Many of the great works of ancient science were lost in the collapse of Greco-Roman civilization, though in the past century a few of these classics have been rediscovered from the dustbin of history in some cases almost miraculously.
In 1900 a Greek sponge boat from the island of Symi anchored off the northern coast of the remote islet of Antikythera to escape a storm. After the storm abated a diver named Elias Stadiatos went down to look for sponges and found a wrecked ship on the sea bottom. He was stunned by what he found, and talked excitedly of having seen a heap of dead naked women, which when they were brought to the surface were found to be Greco-Roman bronze statues. The other contents of the ship included jewelry, bronzes, pottery, furniture, and amphorae filled with wine. The ship was dated to the first century B.C. and is thought to have been on its way from Rhodes to Italy. One of the bronze statues, known as the Ephebe of Antikythera, is now in the National Archaeological Museum in Athens. It represents a nude youth thought to be Paris, the son of King Priam of Troy, possibly done by the renowned sculptor Euphranor, who worked in Athens in the mid-fourth century B.C.
Almost overlooked in the objects brought up from the wreck was a wooden box about the size of a book, which when opened proved to contain a complex arrangement of bronze gears and dials, all heavily eroded into shapeless lumps of green metal. The wooden box soon disintegrated into dust, but the bronze gears and dials survived and were eventually subjected to an X-ray analysis to determine their function. The device, now known as the Antikythera computer, proved to be an elaborate clockwork mechanism, which Derek De Solla Price, a historian of science at Yale, showed to be an astronomical device that reproduced the motions of the sun, moon, and visible planets. Such a device, now called an orrery, or planetarium, was known to the Greeks as asphairopoiia, a mechanism that modeled the motions of the celestial bodies. A more recent analysis by Michael Wright, curator at the Science Museum in London, has shown that the gear system reproduced the epicycle theory for planetary motions derived by Apollonius of Perge and used by Ptolemy of Alexandria.
Several classical sources credit Archimedes with the invention of two mechanical devices in bronze that reproduced the motions of the heavenly bodies. One of these sources, the mathematician Pappus of Alexandria, says that Archimedes wrote a treatise, now lost, called Peri Spairopoiias (On Sphere-Making), describing a celestial globe that he made to represent the motions of the sun and moon and demonstrate both solar and lunar eclipses. The other Archimedean invention was an orrery that reproduced the motions of the planets as well as those of the sun and moon.
Cicero reports that the Roman general Marcellus took both of these Archimedean devices back to Rome as booty after his sack of Syracuse in 212 B.C. He set up the celestial globe in the temple of Vesta for all to see. The poet Ovid, writing circa A.D. 8, describes the globe in his verses on the goddess and her sanctuary: “There stands a globe hung by Syracu-san art in closed air, a small image of the vast vault of heaven, and the Earth is equally distant from the top and bottom. That is brought about by its round shape.”
The orrery eventually came into the possession of a grandson of Marcellus, who showed it to the astronomer Gaius Sulpicius Gallus. Gallus used the orrery to predict a lunar eclipse on 21 June 168 B.C., and Cicero says that he demonstrated solar eclipses as well, though obviously he could not predict whether these would be visible in Rome.
Other astronomers followed the lead of Archimedes in constructing these mechanical devices for demonstrating celestial motions, and examples survive from the Byzantine, Islamic, and late medieval European eras. Cicero writes, “Our friend Posidonius as you know has recently made a globe which in its revolution shows the movement of the sun and stars and planets, by day and night, just as they appear in the sky.” Cicero would have seen this orrery, for as a young man he had attended the lectures of Posidonius at Rhodes. This suggests the possibility that the Antikythera computer may have been the orrery made by Posidonius, since the ship in which it was found is thought to have been heading from Rhodes to Italy.
The finding of the Antikythera computer was the first of two dramatic discoveries of lost works of Greek science made at the beginning of the twentieth century. The second came in 1906, when the Danish scholar John Ludwig Heiberg discovered a copy of a lost work of Archimedes’ in a Greek church in Istanbul, along with other ancient manuscripts. Heiberg's remarkable finding was reported on the front page of The New York Times on 16 July 19 07, though it failed to receive any mention in The Times of London.
Heiberg made his discovery in the church of Agios Giorgios (St. George) Metochi in the old Greek quarter of the Fener on the Golden Horn. Agios Giorgios is a metochion, or daughter church, of the Monastery of the Holy Sepulchre in Jerusalem and belongs to the Jerusalem Patriarchate rather than to the Patriarchate of Constantinople, which has had its headquarters in the Fener since the end of the sixteenth century. Among the manuscripts discovered there was one now known as Codex C, Archimedes’ thesis On the Method, which had been lost for some two thousand years.
Newton and all of his European predecessors were aware of their great debt to Archimedes, who is mentioned more than a hundred times by Galileo, for his rigorously mathematical approach to the study of nature became the model for the new science that replaced the moribund Aristotelianism of the medieval era. As Marshall Clagett writes regarding the influence of Archimedes on Galileo and his contemporaries: “Archimedes’ significance for these founders of early modern science lay in the use of mathematics in the treatment of physical problems as well as in the originality and fertility of his mathematical techniques.”
Given the crucial importance of Archimedes in the Scientific Revolution, it is somewhat of a miracle that his extant writings survived at all. Carl B. Boyer writes in his history of mathematics, “Unlike the Elements of Euclid, which have survived in many Greek and Arabic manuscripts, the treatises of Archimedes have reached us through a slender thread. Almost all copies are from a single Greek original which was in existence in the sixteenth century and itself copied from an original of about the ninth or tenth century.” One of the most important of Archimedes’ works, his treatise On the Method, was believed to have been lost in late antiquity; though its rediscovery by Heiberg caused a sensation, it disappeared from sight again a few years later. Finally, it reemerged at the end of the twentieth century, under circumstances reminiscent of an Eric Ambler novel.
Early in the sixth century only three of the many works of Archimedes were generally known, those that appeared in the collection edited by Eutocius of Ascalon: On the Equilibrium of Planes, On the Sphere and the Cylinder, and the incomplete On the Measurement of the Circle In the ninth century, Leo the Mathematician added to these the works On Conoids and Spheroids, On Spirals, On the Quadrature of the Parabola, the Book of Lemmas, and The Sand Reckoner Leo's collection, known as Codex A, thus contained all of the works of Archimedes’ in Greek now known, except On Floating Bodies, On the Method, Stomachion, and the Cattle Problem This was one of two manuscripts available to William of Moerbeke when he made his translations of Archimedes in 1269. The other, known as Codex B, also called the Codex Mechanicorum, which contained only the mechanical works—On the Equilibrium of Planes, On the Quadrature of the Parabola, and On Floating Bodies (and possibly On Spirals)—was last referred to in the early fourteenth century; it then disappeared. Thus, as Marshall Clagett remarks, Codex A “was the source, directly or indirectly, of all the Renaissance copies of Archimedes.”
Clagett also notes that it seems unlikely that Arab mathematicians possessed any collection of the works of Archimedes as complete as Codex A. According to Clagett, the writings of Archimedes available to the Arabs consisted of the following works: On the Sphere and the Cylinder, in an early-ninth-century translation revised in turn by Ishaq ibn Hunayn and Thabit ibn Qurra and reedited by Nasir ad-Din al-Tusi; On the Measurement of the Circle, translated by Thabit ibn Qurra and reedited by al-Tusi; a fragment ofOn Floating Bodies; possibly On the Quadrature of the Parabola, as evidenced by research on this work by Thabit ibn Qurra; some indirect material of On the Equilibrium of Planes, as indicated in Greek mechanical works translated into Arabic; and other writings attributed to Archimedes by Arab mathematicians for which there is no extant Greek text, such as the Book of Lemmas, the Book on the Division of the Circle into Seven Equal Parts, and On the Properties of the Right Triangle
Western Europe acquired its knowledge of Archimedes solely from Byzantium and Islam, for there is no trace of the earlier translations that Cassiodorus attributes to Boethius. The translation of Archimedean texts from the Arabic began in the twelfth century with On the Measurement of the Circle, a defective rendering that may have been done by Plato of Tivoli. A much superior translation of the same work was done by Gerard of Cremona, using an Arabic text due to Thabit ibn Qurra. The earliest-known Arabic translations of Archimedes are those of Thabit ibn Qurra. These comprise all the works of Archimedes’ that have not been preserved in Greek, including the Book of Lemmas, On Touching Circles, and On Triangles
The texts used by William of Moerbeke in his 1269 translations—Codices A and B—had come to the papal library in the Vatican from the collection of the Norman kings of the Two Sicilies. William translated all the works included in Codices A and B, except for The Sand Reckoner and Eutocius's Commentary on the Measurement of the Circle William's translations did not include On the Method, the Cattle Problem, or the Stom-achion, since these works were not in Codices A and B.
A new Latin translation of the works of Archimedes was done circa 1450 by James of Cremona, sponsored by Pope Nicholas V. James worked entirely from Manuscript A, and so his translation did not include On Floating Bodies, but it did have the two works in Codex A omitted by William of Moerbeke, The Sand Reckoner and Eutocius's Commentary on the Measurement of the Circle Soon after James completed his translation the pope sent a copy to Nicholas of Cusa, who made use of it in his De Mathematicis Complementis, written in 1453-54. There are at least nine extant copies of this translation, one of which was corrected by Regiomontanus.
Codex A itself was copied several times, one copy being made by Cardinal Bessarion in the period 1449-68, and another by the humanist Georgio Valla, who used it in his Outline of Knowledge, printed at Venice in 1501. Copernicus, as we have learned, had a copy of the Outline of Knowledge, in which he would have read Archimedes’ account in The Sand Reckoner of the heliocentric theory proposed by Aristarchus of Samos, which preceded the Copernican theory by eighteen centuries.
Interest in Archimedes intensified from the mid-sixteenth century onward, and his influence can be seen in the works of Commandino, Simon Stevin, Kepler, Galileo, Torricelli, Leibniz, Newton, and many others. Translations were made into Italian, French, and German, and a new Latin edition was published in London in 1675 by Isaac Barrow, Newton's predecessor as Lucasian professor of geometry at Cambridge. At the end of the eighteenth century a new edition of the Greek text with Latin translation was prepared by the Italian mathematician Joseph Torelli (1721-81), published at Oxford after his death by Abram Robertson.
Nevertheless, a number of Archimedean writings remained missing, most notably the work On the Method, the existence of which was known only from references by Hero of Alexandria and the tenth-century Byzantine writer Suidas, who says that Theodosius of Bithynia wrote a commentary on it, though that was also lost.
The manuscripts Heiberg discovered were part of a palimpsest, in this case a euchologion, or prayer book, made up from recycled parchment leaves whose original contents had been scraped away and then written over with the new liturgical document. Heiberg's attention had been drawn to the euchologion through a report published in 1899 by the Greek scholar A. Papadopoulos-Kerameus, a catalog description of a manuscript collection in Istanbul belonging to the Metochion of the Holy Sepulchre, the daughter house of a famous monastery in Jerusalem. Papadopoulos-Kerameus had noted that the underlying script of the palimpsest MS 355 included a mathematical text, of which he printed a few lines in his catalog. Heiberg, who at the time was revising his edition of Archimedes, recognized the lines as being from an Archimedean work. He went to Istanbul and examined the palimpsest, first in 1906 and then again two years later, when he photographed the manuscript using the newly invented ultraviolet lamp. He reported on his discovery in 1907 in a long article in the scholarly journal Hermes; and in 1910-15 he incorporated his findings in the second edition of his three-volume opus on the works of Archimedes, upon which all subsequent Archimedean studies have been based. Meanwhile T. L. Heath translated On the Method into English, including it as a supplement to a new edition of his book The Works of Archimedes, published in 1912, which I used when I first began studying Archimedes.
Papadopoulos-Kerameus noted that the palimpsest contained a sixteenth-century inscription recording that it belonged to the ancient Palestinian monastery of St. Savas, known in Arabic as Mar Saba, founded in 483 a few miles east of Bethlehem on the west bank of the Jordan. The monastery had a renowned scriptorium for the copying and preservation of ancient manuscripts, of which its collection included more than a thousand works. Mar Saba was in ruins in 1625 when it was purchased by the Greek Orthodox Patriarchate of Jerusalem, which began a restoration in 1688. It has been suggested that in the early nineteenth century the euchologion and other ancient manuscripts in Mar Saba were taken for safekeeping to Istanbul; there they were preserved in the Jerusalem patriarchate's Metochion of the Holy Sepulchre, which was under the jurisdiction of the Patriarchate of Constantinople on the shore of the Golden Horn in the Fener quarter.
The German biblical scholar Constantine Tischendorf visited the Metochion in the early 1840s. He described the Metochion in his Reise in den Orient (Leipzig, 1846), where he says he found nothing of particular interest except a palimpsest whose pages included some mathematics. He appears to have stolen a page from the euchologion, for in 1879 a leaf from the palimpsest was sold from his estate to the Cambridge University Library. Nigel Wilson, a professor at Lincoln College, Oxford, examined this leaf in 1971 and identified it as being part of what by then had come to be known as the Archimedes Palimpsest.
The palimpsest disappeared from the Metochion not long after Heiberg's discovery, probably stolen in the chaos surrounding the fall of the Ottoman Empire and the creation of the new Republic of Turkey in 1923. Early in the 1920s the palimpsest was acquired by Marie Louis Sirieix, a French businessman and civil servant. In 1946 Sirieix gave the palimpsest as a wedding present to his daughter Anne Guersan, who had the euchologion restored and, as it appears, “embellished,” by the addition of what proved to be forged images of the four Evangelists. Nigel Wilson stated that the images were “a disastrously misguided attempt to embellish the manuscript, presumably to enhance its value in the eyes of a prospective purchaser.” In any event, the Guersan family put the palimpsest up for sale, and on 29 October 1998 it was auctioned at Christie's in New York, where it was purchased for $2 million by an anonymous buyer. The Jerusalem Patriarchate contested the auction in a lawsuit in New York, but the court ruled that the sale was legal.
Meanwhile, the anonymous buyer deposited the palimpsest with the Walters Art Museum in Baltimore, Maryland, in January 1999, providing funds for conservation, imaging, and scholarly study of the manuscript. A team of scientists from the Rochester Institute of Technology and Johns Hopkins University used computer processing of digital images of the underlying text of the palimpsest photographed in ultraviolet, infrared, and visible light. In May 2005 the palimpsest was irradiated with highly focused X-rays produced at the Stanford University Linear Accelerator Center in Menlo Park, California, which made it possible to read parts of the underlying text that had previously been undecipherable.
In 2002, John Lowden, a professor at the Courtauld Institute in London, deciphered a colophon on the palimpsest giving the date 13 April 1229, when the euchologion was dedicated after the recycling of the ancient manuscripts on which it was written. The palimpsest, now referred to as Codex C, contains parts of seven treatises by Archimedes as well as pages of four other works, including those of the fourth-century B.C. Attic orator Hyperides. The Archimedean writings include an almost complete text of the previously unknown work On the Method; a substantial part of On Floating Bodies, whose original Greek text had been lost; a page from the Stomachion, another unknown work of Archimedes’; and fragments of On the Sphere and the Cyclinder, On Spirals, On the Measurement of the Circle, and On the Equilibrium of Planes Thus Codex C overlaps with Codices A and B for several works: together with A, it has a text of On Spirals, On the Sphere and the Cylinder, and On the Measurement of the Circle; along with B, it has a text of On Floating Bodies Studies have shown that the Archimedean texts in the palimpsest, Codex C, were written in the second half of the tenth century, almost certainly in Constantinople.
The most important of these works by far is On the Method, whose full title is On the Method Treating of Mechanical Problems, Dedicated to Eratosthenes Addressing Eratosthenes, the head of the Library of Alexandria, Archimedes explains the method by which he arrived at the propositions from which he deduced his theorems: “I thought fit to write out for you and explain in detail in the same book the peculiarity of a certain method, by which it will be possible for you to get a start to investigate some of the problems in mathematics by means of mechanics.”
The mechanical method used by Archimedes was one in which he mathematically balanced geometrical figures as if they were weights on a scale, comparing a figure of unknown area with one whose area was already known. Then, using the law of the lever, he determined the unknown area from the area that was known. He then extended his method to three dimensions so as to determine volumes, as he shows in Proposition 2 of the On the Method: “The cylinder with base equal to a great circle of the sphere and height equal to the diameter is 3/2; times the sphere [in area].” This was Archimedes’ favorite theorem; he had it represented in a relief carved on his tombstone. Cicero found this monument in 75 B.C., when he was quaester of Sicily and the relief was still visible.
Archimedes used his mechanical method to determine the volumes of three solids of revolution—the ellipsoid, the paraboloid, and the hyperboloid—as well as the centers of gravity of the paraboloid and the hemisphere. He concluded the thesis by finding the volume of two solids, first the wedge cut from a right circular cylinder by two planes, and second the volume common to two equal right cylinders intersecting at right angles. All of these were results first achieved in western Europe only after the invention of the calculus by Newton and Leibniz, more than nineteen centuries after Archimedes had done the same in his lost work On the Method
The treatise On Floating Bodies, which contains Archimedes’ famous principal of buoyancy, the basis of hydrostatics, was previously known only from the Latin translation done in 1269 by William of Moerbeke, the Greek original having been lost. The text of this thesis in the Archimedes Palimpsest has considerable lacunae, so that William's translation is still used to give the undecipherable and missing parts of the Greek text.
Only a single page of the Stomachion was used in the palimpsest, the first page of the thesis, which became the last page in the euchologion. Otherwise, the only source for this thesis is a brief passage in an Arabic text published in Berlin in 1899, said to derive from a work by Archimedes entitled Stomachion Together the two sources are not enough to enable one to understand Archimedes’ motive in writing this thesis, which appears to be about a type of geometrical game. The name Stomachion means literally “that which relates to the stomach,” and it has been suggested that the game was so called because its difficulties were so great that it could give one a bellyache. E. J. Dijksterhuis, in his definitive book on Archimedes, concluded from his study of the ancient sources that the Stomachion “is a kind of game, played with bits of ivory in the form of simple planimetrical figures, the object being to fit these bits together in such a way that the various shapes of human beings, animals or different objects were imitated.” He notes that the game board was apparently known to the Romans as the loculus Archimedius, or Archimedian box, and that “it consisted of fourteen bits of ivory of different forms, that these bits together formed a square, that it was possible to compose from them all sorts of figures (a ship, a sword, a tree, a helmet, a dagger, a column), and that this game was considered very instructive for children, because it strengthened the memory.”
The single page of the Stomachion preserved in the palimpsest indicates that Archimedes was the first author in a field of mathematics now known as combinatorics, in this case finding the number of possible ways of fitting together the pieces on the game board. A recent analysis has shown that if there are 14 pieces on the board then there are 17,152 ways of arranging them. It is not known whether Archimedes found this solution, but anyone familiar with his work would bet that he did.
The Archimedes Palimpsest presently comprises 174 pages, three less than when Heiberg examined it, the missing pages probably removed when the euchologion was stolen from the Metochion. All but fifteen pages of the underlying text of the palimpsest have been deciphered, and these are now being analyzed at the Stanford Linear Accelerator Center. It takes about twelve hours to scan one page using an X-ray beam about the width of a human hair. Once each new page is analyzed it is posted online for the general public to examine.
After I viewed the most recent page posted on the Internet I went to visit the church of Agios Giorgios Metochi again, for I had not been there since before the 1998 rediscovery of the Archimedes Palimpsest. First I had to go to the office of the Jerusalem Patriarch in the headquarters of the Ecumenical Patriarch of Constantinople; after finding my way through a labyrinth of Byzantine bureaucracy I finally received a document allowing me to visit the Metochion.
The church is within an extensive walled enclosure on a hillside above the Golden Horn, completely isolated from the tumultuous city around it, its main entryway closed by an enormous iron-barred wooden gate. I rang the bell several times, and when no one answered I picked up a stone and hammered on the door until finally the gate creaked open and a white-bearded old priest stuck his head out. I addressed him in Greek and showed him the document I had obtained from the patriarchate, whereupon he let me in and went to find the keys to the church.
The church itself had been restored since I'd last seen it, but everything else within the enclosure was in utter ruins, with a flock of goats grazing among the fallen columns and the other architectural fragments of the buildings that had once stood there. This was once the site of the palace of the Cantacuzenos family, Greeks from the Fener who ruled as hospodars of the trans-Danubian principalities of Moldavia and Wallachia under the aegis of the Ottoman sultan. All that remained of the palace was the shell of its chapel, dedicated to the Virgin, which had been the site of the Patriarchate of Constantinople in the late sixteenth century.
After seeing the church I sat for a while in the courtyard with the priest, who told me that he had been born in Istanbul but that after joining the priesthood he had been sent to the Monastery of the Holy Sepulchre in Jerusalem. He had returned only recently, he said, and was now living in virtual retirement in the Patriarchate of Constantinople, his only duty being to look after the church of Agios Giorgios Metochi. The church itself was all that was left of the Metochion of the Holy Sepulchre, the ruins of its other buildings indistinguishable from those of the Palace of the Hospodars of Moldavia and Wallachia. The survival of the church itself was an unexplained miracle, as was the preservation of its collection of ancient manuscripts until the early 1920s, when those that had not been stolen were removed to Athens for safekeeping.
The priest was not a learned man, and he knew nothing of the ancient manuscripts that had been preserved here. I showed him the copy of the latest page of the Archimedes Palimpsest that I had recently downloaded, and he was able to read some of it, though he had no idea what it meant since it was all in mathematical language. I told him that the works of Archimedes were written in the third century before Christ, and that a thousand years ago those that were in this palimpsest were copied by scribes in Constantinople, after which they had been preserved at the Mar Saba monastery in Palestine before being brought here to the Metochion early in the nineteenth century. The priest knew Mar Saba well, since he had been there several times during his years in Jerusalem. I then told him of the rediscovery of the manuscripts in the past century, and of the recovery of their underlying text at research laboratories in the United States. The priest shook his head in wonder after he heard my story, and he said that it must have been the will of God that these masterpieces of ancient science should have been preserved. I nodded as if I agreed with him, and then I thought of Archimedes himself, recalling the words with which he begins his treatise On the Method:
Archimedes to Eratosthenes greeting.
I sent you on a former occasion some of the theorems discovered by me, merely writing out the enunciations and inviting you to discover the proofs, which at the moment I do not give. The enunciation of the theorems which I sent were as follows.
And so now we have the proofs of Archimedes’ theorems, lost for more than two thousand years, uncovered from a palimpsest that connects his time with ours through the intervening layers of the medieval Byzantine, Islamic, and Latin worlds.
Greetings, Archimedes; we await the next page from your manuscript.