Consider two circles and associated inscribed squares (Figure 1). We know from Euclid XII.1 that the measures of the squares, and for that matter any pair of similar inscribed polygons, are in the same ratio as the duplicate (square) ratio of the respective diameters. Euclid assumes, for the purpose of ultimate contradiction, that the circles are not in that ratio, but are in a ratio either smaller or larger. In Part 1, he assumes that the ratio of circles, EFGH : ABCD, is a smaller ratio, and that makes the area of circle EFGH smaller than "expected." Call its area S.
The squares fall short of their circles' area by a certain amount, but when we change from squares to inscribed regular octagons, more than half of the uncovered area is made up (see this enlarged in Figure 2). This fact satisfies the premise of Euclid X.1 which implies that if we continue this process with both polygons, then they will get as close as we want to the areas of their circles. Remember though, that these polygons are maintaining the duplicate diameter ratio. By X.1 (sometimes refered to as the Axiom of Archimedes) at some point, the area of the polygon in EFGH will surpass S, and give us the required contradiction.
Euclid then argues that the ratio of circles cannot be larger than the duplicate diameter ratio. He changes strategy by using inverse ratios and the facts of Part 1. I will present Heath's alternative argument, which is more in keeping with the line of reasoning used in Part 1.
Here we assume that the measure of one circle, say S', is greater than that needed for the duplicate ratio of the respective diameters. In Part 2 we use circumscribed polygons, beginning with squares, and doubling the number of sides at each step in the "passage to limit." As the circles get "shrink-wrapped" we reach a contradiction when the S' circle's polygon becomes less than S', while simultaneously preserving the expected duplicate ratio (see Figure 3). Euclid's method of proof for this proposition is known as double reductio ad absurdum. The ratio of circles is neither less than nor more than, so it must equal, the duplicate ratio of diameters,
