## Cm2 Cb225

and, hence, that | CA | /1 CM| > 7/5, which in turn implies that

(Note that here Aristarchus has used the well-known Pythagorean approximation to ^/2, namely 7/5.) Putting Eqns (2.1) and (2.2) togetherwe obtain |AM| _ |AM| |CM| 12 15 _

But |ES| > |MS| = \AM\, and so we arrive finally at the desired result dS | ES| — = --- > 18.

Having determined the relative distances of the Sun and Moon, Aristarchus immediately could deduce the relative sizes of the Sun and Moon since (from the observed 'fact' that the Moon exactly eclipses the Sun) he assumed that they have the same angular diameter. Hence, the radii of the Sun and Moon, rS and rM, for example, satisfy

Aristarchus' geometry based on the bisected Moon is flawless, but the inaccuracies in his observations mean that his results are not. The correct value

Fig. 2.9. A simplified illustration of Aristarchus' method for determining the relative sizes of the Sun and Moon.

of rS/rM is about 400. However, the conclusion that the distance to the Sun is about 19 times greater than the distance to the Moon appears frequently in astronomical writings over the course of the 2000 years following Aristarchus.

Aristarchus then went on to determine the sizes of the Sun and Moon relative to the Earth. His method was fairly complicated, but a simplified version is given

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