Authors/Thomas Aquinas/physics/L6/lect11

Lecture 11 Zeno's arguments excluding all motion are resolved

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	Lecture 11 Zeno's arguments excluding all motion are resolved
lib. 6 l. 11 n. 1 Postquam philosophus determinavit de divisione motus et quietis, hic excludit quaedam, ex quibus errabant aliqui circa motum. Et circa hoc tria facit: primo solvit rationes Zenonis, negantis totaliter motum esse; secundo ostendit quod indivisibile non movetur, contra Democritum, qui ponebat indivisibilia moveri semper, ibi: ostensis autem his etc.; tertio ostendit mutationem omnem esse finitam, contra Heraclitum, qui ponebat omnia moveri semper, ibi: mutatio autem et cetera. Circa primum duo facit: primo ponit quandam rationem Zenonis et solvit eam, quae pertinet ad id quod immediate de motu praemiserat; secundo explicat omnes rationes eius per ordinem, ibi: quatuor autem sunt rationes et cetera.	860. After finishing with the division of motion and of rest, the Philosopher now refutes certain opinions that have been the source of error in regard to motion. About this he does three things: First he answers the arguments of Zeno who absolutely denies that motion exists; Secondly, he shows that an indivisible is not moved, against Democritus, who said that they are always in motion, at L. 12; Thirdly, he shows that all change is finite, against Heraclitus, who said that all things are eternally moved, at L. 13. About the first he does two things: First he gives and rejects one of Zeno’s arguments, which pertains to what Zeno had accepted about motion; Secondly, he explains all his arguments in order, at 863.
lib. 6 l. 11 n. 2 Dicit ergo primo quod Zeno male ratiocinabatur, et apparenti syllogismo utebatur ad ostendendum quod nihil movetur, etiam illud quod videtur velocissime moveri, sicut sagitta quae fertur. Et erat ratio sua talis. Omne quod est in loco sibi aequali, aut movetur aut quiescit: sed omne quod fertur, in quolibet nunc est in aliquo loco sibi aequali: ergo et in quolibet nunc aut movetur aut quiescit. Sed non movetur: ergo quiescit. Si autem in nullo nunc movetur, sed magis videtur quiescere, sequitur quod in toto tempore non moveatur, sed magis quiescat. Posset autem haec ratio solvi per id quod supra ostensum est, quod in nunc neque movetur neque quiescit. Sed haec solutio intentionem Zenonis non excluderet: sufficit enim Zenoni, si ostendere possit quod in toto tempore non movetur; quod videtur sequi ex hoc quod in nullo nunc eius movetur. Et ideo Aristoteles aliter solvit, et dicit falsum esse quod ratio concludit, et non sequi ex praemissis. Ad hoc enim quod aliquid moveatur in tempore aliquo, oportet quod moveatur in qualibet parte illius temporis: ipsa autem nunc non sunt partes temporis; non enim componitur tempus ex nunc indivisibilibus, sicut neque aliqua magnitudo componitur ex indivisibilibus, ut supra probatum est: unde non sequitur quod in tempore non moveatur aliquid, ex hoc quod in nullo nunc movetur.	861. He says therefore first (660 239 b5) that Zeno reasoned badly and used what had only the appearance of a syllogism to show that nothing is being moved, even what seems to be in rapid motion, as an arrow in flight. And this was his argument: Anything that is in a place equal to itself is either being moved or is at rest. But whatever is being moved is at each instant in a place equal to itself. Therefore, even at each instant it is either in motion or at rest. But it is not in motion, Therefore, it is at rest. But if it is not in motion at any instant but at rest, as it seems, then throughout the entire time it is at rest and not in motion. Now this argument could be answered by appealing to something already proved; namely, that in an instant there is neither motion nor rest. But such a solution would not cripple Zeno’s intention, for he is satisfied to show that through the entire time there is no motion&8212;a fact that seems to follow, if there is no motion at any instant of the time. Therefore Aristotle answers in a different manner and says that the conclusion is both false and does not follow from the premises. For in order that something be moved in a given period of time, it has to be moved in each part of the time. But instants are not parts of time, for time is not made up of instants any more than a magnitude is made of points, as we have already proved. Hence it does not follow that a thing is not in motion in a given time, just because it is not in motion in any instant of that time.
lib. 6 l. 11 n. 3 Deinde cum dicit: quatuor autem sunt rationes etc., ponit seriatim omnes rationes Zenonis, quibus utebatur ad destruendum motum. Et circa hoc tria facit: primo manifestat quomodo destruebat per suas rationes motum localem; secundo quomodo destruebat alias species mutationum, ibi: neque igitur secundum mutationem etc.; tertio quomodo destruebat specialiter motum circularem, ibi: iterum autem in circulo et cetera.	862. Then at (661 239 b9) he lists in order all the arguments that Zeno used for destroying motion. About this he does three things: First he shows how he destroyed local motion with his arguments; Secondly, how he destroyed the other types of change, at 870; Thirdly, how in particular he destroyed circular motion, at 871.
lib. 6 l. 11 n. 4 Circa primum quatuor rationes ponit: et hoc est quod dicit, quod Zeno utebatur quatuor rationibus contra motum, quae ingerebant difficultatem multis eas solvere volentibus. Quarum prima talis est. Si aliquid movetur per totum aliquod spatium, oportet quod prius pertranseat medium quam perveniat ad finem: sed cum illud medium sit divisibile, oportebit quod etiam prius pertranseat medium illius medii, et sic in infinitum, cum magnitudo sit in infinitum divisibilis: infinita autem non est transire in aliquo tempore finito: ergo nihil potest moveri. Dicit ergo Aristoteles quod superius, circa principium huius sexti libri, solvit istam rationem per hoc, quod similiter tempus in infinita dividitur, sicut et magnitudo. Quae quidem solutio est magis ad interrogantem si infinita contingat transire in tempore finito, quam ad interrogationem, ut dicet in octavo; ubi solvit hanc rationem per hoc, quod mobile non utitur infinitis quae sunt in magnitudine, quasi in actu existentibus, sed ut in potentia existentibus. Tunc autem aliquo puncto spatii utitur quod movetur ut in actu existenti, quando utitur eo ut principio et ut fine: et tunc necesse est quod ibi stet, ut ibi ostendetur. Et si sic oporteret transire infinita quasi in actu existentia, numquam veniretur ad finem.	863. In regard to the first he lists four reasons, and this is what he says: Zeno used against motion four arguments which have caused difficulty for many of those who tried to answer them. The first of which is this: If anything is being moved through a certain space, it must reach the middle before it reaches the end. But since the first half is divisible, half of it must be first traversed and so on indefinitely, since a magnitude can be divided ad infinitum. Infinites, however, cannot be traversed in finite time. Therefore, nothing can be moved. Therefore Aristotle says that he has already answered this argument (in the beginning of this Book VI), when he proved that time is divided ad infinitum in the same way as a magnitude is. This answer is directed more to one who asks whether infinites can be traversed in finite time than to the question, as he will say in Book VIII, (L. 17) where he answers this argument by showing that a mobile does not use the infinites which exist in a magnitude as though they were actually existing, but only as existing potentially. For a thing in motion uses a point in space as actually existing, when it uses it as a beginning and as an end, and it is then that the mobile must be at rest, as will be explained in Book VIII. But if it had to traverse infinites that were actually existing, then it would never reach the end.
lib. 6 l. 11 n. 5 Secundam rationem ponit ibi: secunda autem vocata etc.; et dicit quod hanc secundam rationem vocabant Achillem, quasi invincibilem et insolubilem. Et erat ratio talis. Quia si aliquid movetur, sequitur quod id quod currit tardius, si incepit primo moveri, nunquam iungetur vel attingetur a quocumque velocissimo. Quod sic probabat. Si tardum incepit moveri ante velocissimum per aliquod tempus, in illo tempore pertransivit aliquod spatium: ante igitur quam velocissimum quod persequitur, possit attingere tardissimum quod fugit, necesse est quod vadat ab illo loco unde movit fugiens, usque ad illum locum quo pervenit fugiens tempore illo quo persequens non movebatur. Sed oportet quod velocissimum illud spatium pertranseat in aliquo tempore, in quo tempore iterum tardius aliquod spatium pertransit, et sic semper. Ergo semper tardius habet aliquid ante, idest aliquod spatium in quo praecedit velocissimum, quod ipsum persequitur: et ita velocius numquam attinget tardius. Hoc autem est inconveniens. Ergo magis dicendum est quod nihil movetur.	864. The second argument is given at (662 239 b14) and he says that they called this one the “Achilles”, as though it were invincible and unanswerable. The argument was this: If anything is being moved, it follows that a slower thing, if it started earlier, will never be caught by anything moving most rapidly. And it was proved in the following way: If a slower object began to be moved for some time before a very swift one, then in that time it has traversed some distance. Therefore, before the very swift one in pursuit could reach the slower, which is still running, it must leave the place first left by the pursued and reach the place which the pursued reached during the time the pursuer was not in motion. But the very fast pursuer must traverse this space in some time, during which the slower has meanwhile traversed a certain space, and so on forever. Therefore, the slower always has “something ahead”, i.e., is always some distance ahead of the most swift pursuer, and so the swifter will never catch the slower. But this is unacceptable. Therefore, it is better to say that nothing is moved.
lib. 6 l. 11 n. 6 Ad solvendum autem hanc rationem dicit, quod haec ratio est eadem cum prima, quae procedebat ex decisione spatii in duo media, quantum ad virtutem medii: sed differt ab ea in hoc, quod aliqua accepta magnitudo spatii non dividitur in duo media, sed dividitur secundum proportionem excessus velocioris ad tardius in motu. Nam in primo tempore, quo movebatur solum tardius, accipitur maior magnitudo; in secundo autem tempore, in quo velocius pertransit praedictum spatium, cum sit minus, accipitur minor magnitudo pertransita a tardiori, et sic semper. Unde quia tempus et magnitudo semper dividuntur, videtur accidere ex hac ratione quod tardius nunquam iungatur a velociori. Sed hoc in idem tendit cum eo quod in prima ratione dicebatur de divisione magnitudinis in duo media: quia in utraque ratione videtur accidere quod mobile non possit adiungere usque ad terminum quendam, propter divisionem magnitudinis in infinitum, quocunque modo dividatur; scilicet vel in duo media, sicut prima ratio procedebat, vel secundum excessum velocioris ad tardius, sicut procedebat secunda ratio. Sed in hac secunda ratione apponitur, quod velocissimum non potest attingere ad tardius dum persequitur ipsum: quod dictum est cum quadam tragoedia, idest cum quadam magnificatione verborum ad admirationem movendam; sed non facit aliquid ad virtutem rationis. Unde patet quod necesse est esse eandem solutionem huius secundae rationis et primae. Sicut enim in prima ratione falsum concludebatur, quod scilicet mobile nunquam perveniret ad terminum magnitudinis, propter infinitam magnitudinis divisionem; ita falsum est quod vult secunda ratio concludere, quod tardius praecedens nunquam iungatur a velociori sequente; quod nihil est aliud quam mobile non pervenire ad aliquem terminum. Verum est enim quod quamdiu praecedit tardius, non coniungitur sibi velocius. Sed tamen quandoque coniungetur sibi, si hoc detur, quod mobile possit pertransire finitam magnitudinem in tempore finito. Pertransibit enim velocius persequens totam illam magnitudinem qua praecedebat ipsum tardius fugiens, et adhuc maiorem, in minori tempore quam tardius moveatur ultra per aliquam determinatam quantitatem: et ita non solum attinget ipsum, sed etiam ultra transibit. Hae igitur sunt duae rationes Zenonis, sic solutae.	865. In solving this argument he says that it is the same as the first, which proceeded by dividing the distance into two halves and then continually halving one part of the remainder. But the difference between them is that in the second the given magnitude of space is not divided into halves but according to the difference between the swift and the slower motion. For in the first period of time in which only the slower was in motion, there is a greater magnitude involved; in the second period (in which the faster traversed the distance covered by the slower between its start and the start of the faster), which is a shorter time period, a smaller magnitude was traversed by the slower, and so on forever. Hence, the time and the magnitude are always being divided and that seems to be the reason why the slower is never caught by the swifter, But this tends to the same thing as what was said of the division of the magnitude into halves; because in both arguments it seems that the mobile cannot reach a certain goal on account of the magnitude’s being infinitely divided, no matter how it happens to be divided, i.e., whether according to halves, as happens in the first argument, or according to the excess of the faster over the slower, as in the second argument. However, in this second argument it is further added that the very swift cannot reach the slower, which it is pursuing. This “tragic” phraseology employs inflated language in order to excite wonder, but it does not do anything to the force of the argument. Hence it is clear that the solution of the two arguments is the same. For just as a false conclusion was reached in the first argument, namely, that the mobile would never reach the end of the magnitude on account of the infinite division of the magnitude, so also what the second argument tries to conclude is false, namely, that the slower will never be caught by the swifter, which is just another way of saying that a mobile will never reach its goal. Now, it is true that as long as the slower is ahead, it is not yet reached by the swifter. But yet it will at some time be reached, if you concede that a finite magnitude can be traversed in finite time. For the swifter pursuing mobile will traverse the whole distance by which the slower is ahead and even more, in less time than the slower was meantime moving farther ahead. Proceeding in this way the swifter will not only catch but pass the slower. These therefore are the solutions to two of Zeno’s arguments.
lib. 6 l. 11 n. 7 Tertiam rationem ponit ibi: tertia autem et cetera. Et dicit quod tertia ratio Zenonis erat illa quam supra posuit antequam inciperet rationes enumerare, scilicet quod sagitta, quando fertur, quiescit. Et sicut supra dictum est, hoc accidere videtur ex eo quod ipse supponit quod tempus componatur ex ipsis nunc. Nisi enim hoc concedatur, non poterit syllogizare ad propositum.	866. The third argument is given at (663 239 b30) and he says that the third argument of Zeno was the one cited above (before he began to give the arguments): namely, that an arrow in flight is always at rest. And as was said above, this seems to happen, because Zeno supposed that time is made up of instants. For unless that be granted, the syllogism fails.
lib. 6 l. 11 n. 8 Quartam rationem ponit ibi: quarta autem et cetera. Circa quam tria facit: primo ponit rationem; secundo solutionem, ibi: est autem deceptio etc.; tertio manifestat per exempla, ibi: ut sint stantes aequales magnitudines et cetera. Dicit ergo primo quod quarta ratio Zenonis procedebat ex aliquibus quae moventur in aliquo stadio, ita quod sint duae magnitudines aequales, quae moveantur iuxta aequalia, idest per aliquod spatium stadii aequale utrique in quantitate: et sit iste motus ex contrarietate, idest ita quod una magnitudinum aequalium moveatur per illud spatium stadii versus unam partem, et alia versus aliam partem: ita tamen quod una magnitudinum mobilium incipiat moveri a fine stadii ei aequalis, alia vero incipiat moveri a medietate stadii, sive spatii in stadio dato: et utraque moveatur aeque velociter. Hac positione facta, opinabatur Zeno quod accideret quod tempus dimidium esset aequale duplo: quod cum sit impossibile, volebat ex hoc ulterius inferre quod impossibile est aliquid moveri.	867. He sets out the fourth argument at (664 239 b33). Concerning which he does three things: First he sets out the argument; Secondly, the solution, at 868; Thirdly, he explains it by an example, at 869. First therefore he says that the fourth argument of Zeno proceeded from some bodies which move in a stadium so that there are two equal magnitudes which are moved in an equal manner, that is, through a space in the stadium equal to both in quantity, and this motion is a contrary one, i.e., one of the equal magnitudes is moved through that space of the stadium toward one part, and the other toward the other part, in such a way, however, that one of the mobile magnitudes begins to move from the terminus of the stadium which is equal to it, and the other begins to move from the middle of the stadium or of a space in the given stadium; both move with equal velocity. This being given, Zeno held that it would result in a half time equalling a double time. Since this is impossible, he intended to infer further that it is impossible for anything to be moved.
lib. 6 l. 11 n. 9 Deinde cum dicit: est autem deceptio etc., ponit solutionem. Et dicit quod Zeno in hoc decipiebatur, quod accipiebat ex una parte mobile moveri iuxta magnitudinem motam, et ex alia parte accipiebat quod moveretur iuxta magnitudinem quiescentem, aequalem magnitudini motae. Et quia supponitur aequalis velocitas mobilium, volebat quod secundum aequale tempus sit motus aeque velocium circa aequales magnitudines, quarum una movetur et alia quiescit. Quod patet esse falsum. Quia cum aliquid movetur iuxta magnitudinem quiescentem, non est ibi nisi unus motus: sed quando aliquid movetur iuxta magnitudinem motam, sunt ibi duo motus. Et si sint in eandem partem, addetur de tempore; si autem sint in oppositas partes, diminuetur de tempore, secundum quantitatem alterius motus. Quia si magnitudo iuxta quam aliquod mobile movetur, in eandem partem moveatur aequali velocitate vel etiam maiori, nunquam mobile poterit eam pertransire: si vero minori velocitate magnitudo moveatur, pertransibit eam quandoque, sed in maiori tempore quam si quiesceret. E contrario autem est si magnitudo moveatur in oppositum mobilis: quia quantum magnitudo velocius movetur, tanto mobile in minori tempore eam pertransit; quia uterque motus operatur ad hoc quod se invicem pertranseant.	868. Then at (665 240 a1) he gives the solution. He says that Zeno was deceived in this, that he held that on the one hand the mobile is moved according to the moved magnitude, and on the other he held that it was moved according to a quiescent magnitude equal to the moved magnitude. Because an equal velocity of the moved bodies is supposed, he wanted to conclude that the motion of equally swift bodies in regard to equal magnitudes, one of which is in motion and the other standing still, is done in equal times This is seen to be false for the following reason: When something is moved in relation to a quiescent magnitude, there is only one motion; but when something is moved in relation to a moving magnitude, there are two motions. If they are moving in the same direction it takes more, if they are moving in opposite directions it takes less time according to the amount of either motion. If the magnitude in relation to which something mobile is moved, is moved in the same direction with an equal velocity or even a greater velocity, the other moving body can never pass it. If the magnitude moves with less speed, it will pass by it at a certain time, but it will take more time than if it were quiescent. It is quite the contrary if the magnitude is moved opposite the direction of the other body. The more swiftly the magnitude moves, the less time the other body takes to pass it, because both motions work together to pass each other.
lib. 6 l. 11 n. 10 Deinde cum dicit: ut sint etc., manifestat quod dixerat in terminis. Ponatur enim quod sint tres magnitudines aequales sibi invicem, in quarum qualibet ponatur a; et sint istae magnitudines stantes, idest non motae; ut si intelligatur quod sit aliquod spatium trium cubitorum, in quorum quolibet describatur a. Et sint aliae tres magnitudines aequales sibi invicem, in quarum qualibet describatur b; ut puta si intelligamus quod sit unum mobile trium cubitorum. Incipiant autem hae magnitudines moveri a medio spatii. Sint etiam tres magnitudines aliae aequales numero et magnitudine et velocitate ipsis b, et scribatur in istis c, et incipiant moveri ab ultimo spatii, scilicet ab ultimo a. Hac ergo suppositione facta, continget quod primum b per suum motum perveniet ad hoc quod sit simul cum ultimo a; et iterum primum c per suum motum perveniet ut sit cum primo a, opposito ultimo: et simul etiam cum hoc erit cum ultimo b, quasi transiens secus invicem motorum, idest iuxta omnia b, quae invicem ei contramoventur. Cum autem hoc factum fuerit, constat quod istud primum c transivit omnia a, sed ipsum b non transivit nisi media. Cum ergo b et c sint aequalis velocitatis, et aeque velox minorem magnitudinem in minori tempore pertranseat; sequitur quod tempus in quo b pervenit ad ultimum a, sit dimidium temporis in quo c pervenit ad primum a oppositum: in aequali enim tempore utrumque, scilicet b et c, est iuxta unumquodque; idest in aequali tempore c et b pertranseunt quandocumque a. Hoc ergo supposito, quod tempus in quo b pervenit ad ultimum a, sit dimidium temporis in quo c pervenit ad primum a oppositum, ulterius considerandum est quomodo Zeno volebat concludere quod hoc dimidium temporis sit aequale illi duplo. Ex quo enim ponitur tempus motus ipsius c, duplum temporis motus ipsius b. Ponatur quod in prima medietate temporis, b quiescebat et c movebatur, et sic c in illa medietate temporis pervenit usque ad medietatem spatii ubi est b; et tunc b incipiat moveri ad unam partem, et c ad aliam. Quando autem b pervenit ad ultimum a, oportet quod pertransierit omnia c, quia simul primum b et primum c sunt in contrariis ultimis, scilicet unum in primo a, et aliud in ultimo; et sicut ipse dicebat, c in aequali tempore fit iuxta unumquodque b, in quanto tempore pertransit unumquodque ipsorum a. Et hoc ideo, quia ambo, scilicet b et c, in aequali tempore pertranseunt unum a: et sic videtur quod si b in aequali tempore pertransit in quanto pertransit ipsum c, quod c in aequali tempore pertransit ipsum b et ipsum a. Ergo tempus in quo c pertransit omnia b, est aequale tempori in quo pertransivit omnia a. Tempus autem in quo c pertransit omnia b, est aequale tempori in quo c vel b pertransit medietatem ipsorum a, ut dictum est. Probatum est autem quod tempus in quo ipsum b pertransit medietatem ipsorum a, est dimidium temporis in quo c pertransit omnia a. Ergo sequitur quod dimidium sit aequale duplo; quod est impossibile. Haec igitur est ratio Zenonis. Sed incidit in falsitatem praedictam: quia scilicet accipit quod c in eodem tempore pertranseat b contra-motum et a quiescens; quod est falsum, ut supra dictum est.	869. Then at (666 240 a4) he makes clear what he said in the latter part. Suppose that there are three magnitudes equal to each other, each designated as A, and these magnitudes are standing still; thus there might be a space of three cubits, each one of which is marked by an A. There are another three magnitudes all equal and designated as B, as there might be one moving unit of three cubits. These magnitudes begin to move from the middle of the space. There are also three other magnitudes, equal in number, size and velocity to B, and designated as 0. These begin to move from the last space, that is from the last A. This being given, it occurs that the first R by its motion arrives at the last A and likewise the first C by its motion arrives at the first A, opposite the last. When this has been done, it is evident that this first C has passed all the A’s, but B has passed by only half. Since, therefore, R and C are equal in velocity, and an equal velocity passes by a smaller magnitude in less time, it follows that the time in which B travels to the last A is half the time in which C arrives at the first A opposite; in equal times C and B pass each section of A. This being supposed, namely, that the time in which B arrives at the last A is half the time in which 0 arrives at the first A opposite, it must be further considered how Zeno wished to conclude that this half time is equal to its double, For from the supposition that the time of the motion of C is double the time of the motion of B, it is supposed that, in the first half of the time, B was still and C moved, and thus C in that half of the time arrived at the middle of the space, where B was; and then B began to move to one part and C to another. When B arrived at the last A it had to pass all the C’s, because at the same time the first B and the first C are at contrary ultimates; namely, one at the first A and the other at the last, and as he said, C is next to each B, in the same amount of time as it takes to pass each one of the A’s. This is so, because both, namely Rand C, pass one A in the same interval of time. Thus it seems that if B covers a time equal to that in which it passes C, that C, in an equal interval of time, passes B and A. Therefore, the interval in which C passes all B’s is equal to the time in which it passed all the A’s. The time in which C passed all the B’s is equal to the time in which C or B passed the middle of the A’s, as was said. But it was proved that the time in which B passed the middle of the A’s is half the time in which C reached all the A’s. Therefore, it follows that the half is equal to the double, which is impossible. This is the argument of Zeno. But he falls into the aforesaid error; namely, he assumes the fact that C in the same interval of time crosses B moving in a counter direction and A quiescent, which is false, as was said above. [In this description of St. Thomas, since 0 is supposed to pass all the A’s, he is careful to have Cc first begin from the last A and come to the middle, while meanwhile B is motionless. This is not in the text of Aristotle.]
lib. 6 l. 11 n. 11 Deinde cum dicit: neque igitur secundum etc., ponit rationem qua Zeno excludebat mutationem quae est inter contradictoria. Dicebat enim sic. Omne quod mutatur, dum mutatur, in neutro terminorum est: quia dum est in termino a quo, nondum mutatur; dum autem est in termino ad quem, iam mutatum est. Si ergo aliquid mutetur de uno contradictorio in aliud, sicut de non albo in album, sequitur quod dum mutatur, neque sit album neque non album; quod est impossibile. Licet autem hoc inconveniens sequatur aliquibus, qui ponunt impartibile moveri, tamen nobis, qui ponimus quod omne quod movetur est partibile, nullum accidit impossibile. Non enim oportet, si non est totum in altero extremorum, quod propter hoc non possit dici aut album aut non album. Contingit enim quod una pars eius sit alba, et alia non alba. Non autem dicitur aliquid album ex eo quod totum sit huiusmodi, sed quia plures et principaliores partes sunt tales, quae magis propriae sunt natae tales esse: quia non idem est non esse in hoc, et non esse totum in hoc, scilicet in albo vel non albo. Et quod dictum est de albo vel non albo, intelligendum est de esse vel non esse simpliciter, et in omnibus quae opponuntur secundum contradictionem, sicut calidum et non calidum, et huiusmodi. Semper enim oportebit quod sit in altero contra oppositorum illud quod mutatur, quia denominabitur ab eo quod principalius inest: sed non sequitur quod semper sit totum in neutro extremorum, ut Zeno putabat. Sciendum est autem quod haec responsio sufficit ad repellendum rationem Zenonis, quod hic principaliter intenditur. Quomodo autem circa hoc se habeat veritas, in octavo plenius ostendetur. Non enim in quolibet verum est, quod pars ante partem alteretur vel generetur, sed aliquando totum simul, ut supra dictum est: et tunc non habet locum haec responsio, sed illa quae ponitur in octavo.	870. Then at (667 240 a19) he gives the argument by which Zeno rejected change between contradictories. For he said: Whatever is being changed is in neither of the extremities while it is being changed, because while it is in the terminus a quo it is not yet being changed, and while it is in the terminus ad quem it has already been changed. Therefore, if something is being changed from one contradictory to another, as from non-white to white, it follows that while it is being changed, it is neither white nor black—which is impossible. Now although this strange conclusion would follow for those who posit that an indivisible can be moved, yet for us who posit that whatever is being moved is divisible, nothing impossible follows. For even though it is not totally in one or other of the extremes, it is not for that reason neither white nor non-white, For one part could be white and the other non-white. For a thing is called white not only when all of it is white but also when very many or its main parts are white, i.e., the parts that are expected to receive whiteness, because it is one thing not to be something at all and another not to be entirely something, for example, white or non-white. And what has been said of white and non-white is to be understood of unqualified being or non-being and of all things that are contradictorily opposed, as hot and non-hot and so on. For what is being changed must always be in one of the opposites, because it is described in terms of whichever opposite predominates in it, But it does not follow that it is always as a whole in neither of the extremities, as Zeno supposed. Now it should be known that this answer is sufficient to refute Zeno’s argument and that is what Aristotle’s main intention is. But the truth of this matter will be more fully given in Book VIII. For it is not true in all cases that part is altered or generated after part, but sometimes the whole comes all at once, as was said above. In that case it is not this answer but the one in Book VIII that would apply.
lib. 6 l. 11 n. 12 Deinde cum dicit: iterum autem in circulo etc., solvit rationem Zenonis, qua destruebat motum sphaericum. Dicebat enim quod non est possibile aliquid circulariter vel sphaerice moveri, vel quocumque alio modo, ita quod aliqua moveantur in seipsis, id est non progrediendo a loco in quo sunt, sed in ipsomet loco. Et hoc probabat tali ratione. Omne illud quod per aliquod tempus secundum totum et partes est in uno et eodem loco, quiescit: sed omnia huiusmodi sunt in eodem loco et ipsa et partes eorum secundum aliquod tempus, etiam dum ponuntur moveri: ergo sequitur quod simul moveantur et quiescant; quod est impossibile. Huic autem rationi obviat philosophus dupliciter. Primo quantum ad hoc quod dixerat, partes sphaerae motae esse in eodem loco per aliquod tempus: contra quod Aristoteles dicit quod partes sphaerae motae in nullo tempore sunt in eodem loco. Zeno enim accipiebat locum totius: et verum est quod dum sphaera movetur, nulla pars exit extra locum totius sphaerae; sed Aristoteles loquitur de proprio loco partis, secundum quod pars potest habere locum. Dictum est enim in quarto quod partes continui sunt in loco in potentia. Manifestum est autem in motu sphaerico, quod pars mutat proprium locum, sed non locum totius: quia ubi fuit una pars, succedit alia pars. Secundo obviat ad praedictam Zenonis rationem, quantum ad hoc quod dixit, quod totum manet in eodem loco per tempus. Contra quod Aristoteles dicit, quod etiam totum semper mutatur in alium locum: quod sic patet. Ad hoc enim quod sint duo loca diversa, non oportet quod unus illorum locorum sit totaliter extra alium; sed quandoque quidem secundus locus est partim coniunctus primo loco, et partim ab eo divisus, ut potest in his considerari quae moventur motu recto. Si enim aliquod cubitale corpus moveatur de ab loco in bc locum, quorum uterque locus sit cubitalis; dum movetur ab uno loco in alium, oportet quod partim deserat unum et subintret alium; sicut si deserat de loco ab quantum est ad, subintrabit in locum bc quantum est be. Manifestum est ergo quod locus de est alius a loco ab, non tamen totaliter ab eo seiunctus, sed partim. Si autem daretur quod illa pars mobilis quae subintrat secundum locum, regrederetur in partem loci quam mobile deserit, essent duo loca, et tamen in nullo ab invicem separata; sed solum differrent secundum rationem, secundum quod principium loci in diversis signis acciperetur, ubi scilicet est principium mobilis, idest aliquod signum quod in mobili accipitur ut principium: et sic erunt duo loca secundum rationem, sed unus locus secundum subiectum. Et sic intelligendum est quod hic dicit, quod non est eadem circulatio secundum quod accipitur ut incipiens ab a, et ut incipiens a b, et ut incipiens a c, vel a quocumque alio signo; nisi forte dicatur eadem circulatio subiecto, sicut musicus homo et homo, quia unum accidit alteri. Unde manifestum est quod semper de uno circulari loco movetur in alterum, et non quiescit, ut Zeno probare nitebatur. Et eodem modo se habet et in sphaera et in omnibus aliis quae infra locum proprium moventur, sicut rota et columna vel quidquid aliud huiusmodi.	871. Then at (668 240 a29) he refutes the argument by which Zeno rejected spherical motion. For he said that it is not possible for anything to be moved circularly or spherically or in such a way that the motion is confined within the space occupied by the mobile. And he proved this with the following argument: Anything that is in its entirety and in respect of its parts in one and the same place for a period of time is not in motion but at rest. But all the above-mentioned fulfill these conditions, even when they are apparently in motion. Therefore, they are at once in motion and at rest—which is impossible. The Philosopher attacks this argument on two points. First, as to the statement that the parts of the moving sphere are in the same place for some time. For Zeno was speaking of the place of the whole, and it is true that while the sphere is in motion, no part passes out of the place of the sphere, but Aristotle speaks of the particular place of each part, according as a part has a place. For it was said in Book IV that the parts of a continuum are in place potentially. But it is evident in spherical motion that a part does change its particular place, although it does not lose the place of the whole, because where one part was, another part succeeds. Secondly, he attacks the statement that the whole remains in the same place for some time. Against this Aristotle says that even the whole is changing its place. For in order that two places be not the same, it is not required that one of them be entirely outside the other, but sometimes the second place is partly joined to part of the first and partly divided from it, as is clear in things moved in a straight line. For let a body of one cubit be moved from place AB to place BC—both places being one cubit each. While the mobile is being moved from one place to the other, it must partly desert one place and enter the other; for example, it could leave the portion AD of AB and enter the portion BE of BC. Therefore, it is clear that the place DE is distinct from AB, although not entirely, but only partly separated from it. But if it were assumed that that part of the mobile which entered the second place re-entered part of the place deserted, there would be two places, yet in no way separated—they would differ not really but only in conception, i.e., in the sense that the beginning of the place might be successively called by different letters each time the mobile re-entered it, namely, where the beginning of the mobile is, i.e., some spot in the mobile which is taken as a beginning, Thus there would be two places conceptually but one and the same in reality. This is how we must understand what Aristotle says here, namely, that it is not the same revolution, when it is taken as beginning at A and as beginning at B and as beginning at C or any other mark, unless you insist that it is the same revolution as to subject, as in the case of “musical man” and “man”, since one happens to the other. Hence it is clear that the mobile is always being moved from one circular place to another and is not at rest as Zeno tried to prove. And it is the same with the sphere and everything else whose motion is confined within the space it occupies, as in the case of a potter’s wheel and a (rotating) pillar or anything of that sort.