Authors/Thomas Aquinas/physics/L6/lect3

Lecture 3 Time follows magnitude in divisibility and conversely

Latin	English
	Lecture 3 Time follows magnitude in divisibility and conversely
lib. 6 l. 3 n. 1 Postquam philosophus ostendit eiusdem rationis esse, quod magnitudo et motus per eam transiens ex indivisibilibus componantur, ostendit etiam idem de tempore et magnitudine. Et dividitur in partes duas: in prima ostendit quod ad divisionem magnitudinis sequitur divisio temporis, et e converso; in secunda ostendit quod ex infinitate unius sequitur infinitas alterius, ibi: et si quodcumque infinitum est et cetera. Circa primum duo facit: primo ponit propositum; secundo demonstrat, ibi: si enim omnis et cetera. Dicit ergo primo quod etiam tempus necesse est similiter esse divisibile et indivisibile, et componi ex indivisibilibus, sicut longitudo et motus.	766. After showing that it is for a same reason that a magnitude and a motion traversing it would be composed of indivisibles, the Philosopher shows the same for time and magnitude. And the treatment falls into two parts: In the first he shows that division of time follows upon division of magnitude, and vice versa; In the second that the infinity of one follows upon the infinity of the other, in L. 4. About the first he does two things: First he states his proposition; Secondly, he demonstrates it, at 767. He says therefore first (578 232 a18) that time, too, is divisible and indivisible, and composed of indivisibles, just as length and motion are.
lib. 6 l. 3 n. 2 Deinde cum dicit: si enim omnis etc., probat propositum tribus rationibus: quarum prima sumitur per aeque velocia; secunda per velocius et tardius, ibi: quoniam autem omnis etc.; tertia per idem mobile, ibi: amplius autem et ex consuetis et cetera. Dicit ergo primo quod de ratione aeque velocis est, quod minorem magnitudinem transeat in minori tempore. Detur ergo aliqua magnitudo divisibilis, quam pertransit aliquod mobile in aliquo tempore dato: sequitur ergo quod mobile aeque velox transeat partem magnitudinis in minori tempore; et sic oportuit tempus datum esse divisibile. Si autem e converso detur quod tempus sit divisibile, in quo mobile datum movetur per magnitudinem aliquam datam, sequitur quod aeque velox mobile in minori tempore, quod est pars totius temporis, moveatur per minorem magnitudinem: et ita sequitur quod magnitudo quae est a sit divisibilis.	767. Then he proves his proposition, giving three reasons: The first of which is based on things equally fast; The second is based on the faster and the slower, at 769; The third uses one and the same mobile, at 776. He says therefore first (579 232 a19) that a mobile which is as fast as another traverses a smaller magnitude in less time. Therefore, let us take a divisible magnitude which a mobile traverses in a given time. It follows that an equally fast mobile traverses part of that magnitude in less time. Consequently, the given time must be divisible. Conversely, if the time is given as divisible and a given mobile is in motion over a given magnitude, it follows that a mobile equally fast traverses a smaller magnitude in less time, which is part of the whole time. Consequently, the magnitude A is divisible.
lib. 6 l. 3 n. 3 Deinde cum dicit: quoniam autem omnis etc., ostendit idem per duo mobilia, quorum unum est velocius et aliud tardius. Et primo praemittit quaedam necessaria ad propositum ostendendum; secundo probat propositum, ibi: quoniam autem omnis quidem motus et cetera. Circa primum duo facit: primo ostendit quomodo velocius se habet ad tardius in hoc quod moveatur per maiorem magnitudinem; secundo quomodo se habeat ad ipsum quantum ad hoc quod est moveri per aequalem magnitudinem, ibi: manifestum autem ex his et cetera. Circa primum duo facit: primo proponit propositum, resumens quoddam ex superioribus, quod est necessarium ad demonstrationes sequentes; secundo demonstrat propositum, ibi: sit enim ipsum et cetera.	768. Then at (580 232 a23) he proves the same thing with two mobiles, one of which is faster and the other slower. But first he lays down some presuppositions to be used in proving his proposition. Secondly, he proves the proposition at 774. About the first he does two things: First he explains how the faster and the slower compare with regard to being moved over a larger magnitude; Secondly, how they compare with regard to being moved over an equal magnitude, at 772. About the first he does two things: First he states his proposition, repeating something mentioned previously but needed for the demonstrations that follow; Secondly, he demonstrates his proposition, at 770.
lib. 6 l. 3 n. 4 Resumit ergo hoc, quod omnis magnitudo sit divisibilis in magnitudines. Et hoc patet per hoc quod ostensum est supra, quod impossibile est aliquod continuum componi ex atomis, idest ex indivisibilibus; et manifestum est quod magnitudo omnis est de genere continuorum. Ex his sequitur quod necesse sit aliquod corpus velocius in aequali tempore per maiorem magnitudinem moveri; et etiam in minori tempore per maiorem magnitudinem moveri. Et hoc modo quidam definierunt velocius, quod plus movetur in aequali tempore et etiam in minori.	769. He repeats therefore (580 232 a23) that every magnitude is divisible into magnitudes. And this is evident from a previous conclusion that it is impossible for a continuum to be composed of atoms, i.e., indivisibles; and every magnitude is a kind of continuum. From these it follows that a faster body is moved through a greater magnitude in equal time and even in less time. Indeed, that is the way in which some have defined the faster, that it is moved more in. equal and even in less time.
lib. 6 l. 3 n. 5 Deinde cum dicit: sit enim ipsum etc., probat duo praemissa. Et primo quod velocius in aequali tempore per maius spatium moveatur; secundo quod etiam in minori tempore per maius spatium movetur, ibi: at vero et in minori et cetera. Dicit ergo primo: sint duo mobilia a et b, quorum a velocius sit quam b; et sit magnitudo cd, quam pertransit a in tempore zi. Moveatur autem b quod est tardius, et a quod est velocius, per eandem magnitudinem, et incipiant simul moveri. His ergo positis, sic argumentatur. Velocius est quod in aequali tempore plus movetur: sed a est velocius quam b: ergo cum a pervenerit ad d, b nondum pervenit ad d, quod est terminus magnitudinis, sed adhuc deficiet, idest distabit ab eo; motum tamen erit in hoc tempore per aliquam partem magnitudinis. Cum ergo omnis pars sit minor toto, relinquitur quod a in tempore zi movetur per maiorem magnitudinem quam b, quod in eodem tempore movetur per partem magnitudinis. Unde sequitur quod velocius in aequali tempore plus de spatio pertransit.	770. Then at (581 232 a27) he proves his two presuppositions: First, that a faster thing is moved a greater distance in equal time; Secondly, that it is moved a greater distance in less time, at 771. He says therefore first (581 232 a27): Let A and B be two mobiles, of which A is faster than B, and let CD be the magnitude traversed by A in time ZI. Now let B, which is slower, and A, which is faster, pass over the same magnitude, and let them start together. Therefore, under these conditions, the following argument is given: The faster is the one moved more in equal time; but A is faster than B. Therefore, when A shall have arrived at D, B will not have arrived at D (which is the terminus of the magnitude) but will be some distance from it; yet it will have covered part of the magnitude. Now, since every part is less than the whole, what remains is that A is moved through a greater distance in time ZI than B, which in the same time has traversed part of the magnitude. Consequently, the faster traverses more distance in equal time.
lib. 6 l. 3 n. 6 Deinde cum dicit: at vero et in minori plus etc., ostendit quod velocius in minori tempore plus de spatio pertransit. Dictum est enim quod in tempore in quo a iam pervenit ad d, b quod est tardius, adhuc distat a d. Detur ergo quod in eodem tempore perveniat usque ad e. Quia igitur omnis magnitudo divisibilis est, ut supra positum est, dividatur residuum magnitudinis, scilicet ed, in quo velocius excedit tardius, in duas partes in puncto t. Manifestum est ergo quod magnitudo ct est minor quam magnitudo cd. Sed idem mobile per minorem magnitudinem movetur in minori tempore. Quia ergo ipsum a pervenit ad d in toto tempore zi, ad punctum t perveniet in minori tempore; et sit illud tempus zk. Inde sic arguitur. Magnitudo ct, quam pertransit a, maior est magnitudine ce, quam pertransit b: sed tempus zk, in quo pertransit a magnitudinem ct, est minus toto tempore zi, in quo b tardius pertransit magnitudinem ce: sequitur ergo quod velocius in minori tempore pertranseat maius spatium.	77l. Then at (582 232 a31) he shows that the faster traverses more space in less time. For it was said that at the time when A arrived at D, B, which is slower, was still distant from D. Let us grant, therefore, that B arrived at E when A arrived at D. Now, since every magnitude is divisible, let us divide the remaining magnitude ED (which is how much the faster exceeds the slower) at T. It is evident that the magnitude CT is less than CD. But one and the same mobile traverses a smaller magnitude in less time. Therefore, since A arrived at D in the total time ZI, it arrived at T in less time. Let that less time be ZK. Then the argument continues: the magnitude CT which A traversed is greater than the magnitude CE which B traversed. But the time ZK in which A traversed CT is less than the whole time ZI, in which the slower B traversed CE. Therefore, it follows that the faster traverses a larger space in less time.
lib. 6 l. 3 n. 7 Deinde cum dicit: manifestum autem etc., ostendit quomodo velocius se habeat ad tardius in moveri per aequalem magnitudinem. Et primo proponit intentum; secundo probat propositum, ibi: quoniam enim et cetera. Dicit ergo primo: quod ex praemissis manifestum esse potest, quod velocius pertransit aequale spatium in minori tempore. Secundo ibi: quoniam enim maiorem etc., probat propositum duabus rationibus. Ad quarum primam duo praemittit: quorum unum iam probatum est, scilicet quod velocius pertranseat maiorem magnitudinem in minori tempore quam tardius; secundum vero est per se manifestum, scilicet quod ipsum mobile secundum seipsum consideratum, in maiori tempore pertransit maiorem magnitudinem quam in minori. Pertranseat enim hoc mobile a, quod est velocius, hanc magnitudinem quae est lm, in pr tempore: et partem magnitudinis, scilicet LX, pertransibit in minori tempore quod est ps; quod est minus quam pr, in quo pertransit lm, sicut et LX est minor quam lm. Ex prima autem suppositione accipit quod totum tempus pr, in quo a pertransit totam magnitudinem lm, sit minus tempore h, in quo b quod est tardius, pertransit minorem magnitudinem, scilicet LX. Dictum est enim quod velocius in minori tempore pertransit maiorem magnitudinem. Ex his procedit sic. Tempus pr est minus tempore h, in quo b quod est tardius, pertransit magnitudinem LX; et tempus ps est minus quam tempus pr; ergo sequitur quod tempus ps sit minus quam tempus h: quia si aliquid est minus minore, etiam ipsum erit minus maiore. Cum ergo datum sit quod in tempore ps velocius movetur per LX magnitudinem, et tardius movetur per eandem in tempore h, sequitur quod velocius movetur in minori tempore per aequale spatium.	772. Then at (583 232 b5) he shows how the faster compares with the slower in regard to being moved through an equal magnitude. First he states his intention; Secondly, he proves his proposition here at 772. He says therefore first (583 232 b5) that from the foregoing it could be clear that a faster thing traverses an equal space in less time. Then he proves this with two arguments, to the first of which he prefaces two facts: one of which has already been proved, namely, that a faster thing traverses a greater magnitude in less time than a slower. The second is per se evident, namely, that one and the same mobile traverses a greater magnitude in a given time than in a shorter time. For let the mobile A, which is faster, traverse the magnitude LM in time PR and the part LX of the magnitude in less time PS, which is less than PR in which it traverses LM just as LX is less than LM. From the first supposition he takes it that the whole time PR in which A traverses the entire magnitude LM is less than time H in which B (which is slower) traverses the smaller magnitude LX. For it was said that a faster object traverses a greater magnitude in less time. With this background he proceeds to his argument: The time PR is less than time H (in which B, which is slower, traverses magnitude LX); moreover, time PS is less than time PR. Therefore, it follows that time PS is less than time H, for what is less than the lesser is less than the greater. Therefore, since it was granted that in the time PS the faster traverses magnitude LX and the slower traverses the same LX in time H, it follows that the faster traverses an equal magnitude in less time.
lib. 6 l. 3 n. 8 Secundam rationem ponit ibi: amplius autem si omne etc.: quae talis est. Omne quod movetur per aequalem magnitudinem cum aliquo alio mobili, aut movetur per eam in aequali tempore aut in minori aut in maiori. Quod autem movetur per aequalem magnitudinem in maiori tempore est tardius, ut supra probatum est: quod autem movetur in aequali tempore per aequalem magnitudinem, est aeque velox, ut per se manifestum est. Cum igitur id quod velocius est, neque sit aeque velox neque tardius, sequitur quod neque in pluri tempore moveatur per aequalem magnitudinem, neque in aequali: relinquitur ergo quod in minori. Sic ergo probatum est quod necesse est velocius pertransire aequalem magnitudinem in minori tempore.	773. Then, after these preliminaries, he gives his second argument, which is this: A thing that traverses an equal magnitude along with another mobile is moved through that magnitude either in equal time or less or more. If it is moved through that equal magnitude in greater time, it is slower, as was proved above; if it is moved in equal time through the equal magnitude, it is equally fast, as is per se evident. Therefore, since what is faster is neither equally fast nor slower, it follows that it is moved through an equal magnitude neither in more time nor in equal time. Therefore, in less time. Thus, we have proved that necessarily the faster traverses an equal magnitude in less time.
lib. 6 l. 3 n. 9 Deinde cum dicit: quoniam autem omnis quidem etc., probat propositum, scilicet quod eiusdem rationis sit tempus et magnitudinem semper dividi in divisibilia, aut etiam ex indivisibilibus componi. Et circa hoc tria facit: primo praemittit quaedam quae sunt necessaria ad sequentem probationem; secundo ponit propositum, ibi: haec autem cum sint etc.; tertio probat, ibi: quoniam enim ostensum est et cetera. Praemittit ergo primo, quod omnis motus est in tempore; et hoc probatum est in quarto: item quod in omni tempore possibile sit moveri; quod ex definitione temporis apparet, quae in quarto data est. Secundum est, quod omne quod movetur, contingit moveri velocius et tardius; idest quod in quolibet mobili est invenire aliquid quod velocius movetur, et aliquid quod tardius. Sed haec propositio videtur esse falsa. Determinatae enim sunt velocitates motuum in natura: est enim aliquis motus ita velox, quod nullus potest esse eo velocior, scilicet motus primi mobilis. Ad hoc ergo dicendum, quod de natura alicuius rei possumus loqui dupliciter: vel secundum rationem communem, vel secundum quod ad propriam materiam applicatur. Et nihil prohibet aliquid, quod non impeditur ex ratione communi rei, impediri ex applicatione ad aliquam materiam determinatam; sicut non impeditur ex ratione formae solis esse plures soles, sed ex hoc quod tota materia speciei sub uno sole continetur. Et similiter ex communi natura motus non prohibetur quin qualibet velocitate data, possit alia maior velocitas inveniri: sed impeditur ex determinatis virtutibus mobilium et moventium. 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. 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. 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.	774. Then at (586 232 b20) he proves the proposition that one and the same reason proves that both time and magnitude are always divided into divisibles, or are composed of indivisibles. About this he does three things: First he lays down premises to be used in the proof; Secondly, he states his proposition at 775; Thirdly, he proves it at 775. Therefore (586 232 b20) he lays down the premises that every motion exists in time—this was proved in Book IV—and that motion is possible in any time—this is evident from the definition of time given in Book IV. Secondly, that whatever is being moved can be moved faster and slower, i.e., among mobiles some are moved faster and some slower. But this statement seems false, because the speeds of motions are fixed in nature; for there is one motion so fast that none could be faster, namely, the motion of the first mobile. In reply it must be said that we can speak of the nature of anything in two ways: either according to its general notion or insofar as it is applied to its proper matter. Now, there is nothing to forbid something which is possible in the light of a thing’s general definition to be prevented from happening when application is made to some definite matter; for example, it is not the general definition of the sun that precludes many suns, but the fact that the total matter of this nature is contained under one sun, Likewise, it is not the general nature of motion that prevents the existence of a speed greater than any given speed; rather it is the particular powers of the mobiles and movers. Now, Aristotle is here discussing motion from the viewpoint of its general nature 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 not necessarily true, if you get down to particular mobiles. Likewise, it is not against the nature of magnitude that. every magnitude be divisible into smaller ones. Therefore, in this 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 I. From these two premises he concludes to a third one, namely, that in any given time, faster and slower motions than a given motion are possible.
lib. 6 l. 3 n. 10 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. 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. Manifestum est ergo quod a quod est velocius, movetur per eandem magnitudinem in minori tempore; et sit tempus illud 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. 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.	775. Then at (587 232 b23) from the foregoing he concludes to his proposition. And he says that since the foregoing are true, time must be a continuum, i.e., 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, end vice versa. Then at (588 232 b26) 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 CB in time ZI. It is plain that A, which is faster, traverses the same magnitude in less time ZT. But again, since A, which is faster, has in time ZT traversed the entire magnitude CD, B, the slower, traversed a smaller magnitude CK in the same time. And because B, the slower, traversed the magnitude CK in time ZT, A, the faster, 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 statement already proved 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. 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 continuous, i.e., divisible into times that are further divisible, and the same for magnitude; for both time and magnitude will receive the same and equal divisions, as we have already shown.
lib. 6 l. 3 n. 11 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 (589 233 a12) 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 ordinary reasons that if time is continuous, i.e., 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 why this happens is that time is divided as magnitude is.