Plato's “House”

“The rectangle ABCD has an area exceeding that of the circle by less than 1½ pro mille”

It is a curious fact that √2 + √3 very nearly approximates π. (Cp. E. Borel, Space and Time, 1926, 1960, p. 216 ... ) The excess is less than 0.0047, i.e. less than 1½ pro mille of π, and a better approximation to π was hardly known at the time. A kind of explanation of this curious fact is that the arithmetical mean of the areas of the circumscribed hexagon and the inscribed octagon is a good approximation of the area of the circle. Now it appears, on the one hand, that Bryson operated with the means of circumscribed and inscribed polygons ... and we know, on the other hand (from the Greater Hippias), that Plato was interested in the adding of irrationals, so that he must have added √2 + √3. There are thus two ways by which Plato may have found out the approximate equation √2 + √3 ≈ π, and the second of these ways seems almost inescapable. It seems a plausible hypothesis that Plato knew of this equation, but was unable to prove whether or not it was a strict equality or only an approximation.

Plato’s Elementary Square, composed of four sub-elementary isosceles rectangular triangles	Plato’s Elementary Equilateral, composed of six sub-elementary scalene rectangular triangles

pp. 252-253

Karl Popper - The Open Society and Its Enemies