The Iron-Gall Prayer Book

Reviel Netz, William Noel, and the X-Ray Archimedes Palimpsest

Cataloged for centuries as Euchologion 329, a burnt, mold-spotted thirteenth-century Byzantine prayer book of 174 goatskin leaves sat on an auction block at Christie's in New York on October 29, 1998. When the gavel fell at two million dollars to an anonymous American technology billionaire, the buyer placed the volume into the hands of curator William Noel at the Walters Art Museum in Baltimore for open scientific study. Researchers soon confronted the most battered crime scene in the history of mathematics.

Ninety-two years earlier, in the summer of 1906, Danish philologist Johan Ludvig Heiberg had traveled to Constantinople to inspect that same prayer book inside the Metochion of the Holy Sepulchre. He had noticed a brief catalog entry by paleographer Athanasios Papadopoulos-Kerameus mentioning mathematical diagrams beneath the prayers. Examining the pages with a magnifying glass, Heiberg realized that the prayer book was a palimpsest (from the Greek palin psao, scraped again). Because parchment was expensive in the medieval Mediterranean, a Christian scribe named Johannes Myronas—who dated his colophon on April 14, 1229, almost certainly in Jerusalem—had recycled seven older Greek manuscripts. He washed and scraped their ink off with weak acid and pumice, cut the sheets in half down their fold, rotated them ninety degrees, and copied liturgical prayers across the faint brown ghosts of the older writing.

The underlying tenth-century manuscript, copied in Constantinople around 975 CE, contained seven treatises by Archimedes of Syracuse (287–212 BCE). Even more critically, this scraped codex held the only surviving Greek text of On Floating Bodies and the only copies anywhere on Earth of two works long thought lost: Stomachion (a treatise on combinatorics) and The Method of Mechanical Theorems. Written as a letter from Archimedes to Eratosthenes at Alexandria, The Method explained how Archimedes discovered geometric theorems by weighing infinitely thin slices on an imaginary balance.

After Heiberg photographed a fraction of the pages in 1906 and 1908, the manuscript vanished after World War I and remained hidden in a French private collection for seventy years. By the time William Noel and Stanford historian Reviel Netz opened the codex in Baltimore in 1999, its condition was catastrophic. Thick mold had perforated the parchment, and modern glue bound the spine too tightly to photograph the inner margins. Worst of all, after 1929 an unscrupulous owner had tried to inflate the book's value by painting four Byzantine-style gold-leaf and lead-white Evangelist icons directly on top of four pages of Archimedes' text.

Working with imaging scientists Roger Easton Jr., Keith Knox, and Bill Christens-Barry, the team conserved and separated every leaf. They imaged the exposed pages under multispectral light, flashing ultraviolet (365 nm) and red (625 nm) diodes so the brown iron-gall ink of Archimedes vanished in the red channel while the black prayer text remained. That contrast allowed digital subtraction of the prayers. Yet multispectral light could not penetrate the solid gold leaf and lead paint of the four forged icons.

In 2004, physicist Uwe Bergmann at the Stanford Linear Accelerator Center (SLAC) proposed a nuclear-physics solution. Both the tenth-century scribe and the thirteenth-century priest had written with iron-gall ink, brewed by mixing crushed oak galls (tannic acid) with ferrous sulfate ($\text{FeSO}_4$). Even when Myronas scraped away the surface crust of the tenth-century letters in 1229, trace iron atoms remained chelated deep inside the collagen fibers. At the Stanford Synchrotron Radiation Lightsource (SSRL), Bergmann aimed a 50-micron beam of synchrotron X-rays tuned to $7.11\text{ keV}$—just above the K-shell binding energy of iron—directly through the gold paintings. As the X-ray beam penetrated thickness $x$ with attenuation coefficient $\mu(E)$, every buried iron atom inside Archimedes' inkstrokes ejected an inner-shell electron and emitted X-ray fluorescence photons at iron's $K\alpha$ spectral line:

Because $6.40\text{ keV}$ X-rays pass back out through gold leaf and parchment—and enabled reading both sides of a leaf using front and back detectors—the synchrotron mapped the iron atoms at 40-micron resolution. The scans also uncovered ten pages of lost speeches by the Athenian orator Hyperides. As the scan lines assembled in August 2006, the forged gold icons vanished and revealed Archimedes calculating the volume of solids with actual infinite sets nineteen centuries before Newton and Leibniz.