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Atranslation is given in Heath (1913).

Further on in the work Aristarchus assumed that the angular diameter of the Sun is the same as that of the Moon, i.e. 2°. Archimedes claimed that Aristarchus later discovered the much more accurate value of 1/2° for the angular diameter of the Sun, in agreement with his own observations (see Shapiro (1975)). It is likely that Aristarchus' figure of 2° for the Moon's angular diameter was not based on measurements at all, but simply assumed for the purposes of his demonstration (see van Helden (1985), p. 8).

Moon

Moon

Fig. 2.7. Aristarchus' method for determining the relative distances from the Earth of the Sun and Moon.

Using the first four of these hypotheses, which are illustrated in Figure 2.7, Aristarchus concluded that the ratio of the distance from the Earth to the Sun, dS, to that of the moon, dM, satisfies the inequality

The derivation of the value of dS/dM from Figure 2.7 would now be a simple exercise in trigonometry (dM/dS = cos 87° = sin 3°) but this had not yet been invented. The reason for the inequality that results from Aristarchus' calculations has nothing to do with an estimate of the accuracy of the experimental data, but instead represents the accuracy with which Aristarchus could determine what we would now write as sin3°. Here, we will demonstrate his argument for showing that dS /dM > 18, which is particularly elegant. In Figure 2.8, E, S and M represent the Earth, Sun and Moon with /ESM = 3°. The line EA is a continuation of ME such that | AM | = | MS | , and SC bisects the angle MSA. The line BC is perpendicular to AS so that | BC| = |CM|. Aristarchus began by noting that

from which it follows, using the equivalent of the fact that tan a/ tan j > a/j if j < a < 90° (a result which was well known in Aristarchus' time; a proof can be found in Euclid's Optics), that

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