The Rediscovery of the Archimedes Palimpsest and His Use of the Infinite

One of the first steps towards the development of the infinitesimal calculus in Europe during the seventeenth century was undertaken by Johannes Kepler (1571-1630). With the help of one of my subscribers, I was able to get a copy of Eberhard Knobloch’s recent English-language translation of Kepler’s Nova Stereometria Doliorum Vinariorum [New Solid Geometry of Wine Barrels¹]. This is how the Preface begins:

In 1615 Johannes Kepler published his Latin treatise New Solid Geometry of Wine Barrels. It is his most important mathematical publication, which made him a precursor of infinitesimal mathematics. As such it already attracted the interest of other mathematicians during Kepler's lifetime. His new interpretation of Archimedes’s results in plane and solid geometry was based on indivisibles considered as infinitely small quantities. He could not know that his famous Greek predecessor had used the same method in his letter to Eratosthenes, known as Approach related to Mechanical theorems rediscovered by Johan Ludvig Heiberg in 1906. [p.7, my emphasis]

So I thought to myself that I should familiarise myself with the mathematical works of Archimedes (c.287-212 BCE), and perhaps also of Apollonius of Perga (c.240-c.190 BCE), before I started to write about seventeenth-century mathematics.

It did not take long for me to come across a fascinating book entitled The Archimedes Codex: How a Medieval Prayer Book Is Revealing the True Genius of Antiquity’s Greatest Scientist², by Reviel Netz and William Noel, which relates the rediscovery and careful study of the Archimedes Palimpsest that Heiberg (1854-1928) had examined in Istanbul in 1906. This project was led by Noel, at the time Curator of Manuscripts and Rare Books at the Walters Art Museum in Baltimore, who passed away in April 2024 after being struck by a van in Edinburgh; as can be seen in this obituary, he had become a champion of open access to historical documents.

According to the Oxford dictionary, a palimpsest is “A parchment or other writing surface on which the original text has been effaced or partially erased, and then overwritten by another”. In this case, the Archimedes Palimpsest is a codex, i.e., a collection of parchments bound with wooden covers, of Archimedes’s writings, prepared in the late tenth century. Some two and half centuries later, it was taken apart, the bifolios, i.e., single sheets of double the size of a folio, were cleaned of most of the original text, cut down the middle, turned ninety degrees, and written upon again to form a prayer codex whose sheets were half the size of the original, and then rebound. As a result, the binding of the new codex covered the middle three lines or so of each of the original pages containing the works of Archimedes. The new codex did not use all of the Archimedes codex, and also used parchments from other codices.

The Palimpsest disappeared from the public eye after Heiberg examined and photographed it in 1906, and reappeared, in a pitiful state, in a Christie’s auction in London in 1998. It was sold to an anonymous bidder for the sum of two million US dollars. The Netz-Noel book mentioned above relates the careful taking apart and study of the Palimpsest, over a period of ten years, by numerous scholars and experts from different disciplines, which ultimately revealed many unknown aspects of Archimedes’s work.

The Palimpsest can now be viewed in a beautiful two-volume set entitled The Archimedes Palimpsest³⁴ published by Cambridge University Press, complete with the transcriptions in Greek. The review in the Times Literary Supplement was stunning:

Cambridge University Press have surpassed themselves: these are, quite simply, among the loveliest printed books produced this century. Both volumes contain hundreds of photographs, all in breathtaking full colour; the type is elegant and spacious, and the page proportions of the layout are beautifully harmonious. It is a joy that serious books of such extraordinary art can still be produced.

A paragraph from the press release from the Walters Art Museum, gives an idea of the efforts undertaken to decipher the original text of the parchments:

Since 1999, intense efforts have been made to retrieve the Archimedes text. Many techniques have been employed. Multispectral imaging, undertaken by researchers at the Rochester Institute of Technology and Johns Hopkins University, has been successful in retrieving about 80% of the text. More recently the project has focused on experimental techniques to retrieve the remaining 20%. One of the most successful of these techniques has proved to be x-ray fl[u]orescence imaging (XRF). In April 2005, at the EDAX company in New Jersey, XRF was able to reveal the iron in the ink of folio 81r of the Archimedes Palimpsest. This was the first image that allowed scholars to read Archimedes’ text underneath a 20th-century forged icon. This month, using one of the most powerful sources of focused electro-magnetic radiation in the world, the Synchrotron Radiation Laboratory, which is part of the Stanford Linear Accelerator Center (SLAC) in California, used a synchrotron x-ray beam to continuously scan the parchment of folio 81r. This has enabled scholars to read large sections of previously hidden text.

But before that imaging work, the codex had to be taken apart, page by page, despite the fact that many pages were damaged by mold, and wood glue had been used to bind them together sometime during the twentieth century. This painstaking work was undertaken by Abigail Quandt, Conservator at the Walters Art Museum, and took a full four years!

The principal scholars reading the text were Reviel Netz, who currently holds the Suppes Professorship in Greek Mathematics and Astronomy (Professor of Classics and by courtesy of Philosophy), Stanford University, and Nigel Wilson, now emeritus fellow and tutor in Classics, Lincoln College, Oxford.

Netz is currently undertaking the translation into English of all of the extant Archimedes works which are available to us from Greek-language sources. Two volumes are already printed by Cambridge University Press: Volume I covers On the Sphere and the Cylinder⁵ and Volume II covers On Spirals⁶. Volume III is forthcoming, and will cover the works that are most affected by the discoveries in the Palimpsest, including the Method, referred to above in Knobloch’s Preface as the Approach related to Mechanical theorems.

Chapter 8 of the Netz-Noel book is written by Netz. It is entitled “Archimedes’ Method, 2001, or Infinity Unveiled”. Therein, Netz recounts how he and Ken Saito, now researcher at the Takakazu Seki Mathematical Research Institute, Yokkaichi University (near Nagoya, Japan), try to read a bifolio from the Method that Heiberg had not been able to decipher. They figure out while reading under ultraviolet light that the word megethos [magnitude] is being used by Archimedes. A month later, once proper scans have been produced, Netz is able to determine that the text is peppered with the words isos plethei [equal in magnitude]. Netz writes:

The expression “equal in multitude” is used in Greek mathematics when discussing the numbers of objects in two separate sets. Suppose I have a set of three triangles and a separate set of three lines. A Greek mathematician would say that the two sets are “equal in multitude,” meaning that they are each made up of three objects. [p.201]

But Archimedes is using this expression to compare two infinite sets, something that no modern researcher believed had been undertaken in Greek mathematics:

He was saying that, with the infinitely many slices produced in the cube, after all the random slices were made and the cube was entirely cut, the triangles, produced in the cube by all the possible random cuts, were “equal in multitude” to the lines in the rectangle. You see? In each random slice there was a triangle in the cube, standing on top of a line in the rectangle. And Archimedes pointed out that the number of triangles of which the prism was made was the same as the number of lines of which the rectangle was made. Surely he meant this to be verified by the fact that there was a one-to-one relationship. Each triangle stood on an individually separate line and each line was at the bottom of an individually separate triangle. [p.201, my emphasis]

Netz makes all of this clear: Archimedes was working with real, existing, infinite sets, and comparing them in the same manner that we do today.

Only, of course, those equalities of number were like nothing else we ever knew from Greek mathematics. The objects Archimedes counted—the sets of triangles and lines—were all infinite. Archimedes was explicitly calculating with infinitely great numbers.

More than this, Archimedes was making his calculations based on a sound principle. He apparently was stating that this infinite set was equal to that infinite set, because there was a one-to-one relationship between the two sets. He did not say so in so many words, but Archimedes was never an explicit author. He always left much of the work for the reader. [p.202, my emphasis]

Netz finishes with the following astounding conclusions:

First, we find that Archimedes did not merely make an “implicit” move from a random slice to the object made up of those random slices. He relied instead on certain principles of summation. This means that he was already making a step toward the modern calculus and was not merely anticipating it in some naive way.

Second, we find that Archimedes calculated with actual infinities in direct opposition to everything historians of mathematics have always believed about their discipline. Actual infinities were known already to the ancient Greeks.

Third, we see that with this concept of infinity—as with so many others—the genius of Archimedes pointed the way toward the achievements of modern science itself. Back in the third century BC, at Syracuse, Archimedes foresaw a glimpse of Set Theory, the product of the mature mathematics of the late nineteenth century. [pp.202-203, my emphasis]

And here we see the importance of the comment made by Knobloch at the beginning of his Preface. I look forward to reading more about this topic, and most especially to the appearance of Netz’s third volume of The Works of Archimedes, so that I can read the translation of the Method.

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1. Johannes Kepler. Nova Stereometria Doliorum Vinariorum. New Solid Geometry of Wine Barrels. Accessit Stereometriæ Archimedeæ Supplementum. A Supplement to the Archimedean Solid Geometry Has Been Added. Edited and translated, with an Introduction, by Eberhard Knobloch. Paris: Les Belles Lettres, 2018.

2. Reviel Netz and William Noel. The Archimedes Codex: How a Medieval Prayer Book Is Revealing the True Genius of Antiquity’s Greatest Scientist. Philadelphia: Da Capo Press, 2007.

3. Reviel Netz, William Noel, Nigel Wilson and Natalie Tchernetska, editors. The Archimedes Palimpsest. Volume I: Catalogue and Commentary. Cambridge University Press, 2011.

4. Reviel Netz, William Noel, Natalie Tchernetska and Nigel Wilson, editors. The Archimedes Palimpsest. Volume II: Images and Transcriptions. Cambridge University Press, 2011.

5. Reviel Netz. The Works of Archimedes. Volume I: The Two Books On the Sphere and the Cylinder. Translated into English, together with Eutocius’ commentaries, with commentary, and critical edition of the diagrams. Cambridge University Press, 2004.

6. Reviel Netz. The Works of Archimedes. Volume II: On Spirals. Translated into English, with commentary, and critical edition of the diagrams. Cambridge University Press, 2017.