Hipparchus and Ptolemy founded their theory of the moon on these observations. Perhaps we cannot consider, as equally certain, the story that, at the time of Alexander's conquest, the Chaldeans possessed a series of observations, which went back 1903 years, and which Aristotle caused Callisthenes to bring to him in Greece. All the Greek observations which are of any value, begin with the school of Alexandria. Aristyllus and Timocharis appear, by the citations of Hipparchus, to have observed the places of stars and planets, and the times of the solstices, at various periods from B.C. 295 to ы.C. 269. Without their observations, indeed, it would not have been easy for Hipparchus to establish either the theory of the sun or the precession of the equinoxes.

In order that observations at distant intervals may be compared with each other, they must be referred to some common era. The Chaldeans dated by the era of Nabonassar, which commenced 749 B.C. The Greek observations were referred to the Calippic periods of 76 years, of which the first began 331 B.C. These are the dates used by Hipparchus and Ptolemy.

CHAPTER III.

INDUCTIVE EPOCH OF HIPPARCHUS.

Sect. 1.-Establishment of the Theory of Epicycles

A

and Eccentrics.

LTHOUGH, as we have already seen, at the

time of Plato, the Idea of Epicycles had been suggested, and the problem of its general application proposed, and solutions of this problem offered by his followers; we still consider Hipparchus as the real discoverer and founder of that theory; inasmuch as he not only guessed that it might, but showed that it must, account for the phenomena, both as to their nature and as to their quantity. The assertion that "he only discovers who proves," is just; not only because, until a theory is proved to be the true one, it has no pre-eminence over the numerous other guesses among which it circulates, and above which the proof alone elevates it; but also because he who takes hold of the theory so as to apply calculation to it, possesses it with a distinctness of conception which makes it peculiarly his.

In order to establish the Theory of Epicycles, it was necessary to assign the magnitudes, distances,

and positions of the circles or spheres in which the heavenly bodies were moved, in such a manner as to account for their apparently irregular motions. We may best understand what was the problem to be solved, by calling to mind what we now know to be the real motions of the heavens. The true motion of the earth round the sun, and therefore the apparent annual motion of the sun, is performed, not in a circle of which the earth is the center, but in an ellipse or oval, the earth being nearer to one end than to the other; and the motion is most rapid when the sun is at the nearer end of this oval. But instead of an oval, we may suppose the sun to move uniformly in a circle, the earth being now, not in the center, but nearer to one side; for on this supposition, the sun will appear to move most quickly when he is nearest to the earth, or in his Perigee, as that point is called. Such an orbit is called an Eccentric, and the distance of the earth from the center of the circle is called the Eccentricity. It may easily be shown by geometrical reasoning, that the inequality of apparent motion so produced, is exactly the same in detail, as the inequality which follows from the hypothesis of a small Epicycle, turning uniformly on its axis, and carrying the sun in its circumference, while the center of this epicycle moves uniformly in a circle of which the earth is the center. This identity of the results of the hypothesis of the Eccentric and the Epicyle is proved by Ptolemy in the third book of the "Almagest."

The Sun's Eccentric.-When Hipparchus had clearly conceived these hypotheses, as possible ways of accounting for the sun's motion, the task which he had to perform, in order to show that they deserved to be adopted, was to assign a place to the Perigee, a magnitude to the Eccentricity, and an Epoch at which the sun was at the perigee; and to show that, in this way, he had produced a true representation of the motions of the sun. This, accordingly, he did; and having thus determined, with considerable exactness, both the law of the solar irregularities, and the numbers on which their amount depends, he was able to assign the motions and places of the sun for any moment of future time with corresponding exactness; he was able, in short, to construct Solar Tables, by means of which the sun's place with respect to the stars could be correctly found at any time. These tables (as they are given by Ptolemy',) give the Anomaly, or inequality of the sun's motion; and this they exhibit by means of the Prosthapheresis, the quantity which, at any distance of the sun from the Apogee, it is requisite to add to or subtract from the arc, which he would have described if his motion had been equable.

The reader might perhaps expect that the calculations which thus exhibited the motions of the sun for an indefinite future period must depend upon a considerable number of observations made at all seasons of the year. That, however, was not the Syntax. 1. iii.

case; and the genius of the discoverer appeared, as such genius usually does appear, in his perceiving how small a number of facts, rightly considered, were sufficient to form a foundation for the theory. The number of days contained in two seasons of the year sufficed for this purpose to Hipparchus. "Having ascertained," says Ptolemy, "that the time from the vernal equinox to the summer tropic is 941 days, and the time from the summer tropic to the autumnal equinox 921 days, from these phenomena alone he demonstrates that the straight line joining the centre of the sun's eccentric path with the centre of the zodiac (the spectator's eye) is nearly the 24th part of the radius of the eccentric path; and that its apogee precedes the summer solstice by 24 degrees nearly, the zodiac containing 360."

The exactness of the Solar Tables, or Canon, which was founded on these data, was manifested, not only by the coincidence of the sun's calculated place with such observations as the Greek astronomers of this period were able to make, (which were indeed very rude,) but by its enabling them to calculate solar and lunar eclipses; phenomena which are a very precise and severe trial of the accuracy of such tables, inasmuch as a very minute change in the apparent place of the sun or moon would completely alter the obvious features of the eclipse. Though the tables of this period were by no means perfect, they bore with tolerable credit this trying