Coaxing Answers from Ancient Puzzles

December 21, 2008

Figure 1. A reconstruction of the Stomachion puzzle. From The Archimedes Codex.

Book Review
James Case

The Archimedes Codex: How a Medieval Prayer Book Is Revealing the True Genius of Antiquity's Greatest Scientist. By Reviel Netz and William Noel, Da Capo Press, New York, 2007, $27.50.

The Forgotten Revolution: How Science Was Born in 300 BC and Why it Had To Be Reborn. By Lucio Russo (with the collaboration of Silvio Levy, translator), Springer, Berlin, 2003, $99 (hardcover), $31.90 (paperback).

The Archimedes Codex describes the unlikely sequence of events that led to the recovery of two previously unknown treatises of Archimedes---Method and Stomachion---from a 13th-century prayer book. The book is a palimpsest, meaning that the works of Archimedes originally written on its (goatskin) parchment pages were subsequently scraped away and overwritten, to avoid the expense of new parchment. The authors calculate that at least twenty-four goatskins were required to produce the original book; it is impossible to determine the exact number, because the beginning, end, and several chunks of the middle of the book are now absent. They may in fact never have been included in this copy of the book, because the medieval scribes who copied and recopied the works of Archimedes plainly knew nothing of mathematics, and could unwittingly have omitted significant portions of the text.

Archimedes, who died in 212 BC, wrote on papyrus scrolls, most of which found their way to the library at Alexandria, where works by a single author could be shelved together and copies made for other libraries. His name was well known, and his most important achievements were frequently cited, but few could read his actual writings. As a result, the scrolls mostly gathered dust during the centuries before scrolls were re-placed by codices.

A codex is essentially a book. The earliest codices were made by cutting scrolls into segments the width of two pages, folding the resulting bifolios in the middle, and binding the result between hard covers. The fact that the ancients typically wrote in columns perpendicular to the length of their scrolls made codices particularly convenient. But papyrus tends to crumble when folded, and most of the earliest codices no longer exist. Most that survive are copies on parchment, which is made of animal skin and is more flexible and durable than papyrus. There is reason to believe that the bulk of Archimedes's writings were transcribed onto parchment at Constantinople during the sixth century AD. But few sixth-century transcriptions have survived, and there is no way to know for sure.

The codices transcribed at the Imperial Palace School in Constantinople at that time were written entirely in upper-case letters (majuscules), without spaces between words and with little or no punctuation. Most of those still in existence---including the Archimedes codex---were recopied during the tenth century, in lower-case letters (minuscules), with spaces between words and rudimentary punctuation; these changes increased the number of characters that could appear on a single page, while improving legibility. Most of the majuscule copies have been lost or destroyed.

Of the tenth-century codices into which Archimedes's writings were in fact transcribed, only three (denoted by scholars as A, B, and C) are currently known. Both A and B were readily available to Western scholars during the Renaissance, and were translated many times---and into a variety of languages---after the invention of printing. Both have since disappeared. Codex C was unknown to Western scholars until 1906, when Johan Ludwig Heiberg, a Danish historian of mathematics, learned from a colleague that the catalogue of a monastic library in Constantinople mentioned a thousand-year-old palimpsest that appeared to contain mathematics. Heiberg traveled to Constantinople, translated as much of it as he could using the techniques at his disposal, and returned to his home in Denmark. During or shortly after World War I, the codex disappeared, resurfacing only in 1998, when Christie's offered it at auction.

Codices A, B, and C all contain Archimedes's treatise Balancing Planes. Quad-rature of the Parabola is found in A and B, Floating Bodies in B and C; Sphere and Cylinder, Measurement of the Circle, and Spiral Lines are found in A and C. A alone includes Conoids and Spheroids and Sand-Reckoner, and C alone contains Method and Stomachion. Sherman Stein's excellent Archimedes: What Did He Do Besides Shout Eureka? (Mathematical Association of America, 1999) summarizes the content of all save the last two, which were not then available.

Historians of mathematics had long known of Archimedes's Method from references to it by other ancient writers. Originally contained in a letter to Eratosthenes, director of the library at Alexandria, the treatise on method describes the use of mechanical analogies---involving imaginary scales, levers, centers of gravity, and the like---as a heuristic for discovering new theorems of (Euclidean) geometry. Archimedes seemed to be confident that propositions so discovered could be deduced in the traditional manner, from Euclid's postulates, perhaps augmented by one or two others, including his own (Archimedean) axiom. The authors of the book under review express surprise at the ease and confidence with which Archimedes dealt with infinite quantities in this (previously unknown) treatise. Traditionally, the Greeks have been thought to distrust such arguments, and to avoid them like the plague.

Stomachion is a different story. Heiberg had failed completely to perceive its actual significance. The name refers to the pictureless puzzle obtained by cutting with care along the lines of the square shown in Figure 1. Archimedes remarked in passing that the resulting triangles and quadrilaterals could be rearranged in many ways to fill the original square. But he didn't say how many ways, and the authors began to wonder how large the total might be. If it were (say) 31, they reflected, Archimedes probably wouldn't have been interested. They began to suspect that he knew the exact answer, that it was quite large, and that a modern expert could discover it without undue difficulty.

The first response to their (privately circulated) Request for Enlightenment came from Bill Cutler, a computer scientist in Illinois, who devised a program to settle the question. His answer was 17,152. But Archimedes, if he actually had a precise answer, couldn't have found it the way Cutler did, and it was natural to wonder how Archimedes might have obtained an answer. Among the curious were Persi Diaconis and Susan Holmes, who got together with Ron Graham and Fan Chung Graham for a long weekend of puzzle solving. Altogether, they found 24 basic families of solutions, each consisting of many distinct individuals obtainable one from another either by rotating appropriate blocks of pieces, or by permuting congruent blocks among themselves. Their answer agreed with Cutler's; significantly, they obtained it with pencil and paper alone, using only methods available to Archimedes!


Archimedes, like Euclid, lived near the beginning of what Lucio Russo terms "the Golden Age of Greek Science." It began with the founding of Alexandria in 331 BC and lasted until perhaps 145 BC, by which time the Roman conquest of the eastern Mediterranean was effectively complete. The Romans prized the practical results of science but showed no interest in its methods. Because Russo's Golden Age (otherwise known as the Hellenistic era) produced no law giver to match Solon, no philosophers to match Plato and Aristotle, and no playwrights to match Aeschylus, Sophocles, and Euripides, students of ancient history have tended to regard it as a somewhat tarnished reflection of the classical Athenian period. Russo and his collaborator/translator Silvio Levy seek to correct this gross misapprehension.

Einstein was repeating conventional wisdom when he remarked, "Development of Western science is based on two great achievements: the invention of the formal logical system (in Euclidean geometry) by the Greek philosophers, and the discovery of the possibility to find out causal relationships by systematic experiment (Renaissance)." Russo's Forgotten Revolution contradicts that so-called wisdom by demonstrating---in myriad different ways---that the Hellenistic Greeks deserve credit for both seminal achievements, along with the development of science-based technology! Copernicus, Galileo, and other Renaissance investigators did not rediscover experimental science on their own, but read about it in Hellenistic documents, many of which have since been lost. What still exists of the evidence is widely scattered, in part because so few of the "tech reports" once stored in the library at Alexandria were ever transcribed into codices.

Lacking original sources, modern students of ancient science are forced to rely on a combination of archeological finds and passing references by non-technical writers, including Pliny, Plutarch, and Proclus. The latter two, for in-stance, separately reported that Archimedes used his theoretical knowledge of mechanics to design a contraption permitting a single man---either himself or King Hiero II, depending on whom one chooses to believe---to drag a ship into the water from dry land in Syracuse harbor. Proclus further specified that the ship chosen for the demonstration was no dinghy, but a fully laden three-masted merchantman. Archimedes's choice of a demonstration project was no doubt inspired by Aristotle, who once postulated (in the course of arguing a quite different point) the impossibility of just such a feat.

Our knowledge of Hellenistic mechanics (literally, "science of machines") begins with Archimedes's treatise On the Equilibrium of Plane Figures (a.k.a. Balancing Planes), which deals mainly with his law of levers and his several methods for locating the barycenters of plane figures. His goal in formulating the law of levers was clearly the calculation of mechanical advantage in the machines he considered basic: the winch, the lever, the pulley, the wedge/ramp, and the screw. Further knowledge of the progress of Hellenistic science and technology comes from ancient writers' descriptions of machines actually built, such as gear-boxes for lifting heavy weights, devices for lifting water, and catapults.

Among the most telling pieces of evidence are a goodly number of water-powered Roman-era mills, including some at which stone was cut by straight saws engaged in up–down reciprocating motion. Because such mills would be all but impossible to design without a workable theory of machines, and because none predate Archimedes, it stands to reason that his theory prompted the spread of such mills. Another significant find was a Roman-era reciprocating pump deep in a Spanish mine, where it was apparently used to expel water from the depths. Here again, it seems unlikely that mere trial and error could have produced a machine so well adapted to its function. More likely, the Romans employed Greek slaves trained in Alexandria to serve as mining engineers and/or technical consultants. Such individuals may even have been able to sell themselves into slavery for the duration of a particular project, the conditions of slavery being far more diverse in the ancient world than in the American South.

Though Archimedes's mechanics was purely static in nature, it incorporated a clear understanding of mechanical advantage in linear, reciprocating, and/or rotational motion. Moreover, as his stunt with the ship makes clear, the transition had clearly been made from the natural philosophy of Plato and Aristotle (wherein conclusions are justified by unadorned logic) to genuine science, in which models require testing and nature itself is the final arbiter of truth.

Mechanics is by no means the only science born during Russo's Golden Age. Aristarchus, as Copernicus was apparently well aware, proposed a heliocentric model of the solar system. Widespread use of accurate balances led to the realization that mass is conserved during chemical changes, and thence to the hypothesis that atoms are indestructible. Botany and zoology were stimulated by Alexander's practice of collecting flora and fauna wherever he went, and sending specimens home for systematic study. Paintings from excavated houses in Pompeii and Herculaneum demonstrate an essentially complete command of perspective, long thought to be a Renaissance invention. Together with Eristratus of Ceos, Herophilus of Chalcedon (who worked at Alexandria) advanced the sciences of anatomy and physiology by dissecting cadavers. The Ptolemies reportedly provided the former with condemned men upon whom to perform vivisections! Although their writings have not survived, it is possible to reconstruct a substantial number of their medical discoveries from fragments and contemporary remarks. Herophilus was particularly interested in the eye, the brain, the nervous system, and mental illness.

Perhaps the most intriguing question raised by Russo's book is why Hellenistic science, after coming so far in so short a time, had to be reinvented in the 17th century. How did something as basic as the experimental method fall into disuse for almost two thousand years? Russo's answer is by no means definitive, and is far too complex to be summarized here. One suspects that the matter will be debated for years to come, and that Russo's book will remain essential background for that debate. Its publication appears to have marked an important turning point in the history of science.

James Case writes from Baltimore, Maryland.

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