How do we know $\pi$ is a constant?

Question

How did the ancient Greeks discover that the ratio of a circle's circumference to its diameter is constant? It does not seem so intuitive. Thanks!

I don't know the answer [hence why I'm posting this as a comment], but I suspect that it wasn't proved rigorously [at least, not at first]. Rather, I suspect they'll have tested it on loads of circles and thought 'wow, this is, like, the same each time, lol'. But I'm open to being corrected. — Clive Newstead, Apr 16 '13 at 20:31
It seems perfectly obvious to me. Like so many important discoveries $-$ Darwin's theory of evolution, for instance $-$ it only needed to be stated. — TonyK, Apr 16 '13 at 20:31
http://math.stackexchange.com/questions/3198/proof-that-pi-is-constant-the-same-for-all-circles-without-using-limits?rq=1 — Amzoti, Apr 16 '13 at 20:31
How do you know the ratio of a square's circonference to its side is 4? — Raskolnikov, Apr 16 '13 at 20:36
It is (and was!) obvious. Archimedes gave what is in essence a formal proof. — André Nicolas, Apr 16 '13 at 20:39
@CliveNewstead The classics would be so much more entertaining if they were interspersed with "lol" and "omg" :) — rschwieb, Apr 16 '13 at 20:56
The idea is that:
Similar polygons inscribed in circles are proportional to the squares on the diameters of the circles. By approximating circles closely by similar polygons, the proportion is carried over to the circles.

This is the way that it was proved by Euclid. — Ambesh, Apr 16 '13 at 21:22

score 1 · Accepted Answer · answered Apr 16 '13 at 20:33

1

Probably they used an intuitive (or maybe exact) idea of geometric similarity.

answered Apr 16 '13 at 20:33

Boris Novikov

1 Answers1