Hipparchus : biography

At the end of his career, Hipparchus wrote a book called Peri eniausíou megéthous ("On the Length of the Year") about his results. The established value for the tropical year, introduced by Callippus in or before 330 BC was 365 + 1/4 days. Speculating a Babylonian origin for the Callippic year is hard to defend, since Babylon did not observe solstices thus the only extant System B yearlength was based on Greek solstices (see below). Hipparchus’ equinox observations gave varying results, but he himself points out (quoted in Almagest III.1(H195)) that the observation errors by himself and his predecessors may have been as large as 1/4 day. He used old solstice observations, and determined a difference of about one day in about 300 years. So he set the length of the tropical year to 365 + 1/4 – 1/300 days (= 365.24666… days = 365 days 5 hours 55 min, which differs from the actual value (modern estimate) of 365.24219… days = 365 days 5 hours 48 min 45 s by only about 6 min).

Between the solstice observation of Meton and his own, there were 297 years spanning 108,478 days. D.Rawlins noted that this implies a tropical year of 365.24579… days = 365 days;14,44,51 (sexagesimal; = 365 days + 14/60 + 44/602 + 51/603) and that this exact yearlength has been found on one of the few Babylonian clay tablets which explicitly specifies the System B month. This is an indication that Hipparchus’ work was known to Chaldeans., volume 1, number 1, pages 49–66; A.Jones, 2001; Thurston, op. cit., page 62

Another value for the year that is attributed to Hipparchus (by the astrologer Vettius Valens in the 1st century) is 365 + 1/4 + 1/288 days (= 365.25347… days = 365 days 6 hours 5 min), but this may be a corruption of another value attributed to a Babylonian source: 365 + 1/4 + 1/144 days (= 365.25694… days = 365 days 6 hours 10 min). It is not clear if this would be a value for the sidereal year (actual value at his time (modern estimate) about 365.2565 days), but the difference with Hipparchus’ value for the tropical year is consistent with his rate of precession (see below).

Orbit of the Sun

Before Hipparchus, astronomers knew that the lengths of the seasons are not equal. Hipparchus made observations of equinox and solstice, and according to Ptolemy (Almagest III.4) determined that spring (from spring equinox to summer solstice) lasted 94½ days, and summer (from summer solstice to autumn equinox) 92½ days. This is inconsistent with a premise of the Sun moving around the Earth in a circle at uniform speed. Hipparchus’ solution was to place the Earth not at the center of the Sun’s motion, but at some distance from the center. This model described the apparent motion of the Sun fairly well. It is known today that the planets, including the Earth, move in approximate ellipses around the Sun, but this was not discovered until Johannes Kepler published his first two laws of planetary motion in 1609. The value for the eccentricity attributed to Hipparchus by Ptolemy is that the offset is 1/24 of the radius of the orbit (which is a little too large), and the direction of the apogee would be at longitude 65.5° from the vernal equinox. Hipparchus may also have used other sets of observations, which would lead to different values. One of his two eclipse trios’ solar longitudes are consistent with his having initially adopted inaccurate lengths for spring and summer of 95¾ and 91¼ days.Thurston, op. cit., page 67, note 16. R.Newton proposed that Hipparchus made an error of a degree in one of the trios’ eclipses. D.Rawlins’s theory (Thurston op. cit.) that Hipparchus analysed the two trios in pairs not threesomes provides a possible explanation for the one degree slip. It was a necessitated by inadequacies of analysing by pairs instead of using the better method Ptolemy applies at Almagest Book 4 Parts 6 and 11. His other triplet of solar positions is consistent with 94¼ and 92½ days,Ibid, note 14; Jones 2001 an improvement on the results (94½ and 92½ days) attributed to Hipparchus by Ptolemy, which a few scholars still question the authorship of. Ptolemy made no change three centuries later, and expressed lengths for the autumn and winter seasons which were already implicit (as shown, e.g., by A. Aaboe).