Authors/Thomas Aquinas/physics/L8/lect17

Lecture 17 Certain doubts resolved

Latin	English
	Lecture 17 Certain doubts resolved.
lib. 8 l. 17 n. 1 Postquam philosophus ostendit quod motus reflexus non potest esse continuus et unus, hic secundum praemissa solvit quasdam dubitationes. Et dividitur in partes tres, secundum tres dubitationes quas ex praemissis solvit: secunda pars incipit ibi: eodem autem modo obviandum est etc.; tertia ibi: manifestum autem et cetera. Circa primum duo facit: primo ponit dubitationem; secundo solvit eam, ibi: non ergo ponendum est et cetera.	1112. After showing that a reflex motion is neither continuous nor one, the Philosopher now settles some doubts on the basis of what has gone before. And it is divided into three parts according to the three doubts he resolves from the foregoing. The second part begins at 1115; The third, at 1119. About the first he does two things: First he sets forth the doubt; Secondly, he solves it, at 1114.
lib. 8 l. 17 n. 2 Dicit ergo primo, quod hoc quod dictum est ad probandum quod motus reflexus non est continuus, potest etiam dici ad solvendum quandam dubitationem. Est enim una talis dubitatio. Sint duae magnitudines aequales, quarum una dicatur e et alia z. Sint etiam duo mobilia aequaliter velocia, quorum unum sit a et aliud sit d; et moveatur a continue ab extremo, idest principio magnitudinis, ad c; d vero feratur ad I. Et ponamus quod in magnitudine quae est e, signetur quoddam signum medium quod est b, quod tantum distet a c, quantum in magnitudine quae est z, distat z ab I. Et ponamus quod simul dum a in suo motu continuo accedit ad b signum, quod d mobile in suo motu continuo recedat a z, et veniat ad I. Cum ergo motus sint regulares et aeque veloces utriusque mobilis, sequetur quod prius veniet d in I, quam a veniat in c: quia quod prius recedit, prius perveniet ad finem aequalis magnitudinis; prius autem recessit d a z, quam a recederet a b, quia d recessit a z quando a pertingebat ad b. Ergo secundum hoc a non simul advenit in b, et recessit ab eo; et ita sequitur quod posterius recesserit quam advenerit: quia si simul adveniret et recederet, non posterius moveri inciperet. Et ita necessitas est quod a, dum continue fertur, quiescat in b: et sic motus continuus erit compositus ex quietibus, sicut Zeno ponebat, ut supra habitum est in sexto.	1113. He says therefore First (874 262 b8) that what was said in order to prove that a reflex motion is not continuous may be applied to solving a certain doubt, which is this: Assume two equal magnitudes, one called E, and the other Z. Let A and D be two equally swift mobiles, such that A is continuously moved from the beginning of the magnitude (E) to C, and D (along Z) to I, And let us assume that in the magnitude E there is an intermediate point B, which is as far from C as a like point Z on Z is distant from I. Let us further assume that at the same time that A in its continuous motion is approaching B, D in its continuous motion is receding from Z and going to I. Now, since these motions are regular and equally swift, it will follow that D will arrive at I before A arrives at C, because the one which starts first will first arrive to the end of an equal distance. But D left Z before A left B, because D left Z when A was arriving at B. Therefore, according to this, A did not simultaneously arrive at B and leave B, and it consequently follows that it departed after it arrived, because if it arrives and departs at the same time, it will not have begun to move later. And so it is necessary that A, while being carried along, rest in B. Therefore a continuous motion will be composed of periods of rest, as Zeno claimed in Book VI.
lib. 8 l. 17 n. 3 Deinde cum dicit: non ergo ponendum etc., solvit motam dubitationem secundum praemissa. Supponebat enim obiectio praedicta quod a, dum continue movetur, accedit ad aliquod signum in medio magnitudinis positum, scilicet ad b, et quod simul dum accedit a ad b, d recedit a quodam alio signo, scilicet a z; quod est contra praemissa. Dictum est enim supra, quod cum aliquid continue fertur, neque potest adesse neque abesse, idest recedere et accedere, a signo medio. Ergo non est ponendum hoc quod obiectio supponebat, quod cum a affuit, idest accessit ad b, ipsum d simul recessit a z: quia si detur quod a accessit ad b, erit pari ratione dare quod recesserit, et quod hoc non fuerit simul, sed in duobus instantibus, ita quod in tempore intermedio quieverit. Sed sicut dictum est prius, cum aliquid continue movebatur, in aliquo signo medio non aberat et aderat, sed simpliciter erat; non quidem per aliquod tempus, quia sic quiesceret, sed in decisione temporis, idest in aliquo nunc, quod dividit tempus. Hoc ergo quod obiectio supponebat, scilicet quod a adesset, et quod d abesset ab aliquo signo medio, impossibile est dicere in motu continuo. Sed in reflexo necesse est ut ita dicatur. Si enim aliquod mobile quod est I, feratur ad punctum quod est d, et iterum reflectatur, manifestum est quod mobile utitur ultimo quod est d, quasi principio et quasi fine, scilicet uno signo ut duobus: unde necesse est quod ibi quiescat. Nec est dicendum quod simul accesserit ad ipsum d, et recesserit ab eodem: quia sequeretur quod simul in eodem instanti esset ibi et non esset. Omne enim quod motum est, est in termino ad quem movebatur; et omne quod incipit moveri, non est in termino a quo incipit moveri: hoc autem significatur, cum dicimus adesse vel accedere, quod est terminari motum ad punctum illud; cum autem dicimus abesse vel recedere, significamus motum incipere. Unde necesse est omne quod accedit vel adest ad aliquod signum, esse in eo: quod autem abest vel abscedit, non esse in eo. Quia ergo impossibile est simul esse et non esse in aliquo signo, per consequens impossibile est quod simul adsit et absit eidem, ut superius pluries est suppositum. Est autem hic attendendum quod aliter utitur hic litteris quam supra. Utitur enim hic I pro mobili, d vero pro termino: supra autem e converso. Non est autem in motu reflexo danda solutio, quae prius data est in motu continuo. Non enim potest dici quod mobile quod est I, sit in termino quod est d, a quo incipit reflecti, solum in decisione temporis, idest in nunc; et quod mobile neque affuerit neque defuerit eidem, sicut dicebatur in motu continuo: quia in motu reflexo necesse est venire ad finem qui est actu finis, et non in potentia tantum, sicut medium in motu continuo erat principium et finis solum in potentia. Illud ergo quod est in medio motus continui, est in potentia tantum principium et finis; sed hoc a quo incipit reflexio, est actu principium et finis: finis quidem motus qui erat deorsum, puta lapidis; principium autem est in actu motus reflexi qui est sursum, dum lapis cadens in terram resilit sursum. Sicut ergo in magnitudine in qua est motus, signum a quo reflectitur est principium et finis in actu; ita et in ipsis motibus est accipere actu finem unius et principium alterius: quod non esset, nisi quies interveniret media. Necesse est ergo quod id quod reflectitur in linea recta, quiescat. Et ita sequitur quod in recta magnitudine non possit esse motus continuus et perpetuus: quia magnitudo recta non est infinita, et ita non posset esse in perpetuum motus rectus continuus, nisi reflecteretur.	1114. Then at (875 262 b15) he resolves this doubt in the light of the foregoing. For the objection supposed that A in its continuous motion arrives at a point B in the magnitude and that at the same time that A arrived at B, D left the point Z—which is against what was had above. For it was said above that when something is being moved continuously, it can neither arrive at, nor depart from, any intermediate point. Therefore, what the objection assumes must not be assumed, i.e., that when A was at, i.e., approached B, D was departing from Z, because if it be granted that A arrived at B, then for the same reason it should be granted that it left B, and that this did not occur simultaneously, but in two instants, so that in the intermediate time between the two instants it was at rest. But as was said previously, when something was being continuously moved, it was neither departing from, nor approaching, a given point, but was simply there—and this not for a time, because then it would have been resting, but in a division of time, i.e., in some “now,” which divides time. Therefore, what the objection assumed, namely, that A arrived at and that D left some intermediate point is impossible to state in a continuous motion. But in a reflex motion this must be stated. For if a mobile I is moved to the point D and is then rebounded, it is plain that the mobile uses the ultimate, which is D, as a beginning and as an end, i.e., the point is used for two things, hence it had to be at rest there. Nor can it be said that it simultaneously arrived at and left D, because then it would have been, and not have been, there in the same instant. For whatever has been moved exists in the terminus to which it was being moved, and whatever begins to be moved is not in the terminus from which it begins to be moved. But when we use the expression “to be at” or “to approach,” we mean that a motion is being terminated at that point, and when we say “to be away from” or “to depart,” we mean that the motion is beginning. Hence, it is necessary that whatever arrives at, or is at, a point, be in it, while what is leaving it or is departing from it, be not in it. Since, therefore, it is impossible to be and not to be in a given point at the same time, it is consequently impossible to be at once at and away from the same, as the objection more than once assumed. It should be noted that here he uses different letters from those used above. Here I is the mobile and D the terminus; above, it was the opposite. But the solution given for continuous motion is not to be used with respect to a reflex motion. For it cannot be said that the mobile I is in the terminus D, from which it began to be reflected, only in the division of time, i.e., only during the “now,” and that the mobile neither arrived at, nor departed from, the same, as was said with respect to a continuous motion. For in a reflex motion an end must be reached that is an actual end, and not merely a potential one, as the intermediate point in a continuous motion was only potentially a beginning and an end. Therefore, that which is an intermediate point of a continuous motion is only potentially a beginning and an end; but the point from which a reflex motion begins is actually a beginning and end. For example, it is the end of the downward motion of a stone, and the beginning of its upward motion, in the case of a stone falling to earth and bouncing upward. Therefore, just as in the magnitude in which a motion is occurring, a point from which the motion is reflexed is both an actual beginning and end, so also in the motions themselves, there is actually an end of one and a beginning of the other. And this would not be so, unless an interval of rest occurred. Therefore, it is necessary that what is reflected in a straight line be at rest. And so it follows that on a straight magnitude there cannot be a continuous and perpetual motion, because no straight magnitude is infinite. And so there could not be perpetual continuous rectilinear motion, unless reflexion is involved.
lib. 8 l. 17 n. 4 Deinde cum dicit: eodem autem modo obviandum etc., ponit secundam dubitationem. Et circa hoc tria facit: primo movet dubitationem; secundo excludit quandam solutionem supra in sexto positam, ibi: in primis igitur etc.; tertio ponit veram solutionem, ibi: sed verum dicendum et cetera. Dicit ergo primo, quod eodem modo per ea quae supra ostensa sunt, possumus obviare ad eos qui ponunt obiectionem Zenonis, qui sic volebant argumentari. Omne quod movetur oportet quod prius pertranseat medium quam perveniat ad finem: sed inter quoslibet duos terminos sunt infinita media, propter hoc quod magnitudo est divisibilis in infinitum; et ita impossibile est transire media, quia infinita non contingit transire: ergo nihil potest movendo ad aliquem terminum pervenire. Vel potest eadem dubitatio aliter formari, sicut quidam eam proponunt. Omne quod pertransit aliquod totum, prius pertransit medietatem: et cum medietas iterum dividatur in medietatem, oportet quod prius pertransierit medietatem medietatis: et ita omne quod movetur, numerat quamlibet medietatem, pertingendo ad ipsam. Sed medietates sic accipi possunt in infinitum: ergo sequitur quod si aliquid pertransit totam magnitudinem, quod numeravit numerum infinitum; quod est manifeste impossibile.	1115. Then at (876 263 a3) he presents the second doubt. About this he does three things: First he mentions the doubt; Secondly, he rejects a solution given in Book VI, at 1116; Thirdly, he gives the true solution, at 1118. He says therefore First (876 263 a3) that by the same method, using the things shown above, one can block those who give the objection of Zeno and wish to argue in the following manner: Whatever is being moved must first cross what is intermediate before arriving at the end; but between any two termini there are infinite intermediates on account of a magnitude’s infinite divisibility; and so it is impossible to traverse the intermediates, because infinites cannot be traversed. Therefore, nothing can by motion arrive at any terminus. Again, the same difficulty can be presented under another form, as some do in fact propose it: Whatever traverses a whole must previously traverse the half; and since the half is again divided in half, half of the half must be first traversed. And thus, whatever is being moved counts off every half as it reaches it. But such halves can be multiplied ad infinitum. Therefore, it follows that if anything traverses an entire magnitude, it has counted off an infinite number, which is plainly impossible.
lib. 8 l. 17 n. 5 Deinde cum dicit: in primis igitur rationibus etc., excludit solutionem quam supra in sexto posuerat ad hanc obiectionem: et primo recitat eam; secundo excludit, ibi: sed haec solutio et cetera. Dicit ergo primo quod praedicta obiectio soluta est supra in sexto, cum de motu in communi agebatur, per hoc quod sicut magnitudo dividitur in infinitum, ita et tempus; et sic eodem modo tempus habet in seipso infinita, sicut et magnitudo. Et ita non est inconveniens si infinita quae sunt in magnitudine, transeat aliquis in infinitis quae sunt in tempore: quia non est inconveniens quod infinita magnitudo transeatur tempore infinito; sed sicut in sexto ostensum est, infinitum eodem modo invenitur in magnitudine et in tempore.	1116. Then at (877 263 a11) he rejects the solution he had presented above in Book VI. First he cites it; Secondly, he sets it aside, at 1117. He says therefore first that the foregoing objection was answered in Book VI, when motion in general was being discussed, on the ground that just as a magnitude is divided infinitely, so also is time. Consequently, time possesses infinities in itself in the same way as a magnitude. And so it is not unfitting if the infinites in a magnitude be traversed in the infinites which are in time. For it is not inconsistent for an infinite magnitude to be traversed in an infinite time. But, as shown in Book VI, the infinite is found in magnitude and in time in the same way.
lib. 8 l. 17 n. 6 Deinde cum dicit: sed haec solutio etc., excludit hanc solutionem. Et dicit quod haec solutio sufficiens est ad obviandum interroganti qui sic interrogabat: an contingeret in tempore finito transire et numerare infinita. Quae quidem interrogatio repellebatur per hoc quod dicitur, quod tempus finitum habet infinita, in quibus possunt transiri infinita quae sunt in magnitudine. Sed ista solutio non sufficit ad rei veritatem: quia si aliquis praetermittat quaerere de magnitudine; et praetermittat interrogare an in tempore finito contingat infinita transire; et faciat hanc eandem interrogationem de ipso tempore, utrum scilicet infinita quae sunt in tempore possint transiri, propter hoc quod tempus in infinitum dividitur: ad hanc interrogationem non sufficiet praedicta solutio, et ideo oportet aliam solutionem quaerere.	1117. Then at (878 263 a15) he sets aside this solution, And he says that this solution is sufficient to answer the questioner who asked whether it was possible in a finite time to traverse and count off infinites. This question was retorted by saying that a finite time possesses infinities in which the magnitudinal infinites can be traversed. But that solution does not reach the truth of the matter, because if someone should omit to ask about the magnitude and whether it is possible to traverse infinities in finite time, but asked rather this same question about time, namely, whether the infinites which are in time can be traversed—since time is divided ad infinitum—then the previous solution would not answer this question. Consequently, another solution must be sought.
lib. 8 l. 17 n. 7 Deinde cum dicit: sed verum dicendum est etc., ponit veram solutionem, secundum ea quae supra praemiserat. Et dicit quod secundum veritatem hoc dicendum est ad solutionem dubitationis motae, illud quod praemisimus in rationibus supra positis proxime, scilicet quod si aliquis dividat continuum in duo media, tunc utitur uno signo, scilicet in quo dividitur continuum, tanquam duobus, quia facit ipsum et principium unius partis, et finem alterius. Facit autem hoc numerando, et in duo media dividendo. Cum autem sic divisum fuerit continuum, iam non erit continuum, sive dividatur magnitudo, ut linea, sive dividatur motus: quia nec motus potest esse continuus nisi sit continui, scilicet et subiecti et temporis et magnitudinis super quam transit motus. Sic ergo dividens numerat, et numerando continuitatem solvit. Sed in continuo dum continuitas durat, sunt infinita media non in actu, sed in potentia: quia si faciat aliquis aliquod medium esse in actu, hoc erit per divisionem, ut dictum est, in quantum accipietur ut principium unius et finis alterius; et sic non remanebit continuum, sed stabit; idest iam media in actu non erunt infinita, sed in eis erit status. Quod maxime accidit in eo qui vult numerare media: quia necesse est ei quod unum signum numeret quasi duo, inquantum est unius medietatis finis, et alterius principium. Et hoc dico quando non numeratur totum continuum ut unum, sed numerantur duae medietates in ipso. Si enim accipietur totum continuum ut unum, tunc iam dictum est quod signum medium non accipitur ut finis et principium in actu, sed in potentia tantum. His ergo visis, respondendum est ad eum qui interrogat an contingat infinita transire sive in tempore sive in magnitudine, quod quodammodo contingit, et quodammodo non contingit. Cum enim sint infinita in actu, non contingit ea transire: cum autem sint infinita in potentia, contingit. Et sic cum in continuo non sint infinita media nisi in potentia, contingit infinita transire: quia illud quod continue movetur, secundum accidens transivit infinita, scilicet in potentia. Per se enim transivit lineam finitam, cui accidit quod insint ei infinita media in potentia; sed ipsa linea secundum substantiam et rationem est alia ab illis mediis infinitis. Non enim linea componitur ex punctis: sed puncta possunt signari in linea, inquantum dividitur.	1118. Then at (879 263 a22) he gives the true solution in the light of his premises above. And he says that the true solution of the present doubt requires us to repeat what was premised in the immediately foregoing arguments, namely, that if someone divides a continuum into two halves, he then uses the one point at which the continuum is divided as two, because he is making it serve both as the beginning of one part and as the end of the other. He does this by numbering, and by dividing into two halves. But when a continuum has been divided in this manner, it is no longer a continuum, whether it be a magnitude, such as a line, that is divided, or a motion, for a motion cannot be continuous unless it is the motion of something continuous, namely, as to subjects and time and magnitude traversed. Therefore, the divided in effect counts and by counting breaks the continuity. But so long as continuity endures in a continuum, there is an infinity of intermediates not in act but in potency, for if someone should make some middle actual, it will be due to division, as has been said, insofar as it is taken as the beginning of one and the end of the other. In that case, the continuum will not remain but will “stop,” i.e., the intermediates that are now in act will not be infinite but one will come to a stop in them. This shows up especially in the case of one who wishes to count the intermediates, because he will have to count one as two, inasmuch as it is the end of one half and the beginning of the other. And this, I say, takes place when the whole continuum is not counted as one, but two halves are counted in it, For if the whole continuum is taken as one, it has already been stated that then an intermediate point will not be taken as an actual end and beginning but potentially only. With these facts in mind, the answer to be given to one who asks whether infinites in time or in a magnitude may be traversed is that in one sense it does happen, and in another it does not happen. For when one has infinites in act, it is impossible that they be traversed, but when they are potentially infinite, they can be traversed. And so, since the intermediates in a continuum are infinite only in potency, it does happen that infinites are traversed, because what is in continuous motion traverses per accidens what is infinite, namely, what is infinite in potency, But per se it has traversed a finite line which happens to have an infinitude of intermediates in potency. The line itself, however, in its nature and definition, is distinct from those infinite intermediates. For a line is not a composite of points, but points may be designated in a line insofar as it is divided.
lib. 8 l. 17 n. 8 Deinde cum dicit: manifestum autem et quia etc., solvit tertiam dubitationem. Et circa hoc tria facit: primo ponit dubitationem et solutionem; secundo manifestat utrumque per exempla, ibi: sit tempus etc.; tertio infert quoddam corollarium ex dictis, ibi: si autem quodcumque et cetera. Ponit ergo dubitationem primo, quae solet fieri in generationibus et corruptionibus. Quod enim generatur, desinit non esse et incipit esse. Oportet autem aliud tempus assignari ei quod est esse rei generatae vel corruptae, et aliud ei quod est non esse: puta si ex aere generetur ignis, in toto tempore ab erat non ignis sed aer; in toto autem tempore bc est ignis. Cum ergo hoc signum temporis quod est b, sit utrique tempori commune, videtur quod in illo instanti communi sit simul esse ignis et non esse eiusdem. Hanc ergo dubitationem philosophus solvens, dicit manifestum esse quod nisi aliquis illud signum temporis, quod dividit tempus prius a posteriori, faciat semper esse posterioris rei, idest quod in illo instanti hoc modo se habeat res sicut in tempore sequenti, sequitur quod idem sit ens et non ens simul, et sequitur quod quando aliquid factum est, sit non ens. Tunc enim factum est, quando generatio terminatur, scilicet in illo nunc quod dividit tempus prius et posterius: si ergo in toto tempore priori erat non ens, in hoc etiam nunc quando iam generatum est, est etiam non ens, quia istud nunc est finis prioris temporis. Quomodo autem ista inconvenientia non sequantur ostendit, subdens quod unum et idem numero signum, scilicet nunc, est commune utrique tempori, scilicet priori et posteriori: sed quamvis sit unum subiecto, non tamen est unum ratione, sed duo; est enim finis prioris temporis et principium posterioris. Sed si accipiatur in ipso nunc quod res est, idest si accipiatur secundum quod est unum re, semper tenet se cum posteriori passione. Vel aliter: quamvis ipsum nunc sit finis temporis prioris et principium posterioris, et sic sit communis utrique; tamen secundum quod est rei, idest secundum quod comparatur ad rem quae movetur, semper est posterioris passionis; quia res quae movetur, in illo instanti est subiecta passioni posterioris temporis.	1119. Then at (880 263 b9) he resolves the third doubt. About this he does three things: First he mentions the doubt and its solution; Secondly, he explains each with an example, at 1120; Thirdly, he draws a corollary from the foregoing, at 1122. First therefore (880 263 b9) he states the doubt that is wont to arise with respect to generation and ceasing-to-be. For what is generated ceases not to be, and begins to be. But the time assigned for the existence of a thing that is generated or has ceased to be, must be different from the one assigned to its non-existence. For example, if from air fire is generated, then in the whole time AB there was not fire but air, but in the entire time BC there is fire. Since, therefore, sign B of the whole time ABC is common to both times, it seems that in that common instant the fire both exists and does not exist. The Philosopher therefore solves this doubt, saying that it is plain that, unless someone holds that the point of time which divides a prior time from a later one, “always belongs to the later,” i.e., that in that instant the thing is in the state which it subsequently has, it follows that the same is simultaneously being and non-being, and that when something has been produced, it is non-being. For it is then produced when generation terminates, namely, in that “now” which divides the prior time and the later. If, therefore, in the entire prior time it was non-being, in that “now” also when it has already been generated, it is also non-being, since this “now” is the end of the prior time. How these impossibilities do not follow he explains by adding that one and the came sign as to number, i.e., the “now” is common to both times, namely, to the prior and to the subsequent. But although it be one as to subject, it is not one in conception but two, for it is the end of the prior time and beginning of the subsequent. But if we take the “now” as it is a thing, i.e., if it be taken as it is one in reality, it always belongs with the subsequent state (passion). Or in other words: Although the “now” is the end of the prior time and the beginning of the subsequent, and is thus common to both, yet accordingly as it belongs to the thing, i.e., insofar as it is compared to the thing which is being moved, it always belongs to the subsequent passion, because the thing being moved is in that instant being subject to the passion of the subsequent time.
lib. 8 l. 17 n. 9 Sic ergo obiectione et solutione posita, manifestat utrumque per exempla: et primo obiectionem, cum dicit: sit tempus et cetera. Dicit ergo: sit tempus acb; res autem quae movetur sit d; quod quidem d in a tempore sit album, in b autem non album. Videtur ergo sequi quod in c sit album et non album. Et quomodo hoc sequatur ostendit subdens: si enim in toto tempore a est album, sequitur quod in quolibet accepto in ipso a sit album; et similiter si in toto tempore b est non album, sequitur quod in quolibet ipsius accepto sit non album: cum ergo c sit acceptum in utroque, quia est huius finis et illius principium, videtur sequi quod in c sit album et non album.	1120. Having given the objection and its solution, he explains both with examples. And first the objection, at (881 263 b15). He says therefore: Let ACB be the time, and D the thing that is being moved, so that, in time A, D is white, and in B it is non-white. It seems therefore to follow that in C it is white and non-white. How this follows he now explains: If it is white in the entire time A, then at any time taken in A it is white; and likewise, if it is non-white in the entire time B, it follows that at any time taken in B it is non-white. Since, therefore, C is taken in both—being both the end of the former and the beginning of the latter—it seems to follow that in C it is white and non-white.
lib. 8 l. 17 n. 10 Secundo ibi: non ergo dandum est etc., manifestat solutionem supra positam. Et dicit quod non est concedendum quod in quolibet accepto in a sit album, sed est excipiendum ultimum nunc, quod est c, quod quidem iam est postremum, idest ultimus terminus mutationis: puta si album vel fiebat vel corrumpebatur in toto a, in c non corrumpitur nec fit album, sed iam factum est et corruptum. Quod autem factum est, est; quod autem corruptum est, non est. Unde manifestum est quod in c primo verum est dicere hoc esse album, si ibi terminetur generatio albi, aut esse non album, si ibi terminetur corruptio albi. Aut si hoc non dicatur, sequentur inconvenientia supra posita, scilicet quod cum aliquid est iam generatum, adhuc est non ens, et cum corruptum est, adhuc est ens. Aut etiam sequitur quod aliquid simul sit album et non album, et universaliter ens et non ens.	1121. Secondly, at (882 263 b20) he illustrates the solution given above. And he says that we must not concede that it is white at any point of time in A, for the ultimate “now,” which is C, must be excepted, for it is already “later,” i.e., it is the ultimate terminus of the change. For example, if the white was coming to be or ceasing to be in the entire time A, in C it is not ceasing to be or becoming white, but already become or ceased to be. But what has already been made, exists, and what has already ceased to be, does not exist. Hence it is clear that in C it is first true to say this is white, if the generation of white has terminated there, or this is not white, if the ceasing-to-be of white has terminated there. Or, if that is not stated, the above-mentioned incompatibilities follow, namely, that when something has been already generated, it is still non-existent, and when it has ceased to be it is still a being. Or, it also follows that something is at once white and non-white, and, universally, being and non-being.
lib. 8 l. 17 n. 11 Deinde cum dicit: si autem quodcumque etc., infert quoddam corollarium ex praemissis, scilicet quod tempus non dividatur in indivisibilia tempora: quia hoc posito, non poterit solvi praemissa dubitatio. Dicit ergo quod necesse est omne quod est prius non ens et postea ens, aliquando fieri ens: et iterum necesse est quod cum aliquid fit, non est. Si autem haec duo quae supponit, sunt vera, impossibile est quod tempus dividatur in indivisibilia tempora. Dividatur enim tempus in indivisibilia tempora: et sit primum tempus indivisibile a; secundum autem, consequenter se habens ad ipsum, sit b. D autem, quod prius non erat album, et postmodum est album, fiebat album in a, et tunc non erat album: oportet autem dare quod sit factum in aliquo tempore indivisibili et habito, idest consequenter se habente, scilicet in b, in quo iam est. Si autem fiebat album in a, sequitur quod in a non erat album: in b autem est album. Cum ergo inter non esse et esse sit generatio media, quia nihil transit de non esse in esse nisi per generationem, sequitur quod inter a et b sit generatio media: ergo erit aliquod tempus medium inter a et b, in quo fiebat album (quia hoc ponitur tempus b, d generationis). Et similiter cum in illo medio tempore indivisibili fiat album, est non album: unde eadem ratione oportebit ponere aliud tempus adhuc medium, et sic in infinitum. Et hoc ideo, quia non potest poni quod in eodem tempore fiat et factum sit. Sed non est eadem ratio si dicatur quod non sunt indivisibilia tempora in quae tempus dividitur. Dicemus enim secundum hoc, quod unum et idem tempus est in quo fiebat et factum est. Sed fiebat et erat non ens in toto tempore praecedenti: est autem factum et ens in ultimo nunc temporis; quod quidem non se habet ad tempus praecedens, sicut habitum aut consequenter, sed sicut terminus eius. Sed si ponantur tempora indivisibilia, necesse est quod consequenter se habeant. Manifestum est autem secundum praemissa, quod non suppositis temporibus indivisibilibus, si aliquid fiat album in toto tempore a, non est maius tempus in quo factum est et fiebat, quam in quo fiebat solum. Quia in toto tempore fit, in ultimo autem termino temporis est factum: tempus autem et terminus temporis non sunt aliquid maius quam tempus tantum, sicut etiam punctum nihil magnitudinis adiicit lineae. Sed si ponantur tempora indivisibilia, manifestum est ex praemissis, quod oportet plus temporis esse in quo fit et factum est, quam in quo fit solum. Ultimo autem epilogando concludit principale intentum, dicens quod praemissae rationes sunt, et similes eis, quibus credendum est tanquam propriis, quod motus reflexus non est continuus.	1122. Then at (883 263 b26) he draws a certain corollary from the foregoing, namely, that time is not divided into indivisible times, because, should one suppose this, it would be impossible to solve the doubt previously mentioned. He says therefore that it is necessary that whatever is first a non-being, and later is a being, come to be at some time; and again, it is necessary that when something is coming to be, it is not existing. Now, if these two assumptions are true, it is impossible for time to be divided into times that are indivisible. For let a time be divided into indivisible times. Then let A be the first indivisible time, and B the second and subsequent time. Now D, which was previously not white and later is white, was becoming white in time A, and at that time was not white. But one must suppose that it has been made white in some indivisible time which is “had,”, i.e., subsequent, to A, namely, in time B in which it is now white. Now, if it was becoming white in A, it follows that in A it was not white; in B, however, it is white. Since, therefore, between non-existence and existence an instance of generation occurs, because nothing passes from non-existence to existence but by generation, it follows that an act of generation occurs between time A and time B. Therefore, there will be between A and B an intermediate time in which it was becoming white (since in time B, D is already generated). And similarly, since in that intermediate indivisible time it is becoming white, it is not white: hence for the same reason it will be necessary to posit still another intermediate time and so on ad infinitum, because we cannot assume that it is becoming white and is white in the same period of time. But the argument is not the same, if one states that the times are not divided into indivisible times. For according to this, we will say that it is one and the same time in which it was coming to be, and was produced. But it was coming to be, and was non-being, in the entire preceding time, and it was produced and a being in the final “now” of the time, which instant is not related to the preceding time as being “had” or subsequent, but as its terminus. But if one assumes indivisible times, they are necessarily (discrete and) consecutive. But it is plain according to the foregoing that, if we do not assume indivisible times, then if something comes to be white in the entire time A, the time in which it was coming to be and was completely made, is no greater than the time in which it was coming to be alone. For it is coming to be in the entire time, but in the ultimate terminus of that time it was completely made. But time plus its terminus is not something greater than the time by itself, any more than a point adds any magnitude to a line. But if indivisible times are assumed, it is clear from the foregoing that there must be more time in coming to be and completely being, than in coming to be alone. Finally, in summary, he concludes to his main intention, saying that the foregoing arguments, and ones like them, are the appropriate ones to convince us that a reflex motion is not continuous.