Authors/Thomas Aquinas/physics/L6/lect6

Lecture 6 Two manners of dividing motion. What things are co-divided with motion

Latin	English
	Lecture 6 Two manners of dividing motion. What things are co-divided with motion
lib. 6 l. 6 n. 1 Praemissis quibusdam quae sunt necessaria ad divisionem motus, hic incipit agere de divisione motus. Et dividitur in partes duas: in prima agit de divisione motus; in secunda ex determinatis excludit quosdam errores circa motum, ibi: Zeno autem male ratiocinatur et cetera. Prima autem pars dividitur in partes duas: in prima determinat de divisione motus; in secunda de divisione quietis, ibi: quoniam autem omne aut movetur et cetera. Prima dividitur in duas: in prima agit de divisione motus; in secunda de finito et infinito circa motum (utrumque enim videtur ad rationem continui pertinere, scilicet divisibile et infinitum), ibi: quoniam autem omne quod movetur, in tempore movetur et cetera. Prima autem pars dividitur in duas: in prima ostendit quomodo motus dividitur; in secunda agit de ordine partium motus, ibi: quoniam autem omne quod mutatur, ex quodam et cetera. Circa primum duo facit: primo ponit duos modos quibus motus dividitur; secundo ostendit quae sunt illa quae simul dividuntur cum motu, ibi: quoniam autem omne quod movetur, in aliquo et cetera. Circa primum duo facit: primo ponit modos quibus motus dividitur; secundo exponit eos, ibi: sit igitur ipsius quidem ab et cetera.	806. Having established the facts needed for dividing motion, he now begins to treat of the division of motion. And the treatment is divided into two parts. In the first he treats of the division of motion; In the second he uses his conclusions to refute errors about motion, at L. 11. The first part is divided into two sections: In the first he discusses division of motion; In the second, division of rest, at L. 10. The first section is divided into two parts: In the first he deals with division of motion; In the second he discusses finite and infinite with respect to motion (for both, namely, “divisible” and “infinite” seem to belong to the continuum), at L. 9. The first is divided into two parts: In the first he shows how motion is divided; In the second he treats of the order of the parts of motion, at L. 7. In regard to the first he does two things: First he lists two ways by which motion is divided; Secondly, he mentions what else is divided when motion is divided, at 812. In regard to the first he does two things: First he mentions the ways in which motion is divided; Secondly, he explains them, at 808.
lib. 6 l. 6 n. 2 Dicit ergo primo, quod duobus modis dividitur motus. Uno modo secundum tempus; quia ostensum est quod motus non est in nunc sed in tempore. Alio vero modo dividitur secundum motus partium mobilis. Sit enim ac mobile, et dividatur: ostensum est enim omne quod movetur divisibile esse. Si ergo ipsum ac totum movetur, necesse est quod moveatur utraque pars eius, scilicet ab et bc. Est autem considerandum, quod divisio motus secundum partes mobilis, potest intelligi dupliciter. Uno modo ut pars post partem moveatur: quod quidem non est possibile in eo quod secundum se totum movetur; quia eius quod secundum se totum movetur, omnes partes simul moventur, non quidem seorsum a toto, sed in ipso toto. Alio modo potest intelligi ista divisio motus secundum partes mobilis, sicut et divisio cuiuslibet accidentis cuius subiectum est divisibile, attenditur secundum divisionem sui subiecti; sicut si totum hoc corpus est album, secundum divisionem corporis dividetur per accidens albedo. Et sic accipitur hic divisio motus secundum partes mobilis; ut sicut utraque pars mobilis simul movetur in toto, ita motus utrarumque partium sint simul. Et per hoc ista divisio motus, quae est secundum partes mobilis, est alia ab illa quae est secundum tempus, in qua duae partes motus non sunt simul. Si tamen motus partis unius comparetur ad motum partis alterius non simpliciter, sed secundum aliquod signum determinatum, sic motus unius partis etiam tempore praecedit motum alterius partis. Si enim mobile abc moveatur in magnitudine efg, ita quod ef sit aequale toti ac, manifestum est quod hoc signum f prius pertransibit bc quam ab: et secundum hoc simul curret divisio motus secundum partes temporis et secundum partes mobilis.	807. He says therefore first (612 234 b21) that motion is divided in two ways. In one way it is divided according to time, because it has been shown that motion occurs not in the “now” but in time. In a second way, it is divided according to the motions of the parts of the mobile. For let the mobile AC be divided, for any mobile can be divided, as we have shown. If therefore the entire mobile AC is being moved, then each of its parts AB and BC is in motion, But notice that the dividing of motion according to the parts of the mobile can be understood in two ways, First of all, that part is being moved after part—which is not possible in that which is in motion per se in its entirety, for in the case of such a mobile all the parts are moved together, not in isolation from the whole, but in the whole. In the second sense, the dividing of motion according to parts of the mobile can be taken in the same sense that the division of an accident whose subject is divisible depends on the division of that subject; for example, if a whole body is white, then as the body is divided, the whiteness will be divided per accidens. And it is in this sense that we are taking division of motion according to the parts of the mobile, i.e., just as both parts of the mobile are in motion at the same time as the whole is, so the motions of both parts occur at the same time. This shows that division of motion according to the parts of the mobile is different from, that which is according to time, in which division two given parts of a motion do not occur at the same time. But if we were to compare the motion of one part to that of another part not absolutely but according to a fixed stage to be reached, then the motion of one part, will precede in time the motion of another part. For if the mobile ABC is moved in the magnitude EFG, so that. EY is equal to length ABC of the mobile, it is clear that BC will reach F before AB does. According to this the division of motion according to the parts of time and according to the parts of the mobile will be concurrent.
lib. 6 l. 6 n. 3 Deinde cum dicit: sit igitur ipsius quidem etc., manifestat positos modos: et primo ostendit quod motu dividatur secundum partes mobilis; secundo quod dividatur secundum partes temporis, ibi: alius autem secundum tempus et cetera. Primum ostendit tribus rationibus: quarum prima talis est. Ex quo moto toto moventur partes, motus illius partis quae est ab, sit de; et motus alterius partis, quae est bc, sit ez. Sicut ergo totum mobile ac componitur ex ab et bc, ita totus motus dz componitur ex de et ez. Cum ergo utraque partium mobilis moveatur secundum utramque partium motus, ita tamen quod neutra pars mobilis movetur secundum motum alterius partis (quia secundum hoc totus motus esset unius partis, quae moveretur motu suo et motu alterius partis), oportet dicere quod totus motus dz sit totius mobilis ac; et sic motus totius dividitur per motum partium.	808. Then at (613 234 b24) he explains these ways of dividing motion: First he shows that motion is divided according to the parts of the mobile; Secondly, that it is divided according to the parts of time, 8-1.71. The first he shows by three arguments, of which the first is this: Since the parts are in motion by the fact of the wholes being in motion, let DE be the motion of the part AB and EZ the motion of the part BC. Therefore, just as the whole mobile is composed of AB and BC, so the whole motion DZ is composed of DE and EZ. Since, therefore, both of the parts of the mobile are being moved in accordance with both of the parts of the motion in such a way that neither part of the mobile is being moved in accordance with the motion of the other part (because then the entire motion would be the motion of one part, which would be moved by its own motion and by the motion of the other part), then it must be admitted that the whole motion DZ is the motion of the whole mobile AC; and thus the motion of the whole is divided by means of the motion of the parts.
lib. 6 l. 6 n. 4 Secundam rationem ponit ibi: amplius autem, si omnis motus etc.: quae talis est. Omnis motus est alicuius mobilis: totus autem motus dz non est alterius partium; quia neutra movetur secundum totum motum, sed utraque movetur secundum partes motus, ut dictum est. Neque iterum potest dici quod sit motus cuiuscumque alterius mobilis separati ab ac: quia si totus iste motus esset totius alterius mobilis, sequeretur quod partes huius motus essent partium illius mobilis; sed partes huius motus qui dicitur dz, sunt partium huius mobilis quae sunt ab, bc, et nullarum aliarum; quia si essent et harum et aliarum, sequeretur quod unus motus esset plurium, quod est impossibile. Relinquitur ergo quod totus motus sit totius magnitudinis, sicut et partes partium; et ita motus totius dividitur secundum partes mobilis.	809. At (614 234 b29) he gives the second argument, which is this: Every motion belongs to some mobile. But the entire motion DZ does not belong to either of the parts, because neither is being moved according to the entire motion, but both are being moved according to the parts of the motion, as we have said. Nor can it be said that the whole motion DZ is the motion of some other mobile separated from AC, because, if the whole of this motion were the motion of some other whole mobile, it would follow that the parts of this motion would belong to the parts of that mobile; whereas we have already agreed that the parts of the motion DZ belong to the parts of the original mobile, which are AB and BC, and to no other parts (for if they belonged to these and to others as well, it would follow that one motion would belong to several things, which is impossible), What remains, therefore, is that the entire motion belongs to the entire magnitude just as the parts of it belong to the parts of the magnitude. And thus the motion of the whole mobile is divided according to the parts off the mobile.
lib. 6 l. 6 n. 5 Tertiam rationem ponit ibi: amplius autem, si est quidem etc.: quae talis est. Omne quod movetur, habet aliquem motum: si igitur totus motus qui est dz, non sit totius mobilis quod est ac, oportet quod aliquis alius motus sit eius; et sit ille motus ti. Ab hoc ergo motu ti auferantur per divisionem motus utrarumque partium, quos oportet esse aequales iis quae sunt dez, hac ratione: quia unius mobilis non est nisi unus motus; unde non potest dici quod motus partium, qui auferuntur a motu ti, qui ponitur esse totius, sint maiores aut minores quam de et ez, qui ponebantur motus earundem partium. Aut ergo motus partium consumunt per divisionem totum ti, aut deficiunt ab eo, aut superexcedunt. Si consumunt totum ti, et non excedunt nec deficiunt, sequitur quod motus ti sit aequalis motui dz, qui est motus partium, et non differat ab eo. Si autem motus partium deficiunt a ti, ita quod ti excedat dz in ki, ista pars motus quae est ki, nullius mobilis erit. Non enim est motus totius ac, neque partium eius; quia unius non est nisi unus motus, et tam toti quam partibus assignatus est iam alius motus. Neque iterum potest dici quod sit alicuius alterius mobilis; quia totus motus ti est quidam motus continuus; et motus continuus oportet quod sit continuorum, ut in quinto ostensum est. Unde non potest esse quod pars huius motus continui, quae est ki, sit alicuius mobilis quod non continuetur cum abc. Similiter etiam sequitur inconveniens, si dicatur quod motus partium excellat secundum divisionem; quia sequetur quod partes excedant totum, quod est impossibile. Si ergo hoc est impossibile, quod excedat vel deficiat, necesse est quod motus partium sit aequalis et idem motui totius. Haec igitur divisio est secundum motus partium; et necesse est quod talis partitio inveniatur in motu, propter hoc quod omne quod movetur est partibile.	810. At (615 234 b34) he gives the third argument, which is this: Everything that is being moved has a position. Therefore, if the whole motion DZ does not belong to the whole mobile AC, then some of the motion does, and let it be TI. Now, from this motion TI take away by division the motions of both parts, which must be equal to the motions that form DEZ, for the following reason: One mobile does not have but one motion, and, consequently, the parts’ motions which are taken away from the motion TI (which is the motion of a whole) cannot be said to be greater or less than DE and EZ, which we agreed are the motions of those same parts. Now the motions of the parts consume the whole motion TI or they are less or greater. If they consume the entire TI and are neither greater nor less, it follows that the motion TI is equal to the motion DZ (which is the motion of the parts) and does not differ from it. But if the motions of the parts are less than TI so that TI exceeds DZ by the amount KI, then the part KI of the motion does not belong to any mobile. For it is neither the motion of AC nor of any of its parts, because one thing has only one motion, and we have already assigned a different motion both to the whole AC and to its parts. Nor can we say that KI belongs to some other mobile, because the entire motion TI is one continuous motion and a continuous motion must belong to a thing that is continuous, as we have shown in Book V. Hence it cannot be that the part KI of this continuous motion belongs to a mobile not continuous with ABC. A like difficulty follows, if it is said that the motion of the parts exceeds the divided motion TI, because it will follow that the parts exceed the whole—which is impossible. Consequently, if it is impossible that the parts either exceed or are less than to the whole, then necessarily the motion of the parts is equal to and is the same as the motion of the whole. And so this division is based on the motions of the parts and such a partition must be found in motion, because everything that is being moved is capable of being divided into parts.
lib. 6 l. 6 n. 6 Deinde cum dicit: alius autem secundum tempus etc., ostendit quod motus dividatur secundum divisionem temporis, tali ratione. Omnis motus est in tempore: et omne tempus est divisibile, ut probatum est. Cum ergo in minori tempore sit minor motus, necesse est quod omnis motus dividatur secundum tempus.	811. Then at (616 235 a10) he shows in the following argument that motion is divided according to the division of time; Every motion occurs in time and every time is divisible, as we have proved. Therefore, since there is less motion in less time, every motion must be capable of being divided according to time.
lib. 6 l. 6 n. 7 Deinde cum dicit: quoniam autem omne quod movetur etc., ostendit quae simul dividantur cum motu. Et circa hoc tria facit: primo ponit quinque quae simul dividuntur; secundo ostendit quod in omnibus praedictis simul invenitur finitum et infinitum, ibi: et in ipso finita esse etc.; tertio ostendit in quo horum primo invenitur divisio et infinitum, ibi: secutum autem maxime est et cetera. Circa primum duo facit: primo proponit quod intendit; secundo manifestat propositum, ibi: accipiatur enim tempus et cetera. Dicit ergo primo, quod quia omne quod movetur, movetur in aliquo, idest secundum aliquod genus vel speciem, et iterum in aliquo tempore; et iterum cuiuslibet mobilis est aliquis motus; necesse est quod ista quinque simul dividantur, scilicet tempus, et motus, et ipsum moveri, et mobile quod movetur, et id in quo est motus, vel locus vel qualitas vel quantitas. Sed tamen non est eodem modo divisio omnium eorum in quibus est motus; sed quorundam quidem per se, quorundam vero per accidens: per se quidem omnium eorum quae pertinent ad genus quantitatis, ut est in motu locali, et etiam in augmento et decremento; per accidens vero in iis quae pertinent ad qualitatem, ut in motu alterationis.	812. Then at (617 235 a13) he shows what other things are divided when motion is divided. About this he does three things: First he mentions five things that are co-divided; Secondly, he shows that if the finite or infinite is found in any of them, it is found in all the others, at 816; Thirdly, he shows in which of them is first found division and infinite, at 817. About the first he does two things: First he states his proposition; Secondly, he explains the proposition, at 813. He says therefore first (617 235 a13) that since everything that is being moved is being moved in respect to some genus or species as well as in time and, moreover, since every mobile is capable of some motion, then necessarily the following five things must be divided at the same time that any one of them is divided: time and motion and the very “act of being moved” and the mobile which is being moved and “the sphere of motion”, i.e., the genus or species in regard to which there is motion, i.e., place or quality or quantity. Nevertheless, the divisions of the “spheres of motion” do not all occur in the same way but in some the division is per se and in others per accidens. The division is per se, if it is in the sphere of quantity, as it is in local motion and also in growth and decrease; but it is per accidens in the sphere of quality, as in the motion called “alteration”.
lib. 6 l. 6 n. 8 Deinde cum dicit: accipiatur enim tempus etc., manifestat quod dixerat. Et primo quantum ad hoc quod tempus et motus simul dividuntur; secundo quod motus et ipsum moveri simul dividuntur, ibi: eodem autem modo etc.; tertio ostendit idem de motu et eo in quo est motus, ibi: similiter autem demonstrabitur et cetera. Circa primum duo facit: primo ostendit quod ad divisionem temporis dividitur motus; secundo quod e converso ad divisionem motus dividitur tempus, ibi: similiter autem et si motus et cetera. Dicit ergo primo: ponatur quod tempus in quo aliquid movetur sit a, et motus qui est in hoc tempore sit b. Manifestum est autem quod si aliquid movetur per totam magnitudinem in toto tempore, quod in medietate temporis movetur per minorem magnitudinem. Idem est autem moveri toto motu, et per totam magnitudinem; et parte motus et per partem magnitudinis. Unde manifestum est quod si in toto tempore movetur toto motu, quod in parte temporis movebitur minori motu: et iterum diviso tempore, invenietur minor motus; et sic semper. Ex quo patet quod secundum divisionem temporis dividitur motus. Deinde cum dicit: similiter autem, et si motus etc., ostendit quod e converso, si motus dividitur, et tempus dividitur. Quia si per totum motum movetur in toto tempore, per medium motus movebitur in medio tempore, et semper minor erit motus in minori tempore, si sit mobile idem vel aeque velox.	813. Then at (618 235 a18) he explains what he has said: First the statement that time and motion are co-divided; Secondly, that motion and the “act of being moved” are, at 814. Thirdly, that motion and the sphere of motion are, at 815, About the first he does two things: First he shows that with division of time, motion is divided; Secondly, vice versa, at 814. He says therefore first (618 235 a18): Let A be the time in which something Is being moved, and let B be the motion occurring in this time. Now it is evident that if something is being moved through an entire magnitude in the whole time A, then in half the time, it will be moved through a smaller magnitude. But to be moved through the entire motion is the same as being moved through the entire magnitude, just as to be moved through part of the motion is the same as being moved through part of the magnitude. Therefore, it is clear that if in the entire time it is moved through the whole motion, then in part of the time it will be moved through a smaller motion. And if the time be again divided, a smaller motion will be found, and so on indefinitely. And so it is evident that according to the division of time, motion is divided. Then at (619 235 a22) he shows that on the other hand, if the motion is divided, the time is divided. Because if it is being moved through the entire time, then through half the motion it will be moved through half the time and so on, as the motion is smaller, the corresponding time is also, provided of course that we are dealing with the same mobile or one equally fast.
lib. 6 l. 6 n. 9 Deinde cum dicit: eodem autem modo etc., ostendit quod motus et moveri simul dividuntur. Et circa hoc duo facit: primo ostendit quod ipsum moveri dividitur secundum divisionem motus; secundo quod motus dividitur secundum divisionem eius quod est moveri, ibi: est autem et ponentem et cetera. Dicit ergo primo, quod eodem modo probatur quod ipsum moveri dividitur secundum divisionem temporis et motus: et ipsum moveri sit c. Manifestum est autem quod non tantum movetur aliquid secundum partem motus, quantum secundum totum motum. Manifestum est ergo quod secundum medium motum, pars eius quod est moveri, erit minor toto ipso moveri, et adhuc minor secundum medietatis medium; et sic semper procedetur. Ergo sicut tempus et motus semper dividuntur, ita et ipsum moveri. Deinde cum dicit: est autem et ponentem etc., probat quod e converso motus dividitur secundum divisionem eius quod est moveri. Sint enim duae partes motus dc et ce, secundum quarum utramque aliquid movetur. Et sic si partibus eius quod est moveri respondent partes motus, oportet dicere quod toti respondeat totum: quia si aliquid plus esset in uno quam in altero, erit hic argumentari de moveri ad motum, sicut supra argumentati sumus, quando ostendimus quod motus totius est divisibilis in motus partium, ita quod nec potest deficere nec excellere. Similiter etiam et partes eius quod est moveri, non possunt excedere partes motus nec deficere: quia enim necesse est accipere secundum utramque partem motus hoc quod est moveri, necesse est quod totum moveri sit continuum, correspondens toti motui. Et ita semper partes eius quod est moveri, respondent partibus motus, et totum toti; et sic unum dividitur secundum alterum.	814. Then at (620 235 a25) he shows that motion and the “act of being moved” are co-divided. Regarding this he does two things: First he shows that “being moved” is divided according to the division of motion; Secondly, that motion is divided in accordance with the division of “being moved”, at 814. He says therefore first (620 235 a25) that in the same way, it is proved that “being moved” is divided in accordance with the division of time and motion. For let “being moved” be C. Now it is evident that a thing is not moved as much according to part of the motion as according to the whole of the motion. Therefore, according to half of the motion, part of the factor called “being moved” will be less than the whole factor and still less according to half of the half, and so on. Therefore, as time and motion are continually subdivided, so also the factor called “being moved”. Then at (621 235 a28) he proves that conversely motion is divided according to the division of “being moved”. For let DC and CE be two parts of a motion, according to both of which something is being moved. Then if the parts of the motion correspond to the parts of “being moved”, then the whole corresponds to the whole, because if there were more in one than in the other, then the same argument would apply here that applied when we proved that the motion of a whole can be divided into motions of the parts in such a way that there is neither excess nor defect. In like manner, the parts of “being moved” can neither be less nor greater than the parts of the motion; for since we must admit a “being moved” for each part of the. motion, then necessarily the entire factor called “being moved” is continuous and corresponds to the entire motion. And thus, the parts of “being moved” correspond to the parts of the motion and the whole to the whole. Consequently, one is divided in accordance with the other.
lib. 6 l. 6 n. 10 Deinde cum dicit: similiter autem demonstrabitur etc., ostendit idem de eo in quo est motus. Et dicit quod eodem modo demonstrari potest, quod longitudo in qua movetur aliquid secundum locum, sit divisibilis secundum divisionem temporis, et motus, et ipsius moveri. Et quod dicimus de longitudine in motu locali, est etiam intelligendum de omni eo in quo est motus: nisi quod quaedam sunt divisibilia per accidens, sicut qualitates in motu alterationis, ut dictum est. Et inde est quod omnia ista sic dividuntur; quia illud quod mutatur est divisibile, ut ostensum est supra. Unde uno horum diviso, oportet quod omnia dividantur.	815. Then at (622 235 a34) he shows the same for the sphere of motion, i.e., for the genus or species in which the motion takes place. And he says that in the same way it can be demonstrated that the length in which something is moved locally can be divided according to the division of time and of motion and of “being moved”. And what we say of the length in local motion is to be understood of every sphere in which there is motion, except that in some spheres the division is per accidens, as in the case of qualities in the motion, of alteration, as was said. And hence it is that all those things are divided, because the subject of change can be divided, as was explained above. Consequently, if one is divided, all the others must.
lib. 6 l. 6 n. 11 Deinde cum dicit: et in ipso finita esse etc., ostendit quod sicut se consequuntur praemissa in divisibilitate, ita se consequuntur in hoc quod est esse finita vel infinita: ita quod si unum horum fuerit finitum, omnia erunt finita; et si infinitum, similiter.	816. Then at (623 235 a37) he says that just as the above-mentioned things follow upon one another in divisibility, so also in being finite or infinite, so that if one of them is finite, all the others are, and if one is infinite, so are all the others.
lib. 6 l. 6 n. 12 Deinde cum dicit: secutum autem maxime etc., ostendit in quo praemissorum primo inveniatur divisibilitas et finitum seu infinitum. Et dicit quod maxime ab ipso quod mutatur, consequitur de omnibus aliis quod dividantur, et quod sint finita vel infinita: quia illud quod est primum naturaliter in motu, est ipsum mobile, et statim ipsi ex sua natura inest esse divisibile, et esse finitum vel infinitum; et sic ex ipso ad alia derivatur divisibilitas vel finitum. Quomodo autem ipsum mobile sit divisibile, et per ipsum alia dividantur, ostensum est prius. Sed quomodo etiam hoc sic se habet de infinito, ostendetur inferius in hoc eodem sexto libro.	817. Then at (624 235 b1) he shows in which of the five above-mentioned things divisibility and finite and infinite are first found. And he says that the subject of change is the first root from which the divisibility and finiteness and infinity of the others flow, because what is naturally first in motion is the mobile, which of its very nature has the properties called “divisibility”, “finiteness” and “infinity”. Hence from it divisibility and finiteness flow to the others. But how the mobile is divisible and how the others are divided through it, we have already shown. How the mobile is infinite will be explained later in this Book VI.