EUDOXUS, CALLIPPUS, ARISTOTLE.

Eudoxus of Cnidos (about 408–355 B.C.) was one of the very greatest of the Greek mathematicians. He was the discoverer and elaborator of the great theory of proportion applicable to all magnitudes whether commensurable or incommensurable which is given in Euclid’s Book V. He was also the originator of the powerful method of exhaustion used by all later Greek geometers for the purpose of finding the areas of curves and the volumes of pyramids, cones, spheres and other curved surfaces. It is not therefore surprising that he should have invented a remarkable geometrical hypothesis for explaining the irregular movements of the planets. The problem was to find the necessary number of circular motions which by their combination would produce the motions of the planets as actually observed, and in particular the variations in their apparent speeds, their stations and retrogradations and their movements in latitude. This Eudoxus endeavoured to do by combining the motions of several concentric spheres, one inside the other, and revolving about different axes, each sphere revolving on its own account but also being carried round bodily by the revolution of the next sphere encircling it. We are dependent on passages from Aristotle and Simplicius for our knowledge of Eudoxus’s system, which he had set out in a work On Speeds, now lost. Eudoxus assumed three revolving spheres for producing the apparent motions of the sun and moon respectively, and four for that of each of the planets. In his hypothesis for the sun he seems deliberately to have ignored the discovery made by Meton and Euctemon some sixty or seventy years before that the sun does not take the same time to describe the four quadrants of its orbit between the equinoctial and solstitial points.

It should be observed that the whole hypothesis of the concentric spheres is pure geometry, and there is no mechanics in it. We will shortly describe the arrangement of the four spheres which by their revolution produced the motion of a planet. The first and outermost sphere produced the daily rotation in twenty-four hours; the second sphere revolved about an axis perpendicular to the plane of the zodiac or ecliptic, thereby producing the motion along the zodiac “in the respective periods in which the planets appear to describe the zodiac circle,” i.e. in the case of the superior planets, the sidereal periods of revolution, and in the case of Mercury and Venus (on a geocentric system) one year. The third sphere had its poles at two opposite points on the zodiac circle, the poles being carried round in the motion of the second sphere; the revolution of the third sphere about the axis connecting the two poles was again uniform and took place in a period equal to the synodic period of the planet, or the time elapsing between two successive oppositions or conjunctions with the sun.

The poles of the third sphere were different for all the planets, except that for Mercury and Venus they were the same. On the surface of the third sphere the poles of the fourth sphere were fixed, and its axis of revolution was inclined to that of the former at an angle constant for each planet but different for the different planets. The planet was fixed at a point on the equator of the fourth sphere. The third and fourth spheres together cause the planet’s movement in latitude. Simplicius explains clearly the effect of these two rotations. If, he says, the planet had been on the third sphere (by itself), it would actually have arrived at the poles of the zodiac circle; but, as things are, the fourth sphere, which turns about the poles of the inclined circle carrying the planet and rotates in the opposite sense to the third, i.e. from east to west, but in the same period, will prevent any considerable divergence on the part of the planet from the zodiac circle, and will cause the planet to describe about this same zodiac circle the curve called by Eudoxus the hippopede (horse-fetter), so that the breadth of this curve will be the maximum amount of the apparent deviation of the planet in latitude. The curve in question is an elongated figure-of-eight lying along and bisected by the zodiac circle. The motion then round this figure-of-eight combined with the motion in the zodiac circle produces the acceleration and retardation of the motion of the planet, causing the stations and retrogradations. Mathematicians will appreciate the wonderful ingenuity and beauty of the construction.

Eudoxus spent sixteen months in Egypt about 381–380 B.C., and, while there, he assimilated the astronomical knowledge of the priests of Heliopolis and himself made observations. The Observatory between Heliopolis and Cercesura used by him was still pointed out in Augustus’s time; he also had one built at Cnidos. He wrote two books entitled respectively the Mirror and the Phænomena; the poem of Aratus was, so far as verses 19–732 are concerned, drawn from the Phænomena of Eudoxus. He is also credited with the invention of the arachne (spider’s web) which, however, is alternatively attributed to Apollonius, and which seems to have been a sun-clock of some kind.

Eudoxus’s system of concentric spheres was improved upon by Callippus (about 370–300 B.C.), who added two more spheres for the sun and the moon, and one more in the case of each of the three nearer planets, Mercury, Venus and Mars. The two additional spheres in the case of the sun were introduced in order to account for the unequal motion of the sun in longitude; and the purpose in the case of the moon was presumably similar. Callippus made the length of the seasons, beginning with the vernal equinox, ninety-four, ninety-two, eighty-nine and ninety days respectively, figures much more accurate than those given by Euctemon a hundred years earlier, which were ninety-three, ninety, ninety and ninety-two days respectively.

With Callippus as well as Eudoxus the system of concentric spheres was purely geometrical. Aristotle (384–322 B.C.) thought it necessary to alter it in a mechanical sense; he made the spheres into spherical shells actually in contact with one another, and this made it almost necessary, instead of having independent sets of spheres, one set for each planet, to make all the sets part of one continuous system of spheres. For this purpose he assumed sets of reacting spheres between successive sets of the original spheres. E.g. Saturn being carried by a set of four spheres, he had three reacting spheres to neutralise the last three, in order to restore the outermost sphere to act as the first of the four spheres producing the motion of the next lower planet, Jupiter, and so on. The change was hardly an improvement.

Aristotle’s other ideas in astronomy do not require detailed notice, except his views about the earth. Although he held firmly to the old belief that the earth is in the centre and remains motionless, he was clear that its shape (like that of the stars and the universe) is spherical, and he had arrived at views about its size sounder than those of Plato. In support of the spherical shape of the earth he used some good arguments based on observation. (1) In partial eclipses of the moon the line separating the dark and bright portions is always circular—unlike the line of demarcation in the phases of the moon which may be straight. (2) Certain stars seen above the horizon in Egypt and in Cyprus are not visible further north, and, on the other hand, certain stars set there which in more northern latitudes remain always above the horizon. As there is so perceptible a change of horizon between places so near to each other, it follows not only that the earth is spherical but also that it is not a very large sphere. Aristotle adds that people are not improbably right when they say that the region about the Pillars of Heracles is joined on to India, one sea connecting them. He quotes a result arrived at by the mathematicians of his time, that the circumference of the earth is 400,000 stades. He is clear that the earth is much smaller than some of the stars, but that the moon is smaller than the earth.

The systems of concentric spheres were not destined to hold their ground for long. In these systems the sun, moon and planets were of necessity always at the same distances from the earth respectively. But it was soon recognised that they did not “save the phenomena,” since it was seen that the planets appeared to be at one time nearer and at another time further off. Autolycus of Pitane (who flourished about 310 B.C.) knew this and is said to have tried to explain it; indeed it can hardly have been unknown even to the authors of the concentric theory themselves, for Polemarchus of Cyzicus, almost contemporary with Eudoxus, is said to have been aware of it but to have minimised the difficulty because he preferred the hypothesis of the concentric spheres to others.

Development along the lines of Eudoxus’s theory being thus blocked, the alternative was open of seeing whether any modification of the Pythagorean system would give better results. We actually have evidence of revisions of the Pythagorean theory. The first step was to get rid of the counter-earth, and some Pythagoreans did this by identifying the counter-earth with the moon. We hear too of a Pythagorean system in which the central fire was not outside the earth but in the centre of the earth itself. The descriptions of this system rather indicate that in it the earth was supposed to be at rest, without any rotation, in the centre of the universe. This was practically a return to the standpoint of Pythagoras himself. But it is clear that, if the system of Philolaus (or Hicetas) be taken and the central fire be transferred to the centre of the earth (the counter-earth being also eliminated), and if the movements of the earth, sun, moon and planets round the centre be retained without any modification save that which is mathematically involved by the transfer of the central fire to the centre of the earth, the daily revolution of the earth about the central fire is necessarily transformed into a rotation of the earth about its own axis in about twenty-four hours.