8 to the establishment of the various parts of the theory of epicycles. It is probable that this theory was adopted with respect to the planets at or before the time of Plato. And Aristotle gives us an account of the system thus devised". "Eudoxus," he says, "attributed four spheres to each planet: the first revolved with the fixed stars (and this produced the diurnal motion); the second gave it a motion along the ecliptic (the mean motion in longitude); the third had its axis perpendicular to the ecliptic (and this gave the inequality of each planetary motion); the fourth produced the oblique motion transverse to this (the motion in latitude.)" He is also said to have attributed a motion in latitude and a corresponding sphere to the sun as well as to the moon, of which it is difficult to understand the meaning, if Aristotle has reported rightly of the theory; for it would be absurd to ascribe to Eudoxus a knowledge of the motions by which the sun deviates from the ecliptic. Calippus conceived that two additional spheres must be given to the sun and to the moon, in order to explain the phenomena : probably he was aware of the inequalities of the motions of these luminaries. He also proposed an additional sphere for each planet, to account, we may 7 Metaph. xi. 8. 8 Aristotle says "has its poles in the ecliptic," but this must be a mistake of his. He professes merely to receive these opinions from the professed astronomers “εκ της οικειοτατης φιλοσοφίας των μαθηματικών.” suppose, for the results of the eccentricity of the orbits. The hypothesis, in this form, does not appear to have been reduced to measure, and was, moreover, unnecessarily complex. The resolution of the oblique motion of the moon into two separate motions, by Eudoxus, was not the simplest way of conceiving it; and Calippus imagined the connexion of these spheres in some way which made it necessary nearly to double their number; in this manner his system had no less than 55 spheres. Such was the progress which the idea of the hypothesis of epicycles had made in men's minds, previously to the establishment of the theory by Hipparchus. There had also been a preparation for this step, on the other side, by the collection of facts. We know that observations of the eclipses of the moon were made by the Chaldeans 367 B. C. at Babylon, and were known to the Greeks; for Hipparchus and Ptolemy found 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, they had 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 B. c. 269. Without their observations, indeed, it would not have been easy for him 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 and Eccentrics. ALTHOUGH, as we have already seen, the idea of epicycles had been suggested, the problem of its general application proposed, at the time of Plato, and the solutions 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 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 centre, 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 centre, 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 centre 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 centre of this epicycle moves uniformly in a circle of which the earth is the centre. This identity of the results of the hypothesis of the eccentric and the epicycle is proved by Ptolemy in the third book of the "Almagest," The Sun's Eccentric.-When Hipparchus had clearly |