Authors/Thomas Aquinas/physics/L6/lect9

Lecture 9 Finite and infinite are found simultaneously in magnitude, time, mobile, and motion

Latin	English
	Lecture 9 Finite and infinite are found simultaneously in magnitude, time, mobile, and motion
lib. 6 l. 9 n. 1 Postquam philosophus determinavit de divisione motus, hic determinat de finito et infinito in motu: sicut enim divisio pertinet ad rationem continui, ita finitum et infinitum. Sicut autem supra ostendit quod divisio simul invenitur in motu, magnitudine, tempore et mobili; ita ostendit nunc idem de infinito. Unde circa hoc tria facit: primo ostendit quod infinitum similiter invenitur in magnitudine et tempore; secundo quod similiter cum his invenitur etiam in mobili, ibi: demonstratis autem his etc.; tertio quod similiter invenitur in motu, ibi: quoniam autem neque finitum et cetera. Circa primum duo facit: primo ostendit quod si magnitudo est finita, tempus non potest esse infinitum; secundo quod e converso si tempus est finitum, quod magnitudo non potest esse infinita, ibi: eadem autem ratio et cetera. Circa primum duo facit: primo proponit quod intendit; secundo probat propositum, ibi: quod igitur si aliquid moveatur et cetera.	841. After determining the division of motion, the Philosopher now determines about the infinite and finite in motion; for just as division pertains to the notion of continuum, so also do finite and infinite. But just as above he said that division is found simultaneously in motion, magnitude, time and mobile, so now he shows that the same is true of the infinite. Hence about this he does three things: First he shows that the infinite is found in the same way in magnitude and in time; Secondly, that it is found in the same way in the mobile, 846; Thirdly, and in motion, at 652. About the first he does two things: First he shows that if a magnitude is finite, the time cannot be infinite; Secondly, that if the time is finite, the magnitude cannot be infinite, at 845. in regard to the first he does two things: First he proposes what he intends; Secondly, he proves his proposition, at 843.
lib. 6 l. 9 n. 2 Primo ergo repetit duo quae sunt necessaria ad propositum ostendendum. Quorum unum est, quod omne quod movetur, in tempore movetur; secundum est, quod in pluri tempore ab eodem mobili pertransitur maior magnitudo. Et ex his duobus suppositis intendit probare tertium, scilicet quod impossibile sit in tempore infinito pertransire magnitudinem finitam. Quod tamen sic intelligendum est, quod non reiteretur illud quod movetur per eandem magnitudinem, aut per aliquam partem eius multoties: sed ita quod in toto tempore moveatur per totam magnitudinem. Et addidit hoc, ut praeservaret se a motu circulari, qui est super magnitudine finita, et tamen potest esse in tempore infinito, ut ipse dicet in octavo.	842. First, therefore, (644 237 b23) he repeats two things that are needed for proving the proposition. One of which is that whatever is being moved is being moved in time. The second is that in more time a greater magnitude is traversed by the same mobile. From these two suppositions he intends to prove a third, namely, that it is impossible to traverse a finite magnitude in infinite time. This is to be understood in the sense that the thing in motion is not to retraverse the same magnitude repeatedly or any part of it, but must be moved through the entire magnitude in the entire time. And he added this to save himself from circular motion over a finite magnitude, which can occur in infinite time, as will be explained in Book VIII.
lib. 6 l. 9 n. 3 Deinde cum dicit: quod igitur etc., probat propositum: et primo si detur mobile quod aeque velociter moveatur per totam magnitudinem; secundo si non uniformiter et regulariter moveatur, ibi: sed si non sit et cetera. Dicit ergo primo, quod si sit aliquod mobile quod aeque velociter moveatur per totum, necesse est, si pertransit finitam magnitudinem, quod hoc sit in tempore finito. Accipiatur enim una pars magnitudinis, quae mensuret totum; puta sit tertia vel quarta pars magnitudinis. Si ergo mobile aeque velociter movetur per totum, et aeque velox est quod aequale spatium in aequali tempore pertransit, sequitur quod in aequalibus temporibus, et tot quot sunt partes magnitudinis, pertranseat mobile totam magnitudinem: puta, si accepta sit quarta pars magnitudinis, eam pertransibit in aliquo tempore, et aliam quartam in alio tempore aequali; et sic totam magnitudinem pertransit in quatuor aequalibus temporibus. Quia ergo partes magnitudinis sunt finitae numero, et unaquaeque etiam est finita secundum quantitatem, et tot modis pertransit omnes partes, idest in totidem temporibus aequalibus; sequitur quod totum tempus in quo pertransit totam magnitudinem, sit finitum. Mensurabitur enim a tempore finito: quia erit toties tantum quantum est tempus in quo pertransit partem, quoties magnitudo tota est tanta quanta est pars. Et sic totum tempus erit multiplicatum secundum multiplicationem partium. Omne autem multiplicatum mensuratur a submultiplici, sicut duplum a dimidio, et triplum a subtriplo, et sic de aliis. Tempus autem quo pertransit partem est finitum: quia si detur quod sit infinitum, sequetur quod in aequali tempore pertranseat totum et partem; quod est contra id quod suppositum est. Et sic oportet quod totum tempus sit finitum; quia nullum infinitum mensuratur a finito.	843. Then at (645 237 b26) he proves his proposition: First by assuming a mobile of equal speed being moved over the whole magnitude; Secondly, if it is not being moved with a regular and uniform motion, at 844. He says therefore first (645 237 b26) that if a mobile of equal speed is traversing a whole, then if the whole is a finite magnitude, it must be traversed in finite time. For we can take one part of the magnitude and make it measure the whole; for example, a part that is one-third or one-fourth of the magnitude, If, therefore, a mobile is moved with equal speed over the whole and if the equally fast is what traverses an equal space in equal time, it follows that in a number of equal times that are determined by the number of parts into which the magnitude was divided, it will traverse the whole magnitude; for example, if one-fourth of the magnitude is taken, it will traverse it in a certain time and another fourth in an equal time, and so it will traverse the entire magnitude in four equal times. Because, therefore, the parts of the magnitude are finite in number and each is finite in quantity, and in a given number of equal times the whole magnitude is traversed, it follows that the whole time in which the entire magnitude is traversed is finite. For it will be measured by a finite time, since it will be as many times as much as the time required to traverse one part, the whole magnitude being as many times as the quantity of each part. And thus the whole time will be the multiplication product of the length multiplied by the number of parts. But every multiplication product is measured by a denominator, as double is measured by half and triple by third, and so on. The time, however, required to traverse a part is finite, because if it were infinite, it would follow that the whole and the part were traversed in equal time, which is against the original assumption. Therefore, the whole time has to be finite, because nothing infinite can be measured by the finite.
lib. 6 l. 9 n. 4 Sed quia posset aliquis dicere, quod licet partes magnitudinis sint aequales, et mensurent totam magnitudinem, tamen potest contingere quod partes temporis non sunt aequales, sicut quando non est aequalis velocitas in toto motu; et sic tempus quo movetur per partem magnitudinis, non mensurabit tempus quo movetur per totam: ideo consequenter ibi: sed si non sit etc., ostendit quod hoc nihil differt quantum ad propositum. Sit enim ab spatium finitum, quod pertransitum sit in tempore infinito quod est cd. Necesse est autem in omni motu, quod prius pertranseatur una pars quam altera: et hoc etiam manifestum est, quod in priori parte temporis et posteriori, altera et altera pars magnitudinis pertransitur. Et ita oportet quod neque duae partes magnitudinis pertranseantur in una et eadem parte temporis; neque quod in duabus partibus temporis pertranseatur una et eadem pars magnitudinis. Et sic oportet, si in aliquo tempore pertransita est aliqua pars magnitudinis, quod in pluri tempore pertranseatur non solum illa pars magnitudinis, sed etiam cum hac et altera: et hoc indifferenter, sive aeque velociter moveatur mobile, sive non; vel per hoc quod velocitas semper magis ac magis intenditur, sicut in motibus naturalibus, vel per hoc quod magis et magis remittitur, sicut in motibus violentis. His igitur suppositis, accipiatur aliqua pars spatii ab, quae quidem pars sit ae, et mensuret totum ab, ita scilicet quod sit aliquota pars eius, vel tertia vel quarta. Haec igitur pars spatii pertransita est in aliquo tempore finito. Non enim potest dari quod sit pertransita in tempore infinito; quia totum spatium pertransitum est in tempore infinito, et in minori pertransitur pars quam totum. Item accipiamus aliam partem spatii quae sit aequalis parti ae, et eadem ratione necesse est quod haec pars pertranseatur in tempore finito, quia totum spatium pertransitur in tempore infinito. Et sic semper accipiendo, accipiam tot tempora finita, quot sunt partes spatii; ex quibus constituetur totum tempus, in quo movetur per totum spatium. Impossibile est autem quod aliqua pars infiniti mensuret totum, neque secundum magnitudinem neque secundum multitudinem: quia impossibile est quod infinitum constet ex partibus finitis numero, quarum etiam unaquaeque sit finita quantitate, sive dicatur quod illae partes sint aequales, sive quod sint inaequales; quia quaecumque mensurantur a quodam uno, sive secundum multitudinem sive secundum magnitudinem, oportet ea esse finita. Ideo autem dico multitudinem et magnitudinem, quia nihil minus mensuratur aliquid per hoc quod habet finitam magnitudinem, sive partes mensurantes sint aequales sive inaequales. Quando enim sunt aequales, tunc pars mensurat totum et multitudine et magnitudine; quando vero sunt inaequales, mensurat multitudine, sed non magnitudine. Sic ergo patet quod omne tempus quod habet partes finitas numero et quantitate, sive sint aequales sive inaequales, est finitum. Sed spatium finitum mensuratur aliquibus finitis, ex quantis contingit componi ab; et oportet esse aequales numero partes temporis et partes magnitudinis, et quaslibet esse finitas quantitate: ergo relinquitur quod per totum spatium moveatur in tempore finito.	844. But someone could say that although the parts of the magnitude are equal and measure the whole magnitude, it could happen that the parts of time are not equal, as when an equal speed is not maintained through the entire motion, and so the time required to traverse a part of the magnitude will not be a measure of the time required to traverse the whole. Therefore at (646 237 b34) He shows that this makes no difference to the proposition. For let AB be a finite space that has been traversed in infinite time CD. Now in every motion, one part must be traversed ahead of another and also one part of the magnitude is traversed in the prior part of time and another part in a subsequent part of time. And so, no two parts of the magnitude are ever traversed in one and the same part of time, and no two parts of time correspond to one and the same part of the magnitude. Consequently, if a certain part of the magnitude is traversed in a certain time, then in more time is traversed not only that part of the magnitude but that part and another. And this will happen whether the mobile maintains constant speed or not, for in natural motions the speed is continually increased, while in compulsory motions it is diminished. With these suppositions in mind, let AE be a part of the space AB and let it be an exact measure, say, one third or one fourth of AB. Therefore, this part of space has been traversed in a finite time. For it cannot be assumed that it was traversed in infinite time, because the whole space was traversed in infinite time, whereas less time is required to traverse a part than to traverse the whole. Likewise, let us take another part of the space and let it equal the part AE. This part, too, must be traversed in finite time, for it is the whole space that is being traversed in infinite time. Proceeding in this manner, let us take, in accordance with the parts of the entire space, a corresponding number of such times. From these will be constituted the whole time in which the entire space is traversed. Now it is impossible that a part of an infinite measure the whole, either in the case of a magnitude or in that of a multitude, because it is impossible for the infinite to be composed of a finite number of parts, each of which is finite in quantity, whether those parts are equal or unequal—for whatever things are measured by some one thing, either according to magnitude or multitude, must be finite. Now, I say “magnitude and multitude”, because a thing of finite magnitude can still be measured, whether the measuring parts are of equal or unequal size. For when they are equal, then any part is a measure of the whole, whether the whole be a magnitude or a multitude; but when they are unequal parts, any part will measure a multitude but not a magnitude. So, therefore, it is evident that any time which has parts finite in number and quantity, whether they be equal or not, is finite. But a finite space is measured by as many finite parts as are necessary to form AB. Moreover, the parts of the time will be equal in number to the parts of the magnitude, and the parts will be finite in quantity. What remains, therefore, is that the entire space is traversed in finite time.
lib. 6 l. 9 n. 5 Deinde cum dicit: eadem autem ratio est etc., ostendit quod e converso, si tempus est finitum, et magnitudo est finita. Et dicit quod per eandem rationem potest ostendi, quod infinitum spatium non potest pertransiri in tempore finito: neque iterum potest quies esse infinita in tempore finito: et hoc indifferenter, sive moveatur aliquid regulariter, idest aeque velociter, sive non regulariter. Quia ex quo tempus ponitur finitum, accipiatur aliqua pars temporis quae mensuret totum tempus, in qua mobile pertransit aliquam partem magnitudinis (non autem totam, quia totam pertransit in toto tempore); et iterum in aequali tempore pertransit aliam partem magnitudinis. Et similiter pro unaquaque parte temporis accipietur aliqua pars magnitudinis: et hoc indifferenter, sive pars magnitudinis secundo accepta, sit aequalis primae parti (quod contingit quando aeque velociter movetur), sive sit inaequalis (quod contingit quando non aeque velociter movetur). Hoc enim nihil differt, dummodo ponatur quod quaelibet pars magnitudinis accepta sit finita: quod oportet dicere; alioquin tantum moveretur in parte temporis, quantum in toto. Sic enim manifestum est quod per divisionem temporis auferetur totum spatium infinitum per aliquam finitam ablationem: quia cum tempus dividatur in partes finitas aequales, et tot oporteat esse partes magnitudinis quot temporis, sequitur quod spatium infinitum consumetur, facta finita ablatione, eo quod tot modis oportet dividi magnitudinem sicut et tempus. Hoc autem est impossibile. Ergo manifestum est quod infinitum spatium non pertransitur in tempore finito. Et hoc indifferenter, sive magnitudo spatii sit infinita ex una parte, sive ex utraque: quia eadem ratio est de utroque.	845. Then at (647 238 a20) he shows that on the other hand, if the time is finite, so too the magnitude, And he says that by the same reasoning it can be shown that infinite space cannot be traversed in finite time, and that rest cannot be infinite in finite time, no matter whether the motion is regular or not. For since the time posited is finite, it is possible to take as a measure of the whole time a part in which the mobile traverses a part of the magnitude but not the whole magnitude, which is traversed in the whole time. Then in an equal time it will traverse another part of the magnitude. And, in like manner, for each part of the time take a corresponding part of the magnitude, and let this be done whether the second part of the magnitude be equal to the first part (which happens when the speed is constant) or not equal to it (which happens when the speed varies). For whether they are equal or not makes no difference, as long as each part you take of the magnitude is finite, which it must be; otherwise as much will be traversed in a part of time as in the whole time. According to this procedure, it is clear that by dividing time the entire infinite space will be exhausted as the finite parts are used up. For since the time is divided into finite equal parts and the number of magnitudinal parts must equal the number of parts of time, it follows that the infinite space will be consumed by making finite subtractions, since the magnitude has to be divided according to the way the time is divided. But this is impossible. Therefore, it is clear that an infinite space cannot be traversed in finite time, whether the magnitude of space be infinite in one direction or more, because in either case the same reason would hold.
lib. 6 l. 9 n. 6 Deinde cum dicit: demonstratis autem his etc., ostendit quod infinitum et finitum similiter invenitur in mobili, sicut in magnitudine et tempore. Et circa hoc tria facit: primo ostendit quod mobile non est infinitum, si tempus et magnitudo sint finita; secundo quod mobile non est infinitum, si magnitudo sit infinita et tempus finitum, ibi: at vero neque infinitum etc.; tertio quod mobile non potest esse infinitum, si magnitudo sit finita et tempus infinitum, ibi: amplius autem et tempore et cetera. Primum ostendit duabus rationibus. Circa quarum primam dicit quod demonstrato quod magnitudo finita non pertransitur tempore infinito, neque infinita finito, manifestum est ex eadem causa, quod neque infinitum mobile potest pertransire finitam magnitudinem in tempore finito. Accipiatur enim aliqua pars temporis finiti. In illa parte spatium finitum pertransibit non totum mobile, sed pars mobilis, et in alia parte temporis similiter, et sic de aliis. Et sic oportebit accipere tot partes mobilis, quot accipiuntur partes temporis. Infinitum autem non componitur ex partibus finitis, ut ostensum est. Ergo sequetur quod mobile quod movetur in toto tempore finito, sit finitum.	846. Then at (648 238 a32) he shows that infinite and finite are found in the mobile in the same way as they are found in magnitude and time. About this he does three things: First he shows that the mobile is not infinite, if the magnitude is finite and the time finite; Secondly, that the mobile is not infinite, if the magnitude is infinite and the time finite, at 848; Thirdly, that the mobile cannot be infinite, if the magnitude is finite and the time infinite at 849. He proves the first point with two arguments. In regard to the first of these he says that, since it has been demonstrated that a finite magnitude is not traversed in infinite time nor an infinite magnitude in finite time, it is clear from the same causes that an infinite mobile cannot traverse a finite magnitude in finite time. For if you take any part of finite time, then during that part of time the finite space will be traversed not by the whole mobile but by a part, and during another part, it will be traversed by another part of the mobile, and so on. And so, it will be necessary to take as many parts of the mobile as parts of time. But the infinite is not composed of finite parts. Therefore, the mobile that is moved in a whole finite time is finite.
lib. 6 l. 9 n. 7 Secundam rationem ponit ibi: quoniam autem finitum et cetera. Et differt haec secunda ratio a priori, quia in priori assumebatur pro principio idem medium ex quo superius demonstrabat: hic autem accipitur pro principio ipsa conclusio superius demonstrata. Ostensum est enim supra, quod mobile finitum non potest pertransire spatium infinitum in tempore finito: unde manifestum est quod eadem ratione nec mobile infinitum potest pertransire spatium finitum in tempore finito. Quia si infinitum mobile pertransit spatium finitum, sequitur quod etiam finitum mobile pertranseat spatium infinitum: quia cum tam mobile quam spatium sit quantum, datis duobus quantis, nihil differt quod eorum moveatur, et quod quiescat. Hoc enim habebitur pro spatio, quod quiescit; et illud pro mobili, quod movetur. Manifestum est enim quod quodcumque ponatur moveri, sequitur quod finitum pertranseat infinitum. Moveatur enim infinitum quod est a, et sit aliqua pars eius finita quae est cd. Quando totum movetur, haec pars finita erit secundum aliquod signum spatii, quod sit b; et continuato motu, iterum alia pars infiniti mobilis fiet iuxta illud spatium, et sic semper. Unde sicut mobile pertransit spatium, ita spatium quodammodo pertransit mobile, inquantum successive alternantur diversae partes mobilis iuxta spatium. Unde patet quod simul accidit infinitum mobile moveri per finitum spatium, et finitum transire infinitum. Non enim aliter est possibile quod infinitum moveatur per spatium finitum, quam quod finitum pertranseat infinitum: aut ita quod finitum feratur per infinitum, sicut quando mobile est finitum et spatium infinitum; aut ita quod saltem finitum metiatur infinitum, sicut cum spatium est finitum et mobile infinitum. Tunc enim, licet finitum non feratur per infinitum, tamen finitum mensurat infinitum, inquantum finitum spatium fit iuxta singulas partes mobilis infiniti. Quia ergo hoc est impossibile, sequitur quod infinitum mobile non pertransit spatium finitum in tempore finito.	847. The second argument is given at (649 238 a36) and it differs from the first, because in the first he took as his principle the same medium that he used in the previous demonstrations, but here he takes as his principle the conclusion reached above. For it has been shown above that a finite mobile cannot traverse an infinite space in finite time, Hence it is clear that for the same reason neither can an infinite mobile traverse a finite space in finite time. For if an infinite mobile traverses a finite space, it follows that a finite mobile can traverse an infinite space, because both the mobile and the space have dimensions. Now when two things having dimensions are involved, it makes no difference which is in motion and which is at rest. For it is clear that whichever is assumed as being in motion, it follows that the finite traverses the infinite. For let A be the infinite that is in motion and let CD be a finite part of it. When the whole is being moved, this finite part will be at the part B of the space, and as the motion continues, another part of the infinite mobile will be at B and so on. Hence, just as the mobile traverses space, so space in a sense traverses the mobile, inasmuch as the various parts of the mobile are successively other and other in regard to the space. Hence it is evident that at the same time that an infinite mobile is being moved through a finite space, something finite is traversing something infinite. For there is no other possible way for an infinite to be moved through finite space than for the finite to traverse infinite space, either by having the finite moved over the infinite, as when the mobile is finite and the space infinite, or by making something finite measure the infinite, as when the space is finite and the mobile infinite. For then, even though the finite is not being moved over the infinite, yet the finite is measuring the infinite, inasmuch as a finite space is placed opposite each of the parts of the infinite mobile. Therefore, because this is impossible, it follows that an infinite mobile does not traverse a finite space in finite time.
lib. 6 l. 9 n. 8 Deinde cum dicit: at vero neque infinitum etc., ostendit quod non potest esse mobile infinitum, spatio existente infinito et tempore finito. Et hoc est quod dicit, quod infinitum mobile non pertransit infinitum spatium in tempore finito. In omni enim infinito est aliquid finitum: si igitur mobile infinitum pertranseat spatium infinitum in tempore finito, sequitur quod pertranseat spatium finitum in tempore finito; quod est contra praeostensa.	848. Then at (650 238 b14) he shows that there cannot be an infinite mobile, if the space is infinite and time finite. And this is what he says: that an infinite mobile cannot traverse an infinite space in finite time. For in every infinite there is something finite. Therefore, if an infinite mobile should traverse an infinite space in finite time, it follows that it traverses a finite space in finite time, which is against a previous conclusion.
lib. 6 l. 9 n. 9 Deinde cum dicit: amplius autem etc., dicit quod eadem demonstratio erit, si accipiatur tempus infinitum et spatium finitum. Quia si pertransit infinitum mobile finitum spatium in tempore infinito, sequitur quod in aliqua parte temporis finiti pertranseat aliquam partem spatii: et ita infinitum pertransibit finitum in tempore finito; quod est contra praeostensa.	849. Then at (651 238 b16) he says that the same demonstration holds if the time be infinite and the space finite. Because if an infinite mobile traverses a finite space in infinite time, it follows that in a part of that time it will traverse a part of the space. Consequently, the infinite will be traversing the finite in finite time, which is also against a previous conclusion.
lib. 6 l. 9 n. 10 Deinde cum dicit: quoniam autem neque finitum etc., ostendit quod finitum et infinitum similiter invenitur in motu, sicut et in praemissis. Et dicit quod quia finitum mobile non pertransit spatium infinitum, neque infinitum mobile finitum spatium, neque infinitum mobile infinitum spatium in tempore finito; sequitur ex his quod non possit esse motus infinitus in tempore finito. Quantitas enim motus accipitur secundum quantitatem spatii: unde non differt motum dicere infinitum aut magnitudinem. Necesse est enim, si unum eorum fuerit infinitum, et alterum infinitum esse, quia non potest esse aliqua pars loci mutationis extra locum.	850. Then at (652 238 b17) ha shows that finite and infinite are found in motion in the way that they are found in mobile, space and time. And he says that a finite mobile does not traverse an infinite space, nor an infinite mobile finite space, nor an infinite mobile infinite space, in finite time, From these facts, it follows that there cannot be an infinite motion in finite time. For the quantity of motion depends on the quantity of space. Hence there is no difference between saying that the motion is infinite and that the magnitude is. For it is necessary that if either is infinite, so is the other, because no part of a local motion can exist outside of a place.