September 4, 1998

square the circle

My brother, Tom in Houston, writes that he has been hearing the phrase "squaring the circle," but doesn't know what it means. Neither do I. Any light you can shed on this phrase would be appreciated.

Literally, to square the circle means 'to construct a square equal in area to a given circle'. Squaring the circle was one of the three great problems of ancient mathematics (the other two were trisecting an angle and duplicating a cube), with the rule that you had to do it using only a compass and unmarked straightedge. To figure it out would be to find a value for pi (the ratio of a circle's circumference to its diameter; the area of a circle is equal to pi times the square of its radius), and much of the early history of pi is concerned with the problem of squaring the circle.

The Ancients and their followers believed that it was possible to square the circle, if they only worked harder to figure it out, but by the sixteenth century or so most mathematicians came to believe that it was impossible. Finally, in 1882, Lindemann proved that pi was transcendental--that it was not a root of any algebraic equation having integral coefficients--thus also proving that squaring the circle is impossible. (Note that it is possible to square the circle if you can use non-Euclidean techniques--with an infinite number of steps, higher-order curves, or the methods of calculus, it can be done, but not with just a compass and unmarked straightedge.)

Thus, to square the circle is used figuratively to mean 'to attempt the impossible'. The expression has been used this way in English for centuries, and this use is still common, despite the stereotypical American mathematical illiteracy that our news media love to bemoan on otherwise slow days. Some recent examples: "Lange hoped to reconcile neoclassical economics with Communist social ideals, as well as to achieve a politics of peaceful coexistence. This endeavor, it turned out, was like squaring the circle" (New Republic, 1991); "Mr. Yeltsin...needs to be able to sell arms, win Western aid, and begin converting military factories to the sale of consumer products. It is going to be a tough circle for him to square" (New York Times, 1992).