HIPPARCHUS

(b. Nicaea, Bithynia [now Iznik, Turkey], first quarter of second century BCE; d. Rhodes [?], after 127 BCE),

astronomy, mathematics, geography. For the original article on Hipparchus see DSB, vol. 15.

Hipparchus has continued to be a subject of controversy among historians of astronomy on account of the fragmentary character of much of our evidence for his writings and for Greek astronomy in general before Ptolemy (second century CE). At stake, particularly, is how his work stands in relation on the one hand to Ptolemy’s astronomical theories and methods, and on the other to the Babylonian mathematical astronomy of the last three centuries BCE. The prevailing view of Hipparchus’s historical place shifted through the great part of the twentieth century away from the assumption that Ptolemy’s Almagest was substantially a recapitulation and completion of his work. Concepts, conventions, observations, and numerical elements of Babylonian origin have become recognized as pervading his astronomy. More recently, however, scholarship has begun to react against portraying Hipparchus as a user and advocate of the approximative arithmetical methodology characteristic of Babylonian astronomy as opposed to the trigonometrical methods of the Almagest.

The original DSB article asserted that Hipparchus constructed and used a trigonometrical chord table based on a standard circle of radius 3,438 units. This was, in fact, a hypothesis resting on a reconstruction of Hipparchus’s calculations of the eccentricity of the Moon’s orbit that its author subsequently repudiated, although it has since been persuasively revived in modified form. That Hipparchus was able to make computations involving plane trigonometry has never been seriously in question. As long ago as 1934 it was shown that a series of terrestrial latitudes that—according to Strabo (64 or 63 BCE–after 23 CE)—Hipparchus associated with specific values for the maximum length of day are in close agreement with

computation by spherical trigonometry, but this evidence was long disregarded. Analemma techniques could have yielded these latitudes, but they do not seem capable of certain calculations in spherical astronomy reported in the extant Commentary on Aratus of Hipparchus. Hence it appears that Hipparchus was in possession of Menelaus’s theorem or some equivalent and did not depend on Babylonian-style arithmetical sequences to approximate spherical functions.

Hipparchus’s familiarity with Babylonian astronomy has given rise to the attractive conjecture that he might have obtained access to this technical knowledge by visiting Babylon. Remarkably, scholars now have firm evidence that the Babylonian scribes of the late second century BCE knew something of Hipparchus’s work on solar theory. The Babylonian cuneiform BM 55555 (published by Neugebauer as ACT No. 210) reports a value for the length of the year that corresponds with high precision to 108,478 days divided by 297. An interval of 297 years turns out to be that between the summer solstice observations of Meton in 432 BCE and Hipparchus in 135 BCE that Ptolemy says Hipparchus discussed in his On the Length of the Year. Interestingly, the year length in BM 55555 is slightly shorter than the tropical year that Ptolemy took over from Hipparchus, which apparently was derived from comparison of a different pair of solstices.

Fresh studies of the Commentary on Aratus necessitate some significant revisions of statements in the original DSB article about Hipparchus’s data for stellar positions and their relationship to the catalog of stars in Books VII and VIII of the Almagest. Heinrich Vogt’s 1925 investigation concluding that the Commentary data were independent of Ptolemy’s positions suffered from serious defects, including limitation of its analysis to the relatively small number of stars for which the Commentary appeared to provide sufficient information to allow reconstruction of both ecliptic coordinates. More careful recent analyses of a much larger body of data lead to the conclusion that Ptolemy’s coordinates for most of the stars that have positional data in the Commentary were derived ultimately from the same measurements as the data in the Commentary, and this was presumably a lost catalog by Hipparchus. This does not necessarily mean that Hipparchus’s catalog was the source for all Ptolemy’s stars, because in particular the Almagest catalog is proportionally much more abundant in dim stars than the Commentary. Now confirmed is the contention of Jean Baptiste Delambre that, at the time that he wrote the Commentary, Hipparchus’s regular method of representing stellar positions was by means of equatorial coordinates, so that the dependence of Ptolemy’s catalog on Hipparchus must have involved a conversion of coordinates, which may for that matter have occurred during the interval between Hipparchus and Ptolemy.

Combinatorics . That Hipparchus did some work on combinatorics (which has generally not been recognized as having been a distinct mathematical subdiscipline in antiquity) is known from a passage (1047CE) in the De Stoicorum Repugnantiis of Plutarch (first and early second centuries CE). The passage asserts that whereas the Stoic Chrysippus (third century BCE) counted “the conjunctions arising from ten assertibles” as exceeding one hundred myriads, Hipparchus proved that “the affirmative yields 103049 conjoined assertibles and the negative 310952.” The first light cast on the enigma of what Hipparchus was counting, and how, was the recognition in the 1990s that 103,049 is the tenth of the series of Schröder numbers, while 310,952 differs only in its final digit (2 instead of 4) from half the sum of the tenth and eleventh Schröder numbers—likely reflecting a textual error in antiquity.

The n th Schröder number is the number of possible bracketings of a string of n symbols. Hipparchus seems to have interpreted Stoic logic as allowing an ordered set of n simple affirmative statements to be combined hierarchically by an “and” conjunction according to rules structurally analogous to bracketing, such that any such combination is counted as distinct from any other. A further rule governs the counting of negative conjunctions, understood as a single negation applied to what amounts to the first bracketed section of a set of affirmative statements. (Whether Hipparchus’s interpretation of these conjunctions was the same as Chrysippus’s is disputable.) From a mathematical point of view, what is most interesting about Hipparchus’s computations is that he succeeded in framing and correctly solving a combinatorical problem that was far from trivial and that his method of calculating the Schröder numbers must have had a recursive character.

SUPPLEMENTARY BIBLIOGRAPHY

Acerbi, Fabio. “On the Shoulders of Hipparchus: A Reappraisal of Ancient Greek Combinatorics.” Archive for History of Exact Sciences 57 (2003): 465–502. Explains the meaning of the numbers reported by Plutarch in relation to Stoic logic and offers a reconstruction of Hipparchus’s method of computing them.

Diller, Aubrey. “Geographical Latitudes in Eratosthenes, Hipparchus, and Posidonius.” Klio 27 (1934): 258–269. On the use of spherical trigonometry in Hipparchus’s geographical and astronomical work.

Duke, Dennis W. “Associations between the Ancient Star Catalogues.” Archive for History of Exact Sciences 56 (2002): 435–450. On the relationship between the Aratus commentary and the Almagest star catalog.

———. “Hipparchus’ Coordinate System.” Archive for History of Exact Sciences 56 (2002): 427–433. Shows the equatorial basis of Hipparchus’s stellar data.

———. “Hipparchus’ Eclipse Trios and Early Trigonometry.” Centaurus 47 (2005): 163–177. Defends Toomer’s reconstruction of the chord table.

Grasshoff, Gerd. The History of Ptolemy’s Star Catalogue. New York: Springer-Verlag, 1990.

Habsieger, Laurent, Maxim Kazarian, and Sergi Lando. “On the Second Number of Plutarch.” American Mathematical Monthly 105 (1998): 446. Supplement to Stanley’s paper cited below.

Jones, Alexander. “In Order that We Should Not Ourselves Appear to be Adjusting Our Estimates ..... to Make Them Fit Some Predetermined Amount.” In Wrong for the Right Reasons, edited by Jed Z. Buchwald and Allan Franklin. Dordrecht, Netherlands: Springer 2005. A discussion of the implications of the new Hipparchian value for the tropical year in BM 55555 for Hipparchus’s and Ptolemy’s solar theory.

Rawlins, D. “Hipparchos’ Ultimate Solar Orbit & the Babylonian Tropical Year.” Dio 1 (1991): 49–66. The new Hipparchian value for the tropical year in BM 55555 is explained.

Sidoli, Nathan. “Hipparchus and the Ancient Metrical Methods on the Sphere.” Journal for the History of Astronomy 35 (2004): 71–84.

Stanley, Richard P. “Hipparchus, Plutarch, Schröder, and Hough.” American Mathematical Monthly 104 (1997): 344–350. Original report of Hough’s identification of the connection between Hipparchus’s combinatorics and Schröder numbers.

Toomer, Gerald J. “Hipparchus and Babylonian Astronomy.” A Scientific Humanist: Studies in Memory of Abraham Sachs, edited by Erle Leichty, Maria deJ. Ellis, and Pamela Gerardi. Philadelphia. Occasional Publications of the Samuel Noah Kramer Fund, 9, 1988. Discusses elements of Babylonian astronomy in Hipparchus’s work and his conjectural sojourn in Babylon.

Alexander Jones