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The Real and the Complex: A History of Analysis in the 19th Century

Jeremy Gray
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Springer Undergraduate Mathematics Series
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The Basic Library List Committee strongly recommends this book for acquisition by undergraduate mathematics libraries.

[Reviewed by
Mark Hunacek
, on

The late Carl B. Boyer, author of the well-known The History of Mathematics, and the professor from whom, as an undergraduate, I first studied this subject, writes in the third edition of his book (revised after his death by Uta Merzbach) that

The nineteenth century deserves to be known as the Golden Age of mathematics. The additions to the subject during these one hundred years far outweigh the total combined productivity of all preceding ages. The century was also one of the most revolutionary in the history of mathematics. The introduction to the mathematician’s repertoire of concepts such as non-Euclidean geometries, n-dimensional spaces, noncommutative algebras, infinite processes and nonquantitative structures all contributed to a radical transformation that changed the appearance, as well as the definitions, of mathematics.

As the above quote makes clear, advances took place in many different areas of mathematics during this period. Jeremy Gray, the author of the book now under review, has previously written Worlds out of Nothing, which chronicles the history of geometry during the nineteenth century, and now, in what can be described as a companion piece to that book, has turned his attention to 19th century developments in analysis.

The text is organized roughly chronologically, and actually begins slightly before the dawn of the nineteenth century with the work of Lagrange, who tried to firm up calculus by reducing it to algebraic manipulation of power series. (See A Historian Looks Back: The Calculus as Algebra and Selected Writings by Judith Grabiner, and its review in this column.)

Other mathematicians whose work at about this time helped provoke future developments include Fourier and Legendre, who worked on trigonometric series and elliptic integrals, respectively. These difficult mathematical ideas forced people to think hard about the underpinnings of analysis. (David Bressoud, for example, begins his A Radical Approach to Real Analysis with a discussion of the “crisis” caused by Fourier’s work: “The edifice of calculus was shaken to its foundations... [W]hile most scientists realized that something had happened, it would take fifty years before the full impact of the event was understood.”) Chapters 2 and 3 of the text discuss Fourier and Legendre as a way of setting the stage for future developments.

There follow three chapters on Cauchy. Chapters 4 and 5 discuss his work on real analysis and chapter 6 talks about his contributions to complex analysis, which in the early 19th century was not even yet recognized as a subject in its own right. After this, there are three chapters discussing the early history of elliptic functions and elliptic integrals, concentrating on the work of Abel, Jacobi and Gauss. We then return to Cauchy and his work on complex function theory in the middle third of the century, and chapter 11 then explains how elliptic function and complex function theory were brought together.

These chapters constitute roughly a third of the book. The second third (chapters 13 through 20) of the text begins with potential theory and moves from there to the further development of complex function theory during the mid-19th century. Riemann’s geometric approach to the subject is discussed in some depth over several chapters, and is compared and contrasted with the more algebraic approach of Weierstrass.

Finally, chapters 22 through 29 address developments in the rigorization of analysis in the latter part of the century. There are chapters here on uniform convergence of series of functions, the fundamental theorem of calculus, and the construction of the real numbers by Dedekind and Cantor. Chapters in this group also discuss Lebesgue theory, Cantor’s set theory and the beginnings of topology. These final chapters “are offered as pointers to some of the ways analysis was to develop in the 20th century, and to branch into new domains, abstract set theory and topology.”

The author discusses not only mathematics but also personalities. Many of the chapters in the book are organized around people, and good biographies of them, complete with photographs, appear throughout. The lives of these mathematicians are placed in historical context, and the author does a good job of conveying the fact that mathematical advances are often the product of false starts and mistaken ideas.

This book, like Worlds out of Nothing, is part of the Springer Undergraduate Mathematics Series and, we are told, derives from a series of lectures given by the author to (British) senior undergraduate students. Consistent with these undergraduate origins, the author has employed a number of useful pedagogical devices. The three thirds of the book that are discussed above are set off from one another by two chapters, 12 and 21, that are called “revisions” and which pause and reflect on what has come before; I confess that I cheated a bit and read these chapters first, figuring that I might as well know in advance what to look out for.

There is also a final (very short) chapter 30, titled “Assesment”, in which Gray describes his class’s end-of-course essay assignment and reproduces the advice on essay writing that he gave his students. The earlier chapter 21 also touched on assessment issues. Analogous chapters on writing (12, 21 and 31) appeared in Worlds.

In addition to these chapters, Gray has, in an Appendix, included translations (all but one done by himself) of portions of important papers, including works by Fourier, Dirichlet, Riemann, and Schwarz.

Notwithstanding these nice features, I doubt this book would prove very successful as a text for undergraduates on this side of the Pond. For one thing, there is the obvious difference between a British undergraduate mathematics education and an American one, and, in addition, the prerequisites for reading this book seem a bit daunting. Unlike other history-minded analysis books such as Bressoud’s aforementioned Radical Approach and How We Got From There to Here: A Story of Real Analysis by Rogers and Boman, a substantial background in analysis seems to be assumed here. (This is presumably because these two books are not really intended as histories of analysis, but instead as introductory courses in analysis from a historical perspective.) In addition, although billed as a text, Gray’s book is curiously devoid of exercises; some appear sporadically throughout the text, but there are very few of them (less than 20 throughout the entire book).

Although I don’t anticipate using this book as the text for a course, I view it as an extremely valuable addition to my library and a book that I am sure I will consult frequently in the future: this is a text that should serve well as a desk reference for a faculty member teaching either analysis (real or complex) or the history of mathematics. Certainly, no good university library should be without it. We are also told in the preface that two other volumes on 19th century mathematics history (on algebra and differential equations, respectively) are planned, and I eagerly await their publication.

Mark Hunacek ( teaches mathematics at Iowa State University.