The Epoch of the Constellations on the Farnese Atlas and Their Origin in Hipparchus's Lost Catalogue

1. The best pictures of the Atlas appear in Stern H., “Classical Antiquity” in the entry titled “Astronomy and astrology” in Encyclopedia of world art (New York, 1960), ii, cols 48–57 and plate 23. Many short descriptions of the statue have appeared, of which the best are G. Aujac, “The foundations of theoretical cartography in archaic and classical Greece”, in History of cartography, ed. by Harley J. B., Woodward D. (Chicago, 1987), 142–3; Evans J., The history & practice of ancient astronomy (New York, 1998), 78–79; Fiorini M., Sfere terrestri e celesti (Rome, 1899), 8–25; Gialanella C., Valerio V., “Atlas Farnese” in Cartes et figures de la Terre (Paris, 1980), 84; Stevenson E. L., Terrestrial and celestial globes (New Haven, 1921), 1–25; Thiele G., Antike Himmelsbilder (Berlin, 1898), 27–45; and Warner D. J., The sky explored, celestial cartography 1500–1800 (New York, 1979), 278–9. A detailed history of the historical writings on the Farnese Atlas appears in Valerio V., “Historiographic and numerical notes on the Atlante Farnese and its celestial sphere”, Der Globusfreund, xxxv/xxxvii (1987), 97–124.

2. Here is the complete list of constellations visible on the Farnese Atlas in the constellation order as given in the Almagest: Dra, Cep, Boo, CrB, Her, Lyr, Cyg, Cas, Per, Aur, Oph, Ser, Aql, Del, Peg, And, Ari, Tau, Gem, Cnc, Leo, Vir, Lib, Sco, Sgr, Cap, Aqr, Psc, Cet, Ori, Eri, Lep, CMa, Argo, Hya, Crt, Crv, Cen, Lup, Ara, and CrA. (Stevenson cites 42 constellations while Gialanella and Valerio [and hence Aujac] cite 43 constellations.) There are a number of constellations from Antiquity that are missing. Ursa Major is missing as it is located entirely within the hole in the top of the globe. Canis Minor is entirely positioned under one of the hands holding the globe. Piscis Austrinus is completely hidden by the shoulder of the Titan. Of the older Greek constellations, the only ones missing without cause are Ursa Minor, Triangulum, and Sagitta. A short, simple line segment near a wing of Cygnus could conceivably be Sagitta, but this has the wrong iconography, the wrong orientation, and is 15° misplaced, and so I conclude that the line segment is not Sagitta. The later Greek constellations of Equuleus, Coma Berenices, and Antinous are all missing and would certainly have been visible if depicted. The only addition that is not part of the set of old constellations is a curious small rectangular structure with three internal lines that appears above the Crab. This is certainly not any known constellation, and the only explanation suggested is that is this represents the comet known as “Throne of Caesar” or a throne of Zeus (Stern, op. cit. (ref. 1); Thiele, op. cit. (ref. 1)).

3. Stern, op. cit. (ref. 1), col. 49.

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

5. Aratus Phaenomena, transl. by Kidd D. (Cambridge, 1997). Another widely available translation is by G. R. Mair and appears in the Loeb Classical Library series titled Aratus (Cambridge, MA, 1921) with many subsequent reprints.

6. MacFarlane R. T., Mills P. S., Hipparchus' commentaries on the Phaenomena of Aratus and Eudoxus, manuscript, 2003. This translation is the first in English and the first in modern times.

7. Condos T., Star myths of the Greeks and Romans: A sourcebook (Grand Rapids, MI, 1997).

8. Toomer G. J., Ptolemy's Almagest (Princeton, 1998).

9. That is, one of the primary statements by Aratus on the cardinal points of the ecliptic is true for a time almost a millennium earlier. This immediately tells us that Aratus (and hence Eudoxus) include lore that was already ancient for their times. A detailed analysis of the lore in Aratus and Eudoxus (in Schaefer, op. cit. (ref. 4)) shows that all the lore is from many centuries before even Homer and Hesiod. A detailed analysis gives a date of 1130 ± 80 b.c.

10. Schaefer B. E., “Latitude and epoch for the formation of the southern Greek constellations”, Journal for the history of astronomy, xxxiii (2002), 313–50. Roughly, the extinction angle plus refraction correction for the bright stars in Centaurus, Crux, and Carina is close to 2′, while the gap for these stars (the angle above the region of invisibility) is certainly greater than 1′ and more likely around 2′ (although it could be larger). Thus, the correction to the simply derived latitude is likely 4′, but the value could be 3′ or larger.

11. Evans, op. cit. (ref. 1), 91–95.

12. Schaefer, op. cit. (ref. 4), Table 7 and Section 6.7.

13. Stevenson, op. cit. (ref. 1), 1–25; Evans, op. cit. (ref. 1), 78–79 and Figure 5.11.

14. The quote is from Neugebauer O., A history of ancient mathematical astronomy (3 vols, Berlin, 1975), 277–80; Evans, op. cit. (ref. 1), 103.

15. Pickering K. A., “Evidence of an ecliptical coordinate basis in the Commmentary of Hipparchos”, Dio, ix (1999), 26–29.

16. Duke D. W., “Hipparchus' coordinate system”, Archive for the history of exact sciences, lvi (2002), 427–33; Delambre J. B. J., Histoire de l'astronomie ancienne (1817; repr. New York, 1965), i, 117, 172, 184.

17. A history of this dispute is given in Schaefer B. E., “The great Ptolemy-Hipparchus debate”, Sky & telescope, ciii (2002), Feb. issue, 38–44.

18. Vogt H., “Versuch einer Wiederstellung von Hipparchs Fixsternverzeichnis”, Astronomische Nachtrichten, ccxxiv (1925), cols 2–54; Grasshoff G., The history of Ptolemy's star catalogue (New York, 1990); Swerdlow N. M., “The enigma of Ptolemy's catalogue of stars”, Journal for the history of astronomy, xxiii (1992), 173–83; and Duke D. W., “The depth of association between the ancient star catalogues”, Journal for the history of astronomy, xxxiv (2003), 227–30.

19. The obliquity of the ecliptic for 125 b.c. is 23.71′, as determined from Meeus J., Astronomical algorithms (Richmond, VA, 1991), 135–6. The mean obliquity of the ecliptic is 23.76′ in 500 b.c. and 23.70′ in 1 a.d.

20. Meeus, op. cit. (ref. 19), 126–8. Proper motions of all stars used in this paper are completely negligible.

21. The conversion from ecliptic to equatorial coordinates is taken from Equations 12.3 and 12.4 in Meeus, op. cit. (ref. 19), 87–92.

22. Technically, the chi-square analysis applies only when the errors have Gaussian distributions. However, in practice, the chi-square technique is still best even for distributions far from Gaussian, even though the derived error bars become more uncertain. Fortunately, the residuals for the analysis in this paper are all good Gaussian distributions.

23. With 70 data items and one variable fit parameter, the total number of degrees of freedom in this chi-square fit is 69. Then, the reduced chi-square is 66.3/69 = 0.96. In general, a reduced chi-square of near unity implies a good fit. However, this conclusion of a good fit does not apply for this case because the size of the uncertainties were selected as the observed scatter which then forces a reduced chi-square of near unity. If alternatively, we had chosen a substantially larger or smaller value for the sigma, then the derived reduced chi-square would have been greatly smaller or larger than unity respectively. If we had any way of independently knowing the typical total uncertainty (which we do not), then we could have used this value and derived a meaningful reduced chi-square that is useful for evaluating the quality of the model. One comforting fact is that the typical scatters (which yield a reduced chi-square near unity) have reasonable sizes, for example judged based on the typical scatters for the lore of Eudoxus (see Schaefer, op. cit. (ref. 4)). The determination of the best date does not depend on any scaling of the adopted uncertainties. However, the derived error bars for the best-fit epoch will depend on the size of the adopted uncertainties for the individual data points. And, indeed, the process can be turned around so as to derive the typical total uncertainty of a single point (based on the average uncertainty so as to make the reduced chi-square near unity). With this derived uncertainty for an individual observation, we will necessarily get an acceptable reduced chi-square and the derived error bars on the epoch will be good. In all, the exact value of the reduced chi-square is not meaningful since the uncertainties have been taken from the observed scatter in the data themselves, but this choice allows for valid error bars on the date.