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.
We can see why this works. Nothing in the apparent motion of planets tells us how far away they are. Hence in the theory of Ptolemy, the apparent motion of any planet in the sky does not depend on the absolute sizes of the epicycle and deferent; it depends only on the ratio of their sizes. If Ptolemy had wanted to he could have expanded the sizes of both the epicycle and the deferent of Venus, keeping their ratio fixed, and likewise of Mercury, so that both planets would have the same deferent, namely, the orbit of the Sun. The Sun would then be the point on the deferent about which the inner planets travel on their epicycles. This is not the theory proposed by Hipparchus or Ptolemy, but it gives the motion of the inner planets the same appearance, because it differs only in the overall scale of the orbits, which does not affect apparent motions. This special case of the epicycle theory is just the same as the theory attributed to Heraclides discussed above, in which Mercury and Venus go around the Sun while the Sun goes around the Earth. As already mentioned, Heraclides’ theory works well because it is equivalent to one in which the Earth and inner planets go around the Sun, the two theories differing only in the point of view of the astronomer. So it is no accident that the epicycle theory of Ptolemy, which gives Mercury and Venus the same apparent motions as the theory of Heraclides, also works pretty well in comparison with observation.
Ptolemy could have applied the same theory of epicycles and deferents to the outer planets—Mars, Jupiter, and Saturn—but to make the theory work it would have been necessary to make the planets’ motion around the epicycles much slower than the motion of the epicycles’ centers around the deferents. I don’t know what would have been wrong with this, but for one reason or another Ptolemy chose a different path. In the simplest version of his scheme, each outer planet goes on its epicycle around a point on the deferent once a year, and that point on the deferent goes around the Earth in a longer time: 1.88 years for Mars, 11.9 years for Jupiter, and 29.5 years for Saturn. Here there is a different sort of fine-tuning—the line from the center of the epicycle to the planet is always parallel to the line from the Earth to the Sun. This scheme agrees fairly well with the observed apparent motions of the outer planets because here, as for the inner planets, the different special cases of this theory that differ only in the scale of the epicycle and deferent (keeping their ratio fixed) all give the same apparent motions, and there is one special value of this scale that makes this model the same as the simplest Copernican theory, differing only in point of view: Earth or Sun. For the outer planets, this special choice of scale is the one for which the radius of the epicycle equals the distance of the Sun from the Earth. (See Technical Note 13.)
Ptolemy’s theory nicely accounted for the apparent reversal in direction of planetary motions. For instance, Mars seems to go backward in its motion through the zodiac when it is on a point in its epicycle closest to the Earth, because then its supposed motion around the epicycle is in the direction opposite to the supposed motion of the epicycle around the deferent, and faster. This is just a transcription into a frame of reference based on the Earth of the modern statement that Mars seems to go backward in the zodiac when the Earth is passing it as they both go around the Sun. This is also the time when it is brightest (as noted in the above quotation from Simplicius), because at this time it is closest to the Earth, and the side of Mars that we see faces the Sun.
The theory developed by Hipparchus, Apollonius, and Ptolemy was not a fantasy that, by good luck, just happened to agree fairly well with observation but had no relation to reality. As far as the apparent motions of the Sun and planets are concerned, in its simplest version, with just one epicycle for each planet and no other complications, this theory gives precisely the same predictions as the simplest version of the theory of Copernicus—that is, a theory in which the Earth and the other planets go in circles at constant speed with the Sun at the center. As already explained in connection with Mercury and Venus (and further explained in Technical Note 13), this is because the Ptolemaic theory is in a class of theories that all give the same apparent motions of the Sun and planets, and one member of that class (though not the one adopted by Ptolemy) gives precisely the same actual motions of the Sun and planets relative to on
e another as given by the simplest version of the Copernican theory.
It would be nice to end the story of Greek astronomy here. Unfortunately, as Copernicus himself well understood, the predictions of the simplest version of the Copernican theory for the apparent motions of the planets do not quite agree with observation, and so neither do the predictions of the simplest version of the Ptolemaic theory, which are identical. We have known since the time of Kepler and Newton that the orbits of the Earth and the other planets are not exactly circular, the Sun is not exactly at the center of these orbits, and the Earth and the other planets do not travel around their orbits at exactly constant speed. Of course, none of that was understood in modern terms by the Greek astronomers. Much of the history of astronomy until Kepler was taken up with trying to accommodate the small inaccuracies in the simplest versions of both the Ptolemaic and the Copernican theories.
Plato had called for circles and uniform motion, and as far as is known no one in antiquity conceived that astronomical bodies could have any motion other than one compounded of circular motions, though Ptolemy was willing to compromise on the issue of uniform motion. Working under the limitation to orbits composed of circles, Ptolemy and his forerunners invented various complications to make their theories agree more accurately with observation, for the Sun and Moon as well as for the planets.*
One complication was just to add more epicycles. The only planet for which Ptolemy found this necessary was Mercury, whose orbit differs from a circle more than that of any other planet. Another complication was the “eccentric”; the Earth was taken to be, not at the center of the deferent for each planet, but at some distance from it. For instance, in Ptolemy’s theory the center of the deferent of Venus was displaced from the Earth by 2 percent of the radius of the deferent.*
The eccentric could be combined with another mathematical device introduced by Ptolemy, the “equant.” This is a prescription for giving a planet a varying speed in its orbit, apart from the variation due to the planet’s epicycle. One might imagine that, sitting on the Earth, we should see each planet, or more precisely the center of each planet’s epicycle, going around us at a constant rate (say, in degrees of arc per day), but Ptolemy knew that this did not quite agree with actual observation. Once an eccentric was introduced, one might instead imagine that we should see the centers of the planets’ epicycles go at a constant rate, not around the Earth, but around the centers of the planets’ deferents. Alas, that didn’t work either. Instead, for each planet Ptolemy introduced what came to be called an equant,* a point on the opposite side of the center of the deferent from the Earth, but at an equal distance from this center; and he supposed that the centers of the planets’ epicycles go at a constant angular rate about the equant. The fact that the Earth and the equant are at an equal distance from the center of the deferent was not assumed on the basis of philosophical preconceptions, but found by leaving these distances as free parameters, and finding the values of the distances for which the predictions of the theory would agree with observation.
There were still sizable discrepancies between Ptolemy’s model and observation. As we will see when we come to Kepler in Chapter 11, if consistently used the combination of a single epicycle for each planet and an eccentric and equant for the Sun and for each planet can do a good job of imitating the actual motion of planets including the Earth in elliptical orbits—good enough to agree with almost any observation that could be made without telescopes. But Ptolemy was not consistent. He did not use the equant in describing the supposed motion of the Sun around the Earth; and this omission—since the locations of planets are referred to the position of the Sun—also messed up the predictions of planetary motions. As George Smith has emphasized,13 it is a sign of the distance between ancient or medieval astronomy and modern science that no one after Ptolemy appears to have taken these discrepancies seriously as a guide to a better theory.
The Moon presented special difficulties: the sort of theory that worked pretty well for the apparent motions of the Sun and planets did not work well for the Moon. It was not understood until the work of Isaac Newton that this is because the Moon’s motion is significantly affected by the gravitation of two bodies—the Sun as well as the Earth—while the planets’ motion is almost entirely governed by the gravitation of a single body: the Sun. Hipparchus had proposed a theory of the Moon’s motion with a single epicycle, which was adjusted to account for the length of time between eclipses; but as Ptolemy recognized, this model did not do well in predicting the location of the Moon on the zodiac between eclipses. Ptolemy was able to fix this flaw with a more complicated model, but his theory had its own problems: the distance between the Moon and the Earth would vary a good deal, leading to a much larger change in the apparent size of the Moon than is observed.
As already mentioned, in the system of Ptolemy and his predecessors there is no way that observation of the planets could have indicated the sizes of their deferents and epicycles; observation could have fixed only the ratio of these sizes for each planet.* Ptolemy filled this gap in Planetary Hypotheses, a follow-up to the Almagest. In this work he invoked an a priori principle, perhaps taken from Aristotle, that there should be no gaps in the system of the world. Each planet as well as the Sun and Moon was supposed to occupy a shell, extending from the minimum to the maximum distance of the planet or Sun or Moon from the Earth, and these shells were supposed to fit together with no gaps. In this scheme the relative sizes of the orbits of the planets and Sun and Moon were all fixed, once one decided on their order going outward from the Earth. Also, the Moon is close enough to the Earth so that its absolute distance (in units of the radius of the Earth) could be estimated in various ways, including the method of Hipparchus discussed in Chapter 7. Ptolemy himself developed the method of parallax: the ratio of the distance to the Moon and the radius of the Earth can be calculated from the observed angle between the zenith and the direction to the Moon and the calculated value that this angle would have if the Moon were observed from the center of the Earth.14 (See Technical Note 14.) Hence, according to Ptolemy’s assumptions, to find the distances of the Sun and planets all that was necessary was to know the order of their orbits around the Earth.
The innermost orbit was always taken to be that of the Moon, because the Sun and the planets are each occasionally eclipsed by the Moon. Also, it was natural to suppose that the farthest planets are those that appear to take the longest to go around the Earth, so Mars, Jupiter, and Saturn were generally taken in that order going away from the Earth. But the Sun, Venus, and Mercury all on average appear to take a year to go around the Earth, so their order remained a subject of controversy. Ptolemy guessed that the order going out from the Earth is the Moon, Mercury, Venus, the Sun, and then Mars, Jupiter, and Saturn. Ptolemy’s results for the distances of the Sun, Moon, and planets as multiples of the diameter of the Earth were much smaller than their actual values, and for the Sun and Moon similar (perhaps not coincidentally) to the results of Aristarchus discussed in Chapter 7.
The complications of epicycles, equants, and eccentrics have given Ptolemaic astronomy a bad name. But it should not be thought that Ptolemy was stubbornly introducing these complications in order to make up for the mistake of taking the Earth as the unmoving center of the solar system. The complications, beyond just a single epicycle for each planet (and none for the Sun), had nothing to do with whether the Earth goes around the Sun or the Sun around the Earth. They were made necessary by the fact, not understood until Kepler’s time, that the orbits are not circles, the Sun is not at the center of the orbits, and the velocities are not constant. The same complications also affected the original theory of Copernicus, who assumed that the orbits of planets and the Earth had to be circles and the speeds constant. Fortunately, this is a pretty good approximation, and the simplest version of the epicycle theory, with just one epicycle for each planet and none for the Sun, worked far better than the homocentric spheres of Eudoxus, Callippus, and Aristotle. If P
tolemy had included an equant along with an eccentric for the Sun as well as for each planet, the discrepancies between theory and observation would have been too small to be detected with the methods then available.
But this did not settle the issue between the Ptolemaic and Aristotelian theories of planetary motions. The Ptolemaic theory agreed better with observation, but it did violence to the assumption of Aristotelian physics that all celestial motions must be composed of circles whose center is the center of the Earth. Indeed, the queer looping motion of planets moving on epicycles would have been hard to swallow even for someone who had no stake in any other theory.
For fifteen hundred years the debate continued between the defenders of Aristotle, often called physicists or philosophers, and the supporters of Ptolemy, generally referred to as astronomers or mathematicians. The Aristotelians often acknowledged that the model of Ptolemy fitted the data better, but they regarded this as just the sort of thing that might interest mathematicians, not relevant for understanding reality. Their attitude was expressed in a statement by Geminus of Rhodes, who flourished around 70 BC, quoted about three centuries later by Alexander of Aphrodisias, who in turn was quoted by Simplicius,15 in a commentary on Aristotle’s Physics. This statement lays out the great debate between natural scientists (sometimes translated “physicists”) and astronomers:
It is the concern of physical inquiry to enquire into the substance of the heavens and the heavenly bodies, their powers and the nature of their coming-to-be and passing away; by Zeus, it can reveal the truth about their size, shape, and positioning. Astronomy does not attempt to pronounce on any of these questions, but reveals the ordered nature of the phenomena in the heavens, showing that the heavens are indeed an ordered cosmos, and it also discusses the shapes, sizes, and relative distances of the Earth, Sun, and Moon, as well as eclipses, the conjunctions of the heavenly bodies, and qualities and quantities inherent in their paths. Since astronomy touches on the study of the quantity, magnitude, and quality of their shapes, it understandably has recourse to arithmetic and geometry in this respect. And about these questions, which are the only ones it promised to give an account of, it has the power to reach results through the use of arithmetic and geometry. The astronomer and the natural scientist will accordingly on many occasions set out to achieve the same objective, for example, that the Sun is a sizeable body, that the Earth is spherical, but they do not use the same methodology. For the natural scientist will prove each of his points from the substance of the heavenly bodies, either from their powers, or from the fact that they are better as they are, or from their coming-to-be and change, while the astronomer argues from the properties of their shapes and sizes, or from quantity of movement and the time that corresponds to it. . . . In general it is not the concern of the astronomer to know what by nature is at rest and what by nature is in motion; he must rather make assumptions about what stays at rest and what moves, and consider with which assumptions the appearances in the heavens are consistent. He must get his first basic principles from the natural scientist, namely that the dance of the heavenly bodies is simple, regular, and ordered; from these principles he will be able to show that the movement of all the heavenly bodies is circular, both those that revolve in parallel courses and those that wind along oblique circles.
To Explain the World: The Discovery of Modern Science Page 10