A close examination of the most important mathematical text for centuries, sold at Christies nine years ago for about £1M, has proved that Archimedes was a genius who discovered the building blocks of calculus nearly 2,000 years before it was invented. The text lay undisturbed in a Paris apartment until the family who owned it put it up for sale, and experts have been working on it ever since.

The text was a palimpsest, originally a papyrus that had been used many times but at some stage was transcribed on to an animal skin parchment, possibly by someone who realized its importance. On it was the Method, the radical thought process by which Archimedes arrived at many of his conclusions. When he was killed by a Roman soldier at Siracuse in 221 BC, still working on his dustboard, Archimedes’ calculations were scattered to the four winds. About 700 years ago the palimpsest was re-used as the parchment for a prayer book and the old writing all but obliterated to make room for the new.

It resurfaced in Constantinople at the beginning of the 20th century and in 1906 a Danish scholar of Greek, Johann Heiberg, examined it with a magnifying glass and deciphered some of its contents. His plan to return was frustrated by the outbreak of the First World War and the palimpsest was again lost, only to be offered to a member of the Parisian family in Constantinople in the early 1920s. He was a keen antiquarian book collector and took the palimpsest back to Paris, where in remained for almost seventy years.

Archimedes is famous for using his bath to determine the volume of an object by measuring the displacement of water. Rather more significantly, he worked out the true value of Pi, essential for calculating the area of the circle, one of the great fundamental mathematical formulae. But both of these were child’s play compared to the work in his head eventually recorded on the palimpsest. To calculate the volume of a sphere, a challenge far beyond mathematical concepts of his time, Archimedes created in his mind’s eye an imaginary set of scales designed to compare the volumes of a series of curved shapes.

The lines that Heiberg could not read have been made legible by modern scientific techniques. They show that Archimedes had created a set of rules for dealing with infinity, and had worked out a solution for dealing with the value of each slice of a curved object, however small, and then adding together an infinite number of them. 

November 2007