Ramble, Part 6: Give Me A Lever...
Nov. 22nd, 2006 09:39 pm![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png)
Archimedes was, I think unquestionably, the greatest mathematician of the classical era - and it was a time when mathematics subsumed astronomy, dynamics, and a variety of other fields. His work on floating bodies, on levers, in number theory, and above all in geometry was first-rate, and in some cases centuries ahead of anyone else. I previously mentioned his remarkable work in computing areas and volumes of curvilinear regions, justified by means of the method of exhaustion. What was lost for two thousand years was the method by which he came up with the values of those magnitudes, before he could attempt justifying them.
In 1910, an old manuscript was discovered in Constantinople. What was written on the parchment was of no great interest, but close examination revealed that it was a palimpsest - that it had been written on, scraped clean, and then rewritten. It eventually became clear that its original content was a letter from Archimedes to Eratosthenes (yes, that Eratosthenes - the man who first computed the circumference of the Earth), in which he described his "Method for Solving Mechanical Problems". The method, remarkably enough, was very close to what we would now call integration, and, almost equally remarkably, it was based on the Lever Law that Archimedes had discovered. This is the fact that two weights on opposite sides of the pivot of a lever will balance if their distances from the pivot are inversely proportional to their weights.
Rather than attempt to describe the method, I will refer you to this site, which demonstrates its application to the computation of the volume of a sphere. (It's hard enough to follow with the helpful diagram; without it, the argument would be well-nigh impossible!) Note that the "piling up" of circles at the point H is, in essence, the computation of a volume by integrating cross-sections (as one learns to do in Calculus II).
Archimedes did not publish his method, except in this letter; since it was based on physical considerations, it did not fit within the rigorous confines of mathematics post-Euclid, and so could not gain footing as anything but a heuristic (and one that, apparently, only Archimedes ever mastered). Furthermore, lurking in the background was yet another limit process, and the Greeks never did get a firm handle on that concept. Mathematical rigor was an invaluable framework, but it may be that it was, in this one case, premature; when similar concepts finally resurfaced in the 17th century AD, it was in a more relaxed atmosphere - but more on that later.
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In 1910, an old manuscript was discovered in Constantinople. What was written on the parchment was of no great interest, but close examination revealed that it was a palimpsest - that it had been written on, scraped clean, and then rewritten. It eventually became clear that its original content was a letter from Archimedes to Eratosthenes (yes, that Eratosthenes - the man who first computed the circumference of the Earth), in which he described his "Method for Solving Mechanical Problems". The method, remarkably enough, was very close to what we would now call integration, and, almost equally remarkably, it was based on the Lever Law that Archimedes had discovered. This is the fact that two weights on opposite sides of the pivot of a lever will balance if their distances from the pivot are inversely proportional to their weights.
Rather than attempt to describe the method, I will refer you to this site, which demonstrates its application to the computation of the volume of a sphere. (It's hard enough to follow with the helpful diagram; without it, the argument would be well-nigh impossible!) Note that the "piling up" of circles at the point H is, in essence, the computation of a volume by integrating cross-sections (as one learns to do in Calculus II).
Archimedes did not publish his method, except in this letter; since it was based on physical considerations, it did not fit within the rigorous confines of mathematics post-Euclid, and so could not gain footing as anything but a heuristic (and one that, apparently, only Archimedes ever mastered). Furthermore, lurking in the background was yet another limit process, and the Greeks never did get a firm handle on that concept. Mathematical rigor was an invaluable framework, but it may be that it was, in this one case, premature; when similar concepts finally resurfaced in the 17th century AD, it was in a more relaxed atmosphere - but more on that later.
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Date: 2006-11-23 05:16 pm (UTC)no subject
Date: 2006-11-27 03:00 am (UTC)