Ancient Infinities

An ancient manuscript long hidden from public view has provided significant, new insights into the way Archimedes (287–212 B.C.) did his mathematical work more than 2,000 years ago.

The manuscript, known as the Archimedes Palimpsest, is the only source of Archimedes’ treatise on the “Method of Mechanical Theorems.” As the oldest surviving Archimedes manuscript, it’s the closest we can get to the mathematician himself, says science historian and classics professor Reviel Netz of Stanford University, who has been studying the relic.

Dating from the 10 century, the Archimedes text survives as writing on parchment that 2 centuries later was cut apart, roughly scraped, and overwritten with a description of a church ritual. The document was first rediscovered in Constantinople in 1906 by the Danish scholar J.L. Heiberg. Aided only by a magnifying glass, however, he could not read every word of the text. The manuscript vanished from view in the 1920s before resurfacing in 1998 and being auctioned off for $2 million to an anonymous buyer. The buyer allowed the palimpsest (a scraped and overwritten parchment) to be conserved, photographed, and displayed at the Walters Art Gallery in Baltimore.

The use of ultraviolet photography and digital imaging–technologies unavailable to Heiberg–has made it possible to read beneath the prayer book’s lines and see important details of Archimedes’ text and diagrams. The geometric diagrams, for example, suggest that Greek mathematicians tended to emphasize qualitative relationships over quantitative accuracy.

Last year, Netz happened to examine a hitherto unread portion of the “Method of Mechanical Theorems.” To his surprise, the text indicated that Archimedes had dealt in an unexpectedly sophisticated way with the concept of infinity–using the idea of infinitely large sets in a mathematical proof.

In the “Method of Mechanical Theorems,” Archimedes blended concepts of straight and curved with physical and geometrical models. In doing so, he anticipated the infinities and infinitesimals of calculus, as developed many centuries later by Isaac Newton (1643–1727) and Gottfried Wilhelm von Leibniz (1646–1716).

“It has always been thought that modern mathematicians were the first to be able to handle infinitely large sets, and that this was something the Greek mathematicians never attempted to do,” Netz wrote in the Nov. 1 Science. “But in the palimpsest we found Archimedes doing just that. He compared two infinitely large sets and stated that they have an equal number of members. No other extant source for Greek mathematics has that.”