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Hipparchus’ selenelion and two pairs of lunar eclipses revisited

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Abstract

Ptolemy reports three dated lunar eclipses observed by Hipparchus, and also refers to two more, without identifying them, which Hipparchus compared with two earlier counterparts (apparently, observed in Mesopotamia) to assess the validity of the Babylonian period relations of the lunar motion. Also, in Pliny the Elder’s Historia naturalis, we are told that a horizontal lunar eclipse (selenelion) at sunrise and moonset was reported (observed?) by Hipparchus. Reviewing a paper by G.J. Toomer in 1980, it is shown that the pairs of the eclipses were, almost certainly, the ones occurring on “31 January 486 b.c. and 27 January 141 b.c.” and “19 November 502 b.c. and 14 November 157 b.c.”; and if Hipparchus observed from St. Stephen’s Hill in Rhodes, the most probable candidate for the selenelion at moonset was the lunar eclipse of 7 February 142 b.c., although he also had the chance to observe any of the four others, occurring on 3 July 150 b.c., 10 April 145 b.c., 26 November 139 b.c., and 15 November 138 b.c., on a sufficiently elevated mountain on the island.

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Notes

  1. Almagest III.1: Toomer [1984] (1998), p. 133.

  2. Almagest V.5: Toomer [1984] (1998), p. 230.

  3. Almagest III.1, VI.5&9: Toomer [1984] (1998), pp. 135, 284, 309. See, also, Hipparchus’ observations quoted by Ptolemy as listed in the chronological order in Pedersen 1974, pp. 413–415, and, for his lunar eclipses, Steele (2000), pp. 93, 101. These are, respectively, the eclipses nos. 04457, 04468, and 04483 in the NASA’s 5MCLE (eclipse.gsfc.nasa.gov/LEcat5/LE-0199-0100.html).

  4. Also noted in Steele (2000), p. 102; see, also, Stephenson 1997, pp. 371–372.

  5. Almagest IV.11: Toomer [1984] (1998), pp. 211–215.

  6. A phrase in Pliny the Elder’s Historia naturalis II.ix.53 (1938–1962, Vol. 1, pp. 202–203), stating “[…] Hipparchus predicted/proclaimed the course (cursum) of each star for six hundred years […]”, is interpreted in Neugebauer (1975), Vol. 1, pp. 319–321, as a misunderstanding of a compilation by Hipparchus of eclipse records for the 600 years preceding his time. This interpretation was emphasized later in Toomer (1988), p. 355. Pliny’s ambiguous expression was given a seemingly more plausible interpretation in Goldstein and Bowen (1995): on the ground of the technical meaning of cursum as the “daily progress of the Moon”, they link it with Hipparchus’ use of the 248-day cycle of the true motion of the Moon, originated in Babylonian astronomy (as indicated in Almagest V.3), where each day is associated with a specific daily progress of the Moon in longitude, and the underlying scheme is a linear zigzag.

  7. Almagest IX.2: Toomer [1984] (1998), p. 421.

  8. Toomer [1984] (1998), pp. 243–244; Bowen and Todd (2004), pp. 131–132; Toomer (1967), p. 147; Swerdlow (1969), pp. 288–289, 303, note 3; Toomer (1974), pp. 126–127; Morrison et al. (2019), pp. 4–5.

  9. Toomer (1967), p. 150, note 14; Toomer (1974), pp. 127–128, gives a history of the modern identification of Hipparchus’ eclipse.

  10. Toomer (1974), esp. pp. 132–137.

  11. According to the reconstructed method, the lunar distance is computed as

    \(D = 2\sin \tfrac{1}{2}(\varphi_{\rm H} - \varphi_{\rm A} ) \times \cos (\tfrac{1}{2}(\varphi_{\rm H} + \varphi_{\rm A} ) - \delta_{\rm M} )/\sin (f \cdot \theta ) + 1\)

    where f · θ stands for the lunar parallax, f = the fraction of the solar disk uncovered at Alexandria = 0.2, θS = the solar apparent angular diameter = 360°/650 ≈ 0;33,14°, φ = the geographical latitude, H = Hellespont and A = Alexandria, reasonably taken as equal, respectively, to 41° and 31° (in actuality, ~ 40;12° and ~ 31;12°), and δM = the lunar declination.

  12. Hipparchus should have acquired an a priori knowledge of the relative distance of the Moon from the Earth at the time of the eclipse from its anomalistic position on the basis of his already-established lunar theory. In the solar eclipse of 15 August 310 b.c., the Moon was near the least distance (in the mean anomaly of about 202°, according to the Almagest and modern theories), and, in both solar eclipses of 14 March 190 b.c. and 20 November 129 b.c., it was at midway between the mean and least distance (respectively, in the mean anomalies of 228° and 132°). So, regardless of which eclipse he used and irrespective of whether Toomer’s method is what Hipparchus performed in reality, the value he obtained for the lunar distance must not have exceeded 74 Earth radii.

  13. Morrison et al. (2019), esp. pp. 11–13.

  14. Toomer [1984] (1998), p. 178.

  15. Pliny (1938–1962), Vol. 1, pp. 204–207.

  16. Bowen and Todd (2004), pp. 159–163. A translation of the passage in question is presented in Heath (1932), pp. 162–166, which is quoted in Cohen and Drabkin (1958), pp. 284–285; see, also, Heath (1921), Vol. 1, pp. 6–7, Vol. 2, p. 238.

  17. For a better view of Cleomedes’ contribution to the optical topics, see Ross (2000); Ross and Plug (2002), pp. 28–29, 55, et passim; Lehn and van der Werf (2005), pp. 5625–5627. It comes as a surprise that in his rather lengthy discussion on the atmospheric refraction in Optics V.23–31, Ptolemy neither refers to the selenelion, nor to the fact that atmospheric refraction can project the Sun above the horizon before it has actually risen (Smith 1996, pp. 238–242; 1999, pp. 134–137; 2003, pp. 101–104). This is true of Ibn al-Haytham in al-Manāẓir VII.4[.28]–[.35], too (Smith 2010, Vol. 2, pp. 270–274). These interrelated topics were apparently laid dormant during the medieval period until it was revived in the early modern period (for a brief outline, see Poole 2007, pp. 251–252).

  18. Toomer (1980).

  19. Toomer (1980), p. 103.

  20. Toomer [1984] (1998), pp. 208–209.

  21. Toomer (1980), p. 103.

  22. See Jones (2016), pp. 82–83; Tihon and Fournet (2016), p. 356.

  23. Toomer (1980), p. 103.

  24. It may seem prima facie that Toomer’s mistake in the case of the lunar eclipse of 30 November 484 b.c. would stem from the use of the eclipse catalogs published in the 19th and the first half of the twentieth centuries, like those of Paul Victor Neugebauer (1878–1940; cf. Neugebauer 1912–1925, 1929, 1934) or the Canon der Finsternisse by Theodor Ritter von Oppolzer (1841–1886; posthumously published in 1887; cf. Oppolzer [1887] 1962), which, in the light of vast investigations on the quantification of the Earth’s clock error in the latter part of the past century, “is of severely limited usefulness for the investigation of both modern and ancient/medieval solar eclipses” (quoted from Morrison and Stephenson 2004, p. 334). But it is not absolutely the case here: as Toomer explains (1980, p. 109, note 18), P.V. Neugebauer had already derived the time of the beginning of the eclipse as 7:12, and was in doubt whether it had begun after sunrise. Toomer changes this time to 6:55, and reaches approximately the same value for the time of sunrise, both on the basis of his calculations from P.V. Neugebauer’s works. P.V. Neugebauer’s values a little differ from those computed on the basis of the today’s theories (respectively, 7:7 and 6:46).

  25. Toomer (1980), p. 104.

  26. Toomer (1980), p. 104–105; Bruin (1966), p. 20.

  27. Toomer (1980), p. 106.

  28. It seems that there is a tendency in the modern scholarship to assume that Hipparchus himself “observed” the selenelion; another example is Bowen and Todd (2004), p. 159, note 15.

  29. www.heywhatsthat.com. It gives a panorama of the horizon for any given locality either by marking it on the underlying Google Maps or by directly entering the values of its longitude and latitude.

  30. Also, www.rhodesprivatetours.com/mount_smith_hill_affordable.htm contains the high-resolution photos of the north and western shores of Rhodes city as seen from the top of St. Stephen’s Hill, in which the far-distant geographical features can be vividly seen and distinguished by comparing with Fig. 2a.

  31. Steel (2001), pp. 56–57.

  32. Nos. 04448, 04459, 04466, 04474, and 04476 in the NASA’s 5MCLE. Note that all of the following considerations are dependent on the accuracy of the 5MCLE, in particular with regard to ΔT, in which case it is on the basis of Morrison and Stephenson (2004). Uncertainty in Morrison and Stephenson (2004), with the standard deviation not more than 6 min for Hipparchus’ time, exerts no undesirable influence on the situation of the five selenelions considered here, as when it is translated into the uncertainty in the altitude and azimuth of the Sun and Moon, the differences remain significantly small and make no dramatic shift in the Luminaries’ position with respect to the surrounding geographical barriers. The same also holds true for other modern determinations, except for Tuckerman (1962, 1964) (which was rendered obsolete long ago; see Houlden and Stephenson 1986). The interested reader can simply check the situations of the five selenelions on the basis of nearly all modern estimations of ΔT with the aid of the Alcyone Ephemeris (http://www.alcyone-ephemeris.info).

  33. See Steel (2001), p. 58, for a historical sketch of it; Poole (2007).

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Acknowledgements

The author wishes to thank Prof. Alexander Jones and Prof. John Steele for their valuable and constructive suggestions on an early draft of this paper.

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  1. Department of History of Astronomy, Research Institute for Astronomy and Astrophysics of Maragha (RIAAM), University of Maragheh, P.O. Box 55136-553, Maragheh, Iran

    S. Mohammad Mozaffari

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Mozaffari, S.M. Hipparchus’ selenelion and two pairs of lunar eclipses revisited. Arch. Hist. Exact Sci. (2024). https://doi.org/10.1007/s00407-024-00330-8

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