778. 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. 778. He says first (233a17) that, if either of these two (time and magnitude) is infinite, so is the other; likewise, both will be infinite in the same manner. 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. He explains this by distinguishing two ways of being infinite, saying that, if time is infinite in respect of its extremities, then 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. 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. 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 to infinity; which, of course, pertains to the definition of a continuum, as was said. Consequently, if time is infinite in both ways, so, too, is length. Et convenienter isti duo modi infiniti contraponuntur: It is fitting that these two ways of being infinite be set in contrast: quia primus modus infiniti accipitur ex parte ultimorum indivisibilium quae privantur; For the first way is taken from the viewpoint of indivisible extremities that are absent; secundus autem modus accipitur secundum indivisibilia quae signantur in medio; dividitur enim linea secundum puncta infra lineam signata. the second is taken from the viewpoint of the indivisibles that are intermediate, for a line is divided according to points within the line. 779. 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. 779. Then, at hence, Zeno’s argument (233a21), 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. 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. For it is clear that there is an infinitude of intermediate points between A and B, since a continuum is divisible to infinity. 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. 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. 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: according to division and according to their extremities. Accordingly, if there were things (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. 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. Hence two things follow: the infinite can be traversed not in finite time, but infinite time; and the infinite points of a magnitude are traversed in the infinite “nows” of time, but not in the finite “nows.” Est autem sciendum quod haec solutio est ad hominem, et non ad veritatem, sicut infra Aristoteles manifestabit in octavo. But it should be noted that this solution is ad hominem and not ad veritatem, as Aristotle will explain in book 8. 780. Deinde cum dicit: neque iam infinitum etc., probat quod supra posuit. 780. Then, at the passage over (233a31), he proves what he stated above as a proposition. Et primo resumit propositum; First, he restates the proposition; secundo probat, ibi: sit enim magnitudo etc. second, he proves it, at this may be shown (233a34). 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. He says first (233a31) that no mobile can traverse an infinite distance over finite time, nor a finite distance over infinite time; rather, if the time is infinite, then the magnitude must be infinite, and vice versa. Deinde cum dicit: sit enim magnitudo etc., probat propositum. Then, at this may be shown (233a34), he proves the proposition: Et primo quod tempus non potest esse infinitum, si magnitudo sit finita; first, that the time cannot be infinite if the magnitude is finite; secundo quod e converso, si longitudo sit infinita, tempus non potest esse finitum, ibi: eadem autem demonstratio est etc. second, that, if the length is infinite, the time cannot be finite, at the same reasoning (233b14; [784]). 781. Primum autem ostendit duabus rationibus: 781. He proves the first part of the proposition with two arguments. 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: The first (233a34) 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 the part of this time that is GD, it will traverse some part, BE, of the magnitude. But, since the magnitude AB is finite and greater than BE, which is finite and less, then either BE is an exact measure of AB or it will be less or greater. 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. These are the only relationships that a lesser finite quantity can bear to a greater finite quantity, as is evident in numbers. For three, which is less than six, measures it twice, but three taken twice does not measure five, which is greater than three, but exceeds it; nor does it measure seven, but is less than seven. But, if three were taken thrice, that product would exceed even seven. 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. 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 exceeds it, if taken a sufficient number of times. Therefore, GD should also 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, since there was a corresponding segment of time for every segment of magnitude. 782. 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. 782. The second reason is given at moreover, if it is (233b7): although it be granted that a mobile traverse the finite magnitude AB in infinite time, it cannot be granted that it will traverse any magnitude at random in infinite time, because we see finite magnitudes being traversed in finite times. 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. So, let BE be the finite magnitude that is traversed in a 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. 783. Differt autem haec ratio a prima; quia in prima ratione be ponebatur pars magnitudinis ab, hic autem be ponitur quaedam alia magnitudo separata. 783. This second reason is different from the first: in the first, BE was taken to be part of the magnitude AB, but here it is taken as a separate magnitude. Necessitatem autem huius secundae rationis positae ostendit cum subdit: quod autem non in infinito etc. 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. Then, he shows the necessity of this second reason when he adds that infinite time (233b11). For someone could cavil by saying that, just as the whole magnitude AB is traversed in infinite time, so would every part of it be, 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 that is traversed in finite time. 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. And he adds that it is clear that the mobile body does not cross the magnitude BE in an infinite time if the time be taken as limited in one direction—that is, if we take some magnitude other than the first one. This other magnitude is called BE and is crossed in a finite time. 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. For if the body crosses part of the magnitude in less time than the whole, then this magnitude BE must be finite, the limit in one direction being given, that is, 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. It is as if he said that, if the time in which BE is traversed is finite and less than the infinite time in which AB is traversed, then BE is necessarily less than AB and must be finite, since AB is finite. 784. 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 the same reasoning (233b14), 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 taken as finite, just as a finite part of infinite time was taken. 785. Deinde cum dicit: manifestum igitur ex dictis etc., probat quod nullum continuum est indivisibile. 785. Then, at it is evident (233b15), he proves that no continuum is indivisible. Et primo dicit quod inconveniens sequitur si hoc ponatur; First, he says that an inconsistency would otherwise follow; secundo ponit demonstrationem ad illud inconveniens ducentem, ibi: quoniam enim in omni tempore etc. second, he gives the demonstrations that lead to that inconsistency, at for since the distinction (233b19; [786]). Dicit ergo primo manifestum esse ex dictis, quod neque linea neque planum, idest superficies, neque omnino aliquod continuum, est atomus, idest indivisibile: He says, therefore (233b15), that it is clear from what has been said that no line or plane, that is, surface, or any continuum is atomic, that is, indivisible: tum propter praedicta, quia videlicet impossibile est aliquod continuum ex indivisibilibus componi, cum tamen ex continuis possit componi continuum; 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; tum etiam quia sequeretur quod indivisibile divideretur. second, because it would follow that an indivisible can be divided.