A History of the Circle

  • Ernest Zebrowski
Rutgers University Press: 1999. 214pp. $28

Popular books on or around mathematics appear quite regularly nowadays. This one nominally takes the circle as its theme, both for its mathematical properties and for its use in the physical sciences. Based upon teaching in American colleges, the level is fairly elementary.

The early chapters treat several manifestations of the circle in antiquity, including the use of rollers in transporting heavy objects and some features of astronomy. But after that the circle gives way to conic sections and closed but non-circular orbits as studied from Kepler to Newton. It comes back later rather indirectly via oscillations, pendular and simple harmonic motions and waves, and associated mathematical theories such as Fourier series. Late chapters skim across relativity theory and quantum mechanics, sandwiching an interlude on ancient architecture.

While the emphasis on applications is well taken, the overall impression is rather incoherent: links to circularity are tenuous in places, and many topics specifically tied to circles are omitted. The most obvious case concerns π. Some properties are stated, but no highlight is given to its four main roles, of relating the diameter of a circle not only to its circumference (hence the eighteenth-century choice of the letter π, for perimetros) but also to its area, and to the surface area and volume of the sphere. It is enlightening, and also not at all obvious, that the same factor π is involved. The enabling theorem for the circle — area = (circumference/2) × (diameter/2) — is a profound result; the author presents one intuitive argument for it but does not register its significance. All sorts of other cultural possibilities are ignored; for example, that the classical problem in Greek geometry of squaring the circle may echo a desire, very evident with the Egyptians, of uniting the circular heavens with the Earth, long symbolized by the square.

As for history, the author states at once that he gives no conventional version. The accuracy of his statement can be fairly judged by a few typical quotations. “It was a long time before anyone had a practical need to measure angles to a precision much finer than one degree.” In his Elements, Euclid “laboriously proved every mathematical relationship that was then known”. “What Descartes did accomplish was to use rectangular co-ordinates in a new way” (he actually introduced analytic geometry). Einstein's creation of special relativity in 1905 was effected by “conduct[ing] his Gedanken experiments through mathematical logic”; indeed, in general “Scientists depend upon mathematical logic to generate their theoretical predictions” (“If only they did,” Bertrand Russell might have sighed). Another circular sign comes to mind: 0 per cent for accuracy.

A good heuristic and historical introduction to parts of mathematics using the circle as its theme would make an excellent contribution to the popular understanding of mathematics. Perhaps somebody can take up the challenge.