Authors/Thomas Aquinas/physics/L7/lect9

Lecture 9 Rules for the comparison of motions

Latin	English
	Lecture 9 Rules for the comparison of motions
lib. 7 l. 9 n. 1 Postquam philosophus ostendit qui motus sint comparabiles ad invicem, hic docet quomodo comparentur. Et primo in motu locali; secundo in aliis motibus, ibi: sic igitur est in alteratione et cetera. Circa primum duo facit: primo ponit ea secundum quae oportet comparari motus locales ad invicem; secundo accipit regulas comparationis secundum praedicta, ibi: si igitur a quod est movens et cetera. Dicit ergo primo, quod movens localiter semper movet aliquod mobile, et iterum in aliquo tempore, et usque ad aliquam quantitatem spatii. Quod ideo oportet esse, quia sicut in sexto probatum est, semper simul aliquid movet et movit. Probatum est enim ibi, quod omne quod movetur, iam est motum per aliquam partem spatii, et per aliquam partem temporis. Unde sequitur quod et illud quod movetur est aliquod quantum et divisibile, et etiam illud per quod movetur, et tempus in quo movetur. Movens autem non omne est quantum, ut in octavo probabitur: sed tamen manifestum est aliquod quantum esse movens; et de hoc movente hic proponit regulas comparationis.	956. After showing which motions are mutually comparable, the Philosopher now teaches how they are compared: First in local motions; Secondly, in other motions, at 962. About the first he does two things: First he mentions the aspects according to which local motions ought to be mutually compared; Secondly, he sets forth the rules of comparison in the light of the foregoing, at 957. He says therefore first (738 249 b27) that the mover in local motion always moves some mobile, in some definite time, and through some quantity of space. And this is required, because, as was proved in Book VI, something always moves and has moved, simultaneously. For it was proved there that whatever is being moved has already been moved through some part of a distance and through some part of time. Hence it follows that what is being moved is something quantitative and divisible, as are the distance and the time involved. However, not every mover is quantified, as will be proved in Book VIII; nevertheless, it is clear that some quantitative things are movers and it is in respect to those that he proposes the following rules of comparison,
lib. 7 l. 9 n. 2 Deinde cum dicit: si igitur a etc., ponit regulas comparationis. Et primo secundum divisionem mobilis; secundo quando movens dividitur, ibi: et si eadem potentia et cetera. Dicit ergo primo: accipiatur aliquod movens quod sit a, et aliquod mobile quod sit b, et longitudo spatii pertransiti quae sit c; et tempus in quo a movet b per c sit d. Si ergo accipiatur aliqua alia potentia movens, aequalis potentiae ipsi a, sequetur quod illa potentia movebit medietatem mobilis quod est b, in eodem tempore per longitudinem quae sit dupla quam c; sed medietatem mobilis movebit per totam longitudinem c, in medietate temporis quod est d. Ex his igitur verbis philosophi duae regulae generales accipi possunt. Quarum prima est, quod si aliqua potentia movet aliquod mobile per aliquod spatium in aliquo tempore, medietatem illius mobilis per duplum spatium movebit vel aequalis potentia in eodem tempore, vel eadem in alio aequali. Alia regula est, quod medietatem mobilis movebit per idem spatium aequalis potentia in medietate temporis. Et horum ratio est, quia sic conservabitur eadem analogia, idest eadem proportio. Manifestum est enim quod velocitas motus est ex victoria potentiae moventis super mobile: quanto autem mobile fuerit minus, tanto potentia moventis magis excedit ipsum: unde velocius movebit. Velocitas autem motus diminuit tempus, et auget longitudinem spatii: quia velocius est quod in aequali tempore pertransit maiorem magnitudinem, et aequalem magnitudinem in minori tempore, ut in sexto probatum est. Ergo secundum proportionem qua subtrahitur a mobili, oportet subtrahi de tempore, vel addi ad longitudinem spatii, dummodo movens sit idem vel aequale.	957. Then at (739 249 b30) he lays down the rules of comparison. First according to division of the mobile; Secondly, when the mover is divided, at 958. He says therefore first (739 249 b30): Let A be a mover, and B a mobile, and C the length of space traversed, and D the time in which A moves B through C. If therefore we take another moving power, equal to the power of A, it will follow that it will move half of the mobile B through a distance twice C in the same time; but in half the time D it will move half of mobile B through the entire length C, From these statements of the Philosopher two general rules may be gathered. The first is that if some power moves a mobile through some certain distance in a given time, then it or an equal power will move half of that mobile through twice the distance in the same time or in an equal time. The other rule is that an equal power will move half the mobile over the same distance in half the time. The reason behind these rules is that the same proportion is being kept. For it is clear that the swiftness of a motion results from the victory of the mover’s power over the mobile, because the weaker the mobile the more the strength of the mover prevails over it and the more swiftly will it move the mobile. The swiftness of a motion cuts down on the time and increases the length traversed, for the swifter is what traverses a greater distance in an equal time or an equal distance in less time, as was proved in Book VI. Therefore, according to the same proportion by which the mobile is diminished, either the time is diminished or the length traversed is increased, provided, of course, that the mover is the same or an equal mover.
lib. 7 l. 9 n. 3 Deinde cum dicit: et si eadem potentia etc., docet comparare motus ex parte moventis: et primo secundum divisionem moventis; secundo secundum oppositam congregationem, ibi: si vero duo et utrumque et cetera. Circa primum tria facit: primo ponit comparationem veram; secundo removet comparationes falsas, ibi: et si e ipsum z etc.; tertio ex hoc solvit rationem Zenonis, ibi: propter hoc Zenonis ratio et cetera. Dicit ergo primo, quod si aliqua potentia idem mobile movet in eodem tempore per tantum spatium, ipsamet movet medietatem mobilis in medietate temporis per idem spatium; vel in eodem tempore movet medium mobilis per duplum spatium; sicut et de aequali potentia dictum est. Et ulterius, si dividatur potentia, media potentia movebit medietatem mobilis per idem spatium in aequali tempore. Sed hoc intelligendum est, quando potentia est talis quae per divisionem non corrumpitur. Loquitur enim secundum considerationem communem, nondum applicando ad aliquam specialem naturam, sicut et in omnibus quae praemisit. Et ponit exemplum. Si enim accipiatur medietas huius potentiae quae est a, et dicatur e; et accipiatur medietas mobilis quod est b, et dicatur z: sicut a movebat b per c in tempore d, ita e movebit z per idem spatium in aequali tempore; quia et hic etiam servatur eadem proportio virtutis motivae ad corpus ponderosum quod movetur. Unde sequitur quod in aequali tempore fiat motus per aequale spatium, sicut dictum est.	958. Then at (740 250 a4) he teaches how motions are to be compared from the viewpoint of the mover, First according to a division of the mover; Secondly, and conversely, according to an assemblage of movers, at 9610 About the first he does three things: First he sets forth a true comparison; Secondly, he rejects some false comparisons, at 959; Thirdly, from this he answers an argument of Zeno, at 960. He says therefore first (740 250 a4) that if a power moves the same mobile through a certain distance in a given time, it moves half the mobile the same distance in half the time, or it moves half the mobile through twice the distance in the given original time, as was said of an equal power. Further, if the power be divided, half the power will move half the mobile through the same distance in the given time. However, this must be understood of a mover that is not destroyed by division, for he has been speaking in a general way without making application to the particular natures involved. And he gives an example: Let E be half of power A and let Z be half of mobile B, then just as A moved B through C in time D, so E will move Z through the same distance in the same amount of time, because the same proportion of motive power to body mass moved is preserved. Hence, it follows that in an equal time the motion will traverse an equal distance, as was said.
lib. 7 l. 9 n. 4 Deinde cum dicit: et si e ipsum z etc., excludit duas falsas comparationes. Quarum prima est, quod addatur ad mobile, et non addatur ad potentiam moventem. Unde dicit quod si e, quod est medietas motivae potentiae, moveat z, quod est medietas mobilis, in tempore d secundum spatium c; non est necessarium quod ipsa potentia dimidiata, quae est e, moveat mobile quod sit in duplo maius quam z, in aequali tempore secundum medietatem spatii quod est c; quia poterit esse quod dimidia potentia duplum mobile nullo modo movere poterit. Sed si posset movere, teneret haec comparatio. Secunda falsa comparatio est, quando dividitur movens, et non dividitur mobile. Et hanc excludit ibi: si vero a etc.: dicens quod si potentia movens quae est a, moveat mobile quod est b, in tempore d, per spatium quod est c; non oportet quod medietas moventis moveat totum mobile quod est b, in tempore d, neque etiam per quamcumque partem spatii c, cuius partis sit proportio ad totum spatium c sicut e converso erat quando comparabamus a ad z, idest totam potentiam motivam ad partem mobilis. Illa enim erat conveniens comparatio, sed hic non: quia potest contingere quod medietas moventis non movebit totum mobile per aliquod spatium. Si enim aliqua tota virtus movet totum mobile, non sequitur quod medietas illius virtutis moveat totum mobile, neque per quantumcumque spatium, neque in quocumque tempore: quia sequeretur quod solus unus homo posset movere navem per aliquod spatium, si potentia trahentium dividatur secundum numerum trahentium, et secundum longitudinem spatii per quod omnes simul trahunt navem.	959. Then at (741 250 a9) he rejects two false comparisons. The first consists in adding to the mobile without adding to the motive power. Hence he says that if E, which is half the motive power, moves Z9 which is half the mobile, a distance C in time D, it is not necessarily true that the halved power E will move a mobile twice Z through half the distance in the given time, for it could happen that the halved power cannot move the doubled mobile at all. But if it can move it, the comparison will hold. The second false comparison occurs when the mover is divided and the mobile is not divided. This he rejects at (742 250 a12), saying that if the motive power A moves the mobile B through distance C in time D, it does not necessarily follow that half the motive power will move the entire mobile B in time D through a part of distance C such that this part of C is related to the entire distance C as A was related to Z in our other example. For when A was compared with Z, it was a suitable comparison, but in the present case it is not, for it can happen that half the motive power will not move the whole mobile any distance. For if some whole power moves some whole mobile, it does not follow that half of it will move the same mobile any distance, no matter how much time is allowed. Otherwise it would follow that a man by himself could move a whole ship a certain distance, if the combined power of the shiphaulers is divided by the number of haulers and the distance they haul it be so divided.
lib. 7 l. 9 n. 5 Deinde cum dicit: propter hoc Zenonis ratio etc., secundum praemissa solvit rationem Zenonis, qui volebat probare quod quodlibet granum milii faciat aliquem sonum, proiectum in terra, quia totus modius milii, quando in terram effunditur, facit aliquem sonum. Sed Aristoteles dicit quod haec Zenonis ratio non est vera, scilicet quod quaelibet pars milii sonet, idest quodlibet granum milii sonum faciat cum cadit in terram: quia nihil prohibet dicere quod granum milii in nullo tempore movet aerem intantum ut faciat sonum, quem aerem movet ad sonum faciendum totius modius cadens. Et ex hoc possumus concludere quod non est necessarium, quod si aliqua quantacumque pars existens in toto, movet, quod separatim per se existens movere possit: quia pars in toto non est in actu, sed in potentia, maxime in continuis. Sic enim aliquid est ens, sicut et unum; unum autem est quod est in se indivisum et ab aliis divisum: pars autem prout est in toto, non est divisa in actu, sed in potentia tantum: unde non est actu ens neque una, sed in potentia tantum. Et propter hoc etiam non agit pars, sed totum.	960. Then at (743 250 a19) he uses the foregoing to answer an argument of Zeno who wished to prove that each grain of millet falling to the earth makes a sound, because an entire bushel of it, when poured to the earth, makes a sound. But Aristotle says that this argument of Zeno is not true, i.e., that each grain of millet makes a sound when it falls to the earth. For there is no reason why any such part should in any length of time move the air to produce a sound, as does the whole bushel in falling. And from this we can conclude that it is not necessary, if a part existing in a whole causes a motion, that this part, now existing in isolation from the whole, can cause a motion. For in the whole the part is not actual but potential, especially in continua. For a thing is a being in the same way that it is one, and “one” is that which is undivided in itself and divided from others. But a part, precisely as existing in a whole, is not actually divided from it but only potentially; hence it is not actually one but only potentially. For this reason, it is not the part but the whole that acts.
lib. 7 l. 9 n. 6 Deinde cum dicit: si vero duo, et utrumque etc., ponit comparationem secundum aggregationem moventium. Et dicit quod si sint duo, et utrumque eorum moveat; quorum utrumque per se moveat tantum mobile in tanto tempore per tantum spatium: quando coniunguntur istae duae potentiae moventium, movebunt illud quod est coniunctum ex ponderibus motis, per aequale spatium in aequali tempore: quia in hoc etiam servatur eadem analogia.	961. Then at (744 250 a25) he sets forth a comparison based on an aggregate of movers and says that if there are two and each of them causes motion and if each by itself is moving its own mobile a certain distance in a given time, then when the two are united they will move the mobiles—which are now joined together—through an equal distance in the same time, because even in this case the same proportion is maintained.
lib. 7 l. 9 n. 7 Deinde cum dicit: sic igitur est in alteratione etc., ponit easdem comparationis regulas in aliis motibus. Et circa hoc tria facit: primo ostendit divisibilitatem eorum secundum quae attenduntur comparationes motuum; secundo ponit comparationes veras, ibi: in duplo duplum etc.; tertio removet comparationes falsas, ibi: si autem alterans et cetera. Dicit ergo primo quantum ad augmentum, quod sunt tria, scilicet augens, et id quod augetur, et tempus: et haec tria habent aliquam quantitatem. Est etiam quarto accipere quantitatem, secundum quam augens auget, et auctum augetur. Et haec etiam quatuor est accipere in alteratione: scilicet alterans, et quod alteratur, et quantitas passionis secundum quam fit alteratio, quae inest secundum magis et minus, et iterum quantitas temporis in quo fit alteratio; sicut et haec quatuor in motu locali inveniebantur.	962. Then at (745 250 a28) he sets forth the same rules of comparison for other motions. About this he does three things: First he shows that the things according to which the comparison of motions must be judged are divisible; Secondly, he sets forth the true comparison, at 963; Thirdly, he rejects some false comparisons, at 964. He says therefore first (745 250 a28), in respect to growth, that there are three things involved, namely, the cause of increase, the thing increased, and the time; and these three have a certain quantity. Also there is a fourth thing to be considered, namely, the quantity of increase produced by the cause and received by the growing thing. And these four things must be considered also in alteration, namely, the cause of alteration, the thing altered, the amount or degree of alteration (which is present according to more and less), and the amount of time. These four, of course, are the same as are involved in local motion.
lib. 7 l. 9 n. 8 Deinde cum dicit: in duplo duplum etc., ponit comparationes veras. Et dicit quod si aliqua potentia secundum hos motus moveat tantum in tanto tempore, in duplo tempore movebit duplum: et si moveat duplum, hic erit in duplo tempore. Et similiter movebit eadem potentia medium in medio tempore: aut si moveat in medio tempore, erit dimidium quod est motum. Aut si sit dupla potentia, in aequali tempore movebit duplum.	963. Then at (746 250 b2) he sets forth the true comparison and says that if a power moves something to a certain amount in a given time according to these motions, then it will move to twice the amount in twice the time; and if it moves to twice the amount, it will be in twice the time. Likewise, the same power will move to half the amount in half the time, or if it moves in half the time, then the motion will be to half the amount, Or if there is twice the power, it will move something to twice the amount in an equal time.
lib. 7 l. 9 n. 9 Deinde cum dicit: si autem alterans etc., excludit falsam comparationem. Et dicit quod si aliqua potentia moveat motu alterationis et augmenti tantum in tanto tempore, non necesse est quod medietas potentiae moveat medietatem in eodem tempore, aut in medio tempore tantundem: sed forte continget quod nihil augmentabit vel alterabit, sicut et in gravi, idest sicut dictum est quod dimidiata potentia non potest movere totum pondus, neque per totum spatium, neque per aliquam eius partem. Est enim intelligendum, quod hoc quod dicit: in medio medium, aut in aequali duplum, ly duplum et medium (quod in accusativo ponitur) non accipitur pro dimidio vel duplo ipsius mobilis, sed pro dimidio et duplo ex parte rei in qua est motus, scilicet qualitatis aut quantitatis, quae ita se habent in istis duobus motibus, sicut longitudo spatii in motu locali: alioquin non similiter esset in istis motibus et in motu locali. In motu enim locali, dictum est quod si tanta potentia movet tantum mobile, medietas movebit medietatem mobilis: hic autem dicitur quod medietas forte nihil movebit. Sed intelligendum est de toto mobili integro: quia virtus motiva dimidiata non movebit ipsum, neque per tantam quantitatem aut qualitatem, neque per eius medium.	964. Then at (747 250 b4) he dismisses a false comparison and says that if what causes alteration or increase causes a certain amount of increase or alteration respectively in a certain amount of time, it does not necessarily follow that half the force will alter or increase half the object or some given amount in half the time; for it may happen that there will be no alteration or increase at all, the case being the same as with the locally mobile that has weight. It should be observed that when Aristotle says “half will be moved in half or double will be moved in an equal,” “double” and “half” (in the accusative case) refer, not to the mobile but to the sphere of motion, i.e., the quality or the quantity, which are related to alteration and growth as length of distance is related to local motion. For in local motion it was said that if a certain power moves a certain mobile, half will move half the mobile, but here it is said that half might not move anything. But it must be understood that we are speaking of an integral mobile whole, which will not be moved by a halved motive power to any amount of quantity or degree of quality, much less to half.