Hipparchus : biography

1. In the first, the Moon would move uniformly along a circle, but the Earth would be eccentric, i.e., at some distance of the center of the circle. So the apparent angular speed of the Moon (and its distance) would vary.
2. The Moon itself would move uniformly (with some mean motion in anomaly) on a secondary circular orbit, called an epicycle, that itself would move uniformly (with some mean motion in longitude) over the main circular orbit around the Earth, called deferent; see deferent and epicycle.

Apollonius demonstrated that these two models were in fact mathematically equivalent. However, all this was theory and had not been put to practice. Hipparchus was the first astronomer we know attempted to determine the relative proportions and actual sizes of these orbits.

Hipparchus devised a geometrical method to find the parameters from three positions of the Moon, at particular phases of its anomaly. In fact, he did this separately for the eccentric and the epicycle model. Ptolemy describes the details in the Almagest IV.11. Hipparchus used two sets of three lunar eclipse observations, which he carefully selected to satisfy the requirements. The eccentric model he fitted to these eclipses from his Babylonian eclipse list: 22/23 December 383 BC, 18/19 June 382 BC, and 12/13 December 382 BC. The epicycle model he fitted to lunar eclipse observations made in Alexandria at 22 September 201 BC, 19 March 200 BC, and 11 September 200 BC.

• For the eccentric model, Hipparchus found for the ratio between the radius of the eccenter and the distance between the center of the eccenter and the center of the ecliptic (i.e., the observer on Earth): 3144 : 327+2/3 ;
• and for the epicycle model, the ratio between the radius of the deferent and the epicycle: 3122+1/2 : 247+1/2 .

The somewhat weird numbers are due to the cumbersome unit he used in his chord table according to one group of historians, who explain their reconstruction’s inability to agree with these four numbers as partly due to some sloppy rounding and calculation errors by Hipparchus, for which Ptolemy criticised him (he himself made rounding errors too). A simpler alternate reconstruction Summarized in Hugh Thurston (2002): Isis 93, 58–69. agrees with all four numbers. Anyway, Hipparchus found inconsistent results; he later used the ratio of the epicycle model (3122+1/2 : 247+1/2), which is too small (60 : 4;45 sexagesimal). Ptolemy established a ratio of 60 : 5+1/4.Toomer, 1967 (The maximum angular deviation producible by this geometry is the arcsin of 5 1/4 divided by 60, or about 5° 1′, a figure that is sometimes therefore quoted as the equivalent of the Moon’s equation of the center in the Hipparchan model.)

Apparent motion of the Sun

Before Hipparchus, Meton, Euctemon, and their pupils at Athens had made a solstice observation (i.e., timed the moment of the summer solstice) on 27 June 432 BC (proleptic Julian calendar). Aristarchus of Samos is said to have done so in 280 BC, and Hipparchus also had an observation by Archimedes. In 158 BC Hipparchus computed from calendars an estimate for summer solstice. He observed the summer solstice in 146 BC and 135 BC both to a few hours, but observations of the moment of equinox were more precise, and he made twenty during his lifetime. Ptolemy gives an extensive discussion of Hipparchus’ work on the length of the year in the Almagest III.1, and quotes many observations that Hipparchus made or used, spanning 162 BC to 128 BC. of Hipparchus’s seventeen equinox observations made at Rhodes shows that the mean error in declination is positive seven arc minutes, nearly agreeing with the sum of refraction by air and Swerdlow’s parallax. The random noise is two arc minutes or more nearly one arcminute if rounding is taken into account which approximately agrees with the sharpness of the eye. Ptolemy quotes an equinox timing by Hipparchus (at 24 March 146 BC at dawn) that differs by 5 hours from the observation made on Alexandria’s large public equatorial ring that same day (at 1 hour before noon): Hipparchus may have visited Alexandria but he did not make his equinox observations there; presumably he was on Rhodes (at nearly the same geographical longitude). He could have used the equatorial ring of his armillary sphere or another equatorial ring for these observations, but Hipparchus (and Ptolemy) knew that observations with these instruments are sensitive to a precise alignment with the equator, so if he were restricted to an armillary, it would make more sense to use its meridian ring as a transit instrument. The problem with an equatorial ring (if an observer is naive enough to trust it very near dawn or dusk) is that atmospheric refraction lifts the Sun significantly above the horizon: so for a northern hemisphere observer its apparent declination is too high, which changes the observed time when the Sun crosses the equator. (Worse, the refraction decreases as the Sun rises and increases as it sets, so it may appear to move in the wrong direction with respect to the equator in the course of the day – as Ptolemy mentions. Ptolemy and Hipparchus apparently did not realize that refraction is the cause.) However, such details have doubtful relation to the data of either man, since there is no textual, scientific, or statistical ground for believing that their equinoxes were taken on an equatorial ring, which is useless for solstices in any case. Not one of two centuries of mathematical investigations of their solar errors has claimed to have traced them to the effect of refraction on use of an equatorial ring. Ptolemy claims his solar observations were on a transit instrument set in the meridian.