To Explain the World: The Discovery of Modern Science (13 page)

In the homocentric models of Eudoxus and Callippus, the Sun, Moon, and planets were each given a separate suite of spheres, all with outer spheres rotating in perfect unison with a separate sphere carrying the fixed stars. This is an early example of what modern physicists call “fine-tuning.” We criticize a proposed theory as fine-tuned when its features are adjusted to make some things equal, without any understanding of why they should be equal. The appearance of fine-tuning in a scientific theory is like a cry of distress from nature, complaining that something needs to be better explained.

A distaste for fine-tuning led modern physicists to make a discovery of fundamental importance. In the late 1950s two types of unstable particle called tau and theta had been identified that decay in different ways—the theta into two lighter particles called pions, and the tau into three pions. Not only did the tau and theta particles have the same mass—they had the same average lifetime, even though their decay modes were entirely different! Physicists assumed that the tau and the theta could not be the same particle, because for complicated reasons the symmetry of nature between right and left (which dictates that the laws of nature must appear the same when the world is viewed in a mirror as when it is viewed directly) would forbid the same particle from decaying sometimes into two pions and sometimes into three. With what we knew at the time, it would have been possible to adjust the constants in our theories to make the masses and lifetimes of the tau and theta equal, but one could hardly stomach such a theory—it seemed hopelessly fine-tuned. In the end, it was found that no fine-tuning was necessary, because the two particles are in fact the same particle. The symmetry between right and left, though obeyed by the forces that hold atoms and their nuclei together, is simply not obeyed in various decay processes, including the decay of the tau and theta.
⁷
The physicists who realized this were right to distrust the idea that the tau
and the theta particles just happened to have the same mass and lifetime—that would take too much fine-tuning.

Today we face an even more distressing sort of fine-tuning. In 1998 astronomers discovered that the expansion of the universe is not slowing down, as would be expected from the gravitational attraction of galaxies for each other, but is instead speeding up. This acceleration is attributed to an energy associated with space itself, known as dark energy. Theory indicates that there are several different contributions to dark energy. Some contributions we can calculate, and others we can’t. The contributions to dark energy that we can calculate turn out to be larger than the value of the dark energy observed by astronomers by about 56 orders of magnitude—that is, 1 followed by 56 zeroes. It’s not a paradox, because we can suppose that these calculable contributions to dark energy are nearly canceled by contributions we can’t calculate, but the cancellation would have to be precise to 56 decimal places. This level of fine-tuning is intolerable, and theorists have been working hard to find a better way to explain why the amount of dark energy is so much smaller than that suggested by our calculations. One possible explanation is mentioned in
Chapter 11
.

At the same time, it must be acknowledged that some apparent examples of fine-tuning are just accidents. For instance, the distances of the Sun and Moon from the Earth are in just about the same ratio as their diameters, so that seen from Earth, the Sun and Moon appear about the same size, as shown by the fact that the Moon just covers the Sun during a total solar eclipse. There is no reason to suppose that this is anything but a coincidence.

Aristotle took a step to reduce the fine-tuning of the models of Eudoxus and Callippus. In
Metaphysics
⁸
he proposed to tie all the spheres together in a single connected system. Instead of giving the outermost planet, Saturn, four spheres like Eudoxus and Callippus, he gave it only their three inner spheres; the daily motion of Saturn from east to west was explained by tying these three spheres to the sphere of the fixed stars. Aristotle also added, inside the three of Saturn, three extra spheres that rotated in
opposite directions, so as to cancel the effect of the motion of the three spheres of Saturn on the spheres of the next planet, Jupiter, whose outer sphere was attached to the innermost of the three extra spheres between Jupiter and Saturn.

At the cost of adding these three extra counter-rotating spheres, by tying the outer sphere of Saturn to the sphere of the fixed stars Aristotle had accomplished something rather nice. It was no longer necessary to wonder why the daily motion of Saturn should precisely follow that of the stars—Saturn was physically tied to the sphere of the stars. But then Aristotle spoiled it all: he gave Jupiter all
four
spheres that had been given to it by Eudoxus and Callippus. The trouble with this was that Jupiter would then get a daily motion from that of Saturn and also from the outermost of its own four spheres, so that
it would go around the Earth twice a day.
Did he forget that the three counter-rotating spheres inside the spheres of Saturn would cancel only the special motions of Saturn, not its daily revolution around the Earth?

Worse yet, Aristotle added only three counter-rotating spheres inside the four spheres of Jupiter, to cancel its own special motions but not its daily motion, and then gave Mars, the next planet, the full five spheres given to it by Callippus, so that Mars would go around the Earth three times a day. Continuing in this way, in Aristotle’s scheme Venus, Mercury, the Sun, and the Moon would in a day respectively go around the Earth four, five, six, and seven times.

I was struck by this apparent failure when I read Aristotle’s
Metaphysics
, and then I learned that it had already been noticed by several authors, including J. L. E. Dreyer, Thomas Heath, and W. D. Ross.
⁹
Some of them blamed it on a corrupt text. But if Aristotle really did present the scheme described in the standard version of
Metaphysics
, then this cannot be explained as a matter of his thinking in different terms from ours, or being interested in different problems from ours. We would have to conclude that on his own terms, in working on a problem that interested him, he was being careless or stupid.

Even if Aristotle had put in the right number of counter-rotating spheres, so that each planet would follow the stars around the Earth just once each day, his scheme still relied on a great deal of fine-tuning. The counter-rotating spheres introduced inside the spheres of Saturn to cancel the effect of Saturn’s special motions on the motions of Jupiter would have to revolve at precisely the same speed as the three spheres of Saturn for the cancellation to work, and likewise for the planets closer to the Earth. And, just as for Eudoxus and Callippus, in Aristotle’s scheme the second spheres of Mercury and Venus would have to revolve at precisely the same speed as the second sphere of the Sun, in order to account for the fact that Mercury, Venus, and the Sun move together through the zodiac, so that the inner planets are never seen far in the sky from the Sun. Venus, for instance, is always the morning star or the evening star, never seen high in the sky at midnight.

At least one ancient astronomer seems to have taken the problem of fine-tuning very seriously. This was Heraclides of Pontus. He was a student at Plato’s Academy in the fourth century BC, and may have been left in charge of it when Plato went to Sicily. Both Simplicius
¹⁰
and Aëtius say that Heraclides taught that the Earth rotates on its axis,
*
eliminating at one blow the supposed simultaneous daily revolution of the stars, planets, Sun, and Moon around the Earth. This proposal of Heraclides was occasionally mentioned by writers in late antiquity and the Middle Ages, but it did not became popular until the time of Copernicus, again presumably because we do not feel the Earth’s rotation. There is no indication that Aristarchus, writing a century after
Heraclides, suspected that the Earth not only moves around the Sun but also rotates on its own axis.

According to Chalcidius (or Calcidius), a Christian who translated the
Timaeus
from Greek to Latin in the fourth century, Heraclides also proposed that since Mercury and Venus are never seen far in the sky from the Sun, they revolve about the Sun rather than about the Earth, thus removing another bit of fine-tuning from the schemes of Eudoxus, Callippus, and Aristotle: the artificial coordination of the revolutions of the second spheres of the Sun and inner planets. But the Sun and Moon and three outer planets were still supposed to revolve about a stationary, though rotating, Earth. This theory works very well for the inner planets, because it gives them precisely the same apparent motions as the simplest version of the Copernican theory, in which Mercury, Venus, and the Earth all go at constant speed on circles around the Sun. As far as the inner planets are concerned, the only difference between Heraclides and Copernicus is point of view—either based on the Earth or based on the Sun.

Besides the fine-tuning inherent in the schemes of Eudoxus, Callippus, and Aristotle, there was another problem: these homocentric schemes did not agree very well with observation. It was believed then that the planets shine by their own light, and since in these schemes the spheres on which the planets ride always remain at the same distance from the Earth’s surface, the planets’ brightness should never change. It was obvious however that their brightness changed very much. As quoted by Simplicius,
¹¹
around AD 200 the philosopher Sosigenes the Peripatetic had commented:

However the [hypotheses] of the associates of Eudoxus do not preserve the phenomena, and just those which had been known previously and were accepted by themselves. And what necessity is there to speak about other things, some of which Callippus of Cyzicus also tried to preserve when Eudoxus had not been able to do so, whether or not Callippus did preserve
them? . . . What I mean is that there are many times when the planets appear near, and there are times when they appear to have moved away from us. And in the case of some [planets] this is apparent to sight. For the star which is called Venus and also the one which is called Mars appear many times larger when they are in the middle of their retrogressions so that in moonless nights Venus causes shadows to be cast by bodies.

Where Simplicius or Sosigenes refers to the size of planets, we presumably should understand their apparent luminosity; with the naked eye we can’t actually see the disk of any planet, but the brighter a point of light is, the larger it
seems
to be.

Actually, this argument is not as conclusive as Simplicius thought. The planets (like the Moon) shine by the reflected light of the Sun, so their brightness would change even in the schemes of Eudoxus et al. as they go through different phases (like the phases of the Moon). This was not understood until the work of Galileo. But even if the phases of the planets had been taken into account, the changes in brightness that would be expected in homocentric theories would not have agreed with what is actually seen.

For professional astronomers (if not for philosophers) the homocentric theory of Eudoxus, Callippus, and Aristotle was supplanted in the Hellenistic and Roman eras by a theory that did much better at accounting for the apparent motions of the Sun and planets. This theory is based on three mathematical devices—the epicycle, the eccentric, and the equant—to be described below. We do not know who invented the epicycle and eccentric, but they were definitely known to the Hellenistic mathematician Apollonius of Perga and to the astronomer Hipparchus of Nicaea, whom we met in
Chapters 6
and
7
.
¹²
We know about the theory of epicycles and eccentrics through the writings of Claudius Ptolemy, who invented the equant, and with whose name the theory has ever after been associated.

Ptolemy flourished around AD 150, in the age of the Antonine
emperors at the height of the Roman Empire. He worked at the Museum of Alexandria, and died sometime after AD 161. We have already discussed his study of reflection and refraction in
Chapter 4
. His astronomical work is described in
Megale Syntaxis
, a title transformed by the Arabs to
Almagest
, by which name it became generally known in Europe. The
Almagest
was so successful that scribes stopped copying the works of earlier astronomers like Hipparchus; as a result, it is difficult now to distinguish Ptolemy’s own work from theirs.

The
Almagest
improved on the star catalog of Hipparchus, listing 1,028 stars, hundreds more than Hipparchus, and giving some indication of their brightness as well as their position in the sky.
*
Ptolemy’s theory of the planets and the Sun and Moon was much more important for the future of science. In one respect the work on this theory described in the
Almagest
is strikingly modern in its methods. Mathematical models are proposed for planetary motions containing various free numerical parameters, which are then found by constraining the predictions of the models to agree with observation. We will see an example of this below, in connection with the eccentric and equant.

In its simplest version, the Ptolemaic theory has each planet revolving in a circle known as an “epicycle,” not about the Earth, but about a moving point that goes around the Earth on another circle known as a “deferent.” The inner planets, Mercury and Venus, go around the epicycle in 88 and 225 days, respectively, while the model is fine-tuned so that the center of the epicycle goes around the Earth on the deferent in precisely one year, always remaining on the line between the Earth and the Sun.