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A new reading of Method proposition 14: Preliminary evidence from the Archimedes Palimpsest. I. (English) Zbl 1006.01002

The authors give a first description of the results of their careful study of the recently resurfaced manuscript of Archimedes’ Method. This manucript had been edited carefully by Heiberg 1906-1910. Then it disappeared and, during the intervening time, deteriorated considerably. It reappeared 1998 and its current owner made possible a new investigation of the text. The authors describe the physical appearance of the manuscript in detail.
Modern imaging systems allowed to read some passages that had been virtually invisible to Heiberg. Apart from these few instances, Heiberg’s text proved to be very accurate. The only things he changed were the figures, which he redrew in a modernized form.
One of the cases where new insights can be derived from now readable lines is the proof of Archimedes’ proposition 14. It concerns the volume of a certain part of a cylinder looking like a wedge. The authors present Archimedes’ proof line by line and discuss it carefully. The crucial point is in the way how Archimedes passes from a finite collection of areas and lines to an infinite one. Then he uses some kind of as yet unclear summation to pass from the areas to volumes and from the lines to areas. Various aspects of this procedure are investigated by the authors, especially the possibility of Archimedes using a kind of indivisibles.
From this and some other observations of the text the authors conclude that the overall structure of the Method has to be reconsidered.
The Greek text of the proof of prop. 14 has been added as an appendix. A new publication of the entire manuscript is in preparation.

MSC:

01A20 History of Greek and Roman mathematics
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