LECTURE 21 (266 a 10-b 26)

A FINITE MOVER CANNOT MOVE IN AN INFINITE TIME. AN INFINITE POWER CANNOT RESIDE IN A FINITE MAGNITUDE. A FINITE POWER CANNOT RESIDE IN AN INFINITE MAGNITUDE

1141. After the Philosopher has explained the nature of the first motion, he here explains the nature of the first mover.

             This discussion is divided into two parts. First he states his intention. Secondly, where he says, 'One of these premises . . .' (266 a 12), he develops his position.

             He says, then, first that since it was shown above that the first mover is immobile, it must now be shown that the first mover is indivisible and has no magnitude, that is, it is completely incorporeal.

             But before we prove this, we ought to determine first the things which are required to prove this.

             1142. Next where he says, 'One of these premises . . .' (266 a 12), he develops his position. First he sets forth certain things which are necessary to prove his main point. Secondly, where he says, 'Now that these points . . .' (267 b 18), he proves his main point.

             Concerning the first part he makes three points. First he shows that infinite power is required for infinite motion. Secondly, where he says, 'It has now to be shown . . .' (266 a 25), he shows that there cannot be infinite power in a finite magnitude. Thirdly, where he says, 'But before proceeding . . .' (266 b 27), he shows that the first mover, which causes eternal and continuous motion, must be one.

             He says, therefore, first that one of the things which must be determined prior to the main point is that it is impossible for that which is finite in power to move through an infinite time. He proves this as follows.

             In any motion there are three things. One is that which is moved; another is the mover itself; the third is time in which the motion occurs.

             Now all of these must be infinite, or else all must be finite, or else certain ones are finite and certain ones infinite, either only one or two.

             Let it be granted that A is a mover, B is a mobile object, and C is an infinite time. And assume that some part of A, which is D, moves some part of B, which is E. Granting these assumptions, we can conclude that D moves E in a time which is not only equal to C, in which A moved B, but in a shorter time.

             For it was proven in Book VI that a whole mobile object crosses through a designated distance in a greater time than does part of it. Since time C is infinite, it follows that the time in which D moves E will not be infinite, but finite. Let that time be F. Thus just as A moves B in the infinite time C, so D moves E in the finite time F. Since, however, D is part of A, then if one subtracts from A and adds to D, A will be completely removed or exhausted, since it is finite. For if the same quantity is always taken, any finite thing is exhausted by subtraction, as was explained in Book III.

             Similarly, B will be exhausted if something is continually subtracted from it and added to E. For B was given as finite. But no matter what is subtracted from time C, even if I subtract the same quantity, the whole of C is not exhausted, because it was given as infinite.

             From this he concludes that the whole of A moves the whole of B in some finite time which is a part of C. This follows from the foregoing because an addition to a mobile object and to a mover requires a proportional addition to the time of the motion. But by subtracting from the whole mobile object and the mover, and by adding to their parts, at some time the whole mobile object and the whole mover will be exhausted such that the whole which was in the whole is added to the part. Therefore it follows that, by proportionally adding to the time, a finite time will result in which the whole mover will move the whole mobile object. Hence, if the mover is finite, and the mobile object is finite, the time must be finite.

             Thus a finite mover cannot move something with an infinite motion in an infinite time. And thus, what was proposed at the beginning is clear, namely, a finite mover does not move in an infinite time.

             1143. Avicenna, however, raises an objection to this proof from Aristotle. The proof does not appear to be universal. For there is a finite mover and mobile object from which it is impossible to subtract or remove anything, namely, a celestial body, which is not excluded from this proof. And so it seems either that the proof is particular, or else it proceeds from a false supposition.

             Averroes answers this objection in his Commentary by saying that although nothing can be subtracted from a celestial body, nevertheless this conditional statement is true: if some part were removed from a celestial body, that part will move, or be moved, in less time than the whole. For a conditional proposition is not prevented from being true if its antecedent is impossible, as is clear in this conditional statement: if a man flies, he has wings. Rather, whatever destroys the truth of a true condition is false, even though the antecedent of the condition be false. Now the truth of the above condition is not consistent with the statement that a finite thing moves in an infinite time, as is clear from Aristotle's deduction. Therefore, from the truth of the above condition, Aristotle concludes that it is impossible for a finite thing to move in an infinite time.

             It can be said more briefly, however, that when Aristotle uses removal or subtraction in his proofs, the dissolution of a continuum by subtraction must not always be inferred, for this is impossible in a celestial body. Rather subtraction may be understood according to a designation. For example, in wood which remains continuous, it is possible to designate some point either by touch or by thought, as if dividing the whole, and in this way to remove some part from the whole and to say that there is less whiteness in the part than in the whole. And in this way, too, it can be said that there is less power to move in a part of a celestial body which is removed by designation than in the whole.

             1144. Another problem is more difficult. It does not appear to be contrary to the nature [ratio] of a finite mover to move in an infinite time. For if a finite thing is incorruptible or impassive in its nature and it does not withdraw from its nature, it will always be related to motion in the same way. And that which is related in the same way always does the same thing. Hence there is no reason why it cannot move afterwards rather than before. And this is apparent to the senses. For we see that the sun can move lower bodies in an infinite time.

             To answer this problem the procedure of the above demonstration must be investigated. For just what conclusion follows from the premises should be clearly understood.

             It should be noted that the time of a motion can be understood in two ways, especially in regard to local motion: first with respect to the parts of the mobile object, and secondly with respect to the parts of the magnitude over which the motion passes. For it is clear that one part of the mobile object passes through some point of the magnitude before the whole mobile object does. Similarly, the whole mobile object passes through one part of the magnitude before it passes through the whole. It is clearly apparent from Aristotle's procedure that he is speaking here of the time of the motion insofar as the time of the motion is taken with respect to the parts of the mobile object and not with respect to the parts of the magnitude. For he says in his proof that a part of the mover moves a part of the mobile object in less time than the whole moves the whole. This would not be true if we took the time of the motion with respect to the parts of the magnitude which is crossed by the motion. For the proportion of the parts of the mover to the parts of the mobile object is the same as the proportion of the whole mover to the whole mobile object. And so a part will always move a part with a velocity equal to that by which the whole moves the whole. Hence, a part of the mobile object, which is moved by a part of the mover, will pass through a magnitude in a time equal to that in which the whole mobile object is moved by the whole mover.

             Or perhaps the whole will be moved in less time than a part. For a unified power is greater than a divided power, and the greater the power of the mover, the swifter the motion and the shorter the time. This must be understood insofar as the time of the motion is considered with respect to the parts of the mobile object. For one part of the mobile object passes through a designated distance in less time than does the whole mobile object. And accordingly it is impossible for it to be moved in an infinite time, unless it is an infinite mobile object. It is impossible, moreover, for an infinite mobile object to be moved by a finite mover, for the power of the mover is always greater than the power of the mobile object. Hence an infinite mobile object must be moved by an infinite mover. From this it follows that it is impossible to hold that a finite mover moves a finite mobile object with a motion which is infinite with respect to the parts of the mobile object. Thus when this inconsistency is removed, one must conclude further that infinite motion is given to an infinite mobile object by an infinite mover.

             1145. But someone may object to this by saying that Aristotle did not prove above that motion is infinite with respect to the parts of the mobile object such that the motion of an infinite body is called infinite. For the whole corporeal universe is finite, as was proven in Book III and as will be proven in De Caelo, I. Hence it does seem that Aristotle's proof is valid for the conclusion of this proposition; namely, the first mover which causes an infinite motion is infinite.

             But it must be said that that which is the first cause of infinite motion must be a per se cause of infinite motion. For a cause which is per se is always prior to one which is through another, as was explained above. The power of a per se cause is directed to a per se effect and not to a per accidens effect. For it was thus that Aristotle taught us in Book II to compare causes to effects. Now a motion may be infinite in two ways, as was said, namely with respect to the parts of the mobile object and with respect to the parts of the magnitude over which the motion passes. Motion is infinite per se from the parts of the mobile object, and infinite per accidens with respect to the parts of the magnitude. For the quantity of motion which is taken with respect to the parts of the mobile object belongs to the motion with respect to its proper subject, and thus is present in it per se. But the quantity of motion which is taken with respect to the parts of the magnitude is taken with respect to the repetition of the motion of the mobile object, such that the whole mobile object which completes its motion over one part of the magnitude crosses another part by repetition. Therefore, that which is the first cause of the infinity of motion has a per se power over the infinity of motion such that it could move an infinite mobile object if there were such an object. Therefore it must be infinite. And even though the first mobile object is finite, it still has a certain similarity to an infinite one, as was explained in Book III. But in order for something to be the cause of a motion which is infinite due to the repetition of a motion (which is per accidens), it is not necessary for it to have infinite power. Rather it is sufficient if it has an immobile finite power. For always remaining the same in power, it could repeat the same effect. For example, the sun has a finite power, and yet it could move the lower elements in an infinite time, if motion were eternal, as Aristotle says. For there is no first cause of the infinity of motion, but it is as if things have been moved to move by another for an infinite time, according to the foregoing position.

             1146. Next where he says, 'It has now to be shown . . .' (266 a 25), he shows that the power in a magnitude must be proportional to the magnitude in which it resides. First he shows that an infinite power cannot reside in a finite magnitude, which is his main point. Secondly, where he says, 'Therefore nothing finite . . .' (266 b 5), he shows that a finite power cannot reside in an infinite magnitude.

             He proves that an infinite power cannot reside in a finite magnitude by setting forth two suppositions. The first is that a greater power produces an equal effect in a shorter time than a lesser power. For example, a greater power of heating produces in less time an equal heat in that on which it acts. And the same is true of the power of sweetening or throwing or of any kind of motion.

             From this supposition he concludes that since an infinite power is greater than a finite one, then if a finite magnitude has an infinite power, it follows that in the same time either one or many patients undergo a greater mutation from such an agent than from another agent of finite power. Or conversely, that which undergoes an equal mutation would be changed by such an agent in less time. Either of these may be understood from his words '. . . in fact to a greater extent than by anything else . . .' (266 a 29).

             The second supposition is that since everything which is moved is moved in time, as was proven in Book VI, it is impossible for a patient to be changed by an infinitely powerful agent in no time. Therefore, it is changed in time.

             From this he proceeds as follows. Let A be the time in which an infinite power heats or drives away. And let the time in which some finite power moves be AB, which is greater than A. Now another greater finite power can be taken. Therefore, if we take another finite power which is greater than the first one, which moves in time AB, it follows that this second power will move in less time. And again, a third still greater finite power will move in still less time. And thus, by always taking a finite power, I shall come to some finite power which moves in time A. For when addition is endlessly made to the finite power, every determinate proportion is exceeded. And at the same time an addition is made to the moving power, there is a subtraction from the time of the motion. For a greater power can move in a shorter time.

             Therefore, it will follow that a finite power completes a motion in a time equal to that of the infinite power which moves in time A. This, however, is impossible. Therefore, no finite magnitude has an infinite power.

             1147. There are numerous difficulties with this argument.

             First, it seems that this argument comes to no conclusion at all. For what per se belongs to a thing cannot be removed from it by any power, however great it may be. For it is not because of a lack of power, or because it is repugnant to an infinite power, that it cannot be said that a man is not an animal. Now to be in time belongs per se to motion. For motion is included in the definition of time, as was shown above in Book IV. Hence, if an infinitely powerful mover is granted, it does not follow that motion does not take place in time, as Aristotle concludes here.

             Further, if the Philosopher's procedure is considered, it seems that he concludes that motion does not take place in time because the moving power is infinite. But an infinite moving power cannot exist in a body. For the same reason, therefore, it follows that if such a power is infinite, it will move in no time. Therefore, from the fact that it is impossible for a thing to be moved in no time, it cannot be concluded that no infinite power resides in a magnitude but only that no moving power is infinite.

             Further, two things seem to pertain to the magnitude of the power, namely, the velocity of the motion, and its duration. And we see that an excess of power produces an excess in both of these. But with regard to the excess of an infinite power, he has shown above that eternal motion results from an infinite power, but not that an infinite power does not reside in a magnitude. Hence also here, with respect to the excess of velocity, one ought not to conclude that no infinite power resides in a magnitude, but that a power which moves in an infinite time also moves in no time because of its infinity.

             Further, the conclusion seems to be false. For the greater the power of a body, the longer can it be preserved in existence. Therefore, if no body has infinite power, no body can endure infinitely. This seems to be false both according to Aristotle's opinion and according to the judgment of the Christian faith which holds that the substance of the world will endure forever.

             An objection could also be raised regarding the division and addition which he uses, since they do not agree with the nature of things. But since this was discussed sufficiently earlier, it is omitted at the present time.

             1148. In answering these difficulties in order, one must reply to the first objection that the Philosopher does not intend here to give a direct demonstration but rather a demonstration which leads to impossibility. In such a proof one concludes that what is initially given is impossible because from it follows an impossibility. Moreover, it is not true that what is initially given together with the conclusion could possibly exist. For example, if it were granted that there is a power which could remove a genus from a species, it would follow that that power could make a man who is not an animal. But since this is impossible, the first statement is also impossible. But it cannot be concluded from this that it is possible for there to be a power which can make a man who is not an animal. And so, from the statement that there is an infinite power in a magnitude, of necessity it follows that motion occurs in no time. But because this is impossible, it is impossible for there to be an infinite power in a magnitude. But it cannot be concluded from this that it is possible for an infinite power to move in no time.

             1149. Averroes answers the second difficulty in his Commentary on this text by saying that Aristotle's argument here deals with the infinity of power. Now the finite and the infinite pertain to quantity, as was shown above in Book I. Hence neither the finite nor the infinite properly pertain to a power which does not reside in magnitude.

             But this answer is contrary both to Aristotle's intention and to the truth.

             It is contrary to Aristotle's intention because in the preceding argument Aristotle proved that a power which moves in an infinite time is infinite. And from this he concludes below that the power which moves the heavens is not a power which resides in a magnitude.

             It is also contrary to the truth. For since every power is active with respect to some form, it follows that magnitude, and consequently the finite and the infinite, pertain to power in the same way as they pertain to form. Now magnitude pertains to form both per se and per accidens: per se with respect to the perfection of the form itself, as when a small amount of snow is said to have much whiteness according to the perfection of its proper nature [ratio]; and per accidens insofar as a form has extension in a subject, as when a thing is said to have much whiteness because of the magnitude of its surface.

             This second magnitude cannot belong to a power which does not reside in a magnitude. But the first magnitude is especially proper to it. For immaterial powers are more perfect and more universal to the extent that they are less contracted by application to matter.

             The velocity of motion, however, is not due to that magnitude of power which is per accidens because of the extension of the magnitude of a subject, but rather is due to that magnitude of power which is per se according to its proper perfection. For to the extent that a being is more perfectly in act, it is to that degree more vigorously active. Hence, although a power which does not reside in a magnitude is not infinite with the infinity of magnitude, which is due to the magnitude of the subject, it cannot be said that for this reason it does not cause an infinite increase of velocity, which is to move in no time.

             Hence this same Commentator answers this difficulty in another way in Metaphysics, XII, where he says that a celestial body is moved by a double mover, namely, by a mover united to it, which is the soul of the heavens, and by a separated mover, which is not moved either per se or per accidens. And since that separated mover has infinite power, the motion of heavens receives eternal duration from it. And since the mover joined to it has finite power, the motion of heavens receives a determinate velocity from it.

             But this answer is not sufficient. For that which moves in an infinite time seems to require infinite power, as the preceding proof concluded, and also that which moves in no time seems to require infinite power, as this proof seems to conclude. Therefore there still remains the difficulty of why the soul of the heavens, which moves in virtue of the infinite separated mover, is caused by that separated mover to move in an infinite time rather than with infinite velocity, that is in no time.

             1150. To answer this difficulty it must be said that every power which is not in a magnitude moves through an intellect. For thus the Philosopher proves in Metaphysics, XII, that the heavens are moved by their own mover. Moreover, no power which is in a magnitude moves as if it has intelligence. For it was proven in De Anima, III, that the intellect is not the power of a body.

             For this is the difference between an intellectual agent and a material agent. The action of a material agent is proportionate to the nature of the agent. For a thing produces heat to the extent that it is itself hot. But the action of an intellectual agent is not proportionate to its nature, but to the apprehended form. For a builder does not build as much as he can, but as much as the intelligibility [ratio] of the conceived form requires.

             Therefore, if there were some infinite power in a magnitude, it would follow that the motion proceeding from it would be according to its proportion, and thus the present proof proceeds. If, however, there is an infinite power which is not in a magnitude, motion would not proceed from it according to the proportion of power, but according to the intelligibility [ratio] of the apprehended form, that is, according to that which is proper to the end and nature of the subject.

             There is another thing to be noted. As was proven in Book VI, nothing is moved unless it has magnitude. And so the velocity of motion is an effect received from a thing which moves in something having magnitude. It is clear, moreover, that nothing which has magnitude can receive an effect equally proportioned to a power which is not in a magnitude. For every corporeal nature is compared to incorporeal nature as something particular to the absolute and the universal. Hence, if there is an infinite power which is not in a magnitude, it cannot be concluded that it produces in some body an infinite velocity as an effect proportioned to such a power, as was said.

             But nothing prevents the reception in a magnitude of an effect of a power which is in a magnitude, since a cause is proportioned to its effect. Hence, if it were granted that some infinite power exists in a magnitude, it would follow that there is a corresponding effect in the magnitude, namely, an infinite velocity. But this is impossible. Therefore, so is the first statement.

             1151. From these remarks the answer to the third difficulty is clear. To be moved in an infinite time is not repugnant to the nature [ratio] of a moved magnitude. Rather this is proper to a circular magnitude, as was shown above. But to be moved with infinite velocity, that is, in no time, is contrary to the nature [ratio] of magnitude, as was proven in Book VI. Hence, according to Aristotle, a motion of infinite duration is caused by a first mover of infinite power. But a motion of infinite velocity is not.

             1152. As Averroes remarks at this point in his Commentary, Alexander answers the fourth difficulty by saying that a celestial body receives its eternity and its eternal motion from a separated mover of infinite power. Hence, it is not moved eternally because of the infinity of the celestial body just as it does not endure eternally because of the infinity of the celestial body. Rather both of these are due to the infinity of a separated mover.

             Averroes tries to disprove this answer both here in his Commentary and in Metaphysics, XII. He says that is impossible for a thing to receive eternity of being from another for it would follow that what is corruptible in itself would become eternal. But a being can acquire eternity of motion from another because motion is the act of a mobile object from the mover. He says, therefore, that a celestial body in itself has no potency for non-being because its substance has no contrary. But there is a potency for rest in a celestial body because rest is contrary to its motion. Hence it does not need to acquire eternity of being from another, but it does need to acquire eternity of motion from another.

             He says that a celestial body has no potency for non-being because he also says that a celestial body is not composed of matter and form, as if composed of potency and act. Rather he says that it is matter existing in act, and its form he calls its soul. However, it is not constituted in being, but only in motion, by its form. And thus he says that there is in it no potency for being, but only potency for place, as the philosopher says in Metaphysics, XII.

             1153. But this answer is contrary to the truth and to Aristotle's intention.

             It is contrary to the truth in many ways, first, because he says that a celestial body is not composed of matter and form. For this is absolutely impossible.

             It is clear that a celestial body is something in act, otherwise it would not be moved. For what is in potency only is not a subject of motion, as was explained in Book VI. And everything which is in act must be either a subsisting form, as are the separated substances, or must have a form in another which is related to form as matter and as potency to act.

             Now it cannot be said that a celestial body is a subsisting form, for then it would be an intellect in act, and would be neither sensible nor quantified. Hence it follows that it is composed of matter and form and of potency and act. And thus potency for non-being is in it in some way.

             But even if it is granted that a celestial body is not composed of matter and form, it still must be admitted that potency for non-being is in it in some way. For it is necessary that each simple subsisting substance either is its own existence or participates in existence. A simple substance which is its own subsisting existence cannot exist except as one, just as whiteness, if it were subsisting, could not exist except as one. Therefore, every substance which is after the first simple substance participates in existence. Moreover, everything that participates is composed of the participant and the participated, and the participant is in potency to the participated. Therefore, in every simple substance after the first simple substance there is potency for being.

             He was deceived, moreover, by the equivocal meaning of potency. For potency is sometimes predicated in relation to an opposite. But this is excluded from celestial bodies and from separated simple substances. For according to Aristotle's intention, there is no potency in them for non-being because of the fact that simple substances are forms only, and existence belongs to form per se. Moreover, the matter of a celestial body is not in potency to another form. For just as a celestial body is related to its own figure, of which it is the subject, as potency to act, and yet it cannot not have such a figure, so likewise the matter of a celestial body is related to such a form as potency to act, and yet it is not in potency to a privation of this form, or to non-being. For not every potency is of opposites, otherwise the possible would not be a consequence of the necessary, as is said in De Interpretatione.

             His position is also contrary to the intention of Aristotle who, in De Caelo, I, uses in a proof the proposition that a celestial body has the potency or power to always exist. Hence it is not possible to avoid inconsistency when he says that there is no potency for being in a celestial body. For this is clearly false and contrary to Aristotle's intention.

             1154. Let us see whether Averroes adequately refutes the answer of Alexander, who says that a celestial body acquires eternity from another. His refutation would indeed be adequate if Alexander had held that a celestial body of itself has potency for being and non-being, and acquires eternal existence from another. And I say this on the supposition that his intention is not to exclude the omnipotence of God through which a corruptible thing can assume incorruptibility. The discussion of this latter point is not pertinent to the proposition. Even granting his intention, Averroes still cannot reach a conclusion contrary to Alexander, who did not hold that a celestial body acquires eternity from another in the sense that it has in itself potency for being and nonbeing, but in the sense that it does not have existence from itself. For everything which is not its own existence, participates in existence from the first cause, which is its own existence. Hence even Averroes admits in De Substantia Orbis that God causes the heavens not only in respect to its motion, but also in respect to its substance. This could not be unless it has its existence from Him. But it has only eternal existence from Him. Therefore, it has its eternity from another. And Aristotle's words are also in agreement with this when he says in Metaphysics, V, and above at the beginning of Book VIII that there are certain necessary things which have a cause of their necessity. Granting this, the solution according to Alexander's intention is clear, namely, just as a celestial body has its motion from another, so also it has its existence from another. Hence, just as eternal motion demonstrates the infinite power of the mover, but not of the mobile object, likewise its eternal duration demonstrates the infinite power of the cause from which it has its existence.

             1155. Nevertheless the potency of a celestial body is not related in exactly the same way to being and to being moved eternally.

             They do not differ as he says, namely, that with respect to being moved there is in a celestial body a potency for the opposites of motion and rest. Rather there is a potency for opposites which are diverse places.

             They differ with respect to something else. Motion in itself occurs in time. But existence in itself does not occur in time, but only insofar as it is subject to motion. Therefore, if there is some existence which is not subject to motion, that existence would in no way occur in time. Hence the potency to be moved in infinite time is related to the infinity of time directly and per se. But the potency to exist in an infinite time, if indeed that existence is mutable, is related to the quantity of time. Therefore a greater power or potency is required for a thing to endure in mutable existence for a greater time. But a potency which is immutable with respect to existence in no way is related to the quantity of time. Hence the magnitude or infinity of time has nothing to do with the magnitude or infinity of the potency for such existence. Therefore, if one grants the impossibility that a celestial body does not have its existence from another, one still could not conclude from its eternity that there is an infinite power in it.

             1156. Next where he says, 'Therefore nothing finite . . .' (266 b 5), he proves that a finite power cannot reside in an infinite magnitude.

             He proves this with two arguments. In the first argument he makes three points.

             First he states his conclusion. He says that just as an infinite power cannot reside in a finite magnitude, likewise a power which is finite with respect to the whole (for if a part of the infinite is finite, it will have finite power) cannot reside in an infinite quantity.

             He introduces this, however, not as though it were essential for the proof of the main proposition, but as closely connected to the conclusion demonstrated above.

             1157. Secondly, where he says, 'It is true that a greater . . .' (266 b 7), he indicates a reason why someone may think it is possible for a finite power to reside in an infinite magnitude. We see that a smaller magnitude has a greater power than a larger magnitude, for example, a small fire has a greater active power than much air. But from this it cannot be held that an infinite quantity has a finite power. For if a still greater magnitude is taken, it will have greater power. For example, air which is greater in respect to quantity has less power than a small fire. But if the quantity of the air is greatly increased, it will have a greater power than the small fire.

             1158. Thirdly, where he says, 'Now let AB . . .' (266 b 8), he gives his proof, which is as follows. Let AB be an infinite quantity; and let BC be a finite magnitude of a different genus which has a certain finite power; and let D be a mobile object which is moved by the magnitude BC in a time which is EF. Now since BC is a finite magnitude, a larger magnitude can be taken. Let a magnitude twice its size be taken. Now the greater the power of the mover, the smaller the time in which it moves, as was explained in Book VII. Therefore, twice BC will move the same mobile object, D, in half the time, which is FG. The time EF is understood to be divided in half at point G. By thus continually adding to BC, the time of the motion is lessened. But no matter how much is added, BC can never equal AB, which exceeds BC without proportion as the infinite exceeds the finite. And since AB has finite power, it moves D in a finite time. Hence by continually diminishing the time in which BC moves, we come to a time which is less than the time in which AB moves. For whatever is finite is crossed by division. It will follow, therefore, that a smaller power moves in less time, which is impossible. It follows, therefore, that there was an infinite power in the infinite magnitude, because the power of an infinite magnitude exceeds every finite power.

             And this was proven by the subtraction of time because it is necessary to hold that every finite power moves in a determinate time. This is clear from the following. If so much power moves in so much time, a greater power will move in a smaller, yet finite and determinate, time according to an inverse proportion, that is, the time is decreased as much as the power is increased. And thus, whatever is added to a finite power, as long as it remains a finite power, will always have a finite time. For there will be a time which is as much less than the earlier time as the power which is increasing by addition is greater than the initially given power.

             But an infinite power of motion surpasses every determinate time, as happens with all infinites. For every infinite, for example, in multitude or magnitude, exceeds every determinate thing of its genus. And thus it is clear that an infinite power exceeds every finite power in which the increase of power over power is similar to the lessening of time from time, as was said. Hence it is clear that the above conclusion, namely, there is infinite power in the infinite magnitude, follows of necessity from the premises.

             1159. Next where he says, 'This point may also . . .' (266 b 20), he gives another proof of the same thing. This proof differs from the first only in that the first one concluded by taking a finite power existing in a finite magnitude of another genus, whereas the second proof proceeds by taking another finite power existing in another finite magnitude of the same genus, the magnitude of which is infinite. For example, if air is infinite in magnitude and has a finite power, we shall take a finite power existing in some finite magnitude of other air.

             Granting this position, it is clear that when the finite power of the finite magnitude is multiplied many times, it will measure the finite power which is in the infinite magnitude. For whatever is finite is measured or exceeded by a smaller finite thing taken many times. Now in a magnitude of the same genus, a greater magnitude must have a greater power, as more air has greater power than less air. Therefore, that finite magnitude which will have the same proportion to the first finite magnitude as the finite power of the infinite magnitude has to the power of the first finite magnitude must have a power equal to the power of the infinite magnitude. For example, if the finite power of the infinite magnitude is one hundred times the finite power of the given finite magnitude, then a magnitude which is one hundred times the finite magnitude must have a power equal to the infinite magnitude. For in a thing of the same genus, magnitude and power are increased proportionately. The conclusion, however, is impossible, for it would necessarily follow either that a finite magnitude would be equal to an infinite magnitude, or else a smaller magnitude of the same genus would have a power equal to a larger one. Since this is impossible, so is the premise from which it follows, namely, an infinite magnitude has a finite power.

             Therefore, in summarizing, he draws two demonstrative conclusions, namely, there cannot be infinite power in a finite magnitude, and there cannot be finite power in an infinite magnitude.