On the positive side, Ptolemy's configuration, physically absurd as it was, could still account rather well for the longitudinal motions of the planets. It could explain the daily progression of the planets from west to east, i.e. contrary to the direction of the primary daily motion of the heavens, and in the direction of the ascending order of the zodiacal signs Aries, Taurus, Gemini, etc., as a result of the planet's own mean motion. The planet's particular motion, said anomalistic motion, took place uniformly on its own epicycle, and could account for the forward and backward motions of the planet as well as account for the stations in between. These positive conditions, when coupled with the ability of the model to predict the positions of the planets rather accurately, for its time, could satisfy the astrological predictions and the like.
On the negative side, the sheer absurdity of the equant concept, in the realm of physical spheres, turned this model into a point of contention to be taken up by every serious astronomer up to and including Copernicus.[251]
In the case of the moon, Ptolemy's model became more complicated, and even more absurd than the two previously discussed models of the sun and the planets. In order to account for the observable motion of the moon, with its variations in the position of eclipses, the apparent motion of the moon on its epicycle without undergoing retrograde motion, and the variation in the size of the epicycle as it appears to the observer on Earth, he could not possibly account for all those variables with a relatively simple model as that of the sun or the planets. Instead he introduced the "sphere of the nodes" (figure 4.3) as an engulfing sphere for the moon and made it concentric with the Earth. He made this sphere responsible for the motion of another sphere inside it that he called the deferent, which was in turn eccentric with respect to the Earth. He made the sphere of the nodes move from east to west, while the engulfed deferent in the opposite direction. The moon was finally moved directly by an epicycle which was carried within the shell of the deferent but which moved in the direction opposite to that of the deferent.
Figure 4.3
Ptolemy's model for the moon where the engulfing sphere that moves the nodes as well as everything contained in it is centered on the Earth at O. The deferent sphere, with center F, is moved by the engulfing sphere from east to west so that its apogee moves to point A. The deferent that carries the epicycle within its shell, here represented by the circle of the deferent itself, moves in the opposite direction to bring the center of the epicycle to point C. Note that the uniform motion of the deferent is measured around the center of the universe O, which means that it does not move uniformly around its own center F, thus producing an equant-like concept of its own. Now the epicycle moves by its own anomalistic motion in the same direction as the engulfing sphere. But its anomalistic motion is measured from the extension of the line that connects the center of the epicycle C to the ever-moving prosneusis point N. Furthermore, note that the distance of the epicyclic center from the Earth, when the epicycle is 90° away from the mean sun is almost half the distance it would have when the epicyclic center is in conjunction or opposition with the sun. That means that a quarter moon should look almost twice as big as a full moon, which is obviously untrue.
But in order to create a variation in the size of the epicycle, especially when the moon was away from the mean sun by 90°, Ptolemy made the deferent move uniformly around the center of the Earth rather than around its own center, thus creating again an equant problem similar to that encountered with the upper planets. Furthermore, and in order to account for the second anomaly of the moon, he stipulated that the moon's own motion on its epicycle should be measured from a line that started at a point, N in the diagram, which was located diametrically opposite from the deferent center, F in the diagram, with respect to the Earth, and at the same distance from the Earth as the deferent center, but in the opposite direction. The point was called the prosneusis point (nuqṭat al-muḥādhāt) and the line joining it to the center of the epicycle, when extended to the circumference of the epicycle, constituted the starting point for the true anomaly of the moon. One should note here that this prosneusis point N was itself in constant motion as it was always defined by its diametrically opposite location from the moving center of the deferent F. That is, it was simply symmetrical to the deferent center with respect to the center of the universe. Thus the line that joined the prosneusis point to the center of the epicycle oscillated back and forth from the line that joined the center of the epicycle to the fixed center of the universe and never completed a full revolution. This oscillating motion was found objectionable as well, since there should not be any non-circular motions in the heavens according to Aristotle, and thus all revolutions should be completed.
For very instrumental reasons, this crank-like model, however, accommodated the observations rather well, as far as the longitude of the moon was concerned. But it failed miserably when it came to the apparent size of the moon. If taken seriously, and with its crank-like operation, the model required the moon to be pulled very close to the Earth when it was at 90° away from the mean sun. Accordingly, the moon's distance from the Earth at that point would become almost half the distance it had when it was at full moon or in conjunction with the sun. This would mean that for an observer, situated on the Earth, when the moon was quarter moon, it would then have to appear twice as big as when it was a full moon. This predictive aspect of the model was obviously untrue, and was fittingly described later on by Ibn al-Shāṭir as untenable since the moon was never seen as such (lam yura kadhālika).
Furthermore, since the position of the prosneusis point itself was determined by the position of the symmetrically opposite position of the deferent center on the other side from the Earth, and since that center was itself moved around the Earth by the engulfing sphere of the nodes, that meant that both the deferent center as well as the prosneusis point, that depended on it, were in constant motion. This also meant that the line that joined the prosneusis point to the center of the epicycle's center was no longer fixed as well as we just said, and thus in 'Urḍī's terms, was not fit to be considered the beginning line for the measurement of the anomalistic motion, since it would not constitute a fixed starting point.
With a deferent that moved around its own center, but measured its uniform motion around another center, now the center of the universe, thus repeating the same equant problem, and with the introduction of the moving prosneusis point that introduced an oscillating line that never completed a full revolution, and with the huge increase in the apparent size of the moon at quadrature, all implied by Ptolemy's model, one can see why this model attracted a large critical literature within Islamic civilization. Its own problem was often referred to as the prosneusis problem, in analogy to the equant problem that was used to describe the difficulties with the model for the upper planets. Several astronomers working in Islamic civilization tried to rectify the situation by creating their own models, some of which were more successful than others. In that tradition, Ibn al-Shāṭir's lunar model was by far the best, not only because it did away with the equant construction when it made all spheres move uniformly in place around axis that passed through their own centers, and reduced the variation in the apparent size of the moon, while keeping the increase of the size of the epicycle, but because it also turned out to be identical to the same model which was proposed by Copernicus (d. 1543) himself about 200 years later.[252]
251
In addition to the objections raised by Ibn al-Haitham and others in the Arabic astronomical tradition note the following statement of Copernicus in his earliest astronomical work, the
252
See the seminal paper of Victor Roberts, "The Solar and Lunar Theory of Ibn al-Shāṭir: A Pre-Copernican Copernican Model",