=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. Those guys tended to talk (and presumably think) mainly in terms of relations *between* geometrical constructs, integers and the like. (Heck, so did *Newton* et al) 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. For example there is no explicit "PI" apparent in the classical statements of the relation between cylinders and spheres, just as there was no explicit "sqrt(2)" apparent in the proportion of between the edges and diagonals of squares. 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)"). Moreover, I've even heard that *one* wasn't considered "a number" in our sense, because they thought of numbers in terms of "a number of things" and a single thing was just that thing, and not a group of things that could be numbered. That may be hard for us moderns to get our heads around, but apparently there are proofs in eg Euclid that special-case out N=1 from N>1 for this reason. Given that context, I can imagine even the most radical Platonist having problems with "zero" (Dude, you say it's "the number of no things", yet itself a thing?!) let alone apprehend some "strange and magical" PI...