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 10 Cf. Aristot. Met. 13.8.17, 20.
This was a Pythagorean survival, cf. Vol. I. Introduction.
xvi.; 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 universe i.e., the (apparent) diurnal revolution of the heavens. (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 treatises Aristot. Physics 8.8, 9, Aristot. De Caelo 1.2, 2.3-8. ); 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. Aristot. Met. 12.7.12, 13.
Thus 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.
Eudoxus Of Cnidus (circa 408 -355 B.C.). He was a pupil of Plato, and a distinguished mathematician. held that the motion of the sun and moon involves in either case three spheres, For a full discussion of the theories of Eudoxus and Callipus see Dreyer, Planetary Systems 87-114; Heath, Aristarchus of Samos 190-224. of which the outermost is that of the fixed stars, Not identical with that of the fixed stars, but having the same motion. the second revolves in the circle which bisects the zodiac, i.e., revolves with its equator in the ecliptic. and the third revolves in a circle which is inclined across the breadth of the zodiac i.e., has the plane of its equator inclined to the plane of the ecliptic. This sphere carries the sun (or moon) fixed to a point in its equator.; 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 same Not the same, but having the same motion. 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. Callippus of Cyzicus (fl. 380 B.C.). Simplicius says ( Simplicius 493.5-8 ) that he corrected and elaborated Eudoxus's theory with Aristotle's help while on a visit to him at Athens. 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, there 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. Aristotle is trying to establish a mechanical relation between the spheres, which Eudoxus and Callipus did not attempt to do. 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 planet The moon.
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, In sect.
11. there will be 47 (?) Either Aristotle has made a slip in his calculations, or we should read E)NNE/A (Sosigenes) for E(PTA/; this would give 49, which appears to be the correct total. For alternative explanations of an error in calculation see Ross ad loc. 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, i.e., the movers of the spheres. —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, See previous note. 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. This paragraph seems to belong to an earlier period of Aristotle's thought. At any rate the argument that plurality involves matter is inconsistent with the view that there are 55 immaterial movers. 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 one The definition or form is one and universal; it is the combination of form with matter that constitutes an individual. Thus a plurality of individuals is caused by the combination of the same form with different matter. ), 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. A 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, This statement is not literally true. The planets do not seem to have been associated with the gods of popular mythology until the fourth general meaning seems to be that the gods were identified with the primary natural forces; and this is substantially true. 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 expedient Cf. Aristot. Met. 2.3.1.; they say that these gods are human in shape or are like certain other animals, e.g. the Egyptian deities.
Zoomorphism in Greek religion is a doubtful quantity. 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.
The subject of Mind involves certain difficulties. Mind is held to be of all phenomena the most supernatural; but the question of how we must regard it if it is to be of this nature involves certain difficulties. If Mind thinks nothing, where is its dignity? It is in just the same state as a man who is asleep. If it thinks, but something else determines its thinking, then since that which is its essence is not thinking but potentiality, i.e., if its thinking is determined by something else, Mind is only a potentiality, and not (as described in Aristot. Met. 12.7.1-9 ) the highest actuality. it cannot be the best reality; because it derives its excellence from the act of thinking. Again, whether its essence is thought or thinking, what does it think? It must think either itself or something else; and if something else, then it must think either the same thing always, or different things at different times. Then does it make any difference, or not, whether it thinks that which is good or thinks at random? Surely it would be absurd for it to think about some subjects. Clearly, then, it thinks that which is most divine and estimable, and does not change; for the change would be for the worse, and anything of this kind would immediately imply some sort of motion. Therefore if Mind is not thinking but a potentiality, (a) it is reasonable to suppose that the continuity of its thinking is laborious Cf. Aristot. Met. 9.8.18.; (b) clearly there must be something else which is more excellent than Mind; i.e. the object of thought; for both thought and the act of thinking will belong even to the thinker of the worst thoughts. If Mind is a potentiality, since a potentiality is of contraries, Mind may think that which is worst. Therefore if this is to be avoided (as it is, since it is better not to see some things than to see them), thinking cannot be the supreme good. Therefore Mind thinks itself, if it is that which is best; and its thinking is a thinking of thinking. Yet it seems that knowledge and perception and opinion and understanding are always of something else, and only incidentally of themselves. And further, if to think is not the same as to be thought, in respect of which does goodness belong to thought? for the act of thinking and the object of thought have not the same essence. The answer is that in some cases the knowledge is the object. In the productive sciences, if we disregard the matter, the substance, i.e. the essence, is the object; but in the speculative sciences the formula or the act of thinking is the object. Therefore since thought and the object of thought are not different in the case of things which contain no matter, they will be the same, and the act of thinking will be one with the object of thought. There still remains the question whether the object of thought is composite; for if so, thought would change in passing from one part of the whole to another. The answer is that everything which contains no matter is indivisible. Just as the human mind, or rather the mind of composite beings, i.e., beings composed of matter as well as form.
Such beings are contrasted with the divine Mind, which is pure form. is in a certain space of time The meaning of this sentence is shown by the definition of Happiness in Aristot.
Nic. Eth. 1098a 16-20. It takes the human mind a lifetime of the highest intellectual activity of which it is capable to attain to happiness; but the divine Mind is always happy. Cf. Aristot. Met. 12.7.9. (for it does not possess the good at this or at that moment, but in the course of a certain whole period it attains to the supreme good, which is other than itself), so is absolute self-thought throughout all eternity. We must also consider in which sense the nature of the universe contains the good or the supreme good; whether as something separate and independent, or as the orderly arrangement of its parts. Probably in both senses, as an army does; for the efficiency of an army consists partly in the order and partly in the general; but chiefly in the latter, because he does not depend upon the order, but the order depends upon him. All things, both fishes and birds and plants, are ordered together in some way, but not in the same way; and the system is not such that there is no relation between one thing and another; there is a definite connection. Everything is ordered together to one end; but the arrangement is like that in a household, where the free persons have the least liberty to act at random, and have all or most of their actions preordained for them, whereas the slaves and animals have little common responsibility and act for the most part at random; for the nature of each class is a principle such as we have described. The free persons correspond to the heavenly bodies, whose movements are fixed by necessity; the servile class to human beings. Each class acts in accordance with its nature, a principle which "produces obedience to duty in the higher creatures, caprice in the lower" ( Ross). I mean, for example, that everything must at least come to dissolution; and similarly there are other respects in which everything contributes to the good of the whole. We must not fail to observe how many impossibilities and absurdities are involved by other theories, and what views the more enlightened thinkers hold, and what views entail the fewest difficulties. All thinkers maintain that all things come from contraries; but they are wrong both in saying "all things" Because there is an eternal substance, which is not derived from contraries ( Aristot. Met. 12.6.1 ). and in saying that they come from contraries, Things are derived from a substrate as well ( Aristot. Met. 12.2.1 ). nor do they explain how things in which the contraries really are present come from the contraries; for the contraries cannot act upon each other.
For us, however, this problem is satisfactorily solved by the fact that there is a third factor. Other thinkers make one of the two contraries matter; e.g., this is done by those See on Aristot. Met.
14.1.4. who make the Unequal matter for the Equal, or the Many matter for the One. But this also is disposed of in the same way; for the one matter of two contraries is contrary to nothing. Further, on their view everything except Unity itself will partake of evil; for "the Bad" The "Bad" was identified with the unequal; cf. Aristot. Met.
1.6.10. is itself one of the elements. The other school See Aristot. Met. 12.7.10 does not even regard the Good and the Bad as principles; yet the Good is in the truest sense a principle in all things. The former school is right in holding that the Good is a principle, but they do not explain how it is a principle— whether as an end or as a moving cause or as form. Empedocles theory is also absurd, for he identifies the Good with Love. Cf. Aristot. Met.
1.4.3. This is a principle both as causing motion (since it combines) and as matter (since it is part of the mixture). Empedocles Fr. 17 (Diels), 18-20. Now even if it so happens that the same thing is a principle both as matter and as causing motion, still the essence of the two principles is not the same. In which respect, then, is Love a principle? And it is also absurd that Strife should be imperishable; strife is the very essence of evil. Cf. Aristot. Met. 9.9.3.
Anaxagoras makes the Good a principle as causing motion; for Mind moves things, but moves them for some end, and therefore there must be some other Good Motion presupposes a final cause, which was not what Anaxagoras meant by "Mind." Cf. Aristot. Met.
1.7.5. —unless it is as we say; for on our view the art of medicine is in a sense health. Aristotle identifies the efficient cause, in a sense, with the final cause. Cf. Aristot. Met.
7.9.3. It is absurd also not to provide a contrary for the Good, i.e. for Mind. In Aristot. Met. 1.6.10 Aristotle describes Anaxagoras as a recognizing contrary principles of good and evil. Moreover, on Aristotle's own showing, evil cannot be a principle ( Aristot. Met. 9.9.3 ).
But all those who recognize the contraries fail to make use of the contraries, unless we systematize their theories. And none of them explains why some things are perishable and others imperishable; for they make all existing things come from the same first principles. Cf. Aristot. Met.
3.4.11-20. Again, some Cf. Aristot. Met.
12.2.2, 3. make existing things come from not-being, while others, The Eleatics. Cf. Aristot. Met. 1.5.10-13.
to avoid this necessity, make all things one. Again, no one explains why there must always be generation, and what the cause of generation is. Moreover, those who posit two principles must admit another superior principle, i.e., an efficient cause. and so must the exponents of the Forms; for what made or makes particulars participate in the Forms? And on all other views it follows necessarily that there must be something which is contrary to Wisdom or supreme knowledge, but on ours it does not.
For there is no contrary to that which is primary, since all contraries involve matter, and that which has matter exists potentially; and the ignorance which is contrary to Wisdom would tend towards the contrary of the object of Wisdom; but that which is primary has no contrary. Further, if there is to be nothing else besides sensible things, there will be no first principle, no order, no generation, and no celestial motions, but every principle will be based upon another, If there is nothing but what is sensible or potential, there can be no prime mover (which is actuality) to excite motion in the universe, and no teleology in causation. For the cosmologists on causation see Aristot. Met. 3.3.11-13. as in the accounts of all the cosmologists and physicists. And if the Forms or numbers are to exist, they will be causes of nothing; or if not of nothing, at least not of motion. Further, how can extension, i.e. a continuum, be produced from that which is unextended? Number cannot, either as a moving or as a formal cause, produce a continuum. Moreover, no contrary can be essentially productive and kinetic, for then it would be possible for it not to exist; and further, the act of production would in any case be posterior to the potentiality. Therefore the world of reality is not eternal. But there are real objects which are eternal. Therefore one of these premisses must be rejected. We have described how this may be done. By assuming an eternal actual mover ( Aristot. Met.
12.6.4 ). Further, in virtue of what the numbers, or soul and body, or in general the form and the object, are one, no one attempts to explain; nor is it possible to do so except on our theory, that it is the moving cause that makes them one. Cf. Aristot. Met.
8.6.
As for those Speusippus and his followers; cf. Aristot. Met. 7.2.4, Aristot. Met. 14.3.8. who maintain that mathematical number is the primary reality, and so go on generating one substance after another and finding different principles for each one, they make the substance of the universe incoherent (for one substance in no way affects another by its existence or non-existence) and give us a great many governing principles. But the world must not be governed badly: The rule of many is not good; let one be the ruler. Hom.
Il.2.204.
We have already explained what the substance of sensible things is, dealing in our treatise on physics The reference is presumably to Aristot. Physics 1. with the material substrate, and subsequently with substance as actuality. In Books 7-9.
Now since we are inquiring whether there is or is not some immutable and eternal substance besides sensible substances, and if there is, what it is, we must first examine the statements of other thinkers, so that if they have been mistaken in any respect, we may not be liable to the same mistakes; and if there is any view which is common to them and us, we may not feel any private self-irritation on this score. For we must be content if we state some points better than they have done, and others no worse. There are two views on this subject.
Some say that mathematical objects, i.e. numbers and lines, are substances; and others again that the Ideas are substances. Now since some This was the orthodox Platonist view; cf. Aristot. Met. 1.6.4.
recognize these as two classes— the Ideas and the mathematical numbers—and others Xenocrates and his followers. regard both as having one nature, and yet others The Pythagoreans and Speusippus. hold that only the mathematical substances are substances, we must first consider the mathematical objects, without imputing to them any other characteristic—e.g. by asking whether they are really Ideas or not, or whether they are principles and substances of existing things or not—and merely inquire whether as mathematical objects they exist or not, and if they do, in what sense; then after this we must separately consider the Ideas themselves, simply and in so far as the accepted procedure requires; for most of the arguments have been made familiar already by the criticisms of other thinkers. And further, the greater part of our discussion must bear directly upon this second question—viz. when we are considering whether the substances and first principles of existing things are numbers and Ideas; for after we have dealt with the Ideas there remains this third question. Now if the objects of mathematics exist, they must be either in sensible things, as some hold; or separate from them (there are some also who hold this view); or if they are neither the one nor the other, either they do not exist at all, or they exist in some other way. Thus the point which we shall have to discuss is concerned not with their existence, but with the mode of their existence. That the objects of mathematics cannot be in sensible things, and that moreover the theory that they are is a fabrication, has been observed already in our discussion of difficulties Cf.
Aristot. Met. 3.2.23-30.
—the reasons being (a) that two solids cannot occupy the same space, and (b) that on this same theory all other potentialities and characteristics would exist in sensible things, and none of them would exist separately. This, then, has been already stated; but in addition to this it is clearly impossible on this theory for any body to be divided. For it must be divided in a plane, and the plane in a line, and the line at a point; and therefore if the point is indivisible, so is the line, and so on. For what difference does it make whether entities of this kind are sensible objects, or while not being the objects themselves, are yet present in them? the consequence will be the same, for either they must be divided when the sensible objects are divided, or else not even the sensible objects can be divided. Nor again can entities of this kind exist separately. For if besides sensible solids there are to be other solids which are separate from them and prior to sensible solids, clearly besides sensible planes there must be other separate planes, and so too with points and lines; for the same argument applies. And if these exist, again besides the planes, lines and points of the mathematical solid, there must be others which are separate; for the incomposite is prior to the composite, and if prior to sensible bodies there are other non-sensible bodies, then by the same argument the planes which exist independently must be prior to those which are present in the immovable solids. Therefore there will be planes and lines distinct from those which coexist with the separately-existent solids; for the latter coexist with the mathematical solids, but the former are prior to the mathematical solids. Again, in these planes there will be lines, and by the same argument there must be other lines prior to these; and prior to the points which are in the prior lines there must be other points, although there will be no other points prior to these. Now the accumulation becomes absurd; because whereas we get only one class of solids besides sensible solids, we get three classes of planes besides sensible planes—those which exist separately from sensible planes, those which exist in the mathematical solids, and those which exist separately from those in the mathematical solids—four classes of lines, and five of points; with which of these, then, will the mathematical sciences deal? Not, surely, with the planes, lines and points in the immovable solid; for knowledge is always concerned with that which is prior. And the same argument applies to numbers; for there will be other units besides each class of points, and besides each class of existing things, first the sensible and then the intelligible; so that there will be an infinite number of kinds of mathematical numbers. Again, there are the problems which we enumerated in our discussion of difficulties Aristot. Met.
3.2.23-27.: how can they be solved? For the objects of astronomy will similarly be distinct from sensible things, and so will those of geometry; but how can a heaven and its parts (or anything else which has motion) exist apart from the sensible heaven? And similarly the objects of optics and of harmonics will be distinct, for there will be sound and sight apart from the sensible and particular objects. Hence clearly the other senses and objects of sense will exist separately; for why should one class of objects do so rather than another? And if this is so, animals too will exist separately, inasmuch as the senses will. Again, there are certain general mathematical theorems which are not restricted to these substances. Here, then, we shall have yet another kind of substance intermediate between and distinct from the Ideas and the intermediates, which is neither number nor points nor spatial magnitude nor time. And if this is impossible, clearly it is also impossible that the aforesaid substances should exist separately from sensible objects. In general, consequences result which are contrary both to the truth and to received opinion if we thus posit the objects of mathematics as definite separately-existent entities. For if they exist in this way, they must be prior to sensible spatial magnitudes, whereas in truth they must be posterior to them; for the incomplete spatial magnitude is in point of generation prior, but in point of substantiality posterior, as the inanimate is to the animate. Again, in virtue of what can we possibly regard mathematical magnitudes as one? Things in this world of ours may be reasonably supposed to be one in virtue of soul or part of the soul, or some other influence; apart from this they are a plurality and are disintegrated. But inasmuch as the former are divisible and quantitative, what is the cause of their unity and cohesion? Again, the ways in which the objects of mathematics are generated prove our point; for they are generated first in the dimension of length, then in that of breadth, and finally in that of depth, whereupon the process is complete. Thus if that which is posterior in generation i.e., in the natural order of development. Thus "generation" ( GE/NESIS ) is used in two different senses in this argument, which therefore becomes invalid (Bonitz). is prior in substantiality, body will be prior to plane and line, and in this sense it will also be more truly complete and whole, because it can become animate; whereas how could a line or plane be animate? The supposition is beyond our powers of apprehension. Further, body is a kind of substance, since it already in some sense possesses completeness; but in what sense are lines substances? Neither as being a kind of form or shape, as perhaps the soul is, nor as being matter, like the body; for it does not appear that anything can be composed either of lines or of planes or of points, whereas if they were a kind of material substance it would be apparent that things can be so composed. Let it be granted that they are prior in formula; yet not everything which is prior in formula is also prior in substantiality. Things are prior in substantiality which when separated have a superior power of existence; things are prior in formula from whose formulae the formulae of other things are compounded. And these characteristics are not indissociable. For if attributes, such as "moving" or "white," do not exist apart from their substances, "white" will be prior in formula to "white man," but not in substantiality; for it cannot exist in separation, but always exists conjointly with the concrete whole—by which I mean "white man." Thus it is obvious that neither is the result of abstraction prior, nor the result of adding a determinant posterior—for the expression "white man" is the result of adding a determinant to "white." Thus we have sufficiently shown (a) that the objects of mathematics are not more substantial than corporeal objects; (b) that they are not prior in point of existence to sensible things, but only in formula; and (c) that they cannot in any way exist in separation. And since we have seen sect. 1-3 above. that they cannot exist in sensible things, it is clear that either they do not exist at all, or they exist only in a certain way, and therefore not absolutely; for "exist" has several senses. The general propositions in mathematics are not concerned with objects which exist separately apart from magnitudes and numbers; they are concerned with magnitudes and numbers, but not with them as possessing magnitude or being divisible. It is clearly possible that in the same way propositions and logical proofs may apply to sensible magnitudes; not qua sensible, but qua having certain characteristics. For just as there can be many propositions about things merely qua movable, without any reference to the essential nature of each one or to their attributes, and it does not necessarily follow from this either that there is something movable which exists in separation from sensible things or that there is a distinct movable nature in sensible things; so too there will be propositions and sciences which apply to movable things, not qua movable but qua corporeal only; and again qua planes only and qua lines only, and qua divisible, and qua indivisible but having position, and qua indivisible only. Therefore since it is true to say in a general sense not only that things which are separable but that things which are inseparable exist, e.g., that movable things exist, it is also true to say in a general sense that mathematical objects exist, and in such a form as mathematicians describe them. And just as it is true to say generally of the other sciences that they deal with a particular subject—not with that which is accidental to it (e.g. not with "white" if "the healthy" is white, and the subject of the science is "the healthy"), but with that which is the subject of the particular science; with the healthy if it treats of things qua healthy, and with man if qua man—so this is also true of geometry. If the things of which it treats are accidentally sensible although it does not treat of them qua sensible, it does not follow that the mathematical sciences treat of sensible things—nor, on the other hand, that they treat of other things which exist independently apart from these.
Many attributes are essential properties of things as possessing a particular characteristic; e.g., there are attributes peculiar to an animal qua female or qua male, although there is no such thing as female or male in separation from animals. Hence there are also attributes which are peculiar to things merely qua lines or planes. And in proportion as the things which we are considering are prior in formula and simpler, they admit of greater exactness; for simplicity implies exactness. Hence we find greater exactness where there is no magnitude, and the greatest exactness where there is no motion; or if motion is involved, where it is primary, because this is the simplest kind; and the simplest kind of primary motion is uniform motion. Aristot. Met.
12.7.6. The same principle applies to both harmonics and optics, for neither of these sciences studies objects qua sight or qua sound, but qua lines and numbers Optics studies lines and harmonics numbers because these sciences are subordinate to geometry and arithmetic ( Aristot. An. Post. 75b 15 ).; yet the latter are affections peculiar to