Aristarchus figured out how to measure the distancesto and sizes of the Sun and the Moon. Because he deduced that the Sun was so much bigger than the moon, he concluded that the Earth must therefore revolve around the Sun.

He figured out how to measure the relative distances from the Earth (E) of the Sun (S) and the Moon (M). When the Moon is exactly half full, the angle E-M-S must be exactly 90 degrees. Therefore, a measurement of the angle M-E-S when the Moon is half full will give the ratio of the Earth-Moon distance to the Earth-Sun distance. Aristarchus measured the angle M-E-S to be 87 degrees, giving the ratio to be 1/19. Actually, the angle is 89 degrees, 51 minutes, giving an actual value of 1/400, that is, the Sun is 400 times further away from the Earth than the Moon is.Aristarchus' measurement was probably off because first, it is hard to determine the exact centers of the Sun and the Moon and second, it is hard to know exactly when the Moon is half full. On the other hand, his estimate showed that the Sun is much further away from us than the Moon is. |

Aristarchus also figured out how to measure the size of the Moon. During a lunar eclipse, he measured the duration of time between the moment when the edge of the Moon first entered the umbra and the moment when the Moon was first totally obscured. He also measured the duration of totality. Because he found the two times to be the same, he concluded that the width of the Earth's shadow at the distance where the Moon crosses it must be twice the diameter of the Moon. .Therefore, the Moon must be about half as big as the Earth. Note that he already knew the approximate size of the Earth. Actually, the Moon is about 1/4 as big as the Earth. |

Aristarchus also reasoned that since the Sun and the Moon have the same angular size, but the Sun is 19 times further (or so he thought), then the Sun must be 19 times bigger than the Moon. . |

While his measurements were not very precise, they nonetheless demonstrate a simple understanding of the sizes and distances of the Earth, Moon and Sun.

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