The Bactra Review: Occasional and eclectic book reviews by Cosma Shalizi   52

Geometry Civilized

History, Culture, and Technique

by J. L. Heilbron

Oxford: Clarendon Press, 1998

Construction of the Rhodian Shore, with Straightedge and Compass

"Aristippus the Socratic philosopher being shipwrecked and cast upon the Rhodian shore noticed some geometric diagrams drawn on the beach and said to his companions, `We can hope for the best for I see signs of men' ": so Vitruvius, in the sixth book of De Architectura. This story, and its moral — that geometry indicates not just humanity, but civilization — have had a long and distinguished career. One incident therein was a famous illustration of the story, commissioned by the Oxford University Press for an edition (1703) of Euclid, which became a success in itself, recycled for numerous geometrical works, and providing both the title (Traces on the Rhodian Shore) and the cover of Clarence Glacken's superb study of Nature and Culture in Western Thought from Ancient Times to the End of the Eighteenth Century. The latest incarnation of story and picture is at the beginning of Heilbron's book, one of the best and most quixotic of the unnumbered editions, translations, commentaries, vulgarizations, cribs, refinements, imitators and purported replacements of Euclid's Elements that have been written in every literary language used by classical, Christian, Islamic or modern civilizations. How appropriate this is, will appear presently.

Geometry Civilized is not really supposed to teach geometry to those without prior knowledge of the art — though Heilbron expresses the hope that the book will provide a "grasp of geometry sufficient to excel on college aptitude tests" (p. 47). This is just as well. A glance at the first chapter's fifty pages of densely footnoted, lovingly written history of geometry and geometrical instruction from the days of Imhotep to the New Math inclusive is enough to see that not even the most owlish and bookish of high school students will plow through it to get to the notion of point in the next chapter. (I was about as bookish and owlish as they come at that age, and I would rather have taken an extra hour of gym each day.)

Rather, this book is a celebration of geometry for those of us already enamored of it, especially those who find disquisitions on (inter alia) constructing window traceries for Gothic cathedrals, surveying in Mozambique, Renaissance military architecture, remodeling Vedic altars, or Chinese cut-and-place proofs absolutely fascinating. Heilbron is certainly among our number, being a distinguished historian of science as well as a geometry-freak. (He was also the vice-chancellor of Berkeley when I was an undergraduate there, and in his preface confesses to a certain distraction from his administrative duties in favor of geometry problems, which explains much.) After the historical and pedagogical introduction, readers are led through, successively, points and lines, triangles (reached through crossing the "bridge of asses"), "many cheerful facts about the square of the hypotenuse," polygons and circles, and final puzzles and applications, among them the burning machines of Archimedes (almost certainly legendary, alas) and Lavoisier (demonstrated before tout le monde). Compass-and-straightedge puritans will shudder at Heilbron's resort to algebra, trigonometry, and even examples where numbers are worked out and approximated; the rest of us will appreciate the happy, anything-if-it-works pragmatic spirit displayed.

This brings us to one of the best things about this book, namely the problems — averaging one for every other page. These range from the easy or even trivial (like the three taken from the Cambridge Tripos examinations) to the fiendish (like many taken from the Ladies Diary [sic], an 18th and 19th century British women's magazine). (There is much food for thought in the fact that the sisters of Cambridge men solved much harder problems, without formal instruction, than their brothers tackled in the most prestigious competition their university held, for which they trained intensively.) Many of them have to do with surveying; very few of the "applied" problems seem contrived; one is attributed to Napoleon; some of them are altogether new. Part of the attraction of Euclidean geometry is its game-like quality — the restrictions on allowable constructions to keep things sporting, the strategy and tactics of getting the better of the problem, the thrill familiar to crossword-puzzle addicts, and the huge range of skills at which it can be played. In fact, I think the puzzles add even more than the pictures, which is saying a great deal.

At this point, in the discussion of a book which sets out to show that "pursuing geometry opens the mind to relationships among learning, its applications, and the societies that support them... which run from high philosophy through surveying," it would be appropriate to talk about the place of Euclidean geometry in the societies which have been its carriers. In large part, this would be the history of a dream, the dream of knowledge which is unshakeable, which is "axiomatic" in the sense which has entered the common language. For what distinguishes the Euclidean tradition from the others, some its equals as practical techniques, is precisely formal proof: the answers don't just work, they are always demonstrably correct. In the words of the poet:

Euclid alone has looked on Beauty bare.
Let all who prate of Beauty hold their peace,
And lay them prone upon the earth and cease
To ponder on themselves, the while they stare
At nothing, intricately drawn nowhere
In shapes of shifting lineage; let geese
Gabble and hiss, but heroes seek release
From dusty bondage into luminous air.
O blinding hour, O holy, terrible day,
When first the shaft into his vision shone
Of light anatomized! Euclid alone
Has looked on Beauty bare. Fortunate they
Who, though once only and then but far away,
Have heard her massive sandal set on stone.
And, indeed, to those who have grasped the notion of proof, the arguments of those who are not so illuminated (or intoxicated) sound little better than the gabble of geese, as the story of Raimbeau and Raoul shows. On the basis of axiomatic proof, and the certainty it seems to convey, Euclidean geometry became the ideal of true knowledge, and everything else that aspired to the same status — theology and physics, jurisprudence and history, politics and ethics — was cast into the same form; intelligent and learned men even tried to deduce these other subjects directly from geometry. In the end, we awoke from the dream, not in any simple repudiation, but by becoming more Euclidean than Euclid himself. Under its influence, we learned rigor, and the detection of fallacies, and tact in drawing inferences, and intellectual conscience: and because of these, we could no longer accept the dream of perfect, timeless, certain knowledge which had inspired us to acquire those talents.

—But something has gone terribly wrong here. I set out to say what an enjoyable and elegant book Geometry Civilized is, and somehow have wandered into dark meditations on the Cunning of Reason and involuntary quotations from Nietzsche. Clearly, it is time to close up Emacs and open the compass on a blank page with one of the Ladies Diary stumpers.

Typos: p. 30, for "2460", read "24/60"; pp. 52--3, for " OV' ", read " O'V' " throughout; p. 121, for "From O draw DE" read "From O draw OE"; p. 241, "It [pi] cannot be expressed as a number, even an irrational one" — a truly awful statement of the fact that pi is transcendental (i.e., it is not the solution of any polynomial equation whose coefficients are all rational numbers); figure captions, persistently give the date of vol. 3 of Science and Civilisation in China as 1970, while the bibliography gives the correct date of 1959.
Education, and Kindred Ways of Moulding the Young / History of Science / Mathematics
309 pp., 8 color plates, hundreds of black and white diagrams and illustrations, footnotes, bibliography, index
Currently in print as a hardback, US$60, ISBN 0-19-850078-5, LoC QA455.H413
17--19 August 1998