-adic modular forms and Galois representations.

Back Issues:

• April 1999: Reprints of "classics", books on applied mathematics.
• May 1999: Books on math and physics, both popular and technical.
• July 1999: Various kinds of geometry, and classics in applied mathematics.
• August 1999: What's happening, Dirichlet, and Cryptonomicon.
• October 1999: Graph theory sources, John von Neumann, and Noncommutative Geometry.
• November 1999: Number theory in nine chapters, mathematics at work, and Jacques Hadamard.
• February 2000: Fallacies, Flaws, and Flimflam, measuring the universe, and primary sources at USMA.
• March 2000: Cardano and Manifold: Time.
• May 2000: Grassmann, Smarandache, and a really big prime.
• June 2000: A problem book, a history book, a collection of biographies, and a historical survey of reciprocity laws.
• August 2000: acrostics, field theory, categories, and partial differential equations.
• September 2000: excursions, history, and some high-level expository writing.
• November 2000: a festival of math videos and the Fourier-analytic proof of quadratic reciprocity.
• December 2000: Analysis problems, mathematical mysteries, and industrial mathematics.

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Read This! is the MAA Online book review column. Contributions are welcome; contact the editor if you'd like to be one of our reviewers. Books for review should be sent to the editor: Fernando Gouvêa, Dept. of Math&CS, Colby College, Waterville, ME 04901. Publishers, please check our reviews information page.