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, i.e. to subtend an angle of about half a degree; and he describes a simple instrument by which he himself found that the angle subtended by the diameter of the sun at the time when it had just risen was less than 1⁄164th part and greater than 1⁄200th part of a right angle. Taking this as the size of the angle subtended at the eye of the observer on the surface of the earth, he works out, by an interesting geometrical proposition, the size of the angle subtended at the centre of the earth, which he finds to be > 1⁄203rd part of a right angle. Consequently the diameter of the sun is greater than the side of a regular polygon of 812 sides inscribed in a great circle of the so-called “universe,” and a fortiori greater than the side of a regular chiliagon (polygon of 1000 sides) inscribed in that circle.

On these assumptions, and seeing that the perimeter of a regular chiliagon (as of any other regular polygon of more than six sides) inscribed in a circle is more than 3 times the length of the diameter of the circle, it easily follows that, while the diameter of the earth is less than 1,000,000 stadia, the diameter of the so-called “universe” is less than 10,000 times the diameter of the earth, and therefore less than 10,000,000,000 stadia.