Two Problems in Aristarchus's Treatise On the Sizes and Distances of the sun and moon

Abstract

The book of Aristarchus of Samos, On the distances and sizes of the sun and moon, is one of the few pre-Ptolemaic astronomical works that have come down to us in complete or nearly complete form. The simplicity and cleverness of the basic ideas behind the calculations are often obscured in the reading of the treatise by the complexity of the calculations and reasoning. Part of the complexity could be explained by the lack of trigonometry and part by the fact that Aristarchus appears unwilling to make some simplifications that could be simply taken for granted. But an important part of the complexities is due to some unnecessary inconsistencies, as recently discovered by Berggren and Sidoli (Arch Hist Exact Sci 61:213–254, 2007). In the first part of this paper, I will try to show that some of these inconsistencies are just apparent. But the complexity of the calculations and reasoning is not the only reason that could disturb a reader of the treatise. The great inaccuracy—even for the measurement methods and instruments available at those times—of one of the three input values of the treatise is really astonishing. In the sixth and last hypothesis, Aristarchus states that the moon’s apparent size is equal to 2 , while the correct value is one-fourth of that. Some attempts have been made in order to explain such a big value, but all of them have problems. In the second part of this paper, I will propose a new speculative but plausible explanation of the origin of this value.