Approximation theory is concerned with the problem of approximating a given function by functions in a special class (for instance, polynomials or trigonometric polynomials). The main goal is to be able to control the approximation error. (Hence, the focus is on global, uniform approximation instead of local, pointwise approximation.) Although approximate calculations have existed since the dawn of mathematics (recall Archimedes's approximation of π), approximation theory is a relatively young branch of mathematics, because it requires a precise notion of function, which only appeared in the end of the 18th century.

This book aims to tell the historical evolution of the methods and results of approximation theory, starting from the work of Euler in 1777 on minimizing distance errors in maps of Russia and of Laplace in 1843 on finding the best ellipsoid for the earth, and ending with the the work of Bernstein. The central character in the book is Chebyshev (1821--1894) and the Saint Petersburg Mathematical School he created.

A typical approximation problem that concerned Chebyshev was the determination of the polynomial of fixed degree that deviates the least from zero in a given interval. This and other similar problems lead to a central result in approximation theory: the alternation theorem, which says that the maximum deviations of the best approximation are equal in absolute value and alternate in sign. Special forms of this theorem already appear in the work of Euler and Laplace mentioned above. One of the main issues discussed in the book is whether Chebyshev actually proved the alternation theorem. Steffens argues that he must have known about it, but that he never proved it, contrary to what some authors claim. The consensus seems to be that the Markov and Borel were the first to prove the alternation theorem (independently). Kirchberger's thesis contained a purported proof, but it had a gap.

Strangely, the book touches only lightly on Weierstrass's theorem. Only Bernstein's proof is given. (A probabilistic proof, not the usual one given in textbooks.) It seems that Chebyshev wasn't interested in Weierstrass's theorem, even if he must have known about it. Apparently Chebyshev was only interested in concrete problems, and existence and uniqueness questions were settled by concrete solutions. He regarded foundational matters as pure philosophy.

Chebyshev had wide interests in addition to approximation theory. He is famous for proving a weak form of the Prime Number Theorem, which gives the asymptotic behavior of the distribution of primes. He also worked on continued fractions, elliptic integrals, and on the foundations of probability.

Steffens says in the preface that most surveys on the history of approximation theory are in Russian, and in some cases may contain contradictory or biased views of the efforts in Russia. However, some of those surveys have appeared in English [1,2] For further references to papers on the subject, including on-line copies of some seminal papers, see [3].

The book contains an extensive bibliography, listing many works in Russian, as expected. However, this list does not seem to have been revised thoroughly. For instance, Pinkus paper is cited as a draft but was published in 2000 [4]. The book also contains an index, but it lists only names of people. An index listing concepts and theorems would make the book even more useful. There are several minor misprints in the text.

One thing that I found distracting is that there are too many footnotes, containing short biographies, quotations, and references. Many of the footnotes are citations in Cyrillic. Some pages even have more footnotes than text. I think it would be less distracting to the reader if the footnotes were moved to the end of each chapter.

The book tries to integrate history, philosophy, and mathematics. I think the author did not completely succeed in this. The book is well written and the history and philosophy parts are good, but the mathematics is not always engaging. Perhaps I've approached this book with the wrong expectations. I expected to learn about the main problems and results in approximation theory in a historical setting. The book naturally gives priority to the history; the organized mathematical theory is available in many books on approximation theory [5,6].

Nevertheless, the book contains much interesting material and has certainly motivated me to learn more about the subject and its history. No book can wish for more and so I recommend the book to anyone interested in approximation theory or in the history of mathematics.

**References:**

[1] Butzer, Paul; Jongmans, François. P. L. Chebyshev (1821--1894). A guide to his life and work. *J. Approx. Theory* **96** (1999), no. 1, 111--138. MR1659404 (99m:01019)

[2] Goncharov, V. L., The Theory of Best Approximation of Functions, *J. Approx. Theory* **106** (2000), 2-57. MR1778073 (2001g:41002) (Followed by a commentary by V. M. Tikhomirov on pages 58-65. MR1778072 (2001g:41003))

[3] History of Approximation Theory, http://www.math.technion.ac.il/hat/

[4] Pinkus, A., Weierstrass and Approximation Theory, *J. Approx. Theory* **107** (2000), 1-66. MR1799549 (2001k:01039)

[5] Cheney, E. W., *Introduction to Approximation Theory* , McGraw-Hill, 1966. (Contains historical notes at the end.) MR0222517 (36 #5568)

[6] Davis, P. J., *Interpolation and approximation* . Dover, 1975. MR0380189 (52 #1089)

Luiz Henrique de Figueiredo is a researcher at IMPA in Rio de Janeiro, Brazil. His main interests are numerical methods in computer graphics, but he remains an algebraist at heart. He is also one of the designers of the Lua language.