To explain the complex motion of Mars, Eudoxus had the second sphere rotate in roughly the opposite direction to the first, and at the same speed. And since they rotated at the same speed, when the first sphere completed one rotation, the second sphere did too, so that Mars would end up back in the same place. But in between, because the spheres were set at an angle to one another, Mars would not remain stationary, but would instead trace out a figure-8 like shape. If you superimposed this figure-8 on the otherwise smooth west-to-east motion of the planets, Eudoxus suggested, you would get a planet that usually moved west to east, but that moved the other way from time to time, describing loops that went first up, then down, then up again, and so on. Because the figure-8 resembled the path of a horse on a fetter, this mechanism was called the hippopedes, which is Greek for “horse-fetter.”
The hippopedes explained, in broad terms, the strange motion of some of the planets, and it had the advantage of involving only the perfect shapes of spheres and circles. But it failed in important ways, too. It didn’t explain why the seasons are of different lengths—that is, why summer is a few days longer than winter. It didn’t match the actual orbits of Mars, Jupiter, and Saturn, even to the limited precision of the observations then available, and it didn’t come close to explaining the motions of Mercury and Venus. (However, it wasn’t a total loss. If we interpret the motions of the two spheres as the Earth orbiting the Sun in one case, and the Earth rotating on its axis in the other, then we get a very good explanation of the figure-8 shape of the analemma, the curious sigil that appears in the South Pacific on some globes. But that’s a matter for another essay.)
A partial step forward was made by the greatest astronomer of antiquity, Hipparchus of Rhodes (c. 190-120 B.C.). Hipparchus seems to have taken as his starting point a work on eccentrics and equants by the Greek geometer, Apollonius of Perga (c. 262-190 B.C.). He then extended and applied it to the question of lunar and solar motion.
The motion of the Moon was particularly difficult to explain. On the whole it goes around the Earth about once a month (hence its name, as described in “The Moon”). However, it doesn’t move uniformly, as you’d expect. Instead, it constantly speeds up and slows down, an effect called the lunar anomaly. This is a result of its elliptical orbit and gravitational perturbations from the Sun, the Earth’s equatorial bulge, and the other planets. However, Hipparchus wasn’t aware of the causes of the anomaly, and in any case he was only interested in a mathematical model of the moon’s behavior. He didn’t care about why the Moon went every which way—he only wanted to explain how it moved.
Here was the model devised by Hipparchus: the Moon was indeed carried around the Earth by a circular orbit. But the Moon lay not on the orbit itself, but rather at the edge of a sort of wheel, or epicycle. It was the center of the epicycle that went around the Earth in uniform circular motion, while the Moon in turn went around the center of the epicycle. When the Moon was moving in the same direction as the epicycle, it appeared to speed up; when it was moving in the opposite direction, it would appear to slow down.
Hipparchus also went on to describe the motion of the Sun in terms of epicycles, and this model worked reasonably well. However, having the sort of integrity that he did, Hipparchus had to admit to himself (and to others) that his lunar model, although it agreed with actual lunar observations when it came to the general effects of the anomaly, it did not agree when it came to the finer aspects.
Although Hipparchus was the greatest astronomer of antiquity, very little of his actual writing survives, unfortunately. The little we do have is a three-volume review of two other Greek works, and only in the second half of the set do we get any of Hipparchus’s own theories. Fortunately, much of that theory was not yet lost at the time of the Greek astronomer and geographer, Ptolemy (c. 85-165, and no relation to the succession of Egyptian kings). Ptolemy collected the various ideas of Hipparchus and synthesized them into a coherent whole, which he called Syntaxis Mathematica, which is Greek for “Mathematical Collection.” Almost immediately, however, that title was enhanced by the appellation, “Megiste,” meaning “greatest.” When the Syntaxis was translated into Arabic, they took heed of that honorific, and we now know the work best by its Arabic name, Almagest.
It used to be thought that Hipparchus was the innovator, and Ptolemy only the poor, benighted compiler who followed everything that Hipparchus said. For example, one of the sections of the Almagest is a star catalogue of about 1,000 stars. It contains, interestingly, a number of stars whose positions only make sense if they were observed during Hipparchus’s time, not Ptolemy’s.
However, in more recent times, it has become evident that Ptolemy was more than just a compiler, and indeed made important contributions to Greek astronomy. For example, Ptolemy made corrections to the epicycle model of Hipparchus in order to account for discrepancies in the Moon’s motion. He also extended the epicycle model to the planets, which Hipparchus did not attempt. Ptolemy showed that if the epicycle turned roughly once a year, and were the appropriate size, the resulting apparent motion would show the correct retrograde motion.
Even if he drew a significant amount of this work from Hipparchus, at the very least Ptolemy can be credited with doing for astronomy what Euclid did for geometry with his Elements—that is, putting everything in order. In fact, Ptolemy put everything in such good order, and predicted the motions of the planets to such a high degree of accuracy, that his work was not surpassed for over a thousand years.
That is not to say it was without its problems. For example, the epicycle model for the Moon that Ptolemy adapted from Hipparchus unfortunately predicted that at times the Moon should be half as far from the Earth than it was at other times. If that were really the case, the Moon should sometimes appear twice as wide as it does. That was very plainly not the case, but it doesn’t appear to have bothered Ptolemy overmuch, since he, like Hipparchus, was only interested in a mathematical model of the motions of the planets.
That thinking also dominated most of Europe for the next 1,000 years or so. One of the trademarks of the Dark Ages was a marked refusal to do anything but refine the old classics. It is clear, in hindsight, that there was something fundamentally wrong with the Ptolemaic model, but what better model was there? There was little inclination for anyone to do the kind of tedious, painstaking accurate observations that were required to propose an alternative.
Even when things changed, they didn’t change much. The Polish cleric Nicolas Copernicus (1473-1543), near the end of his life, published De Revolutionibus, in which he described an alternative model: that the Earth, along with all of the other planets (except for the Moon and the Sun), revolved around the Sun. But even then, Copernicus wasn’t spurred on by improved observations of the planets. Those wouldn’t come around until the time of the Danish astronomer Tycho Brahe (1546-1601) half a century later. Instead, Copernicus was motivated by the fundamentally ad hoc nature of the Ptolemaic system. It involved a large number of parameters, each of which could be determined by observation alone, and weren’t tied to each other in any systematic way. It was his goal to simplify the foundations of celestial mechanics.
There is a persistent myth that in proposing his solution, Copernicus somehow cut the Gordian knot of celestial mechanics that Ptolemy had left tied, that he made computations orders of magnitude easier than they had been before. Actually, computations based on the Copernican model were just as involved as they had been before. Like the Ptolemaic model, the Copernican model used circular orbits, so it retained much of the complicated machinery of the Ptolemaic model. However, what was simpler were the underlying relationships. The various parameters of the Ptolemaic system were reduced to the tilt of the Earth’s axis, the rotation of the Earth on that axis, and its revolution around the Sun (along with the other planets). Everything else could essentially be determined from these three basic factors.
Also, Copernicus wasn’t the first to suggest that the Earth went around the Sun. The Greek astronomer Aristarchus (c. 310-230 B.C.) also proposed this idea (as discussed in “Double Vision”), as did the German cardinal and philosopher, Nicholas of Cusa (1401-1464). However, Copernicus was the first to set down this model in full, mathematical detail. In his first draft, Copernicus credited Aristarchus in his preface, but in the final version, either he or his printer suppressed the reference—quite likely the latter, in my opinion. After Copernicus’s death, the printer was the sole party remaining as a target for persecution. Eager to avoid being singled out, he added a notice to Copernicus’s book, that the work was not intended to assert celestial fact, but only to facilitate computation.