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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Many ancient societies knew important mathematical facts, but only one discovered mathematics—which is not a collection of accurate rules of thumb, but a body of knowledge organized deductively, by the radical notion of proof. And Euclid is its prophet. 

'Portrait of Fra Luca Pacioli with a student' by Jacapo de' Barbari (1495)

'Portrait of Fra Luca Pacioli with a student' by Jacapo de' Barbari (1495)

The Elements, composed in about 300 b.c., is a landmark of human thought. It has exerted a profound influence on science and philosophy by serving as both a basic geometry textbook (widely used for more than 2,000 years) and an archetype of rigorous knowledge. Accordingly, the ultimate subject of The King of Infinite Space is the life of the mind. Written with David Berlinski’s characteristic mix of hothouse prose and standup comedy, it is aimed at the Intelligent General Reader—though I often found myself wondering what that admirable creature might make of it.

Berlinski begins with Euclid’s foundations—definitions, axioms, and common notions (general logical principles)— and devotes much attention to their shortcomings. For example, Euclid appeals to the notion of figures “coinciding,” meaning that one figure can be placed on another so that their sides match up exactly. Are we now in a bind?

If the figures in question are physical objects, we understand what it means to move one and place it on another; but physical objects will never match exactly. If, on the other hand, they are nonphysical, perceived only by the mind, then the notions of moving and placing are metaphorical. So what, exactly, do they mean? Euclid shows, Berlinski says, “a willingness to repose his confidence in things he could neither explain nor justify.”

Is the author condescending to his subject? Is he suggesting that an interest in Euclid is some embarrassing affectation, like an enthusiasm for Renaissance fairs (excuse me, faires)? In fact, the point of this discussion is to show how bold Euclid’s project was and how remarkable was its success. Without benefit of certain refined logical tools and distinctions (that would be unavailable for another 2,200 years), Euclid erected an enduring architectural marvel: an axiomatic system, the first, a rich body of results supported on a tidy footing of “evident” principles by timbers of the strongest possible kind, proofs that compel belief.

We are next led through a sample of these proofs, taken from the first of Euclid’s 12 books—including the proof of its most famous result, the Pythagorean Theorem. To understand the life of a certain kind of mind, we must see that mind in action. Berlinski adds an ahistorical gloss claiming that Euclid didn’t really “get” the Pythagorean Theorem, which ought to be understood as an algebraic statement about the numerical lengths of the sides of a right triangle: a2+b2=c2. Euclid’s formulation, about the areas of squares drawn on the sides of the triangle, is a clumsy second best.

Berlinski does acknowledge that a notion of number applicable not only to the counting numbers (such as 2 and 12), but also to lengths and areas and volumes as well, wasn’t developed until nearly two millennia after Euclid, and that a few additional centuries were then required to make rigorous sense of the maneuver. But that concession feels grudging.

The remarkable fact, which deserves celebration, is that Euclid’s account of the quantitative aspects of geometry was rigorous from the get-go. For Euclid, multitudes (collections of things that can be numbered) are distinct from magnitudes (continuously variable entities, such as lines). And magnitudes themselves are of different kinds. We can add a number to a number but not to amagnitude. We can add like magnitudes (line to line, or area to area) but not unlike magnitudes. We can “multiply” a line by a number—by adding together two or three or four copies of it—but we cannot multiply a length by a length. Multiplicity is reserved for multitudes. This makes perfect sense.

The need for these careful distinctions was reaffirmed by the surprising and unwelcome discovery that the side and the diagonal of a square have no “common measure”: There exists no unit of length of which both are exact multiples. (An algebraist would say that the square root of two is irrational.)This caused metaphysical problems for the Pythagoreans, who seem to have believed that numbers were in some sense the ultimate constituents of the world. It also presented a serious obstacle to the creation of a rigorous account of geometry.