## Sunday, May 22, 2005

### SLAC deciphers Archimedes

The physicist Uwe Bergmann is gonna use the gadgets at the Stanford Linear Accelerator Center (SLAC) to decipher Archimedes' notes - created in the 2nd century B.C. and copied in the 10th century A.D. - revealing how he used mechanical devices to derive various mathematical theorems.

A monk in the 12th century vandalized Archimedes' notes and used them as a prayer book. Today, the X-rays should make the iron glow and make the text readable.

1. Happy to hear we can help. You can find a detailed chronology of Archimedes Palimpsest as an appendix 2 of http://arxiv.org/abs/physics/0006065
(sorry it is in Spanish; v1 is a previous English presentation of the main text only)

2. Why is the palimsest so important? Well, the whole text and diagrams could be the oldest surviving greek version of Archimedes works. Part of it was incorporated into the standard editon by Heiberg before the new disappearance of the book, but it was a fast work and mainly from photographies. The diagrams, it seems, were not studied in a detailled way, and Heiberg was never able to open fully the book to see inside the spine. But of course the more important surprise is the "Letter about the Method". There, Archimedes avoids exhaustion principle favouring the non-rigorous (then) methods of decomposition in infinitesimal lines. To my taste, there are three important points in this treaty:

a) Proposition 14, which is the main theme of worry between the actual students of the book. The new published reading proposes that it explicitly mentions two infinite quantities as being "equal in magnitude", a move never done explicitly in the mathematics of the Old Age. But proposition 14 is the last extant page of the book, and it is very difficult to read. So imaging it is very important to stablish this result beyond doubt.

b) The introduction, where, about the volume of cone and cylinder, Archimedes says that "no small share of the credit should be given to Democritus, who was the first to make the assertion with regard to the said figure, though without proof". This is to be joined with a small note in Plutarch, Common Notions 39.3 1079 E where a Zeno paradox is recasted by Democritos into a cone slicing procedure [arxiv:math/9904021].

(that the volume of the pyramid can not be calculated by finite slicing, and thus it needs of some infinitesimal procedure, was the theme of Hilbert's Third Problem See also the ancient chinese version.)

c) The lacking part. According the introduction, the lacking part was about a intersection of two cylinders inside a cube. Tom M. Apostol calls this figure a N=4 Archimedan Dome (Amer. Math. M. Jun/Jul 2004) . Now the funny part is that the N=infinity Archimedan Dome is the figure that Cavalieri and Galileo discuss to stablish the infinitesimal calculis (or if you prefer, "atomic calculus", following a suggestion of Galileo to Cavalieri). It is the infamous "Galilean Scudella" in the "Dialogue about two new sciences". So, Apostol mediating, the Old Age slips naturally into the Modern Age of mathematics.

3. It is always good to see ideas manifest from abstract maths to mechanizations.