In his usual fashion, Ptolemy said nothing about the difficulties of his model, and did not even draw attention to the fact that his model directly contradicted the real apparent size ever so blatantly. But things were moving from worse to worst, as the next model of Mercury and the models for the latitudinal motions of the planets were worst still.
Because of the high speed of this planet, and because of its proximity to the sun, and thus the difficulty in observing it in a reliable fashion, Ptolemy's model for the motions of this planet reflected the faulty conditions of the observations.[253] As in the case of the moon, where Ptolemy's crank-like model predicted that the moon would come closest to the Earth twice during its monthly revolution (when the moon reached 90° or 270° from the mean sun, or when the moon was at the first or third quarter of its revolution), so was the case with Mercury, which was supposed to come closest to the Earth at two points during its revolution: when Mercury was 120° away from the apogee, on either side of the apsidal line. This meant that Ptolemy's model for Mercury would mimic some of the features of the lunar model.
In Almagest IX, Ptolemy proposed a model for the planet Mercury that had an engulfing eccentric sphere called the director (centered at B in figure 4.4), which in turn carried another eccentric sphere called the deferent (here centered at G). Needless to say, both eccentrics were in direct violation of the Aristotelian presuppositions. The director moved around its own center, in the direction opposite to the succession of the signs, i.e. from east to west, and carried with it the deferent in the same direction. The deferent, however, moved in place, inside the director, by its own motion but in the opposite direction, thus producing a crank-like mechanism that was similar to that which was employed in the lunar model. And like the lunar deferent, this one too did not move uniformly around its own center G, but around a center E, also called the equant as in the model of the upper planets, but placed, again without any proof, half way between the center of the Earth and the center of the director, instead of being on the far side as was the case with the upper planets. The epicycle, which carried the planet Mercury with its own anomalistic motion, moved in the same direction as that of the deferent, and was itself carried by the motion of the deferent in the direction of the succession of the signs.
Thus, in addition to two eccentrics (which one may have thought that Ptolemy could explain away in the same way he used the Apollonius theorem to explain the solar eccentric away) and one epicycle (unavoidable on account of the second anomaly), there was the same additional absurdity which had appeared twice before: the absurdity of having a sphere move uniformly, in place, around an axis that did not pass through its center. And as in the case of the model for the upper planets, there was the additional unproved statement of Ptolemy that the equant laid half way between the center of the world and the center of the director. One can see why such accumulated technical considerations would make Ptolemaic astronomy subject to the kind of severe criticism that was leveled against it once it came into Islamic civilization.
Figure 4.4
Ptolemy's model for Mercury. The observer is at point O, the center of the universe. The planet M is carried by the epicycle with center C, which is itself moved by the deferent with center G. The deferent, which is also moved by an engulfing sphere called the director and whose center is B, moves in the same direction as the epicycle, but measures its equal motion around the equant E rather than its own center G. The equant E is halfway between the center of the universe O and the center of the director B. For the observer at point O, Mercury's epicycle will appear at its largest when it is closest to Earth, at ±120° away from the apogee A, and not at quadrature, when it is only 90° away from the same apogee, as was thought by Copernicus. The two elongations are represented here by angles drawn with dotted and continuous lines.
To make matters worse, the Ptolemaic models for the latitudinal motion of the planets further introduced some absurdities of their own. In this instance, and for purposes of computing the latitudinal component of the planetary positions, Ptolemy made a distinction between two groups of planets: He grouped Saturn, Jupiter, and Mars in one group and described their latitudinal motion with one model, and grouped Venus and Mercury in another group that was the subject of a different, and quite offending, model. It should be stressed at this point, that in terms of longitudinal motions the models described by Ptolemy still yielded quite reasonable predictive results despite their physical absurdities. Those results were at least convincing enough to allow Ptolemy to make his pragmatic claim that he must have been following some correct conjecturing in configuring them out although he did not know rigorously enough how they worked.
Figure 4.5
Ptolemy's model for the latitude of the upper planets. The observer is at point O, and the inclined deferent plane has a fixed inclination. The epicycle however had its own deviation from the deferent plane, whose value depends on the position of the epicycle along the deferent.
For the upper planets (figure 4.5), Ptolemy proposed a model that included an observer at the center of the world O, that is the center of the ecliptic. He then proposed an inclined, eccentric deferent plane that intersected the plane of the ecliptic at a fixed angle. The line of the intersection between the two planes passed through the position of the observer and marked the two nodes. The epicycle that was carried by the inclined plane had its own deviation from that plane as well, but this deviation varied depending on the longitudinal position of the epicycle. At the northernmost end of the deferent the epicycle would have its maximum deviation, but as soon as the epicycle reached the position of the nodes it would lie flat in the plane of the ecliptic. At the southern end of the deferent it will have the same phenomenon of maximum deviation, but in the opposite direction. And although both deviations had the same value, the southern one simply looked bigger since it was closer to the observer.
253
The same chagrin was expressed centuries later by Copernicus in his Commentariolus: "But of all things in the heavens the most remarkable is the motion of Mercury which passes through nearly untraceable points so that it cannot easily be investigated." Swerdlow,