December 21, 2007

Archimedes Palimpsest

Stomachion is a dissection puzzle in the Archimedes Palimpsest
Stomachion is a dissection puzzle in the Archimedes Palimpsest

The Archimedes Palimpsest is a palimpsest on parchment in the form of a codex which originally was a copy of an otherwise unknown work of the ancient mathematician, physicist, and engineer Archimedes of Syracuse and other authors. Archimedes lived in the third century BC, but the copy was made in the 10th century by an anonymous scribe. In the 12th century the codex was unbound and washed, in order that the parchment leaves could be folded in half and reused for a Christian liturgical text. It was a book of nearly 90 pages before being made a palimpsest of 177 pages; the older leaves folded so that each became two leaves of the liturgical book. The erasure was incomplete, and Archimedes' work is now readable using digital processing of ultraviolet, X-ray, and visible light.

In 1906 it was briefly inspected in Constantinople and was published, from photographs, by the Danish philologist Johan Ludvig Heiberg; shortly thereafter Archimedes' Greek text was translated into English by Thomas Heath. Before that it was not widely known among mathematicians, physicists, or historians. It contains

  • "Equilibrium of Planes"
  • "Spiral Lines"
  • "The Measurement of the Circle"
  • "Sphere and Cylinder"
  • "On Floating Bodies" (only known copy in Greek)
  • "The Method of Mechanical Theorems" (only known copy)
  • "Stomachion" (only known copy)

The palimpsest also contains speeches by the 4th century BC politician Hypereides, and a commentary on Aristotle's Categories by Alexander of Aphrodisias.[1]

Mathematical content

The most remarkable of the above works is The Method, of which the palimpsest contains the only known copy. In his other works, Archimedes often proves the equality of two areas or volumes with his method of double contradiction: assuming that the first is bigger than the second leads to a contradiction, as does the assumption that the first be smaller than the second; so the two must be equal. These proofs, still considered to be rigorous and correct, used what we might now consider secondary-school geometry with rare brilliance. Later writers often criticized Archimedes for not explaining how he arrived at his results in the first place. This explanation is contained in The Method.

Essentially then, the method consists in dividing the two areas or volumes in infinitely many stripes of infinitesimal width, and "weighing" the stripes of the first figure against those of the second, evaluated in terms of a finite Egyptian fraction series. He considered this method as a useful heuristic but always made sure to prove the results found in this manner using the rigorous arithmetic methods mentioned above.

He was able to solve problems that would now be treated by integral calculus, which was formally invented in the 17th century by Isaac Newton and Gottfried Leibniz, working independently. Among those problems were that of calculating the center of gravity of a solid hemisphere, the center of gravity of a frustum of a circular paraboloid, and the area of a region bounded by a parabola and one of its secant lines. Contrary to exaggerations found in some 20th century calculus textbooks, he did not use anything like Riemann sums, either in the work embodied in this palimpsest or in any of his other works. (For explicit details of the method used, see How Archimedes used infinitesimals.)

A problem solved exclusively in the Method is the calculation of the volume of a cylindrical wedge, a result that reappears as theorem XVII (schema XIX) of Kepler's Stereometria.

Some pages of the Method remained unused by the author of the Palimpsest and thus they are still lost. Between them, an announced result concerned the volume of the intersection of two cylinders, a figure that Apostol and Mnatsakian have renamed n = 4 Archimedean globe (and the half of it, n = 4 Archimedean dome), whose volume relates to the n-polygonal pyramid.

In Heiberg's time, much attention was paid to Archimedes' brilliant use of infinitesimals to solve problems about areas, volumes, and centers of gravity. Less attention was given to the Stomachion, a problem treated in the Palimpsest that appears to deal with a children's puzzle. Reviel Netz of Stanford University has argued that Archimedes discussed the number of ways to solve the puzzle. Modern combinatorics leads to the result that this number is 17,152. Due to the fragmentary state of the palimpsest it is unknown whether or not Archimedes came to the same result. This may have been the most sophisticated work in the field of combinatorics in Greek antiquity.

Modern history

From the 1920s, the manuscript lay unknown in the Paris apartment of a collector of manuscripts and his heirs. In 1998 the ownership of the palimpsest was disputed in federal court in New York in the case of the Greek Orthodox Patriarchate of Jerusalem versus Christie's, Inc. At some time in the distant past, the Archimedes manuscript had lain in the library of Mar Saba, near Jerusalem, a monastery bought by the Patriarchate in 1625. The plaintiff contended that the palimpsest had been stolen from one of its monasteries in the 1920s. Judge Kimba Wood decided in favor of Christie's Auction House on laches grounds, and the palimpsest was bought for $2 million by an anonymous information technology person.

The palimpsest is now at the Walters Art Museum in Baltimore, where conservation continues (as it had suffered considerably from mold).

A team of imaging scientists from the Rochester Institute of Technology and Johns Hopkins University has used computer processing of digital images from various spectral bands, including ultraviolet and visible light, to reveal more of Archimedes' text. Dr. Reviel Netz [2] of Stanford University has been trying to fill in gaps in Heiberg's account with these images.

Sometime after 1938, one owner of the manuscript forged four Byzantine-style religious images in the manuscript in an effort to increase its value. It appeared that these had rendered the underlying text forever illegible. Then, in May 2005, highly-focused X-rays produced at the Stanford Linear Accelerator Center in Menlo Park, California, were used to begin deciphering the parts of the 174-page text that have not yet been revealed. The production of x-ray fluorescence was described by Keith Hodgson, director of SSRL. "Synchrotron light is created when electrons traveling near the speed of light take a curved path around a storage ring—emitting electromagnetic light in X-ray through infrared wavelengths. The resulting light beam has characteristics that make it ideal for revealing the intricate architecture and utility of many kinds of matter—in this case, the previously hidden work of one of the founding fathers of all science." [3]

In April 2007 it was announced that a new text had been found in the palimpsest, which was a commentary on the work of Aristotle attributed to Alexander of Aphrodisias. Doctor William Noel, the curator of manuscripts at the Walters Art Museum, said in an interview: "You start thinking striking one palimpsest is gold, and striking two is utterly astonishing. But then something even more extraordinary happened." This referred to the previous discovery of a text by Hypereides, an Athenian politician from the 4th century BC, which has also been found within the palimpsest.

Greek mathematics

Greek mathematics, as that term is used in this article, is the mathematics written in Greek, developed from the 6th century BC to the 5th century AD around the Eastern shores of the Mediterranean.


Classical Greek mathematics refers to the mathematics studied before the Hellenistic period, when Greek mathematics was mostly limited to the Greek city-states in ancient Greece, Asia Minor, Libya, and Sicily.

Greek mathematics studied from the time of the Hellenistic period onwards (from 323 BC) refers to all mathematics of those who wrote in the Greek language, since Greek mathematics was now not only written by Greeks but also non-Greek scholars throughout the Hellenistic world, which was spread across the Eastern end of the Mediterranean. Greek mathematics from this point merged with Egyptian and Babylonian mathematics to give rise to the latter phase of Greek mathematics known as Hellenistic mathematics. The most important centre of learning during this period was Alexandria in Egypt, which attracted scholars from across the Hellenistic world, including mostly Greek and Egyptian scholars, as well as Jewish, Persian, Phoenician and even Indian scholars.[1]

Most of the mathematical texts written in Greek were found in Greece, Egypt, Asia Minor, Mesopotamia, and Sicily.


Greek mathematics constitutes a major period in the history of mathematics, fundamental in respect of geometry and the idea of formal proof. Greek mathematics also contributed importantly to ideas on number theory, mathematical analysis, applied mathematics, and, at times, approached close to integral calculus.

Well-known figures in Greek mathematics include Pythagoras, a shadowy figure from the isle of Samos associated partly with number mysticism and numerology, but more commonly with his theorem, and Euclid, who is known for his Elements, a canon of geometry for many centuries.

The most characteristic product of Greek mathematics may be the theory of conic sections, largely developed in the Hellenistic period. The methods used made no explicit use of algebra, nor trigonometry.


Greek mathematics has origins that are presumed to go back to the 7th century BC, but are not easily documented. It is generally believed that it built on the computational methods of earlier Babylonian and Egyptian mathematics, and it may well have had Phoenician influences.

Greek mathematics proper is thought to have begun from the late 500s BC, when Thales and Pythagoras brought knowledge of Egyptian and Babylonian mathematics to Greece. Thales used geometry to solve problems such as calculating the height of pyramids and the distance of ships from the shore. Pythagoras stated the Pythagorean theorem and constructed Pythagorean triples algebraically, according to Proclus' commentary on Euclid.

The high period

Mathematical developments took place in Greek-speaking centres as far apart as Egypt and Sicily, and with a high estimation of the intellectual and cultural status of mathematics (for example in the school of Plato). The Sand Reckoner by Archimedes, a resident of Syracuse, bespeaks a man who made major discoveries, and whose originality and accomplishments are commonly reckoned to be on par with those of Isaac Newton and C. F. Gauss.

Greek mathematics and astronomy reached a rather advanced stage during Hellenism, with scholars such as Hipparchus, Posidonius and Ptolemy, capable of the construction of simple analogue computers such as the Antikythera mechanism.

Transmission and the manuscript tradition

Although the earliest Greek language texts on mathematics that have been found were written after the Hellenistic period, many of these are considered to be copies of works written during and before the Hellenistic period. Nevertheless, the dates of Greek mathematics are more certain than the dates of earlier mathematical writing, since a large number of chronologies exist that, overlapping, record events year by year up to the present day. Even so, many dates are uncertain; but the doubt is a matter of decades rather than centuries.

During the Middle Ages, Europe derived much of its knowledge of Greek mathematics via Islamic mathematics. The texts of Greek mathematics were for the most part preserved and transmitted in the Muslim world.