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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 perhaps saying 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:

If two plane curves C and D with the same endpoints are concave in the same direction, and C is included between D and the straight line joining the endpoints, then the length of C is less than the length D.

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<P where 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 then p/d < C/d < P/d and, 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.

Two notes: Archimedes bases the inequalities (according to Richman) on the principle that

If two plane curves C and D with the same endpoints are concave in the same direction, and C is included between D and the straight line joining the endpoints, then

Here we see the length idea of C is less than the length D.

This is intuitive but not obvious (it seems like a new axiom). We are running into defining 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 pathspaths.)

Archimedes accomplishment may have been in finding the ratio of area to $r^2$ is the same of that of C to d rather than merely that it exists.

It should also be noted that C/d means something numeric to us and something geometric to the classic Greeks. To us it would seem clear that if we had measured C and d in feet and then switched to inches, both would be 12 times as large preserving C/d. That might not persuade them.

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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.

Archimedes is perhaps saying that for a circle of diameter d, p/d < C/d < P/dthe circumference C is something such that p<C<P where 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 then p/d < C/d < P/d and, because we know the bounds are independent of d (thanks to similarity of polygons), we implicitly have that his bounds are independent. Implicitly, letting the number of sides increase, have that C/d must be similarly independent and at any rate have his bounds are independent.

 Two notes: The Archimedes bases the inequalities depend (claims according to Richman) on the assertion principle that a convex curve If two plane curves C and D with the same endpoints AC and enclosed are concave in a triangle ABC has the same direction, and C is included between D and the straight line joining the endpoints, then the length more than AC but of C is less than the sum of the other two sides, this length D. This is intuitive but not obvious (it seems like a new axiom). We are running into defining arc length via limits(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) Archimedes accomplishment may have been in finding the ratio of area to $r^2$ is the same of that of C to d rather than merely that it exists. It should also be noted that C/d means something numeric to us and something geometric to the classic Greeks. To us it would seem clear that if we had measured C and d in feet and then switched to inches, both would be 12 times as large preserving C/d. That might not persuade them. 
 
 
 
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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.

Archimedes is perhaps saying that for a circle of diameter d, p/d < C/d < P/d where p and P are the perimeters of polygons (of some number of sides) inscribed and circumscribed about a fixed circle. If this is granted then, because we know the bounds are independent of d (thanks to similarity of polygons), we implicitly have that C/d must be similarly independent and at any rate have his bounds are independent.

Two notes: The inequalities depend (claims Richman) on the assertion that a convex curve with endpoints AC and enclosed in a triangle ABC has length more than AC but less than the sum of the other two sides, this is intuitive but not obvious. We are running into defining arc length via limits

Archimedes accomplishment may have been in finding the ratio of area to $r^2$ is the same of that of C to d rather than merely that it exists.

It should also be noted that C/d means something numeric to us and something geometric to the classic Greeks. To us it would seem clear that if we had measured C and d in feet and then switched to inches, both would be 12 times as large preserving C/d. That might not persuade them.