The Prayer Book That Was a Math Textbook

In the spring of 1906, a Danish classical philologist named Johan Ludvig Heiberg arrived in Constantinople to look at a particular medieval prayer book at the library of the Metochion of the Holy Sepulchre, a Greek Orthodox monastery in the city’s central districts. Heiberg, who was one of the most accomplished editors of ancient Greek scientific texts of his generation, had been told by an Athenian colleague that the prayer book showed traces of older writing underneath — what scholars call a palimpsest, a parchment that has been scraped clean and reused for a new text — and that the underwriting appeared to be Greek mathematical prose.

Heiberg examined the manuscript at the monastery library for several days, with a magnifying glass and an oil lamp. He read what he could of the underwriting. He realized, with mounting astonishment, that he was looking at three texts he had already published critical editions of — Archimedes’s On the Sphere and Cylinder, On Spiral Lines, and On the Equilibrium of Planes — in versions that were 700 years older than the manuscripts he had used as his sources. He continued reading. He realized he was also looking at three texts that had been lost since late antiquity: a complete version of On Floating Bodies (previously known only in a flawed thirteenth-century Latin translation), and two works that had been considered entirely lost — The Method of Mechanical Theorems and Stomachion.

He spent the rest of the summer of 1906 transcribing what he could from the palimpsest with a magnifying glass. He published the discovery in a 1907 article in Hermes. The article identified the manuscript as a Byzantine prayer book of the thirteenth century made by scraping the ink off an older parchment codex — the older codex containing what we now think was the most complete single collection of Archimedes ever assembled in the ancient or medieval world.

It was, in the careful summary by William Noel a century later, “the most important manuscript ever discovered in the field of the history of science.”

What Heiberg actually read

The palimpsest is a tenth-century Byzantine codex on parchment, originally containing seven treatises by Archimedes (third century BC) copied by an unknown Constantinopolitan scribe sometime around 950 AD. The parchment was scraped clean of the Archimedes writing in the late twelfth or early thirteenth century, cut down to a smaller format, and reused to copy a Byzantine prayer book — a euchologion, used in monastic services. The original Archimedes text was not fully erased, just made very faint; the medieval scraper’s job was to make the parchment usable for new ink, not to obliterate the old.

The new prayer book was written perpendicular to the original Archimedes text. The result is a manuscript that, viewed normally, looks like a perfectly ordinary Greek prayer book with some faint smudges between the lines. Viewed under raking light, or with the page held at the right angle, the ghost of older writing becomes visible — running sideways across the page, in a fainter ink, in a different hand. This is the Archimedes text.

Of the seven treatises in the original tenth-century codex, two were known nowhere else. The Method of Mechanical Theorems is Archimedes’s own description of how he discovered his geometrical results — using physical thought experiments involving levers and balance points, then proving them rigorously afterward. The text reveals that the Greeks had a clear conception of what we would now recognize as integral calculus, almost two thousand years before Newton and Leibniz. The Stomachion is a treatise on a fourteen-piece dissection puzzle and is now believed to contain the first known work in combinatorial mathematics — counting the number of distinct ways the pieces can be arranged to form a square.

Both works survive only in the palimpsest. There is no other copy. If the prayer book had been lost or burned or destroyed at any point in the eight centuries between its making and Heiberg’s reading, the two texts would have been gone permanently.

What happened to it

Heiberg published his transcription of as much of the palimpsest as he could read between 1910 and 1915, in revised editions of his standard Opera Omnia of Archimedes. The transcription was incomplete and in places conjectural; the readings were limited by the available technology of a magnifying glass and natural light.

Then the manuscript vanished. The Greek Orthodox patriarchate of Jerusalem, which owned the Metochion, was caught up in the wars and upheavals of the First World War and the Greco-Turkish War of 1919-1922. The library at the Metochion was dispersed. The Archimedes palimpsest was, at some point between 1906 and 1922, removed from the monastery library — by whom, when, or how is not documented — and disappeared from public knowledge for the next seventy-six years.

It surfaced again in 1998. A French family in Paris brought a Byzantine prayer book to Christie’s auction house in New York for valuation. The book had been in the family since the 1920s and was — according to the family — a relatively minor religious manuscript that they wanted to sell. The Christie’s specialists immediately recognized what it was. They contacted the appropriate authorities. The Greek Orthodox Patriarchate filed a legal claim asserting that the manuscript had been stolen from its library in Constantinople in the early twentieth century and was therefore not legitimate property of the French family. The legal claim was, after considerable argument, rejected by U.S. courts on the grounds that the chain of custody could not be definitively established.

The palimpsest was sold at auction on 29 October 1998 for $2.2 million. The buyer, identified only as “an anonymous American billionaire” (later confirmed to be the financier David Solomon), gave the manuscript on permanent loan to the Walters Art Museum in Baltimore. It has been there since.

What modern technology found

Between 1999 and 2008, a team led by William Noel at the Walters carried out a decade-long imaging project on the manuscript. The team used multispectral photography (taking images at wavelengths across the visible and near-infrared and ultraviolet spectra), X-ray fluorescence imaging, and a custom synchrotron beam at the Stanford Linear Accelerator Center.

The results made visible what Heiberg had not been able to see. About 80% of the original Archimedes text was recovered — including substantial portions of The Method and Stomachion that Heiberg had given up on. The recovered text included the previously-missing first half of The Method’s critical Proposition 14, in which Archimedes explicitly compares the volume of a parabolic cylinder to a triangular prism using infinite slicing — the closest thing to an integral that survives from antiquity.

The team also recovered, on previously unreadable folios at the end of the codex, traces of two completely different texts that had nothing to do with Archimedes. One was a previously unknown speech by the Athenian orator Hyperides (fourth century BC), of which fragments had been known but the bulk was lost. The other was a commentary on Aristotle’s Categories attributed to Alexander of Aphrodisias (third century AD) — a work known by title from medieval references but never before recovered.

The original Byzantine scribe of the 950s, in other words, had bound together a single codex containing not only the Archimedes treatises but also previously unknown works of two other ancient authors. The choice of texts is the closest thing we have to a personal library decision by an educated tenth-century Constantinopolitan scholar. Whoever made the codex was interested in mathematics, philosophy, and Athenian rhetoric. We do not know his name.

The palimpsest is on permanent display in a climate-controlled case at the Walters Art Museum in Baltimore. The display is open to the public. The manuscript looks, to a visitor walking past, like a fairly ordinary medieval prayer book in Greek. The Archimedes text is invisible to the naked eye and remains so. The full multispectral data is published, free, on a website maintained by the Walters. Anyone in the world with an internet connection can read the recovered text of The Method of Mechanical Theorems, in the only manuscript copy that survives.