The Archimedes Palimpsest

The Diagrams as Floating Bodies
by Reviel Netz of Stanford University

The Archimedes Palimpsest is the closest we get to the hand of Archimedes. It is not his handwriting, indeed he would be unable to read it. Greek letters have changed so dramatically at around the Eighth century A.D. that an ancient author, such as Archimedes, would no longer be able to recognize his own words. As far as the text is concerned, then, the Palimpsest provides us with the content, not the form: we know what Archimedes has written down, but not how he written it down. The case may be different with the diagrams. It is completely feasible that the diagrams we possess in the Palimpsest are essentially the same as those drawn by Archimedes in the Third century BC. This would have more than a mere antiquarian interest, i.e. this is more than just satisfying our curiousity concerning Archimedes' writing. Instead, knowledge of Archimedes' diagrams would be crucial to our understanding of his scientific personality. This is because in Greek mathematics, unlike modern mathematics, the diagrams are much more than mere illustrations. Rather, Greek diagrams serve as a crucial part of the logic of the argument. Ancient mathematicians were mostly geometers, studying drawn forms. They thought in diagrams. The Archimedes Palimpsest, by bringing us closest to Archimedes' hand, also brings us closest to his mind.

How do we tell that the diagrams in the Palimpsest are, as it were, "originals"? The short answer is, we cannot tell for sure. It would be much easier to argue that diagrams are not originals. We could do this by comparing the Palimpsest to other surviving manuscripts. If the diagrams at the various manuscripts are radically different, this would raise the suspicion that medieval scribes have ignored the diagrams present in their originals, and have instead simply drawn their own diagrams. In fact, this is – somewhat shockingly – how modern editors and translators have systematically proceeded. That is, Heiberg not only did not attempt to draw his diagrams based on manuscript authority, but also he did not report how the diagrams in his manuscripts look like. While his edition from 1915 furnishes us with important information concerning the text contained in the Palimpsest, we had no information at all concerning the diagrams contained in the Palimpsest, prior to the resurfacing of the manuscript in 1998.

However, medieval scribes were not like that. For all the treatises preserved both in the Palimpsest as well as in the remaining manuscript traditions, we can see that the diagrams are in fact very closely allied.

They "share DNA". The medieval scribes did not re-invent their diagrams, but instead copied them, as best as they could, from their sources. The reason for this is in fact not hard to come by: medieval scribes, unlike modern editors such as Heiberg, did not understand any mathematics and certainly did not follow the contents of Archimedes' text (for which the best evidence are the great many errors concerning labeling of figures in both text and diagrams). To the ignorant, there is safety in faithfulness.

The medieval scribe could not re-work Archimedes: instead, he has copied him. We should be grateful for that.

The one major exception appears to have been William of Moerbeke. His Latin translation of the works of Archimedes, dating from 1269, is the most significant effort by any Medieval European author to have come to terms with the science of Archimedes. Moerbeke did not always succeed (some of his translation betrays elementary errors), but he tried hard and usually creditably. Not surprisingly, his diagrams tend to differ from those preserved elsewhere: as part of his translation and his general effort to re-work his material, he has also re-drawn his diagrams in relative freedom from the sources he had.

This is most important for the case of Floating Bodies. For this work, two sources survive: the Latin translation by Moerbeke – as well as the Palimpsest. To the extent that the two differ, we should assume that the Palimpsest provides us with the more accurate source to the original form of the ancient diagrams. In point of fact, the manuscript diagrams often differ between the two traditions. While scholars have yet to study the detail of such discrepancies so as to discover their precise meaning, the fundamental issue seems to be clear. In this treatise, Archimedes considers several hydrostatic configurations, with solids of different shapes and weights immersed into water in different arrangements.

The question is throughout: given the precise configuration, will the figure stay put, or will it collapse through the combined action of gravity and buoyancy? Archimedes specifies, textually, the key components of the configuration, but some remain to be decided by the diagram (e.g., is the object immersed face-down, or face-up?). Whenever the Palimpsest and the Moerbeke manuscript differ on such points, then, we have a substantial piece of the science of hydrostatics, riding on our interpretation of the manuscript tradition. Without the Palimpsest, we could not even address such questions. Thanks to the Palimpsest, such questions are likely to be resolved over the next few years, as the diagrams are edited and studied.