that the centre (a mathematical point) can have no ratio whatever to the surface of the sphere, and that we must therefore take Aristarchus to mean that the size of the earth is to that of the so-called “universe” as the size of the so-called “universe” is to that of the real universe in the new sense.

Next, he has to assume certain dimensions for the earth, the moon and the sun, and to estimate the angle

subtended at the centre of the earth by the sun’s diameter; and in each case he has to exaggerate the probable figures so as to be on the safe side. While therefore (he says) some have tried to prove that the perimeter of the earth is 300,000 stadia (Eratosthenes, his contemporary, made it 252,000 stadia, say 24,662 miles, giving a diameter of about 7,850 miles), he will assume it to be ten times as great or 3,000,000 stadia. The diameter of the earth, he continues, is greater than that of the moon and that of the sun is greater than that of the earth.

Of the diameter of the sun he observes that Eudoxus had declared it to be nine times that of the moon, and his own father, Phidias, had made it twelve times, while Aristarchus had tried to prove that the diameter of the sun is greater than eighteen times but less than twenty times the diameter of the moon (this was in the treatise of Aristarchus On the Sizes and Distances of the Sun and Moon, which is still extant, and is an admirable piece of geometry, proving rigorously, on the basis of certain assumptions, the result stated). Archimedes again intends to be on the safe side, so he takes the diameter of the sun to be thirty times that of the moon and not greater.

Lastly, he says that Aristarchus discovered that the diameter of the sun appeared to be about 1⁄720th part of the zodiac circle,