Hic autem Aristoteles determinat de motu secundum communem rationem motus, nondum applicando motum ad determinata moventia et mobilia: et ideo frequenter talibus propositionibus utitur in hoc sexto libro, quae sunt verae secundum considerationem communem motus, non autem secundum applicationem ad determinata mobilia. Now, Aristotle is here discussing motion from the viewpoint of its general account, without application to particular movers and mobiles. Indeed, he frequently uses such propositions in this sixth book, and they are true if you limit yourself to a general consideration of motion, but they are not necessarily true if applied to particular mobiles. Et similiter non est contra rationem magnitudinis, quod quaelibet magnitudo dividatur in minores: et ideo utitur in hoc libro, ut accipiat qualibet magnitudine data aliam minorem; licet applicando magnitudinem ad determinatam naturam, sit aliqua minima magnitudo; quia quaelibet natura requirit determinatam magnitudinem et parvitatem, ut etiam in primo dictum est. Likewise, it is not contrary to the account of magnitude that every magnitude be divisible into smaller ones. Therefore, in the present book, he goes on the assumption that it is possible to take a magnitude smaller than any given magnitude, even though, in every particular nature, there is always a minimum magnitude, since each nature has limits of largeness and smallness, as was mentioned even in book 1. Ex duobus autem praemissis concludit tertium, scilicet quod in omni tempore dato contingit et velocius et tardius moveri, quam sit motus datus in tali tempore. From these two premises, he infers a third one: in any given time, motions faster and slower than a given motion are possible. 775. Deinde cum dicit: haec autem cum sint etc., ex praemissis concludit propositum. Et dicit quod cum praemissa sint vera, necesse est quod tempus sit continuum, idest divisibile in semper divisibilia. Supposito enim quod haec sit definitio continui, necesse est quod tempus sit continuum, si magnitudo est continua; quia ad divisionem magnitudinis sequitur divisio temporis, et e converso. 775. Then, at and this being so (232b23), from the foregoing, he argues for his proposition. And he says that, since the foregoing are true, time must be a continuum, and thus divisible into parts that are further divisible. For if that is the definition of a continuum, then, if a magnitude is a continuum, time must be continuous, because the division of time follows upon division of magnitude, and vice versa. Deinde cum dicit: quoniam enim ostensum est etc., ostendit propositum, scilicet quod similiter dividatur tempus et magnitudo. Quia enim ostensum est quod velocius pertransit aequale spatium in minori tempore, ponatur quod a sit velocius et b sit tardius, et moveatur b tardius per magnitudinem quae est cd, in tempore zi. Then, at for since it has been (232b26), he proves the proposition, namely, that time and magnitude are divided in a similar way. For since we have shown that a faster thing traverses an equal space in less time, let A be the faster and B the slower, and let B be moved more slowly through magnitude CD in time ZI. Manifestum est ergo quod a quod est velocius, movetur per eandem magnitudinem in minori tempore; et sit tempus illud zt. It is plain that A, which is faster, traverses the same magnitude in less time, ZT. Iterum autem quia a quod est velocius, in tempore zt pertransivit totam magnitudinem quae est cd, b quod est tardius, in eodem tempore pertransit minorem magnitudinem, quae sit ck. Et quia b quod est tardius, pertransit magnitudinem ck in tempore zt, a quod est velocius, pertransibit eandem magnitudinem adhuc in minori tempore; et sic tempus zt iterum dividetur. Et eo diviso, secundum eandem rationem dividetur magnitudo ck; quia tardius in parte illius temporis movetur per minorem magnitudinem. Et si dividitur magnitudo, iterum dividetur et tempus; quia illam partem magnitudinis velocius transibit in minori tempore. Et sic semper procedetur, accipiendo post motum velocioris aliquod mobile tardius, et post tardius iterum velocius; et utendo eo quod demonstratum est, scilicet quod velocius pertranseat aequale in minori tempore, et tardius in aequali tempore minorem magnitudinem. Sic enim accipiendo id quod est velocius, dividemus tempus; et accipiendo id quod est tardius, dividemus magnitudinem. But again, since A, which is faster, has traversed the entire magnitude CD in time ZT, then B, the slower, traversed a smaller magnitude CK in the same time. And, because B traversed the magnitude CK in time ZT, A traversed the same magnitude in even less time. Thus, the time ZT will be further divided. And when it is, the magnitude CK will also be divided, because the slower traverses less space in part of that time. And, if the magnitude is divided, the time also is divided, because the faster will cover that part of the magnitude in less time. So, we continue in this manner, taking a slower mobile after the motion of the faster, and after the slower taking the faster, and making use of the statements already proved, namely, that the faster traverses an equal space in less time and that the slower traverses a smaller magnitude in equal time. For by thus taking what is faster, we will divide the time, and by taking what is slower, we will divide the magnitude. Si ergo hoc verum est, quod semper possit talis conversio fieri, procedendo a velociori in tardius et a tardiori in velocius; et facta tali conversione semper fit divisio magnitudinis et temporis; manifestum erit quod omne tempus est continuum, idest divisibile in semper divisibilia, et similiter omnis magnitudo; quia per easdem et aequales divisiones dividitur tempus et magnitudo, ut ostensum est. Therefore, it is true that such a conversion can be made by going from the faster to the slower and from the slower to the faster. And if such switching causes the magnitude and then the time to be divided, then it will be clear that time is a continuum, and thus divisible into times that are further divisible, and the same for magnitude, since both time and magnitude will receive the same and equal divisions, as we have already shown. 776. Deinde cum dicit: amplius autem et ex consuetis etc., ponit tertiam rationem ad ostendendum quod magnitudo et tempus similiter dividuntur, ex consideratione unius et eiusdem mobilis. Et dicit quod manifestum est etiam per rationes quae consueverunt dici, quod si tempus est continuum, idest divisibile in semper divisibilia, quod et magnitudo eodem modo continua est: quia unum et idem mobile regulariter motum, sicut in toto tempore pertransit totam magnitudinem, ita in medio tempore medium magnitudinis, et universaliter in minori tempore minorem magnitudinem. Et hoc ideo contingit, quia similiter dividitur tempus sicut et magnitudo. 776. Then, at moreover, the current (233a10), he gives a third reason to show that magnitude and time are correspondingly divided. But this time we shall consider one and the same mobile. And he says that it is clear from the reasons customarily given that, if time is a continuum—that is, divisible into parts that are further divisible—then a magnitude is likewise divisible, because one and the same mobile in uniform motion, since it traverses the whole magnitude in a given time, will traverse half in half the time, and a smaller part in less than half the time. And the reason that this happens is that time is divided as magnitude is. Lectio 4 Lecture 4 Finitum et infinitum similiter inveniuntur in magnitudine et in tempore—nullum continuum indivisibile esse demonstratur No continuum is indivisible Et si quodcumque infinitum est, et alterum; et sicut alterum, et alterum est: ut si quidem ultimis infinitum est tempus, et longitudo ultimis; si vero divisione, divisione et longitudo; si autem in utrisque tempus, in utrisque et longitudo. And, if either is infinite, so is the other, and the one is so in the same way as the other—that is, if time is infinite in respect of its extremities, length is also infinite in respect of its extremities; if time is infinite in respect of divisibility, length is also infinite in respect of divisibility; and if time is infinite in both respects, magnitude is also infinite in both respects. Unde et Zenonis ratio falsum opinatur, quod non est possibile infinita pertransire, aut tangere infinita secundum unumquodque, in finito tempore. Dupliciter enim dicitur et longitudo et tempus infinitum, et omnino omne continuum; aut secundum divisionem, aut in ultimis. Hence Zeno’s argument makes a false assumption in asserting that it is impossible for a thing to pass over or severally to come in contact with infinite things in a finite time. For there are two senses in which length and time, and generally anything continuous, are called “infinite”: they are called so either in respect of divisibility or in respect of their extremities. Infinitis quidem igitur secundum quantitatem non contingit sese tangere in finito tempore: eis autem quae sunt secundum divisionem, contingit; et namque ipsum tempus sic infinitum est. Quare et in infinito tempore, et non finito, accidit transiri infinitum; et tangere infinita infinitis, et non finitis. So, while a thing in a finite time cannot come in contact with things quantitatively infinite, it can come in contact with things infinite in respect of divisibility, for in this sense, the time itself is also infinite; and so we find that the time occupied by the passage over the infinite is not a finite time, but an infinite one, and the contact with the infinites is made by means of moments not finite in number, but infinite. Neque iam infinitum potest in finito tempore transire; neque in infinito, finitum: sed si tempus infinitum sit, et magnitudo erit infinita; et si magnitudo, et tempus. The passage over the infinite, then, cannot occupy a finite time, and the passage over the finite cannot occupy an infinite time: if the time is infinite, the magnitude must be infinite also, and if the magnitude is infinite, so also is the time. Sit enim magnitudo finita in quo AB, tempus autem infinitum in quo est G. Accipiatur igitur temporis aliquid finitum in quo GD. This may be shown as follows. Let AB be a finite magnitude, and let us suppose that it is traversed in infinite time G, and let a finite period GD of the time be taken. In hoc igitur transibit aliquid magnitudinis; et sit quod transitum est in quo BE. Hoc autem aut mensurabit in quo est AB, aut deficiet, aut excellet: differt enim nihil. Now, in this period, the thing in motion will pass over a certain segment of the magnitude: let BE be the segment that it has thus passed over. (This will be either an exact measure of AB or less or greater than an exact measure; it makes no difference which it is.) Si enim semper aequalem ei quae est in BE magnitudinem in aequali tempore transibit, hoc autem mensurat totum; finitum erit omne tempus, in quo transibit. In aequalia enim dividetur, sicut et magnitudo. Then, since a magnitude equal to BE will always be passed over in an equal time, and BE measures the whole magnitude, the whole time occupied in passing over AB will be finite, for it will be divisible into periods equal in number to the segments into which the magnitude is divisible. Amplius autem, si non omnem magnitudinem in infinito tempore transibit, sed contingit aliquam et in finito tempore transire, ut quae est BE: haec autem mensurabit totam, et aequalem in aequali transibit. Quare finitum erit et tempus. Moreover, if it is the case that infinite time is not occupied in passing over every magnitude, but it is possible to pass over some magnitude, say BE, in a finite time, and if this BE measures the whole of which it is a part, and if an equal magnitude is passed over in an equal time, then it follows that the time, like the magnitude, is finite. Quod autem non in infinito tempore transit quod est BE, manifestum est, si accipiatur in altera finitum tempus. Si enim in minori partem pertransit, hanc necesse est finitam esse, altero termino existente. That infinite time will not be occupied in passing over BE is evident if the time be taken as limited in one direction; for as the part will be passed over in less time than the whole, the time occupied in traversing this part must be finite, the limit in one direction being given. Eadem autem demonstratio est, et si longitudo infinita sit, tempus autem finitum. The same reasoning will also show the falsity of the assumption that infinite length can be traversed in a finite time. Manifestum igitur ex dictis est quod neque linea neque planum neque omnino ullum continuorum erit atomus: non solum propter id quod nunc dictum est; sed quia accidit dividi indivisibile. It is evident from what has been said, then, that neither a line nor a plane, nor in fact anything continuous, can be atomic. This conclusion follows not only from the present argument but also from the consideration that the opposite assumption implies the divisibility of the indivisible. Quoniam enim in omni tempore velocius et tardius est; velocius autem plus transit in aequali tempore; For since the distinction of quicker and slower may apply to motions in any period of time, but in an equal time, the quicker passes over a greater length, contingit autem et duplam et hemioliam transire longitudinem (sit enim haec ratio velocis): it may happen that it will pass over a length twice as great, or in the ratio of three to two, since their respective velocities may be in this proportion. adducatur igitur velocius secundum hemiolium in eodem tempore, et dividantur magnitudines quae quidem velocioris sunt, AB, BC, CD, in tres atomos; quae vero sunt tardioris, in duos in quibus sunt EZ, ZI. Suppose, then, that the quicker has in the same time been carried over a length one and a half times as great as that traversed by the slower, and that the respective magnitudes are divided: that of the quicker, the magnitude ABCD, into three indivisibles, and that of the slower into the two indivisibles: EZ and ZH. Itaque et tempus dividetur in tria atoma: aequale enim in aequali tempore transibit. Dividatur igitur tempus in ea quae sunt KL, LM, MN. Then the time may also be divided into three indivisibles, for an equal magnitude will be passed over in an equal time. Suppose, then, that it is thus divided into KL, LM, and MN. Iterum autem, quoniam deductum est tardius per EZ, ZI, et tempus secabitur in gemina. Dividetur ergo atomum et impartibile: non enim in atomo transit, sed in pluri. Manifestum ergo est quod nihil continuorum impartibile est. Again, since the slower has been carried over EZ and ZH in the same time, the time may also be similarly divided into two. Thus, the indivisible will be divisible, and that which has no parts will be passed over not in an indivisible time, but in a greater one. It is evident, therefore, that nothing continuous is without parts. 777. Postquam ostendit quod magnitudo et tempus similiter dividuntur, hic ostendit quod finitum etiam et infinitum similiter inveniuntur in magnitudine et tempore. 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. Et circa hoc tria facit: About this, he does three things: primo ponit propositum; first, he states the proposition; secundo ex hoc solvit dubitationem, ibi: unde et Zenonis ratio etc.; second, from this, he settles a doubt, at hence Zeno’s argument (233a21; [779]); tertio probat propositum, ibi: neque iam infinitum etc. third, he proves the proposition, at the passage over (233a31; [780]). 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: