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Mathematical masterpieces. Further chronicles by the explorers. (English) Zbl 1140.00002

Undergraduate Texts in Mathematics. Readings in Mathematics. New York, NY: Springer (ISBN 978-0-387-33060-0/hbk; 978-0-387-33061-7/pbk; 978-0-387-33062-4/ebook). xii, 333 p. (2007).
This book consists of four chapters, each of which presents a “sequence of selected primary sources” leading up to a “masterpiece of mathematical achievement”. Chapter 1 deals with the bridge between continuous and discrete and leads the readers from Archimedes’ quadrature of the spiral to Euler’s solution of the Basel problem, the summation of the series \(1 + \frac14 + \frac19 + \cdots = \pi^2/6\). The second chapter deals with solving equations numerically: from Qin’s method of solving quartic equations to Newton’s and Simpson’s methods for finding roots of equations all the way to Smale’s discussion of these methods and the connection to fractals. Chapter 3 discusses the concept of curvature in the works of Euler, Gauss, and Riemann. The last chapter finally starts with Euler’s discovery of the patterns that govern the primes dividing quadratic forms \(x^2 - ay^2\), his version of the quadratic reciprocity law, Legendre’s modern formulation, and proofs by Gauss and Eisenstein.
Each chapter contains selections of original sources translated into English, with explanations in between, and lots of historical comments sketching the further development of the topic. There are also a lot of exercises.
The preface starts with a quotation from Stephen J. Gould: “I can attest that all major documents of science remain chockful of distinctive and illuminating novelty, if only people will study them – in full and in the original editions.” It is unfortunate that the authors have not followed Gould’s advice. Of course it is not easy to incorporate the often lengthy discussions in the original sources, but the bits and pieces quoted in the book sometimes give the reader the impression that he is being spoonfed those parts of the manuscripts that the authors think are important. But this is not what Gould had in mind. And in fact, the gold nuggets in the original sources most likely are not contained in those passages that everybody understands but in those parts that nowadays appear to be somewhat cryptic.
A good example is the “brief and lamentably cryptic” remark at the end of a paper by Eisenstein on quadratic reciprocity that the authors did include on p. 299: there, Eisenstein sketches a proof of Leibniz’s result that \(\pi = 4(1 - \frac13 + \frac15 - \frac17 \pm \cdots)\), which is closely related to a proof found in the posthumous papers of Gauss related to the class number formula for quadratic fields. In fact, Eisenstein mentions connections of higher dimensional analogues to the theory of higher forms and promises to return to this topic on another occasion, which he unfortunately did not. Dirichlet was working on similar topics at the time, but left the field to Kummer, whose language of ideal primes turned out to be superior to that of forms, and in fact Kummer managed to derive class number formulas for cyclotomic fields using methods similar to those hinted at in Eisenstein’s remark. The general class number formula for number fields was eventually proved by Dedekind. The whole development makes a great story, but the reader is only given comments by Cayley (who also failed to see the big picture) and a remark that Eisenstein’s geometric approach eventually led to Minkowski’s geometry of numbers.
Another problem that the authors had to face was providing the background to the original sources that is necessary for understanding them properly. As Umar Khayyam wrote at the beginning of his work on solving cubics, “this treatise cannot be understood except by one who has mastered Euclid’s Elements”. When Umar later uses proportions without comments, the authors’ explanation essentially consists of two exercises where ratios are identified with fractions of magnitudes of the same dimension. Perhaps one of the greatest accomplishments of Greek mathematics, namely the theory of ratios, should have been given a little bit more room.
Among the few inaccuracies I have noticed, here’s one: on p. 276 we read that Lagrange “is unable to determine whether a given prime \(8n+1\) is a nontrivial divisor of a number of the form \(x^2 + 2y^2\), because he does not yet have complete quadratic reciprocity in hand.” Similar claims, though less explicitly, are made about the form \(x^2 + 3y^2\). Lagrange, however, knew enough special cases of the reciprocity law for treating these two forms (and a few others) completely. Also, the claim that Legendre’s problems with making his composition of forms unique was solved by Gauss through the introduction of proper equivalence (p. 304) is not correct; but quoting a few pages from the Disquisitiones is not nearly enough to understand the problem at hand and to see how Gauss solved it.
This is a well written and entertaining book that can (and should) be used in seminars or reading courses. My only complaint is that each of the four chapters would have deserved a book of its own.

MSC:

00A05 Mathematics in general
11-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to number theory
40-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to sequences, series, summability
53-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to differential geometry
65-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to numerical analysis
97-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to mathematics education
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