The Magazine

Here's Looking at Euclid

Why geometry matters in the life of the mind.

Feb 18, 2013, Vol. 18, No. 22 • By DAVID GUASPARI
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To explain the concept of similarity—of having the same shape, if not the same size—we need an account of ratios and proportions. This is easily provided for multitudes—or for magnitudes with a common measure, so that proportions among them can be understood as proportions among multitudes of that measure. The solution to this very difficult problem, presented in Book V, is the most brilliant technical achievement in the Elements. And it is attributed not to Euclid but to Eudoxus.

A bit more than half of The King of Infinite Space is spent chewing slowly on Book I. The rest ranges beyond Euclid proper, the pace accelerating through discussions of geometry and algebra, non‑Euclidean and Riemannian geometries, Gauss’s intrinsic curvature, Hilbert’s reformulation of Euclidean geometry, groups and fields (ordered Archimedean fields, to be precise), and whips past, at the speed of light, Felix Klein’s “Erlangen Program.”

These are deep subjects. I fear that readers will find much of the account gnomic, partly because so much is compressed into so small a space, and partly because of the author’s strenuous manufacture of lapidary phrases. Many succeed: It is apt to call mathematics a form of “mountain‑climbing pastoral,” a quest for worlds of ideal beauty accessible only by grueling and dangerous journeys. Some seem merely to have had tailfins slapped on, as when railroad tracks in a perspective drawing are said to “converge toward a distant, soundlessly spinning point.” Would any insight be lost if that point were “angrily motionless” or “mysteriously aloof”— or left unadorned? And some want to conjure drama where none exists: Noting that a proposition proven in Book I is equivalent to one of Euclid’s axioms, Berlinski says that, in proving it, “Euclid had demonstrated what he had already assumed.” From that point of view, every proposition Euclid proves is merely a restatement of something already assumed, since it follows from an axiom.

Non-Euclidean geometries occupy much of the book’s latter half, and their discovery is a fascinating episode of intellectual history. Euclid’s parallel postulate amounts to saying that, given a line and a point, exactly one line can be drawn parallel to that line through the point. (Euclid formulates it differently.) To succeeding generations, this seemed both “evidently” true and more complicated than Euclid’s other axioms. And centuries of futile effort were spent trying to prove it from the others. In the 19th century, Janos Bolyai and Nikolai Lobachevsky independently developed consistent and rich theories of “geometry” in which the parallel postulate is false. They made themselves at home in a new mental world—one that was, from some points of view, literally unthinkable. Euclid’s geometry had been the paradigm of certain knowledge. How could denying one of its basic principles lead to anything other than incoherence? What exactly was geometry true of ?

The ensuing story satisfies Berlinski’s taste for drama. He quotes a letter to Bolyai from his father, who had himself spent many years trying to prove the parallel postulate and was alarmed by his son’s growing interest in the problem:

Do not in any case have anything to do with the parallels. I know every twist and turn in this business and I have myself wandered in its fathomless night, which has extinguished every light and joy in my life. I beg you in the name of God. Leave the parallels in peace.

It has been a long time since any professional mathematician worked in classical Euclidean geometry, searching for theorems that Euclid overlooked. So why study it? Once a theorem is proven, why should anyone pore over the proof and learn how to prove it all over again? According to a famous story, Euclid believed the work to be its own reward, dismissing a request from the pharaoh for a less arduous way to learn mathematics, telling him that “there is no Royal Road to geometry.” What Euclid offers, Berlinski says, is “a method of proof and so a way of life.”

Could it become obsolete? Berlinski cites a major achievement of modern mathematics, officially called the “classification of finite simple groups.” It is a consequence of results published over decades by more than a hundred mathematicians and totaling more than 10,000 densely packed journal pages. It seems hard to believe that those pages are entirely free from error. (Errors have been found, but have turned out to be correctible.) The authors are retiring and dying off. A generation from now, will there be anyone who understands the proof? Is this the wave of the future?