Apart from individual astronomical discoveries, his chief services to astronomy may be put under four heads. He invented or greatly developed a special branch of mathematics,[21] which enabled processes of numerical calculation to be applied to geometrical figures, whether in a plane or on a sphere. He made an extensive series of observations, taken with all the accuracy that his instruments would permit. He systematically and critically made use of old observations for comparison with later ones so as to discover astronomical changes too slow to be detected within a single lifetime. Finally, he systematically employed a particular geometrical scheme (that of eccentrics, and to a less extent that of epicycles) for the representation of the motions of the sun and moon. 38. The merit of suggesting that the motions of the heavenly bodies could be represented more simply by combinations of uniform _circular_ motions than by the revolving _spheres_ of Eudoxus and his school (§ 26) is generally attributed to the great Alexandrine mathematician _Apollonius_ of Perga, who lived in the latter half of the 3rd century B.C., but there is no clear evidence that he worked out a system in any detail. On account of the important part that this idea played in astronomy for nearly 2,000 years, it may be worth while to examine in some detail Hipparchus’s theory of the sun, the simplest and most successful application of the idea. We have already seen (chapter I., § 10) that, in addition to the daily motion (from east to west) which it shares with the rest of the celestial bodies, and of which we need here take no further account, the sun has also an annual motion on the celestial sphere in the reverse direction (from west to east) in a path oblique to the equator, which was early recognised as a great circle, called the ecliptic. It must be remembered further that the celestial sphere, on which the sun appears to lie, is a mere geometrical fiction introduced for convenience; all that direct observation gives is the change in the sun’s direction, and therefore the sun may consistently be supposed to move in such a way as to vary its distance from the earth in any arbitrary manner, provided only that the alterations in the apparent size of the sun, caused by the variations in its distance, agree with those observed, or that at any rate the differences are not great enough to be perceptible. It was, moreover, known (probably long before the time of Hipparchus) that the sun’s apparent motion in the ecliptic is not quite uniform, the motion at some times of the year being slightly more rapid than at others.