Ptolemys Refined Geocentric Model

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These modifications culminated some 1800 years ago, about the year 150, when the Hellenistic astronomer Claudius Ptolemy, in Alexandria, Egypt, published an extensive treatise, The Great Syntaxis, on the calculations of the motions of the Sun, Moon, and planets. With the decline of secular learning accompanying the collapse of the Roman Empire, this work was temporarily lost to the Western world. It was translated into Arabic, however, and eventually reintroduced into Europe under the title of the Almagest, which means, "the Majestic" or "the Great."

Ptolemy abandoned Aristotle's attempt to link all the motions of the Sun, Moon, and planets to each other, dismissing physics as being too speculative. He felt it was sufficient to develop the mathematical schemes to calculate accurately the celestial motions, without being concerned about the reasons for such motions. The only criterion for judging the quality or validity of a particular scheme of calculation was that it should give correct results and "save the appearances." It was necessary to do this because the available data indicated that the appearances of the heavens were gradually changing. In order to be able to establish calendars and navigational tables, it was important to have accurate calculations, regardless of how the calculations were justified.

To this end, he made use of several devices (as the modifications were called), which had been suggested by other astronomers, as well as some of his own devising. Nevertheless, he preserved the concept of circular motion. Figure 2.5a shows one such device, the eccentric. This simply means that the sphere carrying the planet is no longer centered at the center of the Earth—it is somewhat off center. Therefore in part of its motion, the planet is closer to the Earth than in the rest of its motion. Then when the planet is closer to the Earth, it will seem to be moving faster. The center of the sphere is called the eccentric. In some of his calculations, Ptolemy even had the eccentric itself moving, although slowly.

Figure 2.5b shows an epicycle, which is a sphere whose center moves around or is carried by another sphere, called the deferent. The planet itself is carried by the epicycle. The epicycle rotates about its center, whereas the center is carried along by its deferent. The center of the deferent itself might be at the center of the Earth, or more likely, it too is eccentric.5

Depending on the relative sizes and speeds of the epicycle and deferent motion, almost any kind of path can be traced out by the planet. As shown in Fig. 2.5c, retrograde motion can be generated, and the planet will be closer to the Earth while undergoing retrograde motion. Combinations of epicycle and deferent motions can be used to generate an eccentric circle or an oval approximating an ellipse, or even an almost rectangular path.

For astronomical purposes, however, it is not enough to generate a particular shape path. It is also necessary that the heavenly object travel along the path at the proper speed; that is, the calculation must show that the planet gets to a particular point in the sky at the correct time. In fact, it became necessary to cause either the planet on the epicycle, or the center of the epicycle on the deferent, to speed up or slow down. Thus it was necessary to introduce yet another device, called the equant, as shown in Fig. 2.5d. Note that the Earth is on one side of the eccentric and the equant an equal distance on the other side of the eccentric. As seen from the equant, the center of the epicycle moves with uniform angular motion, thereby saving the appearance of uniform motion. Uniform angular motion means that a line joining the equant to the center of the epicycle rotates at a constant number of degrees per hour (e.g., like a clock hand). However,

5One can visualize the motion as being like that of certain amusement park rides, which have their seats mounted on a rotating platform that is itself mounted at the end of a long rotating arm. The other end of the arm is also pivoted and can move (see Fig. 2.6).

c

Figure 2.5. Devices used in Ptolemy's geocentric model, (a) Eccentric, (b) Schematic of epicycle on deferent, (c) Generation of retrograde motion by epicycle, (d) Equant.

Figure 2.5. Devices used in Ptolemy's geocentric model, (a) Eccentric, (b) Schematic of epicycle on deferent, (c) Generation of retrograde motion by epicycle, (d) Equant.

Figure 2.6. Amusement park ride involving epicycles. (Charles Gupton/Stock Boston.)

uniform angular motion with respect to the equant is not uniform motion around the circumference of the circle. Because the length of the line from the equant to the planet changes, the circumferential speed (say, in miles per hour) changes proportionally to the length of the line.

Ptolemy made use of these various devices, individually or in combination, as needed, to calculate the positions of the planets. He sometimes used one device to calculate the speed of the Moon, for example, and another to calculate the change of its distance from the Earth. He was not concerned about using a consistent set of devices simultaneously for all aspects of the motion of a particular heavenly object. His primary concern was to calculate the correct position and times of appearances of the various heavenly bodies. In this respect, he was like a student who knows the answer to a problem, and therefore searches for a formula that will give him the correct answer without worrying about whether it makes any sense to use that formula.

Some later astronomers and commentators on Ptolemy's system did try to make the calculations agree with some kind of reasonable physical system. Thus they imagined that the epicycles actually rolled around their deferents. This meant, of course, that the various spheres had to be transparent, so that it would be possible to see the planets from the Earth. Thus the spheres had to be of some crystal-like material, perhaps a thickened ether. The space between the spheres was also thought to be filled with ether because it was believed that there was no such thing as a vacuum or empty space.

To the modern mind, Ptolemy's system seems quite far-fetched; however, in his time and for 1400 years afterward, it was the only system that had been worked out and that was capable of producing astronomical tables of the required accuracy for calendar making and navigation. Ptolemy himself, as already mentioned, had one primary goal in mind: the development of a system and mathematical technique that would permit accurate calculations. Philosophical considerations—and physics was considered a branch of philosophy—were irrelevant for his purposes. In short, Ptolemy's scheme was widely used for the very pragmatic and convincing reason that it worked!

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