Dicit ergo primo quod eiusdem rationis est quod magnitudo et tempus et motus componantur ex indivisibilibus et dividantur in indivisibilia, vel nihil horum: quia quidquid dabitur de uno, ex necessitate sequetur de alio. He says therefore first (231b18) that any argument that shows that a magnitude is composed or not composed of indivisibles, and divided or not divided into indivisibles, applies also to time and motion, since whatever is granted in regard to any of them would necessarily be true of the others. 759. Deinde cum dicit: manifestum est autem ex his etc., probat propositum: 759. Then, at this may be made (231b20), he proves this proposition: et primo quantum ad magnitudinem et motum; first, in regard to magnitude and motion; secundo quantum ad tempus et magnitudinem, ibi: similiter autem necesse etc. second, in regard to time and magnitude, at and, if length and motion (232a18; [766]). Circa primum tria facit: About the first, he does three things: primo ponit propositum; first, he presents his proposition; secundo exemplificat, ibi: ut si ipsa ABC etc., second, he gives an example, at for instance, if the magnitude (231b22; [760]); tertio probat, ibi: si igitur praesentis motus etc. third, he proves his proposition, at therefore, since where there is (231b25; [761]). Propositum est istud: si magnitudo ex indivisibilibus componitur, et motus qui transit per magnitudinem, componetur ex indivisibilibus motibus, aequalibus numero indivisibilibus ex quibus componitur magnitudo. The proposition is this. If a magnitude is composed of indivisibles, then the motion that traverses it will likewise be composed of indivisible motions equal in number to the indivisibles of which the magnitude is composed. 760. Exemplificat autem sic. Sit linea abc, quae componatur ex tribus indivisibilibus, quae sunt a et b et c; et sit o mobile quod movetur in spatio lineae abc, et motus eius sit dez: oportebit quod si partes spatii vel lineae sint indivisibiles, quod etiam partes praedicti motus sint indivisibiles. 760. Of this, he gives the following example (231b22). Let the line ABC be composed of the three indivisibles A, B and C, and let O be an object in motion over the distance of the line ABC, so that DEZ is its motion. Now, if the parts of the distance or of the line are indivisibles, then the parts of the motion are indivisibles. Deinde cum dicit: si igitur praesentis motus etc., probat propositum. Then, at therefore, since where there is (231b25; [761]), he proves his proposition. Et circa hoc tria facit: About this, he does three things: primo praemittit quaedam necessaria ad propositi probationem; first, he lays down some premises necessary for his proof; secundo probat quod si magnitudo componitur ex punctis, quod motus componitur non ex motibus, sed ex momentis, ibi: secundum a igitur etc.; second, he proves that, if a magnitude is composed of points, then the motion is composed not of motions, but of instants, at and, as we saw (232a1; [762]); tertio ostendit esse impossibile quod motus componatur ex momentis, ibi: et motum esse aliquid etc. third, he shows that it is impossible for motion to be composed of instants, at and the motion will take place (232a9; [763]). 761. Praemittit ergo primo duo. 761. Therefore, he first lays down two presuppositions. Primum est quod secundum quamcumque partem praesentis motus necesse est aliquid moveri; et e converso, si aliquid movetur, necesse est quod adsit sibi aliquis motus. Et si hoc est verum, oportet quod mobile o moveatur per a, quae est pars totius magnitudinis, ea parte motus quae est d; et secundum b, aliam partem magnitudinis, moveatur alia parte motus quae est e; et secundum c, tertiam partem magnitudinis, moveatur tertia parte motus quae est z; ita quod singulae partes motus respondeant singulis partibus magnitudinis. The first (231b25) is that, according to each part of the motion under consideration, something must be in motion, and conversely, if something is in motion, a motion must be in it. Now, if this is true, then the mobile O is being moved through A, which is part of the entire magnitude, by means of that part of the motion that is D, and through B (another part of the magnitude) by that part of the motion that is E, and through C (the third part of the magnitude) by that part of the motion that is Z. In other words, single parts of motion correspond to single parts of the magnitude. Secundum proponit, ibi: si igitur necesse est etc.: et dicit quod necesse est id quod movetur ab uno termino in alium, non simul moveri et motum esse, inquantum movetur et quando movetur; sicut si aliquis vadit Thebas, impossibile est haec duo simul esse, scilicet ire Thebas et ivisse Thebas. He sets out the second presupposition at now, if a thing that is (231b28), and he says that what is being moved from one terminus to another is not, at the same time, both being moved and finished moving, anymore than a man going to Thebes is, at the time while he is going, already there. Haec autem duo supponit quasi per se manifesta. Nam quod necesse sit moveri ad praesentiam motus, apparet etiam in omnibus accidentibus et formis: quia ad hoc quod aliquid sit album, necesse est habere albedinem; et e converso, si albedo adsit, necesse est quod sit album. He presupposes these two statements as per se evident. As to the statement that, when motion is present, something must be in the state of being moved, a like situation is apparent in all accidents and forms, for in order that something be white, it must have whiteness, and conversely, if whiteness exists, something is white. Quod vero non simul sit moveri et motum esse, apparet ex ipsa motus successione: quia impossibile est aliqua duo temporis simul esse, ut in quarto habitum est: unde impossibile est quod simul sit motum esse, quod est terminus motus, cum ipso moveri. As to the statement that being moved and having been moved are not simultaneous, we appeal to the very successive nature of motion, since it is impossible that any two elements of time coexist, as we explained in book 4. Hence, it is impossible that having been moved, which is the terminus of motion, be simultaneous with being in motion. 762. Deinde cum dicit: secundum A igitur etc., probat propositum ex praemissis. Si enim praesente aliqua parte motus necesse est aliquid moveri, et si movetur necesse est adesse motum; si mobile quod est o, movetur secundum impartibilem partem magnitudinis quae est a, oportet quod adsit ei aliquis motus qui est d. Aut ergo o simul movetur per a et motum est, aut non simul. 762. Then, when he says, and, as we saw (232a1), he uses these presuppositions to prove his proposition. For if it is true that, whenever a part of motion is present, something has to be in motion, and if it is in motion, there must be motion present, then, if the mobile O is in motion with respect to an indivisible part of the magnitude—namely, A—there is in O that part of the motion we called D. Accordingly, O is being moved through A and has completed its motion either at the same time or not at the same time. Si autem non simul, sed posterius devenerit quam venit, idest sed posterius motum est quam movetur, sequitur quod a sit divisibilis: quia cum veniret, idest dum erat in ipso moveri, neque quiescebat in a, quiete scilicet praecedente motum, neque transierat totum ipsum a, quia iam non moveretur per a (nihil enim movetur per spatium per quod iam pertransivit); sed oportet quod medio modo se habeat. Ergo cum movetur per a, partem eius iam transivit et in parte eius adhuc manet: et ita sequitur quod a sit divisibilis; quod est contra suppositum. If not at the same time but after being in motion, it follows that A is divisible, for while O was in motion, it neither was resting at A (with the rest preceding motion) nor had passed through the entire distance A, since it then would not still be in motion through A, as nothing is in motion through a distance it has already traversed. Consequently, it must be midway. Therefore, when it is in motion through A, it has already passed through part of A and is now in another part of A. Consequently, A is divisible, contrary to our supposition. Si vero simul venerit et venit, idest si simul motum est et movetur per a, sequitur quod cum veniens venit, erit ibi ventum, et erit motum ubi movetur: quod est contra secundam suppositionem. But if it is in motion through A and has completed its motion at the same time, it follows that it arrived while it was coming, and it will have completed its motion while it was being moved, which is against the second presupposition. Sic igitur patet quod secundum impartibilem magnitudinem non potest aliquid moveri: quia vel oporteret quod simul esset moveri et motum esse, vel quod magnitudo divideretur. From this, it is clear that no motion is possible when the magnitude is indivisible, for there are only two choices: either things can be in motion at the same time that their motion is over, or the magnitude must be divisible. Supposito ergo quod per a impartibile nihil moveri possit, si aliquis dicat quod mobile movetur per totam magnitudinem quae est abc, et motus totus quo per eam movetur est dez, ita quod secundum a impartibile nihil moveatur, sed tantum motum sit, sequitur quod motus non sit ex motibus, sed ex momentis. Therefore, assuming that nothing can be in motion through the indivisible A, if someone should say that a mobile is in motion through the entire magnitude ABC and that the whole motion by which it is in motion is DEZ, and moreover that nothing can be in motion, but only in the state of completed motion through the indivisible A, it follows that the motion consists not of motions, but of instants. Ideo autem sequitur quod non sit ex motibus, quia cum pars motus qui est d, respondeat parti magnitudinis quae est a, si d esset motus, oporteret quod per a moveretur, quia praesente motu mobile movetur: sed probatum est quod secundum a impartibile non movetur, sed solum motum est, quando scilicet pertransitum est hoc indivisibile. Now, the reason we say that it follows that the motion is not composed of motions is that, since the part of the motion that is D corresponds to the part of the magnitude that is A, if D were a motion, the mobile should be in motion through A, for when motion is present, the mobile is being moved. But it was proved that the mobile is not in motion through A as indivisible, but in the state of having completed its motion when it had traversed this indivisible. Ergo relinquitur quod d non sit motus, sed sit momentum, a quo denominatur motum esse, sicut a motu denominatur moveri; et quod ita se habet ad motum, sicut punctum indivisibile ad lineam. Et eadem ratio est de aliis partibus motus et magnitudinis. Ex necessitate ergo sequitur, si magnitudo componitur ex indivisibilibus, quod motus ex indivisibilibus componatur, idest ex momentis. Et hoc est quod demonstrare intendebat. Consequently, what remains is that D is not a motion, but a moment. (The state of completed motion is called a “moment,” just as being moved is called “motion”; moreover, the moment is related to motion as point is related to line.) And the same holds for the other parts of the motion and of the magnitude. Consequently, it follows necessarily that, if a magnitude is composed of indivisibles, then a motion is composed of indivisibles, that is, of instants; and this is what he intended to show. 763. Sed quia hoc est impossibile, quod motus componatur ex momentis, sicut impossibile est quod linea componatur ex punctis, ideo consequenter cum dicit: et motum esse aliquid etc., ostendit huiusmodi impossibilitatem, ducendo ad tria inconvenientia. 763. But, since it is not possible for a motion to be composed of instants any more than for a line be composed of points, at and the motion will take place (232a9), he exposes this impossibility by concluding to three absurdities. Quorum primum est, quod si motus componatur ex momentis et magnitudo ex indivisibilibus, ita quod per indivisibilem partem magnitudinis non moveatur sed motum sit, sequetur quod aliquid sit motum non motum, idest quod prius non movebatur: quia ponitur quod secundum indivisibile transivit, idest motum est, non transiens; quia in eo moveri non poterat. Unde sequitur aliquid esse transitum absque hoc quod aliquando iret: quod est impossibile, sicut impossibile est quod aliquid sit praeteritum, quod nunquam fuerit praesens. The first of these is that, if motion is composed of instants and a magnitude of indivisibles in such a way that, through an indivisible part of a magnitude, things are not in motion, but in the state of completed motion, it will follow that something has completed a motion without having been in motion, because it was assumed that, in regard to the indivisible, something arrived without going, because it was not able to be in motion in that indivisible. Hence, it follows that something has finished a motion without previously being in motion. But this is no more possible than for an event to be past without having been present. 764. Sed quia hoc inconveniens posset concedere ille qui diceret motum componi ex momentis, ducit ad secundum inconveniens, ibi: si igitur necesse est etc., tali ratione. Omne quod natum est moveri et quiescere, necesse est quod vel quiescat vel moveatur. 764. But, because a person who claimed that motion is composed of instants might grant this strange state of affairs, Aristotle gives another impossibility in the following argument, at since, then, everything must be (232a12). Anything capable of being in motion and at rest must be either in motion or at rest. Sed dum mobile est in a, non movetur, et similiter dum est in b, et similiter dum est in c: ergo dum est in a et dum est in b et dum est in c, quiescit. Ergo sequitur quod aliquid simul continue quiescat et moveatur. But, in our original example, while the mobile is in A, it is not being moved (and likewise when it is at B, and when it is at C); therefore, it must be at rest while at A and while at B and while at C. Therefore, it follows that a thing is at the same time continually at rest and continually in motion. Et quod hoc sequatur, sic probat. Positum est enim quod moveatur per totam longitudinem quae est abc; et iterum positum est quod quiescat secundum quamlibet partem: sed quod quiescit per quamlibet partem, quiescit per totum; ergo sequitur quod quiescat per totam magnitudinem. Et ita sequitur quod per totam magnitudinem continue moveatur et quiescat: quod est omnino impossibile. He now proves that this follows. We have agreed that it is in motion throughout the entire length ABC, and again that it is at rest in relation to each part. But what is at rest in relation to each and every part is at rest throughout the whole. Consequently, it is at rest throughout the entire length. Thus, it follows that, throughout the entire length, it is continually in motion and continually at rest—which is wholly impossible. 765. Tertium inconveniens ponit ibi: et si indivisibilia etc., tali ratione. Ostensum est quod si magnitudo componitur ex indivisibilibus, quod etiam motus: aut ergo illa indivisibilia motus, quae sunt d et e et z, ita se habent quod quodlibet eorum est motus, aut non. Si quodlibet eorum est motus, cum quodlibet eorum respondeat indivisibili parti magnitudinis in qua non movetur sed motum est, sequetur quod praesente motu mobile non moveatur, quod est contra primam suppositionem, sed quiescat. Si vero non sunt motus, sequitur quod motus componatur ex non motibus: quod videtur impossibile, sicut et quod linea componatur ex non lineis. 765. He gives the third absurdity at moreover, if the indivisibles (232a15). It has been shown that, if a magnitude is composed of indivisibles, so also the motion. Now, those indivisibles of motion—namely, D and E and Z—are such that each of them is either a motion or not. If each is a motion, then, since each of them corresponds to an indivisible part of the magnitude (in which something is not in motion, but in the state of completed motion), it will follow that a mobile is not in motion, but at rest, even though a motion exists—which is against the first presupposition. If each is not a motion, it follows that motion is composed of non-motions, which is no more possible than that a line be composed of non-lines. Lectio 3 Lecture 3 Tempus in divisibilitate magnitudinem sequitur et e converso The divisibility of time and of magnitude follow on one another Similiter autem necesse longitudini et motui indivisibile esse tempus, et componi ex ipsis nunc existentibus indivisibilibus. And, if length and motion are thus indivisible, it is neither more nor less necessary that time also be similarly indivisible—that is to say, be composed of indivisible moments.