Hipparchus's Computations of Solar Longitudes

Bernsen [1969] Bernsen L. , “On the construction of Column B in System A of the astronomical cuneiform texts”, Centaurus , xiv (1969), 23–28.

Bowen-Goldstein [1988] Bowen A. C. Goldstein B. R. , “Meton of Athens and astronomy in the late fifth century b.c.”, in A scientific humanist: Studies in memory of Abraham Sachs , ed. by Leichty E. Ellis M. de J. Gerardi P. (Philadelphia, 1988), 39–81.

Britton [in press] Britton J. , The quality of Ptolemy's solar and lunar observations and parameters (New York and London, in press).

Heiberg (See Ptolemy.).

Hipparchus (Manitius) Hipparchi in Arati et Eudoxi Phenomena Commentarium , ed. by Manitius K. (Leipzig, 1894).

Jones [1983] Jones A. , “The development and transmission of 248 day schemes for lunar motion in ancient astronomy”, Archive for history of exact sciences , xxix (1983), 1–36.

Kugler [1900] Kugler F. X. , Die Babylonische Mondrechnung (Freiburg im Breisgau, 1900).

Neugebauer [1955] Neugebauer O. , Astronomical cuneiform texts (3 vols, London, 1955).

Neugebauer [1975] Neugebauer O. , A history of ancient mathematical astronomy (New York, etc., 1975).

Neugebauer [1988] Neugebauer O. , “A Babylonian lunar ephemeris from Roman Egypt”, in A scientific humanist … (see Bowen-Goldstein [1988]), 301–4.

Pappus Pappi Alexandrini Collectionis quae supersunt , ed. by Hultsch F. (3 vols, Berlin, 1876–78).

Pingree [1978] Pingree D. , “History of mathematical astronomy in India”, Dictionary of scientific biography , xv (New York, 1978), 533–633.

Ptolemy (Heiberg) Heiberg J. L. (ed.), Claudii Ptolemaei Opera quae exstant omnia, i: Syntaxis mathematica (2 parts, Leipzig, 1898–1903).

Ptolemy (Toomer) Toomer G. J. (transl.), Ptolemy's Almagest (London/New York etc., 1984).

Rome [1950] Rome A. , “The calculation of an eclipse of the Sun according to Theon of Alexandria”, Proceedings of the International Congress of Mathematicians , Cambridge, Mass., U.S.A., Aug. 30–Sept. 6, 1950, i, 209–19.

Swerdlow [1980] Swerdlow N. M. , “Hipparchus's determination of the length of the tropical year and the rate of precession”, Archive for history of exact sciences , xxi (1980), 291–309.

Tannery [1893] Tannery P. , Recherches sur l'histoire de l'astronomie ancienne (Paris, 1893).

Theon (Dupuis) Theon de Smyrne, philosophe platonicien: Exposition des connaissances mathématiques utiles pour la lecture de Platon , ed. & transl. by Dupuis J. (Paris, 1892).

Theon (Hiller) Theonis Smyrnaei philosophi platonici expositio rerum mathematicarum ad legendum Platonem utilium , ed. by Hiller E. (Leipzig, 1878).

Toomer (See also Ptolemy.).

Toomer [1967] Toomer G. J. , “The size of the lunar epicycle according to Hipparchus”, Centaurus , xii (1967), 145–50.

Toomer [1973] Toomer G. J. , “The chord table of Hipparchus and the early history of Greek trigonometry”, Centaurus , xviii (1973), 6–28.

Toomer [1974] Toomer G. J. , “Hipparchus on the distances of the Sun and Moon”, Archive for history of exact sciences , xiv (1974), 126–42.

Toomer [1978] Toomer G. J. , “Hipparchus”, Dictionary of scientific biography , xv (New York, 1978), 207–24.

Toomer [1988] Toomer G. J. , “Hipparchus and Babylonian astronomy”, A scientific humanist … (see Bowen-Goldstein [1988]), 353–62.

Van der Waerden [1988] Van der Waerden B. L. , Die Astronomie der Griechen (Darmstadt, 1988).

Vettius Valens (Kroll) Vettii Valentis Anthologiarum libri , ed. by Kroll W. (Berlin, 1908).

Vettius Valens (Pingree) Vettii Valentis Antiocheni Anthologiarum libri novem , ed. by Pingree D. (Leipzig, 1986).

Almagest I, 2 ( Toomer , 37; Heiberg I , 8–9) and III, Preface ( Toomer , 131; Heiberg I , 191).

Almagest III, 1 ( Toomer , 139; Heiberg I , 206–8).

According to Theon of Smyrna ( Dupuis , 256–7; Hiller , 127), this was Hipparchus's “On the sizes and distances”; but it is difficult to see how the determination of the solar model's parameters would fit in with the known contents of that work (for which see Toomer [1974]).

Almagest III, 4 ( Toomer , 153; Heiberg I , 232–3).

See for example Neugebauer [1975], 306–7, 313, and Toomer [1978], 211. Rome [1950], 214–15, went so far as to maintain that even the exact format of the Almagest tables went back to Hipparchus.

Vettius Valens IX, 12 ( Pingree , 339; Kroll , 354).

For the chronology of Hipparchus's discovery of precession, see Toomer [1978], 217–18, and Swerdlow [1980].

Almagest III, 1 ( Toomer , 136; Heiberg I , 200–1).

Almagest III, 5 ( Toomer , 165; Heiberg I , 251), and see also III, 1 ( Toomer , 140; Heiberg I , 208).

Almagest III, 1 ( Toomer , 135–6; Heiberg I , 198–200) and VII, 2 ( Toomer , 327; Heiberg II , 12–13).

Almagest III, 1 ( Toomer , 135; Heiberg I , 199). All passages of the Almagest quoted in this article are from Toomer's translation.

Almagest III, 1 ( Toomer , 136; Heiberg I , 200).

Almagest V, 3 ( Toomer , 224; Heiberg I , 363–4) and V, 5 ( Toomer , 227–8 and 230; Heiberg I , 369 and 374–5).

For how this could be done if one were using an armillary sphere, see Toomer's note 4 to Almagest V, 1 (p. 219).

Theon of Smyrna ( Dupuis , 268–9 and 298–9; Hiller , 166 and 185). Claim Tannery's ([1893], 60, since revived by Van der Waerden (e.g. [1988], 180–1), that Hipparchus did not understand the easy geometrical proof of this equivalence derives from a mistranslation of the former passage of Theon.

Almagest IV, 5 ( Toomer , 181; Heiberg I , 294–5).

Almagest IV, 11 ( Toomer , 211–15; Heiberg I , 340–7).

Ptolemy ‘rounds’ this to 13h.

The possibility of errors in the textual transmission of the figures can be ruled out. The numerical consistency of Ptolemy's discussion of Hipparchus's intervals guarantees that we are reading exactly the same numbers that Ptolemy read in Hipparchus's book; while the fact that the intervals ascribed to Hipparchus lead (approximately) to Hipparchus's attested values for the ratio of radii assures us that these really were Hipparchus's numbers (cf. Toomer's note 75 to Almagest IV, 11 (p. 215)).

For example, Toomer [1973], 26, note 10 writes: “One might, with difficulty, explain Hipparchus's errors in the time-intervals by supposing that he neglected or miscalculated the equation of time, but his errors in the longitude intervals are completely inexplicable to me.” To my knowledge only Britton [in press] (chap. 2, “Ptolemy's solar tables”) has said outright that the discrepancies imply a systematic difference between the ways that Hipparchus and Ptolemy computed solar longitudes.

There is a particular difficulty in the case of eclipse A3, where Ptolemy interprets a phrase that seems unambiguously to mean “Four hours past sunset” as if it meant “during the fourth hour”, and therefore counts it as 3½ hours. I am inclined to attribute the inconsistency to a textual corruption (Toomer's note 68 to Almagest IV, 11 (p. 213)).

The fundamental theorems of spherical trigonometry are supposed to have first appeared in Menelaus's Spherics about a.d. 100. Pappus of Alexandria (Collection 6.109) informs us that Hipparchus used arithmetical functions for oblique ascensions.

For a detailed discussion of the equation of time, see Neugebauer [1975], 61–68.

4 equinoctial hours is approximately the maximum duration of a lunar eclipse (cf. Toomer's note 30 to Almagest IV, 6 (p. 191)). A duration of 3½^h could follow from Hipparchus's measurement of the Earth's shadow as 2½ times the Moon's disk, combined with the crude assumption that the Moon travels the breadth of its own disk in about 1^h with respect to the shadow.

I do not know of any direct evidence of knowledge of the equation of time before Ptolemy (cf. Neugebauer [1975], 61). In Indian astronomy, a partial correction for the inequality of days due to solar anomaly appears first in the seventh century in the astronomy of Brahmagupta ( Pingree [1978], 569; cf. also 582).

Based on the difference between the maximum and mean solar motion per synodic month in Column A (see Neugebauer [1955], i, 70–71). The equations implicit in the System B solar scheme are actually about two-thirds of what they should be.

Neugebauer [1955], i, 45–46.

Bernsen [1969].

Almagest IV, 11 ( Toomer , 211; Heiberg I , 338–9).

The argument still holds if (as I am unable to believe) Hipparchus did not know that the hypotheses were geometrically equivalent (cf. ref. 15 above).

Toomer [1967]. Ptolemy's account excludes the possibility that the measurement according to the epicyclic model preceded the measurement according to the eccentric model.

Although f and s vary in their seconds place depending on the choice of initial longitude λ₁, the year length resulting from them does not vary by more than 1 second for 175° ≤ λ₁ ≤ 177°.

Almagest V, 3 ( Toomer , 224; Heiberg I , 363–4).

Almagest V, 5 ( Toomer , 227; Heiberg I , 369).

Almagest V, 5 ( Toomer , 230; Heiberg I , 374–5).

Almagest III, 1 ( Toomer , 134; Heiberg I , 196).

Toomer [1978], 219 has already suggested that at least the longitude for −126 July 7 was observed.

Kugler [1900], 111. For a survey of how much Hipparchus knew of Babylonian astronomy in general, see Toomer [1988].

Neugebauer [1988].

Hipparchus I, 9 ( Manitius , 88–90).

Bowen-Goldstein [1988], 68–69. Kugler [1900], 84–86 had already pointed out that Hipparchus's values for the lengths of both astronomical spring and astronomical summer are strikingly close to the values implied by System A.

The two passages that seem to speak of solstice observations (III, 1; Toomer , 133 and 139; Heiberg I , 194–5 and 206–7) can be interpreted as referring to solstices calculated from observed equinoxes.

Almagest III, 1 ( Toomer , 133–4; Heiberg I , 195–6).

Almagest III, 4 ( Toomer , 153; Heiberg I , 233).

An interval of 90 days for astronomical winter is a ‘fringe benefit’ of the determination of the solar model's parameters; and this value is expressly attributed to Hipparchus by Ptolemy (III, 4; Toomer , 155–6; Heiberg I , 237–8).

Almagest IV, 11 ( Toomer , 211; Heiberg I , 338). See Toomer [1973] for a hypothetical reconstruction of Hipparchus's calculation of this ratio.

In fact the ratio that Hipparchus extracted from his first eclipse trio was significantly different from the exact ratio that one can recompute from his data (e.g. Toomer [1973], 12–15 recomputes 338:3134).

Hipparchus's confirmation of the period of latitude used his observation of the eclipse of −140 January 27 ( Almagest VI, 9; Toomer , 309; Heiberg I , 526). Toomer [1978] has identified the Hipparchian eclipses used for the confirmation of the anomalistic period ( cf. Almagest IV, 2; Toomer , 178; Heiberg I , 276) as those of −140 January 27 (again) and −138 November 26.

E.g. the eclipse of −145 April 21, referred to in Almagest III, 1 ( Toomer , 135 and note 14; Heiberg I , 199).

For Hipparchus on the planets, see Almagest IX, 2 ( Toomer , 421; Heiberg II , 210). It is customary to interpret the observations of lunar elongations in −127/126 as evidence that Hipparchus suspected that his lunar model did not predict accurate longitudes away from syzygies; but of course without an accurate epicycle radius, it would not have worked well at the syzygies either.

Hipparchus's report of his observation on −127 August 5 of the lunar elongation provides a clue to what kind of lunar table he used at this late date in his career, and it was a Babylonian one. See Jones [1983], 25–26.