Abstract
In Books VII and VIII of the Almagest Ptolemy interrupts the development of the planetary theories in order to deal with a number of problems connected with the fixed stars. The reason for this is explained in the prologue to this second part of the Almagest [VII, 1; Hei 2, 2], where Ptolemy tells his friend or protector Syrus (see page 26) that the further development of planetary theory presupposes a chapter on the so-called fixed stars. The reason is that while the solar theory was founded upon simple observations of solstitial and equinoctial times, and the lunar theory upon eclipses, the theories of the remaining five planets are, to a great extent, founded upon exact determinations of planetary longitudes derived from the distance of the planet in question from a fixed star, measured with the astrolabon (see page 183). Thus a position of Mercury is referred to Aldebaran (α Tauri) [IX, 7; Hei 2, 262] or to Regulus (a Leonis) [ibid.; Hei 2, 263] just as a longitude of Venus is found relative to that of Antares (a Scorpii) [X, 3; Hei 2, 303].
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- 1.
Egyptian astronomers and surveyers had a particular instrument consisting of a slit palm-leaf (merkhet) and plumb-line for observing the passage of a star across the meridian (see the illustration in L. Borchardt, Altägyptische Zeitmessung, PL XVI, Berlin, 1920). This may have accustomed thsm to connect the stars by a string held in front of the eyes. It is interesting to notice that the method of alignment was used as late as A.D. 1604 when Galileo proved that the nova of that year was a fixed star (see the Opera Omnia II, p. 279).
- 2.
On the stellar globe as a measuring instrument, see Vogt 1920, pp 45–51.
- 3.
Copernicus (De Rev. II, 14) quotes the Regulus observation by means of which the way was laid open to the other fixed stars (fol. 46 r). But he has the time wrong (5 hours after noon instead of 5½ hours) and gives the position of the Sun as 31/24° Pisces instead of about 3°. Tycho Brahe gave a general, critical analysis of Ptolemy’s method, underlining, among other things, Ptolemy’s disregard of atmopheric refraction (see Progym. I, 139 fT. — Opera Omnia II, 151 ff.). - Delambre (II, p. 248 f.) got some of the fractions wrong, working from a faulty MS. - Manitius (1905, p. 406) underlines the importance of the Regulus observation as a school example of the use of the astrolabon.
- 4.
Manitius maintained that the parallax in longitude was positive East of the meridian (first observation), but negative West of it (second observation); see his commentary to the German translation of the Almagest (vol. 2, p. 397–399). This would give π2 − π1 ≈ −0°;8,30 and thus explain the difference; but in the given situation both m and %2 must be negative (west-wards). Furthermore, he read the zenith distance off a celestial globe and found it to be circa 15°. From this value he deduced π1 = +10’ and π2 = −5’, without determining the angles between the ecliptic and the two verticals through L1 and L2.
- 5.
Pannekoek has shown (1955, p. 63) that if Ptolemy had used the exact formula
$$\Delta {\rm{\delta }} = {\rm{number of years}} \cdot {\rm{p}} \cdot \sin {\rm{\varepsilon }} \cdot \cos {\rm{\alpha }}$$for the change of declination of a star with the right ascension α he would have found the rate of precession p = 38" per year using only the six stars listed above. The remaining twelve stars would have led to p = 52" per year. Thus Ptolemy may have selected the six stars because they confirmed the rate of precession (i.e. p ≈ 36" per year) already found from the change of longitude; nevertheless, he did not suppress the other evidence.
- 6.
There were several forms of the theory of trepidation. They all assumed that the total displacement of the fixed stars over a number of years contained a periodic function of time. The theory was advanced by Thabit ben Qurra in his treatise On the Mjtion of the Eighth Sphere (ed. Carmody, 1960, pp. 84–113). - Cf. Goldstein (1965) and Hartner (1971).
- 7.
7) The idea of a 9th sphere containing the sphere of the fixed stars is already introduced by Ptolemy in the Hypotheses (ed. Nix, pp. 122 and 125). It serves as a ’mover’ of the starry spheres, but seems to have no direct connection with the theory of precession.
- 8.
Bjørnbo (1901) infers from statements by al-Sūfī (903–986) and al-Battānī (died 929) that Ptolemy relied upon a catalogue of the fixed stars made by Menelaos. There is not sufficient evidence that Menelaus ever compiled such a catalogue, although he may have determined a considerable number of stellar longitudes.
- 9.
Dreyer changed his opinion on this point. In his History (1906, p. 202) he considered Ptolemy’s catalogue as nothing but the catalogue of Hipparchus brought down to his own time with an erroneous value of the constant of precession. But in his two papers in the Monthly Notices of the Roy. Astr. Soc. (1917 and 1918) he arrived at the result that there is no reason to disbelieve his positive statement, that he had made extensive observations of fixed stars (1917, p. 539).
- 10.
The hypothesis that the errors of Ptolemy’s observations might be due to an eccentricity of the ecliptic ring of the astrolabon has been examined in great detail by A. Czwalina (1958, p. 287 ff.). Also Dreyer (1917) called attention to the possible errors of the instrument, and to the disastrous consequences of ignoring refraction when observing the Sun at the horizon.
- 11.
Ptolemy’s stellar magnitudes have been examined by Lundmark (1926) who compared them with those of al-Sūfī (10th century) and Tycho Brahe. He found that they are not sufficiently precise to prove the existence of secular variations in brightness since Antiquity. See also Boll (1916).
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Pedersen, O. (2011). The Fixed Stars. In: Jones, A. (eds) A Survey of the Almagest. Sources and Studies in the History of Mathematics and Physical Sciences. Springer, New York, NY. https://doi.org/10.1007/978-0-387-84826-6_8
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