In this lovely, thoroughly enjoyable book, Gillian Bradshaw tells a tale of young Archimedes returning to Syracuse from Alexandria in 264 B.C., just in time to bid farewell to his dying father and to save Syracuse from a Roman siege. The siege in question, then, is not the famous one. Bradshaw has cleverly transposed the Archimedes we are familiar with — the venerable sage of the Second Punic War, heroic and doomed in 212 B.C. — into a younger and altogether more upbeat Archimedes. The transposition also allows, paradoxically, for greater historical accuracy: since Bradshaw tells of an Archimedes that is completely unknown, her guesses are as good as that of any professional historian. Generally speaking, she plays the Archimedes legend for whatever its worth, but is also very careful to stick to the plausible or, at least, to the feasible. And here comes a major advantage of avoiding the great siege: otherwise, Bradshaw would have had to address the issue of *burning mirrors*, and so would have had to sacrifice either sobriety (by allowing them in) or excitement (by keeping them out). As it is, her exciting and sober romance can be confined to catapults.

This being a romance — which I mean as praise — literary criticism is somewhat inappropriate. The reader should know what to expect. Bardshaw's style is fully under control and there is nothing to make you wince. The book is written in the functional prose of simple sentences with little of either word play or metaphor. The curious result is that the careful accumulation of verisimilitudes does not quite bring a reflected universe into existence: all the characteristics are real, but not the characters themselves. Perhaps, to achieve the full magic of an invented reality, the irrationality of language must be allowed to match the irrationality of the world. More important, reality cannot be made out of the generalities of plausible characteristics: it derives from precise, surprising observations. Bradshaw's descriptions are usually minimal and her eye does not wander off the storyline. We are offered the conventional psychology of white or (very frequent) red faces. Bradhsaw's best strength, though — an important one - is sartorial description: tunics and cloaks are very convincingly rendered and the author never forgets what a character wears or how wool feels in the Mediterranean summer.

At any rate, it is unfair to criticize Bradshaw on account of her style and description. These are subservient to an inventive and, broadly speaking, convincing story. It appears that, for Bradshaw, the greater part of the art of storytelling is the orchestration of subplots; she is very good about that. Each subplot belongs to a clearly distinct type — coming of age, choice of fate, war, devotion and sacrifice, manipulation, courtship. These are all neatly interdependent. Typically, when X goes to meet Y in the service of subplot A, he happens to come across Z, which serves subplot B. (It helps that this all takes place on the small stage of a City State under siege.) Trails are pursued and put aside with great precision, making sure the reader is always curious but never frustrated. Climaxes and resolutions progress in a crescendo of human interest. The overall experience is like watching an excellent made-for-TV film. The stock situations give rise to stock reactions as we are momentarily moved and intrigued; when the novel ends, this is all forgotten. Apparently this is as much as Bradshaw wanted to achieve and the novel is clearly successful.

Bradshaw is at ease in the Greek world and whatever anachronisms there are seem to be the outcome of a deliberate policy to lull the reader into accepting a world which is only superficially unfamiliar. Modern middle-class anxieties are translated from dollars to drachmas; the love interest is strictly heterosexual. Bradshaw is apologetic, in an appended historical note, about not being a geometer. She has in fact made a very impressive effort. Practically all the extant works of Archimedes are alluded to, going beyond the mere evocation of title to a sense of the discoveries themselves. (I have found allusions to all the treatises, with the exception of *Conoids and Spheroids*: did I miss anything?) Typically, Archimedes would react to some chance observation and then go off a mathematical tangent: a spiral pattern on a dress here, a dead fish floating there. This game is very ingeniously played by Bradshaw.

There are very few mathematical mistakes to notice. I do not know what Bradshaw means when she has Archimedes say that a triangle and a parabola — by which is meant a parabolic segment — do not balance (this is said in the course of a rather contrived simile about Archimedes' relationship with the king). Of course, every pair of homogeneous magnitudes balances somewhere, and, for this particular pair, Archimedes had proved that it balances in a surprisingly straightforward way. A seemingly innocuous anachronism is the symbol pi used by Archimedes to denote the ratio of the circumference to the diameter. (The symbol is of course modern.). In fact, this anachronism is surprisingly significant for the history of mathematics: pi makes you think of a kind of *number* and it is crucial that, in Greek mathematics, ratios are not numerical values. Consider how the same mistake is repeated in a dialogue towards the end of the novel:

"Dearest?" she said gently. — He raised his head and beamed at her: "it's three halves!" he told her. ... "The ratio is?" she asked, trying to take an interest. — He nodded... "it all comes out so perfectly," he wondered. "A rational number, after all that. So exact, so... perfect!".

Archimedes would obviously be excited to discover the ratio of a cylinder to an enclosed sphere, but it would be a *ratio* — 'half as much as great' — and not the so-called *rational number* 'three halves'.

This arithmetization of geometry is related to two constant misperceptions. First, Bradshaw believes — wrongly, in my view — that, in the ancient world, pure mathematics would have had direct consequences for engineering. As a mitigating factor one can mention that this misperception is explained by the narrative need to tie Archimedes' catapults to his mathematical genius; anyway, one can have genuine historical debate about this question. More jarring to my mind is the second misperception (deriving from the arithmetization of geometry, and tied to the mathematization of engineering). Bradshaw keeps referring to mathematicians *calculating* — as if it was all a matter of multiplying and manipulating *numbers*. (Incidentally, Singh makes the same mistake in his popular history, *Fermat's Last Theorem*). Greek mathematicians did not *calculate*, they *thought*: the two are nearly opposites. Thus Bradshaw misses something central about the very soul of Greek mathematics — ultimately, all because she confuses 'half as much as great' with 'three halves'. She cannot be blamed for this very subtle mistake (made, in fact, by several past historians of Greek mathematics). But this is an instructive lesson for us: the reality of works of mathematics — as of all living things — can be brought out only by the precise observation of detail.

Reviel Netz is Assistant Professor of Ancient Science at Stanford University. His books include

The Shaping of Deduction in Greek Mathematics: a Study in Cognitive History (CUP 1999) and

*Adayin Bahuc* (a volume of Hebrew poetry: Shufra, Tel-Aviv, 1999). The first volume of his

*Translation with Commentary of the Works of Archimedes* is in press with CUP, and he is currently at work on the edition of the

Archimedes Palimpsest.