Abel on Analysis

It surprises me that it has taken me ten years to become aware of this excellent book. Published by Kendrick Press in 2007, it deserves to be much better known and widely read by those interested in elliptic and modular functions and their history.

Philip Horowitz has collected here the translations of a selection of Niels Henrik Abel’s papers. The title describes them as papers “on analysis,” I presume to warn the reader that the famous papers on algebra (dealing with the unsolvability of the quintic and with the conditions under which algebraic equations are solvable) are not part of the selection. The true focus, however, is on Abel’s work on elliptic functions and their generalizations.

Two papers don’t quite fit this rubric: one is Abel’s famous paper on the binomial series, in which he investigates for the first time the conditions under which the series converges. It is an important manifesto in favor of a rigorous approach to analysis, but not one that Abel ever pushed further. The other is a short paper on a proposed criterion for convergence which Abel shows is actually a necessary but not sufficient condition.

The meat of the book are two long sections, one entitled “Abel’s Theorem” and the other “Elliptic Functions”. These include several long and important papers dealing with what later would come to be known as the theory of elliptic curves and of Abelian varieties. They are not easy to read, but they are of fundamental historical importance.

The book is enriched with commentary by the translator, which is often of great interest. For example, his comments on Abel’s almost-lost Paris memoir on Abelian functions include some remarks about the techniques Abel had to use because he did not know Cauchy’s residue theorem. This immediately clarifies some of the paper; the translator also points us to an account of the same theory that does use Cauchy’s theorem, making it easy for readers to compare the two approaches.

Horowitz has also provided an interesting and useful annotated bibliography. His annotations range from critical (e.g., he remarks on one of Abel’s biographies that “What is said on the mathematics tends to be unsound.”) to congratulatory (“a very rich paper…”) to revelatory (of a book on Abel’s Parisian memoir, he points out that it “contains a gorgeous facsimile” — I want one!).

But that’s not all. The editor has also included a generous selection of correspondence. Indeed, there are two selections. The first collects mathematical letters to and from Abel. Horowitz says that “Letters with no mathematical significance are not included in this selection but those which are here are translated in full,” which means that we can also get a sense of Abel’s relationships with mathematical friends such as Holmboe and Crelle.

The second letter collection is perhaps even more fascinating. It contains the correspondence between Legendre and Jacobi on elliptic functions. Legendre, much the older man, was the author of the first major treatise on the subject, which he says was not popular in his home country. Jacobi was the young German upstart making new advances on the subject and daring to tell Legendre what he is up to. The two exchange many letters, and of course “M. Abel” is often mentioned as the other big player and Jacobi’s main competitor in the field. (It must be said that Jacobi is clear that he thinks Abel has the deeper insights.) The letters extend until the 1830s, which means they include both men commenting on Abel’s tragic early death.

The one sad note is that Horowitz mentions a forthcoming volume with a collection of translations of papers on the theory of equations, which was to also include Abel’s papers on the subject. I don’t think that volume ever appeared. Too bad; I would have enjoyed that book, and it would have helped me in my own research.

While the material on elliptic and Abelian functions is quite difficult, the two papers on series and some of the discussions of that topic in Abel’s letters are accessible to strong undergraduate students. In particular, one should call attention to one of Abel’s questions about foundational issues. He argued that the theory of series, in particular, was in terrible shape, with no adequate foundations; nevertheless, he recognized that the lack of rigorous foundations did not seem to have led to errors. How could one explain that? Abel himself does not seem to have ever proposed an answer to that question, but it is a good one.

Note that Kendrick Press books can be ordered from their web site at http://www.kendrickpress.com/.

Fernando Q. Gouvêa hopes someone will do a book like this one for Kronecker.