Aristarchus used simple geometric ideas to deduce that the Earth is larger than the Moon and that the Sun is larger than the Earth. They are all based on the Greek understanding that we see the Moon in reflected sunlight. By observing the lighting and geometric relations between bodies in the sky, we can deduce their positions in three-dimensional space. This led Aristarchus to the idea of a Sun-centered or heliocentric cosmology.
Geocentric model and Heliocentrism. Click here for original source URL
Aristarchus used several logical steps to conclude that the Sun is larger than the Earth. If the Moon is a sphere illuminated by the Sun, then we can think about how that illumination should look to us for different arrangements of Earth, Sun, and Moon. Consider the situation on a day when the Moon is located 90° from the Sun: for example, when the Sun is setting and the Moon is just crossing the meridian. If the Moon were much farther from Earth than the Sun is, then the Moon would look nearly fully illuminated. But if the Moon were much closer to Earth than the Sun is, then the Moon would be half illuminated, which is true.
Imagine a lunar eclipse in progress. Since the Sun is so far away from the Earth, it casts a shadow with nearly parallel edges (they actually converge by ½° since that is the angular size of the Sun in the sky). Aristarchus knew that the Moon looked about one third as big as the Earth's shadow and concluded that the Moon was one third as big as the Earth. The actual fraction is one quarter; it is difficult to measure it accurately by eye. We also know that the Moon subtends an angle of ½° as seen from the Earth. Using the small angle equation
D = 206,265 d / a = (2.1 x 105 x d) / 1.8 x 103 = 120 d
Scale model?of the?Earth?and the?Moon, with a?beam of light?traveling between them at the?speed of light. It takes approximately 1.26 seconds. Click here for original source URL.
So the distance to the Moon is more than a hundred times larger than the Moon's diameter. It is also 120 (1/3) = 40 times further away than the Earth's diameter.
10th century CE Greek copy of? Aristarchus Samos's 2nd century BCE calculations of the relative sizes of the Sun, Moon and the Earth. Click here for original source URL.
The lunar phase depends on the Moon's position in orbit around the Earth and the Earth's position in orbit around the sun. This diagram looks down on Earth from north. Earth's rotation and the Moon's orbit are both counter-clockwise here. Sunlight is coming in from the right, as indicated by the yellow arrows. From this diagram we can see, for example, that the full moon will always rise at sunset and that the waning crescent moon is high overhead around 9:00 AM local time. Click here for original source URL.
Aristarchus then looked at the relationship between the first and last quarter phases of the Moon. If the Sun was enormously far away compared to the Moon, then the Sun's rays would come in parallel and there would be the same amount of time between a first and last quarter Moon as between a last and first quarter Moon. But if the Sun is somewhat closer, then the angle between the Sun and Moon at first and third quarters is slightly less than 90° Aristarchus deduced from this triangle that the Sun is 20 times further from the Earth than the Moon is. This is a difficult measurement and the true ratio is a factor of 390. Nevertheless the logic is sound.
The Sun and the Moon subtend the same angle in the sky. The small angle equation tells us that if the Sun is 20 times further away than the Moon then it is also 20 times larger than the Moon. We can combine the two previous results
DSun / DEarth = (DSun / DMoon) × (DMoon / DEarth) = 20 × (1/3) = 7
So the Sun is larger than the Earth (the true ratio is larger because the factor 20 should be 390). Aristarchus went further, surmising that it was more reasonable for the smaller body — the Earth — to go around the larger body — the Sun — than the other way around. He likened the situation to that of a hammer thrower. It is reasonable for someone to swing a small object, but it is impossible for someone the swing an object that is much larger than they are! Aristarchus correctly put the Sun in the middle of his system with the smaller, spherical Earth going around it.