Diagrams in the Archimedes Palimpsest

This paper discusses the diagrams in the Archimedes Palimpsest, which provides among other texts the only extant witness to Archimedes’ Method of Mechanical Theorems. I argue that the diagrams should not be attributed wholly to scribal intervention but are an integral part of the text, and that they are not meant to represent mathematical objects faithfully, but to visualize and magnify information in the text.

Latour’s advocation to explore how writing and imaging shapes scientific practices has influenced the recent studies of science and technology. Meanwhile, visual thinking in mathematical reasoning has emerged as a new direction of research in the philosophy of mathematics in the past two decades. Topics explored include the roles of visualization in logic and mathematics, its relationship with mathematical style and discovery, and the significance of pictures in proofs (Mancosu, Giaquinto, Brown, etc.). When it comes to images in works of ancient sciences and mathematics, however, the question of authenticity looms large in the first place. Images in extant manuscripts are likely to be a scribe’s adaptations, or even creations. This is immediately obvious with cases where images are introduced in the scholia. Images in manuscripts can thus be an accumulation of scribal interventions through textual transmission. Distrusting images in manuscripts is a visible practice in modern text editing; many editors and translators dismiss images found on manuscripts and draw new ones by contemporary standards.

The first part of my paper argues that the text of the twelfth proof in Archimedes’ Method requires the exact number of diagrams drawn from the exact perspectives shown in the Archimedes Palimpsest. My argument is not that the diagrams in the palimpsest are entirely faithful to the original, but that the text itself demands the existence of diagrams fulfilling certain restrictions, which are showcased by the diagrams presented in the palimpsest.

In the second part I examine the role of diagrams in the Method, judging from recent digitization and the transcription by Netz and Wilson. Netz et al. 2001 notes that the diagram in Method 14 represents a parabolic segment with straight lines, in contrast to the curved representation of polygons in On the Sphere and Cylinder. I argue that the function of the diagrams is to signal and even magnify points, lines, and planes designated by the text with bare letters. As evidence, I evoke the diagrams in Method 12. The two diagrams provide top and side views of a cylinder inscribed in a rectangular prism. While a line is represented twice in the two diagrams by different letters, Archimedes expresses the identity of the two lines with the language of equality (ἴσην), as if the two representations of the same line become two different lines equal in length. Both the text and the diagrams do not reflect the truth that the two lines are identical, nor are they meant to mislead readers into thinking otherwise, but the collaboration between the text and the diagrams puts under spotlight a line with significant mathematical relations in both diagrams.