We must not disregard the question whether we should hold that there is one substance of this kind or more than one, and if more than one, how many; we must review the pronouncements of other thinkers and show that with regard to the number of the substances they have said nothing that can be clearly stated. The theory of the Ideas contains no peculiar treatment of the question; for the exponents of the theory call the Ideas numbers, and speak of the numbers now as though they were unlimited and now as though they were limited by the number 10¹; but as for why there should be just so many numbers, there is no explanation given with demonstrative accuracy. We, however, must discuss the question on the basis of the assumptions and distinctions which we have already made. The first principle and primary reality is immovable, both essentially and accidentally, but it excites the primary form of motion, which is one and eternal. Now since that which is moved must be moved by something, and the prime mover must be essentially immovable, and eternal motion must be excited by something eternal, and one motion by some one thing; and since we can see that besides the simple spatial motion of the universe²(which we hold to be excited by the primary immovable substance) there are other spatial motions—those of the planets—which are eternal (because a body which moves in a circle is eternal and is never at rest—this has been proved in our physical treatises³); then each of these spatial motions must also be excited by a substance which is essentially immovable and eternal. For the nature of the heavenly bodies is eternal, being a kind of substance; and that which moves is eternal and prior to the moved; and that which is prior to a substance must be a substance. It is therefore clear that there must be an equal number of substances, in nature eternal, essentially immovable, and without magnitude; for the reason already stated.⁴ 1073bThus it is clear that the movers are substances, and that one of them is first and another second and so on in the same order as the spatial motions of the heavenly bodies. As regards the number of these motions, we have now reached a question which must be investigated by the aid of that branch of mathematical science which is most akin to philosophy, i.e. astronomy; for this has as its object a substance which is sensible but eternal, whereas the other mathematical sciences, e.g. arithmetic and geometry, do not deal with any substance. That there are more spatial motions than there are bodies which move in space is obvious to those who have even a moderate grasp of the subject, since each of the non-fixed stars has more than one spatial motion. As to how many these spatial motions actually are we shall now, to give some idea of the subject, quote what some of the mathematicians say, in order that there may be some definite number for the mind to grasp; but for the rest we must partly investigate for ourselves and partly learn from other investigators, and if those who apply themselves to these matters come to some conclusion which clashes with what we have just stated, we must appreciate both views, but follow the more accurate. Eudoxus⁵ held that the motion of the sun and moon involves in either case three spheres,⁶ of which the outermost is that of the fixed stars,⁷ the second revolves in the circle which bisects the zodiac,⁸ and the third revolves in a circle which is inclined across the breadth of the zodiac⁹; but the circle in which the moon moves is inclined at a greater angle than that in which the sun moves. And he held that the motion of the planets involved in each case four spheres; and that of these the first and second are the same¹⁰ as before (for the sphere of the fixed stars is that which carries round all the other spheres, and the sphere next in order, which has its motion in the circle which bisects the zodiac, is common to all the planets); the third sphere of all the planets has its poles in the circle which bisects the zodiac; and the fourth sphere moves in the circle inclined to the equator of the third. In the case of the third sphere, while the other planets have their own peculiar poles, those of Venus and Mercury are the same. Callippus¹¹ assumed the same arrangement of the spheres as did Eudoxus (that is, with respect to the order of their intervals), but as regards their number, whereas he assigned to Jupiter and Saturn the same number of spheres as Eudoxus, he considered that two further spheres should be added both for the sun and for the moon, if the phenomena are to be accounted for, and one for each of the other planets. But if all the spheres in combination are to account for the phenomena, 1074athere must be for each of the other planets other spheres, one less in number than those already mentioned, which counteract these and restore to the same position the first sphere of the star which in each case is next in order below.¹² In this way only can the combination of forces produce the motion of the planets. Therefore since the forces by which the planets themselves are moved are 8 for Jupiter and Saturn, and 25 for the others, and since of these the only ones which do not need to be counteracted are those by which the lowest planet¹³ is moved, the counteracting spheres for the first two planets will be 6, and those of the remaining four will be 16; and the total number of spheres, both those which move the planets and those which counteract these, will be 55. If we do not invest the moon and the sun with the additional motions which we have mentioned,¹⁴ there will be 47 (?)¹⁵ spheres in all. This, then, may be taken to be the number of the spheres; and thus it is reasonable to suppose that there are as many immovable substances and principles,¹⁶—the statement of logical necessity may be left to more competent thinkers. If there can be no spatial motion which is not conducive to the motion of a star, and if moreover every entity and every substance which is impassive and has in itself attained to the highest good should be regarded as an end, then there can be no other entity besides these,¹⁷ and the number of the substances must be as we have said. For if there are other substances, they must move something, since they are the end of spatial motion. But there can be no other spatial motions besides those already mentioned. This is a reasonable inference from a general consideration of spatial motion. For if everything which moves exists for the sake of that which is moved, and every motion for the sake of something which is moved, no motion can exist for the sake of itself or of some other motion, but all motions must exist for the sake of the stars. For if we are to suppose that one motion is for the sake of another, the latter too must be for the sake of something else; and since the series cannot be infinite, the end of every motion must be one of the divine bodies which are moved through the heavens. It is evident that there is only one heaven.¹⁸ For if there is to be a plurality of heavens (as there is of men), the principle of each must be one in kind but many in number. But all things which are many in number have matter (for one and the same definition applies to many individuals, e.g. that of “man”; but Socrates is one¹⁹), but the primary essence has no matter, because it is complete reality. Therefore the prime mover, which is immovable, is one both in formula and in number; and therefore so also is that which is eternally and continuously in motion. Therefore there is only one heaven. 1074bA tradition has been handed down by the ancient thinkers of very early times, and bequeathed to posterity in the form of a myth, to the effect that these heavenly bodies are gods,²⁰ and that the Divine pervades the whole of nature. The rest of their tradition has been added later in a mythological form to influence the vulgar and as a constitutional and utilitarian expedient²¹; they say that these gods are human in shape or are like certain other animals,²² and make other statements consequent upon and similar to those which we have mentioned. Now if we separate these statements and accept only the first, that they supposed the primary substances to be gods, we must regard it as an inspired saying and reflect that whereas every art and philosophy has probably been repeatedly developed to the utmost and has perished again, these beliefs of theirs have been preserved as a relic of former knowledge. To this extent only, then, are the views of our forefathers and of the earliest thinkers intelligible to us.