## Ptolemys Exposition Of Mathematical Astronomy

Hundreds of years of Greek geometrical astronomy was systematized and quantified with rigorous geometrical demonstrations and proofs by Claudius Ptolemaeus around a.d. 140. He did for astronomy what Euclid had done for geometry and earned a reputation as the greatest astronomer of the ancient world.

Ptolemy's mathematical systematic treatise of astronomy, The Mathematical Syntaxis, soon attracted the appellation megiste, Greek for "greatest." This was transliterated into Arabic and preceded by al, Arabic for "the." Ptolemy's complete exposition of mathematical astronomy became, upon passing from Arabic into medieval Latin in a.d. 1175, the Almagest. This Latin translation became, in a.d. 1515, the first printed version of the Almagest.

The Almagest proceeds in logical order, beginning with a brief introduction to the nature of astronomy and a presentation of the necessary trigonometric theory and spherical astronomy. Then come theories of the Sun and the Moon, an account of eclipses (requiring knowledge of the Sun and the Moon), and discussion of the fixed stars (some of their positions determined with respect to the Moon). The final sections treat the planets, observations of them made largely with respect to the fixed stars.

Ptolemy's motivation is revealed in an epigram appearing in some manuscripts of the Almagest. Attribution to Ptolemy is plausible, but not certain. The epigram reads: "I know that I am mortal and the creature of a day; but when I search out the massed wheeling circles of the stars, my feet no longer touch the Earth, but, side by side with Zeus himself, I take my fill of ambrosia, the food of the gods" (Gingerich, The Eye of Heaven, 4).

At the beginning of the Almagest, Ptolemy distinguished between mathematics and physics, and also between them and theology. He would cultivate mathematics, particularly with respect to divine and heavenly things: "Those who have been true philosophers . . . have very wisely separated the theoretical part of philosophy from the practical. . . . We accordingly thought . . . to train our actions . . . upon the consideration of their [whatever things we happen upon] beautiful and well-ordered disposition, and to indulge in meditation mostly for the exposition of many beautiful theorems and especially of those specifically called mathematical" (Almagest, I 1).

Physics dealt with the changeable, corruptible world below the Moon. Astronomy, called mathematics by the Greeks, dealt with the eternal and ethereal world of the Moon, the Sun, the planets, and the stars. Mathematics was "the kind of science which shows up quality with respect to forms and local motions, seeking figure, number, and magnitude, and also place, time, and similar things . . . It can be conceived both through the senses and without the senses" (Almagest, I 1). Motion of the planets from place to place (local motion) was the subject of mathematics.

Only indisputable geometrical proof could provide sure knowledge. According to Ptolemy: "Only the mathematical, if approached enquiringly, would give its practitioners certain and trustworthy knowledge with demonstration both arithmetic and geometric resulting from indisputable procedures" (Almagest, I 1).

Ptolemy linked divine and heavenly things with the unchanging and with the discipline of mathematics, in contrast to physics, which dealt with the changing, or corruptible. Thus, as Ptolemy explained: "And especially were we led to cultivate that discipline [mathematics] developed in respect to divine and heavenly things as being the only one concerned with the study of things which are always what they are . . . eternal and impassible" (Almagest, I 1). Corruptible was linked with straight movements and incorruptible with circular movements: straight on Earth and circular in the heavens.

Continuing his introduction to the Almagest, Ptolemy echoed Plato's concern with education. He cited above all other topics the potential of astronomy— dealing as it does with the constancy, order, symmetry, and calm associated with the divine—to make its followers lovers of this divine beauty and to reform their nature and spiritual state. Ptolemy wrote: "And indeed this same discipline would more than any other prepare understanding persons with respect to nobleness of actions and character by means of the sameness, good order, due proportion, and simple directness contemplated in divine things, making its followers lovers of that divine beauty, and making habitual in them, and as it were natural, a like condition of the soul" (Almagest, I 1). This value judgment regarding unchanging versus changing is repeated in the concluding paragraph of the Almagest's preface.

Ptolemy also stated his intention to record everything that had already been discovered and to add his own original contributions: "And so we ourselves try to increase continuously our love of the discipline of things which are always what they are, by learning what has already been discovered in such sciences . . . and also by making a small original contribution . . . we shall only report what was rigorously proved by the ancients, perfecting as far as we can what was not fully proved or not proved as well as possible" (Almagest, I 1).

He would begin with reliable observations and then attach to this foundation a structure of ideas to be confirmed using geometrical proofs. Concluding his preface and moving on to the order of the theorems, Ptolemy wrote: "And we shall try and show each of these things using as beginnings and foundations for what we wish to find, the evident and certain appearances from the observations of the ancients and our own, and applying the consequences of these conceptions by means of geometrical demonstrations" (Almagest, I 2).

Replacing the geometrical astronomy he had inherited with an inductive, observational science would have been no less than a scientific revolution. Determining the reliability of observations, however, other than from their agreement with the very theory they were to confirm, would prove a major problem for Ptolemy.

Ptolemy presented preliminary astronomical concepts in Book I of the Almagest, including the spherical motion of the heavens and the nature of ethereal bodies to move in a circular and uniform fashion. He also stated that the Earth is spherical, in the middle of the heavens, and very small relative to the size of the universe. Ptolemy appealed primarily to observations rather than logically deducing these concepts from physical principles.

Ptolemy also presented necessary mathematical techniques. How to determine the length of a chord between two points on a great circle, along with a table of chord lengths for each half degree of arc, comprised the remainder of Book I. All of Book II was devoted to the application of trigonometric techniques to an oblique sphere, such as the Earth.

In Book III, Ptolemy took up the problem of the Sun's motion. He described previous observations of the length of the year and summarized in a table the results of the Sun's regular movement. The next task, he wrote, was to explain the apparent irregularity of the Sun's motion as a combination of regular circular motions. Both the eccentric and the epicycle hypotheses produced the observed motion.

Book IV of the Almagest is concerned with motions of the Moon, which Ptolemy presented in a table. Complex irregularities called for ingenious solutions were the lunar observations to be saved by a combination of regular circular motions.

Ptolemy determined the variation of the Moon in latitude from three ancient eclipses observed in Babylon and from three among those "most carefully observed" by himself in Alexandria. (Eclipse observations are particularly convenient to work with, because at these times the Moon is known to be on the ecliptic and at a longitude 180 degrees from the Sun.)

Ptolemy reproduced the lunar anomaly in a qualitative way simply by inclining the plane of the lunar deferent to the plane of the ecliptic. More quantitatively, the Moon was observed neither to cross the ecliptic at the same longitude after each revolution nor to return to the same latitude in equal intervals of time. It would have done so had its epicycle revolved counterclockwise around the deferent with the same angular speed with which the Moon moved clockwise around its epicycle. Ptolemy slowed the speed of the Moon around

Figure 9.1: Ptolemy's Lunar Theory. top: To produce a variation in latitude, Ptolemy inclined the Moon's deferent at an angle of about 5 degrees to the ecliptic, thus moving the Moon (M) above and below the Sun (S) as observed from the Earth (E). bottom: A further variation was introduced by reducing the clockwise speed of the Moon around its epicycle. This speed was made less than the counterclockwise speed of rotation of the epicycle itself around the deferent. In the time it took for the epicycle to move through 360 degrees and so return to its original position on the deferent, the Moon, which began at point 1 on its epicycle, moved only to point 2, a clockwise motion of less than 360 degrees.

Figure 9.1: Ptolemy's Lunar Theory. top: To produce a variation in latitude, Ptolemy inclined the Moon's deferent at an angle of about 5 degrees to the ecliptic, thus moving the Moon (M) above and below the Sun (S) as observed from the Earth (E). bottom: A further variation was introduced by reducing the clockwise speed of the Moon around its epicycle. This speed was made less than the counterclockwise speed of rotation of the epicycle itself around the deferent. In the time it took for the epicycle to move through 360 degrees and so return to its original position on the deferent, the Moon, which began at point 1 on its epicycle, moved only to point 2, a clockwise motion of less than 360 degrees.

Figure 9.1: (Continued) above: Yet further variation was produced by making the Moon's deferent, with its center at C, eccentric to the Earth, at E. This stratagem varied the distance of the Moon's epicycle from the Earth and thus also the apparent angular speed of the Moon's epicycle, as viewed from the Earth. However, Ptolemy kept the Moon's deferent rotating with uniform circular motion about the Earth at E, rather than about the actual center of the deferent, at C.

Figure 9.1: (Continued) above: Yet further variation was produced by making the Moon's deferent, with its center at C, eccentric to the Earth, at E. This stratagem varied the distance of the Moon's epicycle from the Earth and thus also the apparent angular speed of the Moon's epicycle, as viewed from the Earth. However, Ptolemy kept the Moon's deferent rotating with uniform circular motion about the Earth at E, rather than about the actual center of the deferent, at C.

its epicycle while maintaining the (now greater) speed of the epicycle around the deferent. The resulting combination of regular circular motions adequately reproduced the lunar positions near new and full moon (when the Moon is in the direction of the Sun, and in the opposite direction, as viewed from the Earth).

Agreement was less satisfactory when the Moon was at first or third quarter (90 degrees from the Sun). Ptolemy sought to eliminate this discrepancy by varying the distance of the Moon's epicycle from the Earth (which in turn would vary the apparent angular speed of the epicycle as viewed from the Earth). He placed the Moon's epicycle on a deferent eccentric to the Earth.

The Moon's epicycle now had a new deferent. Ptolemy, however, kept the epicycle's center rotating with uniform circular motion about the center of its previous, geocentric deferent. He did this in order to preserve the match between observation and theory already achieved. The eccentric deferent was also made to rotate with a uniform circular motion about the center of the ecliptic rather than about its own center.

Ptolemy obscured the violation of regular circular motion by using in his example an eccentric circle inside a circle that was itself concentric to the

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