* 42] Hipparchus 55 



equinoctial point at T arrives at the end of a tropical 

 year at the new equinoctial point at T'; but the sidereal 

 year is only complete when the sun has furiher described 

 the arc r'r and returned to its original starting-point T. 

 Hence, taking the modern estimate 50" of the arc T T', 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 , days or about 20 



1296000 



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 astrono- 

 mers 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 365 \ 

 days for the length of the year had to be diminished 

 by g-J^th 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 observa- 

 tions, and concludes that the time of a solstice may be 

 erroneous to the extent of about f day, while that of an 

 equinox may be expected to be within \ day of the truth. 

 In the illustration given, this would indicate a possible 

 error of i| 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 



