8
The Problem of the Planets
The Sun and Moon are not alone in moving from west to east through the zodiac while they share the quicker daily revolution of the stars from east to west around the north celestial pole. In several ancient civilizations it was noticed that over many days five “stars” travel from west to east through the fixed stars along pretty much the same path as the Sun and Moon. The Greeks called them wandering stars, or planets, and gave them the names of gods: Hermes, Aphrodite, Ares, Zeus, and Cronos, translated by the Romans into Mercury, Venus, Mars, Jupiter, and Saturn. Following the lead of the Babylonians, they also included the Sun and Moon as planets,* making seven in all, and on this based the week of seven days.*
The planets move through the sky at different speeds: Mercury and Venus take 1 year to complete one circuit of the zodiac; Mars takes 1 year and 322 days; Jupiter 11 years and 315 days; and Saturn 29 years and 166 days. All these are average periods, because the planets do not move at constant speed through the zodiac—they even occasionally reverse the direction of their motion for a while, before resuming their eastward motion. Much of the story of the emergence of modern science deals with the effort, extending over two millennia, to explain the peculiar motions of the planets.
An early attempt at a theory of the planets and Sun and Moon was made by the Pythagoreans. They imagined that the five planets, together with the Sun and Moon and also the Earth, all revolve around a central fire. To explain why we on Earth do not see the central fire, the Pythagoreans supposed that we live on the side of the Earth that faces outward, away from the fire. (Like almost all the pre-Socratics, the Pythagoreans believed the Earth to be flat; they thought of it as a disk always presenting the same side to the central fire, with us on the other side. The daily motion of the Earth around the central fire was supposed to explain the apparent daily motion of the more slowly moving Sun, Moon, planets, and stars around the Earth.)1 According to Aristotle and Aëtius, the Pythagorean Philolaus of the fifth century BC invented a counter-Earth, orbiting where on our side of the Earth we can’t see it, either between the Earth and the central fire or on the other side of the central fire from the Earth. Aristotle explained the introduction of the counter-Earth as a result of the Pythagoreans’ obsession with numbers. The Earth, Sun, Moon, and five planets together with the sphere of the fixed stars made nine objects about the central fire, but the Pythagoreans supposed that the number of these objects must be 10, a perfect number in the sense that 10 = 1 + 2 + 3 + 4. As described somewhat scornfully by Aristotle,2 the Pythagoreans
supposed the elements of numbers to be the elements of all things, and the whole heaven to be a musical scale and a number. And all the properties of numbers and scales which they could show to agree with the attributes and parts and the whole arrangement of the heavens, they collected and fitted into their scheme, and if there was a gap anywhere, they readily made additions so as to make their whole theory coherent. For example, as the number 10 is thought to be perfect and to comprise the whole nature of numbers, they say that the bodies which move through the heavens are ten, but as the visible bodies are only nine, to meet this they invent a tenth—the “counter-Earth.”
Apparently the Pythagoreans never tried to show that their theory explained in detail the apparent motions in the sky of the Sun, Moon, and planets against the background of fixed stars. The explanation of these apparent motions was a task for the following centuries, not completed until the time of Kepler.
This work was aided by the introduction of devices like the gnomon, for studying the motions of the Sun, and other instruments that allowed the measurement of angles between the lines of sight to various stars and planets, or between such astronomical objects and the horizon. Of course, all this was naked-eye astronomy. It is ironic that Claudius Ptolemy, who had deeply studied the phenomena of refraction and reflection (including the effects of refraction in the atmosphere on the apparent positions of stars) and who as we will see played a crucial role in the history of astronomy, never realized that lenses and curved mirrors could be used to magnify the images of astronomical bodies, as in Galileo Galilei’s refracting telescope and the reflecting telescope invented by Isaac Newton.
It was not just physical instruments that furthered the great advances of scientific astronomy among the Greeks. These advances were made possible also by improvements in the discipline of mathematics. As matters worked out, the great debate in ancient and medieval astronomy was not between those who thought that the Earth or the Sun was in motion, but between two different conceptions of how the Sun and Moon and planets revolve around a stationary Earth. As we will see, much of this debate had to do with different conceptions of the role of mathematics in the natural sciences.
This story begins with what I like to call Plato’s homework problem. According to the Neoplatonist Simplicius, writing around AD 530 in his commentary on Aristotle’s On the Heavens,
Plato lays down the principle that the heavenly bodies’ motion is circular, uniform, and constantly regular. Therefore he sets the mathematicians the following problem: What circular motions, uniform and perfectly regular, are to be admitted as hypotheses so that it might be possible to save the appearances presented by the planets?3
“Save (or preserve) the appearances” is the traditional translation; Plato is asking what combinations of motion of the planets (here including the Sun and Moon) in circles at constant speed, always in the same direction, would present an appearance just like what we actually observe.
This question was first addressed by Plato’s contemporary, the mathematician Eudoxus of Cnidus.4 He constructed a mathematical model, described in a lost book, On Speeds, whose contents are known to us from descriptions by Aristotle5 and Simplicius.6 According to this model, the stars are carried around the Earth on a sphere that revolves once a day from east to west, while the Sun and Moon and planets are carried around the Earth on spheres that are themselves carried by other spheres. The simplest model would have two spheres for the Sun. The outer sphere revolves around the Earth once a day from east to west, with the same axis and speed of rotation as the sphere of the stars; but the Sun is on the equator of an inner sphere, which shares the rotation of the outer sphere as if it were attached to it, but that also revolves around its own axis from west to east once a year. The axis of the inner sphere is tilted by 23½° to the axis of the outer sphere. This would account both for the Sun’s daily apparent motion, and for its annual apparent motion through the zodiac. Likewise the Moon could be supposed to be carried around the Earth by two other counter-rotating spheres, with the difference that the inner sphere on which the Moon rides makes a full rotation from west to east once a month, rather than once a year. For reasons that are not clear, Eudoxus is supposed to have added a third sphere each for the Sun and Moon. Such theories are called “homocentric,” because the spheres associated with the planets as well as the Sun and the Moon all have the same center, the center of the Earth.
The irregular motions of the planets posed a more difficult problem. Eudoxus gave each planet four spheres: the outer sphere rotating once a day around the Earth from east to west, with the same axis of rotation as the sphere of the fixed stars and the outer spheres of the Sun and Moon; the next sphere like the inner spheres of the Sun and Moon revolving more slowly at various speeds from west to east around an axis tilted by about 23½° to the axis of the outer sphere; and the two innermost spheres rotating, at exactly the same rates, in opposite directions around two nearly parallel axes tilted at large angles to the axes of the two outer spheres. The planet is attached to the innermost sphere. The two outer spheres give each planet its daily revolution following the stars around the Earth and its average motion over longer periods through the zodiac. The effects of the two oppositely rotating inner spheres would cancel if their axes were precisely parallel, but because these axes are supposed to be not quite parallel, they superimpose a figure eight motion on the average motion of each plane
t through the zodiac, accounting for the occasional reversals of direction of the planet. The Greeks called this path a hippopede because it resembled the tethers used to keep horses from straying.
The model of Eudoxus did not quite agree with observations of the Sun, Moon, and planets. For instance, its picture of the Sun’s motion did not account for the differences in the lengths of the seasons that, as we saw in Chapter 6, had been found with the use of the gnomon by Euctemon. It quite failed for Mercury, and did not do well for Venus or Mars. To improve things, a new model was proposed by Callippus of Cyzicus. He added two more spheres to the Sun and Moon, and one more each to Mercury, Venus, and Mars. The model of Callippus generally worked better than that of Eudoxus, though it introduced some new fictitious peculiarities to the apparent motions of the planets.
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 Metaphysics8 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 Simplicius10 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:
To Explain the World: The Discovery of Modern Science Page 9