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 36514 X 501296000 days or 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.
The length of each kind of year was also fixed by Hipparchus with considerable accuracy. That of the tropical year was obtained by comparing the times of solstices and equinoxes observed by earlier astronomers with those observed by himself. He found, for example, by comparison of the date of the summer solstice of 280 b.c, observed by Aristarchus of Samos, with that of the year 135 b.c., that the current estimate of 36514 days for the length of the year had to be diminished by 1300th of a day or about five minutes, an estimate confirmed roughly by other cases. It is interesting to note as an illustration of his scientific method that he discusses with some care the possible error of the observations, and concludes that the time of a solstice may be erroneous to the extent of about 34 day, while that of an equinox may be expected to be within 14 day of the truth. In the illustration given, this would indicate a possible error of 112 days in a period of 145 years, or about 15 minutes in a year. Actually his estimate of the length of the year is about six minutes too great, and the error is thus much less than that which he indicated as possible. In the course of this work he considered also the possibility of a change in the length of the year, and arrived at the conclusion that, although his observations were not precise
