Authors/Thomas Aquinas/physics/L6/lect4

Lecture 4 Proof that no continuum is indivisible

Latin	English
	Lecture 4 Proof that no continuum is indivisible
lib. 6 l. 4 n. 1 Postquam ostendit quod magnitudo et tempus similiter dividuntur, hic ostendit quod finitum etiam et infinitum similiter inveniuntur in magnitudine et tempore. Et circa hoc tria facit: primo ponit propositum; secundo ex hoc solvit dubitationem, ibi: unde et Zenonis ratio etc.; tertio probat propositum, ibi: neque iam infinitum et cetera.	777. After showing that magnitude and time are subject to similar divisions, the Philosopher now shows that if either is finite or infinite, so is the other. About this he does three things: First he states the proposition; Secondly, from this he settles a doubt at 779; Thirdly, he proves the proposition at 780.
lib. 6 l. 4 n. 2 Dicit ergo primo, quod si quodcumque horum duorum, scilicet temporis et magnitudinis, sit infinitum, et alterum est infinitum; et eo modo quo alterum est infinitum et alterum. Et hoc exponit distinguendo duos modos infiniti; dicens quod si tempus est infinitum in ultimis, et magnitudo est infinita in ultimis. Dicitur autem tempus et magnitudo esse infinita in ultimis, quia scilicet ultimis caret; sicut si imaginaremur lineam non terminari ad aliqua puncta, vel tempus non terminari ad aliquod primum aut ultimum instans. Et si tempus sit infinitum divisione, et longitudo erit divisione infinita. Et est hic secundus modus infiniti: dicitur enim divisione infinitum, quod in infinitum dividi potest; quod est de ratione continui, ut dictum est. Et si tempus esset utroque modo infinitum, et longitudo esset utroque modo infinita. Et convenienter isti duo modi infiniti contraponuntur: quia primus modus infiniti accipitur ex parte ultimorum indivisibilium quae privantur; secundus autem modus accipitur secundum indivisibilia quae signantur in medio; dividitur enim linea secundum puncta infra lineam signata.	778. He says therefore first (590 233 a17) that if either of these two, namely, time and magnitude, is infinite, so is the other; likewise, both will be infinite in the same manner. He explains this by distinguishing two ways of being infinite, saying that if time is infinite in respect of its extremities, the magnitude, too, is infinite in that way. Now time and magnitude are said to be infinite in their extremities, because they lack extremities. It is as though we imagined that a line is not terminated at any points, or that time is not terminated at a first or final instant. Moreover, if time is infinite through division, so also is a length. And this is the second way in which something is infinite. But something is said to be infinite through division, because it can be divided ad infinitum; which, of course, pertains to the definition of a continuum, as was said. Consequently, if time is infinite both ways, so, too, is length. It is fitting that these two ways of being infinite be set in contrast: for the first way is taken from the viewpoint of indivisible extremities that are absent; the second is taken from the viewpoint of the indivisibles which are intermediate, for a line is divided according to points within the line.
lib. 6 l. 4 n. 3 Deinde cum dicit: unde et Zenonis etc., ex praemissis removet dubitationem Zenonis Eleatis, qui volebat probare quod nihil movetur de uno loco ad alium, puta de a in b. Manifestum est enim quod inter a et b sunt infinita puncta media, cum continuum sit divisibile in infinitum. Si ergo movetur aliquid de a in b, oportet quod pertranseat infinita, et quod tangat unumquodque infinitorum; quod non est possibile fieri in tempore finito. Ergo in nullo tempore quantumcumque magno, dummodo sit finitum, aliquid potest moveri per quantumcumque parvum spatium. Dicit ergo philosophus quod ista ratio procedit ex falsa existimatione; quia longitudo et tempus, et quodcumque continuum, dupliciter dicitur esse infinitum, ut dictum est; scilicet secundum divisionem et in ultimis. Si igitur essent aliqua, scilicet mobile et spatium, infinita secundum quantitatem, quod est esse infinitum in ultimis; non contingeret quod se invicem tangerent in tempore finito. Si vero sint infinita secundum divisionem, hoc contingit; quia etiam tempus quod est finitum secundum quantitatem, est sic infinitum, scilicet secundum divisionem. Unde sequitur quod infinitum transeatur, non quidem in tempore finito, sed in tempore infinito; et quod infinita puncta magnitudinis transeantur in infinitis nunc temporis, non autem in nunc finitis. Est autem sciendum quod haec solutio est ad hominem, et non ad veritatem, sicut infra Aristoteles manifestabit in octavo.	779. Then at (591 233 a21) he uses these facts to refute Zeno, who tried to prove that nothing is moved from one place to another, for example, from A to B. For it is clear that between A and B there is an infinitude of intermediate points, since a continuum is divisible ad infinitum. Therefore, if something were to be moved from A to B, it would have to bridge the infinite and touch each of the infinites, and this cannot be done in finite time. Therefore, nothing can be moved through even the smallest distance during a period of finite time, however great. The Philosopher, therefore, says that this argument is based on a false opinion, for length and time and any magnitude are said to be infinite in two ways, as we have said; namely, according to division and according to their extremities. Accordingly, if there were things (namely, a mobile and a distance) infinite in regard to quantity, which is to be infinite at the extremities, they could not touch one another in finite time. But if they are infinite in respect of division, they will touch, because time also, which is finite in respect of quantity, is infinite in respect of division. Hence two things follow: that the infinite can be traversed not in finite but in infinite time, and that the infinite points of a magnitude are traversed in the infinite “now’s” of time but not in the finite “now’s”. But it should be noted that this solution is ad hominem and not ad veritatem, as Aristotle will explain in Book VIII, L. 17. 78C.
lib. 6 l. 4 n. 4 Deinde cum dicit: neque iam infinitum etc., probat quod supra posuit. Et primo resumit propositum; secundo probat, ibi: sit enim magnitudo et cetera. Dicit ergo primo quod nullum mobile potest transire infinitum spatium in tempore finito, neque finitum spatium in tempore infinito; sed oportet, si tempus est infinitum, quod magnitudo sit infinita, et e converso. Deinde cum dicit: sit enim magnitudo etc., probat propositum. Et primo quod tempus non potest esse infinitum, si magnitudo sit finita; secundo quod e converso, si longitudo sit infinita, tempus non potest esse finitum, ibi: eadem autem demonstratio est et cetera.	780. Then at (592 233 a31) he proves what he stated above as a proposition. First he restates the proposition; Secondly, he proves it at 781. He says therefore first (592 233 a31) that no mobile can traverse an infinite distance in finite time nor a finite distance in infinite time; rather, if the time is infinite, then the magnitude must be infinite, and vice versa. Then at (593 233 a34) he proves the proposition: First that the time cannot be infinite, if the magnitude is finite; Secondly, that if the length is infinite, the time cannot be finite at 784.
lib. 6 l. 4 n. 5 Primum autem ostendit duabus rationibus: quarum prima talis est. Sit magnitudo finita quae est ab, et sit tempus infinitum quod est g. Accipiatur autem huius infiniti temporis aliqua pars finita quae sit gd. Quia igitur mobile per totum tempus g pertransit totam magnitudinem ab, oportet quod in hac parte temporis quae est gd, pertranseat aliquam partem illius magnitudinis, quae quidem sit be. Cum autem ab magnitudo sit finita et maior, be autem finitum et minus, necesse est quod be aut mensuret totum ab, aut deficiet aut excellet in mensurando, si multoties sumatur be: sic enim omne finitum minus se habet ad finitum maius, ut patet in numeris. Ternarius enim, qui est minor senario, bis acceptus mensurat ipsum: quinarium vero, qui etiam est maior, non mensurat bis acceptus, sed excedit; plus enim est bis tria quam quinque. Similiter etiam et septenarium bis acceptus non mensurat, sed deficit ab eo: minus enim est bis tria quam septem. Sed tamen si ternarius ter accipiatur, excedet etiam septenarium. Nihil autem differt quocumque modo horum trium be se habeat ad ab: quia idem mobile semper pertransibit magnitudinem aequalem ei quod est be, in tempore aequali ei quod est gd. Sed be mensurat totum ab vel excedit ipsum, si multoties sumatur. Ergo et gd mensurabit totum tempus g vel excedit ipsum, si multoties sumatur; et sic oportet quod totum tempus g sit finitum, in quo pertransit totam magnitudinem finitam: quia oportet quod in aequalia secundum numerum dividatur tempus, sicut et magnitudo.	781. He proves the first part of the proposition with two reasons, the first of which (593 233 a34) is this: Let AB be a finite magnitude and let G be an infinite time. Take GD as a finite part of this infinite time. Now, since the mobile traverses the entire magnitude AB in the entire time G, then in part of this time, which is GD, it will traverse the part BE of the magnitude. But since the magnitude AB is finite and greater than BE, which is finite and less, then BE is either an exact measure of AB or it will be less or greater. (These are the only relationships that a lesser finite quantity can bear to a greater finite quantity, as is evident in numbers. For 3, which is less than 6, measures it twice, but 3 taken twice does not measure 5, which is greater than 3, but exceeds it, nor does it measure 7, but is less than 7. But if 3 were taken thrice, that product would exceed even 7). Now it makes no difference in which of these three ways BE is related to AB, for the same mobile will always traverse a magnitude equal to BE in a time equal to GD. But BE is either an exact measure of AB or will exceed it, if taken a sufficient number of times. Therefore, also GD should exactly measure the entire time G or exceed it, if GD is repeated frequently enough. Consequently, the whole time G (in which the entire finite magnitude was traversed) must be finite; because for every segment of magnitude there was a corresponding segment of time.
lib. 6 l. 4 n. 6 Secundam rationem ponit ibi: amplius autem etc.: quae talis est. Quamvis enim detur quod magnitudinem finitam quae est ab, pertranseat aliquod mobile in tempore infinito, non tamen potest dari quod omnem magnitudinem pertranseat in tempore infinito: quia videmus quod multae magnitudines finitae temporibus finitis pertranseuntur. Sit igitur magnitudo finita quae est be, quae pertranseatur tempore finito. Sed be, cum sit finita, mensurat ab, quae est etiam finita. Sed idem mobile pertransibit aequalem magnitudinem ei quae est be, in aequali tempore finito in quo ipsam pertransibat: et ita quot accipiebantur magnitudines aequales be ad constituendam totam ab, tot tempora finita aequalia accipientur ad mensurationem vel constitutionem totius temporis. Unde sequitur quod totum tempus sit finitum.	782. The second reason is given at (782). It is this: Although it be granted that a mobile traverse the finite magnitude AB in infinite time, it cannot be granted that it will traverse any magnnitude at random in infinite time, because we see finite magnitudes being traversed in finite times. So let BE be the finite magnitude which is traversed In finite time. But BE, since it is finite, will measure AB, which is also finite. Now, the same mobile will traverse a magnitude equal to BE in a finite time equal to that in which it traversed BE. Thus the number of magnitudes equal to BE that will form AB corresponds to the number of equal times required to form the entire time consumed. Hence the entire time was finite.
lib. 6 l. 4 n. 7 Differt autem haec ratio a prima; quia in prima ratione be ponebatur pars magnitudinis ab, hic autem be ponitur quaedam alia magnitudo separata. Necessitatem autem huius secundae rationis positae ostendit cum subdit: quod autem non in infinito et cetera. Posset enim aliquis contra primam rationem cavillando dicere, quod sicut totam magnitudinem ab pertransit in tempore infinito, ita et quamlibet partem eius; et sic partem be non pertransibit in tempore finito. Sed quia non potest dari quod quamlibet magnitudinem pertranseat tempore infinito, oportuit inducere secundam rationem, quod be sit quaedam alia magnitudo, quam tempore finito pertranseat. Et hoc est quod subdit, quod manifestum est quod mobile non pertransit magnitudinem quae est be in infinito tempore, si accipiatur in altera finitum tempus, idest si accipiatur aliqua alia magnitudo a prima, quae dicatur be, quam pertransit tempore finito. Si enim in minori tempore pertransit partem magnitudinis quam totum, necesse est hanc magnitudinem quae est be, finitam esse, altero termino existente finito, scilicet ab. Quasi dicat: si tempus in quo pertransit be, est finitum, et minus tempore infinito in quo pertransit ab, necesse est quod be sit minor quam ab; et ita quod be sit finita, cum ab finita sit.	783. This second reason is different from the first, because in the first, BE was taken to be part of the magnitude AB, but here it is taken as a separate magnitude. Then at (595233 b11) he shows the necessity of this second reason. For someone could cavil by saying that just as the whole magnitude AB is traversed in infinite time, so would every part of it, and thus the part BE would not be traversed in finite time. But because it cannot be granted that any magnitude at random is traversed in infinite time, it was necessary to present the second reason in which BE is a different magnitude which is traversed in finite time. For if the time in which BE is traversed is finite and less than the infinite time in which AB is traversed, then necessarily, BE is less than AB, and must be finite, since AB is finite.
lib. 6 l. 4 n. 8 Deinde cum dicit: eadem autem demonstratio etc., ponit quod eadem demonstratio est ducens ad impossibile, si dicatur quod longitudo sit infinita et tempus finitum. Quia accipietur aliquid longitudinis infinitae, quod erit finitum; sicut accipiebatur aliquid temporis infiniti, quod est finitum.	784. Then at (596 233 b14) he posits that the same proof leads to an impossibility if the length is said to be infinite and the time finite, because a part of the infinite length will be taker, as finite, just as a finite part of infinite time was taken,
lib. 6 l. 4 n. 9 Deinde cum dicit: manifestum igitur ex dictis etc., probat quod nullum continuum est indivisibile. Et primo dicit quod inconveniens sequitur si hoc ponatur; secundo ponit demonstrationem ad illud inconveniens ducentem, ibi: quoniam enim in omni tempore et cetera. Dicit ergo primo manifestum esse ex dictis, quod neque linea neque planum, idest superficies, neque omnino aliquod continuum, est atomus, idest indivisibile: tum propter praedicta, quia videlicet impossibile est aliquod continuum ex indivisibilibus componi, cum tamen ex continuis possit componi continuum; tum etiam quia sequeretur quod indivisibile divideretur.	785. Then at (597 233 b15) he proves that no continuum is indivisible. First he says that an inconsistency would otherwise follow; Secondly, he gives the demonstrations that lead to that inconsistency, at 786. He says therefore first (597 233 b15) that it is clear from what has been said that no line or plane or any continuum is indivisible: first of all on account of the foregoing, namely, that it is impossible for any continuum to be composed of indivisibles, although a continuum can be composed of continua; secondly, because it would follow that an indivisible can be divided.
lib. 6 l. 4 n. 10 Deinde cum dicit: quoniam enim in omni tempore etc., ponit demonstrationem ad hoc inconveniens ducentem: in qua primo praesupponit quaedam superius manifestata. Quorum unum est, quod in omni tempore contingat velocius et tardius moveri. Secundum est quod velocius plus pertransit de magnitudine in aequali tempore. Tertium est quod contingit esse excessum velocitatis ad velocitatem, et longitudinis pertransitae ad longitudinem, secundum diversas proportiones: puta secundum duplicem, quae est proportio duorum ad unum; et secundum hemioliam, quae habet totum et dimidium, quae alio nomine dicitur sexquialtera, ut proportio trium ad duo; vel secundum quantamcumque aliam proportionem. Ex his autem suppositis sic procedit. Sit haec proportio velocis ad velox, ut inveniatur aliquid velocius altero secundum hemiolium, idest sexquialteram proportionem; et sit ita, quod velocius pertranseat unam magnitudinem quae sit abcd, compositam ex tribus magnitudinibus indivisibilibus, quarum una sit ab, alia bc, tertia cd. In eodem autem tempore oportet quod tardius secundum praedictam proportionem pertranseat magnitudinem compositam ex duabus indivisibilibus magnitudinibus, quae sit magnitudo ezi. Et quia tempus dividitur sicut et magnitudo, necesse est quod tempus in quo velocius pertransit tres indivisibiles magnitudines, dividatur in tria indivisibilia; quia oportet quod aequale in aequali tempore pertranseat. Sit ergo tempus klmn divisum in tria indivisibilia. Sed quia tardius in eodem tempore movetur per ezi, quae sunt duae magnitudines indivisibiles, necesse est quod tempus dividatur in duo media: et sic sequetur quod indivisibile dividatur. Oportebit enim quod tardius unam magnitudinem indivisibilem pertranseat in uno indivisibili tempore et dimidio. Non enim potest dici quod unam indivisibilem magnitudinem transeat in uno indivisibili tempore; quia sic non prius moveretur velocius quam tardius. Ergo relinquitur quod tardius pertranseat indivisibilem magnitudinem in pluri quam in uno indivisibili tempore, et in minori quam in duobus; et sic oportebit unum indivisibile tempus dividi. Et eodem modo sequitur quod indivisibilis magnitudo dividatur, si ponatur quod tardius moveatur per tres indivisibiles magnitudines, in tribus indivisibilibus temporibus. Velocius enim in uno indivisibili tempore movebitur per plus quam per unam indivisibilem magnitudinem, et per minus quam per duas. Unde manifestum fit, quod nullum continuum potest esse indivisibile.	786. Then at (598 233 b19) he gives the proof which leads to this inconsistency. In this proof he makes use of certain facts already established. One of these is that in any finite time the faster and the slower can be in motion. The second is that the faster will traverse more distance in equal time. The third is that there can be excess of speed over speed and of length traversed over length traversed according to varying proportions; for example, according to the proportion of 2 to 1, or 3 to 2, or any other proportion. With these presuppositions he proceeds thus: Let this be the ratio of the faster to the fast, that the one is faster in the ratio of 3 to 2; and let the faster traverse one magnitude ABCD composed of 3 indivisible magnitudes AB, BC and CD. During the same time according to the given ratio, tae slower will traverse a magnitude of two indivisible magnitudes, which form the magnitude EZI. And because the time is divided as the magnitude, the time in which the faster traverses the 3 indivisible magnitudes must be divided into 3 indivisibles, because the equal magnitude must be traversed in equal time. So let the time be KLMN divided into 3 indivisibles. But because the slower, during that time, traverses EZI, which are 2 indivisible magnitudes, the time can be divided into 2 halves. Consequently, it follows that an indivisible has been divided. For the slower had to traverse one indivisible magnitude in 1 and a half indivisibles of time, since it cannot be said that it traverses one indivisible magnitude in one indivisible time, for then the faster would not have been moved ahead of the slower. Therefore what remains is that the slower traverses an indivisible magnitude in more than one indivisible and less than two indivisibles of time. Thus the indivisible time will have had to be divided. In like manner, it follows that an indivisible magnitude is divided, if the slower manages to move through three indivisible magnitudes in three indivisible times. For the faster will in one indivisible time be moved through more than one indivisible of magnitude and less than two. Therefore, it is clear that no continuum can be indivisible.