On 1 May 2008 at 10:41, Marc LeBrun wrote:
=Bernie Cosell So it *is* likely that [Archimedes] _did_ know that there was this single, strange and magical constant that underlay lots of things in geometry,
Well... I'd advise being very careful about jumping to modern-colored conclusions about how earlier folks thought about things.
For example you might first want to establish that Archimedes even *had* an idea of "a constant" in anything close to the modern sense.
I would guess that he did. I would have thought that he would think that *ALL* spheres and cylinders had volumes in the same ratio. That ALL circles had the same ratio of radius to circumference.
For ancients I suspect "PI" might seem inescapably to just be an abbreviation for "the proportion between a circle's circumference and its diameter" and the idea of trying to separate it out so that "PI" somehow had an existence on its own independently from a relationship might have seemed weird.
Right, but I think they would have noticed that this OTHER thing happened to have the "same proportion". They may not have had a bent to use the abstraction-machinery we do [to instantly give it a greek letter..:o) And hey: If Archimedes WERE inclined to do that, what alphabet would he have used for his constants...:o)] but I think it is pretty clear that they'd recogize "similar proportions". (e.g., that the ratio of the circumference to the radius of a circle is just twice the ratio of the area of the circle to the square of it radius). *THEN* you get the millennium-and-a-half hiatus until the proportions are abstracted and anlyzed on their own...
...Moreover, they had no reason to assume that there might not be some as-yet-undiscovered tombstone-worthy nice relationship between, say, diagonals and circumferences (that is, in modern terms, between "PI" and "sqrt(2)").
Just so!! That'll take that millennium-and-a-half before THAT kind of thing gets resolved..:o). But I still think, having now seen what Archimedes was able to do and figure out, that he would have recognized that the *same* proportionality showed up in several places. I don't need to truly nail it down but I was just trying to get a feel for where things stood until we see an actual proof that Pi is irrational... /Bernie\ -- Bernie Cosell Fantasy Farm Fibers mailto:bernie@fantasyfarm.com Pearisburg, VA --> Too many people, too few sheep <--