Aristarchus of Samos lived from approximately 310 BCE to 230 BCE. He was a Greek mathematician and astronomer. He is best known today as the first person known to propose a heliocentric model of the Solar System (effectively the Universe for ancient times). Aristarchus believed that the angle made between the center of the Earth, the center of the Sun, and the terminator of the First Quarter Moon was about 87 degrees. Since the Sun and Moon had about the same angular size (witness the Moon covering the Sun perfectly during a Total Solar Eclipse), this angle could be used to gemetrically determine the ratio of the Earth-Moon distance to the Earth-Sun distance. Using the angle of 87 degrees, Aristarchus concluded that this distance ratio is about 19, thereby suggesting that the Sun's diameter must be at least seven times greater than the Earth's.Today we know that this distance ratio is actually close to 400 (and that the above mentioned angle is about 89.5 degrees), and that the Sun's diameter is almost 110 times greater than Earth's. It was from this "distance ratio reasoning" that Aristarchus correctly concluded that the Sun must be the most massive object in the Solar System, and that it must therefore be at the center of the Solar System. Unfortunately, a lack of understanding amongst his contemporaries of how truly far away the so-called "fixed" stars were, and why they therefore did not exhibit at least some visible parallax relative to each other, caused them to reject Aristarchus' heliocentric explanation of the Solar System. It would be another 1800 years before the Polish cleric Copernicus would revisit his idea.
To the left is a drawing that illustrates Aristarchus' idea of how the "Sun to Earth to First Quarter Moon" angle could be used to find the above mentioned distance ratio. Click on the image to enlarge it, so that you can study it.Note that the inverse of the sine of 3 degrees is about 19.