[Illustration: FIG. 33.—The precession of the equinoxes.] The annual motion ♈ ♈′ was, as has been stated, estimated by Hipparchus as being at least 36″ (equivalent to one degree in a century), and probably more. Its true value is considerably more, namely about 50″. [Illustration: FIG. 24.—The precession of the equinoxes.] An important consequence of the motion of the equator thus discovered is that the sun in its annual journey round the ecliptic, after starting from the equinoctial point, returns to the new position of the equinoctial point a little before returning to its original position with respect to the stars, and the successive equinoxes occur slightly earlier than they otherwise would. From this fact is derived the name =precession of the equinoxes=, or more shortly, =precession=, which is applied to the motion that we have been considering. Hence it becomes necessary to recognise, as Hipparchus did, two different kinds of year, the =tropical year= or period required by the sun to return to the same position with respect to the equinoctial points, and the =sidereal year= or period of return to the same position with respect to the stars. If ♈ ♈′ denote the motion of the equinoctial point during a tropical year, then the sun after starting from the equinoctial point at ♈ arrives—at the end of a tropical year—at the new equinoctial point at ♈′; but the sidereal year is only complete when the sun has further described the arc ♈′ ♈ and returned to its original starting-point ♈. Hence, taking the modern estimate 50″ of the arc ♈ ♈′, the sun, in the sidereal year, describes an arc of 360°, in the tropical year an arc less by 50″, or 359° 59′ 10″; the lengths of the two years are therefore in this proportion, and the amount by which the sidereal year exceeds the tropical year bears to either the same ratio as 50″ to 360° (or 1,296,000″), and is therefore (365-1∕4 × 50)∕1296000 days of about 20 minutes. Another way of expressing the amount of the precession is to say that the equinoctial point will describe the complete circuit of the ecliptic and return to the same position after about 26,000 years.