Did Euclid prove that $\pi$ is constant?

Pi is defined the ratio of the circumference of a circle to its diameter, but of course different circles have different circumferences and diameters, so in order for it to be well-defined we need to show that the ratios for any two circles is the same. This is fairly trivial if we approximate the circles by regular n-gons and take the limit as n goes to infinity, what the ancients called Eudoxus' method of exhaustion. Archimedes used this method with great success, in finding the circumference and area of a circle, the volume and surface area of a sphere, area bounded by a parabola, etc.

My question is, did Euclid ever prove that Pi is constant in his Elements? In Book XII Proposition II, he proves that the ratio of the area of a circle to the square of its diameter is the same for all circles, but does he ever prove that the ratio of a circumference of a circle to its diameter is the same for all circles? In Book VI Proposition 33 he proves that for two circles of equal diameter, the length of an arc is proportional to the angle that subtends it, but does he ever relate the lengths of arcs on unequal circles to each other?

Even if Euclid didn't prove this result, is it at least an easy corollary of something he did prove?

Any help would be greatly appreciated.

Thank You in Advance.

EDIT: In this thread on MathOverflow, it's claimed that the result follows immediately from Book III Proposition 34 and Book VI Proposition 33, but I don't see how it follows at all. As I said above, Book VI Proposition 33 is about arc lengths for circles of equal diameter, so how do you get from that to a result about arc lengths for circles of unequal diameter? Book III Proposition 34, which is just about transferring angles from one circle to another, doesn't seem like it would suffice.

EDIT 2: I think there's a proposition that the result is even more likely to follow from than Book VI Proposition 33: Book III Proposition 27, which says that equal arcs on equal circles correspond to equal angles. Is there any way to use that proposition to prove that arcs on two UNequal circles corresponding to equal angles are proportional to the diameters of the circles? That is to say, if S1 and S2 are arcs subtended by equal angles on circles of diameter D1 and D2 respectively, then S1/S2 = D1/D2.

EDIT 3: I should make clear that Euclid may not have viewed "the ratio of a circumference of a circle to its diameter" as meaningful, but I think he would have found meaningful the statement I gave in my previous edit: if S1 and S2 are arcs subtended by equal angles on circles of diameter D1 and D2 respectively, then the ratio of S1 to S2 is equal to the ratio of D1 to D2. I should also mention that Euclid's definition of the equality of two ratios is Eudoxus' theory of proportion, a precursor to the Dedekind cut construction of the real numbers.

EDIT 4: It occurs to me that just as Euclid believed that a straight line and a circular arc were not magnitudes of the same kind, he may have believed the same thing about circular arcs in circles of unequal diameter, i.e. he may have thought that it's meaningless to ask whether a circular arc from one circle is longer or shorter than a circular arc from another circle if the circles don't have the same diameter. Can anyone confirm or deny this?

Solution 1:

"This is fairly trivial if we approximate the circles by regular $n$-gons and take the limit as n goes to infinity." You describe one of the most sophisticated techniques in ancient Greek mathematics. It was most assuredly not fairly trivial.

Study the Elements closer and you will learn Euclid's style. Ancient Greek geometry had a completely different view of mathematics, from the foundations up. Any of the ancient Greeks would find the modern notion of a limit absurd. They would reject it on the grounds of a completed infinity, and it would never appear in any of their polished works. I'm not saying I side with them; but that's what they'd say.

The answer, though anticlimactic, is simple: The question you ask is totally alien to the language and spirit of the Elements. In Euclid's work, $\pi$ does not yet appear. Sure you can tease it out using modern interpretations. But overall, Euclid is far more interested in the special ratios involved in constructing structures, rather than describing them.

Studying the history of mathematics, the unprepared student is particularly prone to hindsight bias. Sadly so are many of the popular expositors of the subject!

Solution 2:

In XI.2 Euclid proved that circles are proportional to squares on their diameters. It follows from alternation (V.14, something Euclid used in XII.2) that the ratio of a circle to the square on its diameter is constant, and so is the ratio of a circle to the square on its radius a constant.

That constant is the same constant that you're calling $\pi$.

Euclid did not show that the ratio of a circle to the square on its radius is equal to the ratio of the circumference of a circle to its diameter. Archimedes did that.

That similar figures, and circles in particular, are proportional to squares on their corresponding parts was a general principle that predates Euclid by over a thousand years. Euclid proved that for triangles and polygons in Book VI, and for circles in Book XII.