## Aristarchos and Eratosthenes

Few of the early sources survive, but an important source is the writing of Aristarchos (310—230 b.c.) of Samos, of the Alexandrian school of astronomers. He made an important estimate of the relative distances of the Earth, Moon, and Sun. Aristarchos's experimental method consisted of measurement of the precise time interval between 3rd and 1st

32 The Republic, Cornford tr., Chapter XXVI, pp. 247-248.

33 The Republic, Cornford tr., Chapter XXV, pp. 227-235.

3rd quarter

Figure 7.14. The geometry of Aristarchos's method to find the ratio of the distances of Moon and Sun from Earth. Drawing by E.F. Milone.

quarters of the Moon. The ratio of this interval to the synodic period of the Moon would be equal to the ratio of the angle subtended at the Earth by the Moon at these two points to 360° (or, in radian measure, 2p) if the orbits were purely circular. The cosine of half this angle is the ratio of the distances of Moon and Sun from Earth. Figure 7.14 illustrates the geometry.

In addition to this experiment, which Aristarchos could not possibly have carried out to the requisite precision over a single lunation—if it was ever carried out at all—he was able to make the distinction between the apparent size as opposed to the actual size of an object.34 This led directly to estimates of the relative sizes of Sun and Moon. Aristar-chos's values compared with modern values (in brackets), in units of Earth diameter, are

Solar distance: 200 [11700] Lunar distance: 10 [30] Lunar diameter: 1/3 [0.272] Solar diameter: 7 [109]

Presumably on the basis of the relative sizes of Earth, Sun, and Moon, he asserted the primacy of the Sun and helio-centricity. Copernicus was later to cite him as one of the ancient astronomers who had come to a similar conclusion about the nature of the solar system.35

The geometrical approach of Greek astronomers that has proven so effective for more than two millennia can be readily illustrated in an important case. Earlier Greek "mathematicians" mentioned by Aristotle had estimated the circumference of the Earth as 400,000 stadia, but Eratosthenes (283-200 b.c.) provided a much more accurate result.

34 A distinction that evidently eluded the 1st-century b.c. writer Lucretius (On the Nature of Things, VI, lines ~360—430).

35 In fact, Gingerich (1993, pp. 186-204) argues that Copernicus was only indirectly aware of Aristarchus's cosmology, and that the one passage in which he mentions his cosmology was deleted by Copernicus prior to De Revolutionibus's publication. It mentions that Philolaus believed that the Earth moved and that "some even say" that

Aristarchus held a like opinion.

to Sun q to Sun

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