## Versus Realism

A new stage in the historical study of ancient exact science began in the nineteenth century with the preparation of reliable textual editions of the extant Greek scientific classics. Of Ptolemy's two notable works on astronomy, the Almagest was a work of positional mathematical astronomy, while the Planetary Hypotheses was devoted to an investigation of the physical structure and dimensions of the celestial system. The Almagest has survived relatively intact and represents a scientific achievement of the highest order; it has received far and away the most attention from historians of astronomy. The Planetary Hypotheses is a comparatively minor work. Only in the 1960s was it realized that an important part of it that is missing in the Greek existed in Arabic translation. A full appreciation of Ptolemy's cosmological conceptions as set forth in this work has been the product of relatively recent historical investigation.

In the nineteenth century the study of ancient Greek astronomy was carried out in a philosophical atmosphere that was strongly influenced by positivism. A prominent writer around 1900 was Pierre Duhem, a leading proponent of positivist physical philosophy and a major contributor to the history of astronomy. Duhem asserted that the primary purpose of ancient Greek astronomy was "to save the phenomena," that is, to devise predictive schemes to account for the motions of the planets. Duhem's point of view was developed by later historians into an interpretation of ancient astronomy that has come to be known as instru-mentalism. According to this view, the geometrical models of Greek astronomy were not regarded by their inventors as real material mechanisms in the heavens but were merely mathematical constructions that were effective in prediction.

Some familiarity with the debate over the status of astronomical entities in ancient astronomy is crucial to any understanding of the rational cosmology of the Greeks. If the positivist-instrumentalists are correct, Ptolemy's astronomical theory as set forth in the Almagest was of limited explanatory import and should not be viewed as expressing a strong commitment to any particular physical arrangement of the universe. Although the Planetary Hypotheses did present an explicit cosmology, it was a minor work in comparison with the Almagest and was less influential in the subsequent history of astronomy.

The instrumentalist position is based first on the fact that it is not possible with naked-eye observation to determine the distances to the planets; with the exception of the Moon, they show no observable parallax. All we are able to observe is their direction in the sky, their positions on the celestial sphere. As historian Derek Price (1959, 200) has explained, all observations and hence all planetary theory was concerned only with the angular motion of the planets. Indeed it was concerned only with their apparent motion on and about the arbitrary unit circle constituted by the ecliptic____We must not therefore make the mistake of thinking that the mathematical astronomers regarded the epicyclic loops traced out by the combination of deferent and epicycle as being in any way the real path of the planet in space. The orbit in space was not a question which could be resolved from observation alone, only by the importation of cosmological ideas not capable of experimental proof or disproof.

In the case of Ptolemy, evidence for his instrumentalism is found in his presentation of different geometrical models to explain the same motion, models that are clearly incompatible if they are regarded as material mechanisms to produce the motion in question. Consider the case of the Sun. As we saw earlier, Ptolemy developed a successful analysis of the variable speed of the Sun along the ecliptic using an eccentric circle. He also presented a second model for the solar motion using the concept of what is known as a secondary epicycle. In figure 3.6 the observer is located at E at the center of the deferent circle ABGD. The Sun lies on the epicycle ZK0H, whose center is A. A moves on the deferent, while the Sun moves on the epicycle; the period of the two motions is the same and is equal to one year. The direction of rotation of the deferent is counterclockwise, while the direction of rotation of the epicycle is clockwise.

We now have two geometrical representations of the motion of the Sun, the eccentric-circle model (figure 3.3) and the secondary-deferent-epicycle model (figure 3.6). Ptolemy was able to show by elementary geometry that if the radius of the eccentric circle is equal to the radius of the deferent, and if the eccentricity of the eccentric circle is equal to the radius of the secondary epicycle, then the two models are fully equivalent: the combined motion of the deferent-epicycle results in the same observed solar motion as the eccentric-circle model. The trajectory of the Sun in the two models is identical, although the motion is given by different geometrical constructions in each case.

In the first section of the 12th book of the Almagest, Ptolemy also introduced equivalent models for the superior planets, one the standard deferent-epicycle model and another model involving a circle—the eccenter—centered on a point that itself moves on a smaller circle, called the concenter. The planet moves on the eccenter with a period equal to its sidereal period, while the center of the eccenter moves on the concenter with a period equal to one year. The Earth is located at a point slightly offset from the center of the concenter circle. The two models are shown to be equivalent by elementary geometry. Ptolemy's presentation of two geometrical models for the motion of the Sun and for the motion of each planet has led some astronomers of the past and many modern historians of astronomy to conclude that his models were only mathematical devices and should not be interpreted as actual physical mechanisms to produce the planetary motion.

The case of the Moon raises even more direct issues because Ptolemy used a model that seemed to be inconsistent with observation. The variation in lunar distance in this model is at odds with his eclipse theory and also leads to values for the Moon's diurnal parallax that do not accord with observed values. It is also clear that if the Moon's distance varied so greatly, its apparent diameter in the sky should change markedly, and this is not observed to be the case. The English historian of astronomy J.L.E. Dreyer, a contemporary of Duhem's, found in the Almagest lunar model compelling evidence for Ptolemy's instru-mentalism. Dreyer (1953, 196), writing in 1905, concluded,

But though Ptolemy cannot have failed to perceive this [change in the apparent size of the Moon that occurs in the model] he takes no notice of it. It had now become a recognized fact, that the epicyclic theory was merely a means of calculating the apparent places of the planets without pretending to represent the true system of the world, and it certainly fulfilled its object satisfactorily, and, from a mathematical point of view, in a very elegant manner.

Although the instrumentalist position was popular among historians during the formative period in the study of ancient Greek astronomy, over the past 40 years, many commentators have shifted to a realist interpretation of Ptolemy's theories. (It is of interest to note that historians of non-Western astronomy still seem very much attracted to an instrumentalist interpretation of Greek astronomy.) While it is certainly true that the considerations raised above should be carefully weighed, these commentators cite counterevidence that is, on balance, compelling. First, and most obviously, there is the extended discussion at the beginning of the Almagest, in which Ptolemy attempted to justify a geocentric conception of the heavens. Ptolemy cited reasons traditional within Aristotelian philosophy and discussed the composition of the superlunary world (the world including and beyond the Moon) from the fifth, or perfect, element, ether.

Although it is true that there are alternative geometric constructions to explain the motion of the Sun, the actual trajectory of the Sun in the two models is identical. The existence of multiple models may caution us against assuming material reality for the parts of the model, but it is still meaningful to speak of a definite trajectory of the planet in three-dimensional space.

In book five of the Almagest Ptolemy included tables of the diurnal parallax of the Moon, and these values, in combination with eclipse data, were used to obtain estimates of the distances to the Moon and the Sun, measured in Earth diameters. The method in question originated with the astronomer Aristarchus of Samos, who lived in the third century b.c. This concern for the dimensions of the celestial system in the work of ancient Greek astronomers seems inconsistent with a purely instrumentalist understanding of planetary models.

From a realist perspective Ptolemy's lunar model may be seen as only a provisional and imperfect attempt to deal with complex irregularities in the Moon's motion. As historian John North (1994, 113) puts it, "If he [Ptolemy] noticed the variation [in the apparent size of the Moon]—and he could hardly have failed to do so—it must have been a great disappointment to him." According to this view, Ptolemy was striving to produce physically correct models, and his inability to find such a model in the case of the Moon would have been regarded by him as a defect to be remedied in some future revision of the theory.

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