In any case the work started by Hipparchus has had a lasting heritage, and was much later updated by Al Sufi (964) and Copernicus (1543). Ulugh Beg reobserved all the Hipparchus stars he could see from Samarkand in 1437 to about the same accuracy as Hipparchus's. The catalog was superseded only in the late 16th century by Brahe and Wilhelm IV of Kassel via superior ruled instruments and spherical trigonometry, which improved accuracy by an order of magnitude even before the invention of the telescope. Hipparchus is considered the greatest observational Astronomer from classical antiquity until Brahe.
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's 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. His other triplet of solar positions is consistent with 94¼ and 92½ days, 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).
As with most of his work, Hipparchus's star catalog was adopted and perhaps expanded by Ptolemy. Delambre, in 1817, cast doubt on Ptolemy's work. It was disputed whether the star catalog in the Almagest is due to Hipparchus, but 1976–2002 statistical and spatial analyses (by R. R. Newton, Dennis Rawlins, Gerd Grasshoff, Keith Pickering and Dennis Duke) have shown conclusively that the Almagest star catalog is almost entirely Hipparchan. Ptolemy has even (since Brahe, 1598) been accused by astronomers of fraud for stating (Syntaxis, book 7, chapter 4) that he observed all 1025 stars: for almost every star he used Hipparchus's data and precessed it to his own epoch 2 ⁄3 centuries later by adding 2°40' to the longitude, using an erroneously small precession constant of 1° per century.
Hipparchus ranked stars in three magnitude very general classes according to their brightness but he did not assign a numerical brightness value to any star. The magnitude system ranging from 1 (brightest) to 6 (faintest) was established by Ptolemy. That system by Ptolemy is effectively still in use today, though extended and made more precise through the introduction of a logarithmic scale by N. R. Pogson in 1856.
Pliny also remarks that "he also discovered for what exact reason, although the Shadow causing the eclipse must from sunrise onward be below the earth, it happened once in the past that the moon was eclipsed in the west while both luminaries were visible above the earth" (translation H. Rackham (1938), Loeb Classical Library 330 p. 207). Toomer (1980) argued that this must refer to the large total lunar eclipse of 26 November 139 BC, when over a clean sea horizon as seen from Rhodes, the Moon was eclipsed in the North West just after the Sun rose in the southeast. This would be the second eclipse of the 345-year interval that Hipparchus used to verify the traditional Babylonian periods: this puts a late date to the development of Hipparchus's lunar theory. We do not know what "exact reason" Hipparchus found for seeing the Moon eclipsed while apparently it was not in exact opposition to the Sun. Parallax lowers the altitude of the luminaries; refraction raises them, and from a high point of view the horizon is lowered.
But some scholars do not believe Āryabhaṭa's sine table has anything to do with Hipparchus's chord table which does not exist today. Some scholars do not agree with this hypothesis that Hipparchus constructed a chord table. Bo C. Klintberg states "With mathematical reconstructions and philosophical arguments I show that Toomer's 1973 paper never contained any conclusive evidence for his claims that Hipparchus had a 3438'-based chord table, and that the Indians used that table to compute their sine tables. Recalculating Toomer's reconstructions with a 3600' radius – i.e. the radius of the chord table in Ptolemy's Almagest, expressed in 'minutes' instead of 'degrees' – generates Hipparchan-like ratios similar to those produced by a 3438' radius. It is therefore possible that the radius of Hipparchus's chord table was 3600', and that the Indians independently constructed their 3438'-based sine table."
Hipparchus could confirm his computations by comparing eclipses from his own time (presumably 27 January 141 BC and 26 November 139 BC according to [Toomer 1980]), with eclipses from Babylonian records 345 years earlier (Almagest IV.2; [A.Jones, 2001]). Already al-Biruni (Qanun VII.2.II) and Copernicus (de revolutionibus IV.4) noted that the period of 4,267 moons is actually about 5 minutes longer than the value for the eclipse period that Ptolemy attributes to Hipparchus. However, the timing methods of the Babylonians had an error of no less than 8 minutes. Modern scholars agree that Hipparchus rounded the eclipse period to the nearest hour, and used it to confirm the validity of the traditional values, rather than try to derive an improved value from his own observations. From modern ephemerides and taking account of the change in the length of the day (see ΔT) we estimate that the error in the assumed length of the synodic month was less than 0.2 seconds in the 4th century BC and less than 0.1 seconds in Hipparchus's time.
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. As shown in a 1991 paper, in 158 BC Hipparchus computed a very erroneous summer solstice from Callippus's calendar. He observed the summer solstice in 146 and 135 BC both accurate to a few hours, but observations of the moment of equinox were simpler, and he made twenty during his lifetime. Ptolemy gives an extensive discussion of Hipparchus's work on the length of the year in the Almagest III.1, and quotes many observations that Hipparchus made or used, spanning 162–128 BC. Analysis 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.
Hipparchus is thought to be the first to calculate a heliocentric system, but he abandoned his work because the calculations showed the orbits were not perfectly circular as believed to be mandatory by the science of the time. As an Astronomer of antiquity his influence, supported by ideas from Aristotle, held sway for nearly 2000 years, until the heliocentric model of Copernicus.
Hipparchus was in the international news in 2005, when it was again proposed (as in 1898) that the data on the celestial globe of Hipparchus or in his star catalog may have been preserved in the only surviving large ancient celestial globe which depicts the constellations with moderate accuracy, the globe carried by the Farnese Atlas. There are a variety of mis-steps in the more ambitious 2005 paper, thus no specialists in the area accept its widely publicized speculation.
Hipparchus applied his knowledge of spherical angles to the Problem of denoting locations on the Earth's surface. Before him a grid system had been used by Dicaearchus of Messana, but Hipparchus was the first to apply mathematical rigor to the determination of the latitude and longitude of places on the Earth. Hipparchus wrote a critique in three books on the work of the geographer Eratosthenes of Cyrene (3rd century BC), called Pròs tèn 'Eratosthénous geografían ("Against the Geography of Eratosthenes"). It is known to us from Strabo of Amaseia, who in his turn criticised Hipparchus in his own Geografia. Hipparchus apparently made many detailed corrections to the locations and distances mentioned by Eratosthenes. It seems he did not introduce many improvements in methods, but he did propose a means to determine the geographical longitudes of different cities at lunar eclipses (Strabo Geografia 1 January 2012). A lunar eclipse is visible simultaneously on half of the Earth, and the difference in longitude between places can be computed from the difference in local time when the eclipse is observed. His approach would give accurate results if it were correctly carried out but the limitations of timekeeping accuracy in his era made this method impractical.
Previously, Eudoxus of Cnidus in the 4th century BC had described the stars and constellations in two books called Phaenomena and Entropon. Aratus wrote a poem called Phaenomena or Arateia based on Eudoxus's work. Hipparchus wrote a commentary on the Arateia – his only preserved work – which contains many stellar positions and times for rising, culmination, and setting of the constellations, and these are likely to have been based on his own measurements.
In the first book, Hipparchus assumes that the parallax of the Sun is 0, as if it is at infinite distance. He then analyzed a solar eclipse, which Toomer (against the opinion of over a century of astronomers) presumes to be the eclipse of 14 March 190 BC. It was total in the region of the Hellespont (and in his birthplace, Nicaea); at the time Toomer proposes the Romans were preparing for war with Antiochus III in the area, and the eclipse is mentioned by Livy in his Ab Urbe Condita Libri VIII.2. It was also observed in Alexandria, where the Sun was reported to be obscured 4/5ths by the Moon. Alexandria and Nicaea are on the same meridian. Alexandria is at about 31° North, and the region of the Hellespont about 40° North. (It has been contended that authors like Strabo and Ptolemy had fairly decent values for these geographical positions, so Hipparchus must have known them too. However, Strabo's Hipparchus dependent latitudes for this region are at least 1° too high, and Ptolemy appears to copy them, placing Byzantium 2° high in latitude.) Hipparchus could draw a triangle formed by the two places and the Moon, and from simple geometry was able to establish a distance of the Moon, expressed in Earth radii. Because the eclipse occurred in the morning, the Moon was not in the meridian, and it has been proposed that as a consequence the distance found by Hipparchus was a lower limit. In any case, according to Pappus, Hipparchus found that the least distance is 71 (from this eclipse), and the greatest 81 Earth radii.
Hipparchus thus had the problematic result that his minimum distance (from book 1) was greater than his maximum mean distance (from book 2). He was intellectually honest about this discrepancy, and probably realized that especially the first method is very sensitive to the accuracy of the observations and parameters. (In fact, modern calculations show that the size of the 189 BC solar eclipse at Alexandria must have been closer to 9/10ths and not the reported 4/5ths, a fraction more closely matched by the degree of totality at Alexandria of eclipses occurring in 310 and 129 BC which were also nearly total in the Hellespont and are thought by many to be more likely possibilities for the eclipse Hipparchus used for his computations.)
Hipparchus's treatise "Against the Geography of Eratosthenes" in three books is not preserved. Most of our knowledge of it comes from Strabo, according to whom Hipparchus thoroughly and often unfairly criticized Eratosthenes mainly for internal contradictions and inaccuracy in determining positions of geographical localities. Hipparchus insists that a geographic map must be based only on astronomical measurements of latitudes and longitudes and triangulation for finding unknown distances. In geographic theory and methods Hipparchus introduced three main innovations. He was the first to use the grade grid, to determine geographic latitude from star observations, and not only from the sun’s altitude, a method known long before him, and to suggest that geographic longitude could be determined by means of simultaneous observations of lunar eclipses in distant places. In the practical part of his work, the so-called "table of climata", Hipparchus listed latitudes for several tens of localities. In particular, he improved Eratosthenes' values for the latitudes of Athens, Sicily, and southern extremity of India. In calculating latitudes of climata (latitudes correlated with the length of the longest solstitial day), Hipparchus used an unexpectedly accurate value for the obliquity of the ecliptic, 23°40′ (the actual value in the second half of the 2nd century BC was approximately 23°43′), whereas all other ancient authors knew only a roughly rounded value 24°, and even Ptolemy used a less accurate value, 23°51′. Hipparchus opposed the view generally accepted in the Hellenistic period that the Atlantic and Indian Oceans and the Caspian Sea are parts of a single ocean. At the same time he extends the limits of the oikoumene, i.e. the inhabited part of the land, up to the equator and the Arctic Circle. Hipparchus’ ideas found their reflection in the Geography of Ptolemy. In essence, Ptolemy's work is an extended attempt to realize Hipparchus’ vision of what geography ought to be.
