### Archimedes uncovered

The That’s Maths column in this week’s Irish Times ( TM012 ) describes the analysis of the ancient codex known as the Archimedes Palimpsest.

Archimedes of Syracuse

Archimedes (Ἀρχιμήδης, 287-212 BC) was a brilliant physicist, engineer and astronomer, and the greatest mathematician of antiquity. He is famed for founding hydrostatics, for formulating the law of the lever, for designing the helical pump that bears his name, for devising engines of war, and for much more.

Archimedes Thoughtful. Painting by Domenico Fetti (1588–1623). Wikimedia Commons.

We have all heard the story of how Archimedes, upon discovering the Principle of Hydrostatics that bears his name, jumped from his bath and ran naked through the streets of Syracuse crying Eureka, I have found it!

Value of π

But it was his mathematical discoveries that Archimedes regarded as his greatest achievements. He estimated the value of π, the ratio of the circumference to the diameter of a circle, to remarkable accuracy using polygons of 96 sides within and around a circle.

And he showed that a sphere inscribed in a cylinder has a volume that is two-thirds of the value for the cylinder (that gives the formula V = (4/3)πr ³). Archimedes asked that a sphere within a cylinder be inscribed on his tombstone. Centuries later, the Roman orator Cicero found such a carving on a grave in Syracuse.

Sphere embedded in cylinder. The volume of the sphere is 2/3 that of the cylinder  (image Wikimedia Commons)

Lost and Found

Many of Archimedes’ writings are lost, being known to us only through references made to them by later writers. Others have reached us by a curious and circuitous route: they were translated into Arabic in the ninth century, and from Arabic into Latin during the Renaissance.

But some of his most important work remained hidden from us until the remarkable discovery of the Archimedes Palimpsest. Palimpsests were works written on parchment that had been scraped clean of earlier writing. This was common practice in the Middle Ages, as vellum was very expensive.

In 1906 a prayer book written in the 13th century AD came to light in Constantinople. Upon close examination, the incompletely erased work underlying the text was recognized as a 10th century copy of several works of Archimedes, which had been thought to have been lost forever.

It is the only source we have of The Method of Mechanical Theorems, in which Archimedes uses infinitesimal quantities to calculate the volumes of various bodies. This foreshadowed integral calculus, invented nearly two thousand years after Archimedes, independently by Newton and Leibniz.

The Archimedes Palimpsest is the earliest extant manuscript of Archimedes’ work; it has copies of the geometric diagrams that Archimedes drew in the sand in the third century BC. It contains several treatises by Archimedes in addition to The Method.

A Cornucopia for Scholars

The Palimpsest was bought at auction in New York in 1998 for \$2 million by (according to Der Spiegel) Jeff Bezos, founder of amazon.com. It has been intensively analysed over the past ten years, using advanced imaging methods.

Page of the palimpsest showing older and more recent writing in orthogonal directions
(from archimedespalimpsest.org)

In Autumn 2011 an exhibition at the Walters Art Museum in Baltimore, Lost and Found: The Secrets of Archimedes, was devoted to the analysis of the palimpsest and to the outcome of the project to study it.

All the images and translations are freely available to scholars on the Digital Palimpsest Web Page (archimedespalimpsest.org), providing a treasure trove for anyone interested in the history of Greek mathematics.