As the longitudes of all the fixed stars are increased by this quantity, the effects of precession are soon detected. It was accordingly discovered by Hipparchus in the year 128 before Christ, from a comparison of his own observations with those of Timocharis 155 years before. In the time of Hipparchus the entrance of the sun into the constellation Aries was the beginning of spring; but since that time the equinoctial points have receded 30°, so that the constellations called the signs of the zodiac are now at a considerable distance from those divisions of the ecliptic which bear their names. Moving at the rate of 50ʺ·1 annually, the equinoctial points will accomplish a revolution in 25,868 years. But, as the precession varies in different centuries, the extent of this period will be slightly modified. Since the motion of the sun is direct, and that of the equinoctial points retrograde, he takes a shorter time to return to the equator than to arrive at the same stars; so that the tropical year of 365^d 5^h 48^m 49^s·7 must be increased by the time he takes to move through an arc of 50ʺ·1, in order to have the length of the sidereal year. The time required is 20^m 19^s·6, so that the sidereal year contains 365^d 6^h 9^m 9^s·6 mean solar days. The mean annual precession is subject to a secular variation; for, although the change in the plane of the ecliptic in which the orbit of the sun lies be independent of the form of the earth, yet, by bringing the sun, moon, and earth into different relative positions from age to age, it alters the direct action of the two first on the prominent matter at the equator: on this account the motion of the equinox is greater by 0ʺ·455 now than it was in the time of Hipparchus. Consequently the actual length of the tropical year is about 4^s·21 shorter than it was at that time. The utmost change that it can experience from this cause amounts to 43 seconds.