D'Arcy Thompson on Plato and Planets

D'Arcy Thompson published Plato's Theory of the Planets in the October 1904 edition of The Observatory. He published it again in the August 1910 edition of The Classical Review. We give here a version of the article. The original contains many Greek quotes. As part of the purpose of the article is to discuss the translation of the Greek text, this gives some difficulty in presenting the article for readers who do not know any Greek. However, we have attempted to make it as understandable as possible while trying to keep the style of the original article. Where we have replaced a Greek word, a Greek phrase, or a Greek passage by a translation, we enclose the material in square brackets and [colour it red].


(The Republic X, 616 E)
D'Arcy Thompson

There is a passage of great interest to the astronomer and of well-known difficulty to the classical scholar, in Plato's allegory of the Vision of Er, where the Spirits survey the heavens and the earth and all the harmonious motions of the universe. They see the heavenly bodies set in revolving whorls, which, whorl within whorl, combine to form the Spinning-whorl on the Spindle of Necessity; and the Goddess holds the spindle on her knee, and spins the thread which the Fates wind, unwind and cut.

The whorls are described as follows:

[Now the first and outermost whorl had the broadest circular rim, that of the sixth was second, and third was that of the fourth, and fourth was that of the eighth, fifth that of the seventh, sixth that of the fifth, seventh that of the third, eighth that of the second.]

Here D'Arcy Thompson is quoting a Greek passage from The Republic. The full description of "The Spindle of Necessity" by Plato is given in the following article where we present two different translations. See Plato on planets

The heavenly bodies, or the spheres or whorls in which they lie, are arranged one within another in the following order:
  1. The Fixed Stars.
  2. Saturn.
  3. Jupiter.
  4. Mars.
  5. Mercury.
  6. Venus.
  7. The Sun.
  8. The Moon.
There is no difficulty here. They are set in the order of their remoteness from the Earth; and if it be only with an approach to correctness, it must be remembered that by no system of geocentric spheres is correctness to be attained. According to some old writers Plato had set Venus to the farther side of Mercury; others, if I mistake not, put the Sun beyond both of these planets. All three ways are equally good, and equally untrue. But we are further told, and here the difficulty begins, that the whorls differ from one another in respect of 'breadth of rim': 'The first and outermost whorl is that which has its circular rim the broadest, and the sixth whorl comes next to it in regard to breadth of rim; and, proceeding in order of breadth, the fourth whorl comes third, and the eighth fourth, and the seventh fifth, and the fifth sixth, and the third seventh, and the second eighth.' Thus we have now a new classification of the heavenly bodies, in the following sequence:
  1. The Fixed Stars.
  2. Venus.
  3. Mars.
  4. The Moon.
  5. The Sun.
  6. Mercury.
  7. Jupiter.
  8. Saturn.
We might expect to elucidate the passage in two ways, either by literal interpretation of the Greek or by seeking a clue in the facts and language of Astronomy. The first hope eludes us, chiefly for the reason that Plato was not merely describing a phenomenon but was composing a riddle; and the second method has proved hitherto unsuccessful, the order given being out of accord with any scheme of classification employed by astronomers. Nevertheless it is my purpose to show that there is a certain astronomical phenomenon which sets all the planets in the precise order given, and that we can trace the knowledge of that phenomenon back to Eudoxus. And astronomical knowledge which reaches back as far as him may well come from even farther back and farther away.

The spheres of Eudoxus have been described and explained by many astronomers and historians of astronomy, but best, and far best of all, by Schiaparelli. Let me epitomize as best I can, and as far as need be, the Italian astronomer's interpretation. To account for the apparent movements of the Sun and Moon Eudoxus postulates three concentric spheres in each case, and four spheres for each of the other planetary bodies. The two outermost spheres are alike in all, Sun and Moon included. The first or outermost is identical, or coaxial, with the sphere of the fixed stars; its poles are the poles of heaven, and all the heavenly bodies share in its uniform rotation, which is that of night and day. The second sphere, sharing in the revolution of the first but with its own proper motion superadded, revolves about the axis of the ecliptic; its poles are fixed within the former sphere at points corresponding with the poles of the ecliptic, and its rotation carries Sun or Moon or planet along its oblique pathway, through the circle of the zodiacal signs. The second spheres, then, are all alike in configuration and orientation, but differ in their speed of rotation: that of the Moon revolving in a month, that of the Sun in a year, and those of the planets according to their own proper orbital periods.

Without recapitulating further details for Sun and Moon, we come to a very great difficulty in the case of the planets, namely to explain, by any simple imaginary mechanism, what are known as their stations and retrogradations. For the planets pursue their apparent course around the zodiac not with an even movement but in a jerky, sinuous way, alternately accelerated and reversed.

Eudoxus, who is said to have been the first to analyse the apparent motion of a planet into a system of interlinked figures-of-eight, and who gave to this figure-of-eight curve the name 'hippopede,' accounted for this peculiar motion in a very masterly way, without departing from the general plan of a system of concentric spheres. In addition to the two spheres already indicated, he conceived each planet to be controlled by two other concentric spheres, and to be set upon the equator of the fourth or innermost. The third sphere had its poles in the equator of the second, that is to say in the plane of the ecliptic, and therefore revolved, approximately, in a north and south direction. The fourth sphere, placed within it and revolving with equal and opposite velocity, counteracted the motion of the third and prevented the planet being carried, as the third would of itself have carried it, to first one and then the other pole of the ecliptic. But the fourth sphere was not coaxial with the third, but was set somewhat obliquely to it (just as the first sphere was to the second), and thus introduced another component in the form of a simple harmonic motion, causing the planet to perform apparently a pendulum-like vibration in the plane of the ecliptic, while all the while it was being carried around that circle by the proper motion of the second sphere. The phase of this vibration is obviously conditioned by the angle which the axis of the fourth sphere bears to that of the third; and Eudoxus tells us that these angles differ for the different planets, though he does not tell us what they severally were. The calculation, a very simple one, has been performed by Schiaparelli, who shows the angles to be as follows:
Venus. . . 46°
Mars . . . 34°
Mercury. . . 23°
Jupiter. . . 13°
Saturn. . . 6°
Obvious as it is, it is worth emphasizing that these are not merely angles concealed, so to speak, in the mechanism of the spheres, but doubled they represent the amplitude of each vibration, or the apparent visible excursion of each planet in its performance of a complete hippopede. As Simplicius says: [ As much as the breadth of the figure, the planet too will appear to shift sideways in breadth.]

Now these numbers place the planets, other than the Sun and Moon, in the precise order of Plato's text. Moreover, though not identical with these motions of the planets properly so called, the excursions of the Sun and Moon to either side of the equator, across the intertropic belt, are in a way comparable to them: so far at least, that they might with equal truth be described as [ motions in breadth], or ascribed to a range of freedom on either side of a theoretic orbit; and the analogy is all the better seen when we think of the identical mechanism of axial obliquities to which in each case, according to Eudoxus, the phenomenon is due. We may be justified in including in our series the angle of the ecliptic for the Sun, that is to say about 23° 28', or rather, if we assume the epoch of Eudoxus (as Dr Copeland suggested to me long ago) about 23° 45': which angle, in the [ artificial sphere] of Eudoxus, is the angle which the axis of the Sun's second sphere makes with that of his first; and in including for the Moon an angle which, as Eudoxus tells us, is in her case somewhat greater than the Sun's, in fact about 5° more. Then, having intercalated these two planets with their appropriate angles, we find the whole seven to follow the precise order of Plato's description; and these seven planets might be arranged in five thousand different ways!

As for the Fixed Stars, they extend over the whole surface of their sphere, or in other words the breadth of their whorl is a zone of 180°. So we at least should say; or perhaps, as the ancients knew nothing of the antarctic stars, which were permanently invisible to them, it might be more reasonable to assume a zone or whorl some 30° less. As our final result, our whole eight whorls with their respective breadths fall as follows into the Platonic order:
1. The Fixed Stars . . . 130° - 180°
2. Venus . . . 46°
3. Mars . . . 34°
4. The Moon . . . 28°
5. The Sun . . . 23° 45'
6. Mercury . . . 23°
7. Jupiter . . . 13°
8. Saturn . . . 6°
And there are forty thousand other ways in which these eight can be arranged!

A chief source of obscurity in the whole passage is the simple circumstance that Plato was talking in riddles and in allegory, after the fashion of antiquity and the East, and therefore did not choose to tell us many things which he must have known. But he was utilizing, and simplifying for the purposes of his allegory, all the astronomic knowledge of his time. Plato never supposed that a single sphere could account for all the complicated motions of each planet. He omitted from his description, I take it, all those spheres, the [spheres, the ones carrying no planets], which merely bore other and dependent spheres, but which themselves bore no heavenly body: even to the vision of the Spirits these spheres were invisible. He suppresses all those portions of the spheres into which the planetary orbits never range, and by so doing he reduces them to his [circular whorl], or 'whorls'.

He draws his picture for us of what the eye does not see but insight discerns - the [true embroidery], the [true figures] of the celestial fabric. [There are also heavens beyond the perceptible heavens]. It is this transcendental heaven which is brought down to our comprehension under the guise of homely allegory: [One must use the model of embroidery of the heavens for the sake of learning these things].

We cannot wholly elucidate what Plato left purposely obscure. We cannot reconstruct his system of spheres or whorls as we can those of Eudoxus or Callippus, for which material is expressly and carefully supplied: except in so far as we are justified in illustrating the Master's words by help of the pupil's. But so reading Plato's riddle in the light of Eudoxus' facts we cannot doubt that we have before us two settings of the same story, nor fail to see veiled in Plato's poetic language the finest technical knowledge of the age.
Greek key

While writing on this subject let me add, in parenthesis, that the very ancient and very decorative 'Greek key-pattern' seems to me to be nothing more nor less than an archaic representation of a planet's apparent course, a series of simplified hippopedes. It is, indeed, intermediate between the typical hippopede of Saturn or of Jupiter and the open one of Mercury, and is just such a figure as would be devised and used to depict the retrogressional course of a planet, prior to its more perfect representation by means of the hippopede itself.

Last Updated April 2007