TITLE: Why are all circles similar? (Why is $\pi$ a constant?)
QUESTION [9 upvotes]: I just know that I'm going to look like a crackpot, but here goes.
The number $\pi$ is defined as the ratio of the circumference of a circle to its diameter. So there is an assumption here that all circles are similar. Using undergraduate calculus (Moise), I can analytically convince myself that this is true. But how would a 5th century geometer prove this synthetically?

REPLY [3 votes]: They're not!!
Circles have circumferences which depend on the radius, except if you happen to be in Euclidean space (a space with zero curvature). For instance, in hyperbolic space:
$$\frac{C}{r}=2\pi \frac{\sinh r}{r}\neq 2\pi$$
So really, the question is how we know (or the ancients knew) that in Euclidean space the ratio is constant. This can be seen by observing triangles, and noting that as long as the angles are the same on a plane, the side lengths scale linearly. Subdividing a circle into many small triangles, shows that this logic then must apply to circles as well - if you increase the radius, the circumference must scale by a constant factor, which happens to be $2\pi$.