B00B7H7M2E EBOK (5 page)

Aristarchus’s fourth and sixth assumptions are both far from accurate. The actual angle at the Earth in Aristarchus’s triangle would be 89° 52’, not 87°, and 89° 52’ is very close to 90°. The
angle
at the Moon in Aristarchus’s triangle
is
90°. That makes lines B and C so close to parallel that, on a drawing, the triangle would close up and be no triangle at all. The portion of one sign of the zodiac that the Moon covers is not
, and it isn’t clear why Aristarchus, who must have known 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 and the instant that it was totally hidden. He then found that that length of time was the same as the length of time during which the Moon was totally hidden. He reasoned that the breadth of the Earth’s shadow where it was crossed by the Moon must therefore be approximately twice the diameter of the Moon itself (
Figure 1.4c
). If, as he thought, the angle formed at the point of the Earth’s shadow was the same as the angular size of the Moon, that gave him only one distance at which to put the Moon where it would cover half of the area of the shadow.

Aristarchus concluded that the Moon was ¼ the size of the Earth, and that the distance to the Moon was about 60 times the radius of the Earth. Both of those values are close to the modern values. Using Eratosthenes’s calculation of the Earth’s radius, Aristarchus arrived at an actual distance to the Moon in stades. He had less success with the distance to the Sun. His earlier estimate – that the Sun’s distance is about 19 times the Moon’s distance – was in error, and a second approach he tried, though it was ingenious and correct, required timing the phases of the Moon with a precision impossible in his day.

It was another of Aristarchus’s ideas that secured his place much more firmly in the annals of astronomy. Hearing of it, one has a chilling sensation of stumbling into a prophetic vision. For Aristarchus suggested, 17 centuries before Copernicus, that the Earth is not the unmoving centre of everything but instead
moves
round the Sun, and that the universe is many times larger than anyone in his time thought – perhaps infinitely large.

For centuries it had been widely assumed that the Earth was the centre of everything. The accepted picture of the cosmos was a series of concentric spheres – spheres embedded one within the other – with the Earth resting motionless at the centre of the system. (
See Figure 1.5.
)

Plato and Euxodus of Cnidus, a younger contemporary of Plato, had introduced this model, and Aristotle’s model of the universe was a further development of it, though he differed from Euxodus as to the number and nature of the spheres. However, it wouldn’t be correct to think that everyone, without
exception
, since the dawn of human thought had agreed that the Earth was the centre and didn’t move. Some Pythagorean thinkers had decided in the fifth century
BC
, largely for symbolic and religious reasons, that the Earth was a planet and that the centre of the universe must be an invisible fire. Heraclides of Pontus, a member of Plato’s Academy under Plato, proposed that the daily rising and setting of all the celestial bodies could be nicely explained if the Earth rotated on its axis once every 24 hours.

Figure 1.5

But Aristarchus went further. Although information about his theory of a Sun-centred cosmos comes second-hand, no one disputes his authorship of the idea because there is plenty of secondary evidence. According to Archimedes:

Aristarchus of Samos brought out a book of certain hypotheses, in which it follows from what is assumed that the universe is many times greater than that now so called. He hypothesizes that the fixed stars and the Sun remain unmoved; that the Earth is borne round the Sun on the circumference of a circle . . . and that the sphere of the fixed stars, situated about [that is, centred on] the same centre as the Sun, is so great that the circle in which he hypothesizes that the Earth revolves bears such a proportion to the distance of the fixed stars as the centre of the sphere does to its surface.

Aristarchus had done no less than move the centre of the cosmos to the Sun. In this astounding turn-about, the Earth moves round the Sun and, rather than the sphere of the fixed stars making a revolution of the heavens once every 24 hours, it is the Earth that turns, rotating on its axis – as Heraclides had suggested. The stars are extremely far away. The implication is, infinitely far.