A ‘Watermark’ of Eudoxan Astronomy

The most systematic account of the standard type of medieval hemispheres is in McGurk P. , “Germanici Caesaris Aratea cum scholiis: A new illustrated witness from Wales”, The National Library of Wales journal , xviii (1973), 197–216, esp. pp. 203–4 and 215, note 33. A publication on the standard medieval hemispheres and planispheres is in preparation by the present author.

McGurk , op. cit. (ref. 1), esp. p. 200, labels these Aberystwyth maps “a trial run” and does not include them in his analysis. The Monza map is described in McGurk P. , Catalogue of astrological and mythological illuminated manuscripts of the Latin Middle Ages, vi: Astrological manuscripts in Italian libraries (other than Rome) (London, 1966), 52.

For example, the zodiac of the summer hemisphere has been clumsily drawn. Cancer is at 90° from the North Pole instead of 66°.

McGurk , op. cit. (ref. 2), 52, states that the last page of the manuscript, the one with the map (fol. 67r), is “not an insertion but part of a quire of 12”.

This phase is described in Goldstein B. R. Bowen A. C. , “A new view of early Greek astronomy”, Isis, lxxiv (1983), 330–40.

The summer hemisphere appears upside-down because south is on top and north is at the bottom.

Neither the arm of Perseus nor the hand of Centaurus is drawn on the Aberystwyth winter hemisphere, but the missing hands in the pictures of these two constellations on the summer hemisphere suggest that they should have been part of the winter hemisphere.

The summer and winter hemispheres in Aberystwyth NLW 735C (fols 3v and 4r) have been re-arranged so that they can be compared directly with the hemispheres in Figure 1.

Note that the location of Centaurus on these common winter hemispheres is astronomically wrong: In 128 b.c. the right ascension of the star in the eastern shoulder (θ Cen) was 3° east of the autumnal colure but most of the body was west of it.

Hipparchus , In Arati et Eudoxi Phaenomena commentariorum libri tres , edited with German translation by Manitius Karl (Leipzig, 1894), i, 11.9–10. I am indebted to Alexander Jones for providing the English translation. A French translation is in Aratos, Phénomènes, ed. and French transl. by Martin Jean (Paris, 1998), 127–8.

Hipparchus , op. cit. (ref. 10), i, 11.17. I am again indebted to Alexander Jones for providing the English translation. A French translation is in Aratos, op. cit. (ref. 10), 127.

Hipparchus , op. cit. (ref. 10), i, 11.11–12.

Hipparchus , op. cit. (ref. 10), i, 11.18.

Hipparchus , op. cit. (ref. 10), i, 11.21.

The colures are among the 24 hour circles described in Hipparchus, op. cit. (ref. 10), iii, 5.

The details of this calculation can be obtained from the author.

These maps were produced with the help of Chris Marriott's SkyMap Pro, Version 8.

According to Martin (Aratos, Phénomènes (ref. 10), 127–9) this conclusion was already drawn in 1630 by Père Denis Petau in his Vranologion. Unfortunately Martin confuses the issue by adding: “In fact this description [of the colures] is sufficiently exact if one accepts that these two circles pass not through the fifth but through the eighth degree of signs, in conformity with common usage.” Martin confuses here two concepts: Conventions by which the vernal equinox is fixed with respect to the signs and epochs for which the vernal equinox is fixed with respect to the stars.

Bowen A. C. Goldstein B. R. , “Hipparchus' treatment of early Greek astronomy: The case of Eudoxus and the length of daytime”, Proceedings of the American Philosophical Society, cxxxv (1991), 233–54, especially p. 245.

Hipparchus , op. cit. (ref. 10), i, 5.11; i, 6.4; and ii, 1.15. Neugebauer O. , A history of ancient mathematical astronomy (3 vols, Berlin and New York, 1975), 599.

Hipparchus , op. cit. (ref. 10), ii, 1.15.

Aratus , Phaenomena, ed. with introduction, English translation, and commentary by Kidd Douglas (Cambridge, 1997), lines 534–9.

The same phenomenon is explained in detail by Géminos, Introduction aux phénomènes , edited with French translation by Aujac Germaine (Paris, 1975), V.18 and VII. 4–8. There is an English translation by James Evans and Lennart Berggren, Geminos's Introduction to the Phenomena: A translation and study of a Hellenistic survey of astronomy (Princeton, NJ, 2006).

Hipparchus , op. cit. (ref. 10), ii, 1.17.

Aratus , op. cit. (ref. 22), lines 149–51.

Hipparchus , op. cit. (ref. 10), ii, 1.18.

Aratus , op. cit. (ref. 22), lines 332–3, would provide support for this.

Martin underlines this point in his comments in Aratos, op. cit. (ref. 10), 221.

The ecliptic intersects the Lion in the breast close to α Leo.

Hipparchus's reliability was questioned in more than one way by Bowen and Goldstein, op. cit. (ref. 19) and by Goldstein B. R. Bowen A. C. , “The introduction of dated observations and precise measurement in Greek astronomy”, Archive for history of exact sciences, xliii (1991), 93–132.

Hipparchus , op. cit. (ref. 10), ii, 1.19; Neugebauer , op. cit. (ref. 20), 600. Evidence for the use of the Ari 0°-convention seems to be presented in the parapegmas appended to the treatise of Geminus (Géminos, op. cit. (ref. 23), 98–108). However, this appendix is clearly a re-working of older documents, in which not much faith can be put.

Newton Isaac , The chronology of ancient kingdoms amended (London, 1728).

Newton's Eudoxan colures are marked in on the celestial globe of a diameter 68 cm (27 inches), published around 1730 by John Senex. See Dekker Elly , Globes at Greenwich: A catalogue of the globes and armillary spheres in the National Maritime Museum, Greenwich (Oxford, 1999), 120–4, esp. Fig. 9.88, and 493–5.

Schaeffer B. E. , “The latitude and epoch for the origin of the astronomical lore of Eudoxus”, Journal for the history of astronomy, xxxv (2004), 161–223.

Duke Dennis , “Statistical dating of the Phenomena of Eudoxus”, http://www.dioi.org/sneak.htm.

Neugebauer , op. cit. (ref. 20), 599.

Neugebauer , op. cit. (ref. 20), 599–600. Evans James , The history and practice of ancient astronomy (New York and Oxford, 1998), 202. In P. Hibeh 27, the vernal equinox is on Tybi 20 which would imply a date around 300 b.c.; 15 days later, on Mekheir 6, the Sun is reported to enter Taurus. The other so-called “Eudoxus Papyrus”, P. Paris 1, reports that according to Democritus and Eudoxus the winter solstice takes place on Athyr 20 or 19, which is correct for a date between 196 and 173 b.c. Indirectly this date for the winter solstice can be shown to rest on the so-called midpoint convention.

Neugebauer , op. cit. (ref. 20), 599.

Manilius , Astronomica, edited with introduction, English translation, and commentary by Goold G. P. (Cambridge, MA, and London, 1977), I. 603–30.

The details of this calculation can be obtained from the author.

In 128 b.c. α and β Lib stood about 15°–20° east of the autumnal colure.

Next to the Hipparchan epoch, Manilius's treatise contains another reminder of Hipparchus: The system of arranging the stars into six classes of magnitudes, the earliest evidence of which is found in Manilius, op. cit. (ref. 39), V. 710–17. This idea is generally attributed to Hipparchus who wrote a now lost treatise on the magnitudes and positions of the stars. See Neugebauer , op. cit. (ref. 20), 291–2.

From the position it is clear that only the right wing can be meant.

Capella Martianus , De nuptiis Philologiae et Mercurii , edited by Dick A. (Leipzig, 1925), VIII, 832–3. The translation is from Martianus Capella and the seven liberal arts, ii, “The marriage of Philology and Mercury”, translated by Stahl W. H. Johnson R. , with Burge E. L. (New York, 1977), 324.

The details of this calculation can be obtained from the author.

In 128 b.c., there was a fair correspondence between signs and constellations. For example, the longitude of the westernmost star of the Ram (γ Ari) was Ari 3.5°. In terms of the Ari 8°-convention, γ Ari would be located at Ari 11.5°.

The Capellan colures coincide with hour circles that are on average 4.9° ± 5.5° east of the true colures in 375 b.c., These circles are about 6° west of the Eudoxan colures at 11° ± 4°.

Boll F. , Sphaera (Berlin 1903), 59–69, esp. p. 64 and n. 1.

Columella , De re rustica , IX. 14. 12, cited in Albert Rehm, Parapegmastudien. Mit einem Anhang Euktemon und das Buch De signis (Munich, 1940), 18. “Nec me fallit Hipparchi ratio, qua docet solstitia et aequinoctia non octavis sed primis partibus signorum confici. Verum in hac ruris disciplina sequor nunc Eudoxi et Metonis antiquorumque fastus astrologorum [namely by placing the solstices and equinoxes in the eighth degree].” Rehm considers it a fact that Eudoxus used the Ari 8°-convention, see his chap. 2, 30–43.

Géminos , op. cit. (ref. 23), I. 3.

Hygin, Astronomie, ed. with French translation by Le Boeuffle A. (Paris, 1983), III. 3.

Note that the colures are said to pass just in front of Aries and Libra.

Hygin, op. cit. (ref. 51), III. 14.

Both features are seen on a recently discovered antique celestial globe. See Cuvigny Hélène , “Une sphère céleste antique en argent ciselé”, in Gedenkschrift Ulrike Horak (P. Horak), ed. by Harrauer Hermann Pintaudi Rosario (Florence, 2004), 345–80, esp. p. 377, Fig. 5.