Mathematicians today form an international community. Mathematical papers often have co-authors who live in different countries. Mathematical ideas and discoveries are exchanged quickly over the Internet and World-Wide Web. Internet discussion groups on particular mathematical topics draw members from many countries. And, every four years, the International Congress of Mathematicians gathers attenders from all over the world.

Few of us, perhaps, stop to wonder, "How did it get this way?" The volume under review, *Mathematics Unbound: The Evolution of an International Mathematical Research Community, 1800–1945*, explores this question. Jointly published by the American Mathematical Society and the London Mathematical Society in a series devoted to the History of Mathematics, it contains the proceedings of a symposium held at the University of Virginia in May 1999. There are 18 chapters, each by a different author or pair of authors.

Karen Hunger Parshall and Adrian C. Rice, the editors of the whole collection, acknowledge in their introductory essay that mathematics and mathematicians have a long history of crossing national and cultural boundaries; they mention Gerbert d'Aurillac (later Pope Sylvester II), Leonardo Fibonacci, and Leonhard Euler, among others. But to speak of mathematics as an international endeavor in the modern sense presupposes the modern nation-state; for this reason, the scope of the symposium is restricted to the time-period 1800–1945.

What will readers of *MAA Online* find of interest in this volume? Well, there is very little mathematics. On the other hand, I suspect that most mathematicians have a desire to know about the community they are a part of; and the question of how this community acquired the international nature it has today seems to me to be an inherently interesting one. An interesting topic does not, alas, guarantee an interesting presentation.

Parshall and Rice begin their essay with a tedious account of the fine distinctions in meaning among the terms 'internationalization', 'internationalism', 'transnationalism', 'supranationalism', 'multinationalism', and 'denationalization'. These are said to represent "analytic concepts". I read this section three times, but I'm still not sure what the point of the discussion is. Several authors of later articles in the collection dutifully refer back to the distinctions made here, but I can't see that they lead anywhere.

How, in fact, can we study the process of internationalization (if that's the correct word to use)? Several of the authors have been able to come up with nothing better than to offer us dreary statistics concerning the percentage of articles published in particular journals by authors of various nationalities. These statistics are mostly presented in the form of numerical tables. These authors might profit from studying the works of Edward Tufte on the visual display of data.

On the other hand, I did find some of the chapters to be quite absorbing. Chikara Sasaki writes of the reception of European mathematics into Japan, and Joseph Dauben of its reception into China. In both of these cases, we see an alien way of thinking about mathematics entering in and eventually displacing a native tradition of great venerability. To me, this is a fascinating story, and deserves to be told at much greater length.

Another topic that I found very thought-provoking is that of the position of German mathematicians in European mathematics during the twentieth century. We think immediately, of course, of the Nazi period, which is the subject of a chapter by Reinhard Siegmund-Schultze. But of equal interest, I think, is the treatment of German mathematicians by the European mathematical community in the aftermath of World War I. This story is partly told by Siegmund-Schultze, as well as in a chapter by Sanford Segal. It also comes into Olli Lehto's account of the history of the International Mathematical Union.

Some of the chapters focus on individual mathematicians, and the accounts of their personalities help to enliven what might otherwise be a rather arid analysis. Thus, Jesper Lützen writes on Joseph Liouville, Thomas Archibald on Charles Hermite, June Barrow-Green on Gösta Mittag-Leffler, Laura Martini on Cesare Arzelà, and Della Dumbaugh Fenster on Leonard Dickson. For the most part, these discussions treat the organizational activity of these mathematicians, rather than their mathematical work.

There is an essay by Aldo Brigaglia on the remarkable place of Palermo, Sicily, in late nineteenth-century mathematics; and an essay by Jeremy Gray which considers the relevance to mathematics of artificial languages such as "Volapük".

Since the chapters in this collection are written by different authors, the writing style is somewhat uneven. On the whole, though, it is the turgid verbal sludge that passes for modern "scholarly" English. Statements are "arguably" the case (p. 2, 4, 373); terms are "utilized" (p. 2); things are "contextualized" (p. 8); "roles" are "accessed" (p. 105); the "process of structuralization" is "conditioned" (p. 108); assimilation and transmission are "facilitated" (p. 124); debates are "situated" (p. 212); *nations* are "role models" (really!: pp. 237, 263, 264); sentiments are "reinforced" (p. 255); combinations are "embraced" (p. 263) (sometimes "firmly", p. 316); contributions have "impact" (p. 311); topics are of course "cutting-edge" (p. 315); roles are "highlighted" (p. 317, 319, 326, 328) (sometimes only "potentially", p. 204) or, alternately, "showcased" (p. 319) or "underscored" (p. 328); reputations are "cemented further" (p. 323). We are told of the idea that teaching and research should "inform" one another by "working in tandem" (p. 361) — I wonder if this author thinks that "in tandem" means "side by side". The ugly legalese "and/or" (pp. 17, 25, 29) could be avoided by following the common mathematical convention of "inclusive or".

Some of the ideas presented are quite remarkable. For example, "sociologists have argued that scientific papers are valuable indicators of scientific activity" (p. 62). Also, "naturally, published works served to transmit ideas from one mathematician to another" (p. 322).

The volume is well-printed and sturdily bound.

I can say that by reading this volume I learned a number of interesting things. To do so, however, I had to slog through a lot of tedious verbiage. No doubt the specialist will scan the volume eagerly, looking for points to dispute. The general reader, however, will likely find it tough going.

Stacy G. Langton (langton@sandiego.edu) is Professor of Mathematics and Computer Science at the University of San Diego. He is particularly interested in the works of Leonhard Euler, a few of which he has translated into English.