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.