Abstract
Having dealt with the theories of the Sun, the Moon, and the superior planets, we are left with the problem of describing the motions of Venus and Mercury. The reason why these planets are called inferior is that according to the usual order of the celestial bodies (see page 261) they are situated below the Sun, or, more precisely, between the orbits of the Sun and the Moon. Now in ancient astronomy the order of the planets is more or less a convention, without any sure foundation upon observable facts. Thus Ptolemy reminds us that there is no perceptible parallax in any of the planets, and that one has never observed a passage of Venus or Mercury before the disc of the Sun. He even maintains that such a passage has never occurred; but on the other hand this latter circumstance is no proof that there are no planets below the Sun since their orbits could be so inclined that a conjunction never results in a passage [IX, 1; Hei 2, 207]. It is clear, however, that it is too superficial to distinguish the inferior from the superior planets by their conventional and essentially arbitrary order.
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Notes
- 1.
For Venus and Mercury text books of astronomy often derive the relation \(\frac{1}{{\rm T}} = \frac{1}{{{{\rm T}_{\rm t}}}} - \frac{1}{{{{\rm T}^{\rm a}}}}\) as the analogy to (9.11). But then the period Tt is not one tropical year, but the tropical period of one of the inner planets as defined in a heliocentric system. This concept is foreign to Ptolemaic astronomy to which the relation above is without interest. Cf. Dreyer (1906, pp. 154 and 197).
- 2.
For earlier conceptions of the motions of the inferior planets, see Saltzer (1970).- Neugebauer (1972) argues that the ascription of the geo-heliocentric system to Heracleides rests upon a mistake by Chalcidius in his commentary on Plato's Timaios.
- 3.
For a general account of the Ptolemaic theories of the inferior planets see Delambre (II, pp. 308-346), Herz (1887, pp. 109 ff.), and Boelk (1911). The most lucid explanation of the theory of Mercury is due to W. Hartner (1955).
- 4.
A critical analysis of the observations of Venus quoted in the Almagest has been given by Czwalina (1959) who calculated the times of maximal elongations and compared them with the Ptolemaic data. His conclusion (p. 17) is that the discrepancies between observed and theoretical values can be explained on the assumption that Ptolemy was unable to measure maximal elongations to an accuracy better than ± 10.6 minutes of arc. - See also Wilson (1972).
- 5.
In Heiberg's edition of the Almagest [X, 1; Hei 2, 297] the date of the observation Vn is given as the 14th year of Antoninus. This year is also found in Delambre (II, p. 333) who only wondered that observations with the naked eye were able to give exactly equal maximal eastern elongations (cf. V5 and V11). However, in his German translation (Vol. 2, p. 412, note 11) Manitius found it difficult to believe that Ptolemy used an observation at so late a date. He proposed accordingly to place Vn in the 4th year of Antoninus, considering the Greek 1δ' (= 14) to be a scribal error for δ' (= 4). This emendation was confirmed by direct calculation by Czwalina (1959, p. 17).
- 6.
A critical examination of the observations of Mercury used by Ptolemy was made by Czwalina (1959) who made it plausible that both M11 and M12 took place in the 18th year of Hadrian (i.e. before M10). His dating of M3 to Mechir 30 instead of Phamenoth 30 (in disagreement with Hei-berg's text [IX, 7: Hei 2, 265]) was already proposed by Ideler and adopted by Manitius (see the German translation, vol. 2, p. 133).
- 7.
Boelk (p. 18) accused Delambre of dating the observation M6 to Nabonassar 564, asserting that Herz followed Delambre. It is true that Herz (p. 133) gives this erroneous date, and yet Delambre (II, p. 520) has the correct value Nabonassar 504.
- 8.
This explains why the theory of Mercury has often been construed in a confused and erroneous way by early authors, beginning with Proclus, Hypotyposis V, 3 B.
- 9.
According to Dreyer (1906, p. 197 f.) the radius of the small circle is 1/21 of that of the deferent. This is not in agreement with the Almagest which has TE = EF = 3P for R = 60p. Dreyer supports his statement with a reference to the Planetary Hypotheses, but here Ptolemy has TE = 3P and EF — 2;30 (see Opera Minora, ed. Heiberg, p. 87). The discrepancy is due to an error in Halma's edition of the Hypotheses used by Dreyer.
- 10.
This was also indicated by D. J. de Solla Price in The Equatorie of the Planetis, Cambridge, 1955, p. 102 which appeared at the same time as Hartner's paper. - Needles to say, Hartner's investigation was concerned with very much more than the derivation of this minimum value.
- 11.
Dr. D. T. Whiteside has kindly provided me with the following analytical treatment of the problem. From (10.35) it follows that the condition dp/dcm = 0 for the minimum of p(cm) is equivalent to
$$\frac{{{\rm ds}}}{{{\rm d}{{\rm c}_m}}} = \frac{{{\rm es}\,\sin \,{{\rm c}_{\rm m}}}}{{{\rm s} + {\rm e}\,\cos \,{{\rm c}_{\rm m}}}}$$The expression (10.34) can be reduced to
$${{\rm R}^2} = {{\rm S}^2} - 2{\rm es}\left( {\cos \,{{\rm c}_{\rm m}} + \cos \,2{{\rm c}_{\rm m}}} \right) + 2{{\rm e}^2}\left( {1 + \cos \,{{\rm c}_{\rm m}}} \right)$$from which follows
$$\frac{{{\rm ds}}}{{{\rm d}{{\rm c}_{\rm m}}}} = \frac{{ - {\rm es}(\sin \,{{\rm c}_{\rm m}} + 2\sin \,2{{\rm c}_{\rm m}}) + {{\rm e}^2}\sin \,{{\rm c}_{\rm m}}}}{{{\rm s} - {\rm e}(\cos \,{{\rm c}_m} + \cos \,2{{\rm c}_{\rm m}})}}$$Equating the two values of ds/dcm and dividing by e sin cm ≠0 we get
$$2{{\rm s}^2}\left( {1 + 2\,\cos \,{{\rm c}_{\rm m}}} \right) + 2\,{\rm es}\,{\rm co}{{\rm s}^{\rm 2}}\,{{\rm c}_{\rm m}} = {{\rm e}^{\rm 2}}\cos \,{{\rm c}_{\rm m}}$$whence
$$\cos \,{{\rm c}_{\rm m}} = \frac{{ - 4{{\rm s}^{\rm 2}} + {{\rm e}^2}\left( \pm \right)\sqrt {{{\left( {4{{\rm {s}^2} - {{\rm e}^{\rm 2}}}} \right)}^2} - 16{\rm e}{{\rm s}^{\rm 3}}} }}{{4{\rm es}}}$$Using e = 0.05 · R and the approximate value s ≈ s (120 ) = 0.9260 · R we get cm=120 ;29,7 which is only slightly different from (10.39).
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Pedersen, O. (2011). The Inferior Planets. 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_10
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