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8. An Original and Persuasive Method
- Baylor University Press
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69 AN ORIGINAL AND PERSUASIVE METHOD VIII Archimedes’ short work entitled The Method on Mechanical Theorems has had a stormy history. Lost for many centuries , it was rediscovered by chance in a manuscript that arrived in Istanbul from Jerusalem’s Monastery of the Holy Sepulcher in 1906. The work was quickly translated into German and published in 1907 by its discoverer, the Danish philologist Johan Ludvig Heiberg. In that same year a large commentary on Archimedes’ work appeared, written by Hieronymus Georg Zeuthen, a historian of mathematics. Just after the First World War, however, the codex disappeared, probably stolen from Munich during a transfer of manuscripts from the Istanbul headquarters of the Orthodox Patriarchate of Jerusalem at the National Library of Athens. Fortunately, this manuscript was rediscovered at the end of the last century; it was put to auction on 29 October 1998 at Christie’s in New York, fetching the remarkable figure of $2,150,000. The new owner, an 70 THE GREAT ARCHIMEDES VIII American billionaire, has entrusted the document for safekeeping to the Walters Art Museum in Baltimore, while 1906 photographs of Heiberg, the manuscript’s discoverer, now faded, are housed in the Royal Library in Copenhagen , Denmark. The Method of Mechanical Theorems is contained in a palimpsest. This document consists of the codex containing Archimedes’ ancient text, copied afresh in the tenth century, which, about two hundred years later, was expunged and rewritten with a series of prayers and Christian texts.1 Aside from the many portions of works already known to have belonged to Archimedes, which it often also manages to integrate—these works include On Floating Bodies, which had been known up to that time only in a medieval Latin translation—the manuscript has enabled us to acquaint ourselves with almost the entire text devoted explicitly to the Method, hitherto lost. This discovery is truly one of the greatest moments in the entire history of mathematics [plate 14]. This brief treatise is more dialogic than the greater part of the scientific treatises of Greek mathematics passed down to us. The work itself is presented as if a letter to Archimedes’ contemporary Eratosthenes, mathematician, astronomer, geographer, and poet, who was at that time director of the Museum at Alexandria: Seeing that you are a diligent and excellent teacher of philosophy, and also given that it is a great thing to appreciate in mathematics the theory that falls to your consideration, I decided to write to you and express in this same book the characteristics of a certain method, through which you will be given the opportunity to address mathematical issues through mechanical means. And I am convinced that this method is no less useful for the demonstration of these purely mathematical theorems . In fact some of the properties that I have first [34.204.181.19] Project MUSE (2024-03-19 04:14 GMT) AN ORIGINAL AND PERSUASIVE METHOD 71 VIII presented by mechanics were later proven to me through geometry, since the research carried out by means of this method is not really a demonstration. For, since this method has already obtained some knowledge of what you seek, it is easier to complete the demonstration after mechanical research than to press on in research without any prior understanding. [Praefatio, Method] In reaching his findings, together with mathematical procedures of the infinitesimal type2 Archimedes uses, in a distinctly original way, arguments about mechanics, which he denies have absolute value. These arguments, however, have value for imagining and producing new discoveries. Thus Archimedes anticipates his mathematical successors: I am confident that some mathematicians present or future, having been shown this method, will find other theorems not yet imagined by us. [Praefatio, Method] The geometric method was used only at a later time to demonstrate rigorously propositions already identified as a mechanically plausible [fig. 24]. The exposition of Archimedes is of great interest, as Lucio Russo has noted, for someone who is trying to communicate not only the proofs of his results but also the mental route that led to them.3 From a philosophical point of view, it is possible to consider the application of mathematics to physics as a necessary corollary, owing to the abstract and general nature of mathematics, but the reverse attribution is far more problematic . For example, one cannot say that the behavior of a lever could influence the natural properties of a line, a triangle, or a parabola. Practically speaking, then, Archimedes’ Method consisted of the application of mechanical arguments to...