On the Origin of the Ptolemaic Star Catalogue: Part 2

Ptolemy , Almagest VII, 2 (Toomer, 327).

Ptolemy , Almagest VII, 1 (Toomer, 135).

I observed the lunar eclipse of 17 July 1981 from the sidewalk outside my home in Seattle, Washington, and improved the opportunity to measure the longitude of λ Sagittarii by Hipparchus's method. This was several years before I began to be concerned with the origin of the Ptolemaic catalogue and I undertook the measurement solely for my own amusement. I measured the angular distance between the star and the shadow's centre using a home-made cross staff. The longitude of the Sun I computed using a Ptolemaic theory with modern elements (the theory is accurate to 1′). I find, on examining my notes from that evening, that my final longitude for λ Sagittarii was too small by about 40′. I remember being quite pleased with my result. Although the error is huge by modern standards, it is no worse than the errors associated with Hipparchus's measurements of the longitude of Spica. Historians of ancient astronomy should perhaps be encouraged to try, very occasionally, similar measurements: Such experiences might temper their judgements regarding the precision achievable with the ancient instruments and techniques.

In principle, the Moon's longitudinal parallax may be calculated at the moment of each observation using special tables that Ptolemy had prepared for this purpose (Almagest V, 18–19). In fact, Ptolemy simply states, without explanation, that the correction for the Moon's parallactic motion during the half-hour between the two observations came to 1/12°. Nor does this figure appear to agree with an exact calculation from Ptolemy's tables. This problem has been discussed by several writers, but without definite result. See Pedersen , A survey of the Almagest, 243–5; see also Toomer's G. J. review of the same book, Archives internationales d'histoire des sciences, xxvii (1977), 137–50, p. 143. The simple explanation offered above appears to solve the mystery.

The figure is P. Rome's reconstruction, based upon Ptolemy's description of his armillary sphere in Almagest V, 1 and the commentary thereon by Pappus of Rome Alexandria. A. , “L'astrolabe et le météoroscope d'après le commentaire de Pappus sur le 5e livre de l'Almageste”, Annales de la Société Scientifique de Bruxelles, série A, xlvii (1927), 2e partie, 77–102.

This possibility has been noted by Gingerich , op. cit. (ref. 14, 1981), 42.

A useful list of dated observations in the Almagest is provided by Pedersen , A survey of the Almagest (ref. 11), 408–22.

The precession was not widely known or accepted in Antiquity. “…precession is never alluded to by Geminus, Kleomedes, Theon of Smyrna, Manilius, Pliny, Censorinus, Achilles, Chalcidius, Macrobius, Martianus Capella! The only writers except Ptolemy who allude to it are Proklus, who flatly denies its existence, and Theon of Alexandria, who accepts the Ptolemaic value of one degree in a hundred years…”, Dreyer J. L. E. , History of the planetary systems from Thales to Kepler (Cambridge, 1906), reprinted as A history of astronomy from Thales to Kepler (New York, 1953), 203–4.

Ptolemy's solar theory and his observations of equinoxes and solstices have given rise to a large literature of analysis. Some of Ptolemy's solar observations are substantially in error, but they nevertheless agree with the solar theory of Hipparchus. The emerging consensus, therefore, is that Ptolemy adopted Hipparchus's solar elements and that Ptolemy's own solar observations were simply intended as demonstration pieces. Depending upon the disposition of the writer, Ptolemy may have deliberately selected observations that agreed with Hipparchus's theory, or doctored the observations a little to secure agreement, or fabricated them outright. Again, Ptolemy's motive in doing so was either the desire to produce a clear and coherent demonstration of the principles involved in the solar theory, or plagiarism and deception. For an introduction to the debate, see the following two works, which differ rather markedly in tone: Britton John P. , “On the quality of solar and lunar observations and parameters in Ptolemy's ‘Almagest’” (unpublished Ph.D. dissertation, Yale University, 1967), 17–67; Newton , The crime of Claudius Ptolemy, 75–94.

Halma N. , Tables manuelles de Ptolemée et de Théon, Part 3 (Paris, 1825), 45–58. The star catalogue of the Handy tables is not independent of that of the Almagest, but was obtained from it by subtracting the longitude of Regulus from the longitudes of all the other stars.

Newton , “On the fractions of degrees in an ancient star catalogue” (ref. 14), 384.

Pedersen provides a list of Ptolemy's reference stars: A survey of the Almagest, 236–7. However, it should be used with caution, for the following reasons. A few stars used in the Almagest are absent from Pedersen's list, e.g., ζ Geminorum and γ Arietis. In at least one case, the stars in question appear to be misidentified. (In observation # 57, the Almagest text points to γ and not α Arietis, and probably μ Ceti rather than 38 Arietis.) Pedersen does not distinguish between stars used in simple alignments and stars used for orienting the armillary sphere, nor does he discriminate between stars used by Ptolemy himself and stars used in the observations of his predecessors.

Almagest IX, 7 (Toomer, 449). The observation is #60 in Pedersen's enumeration (ref. 54).

Almagest IX, 8 (Toomer, 453–4).

Almagest X, 2 (Toomer, 471). This is potentially a rather precise method of making naked-eye observations if one has a good star chart to work with. I always direct my beginning students to make observations of the Moon and planets in just this way.

Of Ptolemy's very few planetary observations by the method of alignments, the most clear-cut are two observations of Venus in Almagest X, 1, one using ζ Geminorum (Toomer, 469) and one using φ and λ Aquarii (Toomer, 470). (These are # 93 and # 77 in Pedersen's ennumeration.) On at least one occasion, Ptolemy used an alignment with a fixed star as a control on a measurement taken with the armillary sphere. Venus was sighted with respect to Spica using the armillary sphere; Venus's proximity to β Scorpii was used as a check. (Almagest X, 4; Toomer, 474; # 80 in Pedersen.) Ptolemy is very clear about the different functions of the two stars in this observation. Also, while Venus was only a fraction of a degree from β Scorpii, it was some 36° from Spica. Sometimes Ptolemy also attempted to control planetary observations taken by means of a reference star and the armillary sphere by making an alignment with the Moon, if the Moon happened to be nearby. This required the use of a lunar position calculated from theory. (For an example, see the observation of Venus just cited.) This use of the Moon should not be confused with the use of the Moon as a connecting link between the Sun and a star, described above, which does not require a theoretical lunar position. Graßhoff G. , op. cit. (ref. 9), 270–323, has produced an instructive set of constellation charts graphically illustrating the patterns of error in each constellation of the catalogue. Graßhoff's charts make it clear that the stars were observed by constellations, for within a single constellation there is often a readily identifiable peculiar error. It does not, however, follow that each cluster of stars with a peculiar error was observed with respect to a different fundamental star. A whole constellation might be observed in quick succession with the orientation of the armillary sphere left almost unchanged. The sphere might then be re-oriented for the next group of stars. Such a procedure could well result in different peculiar errors even for star groups observed with respect to the same fundamental star.

Włodarczyk J. has discussed the results of an extensive series of observations taken in the ancient manner with an armillary sphere of recent construction. See his “Examining the armillary sphere”, Journal for the history of astronomy, xviii (1987), 173–95. Wlodarczyk made the difficult and laborious observations of absolute longitudes for two reference stars only, α Ari and α Tau. These stars were then used to measure the longitudes of eighteen secondary stars. In a few cases the longitudinal distance between the fundamental and the secondary star was as large as 150°. Apparently this caused no serious difficulty.

The figure is after the graph in Peters and Knobel, op. cit. (ref. 3), 6. The error in latitude, Δβ = 0°.31 cos(λ + 63°), is the result of a least-squares fit to Peters and Knobel's Table II (p. 17). The periodic error in longitude, similarly determined, is Δλ = 0°.21 cos(λ + 1°). It should be pointed out that Peters and Knobel define the error as the computed position minus the Ptolemaic position, which is opposite to the convention adopted here.

Rawlins , op. cit. (ref. 16), 366–70 has attempted to uncover the amplitudes and phases of several periodic errors in the catalogue by a least squares fit. The logical fallacy of such a procedure has been discussed above. To Rawlins's credit, he has tried to take into account several different possible sources of error. Unfortunately, this involves the simultaneous variation of no fewer than five parameters. Despite the relatively large number of parameters, a certain arbitrariness remains in the postulated forms and physical causes of the errors. Not surprisingly, the results of the variation of parameters appear to be unstable. Thus, the phase of a certain periodic error of unspecified origin turns out to be quite different, depending on whether longitude or latitude residuals are used to determine it. Further, the rules of the game are not very well specified: One might wish that different samples of stars would yield similar results; but when two samples yield different values for the error in the obliquity of the ecliptic ring, Rawlins concludes that Hipparchus used two different values for the obliquity.

Schoenberg E. , “Theoretische Photometrie”, in Handbuch der Astrophysik, ed. by Eberhard G. , ii/1 (Berlin, 1929), 1–280. Tables for the air mass are found on pp. 268–73.

The astronomical almanac for the year 1985 (Washington: U.S. Government Printing Office; London: Her Majesty's Stationery Office).

Hoffleit , The bright star catalogue (ref. 46).

Explanatory supplement to the Astronomical ephemeris and the American ephemeris and nautical almanac (London, 1961), 30–31.

See Woolard Edgar W. and Clemence Gerald M. , Spherical astronomy (New York and London, 1966), 302–7.

Hellwege K. H. (editor-in-chief), Landolt-Börnstein, n.s., VI/i (Berlin, 1965), 49. As is the case with most refraction tables, Landolt-Börnstein tabulates the refraction with apparent (not true) zenith distance as argument. Some care in interpolation is therefore required.

Because there occasionally has been confusion on this point, it is worth remarking that there is no evidence that Hipparchus ever observed at Alexandria. All the observations of Hipparchus for which the places are known were made in Rhodes. The only exception to this appears to be Hipparchus's collection of weather observations, said by Ptolemy in the Phaseis to be valid for the parallel through Bithynia, which was Hipparchus's native land. See the discussion by Dicks D. R. , The geographical fragments of Hipparchus (London, 1960), 2–6. See also Toomer , Ptolemy's Almagest, 134.

In Hipparchus's commentary on Aratus, β and γ Arae are not accorded separate status. They are there referred to jointly as “the double star that is the southernmost of the stars in the rim”. Commentary III, ii, 6; Manitius, 236.

For stars that culminated at apparent altitudes above 5° at Rhodes, the difference between the extinction calculated for Rhodes and the extinction calculated for Alexandria is less than one magnitude. Since each Ptolemaic magnitude is spread over about two photometric magnitudes, a differential extinction of less than a magnitude will not permit us to distinguish between alternatives. Thus, the magnitudes assigned to higher-altitude stars may be found to be equally consistent with their having been observed either at Rhodes or at Alexandria. This was already the case with β Arae. The six next most southern usable stars are ζ, Ara, β Cen, N Vel, β Cru, δ Cru, and κ Vel. Their calculated magnitudes do appear slightly to favour Ptolemy, but much less decisively than was the case with the first six. There is some bad luck in the fact that many of the most southerly stars of the catalogue had longitudes near 180° in Antiquity. The precession between the times of Hipparchus and Ptolemy caused these stars' declinations to decrease by nearly 1 ½°, thus substantially reducing the differential extinction between Hipparchus's time at Rhodes and Ptolemy's at Alexandria.

King Ivan , “Effective extinction values in wide-band photometry”, Astronomical journal, lvii (1952), 253–8. Useful discussion of King's equation may be found in the two following works: Young Andrew T. , “Observational technique and data reduction”, in Marton L. (editor-in-chief), Methods of experimental physics, xii/A (New York and London, 1974), especially 154–7; Taylor B. J. , “Two methods for computing monochromatic extinction from BV measurements”, Astronomical journal, lxxviii (1973), 61–66. Equation (2), like Equation (1), involves the assumption that the extinction is proportional to the actual mass of air traversed by the ray. In such a case one is justified in using Bemporad's tables to find the air mass. For the Rayleigh component of the extinction, this condition is rigorously satisfied. Aerosols and ozone, however, are not mixed uniformly through the atmosphere. Aerosols usually are concentrated near the ground, with a scale height on the order of one kilometre, while ozone is mostly concentrated near a height of 30km. The ‘effective air masses’ for aerosol and ozone extinction therefore differ from one another and from Bemporad's tabulated air mass. Fortunately, the deviations from Bemporad's theory due to ozone and to aerosols tend to compensate. See the discussion by Young, pp. 146–52. In ordinary situations, aerosol extinction at 0.55μm is greater than the extinction due to ozone. Bemporad's theory tends to underestimate the effective air mass for aerosol extinction. Thus, on balance, in our situation, Equation (2) is conservative in the sense that at large air masses it is more likely to underestimate than to overestimate the total extinction. Since the calculations presented in the main text appear to eliminate Rhodes as a place of observation, it is important that the total extinction at Rhodes be not overestimated.

The response functions of the standard V-system and the normal eye are tabulated by Allen C. W. , Astrophysical quantities, 3rd edn (London, 1973), 201.

Taylor , “Two methods for computing monochromatic extinction from BV measurements” (ref. 76), 62.

Laulainen N. S. , Taylor B. J. and Hodge P. W. , “Analyses of atmospheric extinction data obtained by astronomers — II. Seasonal variations in astronomical extinction”, Atmospheric environment, xi (1977), 21–27.

Allen (ref. 77), 126.

Hellwege (ed.), Landolt-Börnstein, n.s., VI/i, 52.

Taylor B. J. , Lucke P. B. and Laulainen N. S. , “Analyses of atmospheric extinction data obtained by astronomers — I. A time-trend analysis of data with internal accidental errors obtained at four observatories”, Atmospheric environment, xi (1977), 1–20.

Laulainen N. S. , “Analyses of atmospheric extinction data obtained by astronomers — III. Compilation of optical depths for 31 observatory sites”, Atmospheric environment, xi (1977), 29–33.

Mahmoud F. M. , unpublished Ph.D. dissertation, University of Thessaloniki, 1978, 74–77.

The Landolt-Börnstein tables refer to an aerosol plus Rayleigh extinction of 0.335 mag./unit air mass at 0.55μm as “slightly hazy”. If we include ozone extinction, this puts the total extinction at about 0.366 for “slightly hazy” conditions.

Rovithis P. and Rovithis-Livaniou H. , “Photometric results for the Kryonerion Astronomical Station”, , liv (1980), 253–63: Table II, on p. 257.

Kontizas E. and Kontiza M. , , liii (1978), 78. Graphical results reproduced by P. Rovithis and H. Rovithis-Livaniou (see ref. 86).

Brosch Noah , Technical Director of the Wise Observatory (private communication).

Taylor , “Two methods for computing monochromatic extinction from BV measurements” (ref. 76), 63.

Allen , Astrophysical quantities (ref. 77), 126.

Taylor , Lucke and Laulainen , “Analyses of atmospheric extinction data obtained by astronomers — I” (ref. 82), 81.