Planetary Hypotheses

The thesis that the astronomy of the Almagest is independent of its ostensible underlying cosmological assumptions is provocative and, in an unqualified form, implausible. In the case of Ptolemy's other astronomical work, the Planetary Hypotheses, there can be no question of the logical priority of cosmology in the theory. This work was composed sometime after the Almagest and consists of two books, each divided into two parts. The first part of the first book survives in Greek, while the rest of the work is available only in an Arabic translation made in the ninth century. Furthermore, it was only in the 1960s that the second part of the first book came to the notice of modern scholars. The work as a whole was intended to explain the physical cosmology underlying the astronomy of the Almagest and to help instrument makers in the construction of models of the planetary system.

Part two of book one is devoted to an investigation of the dimensions of the planetary system. The planetary order adopted is the same as that of the Almagest. Moving out from the Earth at the center, one encounters successively the Moon, Mercury, Venus, the Sun, Mars, Jupiter, and Saturn. The dimensions of the planetary system follow from a nesting principle, according to which there can be no gaps, or empty space, between the spherical shells within which the planetary bodies move. Thus the maximum distance of the Moon from the Earth is equal to the minimum distance of Mercury, the maximum distance of Mercury is equal to the minimum distance of Venus, and so on.

Ptolemy's adherence to the nesting principle was based in part on the Aristotelian doctrine that there are no vacua in nature, that space without matter is quite literally inconceivable. More to the point, perhaps, the principle gave him a method for establishing the distance to each of the planets and thereby for determining the dimensions of the planetary system. The first step is to derive the distance to the Moon. The Moon is close enough to the Earth that it exhibits a diurnal parallax. It revolves about the center of the Earth, while it is observed from the surface of the Earth. The positions of the Moon given by tables derived from the lunar model are for a hypothetical observer situated at the Earth's center. The line from the observer to the Moon and the line from the center of the Earth to the Moon make a slight angle, the angle of parallax. This angle will evidently depend on the relative positions of the center, observer, and Moon during each 24-hour revolution of the Moon and is referred to as diurnal (or daily) parallax. The maximum value of this angle is called the horizontal parallax. Measuring the difference between the observed and predicted positions of the Moon, Ptolemy obtained a value of 1 degree, 26 minutes for the horizontal parallax of the Moon. (This value is larger than the true value because of inaccuracies of naked-eye observation and limitations of the theory.) By an elementary trigonometric calculation, he then determined that the distance to the Moon is 39.75 Earth radii. The radius of the Earth was known according to an established method originating several centuries earlier with Eratosthenes, which involved measuring the altitude of the Sun at noon at two points a known distance apart that lie on the same line of longitude. Hence the distance to the Moon was given absolutely in terms of a standard surveyor's unit of distance as defined by an observer stationed in Alexandria.

Beginning with the distance to the Moon, one may use the nesting principle and planetary parameters known from observation to obtain the distance to each of the planets and the width of the shell within which it moves. For example, suppose we have two adjacent planets. In the case of the planet closer to the Earth, suppose we know the radii of the outside and inside spheres making up the shell within which it moves. Consider the planet farther from the Earth, and suppose that the ratio of its epicycle radius r to its deferent radius R is e so that r = eR. (The value of this ratio is a parameter given from observation in the Ptolemaic model.) Then, the planet moves within the shell formed by spheres of radii R +eR and R — eR. The inner sphere will coincide with the outside sphere of the planet closer to the Earth, and by assumption we know the value R* of the radius of this sphere. By equating R - eR to R* we are able to calculate R. Using this value, we proceed to the next planet out from the Earth and calculate the value of its deferent radius. The set of values obtained in this way must be fine-tuned to account for the eccentricities of the deferents. Table 3.1 gives the resulting dimensions of the planetary system in units of Earth radii. Note that the maximum distance of each planet is equal to the minimum distance of the next planet beyond it. The slight discrepancy in the case of Venus and the Sun, which would seem to allow for an empty space between them and a violation of the nesting principle, may be accounted for by rounding errors or slight adjustments that need to be made to the distances to the Moon and Mercury.

The value for the ratio of the distance to the Sun to the distance to the Moon is the same value as the one derived in the Almagest using the eclipse method of Aristarchus. (It is also equal to the value Aristarchus obtained using another method known as the method of lunar dichotomy.) The consistency of these results presumably strengthened Ptolemy's confidence in the system set out in the Planetary Hypotheses. However, the true value of the ratio of solar and lunar distances is some 16 times Ptolemy's value. The fact that he obtained the same wildly incorrect value by different and independent methods would seem to indicate that he manipulated the data to conform to theoretical desiderata. This tendency of Ptolemy's, which would be considered unacceptable today (in principle, if not in practice), was present in other parts of his astronomical work and is a well-documented aspect of his science.

The second book of the Planetary Hypotheses is devoted to a discussion of the physical structure of the planetary system. Each planet revolves within its epicyclic shell, while the epicyclic shell itself revolves within a deferent shell. Ptolemy indicated that it would be possible to replace the spheres composing the shells by tambourine-like disks. Although this would not be possible for the sphere of the fixed stars, which contains stars throughout its surface, the planets move in a fairly narrow disk aligned to the plane of the ecliptic. Whereas in the Almagest Ptolemy had presented two different models to describe the same motion, in the Planetary Hypotheses, only one of these models is given.

Table 3.1: Ptolemaic Planetary Dimensions. Distances given in Earth radii. From Vanltelden (1985, p. 27).

Body

Least distance

Average distance

Least distance

Moon

33

64

48

Mercury

64

166

115

Venus

166

1,079

622.5

Sun

1,160

1,260

1,210

Mars

1,260

8,820

5,040

Jupiter

8,820

14,187

11,503

Saturn

14,187

19,865

17,026

Fixed stars

20,000

For example, the eccentric-circle construction for the Sun's motion is described using a physical model involving an eccentrically placed shell. The alternative description in terms of a concentric deferent and secondary epicycle is not mentioned, either as a mathematical or physical possibility.

Ptolemy rejected the Aristotelian doctrine according to which motion was transferred from the outer parts of the planetary system to the inner parts by means of an intervening set of rolling spheres. In addition to the question of mechanical difficulties in how such a transfer would be made, the conception did not fit easily within the deferent-epicycle theory developed by Ptolemy. The concept of a prime mover as the thing causing the motion of the outermost celestial sphere was also rejected. To explain the motion of each of the planets, Ptolemy instead made reference to the concept of an active planetary soul or intelligence that served to power and guide the planet's somewhat complicated motion. In the flight of a bird, messages in the form of sensations or impressions pass from the mental faculty of the bird through its nerves to its wings. The bird flies without any assistance or interaction with other creatures. Similarly, a planet moves itself. It possesses a soul or intelligence, and instructions on how to move the epicycle, deferent, and other circles involved in its motion are transmitted from the planet to the corresponding spheres.

Today, the idea of a planetary soul that determines or guides the motion of the planet may seem rather farfetched, but it is an idea that has much to offer. In the absence of any physical theory, such as gravitation, it explains how it happens that the planet comes to execute the many different motions that combine in exactly the right way to produce the observed motion of the planet. The idea was entertained seriously by Kepler in his astronomical research and may be regarded as a natural step in the sequence of ideas leading to a physical explanation of planetary motion. Consider the example of the Ptolemaic eccentric-circle model of the Sun's motion. The Sun is revolving in a circle in which the Earth is offset from the center. How is the Sun able to guide its motion in this circle since the center of the latter is an empty point in space? How does one coordinate the motion of a sphere about an empty point? One possibility would be to suppose that the Sun uses the apparent diameter of the Earth as a point of reference to calculate this center. Considerations such as these, and the difficulties and possibilities suggested by them, are the first step of an investigation leading to a study of the physical causes of planetary motion.

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