Plato and Eudoxus on the Planetary Motions

The myth appears on pp. 614b–621d (here and throughout I will cite passages from the Republic and Timaeus according to the standard pagination, as in Burnet J. , Platonis opera , iv (Oxford, 1902)). On the astronomical section, presented at 616b–617d, see, for instance, the annotations in Adam's J. edition, The Republic of Plato (Cambridge, 1902), ii, 441–53, 470–9; Guthrie W. K. C. , A history of Greek philosophy (Cambridge, 1962–81), iv, 557–8; Dicks D. R. , Early Greek astronomy to Aristotle (Bristol, 1970), 109–13; Heath T. L. , Aristarchus of Samos: The ancient Copernicus (Oxford, 1913), 148–58; Lee D. , Plato: The Republic , 2nd edn (Harmondsworth, Middlesex, 1974), 460–4.

For an illustration, see Adam , op. cit. (ref. 1), 444, and Lee , op. cit. (ref. 1), 464.

See Singer C. , A history of technology (Oxford, 1954–78), ii, 201; cf. also i, 281. Illustrations of later Roman spindles, shafts and whorls (ii, 202) suggest elements of Plato's description.

On the astronomy in the Timaeus, see Taylor A. E. , A commentary on Plato's Timaeus (Oxford, 1928), 146–74, 191–245; Cornford F. M. , Plato's cosmology: The Timaeus of Plato (London, 1937; repr. Indianapolis/New York, n.d.), 72–93, 105–37; Guthrie , op. cit. (ref. 1), v, 295–6; Dicks , op. cit. (ref. 1), 116–37; Vlastos G. , Plato's universe (Seattle, 1975), Part I.

My proposal of two annular disks for the scheme in the Timaeus can draw support from the parallel in the Republic, where each planet is assigned an annular figure, being the upper surface of a hemispherical shell, such that the whole system of orbits forms “a continuous back of a single whorl”, corresponding to the circular plane of the equator (616d-e). But Taylor construes the Timaeus figure as if of two circles, inscribed on the surface of the celestial sphere ( op. cit. (ref. 4), 147); Cornford seems to intend much the same idea, inasmuch as he infers from Timaeus's description the form of an actual armillary sphere ( op. cit. (ref. 4), 74–79); Dicks also takes the figures as the intersecting circles of ecliptic and equator, but he criticizes Cornford's effort to infer a more elaborate form of planetarium ( op. cit. (ref. 1), 119–21).

The inner four orbits are explicitly identified (38c-d, 39b-c), while the outer three can be inferred indirectly from the principle of relative speeds (39a-b); see below.

The precise meaning of this term is debated; see ref. 11.

Cf. the account by Aristotle in De caelo , II, 9; for the text and translation, see, for instance, Kirk G. S. Raven J. E. Schofield M. , The Presocratic philosophers , 2nd edn (Cambridge, 1983), 344–5.

Cornford and Taylor assume a physical model of some sort, although the view is contested by Dicks; cf. ref. 5 above.

In Ptolemy the motions are compared to the diurnal rotation, so that retrograde motion in the ecliptic is termed “toward the leading parts” (), while forward motion becomes “toward the rear” (); cf. Toomer G. J. , Ptolemy's Almagest (New York etc., 1984), 20. In his commentary on this passage of the Timaeus, Proclus introduces for Timaeus's term the synonym ; cf. In Platonis Timaeum commentaria, ed. by Diehl E. (3 vols, Leipzig, 1903–6), iii, 146. Theon of Smyrna uses in a related context; cf. Expositio rerum mathematicarum ad legendum Platonem utilium, ed. by Hiller E. (Leipzig, 1873), 147, line 11.

The modern commentators generally agree that the term in the Republic refers to the retrograde motion, but differ on the precise interpretation, in particular, why Mars alone is so designated (cf. Dicks, op. cit. (ref. 1), 112–13). I think the key lies in Plato's phrase “as it appeared to them [sc. the observers]”. Unlike the schematic account of the Timaeus, the myth in the Republic presumes to describe a specific time at which the heavenly system was seen; thus, the narrator would be reporting how at this particular time Mars happened to be in retrograde, while the other planets were moving with their forward motions. To suppose, as does Dicks, some special aspect of the planetary scheme for Mars thus becomes unnecessary.

To similar effect, Timaeus has earlier described how the cosmic Creator “ordained the circles to move in the contrary sense () to each other, three moving similarly in speed, the four [others] dissimilarly both to the three and to each other, though in proportion” (36d).

Cf. Vlastos, op. cit. (ref. 4), 107–8, who reviews alternative suggestions by Martin T. H. , and others; Taylor , op. cit. (ref. 4), 202 is similar to Vlastos in his view. On a very different proposal by van der Waerden, see ref. 40.

Note that Timaeus does not at all suggest that the Sun and Moon might move nonuniformly.

To be sure, he admits that the phenomena display “a hopeless multiplicity and are wondrously intricate” (39d). This ostensible discrepancy with the simplicity of the underlying scheme will be discussed further below.

Cf. Vlastos, op. cit. (ref. 4), 101–2: “Plato could not have held up the movements of the planets as an ‘image of eternity revolving according to number’ (39A7–8) unless he believed that they instantiate periodicities of the most exact and dependable form which can be realized in the physical universe”.

Modern accounts of the Eudoxean system stem from the reconstruction proposed by Schiaparelli (1873), explicating the account by Simplicius; for summaries, see Neugebauer O. , Exact sciences in Antiquity , 2nd edn (Providence, 1957), 153–5, 182–3; A history of ancient mathematical astronomy (Berlin etc., 1975), 675–85; and Heath , op. cit. (ref. 1), 193–224. For an elaborate, highly speculative reconstruction, see Maula E. , Studies in Eudoxus' homocentric spheres (Helsinki, 1974).

The dating of the Timaeus is contested. In a controversial study, Owen G. E. L. proposes setting it only shortly after the Republic, that is, in the 370s; cf. his “The place of the Timaeus in Plato's dialogues”, Classical quarterly , n.s., iii (1953), 79–95 (repr. in his collected papers, Logic, science and dialectic , ed. by Nussbaum M. C. (Ithaca, 1986), 65–84). But a later dating, not much before the Laws (that is, c. 350), was the prevalent view among scholars early in our century and retains its hold with many, despite Owen's counterproposal. As Owen observes, the traditional dating of Eudoxus, 408/5-355/2, must imply that Eudoxus had developed his astronomical system well before Plato wrote the Timaeus, if that were in the late 350s, so that Plato's failure even to hint at the Eudoxean system in the dialogue becomes a real puzzle ( op. cit. , 86–87); cf. ref. 26. But significantly later dates for Eudoxus, e.g. c. 390–c. 340, have emerged through recent re-examinations of the chronological evidence; cf. Lasserre F. , Die Fragmente des Eudoxos von Knidos (Berlin, 1966), 137–9. On this view, it becomes probable, regardless of where one sets the Timaeus, that both dialogues had appeared before Eudoxus worked out his planetary system.

Neugebauer speculates that the heuristic inspiration of Eudoxus's device lay in considering the general characteristics of the two-sphere composition, as suggested first by the system of equator and ecliptic ( Exact sciences (ref. 17), 153–4). In support of his idea one can cite Timaeus's observation that each planet actually moves along a spiral, as its proper motion in the ecliptic is superimposed over its diurnal motion parallel to the equator (Timaeus 39a-b). Conceivably, a remark of this sort could have provoked an astronomer like Eudoxus to inquire how the shape of a planet's spiral would depend on the relative periods of the two spherical motions; in the limiting case, when the periods are equal, the spiral is changed into the hippopede.

Goldstein B. R. Bowen A. C. , “A new view of early Greek astronomy”, Isis , lxxiv (1983), 330–40, p. 338.

In Aristotelis De caelo , ed. by Heiberg J. L. (Berlin, 1894), 488 (commenting on De caelo , II, 12). (I have inserted the bracketed numerals for convenience of reference.) This passage is prominent in most accounts of Plato's view of astronomy and of his relation to Eudoxus; see, for instance, Duhem P. , To save the phenomena (Chicago, 1969; translated from the French edition of 1908), 5–6; Heath , op. cit. (ref. 1), 140–1; Lasserre F. , op. cit. (ref. 18), 67, 200–1; Guthrie , op. cit. (ref. 1), v, 296n. For an extensive discussion, to which I return below, see Vlastos , op. cit. (ref. 4), 59–61, 110–11.

On Sosigenes , see the article by Rehm in Paulys Realencyclopädie der classischen Altertumswis-senschaft , ed. by Wissowa G. , ser. 2, v, 1157–9.

Tannery P. , for instance, maintains that Proclus and other commentators cited Eudemus's History of geometry only through secondary sources; for remarks and references, see my Textual studies in ancient and medieval geometry (Boston etc., 1989), 126, note 124. The most extensive fragment from this Eudemean work, the account of the crescent quadratures by Hippocrates of Chios, is transmitted by Simplicius, who claims to be quoting it “verbatim” (); for a thorough discussion, see Lloyd G. E. R. , “The alleged fallacy of Hippocrates of Chios”, Apeiron , xx (1987), 103–28. Even here, however, the suspicion arises for me that Simplicius depends on secondary authorities for his text. For in precisely the same context, Simplicius's contemporary, Eutocius of Ascalon, cites not only the History of Eudemus, but also the Aristotelian honeycombs of Sporus of Nicaea, a commentator of the second or third century a.d (for references, see Knorr , loc. cit. ). Sporus thus emerges as a possible transmitter of Eudemean fragments to later writers like Eutocius and Simplicius.

That Sosigenes, rather than Eudemus, is the actual source here has been argued already by Grote G. ( Plato (London, 1865), i, 124–5), although he supposes that Sosigenes had access to a legitimate tradition for it, perhaps deriving from Plato's oral teachings. By contrast, most discussions assume uncritically that Eudemus is source. Vlastos explores certain conditions that he maintains would have to hold if one wished to “underwrite categorically” the doubts about Eudemean authority, and concludes that the validity of these conditions remains open ( op. cit. , (ref. 4), 110–11). But even this limited effort at rehabilitation depends on intricate assumptions about Simplicius's style and motives. Vlastos himself finally observes that “no historical argument could be grounded on the assumption … that Sosigenes' report about Plato … represents historical fact”.

Note that most accounts of Plato's scheme fail to recognize its violation of the principle of uniformity; cf. Vlastos , ibid. , 54, 61; Taylor , op. cit. (ref. 4), 209.

To be sure, some commentators have insisted that Plato did indeed know Eudoxus's system; but even here, the absence of signs of this knowledge in the Timaeus is admittEd. Taylor , for instance, explains that Plato would have been guilty of anachronism, had he ascribed the Eudoxean ideas to the fifth-century figure Timaeus ( ibid. , 209–12). Dicks ( op. cit. (ref. 1), 241, n. 200) and Lasserre ( op. cit. (ref. 18), 181–2) criticize Ross, Guthrie and others for presuming to detect “hints” of Eudoxus's scheme in Plato.

Dicks , op. cit. (ref. 1), 138–40; Cornford , op. cit. (ref. 4), 90. Vlastos vigorously defends the same position, op. cit. (ref. 4), 99–100.

Some have maintained that in this passage of the Laws, as also in a related passage of the Timaeus (40b–c), Plato means to assign a daily axial rotation to the Earth, thus to render merely apparent the rotation of the celestial sphere. That Plato adopts this doctrine seems dubious to me, but I think the proposal cannot be dismissed as trivial. For discussions of the issue see Taylor , op. cit. (ref. 4), 226–39; Cornford , op. cit. (ref. 4), 120–34; Heath , op. cit. (ref. 1), 181–9; and Dicks , op. cit. (ref. 1), 132–40.

See, for instance, Pangle T. L. , The Laws of Plato (Chicago, 1980), 511, n. 2, who cites the arguments of Strauss L. and, in a related context, Post L. A. ( ibid. , 517, n. 45). Although Aristotle seems tacitly to identify the “Stranger” as Socrates ( Politics , II, vi), he does not explicitly assert this.

The alternative view, that the “Stranger” is simply Plato, is maintained by Cicero ( Laws , I, v, 15). But it was already questioned in Antiquity, as by Diogenes Laertius, who asserts that “the Athenian Stranger is not, as some have supposed, Plato, but an anonymous fiction” ( Lives of the philosophers , iii, 52). This seems to be the prevalent modern view. Taylor, for instance, observes that the character is in certain general respects modelled on Plato, and to this extent represents him, “though we have no reason to suppose that he is drawn with any deliberate intention of self-portraiture” ( Taylor A. E. , Plato: The man and his work , 6th ed. (New York, 1952; repr. Cleveland , 1963), 465). Guthrie calls the character “Plato's own mouthpiece” ( op. cit. (ref. 1), v, 324); T. J. Saunders describes him as “in effect Plato himself” ( Plato: The laws (Harmondsworth, Middlesex, 1970), 39). No one, I trust, would presume to infer on the basis of the dialogue's setting that Plato (or even less, Socrates) had ever actually visited Crete.

This view is maintained also by Cornford , op. cit. (ref. 4), 92, and by Dicks , op. cit. (ref. 1), 148–9.

There is apparently a shift in perspective from the Timaeus (36c), where the primary diurnal rotation is “rightward” and the planetary motions are “to the left”. For an attempt to explain the discrepancy, see Dicks , op. cit. (ref. 1), 121–2, 146, and 245, n. 223.

I thus reject the suggestion by Burnet , who against the manuscript evidence inserts Oμκ, so to read, “which does not impel the others …”. For discussion, see Dicks , ibid. , 146. Note that by calling the celestial sphere the “eighth” orbit, the Epinomis reverses the count as in the Republic, where it is the first.

Note that in the Timaeus (38d-e) a similar remark cuts off the description of planetary motions after Mercury and Venus (see the quotation at the end of Section I above).

Metaphysics , XII, ch. 8 (1073b1–74a13).

See ref. 18 above.

Bulmer-Thomas I. , “Plato's astronomy”, Classical quarterly , n.s., xxxiv (1984), 107–12. Bulmer-Thomas surveys prior views on this much-discussed passage (528e–530c). I accept his basic reading of it: That by assigning priority to geometric theory over observation, Plato has not lapsed, as critics often maintain, into a purely speculative astronomy, utterly divorced from the phenomena. But Bulmer-Thomas wishes to restrict the observational activity Plato criticizes to the simple project of grouping stars into constellations, by construing , “the [things] in the sky” in 530b as an ellipsis for “the embroideries (πoiκλματα) in the sky”. I do not think this restriction is necessary, or even desirable. To be sure, the expanded phrase occurs earlier, at 529c, where it must indeed refer to the intricate tracery of the constellations. In the myth of Er, for instance, the sphere of the fixed stars alone is described as πoiκλoν (616e). But since the elliptical phrase at 530b is perfectly acceptable of itself to denote the general celestial phenomena, one could not, I think, be expected to supply a missing noun from the earlier passage at 529c, so to obtain the more restrictive sense. More important, the restriction has the effect of trivializing the significance of Plato's methodological observation at 530b, for it is hard to suppose that the mere grouping and naming of constellations would ever have been viewed by anyone as a profound scientific undertaking.

Van der Waerden B. L. , “On the motion of the planets according to Heraclides of Pontus”, Archives internationales d'histoire des sciences , xxviii (1978), 167–82.

Ibid. , 171; quoted from the translation by Heath , op. cit. (ref. 1), 269, from Manitius's edition of the Greek.

Van der Waerden argues from these passages that Plato and the Pythagorean astronomers introduced epicycles for explaining the planetary motions; cf. op. cit. (ref. 38), 170; “The earliest form of the epicycle theory”, Journal for the history of astronomy , v (1974), 175–85; “The motion of Venus, Mercury and the Sun in early Greek astronomy”, Archive for history of exact sciences , xxvi (1982), 99–113; and elsewhere (references given in the cited articles). His claims to have established this ambitious thesis, however, must be tempered by awareness of the possibility, as I have argued, of the much simpler notion of oscillations. Further, Van der Waerden's evidence for early epicyclic theories rests primarily on the testimony of late authors (like Geminus, Dercyllides, Theon of Smyrna, Proclus, Simplicius); these authorities could easily have drawn on their own familiarity with the mature astronomical theories of Hipparchus and (in the case of Proclus and Simplicius) of Ptolemy, and projected them back anachronistically into their reading of Plato and the older Pythagoreans, just as, in effect, van der Waerden himself has done. The notion that passages like that quoted above from Geminus indicate a Pythagorean theory of epicycles had long ago been proposed by Schiaparelli, but dismissed by Zeller E. , by Heath ( op. cit. (ref. 1), 270–1), and more recently by Neugebauer ( History (ref. 17), 695–6; partial rejoinder by Van der Waerden , op. cit. (ref. 38), 168–9).