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)?
Or equivalently:
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).
Best Answer
I suggest the article A Circular Argument (Fred Richman, The College Mathematics Journal Vol. 24, No. 2 (Mar., 1993), pp. 160-162.) It may be relevant to your questions. It suggests that (a variant of) the limit $\lim_{x\to 0}\frac{\sin{x}}{x}=1$ is important to the area result of Archimedes which you mention and that the reasoning may be ... circular. Here is: a freely available version.
revised version I think that it is a bit subtle. The right question might be: Who first treated the question as one which could make sense. The answer to that is probably Archimedes. Once you have that (in an acceptably defined way) the result may not be that hard.
Consider first questions simply of inequalities. If a circle is inscribed in a square the Euclid would agree that the area of the circle is less than that of the square because the whole is greater than the part. But Euclid never says that the perimeter is greater than the circumference because they are different kinds of things. Mark Saphir notes that in Book VI Proposition 33, Euclid proves that in circles of equal radii the lengths of two arcs are in equal proportion to the (central) angles cutting them off. Just sticking to one circle for now with center $O$ we understand what it would mean to say that $\angle AOB < \angle COD$ or that $\stackrel{\frown}{AB} < \stackrel{\frown}{CD}$ and also what it would mean to say that one is twice the other. And hence we have that proposition: $\frac{\angle AOB}{\angle COD}=\frac{\stackrel{\frown}{AB}}{\stackrel{\frown}{CD}}$ (But $\frac{\angle AOB}{\stackrel{\frown}{AB}}=\frac{\angle COD}{\stackrel{\frown}{CD}}$ would not make sense.) Again, Euclid could describe the situation that the radius of one circle is twice that of another. And would even agree that the area of the second is four times that of the first. However he would not say that the circumference of the second was larger than that of the first (let alone twice as much.)
Archimedes introduces the concept of concavity and the postulate:
This is intuitive (as befits a postulate) but is not obvious. With this in hand he can say that for a circle of diameter d, the circumference C is something such that
p<C<Pwhere p and P are the perimeters of polygons (of some number of sides, he used 96) inscribed and circumscribed about a fixed circle. If this is granted thenp/d < C/d < P/dand, because we know the bounds are independent of d (thanks to similarity of polygons), we have that his bounds are independent. Implicitly, letting the number of sides increase, we have that C/d must be similarly independent.Here we see the idea of arc length (for convex curves) as the limit of the length of inscribed polygonal paths (or perhaps the common limit, if it can be demonstrated, of inscribed and tangential paths.)