One day in 2025, Victor Gysembergh, research director at the Centre Léon Robin (CNRS/Sorbonne Université), a specialist in ancient thought, joked with a colleague: "What if we looked for a palimpsest in Blois?" The joke turned into a search in Arca, the online catalogue of digitized manuscripts. And there, astonishment. A Greek leaf. Geometric figures. Copying errors that matched, line for line, the photographs taken in 1906 by the philologist Johan Ludvig Heiberg. Victor Gysembergh had found a page of the Archimedes palimpsest lost for more than a century — hidden at the Musée des Beaux-Arts in Blois, two hours from Paris.
The discovery, published on 6 March 2026 in the journal Zeitschrift für Papyrologie und Epigraphik (vol. 236, pp. 23-26), is short — four pages — but its significance is immense. It is leaf number 123 of a manuscript that historians of mathematics regard as one of the most precious documents ever recovered.
A reused parchment, an erased treasure
To understand why this leaf matters so much, we must go back to the 10th century in Constantinople. A Byzantine scribe transcribes several treatises by Archimedes of Syracuse — the greatest mathematician of antiquity — onto animal-skin parchment. Two centuries later, in the 12th century, a monk needs material for a prayer book. The parchment is too precious to waste. So the mathematical writing is scraped away, the sheets are turned 90 degrees, and new text is written over it. This is what is called a palimpsest — from the Greek palin (again) and psao (to scrape).
The result: a medieval prayer book that conceals, beneath its religious lines, Archimedes' geometric proofs. The mathematical text has not entirely disappeared — the old ink has soaked deep into the parchment — but it has become almost invisible to the naked eye.
In 1906, Heiberg manages to photograph the codex and transcribe a good part of it. The manuscript then passes from hand to hand, stays in Jerusalem, resurfaces in Paris, and is sold at auction in 1998 to a private collector who entrusts it to the Walters Art Museum in Baltimore for conservation and study. Three leaves visible in the 1906 photographs have since disappeared. Until now, they were considered lost.
The hidden side: a painting of the prophet Daniel
The Blois leaf is in a paradoxical state. On one side, the medieval prayer text still sits alongside Archimedes' geometric figures and a legible passage from the treatise On the Sphere and the Cylinder, Book I, propositions 39 to 41 — as confirmed by Sorbonne Université in its press release. On the other side, however, someone added an illumination in the 20th century depicting the prophet Daniel flanked by two lions. According to the AFP news agency, this painting was probably added to increase the document's market value. As a result, Archimedes' mathematical text is entirely hidden on the Daniel side.
This is where modern mathematics enters the picture — no longer to prove theorems about spheres, but to read what the eye cannot see.
Reading the invisible: when algorithms do the work of the eye
Two techniques are being considered to decipher the hidden side, subject to the necessary authorizations. The first, multispectral imaging, involves photographing the leaf under many different wavelengths — ultraviolet, infrared, visible light — and then combining these images computationally. The idea, as described in a study by Roger L. Easton Jr. and colleagues presented at EUSIPCO, is that "images at different wavelengths are combined arithmetically" to bring out contrasts invisible under ordinary light.
The second technique is even more powerful for areas covered with opaque paint: X-ray fluorescence produced by a synchrotron. Each atom in the parchment, excited by a beam of X-rays, re-emits energy at a characteristic wavelength. Archimedes' medieval ink is rich in iron; the 20th-century paint contains other elements. By mapping the distribution of iron across the entire surface of the leaf, the pen strokes of a 10th-century scribe can be made to reappear beneath a modern illumination. In previous imaging campaigns on the palimpsest, this method recovered an entire column of text in about 24 hours of acquisition.
During the first imaging campaigns carried out on the palimpsest in the early 2000s, and then in a new session in 2007 using a high-resolution camera under spectral LED lighting, researchers had managed to read about 80% of the original text. The Blois leaf revives this project, with tools far more refined than those available at the time.
What Archimedes wanted to prove — and why it is still beautiful
Propositions 39 to 41 of Book I of On the Sphere and the Cylinder deal with complex geometric proofs concerning volumes. To grasp what is at stake, here is the central result of the entire treatise: if a sphere is inscribed in a cylinder whose base has the same radius and whose height equals the sphere's diameter, then the volume of the cylinder is exactly one and a half times that of the sphere.
But how did Archimedes prove this, without integral calculus, without algebraic notation? Through the method of exhaustion: the unknown figure (a sphere, a spherical segment) is bounded by inscribed and circumscribed figures whose volumes are known, and the bounds are then tightened until equality is forced. It is the idea of a limit, without the name. As Fernando Q. Gouvêa notes in his review for the Mathematical Association of America, the difficulty lies in not projecting modern calculus onto these proofs: Archimedes reasons with particular figures, precise diagrams, a mathematical language all his own. What the manuscripts pass down is not only results — it is ways of thinking.
This is why every recovered leaf matters. As Victor Gysembergh told AFP, until this discovery, "we had no reason to hope we would ever find them". And he hopes the event will encourage other institutions and private collectors to check their collections: two missing leaves have yet to be located.
Key concepts
The method of exhaustion: bounding infinity with polygons
For readers who want to understand how Archimedes proved his results on volumes without integral calculus, here is the structure of his reasoning — as it can be reconstructed from the complete edition of Archimedes' works translated by Thomas L. Heath, published by Cambridge University Press in 1897 and now in the public domain.
The method of exhaustion, formalized before Archimedes by Eudoxus of Cnidus (4th century BC), rests on a principle simple to state and formidable to wield: to prove that two quantities A and B are equal, you show that neither can be greater than the other, nor smaller. The argument proceeds by contradiction in both directions.
Concretely, to calculate the volume of a sphere of radius r, Archimedes inscribes a polygonal-base pyramid within the sphere, then circumscribes another one around it. He progressively increases the number of faces. The greater the number of faces, the more closely the two pyramids bound the sphere — from above and from below. If you assume that the volume of the sphere is strictly greater than (4/3)πr³, an inscribed pyramid suffices to contradict this hypothesis. If you assume it is strictly smaller, a circumscribed pyramid contradicts the reverse hypothesis. Only one possibility remains: equality.
In its logical structure, this argument is the direct ancestor of the notion of a limit. It says: whatever precision ε > 0 you demand, I can find a polygonal figure that approaches the sphere to within ε. The difference from modern integral calculus is that this idea remains encapsulated within a proof by contradiction, without ever explicitly naming the limit or passing to the limit.
The Blois leaf, with its propositions 39 to 41, deals with intermediate results within this same edifice — geometric lemmas on spherical segments that make it possible to reach the general results. To read these propositions in the original manuscript is to see the foundations of a line of reasoning that waited nineteen centuries before being formalized as differential and integral calculus.
