In many ways people interested in mathematics have never had it so good. New expository and historical books on mathematics abound. When else have we seen two biographies of a twentieth-century mathematician appear within a matter of months? (See Hoffman's The Man Who Loved Only Numbers and Schechter's My Brain Is Open.) Just recently, John Wallis played a significant role in Iain Pears' An Instance of the Fingerpost. And now we see in quick succession a novel about Galois (Petsinis's The French Mathematician) and this book, a novel ostensibly about D'Alembert.
Novels based on actual mathematical events or characters create interesting opportunities but also pose some special problems. They have the potential for offering insights into the thinking and the interactions with others that went into the discovery of great mathematical results. The form of the novel allows the author to reconstruct conversations, whether accurately or not we'll never know. We avoid the annoying experience of reading in a biography a conversation that we know has to have been the work of the biographer's imagination because there could be no tapes or transcriptions of actual conversations. But in either case, the conversations, the influences, the moments of inspiration, and the insights are the constructions of the novelist or the biographer. Why a novel and not a straight biography?
In the present rather slim volume only the first of the three sections deals primarily with the mathematician-physicist-philosopher Jean le Rond D'Alembert (1717-1783). The narrative is attractive, with some beautiful writing in a rather grand romantic style. It is not for everyone, however. First of all, for a mathematical audience it may not be immediately attractive to some because D'Alembert's contributions were largely in what we would today consider to be physics or very traditional branches of applied mathematics. He made some contributions to the development of the calculus in its early stages and is known for a number of discoveries in mechanics, largely appearing in his classic work, the Traité de Dynamique. Principal among these is the relationship between dynamics and statics, the so-called D'Alembert Principle of the title. Of course, he is probably best remembered these days as Diderot's collaborator in putting together that great monument of the Enlightenment, the Encyclopédie. So those looking for nice bits of mathematics and insights into their discovery will probably be disappointed.
Even the style of the first, most easily accessible "panel" takes some getting used to, because it alternates between seemingly autobiographical passages in the first person by D'Alembert and third person narratives on life in the D'Alembert household by his maid, Justine. But it's not that simple. There are interludes of letters ostensibly from D'Alembert; from the object of his longtime romantic obsession, the historical figure Julie de L'Espinasse, hostess of a prominent Parisian salon; from the object of her longtime romantic obsession, the Comte de Mora; from the later object of her affection (after the death of the Comte de Mora), the Comte de Guibert; along with a number of others. (Yes, it does remind one at times of Schnitzler's La Ronde, though it's not as racy--a complicated string of romantic relationships and efforts to conceal them.) And with all of this, is there interesting mathematics? Well, not much.
Still, when the author does have the characters talk about mathematics, it can be quite beautiful. Early on D'Alembert describes how, in his dream, he saw
that first manifestation of natural geometry, as it appeared to me all those years ago. I see myself as a young child (no more than three years old) sitting on the floor, while the sunlight falls crumpled from the flawed pane in the window above me. I watch the pattern which the light makes upon the floor; ripples of brightness, where the sunshine has been distorted in its flight. By some mysterious process,the falling light is creating an image, or at least an interpretation, of the imperfect pane through which it has passed. It must have been then that my passion for understanding the ways of nature first took root.
A later passage tells us movingly of the moment in class when the schoolmaster introduced him to the notion of Zeno's Paradox and he got his first glimpse of the calculus.
When D'Alembert tries to pass along to Julie some of his passion for mathematics, she replies that she will "never be able to understand your wonderful theories; they are as incomprehensible to me as the language of the Chinese." He replies,
No, Julie--mathematics is the easiest of all knowledge to comprehend. If I want to learn Chinese, then I must patiently learn a system of rules and symbols which have been developed by that race over many thousands of years--a system which is, in fact, wholly arbitrary, and fashioned only by convention and the mutual agreement of all those who use it. But in mathematics there is nothing which is arbitrary. If no-one taught you the rules of mathematics then you could still, if you were patient enough, discover them all for yourself.
That's debatable, but the comparison to a foreign language is provocative.
In addition to the mathematics and the story of D'Alembert's unrequited passion for Julie de L'Espinasse, there is a fair smattering of philosophical speculation and gossip of the salons to provide entertainment. The cast of characters is formidable: Rousseau, Diderot, Voltaire, Condorcet, among others.
The author's language is usually elegant, but there are those exceptional moments. When Diderot talks about his foster mother, Madame Rousseau, and comments that her husband has "since passed away", it's a surprise to see such a tired euphemism in otherwise believable eighteenth century prose. On another page we see: "I mentioned Mademoiselle de Mais, whom I know has shown considerable interest..."--a typo, we hope.
The curious structure of the book--a triptych motivated by Diderot's organization of the Encyclopédie: memory, reason and imagination--will probably not be appreciated by most readers. The second section, "The Cosmography of Magnus Ferguson", has only a tenuous connection to D'Alembert and is concerned with interplanetary travel. It has sections about each of the planets and, though it contains a few suggestions of probabalistic games, it has an overall dream-like quality that will discourage many readers. The section redeems itself by being mercifully short.
The third section, "Tales from Rreinnstadt" [sic] is a Borges-like account of some rather strange, albeit tantalizing, tales, seemingly far removed from D'Alembert, and really an extension of one of the author's earlier novels.
Advice: read the first section on D'Alembert (half of the book), then be prepared to call it quits.
G. L. Alexanderson is Valeriote Professor of Science and Chair of the Mathematics and Computer Science Department at Santa Clara University. A former editor of Mathematics Magazine and secretary of the MAA, he has recently completed a term as MAA president.