You're probably wondering two things: Is this a good book? Would I enjoy reading it? The answer to the first question is unequivocally yes. It is a very good book. I am afraid that most mathematicians probably would not enjoy reading it, at least not cover to cover. I found parts of it to be fascinating, and other parts much less so.

The book is about negative numbers and infinitely small quantities in the mathematical literature of the 17th to 19th centuries, principally the French and German literature, with special attention to textbooks. (Many authors in this period distinguished between number and quantity, although the nature of the distinction varied.) The author is extraordinarily well-read, and he presents and contrasts the work of a great many writers, some well-known and others quite obscure. Unfortunately, one of the results of this industry is page after page of tedious metaphysical reflections by so-called mathematicians — one wishes for a word like "poetaster" — who were not fit to erase Euler's slate.

The material on negative numbers is particularly exasperating, but perhaps only in this way can we fully appreciate how slowly they came to be accepted by rank and file mathematicians, even into the 1800s. On p. 5 of Euler's *Elements of Algebra* is the seemingly unremarkable statement that by continually subtracting 1 from 0 we get the series [we would now say sequence] of negative numbers 0, –1, –2, –3, –4,..., which consequently are less than nothing. As the author says, and then more than amply demonstrates, this "was to provoke sharp reactions in France" (p. 259), and "continued to be rejected over and over again" (p. 258). It was felt that negative numbers (or quantities) do not exist because — well, they just don't, that's all; it is impossible to imagine a quantity less than nothing. The debt interpretation goes back at least to Brahmagupta in the 7th century (p. 37), and at this remove it is hard to understand why it failed to satisfy so many people. In our day, the value to society of Paris Hilton and Tom DeLay are other convenient examples, or Ashlee Simpson's musicianship. By the time I reached the bottom of p. 367, where "Laplace simply assumes the existence of negative quantities", I was tempted to stand up and cheer.

The book is a translation of the author's Habilitationsschrift at the University of Bielefeld in 2002, with one significant difference: a chapter of the earlier work has been published separately (in German), and is only summarized here. A team of four translators was used to "accelerate" (p. VI) the process, and the result looks rushed. It is riddled with misprints that the most rudimentary spellchecker would have caught, such as "roblem", "pricese" and "presentend" within the space of four pages in section 2.9.3. We also find "und" in place of "and" a few times, generally between two proper names. I cringed when the phrase "continuous differentiation" was used in a non-mathematical sense four times on p. 15, but this ceases almost as abruptly. There are many awkward sentences (the long one in the middle of p. 307 is noteworthy) and few elegant ones, and I am inclined to place some blame for this as well on the translators, since the preface shows what the author is capable of. As historiography the book is excellent, but as a specimen of English prose it reads like a first draft. Anyone fluent in German would surely do better, if possible, to read the original.

The references run to 39 pages, the 38 references to the author's own work occupying almost 2 1/2 pages. Still, they are far from being exhaustive. It would have been natural to mention Frank Smithies's excellent book *Cauchy and the Creation of Complex Function Theory* on p. 476, or Kathryn Olesko's *Physics as a Calling* in the brief discussion of the seminar system on p. 486, or Thomas Hankins's biography of d'Alembert in section 2.9.3. The recent article by Ivor Grattan-Guinness (*American Mathematical Monthly*, March 2005, pp. 233--250) could be read in conjunction with the material of chapters 4 and 5 on the teaching of analysis at the École Polytechnique, which, except for chapter 6 on Cauchy, I found the most interesting.

Even if I were inclined to do so, it would be silly of me to dispute the author's judgments of figures like Francois-Daniel Porro and Johann Georg Tralles, whose work he must know better than anyone else. In the case of Cauchy, however, I will be a little more courageous. The author is not shy about criticizing other historians, and already on pp. 3–4 he takes Detlefs Laugwitz and Spalt to task for not examining Cauchy's communication with other mathematicians. On pp. 442–444 (see also Appendix C) he undertakes this himself, which is admirable, but every time I look at these pages I can't refrain from laughing because the results are so meager. It seems unlikely to me that Cauchy wrote many letters of scientific interest after he became a professor. He did send some letters to Libri and others during his exile in the 1830s. However, this was not because he wanted them to be read or replied to, but because he wanted them to be published. When he wrote something, he wanted everybody to read it. At several stages of his career he was publishing a journal, the *Exercices de Mathématiques*, consisting solely of articles by himself.

As great as Cauchy was, the second half of his career, after his return to Paris in 1838, would have been more successful if there had been someone he felt comfortable discussing mathematical ideas with on level terms; he would have published less, and better. There were several good candidates — Liouville, Dirichlet, even Gauss — but I think Jacobi would have been the best, for they had many common interests. The book has two wonderful quotations of Jacobi (who did write interesting letters), one to Dirichlet about Cauchy on p. 430, and one to von Humboldt praising Dirichlet on p. 558. These incidentally give some sense of what Cauchy was missing. He had two close mathematical friends, Andre Ampère and Jacques Binet, both Jesuits. Ampère taught him at the École Polytechnique and was later his colleague there, while Binet was in the class before Cauchy's at the École, and his tenure as Director of Studies there coincided with Cauchy's as professor. Ampère certainly had some influence on the young Cauchy, and perhaps Binet did too: they both presented papers on determinants (which Binet, following Laplace, called resultants) to the same body on November 30, 1812. (For a discussion of these two papers see chapter 4 of the first volume of Thomas Muir's *The Theory of Determinants in the Historical Order of Development*. This, Cauchy's first great paper, was also the first work to use the word determinant in the modern sense.) But neither Ampère nor Binet was quite a strong enough mathematician to be what the mature Cauchy really needed.

It is not so clear to me that Cauchy's "years as a professor at the École were a painful experience for everybody involved" (p. 429). Much as we may admire his *Analyse Algébrique* and his lectures on calculus, most of us will be glad not to have been among Cauchy's students. But I doubt that there was anything he enjoyed more than (not teaching, exactly, but) lecturing on mathematics, unless it was (not doing it, exactly, but) writing it. In spite of the constant complaints he endured, I suspect that Cauchy found his years teaching at the École Polytechnique, before the political situation deteriorated in advance of the July Monarchy of 1830, less painful than most other periods of his life.

The author discusses at some length whether the work of Cauchy and others with infinitely small quantities ought to be considered non-standard analysis, in the sense of Abraham Robinson and the calculus textbook of Jerome Keisler. (He thinks it should not.) This is some of the most interesting material in the book, and I find his criticisms of Laugwitz and Spalt generally persuasive despite the lack of support from Cauchy's correspondence. Unlike Spalt, I cannot see Cauchy as a follower of Lacroix (p. 433), even though Lacroix wrote a great calculus book and was Cauchy's teacher. I am sure that Cauchy did not see himself as a follower of Lacroix or anyone else, although this is not quite the same thing. The author makes an interesting criticism of Judith Grabiner regarding Cauchy and Lacroix on p. 30, although the first sentence of this paragraph seems to contradict the third one, and I am more inclined to agree with Grabiner in this case.

On p. 435 the author criticizes Spalt again: "This is why Spalt confirms the solipsistic position here as well: Cauchy apparently found himself in a historical situation in which it was possible for 'a creative mind to generate its own theory all by itself. With Cauchy, we have such a person before us.'" How well this describes Cauchy's real analysis can be debated, but isn't this exactly what he did in complex analysis? (This story is very well told by Smithies.) I think Spalt has expressed very well here what Cauchy was striving to do, in all of his mathematics — if not to generate, then at least to regenerate the theory all by himself.

There are interesting asides scattered throughout the book, often in footnotes. On p. 90, footnote 53 criticizes Wallis for slighting French mathematicians in favor of English ones and suggests that this may be the first instance of nationalist behavior by a mathematician. There are perhaps half a dozen statements of the author that could be construed as German nationalism, if one wanted to be uncharitable. He makes an excellent point on p. 386: some things in French mathematics around 1800 would be clearer if the authors had used subscripts, which were extensively employed by the Hindenburg school of combinatorial analysis in Germany at this time. (This point comes up again on p. 471.) He observes that Lacroix, who was very widely read, did use them a few times; he might have used them more often if he had not spent so much of his career describing other people's work. The Hindenburg school has not had such a good press, but they deserve a little credit here.

There is a mathematical error on p. 594, for which I don't know whether to blame the author or Paul Mansion, whose work he is describing there. (Presumably the translators are not at fault in this case.) It is claimed that the series 1 + 1/e + 2 exp(–4) + 3 exp(–9) + ... has sum e/(e–1). That is what the sum would be if the series continued geometrically from the first two terms, but it is not hard to see that the given series is less than this geometric series. One could obtain a very complicated expression for it by differentiating Jacobi's triple product identity. On p. 340, f – g should presumably go to zero, not one, and contrary to footnote 14 on p. 37, I think the quotation there refers to the 4 in the equation 4 = 4x + 20, not the 4x.

While the quality of the reading experience is highly non-uniform, there are many rewards here for a diligent and historically-minded reader with a passing interest in infinitesimals, ancient or modern. Some readers may also have more patience for the existential discussions about negative numbers than I did.

Warren Johnson (warren.johnson@conncoll.edu) is visiting assistant professor of mathematics at Connecticut College.