Authors/Thomas Aquinas/physics/L6/lect7

Lecture 7 The time in which something is first changed is indivisible. How a first may, and may not, be taken in motion

Latin	English
	Lecture 7 The time in which something is first changed is indivisible. How a first may, and may not, be taken in motion
lib. 6 l. 7 n. 1 Postquam philosophus ostendit qualiter dividatur motus, hic determinat de ordine partium motus. Et primo inquirit an sit primum in motu; secundo ostendit quomodo ea quae sunt in motu, praecedunt se invicem, ibi: quoniam autem omne quod mutatur, in tempore mutatur et cetera. Circa primum duo facit: primo ostendit quod id in quo primum mutatum est, est indivisibile; secundo ostendit quomodo in motu possit inveniri primum, et quomodo non possit, ibi: dicitur autem in quo primo mutatum est et cetera. Circa primum duo facit: primo praemittit quoddam quod est necessarium ad propositi ostensionem; secundo ostendit propositum, ibi: in quo autem primo mutatum est et cetera. Circa primum duo facit: primo proponit quod intendit; secundo probat propositum, ibi: quod mutatur enim et cetera.	818. After explaining how motion is divided, the Philosopher now discusses the order of the parts of motion. First he asks whether there is a first in motion; Secondly, he shows how the factors involved in motion precede one another, in L. 8. About the first he does two things: First he shows that that into which something is first changed is indivisible; Secondly, how in motion a first can and cannot be found, 822. About the first he does two things: First he mentions facts to be used in explaining the proposition; Secondly, he proves the proposition, at 821. About the first he does two things: First he mentions his proposition; Secondly, he proves it, at 819.
lib. 6 l. 7 n. 2 Dicit ergo primo, quod quia omne quod mutatur, mutatur de uno termino in alium; necesse est omne quod mutatur, quando iam mutatum est, esse in termino ad quem. Deinde cum dicit: quod mutatur enim etc., probat propositum duabus rationibus; quarum prima est particularis, secunda universalis. Prima ratio talis est. Omne quod mutatur, oportet quod aut distet a termino a quo mutatur, sicut patet in motu locali, in quo locus a quo mutatur remanet, et mobile per motum fit distans ab eo; aut oportet quod ipse terminus a quo deficiat, sicut est in motu alterationis: cum enim ex albo fit nigrum, ipsa albedo deficit. Et ad huius propositionis manifestationem subiungit, quod vel mutari est idem quod deficere; vel ad hoc quod est mutari sequitur ipsum deficere, et ad hoc quod est mutatum esse sequitur defecisse, scilicet a termino a quo. Manifestum est autem quod sunt idem subiecto, sed differunt ratione. Nam deficere dicitur per respectum ad terminum a quo, mutatio autem magis denominatur a termino ad quem. Et ad manifestationem eius quod dixerat, subdit quod similiter utrumque se habet ad utrumque, idest sicut se habet deficere ad mutari, ita defecisse ad mutatum esse. Ex praemissis autem argumentatur ad propositum ostendendum in una specie mutationis, quae scilicet est inter contradictorie opposita, scilicet inter esse et non esse, ut patet in generatione et corruptione. Patet enim ex praemissis, quod omne quod mutatur deficit a termino a quo, et quod mutatum est iam defecit. Quando ergo aliquid mutatum est a non esse in esse, iam defecit a non esse; sed de quolibet verum est dicere, quod aut est aut non est: quod ergo mutatum est de non esse in esse, quando mutatum est, est in esse: et similiter quod mutatum est de esse in non esse, oportet quod sit in non esse. Manifestum est ergo quod in mutatione quae est secundum contradictionem, quod mutatum est, est in eo ad quod mutatum est. Et si est verum in ista mutatione, pari ratione est verum in aliis mutationibus. Ex quo patet id quod primo propositum est.	819. He says therefore first (625 235 b6) that because whatever is being changed is being changed from one term to the other, then when the subject of change has now been changed, it has to be in the terminus ad quem. Then at (626 235 b8) he proves this proposition with two arguments, the first of which is particular and the second universal. The first argument is this: Everything being changed must either (1) be distant from the term at which the change starts, as is evident in local motion, in which the place from which the motion starts remains and the mobile gets to be distant from it; or (2) the terminus a quo must cease to be, as in the motion called alteration: for when something white becomes black, the whiteness ceases to be. In order to explain this proposition he adds that either the process of being changed is the same as departing, or the latter is a consequence of change and, therefore, “to have departed” (from the terminus a quo) is a consequence of having been changed. But it is evident that they are the same in reality but different in conception. For “departing” is spoken of in relation to the terminus a quo, whereas “change” gets its name from the terminus ad quem. And in explanation of this, Aristotle adds that “both are related to both in a similar way”, i.e., as “departing” is related to “being changed”, so “having departed” is related to “having been changed”. From these premises he argues to the conclusion, using as his example the species of change that involves terms contradictorily opposed, where the transition is between being and non-being, as in generation and ceasing-to-be. For it is evident from the foregoing that whatever is being changed departs from the terminus a quo and that whatever has been changed has already departed. When, therefore, something has been changed from non-being to being, it has already departed from non-being. But of anything at all it is true to say that it either is or is not. Therefore, what has been changed from non-being to being is in being, when the change is over. Likewise, what has been changed from being to non-being must be in non-being. Therefore, it is evident that in the change which involves contradictories, the thing which has been changed exists in that into which it has been changed. And if it is true in that type of change, then for an equal reason it is true in other changes. From this the first proposition is clear.
lib. 6 l. 7 n. 3 Secundam rationem generalem ponit ibi: amplius autem etc.: et dicit quod hoc idem potest esse manifestum considerando secundum unamquamque mutationem. Et manifestat in mutatione locali. Omne enim quod mutatum est, necesse est esse alicubi, vel in termino a quo vel in aliquo alio. Sed quia illud quod mutatum est, iam defecit ab eo ex quo mutatum est, necesse est quod sit alibi. Aut igitur necesse est quod sit in hoc de quo intendimus, scilicet in termino ad quem, aut in alio. Et si est in hoc, habetur propositum: si autem in alio, ponamus quod aliquid moveatur in b, et quando mutatum est non sit in b sed in c. Tunc oportebit dicere quod etiam de c mutetur in b; quia c et b non sunt habita, idest consequenter se habentia. Oportet enim quod tota huiusmodi mutatio sit continua; et in continuis unum signum non est consequenter se habens ad alterum, quia necesse est quod cadat in medio aliquid sui generis, ut supra probatum est. Unde sequetur, si illud quod mutatum est, quando mutatum est, sit in c, et de c mutetur in b, quod est terminus ad quem, quod quando mutatum est, tunc mutatur in quod mutatum est; quod est impossibile. Non enim simul est mutari et mutatum esse, ut supra dictum est. Nihil autem differt si huiusmodi termini c et b accipiantur in motu locali, vel in quacumque alia mutatione. Necesse est ergo universaliter verum esse, quod id quod mutatum est, quando mutatum est, est in hoc ad quod mutatum est, idest in termino ad quem. Et ex hoc ulterius concludit, quod illud quod factum est, quando factum est, habet esse; et quod corruptum est, quando corruptum est, est non ens. Ostensum est enim universaliter hoc de omni mutatione, et maxime manifestum est in mutatione, quae est secundum contradictionem, ut ex dictis patet. Sic igitur manifestum est, quod id quod mutatum est, cum primo mutatum est, est in illo ad quod mutatum est. Addit autem primo; quia postquam mutatum est ad aliquid, posset exinde moveri, et ibi non esset; sed quando primo mutatum est, oportet quod sit ibi.	820. Then at (627 235 b18) the second argument, a general one, is given, And he says that the same conclusion can be proved by considering any change at all. And he picks local motion; Whatever has been changed must be somewhere, i.e., either in the terminus a quo or in some other. But since what has been changed has already departed from that from which it has been changed, it must be elsewhere. Therefore, it must be either in that in which we are trying to prove it is, i.e., in the terminus ad quem or elsewhere. If it is in the former, our point is proved; if not, then let us suppose that something is being moved into B and when the change is finished the thing is not in B but in C. Then we must say that from C it is also changed into B, because B and C are not consecutive. For a change of the type under discussion is continuous, and in continua one part is not consecutive to another, because between two parts there occurs a part that is similar to those two, as was proved above. Hence, it will follow, if that which has been changed is in C when it has been changed and from C it is being changed to B (which is the terminus ad quem), that when it has been changed, it is also being changed into what it has already become—which is impossible, For “being changed” and “having been changed” are never simultaneous, as we have shown above. Now it makes no difference whether the termini C and B are applied to local motion or to any other change. Consequently, it is universally true that what has been changed is (when it has been changed) in that into which it has been changed, i.e., in the terminus ad quem. From this he further concludes that what has been changed is, as soon as it has been changed, in that into which it has been changed. He added “as soon as”, because after it has been changed into something, it could depart from it and not be there; but as soon as it has been changed, it must be there.
lib. 6 l. 7 n. 4 Deinde cum dicit: in quo autem primo mutatum est etc., ostendit quod mutatum esse primo et per se est in indivisibili: et dicit quod illud tempus in quo primo mutatum est quod mutatum est, necesse est quod sit atomum, idest indivisibile. Quare autem addit primo, exponit subdens quod in illo primo dicitur aliquid mutatum esse, in quo non dicitur esse mutatum ratione alicuius suae partis: sicut si dicatur aliquod mobile mutatum esse in die, quia mutatum est in aliqua parte illius diei; non enim primo mutatur in die. Quod autem illud temporis in quo primo mutatum est sit indivisibile, sic probat. Si enim sit divisibile, sit ac, et dividatur secundum b: necesse est dicere quod aut in utraque mutatum sit, aut in utraque parte mutetur, aut in una parte mutetur et in alia sit mutatum. Sed si in utraque parte mutatum est, non primo mutatum est in toto, sed in parte. Si vero detur quod transmutetur in utraque parte, oportebit dicere quod transmutetur in toto: sic enim dicitur aliquid in toto tempore mutari, quia mutatur in qualibet eius parte. Hoc autem est contra positum: positum enim erat quod in toto ac erat mutatum. Si autem detur quod in una parte mutetur et in alia sit mutatum, sequitur idem inconveniens, scilicet quod non sit primo mutatum in toto; quia cum pars sit prior toto, et prius mutetur aliquid in parte temporis quam in toto, sequetur quod sit aliquid prius primo, quod est impossibile. Oportet ergo dicere quod illud temporis in quo primo aliquid mutatum est, sit indivisibile. Ex hoc autem ulterius concludit, quod omne quod corruptum est, et omne quod factum est, est in indivisibili temporis factum et corruptum; quia generatio et corruptio sunt termini alterationis. Unde si quilibet motus terminatur in instanti (idem est enim primo mutatum esse, quod terminari motum), sequitur quod generatio et corruptio sint in instanti.	821. Then at (628 235 b30) he shows that “to have been changed” is first and per se in an indivisible; and he says that that time in which what has been changed was first changed must be indivisible. ‘Why he adds “first” he explains by saying that A is said to have been first changed as soon as it is not said to have been changed merely by reason of any of its parts. For example, if we say that a mobile has been changed in a day, because it was changed in some part of the day. in that case it was not first changed in the day. But that the time in which something has been first changed is indivisible he now proves: If the said time were divisible, let it be AC and let it be divided at B. Now three things are possible: either (1) the change is over in each part or (2) it is going on in each part or (3) in one part it is going on and in the other it is over. Now, if in each part it is over, then it was first completely changed not in the whole but in the part; but if it is being changed in each part, then it is also being changed in the whole (for the reason why something is said to be changing in a whole period of time is that the change was going on during each part of the whole time). But this is against our assumption that in the whole of AC it had been changed. On the other hand, if it be supposed that in one part of the time it is being changed and in the other part it has been changed, the same difficulty ensues; namely, that it was not first changed in the whole time, because since the part is prior to the whole and something is in motion in a part of time before it is moved in the entire time, it follows that there was something prior to the first, which is impossible. Consequently, it must be admitted that the time in which the thing was first completely changed is indivisible, From this he further concludes that everything that has ceased to be and everything that has been completely made, was made and ceased to be in an indivisible of time, because generation and ceasing to be are the termini of alteration. Consequently, if a motion is terminated in an instant (for these two things are the same, i.e., the termination of a motion and to have been first changed), it follows that generation and ceasing-to-be occur in an instant.
lib. 6 l. 7 n. 5 Deinde cum dicit: dicitur autem in quo primo etc., ostendit quomodo in motu possit accipi primum. Et circa hoc duo facit: primo proponit veritatem; secundo probat, ibi: sit enim primum et cetera. Dicit ergo primo, quod hoc quod dicitur in quo primo mutatum est aliquid, potest intelligi dupliciter. Uno modo in quo primo mutatio est perfecta vel terminata: tunc enim verum est dicere quod mutatum est, quando iam mutatio est perfecta. Alio modo potest intelligi in quo primo mutatum est, idest in quo primo incepit mutari, non in quo primo fuit verum dicere quod iam mutatum esset. Primo igitur modo accipiendo, scilicet secundum terminationem mutationis, dicitur in motu, et est in eo quod primo mutatum est. Contingit enim aliquando primo terminari mutationem, quia cuiuslibet mutationis est aliquis terminus. Et hoc modo intelleximus quod primo mutatum est esse indivisibile; et ostensum est hoc hac ratione: quia est finis, idest terminus motus; omnis autem terminus continui indivisibilis est. Sed si accipiatur quod primo mutatum est secundo modo dicendi, scilicet secundum principium, idest secundum primam partem motus, sic non est in quo primo mutatum est. Non enim est accipere aliquod principium mutationis, idest aliquam primam partem mutationis, quam non praecedat alia pars. Similiter etiam non est accipere aliquid primum in tempore, in quo primo mutetur.	822. Then at (629 236 a7) he shows how to discern in a motion, that which is first. About this he does two things: First he proposes the truth; Secondly, he proves it at 823. He says therefore first (629 236 a7) that the expression “in which something has been first changed” has two interpretations: first, it can mean that in which the change is first complete or terminated —in which case it is true to say that something has been changed, when the change is now over. Secondly, it can mean that in which it first began to be changed, and not that in which it was first true to say that it has been changed. Taken in the first sense, namely, according to the termination of the change, it is applied to instances of motion in which there exists a first in which something has been changed. For a change can be first terminated some time, because every change has a termination. It was in this sense that we understood that “that in which something was first changed” is an indivisible—which was proved on the ground that it is the end, i.e., the terminus, of the motion—and we know that every terminus of a continuum is an indivisible. But if it is taken in the second sense, namely, according to the beginning of the change, i.e., according to the first part of the motion, then there is no first in which something has been changed. For no beginning of a change can be definitely pointed out, i.e., no part that is not preceded by some other part. In like manner, it is not possible to isolate a first time in which something is first being moved.
lib. 6 l. 7 n. 6 Deinde cum dicit: sit enim primum etc., probat quod non est accipere primum in quo mutatum est, ex parte principii. Et primo ratione accepta ex parte temporis; secundo ex parte mobilis, ibi: neque igitur in eo quod mutatum est etc.; tertio ex parte rei in qua est motus, ibi: ipsum autem quod mutatur et cetera. Circa primum ponit talem rationem. Si est aliquod temporis in quo primo mutatum est, sit illud ad. Hoc igitur aut est divisibile aut indivisibile. Si est indivisibile, sequuntur duo inconvenientia: quorum primum est, quod ipsa nunc in tempore sint habita, idest consequentia. Quod quidem inconveniens hac ratione sequitur, quia tempus dividitur sicut et motus, ut supra ostensum est. Si autem aliqua pars motus fuerit in ad, necesse est dicere quod ad sit aliqua pars temporis; et ita tempus erit compositum ex indivisibilibus. Indivisibile autem temporis est ipsum nunc: sequetur ergo quod ipsa nunc consequenter se habeant in tempore. Secundum inconveniens est. Ponamus enim quod in tempore quod praecedit ipsum ad, quod est ca, idem mobile quod ponebatur moveri in ad, totaliter quiescat. Si ergo in toto ca quiescit, sequitur quod quiescat in a, quod est aliquid eius. Si ergo ad est indivisibile, ut datum est, sequetur quod simul aliquid quiescat et moveatur: conclusum est enim quod quiescit in a, et positum erat quod in ad moveretur. Idem autem est a et ad, si ad sit indivisibile. Sequetur ergo quod in eodem quiescat et moveatur. Sed advertendum est, quod non sequitur si aliquid quiescit in toto tempore, quod quiescat in ultimo eius indivisibili: quia ostensum est supra, quod in nunc neque movetur aliquid neque quiescit. Sed Aristoteles hoc concludit hic ex hoc quod ponitur ab adversario: quod id temporis in quo primo movetur, est indivisibile. Et si contingit moveri in indivisibili temporis, contingit eadem ratione in indivisibili temporis quiescere. Remoto ergo quod ad, in quo dicitur primo moveri, sit impartibile, relinquitur quod necesse sit illud esse divisibile: et ex quo in ad ponitur primo moveri, sequitur quod in quolibet eius moveatur. Quod sic probat. Dividatur enim ipsum ad in duas partes: aut igitur in neutra parte mutatur, aut in ambabus, aut in altera parte tantum. Si in neutra mutatur, sequitur quod neque in toto: sed si mutetur in ambabus partibus, tunc poterit poni quod mutatur in toto: sed si in altera tantum moveatur, sequetur quod moveatur in toto, sed non primo, sed ratione partis. Quia igitur primo ponitur moveri in toto, oportet hoc accipere, quod in qualibet parte eius moveatur. Sed tempus dividitur in infinitum, sicut et quodlibet continuum; et ita semper est accipere partem minorem ante partem maiorem; sicut si acciperem diem ante mensem, et horam ante diem. Manifestum est ergo quod non est accipere aliquid temporis in quo primo moveatur; ita scilicet quod non sit accipere aliquam partem eius, in qua primo moveatur. Sicut si daretur quod dies est in quo primo aliquid movetur, hoc non potest esse; quia in parte eius, scilicet in prima hora diei, primo movetur quam in toto die.	823. Then at (630 236 a17) he proves that if one looks at the beginning of a motion, it is not possible to assign “a first in which something has been changed”. First with an argument from time; Secondly, with an argument from the mobile, at 824; Thirdly, with an argument from the sphere in which the motion occurs, at 825. As to the first he gives this reason: If there is any element of time in which something has been first changed, let it be AD. Now AD must be either divisible or indivisible. If the latter, two difficulties ensue. The first is that the “now’s” in time are consecutive. This difficulty follows from the fact that time is divided just like motion, as was shown above. But if any part of the motion was present in AD, then AD must have been a part of time and, consequently, time will be composed of indivisibles. However, the indivisibles of time are the “now’s”. It will follow, therefore, that the “now’s” are consecutive in time. And there is a second difficulty. Let us suppose that in the time CA, which preceded AD, the same mobile that was being moved in time AD was entirely at rest. If, therefore, it was at rest in the entire time CA, it was at rest in A, which is an element of the time CA. If, therefore, (as we supposed) AD is indivisible, it’ will follow that a thing is at rest and in motion at the same time; for we have already concluded that it was at rest in A and assumed that it was in motion in AD. But if AD is indivisible, then A is the same as AD. It will follow, therefore, that a thing is at rest and in motion in the same time. It should be noted, however, that if a thing was at rest throughout an entire time, it does not follow that it was at rest in the last indivisible of that time; for we have already shown that in the “now” things are neither at rest nor in motion. But Aristotle concludes this here by arguing from what his adversary has proposed, namely, that the element of time in which the object was first being moved is an indivisible. And if it can be in motion in an indivisible of time, there is no reason why it could not also be at rest. Therefore, having rejected the indivisibility of time AD, we are left with the fact that it is divisible. And since it is in AD that the object is said to be first moved, then it is being moved in any part of AD. This he now proves: Let AD be divided into two parts, Then the object is being moved either in neither part or in both parts or in one part only. If in neither part, then not in the whole time. If in both parts, then it could be granted that it is being moved in the whole time. But if in one part only, it will follow that it is being moved in the whole time but not first, but by reason of the part. Therefore, since it is agreed to be moving in the whole time, it has been in motion in each part of the whole time. But time is divided infinitely just like any continuum; consequently, it is possible always to consider a part smaller than a previous one; for example, a day before a month and an hour before the day. Therefore, it is evident that it is impossible to find a time in which it is first being moved so that a previous could not be found. For if you were to assume that it is in a day that the object is first moved, that assumption would not be true, because it would have been first moved in the first part of the day, before it was moved in the whole day.
lib. 6 l. 7 n. 7 Deinde cum dicit: neque igitur in eo quod mutatum est etc., ostendit idem ex parte mobilis; concludens ex praemissis quod neque in ipso quod mutatur est accipere aliquid quod primo mutetur. Quod quidem intelligendum est secundum quod per motum totius vel partis aliquod determinatum signum pertransitur: manifestum est enim quod primo pertransit aliquid determinatum prima pars mobilis, et secundo secunda, et sic deinceps. Alioquin si intelligeretur de motu absolute, non haberet locum quod hic dicitur: manifestum est enim quod simul movetur totum et omnes partes eius: sed non simul pertransit aliquid determinatum, sed semper pars ante partem. Unde sicut non est accipere primam partem mobilis, ante quam non sit alia pars minor; ita non est accipere aliquam partem mobilis, quae primo moveatur. Et quia tempus et mobile similiter dividuntur, ut supra ostensum est, convenienter ex eo quod demonstratum est de tempore, concludit idem de mobili: et probat sic. Sit mobile ipsum de: et quia omne mobile divisibile est, ut supra probatum est, sit pars eius quae primo movetur dz. Et moveatur dz pertranseundo aliquod determinatum signum in tempore quod sit ti. Si igitur dz mutatum est in toto hoc tempore, sequitur quod illud quod mutatum est in medio temporis, sit minus et prius motum quam dz; et eadem ratione erit aliud prius isto, et iterum aliud prius illo, et sic semper; quia tempus in infinitum dividitur. Manifestum est ergo quod in mobili non est accipere aliquid quod primo mutatum est. Et sic patet quod primum in motu non potest accipi neque ex parte temporis neque ex parte mobilis.	824. Then at (631 236 a27) he establishes the same point by considering the mobile, and he concludes from the foregoing that neither in that which is being changed is it possible to take something that is first changed. Now this is to be understood in the sense that some definite point is to be crossed ‘through ‘the motion of the whole or of the part: for it is evident that the first part of the mobile will first pass a given point, and a second part will pass it after that, and so on. Otherwise, if it were understood in the sense of the absolute nature of motion, what we have to say would not be ad rem: for it is clear that the whole is being moved at the same time as all the parts, but the whole does not pass a certain point all at once but part before part continuously. Hence, just as it is impossible to find a first part of the mobile than which there is not a previous smaller part, so also is it impossible to isolate a part of the mobile that would be first moved. And because time and mobile are correspondingly divided, as we have shown above, then what was concluded about time, he now concludes about the mobile. Here is his proof: Let DE be a mobile and (because every mobile can be divided, as was proved above) let DZ be the part that is first being moved. And let DZ be moved so that it passes a definite point in the time TI. if, therefore, DZ has been changed in this whole time, it follows that what has been changed in half the time is both less than DZ and moved prior to DZ. And for the same reason there will be something prior to that and so on forever, because time can be divided infinitely, it is evident, therefore, that in the mobile one cannot find something that has been first changed. Hence it is clear that a first cannot be found in motion, whether we consider the time or the mobile.
lib. 6 l. 7 n. 8 Deinde cum dicit: ipsum autem quod mutatur etc., ostendit idem ex parte rei in qua est motus. Praemittit tamen quod non similiter se habet de eo quod mutatur, vel ut melius dicatur secundum quod mutatur, sicut de tempore et mobili. Cum enim sit tria accipere in mutatione, scilicet mobile quod mutatur, ut homo; et in quo mutatur, ut tempus: et in quod mutatur, ut album; horum duo, scilicet tempus et mobile, sunt semper divisibilia. Sed de albo est alia ratio: quia album non est divisibile per se, sed tamen tam ipsum quam omnia alia huiusmodi, sunt divisibilia per accidens, inquantum scilicet illud cui accidit album vel quaecumque alia qualitas, est divisibile. Divisio autem albi per accidens potest esse dupliciter. Uno modo secundum partes quantitativas; sicut si superficies alba dividatur in duas partes, album per accidens divisum erit. Alio modo secundum intensionem et remissionem: quod enim una et eadem pars sit magis vel minus alba, non est ex ipsa ratione albedinis (quia si esset separata, non diceretur secundum magis et minus; sicut neque substantia suscipit magis et minus): sed est ex diverso modo participandi albedinem ex parte subiecti divisibilis. Praetermisso igitur hoc quod dividitur per accidens, si accipiamus ea secundum quae est motus, quae dividuntur per se et non per accidens, neque etiam in his erit primum. Et manifestat hoc primo in magnitudinibus, in quibus est motus localis. Sit enim magnitudo spatii in quo est ab, et dividatur in c: detur ergo quod ex b in c aliquid primo moveatur. Aut igitur bc est divisibile, aut indivisibile. Si indivisibile, sequitur quod impartibile erit coniunctum impartibili; quia eadem ratione secunda pars motus erit in impartibili; sic enim oportet dividere magnitudinem, sicut et motum, ut supra de tempore dictum est. Si autem bc sit divisibile, erit accipere aliquod signum prius, idest propinquius ipsi b, quam c; et sic prius mutabitur ex b in illud, quam in c: et iterum illo erit accipere aliud prius, et sic semper, quia divisio magnitudinis non deficit. Patet ergo quod non est accipere aliquod primum in quod mutatum sit motu locali. Et similiter manifestum est in mutatione quantitatis, quae est augmentum et decrementum: quia haec etiam mutatio est secundum aliquod continuum, scilicet secundum quantitatem accrescentem vel subtractam; quae cum sit in infinitum divisibilis, non est in ea accipere primum. Et sic manifestum est, quod in sola mutatione quae est secundum qualitatem, contingit aliquid esse indivisibile per se. Inquantum tamen est divisibile per accidens, similiter non est accipere primum in mutatione tali: sive accipiatur successio mutationis inquantum pars post partem alteratur (manifestum est enim quod non erit accipere primam partem albi, sicut nec primam partem magnitudinis); sive accipiatur successio alterationis secundum quod aliquid idem est albius vel minus album; quia subiectum infinitis modis potest variari secundum magis album et minus album. Et sic motus alterationis potest esse continuus, et non habens aliquid primum.	825. Then at (632 236 a35) he proves the same thing by considering the sphere in which the motion occurs. But first he mentions that the situation with respect to the sphere in which the motion occurs is not exactly the same as it was with respect to time and the mobile. For since there are three things to be considered in change; namely, the mobile which is being changed (for example, a man) and that in which it is being changed, i.e., the time, and that into which it is being changed (for example, into white), two of these, namely, the time and the mobile are always divisible. But with white it is another story, because a white thing is not divisible per se, but it, and things like it, are divisible per accidens, inasmuch as the subject of whiteness or of any other quality is divisible. Now the per accidens division of white can take place in two ways. In one way according to the quantitative parts, as when a white surface is split into two parts, the white will be divided per accidens. In another way, according to greater or less intensity, for the fact that one and the same part is whiter or less white is not due to the nature of whiteness (because if it existed in isolation, whiteness would be constant and never subject to more and less, any more than a substance is susceptible of more and less) but to the varying degrees in which a divisible subject participates whiteness. Therefore, neglecting what is divided per accidens in the sphere of motion and considering only what is divided per se in those spheres, it is impossible to find a first. And he proves this first of all in magnitudes in which there is local motion. Let the magnitude AC be divided at B, and suppose that C is that into which something is first moved from B. Now BC is either divisible or indivisible. If the latter, it follows that an indivisible will be touching an indivisible, for there is no reason why the second part of the motion will not be into an indivisible, since we can divide a magnitude just as the motion was divided, and as time was. But if BC is divisible, it is possible to take a stage nearer to B than to C, and so the thing will be changed from B into that stage before it is changed into C and into a stage prior to that one, and so on, because there is no limit to the division of a magnitude. It is therefore evident that it is impossible to find a first stage into which a thing has been changed in local motion. The same is true in change of quantity, i.e., growing and decreasing. For even these changes are in terms of a continuum, i.e., in terms of added quantity or subtracted quantity, in which no first is to be found, since there can be division ad infinitum. And so it is clear that it is only in qualitative change that something is per se indivisible. But inasmuch as in this per accidens divisibility is found, likewise no first is discernible in such change. This is true whether the succession consists in part being altered after part (for it is evident that no first part of white can be found any more than a first part of magnitude can) or whether the succession is based on one and the same thing becoming more and more white or less and less white, for a subject can be modified in an infinite number of ways with regard to degrees of whiteness, Thus the motion involved in alteration can be continuous and not possess a first.