With respect to the first he does two things:

About the first he does two things:

As to the first he does two things:

About the first he does three things:

As to the first he does two things:

With regard to the first [2], we must consider that the continuum is found defined in two ways by the Philosopher. In one way with a formal definition, where it is said in the Predicaments (c.4) that the continuum is "that whose parts are joined at one common term"; for the unity of a continuum is, as it were, its form. In another way, with a material definition taken from the parts, which have the aspect of matter, as is said in Physics II - and it is thus that the continuum is defined here, namely, as "what is divisible into parts always divisible." For no part of a continuum can be indivisible, because no continuum is composed of indivisibles, as is proved in Physics VI. And it is fitting that this latter definition be used here, and the other one in the Predicaments, because the consideration of natural science is concerned with matter, while that of logic is concerned with notions and species.

10.   Then at [3] he defines "body."

Body is divided according to three dimensions. But what is divided according to three dimensions is divided according to all. Therefore, body is divisible according to all the dimensions.

First, therefore, he explains the minor proposition as though by division. For among magnitudes there is one which is divided with respect to one dimension, and this is called "line"; another is divided with respect to two dimensions, and this is called "plane," i.e., a surface; still another is divided according to three dimensions, and since such a magnitude is neither line nor surface, it follows that it is body.

The major proposition he gives at [5]. First he mentions it and says that, besides these magnitudes or dimensions, there is no other magnitude or dimension, on the ground that "three" has the property of being all, because it implies a certain totality, and because whatever is thrice seems to be "in all ways" and "entirely," i.e., according to every mode.

11. Secondly, at [6] he proves what he had said in three ways.

First, according to the teaching of Pythagoras who said that what is called "whole" and "all" is determined by the number 3. For the beginning and the middle and the "consummation," i.e., the end, have a number which befits what is "whole" and "all" - for in things divisible, the first part is not enough to complete the whole, which is completed by the ultimate that is reached by passing from the beginning through the middle. But these three, namely, beginning, middle and end, have 3 as their number. Consequently, it is clear that the number 3 belongs to the "all" and "whole."

12. Secondly, at [7] he proves the same by means of what is observed in divine worship. For we use this number 3 "in the worship of the gods" (whom, namely, the gentiles worshipped), i.e., in sacrifices and praises for them, as though we should receive from nature its laws and rules: just as nature completes all things with the number 3, so those who established the divine worship have, in their desire to attribute to God everything per­fect, attributed to Him the number 3.

13. Thirdly, he proves at [8] the same by appealing to the general way we speak. And he says that we even assign names to things according to the aforementioned method, in which perfection agrees with the number 3. For when there are two things, we say "both," - thus we speak of two men as "both" - but we do not say "all," which we use for the first time in the case of three. And we all in general use this way of speaking, because na­ture so inclines us. For whatever is peculiar to individuals in their way of speaking seems to arise from the particular conceptions of each, but what is generally observed among all would seem to arise from natural inclination.

14. Now, it should be noted that nowhere else does Aristotle either use the arguments of Pythagoras to explain a proposition, or from the properties of numbers conclude anything about things. And perhaps he does so here on account of the affinity of numbers to magnitudes, which he is now considering.

Be that as it may, the proof here given does not seem valid, for it does not seem, if 3 is the number corresponding to "whole" and "all" that it follows there are three dimensions. Otherwise, it would follow according to the same reasoning that there would be only three elements or only three fingers on the hand.

But it should be known that, as Simplicius says in his Commentary 13, Aristotle is not here proceeding demonstratively but according to probability, and this is sufficient after previous demonstrations or ones supposed from another science. Now, it is plain that the task of deciding about the dimensions of bodies as such pertains to mathematics; and whatever the natural philosopher considers with dimensions, he takes from mathematics. Therefore, to prove demostratively that there are just three dimensions pertains to mathematics - thus Ptolemy proves it by showing that it is impossible for more than three perpendicular lines to meet at the same point, while each dimension is measured according to a perpendicular line. Supposing such a demonstration from mathematics, Aristotle here uses testimony and signs, just as he customarily does after his own demonstrations.

15. Then at [9] he goes on to manifest the main proposition from what has been shown. And he says that these three, namely, "all," "whole," and "per­\fect," do not differ from one another according to species, i.e., according to their formal notion, because all imply a certain completeness; but if they do differ in any way, it is in matter and subject, insofar as they are said of diverse things. For we use "all" in discrete things, as we say "all men"; we use it also with respect to continua which are easily divided, as we say "all water" and "all air." "Whole," however, is used both with these and with all continua, as we say "the whole people" and"the whole world." But "perfect" is used with respect to these and forms: for we say "perfect whiteness" and "perfect virtue." Therefore, because "all" and "perfect" are the same, the consequence is that among magnitudes the perfect one is body, because only a body is determined by three dimensions, and this carries with it the notion of "all," as has been shown above, for since it is divisible in three ways, it follows that it is divisible in every way, i.e., according to every dimension. But among other magnitudes, there is one divisible accord­ing to two dimensions, namely, a surface; and another according to one, namely, a line. "Now according to the number that it has," i.e., the number of dimensions that a magnitude has, so is it divisible and continuous. Thus one magnitude is continuous in one way, namely, a line; another in two ways, namely, a surface; but a body is continuous in every way. Hence it is plain that body is a perfect magnitude, as possessing all ways of being continuous.

16. Then at [10] he shows what is or is not plain from the foregoing. And he mentions three things. The first of these is plain in itself, namely, that any magnitude that is divisible is continuous; for if it were not con­tinuous, it would not be a magnitude but a number. The second is the converse of this, namely, that every continuum is divisible, as was indicated in the definition. And this is plain from what was proved in Physics VI, as was said above. But it is not plain from what was just said, however, because here he supposes, but does not prove, that a continuum is divisible. The third thing is plain from the foregoing, namely, that unlike the passing from length to surface and from surface to body, there is no passing from body to another kind of magnitude. And he uses a way of speaking employed by geometers imagining that a point in motion makes a line, and a line in motion a surface, and a surface a body. But from body there is no transition to another magnitude, because such a passing, i.e., to another kind of magnitude is due to a defect in that from which the process beings - that is why natural motion is the act of an imperfect thing . But it is not possible that body, which is perfect magnitude, should be defective in this way, because it is continuous in every way. Consequently, no transition from body to another kind of magnitude is possible.

17. Then at [113 he manifests the proper perfection of the universe based on its difference from particular bodies. First he mentions how particular bodies are related to perfection. And he says that each particular body, according to the common notion of body, is such, i.e., perfect, inasmuch as it has three dimensions; nevertheless, it is terminated at an adjacent body, inasmuch as it touches it. And thus every such body is in a certain way "many," i.e., perfect, in having three dimensions, but imperfect in having another body outside it at which it is terminated. Or it is "many" according to contact with diverse bodies; or it is "many" because there are more than one in one species due to imperfection, whereas such is not the case with the universe.

18. Secondly at [12j he shows how the universe is related to perfection. And he says that "the whole," i.e., the universe, which has particular bodies as its parts, must be perfect in all ways, for the word "universe" signifies perfect "in all ways," and not in one way to the exclusion of some other way, and it both has all the dimensions, and includes in itself all bodies.

With respect to the first [13] it should be considered that, as is said in Physics III, the ancients described the infinite as "that outside of which there is nothing." Now, since he has proved that the universe is perfect on the ground that nothing is outside it, but that it embraces all things, one might think it to be infinite. Accordingly, meeting this opinion, he concludes by adding that later on, in discussing the nature of the whole universe, there will be treated the question of whether it is infinite in magnitude, or finite with respect to its total mass. But meanwhile, before treating of this, something must be said about those parts of it that are "according to the species," namely, those parts in which the integrity of its species consists, and which are the simple bodies. For animals and plants and other such are its secondary parts, and pertain more to the well-being of the universe than to its basic integrity. And we shall begin this consideration from a principle given below.

20. Then at [14] he starts to manifest the proposition stating of which principal parts the perfect species of the universe is made.

About the first he does two things:

With respect to the first he does two things:

About the first he does two things:

About the first he does two things:

21. He says therefore first [14] that all physical, i.e., natural, bodies are said to be mobile with respect to place according to themselves, i.e., according to their very natures, and the same is true for other natural magnitudes, e.g. planes and lines, insofar as they are the boundaries of natural bodies. And this is true in the sense that bodies are moved per se, but the other magnitudes per accidens, when the bodies are moved. In proof of this he adduces the definition of nature, which is "the principle of motion in those things in which it exists," as is said in Physics II. From this he argues thus: Natural bodies are ones that have a nature, but nature is a principle of motion in things in which it is; therefore, natural bodies have a principle of motion in them. But whatever is moved with any sort of motion is moved locally, but not conversely, as is plain in Physics VII, because local motion is the first of motions. Therefore all natural bodies are naturally moved with a local motion, but not all of them with all of the other motions.

This, however, seems to be false: for the heavens are a natural body, but their motion seems to be due, not to nature but to intellect, as is plain from what has been determined in Physics VIII and Metaphysics XII.

But it must be said that there are two kinds of principles of motion: one is active, i.e., the mover, as the soul is the active principle of the motion of animals; the other is a passive principle of motion, i.e., a principle according to which a body has an aptitude to be thus moved, and such a prin­ciple of motion exists in the heavy and the light. For these are not composed of a mover and a moved, because, as the Philosopher says in Physics VIII, "it is plain that none of these - i.e., the heavy and the light - moves itself, but each has, with respect to its motion, a principle not of causing motion or of acting, but of being acted upon." Consequently, it must be said that the active principle of the motion of heavenly bodies is an intel­lectual .stance, but the passive principle is that body's nature according to which it is apt to be moved with such a motion. And the same situation would prevail in us, if the soul did not move our body in any way other than according to its natural inclination, namely, down.

23. Then at [15] he distinguishes local motions.

With respect to the first he does two things:

First at [15] he proposes what he intends, namely, that every local motion - which is called latio - is either circular, or straight, or composed of these, as is the oblique motion of things that are borne this way and that.

Secondly, at [16] he proves what he had said, on the ground that there are just two simple motions, the straight and the circular. And the reason for this, he says, is that there exist just two simple magnitudes, namely, the straight and the circular: but local motion is specified according to places, just as every other motion is specified according to its termini.

24. But it seems that Aristotle's proof is not suitable, because, as is said in Post. Anal. I, one does not demonstrate who crosses into another genus. Consequently, it seems unfitting to use the division of magnitudes, which pertain to mathematics, in order to reach a conclusion about motion, which pertains to natural science.

But it must be said that a science which is by addition to some other science uses the latter's principles in demonstrating, as geometry uses the principles of arithmetic - for magnitude adds position to number, and thus a point is said to be "a positioned unit." In like manner, natural body adds sensible matter to mathematical magnitude. Consequently, it is not unfitting for the natural philosopher in his demonstrations to use the principles of mathematics - for the latter is not of a completely different genus but is in a certain way contained under the former.

Likewise, it seems to be false that only two magnitudes are simple, namely, the straight and the circular. For a helix [spiral] seems to be one simple line, because every one of its parts is uniform, and yet a helical line [such as a screw thread] is neither straight nor circular.

But it must be said that a helix, if one considers its origin, is not a simple line, but a combination of straight and circular. For a helix is produced by two imaginary motions, one of which is the motion of a line moving round a cylinder, and the other of a point moving through the line: if two such motions take place in a regular manner at the same time, a helix will be formed by the motion of the point in the moving line.

Likewise, it seems that circular motion is not simple. For the parts of a sphere that is in circular motion are not in uniform motion but the parts near the poles or near the center are moved more slowly, because they traverse a smaller circle in a given time; consequently, the motion of a sphere seems to be composed of fast and slow motions.

But it must be said that a continuum does not have parts in act but only in potency. Now, what is not in act is not in actual motion. Hence the parts of a sphere, since they are a continuous body, are not actually being moved. Hence it does not follow that, in a spherical or circular motion, there is actual diversity, but this is only potentially. This does not conflict with the simplicity about which we are now speaking, for every magnitude possesses potential plurality.

27. Then at [17] he distinguishes simple motions.

He says therefore first [17] that circulation, i.e., circular motion is around the middle. And this is to be understood as around the middle of the world: for a wheel which is in motion around its own middle is not in circular motion in the proper sense of the word, but its motion is composed of ups and downs.

But it seems according to this that not all heavenly bodies are in circular motion: for according to Ptolemy, the motion of the planets is in ec­centrics and epicycles, which are motions, not around the middle of the world, which is the earth's center, but around certain other centers.

But it must be said that Aristotle was not of this opinion, but thought that all motions of the heavenly bodies are about the center of the earth, as did all the astronomers of his time. But later, Hipparchus and Ptolemy hit upon eccentric and epicyclic motions to save what appears to the senses in heavenly bodies. Hence this is not a demonstration, but a certain as­sumption. Yet if it be true, all the heavenly bodies are still in motion about the center of the world with respect to the diurnal motion, which is the motion of the supreme sphere that revolves the entire heaven.

29. Then at [18] he distinguishes straight motion into two: namely, one which is up, and one that is down, and describes each in relation to the middle of the world, as he had described circular motion, in order to keep the description uniform. And he says that an upward motion is one from the middle of the world, but a downward motion is one to the middle of the world. The first of these is the motion of light things, the second of heavy things.

30. Then at [19] he concludes to the number of simple motions. First he expresses the conclusion he intended, and says that as to simple latio, i.e., simple local motion, one must be from the middle, and this is the upward motion of light bodies; another must be to the middle, and this is the downward motion of heavy bodies; still another must be about the middle, and such is the circular motion of heavenly bodies.

31. Secondly, at [20] he shows that this conclusion agrees with what has been said above. And he says that what has just been said about the number of simple motions seems to be a consequence of what was said above about the perfection of body, for just as the perfection of body consists in three di­mensions, so the simple motions of body are distinguished into three kinds. But he says that this is "according to reason," i.e., according to a certain probability: for three motions are not properly equated to three dimensions.

32. Then at [21] he gives some reflections about mobile bodies. In regard to this it must be known that, as was stated in Physics III, motion is an act of a mobile. Now an act is proportionate to the thing to be perfected. Hence motions ought to be proportionate to mobile bodies. But some bodies are simple, some composite. A simple body is one that has a principle of some natural motion in it, as is plain in the case of fire, which is light simply, and in that of earth, which is heavy simply, and in their species - as a flame is said to be a species of fire, and bitumen a species of earth. He adds the phrase, "and those related to them," on account of the intermediate elements, of which air has a greater affinity to fire, and water to earth. As a consequence, a mixed body must be one that, according to its proper nature, does not have in itself the principle of some simple motion.

And from this he concludes that some motions must be simple and some mixed: whether the mixed motion is not one but has diverse parts, as one composed of elevation and depression, or of a push and a pull, or whether the mixed motion is one, as is plain in oblique motion and motion upon a helical line. Accordingly, the motions of simple bodies must be simple and those of mixed bodies mixed, as seen in the motion of rain, or any body of this kind in which neither heaviness nor lightness totally predominates. And if it sometimes happens that a mixed body is moved with a simple motion, that will be due to the element predominant in it, as iron is moved downwards accord­ing to the motion of earth which is predominant in its composition.

He says therefore first [28] that, since some of the foregoing statements were supposed (namely, that one thing has one contrary, and that there are but two simple magnitudes, the straight line and the circle, and any other such suppositions) and others were demonstrated from certain premises (for example, that there are three simple motions, and that circular motion is natural to some body which is different in nature from the bodies that exist here), it can be plain from the foregoing that that entire body which is being moved circularly has neither heaviness nor lightness, which are principles of certain local motions.

52. Then at [29] he manifests his proposition. And because the principle of demonstration is "that which something is," as is said in Post. Anal. II,

He says therefore first [29] that in order to prove the proposition we ought to suppose what it is that we call "heavy" and what "light." And he says "suppose" because he is not perfectly investigating their definitions here, but he uses them as suppositions to the extent that the present demonstration requires. But they will be considered more carefully in Book IV, where their "substance," or nature, will be explained. Accordingly, he defines heavy as "That which is apt to be moved to the middle," and the light as "that which is apt to be moved from the middle."

53. He uses this mode of defining in order to keep himself from the contrary position of Plato, who said that in the world according to itself there is no "up" and "down," on account of the rotundity of the world: for a rotund body is everywhere uniform. He said that there is "up" and "down" in the world only with respect to us, who call "up" that which is above our head, and "down" that which is below our feet, so that if we were contrarily situated, we would call "up" and "down" in a contrary manner. Consequently, Plato does not admit an "up" and "down" in the very nature of things but only with respect to us.

Aristotle, however, uses these names according to the common way of speaking, in keeping with his statement in Topics II, that names are to be used as they are used for the most part; hence he calls "up" and "down" in the world what are generally called "up" and "down" by men. Yet they are distinct not only with respect to us, but also according to nature. For just as we distinguish "right" and "left" in ourselves according to the diverse relationship to animal motion which is with respect to place, so too "up" and "down" in the world are distinguished with relation to the motions of the simple bodies which are the principal parts of the world. On this account he says that "up" is the place where light things are carried, and "down" the place where heavy things are carried. And this is reasonable: for just as in us the nobler part is that which is above, so in the world, light bodies are more noble, as if more formal. But here in order to proceed without calumny to the proof of the proposition, he defines "heavy" and "light" by their relation to the middle.

54. Then at [30] he defines "heaviest" and "lightest." And he says that the heaviest is "that which stands under all things that are carried downward," while the lightest is "that which is at the top of all things that are carri­ed up." And this must be understood as concerning those things that are carried upward and downward - for the heaven is not the lightest, even thcugh it is above all, because it is not carried upward. Now it should be recognized that here he is already using "up" and "down" as though "up" and "down" arise as being where a motion from the middle, or to the middle, is terminated.

55. Then at [31] he proves his proposition from the foregoing, and says that every body carried up or down must have heaviness absolutely, as does the heaviest, namely, earth, which stands under all, or must have lightness absolutely, as does fire, which is above all, or must have both, not in respect to the same, but in respect to diverse things. For the intermediate elements, namely, air and water, are mutually heavy and light, as air is light with respect to water, because it is carried above it, and the same is true of water with respect to earth; meanwhile, air with respect to fire is heavy, because it exists under it, and similarly water with respect to air. But the body that is moved circularly can have neither heaviness nor lightness. For it cannot be moved to the middle or from the middle, either according to nature, or outside nature.

And, that it cannot be so moved according to nature, is clear from the fact that a straight motion, which is to the middle, or from the middle, is nat­ural to the four elements. But it was said above that one motion is natural to one of the simple bodies. Therefore it would follow that the body which is moved circularly would be of the same natureas one of the bodies that is moved in a straight line, the contrary of which was proved above. Similarly it cannot be said that a straight motion outside nature belongs to the body that is moved circularly. For if one of a pair of contrary motions is present in a body outside its nature, the other will be for it according to nature, as is plain from what has been said above. Therefore, if downward motion is outside nature for the fifth body, upward motion will be for it according to nature, and conversely. But both are false, as is plain from the preceding argument. It follows therefore that the fifth body, which is carried circularly, is not carried from the middle or to the middle, either according to nature or outside its nature. But every body having lightness or heaviness is moved according to nature by one of these motions, and outside its nature by the other. Therefore, the fifth body has neither heaviness nor lightness.

56. Then at [32] he excludes a certain objection. For some said that the parts of the elements are perishable, so that when existing outside their proper place they are naturally moved with a straight motion, while the elements themselves according to their totality are imperishable and cannot ever be outside their proper place - whence they are being moved circularly in their places. Consequently a body that is being moved circularly in its place according to its totality need not lack heaviness and lightness.

To exclude this the Philosopher proposes that part and whole are naturally carried to the same place, as, for example, in the case of the whole earth and one clod. And this is clear from the state of rest: for each thing is naturally moved to the place in which it is naturally at rest, and it is in the same place that the whole earth and part of it naturally rest. Hence it is clear that the whole earth has a natural inclination to be moved to the center, should it be outside its own place.

Therefore from the foregoing two things follow: The first of these is that the whole fifth body has no lightness or heaviness - for, as is clear from the aforesaid reason, it would be moved naturally to or from the middle. Secondly, it follows from the supposition now introduced that, if any part were detached from a heavenly body it would be moved neither up nor down, for, since the whole and part are of the same nature, it does not befit either the entire fifth body, or any part of it, to be moved either according to its nature or outside it with any motion other than the circular.

With respect to the first he does two things:

First he proposes what he intends [33] and says that just as it has been pointed out above that the fifth body lacks heaviness and lightness, in like manner it is reasonable to believe that it is unproduced and imperishable, and incapable of increase and alteration, i.e., that it is not subject to generation and ceasing-to-be, or to growth or alteration.

Secondly [34], he proves the proposition:

59. With regard to the first he presents the following argument: Whatever can be generated comes to be from a contrary and a certain subject or matter - for something comes to be from a contrary as from something non-permanent, but from a subject as from something permanent, as is plain in Physics I. Likewise, every body that is perishable ceases to be while some subject [continues to] exist Also every case of ceasing-to-be is from a contrary active principle, for every ceasing-to-be is terminated at a contrary, as was said in the first discussions, i.e., in Physics I. But nothing is contrary to the fifth body. Therefore, it can be neither generated nor destroyed. He proves the middle [minor] proposition through the fact that the motions of contraries are contrary, as the light is moved upward and the heavy downward; but the fifth body's natural motion, which is circular motion, has no contrary motion, as will be proved later. Therefore nothing is contrary to this body. Thus nature seems to have acted rightly, exempting this body from contrariety as destined to be, i.e., having to be, unproduced and imperishable.

60. But two thoughts come to mind regarding what Aristotle says here: one is about his assumption that the body of the heaven is incapable of being generated and destroyed; the other is about the reason for it.

Now it should be known, with regard to the first, that some supposed the body of the heaven to be generable and perishable according to its very nature, as did John the Grammarian, called Philoponus. And in support of his contention he uses first the authority of Plato who supposed that the heavens and the entire world were generated. Secondly, he presents this argument: Every power of a finite body is finite, as was proved in Physics VIII; but a finite power cannot extend itself unto infinite duration (that is why something cannot be moved for an infinite time through a finite power, as was proved in the same book); therefore, a heavenly body does not have the power to be infinite in time. Thirdly, he forms the following objection: In every natural body there is matter and privation, as is plain from Physics I; but wherever there is matter with privation, there is potency to cease to be; therefore, the heavenly body is perishable. And if anyone says that the matter of heavenly bodies is not the same as that of inferior bodies, he objects to the contrary - for, according to this, matter would have to be composite, made out of what is common to both matters, and out of what produces diversity between matters.

61. But these statements lack necessity. For the fact that Plato posited the heavens as generated was not drawn from an understanding that they were subject to generation, which Aristotle intends here to deny, but because it was necessary for them to have their existence from a higher cause, as composed of parts multiple and extended - which meant that their existence was caused by some one first thing, from which all multiplicity must be caused.

62. The objection that the power of a heavenly body is finite Averroes solved by saying that in a heavenly body there is a power for local motion, but no power, either finite or infinite, respecting existence.

But in this he is clearly going against Aristotle who later on in the same book supposes in sempiternal things a power to exist forever. But Averroes was deceived by supposing that the power respecting existence pertains solely to the passive power, which is the potency of matter; but the truth is that it pertains more to the power of the form, because everything exists through its form. Hence a thing has as much and as long an existence as the power of its form. Thus there is a power to exist forever, not only in heavenly bodies, but also in separated substances.

Therefore it should be said that whatever requires infinite power must be infinite. But the infinite, according to the Philosopher in Physics I, pertains to quantity, so that what lacks quantity is neither finite nor infinite. Now motion does have a quantity that is measured by time and magnitude, as is plain in Physics VI, and therefore the power which is capable of eternal motion is capable of an infinite effect - and consequently such a power must be infinite. But a thing's existence considered in itself is not a quantity, for it has no parts, but is entire and all at once. Rather it is accidental to it that it is quantified in one sense according to duration, insofar as it is subject to motion, and consequently to time, just as is the existence of changeable things. That is why the power of any bodily thing whose existence is subject to change cannot go beyond a finite duration. In another way the existence of a thing can be called quantified per accidens on the part of the subject, which has a definite quantity. Therefore it must be said that the existence of the heavens is not subject either to variation or time; hence it is not quantified by a quantity of duration, and consequently is neither finite nor infinite in this respect. But it is quantified according to the quantity of an extended body, and in this respect it is finite. Consequently. it must be said that the power of existing of a heavenly body is finite, but that does not mean that it is limited to existing in a finite time, because temporal finiteness or infinity are accidental to a thing's existence, which is not subject to the variation of time. Nevertheless a power of this kind could not cause existence in an infinite magnitude nor even in a magnitude greater than the magnitude of the heavenly body.

Similarly Averroes solves the third objection by destroying it. For he denies that a heavenly body has matter, but says that a heavenly body is a subject that is actual being, to which its soul is compared as form to matter. Now if in stating that a heavenly body does not have matter he should mean matter in relation to motion or change, then it is true - for thus does Aristotle also say in Physics VIII and Metaphysics XII, namely, that a heav­enly body has matter not with respect to existence but to "where," for the simple reason that this matter is not subject to a change according to being but to one according to "where." But if he means that a heavenly body has matter in no way at all or no subject at all, then plainly he is wrong. For it is clear that that body is a being in act; otherwise it would not act on the lower bodies. But whatever is a being in act is either act itself, or has act. Now it cannot be said that a heavenly body is act, for then it would be a subsistent form, and something understood in act but not apprehended by sense. Therefore in a heavenly body there must be something which is the subject of its actuality.

However this subject or matter does not need to have privation, for privation is nothing but the absence of a form which is apt to exist in the matter; but in this matter or subject there is no other form apt to be - rather its form fills out the entire potentiality of the matter, since it is a certain total and universal perfection. And this is clear from the fact that its active power is universal, and not particular like the power of the lower bodies, whose forms as being particular cannot exhaust the entire potentiality of the matter; hence, together with one form there remains in matter the privation of another form which is apt to be in it. Similarly, we see that the lower bodies are subject to diverse shapes, but the heavenly body not. Accordingly, in a heavenly body there is not privation of any form but only privation of some "where." Consequently, it is not changeable with respect to form through generation and ceasing to be, but only with respect to "where." From this it is plain that the matter of the heavenly body is distinct and of a different nature from the matterof lower bodies, not on account of some composition, as Philoponus supposed, but on account of their relationship to diverse forms, of which one is total and the other partial - for thus potencies are diversified, namely, according to the diversity of acts to which they are in potency.

64. Therefore it is manifest from the foregoing that the body of the heavens according to its nature is not subject to generation and ceasing-to-be, as being first in the genus of mobiles, and the closest to immobile things. That is why it has a minimum of motion. For it is moved only with local motion, which varies nothing intrinsic to a thing. And among local motions it has a circular motion, which also has a minimum of variation, because in spherical motion the whole does not vary its "where" as to subject, but only in conception, as was proved in Physics VI; but the parts change their "where" even as to subject.

However, we do not say according to the Catholic faith that the heavens always existed, although we say that they will endure forever. Nor is this against Aristotle's demonstration here, for we do not say that they began to be through generation, but through an efflux from the first principle, by whom is perfect the entire existence of all things, as even the philosophers posited. From whom, however, we differ in this, that they suppose God to have produced the heavens co-eternal to Himself, but we posit that the heavens were produced by God according to their whole substance at some definite beginning of time.

65. Against this, however, Simplicius, a commentator on Aristotle, at this passage, objects on three counts. First, since God produced the heavens, therefore, through His essence and not through something added, since His essence is eternal and unchanging, the heavens have always proceeded from Him. Again, if the goodness of God is the cause of things, the goodness of God would have been idle and disengaged before the world existed, if the latter began to exist from some definite beginning of time. Again, whatever begins to exist in some determined part of time after previously not existing, this happens to it from the ordination of some higher motion from which it happens that this being begins now and not before, as a man begins to be now and not previously, according to the order of the revolution of the heavenly body. But there is no higher revolution or motion beyond the heavenly body. Therefore it cannot be said that the body of the heavens began to be now, so as not to have been before.

But these lack necessity. For the first statement that God acts through His essence and not through something superadded is true, but His essence is not distinct from His understanding, as in us, nor from His willing. Hence He produces according to His understanding and His willing. Now in things produced by an agent acting in virtue of his understanding and will, that which is produced must be as it was understood by the producer, and not as the producer is in his being. Hence, just as what is produced by God acting through His essence does not have to be, in other respects, in the same way as the divine essence, but such as it is determined by His understanding, so too it is not necessary that what is produced by God be as long-lasting as God, but only to the degree determined by His understanding.

And this applies also to the dimensive quantity of the heavens. For the fact that the heavens have such-and-such a quantity, and no greater, is a result of a determination of the divine intellect determining such a quantity for them, and adapting to them a nature proportionate to such quantity, just as He frees them from contraries so that they may be ungenerated and incorrupt­ible, as stated in the text. The phrase in the text that "nature acted rightly" implies the action of an intellect acting for an end, for it is no nature other than the divine that has freed them from contraries.

Similarly, the statement that the divine goodness would have been idle and disengaged before the production of the world does not have any weight. For a thing is called "idle" that does not attain the end for which it is. But the goodness of God is not for the sake of creatures. Hence creatures would be idle if they did not attain to the divine goodness, but the divine goodness would not be idle even if It never produced a creature.

Again, the third objection applies to a particular agent, which supposes time and works in time. In this way what comes to be must be proportioned by the agent both to some part of time, and to the whole of time, or even to the cause of the whole of time. But we are dealing now with a universal agent who produces the whole time together with the things in time. So there is no place here for the question of why now and not before, as though there were presupposed some other preceding part of time, or some more general cause producing all of time. But the pertinent question here is why the uni­versal agent, namely, God, willed time and the things in time not always to exist. And this depends on a determination of His intellect, just as in a house the artisan determines the size of one part of the house in relation to another part or to the whole house, but the size of the entire house he himself determines according to his understanding and will.

67. Another point remains to be considered about Aristotle's demonstration against which John the Grammarian objects: if nothing but what has a contrary can be generated and cease to be, then since there is no contrary of a substance, as is plain in animals and plants (similarly, nothing is contrary to a figure or a relation), none of these will be generated and cease to be.

To this Simplicius responds that this is to be understood about a contrary in the general sense as including even contrariety of privation and species, for that is Aristotle's meaning when he speaks of contraries in Physics I. And that is the way in which contrariety is found in all the foregoing, as the unformed is contrary to the formed, and the unfigured to the figured; but privation has no place in heavenly bodies, as has been said.

But this response, although true, is not ad rem, For Aristotle says that contrariety of local motions corresponds to contrariety of bodies; and it is certain that no local motion corresponds to a privation. Consequently, it must be said that, as he himself will say later, nothing is contrary to substance with respect to its being a composite, or according to matter or substantial form; but there is something contrary to it according to its proper disposition to such a form, as fire is said to be contrary to water by reason of the contrariety of hot and cold. And such contrariety is required in all things that are generated and cease to be. But it is upon such contrariety that contrariety of motions according to heavy and light follow: through the absence of which, a heavenly body is understood to be free of all the contraries that accompany the heavy and the light.

68. Likewise, since he says that contrariety of motions corresponds to contrariety of bodies, it seems that fire is contrary more to earth than to water, because fire agrees with the former in respect of one quality, namely, dryness.

And it must be said that in this book the Philosopher is discussing simple bodies with respect to their position; for it is under this aspect that they are parts making up the universe. And according to this, the contrariety of fire to earth is greater than its contrariety to water. Yet it is true that fire has a greater contrariety to water from the viewpoint of active and passive qualities, which consideration belongs to the book On Generation.

69. Again, it does not seem to follow of necessity that nothing is contrary to a heavenly body just because nothing is contrary to the circular motion with which it is moved, because fire also in its own sphere, and the upper region of air, are moved circularly, as is said in Meteorology I, and yet there is a contrary to fire and air.

But it should be said that fire and air are not moved circularly as though by their own motion; rather they are carried along by the motion of the heavens. The heavenly bodies, however, are moved circularly by their own motion; consequently, the case is not the same.

70. Again, it seems that contrariety of motions does not attest to contrariety of mobiles. For the same numerical substance, which is not contrary to itself, is subject to contraries, as is said in the Predicaments; thus it is moved by contrary motions which are terminated at contraries: for example, a substance is moved by whitening and blackening and similar motions. Moreover, air existing in the place of water is moved upward, but in the place of fire downwards. Therefore the same thing is moved by contrary motions, and, consequently, contrariety of motions does not follow upon contrariety of mobiles. Furthermore, we see that the same soul is moved by the motions of vice and virtue, which are contrary motions.

With regard to this it must be considered that the Philosopher uses this proposition, namely, that if motions are not contrary, the mobiles also are not contrary. But he does not state the converse, that if the mobiles are not contrary the motions are not contrary (because someone could say that the motions of all bodies having contrariety are contrary, but not that all contrary motions involve contrary things): against which the foregoing objection is directed. Yet in truth contrariety of natural motions follows upon what is proper to the active or formal principles (which the motion follows upon), and not upon the contrariety of the passive or material principles, because the same matter is subject to contraries. And therefore nothing prevents the same subject from being affected by alterations caused by extrinsic principles, even though such alterations be contrary. But if an alteration arises from an intrinsic principle, as when health is restored by the nature, then the contrariety of such alterations follows upon the contrariety of the mobiles. And the same holds for local motions, which we are now discussing: for such motions follow upon intrinsic formal principles

Now, regarding the objection about air, it must be said that the contradiction which is included in all opposites requires in its very notion that it be with respect to the same thing and according to the same aspect. But the natural motion of air is not up and down with respect to the same thing; rather it is upward with respect to water and earth, and downward with respect to fire. Consequently such motions are not contrary, for they are not tending to contrary places but to the same place, i.e., the place which is above water and below fire.

What is said about the motion of the soul according to virtue and vice is not ad rem - for such motions are not natural but voluntary.

He says therefore first [40] that from what was said in proving that there exists a fifth body in addition to heavy and light bodies, it can be shown that it is impossible for a greater number of simple bodies to exist. For as was said above, for each simple body there must be some simple motion.

But there is no simple motion other than the ones previously mentioned: one of which is circular and the other straight, the latter being divided into two kinds, one of which is from the middle and is called "upward motion" and the other toward the middle and is called "downward motion." Of the latter two, the one which is toward the middle belongs to a heavy body, namely, to earth and water, while the one from the middle belongs to a light body, namely, to fire and air. Finally, the circular motion is assigned to the first and supreme body. Hence what remains is that there is no other simple body besides the ones mentioned. Consequently, the wholeness of the universe consists of these five bodies.

79. Then at [41] he proves something he had assumed, namely, that there is not a motion contrary to circular motion. This he had assumed in the discussion in which he proved that the body of the heavens is not subject to generation and corruption. But the reason why he did not prove it right away, but waited until now, is that it is also useful in proving that there is not a greater number of simple bodies. For if there were a motion contrary to circular motion, it could be held that just as there are two bodies moved with straight motion on account of the contrariety of this motion, so there are also two bodies moved with circular motion. But this will not occur if it is plain that there is no motion contrary to circular motion. Therefore, on this point,

First he proposes what he intends, and says that there are many reasons to induce one to believe that there is not a circular motion contrary to circular motion.

80. Secondly, he establishes the proposition. In regard to this it must be noted that if there exists contrariety in circular motion, it must be in one of three ways: one is that a straight motion be contrary to circular motion; the second is that there be some sort of contrariety in the parts themselves of circular motion; the third is that one circular motion have some other circular motion contrary to it.

First therefore he shows that a straight motion is not contrary to circular motion;

81. He says therefore first [42] that what seems most opposite to something circular is something straight. For a straight line has no break, while an angular line does have a break, not through the whole, but in the angles; meanwhile a circular figure seems to have breaks throughout, as if the whole were an angle. According to this the straight and the circular seem to be contraries, as though at the farthest extremes.

And because someone could say that it is not the straight that is opposed to the circular, but rather the convex or "gibbous" which is opposed to the concave, to reject this objection, he adds that concave and "gibbous," i.e., convex, are seen to be opposed not only to one another, but to the straight as well. They seem to be mutually opposed after the manner of the combined and the juxtaposed, i.e., in terms of relation: for "concave" is said in relation to things that are inside [a circle or sphere], but "gibbous" with respect to things outside. Consequently, from every aspect, the straight is contrary to the circular, whether taken as concave or as convex.

And because the contrariety of motions is seen to follow the contrariety of the things in which the motion is, the consequence seems to be that if there is a motion contrary to circular motion, it should be most of all straight motion which, namely, is over a straight line. But straight motions are con­trary to one another because of contrary places - for upward motion is contrary to downward because "up" and "down" imply a difference and contrariety of place. Consequently, one straight motion will have as its contrary some other straight motion, and a circular one. This, however, is impossible, for to one thing there is one contrary. Therefore, it is impossible for any motion to be contrary to circular.

82. But someone could object to the statement that the straight is most contrary to the circular. For it is stated in the Predicaments that nothing is contrary to figure, whereas "straight" and "circular" are differences in figure.

But it can be said that the Philosopher is here speaking hypothetically and not categorically. For if anything were contrary to the circular, it would be the straight most of all, for the reason given above.

It can also be said that in every genus there is found a contrariety of differences, as is plain from Metaphysics X, although there is not a contrariety of species in every genus: for although "rational" and"irrational" are contrary differences, "man" and "ass" are not contrary species. Consequently, there is a contrariety between straight and circular not as between species, but as between differences of the same genus. Such contrariety, which can be discerned in motions on the basis of the difference between straight and circular, is not a corruptive contrariety, of the sort, namely, which the Philosopher here intends to exclude from the heavenly body, such as is the contrariety of hot to cold. But nothing forbids contrariety according to the differences of certain genera from being in a heavenly body, for example, that of equal and unequal, or something of that kind.

John the Grammarian, however, objects against the Philosopher's seeming to state that concave and gibbous are opposed according to a relation: because relative things seem to be co-existent, but concave and gibbous are not necessarily together, for a spherical body can be exteriorly convex without beinginteriorly concave. But in this he has been deceived, for the Philosopher is here speaking of concave and convex as found in a circular line, and not as found in a spherical body, in which latter one can indeed exist without the other, but not in a line.

83. Then at [43] he shows that there is no contrariety in the parts of circular motion.

Regarding the first he does three things:

He says therefore first [43] that someone could think that the aspect of contrariety in motion upon a circular line, and that in motion upon a straight line, are the same. For if one straight line between two points, A and B, be designated, it is evident that the local motion occurring on the straight line from A to B will be contrary to the local motion from B to A. But the notion is not the same if a circular line be described through the two points, A and B, because between two points there can be but one straight line, but an infinity of curved lines, which are diverse portions of circles. Therefore it would follow that, if the motion from A to B over a circular line were contrary to the one which is from B to A over a circular line, an infinitude of motions would be contrary to one.

But it should be observed that, in place of what he ought to have said, namely, that the straight line between two points is one, he said that straight lines are "finite" - because if we take two points in diverse places, there will be between them finite straight lines [i.e., in finite number], but between any two points there could be described an infinitude of curved lines.

84. Against this argument John the Grammarian objects, since it does not seem to follow that to one motion there is an infinitude of contrary motions, but that to an infinite number there is. For with respect to each portion of the circle described between two points there will be two motions contrary one to the other. Likewise, the same difficulty seems to follow from the contrariety of straight motions. For it is manifest that just as an infinitude of curved lines can be described between two points, so from the center of the world to the circumference there can be described an infinitude of straight lines.

But in regard to the first it must be said that if the contrariety of motions that occur through curved lines is to be according to the contrariety of the termini as happens in straight motions, then, from this supposition it follows that every motion from B to A through any of the curved lines is con­trary to a motion from A to B. Thus it will follow not only that there is an infinitude of motions contrary to one motion, but also that to each of the infinite motions starting from one end there will be contrary the infinitude of motions beginning from the contrary end.

In regard to the second it must be said that all the infinitude of straight lines from the center to the circumference are equal, and therefore designate the same distance between contrary termini - therefore in all of them is present the same aspect of contrariety, which implies maximum distance. But all the infinitude of curved lines described between the same points are un­equal; hence the same aspect of contrariety is not present in them, for the distance taken with regard to the quantity of the curved line is not the same in every case.

85. Then at [44] he shows that there is not contrariety in circular motion according to one and the same semicircle. For someone could say that the motion upon one curved line from A to B has as its contrary not a motion from R to A through just any curved line but through one and the same - for example, through one semicircle. Let GD be that semicircle, such that the motion through it from G to D is contrary to the one through it from D to G.

But Aristotle proceeds againstthis on the ground that the semicircular distance from G to D is computed in terms of the diametric distance, not in the sense that the semicircle is equal to the diameter, but because we measure every distance by a straight line. The reason for this is that every measure ought to be certain and determinate and the smallest. Now between two points the length of a straight line is certain and determinate, because it can be but one, and it is the smallest of all the lines between the two points. But an infinitude of curved lines can be drawn between two points, and all are greater than the straight line drawn between the two given points. Hence the distance between two points is measured by a straight line, and not by the curved line of a semicircle or any other portion of the circle, either of a larger or a smaller circle. Therefore, since it belongs to the very notion of contrariety that it have maximum distance, as is said in Metaphysics X, then, since the distance between two points is not measured according to a curved line but according to a straight, the consequence is that a contrari­ety of termini does not bring about a contrariety in motions upon a semi-circle, but only in motions upon the diameter.

86. But John the Grammarian objects against this, because not only do geometers and astronomers reckon the quantity of a curved line by a straight line, but they also do the converse: for they prove the quantity of a chord by means of the arc and that of the arc by the chord.

But in this he departs from the intent of Aristotle. For Aristotle does not intend to maintain that a curved line is measured by a straight, but the distance between any two given points is measured by a straight line, for the reason just given.

He [John] objects too that in the heavens there is a greatest distance between two opposite points: for example, between the beginning of Aries and the beginning of Libra; consequently, if contrariety is the greatest distance, then according to this distance, contrariety can be found in circularmotion.

But to this it should be said that that greatest distance is reckoned according to the quantity of the diameter and not according to the quantity of the semicircle - otherwise the beginning of Aries would be farther from the beginning of Sagittarius, to which it has a trinary aspect, than from the beginning of Libra to which it has the aspect of right opposition.

87. Then at [45] he shows that there is not contrariety in circular motion according to two semicircles. And he says that the reasoning is similar to describing a whole circle and positing that the motion in one semicircle is contrary to a motion in the other. For let a circle have a diameter EZ dividing it into two semicircles called I and T respectively. Now someone could say that a motion from E to Z through semicircle I is contrary to the motion from Z to E through semicircle T. But this is disproved by the same argument as the first case: namely, because the distance between E and Z is not measured by a semicircle but by the diameter. But there is still another reason: namely, the motion which begins at E and proceeds to Z through I, and then returns from Z to E through semicircle T, is one continuous motion; but two motions that are contrary cannot be continuous with one another, as is plain in Physics VIII.

88. Then at [46] he shows that even if those parts of circular motions were contrary, that would be no reason for concluding that there would be contrariety in circular motions as a whole; for contrariety of parts is no proof for the contrariety of the whole. Consequently, it is plain that what the Philosopher has just showed about contrariety of the parts of circular motion has been done for added measure in order to exclude contrariety entirely from circular motion.

89. Then at [46 bis] he shows that to one complete circular motion there is not another circular motion contrary: and this for two reasons. The first of these is based on considering circular motion in general. Therefore, take a circle upon which A, B and G are described at three points. Suppose two circular motions occur upon this circle, one beginning at A through B to G and back to A; conversely, let the other start at A through G to B and back to A. He says then that these two motions are not contrary. For each begins at the same term A and terminates at the same term, namely, A; consequently, they neither begin at terms that are contrary nor end at terms that are contrary. But a contrary local motion is one that goes from contrary to con­trary. Therefore, the two circular motions in question are not contrary.

90. The objector against this is once more John the Grammarian. First on the ground that the notion of contrariety in diverse things is seen to be diverse. For to be moved from contrary to contrary determines contrariety in straight motions; hence it is not necessary, if such contrariety is not present in circular motions, that on this account no contrariety may exist therein. Likewise, just as it is of the very nature of contrary motion in straight motions to be from contrary to contrary, so it is of the very nature of motion to be from one thing to another. Now, by the very fact that cir­cular motion is from the same to the same, not only is it not from contrary to contrary, but it is not from one thing to another. Therefore there is excluded from circular motions not only that they be contrary, but that they be motions at all.

To the first objection it should be replied that to be from contrary to con­trary is not a special property of the contrariety found in local motions in a straight line, but it is a common property of contrariety in all motions, as is plain in Physics V. And the reason for this is that contrariety is a difference according to form, as is shown in Metaphysics X. Now a motion possesses form or species from its terminus. Therefore, there can be con­trariety in no motion, unless there is contrariety of termini.

To the second it must be said that circular motion, because it is the first of motions, has a minimum of diversity and a maximum of uniformity. And this even appears proportionally in the mobile and in the motion. In the mobile, indeed, because it does not change its "where" with respect to the whole subject, but only in conception, whereas each part changes its "where" even as to subject, as was shown in Physics VI. And similarly a part of a circular motion is from one to another with a difference as to subject; but the whole circular motion is indeed from the same to the same according to subject, but from one thing to another that differs only in conception. For if we take one circular motion from A to A, the A which is the terminus a quo and the terminus ad quem is the same as to subject, but differs in conception, insofar as it 22.is taken now as beginning and now as end. And therefore, because circular motion has the most unity, its nature is very far from contrariety, which is a maximum distance. That is why such motion belongs to the first bodies which are the nearest to the simple substances which completely lack contrariety.

91. The second argument is at [47], and this argument is based on applying circular motion to natural bodies. And this is the argument: If one circular motion were contrary to another, then one of them would have to be in vain. But nothing in nature is in vain. Therefore, there are not two contrary circular motions.

The truth of the conditional proposition he proves in the following manner: If there were two contrary circular motions, then the bodies subject to them ought to pass through the same signs marked on a circle. The reason for this is that contrariety of local motion demands contrariety of the places, which affect both mobiles. Consequently, if there were contrary circular motions, then contrary places should be able to be designated on the circle. Now on a straight line only two contrary places are designated, namely, those the greatest distance apart, while other places designated on that line, since they are within the extreme places, are not contrary to one another. But on a circle any point at random can be at a greatest distance from some other point on the circle: because from any point on the circle a diameter can be drawn, which is the greatest of the straight lines falling in the circle. And it has been said that every distance is measured according to a straight line. Therefore, because things in contrary motions must reach contrary places, then if circular motions are contrary, it is necessary that each body in circular motion, no matter from which point of the circle its motion begins, reach all the places of the circle, all of which are contrary. (Nor is it unfitting that in a circle places be marked as in every way contrary - for contrariety of place is taken not only with respect to up and down, but according to ahead and to the rear, and left and right.) But it has been said that the contraries of local motion are based on contrariety of places. And thus, if circular motions are contrary, the contrarieties in the circle must be taken according to the forementioned.

Now from all this it follows that one of the motions or of the bodies would be in vain. For if the magnitudes moved were equal, i.e., of equal power, neither would be moved, because one would totally obstruct the other, since both would have to traverse the same places. But if one motion dominated on account of a greater power in one of the mobiles or movers, then the other motion could not exist, because it would be totally obstructed by the stronger motion. Therefore, if both were bodies apt to be moved with contrary circular motions, one of them would exist in vain, for it could not be moved with that motion which was obstructed by the stronger. For we say that a thing is "in vain" when it does not realize its usefulness, as we say that a shoe is in vain if no one can wear it. In like manner, a body would be in vain, if it could not be moved with its proper motion; and likewise a motion would be in vain if nothing could be moved with it.

Consequently, it is plain that if there are two contrary circular motions, there would have to be something in vain in nature. But that this is impossible he now proves: Whatever exists in nature is either from God, as are the first natural things, or from nature as from a second cause, as, for example, lower effects. But God makes nothing in vain, because, since He is a being that acts through understanding, He acts for a purpose. Likewise nature makes nothing in vain, because it acts as moved by God as by a first mover, just as an arrow is not moved in vain, inasmuch as it is shot by the bowman at some definite thing. What remains, therefore, is that nothing in nature is in vain.

It should be noted that Aristotle here posits God to be the maker of the celestial bodies, and not just a cause after the manner of an end, as some have said.

92. John the Grammarian objects against this argument that, for the same reason, someone could conclude that there is no contrariety in straight motions, because contrary mobiles obstruct one another.

But it should be said that the case with straight motions is different from that of circular, for two reasons. First, because two bodies are moved with contrary straight motions without mutually obstructing one another, for in straight motions contrariety is not reckoned except with respect to the extremes of straight lines, for example, with respect to the center of the world and its circumference. Now from the center to the circumference an infinitude of lines can be drawn so that what is moved upward through one of them does not obstruct what is being moved downward th