The Archimedes Palimpsest

The prayer book was made from leaves of at least seven older manuscripts: the Archimedes book, a book of Hyperides, a commentary on Aristotle's Categories, two Byzantine liturgical books and two that have not been identified⁠[9]. Three Archimedes leaves seen by Heiberg are missing, the book was damaged by mould, and four full-page paintings of the Evangelists were added over the text in the twentieth century, after 1929⁠[1]. A detached leaf is at Cambridge University Library, and a negative of folio 57 verso survives at the University of Chicago; images of both are included in the digital release⁠[5].

The digital release is organised by folio. For each leaf it gives registered 8-bit TIFF images in each band of light, processed pseudocolour images, registered copies of Heiberg's 1906 photographs where they exist, X-ray fluorescence maps for iron, calcium, zinc and other elements, embedded and XML metadata, MD5 checksums and line-by-line transcriptions mapped to the images in TEI XML⁠[5]⁠[10]. The original 16-bit images are mirrored at the Rochester Institute of Technology⁠[5].

The Archimedes text includes Equilibrium of Planes, Spiral Lines, Measurement of a Circle, Sphere and Cylinder, On Floating Bodies, The Method and the Stomachion⁠[11]. Before the palimpsest was found, the Greek tradition of Archimedes rested on two lost manuscripts, known as Codex A and Codex B, reconstructed from Renaissance copies and from William of Moerbeke's Latin translation of 1269; the palimpsest, Codex C, is the only Archimedes manuscript older than the thirteenth century to survive⁠[7].

The new imaging extended Heiberg's readings. Reviel Netz and colleagues argued from newly read passages of The Method that Archimedes reasoned with collections of infinitely many objects as completed wholes, and that the Stomachion was a study in counting the ways a dissected square can be reassembled rather than a simple puzzle⁠[11]. Natalie Tchernetska identified the Hyperides speeches in 2002, the first direct copy of that orator outside papyri from Egypt, and in 2005 Netz and Nigel Wilson began reading the Categories commentary, which dates from the second or third century AD⁠[8]. A later study proposed that the commentary is by Porphyry and preserves fragments of the earlier commentator Boethus⁠[12].

Imaging was tested leaf by leaf. Multispectral cameras recorded each leaf under narrow bands of ultraviolet, visible and infrared light; combining bands separates the reddish-brown erased ink from the darker prayer-book ink and the parchment, and the combinations were rendered as pseudocolour images for the readers⁠[13]⁠[5]. Where paint covered the text, light could not reach it. In 2003 Uwe Bergmann proposed X-ray fluorescence, because both inks contained iron; trials in 2004 showed iron maps revealed the Archimedes text, and in May 2005 the SSRL beamline produced a whole column of Equilibrium of Planes from under a forged painting in about a day⁠[3]⁠[4]⁠[14].

The transcriptions themselves have been tested by reviewers. Fabio Acerbi's review of the 2011 edition praised the images and conservation but found errors in its treatment of abbreviations, diagrams and the Archimedean tradition, and judged the edition less satisfying for scholarly use⁠[7]. Netz replied in the same journal⁠[15].

On the imaging question there is no live rival: the combination of multispectral imaging and X-ray fluorescence recovered the Archimedes undertext wherever the parchment survives, and the data are open for anyone to check⁠[5]⁠[8]. The open questions are smaller. The first is how far the new readings change what Archimedes did. Netz's claim that The Method shows reasoning with actual infinity, and his combinatorial reading of the Stomachion, are argued from short passages and remain interpretations rather than settled results⁠[11]⁠[7]. The second is what remains unread: two erased manuscripts are unidentified, three Archimedes leaves are lost, and some passages under the gutter, mould or paint remain uncertain⁠[9]⁠[1]. The third is the authorship of the Categories commentary⁠[12].

The residual problems are now problems of reading, not of imaging. Applying newer methods, such as machine-learning separation of undertext from overtext or hyperspectral reimaging, to the released image stacks could test whether the uncertain passages and the two unidentified manuscripts can be read. A search of collections for the three missing Archimedes leaves could find them, as the Cambridge leaf was found. The mathematical readings are settled only by argument among historians of mathematics.