There is no reason to doubt that in his heliocentric system Aristarchus retained the moon as a satellite of the earth revolving round it as centre; hence even in his system there was one _epicycle_.
The treatise _On sizes and distances_ being the only work of Aristarchus which has survived, it will be fitting to give here a description of its contents and special features.
The style of Aristarchus is thoroughly classical as befits an able geometer intermediate in date between Euclid and Archimedes, and his demonstrations are worked out with the same rigour as those of his predecessor and successor. The propositions of Euclid’s _Elements_ are, of course, taken for granted, but other things are tacitly assumed which go beyond what we find in Euclid. Thus the transformations of ratios defined in Euclid, Book V, and denoted by the terms _inversely_, _alternately_, _componendo_, _convertendo_, etc., are regularly used in dealing with _unequal_ ratios, whereas in Euclid they are only used in proportions, i.e. cases of _equality_ of ratios. But the propositions of Aristarchus are also of particular mathematical interest because the ratios of the sizes and distances which have to be calculated are really trigonometrical ratios, sines, cosines, etc., although at the time of Aristarchus trigonometry had not been invented, and no reasonably close approximation to the value of π, the ratio of the circumference of any circle to its diameter, had been made (it was Archimedes who first obtained the approximation 22/7). Exact calculation of the trigonometrical ratios being therefore impossible for Aristarchus, he set himself to find upper and lower limits for them, and he succeeded in locating those which emerge in his propositions within tolerably narrow limits, though not always the narrowest within which it would have been possible, even for him, to confine them. In this species of approximation to trigonometry he tacitly assumes propositions comparing the ratio between a greater and a lesser _angle_ in a figure with the ratio between two _straight lines_, propositions which are formally proved by Ptolemy at the beginning of his _Syntaxis_. Here again we have proof that textbooks containing such propositions existed before Aristarchus’s time, and probably much earlier, although they have not survived.