Archimedes and the Area of a Circle

In this post, we claim that the method of exhaustion developed by Archimedes to calculate the area of a circle is in fact sufficiently rigorous.

Before the formulation of the Riemann integral and modern integral calculus, Archimedes developed an argument for the area of a circle being by using the method of exhaustion. This method of reasoning relies on proof by contradiction: showing that the area couldn’t possibly be greater than or less than , thus it must be equal. This is in turn done by showing that successive approximations to the area of the circle get within any positive difference from .

However, this last part is really just a restatement of the delta-epsilon definition! Specifically, it is the “limit of a sequence” () definition for the limit of the areas of the successive approximations equaling the area of the circle. In fact, this example could be seen as motivation for the eventual rigorous definition of a limit. Thus, it is appropriate that the method of exhaustion be seen as a direct precursor to modern rigorous calculus, and indeed it standardly is, as stated in this Wikipedia article.

Let’s further compare the argument made by Archimedes to a modern derivation via say a Riemann integral. Here, the article claims that Archimedes’ argument is not rigorous by modern standards, since certain assumptions were made in Archimedes’ proof that were taken to be “geometrically evident” but are actually not so evident.

Now, as a consequence of my post on the definition of mathematical rigor, ultimately math must rest on a set of assertions that are “physically proven” and/or “geometrically proven.” Thus, the claim here is that some of the steps done in Archimedes’ derivation, like comparing the arc lengths of circles to segment lengths, are not “geometrically justifiable.” Instead, today, arc length is defined via a suitable integral — specifically, we define the arc length to be the limit of sums of lengths of segments approximating the arc length. Then, we use integral calculus to justify formulas involving the circumference and area of the circle.

However, I would argue that, philosophically, there is actually no change in rigor here. (Note that a discussion of this type must necessarily be philosophical, not mathematical or based on formal logic.) Specifically, though arc length in general should be defined via a limit/integral procedure as is standardly done today (in order to account for all sorts of curves which could potentially be wonky), intuitively it does not seem necessary to do so for a curve as familiar as the circle, and it does in fact seem geometrically evident to me to compare the lengths of curves as Archimedes and Euclid did (see this writeup.) Furthermore, from an education angle, when I was in school we never saw any sort of reasoning to justify the formula for the area of a circle, even though there was never any confusion about comparing the lengths or areas of certain types of “familiar” regions, like those related to the circle. I would have loved to see a proof of the area of a circle by the method of exhaustion then, as opposed to waiting until calculus to see the justification of that formula.

Thus, in my mathematical framework, I’d consider either derivation, whether via integral calculus as is done in modern times or via the method of exhaustion, to be sufficiently rigorous.