Maria Georgiadou's huge biography of the famous early twentieth century analyst, Constantin Carathéodory, is a truly herculean undertaking. *Constantin Carathéodory: Mathematics and Politics in Turbulent Times*, is a work which attempts to do justice to the complex life of a major scholar, an analyst educated in the Berlin-Göttingen tradition, who was at the same time a player in various European political circles of that era. Carathéodory was a Greek cosmopolitan and a socialite in many different locales, even beyond Greece and Germany, and he was of course an analyst whose work has left a measurable imprint on modern analysis. Thus, in addressing and analyzing both the scientific and political aspects and features of Carathéodory's life, Georgiadou's chosen objective is ambitious indeed.

Georgiadou's academic credentials include that she is a theoretical astrophysicist as well as a cultural historian with a focus on Hellenic studies (the latter by avocation and then some, it appears). She was originally educated at Thessaloniki (which possesses a particular connection to Carathéodory) and went on to advanced studies at Tübingen. These credentials go far to explain the author's painstaking attention to detail and her admirable conscientiousness.

It is worth noting at the outset that Georgiadou's presentation of specifically mathematical material is impressively accurate and complete, especially given that, when all is said and done, she is a physicist and not a mathematician. The book contains very long sections devoted to a play-by-play account and evaluation of Carathéodory's mathematical researches, arranged largely chronologically (and geographically). Admittedly these pages will likely be glossed over, or skipped altogether, by the reader who comes to this book with a dominant interest in relatively recent Greek political history; however, I should think that this describes only a minority of the readership. Mathematicians will without doubt comprise the vast majority of the audience and will find familiar things in these accounts, if only because of Carathéodory's influence on subjects that are now part of every PhD's qualifying examination experience. It is terrific fun to go through the corresponding history, be it in connection with Schwarz' Lemma or with Carathéodory outer measure, to name but two topics. Beyond this, Georgiadou's research provides a treasure-trove of minutiae that cannot fail to satisfy hard analysts and mathematical historians of the indicated disposition: a truly impressive achievement.

The complement of mathematicians coming to this book may find this level of attention to detail a bit tedious, of course, but continuity is not essential to the proper consumption of the whole tale. And this is a good thing, since, for better or worse, most of us already have a sense of what things must have been like in those days in Göttingen and Berlin. It is impossible not to relate many of Georgiadou's accounts of events in Carathéodory's life to counterparts as rendered by, for example, Constance Reid in *Hilbert* and in *Courant in Göttingen and New York*, or, to a lesser extent, in any number of biographies of Einstein (such as Ronald W. Clark's *Einstein, the Life and Times*).

Accordingly I think the reader would do well to come to this book with some history under his belt already: Reid's *Hilbert* is sufficient and perhaps necessary, and the recent *Mathematical Intelligencer* article on Frobenius comes to mind also. Without such a pre-established context it would be difficult to keep track of the comings and goings of some of the players. But with some *a priori* historical background present the reader has the benefit of now being able to add a wealth of biographical detail concerning old heroes to his existing store. This, I think, is the most attractive feature of Georgiadou's book: its most potent appeal consists in a very successful and evocative description of scientific and mathematical life in Europe and in particular in Germany (Heidelberg, Bonn, Berlin, and especially Göttingen, and, later, Munich, of course) during the heroic era of the early and middle decades of the last century. Carathéodory's life is integrally interwoven in this fabric.

Carathéodory did not decide on mathematics as his vocation until he was 27, having been originally educated as an engineer, primarily at the Belgian equivalent of the *École Polytechnique*. He took measurements of the Great Pyramid at Cheops and worked for the British on the first Aswan dam project in Egypt; it was apparently at this time that he developed his passion for mathematics proper. But it was a lecture by Fejér, given in Berlin in 1900, on a geometrical theorem by Hermann Amandus Schwarz, that, at least according to legend, caused Carathéodory to give up engineering for mathematics once and for all. He studied at Berlin (because he had too many relatives in Paris!), and included a lot of physics in his curriculum. After a few years he moved to Göttingen where he quite naturally fell under the spell of Klein and Hilbert; interestingly, Georgiadou claims that it was because he was so much in awe of these two scholars that Carathéodory submitted his doctoral dissertation to Minkowski. So it was that he took his doctorate in 1904 with a thesis on the calculus of variations. We also learn the marvelous bit of trivia that none other than Ernst Zermelo, generally associated all but exclusively with the foundations of mathematics and mathematical logic, also started off in the calculus of variations.

Upon not being offered a suitable post in Greece (in whose higher educational system he always maintained a vivid and vested interest), Carathéodory embarked on a career as a German academic, first at the Technical University of Breslau. He did fundamental work on thermodynamics that was championed by his friend, Max Born, the Nobel-prize winning physicist (and grandfather of Olivia Newton-John!); he married a distant aunt who was actually 11 years his junior, and settled into more-or-less normal academic life. Georgiadou observes that "[f]rom now on his life would follow twin tracks, dedicated to mathematics and to politics." It bears noting that the major proportion of Carathéodory's politics dealt with the world of mathematics and science, as would befit some one with his unusual and uncommon connections.

At Breslau, while occupying himself with the education of future engineers, Carathéodory turned his attention to a host of subjects in classical analysis, including in particular Picard's Great Theorem which Georgiadou discusses very well. It is also very striking to encounter a number of surprising names among Carathéodory's analysis contemporaries and interlocutors at this time: Paul Bernays, who is now known as a logician, Otto Toeplitz, a later topologist, and the later algebraists Issai Schur and none other than Frobenius himself. Truly, borders between mathematical fields are artificial (or at least they weren't controlled as strictly in those days).

Georgiadou includes a good deal of fascinating information here concerning Carathéodory's personal life: a wealth of "gossip" of the innocuous variety. To wit: he was possessed of great will-power and truly Spartan self-discipline, as attested to by the fact that when deprived of the use of his right hand because of a broken collar-bone, Carathéodory simply taught himself to write left-handed. As far as his daily work-regimen went, he labored very intensively: in Munich, where he eventually settled, he worked in two connected rooms, alternating between them as a function of the density of cigar-smoke. It was later claimed insistently by his daughter that it was this very heavy cigar smoking which ultimately claimed Carathéodory's life in 1950, although the facts favor prostatitis and uraemia.

Even Georgiadou's detailed discussions of Carathéodory's mathematical activities at this time are peppered with interesting historical asides. We learn, for instance, that Lars Ahlfors observed that it was greatly due to Carathéodory's generosity that an obscure note by H. A. Schwarz should have led to Schwarz' Lemma becoming the profound analytical tool that it is. J. E. Littlewood once wrote a paper titled, "On the conformal representation of the mouth of a crocodile." Einstein once enlisted Carathéodory's aid as regards the Hamilton-Jacobi equation. And here is my favorite passage in the whole book: "Except for his tireless mathematical research and conscientious lecturing, [Edmund] Landau cultivated two hobbies, namely stamp collecting and passionate reading of detective stories, *and showed contempt for everything even remotely practical.*" [My emphasis.]

There is also a very nice discussion of Hilbert's treatment of Kirchhoff's Law and the theory of radiation in relation to the work of Pringsheim and Planck, modulo kibitzing by Carathéodory. Hilbert was at odds with both, made peace with Planck, but remained at war with Pringsheim, so to speak. Again, what is particularly noteworthy, of course, is the fact that these old scholars were so multifaceted: they all knew their theoretical or mathematical physics. This is consonant with the style prevalent at Göttingen as described by Reid in *Hilbert*.

In 1913 Carathéodory succeeded Felix Klein in Göttingen; in 1918 he moved to Berlin, succeeding Frobenius who died in 1917; Carathéodory was unhappy in Berlin, however, and departed again in 1919, heading for Greece. He landed in a country suffering immense turmoil and was presently charged with the ultimately misbegotten task of starting a university at Smyrna to compete with the University of Athens. Georgiadou, true to her objective of doing justice to history and politics as well as mathematics, presents a wealth of detail in connection with Carathéodory's ultimately unsuccessful work as a would-be university administrator in Greece. The last word on the matter is that Carathéodory lost his belongings "in the flames of the Turkish invasion [of Smyrna]." He headed back to Germany in the early 1920's.

From my point of view this section of the book, and others like it, dealing with politics, for lack of a better word, comprise the least interesting part of the work. Georgiadou's thoroughness works against her in the sense that the mathematical reader, for one, is here quickly apt to become bored and will skip ahead to the next part that deals with mathematics or perhaps Carathéodory's personal life.

It is interesting, for instance, to learn that in 1918 Carathéodory stayed in Berlin by himself, while his wife and two children remained in Göttingen. Carathéodory's son, Stephanos, suffered from polio since he was five and contracted pneumonia at this time, on a visit in Halle, and one can but speculate about the effect these events must have had on the mathematician. But to my sensibilities the role played by Carathéodory in Greek academic politics are of very little interest and no entertainment value. I emphasize that this is a myopic perspective: certain Greek historians would have an altogether different opinion.

In any case, both on a personal level and on a professional one, Carathéodory's life now unfolds along more-or-less predictable paths as in the early 1920's he heads for a professorship at the University of Munich where, except for travel, he spent the rest of his life. Perron and Tietze were his office-mates there and we meet Pringsheim again. It was in Munich that Carathéodory perfected his routine of working in two rooms while essentially chain-smoking cigars, i.e. working modulo smoke-density. His avocation was the composition of love poems in French. Apparently, Carathéodory's wife, Euphrosyne, was quite partial to French sentimental novels, too. We find out more about Carathéodory's status as a member of the local Greek-Orthodox Church: he was a member of the church council which, in 1927, was led by the Greek Consul-General in Munich. And then there is the story of Pringsheim's duel with a music critic...

It would be a straightforward matter to continue along these lines, sketching various aspects of Carathéodory's life as presented by Georgiadou in this scholarly tome of 651 pages (only 456 of which comprise the narrative proper; there are five appendices and another handful of indices and such), but that would be unforgivably cumbersome. The observations presented above convey the tenor and style of the book, as well as its orientations. Suffice it to say, then, that as a senior academic in Munich for the remainder of his natural life, Carathéodory found himself part of the well-known and well-documented events characterizing Germany in the middle part of the twentieth century. Chapter 4 of the book, titled "A Scholar of World Reputation," covers in nearly 100 pages the solidification and amplification of Carathéodory's reputation and position as a major academic player. It is followed by Chapter 5, "National Socialism and War," which addresses a subject of such scope that a separate review might be indicated for it (and I will forego). In addition to what Carathéodory went through as a German academic during this demonic era, Georgiadou discusses his involvement with the difficult issues of the fate of refugee scientists and how to foster some measure of sanity and humanity through it all. It reminds us in this day and age of the enthronement not so long ago of a horrific madness.

It is in connection with a specific aspect of Georgiadou's treatment of World War II, however, that I must offer my most severe criticism. The events she addresses move her to editorialize about the role played by Pope Pius XII and the Vatican in condemning Nazism and pitting the Church against this great evil. I am chagrined that, according to her references and bibliography, Georgiadou bases her entire analysis, which is categorically critical if not a condemnation of the pope's actions (on pp. 333-334), on the fatally flawed book, *Hitler's Pope*, by J. Cornwell. The English historian, Anthony Rhodes, in his book, *The Vatican in the Age of the Dictators*, provides a definitive account of how things really were in those dark days, effectively contradicting and thus debunking Cromwell's assertions. Additionally, in First Things (November 2004, no. 147) one finds a review, by W. Doino, Jr., and J. Bottum, of Harold H. Tittmann, Jr.'s recent book, *Inside the Vatican of Pius XII: The Memoir of an American Diplomat*, which altogether exposes Cromwell's claims as erroneous and therefore, at best, unwarranted (see http://www.firstthings.com/ftissues/ft0411/reviews/doino.htm).

One last word about Carathéodory during the War. The speech he had prepared as an oration at Hilbert's funeral, on behalf of all German mathematicians, was delivered for him by Gutsav Herglotz, apparently in tears. A depiction of the first page of Carathéodory's hand-written notes for the speech is presented on p. 387. It is this sort of attention to the details of the scholar's life that constitutes the best aspect of this book, and, with the caveats expressed above, I recommend it to all mathematicians interested in the heroic generation to which Carathéodory belonged.

Michael Berg (mberg@lmu.edu) is professor of mathematics at Loyola Marymount University.