In Bourbaki’s historical note on the calculus, it is said that the history of mathematics should proceed in the same way as the musical analysis of a symphony. There are a number of themes. You can more or less see when a given theme occurs for the first time. Then it gets mixed up with the other themes, and the art of the composer consists in handling them all simultaneously. Sometimes the violin plays one theme, the flute plays another, then they exchange, and this goes on.

The history of mathematics is just the same. You have a number of themes; for instance, the zeta-function; you can state exactly when and where this one started, namely with Euler in the years 1730 to 1750, as we saw yesterday. Then it goes on and eventually gets inextricably mixed up with the other themes. It would take a long volume to disentangle the whole story.

(André Weil, “Two lectures on number theory, past and present” *Enseignement Math.* (2) **20** (1974), 87–110.)

Celebrated historians of mathematics Umberto Bottazzini and Jeremy Gray have completed a magisterial tome on the history of complex function theory, which in their estimation is “the first ever to be written exclusively on this subject”. They ambitiously claim: “In fact, we provide here the first full treatment of the work of several major mathematicians in the context of complex function theory.” The distinguished list includes Gauss and the founding fathers of complex analysis: Cauchy, Riemann and Weierstrass. The reader of *Hidden Harmony — Geometric Fantasies* will be treated to over 700 pages of Bottazzini and Gray’s descriptions, many in exquisite detail, of a myriad strands from both the history and the heritage of this fascinating subject.

Though the authors have deftly disentangled a number of such strands, discerning themes and uncovering hidden harmonies will remain an arduous challenge for the tireless reader. Furthermore, the book contains a wealth of socio-political context within which the authors seamlessly expose the highly non-linear development of the concomitant mathematical ideas. To quote them:

The rise of complex function theory cannot … be understood solely as a history of ideas, and we have not only traced it as a sequence of ideas, theorems, and theories, but we have also paid attention to the historical context, the biographies of the main actors, the different reactions and developments in various countries depending on the national traditions, and so on.

There is much in this book that will educate, be appreciated by, and no doubt provoke mathematicians as well as historians of mathematics and of science.

The authors have published on various aspects of the subject from the late 1970s and early 1980s and continue to do so right up to the present day. Two notable works include Gray’s 1981 PhD thesis that was later reworked into *Linear Differential Equations and Group Theory from Riemann to Poincaré* (Birkhäuser, 1986) and Bottazzini’s *The Higher Calculus. A History of Real and Complex Analysis from Euler to Weierstrass*. It is also worth pointing out Gray’s recent scientific biography of Poincaré, which discusses some of the themes explored in the latter third of the book. For readers unable to invest the time in studying the gamut of Bottazzini-Gray’s rich combined scholarship, there are two shorter articles by the authors that cover the development of analytic function theory over a somewhat larger period than that covered in the book under review: Bottazzini’s “Complex function theory, 1780–1900”, which appears on pages 213–259 of *A history of analysis*, edited by H. N. Jahnke, and Bottazzini-Gray’s “Complex function theory from Zurich (1897) to Zurich (1932)”, *Suppl. Rend. Palermo* (2) **44**, (1996), 85–111. Readers familiar with the authors’ earlier papers and monographs will find substantial textual and thematic overlap with their new tome, a large part of which must be treated as a work of synthesis. On the other hand, the authors take this opportunity to challenge a number of theses established in the work of historians (e.g., Smithies, Kline, Belhoste and Darrigol) and mathematicians (e.g., Remmert and Markushevich). These provocative claims are sometimes hidden in footnotes, but are just as often left peering defiantly out of the text proper.

**Outline**

The book begins with Legendre’s account of the theory of elliptic integrals (in particular, the computation of their values), by then a well-established topic studied since the time of Netwon. The subject was taken up nearly simultaneously in 1827 by Abel and Jacobi, whose groundbreaking reformulation of the theory both unified and clarified more than a century’s worth of previous mathematical research. By inverting the elliptic integrals and treating the variables as complex, Abel and Jacobi created elliptic functions that were at once both familiar (via analogies to the trigonometric or circular functions) and mysterious. The establishment of the theory of elliptic functions forms the first chapter.

Elliptic functions turned out to be a great wellspring for mathematical research over the next century, leading to the study of theta functions, modular functions, and Abelian functions — all major tributaries that run through large parts of the book. The next couple of chapters are devoted to Cauchy’s work. We are first thrown back to the study of eighteenth-century methods for evaluating integrals, leading up to the work of Laplace and Poisson and their debate regarding the “the passage from the real to the imaginary” in the early 1800s. Much of this fascinating story could easily be treated, albeit whiggishly, as witchcraft. There were, in the wake of the Laplace-Poisson debate, growing concerns about the legitimacy of formal methods: in Poisson’s words, the need for “direct and rigorous” methods to justify such “means of discovery”.

It was left to the prodigious talents of Laplace’s protégé Cauchy to introduce such rigor in his 1814 memoir on definite integrals. In retrospect, Cauchy’s early work seems clearly ahead of its time. Defining the definite integral as limits of partial sums, he made a decisive break with the orthodoxy that had understood the definite integral via antiderivatives. Students familiar with Cauchy’s integral theorems and residue theory will find it interesting to read about its precursors in his study of singular integrals and principal values. Bottazzini and Gray then go on to describe, at times in painful detail, Cauchy’s tortuous path from the contemporary theory of real variable functions to his own ideas for a theory of functions from \(\mathbb{C}\) to \(\mathbb{C}\). For all the inherent obscurities and limitations that are to be found in Cauchy’s journey (e.g., his avoidance of the geometric representation of complex numbers until he was 60 years old!) there remains much to be gleaned for the curious reader. A short fourth chapter returns to Jacobi and his introduction of theta functions, and then proceeds to describe some of the growing connections being made with other ares, viz. mechanics and number theory.

The fifth chapter is devoted to understanding Riemann’s revolutionary visions. Whereas Cauchy started with integrals, Riemann’s complex function theory began (in his 1851 dissertation) with the definition of complex differentiability and its natural consequence, what are today called the Cauchy-Riemann equations. Riemann’s key to the multi-valuedness of a complex function was to imagine a surface (later called a Riemann surface) spread over the complex plane with as many sheets as the branches of the function, which was connected in an intricate way yielding a single-valued function on the surface. This led, via the potential theory of Gauss and Dirichlet that Riemann was well versed in, to the fundamental role of harmonic functions in studying complex functions defined on any two-dimensional surface, and also to his much berated use of Dirichlet’s principle. His ideas also charted the study of complex functions independent of their analytical expressions, the first of a number of methodological divergences from the Weierstrassian credo.

Riemann’s location in Göttingen kept him close to strong traditions in physics and philosophy that played a significant role in the development of his seminal ideas. Mathematicians would take close to a century to fully comprehend, flesh out rigorously and further deepen his insights. Where his contemporaries struggled, the modern reader can glimpse the vast, nascent topological and geometric vistas uncovered by his singular vision. In our opinion, Riemann’s work deserves further study. There remain several mysteries regarding the context in which his ideas developed, and its crucial relationship to the histories of topology and algebraic geometry.

The last major hero of the book Karl Weierstrass’s lifelong ambition was to build a theory of Abelian functions that were functions of several complex variables. Unlike Cauchy and Riemann, he started with power series expansions, algebraic methods and convergence arguments; all of which naturally generalized to higher dimensions more easily than Cauchy and Riemann’s foundations. The study of Weierstrass’s legacy, from his early education under Gudderman right up to his last lecture, takes close to the next 150 pages.

The seventh chapter returns to further applications and interactions of complex function theory with other areas of mathematics via differential equations, e.g. Dirichlet’s principle, hypergeometric equation, and minimal surfaces. The last chapters cover advanced topics from the 1880s to around around 1910 (e.g., the uniformisation theorem, Montel’s theory of normal families, and the Fatou-Julia theory of rational iteration); a very brief account of the budding field of several complex variables (in whose modern development one finds a vindication of Riemann’s vision); and an interesting final chapter that contains a comparative analysis of over 60 textbooks on complex analysis written before 1940. The book ends with a very useful bibliography, and subject and author indices.

**Audience**

Many historians will probably find the book too mathematical, while mathematicians interested in learning about the heritage of their subject will have difficulties with deciphering the exposition of the mathematics involved. Such barriers are made worse by the inevitable array of misprints in a monograph of this size. Our expert authors know, all too well, the slippery slopes that must be maneuvered in exposing mathematical ideas in the process of gestation. In their words:

To say nothing is to produce confusion. To silently bring them into line with modern standards not only introduces anachronisms but also brings in historical falsehoods and nullifies the purpose of a history. To correct them in more than the most egregious cases is to encumber genuine advances with the admission of genuine blunders and thereby diminish the work of major mathematicians. The best policy is to read on in a spirit of dialogue with the earlier authors, aware, as one might be, of the limitations and false implications of their papers and books, and waiting to see when, if at all in the period, a better light was shone on the subject. In this way one can grapple with more of the complexity, and the drama, of the past.

Both writing and reading the history of mathematics demands a more than superficial knowledge of and a certain degree of fluency with the mathematics involved, at least to be able to “to read on in a spirit of dialogue with the earlier authors”. One is reminded of the Klein quote that prefaces the penultimate chapter:

When I was a student, Abelian functions — as a consequence of the Jacobian tradition — stood as the undisputed summit of mathematics, and every one of us had the self-evident ambition to go further in that direction themselves. And now? The younger generation scarcely know Abelian functions any more.

The same could be said of much of the mathematics under discussion in Bottazzini-Gray. For example, the major emphasis placed on elliptic functions and elliptic integrals as the great source around which complex function theory grew and developed will be lost on most of today’s “younger generation.” Such subjects, and many others, are considered far from central and often given hardly more than lip-service in a first graduate course in complex analysis. The authors, having spent a significant portion of their lives researching these paths, often lapse into “talking among themselves” and thereby exclude readers who are unfamiliar with the notation and rigor of a previous mathematical epoch. For a project of this scope and importance, the authors would have been helped by having a few active mathematicians who are involved in teaching and research in complex analysis to carefully review the text in its entirety before publication. Such input would make the mathematical content far more accessible to a larger audience within mathematics, thereby extending its didactic scope.

I must admit to a certain selfishness in believing that mathematicians have the most to benefit from Bottazzini-Gray’s grand endeavor. To quote Leibniz:

Its use is not just that History may give everyone his due and that others may look forward to similar praise, but also that the art of discovery be promoted and its method known through illustrious examples.

A revised edition, keeping in mind the needs of such an audience of “creative or would-be creative scientists”, would be most welcome. For mathematicians reading this review, I point to the example of the French collective Henri Paul de Saint Gervais and “his” recently published first book *Uniformisation des surfaces de Riemann. Retour sur un théorème centenaire*, which grew out of a highly collaborative exercise in understanding the mathematics behind the uniformisation theorem. Saint Gervais’s book is far from being a purely historical work aimed at historians, but rather a creative engagement with classical mathematics by mathematicians of today. It is certainly a model that many younger mathematicians and students would appreciate seeing emulated.

**Minor quibbles: Authorial Carelessness**

There are, as alluded to, considerable difficulties in tracing the development of a certain theme within the rich texture of Bottazzini-Gray’s narrative. However, this is not helped by gaps in the index, e.g., the reader interested in the *Schwarzian derivative* would have to do better than simply following up the two instances listed by the index, neither of which are particularly insightful. There are two other places where the Schwarzian derivative does show up in the book. Its mathematical form is recognizable in the earlier of these instances, though readers may find the surrounding mathematical exposition lacking sufficient clarity.

As mentioned before, typos must surely abound in such a large book. Folland, in his excellent review of *Hidden Harmony — Geometric Fantasies* that appeared earlier this year in the Monthly (accessible on JSTOR) has compiled a short sample list of mathematical inconsistencies and typographical errors. I lost interest in maintaining such a list when it predictbly grew out of bounds due to the size of the book. Typos aside, it is also easy to discern examples where the authors appear to have written sections independently without looking over each other’s work with regard to continuity and accuracy. It was surprising that such slips were overlooked by the referees, copy-editors, not to mention the authors themselves. What follows are a few such examples that are easier to present without getting into technicalities.

As an example regarding discontinuity: in the section on Riemann’s famous 1851 doctoral dissertation, we read on p. 270 that Riemann emphasised the novelty and generality of his approach to complex functions by stating that

Previous methods for handling these functions have always given as the definition an expression of the function in which its value was given for every value of the argument. Through our research it has been shown that, as a consequence of the general character of a function of a complex variable, in a definition of this kind a part of the determining elements is a consequence of the rest, namely the range of the determining elements is traced back to those necessary for determining [the function]. (Riemann 1851, Sect. 20).

Three pages later we again read that “Riemann then sketched, in Sect. 20, how the new approach to complex functions could go:

Previous methods of treating these functions always based the definition of the function on an expression that yields its value for each value of the argument. Our study shows that, because of the general nature of a function of a complex variable, a part of the determination through a definition of this kind yields the rest. Indeed, we reduce this part of the determination to that which is necessary for complete determination of the function. This essentially simplifies the discussion. …

The second quote, taken from a more recent (and in my opinion more readable) edition of Riemann’s complete works, goes on for another two paragraphs having invoked an uncomfortable déjà-vu.

Regarding inaccuracy: I was surprised to read on p. 264 that “Despite the steady decline in his health, Gauss also discussed mathematics with Riemann: as Weil showed, quoting a letter from Betti to Tardy, the idea of cutting a surface came from a conversation with Gauss on mathematical physics.” I was aware of the source that was cited for this claim (A. Weil, “Riemann, Betti and the birth of topology” *Arch. Hist. Exact Sci.* **20** (1979), 91–96) and did not recall any mention of mathematical physics in Betti’s letter. Later, on p. 330, we read again that “the origin of the idea of cross-cuts … came to Riemann’s mind because of a definition Gauss gave him of them during a conversation on another subject” The second time round, as in Weil’s article, there is no mention of physics. One should probably not make too much of this solitary piece of evidence regarding the extent of Riemann’s direct interactions with Gauss, in particular with respect to their mathematical content.

Often the authors forget to reference relevant sources that could help clarify the mathematics being discussed. For example, consider the following lines from the section on *Poincaré and Automorphic Functions* that describes Poincaré and Klein’s competition with regard to the uniformisation theorem:

Poincaré was struck by the connection between non-Euclidean polygons and Riemann surfaces. … It is easy to go from polygons and groups to surfaces, but to go from surfaces to groups is not so straightforward. Can one be sure that this way round is a surjective map? In what sense, if any, is it analytic? Writing in 1884, Poincaré pointed out that an analytic map between complex varieties of the same dimension will be an onto map provided the surfaces have no boundaries. But what if the space of all groups has a boundary? “This is a difficulty one cannot overcome in a few lines” (Poincaré 1884a, 332).

The authors move on without any further remarks on “Poincaré 1884a” which refers to the fourth of the five famous memoirs on Fuchsian and Kleinian functions and groups that appeared in *Acta Mathematica* between 1882 and 1884. The study of the boundary of “the space of Fuchsian groups” that Poincaré was alluding to in this paper would take mathematicians over half a century to begin in earnest, and continues to challenge current research. The reader unfamiliar with such material would appreciate a nod to the theory of Teichmüller spaces and possibly to Thurston’s revolutionary work in the area. As a result, Poincaré’s incredible foresight is sadly downplayed.

There are also a number of examples of careless scholarship, many of them minor, though unexpected of experts. For example, the authors end their four-sentence discussion of KAM theory with: “The amusing hypothesis on \(F\) is that it be 354-times differentiable”, with neither any references to the literature nor to the number being quoted. For a historian of science, or someone unfamiliar with this deeply ploughed area of sustained research, such a statement will be entirely unhelpful. (The non-specialist mathematically inclined reader may read this for a more precise account and attributions.}

The title of the book, *Hidden Harmony — Geometric Fantasies*, may strike readers, as it struck Folland, “as a triumph of imagination over lucidity”. This is unsurprising due to the authors’ lack of explanation. The phrase “geometric fantasies” probably refers to the following remark (my emphasis):

Weierstrass claimed that “he understood Riemann, because he already possessed the results of his [Riemann’s] research”. *As for Riemann surfaces, they were nothing other than “geometric fantasies”*. According to Weierstrass, “Riemann’s disciples are making the mistake of attributing everything to their master, while many [discoveries] had already been made by and are due to Cauchy, etc.; Riemann did nothing more than to dress them in his manner for his convenience”. (U. Bottazzini, *“Algebraic truths” vs “geometric fantasies”: Weierstrass’s response to Riemann*, Proc. ICM, Vol. III (Beijing, 2002), 923-934.)

Curiously, though these snippets of Weierstrassian commentary are repeated in Section 6.4.1 (Casorati’s Notes, p. 383) of the book, the line “As for Riemann surfaces, they were nothing other than “geometric fantasies”,” is no longer present. What “hidden harmony” may refer to is left to the reader’s imagination.

I should re-emphasize that most of the above are but minor quibbles, that may easily be rectified in what would be a very welcome revised edition. More serious effort may be required in making the mathematics approachable to an advanced undergraduate or beginning graduate student audience. This would certainly extend the book’s didactic scope and enhance its reception. One hopes that such ideas may lead to a separate book aimed at students, e.g., in a similar spirit to Saint Gervais’ book, or to Gray’s *Worlds out of nothing. A course in the history of geometry in the 19th century*.

**Major quibble: Springer’ Slovenliness**

I cannot end this review without writing about the disgrace into which Springer’s long tradition of publishing excellence has fallen. In the past, for those who often roamed the stacks, it was relatively easy to admire the mathematical content that came almost hand-in-hand with the great care that went into the production of Springer monographs. There was excellence with regard to the paper, the ink, the printing of the text, the fonts, the reproduction of figures, and the binding, to say nothing of the unsung but scrupulous attention to detail paid by the copy-editors. All such aspects of production quality have all but disappeared in recent Springer publications.

*Hidden Harmony — Geometric Fantasies* has a list price of USD 189.00, and yet comes in a cheap and crude print-on-demand format! A print-out from a pdf onto low-quality paper that seems to be almost “stuck” together with some haste and then mailed out to the unsuspecting customer. The resulting quality is terrible. There are numerous pages where the printer has not left a proper impression on the page, with the typeface considerably thinner and less visible than on others. The reproduction of figures is appalling. It is painful to see such ugly low-resolution scans of the original images, the reproduction of which are further worsened by the bad quality of printing. The entire publishing house should be taken to task for allowing their standards to plummet while their prices continue to rise. Such steep prices may not be entirely unreasonable for a specialist monograph with limited readership, but certainly not at the expense of abysmal publishing standards! Sadly, this admirable book by two celebrated researchers of mathematical history affords us with yet another example of a travesty (that is part of a larger tragedy) of the fine art of mathematical publishing.

**Coda**

For all its imperfections, Bottazzini-Gray’s magnum opus has been long overdue. It stands its ground as a scholarly treatise that fills many lacunae in the extant historical literature. It will surely provoke further debate and research. As a bonus, it comes filled with treasures for both the specialist and the novice. I cannot recommend it strongly enough to students, teachers and researchers of mathematics who are curious about their complex analytic heritage.

Tushar Das is an Assistant Professor of Mathematics at the University of Wisconsin–La Crosse.