Aristarchus of Samos, an early Greek astronomer (about 310 to 230 BC), was the first to suggest that the Earth revolved around the Sun, rather than the other way around. He gave the first estimate of the distance of the Moon (See the chapter “Estimating the Distance to the Moon" in the From Stargazers to Starships FlexBook on www.ck12.org), and it was his careful observation of a lunar eclipse--pin-pointing the Sun's position on the opposite side of the sky--that enabled Hipparchus, 169 years later, to deduce the precession of the equinoxes (See the chapter “Precession" in the From Stargazers to Starships FlexBook on www.ck12.org).
Except for one calculation --- an estimate of the distance and size of the Sun --- no work of Aristarchus has survived. However, one could guess why he believed that the Sun, not the Earth, was the central body around which the other one revolved. His calculation suggested that the Sun was much bigger than the Earth --- a watermelon, compared to a peach --- and it seemed unlikely that the larger body would orbit one so much smaller.
Here we will develop a line of reasoning somewhat like the one Aristarchus used (for his actual calculation, see reference at the end). Aristarchus started from an observation of a lunar eclipse (See the chapter “Estimating the Distance to the Moon" in the From Stargazers to Starships FlexBook on www.ck12.org). At such a time the Moon moves through the Earth's shadow, and what Aristarchus saw convinced him that the shadow was about twice as wide as the Moon. Suppose the width of the shadow was also the width of the Earth (actually it is less --- see below, also here). Then the diameter of the Moon would be half the Earth's.
Illustration of Aristarchus' calculation.
If the Moon's diameter is half the size of the Earth's, the Sun must be 19/2 or nearly 10 times wider than the Earth. The effect described in the figures of the next section modifies this argument somewhat (http://www.phy6.org/stargaze/Sshadow.htm), making the Earth 3 times wider than the Moon, not twice. If Aristarchus had observed correctly, that would make the Sun's diameter 19/3 times --- a bit more than 6 times --- than the Earth.
Good logic, but few accepted it, not even Hipparchus and Ptolemy. In fact, the opposite argument was made: if the Earth orbited the Sun, it would be on opposite sides of the Sun every 6 months. If that distance was as big as Aristarchus claimed it to be, would not the positions of the stars differ when viewed from two spots so far apart? We now know the answer: the stars are so far from us, that even with the two points 20 times further apart than Aristarchus had claimed, the best telescopes can barely observe the shift of the stars. It took nearly 18 centuries before the ideas of Aristarchus were revived by Copernicus.