No continuum is composed of indivisibles
Chapter 1 Εἰ δ' ἐστὶ συνεχὲς καὶ ἁπτόμενον καὶ ἐφεξῆς, ὡς διώρισται πρότερον, συνεχῆ μὲν ὧν τὰ ἔσχατα ἕν, ἁπτόμενα δ' ὧν ἅμα, ἐφεξῆς δ' ὧν μηδὲν μεταξὺ συγγενές, ἀδύνατον ἐξ ἀδιαιρέτων εἶναί τι συνεχές, οἷον γραμμὴν ἐκ στιγμῶν, εἴπερ ἡ γραμμὴ μὲν συνεχές, ἡ στιγμὴ δὲ ἀδιαίρετον. Now if the terms 'continuous', 'in contact', and 'in succession' are understood as defined above things being 'continuous' if their extremities are one, 'in contact' if their extremities are together, and 'in succession' if there is nothing of their own kind intermediate between them—nothing that is continuous can be composed 'of indivisibles': e.g. a line cannot be composed of points, the line being continuous and the point indivisible. οὔτε γὰρ ἓν τὰ ἔσχατα τῶν στιγμῶν (οὐ γάρ ἐστι τὸ μὲν ἔσχατον τὸ δ' ἄλλο τι μόριον τοῦ ἀδιαιρέτου), οὔθ' ἅμα τὰ ἔσχατα (οὐ γάρ ἐστιν ἔσχατον τοῦ ἀμεροῦς οὐδέν· ἕτερον γὰρ τὸ ἔσχατον καὶ οὗ ἔσχατον). For the extremities of two points can neither be one (since of an indivisible there can be no extremity as distinct from some other part) nor together (since that which has no parts can have no extremity, the extremity and the thing of which it is the extremity being distinct). ἔτι δ' ἀνάγκη ἤτοι συνεχεῖς εἶναι τὰς στιγμὰς ἢ ἁπτομένας ἀλλήλων, ἐξ ὧν ἐστι τὸ συνεχές· ὁ δ' αὐτὸς λόγος καὶ ἐπὶ πάντων τῶν (231b.) ἀδιαιρέτων. συνεχεῖς μὲν δὴ οὐκ ἂν εἶεν διὰ τὸν εἰρημένον λόγον· ἅπτεται δ' ἅπαν ἢ ὅλον ὅλου ἢ μέρος μέρους ἢ ὅλου μέρος. ἐπεὶ δ' ἀμερὲς τὸ ἀδιαίρετον, ἀνάγκη ὅλον ὅλου ἅπτεσθαι. ὅλον δ' ὅλου ἁπτόμενον οὐκ ἔσται συνεχές. τὸ γὰρ συνεχὲς ἔχει τὸ μὲν ἄλλο τὸ δ' ἄλλο μέρος, καὶ διαιρεῖται εἰς οὕτως ἕτερα καὶ τόπῳ κεχωρισμένα. Moreover, if that which is continuous is composed of points, these points must be either continuous or in contact with one another: and the same reasoning applies in the case of all indivisibles. Now for the reason given above they cannot be continuous: and one thing can be in contact with another only if whole is in contact with whole or part with part or part with whole. But since indivisibles have no parts, they must be in contact with one another as whole with whole. And if they are in contact with one another as whole with whole, they will not be continuous: for that which is continuous has distinct parts: and these parts into which it is divisible are different in this way, i.e. spatially separate. ἀλλὰ μὴν οὐδὲ ἐφεξῆς ἔσται στιγμὴ στιγμῇ ἢ τὸ νῦν τῷ νῦν, ὥστ' ἐκ τούτων εἶναι τὸ μῆκος ἢ τὸν χρόνον· ἐφεξῆς μὲν γάρ ἐστιν ὧν μηθέν ἐστι μεταξὺ συγγενές, στιγμῶν δ' αἰεὶ [τὸ] μεταξὺ γραμμὴ καὶ τῶν νῦν χρόνος. Nor, again, can a point be in succession to a point or a moment to a moment in such a way that length can be composed of points or time of moments: for things are in succession if there is nothing of their own kind intermediate between them, whereas that which is intermediate between points is always a line and that which is intermediate between moments is always a period of time. ἔτι διαιροῖτ' ἂν εἰς ἀδιαίρετα, εἴπερ ἐξ ὧν ἐστιν ἑκάτερον, εἰς ταῦτα διαιρεῖται· ἀλλ' οὐθὲν ἦν τῶν συνεχῶν εἰς ἀμερῆ διαιρετόν. Again, if length and time could thus be composed of indivisibles, they could be divided into indivisibles, since each is divisible into the parts of which it is composed. But, as we saw, no continuous thing is divisible into things without parts. ἄλλο δὲ γένος οὐχ οἷόν τ' εἶναι μεταξὺ [τῶν στιγμῶν καὶ τῶν νῦν οὐθέν]. ἢ γὰρ [ἔσται, δῆλον ὡς ἤτοι] ἀδιαίρετον ἔσται ἢ διαιρετόν, καὶ εἰ διαιρετόν, ἢ εἰς ἀδιαίρετα ἢ εἰς ἀεὶ διαιρετά· τοῦτο δὲ συνεχές. Nor can there be anything of any other kind intermediate between the parts or between the moments: for if there could be any such thing it is clear that it must be either indivisible or divisible, and if it is divisible, it must be divisible either into indivisibles or into divisibles that are infinitely divisible, in which case it is continuous. φανερὸν δὲ καὶ ὅτι πᾶν συνεχὲς διαιρετὸν εἰς αἰεὶ διαιρετά· εἰ γὰρ εἰς ἀδιαίρετα, ἔσται ἀδιαίρετον ἀδιαιρέτου ἁπτόμενον· ἓν γὰρ τὸ ἔσχατον καὶ ἅπτεται τῶν συνεχῶν. Moreover, it is plain that everything continuous is divisible into divisibles that are infinitely divisible: for if it were divisible into indivisibles, we should have an indivisible in contact with an indivisible, since the extremities of things that are continuous with one another are one and are in contact.
Postquam philosophus determinavit de divisione motus in suas species, et de unitate et contrarietate motuum et quietum, in hoc sexto libro intendit determinare ea quae pertinent ad divisionem motus, secundum quod dividitur in partes quantitativas. 750. After the Philosopher has finished dividing motion into its species and discussing the unity and contrariety of motions and of states of rest, he proposes in this Sixth Book to discuss the things that pertain to the division of motion precisely as it is divisible into quantitative parts. Et dividitur in partes duas. In prima ostendit motum, sicut et omne continuum, esse divisibilem; in secunda ostendit qualiter motus dividatur, ibi: necesse est autem et ipsum nunc et cetera. The whole book is divided into two parts. In the first he shows that motion, as every continuum, is divisible; In the second he shows how motion is divided, at L. 5. Prima autem pars dividitur in duas: in prima ostendit nullum continuum ex indivisibilibus componi; in secunda ostendit nullum continuum indivisibile esse, ibi: manifestum igitur ex dictis est et cetera. The first part is subdivided into two sections: In the first he shows that no continuum is composed solely of indivisibles; at L. 4. In the second that no continuum is indivisible, near the end Prima autem pars dividitur in duas: in prima ostendit nullum continuum ex indivisibilibus componi; in secunda parte (quia probationes praemissae magis ad magnitudinem pertinere videntur) ostendit quod eadem ratio est de magnitudine, motu et tempore, ibi: eiusdem autem rationis est et cetera. The first is further subdivided into two parts: In the first he shows that no continuum is composed of indivisibles only; In the second (because the proofs for the first seem to be applicable mainly to magnitudes) he shows that the same proofs apply to magnitudes, to motion and to time, at L, 2. Circa primum duo facit: primo resumit quasdam definitiones supra positas, quibus nunc utitur ad propositum demonstrandum; secundo probat propositum, ibi: neque enim unum sunt et cetera. In regard to the first part he does two things: First he recalls some definitions previously given, with a view to using them in demonstrating his proposition; Secondly, he proves the proposition, at 752. Dicit ergo primo quod si definitiones prius positae continui, et eius quod tangitur, et eius quod est consequenter, sunt convenientes (scilicet quod continua sint, quorum ultima sunt unum: contacta, quorum ultima sunt simul: consequenter autem sint, quorum nihil est medium sui generis), ex his sequitur quod impossibile sit aliquod continuum componi ex indivisibilibus, ut lineam ex punctis; si tamen linea dicatur aliquid continuum, et punctum aliquid indivisibile. Addit autem hoc, ne aliquis nomine lineae et puncti aliter uteretur. 751. He says therefore first (562) that if the previously given definitions of continuum, of that which is touched, of that which is consecutive to are correct (namely, that continua are things whose extremities are one; contigua are things whose extremities are together; consecutive things are those between which nothing of the same type intervenes), then it would follow that it is impossible for any continuum to be composed solely of indivisibles; i.e., it is impossible, for example, for a line to be composed of points only, provided, of course, that a line is conceded to be a continuum and that a point is an indivisible. This proviso is added to prevent other meanings being attached to point and line. Deinde cum dicit: neque enim unum sunt etc., probat propositum. Et primo inducit rationes duas ad probandum propositum; secundo manifestat quaedam quae poterant esse dubia in suis probationibus, ibi: nullum autem aliud genus et cetera. 752. Then at (563) he proves the proposition: First he gives two proofs of the proposition; Secondly, he explains things that might be misunderstood in his proofs, at 756. Circa primam rationem duo facit: primo ostendit quod ex indivisibilibus non componitur aliquod continuum, neque per modum continuationis, neque per modum contactus; secundo quod neque per modum consequenter se habentium, ibi: at vero neque consequenter et cetera. In regard to the first proof he does two things: First he shows that no continuum is composed solely of indivisibles, either after the manner of continuity or of contact; Secondly, or after the manner of things that are consecutive, at 754. Circa primum ponit duas rationes, quarum prima talis est. Ex quibuscumque componitur aliquid unum, vel per modum continuationis, vel per modum contactus, oportet quod habeant ultima quae sint unum, vel quae sint simul. Sed ultima punctorum non possunt esse unum: quia ultimum dicitur respectu alicuius partis; in indivisibili autem non est accipere aliquid quod sit ultimum, et aliud quod sit aliqua alia pars. Similiter non potest dici quod ultima punctorum sunt simul: quia nihil potest esse ultimum rei impartibilis, cum semper alterum sit ultimum et illud cuius est ultimum; in impartibili autem non est accipere aliud et aliud. Relinquitur ergo quod linea non potest componi ex punctis, neque per modum continuationis, neque per modum contactus. In regard to the first he gives two reasons, of which the first is: Whatever things a unit is composed of, either after the manner of continuity or of contact, the extremities must either be one or they must be together. But the extremities of points cannot be one, because an extremity is spoken of in relation to a part, whereas in an indivisible it is impossible to distinguish that which is an extremity and something else that is a part. Similarly, it cannot be said that the extremities are together, because nothing can be the extremity of a thing that cannot be divided into parts, whereas an extremity must always be distinct from that of which it is the extremity. But in a thing that cannot be divided into parts, there is no way of distinguishing one thing and another. It follows therefore that a line cannot be composed of points either after the manner of continuity or after the manner of contact. Secundam rationem ponit ibi: amplius necesse est etc. quae talis est. Si ex punctis constituitur aliquod continuum, necesse est quod aut sint continua ad invicem, vel se tangant: et eadem ratio est de omnibus aliis indivisibilibus, quod ex eis non componatur continuum. 753. The second reason is given at (564). If a continuum is composed solely of points, they must be either continuous with one another or touch (and the same is true of all other indivisibles, i.e., that no continuum is composed solely of them). Ad probandum autem quod indivisibilia non possunt sibi invicem esse continua, sufficiat ratio prima. To prove that they are not continuous with one another, the first argument suffices. Sed ad probandum quod non possunt se tangere, inducitur alia ratio, quae talis est. Omne quod tangit alterum, aut totum unum tangit totum aliud, aut pars unius partem alterius, aut pars unius totum aliud. Sed cum indivisibile non habeat partem, non potest dici quod pars unius tangat partem alterius, aut pars totum; et sic necesse est, si duo puncta se tangunt, quod totum tangat totum. Sed ex duobus, quorum unum totum tangit aliud totum, non potest componi continuum; quia omne continuum habet partes seiunctas, ita quod haec sit una pars, et haec alia; et dividitur in partes diversas et distinctas loco, idest positione, in his quae positionem habent: quae autem se secundum totum tangunt, non distinguuntur loco vel positione. Relinquitur ergo quod ex punctis non possit componi linea per modum contactus. But to prove that they cannot touch one another, another argument is adduced, which is the following: Everything that touches something else does so either by the whole touching the other wholly, or by a part of one touching a part of the other or the whole of the other. But since an indivisible does not have parts, it cannot be said that part of one touches either a part or the whole of the other. Hence if two points touch, the whole point touches another whole point. But when a whole touches a whole, no continuum can be formed, because every continuum has distinct parts so that one part is here and another there, and is divisible into parts that are different and distinct in regard to place, i.e., position (in things that have-position)—whereas things that touch one another totally are not distinguished as to place or position, It therefore follows that a line cannot be composed of points that are in contact. Deinde cum dicit: at vero neque etc., probat quod continuum non componatur ex indivisibilibus per modum eius quod est consequenter. Non enim punctum consequenter se habebit ad aliud punctum, ita quod ex eis constitui possit longitudo, idest linea; aut unum nunc alteri nunc, ita quod ex eis possit componi tempus: quia consequenter est unum alteri, quorum non est aliquid medium eiusdem generis, ut supra expositum est. Sed inter duo puncta semper est linea media: et sic si linea composita est ex punctis, ut tu das, sequitur quod semper inter duo puncta sit aliud punctum medium. Et similiter inter duo nunc est tempus medium. Non ergo linea componitur ex punctis, aut tempus ex nunc, sicut consequenter se habentibus. 754. Then at (565) he shows that no continuum is composed of indivisibles after the manner of things that are consecutive. For no point will be consecutive to another so as to form a line; and no “now” is consecutive to another “now” so as to form a period of time, because consecutive things are by definition such that nothing of the same kind intervenes between any two. But between any two points there is always a line, and so, if a line is composed of points only, it would follow that between any two points there is always another, mediate, point. The same is true for the “now’s’”. if a period of time is nothing but a series of “now’s”, then between any two “now’s” there would be another “now”. Therefore, no line is composed solely of points, and no time is composed solely of “now’s”, after the manner of things that are consecutive. Secundam rationem principalem ponit ibi: amplius dividerentur etc., quae sumitur ex alia definitione continui, quam supra posuit in principio tertii, scilicet quod continuum sit quod est in infinitum divisibile: et est ratio talis. Ex quibuscumque componitur vel linea vel tempus, in ipsa dividitur: si igitur utrumque istorum componitur ex indivisibilibus, sequitur quod in indivisibilia dividatur. Sed hoc est falsum, cum nullum continuorum sit divisibile in impartibilia: sic enim non esset divisibile in infinitum. Nullum igitur continuum componitur ex indivisibilibus. 755. The second reason is given at (566) and is based on a different definition of continuum—the one given at the beginning of Book III—that a continuum is “that which is divisible ad infinitum ”. Here is the proof: A line or time can be divided into whatsoever they are composed of. If, therefore, each of them is composed of indivisibles, it follows that each is divided into indivisibles. But this is false, since neither of them is divisible into indivisibles, for that would mean they would not be divisible ad infinitum. No continuum, therefore, is composed of indivisibles. Deinde cum dicit: nullum autem aliud etc., manifestat duo quae supra dixerat. Quorum primum fuit, quod inter duo puncta sit linea media, et inter duo nunc, tempus. Et hoc manifestat sic. 756. Then at (567) he explains two statements he made in the course of his proofs. The first of these was that between two points there is always a line and that between two “now’s” there is always time. He explains it thus: Si sunt duo puncta, oportet quod differant secundum situm: alias non essent duo sed unum. Non autem possunt se contingere, ut supra ostensum est: unde relinquitur quod distent, et sit aliquod medium inter ea. Sed nullum aliud medium potest esse inter ea quam linea inter puncta, et tempus inter nunc. Quod sic probat: quia si inter puncta esset aliud medium quam linea, manifestum est aut illud medium esse indivisibile aut divisibile. Si autem sit indivisibile, oportet quod sit distinctum ab utroque in situ; et cum non tangat, oportet iterum quod sit aliquod alterum medium inter indivisibile quod ponitur medium et extrema, et sic in infinitum, nisi ponatur medium divisibile. Si autem medium duorum punctorum fuerit divisibile, aut erit divisibile in indivisibilia, aut in semper divisibilia. Sed non potest dici quod dividatur in indivisibilia, quia tunc redibit eadem difficultas, quomodo ex indivisibilibus possit componi divisibile. Relinquitur igitur quod illud medium sit divisibile in semper divisibilia. Sed haec est ratio continui: ergo illud medium erit quoddam continuum. Nullum autem aliud continuum potest esse medium inter duo puncta quam linea: ergo inter qualibet duo puncta est linea media. Et eadem ratione inter qualibet duo nunc, tempus; et similiter in aliis continuis. If two points exist, they must differ in position; otherwise, they would not be two, but one. But they cannot touch one another, as was shown above; hence they are distant, and something is between them. But no other intermediate is possible, except a line between two points, and time between “now’s”: for if the intermediate between two points were other than a line, that intermediate must be either divisible or indivisible. If indivisible, it must be distinct from the two points—at least in position—and, since it touches neither, there must be another intermediate between that indivisible and the original extremities and so on ad infinitum, until a divisible intermediate is found. However, if the intermediate is divisible, it will be divisible into indivisibles or into what are further divisible. But it cannot be divided into indivisibles only, because then the same difficulty returns—how a divisible can be composed solely of indivisibles. It must be granted, then, that the intermediate is divisible into what are further divisible. But that is what a continuum is. Therefore, that intermediate will be a continuum. But the only continuous intermediate between two points is a line. Therefore, between any two points there is an intermediate line. Likewise, between two “now’s” there is time; and the same for other types of continua. Deinde cum dicit: manifestum autem etc., manifestat secundum, quod supposuerat, scilicet quod omne continuum sit divisibile in divisibilia. Quia si daretur quod continuum esset divisibile in indivisibilia, sequeretur quod duo indivisibilia se contingerent, ad hoc quod possent constituere continuum. Oportet enim quod continuorum sit unum ultimum, ut ex definitione eius apparet, et quod partes continui se tangant: quia si ultima sunt unum, sequitur quod sint simul, ut in quinto dictum est. Cum igitur sit impossibile duo indivisibilia se contingere, impossibile est quod continuum in indivisibilia dividatur. 757. Then at (568) he explains the second statement referred to at the beginning of 756, that every continuum is divisible into divisibles. For on the supposition that a continuum is divisible solely into indivisibles, it would follow that two indivisibles would have to be in contact in order to form the continuum. For continua have an extremity that is one, as appears from the definition thereof; moreover, the parts of a continuum must touch, because if the extremities are one, they are together, as was stated in Book V. Therefore, since it is impossible for two indivisibles to touch, it is impossible for a continuum to be divided into indivisibles.
Motion composed of indivisibles follows a continuum composed of indivisibles—impossibility of the former
Chapter 1 cont. τοῦ δ' αὐτοῦ λόγου μέγεθος καὶ χρόνον καὶ κίνησιν ἐξ ἀδιαιρέτων συγκεῖσθαι, καὶ διαιρεῖσθαι εἰς ἀδιαίρετα, ἢ μηθέν. The same reasoning applies equally to magnitude, to time, and to motion: either all of these are composed of indivisibles and are divisible into indivisibles, or none. δῆλον δ' ἐκ τῶνδε. εἰ γὰρ τὸ μέγεθος ἐξ ἀδιαιρέτων σύγκειται, καὶ ἡ κίνησις ἡ τούτου ἐξ ἴσων κινήσεων ἔσται ἀδιαιρέτων, This may be made clear as follows. If a magnitude is composed of indivisibles, the motion over that magnitude must be composed of corresponding indivisible motions: οἷον εἰ τὸ ΑΒΓ ἐκ τῶν Α Β Γ ἐστὶν ἀδιαιρέτων, ἡ κίνησις ἐφ' ἧς ΔΕΖ, ἣν ἐκινήθη τὸ Ω ἐπὶ τῆς ΑΒΓ, ἕκαστον τὸ μέρος ἔχει ἀδιαίρετον. e.g. if the magnitude ABG is composed of the indivisibles A, B, G, each corresponding part of the motion DEZ of O over ABG is indivisible. εἰ δὴ παρούσης κινήσεως ἀνάγκη κινεῖσθαί τι, καὶ εἰ κινεῖταί τι, παρεῖναι κίνησιν, καὶ τὸ κινεῖσθαι ἔσται ἐξ ἀδιαιρέτων. τὸ μὲν δὴ Α ἐκινήθη τὸ Ω τὴν τὸ Δ κινούμενον κίνησιν, τὸ δὲ Β τὴν τὸ Ε, καὶ τὸ Γ ὡσαύτως τὴν τὸ Ζ. Therefore, since where there is motion there must be something that is in motion, and where there is something in motion there must be motion, therefore the being-moved will also be composed of indivisibles. So O traversed A when its motion was D, B when its motion was E, and G similarly when its motion was Z. εἰ δὴ ἀνάγκη τὸ κινούμενον ποθέν ποι μὴ ἅμα κινεῖσθαι καὶ κεκινῆσθαι οὗ ἐκινεῖτο ὅτε ἐκινεῖτο (οἷον εἰ Θήβαζέ τι βαδίζει, ἀδύνατον ἅμα βαδίζειν Θήβαζε καὶ βεβαδικέναι (232a.) Θήβαζε), Now a thing that is in motion from one place to another cannot at the moment when it was in motion both be in motion and at the same time have completed its motion at the place to which it was in motion: e.g. if a man is walking to Thebes, he cannot be walking to Thebes and at the same time have completed his walk to Thebes: τὴν δὲ τὸ Α τὴν ἀμερῆ ἐκινεῖτο τὸ Ω, ᾗ ἡ τὸ Δ κίνησις παρῆν· ὥστ' εἰ μὲν ὕστερον διεληλύθει ἢ διῄει, διαιρετὴ ἂν εἴη (ὅτε γὰρ διῄει, οὔτε ἠρέμει οὔτε διεληλύθει, ἀλλὰ μεταξὺ ἦν), εἰ δ' ἅμα διέρχεται καὶ διελήλυθε, τὸ βαδίζον, ὅτε βαδίζει, βεβαδικὸς ἐκεῖ ἔσται καὶ κεκινημένον οὗ κινεῖται. εἰ δὲ τὴν μὲν ὅλην τὴν ΑΒΓ κινεῖταί τι, καὶ ἡ κίνησις ἣν κινεῖται τὰ Δ Ε Ζ ἐστι, τὴν δ' ἀμερῆ τὴν Α οὐθὲν κινεῖται ἀλλὰ κεκίνηται, εἴη ἂν ἡ κίνησις οὐκ ἐκ κινήσεων ἀλλ' ἐκ κινημάτων and, as we saw, O traverses a the partless section A in virtue of the presence of the motion D. Consequently, if O actually passed through A after being in process of passing through, the motion must be divisible: for at the time when O was passing through, it neither was at rest nor had completed its passage but was in an intermediate state: while if it is passing through and has completed its passage at the same moment, then that which is walking will at the moment when it is walking have completed its walk and will be in the place to which it is walking; that is to say, it will have completed its motion at the place to which it is in motion. And if a thing is in motion over the whole Kbg and its motion is the three D, E, and Z, and if it is not in motion at all over the partless section A but has completed its motion over it, then the motion will consist not of motions but of starts, καὶ τῷ κεκινῆσθαί τι μὴ κινούμενον· τὴν γὰρ Α διελήλυθεν οὐ διεξιόν. ὥστε ἔσται τι βεβαδικέναι μη δέποτε βαδίζον· ταύτην γὰρ βεβάδικεν οὐ βαδίζον ταύτην. and will take place by a thing's having completed a motion without being in motion: for on this assumption it has completed its passage through A without passing through it. So it will be possible for a thing to have completed a walk without ever walking: for on this assumption it has completed a walk over a particular distance without walking over that distance. εἰ οὖν ἀνάγκη ἢ ἠρεμεῖν ἢ κινεῖσθαι πᾶν, ἠρεμεῖ καθ' ἕκαστον τῶν Α Β Γ, ὥστ' ἔσται τι συνεχῶς ἠρεμοῦν ἅμα καὶ κινούμενον. τὴν γὰρ ΑΒΓ ὅλην ἐκινεῖτο καὶ ἠρέμει ὁτιοῦν μέρος, ὥστε καὶ πᾶσαν. Since, then, everything must be either at rest or in motion, and O is therefore at rest in each of the sections A, B, and G, it follows that a thing can be continuously at rest and at the same time in motion: for, as we saw, O is in motion over the whole ABG and at rest in any part (and consequently in the whole) of it. καὶ εἰ μὲν τὰ ἀδιαίρετα τῆς ΔΕΖ κινήσεις, κινήσεως παρούσης ἐνδέχοιτ' ἂν μὴ κινεῖσθαι ἀλλ' ἠρεμεῖν· εἰ δὲ μὴ κινήσεις, τὴν κίνησιν μὴ ἐκ κινήσεων εἶναι. Moreover, if the indivisibles composing DEZ are motions, it would be possible for a thing in spite of the presence in it of motion to be not in motion but at rest, while if they are not motions, it would be possible for motion to be composed of something other than motions.
Quia rationes supra positae manifestiores sunt in linea et aliis continuis quantitatibus positionem habentibus, in quibus proprie invenitur contactus, vult hic ostendere quod eadem ratio est de magnitudine et tempore et motu. Et dividitur in partes duas: primo proponit intentum; secundo probat propositum, ibi: manifestum est autem ex his et cetera. 758. Because the arguments presented in the previous lecture clearly apply to lines and other continua having position, in which continua contact is properly found, the Philosopher now wishes to show that the same reasoning applies to magnitudes and time and motion. And it is divided into two parts: First he proposes his intention; Secondly, he proves his proposition at 759. 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 (569) that any argument which shows that a magnitude is composed or not composed of indivisibles, and divided or not divided into indivisibles, applies also to time and motion; for whatever is granted in regard to any of them would necessarily be true of the others. Deinde cum dicit: manifestum est autem ex his etc., probat propositum: et primo quantum ad magnitudinem et motum; secundo quantum ad tempus et magnitudinem, ibi: similiter autem necesse et cetera. 759. Then at (570) he proves this proposition: First in regard to magnitude and motion; Secondly in regard to time and magnitude, in L. 3. Circa primum tria facit: primo ponit propositum; secundo exemplificat, ibi: ut si ipsa abc etc., tertio probat, ibi: si igitur praesentis motus et cetera. About the first he does three things: First he presents his proposition; Secondly, he gives an example, at 760; Thirdly, he proves his proposition, at 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, likewise the motion that traverses it will be composed of indivisible motions, equal in number to the indivisibles of which the magnitude is composed. 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 at (571): Let the line ABC be composed of the 3 indivisibles A, B and C, and let 0 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. Et circa hoc tria facit: primo praemittit quaedam necessaria ad propositi probationem; secundo probat quod si magnitudo componitur ex punctis, quod motus componitur non ex motibus, sed ex momentis, ibi: secundum a igitur etc.; tertio ostendit esse impossibile quod motus componatur ex momentis, ibi: et motum esse aliquid et cetera. Then at (572) he proves his proposition. About which he does three things: First he lays down some premises necessary for his proof; Secondly, he proves that if a magnitude is composed of points, then the motion is composed not of motions but of moments, at 762; Thirdly, he shows that it is impossible for motion to be composed of moments, at 763. Praemittit ergo primo duo. 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. 761. Therefore first he lays down two presuppositions. The first at (572) 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 0 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. The second presupposition at (573) is that what is being moved from one terminus to another is not at the same time being moved and finished moving, any more 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. Quod vero non simul sit moveri et motum esse, apparet ex ipsa motus successione: quia impossibile est aliqua duo tempora simul esse, ut in quarto habitum est: unde impossibile est quod simul sit motum esse, quod est terminus motus, cum ipso moveri. He presupposes these two statements as per se evident. For, 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. As to the statement that “being moved” and “having been moved” are not simultaneous, we appeal to the very successive nature of motion; for it is impossible that any two elements of time co-exist, as we explained in Book IV. Hence it is impossible that “having been moved”, which is the terminus of motion, be simultaneous with “being in motion”. 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. 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. 762. Then at (574) 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 0 is in motion with respect to an indivisible part of the magnitude, namely, A, there is in 0 that part of the motion we called D. Accordingly, 0 is being moved through A and has completed its motion, either at the same time or not at the same time. If not at the same time but later, it follows that A is divisible; because while 0 was in motion, it was neither resting at A (with the rest preceding motion) nor had passed through the entire distance A—for then it would not still be in motion through A, since 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 in the state of completed 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. 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. 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. 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 moments. Now the reason why 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, then if D were a motion, the mobile should be in motion through A, because 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. Consequently, what remains is that D is not a motion but a moment. (The state of completed motion is called “moment”, just as being moved is called “motion”; moreover, 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, i.e., of moments; and this is what he intended to show. 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. 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. 763. But since it is not possible for a motion to be composed of moments any more than a line be composed of points, then at (575) he exposes this impossibility by concluding to three impossibilities. The first of these is that if motion is composed of moments, 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. For it was assumed that in regard to the indivisible, something arrived without going, because it was not able to be in motion at 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. 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. 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. 764. But because a person who claimed that motion is composed of moments might grant this strange state of affairs, Aristotle concludes to another impossibility, in the following argument: Anything capable of being in motion and at rest must be either in motion or at rest. But in our original example, while the mobile is in A, it is not being moved; 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. That this follows, he now proves; 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. 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 impossibility at (577): 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.
Time follows magnitude in divisibility and conversely
Chapter 1 cont. ὁμοίως δ' ἀνάγκη τῷ μήκει καὶ τῇ κινήσει ἀδιαίρετον εἶναι τὸν χρόνον, καὶ συγκεῖσθαι ἐκ τῶν νῦν ὄντων ἀδιαιρέτων· 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: εἰ γὰρ πᾶσα διαιρετός, ἐν τῷ ἐλάττονι δὲ τὸ ἰσοταχὲς δίεισιν ἔλαττον, διαιρετὸς ἔσται καὶ ὁ χρόνος. εἰ δ' ὁ χρόνος διαιρετὸς ἐν ᾧ φέρεταί τι τὴν Α, καὶ ἡ τὸ Α ἔσται διαιρετή. for if the whole distance is divisible and an equal velocity will cause a thing to pass through less of it in less time, the time must also be divisible, and conversely, if the time in which a thing is carried over the section A is divisible, this section A must also be divisible. Chapter 2 Ἐπεὶ δὲ πᾶν μέγεθος εἰς μεγέθη διαιρετόν (δέδεικται γὰρ ὅτι ἀδύνατον ἐξ ἀτόμων εἶναί τι συνεχές, μέγεθος δ' ἐστὶν ἅπαν συνεχές), ἀνάγκη τὸ θᾶττον ἐν τῷ ἴσῳ χρόνῳ μεῖζον καὶ ἐν τῷ ἐλάττονι ἴσον καὶ ἐν τῷ ἐλάττονι πλεῖον κινεῖσθαι, καθάπερ ὁρίζονταί τινες τὸ θᾶττον. And since every magnitude is divisible into magnitudes—for we have shown that it is impossible for anything continuous to be composed of indivisible parts, and every magnitude is continuous—it necessarily follows that the quicker of two things traverses a greater magnitude in an equal time, an equal magnitude in less time, and a greater magnitude in less time, in conformity with the definition sometimes given of 'the quicker'. ἔστω γὰρ τὸ ἐφ' ᾧ Α τοῦ ἐφ' ᾧ Β θᾶττον. ἐπεὶ τοίνυν θᾶττόν ἐστιν τὸ πρότερον μεταβάλλον, ἐν ᾧ χρόνῳ τὸ Α μεταβέβληκεν ἀπὸ τοῦ Γ εἰς τὸ Δ, οἷον τῷ ΖΗ, ἐν τούτῳ τὸ Β οὔπω ἔσται πρὸς τῷ Δ, ἀλλ' ἀπολείψει, ὥστε ἐν τῷ ἴσῳ χρόνῳ πλεῖον δίεισιν τὸ θᾶττον. Suppose that A is quicker than B. Now since of two things that which changes sooner is quicker, in the time ZH, in which A has changed from G to D, B will not yet have arrived at D but will be short of it: so that in an equal time the quicker will pass over a greater magnitude. ἀλλὰ μὴν καὶ ἐν τῷ ἐλάττονι πλεῖον· ἐν ᾧ γὰρ τὸ Α γεγένηται πρὸς τῷ Δ, τὸ Β ἔστω πρὸς τῷ Ε τὸ βραδύτερον ὄν. οὐκοῦν ἐπεὶ (232b.) τὸ Α πρὸς τῷ Δ γεγένηται ἐν ἅπαντι τῷ ΖΗ χρόνῳ, πρὸς τῷ Θ ἔσται ἐν ἐλάττονι τούτου· καὶ ἔστω ἐν τῷ ΖΚ. τὸ μὲν οὖν ΓΘ, ὃ διελήλυθε τὸ Α, μεῖζόν ἐστι τοῦ ΓΕ, ὁ δὲ χρόνος ὁ ΖΚ ἐλάττων τοῦ παντὸς τοῦ ΖΗ, ὥστε ἐν ἐλάττονι μεῖζον δίεισιν. More than this, it will pass over a greater magnitude in less time: for in the time in which A has arrived at D, B being the slower has arrived, let us say, at E. Then since A has occupied the whole time ZH in arriving at D, will have arrived at O in less time than this, say ZK. Now the magnitude GO that A has passed over is greater than the magnitude GE, and the time ZK is less than the whole time ZH: so that the quicker will pass over a greater magnitude in less time. φανερὸν δὲ ἐκ τούτων καὶ ὅτι τὸ θᾶττον ἐν ἐλάττονι χρόνῳ δίεισιν τὸ ἴσον. And from this it is also clear that the quicker will pass over an equal magnitude in less time than the slower. ἐπεὶ γὰρ τὴν μείζω ἐν ἐλάττονι διέρχεται τοῦ βραδυτέρου, αὐτὸ δὲ καθ' αὑτὸ λαμβανόμενον ἐν πλείονι χρόνῳ τὴν μείζω τῆς ἐλάττονος, οἷον τὴν ΛΜ τῆς ΛΞ, πλείων ἂν εἴη ὁ χρόνος ὁ ΠΡ, ἐν ᾧ τὴν ΛΜ διέρχεται, ἢ ὁ ΠΣ, ἐν ᾧ τὴν ΛΞ. ὥστε εἰ ὁ ΠΡ χρόνος ἐλάττων ἐστὶν τοῦ Χ, ἐν ᾧ τὸ βραδύτερον διέρχεται τὴν ΛΞ, καὶ ὁ ΠΣ ἐλάττων ἔσται τοῦ ἐφ' ᾧ Χ· τοῦ γὰρ ΠΡ ἐλάττων, τὸ δὲ τοῦ ἐλάττονος ἔλαττον καὶ αὐτὸ ἔλαττον. ὥστε ἐν ἐλάττονι κινήσεται τὸ ἴσον. For since it passes over the greater magnitude in less time than the slower, and (regarded by itself) passes over LM the greater in more time than LX the lesser, the time PRh in which it passes over LM will be more than the time PS, which it passes over LX: so that, the time PRh being less than the time PCh in which the slower passes over LX, the time PS will also be less than the time PX: for it is less than the time PRh, and that which is less than something else that is less than a thing is also itself less than that thing. Hence it follows that the quicker will traverse an equal magnitude in less time than the slower. ἔτι δ' εἰ πᾶν ἀνάγκη ἢ ἐν ἴσῳ ἢ ἐν ἐλάττονι ἢ ἐν πλείονι κινεῖ σθαι, καὶ τὸ μὲν ἐν πλείονι βραδύτερον, τὸ δ' ἐν ἴσῳ ἰσοταχές, τὸ δὲ θᾶττον οὔτε ἰσοταχὲς οὔτε βραδύτερον, οὔτ' ἂν ἐν ἴσῳ οὔτ' ἐν πλείονι κινοῖτο τὸ θᾶττον. λείπεται οὖν ἐν ἐλάττονι, ὥστ' ἀνάγκη καὶ τὸ ἴσον μέγεθος ἐν ἐλάττονι χρόνῳ διιέναι τὸ θᾶττον. Again, since the motion of anything must always occupy either an equal time or less or more time in comparison with that of another thing, and since, whereas a thing is slower if its motion occupies more time and of equal velocity if its motion occupies an equal time, the quicker is neither of equal velocity nor slower, it follows that the motion of the quicker can occupy neither an equal time nor more time. It can only be, then, that it occupies less time, and thus we get the necessary consequence that the quicker will pass over an equal magnitude (as well as a greater) in less time than the slower. ἐπεὶ δὲ πᾶσα μὲν κίνησις ἐν χρόνῳ καὶ ἐν ἅπαντι χρόνῳ δυνατὸν κινηθῆναι, πᾶν δὲ τὸ κινούμενον ἐνδέχεται καὶ θᾶττον κινεῖσθαι καὶ βραδύτερον, ἐν ἅπαντι χρόνῳ ἔσται τὸ θᾶττον κινεῖσθαι καὶ βραδύτερον. And since every motion is in time and a motion may occupy any time, and the motion of everything that is in motion may be either quicker or slower, both quicker motion and slower motion may occupy any time: τούτων δ' ὄντων ἀνάγκη καὶ τὸν χρόνον συνεχῆ εἶναι. λέγω δὲ συνεχὲς τὸ διαιρετὸν εἰς αἰεὶ διαιρετά· τούτου γὰρ ὑποκειμένου τοῦ συνε χοῦς, ἀνάγκη συνεχῆ εἶναι τὸν χρόνον. and this being so, it necessarily follows that time also is continuous. By continuous I mean that which is divisible into divisibles that are infinitely divisible: and if we take this as the definition of continuous, it follows necessarily that time is continuous. ἐπεὶ γὰρ δέδεικται ὅτι τὸ θᾶττον ἐν ἐλάττονι χρόνῳ δίεισιν τὸ ἴσον, ἔστω τὸ μὲν ἐφ' ᾧ Α θᾶττον, τὸ δ' ἐφ' ᾧ Β βραδύτερον, καὶ κεκινήσθω τὸ βραδύτερον τὸ ἐφ' ᾧ ΓΔ μέγεθος ἐν τῷ ΖΗ χρόνῳ. δῆλον τοίνυν ὅτι τὸ θᾶττον ἐν ἐλάττονι τούτου κινήσεται τὸ αὐτὸ μέγεθος· καὶ κεκινήσθω ἐν τῷ ΖΘ. πάλιν δ' ἐπεὶ τὸ θᾶττον ἐν τῷ ΖΘ διελήλυθεν τὴν ὅλην τὴν ΓΔ, τὸ βραδύτερον ἐν τῷ αὐτῷ χρόνῳ τὴν ἐλάττω δίεισιν· ἔστω οὖν ἐφ' (233a.) ἧς ΓΚ. ἐπεὶ δὲ τὸ βραδύτερον τὸ Β ἐν τῷ ΖΘ χρόνῳ τὴν ΓΚ διελήλυθεν, τὸ θᾶττον ἐν ἐλάττονι δίεισιν, ὥστε πάλιν διαιρεθήσεται ὁ ΖΘ χρόνος. τούτου δὲ διαιρουμένου καὶ τὸ ΓΚ μέγεθος διαιρεθήσεται κατὰ τὸν αὐτὸν λόγον. εἰ δὲ τὸ μέγεθος, καὶ ὁ χρόνος. καὶ ἀεὶ τοῦτ' ἔσται μεταλαμβάνουσιν ἀπὸ τοῦ θάττονος τὸ βραδύτερον καὶ ἀπὸ τοῦ βραδυτέρου τὸ θᾶττον, καὶ τῷ ἀποδεδειγμένῳ χρωμένοις· διαιρήσει γὰρ τὸ μὲν θᾶττον τὸν χρόνον, τὸ δὲ βραδύτερον τὸ μῆκος. εἰ οὖν αἰεὶ μὲν ἀντιστρέφειν ἀληθές, ἀντιστρεφομένου δὲ αἰεὶ γίγνεται διαίρεσις, For since it has been shown that the quicker will pass over an equal magnitude in less time than the slower, suppose that A is quicker and B slower, and that the slower has traversed the magnitude GD in the time ZH. Now it is clear that the quicker will traverse the same magnitude in less time than this: let us say in the time ZO. Again, since the quicker has passed over the whole D in the time ZO, the slower will in the same time pass over GK, say, which is less than GD. And since B, the slower, has passed over GK in the time ZO, the quicker will pass over it in less time: so that the time ZO will again be divided. And if this is divided the magnitude GK will also be divided just as GD was: and again, if the magnitude is divided, the time will also be divided. And we can carry on this process for ever, taking the slower after the quicker and the quicker after the slower alternately, and using what has been demonstrated at each stage as a new point of departure: for the quicker will divide the time and the slower will divide the length. If, then, this alternation always holds good, and at every turn involves a division, it is evident that all time must be continuous. And at the same time it is clear that all magnitude is also continuous; for the divisions of which time and magnitude respectively are susceptible are the same and equal. φανερὸν ὅτι πᾶς χρόνος ἔσται συνεχής. ἅμα δὲ δῆλον καὶ ὅτι μέγεθος ἅπαν ἐστὶ συνεχές· τὰς αὐτὰς γὰρ καὶ τὰς ἴσας διαιρέσεις ὁ χρόνος διαιρεῖται καὶ τὸ μέγεθος. Moreover, the current popular arguments make it plain that, if time is continuous, magnitude is continuous also, inasmuch as a thing asses over half a given magnitude in half the time taken to cover the whole: in fact without qualification it passes over a less magnitude in less time; for the divisions of time and of magnitude will be the same.
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. 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. Circa primum duo facit: primo ponit propositum; secundo demonstrat, ibi: si enim omnis et cetera. About the first he does two things: First he states his proposition; Secondly, he demonstrates it, at 767. Dicit ergo primo quod etiam tempus necesse est similiter esse divisibile et indivisibile, et componi ex indivisibilibus, sicut longitudo et motus. He says therefore first (578) that time, too, is divisible and indivisible, and composed of indivisibles, just as length and motion are. 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. 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. 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. He says therefore first (579) 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. 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. 768. Then at (580) 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. 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. 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. 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. 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. 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) 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. 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. 770. Then at (581) 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. 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. He says therefore first (581): 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. 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. 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. 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; 77l. Then at (582) 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. 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. 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. Deinde cum dicit: manifestum autem etc., ostendit quomodo velocius se habeat ad tardius in moveri per aequalem magnitudinem. 772. Then at (593) 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. 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. He says therefore first (583) 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. 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. 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. 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. 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. 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. 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. Sic ergo probatum est quod necesse est velocius pertransire aequalem magnitudinem in minori tempore. Thus, we have proved that necessarily the faster traverses an equal magnitude in less time. 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. 774. Then at (586) 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, 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. Therefore (586) 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 775. Then at (587) 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. 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. Then at (588) 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. Manifestum est ergo quod a quod est velocius, movetur per eandem magnitudinem in minori tempore; et sit tempus illud zt. It is plain that A, which is faster, traverses the same magnitude in less time 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. 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. 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. 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. 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) 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.
Proof that no continuum is indivisible
Chapter 2 cont. ἔτι δὲ καὶ ἐκ τῶν εἰωθότων λόγων λέγεσθαι φανερὸν ὡς εἴπερ ὁ χρόνος ἐστὶ συνεχής, ὅτι καὶ τὸ μέγεθος, εἴπερ ἐν τῷ ἡμίσει χρόνῳ ἥμισυ διέρχεται καὶ ἁπλῶς ἐν τῷ ἐλάττονι ἔλαττον· αἱ γὰρ αὐταὶ διαιρέσεις ἔσονται τοῦ χρόνου καὶ τοῦ μεγέθους. καὶ εἰ ὁποτερονοῦν ἄπειρον, καὶ θάτερον, καὶ ὡς θάτερον, καὶ θάτερον, οἷον εἰ μὲν τοῖς ἐσχάτοις ἄπειρος ὁ χρόνος, καὶ τὸ μῆκος τοῖς ἐσχάτοις, εἰ δὲ τῇ διαιρέσει, τῇ διαιρέσει καὶ τὸ μῆκος, εἰ δὲ ἀμφοῖν, ἀμφοῖν καὶ τὸ μέγεθος. And if either is infinite, so is the other, and the one is so in the same way as the other; i.e. if time is infinite in respect of its extremities, length is also infinite in respect of its extremities: if time is infinite in respect of divisibility, length is also infinite in respect of divisibility: and if time is infinite in both respects, magnitude is also infinite in both respects. διὸ καὶ ὁ Ζήνωνος λόγος ψεῦδος λαμβάνει τὸ μὴ ἐνδέχεσθαι τὰ ἄπειρα διελθεῖν ἢ ἅψασθαι τῶν ἀπείρων καθ' ἕκαστον ἐν πεπερασμένῳ χρόνῳ. διχῶς γὰρ λέγεται καὶ τὸ μῆκος καὶ ὁ χρόνος ἄπειρον, καὶ ὅλως πᾶν τὸ συνεχές, ἤτοι κατὰ διαίρεσιν ἢ τοῖς ἐσχάτοις. τῶν μὲν οὖν κατὰ τὸ ποσὸν ἀπείρων οὐκ ἐνδέχεται ἅψασθαι ἐν πεπερασμένῳ χρόνῳ, τῶν δὲ κατὰ διαίρεσιν ἐνδέχεται· καὶ γὰρ αὐτὸς ὁ χρόνος οὕτως ἄπειρος. ὥστε ἐν τῷ ἀπείρῳ καὶ οὐκ ἐν τῷ πεπερασμένῳ συμβαίνει διιέναι τὸ ἄπειρον, καὶ ἅπτεσθαι τῶν ἀπείρων τοῖς ἀπείροις, οὐ τοῖς πεπερασμένοις. Hence Zeno's argument makes a false assumption in asserting that it is impossible for a thing to pass over or severally to come in contact with infinite things in a finite time. For there are two senses in which length and time and generally anything continuous are called 'infinite': they are called so either in respect of divisibility or in respect of their extremities. So while a thing in a finite time cannot come in contact with things quantitatively infinite, it can come in contact with things infinite in respect of divisibility: for in this sense the time itself is also infinite: and so we find that the time occupied by the passage over the infinite is not a finite but an infinite time, and the contact with the infinites is made by means of moments not finite but infinite in number. οὔτε δὴ τὸ ἄπειρον οἷόν τε ἐν πεπερασμένῳ χρόνῳ διελθεῖν, οὔτ' ἐν ἀπείρῳ τὸ πεπερασμένον· ἀλλ' ἐάν τε ὁ χρόνος ἄπειρος ᾖ, καὶ τὸ μέγεθος ἔσται ἄπειρον, ἐάν τε τὸ μέγεθος, καὶ ὁ χρόνος. The passage over the infinite, then, cannot occupy a finite time, and the passage over the finite cannot occupy an infinite time: if the time is infinite the magnitude must be infinite also, and if the magnitude is infinite, so also is the time. ἔστω γὰρ πεπερασμένον μέγεθος ἐφ' οὗ ΑΒ, χρόνος δὲ ἄπειρος ἐφ' ᾧ Γ· εἰλήφθω δέ τι τοῦ (233b.) χρόνου πεπερασμένον, ἐφ' ᾧ ΓΔ. ἐν τούτῳ οὖν δίεισί τι τοῦ μεγέθους, καὶ ἔστω διεληλυθὸς ἐφ' ᾧ ΒΕ. τοῦτο δὲ ἢ καταμετρήσει τὸ ἐφ' ᾧ ΑΒ, ἢ ἐλλείψει, ἢ ὑπερβαλεῖ· διαφέρει γὰρ οὐθέν· εἰ γὰρ ἀεὶ τὸ ἴσον τῷ ΒΕ μέγεθος ἐν ἴσῳ χρόνῳ δίεισιν, τοῦτο δὲ καταμετρεῖ τὸ ὅλον, πεπερασμέ νος ἔσται ὁ πᾶς χρόνος ἐν ᾧ διῆλθεν· εἰς ἴσα γὰρ διαιρεθήσεται καὶ τὸ μέγεθος. This may be shown as follows. Let AB be a finite magnitude, and let us suppose that it is traversed in infinite time G, and let a finite period GD of the time be taken. Now in this period the thing in motion will pass over a certain segment of the magnitude: let BE be the segment that it has thus passed over. (This will be either an exact measure of AB or less or greater than an exact measure: it makes no difference which it is.) Then, since a magnitude equal to BE will always be passed over in an equal time, and BE measures the whole magnitude, the whole time occupied in passing over AB will be finite: for it will be divisible into periods equal in number to the segments into which the magnitude is divisible. ἔτι δ' εἰ μὴ πᾶν μέγεθος ἐν ἀπείρῳ χρόνῳ δίεισιν, ἀλλ' ἐνδέχεταί τι καὶ ἐν πεπερασμένῳ διελθεῖν, οἷον τὸ ΒΕ, τοῦτο δὲ καταμετρήσει τὸ πᾶν, καὶ τὸ ἴσον ἐν ἴσῳ δίεισιν, ὥστε πεπερασμένος ἔσται καὶ ὁ χρόνος. Moreover, if it is the case that infinite time is not occupied in passing over every magnitude, but it is possible to ass over some magnitude, say BE, in a finite time, and if this BE measures the whole of which it is a part, and if an equal magnitude is passed over in an equal time, then it follows that the time like the magnitude is finite. ὅτι δ' οὐκ ἐν ἀπείρῳ δίεισιν τὸ ΒΕ, φανερόν, εἰ ληφθείη ἐπὶ θάτερα πεπερασμένος ὁ χρόνος· εἰ γὰρ ἐν ἐλάττονι τὸ μέρος δίεισιν, τοῦτο ἀνάγκη πεπεράνθαι, θατέρου γε πέρατος ὑπάρχοντος. That infinite time will not be occupied in passing over BE is evident if the time be taken as limited in one direction: for as the part will be passed over in less time than the whole, the time occupied in traversing this part must be finite, the limit in one direction being given. ἡ αὐτὴ δὲ ἀπόδειξις καὶ εἰ τὸ μὲν μῆκος ἄπειρον ὁ δὲ χρόνος πεπερασμένος. The same reasoning will also show the falsity of the assumption that infinite length can be traversed in a finite time. φανερὸν οὖν ἐκ τῶν εἰρημένων ὡς οὔτε γραμμὴ οὔτε ἐπίπεδον οὔτε ὅλως τῶν συνεχῶν οὐθὲν ἔσται ἄτομον, It is evident, then, from what has been said that neither a line nor a surface nor in fact anything continuous can be indivisible. οὐ μόνον διὰ τὸ νῦν λεχθέν, ἀλλὰ καὶ ὅτι συμβήσεται διαιρεῖσθαι τὸ ἄτομον. ἐπεὶ γὰρ ἐν ἅπαντι χρόνῳ τὸ θᾶττον καὶ βραδύτερον ἔστι, τὸ δὲ θᾶττον πλεῖον διέρχεται ἐν τῷ ἴσῳ χρόνῳ, ἐνδέχεται δὲ καὶ διπλάσιον καὶ ἡμιόλιον διιέναι μῆκος (εἴη γὰρ ἂν οὗτος ὁ λόγος τοῦ τάχους), ἐνηνέχθω οὖν τὸ θᾶττον ἡμιόλιον ἐν τῷ αὐτῷ χρόνῳ, καὶ διῃρήσθω τὰ μεγέθη τὸ μὲν τοῦ θάττονος εἰς τρία ἄτομα, ἐφ' ὧν ΑΒ ΒΓ ΓΔ, τὸ δὲ τοῦ βραδυτέρου εἰς δύο, ἐφ' ὧν ΕΖ ΖΗ. οὐκοῦν καὶ ὁ χρόνος διαιρεθήσεται εἰς τρία ἄτομα· τὸ γὰρ ἴσον ἐν τῷ ἴσῳ χρόνω δίεισιν. διῃρήσθω οὖν ὁ χρόνος εἰς τὰ ΚΛ ΛΜ ΜΝ. πάλιν δ' ἐπεὶ τὸ βραδύτερον ἐνήνεκται τὴν ΕΖΗ, καὶ ὁ χρόνος τμηθήσεται δίχα. διαιρεθήσεται ἄρα τὸ ἄτομον, καὶ τὸ ἀμερὲς οὐκ ἐν ἀτόμῳ δίει σιν ἀλλ' ἐν πλείονι. φανερὸν οὖν ὅτι οὐδέν ἐστι τῶν συνεχῶν ἀμερές. This conclusion follows not only from the present argument but from the consideration that the opposite assumption implies the divisibility of the indivisible. For since the distinction of quicker and slower may apply to motions occupying any period of time and in an equal time the quicker passes over a greater length, it may happen that it will pass over a length twice, or one and a half times, as great as that passed over by the slower: for their respective velocities may stand to one another in this proportion. Suppose, then, that the quicker has in the same time been carried over a length one and a half times as great as that traversed by the slower, and that the respective magnitudes are divided, that of the quicker, the magnitude ABGD, into three indivisibles, and that of the slower into the two indivisibles EZ, ZH. Then the time may also be divided into three indivisibles, for an equal magnitude will be passed over in an equal time. Suppose then that it is thus divided into KL, Lm, MN. Again, since in the same time the slower has been carried over Ez, ZH, the time may also be similarly divided into two. Thus the indivisible will be divisible, and that which has no parts will be passed over not in an indivisible but in a greater time. It is evident, therefore, that nothing continuous is without parts.
Postquam ostendit quod magnitudo et tempus similiter dividuntur, hic ostendit quod finitum etiam et infinitum similiter inveniuntur in magnitudine et tempore. Et circa hoc tria facit: primo ponit propositum; secundo ex hoc solvit dubitationem, ibi: unde et Zenonis ratio etc.; tertio probat propositum, ibi: neque iam infinitum et cetera. 777. After showing that magnitude and time are subject to similar divisions, the Philosopher now shows that if either is finite or infinite, so is the other. About this he does three things: First he states the proposition; Secondly, from this he settles a doubt at 779; Thirdly, he proves the proposition at 780. Dicit ergo primo, quod si quodcumque horum duorum, scilicet temporis et magnitudinis, sit infinitum, et alterum est infinitum; et eo modo quo alterum est infinitum et alterum. 778. He says therefore first (590) that if either of these two, namely, time and magnitude, is infinite, so is the other; likewise, both will be infinite in the same manner. Et hoc exponit distinguendo duos modos infiniti; dicens quod si tempus est infinitum in ultimis, et magnitudo est infinita in ultimis. Dicitur autem tempus et magnitudo esse infinita in ultimis, quia scilicet ultimis caret; sicut si imaginaremur lineam non terminari ad aliqua puncta, vel tempus non terminari ad aliquod primum aut ultimum instans. Et si tempus sit infinitum divisione, et longitudo erit divisione infinita. Et est hic secundus modus infiniti: dicitur enim divisione infinitum, quod in infinitum dividi potest; quod est de ratione continui, ut dictum est. Et si tempus esset utroque modo infinitum, et longitudo esset utroque modo infinita. He explains this by distinguishing two ways of being infinite, saying that if time is infinite in respect of its extremities, the magnitude, too, is infinite in that way. Now time and magnitude are said to be infinite in their extremities, because they lack extremities. It is as though we imagined that a line is not terminated at any points, or that time is not terminated at a first or final instant. Moreover, if time is infinite through division, so also is a length. And this is the second way in which something is infinite. But something is said to be infinite through division, because it can be divided ad infinitum; which, of course, pertains to the definition of a continuum, as was said. Consequently, if time is infinite both ways, so, too, is length. Et convenienter isti duo modi infiniti contraponuntur: quia primus modus infiniti accipitur ex parte ultimorum indivisibilium quae privantur; secundus autem modus accipitur secundum indivisibilia quae signantur in medio; dividitur enim linea secundum puncta infra lineam signata. It is fitting that these two ways of being infinite be set in contrast: for the first way is taken from the viewpoint of indivisible extremities that are absent; the second is taken from the viewpoint of the indivisibles which are intermediate, for a line is divided according to points within the line. Deinde cum dicit: unde et Zenonis etc., ex praemissis removet dubitationem Zenonis Eleatis, qui volebat probare quod nihil movetur de uno loco ad alium, puta de a in b. 779. Then at (591) he uses these facts to refute Zeno, who tried to prove that nothing is moved from one place to another, for example, from A to B. Manifestum est enim quod inter a et b sunt infinita puncta media, cum continuum sit divisibile in infinitum. Si ergo movetur aliquid de a in b, oportet quod pertranseat infinita, et quod tangat unumquodque infinitorum; quod non est possibile fieri in tempore finito. Ergo in nullo tempore quantumcumque magno, dummodo sit finitum, aliquid potest moveri per quantumcumque parvum spatium. For it is clear that between A and B there is an infinitude of intermediate points, since a continuum is divisible ad infinitum. Therefore, if something were to be moved from A to B, it would have to bridge the infinite and touch each of the infinites, and this cannot be done in finite time. Therefore, nothing can be moved through even the smallest distance during a period of finite time, however great. Dicit ergo philosophus quod ista ratio procedit ex falsa existimatione; quia longitudo et tempus, et quodcumque continuum, dupliciter dicitur esse infinitum, ut dictum est; scilicet secundum divisionem et in ultimis. Si igitur essent aliqua, scilicet mobile et spatium, infinita secundum quantitatem, quod est esse infinitum in ultimis; non contingeret quod se invicem tangerent in tempore finito. Si vero sint infinita secundum divisionem, hoc contingit; quia etiam tempus quod est finitum secundum quantitatem, est sic infinitum, scilicet secundum divisionem. The Philosopher, therefore, says that this argument is based on a false opinion, for length and time and any magnitude are said to be infinite in two ways, as we have said; namely, according to division and according to their extremities. Accordingly, if there were things (namely, a mobile and a distance) infinite in regard to quantity, which is to be infinite at the extremities, they could not touch one another in finite time. But if they are infinite in respect of division, they will touch, because time also, which is finite in respect of quantity, is infinite in respect of division. Unde sequitur quod infinitum transeatur, non quidem in tempore finito, sed in tempore infinito; et quod infinita puncta magnitudinis transeantur in infinitis nunc temporis, non autem in nunc finitis. Hence two things follow: that the infinite can be traversed not in finite but in infinite time, and that the infinite points of a magnitude are traversed in the infinite “now’s” of time but not in the finite “now’s”. Est autem sciendum quod haec solutio est ad hominem, et non ad veritatem, sicut infra Aristoteles manifestabit in octavo. But it should be noted that this solution is ad hominem and not ad veritatem, as Aristotle will explain in Book VIII, L. 17. Deinde cum dicit: neque iam infinitum etc., probat quod supra posuit. Et primo resumit propositum; secundo probat, ibi: sit enim magnitudo et cetera. 78C. Then at (592) he proves what he stated above as a proposition. First he restates the proposition; Secondly, he proves it at 781. Dicit ergo primo quod nullum mobile potest transire infinitum spatium in tempore finito, neque finitum spatium in tempore infinito; sed oportet, si tempus est infinitum, quod magnitudo sit infinita, et e converso. Deinde cum dicit: sit enim magnitudo etc., probat propositum. He says therefore first (592) that no mobile can traverse an infinite distance in finite time nor a finite distance in infinite time; rather, if the time is infinite, then the magnitude must be infinite, and vice versa. Et primo quod tempus non potest esse infinitum, si magnitudo sit finita; secundo quod e converso, si longitudo sit infinita, tempus non potest esse finitum, ibi: eadem autem demonstratio est et cetera. Then at (593) he proves the proposition: First that the time cannot be infinite, if the magnitude is finite; Secondly, that if the length is infinite, the time cannot be finite at 784. Primum autem ostendit duabus rationibus: quarum prima talis est. Sit magnitudo finita quae est ab, et sit tempus infinitum quod est g. Accipiatur autem huius infiniti temporis aliqua pars finita quae sit gd. Quia igitur mobile per totum tempus g pertransit totam magnitudinem ab, oportet quod in hac parte temporis quae est gd, pertranseat aliquam partem illius magnitudinis, quae quidem sit be. Cum autem ab magnitudo sit finita et maior, be autem finitum et minus, necesse est quod be aut mensuret totum ab, aut deficiet aut excellet in mensurando, si multoties sumatur be: sic enim omne finitum minus se habet ad finitum maius, ut patet in numeris. Ternarius enim, qui est minor senario, bis acceptus mensurat ipsum: quinarium vero, qui etiam est maior, non mensurat bis acceptus, sed excedit; plus enim est bis tria quam quinque. Similiter etiam et septenarium bis acceptus non mensurat, sed deficit ab eo: minus enim est bis tria quam septem. Sed tamen si ternarius ter accipiatur, excedet etiam septenarium. Nihil autem differt quocumque modo horum trium be se habeat ad ab: quia idem mobile semper pertransibit magnitudinem aequalem ei quod est be, in tempore aequali ei quod est gd. Sed be mensurat totum ab vel excedit ipsum, si multoties sumatur. Ergo et gd mensurabit totum tempus g vel excedit ipsum, si multoties sumatur; et sic oportet quod totum tempus g sit finitum, in quo pertransit totam magnitudinem finitam: quia oportet quod in aequalia secundum numerum dividatur tempus, sicut et magnitudo. 781. He proves the first part of the proposition with two reasons, the first of which (593) is this: Let AB be a finite magnitude and let G be an infinite time. Take GD as a finite part of this infinite time. Now, since the mobile traverses the entire magnitude AB in the entire time G, then in part of this time, which is GD, it will traverse the part BE of the magnitude. But since the magnitude AB is finite and greater than BE, which is finite and less, then BE is either an exact measure of AB or it will be less or greater. (These are the only relationships that a lesser finite quantity can bear to a greater finite quantity, as is evident in numbers. For 3, which is less than 6, measures it twice, but 3 taken twice does not measure 5, which is greater than 3, but exceeds it, nor does it measure 7, but is less than 7. But if 3 were taken thrice, that product would exceed even 7). Now it makes no difference in which of these three ways BE is related to AB, for the same mobile will always traverse a magnitude equal to BE in a time equal to GD. But BE is either an exact measure of AB or will exceed it, if taken a sufficient number of times. Therefore, also GD should exactly measure the entire time G or exceed it, if GD is repeated frequently enough. Consequently, the whole time G (in which the entire finite magnitude was traversed) must be finite; because for every segment of magnitude there was a corresponding segment of time. Secundam rationem ponit ibi: amplius autem etc.: quae talis est. Quamvis enim detur quod magnitudinem finitam quae est ab, pertranseat aliquod mobile in tempore infinito, non tamen potest dari quod omnem magnitudinem pertranseat in tempore infinito: quia videmus quod multae magnitudines finitae temporibus finitis pertranseuntur. 782. The second reason is given at (782). It is this: Although it be granted that a mobile traverse the finite magnitude AB in infinite time, it cannot be granted that it will traverse any magnnitude at random in infinite time, because we see finite magnitudes being traversed in finite times. Sit igitur magnitudo finita quae est be, quae pertranseatur tempore finito. Sed be, cum sit finita, mensurat ab, quae est etiam finita. Sed idem mobile pertransibit aequalem magnitudinem ei quae est be, in aequali tempore finito in quo ipsam pertransibat: et ita quot accipiebantur magnitudines aequales be ad constituendam totam ab, tot tempora finita aequalia accipientur ad mensurationem vel constitutionem totius temporis. Unde sequitur quod totum tempus sit finitum. So let BE be the finite magnitude which is traversed In finite time. But BE, since it is finite, will measure AB, which is also finite. Now, the same mobile will traverse a magnitude equal to BE in a finite time equal to that in which it traversed BE. Thus the number of magnitudes equal to BE that will form AB corresponds to the number of equal times required to form the entire time consumed. Hence the entire time was finite. Differt autem haec ratio a prima; quia in prima ratione be ponebatur pars magnitudinis ab, hic autem be ponitur quaedam alia magnitudo separata. 783. This second reason is different from the first, because in the first, BE was taken to be part of the magnitude AB, but here it is taken as a separate magnitude. Necessitatem autem huius secundae rationis positae ostendit cum subdit: quod autem non in infinito et cetera. Posset enim aliquis contra primam rationem cavillando dicere, quod sicut totam magnitudinem ab pertransit in tempore infinito, ita et quamlibet partem eius; et sic partem be non pertransibit in tempore finito. Sed quia non potest dari quod quamlibet magnitudinem pertranseat tempore infinito, oportuit inducere secundam rationem, quod be sit quaedam alia magnitudo, quam tempore finito pertranseat. Et hoc est quod subdit, quod manifestum est quod mobile non pertransit magnitudinem quae est be in infinito tempore, si accipiatur in altera finitum tempus, idest si accipiatur aliqua alia magnitudo a prima, quae dicatur be, quam pertransit tempore finito. Si enim in minori tempore pertransit partem magnitudinis quam totum, necesse est hanc magnitudinem quae est be, finitam esse, altero termino existente finito, scilicet ab. Quasi dicat: si tempus in quo pertransit be, est finitum, et minus tempore infinito in quo pertransit ab, necesse est quod be sit minor quam ab; et ita quod be sit finita, cum ab finita sit. Then at (595) he shows the necessity of this second reason. For someone could cavil by saying that just as the whole magnitude AB is traversed in infinite time, so would every part of it, and thus the part BE would not be traversed in finite time. But because it cannot be granted that any magnitude at random is traversed in infinite time, it was necessary to present the second reason in which BE is a different magnitude which is traversed in finite time. For if the time in which BE is traversed is finite and less than the infinite time in which AB is traversed, then necessarily, BE is less than AB, and must be finite, since AB is finite. Deinde cum dicit: eadem autem demonstratio etc., ponit quod eadem demonstratio est ducens ad impossibile, si dicatur quod longitudo sit infinita et tempus finitum. Quia accipietur aliquid longitudinis infinitae, quod erit finitum; sicut accipiebatur aliquid temporis infiniti, quod est finitum. 784. Then at (596) he posits that the same proof leads to an impossibility if the length is said to be infinite and the time finite, because a part of the infinite length will be taker, as finite, just as a finite part of infinite time was taken, Deinde cum dicit: manifestum igitur ex dictis etc., probat quod nullum continuum est indivisibile. Et primo dicit quod inconveniens sequitur si hoc ponatur; secundo ponit demonstrationem ad illud inconveniens ducentem, ibi: quoniam enim in omni tempore et cetera. 785. Then at (597) he proves that no continuum is indivisible. First he says that an inconsistency would otherwise follow; Secondly, he gives the demonstrations that lead to that inconsistency, at 786. Dicit ergo primo manifestum esse ex dictis, quod neque linea neque planum, idest superficies, neque omnino aliquod continuum, est atomus, idest indivisibile: tum propter praedicta, quia videlicet impossibile est aliquod continuum ex indivisibilibus componi, cum tamen ex continuis possit componi continuum; tum etiam quia sequeretur quod indivisibile divideretur. He says therefore first (597) that it is clear from what has been said that no line or plane or any continuum is indivisible: first of all on account of the foregoing, namely, that it is impossible for any continuum to be composed of indivisibles, although a continuum can be composed of continua; secondly, because it would follow that an indivisible can be divided. Deinde cum dicit: quoniam enim in omni tempore etc., ponit demonstrationem ad hoc inconveniens ducentem: in qua primo praesupponit quaedam superius manifestata. Quorum unum est, quod in omni tempore contingat velocius et tardius moveri. Secundum est quod velocius plus pertransit de magnitudine in aequali tempore. Tertium est quod contingit esse excessum velocitatis ad velocitatem, et longitudinis pertransitae ad longitudinem, secundum diversas proportiones: puta secundum duplicem, quae est proportio duorum ad unum; et secundum hemioliam, quae habet totum et dimidium, quae alio nomine dicitur sexquialtera, ut proportio trium ad duo; vel secundum quantamcumque aliam proportionem. 786. Then at (598) he gives the proof which leads to this inconsistency. In this proof he makes use of certain facts already established. One of these is that in any finite time the faster and the slower can be in motion. The second is that the faster will traverse more distance in equal time. The third is that there can be excess of speed over speed and of length traversed over length traversed according to varying proportions; for example, according to the proportion of 2 to 1, or 3 to 2, or any other proportion. Ex his autem suppositis sic procedit. Sit haec proportio velocis ad velox, ut inveniatur aliquid velocius altero secundum hemiolium, idest sexquialteram proportionem; et sit ita, quod velocius pertranseat unam magnitudinem quae sit abcd, compositam ex tribus magnitudinibus indivisibilibus, quarum una sit ab, alia bc, tertia cd. In eodem autem tempore oportet quod tardius secundum praedictam proportionem pertranseat magnitudinem compositam ex duabus indivisibilibus magnitudinibus, quae sit magnitudo ezi. Et quia tempus dividitur sicut et magnitudo, necesse est quod tempus in quo velocius pertransit tres indivisibiles magnitudines, dividatur in tria indivisibilia; quia oportet quod aequale in aequali tempore pertranseat. Sit ergo tempus klmn divisum in tria indivisibilia. Sed quia tardius in eodem tempore movetur per ezi, quae sunt duae magnitudines indivisibiles, necesse est quod tempus dividatur in duo media: et sic sequetur quod indivisibile dividatur. Oportebit enim quod tardius unam magnitudinem indivisibilem pertranseat in uno indivisibili tempore et dimidio. Non enim potest dici quod unam indivisibilem magnitudinem transeat in uno indivisibili tempore; quia sic non prius moveretur velocius quam tardius. Ergo relinquitur quod tardius pertranseat indivisibilem magnitudinem in pluri quam in uno indivisibili tempore, et in minori quam in duobus; et sic oportebit unum indivisibile tempus dividi. With these presuppositions he proceeds thus: Let this be the ratio of the faster to the fast, that the one is faster in the ratio of 3 to 2; and let the faster traverse one magnitude ABCD composed of 3 indivisible magnitudes AB, BC and CD. During the same time according to the given ratio, tae slower will traverse a magnitude of two indivisible magnitudes, which form the magnitude EZI. And because the time is divided as the magnitude, the time in which the faster traverses the 3 indivisible magnitudes must be divided into 3 indivisibles, because the equal magnitude must be traversed in equal time. So let the time be KLMN divided into 3 indivisibles. But because the slower, during that time, traverses EZI, which are 2 indivisible magnitudes, the time can be divided into 2 halves. Consequently, it follows that an indivisible has been divided. For the slower had to traverse one indivisible magnitude in 1 and a half indivisibles of time, since it cannot be said that it traverses one indivisible magnitude in one indivisible time, for then the faster would not have been moved ahead of the slower. Therefore what remains is that the slower traverses an indivisible magnitude in more than one indivisible and less than two indivisibles of time. Thus the indivisible time will have had to be divided. Et eodem modo sequitur quod indivisibilis magnitudo dividatur, si ponatur quod tardius moveatur per tres indivisibiles magnitudines, in tribus indivisibilibus temporibus. Velocius enim in uno indivisibili tempore movebitur per plus quam per unam indivisibilem magnitudinem, et per minus quam per duas. In like manner, it follows that an indivisible magnitude is divided, if the slower manages to move through three indivisible magnitudes in three indivisible times. For the faster will in one indivisible time be moved through more than one indivisible of magnitude and less than two. Unde manifestum fit, quod nullum continuum potest esse indivisibile. Therefore, it is clear that no continuum can be indivisible.
The “now” as the indivisible of time.
Everything that moves is divisible.
Chapter 3 Ἀνάγκη δὲ καὶ τὸ νῦν τὸ μὴ καθ' ἕτερον ἀλλὰ καθ' αὑτὸ καὶ πρῶτον λεγόμενον ἀδιαίρετον εἶναι, καὶ ἐν ἅπαντι τὸ τοιοῦτο χρόνῳ ἐνυπάρχειν. The present also is necessarily indivisible—the present, that is, not in the sense in which the word is applied to one thing in virtue of another, but in its proper and primary sense; in which sense it is inherent in all time. ἔστιν γὰρ ἔσχατόν τι τοῦ (234a.) γεγονότος, οὗ ἐπὶ τάδε οὐθέν ἐστι τοῦ μέλλοντος, καὶ πάλιν τοῦ μέλλοντος, οὗ ἐπὶ τάδε οὐθέν ἐστι τοῦ γεγονότος· ὃ δή φαμεν ἀμφοῖν εἶναι πέρας. τοῦτο δὲ ἐὰν δειχθῇ ὅτι τοιοῦτόν ἐστιν [καθ' αὑτὸ] καὶ ταὐτόν, ἅμα φανερὸν ἔσται καὶ ὅτι ἀδιαίρετον. For the present is something that is an extremity of the past (no part of the future being on this side of it) and also of the future (no part of the past being on the other side of it): it is, as we have said, a limit of both. And if it is once shown that it is essentially of this character and one and the same, it will at once be evident also that it is indivisible. ἀνάγκη δὴ τὸ αὐτὸ εἶναι τὸ νῦν τὸ ἔσχατον ἀμφοτέ ρων τῶν χρόνων· Now the present that is the extremity of both times must be one and the same: εἰ γὰρ ἕτερον, ἐφεξῆς μὲν οὐκ ἂν εἴη θάτερον θατέρῳ διὰ τὸ μὴ εἶναι συνεχὲς ἐξ ἀμερῶν, εἰ δὲ χωρὶς ἑκάτερον, μεταξὺ ἔσται χρόνος· πᾶν γὰρ τὸ συνεχὲς τοιοῦτον ὥστ' εἶναί τι συνώνυμον μεταξὺ τῶν περάτων. ἀλλὰ μὴν εἰ χρόνος τὸ μεταξύ, διαιρετὸν ἔσται· πᾶς γὰρ χρόνος δέδεικται ὅτι διαιρετός. ὥστε διαιρετὸν τὸ νῦν. for if each extremity were different, the one could not be in succession to the other, because nothing continuous can be composed of things having no parts: and if the one is apart from the other, there will be time intermediate between them, because everything continuous is such that there is something intermediate between its limits and described by the same name as itself. But if the intermediate thing is time, it will be divisible: for all time has been shown to be divisible. Thus on this assumption the present is divisible. εἰ δὲ διαιρετὸν τὸ νῦν, ἔσται τι τοῦ γεγονότος ἐν τῷ μέλλοντι καὶ τοῦ μέλλοντος ἐν τῷ γεγονότι· But if the present is divisible, there will be part of the past in the future and part of the future in the past: for past time will be marked off from future time at the actual point of division. καθ' ὃ γὰρ ἂν διαιρεθῇ, τοῦτο διοριεῖ τὸν παρήκοντα καὶ τὸν μέλλοντα χρόνον. Also the present will be a present not in the proper sense but in virtue of something else: for the division which yields it will not be a division proper. ἅμα δὲ καὶ οὐκ ἂν καθ' αὑτὸ εἴη τὸ νῦν, ἀλλὰ καθ' ἕτερον· ἡ γὰρ διαίρεσις οὐ καθ' αὑτό. πρὸς δὲ τούτοις τοῦ νῦν τὸ μέν τι γεγονὸς ἔσται τὸ δὲ μέλλον, καὶ οὐκ ἀεὶ τὸ αὐτὸ γεγονὸς ἢ μέλλον. οὐδὲ δὴ τὸ νῦν τὸ αὐτό· πολλαχῇ γὰρ διαιρετὸς ὁ χρόνος. ὥστ' εἰ ταῦτα ἀδύνατον ὑπάρχειν, ἀνάγκη τὸ αὐτὸ εἶναι τὸ ἐν ἑκατέρῳ νῦν. Furthermore, there will be a part of the present that is past and a part that is future, and it will not always be the same part that is past or future: in fact one and the same present will not be simultaneous: for the time may be divided at many points. If, therefore, the present cannot possibly have these characteristics, it follows that it must be the same present that belongs to each of the two times. ἀλλὰ μὴν εἰ ταὐτό, φανερὸν ὅτι καὶ ἀδιαίρετον· εἰ γὰρ διαιρετόν, πάλιν ταὐτὰ συμβήσεται ἃ καὶ ἐν τῷ πρότερον. ὅτι μὲν τοίνυν ἔστιν τι ἐν τῷ χρόνῳ ἀδιαίρετον, ὅ φαμεν εἶναι τὸ νῦν, δῆλόν ἐστιν ἐκ τῶν εἰρημένων· But if this is so it is evident that the present is also indivisible: for if it is divisible it will be involved in the same implications as before. It is clear, then, from what has been said that time contains something indivisible, and this is what we call a present. ὅτι δ' οὐθὲν ἐν τῷ νῦν κινεῖται, ἐκ τῶνδε φανερόν ἐστιν. εἰ γάρ, ἐνδέχεται καὶ θᾶττον κινεῖσθαι καὶ βραδύτερον. ἔστω δὴ τὸ νῦν ἐφ' ᾧ Ν, κεκινήσθω δ' ἐν αὐτῷ τὸ θᾶττον τὴν ΑΒ. οὐκοῦν τὸ βραδύτερον ἐν τῷ αὐτῷ ἐλάττω τῆς ΑΒ κινηθήσεται, οἷον τὴν ΑΓ. ἐπεὶ δὲ τὸ βραδύτερον ἐν ὅλῳ τῷ νῦν κεκίνηται τὴν ΑΓ, τὸ θᾶττον ἐν ἐλάττονι τούτου κινηθήσεται, ὥστε διαιρεθήσεται τὸ νῦν. ἀλλ' ἦν ἀδιαίρετον. οὐκ ἄρα ἔστιν κινεῖσθαι ἐν τῷ νῦν. We will now show that nothing can be in motion in a present. For if this is possible, there can be both quicker and slower motion in the present. Suppose then that in the present N the quicker has traversed the distance AB. That being so, the slower will in the same present traverse a distance less than AB, say AG. But since the slower will have occupied the whole present in traversing AG, the quicker will occupy less than this in traversing it. Thus we shall have a division of the present, whereas we found it to be indivisible. It is impossible, therefore, for anything to be in motion in a present. ἀλλὰ μὴν οὐδ' ἠρεμεῖν· ἠρεμεῖν γὰρ λέγομεν τὸ πεφυκὸς κινεῖσθαι μὴ κινούμενον ὅτε πέφυκεν καὶ οὗ καὶ ὥς, ὥστ' ἐπεὶ ἐν τῷ νῦν οὐθὲν πέφυκε κινεῖσθαι, δῆλον ὡς οὐδ' ἠρεμεῖν. Nor can anything be at rest in a present: for, as we were saying, only can be at rest which is naturally designed to be in motion but is not in motion when, where, or as it would naturally be so: since, therefore, nothing is naturally designed to be in motion in a present, it is clear that nothing can be at rest in a present either. ἔτι δ' εἰ τὸ αὐτὸ μέν ἐστι τὸ νῦν ἐν ἀμφοῖν τοῖν χρόνοιν, (234b.) ἐνδέχεται δὲ τὸν μὲν κινεῖσθαι τὸν δ' ἠρεμεῖν ὅλον, τὸ δ' ὅλον κινούμενον τὸν χρόνον ἐν ὁτῳοῦν κινηθήσεται τῶν τούτου καθ' ὃ πέφυκε κινεῖσθαι, καὶ τὸ ἠρεμοῦν ὡσαύτως ἠρεμήσει, συμβήσεται τὸ αὐτὸ ἅμα ἠρεμεῖν καὶ κινεῖσθαι· τὸ γὰρ αὐτὸ ἔσχατον τῶν χρόνων ἀμφοτέρων, τὸ νῦν. Moreover, inasmuch as it is the same present that belongs to both the times, and it is possible for a thing to be in motion throughout one time and to be at rest throughout the other, and that which is in motion or at rest for the whole of a time will be in motion or at rest as the case may be in any part of it in which it is naturally designed to be in motion or at rest: this being so, the assumption that there can be motion or rest in a present will carry with it the implication that the same thing can at the same time be at rest and in motion: for both the times have the same extremity, viz. the present. ἔτι δ' ἠρεμεῖν μὲν λέγομεν τὸ ὁμοίως ἔχον καὶ αὐτὸ καὶ τὰ μέρη νῦν καὶ πρότερον· ἐν δὲ τῷ νῦν οὐκ ἔστι τὸ πρότερον, ὥστ' οὐδ' ἠρεμεῖν. ἀνάγκη ἄρα καὶ κινεῖσθαι τὸ κινούμενον ἐν χρόνῳ καὶ ἠρεμεῖν τὸ ἠρεμοῦν. Again, when we say that a thing is at rest, we imply that its condition in whole and in part is at the time of speaking uniform with what it was previously: but the present contains no 'previously': consequently, there can be no rest in it. It follows then that the motion of that which is in motion and the rest of that which is at rest must occupy time. Chapter 4 Τὸ δὲ μεταβάλλον ἅπαν ἀνάγκη διαιρετὸν εἶναι. ἐπεὶ γὰρ ἔκ τινος εἴς τι πᾶσα μεταβολή, καὶ ὅταν μὲν ᾖ ἐν τούτῳ εἰς ὃ μετέβαλλεν, οὐκέτι μεταβάλλει, ὅταν δὲ ἐξ οὗ μετέβαλλεν, καὶ αὐτὸ καὶ τὰ μέρη πάντα, οὔπω μεταβάλλει (τὸ γὰρ ὡσαύτως ἔχον καὶ αὐτὸ καὶ τὰ μέρη οὐ μεταβάλλει), ἀνάγκη οὖν τὸ μέν τι ἐν τούτῳ εἶναι, τὸ δ' ἐν θατέρῳ τοῦ μεταβάλλοντος· οὔτε γὰρ ἐν ἀμφοτέροις οὔτ' ἐν μηδετέρῳ δυνατόν. λέγω δ' εἰς ὃ μεταβάλλει τὸ πρῶτον κατὰ τὴν μεταβολήν, οἷον ἐκ τοῦ λευκοῦ τὸ φαιόν, οὐ τὸ μέλαν· οὐ γὰρ ἀνάγκη τὸ μεταβάλλον ἐν ὁποτερῳοῦν εἶναι τῶν ἄκρων. φανερὸν οὖν ὅτι πᾶν τὸ μεταβάλλον ἔσται διαιρετόν. Further, everything that changes must be divisible. For since every change is from something to something, and when a thing is at the goal of its change it is no longer changing, and when both it itself and all its parts are at the starting-point of its change it is not changing (for that which is in whole and in part in an unvarying condition is not in a state of change); it follows, therefore, that part of that which is changing must be at the starting-point and part at the goal: for as a whole it cannot be in both or in neither. (Here by 'goal of change' I mean that which comes first in the process of change: e.g. in a process of change from white the goal in question will be grey, not black: for it is not necessary that that that which is changing should be at either of the extremes.) It is evident, therefore, that everything that changes must be divisible.
Postquam ostendit philosophus quod nullum continuum ex indivisibilibus componitur, neque indivisibile esse, ex quibus apparet motum esse divisibilem; hic determinat de divisione motus. Et primo praemittit quaedam necessaria ad motus divisionem; secundo de ipsa motus divisione determinat, ibi: motus autem est divisibilis dupliciter et cetera. 787. After showing that no continuum is composed of indivisibles and that no continuum is indivisible, thus making it seem that motion is divisible, the Philosopher now determines about the division of motion. First he states certain facts necessary for the division of motion; Secondly, he treats of the division of motion, L. 6. Circa primum duo facit: primo ostendit quod in indivisibili temporis non contingit esse motum neque quietem; secundo ostendit quod indivisibile non potest moveri, ibi: quod mutatur autem omne et cetera. About the first he does two things: First he shows that in an indivisible of time, there is neither motion nor rest; Secondly, that an indivisible cannot be moved, at 796. Circa primum duo facit: primo ostendit quod indivisibile temporis est ipsum nunc; secundo quod in nunc nihil movetur aut quiescit, ibi: quod autem nihil in ipso nunc movetur et cetera. About the first he does two things: First he shows that the indivisible of time is the “now”; Secondly that in the “now” nothing is being moved or is at rest, at 794. Circa primum tria facit: primo ponit quod intendit; secundo ponit ea ex quibus probari potest propositum, ibi: est enim aliquid ultimum eius etc.; tertio ostendit id quod ad haec consequitur, ibi: necesse est igitur et cetera. About the first he does three things: First he states his intention; Secondly, he states facts from which his proposition can be reached, at 789. Thirdly, he shows what follows from his proposition, ?90. Circa primum considerandum est, quod aliquid dicitur nunc secundum alterum, et non secundum seipsum: sicut dicimus nunc agi quod in toto praesenti die agitur; tamen totus dies praesens non dicitur praesens secundum seipsum, sed secundum aliquid sui. Manifestum est enim quod totius diei aliqua pars praeteriit, et aliqua futura est: quod autem praeteriit vel futurum est, non est nunc. Sic ergo patet quod totus dies praesens non est nunc primo et per se, sed per aliquid sui: et similiter nec hora, nec quodcumque aliud tempus. 788. About the first (599) we must take into account that something is called “now” in relation to something else and not in relation to itself; for example, we say that what is being done in the course of a whole day is being done “now”, yet the whole day is not said to be present according to its entirety but according to some part of itself. For it is evident that part of a whole day has passed and part is still to come, and neither of them is “now”. Thus it is evident that the entire present day is not a “now” primarily and per se but only according to something of itself—and what is true of the day is true of an hour or any period of time. Dicit ergo quod id quod dicitur nunc primo et per se, et non secundum alterum, ex necessitate est indivisibile, et iterum ex necessitate est in omni tempore. He says therefore that what is “now” primarily and per se and not according to something else is necessarily indivisible and present in every time. Deinde cum dicit: est enim aliquid etc., probat propositum. Manifestum est enim quod cuiuslibet continui finiti est accipere aliquod ultimum, extra quod nihil est eius cuius est ultimum; sicut nihil lineae est extra punctum, quod terminat lineam. Tempus autem praeteritum est quoddam continuum finitum ad praesens. Est ergo accipere aliquod ultimum eius quod factum est, idest praeteriti, extra quod nihil est praeteriti, et infra quod nihil est futuri. Et similiter erit accipere aliquod ultimum futuri, infra quod nihil est praeteriti. Et illud ultimum est terminus utriusque, scilicet praeteriti et futuri; quia cum totum tempus sit continuum, oportet quod praeteritum et futurum ad unum terminum copulentur. Si igitur de aliquo demonstretur quod ipsum sit tale per seipsum, quod est esse nunc per seipsum et non per aliquid sui, simul cum hoc manifestum erit quod sit indivisibile. 789. Then at (600) he proves his proposition, For it is evident that it is possible in regard to any finite continuum to take an extremity outside of which there is existing nothing of that of which it is the extremity, just as nothing of a line is outside the point which terminates the line. But past time is a continuum which is terminated at the present. Therefore it is possible to take something as the extremity of the past, so that beyond it there is nothing of the past, and previous to it nothing of the future. In like manner, it is possible to take an extremity of the future, beyond which there is nothing of the past. Now that extremity will be the limit of both, i.e., of the past and of the future; for since the totality of time is a continuum, the past and the future must be joined at one term. And if the “now” fits the description just given, it is clear that it is indivisible. Deinde cum dicit: necesse est igitur etc., ostendit quoddam consequens ad praemissa. Et circa hoc duo facit: primo ostendit, supposito quod nunc sit indivisibile, quod oporteat idem nunc esse quod est terminus praeteriti et terminus futuri; secundo ostendit quod e converso, si est idem utrumque nunc, oportet quod nunc sit indivisibile, ibi: at vero si idem est et cetera. 790. Then he shows what follows from these premises. About this he does two things: First he shows that on the supposition that the “now” is indivisible, the limit of the past and the limit of the future must be one and the same “now”. Secondly, that on the other hand, if each is the “now”, then the “now” must be indivisible, at 79-3. Circa primum duo facit: primo concludit ex dictis, quod necesse est esse idem nunc, quod est ultimum utriusque temporis, scilicet praeteriti et futuri. About the first he does two things: First at (601) he concludes from the foregoing that it must be the same “now” which is the limit of the past and of the future. Secundo ibi: si enim alterum est etc., probat tali ratione. Si est alterum nunc quod est principium futuri, et alterum quod est finis praeteriti, oportet quod haec duo nunc vel sint consequenter ad invicem, ita quod immediate sibi succedant; vel oportet quod unum sit seorsum ab altero, distans ab eo. Sed non potest dici quod unum consequenter se habeat ad alterum; quia sic sequeretur tempus componi ex nunc aggregatis; quod non potest esse propter id quod nullum continuum componitur ex impartibilibus, ut supra ostensum est. Nec etiam dici potest quod unum nunc sit seorsum ab altero, distans ab eo, quia tunc oporteret quod inter illa duo nunc esset tempus medium. Haec est enim natura omnis continui, quod inter quaelibet duo indivisibilia sit continuum medium, sicut inter quaelibet duo puncta, linea. 791. Secondly, at (602) he proves this statement with the following argument: If the “now” which is the beginning of the future is other than the “now” which is the end of the past, then either these two “now’s” are consecutive and immediately follow one upon the other or one is apart from and distinct from the other. But it cannot be that they are immediately consecutive, because then it will follow that time is composed of an aggregate of “now’s”—which cannot be, because no continuum is composed of indivisible parts, as was said above. Neither can it be that one “now” is apart from the other and distant from it, because then there would have to be a time between those two “now’s”. For it is the very nature of a continuum that there is something continuous between any two given indivisibles, just as there is line between any two given points of a line. Quod autem hoc sit impossibile ostendit dupliciter. Primo quia si aliquod esset tempus medium inter praedicta duo nunc, sequeretur quod aliquod univocum, idest eiusdem generis, esset medium inter duos terminos; quod est impossibile. Non enim est possibile quod inter extrema duarum linearum se tangentium vel consequenter se habentium, sit aliqua linea media. Hoc enim esset contra rationem eius quod est consequenter: quia consequenter sunt, ut supra dictum est, quorum nihil est medium proximi generis. Et sic, cum tempus futurum consequenter se habeat ad praeteritum, impossibile est quod inter terminum futuri et terminum praeteriti cadat aliquod tempus medium. Alio modo ostendit idem sic. But that this is impossible, he proves in two ways. First of all, because if there were a period of time between the two “now’s” in question, it would follow that something of the same kind would exist between the two, which is impossible, for it is not possible that between the extremities of two lines that touch or are consecutive, there be a line between. For that is against the nature of consecutive things, which were defined as things between which nothing like them occurs. And so, since future time is consecutive to past time, it is impossible that between the end of the past and the beginning of the future there be an intervening time. Quidquid est medium inter praeteritum et futurum, dicitur nunc: si igitur tempus aliquod sit medium inter extrema temporis praeteriti et futuri, sequetur quod totum illud dicatur nunc. Sed omne tempus est divisibile, ut ostensum est. Ergo sequetur quod ipsum nunc sit divisibile. He proves the same point in another way: Whatever is intermediate between the past and the future is called “now”. If, therefore, there is any time between the limits of the past and future, it will follow that that will also be called a “now.” But all time is divisible, as has been proved. Consequently, it would follow, that the “now” is divisible. Et quamvis supra posuerit principia ex quibus probari potest quod nunc sit indivisibile; quia tamen conclusionem non deduxerat ex principiis, hic consequenter ostendit quod nunc sit indivisibile, ibi: si autem divisibile est et cetera. Et hoc triplici ratione. 792. Although in the immediately foregoing he had laid down the principles from which it could be proved that the Now is indivisible, yet because he had not derived the conclusion from these principles, he now shows that the Now is indivisible at (603). And he does this with three arguments. Quarum prima est, quia si nunc sit divisibile, sequetur quod aliquid de praeterito sit in futuro, et aliquid futuri sit in praeterito. Cum enim nunc sit extremum praeteriti et extremum futuri; omne autem extremum est in eo cuius est extremum, sicut punctum in linea; necesse est quod totum nunc et sit in tempore praeterito ut finis, et in tempore futuro ut principium. Sed si nunc dividatur, oportet quod illa divisio determinet praeteritum et futurum. Omnis enim divisio in tempore facta, distinguit praeteritum et futurum; cum omnium partium temporis una comparetur ad aliam ut praeteritum ad futurum. Sequetur ergo quod ipsius nunc aliquid sit praeteritum, et aliquid futurum. Et ita cum nunc sit in praeterito et in futuro, sequetur quod aliquid futuri sit in praeterito, et aliquid praeteriti sit in futuro. The first of these is that if Now be divisible, it will follow that something of the past is in the future and something of the future in the past. For since the Now is the extremity of the past and the extremity of the future, and every extremity is in that of which it is the extremity, as a point in a line, then necessarily the entire Now is both in the past as its end and in the future as its beginning. But if the Now be divided, that division must determine the past and the future. For any division made in time distinguishes past and future, since among any parts of time taken at random, one is related to the other as past to future. It will follow, therefore, that part of the Now is past and part future. And so, since the Now is in the past and in the future, it will follow that something of the future is in the past and something of the past in the future. Secundam rationem ponit ibi: simul autem etc.: quia si nunc sit divisibile, non erit nunc secundum seipsum, sed secundum alterum. Nullum enim divisibile est sua divisio qua dividitur: ipsa autem divisio temporis est nunc. Nihil enim est aliud divisio continui quam terminus communis duabus partibus: hoc autem intelligimus per nunc, quod est terminus communis praeteriti et futuri. Sic ergo manifestum est quod id quod est divisibile, non potest esse nunc secundum seipsum. The second argument he gives at (604): If the Now be divisible, it will be such, not according to itself, but according to something else. For no divisible is the very division by which it is divided. But the division of time is the Now. For that by which a continuum is divided is nothing but a term common to two parts. But that is what we understand by the Now, that it is a term common to the past and future. Thus, therefore, it is clear that what is divisible cannot be the Now according to itself. Tertiam rationem ponit ibi: adhuc autem ipsius nunc etc.: quae talis est. Semper, facta divisione temporis, una pars est praeterita, et alia futura. Si igitur et nunc dividatur, oportet quod aliquid eius sit praeteritum, et aliquid futurum. Sed non idem est praeteritum et futurum: sequetur ergo quod ipsum nunc non sit idem sibi ipsi, quasi totum simul existens (quod est contra rationem eius quod dicitur nunc: cum enim dicimus nunc, intelligimus simul in praesenti esse); sed oportebit multam diversitatem esse in nunc et successionem, sicut et in tempore, quod multipliciter est divisibile. The third argument is given at (605): Whenever time is divided, one part is always past and the other future. If, therefore, the Now is divided, necessarily part of it will be past and part future. But past and future are not the same. It will follow, therefore, that the Now is not the same as itself, i.e., something existing as a whole all at once (which is against the definition of the Now: for when we speak of the Now, we consider it as existing completely in the present); rather there will be much diversity and even succession in the Now, just as there is in time, which can be divided any number of times. Sic ergo ostenso quod nunc sit divisibile, quod erat consequens ad hoc quod dicebatur non esse idem nunc quod est extremum praeteriti et futuri, et destructo consequente, concludit destructionem antecedentis. Et hoc est quod dicit, quod si hoc est impossibile inesse ipsi nunc, scilicet quod sit divisibile, necesse est dicere quod idem sit nunc quod est extremum utriusque temporis. 793. Therefore, having thus shown that the Now is divisible (as a consequence of supposing that the Now which is the extremity of the past and of the future is not identical), and having rejected this consequent, he concludes to the rejection of the antecedent. And that is what he says: If it is impossible for the Now to be divisible, then it must be admitted that the Now which is the extremity of the past and of the future is one and the same. Deinde cum dicit: at vero si idem etc., ostendit quod e contrario, si idem est nunc praeteriti et futuri, necesse est quod nunc sit indivisibile: quia si esset divisibile, sequerentur omnia praedicta inconvenientia. Et sic ex quo non potest dici quod nunc sit divisibile, quasi existente altero nunc praeteriti et altero nunc futuri: nec etiam est divisibile si ponatur idem; concludit manifestum esse ex dictis, quod necesse est in tempore esse aliquid indivisibile, quod dicitur nunc. Then at (606) he shows that conversely, if the Now of the past and of the future is the same, then it must be indivisible; because if it were divisible, all the aforementioned inconsistencies would follow, And so, from the fact that the Now cannot be admitted to be divisible (as though the Now of the past were something distinct from the Now of the future) and is indeed not divisible, if the Now of the present is the same as the Now of the future, he concludes from the foregoing that it is clear that in time there must be something indivisible which is called the Now. Deinde cum dicit: quod autem nihil in ipso nunc etc., ostendit quod in nunc non potest esse nec motus nec quies. Et primo ostendit de motu; secundo de quiete, ibi: at vero neque quiescere et cetera. 794. Then at (607) he shows that in the Now there can be neither motion nor rest. First he shows it for motion; Secondly, for rest, at 795. Dicit ergo primo, manifestum esse ex iis quae sequuntur, quod in nunc nihil possit moveri: quia si aliquid potest moveri in nunc, continget in nunc moveri duo mobilia, quorum unum sit velocius, et aliud tardius. Sit ergo ipsum nunc n, et aliquod corpus velocius moveatur in n per ab magnitudinem. Sed tardius in aequali minus movetur: ergo tardius in hoc instanti movetur per minorem magnitudinem quae est ag. Sed velocius idem spatium pertransit in minori quam tardius. Quia ergo corpus tardius movebatur per ag magnitudinem in toto ipso nunc, sequitur quod velocius moveatur per eandem magnitudinem in minori quam nunc: ergo nunc dividitur. Sed ostensum est quod nunc est indivisibile: ergo non potest aliquid moveri in nunc. He says therefore first (607) that it is clear from what follows that in the Now nothing can be in motion, for if anything were in motion in the Now, two things could be in motion then, one of which is faster than the other. So let N be the Now, and let there be a faster body being moved in N through the magnitude AB. In an equal time, a slower body is moved a smaller distance. Therefore, in this instant, it traverses the smaller magnitude AG. But the faster will. cover the same distance in less time than the slower. Therefore, because the slower body traversed the magnitude AG in the very Now, the faster traversed the same magnitude in less than the Now. Hence the Now is divided. But it was already proved that the Now cannot be divided. Therefore, nothing can be moved in a Now. Deinde cum dicit: at vero neque quiescere etc., ostendit idem de quiete tribus rationibus. Quarum prima talis est. Dictum est enim in quinto, quod illud quiescit quod est aptum natum moveri et non movetur quando aptum natum est moveri, et secundum illam partem qua natum est moveri, et eo modo quo natum est moveri. Si enim aliquid caret eo quod non est natum habere, ut lapis visu; aut eo tempore quando non natum est habere, ut canis ante nonum diem; aut in ea parte qua non natum est habere, sicut in pede vel in manu; aut eo modo quo non natum est habere, ut si homo non videat ita acute ut aquila: non propter hoc dicitur esse privatum visu. Quies autem est privatio motus: unde nihil quiescit nisi quod est aptum natum moveri, et quando et sicut natum est moveri. Sed ostensum est quod nihil aptum natum est moveri in ipso nunc. Ergo manifestum est quod nihil quiescit in nunc. 795. Then at (608) he proves the same thing for rest, giving three arguments. The first of which is this: It was said in Book V that an object at rest is one that is naturally capable of being in motion, but is not in motion when it is capable of being in motion and in respect to the part by which it is capable of being in motion and in the manner in which it is apt to be in motion. For if a thing lacks what it is not naturally capable of having (as a stone lacks sight) or lacks it when it is not naturally due to have it (as a dog lacks sight before the ninth day) or in the part in which it is not naturally capable of having it (as sight in the foot or in the hand) or in the way in which it is not apt to have it (as for a man to have sight as keen as an eaglets), none of these reasons is sufficient for saying that a thing is deprived of sight. Now rest is privation of motion. Hence nothing is at rest except what is apt to be moved and when and as it is apt. But it has been shown that nothing is naturally capable of being moved in the very Now. Therefore, it is clear that nothing is at rest in the Now. Secundam rationem ponit ibi: amplius si idem etc.: quae talis est. Illud quod movetur in toto aliquo tempore, movetur in quolibet illius temporis in quo natum est moveri: et similiter quod quiescit in aliquo toto tempore, quiescit in quolibet illius temporis in quo natum est quiescere. Sed idem nunc est in duobus temporibus, in quorum uno toto quiescit, et in altero toto movetur; sicut apparet in eo quod post quietem movetur, et post motum quiescit. Si ergo in nunc aliquid natum est quiescere et moveri, sequeretur quod aliquid simul quiesceret et moveretur; quod est impossibile. The second argument is given at (609): That which is being moved in an entire period of time is being moved in each part of that time, in which it is apt to be moved; likewise, what is at rest in a given period of time is at rest in each period of that time in which it is apt to be at rest. But the same Now is in two periods of time, in one of which the mobile is totally at rest and in the other of which it is totally in motion (as appears in that which is in motion after rest or at rest after motion). Therefore, if in the Now something is apt to rest and be in motion, it will follow that something is at once in motion and at rest which is impossible. Tertiam rationem ponit ibi: amplius autem quiescere etc.: quae talis est. Illud dicimus quiescere, quod se habet similiter et nunc et prius, et secundum se totum et secundum partes suas. Ex hoc enim aliquid dicitur moveri, quod nunc et prius dissimiliter se habet, vel secundum locum vel secundum quantitatem vel secundum qualitatem. Sed in ipso nunc non est aliquid prius; quia sic nunc esset divisibile quia ly prius pertinet ad praeteritum: ergo non contingit in nunc quiescere. The third argument is given at (610): Rest is said of things which maintain themselves now just as they were previously, but in their entirety and in respect of all their parts. For it is on this account that a thing is said to be in motion, that now it is different from what it was previously, either in respect to place or quantity or quality. But in the Now itself, there is nothing previous; otherwise, the Now would be divisible, because the word “previous” refers to the past. Therefore, it is impossible to rest in the Now. Ex hoc autem ulterius concludit, quod necesse est omne quod movetur, et omne quod quiescit, moveri et quiescere in tempore. From this he further concludes that necessarily anything that is being moved and anything that is at rest, is being moved and is at rest in time. Deinde cum dicit: quod mutatur autem omne etc., ostendit quod omne quod movetur est divisibile, tali ratione. Omnis mutatio est ex quodam in quiddam: sed quando aliquid est in termino ad quem mutatur, ulterius non mutatur, sed iam mutatum est; non enim simul aliquid movetur et mutatum est, ut supra dictum est. Quando vero est aliquid in termino ex quo mutatur, secundum se totum et secundum omnes partes suas, tunc non mutatur: dictum est enim quod illud quod similiter se habet et ipsum et omnes partes eius, non mutatur, sed magis quiescit. Addit autem et omnes partes eius; quia cum aliquid incipit mutari, non simul totum egreditur de loco quem prius occupabat, sed pars post partem. 796. Then at (611) he shows that whatever is in motion is divisible: For every change is from this to that. But when something is at the goal, it is no longer being changed but has been changed, for nothing can be at the same time in the state of being changed and having been changed, as was said above. But when something is at the starting-point of change both in its entirety and in regard to all its parts, then it is not being changed; for it was said above that whatever maintains itself constant in its entirety and in regard to all its parts is not being changed but is at rest. He adds “In regard to all its parts”, because when a thing is beginning to be changed, it does not emerge in its entirety from the place it previously occupied, but part emerges after part. Neque iterum potest dici quod sit in utroque termino secundum se totum et secundum partes suas, dum movetur: sic enim aliquid esset simul in duobus locis. Moreover, it cannot be said that it is in both terms in its entirety and in regard to its parts, while it is being moved; for then something would be in two places at one time. Neque iterum potest dici quod in neutro terminorum sit: loquimur enim nunc de proximo termino in quem mutatur, et non de ultimo extremo; sicut si ex albo aliquid mutetur in nigrum, nigrum est ultimum extremum, fuscum vero est proximum. Et similiter si sit una linea divisa in tres partes aequales, scilicet linea abcd, manifestum est quod mobile, quod in principio motus est in parte ab sicut in loco sibi aequali, contingit in aliqua parte sui motus non esse neque in ab neque in cd: quandoque enim est totum in bc. Nor, again, can it be said that it is in neither of the terms: for we are now speaking of the nearest goal into which a thing is being changed and not of the remotest; for example, if something is being changed from white to black, black is the remote goal, but grey is the nearer one. In like manner, if a line ABCD is divided into three equal parts, it is clear that a mobile, which in the beginning of the motion was in AB as in a place equal to itself, can during the motion be neither in AB nor in CD; for at some time it is in its entirety in BC. Cum ergo dicitur quod illud quod mutatur, quando mutatur, non potest in neutro esse, accipitur non extremus terminus, sed proximus. Therefore, when it is said that what is being moved cannot happen to be in neither extremity while it is being moved, must be understood as referring not to the remotest extremity but to a nearer one. Relinquitur ergo quod omne quod mutatur, dum mutatur, secundum aliquid sui est in uno, et secundum aliquid sui est in altero; sicut cum aliquid mutatur de ab in bc, in ipso moveri pars egrediens de loco ab, ingreditur locum bc; et quod movetur de albo in nigrum, pars quae desinit esse alba, fit fusca vel pallida. What is left, therefore, is that whatever is being changed is, while it is being changed, partly in one and partly in the other; for example, when something is being changed from AB to BC, then during the motion, the part leaving the place AB is entering the place BC; likewise, when something is being moved from white to black, the part which ceases to be white becomes grey or light grey. Sic igitur manifestum est quod omne quod mutatur, cum sit partim in uno et partim in altero, est divisibile. Consequently, it is clear that anything that is being moved, since it is partly in one and party in the other, is divisible. Sciendum est autem quod Commentator in hoc loco movet dubitationem de hoc, quod si Aristoteles non intendit hic demonstrare quod mobile sit divisibile, nisi de mobili quod movetur motu quem dixit esse in solis tribus generibus, scilicet quantitate, qualitate et ubi, demonstratio sua non erit universalis, sed particularis: quia illud etiam quod mutatur secundum substantiam, divisibile invenitur. Unde videtur quod intelligat de eo quod transmutatur secundum quamcumque transmutationem, ut includatur generatio et corruptio in substantia. Et hoc etiam ex ipsis verbis eius apparet: non enim dicit quod movetur sed quod mutatur. 797. But it should be mentioned that the Commentator here raises the problem that if Aristotle does not intend in this place to demonstrate that every mobile is divisible but only what is mobile in regard to motion (which he said is present in only three genera; namely, quantity, quality and where), then his demonstration will not be universal but particular; because even the subject of substantial change is found to be divisible. Hence, he seems to be speaking of what is subject to any and every type of change, including even generation and ceasing-to-be in the genus of substance. And this is evident from his very words: for he does not say “what is being moved”, but “what is being changed”. Sed tunc videtur sua demonstratio non valere: quia aliquae transmutationes sunt indivisibiles, sicut ipsa generatio substantialis et corruptio, quae non sunt in tempore; et in huiusmodi transmutationibus non est verum, quod illud quod mutatur, sit partim in uno et partim in alio; non enim cum ignis generatur, partim est ignis et partim non ignis. But in that case his demonstration has no value, because some changes are indivisible, such as generation and ceasing-to-be of substance, which do not consume time. In such changes it is not true that what is being changed is partly in one extremity and partly in the other, for when fire is generated, it is not partly fire and partly non-fire. Et inducit ad hoc plures solutiones: quarum una est Alexandri, dicentis quod nulla transmutatio est indivisibilis, aut in non tempore. Sed hoc reprobatur; quia per hoc destruitur quoddam probabile et famosum apud Aristotelem et omnes Peripateticos, scilicet quod aliquae transmutationes sint in non tempore, ut illuminatio et alia huiusmodi. 798. In the face of this problem he proposes a number of solutions, one of which is Alexander’s, who says that no change is indivisible or in non-time, But this must be rejected, because it conflicts with an opinion that is held as probable and famous with Aristotle and all Peripatetics, namely, that certain changes are in non-time, such as illumination and the like. Inducit etiam solutionem Themistii, dicentis quod etsi sit aliqua transmutatio in non tempore, tamen hoc latet; et Aristoteles utitur eo quod est manifestum, scilicet quod transmutatio sit in tempore. Sed hoc reprobat; quia eodem modo se habet de divisione mutationis et mutabilis; et adhuc videtur latentius divisibilitas mobilis quam mutationis. Unde demonstratio Aristotelis non esset efficax: quia posset aliquis dicere, quod licet ea quae mutantur mutationibus manifeste divisibilibus, sint divisibilia, sunt tamen aliqua mutabilia latentia, quae sunt indivisibilia.  In Physic., lib. 6 l. 5 n. 13 He mentions also the solution of Themistius, who says that even if there be changes in non-time, they are hidden, whereas Aristotle appeals to what is evident, namely, that change occurs in time. But this he also rejects, because change and the changeable are divided in the same way, and the divisibility of a mobile is more hidden than the divisibility of change. Hence Aristotle’s demonstration would not be valid, because someone could say that although things which changed by changes evidently divisible are themselves divisible, yet there are some hidden changeable beings which are indivisible. Ponit etiam solutionem Avempacis, dicentis quod hic non agitur de divisione mutabilis secundum quantitatem, sed de divisione mutabilis secundum quod subiectum dividitur per accidentia contraria, de quorum uno mutatur in alterum. He mentions, too, the solution of Avempace [Ibn-Bajja], who says that the problem here is not about the quantitative division of the things capable of change but of that division whereby the subject is divided by contrary accidents, one of which is changed into the other. Et addit postea suam solutionem, quod illae mutationes quae dicuntur fieri in non tempore, sunt termini quorundam motuum divisibilium. Accidit ergo aliquid transmutari in non tempore, inquantum scilicet quilibet motus terminatur in instanti. Et quia illud quod est per accidens praetermittitur in demonstrationibus, ideo illo Aristoteles in hac demonstratione utitur, ac si omnis mutatio sit divisibilis et in tempore. 799. Then the Commentator adds his own solution: namely, that those changes which are said to occur in non-time are the extremities of certain divisible motions, It happens, therefore, that something should be changed in non-time, insofar as every motion is terminated in an instant. And because what is accidental is ignored when it comes to demonstrating, for that reason Aristotle proceeds in this demonstration as though every change were divisible and in time. Sed si quis recte consideret, haec obiectio non est ad propositum. Non enim Aristoteles in sua demonstratione utitur quasi principio, quod omnis mutatio sit divisibilis; cum magis e converso ex divisione mobilis procedat ad divisionem mutationis, ut infra patebit. Et sicut ipse post dicit, divisibilitas per prius est in mobili quam in motu vel mutatione. Sed utitur principiis per se notis, quae necesse est concedere in quacumque mutatione: scilicet quod illud quod mutatur, quando est secundum totum et partes in termino a quo mutatur, nondum mutatur secundum illam mutationem; et quod quando est in termino ad quem, non mutatur sed mutatum est; et quod non potest esse nec in utroque totum, nec in neutro, sicut expositum est. Unde ex necessitate sequitur quod in qualibet mutatione, illud quod mutatur, dum mutatur, sit partim in uno termino et partim in alio. 800. But if you consider the matter correctly, you will see that this objection is not to the point. For in his demonstration Aristotle does not use as his principle the statement that every change is divisible (since he proceeds rather from the divisibility of the mobile to the division of change, as will be clear later, for as he says later, divisibility is first in the mobile, before it is in motion or change). Rather he uses principles that are evident and which must be conceded in any and every case of change; namely, (1) that what is being changed in regard to a certain matter is not being changed in regard to that matter as long as it is totally and according to all its parts still in the starting point, and (2) when it is in the goal, it is not being changed but has been changed, and (3) that it cannot be entirely in both terms or entirely in either of them, as was explained. Hence, it necessarily follows that in any change whatsoever, what is being changed is, during the change, partly in one extremity and partly in the other. Sed hoc diversimode invenitur in diversis mutationibus. Nam in illis mutationibus, inter quarum extrema est aliquod medium, contingit quod id quod mutatur, dum mutatur, partim sit in uno extremo et partim in alio, secundum ipsa extrema. In illis vero inter quarum terminos non est aliquod medium, id quod mutatur non est secundum diversas partes suas in diversis extremis secundum ipsa extrema, sed secundum aliquid eis adiunctum. Sicut cum materia mutatur de privatione ad formam ignis, dum est in ipso mutari, est quidem sub privatione secundum seipsam; sed partim est sub forma ignis non secundum seipsam, sed secundum aliquid ei adiunctum, scilicet secundum dispositionem propriam ignis, quam partim recipit antequam formam ignis habeat. Unde infra probabit Aristoteles quod etiam generatio et corruptio sunt divisibiles: quia quod generatur, prius generabatur; et quod corrumpitur, prius corrumpebatur. But this occurs in various ways in various changes. For in changes between whose extremities there is something intermediate, it can happen that the mobile is, during the change, partly in one extreme and partly in the other extreme, precisely as extremes. But in those between whose extremes there is nothing intermediate, that which is being changed does not have different parts in different extremities precisely as extremities, but by reason of something connected with the extremities. For example, when matter is being changed from privation of fire to the form of fire, then while it is in the state of being changed, it is indeed under privation as to itself, but yet it is partly under the form of fire, not inasmuch as it is fire, but according to something connected with it, i.e., according to the particular disposition for fire, which disposition it partly receives before it has the form of fire. That is why Aristotle will later prove that even generation and ceasing-to-be are divisible, because what is generated was previously being generated, and what ceases-to-be was previously ceasing-to-be. Et forte hoc modo intellexit Alexander quod omnis transmutatio est divisibilis, scilicet vel secundum seipsam vel secundum motum ei adiunctum. Sic etiam intellexit Themistius quod Aristoteles assumpsit id quod erat manifestum, et praetermisit id quod erat latens: quia nondum erat locus tractandi de divisibilitate vel indivisibilitate mutationum; sed hoc reservatur in posterum. Perhaps this was the sense in which Alexander understood the statement that every change is divisible; namely, either according to itself or according to a motion connected with it. So also Themistius understood by the statement that Aristotle took what was evident and abstracted from what was hidden, that the proper place for treating of the divisibility or indivisibility of changes would not be reached until later. In omnibus tamen vel divisibilibus vel indivisibilibus salvatur quod Aristoteles hic dicit: quia etiam quae dicuntur indivisibiles mutationes, sunt quodammodo divisibiles, non secundum propria sua extrema, sed per ea quae eis adiunguntur. Et hoc est quod Averroes dicere voluit, quod hoc est per accidens, aliquas mutationes esse in non tempore. Nevertheless, in all divisibles and indivisibles, what Aristotle says here is true: because even changes that are called indivisible are in a sense divisible, not by reason of their extremities but by reason of something connected to them. And this is what Averroes wanted to say when he said that it is per accidens that some changes occur in non-time. Est etiam hic alia dubitatio. Non enim videtur hoc verum in motu alterationis, quod id quod alteratur, partim sit in uno termino et partim in altero, dum alteratur. Non enim sic procedit motus alterationis, quod prius una pars alteretur et postea altera: sed totum prius est minus calidum, et postea magis calidum. Unde etiam Aristoteles in libro de sensu et sensato dicit, quod non similiter se habet in alteratione sicut in latione. Lationes namque rationabiliter in medio prius attingunt: quaecumque vero alterantur, non adhuc similiter. Contingit enim simul alterari, et non dimidium prius; velut aquam simul omnem coagulari. 801. However, there is here another difficulty. For when it comes to alteration, it does not seem to be true that what is being altered is partly in one term and partly in the other, during the alteration. For the motion of alteration does not take place in such a way that first one part and then another is altered; rather the entire thing that was less hot becomes hotter. For which reason Aristotle even says in the book On Sense and the Thing Sensed that alterations are not like local motions. For in the latter, the subject reaches the intermediate before the goal, but such is not the case with things that are altered; for some things are altered all at once and not part by part, for it is the entire water that all at once freezes. Est autem ad hoc dicendum quod Aristoteles in hoc sexto libro agit de motu secundum quod est continuus. Continuitas autem primo et per se et proprie invenitur in motu locali tantum, qui solum potest esse continuus et regularis, ut ostendetur in octavo. Et ideo demonstrationes in hoc libro positae, pertinent quidem ad motum localem perfecte, ad alios autem motus non totaliter, sed secundum quod aliquid continuitatis et regularitatis participant. 802. But to this it must be replied that in this Sixth Book Aristotle is treating of motion as continuous. And continuity is primarily and per se and strictly found only in local motion, which alone can be continuous and regular, as will be shown in Book VIII. Therefore, the demonstrations given in Book VI pertain perfectly to local motion but imperfectly to other motions, i.e., only to the extent that they are continuous and regular. Sic ergo dicendum est quod mobile secundum locum semper prius subintrat locum in quem tendit secundum partem quam secundum totum: in alteratione autem est quidem ut sic, est autem ut non. Manifestum est enim quod omnis alteratio fit per virtutem agentis quod alterat, cuius virtus quanto fuerit maior, tanto maius corpus alterare potest. Quia ergo alterans est finitae virtutis, usque ad determinatam quantitatem corpus alterabile subditur eius virtuti, et simul recipit impressionem agentis; unde simul alteratur totum, non pars post partem. Sed illud alteratum iterum alterat aliquid aliud sibi coniunctum: est tamen minoris efficaciae in agendo. Et sic inde quousque deficiat virtus alterativa; sicut ignis calefacit unam partem aeris statim, et illa calefacta calefacit aliam: et sic pars post partem alteratur. Unde et Aristoteles in libro de sensu et sensato, post verba praemissa subiungit: attamen si multum fuerit quod calefit aut coagulatur, habitum ab habito patitur. Primum autem ab ipso faciente transmutari necesse est, et simul alterari et subito. Consequently, it must be said that what is mobile in respect of place always enters a new place part by part before it is there in its entirety; but in alteration, that is only partially true. For it is clear that every alteration depends on the power of the agent that causes the alteration—as its power is stronger it is able to alter a greater body. Therefore, since the cause of the alteration has finite power, a body capable of being altered is subject to its power up to a certain limit of quantity, which receives the impression of the agent all at once; hence the whole is altered all at once, and not part after part. Yet that which is altered can in turn alter something else conjoined to it, although its power in acting will be less forceful, and so on, until the power involved in the series of alterations is depleted. An example of this is fire which all at once heats one section of air, which in turn heats another, and thus part after part is altered. Hence in the book On Sense and the Thing Sensed, after the above-quoted passage, Aristotle goes on to say: “Yet if the object heated or frozen is large, part after part will be affected. But the first part had to be altered all at once and suddenly by the agent”. Verumtamen et in hoc ipso quod simul alteratur, est quandam successionem considerare; quia cum alteratio fiat per contactum alterantis, partes alterati quanto magis appropinquant ad corpus alterans, perfectius a principio recipiunt impressionem alterantis: et sic successive secundum ordinem partium ad perfectam alterationem pervenitur; et maxime quando in corpore alterabili est aliquid contra resistens alteranti. Yet even in things that are altered all at once, it is possible to discover some kind of succession, because since alteration depends on contact with the cause which alters, the parts closer to the body that causes the alteration will more perfectly receive at the very beginning an impression from the agent: and thus the state of perfect alteration is reached successively according to an order of parts. This is especially true when the body to be altered has something which resists the power of the altering cause. Sic ergo id quod concludit, quod videlicet id quod mutatur, dum mutatur partim est in termino a quo et partim in termino ad quem, quasi una pars prius perveniat ad terminum ad quem quam alia, simpliciter et absolute verum est in motu locali: in motu autem alterationis aliqualiter, ut dictum est. Consequently, the conclusion (that what is being changed, is, while it is being changed, partly in the terminus a quo and partly in the terminus ad quem, in the sense that one part reaches the terminus ad quem before another does) is unqualifiedly and absolutely true in local motion. But in alteration it is qualifiedly true, as we have said. Quidam vero e converso dixerunt quod hoc quod hic dicitur, magis habet veritatem in motu alterationis quam in motu locali. Dicunt enim quod hoc quod dicitur, quod id quod mutatur partim est in termino a quo et partim in termino ad quem, non sic est intelligendum, quod una pars eius quod movetur sit in uno termino et alia in alio, sed est referendum ad partes terminorum: quia scilicet id quod movetur partem habet de termino a quo et partem de termino ad quem; sicut illud quod movetur de albedine in nigredinem, primo non habet perfecte albedinem nec perfecte nigredinem, sed aliquid participat imperfecte de utroque. In motu autem locali hoc non videtur verum nisi secundum quod id quod movetur, dum est in medio duorum terminorum, quodammodo aliquid participat de utroque extremo. Sicut si terra moveatur ad locum ignis, dum est in loco aeris in suo moveri, partem habet utriusque termini; inquantum scilicet locus aeris et est sursum respectu loci terrae, et deorsum respectu loci ignis. 803. Some on the other hand have held that the present doctrine is truer when applied to alteration than when applied to local motion, For they hold that the statement “what is being changed is partly in the terminus a quo and partly in the terminus ad quem is not to be interpreted as meaning that one part of the thing in motion is in one term and another in the other, but that reference is being made to the parts of the termini, i.e., that what is being moved has part of the terminus a quo and part of the terminus ad quem, as something in motion from white to black, is at the very beginning neither perfectly white nor perfectly black, but imperfectly partakes of both; whereas in local motion this does not seem to be true, except in the sense that the thing in motion, while it is between the two extremities, somehow partakes of both extremities. For example, if earth were to be moved to the place normal to fire, then while it was in the region proper to air, it would have a part of each extremity, (i.e., earth and fire), in the sense that the place of air is above that of earth, and below that of fire, Haec autem expositio extorta est, et contra opinionem Aristotelis. Et primo quidem apparet hoc ex ipsis verbis Aristotelis. Concludit enim: necesse igitur hoc quidem aliquid in hoc esse, aliud vero in altero mutantis, idest eius quod mutatur. Loquitur ergo de partibus mobilis, non de partibus terminorum. 804. But this is a forced explanation and against Aristotle’s opinion. For in the first place we need only look at the very words of Aristotle. For he says as a conclusion: “it follows therefore that part of that which is being changed must be at the starting-point and part at the goal”. He is speaking therefore about the parts of the mobile and not about the parts of the termini. Secundo ex eius intentione. Inducit enim ad probandum id quod mutatur esse divisibile: quod non posset concludi ex praemissis. Unde et Avempace dixit, quod non intendit hic probare quod mobile sit divisibile in partes quantitativas, sed secundum formas: inquantum scilicet id quod mutatur de contrario in contrarium, dum est in ipso mutari, habet aliquid de utroque contrario. Sed intentio Aristotelis est expresse, ostendere quod mobile est divisibile in suas partes quantitativas, sicut et alia continua. Et sic utitur in sequentibus demonstrationibus. In the second place it is against Aristotle’s intention, For Aristotle brings to light facts that will prove that what is being changed is divisible—a statement that could not be proved, if you held to the interpretation given. Hence Avempace said that Aristotle does not intend here to prove that a mobile can be divided into quantitative parts but according to forms, in the sense that what is being moved from contrary to contrary has, while it in being changed, something from each contrary. But the intention of Aristotle is expressly to show that a mobile can be divided into its quantitative parts, just as any continuum, for he makes use of that fact in the demonstrations that will follow. Nec hoc videtur esse conveniens quod dicunt quidam, quod per hoc probatur etiam divisibilitas mobilis secundum continuitatem. Quia per hoc quod mobile, dum movetur, participat utrumque terminum, et non statim habet perfecte terminum ad quem, manifestum apparet mutationem esse divisibilem secundum continuitatem: et ita, cum divisibile non possit esse in indivisibili, sequitur quod etiam mobile sit divisibile ut continuum. Manifeste enim Aristoteles in subsequentibus ostendit divisionem motus ex divisione mobilis. Unde si intenderet concludere divisionem mobilis per divisionem motus, esset demonstratio circularis. Nor can we heed the opinion that such an interpretation will help to prove that a mobile can be divided on the basis of continuity. Because the very fact that a mobile, while it is being moved, partakes of each terminus and does not perfectly possess the terminus ad quem all at once, reveals that change is divisible on the basis of continuity. And thus, since a divisible cannot exist in an indivisible, it follows that the mobile also can be divided as a continuum. For in the matters to follow, Aristotle will clearly prove that motion is divisible, because the mobile is divisible. Hence, if he intended to conclude that a mobile is divisible because motion is divisible, he would be arguing in a circle. Tertio apparet hanc expositionem esse inconvenientem ex ipsa expositione Aristotelis, cum dicit: dico autem in quod mutatur primum secundum mutationem. Ex quo apparet quod non intendit dicere quod partim sit in termino a quo et partim in termino ad quem, propter hoc quod sit in medio, quasi participans utrumque extremum; sed quia secundum unam partem sui est in uno extremo, et secundum aliam in medio. Thirdly, such an interpretation appears to conflict with Aristotle’s own interpretation at (611 bis) where he says “here by ‘goal’ of change I mean that into which it is first changed during the process of change”. This shows that he does not intend to say that it is partly in the terminus a quo and partly in the terminus add quem just because it is midway and, as it were, sharing in both extremities, but because in regard to one part of itself it is in one extreme, and according to another part in what is midway. Sed circa hanc expositionem Aristotelis dubium esse videtur quod dicit in quod primum mutatur. Non enim videtur posse accipi in quod primum mutatur, propter divisibilitatem magnitudinis in infinitum. 805. But with respect to this explanation of Aristotle, one might wonder why he says “that into which it is first changed” for it seems impossible to discover that into which it is first changed, since a magnitude can be divided ad infinitum. Et ideo dicendum est, quod id in quod primum mutatur in motu locali, dicitur locus qui contingit locum a quo mutatur, ita quod nihil est eius. Si enim acciperetur secundus locus qui haberet aliquid primi, non esset accipere primum locum in quem mutatur. Quod sic patet. Sit locus unde mutatur aliquod mobile ab, et locus ei contactus aequalis sit bc. Quia enim ab divisibile est, dividatur in puncto d, et sumatur de loco bc versus c, quod sit aequale ei quod est bd; et sit illud gc. Manifestum est igitur quod mobile prius mutatur ad locum dg quam ad locum bc. Et iterum, cum ad sit divisibile, erit accipere alium locum priorem; et sic infinitum. Therefore, it must be said that “That into which it is first changed” in local motion is the place next to but not part of the place from which the local motion starts. For if we took it to mean a place that included part of the original place, we would not be assigning the first place into which it is being moved. The following example will illustrate this: Let AB be the place whence a mobile is being moved, and let BC be the adjacent place equal to AB. Now, since AB can be divided, let it be divided at D and take a point G near C so that the place GC is equal to BD. It is clear that the mobile will arrive at DG before it reaches BC. Moreover, since AD can be divided, a place prior to DG can be take n, and so on ad infinitum. Et similiter in motu alterationis accipiendum est primum in quod mutatur, medium alterius speciei; sicut cum mutatur de albo in nigrum, accipi debet fuscum, non autem minus album. Similarly, in regard to alteration, “the first into which something is changed” must be considered to be an intermediate; for example, when something is changed from white to black, the first into which the subject is changed is into grey, not into less white.
Two manners of dividing motion.
What things are co-divided with motion
Chapter 4 cont. κίνησις δ' ἐστὶν διαιρετὴ διχῶς, ἕνα μὲν τρόπον τῷ χρόνῳ, ἄλλον δὲ κατὰ τὰς τῶν μερῶν τοῦ κινουμένου κινήσεις, οἷον εἰ τὸ ΑΓ κινεῖται ὅλον, καὶ τὸ ΑΒ κινήσεται καὶ τὸ ΒΓ. Now motion is divisible in two senses. In the first place it is divisible in virtue of the time that it occupies. In the second place it is divisible according to the motions of the several parts of that which is in motion: e.g. if the whole AG is in motion, there will be a motion of AB and a motion of BG. ἔστω δὴ τοῦ μὲν ΑΒ ἡ ΔΕ, τοῦ δὲ ΒΓ ἡ ΕΖ κίνησις τῶν μερῶν. ἀνάγκη δὴ τὴν ὅλην, ἐφ' ἧς ΔΖ, τοῦ ΑΓ εἶναι κίνησιν. κινήσεται γὰρ κατὰ ταύτην, ἐπείπερ ἑκάτερον τῶν μερῶν κινεῖται καθ' ἑκατέραν· οὐθὲν δὲ κινεῖται κατὰ τὴν ἄλλου κίνησιν· ὥστε ἡ ὅλη κίνησις τοῦ ὅλου ἐστὶν μεγέθους κίνησις. That being so, let DE be the motion of the part AB and EZ the motion of the part BG. Then the whole Dz must be the motion of AG: for DZ must constitute the motion of AG inasmuch as DE and EZ severally constitute the motions of each of its parts. But the motion of a thing can never be constituted by the motion of something else: consequently the whole motion is the motion of the whole magnitude. ἔτι δ' εἰ πᾶσα μὲν κίνησις τινός, ἡ δ' ὅλη κίνησις ἡ ἐφ' ἧς ΔΖ μήτε τῶν μερῶν ἐστιν μηδετέρου (μέρους γὰρ ἑκατέρα) μήτ' ἄλλου μηδενός (οὗ γὰρ ὅλη ὅλου, καὶ τὰ μέρη τῶν μερῶν· τὰ δὲ μέρη τῶν ΑΒ ΒΓ καὶ οὐδένων ἄλλων· πλειόνων γὰρ οὐκ ἦν μία κίνησις), κἂν ἡ ὅλη κίνησις εἴη τοῦ ΑΒΓ μεγέθους. Again, since every motion is a motion of something, and the whole motion DZ is not the motion of either of the parts (for each of the parts DE, EZ is the motion of one of the parts AB, BG) or of anything else (for, the whole motion being the motion of a whole, the parts of the motion are the motions of the parts of that whole: and the parts of DZ are the motions of AB, BG and of nothing else: for, as we saw, a motion that is one cannot be the motion of more things than one): since this is so, the whole motion will be the motion of the magnitude ABG. ἔτι δ' εἰ ἔστιν ἄλλη τοῦ ὅλου κίνησις, οἷον ἐφ' ἧς ΘΙ, ἀφαιρεθήσεται ἀπ' αὐτῆς (235a.) ἡ ἑκατέρων τῶν μερῶν κίνησις· αὗται δ' ἴσαι ἔσονται ταῖς ΔΕ ΕΖ· μία γὰρ ἑνὸς κίνησις. ὥστ' εἰ μὲν ὅλη διαιρεθήσεται ἡ ΘΙ εἰς τὰς τῶν μερῶν κινήσεις, ἴση ἔσται ἡ ΘΙ τῇ ΔΖ· εἰ δ' ἀπολείπει τι, οἷον τὸ ΚΙ, αὕτη οὐδενὸς ἔσται κίνησις (οὔτε γὰρ τοῦ ὅλου οὔτε τῶν μερῶν διὰ τὸ μίαν εἶναι ἑνός, οὔτε ἄλλου οὐθενός· ἡ γὰρ συνεχὴς κίνησίς ἐστι συνεχῶν τινῶν), ὡσαύτως δὲ καὶ εἰ ὑπερβάλλει κατὰ τὴν διαίρεσιν· ὥστ' εἰ τοῦτο ἀδύνατον, ἀνάγκη τὴν αὐτὴν εἶναι καὶ ἴσην. αὕτη μὲν οὖν ἡ διαίρεσις κατὰ τὰς τῶν μερῶν κινήσεις ἐστίν, καὶ ἀνάγκη παντὸς εἶναι τοῦ μεριστοῦ αὐτήν· Again, if there is a motion of the whole other than DZ, say the the of each of the arts may be subtracted from it: and these motions will be equal to DE, EZ respectively: for the motion of that which is one must be one. So if the whole motion OI may be divided into the motions of the parts, OI will be equal to DZ: if on the other hand there is any remainder, say KI, this will be a motion of nothing: for it can be the motion neither of the whole nor of the parts (as the motion of that which is one must be one) nor of anything else: for a motion that is continuous must be the motion of things that are continuous. And the same result follows if the division of OI reveals a surplus on the side of the motions of the parts. Consequently, if this is impossible, the whole motion must be the same as and equal to DZ. This then is what is meant by the division of motion according to the motions of the parts: and it must be applicable to everything that is divisible into parts. ἄλλη δὲ κατὰ τὸν χρόνον· ἐπεὶ γὰρ ἅπασα κίνησις ἐν χρόνῳ, χρόνος δὲ πᾶς διαιρετός, ἐν δὲ τῷ ἐλάττονι ἐλάττων ἡ κίνησις, ἀνάγκη πᾶσαν κίνησιν διαιρεῖσθαι κατὰ τὸν χρόνον. Motion is also susceptible of another kind of division, that according to time. For since all motion is in time and all time is divisible, and in less time the motion is less, it follows that every motion must be divisible according to time. ἐπεὶ δὲ πᾶν τὸ (13) κινούμενον ἔν τινι κινεῖται καὶ χρόνον τινά, καὶ παντὸς ἔστι κίνησις, ἀνάγκη τὰς αὐτὰς εἶναι διαιρέσεις τοῦ τε χρόνου καὶ τῆς κινήσεως καὶ τοῦ κινεῖσθαι καὶ τοῦ κινουμένου καὶ ἐν ᾧ ἡ κίνησις (πλὴν οὐ πάντων ὁμοίως ἐν οἷς ἡ κίνησις, ἀλλὰ τοῦ μὲν τόπου καθ' αὑτό, τοῦ δὲ ποιοῦ κατὰ συμβεβηκός). And since everything that is in motion is in motion in a certain sphere and for a certain time and has a motion belonging to it, it follows that the time, the motion, the being-in-motion, the thing that is in motion, and the sphere of the motion must all be susceptible of the same divisions (though spheres of motion are not all divisible in a like manner: thus quantity is essentially, quality accidentally divisible). εἰλήφθω γὰρ ὁ χρόνος ἐν ᾧ κινεῖται ἐφ' ᾧ Α, καὶ ἡ κίνησις ἐφ' ᾧ Β. εἰ οὖν τὴν ὅλην ἐν τῷ παντὶ χρόνῳ κεκίνηται, ἐν τῷ ἡμίσει ἐλάττω, καὶ πάλιν τούτου διαιρεθέντος ἐλάττω ταύτης, καὶ ἀεὶ οὕτως. For suppose that A is the time occupied by the motion B. Then if all the time has been occupied by the whole motion, it will take less of the motion to occupy half the time, less again to occupy a further subdivision of the time, and so on to infinity. ἄλλη δὲ κατὰ τὸν χρόνον· ἐπεὶ γὰρ ἅπασα κίνησις ἐν χρόνῳ, χρόνος δὲ πᾶς διαιρετός, ἐν δὲ τῷ ἐλάττονι ἐλάττων ἡ κίνησις, Again, the time will be divisible similarly to the motion: for if the whole motion occupies all the time half the motion will occupy half the time, and less of the motion again will occupy less of the time. ἀνάγκη πᾶσαν κίνησιν διαιρεῖσθαι κατὰ τὸν χρόνον. ἐπεὶ δὲ πᾶν τὸ (13) κινούμενον ἔν τινι κινεῖται καὶ χρόνον τινά, καὶ παντὸς ἔστι κίνησις, ἀνάγκη τὰς αὐτὰς εἶναι διαιρέσεις τοῦ τε χρόνου καὶ τῆς κινήσεως καὶ τοῦ κινεῖσθαι καὶ τοῦ κινουμένου καὶ ἐν ᾧ ἡ κίνησις (πλὴν οὐ πάντων ὁμοίως ἐν οἷς ἡ κίνησις, ἀλλὰ τοῦ μὲν τόπου καθ' αὑτό, τοῦ δὲ ποιοῦ κατὰ συμβεβηκός). εἰλήφθω γὰρ ὁ χρόνος ἐν ᾧ κινεῖται ἐφ' ᾧ Α, καὶ ἡ κίνησις ἐφ' ᾧ Β. εἰ οὖν τὴν ὅλην ἐν τῷ παντὶ χρόνῳ κεκίνηται, ἐν τῷ ἡμίσει ἐλάττω, καὶ πάλιν τούτου διαιρεθέντος ἐλάττω ταύτης, καὶ ἀεὶ οὕτως. Motion is also susceptible of another kind of division, that according to time. For since all motion is in time and all time is divisible, and in less time the motion is less, it follows that every motion must be divisible according to time. And since everything that is in motion is in motion in a certain sphere and for a certain time and has a motion belonging to it, it follows that the time, the motion, the being-in-motion, the thing that is in motion, and the sphere of the motion must all be susceptible of the same divisions (though spheres of motion are not all divisible in a like manner: thus quantity is essentially, quality accidentally divisible). For suppose that A is the time occupied by the motion B. Then if all the time has been occupied by the whole motion, it will take less of the motion to occupy half the time, less again to occupy a further subdivision of the time, and so on to infinity. Again, the time will be divisible similarly to the motion: for if the whole motion occupies all the time half the motion will occupy half the time, and less of the motion again will occupy less of the time. ὁμοίως δὲ καί, εἰ ἡ κίνησις διαιρετή, καὶ ὁ χρόνος διαιρετός· εἰ γὰρ τὴν ὅλην ἐν τῷ παντί, τὴν ἡμίσειαν ἐν τῷ ἡμίσει, καὶ πάλιν τὴν ἐλάττω ἐν τῷ ἐλάττονι. τὸν αὐτὸν δὲ τρόπον καὶ τὸ κινεῖσθαι διαιρεθήσεται. In the same way the being-in-motion will also be divisible. For let G be the whole being-in-motion. Then the being-in-motion that corresponds to half the motion will be less than the whole being-in-motion, that which corresponds to a quarter of the motion will be less again, and so on to infinity. ἔστω γὰρ ἐφ' ᾧ Γ τὸ κινεῖσθαι. κατὰ δὴ τὴν ἡμίσειαν κίνησιν ἔλαττον ἔσται τοῦ ὅλου, καὶ πάλιν κατὰ τὴν τῆς ἡμισείας ἡμίσειαν, καὶ αἰεὶ οὕτως. ἔστι δὲ καὶ ἐκθέμενον τὸ καθ' ἑκατέραν τῶν κινήσεων κινεῖσθαι, οἷον κατά τε τὴν ΔΓ καὶ τὴν ΓΕ, λέγειν ὅτι τὸ ὅλον ἔσται κατὰ τὴν ὅλην (εἰ γὰρ ἄλλο, πλείω ἔσται κινεῖσθαι κατὰ τὴν αὐτὴν κίνησιν), ὥσπερ ἐδείξαμεν καὶ τὴν κίνησιν διαιρετὴν εἰς τὰς τῶν μερῶν κινήσεις οὖσαν· ληφθέντος γὰρ τοῦ κινεῖσθαι καθ' ἑκατέραν συνεχὲς ἔσται τὸ ὅλον. Moreover by setting out successively the being-in-motion corresponding to each of the two motions DG (say) and GE, we may argue that the whole being-in-motion will correspond to the whole motion (for if it were some other being-in-motion that corresponded to the whole motion, there would be more than one being-in motion corresponding to the same motion), the argument being the same as that whereby we showed that the motion of a thing is divisible into the motions of the parts of the thing: for if we take separately the being-in motion corresponding to each of the two motions, we shall see that the whole being-in motion is continuous. ὡσαύτως δὲ δειχθήσεται καὶ τὸ μῆκος διαιρετόν, καὶ ὅλως πᾶν ἐν ᾧ ἐστιν ἡ μεταβολή (πλὴν ἔνια κατὰ συμβεβηκός, ὅτι τὸ μεταβάλλον ἐστὶν διαιρετόν)· ἑνὸς γὰρ διαιρουμένου πάντα διαιρεθήσεται. The same reasoning will show the divisibility of the length, and in fact of everything that forms a sphere of change (though some of these are only accidentally divisible because that which changes is so): for the division of one term will involve the division of all. καὶ ἐπὶ τοῦ (235b.) πεπερασμένα εἶναι ἢ ἄπειρα ὁμοίως ἕξει κατὰ πάντων. So, too, in the matter of their being finite or infinite, they will all alike be either the one or the other. ἠκολούθηκεν δὲ μάλιστα τὸ διαιρεῖσθαι πάντα καὶ ἄπειρα εἶναι ἀπὸ τοῦ μεταβάλλοντος· εὐθὺς γὰρ ἐνυπάρχει τῷ μεταβάλλοντι τὸ διαιρετὸν καὶ τὸ ἄπειρον. τὸ μὲν οὖν διαιρετὸν δέδεικται πρότερον, τὸ δ' ἄπειρον ἐν τοῖς ἑπομένοις ἔσται δῆλον. And we now see that in most cases the fact that all the terms are divisible or infinite is a direct consequence of the fact that the thing that changes is divisible or infinite: for the attributes 'divisible' and 'infinite' belong in the first instance to the thing that changes. That divisibility does so we have already shown: that infinity does so will be made clear in what follows.
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. 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. 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. 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. 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. 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. 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. 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. 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. 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. Circa primum duo facit: primo ponit modos quibus motus dividitur; secundo exponit eos, ibi: sit igitur ipsius quidem ab et cetera. 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. 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. 807. He says therefore first (612) 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, 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. 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. 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. 808. Then at (613) 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. 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. 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. 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) 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. 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. 810. At (615) 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. 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. 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. Haec igitur divisio est secundum motus partium; et necesse est quod talis partitio inveniatur in motu, propter hoc quod omne quod movetur est partibile. 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. 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) 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. 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. 812. Then at (617) 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. Circa primum duo facit: primo proponit quod intendit; secundo manifestat propositum, ibi: accipiatur enim tempus et cetera. About the first he does two things: First he states his proposition; Secondly, he explains the proposition, at 813. 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. He says therefore first (617) 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. 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. 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”. 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. 813. Then at (618) 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, 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. About the first he does two things: First he shows that with division of time, motion is divided; Secondly, vice versa, at 814. 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. He says therefore first (618): 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. 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. Then at (619) 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. 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. 814. Then at (620) 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. 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. He says therefore first (620) 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”. 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. Then at (621) 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. 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) 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. 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) 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. 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. 817. Then at (624) 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. 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. 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.
The time in which something is first changed is indivisible.
How a first may, and may not, be taken in motion
Chapter 5 Ἐπεὶ δὲ πᾶν τὸ μεταβάλλον ἔκ τινος εἴς τι μεταβάλλει, ἀνάγκη τὸ μεταβεβληκός, ὅτε πρῶτον μεταβέβληκεν, εἶναι ἐν ᾧ μεταβέβληκεν. Since everything that changes changes from something to something, that which has changed must at the moment when it has first changed be in that to which it has changed. τὸ γὰρ μεταβάλλον, ἐξ οὗ μεταβάλλει, ἐξίσταται ἢ ἀπολείπει αὐτό, καὶ ἤτοι ταὐτόν ἐστι τὸ μεταβάλλειν καὶ τὸ ἀπολείπειν, ἢ ἀκολουθεῖ τῷ μετα βάλλειν τὸ ἀπολείπειν. εἰ δὲ τῷ μεταβάλλειν τὸ ἀπολείπειν, τῷ μεταβεβληκέναι τὸ ἀπολελοιπέναι· ὁμοίως γὰρ ἑκάτερον ἔχει πρὸς ἑκάτερον. ἐπεὶ οὖν μία τῶν μεταβολῶν ἡ κατ' ἀντίφασιν, ὅτε μεταβέβληκεν ἐκ τοῦ μὴ ὄντος εἰς τὸ ὄν, ἀπολέλοιπεν τὸ μὴ ὄν. ἔσται ἄρα ἐν τῷ ὄντι· πᾶν γὰρ ἀνάγκη ἢ εἶναι ἢ μὴ εἶναι. φανερὸν οὖν ὅτι ἐν τῇ κατ' ἀντίφασιν μεταβολῇ τὸ μεταβεβληκὸς ἔσται ἐν ᾧ μεταβέβληκεν. εἰ δ' ἐν ταύτῃ, καὶ ἐν ταῖς ἄλλαις· ὁμοίως γὰρ ἐπὶ μιᾶς καὶ τῶν ἄλλων. For that which changes retires from or leaves that from which it changes: and leaving, if not identical with changing, is at any rate a consequence of it. And if leaving is a consequence of changing, having left is a consequence of having changed: for there is a like relation between the two in each case. One kind of change, then, being change in a relation of contradiction, where a thing has changed from not-being to being it has left not-being. Therefore it will be in being: for everything must either be or not be. It is evident, then, that in contradictory change that which has changed must be in that to which it has changed. And if this is true in this kind of change, it will be true in all other kinds as well: for in this matter what holds good in the case of one will hold good likewise in the case of the rest. ἔτι δὲ καὶ καθ' ἑκάστην λαμβάνουσι φανερόν, εἴπερ ἀνάγκη τὸ μεταβεβληκὸς εἶναί που ἢ ἔν τινι. ἐπεὶ γὰρ ἐξ οὗ μεταβέβληκεν ἀπολέλοιπεν, ἀνάγκη δ' εἶναί που, ἢ ἐν τούτῳ ἢ ἐν ἄλλῳ ἔσται. εἰ μὲν οὖν ἐν ἄλλῳ, οἷον ἐν τῷ Γ, τὸ εἰς τὸ Β μεταβεβληκός, πάλιν ἐκ τοῦ Γ μεταβάλλει εἰς τὸ Β· οὐ γὰρ ἦν ἐχόμενον τὸ Β, ἡ δὲ μεταβολὴ συνεχής. ὥστε τὸ μεταβεβληκός, ὅτε μεταβέβληκεν, μεταβάλλει εἰς ὃ μεταβέβληκεν. τοῦτο δ' ἀδύνατον· ἀνάγκη ἄρα τὸ μεταβεβληκὸς εἶναι ἐν τούτῳ εἰς ὃ μεταβέβληκεν. φανερὸν οὖν ὅτι καὶ τὸ γεγονός, ὅτε γέγονεν, ἔσται, καὶ τὸ ἐφθαρμένον οὐκ ἔσται· καθόλου τε γὰρ εἴρηται περὶ πάσης μεταβολῆς, καὶ μάλιστα δῆλον ἐν τῇ κατ' ἀντίφασιν. ὅτι μὲν τοίνυν τὸ μεταβεβληκός, ὅτε μεταβέβληκε πρῶτον, ἐν ἐκείνῳ ἐστίν, δῆλον· Moreover, if we take each kind of change separately, the truth of our conclusion will be equally evident, on the ground that that that which has changed must be somewhere or in something. For, since it has left that from which it has changed and must be somewhere, it must be either in that to which it has changed or in something else. If, then, that which has changed to B is in something other than B, say G, it must again be changing from G to B: for it cannot be assumed that there is no interval between G and B, since change is continuous. Thus we have the result that the thing that has changed, at the moment when it has changed, is changing to that to which it has changed, which is impossible: that which has changed, therefore, must be in that to which it has changed. So it is evident likewise that that that which has come to be, at the moment when it has come to be, will be, and that which has ceased to be will not-be: for what we have said applies universally to every kind of change, and its truth is most obvious in the case of contradictory change. It is clear, then, that that which has changed, at the moment when it has first changed, is in that to which it has changed. ἐν ᾧ δὲ πρώτῳ μεταβέβληκεν τὸ μεταβεβληκός, ἀνάγκη ἄτομον εἶναι. λέγω δὲ πρῶτον ὃ μὴ τῷ ἕτερόν τι αὐτοῦ εἶναι τοιοῦτόν ἐστιν. ἔστω γὰρ διαιρετὸν τὸ ΑΓ, καὶ διῃρήσθω κατὰ τὸ Β. εἰ μὲν οὖν ἐν τῷ ΑΒ μετα βέβληκεν ἢ πάλιν ἐν τῷ ΒΓ, οὐκ ἂν ἐν πρώτῳ τῷ ΑΓ με ταβεβληκὸς εἴη. εἰ δ' ἐν ἑκατέρῳ μετέβαλλεν (ἀνάγκη γὰρ (236a.) ἢ μεταβεβληκέναι ἢ μεταβάλλειν ἐν ἑκατέρῳ), κἂν ἐν τῷ ὅλῳ μεταβάλλοι· ἀλλ' ἦν μεταβεβληκός. ὁ αὐτὸς δὲ λόγος καὶ εἰ ἐν τῷ μὲν μεταβάλλει, ἐν δὲ τῷ μεταβέβληκεν· ἔσται γάρ τι τοῦ πρώτου πρότερον· ὥστ' οὐκ ἂν εἴη διαιρετὸν ἐν ᾧ μεταβέβληκεν. φανερὸν οὖν ὅτι καὶ τὸ ἐφθαρμένον καὶ τὸ γεγονὸς ἐν ἀτόμῳ τὸ μὲν ἔφθαρται τὸ δὲ γέγονεν. We will now show that the 'primary when' in which that which has changed effected the completion of its change must be indivisible, where by 'primary' I mean possessing the characteristics in question of itself and not in virtue of the possession of them by something else belonging to it. For let AG be divisible, and let it be divided at B. If then the completion of change has been effected in AB or again in BG, AG cannot be the primary thing in which the completion of change has been effected. If, on the other hand, it has been changing in both AB and BG (for it must either have changed or be changing in each of them), it must have been changing in the whole AG: but our assumption was that AG contains only the completion of the change. It is equally impossible to suppose that one part of AG contains the process and the other the completion of the change: for then we shall have something prior to what is primary. So that in which the completion of change has been effected must be indivisible. It is also evident, therefore, that that that in which that which has ceased to be has ceased to be and that in which that which has come to be has come to be are indivisible. λέγεται δὲ τὸ ἐν ᾧ πρώτῳ μεταβέβληκε διχῶς, τὸ μὲν ἐν ᾧ πρώτῳ ἐπετελέσθη ἡ μεταβολή (τότε γὰρ ἀληθὲς εἰπεῖν ὅτι μεταβέβληκεν), τὸ δ' ἐν ᾧ πρώτῳ ἤρξατο μεταβάλλειν. τὸ μὲν οὖν κατὰ τὸ τέλος τῆς μεταβολῆς πρῶτον λεγόμενον ὑπάρχει τε καὶ ἔστιν (ἐνδέχεται γὰρ ἐπιτελεσθῆναι μεταβολὴν καὶ ἔστι μεταβολῆς τέλος, ὃ δὴ καὶ δέδεικται ἀδιαίρετον ὂν διὰ τὸ πέρας εἶναι)· τὸ δὲ κατὰ τὴν ἀρχὴν ὅλως οὐκ ἔστιν· οὐ γὰρ ἔστιν ἀρχὴ μεταβολῆς, οὐδ' ἐν ᾧ πρώτῳ τοῦ χρόνου μετέβαλλεν. But there are two senses of the expression 'the primary when in which something has changed'. On the one hand it may mean the primary when containing the completion of the process of change— the moment when it is correct to say 'it has changed': on the other hand it may mean the primary when containing the beginning of the process of change. Now the primary when that has reference to the end of the change is something really existent: for a change may really be completed, and there is such a thing as an end of change, which we have in fact shown to be indivisible because it is a limit. But that which has reference to the beginning is not existent at all: for there is no such thing as a beginning of a process of change, and the time occupied by the change does not contain any primary when in which the change began. ἔστω γὰρ πρῶτον ἐφ' ᾧ τὸ ΑΔ. τοῦτο δὴ ἀδιαίρετον μὲν οὐκ ἔστιν· συμβήσεται γὰρ ἐχόμενα εἶναι τὰ νῦν. ἔτι δ' εἰ ἐν τῷ ΓΑ χρόνῳ παντὶ ἠρεμεῖ (κείσθω γὰρ ἠρεμοῦν), καὶ ἐν τῷ Α ἠρεμεῖ, ὥστ' εἰ ἀμερές ἐστι τὸ ΑΔ, ἅμα ἠρεμήσει καὶ μεταβεβληκὸς ἔσται· ἐν μὲν γὰρ τῷ Α ἠρεμεῖ, ἐν δὲ τῷ Δ μεταβέβληκεν. ἐπεὶ δ' οὐκ ἔστιν ἀμερές, ἀνάγκη διαιρετὸν εἶναι καὶ ἐν ὁτῳοῦν τῶν τούτου μεταβεβληκέναι· διαιρεθέντος γὰρ τοῦ ΑΔ, εἰ μὲν ἐν μηδετέρῳ μεταβέβληκεν, οὐδ' ἐν τῷ ὅλῳ· εἰ δ' ἐν ἀμφοῖν μεταβάλλει καὶ ἐν τῷ παντί, εἴτ' ἐν θατέρῳ μεταβέβληκεν, οὐκ ἐν τῷ ὅλῳ πρώτῳ. ὥστε ἀνάγκη ἐν ὁτῳοῦν μεταβεβλη κέναι. φανερὸν τοίνυν ὅτι οὐκ ἔστιν ἐν ᾧ πρώτῳ μεταβέβληκεν· ἄπειροι γὰρ αἱ διαιρέσεις. For suppose that AD is such a primary when. Then it cannot be indivisible: for, if it were, the moment immediately preceding the change and the moment in which the change begins would be consecutive (and moments cannot be consecutive). Again, if the changing thing is at rest in the whole preceding time GA (for we may suppose that it is at rest), it is at rest in A also: so if AD is without parts, it will simultaneously be at rest and have changed: for it is at rest in A and has changed in D. Since then AD is not without parts, it must be divisible, and the changing thing must have changed in every part of it (for if it has changed in neither of the two parts into which AD is divided, it has not changed in the whole either: if, on the other hand, it is in process of change in both parts, it is likewise in process of change in the whole: and if, again, it has changed in one of the two parts, the whole is not the primary when in which it has changed: it must therefore have changed in every part). It is evident, then, that with reference to the beginning of change there is no primary when in which change has been effected: for the divisions are infinite. οὐδὲ δὴ τοῦ μεταβεβληκότος ἔστιν τι πρῶτον ὃ μεταβέβληκεν. ἔστω γὰρ τὸ ΔΖ πρῶτον μεταβεβληκὸς τοῦ ΔΕ· πᾶν γὰρ δέδεικται διαιρετὸν τὸ μεταβάλλον. ὁ δὲ χρόνος ἐν ᾧ τὸ ΔΖ μεταβέβληκεν ἔστω ἐφ' ᾧ ΘΙ. εἰ οὖν ἐν τῷ παντὶ τὸ ΔΖ μεταβέβληκεν, ἐν τῷ ἡμίσει ἔλαττον ἔσται τι μεταβεβληκὸς καὶ πρότερον τοῦ ΔΖ, καὶ πάλιν τούτου ἄλλο, κἀκείνου ἕτερον, καὶ αἰεὶ οὕτως. ὥστ' οὐθὲν ἔσται πρῶτον τοῦ μεταβάλλοντος ὃ μεταβέβληκεν. ὅτι μὲν οὖν οὔτε τοῦ μεταβάλλοντος οὔτ' ἐν ᾧ μεταβάλλει χρόνῳ πρῶτον οὐθέν ἐστιν, φανερὸν ἐκ τῶν εἰρημένων· So, too, of that which has changed there is no primary part that has changed. For suppose that of AE the primary part that has changed is Az (everything that changes having been shown to be divisible): and let OI be the time in which DZ has changed. If, then, in the whole time DZ has changed, in half the time there will be a part that has changed, less than and therefore prior to DZ: and again there will be another part prior to this, and yet another, and so on to infinity. Thus of that which changes there cannot be any primary part that has changed. It is evident, then, from what has been said, that neither of that which changes nor of the time in which it changes is there any primary part. (236b.) αὐτὸ δὲ ὃ μεταβάλλει ἢ καθ' ὃ μεταβάλλει, οὐκέθ' ὁμοίως ἕξει. τρία γάρ ἐστιν ἃ λέγεται κατὰ τὴν μεταβολήν, τό τε μεταβάλλον καὶ ἐν ᾧ καὶ εἰς ὃ μεταβάλλει, οἷον ὁ ἄνθρωπος καὶ ὁ χρόνος καὶ τὸ λευκόν. ὁ μὲν οὖν ἄνθρωπος καὶ ὁ χρόνος διαιρετοί, περὶ δὲ τοῦ λευκοῦ ἄλλος λόγος. πλὴν κατὰ συμβεβηκός γε πάντα διαιρετά· ᾧ γὰρ συμβέβηκεν τὸ λευκὸν ἢ τὸ ποιόν, ἐκεῖνο διαιρετόν ἐστιν· ἐπεὶ ὅσα γε καθ' αὑτὰ λέγεται διαιρετὰ καὶ μὴ κατὰ συμβεβηκός, οὐδ' ἐν τούτοις ἔσται τὸ πρῶτον, οἷον ἐν τοῖς μεγέθεσιν. ἔστω γὰρ τὸ ἐφ' ᾧ ΑΒ μέγεθος, κεκινήσθω δ' ἐκ τοῦ Β εἰς τὸ Γ πρῶτον. οὐκοῦν εἰ μὲν ἀδιαίρετον ἔσται τὸ ΒΓ, ἀμερὲς ἀμεροῦς ἔσται ἐχόμενον· εἰ δὲ διαιρετόν, ἔσται τι τοῦ Γ πρότερον, εἰς ὃ μεταβέβληκεν, κἀκείνου πάλιν ἄλλο, καὶ ἀεὶ οὕτως διὰ τὸ μηδέποτε ὑπολείπειν τὴν διαίρεσιν. ὥστ' οὐκ ἔσται πρῶτον εἰς ὃ μεταβέβλη κεν. ὁμοίως δὲ καὶ ἐπὶ τῆς τοῦ ποσοῦ μεταβολῆς· καὶ γὰρ αὕτη ἐν συνεχεῖ ἐστιν. φανερὸν οὖν ὅτι ἐν μόνῃ τῶν κινήσεων τῇ κατὰ τὸ ποιὸν ἐνδέχεται ἀδιαίρετον καθ' αὑτὸ εἶναι. With regard, however, to the actual subject of change—that is to say that in respect of which a thing changes—there is a difference to be observed. For in a process of change we may distinguish three terms—that which changes, that in which it changes, and the actual subject of change, e.g. the man, the time, and the fair complexion. Of these the man and the time are divisible: but with the fair complexion it is otherwise (though they are all divisible accidentally, for that in which the fair complexion or any other quality is an accident is divisible). For of actual subjects of change it will be seen that those which are classed as essentially, not accidentally, divisible have no primary part. Take the case of magnitudes: let AB be a magnitude, and suppose that it has moved from B to a primary 'where' G. Then if BG is taken to be indivisible, two things without parts will have to be contiguous (which is impossible): if on the other hand it is taken to be divisible, there will be something prior to G to which the magnitude has changed, and something else again prior to that, and so on to infinity, because the process of division may be continued without end. Thus there can be no primary 'where' to which a thing has changed. And if we take the case of quantitative change, we shall get a like result, for here too the change is in something continuous. It is evident, then, that only in qualitative motion can there be anything essentially indivisible.
Postquam philosophus ostendit qualiter dividatur motus, hic determinat de ordine partium motus. Et primo inquirit an sit primum in motu; secundo ostendit quomodo ea quae sunt in motu, praecedunt se invicem, ibi: quoniam autem omne quod mutatur, in tempore mutatur et cetera. 818, After explaining how motion is divided, the Philosopher now discusses the order of the parts of motion. First he asks whether there is a first in motion; Secondly, he shows how the factors involved in motion precede one another, in L. 8. Circa primum duo facit: primo ostendit quod id in quo primum mutatum est, est indivisibile; secundo ostendit quomodo in motu possit inveniri primum, et quomodo non possit, ibi: dicitur autem in quo primo mutatum est et cetera. About the first he does two things: First he shows that that into which something is first changed is indivisible; Secondly, how in motion a first can and cannot be found, 822. Circa primum duo facit: primo praemittit quoddam quod est necessarium ad propositi ostensionem; secundo ostendit propositum, ibi: in quo autem primo mutatum est et cetera. About the first he does two things: First he mentions facts to be used in explaining the proposition; Secondly, he proves the proposition, at 821. Circa primum duo facit: primo proponit quod intendit; secundo probat propositum, ibi: quod mutatur enim et cetera. About the first he does two things: First he mentions his proposition; Secondly, he proves it, at 819. Dicit ergo primo, quod quia omne quod mutatur, mutatur de uno termino in alium; necesse est omne quod mutatur, quando iam mutatum est, esse in termino ad quem. 819. He says therefore first (625) that because whatever is being changed is being changed from one term to the other, then when the subject of change has now been changed, it has to be in the terminus ad quem. Deinde cum dicit: quod mutatur enim etc., probat propositum duabus rationibus; quarum prima est particularis, secunda universalis. Then at (626) he proves this proposition with two arguments, the first of which is particular and the second universal. Prima ratio talis est. Omne quod mutatur, oportet quod aut distet a termino a quo mutatur, sicut patet in motu locali, in quo locus a quo mutatur remanet, et mobile per motum fit distans ab eo; aut oportet quod ipse terminus a quo deficiat, sicut est in motu alterationis: cum enim ex albo fit nigrum, ipsa albedo deficit. The first argument is this: Everything being changed must either (1) be distant from the term at which the change starts, as is evident in local motion, in which the place from which the motion starts remains and the mobile gets to be distant from it; or (2) the terminus a quo must cease to be, as in the motion called alteration: for when something white becomes black, the whiteness ceases to be. Et ad huius propositionis manifestationem subiungit, quod vel mutari est idem quod deficere; vel ad hoc quod est mutari sequitur ipsum deficere, et ad hoc quod est mutatum esse sequitur defecisse, scilicet a termino a quo. Manifestum est autem quod sunt idem subiecto, sed differunt ratione. Nam deficere dicitur per respectum ad terminum a quo, mutatio autem magis denominatur a termino ad quem. Et ad manifestationem eius quod dixerat, subdit quod similiter utrumque se habet ad utrumque, idest sicut se habet deficere ad mutari, ita defecisse ad mutatum esse. In order to explain this proposition he adds that either the process of being changed is the same as departing, or the latter is a consequence of change and, therefore, “to have departed” (from the terminus a quo) is a consequence of having been changed. But it is evident that they are the same in reality but different in conception. For “departing” is spoken of in relation to the terminus a quo, whereas “change” gets its name from the terminus ad quem. And in explanation of this, Aristotle adds that “both are related to both in a similar way”, i.e., as “departing” is related to “being changed”, so “having departed” is related to “having been changed”. Ex praemissis autem argumentatur ad propositum ostendendum in una specie mutationis, quae scilicet est inter contradictorie opposita, scilicet inter esse et non esse, ut patet in generatione et corruptione. Patet enim ex praemissis, quod omne quod mutatur deficit a termino a quo, et quod mutatum est iam defecit. Quando ergo aliquid mutatum est a non esse in esse, iam defecit a non esse; sed de quolibet verum est dicere, quod aut est aut non est: quod ergo mutatum est de non esse in esse, quando mutatum est, est in esse: et similiter quod mutatum est de esse in non esse, oportet quod sit in non esse. Manifestum est ergo quod in mutatione quae est secundum contradictionem, quod mutatum est, est in eo ad quod mutatum est. Et si est verum in ista mutatione, pari ratione est verum in aliis mutationibus. Ex quo patet id quod primo propositum est. From these premises he argues to the conclusion, using as his example the species of change that involves terms contradictorily opposed, where the transition is between being and non-being, as in generation and ceasing-to-be. For it is evident from the foregoing that whatever is being changed departs from the terminus a quo and that whatever has been changed has already departed. When, therefore, something has been changed from non-being to being, it has already departed from non-being. But of anything at all it is true to say that it either is or is not. Therefore, what has been changed from non-being to being is in being, when the change is over. Likewise, what has been changed from being to non-being must be in non-being. Therefore, it is evident that in the change which involves contradictories, the thing which has been changed exists in that into which it has been changed. And if it is true in that type of change, then for an equal reason it is true in other changes. From this the first proposition is clear. Secundam rationem generalem ponit ibi: amplius autem etc.: et dicit quod hoc idem potest esse manifestum considerando secundum unamquamque mutationem. Et manifestat in mutatione locali. Omne enim quod mutatum est, necesse est esse alicubi, vel in termino a quo vel in aliquo alio. Sed quia illud quod mutatum est, iam defecit ab eo ex quo mutatum est, necesse est quod sit alibi. Aut igitur necesse est quod sit in hoc de quo intendimus, scilicet in termino ad quem, aut in alio. Et si est in hoc, habetur propositum: si autem in alio, ponamus quod aliquid moveatur in b, et quando mutatum est non sit in b sed in c. Tunc oportebit dicere quod etiam de c mutetur in b; quia c et b non sunt habita, idest consequenter se habentia. Oportet enim quod tota huiusmodi mutatio sit continua; et in continuis unum signum non est consequenter se habens ad alterum, quia necesse est quod cadat in medio aliquid sui generis, ut supra probatum est. Unde sequetur, si illud quod mutatum est, quando mutatum est, sit in c, et de c mutetur in b, quod est terminus ad quem, quod quando mutatum est, tunc mutatur in quod mutatum est; quod est impossibile. Non enim simul est mutari et mutatum esse, ut supra dictum est. Nihil autem differt si huiusmodi termini c et b accipiantur in motu locali, vel in quacumque alia mutatione. Necesse est ergo universaliter verum esse, quod id quod mutatum est, quando mutatum est, est in hoc ad quod mutatum est, idest in termino ad quem. Et ex hoc ulterius concludit, quod illud quod factum est, quando factum est, habet esse; et quod corruptum est, quando corruptum est, est non ens. Ostensum est enim universaliter hoc de omni mutatione, et maxime manifestum est in mutatione, quae est secundum contradictionem, ut ex dictis patet. Sic igitur manifestum est, quod id quod mutatum est, cum primo mutatum est, est in illo ad quod mutatum est. 820. Then at (627) the second argument, a general one, is given, And he says that the same conclusion can be proved by considering any change at all. And he picks local motion; Whatever has been changed must be somewhere, i.e., either in the terminus a quo or in some other. But since what has been changed has already departed from that from which it has been changed, it must be elsewhere. Therefore, it must be either in that in which we are trying to prove it is, i.e., in the terminus ad quem or elsewhere. If it is in the former, our point is proved; if not, then let us suppose that something is being moved into B and when the change is finished the thing is not in B but in C. Then we must say that from C it is also changed into B, because B and C are not consecutive. For a change of the type under discussion is continuous, and in continua one part is not consecutive to another, because between two parts there occurs a part that is similar to those two, as was proved above. Hence, it will follow, if that which has been changed is in C when it has been changed and from C it is being changed to B (which is the terminus ad quem), that when it has been changed, it is also being changed into what it has already become—which is impossible, For “being changed” and “having been changed” are never simultaneous, as we have shown above. Now it makes no difference whether the termini C and B are applied to local motion or to any other change. Consequently, it is universally true that what has been changed is (when it has been changed) in that into which it has been changed, i.e., in the terminus ad quem. Addit autem primo; quia postquam mutatum est ad aliquid, posset exinde moveri, et ibi non esset; sed quando primo mutatum est, oportet quod sit ibi. From this he further concludes that what has been changed is, as soon as it has been changed, in that into which it has been changed. He added “as soon as”, because after it has been changed into something, it could depart from it and not be there; but as soon as it has been changed, it must be there. Deinde cum dicit: in quo autem primo mutatum est etc., ostendit quod mutatum esse primo et per se est in indivisibili: et dicit quod illud tempus in quo primo mutatum est quod mutatum est, necesse est quod sit atomum, idest indivisibile. Quare autem addit primo, exponit subdens quod in illo primo dicitur aliquid mutatum esse, in quo non dicitur esse mutatum ratione alicuius suae partis: sicut si dicatur aliquod mobile mutatum esse in die, quia mutatum est in aliqua parte illius diei; non enim primo mutatur in die. 821. Then at (628) he shows that “to have been changed” is first and per se in an indivisible; and he says that that time in which what has been changed was first changed must be indivisible. ‘Why he adds “first” he explains by saying that A is said to have been first changed as soon as it is not said to have been changed merely by reason of any of its parts. For example, if we say that a mobile has been changed in a day, because it was changed in some part of the day. in that case it was not first changed in the day. Quod autem illud temporis in quo primo mutatum est sit indivisibile, sic probat. Si enim sit divisibile, sit ac, et dividatur secundum b: necesse est dicere quod aut in utraque mutatum sit, aut in utraque parte mutetur, aut in una parte mutetur et in alia sit mutatum. Sed si in utraque parte mutatum est, non primo mutatum est in toto, sed in parte. Si vero detur quod transmutetur in utraque parte, oportebit dicere quod transmutetur in toto: sic enim dicitur aliquid in toto tempore mutari, quia mutatur in qualibet eius parte. Hoc autem est contra positum: positum enim erat quod in toto ac erat mutatum. But that the time in which something has been first changed is indivisible he now proves: If the said time were divisible, let it be AC and let it be divided at B. Now three things are possible: either (1) the change is over in each part or (2) it is going on in each part or (3) in one part it is going on and in the other it is over. Now, if in each part it is over, then it was first completely changed not in the whole but in the part; but if it is being changed in each part, then it is also being changed in the whole (for the reason why something is said to be changing in a whole period of time is that the change was going on during each part of the whole time). But this is against our assumption that in the whole of AC it had been changed. Si autem detur quod in una parte mutetur et in alia sit mutatum, sequitur idem inconveniens, scilicet quod non sit primo mutatum in toto; quia cum pars sit prior toto, et prius mutetur aliquid in parte temporis quam in toto, sequetur quod sit aliquid prius primo, quod est impossibile. Oportet ergo dicere quod illud temporis in quo primo aliquid mutatum est, sit indivisibile. On the other hand, if it be supposed that in one part of the time it is being changed and in the other part it has been changed, the same difficulty ensues; namely, that it was not first changed in the whole time, because since the part is prior to the whole and something is in motion in a part of time before it is moved in the entire time, it follows that there was something prior to the first, which is impossible. Consequently, it must be admitted that the time in which the thing was first completely changed is indivisible, Ex hoc autem ulterius concludit, quod omne quod corruptum est, et omne quod factum est, est in indivisibili temporis factum et corruptum; quia generatio et corruptio sunt termini alterationis. Unde si quilibet motus terminatur in instanti (idem est enim primo mutatum esse, quod terminari motum), sequitur quod generatio et corruptio sint in instanti. From this he further concludes that everything that has ceased to be and everything that has been completely made, was made and ceased to be in an indivisible of time, because generation and ceasing to be are the termini of alteration. Consequently, if a motion is terminated in an instant (for these two things are the same, i.e., the termination of a motion and to have been first changed), it follows that generation and ceasing-to-be occur in an instant. Deinde cum dicit: dicitur autem in quo primo etc., ostendit quomodo in motu possit accipi primum. Et circa hoc duo facit: primo proponit veritatem; secundo probat, ibi: sit enim primum et cetera. 822. Then at (629) he shows how to discern in a motion, that which is first. About this he does two things: First he proposes the truth; Secondly, he proves it at 823. Dicit ergo primo, quod hoc quod dicitur in quo primo mutatum est aliquid, potest intelligi dupliciter. Uno modo in quo primo mutatio est perfecta vel terminata: tunc enim verum est dicere quod mutatum est, quando iam mutatio est perfecta. Alio modo potest intelligi in quo primo mutatum est, idest in quo primo incepit mutari, non in quo primo fuit verum dicere quod iam mutatum esset. He says therefore first (629) that the expression “in which something has been first changed” has two interpretations: first, it can mean that in which the change is first complete or terminated —in which case it is true to say that something has been changed, when the change is now over. Secondly, it can mean that in which it first began to be changed, and not that in which it was first true to say that it has been changed. Primo igitur modo accipiendo, scilicet secundum terminationem mutationis, dicitur in motu, et est in eo quod primo mutatum est. Contingit enim aliquando primo terminari mutationem, quia cuiuslibet mutationis est aliquis terminus. Et hoc modo intelleximus quod primo mutatum est esse indivisibile; et ostensum est hoc hac ratione: quia est finis, idest terminus motus; omnis autem terminus continui indivisibilis est. Taken in the first sense, namely, according to the termination of the change, it is applied to instances of motion in which there exists a first in which something has been changed. For a change can be first terminated some time, because every change has a termination. It was in this sense that we understood that “that in which something was first changed” is an indivisible—which was proved on the ground that it is the end, i.e., the terminus, of the motion—and we know that every terminus of a continuum is an indivisible. Sed si accipiatur quod primo mutatum est secundo modo dicendi, scilicet secundum principium, idest secundum primam partem motus, sic non est in quo primo mutatum est. Non enim est accipere aliquod principium mutationis, idest aliquam primam partem mutationis, quam non praecedat alia pars. Similiter etiam non est accipere aliquid primum in tempore, in quo primo mutetur. But if it is taken in the second sense, namely, according to the beginning of the change, i.e., according to the first part of the motion, then there is no first in which something has been changed. For no beginning of a change can be definitely pointed out, i.e., no part that is not preceded by some other part. In like manner, it is not possible to isolate a first time in which something is first being moved. Deinde cum dicit: sit enim primum etc., probat quod non est accipere primum in quo mutatum est, ex parte principii. Et primo ratione accepta ex parte temporis; secundo ex parte mobilis, ibi: neque igitur in eo quod mutatum est etc.; tertio ex parte rei in qua est motus, ibi: ipsum autem quod mutatur et cetera. 823. Then at (630) he proves that if one looks at the beginning of a motion, it is not possible to assign “a first in which something has been changed”. First with an argument from time; Secondly, with an argument from the mobile, at 824; Thirdly, with an argument from the sphere in which the motion occurs, at 825. Circa primum ponit talem rationem. Si est aliquod temporis in quo primo mutatum est, sit illud ad. Hoc igitur aut est divisibile aut indivisibile. Si est indivisibile, sequuntur duo inconvenientia: quorum primum est, quod ipsa nunc in tempore sint habita, idest consequentia. Quod quidem inconveniens hac ratione sequitur, quia tempus dividitur sicut et motus, ut supra ostensum est. Si autem aliqua pars motus fuerit in ad, necesse est dicere quod ad sit aliqua pars temporis; et ita tempus erit compositum ex indivisibilibus. Indivisibile autem temporis est ipsum nunc: sequetur ergo quod ipsa nunc consequenter se habeant in tempore. As to the first he gives this reason: If there is any element of time in which something has been first changed, let it be AD. Now AD must be either divisible or indivisible. If the latter, two difficulties ensue. The first is that the “now’s” in time are consecutive. This difficulty follows from the fact that time is divided just like motion, as was shown above. But if any part of the motion was present in AD, then AD must have been a part of time and, consequently, time will be composed of indivisibles. However, the indivisibles of time are the “now’s”. It will follow, therefore, that the “now’s” are consecutive in time. Secundum inconveniens est. Ponamus enim quod in tempore quod praecedit ipsum ad, quod est ca, idem mobile quod ponebatur moveri in ad, totaliter quiescat. Si ergo in toto ca quiescit, sequitur quod quiescat in a, quod est aliquid eius. Si ergo ad est indivisibile, ut datum est, sequetur quod simul aliquid quiescat et moveatur: conclusum est enim quod quiescit in a, et positum erat quod in ad moveretur. Idem autem est a et ad, si ad sit indivisibile. Sequetur ergo quod in eodem quiescat et moveatur. And there is a second difficulty. Let us suppose that in the time CA, which preceded AD, the same mobile that was being moved in time AD was entirely at rest. If, therefore, it was at rest in the entire time CA, it was at rest in A, which is an element of the time CA. If, therefore, (as we supposed) AD is indivisible, it’ will follow that a thing is at rest and in motion at the same time; for we have already concluded that it was at rest in A and assumed that it was in motion in AD. But if AD is indivisible, then A is the same as AD. It will follow, therefore, that a thing is at rest and in motion in the same time. Sed advertendum est, quod non sequitur si aliquid quiescit in toto tempore, quod quiescat in ultimo eius indivisibili: quia ostensum est supra, quod in nunc neque movetur aliquid neque quiescit. Sed Aristoteles hoc concludit hic ex hoc quod ponitur ab adversario: quod id temporis in quo primo movetur, est indivisibile. Et si contingit moveri in indivisibili temporis, contingit eadem ratione in indivisibili temporis quiescere. It should be noted, however, that if a thing was at rest throughout an entire time, it does not follow that it was at rest in the last indivisible of that time; for we have already shown that in the “now” things are neither at rest nor in motion. But Aristotle concludes this here by arguing from what his adversary has proposed, namely, that the element of time in which the object was first being moved is an indivisible. And if it can be in motion in an indivisible of time, there is no reason why it could not also be at rest. Remoto ergo quod ad, in quo dicitur primo moveri, sit impartibile, relinquitur quod necesse sit illud esse divisibile: et ex quo in ad ponitur primo moveri, sequitur quod in quolibet eius moveatur. Quod sic probat. Therefore, having rejected the indivisibility of time AD, we are left with the fact that it is divisible. And since it is in AD that the object is said to be first moved, then it is being moved in any part of AD. This he now proves: Dividatur enim ipsum ad in duas partes: aut igitur in neutra parte mutatur, aut in ambabus, aut in altera parte tantum. Si in neutra mutatur, sequitur quod neque in toto: sed si mutetur in ambabus partibus, tunc poterit poni quod mutatur in toto: sed si in altera tantum moveatur, sequetur quod moveatur in toto, sed non primo, sed ratione partis. Quia igitur primo ponitur moveri in toto, oportet hoc accipere, quod in qualibet parte eius moveatur. Sed tempus dividitur in infinitum, sicut et quodlibet continuum; et ita semper est accipere partem minorem ante partem maiorem; sicut si acciperem diem ante mensem, et horam ante diem. Manifestum est ergo quod non est accipere aliquid temporis in quo primo moveatur; ita scilicet quod non sit accipere aliquam partem eius, in qua primo moveatur. Sicut si daretur quod dies est in quo primo aliquid movetur, hoc non potest esse; quia in parte eius, scilicet in prima hora diei, primo movetur quam in toto die. Let AD be divided into two parts, Then the object is being moved either in neither part or in both parts or in one part only. If in neither part, then not in the whole time. If in both parts, then it could be granted that it is being moved in the whole time. But if in one part only, it will follow that it is being moved in the whole time but not first, but by reason of the part. Therefore, since it is agreed to be moving in the whole time, it has been in motion in each part of the whole time. But time is divided infinitely just like any continuum; consequently, it is possible always to consider a part smaller than a previous one; for example, a day before a month and an hour before the day. Therefore, it is evident that it is impossible to find a time in which it is first being moved so that a previous could not be found. For if you were to assume that it is in a day that the object is first moved, that assumption would not be true, because it would have been first moved in the first part of the day, before it was moved in the whole day. Deinde cum dicit: neque igitur in eo quod mutatum est etc., ostendit idem ex parte mobilis; concludens ex praemissis quod neque in ipso quod mutatur est accipere aliquid quod primo mutetur. Quod quidem intelligendum est secundum quod per motum totius vel partis aliquod determinatum signum pertransitur: manifestum est enim quod primo pertransit aliquid determinatum prima pars mobilis, et secundo secunda, et sic deinceps. Alioquin si intelligeretur de motu absolute, non haberet locum quod hic dicitur: manifestum est enim quod simul movetur totum et omnes partes eius: sed non simul pertransit aliquid determinatum, sed semper pars ante partem. Unde sicut non est accipere primam partem mobilis, ante quam non sit alia pars minor; ita non est accipere aliquam partem mobilis, quae primo moveatur. Et quia tempus et mobile similiter dividuntur, ut supra ostensum est, convenienter ex eo quod demonstratum est de tempore, concludit idem de mobili: et probat sic. 824. Then at (631) he establishes the same point by considering the mobile, and he concludes from the foregoing that neither in that which is being changed is it possible to take something that is first changed. Now this is to be understood in the sense that some definite point is to be crossed ‘through ‘the motion of the whole or of the part: for it is evident that the first part of the mobile will first pass a given point, and a second part will pass it after that, and so on. Otherwise, if it were understood in the sense of the absolute nature of motion, what we have to say would not be ad rem: for it is clear that the whole is being moved at the same time as all the parts, but the whole does not pass a certain point all at once but part before part continuously. Hence, just as it is impossible to find a first part of the mobile than which there is not a previous smaller part, so also is it impossible to isolate a part of the mobile that would be first moved. And because time and mobile are correspondingly divided, as we have shown above, then what was concluded about time, he now concludes about the mobile. Here is his proof: Sit mobile ipsum de: et quia omne mobile divisibile est, ut supra probatum est, sit pars eius quae primo movetur dz. Et moveatur dz pertranseundo aliquod determinatum signum in tempore quod sit ti. Si igitur dz mutatum est in toto hoc tempore, sequitur quod illud quod mutatum est in medio temporis, sit minus et prius motum quam dz; et eadem ratione erit aliud prius isto, et iterum aliud prius illo, et sic semper; quia tempus in infinitum dividitur. Manifestum est ergo quod in mobili non est accipere aliquid quod primo mutatum est. Let DE be a mobile and (because every mobile can be divided, as was proved above) let DZ be the part that is first being moved. And let DZ be moved so that it passes a definite point in the time TI. if, therefore, DZ has been changed in this whole time, it follows that what has been changed in half the time is both less than DZ and moved prior to DZ. And for the same reason there will be something prior to that and so on forever, because time can be divided infinitely, it is evident, therefore, that in the mobile one cannot find something that has been first changed. Et sic patet quod primum in motu non potest accipi neque ex parte temporis neque ex parte mobilis. Hence it is clear that a first cannot be found in motion, whether we consider the time or the mobile. Deinde cum dicit: ipsum autem quod mutatur etc., ostendit idem ex parte rei in qua est motus. Praemittit tamen quod non similiter se habet de eo quod mutatur, vel ut melius dicatur secundum quod mutatur, sicut de tempore et mobili. Cum enim sit tria accipere in mutatione, scilicet mobile quod mutatur, ut homo; et in quo mutatur, ut tempus: et in quod mutatur, ut album; horum duo, scilicet tempus et mobile, sunt semper divisibilia. Sed de albo est alia ratio: quia album non est divisibile per se, sed tamen tam ipsum quam omnia alia huiusmodi, sunt divisibilia per accidens, inquantum scilicet illud cui accidit album vel quaecumque alia qualitas, est divisibile. 825. Then at (632) he proves the same thing by considering the sphere in which the motion occurs. But first he mentions that the situation with respect to the sphere in which the motion occurs is not exactly the same as it was with respect to time and the mobile. For since there are three things to be considered in change; namely, the mobile which is being changed (for example, a man) and that in which it is being changed, i.e., the time, and that into which it is being changed (for example, into white), two of these, namely, the time and the mobile are always divisible. But with white it is another story, because a white thing is not divisible per se, but it, and things like it, are divisible per accidens, inasmuch as the subject of whiteness or of any other quality is divisible. Divisio autem albi per accidens potest esse dupliciter. Uno modo secundum partes quantitativas; sicut si superficies alba dividatur in duas partes, album per accidens divisum erit. Alio modo secundum intensionem et remissionem: quod enim una et eadem pars sit magis vel minus alba, non est ex ipsa ratione albedinis (quia si esset separata, non diceretur secundum magis et minus; sicut neque substantia suscipit magis et minus): sed est ex diverso modo participandi albedinem ex parte subiecti divisibilis. Praetermisso igitur hoc quod dividitur per accidens, si accipiamus ea secundum quae est motus, quae dividuntur per se et non per accidens, neque etiam in his erit primum. Now the per accidens division of white can take place in two ways. In one way according to the quantitative parts, as when a white surface is split into two parts, the white will be divided per accidens. In another way, according to greater or less intensity, for the fact that one and the same part is whiter or less white is not due to the nature of whiteness (because if it existed in isolation, whiteness would be constant and never subject to more and less, any more than a substance is susceptible of more and less) but to the varying degrees in which a divisible subject participates whiteness. Therefore, neglecting what is divided per accidens in the sphere of motion and considering only what is divided per se in those spheres, it is impossible to find a first. Et manifestat hoc primo in magnitudinibus, in quibus est motus localis. Sit enim magnitudo spatii in quo est ab, et dividatur in c: detur ergo quod ex b in c aliquid primo moveatur. Aut igitur bc est divisibile, aut indivisibile. Si indivisibile, sequitur quod impartibile erit coniunctum impartibili; quia eadem ratione secunda pars motus erit in impartibili; sic enim oportet dividere magnitudinem, sicut et motum, ut supra de tempore dictum est. And he proves this first of all in magnitudes in which there is local motion. Let the magnitude AC be divided at B, and suppose that C is that into which something is first moved from B. Now BC is either divisible or indivisible. If the latter, it follows that an indivisible will be touching an indivisible, for there is no reason why the second part of the motion will not be into an indivisible, since we can divide a magnitude just as the motion was divided, and as time was. Si autem bc sit divisibile, erit accipere aliquod signum prius, idest propinquius ipsi b, quam c; et sic prius mutabitur ex b in illud, quam in c: et iterum illo erit accipere aliud prius, et sic semper, quia divisio magnitudinis non deficit. Patet ergo quod non est accipere aliquod primum in quod mutatum sit motu locali. But if BC is divisible, it is possible to take a stage nearer to B than to C, and so the thing will be changed from B into that stage before it is changed into C and into a stage prior to that one, and so on, because there is no limit to the division of a magnitude. It is therefore evident that it is impossible to find a first stage into which a thing has been changed in local motion. Et similiter manifestum est in mutatione quantitatis, quae est augmentum et decrementum: quia haec etiam mutatio est secundum aliquod continuum, scilicet secundum quantitatem accrescentem vel subtractam; quae cum sit in infinitum divisibilis, non est in ea accipere primum. The same is true in change of quantity, i.e., growing and decreasing. For even these changes are in terms of a continuum, i.e., in terms of added quantity or subtracted quantity, in which no first is to be found, since there can be division ad infinitum. Et sic manifestum est, quod in sola mutatione quae est secundum qualitatem, contingit aliquid esse indivisibile per se. Inquantum tamen est divisibile per accidens, similiter non est accipere primum in mutatione tali: sive accipiatur successio mutationis inquantum pars post partem alteratur (manifestum est enim quod non erit accipere primam partem albi, sicut nec primam partem magnitudinis); sive accipiatur successio alterationis secundum quod aliquid idem est albius vel minus album; quia subiectum infinitis modis potest variari secundum magis album et minus album. Et sic motus alterationis potest esse continuus, et non habens aliquid primum. And so it is clear that it is only in qualitative change that something is per se indivisible. But inasmuch as in this per accidens divisibility is found, likewise no first is discernible in such change. This is true whether the succession consists in part being altered after part (for it is evident that no first part of white can be found any more than a first part of magnitude can) or whether the succession is based on one and the same thing becoming more and more white or less and less white, for a subject can be modified in an infinite number of ways with regard to degrees of whiteness, Thus the motion involved in alteration can be continuous and not possess a first.
Before every “being moved” is a “having been moved,” and conversely
Chapter 6 Ἐπεὶ δὲ τὸ μεταβάλλον ἅπαν ἐν χρόνῳ μεταβάλλει, λέγεται δ' ἐν χρόνῳ μεταβάλλειν καὶ ὡς ἐν πρώτῳ καὶ ὡς καθ' ἕτερον, οἷον ἐν τῷ ἐνιαυτῷ ὅτι ἐν τῇ ἡμέρᾳ μεταβάλλει, ἐν ᾧ πρώτῳ χρόνῳ μεταβάλλει τὸ μεταβάλλον, ἐν ὁτῳοῦν ἀνάγκη τούτου μεταβάλλειν. δῆλον μὲν οὖν καὶ ἐκ τοῦ ὁρισμοῦ (τὸ γὰρ πρῶτον οὕτως ἐλέγομεν), οὐ μὴν ἀλλὰ καὶ ἐκ τῶνδε φανερόν. ἔστω γὰρ ἐν ᾧ πρώτῳ κινεῖται τὸ κινούμενον ἐφ' ᾧ ΧΡ, καὶ διῃρήσθω κατὰ τὸ Κ· πᾶς γὰρ χρόνος διαιρετός. ἐν δὴ τῷ ΧΚ χρόνῳ ἤτοι κινεῖται ἢ οὐ κινεῖται, καὶ πάλιν ἐν τῷ ΚΡ ὡσαύτως. εἰ μὲν οὖν ἐν μηδετέρῳ κινεῖται, ἠρεμοίη ἂν ἐν τῷ παντί (κινεῖσθαι γὰρ ἐν μηθενὶ τῶν τούτου κινούμενον ἀδύνατον)· εἰ δ' ἐν θατέρῳ μόνῳ κινεῖται, οὐκ ἂν ἐν πρώτῳ κινοῖτο τῷ ΧΡ· καθ' ἕτερον γὰρ ἡ κίνησις. ἀνάγκη ἄρα ἐν ὁτῳοῦν τοῦ ΧΡ κινεῖσθαι. Now everything that changes changes time, and that in two senses: for the time in which a thing is said to change may be the primary time, or on the other hand it may have an extended reference, as e.g. when we say that a thing changes in a particular year because it changes in a particular day. That being so, that which changes must be changing in any part of the primary time in which it changes. This is clear from our definition of 'primary', in which the word is said to express just this: it may also, however, be made evident by the following argument. Let ChRh be the primary time in which that which is in motion is in motion: and (as all time is divisible) let it be divided at K. Now in the time ChK it either is in motion or is not in motion, and the same is likewise true of the time KRh. Then if it is in motion in neither of the two parts, it will be at rest in the whole: for it is impossible that it should be in motion in a time in no part of which it is in motion. If on the other hand it is in motion in only one of the two parts of the time, ChRh cannot be the primary time in which it is in motion: for its motion will have reference to a time other than ChRh. It must, then, have been in motion in any part of ChRh. δεδειγμένου δὲ τούτου φανερὸν ὅτι πᾶν τὸ κινούμενον ἀνάγκη κεκινῆσθαι πρότερον. εἰ γὰρ ἐν τῷ ΧΡ πρώτῳ χρόνῳ τὸ ΚΛ κεκίνηται μέγεθος, ἐν τῷ ἡμίσει τὸ ὁμοταχῶς κινούμενον καὶ ἅμα ἀρξάμενον τὸ ἥμισυ ἔσται κεκινημένον. εἰ δὲ τὸ (237a.) ὁμοταχὲς ἐν τῷ αὐτῷ χρόνῳ κεκίνηταί τι, καὶ θάτερον ἀνάγκη ταὐτὸ κεκινῆσθαι μέγεθος, ὥστε κεκινημένον ἔσται τὸ κινούμενον. And now that this has been proved, it is evident that everything that is in motion must have been in motion before. For if that which is in motion has traversed the distance KL in the primary time ChRh, in half the time a thing that is in motion with equal velocity and began its motion at the same time will have traversed half the distance. But if this second thing whose velocity is equal has traversed a certain distance in a certain time, the original thing that is in motion must have traversed the same distance in the same time. Hence that which is in motion must have been in motion before. ἔτι δὲ εἰ ἐν τῷ παντὶ χρόνῳ τῷ ΧΡ κεκινῆσθαι λέγομεν, ἢ ὅλως ἐν ὁτῳοῦν χρόνῳ, τῷ λαβεῖν τὸ ἔσχατον αὐτοῦ νῦν (τοῦτο γάρ ἐστι τὸ ὁρίζον, καὶ τὸ μεταξὺ τῶν νῦν χρόνος), κἂν ἐν τοῖς ἄλλοις ὁμοίως λέγοιτο κεκινῆσθαι. τοῦ δ' ἡμίσεος ἔσχατον ἡ διαίρεσις. ὥστε καὶ ἐν τῷ ἡμίσει κεκινημένον ἔσται καὶ ὅλως ἐν ὁτῳοῦν τῶν μερῶν· ἀεὶ γὰρ ἅμα τῇ τομῇ χρόνος ἔστιν ὡρισμένος ὑπὸ τῶν νῦν. εἰ οὖν ἅπας μὲν χρόνος διαιρετός, τὸ δὲ μεταξὺ τῶν νῦν χρό νος, ἅπαν τὸ μεταβάλλον ἄπειρα ἔσται μεταβεβληκός. Again, if by taking the extreme moment of the time—for it is the moment that defines the time, and time is that which is intermediate between moments—we are enabled to say that motion has taken place in the whole time ChRh or in fact in any period of it, motion may likewise be said to have taken place in every other such period. But half the time finds an extreme in the point of division. Therefore motion will have taken place in half the time and in fact in any part of it: for as soon as any division is made there is always a time defined by moments. If, then, all time is divisible, and that which is intermediate between moments is time, everything that is changing must have completed an infinite number of changes. ἔτι δ' εἰ τὸ συνεχῶς μεταβάλλον καὶ μὴ φθαρὲν μηδὲ πεπαυμένον τῆς μεταβολῆς ἢ μεταβάλλειν ἢ μεταβεβληκέναι ἀναγκαῖον ἐν ὁτῳοῦν, ἐν δὲ τῷ νῦν οὐκ ἔστιν μεταβάλλειν, ἀνάγκη μεταβεβληκέναι καθ' ἕκαστον τῶν νῦν· ὥστ' εἰ τὰ νῦν ἄπειρα, πᾶν τὸ μεταβάλλον ἄπειρα ἔσται μεταβεβληκός. Again, since a thing that changes continuously and has not perished or ceased from its change must either be changing or have changed in any part of the time of its change, and since it cannot be changing in a moment, it follows that it must have changed at every moment in the time: consequently, since the moments are infinite in number, everything that is changing must have completed an infinite number of changes. οὐ μόνον δὲ τὸ μεταβάλλον ἀνάγκη μεταβεβληκέναι, ἀλλὰ καὶ τὸ μεταβεβληκὸς ἀνάγκη μεταβάλλειν πρότερον· And not only must that which is changing have changed, but that which has changed must also previously have been changing, ἅπαν γὰρ τὸ ἔκ τινος εἴς τι μεταβεβληκὸς ἐν χρόνῳ μεταβέβληκεν. ἔστω γὰρ ἐν τῷ νῦν ἐκ τοῦ Α εἰς τὸ Β μεταβεβληκός. οὐκοῦν ἐν μὲν τῷ αὐτῷ νῦν ἐν ᾧ ἐστιν ἐν τῷ Α, οὐ μεταβέβληκεν (ἅμα γὰρ ἂν εἴη ἐν τῷ Α καὶ ἐν τῷ Β)· τὸ γὰρ μεταβεβληκός, ὅτε μεταβέβληκεν, ὅτι οὐκ ἔστιν ἐν τούτῳ, δέδεικται πρότερον· εἰ δ' ἐν ἄλλῳ, μεταξὺ ἔσται χρόνος· οὐ γὰρ ἦν ἐχόμενα τὰ νῦν. since everything that has changed from something to something has changed in a period of time. For suppose that a thing has changed from A to B in a moment. Now the moment in which it has changed cannot be the same as that in which it is at A (since in that case it would be in A and B at once): for we have shown above that that that which has changed, when it has changed, is not in that from which it has changed. If, on the other hand, it is a different moment, there will be a period of time intermediate between the two: for, as we saw, moments are not consecutive. ἐπεὶ οὖν ἐν χρόνῳ μεταβέβληκεν, χρόνος δ' ἅπας διαιρετός, ἐν τῷ ἡμίσει ἄλλο ἔσται μεταβεβληκός, καὶ πάλιν ἐν τῷ ἐκείνου ἡμίσει ἄλλο, καὶ αἰεὶ οὕτως· ὥστε μεταβάλλοι ἂν πρότερον. Since, then, it has changed in a period of time, and all time is divisible, in half the time it will have completed another change, in a quarter another, and so on to infinity: consequently when it has changed, it must have previously been changing. ἔτι δ' ἐπὶ τοῦ μεγέθους φανερώτερον τὸ λεχθὲν διὰ τὸ συνεχὲς εἶναι τὸ μέγεθος ἐν ᾧ μεταβάλλει τὸ μεταβάλλον. ἔστω γάρ τι μεταβεβληκὸς ἐκ τοῦ Γ εἰς τὸ Δ. οὐκοῦν εἰ μὲν ἀδιαίρετόν ἐστι τὸ ΓΔ, ἀμερὲς ἀμεροῦς ἔσται ἐχόμενον· ἐπεὶ δὲ τοῦτο ἀδύνατον, ἀνάγκη μέγεθος εἶναι τὸ μεταξὺ καὶ εἰς ἄπειρα διαιρετόν· ὥστ' εἰς ἐκεῖνα μεταβάλλει πρότερον. Moreover, the truth of what has been said is more evident in the case of magnitude, because the magnitude over which what is changing changes is continuous. For suppose that a thing has changed from G to D. Then if GD is indivisible, two things without parts will be consecutive. But since this is impossible, that which is intermediate between them must be a magnitude and divisible into an infinite number of segments: consequently, before the change is completed, the thing changes to those segments. Everything that has changed, therefore, must previously have been changing: ἀνάγκη ἄρα πᾶν τὸ μεταβεβληκὸς μεταβάλλειν πρότερον. ἡ γὰρ αὐτὴ ἀπόδειξις (237b.) καὶ ἐν τοῖς μὴ συνεχέσιν, οἷον ἔν τε τοῖς ἐναντίοις καὶ ἐν ἀντιφάσει· ληψόμεθα γὰρ τὸν χρόνον ἐν ᾧ μεταβέβληκεν, καὶ πάλιν ταὐτὰ ἐροῦμεν. for the same proof also holds good of change with respect to what is not continuous, changes, that is to say, between contraries and between contradictories. In such cases we have only to take the time in which a thing has changed and again apply the same reasoning. ὥστε ἀνάγκη τὸ μεταβεβληκὸς μεταβάλλειν καὶ τὸ μεταβάλλον μεταβεβληκέναι, καὶ ἔσται τοῦ μὲν μεταβάλλειν τὸ μεταβεβληκέναι πρότερον, τοῦ δὲ μεταβεβληκέναι τὸ μεταβάλλειν, καὶ οὐδέποτε ληφθήσεται τὸ πρῶτον. αἴτιον δὲ τούτου τὸ μὴ εἶναι ἀμερὲς ἀμεροῦς ἐχόμενον· ἄπειρος γὰρ ἡ διαίρεσις, καθάπερ ἐπὶ τῶν αὐξανομένων καὶ καθαιρουμένων γραμμῶν. So that which has changed must have been changing and that which is changing must have changed, and a process of change is preceded by a completion of change and a completion by a process: and we can never take any stage and say that it is absolutely the first. The reason of this is that no two things without parts can be contiguous, and therefore in change the process of division is infinite, just as lines may be infinitely divided so that one part is continually increasing and the other continually decreasing. φανερὸν οὖν ὅτι καὶ τὸ γεγονὸς ἀνάγκη γίγνεσθαι πρότερον καὶ τὸ γιγνό μενον γεγονέναι, ὅσα διαιρετὰ καὶ συνεχῆ, οὐ μέντοι αἰεὶ ὃ γίγνεται, ἀλλ' ἄλλο ἐνίοτε, οἷον τῶν ἐκείνου τι, ὥσπερ τῆς οἰκίας τὸν θεμέλιον. ὁμοίως δὲ καὶ ἐπὶ τοῦ φθειρομένου καὶ ἐφθαρμένου· εὐθὺς γὰρ ἐνυπάρχει τῷ γιγνομένῳ καὶ τῷ φθειρομένῳ ἄπειρόν τι συνεχεῖ γε ὄντι, καὶ οὐκ ἔστιν οὔτε γί γνεσθαι μὴ γεγονός τι οὔτε γεγονέναι μὴ γιγνόμενόν τι, ὁμοίως δὲ καὶ ἐπὶ τοῦ φθείρεσθαι καὶ ἐπὶ τοῦ ἐφθάρθαι· αἰεὶ γὰρ ἔσται τοῦ μὲν φθείρεσθαι τὸ ἐφθάρθαι πρότερον, τοῦ δ' ἐφθάρθαι τὸ φθείρεσθαι. φανερὸν οὖν ὅτι καὶ τὸ γεγονὸς ἀνάγκη γίγνεσθαι πρότερον καὶ τὸ γιγνόμενον γεγονέναι· πᾶν γὰρ μέγεθος καὶ πᾶς χρόνος ἀεὶ διαιρετά. ὥστ' ἐν ᾧ ἂν ᾖ, οὐκ ἂν εἴη ὡς πρώτῳ. So it is evident also that that that which has become must previously have been in process of becoming, and that which is in process of becoming must previously have become, everything (that is) that is divisible and continuous: though it is not always the actual thing that is in process of becoming of which this is true: sometimes it is something else, that is to say, some part of the thing in question, e.g. the foundation-stone of a house. So, too, in the case of that which is perishing and that which has perished: for that which becomes and that which perishes must contain an element of infiniteness as an immediate consequence of the fact that they are continuous things: and so a thing cannot be in process of becoming without having become or have become without having been in process of becoming. So, too, in the case of perishing and having perished: perishing must be preceded by having perished, and having perished must be preceded by perishing. It is evident, then, that that which has become must previously have been in process of becoming, and that which is in process of becoming must previously have become: for all magnitudes and all periods of time are infinitely divisible. Consequently no absolutely first stage of change can be represented by any particular part of space or time which the changing thing may occupy.
Postquam philosophus ostendit qualiter sit accipere primum in mutatione et qualiter non, hic ostendit ordinem eorum quae in motu inveniuntur ad invicem: et primo praemittit quoddam necessarium ad propositum ostendendum; secundo ostendit propositum, ibi: ostenso autem hoc et cetera. 826. After explaining how a first is to be taken in motion and how not, the Philosopher now explains the order of precedence among the things present in motion. First he premises facts needed for explaining the proposition; Secondly, he explains the proposition, at 828. Dicit ergo primo, quod omne quod mutatur, mutatur in tempore, ut supra ostensum est: sed in tempore aliquo dicitur aliquid mutari dupliciter; uno modo primo et per se, alio modo secundum alterum, idest ratione partis, sicut dicitur aliquid mutari in anno, quia mutatur in die. 827. He says therefore first (633) that whatever is being changed is being changed in time, as we have explained. But something is being changed in a time in two ways: in one way, first and per se; in another way, by reason of something else, i.e., by reason of a part, as when something is said to be changed in a year, because it is being changed in a day. Hac ergo distinctione praemissa, proponit quod intendit probare: scilicet, si aliquid mutatur primo in aliquo tempore, necesse est quod mutetur in qualibet parte illius temporis. Et hoc probat dupliciter. With this distinction in mind, he states what he intends to prove: namely, that if something is being first moved in a time, it is necessarily being moved in some part of that time. This he proves in two ways: Primo quidem ex definitione eius quod dicitur primum: hoc enim dicitur primo alicui convenire, quod convenit ei secundum quamlibet suam partem, ut in principio quinti dictum est. First, from the definition of “first”, for here something is said to be in a thing “first”, if it belongs to it by reason of each and every part, as was said in the beginning of Book V. Secundo probat idem per rationem. Sit enim tempus in quo primo aliquid movetur xr: et quia omne tempus est divisibile, dividatur secundum k. Necesse est ergo dicere quod in parte temporis quae est xk, aut moveatur aut non moveatur; et similiter de parte quae est kr. Si ergo detur quod in neutra harum partium movetur, sequitur quod neque in toto xr moveatur, sed quiescat in eo: quia impossibile est quod aliquid moveatur in tempore, in cuius nulla parte movetur. Si autem detur quod in una parte temporis moveatur et non in alia, sequetur quod non primo moveatur in xr tempore: quia oporteret quod secundum utramque partem moveretur, et non secundum alteram tantum. Necesse est ergo dicere quod moveatur in qualibet parte temporis quod est xr. Et hoc est quod demonstrare volumus; scilicet quod in quo primo tempore aliquid movetur, in qualibet parte eius movetur. Secondly, he proves the same thing with an argument: Let XR be the time in which something is being first moved and, since time is divisible, let XR be divided at K. Then of necessity in the part XK of the time, the object is either being moved or not, and likewise for the part KR. Now if it be said that it is being moved in neither of those parts, it follows that it is not being moved in the whole time but is at rest throughout that time, for it is impossible for a thing to be in motion in a time without being in motion in some part of it. But if it be supposed that it is being moved in just one part of the time, it will follow that it is not being first moved in the time called XR; because that would require motion in respect to both parts and not in respect to just one. Therefore, of necessity, it must be in motion in each part of the time XR. And that is what we want to demonstrate: namely, that if something is being first moved in a time, it is being moved in every part of it. Deinde cum dicit: ostenso autem hoc etc., procedit ad principale propositum ostendendum. Et circa hoc duo facit: primo inducit demonstrationes ad propositum ostendendum; secundo concludit veritatem determinatam, ibi: quare necesse est et cetera. 828. Then at (634) he sets about proving the main proposition. And about this he does two things: First he introduces the proofs of the proposition; Secondly, he concludes to the truth, at 838. Circa primum duo facit: primo ostendit quod ante omne moveri praecedit mutatum esse; secundo quod e converso ante quodlibet mutatum esse praecedit moveri, ibi: non solum autem quod mutatur et cetera. About the first he does two things: First he shows that before each state of being moved there was a state of completed motion; Secondly, that, conversely, before each state of completed motion there was a state of being moved, at 832. Primum ostendit tribus rationibus: quarum prima talis est. Detur quod in xr primo tempore aliquod mobile motum sit per kl magnitudinem: manifestum est quod si accipiatur aliud mobile aeque velox, quod simul inceptum est moveri cum ipso, in medietate temporis motum erit per medium magnitudinis. Cum ergo sit aeque velox illud mobile quod ponitur moveri per totam magnitudinem, sequitur quod etiam ipsum in eodem tempore, scilicet in medietate temporis xr, motum est iam per eandem magnitudinem, quae scilicet est pars totius magnitudinis kl. Sequetur ergo quod illud quod movetur, prius est mutatum. 829. He proves the first with three arguments, of which the first is: Let KL be the magnitude through which a mobile has been moved in the first time XR. It is clear that an equally fast mobile, which began its motion with the first one, will have covered half the magnitude in half the time. Since the first mobile (which we have said covers the entire magnitude) is as fast as the second, it follows that even it has in half the time already been moved through, half the magnitude KL. It will follow, therefore, that what is being moved has been previously moved. Ut autem illud quod hic dicitur manifestius intelligatur, considerandum est quod sicut punctus nominat terminum lineae, ita mutatum esse nominat terminum motus. Quamcumque autem lineam vel partem lineae accipias, semper est dicere quod ante consummationem lineae totius, sit accipere aliquod punctum, secundum quod linea dividatur. Et similiter ante quemlibet motum, et ante quamcumque partem motus, est accipere aliquod mutatum esse: quia dum mobile est in moveri ad aliquem terminum, iam pertransivit aliquod signum, respectu cuius iam dicitur mutatum esse. Sed sicut punctum infra lineam est in potentia ante lineae divisionem, in actu autem quando iam linea est divisa, cum punctum sit ipsa lineae divisio; similiter hoc quod dico mutatum esse infra motum, est in potentia quando motus non ibi terminatur: sed si ibi terminetur, erit in actu. Et quia quod est in actu est notius eo quod est in potentia, ideo Aristoteles probavit quod illud quod continue movetur, iam mutatum est aliquid, per aliud mobile aeque velox, cuius motus iam terminatus est: sicut si quis probaret quod in aliqua linea esset punctum in potentia, per hoc quod alia linea eiusdem rationis esset divisa in actu. To get a better understanding of what we mean, it must be considered that just as “point” is a name for the terminus of a line, so “completed motion” is a name for the terminus of a motion. Now, no matter what line or what part of a line you take, it is always true that before the consummation of the whole line, you can take a point according to which the line can be divided. Likewise, before any motion or part of a motion, you can take a “state of completed motion”; because while the mobile is being moved to its terminus, it has already passed a certain stage in respect to which the mobile is said to have been already changed. But just as a point within a line is in potency before the line is actually divided (for a point is the very division of a line), so also the thing called “completed motion” (within a motion) is in potency as long as the motion does not stop there; but if it does stop there, it will be actual. And since what is in act is better known than what is in potency, therefore Aristotle proves his proposition (that what is being continually moved has already been moved) by referring to an equally fast mobile whose motion has already been completed. This is like proving that in a certain line there is a point in potency by showing that a like line has been actually divided. Secundam rationem ponit ibi: amplius autem et si in omni etc.: quae talis est. In toto tempore xr, vel in quocumque alio, dicitur aliquid mutatum esse, per hoc quod accipitur ultimum nunc ipsius temporis: non quod in nunc moveatur aliquid, sed quia in nunc terminatur motus. Unde hic non accipit mutatum esse pro eo quod est aliquando moveri, sed pro eo quod est terminari motum. Ideo autem necesse est terminari motum in ultimo nunc temporis mensurantis motum, quia ipsum nunc determinat tempus, idest est terminus ipsius, sicut punctum lineae; et oportet omne tempus esse medium inter duo nunc, sicut linea est inter duo puncta. Quia ergo moveri est in tempore, sequitur quod motum esse sit in nunc, quod est terminus temporis. Et si ita est de motu qui est in toto tempore, oportet etiam quod similiter dicatur de partibus motus, quae sunt in partibus temporis. Iam enim ostensum est quod si aliquid movetur primo in toto tempore, quod movetur in qualibet parte temporis. Quaelibet autem pars temporis accepta terminatur ad aliquod nunc. Oportet enim quod ultimum medietatis temporis sit divisio, idest ipsum nunc, quod dividit inter duas partes temporis. Quare sequitur quod illud quod movetur per totum, sit prius motum in medio, propter nunc quod determinat medium. Et eadem ratio est de qualibet alia parte temporis. Qualitercumque enim dividatur tempus, semper invenietur quaelibet pars temporis determinari a duobus nunc: et post primum nunc temporis mensurantis motum, quodcumque aliud nunc accipiatur, in eo iam motum est; quia illud nunc, quodcumque accipiatur, est terminus temporis mensurantis motum. 830. The second argument, which he gives at (635), is this: In the whole time XR or in any other, something is said to have been changed by the very fact that a final “now” of the time is taken, not that something is being moved in that “now”, but that the motion is terminated then. Hence “having been moved” is taken here not for that which is at some time being moved but for the fact that the motion is ended. Now the reason why the motion must be terminated in the final “now” of the time that measures the motion is that that “now” terminates the time, just as a point terminates a line. And all time is midway between two “now’s”, just as a line is between two points. Therefore, since “being moved” occurs in time, it follows that “having been moved” occurs in the “now” which is the terminus of time. And if that is the case with a motion in a whole period of time, the same must be true of the parts of motion that occur in the parts of time. Now, we have already shown that if something is being first moved in the whole time, it is being moved in each part of the time. But whichever part of time you take, it is terminated at some “now”. For the terminus of half of the time is the “now” which divided the time into two parts. Therefore, it follows that what is being moved through the whole is previously moved at the middle of time, on account of the “now” which determines the middle. And the same reasoning applies to any part of time. For no matter how the time is divided, it will always be found that each part of the time is determined by two “now’s”, and after the first “now” of the time measuring the motion, no matter which other “now” is taken, the object has already been moved in that part of the time, for that “now”—whichever it is—is the terminus of the time measuring the motion. Quia ergo omne tempus divisibile est in tempora; et omne tempus est medium inter duo nunc; et in omni nunc, quod est ultimum temporis mensurantis motum, aliquid motum est, sicut probatum est: sequitur quod omne quod mutatur sit infinities mutatum; quia mutatum esse est terminus motus, sicut punctum lineae et nunc temporis. Now, because every period of time is divisible into times and each period exists between two “now’s”, and because in any “now” that happens to be the ending of a time measuring the motion, something has been moved, it follows that whatever is being changed has been changed an infinite number of times, because “having been changed” is the terminus of a motion, just as a point is of a line and a “now” is of a time. Sicut ergo in qualibet linea est signare infinities punctum ante punctum, et in quolibet tempore infinities nunc ante nunc, propter hoc quod utrumque est divisibile in infinitum; ita in quolibet moveri est signare infinities mutatum esse, quia motus est in infinitum divisibilis, sicut linea et tempus, ut supra probatum est. Therefore, just as it is possible in any line to pick out point ahead of point ad infinitum and in any period of time “now” before “now” (because both line and time are divisible ad infinitum), so in any “being moved” it is possible to pick out infinitely many “having been moved’s”, because motion, too, is divisible ad infinitum, just as the line and time, as was previously proved. Tertiam rationem ponit ibi: amplius autem si id quod continue mutatur etc.: quae talis est. Omne quod mutatur, si non corrumpitur neque pausat a mutatione, idest neque desinit moveri, quasi continue mutatum, necesse est quod in quolibet nunc temporis in quo movetur, vel mutetur vel sit mutatum. Sed in nunc non mutatur, ut supra ostensum est: ergo necesse est quod in quolibet nunc temporis mensurantis motum continuum sit mutatum. Sed in quolibet tempore sunt infinita nunc, quia nunc est divisio temporis, et tempus est in infinitum divisibile: ergo omne quod mutatur est infinities mutatum. Et ita sequitur quod ante omne moveri sit mutatum esse, non quasi extra ipsum moveri existens, sed in ipso, ut terminans aliquam partem eius. 831. The third argument is in (636): In the case of anything that is being changed (if it is not ceasing-to-be and does not cease to be moved, but is being continually changed), it is necessary that in each “now” of the time in which it is being moved, it is being changed or has been changed. But in the “now” nothing is being changed, as we have shown. Therefore, in each “now” of the time which measures continuous motion, the object has been changed, But in any portion of time there are an infinitude of “now’s”, because the “now” divides time, and time is infinitely divisible. Therefore, everything that is being changed has been changed an infinite number of times. And so it follows that before every state called “being moved” is a state called “having been moved”, which, however, does not exist outside the state of “being changed” but is in it and terminates a part of it. Deinde cum dicit: non solum autem quod mutatur etc., probat quod e converso ante omne mutatum esse, praecedat mutari. Et primo ex parte temporis; secundo ex parte rei secundum quam est motus, ibi: amplius autem in magnitudine et cetera. 832. Then at (637) he proves that on the other hand a state of “being changed” precedes each state of “having been changed”. First he proves it from the viewpoint of the time; Secondly, from the viewpoint of the sphere in which the motion occurs, at 836. Circa primum tria facit: primo proponit propositum; secundo demonstrat quoddam necessarium ad probandum propositum, ibi: omne enim quod ex quodam etc.; tertio inducit probationem principalis propositi, ibi: quoniam igitur et cetera. About the first he does three things: First he states the proposition; Secondly, he proves certain things needed for proving the proposition, at 833. Thirdly, he gives the proof of the main proposition, at 835. Dicit ergo primo quod non solum omne quod mutatur necesse est mutatum esse iam, sed etiam omne quod mutatum est necesse est prius mutari: quia mutatum esse est terminus eius quod est moveri. Unde oportet quod ante mutatum esse praecedat moveri. He says therefore first that not only is it true that whatever is being changed had already been changed, but that every state of “having been changed” must be preceded by a state of “being changed”, because the former is the terminus of the latter. Therefore, every “having been changed” must be preceded by a “being changed”. Deinde cum dicit: omne enim etc., ponit quoddam necessarium ad propositi probationem, scilicet quod omne quod mutatur ex quodam in quiddam, sit mutatum in tempore. Sed advertendum quod hic mutatum esse non est idem quod terminari motum: supra enim ostensum est quod illud temporis, in quo primo dicitur mutatum esse, est indivisibile. Sed accipitur hic mutatum esse, secundum quod significat quod aliquid prius movebatur; quasi dicat: omne quod movebatur, movebatur in tempore. 633. Then at (638) he states something needed for his proof of the proposition, i.e., that whatever is being changed from something to something was changed in time. But note carefully that here the words, “was changed”, do not refer to the termination of motion, for it was explained above that the time in which a thing “was changed” is an indivisible. But here “was changed” signifies that something was previously being moved, as though he said: “Whatever was being moved was being moved in time”. Et hoc probat sic. Si hoc non est verum, sit aliquid mutatum ex a in b, idest ex uno termino in alterum, in ipso nunc. Hoc posito, sequitur quod quando est in ipso a, idest in termino a quo, in eodem nunc nondum est mutatum: quia iam supra ostensum est, quod illud quod mutatum est, quando mutatum est, non est in termino a quo, sed magis in termino ad quem; sequeretur ergo quod simul esset in a et in b. Oportet ergo dicere quod in alio nunc sit in a, et in alio nunc sit mutatum. Sed inter quaelibet duo nunc est tempus medium, quia duo nunc non possunt esse sibi coniuncta immediate, ut supra ostensum est. Relinquitur ergo quod omne quod mutatur, mutatur in tempore. This he now proves: If our proposition is not true, then let there be something that was changed from A to B, i.e., from one term to another, in a “now”. From this it follows that when it is in A, i.e., in the terminus a quo in the same “now” it was not yet changed, because it has already been proved that what was changed, when it was being changed is not in the terminus a quo but more in he terminus ad quem. Otherwise, it would follow that it was at once in A and in B. Therefore, it is necessary to say that in one “now” it is in A, and in another it was being changed. But between two “now’s” there is a time, because two “now’s” cannot be immediately connected, as we have shown. What remains, therefore, is that whatever is being changed is being changed in time. Videtur autem quod hic concluditur habere instantiam in generatione et corruptione, inter quorum terminos non est aliquod medium. Si enim inter nunc in quo est in termino a quo, et inter nunc in quo est in termino ad quem, sit tempus medium, sequetur quod aliquid sit medium inter esse et non esse; quia in illo medio tempore, id quod mutatur neque esset ens, neque non ens. 334. But it seems that this conclusion has no application in generation and ceasing-to-be, between whose two termini there is nothing intermediate. For if between the “now” in which something is at the terminus a quo and the “now” in which it is at the terminus ad quem a period of time occurs, it will follow that there is something between being and non-being, because in that intermediate time the subject of change would be neither being nor non-being. Sed quia ratio quae hic ponitur demonstrativa est, oportet quod hic dicitur aliquo modo etiam in generatione et corruptione salvari: ita tamen quod aliquo modo etiam huiusmodi mutationes sint momentaneae, cum non possit esse aliquod medium inter extrema earum. Nevertheless, because the argument which Aristotle gives here is demonstrative, it must be said that it applies somehow even to generation and ceasing-to-be but in the sense that such changes are also instantaneous, since there can be no medium between the termini. Est igitur dicendum quod illud quod mutatur de non esse in esse, vel e converso, non est simul in non esse et esse: sed sicut in octavo dicetur, non est dare ultimum instans, in quo id quod generatur sit non ens; sed est dare primum instans in quo est ens, ita quod in toto tempore praecedenti illud instans, est non ens. Inter tempus autem et instans quod terminat motum, non est aliquod medium: et sic non oportet quod sit medium inter esse et non esse. Sed quia tempus quod praecedit instans in quo primo est quod generatur, mensurat aliquem motum, sequitur quod sicut illud instans in quo primo est quod generatur, est terminus praecedentis temporis mensurantis motum, ita incipere esse est terminus praecedentis motus. Si ergo generatio dicatur ipsa inceptio essendi, sic est terminus motus, et sic est in instanti: quia terminari motum, quod est mutatum esse, est in indivisibili temporis, ut supra ostensum est. So it must be said that whatever is being changed from non-being to being or vice versa is not in being and non-being at the same time. But, as will be said in Book VIII, there is no final instant in which what is generated is a non-being, but there is a first instant in which it is a being, so that in the entire time preceding that instant, it is non-being. However, between that “now” and the time preceding, there is nothing intermediate, so that between being and non-being there is no medium. Now, since the time which precedes the instant in which something is generated first, is the measure of some motion, it follows that just as that instant in which something is first generated is the terminus of the preceding time that measures the motion, so the first instant of the being of the thing generated is the terminus of a preceding change. If, therefore, generation is said to be the very beginning of being, it must be the terminus of a motion, and thus it takes place in an instant, because a motion’s being terminated—which is the same as having been changed—occurs in an indivisible of time, as we have shown. Si autem generatio accipiatur ipsa inceptio essendi cum toto motu praecedente cuius est terminus, sic non est in instanti, sed in tempore: ita quod in toto tempore praecedenti est non ens illud quod generatur, et in ultimo instanti est ens. Et similiter dicendum est de corruptione. However, if generation is taken as the very beginning of being plus the entire preceding motion of which it is the terminus, then it occurs not in an instant but in time, so that what is being generated is a non-being during the entire preceding time and a being in the final instant. And the same applies to ceasing-to-be. Deinde cum dicit: quoniam igitur in tempore etc., probat principale propositum tali ratione. Omne quod mutatum est, in tempore mutabatur, ut probatum est: omne autem tempus est divisibile: quod autem in aliquo tempore mutatur, in qualibet parte illius temporis mutatur: ergo oportet dicere, quod illud quod mutatum est in toto aliquo tempore, mutabatur prius in medietate temporis, et iterum in medietate medietatis: et sic semper procedetur, propter hoc quod tempus est in infinitum divisibile. Ergo sequitur quod omne quod mutatum est, prius mutabatur: et ita ante omne mutatum esse praecedit mutari. 835. Then at (639) he proves the main proposition with the following reason: Whatever has been changed was being changed in time, as we have proved; but time is divisible and whatever is being changed in time is being changed in part of time. Therefore, it is necessary to say that what has been changed in some entire period of time was previously being changed during half of the time and again during half of that half and so on, because time is divisible infinitely. Therefore, it follows that what has been changed was previously being changed. Consequently, before every state of “having been changed” there is a previous state of “being changed”. Deinde cum dicit: amplius autem in magnitudine etc., ostendit idem, ratione accepta ex parte eius secundum quod mutatur. 836. Then at (640) he proves the same point with an argument based on the sphere of motion. First as to motions in quantity; Secondly, as to other changes, at 837. Et primo quantum ad motus qui sunt in quantitate; secundo quantum ad alias mutationes, ibi: eadem enim demonstratio est et cetera. Dicit ergo primo, quod hoc quod dictum est ex parte temporis, communiter ad omnem mutationem, manifestius potest accipi ex parte magnitudinis: quia magnitudo manifestior est quam tempus, et magnitudo continua est sicut et tempus, et in ea aliquid mutatur, scilicet illud quod movetur secundum locum, vel quod movetur secundum augmentum et decrementum. Sit ergo aliquid mutatum ex c in d. Non autem potest dici quod totum quod est cd sit indivisibile; quia oportet quod cd sit pars alicuius magnitudinis, sicut motus qui est ex c in d est pars totius motus: similiter enim dividitur magnitudo et motus, ut supra ostensum est. Si autem aliquod indivisibile sit pars magnitudinis, sequitur quod duo impartibilia erunt immediate coniuncta; quod est impossibile, ut supra ostensum est. Non ergo potest dici quod totum cd sit indivisibile. Ergo necesse est quod illud quod est inter c et d, sit quaedam magnitudo, et per consequens quod in infinitum dividi possit. Sed semper prius mutatur in parte magnitudinis, quam sit mutatum per totam magnitudinem. Ergo necesse est omne quod mutatum est, prius mutari; sicut necesse est quod ante quamlibet magnitudinem totam, sit pars eius. He says therefore first (640) that what was said, from the viewpoint of time, to be common to every change, becomes clearer from the viewpoint of magnitude, for magnitude is better known than time, and magnitude is continuous, as a line, and in it something is changed, namely, that which is according to place, or according to increase and decrease. Therefore, consider something changed from C to D. Now, it cannot be said that the whole of CD is indivisible, because CD has to be part of a magnitude, just as the motion from C to D is part of a whole motion, for there is a correspondence between division of magnitude and division of motion, as we have shown. But if an indivisible is a part of a magnitude, it follows that two indivisibles are immediate neighbors—which is impossible, as we have shown. Therefore, the whole CD cannot be an indivisible, Consequently, that which is between C and D is a magnitude and can be infinitely divided. And something is always first changed in part of a magnitude before it has been changed throughout the entire magnitude. Therefore, anything that has been changed was previously being changed, just as before any whole magnitude there are its parts. Deinde cum dicit: eadem enim demonstratio etc., ostendit quod idem necesse est esse in illis mutationibus, quae non sunt secundum aliqua continua; sicut de alteratione, quae est inter contrarias qualitates, et de generatione et corruptione, quae sunt inter contradictorie opposita. Licet enim in his non possit hoc demonstrari ex parte rei secundum quam est motus, accipietur tamen tempus in quo sunt huiusmodi mutationes, et eodem modo procedetur. 837. Then at (641) he shows that the same point is true in those changes which do not take place in terms of a continuum; for example, alteration, which is between contrary qualities, and generation and ceasing-to-be, which are between contradictories. And although in those changes the demonstration is not derived from things in which the motion is, yet it is possible to take the time in which the changes occur, and then the demonstration will proceed the same way. Sic igitur in tribus mutationibus, scilicet alteratione et corruptione et generatione, habet locum sola prima ratio: in aliis autem tribus, scilicet augmento et decremento et loci mutatione, habet locum utraque. Thus in the three changes, which are alteration, generation and ceasing-to-be, only the first argument holds, while in the other three, namely, growth, decrease and local motion, both arguments hold. Deinde cum dicit: quare necesse etc., concludit principale propositum: et primo in communi; secundo specialiter quantum ad generationem et corruptionem, ibi: manifestum igitur et cetera. 838. Then at (642) he concludes to the main proposition: First, in general; Secondly, with special application to generation and ceasing-to-be, at 839. Concludit ergo primo ex praemissis, quod necesse est omne mutatum prius mutari, et omne quod mutatur prius esse mutatum. Et sic verum est dicere quod hoc ipso quod est mutari, prius est mutatum esse: et iterum, hoc ipso quod est mutatum esse, est prius mutari. Et ita manifestum fit quod nullo modo comprehenditur aliquid primum. He concludes therefore first (642) from the foregoing that everything which has been changed was previously being changed, and that everything which is being changed has previously been changed. Consequently, it is true that a state of “having been changed” preceded a state of “being changed”, and vice versa. And so it is clear that a first something cannot be definitely pointed to. Et huius causa est, quia in motu non coniungitur impartibile impartibili, ita quod totus motus componatur ex impartibilibus: quia si hoc esset, esset accipere aliquod primum. Hoc autem non est verum: quia motus est divisibilis in infinitum, sicut etiam et lineae, quae in infinitum diminuuntur per divisionem, et in infinitum augmentantur per additionem oppositam diminutioni; dum scilicet quod subtrahitur ab uno, alteri additur, ut in tertio est ostensum. Manifestum est enim in linea, quod ante quamlibet partem lineae est accipere punctum in medio illius partis; et ante illud punctum medium est accipere aliquam partem lineae; et sic in infinitum. Non tamen linea est infinita; quia ante primum punctum lineae non est aliqua pars lineae. The reason for this is that in motion an indivisible is not joined to an indivisible so as to make a motion be composed of indivisibles, because, if that were the case, we could discover a first. But it is not true, for motion is infinitely divisible just as a line is, which can be infinitely decreased by division and increased by addition opposite to the decrease, in the sense that what is taken from one is being added to another, as was shown in Book III. For it is evident that in a line, before each part of a line, one can take a point in its midst, and before that midpoint is a part of the line, and so on ad infinitum. However, the line is not infinite, because no part of the line is in front of the first point of the line. Et similiter considerandum est in motu: quia cum quaelibet pars motus sit divisibilis, ante quamlibet partem motus est accipere indivisibile aliquid in medio illius partis, quod est mutatum esse; et ante illud indivisibile est accipere partem motus; et sic in infinitum. Non tamen sequitur quod motus sit infinitus: quia ante primum indivisibile motus, non est aliqua pars motus. Illud tamen primum indivisibile non dicitur mutatum esse, sicut nec primum punctum lineae dicitur divisio. Well, the same thing is true of motion. For since each part of motion is divisible, before each part of the motion there is in the midst of that part an indivisible, which is called “having been changed”, and before that indivisible there is a part of the motion, and so on ad infinitum. Yet it does not follow that the motion is infinite, cause in front of the first indivisible of motion there was no part of motion. But note that the first indivisible is not one called “having been changed”, any more than the first point of a line is a dividing point. Deinde cum dicit: manifestum igitur etc., concludit idem specialiter in generatione et corruptione. Et hoc ideo, quia aliter se habet mutatum esse ad mutari in generatione et corruptione, et aliter in aliis. 839. ‘Then at (643) he comes to the same conclusion with reference to generation and ceasing-to-be. And he makes a special point of these changes, because the relation of “having been changed” to “being changed” in generation and ceasing-to-be is not the same as it is in other changes. In aliis enim mutatum esse et mutari est secundum idem, sicut alteratum esse et alterari est secundum album. Nam alterari est mutari secundum albedinem, alteratum autem esse est mutatum esse secundum albedinem: et idem dicendum est in motu locali, et augmento et decremento. Sed in generatione secundum aliud est mutatum esse, et secundum aliud mutari. Nam mutatum esse est secundum formam: mutari vero non est secundum negationem formae, quae non suscipit magis et minus secundum se; sed mutari est secundum aliquid adiunctum negationi, quod suscipit magis et minus, quod est qualitas. Et ideo generatum esse est terminus eius quod est alterari, et similiter corruptum esse. Et quia motus denominatur a termino ad quem, ut in principio quinti dictum est, ipsum alterari, quia habet duos terminos, scilicet formam substantialem et qualitatem, dupliciter nominatur; quia potest dici et alterari, et fieri et corrumpi. For in the others, the state of “having been changed” and the state of “being changed” occur in respect to the same thing; for example, to whiteness, in the case of alteration. For “to be being altered” is to be being changed in respect to whiteness, and “to have been altered” is to have been changed in regard to whiteness; and the same is true in local motion, in growth and in decrease. But in generation “having been changed” refers to one thing and “being changed” to another. For the former is based on the form, but the latter, though not based on negation of a form (which is not of itself susceptible of more and less) is based on something joined to such a negation, something, that is, which is susceptible of more and less, namely, a quality. Therefore, “to have been generated” is the terminus of “being altered” and the same is true of “having been corrupted”. And because motions get their name from the terminus ad quem, as we have said in the beginning of Book V, “to be altered” (since it has two termini, namely, substantial form and quality) has two names: for it can be called “to be altered”, and “to come to be and cease to be”. Et hoc modo accipit hic fieri et corrumpi pro ipso alterari, secundum quod terminatur ad esse vel non esse. Unde dicit quod illud quod factum est, necesse est prius fieri, et illud quod fit, necesse est factum esse, quaecumque tamen sunt divisibilia et continua. Quod quidem ponitur, ut Commentator dicit, ad excludendum quaedam quae indivisibiliter fiunt absque motu continuo, sicut intelligere et sentire: quae etiam non dicuntur motus nisi aequivoce, ut in tertio de anima dicitur. Vel potest dici aliter, quod hoc philosophus addidit ut accipiatur generatio cum toto motu continuo praecedente. And this is the sense in which coming-to-be and ceasing-to-be are substituted for “being altered”, i.e., because the alteration terminates at being or non-being. And consequently, Aristotle says that what has been made was previously being made, and what is being made must necessarily have been made, provided that divisible and continuous things are involved. And Aristotle makes that addition (as the Commentator says) in order to exclude things that indivisibly come to be without continuous motion; for example, understanding and sensing, which are motions only in an analogous sense, as will be shown in Book III of On the Soul. But it could be that Aristotle made this addition in order to show that generation should include the entire continuous motion that precedes it. Sed id quod fit, prius factum esse, diversimode invenitur in diversis. Quaedam enim sunt simplicia, quae habent simplicem generationem, sicut aer aut ignis: et in istis non generatur pars ante partem, sed simul generatur et alteratur totum et partes. Et in talibus id quod factum est, ipsummet prius fiebat; et quod fit, ipsummet prius factum est, propter continuitatem alterationis praecedentis. 840. But the statement “what is being made has been previously made” applies in different ways to different things. For some things, such as air and water, are simple and have simple generation—in these cases, part is not generated after part, but the whole and the parts are altered and generated at once. And it is in such that what has been made was previously being made and what is being made has been previously made, on account of the preceding alteration being continuous. Quaedam vero sunt composita ex dissimilibus partibus, quorum pars generatur post partem, sicut in animali prius generatur cor, et in domo fundamentum: et in istis quod fit, prius factum est, non ipsummet, sed aliquid eius. Et hoc est quod subdit, quod non semper id quod fit, prius ipsummet factum est, sed aliquando aliquid eius factum est, sicut fundamentum domus. Sed quia oportet devenire ad aliquam partem quae tota simul fit, oportet quod in aliqua parte id quod fit, factum sit secundum aliquem terminum acceptum in alteratione praecedenti: sicut dum generatur animal iam factum est cor, et dum generatur cor iam factum est aliquid; non quidem aliqua pars cordis, sed aliqua alteratio facta est, ordinata ad generationem cordis. But other things are composites of unlike parts. In these cases, part is generated after part, as in an animal the heart is first generated, and in a house the foundation. In such things what is being made was not itself previously made, but a part was. And this is what he adds, namely, that it is not always so that what is being made has been itself previously made but something pertaining to it has been made, as the foundation of a house. But since we must come to a part that is entirely being made at once, then in some part, that which is being made has been made in relation to a terminus taken in the preceding alteration; for example, in the generation of an animal, the heart has already been made and while the heart is being generated, something has already been made—not indeed that there has been made some part of the heart, but some alteration ordained to the generation of the heart. Et sicut dictum est de generatione, ita intelligendum est de corruptione. Statim enim ei quod fit et corrumpitur, inest quoddam infinitum, cum sit continuum; quia ipsum fieri et ipsum corrumpi continuum est. Et ideo non est fieri, nisi aliquid factum sit prius: neque est aliquid factum esse, nisi fiat prius. Et similiter dicendum est de corrumpi et de corruptum esse. Semper enim corruptum esse est prius ipso corrumpi, et corrumpi est prius hoc quod est corruptum esse. And what has been said of generation is to be understood with regard to ceasing-to-be. For immediately there is in something that is produced in being and is corrupted, something infinite, since it is continuous. For the very coming-to-be and the ceasing-to-be are continuous. Therefore, there is no “being produced in being”, unless something has been previously made, and nothing has been made unless it was previously being produced in being. And the same is true of ceasing-to-be and having ceased-to-be. For a “having-ceased-to-be” is always prior to a “ceasing-to-be” and a “ceasing-to-be” prior to a “having ceased-to-be”. Unde manifestum est quod omne quod factum est, necesse est prius fieri; et omne quod fit, necesse est prius factum esse aliquo modo. Et hoc ideo, quia omnis magnitudo et omne tempus sunt in infinitum divisibilia. Et ideo in quocumque tempore fit aliquid, hoc non erit sicut in primo, quia erit accipere partem priorem. Et hoc quod dictum est de generatione et corruptione, intelligendum est etiam de illuminatione, quae est terminus motus localis corporis illuminantis, sicut generatio et corruptio est terminus alterationis. From this it is evident that whatever has been made was previously being made, and that all that is being made has in some way previously been made. And the reason is that every magnitude and every period of time are infinitely divisible. Consequently, in whatever period of time something comes to be, it is not coming to be in that time as in a first time, because it always possible to find a period previous. And what we have said of generation and ceasing-to-be is true also of illumination, which is the termination of the local motion of the illuminating body, just as generation and ceasing-to-be is the terminus of an alteration.
Finite and infinite are found simultaneously in magnitude, time, mobile, and motion
Chapter 7 Ἐπεὶ δὲ πᾶν τὸ κινούμενον ἐν χρόνῳ κινεῖται, καὶ ἐν τῷ πλείονι μεῖζον μέγεθος, ἐν τῷ ἀπείρῳ χρόνῳ ἀδύνατόν ἐστιν πεπερασμένην κινεῖσθαι, μὴ τὴν αὐτὴν αἰεὶ καὶ τῶν ἐκείνης τι κινούμενον, ἀλλ' ἐν ἅπαντι ἅπασαν. Now since the motion of everything that is in motion occupies a period of time, and a greater magnitude is traversed in a longer time, it is impossible that a thing should undergo a finite motion in an infinite time, if this is understood to mean not that the same motion or a part of it is continually repeated, but that the whole infinite time is occupied by the whole finite motion. ὅτι μὲν οὖν εἴ τι ἰσοταχῶς κινοῖτο, ἀνάγκη τὸ πεπερασμένον ἐν πεπερασμένῳ κινεῖσθαι, δῆλον (ληφθέντος γὰρ μορίου ὃ καταμετρήσει τὴν ὅλην, ἐν ἴσοις χρόνοις τοσούτοις ὅσα τὰ μόριά ἐστιν, τὴν ὅλην κεκίνηται, ὥστ' ἐπεὶ ταῦτα πεπέρανται καὶ τῷ πόσον ἕκαστον καὶ τῷ ποσάκις ἅπαντα, καὶ ὁ χρόνος ἂν εἴη πεπερασμένος· τοσαυτάκις γὰρ ἔσται τοσοῦτος, ὅσος ὁ τοῦ μορίου χρόνος πολλαπλασιασθεὶς τῷ πλήθει τῶν μορίων)· In all cases where a thing is in motion with uniform velocity it is clear that the finite magnitude is traversed in a finite time. For if we take a part of the motion which shall be a measure of the whole, the whole motion is completed in as many equal periods of the time as there are parts of the motion. Consequently, since these parts are finite, both in size individually and in number collectively, the whole time must also be finite: for it will be a multiple of the portion, equal to the time occupied in completing the aforesaid part multiplied by the number of the parts. ἀλλὰ δὴ κἂν μὴ ἰσοταχῶς, διαφέρει οὐθέν. ἔστω γὰρ ἐφ' ἧς τὸ ΑΒ διάστημα πεπερασμένον, ὃ κεκίνηται (238a.) ἐν τῷ ἀπείρῳ, καὶ ὁ χρόνος ἄπειρος ἐφ' οὗ τὸ ΓΔ. εἰ δὴ ἀνάγκη πρότερον ἕτερον ἑτέρου κεκινῆσθαι (τοῦτο δὲ δῆλον, ὅτι τοῦ χρόνου ἐν τῷ προτέρῳ καὶ ὑστέρῳ ἕτερον κεκίνηται· ἀεὶ γὰρ ἐν τῷ πλείονι ἕτερον ἔσται κεκινημένον, ἐάν τε ἰσοταχῶς ἐάν τε μὴ ἰσοταχῶς μεταβάλλῃ, καὶ ἐάν τε ἐπιτείνῃ ἡ κίνησις ἐάν τε ἀνιῇ ἐάν τε μένῃ, οὐθὲν ἧττον), εἰλήφθω δή τι τοῦ ΑΒ διαστήματος, τὸ ΑΕ, ὃ καταμετρήσει τὴν ΑΒ. τοῦτο δὴ τοῦ ἀπείρου ἔν τινι ἐγένετο χρόνῳ· ἐν ἀπείρῳ γὰρ οὐχ οἷόν τε· τὸ γὰρ ἅπαν ἐν ἀπείρῳ. καὶ πάλιν ἕτερον δὴ ἐὰν λάβω ὅσον τὸ ΑΕ, ἀνάγκη ἐν πεπερασμένῳ χρόνῳ· τὸ γὰρ ἅπαν ἐν ἀπείρῳ. καὶ οὕτω δὴ λαμβάνων, ἐπειδὴ τοῦ μὲν ἀπείρου οὐθὲν ἔστι μόριον ὃ καταμετρήσει (ἀδύνατον γὰρ τὸ ἄπειρον εἶναι ἐκ πεπερασμένων καὶ ἴσων καὶ ἀνίσων, διὰ τὸ καταμετρηθήσεσθαι τὰ πεπερασμένα πλήθει καὶ μεγέθει ὑπό τινος ἑνός, ἐάν τε ἴσα ᾖ ἐάν τε ἄνισα, ὡρισμένα δὲ τῷ μεγέθει, οὐθὲν ἧττον), τὸ δὲ διάστημα τὸ πεπερασμένον ποσοῖς τοῖς ΑΕ μετρεῖται, ἐν πεπερασμένῳ ἂν χρόνῳ τὸ ΑΒ κινοῖτο (ὡσαύτως δὲ καὶ ἐπὶ ἠρεμήσεως)· ὥστε οὔτε γίγνεσθαι οὔτε φθείρεσθαι οἷόν τε ἀεί τι τὸ αὐτὸ καὶ ἕν. But it makes no difference even if the velocity is not uniform. For let us suppose that the line AB represents a finite stretch over which a thing has been moved in the given time, and let GD be the infinite time. Now if one part of the stretch must have been traversed before another part (this is clear, that in the earlier and in the later part of the time a different part of the stretch has been traversed: for as the time lengthens a different part of the motion will always be completed in it, whether the thing in motion changes with uniform velocity or not: and whether the rate of motion increases or diminishes or remains stationary this is none the less so), let us then take AE a part of the whole stretch of motion AB which shall be a measure of AB. Now this part of the motion occupies a certain period of the infinite time: it cannot itself occupy an infinite time, for we are assuming that that is occupied by the whole AB. And if again I take another part equal to AE, that also must occupy a finite time in consequence of the same assumption. And if I go on taking parts in this way, on the one hand there is no part which will be a measure of the infinite time (for the infinite cannot be composed of finite parts whether equal or unequal, because there must be some unity which will be a measure of things finite in multitude or in magnitude, which, whether they are equal or unequal, are none the less limited in magnitude); while on the other hand the finite stretch of motion AB is a certain multiple of AE: consequently the motion AB must be accomplished in a finite time. Moreover it is the same with coming to rest as with motion. And so it is impossible for one and the same thing to be infinitely in process of becoming or of perishing. ὁ αὐτὸς δὲ λόγος καὶ ὅτι οὐδ' ἐν πεπερασμένῳ χρόνῳ ἄπειρον οἷόν τε κινεῖσθαι οὐδ' ἠρεμίζεσθαι, οὔθ' ὁμαλῶς κινούμενον οὔτ' ἀνωμάλως. ληφθέντος γάρ τινος μέρους ὃ ἀναμετρήσει τὸν ὅλον χρόνον, ἐν τούτῳ ποσόν τι διέξεισιν τοῦ μεγέθους καὶ οὐχ ὅλον (ἐν γὰρ τῷ παντὶ τὸ ὅλον), καὶ πάλιν ἐν τῷ ἴσῳ ἄλλο, καὶ ἐν ἑκάστῳ ὁμοίως, εἴτε ἴσον εἴτε ἄνισον τῷ ἐξ ἀρχῆς· διαφέρει γὰρ οὐδέν, εἰ μόνον πεπερασμένον ἕκαστον· δῆλον γὰρ ὡς ἀναιρουμένου τοῦ χρόνου τὸ ἄπειρον οὐκ ἀναιρεθήσεται, πεπερασμένης τῆς ἀφαιρέσεως γιγνομένης καὶ τῷ ποσῷ καὶ τῷ ποσάκις· ὥστ' οὐ δίεισιν ἐν πεπερασμένῳ χρόνῳ τὸ ἄπειρον. οὐδέν τε διαφέρει τὸ μέγεθος ἐπὶ θάτερα ἢ ἐπ' ἀμφότερα εἶναι ἄπειρον· ὁ γὰρ αὐτὸς ἔσται λόγος. The reasoning he will prove that in a finite time there cannot be an infinite extent of motion or of coming to rest, whether the motion is regular or irregular. For if we take a part which shall be a measure of the whole time, in this part a certain fraction, not the whole, of the magnitude will be traversed, because we assume that the traversing of the whole occupies all the time. Again, in another equal part of the time another part of the magnitude will be traversed: and similarly in each part of the time that we take, whether equal or unequal to the part originally taken. It makes no difference whether the parts are equal or not, if only each is finite: for it is clear that while the time is exhausted by the subtraction of its parts, the infinite magnitude will not be thus exhausted, since the process of subtraction is finite both in respect of the quantity subtracted and of the number of times a subtraction is made. Consequently the infinite magnitude will not be traversed in finite time: and it makes no difference whether the magnitude is infinite in only one direction or in both: for the same reasoning will hold good. ἀποδεδειγμένων δὲ τούτων φανερὸν ὅτι οὐδὲ τὸ πεπερασμένον μέγεθος τὸ ἄπειρον ἐνδέχεται διελθεῖν ἐν πεπερασμένῳ διὰ τὴν αὐτὴν αἰτίαν· ἐν γὰρ τῷ μορίῳ τοῦ χρόνου πεπερασμένον δίεισι, καὶ ἐν ἑκάστῳ ὡσαύτως, ὥστ' ἐν τῷ παντὶ πεπερασμένον. This having been proved, it is evident that neither can a finite magnitude traverse an infinite magnitude in a finite time, the reason being the same as that given above: in part of the time it will traverse a finite magnitude and in each several part likewise, so that in the whole time it will traverse a finite magnitude. ἐπεὶ δὲ τὸ πεπερασμένον οὐ δίεισι τὸ ἄπειρον (238b.) ἐν πεπερασμένῳ χρόνῳ, δῆλον ὡς οὐδὲ τὸ ἄπειρον τὸ πεπερασμένον· εἰ γὰρ τὸ ἄπειρον τὸ πεπερασμένον, ἀνάγκη καὶ τὸ πεπερασμένον διιέναι τὸ ἄπειρον. οὐδὲν γὰρ διαφέρει ὁποτερονοῦν εἶναι τὸ κινούμενον· ἀμφοτέρως γὰρ τὸ πεπερασμένον δίεισι τὸ ἄπειρον. ὅταν γὰρ κινῆται τὸ ἄπειρον ἐφ' ᾧ τὸ Α, ἔσται τι αὐτοῦ κατὰ τὸ Β τὸ πεπερασμένον, οἷον τὸ ΓΔ, καὶ πάλιν ἄλλο καὶ ἄλλο, καὶ αἰεὶ οὕτως. ὥσθ' ἅμα συμβήσεται τὸ ἄπειρον κεκινῆσθαι τὸ πεπερασμένον καὶ τὸ πεπερασμένον διεληλυθέναι τὸ ἄπειρον· οὐδὲ γὰρ ἴσως δυνατὸν ἄλλως τὸ ἄπειρον κινηθῆναι τὸ πεπερασμένον ἢ τῷ τὸ πεπερασμένον διιέναι τὸ ἄπειρον, ἢ φερόμενον ἢ ἀναμετροῦν. ὥστ' ἐπεὶ τοῦτ' ἀδύνατον, οὐκ ἂν διίοι τὸ ἄπειρον τὸ πεπερασμένον. And since a finite magnitude will not traverse an infinite in a finite time, it is clear that neither will an infinite traverse a finite in a finite time. For if the infinite could traverse the finite, the finite could traverse the infinite; for it makes no difference which of the two is the thing in motion; either case involves the traversing of the infinite by the finite. For when the infinite magnitude A is in motion a part of it, say GD, will occupy the finite and then another, and then another, and so on to infinity. Thus the two results will coincide: the infinite will have completed a motion over the finite and the finite will have traversed the infinite: for it would seem to be impossible for the motion of the infinite over the finite to occur in any way other than by the finite traversing the infinite either by locomotion over it or by measuring it. Therefore, since this is impossible, the infinite cannot traverse the finite. ἀλλὰ μὴν οὐδὲ τὸ ἄπειρον ἐν πεπερασμένῳ χρόνῳ τὸ ἄπειρον· δίεισιν· εἰ γὰρ τὸ ἄπειρον, καὶ τὸ πεπερασμένον· ἐνυπάρχει γὰρ τῷ ἀπείρῳ τὸ πεπερασμένον. Nor again will the infinite traverse the infinite in a finite time. Otherwise it would also traverse the finite, for the infinite includes the finite. ἔτι δὲ καὶ τοῦ χρόνου ληφθέντος ἡ αὐτὴ ἔσται ἀπόδειξις. We can further prove this in the same way by taking the time as our starting-point. ἐπεὶ δ' οὔτε τὸ πεπερασμένον τὸ ἄπειρον οὔτε τὸ ἄπειρον τὸ πεπερασμένον οὔτε τὸ ἄπειρον τὸ ἄπειρον ἐν πεπερασμένῳ χρόνῳ κινεῖται, φανερὸν ὅτι οὐδὲ κίνησις ἔσται ἄπειρος ἐν πεπερασμένῳ χρόνῳ· τί γὰρ διαφέρει τὴν κίνησιν ἢ τὸ μέγεθος ποιεῖν ἄπειρον; ἀνάγκη γάρ, εἰ ὁποτερονοῦν, καὶ θάτερον εἶναι ἄπειρον· πᾶσα γὰρ φορὰ ἐν τόπῳ. Since, then, it is established that in a finite time neither will the finite traverse the infinite, nor the infinite the finite, nor the infinite the infinite, it is evident also that in a finite time there cannot be infinite motion: for what difference does it make whether we take the motion or the magnitude to be infinite? If either of the two is infinite, the other must be so likewise: for all locomotion is in space.
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. 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. 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. 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. Circa primum duo facit: primo proponit quod intendit; secundo probat propositum, ibi: quod igitur si aliquid moveatur et cetera. in regard to the first he does two things: First he proposes what he intends; Secondly, he proves his proposition, at 843. 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) 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. 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. 843. Then at (645) 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. 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. He says therefore first (645) 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. 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. 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. 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: 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. 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. Therefore at (646) 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. 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. 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. 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. 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. 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. 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. 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) 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. 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. 846. Then at (648) 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. 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. 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. 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) 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. 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) 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. 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) 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. 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) 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.
Things pertaining to the division of “coming to a stand” and “rest”
Chapter 8 8 Ἐπεὶ δὲ πᾶν ἢ κινεῖται ἢ ἠρεμεῖ τὸ πεφυκὸς ὅτε πέφυκε καὶ οὗ καὶ ὥς, ἀνάγκη τὸ ἱστάμενον ὅτε ἵσταται κινεῖσθαι· εἰ γὰρ μὴ κινεῖται, ἠρεμήσει, ἀλλ' οὐκ ἐνδέχεται ἠρεμίζεσθαι τὸ ἠρεμοῦν. Since everything to which motion or rest is natural is in motion or at rest in the natural time, place, and manner, that which is coming to a stand, when it is coming to a stand, must be in motion: for if it is not in motion it must be at rest: but that which is at rest cannot be coming to rest. τούτου δ' ἀποδεδειγμένου φανερὸν ὅτι καὶ ἐν χρόνῳ ἵστασθαι ἀνάγκη (τὸ γὰρ κινούμενον ἐν χρόνῳ κινεῖται, τὸ δ' ἱστάμενον δέδεικται κινούμενον, ὥστε ἀνάγκη ἐν χρόνῳ ἵστασθαι)· ἔτι δ' εἰ τὸ μὲν θᾶττον καὶ βραδύτερον ἐν χρόνῳ λέγομεν, ἵστασθαι δ' ἔστιν θᾶττον καὶ βραδύτερον. From this it evidently follows that coming to a stand must occupy a period of time: for the motion of that which is in motion occupies a period of time, and that which is coming to a stand has been shown to be in motion: consequently coming to a stand must occupy a period of time. Again, since the terms 'quicker' and 'slower' are used only of that which occupies a period of time, and the process of coming to a stand may be quicker or slower, the same conclusion follows. ἐν ᾧ δὲ χρόνῳ πρώτῳ τὸ ἱστάμενον ἵσταται, ἐν ὁτῳοῦν ἀνάγκη τούτου ἵστασθαι. διαιρεθέντος γὰρ τοῦ χρόνου εἰ μὲν ἐν μηδετέρῳ τῶν μερῶν ἵσταται, οὐδ' ἐν τῷ ὅλῳ, ὥστ' οὐκ ἂν ἵσταιτο τὸ ἱστάμενον· εἰ δ' ἐν θατέρῳ, οὐκ ἂν ἐν πρώτῳ τῷ ὅλῳ ἵσταιτο· καθ' ἕτερον γὰρ ἐν τούτῳ ἵσταται, καθάπερ ἐλέχθη καὶ ἐπὶ τοῦ κινουμένου πρότερον. And that which is coming to a stand must be coming to a stand in any part of the primary time in which it is coming to a stand. For if it is coming to a stand in neither of two parts into which the time may be divided, it cannot be coming to a stand in the whole time, with the result that that that which is coming to a stand will not be coming to a stand. If on the other hand it is coming to a stand in only one of the two parts of the time, the whole cannot be the primary time in which it is coming to a stand: for it is coming to a stand in the whole time not primarily but in virtue of something distinct from itself, the argument being the same as that which we used above about things in motion. ὥσπερ δὲ τὸ κινούμενον οὐκ ἔστιν (239a.) ἐν ᾧ πρώτῳ κινεῖται, οὕτως οὐδ' ἐν ᾧ ἵσταται τὸ ἱστάμενον· οὔτε γὰρ τοῦ κινεῖσθαι οὔτε τοῦ ἵστασθαί ἐστίν τι πρῶτον. ἔστω γὰρ ἐν ᾧ πρώτῳ ἵσταται ἐφ' ᾧ τὸ ΑΒ. τοῦτο δὴ ἀμερὲς μὲν οὐκ ἐνδέχεται εἶναι (κίνησις γὰρ οὐκ ἔστιν ἐν τῷ ἀμερεῖ διὰ τὸ κεκινῆσθαί τι ἂν αὐτοῦ, τὸ δ' ἱστάμενον δέδεικται κινούμε νον)· ἀλλὰ μὴν εἰ διαιρετόν ἐστιν, ἐν ὁτῳοῦν αὐτοῦ τῶν μερῶν ἵσταται· τοῦτο γὰρ δέδεικται πρότερον, ὅτι ἐν ᾧ πρώτῳ ἵσταται, ἐν ὁτῳοῦν τῶν ἐκείνου ἵσταται. ἐπεὶ οὖν χρόνος ἐστὶν ἐν ᾧ πρώτῳ ἵσταται, καὶ οὐκ ἄτομον, ἅπας δὲ χρόνος εἰς ἄπειρα μεριστός, οὐκ ἔσται ἐν ᾧ πρώτῳ ἵσταται. And just as there is no primary time in which that which is in motion is in motion, so too there is no primary time in which that which is coming to a stand is coming to a stand, there being no primary stage either of being in motion or of coming to a stand. For let AB be the primary time in which a thing is coming to a stand. Now AB cannot be without parts: for there cannot be motion in that which is without parts, because the moving thing would necessarily have been already moved for part of the time of its movement: and that which is coming to a stand has been shown to be in motion. But since Ab is therefore divisible, the thing is coming to a stand in every one of the parts of AB: for we have shown above that it is coming to a stand in every one of the parts in which it is primarily coming to a stand. Since then, that in which primarily a thing is coming to a stand must be a period of time and not something indivisible, and since all time is infinitely divisible, there cannot be anything in which primarily it is coming to a stand. οὐδὲ δὴ τὸ ἠρεμοῦν ὅτε πρῶτον ἠρέμησεν ἔστιν. ἐν ἀμερεῖ μὲν γὰρ οὐκ ἠρέμησεν διὰ τὸ μὴ εἶναι κίνησιν ἐν ἀτόμῳ, ἐν ᾧ δὲ τὸ ἠρεμεῖν, καὶ τὸ κινεῖσθαι (τότε γὰρ ἔφαμεν ἠρεμεῖν, ὅτε καὶ ἐν ᾧ πεφυκὸς κινεῖσθαι μὴ κινεῖται τὸ πεφυκός)· ἔτι δὲ καὶ τότε λέγομεν ἠρεμεῖν, ὅταν ὁμοίως ἔχῃ νῦν καὶ πρότερον, ὡς οὐχ ἑνί τινι κρίνοντες ἀλλὰ δυοῖν τοῖν ἐλαχίστοιν· ὥστ' οὐκ ἔσται ἐν ᾧ ἠρεμεῖ ἀμερές. εἰ δὲ μεριστόν, χρόνος ἂν εἴη, καὶ ἐν ὁτῳοῦν αὐτοῦ τῶν μερῶν ἠρεμήσει. τὸν αὐτὸν γὰρ τρόπον δειχθήσεται ὃν καὶ ἐπὶ τῶν πρότερον· ὥστ' οὐθὲν ἔσται πρῶτον. τούτου δ' αἴτιον ὅτι ἠρεμεῖ μὲν καὶ κινεῖται πᾶν ἐν χρόνῳ, χρόνος δ' οὐκ ἔστι πρῶτος οὐδὲ μέγεθος οὐδ' ὅλως συνεχὲς οὐδέν· ἅπαν γὰρ εἰς ἄπειρα μεριστόν. Nor again can there be a primary time at which the being at rest of that which is at rest occurred: for it cannot have occurred in that which has no parts, because there cannot be motion in that which is indivisible, and that in which rest takes place is the same as that in which motion takes place: for we defined a state of rest to be the state of a thing to which motion is natural but which is not in motion when (that is to say in that in which) motion would be natural to it. Again, our use of the phrase 'being at rest' also implies that the previous state of a thing is still unaltered, not one point only but two at least being thus needed to determine its presence: consequently that in which a thing is at rest cannot be without parts. Since, then it is divisible, it must be a period of time, and the thing must be at rest in every one of its parts, as may be shown by the same method as that used above in similar demonstrations. So there can be no primary part of the time: and the reason is that rest and motion are always in a period of time, and a period of time has no primary part any more than a magnitude or in fact anything continuous: for everything continuous is divisible into an infinite number of parts. ἐπεὶ δὲ πᾶν τὸ κινούμενον ἐν χρόνῳ κινεῖται καὶ ἔκ τινος εἴς τι μεταβάλλει, ἐν ᾧ χρόνῳ κινεῖται καθ' αὑτὸν καὶ μὴ τῷ ἐν ἐκείνου τινί, ἀδύνατον τότε κατά τι εἶναι πρῶτον τὸ κινούμενον. And since everything that is in motion is in motion in a period of time and changes from something to something, when its motion is comprised within a particular period of time essentially—that is to say when it fills the whole and not merely a part of the time in question—it is impossible that in that time that which is in motion should be over against some particular thing primarily. τὸ γὰρ ἠρεμεῖν ἐστιν τὸ ἐν τῷ αὐτῷ εἶναι χρόνον τινὰ καὶ αὐτὸ καὶ τῶν μερῶν ἕκαστον. οὕτως γὰρ λέγομεν ἠρεμεῖν, ὅταν ἐν ἄλλῳ καὶ ἄλλῳ τῶν νῦν ἀληθὲς ᾖ εἰπεῖν ὅτι ἐν τῷ αὐτῷ καὶ αὐτὸ καὶ τὰ μέρη. εἰ δὲ τοῦτ' ἔστι τὸ ἠρεμεῖν, οὐκ ἐνδέχεται τὸ μεταβάλλον κατά τι εἶναι ὅλον κατὰ τὸν πρῶτον χρόνον· ὁ γὰρ χρόνος διαιρετὸς ἅπας, ὥστε ἐν ἄλλῳ καὶ ἄλλῳ αὐτοῦ μέρει ἀληθὲς ἔσται εἰπεῖν ὅτι ἐν ταὐτῷ ἐστιν καὶ αὐτὸ καὶ τὰ μέρη. εἰ γὰρ μὴ οὕτως ἀλλ' ἐν ἑνὶ μόνῳ τῶν νῦν, οὐκ ἔσται χρόνον οὐδένα κατά τι, ἀλλὰ κατὰ τὸ πέρας τοῦ χρόνου. ἐν δὲ τῷ νῦν ἔστιν μὲν ἀεὶ κατά τι μὲν (239b.) ὄν, οὐ μέντοι ἠρεμεῖ· οὔτε γὰρ κινεῖσθαι οὔτ' ἠρεμεῖν ἔστιν ἐν τῷ νῦν, ἀλλὰ μὴ κινεῖσθαι μὲν ἀληθὲς ἐν τῷ νῦν καὶ εἶναι κατά τι, ἐν χρόνῳ δ' οὐκ ἐνδέχεται εἶναι κατά τι ἠρεμοῦν· συμβαίνει γὰρ τὸ φερόμενον ἠρεμεῖν. For if a thing—itself and each of its parts—occupies the same space for a definite period of time, it is at rest: for it is in just these circumstances that we use the term 'being at rest'—when at one moment after another it can be said with truth that a thing, itself and its parts, occupies the same space. So if this is being at rest it is impossible for that which is changing to be as a whole, at the time when it is primarily changing, over against any particular thing (for the whole period of time is divisible), so that in one part of it after another it will be true to say that the thing, itself and its parts, occupies the same space. If this is not so and the aforesaid proposition is true only at a single moment, then the thing will be over against a particular thing not for any period of time but only at a moment that limits the time. It is true that at any moment it is always over against something stationary: but it is not at rest: for at a moment it is not possible for anything to be either in motion or at rest. So while it is true to say that that which is in motion is at a moment not in motion and is opposite some particular thing, it cannot in a period of time be over against that which is at rest: for that would involve the conclusion that that which is in locomotion is at rest.
Postquam philosophus determinavit de iis quae pertinent ad divisionem motus, hic determinat de iis quae pertinent ad divisionem quietis. Et quia statio est generatio quietis, ut in quinto dictum est, primo determinat ea quae pertinent ad stationem; secundo ea quae pertinent ad quietem, ibi: neque igitur quiescens et cetera. 851. After finishing the things that pertain to the division of motion, the Philosopher now determines about things that pertain to the division of rest. And because coming to rest is generation of rest, as we have said in Book V. First he determines the things that pertain to coming to rest; Secondly, the things that pertain to rest, at 856. Circa primum tria facit: primo ostendit quod omne quod stat, movetur; secundo quod omne quod stat, stat in tempore, ibi: hoc autem demonstrato etc.; tertio ostendit quomodo primum dicatur in statione, ibi: in quo autem tempore et cetera. About the first he does three things: First he shows that whatever is coming to rest is being moved! Secondly, whatever is coming to rest does so in time, at 853; Thirdly, how a first is spoken of in coming to rest, at 854. Primum ostendit sic. Omne quod natum est moveri, eo tempore quando natum est moveri, et secundum illud et eo modo prout natum est, oportet quod moveatur vel quiescat: sed illud quod stat, idest tendit ad quietem, nondum quiescit; quia contingeret quod aliquid simul quiescens, idest in quietem tendens, quiesceret, idest in quiete esset: ergo omne quod stat, idest in quietem tendit, movetur quando stat. 852. He shows the first at (653): Everything apt to be moved must be either in motion or at rest at the time when it is apt to be moved and in the place in which it is apt to be moved and in the way in which it is apt to be moved. But what is coming to rest is not yet at rest—otherwise, it would happen that a thing would be at the same time tending to rest and actually resting. Therefore, whatever is coining to rest is in motion, when it is coming to rest. Deinde cum dicit: hoc autem demonstrato etc., probat quod omne quod stat, stat in tempore, duabus rationibus: quarum prima talis est. Omne quod movetur, movetur in tempore, ut supra probatum est: sed omne quod stat movetur, ut nunc probatum est: ergo omne quod stat, stat in tempore. 853. Then at (654) he proves by two arguments that whatever is coming to rest is doing so in time. For whatever is being moved is being moved in time, as has been proved. But whatever is coming to rest is being moved, as we have just proved. Therefore, whatever is coming to rest is coming to rest in time. Secunda ratio est, quia velocitas et tarditas determinantur secundum tempus: sed contingit aliquid velocius et tardius stare, idest in quietem tendere: ergo omne quod stat, stat in tempore. The second argument is that swiftness and slowness are determined according to time. But it can happen that something comes to rest either more swiftly or more slowly. Therefore, whatever is coming to rest does so in time. Deinde cum dicit: in quo autem tempore etc., ostendit qualiter dicatur primum in statione. Et circa hoc duo facit: primo ostendit qualiter dicatur aliquid stare in aliquo tempore primo, secundum quod primum opponitur ei quod dicitur secundum partem; secundo ostendit quod in statione non est accipere aliquam primam partem, ibi: sicut autem quod movetur et cetera. 854. Then at (655) he shows how “first” is spoken of in coming to rest. About this he does two things: First he shows how something is said to be “first” coming to rest in a given time, where “first” is opposed to what is spoken of in reference to a part; Secondly, he shows that in coming to rest, it is not possible to discern a first part, at 855. Dicit ergo primo quod si in aliquo tempore dicatur aliquid stare primo et per se, et non ratione partis, necesse est quod stetur in qualibet parte illius temporis. Dividetur enim tempus in duas partes; et si dicatur quod in neutra parte stet, sequetur quod non stet in toto, in quo tamen ponebatur stare; ergo stans non stat. Neque etiam potest dici quod in altera tantum parte stet: quia sic non primo staretur in toto tempore, sed solum ratione partis. Unde relinquitur quod stet in utroque. Sic enim dicitur primo stare in toto, quia stat in utraque parte, sicut dictum est supra de eo quod movetur. He says therefore first (655) that if at a certain time something is said to be coming to rest first and per se and not by reason of a part, then it must be coming to rest in each part of that time. For time can be divided into two parts, and if it is said that it is coming to rest in neither, it will follow that it is not coming to rest in the whole time, in which it was assumed to be coming to rest. Therefore, something coming to rest is not coming to rest. Nor can it be said that it is coming to rest in only one of the parts, because then it would not be coming to rest first, but only by reason of a part. Hence it will remain that it is coming to rest in both. For it is said to be coming to rest in the whole time only because it is coming to rest in each part, as was said above about things in motion. Deinde cum dicit: sicut autem quod movetur etc., ostendit quod non est accipere aliquam primam partem in statione. Et dicit quod sicut non est accipere aliquam primam partem temporis, in qua aliquod mobile movetur, ita etiam est in statione; quia neque in ipso moveri, neque in ipso stare potest esse aliqua prima pars. 855. Then at (656) he shows that there is no first part in coming to rest. And he says that just as it is not possible to find in time a first part in which a mobile is being moved, so also in regard to coming to rest, because in neither case can there be a first part. Quod si non concedatur, sit prima pars temporis, in qua statur, ab: quae quidem non potest esse impartibilis, quia ostensum est supra quod motus non est in impartibili temporis, eo quod semper quod movetur, iam per aliquid motum est, ut supra ostensum est; demonstratum est etiam nunc, quod omne quod stat, movetur. Unde relinquitur quod ab sit divisibile. Ergo in qualibet parte eius statur: iam enim ostensum est quod quando in aliquo tempore dicitur stari primo et per se, et non ratione partis, in qualibet parte illius statur. Ergo cum sit pars prior toto, non erat ab primum in quo statur. Et quia omne illud in quo statur, est tempus, et non est aliquid indivisibile temporis; omne autem tempus est divisibile in infinitum: sequitur quod non erit accipere primum in quo stetur. If this is denied, then let AB be the first part of time in which something is coming to rest. This part cannot be indivisible, because it has been shown above that motion does not occur in an indivisible of time (for it is always true that whatever is being moved has already been moved, as we have shown above) and, moreover, whatever is coming to rest is being moved, as we have just now proved. Hence AB must be divisible. Therefore, there is a coming to rest in each part of it, for we have just shown that when in a given time something is coming to rest first and per se and not by reason of a part, it is coming to rest in each part of that given time. Therefore, since the part is prior to the whole, AB was not, the first in which there was a coming to rest. And because that in which something is coming to rest is a time and all time is divisible ad infinitum, it follows that it is impossible to find a first in which something is coming to rest. Deinde cum dicit: neque igitur quiescens etc., ostendit idem de quiete. Et circa hoc duo facit: primo ostendit quod non est accipere primum in quiete; secundo ponit quandam considerationem, per quam motus a quiete distinguitur, ibi: quoniam autem omne quod movetur et cetera. 856. Then at (657) he shows the same thing is true for rest. About this he does two things: First he shows that there is no first in rest; Secondly, he gives a method to distinguish motion from rest, Et quia eadem ratio est quare non sit primum in motu, statione et quiete, ideo ex his quae supra dicta sunt de motu et statione, concludit idem in quiete. Et dicit quod non est accipere aliquod primum in quo quiescens quieverit. Et ad hoc probandum resumit quoddam quod supra probatum est, scilicet quod nihil quiescat in impartibili temporis; et resumit etiam duas rationes quibus hoc supra probatum est. Quarum prima est, quod motus non est in indivisibili temporis: in eodem autem est quiescere et moveri; quia non dicimus quiescere, nisi quando id quod aptum natum est moveri, non movetur tunc quando aptum natum est moveri et secundum id secundum quod natum est moveri, puta qualitatem aut locum, aut aliquid huiusmodi. Unde relinquitur quod nihil quiescat in impartibili temporis. And because it is for the same reason that no first is found in notion and in coming-to-rest and in rest, therefore, he concludes the same thing for rest as he concluded for motion and coming-to rest. And he says that there is no first in which a thing at rest has been at rest. To prove this he repeats something previously proved, namely, that nothing is at rest in an indivisible of time. Likewise, he repeats the two reasons he used when he proved this. The first of which is that there is no motion in an indivisible of time. But to rest and to be in motion are in the same: because we do not say that something is resting, unless what is capable of being moved is not being moved when it is apt to be moved and in the sphere in which it is apt to be moved; for example, quality or place or something of this sort. Hence it remains that nothing is at rest in an indivisible of time. Secunda ratio est, quia tunc dicimus aliquid quiescere, quando similiter se habet nunc sicut prius; ac si non diiudicemus quietem per aliquod unum tantum, sed per comparationem duorum ad invicem, ex eo scilicet quod similiter se habet in duobus. Sed in impartibili non est accipere nunc et prius, neque aliqua duo: ergo illud temporis in quo aliquid quiescit, non est impartibile. The second reason is that it is then that we say something is at rest when it maintains itself as it was previously: as if to say that we do not judge rest by reason of one factor only but by comparing two things to one another and seeing that there is a similar situation in both. But it is impossible to find in something indivisible a “now” and something previous, or any two things. Therefore, that element of time in which something is at rest is not indivisible. Isto autem probato, procedit ulterius ad principale propositum ostendendum. Si enim illud in quo aliquid quiescit est partibile, habens in se prius est posterius, sequitur quod sit tempus: haec est enim ratio temporis. Et si est tempus, oportet quod in qualibet partium eius quiescat. Et hoc demonstrabitur eodem modo, sicut et supra monstratum est in motu et statione: quia scilicet si non quiescit in qualibet parte, aut ergo in nulla, aut in una tantum. Si in nulla, ergo neque in toto: si in una tantum, ergo in illa primo et non in toto. Si vero in qualibet parte temporis quiescit, non erit aliquid accipere primum in quiete, sicut neque in motu. Having established this, he proceeds further to prove the main. proposition. For if that in which something is at rest is divisible into parts that possess a prior and a subsequent, it follows that it is a time; for this is the very nature of time. And if it is time, then it must be resting in each part of it. And this will be demonstrated in the same way that it was demonstrated in motion and in coming to rest; namely, that if it is not at rest in each part, it will be at rest in no part or in one only. If in no part, then not in the whole; if in one only, then in that part first and not in the whole first. But if it is at rest in each part of the time, it will not be possible to discover a first in rest any more than in motion. Et huius causa est, quia unumquodque quiescit et movetur in tempore; sed in tempore non est accipere aliquod primum, sicut neque in magnitudine, neque in aliquo continuo, propter hoc quod omne continuum divisibile est in infinitum, et sic semper est accipere partem minorem parte. Et inde est quod neque in motu, neque in statione, neque in quiete est aliquid primum. The reason for this is that things are at rest and in motion in time. But in time there is no first any more than in a magnitude or in any continuum, for every continuum is divisible ad infinitum and, consequently, it is always possible to find a part smaller than another. And that is why there is no first in motion or in coming-to-rest or in rest. Deinde cum dicit: quoniam autem omne quod movetur etc., ponit quandam considerationem, per quam distinguitur id quod movetur ab eo quod quiescit. Et primo ponit eam; secundo probat, ibi: quiescere enim est et cetera. 857. Then at (658) he gives a way through which what is in motion is distinguished from what is at rest. First he mentions it; Secondly, he proves it, at 858. Circa primum praemittit duas suppositiones: quarum una est, quod omne quod movetur, movetur in tempore; secunda est, quod omne quod mutatur, mutatur ex uno termino in alium. Et ex his duobus intendit concludere tertium, scilicet quod si accipiatur aliquod mobile quod primo et per se moveatur, et non solum ratione suae partis, impossibile est quod sit secundum aliquid unum et idem illius rei in qua est motus, puta in uno et eodem loco vel in una et eadem dispositione albedinis, in aliquo tempore, ita quod accipiamus in tempore esse secundum se, et non ratione alicuius quod in tempore sit. In regard to the first he premises two suppositions, the first of which is that whatever is being moved is being moved in time. The second is that whatever is being changed is being changed from one terminus to another. From these two facts he intends to conclude a third; namely, that if you take a mobile, which is being moved first and per se and not by reason of its part only, it cannot remain one and the same with respect to that in which the motion is—for example, it cannot remain in one and the same place, or retain one and the same degree of whiteness—during a given period of time, provided that you take it as being in time according to itself and not according to something which is in time. Ideo autem oportet quod accipiatur mobile quod primum movetur, quia nihil prohibet aliquid moveri secundum partem, et tamen ipsum manet per totum tempus in uno et eodem loco, sicut cum homo sedens movet pedem. The reason why you must take a mobile which is being moved first and per se is that there is nothing to prevent a thing from being moved according to a part even though it remains in one and the same place throughout the entire time, as when a man sitting down moves his foot. Ideo autem dicit ex parte temporis, in quo tempore movetur secundum se, et non quo in illius aliquo: quia aliquid, dum movetur, potest dici quod est in aliquo uno et eodem loco in tali die; sed hoc dicitur quia fuit in illo loco non in toto die, sed in aliquo nunc illius diei. And the reason why he speaks of a time throughout which something is being moved per se and not by reason of some element of time is that while a thing is being moved it can be said that on such and such a day it is in one and the same place; but this would be said, because it was in that place not throughout the day but in some “now” of that day. Deinde cum dicit: quiescere enim est etc., probat propositum. Et dicit quod si id quod mutatur, sit per totum aliquod tempus in aliquo uno et eodem, puta in uno loco, sequitur quod quiescat; propter hoc quod in quodam tempore est in uno et eodem loco et ipsum et quaelibet pars eius, et iam supra dictum est quod hoc est quiescere, cum verum sit dicere de aliquo quod ipsum et partes eius sunt in uno et eodem in diversis nunc. Si ergo haec est definitio eius quod est quiescere, et non contingit aliquid simul quiescere et moveri; sequitur quod non contingat id quod movetur esse totum secundum aliquid, idest in aliquo, puta in uno et eodem loco, secundum primum tempus, idest secundum aliquod totum tempus, et non tantum secundum aliquid eius. 858. Then at (659) he proves the proposition. And he says that if what is being changed is throughout a definite period of time in one and the same state—for example, in one place—it follows that it is at rest, due to the fact that in that time there is present in one and the same place the entire mobile and each part of it; for we have already said that to be at rest means to be able to say of something that it and its parts are in one and the same state in different “now’s”, If, therefore, this is the definition of being at rest and if nothing can be at rest and in motion at the same time, it follows that the whole which is being moved cannot be totally in one state, e.g., in one and the same place, during the whole time and not only in something of it. Et quare hoc sequatur ostendit. Quia omne tempus est divisibile in diversas partes, quarum una est prior altera: unde si per totum tempus sit in aliquo uno, verum erit dicere quod in alia et in alia parte temporis ipsum mobile et partes eius sint in uno et eodem, puta loco; quod est quiescere. Quia si dicatur quod non est in diversis partibus temporis in uno et eodem, sed solum in uno et eodem est per unum nunc, non sequitur quod sit tempus in quo est secundum aliquid, idest in aliquo uno et eodem, sed quod sit in uno et eodem secundum terminum temporis, idest secundum nunc. Why this follows he now explains. Every period of time is divisible into diverse parts, of which one is prior to another. Hence if something is in one state throughout the entire period, it will be true to say that in one and in another part of the time the whole mobile and its parts are in one and the same state, e.g., place—and this is to be at rest. For if it is said to be in one and the same state not in different parts of time but throughout one “now”, it does not follow that there is a time in which it is in one and the same state, but that there is a “now” in which it is in one and the same state. Licet autem ex hoc quod est aliquid esse in tempore in uno et eodem, sequatur quod quiescat, hoc tamen non sequitur de nunc, si sit ibi in uno solo nunc. Quia omne quod movetur, in quolibet nunc temporis in quo movetur, semper est manens, idest existens, secundum aliquid rei in qua est motus, puta secundum locum aut qualitatem aut quantitatem: non tamen quiescit, quia iam ostensum est quod neque quiescere neque moveri contingit in ipso nunc. Sed verum est dicere quod in ipso nunc aliquid non movetur, et quod in ipso nunc est alicubi, vel secundum aliquid, etiam illud quod movetur. Sed non contingit illud quod movetur, esse quiescens in tempore secundum aliquid: accideret enim quod aliquid, dum fertur, quiesceret; quod est impossibile. Relinquitur ergo quod omne quod movetur, quamdiu movetur, nunquam est in uno et eodem per duo nunc, sed per unum solum. For although from the fact that if something remains in one and the same state during a period of time, the conclusion can be drawn that it is at rest, that conclusion cannot be drawn if it remains in one and the same state in just one “now”. For whatever is being moved is always stable, i.e., existing, vis-a-vis something of that in which it is being moved in each “now” of the time in which it is being moved; for example, place or quality or quantity, Yet it is not for that reason at rest, because it has already been proved that neither rest nor motion can occur in a “now”. But it is true to say that in the very “now” something is not being moved and that in the “now” even what is being moved is somewhere or according to something. But what is being moved in time cannot be under any aspect at rest, for then it would happen that something is at rest while it is in motion—which is impossible. What remains, therefore, is that whatever is being moved is never, as long as it is being moved, in one and the same state for two “now’s” but for only one. Et hoc patet in motu locali. Sit enim magnitudo ac, et dividatur in duo media in puncto b, et accipiatur aliquod corpus quod sit o, aequale utrique, scilicet ab et bc, et moveatur de ab in bc. Si autem accipiantur loca totaliter ab invicem distincta, non est hic accipere nisi duo loca: sed manifestum est quod mobile non simul sed successive deserit primum locum et subintrat secundum; unde secundum quod locus est divisibilis in infinitum, secundum hoc multiplicantur loca in infinitum. Quia si dividatur ab in duo media in puncto d, et bc in duo media in puncto e, manifestum est quod de erit alius locus ab utroque. Et similiter semper divisione facta, fiet alius locus. 859. And this point is clear in local motion. For let AC be a magnitude divided in half at B and let 0 be a body equal to each half, i.e., to AB and to BC, and let that body be moved from AB to BC. If no part of one of these two places can be a part of the other, there will be only two places for that body on AC. But it is evident that 0 does not relinquish its first place and enter the second all at once but successively. Hence, because place is divisible ad infinitum, the places also are multiplied ad infinitum. For if the half part AB is again halved at D and the other half part BC at E, it is evident that DE will be a place distinct from both AB and BC. By continuing such divisions other and other places will be found. Et idem etiam manifestum est in alteratione: quia quod de albo transit in nigrum, per infinitos gradus albedinis et nigredinis et mediorum colorum pertransit. The same point is clear in alteration. For what passes from white to black passes through an infinitude of shades of whiteness and blackness and intermediate colors. Non tamen sequitur quod cum sint infinita media, quod nullo modo possit perveniri ad ultimum; quia huiusmodi media loca non sunt infinita in actu, sed in potentia tantum; sicut et magnitudo non est divisa actu in infinitum, sed in potentia divisibilis. However, it does not follow that since there are an infinitude of intermediates, the ultimate cannot be reached, because these intermediate places are infinite not in act but only in potency, just as a magnitude is not actually divided infinitely but is potentially divisible.
Zeno's arguments excluding all motion are resolved
Chapter 9 9 Ζήνων δὲ παραλογίζεται· εἰ γὰρ αἰεί, φησίν, ἠρεμεῖ πᾶν [ἢ κινεῖται] ὅταν ᾖ κατὰ τὸ ἴσον, ἔστιν δ' αἰεὶ τὸ φερόμενον ἐν τῷ νῦν, ἀκίνητον τὴν φερομένην εἶναι ὀϊστόν. τοῦτο δ' ἐστὶ ψεῦδος· οὐ γὰρ σύγκειται ὁ χρόνος ἐκ τῶν νῦν τῶν ἀδιαιρέτων, ὥσπερ οὐδ' ἄλλο μέγεθος οὐδέν. Zeno's reasoning, however, is fallacious, when he says that if everything when it occupies an equal space is at rest, and if that which is in locomotion is always occupying such a space at any moment, the flying arrow is therefore motionless. This is false, for time is not composed of indivisible moments any more than any other magnitude is composed of indivisibles. τέτταρες δ' εἰσὶν οἱ λόγοι περὶ κινήσεως Ζήνωνος οἱ παρέχοντες τὰς δυσκολίας τοῖς λύουσιν, πρῶτος μὲν ὁ περὶ τοῦ μὴ κινεῖσθαι διὰ τὸ πρότερον εἰς τὸ ἥμισυ δεῖν ἀφικέσθαι τὸ φερόμενον ἢ πρὸς τὸ τέλος, περὶ οὗ διείλομεν ἐν τοῖς πρότερον λόγοις. Zeno's arguments about motion, which cause so much disquietude to those who try to solve the problems that they present, are four in number. The first asserts the non-existence of motion on the ground that that which is in locomotion must arrive at the half-way stage before it arrives at the goal. This we have discussed above. δεύτερος δ' ὁ καλούμενος Ἀχιλλεύς· ἔστι δ' οὗτος, ὅτι τὸ βραδύτατον οὐδέποτε καταληφθήσεται θέον ὑπὸ τοῦ ταχίστου· ἔμπροσθεν γὰρ ἀναγκαῖον ἐλθεῖν τὸ διῶκον ὅθεν ὥρμησεν τὸ φεῦγον, ὥστε ἀεί τι προέχειν ἀναγκαῖον τὸ βραδύτερον. ἔστιν δὲ καὶ οὗτος ὁ αὐτὸς λόγος τῷ διχοτομεῖν, διαφέρει δ' ἐν τῷ διαιρεῖν μὴ δίχα τὸ προσλαμβανόμενον μέγεθος. τὸ μὲν οὖν μὴ καταλαμβάνε σθαι τὸ βραδύτερον συμβέβηκεν ἐκ τοῦ λόγου, γίγνεται δὲ παρὰ ταὐτὸ τῇ διχοτομίᾳ (ἐν ἀμφοτέροις γὰρ συμβαίνει μὴ ἀφικνεῖσθαι πρὸς τὸ πέρας διαιρουμένου πως τοῦ μεγέθους· ἀλλὰ πρόσκειται ἐν τούτῳ ὅτι οὐδὲ τὸ τάχιστον τετραγῳδημένον ἐν τῷ διώκειν τὸ βραδύτατον), ὥστ' ἀνάγκη καὶ τὴν λύσιν εἶναι τὴν αὐτήν. τὸ δ' ἀξιοῦν ὅτι τὸ προέχον οὐ καταλαμβάνεται, ψεῦδος· ὅτε γὰρ προέχει, οὐ καταλαμβάνεται· ἀλλ' ὅμως καταλαμβάνεται, εἴπερ δώσει διεξιέναι τὴν πεπερασμένην. οὗτοι μὲν οὖν οἱ δύο λόγοι, The second is the so-called 'Achilles', and it amounts to this, that in a race the quickest runner can never overtake the slowest, since the pursuer must first reach the point whence the pursued started, so that the slower must always hold a lead. This argument is the same in principle as that which depends on bisection, though it differs from it in that the spaces with which we successively have to deal are not divided into halves. The result of the argument is that the slower is not overtaken: but it proceeds along the same lines as the bisection-argument (for in both a division of the space in a certain way leads to the result that the goal is not reached, though the 'Achilles' goes further in that it affirms that even the quickest runner in legendary tradition must fail in his pursuit of the slowest), so that the solution must be the same. And the axiom that that which holds a lead is never overtaken is false: it is not overtaken, it is true, while it holds a lead: but it is overtaken nevertheless if it is granted that it traverses the finite distance prescribed. These then are two of his arguments. τρίτος δ' ὁ νῦν ῥηθείς, ὅτι ἡ ὀϊστὸς φερομένη ἕστηκεν. συμβαίνει δὲ παρὰ τὸ λαμβάνειν τὸν χρόνον συγκεῖσθαι ἐκ τῶν νῦν· μὴ διδομένου γὰρ τούτου οὐκ ἔσται ὁ συλλογισμός. The third is that already given above, to the effect that the flying arrow is at rest, which result follows from the assumption that time is composed of moments: if this assumption is not granted, the conclusion will not follow. τέταρτος δ' ὁ περὶ τῶν ἐν τῷ σταδίῳ κινουμένων ἐξ ἐναντίας ἴσων ὄγκων παρ' ἴσους, τῶν μὲν ἀπὸ τέλους τοῦ σταδίου τῶν δ' ἀπὸ μέσου, ἴσῳ τάχει, ἐν ᾧ συμβαίνειν (240a.) οἴεται ἴσον εἶναι χρόνον τῷ διπλασίῳ τὸν ἥμισυν. The fourth argument is that concerning the two rows of bodies, each row being composed of an equal number of bodies of equal size, passing each other on a race-course as they proceed with equal velocity in opposite directions, the one row originally occupying the space between the goal and the middle point of the course and the other that between the middle point and the starting-post. This, he thinks, involves the conclusion that half a given time is equal to double that time. ἔστι δ' ὁ παραλογισμὸς ἐν τῷ τὸ μὲν παρὰ κινούμενον τὸ δὲ παρ' ἠρεμοῦν τὸ ἴσον μέγεθος ἀξιοῦν τῷ ἴσῳ τάχει τὸν ἴσον φέρεσθαι χρόνον· τοῦτο δ' ἐστὶ ψεῦδος. The fallacy of the reasoning lies in the assumption that a body occupies an equal time in passing with equal velocity a body that is in motion and a body of equal size that is at rest; which is false. οἷον ἔστωσαν οἱ ἑστῶτες ἴσοι ὄγκοι ἐφ' ὧν τὰ ΑΑ, οἱ δ' ἐφ' ὧν τὰ ΒΒ ἀρχόμενοι ἀπὸ τοῦ μέσου, ἴσοι τὸν ἀριθμὸν τούτοις ὄντες καὶ τὸ μέγεθος, οἱ δ' ἐφ' ὧν τὰ ΓΓ ἀπὸ τοῦ ἐσχάτου, ἴσοι τὸν ἀριθμὸν ὄντες τούτοις καὶ τὸ μέγεθος, καὶ ἰσοταχεῖς τοῖς Β. συμβαίνει δὴ τὸ πρῶτον Β ἅμα ἐπὶ τῷ ἐσχάτῳ εἶναι καὶ τὸ πρῶτον Γ, παρ' ἄλληλα κινουμένων. συμβαίνει δὲ τὸ Γ παρὰ πάντα [τὰ Β] διεξεληλυθέναι, τὸ δὲ Β παρὰ τὰ ἡμίση· ὥστε ἥμισυν εἶναι τὸν χρόνον· ἴσον γὰρ ἑκάτερόν ἐστιν παρ' ἕκαστον. ἅμα δὲ συμβαίνει τὸ πρῶτον Β παρὰ πάντα τὰ Γ παρεληλυθέναι· ἅμα γὰρ ἔσται τὸ πρῶτον Γ καὶ τὸ πρῶτον Β ἐπὶ τοῖς ἐναντίοις ἐσχάτοις, [ἴσον χρόνον παρ' ἕκαστον γιγνόμενον τῶν Β ὅσον περ τῶν Α, ὥς φησιν,] διὰ τὸ ἀμφότερα ἴσον χρόνον παρὰ τὰ Α γίγνεσθαι. ὁ μὲν οὖν λόγος οὗτός ἐστιν, συμβαίνει δὲ παρὰ τὸ εἰρημένον ψεῦδος. For instance (so runs the argument), let A, A...be the stationary bodies of equal size, B, B...the bodies, equal in number and in size to A, A...,originally occupying the half of the course from the starting-post to the middle of the A's, and G, G...those originally occupying the other half from the goal to the middle of the A's, equal in number, size, and velocity to B, B....Then three consequences follow: First, as the B's and the G's pass one another, the first B reaches the last G at the same moment as the first G reaches the last B. Secondly at this moment the first G has passed all the A's, whereas the first B has passed only half the A's, and has consequently occupied only half the time occupied by the first G, since each of the two occupies an equal time in passing each A. Thirdly, at the same moment all the B's have passed all the G's: for the first G and the first B will simultaneously reach the opposite ends of the course, since (so says Zeno) the time occupied by the first G in passing each of the B's is equal to that occupied by it in passing each of the A's, because an equal time is occupied by both the first B and the first G in passing all the A's. This is the argument, but it presupposed the aforesaid fallacious assumption. οὐδὲ δὴ κατὰ τὴν ἐν τῇ ἀντιφάσει μεταβολὴν οὐθὲν ἡμῖν ἔσται ἀδύνατον, οἷον εἰ ἐκ τοῦ μὴ λευκοῦ εἰς τὸ λευκὸν μετα βάλλει καὶ ἐν μηδετέρῳ ἐστίν, ὡς ἄρα οὔτε λευκὸν ἔσται οὔτε οὐ λευκόν· οὐ γὰρ εἰ μὴ ὅλον ἐν ὁποτερῳοῦν ἐστιν, οὐ λεχθήσεται λευκὸν ἢ οὐ λευκόν· λευκὸν γὰρ λέγομεν ἢ οὐ λευκὸν οὐ τῷ ὅλον εἶναι τοιοῦτον, ἀλλὰ τῷ τὰ πλεῖστα ἢ τὰ κυριώτατα μέρη· οὐ ταὐτὸ δ' ἐστὶν μὴ εἶναί τε ἐν τούτῳ καὶ μὴ εἶναι ἐν τούτῳ ὅλον. ὁμοίως δὲ καὶ ἐπὶ τοῦ ὄντος καὶ ἐπὶ τοῦ μὴ ὄντος καὶ τῶν ἄλλων τῶν κατ' ἀντίφασιν· ἔσται μὲν γὰρ ἐξ ἀνάγκης ἐν θατέρῳ τῶν ἀντικειμένων, ἐν οὐδετέρῳ δ' ὅλον αἰεί. Nor in reference to contradictory change shall we find anything unanswerable in the argument that if a thing is changing from not-white, say, to white, and is in neither condition, then it will be neither white nor not-white: for the fact that it is not wholly in either condition will not preclude us from calling it white or not-white. We call a thing white or not-white not necessarily because it is be one or the other, but cause most of its parts or the most essential parts of it are so: not being in a certain condition is different from not being wholly in that condition. So, too, in the case of being and not-being and all other conditions which stand in a contradictory relation: while the changing thing must of necessity be in one of the two opposites, it is never wholly in either. πάλιν δ' ἐπὶ τοῦ κύκλου καὶ ἐπὶ τῆς σφαίρας καὶ ὅλως τῶν ἐν αὑτοῖς κινουμένων, ὅτι συμβήσεται αὐτὰ ἠρεμεῖν· ἐν γὰρ τῷ αὐτῷ τόπῳ χρόνον τινὰ ἔσται καὶ αὐτὰ καὶ τὰ μέρη, ὥστε ἠρεμήσει ἅμα καὶ κινήσεται. πρῶτον μὲν γὰρ τὰ μέρη οὐκ ἔστιν ἐν τῷ αὐτῷ οὐθένα χρό (240b.) νον, εἶτα καὶ τὸ ὅλον μεταβάλλει αἰεὶ εἰς ἕτερον· οὐ γὰρ ἡ αὐτή ἐστιν ἡ ἀπὸ τοῦ Α λαμβανομένη περιφέρεια καὶ ἡ ἀπὸ τοῦ Β καὶ τοῦ Γ καὶ τῶν ἄλλων ἑκάστου σημείων, πλὴν ὡς ὁ μουσικὸς ἄνθρωπος καὶ ἄνθρωπος, ὅτι συμβέβηκεν. ὥστε μεταβάλλει αἰεὶ ἡ ἑτέρα εἰς τὴν ἑτέραν, καὶ οὐδέποτε ἠρεμήσει. τὸν αὐτὸν δὲ τρόπον καὶ ἐπὶ τῆς σφαίρας καὶ ἐπὶ τῶν ἄλλων τῶν ἐν αὑτοῖς κινουμένων. Again, in the case of circles and spheres and everything whose motion is confined within the space that it occupies, it is not true to say the motion can be nothing but rest, on the ground that such things in motion, themselves and their parts, will occupy the same position for a period of time, and that therefore they will be at once at rest and in motion. For in the first place the parts do not occupy the same position for any period of time: and in the second place the whole also is always changing to a different position: for if we take the orbit as described from a point A on a circumference, it will not be the same as the orbit as described from B or G or any other point on the same circumference except in an accidental sense, the sense that is to say in which a musical man is the same as a man. Thus one orbit is always changing into another, and the thing will never be at rest. And it is the same with the sphere and everything else whose motion is confined within the space that it occupies.
Postquam philosophus determinavit de divisione motus et quietis, hic excludit quaedam, ex quibus errabant aliqui circa motum. Et circa hoc tria facit: primo solvit rationes Zenonis, negantis totaliter motum esse; secundo ostendit quod indivisibile non movetur, contra Democritum, qui ponebat indivisibilia moveri semper, ibi: ostensis autem his etc.; tertio ostendit mutationem omnem esse finitam, contra Heraclitum, qui ponebat omnia moveri semper, ibi: mutatio autem et cetera. 860. After finishing with the division of motion and of rest, the Philosopher now refutes certain opinions that have been the source of error in regard to motion. About this he does three things: First he answers the arguments of Zeno who absolutely denies that motion exists; Secondly, he shows that an indivisible is not moved, against Democritus, who said that they are always in motion, at L. 12; Thirdly, he shows that all change is finite, against Heraclitus, who said that all things are eternally moved, at L. 13. Circa primum duo facit: primo ponit quandam rationem Zenonis et solvit eam, quae pertinet ad id quod immediate de motu praemiserat; secundo explicat omnes rationes eius per ordinem, ibi: quatuor autem sunt rationes et cetera. About the first he does two things: First he gives and rejects one of Zeno’s arguments, which pertains to what Zeno had accepted about motion; Secondly, he explains all his arguments in order, at 863. Dicit ergo primo quod Zeno male ratiocinabatur, et apparenti syllogismo utebatur ad ostendendum quod nihil movetur, etiam illud quod videtur velocissime moveri, sicut sagitta quae fertur. Et erat ratio sua talis. Omne quod est in loco sibi aequali, aut movetur aut quiescit: sed omne quod fertur, in quolibet nunc est in aliquo loco sibi aequali: ergo et in quolibet nunc aut movetur aut quiescit. Sed non movetur: ergo quiescit. Si autem in nullo nunc movetur, sed magis videtur quiescere, sequitur quod in toto tempore non moveatur, sed magis quiescat. 861. He says therefore first (660) that Zeno reasoned badly and used what had only the appearance of a syllogism to show that nothing is being moved, even what seems to be in rapid motion, as an arrow in flight. And this was his argument: Anything that is in a place equal to itself is either being moved or is at rest. But whatever is being moved is at each instant in a place equal to itself. Therefore, even at each instant it is either in motion or at rest. But it is not in motion, Therefore, it is at rest. But if it is not in motion at any instant but at rest, as it seems, then throughout the entire time it is at rest and not in motion. Posset autem haec ratio solvi per id quod supra ostensum est, quod in nunc neque movetur neque quiescit. Sed haec solutio intentionem Zenonis non excluderet: sufficit enim Zenoni, si ostendere possit quod in toto tempore non movetur; quod videtur sequi ex hoc quod in nullo nunc eius movetur. Et ideo Aristoteles aliter solvit, et dicit falsum esse quod ratio concludit, et non sequi ex praemissis. Now this argument could be answered by appealing to something already proved; namely, that in an instant there is neither motion nor rest. But such a solution would not cripple Zeno’s intention, for he is satisfied to show that through the entire time there is no motion&8212;a fact that seems to follow, if there is no motion at any instant of the time. Therefore Aristotle answers in a different manner and says that the conclusion is both false and does not follow from the premises. Ad hoc enim quod aliquid moveatur in tempore aliquo, oportet quod moveatur in qualibet parte illius temporis: ipsa autem nunc non sunt partes temporis; non enim componitur tempus ex nunc indivisibilibus, sicut neque aliqua magnitudo componitur ex indivisibilibus, ut supra probatum est: unde non sequitur quod in tempore non moveatur aliquid, ex hoc quod in nullo nunc movetur. For in order that something be moved in a given period of time, it has to be moved in each part of the time. But instants are not parts of time, for time is not made up of instants any more than a magnitude is made of points, as we have already proved. Hence it does not follow that a thing is not in motion in a given time, just because it is not in motion in any instant of that time. Deinde cum dicit: quatuor autem sunt rationes etc., ponit seriatim omnes rationes Zenonis, quibus utebatur ad destruendum motum. Et circa hoc tria facit: primo manifestat quomodo destruebat per suas rationes motum localem; secundo quomodo destruebat alias species mutationum, ibi: neque igitur secundum mutationem etc.; tertio quomodo destruebat specialiter motum circularem, ibi: iterum autem in circulo et cetera. 862. Then at (661) he lists in order all the arguments that Zeno used for destroying motion. About this he does three things: First he shows how he destroyed local motion with his arguments; Secondly, how he destroyed the other types of change, at 870; Thirdly, how in particular he destroyed circular motion, at 871. Circa primum quatuor rationes ponit: et hoc est quod dicit, quod Zeno utebatur quatuor rationibus contra motum, quae ingerebant difficultatem multis eas solvere volentibus. Quarum prima talis est. Si aliquid movetur per totum aliquod spatium, oportet quod prius pertranseat medium quam perveniat ad finem: sed cum illud medium sit divisibile, oportebit quod etiam prius pertranseat medium illius medii, et sic in infinitum, cum magnitudo sit in infinitum divisibilis: infinita autem non est transire in aliquo tempore finito: ergo nihil potest moveri. 863. In regard to the first he lists four reasons, and this is what he says: Zeno used against motion four arguments which have caused difficulty for many of those who tried to answer them. The first of which is this: If anything is being moved through a certain space, it must reach the middle before it reaches the end. But since the first half is divisible, half of it must be first traversed and so on indefinitely, since a magnitude can be divided ad infinitum. Infinites, however, cannot be traversed in finite time. Therefore, nothing can be moved. Dicit ergo Aristoteles quod superius, circa principium huius sexti libri, solvit istam rationem per hoc, quod similiter tempus in infinita dividitur, sicut et magnitudo. Quae quidem solutio est magis ad interrogantem si infinita contingat transire in tempore finito, quam ad interrogationem, ut dicet in octavo; ubi solvit hanc rationem per hoc, quod mobile non utitur infinitis quae sunt in magnitudine, quasi in actu existentibus, sed ut in potentia existentibus. Tunc autem aliquo puncto spatii utitur quod movetur ut in actu existenti, quando utitur eo ut principio et ut fine: et tunc necesse est quod ibi stet, ut ibi ostendetur. Et si sic oporteret transire infinita quasi in actu existentia, numquam veniretur ad finem. Therefore Aristotle says that he has already answered this argument (in the beginning of this Book VI), when he proved that time is divided ad infinitum in the same way as a magnitude is. This answer is directed more to one who asks whether infinites can be traversed in finite time than to the question, as he will say in Book VIII, (L. 17) where he answers this argument by showing that a mobile does not use the infinites which exist in a magnitude as though they were actually existing, but only as existing potentially. For a thing in motion uses a point in space as actually existing, when it uses it as a beginning and as an end, and it is then that the mobile must be at rest, as will be explained in Book VIII. But if it had to traverse infinites that were actually existing, then it would never reach the end. Secundam rationem ponit ibi: secunda autem vocata etc.; et dicit quod hanc secundam rationem vocabant Achillem, quasi invincibilem et insolubilem. Et erat ratio talis. Quia si aliquid movetur, sequitur quod id quod currit tardius, si incepit primo moveri, nunquam iungetur vel attingetur a quocumque velocissimo. Quod sic probabat. Si tardum incepit moveri ante velocissimum per aliquod tempus, in illo tempore pertransivit aliquod spatium: ante igitur quam velocissimum quod persequitur, possit attingere tardissimum quod fugit, necesse est quod vadat ab illo loco unde movit fugiens, usque ad illum locum quo pervenit fugiens tempore illo quo persequens non movebatur. Sed oportet quod velocissimum illud spatium pertranseat in aliquo tempore, in quo tempore iterum tardius aliquod spatium pertransit, et sic semper. Ergo semper tardius habet aliquid ante, idest aliquod spatium in quo praecedit velocissimum, quod ipsum persequitur: et ita velocius numquam attinget tardius. Hoc autem est inconveniens. Ergo magis dicendum est quod nihil movetur. 864. The second argument is given at (662) and he says that they called this one the “Achilles”, as though it were invincible and unanswerable. The argument was this: If anything is being moved, it follows that a slower thing, if it started earlier, will never be caught by anything moving most rapidly. And it was proved in the following way: If a slower object began to be moved for some time before a very swift one, then in that time it has traversed some distance. Therefore, before the very swift one in pursuit could reach the slower, which is still running, it must leave the place first left by the pursued and reach the place which the pursued reached during the time the pursuer was not in motion. But the very fast pursuer must traverse this space in some time, during which the slower has meanwhile traversed a certain space, and so on forever. Therefore, the slower always has “something ahead”, i.e., is always some distance ahead of the most swift pursuer, and so the swifter will never catch the slower. But this is unacceptable. Therefore, it is better to say that nothing is moved. Ad solvendum autem hanc rationem dicit, quod haec ratio est eadem cum prima, quae procedebat ex decisione spatii in duo media, quantum ad virtutem medii: sed differt ab ea in hoc, quod aliqua accepta magnitudo spatii non dividitur in duo media, sed dividitur secundum proportionem excessus velocioris ad tardius in motu. Nam in primo tempore, quo movebatur solum tardius, accipitur maior magnitudo; in secundo autem tempore, in quo velocius pertransit praedictum spatium, cum sit minus, accipitur minor magnitudo pertransita a tardiori, et sic semper. Unde quia tempus et magnitudo semper dividuntur, videtur accidere ex hac ratione quod tardius nunquam iungatur a velociori. 865. In solving this argument he says that it is the same as the first, which proceeded by dividing the distance into two halves and then continually halving one part of the remainder. But the difference between them is that in the second the given magnitude of space is not divided into halves but according to the difference between the swift and the slower motion. For in the first period of time in which only the slower was in motion, there is a greater magnitude involved; in the second period (in which the faster traversed the distance covered by the slower between its start and the start of the faster), which is a shorter time period, a smaller magnitude was traversed by the slower, and so on forever. Hence, the time and the magnitude are always being divided and that seems to be the reason why the slower is never caught by the swifter. Sed hoc in idem tendit cum eo quod in prima ratione dicebatur de divisione magnitudinis in duo media: quia in utraque ratione videtur accidere quod mobile non possit adiungere usque ad terminum quendam, propter divisionem magnitudinis in infinitum, quocunque modo dividatur; scilicet vel in duo media, sicut prima ratio procedebat, vel secundum excessum velocioris ad tardius, sicut procedebat secunda ratio. Sed in hac secunda ratione apponitur, quod velocissimum non potest attingere ad tardius dum persequitur ipsum: quod dictum est cum quadam tragoedia, idest cum quadam magnificatione verborum ad admirationem movendam; sed non facit aliquid ad virtutem rationis. But this tends to the same thing as what was said of the division of the magnitude into halves; because in both arguments it seems that the mobile cannot reach a certain goal on account of the magnitude’s being infinitely divided, no matter how it happens to be divided, i.e., whether according to halves, as happens in the first argument, or according to the excess of the faster over the slower, as in the second argument. However, in this second argument it is further added that the very swift cannot reach the slower, which it is pursuing. This “tragic” phraseology employs inflated language in order to excite wonder, but it does not do anything to the force of the argument. Unde patet quod necesse est esse eandem solutionem huius secundae rationis et primae. Sicut enim in prima ratione falsum concludebatur, quod scilicet mobile nunquam perveniret ad terminum magnitudinis, propter infinitam magnitudinis divisionem; ita falsum est quod vult secunda ratio concludere, quod tardius praecedens nunquam iungatur a velociori sequente; quod nihil est aliud quam mobile non pervenire ad aliquem terminum. Hence it is clear that the solution of the two arguments is the same. For just as a false conclusion was reached in the first argument, namely, that the mobile would never reach the end of the magnitude on account of the infinite division of the magnitude, so also what the second argument tries to conclude is false, namely, that the slower will never be caught by the swifter, which is just another way of saying that a mobile will never reach its goal. Verum est enim quod quamdiu praecedit tardius, non coniungitur sibi velocius. Sed tamen quandoque coniungetur sibi, si hoc detur, quod mobile possit pertransire finitam magnitudinem in tempore finito. Pertransibit enim velocius persequens totam illam magnitudinem qua praecedebat ipsum tardius fugiens, et adhuc maiorem, in minori tempore quam tardius moveatur ultra per aliquam determinatam quantitatem: et ita non solum attinget ipsum, sed etiam ultra transibit. Hae igitur sunt duae rationes Zenonis, sic solutae. Now, it is true that as long as the slower is ahead, it is not yet reached by the swifter. But yet it will at some time be reached, if you concede that a finite magnitude can be traversed in finite time. For the swifter pursuing mobile will traverse the whole distance by which the slower is ahead and even more, in less time than the slower was meantime moving farther ahead. Proceeding in this way the swifter will not only catch but pass the slower. These therefore are the solutions to two of Zeno’s arguments. Tertiam rationem ponit ibi: tertia autem et cetera. Et dicit quod tertia ratio Zenonis erat illa quam supra posuit antequam inciperet rationes enumerare, scilicet quod sagitta, quando fertur, quiescit. Et sicut supra dictum est, hoc accidere videtur ex eo quod ipse supponit quod tempus componatur ex ipsis nunc. Nisi enim hoc concedatur, non poterit syllogizare ad propositum. 866. The third argument is given at (663) and he says that the third argument of Zeno was the one cited above (before he began to give the arguments): namely, that an arrow in flight is always at rest. And as was said above, this seems to happen, because Zeno supposed that time is made up of instants. For unless that be granted, the syllogism fails. Quartam rationem ponit ibi: quarta autem et cetera. Circa quam tria facit: primo ponit rationem; secundo solutionem, ibi: est autem deceptio etc.; tertio manifestat per exempla, ibi: ut sint stantes aequales magnitudines et cetera. 867. He sets out the fourth argument at (664). Concerning which he does three things: First he sets out the argument; Secondly, the solution, at 868; Thirdly, he explains it by an example, at 869. Dicit ergo primo quod quarta ratio Zenonis procedebat ex aliquibus quae moventur in aliquo stadio, ita quod sint duae magnitudines aequales, quae moveantur iuxta aequalia, idest per aliquod spatium stadii aequale utrique in quantitate: et sit iste motus ex contrarietate, idest ita quod una magnitudinum aequalium moveatur per illud spatium stadii versus unam partem, et alia versus aliam partem: ita tamen quod una magnitudinum mobilium incipiat moveri a fine stadii ei aequalis, alia vero incipiat moveri a medietate stadii, sive spatii in stadio dato: et utraque moveatur aeque velociter. Hac positione facta, opinabatur Zeno quod accideret quod tempus dimidium esset aequale duplo: quod cum sit impossibile, volebat ex hoc ulterius inferre quod impossibile est aliquid moveri. First therefore he says that the fourth argument of Zeno proceeded from some bodies which move in a stadium so that there are two equal magnitudes which are moved in an equal manner, that is, through a space in the stadium equal to both in quantity, and this motion is a contrary one, i.e., one of the equal magnitudes is moved through that space of the stadium toward one part, and the other toward the other part, in such a way, however, that one of the mobile magnitudes begins to move from the terminus of the stadium which is equal to it, and the other begins to move from the middle of the stadium or of a space in the given stadium; both move with equal velocity. This being given, Zeno held that it would result in a half time equalling a double time. Since this is impossible, he intended to infer further that it is impossible for anything to be moved. Deinde cum dicit: est autem deceptio etc., ponit solutionem. Et dicit quod Zeno in hoc decipiebatur, quod accipiebat ex una parte mobile moveri iuxta magnitudinem motam, et ex alia parte accipiebat quod moveretur iuxta magnitudinem quiescentem, aequalem magnitudini motae. Et quia supponitur aequalis velocitas mobilium, volebat quod secundum aequale tempus sit motus aeque velocium circa aequales magnitudines, quarum una movetur et alia quiescit. Quod patet esse falsum. 868: Then at (665) he gives the solution. He says that Zeno was deceived in this, that he held that on the one hand the mobile is moved according to the moved magnitude, and on the other he held that it was moved according to a quiescent magnitude equal to the moved magnitude. Because an equal velocity of the moved bodies is supposed, he wanted to conclude that the motion of equally swift bodies in regard to equal magnitudes, one of which is in motion and the other standing still, is done in equal times This is seen to be false for the following reason: Quia cum aliquid movetur iuxta magnitudinem quiescentem, non est ibi nisi unus motus: sed quando aliquid movetur iuxta magnitudinem motam, sunt ibi duo motus. Et si sint in eandem partem, addetur de tempore; si autem sint in oppositas partes, diminuetur de tempore, secundum quantitatem alterius motus. Quia si magnitudo iuxta quam aliquod mobile movetur, in eandem partem moveatur aequali velocitate vel etiam maiori, nunquam mobile poterit eam pertransire: si vero minori velocitate magnitudo moveatur, pertransibit eam quandoque, sed in maiori tempore quam si quiesceret. E contrario autem est si magnitudo moveatur in oppositum mobilis: quia quantum magnitudo velocius movetur, tanto mobile in minori tempore eam pertransit; quia uterque motus operatur ad hoc quod se invicem pertranseant. When something is moved in relation to a quiescent magnitude, there is only one motion; but when something is moved in relation to a moving magnitude, there are two motions. If they are moving in the same direction it takes more, if they are moving in opposite directions it takes less time according to the amount of either motion. If the magnitude in relation to which something mobile is moved, is moved in the same direction with an equal velocity or even a greater velocity, the other moving body can never pass it. If the magnitude moves with less speed, it will pass by it at a certain time, but it will take more time than if it were quiescent. It is quite the contrary if the magnitude is moved opposite the direction of the other body. The more swiftly the magnitude moves, the less time the other body takes to pass it, because both motions work together to pass each other. Deinde cum dicit: ut sint etc., manifestat quod dixerat in terminis. Ponatur enim quod sint tres magnitudines aequales sibi invicem, in quarum qualibet ponatur a; et sint istae magnitudines stantes, idest non motae; ut si intelligatur quod sit aliquod spatium trium cubitorum, in quorum quolibet describatur a. Et sint aliae tres magnitudines aequales sibi invicem, in quarum qualibet describatur b; ut puta si intelligamus quod sit unum mobile trium cubitorum. Incipiant autem hae magnitudines moveri a medio spatii. Sint etiam tres magnitudines aliae aequales numero et magnitudine et velocitate ipsis b, et scribatur in istis c, et incipiant moveri ab ultimo spatii, scilicet ab ultimo a. 869. Then at (666) he makes clear what he said in the latter part. Suppose that there are three magnitudes equal to each other, each designated as A, and these magnitudes are standing still; thus there might be a space of three cubits, each one of which is marked by an A. There are another three magnitudes all equal and designated as B, as there might be one moving unit of three cubits. These magnitudes begin to move from the middle of the space. There are also three other magnitudes, equal in number, size and velocity to B, and designated as 0. These begin to move from the last space, that is from the last A. Hac ergo suppositione facta, continget quod primum b per suum motum perveniet ad hoc quod sit simul cum ultimo a; et iterum primum c per suum motum perveniet ut sit cum primo a, opposito ultimo: et simul etiam cum hoc erit cum ultimo b, quasi transiens secus invicem motorum, idest iuxta omnia b, quae invicem ei contramoventur. Cum autem hoc factum fuerit, constat quod istud primum c transivit omnia a, sed ipsum b non transivit nisi media. Cum ergo b et c sint aequalis velocitatis, et aeque velox minorem magnitudinem in minori tempore pertranseat; sequitur quod tempus in quo b pervenit ad ultimum a, sit dimidium temporis in quo c pervenit ad primum a oppositum: in aequali enim tempore utrumque, scilicet b et c, est iuxta unumquodque; idest in aequali tempore c et b pertranseunt quandocumque a. This being given, it occurs that the first R by its motion arrives at the last A and likewise the first C by its motion arrives at the first A, opposite the last. When this has been done, it is evident that this first C has passed all the A’s, but B has passed by only half. Since, therefore, R and C are equal in velocity, and an equal velocity passes by a smaller magnitude in less time, it follows that the time in which B travels to the last A is half the time in which C arrives at the first A opposite; in equal times C and B pass each section of A. Hoc ergo supposito, quod tempus in quo b pervenit ad ultimum a, sit dimidium temporis in quo c pervenit ad primum a oppositum, ulterius considerandum est quomodo Zeno volebat concludere quod hoc dimidium temporis sit aequale illi duplo. Ex quo enim ponitur tempus motus ipsius c, duplum temporis motus ipsius b. Ponatur quod in prima medietate temporis, b quiescebat et c movebatur, et sic c in illa medietate temporis pervenit usque ad medietatem spatii ubi est b; et tunc b incipiat moveri ad unam partem, et c ad aliam. Quando autem b pervenit ad ultimum a, oportet quod pertransierit omnia c, quia simul primum b et primum c sunt in contrariis ultimis, scilicet unum in primo a, et aliud in ultimo; et sicut ipse dicebat, c in aequali tempore fit iuxta unumquodque b, in quanto tempore pertransit unumquodque ipsorum a. Et hoc ideo, quia ambo, scilicet b et c, in aequali tempore pertranseunt unum a: et sic videtur quod si b in aequali tempore pertransit in quanto pertransit ipsum c, quod c in aequali tempore pertransit ipsum b et ipsum a. Ergo tempus in quo c pertransit omnia b, est aequale tempori in quo pertransivit omnia a. Tempus autem in quo c pertransit omnia b, est aequale tempori in quo c vel b pertransit medietatem ipsorum a, ut dictum est. Probatum est autem quod tempus in quo ipsum b pertransit medietatem ipsorum a, est dimidium temporis in quo c pertransit omnia a. Ergo sequitur quod dimidium sit aequale duplo; quod est impossibile. This being supposed, namely, that the time in which B arrives at the last A is half the time in which 0 arrives at the first A opposite, it must be further considered how Zeno wished to conclude that this half time is equal to its double, For from the supposition that the time of the motion of C is double the time of the motion of B, it is supposed that, in the first half of the time, B was still and C moved, and thus C in that half of the time arrived at the middle of the space, where B was; and then B began to move to one part and C to another. When B arrived at the last A it had to pass all the C’s, because at the same time the first B and the first C are at contrary ultimates; namely, one at the first A and the other at the last, and as he said, C is next to each B, in the same amount of time as it takes to pass each one of the A’s. This is so, because both, namely Rand C, pass one A in the same interval of time. Thus it seems that if B covers a time equal to that in which it passes C, that C, in an equal interval of time, passes B and A. Therefore, the interval in which C passes all B’s is equal to the time in which it passed all the A’s. The time in which C passed all the B’s is equal to the time in which C or B passed the middle of the A’s, as was said. But it was proved that the time in which B passed the middle of the A’s is half the time in which C reached all the A’s. Therefore, it follows that the half is equal to the double, which is impossible. Haec igitur est ratio Zenonis. Sed incidit in falsitatem praedictam: quia scilicet accipit quod c in eodem tempore pertranseat b contra-motum et a quiescens; quod est falsum, ut supra dictum est. This is the argument of Zeno. But he falls into the aforesaid error; namely, he assumes the fact that C in the same interval of time crosses B moving in a counter direction and A quiescent, which is false, as was said above. [In this description of St. Thomas, since 0 is supposed to pass all the A’s, he is careful to have Cc first begin from the last A and come to the middle, while meanwhile B is motionless. This is not in the text of Aristotle.] Deinde cum dicit: neque igitur secundum etc., ponit rationem qua Zeno excludebat mutationem quae est inter contradictoria. Dicebat enim sic. Omne quod mutatur, dum mutatur, in neutro terminorum est: quia dum est in termino a quo, nondum mutatur; dum autem est in termino ad quem, iam mutatum est. Si ergo aliquid mutetur de uno contradictorio in aliud, sicut de non albo in album, sequitur quod dum mutatur, neque sit album neque non album; quod est impossibile. 870, Then at (667) he gives the argument by which Zeno rejected change between contradictories. For he said: Whatever is being changed is in neither of the extremities while it is being changed, because while it is in the terminus a quo it is not yet being changed, and while it is in the terminus ad quem it has already been changed. Therefore, if something is being changed from one contradictory to another, as from non-white to white, it follows that while it is being changed, it is neither white nor black—which is impossible. Licet autem hoc inconveniens sequatur aliquibus, qui ponunt impartibile moveri, tamen nobis, qui ponimus quod omne quod movetur est partibile, nullum accidit impossibile. Non enim oportet, si non est totum in altero extremorum, quod propter hoc non possit dici aut album aut non album. Contingit enim quod una pars eius sit alba, et alia non alba. Non autem dicitur aliquid album ex eo quod totum sit huiusmodi, sed quia plures et principaliores partes sunt tales, quae magis propriae sunt natae tales esse: quia non idem est non esse in hoc, et non esse totum in hoc, scilicet in albo vel non albo. Now although this strange conclusion would follow for those who posit that an indivisible can be moved, yet for us who posit that whatever is being moved is divisible, nothing impossible follows. For even though it is not totally in one or other of the extremes, it is not for that reason neither white nor non-white, For one part could be white and the other non-white. For a thing is called white not only when all of it is white but also when very many or its main parts are white, i.e., the parts that are expected to receive whiteness, because it is one thing not to be something at all and another not to be entirely something, for example, white or non-white. Et quod dictum est de albo vel non albo, intelligendum est de esse vel non esse simpliciter, et in omnibus quae opponuntur secundum contradictionem, sicut calidum et non calidum, et huiusmodi. Semper enim oportebit quod sit in altero contra oppositorum illud quod mutatur, quia denominabitur ab eo quod principalius inest: sed non sequitur quod semper sit totum in neutro extremorum, ut Zeno putabat. And what has been said of white and non-white is to be understood of unqualified being or non-being and of all things that are contradictorily opposed, as hot and non-hot and so on. For what is being changed must always be in one of the opposites, because it is described in terms of whichever opposite predominates in it, But it does not follow that it is always as a whole in neither of the extremities, as Zeno supposed. Sciendum est autem quod haec responsio sufficit ad repellendum rationem Zenonis, quod hic principaliter intenditur. Quomodo autem circa hoc se habeat veritas, in octavo plenius ostendetur. Non enim in quolibet verum est, quod pars ante partem alteretur vel generetur, sed aliquando totum simul, ut supra dictum est: et tunc non habet locum haec responsio, sed illa quae ponitur in octavo. Now it should be known that this answer is sufficient to refute Zeno’s argument and that is what Aristotle’s main intention is. But the truth of this matter will be more fully given in Book VIII. For it is not true in all cases that part is altered or generated after part, but sometimes the whole comes all at once, as was said above. In that case it is not this answer but the one in Book VIII that would apply. Deinde cum dicit: iterum autem in circulo etc., solvit rationem Zenonis, qua destruebat motum sphaericum. Dicebat enim quod non est possibile aliquid circulariter vel sphaerice moveri, vel quocumque alio modo, ita quod aliqua moveantur in seipsis, id est non progrediendo a loco in quo sunt, sed in ipsomet loco. Et hoc probabat tali ratione. Omne illud quod per aliquod tempus secundum totum et partes est in uno et eodem loco, quiescit: sed omnia huiusmodi sunt in eodem loco et ipsa et partes eorum secundum aliquod tempus, etiam dum ponuntur moveri: ergo sequitur quod simul moveantur et quiescant; quod est impossibile. 871. Then at (668) he refutes the argument by which Zeno rejected spherical motion. For he said that it is not possible for anything to be moved circularly or spherically or in such a way that the motion is confined within the space occupied by the mobile. And he proved this with the following argument: Anything that is in its entirety and in respect of its parts in one and the same place for a period of time is not in motion but at rest. But all the above-mentioned fulfill these conditions, even when they are apparently in motion. Therefore, they are at once in motion and at rest—which is impossible. Huic autem rationi obviat philosophus dupliciter. Primo quantum ad hoc quod dixerat, partes sphaerae motae esse in eodem loco per aliquod tempus: contra quod Aristoteles dicit quod partes sphaerae motae in nullo tempore sunt in eodem loco. Zeno enim accipiebat locum totius: et verum est quod dum sphaera movetur, nulla pars exit extra locum totius sphaerae; sed Aristoteles loquitur de proprio loco partis, secundum quod pars potest habere locum. Dictum est enim in quarto quod partes continui sunt in loco in potentia. Manifestum est autem in motu sphaerico, quod pars mutat proprium locum, sed non locum totius: quia ubi fuit una pars, succedit alia pars. The Philosopher attacks this argument on two points. First, as to the statement that the parts of the moving sphere are in the same place for some time. For Zeno was speaking of the place of the whole, and it is true that while the sphere is in motion, no part passes out of the place of the sphere, but Aristotle speaks of the particular place of each part, according as a part has a place. For it was said in Book IV that the parts of a continuum are in place potentially. But it is evident in spherical motion that a part does change its particular place, although it does not lose the place of the whole, because where one part was, another part succeeds. Secundo obviat ad praedictam Zenonis rationem, quantum ad hoc quod dixit, quod totum manet in eodem loco per tempus. Contra quod Aristoteles dicit, quod etiam totum semper mutatur in alium locum: quod sic patet. Ad hoc enim quod sint duo loca diversa, non oportet quod unus illorum locorum sit totaliter extra alium; sed quandoque quidem secundus locus est partim coniunctus primo loco, et partim ab eo divisus, ut potest in his considerari quae moventur motu recto. Si enim aliquod cubitale corpus moveatur de ab loco in bc locum, quorum uterque locus sit cubitalis; dum movetur ab uno loco in alium, oportet quod partim deserat unum et subintret alium; sicut si deserat de loco ab quantum est ad, subintrabit in locum bc quantum est be. Manifestum est ergo quod locus de est alius a loco ab, non tamen totaliter ab eo seiunctus, sed partim. Secondly, he attacks the statement that the whole remains in the same place for some time. Against this Aristotle says that even the whole is changing its place. For in order that two places be not the same, it is not required that one of them be entirely outside the other, but sometimes the second place is partly joined to part of the first and partly divided from it, as is clear in things moved in a straight line. For let a body of one cubit be moved from place AB to place BC—both places being one cubit each. While the mobile is being moved from one place to the other, it must partly desert one place and enter the other; for example, it could leave the portion AD of AB and enter the portion BE of BC. Therefore, it is clear that the place DE is distinct from AB, although not entirely, but only partly separated from it. Si autem daretur quod illa pars mobilis quae subintrat secundum locum, regrederetur in partem loci quam mobile deserit, essent duo loca, et tamen in nullo ab invicem separata; sed solum differrent secundum rationem, secundum quod principium loci in diversis signis acciperetur, ubi scilicet est principium mobilis, idest aliquod signum quod in mobili accipitur ut principium: et sic erunt duo loca secundum rationem, sed unus locus secundum subiectum. But if it were assumed that that part of the mobile which entered the second place re-entered part of the place deserted, there would be two places, yet in no way separated—they would differ not really but only in conception, i.e., in the sense that the beginning of the place might be successively called by different letters each time the mobile re-entered it, namely, where the beginning of the mobile is, i.e., some spot in the mobile which is taken as a beginning, Thus there would be two places conceptually but one and the same in reality. Et sic intelligendum est quod hic dicit, quod non est eadem circulatio secundum quod accipitur ut incipiens ab a, et ut incipiens a b, et ut incipiens a c, vel a quocumque alio signo; nisi forte dicatur eadem circulatio subiecto, sicut musicus homo et homo, quia unum accidit alteri. Unde manifestum est quod semper de uno circulari loco movetur in alterum, et non quiescit, ut Zeno probare nitebatur. Et eodem modo se habet et in sphaera et in omnibus aliis quae infra locum proprium moventur, sicut rota et columna vel quidquid aliud huiusmodi. This is how we must understand what Aristotle says here, namely, that it is not the same revolution, when it is taken as beginning at A and as beginning at B and as beginning at C or any other mark, unless you insist that it is the same revolution as to subject, as in the case of “musical man” and “man”, since one happens to the other. Hence it is clear that the mobile is always being moved from one circular place to another and is not at rest as Zeno tried to prove. And it is the same with the sphere and everything else whose motion is confined within the space it occupies, as in the case of a potter’s wheel and a (rotating) pillar or anything of that sort.
What is indivisible according to quantity is moved only per accidens
Chapter 10 Ἀποδεδειγμένων δὲ τούτων λέγομεν ὅτι τὸ ἀμερὲς οὐκ ἐνδέχεται κινεῖσθαι πλὴν κατὰ συμβεβηκός, οἷον κινουμένου τοῦ σώματος ἢ τοῦ μεγέθους τῷ ἐνυπάρχειν, καθάπερ ἂν εἰ τὸ ἐν τῷ πλοίῳ κινοῖτο ὑπὸ τῆς τοῦ πλοίου φορᾶς ἢ τὸ μέρος τῇ τοῦ ὅλου κινήσει. (ἀμερὲς δὲ λέγω τὸ κατὰ ποσὸν ἀδιαίρετον.) καὶ γὰρ αἱ τῶν μερῶν κινήσεις ἕτεραί εἰσι κατ' αὐτά τε τὰ μέρη καὶ κατὰ τὴν τοῦ ὅλου κίνησιν. ἴδοι δ' ἄν τις ἐπὶ τῆς σφαίρας μάλιστα τὴν διαφοράν· οὐ γὰρ ταὐτὸν τάχος ἐστὶ τῶν τε πρὸς τῷ κέντρῳ καὶ τῶν ἐκτὸς καὶ τῆς ὅλης, ὡς οὐ μιᾶς οὔσης κινήσεως. καθάπερ οὖν εἴπομεν, οὕτω μὲν ἐνδέχεται κινεῖσθαι τὸ ἀμερὲς ὡς ὁ ἐν τῷ πλοίῳ καθήμενος τοῦ πλοίου θέοντος, καθ' αὑτὸ δ' οὐκ ἐνδέχεται. Our next point is that that which is without parts cannot be in motion except accidentally: i.e. it can be in motion only in so far as the body or the magnitude is in motion and the partless is in motion by inclusion therein, just as that which is in a boat may be in motion in consequence of the locomotion of the boat, or a part may be in motion in virtue of the motion of the whole. (It must be remembered, however, that by 'that which is without parts' I mean that which is quantitatively indivisible (and that the case of the motion of a part is not exactly parallel): for parts have motions belonging essentially and severally to themselves distinct from the motion of the whole. The distinction may be seen most clearly in the case of a revolving sphere, in which the velocities of the parts near the centre and of those on the surface are different from one another and from that of the whole; this implies that there is not one motion but many). As we have said, then, that which is without parts can be in motion in the sense in which a man sitting in a boat is in motion when the boat is travelling, but it cannot be in motion of itself. μεταβαλλέτω γὰρ ἐκ τοῦ ΑΒ εἰς τὸ ΒΓ, εἴτ' ἐκ μεγέθους εἰς μέγεθος εἴτ' ἐξ εἴδους εἰς εἶδος εἴτε κατ' ἀντίφασιν· ὁ δὲ χρόνος ἔστω ἐν ᾧ πρώτῳ μεταβάλλει ἐφ' οὗ Δ. οὐκοῦν ἀνάγκη αὐτὸ καθ' ὃν μεταβάλλει χρόνον ἢ ἐν τῷ ΑΒ εἶναι ἢ ἐν τῷ ΒΓ, ἢ τὸ μέν τι αὐτοῦ ἐν τούτῳ τὸ δ' ἐν θατέρῳ· πᾶν γὰρ τὸ μεταβάλλον οὕτως εἶχεν. ἐν ἑκατέρῳ μὲν οὖν οὐκ ἔσται τι αὐτοῦ· μεριστὸν γὰρ ἂν εἴη. ἀλλὰ μὴν οὐδ' ἐν τῷ ΒΓ· μεταβεβληκὸς γὰρ ἔσται, ὑπόκειται δὲ μεταβάλλειν. λείπεται δὴ αὐτὸ ἐν τῷ ΑΒ εἶναι, καθ' ὃν μεταβάλλει χρόνον. ἠρεμήσει ἄρα· τὸ γὰρ ἐν τῷ αὐτῷ εἶναι χρόνον τινὰ ἠρεμεῖν ἦν. ὥστ' οὐκ ἐνδέχεται τὸ ἀμερὲς κινεῖσθαι οὐδ' ὅλως μεταβάλλειν· μοναχῶς γὰρ ἂν οὕτως ἦν αὐτοῦ κίνησις, εἰ ὁ χρόνος ἦν ἐκ τῶν νῦν· αἰεὶ γὰρ ἐν τῷ νῦν κεκινημένον ἂν ἦν καὶ μετα(241a.) βεβληκός, ὥστε κινεῖσθαι μὲν μηδέποτε, κεκινῆσθαι δ' ἀεί. τοῦτο δ' ὅτι ἀδύνατον, δέδεικται καὶ πρότερον· οὔτε γὰρ ὁ χρόνος ἐκ τῶν νῦν οὔθ' ἡ γραμμὴ ἐκ στιγμῶν οὔθ' ἡ κίνησις ἐκ κινημάτων· οὐθὲν γὰρ ἄλλο ποιεῖ ὁ τοῦτο λέγων ἢ τὴν κίνησιν ἐξ ἀμερῶν, καθάπερ ἂν εἰ τὸν χρόνον ἐκ τῶν νῦν ἢ τὸ μῆκος ἐκ στιγμῶν. For suppose that it is changing from AB to BG—either from one magnitude to another, or from one form to another, or from some state to its contradictory—and let D be the primary time in which it undergoes the change. Then in the time in which it is changing it must be either in AB or in BG or partly in one and partly in the other: for this, as we saw, is true of everything that is changing. Now it cannot be partly in each of the two: for then it would be divisible into parts. Nor again can it be in BG: for then it will have completed the change, whereas the assumption is that the change is in process. It remains, then, that in the time in which it is changing, it is in Ab. That being so, it will be at rest: for, as we saw, to be in the same condition for a period of time is to be at rest. So it is not possible for that which has no parts to be in motion or to change in any way: for only one condition could have made it possible for it to have motion, viz. that time should be composed of moments, in which case at any moment it would have completed a motion or a change, so that it would never be in motion, but would always have been in motion. But this we have already shown above to be impossible: time is not composed of moments, just as a line is not composed of points, and motion is not composed of starts: for this theory simply makes motion consist of indivisibles in exactly the same way as time is made to consist of moments or a length of points. ἔτι δὲ καὶ ἐκ τῶνδε φανερὸν ὅτι οὔτε στιγμὴν οὔτ' ἄλλο ἀδιαίρετον οὐθὲν ἐνδέχεται κινεῖσθαι. ἅπαν γὰρ τὸ κινούμενον ἀδύνατον πρότερον μεῖζον κινηθῆναι αὑτοῦ, πρὶν ἢ ἴσον ἢ ἔλαττον. εἰ δὴ τοῦτο, φανερὸν ὅτι καὶ ἡ στιγμὴ ἔλαττον ἢ ἴσον κινηθήσεται πρῶτον. ἐπεὶ δὲ ἀδιαίρετος, ἀδύνατον ἔλαττον κινηθῆναι πρότερον· ἴσην ἄρα αὑτῇ. ὥστε ἔσται ἡ γραμμὴ ἐκ στιγμῶν· αἰεὶ γὰρ ἴσην κινουμένη τὴν πᾶσαν γραμμὴν στιγμὴ καταμετρήσει. εἰ δὲ τοῦτο ἀδύνατον, καὶ τὸ κινεῖσθαι τὸ ἀδιαίρετον ἀδύνατον. Again, it may be shown in the following way that there can be no motion of a point or of any other indivisible. That which is in motion can never traverse a space greater than itself without first traversing a space equal to or less than itself. That being so, it is evident that the point also must first traverse a space equal to or less than itself. But since it is indivisible, there can be no space less than itself for it to traverse first: so it will have to traverse a distance equal to itself. Thus the line will be composed of points, for the point, as it continually traverses a distance equal to itself, will be a measure of the whole line. But since this is impossible, it is likewise impossible for the indivisible to be in motion. ἔτι δ' εἰ ἅπαν ἐν χρόνῳ κινεῖται, ἐν δὲ τῷ νῦν μηθέν, ἅπας δὲ χρόνος διαιρετός, εἴη ἄν τις χρόνος ἐλάττων ὁτῳοῦν τῶν κινουμένων ἢ ἐν ᾧ κινεῖται ὅσον αὐτό. οὗτος μὲν γὰρ ἔσται χρόνος ἐν ᾧ κινεῖται διὰ τὸ πᾶν ἐν χρόνῳ κινεῖσθαι, χρόνος δὲ πᾶς διαιρετὸς δέδεικται πρότερον. εἰ δ' ἄρα στιγμὴ κινεῖται, ἔσται τις χρόνος ἐλάττων ἢ ἐν ᾧ αὑτὴν ἐκινήθη. ἀλλὰ ἀδύνατον· ἐν γὰρ τῷ ἐλάττονι ἔλαττον ἀνάγκη κινεῖσθαι. ὥστε ἔσται διαιρετὸν τὸ ἀδιαίρετον εἰς τὸ ἔλαττον, ὥσπερ καὶ ὁ χρόνος εἰς τὸν χρόνον. μοναχῶς γὰρ ἂν κινοῖτο τὸ ἀμερὲς καὶ ἀδιαίρετον, εἰ ἦν ἐν τῷ νῦν κινεῖσθαι δυνατὸν τῷ ἀτόμῳ· τοῦ γὰρ αὐτοῦ λόγου ἐν τῷ νῦν κινεῖσθαι καὶ ἀδιαίρετόν τι κινεῖσθαι. Again, since motion is always in a period of time and never in a moment, and all time is divisible, for everything that is in motion there must be a time less than that in which it traverses a distance as great as itself. For that in which it is in motion will be a time, because all motion is in a period of time; and all time has been shown above to be divisible. Therefore, if a point is in motion, there must be a time less than that in which it has itself traversed any distance. But this is impossible, for in less time it must traverse less distance, and thus the indivisible will be divisible into something less than itself, just as the time is so divisible: the fact being that the only condition under which that which is without parts and indivisible could be in motion would have been the possibility of the infinitely small being in motion in a moment: for in the two questions—that of motion in a moment and that of motion of something indivisible—the same principle is involved.
Postquam philosophus solvit rationes Zenonis improbantis motum, hic intendit ostendere quod impartibile non movetur. Per quod destruitur opinio Democriti, ponentis atomos per se mobiles. Et circa hoc duo facit: primo proponit intentionem; secundo probat propositum, ibi: mutetur enim ex ab in bc et cetera. 872. After answering the arguments of Zeno who tried to disprove motion, the Philosopher now intends to show that a thing incapable of being divided into parts cannot be moved. This will answer the opinion of Democritus, who posited atoms that are per se mobile. About this he does two things: First he proposes his intention; Secondly, he proves his proposition, at 876. Dicit ergo primo, quod suppositis his quae supra ostensa sunt, dicendum est quod impartibile non potest moveri, nisi forte per accidens, sicut punctum movetur in toto corpore, vel quacumque alia magnitudine in qua est punctum, scilicet linea vel superficie. He says therefore first (669) that assuming what we have proved above, it must be said that a thing incapable of being divided into parts cannot be moved, except perchance per accidens, as a point is moved in a whole body in which there is a point, for example, in a line or a surface. Moveri autem ad motum alterius contingit dupliciter. Uno modo quando illud quod movetur ad motum alterius, non est aliqua pars eius; sicut illud quod est in navi movetur ad motum navis, et albedo etiam movetur ad motum corporis, cum non sit pars eius: alio modo sicut pars movetur ad motum totius. 873. To be in motion as a result of something else being in motion can occur in two ways. In one way, when what is moved as the result of something else being moved is not part of the latter, as what is on a ship is being moved when the ship is being moved, and as whiteness is moved with the motion of body, since it is not of the body. In a second way, as a part is moved when the whole is moved. Et quia impartibile dicitur multipliciter, sicut et partibile, ostendit quomodo accipiat hic impartibile: et dicit quod impartibile hic dicitur illud quod est indivisibile secundum quantitatem. Dicitur enim et aliquid impartibile secundum speciem, sicut si dicamus ignem impartibilem aut aerem, quia non potest resolvi in plura corpora specie diversa. Sed tale impartibile nihil prohibet moveri: intendit ergo excludere motum ab impartibili secundum quantitatem. And because “what is incapable of being divided into parts” has many senses, just as what is capable of being divided into parts” has, he shows how he uses the phrase here and says that here it means what is indivisible in respect of quantity. For some things are indivisible according to species, as when we say that fire or air are indivisible, because they cannot be further resolved into several bodies that differ in species. But in regard to such an indivisible there is nothing to prevent it from being moved. Consequently, Aristotle intends to exclude motion from what is indivisible according to quantity. Et quia dixerat quod pars movetur ad motum totius, et aliquis posset dicere quod pars nullo modo movetur, subiungit quod sunt aliqui motus partium, inquantum sunt partes, qui sunt diversi a motu totius, inquantum est motus totius. 874. Because he had said that the part is being moved when the whole is, and someone might say that the part is not moved at all, he adds that there are some motions of parts precisely as parts, that are diverse from the motion of the whole, as a motion of the whole. Et hanc differentiam aliquis maxime potest considerare in motu sphaerico: quia non est eadem velocitas partium quae moventur circa centrum, et partium quae sunt extra, idest versus superficiem exteriorem sphaerae, et quae est etiam velocitas totius: ac si motus iste non sit unius sed diversorum. Manifestum est enim quod velocius est, quod in aequali tempore pertransit maiorem magnitudinem. Dum autem sphaera movetur, manifestum est quod maiorem circulum pertransit pars exterior sphaerae quam pars interior; unde maior est velocitas partis exterioris quam interioris. Tamen velocitas totius est eadem cum velocitate interioris et exterioris partis. This difference is particularly clear in the motion of a sphere, because the speed of the parts near the center is not the same as that of those outside, i.e., on the exterior surface of the sphere, the speed of whose parts is considered to be the speed of the whole. It is as if there is not just one motion but the motions of many parts involved. For it is evident that whatever traverses a larger magnitude in an equal time is faster. Now, while the sphere is rotating, it is clear that an external part describes a larger circle than an interior part; hence the velocity of the external part is greater than that of an interior part. Yet the velocity of the whole sphere is the same as the velocity of the interior and exterior part. Ista autem diversitas motuum intelligenda est secundum quod partibus continui convenit moveri, scilicet in potentia. Unde actu est unus motus totius et partium: sed potentia sunt diversi motus partium, et ad invicem, et a motu totius. Et sic cum dicitur pars moveri per accidens ad motum totius, est tale per accidens, quod est in potentia per se: quod non est de motu per accidens, secundum quod dicuntur accidentia vel formae per accidens moveri. But this diversity of motions is to be understood in the sense in which motion is ascribed to parts of a continuum, i.e., in a potential sense, Hence, actually there is one motion of the whole and of the parts, but potentially there are diverse motions: those of the parts being different from one another and from the motion of the whole. And so, when it is said that a part is being moved per accidens with the motion of the whole, it is a per accidens which is in potency per se —which is something not true of motion per accidens, when it is taken in the sense that accidents or forms are said to be moved per accidens. Posita igitur distinctione eius quod movetur, explicat suam intentionem. Et dicit quod id quod est impartibile secundum quantitatem, potest moveri quidem ad motum corporis per accidens: non tanquam pars, quia nulla magnitudo componitur ex indivisibilibus, ut ostensum est; sed sicut movetur aliquid ad motum alterius quod non est pars eius, sicut sedens in navi movetur ad motum navis. Sed per se non contingit impartibile moveri. 875. Having made a distinction among things that are moved, he explains his intention. And he says that what is indivisible in respect of quantity can indeed be moved per accidens when something else is moved, but it is not moved as a part, for no magnitude is made up of indivisibles, as we have proved. Now, something not a part of another is moved along with the other in the same way that one sitting in a ship is moved along with the motion of a ship. But per se the indivisible cannot be moved. Hoc autem idem supra probavit, non ex principali intentione, sed incidenter. Unde praeter rationem supra positam, hic magis explicat veritatem, et rationes addit efficaces ad propositum ostendendum. He had proved this point previously, not as a main proposition but incidentally. Hence, in addition to the reason cited earlier, he now gives a further explanation of the truth and adds reasons that are strong enough to prove the proposition. Deinde cum dicit: mutetur enim etc., probat propositum tribus rationibus. Quarum prima talis est. Si ponatur quod impartibile movetur, moveatur ex ab in bc. Nec differt quantum ad hanc rationem, utrum ista duo, scilicet ab et bc, sint duae magnitudines, sive duo loca, ut in motu locali et augmenti et decrementi; vel duae species, idest duae qualitates, sicut in motu alterationis; vel sint duo contradictorie opposita, ut in generatione et corruptione. Et sit tempus ed in quo aliquid mutatur de uno termino in alterum primo, idest non ratione partis. In hoc ergo tempore necesse est quod id quod mutatur, aut sit in ab, idest in termino a quo; aut in bc, idest in termino ad quem; aut aliquid eius est in uno termino, alia vero pars eius est in altero. Omne enim quod mutatur, oportet quod aliquo horum trium modorum se habeat, sicut supra dictum est. 876. Then at (670) he proves his point with three arguments. The first of which is this: If it is insisted that an indivisible can be moved, let it be moved from AB into BC. (In this argument it makes no difference whether AB and BC are two magnitudes or two places, as in local motion and growth and decrease, or whether they are two qualities, as in the motion of alteration, or two things that are contradictorily opposed, as in generation and ceasing-to-be.) Let ED be the time in which something is changed from one terminus to the other first, i.e., not by reason of a part. In this time, then, it is necessary that what is being moved be either in AB, i.e., in the terminus a quo, or in BC, i.e., in the terminus ad quem; or else a part is in one terminus and a part in the other. For anything being moved must be in one of these three ways, as was said above. Non autem potest dari tertium membrum, scilicet quod sit in utroque secundum diversas partes sui: quia sic sequeretur quod esset partibile, et positum erat quod esset impartibile. Sed similiter non potest dari secundum membrum, scilicet quod sit in bc, idest in termino ad quem: quia quando est in termino ad quem, tunc iam est mutatum, ut ex superioribus patet; ponebatur autem quod in hoc tempore mutaretur. Relinquitur ergo quod in toto tempore in quo mutatur indivisibile, sit in ab, idest in termino a quo. Ex quo sequitur quod quiescat: nihil enim est aliud quiescere, quam quod aliquid sit in uno et eodem per totum aliquod tempus. Cum enim in quolibet tempore sit prius et posterius, si tempus est divisibile, quidquid per aliquod tempus est in uno et eodem, similiter se habet nunc et prius; quod est quiescere. Sed hoc est impossibile, quod aliquid dum mutatur quiescat. Now the third situation is impossible; namely, that it be in each term according to its various parts, because then, the mobile would be divided into parts, and we have assumed that it is an indivisible mobile. Likewise, it cannot be the second alternative, i.e., that it be in BC, i.e., in the terminus ad quem, for when it is in the terminus ad quem, it has been already changed (as is clear from what we have said above), whereas we are assuming that it is being changed. What remains, therefore, is that in the entire time that the indivisible is being changed it remains at AB, i.e., in the terminus a quo, From which it follows that it is at rest, for resting is nothing more than to be in one and the same state throughout a definite period of time. For since there is a prior and a subsequent in time, if time is divisible, whatever for a period of time is in one and the same state keeps itself the same; namely, as it was previously—which is to rest. But it is impossible that a thing is at rest while it is being changed. Therefore, it cannot be that an indivisible is moved or changed in any way whatsoever. Relinquitur ergo quod non contingit impartibile moveri, neque aliquo modo mutari. Hoc enim solo modo posset esse aliquis motus rei indivisibilis, si tempus componeretur ex ipsis nunc: quia in nunc semper est motum esse vel mutatum. The only way in which there could be motion of an indivisible thing is to have the time composed of “now’s”, because in the “now” there is always a condition called “having been moved” or “having been changed”. Et quia quod motum est, inquantum huiusmodi, non movetur, sequitur quod in nunc nihil movetur, sed sit motum. Sic igitur posset poni indivisibile moveri in aliquo tempore, si tempus ex ipsis nunc componeretur: quia posset dari quod in quolibet ipsorum nunc ex quibus componitur tempus, esset in uno, et in toto tempore, idest in omnibus nunc, esset in multis; et sic in toto tempore moveretur, non autem in aliquo nunc. Sed quod hoc sit impossibile, scilicet tempus componi ex ipsis nunc, ostensum est prius. And because what has been moved, precisely as such is not now being moved, it follows that in the “now” nothing is being moved, but has been moved. But if time were made up of “now’s”, there would be a way in which motion could be posited in an indivisible, because it could be granted that in each of those “now’s” of which time is composed, it would be in one, and in the whole time, i.e., in all the “now’s”, it would be in many. And thus it would be in motion throughout the entire time, but not in one “now”. Ostensum est enim supra quod neque tempus componitur ex ipsis nunc, neque linea ex ipsis punctis, neque motus componitur ex momentis (ut per momentum intelligamus hoc quod est mutatum esse). Qui enim hoc dicit, quod indivisibile movetur, aut quod motus componatur ex indivisibilibus, nihil aliud facit quam quod tempus componatur ex nunc, aut magnitudo ex punctis; quod est impossibile. Ergo et impossibile est impartibile moveri. But it has been proved above that it is impossible for time to be made up of “now’s”. Indeed, we have proved that neither is time composed of now’s nor a line of points, nor a motion of moments (where “moments” refers to states called “having been changed”). For anyone who says that an indivisible is being moved or that motion is composed of indivisibles is making time be composed of “now’s” or a magnitude of points—which is impossible. Therefore, it is also impossible that a thing incapable of being divided into parts be moved. Secundam rationem ponit ibi: amplius autem ex his etc.: et dicit quod ex his quae sequuntur, potest esse manifestum quod neque punctum, neque aliud quodcumque indivisibile potest moveri. Et ista ratio specialis est de motu locali. Omne enim quod movetur secundum locum, impossibile est quod prius pertranseat maiorem magnitudinem ipso mobili, quam aequalem vel minorem; sed semper mobile prius pertransit magnitudinem aequalem sibi aut minorem, quam maiorem. Si ergo hoc ita se habet, manifestum est quod et punctum, si movetur, prius pertransibit aliquid minus se aut aequale sibi, quam longitudinem maiorem se. Sed impossibile est quod pertranseat aliquid minus se, quia est indivisibile. Relinquitur ergo quod pertransibit aliquid aequale sibi. Et ita oportet quod numeret omnia puncta quae sunt in linea: quia semper punctum, cum moveatur motu aequali lineae, propter hoc quod movetur per totam lineam, sequitur quod totam lineam mensuret; hoc autem facit numerando omnia puncta. Ergo sequitur quod linea sit ex punctis. Si ergo hoc est impossibile, impossibile est quod indivisibile moveatur. 877. The second argument is given at (671). He says that if we look at the consequences, it is clear that neither a point nor any indivisible can be moved. And this special argument applies to local motion. For whatever is being moved according to place cannot traverse a distance greater than the mobile itself before traversing one that is equal to or less than it; rather, a mobile always traverses a magnitude equal to itself or less than itself before one greater than itself. If this is so, then it is clear that a point, if it is being moved will first traverse a length less than or equal to itself, before it traverses one greater than itself. But it is impossible for it to traverse something less than itself, since it is indivisible. So it has to traverse a length equal to itself. Consequently, it must number all the points in the line; for the point, since it is being moved through a motion equal to a line, is by that very fact being moved through the whole line, and, consequently, is always measuring the whole line—and this it does by counting all the points. Therefore, it follows that a line arises from points. Therefore, if this is impossible, it is impossible for an indivisible to be moved. Tertiam rationem ponit ibi: amplius autem si omne etc.: quae talis est. Omne quod movetur, movetur in tempore, et nihil movetur in ipso nunc, ut supra probatum est. Ostensum est autem supra quod omne tempus est divisibile. Ergo in quolibet tempore in quo aliquid movetur, erit accipere minus tempus, in quo movetur aliquod minus mobile: quia manifestum est quod supposita eadem velocitate, in minori tempore pertransit minus mobile aliquod signum datum, quam mobile maius, sicut in minori tempore pars quam totum, ut ex superioribus patet. Si ergo punctum movetur, erit accipere aliquod tempus minus tempore in quo ipsum movetur. Sed hoc est impossibile: quia sequeretur quod in illo minori tempore moveretur aliquid minus quam punctum; et sic indivisibile esset divisibile in aliquod minus, sicut tempus dividitur in tempus. Hoc enim solo modo posset moveri indivisibile, si esset possibile aliquid moveri in nunc indivisibili: quia sicut non esset accipere aliquod minus ipso nunc in quo movetur, ita non oporteret accipere aliquod minus mobili. 878. The third argument is at (672) and is this: Since motion is always in a period of time and never in a “now”, and since all time is divisible, as was shown above, then in every time in which something is. moved, there must be a lesser time in which a lesser mobile is moved. For, supposing the same speed, it is plain that in a lesser time the lesser mobile crosses a given mark than does a greater mobile, as in a lesser time the part than the whole, as is evident from what is above. If, therefore, a point is in motion, there must be a time less than that in which it is moved. But this is impossible, for it would follow that in that lesser time something less than a point would be moved, and thus the indivisible would be divisible into something less, just as time is divisible. This would be the only condition under which the indivisible could be in motion, namely, if it were possible for something to be moved in an indivisible “now”, for just as there is nothing smaller than the “now” in time, so one cannot take a smaller mobile. Et sic patet quod eiusdem rationis est, quod fiat motus in nunc, et quod indivisibile aliquod moveatur. Hoc autem est impossibile, quod in nunc fiat motus. Ergo impossibile est quod indivisibile moveatur. And so it is evident that in the two questions—that of motion in a “now” and that of an indivisible being moved—the same principle is involved. But it is impossible for motion to occur in a “now”. Therefore, it is impossible for an indivisible to be moved.
By nature, no change is infinite. How motion may be infinite in time
Chapter 10 cont. μεταβολὴ δ' οὐκ ἔστιν οὐδεμία ἄπειρος· ἅπασα γὰρ ἦν ἔκ τινος εἴς τι, καὶ ἡ ἐν ἀντιφάσει καὶ ἡ ἐν ἐναντίοις. ὥστε τῶν μὲν κατ' ἀντίφασιν ἡ φάσις καὶ ἡ ἀπόφασις πέρας (οἷον γενέσεως μὲν τὸ ὄν, φθορᾶς δὲ τὸ μὴ ὄν), τῶν δ' ἐν τοῖς ἐναντίοις τὰ ἐναντία· ταῦτα γὰρ ἄκρα τῆς μεταβολῆς, ὥστε καὶ ἀλλοιώσεως πάσης (ἐξ ἐναντίων γάρ τινων ἡ ἀλλοίωσις), ὁμοίως δὲ καὶ αὐξήσεως καὶ φθίσεως· αὐξήσεως μὲν γὰρ τὸ πέρας τοῦ (241b.) κατὰ τὴν οἰκείαν φύσιν τελείου μεγέθους, φθίσεως δὲ ἡ τούτου ἔκστασις. Our next point is that no process of change is infinite: for every change, whether between contradictories or between contraries, is a change from something to something. Thus in contradictory changes the positive or the negative, as the case may be, is the limit, e.g. being is the limit of coming to be and not-being is the limit of ceasing to be: and in contrary changes the particular contraries are the limits, since these are the extreme points of any such process of change, and consequently of every process of alteration: for alteration is always dependent upon some contraries. Similarly contraries are the extreme points of processes of increase and decrease: the limit of increase is to be found in the complete magnitude proper to the peculiar nature of the thing that is increasing, while the limit of decrease is the complete loss of such magnitude. ἡ δὲ φορὰ οὕτω μὲν οὐκ ἔσται πεπερασμένη· οὐ γὰρ πᾶσα ἐν ἐναντίοις· ἀλλ' ἐπειδὴ τὸ ἀδύνατον τμηθῆναι οὕτω, τῷ μὴ ἐνδέχεσθαι τμηθῆναι (πλεοναχῶς γὰρ λέγεται τὸ ἀδύνατον), οὐκ ἐνδέχεται τὸ οὕτως ἀδύνατον τέμνεσθαι, οὐδὲ ὅλως τὸ ἀδύνατον γενέσθαι γίγνεσθαι, οὐδὲ τὸ μεταβαλεῖν ἀδύνατον ἐνδέχοιτ' ἂν μεταβάλλειν εἰς ὃ ἀδύνατον μεταβαλεῖν. εἰ οὖν τὸ φερόμενον μεταβάλλοι εἴς τι, καὶ δυνατὸν ἔσται μεταβαλεῖν. ὥστ' οὐκ ἄπειρος ἡ κίνησις, οὐδ' οἰσθήσεται τὴν ἄπειρον· ἀδύνατον γὰρ διελθεῖν αὐτήν. ὅτι μὲν οὖν οὕτως οὐκ ἔστιν ἄπειρος μεταβολὴ ὥστε μὴ ὡρίσθαι πέρασι, φανερόν. Locomotion, it is true, we cannot show to be finite in this way, since it is not always between contraries. But since that which cannot be cut (in the sense that it is inconceivable that it should be cut, the term 'cannot' being used in several senses)—since it is inconceivable that that which in this sense cannot be cut should be in process of being cut, and generally that that which cannot come to be should be in process of coming to be, it follows that it is inconceivable that that which cannot complete a change should be in process of changing to that to which it cannot complete a change. If, then, it is to be assumed that that which is in locomotion is in process of changing, it must be capable of completing the change. Consequently its motion is not infinite, and it will not be in locomotion over an infinite distance, for it cannot traverse such a distance. It is evident, then, that a process of change cannot be infinite in the sense that it is not defined by limits. ἀλλ' εἰ οὕτως ἐνδέχεται ὥστε τῷ χρόνῳ εἶναι ἄπειρον τὴν αὐτὴν οὖσαν καὶ μίαν, σκεπτέον. μὴ μιᾶς μὲν γὰρ γιγνομένης οὐθὲν ἴσως κωλύει, οἷον εἰ μετὰ τὴν φορὰν ἀλλοίωσις εἴη καὶ μετὰ τὴν ἀλλοίωσιν αὔξησις καὶ πάλιν γένεσις· οὕτω γὰρ αἰεὶ μὲν ἔσται τῷ χρόνῳ κίνησις, ἀλλ' οὐ μία διὰ τὸ μὴ εἶναι μίαν ἐξ ἁπασῶν. ὥστε δὲ γίγνεσθαι μίαν, οὐκ ἐνδέχεται ἄπειρον εἶναι τῷ χρόνῳ πλὴν μιᾶς· αὕτη δ' ἐστὶν ἡ κύκλῳ φορά. But it remains to be considered whether it is possible in the sense that one and the same process of change may be infinite in respect of the time which it occupies. If it is not one process, it would seem that there is nothing to prevent its being infinite in this sense; e.g. if a process of locomotion be succeeded by a process of alteration and that by a process of increase and that again by a process of coming to be: in this way there may be motion for ever so far as the time is concerned, but it will not be one motion, because all these motions do not compose one. If it is to be one process, no motion can be infinite in respect of the time that it occupies, with the single exception of rotatory locomotion.
Postquam philosophus ostendit quod impartibile non movetur, hic intendit ostendere quod nulla mutatio est infinita; quod est contra Heraclitum, qui posuit omnia moveri semper. Et circa hoc duo facit: primo ostendit quod nulla mutatio est infinita secundum propriam speciem; secundo ostendit quomodo possit esse infinita tempore, ibi: sed si sic contingit et cetera. 878. After showing that things which cannot be divided into parts are not moved, the Philosopher now intends to show that no change is infinite, This is against Heraclitus, who supposed that things are always in motion. About this he does two things: First he shows that no change is infinite according to its own species; Secondly, how there can be infinites in time, at 883, Circa primum duo facit: primo ostendit quod mutatio non est infinita secundum speciem in aliis mutationibus praeter motum localem; secundo ostendit idem in motu locali, ibi: loci autem mutatio et cetera. About the first he does two things: First he shows for all changes except local motion that no change is infinite according to its species; Secondly, he shows the same thing for local motion, at 881. Prima ratio talis est. Supra dictum est quod omnis mutatio est ex quodam in quiddam. Et in quibusdam quidem mutationibus, quae scilicet sunt inter contradictorie opposita, ut generatio et corruptio, vel inter contraria, ut alteratio, et augmentum et decrementum, manifestum est quod habent praefixos terminos. Unde in his mutationibus quae sunt inter contradictorie opposita, terminus est vel affirmatio vel negatio, sicut terminus generationis est esse, corruptionis vero non esse. 880. The first reason is this: Every change is from something to something. Indeed, in some changes, namely, those which occur between contradictories, as do generation and ceasing-to-be, or between contraries, as do alteration and growing and decreasing, it is evident that they have pre-defined termini. Hence in changes that occur between contradictory termini, the terminus is either affirmation or negation, as the terminus of generation is a being, and that of ceasing-to-be, non-being. Similiter illarum mutationum quae sunt inter contraria, ipsa contraria sunt termini ad quos, sicut ad quaedam ultima, mutationes huiusmodi terminantur. Unde sequitur quod, cum omnis alteratio sit de contrario in contrarium, quod omnis alteratio habeat aliquem terminum. Likewise, in regard to changes that are between contraries, the contraries are termini at which, as at ultimate goals, changes of this kind are terminated. Hence it follows, since every alteration is from contrary to contrary, that every alteration has some terminus. Et similiter dicendum est in augmento et decremento: quia terminus augmenti est perfecta magnitudo (et dico perfectam secundum conditionem propriae naturae: alia enim perfectio magnitudinis competit homini et alia equo); terminus autem decrementi est id quod contingit esse in tali natura maxime remotum a perfecta magnitudine. The same must be said for growth and decrease, for the terminus of growth is perfect magnitude (and I say “perfect” in respect of the nature, for a different perfection of magnitude befits man from the one that befits a horse), and the terminus of decrease is the one that happens to a definite nature to be most removed from perfect magnitude. Et sic patet quod quaelibet praedictarum mutationum habet aliquid ultimum in quod terminatur: nihil autem tale est infinitum: ergo nulla praedictarum mutationum potest esse infinita. Consequently, it is evident that each of the above-mentioned changes has a goal at which it is terminated. But such a situation precludes the infinite. Therefore, none of these changes can be infinite. Deinde cum dicit: loci autem mutatio etc., procedit ad loci mutationem. Et primo ostendit quod non est similis ratio de loci mutatione et aliis mutationibus. Non enim potest sic probari quod loci mutatio sit finita, sicut probatum est de aliis mutationibus, per hoc quod terminantur ad aliqua contraria, vel contradictorie opposita: quia non omnis loci mutatio est inter contraria simpliciter. Dicuntur enim contraria quae maxime distant. 881. Then at (674) he proceeds to local motion. And first he shows that the argument in regard to local motion is not the same as for the other changes. For it cannot be proved that local motion is finite (as we have proved other motions are finite), because it is terminated at something contrary or contradictory, for not every local motion is between strict contraries, where contraries refer to things most distant. Maxime autem distantia simpliciter accipitur quidem in motibus naturalibus gravium et levium: locus enim ignis a centro terrae habet maximam distantiam, secundum distantias determinatas talibus corporibus in natura. Unde tales mutationes sunt inter contraria simpliciter. Unde de huiusmodi mutationibus posset ostendi quod non sunt infinitae, sicut et de aliis. There is a maximum distance in the strict sense in the natural motions of heavy and light bodies, for the place of fire is at a maximum distance from the center of the earth, in accordance with the distance that nature determines for such bodies. Hence, such changes are between strict contraries, Hence, it can be proved of such changes that they are not infinite any more than the others were. Sed maxima distantia in motibus violentis aut voluntariis, non accipitur simpliciter secundum aliquos terminos certos; sed secundum propositum aut violentiam moventis, qui aut non vult, aut non potest ad maiorem distantiam movere. Unde est ibi secundum quid maxima distantia, et per consequens contrarietas, non autem simpliciter. Et ideo non poterat ostendi per terminos, quod nulla mutatio localis esset infinita. But maximum distance in compulsory or voluntary motions does not depend strictly on certain definite termini but on the intention or energy of the one causing the motion, who either does not desire to or cannot physically move something any farther. Hence, it is only in a qualified sense., ere is maximum distance and a consequent contrariety. Hence, if you stick with the termini, it cannot be proved that no local motion is infinite. Unde consequenter hoc ostendit alia ratione, quae talis est. Illud quod impossibile est esse decisum, non contingit decidi. Et quia multipliciter dicitur aliquid impossibile, scilicet quod omnino non contingit esse, et quod non de facili potest esse; ideo interponit de quo impossibile hic intelligat. Intelligit enim de eo quod sic est impossibile, quod nullo modo contingit esse. Et eadem ratione id quod est impossibile factum esse, impossibile est fieri; sicut si impossibile est contradictoria esse simul, impossibile est hoc fieri. Et pari ratione illud quod impossibile est mutatum esse in aliquid, impossibile est quod mutetur in illud; quia nihil tendit ad impossibile. Sed omne quod mutatur secundum locum, mutatur in aliquid. Ergo possibile est per motum pervenire in illud. Sed infinitum non potest pertransiri. Non ergo fertur aliquid localiter per infinitum. Sic ergo nullus motus localis est infinitus. 882. Consequently, this must be proved by another argument, which is this: What is impossible to exist divided cannot be divided, And because things are said to be impossible in many senses, name, what never can occur or what cannot occur except with great difficulty, he therefore explains his meaning of “impossible” here. And he means it in the sense of that which cannot happen at all. For the same reason, what is impossible to have been made, is impossible to make; for example, if it is impossible that contradictories be together, it is impossible that this be brought about. For the same reason, what is impossible to have been changed into something cannot be changed into it, because nothing tends toward the impossible. But everything that is being changed according to place is being changed into something. Therefore, it is possible through that motion to arrive at something. But the infinite cannot be gone through. Therefore, nothing is moved through the infinite. Thus, therefore, no local motion is infinite. Et ita universaliter patet quod nulla mutatio potest esse sic infinita, ut non finiatur certis terminis, a quibus speciem habet. And so it is universally evident that no change can be infinite in such a way that it be not terminated by definite termini from which it derives its species. Deinde cum dicit: sed si sic contingit etc., ostendit quomodo motus possit esse infinitus tempore. Et dicit quod considerandum est utrum sic contingat motum esse infinitum tempore, ut semper maneat unus et idem numero. Quod enim motus duret per infinitum tempus, non existente uno ipso motu, nihil prohibet: quod sub dubitatione dicit, addens forte, quia posterius de hoc inquiret. Et ponit exemplum: sicut si dicamus quod post loci mutationem est alteratio, et post alterationem est augmentum, et post augmentum iterum generatio, et sic in infinitum. Sic enim semper posset motus durare tempore infinito. Sed non esset unus secundum numerum; quia ex huiusmodi motibus non fit unum numero, ut in quinto ostensum est. Sed quod motus duret tempore infinito, ita quod semper maneat unus numero, hoc non contingit nisi in una specie motus: motus enim circularis potest durare unus et continuus tempore infinito, ut in octavo ostendetur. 883. Then at (675) he shows how motion can be infinite in time. And he says that we must consider whether motion can be infinite in time in such a way that it remains numerically one and the same motion. For there is nothing to prevent motion from enduring through infinite time as long as it is not one and the same motion. But he leaves that matter in doubt when he adds “perhaps”, because he will settle the matter later. And he gives an example: Let us say that after a local motion there is an alteration, and after that a growing, and after that generation, and so on ad infinitum. In this way motion could always endure throughout infinite time. And it would not be one and the same numerical motion, because a series of such motions as are given in the example do not form one numerical motion, as we have proved in Book V. But that motion endure throughout infinite time in such a way that it remain one numerical motion can occur in only one species of motion, for a circular motion can endure as one and continuous throughout infinite time, as will be proved in Book VIII.