I have an elementary question about the history of $\pi$. I thought the answer would be easy to find. But, to the contrary, after quite a bit of searching and after consulting math historians, I have been unable to find a satisfactory answer.
Who first proved that $C/d$ is independent of the choice of circle ($C$ and $d$ are the circumference and diameter, respectively)?
Who first proved that given two circles with circumferences $C_1$ and $C_2$ and diameters $d_1$ and $d_2$, that $C_1/C_2=d_1/d_2$? (Or, as I imagine Euclid would have written it: the circumferences of circles are to one another as their diameters.)
Most accounts of the history of $\pi$ spend a lot of time talking about how this fact has been "known" for a long time (giving Egyptian, Babylonian, biblical, etc. approximations to the value). But they never say who first proved it. I expected it to be in Euclid's Elements, but was surprised to find that it isn't. Can I take that to mean that it hadn't been proved by then? I would be very surprised if the proof was known to Euclid and he had not included it in Elements.
Note: Euclid does contain Eudoxus's proposition that $A_1/A_2=d_1^2/d_2^2$, where the $A_i$ are the areas of the two circles (Elements XII.2: Circles are to one another as the squares on their diameters.). This implies that the value of $A/d^2$ is independent of the choice of circle.
If we jump ahead a few years from Euclid we find the fact that $C/d$ is constant given implicitly in Archimedes's Measurement of the Circle. First of all, he finds bounds for $C/d$ (it being between $223/71$ and $22/7$). So presumably he knew that it was a constant. But also, it follows logically from his result that $A=rC/2$, where $r$ is the radius of the circle (Archimedes says that the area of a circle is equal to the area of a triangle with height $r$ and base $C$): if we take Eudoxus's proposition as saying $A=kd^2$ (for some constant $k$) and Archimedes's result as $A=dC/4$, then setting them equal we get $kd^2=dC/4$, or equivalently $C/d=4k$ (i.e., $k=\pi/4$).
So, my question is: who first prove this fact? Was it Archimedes? I've read that the version of the Measurement of the Circle that we have may be only a part of what Archimedes actually wrote. Do people conjecture that it was proved and stated explicitly in the missing part of this document?
This all seems very mysterious to me. I would be a little surprised to discover that the answer to this question is lost to history since it is such a major mathematical result (but maybe that is so). I would be surprised if it took until Archimedes to get a proof of this; if it was "known" empirically for the entire Greek period (which I assume it was), one would imagine that a rigorous proof would be highly sought after. One imagines a proof would have been within Eudoxus's reach. Finally, whether the answer the answer to the question is known or not known, I have been very surprised that no one has written about this fact (or at least not that I've found).