translated by Fabian R.Larcher and Pierre H. Conway
INTRODUCTION OF SAINT THOMAS
BOOK I Introduction by Thomas Aquinas Lecture 1 The things it pertains to natural science to consider Lecture 2 The perfection of the universe both as body and as containing all Lecture 3 Preliminary notions for showing the parts perfecting the universe Lecture 4 Five reasons why, besides the elements, there must be another simple body Lecture 5 Difference of the body moved circularly as to light and heavy Lecture 6 The fifth body not subject to other motions Lecture 7 The heavenly body is not subject to growth and decrease, or to alteration Lecture 8 Only five simple bodies required. No motion contrary to circular Lecture 9 The need for treating of the infinity of the universe Lecture 10 The second and third reasons proving the circularly moved body not infinite Lecture 11 Three additional reasons why the body moving circularly cannot be infinite. Lecture 12 Various reasons why a body moving in a straight line is not infinite Lecture 13 A natural and demonstrative argument showing no natural body can be infinite Lecture 14 No sensible body is infinite - from action and passion, which follow upon motion Lecture 15 Logical reasons why no body is infinite Lecture 16 Two arguments for one universe, taken from lower bodies Lecture 17 A third argument from lower bodies. Natural bodies have determinate places Lecture 18 Exclusion of the opinion that natural bodies are not moved naturally to determined places. Unity of the world from higher bodies Lecture 19 Solution of the argument seeming to justify several worlds Lecture 20 The universe shown to consist of every natural and sensible body as its matter Lecture 21 Outside the heaven there is no place, time etc., consequent upon sensible bodies Lecture 22 Whether the universe is infinite by eternal duration Lecture 23 A Platonic evasion rejected. Two remaining opinions disproved Lecture 24 Various meanings of "generable" and "ungenerable," "corruptible" and "incorruptible" Lecture 25 How something is said to be "possible" and "impossible" Lecture 26 Everything eternal is indestructible and ungenerated Lecture 27 Nothing eternal generated and corrupted, and conversely Lecture 28 Generated and corruptible, ungenerated and incorruptible, follow on each other Lecture 29 Refutation of corruptible ungenerated and incorruptible generated. Argument from natural science BOOK II Lecture 1 The heaven is eternal and its motion endless and without labor. Contrary opinions excluded Lecture 2 Diversity of parts of the heaven as to position. Opinion of Pythagoras Lecture 3 How the differences of position befit the parts of the heaven according to the Philosopher's opinion Lecture 4 The reason why there are in the heaven several spheres moved with a circular motion Lecture 5 The spherical shape of the heaven shown from the fact that it is the first of figures Lecture 6 The heaven must be spherical, because this shape is most fitting Lecture 7 Why the circular motion of the heaven is in one direction rather than another Lecture 8 The regularity, or uniform velocity, of the heaven's motion shown by two arguments Lecture 9 Two other arguments proving no irregularity in the motion of the heaven Lecture 10 On the nature of the stars Lecture 11 Proof that the stars move, not of themselves, but as carried by the motion of the spheres, from a comparison with their circles Lecture 12 That the stars do not move themselves concluded from the motions proper to the spherical shape Lecture 13 From their shape the stars shown not to move themselves. No sense power in the heavenly bodies Lecture 14 Indirect and direct proof that heavenly bodies do not produce sounds Lecture 15 Swiftness and slowness in the motion of the planets is proportionate to their distance from the first sphere and the earth Lecture 16 By reason, and by what sensibly appears, the stars are proved to be spherical in shape Lecture 17 Two difficulties proposed in connection with what has been determined about the stars Lecture 18 The first difficulty, concerning the number of motions of the stars, is solved. The number shown to agree with modern astronomers Lecture 19 The second difficulty of Lecture 17 is resolved Lecture 20 Opinions of the philosophers as to the site of the earth. Pythagorean theory of fire in the center is rejected Lecture 21 Different opinions of the motion, rest, and shape of the earth Lecture 22 The problem about the earth's rest Lecture 23 The cause of the earth's rest is not supporting air Lecture 24 Earth's rest not from gyration of the heaven Lecture 25 Earth's rest not explained by supposing that all directions being alike to earth, nothing induces it to be moved in one direction rather than another Lecture 26 Proof of the earth's rest in the middle Lecture 27 Proof of the earth's spherical shape, from motion Lecture 28 Proofs of the earth's sphericity from the angle of motion of its parts, and from astronomy BOOK III Lecture 1 What has gone before and what remains to be treated Lecture 2 Opinions of the ancients on the generation of things Lecture 3 Bodies not generated from surfaces, proved mathematically and naturally Lecture 4 Other natural arguments against Plato's opinion. Pythagorean opinion refuted Lecture 5 Natural motion in natural bodies. Leucippus & Democritus Lecture 6 Refutation of Plato's opinion of disordered motion before the world Lecture 7 Every body moving naturally in a straight line has gravity or lightness. Natural and violent motions Lecture 8 Everything not generated. Elements and their existence
[The numbers in brackets refer to the passages in the text of Aristotle.]
INTRODUCTION BY SAINT THOMAS
Subject matter of this book
and its relation to the subject matter of natural science in general
1. As the Philosopher says in Physics I, "We judge that we know a thing when we know the first causes and the first. principles down to the elements." Plainly from this the Philosopher shows that in sciences there is an orderly process, a procedure from first causes and principles to the proximate causes, which are the elements constituting the essence of a thing. And this is reasonable: For the method pursued in sciences is a work of reason, whose prerogative it is to establish order; wherefore, in every work of reason is found some order according to which one goes from one thing to another. And this shows up not only in the practical reason, which considers things that we make, but in the speculative reason as well, which considers things made by some other source.
2. The process from prior to subsequent is found in the act of the practical reason with respect to a fourfold order: first, according to the order of apprehension, inasmuch as an artisan first apprehends the form of a house absolutely and then realizes it in matter; secondly, according to the order of intention, inasmuch as an artisan intends to complete the house and for that purpose does whatever he does to the parts of the house; thirdly, according to the order of combining, inasmuch as he first trims the stones and then joins them into one wall; fourthly, according to the order of supporting the edifice, inasmuch as the artisan first lays the foundation, upon which the other parts of the house are supported.
In like manner, a fourfold order is found in the consideration of speculative reason. First, because there is a process from the general to the less general.. And this order corresponds to the first order which we have called "the order of apprehension," for universals are considered according to an absolute form, but particulars by applying form to matter, as the Philosopher in On the Heavens says, that the word "heaven" signifies a form, and "this heaven" signifies a form in matter.
The second order is that according to which one goes from the whole to the parts. And this corresponds to "the order of intention," inasmuch as, namely, the whole is considered prior to the parts, not just any parts but parts which are according to matter and which are of the individual - as in the case of a semi-circle, in the definition of which "circle" is used (for it is "half a circle") and of an acute angle, in the definition of which "right angle" is used (for an acute angle is an angle "less than a right angle"). To be divided in that manner is incidental to a circle and to a right angle; hence, neither is a part of the species of a circle or right angle. For parts of this sort [i.e. parts of the species] are prior in consideration to the whole and are used in the definition of the whole, as are flesh and bones in the definition of man, as is said in Metaphysics VII.
The third order is that according to which one goes from the simple to the combined, inasmuch as composites are known in terms of the simple, as through their principles. And this order is compared to the third order, which is the "order of combining." But the fourth order is the one that calls for the principal parts to be considered first, as are the heart and liver before the arteries and blood. And this corresponds in the practical order to that order according to which the foundation is laid first.
This fourfold order is also considered in the procedure of natural science. For, first of all, things common to nature are determined in the book of the Physics, in which mobile being is treated insofar as it is mobile. Hence what remains in the other books of natural science is to apply these common things to their proper subjects. The subject of motion, however, is a magnitude and body, because nothing is moved except what is quantified.
Now it is in bodies that the three other orders are considered: in one way, insofar as the entire corporeal universe is prior in consideration to its parts; in another way, insofar as simple bodies are considered before the mixed; thirdly, insofar as, among the simple bodies, the first must be considered first, i.e., the heavenly body, through which all the others are sustained. And these three are treated in this book, which the Greeks entitle On the Heavens. For in this book are treated certain things that pertain to the entire universe, as is plain in Book I; and things that pertain to the heavenly body, as is plain in Book II; and things that pertain to the simple bodies, as is plain in Books III-IV. Consequently, it is with good reason that this book is first in order after the book of the Physics. For this reason the first topic of discussion in the very beginning of this book is body, to which must be applied all that was set forth about motion in the Physics.
Because diverse things are treated in this book, there was among the early expositors of Aristotle a question about the subject of this book. For Alexander believed that the subject principally treated herein is the universe. Hence, since "the heavens" is subject to a threefold meaning - for sometimes it refers to the outermost sphere, sometimes to the whole body moved circularly, and sometimes to the entire universe - he asserts that this book is entitled On the Heavens as though meaning On the Universe or On the World. In asserting this he assumes that the Philosopher is here determining certain matters pertaining to the entire universe, for example, that it is finite, that it is unique, and things of this sort.
On the other hand, it seems to some that the main subject handled in this book is the heavenly body which is moved circularly, for which reason it is entitled On the Heavens. Other bodies, however, are discussed therein consequentially, insofar as they are contained by the heavens and influenced by them, as Iamblichus said; or only incidentally, insofar as a knowledge of other bodies is assumed in order to explain what is being said of the heavens, as Syrianus says. But this does not seem probable, for after the Philosopher has finished his discussion of the heavens in Book II, he treats in Books III and IV of the other simple bodies as though they were his main subject. Now the Philosopher is not wont to assign a principal part in some science to things that are brought up only incidentally.
Therefore it seemed to others, as Simplicius said, that the intention of Aristotle in this book is to determine about simple bodies inasmuch as they share in the common notion of simple body; and because among simple bodies. The chief is the heavens, on which the others depend, the entire book gets its name from the heavens. And, so he says, it makes no difference that in this book things pertaining to the whole universe are considered, for the conditions in question belong to the universe insofar as they belong to the heavenly body, i.e., to be finite and eternal, and so on. But if the principal intention of the Philosopher were to determine about the universe or the world, then he would have had to extend his consideration to all the parts of the world, even down to plants and animals, as Plato does in the Timaeus.
But the same argument could be used against Simplicius, because if Aristotle in this book intended to treat principally of the simple bodies, then in this book he would have had to mention everything that pertains to the simple bodies, whereas he discusses only what pertains to their lightness and heaviness, while he treats the other aspects in the book, On Generation.
5. Accordingly, the opinion of Alexander appears more reasonable, i.e., that the subject of this book is the universe itself, which is called "the heavens" or "the world," and that determination is made concerning simple bodies in this book accordingly as they are parts of the universe. Now, the corporeal universe is composed of its parts according to an order of position [situs]; consequently this book determines only concerning those parts of the universe that primarily and per se have position in the universe, namely, the simple bodies. That is why the four elements are not dealt with in this book from the aspect of their being hot or cold or something of that sort, but only with respect to their heaviness and lightness, from which their position in the universe is determined. Other parts of the universe, such as stones, plants and animals, have a determined place [situs] in the universe not according to what they are in themselves but according to the simple bodies; consequently, they are not treated in this book. And this agrees with what is usually said among the Latins, that this book discusses body that is mobile with respect to position or place, such motion being common to all the parts of the universe.
Lecture 1: The things it pertains to natural science to consider.
6. In this first book Aristotle begins for the first time to apply to bodies the things that were said about motion in a general way in the book of the Physics. For that reason he first shows by way of introduction that it pertains to natural science to determine about bodies and magnitudes; Secondly, he begins to carry out his proposal (Lecture 2).
With respect to the first he presents this argument: Natural things are bodies and magnitudes and whatever pertains to these. But natural science is about natural things. Therefore, natural science consists in treating of bodies and magnitudes.
7. First  therefore, he posits the conclusion, saying that the science which treats of nature seems to be "for the most part" concerned with bodies, and "magnitudes," i.e., lines and surfaces. However, the natural philosopher considers these in a different way from the geometer. For the former treats of bodies insofar as they are mobile, and of surfaces and lines insofar as they are the boundaries of mobile bodies; the geometer, on the other hand, considers them insofar as they are measurable quantities. And because a science should consider not only subjects but also their passions, as is said in Post. Anal. I, he therefore adds that natural science is concerned with the passions and motions of the aforesaid - by "passions" meaning alterations and other consequent motions, with respect to which something is altered in the substance of a thing; and he adds, "and motions," as though going from the particular to the general. Or perhaps by "motions" he specifically understands local motions, which are the more perfect in the genera of motions. Or by "passions" is meant the properties, and by "motions" the operations of natural things, which do not occur without motion. And because, in every science, principles must be considered, he adds that natural science is concerned with any and all the principles of the afore-mentioned substance, namely, mobile corporeal substance. By this we are given to understand that it pertains to natural science primarily to consider body insofar as it is in the genus of substance, for it is in this respect that it is the subject of motion; whereas it pertains to the geometer to consider it insofar as it is in the genus of quantity, for thus it is measured.
Since the minor premiss is plain, namely, that natural science is concerned with natural things, he adds the major, saying that the reason why natural science is concerned with the aforementioned is that among things which are according to nature, some are bodies and magnitudes, e.g. stones and other inanimate things; and some have body and magnitude, as do plants and animals, whose principal part is the soul (hence they are what they are more with respect to soul than with respect to body); finally, some things are principles of things having body and magnitude - for example, the soul, and universally form, and matter.
From this is clear why he said that the science of nature is "for the most part" concerned with bodies and magnitudes: for one part of this science is concerned with things having body and magnitude; it is also concerned with the principles of these; it is further concerned with some things which do not exist in nature but which some have attributed to bodies and magnitudes, namely, the void and the infinite.
Lecture 2: The perfection of the universe both as body and as containing all.
After showing by way of introduction that bodies and magnitudes are to be studied in natural science, the Philosopher here begins to carry out his main resolve. And because, as was said above, Aristotle in this book is mainly concerned with determining about the corporeal universe and its principal parts which are the simple bodies, among which the most important is the heavenly body, the book therefore is divided into three parts:
In the first he determines concerning the corporeal universe; In the second concerning the heavenly body, in Book II; In the third about other simple bodies, i.e., heavy and light, in Book III.
With respect to the first he does two things:
First he shows the perfection of the universe; Secondly, he determines certain of its conditions or properties (L. 13. 9).
About the first he does two things:
First he shows the perfection of the universe; Secondly, he explains of what parts it is composed (L. 13).
As to the first he does two things:
First he shows the perfection which the universe has in virtue of the common notion of its genus, i.e., inasmuch as it is a body, at 9. Secondly, he proves the perfection proper to it, at 18.
About the first he does three things:
First he explains the definition of body, to be used in proving his proposition, at 10. Secondly, he proves the proposition, at 15; Thirdly, he shows what could be clear from the foregoing, at 16.
As to the first he does two things:
First he defines "continuum," which is the genus of body; Secondly, he clarifies the definition of body, at 10.
With regard to the first , 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  he defines "body."
First he proposes the definition, saying that body is "a continuum which is divisible in every way," i.e., at every part or according to every dimension. Secondly, at  he proves the proposed definition with this argument:
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 . 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  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  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  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  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  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.
Lecture 3: Preliminary notions for showing the parts perfecting the universe.
After showing that the universe is perfect by reason both of its corporeity and its universalness, the Philosopher here shows from which parts its perfection is made up.
First he expresses his intention; Secondly, he proves his proposition, at 20.
With respect to the first  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  he starts to manifest the proposition stating of which principal parts the perfect species of the universe is made.
First he shows that in addition to the four elements, another simple body must exist; Secondly, that there is no simple body other than these five (L. 8).
About the first he does two things:
First he shows that there is a fifth body besides the four elements; Secondly, how it differs from the four elements (L. 5).
With respect to the first he does two things:
First he mentions some preliminary facts needed in proving his proposition; Secondly, he argues to the proposition (L. 4).
About the first he does two things:
First he premises facts regarding motion; Secondly, facts pertaining to mobile bodies, at 32.
About the first he does two things:
First he mentions the connection between local motion and mobile bodies; Secondly, he distinguishes the kinds of local motion, at 23.
21. He says therefore first  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  he distinguishes local motions.
First he distinguishes in a general way both composite and simple local motions; Secondly, he distinguishes simple motions, at 27.
With respect to the first he does two things:
First at  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  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  he distinguishes simple motions.
First he mentions one, namely, the circular; Secondly, he mentions two that are straight, at 29; Thirdly, he concludes that the number of simple motions is three, at 30.
He says therefore first  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  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  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  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  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.
Lecture 4: Five reasons why, besides the elements, there must be another simple body
33. After stating in advance certain things necessary for showing the proposition, the Philosopher here begins to reason toward the propogition, and this with five arguments. The first  is this: Circular motion is a simple motion. But a simple motion belongs primarily and per se to a simple body - because even though a simple motion might occur in a composite body, this will be with respect to the simple body that is predominant in it; for example, in a stone, earth predominates, according to whose nature it is moved down. Therefore, there must be a simple body which is naturally moved according to a circular motion.
Now, someone could object to this argument and say that, although a simple motion belongs to a simple body, yet that simple body which is circularly moved would not necessarily be different from the simple body that is moved with a simple straight motion. Accordingly, he rejects this by adding that nothing prevents diverse bodies from being moved unnaturally with some one motion, as when one body might be moved violently with the motion of another; but that one body be moved according to nature with the natural motion of some other body is impossible. For one simple natural motion must belong to one simple body, and diverse to diverse. Hence, if circular motion is simple and distinct from straight motions, then it must belong to a natural simple body that is different from the simple bodies that are moved with a straight motion.
34. But this seems to be false, namely, that one simple motion belongs to just one simple body, for downward motion is natural to both water and earth, and upward motion to fire and air.
But it must be said that local motion is attributed to the elements, not according to hot and cold, moist and dry, with respect to which the four elements are distinguished - as is plain in On Generation II - for these four properties are principles of alterations. But local motion is attributed to the elements with respect to heaviness and lightness. Hence the two heavy bodies are compared to local motion as one body; and the same for the two light bodies. For moist and dry, according to which earth and water, or fire and air, differ, have an incidental relationship to local motion. Yet in the realm of heavy and light there is a difference, for fire is light simply and absolutely, and earth heavy; while air is light compared to two elements and likewise water is heavy. Hence the motions of water and earth, or fire and air, are not completely the same according to species, because the termini according to which their motions are specified are not the same: for air is apt to be moved to a place below fire, and water to a place above earth.
35. Likewise it seems not necessary, if of one simple body there is one simple motion, that on this account any simple motion should belong to some [different?] simple body, any more than it is necessary that there be as many composite bodies as there are composite motions, which are infinitely diverse.
But it must be said that just as simple local motion does not correspond to a simple body with respect to hot and cold, and moist and dry, so neither does composite motion correspond to mixed body according to the degrees of mixture of those qualities, but rather according to a composition of heavy and light, according to the diversity of which is diversified the obliquity of a mixed body from the simple motion of the heavy or light. Neither of these diversities tends to infinity with respect to species, but only with respect to number.
36. Likewise it seems that according to this there are many simple bodies. For just as motions upward and downward seem to be simple motions, so too motions to the right and to the left, and those ahead and to the rear.
It must be stated therefore that, since simple bodies are the essential and first parts of the universe, the simple motions which are natural to simple bodies must be considered in relation to the condition of the universe. Since this latter is spherical, as will be proved later, its motion must be considered in relation to the middle, which is immobile, because every motion is founded upon something immobile, as is stated in the book, On the Cause of the Motion of Animals. Consequently, there must be but three simple motions, according to their diverse relations to the middle [center]: i.e., one which is from the center, one which is to the center, and one which is around the center. To the right and left, ahead and to the rear, are considered in animals but not in the whole universe, except in the sense that" they are placed in the heavens, as will be said in Book II. And according to this the circular motion of the heavens is with respect to right and left, ahead and to the rear.
37. In like manner it seems that straight motion and circular are not of the same kind. For a straight motion belongs to a body not yet having its completeness of species, as will be said in Book IV, and existing outside its proper place, while a circular motion belongs to a body that has completeness of species and is existing in its proper place. Hence simple bodily motions do not seem to belong to simple bodies according to a same notion, but some motions seem to belong to bodies inasmuch as they are coming into being, while circular motion insofar as they have complete existence.
But it must be said that, since a motion is proportionate to the mobile as being its act, it is fitting that a body which is separated from generation and corruption and cannot be expelled from its proper place by violence should have a circular motion, which is proper to a body existing in its own place; but to other bodies that can be generated and corrupted there belongs a motion outside their proper place and which is incomplete in species. But this is not in the sense that a body which is naturally moved with a straight motion lacks the first complement of its species, namely, form, for it is the form that such a motion is consequent upon; but in the sense that it does not have its final complement which consists in attaining the end, which is a place that agrees with it and conserves it.
38. The second argument he gives at  and in it he presupposes two principles: one of which is that a motion which is outside nature, i.e., violent, is contrary to a natural motion, as earth is according to nature moved downward but upward against its nature. The second principle is that one thing is contrary to one thing, as is proved in Metaphysics X. A third also must be presupposed from sense experience, namely, that there exists a body which is moved circularly. Now, if that motion is natural to that body, we have the proposition, in keeping with the previously given reason, namely, that that body which is moved in a circle naturally is distinct from the four simple elements. But if such a motion is not natural to it, it must be against its nature.
Let us therefore first assume that that body in circular motion is fire, as some claim, or any of the other four elements. Then the natural motion of fire, which is to be moved upward, will have to be contrary to the circular motion. But this cannot be, for to one thing, one thing is contrary, and the motion contrary to an upward motion is a downward one; consequently, circular motion cannot be contrary to it. And the same holds for the other three elements. Likewise, if it be assumed that the body which is being moved circularly against its nature is a body other than the four elements, it would have to have some other natural motion. But this is impossible, because if its natural motion is up, it will be fire or air; if its motion is down, it will be water or earth. But we supposed that it is not one of the four elements. Accordingly it must be that the body moved in circular motion is being moved naturally with this motion.
Now according to what he says here Aristotle seems to be contrary to Plato who assumed that the body which is circularly moved is fire. But with respect to the truth, the opinion of both philosophers is the same on this point. For Plato calls the body which is being circularly moved "fire" on account of light, which is posited as a form of fire, but not as being of the nature of elemental fire. Hence he posited five bodies in the universe, and to these he adapted five bodily figures which geometers teach, calling the fifth body "aether."
39. But further, what is said here, namely, that for fire to be moved circularly is outside nature seems to be contrary to what is said in Meteorology I, where Aristotle himself sets forth that hypeccauma, i.e., fire, and the upper portion of the air, are carried along circularly by the motion of the firmament, as is plain in the motion of a comet.
But it must be said that that circulation of fire or air is not natural to them, because it is not caused from an intrinsic principle. Neither is it through violence or against nature, because such a motion is in them from the influence of a higher body, whose motion fire and air follow according to a complete circulation because these bodies are closer to the heavens, but water according to an incomplete circulation, i.e., according to the ebb and flow of the sea. Earth, however, as being most remote from the heavens, suffers no such change except with respect to the sole alteration of its parts. Now whatever is present in lower bodies from the impression of the higher is not violent for them or against nature, for they are naturally apt to be moved by the higher body.
40. Likewise it seems to be false, as here stated, that to one thing one thing is contrary, for to one vice both a virtue and the opposite vice are contrary, as to illiberality both prodigality and liberality are opposed.
But it must be said that there is only one contrary to one thing according to the same aspect, although from different aspects nothing forbids one thing from having several contraries: thus, if the same subject is sweet and white, black and bitter will be contrary to it. Accordingly, the virtue of liberality is contrary to illiberality as what is well ordered to what is disordered, but prodigality is contrary to it as superabundance is to defect. Now, it cannot be said that both motions, namely, the one that is upward and the one that is downward, are contrary to circular motion according to the common aspect of straightness. For straight and circular are not contrary, for they pertain to figure, to which nothing is contrary.
41. He gives the third argument at . With regard to this he first shows that circular motion is the first of local motions. For circular motion is related to straight motion, such as up or down, as circle is compared to straight line. A circle, i.e., a circular line, is proved to be prior to a straight line because the perfect is naturally prior to the imperfect. But a circle, or circular line, is perfect, because whatever is taken in it is a beginning and middle and end. Hence it does not suffer the addition of anything from without. But no straight line is perfect, whether it be an infinite line, which is imperfect because it lacks an end, from which things are called perfect in Greek, or a finite line, because every finite line can be increased, i.e., receive more quantity and so there is something outside it. Consequently a circular line is naturally prior to the straight. Therefore circular motion, too, is naturally prior to straight motion.
But a prior motion naturally belongs to a prior body. Now straight motion naturally belongs to some one or other of the simple bodies, as fire is moved upward and earth downward and toward the middle. And if it happens that a straight motion is found in mixed bodies, that will be due to the nature of the simple body predominant in it. Since, therefore, a simple body is naturally prior to the mixed, the consequence is that circular motion is proper and natural to some simple body which is prior to the elementary bodies that exist here among us. Thus it is clear from these facts that besides the bodily substances that exist here among us, there must be some bodily substance which is nobler and prior to all the bodies that exist among us.
42. But the assertion that no straight line is perfect seems to be false. For if the perfect is what has a beginning, middle and end, as we held above, it seems that a straight finite line, which has beginning, middle and end, is perfect.
But it should be stated that in order for something to be partially perfect it must have the beginning, middle and end in itself; but to be completely perfect it is required that there be nothing outside it. And this mode of perfection belongs to the first and supreme body which contains all bodies; and with respect to this mode a straight line is said to be imperfect and a circular line perfect.
Yet it seems that even according to this mode some straight lines are perfect, because the diameter of a circle cannot suffer addition.
But it must be said that this happens to it insofar as it is in such and such a matter, and not insofar as it is a straight line, from which aspect there is nothing to prevent additions being made. But a circle, precisely as circle, cannot suffer such addition.
43. But it seems that, if this is so, one cannot conclude that circular motion is perfect, because it does receive addition, since it is continuous and eternal, according to Aristotle.
To this it should be said that one revolution is complete in species when it returns to the beginning from which it started. Hence no addition is being made to the same revolution, but whatever follows pertains to another revolution.
Yet if only a thing to which no addition can be made is to be called perfect, it follows that neither man nor any finite thing in bodies is perfect, since additions can be made to them.
And it should be answered that things of this kind are said to be perfect with respect to their species, inasmuch as they can suffer no addition of anything pertaining to the notion of their species; but to a straight line can be added something that pertains to its species, and to that extent it is said to be imperfect insofar as it is a line.
But still it seems that a circle is not perfect. For a perfect thing among magnitudes is something having three dimensions; which our circular line certainly lacks.
To this it should be responded that a circular line is not an absolutely perfect magnitude, because it does not have everything that pertains to the notion of a magnitude, Yet it is perfect in the realm of lines, because linearly something cannot be added to it.
44. It also seems false that the perfect is prior to the imperfect. For the simple is prior to the composite and yet the latter is to the former as perfect to imperfect. To this it must be said that perfect is to imperfect as act to potency, and simply speaking, act is prior to potency in things that are diverse, although in one and the same thing that is moved from potency to act, potency is prior to act in time, but act is prior to potency according to nature, for this is what nature intends first and principally. Now the Philosopher does not mean here that the perfect is prior to the imperfect in one and the same thing, but in diverse things, nor does he intend to say that it is prior in time but in nature, as he expressly states.
45. Moreover it seems that the Philosopher is arguing in an unsuitable manner. For he proceeds from the perfection of a circular line to prove the perfection of a circular motion, and from the latter perfection he goes on to prove the perfection of a circular body. And so his proof seems to be circular, because a circular line does not seem to be anything other than that of the very body that is being moved circularly. And it should be said that a circular motion is proved to be perfect on account of the perfection of the circular line absolutely; then from the perfection of circular motion in common one proves that this body which is moved circularly is perfect. Thus one does not go from the same to the same, but from common to proper.
46. The fourth argument is given at , and it proceeds from two assumptions. The first is that every simple motion is either according to nature or outside nature. The second is that a motion which is outside nature for one body is according to nature for another, as is clear in the upward motion which, for fire, is according to nature, and for earth is outside nature; and in the downward motion which is natural to earth, but outside nature for fire. Now it is manifest that a circular motion is present in some body, which the senses observe is moved circularly. And if such a motion is natural to it, we will have the conclusion, namely, that, besides the four elements, there is an additional body which is moved circularly. But if the circular motion is outside the nature of the body that is moved circularly, it follows from the foregoing assumption that for some other body it is according to nature, which body, consequently, will be of a different nature from the four elements.
47. Aristotle here seems to be at odds with himself, for above he proved that circular motion is not outside the nature of the body in circular motion, but here he supposes the contrary.
Accordingly some say that above the Philosopher was taking "outside nature" in the sense of "against nature" - for then a motion against the nature of some body would also be contrary to its natural motion, as he proceeded above. But here he takes "outside nature" in the more general sense of "not according to nature." Thus it includes both what is against nature and what is above nature, and it is in this sense that he assumes here that a body can be moved circularly outside its nature, just as it was said above that fire in its sphere is moved circularly outside its nature under the influence of the motion of the heavens.
But this seems to be against the intention of Aristotle. For he seems to take "outside nature" in the same sense in both cases, because both here and above he uses the example of motion which is upward and downward, which is against nature for one body, and according to nature for another. Therefore it is better to say that Aristotle in the first argument proved that some body is being moved circularly according to nature. And because someone could say that that body which is seen to be moved circularly is being moved against nature by this movement, he argues against this in two ways: in one way, by showing that that motion is not against nature, as is clear in the second argument and also in the third; in another way, by showing that, even if it is being moved against nature, it still follows that there is some other body which is moved circularly according to nature. Consequently what he denied above when speaking according to the truth of his own opinion, he here denies by using, so to speak, the assumptions of his adversaries.
48. Likewise it does not seem to follow that, if some motion is outside nature for one body, it is natural to some other body. For fire or any other body can be moved in a number of ways - yet this does not prove that such motions are natural to certain bodies.
But it should be noted that the Philosopher is here speaking of simple motion, to which the nature of a simple body is inclined as to one definite thing, whereas motions diversely various seem to be rather brought about by art, which can be a principle of diverse things. It should also be considered that, although a motion which for one body is beside nature is according to nature for another, yet it is not necessary that every body for which some motion is natural should have a motion that is beside nature: for every body which can suffer an impression from without has something proper and connatural to it, yet not every body can receive an impression from without so as to be able to have a natural motion.
49. The fifth argument is at , and it is this. The conclusion of the foregoing argument was that if a body observed to be in circular motion is being moved outside its nature, then such a motion must be according to nature for some other body. And if this is granted, namely, that circular motion is according to nature for some body, then it is clear that there will be some first and simple body which is being moved circularly, on account of the simplicity and priority of circular motion, as is plain from the foregoing arguments, just as fire is moved upward and earth downward. But if the procedure of the foregoing argument is not admitted, and it is stated rather that all things in circular motion with respect to a periphery, i.e., a circumference, are being moved outside their nature, in such a way that this motion is not natural to any body, then such a thing seems to be marvelous and, indeed, wholly unreasonable. For it was proved in Physics VIII that only circular motion can be continuous and eternal. Now it is unreasonable that what is eternal should be outside nature, and that a non-eternal motion should be according to nature. For we see that things which are outside nature quickly pass and cease to be, as in the case of the heat of water and the projecting of a stone into the air, while things that are according to nature are seen to last a longer time. Thus it is wholly necessary that circular motion be natural to some body.
If therefore the body which is observed to be carried along circularly is of the nature of fire, as some say, that motion will be beside its nature, just as a downward motion is. For we see that the natural motion of fire is upward according to a straight line. Accordingly, just as a downward motion is natural for another body, namely, earth, so a circular motion will be natural to some other body.
50. Finally, in summary, he concludes that if someone should reason from all the foregoing in the aforesaid manner, he will believe, i.e., firmly assent, that there is a body over and above the bodies which exist among us (i.e., the four elements and composites of them), a body that is separated from them and of a nature that is more noble than they to the extent that it is farther separated from them in space. For in the universe the bodies that contain are to contained bodies as form to matter, and act to potency, as was said in Physics IV.
Lecture 5. Difference of the body moved circularly as to light and heavy
51. After showing that there is a body distinct from those that are here, namely, from the four elements, and from things composed of them, the Philosopher here shows the difference of this body from those which exist here.
First by comparing them with respect to local motion; Secondly, with respect to other motions (L. 6);
About the first he does three things:
First he proposes what he intends; Secondly, he proves the proposition, at 52; Thirdly, he dismisses an objection, at 56.
He says therefore first  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  he manifests his proposition. And because the principle of demonstration is "that which something is," as is said in Post. Anal. II,
he first supposes the definitions of heavy and light, at 52; Secondly, from these he argues to his proposition, at 54.
He says therefore first  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  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  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  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.
Lecture 6: The fifth body not subject to other motions
58. After having shown the difference between the fifth body and the other bodies that exist here from the standpoint of lightness and heaviness, according to which bodies have an inclination to local motion, the Philosopher here shows how the fifth body differs from bodies that exist here from the standpoint of other motions, and shows that the former is not subject to the other motions to which these bodies are subject.
First he shows this by an argument; Secondly, by signs (L. 7);
With respect to the first he does two things:
First he proposes what he intends  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 , he proves the proposition:
First he shows that the heavenly body is incapable of being generated or corrupted; Secondly, that it cannot be increased (L. 7); Thirdly, that it cannot be altered (also in L. 7).
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.
Lecture 7: The heavenly body is not subject to growth and decrease, or to alteration.
After showing that the fifth body is not subject to generation and corruption, the Philosopher here shows that it is not subject to increase and diminution  and uses this argument: Every augmentable body is, with respect to something, subject to generation and corruption. To explain this, he proposes that every augmentable body is increased by the addition of something connatural that comes to it. This, indeed, while being first unlike, has become like by being resolved into its proper matter which, doffing its previous form, has assumed the form of the body to be increased - as bread, after being resolved into matter, receives the form of flesh, and thus, through being added to pre-existing flesh, produces increase. Hence wherever there is growth there must be generation and corruption into something. But there is nothing from which a heavenly body can be generated, as has been shown. Therefore it cannot be augmentable or decreasable.
72. Then at  he shows that it is not subject to alteration. Now it might seem to someone that an easy way to remove alteration from the heavenly body would be by removing contrariety, for just as generation occurs from contraries, so too, does alteration. But it should be observed that Aristotle removed contrariety from the fifth body by removing from it contrariety of motion. Alteration, however, seems to occur not only according to the contrariety to which contrary local motions correspond, namely, heavy and light and whatever results from them, but also according to other contraries which do not pertain to this, for example, according to black and white. Accordingly, he uses another way, based on increase.
And he says that it is for the same reason that we estimate a heavenly body not to be alterable and not to be augmentable or perishable. For alteration is a motion affecting quality, as has been said in Physics V. But alteration, as was shown in Physics VII, properly takes place according to the third species of quality, which is "passion and passible quality": for although "habit and disposition" pertain to [the first species of] the genus of quality, they are not produced without a change made according to the passions, just as health and languor result from a change of cold and hot, moist and dry. Now all natural bodies that are changed with respect to passion or passible quality seem as a consequence to have growth and decrease, as is clear from the bodies of animals and their parts and even of plants, in which growth properly exists. The same applies also to the elements, which rarefy and condense with respect to a change in hot and cold, from which results a change into larger or smaller quantity which is in a sense the same as being increased and decreased. Thus it is plain that if a body which is moved circularly is not subject to increase or decrease it is not subject to alteration.
Finally, in summary he concludes that it is plain from the foregoing - if anyone wants to assent to the previous demonstrations without wantonly contradicting - that the first body, which, namely, is moved with the first and perfect motion, i.e., circular motion, is sempiternal (as not being subject to generation and corruption), that it undergoes neither increase nor decrease, and that it is not subject to aging or alteration or passion.
73. Nevertheless objections can be leveled against this argument of Aristotle on two counts. First of all against the conclusion. For it seems to be false that a heavenly body is not altered, for it is plainly evident that the moon is illumined by the sun and obscured by the shadow of the earth.
But it must be said that alteration is of two kinds. One is passive and according to it things are so added that something else is cast off, as, when something is altered from hot to cold, it loses heat and receives coldness. It is that kind of alteration, which takes place according to passions, that the Philosopher is here excluding from heavenly body. But there is another kind of alteration which is perfecting, which occurs insofar as something is perfected by something else without loss to the former - this is the kind of alteration that the Philosopher in On the Soul II posits even in a sense power. Such an alteration nothing prevents from being in heavenly bodies, some of which receive virtues from others according to conjunctions and various aspects, but without any of them losing their own virtue.
74. The second objection is directed against the procedure of his argument: for it does not seem to be true that whatever is altered receives increase and decrease. For these result from the addition of something that is converted into the substance of what is increased, as is said in the book On Generation and in On the Soul II, and as was said above. Now the motion of increase does not exist except in animals and plants, for things that rarefy and condense are not increased by the addition of anything, as was proved in Physics V. Consequently, it seems unsuitable for Aristotle here to attribute the motion of increase not only to animals and plants and their parts, but to the elements as well.
But it should be said that Aristotle is here speaking of increase in the sense of any motion by which something proceeds to greater quantity. For he has not yet perfectly explained the nature of the motion of increase and it is his custom, before he has shown the true view, to use common opinions. But the force of his proof is not impeded by his having excluded increase from a heavenly body by excluding addition of a body changed into what is increased: for just as anything increased by addition is not utterly free of generation and corruption, so, too, what is increased by rarefaction.
However it is to be noted that in this proof he makes mention of Physical bodies advisedly, because in mathematical bodies increase can occur without alteration - for example, a square grows by adding to it a gnomon, but it is not altered, as is said in the Predicaments; conversely, a thing can be altered without being increased, as when a triangle is made equal to a square.
75. Then at  he manifests the proposition through signs. And he says that both reason and things that appear to be probable seem to support one another on this point. And he gives three signs. The first of which is taken from the general opinion of men, who posit many gods or one God, whom the other separated substances serve. All who believe thus, whether Greeks or barbarians, assign the highest place, namely, the heavenly, to God, namely, all those who believe there are divine beings. But they assign the heavens to the divine substances as though adapting an immortal place to immortal and divine beings. In this way God's habitation in the heavens is understood as appropriate according to likeness, that is, that among all other bodies this body more closely approaches to a likeness to spiritual and divine substances. For it is impossible for the habitation of the heavens to be assigned to God for any other reason, as though He should need a bodily place by which He is comprehended. If therefore divine beings are to be posited, and since, indeed, they certainly must, the consequence is that the statements made about the first bodily substance, namely, the heavenly body, were well made, namely, that the heavenly body is ungenerated and unalterable.
Although men suppose that temples are the place of God, they do not suppose this from God's viewpoint but from that of the worshippers, who must worship Him in some place. That is why perishable temples are proportioned to perishable men, but the heavens to the divine imperishability.
76. The second sign he gives at  and it is taken from long experience. And he says that what has been proved by reason and common opinion occurs, i. e., follows, sufficiently - i.e., not absolutely but to the extent of human faith, i.e., so far as men can testify to what they have seen for a short time and from afar. For according to the tradition which astronomers have passed on concerning their observations of the dispositions and motions of heavenly bodies, in the whole time past there does not seem to have been any change affecting either the entire heavens or any of its own parts. Now this would not be, if the heaven were generable or perishable - for things subject to generation and corruption arrive at their perfect state little by little and step by step, and then gradually depart from that state, and this could not have been concealed in the heavens for such a long time, if they were naturally subject to generation and corruption.
However, this is not necessary but probable. For the more lasting something. is, the greater the time required for its change to be noted, just as change in a man is not noticed in two or three years, as it is in a dog or other animals having a shorter life-span. Consequently someone could say that, even though the heavens are naturally corruptible, nevertheless they are so lasting that the whole extent of human memory is not sufficient to observe their change.
77. The third sign is given at  and is based on a name given by the ancients, which endures to the present, and which gives us to understand that they thought the heaven to be imperishable just as we do. And lest anyone object that some before their time thought the heavens were subject to generation and corruption, he adds that true opinions are revived according to diverse times not once or twice but infinitely, supposing that time is infinite. For the studies of truth are destroyed by various changes occurring in these lower things, but because the minds of men are naturally inclined to truth, then when obstacles are removed, studies are renewed and men at last arrive at the true opinions which previously flourished, but false opinions need not be revived.
Consequently the ancients, supposing that the first body, namely, the heaven, to be of a nature different from the four elements, named the highest place of the world the "aether," thus applying to it a name based on the fact that it always runs for an eternity of time - for thein in Greek is the same as "to run." But Anaxagoras misinterpreted this name, attributing it to fire, as though the heavenly body were fiery - for aether in in Greek is the same as "to burn," which is proper to fire. But that a heavenly body is not of fire is plain from what has been said above [in L. 4].
Lecture 8: Only five simple bodies required. No motion contrary to circular.
78. After showing the necessity of some body besides the four elements, the Philosopher here shows that the integrity of the universe requires no other body besides these five.
First he shows his proposition; Secondly, he proves something he had assumed, at 79.
He says therefore first  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  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;
Secondly, he shows that there is no contrariety in the parts of circular motion, at 10.83; Thirdly, that there is no contrariety between complete circular motions, i.e., of one to another, at 89.
81. He says therefore first  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  he shows that there is no contrariety in the parts of circular motion.
First he excludes contrariety from the parts of this motion; Secondly, he shows that contrariety of parts would not be enough for contrariety of the whole, at 88.
Regarding the first he does three things:
First he shows that there is not contrariety in the parts of circular motion if the parts are taken according to diverse portions of the circle which are designated between two points; Secondly, he shows that there is not contrariety in the parts of circular motion, if the parts are taken according to the same semicircle, at 85; Thirdly, if the parts are taken according to two semicircles, at 87.
He says therefore first  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  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  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  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 , 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 through another. But in circular motion the same aspect of contrariety is present in all parts of the circle. Therefore it will be necessary that both move through the same places of the circle. And so of necessity contrary circular motions would have to obstruct one another.
Secondly, in the two cases the aspect is different - for in the case of a body that is being naturally moved with a straight motion, just as it is naturally apt to be corrupted, so it is naturally apt to be obstructed. Hence, if it is obstructed, this is no more in vain than if it be corrupted. But a body circularly moved is naturally incorruptible: hence it is not apt to be obstructed. Hence if there were in nature something to impede it, that impediment would be useless.
93. Likewise, it can be objected about the motion of the planets which are moved with their own motions from west to east, which seems to be contrary to the motion of the firmament which in its diurnal motion is from east to west.
But it must be said that such motions have indeed a certain mutual diversity which somehow designates the diverse natures of the mobiles. But, for three reasons, there is no contrariety:
First, this is so, because diversity of this kind is not based on contrary termini but on contrary ways of reaching the same terminus: for example, because the firmament is moved from the eastern point to the western point through the upper hemisphere, and returns to the eastern point through the lower hemisphere, while a planet is moved from a western point to the east through another hemisphere.
But to be moved to the same end by diverse routes does not make for contrariety of actions or passions, but pertains to the diverse order of the motions and mobiles - for what reaches its terminus by a nobler route is nobler, just as a better doctor is one who induces health by a more efficacious way. Hence the first motion of the firmament is nobler than the second motion, i.e., that of the planets, just as the supreme orb is nobler. Wherefore, the orbs of the planets are moved with the motion of the first orb without their being impeded from their own proper motions.
The second reason is that, although each motion is over the same center, nevertheless they are over other and other poles; hence they are not contrary.
The third reason is that they are not in the same circle, but the motions of the planets are in the lower circles.
But contrariety must be reckoned with respect to the same distance, as is plain in straight motions, the contrariety of which consists in the distance from the center to the circumference.
Lecture 9: The need for treating of the infinity of the universe.
94. After explaining the perfection of the universe and pointing out the parts that make it complete, the Philosopher here begins to inquire into its infinity, because, as is said in Physics III, some have attributed the notion of "perfect" to the infinite.
Now something can be said to be infinite in three ways: in one way with respect to magnitude; in another with respect to number, and in a third way with respect to duration.
First, then, he asks whether the universe is infinite according to magnitude; Secondly, whether according to multitude, i.e., whether there is just one world, or an infinitude, or many (L. 16); Thirdly, whether it is infinite in duration, as though ever existing (L.22).
About the first he does two things:
First he speaks in a prefatory manner about his intention; Secondly, he carries out his proposal, at 99.
About the first he does three things:
First he states his intention; Secondly, he assigns the reason for his intention, at 96; Thirdly, he decides upon a method of treatment, at 98.
95. He says therefore first  that because it is now clear with respect to the foregoing, namely, that there is no motion contrary to circular motion, and as to the other things mentioned, we must now direct our attention to what remains. And first we must inquire whether there exists any body infinite in act with respect to magnitude, as very many of the early philosophers thought (i.e., all those who posited one material principle, such as fire, or air, or water, or something intermediate); or whether it is impossible that there be a body infinite in act, as was proved in Physics III, supposing, however, that there is no body other than the four elements, according to the opinion of others. Since, however, he has just now proved that there is another body besides the four elements, he therefore repeats this consideration in order that the search for the truth may be more universal.
Then at  he gives a reason for his intention, from the diversity that happens on account of the aforesaid position. And first he mentions this con sequent diversity, and says that it makes no slight difference to the speculation of truth in natural philosophy whether things are this way or that, i.e., whether or not there exists a body that is infinite according to magnitude. Rather, it does make a difference with respect to the whole universe and every natural consideration. For what has just been said, was in the past, and will be in the future, the source of almost all the contradictions between those who have put forth anything about the whole nature of things. For those who posited one infinite principle assumed that all things come to be by a kind of separation from that principle: thus, on account of the infinitude of that principle, they said that the generation of things does not fail. It is as though someone said that from an infinite mass of dough, loaves of bread could be made ad infinitum. But those who posited finite principles said that things come to be ad infinitum through a reciprocal commingling and separating of the elements.
97. Then at  he assigns the cause why such diversity follows from this: it is because one who makes a slight departure from the truth in his principles gets 10,000 times farther from the truth as he goes on. This is so because all things that follow depend on their principles. This is especially clear in an error at the crossroads: for one who at the beginning is only a slight distance from the right road gets very far away from it later on. And he gives, as an example of what he is talking about, the case of those who posited a smallest magnitude, as Democritus posited indivisible bodies. By thus introducing a least quantity, he overthrew the most important propositions of mathematics - for example, that any given line can be cut into two halves. The reason for this effect is that a principle, though small in stature, is nevertheless great in power, just as from a small seed a mammoth tree is produced. Hence it is that what is small in the beginning becomes multiplied in the end, because it reaches unto all that to which the power of the principle extends, whether this be true or false. Now the infinite has the nature of a principle (for all who have spoken about the infinite considered it a principle, as was said in Physics III); besides, the infinite has the greatest force with respect to quantity, because it exceeds every given quantity. If, therefore, a principle which is the least in quantity makes a great difference in what follows from it, then much more is this so of the infinite, which is outstanding not only in virtue of being a principle but also in quantity. Consequently, it is neither inappropriate nor unreasonable that a remarkable difference should follow in natural science from the assumption that some body is infinite. And therefore it must be discussed by resuming our consideration from the principle which we accepted above about the difference between simple and composite bodies, he points out what order must be followed, and says that of necessity every body is either a member of the simple group or of the composite group. Consequently an infinite body must be one or the other. Again, it is plain that if simple bodies are finite in multitude and magnitude, so too must composite body be - for a composite body has as much quantity as the quantity of the simple bodies of which it is composed. However, it has been shown above that simple bodies are finite in multitude, because there is no body other than the ones mentioned. It remains, therefore, to see whether any of the simple bodies is infinite in magnitude, or whether this is impossible. And this we shall show by first arguing from the first body, i.e., the one that is moved circularly; then we shall consider the remaining bodies, namely, those moved with a straight motion.
99. Then at  he shows that there is not an infinite body:
First with reasons proper to the individual bodies; Secondly, with three general reasons applying to all, (L. 13).
As to the first he does two things:
First he proves the proposition as to the body moved circularly; Secondly, as to the bodies moved with a straight motion, (L. 12).
About the first he does two things:
First he proposes his intention and says that it is plain from what will be said that every circularly moved body must be finite (for this is the first of bodies). Then at  he proves his proposition with six arguments, the first of which is this: If any body is infinite, it cannot be moved circularly; but the first body is moved circularly. Therefore, it is not infinite.
First, then, he proves the conditional proposition as follows: If a circularly moved body is infinite, then the straight lines proceeding from its center are infinite, for they are extended as far as the quantity of the body. But the distance between the infinite lines is infinite.
Now someone might say that even if there are infinite lines from the center, yet the distance between them is finite, because every distance is measured according to a straight line, and a finite line can be drawn between two such radii, for example, very close to the center. But it is clear that beyond that line a greater straight line can be drawn between the lines we first mentioned. And therefore he says that he is not speaking of the distance that such lines measure, but that that distance is infinite which is measured by a line beyond which no greater line can be taken, and which touches each of the first lines.
That this distance is infinite he proves in two ways. First, because every such distance between any finite lines proceeding from the center is finite; for the ends of the lines proceeding from the center and of the finite line measuring the greatest distance between them must coincide.
Secondly, he proves the same point because it is possible, if the distance between two measured lines proceeding from the center is given, to take another distance which is greater, just as it is possible to take a number greater than a given number. Hence, just as the infinite is in numbers, so is it in this distance under discussion.
From this he argues as follows: The infinite cannot be traversed, as was proved in Physics VI. But if a body be infinite, the distance between the lines proceeding from the center must be infinite, as was proved. But in order that circular motion occur, one line proceeding from the center must reach the position of another. Consequently, it could never happen that anything be moved circularly.
101. Secondly, at  he proves in two ways the destruction of the consequent. First, because it is evident to sense that the heavens are moved circularly; secondly, because it was proved above by reason that circular motion belongs to some body. What remains, therefore, is that it is impossible for the circularly moved body to be infinite.
Lecture 10: The second and third reasons proving the circularly moved body not infinite
102. After setting forth the first argument showing that the circularly moved body is not infinite, on the ground that the distance between two lines proceeding from the center will be infinite and untraversable, the Philosopher now presents the second argument, based on the fact that imaginary lines described in an infinite body, or in its place, cannot intersect.
And in this argument he sets forth the principle that if a finite time is subtracted from a finite time, the remainder will be finite, because part of a finite cannot be infinite; otherwise the whole would be less than the part. And if that remainder of time is finite, it has a beginning, for we say a time is said to be finite, if it has a beginning and end. But it has been demonstrated in Physics VI that time and motion and mobile follow one another in respect to being finite or infinite. Hence if the time which measures a starting out or motion is finite and has a beginning, then the motion must be finite and have a beginning, and so also must be the magnitude moved. And just as this applies to celestial motion, so too to other motions and mobiles.
Having set these things down as principles, he proceeds to demonstrate the proposition. Suppose that, from the center of an infinite body A, there is drawn the line AGE, which is infinite in one direction, namely, toward E; and let that line be revolving with the motion of the whole body, and that, with respect to the point G its motion describes a circle. Let us imagine also, in that imaginary space in which the infinite body is revolved, a certain immobile fixed line BB which does not cross the center but is nevertheless infinite. If then, as has been said, the line AGE by its motion describes a circle from G, i.e., whose radius is AG, it will turn out that the line AGE in making a revolution will cross the entire line BB in finite time. For it is manifest that the radius of a circle cannot be revolved in its circuit without covering or cutting successively the whole fixed immobile line imagined to be in the circle and not passing through the center. And that it is in a finite time that the line drawn from the center cuts the infinite line not passing through the center is manifest from the fact that the whole time in which the heaven is moved is finite, as is evident to our senses. Consequently, a part of that time which is subtracted from the whole time is finite, namely, the time in which AGE falls on line BB. Or rather it follows that that time is finite in which that cutting line is moved to the line which is cut; and this is the time that must be subtracted from all of finite time, so that the remaining time has a beginning, in keeping with the principle enunciated above. It follows, therefore, that the time in which AGE begins to cut BB has a beginning. However, this is impossible, because since it cuts one part before another, then if there is a beginning of the time in which it begins to cut, there would be a beginning in the infinite line, and that is contrary to the notion of infinite.
In this way, then, it is plain that an infinite body cannot be revolved circularly. Hence if the world is infinite, it follows that it is not moved circularly. However, we do observe that the firmament is moved circularly. Hence it is not infinite.
104. The third argument isgiven at  and is based on the infinity of the whole body which is posited as moving circularly. He says, therefore, that also from what follows it is manifestly impossible for an infinite body to be moved circularly. As a premise he says that if A and B are two finite lines so that A is in motion beside B which is stationary, it follows of necessity that as A moves along it departs from the stationary line B, and conversely that the stationary line B is separated from the moved line A. The reason for this is that each of them overlaps the other to the same extent. But now if both are moved in contrary directions, the lines will separate more quickly. If, however, one is in motion beside the other which is stationary, they will separate more slowly - provided, of course, that they have the same speeds when both are in a separating motion and when one alone is in motion. The reason for presenting this as a premise is that the time in which one line traverses the other is the same as that in which the other traverses it.
After manifesting this point with respect to finite lines, he applies it to the infinite lines he is discussing. And he says that it is manifestly impossible for an infinite line to be traversed by a finite line in finite time. Hence it remains that a finite line traverses an infinite line in infinite time, and this was shown previously in the treatise on motion, i.e., in Physics VI. But as appears from what has been said about finite lines, it makes no difference whether it is a finite line being moved through an infinite or an infinite line being moved over a finite, for when an infinite line is being moved through a finite line, the same reasoning holds, whether the finite line is being moved or not. However, it is manifest that if the finite line is being moved as well as the infinite, each traverses the other.
Hence it is manifest that even if the finite line is not being moved, being traversed by the infinite line will be similar to traversing it.
But because he had said that the situation is similar whether the other is moved or not, he now shows wherein there could be a difference: if each of the lines is being moved in a contrary direction, they will separate more swiftly. But this must be understood if the speed is the same, as was said above. For sometimes nothing prevents the line which is being moved next to a stationary one from traversing it more quickly than if it were moved next to a line in contrary motion; for example, when the two lines in contrary motion would have a slow motion, while the one in motion next to the stationary one would have a swift motion. Accordingly, it is no obstacle, so far as the argument is concerned, that the infinite line be moved next to a stationary finite line - since it happens that the moving line A more slowly traverses the moving line B than if the latter were not in motion, provided, of course, that in this second case, while B is stationary, line A is being moved more swiftly.
105. Thus, having shown that it makes no difference whether the infinite line is moved next to a stationary finite line, or whether the finite line is moved against the infinite, he argues from this that if the time in which the finite line traverses the infinite line is infinite, the consequence is that the time in which an infinite line is moved through a finite line is also infinite. Accordingly, it is plainly impossible for an entire infinite body to be moved through an entire infinite space - in which we imagine its motion to occur - in finite time: because if the infinite were moved even through the slightest finite space, it would follow that the time would be infinite, for it has been proved that the infinite is moved through the finite in infinite time, just as the finite through the infinite. But we observe that the heaven circles all its space in finite time. Hence it is manifest that it traverses some finite line in finite time, for example, the line containing within itself the whole circle described about its center, namely, the line AB. Now this would not happen if it were infinite. It is impossible, therefore, that the circularly moved body be infinite.
Lecture 11: Three additional reasons why the body moving circularly cannot be infinite.
106. Having given three arguments to prove that the body in circular motion cannot be infinite, he now gives a fourth  which is this. It is impossible for a line having an end to be infinite, unless it have an end at one extremity and be infinite at the other. The same is true of a surface: if it has an end at one part, it is not infinite at that part. But when it is limited from every part, it is in no sense infinite. Thus, it is clear that no tetragon, i.e., square, is infinite, nor is a circle which is a plane figure, nor a sphere which is a solid - for these are names of figures, and a figure is something bounded by a terminus or by termini. Thus it is clear that no figured plane is infinite. If, therefore, neither a sphere nor a square nor a circle is infinite, it is clear that there cannot be circular motion that is infinite. For just as there can be no circular motion unless there is a circle, so, if there is no infinite circle, there cannot be an infinite circular motion. But if an infinite body were moved circularly, there would have to be a circular motion that is infinite. Therefore, it is not possible for an infinite body to be moved circularly.
107. The fifth argument is presented at  and it is this. Let G be the center of the infinite body in circular motion. Then through this center let a line AB be drawn which is infinite in both directions; then draw another infinite line not passing through the center but perpendicular to BA at E. Imagine these two lines as stationary in the space in which the infinite body is moved circularly. Draw a third line DG from the center and let it be infinite in the direction of D - for in the direction G it has to be finite. Finally suppose that this third line is in motion by the motion of the body. Because the line E is infinite, it will never be separated from it, because it cannot traverse it, since it is infinite; rather it will always maintain itself as GE, i.e., it will always touch or cut line E just as it cut it in the beginning when it began to be moved - for example, when the line GD was superimposed on the line BA and cut the line E perpendicularly at point E. For leaving this position it will cut the line E at the point Z, and so it will cut point after point in it; yet it will never be able to be entirely separated from it. It is impossible, however, for the circular motion to be completed, unless the line GD departs from the line E: because before the circular motion can be completed, the line GD will have to traverse that part of the whole that is opposite to the line E. And so it is plain that an infinite line can in no way traverse the circle in such a way that the entire circular motion be completed. Consequently, an infinite body cannot be moved circularly.
The sixth argument is presented at  and he forms his argument in two ways. The first is by leading to an impossibility as follows: Suppose, as you say, that the heaven is infinite. Now it is manifest to us that it moves around in finite time - for we see that its revolution is completed in 24 hours. Therefore it will follow that the infinite is traversed in a finite time. This is so because it is necessary to imagine a space equal to the heaven in which the heaven is moved. But we imagine this space as stationary: thus there will have to be an infinite space in which the heaven is moved and a heavenly body equal to the space in which it is moved, because the body must be equal to the space in which it is. If, then, the infinite heaven has been circularly moved in finite time, the consequence is that it traversed the infinite in finite time. But this is impossible, i.e., to traverse the infinite in finite time, as was proved in Physics VI. It is, therefore, impossible for an infinite body to be moved circularly.
109. Then at  he forms his argument conversely in order to make it an ostensive proof. And he says that we can say conversely that, from the fact that the time in which the heaven is revolved is finite (as is plain to the senses), it follows that the magnitude traversed is finite. Now it is plain that the space traversed is equal to the body traversing it. Therefore, the body which is moved circularly is finite.
Therefore he concludes in summary that it is plain that the body which is being moved circularly is not unterminated, i.e., it does not lack a terminus as though it were devoid of shape. Consequently, it is not infinite, but has an ending.
Lecture 12: Various reasons why a body moving in a straight line is not infinite.
110. After showing that the circularly moved body is not infinite, the Philosopher here shows the same for the body moved with a straight motion, whether from the middle [center] or to the middle [center].
First he proposes what he intends and says that just as the circularly moved body cannot be infinite, so, too, the body which is moved with a straight motion, whether from the middle or to the middle, cannot be infinite; Secondly, he shows the proposition, at 111, and this: First on the part of the places which are proper to such bodies; Secondly, on the part of heaviness and lightness, through which such bodies are moved to their proper places, at 114.
About the first he does two things:
First he shows the proposition as to the extreme bodies, of which one is absolutely heavy, namely, earth, and the other absolutely light, namely, fire, at 111; Secondly, as to the intermediate bodies, which are air and water, 112.
111. He proposes therefore first  that motions of the kind that are up and down, or from the middle and to the middle, are contrary motions. For contrary local motions are ones to contrary places, as has been said, and as was shown in Physics V. It remains, therefore, that the proper places to which such bodies are carried are contrary.
Now, we could have at once concluded from this that such places are determinate: for contraries are things which are most distant; but places that are the greatest distance apart are determinate, for the greatest distance is such that none is greater, whereas in infinites a greater and greater distance is always possible. Hence if the places were infinite, contrariety of places would cease. However, Aristotle passes over this argument as manifest and proceeds by another tack. For it is true that if one contrary is determinate, so too the other, because contraries are members of one genus. But the middle of the world which is the midway terminus of a downward motion is determinate - for from whatever part of the heavens something is moved downward (which exists under the upper part facing the heavens) it can travel no farther in its journey from the heavens than the middle: for if it should go beyond the middle, it would now get closer to the heavens and thus would be moved upward. Accordingly, it is clear that the middle place is determinate. It is likewise clear from the aforesaid that the middle having been determined, i.e., the downward place, then the upward place is also necessarily determinate, because they are contraries. And if both are determinate, then the bodies which are apt to be in these places must be finite. Hence it is clear that the extreme bodies subject to straight motion are finite.
112. Then at 13  he shows the same thing for the intermediate bodies.
First he proposes a conditional, namely, that if up and down are determinate, the intermediate place must be determinate. And he proves this with two arguments, the first of which is this: If, when the extremes were determinate, the intermediate should not be determinate, it would follow that a motion from one extreme to the other would be infinite, on account of the infinite intermediate. But that this is impossible has been shown previously in the discussion about circular motion where it was pointed out that motion through the infinite cannot be completed. Consequently, the intermediate place is determinate. Thus, since the thing in place is commensurate with the place, it follows that the body actually existing in this place or that can exist there, is finite.
113. He gives the second argument at  and it is this: A body that is moved up or down can reach the state of existing in such a place. This is clear from the fact that such a body is apt to be moved from the middle or to the middle, i.e., it has a natural inclination to this or that place. Now a natural inclination cannot be in vain, because God and nature do nothing in vain, as was had above. Consequently whatever is naturally moved upward or downward can have its own motion terminated so as to be up or down. But this could not be, if the intermediate place were infinite. Consequently, the intermediate place is finite; so, too, is the body existing in it.
Therefore in summary he concludes from the foregoing that it is clear that no body can be infinite.
Then at  he shows that there is no infinite heavy or light body by an argument based on heaviness or lightness. It is this: If a heavy or a light body be infinite, then heaviness or lightness must be infinite. But this is impossible. Therefore, the first supposition [of non-infinity] must be true.
With respect to this, then, he does two things:
First he proves the conditional proposition, at 114; Secondly, he proves the destruction of the consequent, at 119.
As to the first he does two things:
First he proposes what he intends and says: If there is no infinite heaviness, none of these, i.e., no heavy body, will be infinite, for the heaviness of an infinite body must be infinite. And the same goes for a light body - for if the heaviness of a heavy body is infinite, the lightness, too, must be infinite, if one supposes some light body carried upward to be infinite.
Secondly, at  he proves what he had supposed.
First he presents the proof, at 115; Secondly, he dismisses some objections, at 116.
First, then, he presents an argument leading to an impossibility  and it is this: If what was said above is not true, then suppose that the heaviness of an infinite body is finite, and let AB be the infinite body and G its finite heaviness. From this infinite body take away a finite part which is the magnitude BD, which is necessarily much less than the whole infinite body. Now the heaviness of this smaller body is less; consequently, the heaviness of ]3D is less than the heaviness G which is the heaviness of the whole infinite body body. Let this lesser heaviness be E. Now let E be a measure of the greater but finite heaviness G - for example, E is a third part of the whole G. Now take from the infinite body a part to be added to the finite body BD, accord ing to the proportion by which G exceeds E, and let this exceeding body be BZ, in such a way that the ratio between the lesser heaviness E to the greater G is the same as that between the body BD and body BZ. That this can be done is proved by the fact that from an infinite body can be taken away as much as is needed, since, as is said in Ph sics]II, the infinite is that whose quantity is such that, as much as is taken away, there always remains something beyond to be taken.
Therefore, with these presuppositions, he now argues to three incompatible consequences. First he reasons in this manner. The ratio of heavy magnitudes is the same as the ratio of their heaviness - for we see that a larger body has more heaviness and a smaller body less. But the ratio of E to G, i.e., of the lesser to the more heavy is the same as that of BD to BZ, i.e., of the smaller body to the larger, was was supposed. Therefore, since E is the heaviness of BD, it will follow that G is the heaviness of the body BZ. But G was assumed to be the heaviness of the whole infinite body. Therefore the numerical value of the heaviness of the finite and of the infinite body will be the same. But this is unacceptable, because it will follow that the whole remainder of the infinity body will have no heaviness. Therefore, the first is impossible, namely, that the heaviness of an infinite body be finite.
Secondly, at  he leads to another unacceptable consequence. For since it is possible to take from an infinite body as much as one wishes, as has been said, let yet another part be taken from the infinite body and added to the body BZ. And let hgl be one finite body greater than the finite body BZ. Now the heaviness of larger body is greater, as was said above. Therefore, the heaviness of BI is greater than the heaviness G which was proved to be the heaviness of the body BZ. But it was assumed in the beginning that G was the heaviness of the whole infinite body. Therefore, the heaviness of a finite body will be greater than that of an infinite body. This is impossible. Therefore, the first is impossible, namely, that the heaviness of an infinite body be finite.
Thirdly, at  he leads to the third incompatibility, namely, that the heaviness of unequal magnitudes would be the same. This clearly follows from the foregoing, because the infinite is not equal to the finite, because it is greater than it. Hence, since these conclusions are impossible, it is impossible for the heaviness of an infinite body to be finite.
116. Then at 14  he dismisses two objections against the foregoing argument:
First, the first; Secondly, the second, at 118.
The first objection is that he had supposed in the preceding argument that the lesser heaviness E is a numerical measure of the greater heaviness G. Now this can be denied, for not every greater is measured by a smaller, because a line of 3 hands' length is not a measure of a line of hands' length.
But the Philosopher excludes this objection in two ways. First, because it makes no difference, so far as the conclusion is concerned, whether the two heavinesses, namely, the greater and the less, in question are commensurate, so that the less measures the greater, or not, for the same reasoning holds in either case. For it is necessary that the lesser, taken a certain number of times, either measure or exceed the greater: for example, the product of 2 taken 3 times measures 6 [for 3 times 2 equals 6], while it does not measure 5, but exceeds it. Accordingly, if the heaviness E does not measure the heaviness G, let E be such that 3 times E measures a heaviness greater than the heaviness G. And so in this case the same impossibility as before results, because, if we had taken from the infinite body three magnitudes of quantity BD, the heaviness of such a magnitude will be 3 times that of heaviness E, which is assumed to be the heaviness of the body BD. But a heaviness that is 3 times E is greater (according to our assumptions) than the heaviness G which is the heaviness of the infinite body. Wherefore, the same impossibility as before follows, namely, that the heaviness of a finite body exceed that of an infinite body.
117. Secondly, at  he excludes the same objection in another way. And he says that we can assume in the demonstration under discussion that the two heavinesses are commensurate, in such a way that E is commensurate to G. For above we first took from the magnitude a part BD whose heaviness we called E, and this was grounds for saying that E does not measure G. But it makes no difference, so far as the proposition is concerned, whether we begin with the heaviness (by taking any part we want) or with the magnitude so taken. For example, we might begin with the heaviness and take a part of it, namely, E, which measures the whole, namely, G, and then we can take from the infinite body a part BD whose heaviness is E and proceed as above, so that as the heaviness E is to heaviness G, so the magnitude BD is to a greater magnitude BZ. This we can do, because, since the magnitude of the whole body is infinite, as much can be taken from it as we please. By taking the parts of the heaviness and of the magnitude in this way, it will follow that the magnitudes and the heavinesses will be mutually commensurate, i.e., the lesser heaviness will measure the greater, and the smaller magnitude the larger.
118. Then at  he excludes a second objection. For he had supposed that the magnitudes are proportional to the heavinesses. Now this is true in bodies having similar parts, for where there is like heaviness throughout the whole, there must be more heaviness in the larger part. But in a body of unlike parts this is not necessarily so, because the heaviness of a smaller part could be greater than that of a larger part, just as a smaller part of earth is heavier than a larger part of water.
This objection he therefore excludes by saying that it makes no difference to the aforesaid demonstration whether the infinite magnitude in question is homogeneous, i.e., of similar parts, or heterogeneous, i.e., of dissimilar parts. For from an infinite body we can take as much as we wish, either adding or subtracting; hence we can assume certain parts to have a heaviness equal to the part taken first, namely BD, whether the parts taken later are larger or smaller in magnitude. For if we should first take BD as having 3 Cubits and having heaviness E, and then take many other parts, for example, of 10 cubits, to make an equal heaviness, it will be the same as if we had taken another equal part having the same heaviness. Consequently, the same impossibility follows.
Therefore, having presented his demonstration and excluded the objections thereto, he concludes from the foregoing that the heaviness of an infinite body cannot be finite. Therefore, it must be infinite. If then, as he will immediately prove, infinite heaviness is impossible, the consequence is that it is impossible for there to be an infinite body.
119. Then at  he proves what he had supposed, namely, that there cannot be infinite heaviness. And in this he destroys the consequent of the previously posited conditional. Concerning this he does two things:
First he proposes what he intends, and says that we must still show from what will follow that infinite heaviness is impossible. Secondly, at  he proves the proposition.
First he lays down certain presuppositions;
Secondly, he uses them in his argument, at 121; Thirdly, he excludes an objection, at 122.
First, then, he presents three suppositions. The first of these  is that, if such a heaviness, i.e., of some certain amount, moves so much, i.e., throughout a definite magnitude of space, in this time, i.e., a determined time, then necessarily as much and more, i.e., a greater heaviness that has as much and something more than a lesser, will move through as great a magnitude in less time - for by as much as a moving power is stronger, by that much is its motion swifter. Consequently, it will traverse an equal distance in less time, as is proved in Physics VI.
The second supposition is at  and follows from the first. For if a greater heaviness moves in less time, then the analogy, i.e., proportion between heavinesses and times is the same, but inversely so, i.e., if half the heaviness moves something in a certain amount of time, then double that amount moves in its half, i.e., in half the time.
The third supposition, at , states that a finite heaviness moves through a finite magnitude of space in a certain finite time.
121. Then at  he argues from these premises. If an infinite heaviness should exist, two contradictories will follow: namely, that something would be moved according to it, and not moved. That it would be moved follows, indeed, from the first supposition - for if a certain heaviness moves in a certain amount of time, a greater will move more swiftly, i.e., in less time. Since, therefore, an infinite weight is greater than a finite, then, if a finite moves through a definite distance in a definite time, as the third supposition says, the consequence is that an infinite heaviness will move as much and more, i.e., either through a greater distance in the same time, or through an equal distance in less time, which is to be moved more swiftly.
But that something is not moved according to infinite heaviness follows from the second supposition. For a thing must be moved in proportion to the greatness of the weight in inverse proportion, i.e., the greater weight will move in less time. But there can be no proportion between an infinite and a finite weight, although there is a proportion between less time and more time, provided the time is finite. Consequently, there can be no time given in which an infinite weight can move, but something will always be able to be taken as moved in less time than the time in which an infinite weight moves, for there can be taken no least time in which an infinite weight can move in the sense that it would be impossible for something to be moved in a lesser time. Now the reason why no such least time can be assumed is that since all time is divisible, as is any continuum, it is always possible to take a time smaller than any given time, i.e., a part of the divided time. Consequently, an infinite heaviness cannot exist.
122. Then at  he excludes an objection. For someone could say that there is a least time, namely, an indivisible time, in which the infinite heaviness moves, just as some have posited certain minimum and indivisible magnitudes. But he excludes this objection:
First he shows that an impossibility follows upon assuming a minimum time and that an infinite heaviness moves in that time; Secondly, he shows that the same impossibility follows if an infinite heaviness should move in any amount of time, even not the minimum, at 123.
He says therefore first  that even if there were a minimum time, it would not help in escaping the impossibility that follows from the assumption of infinite heaviness. For although we posit a minimum time, we do not exclude a ratio of this time to a greater time, for this minimum time will be a part of a greater time, just as one is part of number, which allows it to have a ratio to every number. But an indivisible which is not part of a divisible has no proportion to it, just as a point is not a part of a line and therefore there is no proportion of a point to a line. So let us take another heaviness which is finite and as much heavier proportionally than the finite heaviness that moved something in more time than the infinite heaviness, as the minimum time of the infinite heaviness is less than the greater time of the other finite heaviness. For example, let E be the infinite heaviness, and B the minimum time in which it moves, and let G be the finite heaviness that moves something in more time D than time B. Then let F be the other heaviness which is greater than G in the proportion that D exceeds B. Then, since the lessening of time corresponds to the increasing of heaviness, it will follow that the heaviness F, which is finite, will move something in the same time as the infinite heaviness. But this is impossible.
It should be noted that, just as there is no proportion between a point and a line, so there is none between an instant and time, because an instant is not a part of time. Consequently, the only way Aristotle's argument could be destroyed, would be by positing that an infinite heaviness should move in an instant. But that is impossible, as was proved in Physics VI, namely, that any motion should occur in an instant.
Then at  he shows that the same impossibility follows in whatever time we assume an infinite heaviness to move. And this is what he says, namely, that if an infinite heaviness should move in any finite time whatever, even though it be not the minimum, it is still necessary that in that time a finite heaviness could move through a finite distance - for one will be able to take an excess of weight corresponding to a lessening of the time, as was said above. Consequently, it is clearly impossible for an infinite heaviness to exist; and the same argument holds for lightness.
Lecture 13: A natural and demonstrative argument showing no natural body can be infinite
124. After showing that no single natural body is infinite, the Philosopher here shows by a general argument that no natural body is infinite - for a proof through a common medium causes more perfect science.
About this, therefore, he does two things:
First he mentions his intention; Secondly, he proves his proposition, at 128.
125. As to the first he does three things:
First he shows (as if summarizing) what has been previously said. And he states that for those who think according to the lines already laid down, it is clear that there is no infinite body, "by a detailed consideration of the various cases," i.e., on account of the reasons applied to the individual parts of the universe, namely, to the body that is moved circularly and to the bodies that are move upward or downward.
126. Secondly, at [801 he shows what immediately remains to be said. And he says that the same thing can be clear if someone looks at it universally, i.e., by a common medium. And this is in addition to those general arguments given in the book of the Physics where the common principles of all natural bodies were discussed - for in Physics III is a universal treatment of the infinite, as to how it exists and how not, it being shown there that the infinite exists in potency but not in act. Now, however, the infinite has to be treated in another way, by showing universally that no sensible body can be infinite in act.
127. Thirdly, at  he shows what must be determined immediately after these questions. And he says that after proving what has been proposed, our aim will be to inquire (on the supposition that the whole body of the universe is not infinite) whether the whole body is of such size that there can be made from it several heavens, i.e., many worlds. For perhaps someone could wonder whether it is possible that, just as our world is established about us, there might be other worlds, i.e., more than one though not an infinitude. But before dealing with that question, we shall speak universally of the infinite and show that from common reasons no body is infinite.
128. Then at  he proves the proposition:
First by natural demonstrative arguments; Secondly, by logical arguments (L. 15).
Now I call "demonstrative" and "natural" those arguments that are taken from the proper principles of natural science, whose consideration concerns motion, and action and passion which reside in motion, as is said in Physics III.
First, therefore, at  he shows that no body is infinite from the side of local motion, which is the first and most common of motions; Secondly, universally on the part of action and passion (L. 14).
As to the first he does two things:
First he presents certain divisions, at 129; Secondly, he examines the members individually, at 130.
129. Therefore he first  presents three divisions: The first of these is that every body must be either finite or infinite. If it is finite, we have our proposition; but if it is infinite, a second division remains, namely, that it be a heterogeneous whole, i.e., having dissimilar parts, as an animal body which is composed of flesh, bones and sinews; or it is a homogeneous whole, i.e., having like parts, such as water, each part of which is water. But if it is a whole of dissimilar parts, a third division remains, namely, whether the species of the parts of such a body are finite or infinite in number. If it is proved that they are not infinite, nor again finite, and further that no body of parts that are alike is infinite, it will have been proved universally that no body is infinite.
130. Then at  he pursues each member. He does three things about this:
First he shows that it is not possible in a body of unlike parts for the species of its parts to be infinite; Secondly, that it is not possible for an infinite body of unlike parts to be such that the species of its parts be finite, at 131; Thirdly, he shows that there can be no infinite body having parts that are alike, at 135.
He says therefore first  that it is plainly not possible for an infinite body to be constituted from an infinite species of parts, so long as one is loyal to the "first hypotheses," i.e., the previously made suppositions that there are only three species of simple motion. For if the first motions, i.e. the simple motions, are finite, then the species of simple bodies must be finite, for the motion of a simple body is itself simple, as was had above. But it was also held above that simple motions are finite: for there are three, namely, motion to the middle, motion from the middle, and motion around the middle. Now the reason why simple bodies are finite, if simple motions are finite, is that every natural body must have its own proper motion - but if there were an infinite species of bodies, while the number of motions was finite, there would have to be some species of bodies without motions, which is impossible.
Consequently, from the fact that simple motions are finite, it is sufficiently proved that the species of simple bodies are finite. Now it is from simple bodies that mixed bodies are composed. Hence, if there were a whole having unlike parts and composed of an infinite species of mixed bodies, the species of the first components would still have to be finite - though it does not even seem possible that mixtures from finite elements should be infinitely diversified. Neither can any compound body be called a mixture of all like parts, because, even if its quantitative parts be specifically alike, as each part of a stone is stone, yet its essential parts are specifically diverse, for the substance of a mixed body is composed of simple bodies.
131. Then at  he shows that it is impossible to have an infinite body of unlike parts, the species of which parts are finite. And he arrives at this with four arguments. The first is that, if a body of unlike parts is infinite and composed of parts that are finite with respect to species, each of the parts would have to be infinite in magnitude. For example, if a mixed body were infinite and composed of elements that were finite, air would have to be infinite, and so would the water and the fire. But this is impossible, because, since each of these is either heavy or light, it would follow according to what was previously said that its heaviness or lightness would be infinite. But it has been proved that no heaviness or lightness can be infinite. Therefore, it is not possible for an infinite body of unlike parts to be composed of a finite species of parts.
However, someone could object that it does not follow from this argument that each of the parts is infinite: for it could be possible for the whole to be infinite if one part were infinite in magnitude and the others finite. But this was rejected in Physics III - for if one part were infinite it would consume the other finite parts on account of its excessive power. Likewise it can be said that even in that case the same impossibility will follow, namely, that there would be an infinite heaviness or lightness. And therefore Aristotle was not concerned with it.
132. The second argument is presented at . For if the parts of an infinite whole were infinite in magnitude, their places would have to be infinite in magnitude, because places are necessarily equal to the things in them. But motion is measured according to the magnitude of the place into which it passes, as is proved in Physics VI. Therefore it follows that the motions of all these parts would be infinite. But this is impossible, if what we supposed above is true, namely, that nothing can be moved downward infinitely, nor upward either - because "down" is determinate, since it is the middle, and for the same reason "up" is determinate (for if one contrary is determinate, so is the other).
And he also proves this by what is common to all motions. For in the transmutation according to substance, we see that it is impossible for a thing to become what it cannot be, as, for example, there cannot be made a rational ass, since it is impossible for an ass to be such. And the same goes for a motion in "such," i.e., with respect to quality and for a motion in "so much," i.e,, with respect to quantity, and for a motion in "where," i.e., with respect to place. For if it is impossible for something black ever to have been made white, as a raven, it is impossible for it ever to become white. And if it is impossible for anything to be a foot long, as an ant, it is impossible for it to be moving toward that; and if it is impossible for something to be in Egypt, as the Danube, it is impossible for it to be moving thither. The reason for this is that nature does nothing in vain. But it would be in vain for a thing to be tending to what is impossible for it to reach. Consequently, it is impossible for a thing to be locally moved to a place where it cannot arrive. But it is impossible to traverse an infinite place. If, therefore, places were infinite, there would be no motion. But since that is impossible, it cannot be that the parts of an infinite body of unlike parts be infinite in magnitude.
133. He presents the third argument at . For someone could say that there is no infinite continuous unit, but that there are yet certain parts, disconnected and not continued, which are infinite, as Democritus posited infinite indivisible bodies, and as Anaxagoras posited infinite parts all similar to each other.
But Aristotle says that this position leads no less to an impossibility: for if infinite parts of fire are not joined, there is nothing to prevent all of them from joining and thus making one infinite fire from all of them.
134. The fourth argument he presents at . For when something is said to be infinite, the term should be taken according to its proper meaning. For example, if we say that a line is infinite, we understand it to be infinite in length; while, if we say that a surface is infinite, we understand that it is infinite in length and width. But a body stretches in every direction, because it has three dimensions, as was said above. Consequently, if a body is said to be infinite, it will have to be infinite in every direction, and so in no direction will there be anything outside it. It is therefore not possible that there be in an infinite body many things that are unlike, each of which is infinite, for according to the foregoing it is not possible for there to be a number of infinites.
135. Then at  he shows that there cannot be an infinite body having like parts - and this with two arguments. The first of these is that every natural body must have some local motion; but there is no other except those mentioned above, one of which is around the middle, another from the middle, and a third to the middle. It follows, therefore, that it has one of these. But this is impossible - for if it moves upward or downward, it will be heavy or light, and, consequently, its heaviness or lightness will be infinite, which is impossible according to what has gone before. Likewise it cannot be moved circularly, because it is impossible for the infinite to turn in a circle. For there is no difference between saying this and saying that the heaven is infinite - which is impossible, as was proved above. Therefore a whole infinite body cannot be homogeneous.
136. The second argument is set down at  and it follows from the common notion of local motion. For if there should be an infinite body of parts that are alike, it follows that it cannot be moved at all. If it is moved, it will be moved either according to nature, or by compulsion. But if it has a compulsory motion, then must be a motion natural to it, because a compulsory motion is contrary to a natural motion, as was had above. But if there is a motion natural to it, it follows that there is a place equal to it, into which it is naturally moved, for natural motion belongs to what is moved to its own place. This, however, is impossible, because it would follow that there would be two infinite corporeal places, which is as impossible as that there should be two infinite bodies, for, just as an infinite body is infinite in every direction, so too is an infinite place. Therefore it is not possible for an infinite body to be moved. But if every natural body is moved, it therefore follows that no natural body is infinite.
It should be noted that this argument applies only to straight motion, for what is moved circularly does not change its place as to subject, but only in conception, as is proved in Physics VI. But that an infinite body cannot be moved circularly has already been proved above in many ways.
Lecture 14: No sensible body is infinite - from action and passion, which follow upon motion.
137. After showing that a sensible body is not infinite with a reason based on local motion, the Philosopher here shows the same thing with a reason based on action and passion, which follow upon every motion. Concerning this he does two things:
First he demonstrates the proposition; Secondly, he excludes an objection, at 144.
138. With regard to the first, he gives the following argument: No infinite body has active or passive power or both; but every sensible body has active or passive power or both. Therefore, no sensible body is infinite.
Then with regard to this he does two things:
First he proves the major premise; Secondly, he presents the minor and conclusion, at 143.
About the first he does two things:
First he proposes what he intends and says that it is clear from what will be said that not only is it impossible for something infinite to be moved locally but that universally it is impossible for something infinite to be acted upon or to act upon a finite body.
Secondly, at  he proves his proposition.
First he shows that the infinite is not acted upon by the finite, 139;
Secondly, that the finite is not acted upon by the infinite, at 140;
Thirdly, that the infinite is not acted upon by the infinite, at 142.
139. He says therefore first  that if an infinite body is acted upon by a finite, let A be an infinite body and B a finite body and, since every motion occurs in time, let G be the time in which B moves or A has been moved. If, therefore, we posit that A, which is the infinite body, is altered by B, which is the finite body, say heated or carried, i.e., moved locally, or affected in any other way, e.g., cooled or moistened, or moved in any way, in time G, let us take one part of the mover B, i.e., a part D (and it makes no difference, so far as the proposition is concerned, if D be some other body less than B). Now it is clear that a smaller body moves a smaller mobile in an equal time (supposing, of course, that there is in the smaller body less power - which must be said, if it is a body of like parts - hence the lesser power moves a smaller body in an equal time). Therefore, let E be a body which is altered or any other way moved by D in the time G, taking E as a part of the infinite whole A. But since both D and B are finite, and since any two finite bodies are mutually proportionate, then, according to the ratio of D to B, let there be taken the proportion of E to any other larger finite body, for example, F.
Having posited these preliminaries, he makes some suppositions. The first of these is that an altering cause which is equal in magnitude and power will alter an equal body in equal time. A second is that a smaller altering body will alter a smaller in equal time, the result being that one moved body will be less than the other moved body according to a ratio of somethinggEeater to something less, i.e., in the same proportion that the larger moving body exceeds the smaller moving body.
From these preliminaries, therefore, he concludes that the infinite cannot be moved by any finite in any time. For something less than the infinite will in an equal time be moved by that body which is less than the body moving the infinite; in other words, E, which is less than A, will be moved by D, which is less than B, according to our suppositions. But what is "analogous" to E, i.e., in the same ratio to E, as B to D, is finite, for it cannot be said that A, which is infinite, is to E, as B is to D, because the infinite has no proportion to the finite. Now on the assumption that something finite is to E as B is to D, then commutatively B is to that finite, as D is to E. But D moves E in time G; therefore B moves the finite in time G. But G was the time in which it was supposed that B moved the infinite whole A. Therefore the finite will move a finite and an infinite in the same time.
140. Then at  he proves that an infinite body does not move a finite body in any time.
First he shows that it does not move it in finite time; Secondly, not in infinite time, at 141.
He says therefore first  that neither will an infinite body move a finite body in any time, namely, determinate time. For if the contrary should be the case, let Abe the infinite body, and B or BZ the finite body moved by it, and G the time in which it is being moved. Let D be a finite part of the infinite body A. And because a lesser moves a smaller in equal time, then a finite body D in time G moves Z, a body smaller than B, but a part of B. Now because the whole BZ is proportionate to Z, let it be taken that the whole BZ is to Z, as E is to D, each of which is part of the infinite. Therefore, commutatively, E is to BZ in the same proportion as D is to Z. But D moves Z in time G; therefore E will move BZ in time G. But in time G, BZ was being moved by the infinite body A. It follows, therefore, than an infinite and a finite are altering or somehow moving one and the mane mobile in the same amount of time. But this is impossible - for it was supposed above that a greater mover moves an equal mobile in less time, because it moves more swiftly. Consequently, it is impossible for the finite to be moved by the infinite in time G; and the same follows no matter what finite time is taken. Hence there is no finite time possible in which the infinite moves the finite.
141. Then at 1 he shows that this cannot occur in infinite time. For it is not possible that in an infinite time something shall have moved or shall have been moved - because infinite time has no end, whereas every action or passion does have an end, for nothing acts or is acted upon except in order to reach some end. What remains, therefore, is that an infinite does not move a finite in infinite time.
142. Then at  he proves that the infinite does not move the infinite. And he says that an infinite cannot undergo anything from an infinite with respect to any species of motion at all. Otherwise let A be the infinite body which is acting, and B the infinite body acted upon, and DG the time in which B underwent something from A, and let E be a part of the infinite mobile B. Now, since the entire B has been modified by A in the entire time DG, it is clear that E, which is part of B, was not being moved in this whole time. For we must suppose that a smaller mobile is moved in less time by the same mover - for to the extent that a mobile is more overcome by a mover, the more swiftly is it moved by it. So let E, which is less than B, be moved by A in a time D which is part of the whole time GD. Now D is proportionate to GD, since both are finite. Let us assume, therefore, that E has the same ratio to some larger part of the infinite mobile as D has to GD. Then that finite mobile greater than E must be moved by A in time GD, for we must suppose that a larger and a smaller mobile are moved in more and less time when the same mover is acting, in such a way that the division of the mobiles corresponds to the ratio of the times. Since, therefore, the ratio of that finite to E equals the ratio of the entire time DG to D, then commutatively, we must say that the ratio of the entire time DG to that larger finite mobile is as the ratio of time D to mobile E. But E is moved by A in time D; therefore, that greater finite mobile will be moved by A in time DG. Hence the finite and the infinite will be moved in the same amount of time - which is impossible. And the same impossibility follows whatever be the finite time assumed. Consequently it is impossible for an infinite to be moved by an infinite in finite time.
It remains, therefore, that if it is moved, it is moved in infinite time. But that, too, is impossible, as was proved above, because infinite time has no end, but everything which is moved has an end to its motion - for although the whole motion of the heaven does not have an end, one revolution does. It is therefore plain that the infinite has neither active nor passive power.
143. Then at , assuming the minor premise, he draws the conclusion and says that every sensible body has active or passive power or both.. He says "sensible body" here to differentiate from "mathematical body," so that the former means every natural body which, as such, is apt to cause motion or be moved. Thus he concludes that it is impossible for a sensible body to be infinite.
144. Then at  he excludes a certain objection - for someone could say that there is outside the heavens an "intelligible body" which is infinite.
And he says that all bodies in place are sensible. For they are not mathematical bodies, because these do not have place except in a metaphorical sense, as is said in On Generation I. Now place is not needed except for motion, as is said in Physics IV, and only sensible and natural bodies are subject to motion - for mathematical things are outside of motion. Consequently, it is plain that all bodies in place are sensible.
From this he concludes that there is not an infinite body outside the heaven; and indeed, more universally, that no body exists outside the heaven, either absolutely, i.e., namely, an infinite body, or in a certain respect (or up to a certain point), i.e., a finite body. Since bodies are either finite or infinite, it follows that no body at all exists outside the heaven. For if you should say that this body is intellectual, it will follow that it is in a place on account of your assuming that it is outside the heaven - because "outside" and "within" imply place. Consequently, it follows that if there is a body outside the heaven, then, whether it is finite or infinite, it is sensible, since there is no sensible body which does not exist in a place -for even the heaven is somehow in place, as is plain from Physics IV.
So it is manifest according to these words that no intelligible body, finite or infinite, is outside the heaven, because "outside of" signifies place, and nothing is in place except a sensible body. It is also manifest that no infinite sensible body exists outside the heavens, for it was shown above that no sensible body is infinite. But the fact that no sensible finite body exists outside the heavens he does not prove here but supposes it, unless perhaps it is proved by the fact that every sensible body is in a place, and all places are contained within the heavens and determined by the three local motions mentioned above, namely, those around the middle, from the middle, and to the middle.
Lecture 15: Logical reasons why no body is infinite.
145. After showing universally with Physical reasons, i.e., with arguments taken from facts proper to natural science, that there is no infinite body, the Philosopher here shows the same thing with logical reasons, i.e., arguments taken from certain common principles or from things that are probable but not necessary. And this is what he says : "It is," i.e., it is possible, "to try," to prove the proposition "more from reason, i.e., more according to the logical mode, "thus," i.e., according to the following arguments. Hence another MS is more plain when it says: "One can argue more logically [i.e., dialectically] as follows."
First he proves the proposition about an infinite continuous body; Secondly, about one that is not continuous, at 150.
146. Concerning the first he does two things:
First he shows that an infinite body of like parts cannot be moved circularly. This he proves on the ground that there is neither a middle nor a boundary in an infinite body. But circular motion is around a middle, as was had above. Therefore....
147. Secondly, with three arguments he shows that it is not possible for such an infinite body to be moved with a straight motion. The first of these is this: Every body that is moved with a straight motion can be moved naturally and through force. Now what is moved by force has a place to which it is forcefully moved, and whatever is moved naturally has a place to which it is moved naturally. But every place is equal to the thing in place. Consequently, it will follow that there are two places as large as the infinite body, to one of which it is forcefully moved, and to the other of which naturally. But it is no more possible that there be two infinite places than that there be two infinite bodies, as was had prove. It remains, therefore, that no natural body is infinite.
Both of these reasons are called "logical," because they proceed from what occurs to an infinite body as infinite; whether it be mathematical or natural, namely, to have no middle and nothing equal to it outside of it. Above he posited similar statements, not, however, as principal premises but as assumptions used to manifest other things.
148. The second argument is given at , and is as follows: Whether it be said that an infinite body is moved naturally with straight motion or by force, in either case there must be posited a power moving the infinite body - for it was shown in Physics VII and VIII that whatever is moved is moved by another, not only in things that are moved by force (where the principle is more evident), but also in things that are moved naturally, as heavy and light bodies are moved by the generator [or agent producing them], or by whatever removes an obstacle. But since the stronger is not moved by the weaker, it is impossible for an infinite, whose power is infinite, to be moved by the finite power of some mover. Hence it remains that the power of the mover must be infinite.
But it is manifest that if a power is infinite, it will belong to an infinite thing; conversely, if a body is infinite, its power must be infinite. Therefore, if an infinite body is being moved, then the body moving it must be infinite. For it was proved "in the discussion on motion," i.e., in Physics VIII, that no finite thing has infinite virtue, and that no infinite has finite virtue. Consequently, it is plain that if an infinite body is being moved with straight motion, it must be being moved by an infinite body.
Now, if we assume that this infinite body can be moved both according to nature and beside its nature, it will likewise happen, with respect to each motion, that there are two infinites, namely, one that moves thus, i.e., causes natural or compulsory motion, and one that is moved. But this is impossible, namely, that there be two infinite bodies, as was proved above. Therefore, it is not possible for an infinite body to be moved with a straight motion.
This argument is called "logical" because it proceeds from a common property of an infinite body, namely, that it does not have outside it another body equal to it.
It can be concluded from this argument not only that there would be two infin ites but more still. For if the infinite body is moved naturally, the body moving it naturally will be infinite; and because it can be moved by force, the body that moves it by force will be infinite. Thus there will be three infinites.
Again, since a motion which is compulsory for one thing, is natural to another, as was stated above, it will follow, too, that there is another infinite body that is moved naturally in the aforesaid way by an infinite power.
149. The third argument he gives at . And this argument is adduced in order to exclude an objection to the preceding argument. For someone could say that an infinite body is naturally moved not by some other body but by itself, as animals are said to move themselves. Consequently, it will not follow that there are two infinite bodies, as the preceding argument concluded.
And therefore he proposes that it is necessary to say, that if there is an infinite body, whatever moves it is distinct from it. For if it moved itself, it would be animate - for it is proper to animals to move themselves. Consequently if the infinite body should move itself, it will be an infinite animal. But this does not seem possible, because every animal has a definite shape and a definite ratio between its parts and the whole, which factors do not belong to an infinite. Consequently, it cannot be said that the infinite moves itself. But if it be said that something else moves it, it will follow that there are two infinites, namely, the mover and the moved. And from this it follows that they differ in kind and in power: because the mover is related to the mobile as act to potency. But this is impossible, as was previously shown.
150. Then at  he shows that there is no infinite which is non-continuous but distinguished by the interposition of voids, as Democritus and Leucippus posited. This he proves with three arguments. With regard to the first he says that if an infinite is not one continuous whole but is, as Democritus and Leucippus maintain, distinguished by an intermediate void - for they posited that the indivisible bodies cannot be mutually joined without an intervening void - then according to their opinion it follows that for all of them there is one motion. For they said that those infinite indivisible bodies are determined, i.e., mutually distinguished, only by their shape, namely, insofar as one is pyramidal, another spherical, another cubic, and so on. Yet they say that all of them are one with respect to their nature, as if, for example, someone said that each of them in isolation had the nature of gold. But if the nature of all is one, then, necessarily, all have one and the same motion in spite of their being the minimal parts of bodies - because the motion of the whole and of the part is the same, as is the motion of the whole earth and one clod, and of all fire and one spark.
Therefore, if all are of the same nature and have the same motion, then all are either moved downward as though having gravity - and thus there will be no body that is absolutely light, since all bodies are said to be composed of these; or else all are moved upward, as though having lightness, and thus no body will be heavy - which is impossible.
151. The second argument, given at , is this: Every heavy body is moved to the middle and every light body to the boundary. If, therefore, some or each of the aforesaid indivisible bodies had heaviness or lightness, it would follow that there would be a boundary and a center of that whole space contained by the indivisible bodies and the intermediate voids. But that is impossible, since all that space is infinite. It remains, therefore, that this position is impossible.
152. And since this argument effectively destroys the infinite howsoever assumed, i.e., whether continuous or non-continuous, he therefore presents this same argument in a more universal way at . And he says that we can say universally that where there is no middle and no extreme boundary, there is no "up," which is the boundary, and no "down," which is the middle. And if these are removed, there is no place where bodies can be moved with straight motion; for they are moved upward or downward. But if place is removed, there will be no motion - for whatever is moved, must be moved either according to its nature or outside its nature, and this is judged by places that are proper and alien - for natural motions are those in which bodies are moved to their proper places, while compulsory motions are those in which they are moved to alien places. But this is impossible, namely, that motion be taken away from bodies. Therefore, it is impossible to posit an infinite.
153. The third argument is given at . And he says that the place to which something is moved outside its nature, or in which it rests outside its nature, must be according to nature for something else which is moved to it naturally and rests in it naturally. Ahd this becomes credible by induction: for earth is moved upward outside its nature but fire according to nature; conversely, fire is moved downward outside its nature but earth according to nature. Now we observe certain things being moved downward and others upward. If the things being moved upward are moved outside their nature, we will be obliged to say that there are other things which are moved upward according to nature; likewise, if the things being moved downward are assumed to be moved outside their nature, it is necessary to posit other things that are moved downward according to nature. Hence not all things have heaviness and not all have lightness according to the foregoing position, but those naturally moved downward have heaviness, while those naturally moved upward do not have it.
Finally in summary he concludes  that it is manifest from the foregoing that there is no infinite body at all, i.e., no infinite that is continuous and none that is distinguished by intervals of void.
And these last arguments are called "logical," because they proceed. from probabilities not yet completely proved.
Lecture 16: Two arguments for one universe, taken from lower bodies.
154. After showing that the universe is not infinite in magnitude, the Philosopher here shows that there are not numerically many worlds, much less an infinitude of them.
First he mentions his intention; Secondly, he pursues his proposition, at 155.
He says therefore first  that because it has been proved that the body of the whole universe is not infinite, there remains for us to say that it is not possible that there be many heavens, i.e., many worlds: for we had already mentioned above that this was to be discussed.
It should be noted that above the Philosopher mentioned that outside the heavens there is no body either finite or infinite; from which it follows that there is not another world besides it, for that would put a body outside the heavens. Consequently, if it were sufficiently proved above that outside the heavens there exists no body either finite or infinite, nothing would remain to be proved. But if someone does not consider that it was proved for bodies universally, namely, that it is impossible for any of them to be outside the world, but considers that the argument given above refers only to bodies assumed infinite, then, according to this, it still remains to be seen whether it is possible that there be many heavens, i.e., many worlds.
155. Then at  he proves his proposition:
First he shows that there is but one world; Secondly, he inquires whether it is possible that there be many worlds (L. 19).
As to the first he does two things:
First he shows that there is only one world and takes his argument from the lower bodies, of which everyone supposed the world to consist, at 156; Secondly, he shows the same with a general argument based on both the lower and the celestial bodies (L. 18).
About the first he does two things:
First he adduces arguments to prove his proposition; Secondly, he proves something he had presupposed (L. 17).
With regard to the first he gives three arguments:
The second one begins at 159; The third one in Lecture 17.
156. Regarding the first he does two things:
First he presents three suppositions. The first is that all bodies rest and are moved both according to nature and according to compulsion. This of course is true in lower bodies which, since they can be generated and corrupted, can not only be transmuted from their species by the power of a stronger agent, but can be removed from their place by a violent motion or by violent rest. But in celestial bodies, since they are incorruptible, nothing can be violent and outside their nature.
The second supposition is that in whatever place certain bodies remain according to nature and not through compulsion, they are moved thither by nature, and into whatever place things are carried by nature they naturally rest there. And the same is to be said about violence: in whatever place things rest through violence, they are carried to that place by violence; conversely, if they are carried to a place through violence, they are at rest there through violence. The reason for this supposition is that since rest in a place is the end of local motion, the motion must be proportionate to the rest, just as the end is proportionate to the means.
The third supposition is that if any change of place is accomplished by violence to a body, the contrary change is according to nature for that body, as is plain from what was said above.
157. Secondly, at  from these suppositions he argues to his proposition. First on the part of motion. For if there are two worlds, there must be earth in both. Therefore the earth in that other world will be moved to the middle of this world either by nature or by compulsion. If by the latter, we shall have to say, according to the third supposition, that the contrary change of place, i.e., from this world to the middle of that world is natural to it. And this is plainly false, since earth is never naturally moved from the middle of this world. Therefore, the first is also false, namely, that there is more than one world.
158. Secondly, at  he argues to the same on the part of rest. For just as it is plain that the nature of earth does not allow being moved naturally from the middle of this world, so, too, the nature of earth has this quality, that it be naturally at rest in the middle of this world. If then earth brought here from that world remains here not by violence but by nature, it follows, according to the second supposition, that it will be brought from that middle to here according to nature. And this is so because there is but one motion, or one change of place, that is according to nature for earth; hence both motions cannot be natural to earth, namely, from that middle to this or from this to that.
Then at  he presents a second argument which excludes a certain defect which someone can claim in the first argument: for someone could answer to the first that the earth in that world is different in nature from that in this world.
First, then, Aristotle dismisses this at 160; Secondly, from this he argues to his proposition, at 162; Thirdly, he excludes an objection, at 163.
He shows that the earth in the other world is of the same nature as that of this world:
First with an argument taken on the part of the world, at Secondly, with one based on motion, at 161.
160. He says therefore first  that if the several worlds posited are of a like nature, they must be composed of the same bodies; further, each of those bodies must have the same virtue as the body of this world. Consequently, fire and earth must have the same virtue in each of those worlds, and the same goes for the intermediate bodies, air and water. For if the bodies that are there in another world are spoken of equivocally in relation to the bodies that exist among us in this world and are not according to the same "idea," i.e., not of the same species, the consequence will be that the entire world consisting of such bodies will be only equivocally called a world. For wholes that are composed of parts diverse in species are themselves diverse. But this does not seem to be the intention of those who posit many worlds; rather they use the word "world" univocally. Hence it follows according to their intention that the bodies in these different worlds possess the same virtue. And thus it is manifest that even in those worlds, just as in this, some one of the bodies constituting the world is apt to be moved from the middle, which belongs to fire, and some other to the middle, which belongs to earth, if it is true that all fire is akin in species to all other fire in whatever world it exists, just as the various parts of fire in this world are of one species. And the same holds for the other bodies.
161. Then at  he shows the same thing with an argument taken from motion. And he says that it is manifestly necessary that things be as we have said concerning the uniformity of the bodies which are in the various worlds; and this from the suppositions which are assumed with respect to motions. And he calls "suppositions" the statements which he uses to show the proposition, because here they are being assumed as principles, although some of them have been previously proved. Now one of the suppositions is that motions are finite, i.e., determinate with respect to species; for there are not infinite species of simple notions, but three only, as was proved above. A second supposition is that each of the elements is described in terms of having a natural tendency toward some one of the motions; as earth is described as heavy on account of its tendency to downward motion, and fire light on account of its aptitude for upward motion.
Hence, since the species of motion are determinate, the same specific motions must exist in every world. And because each of the elements is described with respect to some motion, it is further necessary that the elements are specifically the same everywhere, i.e., in each world.
162. Then at  from these premises he argues to the proposition. For if the bodies in every world are of the same species, and we see that all the parts of earth in this world are carried to the middle of this world, and all parts of fire to its boundary, then the consequence will be that also all the parts of earth in any other world are moved to the middle of this world, and all the parts of fire in any other world to the boundary of this world. But this is impossible. For if this should happen, the earth in another world would have to be carried upward in its own world and fire in that world would have to be carried to its middle. Similarly, the earth in this world would be naturally carried from the center of this world to the center of that world.
And this must follow on account of the disposition of the worlds which have such a position that the middle of one world is at a distance from the middle of another; consequently, earth cannot be moved to the middle of another world without leaving the middle of its own world and moving to the boundary, which is to be moved upward. Likewise, because the boundaries of various worlds have different positions, then if fire is to be carried to the boundary of another world, it must leave the boundary of its own world, which is to be moved downward in its own world. But all these things are untenable - for either we must posit that the natures of the simple bodies are not the same in the several worlds (which was disproved above), or, if we say that they are of the same nature and wish to avoid the aforesaid inconsistencies which follow upon a diversity of middles and boundaries, we must admit but one middle to which all heavy bodies, wherever they are, are moved, and one boundary to which are moved all light things wherever they be. On this assumption, it is impossible that there be many worlds, because one middle and one boundary imply one circle or sphere.
163. Then at  he excludes an objection, since someone could say that the bodies in another world are not moved to the center and boundaries of this world on account of the distance.
But he rejects this and says that it is unreasonable to accept the postulate that the natures of simple bodies vary on the ground of their being more or less distant from their places, so as to be moved to their places when they are near but not when they are far away. For it does not seem to make any difference to the nature of the body whether it is this far or that far from its place, because mathematical differences do not vary the nature. For it is according to reason that the closer a body gets to its place the more swiftly is it moved, but yet the species of its motion and of the mobile are not varied. For a difference in velocity is according to quantity, not according to species, just as is a difference in length.
Lecture 17: A third argument from lower bodies. Natural bodies have determinate places
164. Having given two arguments showing that the world is one, Aristotle here gives a third argument for the same. And this argument adds something which seemed to be lacking in the first argument. For someone could say that it is not inherent in bodies to be naturally moved to certain definite places, or, if they are moved to definite places, those that are one in species and diverse in number are moved to numerically diverse places, which agree in species. But they are not moved to the same numerical place as the first argument supposed. Therefore, in order to make these things sure, Aristotle adduces this third argument. With respect to this he does three things:
First he gives the argument, at165; Secondly, he excludes an objection, at 166; Thirdly, he infers the main conclusion, at 169.
165. He says therefore first  that the above-mentioned bodies must have some motion. For it is manifest that they are moved - this, indeed, is evident to sense and to reason, because such are natural bodies, i.e., bodies which it befits to be moved. Therefore there can remain the doubt whether it is to be said that natural bodies are moved violently with all the motions with which they are moved, even if they are contrary motions - for example, that fire is moved both upward and downward by compulsion. But this is impossible, because what is not apt to be moved at all, i.e., what of its nature has no motion cannot be moved by compulsion. For we say that a thing suffers compulsion if it is removed from its proper inclination by the force of a stronger agent. If, therefore, there is not a natural inclination to certain motions in bodies, compulsion has no place in them - any more than blindness would be attributed to an animal if it had no capacity to see. Consequently, we must admit that those bodies which are parts of the world have a motion according to nature, and among the bodies having a nature, the motion is one. Now motion is called "one" inasmuch as it is to one terminus, as is plain in Physics V. Therefore the motion of each thing belonging to the same species must be to one numerical place: namely, if they are heavy, it is to the middle, which is of this world; if they are light, it is to the boundary which is of this world. And upon this it follows that there is one world.
166. Then at  he excludes an objection. For someone could say that all bodies having the same natural motion are moved to places that are the same in species, but several numerically - since even the singulars, i.e., the individual parts of one natural body, e.g., earth or water, are numerically many but do not differ in species. But oneness of nature in the mobiles that are of the same species does not seem to require any more than that their motion be one in species; in keeping with this, it would seem to be enough if the places at which the motion is terminated were alike in species.
167. But in order to exclude this he says that such an accident, namely, being moved to the same specific places, does not seem to be congruent to one set of parts and not to another (i.e., such that some parts alike in species would be moved to the same numerical place and others to the same specific place); rather it should be congruent to all alike (i.e., either all the parts alike in species be moved to the same numerical place, or all such parts be moved to one place specifically similar but numerically different) - for all such parts are alike in not differing specifically, but each differs from the other numerically. The reason he says this is that the parts of any body, for example, earth, which are in this world are similarly related both to the parts of earth in this world and to the parts in another world, since earth here and earth there are specifically the same. If, then, a part, e.g., of earth, be taken hence, i.e., from this world, it makes no difference whether it is compared to parts in some other world or to parts in this world; rather the relationships are the same in both cases. For the parts of earth in this world and those in some other world do not differ in species. And the same holds for other bodies. But we see that all parts of earth in this world are moved to one numerical place; similarly for other bodies. Therefore all the parts of earth in whatever world they exist are naturally moved to this middle of this world.
168. Therefore the very natural inclination of all heavy bodies to one numerical middle, and of all light bodies to numerically one boundary, manifests the unity of the world. For it cannot be said that in the many worlds, bodies would be arranged according to diverse middles and boundaries, as happens in the case of men in whom the centers and boundaries are numerically diverse but specifically the same. For the nature of man's members or those of any other animal is not determined with respect to their relationship to some place but rather with respect to their relationship to some act; indeed, the position occupied by the parts of animals is in keeping with a suitable operation of the members. But the nature of heavy and of light things is determined to definite places, such that all having the same nature also have numerically one natural inclination to numerically one place.
169. Then at  he infers the principal conclusion. For when a conclusion according to due form is inferred from premises, either the conclusion must be concluded or the premises denied. He concludes, therefore, that either it is necessary to deny these suppositions, i.e., the principles from which he concluded the proposition, or to concede the conclusion, namely, that there is one middle to which all heavy things are moved, and one boundary to which all light things are carried. If this is true, then it is necessary as a consequence that there be one heaven, i.e,, one world and not several, and this on account of the above-given "arguments," i.e., signs and "necessities," i.e., necessary arguments.
170. Then at  he proves something he had assumed, namely, that natural bodies have definite places to which they are naturally borne.
First he proves the proposition; Secondly, he rejects a contrary opinion (L. 18).
About the first he does two things:
First he shows the proposition by a natural argument; Secondly, by a sign, at 173.
As to the first he does three things:
First he proposes what he intends, and says it is clear from other arguments than the foregoing - or even from other motions - that there is a definite place whither earth is naturally borne. And the same is to be said of water and of any of the other bodies.
Secondly, at  he gives his argument, saying that entirely, i.e., universally, this is true, that whatever is moved is changed from something determinate to something determinate: for it is said in Physics I that something white comes to be, not from any non-white at random, but from black. Now these two factors, namely, that from which a motion proceeds, and that into which it is terminated, differ in species - for they are contrary, as is plain in Physics V; but contrariety is a difference respecting form, as is said in Metaphysics X.
He proves what he has said by the fact that every change is finite, as was proved in Physics VI, and also by the facts cited above, namely, that nothing is moved to what it cannot attain; but nothing can attain to the infinite; hence every change must be finite. But if there were not something definite toward which a motion tends and something specifically different from that, at which it begins, the motion would have to be infinite; for there would be no reason why the motion should end here rather than elsewhere, but, for the same reason that it began to be moved thence, it would also begin to be moved hence.
He also explains what was said, by an example. For what is healed is moved from sickness to health; what is increased is moved from small to large. Hence, too, what is carried, i.e., moved according to place, is moved from something definite to something definite, and these are the place at which a motion begins, and the place to which it tends. Consequently, there must be a specific difference between the place from which something is locally moved and the place into which it is naturally borne, just as what is healed does not tend to just anything at random, as though by chance, or solely according to the will of the mover, but to something definite, to which it is inclined by nature. In the same way, therefore, fire and earth and other natural bodies are not borne ad infinitum, i.e., to something indefinite, as Democritus held; rather they are borne to places opposite to those in which they previously found themselves. But "up" is contrary to "down" in the realm of place. It follows, therefore, that "up" and "down" are the termini of the natural motions of simple bodies.
172. Then at [118 he excludes an objection by which someone could object that circular motion does not seem to be from opposite to opposite, but more from the same to the same.
But he says that even circular motion somehow involves opposition of termini. He says "somehow" for two reasons. First, because opposition in circular motion is not found with respect to points designated on the circle insofar as they are points of the circle, but only insofar as they are the extremities of the diameter - on the basis of which a maximum distance is reckoned in a circle, as was said above. Hence he adds: "What are according to the diameter," i.e., the extremities of the diameter, "are opposite." Secondly, because just as the whole spherical body does not change place as to subject but only in conception, although the parts change their place even as to subject, so also, if the entire circular motion is taken, there is no opposition in termini, except conceptually, namely, in the sense that the same [point], from which and to which circular motion is, is taken now as the beginning and now as the end. But if we take the parts of circular motion, we find opposition with respect to a straight line, as has been said. And therefore he adds that there is nothing contrary to a whole revolution. Consequently, it is plain that even in things circularly moved, the change is in a certain way toward things opposite and determinate.
Thus he concludes universally to what he intended, namely, that there is necessarily an end involved in local motion and that a natural body is not moved in infinitum [i.e., to nothing definite], as Democritus posited about the motion of atoms.
173. Then at  he proves the same thing through a sign. This proof he calls an "argument" in the sense that it is, so to speak, conjectural. And he says that the argument for claiming that a natural body is not moved to infinity but to something certain is that earth, the closer it approaches the middle, the more swiftly it is moved (which can be perceived from its greater impetus, namely, as something is more strongly impelled by the heavy in its fall as it nears the terminus of its motion); and the same holds for fire whose motion is swifter, the closer it approaches an upward place. If, therefore, earth or fire were moved to infinity, their speed could increase indefinitely.
And from this he concludes that the heaviness or lightness of a natural body could be increased infinitely. For just as the speed of a heavy body is greater according as the heavy body descends farther ( and a heavy body is swift on account of its heaviness), so, too, an indefinite addition could be made to the speed if an infinite addition were made to heaviness or lightness. But it was shown above that there cannot be an infinite heaviness or lightness, and that nothing can be moved toward what it cannot attain. Consequently, addition of heaviness ad infinitum cannot occur, and, as a result, neither can addition of speed. Hence neither can the motion of natural bodies be tending toward what is infinite.
174. It should be noted that the cause of this accident that earth is moved more swiftly the more it descends was explained by Hipparchus in terms of an agent causing motion by compulsion. The farther the motion is from such an agent the less remains of that agent's power, so that the motion becomes slower. Hence in the beginning, a compulsory motion is intense but in the end it is weakened, until finally the heavy body can no longer be borne upward, but begins to be moved downward due to the small amount of the agent's virtue that remains, which, the less it becomes, so much the swifter becomes the contrary motion.
But this explanation is applicable only to things that are moved naturally after a compulsory motion; it does not apply to things that are moved naturally on account of being generated outside their proper places.
Others explained this phenomenon in terms of the amount of the medium through which the motion takes place (for example, the amount of air): in such a motion, if it is natural, the farther a thing has been moved, the less is the amount remaining - and, therefore, the less is it able to impede a natural motion. But this explanation also, applies no less to compulsory motions than to natural motions, in which, nevertheless, the contrary happens, as will be said below.
Therefore, it must be said with Aristotle that the cause of this phenomenon is that, to the extent that a heavy body descends more, to that extent is its heaviness the more strengthened on account of its proximity to its proper place. And therefore he argues that if the speed increased infinitely, the heaviness, too, would increase indefinitely. And the same holds for lightness.
Lecture 18: Exclusion of the opinion that natural bodies are not moved naturally to determined places. Unity of the world from higher bodies.
175. After showing that natural bodies are by nature moved to definite places, the Philosopher here excludes a contrary opinion.
First he proposes what he intends; Secondly, he proves his proposition, at 176.
Now, since truth is established by excluding falsehood, the Philosopher here induces the exclusion of an error as a certain demonstration of the truth. He says, therefore, that what has been said is manifested by the fact that natural bodies are not borne upward and downward as though moved by some external agent.
By this is to be understood that he rejects an external mover which would move these bodies Eer se after they obtained their specific form. For light things are indeed moved upward, and heavy bodies downward, by the generator inasmuch as it gives them the form upon which such motion follows, but they are moved per accidens, and not per se, by whatever removes an obstacle to their motion. However, some have claimedthat after bodies of this kind have received their form, they need to be moved per se by something extrinsic. It is this claim that the Philosopher rejects here.
Neither should it be said that these bodies are moved by compulsion, which is the opinion of those who said that they are moved by a certain "extrusion," in the sense that one body is displaced by another, stronger, one. For they assumed that there was one motion natural to all bodies, but since some are given momentum by others, it comes to pass that a certain number are moved upward and a certain number downward.
176. Then at  he proves his proposition with three arguments. The first of these is adduced mainly to show that bodies of this kind in their natural motions are not moved by external movers. For it is clear that a motion is slower to the extent that the mover overcomes the mobile less. But a given virtue of the mover overcomes a larger mobile less than a smaller.
If, then, these bodies were moved by an external mover, a greater amount of fire would be moved upward more slowly and a larger amount of earth downward more slowly. But just the opposite happens, for a greater quantity of fire and a greater quantity of earth are moved more swiftly to their places. This gives us to understand that these bodies have the principles of their motion within themselves, and their motive powers are greater according as the bodies are greater, and that is why they are moved more swiftly. Consequently, it is plain that such bodies in their natural motions are not moved by an exterior power but by an intrinsic one, which they have received from their generator.
177. At  he gives a second argument which is adduced mainly to show that motion of these bodies is not through compulsion. For we see that all things moved by compulsion are moved more slowly according as their distance from the mover increases, as is plain in projectiles, whose motion slackens near the end and finally fails. If, then, heavy and light bodies were moved by compulsion as though mutually pushing one another, it would follow that their motion toward their proper places would not be faster but slower in the end. But the contrary of this is plain to our senses.
178. He gives at  the third argument which can regard both. For we see that no body is moved by violence to a place whence it can be removed by violence. For it is because a body is apt to be in a certain place that it can be moved thence by violence; hence it was originally brought there naturally and not by violence. If, therefore, it is assumed that some motions of heavy and light bodies are violent by which they are moved from certain places, it cannot be said that the contrary motions which brought them there are violent. Thus it is not true that all the motions of these bodies are caused by another and by violence.
He concludes from the foregoing, in summary, that speculation on these points will testify to the truth of what has been said.
179. Then at  he shows through the higher bodies which are moved circularly that the world is one:
First in a special way by the higher bodies; Secondly, in a general way by the higher and the lower, at 181.
He says therefore first  that there is still another way of proving that there is but one world, by arguments taken from first philosophy, i.e., by using what has been determined in the Metaphysics, and from what has been shown in Physics VIII, namely, that circular motion is eternal, which, both in this and in other worlds, has a natural necessity. For the Philosopher concluded to the eternity of celestial motion in Physics VIII by considering the order between mobiles and movers, which must be similar in any world, if "world" is taken univocally. Now if celestial motion is eternal, it must be moved by an infinite power, such as cannot exist in a magnitude, as was proved in Physics VIII. Such a power is non-material and consequently numerically one, since it is a form and species only, whereas it is through matter that individuals are multiplied in the same species. Consequently, the power that moves the heavens must be numerically one. Hence the heavens too must be numerically one, and, consequently, the whole world.
180. But someone can say that this argument does not conclude with necessity. For the first mover moves the heaven as that which is desired, as is said in Metaphysics XII. But there is nothing to prevent the same thing from being accidentally many. So it seems that we cannot from the unity of the first mover ccnclude necessarily to the unity of the heavens.
But it must be said that many can desire one thing, but not indeed in an identical way, since an absolute multitude is not joined immediately to one thing that is first; but many things can desire one thing according to a certain order, some being closer and some more remote, the coordination of which to one ultimate objective makes the unity of the world.
Then at  he proves his proposition with an argument taken generally from higher and lower bodies. And he says that even the following consideration will show that it is necessary for the heaven, i.e., the world, to be one. To prove this he assumes that, just as there are three bodily elements, namely, heaven and earth and an intermediate, so there are three places corresponding to them: one is the place about the middle, that of the subsisting body, i.e., the place of the heaviest body which supports all, namely, earth; another is the place which is the highest in altitude, that of the circularly moved body; the third place is intermediate and corresponds to the intermediate body.
With regard to these words it should be noted that Aristotle here reckons the heaven among the elements, although an element is something out of which things are composed, as is said in Metaphysics V.
However the heaven, even though it does not enter into the composition of a mixed body enters into the composition of the whole universe, as being a part of it. Or he is using the word "element" in a wide sense to designate any of the simple bodies which he calls "bodily elements" to distinguish them from prime matter, which, though an element, is not a bodily element, for considered in itself it is without any form.
Secondly, we should consider his statement that there are three places. Now since place is the boundary of a containing body, as is said in Physics IV, it can be clear what the place of the intermediate element is - for it is the surface of the supreme, body containing it. How the first body is in place has been explained in Physics IV. But how the middle [i.e., the center], which seems to be not a container but a contained, is the place of the heavy body seems to offer difficulty.
But it should be said that, as has been said in Physics IV, the surface of the containing body does not have the notion of place because it is the surface of such a body but with respect to the position it has in relation to the first container accordingly, namely, as it is nearer or farther from it. Now the heavy body in its nature is at a maximum distance from the celestial body on account of its materiality; therefore there is due it a place farthest from the first container and nearest to the middle. Consequently the surface containing the heavy body is called its place according to its nearness to the center. Hence he said advisedly that the place located around the middle is the place of the subsisting body.
182. From what has been set forth he goes on to prove his proposition from a light body, just as above he had proceeded from a heavy body. For it is necessary that a light body which is borne upwards be in this intermediate place: because, since every body is in some place, if the light body were not in this intermediate place, it would be outside it. But that is impossible, because outside this intermediate place there is, on the one side, celestial body which has no heaviness or lightness, and on the other side, terrestrial body which has heaviness. Now it cannot be said that there is a place more downward than the place of the body having heaviness, because the place about the middle is proper to it. But from this it is plainly impossible for another world to exist, because some light body would have to be there and thus, if that world were above this world, a light body would exist above the place of the heavens; if that world were below this world, a light body would be below the place of the heavy body - which is impossible.
183. But to this argument someone could object that the light body would be outside this intermediate place not according to nature but outside its nature. To exclude this he adds that not even outside its nature can a light body be outside this intermediate place. Because every place that is outside nature for some body is according to nature for some other body. For neither God nor nature has made any place in vain, i.e., a place in which no body is apt to be. Now, no other body is found in nature except the three mentioned and to which the aforesaid places are deputed, as is plain from what has been said. Hence neither according to nature nor beside nature can a light body exist outside this intermediate place. Consequently, it is impossible that there be many worlds.
Since he had spoken of an intermediate element as if it were one certain body, he adds that later, i.e., in the third and fourth books, he will speak about; the differences in that intermediate. For it is divided into fire, air and water, which is also light in relation to earth.
Finally in summary he concludes that from the foregoing it is manifest about the bodily elements, which and how many they are, and what is the place of each of them and, in general, how many bodily places exist.
Lecture 19: Solution of the argument seeming to justify several worlds.
184. After showing that there is but one world, the Philosopher here shows that it is impossible for there to be many. And it was necessary to prove this, because nothing prevents the possibility of something's being false [now] which can yet be true [later]. Concerning this he does three things:
First he presents an objection which seems to show that it is possible that many worlds exist; Secondly, he answers it, at 194; Thirdly, he proves something he had presupposed in his answer (L. 20).
About the first he does two things:
First he states his intention and his plan of treatment; Secondly, he begins to prove his proposition, at 186.
185. He says therefore first  that after the foregoing, we must still prove that not only is there one world but that it is impossible for there to be more, and further that the world is eternal, so as to be imperishable, i.e. never ceasing to be, and unborn, i.e., never beginning to be, according to his opinion. He states this because the first consideration seems somehow to depend on the second. For if the world were generable and perishable by union and separation, according to friendship and strife, as Empedocles said, many worlds would be possible in the sense that when one had perished another would be generated later, as Empedocles believed. And because the truth is truly known when the difficulties which seem to be contrary to it are solved, therefore the first thing to do is bring forth the difficulties concerning this, i.e., which seem to indicate that there are or can be many worlds - for the solution to this difficulty will confirm the truth.
186. Then at  he presents the argument that could lead one to question whether it is not possible for more than one world to exist. Hence he prefaces the remark that, for those who hold this point of view, i.e., the one coinciding with the argument to follow, it will appear impossible that it, namely, the world, be one and unique, i.e., that there be necessarily just one world. For the following argument does not prove that it is necessary that there be several worlds, which is equivalent to its being impossible that there be but one; rather it proves that it is possible that there be more than one world, which is equivalent to its not being necessary that there be but one. Now in order to show this he induces an argument containing two syllogisms:
The first of these is at 187; The second at 190.
The first syllogism is this: In all sensible things that come to be by art or by nature, the consideration of the form considered in itself is one thing and the consideration of the form insofar as it is in matter is another. But the heaven is a sensible thing having a form in matter. Therefore, the absolute consideration of its form, i.e., as considered universally, is one thing, and the consideration of its form in matter, i.e., as considered in particular is another.
First, therefore, he presents the major, at 187; Secondly, the minor, at 188; Thirdly, he draws the conclusion, at 189.
187. He says therefore first  that in all things that exist and were generated, i.e., made, either by nature or by art, the form considered according to itself is one thing according to our consideration, and the form mixed with matter, i.e., the form taken as joined with matter, is another.
He first explains this by an example in mathematical objects in which it is more evident, because sensible matter does not enter therein. For the species of a sphere is according to our consideration other than the form of the sphere in sensible matter, which is denoted when a sphere is called "golden" or "bronze"; similarly, the form of a circle is one thing, and what is meant by a golden or bronze circle is another. And this is evident, because when we give the quod quid erat ease, i.e., the defining notion, of a circle or a sphere, we make no mention therein of gold or bronze. This implies that to be "golden" or "bronze" does not pertain to their substance [essence], which the definition signifies.
But there seems to be a difficulty in natural things, whose forms cannot exist or be understood without sensible matter, as "snub" cannot exist and be understood without "nose." Natural forms, however, although they cannot be understood without sensible matter in common, can be understood without signed sensible matter, which is the principle of individuation and of singularity. Thus, "foot" cannot be understood without flesh and bones, but it can be understood without this flesh and these bones. And therefore he adds that if we cannot understand and accept in our consideration anything outside the singular, i.e., outside the matter which is included in the notion of the in dividual, namely, as it is signate - because sometimes there is nothing to prevent this from happening (namely, that a form be able to be understood without sensible matter) in the same way that we understood a circle without sensible matter; nevertheless, in natural things, in which forms are not understood without matter, the notions of the thing taken in common and taken in the singular are not the same, any more than the notion of "man" and of "this man" are the same. Thus the essence of "circle" and "this circle," i.e., of the notions defining a circle, and this circle, are different. For the notion of a thing in common is the species, i.e., the notion of the species, but the notion of a particular thing signifies the notion of the species as found in determinate matter, and pertains to the singular.
188. Then at  he presents the minor of his syllogism. And he says that since the heaven, i.e., the world, is something sensible, it must be among the singulars, for every sensible thing exists in matter. For a form not in matter is not sensible but intelligible only - for sensible qualities are characteristics of matter.
189. Then at  he presents the conclusion and says that if the heaven, i.e., the world, belongs among the singulars, as has been shown, its notion as a singular will differ from its notion absolutely, i.e., taken universally the two notions will differ. Consequently, it follows that "this heaven" taken singularly will be different in consideration from "heaven" taken universally, i.e., this latter heaven taken universally will be as a species and form, while the other, namely, that taken singularly, will be as form joined to matter. However, this is not to be taken as implying that in the definition of a natural thing taken universally no matter is mentioned at all, but rather that individual matter is not mentioned.
190. Then at  he presents the second syllogism, as follows: Whatever things have their forms in matter, are, or are able to be, several individuals of one species. But "this heaven" signifies a form in matter, as was said. Therefore, there either are, or can be, many heavens.
Now in regard to this he first presents the major; Secondly, he explains it, at 191; Thirdly, [having taken the minor from the previous syllogism], he draws the conclusion at 192.
He says therefore first  that all things of which there is a form and species, i.e., which are not themselves forms and species, but have forms and species, are either many individuals of one species or many can exist. But things that are themselves forms and subsistent species, as are separated substances, cannot have several members of one species.
191. Then at  he explains the foregoing both according to Plato's opinion and according to his own. And he says that whether there are "species," i.e., separated ideas, as the Platonists assume, then this must happen, i.e., there must be several individuals of one species - because the separated species is posited as the exemplar of a sensible thing and it is possible to make many copies according to one exemplar; or whether no such species exist separately, there can still be several individuals of one species. For we see this happen in all things whose substance (i.e., whose essence, which is signified by the definition) exists in signate matter, namely, that there are several individuals, or even an infinitude of individuals, of one species. The reason for this is that, since signate matter does not enter the notion of the species, the notion of the species can be indifferently verified in this individual matter and in that; consequently, there can be many individuals of one species.
192. Then at  he draws the intended conclusion, namely, that either there are many worlds or many worlds can be made.
Finally he says in summary that from the foregoing someone can conjecture that either there are, or can be, many worlds.
193. But there seems to be a conflict here between Aristotle and Plato. For Plato in the Timaeus proved the oneness of the world from the oneness of the exemplar; but here Aristotle from the oneness of the separated species concludes to the possibility of several worlds.
But two answers can be given to this. First on the part of the exemplar, which, if it is one in such a way that oneness is its essence, then the copy must imitate the exemplar in this oneness. But the first separated exemplar is such. Hence also the world, which is the first copy thereof, must be one. This was the way Plato proceeded in his proof. But if oneness is not of the essence of the exemplar but is outside its essence, then the copy could be like the exemplar in respect to what belongs to its species - for example, in the notion of man or horse - but not in respect to oneness. And it is in this way that Aristotle's reasoning proceeds.
Or it can be answered from the viewpoint of the copy, which is more perfect to the extent that it is more faithful to the exemplar. Therefore, some copies are like one exemplar in respect to oneness of species, but not in respect to numerical oneness. But the heaven, which is a perfect copy, is like its exemplar with respect to numerical oneness.
194. Then at  he solves this objection:
First he gives the solution; Secondly, he explains it, at 195.
He says therefore first that in order to settle the above doubt we must once more consider what was said well and what not well. For if all the premises are true, the conclusion is necessarily true. He says, therefore, that it was correct to say that the notion of form differs, namely, in the case of that which is without matter and in the case of that which is with matter.
This is to be granted as true. Consequently, the first conclusion which is the minor of the second syllogism is conceded. But from this it does not follow of necessity either that there are several worlds, or that there can be several, if it is true that this world consists of all its matter, as is true and as will be proved below. For the major proposition of the second syllogism, namely, that things which have a form in matter can be numerically many in one species, is not true except in things that do not consist of their entire matter.
195. Then at  he explains what he had said with an example.
First he gives the examples; Secondly, he adapts them to his proposition, at 196.
He says therefore first  that what has been said will become clearer from what will be said. For snub-nosedness is curvature in a nose or in flesh; thus flesh is the matter of snub-nosedness. If then from all flesh one flesh were to be made, namely, the flesh of one nose, and snub-nosedness existed in it, nothing else would be snub-nosed nor could be. And the same holds for man, since flesh and bones are the matter of man: if one man were formed from all the flesh and all the bones, so that he could now not be destroyed, there could be no more than one man - but if he could be destroyed, it would be possible, after his corruption, for another man to exist, just as when a box is destroyed, another can be made from the same wood. And the same is true for other things. And the reason for this he assigns, namely, that none of the things whose form is in matter can come into being if the proper matter is not at hand, any more than a house could be made if there were not stones and wood. Consequently, if there were no bones and flesh other than t Rase of which the one man is composed, no other man could come into being but him.
196. Then at  he adapts this to his proposition. And he says it is true that the heaven is a singular thing and one constituted of matter. But it is not constituted out of part of its matter, but out of all of it. And therefore, although there is a difference between the notions of "heaven" and "this heaven," there neither is, nor can be, another heaven, due to the fact that all the matter of heaven is comprehended under this heaven.
197. However, it should be realized that some prove the possibility of many worlds in other ways. In one way, as follows: The world was made by God; but the power of God, since it is infinite, is not limited to this world alone. Therefore it is not reasonable to say that He cannot make yet other worlds.
To this it must be said that if God were to make other worlds, He would make them either like or unlike this world. If entirely alike, they would be in vain - and that conflicts with His wisdom. If unlike, none of them would comprehend in itself every nature of sensible body; consequently no one of them would be perfect, but one perfect world would result from all of them.
In another way, as follows: To the extent that something is more noble, to that extent is its species more powerful. But the world is nobler than any natural thing existing here. Therefore, since the species of a natural thing existing here, for example, of a horse or cow, could perfect many individuals, much more so can the species of the whole world perfect many individuals.
But to this it must be answered that it takes more power to make one perfect than to make several imperfect. Now the single individuals of natural things which exist here are imperfect, because no one of them comprehends within itself the total of what, pertains to its species. But it is in this way that the world is perfect; hence, from that very fact its species is shown to be more powerful.
Thirdly, one objects thus: It is better for the best to be multiplied than for things not so good. But the world is the best. Therefore, it is better to have many worlds than many animals or many plants.
To this it must be said that here it pertains to the goodness of the world to be one, because oneness possesses the aspect of goodness. For we see that through being divided some things lose their proper goodness.
Lecture 20: The universe shown to consist of every natural and sensible body as its matter
198. Having presented the solution brought forward, the Philosopher here proves what he had presupposed, namely, that the world consists of all its matter.
First he tells his intention and order of procedure  and says that in order to complete the preceding solution, we must show that the world consists of every natural and sensible body, which is its matter. But before showing this, it is necessary to explain what is meant by this word "heaven," and in how many senses it is used, so that our question can be answered more clearly.
199. Secondly he proves his proposition:
First he shows the various senses of the word "heaven"; Secondly, he proves the main proposition, at 200.
With regard to the first  he gives three senses of heaven. In one way the heaven is called "the substance of the extreme circulation of the whole," i.e., that which is at the boundary of the whole universe and is moved circularly. And because he had explained the meaning of the word in terms of "substance," whose notion transcends natural philosophy, since it pertains to Metaphysics, he adds another explanation having the same meaning, saying that the heaven is "the natural body whose place is at the extreme circumference of the world," which explanation is more befitting to natural science.
He proves this meaning from the way people speak - since words are to be used in the sense most people use them, as is said in Topics II. For men are more likely to call "heaven" that which is the extreme of the entire world and which is most up, not, indeed, as "up" is taken in natural science, i.e., as being the terminus of the motion of light things (for in this sense nothing is farther "up" than the place to which fire is borne) but as taken according to common parlance, where "up" designates that which is farther from the middle. "Up" also refers to the place of all divine beings (where "divine" signifies not celestial bodies - not all of which are in the outermost sphere - but non-material and incorporeal substances), for it has been said above that all men attribute to God a place that is up.
In a second way "heaven" means not only the outermost sphere but "the whole body continuous with the extreme circumference of the whole universe," i.e., all the spheres of celestial bodies, in which exist the moon and sun and certain of the stars, namely, the other five planets (for the fixed stars are in the supreme sphere according to the opinion of Aristotle, who did not posit another sphere above that of the fixed stars).
And he proves this meaning also on the basis of common parlance: for we say that the sun and moon and other planets exist in the heaven. Now these bodies are said to be continuous with the extreme sphere, because they are alike in nature, i.e., they are imperishable and movable circularly, and not because one continuous body is formed from all of them - for then they could not have several and different motions, a continuum being something whose motion is one, as is said in Metaphysics V.
In a third way "heaven" means "the whole body contained within the extreme circumference," i.e., by the extreme sphere. This, too, he proves from the common use of the word - since we are wont to call the whole world and the totality, i.e., the universe, the "heaven."
It should be noted that "heaven" is here used in these three ways not equivocally but analogically, i.e,, in relation to one first. For it is the supreme sphere that is first and principally called "heaven"; secondly, the other celestial spheres from the continuity they have with the supreme sphere; thirdly, the universe of bodies insofar as they are contained by the extreme sphere.
200. Then at  he proves the proposition.
First he shows that there is no sensible body outside the heaven taken in the third sense, i.e., outside this world; Secondly, he shows that there is not outside it any of the things that are normally consequent upon natural bodies (L. 21).
As to the first he does three things:
First he proposes what he intends; Secondly, he proves his proposition, at 201; Thirdly, he concludes to his main proposition, at 206.
He says therefore first  that whereas "heaven" is said in three ways, we shall be discussing it now in its third sense, where heaven is taken as "the whole contained by the extreme circumference." Concerning this heaven it is necessary that it consist of every sensible and natural body - which is its matter, and thus it consists of all its matter - due to the fact that outside this heaven no body exists, nor can exist.
201. Then at  he proves the proposition.
First he shows that there is no body outside the heaven; Secondly, that none can be there, at 205.
About the first he does two things:
First he presents a division through which he manifests the proposition; Secondly, he excludes each member of the division, at 202.
He says therefore first  that if there is a Physical, i.e., natural, body outside the extreme periphery , i.e., circumference, it has to be either of the number of simple bodies, or of the number of composite bodies. Moreover, it must exist there according to nature, or outside its nature.
202. Then at  he eliminates each member of this division.
First he shows that outside the extreme sphere no simple body exists according to nature. For simple bodies are such that one is moved circularly, one from the middle, and one is moved to the middle and in the middle supports all the others, as was had above. But none of these bodies can exist outside the extreme circumference. For it has been shown above in Physics VI that the circularly moved body does not as to its whole being change its place except in conception. Consequently, it is not possible for that body which is moved circularly to be transferred to a place outside of that in which it exists. But this would follow, if there were a circularly moved body existing outside the extreme circumference as in its natural place. Since the reason that it would be natural to that circularly moved body would also make it natural to the body circularly moved in this world, and every body is naturally borne to its natural place, it would follow, therefore, that that latter circularly moved body would be transferred outside its proper place to another place - which is impossible.
Similarly it is not possible for a light body which is moved from the center to be outside the extreme circumference or for a heavy body which supports the other bodies in the center. For if it is maintained that they exist naturally outside the extreme circumference, such a thing cannot be, since they have other natural places, namely, within the extreme circumference of the whole. For it was shown above that there is one numerical place for all heavy bodies and one for all light bodies. Hence it is not possible that those bodies be naturally outside the extreme circumference of the whole.
And it should be noted that this argument, both as to the body circularly moved, and as to the body moved with straight motion, possesses necessity on account of what was proved above, namely, that there is but one extreme and one middle.
203. Secondly, at  he shows that no simple body is outside the heaven outside its nature. For if it were there in that manner that place would be natural to some other body; for a place outside nature for one body must be according to nature for some other - if a proper body were lacking to a place, that place would exist in vain. But it cannot be said that that place is natural to any body: for it is not natural to a circularly moved body, nor to a light or heavy body. But it has been shown above that there are no other bodies besides these. Consequently, it is plain that no simple body exists outside the heaven, either according to nature or outside nature.
204. Thirdly, at  he proves that there is no mixed body there. For if none of the simple bodies exists there, it follows that no mixed body is. Wherever there is a mixed body, simple bodies must be there, due to the fact that simple bodies are present in the mixed; and a mixed body gets its natural place according to the simple body predominant in it.
205. Then at  he shows that outside the heaven there cannot be any body. Hence he says that it is not possible for a body to come to be outside the heaven. For it would be there either according to nature or outside nature; again, it would be either simple or mixed. But no matter which of these is given, the same situation as above would prevail. For according to the above-stated reasons, it makes no difference whether the question concerns the existence of a body outside the heaven, or the possibility of its coming to be there, since the foregoing arguments conclude both, and since in sempiternal things tb be and to be able to be do not differ, as is said in Physics III.
206. Then at  he draws the conclusion mainly intended. And he says it is manifest from what has been said that outside the heaven no mass of any sort of body exists, nor can exist, since the whole world consists of its entire proper matter and the matter of the world is the sensible natural body.
However, it should be not understood that he wishes to prove that no sensible body exists outside the heaven on the ground that it consists of the totality of its matter; but rather the converse. Nevertheless, he uses that manner of speaking because the two are mutually convertible.
He concludes, therefore, that there are not many worlds at present, nor were there many in the past, nor will there ever be able to be in the future. Rather the heaven is one and unique and perfect in the sense of consisting of all its parts or of its total matter.
Lecture 21: Outside the heaven there is no place, time etc., consequent upon sensible bodies.
After showing that there neither is, nor can be, any sensible body outside the heaven, the Philosopher here shows that outside the heaven there is none of the things that follow upon sensible bodies.
First he proves the proposition; Secondly, he describes the things that do exist outside the heaven, at 213.
About the first he does three things:
First he proposes what he intends; Secondly, he proves the proposition, at 208; Thirdly, he draws the intended conclusion, at 212.
He says therefore first  that with the proof that outside the heaven there is no sensible body, it is also manifest that outside the heaven there is neither place nor void nor time - for these three things were discussed as being concomitants of natural bodies in Physics IV.
208. Then at  he proves the proposition.
First, as to place: In every place it is possible for a body to exist, otherwise it would be in vain. But outside the heaven it is not possible for any body to exist, as was proved. Therefore, outside the heaven there is no place.
Secondly, at  he proves that outside the heaven there is not a void: Those who posit a void define it to be a place in which a body is not existing but can exist. But outside the heaven it is not possible for a body to exist, as has been shown. Therefore, outside the heaven there is not a void.
209. But it should be noted that the Stoics posited an infinite void, in one part of which the world exists. Consequently, according to them, there is a void outside the heaven. They wanted to prove this with the following fantasy: If someone were on the extreme circumference of the heaven, he could either extend his hand beyond or not. If not, then it is being impeded by something existing beyond. The same question will return regarding that thing existing beyond, if anyone could, while on the extremity, reach out his hand beyond. Consequently we must go on infinitely, or come to an extreme body beyond which a man existing there could reach out his hand. In that case it follows that beyond that a body could exist and does not. Hence there will be a void beyond.
To this Alexander responds that the position is impossible. For since the body of the heaven cannot undergo anything, it cannot receive anything extraneous. Hence, if from this impossible assumption, something against the thesis follows, one should pay it no heed.
But this answer does not seem to be sufficient - since the impossibility of this position is not on the part of something outside the heaven but on the part of the heaven itself. But now we are dealing with what is outside the heaven. Hence it is the same argument if the whole universe were the earth, on whose boundary a man could exist. Consequently, we must state otherwise, just as he says, that a man situated on the extreme circumference could not extend his hand beyond, not because of something outside impeding it, but because it is of the very nature of all natural bodies that they be contained within the extreme circumference of the heaven - otherwise the heaven would not be the universe. Hence if there were a body not depending on the body of the heaven as on a container, there would be nothing to prevent it from existing outside the heaven, as in the case of the spiritual substances, as will be said below.
210. But that there is no void outside the heaven Alexander proves on the ground that such a void is either finite or infinite: If finite, then it is terminated somewhere and the same question will return: Could a person extend his hand beyond that? If it is infinite, it will be capable of receiving an infinite body: then either that power of the void will be in vain or it will be necessary to posit an infinite body capable of being received into the void of the infinite.
Likewise, if there is a void outside the world, the world will be related to each part of the void in exactly the same way, because in a void there are no differences. Consequently, this part of the void in which the world exists is not its proper place. Therefore there is no cause why it should remain in this part of the void. But if the world is in motion, it will not be moved to one part rather than to another, because in the void there are no differences. Therefore, it will be moved in every direction; and thus the world will be torn asunder.
211. Thirdly, at  he proves that outside the heaven there is no time. For time is the number of motion, as is plain in Physics IV. But motion cannot exist without a natural body, and a natural body neither exists nor can exist outside the heaven, as has been proved. Therefore, outside the heaven there neither is, nor can be, time.
212. Then at  he draws the conclusion intended, and concludes that it is manifest from the foregoing that outside the whole heaven there is neither place nor void nor time.
213. Then at  he describes what type of things are outside the heaven. About this he does two things:
First he concludes their condition from the foregoing; Secondly, he shows the same from common opinion, at 217.
About the first he does two things:
First he removes from them the condition of things that exist here; Secondly, he describes their proper condition, at 214.
He says therefore first  that because there is no place outside the heaven, it follows that things by nature apt to be there do not exist in place. And Alexander says that this can be understood about the heaven itself, which is not in place as a whole but with respect to its parts, as is proved in Physics IV.
Again, because time does not exist beyond the heaven, it follows that they do not exist in time; consequently, time does not make them grow old. And this, too, according to Alexander, can belong to the heaven, which, indeed, is not in time in the sense that to be in time consists in being measured by some part of time, as is said in Physics IV. Not only do such beings not grow old in time, but no change affects those things which lie "beyond the outermost motion [lationem]," i.e., beyond the local motion of light bodies - for he is accustomed to call rectilinear motion latio.
But it does not seem to be true that no change affects heavenly bodies, since they are moved locally, unless perhaps we limit "change" to one affecting the substance. But this seems to be a forced explanation, since the Philosopher excludes all change universally. Likewise, it cannot be properly said that the heaven is there, i.e., outside the heaven. Consequently, it is better to understand his words as applying to God and separated substances which plainly are not contained by time, nor place, since they are separated from all magnitude and motion. Such substances are said to be "there," i.e., outside the heaven, not as in a place, but as not contained nor included under the containment of bodily things, and as exceeding all of corporeal nature. It is such beings that the expression befits, namely, that they undergo no change; because they lie beyond the extreme motion, namely, that of the farthest sphere, which is ordered as extrinsic to and containing all change.
214. Then at  he explains the qualities of these beings.
First he describes their condition; Secondly, he explains a word he used, at 215; Thirdly, he shows the influence of these beings on others, at 216.
He says therefore first  that those beings which are outside the heaven are unalterable and wholly impassible. They lead the best of lives, inasmuch as their life is not mingled with matter as is the life of corporeal beings. They also have a life that is most self-sufficient, inasmuch as they do not need anything in order to conserve their life or to perform the works of life, They have a life, too, which is not temporal but in total eternity.
Now, among the qualities here listed some can be attributed to heavenly bodies - for example, that they are impassible and unalterable. But the other two cannot belong to them, even if they are alive. For they do not have the best life, since their life would be one resulting from the union of a soul to a celestial body; neither do they have a most self-sufficient life, since they attain their good through motion, as will be said in Book II.
215. Then at  he explains the word "eternal" which he had used. And he says that the ancients pronounced this word as divine, i.e., as befitting divine things. Now this word has two meanings.
In one way it is used in a qualified sense as meaning the eternity or age [saeculum] of a thing: for in Greek the same word signifies both. He says, therefore, that the eternity or age of a thing is called an end, i.e., a certain terminal measure which contains the time of any thing's life, in such a way that no time of the life belonging to the thing according to nature exists outside that end or measure. It is like saying that the span of 100 years is the "age" or "eternity" of a man.
In another way "eternity" is used in an absolute sense as comprehending and containing all duration. And this is what he says, namely, that according to the same notion, eternity is called the end of the entire heaven, i.e., it is the span containing the entire duration of the heaven, i.e., the span of all of time. In this sense, eternity refers to a certain perfection which contains all time and the entire infinitude of duration - not as though this eternity is stretched out according to the succession of past and future, as in the case of any span of time, because such succession follows upon motion, whereas the things he described as having life in eternity are completely immobile, but this eternity is a whole existing all at once and comprehending all time and all infinitude. (The Greek word [in English "aeon"] is derived from the words for "always existing".) Such an end, which is called "eternal" is immortal, because that life is not ended by death, and "divine," because it is beyond all matter, quantity, and motion.
216. Then at  he shows the influence of these things on others. Now it is manifest that from what is most perfect there is a flowing to others that are less perfect, just as heat flows from fire to other things that are less hot, as is said in Metaphysics II. Hence, since those beings possess the best and most self-sufficient life and eternal existence, it is from them that existence and life are communicated to other things. But not equally to all; rather, to some "more luminously," i.e., more evidently and more perfectly, namely, to those that have individual eternal existence and to those that have rational life; to others "more darkly," i.e., in a lesser and more imperfect way, namely, to those things that are eternal, not in the same individuals, but according to sameness of species, and which have sense or nutritive life.
217. Then at  he manifests what he had said about the condition of the aforesaid beings that exist outside the heaven.
First he proposes what he intends; Secondly, he presents reasons, at 218.
With respect to the first  it should be known that among the philosophers there were two kinds of teachings. For there were some which from the very beginning were proposed according to the order of doctrine to the multitude and these were called "encyclia"; others were more subtle, and were proposed to the more advanced hearers and were called "syntagmatica," i.e., co-ordinal, or "acromatica," i.e., hearable, teachings. The dogmas of the philosophers are called "philosophemata."
He says, therefore, that in the "encyclic" [or popular] philosophic discussions concerning divine things the philosophers very often in their arguments showed that everything divine must be "untransmutable," as not subject to motion, and "first," as not subject to time, and "highest," as not contained by place. And they called every separated substance "divine." And this confirms what has been said about such beings.
218. Then at  he gives reasons to manifest what he had said, namely, that the first and highest is untransmutable.
First he manifests the proposition; Secondly, he draws a conclusion, at 220.
In regard to the first he gives two arguments, the first of which is as follows: What is always causing motion and acting is better than what is moved and acted upon. But there is nothing better than the first and highest divinity, so as to be able to move it, because such a mover would be more divine. Therefore, the first divine being is not moved, since whatever is moved must be moved by another, as is proved in Physics VII and VIII.
219. The second argument is at : Whatever is moved is moved either to avoid an evil or acquire a good. But what is first has no evil to avoid and lacks no good that it could acquire, because it is most perfect. Therefore, the first is not moved.
The argument could also be presented in the following way: Whatever is moved is moved either to better or to worse. But neither of these can belong to God according to what is said here. Therefore, God, is in no way moved. And one should note that this second argument may be introduced to show that He is not moved by Himself.
220. Then at  from the foregoing he draws a conclusion. And he says it "reasonably," i.e., probably, follows that that first mover of the first mobile acts with unceasable motion. For whatever things, after having been moved, rest, these do so when they reach their appropriate place, as is clear in heavy and light bodies. But this cannot be said of the first mobile which is moved circularly, because where its motion starts is the same as where it ends. Therefore, the first mobile is moved by the first mover with an unceasing motion.
And it should be noted that this argument does not conclude of necessity. For it can be said that the motion of the heaven does not cease, not on account of the nature of the place, but on account of the will of the mover. Therefore, he does not present this as a necessary, but as a probable, conclusion.
Lecture 22: Whether the universe is infinite by eternal duration.
221. After the Philosopher showed that the body of the whole universe is not infinite, and that it is not multiple in number, here he inquires whether it is infinite by eternal duration.
And first he give the opinions of others. Secondly, he settles the question according to his own opinion
Regarding the first, he does three things:
First he declares his intention. Secondly he gives the opinions, at . Thirdly he refutes them.
222. Regarding the first, he does two things. First he states his intention and the order of procedure. And he says that after determining the previous matters, he must go on to say whether the universe is ungenerated or generated, that is whehter it begin to exist at some beginning point of time or not, and whether it is incorruptible or corruptible, that is whether after some time it will cease to exist through corruption, or not. But before treating these matters according to our opinion, we must first briefly review the surmisals of others, that is the opinions of other philosphers on this matter. He calls them surmisals [suspiciones], because they were moved to these opinions by frivolous reasons. For it is difficult to adduce efficacious reasons; thus Aristotle said in Topics I that there are some problems about which we do not have reasons, such as whether the world is eternal or not.
223. Secondly, at "of contrary things", he assigns three reasons why here and elsewhere he reviews the opinions of others. The first reason is that "demonstrations", that is proofs, "of contrary things", that is, of contrary opinions, are critiques of "contraries", that is, contrary opinions. That is, they are objections against contrary opinions. Whoever wishes to know any truth, must know the critiques against that truth, because the solution of doubts is the finding of truth, as is said in III Metaphysica. And thus, to know the truth, it is very important to see the reasons for contrary opinions.
224. As for the second reason, he says that there is an additional reason. Because what must be said is made more credible to people who first hear the justification or defense of disputed opinions, that is solutions of the reasons which gave rise to the dispute. For as long as a man is in doubt, before his doubt is resolved, his mind is like someone bound, who cannot move.
225. The third reason is where he says "to condemn without reason" etc. And he says that when we cite the opinions of others and examine and solve their reasons, and give reasons for the contrary, we will not appear guilty of condemning the opinions of others gratuitously, that is, without prope reason, like those who condemn the opinions of others out of mere hatred. This is not becoming to philosophers, who profess to be searchers of the truth. For those who wish to be adequate judges of the truth must not show themselves enemies of those whose statements are to be judged, but as arbiters, and inquirers for both sides.
226. Then at  he gives the opinions of others. First he shows in what they all agree, and says that all who were before him stated that the world is generated, i.e., at a certain beginning of time it began to exist through generation.
227. Secondly, he shows in what they differ. And he touches on three opinions. First of all, some said that, although it began to be at a certain beginning of time, yet it will endure forever, as first was said by certain poets, such as Orpheus and Hesiod, who are called "theologians" because they presented divine things under the form of poetry and myths. Plato followed them in this position, holding the world to be generated but indestructible.
The second opinion was that of certain others, that the world is destructible in the same way as any other generated thing composed of many parts, and that after being destroyed, it will never be repaired, just as Socrates, once corrupted, is never restored by nature. And this was the opinion of Democritus, who declared the world to be generated by a fortuitous gathering together of atoms ever mobile, and likewise to be destined to be dissolved at some time by the separation of these atoms.
The third opinion is that of those who say the world is alternately generated and destroyed, and that this alternation has always endured and will always last. Such was the opinion of Ebipedocles of Agrigenta, for he posited that with friendship assembling the elements and strife separating them, the world was [continuously] generated and destroyed. This, too, was the opinion of Heraclitus of Ephesus, who posited that at some time the world would be consumed by fire and after a certain lapse of time would again be generated by fire, which he supposed was the principle of all things.
Now, some claim that these poets and philosophers, and especially Plato, did not understand these matters in the way their words sound on the surface, but wished to conceal their wisdom under certain fables and enigmatic statements. Moreover, they claim that Aristotle's custom in many cases was not to object against their understanding, which was sound, but against their words, lest anyone should fall into error on account of their way of speaking. So says Simplicius in his Commentary. But Alexander held that Plato and the other early philosophers understood the matter just as the words sound literally, and that Aristotle undertook to argue not only against their words but against their understanding as well. Whichever of these may be the case, it is of little concern to us, because the study of philosophy aims not at knowing what men feel, but at what is the truth of things.
229. Then at  he refutes these opinions:
First, the first one; Secondly, the third one, at 234; Thirdly, the second one, at 235 (for the second has less of an argument).
About the first he does two things:
First he refutes the opinion; Secondly, he rejects an excusing of it, at 231.
With respect to the first he presents two arguments, in the first of which he says that it is impossible for the world to have been made or generated from a certain beginning of time and then afterwards to endure forever. For when we want to assume something "reasonably," i.e., probably, without a demonstration, we must posit what we observe to be true in all or in many cases, for this is the very nature of the probable. But in this case the contrary happens, because all things that are generated we see to corrupt. Therefore one should not lay down that the world is generated and indestructible.
230. He gives the second argument at . And first he states a principle and says that if a thing is such that it does not have within itself a pot ency which is a principle of its being thus and otherwise, but it is impossible for it to have been otherwise throughout all preceding ages, then such a thing cannot be transmuted. This he proves by leading to an impossibility. For if such a thing should be transmuted, it would be when it is transmuted by some cause producing its transmutation, i.e., by its potency to transmutation. This potency, if it had existed before, would have made it possible for that thing to be other than it was, which thing, however, was assumed to be incapable of being otherwise. But if it previously lacked this potency to be otherwise, and later has it, that itself would be a transmutation of that thing. Consequently, even before it had the potency to be changed, it was able to be changed, namely, by receiving the power to be changed.
From this he argues thus to his proposition; If the world was made from certain things which, before the world was made, were otherwise constituted, then if it is true that those things from which the world was formed were never otherwise than they always were, and could never be otherwise, the world could not have been formed from them. But if the world was formed from them, then, necessarily, those things from which it was formed could be otherwise and do not remain always the same. Hence it follows that even as constituents, i.e., after being united to form the world, they can be separated again;and, when dispersed, they have been previously united, and they alternated thus infinitely, or could have. And if this is true, it follows that the world is not imperishable, nor ever will be imperishable, if the things of which the world consists were at one time otherwise, or even could have been: for in either case it follows that even now it is possible that they be otherwise.
Lecture 23: A Platonic evasion rejected. Two remaining opinions disproved.
231. After presenting the arguments against Plato, the Philosopher here rejects a certain excusing of the aforesaid opinion, which Xenocrates and other Platonists proposed. About this he does two things:
First he proposes the explanation; Secondly, he rejects it, at 232.
He says, therefore, first  that there is no truth in that "help," i.e., that excusing, by which some Platonists seek to justify their assertion -that the world is imperishable, but yet made or generated - and make it appear not unreasonable. For they say that their description of the world's generation was after the manner of those who describe geometric figures by first drawing certain parts of the figure, e.g., of a triangle, and later other parts, not implying that these parts existed before the figure was formed of them, but doing this in order to demonstrate more explicitly what things are required for the figure. They say that Plato in like manner declared that the world was made from elements, not as though the world was generated at some definite time, but for the purpose of presenting his doctrine, so that, namely, his hearers would be more easily instructed about the nature of the world, if first the parts of the world were demonstrated to them and what these parts possessed of themselves, and later the composition they had from the cause of the world, which is God. Consequently they look on, i.e., consider, the world as generated in the manner of the description which geometers use in describing figures.
232. Then at  he disproves this explanation. And he says that the way the generation of the world is described by them is not in the same manner as the descriptions of figures made by geometers, as will be clear from what we shall now say. For in geometric descriptions the same thing happens whether all the parts are considered together as constituting the figure, or whether they are not taken together. When they are taken separately, no more is said about them than that they are lines or angles, which is also true of them when they are taken all together in the figure made out of them. But in the demonstrations presented by those who posit the generation of the world, the same thing is not taken when the parts are considered together and when they are not. Rather, it is impossible that the same be taken in both instances, just as it is impossible for opposites to be together - for the things taken first, i.e., before the establishing of the world, and those taken later, i.e., after the world is now established, are "subcontraries," i.e., have a certain conjoined and latent contrariety.
For they say that out of unordered elements, ordered things were made, God reducing the disorder among the elements to order, as Plato says in the Timaeus. But geometers do not say that a triangle is composed out of separated lines but out of lines. The situation would be similar if those in question solely said that the world results from elements, but what they say is that the orderly world came about from disordered elements. Now it is not possible for something to be at once ordered and disordered, but a process of generation is required through which one is separated from the other, so that before generation it is disordered, and after generation ordered. Consequently it is necessary to suppose some time distinguishing the two. But no such distinction of time is required in the descriptions of figures - for it is not necessary that a line and a triangle be distinguished in the order of time as ordered and disordered are.
233. Still others desire to excuse Plato on the ground that he did not teach that there was a prior disorder in the elements which subsequently, at a later time, began to be ordered, but rather disorder is always present under some aspect in the elements of the world, although under another aspect there is order, as Aristotle himself posits that matter always has a concomitant privation, although it is always in some respect under form. It is also possible to interpret Plato as stressing what the elements would be of themselves if they had not been put in order by God, not that there was ever a time in which they existed disordered.
But whatever Plato may have understood about the matter, Aristotle, as has been said, objected against what Plato's words express. He concludes, therefore, from the foregoing that it is impossible for the world to have been generated and yet able to go on forever.
234. Then at  he takes up the opinion of Empedocles which is the third one mentioned. And he says that those who maintain that the world alternates between being assembled and dissolved do nothing more than assert the substantial permanence of the world but its transmutability with respect to its form or its arrangement. It is as though someone seeing a boy becoming a man, if it should be posited that he sees the same person becoming from a man a boy again, should reckon this person as [alternately] at one time coming into existence and at one time ceasing to be. That the opinion of Empedocles is tantamount to positing the substance of the world as eternal, he manifests by the fact that after the elements shall have been separated by strife and later reassembled, it is not just any order and any new arrangement that will ensue but the very same one that now exists.
And this is made clear "in another way," i.e., by reason, because the very same cause, namely, friendship, will assemble the elements which previously assembled them; consequently, the same arrangement of the world will result. And this is plain also from the teachings of those who hold this position and assert that friendship and strife are contrary and the causes of a contrary disposition in the elements, so that at one time they are assembled and at another separated. Hence he concludes that if the entire body of the world, while remaining "continuous," i.e., conjoined, is now disposed and arranged in one way and later in another way, then, since it is the "combination," or substance, of all bodies that is called the world or heaven, it follows that the world is not generated and destroyed but only its arrangements are.
235. Then at  he takes Democritus' opinion, which was the second one mentioned.
First he explains this opinion; Secondly, he shows what will later be clear about it, at 236.
He says therefore first  that if someone should maintain that the world was made, and entirely ceases to be without returning, in such a way, namely, that it will never be restored again, such a thing is impossible, if there is but one world. The reason is that if there is but one world, made at some time, then, since it was not made from nothing, there was, previous to its being made, a substance which existed before it. Either we hold that that substance which pre-existed before the world could have been subject to generation, or that it could not. If not, then the world could not have been made from it. And this is what he says, namely, that if it was not made, or not generated, i.e., not subject to generation, we say it to be impossible of transmutation, i.e., not able to be transmuted in order for the world to be made out of it. But if it possessed in its nature the power to be transmuted, so that the world could be made from it, then also after the destruction of the world it could be transmuted and a world made again from it.
But if someone posits infinite worlds, in the sense that from atoms arranged in one way this world comes to be, and from the same or other atoms differently arranged another world comes to be, and so on ad infinitum, such a position would be a better foundation for what was said, namely, that the world once destroyed is never again regenerated, because from the assumption that other worlds are possible, another world could be arranged from those atoms. However, if there could be but one world, something incompatible with the theory followa: the matter into which the world dissolved would still be in potency to have a world made from it. Hence if a different world were impossible, the very same one would have to be produced again.
236. Then at  he shows what remains to be said, and says that from what will follow, it will be clear whether this is possible or impossible. And if "this" refers to what was just said of the opinion about infinite worlds, the phrase "what will follow" refers, not to what follows immediately, in which nothing is said about this opinion, but to what will be said about the opinion of Democritus in On the Heavens III and in On Generation I. But if "this" refers to the whole preceding section, where there is treated the opinion of those who posit that the world was generated, then the phrase "what will follow" refers to what immediately follows.
And this is confirmed by what he at once adds. For there are some who conceive it possible for something which was never generated to perish at some time, and for something newly generated to remain incorruptible, as Plato says in the Timaeus that the heaven was produced in being, but will nevertheless endure for eternity. Thus he posits both statements: that disarranged matter, which never became disarranged, at some time ceases to be, and that the world began, and never ceases to be. Against those who thus posit that the world began through generation, Aristotle argued above near the beginning of this book with natural reasons solely to the effect that the heaven was proved ungenerated and indestructible, on the ground that it has no contrary. But now this will be shown by a universal consideration of all beings.
Lecture 24: Various meanings of "generable" and "ungenerable," "corruptible" and "incorruptible"
237. After discussing others' opinions about whether the world is generated and destructible [corruptible], the Philosopher here pursues this question according to his own opinion.
First he presents pre-notes needed in his investigation of the question; Secondly, he pursues the question (L. 26).
About the first he does two things:
First he distinguishes various senses of the following words used in the question: namely, "generated" and "ungenerated," "destructible" and "indestructible"; Secondly, he distinguishes various senses of certain words used in the definitions of the foregoing: namely, "possible" and "impossible" (L. 25).
About the first he does two things:
First he reveals his intention; Secondly, he carries it out, at 239.
About the first he does two things:
First he reveals his intention  and says that in investigating the foregoing question it is first of all necessary to distinguish the various ways in which things are said to be "generable" and "ungenerable," "destructible" and "indestructible."
238. Secondly, at  he reveals the reason for his intention and says that when things are said in a number of ways, it sometimes happens that this multiplicity produces no difference with regard to the argument proposed, i.e., when a particular word is restricted to one meaning in the course of the argument. But when a particular word is used with different meanings, such a multiplicity does make a difference. Even where there is no difference as to the argument, the intellect of the hearer becomes confused, if someone uses a word which can be distinguished in many ways as though it could not - for when someone uses a word of multiple meaning, it is not evident according to which signified essence the conclusion occurs.
239. Then at  he distinguishes the aforesaid words:
First, "ungenerated" and "generated"; Secondly, "destructible" and "indestructible," at 243.
About the first he does two things:
First he distinguishes the word "ungenerated"; Secondly, the word "generated," at 241.
He declares first  that this word "ungenerated" is used in three ways. The first of these is when something is called "ungenerated" which now exists but previously did not, yet this occurs without its having been begotten or transmuted. Some give the example of being touched or moved: for they assert that contact and motion are not generated. And this was proved in Physics V because, since generation is a kind of motion or transmutation, if motion were generated, it would follow that there would be a change of a change. Consequently, contact and motion, although they begin to be, are called "ungenerated," because they are not generated and are not apt to be generated.
In a second way something is said to be "ungenerated," if it is able to either come to be or not come to be and still it has not yet come to be. For example, a man to be born tomorrow is able, as far as the future is concerned, to come to be and not come to be, and yet he is said to be "ungenerated," because he has not yet been born. For "ungenerated," in the sense of "not generated," can be applied similarly to what is able to come to be, because it is not yet generated, and to what is not able to be generated.
In a third way something is said to be "ungenerated," when it is entirely impossible for it, through generation or any other way, to come into existence as being able either to exist or not exist. In this sense the word "ungenerated" describes things that cannot be or things that cannot not be. Now this third way is distinguished into two other ways, for there are two ways in which something is "impossible" to be or become: first, absolutely, when it is in no sense true to say that this may at some time come to be; secondly, when a thing is described as impossible to come about because it is not easy for it to come about, either because it does not come about quickly or because its coming into existence cannot be conveniently managed, as when we say that bad iron is not easy to fashion.
240. In order to understand these three ways it should be noted that generation has the common note of something's beginning to exist, and also implies a definite way of existing, namely, through transformation. Therefore the negation implied by the word "ungenerated" may either negate both, namely, both a beginning and the way of beginning, or it can negate only the way of beginning. And both can occur in two ways: in one way in the sphere of act, and in the other in the sphere of potency. Therefore, if the negation does not deny a beginning but merely the manner of beginning, we have the first meaning of the word according to which something is said to be "ungenerated," if it can begin to be but not through generation. But if it does not deny the possibility but merely the actual state, for example, because it can begin to be and can be generated, but has not yet begun to be or been generated, then it is the second sense of the word. But if it does not only deny the manner of its beginning, as in the first sense, or only the actual state of existence, as in the second sense, but both the manner of its beginning and the very beginning itself, both as to its actual state and even its possibility, then it is the third and most perfect sense, according to which something is said to be ungenerated in the strict and absolute sense. This sense, however, is still distinguished according to whether something is said to be "possible" either absolutely or in a qualified sense.
241. Then at  he distinguishes the meanings of the word "generated," and says that it is also in three senses that "generated" is used. The first of these occurs if something previously did not exist and later began to exist, either through generation, as man, or without generation, as contact, provided that the thing described as generated is something that one time is not and later is.
In a second way, something is described as "generated," if it is possible for it to begin to exist, where "possible" refers either to the truth, i.e., to what can exist, or to what is easy, i.e., can easily be made to exist.
In the third way, something is described as "generated," if it can be the subject of generation and proceeds thus from non-existence to existence. In this third sense it makes no difference whether the thing has already begun to be, and this by being made, i.e., through the process of generation, or whether it has not yet begun to be, but may come to be through generation.
In keeping with what has been said, the notion of these ways is apparent. Because when something is called "generated" in the first sense, its actual inception is asserted but not a definite mode of inception that the word "generation" signifies. But in the second sense the possibility of inception is asserted without asserting the definite way it began, which sense is distinguished according to the way "potency" is distinguished. However, the third way asserts not only inception but a definite kind of inception. And this third way can be further distinguished into two: for it asserts either a definite kind of inception that is actual, as when something is already generated, or one that is potential, as indicating something is naturally apt to be generated.
242. Now if anyone rightly considers the senses he has set down of the word "generated," he will see that they differ from the senses of "ungenerated" in two ways: first with respect to distinction, and, secondly, with respect to order.
With respect to distinction: In distinguishing the senses of "ungenerated," the denial of a definite kind of inception as possible was included under one sense and the denial of the same kind of inception as actual, was included under another sense - for in the first sense "ungenerated" referred to what could not begin to be through generation, but in the second it referred to what could begin to be through generation but had not yet been generated.
But in regard to the denial of inception in common, both the possibility and the actuality of inception are included under the same sense - for the third sense of "ungenerated" referred to what has both not begun to be and cannot begin to be. But conversely, in the senses of "generated" it is on the part of a beginning in common that he distinguishes the modalities according to potency and act - for the first sense refers to what actually begins to be in any way whatever, while the second sense refers to what can begin in any way, although it has not yet begun. However, with regard to a definite kind of inception, the actuality and possibility are included under one mode -for in the third sense something is described as "generated" which either has been generated or can be generated. Thus it is plain that the last three senses are not exactly parallel to the first three, because what was distinguished in the first remains undistinguished in the second, and vice versa.
With respect to order these senses are different: For in presenting the modes of "ungenerated," that which pertains to a definite kind of inception was placed before that which pertains to inception in common, whereas in presenting the modes of "generated," that which pertains to inception in common was mentioned first. And Aristotle had a subtle reason for so doing. For he wanted to list the imperfect senses first and the perfect ones last. Now denial and affirmation are related to the proper and to the common in different ways: for a denial of what is proper is imperfect, but a denial of what is common is perfect, because when the common is denied, the proper is denied. Consequently, the last sense of "ungenerated" is presented as the perfect sense, because it denies inception in general. And because the denial of a particular kind of inception is imperfect, he presents the partial modes as distinguished according to potency and act.
But the affirming of what is proper is perfect, because in affirming what is proper that which is common is also affirmed, while the affirming of what is common is imperfect. Accordingly, the last sense of "generated" is presented as the perfect one, namely, when something begins to be through generation, and he includes under this sense, as under the perfect sense, both the possibility and the actuality. However, the senses pertaining to inception in general are presented first, as the imperfectsenses: for a thing is not said to be "generated" in the perfect sense just because it has begun to be. For this reason he distinguished these modes, as partial, into one referring to possibility and another referring to actuality.
243. Then at  he distinguishes the senses of "destructible" and "indestructible":
First, "destructible"; Secondly, "indestructible," at 245.
He says therefore first  that "destructible" and "indestructible" are also said in many senses, and he presents three senses of "destructible." Now it should be noted that, just as generation implies inception in a definite way, so, too, destruction implies extinction in a definite way, namely, through transmutation. Consequently, the first sense of "destruction" is that of extinction in common without any distinction between possibility and actuality. And the reason for this order is the same as that used above for the word "generated": just as a thing is not said perfectly to be "generated" just because it begins to be, so a thing is not said perfectly to be "destroyed" just because it ceases to be, nor "destructible" just because it can cease to be.
Therefore the first sense in which we describe something as "destructible" is when it previously existed but later it either is not, or is able not to be, whether this is due to perishing and transmutation, as a man is perishable, or not through perishing and transmutation, as contact and motion cease to be.
In the second sense we describe something as "destructible," if it can not be, i.e., able at some time to cease to be, on account of a specific way of ceasing to be.
In the third sense, something is said to be "destructible," because it is easily destroyed, and can be called "euphtharton," i.e., well destructible.
244. It should be observed that although the senses of "destructible" agree with those of "generated" as far as the order is concerned for just as in the latter there is placed first inception in general, so here there is placed first destruction in general, there is a difference in the way their modes are distinguished. For there the modes were distinguished according to possibility and actuality, but here they are distinguished according to absolute, and perfect, possibility, which latter is the last, as the most perfect mode - for the most perfectly destructible is what is easily destroyed. The reason for this is that "generated" is said according to act, while "destructible" according to potency. Hence "generated" can refer to both actuality and possibility, but "destructible" to possibility only.
The reason he set down "generated," which is according to act, and "corruptible," which is according to potency, is this: Since generation is from non-being to being, and corruption from being to non-being, that which is "generable" is not yet a being, but only that which has been "generated" is; on the other hand, that which is "corruptible" is a being, but that which has been "corrupted" is no longer a being. Now the intention of the Philosopher is to discuss a question, not of non-beings but of beings. And that is why he employs the words "generated" and "corruptible."
245. Then at  he distinguishes the senses of "indestructible." And he presents three senses. The first of these denies a definite kind of extinguishing process, insofar as that is said to be "indestructible" which can cease to be in such a way that at one time it exists and later does not, but this without corruption. Examples are contact and motion which, after first existing do not later, but this is without their corruption, since things are subject neither to generation nor corruption. Consequently, this sense corresponds to the first sense of "ungenerated."
In a second sense something is called "indestructible," when extinction in common is denied. And he says that "indestructible" in this sense refers to what is now a being and it is impossible for it later not to be a being, or not to be in the future. This kind of indestructibility does not belong to any thing that can cease to be through corruption. For you, who can cease to be through corruption, exist now, and so does contact, which can cease to be, but not by corruption - yet both of these are called "corruptible" in a certain way, since a time will come when it will not be true to say that you exist, or that this is in contact. And, therefore, that is most properly called "indestructible" which is, indeed, a being but cannot be destroyed in such a way that, while it is a being now, later it will not be, or is able not to be, and although not yet destroyed can, nevertheless, eventually become non-existent. What is not so constituted is properly called "indestructible."
In a third sense, something is called "indestructible" which is not destroyed easily. And this corresponds to the third sense of "destructible," just as the second corresponds to the second, and the first to the first.
Lecture 25: How something is said to be "possible" and "impossible".
246. After pointing out the various senses of "generated" and "ungenerated," "destructible" and "indestructible," the Philosopher here explains the meaning of what is called "possible" and "impossible."
First he reveals his intention; Secondly, he executes his plan, at 248.
About the first he does two things:
First he states what his intention is concerned with , and says that since the situation is thus with regard to the meanings of "generated" and "ungenerated," "destructible" and "indestructible," it is necessary to consider how something is said to be "possible" and "impossible."
247. Secondly, at  he gives the reason for his intention, namely, that "possible" and "impossible" are included in the definition of the aforesaid. For, as has been said above, that is most appropriately called "indestructible" which not only cannot be destroyed but can in no way exist at one time and later not exist. Similarly, "ungenerated" is appropriately applied to what is impossible, namely, to be and not to be, and which cannot come to be in any way such that previously it does not exist and later does exist. Thus, for the diagonal of a square to be symmetric, i.e., commensurate, to the side, is ungenerated, because it never can begin to be.
Then at  he shows how something is described as possible and impossible. And it should be noted that, as the Philosopher says in Metaphysics V, possible and impossible are said in one way absolutely, namely, because in themselves they can be true or cannot be true by reason of the relationship existing between the terms; in another way a thing is said to be possible or impossible to something, namely, what it is able for with respect to its active or passive power. And it is in this sense that "possible" and "impossible" are taken here, namely, as what is, or is not, within the power of an agent or patient - for this is the meaning that is most appropriate to natural things.
First, then, he shows how something is said to be "possible" or "impossible"; Secondly, he excludes an objection, at 251.
With respect to the first he does two things:
First he manifests how some thing is said to be "possible"; Secondly, how something is called "impossible," at 250.
249. To explain the first  he says that if a thing is capable of something great, for example, if a man can walk 100 stades or can lift a great weight, we always determine or describe his power in terms of the most he can do. For example, we say that the power of this man is that he can lift a weight of 100 talents or can walk a distance of 100 stades, even though he is capable of all the partial distances included in that quantity, since he can do what exceeds. But his power is not described by these parts - we do not determine his power as being able to carry 50 talents or walk 50 stades, but by the most he can do. Consequently, the power of each thing is described with respect to the end, i.e., with respect to the ultimate, and to the maximum of which it is capable, and with respect to the strength of its excellence. Thus, too, the size of a thing is determined by what is greatest -for example, in describing the size of something that is three cubits, we do not say that it is two cubits. Similarly, we assign as the notion of man that he is rational, not that he is sensible, because what is the ultimate and greatest in a thing is what completes it and puts upon it the stamp of its species.
Consequently, it is plain that one who can do what exceeds, necessarily can do what is less. For example, if a person can carry 100 talents, he can also carry two, and if he can walk 100 stades, he can also walk two; yet it is to what is excelling that the virtue of a thing is attributed, i.e., the virtue of a thing is gauged in terms of what is most excellent of all the things that can be done.
This is what is said in another translation, "the virtue is the limit of a power," because, namely, the virtue of a thing is determined according to the ultimate it can do. And this applies also to the virtues of the soul: for a human virtue is that through which a man is capable of what is most excellent in human actions, i.e., in an action which is in accordance with reason.
250. Then at  he tells how something is said to be "impossible" to a thing. And he says that if some amount is impossible to someone if one takes what excels, it is plain that it will be impossible for him to carry or do more. For example, a person who cannot walk 100 stades clearly cannot walk 101. Hence, it is plain that just as the possibility is determined by the greatest that a thing can do - which determines its virtue - so what is impossible is determined by the least that it cannot do, and this determines its weakness. For example, if the most that someone can do is to go 20 stades, the least that he cannot go is 21 - and it is from this that his weakness is to be determined, and not from his inability to walk 100 or 1,000 stades.
251. Then at  he excludes an objection.
First he proposes it; Secondly, he solves it, at 252.
He says therefore first  that nothing should disturb us in connection with the fact that what is properly called "possible" is determined according to the limit of excellence. Someone could, indeed, object that what has been said is not necessary in all matters - for there seems to be an objection in the case of sight and other senses. For a person who sees some large quantity, for example, something a stadium long, cannot for that reason see magnitudes of smaller size contained below that quantity. Rather it is more the opposite that occurs - for one who can see a "point," i.e., some smallest thing perceptible to sense, or hear a faint sound, can sense what is greater.
252. Then at  he answers this objection and says that what was stated does not affect the argument whereby it is shown that the possible is determined by excellence. For the excellence according to which the virtue of a thing is measured can be determined according to the virtue, or according to the thing. According to the thing, when there is an excellence right in it, as was said about the 100 stades or the 100 talents. According to this excellence the active virtue is determined, because whatever can act on a greater thing can act on a lesser. But with respect to virtue, there is excellence when something which is not outstanding in quantity requires an excellence of virtue. And this is seen to occur especially with passive powers - for the more a thing is passible, the more it can be moved by what is less. And since the senses are passive powers, it happens in sensible things that one who can sense what is less can sense what is greater.
What he has just said he explains as follows: a vision capable of perceiving a smaller body excels in virtue; hence the excellence in this case is an excellence in the virtue and not in the thing. But that speed is more excellent which is of a greater magnitude - for that is speedier which can traverse a greater distance in the same time - and such excellence is an excellence not only in the virtue, but also in the thing.
Lecture 26: Everything eternal is indestructible and ungenerated
253. After explaining the meanings of the words proposed in the question, the Philosopher here begins to argue on the question proposed, namely, whether something can be generated and indestructible, or ungenerated and destructible.
First he shows with general arguments that this is impossible; Secondly, with arguments proper to natural science, at 286 (L. 29).
About the first he does two things:
First he shows what follows from the preceding with respect to the present question; Secondly, he begins to argue to his proposition, at 255.
254. He says therefore first  that, the preceding having been determined as to the meanings of certain words, it is now time to state what follows in this treatment. For it has been said above that the "possible" is described in terms of something definite - for example, someone's power to run is described in terms of 100 stades. But there exist in external reality some things that can both exist and not exist. Therefore it is necessary according to the foregoing that there be determined some maximum time both affecting existence, such that it is not possible to exist for a greater time, and affecting non-existence, such that it is not possible not to exist for a greater time.
And lest this be understood as applying only to substantial existence, he adds that, when we say that it is possible or not possible for a thing to exist, or that which is able not to exist, such expressions can be understood with regard to any predication, i.e., with regard to any predicament: for example, that a man exist or not exist, which pertains to the genus of substance; or that white exist or not exist, which pertains to the genus of quality; or that "two cubits" exist or not exist, which pertains to the genus of quantity; or any other similar thing.
That when something is said to be able to be or not be, that expression must be understood in terms of some determinate time, he now proves by leading to an impossibility. For, as he says, if there is not a time of determinate quantity in which it could be or not be, but a time greater than a given time is always assumed (for example, if it can be for fifty years and then more and again still more), and no limit is reached with respect to which every time in which it can be is less, then, since it is the same thing that can be and not be, as was said, it follows that the same thing can be for an infinite time, and not be for an infinite time, because the same reasoning applies to the non-existence as applies to the existence.
This does not mean that the time in respect to which something is able not to be, and which was concluded to be infinite, is the same as the time in respect to which the thing is able to be - because then the same thing would be able to be and not be during the same time, which is impossible, as will be said below. It means rather that there is one infinite time for the thing as non-existing, and another for it as existing. Now this is impossible: for there cannot be two infinite times, because then two times would be simultaneous. But this impossibility follows from saying that the possibility to be or the possibility not to be are not reckoned with respect to some determinate time. Therefore the first thing that must be clear is that the possibility of being is said with respect to a determinate time, and similarly the possibility of not being. And this agrees with what has been already laid down about the meaning of"possible."
255. Then at  he begins to argue to his proposition. About this he does two things:
First he argues to it with general reasons; Secondly, with an argument proper to natural science, at 286 (L. 29).
About the first he does two things:
First he shows the truth, namely, that indestructible and unproduced follow one upon the other; and likewise, destructible and generated; Secondly, he disproves the contrary of this (L. 29).
Regarding the first he does two things:
First he proves his proposition by showing how what is eternal is related to the ungenerated and indestructible, and to the generated and destructible; Secondly, how they are related to one another (L. 28).
About the first he does three things:
First he shows that everything eternal is indestructible and ungenerated; Secondly, that nothing eternal is generated or destructible, nor conversely (L. 27); Thirdly, he concludes that everything ungenerated and indestructible is eternal, at 265 (L. 27).
About the first he does two things:
First he presents some needed pre-notes; Secondly, he argues to his proposition, at 257.
256. He says therefore first  that in order to prove the proposition, it is necessary to start from the fact that "impossible" and "false" do not mean the same.
Regarding this he posits four reflections. The first is that both "possible" and "impossible," as well as "true" and "false," are used in two ways. In one way, conditionally, i.e., in the sense that a thing must be true or false, possible or impossible, if certain things are assumed; for example, a triangle must in fact have three angles equal to two right angles, but nevertheless such a property is impossible if certain things are assumed - thus, if we should suppose a triangle to be a square, it would follow that a triangle would have four right angles. In like manner, it will follow that the diagonal of a square is commensurate to the side if certain assumptions are true - for example, if we should assume that the square of the diagonal is 4 times the square of the side - for then it will always follow that the ratio of the diagonal to the side is a numerical proportion, which is a commensurable ratio. In a second way things are said to be