Beautiful Geometry by Maor Eli; Jost Eugen;

Author:Maor, Eli; Jost, Eugen;
Language: eng
Format: epub
Publisher: Princeton University Press
Published: 2013-03-11T16:00:00+00:00

NOTE:

1. For a full account of Archimedes’s method, see The Works of Archimedes, edited by T. L. Heath (New York: Dover. 1953), pp. 91–98.

28

The Digit Hunters

In the second century BCE, during Archimedes’s lifetime, the Hindu-Arabic numeration system was still more than a thousand years in the future. So Archimedes had to do all his calculations in a strange hybrid of the Babylonian sexagesimal (base 60) system and the Greek system, in which each letter of the alphabet had a numerical value (alpha = 1, beta = 2, and so on). Today, of course, we associate the value of π with its decimal expansion—a nonrepeating, seemingly random string of digits that goes on forever. Terminate this expansion after any number of digits, and you’ll get only an approximation of π.

So, where should we stop? For many daily practical tasks, the simple fraction will suffice, differing from π by just 0.04 percent. The Chinese mathematician Zu Chongzhi (Tsu Ch’ung-Chih, 429–501) around 480 CE discovered the nice approximation , accurate to six places—that is, to the nearest millionth. The Dutch-German mathematician Ludolph van Ceulen (1540–1610) computed π to 20 decimal places, using Archimedes’s method with polygons of 60 × 229 sides and spending much of his professional life on the task (he later improved his calculations to 35 places). After Van Ceulen’s death, his widow reportedly had the number inscribed on his tombstone in Leiden, but all traces of it have been lost. Here it is:

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