Symmetry A Human and Scientific Value

Another difficulty with equant points, in addition to their sheer physical improbability, was that Mercury's equant point was located differently than were the equant points for all the other planets.

The epicycles were borne with uniform motion on circles the same size as the eccentric circles but with other centers. Ptolemy wrote: "And these centers, in the case of all except Mercury, bisect the straight lines between the centers of the eccentrics . . . and the center of the ecliptic. But in the case of Mercury alone, this other center [equant point] is the same distance from the center revolving it as this center [center of the eccentric] . . . is in turn from the center effecting the anomaly on the side of the apogee. . . ." (Almagest, IX 5). Mercury's equant point was located not between the centers of the ecliptic and the eccentric, but beyond the center of the eccentric, in the direction of the apogee, at the same distance from the center of the eccentric as the distance between the centers of the eccentric and the ecliptic. Furthermore, Ptolemy noted: "In the case of Mercury alone, we find the eccentric circle revolved by the aforesaid center, contrariwise to the epicycle, back westward one revolution in a year's time" (Almagest, IX 5).

The violation of symmetry between Mercury's motion and the motions of all the other planets would disturb the sensibilities of later astronomers, regardless of the good agreement of Ptolemy's system with observations. Indeed, it was Ptolemy's insistence upon accurately accounting for all the known motions that forced him to compromise on the aesthetics of his system.

Actually, Ptolemy need not have given Mercury a unique mechanism. Often obscured by the Sun's glare, Mercury is the most difficult to observe of the planets visible to the naked eye. Ptolemy was unable to observe Mercury on a crucial occasion because it was too close to the horizon. Assuming a symmetry that does not exist, he erroneously substituted for the unmeasured position another observation made at a different time. And then, trying to accommodate this false data point in his theory, he was led to a unique mechanism for Mercury.

Ptolemy would have been better off had he assumed that all the planets had the same mechanism and then fit his observations of Mercury to such a mechanism as best he could. In this instance, Ptolemy placed too much faith in his own crude observations and too little faith in aesthetic considerations.

general theory of relativity in the fact that it has predicted a few minute observable facts, but rather in the simplicity of its foundation and in its logical consistency" (Hetherington, "Plato's Place," 109). The themata of simplicity was the foundation of Einstein's quasi-aesthetic faith in relativity theory more so than were a few confirming observations. The astronomer Arthur Eddington, purportedly one of only three people who really understood relativity theory, elaborated: "For those who have caught the spirit of the new ideas the observational predictions form only a minor part of the subject. It is claimed for the theory that it leads to an understanding of the world of physics clearer and more penetrating than that previously attained" (Hetherington, "Plato's Place," 109). (On hearing that he was one of only three people to really understand Einstein's relativity theory, Eddington is said to have asked who was the third.)

Perhaps Eudoxus believed his vision of the universe so penetrating that observation was rightly consigned to a minor role. If he did, can modern sci entists, given their own heritage in Einstein and Eddington, reject such an attitude as nonscientific? And is it any wonder that Eudoxus defies neat classification as either an instrumentalist or a realist?

Ptolemy, too, has been the subject of attempts to classify him as either an instrumentalist or a realist. Although he compounded many motions to determine the trajectories of planets, he refused to impose contraptions of wood or metal on the motions. His constructions had no physical reality; only the resultant motion was produced in the heavens. Nowhere in the Almagest, however, is there any explicit suggestion by Ptolemy that he understood his work as inventing merely mathematical fictions to save the phenomena. Instead he was after reality. Side by side with Zeus himself, so Ptolemy had written, he searched out the massed wheeling circles of the planets.

The equant point may be the final, culminating fiction for readers forced from a realist to an instrumentalist characterization of Ptolemy's work. Yet his need to invent and resort to equant points illustrates his insistence on accurately accounting for all the known planetary motions. The more improbable the equant point is, the more strong the argument that Ptolemy was some kind of realist.

Continuing in the Almagest beyond the introduction of equant points, even more material is found to fuel debates over whether Ptolemy was a realist or an instrumentalist. He had begun by treating all the planets as if their orbits were restricted to the ecliptic (the plane of the Earth's orbit around the Sun). There are, however, variations in the latitudes of the planets. Inclining the planets' eccentric circles to the plane of the ecliptic took care of much, but not all, of the observed variations in latitude (north-south, above and below the plane of the ecliptic).

To produce further variation, Ptolemy invented little circles to vary the inclinations of the epicycles to their eccentric circles. He explained that: "In the case of Venus and Mercury . . . those diameters of the epicycles . . . are carried aside by little circles . . . proportionate in size to the latitudinal deviation, perpendicular to the eccentrics' planes, and having their centers on them. These little circles are revolved regularly and in accord with the longitudinal passages" (Almagest, XIII 2)

There were yet further complications:

Now, concerning these little circles by which the oscillations of the epicycles are effected, it is necessary to assume that they are bisected by the planes about which we say the swayings of the obliquities take place; for only in this way can equal latitudinal passages be established on either side of them. Yet they do not have their revolutions with respect to regular movement effected about the proper center, but about another which has the same eccentricity for the little circle as the star's longitudinal eccentricity for the ecliptic. (Almagest, XIII 2).

In other words, each little circle had its own equant point.

Realizing that his readers were likely to react unfavorably to these added intricacies, Ptolemy pleaded that when it was not possible to fit the simpler hypotheses to the movements in the heavens, it was proper to try any hypoth eses. He wrote: "Let no one, seeing the difficulties of our devices, find troublesome such hypotheses. For it is not proper to apply human things to divine things . . . But it is proper to try and fit as far as possible the simpler hypotheses to the movements in the heavens; and if this does not succeed, then any hypotheses possible" (Almagest, XIII 2). Ptolemy continued:

Once all the appearances are saved by the consequences of the hypotheses, why should it seem strange that such complications can come about in the movements of heavenly things? For there is no impeding nature in them, but one proper to the yielding and giving way to movements according to the nature of each planet, even if they are contrary, so that they can all penetrate and shine through absolutely all the fluid media; and this free action takes place not only about the particular circles, but also about the spheres themselves and the axes of revolution. (Almagest, XIII 2)

Had Ptolemy been an instrumentalist interested only in the resulting motion, he would not have had to bother himself about the complicated nature of his devices. Clearly, Ptolemy envisioned actual physical structures in the heavens carrying around the planets, controlling the motions of the planets. The structures were not made of wood or metal or other earthly material, but rather of some divine celestial material offering no obstruction ("there is no impeding nature in them") to the passage of one part of the construction through another.

The argument from planetary latitudes for Ptolemy as a realist is strong but not conclusive, and examination of his lunar theory finds an equally strong argument for him as an instrumentalist. While it predicted the Moon's positions in longitude and latitude accurately, Ptolemy's theory greatly exaggerated the monthly variation in the Moon's distance from the Earth. Therefore—so the argument goes—Ptolemy could not have intended that the theory be interpreted realistically. He had measured the variation in the observed angular diameter of the Moon (a result of variation in distance) and must have known that his theory failed in this aspect. The astronomer and historian of astronomy J.L.E. Dreyer concluded: "It had now become a recognized fact, that the epicycle theory was merely a means of calculating the apparent places of the planets without pretending to represent the true system of the world, and that it certainly fulfilled its object satisfactorily, and, from a mathematical point of view, in a very elegant manner" (Dreyer, History of the Planetary Systems, 196).

Unlike Dreyer, Ptolemy is silent on the matter. His silence on this issue could be interpreted as an implicit understanding that his theory was merely a calculating device, but Ptolemy's silence does not prove beyond reasonable doubt that he had set aside the problem of the variation in the Moon's distance as one of purely mathematical complexity. He might instead have silently continued seeking a realistic physical solution, unsuccessfully battling the physical complexity of the situation. A disappointed or temporarily defeated realist is not necessarily an instrumentalist.

Proponents of Ptolemy as a realist cannot entirely deny the argument from lunar distances. They must try, instead, to dilute it or overwhelm it with a mass of contrary evidence. Ptolemy addressed at great length in the concluding section of the Almagest the issue of the physical complexity of his system. Furthermore, in one of his other books, the Planetary Hypotheses, Ptolemy nested the mechanism of epicycles and deferents for each planet inside a spherical shell between adjoining planets. Even more so than in the Almagest, in the Planetary Hypotheses Ptolemy revealed his concern with the physical world.

Proponents of Ptolemy as an instrumentalist must resort to a schizophrenic Ptolemy. In the Planetary Hypotheses, he is a realist; but earlier, in the Almagest, an instrumentalist.

Ptolemy did not push Plato's analogy of the cave to the extreme conclusion that reality exists only in the mind, or that reality is to be found beyond the visible façade of the phenomena in the mathematical structures that generate them. In changing the location of reality from the mind to the heavens, Ptolemy was more of a realist than were Plato and Eudoxus. But if Ptolemy's reality existed in the heavens, it was not a materialistic reality. It was an ethereal reality corresponding to nothing on the Earth.

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