22 January 2009

The square root of 3 and mock theta functions. (No, they're not connected.)

Mark Dominus writes about why it really isn't so mysterious that Archimedes had the approximation √3 = 265/153 (which is correct to four decimal places). Apparently historians of mathematics have been mystified by this. Dominus points out that tabulating n2 and 3n2 for the first few hundred integers would be enough. And it might even be enough to go up to 100 or so, observing where n2 and 3m2 are close to each other (which gives an approximation √3 ~ n/m), guess the pattern (it comes from the continued fraction of √3, but Archimedes didn't need to know that), and extrapolate. He suggests that's because the historians themselves weren't so good at arithmetic. Many of these historians date from the late 19th and early 20th century, after when mathematics generally turned more abstract and before computers existed, so it's plausible. If I were a historian I'd have something serious and insightful to say about this.

A more general question: if you're trying to work out the history of mathematics by examining the original sources, how important is it to be a good mathematician? I saw a lecture by George Andrews last week on Ramanujan's lost notebook; he and Bruce Berndt are working on an edited version of it (first volume, second volume, review of first volume in the October 2006 Bulletin of the AMS). Andrews happened to be looking through some papers at the library of Trinity College, Cambridge, when he came across these papers. The manuscript conventionally called "Ramanujan's lost notebook" consists of many pages of formulas and almost no words and is concerned with mock theta function; Andrews claims that he would not have recognized the significance of what he was looking at had he not wrote a PhD thesis on mock theta functions.


Anonymous said...

I wrote the following in an email to Mark this morning:

I have a slightly different reading of Rouse Ball and Heath's puzzlement. I don't read them as asking, "how could he possibly have done this?" but instead, "how did he actually do this?" They're aware of methods of extracting roots, and even aware of ones that Archimedes could plausibly have used. But a more interesting question to an historian is what he *actually* did.

Different methods show hints of different theories, and while the theories might not be worked out in gory detail, they show different patterns of thought about the same problem. Compare the dozens of proofs of the Pythagorean theorem and you'll see a number of different ways of thinking about the situation. Which one a particular author uses tells you something about how he thinks about not only this problem, but mathematics in more generality.

Ancient Greek mathematics was usually very geometrical in spirit. The question I'd be interested in is, "did Archimedes have a geometric approximation in mind when deriving these estimates, and if so, which one?"

Mark Dominus said...

I replied that I don't think that interpretation holds up in conjunction with Rouse Ball's remark that "It would seem...that [Archimedes] had some (at present unknown) method of extracting the square root of numbers approximately."

I'm sitting in Van Pelt right now, and if I gather enough motivation before lunch time I'll see if I can hunt up the original sources. Having the library catalog on my lap doesn't hurt the chances.