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this from observation, chose that value.
    Figure 1.3
    Aristarchus’s measurement of the relative distances to the Moon and the Sun: when the Moon is a half Moon, the angle at the Moon (in this triangle) must be 90°. So a measurement of the angle at the Earth determines the ratio of the Earth–Moon line to the Earth–Sun line; in other words, the ratio of the Moon’s distance to the Sun’s distance.
    Aristarchus’s results are not what we now measure these relative distances to be. By his calculation, the distance to the Sun is about 19 times the distance to the Moon and the Sun is 19 times as large as the Moon. The modern ratio between their distances is 400 to one. The measurement Aristarchus was trying to make was extremely difficult with the instruments available to him. It is no simple undertaking to determine the precise centres of the Sun and the Moon or to know when the Moon is exactly a half Moon. Aristarchus chose the smallest angle that would accord with his observations, perhaps to keep the ratio believable. Throughout antiquity and the Middle Ages, estimates of the relative distances to the Sun and Moon would continue to be too small.
    Aristarchus didn’t stop with estimating the ratios, but found ways of converting them into actual numerical distances to the Sun and Moon and diameters for both bodies. He could see that the
apparent
size of the Moon and the Sun (meaning the size they appear to be when viewed from Earth) are about the same. During a solar eclipse, the Moon just about exactly covers the Sun. To put that in more technical language: they both have approximately the same ‘angular size’. Angular size tells how much of the sky a body ‘covers’ and is measured in ‘degrees of arc’. Both the Sun and the Moon have angular sizes of about one half of a ‘degree of arc’. (For a fuller explanation of those terms, see Figure 4.4. ) For that to be true, the two bodies don’t actually have to be the same size, for how large they appear when viewed from Earth also depends on how distant they are. ( See Figure 1.4a. ) Aristarchus assumed that the Sun is much larger than the Earth, and that it was safe to assume also that the shadow cast by the Earth has about the same angular size as the Sun and the Moon (½ a degree of arc). ( See Figure 1.4b. )
    Figure 1.4 Aristarchus’s calculation of the size and distance of the Moon.
    a. Surprisingly, all three of these bodies look the same size when viewed from Earth. We observe the ‘angular size’ of a body like the Moon or Sun, not its true size. It could be small and close or large and far away and still have the same ‘angular size’. Aristarchus saw that the Moon and the Sun have about the same angular size; that is, they
look
the same size when viewed from Earth, but he knew they are not the same true size.
    b. (The angles shown in this drawing are much larger than those that really exist.)
    Aristarchus assumed that the Sun is much larger than the Earth. If that is true, then the angle at the point of the Earth’s shadow is about equal to the angular size of the Sun as viewed from Earth.
    c. Observing an eclipse, Aristarchus concluded that the breadth of the Earth’s shadow where the Moon crossed it was approximately twice the diameter of the Moon. He knew the angle formed at the point of the Earth’s shadow and also the angular size of the Moon. There was only one distance to put the Moon where it would cover half the area of the shadow.
    Note: These drawings are not to scale.
    Aristarchus arrived at his fifth ‘hypothesis’ above – the breadth of the Earth’s shadow (at the distance where the Moon passes through it during an eclipse of the Moon) is the breadth of two Moons – by observing a lunar eclipse of maximum duration, which means an eclipse in which the Moon passes through the exact centre of the Earth’s shadow. He measured the time that elapsed between the instant that the Moon first touched the edge of the Earth’s shadow